Pedestrian localisation for indoor environments
A dissertation submitted for the degree
of Doctor of Philosophy
Oliver J. Woodman
St Catharine’s College September 9, 2010
DIGITAL TECHNOLOGY GROUP
Computer Laboratory
University of Cambridge
c⃝ 2010 Oliver J. Woodman
All rights reserved
Typeset in Times with LATEX
This dissertation is the result of my own work
and includes nothing that is the outcome of
work done in collaboration except where
specifically indicated in the text.
No part of this dissertation has already been,
or is currently being submitted by the author
for any other degree or diploma or other
qualification.
This dissertation does not exceed 60,000
words including tables and footnotes, but
excluding appendices, bibliography,
photographs and diagrams.
This work was supported by the EPSRC.
Abstract
Pedestrian localisation for indoor environments
Oliver J. Woodman
Ubiquitous computing systems aim to assist us as we go about our daily lives, whilst at the
same time fading into the background so that we do not notice their presence. To do this
they need to be able to sense their surroundings and infer context about the state of the world.
Location has proven to be an important source of contextual information for such systems. If a
device can determine its own location then it can infer its surroundings and adapt accordingly.
Of particular interest for many ubiquitous computing systems is the ability to track people
in indoor environments. This interest has led to the development of many indoor location
systems based on a range of technologies including infra-red light, ultrasound and radio. Un-
fortunately existing systems that achieve the kind of sub-metre accuracies desired by many
location-aware applications require large amounts of infrastructure to be installed into the
environment.
This thesis investigates an alternative approach to indoor pedestrian tracking that uses on-
body inertial sensors rather than relying on fixed infrastructure. It is demonstrated that general
purpose inertial navigation algorithms are unsuitable for pedestrian tracking due to the rapid
accumulation of errors in the tracked position. In practice it is necessary to frequently correct
such algorithms using additional measurements or constraints. An extended Kalman filter
is developed for this purpose and is applied to track pedestrians using foot-mounted inertial
sensors. By detecting when the foot is stationary and applying zero velocity corrections a
pedestrian’s relative movements can be tracked far more accurately than is possible using
uncorrected inertial navigation.
Having developed an effective means of calculating a pedestrian’s relative movements, a local-
isation filter is developed that combines relative movement measurements with environmental
constraints derived from a map of the environment. By enforcing constraints such as impass-
able walls and floors the filter is able to narrow down the absolute position of a pedestrian
as they move through an indoor environment. Once the user’s position has been uniquely
determined the same filter is demonstrated to track the user’s absolute position to sub-metre
iii
accuracy.
The localisation filter in its simplest form is computationally expensive. Furthermore symme-
try exhibited by the environment may delay or prevent the filter from determining the user’s
position. The final part of this thesis describes the concept of assisted localisation, in which
additional measurements are used to solve both of these problems. The use of sparsely de-
ployed WiFi access points is discussed in detail.
The thesis concludes that inertial sensors can be used to track pedestrians in indoor environ-
ments. Such an approach is suited to cases in which it is impossible or impractical to install
large amounts of fixed infrastructure into the environment in advance.
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Acknowledgements
Many people have helped me to complete the research presented in this thesis. In particular I
wish to thank Andy Hopper for his insight, guidance and financial assistance. I also wish to
thank Robert Harle for giving up a great deal of his time to help with my research. Similarly
Andrew Rice, Ripduman Sohan, Alistair Beresford and Brian Jones have all contributed their
time and knowledge to assist me during the last three years.
In addition to the individuals named above, the Digital Technology Group as a whole has
provided a friendly and vibrant atmosphere in which it was a pleasure to work. Their diverse
knowledge has helped to broaden my education and their feedback has helped to shape the
direction of my research.
I wish to thank my examiners Peter Robinson and Paul Groves for their comments and sug-
gestions, which have enabled me to make significant improvements to the quality of this
manuscript.
Finally I wish to thank my parents, grandparents, family and friends for their love and support.
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Publications
Oliver Woodman. An Introduction to Inertial Navigation. Technical Report 696, University
of Cambridge, Computer Laboratory, August 2007.
Oliver Woodman and Robert Harle. Pedestrian Localisation for Indoor Environments. In
Proceedings of the 10th International Conference on Ubiquitous Computing (Ubicomp 2008),
Seoul, South Korea, September 2008.
Oliver Woodman and Robert Harle. Rf-based Initialisation for Inertial Pedestrian Tracking.
In Proceedings of the 7th International Conference on Pervasive Computing (Pervasive 2009),
Nara, Japan, May 2009.
Oliver Woodman and Robert Harle. Concurrent scheduling in the Active Bat location system.
In Proceedings of the Sixth IEEE International Workshop on Sensor Networks and Systems
for Pervasive Computing (PerSeNS 2010), Mannheim, Germany, March 2010.
Andrew Rice and Oliver Woodman. Crowd-sourcing World Models with OpenRoomMap.
In Proceedings of the Eighth IEEE International Conference on Pervasive Computing and
Communications (PerCom 2010), Work in Progress, Mannheim, Germany, March 2010.
vii
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Contents
1 Introduction 1
1.1 The potential of inertial sensors . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Limitations of scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Positioning systems, Bayesian filters and localisation 7
2.1 Absolute positioning technologies . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Outdoor positioning . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.2 Indoor positioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Relative positioning technologies . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Inertial navigation . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.2 Pedestrian dead reckoning . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Bayesian filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3.1 Kalman filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Linearised and extended Kalman filters . . . . . . . . . . . . . . . . 27
2.3.3 Multi-hypothesis filters . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.4 Grid-based filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.3.5 Particle filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Bayesian filters in positioning systems . . . . . . . . . . . . . . . . . . . . . 30
2.4.1 Sensor fusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.2 Constraints as pseudo-measurements . . . . . . . . . . . . . . . . . 32
2.5 Robot localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.5.1 Dealing with unmapped obstructions . . . . . . . . . . . . . . . . . . 36
2.6 Pedestrian localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.1 Differences to robot localisation . . . . . . . . . . . . . . . . . . . . 37
2.6.2 Map availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6.3 Concurrent work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
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3 Inertial navigation 41
3.1 Sources of error in MEMS inertial sensors . . . . . . . . . . . . . . . . . . . 43
3.1.1 Gyroscope error sources . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.2 Accelerometer error sources . . . . . . . . . . . . . . . . . . . . . . 46
3.2 Strapdown inertial navigation . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Error propagation in inertial navigation systems . . . . . . . . . . . . . . . . 51
3.3.1 Xsens Mtx example . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4 INS error correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.4.1 Error-state definition . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4.2 Error-state propagation . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.3 Error-state correction . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.4 Error-state transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.5 Xsens Mtx example . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4 Pedestrian dead reckoning 67
4.1 INS-based PDR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1.1 Zero velocity updates . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.1.2 Step segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.3 The cascading filter problem . . . . . . . . . . . . . . . . . . . . . . 71
4.1.4 Filter initialisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.1.5 Alternative filter formulation . . . . . . . . . . . . . . . . . . . . . . 74
4.2 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2.1 Previous testing methodologies . . . . . . . . . . . . . . . . . . . . 76
4.2.2 Ground truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3 Use of magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5 Enforcing environmental constraints with particle filters 87
5.1 Representing building constraints . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2 Localisation filter design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.2.1 Particle propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.2.2 Particle correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.2.3 Re-sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.3 Initialisation and localisation . . . . . . . . . . . . . . . . . . . . . . . . . . 98
5.3.1 Environmental Symmetry . . . . . . . . . . . . . . . . . . . . . . . 99
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5.3.2 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.4 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.4.1 Tracking accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6 Efficient localisation and tracking 115
6.1 Adaptive re-sampling schemes . . . . . . . . . . . . . . . . . . . . . . . . . 115
6.1.1 Likelihood-based adaptation . . . . . . . . . . . . . . . . . . . . . . 117
6.1.2 Kullback-Leibler distance adaptation . . . . . . . . . . . . . . . . . 118
6.1.3 Application to pedestrian localisation . . . . . . . . . . . . . . . . . 121
6.2 Detecting convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2.1 Clustered particle filtering . . . . . . . . . . . . . . . . . . . . . . . 127
6.3 Dependence on environmental constraints . . . . . . . . . . . . . . . . . . . 128
6.3.1 Open plan environments . . . . . . . . . . . . . . . . . . . . . . . . 128
6.3.2 Additional constraints . . . . . . . . . . . . . . . . . . . . . . . . . 130
6.3.3 Constraint accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.3.4 Tracking error characteristics . . . . . . . . . . . . . . . . . . . . . . 133
6.4 Dependence on user motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
7 Assisted localisation 139
7.1 WiFi-assisted localisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
7.1.1 Automatic radio map construction . . . . . . . . . . . . . . . . . . . 142
7.1.2 Prior generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
7.1.3 Containment measurements . . . . . . . . . . . . . . . . . . . . . . 148
7.1.4 Robustness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.1.5 Localisation performance . . . . . . . . . . . . . . . . . . . . . . . . 150
7.2 Mapping other signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.3 Assisted tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8 Conclusions 157
8.1 Research contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
8.1.1 Research question 1: Unconstrained inertial navigation . . . . . . . . 158
8.1.2 Research question 2: Constrained pedestrian dead reckoning . . . . . 159
8.1.3 Research question 3: Determining absolute location and heading . . . 160
8.1.4 Research question 4: Use of environmental constraints . . . . . . . . 160
8.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
8.2.1 Algorithmic enhancements . . . . . . . . . . . . . . . . . . . . . . . 161
8.2.2 Real-world deployment . . . . . . . . . . . . . . . . . . . . . . . . . 165
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8.3 Final words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
A Analysing noise in MEMS inertial sensors 169
A.1 Xsens Mtx analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
A.2 Conversion to sigma values . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
B Rotation matrices 175
B.1 Small angle approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.2 Rotation about an arbitrary axis . . . . . . . . . . . . . . . . . . . . . . . . . 176
B.3 Aligning arbitrary vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
C Kalman filters 179
C.1 Linearised Kalman filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
C.2 Extended Kalman filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
C.2.1 Alternative formulation . . . . . . . . . . . . . . . . . . . . . . . . . 183
D Ultrasonic correction of inertial navigation systems 185
D.1 Applying range measurements . . . . . . . . . . . . . . . . . . . . . . . . . 185
D.2 Filtering range measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 186
D.3 Unilateral versus multilateral positioning . . . . . . . . . . . . . . . . . . . . 187
D.4 PDR correction using the Active Bat system . . . . . . . . . . . . . . . . . . 188
D.4.1 Time synchronisation . . . . . . . . . . . . . . . . . . . . . . . . . . 189
D.4.2 Filtering range measurements . . . . . . . . . . . . . . . . . . . . . 189
E Error ellipses 191
F Glossary 193
References 195
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List of figures
1.1 The components of a pedestrian tracking system. . . . . . . . . . . . . . . . 5
2.1 The Location Stack model for location-aware computing. . . . . . . . . . . . 7
2.2 The Active Badge and Active Bat. . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 The correlation between accuracy and infrastructure requirements for indoor
positioning technologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Strapdown and stable platform inertial navigation systems. . . . . . . . . . . 20
2.5 Approximate drift rates for different grades of IMU. . . . . . . . . . . . . . . 21
2.6 The Xsens Mtx IMU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 A Markov process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.8 The path followed by a mobile robot. . . . . . . . . . . . . . . . . . . . . . . 34
2.9 An example of robot localisation. . . . . . . . . . . . . . . . . . . . . . . . . 35
2.10 Cascaded stacks in a pedestrian localisation system. . . . . . . . . . . . . . . 37
2.11 The backtracking particle filter. . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.1 The body and global frames-of-reference. . . . . . . . . . . . . . . . . . . . 41
3.2 The strapdown inertial navigation algorithm. . . . . . . . . . . . . . . . . . . 42
3.3 The mean drift incurred by an inertial navigation system (INS) using an Xsens
Mtx IMU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 The mean drifts incurred when error sources were selectively removed. . . . . 53
3.5 INS correction using an error-state Kalman filter. . . . . . . . . . . . . . . . 63
3.6 The mean drift incurred by an INS when uncorrected, naı¨vely corrected by
zero velocity updates (ZVUs) and corrected by ZVUs using an error-state
Kalman filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.7 The mean drift incurred when the INS was corrected using the naı¨ve method. 66
3.8 The mean drift incurred when the INS was corrected using an error-state
Kalman filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1 The magnitude of the angular velocity of a pedestrian’s foot during walking. . 68
4.2 The segmentation of a pedestrian’s relative movement into step events. . . . . 70
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4.3 A step event is generated 0.5 s into a stance phase, or at the end of a stance
phase lasting less than 0.5 s. . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4 An example of the fine-grained motion calculated by an INS (with ZVU cor-
rections) and the corresponding step events for a pedestrian walking down a
flight of stairs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 The problem of delayed drift correction. . . . . . . . . . . . . . . . . . . . . 73
4.7 The final drift in position of a PDR system can be a poor indication of its true
performance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.8 An Active Bat mounted above an Xsens IMU on the user’s foot. . . . . . . . 76
4.9 Paths generated by a PDR filter when using different constraints and measure-
ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.10 The 3-dimensional drift in position incurred by the PDR filter during an ex-
ample walk. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.11 Eight more example paths generated by the PDR filter. . . . . . . . . . . . . 82
4.12 The error in heading incurred by the PDR filter during nine walks. . . . . . . 83
4.13 The drift in horizontal position incurred by the PDR filter during nine walks. . 83
4.14 The error in vertical displacement incurred by the PDR filter during nine walks. 83
4.15 A path of step events calculated by the PDR filter as a user walked throughout
the William Gates building. . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.16 Magnetic field vectors measured by a foot-mounted IMU. . . . . . . . . . . . 85
5.1 The pedestrian localisation problem. . . . . . . . . . . . . . . . . . . . . . . 87
5.2 The structure of a pedestrian localisation and tracking system. . . . . . . . . 88
5.3 A lecture theatre and its 2.5-dimensional representation. . . . . . . . . . . . . 90
5.4 The 2.5-dimensional map data-structure and its representation in memory. . . 90
5.5 An example localisation in a three storey building. . . . . . . . . . . . . . . . 100
5.6 The environmental symmetry problem. . . . . . . . . . . . . . . . . . . . . . 101
5.7 The percentage of localisation failures as a function of the number of particles
used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.8 The average time required to process a step event when different numbers of
particles are used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.9 The 95% error ellipse calculated for a cloud of particles. . . . . . . . . . . . 106
5.10 A path of step events calculated by the PDR filter and the corrected path cal-
culated by the localisation filter. . . . . . . . . . . . . . . . . . . . . . . . . 107
5.11 Tracking using the localisation filter. . . . . . . . . . . . . . . . . . . . . . . 110
5.12 The percentage of failures during tracking as a function of the number of
particles used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.13 Paths calculated by the localisation filter and corresponding positions obtained
from the Active Bat system. . . . . . . . . . . . . . . . . . . . . . . . . . . 111
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5.14 Euclidean distances between matched Active Bat and localisation filter posi-
tions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6.1 The problem with likelihood-based adaptation. . . . . . . . . . . . . . . . . 118
6.2 The number of particles generated by KLD-adaptation for each update during
an example localisation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.3 The time required to process each step event during an example localisation. . 124
6.4 The cumulative distribution of the Euclidean distances between matched Ac-
tive Bat and localisation filter positions. . . . . . . . . . . . . . . . . . . . . 125
6.5 Clustered particles during localisation. . . . . . . . . . . . . . . . . . . . . . 125
6.6 Paths calculated by the localisation filter when using different types of con-
straint. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.7 Tracking accuracy with and without interior wall and furniture constraints. . . 130
6.8 The OpenRoomMap editor. . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.9 Bias errors caused by nearby environmental constraints. . . . . . . . . . . . . 133
6.10 Positions calculated by the localisation filter are biased towards the centre-line
of a corridor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.11 A Voronoi diagram of an office environment. . . . . . . . . . . . . . . . . . 135
6.12 Tracking a user as they give a 30 minute talk. . . . . . . . . . . . . . . . . . 137
7.1 Multiple clusters formed due to environmental symmetry. . . . . . . . . . . . 140
7.2 Dividing the 2.5-dimensional representation of a lecture theatre into cells. . . 144
7.3 Automated radio map construction. . . . . . . . . . . . . . . . . . . . . . . . 145
7.4 WiFi-assisted localisation using constrained priors and containment measure-
ments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
7.5 The cumulative distribution of ndist(풛푡, 풄푖) for cells overlapped by a cluster
of particles corresponding to the user’s true location. . . . . . . . . . . . . . 151
7.6 Localisation filter performance during a WiFi-assisted localisation. . . . . . . 151
7.7 Mapping magnetic field deviations on the ground floor of the William Gates
building. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
8.1 An example path calculated by an extended version of the localisation filter
that supports elevators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
A.1 A possible log-log plot of Allan Deviation . . . . . . . . . . . . . . . . . . . 170
A.2 Allan Deviation curves for the gyroscopes of an Xsens Mtx. . . . . . . . . . 171
A.3 Allan Deviation curves for the accelerometers of an Xsens Mtx. . . . . . . . 172
C.1 The Kalman filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
C.2 The linearised Kalman filter. . . . . . . . . . . . . . . . . . . . . . . . . . . 181
C.3 The extended Kalman filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
xv
C.4 The extended Kalman filter (alternative formulation). . . . . . . . . . . . . . 184
E.1 An error ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
xvi
List of tables
2.1 A summary of indoor absolute positioning systems. . . . . . . . . . . . . . . 17
2.2 A summary of commonly used Bayesian filters. . . . . . . . . . . . . . . . . 31
3.1 A summary of MEMS gyroscope error sources. . . . . . . . . . . . . . . . . 46
3.2 A summary of MEMS linear accelerometer error sources. . . . . . . . . . . . 47
4.1 The application of measurements, pseudo-measurements and error-covariance
resets during testing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.1 Coverage and mean scans per cell for WiFi radio maps constructed using dif-
ferent cell granularities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.2 A summary of WiFi-assisted localisation performance in the William Gates
building. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
A.1 Gyroscope noise measurements for the Xsens Mtx. . . . . . . . . . . . . . . 171
A.2 Accelerometer noise measurements for the Xsens Mtx. . . . . . . . . . . . . 172
A.3 The 1휎 values of white noise sequences that underlie bias instability and ran-
dom noise processes for an Xsens Mtx. . . . . . . . . . . . . . . . . . . . . . 173
xvii
xviii
Chapter 1
Introduction
As computers become smaller, cheaper and more powerful, they are becoming increasingly
common in our everyday environment. Computers can now be found in light switches, door
locks, coffee machines, microwaves and fridges. There are multiple computers in most mod-
ern cars, controlling everything from a vehicle’s traction and braking to its interior lights and
sound system. These computers assist us as we go about our lives, often without us noticing
their presence. This is known as ubiquitous computing, a term first used by Mark Weiser in the
late 1980s. In his seminal paper on the subject he states that “The most profound technologies
are those that disappear. They weave themselves into the fabric of everyday life until they are
indistinguishable from it.” [1].
For ubiquitous computers to disappear in the way envisaged by Mark Weiser, they must be
able to infer some context about the environment into which they are placed. For example a
computer controlled light switch is no better than a physical one if it is unable to sense whether
there are people in the room or whether it is light or dark. In contrast a light switch that is able
to sense these properties can automatically switch on when a person enters a darkened room
and off once the room has been empty for a certain period of time.
There are many types of contextual information that can be used by ubiquitous computers.
Location information is one of the most important, particularly now that mobile computing
platforms are cheap and readily available. A computer that is able to determine its location can
adapt to its local environment. For example if a mobile telephone senses that it is in a lecture
theatre then it may choose to automatically switch to silent mode, avoiding a disturbance
should an incoming call be received. Location information alone is not sufficient for this
task. It is also necessary to have access to a world model that can be used to give meaning
to a location [2]. In this example a suitable world model consists of a map of the building
annotated with spatial zones that indicate regions within which audible disturbances should
be suppressed.
1
The branch of ubiquitous computing in which the primary source of context is location (to-
gether with a world model) is often referred to as location-aware computing [3]. Perhaps
the most widespread example of location-aware computing today is satellite navigation, in
which location information obtained using a global navigation satellite system (GNSS) is
used together with a map of the road network in order to provide turn-by-turn navigation for
a vehicle [4].
Whilst GNSSs allow devices to position themselves when outdoors, they rarely support indoor
positioning because the satellite signals used are too weak to penetrate most buildings. As a
result indoor positioning systems have been developed based on a variety of other technolo-
gies including infra-red [5], ultrasound [6, 7, 8] and radio [9, 10, 11, 12]. Of particular interest
for indoor location-aware applications has been the tracking of people, which is also the focus
of this thesis. For example the Active Badge location system was used to track employees
at Olivetti research in Cambridge [5]. This allowed telephone calls to be automatically for-
warded to the rooms within which the intended recipients were located, as well as allowing
computer terminals to automatically retrieve the preferences of whoever sat in front of them.
Despite the fact that indoor positioning systems enable the development of these and many
more location-aware applications, their adoption to date has been extremely limited.
One of the biggest problems preventing the widespread adoption of indoor positioning sys-
tems is the correlation that currently exists between the accuracy of such systems and their cost
and infrastructure requirements. Nearly all indoor location-aware applications need to know
the room within which a person or device is located. For pedestrian tracking metre-level ac-
curacy is often required so that an application can determine which objects in the environment
are close enough to the user for a physical interaction to take place (e.g. whether a user is
positioned close enough to a computer to be using it). Unfortunately existing indoor position-
ing systems that are able to achieve such accuracies require large amounts of infrastructure
to be installed into the environment. For example some ultrasound-based systems are able to
position devices to within a few centimetres [6], but remain prohibitively expensive and rely
on large amounts of fixed infrastructure. Such infrastructure is not only labour intensive to
install, but is also expensive to maintain. At the other end of the scale are systems based on
technologies such as WiFi [10, 11, 12]. Although such systems are far cheaper and require
less infrastructure, they are often unable to reliably determine even the room within which a
person or device is located in real world deployments. Before indoor positioning systems can
be widely adopted it will be necessary to develop positioning technologies that are accurate
enough for the majority of location-aware applications whilst at the same time having minimal
cost and infrastructure requirements.
2
1.1 The potential of inertial sensors
In theory it is possible for an object to keep track of changes in its own position using an
inertial measurement unit (IMU) containing linear accelerometers and gyroscopes. The basic
principle is to track changes in the device’s orientation using the gyroscopes, which can be
used to project the accelerations measured locally by the device into a global frame of ref-
erence. These accelerations can then be double integrated to track changes in the object’s
position [13]. In practice however, errors rapidly accumulate in the tracked position due to the
propagation of measurement errors through the projection and integration calculations. Such
errors are collectively referred to as drift. Despite the accumulation of drift, inertial navigation
systems are commonly used in aeroplanes, missiles and submarines. Aeroplanes and missiles
can use GNSSs to periodically correct drift, however this is not possible for a submerged sub-
marine because the satellite signals used do not penetrate far underwater. Hence extremely
accurate inertial measurement units are required. Those deployed in modern submarines are
accurate enough to limit drift to within a few nautical miles after one day of operation, how-
ever such devices are large and heavy, in some cases measuring metres across.
Until recently it was not possible to manufacture IMUs that were small and light enough to be
used for pedestrian tracking, however recent advances in the construction of micro-machined
electromechanical systems (MEMS) have now made it possible to manufacture small inertial
sensor chips from which such devices can be built. MEMS inertial sensors are not as accurate
as the large mechanical devices used in submarines, but they are far cheaper and contain very
few moving parts, making them extremely durable. The quality of MEMS inertial sensors is
improving rapidly and their price continues to fall, partly driven by their mass manufacture
for use in many of today’s modern mobile telephones (as user input devices) and laptops (to
protect hard disks by detecting sudden accelerations). This has allowed the development of
relative positioning systems that are able to track the movement of pedestrians relative to their
initial positions and headings [14, 15, 16]. Such systems are able to operate both outside and
indoors, however without periodic correction their accuracy degrades over time. This thesis
investigates the use of environmental constraints as a means of providing such corrections
when tracking indoors. In particular the movement of a pedestrian is naturally constrained by
walls, floors and ceilings that are present in nearly all indoor environments. By using such
constraints it may be possible to track a pedestrian indefinitely in an indoor environment, with
no degradation in accuracy and without the need to install any fixed infrastructure. Hence
this approach could enable the development of new indoor location systems that are suitable
for use when the installation of such infrastructure is either uneconomical, impractical or
impossible.
This thesis investigates the development of an indoor pedestrian location system based on the
ideas outlined above. In doing so it aims to answer the following questions:
3
1. How rapidly does drift accumulate in an inertial navigation system when using
current state-of-the-art MEMS inertial sensors? The aim here is to determine the
extent to which it is possible to track the relative movements of an object or person
using modern MEMS inertial sensors together with general purpose inertial navigation
algorithms.
2. How accurately can the relative movement of a pedestrian be measured using
MEMS inertial sensors? This question differs to the one above because it deals specif-
ically with the tracking of people. It may be possible to exploit domain specific knowl-
edge in order to do better than the general case.
3. What is the minimum infrastructure or knowledge required to determine a pedes-
trian’s absolute location and heading? A relative tracking system based on inertial
sensors can only be used to track a user’s absolute position if it is provided with their
initial position and heading. The aim is to find a way in which this initial state can be
determined that does not rely on the installation of large amounts of fixed infrastructure
(and preferably none at all).
4. Can environmental constraints be used to prevent the accumulation of drift over
time? This would allow the position of an individual to be tracked for an indefinite
period of time. If this is possible, to what accuracy can the individual’s position be
tracked?
1.2 Thesis outline
The structure of the remainder of this thesis is outlined below. An indoor pedestrian tracking
system is developed throughout in order to answer the questions posed above. The major
components of the resulting system are shown in Figure 1.1, which is annotated to indicate
the chapters within which each of the components are described.
Chapter 2 outlines related work and motivates the remainder of this thesis. Existing po-
sitioning technologies, robotics research and concurrent work on pedestrian localisation are
described.
Chapter 3 describes the error characteristics of MEMS inertial sensors, introduces the funda-
mentals of inertial navigation and outlines the propagation of errors through inertial navigation
systems. A general purpose extended Kalman filter is presented that is able to correct such
systems using external measurements and constraints.
Chapter 4 describes the application of inertial navigation techniques to the problem of pedes-
trian dead reckoning, in which the aim is the track a pedestrian’s relative movements over
4
Pedestrian dead reckoning filter
Localisation particle filter
Step events
Absolute positions
Environmental constraints
Inertial measurement unit
Additional measurements
Chapter 3
Chapters 5 and 6 Chapter 7
Chapter 4
Figure 1.1: The components of the pedestrian tracking system developed throughout this
thesis. Annotations indicate the chapter(s) in which each component is described.
time. A pedestrian dead reckoning filter is developed based on the extended Kalman filter
presented in Chapter 3.
Chapter 5 investigates how knowledge of environmental constraints can be combined with
relative movement information in order to determine and subsequently track the absolute po-
sition of a pedestrian. A particle filter implementation is developed, which is able to localise
and subsequently track a user by combining relative movement information with constraints
defined by a map of the environment.
Chapter 6 describes a number of optimisations that can be made to the algorithms developed
in Chapter 5. The performance and error characteristics of the resulting system are investi-
gated in detail.
Chapter 7 introduces the concept of assisted localisation, in which additional measurements
and constraints can be used to reduce the cost of localisation. WiFi-assisted localisation is
described in detail and is implemented as an extension to the algorithms developed in Chapters
5 and 6.
Chapter 8 revisits the research questions posed earlier in this chapter, outlines possible av-
enues for future research and summarises the main contributions of this thesis.
1.3 Limitations of scope
Although a pedestrian tracking system is developed throughout this thesis, it is important to
note that the aim is not to develop a complete system that is suitable for use in a practical
scenario. Instead this thesis focuses on the development and evaluation of the system’s core
components and algorithms, with the aim of answering the research questions posed above.
Hence topics such as scaling the system to support many users, location privacy and the dis-
tribution of computational tasks over multiple devices are not discussed. Possible avenues of
5
research in these areas are described in Section 8.2.2. In addition the following limitations of
scope also apply:
∙ This thesis is concerned with the problem of tracking people in an indoor environment.
Tracking other objects is not considered, although the algorithms developed in Chapters
5, 6 and 7 can be applied to determine and subsequently track the position of any object
that is able to measure its own relative movements to a sufficient accuracy. Algorithms
that achieve this requirement for pedestrians are presented in Chapter 4, however they
make use of domain specific knowledge and are unsuitable for tracking arbitrary objects.
∙ The algorithms developed in this thesis assume that walking is the only means by which
a pedestrian can move in an indoor environment. This is a valid assumption in most
buildings (assuming normal user behaviour), however it is violated by moving floor
surfaces such as elevators and escalators. Ways in which the algorithms presented in this
thesis could be extended to handle moving floor surfaces are outlined in Section 8.2.1.1,
however the implementation and testing of such extensions is left as future work. The
assumption is also violated by alternative modes of transportation such as wheelchairs.
To support such cases it would be necessary to develop dedicated algorithms to track
relative movement for each mode of transportation.
6
Chapter 2
Positioning systems, Bayesian filters and
localisation
A wide range of different sensor technologies have been used to develop indoor location
systems. Systems developed in the late 80s and early 90s were nearly all based on a single type
of sensor. For example the Active Badge system was based solely on infra-red sensors [5],
whilst its successor, the Active Bat system, used only ultrasonic sensors [6]. Other positioning
systems were developed based on optical, narrowband and wideband radio technologies. In
a summary of such technologies, Welch and Foxlin concluded that a single technology that
performs well in all situations has yet to be discovered [17].
The lack of a single silver bullet technology for indoor positioning has led to a more recent
shift towards the development of indoor positioning systems that use multiple sensor types (an
approach that has been used for outdoor positioning since the 1960s and is commonly known
as integrated navigation [18]). Such systems combine measurements from different sensors
in a process known as sensor fusion. The idea is to combine measurements in such a way
that the resulting system retains the benefits of each individual sensor technology whilst at the
same time mitigating their individual weaknesses. One example is the Constellation system,
in which both ultrasonic and inertial sensors were used [19].
Sensors
Measurements
Fusion
Arrangements
Contextual Fusion
Activities
Figure 2.1: The Location Stack model for location-aware computing.
7
The components of positioning systems that use both single and multiple sensor types can
be defined in the context of the bottom three layers of the Location Stack, as shown in Fig-
ure 2.1. The Location Stack was initially proposed by Hightower et al. as an abstraction for
location-aware computing analogous to the Open System Interconnect (OSI) model for com-
puter networks [20, 21]. The upper three layers of the Location Stack abstract services that
might be built on top of location systems for use by location-aware applications. The bottom
three layers abstract the components of location systems themselves and hence are of direct
relevance to this thesis:
∙ The sensors layer: Defines the physical sensors of a location system.
∙ The measurements layer: Defines the types of measurements obtained from the physical
sensors.
∙ The fusion layer: Defines the algorithms that combine measurements in order to esti-
mate the positions of tracked objects.
Location systems that use a single sensor type often apply relatively simple deterministic al-
gorithms in their fusion layers in order to estimate the position of a device. For example
multi-lateration and multi-angulation algorithms are often used to calculate the position of
an object based on multiple distance or angle-of-arrival measurements [22]. In systems that
use multiple sensor types, fusion algorithms have the more difficult job of estimating the po-
sition of a tracked object based on measurements of different types and with different error
characteristics. Furthermore different sensors may have different sampling rates and are of-
ten unsynchronized. Bayesian filters provide a mathematical framework that supports sensor
fusion in spite of these difficulties [23]. Rather than computing a single most likely posi-
tion estimate, Bayesian filters maintain a probability distribution that describes not only the
most likely position of a device, but also the uncertainty in that position given the measure-
ments that have been received. This distribution is evolved over time based on a model of
the object’s dynamics and is corrected by measurements as they are received, taking the error
characteristics of each measurement into account.
The first part of this chapter describes the topics outlined above in more detail. Section 2.1
describes underlying sensor technologies that have been used to develop previous absolute
positioning systems. Sensors for tracking the relative position of a device are described in
Section 2.2. Bayesian filtering is introduced in its general form in Section 2.3 and its appli-
cation within positioning systems is described in Section 2.4. The second half of this chapter
describes work in the related field of robotics. Section 2.5 describes algorithms that have
been developed to solve the robot localisation problem, in which the goal is to allow a robot
to determine its absolute position in a known environment using only measurements made
by on-board sensors. Finally, the possibility of applying robot localisation techniques to the
problem of pedestrian tracking is discussed in Section 2.6.
8
2.1 Absolute positioning technologies
Many technologies have been used to develop positioning systems for both outdoor and indoor
use. Although not directly relevant to this thesis, widely used outdoor positioning technologies
are outlined in Section 2.1.1. Technologies that have been developed specifically for indoor
positioning are described in Section 2.1.2.
2.1.1 Outdoor positioning
2.1.1.1 Global navigation satellite systems
Global navigation satellite systems (GNSSs) such as the United States’ Global Positioning
System (GPS) and Russia’s GLONASS allow mobile devices to position themselves world-
wide when outdoors [4]. To do this a receiver must track signals from four or more satellites,
from which pseudo-ranges (distances to satellites that include possible errors due to an offset
in the receiver’s clock) can be obtained. Pseudo-multi-lateration algorithms (multi-lateration
in which the receiver’s clock offset is an additional unknown) are then used to resolve the po-
sition and clock offset of the receiver. The accuracy to which a GNSS receiver can compute its
position is highly dependent on the local environment. With a clear view of the sky and when
tracking all visible satellites in the GPS constellation, 1휎 position errors of approximately
7 m can be achieved by single frequency GPS receivers [18]. This accuracy is significantly
degraded when large parts of the sky are blocked from view, for example in an urban canyon.
In such an environment the number of visible satellites is reduced, whilst multipath effects
caused by reflections from nearby obstructions (e.g. buildings and trees) introduce additional
error.
In addition to GPS and GLONASS, there are several GNSSs that are currently under develop-
ment. Europe’s Galileo1 GNSS is due to reach full operational capability between 2016 and
2018. It will be interoperable with GPS and should allow receivers to calculate their positions
to within 5 m horizontally and 8 m vertically 95% of the time [24]. Galileo satellites will
broadcast signals with 5 dB more power and over wider frequency bands compared to GPS
satellites, making their signals easier to detect and more resistant to multipath effects that
significantly degrade the accuracy of GPS in urban areas. China is also developing a GNSS
called Compass, which is set to provide global coverage by around 2020.
1http://ec.europa.eu/transport/galileo
9
2.1.1.2 WiFi
WiFi access points are now densely deployed in many urban environments. A device can
estimate its position by searching a database containing the locations of such access points
for ones that are visible from its current location. The accuracy of such an approach is highly
dependent on the density of access points, however positioning errors are at worst equal to
the range of an access point (typically 50 − 100 m). When multiple access points are visible
their signal strengths can be used to estimate a more accurate position. In the Placelab project
devices were able to use WiFi access points to position themselves with a median accuracy of
20.5 m in an urban environment [25].
One benefit of WiFi-based positioning is that the same techniques can be applied in both out-
door and indoor environments. The major disadvantages are that coverage is currently limited
to urban environments (since WiFi access points are not yet common in rural areas) and that a
database of access points (often called a radio map) must be built for the areas in which such
systems are to be used. Furthermore such databases need to be frequently updated as access
points are added, moved and removed from the environment. Several commercial companies
currently provide WiFi-based positioning solutions. These include Skyhook2, whose service
combines WiFi-based positioning, GPS and position estimation based on signals received
from cellular networks.
2.1.1.3 Cellular telephony networks
Cellular network infrastructure for telephony is now widespread throughout large parts of the
world. A wide range of measurements can be obtained to estimate the position of any device
that is connected to such a network, including angle-of-arrival (AOA), time-of-arrival (TOA)
and time-difference-of-arrival (TDOA) measurements of radio signals travelling between the
device and multiple cell towers [22, 26]. The simplest approach is to estimate the position of
a device as the position of the cell tower to which it is associated, with the accuracy depen-
dent solely on the dimensions of the cell. More accurate approaches have received increased
attention since the United States Federal Communications Commission introduced an E-911
mandate that required mobile carriers to provide the approximate positions of mobile devices
making emergency calls3. By 2012 carriers must provide positions that are accurate to within
50 m for 67% of calls and within 150 m for 95% of calls (a deadline that has already been
extended by four years from the original deadline of 2008).
It is also possible for a mobile device to estimate its own position based on cell tower signal
strengths. This approach is very similar to that commonly used to implement WiFi-based
2http://www.skyhookwireless.com
3http://www.wirelessdevnet.com/e911/factsheet requirements 012001.pdf
10
positioning systems such as those described in Section 2.1.1.2. Cellular networks provide
wider coverage, however the accuracy with which a device can position itself using such
networks is worse than can typically be achieved using WiFi access points. The Placelab
project demonstrated a median accuracy of 107.2 m when signals from GSM towers were
used to position devices in an urban environment [25].
2.1.2 Indoor positioning
Satellite positioning systems such as GPS do not work well (if at all) in indoor environments.
To track GPS signals indoors typically requires a receiver capable of tracking signals with
power levels ranging from −160 dBW to −200 dBW, however a typical receiver has a noise
floor of around −131 dBW [27]. Multipath effects are likely to cause a degradation in accu-
racy even if a receiver is able to track signals from a sufficient number of satellites. Unlike
in outdoor environments, reflected signals are often stronger than those received via direct
line-of-sight when indoors. Hence it is difficult for a receiver to identify and track the correct
(i.e. direct) signals. Although the Galileo system will address some of these problems by
transmitting signals at higher power and over wider frequency ranges, its accuracy will not
be sufficient to reliably determine the floor of a building on which a device is located. Hence
such systems are not accurate enough for indoor location-aware applications, nearly all of
which require at least room-level accuracy. Cellular-based positioning systems are similarly
inaccurate. WiFi-based positioning systems are able to achieve accuracies sufficient for some
location-aware applications. Despite this they have many drawbacks when applied in indoor
environments, as will be described in Section 2.1.2.4. As a result other technologies have been
developed specifically for indoor positioning, as described below.
2.1.2.1 Infra-red
Infra-red light has been used to provide room-level location information (i.e. the room within
which each tracked object is located) [5]. If an infra-red signal transmitted from one device
is received by another then it is highly likely that they are located in the same room because
infra-red light does not pass through walls. Furthermore infra-red light is strongly reflected
by walls, meaning that a line-of-sight between the two devices is not required. Infra-red
transmitters and sensors are cheap, power efficient and readily available, making them ideal
for providing room-level location information.
The Active Badge system used a network of infra-red receivers to detect badges such as the
one shown in Figure 2.2(a), each of which regularly emitted a unique infra-red code [5].
Badges were worn by employees at the Olivetti research laboratory in which the system was
deployed. The initial application of the system was to aid receptionists, who were able to
11
(a) (b)
Figure 2.2: (a) An active badge. (b) An active bat.
route incoming telephone calls to rooms within which their intended recipients were located.
After an initial two week trial period it was found that the number of calls that did not reach
their intended recipients had dropped substantially.
2.1.2.2 Ultrasound
Ultrasonic positioning systems are able to estimate the positions of tracked objects in three di-
mensions, with accuracies in the order of centimetres. The Active Bat system positions small
narrowband ultrasound transmitters known as bats [28, 6]. Figure 2.2(b) shows an example
of a bat designed to be carried by a human user, however this is only one of a number of
different designs that were manufactured. To position a bat it is queried over a radio chan-
nel. When a bat receives a query it transmits an ultrasonic pulse that is received by multiple
ceiling-mounted receivers. The time-of-flight of the signal to each receiver is measured by us-
ing the radio query to synchronise the bat and the static receivers. This is possible because the
speed of the radio query (≈ 3× 108 ms−1) is far greater than the speed of the ultrasonic pulse
(≈ 340 ms−1). When three or more times-of-flight are obtained a multi-lateration algorithm
is applied in order to calculate the bat’s position. The positions calculated by the system are
accurate to within 3 cm 95% of the time [6].
One problem with the Active Bat system is that ultrasonic noise can prevent a bat from being
positioned. Such noise is emitted by a variety of everyday objects including jingling keys and
rustling crisp packets. A second problem is that bats in a given area have to be positioned
one at a time in order to prevent collisions between their signals. Hence the system does
not scale to track large numbers of bats in small areas. The more recent Dolphin positioning
system makes use of broadband ultrasound to overcome these limitations [7]. Dolphin is able
to position multiple tags simultaneously and in the presence of ultrasonic noise. Its accuracy
12
is similar to that of the Active Bat system.
The Active Bat and Dolphin systems are both multilateral positioning systems, meaning that
the position of a tag is calculated by some computational infrastructure in the environment
rather than by the tag itself. In contrast Cricket is a unilateral4 ultrasonic positioning sys-
tem in which mobile tags calculate their own positions by timing ultrasonic pulses that are
transmitted by beacons in the environment [8, 29]. To prevent signal collisions beacons in a
given area transmit one at a time. This makes positioning a moving tag more difficult than in
the multilateral case, since it is not possible to obtain multiple times-of-flight corresponding
to the same point in time. As a result the accuracy is significantly worse relative to that of
a comparable multilateral system [30]. This problem can be addressed by using broadband
ultrasound, as was demonstrated by a prototype system based on the Dolphin hardware [31].
Although unilateral ultrasound positioning systems are less accurate than their multilateral
counterparts, they do have several advantages. Multiple tags can calculate their positions
based on the same beacon transmissions, meaning such systems scale well to track large
numbers of tags in small areas. A second benefit is location privacy, since the only object
that can calculate the position of a tag is the tag itself (which can then choose whether or not
to share its position).
The main drawback that affects all ultrasound-based positioning systems is that they require
large amounts of infrastructure (e.g. receivers or beacons, data cables, power cables and so
on) to be installed into the environment. Each tag must have a line-of-sight to at least three
receivers (or beacons in a unilateral system) if it is to be successfully positioned. If a user
wears a tag around their neck then their body will typically occlude half of the receivers in
a given room, meaning that in practice more than three receivers need to be installed in each
room in order to achieve acceptable coverage. For example the current Active Bat deployment
in the William Gates building (Cambridge) uses 12 receivers to cover a small (4.5 m × 3 m)
office. The number of receivers that are needed could be reduced by requiring that tags are
mounted on the users’ heads (e.g. attached to hats or helmets), however this would be less
acceptable to most users.
2.1.2.3 Ultra-wideband radio
Ultra-wideband (UWB) radio has been used to develop several commercial indoor positioning
systems. Ubisense5 develop systems that are able to position small tags that emit pulses of
UWB radio to within 0.13 m of their true positions 95% of the time in unobstructed line-of-
sight conditions (i.e. when there are unobstructed lines-of-sight between tags and receivers).
4The Cricket hardware can be deployed to form both multilateral and unilateral positioning systems, however
it was originally developed and deployed as a unilateral system.
5http://www.ubisense.net
13
This accuracy degrades to nearer 1 m when some of the signals have passed through walls.
The Ubisense positioning system uses both TDOA and AOA measurements to position tags.
TimeDomain6 have developed a similar UWB positioning system called PLUS in which only
TDOA measurements are used.
The main benefit of UWB is that by utilising a wide frequency range it is possible for tags
to transmit pulses that are extremely short in duration, making it easier to detect and discard
reflected signals [22]. A secondary benefit is that the transmitted signals are able to pass
through a wide range of materials, meaning that visible line-of-sight is not required between
mobile tags and static receivers (although better accuracies are usually obtained when this
is the case). As a result the number of receivers that must be placed into an environment is
lower than would be required for an equivalent ultrasound-based system. Unfortunately UWB
positioning systems remain very expensive, in part due to the high precision clocks that are
required to measure the time differences between radio signals arriving at different receivers
in the environment.
2.1.2.4 WiFi
Positioning using received signal strength indication (RSSI) measurements of signals received
from WiFi access points has received much attention due to the number of access points that
are already installed in many indoor environments. Such systems usually fall into one of two
basic classes [22]. In the first class signal strengths are used in conjunction with a radio prop-
agation model in order to estimate distances between the receiver and multiple base stations
in the environment. Multi-lateration algorithms can then be applied to estimate the receiver’s
position. The second class requires the construction of a radio map, in which surveyed sig-
nal strengths are recorded at known positions throughout the environment. During tracking a
receiver can estimate its position by comparing observed signal strengths with entries in the
map. This is known as fingerprinting and is the more accurate of the two approaches, however
requires more effort because it is necessary to construct and subsequently maintain a radio
map of the environment. In addition to the two classes already described, it is possible to
define a third class of system in which a radio propagation model is used to derive some or all
of the entries in a radio map.
Radio propagation in indoor environments is dependent on many factors such as the wall
and floor materials, which are difficult to model accurately. Complex propagation models
have been devised that attempt to take such factors into account [32], however RF-location
systems that rely on such models (either to estimate distances between a device and multiple
base stations or to derive a radio map) have been unable to match the accuracies achieved by
6http://www.timedomain.com
14
systems in which manually surveyed radio maps are used [10].
A number of different algorithms exist for estimating the position of a device based on a scan
of visible access points and a pre-constructed radio map. The RADAR system calculates the
Euclidean distance between the vector of signal strengths observed by the device and each
entry in the radio map [10]. The position of the device is then estimated as the position in the
radio map for which the smallest distance was calculated. This approach results in an error
of less than 9 m 95% of the time. Wang et al. [11] used a similar algorithm to obtain a mean
error of 6.44 m. Both of these systems are deterministic, meaning that they compute a single
best guess of the position of a device. An alternative approach is to compute a probability
distribution of the device’s position. The Horus system uses such an approach, with a reported
accuracy of 1.4 m 95% of the time [12].
One of the main problems that limits the accuracy of all WiFi-based positioning systems is that
radio propagation is not only dependent on static factors such as building materials, but also
on dynamic factors such as whether doors are open or closed and whether humans are present
in the environment. The presence of humans in an environment is a particular problem, since
a human body obstructing the line-of-sight to an access point causes signal attenuation that
can result in a reduction of up to 9 dBm in the measured RSSI [33]. In practice it is only
possible to achieve the kind of accuracies reported by the developers of the Horus system
if these dynamic factors are controlled, which is not practical for real-world deployments.
A further problem specific to map-based systems is that the manual construction of radio
maps is very time consuming [34]. In one of the few large-scale deployments of a WiFi
location system, 28 man-hours were required to construct a radio map covering a 12,000 m2
building [35]. Furthermore this process must be repeated whenever an access point is added,
moved or removed from the environment if a degradation in the accuracy of the system is to
be avoided. The use of radio propagation models to derive some or all of the entries in a radio
map can reduce the time required, however the accuracy of the resulting system will also be
reduced.
Although the initial interest in WiFi-based positioning was in part due to the fact that access
points were already deployed in many buildings, all of the systems described above were
tested using a density of access points far higher than would have been installed for the sole
purpose of providing wireless connectivity. This is because multiple access points need to
be visible to position a device to a reasonable level of accuracy, whilst only one access point
needs to be visible for communication. Hence a single access point is sufficient to provide
wireless connectivity throughout a small building (e.g. a typical house), whilst at least three
are required in order to achieve anything close to the positioning accuracies quoted above.
15
2.1.2.5 Vision
A wide range of indoor positioning systems have been developed in which cameras are used
to track objects in the environment. Tracking pedestrians using multiple camera views is a
well researched problem, with techniques such as background subtraction and blob detection
commonly used [36, 37]. The EasyLiving project used such algorithms together with two
stereo cameras (i.e. four views in total) in order to track the users of an intelligent environment
to an accuracy of around 10 cm in the horizontal plane [38]. This was achieved without
requiring a user to wear a tracking device or visual marker. Although this is a highly desirable
property, it makes the identification of each user far more difficult. EasyLiving maintained
colour histograms of each user to ensure consistent assignment of unique identifiers to users
during tracking, however these identifiers did not relate to the absolute identities of the users.
A user previously tracked by the system would be assigned a new identifier when re-entering
the environment after a sufficiently long absence. Camera-based tracking systems such as
the one used in the EasyLiving project often perform poorly when users are partially or fully
occluded and also when there are rapid changes in illumination (e.g. when a light is turned on
or off, or when clouds obscure the sun on a sunny day).
Camera-based positioning systems span a wide range of accuracies and deployment costs. At
the other extreme to the EasyLiving project are expensive motion capture systems, in which
multiple cameras are used to track small passive or active markers. Commercial motion cap-
ture systems such as Vicon7 MX can track passive markers to sub-millimetre accuracies at
rates in excess of 500 Hz. The markers themselves are designed to be easy to detect, however
marker identification is a difficult problem. The IMPULSE motion capture system developed
by Phasespace8 uses active LED markers, which are modulated at unique frequencies to allow
identification. Vicon MX allows an operator to define constraints that exist between different
markers (e.g. the layout of markers that are attached to a rigid object). When possible this
model is used to automatically infer the identity of each marker during tracking.
2.1.2.6 Summary
Table 2.1 summarises the indoor positioning systems described above, although this is by no
means an exhaustive list. Other technologies that have been used for indoor positioning in-
clude Bluetooth [39], audible sound [40] and signals injected into the power lines running
through indoor environments [41]. One property that can be observed for all indoor position-
ing systems is a strong correlation between accuracy and the amount of infrastructure that
must be installed into the environment, as illustrated by Figure 2.3. Whilst ultrasound and
7http://www.vicon.com
8http://www.phasespace.com
16
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17
Location accuracy
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Figure 2.3: The correlation between accuracy and infrastructure requirements for indoor
positioning technologies.
vision-based systems are able to achieve very high accuracies, they also have huge infrastruc-
ture requirements. Since the amount of infrastructure is correlated to the cost of the system’s
hardware, it is often prohibitively expensive to deploy accurate positioning systems over large
areas. More importantly, the installation and maintenance costs of such systems are also high.
These costs (in particular the time required for installation) cannot be expected to fall simply
due to increased production levels and improved manufacturing techniques. Hence it is still
highly desirable to develop new indoor positioning systems that are located further towards
the bottom left of Figure 2.3.
2.2 Relative positioning technologies
Relative positioning systems (also known as dead reckoning systems) track the positions of
objects relative to their initial locations and orientations. As a simple example consider a
pedestrian who counts his steps and remembers any changes in his direction. The pedestrian
can use this information together with an estimate of his average stride length in order to
track his approximate position relative to his starting point. Note that over time the position
that he estimates will become less accurate. This is because he is concatenating estimates
of his relative movements (corresponding to steps), each of which will contain an error due
to variations in his actual stride length, errors in his estimation of heading changes and so
on. These errors accumulate in the estimated position, causing it to become less accurate the
further he walks. This accumulation of error is commonly known as drift and is common to all
relative positioning systems, however the rate at which drift grows and the magnitudes of the
errors incurred vary significantly between systems. For some applications drift is acceptable
because the target need only be tracked for a short period of time. For others it may be
possible to periodically reset the position based on measurements obtained from an absolute
positioning system.
18
2.2.1 Inertial navigation
Inertial navigation is a relative positioning technique in which an inertial measurement unit
(IMU) is tracked relative to its initial starting point, orientation and velocity. An IMU that
can be tracked in 3-dimensions usually consists of three orthogonal accelerometers and three
gyroscopes that are aligned with the accelerometers, as shown in Figure 2.4(a) [18]. An iner-
tial navigation system (INS) consists of an IMU together with a navigation processor, which
uses measurements from the IMU in order to track its orientation, position and velocity in
some frame of reference in which they are desired (the reference frame). The processor uses
angular rate measurements obtained from the gyroscopes in order to track the orientation of
the IMU relative to this frame. The known orientation is then used to transform the specific
force (acceleration due to all forces except for gravity) measured by the accelerometers into
this frame. Acceleration due to gravity is then added to the specific force to obtain the accel-
eration of the device. Finally, this acceleration is integrated once to track velocity and once
more to track the device’s position in the reference frame.
The type of INS described above is a strapdown system, in which the accelerometers measure
specific force in the IMU’s own frame of reference (the body frame). In a stable platform INS
the accelerometers and gyroscopes are instead mounted on a platform that is held in physical
alignment with the reference frame. This is achieved by mounting the platform using gimbals
(frames) that allow it freedom in all three axes, as shown in Figure 2.4(b). The gyroscopes are
used to detect rotations made by the platform relative to the reference frame. Signals are fed
directly from the gyroscopes to torque motors, which rotate the gimbals in order to counteract
the rotation and hence keep the platform correctly aligned. Thus acceleration due to gravity
can be added directly to the output of the accelerometers in order to obtain the acceleration
of the device in the reference frame, which can then be double integrated to track position.
The orientation of a stable platform INS relative to the reference frame can be obtained using
angle pick-offs that measure the angles between the gimbals of the device.
Stable platform and strapdown INSs are both based on the principle of double integrating ac-
celeration in the reference frame. Strapdown systems have reduced mechanical complexity
and as a result tend to be physically smaller and lighter than stable platform systems. These
benefits are achieved at the cost of increased computational complexity, however the require-
ments are now trivial relative to the computational power of modern processors. As a result
strapdown systems have become the dominant type of INS.
Inertial navigation makes no assumptions about the movement of the tracked device, except
that its accelerations and angular velocities remain within the operating range of the individual
sensors. The cost of this generality is that drift in position tends to grow at a rate that is
proportional to 푡3, where 푡 is the duration for which the device has been tracked [14]. This
cubic error growth is due to the propagation of gyroscope bias errors through the computations
19
Accelerometers
Gyroscopes
(a)
Accelerometers
Torque
motors
Gyroscopes
Stable
platform
Gimbals
Angle pick-offs
(b)
Figure 2.4: (a) A strapdown IMU. (b) A stable platform INS.
performed by the navigation processor, and is described in more detail in Section 3.3. The
average error that can be expected after a given duration depends on the quality of the IMU that
is used. Foxlin categorises IMUs into several grades, which broadly correspond to different
inertial sensor technologies [42]. Figure 2.5 gives lower bounds on the growth of drift for
different grades of device. The results shown were obtained by a simulation that considered
only random error sources and assumed a stationary device [42]. Note that in practice drift
will grow more rapidly due to additional systematic and dynamic errors.
Although a description of the different inertial sensor technologies is beyond the scope of this
thesis (such descriptions can be found in [13]) it is worth noting that the sensors required for
navigation-grade and strategic-grade IMUs are currently far too large and heavy to attach to
a pedestrian. Such devices are typically used in missiles, aircraft and submarines. Note that
even a perfect IMU is not by itself sufficient to track position indefinitely without the accumu-
lation of drift. This is because local gravitational conditions continually fluctuate due to tides,
seismic activity and the minute gravitational pull of nearby objects. The geophysical limit
shown in Figure 2.5 is the growth of drift when using a perfect IMU but without accounting
for such fluctuations.
Until relatively recently it was not possible to manufacture IMUs that were small and light
enough to use for pedestrian tracking, however recent advances in the construction of micro-
machined electromechanical systems (MEMS) have now made it possible to manufacture
small inertial sensor chips from which such devices can be built. For example the Xsens9 Mtx
shown in Figure 2.6 is 38× 53× 21 mm, weighs 30 g and reports its acceleration and angular
9http://www.xsens.com
20
 0
 1
 2
 3
 4
 5
 1  10  100  1000
D
rif
t (
cm
)
Time (s)
Commercial Tactical Navigational Strategic
Geophysical 
limit
Figure 2.5: Approximate drift due to random error sources when using a commercial-
grade, tactical-grade, navigation-grade, strategic-grade or ‘perfect’ IMU. This graph is
based on a graph of simulation results that is presented in [42].
velocity at frequencies as high as 512 Hz. Similar products are available from a number of
other manufacturers including Intersense10 and MemSense11.
In terms of quality, it is not yet possible to manufacture MEMS IMUs that are of sufficient
quality to track the position of a pedestrian to metre-level accuracy for minutes at a time
using (uncorrected) inertial navigation. MEMS IMUs are available at tactical-grade perfor-
mance levels (e.g. the Honeywell12 HG1900 series), however such devices currently cost
several thousands of pounds. They are however rapidly decreasing in price, partly due to the
increased use of MEMS inertial sensors in commodity hardware such as mobile telephones
and interactive entertainment systems such as the Nintendo13 Wii. The Xsens Mtx that is
used in this thesis currently retails for around £1200 (GBP), however much of this cost can
be attributed to additional features such as the inclusion of magnetometers and an embedded
Bayesian filter that estimates the absolute orientation of the device. A minimal MEMS IMU
of the same quality would currently retail at between £250 and £350. Note that the Xsens
Mtx and similar products available from Intersense and MemSense are examples of calibrated
IMU packages. It is also possible to purchase uncalibrated MEMS sensors (often sold as sep-
arate gyroscope and accelerometer chips). These are typically an order of magnitude cheaper,
but also an order of magnitude worse in terms of performance.
10http://www.intersense.com
11http://www.memsense.com
12http://www.honeywell.com
13http://www.nintendo.com
21
(a)
Gyroscopes
Accelerometers
Magnetometer
(b)
Figure 2.6: (a) The Xsens Mtx IMU. (a) Its internal circuitry (image provided by Xsens).
2.2.2 Pedestrian dead reckoning
Although it is not possible to track the relative movements of a pedestrian (at least to any
reasonable level of accuracy) using general purpose inertial navigation, it is possible to do so
by using more specialised algorithms. Drift errors grow less rapidly in such algorithms due to
their use of domain specific knowledge (e.g. using the known shape of the human gait cycle
to detect steps taken by the user). Previous research on pedestrian dead reckoning (PDR) has
focused mainly on the problem of tracking emergency and military personnel after they have
entered a building. It is not possible to use absolute positioning technologies in such scenarios,
since the necessary infrastructure cannot be deployed within the building in advance. Some
attempts have been made to develop absolute positioning systems that can be rapidly deployed
in such scenarios, for example the Precision Personnel Locator [43], however these have yet to
be proven in real environments. Such systems also require the rapid deployment of carefully
positioned beacons around the building prior to entry, which may not always be feasible.
Existing PDR algorithms can be split into two main groups: those based on step detection and
those based on (constrained) inertial navigation. Both use data from body-mounted MEMS
inertial sensors to estimate the relative movements of the user. Step-based algorithms can use
sensors mounted in a variety of positions on the body, as described in Section 2.2.2.1. In con-
trast those based on inertial navigation require foot-mounted sensors in order to be effective,
as described in Section 2.2.2.2. Note that in most previous literature the term “pedestrian dead
reckoning” has been used to refer exclusively to step-based algorithms, however at a funda-
mental level this term simply means “the tracking of a pedestrian’s relative position”. Hence
in this thesis the term is used to refer to both step-based systems and those based on inertial
navigation.
22
2.2.2.1 Step-based PDR
Step-based PDR algorithms consist of three phases: step detection, step length estimation
and step heading estimation. Each time a step is detected its estimated length and heading
are used to update the position of the tracked user. Step detection can be done using an ac-
celerometer mounted on the subject’s waist [44], foot [45], backpack [15] or even helmet [46].
For example signals from waist-mounted accelerometers exhibit a double-peaked oscillatory
pattern during walking [47]. This allows steps to be identified using simple techniques such
as the detection of zero-crossings [15] or peaks [48] in the signals. Step detection using a
foot-mounted accelerometer is even easier, since it is trivial to determine when the foot is in
contact with the ground by detecting stationary regions in the signals.
Estimating the length of each step is more difficult. The simplest approach is to use a fixed step
length that can be set for each user. This approach will give a reasonable level of performance
when the user is walking at a constant speed, but will perform poorly otherwise. This is
because step lengths for a given person can vary by up to±50% depending on walking speed,
with a typical person taking smaller steps when walking slowly compared to when they walk
at a brisk pace [49]. Several more advanced algorithms have been developed for estimating
step lengths [11, 44]. For example Alvarez et al. proposed the estimator
푙 =
푘1
푡step
+ 푘2 (2.1)
where 푙 is the estimated step length, 푡step is the measured step duration and 푘1 and 푘2 are
constants that must be determined on a per-user basis [44]. Neural networks have also been
used to predict step lengths based on feature vectors extracted from accelerations that are
measured during each step [46]. This approach was shown to estimate step lengths to within
a few percent, however it is necessary to train a separate neural network for each user. This
requires acceleration data to be obtained for which the corresponding step lengths are known,
which is not always practical and can be time consuming.
Step headings can be estimated by using a magnetometer to measure the user’s absolute head-
ing, which can then be taken as the direction of the step. One problem with this approach is
that it assumes that the user is always walking forwards. Hence an error is introduced if the
user steps sideways or backwards. Magnetometers are also affected by magnetic disturbances
which in some cases render their measurements useless [50]. Some systems use gyroscopes in
addition to magnetometers in order to detect and compensate for such disturbances [46]. Oth-
ers use only gyroscopes, however such systems are only able to measure relative changes in
heading over successive steps (which causes drift to accumulate in the tracked orientation as
well as in the position). All of these approaches usually require sensors to be rigidly attached
to the user. When magnetometers are used to calculate absolute headings, the orientation be-
23
tween the sensor and the user’s body must also be known. Recent research has attempted to
relax these constraints, however the accuracies of the resulting systems have been significantly
worse than those in which fixed mounting points are used [48].
PDR systems based on step detection have the advantage that they can operate with low grade
inertial sensors such as those found in many mobile telephones. Unfortunately step-based
PDR systems also have a number of drawbacks. They must be calibrated for each user and
most require a sensor to be rigidly mounted (and in some cases correctly aligned) with the
user’s body in order to obtain accurate heading measurements. Well calibrated step-based
PDR systems accumulate drift that is in the order of 3% of the distance travelled, however
this is only achievable if the user does not make any unusual movements such as walking
backwards or to the side [51]. Such movements violate the assumptions made by the under-
lying algorithms and hence introduce errors. One final problem that is of particular note for
indoor tracking is that most step-based PDR systems are unable to detect changes in vertical
displacement, for example when a user ascends or descends a flight of stairs.
2.2.2.2 Inertial navigation based PDR
In a PDR system based on inertial navigation techniques a full INS is used to track the position
of the user. The rapid accumulation of drift in the tracked position is prevented by frequently
correcting the INS. This is achieved by placing the IMU on the user’s foot, which has a known
velocity of zero whenever the foot is grounded. Furthermore it is straightforward to identify
periods during which the foot is grounded, since the acceleration and angular velocity signals
reported by the device during such periods are near-static. This allows zero velocity updates
(ZVUs) to be applied in order to correct the INS [14, 16]. As well as correcting any errors that
have accumulated in the tracked velocity, the application of ZVUs can also be used to correct
tilt errors in the tracked orientation and partially correct drift errors that have accumulated
in the tracked position during the previous step. This is possible due to correlations that
exist between accumulating errors in different components of the INS state [14]. Bayesian
filters provide an ideal framework to track such correlations and to apply ZVUs in a way that
allows correlated errors in all components of the state of the INS to be corrected. In particular
extended Kalman filters have been widely used [15, 14, 16].
INS-based PDR systems have both advantages and disadvantages compared to step-based
systems. Their biggest advantage is that they make far fewer assumptions about the user’s
motion. In particular they do not assume that the user is walking forwards. Hence such sys-
tems correctly handle unusual movements such as sidesteps and walking backwards. The
user’s vertical displacement is also tracked correctly when he or she ascends or descends a
flight of stairs. INS-based PDR systems tend to be more accurate than step-based systems.
For example the NavShoe system (an INS-based system) was shown to accumulate drift in
24
z1 z2 z3 z5z4
x1 x2 x3 x5x4
Figure 2.7: A Markov process.
the order of 0.3% of the total distance travelled [14] (albeit with the additional use of mag-
netometers to correct heading errors), compared to around 3% achieved by good step-based
systems. The main disadvantage of INS-based systems is that they always require a full 3-
dimensional IMU. In contrast it is possible to construct a step-based system using only a single
axis accelerometer together with a single axis magnetometer. Hence step-based systems can
be significantly cheaper. Another disadvantage of INS-based systems is that the IMU must be
mounted on the foot, which is less acceptable to most users compared to a mounting point on
the waist.
2.3 Bayesian filters
The inherent uncertainty of the raw measurements obtained from sensors, together with the
desire to use multiple sensor types, has led to the widespread use of Bayesian filters in posi-
tioning systems. Bayesian filters provide a framework for probabilistically estimating the state
of dynamic systems based on noisy measurements [52]. For pedestrian tracking the dynamic
system is (in some sense) the pedestrian and its state is the pedestrian’s position along with
any additional information such as their heading or velocity. Bayesian filters are deployed
in the fusion layer of the Location Stack model of positioning systems shown in Figure 2.1,
combining measurements obtained from individual sensors to form an estimate of the user’s
position [21]. In this section Bayesian filters are introduced in their general form. The use of
Bayesian filters in positioning systems is described in Section 2.4.
A Bayesian filter represents the state of a dynamic system at time 푡 as a probability distribution
over the space of all possible states
Bel(풙푡) = 푝(풙푡∣풛1, ...,풛푡) (2.2)
where 풛1, ...,풛푡 are noisy measurements made up to (and including) time 푡. Representing the
state as a probability distribution allows a Bayesian filter to provide both an estimate of the
true state of the system and a measurement of the uncertainty in that estimate. More generally,
the belief Bel(풙푡) can be queried to obtain the probability that a state 풙푡 is the true state of the
system, given all of the measurements made up until time 푡.
25
In general the complexity of computing Bel(풙푡) increases exponentially as the number of
measurements increases [23]. To make the computation tractable Bayesian filters assume that
the dynamic system is a Markov process, as shown in Figure 2.7, in which the next state
depends only on the current state and in which a measurement made at time 푡 is dependent
only on the state 풙푡 at that time. Such systems are defined by two distributions: a propagation
model 푝(풙푡∣풙푡−훿푡) and a measurement model 푝(풛푡∣풙푡). The propagation model defines how
the system’s state changes over time, whilst the measurement model defines the probability of
making an observation 풛푡 given the current state 풙푡.
Under the Markov assumption Bel(풙푡) can be calculated by updating the previous distribution
Bel(풙푡−훿푡), allowing a Bayesian filter to track the state of a dynamic system through time. To
perform the update the belief is propagated forward according to the system’s propagation
model to obtain the estimated belief, or prior distribution
Bel−(풙푡) =
∫
푝(풙푡∣풙푡−훿푡)Bel(풙푡−훿푡) 푑풙푡−훿푡 . (2.3)
The prior is then corrected using the measurement 풛푡 to obtain the new belief, or posterior
distribution
Bel(풙푡) = 훼푡푝(풛푡∣풙푡)Bel−(풙푡) (2.4)
where 훼푡 is a normalisation factor. In some cases it is necessary to update the state even when
no new measurements are available. In this case the prior is calculated and is taken as the new
posterior distribution. It is also possible to apply multiple measurements in a single update by
using the previously calculated posterior as the prior in successive iterations of the correction
step. Different measurement models can be used to incorporate different measurement types.
Hence Bayesian filters naturally support sensor fusion, in which measurements of different
types are combined to form an overall estimate of the state.
Bayesian filters provide a framework for probabilistically estimating the state of a dynamic
system, however their definition does not specify how such a filter should be implemented. In
particular a concrete implementation must choose how to represent the belief Bel(풙푡). A range
of different implementations have been developed with varying computational requirements
and underlying assumptions. The most widely used types of Bayesian filter are described
below.
2.3.1 Kalman filters
Kalman filters are the most widely used variant of Bayesian filters due to their computational
efficiency. They assume that Bel(풙푡) is Gaussian distributed for all 푡 [53]. Hence the belief
can be represented simply by a mean and covariance. It is also assumed that the state of the
26
system evolves according to a linear equation of the form
풙푡 = 푭푡풙푡−훿푡 +풘푡 (2.5)
where 푭푡 is a state transition matrix that defines how the state evolves over time and 풘푡 =
풩0,푸푡 is zero mean Gaussian distributed noise with a known covariance 푸푡. Each measure-
ment must be related to the state by a linear function of the form
풛푡 = 푯푡풙푡 + 풗푡 (2.6)
where푯푡 is a matrix that defines the relationship between the measurement 풛푡 and the current
state. The measurement noise 풗푡 = 풩0,푹푡 is assumed to be zero mean Gaussian distributed
with a known covariance 푹푡. Kalman filters are optimal Bayesian filters given these assump-
tions, meaning that they compute Equations 2.3 and 2.4 exactly [54]. Furthermore, they do so
using computationally efficient matrix operations (given in Appendix C). Hence Kalman fil-
ters are the preferred type of Bayesian filter for linear or near-linear systems where the belief
will always be uni-modal. Their computational efficiency makes it possible to usefully deploy
Kalman filters on embedded and mobile devices.
2.3.2 Linearised and extended Kalman filters
Kalman filters as described above can only be applied to near-linear systems and measure-
ments. Linearised and extended Kalman filters (EKFs) relax these constraints by linearising
the more general propagation and measurement functions
풙푡 = 퓕 푡(풙푡−훿푡) +풘푡 (2.7)
풛푡 = 퓗푡(풙푡) + 풗푡 (2.8)
using first-order Taylor series expansions [53], where 퓕 푡 and 퓗푡 are differentiable (but pos-
sibly non-linear) functions. The linearised Kalman filter performs the linearisation about an
approximate state trajectory that must be known in advance. For example the intended flight
path of a missile aimed at a stationary target could be used as an approximate trajectory when
tracking the missile’s true position.
The EKF is similar to the linearised Kalman filter except that the linearisation takes place
about the filter’s estimated trajectory rather than a pre-computed one. Hence it is suitable
for problems where an approximate trajectory is not known in advance, as is often the case.
Their main disadvantages are that they are still only able to represent Gaussian belief and that
they are more likely to diverge from the true state in unusual situations than both regular and
linearised Kalman filters [53].
27
2.3.3 Multi-hypothesis filters
A Multi-hypothesis filter consists of multiple Kalman filters (or EKFs), each of which tracks
one of a number of possible hypotheses. The overall belief is a mixture of the Gaussian
distributions maintained by the individual Kalman filters, which are weighted according to
how closely they predict the observed measurements [23]. The main benefit of using a multi-
hypothesis filter is that it is possible to represent multi-modal distributions. The main draw-
back is increased computational complexity. It is also necessary to implement heuristic algo-
rithms in order to determine when to remove or introduce hypotheses. Finally, note that the
tracking of each individual hypothesis is still subject to the restrictions and assumptions that
are imposed by the type of Kalman filter used (e.g. zero mean Gaussian distributed noise).
2.3.4 Grid-based filters
Grid-based filters are optimal Bayesian filters for systems in which the state space is both
discrete and finite [54]. Given this assumption the belief can be represented by a discrete set
of weights {푤푖푡 ∣ 푖 = 1, ..., 푁}, where 푤푖푡 is the probability of state 풙푖푡 and {풙푖푡 ∣ 푖 = 1, ..., 푁}
is the set of possible states at time 푡. The belief is given by
퐵푒푙(풙푡) =
푁∑
푖=1
푤푖푡 훿(풙푡 − 풙푖푡) (2.9)
where 훿(푥) is the Dirac delta function. No assumptions are made about the form of the prop-
agation and measurement models 푝(풙푡∣풙푡−훿푡) and 푝(풛푡∣풙푡).
Grid-based filters can also be applied to systems in which the state space is continuous (but
still finite) by dividing the state space into 푁 discrete cells {풙푖푡 ∣ 푖 = 1, ..., 푁}. The weight 푤푖푡
is then the probability that the true state of the system lies within the cell 풙푖푡 at time 푡.
The main advantages of grid-based filters are that they can represent arbitrary probability
distributions across the set of discrete states and that they do not place any restrictions on the
types of propagation and measurement models that can be used. The main disadvantages are
that they require 풪(푁) memory and to update the state requires 풪(푁2) computation (since
each of the 푁 weights in the new state is calculated by a summation over all 푁 states [54]).
Hence they do not scale well to large state spaces. In practice this means that grid-based
filters are restricted to problems of low dimensionality, since the number of states in a space
normally grows exponentially with the number of dimensions. When a continuous state space
is discretised in order to apply a grid-based filter, there is a trade-off between using larger cells
to reduce computational overheads and using smaller cells to minimise discretisation errors
that are introduced as a result.
28
2.3.5 Particle filters
Particle filters approximate the belief as a set of weighted samples, or particles
풮푡 = {⟨풙푖푡, 푤푖푡⟩ 푖 = 1, ..., 푁} (2.10)
where 풙푖푡 is the state of the 푖
th particle and 푤푖푡 is its weight. The weight of a particle can
be thought of as the probability that the particle’s state corresponds to the true state of the
system. Various algorithms exist for updating the belief, almost all of which are based on the
Bootstrap filter [55] (also known as the Sampling-Importance-Resampling (SIR) filter). The
Bootstrap filter’s update procedure consists of three steps. First, each particle is propagated
according to the system’s propagation model. Their weights are then adjusted according to
any measurements, each of which must have a corresponding measurement model. Finally the
particles are re-sampled in proportion to their weights to obtain a new set of equally weighted
particles, which approximates the new posterior distribution. The re-sampling process causes
more particles to be generated in areas of the state space that are likely to correspond to the
true state of the system. Hence unlike grid-based filters, particle filters automatically focus
computation where it is most needed.
The main advantage of particle filters is that they do not place any restrictions on the form
of the propagation and measurement models. Like grid-based filters they are able to repre-
sent arbitrary multi-modal distributions, however unlike grid-based filters they do not require
a discrete or finite state space. Hence particle filters are well suited to problems in which
the state space is continuous and the belief is multi-modal. A particle filter requires 풪(푁)
memory and to update the state requires 풪(푁 log푁) time (although by using more efficient
re-sampling techniques this can be reduced to풪(푁) [56]), where푁 is the number of particles.
There exist methods for dynamically varying the number of particles and hence the computa-
tional costs [57, 58]. Such algorithms typically generate fewer particles when uncertainty is
low relative to when it is high.
2.3.6 Summary
Table 2.2 gives a summary of the types of Bayesian filter described above. When selecting a
Bayesian filter it is necessary to trade off the computational requirements of the filter with the
restrictions imposed on the belief distribution and the propagation and measurement models.
Both regular and extended Kalman filters have been applied to many systems in which belief
is uni-modal. They can often be applied successfully in practice to systems that violate their
assumptions, for example the Gaussian nature of process and measurement noise. More com-
putationally expensive filters such as particle and grid-based filters are usually only required
29
when it is necessary to represent more complex multi-modal beliefs.
2.4 Bayesian filters in positioning systems
Bayesian filters have been widely used in the fusion layer of positioning systems. In most
cases such filters are used to implement a technique known as Single-Constraint-At-A-Time
(SCAAT) tracking [59], in which measurements from individual sensors are incorporated one
at a time into the estimated state of the tracked object(s). This approach is fundamentally
different from the traditional lateration and angulation algorithms used by many of the posi-
tioning systems described in Section 2.1, in which a certain number of measurements must be
received (e.g. three in the case of lateration) before the object’s position can be calculated. In
SCAAT tracking individual measurements can be used to update the estimated state as soon as
they are received by the filter. Hence a measurement can be used to update the estimated state
of a device even if an insufficient number of measurements are available to fully determine its
position. This ability is particularly useful for unilateral ultrasound-based positioning systems
such as Cricket, in which mobile tags obtain range measurements to beacons in the environ-
ment one at a time. Cricket tags are not able to simultaneously measure the range to multiple
beacons, making the application of traditional lateration algorithms difficult and inaccurate.
In contrast a Bayesian filter can use SCAAT tracking to incorporate the individual range mea-
surements into an estimate of the tag’s true position, updating the estimate whenever a new
measurement is received [30].
The HiBall tracker is another example of a positioning system in which a SCAAT approach
is used [60, 61]. A HiBall consists of a compact vision-based system that detects flashing
LEDs mounted in the ceiling of the environment. SCAAT is used to incorporate individual
measurements into an estimate of the HiBall’s state (i.e. its position and orientation). Using
this approach state estimates can be generated at 2000 Hz, with less than 1 ms latency and
with maximum errors in position and orientation that are of the order of 1 mm and 0.05∘
respectively. The system was designed for virtual reality applications, where rapid updates,
low latencies and high accuracies are important.
2.4.1 Sensor fusion
Although the ability to incorporate individual measurements into the estimated state is a major
benefit, it is not the main reason why Bayesian filters are so often used in location systems.
The most important property of Bayesian filters for location systems is their ability to perform
sensor fusion, in which measurements from different types of sensor are combined to form a
single estimate of the state of a tracked device. Recall from Section 2.3 that a Bayesian filter
30
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31
estimating the time-varying state 풙푡 of a system can incorporate any measurement 풛푡 at time
푡, provided that a corresponding measurement model 푝(풛푡∣풙푡) is available in a form suitable
for the type of Bayesian filter that is being used. Hence Bayesian filters are able to combine
measurements from different types of sensor into a single position estimate, provided that
measurement models for each type of measurement are available. Each measurement model
encodes the accuracy and error characteristics of the underlying sensor, which are taken into
account by the Bayesian filter when updates are performed. Note that a SCAAT approach is
particularly useful when multiple sensor types are used, since different sensors often generate
measurements at different rates and at different points in time.
Sensor fusion is most useful when complementary sensor types are used (i.e. when a weak-
ness of one sensor type is a strength of another). A common example is inertial-GPS, in which
a GPS receiver is enhanced by the addition of an IMU. Inertial-GPS devices are able to track
their position and orientation outdoors at a high update rate (a benefit provided by the IMU)
without incurring drift (a benefit provided by GPS) at an accuracy that is better than can be
achieved using GPS alone [13]. Applications include machine control (e.g. farming machin-
ery), surveying and vehicular positioning [62], missile guidance (often in conjunction with
radar) [63] and the accurate registration of video captured from moving platforms [64].
Sensor fusion is even more relevant to indoor positioning due to the lack of anything approach-
ing a single sensor technology that is accurate and reliable enough for all situations [17]. The
Constellation tracking system uses a Kalman filter to combine measurements from an IMU
with range measurements to ultrasonic beacons in the ceiling of the environment [19]. Con-
stellation is able to estimate the position and orientation of the tracked device to within 8 mm
and 1∘ of its true state. Existing deployments are relatively small (e.g. covering a single
room), in part due to the high infrastructure requirements.
An implementation of the Location Stack used sensor fusion for more coarse grained tracking
of user’s in indoor environments, using particle filters to combine a variety of measurements
including ultrasonic range measurements and infra-red proximity measurements [65]. A de-
ployment in a 900 m2 office was able to position users to within 2 m 90% of the time, compared
to approximately 3 m and 4 m when deterministic algorithms were applied to measurements
from the individual infra-red and ultrasonic systems respectively.
2.4.2 Constraints as pseudo-measurements
In addition to combining measurements from multiple sensor types, Bayesian filters are also
able to factor in constraints that are known to apply to the state of a tracked device. Such
constraints are enforced through the application of pseudo-measurements; imaginary mea-
surements whose corresponding measurement models describe the nature of the constraint.
32
For example consider a device whose position and velocity is tracked by a Bayesian filter. If
the device is known to have a maximum speed of 5 ms−1 then this can be enforced by applying
pseudo-measurements with the measurement model
푝(MaxSpeed∣풙푡) =
{
0.2 ∣풗푡∣ < 5
0 otherwise
(2.11)
after each propagation step, where 풗푡 is the velocity component of the state. Note that since
pseudo-measurements are treated just like regular measurements in the Bayesian framework,
their measurement models must be in a form suitable for the type of Bayesian filter used. For
example MaxSpeed constraints could not be applied by a Kalman filter, since they violate the
Kalman filter’s linear and Gaussian assumptions. Such constraints could however be applied
by a particle or grid-based filter.
Using pseudo-measurements to enforce constraints has been widely applied within Bayesian
implementations of inertial navigation based PDR systems, as described in Section 2.2.2.2.
In such systems an IMU attached to the user’s foot is known to be stationary whilst the
user’s foot is grounded. This constraint can be enforced by applying zero velocity pseudo-
measurements [14].
2.5 Robot localisation
The indoor absolute positioning systems described in Section 2.1.2 rely on the installation
of fixed infrastructure such as ultrasonic beacons and WiFi access points. In some situations
it is desirable to track the location of pedestrians in an indoor environment, however it is
undesirable to have to install such infrastructure in advance. Sometimes it is impossible to do
so, for example when tracking emergency services personnel as they enter a building to which
they have just been called. This has led to the development of PDR systems as described in
Section 2.2.2, however drift causes the accuracy of such systems to degrade over time. The
initial position and orientation of the user must also be known.
One related area of research in which the problems outlined above have been addressed is
that of robot localisation. Robot localisation is the problem of determining and subsequently
tracking the position and orientation of a robot that is initially placed at an unknown location
within a known environment. This must be done using only sensors that are attached to the
robot and a map of the environment. The robots used to investigate this problem are typically
equipped with laser or ultrasonic range-finders [66, 57], which are used to measure distances
to nearby obstructions. Upward facing cameras have also been used [67].
In addition to the sensors described above, the robot’s wheels are typically equipped with
33
(a) (b)
Figure 2.8: (a) The path followed by a mobile robot, as estimated by odometry. (b) The
corrected path. Adapted with permission from [66].
sensors (usually optical rotary encoders) that measure wheel rotation. Measurements from
such sensors can be used to estimate changes in the robot’s position and orientation. The
technique of tracking the relative movement of a robot in this way is known as odometry [68].
As with any dead reckoning technique, errors will accumulate in the tracked position over
time. Sources of error include wheel slippage, floor roughness and the discretised sampling of
wheel rotation [69]. Figure 2.8(a) shows an example of a path estimated by odometry, which
has been aligned with the map to have the correct starting position and orientation. The actual
path followed by the robot is shown in Figure 2.8(b).
It is possible to determine the absolute position of a robot in a known environment (and hence
solve the robot localisation problem) because only some paths through that environment will
be consistent with both the measurements made by the robot and its path as estimated by
odometry. As the robot senses more of the environment its position can be narrowed down
further as more paths are discounted. This process continues until the robot’s position has been
uniquely determined, as shown in Figure 2.9. Bayesian filters provide an ideal framework for
robot localisation, since they are able to represent the changing belief in the robot’s position
throughout the localisation process. The belief is typically multi-modal and non-Gaussian
during localisation, as shown in Figures 2.9(a) and 2.9(b). Grid-based filters and particle
filters are both able to represent such distributions and have been successfully applied to the
robot localisation problem (e.g. [70] and [71] respectively).
Whichever filter is used, it is necessary to specify a propagation model and a measurement
model for each type of measurement. The propagation model 푝(풙푡∣풙푡−1,풖푡) updates the
belief to account for the movement of the robot since the last update, where 풖푡 is is the robot’s
estimated relative movement obtained by odometry. For a range measurement 푟푡 in direction
휃푡, a common measurement model for 풛푡 = (푟푡, 휃푡) is given by
푝(풛푡∣풙푡) = 풩0,휎(푟푡 − range(map,풙푡, 휃푡)) (2.12)
34
Robot position
(a)
Robot position
(b)
Robot position
(c)
Figure 2.9: Density plots after incorporating (a) 5, (b) 18 and (c) 24 sonar scans during
robot localisation. In this example a particle filter is used to perform the localisation. Darker
positions are more likely. Adapted with permission from [66].
where range(map,풙푡, 휃푡) is the range to the nearest obstruction in the map of the environment
from the position defined by 풙푡 in the direction 휃푡. The nearest obstruction can be found by
ray-tracing [66].
Early solutions to the robot localisation problem used position probability grids (a type of
grid-based Bayesian filter), in which the state space of all possible positions and orientations
is divided into discrete cells [70, 72]. Spatial and angular resolutions of around 20 cm and 5∘
were typically used. This approach was shown to allow successful robot localisation and sub-
sequent tracking with a mean position error of 5 cm for a robot equipped with a laser-range
finder [66]. Despite this there were several notable drawbacks. The memory and compu-
tational requirements were proportional to the number of cells and hence to the resolution
of the grid, with a finer grid requiring more resources. Reducing the resolution of the grid
lowered the requirements but also increased discretisation errors, lowering the probability of
successful localisation and decreasing the accuracy during tracking.
Pre-computing the sensor model 푝(풛푡∣풙푡) and performing selective updates have both been
used as techniques for reducing the computational requirements of grid-based localisation
algorithms [66]. Pre-computing the sensor model for every possible cell and measurement
combination avoids the need to ray-trace values of range(map,풙푡, 휃푡) on-line, reducing the
computational cost of the algorithm by a constant factor. Selectively updating only cells that
are actually likely to correspond to the true position of the robot can drastically reduce the
computational requirements of the filter when uncertainty in the robot’s position is low. This
approach was demonstrated to reduce the computational load by over 95% after a robot’s po-
sition was determined in a test localisation, since only a few cells corresponded to the robot’s
known position [66]. The main disadvantage of such an approach is that the complexity of
implementing the algorithm is significantly increased.
More recently particle filters have been preferred to grid-based filters for robot localisation.
Particle filters do not require a discretisation of the state space and automatically focus com-
putation where needed, since re-sampling causes more particles to be generated in areas where
35
the robot is likely to be positioned. The number of particles can also be adapted, so that more
particles are used when the uncertainty is high relative to when the robot’s position has been
determined [57, 58]. The effect of adaptive re-sampling is similar to that of selective updates
in grid-based filters, however it is far easier to implement. Particle filter implementations have
been shown to achieve superior accuracy relative to grid-based approaches, whilst requiring
an order of magnitude less computation and memory [71].
Once a robot localisation algorithm has determined the position of the robot, it can continue
to track the robot’s position indefinitely simply by applying the propagation and measure-
ment models in the same way as during localisation. The relative movement measurements
provided by the robot’s odometry data are used to propagate the robot’s position, whilst the
measurements from sensors attached to the robot are used together with the map of the en-
vironment in order to prevent drift that would otherwise accumulate in the tracked position.
Figure 2.8(b) shows the corrected path calculated by a localisation filter for the robot whose
raw odometry data is shown in Figure 2.8(a).
2.5.1 Dealing with unmapped obstructions
One of the major problems when deploying robots in realistic environments is that of dynamic
obstructions such as pedestrians. Robot localisation as presented above assumes that any mea-
surement (e.g. a range measurement) is of the mapped environment. Hence a measurement
of a dynamic obstruction will be deemed inconsistent with the robot’s true state, causing a
tracking error or localisation failure.
One solution to this problem is to pre-filter measurements, discarding any that are likely to be
of dynamic obstructions. Fox et al. describe filtering techniques based on both entropy and
distance [66]. Their entropy filter applies a measurement only if it will result in a reduction
in the entropy of the belief (which is a measure of uncertainty). Hence a measurement is only
used if it re-enforces the current hypothesis. Their distance filter rejects range measurements
that are very likely to be shorter than expected given the current belief. These algorithms
were tested using robots deployed in museum environments through which many pedestrians
moved. Both reduced the duration for which the tracking error was over a threshold of 45 cm
from 26.8% of the time to just over 1%. The entropy approach has the advantage that it
can be applied equally to any measurement type (not just range measurements), however
takes longer to recover when a tracking error does occur. It is unclear whether either method
would be effective during localisation, since it is much harder to determine whether or not a
measurement should be filtered when uncertainty in the robot’s position is high.
36
Measurements
Localisation filter
Sensors
Fusion
Relative movement measurements
Map of environment
Sensors
Measurements
Fusion
IMU
PDR filter
Acceleration / Angular velocity
Figure 2.10: Cascaded stacks in a pedestrian localisation system.
2.6 Pedestrian localisation
One of the goals of this thesis is to determine a way in which the absolute position and head-
ing of a person can be determined in an indoor environment, without having to install large
amounts of infrastructure. The algorithms outlined in Section 2.5 are able to determine the ab-
solute position and heading of a mobile robot without the need for any infrastructure, provided
that a map of the environment is available. It should be possible to develop similar algorithms
for pedestrians, combining relative movement measurements obtained from a PDR system
with constraints derived from a map of the environment in order to determine and subse-
quently track a user’s absolute position. During tracking the same algorithms could be used
to limit drift that would otherwise accumulate in the tracked position.
In the context of the Location Stack representation for positioning systems shown in Figure
2.1, the approach described above can be thought of as embedding the bottom three layers of
one stack inside the sensors layer of another higher level stack. The lower level stack corre-
sponds to the PDR system itself, defining the inertial sensors, measurements and algorithms
that are used to estimate the relative movements of the pedestrian. This stack is then treated
as a sensor in the sensors layer of the higher level stack, which combines the measurements
provided by the PDR system with a map of the environment as shown in Figure 2.10.
2.6.1 Differences to robot localisation
There are many similarities between the pedestrian localisation problem and the robot locali-
sation problem described in Section 2.5. There are also several important differences. Firstly,
existing solutions to the robot localisation problem have only considered environments con-
sisting of a single floor. Supporting environments with multiple floors was not necessary since
the robots that were used to test such implementations were unable to climb stairs. The extra
mobility of a pedestrian makes it necessary to support such environments.
Secondly, existing robot localisation systems use robots that provide both relative movement
measurements (usually obtained using wheel encoders) and measurements from additional
37
sensors (usually range measurements to nearby obstructions obtained using laser or ultrasonic
range finders). For pedestrian localisation it is desirable to keep the number of sensors to a
minimum so as to minimise the inconvenience to the tracked user. Furthermore the lack of
rigid mounting points on the human body makes it difficult to use sensors that are typically
used for robot localisation. In this thesis a solution to the pedestrian localisation problem
is presented that uses only a small IMU to measure the approximate relative movement of
the user. In effect the user is used as a sensor that detects the absence of obstacles such as
walls in the environment (since a person cannot pass through such obstacles). This task is
significantly more difficult because less information is available. The error characteristics of
the positions calculated by the resulting system are significantly different to those of a typical
robot localisation system, as will be described in Chapter 6.
A final but important difference is that a pedestrian localisation system cannot exert control
over the movement of the user. A robot localisation system can direct a robot to move in a way
that helps the localisation process. For example if there are two distinct hypotheses about the
position of the robot then it can be directed to move towards an asymmetry in the environment
that will allow one hypothesis to be discounted and the other confirmed. This is known as
active rather than passive localisation [73]. In theory a pedestrian localisation system could
request that the user move in a certain direction, but this would be highly undesirable in
practice.
2.6.2 Map availability
Whilst maps of outdoor environments are widely available, maps of indoor environments are
not. This currently prevents robot localisation techniques from being widely applied for the
purpose of pedestrian localisation and tracking. Fortunately the availability of such maps
is set to increase rapidly over the next few years, with several companies already building
such maps for use in navigation and location-based applications. Micello14 claim to be able
to construct a detailed plan of a large shopping centre in only four hours and have already
released applications for mobile telephones that allow users to search and navigate within
some buildings. NavTeq15 are also mapping indoor environments such as airports, shopping
centres and train stations, using golf buggies and backpacks equipped with a combination of
cameras, laser range finders and inertial sensors in order to map quickly and efficiently. They
plan to release their first maps of indoor environments in the fourth quarter of 2010 [74]. It
is expected that many other mapping companies will also start to map indoor environments,
due in part to the perceived demand for location-based services and applications. Established
mapping companies such as NavTeq have also been driven to develop new revenue streams
14http://www.micello.com
15http://www.navteq.com
38
= Estimated object state
= Ground truth
Invalid particles
t
(a)
= Backtracking invalid trajectories
(b)
= Backtracking estimated state
(c)
Figure 2.11: The backtracking particle filter. (a) Particles are invalidated as they pass
through a wall. (b) The tail states of invalidated (red) particles. (c) The corrected state
estimates. Adapted with permission from [77].
since other companies, such as Google, began offering free turn-by-turn satellite navigation
applications for mobile telephones. For these reasons it is expected that a large proportion of
public buildings in the developed world will be mapped within the next five to ten years.
2.6.3 Concurrent work
Several pedestrian localisation and tracking systems have been developed concurrently to the
one presented in this thesis. Krach and Robertson describe a localisation system consisting of
two Bayesian filters, an EKF and a particle filter [75]. The EKF extracts relative movement
measurements from a foot-mounted IMU, using zero velocity updates as described in Section
2.2.2.2. A second IMU attached to the other foot can be used to improve the accuracy of these
measurements [76]. The particle filter fuses such measurements with a map of the environ-
ment, a movement model and measurements received from other devices such as an altimeter
or digital compass. It is unclear how much of the system has actually been implemented since
(at the time of writing) detailed results have yet to be published. In particular their current
work does not discuss handling multi-storey buildings or scalability to large environments. At
present the only published results are in the form of particle plots, which show the localisation
of a pedestrian moving through a single-storey 1000 m2 office environment [75].
Widyawan et al. describe a backtracking particle filter for combining relative movement mea-
surements from an IMU with a building plan [77, 78]. Each particle in the backtracking filter
contains not only its current state, but also a tail of previous states. This allows previous state
39
estimates to be re-calculated from the tail states of surviving particles each time an update
occurs. The re-calculated states are not weighted by particles whose trajectories have been
subsequently invalidated, improving the previous estimates as shown in Figure 2.11. Back-
tracking was shown to reduce the mean positioning error from 1.553 m to 1.321 m during test
runs in a 2704 m2 building for which a detailed map was available [77]. When only the exter-
nal walls of the building were present in the map the mean error was reduced from 1.892 m to
1.819 m.
2.7 Conclusions
Whilst satellite positioning systems allow devices to position themselves anywhere in the
world when outdoors, they do not work well in indoor environments. This has led to the
development of indoor positioning technologies, however the infrastructure requirements of
existing indoor location systems make them expensive and time consuming to deploy and
maintain. As a result they are not well suited to certain usage scenarios, such as when emer-
gency personnel enter a building to which they have just been called. Temporary deployments
(e.g. for a few weeks) of existing indoor positioning systems are impractical due to the high
cost of installation. They are also uneconomical for deployment in large, sparsely populated
areas, since the cost of such systems is largely dependent on the coverage area rather than the
number of tracked users.
In contrast to existing absolute positioning systems, robot localisation algorithms allow robots
to determine and track their absolute positions in indoor environments without the need for any
fixed infrastructure. Such techniques can in theory be applied to track pedestrians, however
they require maps of indoor environments to be available. Until now this has not generally
been the case, however increased mapping of such environments by commercial mapping
companies is likely to address this problem in the future.
40
Chapter 3
Inertial navigation
The pedestrian localisation and tracking system developed in this thesis is based on a micro-
machined electromechanical systems (MEMS) inertial measurement unit (IMU). This chapter
examines these sensors at a low level, develops the theory of inertial navigation, discusses the
propagation of errors through inertial navigation systems and finally introduces a filter that is
able to correct such systems using external measurements.
MEMS IMUs contain three orthogonal gyroscopes and accelerometers. The gyroscopes mea-
sure the angular velocity of the IMU body with respect to inertial space. In other words they
measure the IMU’s angular velocity with respect to any inertial coordinate frame, which is
any frame that does not accelerate or rotate with respect to the rest of the universe [18]. The
accelerometers measure the specific force acted upon the IMU, also with respect to inertial
space. Since the inertial sensors are rigidly attached to the body of the IMU, all measurements
are made in the IMU’s own frame of reference. This is known as the body frame, as shown in
local frame
body fram
e
Figure 3.1: An Xsens Mtx together with body and local frames of reference.
41
RProject 
specific 
force onto 
ref. axes
Add 
acceleration 
due to 
gravity
Orientation
Velocity PositionRR
Initial 
velocity
Initial 
position
Accel
signals
Gyro 
signals
Initial 
orientation
Accel
Figure 3.2: The strapdown inertial navigation algorithm.
Figure 3.1. Hence IMUs generate measurements of the form
흎푏푡 = (휔
푏푥
푡 , 휔
푏푦
푡 , 휔
푏푧
푡 )
푇 (3.1)
풇 푏푡 = (푓
푏푥
푡 , 푓
푏푦
푡 , 푓
푏푧
푡 )
푇 (3.2)
where 흎푏푡 and 풇
푏
푡 are 3-dimensional angular velocity and specific force measurements respec-
tively and 푡 is the time at which the measurements are made. The superscript 푏 indicates that
the measurements are in the body frame. Superscripts 푏푥, 푏푦 and 푏푧 are used to indicate quan-
tities that are measured along the individual axes of the that frame. This superscript notation
is adopted throughout the remainder of this chapter.
An inertial navigation system (INS) tracks its orientation, velocity and position in some ref-
erence frame that is suitable for the target application. Figure 3.2 gives an overview of the
strapdown inertial navigation algorithm that is used. The angular velocity measurements are
integrated to track the orientation of the IMU relative to the reference frame. Specific force
measurements made by the IMU are projected into this frame using the tracked orientation.
Acceleration due to gravity is then added to obtain the acceleration of the IMU in the refer-
ence frame. This acceleration is then integrated once to track the velocity of the device and
again to track its position [13]. Note that the algorithm must be initialised with an initial state
(i.e. orientation, velocity and position).
In the simplest case the reference frame used by an INS is an inertial one, however in practice
this is not usually the case. This thesis is concerned with local (i.e. within-building) naviga-
tion, for which a suitable reference frame is one whose 푥 and 푦 axes are fixed with respect to
north and east and whose 푧 axis is locally vertical. Such frames rotate with respect to inertial
space due to the rotation of the earth, which must be accounted for by an INS if it is to achieve
an accurate navigation solution. Nevertheless there are situations where it is appropriate to ig-
nore such rotation (i.e. assume that a local navigation frame is an inertial frame of reference).
42
These include situations where the expected gyroscope errors are significantly larger than the
error that is introduced as a result of the earth’s rotation. It is also appropriate when tracking
for short periods of time (e.g. several minutes or less, depending on accuracy requirements)
because the earth’s rotation will be small during such periods. Another scenario is when the
INS is frequently corrected using external measurements that effectively correct for errors that
accumulate due to the earth’s rotation. Finally, if the INS is used as a purely relative tracking
system (i.e. it is not initialised with an absolute position and orientation relative to the earth)
then it is not possible to compensate for the earth’s rotation, since knowledge of the latitude
and absolute orientation of the device is required (see [13] for details).
The tracking system developed in this thesis uses inertial navigation as a purely relative track-
ing technique in order to measure changes in a pedestrian’s position and orientation. The
absolute position and heading of the user are (at least initially) unknown and hence it is not
possible to compensate for the earth’s rotation. Furthermore the errors introduced by the
Xsens Mtx IMU that is used are significantly larger than those introduced by ignoring the
earth’s rotation, as will be shown in Section 3.3.1. As a result local navigation frames are ap-
proximated as inertial frames of reference in this thesis. Quantities measured in a local frame
are indicated by a superscript 푔. All frames of reference are assumed to be right-handed1, as
shown in Figure 3.1.
The remainder of this chapter is arranged as follows. Section 3.1 describes error sources
that perturb measurements from the individual sensors contained within a MEMS IMU. Sec-
tion 3.2 develops the strapdown inertial navigation algorithm outlined in Figure 3.2. The prop-
agation of errors through this algorithm is discussed in Section 3.3. Finally, a Kalman filter
that can correct an INS using external measurements is developed and tested in Section 3.4.
3.1 Sources of error in MEMS inertial sensors
Various types of error can perturb measurements generated by the individual gyroscopes and
accelerometers that are contained within an IMU. The main types of error that affect mea-
surements made by MEMS sensors are listed below. Since an inertial navigation system must
integrate signals from such sensors in order to track the state of an IMU, the effect of inte-
grating signals containing each type of error is also considered. Note that in this section these
integrations are considered only for a single axis of the device. A more general discussion
of errors and how they propagate through the complete inertial navigation algorithm can be
found in Section 3.3.
1As opposed to left-handed, in which the 푥 and 푦 axes (or indeed any other pair of axes) are exchanged.
43
3.1.1 Gyroscope error sources
3.1.1.1 Constant bias
The bias of a gyroscope is the average output from the gyroscope when it is not undergoing
any rotation (i.e. the difference between the output and the true value), usually specified in
∘/h. A constant bias error of 휖, when integrated, causes an angular error that grows linearly
with time
휃(푡) = 휖 ⋅ 푡. (3.3)
The constant bias of a gyroscope can be estimated by taking a long term average of its output
when it is not undergoing rotation. This estimate can then be subtracted from any subsequent
measurements. Note however that the presence of other errors means that it is impossible to
determine the bias of a gyroscope exactly.
3.1.1.2 Random noise / angle random walk
MEMS gyroscopes are perturbed by random noise from a number of sources, including
themo-mechanical and electrical noise. The spectrum of such noise is approximately white
for frequencies below 1 Hz, meaning that the standard deviation of the average angular ve-
locity noise is inversely proportional to the square root of the averaging time [18]. Thus
gyroscope sensor noise is usually specified in root power spectral density (PSD) form with
units ∘/h/
√
Hz, ∘/
√
h or ∘/s/
√
Hz (which is equivalent to ∘/
√
s) for less accurate sensors.
The relationship between root PSD 푛 (expressed in each of the units above) and the standard
deviation 휎 of the individual measurements is given by
휎 (∘/푠) =
푛 (∘/h/
√
Hz)
60 ⋅ 60 ⋅√훿푡 (s) (3.4)
=
푛 (∘/
√
h)
60 ⋅√훿푡 (s) (3.5)
=
푛 (∘/s/
√
Hz)√
훿푡 (s)
(3.6)
where 훿푡 is the period between successive samples (e.g. 0.01 s if the sampling rate is 100 Hz).
The result of integrating the random noise that perturbs MEMS gyroscopes is a random walk
in angle. The mean of the resulting random walk is zero, but its standard deviation grows
proportional to
√
푡. For example if the root PSD is 0.2∘/
√
h then the standard deviation of the
angle random walk will be 0.2∘ after one hour,
√
2 ⋅ 0.2 = 0.28∘ after two hours and so on.
44
3.1.1.3 Calibration errors
Calibration errors include errors in the scale factors, alignments, and linearities of the gyro-
scopes. Such errors tend to produce systematic errors that are only observed whilst the device
is turning and cause the accumulation of errors in the integrated signal that are proportional
to the rate and duration of the motions [42]. It is usually possible to measure and correct such
calibration errors. The Xsens Mtx applies its own internal corrections to reduce the effect of
calibration errors.
3.1.1.4 Temperature effects
Temperature fluctuations due to changes in the environment and sensor self-heating can in-
duce small fluctuations in the sensor bias. Any residual bias introduced due to a change in
temperature will cause an error in orientation which grows linearly with time. The relationship
between bias and temperature is often highly non-linear for MEMS sensors. Like many other
IMUs the Xsens Mtx contains a temperature sensor that it uses to minimise bias fluctuations
induced by changes in temperature.
3.1.1.5 Bias instability
The bias of a MEMS gyroscope tends to wander slowly over time even when the temperature
of the device is kept constant. Such fluctuations are often modelled as a random walk [42].
They are quantified by a bias instability measurement that defines how the bias may change
over a specified period of time, typically around 100 seconds, in fixed conditions. Most man-
ufacturers provide a 1휎 value in ∘/h, or ∘/s for less accurate devices. If 퐵푡 is the known bias
at time 푡, then a 1휎 bias instability of BS휎 = 0.01∘/h over 100 seconds means that the bias
at time (푡 + 100) seconds is a random variable with expected value 퐵푡 and standard devia-
tion 0.01∘/h. Under the random walk model this standard deviation grows proportional to the
square root of time. For this reason bias instability is occasionally specified as a bias random
walk measurement
BRW (∘/
√
h) =
BS휎 (
∘/h)√
푡 (h)
(3.7)
where 푡 is the timespan for which BS휎 is defined.
Under the assumption that bias instabilities can be modelled as a random walk, the resulting
error in the integrated signal is a second-order random walk2 in angle. Note however that
this model is unrealistic for running times that are much larger than the timespan over which
BS휎 is defined, since uncertainty in the bias grows without bound in the random walk model
2The integral of a first-order random walk.
45
Error type Description Result of single integration
Bias A constant bias offset A linearly growing error in angle
Random noise Random noise from a variety
of sources (usually specified
in terms of its root PSD)
An angle random walk whose stan-
dard deviation grows proportional to
the square root of time
Calibration Deterministic errors in scale
factors, alignments and gyro
linearities
Orientation drift proportional to the
rate and duration of motion
Temperature
effects
Temperature dependent resid-
ual bias
Any residual bias is integrated into the
orientation, causing an orientation er-
ror which grows linearly with time
Bias instability Bias fluctuations, usually
modelled as a bias random
walk
A second-order random walk
Table 3.1: A summary of MEMS gyroscope error sources.
whereas in reality the bias will always remain within some range around zero. More accurate
(but more processor intensive) approaches include modelling bias instability using Markov
processes and representing sensor biases using second-order and third-order autoregressive
models [18].
A summary of the different error sources described above is shown in Table 3.1.
3.1.2 Accelerometer error sources
The main types of error that perturb measurements from MEMS accelerometers are the same
as those described above for MEMS gyroscopes. The primary difference is that errors in ac-
celeration measurements are integrated twice to track the position of an IMU, whereas angular
velocity measurements are only integrated once to track the estimated orientation. As a result
bias errors cause errors in position that grow proportional to 푡2, white noise causes a second-
order random walk in position and so on. A summary of the different error sources can be
found in Table 3.2. More detailed information can be found in [79].
3.2 Strapdown inertial navigation
This section develops the strapdown inertial navigation algorithm that is illustrated by Figure
3.2. The equations necessary to track the state of an IMU over time are first derived for the
theoretical case in which continuous angular velocity and acceleration signals are used. The
real-life case in which an IMU provides samples of these signals at discrete points in time is
discussed in Section 3.2.2. The derivations in both sections are based on those in [13]. Note
46
Error type Description Result of double integration
Bias A constant bias offset A position error that grows propor-
tional to 푡2
Random noise Random noise from a variety
of sources (usually specified
in terms of its root PSD)
A second-order random walk with a
zero mean and a standard deviation
that grows proportional to 푡3/2.
Calibration Deterministic errors in scale
factors, alignments and ac-
celerometer linearities
Position drift proportional to the
squared rate and duration of acceler-
ation
Temperature
effects
Temperature dependent resid-
ual bias
Any residual bias causes an error in
position that grows quadratically with
time
Bias instability Bias fluctuations, usually
modelled as a bias random
walk
A third-order random walk in position
Table 3.2: A summary of MEMS linear accelerometer error sources.
that the local navigation frame is approximated as an inertial frame of reference in this section.
Hence compensation terms due to the rotation of the earth are omitted.
3.2.1 Theory
The orientation, or attitude, of an IMU relative to a local reference frame can be tracked by
integrating its angular velocity
흎푏(푡) = (휔푏푥(푡), 휔푏푦(푡), 휔푏푧(푡))푇 . (3.8)
The orientation of the device can be specified using several different representations, including
Euler angles, quaternions and direction cosines. The direction cosines representation is used
in the derivations below. Equivalent derivations using both Euler angles and quaternions can
be found in [13].
In the direction cosines representation the orientation of the IMU’s body frame relative to the
local frame is specified by a 3 × 3 rotation matrix 푪, in which each column is a unit vector
along one of the body axes specified in terms of the local axes. A vector quantity 풗푏 defined
in the body frame is equivalent to the vector
풗푔 = 푪풗푏 (3.9)
defined in the local frame. The inverse transformation is given by
풗푏 = 푪푇풗푔 (3.10)
47
since the inverse of a rotation matrix is equivalent to its transpose.
If the orientation of the IMU at time 푡 is given by 푪(푡) then the rate of change of 푪 at 푡 is
given by
푪˙(푡) = lim
훿푡→0
푪(푡+ 훿푡)−푪(푡)
훿푡
(3.11)
in which 푪(푡+ 훿푡) can be written as the product of two matrices
푪(푡+ 훿푡) = 푪(푡)푨(푡) (3.12)
where 푨(푡) is the rotation matrix that relates the body frame at time 푡 to the body frame at
time (푡+ 훿푡) [13]. If 훿휙, 훿휃 and 훿휓 are the small clockwise rotations through which the body
frame has rotated between times 푡 and (푡 + 훿푡) about its 푥, 푦 and 푧 axes respectively, then
using the small angle approximation (see Appendix B.1)푨(푡) can be written as
푨(푡) = 푰 + 휹Ψ (3.13)
where
휹Ψ =
⎛⎜⎝ 0 −훿휓 훿휃훿휓 0 −훿휙
−훿휃 훿휙 0
⎞⎟⎠ . (3.14)
Hence by substitution
푪˙(푡) = lim
훿푡→0
푪(푡+ 훿푡)−푪(푡)
훿푡
(3.15)
= lim
훿푡→0
푪(푡)푨(푡)−푪(푡)
훿푡
(3.16)
= lim
훿푡→0
푪(푡)(푰 + 휹Ψ)−푪(푡)
훿푡
(3.17)
= 푪(푡) lim
훿푡→0
휹Ψ
훿푡
. (3.18)
In the limit 훿푡→ 0 the small angle approximation is valid, and
lim
훿푡→0
휹Ψ
훿푡
= Ω(푡) (3.19)
where
Ω(푡) =
⎛⎜⎝ 0 −휔
푏푧(푡) 휔푏푦(푡)
휔푏푧(푡) 0 −휔푏푥(푡)
−휔푏푦(푡) 휔푏푥(푡) 0
⎞⎟⎠ (3.20)
is the skew symmetric form of the angular velocity 흎푏(푡). Hence in order to track the orienta-
48
tion it is necessary to solve the differential equation
푪˙(푡) = 푪(푡)Ω(푡) (3.21)
which has the solution
푪(푡) = 푪(0) ⋅ exp
(∫ 푡
0
Ω(푡) dt
)
(3.22)
where 푪(0) is the initial orientation of the device [13].
To track the position of an IMU the specific force
풇 푏(푡) = (푓 푏푥(푡), 푓 푏푦(푡), 푓 푏푧(푡))푇 (3.23)
is first projected into the local frame of reference
풇 푔(푡) = 푪(푡)풇 푏(푡) . (3.24)
Acceleration due to gravity is then added to obtain the acceleration of the IMU, which is
integrated once to obtain velocity and again to obtain displacement
풂푔(푡) = 풇 푔(푡) + 품푔 (3.25)
풗푔(푡) = 풗푔(0) +
∫ 푡
0
풂푔(푡) dt (3.26)
풔푔(푡) = 풔푔(0) +
∫ 푡
0
풗푔(푡) dt (3.27)
where 풗푔(0) is the initial velocity of the device, 풔푔(0) is the initial displacement and 품푔 is
acceleration due to gravity specified in the local frame of reference.
3.2.2 Implementation
Since an IMU only provides fixed-rate samples of the continuous signals 흎푏(푡) and 풂푏(푡),
integration schemes must be used to apply Equations 3.22, 3.26 and 3.27 in practice [13]. The
choice of scheme is application dependent. For short timespan and low accuracy applications
a low order integration scheme such as the rectangular rule is sufficient. For more demanding
applications a third or fourth-order scheme may be more appropriate. This section develops
an INS implementation that uses the rectangular rule, since it is computationally simple and
is adequate for use given the current quality of MEMS inertial sensors.
Let the period between successive samples be 훿푡. For a single sampling period (푡− 훿푡, 푡) the
49
solution to Equation 3.21 can be written as
푪푡 = 푪푡−훿푡 ⋅ exp
(∫ 푡
푡−훿푡
Ω(푡) dt
)
. (3.28)
The rectangular rule gives ∫ 푡
푡−훿푡
Ω(푡) dt = 푩푡 (3.29)
where
푩푡 =
⎛⎜⎝ 0 −휔
푏푧
푡 훿푡 휔
푏푦
푡 훿푡
휔푏푧푡 훿푡 0 −휔푏푥푡 훿푡
−휔푏푦푡 훿푡 휔푏푥푡 훿푡 0
⎞⎟⎠ (3.30)
in which 흎푏푡 = (휔
푏푥
푡 , 휔
푏푦
푡 , 휔
푏푧
푡 )
푇 is the angular velocity measurement corresponding to the
update period. Letting 휎 = ∣흎푏푡훿푡∣, substituting Equation 3.29 into Equation 3.28 and finally
performing the Taylor expansion of the exponential term gives
푪푡 = 푪푡−훿푡
(
푰 +푩푡 +
푩2푡
2!
+
푩3푡
3!
+
푩4푡
4!
+ ...
)
(3.31)
= 푪푡−훿푡
(
푰 +푩푡 +
푩2푡
2!
− 휎
2푩푡
3!
− 휎
2푩2푡
4!
+ ...
)
(3.32)
= 푪푡−훿푡
(
푰 +
(
1− 휎
2
3!
+
휎4
5!
...
)
푩푡 +
(
1
2!
− 휎
2
4!
+
휎4
6!
...
)
푩2푡
)
(3.33)
= 푪푡−훿푡
(
푰 +
sin휎
휎
푩푡 +
1− cos휎
휎2
푩2푡
)
. (3.34)
This is the orientation update equation that is applied by an INS each time a new measurement
is received from the IMU [13].
To update the tracked position of the IMU the specific force measurement 풇 푏푡 is first projected
into the local frame of reference
풇 푔푡 = 푪푡풇
푏
푡 . (3.35)
The velocity and displacement components of the state are then updated. Using the rectangular
rule the update Equations 3.25, 3.26 and 3.27 become
풂푔푡 = 풇
푔
푡 + 품
푔 (3.36)
풗푔푡 = 풗
푔
푡−훿푡 + 훿푡 ⋅풂푔푡 (3.37)
풔푔푡 = 풔
푔
푡−훿푡 + 훿푡 ⋅풗푔푡 . (3.38)
Equations 3.34, 3.35, 3.36, 3.37 and 3.38 are all that an INS must implement to update the
estimated orientation, velocity and displacement of the tracked IMU each time a new mea-
surement is received. Note however that the INS must still be provided with the initial state
50
(푪0,풗
푔
0 , 풔
푔
0) of the device.
3.3 Error propagation in inertial navigation systems
When using an INS the most important property that must be considered is the rate at which
drift errors accumulate in the calculated position. The most obvious cause of drift is the prop-
agation of errors perturbing the acceleration measurements through the double integration, as
summarised in Table 3.2. For example a constant bias error affecting a single accelerometer
will cause drift in position that grows proportional to 푡2. Although this is the most obvious
cause of drift, it is not the most severe except during the early stages of inertial navigation.
Over longer time periods errors in orientation that arise due to errors perturbing the angular
velocity measurements account for a greater proportion of drift (with the crossover point de-
pendent on the relative quality of the accelerometers and gyroscopes used). This is due to
the use of the calculated orientation 푪푡 to project specific force measurements into the global
frame of reference. Suppose that errors in the angular velocity measurements cause a small
tilt error 휖 in 푪푡. After acceleration due to gravity is added in the local frame, 풂
푔
푡 will contain
a residual bias of magnitude 푔 ⋅ sin(휖) in the (푥, 푦) plane and 푔 ⋅ (1− cos(휖)) along the 푧 axis.
These errors will then be integrated into both the velocity and displacement components of
the INS state. Note that if 휖 is small then cos(휖) ≈ 1 and sin(휖) ≈ 휖. Hence the majority of
drift caused by small tilt errors occurs in the (푥, 푦) plane.
Note that even a small error in orientation causes a rapidly growing error in the calculated
position. For example consider a constant tilt error of just 0.05∘. This error will cause a
component of acceleration due to gravity with magnitude 0.0086 m/s2 to be projected onto
the horizontal axes. This residual bias will in turn cause drift in the (푥, 푦) plane that grows
quadratically to 7.7 m after only 30 seconds. In practice such tilt errors are often introduced
by small bias errors in the gyroscope signals. In such cases the tilt errors are themselves
growing linearly with time, meaning that the resulting drift in position will grow proportional
to 푡3. Note also that if the local frame is approximated as an inertial frame of reference (as is
the case in this thesis), then the earth’s rotation will also introduce an error in orientation that
grows linearly with time.
3.3.1 Xsens Mtx example
To demonstrate the relative importance of different error sources and their propagation through
the INS equations, the performance of an INS based on an Xsens Mtx IMU was analysed
for the stationary case. The INS navigation processor, which applied the equations given in
Section 3.2.2 to update the estimated state, was implemented within an event-based framework
51
written in the Java programming language. A Java interface to the Xsens Mtx was also written
within this framework to allow measurement data from the device to be processed in real time.
Event logging and replay facilities allowed event streams to be recorded and replayed at a later
date.
To test the performance of the INS it was applied to 100 streams of data, each obtained from
a stationary Mtx device with a sampling frequency of 100 Hz. Each stream was 135 seconds
in length with the first 15 s (from 푡 = −15 s to 푡 = 0 s) used for initialisation in each case.
The first 10 seconds of the initialisation process was used to calculate the average specific
force vector measured by the IMU in the body frame. The INS attitude was then initialised
at 푡 = −5 s such that this vector was aligned3 with the 푧-axis of the local frame of reference.
The heading component of the orientation was initialised to an arbitrary value. Note that this
levelling process is not exact, since the average specific force vector will be affected by bias
errors perturbing the accelerometer signals.
If the INS were run after levelling as described above, drift arising due to the accelerometer
biases would accumulate in the opposite direction to drift caused by the initial tilt error (which
is introduced by the levelling process as a result of the same accelerometer biases). Hence the
two would tend to cancel out, resulting in an overly optimistic estimate of true performance.
To address this problem the IMU was rotated by 90∘ between 푡 = −5 s and 푡 = 0 s of each
run. This causes the two error sources to act at 90∘ to one another rather than in opposite
(cancelling) directions. At 푡 = 0 s of each run the global displacement and velocity were
initialised to zero
풔푔0 = 풗
푔
0 = (0, 0, 0)
푇 . (3.39)
The INS was then used to track the position of the IMU whilst it was kept stationary from
푡 = 0 s until 푡 = 120 s.
Since the IMU was stationary during each run (excluding the initialisation phase), the total
drift in position at time 푡 during a particular run of the INS is given by ∣풔푔푡 ∣. Figure 3.3 shows
how this drift (averaged over all 100 runs) increased over time. After 60 s the average drift
was 170 m and by 120 s it had grown to 1052 m. The average drift along the global 푧-axis
at 푡 = 120 s was ∣푠푔푧푡 ∣ = 168.7 m, compared to a much larger average drift of 1031 m in the
(푥, 푦) plane. This is typical of drift caused by primarily by the propagation of small tilt errors,
as described in Section 3.3. Such tilt errors arise due to both gyroscope errors and the rotation
of the earth, which is not compensated for by the INS.
To analyse the underlying causes of the accumulating drift, each of the streams of data were
reprocessed by the INS whilst selectively removing error sources. This was possible because
it was known that the IMU remained stationary during each run. Accelerometer errors were
3An algorithm for deriving a rotation matrix that aligns two vectors can be found in Appendix B.3.
52
 0
 200
 400
 600
 800
 1000
 1200
 1400
 1600
 0  20  40  60  80  100  120
M
ea
n 
dr
ift
 (m
)
Time (s)
Mean drift
Figure 3.3: The mean drift incurred by an INS during 100 test runs using an Xsens Mtx
IMU. Each run was of length 120 s.
 0
 200
 400
 600
 800
 1000
 1200
 0  20  40  60  80  100  120
M
ea
n 
dr
ift
 (m
)
Time (s)
All errors
Gyro & earth rotation errors only
Accel errors only
Figure 3.4: The mean drifts incurred when error sources were selectively removed.
53
removed by setting every specific force measurement from 푡 = 0 s to 푡 = 120 s to
풇 푏푡 = −푪푇0 품푔 (3.40)
where푪푇0 is the orientation of the IMU after initialisation and 품
푔 = (0, 0,−9.807)푇 is acceler-
ation due to gravity. Errors arising from the gyroscopes (and as a result of the earth’s rotation)
were removed by setting every angular velocity measurement from 푡 = 0 s to 푡 = 120 s to
흎푏푡 = (0, 0, 0)
푇 . (3.41)
Figure 3.4 shows the average drift curves obtained when the error sources were selectively
removed. Note that the accelerometers are the primary cause of drift up until 푡 = 62 s, after
which the drift arising due to the combined effect of gyroscope errors and the earth’s rotation
become dominant. The curve corresponding to errors perturbing the accelerometers grows
almost exactly in proportion to 푡2, which is the rate expected due to the propagation of ac-
celerometer bias errors through the INS. The curve corresponding to errors perturbing the
gyroscopes and the earth’s rotation grows almost exactly in proportion to 푡3, which is the rate
expected due to the propagation of gyroscope bias errors through the INS. At 푡 = 120 s the
average drift caused by the accelerometers was 547 m, whilst the average drift caused by the
gyroscopes and the earth’s rotation was 1012 m.
In the case of drift caused by errors in orientation, it is possible to quantify how much of the
drift is caused by the earth’s rotation as opposed to the gyroscopes. This can be achieved by
initialising the INS at 푡 = 0 s with
풗푔0 = (0, 0, 0)
푇 (3.42)
풔푔0 = (0, 0, 0)
푇 (3.43)
푪0 = 푰 (3.44)
where 푰 is the 3 × 3 identity matrix. The INS is then run for 120 s during which every
measurement is given by
풇 푏푡 = −품푔 (3.45)
흎푏푡 = 푪
푇
0 ⋅ (Ω cos퐿, 0,−Ω sin퐿)푇 (3.46)
in which 퐿 = 52.2∘ is the latitude of the location where the tests were performed (Cambridge,
UK) and Ω = 7.2921 × 10−5 rad/s is the earth’s rate of rotation [13]. This simulates a
stationary INS in which the only error source is a growing error in orientation caused by the
rotation of the earth. After 120 s the drift in position calculated by the simulation is 125 m.
Hence the vast majority of the 1012 m drift caused by orientation errors in the example above
54
can be attributed to errors perturbing the gyroscope signals rather than to the rotation of the
earth. This means that even if the INS compensated for the earth’s rotation, the drift in position
would only be reduced by a small fraction of its total value.
There are two main conclusions that can be drawn from the results above. The first is that
errors perturbing the accelerometer signals are the primary source of drift in the INS for
just over the first minute of operation, after which errors perturbing the gyroscope signals
become dominant. The second and most important conclusion is that MEMS IMUs are not
yet sufficiently accurate to track the position of a device or person to sub-metre accuracy
for extended periods of time (any period greater than 5 s for the tested IMU) using inertial
navigation alone. Although MEMS IMUs are improving rapidly, the 푡3 long term error-growth
means that even major improvements in the quality of such sensors will yield only modest
increases in the durations for which tracking can be performed before error tolerances are
exceeded. Finally, note that drift in position will be significantly worse when tracking moving
objects rather than stationary ones. This is largely due to errors in the tracked heading, which
only propagate to errors in position when the IMU is moving. In addition gyroscope non-
linearities and scale factor errors are only observed when the IMU is rotating. These will
cause additional errors to accumulate in the orientation, which will in turn result in additional
drift in the tracked position.
3.4 INS error correction
One technique to allow a MEMS-based INS to track the position of an object or person for
an extended period of time is to regularly correct its state using external measurements. For
example positions from an absolute positioning system can be used to periodically correct
errors that would otherwise accumulate in the calculated position. The naı¨ve way to apply
such measurements is to directly correct the INS state. For example an absolute position
measurement can be applied by simply setting the displacement component of the state to
equal the displacement reported by the measurement. Although the simplicity of this approach
is attractive, it suffers from two major problems:
∙ External measurements that can be used to correct the system are usually inexact. The
naı¨ve approach to INS correction does not account for uncertainty in external measure-
ments.
∙ The only part of the INS state to be corrected by a measurement is the part that it di-
rectly observes. For example an absolute position measurement is only used to correct
the position component of the state. An INS that is periodically corrected in this way
will still accumulate errors in the orientation and velocity components, which will not
55
be corrected. It should be possible to do much better than this because the propaga-
tion of errors through the INS equations causes correlations between errors in different
components of the state. In particular tilt errors in the orientation are strongly corre-
lated to errors in both velocity and displacement. By keeping track of such correlations
it should be possible to use measurements that observe one component of the state to
correct correlated errors in other components that are not directly observed.
Both these problems can be addressed by using a Bayesian filter to track the state of an INS
and to incorporate external measurements. Recall from Section 2.3 that a Bayesian filter
takes the uncertainty of measurements into account. Bayesian filters are also able to track
correlations between errors that accumulate in different parts of the state. This is possible
because a Bayesian filter must be provided with a propagation model that describes how the
state of the system evolves over time, from which such correlations can be deduced.
This section describes a simple Kalman filter that is able to correct an INS using external
measurements. The Kalman filter is well suited to this task when compared to the other types
of Bayesian filter described in Section 2.3. Unlike both grid-based filters and particle filters
they are sufficiently computationally efficient to support the high update rates required without
the need for expensive hardware. A uni-modal representation of belief is sufficient and hence
the added complexity of Gaussian mixture models is unnecessary.
Rather than tracking the total state (i.e. orientation, velocity and position) of the IMU directly,
the Kalman filter described below maintains a probability distribution representing the error
in the state calculated by a regular INS. Each time a new measurement (흎푏푡 ,풇
푏
푡 ) is received
from the IMU it is used to update the INS state as normal. The Kalman filter’s belief of the
error-state is also propagated forward to the current time 푡. When an external measurement is
received it is used to update the estimated error-state during the filter’s correction step. This
estimate is then immediately transferred to the underlying INS before being reset to zero (this
is known as closed-loop correction). Using an error-state filter together with a separate INS
is mathematically equivalent to an extended Kalman filter (EKF) that tracks the total state
directly (see Appendix C). It is a convenient approach because it allows existing INS software
components such as those developed and tested in Section 3.3.1 to be re-used with minimal
effort, rather than having to replicate the functionality within an equivalent total-state filter.
The remainder of this section describes the error-state Kalman filter and its interaction with
the underlying INS in more detail. Section 3.4.1 defines the error-state and its relationship
with the inertial navigation solution estimated by the INS. The filter’s propagation step, which
is executed whenever a new measurement is received from the IMU, is outlined in Section
3.4.2. The correction step that is executed whenever an external measurement is received is
described in Section 3.4.3. Finally, Section 3.4.4 describes how the estimated error-state is
transferred from the filter to the underlying INS following each correction step.
56
3.4.1 Error-state definition
The true state of an IMU at time 푡 can be defined in terms of its orientation, displacement and
velocity
(푪푡,풗푡, 풔푡) . (3.47)
The rotation matrix 푪푡 defines the orientation of the IMU relative to the local frame of refer-
ence, as described in Section 3.2. The displacement and velocity are both defined in the local
frame. Superscripts indicating the frame of reference in which each quantity is specified are
intentionally omitted in this section in order to avoid notational clutter.
It is not possible for an INS to calculate the true state of its IMU (if it could then it would be
able to accurately track its position indefinitely without the need for external correction). The
inertial navigation solution calculated by the INS diverges from the true state of the IMU due
to the propagation of errors through the inertial navigation equations, as described in Section
3.3. In this section the solution calculated by the INS is denoted
(푪ˆ푡, 풗ˆ푡, 풔ˆ푡) (3.48)
where hat symbols are used to distinguish INS estimates from the true state of the IMU. The
error-state is the difference between the true state and the solution calculated by the INS. It is
defined in column vector form as
훿풙푡 = (훿흓
푇
푡 , 훿풗
푇
푡 , 훿풔
푇
푡 )
푇 (3.49)
where 훿풗푡 and 훿풔푡 are errors in velocity and displacement respectively. The error in orientation
is expressed such that a clockwise rotation of magnitude ∣훿흓푡∣ about 훿흓푡 in the local frame of
reference will correct the error. Formally, the error-state is defined such that
푪푡 = Φ푡푪ˆ푡 (3.50)
풗푡 = 풗ˆ푡 + 훿풗푡 (3.51)
풔푡 = 풔ˆ푡 + 훿풔푡 (3.52)
where Φ푡 is the matrix form of the rotation defined by 훿흓푡 = (훿휙
푔푥
푡 , 훿휙
푔푦
푡 , 훿휙
푔푧
푡 ). If the ro-
tation is small (as will be the case if the INS solution is frequently corrected using external
measurements) then Φ푡 can be approximated as
Φ푡 ≈
⎛⎜⎝ 1 −훿휙
푔푧
푡 훿휙
푔푦
푡
훿휙푔푧푡 1 −훿휙푔푥푡
−훿휙푔푦푡 훿휙푔푥푡 1
⎞⎟⎠ (3.53)
57
in which small angle approximations are used. An exact form suitable for larger rotations can
be found in Appendix B.2.
The error-state Kalman filter maintains a probability distribution Bel(훿풙푡) that represents the
distribution of possible error-states. Recall from Section 2.3.1 that this distribution is assumed
to be Gaussian distributed and hence is represented by its mean 훿풙ˆ푡 and covariance 푷푡.
3.4.2 Error-state propagation
The Kalman filter propagates its belief of the error-state whenever a new measurement (흎푏푡 ,풇
푏
푡 )
is received from the IMU. The standard Kalman filter update equation (see Appendix C) that
is used to propagate the mean error-state is given by
훿풙ˆ−푡 = 푭푡훿풙ˆ푡−훿푡 (3.54)
however for the error-state filter it is always the case that 훿풙ˆ푡−훿푡 = 0 when entering the
propagation step (because the mean error-state is reset to zero when it is transferred to the
underlying INS immediately after each correction step). Hence 훿풙ˆ−푡 = 0 = 훿풙ˆ푡−훿푡 and so
it is not necessary to explicitly update the mean. It is however necessary to propagate the
error-state covariance. This is done using the standard Kalman filter update equation
푷−푡 = 푭푡푷푡−훿푡푭
푇
푡 +푸푡 (3.55)
where 푭푡 is the error-state transition matrix and 푸푡 is the process noise covariance. The
error-state transition matrix is a simplified4 version of the one used in [14]. It contains both
first-order and small angle approximations, however these are almost certainly valid at the
100 Hz sampling rate used. It is given by
푭푡 =
⎛⎜⎝ 푰 0 0훿푡푺(풇 푔푡 ) 푰 0
0 훿푡푰 푰
⎞⎟⎠ (3.56)
where 푰 is the 3×3 identity matrix, 0 is the 3×3 matrix containing all zeros, 훿푡 is the elapsed
time since the previous update (e.g. 훿푡 = 0.01 s for an IMU sampling at 100 Hz) and
푺(풇 푔푡 ) =
⎛⎜⎝ 0 −푓
푔푧
푡 푓
푔푦
푡
푓 푔푧푡 0 −푓 푔푥푡
−푓 푔푦푡 푓 푔푥푡 0
⎞⎟⎠ (3.57)
4The filter in [14] contains additional accelerometer and gyroscope bias states, which are propagated by the
transition matrix into velocity and orientation errors respectively. These additional states were not found to be
necessary for the filter’s usage in this thesis.
58
is the skew-symmetric cross-product operator matrix formed from the current specific force
vector 풇 푔푡 = (푓
푔푥
푡 , 푓
푔푦
푡 , 푓
푔푧
푡 ). The 훿푡푺(풇
푔
푡 ) sub-block of 푭푡 causes errors in orientation to prop-
agate into velocity errors in the local frame of reference. Errors in velocity are subsequently
propagated into displacement errors by the 훿푡푰 sub-block in the third row and second column
of 푭푡. The filter uses the error propagation information that is encoded in the transition matrix
to track correlations between errors in different components of the state, storing them in the
covariance matrix 푷푡. As a result correlated errors in all components of the state are corrected
whenever a measurement is received that observes only one of them.
The process noise covariance matrix 푸푡 describes the error characteristics of the underlying
inertial sensors. Its general form is given by
푸푡 =
⎛⎜⎝ 푻푡 0 00 푼푡 0
0 0 0
⎞⎟⎠ (3.58)
in which 푻푡 and 푼푡 model gyroscope and accelerometer errors respectively, as described be-
low.
3.4.2.1 Modelling random noise
Random noise is modelled as white noise consisting of a sequence of uncorrelated, Gaussian
distributed random variables that perturb the measurements received from the IMU. For the
gyroscopes this noise process can be represented exactly by
푻푡 = 훿푡
2
(
푪ˆ−푡
)푇
(푾푡)
(
푪ˆ−푡
)
(3.59)
in which
푾푡 =
⎛⎜⎝ (휎
푏푥
휔 )
2 0 0
0 (휎
푏푦
휔 )2 0
0 0 (휎푏푧휔 )
2
⎞⎟⎠ (3.60)
is a matrix containing the variances of the random variables perturbing measurements from
each of the three gyroscopes. White noise perturbing the acceleration measurements can be
represented similarly in푼푡. The variances required in both matrices can be measured directly
from signals obtained from an IMU. This can be done using the Allan Variance technique
described in Appendix A.
59
3.4.2.2 Modelling bias instabilities
One way in which bias instability effects can be modelled is to extend the error-state Kalman
filter to estimate additional gyroscope and accelerometer bias states [14]. Bias instability ef-
fects can then be modelled as random walks in which each bias state is perturbed by additive
Gaussian noise, whilst initial uncertainty in the sensor biases can be modelled in 푷0. Extend-
ing the filter in this way was not found to be necessary for the pedestrian dead reckoning filter
that is developed in Chapter 4 (and which motivates the development of the error-state filter
here) and hence is not described further. A simpler alternative is to use white noise in order
to approximate the uncertainty that arises as a result of such errors. Although this approach is
theoretically unsound, it achieves good results in practice whilst avoiding the need to extend
the state vector (and hence avoiding an increase in the filter’s computational requirements). It
is preferable to simply ignoring the error sources because underestimating the uncertainty in
the state of the underlying INS can limit the filter’s ability to apply corrections of sufficient
magnitude.
3.4.2.3 Modelling calibration errors
Sensor calibration errors (i.e. non-linearities and scale factor errors) are systematic errors that
are not naturally described in terms of additive Gaussian noise. Nevertheless it is helpful to
model such error sources so that the filter does not underestimate the amount of uncertainty in
the state. Again this can be done by adding additional white noise, whose magnitude results
in uncertainty that is roughly equivalent to that expected from the error sources themselves.
For example Foxlin used this approach to model non-linearities and temperature dependent
scale factor variations in gyroscope signals [80]. A similar approach is used here, defining 푻푡
to be as in Equation 3.59, but with
푾푡 =
⎛⎜⎝ (휎
푏푥
휔 )
2 + (푠푏푥휔 휔
푏푥
푡 )
2 0 0
0 (휎
푏푦
휔 )2 + (푠
푏푦
휔 휔
푏푦
푡 )
2 0
0 0 (휎푏푧휔 )
2 + (푠푏푧휔 휔
푏푧
푡 )
2
⎞⎟⎠ . (3.61)
The first term in each diagonal component models random noise as described in Section
3.4.2.1 and can be boosted to model otherwise un-modelled error sources as described in
Section 3.4.2.2. The second terms model both gyroscope non-linearities and scale factor
errors as white noise that is varied dynamically based on the measured angular velocity
흎푏푡 = (휔
푏푥
푡 , 휔
푏푦
푡 , 휔
푏푧
푡 ). The result is that uncertainty grows more rapidly when the IMU is
rotating compared to when it is stationary. This is intended behaviour, since calibration errors
are only visible in non-zero measurements. The vector 풔푏휔 = (푠
푏푥
휔 , 푠
푏푦
휔 , 푠푏푧휔 ) determines the
magnitude of the resulting uncertainty. Calibration errors perturbing measurements from the
60
accelerometers are modelled similarly in 푼푡.
The filter implementation used in this thesis models random noise and calibration errors in
the way described above. This is done for both the gyroscope and the accelerometer signals.
Suitable 휎 values for modelling random noise were determined for the Xsens Mtx IMU using
the Allan Variance technique, as described in Appendix A. The values obtained can be found
in Table A.35, however they were (somewhat arbitrarily) doubled for use in the filter in order
to account for otherwise un-modelled error sources. The vectors 풔푏휔 = (0.001, 0.001, 0.001)
and 풔푏푎 = (0.002, 0.002, 0.002) were set according to the manufacturer’s sensor linearity spec-
ifications.
3.4.3 Error-state correction
If an external measurement is made at time 푡 then it is used to correct the estimated error-state.
The filter is able to incorporate any measurement 풛푡 that can be related to the true state 풙푡 of
the IMU by an equation of the form
풛푡 = 퓗푡(풙푡) + 풗푡 . (3.62)
where퓗푡 is a differentiable (but possibly non-linear) function and 풗푡 = 풩0,푹푡 . To apply such
a measurement it is first converted into the equivalent error-state measurement 풚푡, given by
풚푡 = 풛푡 −퓗푡(풙ˆ−푡 ) (3.63)
where 풙ˆ−푡 is the current inertial navigation solution as estimated by the underlying INS. The
minus superscript is used to indicate that the estimate has not yet been corrected by the external
measurements. The error-state measurement is used to calculate the posterior belief (훿풙ˆ푡,푷푡)
using
푲푡 = 푷
−
푡 푯
푇
푡 (푯푡푷
−
푡 푯
푇
푡 +푹푡)
−1 (3.64)
훿풙ˆ푡 = 푲푡풚푡 (3.65)
푷푡 = (푰 −푲푡푯푡)푷−푡 (3.66)
where푹푡 is the covariance matrix describing uncertainty in the measurement and
푯푡 =
[
∂퓗푡
∂풙
]
풙ˆ−푡
. (3.67)
5Note that it is necessary to convert the gyroscope 휎 values into rad/s before they can be used in푾푡.
61
These are the standard Kalman filter correction equations (see Appendix C) except for 3.65,
which is a simplified version of the usual equation that arises for the error-state filter because
it is always the case that 훿풙ˆ−푡 = 0 when entering the correction step.
In practice it will usually be the case that the frequency at which measurements arrive from
the IMU is far greater than the frequency at which external measurements are received. Hence
there will be time-steps at which no external measurements are available. In this case the pos-
terior covariance 푷푡 at time 푡 is simply equal to the prior 푷−푡 as calculated by the propagation
step. It may also be the case that more than one external measurement is made within a single
time-step. In this case the correction step can be executed multiple times, transferring the
error-state to the underlying INS between each execution and using the posterior covariance
푷푡 from one step as the prior covariance 푷−푡 in the next.
3.4.4 Error-state transfer
Immediately after each correction step the estimated error-state 훿풙ˆ푡 is transferred to the un-
derlying INS. Errors in velocity and displacement are transferred by straightforward addition
풗ˆ푡 = 풗ˆ
−
푡 + 훿풗ˆ푡 (3.68)
풔ˆ푡 = 풔ˆ
−
푡 + 훿풔ˆ푡 . (3.69)
The rotation defined by the error vector 훿흓ˆ푡 = (훿휙ˆ
푔푥
푡 , 훿휙ˆ
푔푦
푡 , 훿휙ˆ
푔푧
푡 ) is converted into its equiva-
lent rotation matrix form
Φˆ푡 =
⎛⎜⎝ 1 훿휙ˆ
푔푧
푡 −훿휙ˆ푔푦푡
−훿휙ˆ푔푧푡 1 훿휙ˆ푔푥푡
훿휙ˆ
푔푦
푡 −훿휙ˆ푔푥푡 1
⎞⎟⎠ (3.70)
in which small angle approximations are used. An exact form can be found in Appendix
B.2, however in practice the difference between the two is negligible provided that frequent
corrections are applied. Having calculated Φˆ푡, the corrected INS attitude is given by
푪ˆ푡 = Φˆ푡푪ˆ
−
푡 . (3.71)
Once the mean error-state has been transferred to the underlying INS it is reset to zero.
An outline of the complete system is shown in Figure 3.5. The downward arrows from the
error-state Kalman filter represent the transfer of the estimated error-state to the INS.
62
v^t s^tat
R
Project 
specific 
force onto 
ref. axes
Add 
acceleration 
due to 
gravity
R
Error-state Kalman filter
 
v^0 s^0
C^0
R
C^t!
b
t
f bt
 
±Á^t ±v^t ±s^t
 
Inertial navigation system
 
yt = zt ¡H(x^t)
+
+ +
Figure 3.5: INS correction using an error-state Kalman filter.
3.4.5 Xsens Mtx example
One way to test the performance of the error-state Kalman filter is to revisit the trials con-
ducted with the Xsens Mtx IMU in Section 3.3.1. Tracking this IMU using an uncorrected
INS resulted in an average drift of 1052 m after 120 s. In this section the same data is pro-
cessed, but the INS is periodically corrected using external measurements. In particular it is
assumed that the device is known to be approximately stationary for 0.5 s periods uniformly
spaced at 2 s intervals from 푡 = 0 s to 푡 = 120 s of each run. When a new sample is received
from the IMU during these periods a corresponding zero velocity update (ZVU) is gener-
ated and is used as a pseudo-measurement by the error-state filter in order to correct the INS.
Note that this hypothetical test case is similar to the application of ZVUs in pedestrian dead
reckoning systems, as described in Section 2.2.2.2.
To illustrate the benefit of the error-state filter, the measurements were applied both naı¨vely
(i.e. just correcting the velocity component of the INS state) and using the filter. In both cases
the INS state was initialised in the same way as in Section 3.3.1. In the latter case the required
measurement function and its derivative are given by
퓗푡(풙푡) = 풗푡 (3.72)
푯푡 = (0, 푰,0) . (3.73)
Each ZVU was assumed to have a small amount of uncertainty that was independent along
63
each axis. Hence the measurement covariance matrix was set to
푹푡 =
⎛⎜⎝ 휎
2 0 0
0 휎2 0
0 0 휎2
⎞⎟⎠ (3.74)
in which a value of 휎 = 0.01 ms−1 was used. The filter was initialised assuming that there
was no uncertainty in the initial displacement and heading, and a small amount of uncertainty
in the tilt and velocity of the device. Hence the initial state covariance was
푷0 =
⎛⎜⎝ 푽 0 00 푾 0
0 0 0
⎞⎟⎠ (3.75)
where
푽 =
⎛⎜⎝ 10
−3 0 0
0 10−3 0
0 0 0
⎞⎟⎠ (3.76)
푾 =
⎛⎜⎝ 10
−5 0 0
0 10−5 0
0 0 10−5
⎞⎟⎠ (3.77)
with units (rad/s)2 and (m/s)2 respectively. Note that in general it may be necessary to
initialise the other diagonal elements of 푷0 to be small positive numbers (rather than zeros)
to ensure that 푷푡 remains positive definite. This is a necessary condition to prevent filter
instabilities and likely divergence of the system away from the true state [53]. Appropriate
values depend on the floating point implementation used. In practice no instabilities were
observed when zeros were used in the tested implementation, which was written in Java using
64-bit floating point arithmetic.
Figure 3.6 shows the mean drift in position (averaged over the same 100 event streams as in
Section 3.3.1) when the INS was uncorrected, naı¨vely corrected and corrected using the error-
state filter. The mean drift when naı¨ve corrections were applied is shown together with error
bars in Figure 3.7. The drift was significantly reduced to an average of 14.3 m after 120 s,
however since the orientation component of the state is left uncorrected drift still accumulates
at a rate that is proportional to 푡2 after gyroscope errors become the dominant source of drift.
This is an improvement on the unconstrained INS in which drift grew proportional to 푡3 during
the same period, however it is possible to do much better. Figure 3.8 shows the mean drift
when the ZVU measurements were applied using the error-state filter. Note that the mean drift
of 0.04 m after 120 s is much lower than when naı¨ve corrections are applied, and that drift now
64
grows at a rate that is roughly linear with respect to time. This improvement is a result of the
filter’s ability to track correlations between errors that accumulate in different components of
the INS state. By using this information the filter is able to correct not only errors in velocity
when applying ZVUs, but also tilt errors in the tracked orientation and much of the drift in
position that has accumulated since ZVUs were last applied. These corrections are visible in
Figure 3.8 as sharp reductions in the average drift that correspond to regions of time during
which ZVUs were applied.
3.5 Conclusions
Inertial navigation can be used to track the position, velocity and orientation of an IMU rel-
ative to its initial state, however drift errors rapidly accumulate in the tracked position. Even
when using a high-end MEMS IMU, drift will typically exceed 170 m after one minute of
operation. Furthermore drift grows proportional to 푡3 (after the point in time at which the
gyroscopes become the dominant source of error), meaning that huge improvements in the
quality of MEMS inertial sensors are required to achieve only modest increases in the dura-
tion for which objects can be tracked to a specified level of accuracy. Hence unconstrained
inertial navigation is not suitable for pedestrian tracking and this will remain the case for
the foreseeable future. Many indoor location-aware applications require metre-level accuracy
over extended periods of time, however even with a perfect IMU drift will exceed 1 m after
less than one hour unless local gravitational fluctuations are taken into account (see Section
2.2). Technologies that compensate for such fluctuations will need to be developed in ad-
dition to small and lightweight inertial sensors that give better than strategic-grade levels of
performance if this requirement is ever to be met.
Although unconstrained inertial navigation cannot be used to track the position of a pedestrian,
constrained inertial navigation in which external measurements and known constraints are
used to frequently correct an INS can be used. Applying such measurements using an error-
state Kalman filter is superior to directly correcting the state because such filters are able to
track correlations between errors that accumulate in different components of the state. As a
result it is possible to correct errors in components of the state that are not observed directly
by the measurements. For example zero velocity measurements can be used to correct the
velocity, tilt and (to some extent) position components of the state of an INS.
65
 0
 2
 4
 6
 8
 10
 12
 14
 16
 0  20  40  60  80  100  120
M
ea
n 
dr
ift
 (m
)
Time (s)
Uncorrected
Naive correction
KF correction
Figure 3.6: The mean drift incurred by an INS when uncorrected, naı¨vely corrected by
ZVUs, and corrected by ZVUs using the error-state Kalman filter.
 0
 5
 10
 15
 20
 25
 0  20  40  60  80  100  120
M
ea
n 
dr
ift
 (m
)
Time (s)
Mean drift
Figure 3.7: The mean drift incurred when the INS was corrected using the naı¨ve method.
 0
 0.01
 0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 0.08
 0  20  40  60  80  100  120
M
ea
n 
dr
ift
 (m
)
Time (s)
Mean drift
Figure 3.8: The mean drift incurred when the INS was corrected using the error-state
Kalman filter.
66
Chapter 4
Pedestrian dead reckoning
This chapter describes the development of a pedestrian dead reckoning (PDR) filter that makes
use of domain specific knowledge in order to address the issues with unconstrained inertial
navigation that were identified in Chapter 3. The approach used is similar to that of existing
inertial navigation based PDR systems such as those described in Section 2.2.2.2, however the
intended use of the PDR filter to provide measurements to a higher level localisation filter (see
Figure 1.1 on page 5) results in some small but important differences. As well as highlighting
these differences, this chapter evaluates the performance of the filter and describes the format
of the measurements that it generates for consumption by the higher level filter.
4.1 INS-based PDR
An INS that is corrected using the error-state Kalman filter described in Section 3.4 forms the
basis of the PDR filter described in this section. It is necessary to provide frequent external
measurements to the Kalman filter so that it is able to reduce the rate at which drift accumu-
lates in the state of the underlying INS. This is achieved by using a foot-mounted IMU and
applying zero velocity update (ZVU) pseudo-measurements when the user’s foot is known to
be stationary. An algorithm for detecting when the foot is stationary and applying ZVUs is
described in Section 4.1.1.
Whilst previous PDR systems have been designed primarily for standalone use, the filter de-
veloped in this chapter is designed specifically to generate relative movement events for use
by a higher level localisation filter, which is described in Chapter 5. This higher level filter
combines relative movement measurements with environmental constraints such as walls in
order to determine and subsequently track the user’s absolute position. The requirements of
the higher level filter result in a number of differences between existing standalone PDR sys-
tems and the one that is developed here. Firstly, relative movement events must be generated
67
 0
 0.2
 0.4
 0.6
 0.8
 1
 3.1  3.2  3.3  3.4  3.5  3.6  3.7  3.8
|ω
 - 
α|
 (r
ad
/s
)
Time (s)
10
 sa
m
pl
es
10
 sa
m
pl
es
Stance phase
(a)
 0
 2
 4
 6
 8
 10
 12
 14
 16
 18
 0  1  2  3  4
|ω
 - 
α|
 (r
ad
/s
)
Time (s)
(b)
Figure 4.1: The magnitude of the angular velocity (a) during a single stance phase, and
(b) during three steps. The shaded regions indicate the detected stance phases within which
ZVUs are applied.
at a frequency that is much lower than the sampling frequency of the IMU. This is because the
update that is performed by the localisation filter whenever a new event is received is com-
putationally expensive and hence can only be performed at a relatively low frequency. Hence
it is necessary to segment the user’s movement into coarse events. An approach in which
the user’s movement is divided into coarse events corresponding to steps taken by the user
is described in Section 4.1.2. Secondly, unlike in standalone PDR systems, it is not always
desirable to track the user’s position as accurately as possible when generating step events for
use by a localisation filter. In particular it is not desirable to generate step events that contain
corrections for errors in previously generated events. Section 4.1.3 describes why this is the
case and presents a simple way in which the generation of such events can be prevented.
4.1.1 Zero velocity updates
In order to apply ZVU pseudo-measurements it is necessary to identify periods of time during
which the user’s foot is stationary (called stance phases). This can be done simply by detecting
when the magnitude of the angular velocity falls below a certain threshold. The following
algorithm has been found to reliably detect stance phases.
∙ A stance phase starts 0.1 s (10 samples) into a period of time during which the bias-
corrected angular velocity has remained below a threshold of 0.5:
∣흎푏푡 − 휶ˆ푡∣ < 0.5 rad/s . (4.1)
68
The delay (0.1 s) and threshold (0.5) values were empirically determined to give reliable
results.
∙ A stance phase ends 0.1 s (10 samples) before the first sample which exceeds the thresh-
old:
∣흎푏푡 − 휶ˆ푡∣ ≥ 0.5 rad/s . (4.2)
Note that this condition requires samples from the IMU to be delayed by 0.1 s during
each stance phase before they can be passed to the INS’s navigation processor. This
delay has a negligible effect on the output of the filter, since the IMU is (by definition)
near-stationary throughout each stance phase.
The detection of a single stance phase is illustrated in Figure 4.1(a). The stance phases de-
tected for three steps taken by a user are shown in Figure 4.1(b).
ZVUs are generated and applied throughout each stance phase. The measurement vector and
model that are required by the filter are given by
풛푡 = (0, 0, 0)
푇 (4.3)
푯푡 = (0, 푰,0) . (4.4)
When a ZVU is applied it is usually the case that the user’s foot is not perfectly station-
ary. This uncertainty is modelled in the measurement covariance matrix 푹푡, in which it is
assumed that any residual velocity occurs independently along each axis and has variance
휎2 = 0.01 (ms−1)2, which was empirically determined to give good performance.
4.1.2 Step segmentation
The INS updates an estimate of the 3-dimensional position and orientation of the user’s foot
(relative to a specified starting position and orientation) 100 times per second. In practice few
systems or applications require updates at such a high frequency (with the notable exceptions
of virtual reality and motion capture applications). In particular the localisation filter that
will be described in Chapter 5 requires events describing the user’s relative movement to be
reported at a much lower frequency (in the order of one event per second1). To provide such
events the relative movement is segmented into step events2 of the form
풆푡 = (푙푡, 훿푧푡, 훿휃푡, 휉푡) (4.5)
1The exact frequency at which events can be processed by the higher level filter is dependent on the size
of the building in which the user is tracked, the computational resources that are available and the degree of
certainty to which the user’s absolute position is known.
2This term is slightly misleading. Each step event actually corresponds to two steps taken by the user because
the filter only detects steps taken by one of the user’s feet.
69
Figure 4.2: The segmentation of a pedestrian’s relative movement into step events. The
parameters (excluding 훿푧푡) corresponding to three step events are shown.
corresponding to steps taken by the foot to which the IMU is attached. A step event param-
eterises the relative movement of the user’s foot during a step, describing the movement in
terms of its length 푙푡 in the horizontal (푥, 푦) plane, the change in vertical displacement 훿푧푡,
the change in the user’s heading 훿휃푡 and the offset 휉푡 between the user’s final heading and the
direction of the step, as illustrated by Figure 4.2. Note that this parametrisation specifies the
user’s heading independently from the direction of the step. Hence a localisation filter receiv-
ing such events can distinguish between a 90∘ turn and a sidestep. As a result such a filter can
correctly estimate the user’s heading (i.e. the direction in which they are facing) in each case.
Step events are generated at the end of each stance phase. This ensures that the state of the INS
has been corrected as much as is possible by the error-state Kalman filter immediately prior to
the generation of each event. The one case in which this approach is not ideal is when the user
stops walking. In this case the event corresponding to the user’s last step will not be reported
until he or she resumes walking, since only then will the corresponding stance phase end. In
practice this is easily solved by generating a step event part way through any stance phase that
is found to exceed a specified duration. Note that a step event generated part way through a
stance phase is generated instead of, rather than in addition to, an event at the end of the same
phase. Generating step events at most 0.5 s into each stance phase was found to work well in
practice. Figure 4.3 shows the points in time at which step events corresponding to five steps
are generated. The parameters of a step event are derived from the state of the underlying
INS together with the state when the previous step event was generated. The algorithm used
to generate each step event is given by Algorithm 1, in which it is assumed that the user’s
heading is aligned with the 푥-axis of the IMU (therefore the IMU should be mounted on the
foot such that this is the case). Note however that a mounting misalignment will not result in
any errors in the path of step events. It will only cause each 휉푡 to be offset from its true value.
In other words the step events will be the same as if the IMU was correctly aligned, but the
user’s heading will be reported as though he or she is walking a little to one side when they
are in fact walking forwards.
70
Algorithm 1 GenerateStep - Generates a step event from the current state of the underlying
INS and its state at time (푡− 훿푡) when the previous step event was generated. Square brackets
are used to index into vector quantities, with the first element in each vector assigned an index
of zero.
1: procedure GENERATESTEP( (푪ˆ푡, 풔ˆ푡, 풗ˆ푡) , (푪ˆ푡−훿푡, 풔ˆ푡−훿푡, 풗ˆ푡−훿푡) )
// Calculate the change in displacement
2: 훿풔ˆ← 풔ˆ푡 − 풔ˆ푡−훿푡
// Assume that the user’s heading (풉) is aligned with the 푥-axis of the IMU
3: 풉푏 ← (1, 0, 0)푇
// Calculate 풉 in the local frame at the start (풉푔푡−훿푡) and end (풉
푔
푡 ) of the step
4: 풉푔푡−훿푡 ← 푪ˆ푡−훿푡풉푏
5: 풉푔푡 ← 푪ˆ푡풉푏
// Calculate the change in the user’s heading
6: 훿휃푡 ← atan2(풉푔푡 [1],풉푔푡 [0])− atan2(풉푔푡−훿푡[1],풉푔푡−훿푡[0])
// Calculate the step offset
7: 휉푡 ← atan2(풉푔푡 [1],풉푔푡 [0])− atan2(훿풔ˆ[1], 훿풔ˆ[0])
// Calculate the step length and vertical displacement
8: 푙푡 ←
√
훿풔ˆ[0] ∗ 훿풔ˆ[0] + 훿풔ˆ[1] ∗ 훿풔ˆ[1]
9: 훿푧푡 ← 훿풔ˆ[2]
// Return the step event
10: return (푙푡, 훿푧푡, 훿휃푡, 휉푡)
11: end procedure
By concatenating a sequence of step events generated by the PDR filter it is possible to re-
construct the relative movement of the user as calculated by the INS, albeit without the fine-
grained motion of the foot during each step. An example of the fine-grained motion calculated
by the INS (with ZVU corrections applied by the error-state Kalman filter) and the path ob-
tained by concatenating the corresponding step events is shown in Figure 4.4. Note that the
paths have been manually aligned with one other and with a model of the stairs that the user
was descending when the data were obtained.
4.1.3 The cascading filter problem
The error-state Kalman filter that is used by the PDR filter is able to track growing uncertainty
in the state of the underlying INS over multiple steps. In theory this can cause an error in
position to grow slowly over multiple steps, before being reduced rapidly at the end of a further
step when filter is able to resolve much of the accumulated uncertainty. For a standalone
PDR system this is a desirable property, since the filter is able to correct errors that have
accumulated during previous steps and hence return to a more accurate estimate of the user’s
position. In contrast such corrections are not desirable when the filter is used to generate step
event measurements for a localisation filter.
71
 0
 2
 4
 6
 8
 10
 12
 14
 16
 18
 0  1  2  3  4  5  6  7  8
|ω
 - 
α|
 (r
ad
/s
)
Time (s)
Figure 4.3: A step event is generated 0.5 s into a stance phase, or at the end of a stance
phase lasting less than 0.5 s. Each stance phase is indicated by a shaded region. Vertical
black lines indicate points in time at which step events are generated.
Figure 4.4: The fine-grained motion calculated by the INS (with ZVU corrections) as a
pedestrian walked down a flight of stairs (dashed green line), and the path of step events
generated by the PDR filter (solid red line). A dot indicates the start of each step event.
Grid size = 1 m2.
72
1 2
3
4
5
6
(a)
Particles after step 6
Particles after step 5
1 2 3 4 5
6
(b)
Figure 4.5: The problem of delayed drift correction. (a) A sequence of step events in which
the sixth step corrects drift that has accumulated during the previous five events. (b) The
corrected path calculated by a localisation filter. A failure occurs when the localisation filter
attempts to process the 6th step.
Consider the hypothetical case shown in Figure 4.5, in which a user is walking down the centre
of a straight corridor (shown from above). A sequence of step events that might be generated
by a PDR filter is shown in Figure 4.5(a), in which drift accumulates over five step events
before being corrected in a sixth. A localisation filter receiving these step events will use the
walls of the corridor to correct the drift reported in each of the first five step events, as shown
in Figure 4.5(b). The sixth step event contains a large error that compensates for drift in the
previous five events, however the localisation filter has already corrected most of this drift and
as a result a localisation failure (where all of the particles that the localisation filter uses to
represent the user’s position are found to violate environmental constraints) occurs. This is
an example of a cascading filter problem in which the second filter (i.e. the localisation filter)
does not correctly model the errors in the first filter’s outputs.
In practice large corrections such as the one shown in Figure 4.5(a) are rare, since if ZVUs
applied during one stance phase are unable to correct drift in position then it is quite unlikely
that ZVUs applied during subsequent stance phases will be able to do any better. Nevertheless
such behaviour is occasionally observed and therefore must be addressed. There are several
existing solutions to the problem of cascading filters. These include the federated zero-reset
architecture, in which the state of the first filter is zeroed following input to the second filter,
with the corresponding elements of the first filter’s covariance matrix reset to their initialisa-
tion values [18]. The approach used here is very similar. Whenever a step event is generated
the error-state covariance matrix 푷푡 is simply reset to its initial value, as given by Equation
3.75 (page 64). This effectively asserts that the current position and heading estimates are
correct (since 푷0 assumes no uncertainty in these components of the state). Hence it is not
possible for the filter to correct errors in previously generated step events. Note that it is not
necessary to zero the state of the underlying INS (as in the zero-reset architecture) because
each step event is calculated as the difference between the state at the start and end of the
corresponding period of time (see Section 4.1.2).
In addition to solving the cascading filter problem, the solution outlined above also prevents
73
elements of the error-state covariance matrix from growing large over time. If the matrix were
not reset then the elements of 푷푡 corresponding to heading and position would grow over time
to represent growing uncertainty in the heading (and hence also the position) of the user. This
could potentially destabilise the PDR filter if it were used for a sufficiently long period of
time. By resetting 푷푡 whenever a step event is generated, this possibility is avoided.
4.1.4 Filter initialisation
The initial state of the PDR filter is initialised in a similar way to that described in Section
3.3.1. The initial velocity and displacement of the underlying INS are both set to zero
풔ˆ푔0 = 풗ˆ
푔
0 = (0, 0, 0)
푇 (4.6)
whilst the initial orientation of the IMU is set such that the global gravity vector 품푔 =
(0, 0,−9.807)푇 is aligned with the average specific force vector measured by the IMU dur-
ing the first stance phase. The heading component of the orientation can be initialised to an
arbitrary value, since the filter calculates relative changes in the user’s heading as opposed
to absolute heading estimates. As in Section 3.4.5 it is assumed that there is no uncertainty
in the initial state except for a small amount of uncertainty in the tilt of the IMU. The initial
covariance matrix 푷0 is given by Equation 3.75 (on page 64).
4.1.5 Alternative filter formulation
The PDR filter described above is one of many filters capable of generating step events of the
form described in Section 4.1.2. Its design is convenient in that it makes use of the standard
INS and error-state Kalman filter components developed in Chapter 3. Since the filter does
not estimate the components of each step event (i.e. the horizontal length, change in vertical
displacement, change in heading and direction offset) directly, it is necessary to perform a
conversion to obtain them from the state of the underlying INS. An alternative (and neater)
approach would be to develop a Kalman filter that estimates the components of each step event
directly. Step events could then be generated directly from the filter’s state, without the need
for a conversion. After each step event is generated the state of such a filter would be reset to
zero and its state covariance would be reset to its initial value, as in the federated zero-reset
architecture described in Section 4.1.3. This approach (and other potential approaches) are
not explored further in this thesis, however it is worth noting that there are many ways of
developing a filter capable of performing the same task as the one described above.
74
OQO
UMPC
Xsens
Mtx
(a)
QO
UMPC
X ens
Mtx
(b)
Figure 4.6: Components of a prototype PDR system. (a) An OQO UMPC. (b) The UMPC
was connected by wire to a foot-mounted IMU.
Figure 4.7: The top down view of a closed path followed by a pedestrian (dashed green
line) and the path calculated by a hypothetical PDR system (solid red line). Dots indicate
the end of each path. The error in position at the end of the path is far smaller than the
errors incurred at either end of the corridor.
4.2 Evaluation
A prototype implementation of the PDR filter described in Section 4.1 was developed using
an OQO ultra mobile PC (UMPC) running Linux (Figure 4.6(a)) and an Xsens Mtx IMU. The
UMPC was carried in the user’s pocket and was connected by wire to the IMU (Figure 4.6(b)).
Tape was used to attach the IMU to the front of the user’s foot during testing. The use of a
UMPC allowed the software components of the PDR system to be developed rapidly using the
Java programming language, whilst the use of a wire to connect it to the IMU ensured reliable
communication between the two devices. Note that in a practical deployment of the filter the
UMPC could be replaced by a much cheaper micro-controller that could be embedded into
the IMU package on the user’s foot.
75
Figure 4.8: An Active Bat mounted above an Xsens IMU on the user’s foot.
4.2.1 Previous testing methodologies
In previous research standalone PDR systems have been evaluated in indoor environments by
tracking a pedestrian around a closed path within a building (e.g. [14]). The tracking accuracy
is then measured as the difference between the initial and final positions calculated by the
system and is often specified as a percentage of the total distance travelled. Unfortunately
this approach does not always give a good indication of the true accuracy of a PDR system,
particularly if errors in the estimated heading are the main cause of drift in position. Figure
4.7 shows a hypothetical example in which a user is tracked walking up and down a corridor
three times, starting and ending at the midpoint of the corridor. Bias errors in the gyroscopes
of the hypothetical system cause a heading error that grows linearly over time. This results in
large position errors when the user is at either end of the corridor, but the final error in position
when the user returns to the midpoint remains small. Hence it can be misleading to use the
final error in position as an indication of the system’s accuracy. Furthermore the final error in
position does not provide any indication of its cause (e.g. in this example errors in position
were caused by drift in the estimated heading).
4.2.2 Ground truth
In order to perform a thorough evaluation of the PDR filter, measurements from the Active
Bat ultrasonic location system were used to calculate a ground truth to which paths of step
events calculated by the filter could be compared. Recall from Section 2.1.2.2 that the Active
Bat system measures ranges between mobile tags (known as a bats) and multiple receivers
installed at known locations in the ceiling. A multi-lateration algorithm is applied to these
range measurements to position bats to within 3 cm 95% of the time [6]. The Active Bat
system is currently installed in one wing of the William Gates building (Cambridge, UK) in
which the evaluation was conducted.
76
To calculate a ground truth for the user’s movement, a bat was mounted above the IMU on
the test subject’s foot as shown in Figure 4.8. This bat was queried at a rate of 25 Hz by the
Active Bat system during testing. The PDR filter was extended so that range measurements
obtained from the Active Bat system could be incorporated directly into the filter, allowing
the construction of a drift free ground truth during each test. Drift does not accumulate in
the state estimated by the extended filter because the absolute position of the user’s foot is
constrained each time range measurements are applied. Furthermore the heading calculated
by the extended filter is also drift free. This is because any error in the estimated heading
causes a correlated error in position (unless the user remains stationary). The filter tracks such
correlations as described in Section 3.4 and hence is able to correct heading errors whenever
range measurements are applied.
Fully describing the extension of the filter to incorporate range measurements requires a sig-
nificant amount of space and is largely irrelevant to the evaluation of the PDR filter itself. A
full description can be found in Appendix D, with the main points summarised below.
∙ The equations used to incorporate range measurements must take into account the offset
between the Active Bat and the origin of the IMU (i.e. the point of the IMU that is
actually tracked by the INS) in order to achieve accurate results. Suitable equations can
be found in Section D.1.
∙ It was necessary to ensure that measurements from the Active Bat system and the IMU
were correctly synchronized. The Chrony3 time daemon was used to synchronize the
UMPC with the machine responsible for logging Active Bat measurements. In addition
Active Bat measurements were only applied if they were made during the stance phase.
This approach is tolerant to small timing errors because the user’s foot is stationary
during each stance phase. More details can be found in Section D.4.1.
∙ It was necessary to detect and filter out range measurements that correspond to reflected
(i.e. indirect) ultrasonic pulses. This can be done by comparing range measurements
with those predicted by the current state estimate, as described in Section D.4.2.
The result of incorporating range measurements from the Active Bat system is a combined
inertial and ultrasonic absolute positioning system that can be expected to position the user’s
foot at least as accurately as the Active Bat system alone following corrections during each
stance phase (i.e. positions calculated at the end of each stance phase should be accurate to
within 3 cm 95% of the time). The ability of the extended filter to also calculate drift free
heading estimates is the main benefit of its use as a ground truth over simply using the Active
Bat system directly. Due to the lack of a more accurate ground truth it was not possible to
determine the accuracy to which the system was able to estimate the user’s heading, however
3http://chrony.tuxfamily.org
77
Table 4.1: The application of Active Bat range measurements (AB), zero velocity updates
(ZVU) and 푷푡 resets (PR) during test runs.
Initialisation During run
AB ZVU PR AB ZVU PR
Unconstrained filter Yes Yes No No No No
PDR filter Yes Yes No No Yes Yes
Ground truth filter Yes Yes No Yes Yes No
simulations for a similar combined inertial and ultrasonic positioning system predict errors of
less than 1∘ [19].
It is important to note that the ground truth filter cannot be used to determine the error in each
individual step event generated by the PDR filter. This is not only because the Active Bat
system is not itself accurate enough to measure the kind of errors that are expected, but also
because the ground truth filter processes the same IMU data as the PDR filter itself. Hence
correlations will exist between errors incurred in the individual step events calculated by the
two filters. The ground truth filter is however suitable for measuring the accumulation of drift
in both heading and position over time.
During testing the PDR filter was compared to both the ground truth filter and an uncorrected
INS. To make such comparisons all three were implemented using the extended version of the
PDR filter described above. During each test run the first 15 steps detected by the filter were
used as an initialisation phase. During this phase all three filters applied Active Bat range
measurements and ZVUs. This ensured that all three filters had the same initial state. Rather
than the initial state containing an initial displacement of 풔ˆ푔0 = (0, 0, 0) and an arbitrary
heading (as is the case in the normal operation of the PDR filter), this approach initialised
each filter with an absolute position and heading that was effectively derived using the range
measurements obtained from the Active Bat system. This allowed the paths calculated by
each filter to be rendered on maps of the environment without any need for manual alignment.
During the initialisation phase none of the three filters reset the error-state covariance matrix
푷푡 at the end of each step, as was described in Section 4.1.3. Such resets effectively assert
that the current position estimated by the filter is correct, however this is counter-productive
when range measurements are themselves being used to correct the estimated position.
After the initialisation phase each of the three filters applied a different combination of mea-
surements. For the uncorrected INS case no measurements were applied, the PDR filter ap-
plied ZVUs and the ground truth filter continued to apply both ZVUs and Active Bat range
measurements. In all three cases the movement was segmented into step events as described
in Section 4.1.2. Only in the PDR filter case was the error-state covariance 푷푡 reset following
the generation of each step event. A summary of the measurements and error-state covariance
resets that were applied in each of the three cases is shown in Table 4.1.
78
79th stepFigure (b)
(a) Top down view (overview). Grid size 25 m2.
Start
(b) Top down view (zoomed). Grid size 1 m2.
Ground levelStart
(c) Side view. Grid size 1 m2.
Figure 4.9: Paths of step events generated by unconstrained inertial navigation (black), the
PDR fitler (red), and the ground truth filter (green).
4.2.3 Results
The results obtained for a single walk in the area covered by the Active Bat system are shown
in Figure 4.9. The initialisation phase in which range measurements were applied by all three
filters is not shown. Note that despite the appearance of Figure 4.9, the PDR filter is only
capable of tracking the relative movements of a pedestrian. In normal operation the PDR
filter has no way of determining its initial state and hence is initialised with a position of
(0, 0, 0) and an arbitrary heading as described in Section 4.1.4.
For the example in Figure 4.9 the user made 800 steps after the initialisation phase (causing
400 step events to be generated by each of the three filters) over a period of just under nine
minutes, walking a distance of approximately 500 m. As expected from the results in Section
3.4.5, the position estimated by the unconstrained INS diverged rapidly from the true position
of the user. Note that Figure 4.9(a) only shows the positions calculated by the INS for the first
80 step events. After 400 step events the total drift in position (measured as the Euclidean
79
 0
 0.5
 1
 1.5
 2
 2.5
 3
 3.5
 4
 0  50  100  150  200  250  300  350  400
3-
di
m
en
sio
na
l d
rif
t (
m)
No. of step events
Figure 4.10: The 3-dimensional drift in position (measured as the Euclidean distance be-
tween the positions calculated by the PDR filter and the ground truth filter) incurred by the
PDR filter during the example shown in Figure 4.9.
distance between the positions estimated by the unconstrained INS and the ground truth filter)
was 43.15 km. The PDR filter achieved far better performance, with a final drift of 3.48 m.
This can be broken down into drift in the (푥, 푦) plane and drift in the vertical direction. Figure
4.9(c) shows that in this example there seems to be a systematic error that caused downward
drift over time, resulting in a final error in vertical displacement of −2.60 m. Figure 4.9(b)
shows that there was also an error in heading, which caused drift in the (푥, 푦) plane that was
largest when the user was at either end of the corridor. This is responsible for the peaks in
Figure 4.10, which plots the drift incurred by the PDR filter over the duration of the run. The
worst case drift at any point during the walk was 3.97 m, however note that in general this
value is dependent on the path followed by the user. If the test were repeated whilst the user
walked up and down a longer corridor then the effect of a growing heading error on the drift
in position would be magnified and hence a larger worst case drift would be expected.
To investigate the error characteristics of the PDR filter eight more walks were completed,
each consisting of 800 steps (excluding the initial alignment phase). The results obtained
for the three cases for each of the walks are shown in Figure 4.11. For the PDR filter the
majority of these examples suffer from heading and vertical displacement errors that grow in
one direction over time, however the direction of these errors and the rate at which they grow
varies from one example to the next. Figure 4.12 shows that the drift in heading behaves as
a random walk about a linearly growing value in most cases, with a worst case final error of
16.8∘. This can be explained by uncorrected bias errors in the angular velocity measurements
obtained from the IMU. Although the biases are estimated when the IMU is switched on, it is
not possible to determine them exactly. Furthermore bias instability effects cause the biases
to change over time. This may explain case (e), in which the direction in which heading
errors were accumulating changed after around 310 step events. Figure 4.13 shows the drift
in the (푥, 푦) plane during each run. Note that the drift is highly dependent on the error in
80
heading. The walks during which only small heading errors were incurred also had relatively
small worst case drifts in horizontal position (e.g. walks (d) and (g)), whereas large heading
errors caused large drifts in horizontal position at either end of the corridor (e.g. walks (b)
and (c)). Hence, at least in this scenario, heading errors are the primary cause of the worst
case errors in horizontal position. The worst case drift in horizontal position recorded in any
of the walks was 6.42 m. Figure 4.14 plots the drift in vertical displacement for each walk.
As with heading errors, the error in vertical displacement increases about a linearly growing
value, however the magnitude and direction of this value is again different for each run. The
worst case error in vertical displacement was 5.47 m. Note that the worst case errors in both
horizontal and vertical drift are small relative to the total distances travelled by the user, which
were approximately 500 m for each of the examples.
Although it is clear that there are systematic errors in the step events generated by the PDR
filter, it is not clear how they can be estimated (and hence removed) without the benefit of
ground truth data. It is likely that systematic errors in the measurements received from the
IMU are partly to blame. Such errors can only be removed if they can be precisely measured,
which is not possible in this case. Systematic errors may also be introduced due to the mount-
ing point of the IMU on the user’s foot. For example the IMU may be mounted at a position
where it moves very slightly during each stance phase. If such a movement occurs in the same
direction during each stance phase then a systematic error will result. To investigate such error
sources in more detail would require a ground truth for the acceleration and angular velocity
of the IMU, which is not available from current technology.
Figure 4.15 shows the path calculated by the PDR filter over a larger area of the William
Gates building. The initial position and orientation were obtained using an initialisation phase
in which range measurements from the Active Bat system were used. The Active Bat system
is installed along the corridor in the upper left corner of Figure 4.15(a) where the path begins.
This walk demonstrates the ability of the PDR filter to measure changes in vertical displace-
ment, such as when a user ascends or descends a flight of stairs. By the end of the path a
small error in the user’s heading had accumulated (visible at the top right of Figure 4.15(b)),
causing a final error of 9.31 m in the (푥, 푦) plane (calculated using the final position obtained
from the Active Bat system as ground truth). The error in the final vertical displacement was
−0.48 m. This example will be revisited throughout Chapters 5 and 6.
4.3 Use of magnetometers
One way to prevent drift in heading is to make use of the magnetometers that are contained
within IMUs such as the Xsens Mtx. The magnetometers measure magnetic field strength
in three dimensions at the same sampling frequency as the gyroscopes and accelerometers (a
81
(a) (b)Daniel1
(c) (d)
(e) (f)
Olly1 Olly5
(g)
Olly1 Olly5
(h)
Figure 4.11: Eight more examples of the paths generated by the unconstrained INS (black),
the PDR fitler (red), and the ground truth filter (green). Each example consists of 400 step
events. Grid size 1 m2.
82
-20
-15
-10
-5
 0
 5
 10
 15
 20
 0  50  100  150  200  250  300  350  400
Er
ro
r i
n 
he
ad
in
g 
(de
gre
es)
Step count
(0)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 4.12: The error in heading incurred by the PDR filter during nine walks. Walks
(a-h) are shown in Figure 4.11. Walk (0) is shown in Figure 4.9.
 0
 1
 2
 3
 4
 5
 6
 7
 0  50  100  150  200  250  300  350  400
D
rif
t i
n 
(x,
y) 
pla
ne
 (m
)
Step count
(0)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 4.13: The drift in horizontal position incurred by the PDR filter during nine walks.
Walks (a-h) are shown in Figure 4.11. Walk (0) is shown in Figure 4.9.
-6
-5
-4
-3
-2
-1
 0
 1
 2
 3
 4
 5
 0  50  100  150  200  250  300  350  400
Er
ro
r i
n 
z 
(m
)
Step count
(0)
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Figure 4.14: The error in vertical displacement incurred by the PDR filter during nine
walks. Walks (a-h) are shown in Figure 4.11. Walk (0) is shown in Figure 4.9.
83
View (d)
View (c)
View (b)
Start
Ground truth end PDR end
(a)
Start Ground truth end
PDR end
(b)
Start Ground truth end
PDR end
(c)
Start Ground truth end
PDR end
(d)
Figure 4.15: A path of step events calculated by the PDR filter as a user walked throughout
the William Gates building (red). The path calculated by unconstrained inertial navigation
is also shown (black). Figure (a) is annotated with the view points of Figures (b-d). Grid
size 10 m2. Diagrams are exploded 10x in the z-axis.
84
Approx north
Figure 4.16: Magnetic field vectors measured in a single room by the IMU and projected
into the global frame of reference. A black dot indicates the point from which each vector
was measured.
sampling frequency of 100 Hz is used in this thesis). In the absence of any magnetic distur-
bances the magnetic field strength vectors obtained from the magnetometers should always
point towards the magnetic north pole, making it possible to determine the absolute heading
of the IMU.
Unfortunately the approach described above does not work well in the William Gates building,
where there are many strong local magnetic disturbances. In particular metallic floor panels
are installed throughout much of the building, causing local disturbances that are far stronger
than the earth’s magnetic field. The close proximity of the foot-mounted IMU to these panels
prevents the magnetometer from determining the absolute heading of the device with any
degree of reliability. This is illustrated by Figure 4.16, in which normalised magnetic field
strength vectors obtained during an example walk are plotted in the global frame of reference
(position and orientation estimates obtained from the ground truth filter described in Section
4.2.2 were used to position each vector and project them correctly into the global frame of
reference). In the absence of any magnetic disturbances all of the vectors should point in the
same direction towards the magnetic north pole, however this is clearly not the case for the
measurements obtained.
It is possible to develop advanced Kalman filters that attempt to detect and adapt to magnetic
disturbances [81], however in this case such techniques are of limited use because strong
magnetic disturbances are nearly always present. Hence this thesis does not consider the use
of magnetometers in this way. Instead algorithms that limit heading drift based on the use of
environmental constraints are presented in Chapter 5. An alternative way in which data from
the IMU’s magnetometer can be used is outlined in Section 7.2.
85
4.4 Conclusions
This chapter has developed and evaluated a PDR filter suitable for providing relative move-
ment information to a higher level localisation filter. The requirements of such a filter differ
from those a standalone PDR system, requiring both the segmentation of the user’s move-
ment into coarse step events and the application of additional position assertions to in order to
prevent the filter from compensating for errors in previously generated events.
It is important to remember that the PDR filter described in this chapter is only able to track the
relative movements of a pedestrian. Both systematic and random errors during each run cause
drift in vertical displacement, as well as drift in heading that in turn causes the accumulation of
errors in the calculated position over time. When using an Xsens Mtx IMU the filter incurred
worst case drifts of 6.42 m horizontally and 5.47 m vertically throughout a number of test
walks, each of which was around 500 m in length. Hence although the PDR filter may be
sufficient for tracking a user for one or two minutes, it cannot provide metre-level accuracy
(as is required by many indoor location-aware applications) for longer periods of time.
Although systematic errors are responsible for the majority of the drift during each run, it
is not known how the magnitude and direction of such errors can be determined in advance.
Heading errors can in theory be corrected using magnetometers, but in practice it is not always
possible to do so due to strong magnetic disturbances in buildings. This thesis assumes that
such errors will always be present and instead develops algorithms that make use of environ-
mental constraints to correct them in a higher level localisation filter, which is described in
the following chapter.
86
Chapter 5
Enforcing environmental constraints with
particle filters
Pedestrian dead reckoning (PDR) techniques such as the one described in the previous chapter
can be used to track the position of a user relative to their initial location and heading. The
positional accuracy of such a system degrades over time as errors in the measured sensor
data propagate through whichever algorithm is used, accumulating as drift in the calculated
position. Furthermore, the accuracy of absolute positions calculated by a PDR system is
dependent on the accuracy of the initial location and heading, which must be provided by
some other means.
One way to limit the drift incurred by a PDR system is to exploit knowledge of environmental
constraints such as walls and floors, which naturally limit the movement of a pedestrian in an
indoor environment. This chapter shows that a map of such constraints can be used not only
to limit drift, but also to automatically infer the user’s absolute position by finding a unique
path through the environment that is consistent with the relative movements reported by the
(a) (b) (c)
Figure 5.1: The pedestrian localisation problem: To determine the absolute location of a
pedestrian given (a) a path of step events describing their relative movement, and (b) a map
of the environment. The solution is shown in (c), in which the pedestrian’s absolute position
has been determined.
87
PDR filter
Localisation particle filter
Step events
Absolute positions
Low level
High level
Environmental constraints
Inertial measurement unit
Figure 5.2: The structure of a pedestrian localisation and tracking system.
PDR system. The task of automatically inferring the user’s position is known as the pedestrian
localisation problem and is illustrated in Figure 5.1.
This chapter describes a solution to the pedestrian localisation problem in which a particle
filter is used to locate and subsequently track the user in an indoor environment. Step events
generated by a PDR filter are used as inputs to the filter, as shown in Figure 5.2. In theory
any PDR filter could be used, provided that the error in each step event does not grow larger
over time (i.e. the error in the 10th step event will be in the same order of magnitude as the
error in the 1000th step event). This is the case for the PDR filter developed in Chapter 4 due
to the application of zero velocity updates and position assertions to effectively reset the filter
during each stance phase.
In this chapter the PDR filter is considered as a black box, reporting step events of the form
풆푡 = (푙, 훿푧, 훿휃, 휉) (5.1)
as described in Section 4.1.2. Each step event has a corresponding error model describing
the measurement’s uncertainty. This model is specified by a distribution ℋ퐿,Θ of the error
in (푙, 훿휃) and 풱푍 of the error in 훿푧. These distributions are specified separately due to the
different way in which individual components of the step event are used by the filter (random
variables are sampled fromℋ퐿,Θ, whereas probability densities are read directly from 풱푍), as
described in Section 5.2. The notation
(푙푒, 훿휃푒) ∽ ℋ퐿,Θ (5.2)
will be used to indicate that (푙푒, 훿휃푒) is sampled from the distributionℋ퐿,Θ. The notation
푝 = 풱푍(훿푧푒) (5.3)
indicates that 푝 is the probability that the error in the measurement 훿푧 is 훿푧푒. In theory the error
model corresponding to each step event could be generated by the PDR filter itself, however
in this chapter a generic error model is assumed for each step, as described in Section 5.3.
88
The remainder of the chapter is set out as follows. Section 5.1 introduces a data-structure that
is suitable for representing buildings with multiple floors. Section 5.2 describes the design
of a particle filter to solve the pedestrian localisation problem. The filter’s localisation and
tracking functionalities are described in Sections 5.3 and 5.4 respectively.
5.1 Representing building constraints
In order to use environmental constraints to help locate and track a pedestrian’s position it is
necessary to represent them in a machine-readable form. In previous work on robot locali-
sation 2-dimensional maps have been used [57, 66]. Supporting environments with multiple
floors was not necessary since the robots used were not able to ascend or descend stairs. Such
robots typically used laser range-finders to scan for obstructions in a single horizontal plane
for which a 2-dimensional representation of the environment was sufficient.
In the pedestrian localisation problem the primary type of ‘measurement’ is to use the move-
ment of a pedestrian to detect the absence of an obstruction. Since a pedestrian is constrained
by gravity to be in contact with the floor, obstructions on a single floor of a building can be
represented by a 2-dimensional map. Unfortunately pedestrians frequently ascend and de-
scend stairs and so a solely 2-dimensional representation is not sufficient. A more suitable
representation is a 2.5-dimensional map, as described below.
A 2.5-dimensional map is defined as a collection of rooms, each consisting of one or more
planar floor polygons. Each floor polygon corresponds to a surface in the building on which
a pedestrian’s feet may be grounded. Each edge of a floor polygon is either the edge of
an impassable obstruction or a connection to the edge of another polygon. A connection
represents an edge through which a pedestrian can pass, such as a doorway. Each connection
is composed of two edges, one associated with each of the connected polygons. These edges
must coexist in the horizontal (푥, 푦) plane, however they may be separated in the vertical (푧)
direction to allow the representation of stairs. It is possible to represent even complex rooms
using this format, such as the lecture theatre shown in Figure 5.3.
Figure 5.4 shows how a 2.5-dimensional map data-structure is stored in memory. Each floor
polygon is defined by a packed array of vertices
푽 = (푥1, 푦1, 푧1, 푥2, 푦2, 푧2, ..., 푥푛, 푦푛, 푧푛) (5.4)
which are specified in clockwise order. This vector representation avoids having to approxi-
mate the map to a specified granularity and allows large maps to be represented efficiently in
89
(a) (b)
Figure 5.3: (a) A lecture theatre. (b) Its 2.5-dimensional representation. The edges sep-
arating adjacent rows of seating are treated as walls (solid blue lines), whereas the edges
between stairs in the aisle are connections (dashed green lines). Grid size = 1 m2.
Room1
Room2
(0,0,0) (1,0,0)
(0,3,0) (1,3,0)
(a)
GetRoomGetRoom
Room1
FloorPolys = {ref}
Room2
FloorPolys = {ref}
FloorPoly1
V = {1,0,0,0,0,0,0,1,0,1,1,0}
C = {null,null,ref,null}
GetDstPoly
GetMap GetMap
FloorPoly2
V = {1,1,0,0,1,0,0,3,0,1,3,0}
C = {ref,null,null,null}
GetDstPoly
Map2.5D
Rooms = {ref,ref}
(b)
Figure 5.4: (a) A simple 2.5-dimensional map. (b) Its representation in memory. Arrows
correspond to references. Where applicable, a label indicates the name of the method which
can be used to obtain the referred object.
90
memory. The edges of a floor polygon are given by
푬1 =
−−−−−−−−−−−−−−→
(푥1, 푦1, 푧1)(푥2, 푦2, 푧2) (5.5)
푬2 =
−−−−−−−−−−−−−−→
(푥2, 푦2, 푧2)(푥3, 푦3, 푧3) (5.6)
...
푬풏 =
−−−−−−−−−−−−−−−→
(푥푛, 푦푛, 푧푛)(푥1, 푦1, 푧1) . (5.7)
Connections are defined by an array of references to other polygons 푪, where 푪[푖] is a ref-
erence to the polygon to which edge 푬풊 connects, or null if that edge is a wall. Note that a
connection between two floor polygons A and B is represented by two references, one from
A to B and another from B to A. Hence in theory it is possible to represent a uni-directional
connection by having an unmatched reference from one polygon to another. This may be
useful in some cases, for example to represent a one-way ticket barrier.
Some of the functions that can be performed on 2.5-dimensional maps are listed below and
are used in the remainder of this chapter.
∙ GetDstPoly(푬) returns the polygon to which the specified edge connects, or null if the
edge is a wall:
GetDstPoly(푬푖) = 푪[푖] (5.8)
∙ Type(푬) returns the type of edge 푬:
Type(푬) =
{
Connection if GetDstPoly(푬) ∕= null
Wall otherwise
(5.9)
∙ GetRoom(poly) returns the room to which the specified floor polygon belongs.
∙ Height(푥, 푦, poly) returns the height of the specified polygon at position (푥, 푦) in the
horizontal plane. Usually the height is constant across the polygon’s surface (since
most floor surfaces in buildings are flat), in which case (푥, 푦) is ignored and the con-
stant height of the polygon is returned. Note however that sloping floor polygons are
permitted (the 2.5-dimensional map data-structure only requires floor polygons to be
planar) and can be used to represent sloping surfaces in the environment. A sloping
floor polygon can also be used to represent a staircase rather than having to define each
step individually.
91
5.2 Localisation filter design
Recall from Section 2.3.5 that particle filters approximate the belief Bel(풙푡) of the state at
time 푡 as a set of weighted samples known as particles
풮푡 = {⟨풙푖푡, 푤푖푡⟩ 푖 = 1, ..., 푁} (5.10)
where 풙푖푡 is the state and 푤
푖
푡 is the weight of the 푖
th particle. A particle filter update consists of
generating the set 풮푡 from the previous posterior 풮푡−훿푡. In this section a pedestrian localisation
filter is developed based on the simple Bootstrap particle filter, in which 풮푡 consisting of 푁
particles is generated from the previous state as follows [55].
1. Propagation: For 푖 = 1, ..., 푁 sample the updated state from the motion model distri-
bution:
풙푖푡 ∽ 푝(풙푡∣풙푖푡−훿푡,풖푡) . (5.11)
2. Correction: For 푖 = 1, ..., 푁 update the importance weight:
푤푖푡 = 푤
푖
푡−훿푡 ⋅ 푝(풛푡∣풙푖푡) (5.12)
3. Re-sampling: Create 푁 new states by randomly sampling the propagated states in
proportion to their importance weights. 풮푡 is then the set of푁 particles with the sampled
states and equal weights 푤 = 1/푁 .
In practice the propagation and correction steps are often combined into a single particle
update step, since this avoids having to iterate twice over the set of particles.
In the pedestrian localisation filter each particle has a state
풙푡 = (푥푡, 푦푡, 푧푡, 휃푡, poly푡) (5.13)
at time 푡, where (푥푡, 푦푡, 푧푡) is the position, 휃푡 is the heading and poly푡 is a reference to the floor
polygon to which the particle is constrained. This reference is crucial to the performance of
the filter since it acts as an index into the 2.5-dimensional map data-structure. This reference
can be used to discover nearby constraints (i.e. those defined by the edges of poly푡) in 풪(1)
time1, with the size of the map having no effect on the lookup time. The position (푥푡, 푦푡, 푧푡)
is constrained to lie on the surface of poly푡. Hence the vertical displacement of the particle is
also given by
푧푡 = Height(푥푡, 푦푡, poly푡) . (5.14)
1An excellent description of 풪-notation (pronounced “big oh notation”) can be found in [82].
92
and does not need to be stored explicitly in the state vector. Despite this it is often useful to do
so in practice, since it allows the state vector to be passed directly into functions that expect a
vector containing the particle’s 3-dimensional position. In particular particle state vectors can
be passed directly as vertices into the functions of a 3-dimensional rendering library.
The following sections outline in detail the propagation, correction and re-sampling steps of
the localisation filter.
5.2.1 Particle propagation
The propagation step generates the updated state 풙푡 of a particle by sampling the motion
model distribution 푝(풙푡∣풙푡−훿푡,풖푡), where 풙푡−훿푡 is the state of the particle at time 푡 − 훿푡 and
풖푡 = (푙, 훿휃, 휉) is the step event 풆푡 excluding the 훿푧 component. By using 풖푡 directly in the
propagation step the particle filter is able to propagate particles in a way that is consistent with
the measured step length 푙, change in heading 훿휃 and heading offset 휉 (the 훿푧 component of
the step event is used separately in the correction step for reasons described below).
An alternative approach would be to assume a generic motion model that does not depend
on the step event (for example one that assumes that the user is equally likely to have moved
up to some maximum distance in any direction) before using all of the components of 풆푡
to weight the resulting particles during the correction step. In this case particles that were
propagated by distances and headings consistent with the step event would be assigned large
weights during the correction step, whilst other particles would be assigned small weights.
A major drawback of this approach is that the majority of the particles would be propagated
in a way that is inconsistent with 풆푡. As a result most of the particles would be assigned
very small weights during the correction step and so would contribute very little towards the
representation of the posterior distribution. It would therefore be necessary to propagate far
more particles than are actually needed to represent the posterior distribution.
The approach used in this thesis ensures that all of the particles are propagated in a way that
is consistent with the components of the step event contained by 풖푡. Note however that the 훿푧
component of the step event is still used in the correction step (as described in Section 5.2.2)
rather than during propagation. This is because particles are by definition constrained to lie
on the surface of the 2.5-dimensional map. Hence the vertical displacement of each particle
must be updated according to Equation 5.14, rather than according to the measured change in
vertical displacement.
The updated state 풙푡 of a particle is calculated from its previous state 풙푡−훿푡 and 풖푡 as fol-
lows. First, the horizontal components of the step event are perturbed according to the step’s
93
uncertainty model
푙′ = 푙 + 푙푒 (5.15)
훿휃′ = 훿휃 + 훿휃푒 (5.16)
where 푙푒 and 훿휃푒 are drawn from the step uncertainty model
(푙푒, 훿휃푒) ∽ ℋ퐿,Θ . (5.17)
The new heading and (푥, 푦) position are then calculated from the previous state and the per-
turbed step parameters
휃푡 = 휃푡−훿푡 + 훿휃′ (5.18)
푥푡 = 푥푡−훿푡 + 푙′ ⋅ cos(휃푡 − 휉) (5.19)
푦푡 = 푦푡−훿푡 + 푙′ ⋅ sin(휃푡 − 휉) (5.20)
where 휉 is the offset between the user’s heading and the step direction as specified in the step
event.
It is also necessary to determine the floor polygon poly푡 to which the propagated particle
is constrained. Initially it is assumed that the floor polygon in which the particle resides is
unchanged by the update
poly푡 = poly푡−훿푡 . (5.21)
In order for the particle to have exited this floor polygon, the step vector must intersect one
of its edges in the (푥, 푦) plane. Let 푨 = (푥푡−훿푡, 푦푡−훿푡) and 푩 = (푥푡, 푦푡) be the start and end
points of the step vector in the (푥, 푦) plane. Secondly, let
Intersect(푨,푩, poly푡) = [푪,푬] (5.22)
be a function that calculates the first intersection between the step vector
−−→
푨푩 and the edges
of poly푡, returning both the point of intersection 푪 and the edge 푬 on which the intersection
occurs. This function is used to update poly푡 according to one of three cases:
1. [푪,푬] = [null, null] : No intersection point is found. The particle must still be con-
strained to the same polygon, so no update is necessary.
2. Type(푬) = Wall : The first intersection is with a wall. In this case the particle’s
weight will be set to zero in the correction step, enforcing the constraint that walls are
impassable. Since this particle will never be re-sampled, it is not necessary to update
poly푡.
94
3. Type(푬) = Connection : The first intersection is with an edge connecting to another
polygon. In this case the polygon must be updated
poly푡 = GetDstPoly(푬) . (5.23)
Since a single step may span multiple floor polygons, the intersection test is repeated
between the remainder of the step vector
−−→
푪푩 and the updated polygon. This process
continues recursively until one of the first two cases applies.
Finally, if the vertical position 푧푡 is to be stored explicitly in the state vector then it is calculated
using Equation 5.14.
5.2.2 Particle correction
The correction step updates the weight 푤푡 of a propagated particle. In the localisation filter
this is used to enforce environmental constraints. If a wall is intersected by the particle during
the propagation step then it is assigned a weight
푤푡 = 푤푡−훿푡 ⋅ 0 = 0 . (5.24)
If a wall is not intersected, the particle’s weight is updated based on how closely the change
in height 훿푧 of the current step event matches the change in height of the particle
훿푧poly = 푧푡 − 푧푡−훿푡 . (5.25)
In this case the particle’s weight is calculated as
푤푡 = 푤푡−훿푡 ⋅ 풱푍(∣훿푧 − 훿푧푝표푙푦∣) (5.26)
where 풱푍 is part of the step event’s error model (see page 88). Note that the change in
vertical displacement reported in the step is used as a measurement in the Bayesian framework.
Particles whose change in height over the step closely matches the change in height reported
in the step event are assigned larger weights.
The combined particle propagation and correction algorithm is given by Algorithm 2. The
algorithm executes in 풪(1) time in practice and hence can be applied to all 푁 particles in
풪(푁) time. Note that the cost of applying Algorithm 2 to a single particle does not depend
on the size of the environment. This is due to the use of poly푡 as an index from the particle
into the corresponding part of the map data-structure. This is crucial for deployments in very
large environments. One part of the algorithm for which 풪(1) complexity is not obvious is
the while loop that updates the floor polygon to which the particle is constrained (lines 10 -
95
21). The number of loop iterations is equal to the number of polygons through which the step
update passes. This value is bounded by a combination of the user’s maximum step length
and the fact that in practice floor surfaces in buildings have a minimum size. Within each
iteration the cost of the Intersect function is dependent on the number of edges of the current
floor polygon, however this value is also bounded in practice since floor surfaces tend to have
a maximum number of edges (i.e. most rooms have four). Hence updating the floor polygon
consists of executing an 풪(1) function 풪(1) times and hence is itself 풪(1).
5.2.3 Re-sampling
Consider a filter where 풮푡 is simply the set of propagated and corrected particles. Over time the
number of particles with non-zero weights will decrease as more and more particles violate
wall constraints. More generally the distribution of particle weights becomes skewed, with
fewer and fewer particles having non-negligible weights. This happens not only as a result
of wall intersections, but also when the user descends or ascends a flight of stairs. In this
case any particles that are not constrained to stair regions will have weights close to zero
after a few updates. Hence only a very small proportion of the 푁 particles contribute to
the representation of Bel(풙푡) and as a result the distribution is not adequately represented.
Furthermore computation is wasted updating particles that have negligible or zero weights.
This is known as the weight degeneracy problem.
Re-sampling solves the weight degeneracy problem by generating a new set of particles from
the updated (i.e. propagated and corrected) set whose weights are less skewed. Many re-
sampling schemes have been proposed [83]. The localisation filter described in this chap-
ter implements the simplest approach, known as multinomial re-sampling. Multinomial re-
sampling generates a new set of 푁 particles with equal weights 푤 = 1/푁 . The state of each
new particle is randomly sampled from the states of the updated particles in proportion to their
weights. Thus states with high weights tend to be replicated whilst states with low weights
tend to be removed.
Multinomial resampling is implemented as follows. First the accumulated weights
Acc = (acc0푡 , ..., acc
푁
푡 ) (5.27)
are calculated in 풪(푁) time, where
acc푖푡 =
푖∑
푛=1
푤푛푡 . (5.28)
96
Algorithm 2 Update - Applies the propagation and correction algorithms to generate a particle
at time 푡 from a particle at time (푡− 훿푡).
1: procedure UPDATE( {(푥푡−훿푡, 푦푡−훿푡, 푧푡−훿푡, 휃푡−훿푡, poly푡−훿푡), 푤푡−훿푡} , 풆푡 = (푙, 훿푧, 훿휃, 휉) )
// Perturb the step event
2: (푙푒, 훿휃푒) ∽ ℋ퐿,Θ
3: 푙′ ← 푙 + 푙푒
4: 훿휃′ ← 훿휃 + 훿휃푒
// Update the particle’s xy-position
5: 휃푡 ← 휃푡−훿푡 + 훿휃′
6: 푥푡 ← 푥푡−훿푡 + 푙′ ⋅ cos(휃푡 − 휉)
7: 푦푡 ← 푦푡−훿푡 + 푙′ ⋅ sin(휃푡 − 휉)
// Initialise the intersection algorithm
8: poly푡 ← poly푡−훿푡 ,푨← (푥푡−훿푡, 푦푡−훿푡) ,푩 ← (푥푡, 푦푡)
9: done← false , kill← false
// Determine the polygon to which the updated particle is constrained
10: while ¬done do
11: [푪,푬]← Intersect(푨,푩, poly푡)
12: if 푪 = null then
13: done← true
14: else if Type(푬) = Wall then
// Note the wall intersection
15: kill← true
16: done← true
17: else if Type(푬) = Connection then
18: poly푡 ← GetDstPoly(푬)
19: 푨← 푪
20: end if
21: end while
// Update the particle’s z-position
22: 푧푡 ← Height(푥푡, 푦푡, poly푡)
// Re-weight the particle
23: if kill then
24: 푤푡 ← 0
25: else
26: 훿푧poly ← 푧푡 − 푧푡−훿푡
27: 푤푡 ← 푤푡−훿푡 ⋅ 풱푍(∣훿푧 − 훿푧푝표푙푦∣)
28: end if
// Return the updated particle
29: return ⟨(푥푡, 푦푡, 푧푡, 휃푡, poly푡), 푤푡⟩
30: end procedure
97
The state of each new particle is then sampled by drawing a random value
푟 ∼ 풰(0, acc푁푡 ] (5.29)
where 풰(0, acc푁푡 ] is the uniform distribution from 0 (inclusive) to acc푁푡 (exclusive). The
sampled state is then the state of the 푗푡ℎ particle, where 푗 is the smallest value such that
푟 < acc푗푡 . This value is found by a binary search for 푟 in Acc which requires 풪(log2푁)
time. Since this is repeated for each of the 푁 new particles, the overall complexity of the
re-sampling step is 풪(푁 log2푁).
Note that it is impossible for the state of a particle with a weight of zero to be sampled by
the algorithm above. Hence the states of any particles found to violate wall constraints are
removed during the subsequent re-sampling step. Note also that when the user descends
or ascends a flight of stairs the particle cloud will quickly converge to stair regions. This
will occur because particles constrained to such regions will be assigned large weights in the
correction step relative to other particles, as described in Section 5.2.3. Hence their states are
more likely to be replicated during re-sampling.
5.3 Initialisation and localisation
Figure 5.5 shows an example in which the localisation particle filter is applied in the William
Gates building. Figure 5.5(a) shows the path of step events generated by the PDR filter that
was developed in Chapter 4. Note that this path is the first part of the one shown in Figure
4.15 (page 84), which is used as a running example throughout this thesis. It has been aligned
with the map to have the correct initial location and orientation (using measurements provided
by the Active Bat system), both of which are unknown to the localisation algorithm.
A simplified step uncertainty model was used, which was assumed to be constant for all
steps. The error of each component in a step event was assumed to be independent, Gaussian
distributed and have a mean of zero2. Hence the uncertainty model was given by
ℋ퐿,Θ(푙푒, 훿휃푒) = 풩0,휎퐿(푙푒) ⋅ 풩0,휎Θ(훿휃푒) (5.30)
풱푍(훿푧푒) = 풩0,휎푍 (훿푧푒) (5.31)
where 풩휇,휎 is a Gaussian distribution with mean 휇 and standard deviation 휎. The deviations
used were 휎퐿 = 0.12 m, 휎Θ = 0.4∘ and 휎푍 = 0.05 m. These deviations were chosen to delib-
erately overestimate the magnitude of errors that are present in the step events reported by the
2Note that in reality these assumptions are extremely unlikely to hold. The resulting uncertainty model is
crude, however works well in practice.
98
PDR filter, however this overestimation ensures that the localisation filter is able to operate re-
liably even when unusually large errors are encountered. In contrast an underestimation could
result in a failure where all of the particles are found to violate environmental constraints.
Given the availability of a highly accurate ground truth positioning system it would have been
possible to measure the errors in individual step events generated by the PDR filter and hence
derive a suitable uncertainty model directly from experimental data. Unfortunately the ground
truth tracking system used in Chapter 4 was not accurate enough for this purpose. Although
it was sufficiently accurate to measure the accumulation of drift errors over time, it was not
accurate enough to measure errors in individual step events.
Figure 5.5(b) shows the initial collection of particles sampled from the prior Bel(풙0), each
drawn as a red dot. Since the initial location of the pedestrian is unknown, a uniform distri-
bution over the entire floor surface is used. The initial heading is also unknown, hence the
particle headings are distributed uniformly in all directions. In this example 푁 = 500, 000
particles were used (although in general more particles are required to ensure reliable localisa-
tion, as discussed in Section 5.3.2). Figure 5.5(c) shows the particles after the pedestrian had
taken eight steps. Figures 5.5(d) and 5.5(e) show the distribution just before and just after the
pedestrian started to descend a flight of stairs. Note how the pedestrian is quickly localised to
stair regions due to the use of the measured change in vertical displacement in the correction
step of the particle filter. Figure 5.5(f) shows the final set of particles, which form a single
cluster around the user’s true position.
The example described above demonstrates that it is possible for the localisation filter to
successfully combine building constraints and relative movement measurements in order to
determine the absolute position of a user. It is important to note that the user must be moving
in order for his or her position to be determined in this way. Even when the user is moving
there are two problems that affect the filter’s performance during localisation: symmetry and
scalability.
5.3.1 Environmental Symmetry
Figure 5.6 highlights one of the problems faced by the localisation filter. If the layout of a
building exhibits translational or rotational symmetry then the relative path described by a
sequence of step events is often consistent with multiple different locations. In such cases
a cluster of particles forms at each possible location, as can be observed in Figure 5.5(d)
and more clearly in Figure 5.6(a). Hence symmetry of the environment can delay or prevent
convergence to a single cluster of particles.
For the localisation in Figure 5.5 the user passes through several asymmetries in the environ-
ment that allow him to be successfully located. The asymmetrical positioning of the building’s
99
Start
(a) (b)
(c) (d)
(e) (f)
Figure 5.5: An example of localisation in a three storey building. (a) A manually aligned
overlay of the path of step events generated by the PDR filter; (b) The prior distribution of
푁 = 500, 000 particles; (c-f) The particle distribution at four points during localisation.
Particles for which 푤 = 0 are coloured black. An arrow indicates the actual position of the
pedestrian in each figure. Grid size 10 m2. Diagrams are exploded 10x in the z-axis.
100
(a) (b)
Figure 5.6: The environmental symmetry problem; (a) Multiple clusters formed due to
symmetry on one floor of the William Gates building. The user has walked in a straight line
before making a 90∘ turn to the left. This movement is consistent with eight different loca-
tions which arise due to both translational and rotational symmetry. Each arrow indicates
the path taken by a cluster to reach its current position. Grid size = 10 m2. (b) A hypothet-
ical building for which symmetry prevents the user’s position from being determined.
staircases relative to its corridors restricts the user to two possible locations as shown in Fig-
ure 5.5(e). The user then passes through a lecture theatre, allowing his position to be uniquely
determined.
Since the filter relies on the user passing through asymmetries, it is clear that it is not always
possible to determine the user’s position. This may be because the user has not yet passed
through a sufficient number of asymmetries, or worse, because the environment is not suffi-
ciently asymmetric to make localisation possible. Figure 5.6(b) shows an example of such an
environment. Each floor is identical, meaning that the environment exhibits vertical transla-
tional symmetry. This makes it impossible to determine the floor on which the user is located
unless they visit each one of them (in which case the limits imposed by the bottom and top
floors allow the correct floor to be determined). Even if the user does this, there will still be
four positions on that floor that could equally correspond to their true position. These arise
due to the rotational symmetry exhibited by the environment. Although such an extreme ex-
ample is unlikely, many buildings do exhibit high degrees of symmetry. In particular it is
common for multi-storey buildings to have very similar layouts on each floor. In such cases it
is necessary to provide more information to the filter to enable it to complete the localisation
process. One option is to use some form of absolute positioning to obtain an approximate
position for the user. This approach is explored in more detail in Chapter 7.
101
5.3.2 Scalability
The time required by the localisation filter to perform an update is 풪(푁 log2푁) due to the
re-sampling step, where 푁 is the number of particles used to represent the state. The amount
of memory required to represent the state is 풪(푁). Hence it is clear that there exists a limit
on the number of particles if the filter is to execute in real-time on a given piece of hardware.
It is therefore important to establish how many particles are required to reliably determine the
user’s position. If many particles are used then they will be distributed with a high density
over the floor polygons that define the environment. Conversely if few particles are used then
the particle density will be low. Using fewer particles will allow the filter to execute on lower
grade hardware, however the probability that there will be a particle whose state is within
some distance of the user’s true position and orientation is also lower. One of two things
may occur if there are no particles sufficiently close the user’s true state at some point during
localisation:
∙ The localisation process can fail before the particles converge to form a single cluster.
This happens if all of the particles are found to violate wall constraints in a given step
update.
∙ The particles may converge to a location that does not correspond to the user’s true po-
sition. This case arises when the user has not passed a sufficient number of asymmetries
for his position to be uniquely determined, however all but one of the possible positions
(including the one corresponding to the user’s true location) are not represented due to
an insufficient density of particles. Note that all particles at the incorrect localisation
will violate environmental constraints if the user later passes an asymmetry that would
otherwise have eliminated that position.
In order to determine the number of particles required for reliable localisation in the William
Gates building, the localisation filter was tested on 10 sets of data recorded as a user walked
along 10 different paths through the building. Each path was known to pass a sufficient num-
ber of asymmetries such that the user’s position could be uniquely determined3. Hence the
result of each execution was either a successful localisation or a failure (i.e. all particles were
found to violate constraints). For each success the resulting particle cloud was visually in-
spected to ensure that it had converged to the user’s true final position. This always occurred
in practice because convergence to an incorrect location was always followed by subsequent
failure before the end of the run.
The filter was executed 10 times on each log for each of a number of different particle counts
ranging from 25, 000 to 1, 400, 000. Figure 5.7 shows the overall failure rate as a function of
3This was determined by processing each log using 2 million particles (which is well in excess of the number
required) and checking that the particles converged to form a single cluster around the user’s true position.
102
 0
 10
 20
 30
 40
 50
 60
 70
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
Fa
ilu
re
 ra
te
 (%
)
Millions of particles (N × 10-6)
Figure 5.7: The percentage of localisation failures as a function of the number of particles
used.
the number of particles used. At least 1, 200, 000 particles were required for reliable localisa-
tion in all cases. Figure 5.8 shows the mean time that was required to process each step event
as a function of the number of particles used. These results were obtained using a Java im-
plementation executing on a 2.6 GHz desktop PC. Note that the time required for each update
scales worse than the theoretical complexity of 풪(푁 log2(푁)) for large 푁 . The variance in
the time required to perform an update also increases. It is believed that increased memory
pressure and an increase in the frequency at which the Java garbage collector was required to
execute could account for both trends.
To update the 1, 200, 000 particles required for reliable localisation in the William Gates build-
ing took the tested implementation on average 2.34 s, however a user moving at a brisk walk
can generate around 55 step events per minute4. Hence it is necessary to process each step
event in 1.09 s if such a user is to be tracked in real time, which requires roughly a factor
of two performance increase. This factor could almost certainly be attained by a more effi-
cient implementation (in particular one that exploits multiple processors, to which the particle
propagation and correction steps are well suited) or by running the existing implementation on
more powerful hardware. Note however that the number of particles required scales linearly
with the floor area of the building such that the particle density remains constant. Hence even
with a faster implementation or more powerful hardware there will still exist a limit on the
size of a building after which real-time tracking is not feasible. It is therefore desirable to find
ways to reduce the computational cost of localisation without reducing the filter’s reliability.
One way of doing this is to dynamically vary the number of particles as the uncertainty in
the user’s position changes, as will be discussed in Section 6.1. This approach means that the
455 step events correspond to 110 steps taken by the user, since events are generated for only one of the user’s
feet.
103
-0.5
 0
 0.5
 1
 1.5
 2
 2.5
 3
 3.5
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
Pr
oc
es
si
ng
 ti
m
e 
pe
r s
te
p 
(s
)
Millions of particles (N × 10-6)
µ ± σ
O(Nlog2N)
Figure 5.8: The average time required to process a step event when different numbers
of particles are used. These results are for a Java implementation running on a 2.6 GHz
desktop PC. The green line indicates the theoretical complexity풪(푁 log2푁) (fit to the first
nine data-points).
cost of an update is reduced with each step event. Another method of reducing the computa-
tional cost is to use another positioning system to provide a rough estimate of the user’s initial
position. The prior can then consist of particles generated only in the area surrounding the
estimate rather than throughout the building. This approach is discussed in Chapter 7.
5.4 Tracking
Once the cloud of particles maintained by the localisation filter has converged to form a single
cluster (hopefully corresponding to the user’s true position) the problem solved by the filter
becomes one of tracking rather than of localisation. Each time a step event is received during
the tracking phase the effect of the particle propagation step is to move the particle cloud to the
user’s new position, expanding it slightly to model the uncertainty in the event. This expansion
is limited by the correction step, since particles that have violated environmental constraints
are assigned zero weights and are therefore removed during re-sampling. By continually
enforcing such constraints the drift that would otherwise accumulate in the position estimates
is limited, allowing the localisation filter to track a user indefinitely as they walk through the
environment.
There are several ways to provide position estimates to location-aware applications during
tracking. The set of particles generated by the localisation filter at a given time 푡 represents
a probability distribution for the user’s state at that time. This set could be provided directly,
allowing applications to make full use of the distribution that it represents (or even display
104
the particle cloud directly to the end user). Although this approach maximises the amount
of information provided to the application, most location-aware applications only require a
simple description of the user’s location. For example applications could be provided with an
interface to query the user’s most likely state and optionally a simple description of the error,
such as a single variance or an error ellipse.
This section describes a simple algorithm for computing an estimate of the user’s position
(푋푡, 푌푡, 푍푡) and orientation Θ푡 during tracking. The accuracy of the estimates is then tested
by comparison with the Active Bat absolute positioning system. Note that the problem of
detecting when the cloud of particles has converged to allow the computation of such estimates
is not considered here, but will be discussed in Section 6.2.
The user’s position (푋푡, 푌푡, 푍푡) at time 푡 is estimated as the weighted average of the particles,
given by
푋푡 =
∑푁
푖=1푤
푖
푡 ⋅푥푖푡∑푁
푖=1푤
푖
푡
(5.32)
푌푡 =
∑푁
푖=1푤
푖
푡 ⋅ 푦푖푡∑푁
푖=1푤
푖
푡
(5.33)
푍푡 =
∑푁
푖=1푤
푖
푡 ⋅ 푧푖푡∑푁
푖=1푤
푖
푡
(5.34)
where 푁 is the number of particles in the cluster. Slightly more care must be taken when
calculating an average heading Θ푡. The naı¨ve approach of adding up the weighted angles and
then dividing by 푁 will not always give a sensible answer. For example consider two equally
weighted particles with headings
휃1푡 = −휋 + 휖 (5.35)
휃2푡 = 휋 − 휖 (5.36)
where 휖 is small. Intuitively the average heading should be −휋 (or equivalently 휋), however
the value calculated by the naı¨ve approach is 0 (i.e. the opposite direction). A better approach
is to calculate the average heading vector (ℎ푡푥, ℎ푡푦) and then use it to determine a value for
Θ푡 [84]:
ℎ푡푥 =
푁∑
푖=1
푤푖푡 ⋅ cos 휃푖푡 (5.37)
ℎ푡푦 =
푁∑
푖=1
푤푖푡 ⋅ sin 휃푖푡 (5.38)
Θ푡 = atan2(ℎ푡푦, ℎ푡푥) . (5.39)
105
Figure 5.9: The 95% error ellipse calculated for a cloud of particles. The long black line
indicates the heading Θ.
The most likely state of the user is then provided to applications as the state (푋푡, 푌푡, 푍푡,Θ푡).
If required, a simple error estimate such as an error ellipse can also be calculated, as shown
in Figure 5.9. Details regarding the calculation of error ellipses can be found in Appendix E.
Note that the use of a weighted average differs from the usual approach when using particle
filters, in which the state of the particle with the largest weight is taken to be the most likely
state. For the localisation filter this approach is not sensible since all of the particles in the
cluster are likely to have very similar (if not identical) weights. Hence the state of the particle
with the largest weight will effectively be a random position within the cluster.
Figure 5.10 returns to the running example used throughout this thesis to illustrate the ability
of the localisation filter to track the absolute position of a pedestrian, continually correcting for
errors in the individual step events generated by the PDR filter. Note that the localisation filter
is not able to track the pedestrian from the very beginning of the example, since it must first
localise the user as described in Section 5.3. Note also how the path of positions calculated by
the localisation filter is constrained to the map of the environment throughout tracking. The
final position calculated by the localisation filter has an error in horizontal position of 0.32 m,
compared to horizontal drift of 9.31 m incurred by the PDR filter alone. The PDR filter also
incurred a vertical displacement error of −0.48 m, which the localisation filter corrected by
enforcing the constraint that all particles are constrained to lie on floor polygons.
5.4.1 Tracking accuracy
To evaluate the accuracy of the localisation filter during tracking it was compared to position-
ing data obtained from the Active Bat system, which is installed in one wing on the second
floor of the William Gates building. To do this an Active Bat was attached to the foot-mounted
IMU worn by the user. The user then walked throughout the building, making multiple sep-
arate passes through the wing of the building in which the Active Bat system is deployed.
Positions from the Active Bat system were logged by a desktop machine that was connected
to the system over a wired LAN. Raw data from the IMU were logged by a UMPC carried by
106
View (d)
View (c)
View (b)
Start
Ground truth end PDR end
(a)
Start Ground truth end
PDR end
(b)
Start Ground truth end
PDR end
(c)
Start Ground truth end
PDR end
(d)
Figure 5.10: A path of step events calculated by the PDR filter as a user walked throughout
the William Gates building (red) and the corrected path calculated by the localisation filter
(black). The dashed portion of the red line corresponds to step events during localisation
(i.e. before the filter had determined the user’s position). Figure (a) is annotated with the
view points of Figures (b-d). Grid size 10 m2. Diagrams are exploded 10x in the z-axis.
107
the user and post-processed on a desktop PC to obtain a set of positions estimated by the local-
isation filter. For each run of the localisation filter the particle cloud was initialised around the
user’s known initial position (i.e. it was assumed that the user had already been successfully
localised).
To compare the measurements obtained from each system it was necessary to ensure that the
two logging machines where time-synchronized. This was achieved using Chrony5, which
synchronises machines using the Network Time Protocol (NTP) and additionally constructs
locally stored models for each clock, which are used to minimise clock drift when the ma-
chines cannot contact one another. The two machines were synchronized over a wired LAN
before each data collection run. Chrony estimated a difference of at most 2 ms between the
two clocks.
During each run various factors can cause synchronisation issues to arise. Firstly, the two
clocks can be expected to drift apart slightly despite the corrections applied by Chrony. To
minimise this effect each data collection run was limited to at most 15 minutes, during which
the drift between the two clocks would be of the order of milliseconds. Secondly, small delays
between each measurement being made and subsequently being timestamped by the logging
application can have an effect. Active Bat data must also propagate over a wired LAN before
reaching the logging machine. To mitigate these issues positions from the Active Bat system
were only matched to localisation filter positions if they were timestamped within a stance
phase, as detected by the PDR filter. This approach allowed for drift of up to 10 ms between
the two data-sets without affecting the matched positions (since the user’s foot is stationary
during the stance phase and during a 10 ms window either side). This is an order of magnitude
greater than the maximum time-stamping error that could be expected.
5.4.1.1 Filtering Active Bat positions
During testing it was found that the Active Bat system occasionally reported spurious posi-
tions, either as a result of ultrasonic noise in the environment or reflected signals from the
bat attached to the user. In order to reduce the probability of such positions being included
in the comparison, positions obtained from the Active Bat system were filtered before being
considered for matching. The filter performed a number of checks to attempt to detect and
discard spurious measurements. A position was discarded if:
∙ It was not located within the building.
∙ It was below the floor level of the wing within which the Active Bat system is deployed.
∙ It was greater than 10 cm above the level of the floor. This discards measurements
5http://chrony.tuxfamily.org
108
that are not consistent with the user’s foot being grounded. These may be spurious
measurements calculated when the user’s foot was actually grounded (which should
be discarded), or non-spurious measurements calculated when the user’s foot was not
grounded (in which case they should not be considered for matching anyway).
∙ It was calculated from fewer than five time-of-flight measurements. When five or more
measurements are available the Active Bat system is able to perform consistency checks
that attempt to identify and exclude6 spurious time-of-flight measurements (i.e. those
caused by signal reflections or ultrasonic noise) from the final position calculation.
Hence the position calculation is more robust and therefore less likely to generate a
spurious position when at least five measurements are used.
∙ A ray from the calculated position to the position of a receiver whose measured time-
of-flight was used to calculate the position intersected a wall.
The last two conditions make use of additional information provided by the Active Bat system
with each calculated position. This additional information includes the individual time-of-
flight measurements, the position of each corresponding receiver and whether each time-of-
flight was accepted (i.e. used to calculate the position) or rejected (i.e. classified as noise
or a reflected signal) by the multi-lateration algorithm. Note that whilst some non-spurious
measurements may be discarded by the filter, this will not affect the results beyond reducing
the number of data-points obtained (i.e. the number of matched positions).
5.4.1.2 Results
In total data obtained during five walks were analysed. Each walk was between 10 and 15 min-
utes in duration and consisted of the user walking through significant portions of the building
(visiting at least two of the building’s three floors), as shown in Figure 5.11(a). As with lo-
calisation, using too few particles resulted in failures where all particles were found to violate
environmental constraints. To determine how many particles were required the logs obtained
from the five walks were processed multiple times using different numbers of particles. The
number of particles used was varied from 푁 = 50 to 500 in increments of 25 and then up to
1000 in increments of 100. Each walk was processed 20 times for each value of 푁 , giving
a total of 100 runs of the filter at each particle count. As expected the number of failures
decreased as the number of particles was increased, as shown in Figure 5.12. No failures were
observed when 275 or more particles were used. Note that this is a tiny fraction of the number
required for reliable localisation and hence the 2.6 Ghz test machine was easily able to track
a user in real time.
6Spurious time-of-flight measurements are usually successfully identified and excluded provided that a
greater number of (and at least four) non-spurious measurements are obtained.
109
(a) (b)
Figure 5.11: (a) Paths extracted by the localisation filter for five different walks through
the William Gates building. 푁 = 500 particles were used. Grid size 10 m2. (b) The paths
through the upper left wing of the building in which the Active Bat system is installed. Grid
size 5 m2.
 0
 10
 20
 30
 40
 50
 60
 70
 0  100  200  300  400  500  600  700  800  900  1000
Fa
ilu
re
 ra
te
 (%
)
Number of particles (N)
Figure 5.12: The percentage of failures during tracking as a function of the number of
particles used.
110
(a)
(b)
Figure 5.13: Two traces through the area covered by the Active Bat system generated using
푛 = 500 particles. The paths calculated by the localisation filter are shown as grey lines.
Matched positions obtained from the Active Bat system are shown as green dots, each of
which is connected by a line to the corresponding position obtained from the localisation
filter. Grid size 1 m2.
 0
 0.2
 0.4
 0.6
 0.8
 1
 1.2
 0  200  400  600  800  1000
Eu
cl
id
ia
n 
di
st
an
ce
 (m
)
Number of particles (N)
95th percentile
75th percentile
50th percentile
Figure 5.14: Euclidean distances between matched Active Bat and localisation filter posi-
tions. The 50푡ℎ, 75푡ℎ and 95푡ℎ percentiles are shown.
111
During each walk the user made multiple passes through the wing of the building in which the
Active Bat system is deployed, as shown in Figure 5.11(b). Figure 5.13 shows two such passes
together with matched positions obtained from the Active Bat system. Figure 5.14 shows the
50푡ℎ, 75푡ℎ and 95푡ℎ percentiles of the Euclidean distances between matched positions obtained
from all successful runs made at each particle count. A minimum of 5, 820 distances were
calculated for a given particle count (푁 = 50). 16, 457 distances were calculated for counts
that observed no failures.
If the Active Bat system is taken as the ground truth (which is reasonable given the relative
accuracies of the two systems) then the results shown in Figure 5.14 can be interpreted as the
tracking accuracy of the localisation filter. Under this assumption the localisation filter is able
to track the position of a user to within 0.38 m 75% of the time and 0.62 m 95% of the time
when 500 particles are used. It is also notable that using more particles than is necessary to
achieve reliable tracking did not significantly improve the accuracy of the system. One other
important observation is that by enforcing environmental constraints, the localisation filter is
not able to calculate paths that would have required the user to ‘jump’ through walls between
adjacent rooms. This property does not hold for the majority of existing indoor location
systems, which do not make use of such constraints. For example a WiFi location system that
does not enforce environmental constraints may position a device on different sides of a wall
in adjacent updates due to small fluctuations in received signal strength measurements. Note
that there is no fundamental reason why such systems are unable to enforce environmental
constraints (indeed there are several examples that do [85, 86]). It is simply that most existing
indoor location systems have not been designed to do so.
5.5 Conclusions
In this chapter a localisation filter has been developed that combines relative movement mea-
surements from a PDR system with knowledge of building constraints. The particle filter
implementation that has been presented is able to determine the absolute position of a pedes-
trian in an indoor environment, provided that they pass a sufficient number of environmental
symmetries. The filter is then able to track their position to an accuracy of 0.62 m 95% of
the time in a typical office environment. This level of accuracy is suitable for many indoor
location-aware applications. In particular it is sufficient to allow an application to determine
which objects in the environment are close enough to the user for a physical interaction to
take place.
Although the basic localisation filter has been successfully demonstrated there remain a num-
ber of outstanding problems that need to be solved in order to develop a functional tracking
system. In particular localisation and tracking have until now been considered separately,
112
however in practice a localisation problem turns into a tracking problem dynamically as the
particles converge to form a single cluster. It is necessary to detect when this occurs, since
only then does it make sense to calculate state estimates using the algorithm outlined in Sec-
tion 5.4. Furthermore the number of particles required for tracking is only a tiny fraction of the
number required for localisation. Thus it is desirable for an implementation to dynamically
reduce the number of particles as uncertainty in the user’s position decreases. Algorithms for
both detecting convergence and dynamically varying the number of particles will be described
in Chapter 6.
There are also two issues that have been identified regarding the localisation performance of
the filter:
∙ Symmetry of the environment can delay or prevent the convergence of the particles to
a single cluster corresponding to the user’s true position. Rotational and translational
symmetry of this kind is often exhibited by buildings, in particular multiple floors of a
building often have very similar layouts.
∙ A large amount of memory and computational power is required to perform reliable
real-time localisation. Furthermore both requirements increase with the size of the en-
vironment due to the fact that the number of particles required is proportional to the
floor area of the building. For 푁 particles 풪(푁) memory and 풪(푁 log2푁) time is
required. As a result it is difficult to scale the filter to very large buildings.
Possible solutions to these problems will be discussed in Chapter 7.
113
114
Chapter 6
Efficient localisation and tracking
In the previous chapter the problems of localisation and tracking were considered separately,
however in practice a localisation problem dynamically changes into a tracking problem when
the particles become clustered around a single position. Considering localisation and tracking
together raises two issues. Firstly the number of particles required for tracking is only a tiny
fraction of those needed during localisation. It is therefore desirable to dynamically reduce the
number of particles as the user’s location becomes more certain. Secondly, it is necessary to
detect when the particles have converged to form a single cloud of limited extent, because only
then does it make sense to extract the user’s position as described in Section 5.4. This chapter
explores solutions to both problems. Section 6.1 describes dynamic re-sampling, a technique
that automatically varies the number of particles during both localisation and tracking. Section
6.2 outlines a clustering algorithm that is able to detect when the particle cloud has converged
to a single location.
The second half of this chapter focuses on the performance of the localisation filter. Section
6.3 explores how the filter performs in different environments and when provided with an
inaccurate map. The filter’s error characteristics are also investigated. The effect of different
user motions on the accuracy of the filter is considered in Section 6.4.
6.1 Adaptive re-sampling schemes
The number of particles needed to represent Bel(풙푡) depends on the complexity1 of the dis-
tribution, which can vary drastically over time [87]. For global localisation problems the
complexity of the uniform prior (in which the tracked object is equally likely to be positioned
1In this context complexity is a vague notion describing how difficult it is to represent a given distribution. It
is not a well defined mathematical quantity.
115
anywhere in the environment) is very high, whilst the complexity of the localised distribution
after convergence is relatively low.
In the related field of robot localisation several schemes have been proposed for dynamically
varying the number of particles based on the notion of distribution complexity. One possible
approach is to update the particles as normal (i.e. by applying the propagation and correction
steps) before using some criterion to determine how many particles should be generated during
the re-sampling step. Unfortunately this approach does not perform well when the number of
particles needs to be increased from one step to the next (which occurs when a measurement
introduces a lot of uncertainty). Consider an example in which the previous state consists
of 10 particles but where 50 are needed to adequately represent the new state. Applying the
update algorithm to the 10 particles from the previous state and then re-sampling 50 times
results in a set of 50 particles as required, however they can only have at most 10 distinct
states because the re-sampling step simply copies states from the updated particles. Hence the
particles suffer from a lack of state diversity and so do not optimally represent the underlying
distribution.
Adaptive schemes developed by the robot localisation community solve the problem described
above by modifying the filter so that updating the state no longer consists of an update (i.e.
propagation and correction) step followed by a re-sampling step. Instead new particles of the
form ⟨풙푡, 푤푡⟩ are generated one at a time by first re-sampling from the previous state
풙ˆ푡−훿푡 ∼ 풮푡−훿푡 (6.1)
and then applying the propagation and correction steps in the usual way
⟨풙푡, 푤푡⟩ ← Update({풙ˆ푡−훿푡, 1}, 풆푡) . (6.2)
Re-sampling is implemented using a binary search into the particle weights as described in
Section 5.2.3. Since the particles are re-sampled in proportion to their weights, they are as-
signed equal weights (nominally equal to one in Equation 6.2, although any positive constant
could be used) before being updated. The process of generating particles continues until the
adaptive scheme determines that a sufficient number of particles have been generated (dis-
cussed below). The particle weights are then normalised to sum to one before the set of
particles is returned as the new state 풮푡. This approach ensures that the number of particles
updated in each step is equal to the number of particles that are actually required, and that the
state of each particle is generated independently.
The job of an adaptive scheme is to determine when a sufficient number of new particles have
been generated using the steps described above. Two such schemes are described in Sections
6.1.1 and 6.1.2. Note that if the cost of checking whether a sufficient number of particles has
116
been generated is 풪(1) then the computational complexity of generating 풮푡 using an adaptive
scheme is 풪(푁푡log2푁푡−훿푡), where 푁푡 and 푁푡−훿푡 are the number of particles in 풮푡 and 풮푡−훿푡
respectively.
6.1.1 Likelihood-based adaptation
Likelihood-based adaptation generates particles until the sum of the updated particle weights
exceeds a pre-specified threshold 휂. In other words particles are generated until
푛∑
푖=1
푤푖푡 > 휂 (6.3)
is true, where 푛 is the total number of particles generated up to that point. Hence large
numbers of particles are generated in steps where the measurements do not closely match the
estimated state. When the measurements are consistent with most of the particles far fewer
are generated. This approach was originally applied to localise robots equipped with laser
range finders [57]. In this example measurements from the laser range finders were usually
less consistent with the state when there was a large amount of uncertainty in the robot’s state
(i.e. its position and orientation). As a result more particles were generated during the early
stages of the localisation process relative to when the robot’s state had been determined.
One problem with likelihood-based adaptation is that it continues to generate particles in-
definitely (or in practice, until the available memory is exceeded) when a localisation failure
occurs. In such cases every particle is found to violate an environmental constraint and is
assigned a weight of zero, meaning that the threshold 휂 is never reached. It is preferable for
the filter to fail gracefully in such cases, for example by reporting an error that could in turn
cause the localisation process to be restarted. This can be achieved in practice by introducing
a maximum limit on the number of particles. If the limit is ever reached then a simple test
can be used to determine whether a localisation failure has actually occurred (i.e. by checking
whether the sum of the particle weights is equal to zero).
A far bigger problem with likelihood-based adaptation is that it relies on the mismatch be-
tween the prior and posterior distributions (which are approximated by the particles before
and after the correction step) being closely correlated to the uncertainty in the state and hence
to the number of particles required [87]. For localisation problems this is not always the case,
as shown in Figure 6.1. The mismatch between the prior and posterior distributions is the
same in both Figure 6.1(b) and Figure 6.1(c), however it is clear that the uncertainty is greater
in the first case. For pedestrian localisation the mismatch between the prior and posterior dis-
tributions is also highly dependent on the length of the most recent step taken by the user. If
the user takes a very small step then very few particles will be assigned zero weights due to
117
(a) (b) (c)
Figure 6.1: Particle clouds representing the possible position of an imaginary robot in
two environments. (a) The obstructions sensed by the robot’s laser range finder in each
case. The dot indicates the robot’s position. (b) The robot could be located in any of the
four corners of this environment. (c) The robot can only be located in one corner of this
environment. Likelihood-based adaptation will use the same number of particles in both
cases.
wall constraints. Hence the two distributions will be very closely matched and few particles
will be generated. This could occur early during the localisation process, when in fact many
particles are required to represent the distribution to a suitable degree of accuracy.
6.1.2 Kullback-Leibler distance adaptation
Kullback-Leibler distance (KLD) adaptation generates a number of particles such that the
approximation error introduced by using a particle-based representation of Bel(풙푡) remains
below a specified threshold 휖. This approach is better suited to localisation problems since it
does not rely on a relationship existing between uncertainty and the mismatch between prior
and posterior distributions [87].
To derive KLD-adaptation it is assumed that the true posterior distribution can be represented
in the form of a multi-dimensional histogram. Such a histogram consists of a number of multi-
dimensional bins of size Δ, each of which covers a portion of the state space within which the
probability density is assumed to be constant. Given this assumption, the error between the
particle-based representation and the true posterior can be measured as the Kullback-Leibler
distance
퐾(푝, 푞) =
∑
푏
푝(푏)log
푝(푏)
푞(푏)
(6.4)
between them, where 푏 ranges over all bins and 푞(푏) and 푝(푏) are the probability densities for
bin 푏 according to the true posterior and the particle-based approximation respectively. It can
be shown that if the number of particles is given by
푛req =
1
2휖
휒2푘−1,1−훿 (6.5)
118
where 휒2푘−1,1−훿 is the upper (1−훿) quantile of the chi-square distribution with (푘−1) degrees
of freedom, then 훿 is the probability that the distance between the two distributions is greater
than 휖 [58]. Hence by choosing a small 훿 (e.g. 훿 = 0.01) and generating 푛 ≥ 푛req particles it
is possible to say with near certainty (99% certainty if 훿 = 0.01) that a sufficient number of
particles have been generated such that the distance between the particle-based representation
and the true posterior is less than the threshold 휖. In practice 푛req can be computed using the
approximation
푛req =
1
2휖
휒2푘−1,1−훿 ≈
푘 − 1
2휖
(
1− 2
9(푘 − 1) +
√
2
9(푘 − 1)푧1−훿
)3
(6.6)
where 푘 is the number of bins that have support (i.e. non-zero probability) in the true posterior
and 푧1−훿 is the upper (1− 훿) quantile of the standard normal distribution.
The complete KLD re-sampling algorithm is shown in Algorithm 3. The algorithm terminates
when the total number of particles 푛 is greater than or equal to both 푛req and a minimum
count 푛min. Since the true posterior distribution is unknown, 푘 is estimated by counting the
number of bins which have support during the re-sampling process. This estimate is updated
each time a newly generated particle falls into an empty bin (line 11 of Algorithm 3). In
the early stages of the algorithm the value of 푘 (and as a result the value of 푛req) increases
with nearly every iteration, since only a few of the bins will already be occupied. As more
bins become occupied the rate at which 푘 increases slows down, at some point allowing the
number of particles 푛 to catch up with the required number 푛req at which point the algorithm
terminates. Termination is guaranteed since there are a finite number of bins and hence there
exists a maximum value which 푛req cannot exceed. Note that termination will occur after
relatively few iterations when the particle cloud occupies only a small portion of the state
space, since in this case the number of bins into which the newly generated particles fall is
small. This means that when applied to localisation problems fewer particles are generated
when the user’s position has been determined (i.e. during tracking) relative to cases in which
the user’s position is unknown (i.e. during the early stages of localisation). Note that this
matches the desired behaviour identified in Chapter 5.
KLD adaptation can also be used to generate prior distributions that contain a sufficient num-
ber of particles. The algorithm for generating a prior 풮0 is identical to that given in Algorithm
3, except that the state of each particle is sampled directly from the prior distribution Bel(풙0)
(which must be provided as an input argument) rather than being calculated by applying the
Update algorithm to a previous state. The weight of each particle is initially set to equal
one, however these weights are normalised (lines 18 - 20 of Algorithm 3) before the prior is
returned.
119
Algorithm 3 KldUpdState - Calculates the new state 풮푡 from the previous state 풮푡−훿푡 and the
current step event 풆푡. The number of particles in the new state is determined using KLD-
adaptation.
1: procedure KLDUPDSTATE( 풮푡−훿푡 = {⟨풙푖푡−훿푡, 푤푖푡−훿푡⟩ 푖 = 1, ..., 푛}, 풆푡 = (푙, 훿푧, 훿휃, 휙) )
// Initialise the algorithm
2: 풮푡 ← {} , 푛← 0 , 훼← 0
3: 푛req ← 0 , 푘 ← 0
// Generate particles until the KLD condition is met
4: repeat
5: 푛← 푛+ 1
// Sample the previous state
6: 풙ˆ푛푡−훿푡 ∼ 풮푡−훿푡
// Apply the propagation and correction steps
7: ⟨풙푛푡 , 푤푛푡 ⟩ ← Update({풙ˆ푛푡−훿푡, 1}, 풆푡)
8: 훼← 훼 + 푤푛푡
// Insert the new particle
9: 풮푡 ← 풮푡 ∪ {⟨풙푛푡 , 푤푛푡 ⟩}
10: if 풙푛푡 is in empty bin 푏 then
11: 푘 ← 푘 + 1
12: 푏←non-empty
13: if 푛 ≥ 푛min ∩ 푘 ≥ 2 then
// Update the number of desired particles
14: 푛req ← 푘−12휖
(
1− 2
9(푘−1) +
√
2
9(푘−1)푧1−훿
)3
15: end if
16: end if
17: until 푛 ≥ 푛req ∩ 푛 ≥ 푛min
// Normalise the weights to sum to one
18: for 푖 = 1 to 푛 do
19: 푤푖푡 = 푤
푖
푡/훼
20: end for
// Return the new state
21: return 풮푡
22: end procedure
120
6.1.3 Application to pedestrian localisation
When applying KLD adaptation it is necessary to specify values for the bin size Δ, the bound-
ing parameters 훿 and 휖 and the minimum number of particles 푛min. All four parameters affect
the number of particles generated during each update and so must be assigned suitable values.
In Chapter 5 it was shown that approximately 1, 200, 000 particles are required for reliable
localisation and that approximately 300 particles are needed during tracking in the William
Gates building. Hence the parameters should be set such that the number of particles gener-
ated when applying KLD adaptation in the same environment is initially close to 1, 200, 000
and is reduced to around 300 by the time the particle cloud has converged. There are many
different sets of values that achieve this variation. Although it would be possible to perform
an exhaustive search of the parameter space in order to find an optimal set of values (i.e. those
that minimise the number of particles generated at every step whilst still performing reliable
localisation and tracking), this task is not considered here. Instead suitable values for 푛min, Δ
and 훿 are estimated using prior knowledge, allowing a suitable value for 휖 to be determined
empirically.
∙ A suitable value for 푛min is 300, since this was the minimum number of particles re-
quired for reliable tracking, as established in Section 5.4. Although it is almost certainly
the case that fewer particles can be used at certain points during tracking, doing so is
not important since the computational cost of tracking is already relatively cheap.
∙ The bin size Δ determines the granularity by which the particle count can be varied. If
a very large bin size is chosen then 푛req can only take on a small number of possible
values when calculated using Equation 6.6. Hence the number of particles generated
will jump between these values rather than being varied smoothly. If a small bin size is
used then the number of particles will be continually adjusted but more memory will be
required to keep track of the occupied bins. Since the particle cloud converges to have
a radius of the order of metres, a bin size of
Δ푥 = Δ푦 = Δ푧 = 2 m (6.7)
is sufficiently fine-grained for pedestrian localisation.
∙ It is also necessary to specify a bin heading granularity Δ휃. In practice this value can be
used to control the ratio between the number of particles for the prior distribution and
the number used during tracking, with a smaller value of Δ휃 resulting in a larger ratio.
This is because a smaller value of Δ휃 will result in more bins being occupied by particles
in the prior, whilst roughly the same number of bins will be occupied during tracking
provided that the Δ휃 is still greater than the accuracy to which the user’s heading can be
determined. A value of Δ휃 = 휋6 was found to achieve a ratio close to the desired ratio
121
of 1, 200, 000 : 300.
∙ From the derivation of KLD-adaptation it is known that 훿 is intended to be small. A
value of 훿 = 0.01 was chosen as it has been shown to give good results when applied to
robot localisation [87, 58]2. Having fixed all other parameters, a suitable value for 휖 can
be determined empirically such that the absolute number of particles generated at each
step is of the order required. A value of 휖 = 0.015 was found to achieve this.
Although the values above have been derived based on observations made whilst using the
localisation system within the William Gates building, it is likely that the same parameters
will also be suitable for other buildings of similar architectural style. It is important to note
that the number of particles in a prior generated by KLD-adaptation is dependent on the size
of the building (more particles are generated for larger buildings). Hence it should not be
necessary to adjust the parameters when applying the filter within buildings of different sizes.
It may however be necessary to adjust 푛min and the bin size if a different PDR system is used
to provide step events to the filter. For example if the PDR system is less accurate then the
particle cloud will be larger on average and as a result a larger bin size and value of 푛min may
be more suitable.
6.1.3.1 Localisation performance
Figure 6.2 shows the variation in particle count during an example localisation using the same
data as shown in Figure 5.5 (page 100). The parameters were set as described above:
푛min = 300 (6.8)
Δ푥 = Δ푦 = Δ푧 = 2 m (6.9)
Δ휃 =
휋
6
(6.10)
훿 = 0.01 (6.11)
휖 = 0.015 . (6.12)
KLD-adaptation was used to select a suitable number of particles to represent the prior dis-
tribution and to adapt the number of particles during each update. The prior distribution itself
was a uniform distribution over the entire floor surface. The user’s initial heading was also
distributed uniformly in all directions. As expected the number of particles decreased as un-
certainty in the user’s position was reduced. Note how the number of particles fell rapidly
between points (d) and (e) as the user was quickly constrained to regions of the map cor-
2[87] contains a typing error in which it states that a value of 훿 = 0.99 was used. This should have read
(1 − 훿) = 0.99, or equivalently 훿 = 0.01. This error was corrected in a more recent paper by the same
author [58].
122
 0  20  40  60  80  100
Step event
(c)
(d)
(e) (f)
point of
convergence
N
um
be
r o
f p
ar
tic
le
s (
N
)
102
103
104
105
106
107
Figure 6.2: The number of particles generated by KLD-adapation for each update during
an example localisation. Annotations (c-f) indicate points during the localisation which
correspond to Figures 5.5 (c-f) (page 100).
responding to staircases in the environment. The particle cloud at point (f) contained 345
particles. Figure 6.3 shows the time taken to process each step event during the same locali-
sation 2.6 GHz desktop PC. Although the first few steps could not be processed in real-time,
the filter was able to catch up as the number of particles was reduced. In this example only the
first five steps required more than 1.09 s to process and the filter had caught up with real-time
by the 12th step3.
The reliability of the adaptive filter during localisation was tested by applying it to the same
data-set used to evaluate the basic filter design in Section 5.3.2. Recall that the data-set con-
sists of logs corresponding to 10 walks through the William Gates building, each of which is
known to pass a sufficient number of asymmetries such that the user’s position can be uniquely
determined. Each log was processed 10 times by the filter, which successfully located the user
in all 100 cases (success is defined as convergence of the particle cloud to form a single cluster
around the user’s known final position). Hence the reliability of the filter during localisation
was not affected by dynamically adapting the number of particles.
6.1.3.2 Tracking performance
The tracking performance of the adaptive filter was measured by applying it to the same log
data used to measure performance when using a fixed number of particles in Section 5.4.1.2.
This dataset consisted of five logs corresponding to the user walking along five different paths
in the William Gates building. Corresponding position measurements obtained using the Ac-
31.09 s is the average time per event that is available if the user is walking at a (brisk) rate of 110 steps per
minute.
123
 0.001
 0.01
 0.1
 1
 10
 0  20  40  60  80  100
Pr
oc
es
si
ng
 ti
m
e 
pe
r s
te
p 
(s
)
Step event
(c)
(d)
(e)
(f)
point of
convergence
Figure 6.3: The time required to process each step event during an example localisation.
These results are for a Java implementation running on a 2.6 GHz desktop PC. Annotations
(c-f) indicate points during the localisation which correspond to Figures 5.5 (c-f) (page
100).
tive Bat location system were logged whenever the user was within its coverage area. The
filter was used to process each of the logs 20 times, using an average of 520 particles during
tracking. The minimum and maximum numbers of particles generated for a single step were
300 and 1300 respectively. No tracking failures were observed for any of the 100 runs. This
was expected, since no failures were observed when processing the same log data using a
fixed number of 300 particles.
Figure 6.4 shows the distribution of Euclidean distances between matched positions calculated
by the adaptive filter and the Active Bat system. Recall that the Active Bat system is several
orders of magnitude more accurate than the filter. Hence it is reasonable to use it as a ground
truth for the purpose of assessing filter accuracy. Given this, the adaptive filter was able to
track a user in the William Gates building to within 0.26 m 50% of the time, 0.40 m 75%
of the time and 0.64 m 95% of the time. These percentiles are similar to those obtained
when using a fixed number of 500 particles. Hence the use of an adaptive scheme does not
noticeably improve or degrade the tracking accuracy of the system when compared to using a
fixed number of particles.
6.2 Detecting convergence
During the early stages of localisation it is not possible to compute a meaningful most likely
position for the user, since there are many particles distributed throughout the building all
of which have essentially the same probability of corresponding to the user’s true location.
Following this early phase the particle cloud often forms a number of localised clusters, as
124
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9  1  1.1  1.2
C
D
F
Euclidean distance between Active Bat and localisation filter positions (m)
Figure 6.4: The cumulative distribution of the Euclidean distances between matched Active
Bat and localisation filter positions.
Figure 6.5: A clustered set of particles generated during localisation. The colour of each
particle indicates the cluster to which it has been assigned.
shown in Figure 6.5. These clusters can be detected by applying a clustering algorithm. Over
time the clusters that do not correspond to the user’s true location are removed as they are
found to violate environmental constraints. When all of the particles are grouped into a single
cluster it is said that the particle cloud has converged. At this point the problem solved by the
filter becomes one of tracking rather than of global localisation. During tracking the user’s
position can be estimated as described in Section 5.4.
The clustering algorithm used here is a variant of one originally described by Ashbrook and
Starner for clustering GPS positions [88, 89]. First a particle is drawn at random and its state
(푥, 푦, 푧, 휃) is taken as the centre-point of a new cluster
(푋푐, 푌푐, 푍푐,Θ푐) := (푥, 푦, 푧, 휃) . (6.13)
All particles that are within a specified distance (discussed below) of the centre-point are
125
added to the cluster. The centre-point is then updated so that it is equal to the weighted
average of the cluster’s constituent particles
푋푐 :=
∑푁
푖=1푤
푖 ⋅푥푖∑푁
푖=1푤
푖
(6.14)
푌푐 :=
∑푁
푖=1푤
푖 ⋅ 푦푖∑푁
푖=1푤
푖
(6.15)
푍푐 :=
∑푁
푖=1푤
푖 ⋅ 푧푖∑푁
푖=1푤
푖
(6.16)
Θ푐 := atan2(ℎ푦, ℎ푥) (6.17)
where 푁 is the total number of particles in the cluster, (푥푖, 푦푖, 푧푖, 휃푖) is the state of the 푖th
particle and 푤푖 is its corresponding weight. The values of ℎ푥 and ℎ푦 are given by
ℎ푥 =
푁∑
푖=1
푤푖 ⋅ cos 휃푖푡 (6.18)
ℎ푦 =
푁∑
푖=1
푤푖 ⋅ sin 휃푖푡 . (6.19)
The process of adding particles that are sufficiently close to the cluster’s centre-point before
updating it is then repeated. This continues until there is an iteration in which no new particles
are added, at which point the cluster is complete. The whole process of generating a cluster
is then repeated on the remaining particles until all of the particles have been assigned to
clusters.
In localisation experiments conducted in the William Gates building, it was found that adding
particles to a cluster that were within an 8 m horizontal radius of (푋푐, 푌푐), a 1 m vertical
displacement of 푍푐 and a 14휋 angle of Θ푐 gave good results. Note that when all of the particles
are assigned to a single cluster the centre-point of that cluster is equal to the user’s estimated
state when calculated as described in Section 5.4. Hence the user’s estimated position and
orientation are calculated for free by the clustering algorithm. When there are multiple clusters
their centre-points can be interpreted as possible states of the user. The sum of the weights of
all particles within a cluster can be interpreted as the probability of that cluster corresponding
to the user’s true position.
The clustering algorithm described above requires 푂(푁2) time, where 푁 is the total number
of particles. As a result it is very expensive to cluster hundreds of thousands of particles.
The solution to this problem is simply to wait until the number of particles falls below a
threshold that occurs naturally during the localisation process (a suitable value for the William
Gates building was found to be 8, 000). Alternatively a clustering algorithm with a lower
computational complexity can be used. The important property of the algorithm described
126
above that makes it suitable for detecting the convergence of the particle cloud is that it does
not require the number of clusters to be known in advance. Any other clustering algorithms
possessing this property could equally well be used.
6.2.1 Clustered particle filtering
The clustering of particles as described above can also be used to construct a clustered particle
filter, in which particles in each cluster are evolved independently according to the usual
propagation, correction and re-sampling steps [90]. The idea of the clustered particle filter
is to prevent the particle cloud from converging too rapidly on a single most-likely position
when there are in fact multiple possible hypotheses consistent with the environment. This
problem can arise if one hypothesis has a much larger probability than the others at a given
time-step. When this occurs some of the less likely hypotheses may not be re-sampled during
an update and will be lost. If the cluster corresponding to the user’s true position is lost then
a localisation failure will subsequently occur.
The clustered particle filter prevents premature convergence by evolving each cluster of par-
ticles independently, effectively using a separate particle filter to track each hypothesis4 [90].
Since particles only compete with other particles in the same cluster during the re-sampling
step, there is no possibility of a hypothesis being lost. The probability of each hypothesis is
tracked separately at a higher level. Hypotheses are discarded when their probabilities fall to
zero.
In practice failures due to the premature convergence of the particle cloud are extremely rare
when applying the localisation filter in the William Gates building. This is probably due to
the fact that the corridors and doorways of the William Gates building are all of similar size.
Hence clusters corresponding to multiple hypotheses tend to be of similar size and have similar
probabilities. As a result the probability of a given cluster dying out during the re-sampling
process is very low. Note that if the environment had one very narrow corridor then many of
the particles in a cluster entering that corridor would be removed, causing the corresponding
hypothesis to be assigned a low probability compared to other hypotheses located in wider
corridors. In this case the hypothesis in the narrow corridor could be lost prematurely. A
clustered particle filter will almost certainly be more robust than the standard localisation
filter when applied to environments possessing such features.
4Note that a single filter must be used up to the point at which the particles are initially clustered.
127
6.3 Dependence on environmental constraints
This section investigates the relationship between the 2.5-dimensional map of an environment
and the performance of the localisation filter within that environment. Section 6.3.1 demon-
strates how the tracking accuracy of the filter is dependent on the density of constraints. Ad-
ditional constraints that can be used in environments such as an open plan office are discussed
in Section 6.3.2. Section 6.3.3 describes problems that arise if the map does not accurately
describe the real environment. Finally, the effect of constraints on the error characteristics of
the filter are discussed in Section 6.3.4
6.3.1 Open plan environments
Until now the tracking accuracy of the localisation filter has been measured in a typical office
environment; in particular the wing of the William Gates building in which the Active Bat
system is installed. This is not however a fair representation of the system’s accuracy in
all environments. In fact the accuracy is highly dependent on the environment, since it is
the environmental constraints that correct the drift errors that would otherwise accumulate
in the tracked position. Hence it is expected that the localisation filter will perform better
in structured environments with many constraints compared to large open plan environments
where there are few.
To test the system’s tracking performance in an open plan environment the filter was applied
to the same log data as in Sections 5.4.1.2 and 6.1.3.2, but with a modified 2.5-dimensional
map of the William Gates building. The map was modified by removing all of the interior
walls in the wing of the building covered by the Active Bat system. This simulated the user
following the same paths, but through a large open plan environment rather than through the
office environment that actually exists. Figures 6.6(a) and 6.6(b) show the result of removing
the interior wall constraints for a single walk through the modified area. In this example
a heading error was propagated from the PDR filter as the user entered from the right. In
Figure 6.6(a) the corridor walls constrained the particle cloud and hence prevented error from
accumulating in the user’s position. Without the interior wall constraints this was not possible,
as shown in Figure 6.6(b). The distribution of positioning errors obtained when only exterior
walls were used is shown in Figure 6.7. As expected the accuracy was degraded, with 95% of
measurements accurate to within 1.58 m (compared to 0.64 m when interior wall constraints
are used).
Open plan environments similar in size to the one simulated in the example above are common
in many buildings. In some cases (e.g. a large hallway) the observed degradation in accuracy
is unavoidable, however in others (e.g. an open plan office) there are often objects other than
128
(a)
(b)
(c)
(d)
Figure 6.6: The paths calculated by the localisation filter for a single walk; (a) With interior
wall constraints. (b) Without interior wall constraints. (c) With both furniture and interior
wall constraints. (d) With furniture constraints but without interior wall constraints. In each
case the cloud of particles is shown for every 10th step. Corresponding positions calculated
by the Active Bat system are shown as green dots. Grid size 1 m2.
129
 0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0.8
 0.9
 1
 0  0.5  1  1.5  2  2.5  3
C
D
F
Euclidean distance between Active Bat and localisation filter positions (m)
Walls [no] furniture [no]
Walls [no] furniture [yes]
Walls [yes] furniture [no]
Walls [yes] furniture [yes]
Figure 6.7: Tracking accuracy with and without interior wall and furniture constraints.
walls that can be used to constrain the user’s movement. In particular items of furniture act as
constraints in most open plan office environments. The use of these additional constraints is
outlined below.
6.3.2 Additional constraints
Apart from walls, furniture such as desks and bookcases often limit the movement of a user
through a typical office environment. If the locations of such objects are known then they
can easily be used as additional constraints by the localisation filter. OpenRoomMap was
developed to provide a mechanism to obtain such information [91]. OpenRoomMap is a web-
based application that allows the occupants of a building to place items such as furniture onto
a floor-plan, as shown in Figure 6.8. Its aim is to allow many individuals to contribute towards
building and maintaining detailed models of indoor environments for use by ubiquitous com-
puting systems. This approach of crowd-sourcing data has already proven to be successful
in a number of other domains, most notably in the construction of the on-line encyclopaedia
Wikipedia5. OpenStreetMap6 is also using crowd-sourcing in an attempt to build an open-
source map of the world [92].
It is not sensible to use all of the furniture items as constraints. Small pieces of furniture such
as chairs and coffee tables are moved frequently in the physical world, whereas their positions
in the OpenRoomMap model are likely to be updated far less frequently, if at all. Hence only
large rarely-moved objects should be used, for example full-sized desks, cupboards and filing
cabinets. The 2.5-dimensional map data-structure is easily extended to allow the representa-
5http://www.wikipedia.org
6http://www.openstreetmap.org
130
Figure 6.8: The OpenRoomMap editor.
tion of such objects by adding a list of overlapping furniture objects to each floor polygon.
Note that if a furniture object spans multiple floor polygons then it should be entered into all
of the corresponding lists. To enforce furniture constraints the localisation filter can simply
treat the edges of all furniture objects in a floor polygon’s furniture list as impassable walls
belonging to that polygon.
Figures 6.6(c) and 6.6(d) show paths calculated by the localisation filter using furniture con-
straints with and without interior wall constraints. When used in addition to interior walls
the accuracy of the calculated path was not significantly altered. In fact across all test runs
the tracking accuracy of the filter was slightly worse when furniture constraints were used
(although the difference is not believed to be significant), as shown in Figure 6.7. In contrast
it is clear from Figures 6.6(b) and 6.6(d) that the use of furniture constraints in the absence
of any interior walls significantly improved the accuracy of the calculated path. Over all test
runs the use of furniture constraints reduced the 95th error percentile from 1.58 m to 0.94 m
in the case where interior walls were not used.
131
6.3.3 Constraint accuracy
When developing localisation filters it is often assumed that the virtual model of the envi-
ronment is an accurate reflection of the real world. In practice this is only achievable in
controlled research environments. The problem of inaccurate constraints is particularly likely
to arise when using crowd-sourced data such as that described in Section 6.3.2. Objects may
be placed at incorrect locations or may be moved (in the physical world) without being up-
dated in the virtual model. Although these problems are particularly applicable to furniture
due to its mobility, they are also true of the map itself. Building walls can also be moved,
added and removed, albeit less frequently.
It is possible to describe inaccuracies in a building model in terms of missing and non-existent
constraints:
1. Missing constraint: A constraint in the real world that is not represented in the model.
2. Non-existent constraint: A constraint in the model that does not exist in the real world.
Note that the third case of an incorrectly placed constraint can be thought of as a non-existent
constraint in conjunction with a missing constraint at the object’s true location.
The effect of a missing constraint is simply one fewer constraint available for use during local-
isation and tracking. During localisation this can delay or prevent localisation if removing the
constraint increases the environmental symmetry of the environment, as described in Section
5.3.1. During tracking the accuracy of the filter in the vicinity of the missing constraint may
be reduced. This effect is demonstrated by the experiment described in Section 6.3.1, where
the interior walls of an area of the William Gates building were removed from the model in
order to measure the localisation filter’s tracking performance in an open plan environment.
Non-existent constraints pose a bigger problem to the localisation filter since they can cause
a failure in which the entire particle cloud is found to violate the constraints. For example if a
user who is being tracked walks through a wall that exists in the model but not in real life then
all of the particles will be assigned zero weights. Hence the localisation filter will believe that
it is impossible for the user to be positioned anywhere in the building, at which point the filter
must be re-started (from scratch) or from a guess of the user’s position calculated using some
ad-hoc recovery algorithm. If the user violates a non-existent constraint during localisation
then the problem is even worse. From that point on there will be no particles corresponding
to the user’s true position, however since there will still be particles located elsewhere an
immediate failure will not occur. The filter will only fail when all other candidate positions
have been eliminated. Note that the particle cloud may well converge before this occurs,
causing the filter to report incorrect positions for the user.
In practice it is often possible to detect non-existent constraints in an environment, provided
132
Start
Figure 6.9: Bias errors caused by nearby environmental constraints. The path of a simu-
lated user is shown in green. The corresponding path calculated by the localisation filter is
shown in black. The particle cloud is shown for every 10th step. Grid size 4 m2.
that they are few in number. This can be achieved by logging which constraints are violated
each time a failure occurs. If a particular set of constraints is found to account for multiple
failures then it is likely that at least one of them does not actually exist. Note that this approach
is not foolproof, since delayed failures as described above usually occur when the particle
cloud violates a valid constraint having previously converged to an incorrect location. It is
however a useful method to flag suspect constraints that can then be checked manually. In
the case of crowd-sourced data the user who last edited a suspect constraint could be asked to
perform the check.
Another idea that may prove useful when dealing with possible model inaccuracies is the
idea of probabilistic constraints, where the probability 푝 of a given constraint is the certainty
that a corresponding object actually exists in the real world. When a particle violates such a
constraint its weight is multiplied by (1− 푝). Hence if a constraint is certain (i.e. 푝 = 1) then
any particles that violate it will be assigned a weight of zero and as a result will be removed
during re-sampling. Conversely if a constraint is definitely non-existent (i.e. 푝 = 0) then it will
have no effect. When a constraint is thought to be responsible for a failure its probability could
be reduced. Hence constraints that are thought to be responsible for multiple failures will have
their weights reduced over time, reducing their effect on the particle cloud during subsequent
runs. The probability of furniture constraints could also decay over time to represent the fact
that they may have moved, with the decay rate dependent on the mobility of the item (e.g.
coffee tables are more mobile than heavy filing cabinets). The weights of such constraints
could then be restored to 푝 = 1 each time their positions are manually confirmed.
6.3.4 Tracking error characteristics
Although constraints are important to prevent the accumulation of drift errors over time, they
also introduce bias into the positions calculated by the localisation filter. In particular if a
user is located near some constraint, then the position calculated by the localisation filter will
tend to be biased away from that constraint. This phenomenon is demonstrated in Figure 6.9.
In this example a simulated user walks in a straight line down one side of a corridor before
crossing to the other side and continuing in the same direction. The user’s initial position
133
 0
 0.05
 0.1
 0.15
 0.2
 0.25
 0  0.2  0.4  0.6  0.8  1  1.2  1.4
Fr
ac
tio
n 
of
 p
os
iti
on
s
Offset from corridor centre-line (m)
Active Bat
Localisation filter
Figure 6.10: Deviations from the corridor centre-line calculated for matched positions
obtained from the localisation filter and Active Bat systems.
and orientation are assumed to be known, hence the prior consists of particles only with this
initial state. The user’s movement was simulated by a sequence of perfect step events (shown
in green) that were fed to the localisation filter. As successive step events are processed by
the filter the particle cloud expands across the width of the corridor, however expansion in the
other direction is prevented by the nearest wall. As a result the positions calculated by the
filter become biased towards the centre of the corridor. When the user crosses to the other
side of the corridor this bias is corrected, however it is re-introduced (this time in the opposite
direction) as the user continues down the corridor. Note that the rate at which the bias is
re-introduced is dependent on the step error model, which is itself dependent accuracy of the
PDR system. If a very good PDR system is used then the particle cloud will expand slowly
because each step event will carry only a small amount of uncertainty. If a poor quality PDR
system is used then the particle cloud will expand rapidly and hence the bias will become
noticeable after fewer steps.
A bias towards the centre-line of the corridor can also be observed in traces calculated from
real measurement data, although the effect is less clear due to noise and other bias errors that
perturb the raw IMU measurements. Consider the set of positions calculated by the localisa-
tion filter that were matched to corresponding Active Bat positions during the tests described
in Section 6.1.3.2. Now consider the subset of these positions for which the corresponding bat
positions were located in the central corridor running through the area covered by the Active
Bat system. It is trivial to calculate the distance from the corridor centre-line of each position
in this set. The distribution of these offsets is shown in Figure 6.10 together with the same
distribution calculated from the corresponding Active Bat positions. Note that the majority of
the positions obtained from both systems were close to the centre-line of the corridor. This is
because a pedestrian will naturally tend to walk down the middle of a corridor unless entering
134
Figure 6.11: A Voronoi diagram of an office environment (generated using VRONI [93]).
or exiting a room. Comparing the two distributions, a greater proportion of the positions cal-
culated by the localisation filter lie very close to the centre-line, whilst the Active Bat system
reported more positions near to the walls of the corridor. Hence the data shows a bias in the
positions calculated by the localisation filter towards the centre of the corridor.
6.3.4.1 Voronoi diagrams
Another way to think about the biasing effect of environmental constraints is to consider
a Voronoi diagram of the environment. A Voronoi diagram consists of a set of points with a
locally maximal distance from all surrounding objects. Such points naturally form the edges of
a graph-like structure, as shown in Figure 6.11. The effect of nearby environmental constraints
can be thought of in terms of introducing a bias error towards the nearest edges of the Voronoi
diagram rather away from nearby constraints. Note that although it is possible to predict the
bias error that will be introduced given the true position of the user, it is not possible to reverse
the error since many points in free space map onto a single point on the graph. Understanding
the biasing effect in this way gives weight to the idea of constraining particles to lie on the
edges of a Voronoi diagram. Such an approach has previously been used to estimate the
position of a user based on measurements from absolute positioning systems [85]. The main
benefit is that since the particles are more constrained, far fewer are required and hence the
computational resources required to run the filter are much lower. The main drawback is that
it is not possible for the particle cloud to represent positions that lie near to constraints (e.g.
at the edge of a corridor), however since the biasing effect makes the correct calculation of
such positions unlikely in any case this limitation may not have a significant effect on the
accuracy of the filter. In fact the overall accuracy may actually improve, since pedestrians
have a natural tendency to move along the edges of a Voronoi diagram (i.e. staying well clear
of nearby obstacles) when moving through an environment.
The use of Voronoi diagrams is not discussed further in this thesis, however it is likely that
such diagrams will prove useful in future research on pedestrian localisation and tracking. In
particular they are likely to be beneficial when using low quality PDR systems (for which the
135
biasing effect is large) or when computational resources are limited (in which case the use of
fewer particles is attractive).
6.4 Dependence on user motion
It is not always the case that users are continually walking from one place to another, as
has been the case in the examples used in Chapter 5 and Sections 6.1 and 6.3. In particular
users may loiter in particular areas, for example when chatting in the corridor of an office
environment. Figure 6.12 shows various points during an example in which the user was
tracked whilst giving a 30 minute presentation. During the presentation 440 step events were
generated by the PDR filter corresponding to small steps taken by the user. Most of these
steps were on-the-spot, although a few were generated by the user moving to point out various
objects on his slides, which were projected onto a nearby wall.
Figure 6.12(a) shows the user entering the room to start the presentation. The cloud after 200
step events is shown in Figure 6.12(b) . Note that the particle cloud has expanded due to the
uncertainty in each of the events. A bias in the calculated position away from nearby walls
is also observed, as described in Section 6.3.4. Figure 6.12(c) shows how particles propagate
into a neighbouring room as the user moves to the left in order to point out an object on
his slides, however by the end of the presentation the particle cloud is once again entirely
within the correct room, as shown in Figure 6.12(d). As the user leaves the room two distinct
clusters are formed, as shown in Figure 6.12(e). This occurs because uncertainty in the user’s
orientation has increased during the talk to such an extent that he could be moving towards
either of the two upper exits. One of these clusters is quickly removed as the user moves
further away from the room, as shown in Figure 6.12(f).
It may seem somewhat surprising that the particle cloud does not expand further than the
bounds of the room (except temporarily as in Figure 6.12(c)). One might expect that the
particle cloud would expand to fill a larger and larger proportion of the floor of the building
throughout the talk7. This does not happen in practice because each particle follows a random-
walk-like path due to uncertainty in each of the on-the-spot step events. As a result any particle
that is in close proximity to a constraint is likely to violate it at some point in the future and
hence be removed. To move into a different room a particle must come into close proximity
with nearby constraints because the doorways are relatively narrow. Hence very few particles
successfully pass into neighbouring rooms. Those that do tend to be removed over the course
of multiple re-sampling steps due to the stochastic nature of re-sampling.
7Note that the particles would not spread to different floors due to the use of 훿푧 as a measurement in the
localisation filter.
136
(a) After 0 steps (b) 200 steps (c) 270 steps
(d) 440 steps (e) 456 steps (f) 462 steps
Figure 6.12: Tracking a user as they give a 30 minute talk. The dashed blue box in (a)
indicates the region within which the user was positioned throughout the talk. In each
figure the black line shows the calculated path up to that point. Each yellow dot indicates
the centre-point of a cluster of particles. Grid size 1 m2.
6.5 Conclusions
This chapter has described algorithms for dynamically adapting the number of particles used
by the localisation filter and for detecting the convergence of the particle cloud to a single loca-
tion. Both algorithms enhance the basic localisation filter and deliver important benefits when
deploying such filters in practice. Adapting the number of particles using KLD-adaptation
allows the filter to generate only the number of particles it requires to represent the belief
at each point in time to a specified level of accuracy. The important consequence of using
such an adaptive scheme is that step updates do not consume more computational resources
than they require. Hence only the initial localisation phase is computationally expensive. The
ability to detect when the particle cloud converges to a single location is also important, since
only then does it make sense to compute the user’s position as the weighted average of the
positions of the individual particles.
The second half of this chapter analysed how the accuracy of the filter depends on the envi-
ronment and the motion of the tracked user. The important results are summarised below.
∙ The accuracy of the localisation filter is dependent on the density of the environmental
constraints, with a higher density of constraints leading to a better accuracy.
137
∙ Additional constraints such as those provided by furniture can be used to increase the
constraint density. This approach is particularly applicable to open plan offices, which
typically have few walls but many pieces of furniture.
∙ Inaccuracies in the map can lead to problems during both localisation and tracking.
They can be characterised in terms of missing constraints and non-existent constraints.
Missing constraints can delay or prevent convergence during localisation and degrade
the system’s tracking accuracy. Non-existent constraints pose a bigger problem since
they can cause failures during both localisation and tracking.
∙ The positions calculated by the filter are biased away from nearby constraints, or equiv-
alently towards nearby edges of a Voronoi diagram representation of the environment.
By constraining particles to lie on such a graph it may be possible to decrease the num-
ber of particles required to track a user whilst having a minimal effect on the system’s
accuracy.
∙ Surprisingly the particle cloud does not tend to spread throughout the environment when
the user takes many small steps whilst loitering in a single location. This is because
particles are unlikely to pass through doorways (which are typically narrow) without
subsequently violating nearby wall constraints.
Although the use of KLD-adaptation automatically reduces the computational requirements as
the filter becomes more certain of the user’s position, it does not reduce the initial cost (i.e. the
computational cost and memory required during the first update). Hence even when varying
the number of particles using such an algorithm it is not feasible to apply the localisation filter
to an arbitrarily large environment. Chapter 7 will investigate methods to reduce this initial
cost by using a range of additional sensors.
138
Chapter 7
Assisted localisation
Whilst tracking using the algorithms developed in Chapter 5 is computationally inexpensive,
the cost of initial localisation is high. The adaptive schemes discussed in Chapter 6 are able to
automatically reduce this cost as the user’s position becomes more certain, however they are
not able to reduce the cost of the first few updates during the localisation process. The com-
putational and memory requirements for an update are 풪(푁 log푁) and 풪(푁) respectively,
where the number of particles 푁 is initially proportional to the floor area of the building
in which the system is deployed. This places a practical limitation on the size of buildings
within which global localisation can be performed. Environmental symmetry can also delay
or even prevent the filter from determining the user’s position in a large building, as described
in Section 5.3.1 and illustrated in Figure 7.1.
Both of the problems outlined above can be solved, at least to some extent, by reducing uncer-
tainty during localisation. This is possible if some additional information is available about
the user’s position or orientation, which could be obtained from additional sensors or provided
manually by the user. When such information is available it can be used to place additional
constraints on the prior (i.e. the initial distribution of particles) and the sets of particles gen-
erated by the filter during localisation. This not only reduces the number of particles that
are required, but can also reduce ambiguity caused by environmental symmetry by removing
clusters of particles that are inconsistent with the constraints.
There are many ways in which the position of a user can be constrained both when generating
the prior distribution and throughout the localisation process. The most suitable approach
is dependent on the scenario in which the localisation system is to be deployed. In some
scenarios it may be reasonable to require the user to enter their approximate location when
the system is first switched on; for example when tracking military or emergency personnel
as they enter a building. In such cases the users may be sufficiently motivated to provide
an approximate initial position in order to enable tracking. For other usage scenarios this
139
Figure 7.1: Multiple clusters formed due to environmental symmetry. Arrows indicate the
paths followed by four of the clusters. The top left cluster corresponds to the user’s true
position. Grid size 10 m2.
approach is less practical. For example employees being tracked in an office environment as
part of a use-of-space experiment will not benefit directly from the correct operation of the
tracking system. Hence they will be less inclined to provide manual input when requested to
do so. In some scenarios the user might be lost, making it impossible for them to enter their
position.
An alternative to requesting user input is to make use of additional sensors in order to constrain
the user’s state during localisation. A wide range of sensor types could be used, some of which
are listed below.
∙ A magnetometer can be used to estimate the user’s initial heading. If the heading can
be estimated to within 10∘ then the number of particles required to represent the prior
can be reduced by a factor of 36 for a given floor area. An initial heading estimate will
also resolve ambiguity due to rotational symmetry in the environment. This approach
to reducing uncertainty is desirable since many commercial inertial measurement units
(IMUs) already contain magnetometers. Unfortunately the use of magnetometers is not
possible in environments that contain sources of strong magnetic fields. In the case of
the William Gates building magnetic disturbances caused by metallic floor panels make
it impossible to reliably determine the user’s orientation using the magnetometers in the
foot-mounted Xsens IMU, as described in Section 4.3. An alternative approach in this
case would be to use a separate magnetometer mounted at a higher position on the body,
such as on the hip.
∙ If a user is being tracked by an outdoor positioning system (e.g. GPS) prior to entering
the building, then it may be possible to use the last positions calculated by such a system
in order to determine the user’s entry point into the building, or a set of possible entry
140
points if the outdoor positioning system is not sufficiently accurate. Particles in the
prior distribution can then be focused around these entry points. Such an approach may
be well suited to military personnel, who are often equipped with GPS receivers for
outdoor use.
∙ In some cases it may be possible to augment the environment with a small amount
of fixed infrastructure. For example some form of beacons (e.g. Bluetooth beacons or
RFID tags) could be mounted near to the entry points of a building. A device attached to
the user could scan for such beacons in order to determine the user’s approximate initial
location when entering the building. The localisation filter could then be initialised
with a cloud of particles only in the vicinity of a beacon detected from the user’s initial
location. The number of particles required to represent the prior is independent of
the size of the building when this approach is used. The main disadvantage is that
a constrained prior can only be generated when a beacon is detected from the user’s
location. Hence if beacons are placed only at entry points to a building then they cannot
be used to generate a constrained prior for a user who is already inside the building
when the system is switched on.
∙ Wireless LANs are already deployed in many buildings. The strengths of signals re-
ceived from nearby WiFi access points can be used to estimate the user’s initial position
and hence constrain the user’s location during localisation. This approach usually re-
quires a pre-constructed radio map of the environment or at the very least the positions
of the access points in the environment. Hence such an approach may not be suitable
for the emergency services scenario in which personnel are typically unable to survey
the building in advance. It is however well suited for tracking in office environments.
The use of WiFi access points during localisation in described below.
7.1 WiFi-assisted localisation
Where available, signals received from WiFi access points in the environment can be used to
assist with the localisation process. The requirements for assisting the localisation filter differ
substantially from those of traditional WiFi positioning systems. As a result there are several
important differences between the algorithms described here and those used in the WiFi po-
sitioning systems described in Section 2.1.2.4. Firstly, the aim of a WiFi positioning system
is to calculate the most likely position of a person or device as accurately as possible. In con-
trast the aim here is to assist the localisation filter by reducing the number of particles that are
required and where possible resolve ambiguities that arise due to environmental symmetry.
Constraining the user’s position to a relatively coarse region of space is usually sufficient to
achieve these goals. Furthermore it is preferable to make such regions large enough to con-
141
tain the user with a very high probability, rather than attempting to position the user more
accurately but in a less robust manner. An important consequence of only requiring coarse
location information is that it is not necessary to install a dense collection of access points in
the environment. Nearly all of the WiFi positioning systems described in Section 2.1.2.4 were
tested using many more access points than would have been required just to provide wireless
coverage in the deployment areas. The algorithms developed below were tested using only the
access points that were already installed in the William Gates building for the sole purpose of
providing wireless connectivity.
WiFi-based positioning algorithms can be divided into those based on propagation models
and those based on radio maps. In the latter, a radio map is constructed during an offline
phase prior to the system’s deployment, in which beacon signal strengths are recorded at
positions throughout the environment. A user is positioned by comparing the results of a
WiFi scan performed at the user’s current location to the results stored in the radio map. This
approach has been shown to outperform model-based algorithms [10], however it has the
major drawback that a radio map must be constructed before the system can be used. Until
now this task has been done manually. To do this a human stands at different locations in the
building, manually marking his position on a map and then performing a beacon scan at each
location. This method is very time consuming and is also prone to error, since the user must
estimate their true position each time a scan is performed. Furthermore, the map would have
to be manually updated if new radio beacons were added, or if existing beacons were moved
to new locations. As a result many deployments have been small [12], or restricted to the
corridors of larger buildings [10]. Castro et al. describe map-building as “tedious” [34]. In
one of the few large-scale deployments of an RF-based location system, 28 man-hours were
required to construct a radio map covering a 12, 000 m2 building [35].
One solution to the problem of building radio maps is to use a robot capable of positioning
itself within the environment (e.g. using the robot localisation techniques discussed in Section
2.5) to build one automatically. This approach has been used to acquire radio maps for use
in robot positioning systems [94]. Crowd-sourcing radio maps has also been proposed [95].
An alternative approach that is made possible by the work in this thesis is to automatically
build a radio map whilst a user is tracked by the localisation and tracking system developed
in Chapters 5 and 6.
7.1.1 Automatic radio map construction
Radio maps of WiFi access points can be constructed quickly and easily by a user who is
being tracked by the localisation system described in Chapters 5 and 6. All that is required is
for the user to be equipped with WiFi hardware that is capable of scanning for visible access
142
points and obtaining corresponding received signal strength indication (RSSI) measurements.
Such hardware is provided in a wide range of mobile devices, including the hip-mounted
UMPC that forms part of the prototype system described in this thesis. Whilst the user is
tracked the UMPC’s WiFi hardware is used to repeatedly scan for visible access points and
their corresponding RSSIs. When the particle filter updates the state, the most recent WiFi
scan to have been completed since the last update (if one exists) is incorporated into the radio
map at the user’s estimated location. Note that radio map construction does not start until
after the user has been localised. If environmental symmetry or the size of the environment
prevents localisation, then the user who is building the radio map may manually select their
approximate initial location so that the radio map can be built.
Depending on the WiFi hardware used it may be necessary to take special care to ensure
that the result of each scan does not contain cached results from any previous scans. Such
caching is implemented as a feature by some WiFi device drivers including the MadWifi1
driver for Atheros chipsets such as the one in the test UMPC. The purpose of such caching
is to avoid having to perform a full scan whenever a list of visible access points is requested,
since this requires scanning multiple channels and therefore temporarily breaks connectivity
to any network to which the device is currently associated. For the tests below a modified
version of the MadWifi driver was developed in which the results of previous scans were not
cached2.
The radio map data-structure (and the amount of data stored within it) can be changed ac-
cording to the desired application. The approach used here divides the 2.5-dimensional map
that is used by the localisation filter into cells, which are regions of space within which radio
measurements are grouped. A specified cell size determines the granularity of the radio map.
For each cell a histogram of measured RSSI values is maintained for each access point that
has, at some point, been visible from the cell. A new scan is added to a cell by updating the
histogram for each access point that was sighted in the scan. If an access point is sighted for
the first time in a cell then a new histogram is created.
Two factors should be considered when deciding how to divide a 2.5-dimensional map into
cells:
1. A radio map is most useful for positioning when the variability of radio measurements
within cells is minimised, and variability between cells is maximised. Walls attenuate
radio signals far more than air, therefore it is sensible to ensure that cells do not span
multiple rooms.
2. To get a complete radio map it is necessary to complete WiFi scans in every cell. Some
1http://madwifi-project.org
2The necessary driver modifications were implemented by Ripduman Sohan, to whom the author wishes to
express his thanks.
143
Figure 7.2: The cells generated for a lecture theatre using a cell granularity of 6 m (each
cell is shaded with a different colour). Grid size 1 m2.
rooms such as the lecture theatre shown in Figure 7.2 are made up of many small floor
polygons in the 2.5-dimensional map data-structure (a description of this data-structure
can be found in Section 5.1). Therefore it should be possible for cells to span multiple
floor polygons within a room.
With these factors in mind, a suitable strategy is to subdivide each room in a 2.5-dimensional
map as follows:
1. Compute the 3-dimensional bounding box of each room.
2. Divide the bounding box into the minimum number of identical cuboids such that no
edge is longer than the specified cell granularity.
3. A cell is defined as the fragments of floor polygon that are contained within a single
cuboid.
The Weiler-Atherton clipping algorithm [96] can be used to compute the fragments of each
floor polygon that are contained by each cell, as shown in Figure 7.2.
To construct a complete radio map it is necessary for a user to walk throughout the build-
ing that is being mapped. In tests conducted in the William Gates building this required 2
hours and 28 minutes (split into three sessions) in which the user travelled a total distance of
8.7 km, as shown in Figure 7.3(a). Unfortunately access could not be obtained to a number of
restricted areas (e.g. plant and server rooms), which are shaded red. In total 40 access points
were observed, although seven of these were actually located in neighbouring buildings and
were only sighted from a few cells. The radio map constructed for one access point located
within the William Gates building is shown in Figure 7.3(b). The overall coverage of the radio
map was 99% when using a cell granularity of 5 m, as shown in Table 7.1. In this context cov-
144
(a)
RSSI (dBm)
-50 -70 -90
AP
(b)
Figure 7.3: Building a radio map. (a) Tracks calculated by the localisation system during
radio map construction. Shaded (red) areas indicate rooms that could not be accessed. (b)
The radio map generated for a single access point. Cell colour indicates the mean RSSI
value.
Table 7.1: The map coverage and mean scans per cell at different cell granularities (ex-
cluding restricted areas).
Cell granularity (m) Coverage (%) Mean scans per cell
1 59 1.3
2 85 4.3
3 96 9.14
4 99 12.9
5 99 18.9
erage is defined as the percentage of the total floor area (excluding restricted areas) covered
by cells in which at least one scan was completed. In a more complete system the user could
have been automatically directed to areas that had yet to be mapped whilst moving through the
building (e.g. by an application running on a mobile device), however in this case the radio
map was completed by manually scanning for visible access points from the few remaining
cells.
There are two stages at which WiFi scans together with a radio map can be used to reduce
uncertainty during localisation. The first is when the initial distribution of particles (the prior)
is generated. The second is during the localisation process itself, in which measurements can
be used to weight particles during the filter’s correction step. One way in which WiFi scans
can be used in both of these stages is through the generation of coarse containment regions
within which the user is almost certainly located. The generation and application of such
145
regions is described below.
7.1.2 Prior generation
During initialisation the WiFi hardware is used to scan for visible access points and ob-
tain corresponding RSSI measurements. Let the vector of observed RSSI values be 풛0 =
(ap1, ap2, ..., ap푛), where ap푛 is the RSSI observed for access point 푛. An initial containment
region 푅prior is then constructed as follows.
1. Calculate the normalised Euclidean distance ndist(풛0, 풄푖) between 풛0 and the vector 풄푖
of mean RSSI values for each cell (indexed by 푖) in the radio map. The calculation of
such distances is discussed below.
2. Construct a prior region consisting of all cells for which the calculated distance is under
a threshold 휖 = 9.5 dBm. This threshold was determined experimentally (see Section
7.1.4) to ensure that the user’s true position is almost certainly contained by the region.
Formally the prior region is given by
푅prior =
∪
푖
Cell푖 ∣ ndist(풛0, 풄푖) < 휖 (7.1)
where Cell푖 is the set of polygon fragments that make up the 푖th cell.
During assisted localisation the initial set of particles is generated uniformly across 푅prior
rather than across the entire floor surface of the building (as is the case in unassisted locali-
sation). The particle headings are distributed uniformly in all directions. KLD-adaptation is
used to determine when a sufficient number of particles have been generated, as described in
Section 6.1.2. The number of particles that are generated by this approach is (on average)
proportional to the total area of the polygon fragments that make up 푅prior. Hence the number
of particles in the prior is proportional to the size of the containment region rather than to the
size of the building.
To construct 푅prior it is necessary to calculate distances between the vector of observed RSSI
values and vectors of mean RSSI values stored in the radio map. When calculating such dis-
tances it should not be assumed that the same set of access points will be visible throughout
the building in which the localisation system is deployed. In fact this is usually not the case
in buildings that are large enough such that the use of WiFi could significantly reduce the cost
of localisation. Hence the calculation used to obtain ndist(풛0, 풄푖) values must give sensible
results when 풛0 and 풄푖 are of different lengths and contain measurements corresponding to
different access points. This is a problem rarely addressed in existing literature, since most
experimental deployments of WiFi-based location systems have been small and have implic-
146
itly assumed that all access points are visible from all locations in the environment. Consider
the following scan and cell vectors, in which a question mark indicates that an access point
was not observed and a more negative value indicates a weaker signal:
풛0 = (−82,−73, ?) (7.2)
풄1 = (−84,−75, ?) (7.3)
풄2 = (−84,−75,−65) . (7.4)
One approach might be to compute the distance between 풛0 and a cell using only the access
points for which measurements are present in both vectors. This approach computes equal
distances to both cells, yet intuitively the distance to 풄1 should be smaller, since an access
point that is strongly visible in 풄2 was not observed from the user’s location. A better solution
is to compute the Euclidian distance over all access points, filling in missing readings in both
the cell and the measurement vectors with the weakest RSSI value that can be measured by
the WiFi hardware (−96 dBm for the Atheros chipset in the OQO UMPC). This approach
yields a larger distance to 풄2 as required. Finally, each distance measurement is normalised
by dividing by the number of access points that are visible from the cell according to the radio
map. This makes it possible to compare distance measurements to cells from which different
numbers of access points are visible. Note that this penalises a missing value in the cell vector
more than a missing value in the measurement vector. For example consider the following
vectors and the normalised distance between them:
풛0a = (−82,−73, ?) (7.5)
풄1a = (−84,−75,−75) (7.6)
ndist(풛0a, 풄1a) = 7.1 dBm . (7.7)
If the cell and measurement vectors are switched then a larger distance is obtained:
풛0b = (−84,−75,−75) (7.8)
풄1b = (−82,−73, ?) (7.9)
ndist(풛0b, 풄1b) = 10.6 dBm . (7.10)
This is actually a desirable property, since each cell vector is typically derived from many
measurements (i.e. all of the scans that were performed from within the cell during radio map
construction). If an access point were visible from a cell then you would expect that it would
have been sighted in at least one of them. In contrast the measurement vector 풛0 corresponds
to a single scan during which an access point that it is possible to sight from the user’s location
may have been obscured by the user’s body or some other object in the environment.
147
(a) (b)
Figure 7.4: The result of applying the localisation filter to the same example as in Figure
7.1 when (a) 푅prior (indicated by the shaded region) is used. (b) When both 푅prior and
containment measurements are used (the shaded area indicates the most recently applied
measurement). An arrow indicates the path followed by each cluster.
Figure 7.4(a) shows the result of using 푅prior in the same example as shown in Figure 7.1.
Note that the prior region covers only a small proportion of the total floor area. Since the
number of particles is proportional to the floor area that they cover, the computational cost of
localisation has been reduced. By using the prior the number of distinct clusters is reduced
from twelve to three, since only three of the clusters shown in Figure 7.1 originate from inside
the prior containment region. Hence uncertainty due to symmetry in the environment has also
been reduced. Despite this the use of the푅prior was not sufficient to unambiguously determine
the user’s true location. In particular the use of a prior containment region is not sufficient to
resolve ambiguity between multiple clusters that have different orientations but similar initial
positions (that are all inside the region). Such clusters often arise due to rotational symmetry
in the environment.
7.1.3 Containment measurements
To remove clusters that arise due to rotational symmetry it is possible to generate and apply
containment regions throughout the localisation process. For a filter update at time 푡 a suitable
containment region is given by
푅푡 =
∪
푖
Cell푖 ∣ ndist(풛푡, 풄푖) < 휖 (7.11)
where 풛푡 is the result of the most recently completed WiFi scan since the previous update.
The region is then applied as a containment measurement during the correction step of the
148
particle filter’s update algorithm. The containment measurement corresponding to the region
푅푡 is given by
푝(풛푡∣풙푡) =
{
1 if 풙푡 is in 푅푡
0 otherwise
(7.12)
where 풙푡 is the state of the particle whose weight is being corrected. This sets the weight of
all particles outside of the containment region to zero, causing them to be removed during re-
sampling. Figure 7.4(b) shows the result of using both 푅prior and containment measurements
during localisation. In this example only a single cluster of particles remains, whose position
corresponds to the user’s true location.
7.1.4 Robustness
In the algorithm presented above WiFi signal strength vectors are used to generate contain-
ment regions within which the probability 푝(풛푡∣풙푡) is assumed to be constant. An alternative
(and more conventional) approach would be to compute a probability 푝(풛푡∣풄푖) for each cell
and then apply a measurement with the probability distribution
푝(풛푡∣풙푡) = 푝(풛푡∣풄푖) 풙푡 is in Cell푖 . (7.13)
For each cell the probability distribution 푝(풛푡∣풄푖) could be calculated directly using the signal
strength histograms stored in the radio map. The main advantage of this approach is that
particles can be assigned weights of any value from 0 to 1, whereas containment measurements
can assign weights of 0 and 1 but nothing in-between. This finer granularity makes it possible
for surviving particles to be weighted based on how consistent they are with measurements
received over multiple updates.
Unfortunately this approach also suffers from a major problem: that the distributions 푝(풛푡∣풄푖)
are not typically constant in a real environment. This is because received signal strengths at
a particular location are affected by changes in the local environment such as the opening or
closing of a door [33]. Such a change will introduce a change in the corresponding probability
distribution that will not be reflected in a pre-constructed radio map. The result is a systematic
error in the radio map that can draw the particle cloud away from the user’s true position.
In the worst case this could cause the entire cloud to violate an environmental constraint,
resulting in a localisation failure. Containment regions avoid this problem by assuming that
the distribution is uniform over the entire region. This avoids introducing a bias either towards
the user’s true position (when the radio map is correct) or away from it (when it is not).
One problem that can arise when applying containment regions is that the cluster correspond-
ing to the user’s true position might fall outside of the region and be removed. To ensure that
this is unlikely to occur the threshold 휖 must be set to a sufficiently large value. In particu-
149
lar the threshold should be large enough such that a containment region will still contain the
true position even when systematic errors are present in the radio map due to small changes
in the environment. A suitable threshold was established for the William Gates building by
processing further data obtained using the prototype system that was distinct from the data
that was used to construct the radio map. The new data consisted of walks lasting a total dura-
tion of just under one hour that were performed a week after the radio map was built (during
which small changes in the environment such as the opening and closing of doors would have
occurred). RSSI measurements obtained during the walks were not used to actively aid the
localisation process, but were instead passively compared to the radio map during tracking.
After each step during tracking the normalised distance ndist(풛푡, 풄푖) was calculated to each
cell that overlapped the single cluster of particles calculated by the filter. The cumulative dis-
tribution of the calculated distances is shown in Figure 7.5. The probability of the user being
located outside of a containment region can be estimated from this distribution. For example
if a containment region is generated using a threshold of 휖 = 3.94 dBm then the probability
that it does not contain the user’s true position can be estimated as 0.5. For a threshold of
휖 = 9.5 dBm this probability falls to 0.00028. When using this value no localisations were
found to require more than 74 steps. Hence for the worst case in which 74 steps are required
and a containment measurement is applied during each update, the overall probability of a
failure during localisation can be estimated3 as
1− (1− 0.00028)74 = 0.021 . (7.14)
In an average case requiring 15 steps the probability of failure is just 0.0042. This failure rate
was deemed acceptable and hence a threshold of 휖 = 9.5 dBm was selected for use during
evaluation. When a failure occurs the localisation process can simply be restarted from the
user’s current location. Note that different thresholds may be required for different buildings,
access point topologies and WiFi hardware.
7.1.5 Localisation performance
In order to quantify the benefit of assisted localisation using WiFi RSSI measurements, a
further dataset was obtained as a user walked continuously for 35 minutes throughout the
William Gates building. The localisation filter was applied repeatedly to the data from a
number of randomly selected starting points. In total 50 random starting points were drawn
uniformly from the first 30 minutes of the walk. Starting points were not drawn from the final
five minutes in order to ensure that there was always sufficient remaining data to allow the
user to be localised. From each starting point the filter was applied three times: once without
3It is assumed that the probability of failure is independent for each containment region. This is almost
certainly not the case, however it is assumed in order to obtain an indicative measure.
150
 0
 0.2
 0.4
 0.6
 0.8
 1
 0  2  4  6  8  10  12
C
D
F
Normalised Euclidean distance (dBm)
Figure 7.5: The cumulative distribution of ndist(풛푡, 풄푖) for cells overlapped by a cluster of
particles corresponding to the user’s true location.
 0
 2
 4
 6
 8
 10
 12
 14
 0  20  40  60  80  100
N
o.
 P
ar
tic
le
s
No. Steps
Unassisted
WiFi-Prior
WiFi-Throughout
(a)
 0
 2
 4
 6
 8
 10
 12
 0  20  40  60  80  100
N
o.
 C
lu
st
er
s
No. Steps
Unassisted
WiFi-Prior
WiFi-Throughout
(b)
Figure 7.6: The data gathered for a single localisation without using the radio map, using
푅prior only, and using both 푅prior and containment measurements. (a) The number of
particles generated at each step. (b) The number of clusters at each step (clustering is only
performed when the number of particles is below 8, 000).
151
Table 7.2: A summary of WiFi-assisted localisation performance in the William Gates
building (excluding failed runs).
Radio map use Mean Std-dev Max Min
Particles in
prior
None 1, 271, 300 9, 124 1, 280, 781 1, 252, 106
Prior only 191, 528 118, 025 459, 514 45, 649
Throughout 191, 457 117, 951 459, 886 45, 963
Steps until
localisation
None 71 27.8 146 24
Prior only 49 24.3 140 9
Throughout 34 14.4 74 7
using the radio map, once using 푅prior to constrain the user’s initial position and once using
both 푅prior and containment measurements throughout the localisation process. Particle and
cluster counts were recorded throughout each run.
The results obtained for a single starting point are shown in Figure 7.6. In this example the
prior contained 1, 277, 890 particles and 82 steps were required to localise the user without
WiFi assistance. When 푅prior was used to constrain the initial location of the user 139, 626
particles were generated and 46 steps were required to determine the user’s location. When
containment measurements were used throughout only 30 steps were required.
Table 7.2 summarises the results that were obtained4. These results exclude one failed run
in each scenario and an additional failed run when containment measurements were used
throughout. One starting position was responsible for a failure in each of the scenarios. This
case was a particular difficult one in which the user passed at a tight angle through a narrow
doorway early during the localisation process. For this unusual case the number of particles
determined by the adaptive scheme was not sufficient. The additional failure when contain-
ment measurements were used throughout was caused by an incorrect containment measure-
ment (i.e. one whose containment region did not contain the user’s true position).
As expected the use of푅prior significantly reduced the number of particles in the priors. There
was however a large variation between the number of particles generated in different runs, with
runs from one starting point generating in the order of 460, 000 particles. This is explained
by the presence of large open plan areas in the William Gates building, within which radio
signals propagate far more freely relative to propagation through areas containing many walls.
When the user is located in such an area the containment region obtained using WiFi signal
strengths is likely to be large, resulting in many particles. The average number of steps taken
by the user before localisation was also reduced by using the constrained prior, however the
worst case of 146 steps was not significantly reduced. Hence 푅prior alone was not always
sufficient to reduce the problem of environmental symmetry.
4The particle counts given here are significantly lower than those presented in previous work [97]. This
is because the results in [97] were obtained using a version of the localisation filter in which KLD-adaptation
parameters had not been tuned to use as few particles as possible (see Section 6.1.3).
152
Using containment measurements in addition to 푅prior reduced the average number of steps
required to locate the user to 34. The worst case was reduced to 74 steps. This demon-
strates that the use of containment measurements helps to resolve ambiguities that arise due
to symmetry in the environment. This is achieved at the cost of the additional computational
overhead required to generate and apply such measurements.
The cost of generating a containment measurement is 풪(퐴) where 퐴 is the floor area of
the building, since generating a containment region requires the calculation of a distance to
each cell in the radio map. Fortunately these computations are very cheap. For the William
Gates building 99% of the generated containment regions were constructed in under 5 ms in
a Java implementation running on a 2.6 GHz desktop PC, which is negligible relative to the
time required to update the numbers of particles that are used during localisation (see Section
5.3.2). The weight of a particle is updated during the correction step by first finding the cell in
the radio map within which it is located and then multiplying its weight by 0 or 1 depending
on whether that cell is part of the containment region. This requires풪(1) time5 and in practice
the overhead is negligible relative to the time required to update the particle’s state. Hence
the overhead of generating and applying containment regions is small even for large buildings
and is nearly always worthwhile.
7.2 Mapping other signals
The techniques developed for WiFi-assisted localisation in Section 7.1 can be applied to other
radio signals. The only requirement is that the measurable signal properties (e.g. RSSI in the
case of WiFi) must be both sufficiently stationary (meaning that similar measurements will
be observed from a given location at different points in time) and vary sufficiently across the
building so that they can be used to constrain the user’s location.
The offline construction and subsequent use of maps to assist the localisation filter is not
only applicable to radio signals. In fact it is possible to construct a map of any property that
can be sensed by devices that are attached to the user. Such a map can then be used to assist
during localisation provided that the sensed property satisfies the two requirements mentioned
above. One particularly interesting example is to map magnetic disturbances throughout the
environment, since many commercial IMUs already contain magnetometers that are able to
sense such disturbances. It was established in Section 4.3 that the magnitude and frequency
of such disturbances make it impossible to determine the user’s heading in the William Gates
building using the magnetometers in a foot-mounted IMU. This property is actually beneficial
when using a pre-constructed map of magnetic disturbances, since it means that there will be
many significant features in the map that can be compared to those currently sensed at the
5The radio cell can be determined in 풪(1) time due to the reference poly푡 in the state vector of the particle.
153
0 1
Magnetic field strength
deviation (arbitrary units)
(a)
Metallic floor panels & Carpet
Granite Ceramic tiles Wood
Unmapped
(b)
Figure 7.7: Mapping magnetic field deviations on the ground floor of the William Gates
building. (a) A map of the mean within-step deviation in magnetic field strength, scaled
such that the maximum calculated value is equal to one. (b) Different floor surface materials
in the building.
user’s location. Furthermore the sources of such features are often static, for example the
metallic floor panels installed throughout large portions of the William Gates building.
Figure 7.7 demonstrates a very simple approach to mapping deviations in magnetic field
strength throughout the ground floor of the William Gates building. During each step the
magnetic field strength was sampled at 100 Hz using the magnetometers contained within the
foot-mounted IMU. At the end of a step the standard deviation of the magnetic field strengths
recorded during that step was calculated and inserted into the map. A large deviation indicates
that the user’s foot passed through a significant magnetic disturbance during the step. Figure
7.7(a) shows the mean deviations recorded in each cell of the map. It is interesting to note
that there is a correlation between the mapped deviations and the use of different floor surface
materials in different parts of the building, as shown in Figure 7.7(b). Large deviations were
recorded in areas where metallic floor panels were used, whereas far smaller deviations were
recorded in parts of the building containing stone, wood and ceramic floor surfaces. Thus
even this simple approach may be sufficient to determine whether the user is currently in a
region of the building in which metallic floor panels are installed. A more sophisticated ap-
proach may allow the user to be further constrained, for example by mapping the direction
of the magnetic field throughout the building. The development of such approaches and the
application of magnetic field maps during localisation is left as future work.
154
7.3 Assisted tracking
Although this chapter has focused on the use of additional measurements during localisation,
such measurements can also be applied during tracking. For example containment measure-
ments generated from WiFi scans can be applied throughout the tracking phase as well as
during the initial localisation phase. Although this is possible, it is rarely beneficial. For
example WiFi containment measurements will not improve the accuracy of the positions cal-
culated during tracking due to their coarse size and uniform nature. Furthermore a tracking
failure may occur whenever a containment measurement is applied, since there is a small but
finite chance that the user’s true position will not be contained by the corresponding contain-
ment region. Hence the application of containment measurements during tracking is actually
counter-productive.
In some cases additional measurements can be applied in order to improve accuracy dur-
ing tracking. For example position measurements from the Active Bat system could be used
whenever a user passes through the wing of the William Gates building in which it is deployed.
Although this is possible, it usually makes more sense to use entirely different tracking algo-
rithms when measurements from relatively accurate positioning systems are available. In this
example an INS together with the error-state Kalman filter that was developed as a ground
truth in Section 3.4 can be used to track the user’s position more efficiently, more accurately
and at a much higher frequency when in the area of the building that is covered by the Active
Bat system. When the user exits this area the localisation filter could then resume tracking,
with an initial set of particles that are clustered around the last position to be calculated by the
INS.
7.4 Conclusions
A wide range of approaches can be used to assist the process of global localisation. All such
approaches require some form of additional information to be provided to the localisation filter
that can be used to reduce uncertainty in the user’s state. There are a range of sources from
which such information can be obtained including manual input, additional on-body sensors
and measurements obtained from sensors in the environment. The most suitable approach is
application dependent. When it is not possible to access a building in advance the user could
be asked to provide their approximate position at the start of localisation. When prior access
is available it is often possible to map signals that propagate through the environment, such as
those emitted by WiFi access points. One of the major advantages of the localisation system
developed in this thesis is that it can be used to construct such maps automatically whilst the
user is tracked moving through the environment.
155
156
Chapter 8
Conclusions
8.1 Research contributions
This thesis has demonstrated that it is possible to track the absolute position of a pedestrian
in an indoor environment without relying on any fixed infrastructure. This has been achieved
by combining pedestrian dead reckoning (PDR) techniques with ideas from the related field
of robot localisation.
The pedestrian localisation and tracking system that has been developed in this thesis consists
of a micro-machined electromechanical systems (MEMS) inertial measurement unit (IMU)
and two Bayesian filters. The first of these is a PDR filter, consisting of an error-state Kalman
filter and an inertial navigation system (INS). This filter processes measurements obtained
from the IMU in order to track the relative movement of the user. Drift errors that would
normally accumulate rapidly in the tracked position are greatly reduced by the application of
zero velocity updates (ZVUs). This approach is made possible by mounting the IMU on the
user’s foot, which is known to be stationary whenever it is grounded. The PDR filter segments
the user’s relative movement into step events that are passed to a second Bayesian filter.
The second Bayesian filter is a localisation particle filter that combines step events calculated
by the PDR filter with a map of the environment. A 2.5-dimensional map data-structure was
developed to provide efficient support for multi-storey buildings, which were not supported
by existing robot localisation algorithms. By enforcing environmental constraints that are
defined by a 2.5-dimensional map, the localisation filter is able to narrow down the position
of a pedestrian as they move through the environment. Once the particles have converged to
form a single cluster around the user’s position, the same filter is then able to track the user’s
absolute position without becoming less accurate over time.
A number of techniques have been presented in this thesis that improve the performance of the
157
basic localisation and tracking system. The most important of these is the dynamic adaptation
of the number of particles used by the localisation filter based on the amount of uncertainty in
the user’s position. This has the effect of automatically reducing the computational require-
ments of the filter as the user’s position becomes more certain. The use of furniture constraints
has also been shown to improve tracking accuracy in open plan environments. Finally, addi-
tional sensors can be used to reduce the localisation filter’s computational requirements and
improve the speed with which it can determine the user’s position.
The purpose of developing the pedestrian localisation and tracking system outlined above was
to answer the research questions posed in Section 1.1. These questions are repeated below
and are addressed in the following sections, which also summarise the main contributions and
conclusions of this thesis.
1. How rapidly does drift accumulate in an inertial navigation system when using current
state-of-the-art MEMS inertial sensors?
2. How accurately can the relative movement of a pedestrian be measured using MEMS
inertial sensors?
3. What is the minimum infrastructure or knowledge required to determine a pedestrian’s
absolute location and heading?
4. Can environmental constraints be used to prevent the accumulation of drift over time?
8.1.1 Research question 1: Unconstrained inertial navigation
One of the most important conclusions of this work is that unconstrained inertial navigation
is an unsuitable technique for tracking the relative movements of a pedestrian given current
inertial sensor technologies. Chapter 3 showed that with a current state-of-the-art MEMS
IMU, drift accumulates in the estimated position that typically exceeds 170 m after one minute
of tracking. Such drift arises due to measurement errors incurred by each of the individual
accelerometers and gyroscopes that are contained within an IMU. These individual errors
propagate through the inertial navigation system equations, causing drift in position that grows
proportional to 푡2 in the short term (when the accelerometers are the primary cause of drift)
and 푡3 in the long term (when the gyroscopes are the dominant error source), where 푡 is the
period of time for which the user is tracked. This rapid error growth means that even big
improvements in the quality of MEMS inertial sensors will only result in modest increases
in the duration for which a user can be tracked before the error in position exceeds a given
threshold.
Given that many indoor location-aware applications require sub-metre accuracy (so that they
can determine whether a person is in close enough proximity to another object for a physical
158
interaction to take place) for extended periods of time, it is expected that the use of uncon-
strained inertial navigation will remain infeasible for indoor pedestrian tracking for the fore-
seeable future. If it is to become feasible it will be necessary not only to develop IMUs that
are many orders of magnitude more accurate than those of today, but also new technologies
for modelling and compensating for local gravitational variations.
8.1.2 Research question 2: Constrained pedestrian dead reckoning
Due to the limitations of unconstrained inertial navigation it is necessary to seek additional
constraints or external measurements that can be used to correct an INS that is used to track
people or objects in an indoor environment. In Chapter 3 a general purpose error-state Kalman
filter was developed for enforcing such constraints. This approach is ideal due to the Kalman
filter’s computational efficiency and its ability to track correlations between growing errors in
different components of the state of an INS. By tracking such correlations the filter is able to
correct components of the state that are not directly observed by measurements. For example
it is able to correct errors in tilt, velocity and displacement when a measurement of the IMU’s
velocity is received. More generally Bayesian filters (of which the Kalman filter is a specific
type) allow uncertainty in each measurement to be taken into account. Within the Bayesian
filter framework a constraint can be enforced by generating a pseudo-measurement; a fake
measurement that specifies the nature of the constraint.
In Chapter 4 an INS was used together with the error-state filter to construct a PDR filter
capable of tracking the relative movements of a pedestrian equipped with a foot-mounted
MEMS IMU. This sensor placement allowed the application of ZVU pseudo-measurements
whenever the user’s foot was known to be in contact with the floor. In one example it was
shown that by using this approach a pedestrian’s relative movements could be tracked to within
3.97 m over a period of just under nine minutes, compared to an error of 43.15 km when using
unconstrained inertial navigation.
It is important to note that the PDR filter is more specialised than general purpose inertial
navigation algorithms. In particular it requires the user to be walking and the IMU to be
mounted on the user’s foot. As the quality of IMUs continues to improve it may be possible
to make use of more generic constraints and thus develop less specialised algorithms. Such
algorithms may be more generally applicable, tracking arbitrary objects rather than being re-
stricted to pedestrians. Furthermore such algorithms will still be suitable for providing relative
movement events to a localisation particle filter such as the one developed in Chapter 5.
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8.1.3 Research question 3: Determining absolute location and heading
The third research question posed at the beginning of this thesis asked what minimal infras-
tructure or knowledge was required to determine the absolute location and heading of a person
in an indoor environment. Determining this state is necessary if relative movement measure-
ments obtained from a PDR filter are to be used to track a user’s absolute position rather than
their position relative to an unknown initial starting point and orientation. Chapter 5 demon-
strated that particle filters could be used to infer the absolute location and heading of a person
given only relative movement measurements and a map of the building. The map provides
constraints such as walls and floors that naturally limit the movement of a person in an in-
door environment. By enforcing these constraints a particle filter is able to narrow down the
location of a user as they move through the environment, until their position can be uniquely
determined. This is known as pedestrian localisation.
The localisation algorithms developed in Chapters 5 and 6 demonstrate that it is possible to
determine a user’s absolute location and heading without the need for any fixed infrastructure,
however there are a number of caveats. Firstly the user is required to move in order for their
location to be narrowed down. Hence it is impossible to locate a stationary pedestrian using
this approach. Secondly the localisation process is computationally expensive, since a large
number of particles are required to represent all of the possible positions and orientations of
the user. Thirdly environmental symmetry exhibited by the environment may delay or prevent
the user’s state from being determined. This is particularly problematic for high rise buildings
in which many of the floors have very similar layouts.
To address the problems outlined above the user’s initial state can be approximated (or at least
bounded to some region of space) using additional measurements. This is known as assisted
localisation. Possible approaches for approximating the user’s initial location include the use
of sparsely deployed beacons such as WiFi access points, using the last location estimate
calculated by an outdoor positioning system as a user moves into an indoor environment and
asking the user to enter their approximate position manually. Chapter 7 showed that by using
WiFi measurements in the William Gates building the average number of particles required
during the first filter update could be reduced from 1, 271, 300 to 191, 500, thus reducing both
the computational and memory requirements of the localisation process. The average number
of steps taken by the user before their position was determined was also reduced from 71 to
34.
8.1.4 Research question 4: Use of environmental constraints
The main contribution of this work has been to demonstrate that it is possible to use envi-
ronmental constraints to prevent the accumulation of drift that would otherwise occur when
160
tracking a pedestrian using inertial navigation techniques. The approach demonstrated in
Chapters 5 and 6 used a particle filter to continually enforce constraints such as walls and
floors in order to maintain a drift free estimate of the user’s state. The computational and
memory costs are low relative to the cost of global localisation, since far fewer particles are
required. It was shown that the absolute position of a pedestrian could be tracked indefinitely
to within 0.62 m 95% of the time in a typical office environment. This level of accuracy is
sufficient for location-aware applications that need to be able to infer when a user is standing
close enough to an object for a physical interaction to take place (e.g. when a user is close
enough to a workstation to be using it).
It is important to note that the accuracy of the localisation and tracking system developed in
this thesis is dependent on the density of environmental constraints. In areas where there are
few constraints the tracking accuracy is worse than in areas where there are many constraints.
For open plan offices furniture such as desks and cubicle dividers can be used as additional
constraints in order to improve tracking performance. OpenRoomMap was developed as a
way of crowd-sourcing a world model of such constraints in the William Gates building. When
using the furniture constraints obtained through OpenRoomMap it was possible to position a
user to within 0.94 m 95% of the time in a simulated open plan environment, compared to
1.58 m 95% of the time when furniture constraints were not used.
8.2 Future work
Pedestrian localisation and tracking in indoor environments is still far from being a solved
problem. There are many avenues of future work that must be explored before complete
systems based on the algorithms developed in this thesis can be deployed. These avenues
can be divided into two broad categories: enhancements to the algorithms themselves and
determining how such algorithms can be deployed in real-world systems. Further work that
has been identified in both of these areas is outlined below.
8.2.1 Algorithmic enhancements
There are numerous possible enhancements that can be made to the data-structures and al-
gorithms presented in this thesis, ranging from minor performance improvements to major
changes that add additional functionality. In the latter case generalisation to support moving
floor surfaces such as those found in elevators is of particular interest. One way in which
this can be achieved is outlined in Section 8.2.1.1. A list of more minor topics and possible
enhancements is given in Section 8.2.1.2.
161
8.2.1.1 Moving floor surfaces
Large buildings often contain areas in which there are moving floor surfaces on which a pedes-
trian may stand. Examples of such areas include elevators, escalators and travelators. It is
necessary to make a number of modifications to the algorithms developed in this thesis in
order to support such cases. One possible approach is outlined below, for which an extremely
simple proof of concept implementation has been developed.
PDR filter generalisation The PDR filter developed in Chapter 4 assumes that the user’s
foot has a velocity of zero whenever it is grounded. This assumption is violated by moving
floor surfaces and hence it is necessary to extend such a filter to be able to detect and adapt to
situations in which a pedestrian walks onto a moving floor surface. One crude approach could
be to look for unexpected changes in velocity. Moving floor surfaces usually have velocities
that are far greater than the drift in velocity that can occur during a short period of time.
Hence it should be possible to reliably disambiguate between drift errors and cases in which
a pedestrian steps onto a moving floor surface. When such a transition occurs zero velocity
constraints should be applied by simply resetting the velocity component of the filter to zero
(so as to avoid incorrectly correcting other components of the state such as the orientation and
displacement), after which the PDR filter will track the user’s relative movement in a moving
frame of reference corresponding to the floor surface.
Map generalisation The 2.5-dimensional map data-structure must be extended so that it is
possible to represent areas in which there are moving floor surfaces. Rather than consisting
of connected of floor polygons, an extended 2.5-dimensional map could consist of connected
spatial zones of different types. For the majority of the building a floor polygon type could
be used, with other types used to represent areas containing moving surfaces. For example an
elevator zone could be defined by a polygon representing the base of the elevator shaft and
a list of heights at which the elevator can stop, each with a connection from an edge of the
elevator shaft into another spatial zone at that height.
Localisation filter generalisation Rather than being constrained to lie on a floor polygon,
each particle in an extended localisation filter could be constrained to lie within a spatial zone.
Different motion models could be applied to the particles in zones of different types to model
the movement of floor surfaces. For example particles in an elevator zone could have their
vertical displacements periodically updated in order to model the possible movement of the
elevator. It may also be necessary to augment the state of each particle in such zones. For
example particles in an elevator could have their states extended to model the state of the
elevator (i.e. whether it is stationary, going up or going down).
162
start
Figure 8.1: An example path calculated using a prototype extended localisation filter. The
two elevator shafts are shaded (red). Grid size = 10 m2.
Figure 8.1 shows the result of tracking a pedestrian using a prototype version of the locali-
sation and tracking system that was extended to support the elevators in the William Gates
building. The user was correctly tracked as he descended two floors in an elevator and later
as he ascended one floor to his final position. In this prototype, the PDR filter was modified
to detect unusual changes in the IMU’s vertical velocity. On detecting such a change, zero
velocity constraints were applied simply by resetting the velocity of the underlying INS to
zero. At the same time an additional measurement was generated to indicate the change to
the localisation filter. The 2.5-dimensional map was extended to include elevator zones as
described above. The state of each particle was augmented on entering an elevator zone, with
the initial augmented state always set to indicate that the elevator was stationary. Particles had
their vertical displacements periodically updated when in the elevator zones based on their
augmented states, which were updated based on the additional measurements reported by the
PDR filter. Although this is only a crude demonstration, it does at least show that the approach
described is a feasible way in which moving floor surfaces can be supported. Far more work
is required in the future to further develop the ideas outlined above into a robust solution that
can be applied to moving floor surfaces of different types.
8.2.1.2 Additional algorithmic enhancements
There are many other topics of research and possible algorithmic enhancements that could be
investigated in the future. Note that the following list is by no means exhaustive.
163
∙ It has been established that the performance of the system developed in this thesis is
highly dependent on the environment (i.e. the amount of symmetry exhibited by the
environment as well as the density of environmental constraints). It will be necessary
to evaluate such systems in a large number of different environments in order to inves-
tigate these dependencies in more detail and to identify any cases that are particularly
problematic.
∙ Variations on the standard bootstrap particle filter may yield minor performance im-
provements. For example the clustered particle filter described in Section 6.2.1 is likely
to reduce the probability of localisation failures in some environments.
∙ There are many ways to assist the localisation process in addition to the use of WiFi
access points, some of which are described at the start of Chapter 7. Many mobile
phones are now equipped with GPS receivers that are able to track the position of users
when they are outdoors. Hence it is of particular interest to develop systems that are able
to seamlessly transition from GPS to indoor positioning. For example the last position
calculated using GPS prior to a user entering a building could be used to determine the
entrances into which the user could have walked. A localisation filter could then be
initialised with particle clouds generated only in the vicinity of these entrances.
∙ In addition to GPS receivers, many mobile phones now contain MEMS inertial sensors.
Thus it is desirable to develop PDR algorithms that are able to use measurements made
by these sensors. Such algorithms should ideally work correctly when user carries their
phone in different ways, for example clipped to their belt or in their pocket, hand or bag.
Although algorithms have already been developed for this purpose they are significantly
less accurate compared to the foot mounted approach. For example the foot mounted
approach has been shown to accumulate drift in the order of 0.3% of the total distance
travelled (see Section 2.2.2.2), whereas even algorithms that require a relatively station-
ary mounting point on the user’s body (as might be the case when using sensors in a
phone clipped to a user’s belt) typically incur positioning errors that are closer to 3% of
the travelled distance (see Section 2.2.2.1). Attempts to perform dead reckoning using
sensors in the pocket have proved even more error prone [48]. The development of bet-
ter dead reckoning algorithms that do not require sensors to be mounted on the user’s
foot may be made possible by further improvements in the quality of MEMS inertial
sensors.
∙ Probabilistic constraints can be used to represent uncertainty in the model of the en-
vironment. Such an approach may be useful for representing constraints imposed by
movable objects such as desks, as described in Section 6.3.3. Further work is needed to
investigate the extent to which they can be used. One particularly interesting question is
whether it is possible to automatically vary the probability of constraints based on their
164
interactions with the particle cloud. For example the probability of a constraint could
be automatically reduced whenever it is responsible for a tracking failure, since such a
failure indicates that the constraint may not be present in the physical environment.
∙ There are many other data-structures in addition to the 2.5-dimensional map that can
be used to represent indoor environments. Of particular interest are graph-based repre-
sentations such as Voronoi diagrams. Such representations may allow the development
of localisation and tracking algorithms that require an order of magnitude less com-
putation, whilst incurring little (if any) loss in tracking accuracy. The use of Voronoi
diagrams was discussed in more detail in Section 6.3.4.1.
∙ Commercial mapping companies are beginning to map indoor environments, however
it is unrealistic to expect that such maps will be kept permanently up to date. This poses
a problem for the localisation algorithms developed in Chapters 5 and 6, which rely on
the availability of an accurate model of the building. A crowd-sourcing approach such
as the one used by OpenRoomMap may allow maps to be updated more frequently than
is viable for commercial mapping companies. Probabilistic constraints may also offer a
partial solution to deal with uncertainty in the map itself. Nevertheless the construction
and maintenance of world models in general remains an open research topic that is
particularly relevant to pedestrian localisation and tracking.
8.2.2 Real-world deployment
A major question that must be addressed in the future is how algorithms such as the ones
developed in this thesis might be deployed in real-world positioning systems. When designing
and deploying such systems it is necessary to consider many practical constraints, such as
the limited computational resources and battery lives of mobile devices. It is also necessary
to consider user concerns such as location privacy and the inconvenience of wearing any
device necessary for the system to function. Finally, real-world positioning systems must
support multiple users. This can be achieved by executing multiple instances of the algorithms
developed in this thesis, however it may be possible to develop algorithms that are optimised
for locating and tracking multiple users.
8.2.2.1 Limited power and computational resources
Mobile devices that can be carried by pedestrians are far less computationally powerful than
servers and desktop PCs. The electrical power that a mobile device has available is also lim-
ited by current battery technology. In a practical implementation of a pedestrian localisation
and tracking system it is necessary to power the sensors, the hardware used to execute the
165
necessary algorithms and in some cases a radio to communicate with other systems (e.g. to
publish the user’s current position).
Current high end IMUs have power requirements that make them unsuitable for continuous
use in mobile devices. For example the Xsens Mtx IMU requires 350 mW. A typical mobile
telephone battery is currently rated at 1400 mAh at 3.7 V1. Such a battery would be exhausted
after less than 15 hours by the IMU alone. Hence it will be necessary to investigate power
management strategies (e.g. turning off the IMU when the user is stationary) as well as devel-
oping more power efficient inertial sensors and higher capacity batteries (both of which are
current industry trends).
Whilst most absolute positioning algorithms perform a near constant amount of computation
per update, localisation algorithms such as the one described in Chapters 5 and 6 require far
more computation (and memory) early during the localisation process relative to when the
user’s position has been determined (i.e. during tracking). Although it is unlikely that real-
world positioning systems will ever perform global localisation, they may well perform some
form of assisted localisation that is still far more expensive than tracking. As a result any
mobile device capable of performing such a task would be vastly underutilised during the
majority of time during which the user is being tracked.
An interesting alternative to performing all of the computation on a mobile device is to con-
sider offloading computationally expensive operations. For example a mobile device could
offload step events to a service provider, whose servers are running a localisation filter on the
user’s behalf and returning the most likely position following each update. There are various
trade-offs associated with such an approach. It may well be desirable to offload computation
during the early stages of localisation if the power saved by the mobile device by performing
less computation outweighs the additional power required to communicate with the server. If
computation is offloaded then it is also the case that the device itself does not need to be able
to perform computationally expensive localisation tasks. Although offloading computation
during localisation may well save power on a mobile device, the opposite may be true during
tracking because far less computation is required to perform each update. In this case the ad-
ditional power required to communicate with the server may be larger than the power saving
that can be achieved by reducing the amount of computation. Hence it may be optimal to mi-
grate a localisation filter from a server to a mobile device at some point during the localisation
process.
It is clear that there are many trade-offs involved when dealing with limited electrical power
and computational resources. Quantifying these trade-offs is an important area of future re-
search that needs to be investigated before practical systems can be developed based on the
algorithms in this thesis.
1This figure is the specified rating of the battery in the recently released HTC Nexus 1.
166
8.2.2.2 User concerns
When deploying practical pedestrian tracking systems it is necessary to consider the concerns
of potential users, such as their location privacy and the inconvenience of wearing on-body
sensors.
Since the algorithms developed in this thesis use only on-body sensors, it is possible for a user
to protect their location privacy by running a localisation filter on their own mobile device.
Note however that care must be taken to avoid leaking their location in unintended ways.
For example a localisation filter running on a mobile device may leak the user’s location if it
requests building plan data from a map provider as it is needed.
For many users, wearing on-body sensors is a major inconvenience. One solution might be to
build such sensors directly into devices that are already carried by many users. For example
foot-mounted sensors could be embedded into the soles of shoes. Inertial sensors are also
being embedded within many of today’s mobile telephones.
8.2.2.3 Supporting multiple users
The simplest application of the algorithms in this thesis to multiple users is to run a separate
localisation filter for each individual. Even when using this approach it may be possible to
reduce the cost of tracking each user by sharing resources. In particular consider a localisation
and tracking system in which a service provider runs a localisation filter for each user. In
normal operation it is likely that at a given time the majority of filters will be tracking users’
positions whilst only a few will be performing localisation tasks. Hence the service provider
should not need to provision sufficient computational resources to allow every user to execute
localisation tasks simultaneously. Further work is needed to establish the extent to which
computational resources can be shared and distributed among many users.
In addition to the sharing of computational resources, there are a number of other areas that
could be researched further when considering multiple users. For example radio maps for
assisted localisation could be built, updated and shared between multiple users. Proximity to
other users whose positions are already known could also be used in order to speed up the
localisation process in systems supporting multiple users.
8.3 Final words
MEMS inertial sensors can be used as an alternative means of tracking people in indoor en-
vironments. It is possible to both determine and subsequently track the absolute position of
167
a pedestrian by combining relative movement information obtained from such sensors with
knowledge of environmental constraints provided in the form of a map. Such an approach
does not rely on any fixed infrastructure and hence can be used in situations where it is im-
practical or impossible to install such infrastructure into the environment in advance. Further-
more this approach has been shown to track users to sub-metre accuracies in a typical office
environment, which is sufficient for many location-aware applications.
The feasibility of applying the techniques presented in this thesis will increase as the quality
of MEMS inertial sensors continues to improve, detailed maps of indoor environments be-
come more commonly available and the computational power of mobile devices continues to
increase. Although it is likely that some form of constraints or external measurements will
always be required to track pedestrians using inertial sensors, it may be possible to use more
general constraints as MEMS inertial sensors continue to improve. In particular it may be
possible to relax the requirement for a sensor to be worn on the user’s foot.
168
Appendix A
Analysing noise in MEMS inertial sensors
Section 3.1 describes a number of noise processes that perturb signals from micro-machined
electromechanical systems (MEMS) inertial sensors. This appendix describes a technique
known as Allan Variance that can be used to detect and quantify such processes. Allan Vari-
ance is a time domain analysis technique originally designed for characterising noise and
stability in clock systems. The technique can be applied to any time-series (a sequence of
data-points measured at times separated by uniform intervals) to determine the characteris-
tics of its underlying noise processes. The Allan Variance of a time-series is a function of
averaging time 푡. For a given 푡, the Allan Variance is calculated as follows:
1. Take a long sequence of data and divide it into bins of length 푡. There must be enough
data for at least nine bins, otherwise the results obtained begin to lose their significance
[98].
2. Average the data in each bin to obtain a list of averages (푎(푡)1, 푎(푡)2, ..., 푎(푡)푛), where
푛 is the number of bins.
3. The Allan Variance is then given by
AVAR(푡) =
1
2 ⋅ (푛− 1)
푛−1∑
푖=1
(푎(푡)푖+1 − 푎(푡)푖)2 . (A.1)
To determine the characteristics of the underlying noise processes, the Allan Deviation
AD(푡) =
√
AVAR(푡) (A.2)
is plotted as a function of 푡 on a log-log scale. Different types of random process cause
slopes with different gradients to appear on the plot, as shown in Figure A.1. Furthermore
different processes usually appear in different regions of 푡, allowing their presence to be easily
169
m = -1
m = -0.5
m = 0
m = 
0.5
m 
= 1
Quantisation
Noise
White Noise / Random Walk
Bias Instability Ra
te Ra
ndom
 Wal
k
Ra
te 
Ra
mpCorrelated
Noise Sinusoidal
Averaging Time  t
A
lla
n 
D
ev
ia
tio
n 
 A
D
(t)
Figure A.1: A possible log-log plot of Allan Deviation [99].
identified. Having identified a process it is then possible to read its numerical parameters
directly from the plot. For a MEMS inertial measurement unit (IMU) such as the Xsens Mtx
the most significant noise processes are white noise processes that cause angle-random-walk
(ARW) and velocity-random-walk (VRW) errors in integrated gyroscope and accelerometer
signals respectively, and bias instability processes that cause the signal biases to wander over
time. These processes can be identified and read from the Allan Deviation plot as follows:
∙ A white noise process appears on the Allan Deviation plot as a slope with gradient−0.5.
The corresponding root power spectral density (PSD) 푛 is read directly from the graph
by fitting a straight line through the slope and reading its value at 푡 = 1.
∙ A bias instability process appears on the plot as a flat region around the minimum. Its
numerical value is the minimum value on the Allan Deviation curve.
A full description of the Allan Variance technique and how it can be used to identify and
quantify other random processes can be found in [99].
A.1 Xsens Mtx analysis
Figure A.2 shows the Allan Deviation curves for the 푥, 푦 and 푧 axis gyroscopes of an Xsens
Mtx. The plot was generated from a 12 hour log of measurement data gathered from the
IMU whilst it was kept stationary. The sampling frequency of the device was set to 100 Hz.
The slopes on the left side of the plot have gradients that are almost exactly equal to −0.5,
indicating the presence of white noise as expected. The numerical values read from the plot
for both white noise and bias instability processes are listed in Table A.1.
Figure A.3 shows the Allan Deviation curves for the three accelerometers in the Mtx. The
푧-axis accelerometer is perturbed by a white noise process that is almost twice that of the
other two accelerometers. Initially it was thought that this may be due to a gravitational
170
10-3
10-2
10-1
100
 0.1  1  10  100  1000  10000
A
lla
n 
de
vi
at
io
n 
(d
eg
/s
)
Averaging time (s)
Gyro X
Gyro Y
Gyro Z
Figure A.2: Allan Deviation curves for the gyroscopes of an Xsens Mtx.
Bias instability Random noise 푛 (root PSD)
X Axis 0.010∘/s (at 620 s) 0.075∘/s/
√
Hz
Y Axis 0.009∘/s (at 530 s) 0.078∘/s/
√
Hz
Z Axis 0.012∘/s (at 270 s) 0.079∘/s/
√
Hz
Table A.1: Gyroscope noise measurements for the Xsens Mtx.
effect (since the log data was obtained with the sensor’s 푧-axis pointing upwards in the global
frame-of-reference), however the same curve was obtained when the analysis was repeated
on data obtained with the 푧-axis aligned horizontally. Hence it was concluded that the 푧-axis
accelerometer was for some reason noisier than the other accelerometers in the tested device.
Table A.2 lists the numerical values read from the Allan Deviation plot.
A.2 Conversion to sigma values
When modelling or simulating angle or velocity random walks and bias instability effects
it is usually necessary to convert noise measurements into the 1휎 values of their underlying
noise processes. The most common situation where this is necessary is when modelling such
processes in a Kalman filter, but it is also necessary when simulating noise in an INS simulator
[79].
The standard deviation of the random noise perturbing each measurement is obtained from
171
10-4
10-3
10-2
 0.1  1  10  100  1000  10000
A
lla
n 
de
vi
at
io
n 
(m
/s
2 )
Averaging time (s)
Accel X
Accel Y
Accel Z
Figure A.3: Allan Deviation curves for the accelerometers of an Xsens Mtx.
Bias instability Random noise 푛 (root PSD)
X Axis 3.0 ⋅ 10−4 m/s2 (at 670 s) 0.0011 m/s2/√Hz
Y Axis 2.8 ⋅ 10−4 m/s2 (at 110 s) 0.0011 m/s2/√Hz
Z Axis 2.6 ⋅ 10−4 m/s2 (at 620 s) 0.0020 m/s2/√Hz
Table A.2: Accelerometer noise measurements for the Xsens Mtx.
the root PSD 푛 by dividing by the root of the sampling period
휎 =
푛√
훿푡
(A.3)
where 훿푡 is the sampling period of the device (훿푡 = 0.01 s for the Xsens Mtx tested in this
appendix). See Section 3.1.1.2 for more detail.
A bias instability of magnitude BS over a duration 푡 is usually modelled as a random walk
perturbing the bias of the sensor. The standard deviation of each random variable in the
underlying white noise sequence is given by
휎 =
BS√
훿푡 ⋅ 푡 . (A.4)
The converted 1휎 values for the Xsens Mtx are shown in Table A.3.
172
Bias instability (휎
value of the under-
lying white noise
sequence)
Random noise (휎
value)
X-Gyro (∘/s) 4.0× 10−3 0.75
Y-Gyro (∘/s) 3.9× 10−3 0.78
Z-Gyro (∘/s) 7.3× 10−3 0.79
X-Accel (m/s2) 1.2× 10−4 0.011
Y-Accel (m/s2) 2.7× 10−4 0.011
Z-Accel (m/s2) 1.0× 10−4 0.020
Table A.3: The 1휎 values of white noise sequences that underlie bias instability and random
noise processes for an Xsens Mtx.
173
174
Appendix B
Rotation matrices
A rotation matrix is any orthogonal matrix whose determinant is equal to 1. The effect of
multiplying a vector by a rotation matrix is to change its direction but not its magnitude. In a
right-handed 3-dimensional Cartesian co-ordinate system, rotations about the 푥, 푦 and 푧 axes
are achieved using the rotation matrices
푹푥 =
⎛⎜⎝ 1 0 00 cos휙 sin휙
0 − sin휙 cos휙
⎞⎟⎠ (B.1)
푹푦 =
⎛⎜⎝ cos 휃 0 − sin 휃0 1 0
sin 휃 0 cos 휃
⎞⎟⎠ (B.2)
푹푧 =
⎛⎜⎝ cos휓 sin휓 0− sin휓 cos휓 0
0 0 1
⎞⎟⎠ (B.3)
which are defined such that a positive rotation about an axis occurs in the clockwise direction
when looking along that axis from the origin [13].
Any rotation푹 can be expressed in the form
푹 = 푹푥푹푦푹푧 (B.4)
with suitable values of 휙, 휃 and 휓. It is however important to note that the order in which
the rotations are applied is important, since in general rotations in a 3-dimensional space are
non-commutative.
175
B.1 Small angle approximation
If 휙, 휃 and 휓 are small, then sin휙 → 휙, sin 휃 → 휃, sin휓 → 휓 and the cosines of 휙,
휃 and 휓 tend to 1. Using these approximations and ignoring the products of angles (since
they are small), the order in which rotations around the individual axes are applied becomes
unimportant and the rotation matrix푹 becomes
푹 = 푹푥푹푦푹푧 ≈
⎛⎜⎝ 1 휓 −휃−휓 1 휙
휃 −휙 1
⎞⎟⎠ . (B.5)
This is known as the small angle approximation.
B.2 Rotation about an arbitrary axis
The rotation matrix that defines a clockwise rotation (when looking along the axis from the
origin) of magnitude 휃 about an arbitrary unit-vector 풓ˆ = (푟푥, 푟푦, 푟푧) that passes through the
origin is given by
푹 =
⎛⎜⎝ 푡푟
2
푥 + 푐 푡푟푥푟푦 + 푠푟푧 푡푟푥푟푧 − 푠푟푦
푡푟푥푟푦 − 푠푟푧 푡푟2푦 + 푐 푡푟푦푟푧 + 푠푟푥
푡푟푥푟푧 + 푠푟푦 푡푟푦푟푧 − 푠푟푥 푡푟2푧 + 푐
⎞⎟⎠ (B.6)
where 푠 = sin 휃, 푐 = cos 휃 and 푡 = 1− cos 휃 [100]. If 휃 is small then small angle approxima-
tions can be used to give
푹 ≈
⎛⎜⎝ 1 푟푧휃 −푟푦휃−푟푧휃 1 푟푥휃
푟푦휃 −푟푥휃 1
⎞⎟⎠ . (B.7)
B.3 Aligning arbitrary vectors
It is sometimes necessary to calculate a rotation matrix 푹 that will rotate some vector 풗1 so
that it becomes aligned with another vector 풗2. Such a matrix can be derived as follows:
1. Calculate the normalised vectors 풗ˆ1 and 풗ˆ2.
2. Calculate the cross-product 풓 = 풗ˆ1 × 풗ˆ2. The resulting vector is perpendicular to both
of the original vectors and has a magnitude
∣풓∣ = sin 휃 (B.8)
176
where 휃 is the angle between them.
3. A suitable value of 푹 is a rotation matrix that corresponds to a clockwise rotation of
magnitude 휃 about 풓ˆ, the exact form of which is given in Appendix B.2.
Note that additional care must be taken to detect and handle the special cases in which 풗1 and
풗2 are parallel (in which case푹 = 푰) and in which 풗1 and 풗2 are anti-parallel (in which case a
suitable value of푹 is any rotation matrix that corresponds to a rotation of magnitude 휋 about
an axis perpendicular to 풗1).
177
178
Appendix C
Kalman filters
Kalman filters represent the Bayesian belief Bel(풙푡) as a Gaussian distribution defined by
its mean1 풙ˆ푡 and covariance 푷푡. They assume that the true state 풙 of the system evolves
according to the linear equation
풙푡 = 푭푡풙푡−훿푡 +풘푡 (C.1)
where풘푡 is process noise which is assumed to be zero mean Gaussian distributed with known
covariance푸푡:
풘푡 = 풩0,푸푡 . (C.2)
Measurements are assumed to occur at discrete points in time according to the linear relation-
ship
풛푡 = 푯푡풙푡 + 풗푡 (C.3)
where 풗푡 is measurement noise, also assumed to be zero mean Gaussian distributed
풗푡 = 풩0,푹푡 . (C.4)
Given these assumptions, the optimal Bayesian filter propagation and correction equations
can be implemented using computationally efficient matrix operations [53, 52]. The estimated
state is represented by its mean 풙ˆ푡 and covariance 푷푡 and is propagated using the equations
풙ˆ−푡 = 푭푡풙ˆ푡−훿푡 (C.5)
푷−푡 = 푭푡푷푡−훿푡푭
푇
푡 +푸푡 . (C.6)
1A hat symbol is used to distinguish the estimated mean state 풙ˆ푡 from the true state of the system 풙푡.
179
Measurement: zt
State estimate: x^t, PtPrior: x^0, P0
Correction
Kt = P
¡
t H
T
t (HtP
¡
t H
T
t + Rt)
¡1
x^t = x^
¡
t + Kt(zt ¡Htx^
¡
t )
Pt = (I ¡KtHt)P
¡
t
x^¡t = Ftx^t¡±t
P¡t = FtPt¡±tF
T
t + Qt
Propagation
Figure C.1: The Kalman filter.
A measurement 풛푡 is used to correct the propagated state using the equations
푲푡 = 푷
−
푡 푯
푇
푡 (푯푡푷
−
푡 푯
푇
푡 +푹푡)
−1 (C.7)
풙ˆ푡 = 풙ˆ
−
푡 +푲푡(풛푡 −푯푡풙ˆ−푡 ) (C.8)
푷푡 = (푰 −푲푡푯푡)푷−푡 . (C.9)
Multiple measurements can be applied at a single point in time by repeating the correction
step with 풙ˆ−푡 = 풙ˆ푡. The complete Kalman filter is shown in Figure C.1.
C.1 Linearised Kalman filters
The standard Kalman filter assumes that both the system being estimated and any measure-
ments can be described by linear equations of the form
풙푡 = 푭푡풙푡−훿푡 +풘푡 (C.10)
풛푡 = 푯푡풙푡 + 풗푡 . (C.11)
A linearised Kalman filter relaxes these assumptions by linearising the more general equations
풙푡 = 퓕 푡(풙푡−훿푡) +풘푡 (C.12)
풛푡 = 퓗푡(풙푡) + 풗푡 . (C.13)
about some approximate trajectory 풙∗푡 , which is assumed to be known [53]. The functions퓕 푡
and 퓗푡 are known state transition and measurement equations, which must be differentiable
but may be non-linear.
The true state of the system can be expressed in terms of the approximate trajectory and its
error
풙푡 = 풙
∗
푡 + 훿풙푡 . (C.14)
180
Measurement: zt
State estimate: x^t, PtPrior: x^0, P0
Correction
Kt = P
¡
t H
T
t (HtP
¡
t H
T
t + Rt)
¡1
±x^t = ±x^
¡
t + Kt(yt ¡Ht±x^
¡
t )
Pt = (I ¡KtHt)P
¡
t
±x^¡t = Ft±x^t¡±t
P¡t = FtPt¡±tF
T
t + Qt
Propagation
yt = zt ¡Ht(x
¤
t )
±x^0 = x^0 ¡ x
¤
0 x^t = x
¤
t + ±x^t
Error-state filter
Figure C.2: The linearised Kalman filter.
Hence Equations C.12 and C.13 can be re-written as
풙∗푡 + 훿풙푡 = 퓕 푡(풙∗푡−훿푡 + 훿풙푡−훿푡) +풘푡 (C.15)
풛푡 = 퓗푡(풙∗푡 + 훿풙푡) + 풗푡 . (C.16)
Assuming 훿풙푡 is small, the functions퓕 푡 and퓗푡 can approximated by the first-order terms of
their Taylor’s series expansions, giving
풙∗푡 + 훿풙푡 ≈ 퓕 푡(풙∗푡−훿푡) +
[
∂퓕 푡
∂풙
]
풙∗푡−훿푡
⋅ 훿풙푡−훿푡 +풘푡 (C.17)
풛푡 ≈퓗푡(풙∗푡 ) +
[
∂퓗푡
∂풙
]
풙∗푡
⋅ 훿풙푡 + 풗푡 . (C.18)
Assuming that for the approximate trajectory 풙∗푡 = 퓕 푡(풙∗푡−훿푡), we can re-arrange Eqs. C.17
and C.18 to obtain the linear equations
훿풙푡 =
[
∂퓕 푡
∂풙
]
풙∗푡−훿푡
⋅ 훿풙푡−훿푡 +풘푡 (C.19)
풛푡 −퓗푡(풙∗푡 ) =
[
∂퓗푡
∂풙
]
풙∗푡
⋅ 훿풙푡 + 풗푡 (C.20)
which are in the form required by the standard Kalman filter (Eqs. C.1 and C.3). Hence a
linearised Kalman filter is implemented as a standard Kalman filter that maintains a belief
Bel(훿풙푡) of the error-state. As in the standard Kalman filter the estimated state is represented
181
by its mean 훿풙ˆ푡 and covariance 푷푡. The error-state transition matrix is given by
푭푡 =
[
∂퓕 푡
∂풙
]
풙∗푡−훿푡
. (C.21)
By comparing Equations C.20 and C.3 it is clear that a measurement 풛푡 of the system’s state
can be incorporated by the filter in the form of an error-state measurement
풚푡 = 풛푡 −퓗푡(풙∗푡 ) (C.22)
with measurement matrix
푯푡 =
[
∂퓗푡
∂풙
]
풙∗푡
. (C.23)
The complete linearised Kalman filter is shown in Figure C.2.
C.2 Extended Kalman filters
The linearised Kalman filter is suitable for non-linear systems where an approximate trajectory
of the state can be precomputed. For example when a missile is to be tracked as it moves to-
wards a static target, its planned trajectory is known prior to its launch. The extended Kalman
filter is suitable for non-linear systems in which an approximate trajectory is not known in
advance. It is identical to the linearlised Kalman filter, except that the linearisation takes place
about the filter’s estimated trajectory 풙ˆ푡, rather than the precomputed trajectory 풙∗푡 [53]. The
estimated trajectory is propagated using
풙ˆ−푡 = 퓕 푡(풙ˆ푡−훿푡) . (C.24)
After each correction step the estimated error-state 훿풙ˆ푡 is used to correct the trajectory
풙ˆ푡 = 풙ˆ
−
푡 + 훿풙ˆ푡 (C.25)
before being reset to zero. Hence 훿풙ˆ푡 is equal to zero when entering both the propagation and
correction steps, allowing the corresponding equations to be simplified as shown in Figure
C.3. Since the filter is linearised about the estimated trajectory, 푭푡 and 푯푡 are the Jacobian
matrices
푭푡 =
[
∂퓕 푡
∂풙
]
풙ˆ푡−훿푡
(C.26)
푯푡 =
[
∂퓗푡
∂풙
]
풙ˆ−푡
. (C.27)
182
Measurement: zt
Correction
Kt = P
¡
t H
T
t (HtP
¡
t H
T
t + Rt)
¡1
±x^t = Ktyt
Pt = (I ¡KtHt)P
¡
t
P¡t = FtPt¡±tF
T
t + Qt
Propagation
yt = zt ¡Ht(x^
¡
t )
x^¡t = Ft(x^t¡±t) x^t = x^
¡
t + ±x^t
Error-state filter
State estimate: x^t, PtPrior: x^0, P0
Figure C.3: The extended Kalman filter.
C.2.1 Alternative formulation
The extended Kalman filter as described above consists of a regular Kalman filter operating on
the error-state whilst the state of the system is propagated separately. It is often more conve-
nient to track the estimated state of the system directly within the filter. This can be achieved
using an alternative formulation of the extended Kalman filter in which the propagation equa-
tions are
풙ˆ−푡 = 퓕 푡(풙ˆ푡−훿푡) (C.28)
푷−푡 = 푭푡푷푡−훿푡푭
푇
푡 +푸푡 (C.29)
and the measurement update equations are
푲푡 = 푷
−
푡 푯
푇
푡 (푯푡푷
−
푡 푯
푇
푡 +푹푡)
−1 (C.30)
풙ˆ푡 = 풙ˆ
−
푡 +푲푡(풛푡 −퓗푡(풙ˆ−푡 )) (C.31)
푷푡 = (푰 −푲푡푯푡)푷−푡 . (C.32)
Note that these equations can be obtained simply by incorporating the three external opera-
tions shown in Figure C.3 directly into the propagation and correction steps. The resulting
filter is shown in Figure C.4.
183
Measurement: zt
State estimate: x^t, PtPrior: x^0, P0
Correction
Kt = P
¡
t H
T
t (HtP
¡
t H
T
t + Rt)
¡1
x^t = x^
¡
t + Kt(zt ¡Ht(x^
¡
t ))
Pt = (I ¡KtHt)P
¡
t
x^¡t = Ft(x^t¡±t)
P¡t = FtPt¡±tF
T
t + Qt
Propagation
Figure C.4: The extended Kalman filter (alternative formulation).
184
Appendix D
Ultrasonic correction of inertial
navigation systems
This appendix describes how range measurements between an inertial measurement unit (IMU)
and known locations in an environment can be used to correct an inertial navigation system
(INS) that is tracking the IMU’s state (i.e. its position, velocity and orientation). Such range
measurements can be obtained by timing ultrasonic pulses, which can either be emitted by
beacons in the environment and detected by a receiver attached to the IMU (unilateral posi-
tioning) [8, 29], or emitted from the IMU and detected by receivers in the environment (multi-
lateral positioning) [28, 6]. For more details about ultrasonic positioning systems see Section
2.1.2.2. Note that the approach presented here is broadly applicable to range measurements
obtained using other positioning technologies, with the caveat that the error characteristics of
such technologies may differ from those of ultrasound.
D.1 Applying range measurements
Section 3.4 describes a general purpose error-state Kalman filter for probabilistically correct-
ing the state of an INS. It is able to use any measurement 풛푡 to correct an INS, provided that
it can be related to the state 풙푡 of the INS by an equation of the form
풛푡 = 퓗푡(풙푡) + 풗푡 . (D.1)
where 퓗푡 is a differentiable (but possibly non-linear) function and 풗푡 = 풩0,푹푡 is zero mean
Gaussian distributed noise with a known covariance 푹푡. Hence to correct an INS using ul-
trasonic range measurements it is sufficient to implement this filter and specify a function퓗푡
and a covariance matrix 푹푡 that are suitable for such measurements. This is the approach
185
adopted below. Note that it is assumed that the reader of this appendix is familiar with both
the Kalman filter described in Section 3.4 and with the notation introduced in Section 3.2.
When a microphone (or transducer for multilateral positioning) is attached to an IMU it will
usually be offset from the IMU’s origin (i.e. the point of the IMU that is tracked by an INS).
The offset between the microphone and the origin is constant in the body frame (i.e. the
IMU’s local frame of reference) and is denoted 풐푏. Given that 풐푏 is known, consider a range
measurement 풛푡 = 푑푡 between the IMU and a beacon (or receiver for multilateral positioning)
at a known location 풓 = (푟푔푥 , 푟푔푦 , 푟푔푧) in the environment. The measurement can be related
to the state of the INS by the function
퓗푡(풙푡) =
√
(푟푔푥 − 푖푡)2 + (푟푔푦 − 푗푡)2 + (푟푔푧 − 푘푡)2 (D.2)
where
(푖푡, 푗푡, 푘푡) = 풔푡 +푪푡풐
푏 (D.3)
is the position of the microphone (or transducer) in the local navigation frame, in which 푪푡
and 풔푡 are the orientation and displacement of the IMU respectively. The Jacobian matrix of
partial derivatives that is required by the filter is given by
푯푡 =
[
∂퓗
∂풙
]
풙ˆ−푡
= (0, 0, 0, 0, 0, 0,
(ˆ푖−푡 − 푟푔푥)
ℋ푡(풙ˆ−푡 )
,
(푗ˆ−푡 − 푟푔푥)
ℋ푡(풙ˆ−푡 )
,
(푘ˆ−푡 − 푟푔푧)
ℋ푡(풙ˆ−푡 )
) (D.4)
in which
(ˆ푖−푡 , 푗ˆ
−
푡 , 푘ˆ
−
푡 )
푇 = 풔ˆ−푡 + 푪ˆ
−
푡 풐 . (D.5)
is the position of the microphone (or transducer) as estimated by the current state of the INS.
All that remains is to specify a suitable covariance matrix 푹푡 that describes uncertainty in
the measurement. For a range measurement this matrix is a 1 × 1 matrix containing a single
variance, which should be set to a value appropriate for the ultrasonic system that is being
used (the accuracy of such systems is dependent on many factors, for example whether they
take air temperature into account when converting a time-of-flight into a corresponding range
measurement).
D.2 Filtering range measurements
One problem that has not been addressed above is that of signal reflections, which cause
range measurements that are larger than the true distance between the IMU and the beacon (or
receiver in multilateral positioning) in the environment. Incorporating such measurements into
the INS solution will degrade its accuracy. To avoid this range measurements that correspond
186
to reflected signals should be filtered out whenever possible. There are two approaches to
filtering out such measurements, which should be used together to obtain best performance:
∙ If multiple signals are detected by a receiver for a single transmitted pulse, then only
the first can correspond to a direct signal. Hence the time-of-flight corresponding to the
first signal should be converted into a range measurement, whilst all of the secondary
signals should be ignored.
∙ Before a range measurement is applied it can be compared to the range that is predicted
by the current INS state estimate. If range measurements are applied frequently then this
estimate should be a good estimate of the true range (since the estimated position of the
IMU should be a good estimate of its true position). Hence if the measurement is much
larger than the predicted range then it is likely that it corresponds to a reflected signal
and hence should be discarded. It is also beneficial to discard range measurements that
are much shorter than expected. Although they cannot be caused by signal reflections,
they can be caused by ultrasonic noise in the environment. The thresholds over which
ranges are discarded should be chosen based on the accuracy of the ultrasonic system
that is used and can also take the uncertainty in the current state estimate into account
(e.g. by allowing the measured range to differ from the predicted range by a larger
amount when there is more uncertainty in the estimated position of the IMU).
D.3 Unilateral versus multilateral positioning
When designing a combined inertial and ultrasonic tracking system from scratch it is always
preferable to design the ultrasonic component to be unilateral, meaning that beacons in the
environment transmit ultrasonic pulses that are detected by a microphone that is attached
to the IMU. The main benefit compared to a multilateral design is that all of the required
measurements end up at the same place (i.e. at the IMU). Hence if the software components
of the system are executed locally on the mobile device, then there is no need to transmit
any measurements from one place to another. In contrast in multilateral positioning range
measurements are calculated by receivers in the environment. Hence it is necessary to transmit
them back to the tracked object, or offload IMU measurements if the software is run elsewhere.
A secondary benefit of a unilateral design is that the range obtained from a time-of-flight is a
measurement of the distance between the IMU and a beacon at the precise moment that the
pulse is received. In multilateral positioning a range calculated by a receiver is a measurement
of this distance at the point in time when the pulse was transmitted from the IMU, which is
always in the past. Since Bayesian filters are only able to incorporate measurements at the
current time, the filter must run with a small delay (equal to the largest time-of-flight that
might be measured) if a multilateral design is used. This additional delay is not needed in the
187
unilateral case and is significant if low latency pose estimates are required (e.g. for use by
virtual reality applications).
D.4 PDR correction using the Active Bat system
This section describes the extension of the pedestrian dead reckoning (PDR) filter described
in Section 4.1 to incorporate ultrasonic range measurements obtained from the Active Bat po-
sitioning system. The extended filter provides a drift free estimate of the position of a tracked
user. Heading estimates calculated by the extended filter are also drift free. This is because
the error-state Kalman filter is able to correct correlated errors in different components of the
INS state, as described in Section 3.4. Heading errors are strongly correlated to errors in po-
sition (an error in heading will always result in an error in position unless the IMU remains
stationary) and hence range measurements prevent drift from accumulating in the estimated
heading as well as in the estimated position. The data obtained from the extended filter was
used as ground truth in the analysis of the (unextended) PDR filter, as described in Section
4.2.2.
The Active Bat system is a multilateral ultrasonic location system that was initially developed
at AT&T laboratories in Cambridge [28, 6]. The system is currently deployed in one wing
of the William Gates building. Ideally a unilateral system would have been used to provide
ground truth measurements, however in this case it was necessary to use what was already
installed in the building. Since the ground truth data obtained was required only for testing
purposes, it was sufficient to log all data and replay it to the filter later. Hence the latency issue
described in Section D.3 was avoided. The problem of getting all of the measurements to the
same place was also avoided by logging IMU measurements on an ultra mobile PC (UMPC)
that was carried by the user and logging Active Bat measurements on a desktop PC that was
connected to the Active Bat system. Both logs were then transferred to the same machine for
processing.
The PDR filter was extended by modifying its error-state Kalman filter to support the appli-
cation of range measurements, as described in Section D.1. The Active Bat system was set to
query the position of a single bat 25 times per second, which was attached to the IMU worn
on the user’s foot directly above the IMU (see Figure 4.8 on page 76). Range measurements
obtained from the Active Bat system were used to correct the INS, provided that they passed
the time synchronisation and filtering conditions outlined below. All range measurements that
were applied were assumed to have equal uncertainty of 휎 = 0.02 m, which was empirically
determined to give good performance (and is roughly equal to the expected accuracy of the
system).
188
D.4.1 Time synchronisation
One major problem encountered when applying Active Bat range measurements was synchro-
nising the mobile device that logged the IMU measurements with the desktop PC that logged
measurements from the Active Bat system. Such synchronisation was necessary to ensure that
measurements from the different systems were replayed into the filter at the correct points in
time. This was achieved using Chrony1, which synchronises machines using the Network
Time Protocol (NTP) and additionally constructs locally stored models for each clock that are
used to minimise clock drift when the machines cannot contact one another. The two logging
machines were synchronized over a wired LAN before each data collection run.
During each run various factors can cause synchronisation issues to arise. Firstly, the two
clocks can be expected to drift apart slightly despite the corrections applied by Chrony. To
minimise this effect each data collection run was limited to at most 15 minutes, during which
the drift between the two clocks would be in the order of milliseconds. Secondly, small delays
between each measurement being made and subsequently being timestamped by the logging
application can have an effect. Active Bat data must also propagate over a wired LAN before
reaching the logging machine. To mitigate these issues range measurements obtained from
the Active Bat system were only used if they occurred during a stance phase, as described in
Section 4.1.1. Since the user’s foot (and hence the attached IMU and Active Bat) is stationary
during such periods, small synchronisation errors have no effect. The way in which stance
phases are detected by the PDR filter means that there is a window of 100 ms either side of a
stance phase during which the user’s foot is stationary (see Figure 4.1 on page 68). Hence this
approach allows for drift between the timestamps recorded in the two logs of up to 100 ms,
which is an order of magnitude greater than the maximum drift that could be expected.
D.4.2 Filtering range measurements
Secondary signals (i.e. those that arrive after another signal corresponding to the same trans-
mitted pulse) are automatically discarded by the Active Bat system. Range measurements
were also filtered based on their consistency with the filter’s current state estimate. For a
given range measurement the predicted range is calculated as the distance between the re-
ceiver and the predicted bat position given by Equation D.5. It was found that rejecting any
range measurement that differed from the predicted range by more than 0.06 m gave good
performance. Note that in a system intended for frequent use a more advanced approach may
be required to ensure good performance in all cases, however for the limited use required here
the simple approach described was found to be sufficient.
1http://chrony.tuxfamily.org
189
190
Appendix E
Error ellipses
An error ellipse is a simple way of graphically representing the uncertainty of the position of
a 2-dimensional point [101]. It is assumed that the uncertainty is Gaussian distributed and
hence can be defined by its mean 흁 = (휇푥, 휇푦) and covariance[
휎2푥 휎푥푦
휎푥푦 휎
2
푦
]
. (E.1)
An error ellipse encodes the maximum and minimum standard deviations of such a distribu-
tion as well as their associated directions. These parameters can be calculated by considering
the value of the covariance matrix in a rotated coordinate frame (푚,푛). In this coordinate
frame the covariance matrix is given by[
휎2푚 휎푚푛
휎푚푛 휎
2
푛
]
=
[
cos 휃 − sin 휃
sin 휃 cos 휃
][
휎2푥 휎푥푦
휎푥푦 휎
2
푦
][
cos 휃 sin 휃
− sin 휃 cos 휃
]
(E.2)
where 휃 is the angle between the original coordinate frame (푥, 푦) and the rotated frame (푚,푛)
in the clockwise direction. When multiplied out Equation E.2 gives
휎2푚 = 휎
2
푥 cos
2 휃 − 2휎푥푦 sin 휃 cos 휃 + 휎2푦 sin2 휃 (E.3)
휎2푛 = 휎
2
푥 sin
2 휃 + 2휎푥푦 sin 휃 cos 휃 + 휎
2
푦 cos
2 휃 . (E.4)
To find the directions of the maximum and minimum standard deviations it is necessary to
find the values of 휃 at the turning points of 휎2푚 or 휎
2
푛. Arbitrarily choosing 휎
2
푛 gives
푑
푑휃
휎2푛 = 0 = 2휎
2
푥 sin 휃 cos 휃 − 2휎2푦 sin 휃 cos 휃 + 2휎푥푦(cos2 휃 − sin2 휃) (E.5)
= (휎2푥 − 휎2푦) sin 2휃 + 2휎푥푦 cos 2휃 . (E.6)
191
σ max
σ min
(µx,µy)
θmax
Figure E.1: An error ellipse.
This can be rearranged to give
tan 2휃 =
2휎푥푦
휎2푦 − 휎2푥
(E.7)
which has two solutions in the range 0∘ − 180∘ that are 90∘ apart . These solutions are the
directions of the maximum and minimum standard deviations. If 휎푥푦 > 0 then the value of
휃 that lies in the range 0∘ − 90∘ is the direction of the maximum deviation. Otherwise the
direction of the maximum deviation is the solution in the range 90∘ − 180∘. The values of the
maximum and minimum standard deviations are found by substituting the two solutions back
into Equation E.4 and taking the square root of each result.
Let 휎max and 휎min be the maximum and minimum deviations with corresponding directions
휃max and 휃min. The standard error ellipse given these values is drawn as shown in Figure E.1.
The probability that the true position of the point actually lies within this ellipse is 39%. An
error ellipse that contains the true position of the point with probability 푝 can be obtained by
multiplying the major and minor axis lengths by a scaling factor [102]. For 2-dimensional
error ellipses the scaling factor is given by
푘 =
√
−2log푒(1− 푝) . (E.8)
For example the 95% error ellipse is obtained by calculating the standard error ellipse and
multiplying the axis lengths by 푘 = 2.447.
In 3-dimensional space it is possible to compute an error ellipsoid to represent the uncer-
tainty of the position of a point [101]. Error ellipsoids can also be generalised to represent
uncertainty in spaces that have more than three dimensions.
192
Appendix F
Glossary
AOA Angle of arrival
ARW Angle random walk
EKF Extended Kalman filter
GBP Great British pound
GNSS Global navigation satellite system
IMU Inertial measurement unit
INS Inertial navigation system
KF Kalman filter
KLD Kullback-Leibler distance
LAN Local area network
LED Light emitting diode
MEMS Micro-machined electromechanical systems
NTP Network time protocol
OSI Open system interconnect
PDR Pedestrian dead reckoning
RF Radio frequency
RFID Radio frequency identification
RSSI Received signal strength indication
SCAAT Single constraint at a time tracking
SIR Sampling importance resampling
TDOA Time difference of arrival
TOA Time of arrival
UMPC Ultra mobile personal computer
UWB Ultra wideband
VRW Velocity random walk
ZVU Zero velocity update
193
194
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