A Study of the Effects of Stochastic Inertial Sensor Errors
in Dead-Reckoning Navigation
Except where reference is made to the work of others, the work described in this thesis
is my own or was done in collaboration with my advisory committee. This thesis does
not include proprietary or classifled information.
John H. Wall
Certiflcate of Approval:
George T. Flowers
Professor
Mechanical Engineering
David M. Bevly, Chair
Assistant Professor
Mechanical Engineering
John Y. Hung
Professor
Electrical and Computer Engineering
Joe F. Pittman
Interim Dean
Graduate School
A Study of the Effects of Stochastic Inertial Sensor Errors
in Dead-Reckoning Navigation
John H. Wall
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulflllment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
August 4, 2007
A Study of the Effects of Stochastic Inertial Sensor Errors
in Dead-Reckoning Navigation
John H. Wall
Permission is granted to Auburn University to make copies of this thesis at its
discretion, upon the request of individuals or institutions and at their
expense. The author reserves all publication rights.
Signature of Author
Date of Graduation
iii
Thesis Abstract
A Study of the Effects of Stochastic Inertial Sensor Errors
in Dead-Reckoning Navigation
John H. Wall
Master of Science, August 4, 2007
(B.S.M.E., Christian Brothers University, 2005)
144 Typed Pages
Directed by David M. Bevly
The research presented in this thesis seeks to quantify the error growth of navigation
frame attitude, velocity, and position as solely derived from acceleration and rotation-
rate measurements from a strapdown Inertial Measurement Unit (IMU). The wide-spread
availability of the Global Positioning System (GPS) and increased technological advances
in Inertial Navigation Systems (INS) technology has made possible the use of increasingly
afiordable and compact GPS/INS navigation systems. While the fusion of GPS and
inertial sensing technology ofiers exceptional performance under nominal conditions, the
accuracy of the provided solution degrades rapidly when traveling under bridges, dense
foliage, or in urban canyons due to loss of communication with GPS satellites. The
degradation of the navigation solution in this inertial dead-reckoning mode is a direct
result of the numerical integration of stochastic errors exhibited by the inertial sensors
themselves. As the accuracy of the GPS/INS combined system depends heavily on the
standalone performance of the INS, flrm quantiflcation of the performance of inertial
dead-reckoning is imperative for system selection and design.
iv
To provide quantiflcation of the accuracy of inertial dead-reckoning, stochastic mod-
els are selected which approximate the noise and bias drift present on a wide variety of
both accelerometers and rate-gyroscopes. The stochastic identiflcation techniques of Al-
lan variance and experimental autocorrelation are presented to illustrate the extraction
of process parameters from experimental data using the assumed model forms. The
selected models are then used to develop analytical expressions for the variance of subse-
quent integrations of the stochastic error processes. The resulting analytical expressions
are validated using Monte Carlo simulations. The analytical analysis is extended to a
simple navigation scenario in which a vehicle is constrained to travel on a planar surface
with no lateral velocity. Monte Carlo simulation techniques are employed to exemplify
and compare the expected results of inertial navigation in higher dynamic scenarios.
v
Acknowledgments
I would like to thank my thesis advisor, Dr. David Bevly, for providing me with the
invaluable opportunity to work as a graduate research assistant in the GPS and Vehicle
Dynamics Lab (GAVLab) at Auburn University. His ever enthusiastic vigor for new
research and his comfortable attitude made my experience as a master?s student most
rewarding. I also give thanks to Dr. George Flowers for his warm welcome to Auburn
and support throughout my time in the program.
I owe much of my success as a student and researcher to all the students in the
GAVLab. My sincere gratitude is due to my fellow labmates for their helping hands
in the theoretical and experimental and the willingness with which they shared their
expertise.
I thank my brother, Michael, for his multi-dimensional support as a roommate,
friend, and technical advisor. I thank my parents, who have given their unconditional
love and support. I thank Julia, for her earnest ear and ardent companionship that has
kept my spirit high throughout this entire experience.
vi
Style manual or journal used Journal of Approximation Theory (together with the
style known as \aums"). Bibliography follows the IEEE Transactions format.
Computer software used The document preparation package TEX (speciflcally
LATEX) together with the departmental style-flle aums.sty.
vii
Table of Contents
List of Figures x
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Prior Art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 Inertial Sensor Modeling 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Simple Sensor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Stochastic Sensor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 Gaussian White Noise . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.2 The Gauss-Markov Process . . . . . . . . . . . . . . . . . . . . . . 12
2.3.3 A Simple Stochastic Model . . . . . . . . . . . . . . . . . . . . . . 15
2.4 Stochastic Identiflcation Techniques . . . . . . . . . . . . . . . . . . . . . 16
2.4.1 The Allan Variance . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4.2 Experimental Autocorrelation . . . . . . . . . . . . . . . . . . . . . 21
2.4.3 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Experimental Quantiflcation and Identiflcation . . . . . . . . . . . . . . . 24
2.5.1 Manufacturer Sensor Speciflcations . . . . . . . . . . . . . . . . . . 24
2.5.2 Experimental Approach . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Covariance Propagation of Stochastic Errors 28
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.2 Simplifled Navigation Scenario: Single Axis . . . . . . . . . . . . . . . . . 29
3.3 General Characterization of Raw Sensor Measurement . . . . . . . . . . . 30
3.4 Characterization of Integrated Sensor Measurement . . . . . . . . . . . . . 31
3.5 Single Axis Stochastic Error Contributions . . . . . . . . . . . . . . . . . . 32
3.5.1 Variance of Integrated Wide Band noise . . . . . . . . . . . . . . . 33
3.5.2 Variance of Double Integrated Wide Band noise . . . . . . . . . . . 35
3.5.3 Variance of 1st order Gauss-Markov process . . . . . . . . . . . . . 36
3.5.4 Variance of Integrated 1st order Gauss-Markov process . . . . . . . 38
3.5.5 Variance of Double Integrated 1st order Gauss-Markov process . . 41
3.5.6 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 Validation of the Error Propagation . . . . . . . . . . . . . . . . . . . . . 46
3.6.1 Propagation of Wide-Band Noise Process . . . . . . . . . . . . . . 47
viii
3.6.2 Propagation of Gauss-Markov Process . . . . . . . . . . . . . . . . 49
3.7 Application Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.8 Illustration of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.8.1 Relative Magnitudes . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.8.2 Efiect of Time Constant . . . . . . . . . . . . . . . . . . . . . . . . 58
3.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4 Two Dimensional Error Propagation 61
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2 Velocity Error in Navigation Frame Under No-Slip Planar Motion . . . . . 65
4.2.1 Mean and Variance of East Velocity . . . . . . . . . . . . . . . . . 67
4.2.2 Mean and Variance of North Velocity . . . . . . . . . . . . . . . . 70
4.2.3 Cross Covariance of North and East Velocity . . . . . . . . . . . . 73
4.2.4 Probabilistic Characterization of Velocity Errors . . . . . . . . . . 75
4.2.5 Validation of the Velocity Error Characterization . . . . . . . . . . 76
4.3 Position Error in No-Slip Planar Motion . . . . . . . . . . . . . . . . . . . 80
4.4 Propagation of Error in Planar Motion with Slip . . . . . . . . . . . . . . 85
4.4.1 Acceleration Error in Navigation Frame for Slip-Case . . . . . . . 85
4.5 Comparison of Slip and No-Slip Mechanizations . . . . . . . . . . . . . . . 89
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5 Six DOF Analysis 96
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.2 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2.2 Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.3 Comparison to Planar Mechanization . . . . . . . . . . . . . . . . . . . . . 100
6 Conclusions 108
6.1 Overall Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2 Di?culties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Bibliography 114
Appendices 117
A Stochastic Parameter Identification with an Automotive-Grade
IMU 118
B Demonstration of 6-DOF Mechanization of Experimental Inertial
Measurements Taken at Talledega Superspeedway 125
ix
List of Figures
2.1 Sample Plot of Wide Band Noise, 2 = 1 . . . . . . . . . . . . . . . . . . 11
2.2 Sample Plot of Gauss-Markov process, 2b = 1 , ? = 100 . . . . . . . . . 14
2.3 Allan Variance of Simulated Data 5 Hz 2rw = 1:2, 2b = 4, ? = 300 . . . 20
2.4 Sample Autocorrelation: 1.7x105 time units, fs = 5 Hz, 2b = 4, ? =
200sec . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1 Standard Deviation of Wide-Band Noise Process: 10Hz, ! = 1, 2000
Monte Carlo iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Standard Deviation of Integrated Wide-Band Noise Process: 10Hz, ! =
1, 2000 Monte Carlo iterations . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Standard Deviation of Double Integrated Wide-Band Noise Process:
10Hz, ! = 1, 2000 Monte Carlo iterations . . . . . . . . . . . . . . . . . 49
3.4 Standard Deviation of Gauss-Markov Process: 10Hz, b = 2, ? = 30, 2000
Monte Carlo iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5 Standard Deviation of Integrated Gauss-Markov Process: 10Hz, b = 2,
? = 200, 2000 Monte Carlo iterations . . . . . . . . . . . . . . . . . . . . 50
3.6 Standard Deviation of Double Integrated Gauss-Markov Process: 10Hz,
b = 2, ? = 200, 2000 Monte Carlo iterations . . . . . . . . . . . . . . . . 51
3.7 Body Constrained to Travel in One Direction . . . . . . . . . . . . . . . . 52
3.8 Simulated Acceleration Proflle: 10Hz, ! = 0.5, b = 0.25, ? = 200 . . . . 54
3.9 Velocity with Bounds from Accel Proflle . . . . . . . . . . . . . . . . . . 55
3.10 Position with Bounds from Accel Proflle . . . . . . . . . . . . . . . . . . 55
3.11 Integrated Sensor 1- Bounds for Ratio from 0.1 to 1 . . . . . . . . . . . . 57
3.12 Integrated Sensor 1- Bounds for Ratio from 0.01 to 0.1 . . . . . . . . . . 57
x
3.13 Integrated Sensor 1- Bounds for Ratio from 0.001 to 0.1 . . . . . . . . . 58
3.14 Integrated Sensor 1- Bounds for Fixed Ratio of 0.1 and ? Between 200
to 1000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.15 Value of Integrated 1- Bounds at time index of 120 for Fixed Ratio of
0.1 and ? Between 200 to 1000 . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Simplifled Coordinate Frame . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.2 Simplifled Coordinate Frame 2D . . . . . . . . . . . . . . . . . . . . . . . 63
4.3 Navigation Relationships for Side-Slip Vehicle . . . . . . . . . . . . . . . . 63
4.4 Navigation Relationships for Vehicle with No-Slip . . . . . . . . . . . . . . 64
4.5 Simulated Position Trajectory . . . . . . . . . . . . . . . . . . . . . . . . 77
4.6 Standard Deviation of East Velocity, 2000 Monte Carlo iterations . . . . 79
4.7 Standard Deviation of North Velocity, 2000 Monte Carlo iterations . . . . 79
4.8 Deflned Position Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.9 Velocity Trajectory With No Side Slip . . . . . . . . . . . . . . . . . . . . 91
4.10 Yaw Angle Trajectory With No Side Slip . . . . . . . . . . . . . . . . . . 91
4.11 3- Bounds on Simulated Yaw Angle . . . . . . . . . . . . . . . . . . . . . 92
4.12 RMS 3- Bounds on Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.13 RMS 3- Bounds on Position . . . . . . . . . . . . . . . . . . . . . . . . . 93
5.1 Simplifled Coordinate Frame . . . . . . . . . . . . . . . . . . . . . . . . . 97
5.2 Mechanization of IMU Measurements . . . . . . . . . . . . . . . . . . . . . 100
5.3 Deflned Position Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4 Velocity Trajectory for Planar Motion . . . . . . . . . . . . . . . . . . . . 102
5.5 Yaw Angle Trajectory for Planar Motion . . . . . . . . . . . . . . . . . . . 102
5.6 3- Bounds on Yaw Angle Mechanization Comparison . . . . . . . . . . . 104
xi
5.7 RMS 3- Bounds on Velocity Mechanization Comparison . . . . . . . . . 104
5.8 RMS 3- Bounds on Position Mechanization Comparison . . . . . . . . . 105
5.9 3- Bounds on Pitch Angle DOF Comparison . . . . . . . . . . . . . . . . 106
5.10 3- Bounds on Roll Angle DOF Comparison . . . . . . . . . . . . . . . . . 106
A.1 Filtered Accelerometer Outputs . . . . . . . . . . . . . . . . . . . . . . . . 119
A.2 Filtered Gyro Outputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
A.3 Raw and Filtered Data Within Selected Section . . . . . . . . . . . . . . . 120
A.4 Filtered Data Within Selected Section . . . . . . . . . . . . . . . . . . . . 121
A.5 Autocorrelation of Filtered Gyro Data . . . . . . . . . . . . . . . . . . . . 122
A.6 Allan Variance of Gyro Data . . . . . . . . . . . . . . . . . . . . . . . . . 123
B.1 Position of Track from Starflre GPS . . . . . . . . . . . . . . . . . . . . . 126
B.2 Velocity from GPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.3 Yaw Angle (Heading) from IMU . . . . . . . . . . . . . . . . . . . . . . . 128
B.4 Roll and Pitch Angles from IMU . . . . . . . . . . . . . . . . . . . . . . . 129
B.5 North and East Component Velocities . . . . . . . . . . . . . . . . . . . . 130
B.6 North and East Component Positions . . . . . . . . . . . . . . . . . . . . 132
xii
Chapter 1
Introduction
1.1 Overview
With the wide-spread availability of the Global Positioning System (GPS) and con-
tinued technological advances in Inertial Navigation Systems (INS), the fleld of navi-
gation continues to exploit these ever expanding resources. One of the most popular
methods of current-day navigation involves the synergy of these two systems and is
commonly known as GPS/INS integration. The basic idea of the GPS/INS navigation
approach is to utilize the complementary strengths and weaknesses of each system in
order to navigate with accuracy superior to that of either component system?s stand
alone performance. GPS provides precise global position, velocity, and heading infor-
mation, but updates at a relatively slow rate, is subject to interference, and requires
a clear view of the sky. The INS most commonly consists of an inertial measurement
unit (IMU) which is typically attached rigidly to the frame of the navigating body. The
IMU is a device in which three accelerometers and three rate-gyroscopes (referred hence-
forth simply as rate-gyros) are orthogonally mounted in a sealed unit. Its purpose is
to measure the body accelerations and rotation rates in the corresponding component
directions and about the corresponding axes as deflned by its alignment on the vehicle.
The inertial sensors which make up an IMU are available in a wide-variety of grades
determined by the basic principle of operation, quality of materials, and integrity of
methods/design. The accelerometers and rate-gyros, as with many electronic sensors,
are corrupted by stochastic-type disturbances (noise) which manifest themselves on the
1
sampled output of the device as random variables. The characteristics of these random
variables depend upon the principle of operation and integrity of materials/methods used
to produce the device. To achieve navigation information from such an INS, the outputs
must be numerically integrated and transformed to attain the desired attitude, velocity,
and position information. Due to the stochastic errors on the sensors, the integral values
exhibit an ever increasing variance. Therefore, in contrast to GPS, Inertial Navigation
Systems provide high update information, but will digress with time without corrections.
In summary, the GPS/INS combined system uses the high update inertial measurements
of the INS to boost the slow rate of the position information from GPS.
The most common GPS/INS approach employs an optimal estimation technique
known as the Kalman Filter [1]. This Navigation Kalman Filter numerically blends the
high-rate, short term accuracy of the INS with the long term accuracy of the GPS in an
optimal way. Many variations and approaches of the GPS/INS Kalman Filtering method
have been researched with the intent of providing the best possible navigation solution.
The loosely-coupled GPS/INS uses the common measurements from a standard receiver
and requires at least four satellites for normal operation. More advanced approaches
such as tightly-coupled and deeply-integrated GPS/INS probe deeper within the GPS
receiver and blend the INS with more of the available GPS information. Both of these
methods intend to improve some of the weaknesses of the common GPS/INS system for
more robust navigation performance [2]. In addition to the GPS/INS only approach, re-
searchers have also developed and implemented navigation systems which include other
aiding measurements such as vision [3, 4], odometry [5], and laser-scanners [6]. These ad-
ditional measurements supplement the existing GPS/INS system to improve the overall
navigation performance, especially during GPS outages.
2
For the case when GPS is unavailable, the INS and other sensors become the sole
means of navigation, and the accuracy of the attitude, velocity and position informa-
tion as derived from body-frame-only measurements degrades with time. This GPS-less
navigation mode is known as dead-reckoning whereby the body in motion navigates only
with information on board itself. The characteristic error growth in the dead-reckoning
mode depends upon the amount of time since the GPS signal outage, the equations used
in providing the navigation solution (kinematics of vehicle motion), and the integrity of
the sensors used to dead-reckon.
The particular focus of the research presented in this text is to perform an analysis of
inertial-only navigation in which both GPS and other sensors are not used. This analysis
provides a quantiflcation of the dead-reckoning accuracy of the GPS/INS system when
GPS is unavailable. As there are many difierent types and grades of inertial measure-
ment units, the navigation system designer is charged with the tasks of selecting an IMU
appropriate for the environment in which it will be employed, the budget of the project,
and speciflc objectives of the flnal system. The typical manufacturer speciflcations of
the raw inertial measurement errors do not fully depict its performance as a navigation
means. That is, they fail to provide the information necessary to flrmly quantify the nav-
igation performance of an IMU within the common scenarios encountered. The common
speciflcations from the sensor manufacturer provide only a rough quantiflcation of the
raw output of the sensors; no information of the propagated error is available. There-
fore, to accurately compare and distinguish the dead-reckoning performance of the many
varieties of IMUs, an analysis of the integrated inertial measurements is imperative.
3
1.2 Prior Art
GPS/INS integration is a very widely implemented and studied scheme as it provides
high rate (100 Hz) and high accuracy positioning (1-10 meters) information to a navigat-
ing vehicle at a reasonable cost [2]. As GPS is susceptible to signal loss, the performance
of the GPS/INS system is often quantifled by its INS-only (i.e. dead-reckoning) perfor-
mance. Approaches to the quantiflcation of the INS-only navigation performance have
been based heavily on experimental results. Techniques such as parametric modeling of
IMU-derived positioning data [7] and comparisons based on the use of vehicle dynamics
in mechanization algorithms [8] all intend to quantify the performance of the GPS/INS
system during GPS outages. These post-processing analyses, while giving precise infor-
mation about the INS-only system performance for particular experimental cases, fail
to provide information in support of the general performance of inertial dead-reckoning.
In support of the more general approach to the quantiflcation of performance of iner-
tial dead-reckoning, research has been conducted in which the analysis commences with
initial investigation of the inertial sensor errors themselves. Such approaches have inves-
tigated the characteristics of the stochastic behavior of the raw inertial sensor outputs
by using experimental identiflcation techniques of Allan variance [9, 10, 11, 12, 13], and
autocorrelation [11, 14]. Based on stochastic error models selected using the identifl-
cation techniques, the research has been extended to provide the exact error growth of
the integral values of the stochastic process [11]. Other researchers following the same
path, have investigated the analytical propagation of error into the navigation solution
when mechanized in simple scenarios [14, 15]. These propagation analyses, however,
have restricted their analytical study to the the efiect of white (wide-band) noise on
4
the navigation solution thereby neglecting signiflcant efiects due to sensor bias-drift [14].
The research presented in this thesis has expanded the previous analytical work to in-
clude analytical results of the in uence of both wide-band noise and sensor drift on the
integral sensor outputs. Additional studies of the sensor errors in more complex vehicle
motion and mechanization methods are presented as well.
1.3 Contributions
To accomplish the goals of this research, this thesis presents a study of the stochastic
inertial sensor error characteristics to provide a practical quantiflcation of the uncertainty
growth when dead-reckoning with inertial measurements. This quantiflcation provides
the a priori information on the performance of GPS/INS/AuxSensor navigation in sup-
port of the system designer?s goal of achieving the best navigation performance within
the criteria of given objectives. The research presented provides quantiflcation of inertial
navigation through:
? Proposing a candidate sensor model su?cient to capture the stochastic behavior of
inertial sensor outputs and detailing the means by which the model parameters may
be identifled.
? Development of expressions for the variance of the integrated values of the inertial
sensor error sources.
? An analysis on the relative in uence of sensor error model parameters on the variance
of integrated outputs.
? A study of the propagation of inertial sensor errors into the navigation equations for
a vehicle kinematically constrained in a two-dimensional scenario.
5
? A comparison of the in uence of the navigation equations on the performance of
inertial navigation under various vehicle motion assumptions.
1.4 Thesis Organization
Chapter 2 introduces a simple inertial error model consisting of the sum of two
stationary Gaussian random processes. The processes model the stochastic behavior of
white noise and sensor bias drift observed on both accelerometers and rate-gyros. The
means of experimental identiflcation of the speciflc error process parameters with the
use of Allan variance and Autocorrelation techniques are presented.
Chapter 3, the heart of this research, utilizes the stochastic models presented in
Chapter 2 to derive analytical expressions for the variance of numerically integrated and
double-integrated error processes. The derived expressions are validated using Monte
Carlo computer simulations with flxed parameters. A simple example is used to illustrate
the applicability of the integrated error variance expressions for a simple single-axis
navigation scenario. This chapter concludes its contribution with a study of the efiects
of the stochastic error process parameters on the integrated sensor error growth.
Chapter 4 applies the information from Chapter 3 to study a simple navigation
scenario in which a vehicle travels on a at plane with no lateral velocity (i.e no side-
slip). Expressions for the variance of 2-D velocity error are derived in closed-form and
the framework for 2-D position error is shown. The analysis is then extended to the
more general 2-D case where a vehicle experiences side-slip and exhibits lateral velocity.
A Monte Carlo computer simulation is used to show the additional error when no-slip
trajectory is processed with the assumed-slip equations.
6
Chapter 5 introduces the six degrees-of-freedom (6-DOF) navigation equations as
required for the inertial navigation of a body experiencing motion in the most general
sense. The characteristics of the error for the 6-DOF inertial navigation scenario are
exemplifled with the use a computer simulation. In the simulation, the inertial navigation
error from the 6-DOF method is compared to the planar navigation method for a planar
vehicle trajectory. The purpose of this chapter is to show the efiect of additional sensors
and transformations when they are not required.
Chapter 6 concludes the work of this thesis with a summary of the results, discus-
sionofthevariousassumptionsandapproximationsemployedinthetext, andsuggestions
for future research.
7
Chapter 2
Inertial Sensor Modeling
2.1 Introduction
A preliminary step in describing the behavior of inertial navigation error is to deter-
mine appropriate error models of the inertial sensors themselves. This chapter presents
a simple inertial sensor model su?cient to describe the approximate stochastic nature of
the outputs of both accelerometers and rate-gyros across the range of available devices.
The identiflcation of the models of two random processes presented in this chapter, have
been well researched by many authors including [9, 11, 16] and the applicable techniques
and limitations of these methods are presented proceeding the models in the following
sections.
2.2 Simple Sensor Models
Simple stochastic sensor models are selected to provide approximate representations
to the behavior of that observed on a wide range of inertial measurement sensors. These
simple models apply to both accelerometers and rate-gyros and lend themselves well
for deriving analytical results which quantify the propagation of sensor errors into their
processed states. Although more advanced stochastic models could be developed, the
simpler models provide approximations in support of the goals of this thesis: closed-form
expressions for the error growth when dead-reckoning with inertial measurements.
The general inertial measurement model used in this study has the form of Equation
(2.1) and is suitable for either an accelerometer or rate-gyro.
8
ymeas = (SF)y +?+c (2.1)
y is the true sensor value
? are stochastic ?static? terms
c is the constant bias
SF is the scale factor
The scale factor, though shown in [17] to exhibit some stochastic behavior through-
out the range of sensor motion and temperature, is assumed to have flxed relationship
between the sensor?s input and output. The bias, c is also a flxed quantity, exhibits a de-
terministic behavior with respect to the motion the sensor is sensing. Simple calibration
techniques can easily ascertain the scale-factor and constant bias and are thus removable
prior to implementation. The stochastic terms however, are not predictable and remain
even on a calibrated sensor output. Therefore, a fully calibrated static sensor output as
measured can be described by Equation (2.2).
ymeas = ? (2.2)
The stochastic terms, ?, are, at this point in the heuristic approach, unknown.
The following section introduces simple stochastic processes which intend to capture the
unpredictable stochastic behavior of the calibrated sensor output.
9
2.3 Stochastic Sensor Models
2.3.1 Gaussian White Noise
Inspection of a short-duration static sensor output reveals that the value of the signal
jumps from one value to the next in an unpredictable, or random, manner. Successive
samples in time of such a signal are described as uncorrelated from one time-step to the
next. A random process whose successive values in time are uncorrelated is known as a
white noise process. While the conceptual abstraction of white noise implies that there
is inflnite frequency content in the process, the sensor output is better described by wide-
band noise. As as the frequency content of a signal is limited by sampling equipment,
materials, and other phenomena, wide-band noise is, like white-noise, uncorrelated, but
for all measurable or applicable frequencies.
Closer observations of the short-duration static output indicate that the values of
the static output at any given time are scattered in higher density about some mean
value and becoming scarcer with increasing value from the mean. A good statistical
model for this scattering of the data is the Gaussian or Normal Distribution, whose
probability density function is given by Equation (2.3).
fX(x) = 1p2?
x
exp
"
?12
x?m
x
x
?2#
(2.3)
mx is the mean of random variable x
x is the standard deviation of random variable x
10
Complete probabilistic characterization of a random variable, x, with a Gaussian
distribution requires knowledge only of the mean value, mx, and variance 2x. Therefore,
for the uncorrelated wide-band Gaussian noise process, the sensor output value at any
timeisdescribedbyaGaussiandistribution withthesamemeanandvariance. Therefore,
each successive time sample of the process is independent (i.e. no relationship exists
between the values from one step to the next). An example of a zero mean, unit-variance
Gaussian wide-band noise process is shown in Figure 2.1.
0 200 400 600 800 1000?5
0
5
Value
Time Index
output
??
+?
Figure 2.1: Sample Plot of Wide Band Noise, 2 = 1
For high grade sensors, the static output for short time durations (order of minutes)
can solely be described by a wide-band noise process. However as the cost of sensor
decreases, the integrity of the its output also decreases and other stochastic phenomena
can be observed on the measurement. The output of lower-grade sensors exhibit both
uncorrelated (wide-band) noise and additional stochastic noises which exhibit varying
levels of time correlation, discontinuity, and other irregularities. The discontinuities and
irregularities pose a problem in stochastic characterization as their presence dominates in
11
less-expensive sensor outputs. As a result, a signiflcant amount of study has been dedi-
cated to the development of sophisticated models that accurately capture such low-grade
sensor behaviors. However, for the purpose of this research, a simple stochastic model is
selected to provide a conservative approximation to the bias drift in accordance with the
practical utility of the modeling approach of this thesis. The dominant phenomena on
the low grade sensors often materialize as a slowly wandering error in the output and are
often referred to as a sensor?s drifting bias, or bias drift. The following section presents
an approximate model for this stochastic drifting behavior.
2.3.2 The Gauss-Markov Process
The 1st-order Gauss-Markov process has been used extensively in the navigation
and estimation community to model the various stochastic drift characteristics present
on many types of navigation system outputs [11, 18]. For the purpose of the research
goals of this thesis, this process provides a conservative approximation to the observed
bias drift on many inertial measurements.
A random process is said to be Markovian if its probability density function (PDF)
at any point in future time can be completely specifled with the knowledge of the pro-
cess PDF at the current time. The Markovian property is analogous to the concept in
linear state-space systems whereby the future states can be ascertained by current states
and inputs. A Gauss-Markov process is a stochastic process whose underlying random
phenomenon that drives the process is as a Gaussian sequence of random variables [19].
A commonly used 1st-order Gauss-Markov process model is simply the output of
a low-pass fllter with a zero-mean white noise input. The governing stochastic linear
difierential equation for such a process is expressed as
12
_b = ?b
? +!b (2.4)
? is the time constant
!b is a zero mean Gaussian random variable with variance, 2!b
The process given by Equation (2.4), in continuous time, is somewhat of an abstrac-
tion of the more applicable discrete version of the Gauss-Markov process (or sequence).
This stochastic sequence is given by the linear stochastic difierence Equation (2.5).
bk = 'kbk?1 +wbk
'k = e??tk? (2.5)
' is the state-transition matrix for the process
wb is a zero mean Gaussian random variable with variance, 2wb
As seen by Equation (2.5) the process output is the sum of the Gaussian driving
noise, !, and past values of itself. The output of the process therefore can also be de-
scribed by a Gaussian distribution and can be completely probabilistically characterized
by its mean and variance functions. In contrast to wide-band noise, however, the Markov
process exhibits a non-zero time correlation for any given realization of the process be-
cause of its dependence on its past values. This correlation characteristic is what causes
13
the process to appear as a slowly drifting bias, which is the desired model. One such
example of the Gauss-Markov process is shown in Figure 2.2.
0 200 400 600 800 1000?2
?1
0
1
2
3
Value
Time Index
output
??
+?
Figure 2.2: Sample Plot of Gauss-Markov process, 2b = 1 , ? = 100
Since the driving process is zero mean and Gaussian, the output of the process is
zero mean and Gaussian with a transformed variance. Since the time constant is a value
greater than the sampling frequency (generally much larger), the characteristic root of
the 1st-order Markov dynamics is inside the unit circle, and thus the process is stable.
The stability of the process indicates that the variance of its output will reach a steady
state value after some initial settling time at which point the process is considered to
be stationary. Once steady state has reached, the process autocorrelation function takes
the form of Equation (2.6).
Rbb(T) = b2e?T=? (2.6)
14
? is the time constant or correlation time
b is the variance of the process 2!k
The simple autocorrelation expression lends itself well to experimental identiflcation
as it is straightforward to extract the magnitude and time constant of the process from
a given plot of (2.6). Due to the form of Equation (2.6), the Gauss-Markov process is
also referred to as exponentially correlated noise [16].
2.3.3 A Simple Stochastic Model
For short time intervals an inertial sensor?s output appears as uncorrelated noise
(white noise); however for longer time intervals the sensor exhibits a correlated noise
(drifting bias). An approximate model of this behavior is simply the sum of the two
random processes introduced in the previous sections. The error due to the stochastic
behavior of an inertial sensor at time step, k, can thus be described by Equation (2.7).
?k = !k +bk (2.7)
!k is uncorrelated wide-band noise with zero mean and variance 2!
bk is a 1st-order Gauss-Markov process with time constant, ? and variance 2b
The model above assumes that both processes are zero-mean and no correlation
exists between the white noise process and the Gauss-Markov process; they are indepen-
dent. The model consists of three parameters which allow for three degrees of freedom
15
in describing the sensor behavior of any given accelerometer or rate-gyro. The following
section introduces methodologies used in determining the best set of parameters from
experimental data of a given inertial sensor.
2.4 Stochastic Identiflcation Techniques
Several techniques exist to process experimental sensor data and determine the
various types of error sources present. Given that the raw sensor output is modeled
by Equation (2.7), such identiflcation techniques allow the extraction of the stochastic
model parameters !, 2b, and ?. Of the array of stochastic modeling techniques, the
Allan variance technique has become widely popular in the inertial navigation community
due to its intuitive implementation, straightforward interpretation, and in most cases
unique indication of various stochastic disturbances [20]. This section shows its use as a
comprehensive means to determine approximate parameters to describe a given inertial
sensor. Following the Allan variance technique, the experimental autocorrelation method
is presented. Supplemental to the Allan variance, the autocorrelation method is used for
speciflc identiflcation of the parameters associated with the Gauss-Markov process, 2b,
and ?.
2.4.1 The Allan Variance
The Allan variance technique was introduced by David Allan in the 1960s to char-
acterize the frequency stability of high-precision atomic clocks [21]. The Allan variance
technique is sometimes analogized to the time domain version of the Fourier transform
as it provides a measure of the dominance of a stochastic process over various ranges of
time. The technique computes the variance of arrangements of successive averages of a
16
time set of data which indicates the contribution or dominance of various error sources
as a function of averaging time. Inspection of plots of the Allan variance versus averag-
ing time indicate the presence of speciflc stochastic processes present on a measurement.
The square root of the Allan variance (root Allan variance) for many common stochastic
processes appear as straight lines on a log-log scale. As a result, simple visual inspec-
tion of the root Allan variance provides immediate information about a sensor?s overall
stochastic behavior.
The precise deflnition of the Allan variance is repeated below as adapted from [9].
Given a set of N inertial measurements, ?, sampled at a rate of fs Hz, deflne a vector
of averaging times, T, ranging from T0 seconds to up to half the total time length of the
data set ( N2fs) as shown in Equation (2.8).
T =
?
T0 T0 +fs T0 +2fs ::: N2fs
?
(2.8)
For each averaging time, T, deflne K = N=M clusters where M is the number of
samples per cluster (M = Tfs). Compute cluster averages with Equation (2.9) where k
is the time index of the raw data.
??(T) = 1
M
MX
i=1
?(k?1)M+i; k = 1;:::;K (2.9)
The Allan variance is then computed using Equation (2.10), with an approximation to
the true ensemble average [22].
2AV (T) = 12(K ?1)
K?1X
k=1
???
k+1(M)? ??k(M)
?2 (2.10)
17
The Allan variance can be expressed as a simple sum of the Allan variance contri-
butions of each of the dominant processes. For example, the Allan variance as a function
the contribution of process 1, 2, 3 and so on is expressed as
2AV = 2p1 + 2p2 + 2p3 +::: (2.11)
Well-known analytical expressions of the power spectral density (PSD) of common
stochastic processes have been related to analytical expressions for the Allan variance
[9, 22]. The result of this relationship has yielded Allan variance equations for each noise
process in terms of its processes parameters and the averaging time, T. The component
Allan variances can be summed using Equation (2.11) to yield the total Allan variance
curve as a function of averaging time. A full description of the method used to compute
the Allan variance from experimental data can be found in [20].
The inertial sensor model introduced in Section 1.3 is the sum of the two stochas-
tic processes: wide-band noise and exponentially correlated noise (Gauss-Markov). The
wide-bandnoiseisquantifledbythe\randomwalk"parameter, rw, whichisthestandard-
deviation of the process normalized to the square-root of the sampling frequency, fs in
Hertz. The Allan variance expression for wide-band noise is thus given by Equation
(2.12).
2rw(T) =
wp
fs
?2
(2.12)
18
The Allan variance expression for a flrst-order Markov process is given in terms of
its time constant, ?, and driving noise variance, 2!b, as shown in Equation (2.13).
2bias(T) = ( !b?)
2
T
h
1? ?2T
?
3?4exp?T? +exp?2T?
?i
(2.13)
The resulting analytical expression for the Allan variance of a static sensor sensor
output with a stochastic model as in Equation (2.7) is given as the sum of equations
(2.12) and (2.13) as shown in equation (2.14).
2AV (T) = 2rwT + ( !b?)
2
T
h
1? ?2T
?
3?4exp?T? +exp?2T?
?i
(2.14)
The root Allan variance as is most commonly plotted is simply the square root of the
summed quantities in Equation (2.11). Figure 2.3 shows a sample root Allan variance
plot as computed from a simulated sensor output with speciflcations as labeled.
19
100 101 102 103 104
10?1
100
Averaging time, T
Allan Root Variance,
? A
V(T)
Data
Equation
Bound
Figure 2.3: Allan Variance of Simulated Data 5 Hz 2rw = 1:2, 2b = 4, ? = 300
The analytical expressions shown above have been used by others to curve-flt ex-
perimental Allan variance data for the purpose of extracting the underlying stochastic
process parameters and thus characterizing the process. [9, 22, 13]. For example, the
random walk parameter is straightforward to extract as it is simply the value of the Allan
variance at the averaging time equal to one second. However, since the expression for the
Markov process in Equation (2.13) is a non-linear function of the process parameters,
estimation becomes di?cult and employment of manual methods may be necessary to
provide an estimate of the process magnitude and time constant.
One such problem in practical utilization of the Allan variance for sensor character-
ization is that its accuracy is limited by the time-length of the experimental data set.
The range of averaging time is computed for half the range of experimental data. Ad-
ditionally, the number of segments of averaged data for which the variance is computed
20
become smaller as the averaging time increases and thus yields statistically less signifl-
cant values. As a result, the bounds on the accuracy of the Allan variance increase with
averaging time at rates dependent upon the total length of the experimental data set.
This behavior can been seen in the sample Allan variance in Figure 2.3. Such bounds are
discussed further in [9]. Due to this requirement, accurate Allan variance identiflcation
of an inertial sensor exhibiting slow drift characteristics requires a sample data set of
length many times greater the time constant of the particular drift of interest. For many
inertial measurements, the times required for reasonable accuracy can be on the order
of several days. Additionally, the time constant for the sensor is simply unknown since
it is to be identifled. Some rules of thumb are suggested for the length and required
sample frequency of the sensor data set to ensure accurate Allan variance identiflcation
for time-varying processes in [20].
2.4.2 Experimental Autocorrelation
The Allan variance technique, while su?cient to extract the parameter associated
with the white noise process, remains a di?cult method for the bias drift characteriza-
tion. As the Gauss-Markov process in steady-state has an autocorrelation in the form of
Equation (2.6), the bias time constant, ?, and magnitude of drift, b, can be extracted
from the experimental autocorrelation. However, since the sensor is modeled as the sum
of the two processes, successful identiflcation of the Markov process requires its isolation.
Approximate isolation can be performed by low-pass flltering the raw sensor output. The
flltering removes the higher-frequencies for which the output is uncorrelated while leav-
ing the correlated low-frequency data of interest. However, despite the attenuation of the
21
higher frequencies of the wide-band noise process, its lower-frequencies still exist as ad-
ditive noise on the drifting bias, increasing the di?culty of identiflcation. Furthermore,
appropriate selection of the isolating fllter cut-ofi frequency requires some knowledge of
the approximate time constant of the drift process. As it is the time constant that is to
be identifled, the autocorrelation technique is an approximate and iterative process.
Equation (2.6) gives the characteristic autocorrelation function of the Gauss-Markov
process, which is a simple exponential decay with an initial magnitude of 2b and time
constant, ?. Figure 2.4 shows a sample autocorrelation plot of a simulated Markov
process and its analytical curve. As is shown by the dotted lines on the plot, the time
constant is found by reading the time-shift for which the autocorrelation value decays to
1
e of its initial magnitude. The initial magnitude of the Gauss-Markov process is simply
the y-intercept, or variance of the flltered process.0 200 400 600 800 1000 1200
?1
0
1
2
3
4
Lags, t
R x
x(t)
Data
Equation
?2b e?1
3?? Bound
Figure 2.4: Sample Autocorrelation: 1.7x105 time units, fs = 5 Hz, 2b = 4, ? = 200sec
22
Unfortunately, the experimental autocorrelation shares the same practical drawback
with the Allan variance: the accuracy of the autocorrelation is dependent upon the length
of the dataset. In general, the data set must be a length su?ciently longer than the time
constant of interest. Experiments performed by the author in an attempt to extract
reasonable parameters from raw gyro and accelerometer data proved very di?cult as long
data sets often carried un-modeled disturbances (such as temperature efiects, long initial
settling time). For a stochastic model with an autocorrelation in the form of Equation
2.6, the upper bound of the variance of the empirically-derived autocorrelation, Rbbexp
at any lag-value can be expressed by Equation (2.15) [23].
VAR[Rbbexp] <= 2
4b?
Td (2.15)
Where, 2b is the variance, ? is the time-constant, and Td is the time-length of the
experimental data set.
The bounds plotted in Figure 2.4 show the large uncertainty in the Gauss-Markov
process experimental autocorrelation for a data set of length much longer than its time
constant. For the speciflc sample of data generated in the flgure, the time length of data
set was more than 150 times the time constant.
2.4.3 Implementation Issues
As discussed for both methods, accurate identiflcation of the process for the slowly-
varying stochastic behavior of inertial sensors requires a data set many times longer than
the time constant of the process. For many inertial sensors the bias drift is a slow process
23
(large time constant) yet still contributes signiflcantly to the output. Efiective identifl-
cation for such a process requires a very long data set, sometimes outside the range of
logging capabilities or a feasible window of time. Additionally, since the sensor parame-
ters are unknown, the required length of the data set is therefore unknown and several
iterations may be necessary. Additionally, the approximate Gaussian models presented
may not provide a su?cient characterization of their observed behavior, especially for
lower grade sensors. The identifler of these irregular sensors must then resort to highly
conservative parameter estimates which simply give rough values comparable to that
supplied by the sensor?s manufacturer. Discussion on the supplied sensor speciflcations
and the need for accurate identiflcation follows in the next section.
2.5 Experimental Quantiflcation and Identiflcation
2.5.1 Manufacturer Sensor Speciflcations
Sensor speciflcation sheets give a simple overview of a sensor?s operating charac-
teristics including measurement range, input power consumption, data format, as well
as expected accuracy. The accuracy of the sensor, as it is limited by the magnitude of
the stochastic processes on the output, is quantifled in varying detail across the broad
spectrum of sensors and their listed speciflcations. The speciflcations often indicate pa-
rameters which coarsely bound the expected accuracy based on two assumed stochastic
characteristics: noise and bias. The noise is quantifled by the random walk parameter as
introduced in the Allan variance identiflcation section. The random walk parameter, rw
is deflned by Equation (2.12). Its units are usually given in two forms shown in Equation
24
(2.16).
rw = [output units]pHz
rw =
R[output units]dt
pHr (2.16)
The random walk parameter is easily extracted from experimental Allan variance
data as shown in the previous section.
A wider array of speciflcations are listed to quantify the bias drift. As a minimum,
sensor manufacturers publish a conservative maximum value or maximum standard devi-
ation within which the properly calibrated sensor is expected to output. Others give bias
drift quantiflcation in terms of a value ascertained from the Allan variance chart. Ref-
erence [24] indicates that the bias variation or bias instability parameter listed in some
sheets is the lowest point on the Allan variance chart. In addition to the magnitude of
the bias drift, some manufacturers of higher grade devices include some indication of the
speed at which this bias drifts from the mean value by a correlation time. In any case,
inertial sensor manufacturers provide very little information in support of full stochastic
characterization of the outputs.
2.5.2 Experimental Approach
Upon review of the available literature, manufacturers provide only conservative
bounds on a sensor?s stochastic characteristics. This information gives only a rough
starting point for the accurate characterization of this research. This section presents
the basic methodology by which su?cient identiflcation can be performed. A general
methodology can be roughly outlined by the following steps:
25
1. Examine available speciflcations by manufacture
2. Determine required sampling frequency and duration of data set based on specs
3. Acquire completely static data set (on level surface)
4. Remove constant bias (subtract the mean)
5. Run the Allan variance of the data set, extract the wide-band noise magnitude
6. Filter(zero-phase fllter) the data to reveal the underlying moving bias
7. Process the flltered data in the autocorrelation and extract time constant
The success of experimental identiflcation is often di?cult in practice due to many
of the reasons discussed in the preceding sections. The main di?culties are that the
sensor drift model is only an approximation, the approximate model parameters are
unknown, and the bias can not be fully isolated. As a result, the general methodology
remains a long and highly iterative process. The general process of stochastic sensor
error parameter identiflcation is demonstrated in Appendix A following the chapters of
this thesis. The appendix presents the use of the techniques of this chapter to identify
the stochastic model parameters with experimental data from an automotive-grade IMU.
2.6 Conclusion
In this chapter, a simple inertial sensor error model that provides an approximation
to the stochastic behavior observed on a a wide-range of inertial sensors grades and types
has been presented. The model consists of the sum of two stationary, Gaussian random
processes which describe the short-term and long-term stochastic behavior of static iner-
tial sensor outputs. The techniques of Allan variance and autocorrelation were presented
26
as means to identify the three parameters required to describe the assumed model form
from experimental sensor data. The experimental procedure was then outlined and dif-
flculties of the method presented. In the following chapters, each of the error processes
as detailed in this chapter are used for the purpose of quantifying the error in position,
velocity, and attitude when dead-reckoning with an IMU in various kinematic scenarios.
27
Chapter 3
Covariance Propagation of Stochastic Errors
3.1 Introduction
Bounded accuracy in inertial navigation depends upon regular position, velocity,
and attitude measurements to compensate for the error growth of the integrated IMU
signals. Many navigation methods employ regular measurements from GPS sensors,
vision, odometry, and other sources of velocity, position, and attitude data. These
measurements are often fused together in a navigation Kalman fllter resulting in an
optimal estimate of the vehicle?s state. As GPS requires an unobstructed line-of-sight
to at least four satellites, it fails to provide accurate data when traveling under bridges,
heavily wooded areas, and in downtown city streets where tall buildings bound the path
of the receiver. Under such conditions when the GPS data becomes unavailable, the
Kalman fllter reduces to a simple algorithm in which the navigation states are derived
solely from the integrated outputs of the inertial sensors initialized to the last \best
estimate". As all inertial sensors are inherently corrupted with stochastic type errors (as
introduced in Chapter 2), the integration of these signals cause the uncertainty in the
resulting navigation states to increase with each step in time. As a result, the error in
the estimated position, velocity, and attitude states grow with time. It is the goal of this
chapter to quantify the the error growth due to the integrated stochastic errors present
on the inertial measurements.
By using the stochastic models from Chapter 2, this quantiflcation is achieved by
deriving expressions for the variance of the integrated sensor errors using a technique
28
modeled after [11]. The variance expressions are then validated using a Monte Carlo
simulation. For a single-axis sensor, the variance describes the error expected when
dead-reckoning in one degree of freedom motion. The chapter concludes with a sensitivity
analysis illustrating the in uence of the stochastic model parameters on the variance of
the integrated sensor.
3.2 Simplifled Navigation Scenario: Single Axis
The motion of a navigating body in the inertial or navigation frame can be derived
from body-flxed inertial measurements of acceleration, a, and rotation rate, g. To attain
the vehicle states of velocity, orientation, and position in the navigation frame in the
general sense, the body-frame measurements are transformed using nonlinear difieren-
tial equation relationships (shown later in Chapter 5). As a building block in providing
an analysis of the general navigation scenario, preliminary attention is flrst turned to
a simple one degree-of-freedom (1-DOF) motion scenario in which a single axis gyro or
single axis accelerometer is integrated to provide the navigation frame states in its com-
ponent direction. Equations (3.1-3.3) show the single-axis vehicle states of orientation,
velocity, and position (?, V, P) as derived from the integrations of the corresponding
inertial measurements.
? =
Z
gdt (3.1)
V =
Z
adt (3.2)
P =
Z
V dt (3.3)
29
It is the task of the following sections to quantify the error resulting from the use
of Equations (3.1-3.3) with inertial sensors modeled as shown in Chapter 2.
3.3 General Characterization of Raw Sensor Measurement
Recall the simple sensor model from Chapter 2 as described by Equation (2.1)
ymeas = (SF)y +?+b
As stochastic terms are assumed zero-mean, the mean value of the sensor output is
simply the deterministic terms.
E[ymeas(t)] = E[(SF)y(t)]+E[?(t)]+E[b]
= (SF)y(t)+b (3.4)
The variance of the sensor output is equal to the variance of the stochastic error.
VAR[ymeas] = E[y2meas(t)]?E[ymeas(t)]2
= E[((SF)y +?(t)+b)((SF)y(t)+?(t)+b)]?((SF)y(t)+?(t)+b)2
ymeas2(t) = ?2(t) (3.5)
Since the error sources are Gaussian and uncorrelated, the variance in the sensor output
can be expressed as the sum of the variances of the two contributing error sources: wide
band noise, !, and Gauss-Markov process, b, as introduced in Chapter 2.
?2(t) = !2(t)+ b2(t) (3.6)
30
3.4 Characterization of Integrated Sensor Measurement
Given a sensor modeled with Equation (2.1) and whose stochastic errors are charac-
terized as the sum of the independent noise sources introduced in Chapter 2, the purely
integrated sensor is characterized as follows. The mean of the integrated sensor is simply
the value of the integrated deterministic terms.
E[
Z
ymeas(t)dt] = E
?Z
[(SF)y(t)+b]dt
?
+E
?Z
?(t)dt
?
=
Z
[(SF)y(t)+b]dt+0
= SF
Z
y(t)dt+bt (3.7)
The variance of the integrated sensor output is the variance of the integrated independent
stochastic error sources.
E[
Z
y2meas(t)dt] = E[
Z
(SF)y +?(t)+bdt
? Z
(SF)y(t)+?(t)+bdt
?
]
?
Z
(SF)y(t)+?(t)+bdt
?2
=
Z
E[?2(t)]dt (3.8)
The variance of the integrated sensor output is equal to the sum of the variances of the
independent integrated error sources.
R ymeas2(k) = R !2(k)+ R b2(k) (3.9)
31
Applying the same methods to the double integrated case yields.
E[
ZZ
ymeas(t)dt] = E
?ZZ
[(SF)y(t)+b]dt2
?
+E
?ZZ
?(t)dt2
?
=
ZZ
[(SF)y(t)+b]dt2 +0
= SF
ZZ
y(t)dt2 +bt2 (3.10)
Performing analogous operations yields the resulting variance for the double-integrated
sensor output.
RR ymeas2(k) = RR !2(k)+ RR b2(k) (3.11)
As shown in Equations (3.9) and (3.11), the variance in the integrated sensor output
is equal to the sum of the variances of the integrated stochastic error sources. In the
following sections the expressions of the individual variance functions of the random
error processes and their integrals are derived.
3.5 Single Axis Stochastic Error Contributions
Assuming that the numerical integration of the sensor is performed using an Euler
approximation, the resulting integral values are simply scaled sums of the inertial values.
As the scaled sum is a linear operation and the stochastic processes are Gaussian, the
integrated stochastic processes are also Gaussian with transformed mean and variance
functions. The following sections in this chapter derive the variance functions of the
integrated and double-integrated wide-band noise and Gauss-Markov processes for nu-
merical integration using the Euler method. The straightforward time-domain technique
32
modeled after [11] is employed here for derivations with the error models presented in
Chapter 2.
3.5.1 Variance of Integrated Wide Band noise
A derivation of the variance of integrated wide-band noise using the technique as
follows has been shown in [11]. It is repeated here as an instructive example of the
methodology by which the proceeding expressions are derived.
Let ?y represent a wide-band noise process with variance 2!.
?y = ! (3.12)
Integrating the ?y yields its integral value.
_y =
Z
!dt (3.13)
The above integration can be approximated by Euler?s method (left hand sum) with the
initial condition _y0 = 0.
_yk = _yk?1 +?t!k?1
= _y0 +?t
k?1X
i=0
!i (3.14)
33
Square both sides to obtain
_yk _yk =
?
?t
k?1X
i=0
!i
!?
?t
k?1X
i=0
!i
!
(3.15)
_yk _yk = ?t2
?k?1X
i=0
!i
!?k?1X
i=0
!i
!
Take the expected value of the squared expression.
E[_yk _yk] = E
"
?t2
?k?1X
i=0
!i
!?k?1X
i=0
!i
!#
(3.16)
The expectation of all of cross-terms of !i are equal to zero, as successive ! values in
time are completely uncorrelated. The expression therefore reduces to
E[_yk _yk] = ?t2E
"k?1X
i=0
!2i
#
= ?t2
k?1X
i=0
E[!i!i] (3.17)
The flnal result is an expression for the variance of integrated wide-band noise as a
function of the variance of the wide-band noise, time index, and sampling interval.
2_y = 2!?t2k (3.18)
34
3.5.2 Variance of Double Integrated Wide Band noise
Double integrating the wide-band noise process, ?y, to yield its double-integral value is
shown below
y =
Z
_ydt =
ZZ
!dt2 (3.19)
Approximating the double-integration by Euler?s method (left hand sum), the following
substitution is made
yk = yk?1 +?t_yk?1
= y0 +?t2
k?1X
j=0
?j?1X
i=0
!i
!
(3.20)
Simpliflcation of the double summation yields a single summation with an indexed coef-
flcient.
yk = ?t2
k?1X
j=0
(k?j ?1)!j (3.21)
Squaring both sides gives
ykyk = ?t4
0
@
k?1X
j=0
(k?j ?1)!j
1
A
0
@
k?1X
j=0
(k?j ?1)!j
1
A (3.22)
Taking the expected value of both sides with knowledge that successive values of !j are
uncorrelated results in
E[ykyk] = ?t4
k?1X
j=0
(k?j ?1)2E[!j!j] (3.23)
35
Expansion of the summation leads to
E[ykyk] = ?t4
0
@(k?1)2
k?1X
j=0
(1)?2(k?1)
k?1X
j=0
j +
k?1X
j=0
j2
1
AE[!j!j] (3.24)
Using the analytic solutions for power series summations the expression reduces to
E[ykyk] = ?t4
k(k?1)2 ?2(k?1)12k(k +1)+ 16k(k +1)(2k +1)
?
E[!j!j] (3.25)
Further simpliflcation yields an expression for the variance of double integrated wide-
band noise as a function of its variance, time index, and sampling interval.
y2 = ?t4 !2
1
3k
3 + 1
2k
2 + 1
6k
?
(3.26)
3.5.3 Variance of 1st order Gauss-Markov process
The difierential equation for the 1st order Gauss-Markov process as given in Equation
(2.4) can be realized using an Euler approximation
bk = bk?1 +?t_bk?1
= bk?1 +?t?bk?1? +?t!bk?1
=
1? ?t?
?
bk?1 +?t!bk?1 (3.27)
For clarity in derivation, deflne A =
1? ?t?
?
to get
bk = Abk?1 +?t!k?1 (3.28)
36
The expression can be written as the following summation where the initial condition
of the process is assumed to be zero, b0 = 0.
bk = Ak?1b0 +?t
k?1X
i=0
Ak?i?1!bi
bk = ?t
k?1X
i=0
Ak?i?1!bi (3.29)
While this zero initial condition assumption simplifles the analysis below, the result-
ing process may take time to settle into steady state depending on the size of the time
constant. As this stochastic process is only an approximation to the observed sensor
phenomenon, the signiflcance of the initial condition is uncertain.
Squaring both sides obtains
bkbk =
?
?t
k?1X
i=0
Ak?i?1!bi
!?
?t
k?1X
i=0
Ak?i?1!bi
!
(3.30)
Applying the expectation operator to both sides with the knowledge that successive !bi
values in time are completely uncorrelated and exhibit an identical variance results in
E[bkbk] = ?t2
k?1X
i=0
A2(k?i?1)E[!bi!bi]
= ?t2A2k?2
k?1X
i=0
A?2iE[!bi!bi]
= ?t2A2k?2
k?1X
i=0
A?2i 2!b (3.31)
37
Using the solution to the geometric series yields the following analytical expression
E[bkbk] = ?t2A2k?2
1?A?2k
1?A?2
?
2!b (3.32)
Further simpliflcation results in the following expression for the variance of a 1st order
Gauss-Markov process as a function of the variance of the driving noise, 2!b, time index,
k, and sampling interval, ?t.
2b = ?t2 2!b
A2k ?1
A2 ?1
?
(3.33)
Note that for positive values of ?, A is less than one and therefore 2b will reach a
steady-state value.
3.5.4 Variance of Integrated 1st order Gauss-Markov process
Let ?x represent the bias drift as modeled by the Gauss-Markov process. Assume
the process is realized by an Euler integration with zero initial condition as in Equation
(3.27).
?xk = bk = Abk?1 +?t!bk?1 (3.34)
38
Next, approximate the integral using Euler?s method with initial condition _x0 = 0
_x =
Z
?xdt =
Z
bdt
= _xk?1 +?t?xk?1
= _x0 +?t?A?xk?2 +?t!bk?2?
= ?t2
k?1X
j=0
?j?1X
i=0
Aj?i?1!bi
!
(3.35)
The summation can be rewritten as
_xk = ?t2
k?2X
j=0
? jX
i=0
AjA?i!bi
!
(3.36)
Expand the summation for k = 5 and collect the !bi terms
_x5 = ?t2
4X
j=0
?j?1X
i=0
Aj?i?1!bi
!
(3.37)
= ?t2?!b0 +A!b0 +!b1 +A2!b0 +A!b1 +!b2 +A3!b0 +A2!b1 +A!b2 +!b3?
= ?t2?!b0?A0 +A1 +A2 +A3?+!b1?A0 +A1 +A2?+!b2?A0 +A1?+!b3?A0??
Investigation of the expansion and rearrangement of Equation (3.36) results in simplifl-
cation to the double summation
_xk = ?t2
k?2X
i=0
!bi
0
@
k?1?iX
j=0
Aj
1
A (3.38)
39
Using the solution of the geometric series, the expression is further reduced to a single
summation
_xk = ?t2
k?2X
i=0
1?Ak?1?i
1?A
?
!bi
= ?t
2
1?A
k?2X
i=0
?
1?Ak?1?i
?
!bi (3.39)
Then both sides are squared to obtain
_xk = ?t
4
(1?A)2
?k?2X
i=0
?
1?Ak?1?i
?
!bi
!?k?2X
i=0
?
1?Ak?1?i
?
!bi
!
(3.40)
Taking the expected value of both sides with knowledge that successive !bi values in
time are completely uncorrelated results in
E[_xk _xk] = ?t
4
(1?A)2
k?2X
i=0
?
1?Ak?1?i
?2
E[!bi!bi]
= ?t
4
(1?A)2
k?2X
i=0
1? 2A
k
AAi +
A2k
A2A2i
?
2!b
= ?t
4
(1?A)2
?k?2X
i=0
(1)? 2A
k
A
k?2X
i=0
A?i + A
2k
A2
k?2X
i=0
A?2i
!
2!b (3.41)
Using the solutions to the geometric series to simplify the summations gives
E[_xk _xk] = ?t
4
(1?A)2
(k?1)? 2A
k
A
1?A1?k
1?A?1 +
A2k
A2
1?A2?2k
1?A?2
?
!2 (3.42)
40
This results in an expression for the variance of single integrated 1st order Gauss-Markov
process in terms of the variance of the driving noise, 2!b, time index, k, and sampling
interval, ?t.
_x2 = ?t4 2!b
1+2A?2Ak ?2A1+k +A2k ?k +kA2
?1+2A?2A3 +A4
?
(3.43)
The above equation can be expressed in a condensed form
_x2 = ?t4 2!b
?
?a1 +a2Ak ?a3A2k +a4k
?
(3.44)
Where the constants of Equation (3.44) are
a1 = 1+2A?1+2A?2A3 +A4
a2 = ?2?2A?1+2A?2A3 +A4
a3 = 1?1+2A?2A3 +A4
a4 = A
2 ?1
?1+2A?2A3 +A4 (3.45)
3.5.5 Variance of Double Integrated 1st order Gauss-Markov process
Let x represent the double integration of the bias drift, b
x =
Z
_xdt =
ZZ
bdt2
As before, Euler integration is used to realize the Markov process, _x, using the initial
condition x0 = 0. The double integration the process is represented below as a series of
41
nested summations
xk = ?t3
k?1X
m=0
m?1X
j=0
j?1X
i=0
Aj?i?1!bi (3.46)
Expanding the summations for k = 5 and collect the ! terms results in
x5 = ?t3
4X
m=0
m?1X
j=0
j?1X
i=0
Aj?i?1!bi
= ?t3?A2!b0 +(!b1 +2!b0)A+!b2 +2!b1 +3!b0?
= ?t3?!b0(A2 +2A1 +3A0)+!b1(A1 +2A0)+!b2(A0)? (3.47)
Investigation of the preceding expansion leads to the simpliflcation of the triple summa-
tion expression into two nested summations in which !i is removed from the innermost
summation
xk = ?t3
k?3X
i=0
!bi
k?3?iX
j=0
(j +1)Ak?3?i?j
= ?t3
k?3X
i=0
!biAk?3?i
k?3?iX
j=0
(jA?j +A?j) (3.48)
By substituting the following derivatives into Equation (3.48), the following expression
is obtained
d
dA
?A?j? = ?1
A
?jA?j?
jA?j = ?A ddA ?A?j?
42
where, the iA?i term can be decoupled as shown below
xk = ?t3
k?3X
i=0
!biAk?3?i
0
@
k?3?iX
j=0
A?j ?A ddA
k?3?iX
j=0
A?j
1
A (3.49)
Using the solution to the geometric series of A?i, the expression becomes
xk = ?t3
k?3X
i=0
!biAk?3?i
1?A?k+2+i
1?A?1
?
?A ddA
1?A?k+2+i
1?A?1
??
(3.50)
Evaluation of the derivative allows further simpliflcation, yielding
xk = ?t3
k?3X
i=0
!biAk?3?i
? 1?A?k+2+i
1?A?1
?
?A
(?k +2+i)A?k+2+i
A(1?A?1) ?
1?A?k+2+i
A2(1?A?1)2
??
= ?t3
k?3X
i=0
!bi (A
k?1?i +A?2?kA+k +iA?i)
(A?1)2 (3.51)
With the knowledge that successive!bi values in time are completely uncorrelated, squar-
ing and applying the expectation operator gives
E[xkxk] = ?t6
k?3X
i=0
E[!bi!bi](A
k?1?i +A?2?kA+k +iA?i)2
(A?1)4
= ?t3E[!bi!bi](A?1)4
k?3X
i=0
A2k?2A?i +(A?2?kA+k)2 +(A?1)i2)+
2Ak?1(A?2?kA+k)A?i +2Ak?1(A?1)Aii+
2(A?2?kA+k)(A?1)i
?
(3.52)
43
A distribution of the above summation yields
E[xkxk] = ?t6E[!bi!bi](A?1)4
?
A2k?2
k?3X
i=0
A?i +(A?2?kA+k)2 +(A?1)2
k?3X
i=0
i2 +
2Ak?1(A?2?kA+k)
k?3X
i=0
A?i +2Ak?1(A?1)
k?3X
i=0
iAi +
2(A?2?kA+k)(A?1)
k?3X
i=0
i
!
(3.53)
Using the solutions to the geometric and power series allows the summations to be
simplifled further
E[xkxk] = ?t3E[!bi!bi](A?1)4
?
A2k?21?A
?2k+4
1?A?2 +(A?2?kA+k)
2(k?2)+
(A?1)216(k?3)(k?2)(2k?5)+
2Ak?1(A?2?kA+k)1?A
?k+2
1?A?1 +
2Ak?1(A?1)(?A) ddA1?A
?k+2
1?A?1 +
(A?2?kA+k)(A?1)(k?3)(k?2)
!
(3.54)
The flnal result is an expression for the variance of a double integrated Gauss-Markov
process in terms of the in terms of the variance of the driving noise, 2!b, time index, k,
44
and sampling interval, ?t.
x2 = ?t6 !b2
?
(?2?4A+2A4 +4A3 +2)k3 +
(9?12A?6A2 +12A3 ?3A4)k2 +
(?13+8A?8A3 +A4 +12Ak ?12A2+k)k +
(6?12A2 ?12Ak +12A2+k +6A2k)
?
(3.55)
The above equation can also be expressed as
x2 = ?t6 !b2
?
c1k3 +c2k2 + (3.56)
(c3 +12Ak ?12A2+k)k +
(c4 ?12Ak +12A2+k +6A2k)
?
where,
c1 = ?2?4A+2A4 +4A3 +2
c2 = 9?12A?6A2 +12A3 ?3A4
c3 = ?13+8A?8A3 +A4
c4 = 6?12A2
3.5.6 Summary of Results
The preceding sections derived the variance expressions for the raw, integrated,
and double integrated stochastic error processes of wide-band noise and exponentially-
correlated noise. Tables 3.1 and 3.2 show the resulting expressions in summary. The
45
left-most column represents the level of integration and the right column indicates the
corresponding variance functions. Recall that k is the time index and ?t is the sample
interval.
Table 3.1: Variance Contributions of Wide-Band Noise Integrals
State Variance, 2(k)
! 2!R
! dt 2!?t2kRR
! dt2 ?t4 !2?13k3 + 12k2 + 16k?
Table 3.2: Variance Contributions of 1st-Order Gauss-Markov Process Integrals
State Variance, 2(k)
b ?t2 !b2
?
A2k?1
A2?1
?
R bdt ?t4 2
!b
??a
1 +a2Ak ?a3A2k +a4k
?
RR bdt2 ?t6
!b2
?
c1k3 +c2k2 +(c3 +12Ak ?12A2+k)k +(c4 ?12Ak +12A2+k +6A2k)
?
3.6 Validation of the Error Propagation
In order to validate the derived expressions for the variance functions of the inte-
grated stochastic processes, a Monte Carlo simulation was employed. The basic idea of
the simulation is to generate a large number of independent stochastic processes using
flxed parameters for a given window of time and then integrate (and double integrate)
each simulated process over the duration. The variance over all the simulated runs is
46
computed for each time step. The computed variance functions represent the variance
versus time of the integral processes. The empirically deduced variances are then com-
pared to the derived expressions in Tables 3.1 and 3.2.
The following plots show the analytical variance functions compared against the
Monte Carlo results for each of the stochastic processes. In each example flgure, the
simulated variance matches well to the analytic expression thus validating the expressions
derived in the preceding sections.
3.6.1 Propagation of Wide-Band Noise Process
Figures 3.1, 3.2, and 3.3 show the validation of standard deviation functions for
a wide-band noise process, its integral, and its double integral, respectively. The plots
show that the equations as derived and listed in the Tables match the variance generated
in the Monte Carlo simulation.
47
0 10 20 30 40 50 60?3
?2
?1
0
1
2
3
Time
?(t)
Sample Run
Equation
MonteCarlo
Figure 3.1: Standard Deviation of Wide-Band Noise Process: 10Hz, ! = 1, 2000 Monte
Carlo iterations 0 10 20 30 40 50 60
?2
?1.5
?1
?0.5
0
0.5
1
1.5
2
Time
?(t)
Sample Run
Equation
MonteCarlo
Figure 3.2: Standard Deviation of Integrated Wide-Band Noise Process: 10Hz, ! = 1,
2000 Monte Carlo iterations
48
0 10 20 30 40 50 60
?80
?60
?40
?20
0
20
40
60
80
Time
?(t)
Sample Run
Equation
MonteCarlo
Figure 3.3: Standard Deviation of Double Integrated Wide-Band Noise Process: 10Hz,
! = 1, 2000 Monte Carlo iterations
3.6.2 Propagation of Gauss-Markov Process
Figures 3.4, 3.5, and 3.6 show the validation of standard deviation functions for a
Gauss-Markov process, its integral, and its double integral. The plots show that the
equations as derived and listed in the Tables match the variance achieved through the
Monte Carlo simulation. Note that for the non-integrated Gauss-Markov process shown
in Figure 3.4, the process variance reaches steady state as determined by Equation (3.33).
49
0 10 20 30 40 50 60?3
?2
?1
0
1
2
3
Time
?(t)
Sample Run
Equation
MonteCarlo
Figure 3.4: Standard Deviation of Gauss-Markov Process: 10Hz, b = 2, ? = 30, 2000
Monte Carlo iterations 0 10 20 30 40 50 60
?40
?30
?20
?10
0
10
20
30
40
Time
?(t)
Sample Run
Equation
MonteCarlo
Figure 3.5: Standard Deviation of Integrated Gauss-Markov Process: 10Hz, b = 2, ? =
200, 2000 Monte Carlo iterations
50
0 10 20 30 40 50 60
?1000
?500
0
500
1000
Time
?(t)
Sample Run
Equation
MonteCarlo
Figure 3.6: Standard Deviation of Double Integrated Gauss-Markov Process: 10Hz, b
= 2, ? = 200, 2000 Monte Carlo iterations
3.7 Application Example
While the results obtained in this chapter are in direct support of the more general
navigation scenarios presented in later chapters, the expressions for the propagation of
the errors in this chapter can be directly applied to a single-axis navigation scenario.
It is the purpose of this section to illustrate the use of such expressions with such an
example.
Suppose a body is constrained to move in a straight line trajectory as depicted by
Figure 3.7. Suppose additionally that the sensitive axis of an accelerometer is coincident
with the traveling direction of the body. The accelerometer speciflcations of random
walk, bias drift variance, time constant, and sample frequency are known and the sensor
has been fully calibrated to remove any constant bias or efiects due to temperature. The
51
output of the calibrated measurement is integrated to obtain the velocity and position
for a given acceleration proflle along a straight line position trajectory. For each time
step, in addition to the values of velocity and position, the results of this chapter can
provide the expected accuracy of the acceleration, velocity, and position.
Figure 3.7: Body Constrained to Travel in One Direction
Assume the sensor speciflcations for the accelerometer are given in Table 3.3. These
speciflcations represent a low grade accelerometer with an exaggerated bias magnitude.
Table 3.3: Sample Accelerometer Speciflcations
Speciflcation Value
fs 10 Hz
2! 0.5 ms2
2b 0.25 ms2
? 200 s
For any acceleration proflle, a, in a single direction, the mean value of the veloc-
ity, V and position P in the same direction are simply the integral values of the true
52
acceleration. The mean values represent the Euler integration of true sensor outputs.
E[V(k)] = ?t
kX
i=0
ai meters per second (3.57)
E[P(k)] = ?t2
kX
j=0
jX
i=0
ai meters (3.58)
The variance of the the velocity, V, is the sum of the variances of the error con-
tributions from integrated wide-band noise and integrated Markov process as listed in
Tables 3.1 and 3.2.
2V (k) = 2R ?(k) (3.59)
= 2R !(k)+ 2R b(k) (3.60)
The variance of the the position, P, is the sum of the variances of the error contri-
butions from double integrated wide-band noise and double integrated Markov process
as listed in Tables 3.1 and 3.2.
2P(k) = 2RR ?(k) (3.61)
= 2RR !(k)+ 2RR b(k) (3.62)
To demonstrate these results, consider the sinusoidal acceleration proflle as shown
in Figure 3.8. This noisy accelerometer when integrated gives the velocity is shown in
Figure 3.9. Another step of integration gives the position shown in Figure 3.9. As is
evident by the plots, the 1- bounds show that the error growth becomes larger in time
and with each level of integration. The bounds shown in the plots can be thought of
53
as a time-dependent corridor in which the integral values of velocity and position are
expected to reside. As all error processes are Gaussian, the 1- bounds plotted in Figures
3.9 and 3.10 specify the region where approximately 66.7 percent of all trajectories are
expected to travel. 0 10 20 30 40 50 60
?2
?1.5
?1
?0.5
0
0.5
1
1.5
2
Time, [s]
Acceleration, [m/s
2 ]
Sensor
True
Figure 3.8: Simulated Acceleration Proflle: 10Hz, ! = 0.5, b = 0.25, ? = 200
54
0 10 20 30 40 50 60
0
5
10
15
20
25
30
Time [s]
Velocity, [m/s]
Sample Run
1?? Bounds
MonteCarlo
Figure 3.9: Velocity with Bounds from Accel Proflle0 10 20 30 40 50 600
200
400
600
800
1000
Time, [s]
Position, [m]
Sample Run
1?? Bounds
MonteCarlo
Figure 3.10: Position with Bounds from Accel Proflle
55
3.8 Illustration of Results
Inertial sensors exhibit varying magnitudes of each of their component error pro-
cesses as well as the characteristic time constant of the drifting component. The follow-
ing sections illustrate the efiect of the component stochastic process parameters on the
growth of the variance of the total integrated sensor values.
3.8.1 Relative Magnitudes
The two stochastic error processes, when integrated, each uniquely contribute to
variance function of the integrated sensor output. A sensor?s wide-band noise component
will dominate the error for short integration intervals while the drifting bias dominates
for longer durations. This relative efiect of each can be investigated by observing the
efiect of adjusting the ratio of the process magnitudes, b w, on the variance function of
the integrated output for a flxed time constant. Setting the wide-band noise standard
deviation to 1, Figures 3.11, 3.12, and 3.13 and show the 1- bounds of the integrated
sensor output for various ranges of the bias drift standard deviation.
Since the rate of increase of the integrated bias drift variance is higher than that of
the integrated wide-band noise for any relative ratio, even small relative bias magnitudes
will cause the bias to eventually dominate the variance growth. The general efiect of
the relative ratio is that an increase Gauss-Markov process magnitude, b, causes it
to dominate sooner, resulting in a larger rate of error growth in the integrated sensor
output.
56
0 20 40 60 80 100 1200
2
4
6
8
10
12
14
16
Time
?(t)
1?? Wide?Band
1?? Sensor
Ratio = 0.1
Ratio = 1
Figure 3.11: Integrated Sensor 1- Bounds for Ratio from 0.1 to 10 20 40 60 80 100 1200
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Time
?(t)
1?? Wide?Band
1?? Sensor
Ratio = 0.01
Ratio = 0.1
Figure 3.12: Integrated Sensor 1- Bounds for Ratio from 0.01 to 0.1
57
0 20 40 60 80 100 1200
0.5
1
1.5
2
2.5
3
Time
?(t)
1?? Wide?Band
1?? Sensor
Ratio = 0.001
Ratio = 0.01
Figure 3.13: Integrated Sensor 1- Bounds for Ratio from 0.001 to 0.1
3.8.2 Efiect of Time Constant
For a flxed relative magnitude of 0.1, Figure 3.14 illustrates the shape of the bounds
for various values of the Markov model time constant, ?. For the range of time constants
shown, larger values of ? cause a slower increase in the rate of error propagation, while
lower values indicate a faster increase.
For a flxed relative magnitude of 0.1, Figure 3.15 illustrates the efiect of a larger
range of ? on the value of the integrated sensor error bounds at a particular time of 120
units. For very small values of ?, the initial conditions of the Markov model dominate
over the input noise. As the time constant increases, the maximum error value peaks
and then levels ofi in a nonlinear fashion. For most inertial sensors, the time constant
is usually much longer than values corresponding to the peak. The result of Figure 3.14
best describes the efiect of time constant within its expected range.
58
0 20 40 60 80 100 1200
1
2
3
4
5
6
7
Time
?(t)
1?? Wide?Band
1?? Sensor
tau = 200
tau = 1000
Figure 3.14: Integrated Sensor 1- Bounds for Fixed Ratio of 0.1 and ? Between 200 to
1000 0 200 400 600 800 1000 1200 1400 1600 1800
3
4
5
6
7
8
9
?
?(120)
Figure 3.15: Value of Integrated 1- Bounds at time index of 120 for Fixed Ratio of 0.1
and ? Between 200 to 1000
59
3.9 Conclusion
In this chapter, analytical variance expressions have been derived which quantify
the error growth of subsequent integrations of inertial sensors exhibiting the assumed
stochastic model forms of Chapter 2. A Monte Carlo simulation of the stochastic pro-
cesses was used to validate the analytical results and further simulations illustrated the
use of the expressions in the quantiflcation of accuracy for the single-axis case. The flnal
sections of this chapter showed the relative and total efiects of the three stochastic model
parameters on the resulting variance functions of the integrated sensor. The results of
this chapter are used in direct support of derived expressions and analysis for the planar
navigation scenario studied in Chapter 4.
60
Chapter 4
Two Dimensional Error Propagation
4.1 Introduction
In order to describe the most general motion of a navigating vehicle, six degrees of
freedom are required. An inertial measurement unit (IMU) attached to the navigating
body takes measurements of these six degrees of freedom which include three orthogo-
nal accelerations (ax,ay,az) and three orthogonal rotation rates (gx,gy,gz). In order to
navigate within a suitable frame of reference such as on the surface of the earth, these
measurements must be transformed and integrated to values of orientation, velocity, and
position in that frame. For vehicles traveling within short ranges on the earth, a suit-
able frame of reference is a simple cartesian coordinate system in which North, East,
Down (NED) axes are aligned according to the right-hand rule. The orientation of the
vehicle in this navigation frame can be described by roll (`), pitch ( ), and yaw (?)
angles as deflned about the North, East, and Down axes, respectively. Figure 4.1 is a
three-dimensional diagram depicting the body frame and navigation frame coordinate
systems.
It the task of the inertial navigator to perform the necessary operations on the body
frame measurements to achieve the desired navigation frame values. This process of
operating on the inertial measurements is referred to as inertial mechanization. For the 6-
DOF scenario the mechanization equations are non-linear and require several calculations
involving multiple measurements (see Chapter 5). If, however, the vehicle operates under
some kinematic constraints, the resulting governing equations may be greatly simplifled.
61
Figure 4.1: Simplifled Coordinate Frame
This chapter focuses its attention on the planar navigation scenario in which a vehicle is
constrained to move on a North-East plane. Figure (4.2) depicts the simplifled motion
of the constrained body. The body can translate in the north and east directions and
rotate only about the direction orthogonal to the plane. The rotation rate sensed by a
gyro about the z-axis on the constrained body is the same as the rotation rate about the
down axis in the navigation frame. The yaw angle, ?, as measured positive from north
about the down axis in the navigation frame requires no transformation to the navigation
frame and is directly related to the sensed yaw rate, gz, by simple integration.
The vehicle in this planar navigation scenario, depending upon its capability and
trajectory, can operate under additional kinematic constraints. For general motion,
the vehicle experiences side-slip, in which the body experiences velocity in both the
direction it is pointing, body frame x known as heading (longitudinal direction), and
the body frame y direction (lateral direction). This is the case on many vehicles in
which high dynamic maneuvers force the body to point in a direction incoincident with
its path of motion. Figure 4.3 shows the dynamic equations for use in navigating a
62
Figure 4.2: Simplifled Coordinate Frame 2D
side-slipping vehicle with inertial measurements. As is evident by the diagram, the gyro
is flrst integrated and then used in transforming the acceleration measurements to the
navigation frame. The transformed accelerations are then integrated once for velocity,
and twice for position.
Figure 4.3: Navigation Relationships for Side-Slip Vehicle
For many four-wheeled vehicles performing more moderate maneuvers, the velocity
of the body can be assumed to be strictly coincident with its heading (body frame x). In
this scenario, the body moves only in the direction it is pointing and is considered to be
operating under the no-slip condition. For the no-slip case described above, the dynamic
63
relationships between the inertial measurements and navigation frame vehicle states of
velocity and position simplify even further. These simplifled relationships for the no-slip
case are shown in Figure 4.4. Similar to the case with side-slip, the navigation frame
values of velocity and position are ascertained from the subsequent integrations. How-
ever, in this no-slip case, the velocity is solely derived from integrating the longitudinal
accelerometer, ax. Consequently, the acceleration, requires no transformation with the
integrated gyro and the lateral accelerometer, ay, is not necessary for navigation. As a
result, inertial navigation for the no-slip case requires one less integration and one less
measurement. This observation is shown later in this chapter to ofier some potential
dead-reckoning improvement when such kinematic assumptions are valid.
Figure 4.4: Navigation Relationships for Vehicle with No-Slip
Using results from and a similar approach to Chapter 3, this chapter presents a
derivation of the propagation of navigation-frame velocity errors for the planar no-slip
case as shown in Figure 4.4. For bodies that operate under such kinematic constraints,
the resulting variance expressions provide the accuracy expected when dead-reckoning
with the applicable dynamic equations and choice of inertial sensors. The resulting
velocity expressions are validated with the same type of Monte Carlo simulations as used
64
in Chapter 3. Following the no-slip case is a discussion of the additional requirements for
the case where a vehicleis expected to slip. Variance expressions for the transformation of
accelerations for the no-slip case are presented. This chapter concludes by demonstrating
the additional error induced when employing the slip equations for a non-slip trajectory.
4.2 Velocity Error in Navigation Frame Under No-Slip Planar Motion
This section derives expressions for the variance of the 2-D velocity as derived from
the minimum amount of inertial measurements necessary. The basic method used below
deflnes the simplest governing inertial navigation relationships, substitutes the inertial
measurement errors, and applies the expectation operator to the squared expressions.
Using small angle approximations, the expressions can be simplifled to show a linear
propagation of sensor errors from the IMU to the vehicle states.
For the navigating body in the planar scenario, the yaw angle of the vehicle is derived
by direct integration of the rotation rate sensed about the axis aligned orthogonal to the
plane, gz.
? =
Z
gz dt (4.1)
The velocity, V, of the body is always tangential to its path under the no-slip
condition, and therefore can be derived from the acceleration sensed along the its x axis
(ax accelerometer) by the integral relationship
V =
Z
ax dt (4.2)
65
Assuming Euler integration of the above quantities, the preceding integrals are re-
duced to summations
Vk = ?t
k?1X
i=0
ax +V0 (4.3)
?k = ?t
k?1X
i=0
gz +?0 (4.4)
The rate gyro and accelerometer inertial measurements are corrupted by stochastic
error sources, ?g, and ?a as introduced and quantifled in earlier chapters. The total
velocity and yaw angle can therefore be expressed as
Vk = ?t
k?1X
i=0
(a+?a)+V0
= ?t
k?1X
i=0
a+?t
k?1X
i=0
?a +V0 (4.5)
?k = ?t
k?1X
i=0
(g +?g)+?0
= ?t
k?1X
i=0
g +?t
k?1X
i=0
?g +?0 (4.6)
Fortheclarityinthederivationsfollowingitishelpfultoredeflnetheaboveequations
with simpler notation. Note that despite the removal of subscript, k, the values are still
66
functions of time.
A = ?t
k?1X
i=0
a+V0 (4.7)
B = ?t
k?1X
i=0
?a (4.8)
fi = ?t
k?1X
i=0
g +?0 (4.9)
fl = ?t
k?1X
i=0
?g (4.10)
In summary, A and fi represent the velocity and yaw angle as derived from Euler
integration of the mean sensor outputs and B and fl represent the error due to the
integrated accelerometer and rate gyro, respectively.
4.2.1 Mean and Variance of East Velocity
The velocity in the east direction under this scenario is computed by taking the
resultant velocity from the accelerometer and transforming it into the east component
direction using the resultant yaw angle from the rate gyro.
VEASTk = V sin(?)
=
?
?t
k?1X
i=0
a+V0 +?t
k?1X
i=0
?a
!
sin(?t
k?1X
i=0
g +?0 +?t
k?1X
i=0
?g)
= (A+B)sin(fi+fl) (4.11)
Using a trigonometric identity, the sine factor is expanded as shown below
VEASTk = (A+B)(sin(fi)cos(fl)+cos(fi)sin(fl)) (4.12)
67
Provided that the integrated gyro error, fl, is su?ciently small, small-angle approx-
imations of Equations (4.13) and (4.14) can be made.
cosfl ? 1 (4.13)
sinfl ? fl (4.14)
In general, this small-angle approximation is valid within the typical range of inter-
est. As the integrated sensor error grows outside the region for which the assumption is
valid (jflj < 10 degrees), the resulting velocity and position errors exceed the range of
accuracy in which this research seeks to quantify.
The small angle approximation results in the simplifled expression for the east ve-
locity
VEASTk = (A+B)(sin(fi)+cos(fi)fl) (4.15)
The mean function of velocity in the east direction is found by taking the expected
value.
E[VEASTk] = E[(A+B)(sin(fi)+cos(fi)fl)]
= Asinfi+sinfiE[B]+AcosfiE[fl]+cosfiE[Bfl] (4.16)
Since the stochastic errors are zero mean, the mean of the east velocity is simply
the true value ascertained from the true part of the inertial measurements.
E[VEASTk] = Acosfi (4.17)
68
To derive the variance of the east velocity, it is flrst squared as shown below
(VEASTk)2 = ?Asinfi+sinfiB +Acosfifl +cosfiBfl?2
= A2 sin2(fi)+A2 cos2 fifl2 +sin2 fiB2 +cosfi2B2fl2
+2A2 sinficosfifl +2ABsin2 fi+4ABsinficosfifl
+2ABcos2 fifl2 +2B2 sinficosfifl (4.18)
Next, the expected value of the squared expression is taken as follows
E?(VEASTk)2? = A2 sin2(fi)+A2 cos2 fiE?fl2?+sin2 fiE?B2?+cosfi2E?B2fl2?
+2A2 sinficosfiE?fl?+2Asin2 fiE?B?+4AsinficosfiE?Bfl?
+2Acos2 fiE?Bfl2?+2sinficosfiE?B2fl? (4.19)
Assuming the two stochastic sources are independent, the expression reduces to
E?(VEASTk)2? = A2 sin2(fi)+A2 cos2 fiE?fl2?
+sin2 fiE?B2?+cos2 fiE?B2?E?fl2? (4.20)
Computing the variance of the expression gives the following result
VAR?(VEASTk)2? = E?(VEASTk)2??E?(VEASTk)?2
= A2 cos2 fiE?fl2?+sin2 fiE?B2?+cos2 fiE?B2?E?fl2? (4.21)
69
Recall from Equations (4.8) and (4.10), that B is the integrated accelerometer error
and B is the integrated gyro error. The variances of the integrated inertial sensors were
derived in Chapter 3 and shown again here
E?B2? = 2R ?a (4.22)
E?fl2? = 2R ?g (4.23)
Where, 2R ?a is the variance of the integrated accelerometer errors and 2R ?g is the
variance of the integrated gyro errors.
The expressions are substituted below to yield the variance function of east velocity
2VEAST(k) = V 2 cos2 ? 2R ?g +sin2 ? 2R ?a +cos2 ? 2R ?a 2R ?g (4.24)
Equation (4.24) shows that the variance is a three-termed expression consisting of
the trajectory (velocity and heading) and the variances of the integrate accelerometer
and integrated gyro. The last term includes the product of the two latter variances
indicating that the variance of east velocity is not Gaussian distributed. However, for
su?ciently large values of velocity, V , the flrst term dominates and the east velocity is
approximately Gaussian.
4.2.2 Mean and Variance of North Velocity
The velocity in the North direction under this scenario is computed by taking the
resultant velocity from the accelerometer and transforming it into the North component
70
velocity using the resultant yaw angle from the rate gyro.
VNORTHk = V cos(?)
=
?
?t
k?1X
i=0
a+V0 +?t
k?1X
i=0
?a
!
cos(?t
k?1X
i=0
g +?0 +?t
k?1X
i=0
?g)
= (A+B)cos(fi+fl) (4.25)
Using a trigonometric identity, the cosine factor is expanded as shown below
VNORTHk = (A+B)(cos(fi)cos(fl)?sin(fi)sin(fl)) (4.26)
Using the same small angle approximations shown before in Equations (4.13) and
(4.14), the expression reduces to
VNORTHk = (A+B)(cos(fi)+sin(fi)fl) (4.27)
The mean function of velocity in the north direction is found by taking the expected
value
E[VNORTHk] = E[(A+B)(cos(fi)+sin(fi)fl)]
= Acosfi+cosfiE[B]+AsinfiE[fl]+sinfiE[Bfl] (4.28)
Since the stochastic errors are zero mean, the mean of the north velocity is simply
the true value ascertained from the true part of the inertial measurements
E[VNORTHk] = Acosfi (4.29)
71
To derive the variance of the north velocity, it is flrst squared as shown below
(VNORTHk)2 = ?Acosfi+cosfiB +Asinfifl +sinfiBfl?2
= A2 cos2(fi)+A2 sin2 fifl2 +cos2 fiB2 +sinfi2B2fl2
+2A2 cosfisinfifl +2ABcos2 fi+4ABcosfisinfifl
+2ABsin2 fifl2 +2B2 cosfisinfifl (4.30)
Next, taking the expected value of the squared expression gives
E?(VNORTHk)2? = A2 cos2(fi)+A2 sin2 fiE?fl2?+cos2 fiE?B2?+sinfi2E?B2fl2?
+2A2 cosfisinfiE?fl?+2Acos2 fiE?B?+4AcosfisinfiE?Bfl?
+2Asin2 fiE?Bfl2?+2cosfisinfiE?B2fl? (4.31)
Assuming the two stochastic sources are independent and zero mean, the expression
reduces to
E?(VNORTHk)2? = A2 cos2(fi)+A2 sin2 fiE?fl2?
+cos2 fiE?B2?+sin2 fiE?B2?E?fl2? (4.32)
Computing the variance of the expression gives the terms
VAR?(VNORTHk)2? = E?(VNORTHk)2??E?(VNORTHk)?2
= A2 sin2 fiE?fl2?+cos2 fiE?B2?+sin2 fiE?B2?E?fl2?(4.33)
72
Where, fi is the integrated accelerometer error and B is the integrated gyro error.
The variances of these integrated inertial sensors are derived in Chapter 3 and shown
again here
E?B2? = 2R ?a (4.34)
E?fl2? = 2R ?g (4.35)
Where, 2R ?a is the variance of the integrated accelerometer errors and 2R ?g is the
variance of the integrated gyro errors.
Substituting the known integrated error variances gives the variance function for
the north velocity.
2VNORTH(k) = V 2 sin2 ? 2R ?g +cos2 ? 2R ?a +sin2 ? 2R ?a 2R ?g (4.36)
As shown above for the east velocity in Equation (4.24) it is evident that for rela-
tively large velocities, the north velocity is approximately Gaussian due to the dominance
of the flrst term of Equation (4.36) over the third term.
4.2.3 Cross Covariance of North and East Velocity
To supplement the characterization of the velocity error for this planar no-slip sce-
nario, the cross-covariance between the north and east components is derived as follows.
The cross covariance is deflned as
COV?VNORTHkVEASTk? =
E?(VNORTHk ?E[VNORTH])(VEASTk ?E[VEAST])? (4.37)
73
Using the linear approximation from Equations (4.13) and (4.14) results in
COV?VNORTHkVEASTk? =
E?(Bcosfi?Asinfifl ?Bsinfifl)(Bsinfi+Acosfifl +Bcosfifl)? (4.38)
Next, the expectation operator is expanded and shown below
COV?VNORTHkVEASTk? = E
h
B2 cosfisinfi+Acos2 fifl +cos2 fiB2fl ?Asin2 fiBfl
?A2 sinficosfifl2 ?2AsinficosfiBfl2 ?sin2 fiB2fl ?
sinficosfiB2fl2
i
= cosfisinfiE?B2?+Acos2 fiE?fl?+cos2 fiE?B2fl?
?Asin2 fiE?Bfl??A2 sinficosfiE?fl2? (4.39)
?2AsinficosfiE?Bfl2??sin2 fiE?B2fl??
sinficosfiE?B2fl2? (4.40)
Since the integrated accelerometer error, B, and integrated gyro error, fl, are zero
mean and uncorrelated, the expression reduces to
COV?VNORTHkVEASTk? = cosfisinfiE?B2??A2 sinficosfiE?fl2??
sinficosfiE?B2fl2? (4.41)
74
The flnal result is an expression for the cross-covariance of the velocity in the north
and east component directions as shown below
COV?VNORTHkVEASTk? = sinficosfi
?
2R ?a ?A2 2R ?g ? 2R ?a 2R ?g
?
(4.42)
4.2.4 Probabilistic Characterization of Velocity Errors
As discussed above, if the velocity is relative large, the north and east velocities
ascertained from the no-slip mechanization are approximately Gaussian. Under this
condition, the mean, variance, and cross covariance functions derived above completely
characterize the propagation of the stochastic sensor errors into the velocity errors for
this planar no-slip scenario. The resulting probabilistic characterization is captured for
each step in time by a two-dimensional Gaussian surface as described by Equation (4.43).
fV (VN;VE) = 12?
VN VE
p1??2 exp
h 1
2(1??2)
?
VN
2VN +
VE
2VE ?
2?VNVE
VN VE
!i
(4.43)
Where, ?, the correlation coe?cient is computed by
? = COV
?V
NORTHkVEASTk
?
VN VE (4.44)
and 2VN and 2VE are the variances of velocity in the north and east component
directions as derived in the previous sections. Note that the Gaussian surface of Equation
(4.43) is centered about zero and thus represents only the deviation from the mean
values, which are known from the analysis of the preceding sections in Equations (4.17)
and (4.29).
75
The expressions derived for the variance of the velocity error in the planar no-slip
scenario indicate that the shape of the Gaussian surface described by Equation (4.43)
is completely characterized at any given time by the instantaneous velocity and yaw
angle. The width of its spread is dependent upon both the parameters of the stochastic
processes present on the measurements and the time since integration. The stochastic
characterization of the two-dimensional no-slip velocity error is therefore, at any given
time, independent of the trajectory history. In other words, the variance in the velocity
error is a function only of (in terms of vehicle trajectory) the instantaneous velocity
and yaw angle of the navigating body. This implies that the velocity error for a vehicle
that is inertially navigating under the no-slip assumption with the same initial and flnal
conditions can be fully characterized by the variance expressions of Chapter 3.
4.2.5 Validation of the Velocity Error Characterization
The same Monte Carlo validation methodology as for the single axis case in Chapter
3 was used to validate the velocity variance expressions of the preceding sections. The
basic idea of the simulation is to flrst deflne a North/East Position trajectory in time and
numerically difierentiate the trajectory to obtain the North/East Velocities. From the
velocity, derive a yaw angle assuming the body points tangential to the path and then
difierentiate to obtain the body frame longitudinal acceleration and yaw rate (under
a no-slip assumption). The stochastic errors are then added to the inertial values to
simulate the body-frame sensor measurements. Next, the simulated measurements are
transformed and integrated to attain the now-corrupted attitude and 2-D velocity. After
collecting a su?cient number of simulated integrations, the mean, covariance, and cross-
covariance values of the outputs are computed for each step in time. These resulting
76
variance functions are compared to the analytical expressions using the known simulation
parameters.
Since the mean of the resulting velocity components is simply the processed value
of the true measurements, only the errors about the mean need be considered. To both
illustrate and validate the expressions derived for the velocity accuracy, a candidate
position trajectory is chosen as shown in Figure 4.5.0 10 20 30 40 50 60
?10
0
10
20
30
40
50
60
70
East Position [m]
North Position, [m]
Start
Finish
Figure 4.5: Simulated Position Trajectory
Sensor speciflcations for the accelerometer and gyro chosen relative to trajectory
and are listed in Table 4.1. These speciflcations are taken from a rough identiflcation of
a 6-DOF Crossbow IMU-400CD, a medium grade ($3K-$4K) inertial measurement unit.
Appendix A demonstrates sensor parameter identiflcation on this particular device.
77
Table 4.1: Simulated Sensor Speciflcations
Gyro Spec Value Accel Spec Value
fs 10 Hz fs 10 Hz
2rw 0.14948 deg/s/sqrt(Hz) 2rw 0.000412 g/sqrt(Hz)
2b 0.0061183 deg/s 2b 0.000100 g
? 1300 seconds ? 500 seconds
The deflned position trajectory is processed as described above, and the standard
deviation of the velocity errors are computed. Figure 4.6 and 4.7 show comparisons of
the computed velocity standard deviation to derived expression for the deflned trajectory
for the east and north component velocities, respectively. The y-axis is the 3? value of
the component velocity representing the corridor of near perfect certainty (greater than
99% probability) within where the velocities are expected to reside.
As can be seen in the plots, the derived variance functions match well with the
simulated data. The velocity variance expressions can be observed in Figures 4.6 and 4.7
to exhibit an oscillating type behavior. As can be seen in the derived Equations (4.24)
and (4.36), the variance of the component velocities depends upon the instantaneous
value of the Velocity, V, as well as the heading, ?. As ? is oscillating according to the
sinusoidal position trajectory, the variance of the component velocities thus exhibits the
efiect above.
78
0 10 20 30 40 50 60
?0.3
?0.2
?0.1
0
0.1
0.2
0.3
Time, [s]
3?
? V
EAST
, [m/s]
Sample Run
Equation
MonteCarlo
Figure 4.6: Standard Deviation of East Velocity, 2000 Monte Carlo iterations0 10 20 30 40 50 60
?0.1
?0.05
0
0.05
0.1
Time, [s]
3?
? V
NORTH
, [m/s]
Sample Run
Equation
MonteCarlo
Figure 4.7: Standard Deviation of North Velocity, 2000 Monte Carlo iterations
79
4.3 Position Error in No-Slip Planar Motion
Due to the added analytical complexity in computing the variance of the planar
position for the no-slip scenario, this analysis has not been not completed in closed-
form. Instead, this section will show by example of the derivation of east position, the
limitations of the current analysis approach and lack of necessary information. The
end results of the example derivation make clear the need for more complete statistical
characterization of the integrated sensor errors, and is an avenue of future work as
described in Chapter 6.
Equation (4.45) below gives the velocity in the east direction using the small angle
approximation from the earlier Equation (4.16).
VEASTk = (A+B)(sin(fi)+cos(fi)fl)
= Asinfi+sinfiB +Acosfifl +cosfiBfl (4.45)
The east position can be derived using the Euler method of numerical integration.
The result is expressed below in summation form with the initial condition PEAST0 = 0.
PEASTk = ?t
k?1X
j=0
(Asinfi+sinfiB +Acosfifl +cosfiBfl) (4.46)
The mean value of the east position is straightforward since the stochastic processes,
B and fl are zero-mean and independent. The mean value is simply the numerical
80
integration of the deterministic terms in the summation.
E[PEASTk] = ?t
k?1X
j=0
(Asinfi+sinfiE[B]+AcosfiE[fl]+cosfiE[Bfl])
= ?t
k?1X
j=0
Asinfi (4.47)
Using the same approach from the previous sections, in order to derive the east
position variance the expression is flrst squared.
PEASTk2 =
0
@?t
k?1X
j=0
(Asinfi+sinfiB +Acosfifl +cosfiBfl)
1
A
2
= ?t2
?k?1X
j=0
Acosfi
k?1X
j=0
Acosfi+
k?1X
j=0
sinfiB
k?1X
j=0
sinfiB +
k?1X
j=0
Acosfifl
k?1X
j=0
Acosfifl +
k?1X
j=0
cosfiBfl
k?1X
j=0
cosfiBfl +
2
k?1X
j=0
Asinfi
k?1X
j=0
sinfiB +2
k?1X
j=0
Asinfi
k?1X
j=0
Acosfifl +
2
k?1X
j=0
Asinfi
k?1X
j=0
cosfiBfl +2
k?1X
j=0
sinfiB
k?1X
j=0
Acosfifl +
2
k?1X
j=0
sinfiB
k?1X
j=0
cosfiBfl +
2
k?1X
j=0
AcosfiB
k?1X
j=0
cosfiBfl
!
(4.48)
81
Since the processes are independent and zero mean taking the expected value of the
squared expression cancels out all the \cross-terms" leaving the following expression
E?PEASTk2? = ?t2
?
E
2
4
k?1X
j=0
Acosfi
k?1X
j=0
Acosfi
3
5+E
2
4
k?1X
j=0
sinfiB
k?1X
j=0
sinfiB
3
5+
E
2
4
k?1X
j=0
Acosfifl
k?1X
j=0
Acosfifl
3
5+
E
2
4
k?1X
j=0
cosfiBfl
k?1X
j=0
cosfiBfl
3
5
!
(4.49)
Taking the variance of the above expression reduces to the following three terms
VAR?PEASTk2? = ?t2
?
E
2
4
k?1X
j=0
sinfiB
k?1X
j=0
sinfiB
3
5+
E
2
4
k?1X
j=0
Acosfifl
k?1X
j=0
Acosfifl
3
5+
E
2
4
k?1X
j=0
cosfiBfl
k?1X
j=0
cosfiBfl
3
5
!
(4.50)
82
Observing that each of the three terms in Equation (4.50) have the same basic form,
a single expansion is illustrated with an expansion of the flrst term for k = 5.
?t2
?
E
2
4
k?1X
j=0
sinfifl
k?1X
j=0
sinfifl
3
5
!
=
?t2
?
sin2 fi1B12 +sin2 fi2B22 +sin2 fi3B32 +sin2 fi4B42 +
2sinfi1 sinfi2B1B2 +2sinfi1 sinfi2B1B3 +2sinfi1 sinfi3B1B4 +
2sinfi1 sinfi2B2B3 +2sinfi1 sinfi2B2B4 +
2sinfi1 sinfi2B3B4
?
(4.51)
Recognizing the pattern of expansion, this flrst term can be generalized to the
following expression
?t2
?
E
2
4
k?1X
j=0
sinfifl
k?1X
j=0
sinfifl
3
5
!
=
?t2E
2
4
k?1X
l=0
sin2 filBl2 +
k?1X
j=0
2sinfij
k?1X
i=j+1
sinfiiBjBi
3
5 (4.52)
Distributing the expectation operator yields the following expression for the expan-
sion of the flrst term of Equation (4.50)
?t2
?
E
2
4
k?1X
j=0
sinfifl
k?1X
j=0
sinfifl
3
5
!
=
?t2
?k?1X
l=0
sin2 filE?Bl2?+2
k?1X
j=0
sinfij
k?1X
i=j+1
sinfiiE[BjBi]
!
(4.53)
83
The variables within the argument of the expectation operators are the integrated
accelerometer error, B. While Chapter 3 provides the integrated sensor error variance,
E?Bl2?intheflrstsummation, noanalysishasbeendonetoascertaintheautocorrelation,
E[BjBi] expression in the second double-summation term. While further simpliflcation
may be possible by substituting the original integrated stochastic model form of B into
the second term summation, the indexed coe?cients still remain. Provided that the
expectations of Equation (4.53) can be ascertained, the simpliflcation of the expression
still remains a di?cult task due to the fact that that summations of the coe?cients sinfi
and sin2 fi may not be able to be simplifled or represented by a closed-form expression. As
the goal of this thesis is to provide closed-form expressions which bound the propagation
of error in terms of ascertainable vehicle states (whether computed or measured), the
above analysis remains to be explored in future work.
By example, the sample expansion of the flrst term of the variance expression of
Equation (4.50) has shown di?culty in achieving a solid closed-form expression for the
planar position for the no-slip case. In future work, other approaches of analysis may
provide a more complete and successful characterization of the errors as propagated
in the dynamic equations. However, with the use of Monte Carlo simulations, much
insight can be gained regarding the behavior of inertial navigation in various scenarios.
The remainder of this thesis will use simulation to exemplify claims resulting from the
analysis of inertial navigation in more complex kinematic scenarios. The next section
extends the current planar motion error analysis to a vehicle experiencing side-slip.
84
4.4 Propagation of Error in Planar Motion with Slip
The above sections have derived variance functions that quantify the propagation
of inertial sensor errors when the no-slip assumption is valid and corresponding dynamic
relationships employed. Under the simplifled motion, the dynamic relationships between
the body frame and navigation frame vehicle states are such that only one accelerometer
and one gyro are required to describe the motion of the vehicle. However, a vehicle expe-
riencing side-slip requires an additional measurement to describe all states of its motion.
For the planar case with side-slip, two accelerometers mounted in the body frame x
and y directions and a single gyro mounted orthogonal to the plane in the z direction
can be used to derive the values of 2-D velocity and position (see Figure 4.2). Due to
the kinematics of the slipping motion, the navigation frame states require a coordinate
transformation of the two accelerometer measurements (see Figure 4.3). As this coor-
dinate transformation is achieved with the use of the integrated gyro measurement, the
resulting navigation frame acceleration components exhibit an unbounded error growth
in time. As the states of velocity and position require additional integrations of the
transformed accelerations, their accuracy will grow at even faster rates than that of the
transformed accelerations. As a result, the slip-case navigation-frame acceleration, ve-
locity, and position will exhibit much higher error growth rates than under the no-slip
assumption.
4.4.1 Acceleration Error in Navigation Frame for Slip-Case
In order to illustrate the error growth when the no-slip assumption can not be
made, the following characterization of the transformed accelerations is shown. The
85
acceleration in the planar North and East directions when the vehicle experiences side-
slip can be derived from body frame inertial measurements ax0 and ay0. The navigation
frame accelerations are related to the measurements by the following relationships.
aNORTH = ax0cos? ?ay0sin? (4.54)
aEAST = ax0sin? +ay0cos? (4.55)
Where, as in the no-slip case, the the yaw angle of the vehicle can be derived by
Euler integration of the gz measurement.
?k = ?t
k?1X
i=0
gz +?0 (4.56)
As the inertial measurements include a true acceleration, a, and a stochastic error
component, ?a, the expressions are expanded as
aNORTH = (ax +?ax)cos? ??ay +?ay?sin? (4.57)
aEAST = (ax +?ax)sin? +?ay +?ay?cos? (4.58)
Using the deflnition for the Euler-integrated yaw rate (Equation (4.9)) and Euler-
integrated gyro error (Equation (4.10)) the expression becomes
aNORTH = (ax +?ax)cos(fi+fl)??ay +?ay?sin(fi+fl) (4.59)
aEAST = (ax +?ax)sin(fi+fl)+?ay +?ay?cos(fi+fl) (4.60)
86
As in the no-slip velocity derivation, the trigonometric factors can be expanded by
using the same identities and small-angle approximation. These operations give
aNORTH = a0x cos(fi)?a0x sin(fi)fl ?a0y sin(fi)?a0y cos(fi)fl (4.61)
aEAST = a0x sin(fi)?a0x cos(fi)fl +a0y cos(fi)?a0y sin(fi)fl (4.62)
In order to calculate the variance of the north velocity, the simplifled expression is
flrst squared
aNORTH2 =
?
a0x cos(fi)?a0x sin(fi)fl ?a0y sin(fi)?a0y cos(fi)fl
?2
= a0x2 cos2 (fi)+a0x2 sin2 (fi)fl2 +a0y2 sin2 (fi)+a0y2 cos2 (fi)fl2 ?
2a0x2 cos(fi)sin(fi)fl +2a0xa0y cos(fi)sin(fi)+
2a0xa0y cos2 (fi)fl +2a0xa0y sin2 (fi)fl +
2a0xa0y cos(fi)sin(fi)fl2 +2a0y2 cos(fi)sin(fi)fl (4.63)
Anticipating the expectation operator, the cross terms are removed from the ex-
pression as the error sources are independent and zero-mean. The accelerometer errors,
?a are substituted and the expectation is applied to the resulting expression.
E?aNORTH2? = ax2 cos2 (fi)+ax2 sin2 (fi)E?fl2?+ay2 sin2 (fi)+ay2 cos2 (fi)E?fl2?+
E??ax2?cos2 (fi)+E??ax2?sin2 (fi)E?fl2?+
E??ay2?sin2 (fi)+E??ay2?cos2 (fi)E?fl2?+ (4.64)
87
Subtracting the mean squared value yields the variance of the acceleration in the
North direction.
E?aNORTH2? = ax2 sin2 (fi)E?fl2?+ay2 cos2 (fi)E?fl2?+
cos2 (fi)E??ax2?+sin2 (fi)E??ax2?E?fl2?+
sin2 (fi)E??ay2?+cos2 (fi)E??ay2?E?fl2?+ (4.65)
The results from Chapter 3 can be substituted to yield the flnal variance of the North
acceleration as resulting from the transformation of the acceleration measurements.
VAR?aNORTH2? =
?
ax2 sin2 (^?)+ay2 cos2 (^?)
?
R ?gz2 +
cos2 (^?) ?ax2 +sin2 (^?) ?ax2 R ?gz2 +
sin2 (^?) ?ay2 +cos2 (^?) ?ay2 R ?gz2 (4.66)
Using the same procedure as above, the variance of the east acceleration reduces to
the following expression
VAR?aEAST2? =
?
ax2 cos2 (^?)+ay2 sin2 (^?)
?
R ?gz2 +
sin2 (^?) ?ax2 +cos2 (^?) ?ax2 R ?gz2 +
cos2 (^?) ?ay2 +sin2 (^?) ?ay2 R ?gz2 (4.67)
Where ^? is the integrated gyro measurement.
In the resulting variance expressions for the slip case accelerations, ?ax2 and ?ay2
are simply the output variances of the accelerometers. While these values are bounded
88
in time, the integrated gyro variance R ?gz2 is not. This unbounded acceleration er-
ror causes an even faster error growth in the integrated velocity and double-integrated
position values. For the no-slip case, no such acceleration transformation is necessary.
Consequently, the no-slip velocity and position error will grow at a slower rate than
that for the slip case. This suggests that when such vehicle constraints are valid, better
dead-reckoning performance is achieved when using the fewest possible measurements
with the fewest possible integrations. The following sections seeks to support this claim
with a simulation example.
4.5 Comparison of Slip and No-Slip Mechanizations
Since for the no-slip case, only a single accelerometer is required and no transfor-
mation is necessary, the velocity and position values derived with the simpler dynamic
equations and therefore exhibit a slower error growth. The conclusion then is that if and
only if the no-slip condition exists, better dead-reckoning performance can be achieved
by using as few measurements as possible and few sensor integrations as possible. By
using computer simulation tools, these claims are demonstrated as follows.
A comparison of the two methods can be realized with the use of a Monte Carlo
simulation as used for the validation of the variance expressions earlier in this chapter.
Within this particular simulation, a position trajectory is deflned and the resultant vehi-
cle orientation, velocities, and accelerations are computed under the no slip assumption.
A Monte Carlo simulation is performed (3000 iterations) in which simulated inertial
errors with assumed sensor error parameters are added to the inertial values of accelera-
tions and rotation rates. The simulated inertial measurements are sent to two calculation
routines: one in which the no-slip condition is assumed, the other in which the side-slip
89
assumption is assumed. For 3000 iterations, the variance of the velocity, position, and
attitude for each time step is calculated and stored for comparison.
Figure 4.8 shows a sample North vs. East position trajectory deflned in the navi-
gation frame. Under the no-slip assumption, the 2-D velocity is computed and shown
in Figure 4.9; the corresponding no-slip yaw angle is shown in Figure 4.10. Using the
same sensor speciflcations as listed in Table 3.1 (with a faster sample rate of 100Hz), the
inertial measurements are simulated and processed for both the no-slip case in Figure
4.4 and the slip case in Figure 4.3 for 2000 iterations. The variance functions of the
yaw angle, 2-D velocity, and 2-D position are then computed for each case over the 3000
iterations. 0 1 2 3 4 5 6 7
0
1
2
3
4
5
6
7
8
9
North Position [m]
East Position [m]
Figure 4.8: Deflned Position Trajectory
90
0 10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
1.2
1.4
Velocity [m/s]
Time [s]
Figure 4.9: Velocity Trajectory With No Side Slip0 10 20 30 40 50 60?50
0
50
100
150
200
Yaw Angle [deg]
Time [s]
Figure 4.10: Yaw Angle Trajectory With No Side Slip
91
Figure 4.11 shows the 3- bounds for the yaw angle under each kinematic assump-
tion. Since the processing is identical, the attitude of the body under each assumption
exhibits identical variance growth curves. However, Figures 4.12 (Velocity) and 4.13 (Po-
sition) show a difierent result. Here, the growth in the integral states derived using the
no-slip mechanization is slower than that using the slip mechanization thus exemplifying
the efiect claimed. 0 10 20 30 40 50 60
0
5
10
15
20
Yaw Angle 3?
? [deg]
Time [s]
No Slip Mechanization
Side Slip Mechanization
Figure 4.11: 3- Bounds on Simulated Yaw Angle
92
0 10 20 30 40 50 600
0.5
1
1.5
Velocity RMS 3?
? [m/s]
Time [s]
No Slip Mechanization
Side Slip Mechanization
Figure 4.12: RMS 3- Bounds on Velocity0 10 20 30 40 50 600
10
20
30
40
50
Position RMS 3?
? [m]
Time [s]
No Slip Mechanization
Side Slip Mechanization
Figure 4.13: RMS 3- Bounds on Position
93
This example shows that when the no-slip assumption is valid, use of the no-slip
dynamic equations results in more accurate dead-reckoning performance. In contrast, use
of the side-slip equations result in a higher rate of error growth due to the unnecessary
acceleration measurement and additional step of integration in the governing equations.
It is emphasized again that the no-slip equations only provide a valid result when such
a no-slip assumption can be made.
4.6 Conclusion
This chapter has presented an analysis of the propagation of the stochastic inertial
sensor errors into the position, velocity, and attitude vehicle states for a body restricted
to planar motion. For this planar case, it is shown that the vehicle?s z axis is always
aligned to the navigation frame ? axis and therefore its errors are simply the integrated
gyro errors. These integrated sensor errors can be quantifled by direct application of the
variance expressions derived in Chapter 3. Within this planar navigation scenario, two
kinematiccasesarestudied: side-slipandnoside-slip. Forthenoside-slipcase, avehicle?s
velocity vector is coincident with its heading, and velocity is ascertained from a single
accelerometer integration. However, in the side-slip case the vehicle points away from
its direction of travel and the velocity must be derived from an additional measurement
and additional integration step. This chapter shows through analytical results and a
simulation example that when a vehicle has no side-slip, better dead-reckoning accuracy
is achieved by employing the simpler equations with fewer measurements for the no-slip
mechanization, as compared to the added measurements of the side-slip mechanization.
This chapter has also shown through the position derivation for the no-slip case, the
di?culty in the current analysis approach. It is suggested that further analytical analysis
94
be performed using techniques which provide more comprehensive statistical information
on the propagated inertial errors to analytically quantify the position errors in planar
mechanization.
95
Chapter 5
Six DOF Analysis
5.1 Introduction
As mentioned in Chapter 4, six degrees of freedom (6-DOF) are required to kine-
matically describe the most general motion of a body in space. In modern-day inertial
navigation these six degrees of freedom are typically measured with an inertial measure-
ment unit (IMU) rigidly attached to the body. The IMU measures three orthogonal
accelerations (ax,ay,az) and three orthogonal rotation rates (gx,gy,gz) for a total of 6
degrees of freedom. For the purpose of navigation, such measurements need to be trans-
formed into a coordinate system suitable for navigating. Figure 5.1 shows a common
navigation-frame cartestian coordinate system as commonly employed on many inertial
navigation systems, especially ground vehicles [25]. In one such cartesian frame, the
navigating vehicle?s state is described by its velocity, and position in terms of coordi-
nates North, East, and Down (NED) and by its roll (`), pitch ( ), and yaw (?) angles
about the NED axes, respectively. The diagram from Chapter 4 is shown again in here
in Figure 5.1 to illustrate the body and navigation frame axes.
This chapter presents the equations necessary to use body frame measurements from
a 6-DOF IMU to describe a vehicle?s state in the navigation frame. This 6-DOF navi-
gation scheme is applicable to all of the scenarios introduced in the previous chapters as
it captures the most general vehicle motion. As Chapters 3 and 4 have illustrated, the
accuracy of any inertial navigation system degrades with time, sensor integrity, trajec-
tory, and dynamic relationships employed without aid from external measurements. In
96
Figure 5.1: Simplifled Coordinate Frame
speciflc regards to the efiect of dynamic relationships, Chapter 4 showed a comparative
example showing the relative decline in inertial navigation performance when unneces-
sary measurements and integrations are used in deriving the navigation-frame states.
This chapter will extend this point with a similar example in which inertial navigation of
a planar trajectory is compared using the planar equations of Chapter 4 and the general
6-DOF method presented in this chapter.
5.2 Equations of Motion
5.2.1 Orientation
Two common conventions are used to describe a body?s orientation in the NED
navigation frame: Euler angles and Quaternions. The Euler angle representation, while
intuitive and straightforward to implement, exhibits a matrix singularity for pitch angles
at 90 degrees. As this particular orientation is rarely encountered on many vehicles which
use the NED frame, Euler angles remain as popular choice for ground vehicles. The
97
Quaternion approach involves a set of four linear difierential equations which describe
orientation in three dimensional space. The method has the advantage of being immune
to any particular singularities and therefore is numerically stable for all orientations.
However, as the quaternion values themselves lack strong physical meaning, they are
commonly transformed from and to Euler angles using a nonlinear relationship. The
advantage of the quaternion approach for ground vehicle applications therefore is mainly
in numerical computation and the relative accuracy compared to the Eulerian angle
method is negligible. A more complete comparison and discussion of the two methods
can be found in [26]. For the simple study in this thesis, the Euler angle representation
is presented as follows.
The Eulerian angular velocities are described in terms of the body frame rotation
rates by the following set of flrst order difierential equations
2
66
66
64
_`
_
_?
3
77
77
75 =
1
cos
2
66
66
64
cos sin`sin cos`sin
0 cos`cos ?sin`cos
0 sin` cos`
3
77
77
75
2
66
66
64
gx
gy
gz
3
77
77
75 (5.1)
where gx, gy, and gz are the rotation rates as aligned to orthogonal axes on the
body and (`), ( ), and (?) are the rotation rates about the navigation frame axes. The
nonlinear relationships of orientation given by Equations (5.1) are numerically integrated
to obtain the resulting Euler angles describing the attitude of the navigating body in
space.
It is instructive to note that when two angles are zero, the angular rate correspond-
ing to the remaining angle holds a one-to-one relationship from the body frame to the
navigation frame. Additionally, if the body frame rotation rates gx and gy are zero then
98
the relationship between the navigation-frame yaw rate, ?, and body-frame rate gz ex-
hibit a one-to-one relationship. The latter situation is precisely the planar case studied
in Chapter 3 in which the orientation of the body was constrained to rotate only about
its z-axis.
5.2.2 Translation
The accelerations as measured in the body-frame, must be transformed into the
navigation frame using the Euler angles obtained from the orientation calculations. The
Euler angles are used to construct the direction-cosine rotation matrix [26], which simply
re-orients the three accelerations as measured in the body frame, to the navigation frame
North, East, Down directions. The relationships between body frame and navigation
frame accelerations are shown by Equation (5.2).
2
66
66
64
aN
aE
aD
3
77
77
75 =
2
66
66
64
cos cos? ?cos`sin? +sin`sin cos? sin`sin? +cos`sin cos?
cos sin? cos`cos? +sin`sin sin? sin`cos? +cos`sin sin?
?sin sin`cos cos`cos
3
77
77
75
2
66
66
64
ax
ay
az
3
77
77
75(5.2)
Once the accelerations are transformed, the gravity component is subtracted from
the down acceleration to yield the kinematic acceleration of the body in the navigation
frame. The resulting velocity and positions as described by the North, East, Down
coordinate system are then derived by direct integration of the transformed accelerations.
This process in which the IMU outputs are transformed and integrated into usable
navigational quantities is known as mechanization. The mechanized IMU rigidly at-
tached to the navigating body is considered the Inertial Navigation System (INS). The
6-DOF IMU mechanization algorithm as introduced above is summarized by Figure 5.2.
99
The simple mechanization shown here neglects the coriolis, centripetal accelerations,
and other efiects experienced by an IMU moving on the rotating earth. For the short
range (short time) for which many ground vehicles travel, these efiects are small and the
mechanization discussed here is su?cient to support the typical requirements.
Figure 5.2: Mechanization of IMU Measurements
See Appendix B for a demonstration of the mechanization equations as presented.
This appendix presents the mechanization of a medium grade IMU 6-DOF and compares
its performance to position and velocity from a high-accuracy difierential GPS receiver.
5.3 Comparison to Planar Mechanization
In Chapter 4 it was shown that for a planar vehicle trajectory where the body
experiences no side-slip, the position and velocity error growth using the assumed-slip
planar mechanization exhibited a faster variance growth than results from the no-slip
planar method. The faster error growth observed with the slip equations was due to two
contributing factors. The slip case required both an acceleration measurement and an
additional step of numerical integration in the computation of its velocity and position
100
values. The 6-DOF navigation equations presented above are a natural extension of the
2-D slip case to the 6-DOF system. When the planar assumption can be made, using
the 6-DOF equations will add unnecessary error into the system. As a result, the error
growth for the 6-DOF case will be much worse. In the following, an example is shown
using the same trajectory of Chapter 4 to show the amount of additional error induced
when the 6-DOF method is employed.
Figures 5.3 5.4 and 5.5 show a sample position, velocity, and yaw angle trajectory.
Like the identical trajectory of the chapter 4 slip/no-slip comparison, the velocity, and
yaw angle are derived from the deflned position trajectory.0 1 2 3 4 5 6 7
0
1
2
3
4
5
6
7
8
9
North Position [m]
East Position [m]
Figure 5.3: Deflned Position Trajectory
101
?0.1 0 0.1 0.2 0.3 0.4
?1
?0.5
0
0.5
1
North Velocity [m/s]
East Velocity [m/s]
Figure 5.4: Velocity Trajectory for Planar Motion0 10 20 30 40 50 60?50
0
50
100
150
200
Yaw Angle [deg]
Time
Figure 5.5: Yaw Angle Trajectory for Planar Motion
102
Using the sensor speciflcations as shown in Table 5.1, a Monte Carlo simulation
was performed by computing 6-DOF position, velocity, and attitude with simulated
acceleration measurements, ax, ay, az and rotation rates gx, gy, gz. Another Monte
Carlo simulation was then performed with the planar side-slip equations of Chapter 4
using simulated ax, ay and gz with the same sensor speciflcations. Both simulations in
this example were performed with 1200 iterations. The resulting standard deviations of
the two dimensional rms velocity, rms position and the attitude angle are then plotted
for comparison.
Table 5.1: Simulated Sensor Speciflcations (Comperable to Crossbow IMU-400C)
Gyro Spec Value Accel Spec Value
fs 100 Hz fs 100 Hz
2rw 0.14948 deg/s/sqrt(Hz) 2rw 0.000412 g/sqrt(Hz)
2b 0.61183 deg/s 2b 0.000100 g
? 1300 seconds ? 500 seconds
Figure 5.6 compares the standard deviation results for the yaw angle computed with
the planar equations and then with the 6-DOF equations. It is evident that the error in
the yaw angle for this trajectory is approximately the same. As the 6-DOF equations
\take out" the efiect of the roll and pitch angles on the yaw angle, the yaw orientation
is the same.
Figures 5.7 and 5.8 compare the standard deviation results for the 2-D rms velocity
and 2-D rms position, respectively. It is clear from these two plots that the additional
measurements in the 6-DOF scheme cause a much more severe error growth as contrasted
to the 3-DOF case.
103
0 10 20 30 40 50 600
5
10
15
20
Yaw Angle 3?
? [deg]
Time [s]
Planar Mechanization
6?DOF Mechanization
Figure 5.6: 3- Bounds on Yaw Angle Mechanization Comparison0 10 20 30 40 50 600
10
20
30
40
50
60
70
80
Velocity RMS 3?
? [m/s]
Time [s]
Planar Mechanization
6?DOF Mechanization
Figure 5.7: RMS 3- Bounds on Velocity Mechanization Comparison
104
0 10 20 30 40 50 600
200
400
600
800
1000
1200
1400
Position RMS 3?
? [m]
Time [s]
Planar Mechanization
6?DOF Mechanization
Figure 5.8: RMS 3- Bounds on Position Mechanization Comparison
Figure 5.9 and 5.10 show the standard deviation results for pitch and roll angles,
respectively. These plots show precisely why the position and velocity growth is so
large: the integrated stochastic errors on the additional roll and pitch measurements
cause the accelerations to be rotated to an incorrect orientation. As a result, the mis-
oriented accelerations are integrated along their incorrect directions to give a much more
inaccurate North/East velocity and position.
105
0 10 20 30 40 50 600
2
4
6
8
10
12
Pitch Angle 3?
? [deg]
Time [s]
Planar Mechanization
6?DOF Mechanization
Figure 5.9: 3- Bounds on Pitch Angle DOF Comparison0 10 20 30 40 50 600
2
4
6
8
10
12
Roll Angle 3?
? [deg]
Time [s]
Planar Mechanization
6?DOF Mechanization
Figure 5.10: 3- Bounds on Roll Angle DOF Comparison
106
In the example above, it is clear that the body constrained to travel on the at
plane not only lacked the beneflt of the extra gyro measurements - they greatly increased
the overall error in the desired states. While this sample simulation has demonstrated
the efiect of unnecessary inertial measurements in dead-reckoning navigation, it should
be re-iterated that if the body was indeed traveling in motion that required a 6-DOF
characterization, no less than six inertial measurements can be employed to correctly
compute the vehicle?s trajectory. In other words, better navigation performance will
most likely not be achieved by employing kinematic relationships which are simpler than
the motion of the navigating body. In conclusion, when a vehicle in motion is under
kinematic constraints, few measurements and integrations as required to completely
describe the vehicle will yield the smallest amount of error.
107
Chapter 6
Conclusions
6.1 Overall Contributions
This thesis has presented an analysis of the expected accuracy of inertial dead-
reckoning in a variety of navigation scenarios. First, stochastic models were proposed
as approximations to the observed behavior of many common types and grades of ac-
celerometers and rate gyros. The selected inertial sensor error model included the sum of
two independent Gaussian processes characterized by three parameters. Parameter iden-
tiflcation methods of Allan variance and experimental autocorrelation were presented as
the means of extracting the model parameters from experimental data. Derivations of
the variance of subsequent integrations of each sensor error component process were per-
formed and validated using Monte Carlo simulations. An application of the integrated
error source variance expressions were demonstrated for the single-axis navigation sce-
nario in which a single accelerometer or single gyro is used to ascertain integral navigation
states in its flxed direction. The single-axis results were then expanded to derive expres-
sions for the propagation of the inertial sensor error into the mechanization equations
used for planar navigation in which a traveling body experiences no side-slip. The re-
sults of the no-slip position were discussed and shown to have limitations due to the
analysis techniques used. The planar no-slip inertial navigation mechanization was then
compared to the planar mechanization with slip for the purpose of showing the errors
induced by additional integrations and measurements required of the slip mechanization.
The flnal chapter concluded the quantiflcation of inertial navigation by presenting the
108
6-DOF kinematic relationships as applied in the navigation of a body unconstrained in
three dimensional space. The concluding point of Chapter 4 was reiterated again with a
simulation example illustrating the additional dead-reckoning error induced by employ-
ing unnecessary additional integrations and measurements in the 6-DOF equations for a
simple planar trajectory.
In summary, this research has provided
1. Approximate stochastic models which capture the necessary behavior of accelerom-
eter and rate-gyroscope outputs as measured from various grade inertial measure-
ment units.
2. Derivationsofthevarianceofthenumericallyintegratedvaluesoftheinertialsensor
error sources of wide-band noise and exponentially correlated noise (Gauss-Markov
process) from the error models.
3. Derivations of the variance of the 2-D (North/East) velocity error for a planar nav-
igation scenario in which a vehicle experiences no side-slip. The position derivation
is performed to show the need for methods of analysis superior to that of this thesis.
4. Comparison of the no-slip and slip and general 6-DOF inertial navigation equations
for the planar case.
5. The six degree of freedom equations used for the most general navigation scenario.
In addition to the derivations, this thesis has demonstrated the claim that the
fewest number of integrations and measurements required to provide valid vehicle states
in an inertial navigation system yields the minimal amount of error. It was shown
through a simulation comparison of the planar no-slip, planar slip, and 6-DOF general
109
motion methods, that the inertial navigation with the simplest system had the best dead-
reckoning performance (when the real vehicle trajectory was described by the simplest
set of equations).
6.2 Di?culties
The quantity and nature of approximations presented in this thesis warrant a dis-
cussion of the limitations of the contributing results. It should flrst be noted that the
variance expressions for the integrated sensor errors in Chapter 3 are derived based on
the assumption that inertial sensors can be modeled with the approximations in Chapter
2. The experimental identiflcation and validation of of simple models with real experi-
mental data is a di?cult process due to the extremely large amount of data required and
the inadequacy of the assumed model forms. Based on these di?culties, the accuracy of
the variance expressions based on the assumed model form should only re ect the accu-
racy of the experimental identiflcation techniques by employing conservative parameter
estimates.
The techniques used to ascertain the variance of the integrated inertial sensor errors
in Chapter 3 proved useful in its straightforward and intuitive approach. However, as
shown for the planar no-slip position variance derivation in Chapter 4, the autocorrela-
tion information was needed to simplify the expression to closed-form. The straightfor-
ward method of derivation in Chapter 3, while yielding the desired variance expressions,
did not allow for simple identiflcation of higher order measures of statistics. Additional
information such as the cross-correlation and autocorrelation are di?cult to attain with
the techniques employed in this thesis. In conclusion, other techniques are suggested for
110
use in quantifying the propagation of stochastic errors through various transformations
and integration.
In Chapter 3, the variance of the integral values of the Gauss-Markov process as-
sumes that the process is realized with a zero-initial condition at the onset of integration.
This zero initial condition for some values of the Markov model parameters may not ac-
curately re ect the desired nature of the process as it may exhibit some initial transients
in reaching its stationary status.
The developed expressions in Chapter 4 for the variance of the planar case errors
were based on small angle linear approximations for trigonometric operations on the
integrated gyro error. For errors outside the range for which the approximations are
valid, the error will propagate non-linearly and the resulting value will no longer be a
Gaussian variable. The variance expression will then be inadequate to fully characterize
the error of the nonlinearly processed inertial errors. However, the size of integrated gyro
errors necessary to break the linear approximation is well outside the range for which a
system designer is interested. Therefore the approximation is rarely an issue for the use
of the variance expressions as practical analysis tools.
6.3 Future Work
In order to more fully and successfully characterize the stochastic outputs of ac-
celeration and rotation rate outputs of many grade IMUs, it is suggested that much
experimental data be taken and studied to ensure the feasibility of the approximations
introduced in this thesis. As the stochastic models introduced in Chapter 2 do not in-
clude terms for temperature or range of motion, a study of such efiects should be done to
ensure that their in uence is negligible. Given that the stochastic models from Chapter 2
111
are su?cient to match the behavior of the sensor errors, their corresponding parameters
should be identifled with higher confldence using appropriately the techniques of Allan
variance and autocorrelation. Once the models are validated and parameters identifled,
the resulting variance expressions in Chapters 3 and 4 should be tested with experimental
data to verify their use in real inertial navigation scenarios.
As discussed above, the methodology used to propagate the sensor errors has room
for improvement. Other more comprehensive statistical analysis techniques may provide
a better and more complete characterization of the inertial system errors and perfor-
mance in dead-reckoning. Since the inertial navigation performance using the 6-DOF
mechanization scheme depends highly on non-linear relationships, the Gaussian sensor
errors do not propagate linearly into the velocity, position, and orientation states and
thus exhibit non-Normal distributions. More complete probabilistic characterizations
of the propagation of sensor errors into the applicable nonlinear equations may provide
broader results to the navigation systems mechanized for general motion.
Using validated stochastic models and improved methods of error propagation anal-
ysis, future research may expand its scope to include analyses of dead-reckoning with
inertial measurements in a larger range of scenarios. As other sensors such as odome-
try, laser scanners, and vision, are continually being fused into the navigation system,
knowledge of the dead-reckoning performance of these systems becomes increasingly de-
sirable. In addition, more sophisticated GPS/INS algorithms such as Tightly-Coupled
GPS/INS allow for dead-reckoning aid from any available satellites when the number
visible are less than that required for a position flx. Firm quantiflcation of the increased
performance from any type of vehicle constraint, GPS, or auxiliary sensors in the large
112
range of applicable mechanizations provides much valuable information to the navigation
community at large.
113
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116
Appendices
117
Appendix A
Stochastic Parameter Identification with an Automotive-Grade IMU
This appendix serves to demonstrate the parameter identiflcation techniques of Al-
lan variance and autocorrelation on automotive-grade inertial measurements. The tech-
niques are used to identify parameters of the assumed stochastic model from accelerom-
eters and rate-gyros from a Crossbow IMU-400C logged using a PC with Windows XP
via an RS-232 serial connection.
The accelerometers and gyros were logged at a sample frequency of fs = 5 Hz for
approximately 48 hours in a climate controlled o?ce while resting on a level desk. The
meanisremovedfromeachsensorlogandtheoutputsarefllteredtorevealanyunderlying
drifting bias. Figure A.1 shows the three Crossbow accelerometers zero-phase flltered
(Matlab flltfllt() function) with a second-order low-pass Butterworth fllter with cutofi
frequency of 0.001 Hz. Figure A.2 shows the zero-phase flltered rate-gyros (second-order
Butterworth with 0.001 Hz cutofi frequency).
118
0 10 20 30 40 50?2
?1.5
?1
?0.5
0
0.5
1
1.5
2x 10
?3
time [hr]
Filtered Output [g]
ax
ay
az
Figure A.1: Filtered Accelerometer Outputs0 10 20 30 40 50?0.08
?0.06
?0.04
?0.02
0
0.02
0.04
time [hr]
Filtered Output [deg/s]
gx
gy
gz
Figure A.2: Filtered Gyro Outputs
119
As can be seen in both sets of flltered outputs, the sensors exhibit a very slow drifting
behavior over the 48 hour logging period. While this slow drift may efiect navigation
system performance for long spans of time, it is of primary interest to understand the
drifting behavior within short time intervals (the intervals for which the sensors will be
used to dead-reckon). In order to employ the identiflcation techniques using the assumed
Gauss-Markov model, focus is turned to the most steady of the outputs: the z-axis gyro,
gz, within its most constant interval from hour 20 to 45.
Figure A.3 shows the raw and flltered data for the gz gyro within the interval
selected. The data within this interval is flltered to remove the high frequency content of
the wide-band noise, leaving the driflng bias for identiflcation with the autocorrelation
method. The flltered data shown in the plot was processed with the same low-pass
fllter as before with a cutofi frequency of 0.0005 Hz, a value determined after multiple
iterations to yield the best identiflcation results with the data used.0 5 10 15 20 25?1.5
?1
?0.5
0
0.5
1
1.5
time [hr]
Gyro Data [deg/s]
RawFiltered
Figure A.3: Raw and Filtered Data Within Selected Section
120
Figure A.3 shows the flltered data by itself. This plot indicates the relatively long
term stability of the selected data, while revealing the characteristic bias drift of interest.0 5 10 15 20 25?0.02
?0.015
?0.01
?0.005
0
0.005
0.01
0.015
Filtered Gyro Data [deg/s]
Time [hr]
Figure A.4: Filtered Data Within Selected Section
121
The sectioned and flltered data is then processed with an autocorrelation function.
The result is shown in Figure A.5. The run reveals an exponential function such as that
exhibited by the Gauss-Markov model. From this data, the time constant is extracted by
selecting the intersection on the 1e horizontal line and the variance from the y-intercept.
Note that these parameters are extracted only with the confldence re ected by the bound
as shown which is referenced around the best-flt line.0 1000 2000 3000 4000 5000
?1
0
1
2
3
4
5x 10
?5
Time [s]
Autocorrelation, R
xx
(t)
Data
Best Fit
?2b e?1
1?? Bound
Figure A.5: Autocorrelation of Filtered Gyro Data
Now that rough Gauss-Markov parameters are identifled, an Allan variance calcula-
tion is performed on the raw experimental data within the section selected. Figure A.6
shows the results of the Allan variance.
122
100 101 102 103 104 105
10?4
10?3
10?2
10?1
Averaging Time, T [s]
Root Allan Variance
? AV
Exp
Fit
3?? Bound
Figure A.6: Allan Variance of Gyro Data
Simple inspection of the Allan variance allows extraction of the random walk pa-
rameter (y-intercept of the graph); this parameter quantifles the wide-band noise. Using
the identifled parameters from the autocorrelation and the extracted random walk pa-
rameter, a best flt line is constructed and plotted to represent the Allan variance of
the experimental gyro data with the assumed model form. The 3- bounds, based on
the averaging time and total length of the data set (25 hours @ 5Hz) are plotted along
with the flt and experimental data. The flt, while showing a fair agreement with the
experimental data, is only a conservative representation as re ected by the large and
spreading bounds.
123
Using the techniques above with the gyro data set, the parameters used to represent
the rate-gyro of the Crossbow IMU-400C are shown in Table A.1.
Table A.1: Results of Crossbow Gyro Identiflcation
Gyro Spec Value
fs 5 Hz
2rw 0.14963 deg/s/sqrt(Hz)
2b 0.0061 deg/s
? 1100 seconds
As shown on the plots of Allan variance and autocorrelation, the bounds on the
accuracy of each are very large. These bounds need to be smaller to re ect high con-
fldence in the parameter estimation. Higher levels of confldence require longer sets of
data. However, as shown by the long-term behavior of the flltered IMU outputs, it is
di?cult to ascertain a long set of static data stable enough to employ the techniques
with the assumed model forms.
124
Appendix B
Demonstration of 6-DOF Mechanization of Experimental Inertial
Measurements Taken at Talledega Superspeedway
This appendix serves to demonstrate the 6-DOF mechanization of an automotive
grade 6-DOF IMU in a medium-sized vehicle traveling around a track at high speeds.
The experimental setup is as follows. An Inflniti G-35 sedan was equipped with a single
antenna Novatel StarflreTM DGPS ( < 10cm accuracy) receiver mounted on a roof rack
roughly in the planar center of gravity (CG) of the car. An automotive-grade Crossbow
IMU-400C 6-DOF IMU was attached rigidly to the console of the vehicle, which is
roughly located at the vehicle?s CG. GPS position, velocity, and course was logged at
5 Hz from the GPS, and inertial measurements were logged at 133Hz from the IMU
via RS-232 serial connections to a PC in the trunk running Windows XP. For a more
complete description of the vehicle test-bed and data acquisition system see [27]. The
data used in this appendix was logged with the experimental vehicle and sensors while
driving 3 laps around the 2.66 mile track at Talledega superspeedway (Talledega, AL)
for approximately 6.5 minutes. The track has bank angles of about 33 degrees in the
turns, 16.5 degrees in the tri-oval, and 3 degrees in the straights. The inertial data logged
during the laps was post-processed using the 6-DOF mechanization equations presented
in Chapter 5 to provide position, velocity, and attitude in the North, East, Down (NED)
coordinate system. The inertially-derived values are compared to the high-accuracy GPS
information to show its performance in dead-reckoning.
125
Figure B.1 shows the North and East position from the 6-DOF mechanization of
the IMU as compared to the GPS position. It is immediately evident that the dead-
reckoning position from the IMU quickly drifts away from the true position of the track
(represented by the sub-10cm GPS position). As time elapses for each lap around the
track, the IMU-derived position, while coarsely resembling the ovular shape of the true
trajectory, drifts with increasing variance from the actual position.
?1000 ?500 0 500 1000
?500
0
500
1000
1500
Position from IMU
North Position [m]
East Position [m]
StarFireCrossbow
StartFinish
Figure B.1: Position of Track from Starflre GPS
126
Figure B.2 shows the velocity of the runs around the track as reported by the GPS
receiver. As is shown, the large track allowed for velocities higher than are generally
allowed on public highways.0 1 2 3 4 5 6 7
0
20
40
60
80
100
120
GPS Velocity, [mph]
Time [min]
Figure B.2: Velocity from GPS
127
Figure B.3 shows a comparison of the yaw angles from the IMU and GPS as the
vehicle travels around the track. For the high speed and moderate banked turns around
the track, the true heading and course are expected to be the same (no vehicle side-slip).
As time progresses however, the heading derived IMU drifts from the GPS-reported
orientation due to the integration of the stochastic errors present on the three gyros
used to compute the yaw angle.0 100 200 300 400 500
0
50
100
150
200
250
300
350
400
Angle [deg]
Time [s]
Course from GPS
Heading from IMU
Figure B.3: Yaw Angle (Heading) from IMU
128
Figure B.4 shows the roll and pitch angles as purely derived from the IMU. As only
a single antenna GPS was logged during the test, no un-biased estimate of these angles
was available and so only the IMU-derived angles are shown. With knowledge of bank
angles of the track, however, it is evident that the values of roll match fairly well to the
slope of the track in the turns and straights.0 50 100 150 200 250 300 350 400
?5
0
5
10
15
20
25
30
35
40
Angle [deg]
Time [s]
roll angle , ?
pitch angle, ?
Figure B.4: Roll and Pitch Angles from IMU
129
Figure B.5 shows of comparison of the North and East component velocities from
the IMU and GPS. As shown, the IMU component velocities resemble the shape of the
trajectory but drift with increasing distance from the GPS values.
0 50 100 150 200 250 300 350 400?40
?20
0
20
40
60
North Velocity [m/s]
Time [s]
Inertial
GPS
0 50 100 150 200 250 300 350 400?50
0
50
East Velocity [m/s]
Time [s]
Inertial
GPS
Figure B.5: North and East Component Velocities
130
Figure B.6 shows of comparison of the North and East component positions from
the IMU and GPS. As shown, the IMU component positions resemble the shape of the
trajectory but drift with increasing distance from the GPS values. It is evident that
when compared to the velocity subplots of Figure B.5, the position error grows at a
faster rate. Due to the additional level of integration (once more than velocity) of the
stochastic sensor errors, this accelerated rate in position error growth is expected.
131
0 50 100 150 200 250 300 350 400?1000
?500
0
500
1000
1500
2000
North Position [m]
Time [s]
Inertial
GPS
0 50 100 150 200 250 300 350 400?1500
?1000
?500
0
500
1000
1500
East Position [m]
Time [s]
Inertial
GPS
Figure B.6: North and East Component Positions
132