MASTER PROJECT Geodetic Engineering Laboratory
Integration of Skew-Redundant MEMS-IMU with GPS for Improved Navigation Performance
By
Stéphane Guerrier
Supervisors
Dr. Jan Skaloud & Adrian Wägli
June 2008
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Table of Contents
1. INTRODUCTION ..................................................................................................................................... 11
1.1. CONTEXT OF THE STUDY...................................................................................................................... 11 1.2. OBJECTIVES......................................................................................................................................... 12 1.3. STRUCTURE OF THE RAPPORT.............................................................................................................. 13
2. THEORETICAL BASIS OF IMU REDUNDANCY................ .............................................................. 14
2.1. INTRODUCTION.................................................................................................................................... 14 2.2. GEOMETRICAL CONFIGURATIONS OF INERTIAL SENSORS....................................................................... 14
2.2.1. Information Filters ........................................................................................................................ 15 2.2.2. Partial Redundancy....................................................................................................................... 17
2.3. NOISE REDUCTION AND ESTIMATION.................................................................................................... 19 2.4. FAULT DETECTION AND ISOLATION......................................................................................................22
2.4.1. Introduction................................................................................................................................... 22 2.4.2. The Parity Space Method .............................................................................................................. 22
2.4.2.1. Measurement Model............................................................................................................................ 22 2.4.2.2. Parity Space Model............................................................................................................................. 23 2.4.2.3. Fault Detection Algorithm................................................................................................................... 24 2.4.2.4. Fault Isolation Algorithm.................................................................................................................... 28
2.5. INTEGRATION OF REDUNDANT IMUS WITH GPS................................................................................... 29 2.5.1. Single IMU Mechanization............................................................................................................ 29 2.5.2. Synthetic Mechanization ............................................................................................................... 29 2.5.3. Extended Mechanization ............................................................................................................... 32 2.5.4. Geometrically-Constrained Mechanization .................................................................................. 33 2.5.5. Algorithm Selection....................................................................................................................... 34
3. IMU ARCHITECTURES ......................................................................................................................... 35
3.1. INTRODUCTION.................................................................................................................................... 35 3.2. ASSESSMENT IN TWO DIMENSIONS........................................................................................................ 35 3.3. OPTIMAL CONFIGURATION WITH THREE SENSORS IN THREE DIMENSIONS.............................................. 36 3.4. OPTIMAL CONFIGURATION WITH FOUR SENSORS IN THREE DIMENSIONS............................................... 37 3.5. OPTIMAL CONFIGURATION WITH N SENSOR TRIADS............................................................................... 40 3.6. NOTE ON THE IMPACT OF SENSOR FAILURES......................................................................................... 42 3.7. CONCLUSIONS..................................................................................................................................... 44
4. EXPERIMENTAL ASSESSMENT OF REDUNDANT IMUS.............................................................45
4.1. INTRODUCTION.................................................................................................................................... 45 4.2. EXPERIMENTAL SYSTEM SETUP............................................................................................................. 45 4.3. IMPROVEMENT IN NAVIGATION PERFORMANCE..................................................................................... 46 4.4. NOISE REDUCTION AND ESTIMATION.................................................................................................... 48
4.4.1. Experimental Noise Reduction ...................................................................................................... 48 4.4.2. Assessment of the Noise Estimation Algorithm ............................................................................. 49
4.5. FAULT DETECTION AND ISOLATION......................................................................................................51 4.5.1. Assessment of the Noise Characteristics of in MEMS-IMUs......................................................... 51 4.5.2. Performance of the Parity Space Method with MEMS-IMUs ....................................................... 52
4.6. CONCLUSIONS..................................................................................................................................... 58
5. EMULATION OF REDUNDANT IMU SETUPS.................................................................................. 59
5.1. INTRODUCTION.................................................................................................................................... 59 5.2. INVERSE STRAPDOWN........................................................................................................................... 59 5.3. ERROR MODEL FOR INERTIAL SENSORS................................................................................................. 60 5.4. VARIANCE ANALYSIS............................................................................................................................. 62 5.5. EVALUATION OF THE NOISE CHARACTERISTICS OF INERTIAL SENSORS...................................................64
5.5.1. Quantization Noise........................................................................................................................ 64
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5.5.2. White Noise ................................................................................................................................... 65 5.5.3. Bias Instability .............................................................................................................................. 66 5.5.4. Random Walk ................................................................................................................................ 67 5.5.5. Rate Ramp..................................................................................................................................... 68 5.5.6. Estimation of the Quality of the Allan Variance ........................................................................... 69
5.6. ESTIMATION OF THE PARAMETERS OF THE NOISE MODEL ..................................................................... 71 5.6.1. Estimation of the Scale Factor and of the Constant Part of the Bias............................................ 71 5.6.2. Estimation of the Stochastic Part of the Bias and of the White Noise Parameter ......................... 71
5.7. VALIDATION OF THE MODEL ................................................................................................................ 76 5.8. INFLUENCE OF THE SENSORS ORIENTATION IN REDUNDANT IMU .......................................................... 77 5.9. NAVIGATION PERFORMANCE IMPROVEMENT......................................................................................... 78 5.10. CONCLUSIONS..................................................................................................................................... 80
6. CONCLUSION AND PERSPECTIVES ................................................................................................. 81
6.1. SYNTHESIS........................................................................................................................................... 81 6.2. PERSPECTIVES..................................................................................................................................... 82
7. REFERENCES .......................................................................................................................................... 83
8. APPENDIX ................................................................................................................................................ 86
8.1. APPENDIX A: PROOF OF EQUATION (11)............................................................................................... 86 8.2. APPENDIX B: PROOF OF EQUATION (13)............................................................................................... 86 8.3. APPENDIX C: PROOF OF EQUATION (19) .............................................................................................. 87 8.4. APPENDIX D: PROOF OF EQUATION (23).............................................................................................. 88 8.5. APPENDIX E: PROOF OF EQUATION (42)............................................................................................... 89 8.6. APPENDIX F: PROOF OF EQUATION (60)............................................................................................... 90 8.7. APPENDIX G: PROOF OF EQUATION (72).............................................................................................. 92
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List of Figures
Figure 1 : Two optimal configurations using five sensors. In the first case (left), the cone half-angle is 54. 74
[deg] (i.e. firstα ) and the sensors are separated by 72 [deg]. In the second case (right), the half-
angle is 65.91 [deg] (i.e. ondsecα ) and 90 [deg] separates the sensors. ........................................... 15 Figure 2 : The Platonic solids [13]................................................................................................................... 17 Figure 3 : Two minutes of static measurements with a MEMS-IMU (Xsens MT-i), after one minute vibrations
were added. The increase in the noise level after increased vibrations can be here observed ........ 19 Figure 4 : Theoretical noise reduction when using redundant sensors............................................................. 20 Figure 5 : Schematic representation of the averaging window ........................................................................ 21 Figure 6 : Graphical representation of the threshold-to-noise ratio – Redundancy - FAP .............................. 26
Figure 7 : Schematic representation of the threshold value T and 22 / nD σ probability density function
(pdf) ................................................................................................................................................. 27 Figure 8: Detector Operating Characteristics (DOCs) for various maximal errors (i.e. B) to 2
nσ ratio.......... 28
Figure 9 : Principle of mechanization based on a synthetic IMU .................................................................... 30 Figure 10 : Principle of extended IMU mechanization ...................................................................................... 33 Figure 11 : Principle of geometrically-constrained IMU mechanization ........................................................... 33 Figure 12 : Configuration of two sensors in planimetry .................................................................................... 35 Figure 13 : Configuration of three sensors in a three dimensional space........................................................... 36 Figure 14 : Influence of α and β on the information volume ........................................................................ 37 Figure 15 : Tetrad configuration proposed by A. Pejsa in [12].......................................................................... 39 Figure 16 : Influence of α and β on the information volume ........................................................................ 40 Figure 17 : Six sensors are placed in two different configurations. The symbol indicates the sensors that
will fail (superscript *). ................................................................................................................... 42 Figure 18 : Influence of λ and ϕ on the standard deviation of *2z ................................................................ 44 Figure 19 : (from left to right) (1) Skew-redundant IMUs placed in a tetrahedron. (2) Stéphane Guerrier
holding the scan2map system. (3) Installation of the base station by Yannick Stebler ................... 45 Figure 20 : Orientation errors after integration of a single sensor compared to................................................. 47 Figure 21 : Position, velocity, and acceleration errors after integration of a single sensor compared to extended
mechanization (four sensors)........................................................................................................... 48 Figure 22 : Comparison of the angular rate measurements of 4 MEMS-IMU to............................................... 49 Figure 23 : Estimation of the noise level for a selected portion of the experiment ............................................ 49 Figure 24 : Comparison between the noise estimates and the “real” noise (based on ln200’s data) calculated
during the experiment ...................................................................................................................... 50 Figure 25 : Orientation error after integration of four sensors (extended mechanization), with and without the
noise estimation algorithm............................................................................................................... 50 Figure 26 : Autocorrelation of errors of the Xsens MT-i’s gyros ....................................................................... 51 Figure 27 : Graphical assessment of the normality of Xsens MT-i errors (sensor 12, gyro ry).......................... 52 Figure 28 : Schematized algorithm for the detection and isolation of errors ..................................................... 53 Figure 29 : D values compare to T ................................................................................................................... 54 Figure 30 : Standford plot with TT ˆ= ............................................................................................................... 55 Figure 31 : D values compared to T value with identification of performance................................................ 55 Figure 32 : D values compared to
01.0T .............................................................................................................. 56 Figure 33 : Standford plot with
01.0TT = ............................................................................................................ 56
Figure 34 : D values compared to T values with identification of performance ............................................. 57 Figure 35 : Schematic representation of the Allan variance............................................................................... 63 Figure 36 : Allan variance of a white noise process........................................................................................... 65 Figure 37 : Example of two series of “bias instabilities” and white noise ......................................................... 66 Figure 38 : Allan variance of the series defined in Figure 37 ............................................................................ 67
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Figure 39 : Allan variance of a random walk process........................................................................................ 68 Figure 40 : Schematic sample representation of Allan variance using analysis results ..................................... 69 Figure 41 : Typical Allan variances of MEMS gyroscope (computed with three hours static data); the spotted
lines represent the standard deviation of the Allan variance values. ............................................... 70 Figure 42 : Influence of the number of sample and the averaging time to the precision of the Allan variance 70 Figure 43 : Allan variance results for the gyroscopes (spotted lines indicate the standard deviation of the
measurements) ................................................................................................................................. 72 Figure 44 : Estimation of noise parameters for the gyroscopes (spotted lines indicate the standard deviation of
the measurements) ........................................................................................................................... 73 Figure 45 : Allan variance results for the accelerometers (spotted lines indicate the standard deviation of the
measurements) ................................................................................................................................. 73 Figure 46 : Estimation of noise parameters for the accelerometers (axis x) (spotted lines indicate the standard
deviation of the measurements) ....................................................................................................... 74 Figure 47 : Estimation of noise parameters for the accelerometers (axis y) (spotted lines indicate the standard
deviation of the measurements) ....................................................................................................... 74 Figure 48 : Estimation of noise parameters for the accelerometers (axis z) (spotted lines indicate the standard
deviation of the measurements) ....................................................................................................... 75 Figure 49 : Reference trajectory of an alpine skier (in Plain-Joux, Switzerland, experiment realized by Adrian
Wägli) .............................................................................................................................................. 76 Figure 50 : Comparison of the position errors with the simulated and real data................................................ 76 Figure 51 : Comparison of the velocity errors with the simulated and real data................................................ 77 Figure 52 : Comparison of the orientation errors with the simulated and real data ........................................... 77 Figure 53 : Position, velocity and orientations errors of two redundant (i.e. four sensors) systems placed
differently in space .......................................................................................................................... 78 Figure 54 : Influence of the redundancy of the position error and the maximal position error.......................... 78 Figure 55 : Influence of the redundancy of the velocity error and the maximal velocity error.......................... 79 Figure 56 : Influence of the redundancy of the orientation error and the maximal orientation error ................. 79
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List of Tables
Table 1 : Average absolute accuracy of the tested GPS/MEMS-SRIMU system........................................... 46 Table 2 : Average and maximum errors of performance resulting from the use of the MEMS-SRIMU
compared to the performance of the individually integrated MEMS-IMU sensors......................... 47 Table 3 : Noise estimation for Xsens MT-i using Allan variance.................................................................... 75
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Abstract
Nowadays, in many sports, differences in skills among competitors tend increasingly to
diminish and the margin between victory and defeat has become a matter of little details.
In this context, the Geodetic Engineering Laboratory of the Swiss Federal Institute of
Technology in Lausanne is developing a low-cost GPS/INS system for performance
analysis in sports. This master project is part of this larger project and aims to improve the
precision of this analysis using redundant IMUs.
In many sports, trajectory analysis is essential, and, as a result, competitive athletes have
turned to GPS techniques to help evaluate and improve their performance. This
technology, by recording athletes’ various positions, allows to analyze and compare
trajectories, velocities or accelerations. GPS satellite signals are, however, obstructed by
various obstacles in athletes’ environments, therefore making the resolution of phase
ambiguities difficult or even impossible. To overcome these difficulties, Inertial
Measurement Units (IMUs) are integrated with GPS, which also enables a precise
calculation of accelerations and orientations. For ergonomic and economical
considerations, the systems considered here are composed of MEMS-IMUS together with
low-cost L1 GPS receivers.
The research for this project has demonstrated the following points. Firstly, redundant
inertial sensors enhance navigation performances. With four sensors, for example, errors
are diminished by 30-50%. Secondly, optimal sensor geometry can be formalized and
proves that, in most cases, the performance of the system is independent of this geometry.
Thirdly, the measurements’ redundancy reduces the noise affecting observations, as well as
estimates variations in the noise level. In addition, the MEMS IMU errors can be detected,
isolated and corrected successfully. Finally, measurements of MEMS sensors can be
simulated with an appropriate error model. Consequently, navigation performances of
redundant systems can be theoretically estimated though simulation.
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Résumé
Actuellement, dans le monde du sport, les écarts entre athlètes tendent petit à petit à
s’amoindrir et bien souvent la différence entre une victoire et une défaite trouve son
explication dans quelques petits détails. Dans ce contexte, le laboratoire de Topométrie de
l’Ecole Polytechnique Fédérale de Lausanne (EPFL) est actuellement en train de
développer un système GPS/INS dédié aux analyses des performances sportives. Ce travail
de master s’intègre dans le cadre de ce grand projet et s’intéresse plus particulièrement à
l’utilisation de capteurs inertiels redondants afin d’améliorer la précision de l’analyse de
ces performances.
Dans tous les sports où l’étude des trajectoires est essentielle, les athlètes s’intéressent au
GPS afin d’évaluer leurs performances. En effet, celui-ci enregistre des postions qui
permettent d’analyser puis de comparer des trajectoires, des vitesses ou des accélérations.
Mais, fréquemment l’environnement des athlètes ne permet pas une utilisation performante
de ce système. Ainsi, il est nécessaire de le coupler à des mesures inertielles qui, de plus,
permettent un calcul précis des accélérations et des orientations, déterminantes dans
certains sports. Pour des raisons économiques et ergonomiques, les systèmes considérés
dans ce travail sont basés sur des capteurs inertiels de type MEMS, ainsi que des
récepteurs GPS monofréquence.
Ce travail permet de mettre en évidence les éléments suivants. Tout d’abord, les capteurs
inertiels redondants améliorent les performances de navigation. Avec quatre capteurs, par
exemple, l’erreur est diminuée de 30-50%. Les considérations liées à la géométrie des
systèmes redondants peuvent être formalisées et montrent que, dans la majorité des cas, la
performance du système est indépendante de cette géométrie. Ensuite, la redondance de
mesures permet de réduire le bruit qui l’entache, celle-ci permet, de plus, une estimation
des niveaux de bruit. De surcroit, les erreurs affectant les mesures des capteurs MEMS
peuvent être détectées, isolées et corrigées améliorant, ce faisant, les performances de
navigation. Enfin, les mesures de capteurs MEMS peuvent être simulées grâce à un modèle
d’erreur adapté. Ainsi, les performances de navigation d’un système redondant peuvent
être théoriquement estimées par simulation.
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Remerciements
Je tiens à remercier ici toutes les personnes qui m’ont aidé à réaliser ce projet mais aussi
toutes celles qui m’ont permis d’effectuer ce parcours.
Tout d’abord, mon professeur, Dr. Jan Skaloud, qui a su me passionner, m’encourager et
me donner l’envie de poursuivre dans cette voie. Je tiens également à le remercier pour sa
grande disponibilité, son aide dans le cadre de ce projet et ses conseils judicieux mais aussi
pour la confiance qu’il m’a accordée en me laissant beaucoup de liberté.
Je tiens à remercier tout particulièrement Adrian Wägli, sans l’aide duquel ce projet
n’aurait pu être réalisé. Nos échanges très intéressants, le plaisir de travailler avec
quelqu’un d’aussi brillant qui, de plus, allie la détermination à l’humour furent une chance
pour moi. Sans oublier tous les bons moments passés ensemble.
Ensuite, je souhaiterais remercier le Prof. Bertrand Merminod, pour son enseignement de
qualité qui a éveillé mon intérêt pour ces disciplines et qui a toujours pris le temps de
répondre à mes questions.
Le Prof. Ishmael Colomina pour son aide, l’intérêt de ses travaux et son hospitalité. Ainsi,
qu’Eulàlila Parés, Pere Molina Mazón, Marino Wis et les collaborateurs de l’Institut de
Geomàtica qui m’ont si bien accueilli à Barcelone et avec qui j’ai eu beaucoup de plaisir à
travailler.
Un grand merci à Yannick Stebler, mon camarde d’étude, pour son esprit vif, les échanges
constructifs que nous avons eu et notre chaleureuse complicité.
J’aimerais aussi remercier les collaborateurs du laboratoire de Topométrie pour la bonne
ambiance et l’esprit d’équipe qui y règnent.
Le Prof. Hannelore Lee-Jahnke pour ses amicaux et judicieux conseils à la rédaction de ce
travail en anglais, langue dont je suis loin de maîtriser toutes les subtilités.
Enfin, Justine et Cédric qui ont pris le temps de relire ce travail ainsi que pour tous les
bons moments que nous avons passés ensemble et leur précieuse amitié.
Finalement, je tiens à remercier mes parents qui m’ont toujours soutenu et aidé. Sans
lesquels je n’aurais pu en arriver là !
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Abbreviations
EKF Extended Kalman Filter
FDI Fault Detection and Isolation
GPS Global Positioning System
IMU Inertial Navigation System
INS Inertial Navigation System
MEMS Micro-Electro-Mechanical System
PDF Probability Density Function
PSD Power Spectral Density
RMS Root Mean Square
SRIMU Skew Redundant Inertial Navigation System
SSE Sum of Squared Errors
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1. Introduction
1.1. Context of the Study
This master project was realized in the Geodetic Engineering Laboratory of the Swiss
Federal Institute of Technology Lausanne (EPFL). It is part of a larger investigation which
aims at developing a low-cost GPS (Global Positioning System) / INS (Inertial Navigation
System) system for performance analysis in sports [1-4].
Nowadays, in many sports the differences of skills between competitors tend increasingly
to diminish. Consequently, training for competition, in which every detail counts, has
become a key factor of success. Traditionally, athletes’ performances have been analyzed
using chronometry or video recordings. However, these techniques are limited by
meteorological conditions and by the difficulty of replicating postures and movements
across trials. Moreover, these only provide few quantitative variables that could be used to
analyze objectively athletes’ performances. New methods are, on the contrary, being
sought that offer precise measurements of positions, velocities, accelerations and forces
[4]. Indeed, satellite-based positioning has already been successfully applied in many
sports such as skiing [2, 4], car racing [5] or rowing [6]. Taking this into consideration,
athletes’ environments are often partially composed of areas that may block or attenuate
satellite signals and therefore make the resolution of phase ambiguities difficult or even
impossible. To overcome these difficulties Inertial Measurement Units (IMUs) are
integrated with GPS, which also enable an accurate determination of accelerations and
orientations. Indeed, in numerous applications the orientation is required. For example, in
motorcycling the correct exploitation of torque and force sensors necessitates knowledge
of the sensors’ orientation. Additionally, the attitude determination also provides essential
data to study the vibratory behavior of the pneumatics [3].
Traditionally, GPS/INS equipment consisting of dual-frequency GPS receivers and
tactical-grade INS, provides high accuracies even for large dynamics (cm for position,
cm/s for velocity and 1/100° for orientation). However, such equipment is bulky (a few kg)
and expensive ( > 40’000€) and is consequently not suitable for most sport applications
[3]. Recently, for ergonomic and economical considerations, smaller and lighter
equipment, composed of low-priced Micro-Electro-Mechanical System (MEMS) IMUs
together with low-cost L1 GPS receivers, were successfully applied in alpine skiing and
motorcycling [3, 4, 7, 8].
The outline of this work is to improve navigation performance (and more specifically
orientation) by using redundant MEMS-IMUs with GPS. Indeed, position and velocity can
12
be improved by using dual-frequency GPS receivers, instead of single-frequency receivers.
This, however, cannot be done for orientation. The assessment of the accuracy of this
technology was measured in the field of sports, the reach of this project, however, goes far
behind the sports world. In fact, in many fields, redundant MEMS-IMUs are already used
(e.g. robotics, virtual reality). Above all, this technology could be implemented in a wide
range of domains (e.g. low-cost reliable navigation systems, pedestrian navigation).
Unfortunately, MEMS-IMU are prone to large systematic errors (e.g. biases, scale factors,
drifts), which limits their applicability in integrated navigation systems [1]. Typically, the
performance of these IMU reaches a position accuracy of 0.5 m, a velocity accuracy of 0.2
m/s, as well as an orientation accuracy of 1 deg for the pitch and roll and 2 deg for the
heading. The sensors used are highly miniaturized and thus offer the possibility of using
numerous IMUs to enhance the determination of attitude. The redundancy of
measurements can improve the performances of the GPS/INS integration at several levels.
Firstly, direct noise estimation can be achieved directly from the data in order to improve
the stochastic model of the Extended Kalman Filter (EKF). Secondly, the noise level can
be reduced and defective sensors, spurious signals and sensor malfunctioning can be
detected and isolated. Finally, sensor error calibration becomes conceivable even during
uniform motion or static initialization. Due to the improved orientation accuracy,
redundant IMUs may bridge the gap in the GPS data more effectively [1].
Redundancy in inertial navigation has already been investigated with higher-order IMUs in
photogrammetry and remote sensing [9]. Several authors have presented results with
MEMS-IMU based on simulations or emulations and have found an accuracy improvement
of 33% with MEMS-IMUs placed on a tetrahedron [10]. Emulations presented by [11]
resulted in performance improvements of 20-34%. The first experimental results with
MEMS-MUs were given in [1] and have shown an improvement of navigation
performance, when using four MEMS-IMUs, of 30-50%.
1.2. Objectives
This research project focuses on the potential benefits of using redundant MEMS-IMUs to
improve GPS/INS integration. The objectives are:
• Determine optimal sensor architecture (i.e. geometry)
• Detect and isolate defective sensors and spurious signals
• Reduce noise and direct noise estimation of sensors
• Calibrate sensor even during uniform motion and initialization
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• Assess the experimental navigation improvements stemming from the use of
redundant MEMS-IMUs.
• Develop simulation tools in order to assess theoretical improvements of redundant
configurations
1.3. Structure of the Rapport
This rapport is structured in four parts.
The first part presents the theoretical basis of IMU redundancy. It details how optimal
sensor architectures can be determined, and describes how noise is reduced and can be
estimated when redundant inertial sensors are employed. In addition to that, it presents one
of the most commonly used algorithm of Fault Detection and Isolation (FDI) and different
mechanization’s methods which can be used with redundant IMUs.
The second part studies sensor geometries and the influence of this geometry on the system
performance.
The third part assesses experimentally the advantages of using multiple IMUs. It shows the
noise reduction as well as its estimation and the subsequent improvements brought to the
navigation solution (i.e. position, velocity and orientation). Finally, the use of redundant
IMUs allows to detect and isolate erroneous measurements. In this study, the performance
of the parity space method with MEMS-IMU is presented.
The last part presents the development of a simulation tool to emulate MEMS-IMU
measurements. It details the used error model and explains how its parameters are
estimated. As a conclusion, the theoretical improvements of various redundant
configurations are evaluated.
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2. Theoretical Basis of IMU Redundancy
2.1. Introduction
This chapter aims to present the theoretical basis of IMU redundancy. It covers
geometrical considerations about the spatial configurations of inertial sensors, as well as
the theoretical basis of noise reduction achieved with sensor redundancy. This chapter also
presents an algorithm that estimates the evolution of noise during the processing. In
addition, it contains a review of the most common FDI algorithm, the parity space
approach. Finally, different mechanization methods which can be applied with redundant
IMUs are presented.
2.2. Geometrical Configurations of Inertial Sensors
Redundant IMUs have been used since the early days of the inertial technologies in safety
critical operations such as in the control of military or space aircrafts. These aircrafts are
designed to be dynamically unstable, in order to increase manoeuvrability, while inertial
accelerations and rotation rates are used to observe the vehicle’s stabilizing parameters
[10]. Clearly, such systems require sensor redundancy. The redundant information was
used to create fault-tolerant systems which were able to detect and isolate defective sensors
(i.e. FDI). If two sensors are placed collinear to each other, it is possible to detect a fault
that occurs in either one of the sensors. To isolate the erroneous device at least three
sensors are required. Therefore, traditionally, nine sensors were used in a full three-
dimensional system (three per axis).
In 1974, [12] proposed a first theory to optimally position any number of sensors which
showed that less than nine sensors (four in theory) were required to isolate faults in a three-
dimensional space. This theory essentially considers two situations: firstly, when sensors
are equally spaced in a cone of half-angle α and, secondly, when one sensor is placed
along the central cone axis while the remaining sensors are positioned equally around a
cone of half-angle α . The optimal half-angle α corresponds to the configuration in which
the variance is minimized. The results for α obtained with the two configurations are
shown in Figure 1. Numerically, it yields to:
15
( )
−⋅−=
=
33
3arccos
arccos
sec
33
n
nond
first
α
α (1)
where n is the number of sensor.
xy
z
90 deg.
α
xy
z
72 deg.
α
Figure 1 : Two optimal configurations using five sensors. In the first case (left), the cone half-angle
is 54. 74 [deg] (i.e. firstα ) and the sensors are separated by 72 [deg]. In the second case (right),
the half-angle is 65.91 [deg] (i.e. ondsecα ) and 90 [deg] separates the sensors
2.2.1. Information Filters
Years later, [10] proposed an other method using information filters and showed that it
could be used to maximize the amount of information in redundant sensors configurations.
This approach provides a better conceptual understanding of how the information is
distributed in space.
In the state space, the observation at time k is given by:
( ) ( ) ( ) ( )kkkk vxHz +⋅= (2)
where ( )kx is the current state, ( )kH is the observation model and ( )kv is the
observation noise with covariance ( )kR .
The information contribution to the states of an observation constitutes the information
observation vector ( )ki defined as:
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( ) ( ) ( ) ( )kkkk T zRHi ⋅⋅= −1 (3)
The amount of certainty associated with this observation is given by the information
matrix,
( ) ( ) ( ) ( )kkkk T HRHI ⋅⋅= −1
(4)
Note that the matrix ( )kI is independent from the observations, since ( )kz is absent from
the equation. Thus, it provides a measure of the certainty of these observations purely
based on the geometry of the system. Moreover, it is often assumed that all sensors are
uncorrelated and of equal variance. Under this assumption, equation (4) can be rewritten
as:
( ) ( ) ( )kkk T HHI ⋅⋅= 20
1σ
(5)
Since the information matrix is positive and semi-definite, its corresponding eigenvectors
are orthogonal to each other. These eigenvectors can, additionally, be seen as the axes of
an ellipsoid in the information space. If the eigenvalues of those eigenvectors are different,
then the larger axis of the ellipsoid represents the direction in which information is
maximal. Thus, by maximizing the volume of the ellipsoid, the volume of information is
also maximized. Since the determinant of the information matrix is directly related to the
volume of the ellipsoid, the goal is to maximize the expression presented in equation (6).
( ) ( ) ( )( )( )kkJ T HH ⋅= detmaxmax (6)
The maximization of J shows that the optimal configuration adopts the shape of a sphere
in the information space. Hence, in this optimal geometry, the amount of information place
for each axis is equal.
This approach was applied to 4, 6, 8, 12 and 20 sensor combinations. The corresponding
optimal configurations are regular polyhedrons (tetrahedron, cube, octahedron,
dodecahedron and icosahedron respectively) which verifies the results of [12]. These solids
are perfectly symmetrical and are known as the Platonic Solids (see Figure 2).
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Figure 2 : The Platonic solids [13]
Flowing from these results, when the configuration is optimal and regardless of the number
of sensors, the eigenvectors of the information matrix ( )kI are equal to the number of
sensors divided by the number of dimension. Thus,
=D
niλ (7)
where n is the number of sensors, D the number of dimensions and 3 2, ,1=i . This
equation suggests that, in the information space, each sensor contributes to a third of its
information to each eigenvectors. Moreover, equation (7) enables to quantify the
maximum volume of information of n sensors placed in an optimal configuration as the
product of the eigenvalues, that is,
3
max
=D
nJ (8)
In short, this approach, based on information filters, allows to analyze optimal
configuration and provides a geometric visual understanding of the distribution of
information across axis. Moreover, it can also be used to create geometries with specific
uneven distributions of information (e.g. more information in planimetry than in altimetry).
2.2.2. Partial Redundancy
Another interesting approach to determine optimal sensor configurations is based on the
concept of reliability. This method is typically applied in geodetic networks and has – to
the author’s knowledge – never been applied in a context similar to that of this study.
18
Indeed, in geodetic networks with redundant observations, the measurements control each
other. In order to assess the “amount” of controllability of each measurement, a iz value
can be computed and associated to every observation. This value, called partial
redundancy, corresponds to the participation of the observation i to the global redundancy
of the system. It varies from 0 to 1, an observation having an associated 0=iz isn’t
controlled at all. To the contrary, a 1=iz implies a “totally” controlled observation. In
geodetic engineering, a network is typically considered good (i.e. technically and
economically) if all measurements have an associated ( )0.25,0.60 ∈iz [14].
To compute the partial redundancy, we first calculate the Z matrix is defined as:
( ) 1−⋅⋅⋅−= RHPHRZ T (9)
where R is the covariance matrix of the observations, H is the design matrix. P is the
covariance matrix of the state vector, defined as:
( ) 11 −− ⋅⋅= HRHP T
(10)
The partial redundancies iz are the diagonal elements of the Z matrix. Note that
equations (9) and (10) show that iz depends solely on the geometry of the system (i.e.
design matrix H ) and on the accuracy of the measurements (i.e. covariance matrix R ).
Moreover, it has been shown that:
∑=
=−=n
iizunr
1
(11)
Where r is the system redundancy, n is the number of measurements and u the number
of parameters. Equation (11) is demonstrated in the Appendix A.
Thus, the geometry of the system will influence the iz but not their summation.
Consequently, an optimal sensor configuration would minimize the differences between
these measurements to approach the optimal case, where observations are equally
controlled. Mathematically, this is equivalent to minimizing of the standard deviation of
the iz , that is:
Best configuration(s): [ ][ ]2 min ii zEzE − (12)
19
2.3. Noise Reduction and Estimation
Combinations of redundant inertial sensors not only decrease noise measurements, but also
offer the possibility to estimate, with the help of adaptive filters, the level of noise during
the processing. Indeed, the noise level can evolve during the processing in response to
particular situations (e.g. increased vibrations). Figure 3 shows an example of such
behaviour.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Time [s]
Ang
ular
rot
atio
n ra
te
[rad
/s]
Figure 3 : Two minutes of static measurements with a MEMS-IMU (Xsens MT-i), after one minute vibrations were added. The increase in the noise level after increased vibrations
can be here observed
When redundant IMUs are used, it is possible to quantify by how much redundancy
reduces the noise. Theoretically, from n independent measures 21 , ... , xx (with their
respective variances 221 , ... , nσσ ), we can compute their best estimate x . Assuming
homogenous measurements (i.e. constant iσ ), its variance 2xσ can be derived as [15, 16]:
nw x
n
iiix
σσσ == ∑=1
22ˆ
(13)
where the iw are weighting factors and iix ∀= σσ . The derivation of equation (13) is
given in Appendix B. This theoretical noise reduction corresponding to various redundant
IMUs configurations is represented in Figure 4
20
1 2 3 4 5 6 7 8 9 10 11 120
10
20
30
40
50
60
70
80
90
100
# of sensors
Noi
se le
vel [
%]
Figure 4 : Theoretical noise reduction when using redundant sensors
Another interesting feature of multiple IMUs is the possibility of using a noise estimation
algorithm during the processing. This information can be used to improve the performance
of the GPS/INS integration by providing a more realistic stochastic model. Moreover, it
can improve the FDI performances which depend on the noise model.
The algorithm we propose to use in this case is laid out below.
Firstly, the norms, which is not sensible to the orientation, of angular rotation rates and
specific forces are computed for every time k and for every sensor i :
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )kfkfkfk
krkrkrk
zyx
zyx
iiii
iiii
222
222
++=
++=
f
r (14)
where ni ,,1K= and n is the number of sensors triads.
Squared differences are then computed to form the d matrices:
21
( )
( )( )
( )( )
−−
−−
=
∑
∑
−
=
=
21
1
2
21
1
1
,
n
k
k
n
k
k
knn
jj
n
n
jj
r
r
r
r
d r M and ( )
( )( )
( )( )
−−
−−
=
∑
∑
−
=
=
21
1
2
21
1
1
,
n
k
k
n
k
k
knn
jj
n
n
jj
f
f
f
f
d f M (15)
An averaging time T (typically a few seconds, see Figure 23) is chosen (see Figure 5).
Assuming all sensors’ errors to be uncorrelated and identical, the noise is directly
estimated:
( ) ( )
( ) ( ) dtkin
Tk
dtkin
Tk
Tk
Tk
n
if
Tk
Tk
n
ir
⋅
=
⋅
=
∫ ∑
∫ ∑
+
−=
+
−=
1
1
,1
,
,1
,
f
r
d
d
σ
σ
(16)
T TT T
( )k time
fr dd /
Figure 5: Schematic representation of the averaging window
22
2.4. Fault Detection and Isolation
2.4.1. Introduction
Fault Detection and Isolation (FDI) algorithms for inertial navigation have been thoroughly
investigated in the past. The most commonly used approach is the parity space method [10,
17-20], but other approaches such as artificial neural networks have also been employed
[21]. The complexity of implementation of an efficient FDI system is increased using
MEMS-IMUs. Indeed, their poor performance (i.e. noise density variations and larger
proportion of systematic errors (colored noise, bias, scale factor) compared to the higher
grade-IMU) creates high risks of false alarm, as well as an increased exposure to
misdetection of faulty measurements [10]. We investigate here the parity space method
with skewed redundant MEMS-IMUs. In the context of this project, focused on non-safety
critical sports applications, the main objective is to increase navigation performances and
to prevent gross errors.
2.4.2. The Parity Space Method
2.4.2.1. Measurement Model
In general, we consider l measurements in an n dimensional state space which is over
determined (i.e., 1+≥ nl ). The system can be defined by equation [20]:
bnxHz ++⋅= (17)
where z is the ( )1×l measurement vector compensated with all prior information; H is
the ( )nl × design matrix which converts the state space into the measurement space; x is
the ( )1×n state vector; n is a ( )1×l vector of Gaussian white noise having a variance of
2nσ and b is a ( )1×l vector of uncompensated measurement biases (i.e. faults).
Consider the case of m IMUs, equation (17) can be rewritten as:
bn
m
f
w
A
A
A
l
l
l
++
⋅
=
oo
oo
oo
b
b
b
m
f
w
bm
bf
bw
i
i
i
(18)
23
where ibwl , ib
fl and ibml represent the observation vectors (respectively for the gyroscopes,
accelerometers and magnetometers), bw , bf and bm represent IMU measurements in the
pre-define body frame of reference. Furthermore, wA , fA and mA are the matrices that
transforms from the reference body frame (superscript b ) into the individual sensor frames
of the IMUs (superscript ib ) (these matrices are defined in equation (38)).
The fault model in this case is a single measurement source failure which results in a step
bias shift. A failure of the thi measurement is modeled by ibb = where ib is a ( )1×l
vector with the magnitude of the error B at the thi position, zero elsewhere. If no faults
occurs then 0b = [20].
Additionally, equation (17) shows that the underlying assumption behind the parity space
method is that non erroneous measurements are only subject to Gaussian white noise
errors. This assumption is acceptable in the case of a calibrated high performance IMU,
but may not realistic with MEMS-IMU which are subject to colored noise.
2.4.2.2. Parity Space Model
The parity space method’s theoretical basis presented here are largely based on [20].
The least square best estimate of the state vector of the equation (17) is given by:
zHx * ⋅=ˆ where ( ) T1T* HHHH ⋅⋅= − (19)
The derivation of this equation is given in Appendix C.
Thus, *H converts measurements space into state space. For any ( )nl × H design matrix
with a rank n , a ( )( )nnl ×− P matrix can be found such that the rank of P is ( )nl − ,
nlT IPP −=⋅ and 0HP =⋅ . So, the matrix P spans the null space of H (i.e. the parity
space).
Moreover, the ( )ll × matrix
=
P
HA
*
can be formed and has a rank l . This matrix
represents a linear transformation of the measurement space into two subspaces, namely
the state space and the parity space.
The parity vector is calculated in the following way:
24
( ) ( )bnPbnPxHPzPp +⋅=+⋅+⋅⋅=⋅= (20)
Thus, this vector is independent from x , its elements are normally distributed and have an
expected value of: [ ] ( )[ ] [ ] [ ] bPbPnPbnPp ⋅=⋅+⋅=+⋅= EEEE . What’s more, the
covariance of the parity vector is defined as: [ ] nlnσCOV −⋅= Ip 2 . Consequently, the
elements of p have an equal variance (i.e. the noise variance) and are not correlated. If no
fault occurs, then 0b = and [ ] 0p =E .
The matrix [ ]TPHA 1 =− (see Appendix D) also has a rank l and represents the inverse
transformation caused by A , therefore it transforms the state and parity spaces into a
measurement space. The conversion of the parity vector into the measurement space is
obtained by:
[ ] zSzPPp
0PHf ⋅=⋅⋅=
⋅= × TnT 1 (21)
where f is the fault vector having the following expected value:
[ ] [ ] [ ] [ ] [ ] bSbSnSxHPPzSf ⋅=⋅+⋅+⋅⋅⋅=⋅= EEEEE T (22)
The matrix PPS T ⋅= can be directly computed from H using the equation:
*l HHIS ⋅−= (23)
The derivation of this equation is given in Appendix D.
2.4.2.3. Fault Detection Algorithm
The first step in any FDI system is to detect erroneous measurements. This step falls within
the general framework of composite hypothesis tests [17]. The fault detection can be
viewed as a choice between two hypotheses, in the absence or in the presence of a faulty
measurement. The hypothesis test is based on a comparison of a decision variable D to a
threshold variable T . The test can be defined as:
25
⇒>⇒<
detected isfault a :H
fault no :H
1
0
TD
TD
where 0H is the null hypothesis and 1H the alternative hypothesis.
The performance of the test is defined by the probabilities of false alarm FAP and of
misdetection MDP , which are expressed by:
( ) ( )( ) ( )iMDMD
FAFA
TDPPTDPP
TDPPTDPP
bb
0b
=<=⇔<=
=>=⇔>=
1
0
H
H (24)
The decision variable D is considered to be the sum of the square magnitude of the fault
vector (i.e. SSE) [17, 20, 22]. Thus, D is defined as:
ff ⋅= TD (25)
The definition of the threshold variable T is, however, more complicated. In the fault-free
case 0b = , [ ] 0p =E and 2/ nD σ has a chi-square distribution (since it is a summation of
normal distributions [23]) with nlr −= degrees of freedom [17, 20, 22, 24]. This implies
that:
= r
TQP
nFA 2σ
(26)
where ( ) ( )rPrQ 22 1 χχ −= and ( )rP 2χ is a chi-square probability function defined as:
( ) ( )[ ] dtetrPtrr
r 2
2
22
2
0
1
22 2 −−
∫−
⋅Γ=χ
χ (27)
Consequently, the probability of false alarm FAP depends on the number of redundant
measurements r and is independent of the design matrix H .
The required threshold can now be determined for any given variables FAP , r , 2nσ using:
( ) ( )nlPQnlPT FAnnFA −⋅⋅=− −122,, σσ (28)
26
where ( )rQ 21 χ− is the inverse function of ( )rQ 2χ .
Thus, there is a relation between the ratio threshold-to-noise, the variance ratio 2/ nT σ , the
probability of false alarm FAP and the redundancy of the system r . Figure 3 shows this
relationship.
00.05
0.10.15
0.2
0
5
10
15
0
5
10
15
20
25
30
35
40
PFA
Redundancy
Thr
esho
ld-t
o-N
oise
Rat
io (
T/
σσ σσn2 )
2.5
5
5
7.5
7.5
1 0
10
12.
5
12.5
12.5
15
15
15
17.
517
.5
17.5
20
20
20
22.
52 2
. 52
52 5
PFA
Red
unda
ncy
0.05 0.1 0.15 0.2
2
4
6
8
10
12
0
5
10
15
20
25
Thr
esho
ld-t
o-N
oise
Rat
io (
T/σσ σσ
n2 )
Figure 6: Graphical representation of the threshold-to-noise ratio – Redundancy - FAP
Further, it is assumed that faults are equally likely to occur in any measurement, which
implies that:
( )∑=
=<⋅=l
iiMD TDP
lP
1
1bb (29)
But if ibb = then [ ] iE bPp ⋅= and 2/ nD σ has non-central chi-square distribution with r
degrees of freedom and a non-centrality parameter defined by [17, 20, 22, 24], then:
iin
i SB ⋅
= 2
2
σθ (30)
27
where iiS represents the thi element of the diagonal of the matrix S (defined in equation
(23)). Hence:
∑=
−=
l
iii
nnMD S
Bnl
TP
lP
12
2
2,
1
σσ (31)
where ( )θχ ,2 rP is a non-central chi-square probability function defined by:
( ) ( ) ( )jrPj
erPj
2!
, 222 2 += − χθχθθ
(32)
Figure 7 shows schematically the relationship between the threshold value T and the
central or non-central 2χ probability distribution function of 22 / nD σ .
PFA
PMD
D2/σσσσ2n (faulty case)
Non-central χχχχ2 pdf
D2/σσσσ2n (fault free case)
central χχχχ2 pdf
Threshold
Figure 7 : Schematic representation of the threshold value T
and 22 / nD σ probability density function (pdf)
The probabilities of false alarm FAP and of misdetection MDP having been explained, the
Detector Operating Characteristics (DOCs) can now be defined. The DOCs are the
graphical representation of MDP vs FAP [20]. In some application a certain MDP is required,
however, the value of the threshold is based solely on FAP . Hence, the DOCs allow to
28
determine from any FAP the corresponding MDP (Figure 8). Equation (33) describes this
relationship.
( ) ( ) ( )∑∑=
−
=
−−⋅=
−=
l
iii
nFA
l
iii
nn
FAFAMD S
BnlnlPQP
lS
Bnl
PTP
lPP
12
21
12
2
2,
1,
1
σσσ (33)
Figure 8 shows examples of DOCs:
10-3
10-2
10-1
100
10-3
10-2
10-1
100
PFA
PM
D
B/σn = 5
B/σn = 7.5
B/σn = 10
B/σn = 12.5
B/σn = 15
B/σn = 17.5
B/σn = 20
Figure 8: Detector Operating Characteristics (DOCs) for various maximal errors (i.e. B) to 2
nσ ratio
2.4.2.4. Fault Isolation Algorithm
The fault identification process is based on the maximum likelihood estimation (MLE)
approach [17]. The isolated erroneous measurement is the value having the highest iii /Sf 2
ratio [20]. Moreover, the probability of misidentification, under the assumption of equally
likely measurement faults is:
∑
<= ibMAX
jj
j
jii
iMI S
f
S
fP
lP
221 (34)
29
2.5. Integration of Redundant IMUs with GPS
This section aims to present the mechanization approach applied for the integration of GPS
with a single IMU. Then, it exposes the three mechanizations approaches described by [9].
2.5.1. Single IMU Mechanization
An extended Kalman filter (EKF) has been implemented in the local level frame
(superscript n ) which makes the interpretation of the state variables straightforward. The
following strapdown equations needs to be solved [1, 15]:
( )( )
+⋅+×+−⋅=
nin
nib
nb
nnnie
nin
bnb
n
nb
n
n
ωωR
gvωωfR
v
R
v
r
&
&
&
(35)
Given the observations bibω and bf , the knowledge of ng , and the initial conditions.
For the inertial measurements, a simplified model was considered. Indeed, assuming that
the misalignments, drifts and constant offsets could not be decorrelated efficiently given
the characteristics of the MEMS-IMU sensors and limited integration periods, only a bias
term is considered [1, 8]. Their associated errors are modeled as first order Gauss-Markov
processes:
bb ll
bb ll wb ++=ˆ
bbbbbb llllllwbb ⋅⋅⋅+⋅−= βσβ 22&
(36)
where bl is the estimated inertial observation (specific force bf or rotation rate bibw ), bl
the inertial measurement, blb the bias of the inertial measurement, blw the measurement
noise, 2bl
σ the covariance at zero time lag and blβ the inverse of the correlation time [1,
25].
2.5.2. Synthetic Mechanization
In the synthetic mechanization approach, redundant inertial data are projected to an
arbitrary non-redundant IMU (i.e. the synthetic IMU), before being introduced in the
30
GPS/INS algorithm based on the standard single IMU mechanization (Figure 9) [1, 9]. In
this section, the mathematical construction of a synthetic IMU composed of n triad of
accelerometers, gyroscopes and magnetometer will be described.
IMU 1IMU 1
IMU 2IMU 2
IMU nIMU n
GPSGPS
Synth. IMUSynth. IMU NAV solNAV solNAV procNAV proc
Figure 9 : Principle of mechanization based on a synthetic IMU
Assume the vectors ibwl , ib
fl and ibml represent the observation vectors for the gyroscopes,
the accelerometers and the magnetometers respectively. For example, lwl can be described
as:
[ ]Tzyxzyxlw nnn
rrrrrr K111
=l
Then, assuming perfect measurements it follows that:
⋅
=
b
b
b
m
f
w
bm, true
bf, true
bw, true
oo
oo
oo
i
i
i
m
f
w
A
A
A
l
l
l
(37)
where bw , bf and bm represent the synthetic IMU “measurements.” wA , fA and
mA are ( )33 ×⋅n matrices that transform the reference body frame (superscript b ) into
the body frames of the sensor triads (superscript ib ). These transformation matrices can be
view as the product of two rotation matrices that can be expressed as:
31
),(λ
),(λ
),(λ
nnnj
j
j
j,j
jj
jj
b
b
b
b
b
b
j
=
=
ϕ
ϕϕ
R
R
R
R
R
R
AMM
22
11
2
1
(38)
where mfwj ,,= and where:
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( )
−⋅−⋅−⋅−−⋅−
=λλ
λλ
λλ
)(λ
sin0cos
sincoscossinsin
coscossincossin
, ϕϕϕϕϕϕ
ϕR
(39)
Moreover, equation (37) can be rewritten as an error equation by introducing residual
vectors i
w
blv , i
f
blv and i
m
blv , which yields to:
oo
oo
oo
b
b
b
m
f
w
bl
bl
bl
bm
bf
bw
i
m
i
f
i
w
i
i
i
⋅
=
+
m
f
w
A
A
A
v
v
v
l
l
l
(40)
Hence, equation (40) transforms the redundant observation vectors into the observation of
the “synthetic” IMU. Now, to determine the synthetic measurements equation (40) has to
be solved for bw , bf and bm . Considering the redundant nature of the problem, the well-
known least-squares estimation seems appropriate (see Appendix C). This yields to:
⋅
=
bm
bf
bw
m
f
w
b
b
b
oo
oo
oo
l
l
l
Π
Π
Π
m
f
w ( )( )( )
RAARAΠ
RAARAΠ
RAARAΠ
⋅⋅⋅⋅=
⋅⋅⋅⋅=
⋅⋅⋅⋅=
−−−
−−−
−−−
111
111
111
where
bm
bm
bm
bm
bf
bf
bf
bf
bw
bw
bw
bw
ll
Tmmll
Tmm
ll
Tffll
Tff
ll
Twwll
Tww
(41)
where bw
bwll
R , bf
bf ll
R and bm
bmll
R are the covariance matrices of the raw measurements vectors
ibwl , ib
fl and ibml respectively. Under the assumption IR ⋅= 2
0σbw
bwill
, equations (41) can be
simplified as follows:
32
∑
∑
∑
=
=
=
⋅⋅=
⋅⋅=
⋅⋅=
n
iz
y
xTb
bb
n
iz
y
xTb
bb
n
iz
y
xTb
bb
i
i
i
im
i
i
i
if
i
i
i
iw
m
m
m
n
f
f
f
n
r
r
r
n
1
1
1
1
1
1
Rm
Rf
Rw
(42)
These equations are derived in Appendix E.
This synthetic approach does not require modifications of the GPS/INS software, because
the synthetic IMU can be considered as a single IMU. Futher, while fusing the IMU data,
defective sensors can be detected and covariance terms can be estimated. The estimated
“compound” biases, however, cannot be back projected to the individual sensor space and
the noise may be inflated by these unknown errors [1, 9].
2.5.3. Extended Mechanization
The extended mechanization estimates individual sensors errors. Thus, equations (35) and
(36) have to be adapted in order to take into account the sensor redundancy [1, 9]:
( )( )
−⋅⋅+×+−⋅⋅=
ninw
nb
nnnie
nin
nbf
n
nb
n
n
l
l
ωΠR
gvωωRΠ
v
R
v
r
w
f
&
&
&
(43)
As for the synthetic approach, the extended mechanization offers the possibility to detect
defective sensors and to estimate noise terms during the integration (see Figure 10). That
said, on the contrary to the synthetic mechanization, systematic errors can be modeled and
estimated for each sensor. However, this approach requires the modification of the
GPS/INS software to accommodate the new form of mechanization equations.
33
IMU 1IMU 1
IMU 2IMU 2
IMU nIMU n
GPSGPS
NAV solNAV solNAV procNAV proc
Figure 10: Principle of extended IMU mechanization
2.5.4. Geometrically-Constrained Mechanization
As for the extended mechanization, the geometrically-constrained mechanization allows to
estimate the individual sensor errors. In this approach, multiple navigation solutions are
computed (one for each IMU) and compared at regular time intervals (Figure 11).
Unfortunately, this approach increases the computational effort and important
modifications of the GPS/INS software are required. In principle, defective sensors cannot
be detected and realistic noise terms cannot be estimated using this approach.
GPSGPS
IMU 1IMU 1
IMU 2IMU 2
IMU nIMU n
NAV sol 1NAV sol 1
NAV procNAV proc
NAV procNAV proc
NAV procNAV proc
NAV sol 2NAV sol 2
NAV sol nNAV sol n
ConstraintConstraint
Figure 11 : Principle of geometrically-constrained IMU mechanization
Consider that two IMUs are employed. Both units are integrated using the standard IMU
mechanization. The relative orientation parameters (relative orientation 2
1
bbR and lever arm
34
2
1
bba ) can be modelled and estimated as random constants, supposing their direct
determination is not accurate enough [1, 9].
0
0
2
1
2
1
=
=bb
bb
a
R
&
&
(44)
At predefined stages, the following relationships can be imposed:
( )( ) 2
1
111
2
12
2
2
11
21
21
bbw
bbib
bni
nn
bb
nb
nn
nb
bb
nb
abVV
aRrr
RRR
⋅Ω+Ω+Ω−=
⋅−=
⋅=
(45)
where Ω is the skew-symmetric form of the misalignment angles.
2.5.5. Algorithm Selection
Each of the three approaches described earlier have their advantages and disadvantages.
The synthetic mechanization requires small additional computational efforts (compared to
a single IMU mechanization) and does not necessitate any modification of the standard
GPS/INS algorithm. However, unlike the extended method, this approach does not allow
the feedback of sensor errors which might yield less optimal navigation performance. On
the other hand, [8] has shown that the MEMS-IMU biases are relatively stable for the short
trajectories encountered in some sports (e.g. lap, downhill). Furthermore, the same
research has shown that the simplified error model was suitable for the considered
application and sensors.
The geometrically-constrained approach represents an interesting option for system
calibration if the relative sensor geometry is insufficiently known. As mentioned, the
computational effort is increased considerably compared to the first two approaches. In
addition, it is more sensitive to sensor failures because errors can only be noticed at the
update stage and the measurement faults can generally not be isolated. Finally, the
geometrically-constrained approach is particularly interesting in situations where several
navigation solutions of different element of a dynamic object are matter of interest (e.g.
different parts of an athlete’s body). However, this is beyond the scope of this research. In
the subsequent sections, the synthetic IMU approach will be compared to the approach
based on extended IMU mechanization.
35
3. IMU Architectures
3.1. Introduction
This chapter aims to identify optimal sensor architecture for redundant IMUs. First, will be
considered optimal configurations of two sensors in a two dimensional space and then of
three sensors in three dimensions to illustrate the methods applied in this project.
Thereafter, optimal configurations using four sensors will be described and compared to
the work of [12]. Finally, the case of n sensor triads will be analyzed.
3.2. Assessment in Two Dimensions
This first section will verify the assumption that the best configuration for two sensors (of
the same precision) in 2D is an orthogonal configuration and that the orientation of the first
sensor is not important. Figure 12 presents the situation.
x
y
α
β
Figure 12 : Configuration of two sensors in planimetry
The design matrix H (see section 2.2) derived from this system is:
( ) ( )( ) ( )
++=
βαβααα
sincos
sincosH
Thus, the information content can be computed as defined by equation (6), that is,
Sensor 1
Sensor 2
36
( ) ( ))cos(21det 21 β⋅−⋅=⋅= HHTJ
The maximization of J yields to:
:maxJℜ∈α
( ) Ζ∈⋅+=⇒=⋅=∂∂
kkJ
,02sin 2 πβββ
π α∀ with 1max == JJ
Thus, configurations are optimal as long as 2πβ = .
3.3. Optimal Configuration with Three Sensors in Three Dimensions
Consider now the case where three sensors (with the same error characteristics) are to
placed in an optimal configuration, in a three dimensional space. Intuitively, this is
achieved when each device is placed orthogonally to each other, regardless of the
orientation of the first sensor. This is verified by the information filter approach. Note that
the partial redundancy method cannot be use here (nor in the previous section) because the
total redundancy of the system is null.
y
z
x
β
α
Figure 13: Configuration of three sensors in a three dimensional space
Figure 13 illustrates the system presented above which yields the H matrix:
( ) ( )( ) ( ) ( ) ( ) ( )
⋅⋅=
ββαβααα
sincossincoscos
0sincos
001
H
37
From this, the information volume can be computed:
( ) ( ) ( ) ( )βαβα 22 sinsindet, +=⋅= HH TJ
The maximization of J yields to:
⇒ℜ∈
Jβα ,
max 2πβα == with 1max == JJ
Hence, this result indicates the system’s optimality. Figure 14 shows the relationship
between J , α and β .
0
1
2
3
01
23
0
0.2
0.4
0.6
0.8
1
αααα [rad]ββββ [rad]
J [-
]
Figure 14: Influence of α and β on the information volume
3.4. Optimal Configuration with Four Sensors in Three Dimensions
This section considers four sensors and evaluates the optimality of configurations proposed
in the literature. Indeed, the work of [10, 12, 26] suggests three possible (optimal)
configurations:
• One sensor placed along the central cone axis while the three remaining sensors
are positioned equally around the cone of half-angle α (Figure 1)
• Equally spaced sensors around a half-angle cone α (Figure 1)
• Tetrad configuration (Figure 15).
The maximum information volume of the system is (equation (8)):
38
3704.23
43
max =
=J
This will be used to assess whether or not a configuration is optimal.
Consider the first cone configuration (Figure 1), the system’s design matrix is the
following:
( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
⋅⋅⋅⋅
=
⋅⋅
⋅⋅
αααααααα
ππ
ππ
cossinsincossin
cossinsincossin
cos0sin
100
34
34
32
32
H
The results using this information approach confirm the results of [12] and show that the
configuration is optimal, indeed,
( ) ( )( )αα 2449 cos31sin ⋅+⋅⋅=J
⇒ℜ∈
Jαmax α =70.4738° for maxJJ =
The same conclusion is also reached with the partial redundancies method. Indeed,
( )( )( ) ( )( ) ( )( ) ( )( )
T
z
⋅+⋅⋅+⋅⋅+⋅⋅+⋅=
ααααα
2222
2
cos313
1
cos313
1
cos313
1
cos31
cos3
( ) ( )( )( )[ ]( )⇒−ℜ∈
2min ααα ii zEzE α =70.4738° for 4 ,3 ,2 ,1 25.0 =≈ izi
with ( ) ( )( )( )[ ]( ) ⇒≈− 02αα ii zEzE optimal configuration.
The property related to redundancy described in equation (11) can be verified with this
example:
( )( )( ) ( )( ) runz
ii =−==
⋅++
⋅+⋅=∑
=
1cos31
1
cos31
cos322
24
1 ααα
Overall, the two methods provide the same results. The information filter allows to
determine the optimal configuration, whereas the partial redundancies approach quantifies
the amount of control for each measurement.
Consider now the second cone configuration (Figure 1), the design matrix becomes:
39
( ) ( )( ) ( )
( ) ( )( ) ( )
−−
=
αααααααα
cossin
cossin0
cos0sin
cossin0
o
H
The results provided by the information filter approach, again, confirm the results obtained
by [12] and yield an optimal configuration of
( ) ( )αα 24 cossin16 ⋅⋅=J
⇒ℜ∈
Jαmax °= 7175.54α with maxJJ =
The partial redundancies method provides the same results with 25.0≈iz for 4 ,3 ,2 ,1=i
and thus, ( ) ( )( )( )[ ] ⇒≈
− 02αα ii zEzE optimal configuration.
Finally, the tetrad configuration is now considered (see Figure 15).
y
z
x
α
β
Figure 15: Tetrad configuration proposed by A. Pejsa in [12]
The design matrix associated to this system is:
( ) ( )( ) ( )( ) ( )
( ) ( )
−−
=
0cossin
0cossin
cos0sin
cos0sin
ββββ
αααα
H
40
The results of the information filter method are:
( ) ( ) ( ) ( )( ) ( ) ( )βαβαβα 2222 coscossinsin8det, ⋅⋅+⋅=⋅= HHTJ
⇒ℜ∈
Jβα ,
max == βα 35.323° for maxJJ =
With the partial redundancies, the result also demonstrate the optimality for
== βα 35.323°, confirming the results of [12], which yields to 25.0≈iz for
4 ,3 ,2 ,1=i . Figure 16 shows the relationship between J , α and β .
0
0.5
1
1.50
0.5
1
1.5
0
1
2
3
αααα [rad]ββββ [rad]
J [-
]
Figure 16 : Influence of α and β on the information volume
Hence, the configurations proposed in the literature, are all equivalent in terms of
optimality.
3.5. Optimal Configuration with n Sensor Triads
The optimal configuration with n sensor triads in a three dimensional space is now
explained. Intuitively, skew-redundant configurations have a higher information volume
and are thus preferable. Indeed, this strategy was adopted in [9, 10]. However, as far as
sensor triads are employed, orientation between triads is unimportant with respect to the
system optimality. Consider a situation of n sensors randomly oriented. The associated
design matrix is expressed as:
41
=
),(λ
),(λ
),(λ
nn ϕ
ϕϕ
R
R
R
HM
22
11
where ),(λ ii ϕR correspond to the product of two rotation matrices (as defined in equation
(39)) and can describe any rotation in a three dimensional space. From the H matrix, the
information volume can be computed, that is:
( ) ( )),(λ),(λ),(λ),(λHHJ nnT
nnTT ϕϕϕϕ RRRR ⋅++⋅=⋅= K1111detdet
For rotation matrices, we have: 3IRR =⋅ iTi [27]. As ),(λ ii ϕR are defined as the product
of two of rotation matrices (i.e. 32 RRR ⋅=),(λ ii ϕ ), it implies that :
33223 IRRRRRR =⋅⋅⋅=⋅ TTii
Tii ),(λ),(λ ϕϕ
Note that this relationship is demonstrated in Appendix E. Hence, J becomes:
( ) ( ) 3333 detdet nnJ =⋅=++= III K
The maximum information volume can be computed as:
33
max 3
3n
nJ =
⋅=
In this way, the configuration is optimal and the information volume is independent of
ii ,λ ϕ (i.e. orientation of the triads).
The same result can be obtained using the partial redundancies method. For this system, Z
matrix can be expressed as:
42
( )( )
⋅
⋅−=
⋅⋅
⋅⋅−=
⋅⋅−=
⋅⋅⋅−=
⋅
⋅
⋅
⋅
311
113
3
11
111111
3
13
3
1
1
IRR
RRI
I
RRRR
RRRR
I
HHI
HHHHIZ
L
MOM
L
L
MOM
L
),(λ),(λ
),(λ),(λ
n
),(λ),(λ),(λ),(λ
),(λ),(λ),(λ),(λ
n
Tnn
nnT
n
nnT
nnT
nn
nnTT
n
Tnn
TTn
ϕϕ
ϕϕ
ϕϕϕϕ
ϕϕϕϕ
implying that:
]3,1[ 1
nin
nzi ⋅∈∀−=
All iz are equal, thus, ( ) ( )( )( )[ ]( ) 02 =− αα ii zEzE . The configurations are therefore
optimal, regardless of the orientation of the triads (this is also proved using simulation in
section 5.8).
3.6. Note on the Impact of Sensor Failures
In the previous section, we demonstrated that the orientation of sensor triads is irrelevant to
a system’s optimality. We now consider whether the failure of one of the sensor could alter
this conclusion and therefore render the triads’ relative orientation important. To answer
this question, consider the system presented in Figure 17.
y
z
x
y
z
x
Figure 17 : Six sensors are placed in two different configurations. The symbol indicates the sensors that will fail (superscript *).
43
The design matrices associated are:
=
3
31 I
IH and
=
)(λ ϕ,3
2 R
IH
First, the two systems yield an equivalent information volume and have equal partial
redundancies:
( ) 82det 321 =⋅== IJJ and ]6;1[ 5.02,1, ∈∀== izz ii
It was shown previously that the configurations are optimal, indeed,
836
3
max21 =
=== JJJ
Assume now that the sensors marked by in Figure 17 fail. Then, the design matrices
then becomes (the superscript * refers to system after the failures):
=
3
*1 100
010
IH and
=
)(λ ϕ,100
010*2
RH
These failures modify the optimality of the system, as well as the information volume,
leading to the following result:
630.43
54
3*max
*2
*1 ≈
=≠== JJJ
Hence, although the configurations are theoretically equivalent in the information approach
(because *2
*1 JJ = ), they are not in reality. This can be can be demonstrated using the
partial redundancy approach, where:
224.0]][[] 0 [ 2*2
12
12
12
1*1
11≈−⇒= zEzEz
( ) ( )( )( ) ( ) ( ) ( )( )( ) ( )( )
−⋅−⋅+⋅−⋅−⋅= λλϕλλϕ 222222*2 cos1
21
cos1coscos21
1coscos121
21
21
z
44
Figure 18 shows the relationship between the standard deviation of *2z and ϕ λ, . This
figure shows that 0.224 (i.e. the standard deviation of *1z ) is close to the maximum value
and that the preferred situations (i.e. skewed configuration, see Figure 18) exists. Thus,
skewed configurations are preferable in the case of sensor failures.
0.05
0.1
0.15
0.2
0.25
θθθθ [rad]λλλλ [rad]
E[(
z-E
(z)2 )]
[-]
0ππππ/4
ππππ/23⋅⋅⋅⋅ππππ/4
ππππ
0ππππ/4
ππππ/23⋅⋅⋅⋅ππππ/4
ππππ
(E[z
-E[z
]2 ])1/
2
Figure 18: Influence of λ and ϕ on the standard deviation of *2z
3.7. Conclusions
To summarize the results obtained in this chapter, we recall the main features:
• The partial redundancies and the information filter approaches give, in most cases,
similar results. That said, the partial redundancies approach is a better indicator of
the system’s optimality in certain cases.
• The best geometrical configuration of three sensors in a three dimensional space is
orthogonal to each other (i.e. sensor triads).
• The relative orientation of n sensor triads is irrelevant as long as no sensor fails,
in such case, skewed configurations are preferable.
45
4. Experimental Assessment of Redundant IMUs
4.1. Introduction
The objectives of this chapter are to assess experimentally the advantages of using multiple
IMUs. Firstly, we will evaluate the improvements in navigation (i.e. position, velocity,
orientation) resulting from the use of redundant IMUs (compared to a single IMU).
Secondly, the noise reduction stemming from the redundancy of measurements will be
experimentally assessed. Then, the algorithm of noise estimation during the processing
will be tested. Finally, the results of FDI, using the parity space method, will be exposed.
4.2. Experimental System Setup
The results presented here are based on an experiment realized with the scan2map system
(see Figure 19) which was augmented by a regular tetrahedron consisting of 4 Xsens MT-i
IMUs (see left part of Figure 19). We decided to utilize this practical skew-redundant
configuration which optimized the controllability of measurements in case of sensor failure
after the result obtained in the third chapter (see section 3.6). In order to investigate the
performance of the multi-IMU system, it was fixed rigidly to a reference system consisting
of a tactical-grade IMU (LN200) and a differential, dual-frequency GPS receiver (Javad
Legacy) [1, 28]. The system was installed as shown in Figure 19.
Tetrahedron with4 Xsens MT-i Tactical grade
IMU (LN200)
Base station
GPS antennas(on the top)
Figure 19 : (from left to right) (1) Skew-redundant IMUs placed in a tetrahedron. (2) Stéphane Guerrier holding the scan2map system. (3) Installation of the base station by Yannick Stebler
46
4.3. Improvement in Navigation Performance
This section explains the improvements brought to navigation performance when four
MEMS-IMUs (placed on a tetrahedron) are utilized (Figure 19). Table 1 summarizes the
navigation performance of the particular GPS/MEMS - Skew redundant IMU (SRIMU)
system. The orientation accuracy of this system drops below 1 deg. The experiment
confirms, on one hand, the findings of [29] where the velocity and orientation accuracies
are not dependent on the accuracy of GPS aiding (e.g. L1 or L1/L2 differential code and
carrier-phase). On the other hand, position accuracy is largely improved using dual-
frequency GPS processing with ambiguity fixing. Because of the relatively long baseline
for single-frequency CP-DGPS, the ambiguities could not be fixed, which resulted in code-
differential positioning accuracy.
L1 L1/L2 L1 L1/l2Position [m]North 0.83 0.03 0.83 0.03East 2.40 0.05 2.40 0.04Down 0.81 0.08 0.81 0.07Velocity [m/s]North 0.07 0.05 0.07 0.03East 0.07 0.07 0.07 0.05Down 0.11 0.10 0.12 0.06Attitude [deg]Roll 0.69 1.09 1.04 0.83Pitch 0.79 1.05 0.92 0.86Heading 0.42 0.68 0.67 0.62
Synthetic Extended
Table 1 : Average absolute accuracy of the tested GPS/MEMS-SRIMU system
Table 2 presents the performance improvement of the GPS/MEMS-SRIMU system,
compared to the average accuracy results obtained with the single MEMS-IMUs. The
synthetic IMU approach shows an average improvement of 30%. The extended
mechanization performs slightly better than the synthetic approach, providing for an
average improvement of 46%. This difference can be explained by the estimation of the
individual biases and the FDI scheme that run parallel to the filter (rather than in cascade
as in the synthetic IMU approach). The navigation performance is, however, not improved
by 100% as could be expected from the noise reduction (see section 4.4.1). Indeed,
residual correlations between the inertial measurements, as well as the correlations
between the filter states, most likely limit the progression of accuracy [1].
47
Table 2 : Average and maximum errors of performance resulting from the use of the MEMS-
SRIMU compared to the performance of the individually integrated MEMS-IMU sensors.
Using this technique, maximum errors are diminished, as shown in Table 2. That is, for the
synthetic IMU approach, it is of 61%, whereas for the extended mechanization maximum
errors are further reduced to a factor of two. Figure 20 illustrates how orientation error
peaks in the single GPS/MEMS-IMU integration are smoothed out by the extended
mechanization.
Figure 20 : Orientation errors after integration of a single sensor compared to extended mechanization (four sensors)
Table 2, moreover, shows a reduction in position and velocity errors, which are explained
by the significant improvement achieved in the determination of accelerations (see Figure
21). Indeed, because less noise is integrated, the computation of positions and velocities is
more precise and is smoothed out, compared to a single GPS/MEMS-IMU integration.
Synthetic Extended Synthetic ExtendedPositionNorth -29% -35% -71% -94%East -41% -51% -77% -94%Down -5% -19% -27% -82%VelocityNorth -37% -61% -82% -95%East -50% -67% -81% -93%Down -7% -44% -25% -74%AttitudeRoll -57% -67% -80% -87%Pitch -28% -41% -59% -80%Heading -21% -27% -45% -69%
RMS Maximum Error
48
Figure 21 : Position, velocity, and acceleration errors after integration of a single sensor compared to extended mechanization (four sensors)
4.4. Noise Reduction and Estimation
4.4.1. Experimental Noise Reduction
In the second chapter, we explained with equation (13) that the theoretical noise reduction
depends on a number of independent measures (i.e. the number of sensors). Therefore, the
use of four identical (MEMS-) IMUs, as in this experiment, implies a theoretical noise
reduction of a factor of two, compared to an individual (MEMS-) IMU. Hence, the
expected noise reduction is of 100%.
This theoretical noise reduction was verified by comparing the differences between the
MEMS-IMUs measurements and their best estimates (combining the measurements) with
the reference measurements provided by a tactical-grade IMU (LN200). However, this was
only realized for the gyroscopes, indeed, the (LN200) accelerometers are highly subject to
quantization noise. Thereafter, a parametric compensation was performed to remove
system errors. The remaining differences were thus considered to be only white noise. The
average noise of the four MEMS-IMUs gyros was estimated at 0.0194 [rad/s], whereas the
noise level of their best estimate at 0.0101 [rad/s]. Hence, the experimental noise reduction
is approximately 92 % which confirms the validity of the theoretical model. Figure 22
illustrates these results graphically.
49
49 49.05 49.1 49.15 49.2 49.25 49.30.9
0.92
0.94
0.96
0.98
1
1.02
1.04
1.06
1.08
1.1
Time [s]
Ang
ular
Rot
atio
n ra
te [r
ad/s
]
LN200Best estimate IMU (σ
x = 0.010 [rad/s])
MTi-G 11 (σ1 = 0.017 [rad/s])
MTi 12 (σ2 = 0.018 [rad/s])
MTi 13 (σ3 = 0.018 [rad/s])
MTi 14 (σ4 = 0.025 [rad/s])
Figure 22 : Comparison of the angular rate measurements of 4 MEMS-IMU to the reference measurements from a tactical-grade IMU
4.4.2. Assessment of the Noise Estimation Algorithm
As mentioned in section 2.3, the use of multiple IMUs allows to estimate noise level
during the processing. Figure 23 shows noise variations for a selected portion of the
experiment.
20 25 30 350
0.01
0.02
0.03
0.04
0.05
0.06
Time [s]
σσ σσA
ngul
ar r
otat
ion
rate
[rad
/s]
20 25 30 350
0.1
0.2
0.3
0.4
0.5
Time [s]
σσ σσS
peci
fic fo
rce
[m/s
2 ]
Averaging time: 0.01 [s]Averaging time: 0.1 [s]Averaging time: 1 [s]Averaging time: 10 [s]
Figure 23: Estimation of the noise level for a selected portion of the experiment
The “real” noise level was estimated using the data of the tactical-grade IMU (LN200). The
results are presented in Figure 24.
50
25 25.5 26 26.5 27 27.5 28 28.5 29 29.5 300
0.01
0.02
0.03
0.04
0.05
0.06
Time [s]
σσ σσA
ngul
ar r
otat
ion
rate
[rad
/s]
39 39.5 40 40.5 41 41.5 42 42.5 43 43.5 44
0
0.1
0.2
0.3
Time [s]
σσ σσS
peci
fic fo
rce
[m/s
2 ]
Averaging time: 0.01 [s]Averaging time: 0.1 [s]Averaging time: 1 [s]Averaging time: 10 [s]Noise level estimated with ln200
Averaging time: 0.01 [s]Averaging time: 0.1 [s]Averaging time: 1 [s]Averaging time: 10 [s]Noise level estimated with ln200
Figure 24: Comparison between the noise estimates and the “real” noise (based on ln200’s data) calculated during the experiment
Hence, the implemented adaptive filter shows results that are relatively close to those
obtained with the tactical-grade IMU approach. It allows to visualize the variation in noise
level during the processing and, consequently, to adapt the stochastic model of the EKF. In
order to test if this algorithm can improve navigation performances, we compared the
results of the GPS/INS integration with and without the noise estimation algorithm
presented above. The results are shown in Figure 25.
13 13.5 14 14.5 151
1.25
1.5
Hea
ding
err
or [d
eg]
Time [s]
with noise estimationwithout noise estimation
Figure 25 : Orientation error after integration of four sensors (extended mechanization), with and without the noise estimation algorithm
51
Orientation errors are slightly reduced, approximately 3%, because the algorithm provides
a more realistic stochastic model for the GPS/INS software. We caution, however, that this
conclusion was obtained with the results of a single experiment (presented in section 4.1),
for which the parameters influencing noise level (i.e. vibrations or other external
environmental) were considered to be constant. Nevertheless, orientation improvement
might be bigger when the noise level is expected to vary.
4.5. Fault Detection and Isolation
This section presents the results obtained with the parity space method introduced in
section 2.4. This method is based on the assumption of strictly Gaussian white noise errors.
However, MEMS-IMUs are known to be subject to systematic errors. As a result, we first
verified the white noise assumption, then, we experimentally assessed the performances of
this method with respect to detection and isolation.
4.5.1. Assessment of the Noise Characteristics of in MEMS-IMUs
This section aims to highlight if colored noise is present in Xsens MT-i (MEMS-IMUs)
measurements. Results were obtained after rotating tactical grade IMU (LN200)
measurements, which can be assumed to be perfect in this context, into each Xsens MT-i
frame and then by comparing the measurements.
-100 -50 0 50 100-1
0
1
2
Sensor 11 (Gyro): rx
Time [s]
Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 100-2
0
2
4
Sensor 11 (Gyro): ry
Time [s]
Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 1000
2
4
6
Sensor 11 (Gyro): rz
Time [s]
Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 100-2
0
2
4
Sensor 12 (Gyro): rx
Time [s]Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 100-2
0
2
4
Sensor 12 (Gyro): ry
Time [s]
Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 100-2
0
2
4
Sensor 12 (Gyro): rz
Time [s]
Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 100-2
0
2
4
Sensor 13 (Gyro): rx
Time [s]
Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 100-2
0
2
4
Sensor 13 (Gyro): ry
Time [s]
Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 100-2
0
2
4
Sensor 13 (Gyro): rz
Time [s]
Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 1000
5
Sensor 14 (Gyro): rx
Time [s]
Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 1000
5
10
Sensor 14 (Gyro): ry
Time [s]
Aut
ocor
rela
tion
[rad
2 /s2 ]
-100 -50 0 50 1000
5
10
Sensor 14 (Gyro): rz
Time [s]Aut
ocor
rela
tion
[rad
2 /s2 ]
Figure 26: Autocorrelation of errors of the Xsens MT-i’s gyros
52
Thereafter, we analyzed the “nature” of these differences using autocorrelation techniques
(Figure 26). This clearly shows that a large portion of the non-faulty noise measurements is
composed of colored noise. Indeed, the autocorrelations below suggest that the errors are
mainly composed of white noise and of random walks.
This experiment also enabled to assess the normality of data. Figure 27 presents a “normal
probability plot” of the sensors errors. The data are plotted against a theoretical normal
distribution in such way that the normally distributed points follow a straight line. This
Figure shows quasi-Gaussian results [23].
To conclude, our simulations showed that the assumption of strictly Gaussian white noise
errors in the parity space approach is not valid in the case of MEMS-IMUs. Nevertheless,
we believe that, in sports applications, this FDI approach could be a valuable additional
feature to the system to detect gross errors and enhanced the navigation performance.
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
0.001
0.003
0.01
0.02
0.05
0.10
0.25
0.50
0.75
0.90
0.95
0.98
0.99
0.997
0.999
Difference vectors for the MTi 12 ry [rad/s]
Pro
babi
lity
Figure 27: Graphical assessment of the normality of Xsens MT-i errors (sensor 12, gyro ry)
4.5.2. Performance of the Parity Space Method with MEMS-IMUs
The performance and the applicability of the parity space method to MEMS-IMUs was
also assessed using the experiment presented in section 4.2. The tactical grade IMU
(LN200) measurements (assumed to be perfect) were compared to the four Xsens MT-i’s
measurements. Thus, twelve different vectors were created (showing a quasi-Gaussian
behavior, Figure 26). Their respective standard deviations were computed and all
differences exceeding 3.5 - their standard deviation - were considered to originate from
erroneous measurements. This approach resulted in a list of “true” errors which allows
knowing, at each sample time, if one (or more) fault occurs, and to identify the defective
53
sensor. The algorithm is schematized in Figure 28. It should be noted that the assessment
of the applicability of the parity space method was only realized with gyros since the
accelerometers are highly subject to quantization noise [30].
Xsens MT-i
(Sensor 11)
Xsens MT-i
(Sensor 12)
Xsens MT-i
(Sensor 13)
Xsens MT-i
(Sensor 14)
ln200
1dσ
2dσ
3dσ
4dσ
Difference(t) > 3.5 1dσ
Difference(t) > 3.5
Difference(t) > 3.5
Difference(t) > 3.5
2dσ
3dσ
4dσ
Fault
Fault
Fault
Fault
NO Fault
NO Fault
NO Fault
NO Fault
Figure 28: Schematized algorithm for the detection and isolation of errors
The decision variable D is computed as follows:
whereff ⋅= TD
( )( ) zAAAAIf l ⋅⋅⋅⋅−= − Twwww
1
where 12=l is the number of measurements, 3=n is the number of independent
parameters, wA the matrix which converts the state space into the measurement space (as
defined in equation (38)) and z represents the vector of measurements.
The threshold T is computed according to equation (28). Two thresholds were computed
with different probabilities of false alarm FAP (i.e. 5% and 1%):
[ ]( ) [ ]22205.0 /srad 0155.0Hzrad/s/ 01.0,9,05.0 ===−= nFA nlPT σ
[ ]( ) [ ]22201.0 /srad 0198.0Hzrad/s/ 01.0,9,01.0 ===−= nFA nlPT σ
Firstly, these thresholds were compared to an optimized threshold which minimizes the
sum of false alarm probability and of misdetection [24]. The result was estimated to be
54
[ ]22/srad 0156.0ˆ =T . Thus, we observe that the theoretical threshold values computed with
the parity space method and the empirical best threshold value are of the same order of
magnitude. This experimentally derived T value can be used for future data sets. Figure
29 represents the values of D and T , and shows when fault and correct measurements
are detected successfully, as well as the occurrence of misdetections and false alarms.
46 48 50 52 54 56 58 60 62 64 66
0.005
0.01
0.015
0.02
0.025
Time [s]
D &
T [r
ad2 /s
2 ]
Decision variable (D)Threshold (T)Successful detection of non faulty measurement (95.4 %)Successful detection of faulty measurement (1.0 %)False alarm (0.4 %)Miss detection (3.2 %)
Figure 29: D values compare to T
Moreover, this figure shows the difficulty of finding a good threshold value. Indeed, even
with the best possible T value, approximately 76% of faults are not detected. The number
of false alarms is also important; the false-alarm-to-fault ratio is roughly 35%. The
performance of this test is ( ) %4.961 =+− MDFA PP . Figure 30 shows the Standford plot
of these results.
55
0 1 2 3 4 5 60
0.005
0.01
0.015
0.02
0.025
Quantile
Dec
isio
n V
aria
ble
D [r
ad2 s
- 2]
0
1
2
3
4
5
6
7
95.4%
False alarms0.4%
Misdetections3.2%
T
Successfullydetected errors
1%
3.5 σerror
Point density (per square)
Figure 30: Standford plot with TT ˆ=
Regarding the performance of the identification algorithm, Figure 31 shows that 74.4% of
faults are correctly isolated and 25.6% are not. The results developed here for T are also
valid for 05.0T .
46 48 50 52 54 56 58 60 62 64 660
0.005
0.01
0.015
0.02
0.025
0.03
Tim e [s]
D &
T [r
ad2 /s
2 ]
Decision variable (D)Threshold (T)Successful detection of non faulty measurement (95. 4 % )Successful detection of faulty measurement (1.0 % )False alarm (0.4 % )Miss detection (3.2 % )Correct isolation of fault (74.4 % )Wrong isolation of fault (25.6 % )
Figure 31: D values compared to T value with identification of performance
56
Considering the second threshold 01.0T , Figure 32 shows the values of D and 01.0T as well
as the overall performance of the test, which is ( ) %8.951 =+− MDFA PP .
45 50 55 60 65 70 75 800
0.005
0.01
0.015
0.02
0.025
0.03
Time [s]
D &
T [r
ad2 /s
2 ]
Decision variable (D)Threshold (T)Successful detection of non faulty measurement (95.8 %)Successful detection of faulty measurement (0.1 %)False alarm (0.01 %)Miss detection (4.1 %)
Figure 32: D values compared to 01.0T
Figure 33 shows the Standford plot for 01.0T .
Number of measurements
0 1 2 3 4 5 60
0.005
0.01
0.015
0.02
0.025
Quantile
Dec
isio
n V
aria
ble
D [r
ad2 s
- 2]
0
1
2
3
4
5
6
7
T
Misdetections4.1%
95.8%
False alarms0.01%
Successfullydetected errors
0.1%
3.5 σerror
Point density (per square)
Figure 33: Standford plot with 01.0TT =
57
Finally, the results of the identification algorithm are given in Figure 34.
45 50 55 60 65 70 75 800
0.005
0.01
0.015
0.02
0.025
0.03
Time [s]
D &
T [r
ad2 /s
2 ]
Decision variable (D)Threshold (T)Successful detection of non faulty measurement (95. 8 % )Successful detection of faulty measurement (0.1 % )False alarm (0.01 % )Miss detection (4.1 % )Correct isolation of fault (89.9 % )Wrong isolation of fault (11.1 % )
Figure 34: D values compared to T values with identification of performance
Thus, 05.0T is very similar to the best empirical threshold T , which indicates that 05.0T is
a more appropriate value than 01.0T . The performance of 05.0T is also higher than the
performance of 01.0T . That said, the main purpose of the FDI in non-safety critical
applications is to increase the navigation performance and to prevent gross errors. This
improvement being linked to the noise reduction, we corrected the detected faults and
assessed if this would enhance the noise reduction. Thus, for all faults detected (with 05.0T
and 01.0T ) a corrected IMU was computed based equation (46) and the erroneous
measurement was corrected.
( )( )( )( )
= ∑≠ij
z
y
xTb
bbbcorrectedi
tr
tr
tr
t
j
j
j
jiRRIMU 3
1 , (46)
Next, the noise reduction of the two sets of corrected data was studied. Without
corrections, the noise is reduced by 92% when four IMUs are used instead of one (see
section 4.4.1). With the corrected sets - at 05.0T - noise reduction was estimated at 88%,
therefore, decreasing navigation performances compared to non corrected data. However,
58
with 01.0T the noise reduction was roughly 93%, resulting in a slight increase in the
navigation performances. Consequently, 01.0T is preferred to 05.0T .
4.6. Conclusions
In the first part of this chapter, we showed that the navigation performance can be
improved by 30-50% when using four MEMS-IMUs, while maximum errors can be
reduced by a factor of two. Two integration approaches have been investigated, namely,
the synthetic and the extended mechanization. The second appears to be more optimal for
the system calibration, because the characteristic errors of individual sensors are addressed
separately. However, in the extended mechanization, where the state vector is increased by
individual biases for each sensor, estimation of biases does not reflect the reality. This
might be explained by the reduced observability of the system [1].
In the second part, we demonstrated that the expected and the experimental noise
reductions, respectively 100% and 92%, correspond. The noise estimation algorithm
additionally appears to be able to efficiently estimate the variations in noise levels. This
algorithm tends to improve slightly navigation performance when expected noise level is
constant (e.g. no vibration increases during the processing). However, it might provide
larger improvements when the noise level is expected to vary.
In the last part of this chapter, we demonstrated that the parity space method can be
successfully applied to MEMS-IMUs, because it can increase navigation performance and
prevent gross errors. It was shown that the theoretical threshold matched the best possible
threshold value, derived with reference data. However, the percentages of undetected
errors (76%), as well as the level of false alarms, remain high, showing that more complex
FDI models need to be developed. The low performance of these results may be explained
by the fact that MEMS-IMUs are largely subject to systematic errors (comparatively to
higher grade IMUs). Indeed, when dealing with MEMS-IMUs, the assumption of strictly
Gaussian white noise errors is incorrect.
59
5. Emulation of Redundant IMU Setups
5.1. Introduction
This chapter aims to present the simulation tool developed for MEMS-IMU measurement
in this project. In the context of redundant IMUs, simulation, or more precisely emulation
(which generates data corresponding precisely to the real dynamic studied), can also be
extremely helpful. Indeed, it is laborious and expensive to experimentally assess the
performance of various IMU configurations. However, this can be achieved through
simulation. In order to obtain realistic results, the following approach was followed. First,
a reference trajectory, based on a tactical-grade IMU (LN200) and a differential dual-
frequency GPS receiver (Javad Legacy), was used to generate (i.e. inverse strapdown) the
assumed perfect measurements. Then, perturbations reflecting characteristic error of IMUs
were added. These results enabled to easily emulate different configurations.
5.2. Inverse Strapdown
This section aims to define the equations that emulate the (assumed) perfect measurements
based on a reference trajectory. These equations are given in the standard notation and are
taken from [7] which presents their derivations, that is:
( ) ( ) ( ) ( )( )nb
nb
knn
ennie
nnkk
bnk
b
bbb
ib t
hRh
vwwgaRf
uw
⋅=
×+⋅+−⋅=
∆=
2
r
(47)
where the navigation frame is the local-level frame (NED) and is abbreviated by the index
n , the body frame is indicated by b , the inertial frame by i , whereas the Earth fixed
Earth centered (EFEC) frame is indexed by e.
60
5.3. Error Model for Inertial Sensors
All gyroscopes and accelerometers are subject to errors which limit the accuracy at which
angular rotations or specific forces can be measured. Detailed information on sensors
errors can be found in [15]. The error model presented here is largely based on the work
presented in [31, 32].
The basic noise model can be expressed as:
( )( ) ffff
wwww
WNBfSFMf
WNBwSFMw
++⋅⋅=
++⋅⋅=ˆ
ˆ (48)
where fw,M represent the skew-symmetric misalignment matrices, the fw,SF are the
scale factors, w and f represent respectively the realistic gyros and accelerometers
measures, w and f the idealistic measures, the fw,B are the bias vectors and fw,WN are
the white noise vectors.
The skew-symmetric misalignement matrices fw,M are defined as:
( ) ( )( ) ( )
( ) ( )
−−
−=
1
1
1
wxwy
wxwz
wywz
w
yx
zx
zy
M and
( ) ( )( ) ( )
( ) ( )
−−
−=
1
1
1
fxfy
fxfz
fyfz
f
yx
zx
zy
M (49)
where yx , zx , xy , zy , xz , yz are the misalignments between axis.
The scale factors fw,SF are defined as:
RWf
GMf
RCf
Cff
RWw
GMw
RCw
Cww
SSSS1SF
SSSS1SF
++++=
++++= (50)
where Cfw,S are the constant components of the scale factors, RC
fw,S are random constants
vectors defined by the stochastic processes: ( )fwfwRC
fw G ,,, ,σµ=S , G being a standard
Gaussian function. GMfw,S are first-order Gauss-Markov stochastic processes defined as:
61
( ) ( ) ( )ttt kGMw
GMw wSS +⋅= β& , with initial condition: ( ) 0lim 0
0
=−∞→
tGMw
tS (51)
where T1=β is the inverse of correlation time, kw is a zero-mean Gaussian white-noise
process. RWfw,S are random walk stochastic processes defined by:
( ) ( )tt kRWw wS =& , with initial condition: ( ) ( )00
RWw
RWw t SS = (52)
The bias B is defined as:
BIf
RWf
GMf
RCf
Cff
idealGSw
BIw
RWw
GMw
RCw
Cww
BBBBBB
fBBBBBBB
++++=
⋅+++++= (53)
where Cfw,B are the constant components of the bias, RC
fw,B are random constants stochastic
processes, GMfw,B are first-order Gauss-Markov stochastic processes and RW
fw,B are random
walk stochastic processes (defined as previously for the scale factors). BIfw,B are the bias
instabilities which we decided to add to original model developed by [31, 32] to modeled
more precisely the noise characteristics observed in section 5.6.2. We defined it as:
( ) ( ) ( )( )
−=⋅
= 1
0,mod
,, otherwiset
Ttiftσt BI
fw
BIkBIBIfw
w,f
B
wB
with the initial condition: ( ) ( )00, =⋅== tσt kBIBI
fw w,fwB
(54)
Finally, GSwB is the gravity sensitivity parameter.
The white noise processes fw,WN are defined as:
( ) ( )tt kfwfw wWN ⋅= ,, σ (55)
where fw,σ is the standard deviations of the white noise processes.
62
5.4. Variance Analysis
The analysis of the noise altering the output signal of random oscillators is a major issue in
various fields. These different types of noise are generally distinguished and each of them
has distinct properties. Indeed, the origins of these noises are often linked with the
oscillator environment (e.g. temperatures changes or vibrations) but they may also be
internal (e.g. thermal noise). Several variance techniques have been developed for the
analysis of these perturbations such as the Allan variance, the Hadamard variance or the
total variance [33-36]. The simplest of these methods is the Allan variance. This method
was successfully applied to the modeling of the inertial sensor errors in [30, 37-39].
The Allan variance is a method where the root mean square random-drift error is
represented as a function of the averaging time. It was invented in 1966 by David Allan
and was originally employed to study the stability of oscillators [40]. In 1998, the IEEE
standard introduced this technique as a noise identification method [41]. This method can
be used to determine the characteristics of the underlying random processes that perturb
data. The Allan variance considers five basic noise processes which can be expressed in
the appropriate notation for inertial-sensors, namely: quantization noise, white noise, bias
intensity, random walk and rate ramp. Moreover, a first-order Gauss-Markov process as
well as a sinusoidal noise can also be identified [30, 37, 42]. The Allan variance is here
used as a tool for modeling of inertial sensor errors.
Assume a data set composed of N consecutive samples, each having an interval time of
0t . From this set, n groups (or cluster) of consecutive data can be formed (with
2Nn < ). For each possible value of n , an averaging time T is associated, which is
equal to otnT ⋅= . The cluster average is defined as:
( ) ( )∫+
⋅=Tt
tk
k
k
dttT
T ΩΩ1
(56)
where ( )tΩ is the instantaneous output rate for the considered inertial sensor and ( )TkΩ
represents the cluster average that starts from the thk element of the data set and which
includes n elements. The average of the next cluster is defined as:
( ) ( )∫++
+
⋅=Tt
tnext
k
k
dttT
T1
1
1ΩΩ (57)
where Ttt kk +=+1 .
63
The Allan variance calculates the variance of the difference of two subsequent cluster
averages. Thus, it is defined as [43]:
( ) ( ) ( )[ ]22
21
TTT knextAV ΩΩσ −= (58)
The brackets of equation (58) represent the averaging operation over the ensemble of
groups. This equation can be rewritten as:
( ) ( ) ( ) ( )( )∑−
=
−⋅−⋅
=nN
kknextAV TT
nNT
2
1
22
22
1ΩΩσ (59)
Figure 35 represents schematically the computational steps of the Allan Variance.
tk+1 tk+2 tk+3 tk+n1 t0
0tnT ⋅=
tk+1+1 tk+1+2 tk+1+3 tk+1+n
Ttt kk +=+1
N
2Nn <
( ) ( )∫+
⋅=Tt
tk
k
k
dttT
T ΩΩ1 ( ) ( )∫
++
+
⋅=Tt
tnext
k
k
dttT
T1
1
1ΩΩ
( ) ( ) ( ) ( )( )∑−
=
−⋅−⋅
=nN
kknextAV TT
nNT
2
1
22
22
1ΩΩσ
Figure 35 : Schematic representation of the Allan variance
As mentioned earlier, the Allan variance measures the stability of the sensor output.
Consequently, it has to be related to statistical properties of the intrinsic random processes,
which affects sensors’ performance. There is a unique relationship between ( )TAV2σ and
the power spectral density ( )fΩS of the intrinsic random processes [43]. This relationship
is the following:
( ) ( ) ( )( )∫ ⋅
⋅⋅⋅⋅⋅⋅= Ω+∞→
t
tAV df
Tf
TffT
0 22 sin
lim4π
πSσ (60)
Its derivation is given in Appendix F.
64
5.5. Evaluation of the Noise Characteristics of Inertial Sensors
The Allan variance is an effective tool to identify different types of noise in random
processes. Indeed, equation (60) allows each noise process to be characterized by a specific
Power Spectral Density (PSD) function. This enables to estimate the contribution (and
parameters) of each noise altering a signal. Five basic noise processes will be considered
here, namely: quantization noise, white noise, bias instability, random walk and rate ramp.
Note that a first-order Gauss-Markov process and a sinusoidal noise can also be identified
([30] for more details).
5.5.1. Quantization Noise
The difference between the real analog value and the encoded digital value is called
quantization error. This error is due to the bit resolution of the analog-to-digital converter,
which is either round or truncated [44].
The angle PSD of such process is given in [45]:
( ) ( )( ) Z
zz
z
zzz T
fQTTf
TfQTf
⋅<⋅≈
⋅⋅⋅⋅⋅⋅=
2
1 ,
sin 22
22
ππ
θS (61)
where zQ is the quantization noise coefficient and zT the sample interval.
The rate PSD is related to the angle PSD through equations [30, 43]:
( ) ( ) ( )fSff ⋅⋅⋅⋅⋅=⋅⋅Ω πππ θ 222 2S (62)
and is defined as:
( ) ( ) ( )z
zzzz
z
TfQTfTf
T
Qf
⋅<⋅⋅⋅⋅≈⋅⋅⋅⋅=Ω 2
1 , 2sin
4 2222
ππS (63)
Substituting this result into equation (60) and performing the integration yields,
( )T
QT zAV
3⋅=σ (64)
65
This equation indicates that the quantization is characterized by a slope of -1 in a
logarithmical scale of ( )TAVσ . The quantization noise coefficient can be read off the
slope line at 3=T .
5.5.2. White Noise
This random process is characterized by a flat PSD which means that every frequency is of
equivalent importance in the process. Thus, this PSD is defined as [43]:
( ) 2WNf σ=ΩS (65)
Substituting equation (65) into equation (60) and performing the integration yields to:
( )T
T WNAV
σ=σ (66)
Equation (66) indicates that a white noise random process has a -1/2 slope on a logarithmic
scale. The value of WNσ can be estimated at 1=T . Figure 36 presents an example of the
Allan variance of a white noise process and the estimation of its magnitude.
10-2
10-1
100
101
102
103
104
10-10
10-9
10-8
10-7
10-6
10-5
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[-]
Original Signal (σsignal
= 4.998⋅10-6)
Best fitted straigth line (σestimated
= 4.996⋅10-6 )
Simulated with estimated parameter
Slope = -1/2
Figure 36 : Allan variance of a white noise process
66
5.5.3. Bias Instability
The bias instability originates from electronic or other sensors’ components susceptible to
random flickering [30]. The rate PSD associated with the noise is given by [43]:
( )
>
≤
⋅=Ω
0
0
2
0
2
ff
fff
BI
πσ
S (67)
where BIσ is the bias instability coefficient.
Substituting equation (60) and integrating it yields to (details in [30]):
( ) ( )0
1
2ln2
fTT BIBIBIAV >>⋅⋅= σ
πσ (68)
Thus, at BIT , AVσ reaches the asymptotic value of ( )
2ln2
BIσπ
⋅⋅ which allows the
estimation of BIσ . Such behavior, however, may be overshadowed by the influence of
other noise terms [43]. Figure 37 and Figure 38 are examples of how bias instabilities can
be detected and how its parameters can be estimated using the Allan variance.
23 23.1 23.2 23.3 23.4 23.5 23.6-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Time [min]
[rad
/s]
Bias Instability (+ white noise), update every 1 [s], σBI
= 3⋅10-3
Bias Instability (+ white noise), update every 10 [s], σBI
= 3⋅10-3
Figure 37 : Example of two series of “bias instabilities” and white noise
67
Figure 38 shows the Allan variance of these two series. Note that the parameters
(i.e. BIσ and BIT ) of this noise can be efficiently found with the Allan variance.
10-2
10-1
100
101
102
103
104
10-4
10-3
10-2
10-1
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[rad
/s]
Allan Variance of the first time serie
Allan Variance of the seconde time serie
Theoretical values of flat region of the allan variance
3⋅⋅⋅⋅10-3 ⋅⋅⋅⋅[2 ⋅⋅⋅⋅ln(2)/ ππππ]1/2 = 1.99⋅⋅⋅⋅10-2
T1 ≈≈≈≈ 0.9 [s] T2 ≈≈≈≈ 10.5 [s]
Slope = 0
Theoretical values of flat region of the allan variance
Allan Variance of the seconde time serie
10-2
10-1
100
101
102
103
104
10-4
10-3
10-2
10-1
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[rad
/s]
Allan Variance of the first time serie
Allan Variance of the seconde time serie
Theoretical values of flat region of the allan variance
3⋅⋅⋅⋅10-3 ⋅⋅⋅⋅[2 ⋅⋅⋅⋅ln(2)/ ππππ]1/2 = 1.99⋅⋅⋅⋅10-2
T1 ≈≈≈≈ 0.9 [s] T2 ≈≈≈≈ 10.5 [s]
Slope = 0
Theoretical values of flat region of the allan variance
Allan Variance of the seconde time serie
Figure 38 : Allan variance of the series defined in Figure 37
5.5.4. Random Walk
Random walk noise can be characterized by a trajectory consisting of taking successive
random steps. This process is controlled by the differential equation defined in the
equation (52).
In terms of PSD, a random walk is characterized by [43]:
( )2
2
⋅⋅=Ω f
f RW
πσ
S (69)
where RWσ corresponds to the random walk parameter.
Substituting equation (69) into equation (60) and performing the integration yields to:
( )3
TT RWAV ⋅= σσ (70)
From this equation, we can conclude that such random processes have a 1/2 slope on a
logarithmic scale and that the magnitude of noise RWσ can be read off a fitted straight line
at T = 3. This is illustrated in Figure 39.
68
10-2
10-1
100
101
102
103
104
10-6
10-5
10-4
10-3
10-2
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[rad/
s]
Original Signal (σsignal = 5⋅10-6)
Best fitted straigth line (σestimated
= 4.993⋅10-6)
Simulated with estimated parameter
Slope = +1/2
Figure 39 : Allan variance of a random walk process
5.5.5. Rate Ramp
Unlike the other types of noises described earlier, the rate ramp is not a random process. It
is defined as [43]:
( ) tCt RRk ⋅=Ω (71)
implying that:
( )2
TRTAV
⋅=σ (72)
This equation is derived in Appendix G.
Equation (72) indicates that the Allan variance of the drift rate ramp noise has a slope of 1
in a logarithmical scale. Additionally, the amplitude of this noise can be estimated at
2=T .
To conclude, Figure 40 synthesizes the elements presented in this section and shows a
schematic visual representation of Allan variance’s results.
69
Averaging Time
Roo
t Alla
n V
aria
nce
σσσσBI/(2⋅⋅⋅⋅ln(2)/ ππππ)1/2
TBI
Qz
σσσσWN
σσσσRW
CRR
21/2 31/2 31
Quantization Noise
Slope = -1 White NoiseSlope = -1/2Bias Instability
Slope = 0
Random Walk
Slope = +1/2
Rate Ramp
Slope = +1/2
Figure 40 : Schematic sample representation of Allan variance using analysis results
5.5.6. Estimation of the Quality of the Allan Variance
With real data, gradual transitions would exist between the different Allan standard
deviation slopes. Likewise, a certain amount of noise would exist due to the uncertainty of
the measured Allan variance. Indeed, the estimation of the Allan variance is computed
based on a limited number of independent clusters, hence, as this number increases, the
confidence of the estimation improves [37].
In order to estimate this confidence, the parameter δ is defined as the percentage error in
the calculation of the Allan standard deviation of the cluster, which is due to the finiteness
of the clusters:
( ) ( )( )T
TMT
AV
AVAV
σ
σσ −=
,δ (73)
where ( )MTAV ,σ denotes the estimation of the root of the Allan variance obtained with
M independent clusters. ( )MTAV ,σ converges to ( )TAVσ as the number of clusters
increases. The percentage of error is equal to [37]:
( )
−⋅=
12
1
n
Nδσ
(74)
70
where N is the total number of samples of the entire trail and n is the number of samples
contained in a cluster.
Since the size of the clusters increases along with T , the results of the Allan variance
deteriorates as the averaging time increases. This behavior can be observed in Figure 41.
10-2
10-1
100
101
102
103
104
10-4
10-3
10-2
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[rad
/s]
σAV
AV incertitude
Figure 41 : Typical Allan variances of MEMS gyroscope (computed with three hours static data); the spotted lines represent the standard deviation of the Allan variance values.
Equation (74) shows that ( )δσ decreases as the total number of samples N increases.
Figure 42 shows the relationship between ( )δσ , N and the averaging time which
determines n .
0 2 4 6 8 10 12 14 16 18 20 220
10
20
30
40
50
60
70
Averaging time [min]
σσ σσer
ror o
f Alla
n V
aria
nce
[%]
Error with 1h of measuresError with 2h of measuresError with 6h of measuresError with 12h of measuresError with 24h of measuresError with 48h of measures
Figure 42 : Influence of the number of sample and the averaging time to the precision of the Allan variance
71
5.6. Estimation of the Parameters of the Noise Model
In this section, we estimate the parameters of the error model, described in section 5.3,
with different approaches. In the first one, the constant part of the bias and of the scale
factor will be estimated using data collected in the experiment described in section 4.2. In
the second approach, the stochastic component of the bias, the bias instability and the
white noise will be approximated using the Allan variance method. Note that some
parameters of the error model are difficult or impossible to determine with the collected
data (i.e. the misalignment matrix and the stochastic part of the scale factor), therefore,
they are neglected.
5.6.1. Estimation of the Scale Factor and of the Constant Part of the Bias
In order to estimate the scale and bias (constant components), we considered the following
model:
( )( ) wwLNMEMS
ffLNMEMS
BSF
BSF
++⋅=
++⋅=
1
1
200
200
ww
ff (75)
where MEMSf and MEMSw correspond to the MEMS’ measurements, 200LNf and 200LNw to
the LN200’s measurements, while wfSF , represents the constant component of the scale
factor and fwB , the constant part of the bias.
A parametric compensation revealed the parameters (i.e. wfSF , and fwB , ) to be
statistically none significant. This result is confirmed by the work of [46].
5.6.2. Estimation of the Stochastic Part of the Bias and of the White Noise Parameter
The Allan variance was used to determine the stochastic component of the bias, the bias
instability and the white noise. We conducted three hours static experiment, during which
static measures of the considered sensor (i.e. Xsens MT-i) where recorded, in order to
provide inputs to the Allan variance. We consider this experiment period long enough to
give accurate results (see section 5.5.6 and Figure 41). This experiment was repeated three
times in various situations, since we assumed the noise to be sensitive to environmental
conditions (e.g. temperature, pressure). Because results may vary from one experiment to
72
another, we consider that it would be unproductive to determine precisely the noise
parameter solely based on a specific experiment. The experimental conditions in our
experiment were the following:
• Temperature ≈ 20°C, no exposure to the sun
• Temperature ≈ 16°C, no exposure to the sun
• Temperature ≈ 24°C, exposure to the sun
First, we computed the Allan variances for the gyroscopes in various environmental
conditions. We group these results in a single graphic (Figure 43).
10-2
10-1
100
101
102
103
104
10-4
10-3
10-2
10-1
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[rad
/s]
σAV,r
x
, exp. conditions ≈ 20 [deg], no sun exposure
σAV,r
x
, exp. conditions ≈ 16 [deg], no sun exposure
σAV,r
x
, exp. conditions ≈ 24 [deg], sun exposure
σAV,r
y
, exp. conditions ≈ 20 [deg], no sun exposure
σAV,r
y
, exp. conditions ≈ 16 [deg], no sun exposure
σAV,r
y
, exp. conditions ≈ 24 [deg], sun exposure
σAV,rz
, exp. conditions ≈ 20 [deg], no sun exposure
σAV,r
z
, exp. conditions ≈ 16 [deg], no sun exposure
σAV,r
z
, exp. conditions ≈ 24 [deg], sun exposure
10-2
10-1
100
101
102
103
104
10-4
10-3
10-2
10-1
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[rad
/s]
σAV,r
x
, exp. conditions ≈ 20 [deg], no sun exposure
σAV,r
x
, exp. conditions ≈ 16 [deg], no sun exposure
σAV,r
x
, exp. conditions ≈ 24 [deg], sun exposure
σAV,r
y
, exp. conditions ≈ 20 [deg], no sun exposure
σAV,r
y
, exp. conditions ≈ 16 [deg], no sun exposure
σAV,r
y
, exp. conditions ≈ 24 [deg], sun exposure
σAV,rz
, exp. conditions ≈ 20 [deg], no sun exposure
σAV,r
z
, exp. conditions ≈ 16 [deg], no sun exposure
σAV,r
z
, exp. conditions ≈ 24 [deg], sun exposure
Figure 43 : Allan variance results for the gyroscopes (spotted lines indicate the standard deviation of the measurements)
This figure shows that all sensors have a similar behavior and that the gyroscopes are
scarcely affected by variations in the environment. As a result, we can determine noise
parameters using only one experimental condition and average the different sensors’
results. To verify this estimated average, we first determined the noise parameters, and
then we created a series with the same noise parameter. We, finally, computed the Allan
variance of the simulated series and compared the results obtained with the sensor values.
The outcome of this simulation is presented in Figure 44.
73
10-2
10-1
100
101
102
103
104
10-6
10-5
10-4
10-3
10-2
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[rad
/s]
σσσσ WN = 0.0075
σσσσ RW = 1.904 ⋅⋅⋅⋅10-6
σσσσ BI =2.44 ⋅⋅⋅⋅10-4/(2 ⋅⋅⋅⋅ln(2)/ ππππ)1/2 = 3.67 ⋅⋅⋅⋅10-4
T ≈≈≈≈ 60 [s]
σAV,rx
σAV,rx
σAV,ry
σAV, rx,y,z
, estimated
Figure 44: Estimation of noise parameters for the gyroscopes (spotted lines indicate the standard deviation of the measurements)
Figure 45 reveals that the Allan variance of the simulated series (in black) is similar to
experimentally obtained variances, which leads us to conclude that our estimation reflects
the reality.
With respect to the accelerometers, we grouped the results obtained with the Allan
variances for different environmental conditions in a single graphic (Figure 45).
10-2
10-1
100
101
102
103
104
105
10-4
10-3
10-2
10-1
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[m/s
2 ]
σσσσAV, fz
, exp. conditions ≈≈≈≈ 20 [deg], no sun exposure
σσσσAV, fz
, exp. conditions ≈≈≈≈ 16 [deg], no sun exposure
σσσσAV, fx
, exp. conditions ≈≈≈≈ 24 [deg], sun exposure
σσσσAV, fx
, exp. conditions ≈≈≈≈ 16 [deg], no sun exposure
σσσσAV, fx
, exp. conditions ≈≈≈≈ 20 [deg], no sun exposure
σσσσAV, fy
, exp. conditions ≈≈≈≈ 20 [deg], no sun exposure
σσσσAV, fy
, exp. conditions ≈≈≈≈ 20 [deg], no sun exposure
σσσσAV, fy
, exp. conditions ≈≈≈≈ 24 [deg], sun exposure
σσσσAV, fz
, exp. conditions ≈≈≈≈ 24 [deg], sun exposure
16 [deg], no sun exposure
Figure 45 : Allan variance results for the accelerometers (spotted lines indicate the standard deviation of the measurements)
74
Figure 45 clearly shows that the accelerometers are affected by environmental variations
and that each sensor has a different behavior. As a result, we estimate each sensor
separately.
10-2
10-1
100
101
102
103
104
10-5
10-4
10-3
10-2
10-1
Averaging T ime [s]Averaging T ime [s]Averaging T ime [s]Averaging T ime [s]
Roo
t Alla
n V
aria
nce
[m/s
2 ]
σσσσ RW = 7.1498 10-5
σσσσ BI = 0.0149/(2 ⋅⋅⋅⋅ln(2)/ ππππ)1/2 =
0.0224
T ≈≈≈≈ 0.6 [s]
σσσσ WN = 0.0719
σAV,f
x
, exp. conditions ≈ 24 [deg], sun exposure
σAV,fx
, exp. conditions ≈ 16 [deg], no sun exposure
σAV,f
x
, exp. conditions ≈ 20 [deg], no sun exposure
σAV,f
x,estimated
Figure 46 : Estimation of noise parameters for the accelerometers (axis x) (spotted lines indicate the standard deviation of the measurements)
10-2
10-1
100
101
102
103
104
10-5
10-4
10-3
10-2
10-1
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[m/s
2 ]
σσσσ RW = 1.28 ⋅⋅⋅⋅10-5
T ≈≈≈≈ 9 [s]
σσσσ WN = 0.0132
σσσσ BI = 8.24 ⋅⋅⋅⋅10-4/(2 ⋅⋅⋅⋅ ln(2)/ ππππ)1/2 = 1.24 ⋅⋅⋅⋅10-3
σAV,fy
, exp. conditions ≈ 24 [deg], sun exposure
σAV,f
y
, exp. conditions ≈ 16 [deg], no sun exposure
σAV,fy
, exp. conditions ≈ 20 [deg], no sun exposure
σAV,fy,estimated
Figure 47 : Estimation of noise parameters for the accelerometers (axis y) (spotted lines indicate the standard deviation of the measurements)
75
10-2
10-1
100
101
102
103
104
10-5
10-4
10-3
10-2
10-1
Averaging Time [s]
Roo
t Alla
n V
aria
nce
[rad
/s]
σσσσ WN = 0.0214
σσσσ BI = 1.31 ⋅⋅⋅⋅10-3/(2 ⋅⋅⋅⋅ln(2)/ ππππ)1/2 = 1.96 ⋅⋅⋅⋅10-3
T ≈≈≈≈ 35 [s]
σAV,f
z
, exp. conditions ≈ 24 [deg], sun exposure
σAV,f
z
, exp. conditions ≈ 16 [deg], no sun exposure
σAV,f
z
, exp. conditions ≈ 20 [deg], no sun exposure
σAV,f
z,estimated
Figure 48 : Estimation of noise parameters for the accelerometers (axis z) (spotted lines indicate the standard deviation of the measurements)
Table 3 summarizes the results obtained in this section and compares them with the noise
characteristics provided by the constructor (i.e. Xsens) [47]. These appear to be relatively
optimistic compared to our results.
White Noise Random Walk
σWN TBI [s] σBI σRW
Gyroscopes [deg/s/ √Hz]Gyros X, Y and Z 4.3·10-2 ± 2.9·10-5 60 2.1·10-3 ± 1.1·10-4 1.1·10-5 ± 1.7·10-6
Estimated by Xsens 1.0·10-2 [ - ] [ - ] [ - ]Accelerometer [m/s 2/√Hz]Accl X 7.2·10-3 ± 4.9·10-6 0.6 2.2·10-3 ± 1.2·10-4 7.5·10-6 ± 1.1·10-6
Accl Y 1.3·10-3 ± 9.0·10-7 9 1.2·10-4 ± 3.1·10-5 1.3·10-6 ± 2.0·10-7
Accl Z 2.1·10-3 ± 1.5·10-6 35 2.0·10-4 ± 7.9·10-5 [ - ]Estimated by Xsens 0.002 [ - ] [ - ] [ - ]
Bias Instability
Table 3 : Noise estimation for Xsens MT-i using Allan variance
76
5.7. Validation of the Model
To verify the model described in the previous sections, we emulated the measurements of a
MEMS-IMU from a reference trajectory (Figure 49).
1.73451.7445
1.75451.7645
x 104
-6.4871
-6.4671
-6.4471
x 10
1700
1750
N [m]E [m]
H [m
]
x 104
x 104
Figure 49 : Reference trajectory of an alpine skier (in Plain-Joux, Switzerland,
experiment realized by Adrian Wägli)
We, then, compared the GPS/INS integration results with the emulated IMU measurements
and with a real MEMS-IMU using the same GPS data. The results are shown in Figures
50, 51 and 52.
0 5 10 15 20 25 30-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Time [s]
Pos
ition
err
or [m
]
North error with simulated dataEast error with simulated dataDown error with simulated dataNorth error with "real" dataEast error with "real" dataDown error with "real" data
Figure 50 : Comparison of the position errors with the simulated and real data
77
0 2 4 6 8 10 12 14 16 18-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Time [s]
Vel
ocity
Err
or [m
/s]
VX error with simulated data
VY error with simulated data
VZ error with simulated data
VX error with "real" data
VY error with "real" data
VZ error with "real" data
Figure 51 : Comparison of the velocity errors with the simulated and real data
0 1 2 3 4 5 6 7 8-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
Time [s]
Orie
ntat
ion
Err
or [r
ad]
pitch error simulated dataroll error simulated dataheading error simulated datapitch error real dataroll error real dataheading error real data
Figure 52: Comparison of the orientation errors with the simulated and real data
These figures show that the magnitudes of the errors are similar for the simulated sensors
as for the real sensors. This enables us to validate the model.
5.8. Influence of the Sensors Orientation in Redundant IMU
In the previous section, we demonstrated that our model could generate MEMS-IMU
measurements. As a result, we decided to use this tool to verify if theoretical results
obtained in section 3.5 for sensor triads (i.e. unimportance of relative orientation) were
verified by simulation. In order to do so, we simulated two sets of measurements using
four sensor triads. In the first set, all sensors have the same orientation, while, in the
second set, the sensors were positioned in the form of a regular tetrahedron (similar to
78
Figure 19). Figure 53 shows the errors resulting from the sets, in terms of position, velocity
and orientation. The results of this simulation demonstrated that sensor orientation is
irrelevant and confirmed the theoretical assumptions of the third chapter.
30 31 32 33 34 35-1.5
-1
-0.5
0
0.5
1
1.5
Time [s]
Orie
ntat
ion
erro
r [de
g]
Pitch errors with 4 sensors (orthogonal configuration)
Roll errors with 4 sensors (orthogonal configuration)
Heading errors with 4 sensors (orthogonal configuration)
Pitch errors with 4 sensors (placed on a tetrahedron)
Roll errors with 4 sensors (placed on a tetrahedron)
Heading errors with 4 sensors (placed on a tetrahedron)
10 12 14 16 18 20 22 24 26 28 300
0.05
0.1
0.15
0.2
0.25
0.3
Time [s]
Pos
ition
err
or [m
]
North errors with 4 sensors (orthogonal configuration)
East errors with 4 sensors (orthogonal configuration)
Down errors with 4 sensors (orthogonal configuration)
North errors with 4 sensors (placed on a tetrahedron)
East errors with 4 sensors (placed on a tetrahedron)
Down errors with 4 sensors (placed on a tetrahedron)
45 46 47 48 49 50-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time [s]
Spe
ed e
rror
[m/s
]
VNorth
errors with 4 sensors (orthogonal configuration)
VEast
errors with 4 sensors (orthogonal configuration)
VDown
errors with 4 sensors (orthogonal configuration)
VNorth
errors with 4 sensors (placed on a tetrahedron)
VEast
errors with 4 sensors (placed on a tetrahedron)
VDown
errors with 4 sensors (placed on a tetrahedron)
10 12 14 16 18 20 22 24 26 28 300
0.05
0.1
0.15
0.2
0.25
0.3
Time [s]
Pos
ition
err
or [m
]
North errors with 4 sensors (orthogonal configuration)
East errors with 4 sensors (orthogonal configuration)
Down errors with 4 sensors (orthogonal configuration)
North errors with 4 sensors (placed on a tetrahedron)
East errors with 4 sensors (placed on a tetrahedron)
Down errors with 4 sensors (placed on a tetrahedron)
45 46 47 48 49 50-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Time [s]
Spe
ed e
rror
[m/s
]
VNorth
errors with 4 sensors (orthogonal configuration)
VEast
errors with 4 sensors (orthogonal configuration)
VDown
errors with 4 sensors (orthogonal configuration)
VNorth
errors with 4 sensors (placed on a tetrahedron)
VEast
errors with 4 sensors (placed on a tetrahedron)
VDown
errors with 4 sensors (placed on a tetrahedron)
Figure 53: Position, velocity and orientations errors of two redundant (i.e. four sensors) systems placed differently in space
5.9. Navigation Performance Improvement
In this last section, we assess the navigation performance of various redundancies using
between one and ten IMUs (based on the trajectory presented in Figure 49). The position,
velocity and orientation errors for these systems are presented in Figures 54, 55 and 56
(note that the experimental results for two sensors were taken from [1]).
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100
# of sensors [-]
Ave
rage
pos
ition
impr
ovem
ent [
%]
RMS of simulations (synthetic)
RMS of simulations (extended)
Experimental results (synthetic)
Experimental results (extended)
Tendency cuvre (synthetic)
Tendency curve (extended)
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100
# of sensors [-]
Max
imum
pos
ition
err
or im
prov
emen
t [%
]
Maximum errors of simulations (synthetic)Maximum errors of simulations (extended)Experimental results (synthetic)Experimental results (extended)Tendency cuvre (synthetic)Tendency curve (extended)
Figure 54 : Influence of the redundancy of the position error (right) and the maximal position error (left)
79
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100
# of sensors [-]
Max
imum
vel
ocity
err
or im
prov
emen
t [%
]
Maximum errors of simulations (synthetic)
Maximum errors of simulations (extended)
Experimental results (synthetic)
Experimental results (extended)
Tendency cuvre (synthetic)
Tendency curve (extended)
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100
# of sensors [-]
Ave
rage
vel
ocity
impr
ovem
ent [
%]
RMS of simulations (synthetic)
RMS of simulations (extended)
Experimental results (synthetic)
Experimental results (extended)
Tendency cuvre (synthetic)
Tendency curve (extended)
Figure 55 : Influence of the redundancy of the velocity error (right) and the maximal velocity error (left)
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100
# of sensors [-]
Ave
rage
orie
ntat
ion
impr
ovem
ent [
%]
RMS of simulations (synthetic)
RMS of simulations (extended)
Experimental results (synthetic)
Experimental results (extended)
Tendency cuvre (synthetic)
Tendency curve (extended)
1 2 3 4 5 6 7 8 9 100
10
20
30
40
50
60
70
80
90
100
# of sensors [-]
Max
imum
orie
ntat
ion
erro
r im
prov
emen
t [%
]
Maximum errors of simulations (synthetic)Maximum errors of simulations (extended)Experimental results (synthetic)Experimental results (extended)Tendency cuvre (synthetic)Tendency curve (extended)
Figure 56 : Influence of the redundancy of the orientation error (right) and the maximal orientation error (left)
As expected, these Figures show an inverse correlation between errors and the number of
sensors. Furthermore, the curve tendencies appear to have a comparable form to the
expected noise reduction shown in Figure 4. However, and as explained earlier, residual
correlations between inertial measurements, as well as correlations between filter states,
limit the reduction of navigation errors (i.e. errors in position, velocity and attitude)
compared to the noise reduction.
80
5.10. Conclusions
This chapter has shown that, with an appropriate error model, MEMS-IMUs measurements
can be successfully emulated from a reference trajectory. For that reason, we consider our
model able to simulate any sensor architecture and to assess its resulting navigation
performance. This could provide helpful information for applications that require a limit
precision by determining the needed number of sensors to fulfill the requirements.
Additionally, we verified, through simulation, the theoretical results obtained for the
orientation of the triads (section 3.5). We identified the relationship between the different
navigation performance parameters and the number of sensors employed. That said, these
results should be considered with caution as they are based on a single trajectory. Further
test would be required to define a useable relation.
81
6. Conclusion and Perspectives
6.1. Synthesis
This master project is part of a larger investigation that aims to develop a low-cost GPS/
INS system for performance analysis in sports activities. The main objective of this project
was to investigate orientation improvement when using redundant inertial sensors.
To summarize the results obtained in this research, we remind the main features:
• A new method (i.e. partial redundancy), originally applied in geodetic networks,
allows to assess the optimality of redundant system. This method has
demonstrated that, in most cases, relative sensor orientation is unimportant for the
optimality of the system, as long as no sensor fails; however, in this case, skewed
configurations are preferable.
• Redundant IMUs can improve the determination of position, velocity, acceleration
and orientation. For example, with four sensors, measurements results gain
approximately 30% in accuracy for the synthetic approach and 50% for the
extended approach.
• We demonstrated and quantified the noise reduction resulting from the
redundancy of measurements. We, then, successfully employed this redundancy to
estimate variations of the noise level during the processing. Experimental results
show that this provides a more realistic error model to the EKF and improves
navigation performance.
• This research has proved that, when applied to sports activities, the parity space
method can advantageously be applied with MEMS-IMUs. On the other hand,
percentages of undetected errors, as well as the level of false alarms, remain high
and, consequently, show a need for more complex FDI models to be developed.
• We have demonstrated in this thesis that MEMS-IMU measurements can be
emulated with an appropriated error model. This allows to theoretically estimate
navigation performance of any redundant systems. This offers a tool to design
systems according to specifications.
To conclude, the increased accuracy provided by redundant IMUs has the potential of
better bridging the gaps in GPS data. In addition, this project, in providing the opportunity
to estimate noise levels and to correct errors, demonstrated why redundant IMUs are
interesting to enhance the performance in many (sports) applications.
82
6.2. Perspectives
The conclusions of this thesis have high potential for a wide range of fields, where a single
IMU is insufficient to provide satisfactory reliability or/and accuracy (e.g. direct
georeferencing, virtual reality, military application, pedestrian navigation). The results
should, nevertheless, be considered with caution as they are based on a single experiment.
Further experimentation should be realized to verify these results.
83
7. References
[1] A. Waegli, S. Guerrier, and J. Skaloud, "Redundant MEMS-IMU integrated with GPS for Performance Assessment in Sports," in IEEE/ION Position Location and Navigation Symposium, Monterey, 2008.
[2] A. Waegli, B. J.-M., and J. Skaloud, "L'analyse de performance sportive à l'aide d'un système GPS/INS low-cost: évaluation de capteurs inertiels de type MEMS," Revue XYZ, vol. 113, 2007.
[3] A. Waegli, A. Schorderet, C. Prongué, and S. J., "Accurate Trajectory and Orientation of a Motorcycle derived from low-cost Satellite and Inertial Measurement Systems," in 7th ISEA 2008, Biarritz, France, 2008.
[4] A. Waegli and J. Skaloud, "Turning Point - Trajectory Analysis for Skiers," InsideGNSS, 2007.
[5] J. How and N. Pohlman, "GPS Estimation Algorithms for Precise Velocity, Slip and Race-track Position Measurements," SAE Motorsports Engineering Conference & Exhibition, 2002.
[6] K. Zhang, R. Deakin, R. Grenfell, Y. Li, J. Zhang, W.N.Cameron, and D. M. Silcock, "GNSS for sports - sailing and rowing perspectives," Journal of Global Positioning Sytems, vol. 3, pp. 280-289, 2004.
[7] A. Waegli, "Performance Assessment in Sports based on Navigation Systems," in Geodetic Engineering Laboratory. vol. phd Thesis: EPFL, 2008 (to be published).
[8] A. Waegli, J. Skaloud, P. Tomé, and J.-M. Bonnaz, "Assessment of the Integration Strategy between GPS and Body-Worn MEMS Sensors with Application to Sports," in ION GNSS, Fort Worth, Texas, 2007.
[9] I. Colomia, M. Giménez, J. J. Rosales, M. Wis, A. Gómez, and P. Miguelsanz, "Redundant IMUs for Precise Trajectory Determination," 2006.
[10] S. Sukkarieh, P. Gibbens, B. Grocholsky, K. Willis, and H. F. Durrant-Whyte, "A Low-Cost Redundant Inertial Measurement Unit for Unmanned Air Vehicles," The Internation Journal of Robotics Reasearch, 2000.
[11] A. Osman, B. Wright, S. Nassar, A. Noureldin, and N. El-Sheimy, "Multi-Sensor Inertial Navigation Systems Employing Skewed Redundant Inertial Sensors," in ION GNSS 19th International Technical Meeting of the Satellite Division,, Fort Worth, Texas, 2006.
[12] A. J. Pejsa, "Optimum Skewed Redundant Inertial Navigators," AIAA Journal, vol. 12, 1974.
[13] http://www.intensiondesigns.com/itd-biotensegrity/biotensegrity/papers/images/geometry_of_anatomy_images/fig_2_platonic_solids.jpg.
[14] H. Dupraz and M. Stahl, Théorie des Erreurs. Lausanne, 1994. [15] D. H. Titterton and J. L. Weston, Strapdown inertial navigation technology. London,
1997. [16] N. F. Zhang, "The uncertainty associated with the weighted means of measurement
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84
[17] K. C. Daly, E. Gai, and J. V. Harrison, "Generalized Likelihood Test for FDI in Redundant Sensor Configurations," Journal of Guidance and Control, vol. 2, 1979.
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86
8. Appendix
8.1. Appendix A: Proof of equation (11)
Theorem: ( ) unztrn
ii −==∑
=1
Z
Proof: ( ) ( )( )1−⋅⋅⋅−= RHPHRZ Ttrtr
( )( )( )( )( )( )
( )( )( )( ) ( )( )( ) ( )
un
trtr
trtr
tr
tr
tr
un
TT
TT
TT
TT
−=−=
⋅⋅⋅⋅⋅−⋅=
⋅⋅⋅⋅⋅−⋅=
⋅⋅⋅⋅⋅−⋅=
⋅⋅⋅⋅⋅−=
−−−−
−−−−
−−−−
−−−
II
HRHHRHRR
HRHHRHRR
HHRHHRRR
RHHRHHR
1111
1111
1111
111
8.2. Appendix B: Proof of equation (13)
Theorem: The variance of the best estimate x is n
w xn
iiix
σσσ == ∑=1
22ˆ assuming
iix ∀= σσ
Proof: The best estimate can be expressed as ∑=
⋅=n
iii xwx
1
ˆ where ∑=
=n
iiw
1
1. The
expected value of x is then given by: ( ) ( )∑=
⋅=n
iii xEwxE
1
ˆ . Moreover, the variance of a
quantity y is defined as ( )( )[ ]22 yEyEdef
y −=σ . Thus the variance of x is:
( )
( )( ) ( )( ) ( )( ) ijxExwxExwxExwE
xEwxwE
n
i
n
jjjjiii
n
iiii
n
iiiiix
≠∀
−⋅⋅−⋅+−⋅=
⋅−⋅=
∑ ∑∑
∑
= ==
=
,1 1
2
1
2
2
1
2ˆσ
87
Since all ix are independent, then the ( )( )ii xEx − are uncorrelated and thus all
( )( ) ( )( )[ ] 0=−⋅− jjii xExxExE [15]. Hence, 2xσ can be expressed as:
( )( )[ ] ∑∑∑===
⋅=⇒⋅=−⋅=n
iiix
n
iii
n
iiiix wwxExEw
1
22ˆ
1
22
1
222ˆ σσσσ
Since it is assumed that iix ∀= σσ then iww ix ∀= , since the iw are proportional to
the precision of the measures. This implies that ∑=
=⇒=⋅=n
ixxi n
wnww1
11 . Thus
nx
x
σσ =ˆ
8.3. Appendix C: Proof of equation (19)
Theorem: The best least square estimate x of the equation bnxHz ++⋅= is zHx ⋅= *ˆ
where ( ) TT* HHHH ⋅⋅= −1
Proof: xHzvvxHbnxHz ⋅−=⇒+⋅=++⋅= where bnv += , the best estimate
x is obtained when sum of squared differences (i.e.v ) is minimized, mathematically:
( ) ( ) ( )( )xHzxHzvv ⋅−⋅⋅−=⋅ TT minmin
The partial differentiation of vv ⋅T yields to the determination of x , indeed,
( ) ( )( ) ( ) ( )( )
( )( ) ( ) ( ) ( )( )
( )( ) ( ) ( )( ) zHHHx
zHxHHHHxzxHzH
xHzx
HxzxHzHxzx
xHzHxzx
xHzxHzx
⋅⋅⋅=⇒
⋅=⋅⋅⇔=−⋅⋅−+⋅−⋅−⇔
=⋅−∂∂⋅⋅−+⋅−⋅⋅−
∂∂⇔
=⋅−⋅⋅−∂∂⇔=⋅−⋅⋅−
∂∂
− TT
TTTTTT
TTTTTT
TTTT
1ˆ
ˆ0ˆˆ
0ˆˆˆˆ
0ˆˆ0ˆˆ
This implies that zHx ⋅= *ˆ where ( ) TT* HHHH ⋅⋅= −1
88
8.4. Appendix D: Proof of equation (23)
Theorem: *l HHIS ⋅−=
Proof: Firstly, [ ]TPHA =−1 has to be proved to be the correct inverse of the matrix
=
P
HA
*
In order to do so, the equality lIAA =⋅ −1 must be true.
[ ]
( ) ( )( )( ) ( )
( ) ( )( )( ) ( )
( )( )
( )( ) ( )
l
nlnnl
nlnn
nlnnl
TTn
nlnnl
TTTTT
T
T**
T*
I
I0
0I
I0
HPHHI
I0
PHHHHHHH
PPHP
PHHH
PHP
HAA
=
=
⋅⋅⋅=
⋅⋅⋅⋅⋅⋅=
⋅⋅⋅⋅
=
⋅
=⋅
−×−
−×
−×−
−
−×−
−−
−
1
11
1
Since A is a square with full rank matrix, then:
[ ] SHHPPHHP
HPHAAI +⋅=⋅+⋅=
⋅=⋅= − *T*
*T
l1
This implies that: *l HHIS ⋅−=
89
8.5. Appendix E: Proof of equation (42)
Theorem: ∑=
⋅⋅=n
iz
y
xTb
bb
i
i
i
iw
r
r
r
n 1
1Rw , the derivation is similar for bf and bm
Proof: ( ) bwll
Twwll
Tw
bww
bbw
bw
bw
bw
lRAARAlΠw ⋅⋅⋅⋅⋅=⋅= −−− 111
Assuming IR ⋅= 20σb
wbwll
, it follows:
( )[ ]( ) b
wTw
bb
Tbb
bb
Tbb
bw
Tww
Tw
b
nwnwww lARRRR
lAAAw
⋅⋅⋅++⋅=
⋅⋅⋅=−
−
1
1
11 L
It can shown that 3IRR =⋅ iwiw bb
Tbb , indeed,
( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )
( ) ( )
( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) 1sinsincoscoscos
0cossincossincoscos
0cossinsinsincossincoscos
0cossincossincoscos
1cossin
0sincossinsincossin
0cossinsincossincoscossin
0cossinsinsincossin
1cossinsincossin
since, ,
100
010
001
sin0cos
sincoscossinsin
coscossincossin
sinsincoscoscos
0cossin
cossinsincossin
2222233
32
2231
23
2222
21
2213
12
2222211
3
333231
232221
131211
=+⋅+⋅=
=⋅⋅−⋅⋅−==⋅−⋅⋅+⋅⋅=
=⋅⋅−⋅⋅==+=
=⋅⋅+⋅⋅−==⋅−⋅⋅+⋅⋅=
=⋅⋅−⋅⋅==+⋅+⋅=
∀=
=
=
−⋅−⋅−⋅−−⋅−
⋅
−⋅−⋅−−
⋅−⋅−=⋅
λϕλϕλϕϕλϕϕλ
λλλϕλλϕλϕϕλϕϕλ
ϕϕϕϕλϕϕλ
λλϕλλϕλλϕϕλϕϕλλϕλϕλ
θλ
λλϕλϕϕλϕλϕϕλ
λϕλϕλϕϕ
λϕλϕλ
a
a
a
a
a
a
a
a
a
I
aaa
aaa
aaa
iwiw bb
Tbb RR
Thus,
[ ]( ) ∑=
−
⋅⋅=⋅⋅++=n
1
133
1
iz
y
xTb
bbw
Tw
b
i
i
i
iw
r
r
r
nRlAIIw L
90
8.6. Appendix F: Proof of equation (60)
Theorem: ( ) ( ) ( )( )∫ ⋅⋅⋅= Ω+∞→
t
tAV df
fT
fTfT
0 22 sin
lim4π
πSσ
Proof : This proof is based on [27, 30, 48]. Firstly, the Allan variance defined by the
equation (58) is transformed as follows:
( ) ( ) ( )[ ]
( ) ( ) ( ) ( )
( ) ( )[ ] ( ) ( )TTTT
TTTT
TTT
knextknext
knextknext
knextAV
2222
2222
22
2
1
22
12
1
ΩΩΩΩ
ΩΩΩΩ
ΩΩσ
⋅−+⋅=
⋅⋅−+=
−=
Secondly, the definition of ( )TkΩ (i.e. the equation (56)) is used to compute ( )Tk2
Ω
and ( )Tnext2
Ω , that is,
( ) ( ) ( ) ( ) tddtttT
TdttT
TTt
t
Tt
tk
Tt
tk
k
k
k
k
k
k
′⋅⋅′⋅=⇔⋅= ∫ ∫∫+ ++
,11
22 RΩΩΩ
where ( ) ( ) ( )tttt ′⋅=′ ΩΩR , is the autocovariance function. Assuming the ( )tΩ to be
stationary ( )tt ′,R can be written as ( ) ( ) ( )τRRR =−′=′ tttt, . Moreover, the basic
relationship for stationary processes between the two sided PSD is [27]:
( ) ( ) ( ) ( )∫ ∫− −
⋅⋅⋅⋅−Ω→∞
⋅⋅⋅⋅−
→∞Ω ⋅⋅=⇔⋅⋅=t
t
t
t
fi
t
fi
tdfefSdtef τπτπ ττ 22 limlim RRS
Thus, it follows that:
( ) ( )
( )
( ) ( ) ( )df
f
Tff
T
dftddtefT
tddtttT
T
t
tt
Tt
t
Tt
t
fit
tt
Tt
t
Tt
tk
k
k
k
k
k
k
k
k
⋅
⋅⋅⋅⋅⋅
=
⋅′⋅⋅⋅⋅=
′⋅⋅′⋅=
∫
∫ ∫∫
∫ ∫
− Ω∞→
+ + ⋅⋅⋅⋅−
− Ω∞→
+ +
2
2
22
22
sinlim
1
lim1
,1
ππ
τπ
S
S
RΩ
Additionally, ( )Tk2
Ω is not a function of time, thus ( ) ( )TT knext22
ΩΩ = . Then, the
term ( ) ( )TT knext22
ΩΩ ⋅ is obtained using equation (56), that is,
91
( ) ( ) ( ) ( ) tddtttT
TdttT
TTt
t
Tt
Ttk
Tt
tk
k
k
k
k
k
k
′⋅⋅′⋅=⇔⋅= ∫ ∫∫+ ⋅+
+
+ 2
22 ,
11RΩΩΩ
Substituting ( )tt ′,R , it yields:
( ) ( )
( ) ( ) ( )df
f
Tfef
T
tddtttT
T
Tfit
tt
Tt
t
Tt
Ttk
k
k
k
k
⋅
⋅⋅⋅⋅⋅⋅
=
′⋅⋅′⋅=
⋅⋅⋅⋅
− Ω→∞
+ ⋅+
+
∫
∫ ∫2
22
2
22
sinlim
1
,1
ππ
πS
RΩ
Consequently,
( ) ( ) ( )[ ] ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
⋅⋅⋅⋅⋅⋅⋅⋅−⋅⋅⋅⋅⋅⋅
=
⋅
⋅⋅⋅−⋅⋅⋅
=
⋅−=
⋅−+⋅=
∫∫
∫
− Ω∞→− Ω∞→
⋅⋅⋅⋅
− Ω∞→
2
2
2
4
2
2
22
222
22222
2sinsinlim
sinlim2
1
sin1lim
1
2
1
f
TfTffi
f
Tff
T
dff
Tfef
T
TTT
TTTTT
t
tt
t
tt
Tfit
tt
knextk
knextknextAV
ππππ
ππ
π
SS
S
ΩΩΩ
ΩΩΩΩσ
Since ( )fΩS is defined as an even matrix [48] the above equation becomes:
( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )( )2
4
0
2
4
2
2
2
2
4
22
sinlim4
sinlim
2
2sinsinlim
sinlim2
1
fT
Tff
f
Tff
T
f
TfTffi
f
Tff
TT
t
t
t
tt
t
tt
t
ttAV
⋅⋅⋅⋅⋅⋅=
⋅⋅⋅⋅⋅
=
⋅⋅⋅⋅⋅⋅⋅⋅−⋅⋅⋅⋅⋅⋅
=
∫
∫
∫∫
Ω∞→
− Ω∞→
− Ω∞→− Ω∞→
ππ
ππ
ππππ
S
S
SSσ
92
8.7. Appendix G: Proof of equation (72)
Theorem: ( )2
TRTAV
⋅=σ
Proof: The equation (71) states: ( ) tRt ⋅=Ω , thus Ω and nextΩ are expressed using the
equations (56) and (57), that is,
[ ]22)(1
)(1
kk
Tt
t
Tt
ttTt
T
RdttR
Tdtt
T
k
k
k
k
−+⋅=⋅⋅⋅=⋅⋅= ∫∫++
ΩΩ
[ ]222)()2(
1)(
1TtTt
T
RdttR
Tdtt
T kk
tt
t
Tt
Ttnext
k
k
k
k
+−⋅+⋅=⋅⋅⋅=⋅⋅= ∫∫++
+ΩΩ
Then using the definition of the Allan variance (i.e. equation (59)), it yields to
( )
[ ]
( )
( )
( )
2
2)2(2
)2(2
42244)2(2
)(2)2()2(2
)2(2
1)(
22
2
24
2
1
222
2
22
1
222222
2
22
1
2222
2
22
1
2
RT
nNnNT
RT
TnNT
R
ttTTttTTtnNT
R
tTtTtnNT
R
nNT
nN
k
nN
kkkkkk
nN
kkkk
nN
knextAV
⋅=
⋅−⋅⋅−⋅⋅
⋅=
⋅⋅−⋅⋅
=
+⋅⋅−⋅−⋅−⋅⋅−⋅+⋅⋅−⋅⋅
=
++⋅−⋅−⋅⋅−⋅⋅
=
−⋅⋅−⋅
=
∑
∑
∑
∑
−
=
−
=
−
=
−
=
ΩΩσ
Thus, ( )2
TRTAV
⋅=σ
93
“Aime la vérite, mais pardonne à l’erreur.”
Voltaire, Discours en vers sur l’homme, Deuxième discours de la liberté
