PRECISE RELATIVE NAVIGATION USING
AUGMENTED CDGPS
a dissertation
submitted to the department of mechanical engineering
and the committee on graduate studies
of stanford university
in partial fulfillment of the requirements
for the degree of
doctor of philosophy
Chan-Woo Park
June 2001
c Copyright by Chan-Woo Park 2001
All Rights Reserved
ii
I certify that I have read this dissertation and that in
my opinion it is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Jonathan P. How
Department of Aeronautics and Astronautics
(Principal Adviser)
I certify that I have read this dissertation and that in
my opinion it is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
J. Christian Gerdes
Department of Mechanical Engineering
I certify that I have read this dissertation and that in
my opinion it is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Per K. Enge
Department of Aeronautics and Astronautics
Approved for the University Committee on Graduate
Studies:
iii
iv
Abstract
Autonomous formation flying of multiple vehicles is a revolutionary enabling technology for many future space and earth science missions that require distributed
measurements, such as sparse aperture radars and stellar interferometry. The techniques developed for the space applications will also have a significant impact on
many terrestrial formation flying missions, such as air-to-air refueling.
One of the key requirements of formation flying is accurate knowledge of the relative positions and velocities between the vehicles. Several researchers have shown
that the Global Positioning System (GPS) is a viable sensor to perform this relative
navigation. However, there are several limitations in the use of GPS because it is limited to locations with adequate visibility to the NAVSTAR constellation. For some
mission scenarios, such as MEO, GEO and tight formation missions where signals
can be blocked by nearby vehicles, the visibility/geometry of the GPS constellation
may not be sufficient to accurately estimate the relative states. Another difficulty is
that there is a fundamental limitation to the accuracy of relative position/velocity
estimates using a GPS-only system. In addition, the GPS system has a very limited
capability to initialize the cycle ambiguity, thereby making it impossible to quickly
achieve centimeter-level accuracy, which is important for highly dynamic environments.
One solution to these problems is to include an RF ranging device (transmitter)
onboard the vehicles in the formation and form a local constellation that augments the
existing NAVSTAR constellation. These local range measurements, when properly
combined with the GPS measurements, can provide a sufficient number of measurements and adequate geometry to solve for the relative position and velocity states.
Furthermore, these RF ranging devices can be designed to provide substantially more
v
accurate measures of the vehicle relative ranges and range rates than the traditional
GPS pseudolites. These highly accurate measurements could then be combined with
the CDGPS solution to obtain a better overall estimate (sub-centimeter position accuracy). The local range measurements also allow relative vehicle motion to be used
to efficiently solve for the cycle ambiguities in real-time.
This dissertation presents the development of an onboard ranging sensor and the
extension of several related algorithms for a formation of vehicles with both GPS and
local transmitters. Key among these are a robust cycle ambiguity estimation method
and a decentralized relative navigation filter. The robust cycle ambiguity estimation algorithm can be used in a real-time autonomous initialization process, which
is experimentally demonstrated. The thesis also presents a decentralized approach
to the GPS-only relative navigation problem. This is extended to an iterative cascade extended Kalman filtering (ICEKF) approach when the vehicles have onboard
transmitters. When compared to a centralized estimation scheme, the decentralized
estimation process is shown to substantially reduce the computational load, eliminate
the numerical problems, and reduce the information flow between the vehicles.
It is also shown in this thesis that a linearization error occurs in the NAVSTAR
measurement equations for long baseline (>1 km) separations between the vehicles.
Correction terms are derived, and it is demonstrated that this error can be efficiently
removed using the information available from the standard pseudorange solution.
Finally, a quasi-optimal constellation geometry selection algorithm is developed for
the case where local ranging signals are also available. This algorithm is important
since there are a limited number of channels on the GPS receivers, which only allows it
to track a subset of the available NAVSTAR signals. Thus it is desirable to select the
subset that provides the best possible geometry. This new selection algorithm provides
near optimal geometries from the combined NAVSTAR and local constellations but
greatly reduces the computational load when compared to the true optimal methods.
Several ground testbeds were developed to demonstrate the feasibility of the augmentation concept and the relative navigation algorithms. The testbed includes the
Stanford Pseudolite Transceiver Crosslink (SPTC), which was developed and extensively tested with a formation of outdoor ground vehicles.
vi
Acknowledgments
I would like to thank several individuals who made this thesis possible. First, I would
like to thank my principal advisor, Jonathan How. He introduced me to the Global
Positioning System and provided me with the opportunity to work on the formation
flying research (blimp project, 1997). His guidance, technical insight, management
skill, and dedication are much appreciated. I would also like to thank my oral examination committee: Professors Christian Gerdes, Per Enge, Tommas Kenny, and Paul
Segall. I have profited from their advice, insightful comments, and helpful ideas. In
particular, I would also like to thank for their support during my stay at Stanford.
I wish to express my gratitude to many former and current colleagues at Stanford
and MIT. I was extremely fortunate to work with the former student Eric Olsen.
We worked together on many aspects of the formation flying blimp project including
receiver design and software development that are inherited by various current GPS
projects. Without the foundation created by his great work and our collaboration,
this research would not have been possible.
There are several students, in particular, who I would like to thank. Franz Busse,
Nick Pohlman, Cory Dixon, and Ung-suok Kim contributed to many aspects of this
project, and the successful demonstration of the research would have never been
possible without their invaluable help. Many hours were spent in various discussions
with Franz Busse, Arthur Richards, Phil Ferguson, Jonathan Stone, Eric Prigge, Ed
LeMaster, Demoz Gebre, Chad Jennings, Tobe Corazzini. Their insights, comments,
and suggestions were extremely valuable.
I gratefully acknowledge the Charles Stark Draper Laboratory for supporting the
early formation blimp project (contract DL-H-505325) and the Lockheed Martin for
vii
funding the formation flying research-with special thanks to Mr. Larry Cappots. This
research would not have been possible without the support from the Mitel Semiconductor regarding the GPS receiver design.
Finally, I would like to thank my family for their support both financially and
emotionally, during my stay at Stanford and throughout my academic life.
viii
Contents
Abstract
v
Acknowledgments
vii
1 Introduction
1.1
1
Formation Flying Technology . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1
. . . . . . . . . . . . . .
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Example Formation Flying Missions
1.2
Motivation
1.3
Previous Work
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
1.3.1
Formation Flying Research
. . . . . . . . . . . . . . . . . . .
13
1.3.2
Augmentation of GPS Constellation with Transmitters . . . .
14
1.3.3
Cycle Ambiguity Estimation
. . . . . . . . . . . . . . . . . .
14
1.4
Research Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
1.5
Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2 Formation State Estimation
20
2.1
Coordinate Systems
2.2
Correction Terms for the Large Separations
2.3
2.4
. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
20
22
2.2.1
Correction Terms for the Linear Relative Position Estimation
22
2.2.2
Correction Terms for the Relative Velocity Estimation . . . .
27
Formation Measurement Equation
. . . . . . . . . . . . . . . . . . .
29
2.3.1
Measurement Equations Using NAVSTAR Satellites
. . . . .
29
2.3.2
Measurement Equation Using Local Transmitters . . . . . . .
33
State and Cycle Ambiguity Estimation . . . . . . . . . . . . . . . . .
36
ix
2.5
2.4.1
WLS State Estimation - EKF Bias Estimation
. . . . . . . .
36
2.4.2
EKF State Estimation - EKF Bias Estimation
. . . . . . . .
40
Decentralized Iterative Cascade EKF . . . . . . . . . . . . . . . . . .
47
2.5.1
Iterative Cascade EKF
. . . . . . . . . . . . . . . . . . . . .
48
2.5.2
Numerical Stability of EKF . . . . . . . . . . . . . . . . . . .
59
2.6
Coarse Navigation Sensor
. . . . . . . . . . . . . . . . . . . . . . . .
61
2.7
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
3 Robust Bias Estimation Algorithm
3.1
3.2
Cycle Ambiguity Estimation
68
. . . . . . . . . . . . . . . . . . . . . .
69
3.1.1
Previous Initialization Algorithm . . . . . . . . . . . . . . . .
70
3.1.2
Robust Initialization Algorithm . . . . . . . . . . . . . . . . .
76
3.1.3
Performance of Robust Initialization Algorithm . . . . . . . .
84
3.1.4
Selection of Initialization Path
. . . . . . . . . . . . . . . . .
89
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4 Quasi-Optimal Satellite Selection
4.1
Satellite Selection
4.2
Optimal Satellite Selection
. . . . . . . . . . . . . . . . . . . . . . .
94
4.3
Sub-optimal Satellite Selection Algorithms . . . . . . . . . . . . . . .
97
4.3.1
Highest Elevation Satellite Selection Algorithm . . . . . . . .
97
4.3.2
Kihara’s Maximum Volume Method
. . . . . . . . . . . . . .
97
4.3.3
Four-step Selection Algorithm . . . . . . . . . . . . . . . . . .
98
4.4
4.5
5
92
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Quasi-optimal Satellite Selection
. . . . . . . . . . . . . . . . . . . .
99
4.4.1
Validation of Quasi-Optimal Algorithm
4.4.2
Computational Load . . . . . . . . . . . . . . . . . . . . . . . 112
4.4.3
Other Cost Functions
. . . . . . . . . . . . 104
4.4.4
Weighted Quasi-optimal Selection Algorithm
. . . . . . . . . . . . . . . . . . . . . . 114
. . . . . . . . . 114
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Mission Simulations
5.1
92
116
LEO Formation with Onboard Transmitters . . . . . . . . . . . . . . 117
x
5.2
5.1.1
Orbit Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.1.2
LEO Simulation Parameters
5.1.3
EKF Bias Estimation in LEO . . . . . . . . . . . . . . . . . . 119
5.1.4
EKF State Estimation - EKF Bias Estimation
. . . . . . . . . . . . . . . . . . 117
Effect of Long Baseline Correction Terms
. . . . . . . . . . . . . . . 124
5.2.1
Corrections for Differential Phase Measurements ∆φ . . . . . 124
5.2.2
Corrections for Differential Doppler Measurements ∆φ̇
5.3
Terrestrial Applications
5.4
Formation Geometry & Improved Transmitters
5.5
5.6
. . . 125
. . . . . . . . . . . . . . . . . . . . . . . . . 128
. . . . . . . . . . . . 133
5.4.1
Line Formation (In-Track) . . . . . . . . . . . . . . . . . . . . 134
5.4.2
Improved Line Formation (In-Track & Cross-track) . . . . . . 136
5.4.3
Ellipse Formation (Single Ellipse) . . . . . . . . . . . . . . . . 137
5.4.4
Double Ellipse Formation . . . . . . . . . . . . . . . . . . . . 138
PowerSail Simulation
. . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.5.1
Relative Motions . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.5.2
Visibility of NAVSTAR Satellite
5.5.3
Effect of Local Transmitters on GDOP . . . . . . . . . . . . . 146
5.5.4
Impact of Operating at Higher Altitudes . . . . . . . . . . . . 147
5.5.5
Impact of Adding Local Transmitters
5.5.6
GEO Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 150
. . . . . . . . . . . . . . . . 144
. . . . . . . . . . . . . 149
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
6 Experimental Results
6.1
. . . . . . . . 121
Experimental Setup
154
. . . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.1.1
Formation Truck Testbed
. . . . . . . . . . . . . . . . . . . . 155
6.1.2
Formation Robot Testbed . . . . . . . . . . . . . . . . . . . . 156
6.1.3
Onboard Ranging and Communication Device . . . . . . . . . 157
6.1.4
SPTC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.2
Multi-vehicle Autonomous Initialization
6.3
Robust Bias Estimator Experiment . . . . . . . . . . . . . . . . . . . 161
6.4
Mobile Formation Center & Multi-Transmitters . . . . . . . . . . . . 163
xi
. . . . . . . . . . . . . . . . 159
6.5
6.6
Formation Maneuvers
. . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.5.1
Truck Formation Maneuver . . . . . . . . . . . . . . . . . . . 166
6.5.2
Robot Formation Maneuver with Multi-Transmitters . . . . . 169
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7 Conclusion
173
7.1
Summary of Contributions
7.2
Future Work
7.3
. . . . . . . . . . . . . . . . . . . . . . . 173
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.2.1
LEO Formation Flying Experiment . . . . . . . . . . . . . . . 176
7.2.2
Transceiver Development
. . . . . . . . . . . . . . . . . . . . 177
7.2.3
UAV Relative Navigation
. . . . . . . . . . . . . . . . . . . . 178
Closing
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
A GPS Attitude Receiver Development and Testing
180
A.1 Receiver Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
A.2 Receiver Software
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
A.3 Onboard Cycle Ambiguity Estimation
. . . . . . . . . . . . . . . . . 183
A.3.1 Computing Initial Guess of States For Two Base Line Platform 183
A.3.2 Batch Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 186
A.3.3 Iterative Information Smoother . . . . . . . . . . . . . . . . . 188
B Formation Ranging Device Development
191
B.1 Laser Pointer Ranging Device . . . . . . . . . . . . . . . . . . . . . . 191
Bibliography
195
xii
List of Tables
1.1
Positioning Accuracy Levels Required for Several Missions
. . . . .
7
1.2
Importance of Onboard Transmitter in Various Application Areas . .
12
2.1
Simulated Doppler Correction Terms (2 km baseline) . . . . . . . . .
28
2.2
Major Data Flow
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
2.3
Minor Data Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.1
Final Position Errors . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
5.1
Orbit Parameters of Simulated Vehicles
5.2
Selected Variables for Simulation in Section 5.1.2 . . . . . . . . . . . 118
5.3
Initial Conditions of Simulated Vehicles: Relative Position and Velocity
. . . . . . . . . . . . . . . . 117
(ECI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4
Performance of EKF and WLS State Estimators
5.5
Effect of ∆φ̇ and ∆φ on Estimation Error (WLS) with Vehicle Separation of 2 km.
. . . . . . . . . . . 124
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.6
Parameter Selected for Air-to-air Refueling Simulation . . . . . . . . 130
5.7
Air-to-air Refueling Simulation Results . . . . . . . . . . . . . . . . . 133
5.8
Orbit Parameters Selected for PowerSail Simulation
6.1
Final Position Estimation Errors
xiii
. . . . . . . . . 143
. . . . . . . . . . . . . . . . . . . . 163
List of Figures
1.1
TechSat-21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2
Autonomous Rendezvous and Docking Mission . . . . . . . . . . . .
4
1.3
Nanosat Constellation Trailblazer mission (ST-5)
5
1.4
Gravity Recovery and Climate Experiment mission (GRACE)
1.5
ST-5
. . . . . . . . . .
. . .
5
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
1.6
Terrestrial Planet Finder (TPF) . . . . . . . . . . . . . . . . . . . .
6
1.7
ISS Docking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
1.8
Aircraft Formation Flying (F-16)
. . . . . . . . . . . . . . . . . . .
8
1.9
Farming Vehicle Formation Operation . . . . . . . . . . . . . . . . .
9
1.10 Air-to-Air Refueling . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
1.11 PowerSail Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.1
Formation Coordinate System. . . . . . . . . . . . . . . . . . . . . .
21
2.2
Relative Positioning of Formation Flying Spacecraft . . . . . . . . .
24
2.3
Long Baseline Spacecraft (Simple Example)
. . . . . . . . . . . . .
25
2.4
Magnitude of Phase Correction
. . . . . . . . . . . . . . . . . . . .
27
2.5
Relative Velocity Measurement of Vehicle Formation . . . . . . . . .
28
2.6
Magnitude of Doppler Correction Term . . . . . . . . . . . . . . . .
29
2.7
Centralized Formulation vs. Decentralized Formulation
. . . . . . .
48
2.8
Computational Load Savings (NAVSTAR-only System) . . . . . . .
49
2.9
Coupled Measurement Equation and Multiple Decoupled Equations
50
2.10 Full Centralized Extended Kalman Filter and Iterative Cascade Kalman
Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xiv
53
2.11 Computation Required for a Single Measurement Update (EKF and
ICEKF)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2.12 Ratio of the Computational Power Used in EKF and ICEKF Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
2.13 RMS Position Errors of 4 Different EKFs . . . . . . . . . . . . . . .
56
2.14 Information Flow Diagrams of Decentalized and Centralized System
Architectures
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
2.15 Maximum Data Rate Required in Centralized and Decentralized Systems
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
2.16 RMS Position Error Comparison (EKF and ICEKF) . . . . . . . . .
61
2.17 Effect of the Sampling Time Slew on the Code Phase Single Difference 64
2.18 Comparison of a Typical Convergence Pattern of Conventional Carrier
Smoothing Filter New Smoothing Filter . . . . . . . . . . . . . . . .
66
2.19 Single Base-Line Positioning Error of the Coarse Navigation Filter .
67
3.1
Simulation of Circular Initialization Maneuver (r=5 m) . . . . . . .
72
3.2
Horizontal Estimation Error During Initialization Maneuver
. . . .
73
3.3
Convergence Map of Previous Initialization Algorithm (r=5 m) . . .
74
3.4
Convergence Map of Previous Initialization Algorithm (r=100 m)
.
75
3.5
Error Contour 100m
. . . . . . . . . . . . . . . . . . . . . . . . . .
75
3.6
Typical Formation Geometry and NAVSTAR Satellite . . . . . . . .
78
3.7
Maximum Magnitude of Error Term
c2
2|X|
as Function of Initial Position
Uncertainty and Vehicle Separation . . . . . . . . . . . . . . . . . .
3.8
Performance of Robust Initialization Algorithm (Estimated Mean and
Variance of Error Term)
3.9
80
. . . . . . . . . . . . . . . . . . . . . . . .
85
Convergence Characteristic Comparison (Typical) . . . . . . . . . .
85
3.10 Convergence Characteristic Comparison (Extreme)
. . . . . . . . .
85
3.11 Convergence Map After 5 m Radius Circular Motion: Previous algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
3.12 Convergence Map After 5 m Radius Circular Motion: Robust algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
87
3.13 Convergence Map (control based on current estimate): Previous method
88
3.14 Convergence Map (control based on current estimate): Robust algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.15 Possible Initialization Paths
88
. . . . . . . . . . . . . . . . . . . . . .
90
3.16 Position Error as a Function of the LOS Change . . . . . . . . . . .
90
3.17 Position Error as a Function of the LOS Change (small initial error)
90
4.1
Combined Local and NAVSTAR Constellation . . . . . . . . . . . .
93
4.2
Number of Visible Satellites in LEO . . . . . . . . . . . . . . . . . .
94
4.3
Minimum and Maximum PDOP Values of Selected Subset of Satellites 95
4.4
Computational Load Required for Optimal Selection of Satellites . .
96
4.5
Cost Function of Quasi-optimal Algorithm
99
4.6
Optimal Selection Example . . . . . . . . . . . . . . . . . . . . . . . 102
4.7
Quasi-optimal Sequential Selection Example
4.8
Polar Plot Showing LOS to Satellites in View and Satellites Selected
. . . . . . . . . . . . . .
. . . . . . . . . . . . . 103
Using Optimal and Quasi-optimal Methods . . . . . . . . . . . . . . 105
4.9
All Possible PDOP values and PDOP of Quasi-optimal Method
4.10 Quasi-optimal Selection (worst case)
. . 105
. . . . . . . . . . . . . . . . . 106
4.11 PDOP Values of Quasi-optimal Selection (worst case) . . . . . . . . 106
4.12 Distribution of ζ. (Selection from Random Geometry) . . . . . . . . 107
4.13 Distribution of ζ when 4 Satellites are Selected. (LEO simulation) . 109
4.14 Distribution of ζ when 6 Satellites are Selected. (LEO simulation) . 110
4.15 Distribution of ζ when 6 Measurements are Selected. (LEO simulation, 5 Vehicle Formation)
. . . . . . . . . . . . . . . . . . . . . . . 111
4.16 Distribution of ζ when 6 Measurements are Selected. (Ground simulation)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.17 Computational Loads Required in Each Method . . . . . . . . . . . 113
5.1
Relative Motions in the Formation Frame . . . . . . . . . . . . . . . 119
5.2
Magnitude of Bias Errors of NAVSTAR measurements and Local
Ranging Measurements in LEO
xvi
. . . . . . . . . . . . . . . . . . . . 121
5.3
Magnitude of Entire Bias State Vector of NAVSTAR-only System and
Augmented System . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.4
Relative Position Error Comparison EKF vs. WLS
. . . . . . . . . 123
5.5
Relative Velocity Error Comparison EKF vs. WLS
. . . . . . . . . 123
5.6
Position Error without Correction (WLS) . . . . . . . . . . . . . . . 126
5.7
Velocity Error with Correction (WLS) . . . . . . . . . . . . . . . . . 126
5.8
Velocity Error without Correction (WLS) . . . . . . . . . . . . . . . 127
5.9
Velocity Error with Correction (WLS) . . . . . . . . . . . . . . . . . 127
5.10 Transmitter Locations on DC-10 Aircraft . . . . . . . . . . . . . . . 129
5.11 Relative Horizontal Motion of Receiver Aircraft
. . . . . . . . . . . 130
5.12 Relative Vertical Position . . . . . . . . . . . . . . . . . . . . . . . . 130
5.13 Line-of-sight Angle during Relative Motion . . . . . . . . . . . . . . 131
5.14 PDOP Values of Each Configuration During Relative Motion . . . . 131
5.15 RMS Position Errors during Refueling Maneuver . . . . . . . . . . . 132
5.16 Statistics of Position Estimation Error
. . . . . . . . . . . . . . . . 133
5.17 Statistics of Velocity Estimation Error
. . . . . . . . . . . . . . . . 133
5.18 In-Track Formation Configuration . . . . . . . . . . . . . . . . . . . 134
5.19 In-Track Formation with Small Cross-track Separation
. . . . . . . 134
5.20 RMS Position Error (In-track separation only) . . . . . . . . . . . . 135
5.21 RMS Position Error (In-track and small cross-track component)
. . 136
5.22 Ellipse Formation on Reference Orbit . . . . . . . . . . . . . . . . . 137
5.23 Double Ellipse Formation on Reference Orbit . . . . . . . . . . . . . 137
5.24 3-D Relative Motion on Single Ellipse . . . . . . . . . . . . . . . . . 138
5.25 Double Ellipse (180◦ Phase)
. . . . . . . . . . . . . . . . . . . . . . 138
5.26 Double Ellipse 90◦ Phase . . . . . . . . . . . . . . . . . . . . . . . . 139
5.27 RMS Position Error (Single ellipse formation)
. . . . . . . . . . . . 140
5.28 RMS Position Error (Double ellipse formation) . . . . . . . . . . . . 140
5.29 PowerSail with 4 Onboard Transmitters . . . . . . . . . . . . . . . . 142
5.30 PowerSail with 2 Onboard Transmitters . . . . . . . . . . . . . . . . 142
5.31 Typical Relative Motion of Host Spacecraft with respect to the Sail
(In-track difference) . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
xvii
5.32 Typical Relative Motion of Host Spacecraft with respect to the Sail
(Eccentricity difference)
. . . . . . . . . . . . . . . . . . . . . . . . 143
5.33 Number of Visible Satellites from Host Spacecraft
5.34 Number of Satellites Blocked by PowerSail
. . . . . . . . . . 144
. . . . . . . . . . . . . . 144
5.35 Maximum GDOP Values vs. Distance between the Host and the Sails 145
5.36 Number of Visible Satellites from Host Spacecraft in 2 Different Altitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.37 Maximum GDOP Values without Transmitters for 2 Different Orbits 147
5.38 Visibility comparison at LEO and MEO
. . . . . . . . . . . . . . . 148
5.39 Maximum GDOP Values at 2 Different Altitudes (4 transmitters are
onboard Sail)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.40 Maximum GDOP Values for Different Numbers of Transmitters Onboard Sail
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.41 Number of Commonly Visible Satellites . . . . . . . . . . . . . . . . 151
5.42 Number of Satellites Blocked by Sail
. . . . . . . . . . . . . . . . . 151
5.43 GDOP Values Using 4 Onboard Transmitters
5.44 Visibility comparison at MEO and GEO
. . . . . . . . . . . . 151
. . . . . . . . . . . . . . . 152
6.1
Formation Truck Testbed . . . . . . . . . . . . . . . . . . . . . . . . 155
6.2
RC Truck with Onboard Electronics . . . . . . . . . . . . . . . . . . 156
6.3
Formation of Outdoor Robots with SPTC
6.4
Pseudolite and Transmit Antenna . . . . . . . . . . . . . . . . . . . 158
6.5
Stanford Pseudolite Transceiver Crosslink (SPTC) . . . . . . . . . . 158
6.6
3 Truck Formation in Front of Durand Building
6.7
Demonstration of Autonomous Formation Intialization Maneuver of
. . . . . . . . . . . . . . 157
. . . . . . . . . . . 160
3 Trucks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.8
Initialization Experiment (Estimated Relative Trajectories around Truck 1)
162
6.9
Roof-top Testbed: Grid and Laser Pointer
. . . . . . . . . . . . . . 164
6.10 Roof-top Experiment: Estimated Absolute Position of 2 Vehicles . . 165
xviii
6.11 Roof-top Experiment: Estimated Position of Second Vehicle Relative
to Reference Vehicle
. . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.12 Roof-top Experiment: Estimation Error Convergence
. . . . . . . . 167
6.13 Roof-top Experiment: Number of Measurements Used in Initialization
Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
6.14 Formation Keeping Maneuver (RC Trucks) . . . . . . . . . . . . . . 168
6.15 Robot Fomation Keeping Maneuver . . . . . . . . . . . . . . . . . . 169
6.16 Validation Result (Open-loop Maneuver) . . . . . . . . . . . . . . . 171
6.17 Validation Result(Closed-loop Control Maneuver)
. . . . . . . . . . 171
7.1
Orion Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.2
UAV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
A.1
Double-board Attitude Receiver . . . . . . . . . . . . . . . . . . . . 181
A.2
Single-board Attitude Receiver . . . . . . . . . . . . . . . . . . . . . 182
A.3
Large angle motion of baseline in 2D
B.1
Laser Pointer and Baseline Mounted on Follower Robot . . . . . . . 192
B.2
Target Board Mounted on Leader Robot
B.3
Geometry of Laser Pointer Ranging Device . . . . . . . . . . . . . . 193
xix
. . . . . . . . . . . . . . . . . 184
. . . . . . . . . . . . . . . 192
Chapter 1
Introduction
This dissertation presents theoretical and experimental results on the use of the Global
Positioning System (GPS) and onboard local ranging devices as a relative navigation
sensor system for a formation of vehicles. Successful control of a formation will require
knowledge of the cluster’s absolute position as well as very precise estimates of the
relative positions and velocities of the vehicles. To this end, standard GPS sensing
techniques (e.g. pseudo-ranging [GPS96]) and advanced techniques, such as carrierphase differential GPS (CDGPS), could be used as part of the navigation sensor.
Although CDGPS techniques can provide a highly accurate navigation solution, there
are several important limitations in a relative navigation sensing system that is solely
based on the NAVSTAR constellation:
1. The availability of the NAVSTAR constellation is limited in some missions
(e.g., GEO, MEO, aerial-refueling, docking) where the NAVSTAR signals are
only partially available or are blocked by nearby vehicles or structures. This
can result in a very poor constellation geometry and poor sensing robustness.
2. The NAVSTAR constellation can provide highly accurate phase measurements,
but this accuracy is limited (∼ 2 cm) due to the relatively long wavelength (∼
19 cm). There are many planned future missions that require a more accurate
relative navigation capability.
1
CHAPTER 1. INTRODUCTION
2
3. Initializing cycle ambiguities in the NAVSTAR signal has been a major problem in the use of CDGPS techniques. This is expecially true for systems that
only use the L1 frequency of the NAVSTAR signal.
These fundamental limitations of NAVSTAR-only systems can be successfully overcome by using onboard ranging devices and related advanced algorithms that will be
presented in this thesis.
This thesis extends previous work and successfully demonstrates the augmentation of the NAVSTAR constellation in an outdoor experiment with 2–3 vehicles
(simulation results are presented for a larger number of vehicles). Combined with the
other extensions in this thesis that address long baseline corrections, satellite selection methods, and distributed estimation algorithms; this thesis presents a “complete
package” that is ready for implementation on-orbit. As part of this research, the
Stanford Pseudolite Transceiver Crosslink (SPTC) was developed along with several
ground-based testbeds to demonstrate the relative navigation approach for a formation. The SPTC consists of an onboard transmitter (pseudolite), a GPS receiver, and
a dedicated communication link. The primary ground testbed consists of 3 commercially available robots that can maneuver in formation and carry the SPTC.
1.1
Formation Flying Technology
Formation flying of multiple vehicles is a revolutionary technology for many future
space and earth science missions involving distributed measurements such as a sparse
aperture radar, earth mapping and stellar interferometry. This novel concept is
based on the idea that a monolithic structure can be distributed into several small,
inexpensive spacecraft that collectively form a virtual spacecraft bus of unlimited
scale, resulting in a significant improvement in the mission capability and flexibility [FB97, FB98, KL96, FB99, AD98]. For example, a separated interferometer can be
formed in space that has a virtual baseline of several kilometers [CAB98], but this
would be impractical to build with other existing technologies. Another example is
the separated spacecraft synthetic aperture radar (SAR) that can obtain an extremely
1.1. FORMATION FLYING TECHNOLOGY
3
Figure 1.1: AFRL Formation Flying Mission (TechSat-21) [AFR, AD98].
high resolution using distributed and coordinated measurements from multiple apertures onboard formation flying spacecraft [TS21] (see Figure 1.1).
1.1.1
Example Formation Flying Missions
Previous Missions
Several spacecraft formation flying missions have demonstrated the relative navigation capability using the code-based differential GPS. A GPS receiver was installed
on the OREFEUS-SPAS [FG99] satellite that was deployed from the space shuttle. A
second GPS receiver was mounted on the shuttle and raw GPS phase measurements
were collected by these two receivers. In this mission, 10-50 m relative positioning
accuracy and the meter/sec-level relative velocity accuracy were achieved by post
processing the data. Surrey Space Center [MU00] also demonstrated a formation
flying experiment of two spacecraft, SNAP-1 and Tsinghua-1, using GPS. These two
CHAPTER 1. INTRODUCTION
4
Figure 1.2: Autonomous Rendezvous and Docking Mission (ETS-VII) [IK99].
spacecraft carried GPS receivers and demonstrated meter-level positioning capabilities using the pseudoranging technique. However, these two spacecraft computed
their absolute positions independently, and relative positioning capability using differential GPS was not demonstrated. NASDA (National Space development Agency
of Japan) successfully performed the autonomous rendezvous and docking mission of
the ETS-VII [IK99] (Figure 1.2) using code-based differential GPS. They achieved
relative position and velocity errors of less than 10 m and 1 cm/sec, respectively.
However, the code-based differential GPS was only used in the coarse approach phase
due to its low accuracy. A laser radar and a proximity image sensor were used in the
final approach and docking phases.
The co-flight of the EO-1 with the Landsat-7 [FB97] demonstrated a coarse formation flying capability of two spacecraft separated by a large in-track distance. The
two spacecraft fly in formation so that the EO-1 imager is viewing Earth through the
same column of the air as the similar imager onboard the Landsat-7. Thus, the results obtained by the imager onboard the Landsat-7 can be validated by EO-1. This
requires formation flying accuracy of on the order of 10-100m, which is achievable
with the code based DGPS sensor or the pseudorange technique.
1.1. FORMATION FLYING TECHNOLOGY
Fig. 1.3: Nanosat Constellation Trailblazer
mission (ST-5) [ST5].
5
Fig. 1.4: Gravity Recovery and
Climate Experiment Mission [GR].
Planned Future Missions
The Air Force Research Laboratory (AFRL) is exploring the formation flying technology to perform a space mission called TechSat21 (Technology Satellite of the 21st
Century) [AFR, AD98], as shown in Figure 1.1. Several application scenarios were
initially under consideration for this mission. Among these, the space-based radar
mission for Ground Moving Target Indication was chosen as the focus of the initial design. The key to this concept is a cluster of microsatellites orbiting in close formation
with highly accurate relative position and velocity knowledge. Each spacecraft has a
radar element to coherently detect the return from its own radar transmitter and the
responses received from the signals of the other satellites. In this mode of operation,
the cluster of spacecraft form a large (but sparse) coherent array that provides a rich
set of independently sampled data. Since the accuracy requirements for this mission
exceed the capability of the GPS sensing system, an onboard inter-spacecraft ranging
device is required to augment the GPS sensing system. The device being developed
for this mission will be more accurate than the standard GPS signal and will also
provide a dedicated communication link between the spacecraft [RZ00].
CHAPTER 1. INTRODUCTION
6
Fig. 1.5: The StarLight Interferometry
mission (ST-3) [SL].
Fig. 1.6: Terrestrial Planet Finder mission (TPF) [TPF].
The Nanosat Constellation Trailblazer mission [ST5] (Figure 1.3) is one of the
advanced missions in NASA’s New Millennium Program (It is also known as Space
Technology 5). The mission will attempt to fly three miniature spacecraft in formation and study the magnetosphere of Earth. Each spacecraft is about 42 cm across
and 20 cm high, and weighs about 22 kg. One of the key technologies that ST-5 will
attempt to test is the formation flying capability and operation of multiple spacecraft
as a single system. A key device that will enable this technology is the Formation
Flying and Communications Instrument (AFFCI) from JPL [KL96]. This device is a
miniature spacecraft communications system that provides a capability to communicate between spacecraft and determine the relative positions of the spacecraft using
the GPS. This system, which is currently under development, will be used to realize
complete constellation autonomy.
The precision relative ranging capability between the formation flying spacecraft
is a fundamental issue in the gravity field mapping missions (Figure 1.4). Scheduled
to launch in November of 2001, Gravity Recovery and Climate Experiment (GRACE)
mission [GR] will accurately map variations in the Earth’s gravity field. The GRACE
mission will have two identical spacecraft flying in formation about 220 km apart in
a polar orbit (500 km altitude). The key technology required in this mission is to
use GPS and K-band crosslink ranging system to accurately determine the variations
1.1. FORMATION FLYING TECHNOLOGY
7
Table 1.1: Positioning Accuracy Levels Required for Several Missions
Missions
ST-3
TPF
GRACE
ST-5
TechSat21
Position Accuracy
0.01 m
0.01 m
50 m
∼100 m
0.001 m
in the relative range between spacecraft. These range measurements are essential to
accurately measure the gravity field of Earth since the variations in the gravity will
cause the relative position of the two spacecraft to change.
StarLight [SL] (Figure 1.5) is a technological pioneer for the Terrestrial Planet
Finder (TPF) mission [TPF]. One possible configuration for TPF is shown in Figure 1.6. This version of TPF is a formation-flying, infrared interferometer mission
that consists of four spacecraft with telescopes that send their collected light to a
fifth combiner spacecraft. TPF should be able to see Earth-sized planets around
other stars, and study them for the signs of life. The space-borne optical interferometry and formation flying technologies of StarLight [SL] are vital to the space
interferometry missions. The StarLight’s two small telescopes can achieve the resolution of a telescope mirror 125 m in diameter. The key technology issue in this
mission is precise formation control of two spacecraft. A new formation flying technology is required that will control the distance between the two spacecraft to within
less than 1 centimeter and the attitude angle to within 3 arc minutes. Since the
NAVSTAR signals are not available for these deep space missions, the Autonomous
Formation Flying (AFF) Sensor [KL96] (onboard ranging device) is under development that will enable this mission. Each spacecraft will carry the AFF transceivers
that will determine the relative positions of the spacecraft in a spherical coordinate
system. The measurements are predicted to be accurate to 2 cm in range and 1.5
arcminute in relative attitude. A laser metrology system will then be used to provide
fine attitude angle information (to the arcsecond level) between spacecraft during the
primary observation mode. Various levels of sensing precision are required for these
planned missions in accordance with the overall goals. These requirements for the
sample missions are listed in Table 1.1 [TC00].
CHAPTER 1. INTRODUCTION
8
Fig. 1.7: ISS Docking
Fig. 1.8: Aircraft Formation Flying
The spacecraft formation flying technology can also be applied to an autonomous
Space Station rendezvous and docking missions (Figure 1.7). Note that, in these
rendezvous missions and some of the terrestrial missions that involve close formation
flying maneuvers, GPS based formation flying sensor could be hindered by the bad
geometry due to signal blockage.
Terrestrial formation flying missions include air-to-air refueling (Figure 1.10) and
aircraft formation flying maneuvers (Figure 1.8). It could be extremely beneficial if
these operations can be automated, especially during poor weather and night operations. To automate these high-risk maneuvers, the navigation systems will require
very accurate and robust sensors. A CDGPS sensor is a possible candidate for this
system, but the slow initialization/re-initialization process of the NAVSTAR signal
can be a issue for these operations. Also, the NAVSTAR signals can be blocked by
nearby vehicles, causing a poor geometry. Further potential ground-based applications include formation maneuvers of farming vehicles (see Figure 1.9) [MOC97].
1.2
Motivation
The ability to accurately and robustly measure (in real-time) the relative position and
velocity between all the vehicles in the formation is vital to successfully accomplish
the formation flying missions. This accuracy limit will determine the navigation
1.2. MOTIVATION
Fig. 1.9: Farming Vehicle formation
9
Fig. 1.10: Air-to-air Refueling
precision of the formation which is a determining factor for most of the formation
flying missions (Table 1.1). Several researchers have shown that GPS is an accurate
relative navigation and attitude sensor for a formation [EO99, TC00, PB97, KL96].
In particular, differential carrier phase measurements of GPS (CDGPS) can obtain
centimeter-level relative position between the vehicles within the formation. However,
many future formation flying missions [AFR, TPF, SL] require extreme precision,
robustness, or operations in an environment that is poorly served by the NAVSTAR
constellation. Thus, it will be hard to meet all of these requirements with a GPS-only
navigation system.
An alternative approach is to equip the vehicles in the formation with onboard
devices that transmit RF ranging signals. This array of transmitters created by the
cluster of vehicles can form a local constellation that augments NAVSTAR constellation. By providing additional local range and Doppler measurements to the navigation solution, these onboard transmitters can assist the use of CDGPS for many
mission scenarios in which adequate visibility/geometry to the NAVSTAR constellation is not available [TC99] (for example, MEO and GEO formation flying missions). These RF ranging devices can be designed to provide far more accurate
measurements of the vehicle relative ranges and range rates than the traditional GPS
pseudolites. These highly accurate measurements, together with the improved geometry/availability, could then be combined with the CDGPS measurements to obtain a
superior overall estimate of the vehicle relative positions and velocities. These devices
CHAPTER 1. INTRODUCTION
10
should ultimately be able to provide sub-centimeter relative navigation accuracy for
a formation of vehicles.
There are several researchers that are currently developing ranging and communication devices that could be used for the CDGPS augmentation. For example,
the Applied Physics Laboratory (APL) is developing the Cross Link Transceiver
(CLT) [DG99]. This device will provide inter-spacecraft communications that will
support formation flying missions at a rate of 5-20 kbps. Its pseudorandom noise
code will also provide range measurements that will enhance relative navigation solution. The Autonomous Formation Flying (AFF) sensor [KL96] is being developed by
JPL as a part of the ST-3 mission (Section 1.1.1). It uses technology that is similar
to current GPS system, operating at 30 GHz with a code rate of 100 M Chips per second1 . NASA/GSFC and Stanford Telecommunications, Inc. are jointly developing a
Low Power Transceiver (LPT). The LPT combines an S-band 2-way communications
system with GPS navigation in a compact, low-power package, which will then be
ideal for small formation flying spacecraft with limited power. Star Ranger, which
is being developed by AeroAstro [RZ], will demonstrate ultra high precision ranging
capability (∼ 1 cm) with its Ku-band transceiver. The 128kbps cross link will provide
a data rate that is sufficient to support all of the information that must be exchanged
between the spacecraft.
Since all of these ranging devices are still under development, this thesis discusses
the Stanford Pseudolite Transceiver Crosslink (SPTC) that was developed and extensively tested in this dissertation. It consists of L1 pseudolite, attitude capable GPS
receiver, and modems. It is demonstrated in this dissertation that it can provide
2-5 cm relative position accuracy.
Note that the RF augmentation discussed above not only provides a communication crosslink and highly accurate ranging signals, but it also allows relative motion
between the vehicles in the formation to be used to initialize the cycle ambiguities
(bias in the differential carrier phase measurements) [BP96]. Initializing biases in the
differential carrier phase measurements is essential to achieve cm-level accuracy in
the position estimate. Without the local augmentation, the vehicles must wait for a
1
NAVSTAR C/A code rate is 1 M Chips per second.
1.2. MOTIVATION
11
Fig. 1.11: Power Sail [TM00] with Onboard Transmitters
line-of-sight change to the NAVSTAR constellation. However, in terrestrial formation
missions, this procedure can take more than ∼15 minutes [DL96]. As will be shown in
this thesis, with onboard transmitters, these biases can be initialized in a few seconds.
The augmentation is also very useful for many types of formation flying mission
where a significant portion of the NAVSTAR satellites could be blocked by nearby
vehicles in the formation. For example, Figure 1.11 shows a possible configuration of
the Power Sail mission [TM00], where the host vehicle flies in tight formation with
a large sail vehicle that generates high power. In this case, a large fraction of the
NAVSTAR constellation can be blocked by the sail, resulting in poor visibility for the
host vehicle. A similar situation can occur in the autonomous docking maneuver to
the Space Station (Figure 1.7) or air-to-air refueling maneuver using GPS. For these
missions, one of the vehicles can carry transmitters to compensate for NAVSTAR
occlusion and to improve the viewing geometry.
Section 1.1.1 showed many types of formation flying and related missions that
could benefit from using an onboard ranging device. These missions typically require
CHAPTER 1. INTRODUCTION
12
Table 1.2: Importance of Onboard Transmitter in Various Application Areas
Bias
Estimation
Geometry
Augmentation
High Performance
Transmitter
Occlusion
High
High
Medium
Low
LEO Formation
Flying
Low
Low
High
Low
MEO Formation
Flying
Low
Medium
High
Low
GEO Formation
Flying
Docking Maneuver,
Refueling,
Power Sail
Low
High
High
Low
High
Medium
Medium
High
General Terrestrial
Formation Flying,
Relative Navigation
the ranging device for one or more of the reasons described above. Table 1.2 summarizes the various application areas and the primary areas of concern where a local
constellation plays an important role.
1.3
Previous Work
GPS based navigation and control has been a major area of research at Stanford and
elsewhere for more than two decades. This research builds on a broad base of earlier
work. Some important extensions in this research are discussed in this section. Others
will be presented within the context of the relevant chapters. The related work can
be broken down into a three areas. The first is formation flying research, the second
is pseudolite augmentation, and the third is cycle ambiguity resolution research.
1.3. PREVIOUS WORK
1.3.1
13
Formation Flying Research
Zimmerman [KZ96] applied CDGPS sensing techniques to an indoor, experimental
testbed using two free-flying robots. These robots were able to maneuver around
on a granite table (3.5m x 2.7m) and an array of six pseudolites was placed around
the testbed. As part of this testbed, an overhead vision system was installed which
was used to (re)initialize the vehicle locations whenever the estimation failed due
to excessive signal loss. Zimmerman demonstrated that GPS was able to sense the
relative attitude and locations of the two vehicles in a frame where the locations of
pseudolites were fixed. However, the main limitation of the testbed was that the scale
of the maneuvers was limited due to the size of the granite table and the vision sensor
had to be frequently used to compensate for the noisy indoor environment.
The work on the 2D testbed used by Zimmerman was extended by Corazzini [TC98] to include an additional vehicle and onboard pseudolites. Similar to the
work done by Zimmerman, the vehicles are constrained to maneuver in 2 dimensions. Corazzini’s work [TC00] validated the feasibility of using a combination of
onboard pseudolites and remote transmitters. Most recently, Olsen [EO99] applied
GPS sensing techniques to a large-scale indoor experimental testbed using blimps
carrying a GPS receiver. He also developed an expandable modular GPS receiver
for this testbed that is being used in many other testbeds, including the testbed
described in this dissertation. His work demonstrated autonomous 3D formation flying maneuvers with the blimps. He also addressed issues such as the optimal path
planning algorithm for bias initialization. Although Olsen and Corazzini successfully
demonstrated the formation flying concept, their testbeds were confined to be indoors
where the NAVSTAR constellation is not available. Therefore, they were unable to
develop/demonstrate the initialization algorithms and their analyzes were limited to
the pure pseudolite systems where only one coordinate system is required, as opposed to the outdoor augmentation system where 2 distinct coordinate systems are
used. Also, appropriate orbital simulation/analysis and the constellation geometry
problems were not considered.
Binning investigated the use of GPS for relative satellite to satellite navigation
using both L1 , and L2 [PB97]. His work did not use experimental vehicles, but used a
CHAPTER 1. INTRODUCTION
14
GPS simulator to provide a “hardware-in-the-loop” demonstration. The concept of a
formation flying “simulator testbed” is currently being extended by GSFC [FDB01].
This testbed has multiple GPS simulators that allow formation flying space missions
to be simulated in hardware. Many of the issues related to transitioning from a
terrestrial application to a mission in low-earth-orbit (LEO) can be addressed and
tested prior to launching the spacecraft. However, these simulators are not capable
of generating the onboard range measurements that may be critical in many planned
formation flying missions.
1.3.2
Augmentation of GPS Constellation with Transmitters
Stone [JS99] developed an outdoor testbed augmented by pseudolites fixed on the
ground. This testbed simulated an open-pit mine environment where visibility to
NAVSTAR constellation is limited by high elevation mask angle. He successfully
demonstrated pseudolite aided CDGPS positioning of vehicles that carried a GPS
receiver. Corazzini [TC99], on the other hand, developed a self-constellation concept
where relative positioning is possible with only the transmitters onboard the vehicles
(also shown indoors). Concurrent to the research in this thesis, LeMaster [EL99]
has also applied a similar concept that is applicable to the problem of remote area
navigation. His testbed consists of a 4 transceiver (GPS receiver + pseudolite) array
(1 mobile and 3 stationary) and he does not use the NAVSTAR constellation. He
showed in his simulation that by moving one transceiver around other fixed transceivers [EL00], it is possible to initialize the cycle ambiguities using a simple batch
method. He also experimentally demonstrated limited relative positioning capabilities using his transceivers. However, only one of the transceivers in the testbed can
be mobile and real-time autonomous initialization has not demonstrated.
1.3.3
Cycle Ambiguity Estimation
Since it is required to have a good estimate of the integers in order to achieve centimeter level relative position accuracy, various integer ambiguity estimation algorithms
1.3. PREVIOUS WORK
15
have been developed by a number of projects. It is also a major issue in the formation
flying missions to successfully perform precise relative navigation.
Many researchers worked on the search methods based on the integer property
of the biases in the measurements. Although the search space can be reduced by
differential code phase measurements and other various algorithms, integrity of these
method was not sufficient to be used in practical high-integrity applications [BP96].
Goad [GC90] developed a filtering method that takes both the carrier and code phase
to estimate a better phase measurement. This method is also called “carrier smoothing”. In practice, however, this filter is susceptible to the large multipath components
in the code phase [MB96] and inherent long convergence time could be a problem for
many types of applications.
Loomis [PL89] and Hwang [PH90] independently showed that the satellite motion could provide the observability to resolve the integer ambiguity without using
an integer search algorithms. Lawrence [DL96] extended this idea and developed
a Kalman filtering scheme that uses the null space of measurement matrix to continuously update the estimate of the ambiguity states. Pervan developed a batch
algorithm for IBLS (Integrated Beacon Landing System) [BP96] and demonstrated
the real-time estimation of biases by moving the vehicles close to the transmitter fixed
on the ground. Although the idea that geometry change can provide observability
to the cycle ambiguity was no different than that of Loomis and Hwang, the biases
could be solved for in a matter of a few seconds as the aircraft quickly passes through
the “bubble”. For this reason, this concept was also used in the autonomous tractor
project [MOC97].
The Kalman filter method developed by Lawrence was adapted to the re-initialization
problem for a formation of vehicles by Corazzini [TC99]. She demonstrated rapid reinitialization of biases by continuously running a Kalman filter for the ambiguity
states. Although this algorithm worked well, it required a fairly good guess of the
bias state. Therefore, it could not be applied to the initialization problem where
good initial guess cannot be guaranteed. The source of this problem is geometrically
analyzed in this dissertation and a new, highly improved, algorithm is developed
that can initialize the biases without good initial estimates. This new algorithm
CHAPTER 1. INTRODUCTION
16
can replace both the batch initialization algorithm [BP96] and the re-initialization
algorithm [TC00].
Several other researchers [KZ00, JJ00] proposed a near instant initialization of
biases using multi-frequency receivers. Although these methods are feasible and very
attractive, those devices or multi-frequency ranging signals are not yet available in
most application areas.
1.4
Research Issues
At the beginning of this research effort, several key challenges were determined to
exist before an augmented GPS system could be robustly used to perform the state
estimation for formation flying vehicles:
1. Techniques were needed to solve for the carrier-phase integer ambiguities. Although many researchers developed initialization algorithms (Section 1.3.3), it
is very important to initialize the cycle ambiguities rapidly and robustly in
real-time since the formation of spacecraft is typically deployed from a single
platform and they form a very tight formation in the initial phase of operation.
Therefore a new real-time algorithm was needed for the formation flying vehicles which could be used in a real-time, autonomous initialization procedure and
prevent collision during the initial deployment phase and the tight formation
missions.
2. As the number of vehicles in the formation increases, computational and numerical issues can arise in the measurement equations if the local range measurements are to be used in the estimation. This is in contrast to the NAVSTARonly systems, which leads to a completely decoupled set of estimation problems.
For the augmented GPS system, new techniques were needed to distribute and
decentralize this computation across the formation.
3. Sensing geometry is also a very important factor to obtain highly accurate
estimates of the relative positions. In a typical scenario, a large number of
ranging signals (NAVSTAR + local ranging signals) will be available for the
1.5. CONTRIBUTIONS
17
spacecraft formation in LEO (e.g., ∼13 + local measurements). However, since
most GPS receivers can only track a limited number of signals at a time, it
is desirable to use the subset of the available measurements that forms the
best geometry. The optimal way of selecting a subset of measurements requires
extremely large amounts of computational power and this amount increases
sharply for large fleets. A fast, efficient satellite selection algorithm was required
since each vehicle pair in the formation may have to select a different set of
measurements due to the different geometries created by the local constellation.
4. Complete orbital simulations of formation flying spacecraft must be conducted
to evaluate the performance and limitation of various aspects of the augmented
GPS sensing system for the formation vehicles. For example, these simulations
were necessary to analyze the effect of formation geometry on the performance
of local transmitter.
1.5
Contributions
The goal of this research was to develop and demonstrate a comprehensive set of
sensing algorithms for a formation of vehicles using the Global Positioning System
augmented with a local constellation created by the transmitters onboard the vehicles
in the fleet. The contributions described in this thesis were made in the following
specific areas:
1. Developed and demonstrated relative position and velocity estimation algorithms for formation flying vehicles using onboard transmitters and NAVSTAR
constellation. An efficient decentralized algorithm was developed for a large
number of vehicles with the onboard transmitters. This algorithm can greatly
reduce the total computational load, and at the same time, distribute the load
over the vehicles. This algorithm includes a simple nonlinear orbit model for
the formation flying spacecraft scenario. It also incorporates correction terms
for the long baseline (>1 km) formation flying.
18
CHAPTER 1. INTRODUCTION
2. Developed a robust real-time cycle ambiguity estimation algorithm for formation flying vehicles using onboard transmitters. In contrast to the various batch
algorithms, the bias states can be improved in real-time as the vehicles undergo
relative motions which enables an autonomous initialization maneuver. With
the extensions provided in this thesis, it is demonstrated that large initial errors can be easily tolerated, having comparable convergence region to that of
the batch algorithms. The high-integrity and robustness of the algorithm are
demonstrated in extensive simulations, and these results greatly increase the
confidence in the autonomous initialization process.
3. Demonstrated autonomous formation initialization and multi-vehicle formationfollowing maneuvers for the first time. A series of experiments were performed
using outdoor robots and RC (radio controlled) trucks. The robust cycle ambiguity estimation algorithm and the state estimation algorithm were implemented in a real-time operating system. The sensing provided centimeter-level
relative positioning and precise attitude estimation for the vehicles in the formation after the autonomous initialization maneuver.
4. Developed a new, very efficient, constellation selection algorithm that provides
near-optimal viewing geometries at a fraction of the computational load. This
algorithm can be applied to a formation of vehicles to help select a near-optimal
subset of measurements from both the NAVSTAR and local constellations. This
algorithm requires a very small computational load (typically 2%) when compared to the optimal method, and thus it is feasible for real-time applications.
5. Developed a prototype transceiver system for the experimental demonstration
of advanced algorithms for the local ranging devices. It consists of a GPS
receiver, a pseudolite, and a radio modem. It provides the full communication,
ranging, and CDGPS capabilities that are being embedded in the future devices
(Section 1.2). An embedded algorithm was developed for the GPS receiver to
compute attitude at a rate of 5 Hz and to initialize attitude biases.
1.5. CONTRIBUTIONS
19
The research in these five areas addresses all of the key challenges that were
identified earlier, and the result is a “complete sensing package” that is ready for
implementation on-orbit in future rendezvous and formation flying missions.
Chapter 2
Formation State Estimation
This chapter discusses the state estimation problem for the general formation flying
problem. This includes a complete discussion of the two main issues associated with
linearizing the GPS measurement equations: nonlinearity in the NAVSTAR satellite
signal measurement, and nonlinearity in the local transmitter measurement. The
formation measurement equations are given for both the relative position and velocity
states, and a general solution to the estimation problem is presented. In addition, a
decentralized approch to the estimation problem is introduced. This new approach
to the formation estimation distributes the computation amongst the vehicles in the
fleet, which greatly improves the flexibility and robustness of the formation flying
system.
2.1
Coordinate Systems
Figure 2.1 shows the coordinate systems used for the relative navigation of a cluster of formation flying vehicles. In principle, there are two separate systems specified. The first system, denoted by the unit triad (X,Y ,Z) is used to indicate the
po sition of the formation in some absolute sense. This frame will generally be fixed
to the Earth (Earth-Centered-Earth-Fixed (ECEF) frame or Earth-Centered-Inertial
(ECI)). The vector XF ormation specifies the position of the formation in the inertial
frame. The “absolute” formation location will, in general, not be developed using
20
2.1. COORDINATE SYSTEMS
Zf
21
X1
XN
Yf
ZI
Xf
XFormation
YI
XII
Figure 2.1: Coordinate System for Formation
differential carrier-phase measurements, but instead one could use standard pseudoranging techniques. This will result in a relatively coarse position estimate (≈100 m
single-point position accuracy with SA on). This could be improved using Kalman
filtering techniques, and differential GPS (DGPS) corrections, such as those available
from WAAS [GPS96] or other beacons.
Figure 2.1 also indicates the coordinate system used to specify the relative vehicle
positions. This coordinate frame is attached to the cluster of vehicles, and is given
by the unit triad (Xf ,Yf ,Zf ). The formation state (relative position and velocity)
will be specified in this local reference frame. The formation state estimates used
to specify the relative vehicle locations will be derived from differential carrier-phase
measurements. The accuracy of the relative position measurements will generally be
better than 10 cm, depending on the geometric dilution of precision (GDOP) and the
number of visible satellites.
CHAPTER 2. FORMATION STATE ESTIMATION
22
2.2
Correction Terms for the Large Separations
Most relative navigation applications use a linear measurement equation which assumes that the far constellation approximation or planar wavefront approximation is
valid [BWP96-1] because the separation between the two receiving antennas is much
smaller than the distance from the users to the satellites. However, the nonlinear measurement model is also widely used in the indoor navigation problems [EO99] as well
as in several surveying applications [PS97]. In space, especially in higher orbits, this
planar wavefront assumption will typically be invalid because of the inherently large
separation between spacecraft and the relatively short distance from the NAVSTAR
satellites.
For close-in formation flying missions, it is natural to use the relative states and
the linearized measurement equations. This is the primary type of mission that we
are focused on. However, it would also be very beneficial to use the same approach as
the vehicle separations increase since the linear measurement model does not require
the iteration processes which are necessary in the nonlinear measurement model. The
problem is that the linearization error grows very rapidly with the separation distance,
so we must correct for this linearization error. In this section, the errors introduced
by the linearization are analyzed and methods to estimate and compensate for these
errors are presented.
2.2.1
Correction Terms for the Linear Relative Position Estimation
Consider the case in Figure 2.2 with antennas on separate vehicles in view of a single
transmitter (GPS satellite). The differential phase between antennas 1 and 2 from
the transmitter is given by
∆φ = |R1 | − |R2 | + β + ν
= |R1 | − |R1 − ∆X| + β + ν
where
(2.1)
2.2. CORRECTION TERMS FOR THE LARGE SEPARATIONS
R1
= position vector from vehicle 1 to the transmitter
R2
= position vector from vehicle 2 to the transmitter
23
∆X = relative position vector from vehicle 1 to vehicle 2
β
= integer cycle ambiguity
ν
= receiver noise
Define θ as the angle between the vector R1 and ∆X,
∆φ = |R1 | −
|R1 |2 + |∆X|2 − 2|R1 ||∆X| cos θ + β + ν
= |R1 | − |R1
|1 +
|∆X|2
|∆X|
−2
cos θ + β + ν
2
|R1 |
|R1 |
(2.2)
(2.3)
Eq. 2.3 can be expressed in the power series [HT97] since |∆X|/|R1 | < 1 is true for
most systems.

|∆X| 1 − cos2 θ
∆φ = |R1 | − |R1 | 1 − cos θ
+
|R1 |
2
|∆X|
|R1 |
2

+ ... + β + ν
(2.4)
Ignoring the second and higher order terms in Eq. 2.4,
∆φ ∼
= cos θ|∆X| + β + ν
(2.5)
The differential phase |R2 | − |R1 | can be approximated as a linear function of the
vector ∆X
|R2 | − |R1 | ∼
= cos θ|∆X|
R1 · ∆X
|∆X|
=
|R1 ||∆X|
= los1 · ∆X
where
los1 ≡
R1
|R1 |
(2.6)
(2.7)
(2.8)
(2.9)
CHAPTER 2. FORMATION STATE ESTIMATION
24
5
/26Â ∆;
α
/26
ε∆Φ
∆Φ
*366DWHOOLWH
/26
5
69
∆;
69
Figure 2.2: Relative Positioning of Formation Flying Spacecraft
This linear approximation is valid when the magnitude of the second order term
in Eq. 2.4 is small, or equivalently
|∆X|2
< Errormax
2 |R1 |
(2.10)
where Errormax is the maximum error magnitude that can be tolerated. In the CDGPS
system, this maximum error is usually the magnitude of the receiver carrier phase
noise [HT97] because this error will not substantially impact the estimate if it is
smaller than the phase noise. In the case of the LEO formation flying spacecraft
(R1 ≈ 2 × 107 m) with a relative distance of ≈10km, this linearization is clearly not
valid since
2
|1 × 104 |
= 2.5m Errormax
2 |2 × 107 |
(2.11)
2.2. CORRECTION TERMS FOR THE LARGE SEPARATIONS
25
SV2
20,000.000625 km
5 km
GPS
SV1
20,000 km
Planar wave front approximation error Æ 62.5 cm
Figure 2.3: Simple Example: Error Magnitude of Planar Wave Front Ap-
proximation
Simple Example
Figure 2.3 shows the effect of the planar wave front assumption on the long baseline
formation flying spacecraft with a simple geometric configuration. The two spacecraft
form a right-angle with the line-of-sight vector to the GPS satellite, thus the differential phase measurement would be zero if the planar wave front assumption had been
used. However, without the approximation, the true differential phase measurement
is 62.5 cm, which is almost two orders of magnitude larger than the carrier phase
noise level.
Solution
Figure 2.2 illustrates the geometry of the problem for a pair of vehicles with a NAVSTAR satellite. The figure shows both the differential carrier phase measurement and
the linear measurement model (los1 · ∆X) when the planar wavefront approximation
is used. The measurement error that occurs when the linear approximation is used is
defined to be ∆φ, which is given by
∆φ = los1 · ∆X − (|R1 | − |R2 |)
(2.12)
CHAPTER 2. FORMATION STATE ESTIMATION
26
By estimating the magnitude of ∆φ with an accuracy that is better than the magnitude of receiver phase noise, the linearized measurement equation can be used regardless of the distance between the spacecraft. The linearized equation can be written
as
∆φ + ∆φ = los1 · ∆X + β + ν
(2.13)
Note that ∆φ can be computed using the geometry between the user vehicles and
the NAVSTAR satellite. For example, from Figure 2.2, ∆φ is given by
∆φ = R2 (1 − cos(α))
= R2 (1 − los1 · los2 )
(2.14)
If this ∆φ term is not compensated for correctly, then it can introduce a significant
error for long baseline separations. However, it is possible to accurately estimate this
quantity throughout the orbit using the SPS position information (Eq. 2.14). Because
R2 can only be obtained from pseudoranging, an error in the estimate of R2 can be
as large as 100 m with SA on. However, since the magnitude of (1 − los1 · los2 ) is
very small for a reasonable separation between the spacecraft (i.e. ≤ 100km), the
magnitude of estimation error of ∆φ can be well under the phase noise level.
A simple orbit simulation1 was performed to verify the magnitude of the ∆φ and
the ability to estimate this term with the pseudorange solutions. The first plot in
Figure 2.4 shows the mean and maximum values2 of ∆φ observable as a function of
the distance between two spacecraft. The mean (maximum) value of the ∆φ is 5 cm
(10 cm) for 2 km baseline, which is significantly larger than the receiver noise level
(∼1 cm). In addition, the mean and maximum estimation errors of the ∆φ were
also computed and stored during the simulation. These values (maximum is 6 mm)
are well under the receiver noise level for the spacecraft separation upto 5 km. The
mean and maximum estimation error values are shown in the second plot of Figure 2.4
(note the very different scales in these two plots).
1
2
Details about the simulation procedures and the selected parameters are given in Chapter 5
During approximately one orbit (6000 sec)
2.2. CORRECTION TERMS FOR THE LARGE SEPARATIONS
27
Magnitude of Phase Correction Term
0.7
Mean
Max
Magnitude (m)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
Distance between spacecraft (km)
4
4.5
5
4
4.5
5
Estimation Error of Phase Correction Term
Magnitude (m)
0.02
0.015
0.01
0.005
0
0
0.5
1
1.5
2
2.5
3
3.5
Distance between spacecraft (km)
Figure 2.4: Maximum Magnitude of ∆φ and Maximum Estimation Error of
∆φ
2.2.2
Correction Terms for the Relative Velocity Estimation
A similar correction term is needed in the linear estimation of relative velocity. Figure 2.5 shows the velocity and line-of-sight vectors that determine the Doppler measurements. The differential Doppler measurement ∆φ̇ is given by
∆φ̇ = los2 · (V2 − VSV ) − los1 · (V1 − VSV )
(2.15)
Although this equation is linear, it is not a direct function of relative velocity state
(∆V = V2 − V1 ) that will be used in the formation estimation. Rewriting Eq. 2.15 in
terms of relative velocity state,
∆φ̇ = (los1 − los2 ) · VSV + los2 · V2 − los1 · V1
= los2 · (V2 − V1 ) + ∆φ̇
= los2 · ∆V + ∆φ̇
(2.16)
CHAPTER 2. FORMATION STATE ESTIMATION
28
Vs v
GPS S atellite
V1
LOS 1
LOS 2
V2
S V1
S V2
Figure 2.5: Relative Velocity Measurement of Formation Flying Spacecraft
where
∆φ̇ = (los2 − los1 ) · (V1 − VSV )
(2.17)
Here, VSV can be directly obtained from the ephemeris data, but the absolute velocity
vector V1 and the line-of-sight vectors must be computed using the standard pseudorange technique. Note that the magnitude of ∆φ̇ can be more significant than ∆φ
as the baseline increases. This is because the magnitude of the vector (los2 − los1 )
grows much faster than the value of (1 − los1 · los2 ) as the separation between the
vehicles increases.
Table 2.1: Simulated Doppler Correction Terms (2 km baseline)
Magnitude of ∆φ̇ (m/s)
mean
max
0.32
0.77
Estimation Errors of ∆φ̇ (mm/s)
mean
max
2.7
5.7
2.3. FORMATION MEASUREMENT EQUATION
29
Magnitude of Doppler Correction Term
2
Magnitude (m/s)
Mean
Max
1.5
1
0.5
0
0
1
2
3
Distance between spacecraft (km)
4
5
4
5
Estimation Error of Doppler Correction Term
0.01
Magnitude (m/s)
0.008
0.006
0.004
0.002
0
0
1
2
3
Distance between spacecraft (km)
Figure 2.6: Maximum Magnitude of ∆φ̇ and Maximum Estimation Error of
∆φ̇
Figure 2.6 shows the simulation results. It shows the mean and maximum values of
∆φ̇ and the mean and maximum estimation errors of ∆φ̇ when it is estimated from
the ephemeris and pseudorange information. Table 2.1 summarizes the magnitude of
the values for the 2 km baseline formation. It is shown that the estimation error of
the Doppler correction term is not a strong function of the distance. More simulation
results are presented in Chapter 5.
2.3
2.3.1
Formation Measurement Equation
Measurement Equations Using NAVSTAR Satellites
For the formation flying applications of interest in this thesis, the estimation of relative
position and velocity is performed using measurements from both the NAVSTAR
satellites and onboard transmitters (pseudolites). Attaching the formation coordinate
CHAPTER 2. FORMATION STATE ESTIMATION
30
system to a master vehicle (designated as vehicle m), the measurements from the
NAVSTAR constellation can then be written in vector form [EO99, TC00] as
 


  1 



 


  1 

X
i
 


 + β s + ν s mi
∆φs mi + Ri 2  .  − diag(Gm GTi ) = Gm 
mi
  . 

τi
  . 

 
1

(2.18)


∆φs mi + ∆φs mi = Gm 

Xi
τi
 + βs
mi
+ ν s mi
(2.19)
where
∆φs mi = differential carrier phase between vehicles m and i using the
NAVSTAR signals

1
 Ri
Ri
=







Ri2
..

Gi

.
Rin









1
 losi 1 


 los2 1 
i


=
.. 
 ..
. 
 .


losni 1

diag(A)= vector consists of diagonal entry of matrix A
Xi
= position of vehicle i relative to vehicle m
τi
= relative clock bias between receivers on vehicles m and i
β s mi
= carrier-phase biases for the single-differences between vehicles m
and i using the NAVSTAR signals
ν
s
mi
= carrier-phase noise for the single-differences between vehicles m
and i using the NAVSTAR signals
Ri is a diagonal matrix where the diagonal elements are the ranges from ith vehicle
to the NAVSTAR satellites 1, 2, . . . , n. Gi is the traditional geometry matrix. The
2.3. FORMATION MEASUREMENT EQUATION
31
components loski are the line-of-sights from the ith user vehicle to the k th NAVSTAR
satellite in the formation coordinate frame.
For an N-vehicle formation, these measurements are combined into one equation


 Gm

∆Φs + ∆Φs =







0 


Gm
..
0
.







Gm 



 
τ1 
 

.. 


+
.  
 

XN −1 
 

X1
τN −1
β s m1
β
s
m2
..
.
β s mN −1
= GX + β s + ν s
where

∆Φs =








∆φs m1
s
∆φ m2
..
.
∆φs mN −1





 + νs



(2.20)
(2.21)


















and ∆Φs =
∆φs m1
∆φ
..
.
s
m2
∆φs mN −1









and it is assumed that the master vehicle m had visibility to all available satellites
and all the vehicle tracks the same set of satellites. In general this may not be the
case, and G may have off-diagonal terms corresponding to the single differences that
can be made between vehicles using NAVSTAR signals not available on the master
vehicle. In addition, not all of the block diagonal entries will be Gm as shown in
Eq. 2.20. For example, if the k th GPS satellite was not visible on vehicle m, but was
visible on vehicles i and j, then the following single difference could be formed
∆φji + Rjk (1 − loski · loskj ) = loski (Xj − Xi ) + τj − τi + βijs + νijs
(2.22)
where Rjk is the range from the j th vehicle to the k th GPS satellite. This measurement
would be added to those using the master vehicle, and would appear in the off-diagonal
elements of G.
Doppler measurements using n NAVSTAR satellite signals can also be represented
in vector form. When differential Doppler measurements are formed between the
CHAPTER 2. FORMATION STATE ESTIMATION
32
master vehicle and vehicle i,

∆φ̇smi
−
∆φ̇smi
= Gi 

∆φ̇smi = (Gi − Gm ) 

Ẋi
τ̇i
Vm
τ̇m
 + ν̇ s
(2.23)
mi



 − diag (Gi
− Gm ) 
1
VSV
n
... VSV
1
τ̇SV
n
τ̇SV
...


(2.24)
where
∆φ̇smi
= differential carrier phase Doppler between vehicles m and i using
the NAVSTAR signals
Vm
= absolute velocity of vehicle m
k
VSV
= absolute velocity of NAVSTAR satellite k
Ẋi
= velocity of vehicle i relative to vehicle m
τ̇i
= relative clock drift rate between receivers on vehicles m and i
τ̇m
= clock drift rate of vehicle m
k
τ̇SV
= clock drift rate of NAVSTAR satellite k
s
ν̇mi
= carrier-phase Doppler noise for the single-differences between
vehicles m and i using the NAVSTAR signals
For an N-vehicle formation, these measurements are combined into one equation

∆Φ̇s + ∆Φ̇s =









G1










GN −1 
0
G2
..
0
= Gv Ẋ + ν̇ s
.


Ẋ1





 + ν̇ s


ẊN −1 


τ̇1
..
.
(2.25)
τ̇N −1
(2.26)
2.3. FORMATION MEASUREMENT EQUATION
where

∆Φ̇s =








∆φ̇sm1
∆φ̇sm2
..
.
∆φ̇smN −1


















and ∆Φ̇s =
33
−∆φ̇sm1
−∆φ̇sm2
..
.
−∆φ̇smN −1









As in the position equation, it is assumed that master vehicle m has visibility to
all available satellites and all of the vehicles track the same set of satellites. In general
this may not be the case, especially if the separation between the spacecraft is large,
and Gv may also have off-diagonal terms corresponding to the single differences that
can be made between vehicles using NAVSTAR signals not available on the master
vehicle. For example, if the k th GPS satellite was not visible on vehicle m, but was
visible on vehicles i and j, then the following Doppler single difference could be formed
k
∆φ̇ij + (loskj − loski )(Vi − VSV
) = loskj Ẋij + τ̇ij + ν̇ijs
Ẋij = Ẋj − Ẋi
(2.27)
(2.28)
This measurement would be added to those using the master vehicle and would appear
in the off-diagonal elements of Gv .
These two velocity and position equations for formation vehicles can also be combined into a single matrix equation


2.3.2
∆Φs
∆Φ̇s


+
∆Φs
∆Φ̇s



=

G
0
0 Gv


X
Ẋ

+
βs
0


+
νs
ν̇ s


(2.29)
Measurement Equation Using Local Transmitters
In addition to the signals generated by the NAVSTAR constellation, each vehicle in
the formation could have onboard transmitters or ranging devices that provide additional measurements. Because the local transmitters are very close, the full nonlinear
measurements equations must be used. The transmitter on vehicle m can be used to
CHAPTER 2. FORMATION STATE ESTIMATION
34
form ranging measurements that are given by
∆φp mi = dmi (Xi ) + τi + β p mi + ν p mi
(2.30)
where
∆φp mi = differential carrier phase between vehicles m and i using the
signal generated on vehicle m
dmi (Xi )= distance between vehicles m and i
τi
= relative clock bias (in meters) between receivers on vehicles m
and i,
β p mi
= carrier-phase biases for the single-difference between vehicles m
and i using the transmitter on vehicle m
ν p mi
= carrier-phase noise for the single-difference between vehicles m
and i using the transmitter on vehicle m
The signal generated on vehicle m is assumed to have been measured on the vehicle
itself. This can be accomplished by running the signal through a splitter to both the
transmit antenna and the receiver. Alternatively, the receiver antenna that receives
the NAVSTAR signals and/or the ranging signals from other vehicles can be used to
track the signal transmitted by itself if the antennas are oriented properly. The latter
method was used in the experimental testbed presented in Chapter 6.
The single difference formed between vehicles i and m, using the transmitter on
vehicle i can be written as
∆φp im = dim (Xi ) − τi + β p im + ν p im
(2.31)
Additionally, measurements can be formed between vehicles i and j (using the transmitter on vehicle i), exclusive of vehicle m. These are written as
∆φp ij = dij (Xi , Xj ) + τj − τi + β p ij + ν p ij
(2.32)
2.3. FORMATION MEASUREMENT EQUATION
35
In this case, the range and relative clock biases between the two vehicles are written
in terms of their positions and the clock biases relative to vehicle m. The single
difference made using the transmitter on vehicle j is derived by interchanging the
indexes in Eq. 2.32. These differences are formed between all of the vehicles in the
system in order to generate a maximal set of independent measurements. Note that
for an N-vehicle formation, there are a total of N × (N − 1) independent inter-vehicle
single differences that can be formed. If double differences were used, there would be
a total of N × (N − 1)/2 independent measurements with out the clock bias terms.
In addition, Doppler measurements from the local transmitters can be included
in the relative velocity measurement equation. For the transmitter on vehicle m,
p
∆φ̇pmi = d˙mi (Xi ) + τ̇i + ν̇mi
Xi
p
=
Ẋi + τ̇i + ν̇mi
dmi (Xi )
(2.33)
(2.34)
where
∆φ̇pmi
= differential carrier Doppler between vehicles m and i using the
signal generated on vehicle m
˙
dmi (Xi )= range rate between vehicle m and vehicle i
τ̇i
= relative clock drift (in m/sec) between the receivers on vehicles
m and i,
p
ν̇mi
= carrier Doppler noise for the single-difference between vehicles
m and i using the transmitter on vehicle m
Similar to the discussion above, measurements can be formed using the signal transmitted from vehicle i and that measurements can be formed between vehicles i and
j, exclusive of vehicle m.
The following sections discuss how these measurement equations are used to solve
for the formation state (relative position and velocities) and carrier phase ambiguities.
CHAPTER 2. FORMATION STATE ESTIMATION
36
2.4
State and Cycle Ambiguity Estimation
This section outlines the general state and bias estimation methods. A weighted least
squares method is used for the state estimation problem if a dynamic model of the
vehicle is not available. A simple orbit model is used within an extended Kalman
filter that is presented for the spacecraft problem. The cycle ambiguity estimation
process is coupled with the state estimation process and it often requires iteration
between the position states and the bias states. In this section, a simple extended
Kalman filtering method is presented that can be used with a reasonably good initial
guess. More advanced cycle ambiguity estimation algorithm and the related analyses
are given in Chapter 3
2.4.1
WLS State Estimation - EKF Bias Estimation
WLS State Estimation
The measurements from NAVSTAR satellites and the local transmitters can be combined to form a single weighted least squares (WLS) equation. Since the measurements from the local transmitter signals are nonlinear, the combined nonlinear measurement equation can be expressed as









∆Φs






+

p 

∆Φ 
 
∆Φ̇s 

∆Φs



∆Φ̇s 

∆Φ̇p



0


= h 



 + 




X
Ẋ
βs







+

p 

β 
 
0 




ν̇ s 

(2.35)

νp 

ν̇ p
0
0
νs
This equation can be linearized about the current estimate of the position and velocity
states








δ∆Φs


 
 

δ∆Φ̇ 
+



δ∆Φp 
 
s
δ∆Φ̇p
∆Φs
s
∆Φ̇
0
0













0

 ∂D(X) 
 ∂X X̂

∂ Ḋ(X) ∂X
G
=
X̂
0
Gv
0
∂ Ḋ(X) ˆ
∂ Ẋ
Ẋ












+




δX
δ Ẋ
βs








+


p 
β  

0 

0
νs



ν̇ s 


νp 

ν̇ p
(2.36)
2.4. STATE AND CYCLE AMBIGUITY ESTIMATION
where


h 

X

Ẋ
=

GX







37


Gv Ẋ 


D(X) 

Ḋ(X)

D(X) =









Ḋ(X) =









dm1 (X1 )
dm2 (X2 )
..
.
dN −2,N −1 (XN −1 , XN −2 )
d˙m1 (X1 )
d˙m2 (X2 )
..
.
d˙N −2,N −1 (XN −2 , XN −1 )




 are



the
N(N − 1)
local range measurements
2
the
N(N − 1)
local Doppler measurements
2





 are



and δ∆Φs = ∆Φs − GX̂, δ∆Φp = ∆Φp − D(X̂).
With a prior estimate of the bias states β̂ s and β̂ p , the position and velocity states
could be updated using a standard WLS approach




δX
δ Ẋ

= (H T W H)−1H T W







δ∆Φs






+

p 

δ∆Φ 
 

δ∆Φ̇s 

∆Φs






−
 
 

∆Φ̇s 

δ∆Φ̇p
0
0

β̂ s




0 



β̂ p 

(2.37)
0
where H is block measurement matrix in the Eq. 2.36.

H=

G



0

 ∂D(X) 
 ∂X X̂

∂ Ḋ(X) ∂X
X̂
0
Gv
0
∂ Ḋ(X) ˆ
∂ Ẋ
Ẋ








(2.38)
CHAPTER 2. FORMATION STATE ESTIMATION
38
The position and velocity states can then be updated as,

+

−


X̂
X̂
δX

 =
 +

ˆ
ˆ
δ Ẋ
Ẋ
Ẋ
(2.39)
˙
The position state X̂ and velocity state X̂ can be updated each time that new measurements are available or when the bias state has been updated using the approach
described below. Note that the position and velocity states are coupled in the local
transmitter measurements, which differs from the general CDGPS position/velocity
estimation schemes.
EKF Bias Estimation
The bias states in Eq. 2.35 need to be estimated and they can be updated in real
time using an Extended Kalman Filter (EKF). The bias state is observable through
the null space of the position measurement matrix [DL96, TC00]. It can be updated
recursively using the EKF. Rewriting the perturbed measurement Eq. 2.36 without
the relative velocity states,


δ∆Φs
δ∆Φp


+
∆Φs



=
0
G

∂D(X) ∂X
X̂

δX

+
βs
βp


+
νs
νp


(2.40)
Define L as the left null space matrix, i.e. the row vectors of L span the null space of
the measurement matrix

L = 
G
LN
∂D(X) ∂X
X̂

(2.41)
where (A)LN is defined as the left null space matrix of the matrix A. Multiplying
both sides of Eq. 2.40 by the matrix L yields a measurement equation for the bias
state since

z = L 
δ∆Φs
δ∆Φp


+
∆Φs
0



= L
βs
βp


+ L
νs
νp


(2.42)
2.4. STATE AND CYCLE AMBIGUITY ESTIMATION
39

A Kalman filter can then be used to update the bias state β̄ = 
βs
βp

.
Iteration
The bias state can be updated each time the relative position state X̂ is updated since
a new observation matrix
∂D(X) ∂X
X̂
can be formed. On the other hand, the position
update occurs each time new measurements are available or when the estimates of the
biases are updated. Therefore, an iterative method is used to solve for the position
state and the bias state. The estimate of β̄ at the k th iteration step (bias estimation
ˆ (k) , k = 0, 1, . . ..
using the k th state estimate X̂ (k) ) is defined as β̄

LN
G
L(k) = 
∂D(X) (k)
∂X
X̂

z (k) = L(k) 
δ∆Φs
δ∆Φp


(2.43)

+

∆Φs
0
(k)
(0)
(0)
= β̄ˆ + K (k) z̄ (k) − L(k) β̄ˆ
β̄ˆ

(2.44)
(2.45)
−1
K(k) = P (0) L(k)T L(k) P (0) L(k)T + R
(2.46)
I − K(k) L(k) P (0)
(2.47)
P (k) =
P (k) = error covariance matrix at k th iteration
K(k) = Kalman Filter gain matrix at k th iteration
where the covariance matrices are defined as
Q = E ww T =

0

R = E  ν sT ν pT 
νs
ν
p



=

ΣSatellite
0
0
ΣTransmitter

(2.48)
(2.49)
Since the bias state should be fairly constant over time, the state propagation matrix
should be an identity matrix and the process noise covariance Q can be assumed to
be zero. In the case where the bias states are variable (due to the line bias variation,
CHAPTER 2. FORMATION STATE ESTIMATION
40
multipath, phase delay), a non-zero covariance matrix can be used to estimate timevarying bias states.
2.4.2
EKF State Estimation - EKF Bias Estimation
Further improvement can be achieved in the position and velocity estimate accuracy
by incorporating the relative dynamics of the spacecraft into the filter. Since the
orbit propagation model is nonlinear in the states, an Extended Kalman Filter must
be used to estimate the position and velocity states. Therefore, in this estimation
scheme, there are two Extended Kalman Filters in the iteration cycle – one for the
state and one for the biases.
EKF State Estimation
In an inertial frame, the unperturbed motion of the spacecraft can be described as
r̈1 = −µ
r1
r13
(2.50)
where µ is the Earth graviational parameter. The other spacecraft in the nearby
orbit would have motions that may be influenced by perturbing forces w (e.g., solar
pressure, aerodynamic drag) and control inputs f (e.g., thruster force)
r̈2 = −µ
r2
+w+f
r23
(2.51)
For simplicity, it is assumed that the vehicles are under free motion (f = 0) in the
following development. The relative position is X = r2 − r1 , and since Ẍ = r̈2 − r̈1 ,
from Eqs. 2.50 and 2.51 [MHK76],
r3
µ
Ẍ = 3 r1 − 13 r2 + w
r1
r2
(2.52)
2.4. STATE AND CYCLE AMBIGUITY ESTIMATION
41
Eq. 2.52 can also be written in terms of the relative state X and the position of the
master vehicle r1
r2
r1 + X
=
3
r2
(r12 + 2r1 · X + X 2 )3
(2.53)
and substituting into Eq. 2.52 gives


r13 (r1 + X)
µ
+w
Ẍ = 3 r1 − r1
(r12 + 2r1 · X + X 2 )3
(2.54)
The continuous relative orbit dynamics are then defined as



Ẋ
Ẍ


= f c 

X
Ẋ

 , w
(2.55)
where the vector X contains position and clock states for all the vehicles in the
formation

X=












X1








XN −1 


τ1
..
.
(2.56)
τN −1
Note that the continuous function f c is linear in the disturbance inputs w, so it can
be expressed as a sum of linear and nonlinear terms



Ẋ
Ẍ


= F c 

X
Ẋ
 + Gw
(2.57)
CHAPTER 2. FORMATION STATE ESTIMATION
42
where


0


 .. 
 . 
G=











0
J
..
.





,






1 0 0







J=



0 1 0 


0 0 1 

(2.58)
0 0 0
J
Using the fourth-order Runge-Kutta method, a discrete orbit propagation model can
be formed from the continuous model



X


Ẋ
= Fnd 
n+1
 
X
  + G n wn
Ẋ
(2.59)
n
where Fnd is the fourth order Runge-Kutta form of the continuous function F c of
Eq. 2.55. The function Fnd can be written as

Fnd 
X
 

 
= 
Ẋ


X

Ẋ
n
n


X

Ẋ


X
Ẋ


X
Ẋ
∗


=
n+1/2
∗∗


n+1
Ẋ

=
n+1/2
∗∗∗

X

X
Ẋ

n

X
Ẋ



c
 + F 
+


1
∆tF c 
2

X
Ẋ
∗∗∗ 



X
Ẋ
∗




n+1/2
(2.60)
n+1
 
X
 
Ẋ
X
half step Euler
n
∗




Ẋ
half step backward Euler
n+1/2
∗∗
X
 + ∆tF c 

=

Ẋ
Ẋ
n

n+1/2
+ 12 ∆tF c 
n

X
n
∗∗


 
X
1 
+ ∆t F c 
6
Ẋ
+2F c 

where

  + 2F c 





n+1/2
full step midpoint rule
2.4. STATE AND CYCLE AMBIGUITY ESTIMATION
43
Following the standard derivation of the discrete equivalent for the noise covariance
matrices [GF90, AG74], we have
Gn G∆t
(2.61)
where ∆t is assumed to be small (typically, ∆t = 1 sec torbit = 5400 sec, which is
the time constant for relative motion in LEO). wn in Eq. 2.59 is the discrete version
of the process noise and the covariance matrix is defined as
T
Qn δnm = E[wn wm
]
(2.62)
The discrete process noise covariance matrix Qn has the following relationship with
the continuous counterpart
Gn Qn GTn GQGT ∆t
(2.63)
where Qδ(t − τ ) = E w(t)w(t − τ )T is the spectral density of the white process
noise. Using Eq. 2.61, the covariance matrix Qn of the discrete white noise can be
approximated as
Qn Q
∆t
(2.64)
To simplify the following discussion, define

χn = 

X

Ẋ
(2.65)
n
In order to use the extended Kalman filter algorithm, the nonlinear propagation model
should be linearized about the current estimate χ̂n|n
∂Fn (χ) Fn =
∂χ χ=χ̂
(2.66)
n|n
With the estimate of current state χ̂n|n , Eq. 2.59 can be approximated as
χn+1 = Fnd (χ̂n|n ) + Fn (χn − χ̂n|n ) + Gn wn
(2.67)
CHAPTER 2. FORMATION STATE ESTIMATION
44
= Fn χn + Fnd (χ̂n|n ) − Fn χ̂n|n +Gn wn
(2.68)
known
which is a linear propagation model for χn . Applying the Kalman filter equations [TK00,
AG74, AB99],
χ̂n+1|n = Fn χ̂n|n + Fnd (χ̂n|n ) − Fn χ̂n|n
(2.69)
= Fnd (χ̂n|n )
(2.70)
The EKF time update equations can be written as
χ̂n+1|n = Fnd (χ̂n|n )
(2.71)
Pn+1|n = Fn Pn|n FnT + Gn Qn GTn
(2.72)
The last term in Eq. 2.72 can be calculated from the continuous process noise covariance matrix Q using Eqs. 2.63 and 2.64.
The measurement update equations can be derived from Eq. 2.353 . Define

yn =







∆Φs







+

p 

∆Φ 


∆Φ̇s 

∆Φ̇p
n
∆Φs











∆Φ̇s 

0
0
,



β̄ =
n
βs











0 


βp 

,
ν̄n =
νs


ν̇ s 

(2.73)

νp 

ν̇ p
0

n
Rewriting Eq. 2.35 with these new definitions,
yn = h(χn ) + β̄ + ν̄
∂h(χ) Hn =
∂χ χ=χ̂n|n−1
(2.74)
(2.75)
The EKF measurement update equations are
χ̂n|n = χ̂n|n−1 + Kn yn − β̄ˆ n|n − hn (χ̂n|n−1)
3
Note that the measurement equation is also nonlinear.
!
(2.76)
2.4. STATE AND CYCLE AMBIGUITY ESTIMATION
Kn = Pn|n−1 HnT Hn Pn|n−1 HnT + Rn
45
−1
(2.77)
Pn|n = (I − Kn Hn ) Pn|n−1
(2.78)
where Rn is the covariance matrix of the discrete white sensing noise.
Ri δij = E[ν̄i ν̄jT ]
(2.79)
EKF State Estimation - EKF Bias Estimation
Since a measurement update of β̄ is available each time a new estimate of χ is available, the following iteration method can be used.
1. Update the bias state using the time updated position/velocity state χ̂n|n−1. Let
(0)
(0)
(0)
χ̂n|n−1 = χ̂n|n−1 and β̄ˆ n|n−1 = β̄ˆ n|n−1, Pn|n−1 = Pn|n−1 . This initial update generates
(0)
β̄ˆ , which can be done before the new measurements are available for the time step
n|n
n. The left null space matrix can be computed from Eq. 2.40


L(0)
= 
n

zn(0) = L(0)
n
=


∂D(X) (0)
∂X
χ̂

Kn(0)
LN
G
n|n−1
δ∆Φs
δ∆Φp
(0)
Pn|n−1 L(0)T
n
(2.80)



+
n
∆Φs
0
(0)
(0)T
L(0)
n Pn|n−1 Ln
 
 
n
(2.81)
−1
+R
(0)
(0)
ˆ (0)
β̄ˆ n|n = β̄ˆ n|n−1 + Kn(0) z̄n(0) − L(0)
n β̄ n|n−1
(2.82)
(2.83)
2. For k = 0, 1, 2, · · · , M an iteration on the position/velocity states and the bias
(0)
states can be done until the update to these states is very small. Let Pn|n−1 = Pn|n−1 ,
then
(0)
(0)
Kn(k) = Pn|n−1Hn(k)T Hn(k) Pn|n−1 Hn(k)T + Rn
−1
(k)
(k)
(0)
(0)
χ̂n|n = χ̂n|n−1 + Kn(k) yn − β̄ˆ n|n − hn (χ̂n|n−1)
!
(2.84)
(2.85)
CHAPTER 2. FORMATION STATE ESTIMATION
46

LN
G

∂D(X) (k)
∂X
χ̂

zn(k+1) = Ln(k+1) 
n|n
δ∆Φs
δ∆Φp
(0)
Pn|n−1L(0)T
n
Kn(k+1) =


Ln(k+1) = 
(2.86)



+
n
∆Φs
 
 
0
n
(0)
Ln(k+1) Pn|n−1 L(0)T
n
(2.87)
−1
+R
(k+1)
(0)
(0)
= β̄ˆ n|n−1 + Kn(k+1) z̄n(k+1) − Ln(k+1) β̄ˆ n|n−1
β̄ˆ n|n
(2.88)
(2.89)
3. At the end of the M steps the covariance matrices for the bias states and the
position and velocity states are updated as follows (Covariance matrices measurement
update)
(M )
Pn|n
=
(M +1)
=
Pn|n
(0)
I − Kn(M ) Hn(M ) Pn|n−1
(0)
+1)
I − Kn(M +1) L(M
Pn|n−1
n
(2.90)
(2.91)
4. The next step is to propagate the position and velocity states as well as the
covariance matrix (Time update)
(M )
(M )
χ̂n+1|n = Fnd (χ̂n|n )
(M )
(2.92)
(M )
Pn+1|n = Fn Pn|n FnT + Gn Qn GTn
(2.93)
5. The states and the covariances can be rewritten for the next update.
(M )
(2.94)
Pn+1|n = Pn+1|n
(M )
(2.95)
ˆ
ˆ (M +1)
β̄
n+1|n = β̄ n|n
(2.96)
(M +1)
Pn|n
(2.97)
χ̂n+1|n = χ̂n+1|n
Pn+1|n =
and the cycle repeats back to step #1. This algorithm is extensively demonstrated
in the LEO simulations in Chapter 5 and the EKF bias estimation algorithm is
2.5. DECENTRALIZED ITERATIVE CASCADE EKF
47
demonstrated in the experimental testbed (Chapter 6) as well as in the simulations
(Chapter 3).
2.5
Decentralized Iterative Cascade EKF
This section analyzes the numerical and computational issues in the EKF state estimation problem and presents a fast, numerically stable decentralized algorithm. A
key issue to be addressed in this section is that the EKF state estimation process
becomes highly cumbersome as the number of vehicles in the formation increases.
Thus it is very important to distribute this estimation process amongst the multiple
vehicles in the fleet (i.e. perform decentralized estimation). A decentralized approach
reduces the computational load per vehicle and provides a robust/flexible architecture that is less sensitive to a single vehicle failure. For example, with a centralized
solution approach, the master vehicle performs most of the calculations for the fleet,
so, unless all vehicles are designed with the same processor as the master vehicle,
the entire system could be susceptible to a single point failure. However, the decentralized approach does not require that one vehicle perform all of the calculations.
Instead, one spacecraft acts as the reference vehicle which serves as the formation
frame center. The estimation calculation is then distributed fairly evenly amongst
all of the vehicles in the formation. As such, the vehicles can be designed to be
identical in terms of their computational capability – and the CPU’s can be sized
to be much smaller. Note that if one vehicle failed, switching the reference vehicle
from one spacecraft to another would be a much simpler task than switching the
master of the centralized approach. Furthermore, the results in this section show
that, when compared to a centralized estimation scheme, the decentralized estimation process substantially reduces the computational load, eliminates many of the
numerical problems, and reduces the information flow between the vehicles.
CHAPTER 2. FORMATION STATE ESTIMATION
48
Centralized Formulation
y2
y3
G2
Decentralized Formulation
X2
G3
y2
G2
X2
y3
G3
X3
yN
GN
XN
X3
XN
yN
GN
Figure 2.7: Centralized Formulation and Decentralized Formulation
2.5.1
Iterative Cascade EKF
The sparse nature of the observation matrix H in Eq. 2.36 provides the basis for
improving estimation processes including reducing the processing time (and yields
better numerical stability). The observation matrix H in Eq. 2.36 is completely block
diagonal without the local range and range rate measurements (NAVSTAR-only system) if the master vehicle has visibility to all the satellites visible to entire formation.
This would be the normal case for a typical spacecraft that carries more than one
antenna4 . In this case, Eq. 2.36 can be divided into N-1 small independent filters
(N is number of vehicles) for each vehicle, and then easily distributed amongst the
vehicles in the formation. This distribution is possible because the relative dynamic
models are also decoupled. The process is illustrated in Figure 2.7.
This distribution of effort greatly reduces the computational load for the NAVSTARonly system, especially when the formation is quite large. Figure 2.8 shows computational advantage of the decentralized system as a function of the number of vehicles (x
axis). It is shown that the amount of computation required in the centralized update
is significantly higher than the total computation required in the decentralized system.
4
Mulitple antennas mounted on a spacecraft facing in different directions increase sky coverage [GL97, JCA99], but could lead to different vehicles using measurments from different satellites.
2.5. DECENTRALIZED ITERATIVE CASCADE EKF
49
Computational Load Comparison
8
10
Decentralized update (Total)
Decentralized update (Per vehicle)
Centralized update
7
Number of Floating Point Operation
10
6
10
5
10
4
10
3
10
2
4
6
8
10
Number of vehicles in Formation
12
14
16
Figure 2.8: Computational Load Comparison (Decentralized Process and
Centralized Process)
In addition, the computation for the decentralized process can be easily distributed
to the multiple vehicles as shown in Figure 2.7. As a result, the computational load
per vehicle is 2-3 orders of magnitude lower than that of the centralized system and
independent of the number of vehicles in the fleet.
Systems with Local Range Measurements
Given these advantages, it would be highly beneficial to distribute the estimation
processes when the local ranging measurements are available in addition to the NAVSTAR measurements. However, the local transmitter measurements from non-master
vehicles are coupled across the vehicles, and they appear in off-diagonal locations of
the measurement matrix. As will be shown, it is important to keep all local range
50
CHAPTER 2. FORMATION STATE ESTIMATION
Centralized formulation
Decentralized formulation
y2
G2
x2
Assuming x3
is given
y3
G3
Coupled Local
Range
Measurements
x3
Assuming x2
is given
Figure 2.9: Coupled Measurement Equation and Multiple Decoupled Equa-
tions of Augmented NAVSTAR System (compared to decoupled
equations of NAVSTAR system in Figure 2.7)
measurements to obtain the best possible estimate, but these off-diagonal measurements complicate the process of distributing the estimation. However, it is possible to
develop a decentralized estimation architecture using the following iterative scheme.
The left hand side diagram in Figure 2.9 is a linearized measurement equation for
a 3-vehicle formation combining all the measurements available in the system. This
equation can be divided into multiple small problems, as shown in Figure 2.9. The
coupled local measurements, which are a function of more than one vehicle state,
can also be divided using an approximation. In this decentralization process, it is
assumed that states for all the other vehicles (Xk , { k = 1, 2, . . . , N − 1 | k = i })
are given for the ith divided process (i = 1, 2, . . . , N − 1)
This decentralization process can be explained in detail by rearranging the previous measurement equations. Instead of stacking all the measurements for all the
vehicle, as in Eqs. 2.25 and 2.20, the measurements for a single vehicle are combined
with the local measurements available to that vehicle.
2.5. DECENTRALIZED ITERATIVE CASCADE EKF
51
For a single vehicle i








∆φsmi








+

p 

∆φi 
 
∆φ̇smi 

∆φsmi



∆φ̇smi 

0
∆φ̇pi





= h 



 + 




X
Ẋ
0








+

p 

βi 
 
0 

0
where


h 
s
βmi

X

Ẋ
=














s
νmi



s

ν̇mi

(2.98)

νip 

ν̇ip
 

Xi
 


τi

 


Ẋi
 

Gi 

τ̇i


Di (X) 


Gm 
Ḋi (X)
Di (X) = ranges between vehicle i and all other vehicles
Ḋi (X) = range rates between vehicle i and all other vehicles
∆φpi
= local range measurements between vehicle i and all other vehicles
∆φ̇pi
= local Doppler measurements between vehicle i and all other vehicles
This equation for ith vehicle can be linearized about the best current estimate of the
position and velocity states of other vehicles Xk and Ẋk { k = 1, 2, . . . , N −1 | k = i }








δ∆φsmi
δ∆φ̇smi
δ∆φpi
δ∆φ̇pi


 
 
 
+
 
 
 
∆φsmi
∆φ̇smi
0
0












0

 ∂Di (X) 
 ∂Xi X̂i

∂ Ḋi (X) ∂X
Gm
=
i
X̂i

0
Gi
0
∂ Ḋi (X) ˆ
∂ Ẋ
Ẋi








δXi








+
 

δ Ẋi 
 
τi 

τ̇i
s
βmi








+



βip 
 
0 

0
s
νmi



s 
ν̇mi


νip 

ν̇ip
(2.99)
Eq. 2.99 can be written in simple form for the nth measurement update step.
δ∆Φn,i + ∆Φn,i = Hn,i χn,i + βi + νi
(2.100)
CHAPTER 2. FORMATION STATE ESTIMATION
52
Note that Eq. 2.99 is the measurement equation for the ith vehicle, so there are only
N-1 local range measurements. Whenever there are new measurements available, the
position and velocity states for vehicles i = 1, 2, . . . , N-1 can be updated sequentially
using Eq. 2.99. After the ith vehicle update, the new estimates of Xi and Ẋi are
available to be used in the next sequence where states for (i+1)th vehicle are updated.
After updating states Xi and Ẋi for i = 1, 2, . . . , N-1, an iteration can be performed
until updates to the states are very small5 . The iteration is necessary because the
linearization process for the ith vehicle in Eq. 2.99 not only requires the estimate of
state Xi and Ẋi but also the estimated states of all the vehicles in the system Xk ’s
and Ẋk ’s (k = 1, 2, . . . , N-1).
In this sequential update scheme, the massive covariance matrix Pn|n−1 in Eq. 2.78
can be replaced by N-1 smaller covariance matrices corresponding to the states of
each vehicle. By updating multiple small covariance matrices (Pn|n,i) instead of the
large covariance matrix (Pn|n−1 ), the amount of computation can be reduced and in
addition, the reduced computational load can be distributed.
For i = 1, 2, . . . , N-1,
Pn|n,i = (I − Kn,i Hn,i)Pn|n−1,i
(2.101)
Pn|n−1,i = covariance matrix for the ith vehicle
Kn,i
= Kalman gain for the ith vehicle
Hn,i
= linearized measurement matrix for the ith vehicle in Eq. 2.99
Algorithm #1 summarizes the ICEKF algorithm in pseudo-code form. Figure 2.10
describes this process for a 4-vehicle formation. The arrows represent the range
measurements used in the estimation process. The diagram on the left describes the
traditional centralized EKF that simultaneously uses all of the local measurements in
one update process. In contrast, the diagram on the right shows the cascade EKF that
uses a subset of the local measurements in a sequential fashion. As shown, the update
process is split into three steps (vehicle 2 update, followed by the vehicle 3 update,
followed by the vehicle 4 update, and then the process is repeated). Only a fraction
5
Usually converges after 3-4 iterations.
2.5. DECENTRALIZED ITERATIVE CASCADE EKF
53
<ICEKF>
2
1
<Centralized EKF>
1
2
4
3
Veh2 Update
2
1
4
4
3
Veh2, 3, 4 Update
3
Veh3 Update
1
2
Local range measurement
Vehicle doing computation
4
Veh4 Update
3
Figure 2.10: Full Extended Kalman Filter and Iterative Cascade Kalman
Filter (One iteration cycle is shown in the diagram)
of the local measurements are used in each step, but all the available measurements
are used in the entire process. Because this method exploits the sparse nature of the
measurement equations, the total computational load required in the measurement
update can be greatly reduced. Moreover, computational load of the ICEKF method
can be distributed to the N-1 vehicles in the formation since both the measurement
update and covariance update process can be divided into N-1 small problems. This
can dramatically reduce the computational requirements of the onboard CPU for each
vehicle.
A series of orbit simulations were conducted to demonstrate the computational
efficiency of the decentralized ICEKF (Iterative Cascade Extended Kalman Filter).
Figure 2.11 compares the number of floating point operations (FLOPS) needed in the
single measurement update for various formation sizes. The results show the total
computation required in the centralized EKF, the total computation required in the
CHAPTER 2. FORMATION STATE ESTIMATION
54
Algorithm #1: ICEKF algorithm in Pseudo-code Form.
while update to χ̂n > tol
for i = 1, 2, . . . , N-1

Linearize measurement Eq. 2.98 for ith

X̂
vehicle using χ̂n =  ˆ 
Ẋ
n
Compute Kalman gain matrix Kn,i for ith vehicle
Update χ̂n,i
Update χ̂n
end
end
for i = 1, 2, . . . , N-1
Update Pn,i using the Kn,i and Hn,i used in the last state update
end
✷
ICEKF (3 iterations), and the computation required per each vehicle in the decentralized ICEKF (3 iterations). As expected, the centralized EKF requires the largest
amount of computation (2–10 times more than the total computation required for
the ICEKF). The computation required per each vehicle in the decentralized ICEKF
method is 1–2 orders of magnitude lower than the total required for the centralized
EKF. Figure 2.12 plots the ratio of the FLOPS required for the decentralized ICEKF
to the flops required for the centralized EKF. It is shown that the ICEKF method
requires only about 5% of the total computation power required in the centralized
EKF if there are 6 vehicles in the formation.
Ratio =
FICEKF /(N − 1)
× 100(%)
FEKF
(2.102)
where
FICEKF =Total computation required for the ICEKF method
FEKF
= Total computation required for the centralized EKF method
N
= Number of vehicle in the formation
2.5. DECENTRALIZED ITERATIVE CASCADE EKF
55
Floating Point Operations Required in Single Measurement Update
9
10
EKF
Iterated Cascade EKF
Iterated Cascade EKF (per vehicle)
8
Number of Floating Point Operation
10
7
10
6
10
5
10
4
10
2
4
6
8
10
12
Number of Vehicle in Formation
14
16
18
Figure 2.11: Computation Required for a Single Update (EKF and ICEKF)
- 3 iterations performed in ICEKF
50
45
(ICEKF flops)/(EKF flops)x100 (%)
40
35
30
25
20
15
10
5
0
2
4
6
8
10
12
Number of Vehicle in Formation
14
16
18
Figure 2.12: Ratio of the Computational Power Used (Decentralized
ICEKF)/(EKF) (%)
CHAPTER 2. FORMATION STATE ESTIMATION
56
Position Errors for 4 Different EKFs
9
8.5
Navstar Only
Navstar & Master PL
All Measurements, Cascade EKF
All Measurements, EKF
RMS Error (mm)
8
7.5
7
6.5
6
5.5
5
2
3
4
5
Vehicle ID
Figure 2.13: RMS Position Errors of 4 Different EKFs
Performance of the ICEKF
The performance of the ICEKF algorithm in terms of the estimation accuracy is
also vital since the algorithm is an approximation. Orbit simulations for a 5–vehicle
formation were performed to compare the steady state position errors of ICEKF
method with those of the centralized EKF. Figure 2.13 compares the RMS (Root
Mean Square) position errors of 4 different estimation schemes. The first (− ◦ −) is
the extended Kalman filter solution using only the NAVSTAR measurements and the
second (−✷−) is the EKF solution but it also uses local range measurements transmitted from the master vehicle (origin of the local coordinate). These two Kalman filters
have block diagonal measurement matrices and can be divided into 4 (N − 1) small
problems without using the approximation used in the ICEKF process. However, the
second algorithm only uses a subset of the total ranging measurements available to
the fleet. In this case, it uses 4 local measurements out of total 10 local measurement
2.5. DECENTRALIZED ITERATIVE CASCADE EKF
57
available. The third and fourth cases use all of the local range measurements and the
NAVSTAR measurements. The centralized EKF (−✸−) uses all the measurements
simultaneously, while the ICEKF (−−) finds the solution iteratively, using all the
measurements sequentially in the multiple cascade EKFs. Comparing cases 1 and
4 in Figure 2.13 shows the benefits of adding the local measurements. Comparing
cases 2 and 4 shows the penalty of ignoring some of the local ranging measurements.
Note that improvements achieved by adding more local range measurements are different from vehicle to vehicle. This is because each vehicle in the formation observes
different local constellation geometries. Figure 2.13 also shows that the centralized
EKF and the ICEKF using all the local measurements returned essentially an identical answer for relative positions of all the vehicles (vehicle ID = 2, 3, 4, 5). This
indicates that the approximations made in the iterative cascade EKF are valid and
the ICEKF algorithm can provide near optimal estimates with substantially reduced
computational requirements.
Information Flow
Another major advantage of the decentralized processing, besides the reduced processing load, is that this algorithm can also relieve the inter-vehicle communication load.
Most formation flying control systems require a communication link between the vehicles in order to exchange data for the relative navigation as well as other operations.
Since the bandwidth of the communication link is limited [RZ00], it is highly beneficial to minimize the amount of the navigation data transmitted. Information flow
diagrams for the centralized and decentralized systems are conceptually illustrated in
Figure 2.14 for a 4-vehicle formation. The major information includes GPS carrier
phase measurements, line-of-sight matrix and the minor information includes a single local transmitter measurement and relative position. The centralized estimation
method (Left) requires the master vehicle to collect all of the information from all of
the vehicles in the formation. This requires a large number of communication channels be available on the reference vehicle and an information bottleneck can occur at
the reference vehicle. On the other hand, the decentralized method (Right) allows
CHAPTER 2. FORMATION STATE ESTIMATION
58
Centralized Processing
Decentralized Processing
0%
100%
0%
0%
33%
33%
0%
33%
Major Data Flow
Minor Data Flow
Figure 2.14: Information Flows for 2 Architectures
one vehicle (usually the reference vehicle) to broadcast a data packet. All other vehicles receive this single data packet from the reference vehicle and exchange a few
minor pieces of data with the other vehicles. Thus the decentralized architecture has
a potential to eliminate the information bottlenecks.
To illustrate the amount of data exchanged in each method, various data packet
sizes for a typical formation system are listed in Table 2.2 and 2.3. Based on these
examples, the expected maximum data rate requirements are computed for a 1 Hz
measurement update. Figure 2.15 shows the required maximum data rate (bps) for
both the centralized and decentralized architectures. As shown, the decentralized
system can substantially reduce the amount of data packet transmitted in the system
and lower the inter-vehicle communication bandwidth requirements.
2.5. DECENTRALIZED ITERATIVE CASCADE EKF
59
Table 2.2: Major Data Flow
(n= number of NAVSTAR satellites)
Carrier Phase
Code Phase
Doppler
Line-of-sight
PRN
Total
4
7
Table 2.3: Minor Data Flow
Bytes
4×n
4×n
4×n
12×n
1×n
25×n
Local Range
Relative Position
Total
Bytes
4
12
16
Maximum Data Rate Required in Two Architectures (1Hz)
x 10
6
Centralized Processing
Decentralized Processing
bit/second (bps)
5
4
3
2
1
0
2
3
4
5
6
7
Number of Vehicles in System
8
9
10
Figure 2.15: Maximum Data Rate Required in 2 Communication Architectures
2.5.2
Numerical Stability of EKF
As the number of vehicles in the formation grows, the measurement update step
(Eqs. 2.77 and 2.78) becomes numerically less stable, especially when the measurement equations include highly redundant linearized local range measurement equations. In this case, there can be numerical difficulties ensuring that the covariance
CHAPTER 2. FORMATION STATE ESTIMATION
60
matrix remains positive-definite and symmetric when Eq. 2.78 is used for the covariance update [RS94, AG74]. The standard approach is to use the Joseph Stabilized
form of the covariance update to assure positive definiteness and symmetry
Pn|n = (I − Kn Hn ) Pn|n−1 (I − Kn Hn )T + Kn Rn KnT
(2.103)
This update is a symmetric equation that will remain positive-definite provided that
Pn|n−1 is positive-definite and Rn is positive-semidefinite. However, Eq. 2.103 requires
more computation than Eq. 2.78. In addition, the redundancy in the linearized measurement matrix Hn in Eq. 2.36 can be better handled by a different formulation of
Kn . Eq. 2.77 can be transformed into
Kn = Pn|n−1−1 + HnT Rn−1 Hn
−1
H T R−1
(2.104)
This filter gain computation also requires more computations than the previous
Eq. 2.77. These Eqs. 2.78 and 2.77 can substantially reduce numerical problems
at the cost of higher computational load. Alternatively, the square-root methods,
which propagate the square-root of the covariance matrix can be used in order to
ensure positive-definiteness of the covariance matrix [TK00].
Numerical Stability of ICEKF
Figure 2.16 compares the numerical stability of the EKF and ICEKF for large formations. As expected, the estimation errors for these two methods are effectively the
same for a small number of vehicles (Centralized EKF is slightly better since it is the
optimal method). However, as the number of vehicles in the formation increases, the
estimation errors of the regular centralized EKF grow and eventually diverge due to
the propagation of the numerical errors.
2.6. COARSE NAVIGATION SENSOR
61
RMS Position Errors Comparison
0.02
EKF
Iterated Cascade EKF
0.018
0.016
RMS Error (m)
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0
2
4
6
8
10
12
Number of Vehicles in Formation
14
16
18
Figure 2.16: RMS Position Error Comparison
2.6
Coarse Navigation Sensor
The differential measurements collected from the NAVSTAR constellation and onboard pseudolites should enable cm-level relative position estimates to be derived
once the carrier-phase biases have been resolved to sufficient accuracy. However,
when the receivers onboard each vehicle first acquire the signals (from both the GPS
constellation as well as the pseudolites mounted on the other vehicles), only a coarse
(meter-level) relative position estimate is available. This initial estimate is derived by
utilizing the measured pseudorange on each vehicle and forming a differential codephase measurement [GPS96]. As this is the only information available at start-up,
the carrier-phase biases have a large uncertainty. Depending on the initial separation
between the vehicles, it may not be prudent to immediately begin maneuvers. This
is especially true if the vehicles are only a few meters apart, as the relative position
estimates derived from the single-point differential code-phase measurement may be
CHAPTER 2. FORMATION STATE ESTIMATION
62
in error by a similar magnitude, depending on the Position Dilution of Precision
(PDOP) [GPS96].
An improved estimate of the carrier-phase biases (and therefore relative position),
utilizing only the signals from the NAVSTAR satellites, can be derived by filtering the
GPS measurements over several minutes. Once the coarse relative position estimates
are sufficiently accurate, the vehicles could then begin undergoing relative motion,
which will enable the biases to be quickly resolved to the desired accuracy.
This section derives the equations for a coarse navigation Kalman Filter that uses
all the available information from the GPS measurements to estimate the carrier-phase
biases. This filter utilizes both the carrier and code phase measurements, as well as
incorporating the information that can be derived from the NAVSTAR constellation
geometry.
The measurements on each vehicle should to be sampled simultaneously in order to
compute the relative position state at any particular time. In order to accomplish this,
the receiver software was modified to slew the hardware sample time at 1 Hz based on
its estimate of true time. These modifications were designed to keep the sample time
as closely aligned to the UTC second as possible. This enables an accurate differential
carrier-phase measurement to be made by simply differencing the measured carrierphase on each vehicle. However, the impact of the software control must be accounted
for if a direct comparison of the differential code and carrier are to be made, since
the hardware slew impacts the clock bias of each of those measurements in a different
way.
The impact of the slew can be explicitly accounted for if an accurate estimate of the
receiver clock frequency is available. However, a similar approach is to incorporate
another state into the estimation to account for the perturbations in the relative
clock bias, τe , between the differential code and carrier measurements. In this case,
the differential measurements can be written as
s
+ τe
∆φscode = ρ + νcode
(2.105)
s
∆φscarrier = ρ + β s + νcarrier
(2.106)
2.6. COARSE NAVIGATION SENSOR
63
where
∆φscode = differential code phase between vehicles
∆φscarrier= differential carrier phase between vehicles
ρ
= single difference range including receiver clock bias
τe
= perturbation to clock bias due to the software control of the
sampling time slew
Figure 2.17 shows the effect of the sampling time slew on the measured single difference code phase (dotted line) and the calibrated single difference code phase (solid
line). The calibrated measurement can be formed by accounting for the estimated
time slew on the raw single difference measurement. However, visible in the plot
is a small drift between the differential carrier-phase and the calibrated differential
code-phase. This is a result of the receiver clock frequency error, but is accounted for
in the estimation by incorporating the additional state, τe .
The information contained in the NAVSTAR constellation can also be incorporated into the estimation [DL96]. Similar to the previous Section 2.4.1, pre-multiply
equation 2.106 by L, where the rows of L form an orthonormal basis for the left null
space of the geometry matrix, G, eliminates ρ to give
s
L∆φscarrier = Lβ s + Lνcarrier
≡ z
(2.107)
The pre-multiplication by L eliminates the position and clock state from the equation 2.106, forming a new measurement. All of the available measurements can be
combined to form









∆φscode 





z
∆φscarrier
=
0 L 0 0
I
0 1 0
I I
0 0









ρ



βs 

 + ν̄
τe 

τ̇e
(2.108)
CHAPTER 2. FORMATION STATE ESTIMATION
64
250
Code phase single difference
Carrier phase single difference
Calibrated code phase single difference
single difference range (m)
200
150
100
50
0
−50
0
1
2
3
4
5
time (sec)
6
7
8
9
10
Figure 2.17: Effect of the Sampling Time Slew on the Code Phase Single
Difference




ν̄ = 




s
Lνcarrier
s
νcode
s
νcarrier





(2.109)
LΣcarrier LT
0
LΣcarrier
0
Σcode
0
Σcarrier LT
0
Σcarrier
R = E ν̄ ν̄ T = 





(2.110)
The time update equation can be written as








ρ











βs 


τe 

τ̇e
n
=

0
I
1 δt
0 0







ρ



βs 


τe 

τ̇e
n−1
+w
(2.111)
2.6. COARSE NAVIGATION SENSOR
65

Q = E ww T =








∞







0
0
(2.112)
στ̇e
where
z
= geometric measurement L∆φscarrier
τ̇e
= time rate of change of τe
δt
= time update interval
In terms of the convergence rate, the estimation approach described here has advantages over the conventional carrier smoothing technique that is widely used [GC90]
since there are additional measurements (Eq. 2.108) that are derived from the NAVSTAR geometry. Figure 2.18 shows the results of a simulation comparing the horizontal
estimation error between two vehicles using both the conventional smoothing filter
and the filter described in this section. As shown, both filters converge, but the new
filter performs significantly better.
Figure 2.19 displays the results from a single-baseline relative positioning experiment. In this experiment, a single GPS antenna was routed to two different receivers,
and the Kalman Filter described in the section was updated at 5 Hz in real-time.
The estimates of the carrier-phase biases were then used to generate a least squares
solution of the relative position. Note that the position estimates converged to an
accuracy of 10-20 cm in about 200 seconds.
This coarse navigation scheme can be used at start-up when carrier phase biases
are not resolved. This algorithm improves the estimate of the bias and position
states without requiring any relative motions. The initial position and bias estimates
obtained from this algorithm will then be used in the initialization process involving
relative motions between the vehicles (Chapter 3).
CHAPTER 2. FORMATION STATE ESTIMATION
66
2.5
Conventional Smoothing Filter
New Smoothing Filter
Horizontal Error (m)
2
1.5
1
0.5
0
0
50
100
150
time (sec)
200
250
300
Figure 2.18: Comparison of a Typical Convergence Pattern of Conventional
Carrier Smoothing Filter with New Smoothing Filter
2.7
Summary
This chapter presents the augmented CDGPS relative position and velocity estimators. Both extended Kalman filter and weighted least squares algorithms are used
to estimate these states. A second EKF for the bias states is introduced, and this
can be used iteratively with the EKF for the position and velocity states. A decentralized ICEKF algorithm is given to address the practical issues associated with
implementing these algorithms with large fleets of vehicles that have both CDGPS
and local ranging devices. When compared to the traditional EKF method, this
ICEKF algorithm is shown to:
1. Achieve a more robust and efficient formation system.
2. Reduce and distribute the computational load associated with the estimation
process.
2.7. SUMMARY
67
Position Errror
1
East(m)
0.5
0
−0.5
−1
0
200
400
600
800
1000
1200
1400
800
1000
1200
1400
800
1000
1200
1400
time (sec)
North (m)
1
0.5
0
−0.5
−1
0
200
400
600
time (sec)
1
Up (m)
0.5
0
−0.5
−1
0
200
400
600
time (sec)
Figure 2.19: Single Base-Line Positioning Error of the Coarse Navigation Filter
3. Reduce the inter-vehicle communication requirements.
4. Eliminate numerical problems typically experienced with the much larger centralized EKF.
The linear phase and Doppler measurement equations were also modified to account
for the nonlinear effects of the long baseline formations. In addition, a coarse navigation sensing scheme is presented that can be used in the initial stage of operation
when the carrier phase bias estimates are not available. This algorithm also addresses
effect of the independent sample clock slew of the receivers.
Chapter 3
Robust Bias Estimation Algorithm
This chapter presents a robust cycle ambiguity estimation algorithm for formation
flying vehicles. A real-time filtering approach was used to estimate the biases. Before
the cycle ambiguities have been initialized, the relative position error can be large and
there can be a large uncertainty in the line-of-sight vector from a local transmitter on
one vehicle to a receiver on a second vehicle. These initial line-of-sight errors can cause
a large bias error in the filtering result. This chapter presents a new algorithm that
significantly improves the convergence of the bias estimate even with these potentially
large initial state errors. With this new development, good estimates of the biases are
available in real-time, which allows the vehicles to undergo small-scale initialization
maneuvers. The real-time feature and the robustness of the new algorithm enables
the autonomous initialization procedure.
Recently, O’Connor [MOC97] and Pervan [BP96] demonstrated several manual
initialization procedures around the local pseudolites with their farming vehicle (Tractor) and aircraft, respectively. They performed a manual pseudolite “bubble1 pass”
and ran a batch algorithm at the bubble exit to resolve the biases. In order to robustly
automate this process, this chapter shows how the batch algorithm can be replaced
by an extended Kalman filter that updates the bias states in real-time.
1
A bubble is defined by the region in which the realtively weak pseudolite signal can be received.
68
3.1. CYCLE AMBIGUITY ESTIMATION
3.1
69
Cycle Ambiguity Estimation
In the CDGPS system, it is required to have a sufficiently accurate estimate of the
bias states in order to use Eq. 2.36 to estimate the position states. These bias states
are observable if there are relative motions (line-of-sight changes) between the vehicles
with the onboard transmitters [DL96, BP96, MOC97, CP00, CP01].
One way to estimate the bias states is to apply a batch algorithm [BP96,MOC97]
to the phase measurements after a relative motion around a local transmitter. In
this method, the vehicles undergo an initialization maneuver (relative motion around
a transmitter) using rough position estimates based on the differential code phase
measurements. During the maneuver, measurements are collected from both the GPS
satellites and the onboard transmitters. After the initialization maneuver, an iterated
batch algorithm is used to obtain the optimal estimate of the carrier phase biases.
Pervan [BP96] describes several batch methods that were used in an aircraft precision
landing system. The system consisted of two fixed pseudolites and an aircraft with
a GPS receiver. O’Connor applied similar methods to his farming vehicle [MOC97].
His system, however, consisted of a single pseudolite fixed on the ground and a tractor
was driven around the pseudolite manually.
While these batch methods are effective, they are not able to provide real-time
estimates of the biases as the vehicle undergoes the initialization maneuver. On the
other hand, an extended Kalman filter can be implemented to update the carrierphase bias and relative position estimates while the vehicles are maneuvering. If
these real-time estimates are reasonably accurate, then they should allow the control
system to autonomously keep the vehicles close to the precomputed optimal [EO99]
initialization trajectory. However, due to the nonlinearity in the local transmitter
measurements, this filter is very sensitive to the initial position errors (initial lineof-sight error) and thus the final position estimate accuracy can be quite poor. One
solution to this problem is to use a large-scale initialization maneuver, which results in
a smaller line-of-sight error for given position error. That approach tends to improve
the convergence of the real-time filter, but it also results in a substantial fuel cost and
it takes much longer to perform the initialization maneuver. This chapter presents an
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
70
update to the initialization algorithm that significantly reduces the sensitivity of the
convergence to the initial line-of-sight errors. As a result, the scale of the initialization
maneuvers can be much smaller, which greatly reduces the fuel/time requirements.
This robust real-time algorithm can be used as part of an autonomous initialization
maneuver which is valuable in many mission scenarios.
3.1.1
Previous Initialization Algorithm
The previous bias estimation algorithms [CP00, TC00, DL96], which used the null
space of the state measurement matrix to observe the bias states, assumed that the
current position estimate from the code based measurements was sufficiently accurate
that the first-order approximation of the nonlinear local measurement equation is
valid. The results in this chapter show that this assumption is not valid for many
cases of interest. Note that the resulting line-of-sight error tends not to be an issue
for batch algorithms that are solved iteratively, but they can be crucial for real-time
algorithms that must be solved sequentially.
The null-space algorithms can be summarized as follows. The bias state is observable through the null space of the position measurement matrix. It can then be
updated using an extended Kalman filter (EKF). Rewriting the perturbed measurement Eq. 2.36 without the relative velocity states


δ∆Φs
δ∆Φp


+
∆Φs
0



=
G

∂D(X) ∂X
X̂

δX

+
βs
βp


+
νs
νp


(3.1)
We define L as the left null space (row vectors of matrix L span the orthonormal null
space of the measurement matrix)

L = 
G
LN
∂D(X) ∂X
X̂

(3.2)
3.1. CYCLE AMBIGUITY ESTIMATION
71
where ALN is the left null space of the matrix A. Multiplying both sides of Eq. 3.1
by L yields the measurement equation for the bias state

z = L 
δ∆Φs
δ∆Φp


+
∆Φs



= L
0
βs
βp


+ L

A Kalman filter is then used to update the bias state β̄ = 
νs
νp
βs
βp


(3.3)


using the noise
covariance matrix
 
R = E L 
νs
ν
p


νs νp


LT  = 

ΣSatellite
0
0
ΣTransmitter

(3.4)
Sample Problem
A sample problem was selected to demonstrate the performance of the filter described
above. Figure 3.1 shows the simulated trajectory of one vehicle relative to a reference
vehicle which was stationary during the maneuver and carried an onboard pseudolite.
The plot is given in an ENU (East North Up) frame centered on the reference vehicle.
The intended trajectory of the mobile vehicle was a 5 m radius counter-clockwise
circle (shown as the dotted line in Figure 3.1) around the reference. The 5 m circular maneuver was chosen since similar scale maneuvers are commonly used in the
land vehicle initialization maneuvers [MOC97] in order to observe rapid line-of-sight
changes within a relatively short period. This distance is smaller than would be used
for general spacecraft formation maneuvers, but the formation of spacecraft is typically deployed from a single platform and they form a very tight group during the
initial phase of operations. Thus, a small scale initialization procedure (5∼100 m)
would be very beneficial for these vehicles. In addition, a radius of 5 m or less could
be an ideal initialization/operational radius for tight station-keeping missions such as
the AERcam [DSS92]. A detailed discussion about the selection of the initialization
path is presented in Section 3.1.4.
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
72
True position
Estimated position
5 m circle
5
4
3
2
North (m)
1
Pseudolite
0
−1
Pseudolite
Turned on
Circular motion
started
−2
−3
−4
−5
−4
−2
0
2
East (m)
4
6
8
Fig. 3.1: Convergence During the 5 m Circular Radius Initialization Maneuver
About the Local Transmitter (Vehicle control is based on current estimate of vehicle
position)
At the start of the maneuver, the mobile vehicle was placed 5 m East of the
reference but this relative position information was not given to the estimator. The
vehicle then estimated its position using the coarse estimator described in Chapter 2.
After the position estimate converged to a sufficient accuracy (1∼2 m), the vehicle
was commanded to perform an autonomous circular maneuver (5 m radius) around
the reference vehicle. Note that the vehicle control was entirely based on the current
estimate of its position. Once the maneuver had started, the pseudolite measurement
was incorporated into the estimation using the extended Kalman filtering algorithm
(Eq. 3.3). The dashed and solid lines in the figure show the true and estimated
position of the mobile vehicle during the maneuver, respectively. Note the estimated
3.1. CYCLE AMBIGUITY ESTIMATION
73
north position estimation error (m)
east position estimation error (m)
Estimation errors
6
Static
4
Initialization maneuver
2
0
−2
−4
0
20
40
60
80
100
time (sec)
120
140
160
180
0
20
40
60
80
100
time (sec)
120
140
160
180
1.5
1
0.5
0
−0.5
Fig. 3.2: Horizontal Estimation Error During the Initialization Maneuver
position rapidly converges to the true position as the line-of-sight from the mobile
vehicle to the reference vehicle changes.
Figure 3.2 shows the estimation error in the East and North directions during
the maneuver shown in Figure 3.1. The position estimates converge to 1–2 cm accuracy in each direction after a complete, 360◦ circular motion about the reference
vehicle with a reasonably good initial position estimate. However, the final position
estimates are typically not as accurate as those from the batch methods [BP96], depending on the initial position error from the coarse position estimator. This is a
result of the pseudolite measurements being highly nonlinear and that the linearized
measurements (Eq. 3.1) can introduce large errors, which, in turn, propagate into the
bias measurements (Eq. 3.3).
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
74
Convergence Map
10
2
0.
0
0.15
3
0.
.4
5
0.
7
9
0.
0.1
1.1
5
4
0.
0
0.3 0.2
0.05
0.9
1.1
North position error (m)
0.1
−5
1.1
0.
0.5
0.4
7
−8
0.
2
3
−10
−10
0.
0.
0.7
15
−6
−4
−2
0
2
East position error (m)
0. 0.5
4
4
0.
9
6
8
10
Figure 3.3: Contours Showing Final Position Errors (r=5 m)
Monte Carlo Simulation
In order to evaluate the impact of the initial position error on the estimation, 2000
Monte Carlo simulations were performed. In the Monte Carlo runs, the mobile vehicle
was placed 5 m East of the reference vehicle and the initial position error was randomly given in both horizontal directions2 . The vehicle then attempted to perform
the same maneuver shown in Figure 3.1. Note that the vehicle position during the
initialization maneuver was perfect (5 m radius circle), but the vehicle’s knowledge of
2
Initial position errors were uniformly distributed between ±10 m along the East and North axes.
3.1. CYCLE AMBIGUITY ESTIMATION
75
Monte Carlo simulation of Filter Performance
Convergence Map
0.2
0.15
0.1
5
5
0.1
5
5
0.1
0.25
0.0
0.15
15
0.15
20
5
10
0.4
North position error (m)
Final horizontal position error (m)
0.15
0.3
0.2
0.1
0
−20
0.05
5
0
−5
20
0.05
−10
10
−10
0
0
−10
10
20
−20
0.05
−15
0.15
0.2
5
North position initial error (m)
−20
−20
East position initial error (m)
−15
−10
−5
0
5
East position error (m)
10
15
20
Fig. 3.4: Convergence Characteristics af- Fig. 3.5: Position Error Contours
ter the Relative Circular Motion Around the (r=100m)
reference Vehicle (r=100m)
its initial position was erroneous. The final horizontal position errors after the completed maneuvers were averaged, and are plotted against the initial position errors,
as shown in Figure 3.3. Note that because the mobile vehicle started to the East of
the reference, the initial line-of-sight error to the reference vehicle is most sensitive
along the Northern axis, and is less sensitive to small errors along the East axis. The
sharp increase in the final estimation error around the region where the initial East
error is close to -5 m is a result of the significant error that would be associated with
the estimated line-of-sight to the reference vehicle.
For comparison, figures 3.4 and 3.5 show the results of 105 Monte Carlo simulations when a 100 m circular radius initialization maneuver was performed. Figure 3.5
has a sensitivity pattern along the East and North axes that is similar to the one
in Figure 3.3, but the centimeter-level convergence region is much larger. This increase is expected, as the line-of-sight error is less sensitive to a fixed position error
when the vehicles are separated further apart. These simulation results show that the
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
76
centimeter-level convergence can be achieved with the accuracy level of the coarse positioning algorithm described in Section 2.6 with a reasonably large vehicle separation
(i.e., greater than 5–10 m).
3.1.2
Robust Initialization Algorithm
As shown in the previous sections [CP00], a key difficulty with the algorithm is its
ability to converge from poor initial conditions when the distance between the vehicles
is relatively short. The reasons for this difficulty can easily be seen in the following
development for 2 vehicles. The general solution for n vehicles follows directly from
these results. The observations for the 2 vehicle case are
∆φs + ∆φs = G X
+ βs + νs
(3.5)
∆φp = d(X) + β p + ν p
(3.6)
The nonlinear Eq. 3.6 can be expanded about the current estimate (X̂) using a Taylor
series
∂d(X) δX + δX T HδX + · · · + β p + ν p
∆φ = d(X̂) +
∂X X̂
p
(3.7)
∂d(X) δX + δX T HδX + · · · + β p + ν p
∆φp − d(X̂) =
∂X X̂
(3.8)
⇒
∂d(X) T
p
p
δ∆φ =
δX + δX HδX + · · · + β + ν
∂X X̂
p
(3.9)
where


1

H =
2

X =
∂ 2 d(X)
∂x21
2
∂ d(X)
∂x2 ∂x1
∂ 2 d(X)
∂x3 ∂x1
∂ 2 d(X)
∂x1 ∂x2
∂ 2 d(X)
∂x22
2
∂ d(X)
∂x3 ∂x2
x1 x2 x3
T
∂ 2 d(X)
∂x1 ∂x3
∂ 2 d(X)
∂x2 ∂x3
∂ 2 d(X)
∂x23






X̂
(3.10)
3.1. CYCLE AMBIGUITY ESTIMATION
77
The sum of the second and higher order terms in Eq. 3.9 is defined as e,
e ≡ δX T HδX + · · ·
(3.11)
The terms in Eq. 3.5 can also be expressed in terms of their perturbations and combined with the linearized Eq. 3.9 to form a single measurement equation


δ∆φs
δ∆φp
where


+
∆φs
0



=


∂d(X) ∂X X̂

δX

+

0
e
 + β̄
+ ν̄
(3.12)

 los1 


 los2 


G =  . ,
 . 
 . 

G
losn


β̄ = 
βs
βp

,

ν̄ = 
νs
νp


(3.13)
Note that the linearization error term e in Eq. 3.12 was neglected in the previous
method in Section 3.1.1. However, this term can be significant when the position
estimation error is large relative to the initial vehicle separation. If ignored, the final
position error after the initialization maneuver can also be very large. In order to
understand the impact of e on the estimation of β̄, the following formation geometry
is considered.
Geometric Interpretation
Figure 3.6 illustrates a 2-vehicle formation geometry when the initial position error is
large. X is the true relative position vector, X̂ is the current relative position estimate
vector, and δX is the difference between X and X̂. A geometric interpretation of the
exact linearization error can be formulated from Figure 3.6. In particular, it can be
shown that
|X| − |X̂| =
X̂
δX + (1 − cos θ)|X|
|X̂|
(3.14)
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
78
Fig. 3.6: Formation Geometry
Noting that
∂d(X) X̂
=
∂X X̂
|X̂|
(3.15)
|X| − |X̂| = δ∆φp
(3.16)
then Eq. 3.14 can be expressed as
∂d(X) δX + (1 − cos θ)|X|
δ∆φ =
∂X X̂
p
(3.17)
By comparing Eq. 3.17 and the local transmitter measurement in Eq. 3.9, the error
term in Eq. 3.12 can be identified geometrically as
e = (1 − cos θ)|X|
(3.18)
We can then substitute Eq. 3.18 into the measurement Eq. 3.12 to get


δ∆φs
δ∆φp


+
∆φs
0



=
G
X̂
|X̂|


δX

+

0
(1 − cos θ)|X|
 + β̄
+ ν̄
(3.19)
3.1. CYCLE AMBIGUITY ESTIMATION
Next define L as

L = 
G
X̂
|X̂|
79
LNR

(3.20)
where ALNR is the left null space of the matrix A obtained from the reduced row
echelon form. The advantage of deriving the left null space matrix using the reduced
row echelon form of the measurement matrix is that we can isolate the additional term
(1 − cos θ)|X|. Finally, multiplying both sides of Eq. 3.19 by L yields a measurement
equation for the bias state

z = L 
δ∆φs
δ∆φp


+
∆φs


0

= L β̄ + L 

= L β̄ + 

0
(1 − cos θ)|X|
 + L ν̄

0
(1 − cos θ)|X|
 + L ν̄
(3.21)
Compared to the previous measurement Eq. 3.3, Eq. 3.21 has an additional term
that defines the error in the bias estimation equation. Note that this term only affects
the last row of the measurement equation and it is a function of the magnitude of
the true state X and the angle θ between the vectors X and X̂ (line-of-sight error).
Eq. 3.21 implies that this error term would have a small effect if the angle θ is small
(θ ≈ 0). If we define the uncertainty in the position estimate as c ( differential code
noise level), then we can develop the following bound for the error term in Eq. 3.21
1 − 1 −

(1 − cos θ)|X| ≤

c2 
c2
|X|
|X|2
2|X|
(3.22)
where it is assumed that c < |X|. It is clear that this upper bound tends to zero as
the magnitude of |X| increases. Figure 3.7 shows the magnitude of the upper bound
in Eq. 3.22 as a function of the vehicle separation |X| and the initial position error
c. It is clear from this figure that a large separation can mitigate the impact of the
error term. For example, if the initial position uncertainty (c) is about 3 m and the
vehicle separation is 10 m, the magnitude of the error term is approximately 0.4 m.
However, if the separation increases (∼50 m), the magnitude of this term reduces
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
80
Impact of Initial Uncertainty (c) and Veh. Separation |X| on Bias Estimator
2
10
1
10
2
Max. Magnitude of the error term c /2|X| (m)
1m inital position uncertainty (c)
3m inital position uncertainty (c)
5m inital position uncertainty (c)
0
10
−1
10
−2
10
0
5
10
15
20
25
30
35
Separation between the Vehicle |X| (m)
Fig. 3.7: Maximum Magnitude of Error Term
Uncertainty c and Vehicle Separation |X|
c2
2|X|
40
45
50
as Function of Initial Position
to less than 0.1 m, although it is still large on the scale on the carrier phase noise.
These results show why the previous method (in Section 3.1.1) required a large-scale
initialization maneuver to mitigate the effects of the error term [CP00]. The results
also indicate that a large linearization error will be included in the bias filter if this
term is ignored and a small-scale relative motion (small |X|) is used to estimate the
bias state.
This analysis shows that in order to obtain the benefits of a small-scale initialization maneuver (Section 3.1), the error term (e) must be accounted for in the filtering
process. Although this measurement Eq. 3.21 is exact and no approximation has been
used in the development, it is impossible to find the exact value of (1 − cos θ)|X|.
It is possible, however, to find an approximate value from the current estimate of
X and its covariance PX . From these values, a rough estimate of the measurement
3.1. CYCLE AMBIGUITY ESTIMATION
81
noise covariance matrix R can be computed and used in the EKF. Assuming e is zero
mean random variable and independent of the measurement noise ν̄, the R can be
expressed as

R = 
where

0
0
0 σ
2

 + L 
σ 2 = E e2

ΣSatellite
0
0
ΣTransmitter

c2
E
|2X̂|
 LT
(3.23)
2 

(3.24)
Robust Initialization Algorithm (Second-order Method)
One way to compensate for the impact of the e term in the bias estimation algorithm
would be to include the second-order term of Eq. 3.9 and ignore the higher order
terms in the expansion. Following Eq. 3.11 and 3.18, we have that
e = δX T HδX + · · ·
(3.25)
= (1 − cos θ)|X|
(3.26)
Assuming that the first term in Eq. 3.25 is a reasonably good approximation of the
term e, ê is expressed as
ê ≡ δX T HδX = Trace (HδXδX T )
(3.27)
As before, define L as the left null space of the measurement matrix and then multiply
both sides of Eq. 3.12 by L to obtain a measurement equation for the bias state

z2 = L 
δ∆φs
δ∆φp

= L β̄ + 


+

0
ê
 + L ν̄
∆φs


0
(3.28)
Again, it is still impossible to compute the exact value of ê, but the mean of ê can be
estimated from the state covariance matrix PX . Assuming the elements of the matrix
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
82
H are deterministic3 ,
¯ê ≡ E [ê] = Trace(HE δXδX T )
(3.29)
= Trace(HPX )
(3.30)
The covariance of ê can be found from the following:
σ 2 (ê) ≡ E (ê − ¯ê)2
2
= E δX T HδX δX T HδX − ¯ê
2
= E (Trace(HδXδX T ))2 − ¯ê
(3.31)
For simplicity, it is assumed that the off-diagonal terms in the matrix H are zero in
the computation of the covariance of ê, which gives

Trace(HδXδX T ) h11 h22


h33 

δx21
δx22
δx23





(3.32)
where hjj is the (j, j)th element of the matrix H and δxi is the ith element of the
vector δX. Then

(Trace(HδXδX T ))2 h11 h22 h33


h11



HX  h22 


(3.33)
h33
where



δx41
HX =  δx22 δx21

δx23 δx21
3
δx21 δx22 δx21 δx23
δx42
δx23 δx22
δx22 δx23
δx43





(3.34)
An assumption that is consistent with the derivation of the extended Kalman filter [AG74].
3.1. CYCLE AMBIGUITY ESTIMATION
83
Then

E (Trace(HδXδX T ))2 h11 h22 h33


h11



E [HX ]  h22 


(3.35)
h33
Kurtosis can be used to estimate the 4th order term in the matrix E [HX ]. Kurtosis
is a measure of a skewness of a distribution. The kurtosis of the normal distribution
is 3. Distributions that are more “outlier-prone” than the normal distribution have a
kurtosis greater than 3 and distributions that are less “outlier-prone” have kurtosis
less than 3. The kurtosis of a distribution (x) is defined as
K=
E[(x − x̄)4 ]
σx4
(3.36)
Assuming δX is zero mean and has a distribution close to the normal distribution
(K = 3),
E δx4i = 3σx4i = 3(PX (i, i))2
(3.37)
In addition, assuming δx2i and δx2j (i = j) are uncorrelated,
E δx2i δx2j E[δx2i ]E[δx2i ] = σx2i σx2j = PX (i, i)PX (j, j)
(3.38)
Substituting all of these values into Eq. 3.31, the cross-terms (σx2i σx2j ) cancel out and
the covariance σ 2 (ê) can be expressed in terms of the diagonal elements of the matrix
PX and the matrix H
σ 2 (ê) 2 h211 σx41 + h222 σx42 + h233 σx43
(3.39)
Using the estimated mean and covariance of ê, we can rewrite the measurement
Eq. 3.28 as








0
δ∆φs
∆φs
0
+
−

z2 = z2 −   = L 
¯ê
¯ê
δ∆φp
0
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
84

= L β̄ + 

0
ê − ¯ê
 + L ν̄
(3.40)
Assuming the zero-mean random variable ê − ¯ê and ν̄ are uncorrelated, then a new
noise covariance matrix R2 is defined

R2 = 

0
0
2
0 σ (ê)

 + L 

ΣSatellite
0
0
ΣTransmitter
 LT
(3.41)
A Kalman Filter can then be used to update the bias state β̄ with the covariance
matrix R2 and a new measurement vector z2 .
A sample simulation was conducted to validate the algorithm. The same simulation parameters were selected to be the same as the previous initialization maneuver
simulation (Figure 3.1). The initial PX was calculated based on the initial position
uncertainty level. Figure 3.8 compares the true value of e = (1 − cos θ)|X| with the
estimated mean ¯ê and the 1-σ envelope during the maneuver. Note that the magnitude of e is large (6 m) during the initial phase of the maneuver when the position
error is large (initial position error 7.45 m). However, the algorithm was able to
estimate the mean and covariance of ê (¯ê and σ(ê)) fairly accurately and compensate
for the error term. The final position error in this simulation was approximately 2 cm.
Under the same conditions, the final position error of the previous algorithm was as
large as 3.19 m since it was not able to account for the large bias error (e).
3.1.3
Performance of Robust Initialization Algorithm
The performance of the robust initialization algorithm has been investigated through
extensive simulations and comparisons with the previous filtering algorithms. The
convergence characteristics of the algorithms in the case where there is large initial
position error are presented. Several Monte Carlo simulations were performed to determine the region of algorithm convergence. It is crucial for an algorithm intended
for use in real-time operations to convergence with a reasonable initial error. Note
that the initial simulation results separate the estimation from the control problem
3.1. CYCLE AMBIGUITY ESTIMATION
85
Estimated Error term
2
10
true e
estimated mean of e
1 σ envelope
Position Error
1
10
0
10
−1
magnitude (m)
10
−2
10
noise floor
−3
10
−4
10
−5
10
−6
10
−7
10
0
20
40
60
80
simulation time (sec)
100
120
140
Figure 3.8: Mean and Covariance of the Term e Estimated By the Robust
Initialization Algorithm are Compared to the True e (Position
Error Also Shown).
Typical Convergence
Typical Convergence
10
Initial Position Guess
6
Robust Algorithm
Truth
Previous Algorithm
True Initial/Final Postion
5
4
Initialization
Path Direction
0
2
y (m)
y (m)
−5
0
−10
True Initial / Final Position
−2
Initial Position Guess
−15
−4
−6
−6
−20
Previous Algorithm
Truth
Robust Algorithm
−4
−2
0
2
4
6
x (m)
Fig. 3.9: Convergence Characteristics
Comparison (Typical)
−25
−30
−25
−20
−15
−10
x (m)
−5
0
5
10
Fig. 3.10: Convergence Characteristics
Comparison (Extreme case)
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
86
Table 3.1: Final Position Errors
Typical
Case
Extreme
Case
Initial Error
Final Error
Initial Error
Final Error
Previous Algorithm
4.217 m
0.805 m
34.361 m
1.319 m
Robust Algorithm
4.217 m
0.017 m
34.361 m
0.011 m
for a precise comparison of the robust initialization algorithm with the previous initialization method. Simulation results at the end of this section consider simulations
wherein the vehicle control commands are calculated based on the current estimate
of the relative position.
Convergence Characteristics
Figure 3.9 compares the convergence characteristics of the robust initialization algorithm with the previous method. The relative motion used in both cases was a
5 m radius circle around a local transmitter and the true initial and final positions
were (5,0)m as shown on the figure. Both algorithms were applied to the same simulated phase measurement as the vehicle underwent the circular motion. The initial
position guesses ((0.26, 4.21) m) that were given to both initialization algorithms,
however, were approximately 4.2 m away from the true position ((5.0, 0.0) m). As
shown in Figure 3.9, the robust initialization algorithm was able to initialize biases
and converge to the correct position from the poor initial guess given, but the previous algorithm was not able to account for the large initial error and converged to
an erroneous answer. Table 3.1 summarizes the simulation results. The final position
error of the robust method is approximately 1.7 cm, but the final error of the previous
method is ∼80 cm which is not acceptable in a CDGPS sensing system. In addition,
Figure 3.10 compares the convergence characteristics when the initial position error
was very large (∼34 m). Even in this case, the robust algorithm was able to initialize
the bias, but the previous algorithm failed to converge to the correct answer.
3.1. CYCLE AMBIGUITY ESTIMATION
87
Convergence Map
40
30
30
20
INCONSISTENT CONVERGENCE
20
10
y position error (m)
y position error (m)
Convergence Map
40
0
ALWAYS CONVERGE
−10
ALWAYS CONVERGE
10
0
−10
Error bound of code DGPS sensor (5m)
Error bound of code DGPS sensor (5m)
−20
−20
INCONSISTENT CONVERGENCE
−30
−40
−40
−30
−30
−20
−10
0
10
x position error (m)
20
30
40
−40
−40
−30
−20
−10
0
10
x position error (m)
20
30
40
Fig. 3.11: Convergence Map After 5 m Fig. 3.12: Convergence Map After 5 m
Radius Circular Motion: Previous algo- Radius Circular Motion: Robust algorithm
rithm
Convergence Maps
In order to evaluate the impact of the initial position error on the improved initialization algorithm, 50,000 Monte Carlo simulations were performed. In the Monte Carlo
runs, the mobile vehicle was placed at (5,0) m relative to the reference vehicle and
the initial position error was randomly given. The vehicle then started a prescribed
5 m radius circular motion. The average final horizontal position estimate errors
were computed at the end of the maneuver. Figures 3.11 and 3.12 show the region
of algorithm convergence (final position error less than 5 cm) with respect to x and
y errors in the initial position estimate. The vehicle position during the initialization
maneuver was perfect; there were no deviations from the designated circular path.
However, the vehicle’s knowledge of its position was erroneous and the size of the
initial error was given by the x and y axes in the figures. Figure 3.11 shows the
convergence region of the previous method which ignores the second order and higher
terms. The convergence region is roughly ∼3 m wide in y and ∼4 m wide in x, which
is barely the size of the navigation sensor error (∼5 m) of code-based single-frequency
DGPS (typical source for the initial position estimate) [BWP96-1]. Figure 3.12, on
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
88
Convergence Map
Convergence Map
10
10
INCONSISTENT CONVERGENCE
INCONSISTENT CONVERGENCE
5
y position error (m)
y position error (m)
5
ALWAYS CONVERGE
0
−5
ALWAYS CONVERGE
0
−5
Error bound of code DGPS sensor (5m)
Error bound of code DGPS sensor (5m)
−10
−10
−8
−6
−4
−2
0
2
x position error (m)
4
6
8
10
−10
−10
−8
−6
−4
−2
0
2
x position error (m)
4
6
8
10
Fig.
3.13: Convergence Map (con- Fig. 3.14: Convergence Map (control
trol based on current estimate): Previous based on current estimate): Robust algomethod
rithm
the other hand, shows the convergence region of the robust algorithm that includes
the second order term. As shown, the convergence region is much larger and well exceeds the accuracy expected from differential code based methods. Both convergence
maps also show the error bound of code based DGPS sensor (5 m radius circle).
Real-Time Control
The previous convergence maps and the convergence characteristics plots were generated based on the perfect initialization trajectory (5 m circle selected) in order to
compare the algorithm characteristics under the exact same conditions (exact same
phase measurements). However, in a real-time system, the vehicle motion would be
determined by the current estimate of the vehicle position, which could have a large
error. Therefore, a second set of Monte Carlo simulations were conducted where the
vehicle controls were solely based on the current knowledge of its position. As a result, the vehicle may not have been able to perform the desired circular motion, and
it often underwent maneuvers that generated very small line-of-sight changes. Therefore, the size of the convergence region of the both algorithms shrunk to a certain
3.1. CYCLE AMBIGUITY ESTIMATION
89
degree. Figures 3.13 and 3.14 show the convergence maps of the previous method and
the robust algorithm, respectively. The robust algorithm has a smaller convergence
region than before, but it maintains a convergence region that is larger than the error
bound of the code based DGPS sensor (5 m radius circle). Therefore, the robust
algorithm can still guarantee the convergence. On the other hand, the convergence
region of the previous initialization algorithm is much smaller than the size of the
code DGPS error bound and clearly cannot guarantee the convergence.
3.1.4
Selection of Initialization Path
In the previous sections, circular initialization maneuvers (360◦ line-of-sight change)
are used in the investigation of the previous and robust algorithms. However, it is
possible to choose different types of motion and there are several issues in the selection
of the initialization paths:
1. It may not be possible to perform full 360◦ motion around the local transmitter
for various reasons.
2. Distance between the vehicles during the maneuver must be chosen to avoid
collisions.
Figure 3.15 shows various types of initialization paths in addition to the 360◦ circular maneuver. In some cases, such as the air-to-air refueling mission, the Power
Sail missions (Chapter 5), and the aircraft landing mission [BP96], less than 360◦
motion around a local transmitter is possible in the initialization process. Thus the
non-circular initialization maneuvers shown in Figure 3.15 are often performed in
real systems [BP96, MOC97]. Depending on the initial uncertainty level, less than
360◦ motion can be sufficient to initialize the cycle ambiguity states (to less than a
specified tolerance). For example, figure 3.16 shows the position error magnitude as
a function of the line-of-sight change. In the simulation, a vehicle was undergoing a
5 m radius circular manuever with random initial errors. The mean initial position
error magnitude was approximately 4 m. With this level of initial uncertainty, about
200◦ motion was required to initialize the bias to a sufficient accuracy (5 cm). On
CHAPTER 3. ROBUST BIAS ESTIMATION ALGORITHM
90
(a) 90 Degree LOS Sweep
(b) 270 Degree LOS Sweep
10
10
5
5
0
0
−5
−5
−10
−10
−5
0
5
−10
−10
10
(c) 360 Degree LOS Sweep
10
5
5
0
0
−5
−5
−5
0
5
0
5
10
(d) 360 Degree Spiral
10
−10
−10
−5
−10
−10
10
−5
0
5
10
Fig. 3.15: Various Possible Initialization Paths
Convergence Characteristics
Convergence Characteristics
5
Mean Position Error
1σ Envelope
4.5
Mean Position Error
1σ Envelope
1.4
4
1.2
Mean Position Error (m)
3.5
Position Error (m)
3
2.5
2
5cm convergence
1.5
1
1
0.8
0.6
5cm convergence
0.4
0.5
0.2
0
−0.5
0
100
200
300
400
Sweep Angle (deg)
500
600
700
0
0
90
180
270
360
450
Sweep Angle (deg)
540
630
720
Fig. 3.16: Position Error as a Function Fig. 3.17: Position Error as a Function
of the LOS change (small initial error)
of the LOS Change
3.2. SUMMARY
91
the other hand, only 120◦ motion was necessary if the initial error magnitude was
approximately 1 m, as shown on Figure 3.17.
In some mission scenarios, it may be desirable to form a tight formation or very
small scale initialization maneuvers [DSS92]. In order to avoid collisions, the vehicle
separation must be greater than the uncertainty level of the relative position knowledge, at all times. In this case, types of initialization maneuver such as the one shown
in Figure 3.15-(d) can be used, in which the vehicle separation decreases as the line-ofsight angle changes and the confidence level of the bias estimates increases. The rate
of the decrease can be determined by the convergence rates shown on Figures 3.16
and 3.17, depending on the initial uncertainty level. For example, if the targeted
final separation is ∼1 m and the initial uncertainty is about 4 m, the vehicles can be
commanded to form a 1 m separation only after the vehicle has finished 120◦ relative
motion (Figure 3.16) since, after this motion, the relative position error should be
less than 1 m and a collision avoidance can be assured.
3.2
Summary
A robust initialization algorithm that accounts for the effect of the large initial error is
presented and it is demonstrated in various simulations that the algorithm converges,
even with the relatively poor initial position estimates available from coarse sensors
such as code DGPS. This feature makes the new algorithm suitable for real-time
applications such as an autonomous land vehicle system [MOC97], aircraft landing
system [BP96], and aerial refueling and spacecraft docking missions where the realtime position estimates, as well as system robustness, are required. This algorithm
was implemented in the real-time system and an autonomous bias initialization maneuver was successfully demonstrated using both RC Trucks and outdoor robots.
Chapter 6 presents the experimental results along with the experimental convergence
characteristics of the algorithm.
Chapter 4
Quasi-Optimal Satellite Selection
This chapter presents a quasi-optimal satellite selection method that is able to select
near-optimal geometries of the NAVSTAR and local constellations with significantly
fewer computations than are typically required by conventional methods. It includes
a comparison of the performance of the quasi-optimal selection method with other
existing selection methods. Finally, computational loads of the optimal and the quasioptimal methods are compared and various simulation results are presented.
4.1
Satellite Selection
In contrast to terrestrial users, spacecraft in LEO typically observe more than 10
visible NAVSTAR satellites throughout their orbit. A typical plot of the number of
visible satellites for approximately one LEO orbit is shown in Figure 4.2. Note that
as many as 13 satellites are visible in this orbit. In a formation with ns vehicles, there
are ns -1 local range measurements available in addition to these 10–13 NAVSTAR
measurements. Figure 4.1 illustrates a spacecraft formation observing the combined
local and NAVSTAR constellations. Using all of the available measurements from
all of the visible satellites may be the simplest choice in the position and velocity
estimation. However, many receivers have a limit on the number of channels available, and there is a significant computational load associated with performing the
estimation with a large number of measurements (Figure 2.7). These computational
92
4.1. SATELLITE SELECTION
93
NAVSTAR GPS Constellation
Spacecraft Formation
Onboard
Transmitter
Figure 4.1: Combined Local and NAVSTAR Constellation
difficulties and hardware limitations can be avoided by selecting a small subset of
the visible satellites. However, this selection must be performed with care, since a
poor selection of the geometry can have a significant impact on the performance of
the estimation. The PDOP (Position Dilution of Precision) is typically used as a
measure of the measurement geometry, and a good rule of thumb [GPS96] is that
Effective noise = PDOP × sensor noise
(4.1)
Thus it is critical to have a measurement geometry that provides a small PDOP. For
example, Figure 4.3 shows typical minimum and maximum PDOP values in LEO
as a function of the number of satellites used in the measurement equation. There
are a total of 13 visible satellites and 4–13 satellites are selected (x-axis). Then
the minimum PDOP (optimal PDOP) and the maximum PDOP (worst case PDOP)
were computed for a given number of selected satellites. The figure shows that, if
the optimal selection can be ensured, then there is a minimal increase in PDOP by
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
94
Number of Visible Satellites
15
14.5
number of satellites
14
13.5
13
12.5
12
0
1000
2000
3000
4000
time (sec)
5000
6000
7000
Figure 4.2: Number of Visible Satellites
selecting only 6–9 satellites instead of using all 13 satellites (< 0.38%). However, the
worst case PDOP is shown to increase dramatically as the number of measurements
used is reduced. These results indicate that it is essential for a LEO spacecraft to
be able to select a subset of the large number of the visible satellites that provides a
good geometry.
4.2
Optimal Satellite Selection
A subset of the visible satellites should be chosen such that it provides good constellation geometry and minimizes the degradation of the estimation accuracy. This
problem can be posed as an optimization where the objective is to select the subset of the satellites that minimizes the PDOP (Position Dilution of Precision). The
4.2. OPTIMAL SATELLITE SELECTION
95
Min. Max. PDOP values
2
10
Minimum PDOP (Optimal)
Maximum PDOP (Worst case)
Total visible satellites = 13
4−13 satellites can be selected.
1
PDOP
10
Minimum PDOP achievable with 6 satellites
0
10
Minimum PDOP achievable with 9 satellites
4
5
6
7
8
9
10
Number of Measurements Used
11
12
13
Figure 4.3: Minimum and Maximum PDOP Values of Selected Subset of
Satellites
PDOP [JJS96] is defined as
"
PDOP =
H =
Trace (H T H)−1










los1 

los2 

.. 
. 

losn
(4.2)
(4.3)

where H is the measurement matrix and n is the number of measurements selected.
However, in order to find the optimal selection, PDOP values for all possible measurement combinations must be computed and compared. For example, if there are
N visible signals and only n satellites are to be selected, the PDOP value must be
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
96
Number of Floating Point Operations Required to Select 6 Satellites
7
10
6
Floating Point Operation Count
10
5
10
4
10
3
10
7
8
9
10
11
12
Number of satellites in view
13
14
15
Fig. 4.4: Computational Load Required for Optimal Selection of Satellites

computed 

N

times. Calculating a single PDOP value requires one matrix muln
tiplication and one matrix inversion. Therefore, the total floating point operations
required to find the optimal solution can be very large and grows rapidly as the
number of the visible satellites increases. Figure 4.4 shows the number of floating operations required to select the optimal subset of the visible satellites when 6 channels
are available for tracking NAVSTAR satellites. Since the lines-of-sight from the formation to the NAVSTAR satellites are changing rapidly in LEO, these computations
must be done very frequently. Moreover, if there are ns vehicles in the formation, the
entire set of computations must be repeated ns −1 times for each vehicle in the formation because the vehicles observe different local constellations. For these reasons, it
is virtually impossible to use the optimal selection method in real-time applications.
4.3. SUB-OPTIMAL SATELLITE SELECTION ALGORITHMS
4.3
97
Sub-optimal Satellite Selection Algorithms
Several existing sub-optimal selection algorithms are introduced and briefly explained
in this section. These methods are capable of selecting a subset of satellites that forms
a geometry close to the optimal geometry, but with certain limitations.
4.3.1
Highest Elevation Satellite Selection Algorithm
The most popular method used in the terrestrial applications is the the highest elevation satellites selection method. In principal, this method selects the subset of
satellites with the highest elevations with respect to the user. The algorithm requires very little computation and provides reasonably good measurement geometries
for terrestrial users since the vertical direction is usually the worst. However, this
scheme can often provide a very poor geometry for LEO spacecraft because it is possible that there are a large number of satellites at high elevations, and these are the
only satellites that would be selected.
4.3.2
Kihara’s Maximum Volume Method
Kihara and Okada developed a selection method [MK83] that selects 4 satellites. The
method is based on the idea that PDOP tends to be inversely proportional to the
volume of the tetrahedron formed by the four line-of-sight vectors from the user to
the satellites. [MK83] has shown that
√
1 A
1
PDOP =
∝
6 V
V
(4.4)
where V is the volume of the tetrahedron and the scalar quantity A is function of the
elements of the line-of-sight vectors. This method selects 4 satellites that maximizes
the volume of the tetrahedron with certain conditions. The method is summarized
as follows.
1. Select the satellite (S1) with the highest elevation.
2. Select the satellite (S2) which has the angle to S1 nearest to 109.5◦ .
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
98
3. Select the satellites (S3 and S4) that maximizes the volume of tetrahedron.
It is shown that this method provides an acceptable geometry with reasonable computational loads [MK83]. However, the volume of the tetrahedron may not correctly
represent the PDOP value since it is also a function of A (Eq 4.4). Also, this method
is not directly applicable to the case where more than 4 satellites are to be selected.
Since the minimum number of satellites required in the relative position and clock bias
estimation is 4, it is desirable to select, in general, more than 4 satellites to increase
estimation robustness and to reduce the degradation in the estimation accuracy by
selecting subsets (Figure 4.3).
4.3.3
Four-step Selection Algorithm
Li developed a four-step satellite selection algorithm [JL99] for the Stanford Gravity
Probe-B spacecraft. The algorithm is able to select 4 satellites that form near optimal
geometries with reduced computational loads. The satellites are sequentially selected
based on the idea that a regular tetrahederon forms the optimal geometry when 4
measurements are to be chosen. The four-step selection algorithm can be summarized
as follows [JL99].
1. Select the satellite (F1) with the highest elevation.
2. Select the second satellite (F2) that has the largest angular separation from F1.
3. Based on F1 and F2, a reference triad can be formed and a regular tetrahedron
can be defined in that triad. The third satellite (F3) is selected that has the
smallest angular separation to either of the two vertices of the tetrahederon not
populated by F1 or F2.
4. From the remaining visible satellites, F4 is selected such that the selected set
of 4 satellites minimizes the PDOP (requires N-3 PDOP computations).
Although this algorithm yields a near optimal subset of satellites with reduced
computational load, it is designed to select only 4 satellites. It is possible to continue
selecting beyond 4 satellites by comparing PDOP values of all possible combinations,
but the performance is expected to be limited since the algorithm is based on the
4.4. QUASI-OPTIMAL SATELLITE SELECTION
99
Cost Function
1
0.8
0.6
0.4
Cost
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
−180
−135
−90
−45
0
45
Angle between the LOS vectors (deg)
90
135
180
Fig. 4.5: Cost Function cos 2θij
tetrahedron. Also, the final step (step 4) is no different than the procedure of the
optimal method with fewer satellites, which incorporates matrix inversions.
4.4
Quasi-optimal Satellite Selection
This section presents a quasi-optimal satellite selection method that yields nearoptimal constellation geometries without restrictions on the number of satellites. The
approach also significantly reduces the computational load.
Cost Function
The intuitive concept that line-of-sight vectors that are close to each other are redundant inspired the simple cost function used in this selection algorithm. Figure 4.5
shows the cost function for the ith and j th measurement pair.
Jij = cos 2θij
(4.5)
100
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
The cost is a function of the angle between the line-of-sight vectors losi and losj , or θij .
This function indicates that the cost is high if the two vectors are close to collinear
(θij ∼
= 0◦ or θij ∼
= ±180◦ ) and the lowest if they form a right-angle (θij ∼
= 90◦ ).
Therefore, the cost for the ith measurement can be defined as a sum of the costs
between the ith measurement and all other measurements (j = 1, 2, . . . , N).
Ji =
N
#
(cos 2θij )
(4.6)
j=1
where N is the total number of measurements available. This cost gives a direct measure of the degree of redundancy of the ith measurement, given the other measurements available. The cost can be computed for all measurements Ji for i = 1, 2, . . . , N,
which can be used to determine the most redundant measurements.
Quasi-optimal Selection Algorithm
Define the N × m measurement line-of-sight matrix H (m is the number of states)
and compute the N × N matrix D as


 los1 


 los2 


H = 

 .. 
 . 

losN
(4.7)

D = HH T =










cos θ11
cos θ12
cos θ21
..
.
cos θ22
..
.
· · · cos θ1N 

· · · cos θ2N 


..
..

.
.

cos θN 1 cos θN 2 · · · cos θN N
(4.8)

Each element in the symmetric matrix D, dij , represents the cosine of the angle
between two line-of-sight vectors losi and losj . The cost for the ith measurement Ji
4.4. QUASI-OPTIMAL SATELLITE SELECTION
101
can be expressed in terms of dij
Ji =
N
#
(cos 2θij ) =
j=1
N
#
(2 cos2 (θij ) − 1)
(4.9)
j=1
=
N
#
(2d2ij − 1)
(4.10)
j=1
The cost for the ith measurement is expressed in terms of a sum of the squares of
all the elements in the ith row of the matrix D. The cost Ji , i = 1, 2, . . . , N can be
computed and the measurement (k1 ) with the highest cost Jk1 identified
Jk1 = maxi {J1 , J2 , . . . , JN }
(4.11)
The procedure is then to remove the k1th measurement since the cost indicates that it
is the most redundant, and thus provides the least additional information. The k1th
row and column are then deleted from the D matrix. The new matrix is named D(k1 ) .
The process is then repeated, computing the new costs Ji ’s for the (N − 1) × (N − 1)
matrix D(k1 ) , and the highest cost Jk2 can be identified
Ji =
N
−1
#
(2d2(k1 ),ij − 1)
(4.12)
Jk 2 = maxi {J1 , J2 , . . . , JN −1 }
(4.13)
j=1
where d(k1 ),ij = (i, j)th element of the matrix D(k1 ) . Then, the k2th row and column can
be removed from the matrix D(k1 ) to form D(k1 ,k2 ) . This sequence is repeated until the
desired number (n) of measurements remain. The final n × m measurement matrix
H n will then have the k1 , k2, . . . , kN −n rows removed. In a practical implementation,
the Ji is not calculated by reformulating the matrix D(k1 ) and summing squares of
the elements as shown in Eq. 4.12. Rather, in order to minimize the computations,
Ji can be computed from the previous cost Ji .
Ji = Ji − (2d2ik1 − 1)
(4.14)
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
102
1.5
θ = 80 °
3
θ = 100 °
4
1
0.5
θ = 5.0 °
2
0
θ1=−5.1 °
−0.5
−1
Line−of−sight vectors
Optimal Selection (PDOP = 2.0147)
−1.5
−1.5
−1
−0.5
0
0.5
1
1.5
Figure 4.6: Optimal Selection (selecting 2 out of 4)
The reason why a sequential/iterative method is chosen to filter out the most redundant measurements rather than selecting multiple measurements from the first set
of costs {J1 , J2 , . . . , JN } is described in the following. The cost of the quasi-optimal
Ji method is based on the cost of a single measurement pair Jij (i.e., Ji is the sum
of the costs calculated pair-wise). Therefore, the measurement corresponding to the
second largest cost is, in general, not the most redundant measurement after Jk1
has been removed. The following simple example compares the sequential method
with the method that removes N − n measurements in a single step without recomputing the cost functions. Figure 4.6 shows four 2-D line-of-sight vectors located at
{−5.1◦ , 5◦ , 80◦ , 100◦ }. The optimal set of 2 vectors are also shown for the case
where 2 measurements are to be selected. Figure 4.7-(a) shows the 4 measurements
with the costs. Figure 4.7-(b) shows the result of one-step selection which removed
4.4. QUASI-OPTIMAL SATELLITE SELECTION
(b) Selected 2 satellites
in one step with lowest cost
(a) Cost of Each measurement
1.5
103
1.5
−0.083
1
−0.085
1
0.5
0.5
0.088
0
0
0.089
−0.5
−0.5
−1
−1
−1.5
−1.5
PDOP= 17.1
−1
0
1
(c) Step 1: Deselected 1 satellite with
the highest cost and calculated new costs
1.5
1
0.7812
0.900
−1
0
1
(d) Step 2: Deselected the second satellite based
on the new costs.
1.5
1
0.5
0.5
−0.858
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−1
0
1
PDOP= 2.0153
−1
0
1
Figure 4.7: Quasi-optimal Selection (sequential vs. one-step method) (se-
lecting 2 out of 4)
the two most redundant (highest cost) measurements. In this case, the resulting
PDOP is about 17, which is much higher than the optimal PDOP (2.0147). On the
other hand, Figures 4.7-(c) and 4.7-(d) illustrate two steps of the sequential selection
scheme. In the first step (Figure 4.7-(c)), the measurement with the highest cost is
removed and new costs are computed for the three remaining measurements. Then,
the measurement with the highest cost is eliminated. The resulting PDOP is 2.0153,
which is only slightly higher (0.3%) than the optimal PDOP (2.0147). The reason
why the quasi-optimal method did not select the optimal set of measurements can
be found in the first step. In that step, the measurement with the highest cost is
the one at −5.1◦ . Thus this measurement is eliminated in the quasi-optimal method.
However, for optimality (2 measurements forming as close to 90◦ as possible), the
104
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
measurement at −5.1◦ should be retained since the measurement at −5.1◦ together
with the one at 80◦ form 85.1◦, which is the nearest pair to 90◦ . However, this simple
example shows that a combination of an appropriate cost function and sequential
selection algorithm provide a very powerful selection algorithm that gets very close
to the optimal geometry.
The following sections show that the quasi-optimal algorithm has been applied to
various measurement geometries to evaluate the performance of this approach.
Selection from Random Geometry
The quasi-optimal algorithm and optimal algorithm are applied to 1000 randomly
generated line-of-sight matrices. The test assumed that there are 10 visible satellites,
but only 6 are to be selected. In this case, there exist 210 combinations that can
be formed from 10 satellites. For each of the 1000 randomly generated matrices,
the quasi-optimal PDOP value is computed, as were the PDOP values for all 210
possible combinations. The optimal set of 6 satellites can be found from these 210
combinations. Figure 4.8 shows a typical result of the process. The 10 randomly
generated line-of-sight vectors (one associated with each of the 10 satellites) and
the 6 vectors selected by the quasi-optimal algorithm are shown in the figure. In this
particular run, the quasi-optimal selection agreed with the optimal. In fact, the quasioptimal algorithm yielded the optimal answer in 473 cases out of the 1000 cases. Note
that the four highest elevation satellites are not selected in either the optimal or the
quasi-optimal method (see Figure 4.8). This indicates that very poor PDOP could
result if the highest elevation selection algorithm had been applied to this geometry.
4.4.1
Validation of Quasi-Optimal Algorithm
This typical result is also presented in terms of the PDOP values. Figure 4.9 shows the
possible PDOP values that correspond to all 210 combinations for this one random
configuration along with the single PDOP value for the satellites selected by the
quasi-optimal method. The possible PDOP values are arranged in ascending order.
4.4. QUASI-OPTIMAL SATELLITE SELECTION
105
Line of Sights of Satellites
90
1
60
120
0.8
Satellites in View
Quasi−optimal Selection
Optimal Selection
150
0.6
30
0.4
0.2
180
0
210
330
240
300
270
Fig. 4.8: Polar Plot Showing Lines-of-sight to Satellites in View and Satellites
Selected Using Optimal and Quasi-optimal Methods (Note that the four highest elevation satellites are not selected)
All Possible PDOP Values and PDOP Value of Quasi−Optimal Method
2
10
PDOP value
PDOP
Quasi−Optimal PDOP
1
10
0
10
0
50
100
150
index of 210 combinations
200
250
Fig. 4.9: All Possible PDOP values and PDOP of Quasi-optimal Method
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
106
Line of Sights of Satellites
90
1
60
120
0.8
0.6
30
150
0.4
0.2
180
0
210
330
Satellites in View
Quasi−optimal Selection
300Selection
Optimal
240
270
Fig. 4.10: Quasi-optimal Selection (worst case)
All Possible PDOP Values and PDOP Value of Quasi−Optimal Method
3
10
PDOP
Quasi−Optimal PDOP
2
PDOP value
10
1
10
0
10
0
50
100
150
200
250
index
Fig. 4.11: PDOP Values of Quasi-optimal Selection (worst case)
4.4. QUASI-OPTIMAL SATELLITE SELECTION
107
Quasi−optimal
Highest Elevation
100
100
Selection from Random
Measurement Geometry
(N=1000)
80
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
0
1
ζ
1.5
=(GDOP
quasi
2
2.5
)/(GDOP
quasi
Selection from Random
Measurement Geometry
(N=1000)
90
%
%
90
3
)
optimal
0
1
ζ
1.5
=(GDOP
highElev
2
2.5
)/(GDOP
highElev
3
)
optimal
Figure 4.12: Distribution of ζ. (Selection from Random Geometry)
To further analyze this performance, Figure 4.10 shows the worst case selection
made by the quasi-optimal method (1000 selections). As can be seen from the figure,
the only significant difference between the quasi-optimal and optimal selections (even
in this worst of 1000 cases) is that the quasi-optimal method selected the satellite
at 160◦ azimuth angle instead the one at 10◦ azimuth. Because an azimuth of 160◦
is equivalent to −20◦ in the cost function, the quasi-optimal selection was close to
the optimal selection (within 30◦ ). The resulting PDOP was approximately 23%
higher than the optimal PDOP (see Figure 4.11). But as can be seen in this figure,
the PDOP value in this worst case from the quasi-optimal method still provides
acceptable performance.
The optimality measure of the algorithm is defined as ratio of the PDOP of suboptimal algorithm to the optimal PDOP (ζquasi ). For the quasi-optimal algorithm,
108
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
ζquasi is defined as
ζquasi =
PDOPquasi
PDOPoptimal
(4.15)
The magnitude of ζ provides an indication of the size of the error ellipse of the algorithm compared to the optimal error ellipse. Figure 4.12 compares the performance
(ζquasi and ζhighest) of the two suboptimal algorithms for all 1000 cases. To generate this figure, the quasi-optimal, highest elevation, and optimal algorithms were
applied to the 1000 random geometries and the distributions of the ζ values are generated (bin size = 0.05). The figure clearly shows that the quasi-optimal algorithm
is very similar to the optimal algorithm (ζquasi ∼
= 1, max(ζquasi ) = 1.234), but the
highest elevation selection method is far from optimal in many cases (ζhighest = 1–
4, max(ζhighest) = 35.853).
Selection of NAVSTAR signals in the LEO Environment
A more realistic set of measurement geometries were generated from the LEO simulations and the selection algorithms were applied to these simulated NAVSTAR
constellation geometries. In this simulation, the LEO spacecraft formation without
the onboard transmitters completes 5 orbit periods (∼450min) and performs 1000
constellation geometry selections using the quasi-optimal, highest-elevation, 4-step,
and optimal algorithms. Figure 4.13 shows the distribution of ζ for each of the suboptimal methods. In this test, 4 satellites are selected from 12–15 visible satellites in
order to compare the quasi-optimal and highest elevation selection algorithms with
the 4-step algorithm that only selects 4 satellites. Figure 4.13 clearly shows that the
quasi-optimal algorithm outperforms the other sub-optimal algorithms, including the
4-step selection algorithm.
Figure 4.14, shows the distribution of ζ’s when 6 satellites are selected from the
same simulated constellation geometries. However, the 4-step selection algorithm only
selected 4 satellites since that is the limit of the algorithm (although it can be forced to
select more). It is clear in the figure that the ζf ourstep value is substantially increased
since the optimal method selected 6 satellites as well. Also note that the highest
elevation selection method performs much better than the case where 4 satellites
4.4. QUASI-OPTIMAL SATELLITE SELECTION
Quasi−optimal
Highest Elevation
80
4−step Method
80
Orbit Simulation
5 orbit (N=1000)
Selected 4 satellites
80
Orbit Simulation
5 orbit (N=1000)
Selected 4 satellites
70
70
60
60
50
50
50
40
%
60
%
%
70
109
40
40
30
30
30
20
20
20
10
10
10
0
1
2
3
(PDOPquasi)/(PDOPoptimal)
0
1
2
3
(PDOPhighElev)/(PDOPoptimal)
Orbit Simulation
5 orbit (N=1000)
Selected 4 satellites
0
1
2
3
(PDOPfourstep)/(PDOPoptimal)
Figure 4.13: Distribution of ζ when 4 Satellites are Selected. (LEO simulation)
are selected from the random geometries. It is because the NAVSTAR constellation
is designed to provide good measurement geometry to the terrestrial users with a
reasonable elevation mask angle. However, the quasi-optimal selection algorithm
again outperformed the other methods, which shows that the ζquasi distributions
are almost independent of the geometries of provided or number of satellites being
selected.
Selection from LEO Spacecraft Formation
A vehicle in an ns spacecraft formation could observe local constellation (ranging
signals from other vehicles) in the formation in additiona to the NAVSTAR constellation. This adds to the total number of measurements to choose from and complicates
the geometry of the overall visible constellation. Again, the quasi-optimal algorithm
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
110
Quasi−optimal
Highest Elevation
100
Orbit Simulation
5 orbit (N=1000)
Selected 6 satellites
90
100
Orbit Simulation
5 orbit (N=1000)
Selected 6 satellites
90
80
80
70
70
70
60
60
60
50
%
80
%
%
90
4−step Method
100
50
50
40
40
40
30
30
30
20
20
20
10
10
10
0
1
(PDOP
2
)/(PDOP
quasi
3
)
optimal
0
1
(PDOP
2
)/(PDOP
highElev
3
)
optimal
Orbit Simulation
5 orbit (N=1000)
Selected 4 satellites
0
1
(PDOP
2
)/(PDOP
fourstep
3
)
optimal
Figure 4.14: Distribution of ζ when 6 Satellites are Selected. (LEO simulation)
and the highest elevation satellite selection algorithms are applied to the simulated
combined local and NAVSTAR constellation. Figure 4.15 shows the distribution of
ζ for each algorithm. Since, from the algorithms’ point of view, including the local
constellation has the same effect as including more measurements, the distributions
are similar to Figure 4.14. In this simulation, the local and NAVSTAR measurements
are treated equally, allowing the local measurements to be eliminated. However, it
may desirable to keep all the local measurements in certain mission scenarios since
they may contain inter-spacecraft data or they may have higher resolution than the
NAVSTAR signals. This situation is better handled by the quasi-optimal selection
structure since the algorithm can be easily modified such that local measurements
are not allowed to be rejected. In this case, selection of the NAVSTAR signals will
consider the fact that there is a pre-selected local constellation. The other selection
4.4. QUASI-OPTIMAL SATELLITE SELECTION
Quasi−optimal
Highest Elevation
100
100
5 Vehicle Formation
Orbit Simulation, 5 orbit
(N=1000)
Selected 6 Measurements
90
80
70
70
60
60
50
50
40
40
30
30
20
20
10
10
1
2
(PDOP
3
)/(PDOP
quasi
5 Vehicle Formation
Orbit Simulation, 5 orbit
(N=1000)
Selected 6 Measurements
90
%
%
80
0
111
4
0
1
)
optimal
2
(PDOP
3
)/(PDOP
highElev
4
)
optimal
Figure 4.15: Distribution of ζ when 6 Measurements are Selected. (LEO
simulation, 5 Vehicle Formation)
methods cannot readily handle this situation and it is possible that they would select NAVSTAR measurements that overlap with the local constellation. Section 4.4.4
generalizes this aspect of the selection problem by enabling weighting factors for the
quasi-optimal selection process.
Ground Users
Users on the surface of the Earth can typically see 6–9 satellites. Because there are
a small number of visible satellites, it is often possible to use the optimal method
or use all of the measurements available (unless there are a large number of local
range measurements in addition to the NAVSTAR constellation). However, to further
examine the performance of the quasi-optimal algorithm, the algorithm is applied to
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
112
Highest Elevation
100
90
90
80
80
70
70
60
60
%
%
Quasi−optimal
100
50
50
40
40
30
30
20
20
10
10
0
1
2
(PDOP
3
)/(PDOP
quasi
4
)
optimal
0
1
2
(PDOP
3
)/(PDOP
highElev
4
)
optimal
Figure 4.16: Distribution of ζ when 6 Measurements are Selected. (Ground
simulation)
a simulated terrestrial environment. The following simulation results (Figure 4.16)
show the distribution of ζ for two algorithms. In this simulation, a user was stationary
at the Equator and selects 6 satellites from 7–10 visible satellites. As expected, the
highest elevation satellite selection method performs slightly better on the ground
than in LEO. However, the quasi-optimal selection method performed extremely well
(max ζquasi ∼
= 1.04). These results indicate that the quasi-optimal selection method
could also be useful for terrestrial applications.
4.4.2
Computational Load
The computational efficiency of the quasi-optimal algorithm is presented in this section. Figure 4.17 compares the number of floating point operations required for the
4.4. QUASI-OPTIMAL SATELLITE SELECTION
113
Number of Floating Point Operations Required to Select 6 Sats
7
10
Quasi−optimal
Optimal
Four−step
6
Floating Point Operation Count
10
5
10
4
10
Four−step Algorithm Selected 4 satellites
3
10
2
10
7
8
9
10
11
12
Number of satellites in view
13
14
15
Figure 4.17: Computational Loads Required in Each Method
optimal, quasi-optimal, and 4-step solutions. It assumes that 6 satellites are to be selected from the given number of visible satellites shown in the x-axis (7–15 satellites).
However, the four-step algorithm only selects 4 satellites from the 7–15 satellites.
The graph shows that the number of floating point operations required for the optimal satellite selection grows rapidly as the number of visible satellites increases.
The quasi-optimal algorithm shows a moderate increase, but it offers 2–3 orders of
magnitude savings in the computational load when compared to the optimal method
for 13 satellites (typical LEO case).
CHAPTER 4. QUASI-OPTIMAL SATELLITE SELECTION
114
4.4.3
Other Cost Functions
Faster Algorithm
A faster version of the quasi-optimal algorithm can be developed by using a different
cost function Jia . By taking the 1-norm of the ith row instead of the sum of the
squares of the elements in the ith row, the overall number of floating point operations
can be reduced by about 30%.
Jia =
N
#
j=1
| cos(θij )| =
N
#
|dij |
(4.16)
j=1
However, the faster algorithm does not perform as well as the regular quasi-optimal
algorithm described in the previous sections. This is because the cost function cos(2θ)
is sharper around θ = 0◦ , ±180◦ and smoother around θ = ±90◦ than the cost function
| cos θ|.
4.4.4
Weighted Quasi-optimal Selection Algorithm
The optimal geometry does not necessarily provide the optimal position and velocity
estimates. There are other factors that determine the accuracy of the estimates.
Certainly, these factors can be accounted for in the position/velocity estimation step.
However, if there are a large number of measurements available, it would also be wise
to consider these factors in the selection process. The quasi-optimal selection process
allows the inclusion of a weighting term for each of measurements that can be based on
factors other than the geometry. These factors could include the quality of the signal
and the covariance of the carrier phase bias estimates. The quality of the measured
signal is determined by the signal to noise ratio or the elevation angle of the signal
since the signals that travel through the earth atmosphere are often degraded/delayed.
The covariance of the bias estimate also indicates the quality of the carrier phase
measurements. In addition, the local range measurements may have higher accuracy
than the NAVSTAR signals and, in some cases, it is highly desirable not to reject
these measurements. These factors must be considered in the measurement selection
4.5. SUMMARY
115
and the quasi-optimal algorithm can incorporate these weighting factors in a very
efficient way. Recall that Ji is the cost for the ith measurement.
Ji =
N
#
(cos 2θij )
(4.17)
j=1
The cost for the ith measurement can be multiplied by the relative weighting factor
wi which is calculated based on the factors discussed above. This gives the weighted
cost Jiw for the ith measurement
Jiw = wi
N
#
(cos 2θij )
(4.18)
j=1
Then, the algorithm eliminates the measurement corresponding to the maximum
weighted cost. The weighted quasi-optimal selection algorithm then provides near
optimal selection for a given geometry and quality of the measurements. It can also
be used to substantially reduce the number of satellites switching that cause the slow
convergence of the carrier phase ambiguity states and less smooth state estimates.
The number of satellite switching is reduced by weighting the cost function with the
covariance of the bias states. For example, signals with good bias estimates are less
likely to be eliminated unless they form a bad geometry.
4.5
Summary
In conclusion, the quasi-optimal method is shown to provide near-optimal satellite
geometry with significantly reduced computational effort than previously published
algorithms. As a result, the quasi-optimal algorithm is well suitable for the real-time
applications that require frequent geometry updates (∼ 1Hz), which is especially true
for LEO formation flying applications where a large number of (local and NAVSTAR)
measurements are available.
Chapter 5
Mission Simulations
A series of simulations were conducted in order to assess the benefits and the limitations of using onboard transmitters on the formation flying vehicles. These simulations involved multiple spacecraft in various Earth orbits and multiple aircraft
performing an aerial rendezvous. The first set of simulations show the improvements
in the state and bias estimates that can be obtained by including a simple orbit model
and the onboard transmitter measurements. The next section simulates long baseline
formation flying spacecraft and evaluates the effect of the long baseline correction
terms on the estimation. The effect of the formation geometry and transmitter accuracy on the position and velocity estimation errors are demonstrated by analyzing
various formation geometries and onboard ranging devices with various levels of precision. Finally, an example mission was selected that can potentially have major
visibility/geometry problems. Extensive simulations are performed for this sample
mission and the results are presented. In addition, a sample terrestrial mission (airto-air refueling mission) was investigated that required a rapid bias initialization. The
results from the simulation indicates that the onboard transmitter can significantly
improve the initialization process in the terrestrial missions. These simulation results
show that the augmented CDGPS algorithms developed in the previous chapters can
improve relative position and velocity estimation process for the space users as well
as the terrestrial users under various circumstances (Chapter 1).
116
5.1. LEO FORMATION WITH ONBOARD TRANSMITTERS
117
Table 5.1: Orbit Parameters of Simulated Vehicles
eccentricity
1e−5
5.1
inclination
35◦
altitude
ascending node
354.860 km
20◦
argument of perigee
60◦
LEO Formation with Onboard Transmitters
This section presents various orbital simulation results for a formation of spacecraft
in LEO with onboard transmitters. It is assumed in the following simulations that
the ranging signals from all the spacecraft in the formation are available throughout the simulation period. It is also assumed that the number of the NAVSTAR
satellites that can be used in the estimation is limited (specific number given in each
simulation). This accounts for the constraint in the number of channels available to
track the NAVSTAR satellites, which is limited by the receiver hardware and the
number of pseudolites in the fleet (assuming pseudolites are used as the local ranging
device). If other types of local ranging devices are used in the fleet, a separate receiver/transceiver would be used and all the channels in the GPS receiver could then
be dedicated to the NAVSTAR signals.
5.1.1
Orbit Selection
Typical orbit parameters were selected for the formation flying spacecraft in LEO, as
shown in Table 5.1. Other variables, such as the number of spacecraft in the formation and the distance between the spacecraft, were changed in the simulations. The
NAVSTAR satellite constellation orbital parameters were obtained from a recent GPS
constellation almanac and the same orbital parameters were propagated analytically
using a simple Keplerian orbit model [EO99].
5.1.2
LEO Simulation Parameters
The position and velocity estimation processes and the bias estimation process are
coupled as shown in Section 2.4.1. An extended Kalman filtering algorithm is developed in Chapter 3 for the bias state estimation. However, both an EKF algorithm
CHAPTER 5.
118
MISSION SIMULATIONS
Table 5.2: Selected Variables for Simulation in Section 5.1.2
Simulation Variables
number of spacecraft
number of channels for NAVSTAR satellites
nominal distance between spacecraft
simulation period
NAVSTAR differential carrier phase noise ν s
NAVSTAR differential carrier Doppler noise ν̇ s
NAVSTAR differential code phase noise
NAVSTAR pseudorange uncertainty
onboard transmitter carrier phase noise ν p
onboard transmitter carrier Doppler noise ν̇ p
4
6
1 km
∼1 orbit period
σ = 0.02 m
σ = 0.005 m/s
σ=2m
σ = 25 m
σ = 0.02 m
σ = 0.005 m/s
Table 5.3: Initial Conditions of Simulated Vehicles: Relative Position and
Velocity (ECI)
Vehicle 2
Vehicle 3
Vehicle 4
x (m)
y (m)
z (m)
vx (m/s)
639.742
-295.776
0
-0.1601
599.926
1275.521 -1418.850 -0.3201
-2818.882 -1026.598
-0.401
0.9612
vy (m/s)
0.4398
0.8797
-2.6387
vz (m/s)
0.3277
0.6554
-1.9663
and a weighted least squares solutions are available for the position/velocity states
depending on the availability of the dynamic model. A detailed algorithm description
is given in Sections 2.4.1 and 2.4.2. In this section, simulation results of the EKF state
estimator are compared to those of the WLS state estimator and the performance of
the EKF bias estimator in LEO is demonstrated. Table 5.2 summarizes the variables
selected for this simulation. Note that the measurement noise levels of the onboard
transmitters are chosen to be the same as that of the NAVSTAR constellation. The
effect of changing this accuracy is investigated in detail in Section 5.4. Typical 3–D
trajectories of the vehicle motion relative to the reference vehicle are shown in Figure 5.1. The plot shows the unforced relative motions of all 3 vehicles around the
reference vehicle in the earth centered inertial directions (ECI) with the given initial
conditions (Table 5.3) for approximately one orbit [MHK76].
5.1. LEO FORMATION WITH ONBOARD TRANSMITTERS
119
Unforced Relative Motion of 3 Vehicles w.r.t. the Reference Vehicle (ECI)
3000
2000
z (m)
1000
0
−1000
−2000
−3000
3000
2000
1000
Formation Center (Reference Vehicle)
0
−1000
−2000
y (m)
−3000
−3000
−2000
−1000
0
1000
2000
3000
x (m)
Figure 5.1: Relative Motions in the Formation Frame (Formation frame
aligned with ECI directions)
5.1.3
EKF Bias Estimation in LEO
Figure 5.2 shows the typical convergence characteristics of the bias states by the EKF
bias estimator. It is assumed in this initial simulation that the local transmitter measurements also have the cycle ambiguities (biases) even though several proposed high
performance local ranging device may not have bias ambiguities in the measurements.
In this scenario, it is required to resolve the ambiguities in both the NAVSTAR and
the local measurements accurately and promptly. Figure 5.2 shows the magnitude of
the bias errors of the 6 NAVSTAR measurements as well as the 6 local transmitter
measurements. As discussed in Chapter 3, rapid changes in the line-of-sight from
the receiver to the signal source is essential in the bias EKF estimator. Although
the line-of-sight vectors to the NAVSTAR constellation change fairly rapidly in LEO,
CHAPTER 5.
120
MISSION SIMULATIONS
these measurements are also acquired and lost quite frequently. Thus, frequent satellite hand-overs (indicated by the spikes) can be seen on the figure. This combination
tends to cause a relatively slow convergence rate, as shown in Figure 5.2. The onboard
transmitter measurements, on the other hand, are generally available to all the vehicles in the formation throughout the orbit and the line-of-sight vectors between the
vehicles are also changing fairly rapidly. As a result, the local transmitter bias state
converges faster, thus the local transmitter measurements tend to be more accurate
measurements in terms of the magnitude of the ambiguity states in the measurements.
Another set of simulations was conducted to compare the bias errors of the NAVSTAR only system with the augmented NAVSTAR system. In the simulations, there
were 4 vehicles in the formation and total of 24 measurements are used for both
systems.
1. Total number of measurements of the NAVSTAR only system: 8 NAVSTAR
measurements × 3 vehicle = 24
2. Total number of measurements of the augmented system: 6 NAVSTAR measurements × 3 vehicle + 6 Local transmitter measurements = 24
Figure 5.3 compares the magnitude of the entire bias state vectors (24×1 vector).
Since the augmented system includes 6 local measurements and the magnitude of
the bias in these measurements is lower than the NAVSTAR measurements due to
the continuity of the local measurements (Figure 5.2), the magnitude of the total
bias state vector (24×1 vector) of the augmented system is smaller than that of the
NAVSTAR only system.
In these simulations, the robust bias initialization algorithm described in Chapter 3 was not needed since the separation between the spacecraft was large (1∼3 km)
compared to the initial uncertainty level (≤5 m) that the nonlinear effect was negligible. Thus, the old (first order) initialization filter [CP00] was used to simplify the
simulation process.
5.1. LEO FORMATION WITH ONBOARD TRANSMITTERS
121
Magnitude of Bias States
1
10
NAVSTAR
Local Transmitter
0
10
Magnitude (m)
−1
10
−2
10
−3
10
−4
10
0
1000
2000
3000
time (sec)
4000
5000
6000
Fig. 5.2: Magnitude of Bias Errors of NAVSTAR measurements and Local Ranging
Measurements in LEO
5.1.4
EKF State Estimation - EKF Bias Estimation
An extended Kalman filter can be used in the state estimation process to improve
the estimates. Section 2.4.2 describes how the orbit dynamic model is incorporated
in the state estimation process. A series of orbit simulations was conducted to verify the EKF state estimation algorithm described in Section 2.4.2. Note that this
extended Kalman filter runs in parallel with the EKF for the bias state. There are
3 vehicles in the formation and the nominal distance between the spacecraft is approximately 1000 m. They all have the onboard transmitters for the local range
measurements. The WLS and extended Kalman filter algorithms were applied to the
simulated NAVSTAR and local measurements. The parameters listed in Table 5.2
were used in the simulation. The process noise magnitude of σ = 1e−5 m/s2 was
CHAPTER 5.
122
MISSION SIMULATIONS
Magnitude of Bias States
2
10
Augmented NAVSTAR system
NAVSTAR system
1
10
Magnitude (m)
0
10
−1
10
−2
10
−3
10
0
1000
2000
3000
time (sec)
4000
5000
6000
Fig. 5.3: Magnitude of Entire Bias State Vector of NAVSTAR-only System and
Augmented System
assumed in the orbit propagator and this information was used in the EKF estimator. Figure 5.4 compares the relative position errors of the WLS estimator with the
EKF estimator. The mean position error of the WLS estimator after the convergence
(t = 3000 ∼ 6000 seconds) is 0.0222 m, but that of the EKF state estimator is only
0.0059 m. The standard deviation of the EKF position estimates (σ = 0.0025 m) is
also lower than the WLS estimates (σ = 0.0112 m). The plot also shows that the
initial convergence rates are very similar for these two estimators, which is a result
of using the same EKF bias estimation algorithm. The improvements are more pronounced in the velocity estimates. Figure 5.5 compares the relative velocity estimate
error of the WLS estimator with the EKF estimator. The mean velocity error of
the EKF estimator is only 0.119 mm/s compared to 11.2 mm/s of WLS estimator.
5.1. LEO FORMATION WITH ONBOARD TRANSMITTERS
123
Relative Position Error
0
10
WLS estimation
EKF estimation with orbit model
−1
Error Magnitude (m)
10
−2
10
−3
10
−4
10
0
1000
2000
3000
time (sec)
4000
5000
6000
Figure 5.4: Relative Position Error Comparison EKF vs. WLS
Relative Velocity Error
2
10
1
Error Magnitude (mm/s)
10
0
10
−1
10
−2
10
WLS estimation
EKF estimation with orbit model
−3
10
0
1000
2000
3000
time (sec)
4000
5000
Figure 5.5: Relative Velocity Error Comparison EKF vs. WLS
6000
CHAPTER 5.
124
MISSION SIMULATIONS
Table 5.4: Performance of EKF and WLS State Estimators
mean
σ
position error (m)
WLS estimator EKF estimator
0.0222
0.0059
0.0112
0.0025
velocity error (mm/s)
WLS estimator EKF estimator
11.1525
0.1194
7.2024
0.0506
Table 5.4 summarizes the improvements achieved by incorporating a simple relative
orbit dynamic models.
5.2
5.2.1
Effect of Long Baseline Correction Terms
Corrections for Differential Phase Measurements ∆φ
As shown in Section 2.2, the planar wave front assumption can introduce a large bias
error in the phase measurements if the distance between the spacecraft in the formation is large. The impact of this term as well as the correction method is presented
in Section 2.2. In order to assess the effect of this error source on the position and
velocity estimator, an orbit simulation was performed with two spacecraft (for simplicity) that had onboard transmitters and a nominal separation of 2000 m. Other
simulation parameters were the same as the previous simulations (Tables 5.1 and 5.2).
Figure 5.6 shows the position error of the WLS estimator when the correction terms
are not applied to the measurement equation.
After the estimate of bias converges (t ≥ 3000sec), the mean value of the position
error is 0.1066 m if the onboard transmitter measurements are included. The mean
position error increased to 0.1889 m if onboard transmitters were not used. The WLS
estimator that includes local transmitter measurements has a smaller position error
because the local range measurements do not need corrections. A second simulation
was conducted under the same conditions, but the estimator included the correction
terms discussed in Section 2.2. In this case, the mean position error values were significantly smaller (0.0354 m with the transmitter and 0.0485 m without the transmitter)
5.2. EFFECT OF LONG BASELINE CORRECTION TERMS
125
Table 5.5: Effect of ∆φ̇ and ∆φ on Estimation Error (WLS) with Vehicle
Separation of 2 km.
Without Correction
With Correction
Mean position error (m)
Mean velocity error (m/s)
w/ onboard w/o onboard w/ onboard w/o onboard
transmitter transmitter transmitter transmitter
0.1066
0.1889
0.6086
0.6346
0.0354
0.0485
0.0193
0.0195
than the case in which the correction terms were not applied. Figure 5.7 shows the
position errors for the second simulation.
5.2.2
Corrections for Differential Doppler Measurements ∆φ̇
It can be expected from Figures 2.4 and 2.6 that the Doppler measurements are more
sensitive to the nonlinear effect. In the figures, the magnitude of the Max. Doppler
correction terms are larger and grow faster than the magnitude of the Max. phase
correction terms for separations under ∼10 km. Figure 5.8 shows the simulated velocity error of the WLS estimator. As expected, the relative velocity estimation error
due to this effect is so large that they are no longer better than the absolute velocity
accuracy. The mean value of the velocity errors are 0.6086 m/s and 0.6346 m/s with
and without onboard transmitters, respectively. It is clear in this figure that the lineof-sight dependent error term ∆φ̇ (Eq. 2.17) dominates the total estimation error
since the error magnitude clearly shows a trend associated with the satellite motions
(line-of-sight change) and the satellite hand-overs. The correction terms can be calculated using a rough estimate of the velocity of the NAVSTAR satellites and the
line-of-sight vectors from each vehicle to the NAVSTAR satellites. Figure 5.9 shows
the velocity error magnitude when the correction terms are applied to the estimator.
The mean velocity errors are approximately 0.019 m/s, which is substantially lower
than the mean values without the corrections (∼0.60 m/s). Table 5.5 summarizes the
effects of including the phase and Doppler correction terms.
CHAPTER 5.
126
MISSION SIMULATIONS
Position Error
3.5
With Onboard Transmitter
Without Onboard Transmitter
3
error (m)
2.5
2
1.5
1
0.5
0
0
1000
2000
3000
time (sec)
4000
5000
6000
Figure 5.6: Position Error (no corrections applied, WLS)
Position Error
2.5
With Onboard Transmitter
Without Onboard Transmitter
2
error (m)
1.5
1
0.5
0
0
1000
2000
3000
time (sec)
4000
5000
Figure 5.7: Position Error (corrections applied, WLS)
6000
5.2. EFFECT OF LONG BASELINE CORRECTION TERMS
127
Velocity Error
1.6
1.4
error (m/sec)
1.2
1
0.8
0.6
0.4
0.2
0
1000
2000
3000
time (sec)
4000
5000
6000
Figure 5.8: Velocity Error (no corrections applied, WLS)
Velocity Error
0.18
0.16
0.14
error (m/sec)
0.12
0.1
0.08
0.06
0.04
0.02
0
0
1000
2000
3000
time (sec)
4000
5000
Figure 5.9: Velocity Error (corrections applied, WLS)
6000
128
5.3
CHAPTER 5.
MISSION SIMULATIONS
Terrestrial Applications
As shown in Table 1.2, the onboard transmitters are valuable in the terrestrial relative
navigation systems as well. It is especially useful in the rapid bias initialization
process and the geometry augmentation. Typical terrestrial formation flying missions
include air-to-air refueling mission (Figure 1.10) and the aircraft formation flying
maneuvers (Figure 1.8). It will be extremely beneficial if these complex operations
can be automated (Chapter 1). The CDGPS sensor can be used for an autonomous
navigation system for these maneuvers, but including an onboard ranging device
solves two major problems with the NAVSTAR-only CDGPS sensor for this type of
missions. The onboard transmitter augmented system, in general, can resolve cycle
ambiguities much faster than the NAVSTAR only system and improve measurement
geometry (PDOP) in case the NAVSTAR signals are blocked by nearby vehicles or
if the NAVSTAR constellation provides a poor geometry. A simulation of air-toair refueling mission was developed in order to evaluate the benefits of an onboard
transmitter in the terrestrial mission scenarios. The simulation included 2 aircraft: A
tanker aircraft (modeled after the Douglas DC-10) and a receiver aircraft. The tanker
was in a level flight at the altitude of 10,000 m with the cruise velocity (relative to the
Earth) of 277.778 m/s. The tanker was also equipped with the onboard transmitter(s).
Figure 5.10 shows two different onboard transmitter configurations at the rear end of
the tanker aircraft. The first transmitter configuration has 3 ranging devices: one at
each wing-tip and one on top of the vertical stabilizer. The second configuration has
only one transmitter on top of the vertical stabilizer.
Additional simulation parameters are listed in Table 5.6. The receiver aircraft then
intercepted the tanker from behind, formed a formation, and gradually decreased the
relative distance to the tanker. The horizontal and vertical relative flight paths of the
receiver aircraft with respect to the tanker in the inertial (ECI) directions are shown
on Figure 5.11 and 5.12, respectively. The final relative distance was approximately
10 m, at which point a contact between the tanker and the receiver aircraft could
be made. This initial relative motion provided a rapid line-of-sight change from the
5.3. TERRESTRIAL APPLICATIONS
129
TX1
17.7 m
TX2
TX3
50.4 m
TX1
Figure 5.10: Transmitter Locations on DC-10 Aircraft
transmitters onboard the tanker to the receiver aircraft in the course of the maneuver,
which is essential in the bias initialization process.
Figure 5.13 shows line-of-sight angles from the each transmitter onboard the tanker
to the receiver aircraft during the 200 sec motion illustrated in Figures 5.11 and 5.12.
Because the receiver aircraft is far away from the tanker in the initial phase and the
maximum separation between the onboard transmitters is less than the wing span of
the tanker (50 m), the line-of-sight angles from the 3 transmitters to the receiver are
almost identical (t ≤ 60 sec). However, the initial line-of-sight angle change enabled
the rapid real-time bias initialization procedure [CP00] when these measurements were
used in the estimation. During the final approach (t ≥ 150 sec), the distance between
the aircraft is comparable to the separation between the transmitters. Therefore, the
line-of-sight vectors from the receiver to the 2 transmitters on the wing-tip expand
out to ±80◦ , providing improved geometry (especially when the NAVSTAR signals
CHAPTER 5.
130
MISSION SIMULATIONS
Table 5.6: Parameter Selected for Air-to-air Refueling Simulation
tanker altitude
tanker velocity
tanker wing span
tanker height
simulation period
number of transmitters on tanker
number of NAVSTAR signals available
transmitter carrier phase noise
transmitter Doppler noise
separation between transmitters at wing-tip
height of the transmitter on vertical
stabilizer w.r.t. wing-tip
10,000 m
277.778 m
50.4 m
17.7 m
200 seconds
0, 1, or 3
6–7
σ = 0.02 m
σ = 0.005 m/s
50 m
10 m
Relative Vertical Motion of the Receiver Aircraft to the Tanker Aircraft
Relative Horizontal Motion of the Receiver Aircraft to the Tanker Aircraft
800
1000
Initial Position of the Receiver
800
600
600
400
400
200
z (m)
y (m)
200
0
0
−200
−400
−200
−600
Tanker (0,0)
−400
−800
−1000
0
200
400
600
800
1000
x (m)
1200
1400
1600
1800
2000
Fig. 5.11: Relative Horizontal Motion of
Receiver Aircraft
−600
0
20
40
60
80
100
time (sec)
120
140
160
180
Fig. 5.12: Relative Vertical Motion of the
Receiver Aircraft
are blocked by the tanker). The PDOP improvements are shown in Figure 5.14. The
figure shows the benefits of using 3 transmitters instead of 1. Additionally, in either
case, the PDOP improvement is the most pronounced near the end of the maneuver.
The improvement in the PDOP and the availability1 are extremely important during
1
200
At least 4 commonly visible measurements are necessary.
5.3. TERRESTRIAL APPLICATIONS
131
Line−of−sight Angle Changes from each Transmitter to Receiver Aircraft
PDOP Values during the Refueling Mission
80
1.8
Transmitter on vertical stabilizer
Transmitter on Right wing−tip
Transmitter on Left wing−tip
60
3 transmitters
1 transmitter
0 transmitter
1.7
40
1.6
1.5
0
PDOP
angle (degree)
20
−20
1.4
−40
1.3
−60
1.2
−80
−100
1.1
0
20
40
60
80
100
time (sec)
120
140
160
180
200
Fig. 5.13: Line-of-sight Angles during
Relative Motion (From each transmitter to
the receiver aircraft)
1
0
20
40
60
80
100
time (sec)
120
140
160
180
200
Fig. 5.14: PDOP Values of Each Configuration During Relative Motion
the final “contact” stage because the highest integrity of the navigation system would
be required at this point of an automated rendezvous.
The WLS relative position and velocity estimator was applied to the simulated
data generated during the relative motion described in Figure 5.11 and the EKF bias
estimator was run simultaneously to estimate the bias state. Figure 5.15 shows the
RMS position errors when various number of the onboard transmitters are used in
addition to the 6–7 available NAVSTAR signals. When no local transmitter measurement is available, the position error decreases very slowly and the final position
error is approximately 50 cm since the cycle ambiguity state cannot be estimated
to a sufficient accuracy within 200 seconds. On the other hand, if 1-3 local measurements are available, the RMS position error converges to under 5 cm within one
minute. Slightly faster initial convergence and smaller final error can be obtained
with 3 onboard transmitters.
Figures 5.16 and 5.17 show the improvement in the position and velocity estimates
(mean and standard deviation) by including 1–3 onboard transmitters. In the velocity estimation, improvements are minimal as the number of the local measurements
increases. This is because the velocity measurements do not have biases and the same
measurement noise level as the NAVSTAR measurements (σ =0.005 m/s) is selected
CHAPTER 5.
132
MISSION SIMULATIONS
RMS Position Errors
1
10
No transmitter
1 Transmitter
3 Transmitters
0
RMS Error (m)
10
−1
10
−2
10
5cm RMS error
−3
10
0
20
40
60
80
100
time (sec)
120
140
160
180
200
Figure 5.15: RMS Position Errors During Refueling Maneuver (0,1,3 on-
board transmitter(s) is(are) used)
for the local Doppler measurements. However, the position estimates improve dramatically by incorporating one onboard transmitter in the system. Including additional
local measurements (to a total of three) improves the viewing geometry and provides
slightly improved estimates over the single local transmitter case (Figure 5.16). Table 5.7 summarizes the results of the simulation. In the case of an occlusion of the
NAVSTAR signals (did not happen in this particular simulation), multiple transmitters onboard the tanker aircraft may be required to retain an adequate geometry.
This aspect of the problem is investigated in detail in Section 5.5.
5.4. FORMATION GEOMETRY & IMPROVED TRANSMITTERS
Position Error Statistics
Velocity Error Statistics
0.015
mean
standard deviation
0.8
133
mean
standard deviation
0.7
0.6
0.01
meter/sec
meter
0.5
0.4
0.3
0.005
0.2
0.1
0
0
0
1
Number of transmitters
3
Fig. 5.16: Position Error Statistics
0
1
Number of transmitters
3
Fig. 5.17: Velocity Error Statistics
Table 5.7: Air-to-air Refueling Simulation Results
Number of onboard
Transmitter
0
1
3
5.4
Position
mean
0.8264
0.0307
0.0222
error (m)
σ
0.2968
0.0114
0.0101
Velocity error (m/s)
mean
σ
0.0150
0.0095
0.0137
0.0085
0.0127
0.0082
Convergence
time (sec)
N/A
41
58
Formation Geometry & Improved Transmitters
The improvements in the estimation discussed in the previous section will depend on
both the noise level in the measurement of the onboard ranging device (ISL: Inter
Spacecraft Link) and also the geometric configuration of the formation (relative local measurement geometry of one vehicle to another). Simulations were developed
to investigate how each of these parameters influence the accuracy of the position
estimates. Typical orbit parameters were selected for the reference vehicle, as shown
in Table 5.1. Initial relative position and velocity of other vehicles are then computed [MHK76] according to the formation geometry selected in each simulation. It
is assumed in this simulation that this high performance ranging transmitter (ISL)
CHAPTER 5.
134
MISSION SIMULATIONS
Fig. 5.18: Satellites with In-track Sep- Fig. 5.19: Satellite with In-track and
(small) Cross-track Separation
aration
does not have biases in the measurements. However, the NAVSTAR carrier phase
measurements will have the cycle ambiguities. The EKF position estimation algorithm developed in Chapter 2 is used in this section. In the following simulations, the
improvement in the relative position estimates are presented. In general, the local
transmitters can be designed to provide more accurate Doppler measurements and
achieve the improved relative velocity estimates, but the simulation results are not
shown in this section.
5.4.1
Line Formation (In-Track)
Figure 5.18 shows the first satellite configuration with four satellites in the same
orbit, but with a 100 m in-track separation between vehicles. This geometry is a
good configuration from the perspective of differential disturbances, and thus is often
selected as a starting point for formation flying [FB97]. However, this offers a very
limited geometry for the local ranging measurements, and it is expected that only the
in-track direction estimation will be influenced by these measurements. The other
two directions (radial and cross-track) will have to rely almost entirely on the GPS
measurements. In order to verify the effect of the geometry, the estimation errors
calculated in the Earth Centered Inertial (ECI) frame were transformed into a local
orbit frame (LVLH). Simulations were run for various noise levels on the local ranging
device. The results in the LVLH (in-track, cross-track, radial) were then plotted in
5.4. FORMATION GEOMETRY & IMPROVED TRANSMITTERS
135
Intrack Orbit
0
RMS error normalizated to GPS
10
−1
10
Radial
In−Track
Cross−Track
Total Error
−2
10
−3
10
−2
−1
10
10
Standard deviation of ISL local ranging noise (m)
0
10
Fig. 5.20: RMS Position Error (In-track separation only)
Figure 5.20. The figure shows the RMS errors of the relative position estimates,
normalized to the RMS error level of the GPS-only system, versus the standard
deviation of the ranging noise of the local ranging transceiver.
As expected, Figure 5.20 shows that the in-track improvement is much more substantial than the other two directions as the noise on the ranging device is reduced.
Also as expected, when the ranging device is relatively noisy, there is no improvement
in the in-track ranging estimation (estimation asymptotes to GPS level of estimation
error). The results also show that the in-track accuracy improves by about a factor of
five when the standard deviation of the ranging noise drops from ≈1 cm to ≈1 mm.
However, the plot also shows that the effect of this in-track improvement on the total
positioning error is quite small - it is still dominated by the cross-track and radial
errors.
CHAPTER 5.
136
MISSION SIMULATIONS
Cross−track Orbit
0
RMS error normalizated to GPS
10
−1
10
Radial
In−Track
Cross−Track
Total Error
−2
10
−3
10
−2
−1
10
10
Standard deviation of ISL local ranging noise (m)
0
10
Fig. 5.21: RMS Position Error (In-track and small cross-track component)
5.4.2
Improved Line Formation (In-Track & Cross-track)
The configuration in Figure 5.19 adds some cross-track motion between the satellites
to improve the geometry. As shown, the first satellite is assumed to maintain a fixed
reference orbit, and the other 3 are initialized to have cross-track relative positions
that are approximately 50% of its in-track separation (i.e., satellite 4 would have a
300 m in-track and 150 m cross-track separation from satellite 1). This initialization
process (small orbit inclination changes) results in a periodic motion of the satellites
in the cross-track direction relative to the reference satellite. The resulting estimation
errors for this case are plotted in Figure 5.21.
As would be expected from the improvements to the geometry, the results in
Figure 5.21 indicate that both the in-track and cross-track directions improve as
the local ranging noise is decreased. However, the cross-track improvement is not
5.4. FORMATION GEOMETRY & IMPROVED TRANSMITTERS
Fig. 5.22: Ellipse Formation on Reference Orbit
137
Fig. 5.23: Double Ellipse Formation on
Reference Orbit
substantial, and a more pronounced reduction in the improvement in the in-track
direction is observed. This is an indication that the geometry is still more or less the
same as the purely in-track separation geometry (line formation). Additionally, the
total position estimation error still only decreases slightly with this minor geometric
improvement compared to that of the line formation.
5.4.3
Ellipse Formation (Single Ellipse)
In order to make a significant enhancement in the total position estimation, the
geometry of the local measurements must be good in all three directions. One way
to accomplish this is to move the satellites off the reference orbit. Thus, the next
simulation setup had the satellites follow an ellipse around the reference orbit. Given
proper initial conditions, the satellites will maintain free elliptical motion relative
to the reference orbit without any thruster usage [MHK76]. Figure 5.22 shows a
visualization of this ellipse configuration.
This formation has the advantage over the previous line formation geometry that
each of the measurement line-of-sight vectors spans multiple directions inside the
ellipse plane (2–D) rather than all being aligned with the in-track direction (1–D).
As the satellites travel around the ellipse, the precise local range measurements will
be available from many directions. Thus, this geometry should be able to improve
estimates for all 3 directions (to some extent) and to reduce the total position error as
CHAPTER 5.
200
200
150
150
100
100
Cross−Track (m)
Cross−Track (m)
138
50
0
−50
MISSION SIMULATIONS
50
0
−50
−100
−100
−150
−150
−200
−200
200
150
150
100
100
50
50
0
0
−50
−100
−150
−200
Intrack (m)
−50
−100
−150
100
50
0
−50
Intrack (m)
−100
Radial (m)
Radial (m)
Fig. 5.24: 3-D Relative Motion on Single Ellipse
100
0
Fig. 5.25: 3-D Relative Motion on Two
Ellipses (180◦ Phase)
the transmitter ranging noise level decreases. The results in Figure 5.27 indicate that
there is an improvement in the radial and cross-track direction estimates as well as
in the in-track direction when compared to Figures 5.20 or 5.21 due to the improved
geometry.
Figure 5.24 shows the actual 2×1×2 (200 m in-track / 100 m radial / 200 m
cross-track semi-axes) elliptical relative motion used in the simulation, as well as
the line-of-sight directions between the vehicles. The poor radial improvements in
Figure 5.27 are consistent with the smaller relative motions in the radial direction in
Figure 5.24.
5.4.4
Double Ellipse Formation
One further geometric improvement can be made for this orbital formation. In the
configuration shown in Figure 5.24, all of the satellites remain inside the same elliptical plane rotating around the reference satellite. In order to provide a full 3–D
5.4. FORMATION GEOMETRY & IMPROVED TRANSMITTERS
139
200
150
Cross−Track (m)
100
50
0
−50
−100
−150
−200
200
100
0
100
50
−100
Intrack (m)
0
−200
−50
Radial (m)
Fig. 5.26: 3-D Relative Motion on Two Ellipses (90◦ Phase)
measurement geometry at any given instant, a fifth satellite was added. Two satellites were placed in a second ellipse with a different inclination angle than the first,
forming a double-ellipse configuration. The two satellites in the same ellipse were
also separated by a 90 degree phase (Figure 5.26) rather than the 180 degrees shown
on Figure 5.25. This prevents the satellites on the same ellipse from being co-linear
with the center satellite. Having both pairs separated by 180 degrees would simply
result in another plane being formed by the 5 satellites, but rotating in two directions
rather than one (Figure 5.25). The improved local constellation geometry resulting
from the 90 degree phasing is shown in Figure 5.26. Note that a “pyramid shaped”
measurement geometry (tetrahederon) is now formed from the satellites in the two
CHAPTER 5.
140
MISSION SIMULATIONS
Single Ellipse
0
RMS error normalizated to GPS
10
−1
10
Radial
In−Track
Cross−Track
Total Error
−2
10
−3
10
−2
−1
0
10
10
Standard deviation of ISL local ranging noise (m)
10
Fig. 5.27: RMS Position Error (Single ellipse formation)
Double Ellipse 90 deg Phase
0
RMS error normalizated to GPS
10
−1
10
Radial
In−Track
Cross−Track
Total Error
−2
10
−3
10
−2
−1
10
10
Standard deviation of ISL local ranging noise (m)
Fig. 5.28: RMS Position Error (Double ellipse formation)
0
10
5.5. POWERSAIL SIMULATION
141
ellipses rather than a simple plane. This configuration takes full advantage of the
precise local ranging measurements between the satellites, so it results in the best
estimation accuracy among the configurations considered in this section, as seen in
Figure 5.28. The phasing of the satellites could be optimized to provide the optimal measurement geometry. However, this optimization may not be necessary since
non-optimal configurations may be required by various mission objectives and science
observations [RZ00]. This plot shows that, with a very precise local ranging device,
the total position error is improved nearly an order of magnitude over the GPS-only
system. These results illustrate that, as expected, separating the vehicles in the fleet
can provide a good viewing geometry for the local transmitters, thereby improving the
relative navigation. However, these results also clearly illustrate that quite complex
fleet configurations would have to be used to obtain this good local geometry. Several
missions plan to use the in-track and single ellipse configurations [RZ00], but few
plan to use the complicated double ellipse that would be required to obtain the full
3D improvements. This double ellipse configuration could complicate the collection
of the science data, and it also would require a more complex fleet control system.
5.5
PowerSail Simulation
This section presents the analysis of an example mission in which adequate visibility
and geometry of the NAVSTAR constellation is not always available. The PowerSail [TM00] concept was proposed for a mission requiring high-power, light-weight,
and low-cost. The PowerSail system consists of a large, but light-weight, flexible sail
that generates power and a Host vehicle that carries out science missions requiring
high power. Although these two vehicles are connected together by an ‘umbilical’, it
will remain “slack” in order to avoid dynamic coupling between the Sail and the Host
satellites. Therefore, this system is closer to formation flying two free-flyers than a
single tethered system. Figure 5.29 shows one of the possible configurations of the
PowerSail system that has been investigated [TM00]. In order to minimize the total
mass of the system, the umbilical length must be minimized. If a CDGPS sensing
systems is used to measure the relative motion of the two vehicles, the large Sail could
142
CHAPTER 5.
Fig. 5.29: Power Sail with 4 Onboard
Transmitters
MISSION SIMULATIONS
Fig. 5.30: Power Sail with 2 Onboard
Transmitters
block a significant number of NAVSTAR GPS satellite signals for the Host spacecraft.
In this case, the commonly visible satellites could provide a bad constellation geometry or even drop below the minimum number needed (4). A similar situation can
occur if the PowerSail system is to operate in a remote location where NAVSTAR
signals are not available or partially available (e.g. GEO, MEO, etc.). In order to
maintain the precise relative position between the spacecraft at all times, onboard
ranging devices may be required to augment the CDGPS sensing system. In this case,
however, more than one transmitter may be required to deal with the case in which
very few NAVSTAR signals are available.
Since the number of satellites blocked by the Sail cannot be analytically computed
in general, a series of orbit simulations were performed. Simulations of the PowerSail
and Host spacecraft in Low Earth Orbit were performed using the FreeFlyer [FF] orbit
simulation package and the performance of relative position/velocity estimator as well
as the visibility were predicted. In the simulations, an Extended Kalman filter using
a simple orbit model and the measurement equation described in Chapter 2 were used
in the position and the velocity estimations. The initialization algorithm described in
Chapter 3 was also used to estimate the integer bias states. Typical orbit parameters
for the LEO spacecraft were selected, as shown in Table 5.8.
5.5. POWERSAIL SIMULATION
143
Typical Relative Motion of Host Spacecraft
30
30
20
20
10
10
z (m)
z
Typical Relative Motion of Host Spacecraft
0
0
−10
−10
−20
−20
−30
30
−30
30
20
30
10
20
0
Power Sail
Relative Motion of Host
20
30
10
20
0
10
0
−10
−10
−20
y
−30
−10
−20
−20
−30
10
0
−10
x
y (m)
−20
−30
−30
x (m)
Fig. 5.31: Typical Relative Motion of Fig. 5.32: Typical Relative Motion of
Host Spacecraft with respect to the Sail Host Spacecraft with respect to the Sail
(In-track difference)
(Eccentricity difference)
5.5.1
Relative Motions
Various relative motions between the Host spacecraft and the Sail are considered in
order to perform the visibility analysis. Figure 5.31 shows the relative motion of
Host spacecraft with respect to the Sail in the ECI frame (or inertial x, y, and z
directions). The Sail and the Host spacecraft in this figure are in the same circular
orbit (h 600km) with a fixed in-track separation (assumed to be maintained by
the control system). The attitude of the Sail is assumed to be fixed and known in the
inertial frame of reference - for simplicity it is assumed to be always pointed at the
Sun. Thus from the sail’s point-of-view (along inertial directions), the Host wraps
around the Sail once every orbit – in the local coordinate frame centered at the Sail,
the Sail would be rotating around its center with the Host separated by fixed in-track
distance from the Sail.
Table 5.8: Orbit Parameters Selected for PowerSail Simulation
Sail
Host
Eccentricity
Inclination
0.0
0.0 deg
7.143e-8 ∼ 2.857e-6
0.0 deg
Semi-major Axis
7000.0km
7000.0km
CHAPTER 5.
144
Number of Visible Satellite (h=600km)
MISSION SIMULATIONS
Number of Satellite Blocked by the Sail (h=600km)
14
12
Maximum number of satellite blocked by the sail
Average number of satellite blocked by the sail
12
10
10
number of satellite
number of satellite
8
8
6
6
4
4
2
2
0
Mean number of visible satellite
minimum number of visible satellite
0
2
4
6
8
10
distance (m)
12
14
16
18
20
0
0
2
4
6
8
10
distance (m)
12
14
16
18
20
Fig. 5.33: Number of Visible Satellites Fig. 5.34: Number of Satellites Blocked
from Host Spacecraft
by PowerSail
Figure 5.32 shows another case of relative motion of the Host spacecraft with
respect to the Sail in the inertial x, y, and z directions when the Sail is in circular
equatorial orbit and the eccentricity of the Host spacecraft is 2.143e−6 with 0 in-track
separation (similar to case considered in Ref. [TM00]). In this case, the minimum
relative distance between the two spacecraft is about 15 m and the maximum relative
distance is about 30 m. Again, the Host is assumed to maintain this relative position using feedback control. The primary question is whether the nearby Sail has a
significant impact on the visibility of the Host to the NAVSTAR constellation.
In the following simulations, only the latter case is considered and the distance
between the Sail and the Host is defined as the minimum distance throughout their
orbit.
5.5.2
Visibility of NAVSTAR Satellite
In general, as many as 10∼15 NAVSTAR satellites are visible to LEO spacecraft
throughout their orbits as shown in Figure 4.2. However, line-of-sights from the Host
to the NAVSTAR satellites can be blocked by the Sail since they are flying in relatively
close formation and the Sail is quite large (20 m×20 m). The number of NAVSTAR
5.5. POWERSAIL SIMULATION
145
Maximum GDOP values (h=600km)
10
Max GDOP without PLs on the sail
Max GDOP with 4 PLs on the sail
9
8
7
GDOP
6
5
4
3
2
1
0
0
2
4
6
8
10
distance (m)
12
14
16
18
20
Fig. 5.35: Maximum GDOP Values vs. Distance between the Host and the Sails
satellites commonly visible to the Host and the Sail, as a result, will strongly depend
on the separation between these two spacecraft.
Figure 5.33 shows the mean and minimum number of visible satellite as a function
of distance between the spacecraft in 600 km altitude orbit. The mean and minimum
values are computed over the data collected for approximately two orbit periods. As
expected, the number of visible satellites decreases as the separation becomes smaller.
In the limit, the minimum number of visible satellites throughout the orbit drops to
zero. Since the minimum number of visible satellites needed to maintain the relative
position and velocity estimates is four, the distance between the spacecraft, according
to Figure 5.33, must be larger than 4 m in order to obtain position and velocity
measurements for all time. Figure 5.34 shows the mean and maximum number of
satellite blocked by the Sail. These results suggest that for separations beyond 810 m, there should be enough satellites visible to obtain a CDGPS measurement
update to the position estimate. The quality of this estimate is determined by the
geometry of the visible NAVSTAR constellation.
CHAPTER 5.
146
5.5.3
MISSION SIMULATIONS
Effect of Local Transmitters on GDOP
GDOP (Geometric Dilution of Precision) is a measure of the size of error ellipse
derived from the constellation geometry [GPS96]. In fact, in an initial analysis, one
can simply consider the magnitude of the estimation error to be proportional to the
GDOP value. The GDOP can be calculated as follows [GPS96]
"
1 2
2
2
2
σx + σy + σz + στ = Tr (GT G)−1
GDOP =
σ


1
1 
 los
G =







(5.1)

los2 1 

..
.. 
.
. 

losn 1
(5.2)

where σ is the variance of the measurement noise. In case the local transmitter measurements are used, the GDOP values can be derived from the linearized measurement
Eq. 2.36.
GDOP =

Tr

GT

T !
G
∂D(X) 
∂D(X)
∂X
X̂
∂X
−1

(5.3)
X̂
The GDOP values are typically less than 3 for LEO spacecraft as a result of
the very large number of visible satellites. However, the Host spacecraft may not
have access to all visible satellites because of the signal blockage by the Sail. As the
number of visible satellites drops, the GDOP values will tend to increase (a GDOP
of 1 is considered to be very good). Without any local transmitters on the Sail, the
GDOP values will reach very high values as the number of visible satellites drops to
4∼5. Figure 5.35 shows maximum GDOP values for spacecraft in 600 km altitude
circular orbit as a function of the distance between the Host and Sail spacecraft.
For comparison, Figure 5.35 also shows the GDOP values when the Sail carries four
transmitters as shown in Figure 5.29. In this case, the maximum GDOP values are
substantially smaller than those computed without any transmitters onboard the Sail.
Note that the transmitters are especially beneficial when the distance between the
spacecraft is less than 10 m, as expected based on the visibility analysis.
5.5. POWERSAIL SIMULATION
147
Minimum Number of Visible Satellite
Maximum GDOP values without PLs on the Sail
12
10
Altitude 600km
Altitude 4500km
9
10
8
7
6
GDOP
number of satellite
8
6
5
4
4
3
2
2
Altitude 600km
Altitude 4500km
0
0
2
4
6
8
10
distance (m)
12
14
16
1
18
20
0
0
2
4
6
8
10
distance (m)
12
14
16
18
20
Fig. 5.36: Number of Visible Satellites Fig. 5.37: Maximum GDOP Values
from Host Spacecraft in 2 Different Or- without Transmitters for 2 Different Orbits
bits
5.5.4
Impact of Operating at Higher Altitudes
The number of total visible satellites decreases rapidly as the spacecraft altitude
increases much beyond 4500 km. This occurs because the transmit antenna of the
NAVSTAR GPS satellite have a beam angle of 21.3◦ (Half cone). However, when
these beam angles were included in an analysis for an orbit altitude of 4500 km, the
minimum number of visible satellites does not change significantly for small Sail/Host
separations (see Figure 5.36). Figure 5.38 compares the visibility at LEO and MEO
to clarify this point. The figure shows the observability cones for the GPS spacecraft
and the cones of exclusion associated with the LEO/MEO spacecraft (masking angles
selected to avoid viewing the NAVSTAR constellation through the Ionosphere). Any
GPS satellite in the top half of the plot would be visible to both the MEO/LEO
spacecraft. The location of GPS1 corresponds to a position that it is not visible to
either LEO or MEO spacecraft. However, as can be seen by GPS2 , there are numerous
GPS satellite locations that would be observable to the MEO spacecraft, but not to
the LEO (i.e., they are below the LEO exclusion line). With the ability to see GPS
satellites at these additional locations, the full 3-D analysis shows that the minimum
number of visible satellites does not change significantly from LEO to MEO. However,
CHAPTER 5.
148
MISSION SIMULATIONS
GPS1 cone
GPS2 cone
MEO
LEO
LEO exclusion
GPS2
MEO exclusion
GPS1
Figure 5.38: Visibility comparison at LEO and MEO. LEO and MEO exclu-
sion Zones are Designed to Avoid Viewing Through Ionosphere.
as one could see by looking at the visibility cones as a GPS satellite traces out the
NAVSTAR orbit, spacecraft at altitudes above 4500 km would receive signals from
very few GPS satellites.
Figure 5.36 compares the minimum number of commonly visible satellites for two
orbit altitudes. As discussed above, in the MEO case there are a large number of the
visible satellites that are located at lower elevations than the Host spacecraft. Thus
the Sail cannot block all of the satellites, even in the worst case situation (e.g., the
Host spacecraft is located between the Sail and Earth). Care must be taken to avoid
using GPS satellites that are at too low an elevation, or else their signal will propagate
through (and be corrupted by) the Ionosphere. Thus a mask angle of ≈ 10◦ was used
to avoid those corrupted measurements.
5.5. POWERSAIL SIMULATION
149
Maximum GDOP values with 4 PLs on the Sail
Maximum GDOP values with various number of PLs on the Sail (h=600km)
10
10
9
9
8
8
7
7
6
6
GDOP
GDOP
Altitude 600km
Altitude 4500km
5
5
4
4
3
3
2
2
1
1
0
0
2
4
6
8
10
distance (m)
12
14
16
18
20
4 PLs
2 PLs
0 PL
0
0
2
4
6
8
10
distance (m)
12
14
16
18
20
Fig. 5.39: Maximum GDOP Values at 2 Fig. 5.40: Maximum GDOP Values for
Different Altitudes When 4 transmitters Different Numbers of Transmitters Onare on Sail
board Sail
For similar reasons, the maximum GDOP values are even lower as the spacecraft
altitude increases. Figure 5.37 compares the maximum GDOP values of the spacecraft formation in the altitude of 600 km and 4500 km (no local transmitters are
used). However, as the spacecraft altitude increase further beyond 5000 km, visibility
degrades dramatically and the GDOP increases as a result.
Figure 5.39 compares the maximum GDOP values at two different altitudes when
4 local transmitters are onboard the Sail. The results show that, with the local
transmitters, the maximum GDOP values are much less sensitive to the spacecraft
altitude.
5.5.5
Impact of Adding Local Transmitters
A series of the Host-Sail formation flying simulations were performed with a different
number of transmitters on the Sail as shown on Figures 5.29 and 5.30. Figure 5.40
shows the maximum GDOP values for the spacecraft formation in 600 km altitude for
each case. With two or less transmitters, the GDOP values increase to infinity as the
separation decreases. With two onboard transmitters, the separation can be as small
as 2 m without losing the position estimates and 4 transmitters would be necessary
CHAPTER 5.
150
MISSION SIMULATIONS
under 2 m separation. However, as is also shown in the figure, the benefit of using
the transmitters (lower GDOP values) are relatively small for separations larger than
10 m unless a larger sail is to be used.
5.5.6
GEO Simulation
GEO operation of the PowerSail may seem favorable because some disturbances to the
large Sail will be much smaller than in LEO. However, the GPS approach to formation
sensing will be greatly impaired by poor NAVSTAR constellation visibility.
For example, Figure 5.41 shows the number of GPS satellites visible to the Host
spacecraft in GEO. As shown, the maximum number of visible satellites is only 4, and
the average number of visible satellites for one orbit period is approximately 1. Note
that these low values are not because the Sail blocks a large number of GPS satellites
(the separation between the vehicles was set at approximately 5 m), but because
the NAVSTAR constellation is located far below GEO and the GPS transmitters are
designed to focus on the Earth. These two factors combine to greatly limit the GPS
signal visibility at GEO. This situation is analyzed in Figure 5.44 which shows the
geometry in 2-D. In this case, the GEO satellite can only see GPS satellites that
are very close to the location given by GPS1 . For all other locations, such as GPS2 ,
which would be visible at MEO and LEO, the GEO satellite falls outside the GPS
cone of visibility. Note that very small changes in the location of GPS1 would push
that vehicle into the GEO exclusion zone.
To compound this poor visibility, the Sail can also block some of the available
satellite signals, as shown on Figure 5.42. Because the number of available satellites
is less than 4 for most of the time, it is impossible to maintain relative position and
velocity estimates without using local transmitters. Figure 5.43 shows the GDOP
values when the 4 onboard transmitter measurements are used in addition to all
the available measurements from NAVSTAR constellation. The GDOP values range
between 2 and 3, which are comparable to some of the best LEO results. These GDOP
values indicate that it might be feasible to use a local constellation of 4 transmitters
on the Sail to provide the formation flying sensing data if the Host/Sail are in a GEO.
5.6. SUMMARY
151
Number of Satellite Blocked by the Sail (d=5m, GEO)
5
4.5
4.5
4
4
3.5
3.5
Number of Satellite
Number Satellite
Number of Common Visible Satellite (d=5m, GEO)
5
3
2.5
2
3
2.5
2
1.5
1.5
1
1
0.5
0.5
0
0
0
5
10
15
20
0
5
10
15
20
time (hour)
time (hour)
Fig. 5.41: Number of Commonly Visi- Fig. 5.42: Number of Satellites Blocked
by Sail
ble Satellites
GDOP values with 4 transmitters (d=5m, GEO)
3
2.5
GDOP
2
1.5
1
0.5
0
0
5
10
15
20
time (hour)
Fig. 5.43: GDOP Values Using 4 Onboard Transmitters
5.6
Summary
An extensive number of simulation results that cover a wide range of different mission scenarios are presented in this chapter. These simulation results demonstrate the
CHAPTER 5.
152
MISSION SIMULATIONS
GEO
GPS1 cone
GPS2 cone
MEO
GPS2
MEO exclusion
GPS1
GEO exclusion
Figure 5.44: Visibility comparison at MEO and GEO. GEO Exclusion Zone
is Designed to Avoid Viewing through Ionosphere.
benefits and limitations of the local transmitter augmented CDGPS techniques for
users in many environments, such as LEO, MEO, GEO, and terrestrial. The results
show that a combination of a high performance ranging device and a good formation
geometry has the potential to substantially improve the relative sensing accuracy in
LEO. The onboard transmitters could also be used to improve the visibility/geometry
of the measurements in MEO, GEO, and missions like PowerSail where the NAVSTAR signals are only partially available and/or are blocked by nearby vehicles or
structures. Extensive visibility analyses/simulations are presented for each scenario
showing that the onboard ranging device can provide accurate sensing capabilities in
these cases. The simulation results of a terrestrial formation flying scenario (air-to-air
5.6. SUMMARY
153
refueling mission) are also presented, in which the onboard transmitters enable rapid
initialization of the cycle ambiguities and improve the local measurement geometry.
Chapter 6
Experimental Results
This chapter presents results from a number of tests that were conducted on several
ground based testbeds. The experimental tests were conducted to demonstrate the
performance of the integrated GPS system. Although numerous simulations have been
conducted (Chapter 5) to demontrate various aspects of the augmented CDGPS system and related formation flying algorithms, the purpose of this chapter is to validate
the concepts and algorithms on real experimental systems. “Hardware-in-the-loop”
tests that used the real-time NAVSTAR and local transmitter measurements were
carried out to validate the system concepts and the real-time, experimental performances of the algorithms on physical systems. Multiple, custom-built, modified GPS
receivers [EO99] and local transmitters were used in these experiments. The performance of the robust initialization algorithm, as well as the previous EKF bias estimation algorithm, are demonstrated on real systems under autonomous control. Results
from formation maneuvering tests using both the local transmitter and NAVSTAR
measurements are also presented.
6.1
Experimental Setup
Several ground based testbeds were constructed as a part of this dissertation. These
testbeds consist of a number of simple ground vehicles that carry the transceivers and
a main computer that processes the phase measurements.
154
6.1. EXPERIMENTAL SETUP
155
Figure 6.1: Formation Truck Testbed
6.1.1
Formation Truck Testbed
The first outdoor formation flying testbed (Figure 6.1) consists of four commercially
available RC trucks, GPS receivers, pseudolites, radio modems, and HC11 microcontrollers for vehicle control. A mobile ground computer and monitoring station is used
to process the GPS data and perform the formation state estimation and control.
Each vehicle is capable of tracking the GPS signals from the NAVSTAR constellation, as well as the signals generated from the pseudolites onboard the other vehicles.
A typical RC truck is shown in Figure 6.2. Our previous testbed consisted of two
lighter-than-air vehicles [EO99] that operated inside a large building at Stanford.
However, the trucks were selected for our outdoor testbed because they are much
easier to maneuver under the typical conditions found outside (e.g. wind, variable
CHAPTER 6. EXPERIMENTAL RESULTS
156
Figure 6.2: RC Truck with Onboard Electronics
temperatures). Of course, the vehicle dynamics are quite different than would be expected for spacecraft on-orbit. But to create a formation of multiple vehicles and test
the augmented CDGPS navigation systems, it is important to develop a simple design
that is easily replicated and easily operated. The RC trucks could easily be made
to move in 3-D by constructing ramps if any vertical motions are required. Note
that each truck carries a two RF front-end attitude-capable GPS receiver [CP00].
Detailed information about the attitude receiver development and testing is given in
Appendix A.
6.1.2
Formation Robot Testbed
The problem with the first truck testbed is that the small RC trucks have barely
enough payload capacity to carry the onboard transmitter and all the electronics and
the batteries. Therefore, only the master truck carried the onboard transmitter and
its motion was very limited. The commercially available all-terrain ATRV-jr robots
6.1. EXPERIMENTAL SETUP
157
Figure 6.3: Formation of Outdoor Robots with SPTC
have more than enough payload capacity (25 kg) to carry all of the equipments.
Therefore, all robots in the formation carried onboard transmitters and they could
perform more complex formation maneuvers than was possible with the RC trucks.
These robots are also equipped with an onboard computer, sonars, and a wireless
ethernet.
6.1.3
Onboard Ranging and Communication Device
The spacecraft in several of the example missions cited in Chapter 1 will require an
onboard ranging device to augment/replace the NAVSTAR constellation and help the
CDGPS initialization process (see Chapter 3). If properly designed, this device could
also be used for the inter-spacecraft communication, which is also required for the
relative navigation. The current testbed uses a COTS pseudolite from Integrinautics
that transmits the L1 C/A code with 50 bps data (1000 bps Max data rate) [IN00].
CHAPTER 6. EXPERIMENTAL RESULTS
158
Fig. 6.4: Pseudolite and Transmit Antenna
Fig. 6.5: Stanford Pseudolite Transceiver
Crosslink (SPTC)
This rate is too low for our applications, so we use RF modems for the inter-vehicle
communication. However, as discussed in Chapter 1, several cross-link transceivers
are under development that have a higher data rate and could also provide a better
ranging accuracy. Figure 6.4 shows the pseudolite and the transmit antenna used in
this research. The Lindenblad transmit antenna was developed by a fellow student
Corazzini [TC00, TC99, CB82] for use in her indoor robot testbed. The antenna
provides an omni-directional RHCP (righ-hand circularly polarized) signals to other
vehicles [TC00]. The gain pattern is designed such that it has the null in its vertical
axis. This gain pattern prevents the transmitter from jamming the GPS receiver
onboard the same vehicle since the NAVSTAR signals are potentially much weaker
than the pseudolite signal. The pseudolite is capable of pulsing the transmitted
signals in order to avoid severe near-far problems [SC97]. The selected duty cycle of
the pulsed signal is approximately 10%. This setting ensured a near-uniform power
reception of the pseudolite signal over a 1–20 m range.
6.1.4
SPTC
Figure 6.5 shows a picture of the ranging transceiver called a Stanford Pseudolite
Transceiver Crosslink (SPTC) developed for formation flying vehicles. It combines
6.2. MULTI-VEHICLE AUTONOMOUS INITIALIZATION
159
a pseudolite, a 12-channel attitude GPS receiver, 2 GPS antennas, 1 Transmit antenna, a radio modem, microcontroller, and an interface board. It is a standalone
device that can transmit and receive local pseudolite signals and receive the NAVSTAR signals. It can also exchange information with other transceivers through the
radio link (38.4 kbps). In addition, the attitude of the transceiver is computed inside
the GPS receiver. The attitude solution is needed to determine the location of the
transmit antenna with respect to the reference point of the SPTC since the transmit
antenna is mounted on top of a pole for better visibility to the other vehicles (Figure 6.3). The interface board [EO99,JCA99] relays incoming data stream to the GPS
receiver as well as to the microcontroller. The measurements from the receiver are
also relayed to the radio modem by the interface board. The microcontroller receives
control commands through the radio modem and interface board, and then converts
them into an analog signal, which is then sent to the robot for the autonomous control.
6.2
Multi-vehicle Autonomous Initialization
In this section, an autonomous formation initialization maneuver involving 3 vehicles
is demonstrated for the first time. It used the first version of the extended Kalman
filter algorithm developed in Section 3.1.1. Figure 6.6 shows the 3-truck formation
undergoing an initialization maneuver around the reference vehicle with the pseudolite. Experimental results for a three vehicle formation undergoing an autonomous
initialization manuever are shown in Figure 6.7. The three vehicles are labeled as
trucks 1, 2 and 3. In this experiment, the three vehicles were placed at unknown
locations, with arbitrary and unknown attitudes. At the start of the initialization,
the intra-vehicle cycle ambiguities were determined by having each vehicle maneuver
in a tight circle. This motion enabled the attitude of each vehicle to be resolved to
an accuracy of less than 1◦ . Appendix A describes this process in detail. After these
maneuvers, the coarse relative position estimator described in Section 2.6 was run in
real-time until a meter-level estimate was obtained.
After the initialization, truck 3 was given a command to move to -4 m north of
truck 1 (reference vehicle) and to start a 4 m radius circular motion around truck 1.
CHAPTER 6. EXPERIMENTAL RESULTS
160
Fig. 6.6: Formation of 3 Trucks
During the 360◦ circular motion, the position and bias estimates were improved using
the extended Kalman filtering algorithm described in Section 3.1.1. At the end of the
maneuver, truck 3 was commanded to move to 3 m east and -3 m north of truck 1.
Truck 2 was also given a command to move to -4.5 m north of truck 1 and to
start a 4 m radius circular motion about truck 1 shortly after truck 3 had started its
maneuver. This ensured that the two vehicles did not collide during the initial phase
of the initialization process. At the completion of truck 2’s maneuver, it was given
a command to move to 3 m east and -4 m north of truck 1 (-1 m north of truck 3),
forming a close formation.
The experimental results demonstrate that the onboard pseudolites combined with
the new real-time algorithm can be used to autonomously initialize the formation in
relatively short periods of time by having the vehicles undergo relative motion about
one another.
6.3. ROBUST BIAS ESTIMATOR EXPERIMENT
161
Relative Position
5
Iinitialization Path (Truck 2)
Iinitialization Path (Truck 3)
4m circle around Truck 1
4
3
2
North (m)
1
.
Truck 1 (Pseudolite)
0
−1
−2
.
.
−3
−4
−5
−8
−6
−4
−2
East (m)
0
2
4
Fig. 6.7: Demonstration of Autonomous Formation Intialization Maneuver of 3
Trucks
6.3
Robust Bias Estimator Experiment
Although the first demonstration of the autonomous formation initialization was successful, it required a relatively large-scale initialization maneuver (4 m) and large
initial separation (∼7 m) compared to the size of the vehicle. As discussed earlier
(see Section 3.1.2), the initial line-of-sight error due to initial uncertainty can cause
large final position errors. A new robust algorithm was developed to enable substantially smaller-scale initialization maneuvers without degrading the final estimation
accuracy. The primary advantage of a smaller scale initialization maneuver is that
they can yield a faster, more fuel efficient bias initialization. The detailed algorithm
is presented in Section 3.1.2.
CHAPTER 6. EXPERIMENTAL RESULTS
162
Initialization Maneuver (Experiment)
2
Estimated Final Position
trajectory
Initial Position Guess
1.5
1
0.5
y (m)
0
−0.5
−1
−1.5
−2
−2.5
−3
−2
−1
0
x (m)
1
2
3
Figure 6.8: Initialization: Truck 2 starts at <, but Estimator Given Wrong
Initial Conditions (◦). Estimated Relative Trajectories Shown
around Truck 1 (At origin).
Figure 6.8 shows a set of experimental results for a two vehicle formation undergoing an extremely small radius initialization maneuver. In this experiment, the mobile
vehicle (labeled as truck 2) was placed at a known location (-2,-1) m with respect
to the reference vehicle (truck 1). However, this initial relative position information
was not given to the estimator. Instead, the initial errors were given randomly, as
shown Figure 6.8. Then truck 2 was manually controlled to move along a ∼1.5m
radius circle around truck 1. Note that truck 1 could also be moved during this initialization maneuver – the position of truck 1 need not be pre-surveyed. In fact,
the absolute position of the truck 1 need only be known to SPS accuracy for this
algorithm to work. During the 360◦ circular motion, the position and bias estimates
were improved using the robust initialization algorithm described in Section 3.1.2. At
6.4. MOBILE FORMATION CENTER & MULTI-TRANSMITTERS
163
Table 6.1: Final Position Estimation Errors
X
Y
Robust Algorithm
Mean error (m) Std (m)
0.012
0.039
0.007
0.023
Previous Algorithm
Mean error (m) Std (m)
-0.088
0.104
0.031
0.064
the end of the maneuver, truck 2 was driven to the final point (2,-1) m with respect to
truck 1. Figure 6.8 shows a set of the estimated position trajectories of 10 runs with
various initial position guesses. Note that the final position estimates are very close
to the true final position (2,-1) m even though the initial errors are large compared
to the distance between the vehicles. Since the rate of convergence depends on the
magnitude of the initial position error and the modified measurement noise covariance
matrix R2 in Section 3.1.2, two of the runs that had the largest initial errors converge
slower than others. The convergence pattern can vary with the initial position guesses
since it is also a function of the estimated state X̂ as shown in Section 3.1.2. The previous initialization algorithm was also applied to the identical receiver measurements
recorded and the final errors are calculated in the same manner.
Table 6.1 summarizes the results from both algorithms. The mean error values
of the robust algorithm are approximately 1 cm for each horizontal direction. These
are 3–4 times smaller than the mean values of the previous method. The standard
deviations of the robust algorithm are also approximately 3 times smaller than the
previous method. These experimental results demonstrate that the onboard transmitter can be used to robustly initialize the formation in short periods of time by
having the vehicles undergo small-scale relative motion around each other without
having a pseudolite in a precisely known location.
6.4
Mobile Formation Center & Multi-Transmitters
In the previous experiments, the reference vehicle, which has the formation frame
attached, was stationary while the other vehicles performed initialization maneuvers.
That may not, however, be the optimal way to initialize carrier phase biases. In
164
CHAPTER 6. EXPERIMENTAL RESULTS
Fig. 6.9: Roof-top Testbed: Transceivers with Laser Pointers over 2’×2’ Grid
many cases, small motions of the reference vehicle could provide large line-of-sight
changes for many other vehicles, which is essential to initialize the biases. In the
following experiment, the reference vehicle (or equivalently, the formation center) and
the second vehicle were moved around, and both vehicles had onboard transmitters.
This experiment was performed on a roof-top testbed where the relative positions in
east and north directions are pre-surveyed in a 2’×2’ grid format. The “vehicles”
were equipped with a laser pointer device that projects the horizontal position of the
antenna center onto the grid (located on the ground). This setup provided the truth
relative positions of the vehicles during the maneuvers. Figure 6.9 shows the setup
for the experiment.
Figure 6.10 shows the estimated absolute positions of the 2 vehicles during the
initialization maneuver. Since only one vehicle was moved at a time and the reference
6.4. MOBILE FORMATION CENTER & MULTI-TRANSMITTERS
165
Initialization Maneuver of 2 vehicles (Earth Fixed Frame)
0.5
0
−0.5
−1
North (m)
−1.5
Motion of Master Vehicle
(Formation Center)
Motion of 2nd Vehicle
True Final Positions
−2
−2.5
−3
−3.5
−4
−4.5
−2
−1
0
1
2
3
4
East (m)
Fig. 6.10: Estimated Absolute Position of 2 Vehicles
Initialization Maneuver (relative position)
1
0
North (m)
−1
−2
Estimated Relative Trajectory
True Position
Estimated Position
−3
−4
−5
−3
−2
−1
0
1
2
3
4
5
East (m)
Fig. 6.11: Estimated Position of Second Vehicle Relative to Reference Vehicle
CHAPTER 6. EXPERIMENTAL RESULTS
166
vehicle was placed at (0, 0) point initially, the absolute position1 could be derived from
the estimated relative position without using the pseudoranging solutions. Figure 6.11
shows the relative position estimate of the second vehicle during the initialization
maneuver.
During the maneuvers, the vehicles were paused on the pre-surveyed points to
demonstrate the convergence characteristics by comparing the truth positions with the
estimated positions. These true and estimated points are marked on Figure 6.11. Note
that the position errors at each point are the distance between the truth point (X)
and the estimated point (+), and the results show that these errors decrease rapidly
as the line-of-sight between the vehicles changes. A typical error convergence plot is
shown in Figure 6.12, where the estimation errors are plotted for each sample point.
This figure also shows the estimated calibration error level in the truth measurement.
The calibration error level (≈3 cm) was estimated by combining all possible error
sources, such as grid attitude error, laser pointer alignment error, and antenna phase
center error. The position error converges close to the calibration error level after a
series of maneuvers. Note that the rate of this convergence largely depends on three
major factors: 1) The amount of line-of-sight change from the local transmitters to the
receiver, 2) The total number of measurements (NAVSTAR + local) available, and 3)
The geometry of the combined (NAVSTAR + local) constellation. Figure 6.13 shows
the number of measurements during the initialization maneuver in Figure 6.11. Due
to the surrounding buildings and structures, measurement drop-outs and reaquisitions
occurred very often. Nevertheless, it only affected the convergence speed, the final
estimation error was not compromised.
6.5
6.5.1
Formation Maneuvers
Truck Formation Maneuver
Autonomous formation maneuvering experiments were performed using the trucks
shown in Figure 6.1. The reference truck (truck 1) was equipped with the onboard
1
Origin of the absolute frame is (0, 0) of the grid
6.5. FORMATION MANEUVERS
167
Convergence of Position Error
0
position error (m)
10
Estimation
Error Bound
Estimation Error
−1
10
Calibration Error Level (3cm)
−2
10
1
2
3
4
5
6
7
sample points
8
9
10
11
12
Fig. 6.12: Estimation Error Convergence
Measurement Drop−outs during the Initialization Maneuver
6
Number of NAVSTAR Measurements
Number of Local Transmitter Measurements
Number of Measurements
5
4
3
2
1
0
0
10
20
30
40
50
time (sec)
60
70
80
90
100
Fig. 6.13: Number of Measurements Used in Initialization Process
CHAPTER 6. EXPERIMENTAL RESULTS
168
Horizontal Error in the Formation Keeping Maneuver
1
0.5
0
y error (m)
−0.5
−1
−1.5
Desired Relative Position (0,−1)
−2
−2.5
−3
−2
−1.5
−1
−0.5
0
x error (m)
0.5
1
1.5
2
Figure 6.14: Formation Keeping Maneuver (RC Trucks)
transmitter and it was able to follow the manual command during the formation maneuvers. Figure 6.14 shows the horizontal errors in the formation keeping maneuver.
In this experiment, truck 2 had completed the autonomous initialization maneuver
and formed a tight formation with truck 1 at (0,-1) m. At this point, truck 1 started
moving towards the positive x axis. Truck 2 was under automatic control and was
commanded to maintain the relative position at (0,-1) m. Because there was no feedforward in the control scheme, there was a noticeable lag in the motion of truck 2
(towards negative x in this plot). At the end of the experiment, truck 1 made a turn
towards the negative y axis, causing the relative position of truck 2 to lag towards
positive y. This test demonstrates the autonomous station keeping formation maneuver, in which the reference vehicle with onboard transmitter is mobile and the follower
vehicle is maintaining the relative position in the moving local coordinate frame.
6.5. FORMATION MANEUVERS
169
Robot Formation Manuever (station keeping)
0.35
0.3
0.25
0.2
East (m)
0.15
0.1
0.05
0
Target Point
−0.05
Robot2 Trajectory
Robot2 Heading
−0.1
−0.15
1.3
1.4
1.5
1.6
South (m)
1.7
1.8
1.9
Figure 6.15: Robot Fomation Keeping Maneuver
6.5.2
Robot Formation Maneuver with Multi-Transmitters
In this section, the experimental data of the autonomous formation maneuvers of
multiple robots are presented. Each robot carried an onboard transmitter for the
initialization and the geometry augmentation.
Figure 6.15 shows the relative position of robot 2 with respect to the robot 1 as
well as the heading direction of the follower (robot 2) in the ENU frame. In this
demonstration, robot 2 had completed the initialization maneuver and formed a tight
formation with robot 1 at (0,-1.6) m (East,North). Robot 2 was under automatic
control and was commanded to maintain the relative position at (0,-1.6) m. Because there was no feedforward in the control scheme and the leading robot (robot 1)
performed very complex maneuvers, there was a noticeable lag in the motion of robot 2. However, the heading angles of robot 2 at the each sampled data point show
that robot 2 is headed towards the target point which is consistent with attempting
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CHAPTER 6. EXPERIMENTAL RESULTS
to minimize the station-keeping error. Note that the scale of the entire formation
maneuver in this particular case was approximately 15 m×15 m. A simple ranging
device was developed in order to validate the CDGPS position estimates during the
formation maneuvers. The detailed design of the laser ranging device is given in
Appendix B.
Validation Results
A video camera was mounted onto the follower robot and aimed at the target. A
series of formation maneuvers (both open-loop and closed-loop) were then performed
and CDGPS position measurements were collected while the video was taken. The
video tape was later examined at a 1 Hz rate, and the location of the image of
the pointers on the target was converted to the relative position using Eq. B.11. A
comparison of the video measurements and CDGPS measurements for open-loop and
closed-loop formation maneuvers are shown in Figures 6.16 and 6.17. In the open-loop
case, the relative position is slowly drifting towards one direction. Two manual path
corrections were made during the maneuver in order to keep the 2 laser intersection
points inside the target board. The position errors are typically less than 5 cm when
the vehicles are following a straight path. However, larger errors can be seen during
the angular motion of the robots. This is because the laser ranging device solutions
are sensitive to the large heading angle change and the attitude estimation errors
(Appendix B). In the closed-loop case, the vehicle maintains its relative position at
approximately 1.9 m north. Because the follower robot had to undergo large turns in
order to regulate the lateral relative position, the laser beams are often aimed away
from the target board and data points are missing for those durations. The position
error is also less than 5 cm during the maneuver when the laser ranging measurements
are available.
6.5. FORMATION MANEUVERS
171
South direction
−0.6
Laser pointer measurement
GPS measurement
−0.7
−0.8
m
−0.9
−1
−1.1
−1.2
−1.3
0
50
100
150
100
150
time (sec)
East direction
0.3
0.25
0.2
m
0.15
0.1
0.05
0
−0.05
0
50
time (sec)
Fig. 6.16: Validation Result (Open-loop Maneuver)
Formation Maneuver Validation Data
−1.7
Laser pointer measurement
GPS measurement
−1.75
south direction relative position (m)
−1.8
−1.85
−1.9
−1.95
−2
−2.05
−2.1
−2.15
−2.2
0
20
40
60
time (sec)
80
100
Fig. 6.17: Validation Result(Closed-loop Control Maneuver)
120
CHAPTER 6. EXPERIMENTAL RESULTS
172
6.6
Summary
In conclusion, the augmented CDGPS sensing system and the algorithms developed
in this dissertation were extensively tested on various terrestrial testbeds. In particular, the robust bias intialization algorithm was extensively tested. The final position
errors after intialization maneuvers as well as the convergence characteristics were
also verified in the experiments. The autonomous initialization maneuvers and autonomous formation maneuvers were demonstrated and the estimation accuracy was
validated using the laser pointer ranging device.
Chapter 7
Conclusion
The Global Positioning System combined with onboard ranging devices has been
shown to provide the means for precise relative navigation for a variety of formation
flying applications. This chapter presents a summary of the theoretical and experimental contributions behind the development of a feasible sensing system for vehicle
formations in various operating environments. In addition, future areas of relative
navigation research are suggested.
7.1
Summary of Contributions
The goal of this research was to develop and demonstrate a comprehensive sensing
system for a formation of vehicles using the augmented CDGPS concept. Contributions presented in the thesis were made in the following areas:
Decentralized Estimation
The ability to distribute the computational loads of the estimation process is imperative for future formation flying spacecraft missions. A decentralized relative position
and velocity estimation algorithm was developed for a large number of vehicles using
a combined (local + NAVSTAR) constellation. This decentralized ICEKF (Iterated
Cascade Extended Kalman Filter) algorithm was shown to substantially reduce total
computational load and also distribute this load over the fleet without significant
173
174
CHAPTER 7. CONCLUSION
degradation in the estimation accuracy when compared to the traditional centralized
approach. The numerical problems associated with adding in the local range measurements were eliminated by the distributed estimation process. Additionally, the
decentralized system was able to reduce the inter-spacecraft communication requirements. The decentralized estimation approach also has the benefits of being robust
to single point failures on the master vehicle, and the estimation process can easily
be reconfigured as vehicles join/leave the fleet.
Long Baseline Corrections
The correction terms for the differential phase and Doppler measurements were introduced for the spacecraft formations with a large vehicle separation (>1 km). It
was demonstrated that the traditional planar wave front assumption made by many
previous researchers is not valid for long-baseline formation flying spacecraft missions.
It was shown in the simulation that these errors (maximum ∼10 cm error in phase,
∼40 cm error in Doppler measurements for 2 km baseline) can be a dominant error
source if they are not compensated for correctly. Although the correction techniques
depend only on knowledge at the pseudorange level, it was shown that the accuracy
of these correction terms typically reduce the measurement errors to below the level
of the phase noise.
Robust Initialization Algorithm
The robust real-time cycle ambiguity estimation algorithm was developed to replace
the previous batch algorithms that were unable to provide bias estimates during the
initialization maneuver around a local transmitter. It was shown that this filtering
algorithm can provide the position and bias estimates in real-time and tolerate relatively large initial errors, having a comparable convergence region to that of the
batch (smoothing) methods. The performance and characteristics of this algorithm
were extensively investigated in various simulations and experimental tests.
7.1. SUMMARY OF CONTRIBUTIONS
175
Quasi-optimal Selection Algorithm
A vehicle in a formation can see a large number of ranging signals from the NAVSTAR
constellation as well as the local constellation. However, due to hardware limitations
on most receivers, only a subset of these measurements can be tracked at one time.
Thus, it is desirable to select a subset of the available measurements that forms the
best possible geometry. A quasi-optimal satellite selection algorithm was presented
that provides a fleet of spacecraft with onboard transmitters with a near-optimal
viewing geometry, but at a very small computational load. As such, the algorithm
should be implementable in real-time, which is necessary for a fleet in LEO. This
algorithm was validated in various simulated environments, including LEO.
Mission Simulations
An extensive number of simulations that cover a wide range of different mission
scenarios were conducted to validate the algorithms developed. These simulation
results also demonstrated the benefits of the augmented CDGPS sensing system for
users in many different environments, such as LEO, MEO, GEO, and terrestrial.
Experimental Demonstration
The first real-time autonomous formation initialization maneuver was demonstrated
using outdoor experimental vehicles. The initialization procedure provided centimeterlevel position accuracy at the end of the maneuver. In addition, autonomous formation maneuvers involving multiple onboard transmitters were successfully demonstrated. These experimental results validated the key algorithms and procedures
developed in this dissertation.
Development of SPTC
A prototype standalone transceiver system (SPTC) was developed for the experimental demonstration of the proposed sensing system and the related algorithms. The
device consists of a combined GPS attitude receiver, a pseudolite, a radio modem,
CHAPTER 7. CONCLUSION
176
Demonstrate closed-loop
formation flying in space
Emerald 2 (Chromium)
Carrier phase GPS data
merald 1 (Beryl)
Two GPS antennas
One GPS receiver
Limited control
Orion
• Six GPS antennas
• Three GPS receivers
• Full 6 DOF control
Figure 7.1: Orion Mission
and a micro-controller. It provided the full communication, ranging, and CDGPS
capabilities that are essential for the formation flying vehicles.
7.2
Future Work
There are a number of additional research areas related to the relative navigation system that can be further explored. This future work can be divided into the following
areas.
7.2.1
LEO Formation Flying Experiment
The ability to perform the autonomous formation flying tasks in space must be validated by an on-orbit demonstration. The formation flying experiment requires researches in the areas of distributed control and estimation, path planning, modular
sensing system design, inter-spacecraft communication, long-term optimal fleet management and control, and high level task management. The Orion project [JH99] is
7.2. FUTURE WORK
177
an ongoing mission, scheduled to launch in 2002. The Orion mission will demonstrate
the first autonomous formation flight of 3 micro-satellites in LEO using the CDGPS
techniques. The Orion spacecraft will carry 3 modular GPS receivers (6 antennas)
and determine the relative position and velocity of the spacecraft with respect to
two Emerald spacecraft flying in formation. The mission is expected to test other
essential formation flying technologies introduced above.
7.2.2
Transceiver Development
In this research, the pseudolites operating in L1 frequency was used for an onboard
transmitter in the experiments because the hardware was readily available. However,
a local transmitter operating in the same frequency introduced severe near-far problems. This problem was mitigated by using the pulsing feature of the pseudolites,
but there still exist potential problems such as a limited operating range, a limited
number of Gold codes, and SNR degradation in the NAVSTAR signal due to the inevitable cross-correlations. An onboard transceiver system that operates in a different
frequency range would be highly beneficial. Moreover, a higher ranging resolution and
a higher data rate can be achieved by using a higher code rate and operating at a
higher frequency. In addition, a proper signal structure design could eliminate the
measurement bias issues that exist in the differential carrier phase measurements of
the NAVSTAR and the pseudolite signals. An ability to track the L1 and L2 GPS
signals could be also added to these devices, realizing a complete relative/absolute
navigation device. Several researchers are currently developing ranging and communication devices that could be used in combination with the CDGPS system. Star
Ranger, which is being developed by MIT in conjunction with AeroAstro [RZ], will
demonstrate ultra-high precision ranging capability (∼1 cm) with its Ku-band transceiver. It will also combine multiple modular L1 GPS receivers [EO99] to include the
CDGPS capability.
CHAPTER 7. CONCLUSION
178
Fig. 7.2: UAV
7.2.3
UAV Relative Navigation
The importance of the Unmanned Aerial Vehicle (UAV) is acknowledged for many
potential critical application areas such as battle field reconnaissance, high altitude
exploration, and unmanned combat air vehicle (UCAV). Ultimately, these vehicles
will be required to perform formation maneuvers and aerial refueling maneuvers as
are common practice in manned aircraft. The autonomous refueling capability of
the UAV will be especially valuable due to the limited size of the aircraft. The
(augmented) CDGPS can be implemented and provide the precise relative navigation
capability1 .
1
Relative to ground station or relative to each other.
7.3. CLOSING
7.3
179
Closing
The formation flying of spacecraft will undoubtedly become a common practice in
the future space exploration and the Global Positioning System augmented by the
onboard ranging/communication devices will play an important role as a relative navigation sensor for the various formation systems. Therefore, the potential performance
and limitation of the system must be fully understood and enhanced if necessary. The
research presented in this dissertation has achieved firm contributions towards this
goal. The achievements made in these early days of formation flying and relative navigation researches should provide a solid foundation for the future formation/relative
navigation systems.
Appendix A
GPS Attitude Receiver
Development and Testing
A.1
Receiver Hardware
As a part of this research, a low cost attitude capable receiver was developed using
the Mitel GP2000 chipset. The GP2000 chipset comprises the GP2010 RF frontends and the GP2021 12-channel correlator. The original Mitel ORIONTM design is a
single antenna receiver on a 3×4 inch printed circuit board. This design was modified
to incorporate dual RF front-ends on a single board connected to a single correlator,
with a capability to use an off-board clock signal to drive the receiver circuitry [EO99].
These modifications allow for an arbitrary number of dual front-ends receiver cards
to be chained together using a common clock. In this configuration any number of
antennas can be used to form a GPS based attitude system. Figure A.1 shows a
picture of the attitude capable 4 RF front-ends GPS receiver [EO99].
The attitude receivers consist of two dual RF front-ends receiver cards mounted on
a single interface board (Figure A.1). This interface board provides power (including
backup from a small NiCAD battery) to the receiver cards, a buffered clock signal to
drive them, and three standard serial ports. There is a dedicated serial connection to
each receiver card, and one serial port that is connected to both cards. This configuration enables information to be transmitted between the two cards (through their
180
A.2. RECEIVER SOFTWARE
181
Fig. A.1: Attitude Receiver
dedicated serial ports), while the interface to the entire receiver can be made through
the joint serial port. Although 4 RF front-ends were available, 3 antenna attitude
algorithms were developed and the 4th RF front-ends was reserved for future onboard
pseudolite input. In addition, a simple single-board design was developed for 2 antenna configuration. This single-board, dual-antenna attitude receiver (Figure A.2)
was able to determine the attitude along 2 axes.
A.2
Receiver Software
GPS Architect Toolkit provides access to the receiver firmware and it allows us to
extensively modify the code and carrier tracking loops [RB97], the signal acquisition
algorithms, the cycle-slip detection routines, the input/output routines, and the frequency search region during startup. Since the two dual front-ends cards mounted on
the interface board (Figure A.1) have their own processors, each card can share the
relatively large amount of computations required to estimate the vehicle attitude. A
Master/Slave configuration is used to distribute the computational load to the two
182APPENDIX A. GPS ATTITUDE RECEIVER DEVELOPMENT AND TESTING
Fig. A.2: Single-board Attitude Receiver
cards. These two cards are connected to each other through their dedicated serial
ports which allows them to share the information. Previously, all the vehicle attitude
and relative position calculations were done off-board by sending the carrier phase
data to the ground computer through the serial ports, but now the new software
uses onboard receiver processor to compute the attitude solutions using the spare
processing power. Benefits of onboard processing of the attitude are
• Onboard attitude computation can minimize the data transfer from vehicle to
vehicle in the formation or device to device in the vehicle.
• Especially if the number of the vehicles increases in the formation, this estimation scheme will relieve computational burden on the main processors.
In the dual-board design, six channels (1 antenna) are assigned to the first frontend of the master card and the processor on the master card computes pseudorange
A.3. ONBOARD CYCLE AMBIGUITY ESTIMATION
183
and amount of sampling time each card needs to slew since the sampling time of
the each receiver is synchronized to the GPS time. Then the master board sends
the carrier phase data of these 6 channels to the slave board including the line-ofsight information. Since the slave card does not compute the pseudorange, but just
tracks satellites on the 12 channels assigned to the 2 antennas (6 channels on each
antenna), it has enough spare processing time to compute the attitude at a 5 Hz
rate. The carrier phase data from the master card arrives every 0.2 seconds (GPS
time) and consequently, the slave card forms 12 single differences using the phase
data from the both master card and itself. The master card also sends sampling time
slew information every 0.9 seconds (GPS time) such that both cards slew the sample
time every seconds. On the other hand, the single board attitude receiver assigns six
channels to each antenna and processes the 6 single differences.
The onboard estimation algorithms consists of the cycle ambiguity estimation
algorithm and the attitude estimation algorithm.
A.3
Onboard Cycle Ambiguity Estimation
Since an accurate estimate of the cycle ambiguity states is required in the attitude
determination, cycle ambiguity estimation algorithms are implemented onboard. Exisiting real-time approachs to cycle ambiguity resolution based on systematic integer
search algorithms have been successfully used in a number of applications and have
substantial advantages from a computational perspective [RH90]. However, because
multiple solutions are possible using these techniques, in this research, a robust motion based least-squares algorithm was implemented [BP96].
A.3.1
Computing Initial Guess of States For Two Base Line
Platform
The motion-based least-squares estimation of cycle ambiguity requires the linearization of the observation equation about the best current estimate. The convergence
184APPENDIX A. GPS ATTITUDE RECEIVER DEVELOPMENT AND TESTING
x(2)
x(1)
x
Fig. A.3: Large angle motion of baseline in 2D [CC92]
characteristics of Gauss-Newton iteration are highly dependent on the quality of the
initial guess of the attitude states provided the batch algorithm.
The following kinematic approach developed by Cohen can provide a very good
initial guess of the states using large baseline motions [CC92]. The solution of this
kinematic method will be used as an initial guess to the batch algorithms described
in the following sections. If the baselines sweep out a large angle, then it is possible
to accumulate a set of displacement vectors that can be used to identify an initial
attitude guess.
The procedure is to rotate the fixed baselines from some unknown initial direction
given by x, we then collect the changes in the displacement of the baselines at several
time steps (Fig A.3). The displacement of the baseline vector at each time step, ∆x,
can be easily calculated by subtracting the differential carrier phase of initial time
step from that of any arbitrary time step (do not need to know the integer biases).
Noting that the baseline length is constant [CC92], then
(x + ∆x)T (x + ∆x) = xT x
(A.1)
A.3. ONBOARD CYCLE AMBIGUITY ESTIMATION
185
where x is initial baseline vector in the local coordinate system. Two terms can be
cancelled out, leaving
2∆xT x = −∆xT ∆x
(A.2)
With ∆x taken at N different times, we can stack the data into a matrix form
2∆X T x = −diag(∆X T ∆X)
where
∆X =
∆x(1) ∆x(2) · · · ∆x(N )
(A.3)
(A.4)
and ∆x(k) can be directly computed from the differential phases. If there are n
satellites in view, then



(k)
los1
∆r1



..
 .. 

 .  ∆x(k) = 
.








(A.5)
∆rn(k)
lson
where
(k)
(k)
(0)
∆r1 = ∆φ1 − ∆φ1
(A.6)
and ∆φkl denotes the differential carrier phase measurement at the k th time step using
the lth satellite. The vector los denotes the line-of-sight vector from the antenna to
the satellite in the local coordinate frame. If there are more than one baseline, then
the information from multiple baselines can be combined (the angles between the
baselines are constant for a rigid body). Between two baselines i and j, we have
(xi + ∆xi )T (xj + ∆xj ) = xTi xj
(A.7)
Again, two terms can be cancelled, leaving
∆xTj xi + ∆xTi xj = −∆xTi ∆xj
(A.8)
186APPENDIX A. GPS ATTITUDE RECEIVER DEVELOPMENT AND TESTING
Invoking the same notation as above, the matrix form can be constucted for the two
baselines case
∆XjT xi + ∆XiT xj = −diag(∆XiT ∆Xj )
where
∆Xi =
(1)
∆xi
(2)
∆xi
···
(N )
∆xi
(A.9)
(A.10)
Using Eq. A.3 and Eq. A.9, the initial guesses for the two baseline vectors (x1 and
x2 ) can be combined into a single, unified least-squares fit





∆X2T
∆X1T
2∆X1T
0
0
2∆X2T







x1
x2




diag(∆X1T ∆X2 )



= −  diag(∆X1T ∆X1 ) 


(A.11)
diag(∆X2T ∆X2 )
The left-hand side matrix is 3N × 6, the unknown vector is 6 × 1. Therefore no a
priori information about the antenna placement is required to unambiguously solve
for both initial baseline vectors. Initial attitude states can be obtained from these
baseline vectors x1 and x2 using the following equations
x1 = Ti/b b1
(A.12)
x2 = Ti/b b2
(A.13)
where b1 denotes the baseline vector in the body frame, and Ti/b denotes the rotation
matrix from the body frame to the local coordinate frame. These initial baseline
vectors can then be propagated using the accumulated ∆X1 and ∆X2 to find all of
the attitude states. These attitude states are then used as an initial guess for a batch
nonlinear bias estimation.
A.3.2
Batch Algorithm
After initial guesses of states are found by the method described in the previous
section, a simple batch gradient search technique can be used to solve for the integer
biases. This approach has been used successfully on several applications [BP96]. We
A.3. ONBOARD CYCLE AMBIGUITY ESTIMATION
187
start with the measurement equations in the form
y = h(X) + β + ν
(A.14)
where ν is the measurement noise vector distributed as νk ∼ N(0, σ 2 ). Collecting
carrier phase data while the baselines are in motion results in a set of measurements
yk = hk (xk ) + β + νk , k = 1, ..., N
(A.15)
where the state vector xk includes the three Euler angles
xk =
T
ψk θk φk
(A.16)
Eq.( A.15) can be linearized about the current best guesses of x̂k and β̂ to give
yk ∼
= hk (x̂k ) + Hk (x̂k )δxk + β̂ + δβ + νk
(A.17)
ρk = yk − hk (x̂k ) − β̂ ∼
= Hk (x̂k )δxk + δβ + νk
(A.18)
Stacking these equations for all k = 1, . . . , N gives












ρ1   H1



ρ2 
H2


=


..  
..
.
.  
ρN


0


δx1 
I 


δx2 



I 
.. 


. 
+ν
.. 



. 
  δXN 


I 
δβ
0
HN
(A.19)
If we define

H̃ =









H1
0
H2
..
0
.


I 
 δx1


 δx2
I 


,
δx
=


..
.. 

.
. 

HN I


δXN








(A.20)
188APPENDIX A. GPS ATTITUDE RECEIVER DEVELOPMENT AND TESTING
where Hk = Hk (x̂k ), then

ρ = H̃ 

δx
δβ
+ν
(A.21)
The weighted least solution is then found by iteratively solving



δx̂
δ β̂

= (H̃ T R−1 H̃)−1 H̃ T R−1 ρ
(A.22)
and,
x̂new = x̂old + δx̂, β̂new = β̂old + δ β̂
A.3.3
(A.23)
Iterative Information Smoother
A brute force application of the above can result in an excessive computational load
since the array sizes become quite large, even for small N. In fact, when the algorithm
was tested on the same processor (ARM60) that was also processing the tracking loop
routines for 12 channels (and handling the i/o routines), the entire batch bias estimation took about 1–2 minutes. An alternative approach is to leverage the fact that
H̃ is sparse, and to apply a sparse matrix batch least squares routine. A second
approach is to use an iterated information smoother. This algorithm was used by
Pervan [BP96] in the estimation of the cycle ambiguity states in the relative position
estimator for the Integrity Beacon Landing System (IBLS). In the development of
the attitude receiver, the algorithm is then modified to estimate the bias states in
the attitude estimation equations. These algorithms is not only faster and uses less
memory, but can also bring measurements on or off-line in an easy manner if signals are lost or acquired during initialization. The iterated information smoother is
formulated as follows [BP96].
The perturbation of Eq. A.14 about the current estimates x̂k and β̂ can be written
as
ρk = Hk (x̂k )δxk + δβ + νk =
Hk I



δxk
δβ
 + νk
(A.24)
A.3. ONBOARD CYCLE AMBIGUITY ESTIMATION
189
If no signals are lost or acquired between the sampled measurements, then the baseline
kinematics can be expressed as a discrete Gauss-Markov process



δx


δβ
=
k+1

δx

δβ
+ wk
(A.25)
k
where the process noise wk is distributed as wk ∼ N(0, W ) and

W = µ→∞
lim 

µIx 0
0

(A.26)
0
The matrix Ix is the identity matrix with dimensions corresponding to the number of
states in the system, since the integer ambiguities are constant from epoch to epoch.
Let the state estimate-error covariance be P̂k−1 , the covariance time update P k is
then

P k = P̂k−1 + W = lim 
µ→∞

µIx P̂xβ
T
P̂xβ
P̂β

(A.27)
k−1
where P̂xβ and P̂β are the right upper, and lower sub-blocks of P̂ , with dimensions
conforming to the number of states, and biases in the system. This can be expressed
in information matrix form as

Sk = 

0
0
0 P̂β−1

(A.28)
k−1
where P̂β is the lower right block of P̂ corresponding to the integers. The information
matrix measurement update is then

Sk = 
0
0
0 P̂β



+
k−1
HkT
I
T


Hk I
(A.29)
190APPENDIX A. GPS ATTITUDE RECEIVER DEVELOPMENT AND TESTING
where the measurement covariance has been normalized by measurement noise variance σ 2 . Eq. A.29 can also be expressed as

Sk = 
0
0
T
Ŝβ−1 Ŝxβ
0 Ŝβ − Ŝxβ



+
k−1
HkT
T
I


Hk I
(A.30)
If we define the information vector as

zk = 

zx
zβ


≡ Sk 
k

δx
δβ

(A.31)
k
then the information vector update can then be written as

ẑk = 
0
T
ẑβ − Ŝxβ
Ŝβ−1 ẑx



+
k−1
HkT
I
T

 δyk
(A.32)
Eqs. A.30 and A.32 can be applied sequentially with initial conditions z 1 =0, and
S 1 =0 until the final epoch N is reached. It is also necessary to perform a backward
smoothing of the collected measurements, which is easily done by interchanging the
k and k − 1 indices in Eqs. A.30 and A.32. In this case, the equations are applied
sequentially with z N =0 and S N =0. The information matrix and vector from both
the forward and backward passes are added together to form
Ŝk = ŜkF + ŜkB , ẑk = ẑkF + ẑkB
(A.33)
The state and bias updates can then be computed from Ŝk−1 ẑk . The entire process is
repeated until δx and δβ are sufficiently small. This algorithm was run after finding
the initial guess of states x̂k and β̂k using the kinematic method. Execution time of the
entire bias estimation algorithm is, typically, 1–2 seconds using an onboard processor.
Also, the amount of memory space required for this estimation was reduced by a factor
of 5 compared to brute force batch algorithm.
Appendix B
Formation Ranging Device
Development
B.1
Laser Pointer Ranging Device
This device consists of a pair of laser pointers and a target board. These devices are
mounted on the follower and leader robots, respectively. Figures B.1 and B.2 show
the laser pointer array and the target board mounted on each vehicle. The pair of
laser pointers form a equilateral triangle with a fixed baseline. The focal length of
laser pointer was accurately surveyed prior to each test.
As the vehicles undergo formation maneuvers, a video camera mounted at the
center of the laser pointer baseline recorded the image of the target. This image
clearly shows the dots where the 2 lasers hit the target board. With this image, it is
possible to calculate 2–D position of follower robot with respect to the leader robot
using the geometry shown in Figure B.3
The coordinate frame is attached to the center of the target and the x, y axes
are aligned with the true south and east direction, respectively. Note that the local
coordinate frame does not rotate with the target or the robot. The goal is to calculate
the relative position of the reference (center) point on the laser array with respect
to the origin of reference frame. The measurements that can be extracted from the
camera are the distance from the center of target to the 2 points where the laser
191
192
APPENDIX B. FORMATION RANGING DEVICE DEVELOPMENT
Fig. B.1: Laser Pointer and Baseline Fig. B.2: Target Board Mounted on
Mounted on Follower Robot
Leader Robot
beams intersect the target. These distances are designated as L and R. Figure B.3
shows the geometry of the laser pointer array and the target. It is assumed that the
attitude of the 2 robots (φ1 and φ2 ) are known accurately1 . The relative position of
the reference point of the laser pointer array with respect to the origin is defined as
(a1 , a2 ) and from the geometry,
θ = arctan
f
(B.1)
B
2
Define the location of the left and right laser sources as (c1 , c2 ) and (d1 , d2 )
B
sin φ2 , a2 +
2
B
(d1 , d2 ) = (a1 + sin φ2 , a2 −
2
(c1 , c2 ) = (a1 −
B
cos φ2 )
2
B
cos φ2 )
2
(B.2)
(B.3)
The laser beam lines l and r can then be written as
1
y = tan (φ1 − θ)x + c1 − tan (φ2 − θ)c2
(B.4)
y = tan (φ1 + θ)x + d1 − tan (φ2 + θ)d2
(B.5)
Provided by the attitude GPS receiver.
B.1. LASER POINTER RANGING DEVICE
193
xl
X
φ1
xr
Y
target
t
R
Target Center,
Reference Point
L
f
r
a1
l
θ φ2
φ2
laser pointer
(d1, d2)
B
(c1, c2)
a2
Fig. B.3: Geometry of Laser Pointer Ranging Device
The line representing the target (t) is
y = tan (φ1 +
π
)x
2
(B.6)
The x direction location of the points where the lines l and r intersect the line t can
be expressed as
c1 − tan (φ2 − θ)c2
tan (φ1 + π2 ) − tan (φ1 − θ)
d1 − tan (φ2 + θ)d2
=
tan (φ1 + π2 ) − tan (φ1 + θ)
xl =
(B.7)
xr
(B.8)
APPENDIX B. FORMATION RANGING DEVICE DEVELOPMENT
194
The intersection points xl and xr can be also expressed in terms of the measurements
L and R
xl = −L sin φ1
(B.9)
xr = −R sin φ1
(B.10)
A matrix equation can be formed by combining Eqs. B.7, B.8, B.9, and B.10, and then
substituting c1 , c2 , d1 , and d2 values using Eqs. B.2 and B.3. The resulting expression
is of the form




− tan (φ2 + θ) 1
− tan (φ2 − θ) 1



a1
a2

=
−L sin φ1 (tan (φ1 + π2 ) − tan (φ1 − θ)) +
−L sin φ1 (tan (φ1 +
π
)
2
− tan (φ1 − θ)) +
B
2
B
2
cos φ2 + tan (φ2 + θ) B2 sin φ2
cos φ2 + tan (φ2 +
θ) B2
sin φ2


(B.11)
By solving the matrix Eq. B.11, the relative position (a1 , a2 ) can be expressed in
terms of the measurements L, R and vehicle heading angles.
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