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Dynamics and Control of Satellite Relative Motion: Designs and
Applications
Soung Sub Lee
Dissertation submitted to the Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
in
Aerospace Engineering
Dr. Christopher D. Hall, Committee Chair
Dr. Craig A. Woolsey, Committee Member
Dr. Cornel Sultan, Committee Member
Dr. Scott L. Hendricks, Committee Member
March 20, 2009
Blacksburg, Virginia
Keywords: Satellite Relative Orbit, Satellite Constellation, Satellite Control
Copyright 2009, Soung Sub Lee
Dynamics and Control of Satellite Relative Motion: Designs and Applications
Soung Sub Lee
(ABSTRACT)
This dissertation proposes analytic tools for dynamics and control problems in the per-
spective of large-scale relative motion without perturbations. Specifically, we develop an
exact and efficient analytic solution of satellite relative motion using a direct geometrical
approach in spherical coordinates. The resulting solution is then transformed into general
parametric equations of cycloids and trochoids. With this transformation, the dissertation
presents new findings for design rules and classifications of closed and periodic parametric
relative orbits. A new observation from the findings states that the orbit shape resulting
from the relative motion dynamics of circular orbit cases in polar views are exactly the same
as the parametric curves of cycloids and trochoids. The dynamics problem of satellite rela-
tive motion is expanded to include the design of satellite constellations for multiple satellite
systems. A Parametric Constellation (PC) is developed to create an identical constellation
pattern, or repeating space track, of target satellites with respect to a base satellite. In
this PC theory, the number of target satellites is distributed using a real number system
for node spacing. While using a base satellite orbit as the rotating reference frame, the PC
theory consists of satellite phasing rules and closed form formulae for designing repeating
space tracks. The evaluation of the PC theory is illustrated through it’s comparison to
the existing Flower Constellation theory in terms of node spacing distribution and constel-
lation design process. For the control problems, the efficient analytic solution is applied
to the reference trajectory of satellite relative tracking control systems for inter-satellite
links. Two types of relative tracking control systems are developed and each is evaluated
to determine which is more appropriate for practical applications of inter-satellite links.
All of the proposed analytic solutions and tools in this dissertation will be useful for the
mission analysis and design of relative motions involving a two or more satellite system.
Dedication
I would like to dedicate this dissertation to my parents in Korea, my wife, Kyungju, and
two sons, Kiwon and Kibum.
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Acknowledgments
I would like to begin my acknowledgements by expressing great appreciation for my advisor,
Dr. Chris D. Hall. I cannot begin to describe how helpful and considerate he has been to
me over the past five years. In particular, I really appreciate his patience as an advisor
and the research ideas he has provided through his deep insight into the spacecraft field.
I also owe many thanks to Dr. Scott L. Hendricks, Dr. Cornel Sultan, and Dr. Craig
A. Woolsey for their encouraging remarks and gratitude, as well as their time devoted to
me. I am also grateful for all the lessons I’ve learned from the outstanding professors who
taught my classes at Virginia Tech. Although the period was brief, I am extremely happy
to have met Scott. A. Kowalchuk, Brian Williams, and the other students in the Space
Systems Simulation Laboratory. Additionally, I am indebted to the Republic of Korea
Airforce (ROKAF) for providing this opportunity to pursue my doctorate degree and for
supporting me financially during my time in America.
Personally, I would like to acknowledge and express my thanks to several people for making
my stay in Blacksburg special and enjoyable. These close friends include Dr. Namheui
Jeong, Daewon Kim, Jinwon Park, Hyunsun Do, Hyunju Jeong and my junior Dongsik
Lee. Another individual I would like to thank is my close American friend, Mark. Finally,
I would like to express my dearest thanks to my parents, my wife, and my two sons for
their continued support, patience, and encouragement throughout my entire effort towards
this dissertation.
iv
Contents
1 Introduction 1
1.1 Dissertation Problem Statements . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Dissertation Objectives and Contributions . . . . . . . . . . . . . . . . . . 2
1.3 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Literature Review 6
2.1 Satellite Relative Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Satellite Constellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 Target Tracking Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Satellite Relative Orbit Designs 12
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Keplerian Orbit in Spherical Coordinate Systems . . . . . . . . . . . . . . 13
3.3 Geometrical Relative Orbit Modeling . . . . . . . . . . . . . . . . . . . . . 15
3.4 Linearized Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 22
v
3.5 Modeling Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5.1 Absolute Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5.2 Relative Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.6 Modeling Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4 Parametric Relative Orbit Designs 37
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 General Parametric Equations and Curves . . . . . . . . . . . . . . . . . . 39
4.3 Parametric Relative Equations . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.4 Characteristics of Parametric Relative Orbits . . . . . . . . . . . . . . . . . 44
4.4.1 Design Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.2 Classifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Parametric Constellations Theory 54
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.3 Parametric Constellations . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3.1 Satellite Phasing Rules . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3.2 Transformation of Satellite Phasing Rules . . . . . . . . . . . . . . 65
5.3.3 Repeating Ground Track Orbits . . . . . . . . . . . . . . . . . . . . 68
5.3.4 Repeating Space Tracks with a Single Orbit . . . . . . . . . . . . . 69
vi
5.4 Closed-form Formulae for PCs . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.5 Evaluation of the PC Theory . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5.1 Node Spacing Discussion . . . . . . . . . . . . . . . . . . . . . . . . 72
5.5.2 Comparison of Constellation Design Process . . . . . . . . . . . . . 74
5.6 Numerical Examples of PC Designs . . . . . . . . . . . . . . . . . . . . . . 78
5.6.1 Inter-satellite Constellation Design . . . . . . . . . . . . . . . . . . 79
5.6.2 Formation Flying Design . . . . . . . . . . . . . . . . . . . . . . . . 81
5.6.3 PC Design with a Single Orbit . . . . . . . . . . . . . . . . . . . . . 83
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Satellite Relative Tracking Controls 86
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Representation of Reference Systems . . . . . . . . . . . . . . . . . . . . . 87
6.3 Attitude Parameterization . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6.3.1 Generalized Symmetric Stereographic Parameters (GSSPs) . . . . . 91
6.3.2 Modified Rodrigues Parameters (MRPs) . . . . . . . . . . . . . . . 93
6.4 Relative Angular Velocity and Acceleration Vectors . . . . . . . . . . . . . 94
6.5 Transformation of Equations of Motion . . . . . . . . . . . . . . . . . . . . 97
6.6 Design of Sliding Mode Tracking Controller . . . . . . . . . . . . . . . . . 98
6.6.1 Dynamics and Kinematics for Satellite Tracking Problem . . . . . . 98
6.6.2 Stabilizing the MRP Kinematics . . . . . . . . . . . . . . . . . . . . 101
6.6.3 Stabilizing the Full System . . . . . . . . . . . . . . . . . . . . . . . 102
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6.7 Satellite Relative Tracking Controls . . . . . . . . . . . . . . . . . . . . . . 106
6.7.1 Body-to-Body Relative Tracking Control . . . . . . . . . . . . . . . 106
6.7.2 Payload-to-Payload Relative Tracking Control . . . . . . . . . . . . 113
6.8 Evaluation of Satellite Relative Tracking Controls . . . . . . . . . . . . . . 118
6.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
7 Conclusions and Recommendations 122
7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
7.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
A Spherical Geometry and Spherical Coordinate System 126
B Unit Sphere Approach 128
C Numerical Design Processes of FCs and PCs 131
Bibliography 144
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List of Figures
3.1 Keplerian orbit elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 Projection of a Keplerian orbit on celestial sphere . . . . . . . . . . . . . . 15
3.3 Geometry for modeling the relative motion on the surface of a sphere . . . 16
3.4 Spherical triangle for computing φB
and φT
. . . . . . . . . . . . . . . . . . 17
3.5 Geometry for computing α and δ with ∆Ω = 0 . . . . . . . . . . . . . . . . 19
3.6 Geometry for the relative phase angle ψ. . . . . . . . . . . . . . . . . . . . 25
3.7 In-track/cross track motion by the relative phase angles ψ (Ay = 0.01km, Az =
3.0km). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.8 Relative separations by the relative phase angle ψ. . . . . . . . . . . . . . . 28
3.9 Absolute relative position errors . . . . . . . . . . . . . . . . . . . . . . . . 30
3.10 Absolute relative velocity errors . . . . . . . . . . . . . . . . . . . . . . . . 31
3.11 Index comparison for various relative distances . . . . . . . . . . . . . . . . 33
3.12 Index comparison for various eccentricities . . . . . . . . . . . . . . . . . . 33
4.1 Commensurable relative orbits of γ = 2.0, 2.2, 2.2142 (3-dimensional view). 38
4.2 Commensurable relative orbits of γ = 2.0, 2.2, 2.2142 (polar view). . . . . 38
4.3 Hypotrochoid motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
ix
4.4 Epitrochoid motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.5 Deltoid and astroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.6 Cardioid and nephroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.7 Geometrical descriptions of parametric relative equation. . . . . . . . . . . 43
4.8 Intersection points of a 10-petal parametric relative orbit (γ = 5/3, e = 0.1). 47
4.9 3-cusped hypocycloid motion . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.10 Velocity components of x and y of the 3-cusped hypocycloid motion . . . . 49
4.11 Curtate hypotrochoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.12 Prolate hypotrochoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.13 Spirographs of parametric relative orbits . . . . . . . . . . . . . . . . . . . 52
5.1 Three identical target satellite orbits and a base satellite circular orbit . . . 56
5.2 A target satellite orbit plane and a base satellite elliptic orbit . . . . . . . 57
5.3 Geometry of target satellite orbits about a base satellite orbit plane. . . . . 60
5.4 4-petaled hypocycloid parametric relative orbit in x− y plane. . . . . . . . 64
5.5 Geometry for relative orbital elements and ECI′ frame . . . . . . . . . . . . 66
5.6 Rational rotation with the three decimal places of√
3 . . . . . . . . . . . . 73
5.7 Irrational rotation of√
3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.8 Flowchart of PC design process. . . . . . . . . . . . . . . . . . . . . . . . . 75
5.9 Repeating ground track orbits in the ECI frame. . . . . . . . . . . . . . . . 76
5.10 Repeating relative orbits in the ECI′ frame. . . . . . . . . . . . . . . . . . 77
5.11 Repeating relative orbits in the ECI′ frame. . . . . . . . . . . . . . . . . . 78
x
5.12 3D view (left) and polar view (right) of (20000,3,20) PC. . . . . . . . . . . 81
5.13 Formation flying design of (9000,1,10) PC. . . . . . . . . . . . . . . . . . . 82
5.14 Orbit elements sets of (9000,1,10) PC. . . . . . . . . . . . . . . . . . . . . 83
5.15 PC design of (7000, 1/10, 10) with a single orbit. . . . . . . . . . . . . . . . 84
6.1 Stereographic projection of quaternion . . . . . . . . . . . . . . . . . . . . 92
6.2 Two rotating reference frames in the base satellite coordinate system . . . 99
6.3 Geometry of sliding mode control . . . . . . . . . . . . . . . . . . . . . . . 103
6.4 Rotations from Fb to Fp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.5 Diagram of B-B relative tracking control . . . . . . . . . . . . . . . . . . . 108
6.6 B-B relative tracking control simulation (||σ||, ||δω||, ||s||) . . . . . . . . . . 112
6.7 Time history of Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.8 Coordinate frames of reference system . . . . . . . . . . . . . . . . . . . . 113
6.9 Diagram of P-P relative tracking control . . . . . . . . . . . . . . . . . . . 115
6.10 P-P relative tracking control simulation (||σ||, ||δω||, ||s||) . . . . . . . . . . 117
6.11 Time history of Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . 117
6.12 Comparison of the tracking errors . . . . . . . . . . . . . . . . . . . . . . . 120
6.13 Comparison of the control torques . . . . . . . . . . . . . . . . . . . . . . . 120
A.1 Spherical triangles and spherical coordinates on the sphere . . . . . . . . . 127
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List of Tables
3.1 Parameters of the orbit elements . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 Parameters of the orbit elements . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 Parameter of the orbit elements . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Comparison of analytic solution efficiency . . . . . . . . . . . . . . . . . . . 35
3.5 Differences of unperturbed and J2 perturbed models (Time step : 0.1 sec) . 36
4.1 Numerical examples of computing γpetal . . . . . . . . . . . . . . . . . . . . 46
4.2 Special cases of hypocycloid and epicycloid . . . . . . . . . . . . . . . . . . 48
5.1 Satellite phasing rules in the ECI frame . . . . . . . . . . . . . . . . . . . . 65
5.2 Satellite phasing rules in the ECI′ frame . . . . . . . . . . . . . . . . . . . 67
5.3 Parameters of the orbit elements (γ = 3) . . . . . . . . . . . . . . . . . . . 79
5.4 Orbit element sets of (20000,3,20) PC (unit:degree) . . . . . . . . . . . . . 80
5.5 Geometrical parameters of (20000,3,20) PC (unit:degree) . . . . . . . . . . 80
5.6 Parameters of the orbit elements (γ = 1) . . . . . . . . . . . . . . . . . . . 82
5.7 Parameters of the orbit elements (γ = 1/10) . . . . . . . . . . . . . . . . . 84
5.8 Orbit element sets of (7000,1/10,10) PC (unit:degree) . . . . . . . . . . . . 85
xii
6.1 The Definitions of CRPs and MRPs . . . . . . . . . . . . . . . . . . . . . . 92
6.2 Orbit elements of the base and target satellites . . . . . . . . . . . . . . . . 110
6.3 Parameter values for numerical simulation . . . . . . . . . . . . . . . . . . 111
C.1 Design parameters of FCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
C.2 Design parameters of PCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
C.3 Design parameters of FCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
C.4 Orbital parameters of the base and target satellites . . . . . . . . . . . . . 135
C.5 True anomalies of initial mean anomalies . . . . . . . . . . . . . . . . . . . 136
C.6 PQW position and velocity vectors . . . . . . . . . . . . . . . . . . . . . . 136
C.7 Position and velocity vectors in the ECI′ frame . . . . . . . . . . . . . . . . 137
C.8 Position and velocity vectors in the ECI frame . . . . . . . . . . . . . . . . 137
C.9 Resulting orbital elements of target satellites (FCs) . . . . . . . . . . . . . 139
C.10 Design parameters of PCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
C.11 Resulting orbital elements of target satellites (PCs) . . . . . . . . . . . . . 141
C.12 Design parameters of PCs . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
C.13 Resulting orbital elements of target satellites (PCs) . . . . . . . . . . . . . 143
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Chapter 1
Introduction
1.1 Dissertation Problem Statements
The relative motion of satellites is defined as a space track or trajectory of one satellite with
respect to another satellite in a gravitational field. The dynamics and control problems of
satellite relative motion in a central gravitational field are highly challenging, compared
to the problems associated with a single satellite system. We can extend the system of
satellite relative motion beyond just two satellites to encompass an unlimited quantity.
Although it is possible to have an infinite number of satellites in a system, the associated
dynamics problems grow increasingly more complex with each satellite added.
For the purpose of this dissertation, the satellite relative motion can be divided into two
parts: large-scale relative motion and small-scale relative motion. In large-scale relative
motion the distances between satellites are relatively large, thus creating a more complex
system of dynamics problems. On the contrary, in small-scale relative motion, the distances
between satellites are much smaller, resulting in considerably simplified equations of rela-
tive motion. The advantage of small-scale relative motion lies in the simplified equations
of relative motion providing simpler dynamics problems. With this simplification and the
potential for more practical applications, previous studies have focused intensively on this
1
2
scale of relative motion. However, in the case of large-scale relative motion, few analytic
studies have been performed because of the increased complexity associated. This com-
plexity causes systems involving relative motion to generally rely on computer simulations
and numerical integrations of the equations of motion.
Satellite relative motion, or how satellites appear to move as seen from an observer satellite,
is important for mission planing and constellation designs as well as for data transmission
purposes by allowing us to understand how to design and point antennas, instruments, or
sensors. Because we are interested in the preliminary planning and designs of practical
applications of dynamics and control problems, we can not rely solely on computer simula-
tions for large-scale relative motion. Instead, we need broadly applicable analytic tools to
examine and analyze the designs of constellations, formation flying, and control systems.
This dissertation, therefore, aims at making research efforts to study the geometrical char-
acteristics and to develop the analytic tools for satellite relative motion.
1.2 Dissertation Objectives and Contributions
The objectives of this dissertation are to develop analytic tools for mission analysis and
designs in the perspective of large-scale relative motion. Furthermore, a portion of the
resulting analytic tools are applied to satellite tracking control systems.
The dissertation is divided into two main problems associated with relative motion: those
of dynamics and those of control. Specifically, in the portion pertaining to the dynamics
problems, several contributions are presented. First, we develop an exact and efficient
analytic solution for the problems of satellite relative motion without perturbations. A
direct geometrical approach using spherical trigonometric solutions is taken to develop
these results. From the evaluations, the resulting solution, with the geometrical approach,
illustrates more efficiency than the existing solutions, providing the exact description of
satellite relative motion. Thus, the proposed analytic solution will be useful as an effective
3
tool for the problems of satellite relative motion.
Second, we go on to find general rules and classifications for designing satellite relative
motion. To do this, the proposed analytic solutions are transformed into the mathematical
formulae of parametric curves. With this transformation, new observations for relative
motion geometry are found. One of the new findings states that the orbit shape resulting
from the relative motion dynamics of circular orbit cases in polar views are exactly the
same as the mathematical models of cycloids and trochoids. Furthermore, satellite relative
motion can be specified by the number of petals or cusps of the cycloids and trochoids
based on the relative orbit frequency. These new findings are important for the process of
the mission analysis and design of satellite relative motion.
Finally, as a primary goal of the portion pertaining to the dynamics problems, we develop
a constellation design theory for multiple-satellite relative motion, using the geometrical
relations of satellite orbits and the periodic conditions of satellite relative orbits. With
this proposed constellation theory, an infinite number of target satellite orbits can be
represented by a single identical constellation pattern as seen by a base satellite. The
proposed constellation theory will be useful as an effective design tool for the complex
design problems associated with multiple satellite constellations.
In the portion of the dissertation covering control problems, we develop relative tracking
control systems of two satellites for inter-satellite links, applying the analytic solution of
satellite relative motions to the reference trajectory for tracking. Two types of relative
tracking controls are developed, and we evaluate the tracking control systems in terms of
convergence rate and control torque. Based on the evaluation, we propose an appropriate
tracking control system for the practical applications of inter-satellite links.
4
1.3 Dissertation Overview
This section gives a brief description of each chapter of the dissertation beginning with
Chapter 2. In Chapter 2, previous literature surveys for dynamics and control problems of
satellite relative motions are presented. The literature surveys review three broad topics:
satellite relative orbit, satellite constellation, and target tracking control.
Chapter 3 derives the relative position and velocity of a target satellite as seen by a
base satellite. The chapter begins by discussing Keplerian orbits in spherical coordinates.
Section 3.4 then derives linearized equations of motion and finds geometrical insight about
cross track motion. We compare and evaluate the resulting equations in terms of modeling
accuracy and efficiency in Sections 3.5 and 3.6.
In Chapter 4, we first introduce general equations of parametric curves, and the resulting
solutions in Chapter 3 are then converted into the general parametric formulas. Section
4.4 finds general rules to design, and provides classifications for, satellite relative orbits.
Chapter 5 proposes a constellation design theory for repeating space tracks of satellite
relative motion. Specifically, Section 5.2 gives the problem statement for the constellation
theory. Section 5.3 develops satellite phasing rules to obtain orbit element sets, while Sec-
tion 5.4 introduces the closed-form formulae to describe constellation patterns of repeating
space tracks. We evaluate the proposed constellation theory in terms of node spacing dis-
tribution and constellation design process in Section 5.5. Finally, we illustrate numerical
examples for several types of repeating space tracks.
Chapter 6 applies the analytic solutions in Chapter 3 to satellite relative tracking control
systems by first defining the various types of reference frames used. We discuss Modified
Rodrigues Parameters for attitude coordinates in Section 6.3, and we derive the relative
angular velocity and acceleration for tracking in Section 6.4. Using the sliding mode
scheme, two types of relative tracking control systems are developed in Section 6.7. Finally,
we evaluate the satellite relative tracking controls in terms of convergence rates and control
5
torques in section 6.8.
The dissertation concludes with Chapter 7, which summarizes all of the findings and conclu-
sions discussed in Chapters 2 through 6, and with Appendixes A through D. In Appendix A,
we discuss spherical trigonometric solutions and spherical coordinates for Chapter 3. Ap-
pendix B introduces the analytic solution of the unit sphere approach for the comparisons
of modeling accuracy and efficiency of the solutions resulting from Chapter 3. Appendix C
and D show the flowchart of the proposed constellation design tool and numerical examples
of the constellation design processes, respectively.
Chapter 2
Literature Review
In this chapter, we review previous works associated with the dynamics and control prob-
lems of satellite relative motion. The literature surveys for the previous works are in-
vestigated for three broad areas: satellite relative orbit, satellite constellation, and target
tracking control.
2.1 Satellite Relative Orbit
The study of satellite relative motion has been pursued by those interested in various
challenging tasks of space missions. The main focus has been on the study of formation
flying and rendezvous and docking maneuvers of satellites. For these applications, theories
of satellite relative motion began with the equations of motion derived by Clohessy and
Wiltshire(CW) in 1960 [1]. The reference satellite orbit was assumed to be circular and
the relative orbit coordinates were small compared with the reference orbit radius so that
the resulting equation of motion was linearized. In 1963 Lawden [2] found an improved
form for relative motion including reference orbit eccentricity, and Carter [3] later extended
Lawden’s solution. Next, Kechichian [4] developed an exact formulation of a general elliptic
orbit to analyze the relative motion in the presence of J2 potential and atmospheric drag.
6
7
In this study however, the resulting equations were required to use numerical integrations
over time. Sedwick et al [5] applied the J2 potential forcing function to the right hand side
of Hill’s equations. Schweighart [6] followed these equations and found analytic solutions.
Melton [7] later developed an approximate solution expanding the state transition matrix
in powers of eccentricity with time explicit representation.
In recent decades, numerous other theories of satellite relative motion have been added
to the literature. A brief survey of relative motion theories of satellites was published
by Alfriend and Yan [8]. This survey compared and evaluated various relative motion
theories: Hill’s equations, Gim-Alfriend State Transition Matrix [9], Small-Eccentricity
State Transition Matrix [8], Non-J2 State Transition Matrix [8], Unit Sphere Approach
[10, 11], and the Alfriend-Yan nonlinear method [12]. Their evaluation of the results showed
that the Unit Sphere Approach and the Yan-Alfriend nonlinear method present the highest
accuracy for all eccentricities and relative orbit sizes. The Unit Sphere Approach was
proposed by Vadali who achieved an exact analytic expression in terms of differential orbital
elements for relative motion problems. Alfriend-Yan applied the geometrical method to
nonlinear relative motion. The method was employed in a long term prediction of mean
orbital elements, including nonlinear J2 effects, and then in transforming the Hill’s frame.
Several studies can be found regarding the understanding of relative orbit geometry and
configurations. Gurfil et al [13] studied manifolds and metrics of relative motion problem.
This paper found that the relative motion geometry evolves on an invariant manifold
representing configuration space. In the case of the first order approximation of relative
position components, the relative orbits remain on the parametric shapes of an elliptic
torus. Jiang et al [14] investigated self intersections on three coordinate planes for the
radial, in-track, and cross-track motions, and designed the relative orbits using special
shapes in the coordinate plane.
8
2.2 Satellite Constellation
The diversity of constellation patterns and methods is the predominant characteristic in the
evolution of satellite constellations. Thus, the categorization of the numerous constellation
patterns and methods is difficult. The literature surveys of satellite constellations in this
section classifies the constellation patterns based on orbit types because it is a critical
factor in determining a satellite’s coverage of the Earth.
The simplest class of constellation types is geosynchronous constellations which are used for
many communications and weather purposes. The earliest idea of satellite constellations
was theorized by Clark [15] who proposed a constellation to provide full equatorial coverage
of the Earth using three geostationary satellites. Due to the nature of the geosynchronous
orbit, and because there currently exist hundreds of satellites utilizing this orbit, available
space is limited. Thus, new innovative constellation patterns known as Tundra orbits have
been studied [16]. The Tundra orbit is a special case of a geosynchronous orbit involving
an inclination and an elliptical shape. Alternative studies were proposed using the Tundra
orbit for commercial services [17, 18].
The next class of constellation types is the streets of coverage constellations which use near
polar orbit planes to provide continuous global coverage of the Earth. Several studies have
been performed for this constellation class. One such study was carried out by Luders [19]
for a street of coverage using circular orbits. By reducing overlap of satellite coverage,
Beste [20] was able to optimize the configuration, reducing the number of satellites required
by 15 percent compared to Luders’s configuration. Lider proposed an analytical solution in
a closed form to compute the minimum number of satellites [21]. Adams [22] later applied
Lider’s study to cases involving continuous coverage at specific latitudes.
The most symmetric, or regular, class of constellation types is the Walker constellation
using circular orbits. Walker [23, 24] used three parameters, total number of satellites, the
number of planes, and the phasing angle, to specify, and thus systematize and simplify,
a constellation pattern. The orbit types resulting from the specified constellation pattern
9
were the star and delta patterns. A type of regular constellation similar to the Walker
pattern is the Rosette constellation which provides the best coverage of the Earth along
with multiple satellites visible from the ground station [25].
While the original papers devoted to Walker constellations focused only on circular orbits,
a number of other papers on elliptical constellations were presented [26, 27]. One method
of constellation design studied the combination of elliptical and geostationary orbits to
achieve desired coverage properties [28, 29]. These studies concluded that the Walker
arrays using elliptical orbits showed a better coverage of a desired target than those that
used circular orbits. Mass [30] proposed constellations that combined the characteristics
of circular, elliptical and geosynchronous orbits which resulted in fairly good coverage and
a period of 8 sidereal hours.
Moreover, the fields of station keeping and optimization for satellite constellations has
been studied producing significant contributions [31, 32, 33, 34]. Using a genetic algo-
rithm, several constellation types were studied on the basis of design and optimization of
the number of satellites [35, 36, 37]. While those studies focused on systems containing
relatively few satellites, some studies were performed for constellations containing up to a
hundred satellites [38, 39, 40].
Most of the previous works have focused on satellite constellation design for coverage of the
Earth in the ECI (Earth-Centered-Inertial) frame. A recently developed satellite constel-
lation is the flower constellation proposed at Texas A&M University. Flower constellations
create repeating ground tracks using periodic dynamics in a Planet-Centered-Planet-Fixed
rotating frame. This dissertation proposes a constellation theory which also uses periodic
dynamics to design repeating space tracks using satellite orbits as the rotating reference
frame.
10
2.3 Target Tracking Control
Numerous research has been conducted on spacecraft attitude regulation and tracking
problems. In this section, we focus our literature surveys on the control problems of
tracking moving objects. The control system design of tracking a moving target has been
studied, however, the derivation of angular velocity and acceleration of the moving target
as a reference trajectory is a complex task. Several studies have been performed showing
the design of tracking control systems implementing angular velocity and acceleration for
tracking.
Hablani [41] developed a precision pointing control system for tracking an arbitrary moving
target. The reference trajectory for tracking uses the position, velocity, and acceleration
obtained from a two-degree-of-freedom telescope. The design of the pointing control system
consists of three modes: a linear rate mode, a linear position mode, and a nonlinear position
mode. In this pointing control system, a stabilization subsystem is utilized to minimize
inertial jitter.
Schaub et al [42] presented a nonlinear feedback control law for the precision pointing
of imaging satellites. The control law was developed by using Lyapunov control design
methods and by using Modified Rodrigues Parameters as attitude coordinates. To establish
the angular velocity and acceleration for the desired motion, the angular velocity history as
a function of time is used. Schaub illustrated the appropriateness of the Modified Rodrigues
Parameters for large angle slew maneuvers.
Matthew [43] investigated a nonlinear tracking control law on formation flying concepts.
Using the rigid body models of any number of axisymmetric wheels for formation flying,
the spacecraft tracking control law is addressed with the attitude coordinates of Modified
Rodrigues Parameters. The reference trajectory for tracking is established with a solar
panel aligned perpendicular to the sun vector direction and with ground target tracking
on the Earth.
11
Chen et al [43] developed a quaternion based PID feedback control for ground target track-
ing on the Earth. For a reference trajectory, the desired angular velocity and acceleration
are obtained by using the Hamiltonian function. This paper suggested a pre-maneuver for
target tracking to reduce initial control efforts required and a rotation maneuver for the
commissioned payload that only uses the reaction wheels of non-payload axes.
A recently presented study proposed multi-target attitude tracking of formation flying [44].
A leader satellite has a camera for tracking a ground target and an antenna for tracking
a follower satellite. To compute angular velocity and acceleration, the paper introduces a
method to increase the efficiency of tracking the camera, while the attitude of the antenna is
measured in the body-fixed frame. The robust tracking controller is developed by deriving a
desired inverse system, which converts the attitude tracking problem into a regular problem,
using sliding mode techniques.
2.4 Summary
We have reviewed previous research efforts for the dynamics and control problems of satel-
lite relative motion. Numerous studies have demonstrated important contributions for
satellite relative motion problems. With these contributions, we proceed to develop ana-
lytic tools for satellite relative motion.
Chapter 3
Satellite Relative Orbit Designs
3.1 Introduction
This chapter develops an analytic solution of satellite relative motion using a direct geo-
metrical approach. The analytic solution of satellite relative motion has been studied by
numerous authors. The most common model used is the analytic solution of Hill’s equa-
tion [1]. Using Hill’s equation, Vaddi, Vadali, and Alfriend [45] derived an analytic solution
while accommodating nonlinearity, and they combined Lawden and Melton’s equations [7]
considering the eccentricity effect. Including the first order gravitational J2 effect in the
right hand side of the Hill’s equation [5], Schweighart and Sedwick derived an analytic so-
lution [6]. Most of the relative motion theories mentioned above are the analytic solutions
of linearized differential equations. Using the differential orbital elements of satellites, Gim
and Alfriend derived state transition matrix with a geometrical method [9]. Karlgarrd and
Lutze developed second-order analytic solutions in terms of initial conditions in spherical
coordinates [46]. The equations based on full-sky spherical geometry were introduced by
Wertz [47] for the relative and apparent motions of satellite constellations at same and
different altitudes. However, the equations using spherical geometry solutions do not take
into account the orbit elements of satellites and are applied only to circular orbits.
12
13
This chapter studies unperturbed satellite relative motion using spherical geometry solu-
tions in spherical coordinates. The results provide a complete analytic solution of satellite
relative motion for formation flying and constellation design. For the derivation of the
equations of motion, the approach geometrically interprets the projected Keplerian orbits
on a sphere applying the spherical trigonometry solutions. The resulting equations are
expressed as azimuth and elevation angles representing the relative angular position of the
target satellite. The azimuth and elevation angles are then transformed into the associated
rectangular coordinates for the relative position and velocity vectors. Using the solution,
we also derive the linearized equations of motion and evaluate the modeling accuracy. The
validity of the proposed model results from modeling accuracy and efficiency in comparison
to the exact analytic solutions of satellite relative motion theories.
3.2 Keplerian Orbit in Spherical Coordinate Systems
The purpose of this study is to develop the equations of satellite relative motion through
direct geometrical interpretation of projected Keplerian orbits on a sphere (celestial sphere,
Earth sphere, or unit sphere). We project a Keplerian orbit on a celestial sphere using
spherical coordinates. The Keplerian orbit is commonly specified by the classical orbital
elements for state representations in space. The six orbital element sets are
[a, e, Ω, i, ω, ν] (3.1)
The semi-major axis, a, and eccentricity, e, listed as the first two elements above describe
the orbit size and shape. The following elements, Ω, i, and ω, define the orbit plane
orientation. The final classical orbital element is the true anomaly, ν, which determines
the object’s current angular position relative to the perigee. Figure 3.1 illustrates the orbit
elements of Ω, i, ω, and ν which are angle related orbit elements to describe the Keplerian
orbit from the center of the Earth.
14
Figure 3.1: Keplerian orbit elements
A spherical coordinate system in space can be used to represent the Keplerian orbit pro-
jected on a sphere. In Fig. 3.2, the elevation angle, δ′, defines the angle between the
straight line from the center of the Earth to O′ and the projection of this line on the I J
plane. The angle between this projection and I axis is defined as the azimuth angle, α′.
If we represent the projected Keplerian orbit by α′ and δ′, the formula of α′ and δ′ can be
expressed in terms of the angle related orbital elements as follows:
α′ = f(ν; i, Ω, ω) (3.2a)
δ′ = g(ν; i, Ω, ω) (3.2b)
These angles α′ and δ′ and the radial distance r of the object determine the spherical
coordinate system in space. The radial distance, r, is written in terms of ν as
r =a(1 − e2)
1 + e cos ν(3.3)
If we represent the position of the object in the rectangular coordinate system, the trans-
formation from the spherical coordinate system to the associated rectangular coordinate
15
Figure 3.2: Projection of a Keplerian orbit on celestial sphere
system (I J K) leads to
rIJK
=
r cos δ′ cosα′
r cos δ′ sinα′
r sin δ′
(3.4)
In the next section, the angles α′ and δ′ are expressed in terms of the angle related orbit
elements using direct geometrical interpretations.
3.3 Geometrical Relative Orbit Modeling
In this section, we geometrically derive the relative position and velocity vectors of a target
satellite relative to a base satellite. The subscript B denotes the base satellite, and subscript
T denotes the target satellite.
The Keplerian orbits of the two satellites are projected on a sphere for geometrical inter-
pretation, as seen in Fig 3.3. The poles PB
and PT
denote the orbit poles of the satellites.
The dotted lines on the sphere represent the projected Keplerian orbits of the two satellites
16
Figure 3.3: Geometry for modeling the relative motion on the surface of a sphere
and the solid line represents an equatorial plane. An intersection point, IP, is defined as
the projected crossing point of the two orbit planes on the surface. The relative position
of the target satellite T with respect to the base satellite B is expressed as the azimuth
angle, α, and elevation angle, δ. The angle α is perpendicular to the angle δ through the
point H .
We introduce the argument of latitudes for the transformation between the classical orbital
elements and the angular positions on the sphere. The argument of latitudes, uB, T
, mea-
sures the arc lengths from the ascending nodes to the current satellite angular position.
On the sphere, uB, T
can be expressed as
uj = φj + θj = ωj + νj j = B, T (3.5)
The arc lengths φj and θj represent the distance from the ascending nodes, Ωj , to the
intersection point, IP, and from I
Pto the satellite’s current angular position, respectively.
For the derivation of satellite relative motion, a key parameter is the relative inclination, iR,
which is the angle between two orbit planes at IP. We use the spherical triangle 4Ω
BΩ
TI
P
17
to compute iR. Because i
Ris not equal to the difference between two inclinations of the
orbits (i.e., iR6= i
T− i
B), we must apply the law of cosines for angles to the triangle:
cos iR
= cos iB
cos iT
+ sin iB
sin iT
cos ∆Ω (3.6)
where the relative ascending node, ∆Ω, is defined as
∆Ω = ΩT− Ω
B(3.7)
We first derive α and δ of the target satellite relative to the base satellite in terms of the
angle-related orbit elements: Ω, i, ω, and ν. The general solutions for spherical triangles
are given in Appendix A. From Fig 3.3, the spherical triangle 4ΩBΩ
TI
Pis taken to solve
φB
and φT. Figure 3.4 shows a detailed view of the spherical triangle. We apply the law
of sines to the spherical triangle to compute sinφB:
sinφB
=sin ∆Ω sin i
T
sin iR
(3.8)
Applying the law of cosines for angles to the spherical triangle 4ΩBΩ
TI
P, we find another
geometrical relationship to compute cosφB:
cosφB
=cos(180 − i
T) + cos i
Bcos i
R
sin iB
sin iR
(3.9)
Figure 3.4: Spherical triangle for computing φB
and φT
18
Dividing Eq. (3.8) by Eq. (3.9) gives
φB
= tan−1[ sin ∆Ω sin i
Bsin i
T
− cos iT
+ cos iB
cos iR
]
(3.10)
To compute sinφT, the law of sines is also applied to the spherical triangle seen in Fig 3.4:
sinφT
=sin ∆Ω sin i
B
sin iR
(3.11)
Using the law of cosine for angles, we also obtain that
cosφT
=cos i
B+ cos(180 − i
T) cos i
R
sin(180 − iT) sin i
R
(3.12)
Dividing Eq. (3.11) by Eq. (3.12) results in
φT
= tan−1[ sin ∆Ω sin i
Bsin i
T
cos iB− cos i
Tcos i
R
]
(3.13)
The quadrant ambiguity problem is avoided by using the atan2 built-in function in com-
puter programming languages.
Now we consider the spherical triangles on the surface of the sphere with ∆Ω = 0. In this
case, we construct a celestial sphere having the pole PB
of the base satellite as a geographical
pole, shown in Fig 3.5. The celestial sphere has two spherical triangles, 4PBP
TT and
4THIP. Note that the angle P
BP
TI
Pis always 90 regardless of the inclination of either
satellite. The angle TPBI
Pis equivalent to the angle θ
Tby applying the law of sines.
Hence, the angle PBP
TT is obtained by subtracting 90 by θ
T. The arcs P
TT and P
BH are
always 90. Thus the arc PBT can be found by subtracting δ from 90.
The elevation angle δ is derived from the spherical triangle 4PBP
TT . Applying the law of
cosines for sides to the spherical triangle, we find that
cos(90 − δ) = cos iR
cos 90 + sin iR
sin 90 cos(90 − θT)
sin δ = sin iR
sin θT
(3.14)
Thus, the angle δ is obtained by
δ = sin−1 [sin iR
sin θT] (3.15)
19
Figure 3.5: Geometry for computing α and δ with ∆Ω = 0
The azimuth angle α is found by applying the law of cosines for sides twice to the spherical
triangle 4THIP , resulting in the following equations:
cos θT
= cos δ cos (θB
+ α) + sin δ sin (θB
+ α) cos 90 (3.16)
and
cos δ = cos θT
cos (θB
+ α) + sin θT
sin (θB
+ α) cos iR
(3.17)
Substituting cos δ from Eq. (3.16) into Eq. (3.17), we have
tan (θB
+ α) =sin θ
Tcos i
R
cos θT
(3.18)
Thus, the angle α is derived by
α = −θB
+ tan−1
[
sin θT
cos iR
cos θT
]
(3.19)
Using the definition of the argument of latitudes in Eq. (3.5), the angles α and δ are
expressed as
α = (φB− ω
B− ν
B) + tan−1 [cos i
Rtan(ω
T+ ν
T− φ
T)] , 0 ≤ α < 360(3.20a)
δ = sin−1 [sin iR
sin(ωT
+ νT− φ
T)] , −90 ≤ δ ≤ 90 (3.20b)
20
where φB, φ
Tare given by
φB
= tan−1[ sin ∆Ω sin i
Bsin i
T
− cos iT
+ cos iB
cos iR
]
(3.21a)
φT
= tan−1[ sin ∆Ω sin i
Bsin i
T
cos iB− cos i
Tcos i
R
]
(3.21b)
For the simple analysis of satellite relative motion, the angles α and δ can be directly used
to determine the angular position of the target satellite with respect to the base satellite.
In Eq. (3.20), ν is the only time dependent variable, and the derivatives of α and δ are
obtained by
α = cos iR(1 + tan2 δ)ν
T− ν
B(3.22a)
δ = sin iR
cos(α + νB
+ ωB− φ
B)ν
T(3.22b)
Taking the derivative of δ in Eq. (3.20) directly results in a singularity at a particular
angle. Thus a trigonometric identity is applied during the derivation of δ for Eq. (3.22) to
avoid the singularity.
The relative motion of the target satellite can be described using the previously calculated
α and δ in the rectangular coordinates. The orbit radius of the base satellite is rB, and
the target satellite orbit radius is rT. We introduce the base satellite rotating frame, F
R,
to describe the relative motion of the target satellite with respect to the base satellite.
The center of the Earth is set as the origin, and the orientation of FR
is given by the
unit vectors e1, e2, e3. The direction of the unit vector e1 is set to the orbit radius of
the base satellite, while e3 is perpendicular to the orbit plane of the base satellite. The
unit vector e2 then is computed by the right-hand rule. Mathematically, the base satellite
rotating frame FR
is described by the unit vectors:
e1 =r
B
|rB| (3.23a)
e3 =r×
Br
B
|r×
Br
B| (3.23b)
e2 = e×
3 e1 (3.23c)
21
The × superscript denotes a skew-symmetric 3× 3 matrix associated with a 3× 1 column
matrix. If x is a 3 × 1 matrix, x = [x1 x2 x3]T , then
x× =
0 −x3 x2
x3 0 −x1
−x2 x1 0
(3.24)
The position vectors of the base and target satellites can be written as the vector compo-
nents in FR:
rB
= (rB
0 0)T (3.25a)
rT
= (rT
cos δ cosα rT
cos δ sinα rT
sin δ)T (3.25b)
The relative position vector r of the target satellite in base satellite centered frame, FC
(with the base satellite as the origin), are derived by vector subtraction of the position
vectors from Eq. (3.25):
r =
x
y
z
=
rT
cos δ cosα− rB
rT
cos δ sinα
rT
sin δ
(3.26)
The relative velocity vector v is obtained by taking the time derivatives of Eq. (3.26):
v =
x
y
z
=
rT
cos δ cosα− rTδ sin δ cosα− r
Tα cos δ sinα− r
B
rT
cos δ sinα− rTδ sin δ sinα + r
Tα cos δ cosα
rT
sin δ + rTδ cos δ
(3.27)
where the derivatives of r and ν of satellites are [48]
rj =
√
µ
aj(1 − e2j )ej sin νj, νj =
√
µaj(1 − e2j)
r2j
, j = B, T (3.28)
The relative equations of motion in Eqs. (3.26) and (3.27) are an exact analytic solutions
for satellite relative motions. The only assumption is that no perturbations are acting on
the satellites.
22
3.4 Linearized Equations of Motion
This section derives the linearized equations for the solutions in the preceding section.
Furthermore, we find a valuable geometrical insight of cross-track motion for formation
flying design from the resulting linearized equations.
The linearization of the relative position vector is easily achieved by assuming small quan-
tities of α and δ in Eq. (3.26):
x = rT− r
B= ∆r (3.29a)
y = (rB
+ ∆r)α = rTα (3.29b)
z = (rB
+ ∆r) δ = rTδ (3.29c)
The expressions of Eq. (3.29) describe that the radial-track motion is the difference of the
orbit radius of satellites, and in-track/cross-track motions are determined by the small
angles α and δ with orbit radius rT. Because α and δ are assumed to be small, a small
relative inclination iR
between two orbit planes results. Applying small approximation of
inclination and ascending node differences from Eq. (3.6), the linearized form of iR
can be
expressed as
iR
=√
∆i2 + sin2 iB∆Ω2 (3.30)
The proposed model with small quantities states that the linearized radial-track motion x
is equal to the orbit radius difference ∆r. The expression of ∆r is derived by taking the
first variation of the orbit radius [48]:
x = ∆r =r
B
aB
∆a− aB
cos νB∆e+
aBe
Bsin ν
B√
1 − e2B
∆M (3.31)
The small angle α of in-track motion y in Eq. (3.29) is obtained by small iR
in Eq. (3.19):
α = −θB
+ tan−1
[
sin θT
cos iR
cos θT
]
= ∆θ (3.32)
23
Using Eq. (3.5), the expression of ∆θ is written as
∆θ = ∆ν + ∆ω − ∆φ (3.33)
The first variation of ν is given by [48]
∆ν =
(
aB
rB
+1
1 − e2B
)
sin νB∆e+
a2B
√
1 − e2B
r2B
∆M (3.34)
Using Fig 3.4, we geometrically derive the expression of ∆φ with small quantities of both
∆φ and ∆Ω, and obtain
∆φ = − tan−1(cos iB
tan∆Ω)
= − cos iB∆Ω (3.35)
Finally, the in-track motion y is expressed as
y =
(
aB
+r
B
1 − e2B
)
sin νB∆e+ r
B[cos i
B∆Ω + ∆ω] +
a2B
√
1 − e2B
rB
∆M (3.36)
With small quantity δ of z in Eq. (3.29) and small relative inclination iR, the cross track
motion z in Eq. (3.29) is written as follows:
z = rBiR
sin(νT
+ ωT− φ
T) (3.37)
Equation (3.37) can be also expressed as
z = rBiR
sin(νB
+ ωB
+y
rT
− φB) (3.38)
Assuming y/rT≈ 0, and making use of φ
B= cos−1 (∆i/i
R) by small approximation in
Eq. (3.37), the linearized cross-track motion z is written as
z = rB
√
∆i2 + sin2 iB∆Ω2 sin
[
νB
+ ωB− cos−1
( ∆i√
∆i2 + sin2 iB∆Ω2
)
]
(3.39)
Taking the time derivatives of Eqs. (3.31), (3.36), and (3.39), the linearized relative velocity
is obtained by
24
x =
(
nBe
Bsin ν
B√
1 − e2B
)
∆a+ nB
√
1 − e2B
(
a3B
r2B
)
sin νB∆e+ n
Be
B
(
a3B
r2B
)
cos νB∆M (3.40a)
y =
[
nB
√
1 − e2B
(
2 + eB
cos νB
1 + eB
cos νB
)(
a3B
r2B
)
cos νB
+a
Bn
Be
Bsin2 ν
B
(1 − eB)
3
2
]
∆e
+a
Bn
Be
Bcos i
Bsin ν
B√
1 − e2B
∆Ω +a
Bn
Be
Bsin ν
B√
1 − e2B
∆ω − nBe
B
(
a3B
r2B
)
sin νB∆M (3.40b)
z =
√
µe2B
sin2 νB
aB(1 − e2
B)
√
∆i2 + sin2 iB∆Ω2 sin
[
νB
+ ωB− cos−1
( ∆i√
∆i2 + sin2 iB∆Ω2
)
]
+
√
µaB(1 − e2
B)
rB
√
∆i2 + sin2 iB∆Ω2 cos
[
νB
+ ωB− cos−1
( ∆i√
∆i2 + sin2 iB∆Ω2
)
]
(3.40c)
The equations of linearized relative position and velocity are evaluated with the exact
solutions in the next section.
Design of in-track and cross-track motions
In this section, we take a closer look at the cross-track motion of the linearized relative
position. The formula in Eq. (3.39) is different from the expression proposed in Ref. [49].
Interestingly, the formula offers geometrical insight for purely cross-track motion with a
direct sinusoidal oscillation representation.
The cross-track motion can be simply expressed in the following form:
z = rBiR
sin
[
νB
+ ωB− cos−1
(∆i
iR
)
]
(3.41)
We geometrically interpret Eq. (3.41), which may offer valuable insight of the cross-track
motion. In Fig 3.6, we establish the angle ψ, which is an angle from the perigee of the base
satellite to IP, and then we define
φL
= ωB
+ ψ (3.42)
25
Figure 3.6: Geometry for the relative phase angle ψ.
Applying the law of cosines to the spherical triangle 4ΩBΩ
TI
P, we have
− cos(iB
+ ∆i) = − cos iB
cos iR
+ sin iB
sin iR
cosφL
(3.43)
Assuming that ∆i and iR
are small angles, Eq. (3.43) leads to
φL
= cos−1
(
∆i
iR
)
(3.44)
Combining Eqs. (3.42) and (3.44) leads to the expression of the angle
ψ = cos−1
(
∆i
iR
)
− ωB
(3.45)
Thus, the cross-track motion in Eq. (3.41) can be rewritten in the following sinusoidal
representation:
z = rBiR
sin(νB− ψ) (3.46)
Consequently, it turns out that the angle ψ, called the relative phase angle, in the pure
cross-track motion is geometrically the offset angle of intersection point IP
from the perigee
26
of the base satellite. For example, if the relative phase angle ψ is zero, the intersection
point IP
is on the perigee of the base satellite orbit.
Using this geometrical insight of the angle ψ, we examine the in-track/cross-track forma-
tion design in terms of the relative phase angle ψ. Purely in-track/cross-track motion is
accomplished by setting ∆a = ∆e = ∆M = 0 in Eq. (3.31). We assume that the eccen-
tricity of the base satellite is small quantity which ignores higher order terms of e, then
the linearized formula of the in-track/cross-track motion are written by
y = Ay cos νB
+ yoff (3.47a)
z = Az sin(νB− ψ) (3.47b)
where
Ay = −aBe
B(∆ω + cos i
B∆Ω) (3.48a)
yoff = aB(∆ω + cos i
B∆Ω) (3.48b)
Az = aB
√
∆i2 + sin2 iB∆Ω2 (3.48c)
In Eq. (3.48), the in-track/cross-track motion are specified by three parameters, ∆i, ∆Ω,
and ∆ω, and the parameters characterize dependently the size and shape of the relative
orbit. For the simple in-track/cross-track formation design, we formulate the parameters
as functions of the desired relative orbit size (Ay, Az) and the relative phase angle ψ:
∆i =Az
aB
cos(ωB
+ ψ) (3.49a)
∆Ω =Az
aB
sin iB
sin(ωB
+ ψ) (3.49b)
∆ω = −(
Ay
aBe
B
+Az cot i
B
aB
sin(ωB
+ ψ)
)
(3.49c)
If the orbit of the base satellite is a circular (eB
= 0), the relative motion of the target
satellite moves along the perpendicular line relative to the base orbit plane (Ay = 0).
The following numerical simulations examine the in-track/cross-track motion and the rel-
ative separations based on the relative phase angle ψ which shifts the intersection point
27
Table 3.1: Parameters of the orbit elements
Orbit elements Value Units
a 7000 km
e 0.001 -
i 30.0 deg
Ω 120.0 deg
ω 0.0 deg
M0 0.0 deg
IP
with respect to the perigee of the base satellite. We choose the relative orbit size for
in-track/cross-track motion of Ay = 3.0km and Az = 0.01km. Table 3.1 shows the orbital
elements of the base satellite. Figure 3.7 shows the in-track/cross-track motions of the
exact and linearized relative orbit. We can see that the linearized relative orbits are close
enough to exact relative orbits. If IP
becomes more distant from the perigee, then the rel-
ative motion ellipse shrinks. When IP
is established at forward 90.0 deg from the perigee
(ψ = 90.0 deg), the relative motion describes a straight line centered in IP.
Figure 3.8 shows the relative separation based on the relative phase angle ψ, which was
computed using the orbital element differences of Eq. (3.49). If IP
is established on the
perigee (ψ = 0.0), the minimum separation occurs at perigee. Going away from the perigee,
the true anomaly ν for the minimum separation is linearly changed with the relative phase
angle ψ. The points for the maximum relative separation are also shown to be a linear
combination with the minimum separation.
28
Figure 3.7: In-track/cross track motion by the relative phase angles ψ (Ay = 0.01km, Az =
3.0km).
Figure 3.8: Relative separations by the relative phase angle ψ.
29
3.5 Modeling Accuracy
We first evaluate the modeling accuracy for the validity of the proposed solution called
GROM. For the evaluation of the modeling accuracy, three relative motion theories are in-
troduced: the solutions of numerical integration [48], Classical Two Body Problem(CTBP)
[50], and Unit Sphere Approach(USA) [10]. See the Appendix B for Unit Sphere Approach.
The analytic solutions provide kinematically exact descriptions of satellite relative motion
in the absence of perturbations. In this section the modeling accuracy of GROM is evalu-
ated by means of the absolute and relative errors in comparison to the three relative motion
solutions.
3.5.1 Absolute Error
In this section, we evaluate the absolute error of GROM in comparison to numerical in-
tegration and CTBP. Numerical integration describes the kinematically exact trajectory
of satellites. However, the result must be numerically considered against the truncation
error that arises from taking a finite number of steps in computation. Here we investigate
the absolute errors of numerical integration and GROM with respect to the reference orbit
model, CTBP, and the truncation error of numerical integration will illustrate the relative
accuracy of the GROM absolute error relative to CTBP.
Table 3.2: Parameters of the orbit elements
Satellites a(km) e i(deg) Ω(deg) ω(deg) M0(deg) days
Base 7000 0.01 30 50 45 10 5
Target 8000 0.001 70 120 20 60 5
Table 3.2 shows the parameter values of the satellite orbit elements. The orbit elements
of the base and target satellites are selected by large scale relative motion. For numerical
30
integration, ODE 45 integration routine in MATLAB uses an AbsTol 1.0×10−16 and RelTol
2.22 × 10−14.
Figures 3.9 and 3.10 show the absolute errors of the relative position and velocity vectors of
GROM and numerical integration with respect to CTBP over 5 days. At the initial stage
of several orbits, the errors of the numerical integration and CTBP are approximately
10−12 km and 10−15 km/sec, while GROM and CTBP also show errors of the same value.
However, the error of the numerical integration relative to CTBP gradually grows by reason
of the truncation error of the numerical algorithm. In contrast, the absolute error of GROM
with respect to CTBP show steady oscillations over long timescale. As a result, the GROM
solution has the same accuracy as CTBP because the absolute error maintains the error
values of the initial several orbits.
0 1 2 3 4 5
10−14
10−12
10−10
10−8
10−6
Time(days)
Pos
ition
err
or(k
m)
Abs Error (CTBP−Numerical Integration)
Abs Error (CTBP−GROM)
Figure 3.9: Absolute relative position errors
31
0 1 2 3 4 5
10−16
10−14
10−12
10−10
10−8
Time(days)
Vel
ocity
err
or(k
m/s
ec)
Abs Error (CTBP−Numerical Integration)
Abs Error (CTBP−GROM)
Figure 3.10: Absolute relative velocity errors
3.5.2 Relative Error
This section evaluates the relative errors of GROM and its linearized solution using the
modeling error index which is an effective tool for evaluating the accuracy of relative motion
theories [8]. In Eq. (3.50), xj and xj represent the relative position and velocity vectors of
the reference and proposed model, respectively:
yj = Wxj, yj = Wxj (3.50)
where j represents each sample point of a relative orbit. The weighting matrix W uses the
Earth-value units as shown in Eq. (3.51):
W = diag
(
1
Re
,1
Re
,1
Re
,1
Ren,
1
Ren,
1
Ren
)
(3.51)
where Re is the radius of the Earth and n is the mean motion of satellites. The modeling
error index is written as follows:
λj =yT
j yj
yTj yj
− 1
λ = maxj=1...m
|λj| (3.52)
32
The modeling error index evaluates the relative errors of GROM and the linearized equa-
tion in comparison to USA, relative to the reference orbit model of CTBP. The analytic
solutions of USA and CTBP describe the kinematically exact relative motion of satellites,
which brings about negligible quantities of relative errors upon simulation. We use k-digit
rounding arithmetic when finding the solutions, yj and yj, in order to ignore computa-
tional uncertainties such as roundoff error. The k-digit rounding arithmetic is obtained by
terminating the value of the solution at k decimal digits.
Table 3.3: Parameter of the orbit elements
Orbit elements Value Units
a 7000 km
e 0.001 -
i 30.0 deg
Ω 120.0 deg
ω 0.0 deg
M0 0.0 deg
For numerical simulations, the orbit elements of the base satellite in Table 3.3 are chosen,
and the orbit element differences, ∆oe, of the target satellite are used as the following
values:
∆oe = [∆a ∆e 0.1 0.2 0.01 0.0] (3.53a)
∆a = [0.0 0.001 0.005 0.01 0.1 0.5 5] (3.53b)
∆e = [0.0 0.00001 0.00005 0.0001 0.0005 0.001 0.05 0.1] (3.53c)
Figures 3.11 and 3.12 show the modeling error index using the 6-digit rounding arithmetic
solution with various relative distances and eccentricities. As shown in the figures, the index
of GROM is exactly the same as that of USA, representing index values of 10−6 with respect
to the reference orbit model. The modeling error index of an order 10−3 is sufficiently small
with reasonable confidence regarding the modeling[51]. Therefore, the GROM solution
33
provides accurate representation for all relative orbit sizes and eccentricities. In the case
of the linearized equation, the solution shows modeling indexes of nearly 10−3 at small
orbit element differences, which means sufficient accuracy for small relative orbit sizes and
eccentricities. As expected, however, the index values gradually grow with increasing orbit
size and eccentricity.
10−3
10−2
10−1
100
101
10−6
10−4
10−2
100
ρ(km)
inde
x
GROMUSALinearized equation
Figure 3.11: Index comparison for various relative distances
10−5
10−4
10−3
10−2
10−1
10−6
10−4
10−2
100
Eccentricity
inde
x
GROMUSALinearized equation
Figure 3.12: Index comparison for various eccentricities
34
3.6 Modeling Efficiency
In this section, the GROM solution demonstrates low computational cost through the
comparison of CPU time to describe satellite relative motion. The relationship between
the CPU time and iteration is approximately, but not exactly, linear. Using linearity, we
build a linear model in terms of CPU time T and iteration N as follows:
Tk = mNk + b, k = 1, . . . , n (3.54)
Determining the best linear approximation is to find the values of m and b to minimize the
total error of the linear model. To find the values, least squares method is a convenient
procedure to compute the solutions of the two variables, m and b. For the evaluation of
the GROM efficiency, we are primarily concerned with the variable m which is used to
evaluate the relative efficiency of the solutions.
In the absence of perturbations, we have three exact analytic solutions for satellite relative
motion: GROM, USA, and CTBP. The solutions have been coded efficiently in computer
program. The numerical simulation computes the CPU times of the solutions with respect
to five iterations with the following time spans(sec):
N = [10 10000 30000 50000 100000] (3.55)
Table 3.4 shows the coefficient m∗ normalized with respect to the values of GROM and
the complexity of the formula. As seen in Table 3.4, GROM is nearly 7% and 25% more
efficient than USA and CTBP, respectively. Furthermore, GROM is comparatively simpler
than the other two solutions.
A numerical example studies the effect of satellite relative motion under the influence of
the J2 perturbations through which we demonstrate the modeling efficiency of GROM.
The use of time explicit orbital elements in the analytic solutions provides a simple way
to investigate the difference of unperturbed and J2 perturbed models for satellite relative
motion. The first-order J2 perturbation effects secular changes in the ascending node,
35
Table 3.4: Comparison of analytic solution efficiency
Method Normalized coefficient(m∗) Formula complexity
GROM 1.0000 Solution is comparatively simple
USA 1.0711Requires an efficient ways for simple
form expressions
CTBP 1.2524Needs to keep track of variables and
functions
Ω, argument of perigee, ω, and mean anomaly, M . The time-explicit representations are
written as [10]
a = a0 (3.56a)
e = e0 (3.56b)
i = i0 (3.56c)
Ω = Ω0 −3nR2
eJ2 cos i
2p2t (3.56d)
ω = ω0 +3nR2
eJ2
4p2(4 − 5 sin2 i)t (3.56e)
M = M0 + nt +3nR2
eJ2
√1 − e2
4p2(3 sin2 i− 2)t (3.56f)
We use the following values in the time-explicit orbit elements: p = a(1 − e2), J2 =
0.00108263.
The numerical simulation, coded on iterating the solutions at each time step, runs over
a period of 20 days with the orbit elements of the base satellite in Table 3.3. The orbit
element differences are chosen as the following values:
∆oe = [0.0 0.0001 0.01 0.02 0.01 0.02] (3.57)
As shown in Table 3.5, the maximum differences of the relative position and velocity of the
solutions are 3.8729 km and 0.0041 km/sec over 20 days, respectively. However, each ana-
36
lytic solution results in different CPU times so that the GROM solution is approximately
1′40′′ and 5′57′′ faster than the USA and CTBP solutions, respectively.
Table 3.5: Differences of unperturbed and J2 perturbed models (Time step : 0.1 sec)
Methods CPU time (minutes)Maximumposition
difference (km)
Maximumvelocity
difference (km/sec)
GROM 23.60 3.8729 0.0041
USA 25.27 3.8729 0.0041
CTBP 29.55 3.8729 0.0041
3.7 Conclusions
In this chapter, we developed the analytic solution for satellite relative motion through
a direct geometrical approach using the spherical geometry without perturbations. The
derivation of this geometrical approach is straightforward, and the resulting equations
provide a complete analytic form of the relative motion avoiding the quadrant ambiguity
problem. For the validity of the proposed GROM solution, we have evaluated the solution
by means of the modeling accuracy and efficiency in comparison to other exact analytic
solutions. The modeling accuracy of the GROM solution is equivalent to the exact rela-
tive motion theories of CTBP and USA. Furthermore, the linearized equations of motion
illustrate sufficient accuracy for small relative orbit sizes and eccentricities by using the
modeling error index. From the evaluation of the modeling efficiency, GROM is approxi-
mately 7% and 25% more efficient than USA and CTBP, respectively. Consequently, the
proposed GROM modeling illustrates the exact and efficient analytic solutions for satellite
relative motion.
Chapter 4
Parametric Relative Orbit Designs
4.1 Introduction
In the previous chapter, we derived an exact and efficient tool, GROM, for satellite relative
motion, so that all of the transitional relative motion can be exactly described without
perturbations. However, understanding the relative motion geometry and designing relative
orbits are complex task, due to the nonlinearity of the relative motion [14]. The complexity
of satellite relative motion depends on the mean motions of the orbits. For Keplerian
motion, we characterize the orbit periodicity by the orbit mean motion, n. Thus, we can
define a relative orbit frequency, γ, between two orbit mean motions:
nT
= γ nB
(4.1)
where nB
and nT
is the mean motions of base and target satellites, respectively.
The relative orbits can be broken down into two groups: commensurable (periodic) and
non-commensurable (quasi-periodic) orbits. If there exists a rational number γ, then the
relative orbit will return to its original position at a finite time, which is considered a
commensurable orbit. Conversely, a non-commensurable orbit with irrational γ will never
return to its original position. Figures 4.1 and 4.2 show commensurable relative orbits
37
38
Figure 4.1: Commensurable relative orbits of γ = 2.0, 2.2, 2.2142 (3-dimensional view).
Figure 4.2: Commensurable relative orbits of γ = 2.0, 2.2, 2.2142 (polar view).
of a target satellite with respect to a base satellite in terms of the scalar variable γ over
3 days. If we select an integer relative orbit frequency γ, the relative orbit will be a
simple periodic closed orbit. On the contrary, as seen in the figures, the complexity of the
relative orbits increases with the number of decimal points of the scalar variable γ. This
observation implies that the relative motion can be characterized in terms of the relative
orbit frequency γ.
In Fig 4.2, we also have an observation of satellite relative orbits. The relative orbits in
polar view represent the same shapes as the parametric curves of hypocycloids. Based on
this geometrical insight, we can find analytic solutions to understand the geometrical struc-
ture of relative motion dynamics. To study the geometrical structure of these dynamics,
this section uses the parametric curves of cycloids and trochoids produced by the motion
of epicycle and deferent circles. Then, we find the rules for designing parametric relative
orbits in terms of the relative orbit frequency γ. The parametric relative orbits are then
39
classified by the parametric curves of cycloid and trochoid motions.
4.2 General Parametric Equations and Curves
Cycloid and trochoid curves are mathematical trajectories described by Spirograph drawing
equipment [52]. These parametric curves are used for practical engineering problems such
as the design of the rotary engine [53]. Mathematically, cycloids and trochoids are divided
into hypocycloids and epicycloids or hypotrochoids and epitrochoids based on whether the
curves have cusps or petals, respectively.
A hypotrochoid is defined by the set of points or trajectory traced out by a fixed point P ,
at a constant distance d′ from the center of a deferent circle, on an epicycle of radius d
that rolls around the inside of a deferent circle of radius D. Figure 4.3 shows a diagram of
these components used to produce hypotrochoid motion. The parametric equations for a
hypotrochoid in an x− y plane are [52]
x = (D − d) cos θ + d′ cos
(
D − d
dθ
)
(4.2a)
y = (D − d) sin θ − d′ sin
(
D − d
dθ
)
(4.2b)
where d′ determines the type of hypotrochoid curve; when d′ < d the curve is called a
curtate hypotrochoid, when d′ > d the curve is called a prolate hypotrochoid, and when
d′ = d a special type of hypotrochoid curve occurs, called a hypocycloid. In the case
of d′ = d, Eq. (4.2) can be transformed into the generalized parametric equation for
hypocycloids. Let radius D = kd, and the parametric equation can be written as
x = d(k − 1) cos θ + d cos(
(k − 1)θ)
(4.3a)
y = d(k − 1) sin θ − d sin(
(k − 1)θ)
(4.3b)
where k ≥ 3 is an integer that represents the number of cusps. Cusps occur at the endpoints
of the extremities of cycloids. A hypocycloid with k = 3 is known as a deltoid and has
40
Figure 4.3: Hypotrochoid motions Figure 4.4: Epitrochoid motions
three cusps while a hypocycloid with k = 4 is known as an astroid and has four cusps.
Figure 4.5 illustrates the shapes of deltoid and astroid.
Figure 4.4 shows a diagram similar to Fig 4.3 with the exception that the epicycle circle
rolls around the outside of the deferent circle and thus describes epitrochoid motion. The
parametric equations for an epitrochoid in an x− y plane are
x = (D + d) cos θ − d′ cos
(
D + d
dθ
)
(4.4a)
y = (D + d) sin θ − d′ sin
(
D + d
dθ
)
(4.4b)
where d′ < d the curve is called a curtate epitrochoid, and where d′ > d the curve is called
a prolate epitrochoid. Like the case of the hypotrochoid, a special curve of the epitrochoid
is an epicycloid that occurs when d′ = d. Equation (4.4) can be transformed into the
generalized parametric equation for an epicycloid by substituting radius D = kd, resulting
in the parametric equations given by
x = d(k + 1) cos θ − d cos(
(k + 1)θ)
(4.5a)
y = d(k + 1) sin θ − d sin(
(k + 1)θ)
(4.5b)
where k ≥ 1 is an integer that represents the number of cusps. A cardioid (k = 1) and a
nephroid (k = 2) are examples of epicycloid curves as shown in Fig 4.6.
41
k = 3 k = 4
Figure 4.5: Deltoid and astroid
k = 1 k = 2
Figure 4.6: Cardioid and nephroid
4.3 Parametric Relative Equations
In this section, we transform the relative position formula of GROM into the general equa-
tions of parametric curves. The resulting orbit, in this dissertation, is named as parametric
relative orbit which is a closed and periodic trajectory describing the motion of one object
relative to another. In the GROM solution, the relative position vector, r, of the target
satellite, as seen by the base satellite, is written in the following form:
r =
x
y
z
=
rT
cos δ cosα− rB
rT
cos δ sinα
rT
sin δ
(4.6)
where rB
and rT
represent the orbit radiuses of the base and target satellites, respectively,
and the azimuth and elevation angles, α and δ, are expressed as
α = (φB− ω
B− ν
B) + tan−1 [cos i
Rtan(ω
T+ ν
T− φ
T)] , 0 ≤ α < 360 (4.7a)
δ = sin−1 [sin iR
sin(ωT
+ νT− φ
T)] , −90 ≤ δ ≤ 90 (4.7b)
42
Substituting α and δ into Eq. (4.6), and using common trigonometric relations, the x, y
components can be rewritten by
x = rT
√
cos2 θT
+ cos2 iR
sin2 θT
[
cos(
tan−1[cos iR
tan θT])
cos θB
+ sin(
tan−1[cos iR
tan θT])
sin θB
]
−rB
(4.8a)
y = rT
√
cos2 θT
+ cos2 iR
sin2 θT
[
sin(
tan−1[cos iR
tan θT])
cos θB
− cos(
tan−1[cos iR
tan θT])
sin θB
]
(4.8b)
where the arc lengths θB
and θT
represent the distances from the intersection point be-
tween two orbit planes to the current angular positions of the base and target satellites,
respectively.
We use the following trigonometric relations to transform the x, y components in Eq. (4.8)
into the forms for the parametric equations of cycloid curves in Eqs. (4.3) and (4.5):
cos(tan−1 x) =1√
1 + x2, sin(tan−1 x) =
x√1 + x2
(4.9)
The resulting transformed equation in the x− y plane, assuming two circular orbits, is
x = rd cos(
∆n−t+ ψxy−)
+re cos(
∆n+t+ ψxy+)
−aB
(4.10a)
y = rd sin(
∆n−t+ ψxy−)
−re sin(
∆n+t+ ψxy+)
(4.10b)
where the amplitudes of rd and re are
rd =a
T
2(1 + cos i
R) (4.11a)
re =a
T
2(1 − cos i
R) (4.11b)
with the terms ∆n−, ∆n+, ψxy− and ψxy+ defined as
∆n− = nT− n
B(4.12a)
∆n+ = nB
+ nT
(4.12b)
ψxy− = (MT0
−MB0
) − (φT− φ
B) (4.12c)
ψxy+ = (MB0
+MT0
) − (φB
+ φT) (4.12d)
43
Figure 4.7: Geometrical descriptions of parametric relative equation.
The formula in Eq. (4.10), called the parametric relative equation, represents the general
parametric form of the cycloids with an origin C(−aB, 0). Thus, we can depict the para-
metric relative equation using an epicycle and deferent system, as shown in Fig 4.7. An
object O traveling in an epicycle of radius re is simultaneously revolving about a deferent
circle of radius rd. Also, the center of the epicycle is revolving counterclockwise at the
rate of ∆n−, while the object is revolving clockwise about the center of the epicycle at the
rate of ∆n+. The epicycle and deferent motions describe the relative motion of the target
satellite with respect to the base satellite.
From the relative position vector in Eq. (4.6), the equation of the z-axis component is
expressed in the following sinusoidal oscillation:
z = Az sin(nTt+ ψz) (4.13)
where the amplitude Az and phase angle ψz are
Az = aT
sin iR
(4.14a)
ψz = MT0
− φT
(4.14b)
44
With the incorporation of the z-axis component, we can rewrite the parametric relative
equation of Eq. (4.10) in terms of γ:
x =a
T
2
[
(1 + cos iR) cos
(
θ + ψxy−)
+(1 − cos iR) cos
(γ + 1
γ − 1θ + ψxy+
)
]
− aB
y =a
T
2
[
(1 + cos iR) sin
(
θ + ψxy−)
−(1 − cos iR) sin
(γ + 1
γ − 1θ + ψxy+
)
]
z = aT
sin iR
sin( γ
γ − 1θ + ψz
)
(4.15)
where θ = (γ− 1)nBt. Note that the fixed point O on the epicycle circle in Fig 4.7 has the
opposite initial position and rotation direction when dealing with the general epicycloid
curves in Eq. (4.5). The different initial position and direction of the point O leads to the
opposite sign of the second cosine term in the x component in Eq. (4.15). However, the
dynamics of both cases are equivalent for epicycloid motions.
Finally, we have the same mathematical form of x and y components as the general equa-
tions of parametric curves in Eqs. (4.3) and (4.5). In Eq. (4.15), the relative orbit frequency
γ assumes a rational number, thus the parametric relative orbits represent closed and pe-
riodic trajectories.
4.4 Characteristics of Parametric Relative Orbits
In the preceding section, the parametric relative equation illustrates the combinations of the
parametric equation of x and y components and the sinusoidal oscillation of z component.
Using the parametric relative equation, we investigate general rules and classifications to
design the parametric relative orbits.
4.4.1 Design Rules
This section finds the rules to design parametric relative orbits using the relationship
between the general equations of the parametric curves in Eqs. (4.3) and (4.5) and the
45
derived parametric relative equations in Eq. (4.15). Since the x, y components of both
equations follow the same mathematical form, the amplitude and angle terms of each
equation can be compared, resulting in the following relation for hypocycloid motion:
γ =k
k − 2, k ≥ 3 (4.16)
and for epicycloid motion:
γ =k
k + 2, k ≥ 1 (4.17)
The parameter k determines the number of cusps for hypocycloids and epicycloids, and
we can design parametric relative orbits in terms of relative orbit frequency γ. For the
hypotrochoid and epitrochoid motions, the curves consist of petals that describe flower-like
shapes. The petals or cusps of parametric relative orbits that characterize the shape of
satellite relative motion can be computed in terms of the relative orbit frequency, γ. From
Eqs. (4.16) and (4.17), the number of petals or cusps, defined as γpetal, of the parametric
relative orbit can be computed by the following two solutions:
γpetal =
γnum if γnum
γden= odd
odd,
2 × γnum if γnum
γden= odd
evenor even
odd
(4.18)
where γnum and γden are the numerator and denominator, respectively, of an irreducible
fraction of γ. The following examples show how to compute γpetal from γ:
γ = 2.0 =2
1→ γpetal = 4, γ = 3.4 =
34
10=
17
5→ γpetal = 17 (4.19)
Using the rules in Eq. (4.18), additional numerical examples of computing γpetal are shown
in Table 4.1.
In Eq. (4.18), the number of petals is the same as the value of the numerator when the irre-
ducible fraction of γ has an odd numerator and denominator. When either the numerator
or denominator is an even value, the number of petals will be equal to twice the numerator
value. Physically, the number of petals is equivalent to the crossing number of the target
46
Table 4.1: Numerical examples of computing γpetal
irreducible fraction irreducible fraction
γ (num/den) γpetal γ (num/den) γpetal
1.1 11/10 22 3.3 33/10 66
1.2 6/5 12 3.5 7/2 14
2.2 11/5 11 4.0 4/1 8
2.4 12/5 24 5.0 5/1 5
2.5 5/2 10 7.0 7/1 7
3.0 3/1 3 10.0 10/1 20
satellite in the base satellite orbit plane. When both the numerator and denominator of
an irreducible fraction of γ are odd values, however, the parametric relative orbit in polar
view has overlapping petals. Because of the overlapping petals, the number of the petals
is computed by the two different rules in Eq. (4.18).
While the previous rules apply only to circular orbits, the following relation can be used
to find the number of petals when dealing with elliptical orbits of target satellites:
γpetal = 2 × γnum (4.20)
Note that the parametric relative orbits with the elliptical orbits do not have overlapping
petals, thus the number of petals is equal to twice the numerator value.
The parametric relative orbits are symmetric with respect to the base satellite orbit plane,
and the trajectory of the z-component simply represents sinusoidal motion. These proper-
ties produce the following corollary:
Corollary 1. The number of petals is not only the same as the number of intersection
points of a parametric relative orbit on the base satellite orbit plane, but is also the same
as the number of vertical tracks of a target satellite as seen by the base satellite.
Figure 4.8 shows a 10-petal relative orbit produced by hypocycloid motion of a target
satellite. Since the relative orbit frequency γ = 53
with eccentricity e = 0.1 is chosen, the
47
Figure 4.8: Intersection points of a 10-petal parametric relative orbit (γ = 5/3, e = 0.1).
parametric relative orbit has 10 intersection points on the base satellite orbit plane, as
stated by the corollary above.
In summary, based on the relative orbit frequency γ between satellites, a parametric relative
orbit is specified by the number of petals or cusps in an x − y plane, and we are able to
identify the number of vertical tracks of a target satellite as seen by a base satellite. This
observation is useful in designing constellation patterns of satellite relative motion.
4.4.2 Classifications
From the design rules of the parametric relative orbits in the previous section, complex
nonlinear relative motion can be characterized in terms of petals or cusps, based on the
relative orbit frequency γ. In this section, we study more about the parametric relative
orbits which are categorized into two types of curves, cycloid and trochoid. In the example
for this section, we are concerned with the hypocycloid and hypotrochoid motions rather
than the epicycloid and epitrochoid motions. Typically, the hypocycloid and hypotrochoid
48
motions can be seen in the LEO (Low Earth Orbit) with respect to the MCO (Medium
Circular Orbit), or the ground track orbit relative to the ECEF frame.
Cycloid Motions
Cycloid motions of parametric relative orbits are produced when the radius of the epicycle
and the distance of the fixed point are equal. From the relationship between the general
parametric equations and the derived parametric relative equations in the previous section,
we can find the conditions to determine the shape of the petals. The two parameters
involved in the conditions used to determine the shape of the petals are the relative orbit
frequency γ and relative inclination iR. The conditions, or the relationship between the
two parameters, for the cycloid motions are given by
iR
= cos−1(1
γ
)
⇒ Hypocycloid (γ > 1) (4.21)
iR
= cos−1 γ ⇒ Epicycloid (γ < 1) (4.22)
Table 4.2: Special cases of hypocycloid and epicycloid
k γ iR
hypocycloid k γ iR
epicycloid
3 3 70.5288 deltoid(3-cusped) 1 13
70.5288 cardioid(1-cusped)
4 2 60.0000 astroid(4-cusped) 2 12
60.0000 nephroid(2-cusped)
5 53
53.1301 5-cusped hypocycloid 3 35
53.1301 3-cusped epicycloid
6 32
48.1897 6-cusped hypocycloid 4 23
48.1897 4-cusped epicycloid
Table 4.2 shows some specific cases of the hypocycloid and epicycloid motions, representing
the simplest forms of the closed relative orbits. An orbit shape to be considered a cycloid
motion must possess cusps as seen in Fig 4.9 which shows a 3-cusped hypocycloid motion. A
cusp occurs at the extremities of the petals where the velocity at the cusp is instantaneously
zero. This condition is shown in Fig 4.10, where the velocity components of x and y
49
−2.5 −2 −1.5
x 104
−8000
−6000
−4000
−2000
0
2000
4000
6000
8000
x (km)
y (k
m)
Cusp
Figure 4.9: 3-cusped hypocycloid motion
−6 −4 −2 0 2 4 6−6
−5
−4
−3
−2
−1
0
1
2
3
4
vx (km/sec)
vy (
km/s
ec)
Cusps points
Figure 4.10: Velocity components of x and y
of the 3-cusped hypocycloid motion
illustrate a zero at the cusps. However, the velocity of the z-component, which is not
pictured in Fig 4.10, will be a maximum value.
Corollary 1. The parametric relative orbits of hypocycloid and epicycloid motions have
cusps, at which the velocities of the x and y components are zero and the z-component has
a maximum velocity.
Trochoid Motions
The parametric relative orbits of cycloid motions are simply designed by choosing the rel-
ative inclination iR
determined by the relative orbit frequency γ. The cycloid motions can
be a reference orbit to design the parametric relative orbits of trochoid motions. Depend-
ing on the value of γ relative to 1, the parametric relative orbits can be categorized as
either hypotrochoid or epitrochoid, both of which have a shape characterized by petals.
The following conditions, showing the relationship between iR
and γ, produce curtate or
50
prolate shapes of hypotrochoid motions:
iR
< cos−1 γ ⇒ Curtate hypotrochoid (4.23a)
iR
> cos−1 γ ⇒ Prolate hypotrochoid (4.23b)
Figures 4.11 and 4.12 show the trajectories of the curtate and prolate hypotrochoid motions
on a sphere as seen from polar views. In the case of the curtate hypotrochoid motion, the
trajectories represent outward curves about the hypocycloid reference trajectory. On the
contrary, the prolate hypotrochoid motions represent inward curves about the hypocycloid
trajectory. These behaviors depend on the relative inclination between satellites.
−2.5 −2 −1.5
x 104
−8000
−6000
−4000
−2000
0
2000
4000
6000
8000iR = 60
iR = 50
x (km)
y (k
m)
Figure 4.11: Curtate hypotrochoid
−2.5 −2 −1.5
x 104
−8000
−6000
−4000
−2000
0
2000
4000
6000
8000iR = 80
iR = 90
x (km)
y (k
m)
Figure 4.12: Prolate hypotrochoid
The epitrochoid motions occur when the orbit mean motion of the base satellite is faster
than the mean motion of the target satellite, which implies that γ < 1. Note that the
contrary concept of the hypotrochoid motions is true, implying that γ > 1. The conditions
to describe the trajectories of the epitrochoid motions are given by
iR
< cos−1(1
γ
)
⇒ Curtate epitrochoid (4.24a)
iR
> cos−1(1
γ
)
⇒ Prolate epitrochoid (4.24b)
51
The trajectories of the curtate and prolate epitrochoid motions follow the same concepts
as the curtate and prolate hypotrochoid motions.
Spirographs of parametric relative orbits
A Spirograph is a geometrical drawing tool that produces parametric curves of cycloids
and trochoids [52]. Mathematicians call these curves spirographs. When dealing with
circular orbits of satellites, the relative motion geometry as seen from the polar view is the
same as the mathematical curves created by a Spirograph. As shown in Fig 4.13, only two
parameters are involved in defining spirographs of the parametric relative orbits: relative
orbit frequency γ which determines the number of petals or cusps, and relative inclination
iR
which determines the shape of the petals.
When considering the eccentricities of satellite orbits, the parametric relative orbits will be
transformed from the circular orbit cases of the spirographs in Fig 4.13. However, the orbit
shapes can still act as effective reference trajectories in space when designing constellation
patterns or understanding the relative motion geometry. For the design of repeating ground
track orbits, the relative inclination iR
can be replaced by the inclination of satellites. The
resulting parametric relative orbits will then imply repeating ground tracks with respect
to the ECEF frame.
52
Figure 4.13: Spirographs of parametric relative orbits
53
4.5 Conclusions
Understanding the nature of relative motion geometry is a complex task. To understand the
relative motion geometry, this chapter studies the geometric structure of relative motions
through the mathematical objects of epicycle and deferent circles. In this study, we define
parametric relative orbits which represent relative orbits with the properties of closed
and periodic orbits, and three important observations for relative motion problems are
found. First, we conclude that the relative motion dynamics of circular orbit cases in polar
views are exactly the same as the mathematical models of cycloids and trochoids. When
dealing with the eccentricities of orbits, the parametric relative orbits can act as effective
reference trajectories in space. This finding is useful for designing constellation patterns
or understanding the relative motion geometry. Second, we conclude that the parametric
relative orbits are specified by the number of petals or cusps based on the relative orbit
frequency γ. The number of petals or cusps can be identified as the number of vertical
tracks of a target satellite as seen by a base satellite. Third, we also conclude that two
design parameters are involved in defining the parametric relative orbits: relative orbit
frequency γ which determines the number of petals or cusps, and relative inclination iR
which determines the shape of the petals. In the next chapter, the parametric relative
orbits can be used for designing repeating space tracks because of the their closed and
periodic properties.
Chapter 5
Parametric Constellations Theory
5.1 Introduction
In the previous chapter, we were concerned with the relative motion problem involved in
a two-satellite system. This chapter proposes a constellation theory for a single repeating
space track system of multiple satellites.
One recent trend in the advances of satellite systems is an increase in the number of
smaller and lower-cost satellites. This trend has led to a rapid increase in the number of
satellite constellations for various space missions. Under this circumstance, the complexity
of dynamic systems between multiple satellites will be a potentially critical problem for
the development of future satellite systems.
The theory proposed in this chapter is motivated by the problem of creating a set of satellite
constellations, called the parametric constellations (PCs), that shows a single identical
constellation pattern of target satellites as seen by a base satellite. In such a constellation,
the dynamics and control problem between satellites is simple and consistent, because all
of the relative orbits represent the same repeating space track. In particular examples, the
gimbal and tracking problem of multiple target satellites for inter-satellite links, and the
54
55
formation design for a fleet of target satellites, will be significantly less complex.
A survey of similar satellite constellation designs began with the flower constellations (FCs)
theory developed at Texas A&M University in 2004. The FC theory uses satellite phasing
rules obtained from a Planet-Centered Planet-Fixed rotating frame. The term “Planet-
Centered Planet-Fixed” refers to a zero inclination rotating reference frame with respect to
a planet. For the design of a particular constellation set, the FC theory is identified by five
integer parameters (Np, Nd, Fn, Fd, Fh) and three common orbit parameters (ω, i, e). The
FC theory has demonstrated some potential applications of satellite constellations with
various satellite phasing schemes. Historical reviews can be found in the Refs. [54-61].
The PC theory leads to a single identical constellation pattern, in other words, a repeating
space track, of target satellites with respect to the base satellite orbit that refers to an
inclined rotating frame. To create this constellation pattern, the solutions are derived from
the geometrical relations of satellite orbits and the periodic condition of the parametric
relative orbits. By studying the relative motion geometry in the previous chapter, using the
characteristics of the parametric curves, we develop rules to design the parametric relative
orbits. In the PC theory, these closed and periodic parametric relative orbits are used
as the constellation patterns of the target satellites. Therefore, all of the target satellites
move in a single identical repeating relative orbit as seen by a base satellite.
To distribute satellites for the repeating space track, the PC set uses a real number system
for node spacing as opposed to the integer number system used by FCs. The use of the
real number system gives mathematical advantages to PCs by providing an infinite orbit
element set with an irrational number and a finite orbit element set with a rational number.
In addition to this advantage, and more importantly, the PC theory provides direct solu-
tions to constellation design for three types of repeating space tracks: repeating ground
track orbit in the ECI frame, repeating relative orbit in the newly-defined ECI frame, and
repeating relative orbit in the ECI frame. Evaluation of the PC theory is illustrated by
comparison of constellation design processes for repeating space tracks using FC and PC
56
theories.
5.2 Problem Statement
A problem associated with the design of satellite constellations is the increased complexity
resulting from the relative motion between multiple satellites. The purpose of this section
is to reduce the complexity of multiple satellite dynamics by creating a repeating space
track from the individual relative orbits of satellites in a rotating reference frame. The
repeating space track will represent the same closed periodic relative trajectory for each
target satellite as seen from a base satellite. The design of the repeating space track for
the multiple target satellites can be achieved by obtaining the orbit element set of each
target satellite.
Let us consider a satellite system containing three target satellites and a base satellite
as seen in Fig 5.1. The three target satellites have identical orbit shapes with the only
difference being the node points on the base satellite orbit plane in the ECI frame. In
this particular example, we assume the inclination of the base satellite with respect to the
Earth’s equator is zero. If the angular rate of the base satellite is constant, representing a
Figure 5.1: Three identical target satellite orbits and a base satellite circular orbit
57
circular orbit, then only M0, based on the ascending node Ω, of each target satellite needs
to be considered when defining the repeating space track, because the four remaining orbit
elements (a, e, i, ω) are identical. The repeating space track can be described by three
values of M0, one for each of the three target satellite orbits.
A more practical example involves an inclined base satellite circular orbit, where the orbit
elements for the repeating space track are no longer functions of M0 and node points.
Now, when defining the repeating space track, the four orbit elements, i,Ω, ω,M0, must
be considered, while the orbit elements a and e related to the orbit shape are identical for
all target satellites.
Another possible choice when designing repeating space tracks is to consider a base satellite
elliptical orbit as a rotating reference frame representing a non-constant angular rate. With
the non-constant angular rate of the base satellite, the target satellites must be distributed
on a single orbit plane as seen in Fig 5.2. In Fig 5.2, a closed relative orbit with the
relationship of 3nT
= nB
can be accomplished with three completed revolutions of the base
satellite during one revolution of the target satellite. The relationship between the mean
motions of the satellites produces three possible values of M0 for the repeating space track.
Figure 5.2: A target satellite orbit plane and a base satellite elliptic orbit
58
The PC theory in this chapter develop analytic solutions to obtain the orbit element sets
and closed-form formulae for the repeating space tracks.
5.3 Parametric Constellations
In establishing a rotating reference frame, we can choose an infinite set of frames in terms
of angular rates. To design repeating space tracks, we choose a base satellite circular orbit,
which is a rotating reference frame having a constant angular rate and an inclined rotating
frame with respect to the Earth. Thus, the resulting trajectories of the target satellites are
repeating relative orbits as seen by the base satellite. Another result from choosing this
frame is the ability to create repeating ground tracks in the ECEF frame by substituting
a zero inclination.
The design of PCs is based on the classical orbit element set. For the repeating space
track of target satellites with respect to the base satellite, two orbit elements, a and e,
of the target satellites are identical. While these two orbit elements are identical, the
other four orbit elements have different values. If elliptical orbits of the target satellites
are considered, we can establish repeating space tracks based on the same mean motions
relative to the base satellite.
With these characteristics for creating repeating space tracks, this section defines para-
metric constellations for the design of repeating constellation patterns of satellites with
respect to a rotating reference frame. The parametric constellations (PCs):
• have the rotating reference frame of a base satellite circular orbit that has a constant
angular rate.
• have identical orbit elements of semi-major axis, a, and eccentricity, e, of target
satellites.
• are characterized by ik, ωk, and Mk0 in terms of a real number system for ascending
59
node spacing, Ωk.
(The subscript k is the target satellite’s number and N is the number of target
satellites, k = 1, 2, ..., N)
As a standardization, a constellation set of PCs is named as a (aB, γ, N) PC:
- Base satellite orbit radius (aB): establishes the base satellite orbit plane.
- Relative orbit frequency (γ): determines the constellation pattern and the target
satellite orbit radius.
- The total number of target satellites (N): represents the number of distinct orbits
and the number of satellites per orbit.
If we design a (20000,3,20) PC, the orbit altitudes of 20 target satellites are determined
with 9615.0 km by relative orbit frequency γ, with respect to the base satellite circular orbit
of 20000 km. The constellation pattern represents a 3-petaled parametric shape for the
circular orbits, or a 6-petaled shape for the elliptical orbits. We use this standardization
to specify a particular constellation set of PCs.
5.3.1 Satellite Phasing Rules
The most important issue to construct PCs is to find the satellite phasing rules which are
used to obtain the orbit element set of satellites for repeating space tracks. This section
proposes methods for deriving the satellite phasing rules, using the geometrical relations
of satellite orbits and the periodic condition of parametric relative orbits. Let us consider
satellite orbits projected on the surface of the Earth, assuming identical orbit elements
of a and e between target satellites. Figure 5.3 illustrates the geometry of the projected
satellite orbits to construct repeating space tracks. To design the repeating space tracks,
a significant geometrical concept for PCs is to make identical inclinations and argument
of perigees of the target satellite orbits with respect to the base satellite orbit plane. The
60
satellite phasing rules obtain the orbit element sets of ik and ωk, satisfying the geometrical
concept, and Mk0, for the periodic condition of the parametric relative orbits, in terms of
ascending node spacing, Ωk.
Figure 5.3: Geometry of target satellite orbits about a base satellite orbit plane.
Ascending Node Spacing, Ωk
The distribution of the target satellites in PCs is determined by ascending node spacing, Ωk,
as a free parameter. There are two ways to arbitrarily distribute Ωk: evenly and unevenly
spaced values. In this dissertation we are concerned with the evenly spaced ascending
nodes which give a regularly distributed numerical sequence of PCs. For the distribution
of evenly spaced ascending nodes, we use the following formula:
Ωk = Ω1 + θΩ(k − 1), k = 1, 2, ..., N (5.1)
where the real number θΩ is an angle between each ascending node.
The node spacing distribution corresponds to the dynamical behavior of rotating a circle.
Mathematically, the rotation of the circle through a series of angles is used to illustrate the
61
possible sequence of points. Thus, the node points can be distributed through an angle θΩ
on the equator plane by the following theorems [54]:
Theorem 1 (rational rotations). If θΩ is a rational multiple of 2π radians, say θΩ =
2πβ with β ∈ (0, 1), then the ascending node distribution is periodic. In other words, if β
is written as a relative prime p/q, then the ascending nodes will repeat periodically after
each q-node sequence.
Theorem 2 (irrational rotations). If θΩ is an irrational multiple of 2π radians, say
θΩ = 2πβ for an irrational number β with β ∈ (0, 1), then the ascending node distribution
is aperiodic and will have an infinite number of points.
In the case of irrational rotations, the sequence of the node spacing is uniformly distributed
on the equator plane by Weyl’s equidistribution theorem [54]. Thus, the node points can
be distributed by equally spaced intervals.
Phasing Rule for Inclination, ik
From the geometry in Fig 5.3, identical inclinations of the target satellite orbits relative
to the base satellite orbit plane can be achieved through the same relative inclinations, iR,
which represent the angles between the orbit planes of target satellites and the orbit plane
of the base satellite.
This section computes the inclinations, ik, of target satellites based on an identical iR. The
following relationship is used to compute ik:
A cosx′ +B sin x′ = R cos(x′ − α′) (5.2)
where,
R =√A2 +B2, tanα′ =
B
A(5.3)
The equation for iR
is expressed as
cos iR
= cos iB
cos ik + sin iB
sin ik cos ∆Ωk (5.4)
62
where ∆Ωk = Ωk − ΩB. Note that an identical i
Ris specified by the orbit elements of the
first target satellite.
In Eq. (5.4), we have the specified values of iB, i
R, and ∆Ωk, and Eq. (5.4) can be trans-
formed into the formula in Eq. (5.2). We then derive the inclinations ik in terms of the
specified values. The resulting inclination ik is the phasing rule for inclinations in PCs:
ik = cos−1( cos i
R√
1 − sin2 iB
sin2 ∆Ωk
)
+ tan−1(tan iB
cos ∆Ωk), k = 1, 2, ..., N (5.5)
Note that the three components, iB, i
R, and ∆Ωk, form a spherical triangle if the following
condition is satisfied:
| sin iB
sin ∆Ωk| ≤ | sin iR| (5.6)
Another approach to solving for ik involves the application of a numerical method to
Eq. (5.4). When applying for the numerical method, Eq. (5.4) must be rewritten as a
polynomial function given by
f = cos iB
cos ik + sin iB
cos ∆Ωk sin ik − cos iR
= 0 (5.7)
Because iB, i
R, and ∆Ωk in Eq. (5.7) are known values, we can solve for ik using Newton’s
iterative method.
Phasing Rule for Argument of Perigee, ωk
For the design of repeating space tracks, when considering elliptical orbits of target satel-
lites, a geometrical relation has identical arguments of perigees of target satellites relative
to the base satellite orbit plane. In Fig 5.3, if the projected perigee points, ωk(k = 1, 2, 3),
of the target satellite orbits correspond to the projected base satellite orbit plane, we find
the following relation:
ωk = φk, k = 1, 2, 3 (5.8)
63
However, in general, we need to consider that ωk does not always correspond with the
projected orbit plane of the base satellite. For this general case, we define a relative
argument of perigee, ωR, based on the first target satellite orbit:
ωR
= ω1 − φ1 (5.9)
where the relative argument of perigee, ωR, is the arc length from the intersection point,
IP, to the projected perigee point, ω1, on the first target satellite orbit. The phasing rule
of ωk is then obtained by adding each subsequent φk by ωR
as seen below:
ωk = φk + ωR, k = 1, 2, 3 (5.10)
where
φk = tan−1[ sin ∆Ωk sin i
Bsin ik
cos iB− cos ik cos i
R
]
(5.11)
From the phasing rule in the previous section, the inclination ik has been computed, and
iB, i
R, and ∆Ωk are known values. Finally, we can obtain the identical arguments of
perigees of the target satellites using the phasing rule in Eq. (5.10).
Phasing Rule for Initial Mean Anomaly, Mk0
The phasing problem of initial mean anomaly is a key issue in designing PCs. This section
derives the phasing rule for initial mean anomaly, Mk0, using the periodicity of parametric
relative orbits. Since the parametric relative orbits are closed and periodic, the function
f(x, y, z), which is the generalized form of the parametric relative equation, is satisfied
with the following relation:
f(x, y, z) + Pt = f(x, y, z) (5.12)
for all possible values of (x, y, z) with period Pt. Because of the periodicity of the parametric
relative orbits, we can design identical orbit shapes with the same orientation depending
on satellite initial positions.
64
Figure 5.4: 4-petaled hypocycloid parametric relative orbit in x− y plane.
Figure 5.4 shows a 4-petaled parametric relative orbit with a period of Pt which represents
the time taken for the point O to make a complete closed orbit. From Fig 5.4, we find the
following periodic condition for the motion resulting from the deferent and epicycle circles
to describe the identical parametric relative orbit:
ψxy+
ψxy−=ψxy+ + ∆n+Pt
ψxy− + ∆n−Pt
= Constant (5.13)
where the initial phase angles ψxy− and ψxy+ are given by
ψxy− = MT0
−MB0
+ ωT− φ
T+ φ
B(5.14a)
ψxy+ = MB0
+MT0
+ ωT− φ
B− φ
T(5.14b)
Note that a base satellite circular orbit is used for the rotating reference frame resulting
in the lack of an argument of perigee in Eq. (5.14). Since the relationship between the
orbit mean motions nB
and nT
can be expressed as the relative orbit frequency, γ, we can
rewrite Eq. (5.13) in terms of γ:
ψxy+
ψxy−=γ + 1
γ − 1(5.15)
65
From Eq. (5.15), we derive the satellite phasing rule of the initial mean anomaly to design
an identical relative orbit. The satellite phasing rule when considering N -target satellites
is obtained in the following form:
Mk0 = M10 + γ(MB0
− φ1(k)) − ωR, k = 1, 2, ..., N (5.16)
where φ1(k) is given by
φ1(k) = tan−1[ sin ∆Ωk sin i
Bsin ik
− cos ik + cos iB
cos iR
]
(5.17)
In Eq. (5.16), M10 is an initial value for the distribution of the initial mean anomalies,
Mk0, of the target satellites.
When dealing with elliptical orbits of the target satellites, the constellation pattern will
represent a transformed shape from the circular orbit cases having the same mean motions.
However, the summarized satellite phasing rules in Table 5.1 are still applied to the elliptical
orbit cases, because the periodic condition for orbits with the same mean motions does not
change.
Table 5.1: Satellite phasing rules in the ECI frame
Orbit element Formulae of phasing rules
Ωk Ω1+ θΩ(k − 1), k = 1, 2, ..., N
ik cos−1(
cos iR√
1−sin2 iB
sin2 ∆Ωk
)
+ tan−1(tan iB
cos ∆Ωk)
ωk φk + ωR
Mk0 M10 + γ(MB0
− φ1(k)) − ωR
5.3.2 Transformation of Satellite Phasing Rules
This section transforms the satellite phasing rules of the PC theory into phasing rules in
terms of relative orbital elements which are defined in a new ECI frame. The transformed
66
phasing rules are useful for the design of repeating relative orbits with respect to a par-
ticular orbit plane. This particular orbit plane can be that of the base satellite and the
orbits of the target satellites will be distributed with respect to this plane.
Let I , J , K be orthogonal reference axes in the original ECI frame. In the ECI system, we
can define an ECI′ frame (orbital reference frame: ox, oy, oz) in a reference circular orbit,
as seen in Fig 5.5. The ECI′ frame has the axis ox pointing toward the initial mean anomaly
(MB0
) of the base satellite, and the axis oz aligned with the orbital angular momentum.
The axis oy then completes the system based on the right-hand rule. In Fig 5.5, we define
new orbital elements [ik, Ωk, ωk, Mk0], called relative orbital elements, in the ECI′ frame.
Figure 5.5: Geometry for relative orbital elements and ECI′ frame
Next, we transform the PC phasing rules into phasing rules in terms of the relative orbital
elements defined in the ECI′ frame. For the transformation of the phasing rules, we find
a geometrical term, (φ1(k) −MB0
), corresponding to the relative ascending node, Ωk, from
67
the above phasing rules in Table 5.1:
Ωk = φ1(k) −MB0
(5.18)
In the phasing rules in Table 5.1, the variables iR
and ωR
are equivalent to the relative
orbital elements ik and ωk, respectively. Thus, we have the following relations:
[(φ1(k) −MB0
), iR, ω
R] = [Ωk, ik, ωk] (5.19)
Now the phasing rules from Table 5.1 can be transformed into phasing rules in terms of
the relative orbital elements (Ωk, ik, ωk) in the ECI′ frame, as shown in Table 5.2. For
the node spacing, these transformed phasing rules are also distributed on the base satellite
orbit plane by using the rational and irrational rotations as the phasing rules in the ECI
frame do.
Table 5.2: Satellite phasing rules in the ECI′ frame
Orbit element Formulae of phasing rules
Ωk Ω1 + θΩ(k − 1), k = 1, 2, ..., N
ik cos−1(
cos iB
cos ik − sin iB
sin ik cos(MB0
+ Ωk))
Ωk ΩB
+ sin−1(
sin(MB0
+Ωk) sin ik
sin ik
)
ωk φk + ωk
Mk0 M10 − γΩk + φk − ωk
Note that in Table 5.2 the inclination, ik, is derived from the geometrical relationship
between satellite orbits projected on the Earth’s surface, in order to be expressed in terms
of Ωk.
Finally, using the phasing rules in Table 5.1 and 5.2, we obtain the orbital element sets for
repeating relative orbits in the ECI and ECI′ frame.
68
5.3.3 Repeating Ground Track Orbits
The design of repeating ground track orbits, using the PC theory, is not as complicated
as that of repeating relative orbits, because in the ECEF frame all of the inclinations
and arguments of perigee of satellites are identical. For the repeating ground track orbit
without perturbations, we define the following relative orbit frequency, γ, which is the ratio
of the satellite mean motion to the Earth’s rotation rate:
n = γ ω⊕ (5.20)
To derive the satellite phasing rules for repeating ground track orbits, the base satellite
orbit plane has zero inclination with respect to the Earth’s equator. When we compute the
phasing rules with the zero inclination (iB
= 0) through the formulae in Table 5.1, the arc
length φ1(k) of Mk0 is not mathematically defined. Thus, we introduce another formula to
compute φ1(k), obtained from a geometrical relationship between base and target satellite
orbit planes on the Earth’s surface:
φ1(k) = sin−1(sin ∆Ωk sin ik
sin iR
)
(5.21)
The arc lengths φ1(k) and φk with zero inclination of the base satellite orbit plane are then
given by
φ1(k) = Ωk, φk = 0 (5.22)
Substituting Eq. (5.22) into the formulae in Table 5.1 leaves us with the initial mean
anomaly as the only value that does not remain identical, and thus the only initial mean
anomaly that can be considered as a phasing rule:
Mk0 = M10 − (γΩk + ω), k = 1, 2, ..., N (5.23)
Consequently, the satellite phasing rules of the repeating ground track orbits only involve
the subset of initial mean anomalies in terms of the ascending node distribution.
69
5.3.4 Repeating Space Tracks with a Single Orbit
Using the elliptical orbit of the base satellite unlike using the circular orbit as a rotating
reference frame, we investigate a constellation set of the PC theory in which any number
of satellites can be distributed on a single orbit in the ECI frame. In Eq. (5.1), the initial
ascending node Ω1 can be an angle defined by multiple values, thus we can rewrite Eq. (5.1)
in the following form:
Ωk = Ω1 + 2πm+ θΩ(k − 1), k = 1, 2, ..., N (5.24)
where m is an integer value.
Substituting Eq. (5.24) into Eq. (5.23), assuming M10 = 0 and ω = 0, the phasing rule can
be expressed as
Mk0 = −γ(Ω1 + 2πm) − γθΩ(k − 1) (5.25)
If we consider the distribution of N -satellites in single orbit, namely θΩ = 0, Eq. (5.25) is
given by
Mk0 = −γΩ1 − γ2πk, k = 1, 2, ..., N (5.26)
As a result, Eq. (5.26) represents a phasing rule which describes the distribution of satellites
in a single orbit, for repeating ground tracks, depending on the relative orbit frequency γ
chosen.
In the same manner, for repeating relative orbits with a single orbit, the formula in
Eq. (5.24) can be substituted into the satellite phasing rules in Table 5.1, with θΩ = 0.
The resulting phasing rule of Mk0 is expressed in the following form:
Mk0 = M10 + γMB0
− γφ′
1(k) − ω′
R, k = 1, 2, ..., N (5.27)
where φ′
1(k) and ω′
Rare the resulting equations after substituting Ωk = Ω1 + 2πk.
The distribution of satellites in the single orbit is determined by the relative orbit fre-
quency γ. Recall that γ represents the relationship of the mean motions between the base
70
and target satellites. The maximum possible number of satellites in the single orbit is
determined based on the γ chosen. For example, when we choose γ = 110
, the maximum
possible number of satellites corresponds to the denominator of γ, 10, because the base
satellite makes 10 revolutions while the target satellite makes 1 revolution for a completed
closed relative orbit. Thus, 10 possible initial points in the target satellite orbit can be
chosen for the repeating relative orbit. In this case, where the target satellites are on a
single orbit plane in the ECI frame, the elliptic orbit of the base satellite can be used for
the rotating reference frame, because of the same orbit comparability between the base
and target satellites.
5.4 Closed-form Formulae for PCs
The preceding sections have proposed the satellite phasing rules to obtain the orbit element
sets for designing the repeating space tracks. This section suggests the closed-formulae to
describe the repeating space tracks of N -satellites in the base satellite centered frame and
ECEF frame.
In the case of the circular orbits of the target satellites, Eq. (4.15) provides the simple
closed-form formulae for the constellation design of N -satellites, where the only variables
that change are the phase angle terms ψxy−k and ψxy+
k while the other variables remain
constant. The phasing angles are determined by the orbit elements obtained from the
satellite phasing rules. When we consider the elliptical orbits of target satellites, the
closed-form formulae must be rewritten with the consideration of the terms involved with
eccentricity. The orbit radiuses, rk, of the target satellites are expressed in terms of the
true anomalies, νk:
rk =a(1 − e2)
1 + e cos νk
, k = 1, 2, ..., N (5.28)
When we compute νk using Kepler’s equation, Mk0 obtained from Eq. (5.16) is used for
71
the mean anomaly set:
Mk = Mk0 + nB(t− t0) (5.29)
Finally, the closed form formula of N -satellite constellations is expressed in the following
equation:
xk =rk
2
[
(1 + cos iR) cos
(
θk + ψxy−k
)
+(1 − cos iR) cos
(
θk + 2n1t+ ψxy+k
)]
− aB
(5.30a)
yk =rk
2
[
(1 + cos iR) sin
(
θk + ψxy−k
)
−(1 − cos iR) sin
(
θk + 2n1t+ ψxy+k
)]
(5.30b)
zk = rk sin iR
sin(
θk + n1t+ ψzk
)
(5.30c)
where θk = νk − nBt, and the phase angles, ψxy−
k , ψxy+k , and ψz
k, are defined as
ψxy−k = ω
R−M
B0+ φ1(k) (5.31a)
ψxy+k = ω
R+M
B0− φ1(k) (5.31b)
ψzk = ω
R(5.31c)
The closed form formula of N -satellite constellations for the repeating ground track is
obtained by substituting iB
= 0 into Eq. (5.30) and considering the Earth’s rotation rate,
ω⊕. The formula of N -satellite constellations is written as
xk =rk
2
[
(1 + cos i) cos(
θk + ψxy−k
)
+(1 − cos i) cos(
θk + 2ω⊕t+ ψxy+k
)]
(5.32a)
yk =rk
2
[
(1 + cos i) sin(
θk + ψxy−k
)
−(1 − cos i) sin(
θk + 2ω⊕t+ ψxy+k
)]
(5.32b)
zk = rk sin i sin(
θk + ω⊕t+ ψzk
)
(5.32c)
where θk = νk − ω⊕t, and the phase angles ψxy−k , ψxy+
k , and ψzk are defined as
ψxy−k = ω + Ωk (5.33a)
ψxy+k = ω − Ωk (5.33b)
ψzk = ω (5.33c)
Using the closed-form formulae with orbit element sets from the phasing rules, the con-
stellation set of the repeating space track will be easily visualized by commercial computer
programs.
72
5.5 Evaluation of the PC Theory
In the preceding sections, we proposed satellite phasing rules and closed form formulae for
designing repeating space tracks. This section evaluates the PC theory in terms of node
spacing distribution and constellation design process, compared to the FC theory.
5.5.1 Node Spacing Discussion
One of the purposes of the PC theory is its use in the potential applications (inter-satellite
links or repeating ground tracks) of repeating space track systems about a rotating ref-
erence frame such as a base satellite centered frame or ECEF. To successively achieve
these objectives, the constellation set is assumed to be uniformly distributed, resulting in
satellites that move at regular interval sequences.
By the rational rotation concept from Theorem 1, the sequence of the node spacing can
be regularly distributed on the Earth’s equator or base satellite orbit plane. If we choose
θΩ = 2πβ with β ∈ (0, 1), β = p/q, then every node spacing returns to its original position
after making p turn(s). In the mathematical description, the rotation number of an orbit
is described by p/q. From the integers p and q, the maximum possible number for the
ascending node distribution is equal to the denominator q for a constellation set.
By the irrational rotation concept from Theorem 2, we can establish infinite node points in
which the ascending nodes never return to their original positions on the Earth’s equator.
To compute the iterates modulo multiples of 1, instead of 2π, we define a mathematical
function frac(θk) which gives the fractional part of an irrational angle θ for the k-th iterate
and provides the same distribution as the whole number:
β = frac(θk) ≡ kθ − bkθc, k = 1, 2, 3, ... (5.34)
where bθc is the floor function which denotes the greatest integer less than or equal to kθ.
Thus the sequence θΩ = 2πβ with β ∈ (0, 1) produces an infinite set of ascending nodes.
73
Let us consider a theoretical constellation set which has a single repeating ground track
orbit for 1000 satellites with a node spacing of the fractional part of√
3 (β = 0.7320...).
Figures 5.6 and 5.7 show, in the spherical coordinate system, the distributions of ascending
nodes in the angular coordinate and initial mean anomalies in the radial coordinate, for
both rational and irrational rotations.
100
200
300
400
30
210
60
240
90
270
120
300
150
330
180 0
Figure 5.6: Rational rotation with the three
decimal places of√
3
100
200
300
400
30
210
60
240
90
270
120
300
150
330
180 0
Figure 5.7: Irrational rotation of√
3
If we simply round√
3 (β ≈ 7321000
) to three decimal places, then the maximum number of
available unique ascending node points will be 250, according to Theorem 1. Figure 5.6
shows this distribution of 250 node points. On the contrary, Fig 5.7 shows the irrational ro-
tation distribution of 1000 node points with the node spacing of√
3. According to Theorem
2, an unlimited number of unique node points are available when considering an irrational
rotation, allowing this distribution of 1000 satellites. An increasingly precise application of
the concept discussed in Theorem 1 involves dealing with a more accurate approximation
of√
3, namely Archimedes’s approximation [55]: β ≈ 1351780
. This approximation gives an
accuracy of six decimal places, and results in a maximum number of 780 satellites that can
be distributed uniquely. As a result, the theoretical constellation set for the combination
of 1000 satellites and of√
3 as a node spacing cannot be achieved within the accuracy of
74
six decimal places when using rational rotation. This comparison illustrates the critical
mathematical difference for constellation designs between rational and irrational rotations.
In the case of the FCs theory, two independent integer parameters of Fn and Fd are freely
chosen for the sequence of orbit ascending nodes. This choice has a limitation in that node
points of satellites can in fact be mathematically distributed through an irrational rotation
as stated in Theorem 2. Thus, we reach the following corollary:
Corollary 1. The phasing parameters of PCs use real number systems while those of FCs
use integer number systems. Thus, the range of the element set of PCs includes the entire
range of the element set of FCs. On the contrary, the range of the element set of FCs does
not include the entire range of the element set of PCs.
5.5.2 Comparison of Constellation Design Process
The concept of existing FCs involves a single, identical relative trajectory with respect to
a frame rotating with the planet (e.g. the Earth Centered, Earth Fixed frame). To find
the satellite phasing rules of repeating relative trajectories, the existing FCs consider the
intersection points of satellite orbits in the ECI and ECEF frame. The phasing rules of these
satellites are then characterized by initial mean anomaly, M0, in terms of ascending node,
Ω, with identical orbit elements of a, e, i, and ω. In the case of PCs, satellite phasing rules
are obtained from the geometrical relations of satellite orbits and the periodic condition of
the parametric relative orbits in the base satellite centered frame. This behavior in PCs
produces satellite phasing rules consisting of two identical orbit elements of a, e and four
variable orbit parameters of i,Ω, ω,M0 for repeating space tracks.
A particular set is named as (aB, γ, N) PC to specify a constellation set in PCs. The
flowchart of the constellation design procedure is outlined in Fig 5.8. For the design of
the particular constellation set, we first choose a rotating reference frame for the desired
repeating space track. After the reference frame has been established, the first satellite
orbit plane can be specified based on the mission requirements.
75
!""
# $
%&'(&)***$
+ "
, " "
+-"
+ "
, " "
Figure 5.8: Flowchart of PC design process.
76
Next, the number of satellites, along with the ascending node spacing, must be determined.
Finally, using the satellite phasing rules, the orbit element sets can be directly obtained.
The satellite phasing rules will differ depending on the relative space track chosen. The
closed form formulae are then used to describe the relative space tracks in the rotating
frame.
The constellation design process can be evaluated through the comparison in the number
of steps between PCs and FCs to find orbit element sets for repeating space tracks. For
this comparison, the repeating space tracks can be divided into three types of repeating
trajectories: repeating ground track orbit about the ECEF frame in the ECI frame, re-
peating relative orbit about an inclined rotating frame in the ECI′ frame, and repeating
relative orbit about an inclined rotating frame in the ECI frame.
Repeating ground track orbit in the ECI frame
The design steps for repeating ground tracks in PCs and FCs are straightforward as seen
in Fig 5.9. For the design of repeating ground track orbits, design parameters of the real
number and integer number system are chosen for PCs and FCs, respectively, along with
the first satellite orbit element set. Then, the satellite phasing rules can be applied to
obtain the orbit element sets for each of the satellites.
Figure 5.9: Repeating ground track orbits in the ECI frame.
77
Repeating relative orbit in the ECI′ frame
A repeating relative orbit of the target satellites with respect to the base satellite can be
designed in the ECI′ frame. When designing the repeating relative orbit in the ECI′ frame,
the FC theory requires additional design steps, compared to the PC theory, to obtain the
orbit element set of the target satellite. The orbit elements of the first target satellite are
chosen in the ECI′ frame using integer phasing parameters. Then, we obtain the relative
orbit element set in the ECI′ frame using the phasing rules of FCs. Next, the relative orbit
element set is transformed through the necessary steps to derive the orbit element set in
the ECI frame, as seen in Fig 5.10.
Figure 5.10: Repeating relative orbits in the ECI′ frame.
Repeating relative orbit in the ECI frame
A common approach to design the repeating relative orbit is to establish the orbit element
set in the ECI frame. After choosing the orbit element set of the first target satellite
for design parameters, the FC theory requires a geometrical transformation of the chosen
78
orbit elements in order to use the satellite phasing rules. Then, the constellation design
process follows the same steps as the repeating relative orbit in the ECI′ frame. Therefore,
designing the repeating relative orbit in the ECI frame is more complicated than the
previous design process involving the ECI′ frame, as shown in Fig 5.11. However, the PC
theory for both cases of the repeating relative orbits is relatively straightforward.
Figure 5.11: Repeating relative orbits in the ECI′ frame.
For the example comparison, the numerical design processes for the three types of repeating
space tracks between PCs and FCs are shown in Appendix C. As seen in the numerical
examples, to obtain orbit element sets for repeating relative orbits, the constellation design
process of PCs is carried out through a very direct approach. The design process of FCs,
however, requires many additional design steps. This relative ease associated with the PC
design process gives a great advantage to the constellation designer. Furthermore, for the
design of repeating ground track orbits, the two constellation theories show an equivalent
design process in regards to the number of steps to obtain the orbit element sets.
5.6 Numerical Examples of PC Designs
This section evaluates the PC theory with the demonstration of constellation designs. One
potential application of the PC theory is to create an identical constellation pattern of
79
target satellites in Low Earth Orbit (LEO) with respect to the base satellite in Medium
Earth Orbit (MEO). This application is a representative example of PCs for inter-satellite
constellations. Another interesting application of PCs is to apply the PC theory to forma-
tion flying design. In specific, a fleet of target satellites relative to a base satellite moves in
close proximity along an identical relative trajectory. Thus, the target satellites contribute
to the mission objective as a single complex system.
5.6.1 Inter-satellite Constellation Design
A numerical example demonstrates a (20000,3,20) PC for inter-satellite links. As we choose
the relative orbit frequency, γ = 3.0, the orbit radius of target satellites is determined by
9615.0 km in LEO. The constellation pattern then shows a 6-petaled shape with e = 0.25.
The simulation parameters of orbit elements are shown in Table 5.3.
Table 5.3: Parameters of the orbit elements (γ = 3)
Satellites a(km) e i (deg) Ω(deg) ω(deg) M0(deg)
Base 20000 0 15 30 0 0
Target 9615 0.25 ik Ωk ωk Mk0
In Table 5.4, the ascending node spacing, Ωk, is chosen by 18 (β = 0.05) evenly spaced
values , and we can see the orbit element sets of the (20000,3,20) PC based on the satellite
phasing rules. In the PC theory, we find four geometrical parameters: iR, φ1(k), φk, and
ωR. The parameters are helpful for understanding the geometrical relationship between
each target satellite orbit. For a particular PC set, iR
and ωR
should be the same value,
and the values of φ1(k) and φk are symmetrical relative to the Earth’s equator, as shown in
Table 5.5. Figure 5.12 shows the demonstration of the (20000,3,20) PC, where all of the
20 target satellites fly on an identical 6-petaled relative trajectory as seen from the base
satellite.
80
Table 5.4: Orbit element sets of (20000,3,20) PC (unit:degree)
Sat] Ωk Mk0 ik ωk Sat] Ωk Mk0 ik ωk
1 120 199.9 85.0 110.0 11 300 30.2 83.4 79.8
2 138 144.4 78.7 109.2 12 318 334.1 88.1 80.5
3 156 90.1 74.5 107.0 13 336 278.3 92.4 82.7
4 174 37.3 71.3 103.7 14 354 223.2 95.8 86.1
5 192 345.8 69.3 99.5 15 12 168.9 97.9 90.2
6 210 295.0 68.6 94.9 16 30 115.0 98.6 94.9
7 228 244.3 69.3 90.2 17 48 61.2 97.9 99.5
8 246 192.8 71.3 86.1 18 66 6.9 95.8 103.7
9 264 140.0 74.5 82.7 19 84 311.8 92.4 107.0
10 282 85.7 78.7 80.5 20 102 256.0 88.1 109.2
Table 5.5: Geometrical parameters of (20000,3,20) PC (unit:degree)
Sat] iR
φ1(k) φk ωR
Sat] iR
φ1(k) φk ωR
1 83.6 91.7 15.0 94.9 11 83.6 -91.7 -15.0 94.9
2 83.6 110.2 14.3 94.9 12 83.6 -73.0 -14.3 94.9
3 83.6 128.3 12.1 94.9 13 83.6 -54.4 -12.1 94.9
4 83.6 145.9 8.8 94.9 14 83.6 -36.0 -8.8 94.9
5 83.6 163.0 4.6 94.9 15 83.6 -17.9 -4.6 94.9
6 83.6 180.0 0.0 94.9 16 83.6 0.0 0.0 94.9
7 83.6 -163.0 -4.6 94.9 17 83.6 17.9 4.6 94.9
8 83.6 -145.9 -8.8 94.9 18 83.6 36.0 8.8 94.9
9 83.6 -128.3 -12.1 94.9 19 83.6 54.4 12.1 94.9
10 83.6 -110.2 -14.3 94.9 20 83.6 73.0 14.3 94.9
81
Figure 5.12: 3D view (left) and polar view (right) of (20000,3,20) PC.
5.6.2 Formation Flying Design
An interesting application of the PC theory is to design a fleet of target satellites in which
the target satellites consistently move on a single, identical relative trajectory with respect
to the base satellite. The general schemes of formation flying design make it hard to find
the orbit element sets creating the identical relative orbit for a fleet of target satellites.
However, the PC theory is able to simply resolve the above problem by using the satellite
phasing rules. As a result, the fleet of target satellites maintains a single and identical
formation pattern.
For the formation flying design of the PC theory, we examine a (9000,1,10) PC set. If
γ = 1.0 is selected, the orbit radius of the target satellites is the same as the orbit radius
of the base satellite as seen in Table 5.6. The ascending node spacing, Ωk, is selected
by a small range of 0.15 (β = 0.00041667) evenly spaced values for 10 target satellites.
Figure 5.13 shows the resulting constellation set of the (9000,1,10) PC. All of the 10 target
satellites consistently move on the identical constellation pattern of ellipse shape, relative
82
to the base satellite. Figure 5.14 represents the orbit element sets for the (9000,1,10) PC.
Although we choose a small range of ascending nodes, we can see that the distributions of
Mk0 and ωk are widely spread, compared to ik.
Table 5.6: Parameters of the orbit elements (γ = 1)
Satellites a(km) e i (deg) Ω(deg) ω(deg) M0(deg)
Base 9000 0 30 120 0 20
Target 9000 0.001 ik Ωk ωk Mk0
−20 −15 −10 −5 0 5 10 15 20
−15
−10
−5
0
5
10
15
←Base satellite
x (km)
←sat10
←sat1
←sat9
←sat2
←sat8
←sat3
←sat7
←sat4
←sat6←sat5
y (k
m)
Figure 5.13: Formation flying design of (9000,1,10) PC.
83
119 120 121 1220
50
100
150
200
250
300
350
400
Ascending node (deg)
Initi
al M
ean
anom
aly
(deg
)
1 2 3 4 5
6 7 8 9 10
119 120 121 12230.7
30.75
30.8
30.85
30.9
30.95
31
31.05
Ascending node (deg)
Incl
inat
ion
(deg
)
1 2 3
4
5
6
7
8
9
10
119 120 121 1220
5
10
15
20
25
30
35
40
45
Ascending node (deg)
Arg
umen
t of p
erig
ee (
deg)
1
2
3
4
5
6
7
8
9
10
Figure 5.14: Orbit elements sets of (9000,1,10) PC.
5.6.3 PC Design with a Single Orbit
This section demonstrates a PC design with a single orbit in the ECI frame. When dealing
with multiple orbit planes, the design of a PC set must involve the use of the base satellite
circular orbit as a rotating reference frame. In the case of a single orbit plane, the repeating
space track can also use an elliptical orbit of the base satellite as a rotating reference frame.
As an example of the PC design with a single orbit, we choose a (7000,1/10,10) PC set.
Thus, the orbit altitudes of 10 target satellites are determined with 32491.0 km based on
the relationship of γ = 1/10. The constellation pattern represents a 2-petaled curtate
epitrochoid shape when considering the eccentricity. However, because a small difference
between the two orbit inclinations is chosen in Table 5.7, the resulting relative orbit will
show the constellation pattern of a nearly circular shape.
In Fig 5.15, we can see that 10 target satellites rotate around the Earth with a circular
pattern from the polar view. Table 5.8 represents the orbit element set for the resulting
repeating relative orbit. Observing Fig 5.15, we also can see that the target satellites
84
are distributed uniformly in a circular pattern as seen by the base satellite. Since the
distribution of the initial mean anomalies in the single orbit is uniform, based on the γ
chosen, the resulting constellation pattern represents the harmonic motion of the target
satellites as seen by the base satellite, regardless of eccentricities.
Table 5.7: Parameters of the orbit elements (γ = 1/10)
Satellites a(km) e i (deg) Ω(deg) ω(deg) M0(deg)
Base 7000 0.01 15 30 0 70
Target 32491 0.001 18 120 45 0
−3
−2
−1
0
1
2
x 104
−3−2
−10
12
3x 10
4
−1
−0.5
0
0.5
1
x 104
x (km)
sat 1
sat 10
sat 2
sat 9
sat 3
y (km)
sat 8
sat 4
sat 7
sat 5
sat 6
z (k
m)
Figure 5.15: PC design of (7000, 1/10, 10) with a single orbit.
85
Table 5.8: Orbit element sets of (7000,1/10,10) PC (unit:degree)
Sat] Mk0 Sat] Mk0
1 259.1 6 79.1
2 223.1 7 43.1
3 187.1 8 1.1
4 151.1 9 331.1
5 115.1 10 295.1
5.7 Conclusions
This chapter proposed a PC theory that has a single identical constellation pattern of target
satellites as seen by a base satellite, or of satellites with respect to the ECEF frame. To
design the identical constellation pattern, the key issue of the PC theory is finding satellite
phasing rules to obtain the orbit element sets which produce repeating space tracks in the
rotating reference frames. The satellite phasing rules are obtained from the geometrical
relations of satellite orbits and the periodic condition of parametric relative orbits.
One of the contributions of the PC theory is using a real number system to distribute node
points on the base satellite orbit plane or Earth’s equator. The use of the real number
system gives a great advantage by allowing node spacing to be mathematically distributed
through an irrational number, compared to the FC theory using an integer number system.
More importantly, the PC theory provides direct solutions to constellation design for the
repeating relative orbits while the existing FC theory requires complicated constellation
design processes in regards to the number of steps. For the design of repeating ground track
orbits, the PC theory shows an equivalent design process with the FC theory. Furthermore,
we found that the PC theory with the base satellite elliptical orbit can create repeating
space tracks using a single target satellite orbit plane. Consequently, the PC theory is an
effective design tool for the general types of the repeating space tracks: relative orbits and
ground track orbits.
Chapter 6
Satellite Relative Tracking Controls
6.1 Introduction
The majority of the studies associated with satellite tracking problems have been concerned
with developing a control system for attitude tracking maneuvers that point the satellite at
the desired target for data collection. The task of the control system is to orient the attitude
and angular velocity of a satellite with that of the target. This system has been addressed
in the previous studies [56, 57, 58, 59, 60, 61]. For the concept of formation flying, satellite
tracking control systems have also been developed with coordinated attitude control of
each satellite for simultaneous pointing and tracking of a target [62, 63, 64].
In this chapter, instead of focusing on attitude tracking maneuvers of satellites, we are con-
cerned with relative tracking control systems that point and track the payloads of satellites
to establish inter-satellite links. Thus, the payloads mounted in the body-fixed frame of
satellites are simultaneously aligned. To develop this tracking control system we must as-
sume that the exact attitude, position and velocity of the satellites is known. For payload
to payload tracking maneuvers, a reference trajectory must first be established. For the
reference trajectory, we use the GROM solution which provides the exact relative position
and velocity without perturbations. Furthermore, we propose a solution for the relative
86
87
acceleration which is required to compute the relative angular velocity and acceleration
vectors for tracking.
To develop the relative tracking control system, we use a sliding mode control scheme.
Since attitude control systems involve nonlinear characteristics of modeling uncertainty
and unexpected external torques, attitude tracking control is a complex task. Sliding mode
control has been successfully applied as a robust control technique for dealing with model
uncertainties [65, 66]. Therefore, the sliding mode control technique guarantees global
stability of the tracking control system for satellite-to-satellite links, where the attitude
maneuvers involve large angle slews.
Attitude coordinates using a Modified Rodrigues Parameters (MRPs) and a quaternion set
have been studied for the attitude tracking problem [65, 67]. MRPs and the quaternion
involve singularities in the kinematic equations. In designing the relative tracking control
systems for satellite-to-satellite links, we use MRPs as the attitude coordinates where a
singularity exists for 360 rotations, which is appropriate for the large angle maneuvers.
Typically, the MRPs is defined by the rotation matrix. Furthermore, this chapter suggests
another type of MRPs definition which is defined by the unit direction vectors. The
quaternion-based tracking controller using the unit direction vector has been applied to
the control system for ground target tracking on the Earth [68].
Using the sliding mode control technique along with the two types of MRPs definitions, this
chapter develops the following relative tracking controllers for satellite-to-satellite links:
Body-to-Body and Payload-to-Payload. Then, the relative tracking control systems are
compared and evaluated in terms of the convergence rate and control torque.
6.2 Representation of Reference Systems
To begin, we introduce several reference systems defined through the use of a set of three
orthogonal, right-handed unit direction vectors. In the chapter, a reference frame is labeled
88
with a script uppercase letter such as F , and its associated unit base vectors are labeled
with subscript lowercase letters such as Fi. In the notation, a capital letter B refers to
a base satellite and T represents a target satellite. The reference systems are defined as
follows:
Fi : the inertial reference frame with base vectors iFo: the orbit reference frame with base vectors oFw: the perifocal frame with base vectors wFb: the body-fixed frame with base vectors bFl: the payload frame with base vectors lFp: the reference frame defined by relative position and velocity with base vectors p.where the three unit base vectors of the reference frames are given by
i =
i1
i2
i3
o =
o1
o2
o3
w =
w1
w2
w3
(6.1)
and
b =
b1
b2
b3
l =
l1
l2
l3
p =
p1
p2
p3
(6.2)
To transform the components in one reference frame into another reference frame, we use
a 3 × 3 rotation matrix. The transformation between two reference frames Fo and Fb can
be seen in the following relation:
b = Rboo (6.3)
One of the fundamental properties of the rotation matrix is successive matrix-multiplications
of each rotation matrix. The composition of two rotation matrices, R and R′, can be
projected into a corresponding orthogonal matrix R′′ = R′R. Using this property, we
introduce a composite rotation matrix between base and target satellites.
RTB = RboT
RoiT
RioB
RobB
(6.4)
89
The composite rotation matrix leads to
bT
= RTBbB
(6.5)
Using Eq. (6.5), we can transform the attitude of the base satellite in the body-fixed frame
into the attitude description in target satellite coordinates.
Here, we specifically discuss about each rotation matrix in Eq. (6.4). The rotation matrix
Roi consists of the composition of two rotation matrices as follows:
Roi = RowRwi (6.6)
where Row is the transformation matrix from Fw to Fo. In Fo, the o1-axis is in the negative
nadir direction, the o3-axis is in the orbit normal direction, and the o2-axis completes the
triad and is in the velocity vector direction. The orbit reference frame Fo only rotates
by a true anomaly ν about the o3-axis relative to the perifocal frame. Thus, the rotation
matrix Row is given by
Row = R(ν) =
c ν s ν 0
−s ν c ν 0
0 0 1
(6.7)
where s and c denote sine and cosine functions, respectively. The rotation matrix Rwi is
written in terms of the R313 Euler angles:
Rpi = R3(ω)R1(i)R3(Ω)
=
cω cΩ − ci sω sΩ ci cΩ sω + cω sΩ si sω
−sω cΩ − ci cω sΩ ci cω cΩ − sω sΩ si cω
si sΩ −si cΩ ci
(6.8)
where i is the inclination, ω is the argument of perigee, and Ω is the ascending node of the
satellite.
90
From Eqs. (6.7) and (6.8), we obtain the rotation matrix Roi as follows:
Roi = RowRwi
=
cu cΩ − su ci sΩ su ci cΩ + cu sΩ su si
−su cΩ − cu ci sΩ cu ci cΩ − su sΩ cu si
si sΩ −si cΩ ci
(6.9)
where u = ω + ν.
To find the kinematic differential equation in terms of the rotation matrix, let us consider
ωbob , the angular velocity vector of Fb relative to Fo expressed in Fb. The kinematic
differential equation of the rotation matrix Rbo is then found to be [60]
Rbo = −[ωbob ]×Rbo (6.10)
In the same manner, the kinematic differential equation of the composite rotation matrix
RTB can be written as
RTB = −[ωTB ]×RTB (6.11)
where ωTB is the angular velocity vector of the base satellite in target satellite coordinates.
6.3 Attitude Parameterization
A popular set of attitude coordinates for rigid bodies is the quaternion set. The use of
the quaternion set, also known as Euler parameters, for spacecraft attitude descriptions
has an advantage in that the kinematic differential equation of the quaternion can avoid
singularities. However, the quaternion set requires an extra parameter because of the
non-uniqueness. Another familiar set of attitude coordinates are the Classical Rodrigues
Parameters (CRPs) and the Modified Rodrigues Parameters (MRPs) which provide a min-
imal three parameter set through the transformation of the redundant Euler parameters.
However, in CRPs and MRPs, singularities exist for the large angle rotation of 180 and
360, respectively.
91
6.3.1 Generalized Symmetric Stereographic Parameters (GSSPs)
Euler’s principal rotation theorem states that the most general motion of a rigid body
is a single rigid rotation through a principal angle, Φ, about the principal axis, a [69].
Through the set (Φ, a) of the principal rotation vector, many sets of attitude coordinates
are produced. Using the principal rotation vector, the quaternion set is defined as [48]
q = a sinΦ
2(6.12a)
q4 = cosΦ
2(6.12b)
The quaternion set obeys the holonomic constraint:
1 = q21 + q2
2 + q23 + q2
4 (6.13)
Note that the quaternion can describe any rotational motion without singularities. How-
ever, the sets (Φ, a) and (−Φ,−a) of the principal rotation elements describe the same
orientations due to the non-uniqueness of the quaternion.
To generalize the attitude descriptions, the transformation from the quaternion to sym-
metric stereographic parameters is derived by a graphical relationship of the following
three-dimensional sphere projected onto a two-dimensional plane [70]. In Fig 6.1, a GSSP
set of the stereographic parameters is defined as
zj =d
q4 − ξqj =
1
q4 − ξqj , j = 1, 2, 3 (6.14)
where ξ is the projection point and d is the distance between the projection point and the
position of the mapping plane on the q4-axis. In Eq. (6.14), the condition of a singularity
is given by
q4 = cosΦ
2= ξ (6.15)
The singularity of the stereographic parameters is determined by the projection point on
the q4-axis.
92
Figure 6.1: Stereographic projection of quaternion
Substituting q4 into Eq. (6.14), zj can be rewritten as
zj =sin Φ
2
cos Φ2− ξ
, j = 1, 2, 3 (6.16)
The inverse transformation from the GSSP set to the quaternion is obtained by
qj =zj(Σ − ξ)
1 + z2, q4 =
Σ + ξz2
1 + z2(6.17)
where Σ is√
1 + z2(1 − ξ2). Using Eq. (6.17), the quaternion set can be expressed in terms
of Rodrigues parameters.
From the GSSP set in Eq. (6.16), CRPs and MRPs are defined as shown in Table 6.1.
Table 6.1: The Definitions of CRPs and MRPs
Attitude
parameters
Transformations
(j=1,2,3)
Expressions in terms
of Φ (Singularity)
CRPs (ξ = 0, d = 1) %j =qj
q4% = a tan Φ
2, (Φ = ±180)
MRPs (ξ = −1, d = 1) σj =qj
q4+1σ = a tan Φ
4, (Φ = ±360)
93
6.3.2 Modified Rodrigues Parameters (MRPs)
In control problems for satellite-to-satellite links, the attitude maneuvers of satellites are
performed within 360 rotation angles for pointing and tracking. For these large angle
maneuvers, the attitude coordinate set of the MRPs is appropriate for the description of
the attitude motions, with a geometric singularity at Φ = ±360. The MRP vector is given
by
σ =qj
1 + q4= tan
Φ
4aj , j = 1, 2, 3 (6.18)
The attractive advantage of the MRPs is considered with the alternate shadow set. In
Eq. (6.18), the MRP vector obviously goes singular at an angle ±360 of Φ where q4 goes
to −1. When dealing with the reversed sign of the q’s, the shadow set describing the same
physical orientation can be obtained in
σs =−qj
1 − q4=
−σj
σ2, j = 1, 2, 3 (6.19)
A switching condition for the transformation between original and shadow sets can be
chosen as the surface σT σ = 1, resulting in the magnitude of the MRP vector being
bounded between 0 ≤ |σ| ≤ 1. In this case, the principal rotation angle will be restricted
within −180 ≤ Φ ≤ 180. Using this switching condition, the MRPs provides the shortest
path of the rotation angle [42]. For example, let us consider a payload of the base satellite,
initially offset at an angle of 270 from the target satellite. Using a control law, the payload
will point and track toward the target satellite. To point the payload toward the target
satellite, however, two possible paths of rotation are involved: one short and one long.
The shadow set of the MRPs results in a shorter rotation path for the payload. With the
switching condition for the shadow set, the payload will perform a −90 maneuver instead
of a −270 maneuver, as the rotation angle.
Next we look at the rotation matrix and kinematic differential equation of the MRP vector.
The rotation matrix in terms of the MRP vector is expressed as [48]
R = [I3×3] +8[σ]2 − 4(1 − σ2)[σ]
(1 + σ2)2(6.20)
94
where σ is the skew-symmetric tilde matrix of σ. The MRP kinematic differential equation
in vector form is written as follows:
σ =1
4
[
(1 − σ2)[I3×3] + 2[σ] + 2σσT]
ω =1
4[S(σ)]ω (6.21)
Note that the inverse transformation of Eq. (6.21) can be defined, thus ω is expressed in
terms of σ and σ.
6.4 Relative Angular Velocity and Acceleration Vec-
tors
When developing the satellite relative tracking control system between satellites, relative
angular velocity and acceleration vectors ωr and ωr, which can be obtained by the relative
movements of the satellites, are required. In Chapter 3 , we developed the exact analytic
formula of satellite relative motions representing relative position and velocity vectors r
and v. This section proposes the relative acceleration vector a of the target satellite as
seen by the base satellite. Using the proposed a with r and v, the desired ωr and ωr can
be obtained.
Taking the derivative of the vector v, the computation of a is straightforward. The resulting
acceleration vector a is derived by
a =
cos δ cosα(
rT− r
T(δ2 + α2)
)
− sin δ cosα(rTδ + 2r
Tδ) − cos δ sinα(r
Tα+ 2r
Tα)
+ sin δ sinα(2rTαδ) − r
B
cos δ sinα(
rT− r
T(δ2 + α2)
)
− sin δ sinα(rTδ + 2r
Tδ) + cos δ cosα(r
Tα + 2r
Tα)
− sin δ cosα(2rTαδ)
rT
sin δ + 2rTδ cos δ + r
T(δ cos δ − δ2 sin δ)
(6.22)
95
where the second derivatives of the angles α and δ and the parameters r and ν are
α = cos iR
sec2 δ(2δνT
tan δ + νT) − ν
B(6.23a)
δ =(
−2rT
rT
− (α + νB) tan(α + ν
B+ ω
B− φ
B))
δ (6.23b)
rj =µej
r2j
cos νj j = B, T (6.23c)
νj = − µ
r3j
2ej sin νj (6.23d)
To derive ωr and ωr, the unit direction vector p of a target satellite with respect to a base
satellite is required. The unit vector p is defined by the vector r as follows:
p =r
|r| (6.24)
and the derivative of p is given by
˙p =1
|r|(
v − (pT v)p)
(6.25)
Now, we have p and ˙p in terms of r and v, and the kinematic differential equation satisfied
by p is found to be [48]
˙p = ωr × p (6.26)
We apply the Hamiltonian principle in Eq. (6.26), and Eq. (6.26) can be expanded as
˙p1 = −ωr3p2 + ωr2
p3 (6.27a)
˙p2 = ωr3p1 − ωr1
p3 (6.27b)
˙p3 = −ωr2p1 + ωr1
p2 (6.27c)
In Eq. (6.27), each component of ωr is not uniquely specified. Thus, we can use a constraint
that minimizes the amplitude of ωr, and a cost function J can be chosen as
J =1
2mωT
r ωr (6.28)
96
where m is a positive constant. Substituting Eqs. (6.27) and (6.28) into the Hamilton-
Jacobi-Bellman equation [71], we find that the Hamiltonian H is
H =1
2mωT
r ωr + λ1( ˙p1 + ωr3p2 − ωr2
p3)
+λ2( ˙p2 − ωr3p1 + ωr1
p3) + λ3( ˙p3 + ωr2p1 − ωr1
p2) (6.29)
Thus, a necessary condition is satisfied with ∂H/∂ωr = 0:
∂H
∂ωr1
= mωr1+ p3λ2 − λ3p2 = 0 (6.30)
∂H
∂ωr2
= mωr2− λ1p3 + λ3p1 = 0 (6.31)
∂H
∂ωr3
= mωr3+ λ1p2 − λ2p1 = 0 (6.32)
From Eqs. (6.30) and (6.31), we can then obtain λ1 and λ2:
λ1 =mωr2
+ λ3p1
p3(6.33a)
λ2 =−mωr1
+ λ3p2
p3(6.33b)
Substituting λ1 and λ2 into Eq. (6.32), we find
ωr1p1 + ωr2
p2 + ωr3p3 = 0 (6.34)
Equation (6.34) leads to the following relation:
p • ωr = 0 (6.35)
Combining Eqs. (6.26) and (6.35), the relative angular velocity ωr as a function of unit
direction vectors is obtained by
ωr = p × ˙p (6.36)
Taking the time derivative of Eq. (6.36), the resulting relative angular acceleration vectors
ωr are expressed as
ωr = − 1
|r|(
2(pTv)ωr − p × a)
(6.37)
Since we use the exact solutions of satellite relative motion in the absence of perturbations,
Eqs. (6.36) and (6.37) provide an exact reference trajectory of relative angular velocity and
acceleration vectors for tracking problems between satellites.
97
6.5 Transformation of Equations of Motion
The design of the relative tracking control system can be considered with the attitude
description relative to Fo instead of Fi. This section transforms the Euler’s equations of
motion in Fi into the equations in Fo. Let I be the rigid body inertia matrix, and u be
some unconstrained control torque vector. We assume that the payload frame Fl is aligned
with the body-fixed frame Fb.
Typically, the equations of motion for a rigid body in Fi, without some unknown external
torque acting on the rigid body, are defined as
Iωbib = −[ωbi
b ]×Iωbib + u (6.38)
where ωbib is the angular velocity of Fb with respect to Fi expressed in Fi. The relationship
of the angular velocity vectors between Fo and Fi can be written by
ωbib = ωbo
b + ωoo3 (6.39a)
ωbib = ωbo
b + ωoo3 = ωbob + ωo[o3]
×ωbob (6.39b)
where ωo is the magnitude of the angular rate of Fo and ωoo3 can be written in
ωoo3 = Rboωo =
Rbo11 Rbo
12 Rbo13
Rbo21 Rbo
22 Rbo23
Rbo31 Rbo
32 Rbo33
0
0
ωo
, i = 1, 2, 3 (6.40)
Thus, ωoo3 is the angular rate vector of Fo with respect to Fi. Assuming a circular orbit
of the base satellite, the derivative of ωoo3 is obtained by
ωbib = ωbo
b + ωoo3 = ωbob + ωo[o3]
×ωbob (6.41)
The transformation of the Euler’s equations of motion into Fo is given by
I(ωbob + ωo[o3]
×ωbob ) = −[ωbo
b + ωoo3]×I(ωbo
b + ωoo3) + u (6.42)
98
Finally, the Euler’s rotational equations of motion of Fb with respect to Fo are obtained
as
Iωbob = −[ωbo
b + ωoo3]×I(ωbo
b + ωoo3) − ωoI[o3]×ωbo
b + u (6.43)
However, the inertia matrix I must be commonly considered with an unmodeled inertia.
We define a nominal inertia matrix I, and then the equations of motion with I can be
rewritten as [72]
ωbob = I−1
(
−[ωbob + ωoo3]
×I(ωbob + ωoo3) − ωoI[o3]
×ωbob
)
+I−1(u + δ) (6.44)
where δ represents the estimated modeling error. Using Eqs. (6.43) and (6.44), the uncer-
tainty dynamics δ is obtained by
δ = [ωbob + ωoo3]
×I(ωbob + ωoo3) + ωoI[o3]
×ωbob
−II−1(
[ωbob + ωoo3]
×I(ωbob + ωoo3) + ωoI[o3]
×ωbob
)
+(II−1 − 1)u (6.45)
Note that the uncertainty δ is a piecewise continuous function in time.
6.6 Design of Sliding Mode Tracking Controller
In this section, we develop the tracking controller of a base satellite to track the reference
trajectory of a target satellite using the sliding mode scheme. For simplicity, we assume
that the payload frame Fl of the base satellite is aligned with Fb, thus the payload is
mounted along one axis in Fb. Based on this proposed tracking controller, we will develop
two types of relative tracking control systems in the next section.
6.6.1 Dynamics and Kinematics for Satellite Tracking Problem
Let us consider a payload b1 fixed in the x-axis of Fb of the base satellite, and a relative
trajectory r and v of the target satellite as seen by the base satellite. A tracking controller
99
of the base satellite can be developed allowing the payload to point and track the relative
trajectory of the target satellite as a point mass. To develop this tracking controller, two
different MRP definitions can be applied. In this particular section, we are concerned with
the most common type of MRP vector which is defined by rotation matrices.
Using the GROM solution, the relative trajectory of the target satellite, as seen by the
base satellite, is obtained. Then, a reference frame, Fp, which is an orthogonal coordinate
system determined by r and v, is defined as follows:
p1 =r
|r| , p3 =r × v
|r × v| , p2 = p3 × p1 (6.46)
Figure 6.2: Two rotating reference frames in the base satellite coordinate system
Figure 6.2 shows Fp along with an additional frame Fb, and the two rotating reference
frames define a rotation matrix Rbp. Using Euler’s principal rotation theorem, the rotation
matrix Rbp is expressed in terms of the principal rotation components of a and Φ. By the
inverse transformation of the rotation matrix, the principal Euler axis, a, and Euler angle,
100
Φ, can be obtained by [48]:
cos Φ =1
2
(
Rbp11 +Rbp
22 +Rbp33 − 1
)
(6.47a)
a =
a1
a2
a3
=1
2 sin Φ
Rbp23 −Rbp
32
Rbp31 −Rbp
13
Rbp12 −Rbp
21
(6.47b)
Next, we define the MRP vector σ in terms of the principal Euler axis, a, and Euler angle,
Φ as follows.
σ = a tanΦ
4(6.48)
Thus, the MRP vector σ measures the attitude error of Fb with respect to Fp. Achieving
a zero MRP vector means that the two reference frames are aligned.
Typically, the MRP vector σ is used to measure the attitude error in the regular problem
of attitude feedback control laws where a rigid body is stabilized about the zero attitude
orientation. However, if we develop feedback control laws for tracking problems, the MRP
vector measures the attitude error of a rigid body to some reference trajectory which is
defined through the relative angular velocity, ωr. In this case, involving a tracking problem,
the MRP rate vector σ, with an angular velocity error vector δω, is written by
σ =1
4[(1 − σ2)I + 2[σ]× + 2σσT ]δω =
1
4[S(σ)]δω (6.49)
where δω is defined as
δω = ωbob −Rboωro
o (6.50)
The derivative of δω as seen by the body frame is given by
δω = ωbob − ωro
b + ωbob
×ωrob (6.51)
For the satellite tracking problem, the dynamic equations of motion in Eq. (6.43), after
substituting in Eq. (6.51), can be transformed into the following equations:
δω = I−1[
−[ωbob + ωoo3]
×I(ωbob + ωoo3) − ωoI[o3]
×ωbob
−Iωrob + Iωbo
b×ωro
b + u]
(6.52)
101
For simplicity, we rewrite Eq. (6.52) in the following form:
δω = I−1(Ωbb + Ωr
b + u) (6.53)
where
Ωbb = −[ωbo
b + ωoo3]×I(ωbo
b + ωoo3) − ωoI[o3]×ωbo
b (6.54a)
Ωrb = −Iωro
b + Iωbob
×ωrob (6.54b)
Note that the terms ωrob and ωbo
b×ωro
b represent the dynamics system of relative trajectory
and the cross coupling term, respectively. If a control law is developed to stabilize the rigid
body about a zero attitude orientation in Fo, the term Ωrb will be zero.
Next, we consider a total inertia matrix I and nominal inertia matrix I of the system,
then the dynamic equations can be rewritten as
δω = I−1(Ωbb+ Ωr
b) + I−1(u + δ) (6.55)
Using Eqs. (6.53) and (6.55), the uncertainty dynamics δ is obtained by
δ = −(Ωbb+ Ωr
b) + II−1(Ωb
b+ Ωr
b) + (II−1 − 1)u (6.56)
Equations (6.55) and (6.56) are the regular forms of dynamics equations for tracking prob-
lems in this chapter.
6.6.2 Stabilizing the MRP Kinematics
We choose the positive definite function as a candidate storage function to derive asymp-
totically stabilizing feedback for the MRP vector σ subsystem:
V (σ) = 2 log(1 + σT σ) (6.57)
Recall that the MRP differential kinematic equation is given by
σ =1
4[(1 − σ2)I + 2[σ]× + 2σσT ]δω =
1
4[S(σ)]δω (6.58)
102
We choose a direct control law in which the kinematic subsystem σ output is strictly
passive from u:
δω = −kpσ + u (6.59)
Then the storage function allows us to show that the kinematic system is asymptotically
stabilizing as follows:
V (σ) = σT δω
= σT (−kpσ + u)
= −kpσT σ + σT u
≤ −kp||σ||2 + σT u (6.60)
Therefore, the system is output strictly passive according to the Lemma 6.5 in Ref. [72],
and the origin is globally asymptotically stable.
Thus, we can choose the function φ(σ) for stabilizing the σ subsystem:
φ(σ) = −kpσ (6.61)
where kp is a scalar gain.
6.6.3 Stabilizing the Full System
Developing an asymptotically stabilizing tracking control law implies that both σ and δω
go to zero. Using the state vectors σ and δω, the sliding manifold can be chosen as
s = δω + kpσ (6.62)
Thus, the control vector u drives σ and s to zero in finite time and to maintain the sliding
surface s = 0. Figure 6.3 describes the geometry of sliding mode control. To maintain a
s = 0 surface, the sliding mode control law consists of two phase dynamics: reaching phase
and sliding phase. In a reaching phase, the dynamic system is driven to stabilize a sliding
manifold (s=0), then the trajectory moves on the sliding manifold in a sliding phase.
103
Figure 6.3: Geometry of sliding mode control
To develop a control law, the dynamic equation s with Eqs. (6.55) and (6.58) can be
expressed as
s = I−1(Ωbb+ Ωr
b) + I−1(u + δ) +
kp
4[S(σ)]δω (6.63)
Assuming the uncertainty term δ = 0, we obtain the equivalent control vector which cancels
the nominal terms. Thus, we have
ueq = −(Ωbb+ Ωr
b) − kp
4I[S(σ)]δω (6.64)
Let us define u as
u = ueq + I v (6.65)
Substituting Eq. (6.65) into Eq. (6.63), the dynamic equation s is expressed in the following
form:
s = v + ∆(σ, δω, v) (6.66)
where
∆(σ, δω, v) = I−1(
Ωbb+ Ωr
b
)
−I−1(Ωbb+ Ωr
b) + (I−1 − I−1)I
(
−kp
4[S(σ)]δω + v
)
(6.67)
104
We define the following unmodeled inertia matrix:
∆I = I − I (6.68)
Note that these matrices are commonly estimated as diagonal. Then the uncertainty
dynamic ∆(σ, δω, v) can be rewritten by
∆(σ, δω,v) =∆I
I
(
−[ωbob + ωoo3]
×(ωbob + ωoo3) − ωo[o3]
×ωbob − ωro
b + ωbob
×ωrob
)
+∆I
I
(
−kp
4[S(σ)]δω + v
)
(6.69)
Using the spectral norm of a matrix and the Euclidean norm of a vector, we can set the
the following bounds:
||∆(σ, δω,v)|| ≤ Σ + k||v|| (6.70)
where
Σ = a||ωbob + ωoo3||2 + b||ωo[o3]|| ||ωbo
b || + c||ωrob || + d||ωbo
b || ||ωrob || + e||δω|| (6.71)
Also where the constants a, b, c, d, e, and k are positive values.
A candidate Lyapunov function to be on the sliding phase can be set as
V =1
2sT s ≥ 0 (6.72)
Treating Vj = 12s2
j (j = 1, 2, 3) of V separately, we obtain
Vj = sj sj (6.73)
≤ sj vj + |sj|(
Σ + k||v||)
(6.74)
We then choose
vj = − Σ
1 − ksign(sj) (6.75)
where
Σ ≥ Σ + b0 (6.76)
105
Then, substituting Eq. (6.75) into Eq. (6.74) gives
Vj ≤ −b0|sj| (6.77)
As a result, the Lyapunov rate function Vj is zero for the sliding manifold s = 0 and
always negative for s 6= 0. Thus, the tracking control law u is globally and asymptotically
stabilizing.
Next, we need to consider the chattering problem due to imperfections in switching delays.
To minimize the chattering in the control torques, the signum function is replaced by the
saturation function. Thus the sliding mode control for uncertainty dynamics are
v = − 1
1 − k(a||ωbo
b + ωoo3||2 + b||ωo[o3]|| ||ωbob || + c||ωro
b ||
+d||ωbob || ||ωro
b || + e||δω||+ b0)sat(s
ε
)
(6.78)
where b0 > 0.
In summary, the desired tracking controller u which consists of the equivalent and sliding
mode control vectors are written as
u = ueq + us (6.79)
where
ueq = −(Ωbb+ Ωr
b) − kp
4I[S(σ)]δω (6.80a)
us = − I
1 − k(a||ωbo
b + ωoo3||2 + b||ωo[o3]|| ||ωbob || + c||ωro
b ||
+d||ωbob || ||ωro
b || + e||δω|| + b0)sat(s
ε
)
(6.80b)
In Eq. (6.79), the proposed tracking control law is not required to be a small uncertainty.
The tracking controller will only be limited by the practical constraint of control torques
regardless of the size of the modeling uncertainties.
106
6.7 Satellite Relative Tracking Controls
In this section we develop two types of satellite relative tracking control based on the
proposed tracking control law with the different definitions of MRPs. In general, the MRP
vector can be defined by the rotation matrix which represents attitudes from one reference
frame to another reference frame. With this MRP definition, the proposed relative tracking
control is called Body-to-Body (B-B) relative tracking control. Moreover, the definition of
MRPs can also be defined by a unit direction vector. The control system developed using
this type of MRP definition is called Payload-to-Payload (P-P) relative tracking control.
At the end of the section, we will compare these two types of relative tracking controls in
terms of convergence rates and control torques for satellite-to-satellite links.
6.7.1 Body-to-Body Relative Tracking Control
The purpose of satellite relative tracking control systems is to align the two payloads of a
base and target satellite. To achieve this objective, we are first concerned with the B-B
relative tracking control. The MRP definition for this control system involves orthogonal
reference frames that are defined by the relative trajectory of the target satellite and the
commissioned payload frame of the base satellite.
MRP vector by rotation matrix
The three orthogonal, right-hand unit direction vectors of a rigid body can be described
using displacements of body-fixed reference frames. Let us consider an arbitrary fixed
payload l1 in Fb. In Fig 6.11, the payload frame Fl, which is the orthogonal reference
frame defined by the payload, can be rotated by the (3-2) Euler angle sequence from Fb.
107
Figure 6.4: Rotations from Fb to Fp
The (3-2) Euler angle sequence for the transformation is written as
Rlb = R2(θ2)R3(θ1) =
cos θ2 cos θ1 cos θ2 sin θ1 − sin θ2
− sin θ1 cos θ1 0
sin θ2 cos θ1 sin θ2 sin θ1 cos θ2
(6.81)
From the preceding section, we have the reference frame Fp determined by the relative
position and velocity of the target satellite as seen by the base satellite. Using these two
reference frames Fl and Fp, we can establish a rotation matrix from Fl to Fp:
Rlp = RlbRboRop (6.82)
The rotation matrix Rlp describes the attitude of Fl relative to Fp. Using the formulae of
a and Φ in Eq. (6.47), we can define the MRP vector σ that measures the attitude error
of the payload frame Fl relative to the reference frame Fp. Thus, achieving a zero MRP
vector means that the frame Fl is aligned with the frame Fp.
108
Base and target satellite tracking controller
For the links between satellites, the payload frame Fl of the base satellite must be aligned
with the reference frame Fp related to the relative movements of the target satellite, as
shown in Fig 6.5.
Figure 6.5: Diagram of B-B relative tracking control
The following MRP vector can be used for the attitude representation of the base satellite
tracking controller:
σB
= eB
tanΦ
B
4(6.83)
For tracking the frame Fp of the target satellite, the angular velocity error δω of Fl with
respect to the angular velocity of Fp defined through ωroo is given by
δω = ωlol −Rloωro
o (6.84)
where ωlol is the angular velocity of Fl relative to Fo expressed in Fl, and Rlo is the rotation
matrix from Fo to Fp. Using the state vectors of σB
and δω, the sliding manifold can be
chosen as follows:
s = δω + kpσB(6.85)
where the parameter kp is a positive scalar gain.
The resulting base satellite tracking controller is expressed as
u = ueq + us (6.86)
109
with each control vector given by
ueq = −(Ωll+ Ωr
l) − kp
4I[S(σ)]δω (6.87a)
us = − I
1 − k(a||ωlo
l + ωoo3||2 + b||ωo[o3]|| ||ωlol || + c||ωro
l ||
+d||ωlol || ||ωro
l || + e||δω|| + b0)sat(s
ε) (6.87b)
where
Ωll
= −[ωlol + ωoo3]
×I(ωlol + ωoo3) − ωoI[o3]
×ωlol (6.88a)
Ωrl
= −Iωrol + Iωlo
l×ωro
l (6.88b)
The base satellite tracking controller above tracks the reference trajectory of the target
satellite represented by Fp. For the satellite-to-satellite links, the target satellite tracking
controller simultaneously aligns the Fl of the target satellite with Fl of the base satellite, as
seen in Fig 6.5. We transform Fl of the base satellite into the target satellite coordinates.
The following composite rotation matrix can be defined for the transformation:
RTB = Rlb
TRbo
TRoi
TRio
BRob
BRbl
B(6.89)
Using the composite rotation matrix, the MRP vector of the target satellite is expressed
as
σT
= aT
tanθ
T
4⇐ RTB = Rlb
TRbo
TRoi
TRio
BRob
BRbl
B(6.90)
The relative angular velocity in target satellite coordinates is defined as
ωTB = ωT
l − RTBωB
l (6.91)
where ωB
l and ωT
l are the payload angular velocities ωlol of the base and target satellites,
respectively. Thus, ωTB is the angular velocity error of the payloads in target satellite
coordinates. Using the MRP vector and relative angular velocity, the sliding manifold
(s = 0) can be chosen as:
s = ωTB + kpσT(6.92)
110
The target satellite tracking controller is obtained by
u = ueq + us
with each control vector given by
ueq = −(ΩT
l+ ΩB
l) − kp
4I[S(σ
T)]ωTB (6.93a)
us = − I
1 − k(a||ωT
l + ωoo3||2 + b||ωo[o3]|| ||ωT
l || + c||ωB
l ||
+d||ωT
l || ||ωB
l || + e||ωTB || + b0)sat(s
ε) (6.93b)
where
ΩT
l= [ωT
l + ωoo3]×I(ωT
l + ωoo3) + ωoI[o3]×ωT
l (6.94a)
ΩB
l= IRTB ωB
l − IωT
l × RTBωB
l (6.94b)
Note that the base and target satellite tracking controllers operate simultaneously in a
closed-loop feedback system.
Numerical Simulations
This section demonstrates a numerical example of B-B relative tracking control. The
parameter values for the numerical simulation are shown in Tables 6.2 and 6.3.
Table 6.2: Orbit elements of the base and target satellites
Satellites a(km) e i(deg) Ω(deg) ω(deg) M0(deg) period(sec)
Base 7000 0.0 10.0 0.0 0.0 10.0 20
Target 8000 0.0 15.0 0.0 0.0 12.0 20
The objective of B-B relative tracking control is to link the payload frames of two satellites
together. For satellite to satellite links, the initial orientation of the satellites is critical,
because the actuator capacity for the control torques is commonly limited. If a base satellite
111
Table 6.3: Parameter values for numerical simulation
Parameter Values Units
I1/I1 15/5 kg · m2
I2/I2 10/5 kg · m2
I3/I3 12/5 kg · m2
ωlol (t0) [0.0 0.0 0.0] rad/s
Pitch, Roll, Yaw [0.0 0.0 0.0] deg
a, b, c, d, e, k 0.5 kg · m2/s2
kp 1.0 kg · m2/s
with an arbitrary initial orientation is immediately commanded to point and track a target
satellite, the limitation can cause a failure of the control system, because a large-angle slew
maneuver may be required. Therefore, a pre-maneuver will be required to coarsely align
the payload of the satellite in the direction of the target satellite in order to reduce the
initial control effort. An example of a pre-maneuver can be seen in Ref. [68], showing a
study of ground target tracking on the Earth.
When examining the relative tracking control between satellites, we can choose from various
scenarios. In this numerical example, we assume that a pre-maneuver of the satellites will
be performed before the tracking controllers are commanded. Thus, the payload frames
of the base and target satellites will be nearly aligned. We expect that the payload of the
target satellite will show a fast convergence when aligning with the payload of the base
satellite.
Figure 6.6 shows the history of the magnitudes of the MRP vector, angular velocity error,
and sliding manifold. As expected, the target satellite tracking controller shows a fast
convergence while the base satellite tracking controller is tracking the reference trajectory
of the target satellite, which is determined by the relative position and velocity vectors.
Figure 6.7 shows the pitch, roll, and yaw angles during the tracking maneuvers of the base
112
and target satellite. Note that the trajectories of the Euler angles describe the attitude
angles relative to each orbit reference frame of the base and target satellite. Since the
payloads of both satellites are coarsely pointing toward the opposite payload, the pitch
and roll angles show small rotations for tracking. However, the yaw angle rotates from 0
initially to around 160 to align with the reference frame Fp of the target satellite.
0 5 10 15 200
0.5
1||σ
||
0 5 10 15 200
0.5
1
||δω|
|
0 5 10 15 200
0.5
1
t (seconds)
||s||
BaseTarget
Figure 6.6: B-B relative tracking control simulation (||σ||, ||δω||, ||s||)
0 5 10 15 20−50
0
50
100
150
200
t (seconds)
Eul
er a
ngle
s (d
eg)
Base
Base
Base
Target
Target
Target
PitchRollYaw
Figure 6.7: Time history of Euler angles
113
6.7.2 Payload-to-Payload Relative Tracking Control
In the preceding section, we discussed B-B relative tracking control using an MRP vector
defined by the rotation matrix of reference frames. This section proposes P-P relative
tracking control using an MRP vector defined by the unit direction vectors.
MRP vector by unit direction vector
Using Euler’s principal rotation, an orthogonal reference frame can be rotated from an
arbitrary initial orientation to a desired final orientation through a principal Euler axis
and Euler angle. In Fig 6.8, the reference frame Fp determined by the relative position
and velocity of the target satellite can be rotated from the frame Fl by a single rotation.
Figure 6.8: Coordinate frames of reference system
We assume that l3 in Fl is defined by an axis perpendicular to the plane of l1 and p1.
Thus, l3 can be a principal rotation axis, but we are only concerned with the axes l1 and
p1 for the alignment. It is not necessary to align with all of the three reference axes. Only
one axis alignment between satellites can be a possible choice for satellite to satellite links.
114
In this case, we can write the following matrix transformation for attitude description:
p = Rpll (6.95)
In Eq. (6.95), however, the rotation matrix cannot uniquely specify the attitude because
we are only concerned with the orientation of the payload l1 while two non-payload axes
are arbitrarily oriented. Using the principal rotation axis a and angle Φ, the orientation
description of the payload can be expressed in terms of the two unit direction vectors:
a = l3 =l1 × p1
|l1 × p1|(6.96)
and
Φ = cos−1(l1 · p1) (6.97)
Here, the unit direction vector l1 of the payload is defined by the first row vector of the
payload frame in Eq. (6.81):
l1 = [cos θ2 cos θ1 cos θ2 sin θ1 − sin θ2] (6.98)
Finally, the MRP vector expressed in terms of the principal Euler axis a and angle Φ is
given by
σ = a tanΦ
4(6.99)
Note that the MRP vector σ describes the tracking error of l1 with respect to p1.
Base and target satellite tracking controller
For satellite-to-satellite links, the P-P relative tracking control system synchronizes the
payloads of the base and target satellite. In Fig 6.9, the vectors lB
and lT
are the unit
direction vectors of the base and target satellite payloads, respectively, and the vector p1
is the unit direction vector of the target satellite as seen by the base satellite.
In the base satellite, the tracking controller aligns the payload vector lB
to the vector p1,
thus the base satellite tracking controller tracks the target satellite as a point mass. For
115
Figure 6.9: Diagram of P-P relative tracking control
this tracking controller, the MRP vector can be written as
σB
= aB
tanΦ
B
4(6.100)
with the principal rotation axis aB
and angle ΦB
given by
aB
=l
B× p1
|lB× p1|
and ΦB
= cos−1[lB· p1] (6.101)
Then, the design processes for the sliding mode tracking controller are the same as those
for the B-B relative tracking control.
Using the base satellite tracking controller, the payload of the base satellite tracks the
reference trajectory of the target satellite as a point mass. Simultaneously, a target satellite
tracking controller tracks the payload of the base satellite. Let us consider the payload
lT
fixed in the −x-axis of the body frame of a target satellite and the payload lB
fixed
in the x-axis of the body frame of a base satellite, as shown in Fig 6.9. The objective of
the target satellite tracking controller is to align the negative direction of the payload lT
with a projected payload l′B
in target satellite coordinates. Using the composite rotation
matrix, the projected payload l′B
in target satellite coordinates can be obtained by
l′B
= RTB lB
(6.102)
The principal rotation axis aT
is then defined as follows:
aT
=−l
T× l′
B
| − lT× l′
B|
(6.103)
116
and the principal rotation angle ΦT
is given by
ΦT
= cos−1[−lT· l′
B] (6.104)
Using the axis aT
and angle ΦT, the MRP vector for the target satellite tracking controller
is expressed as
σT
= aT
tanΦ
T
4(6.105)
In general, the commissioned payload such as an antenna or instrument is mounted along
one of the three axes in the body frame. For tracking or pointing, the mounted payload
axis can be achieved using two reaction wheels of non-payload axes. One example shows a
study for the ground target tracking problem of the z-axis payload using only the reaction
wheels along the x-and y-axes [68]. This study uses the quaternion defined by unit direction
vectors.
Numerical Simulations
This section examines numerical simulations for P-P relative tracking control. The pa-
rameter values of the numerical simulation are chosen to be the same as the values for
the examples of the B-B relative tracking control. Thus, the payloads of the satellites are
relatively coarsely aligned. The reason that the parameter values are chosen to be the
same is for the purpose of comparison between the two relative tracking control systems.
Figure 6.10 shows the results of the numerical simulation for the P-P relative tracking
control system. In Fig 6.10, the base and target satellites’ trajectories are shown with
respect to three different parameters as a function of time where the solid line represents
the trajectory of the base satellite and the dotted line represents that of the target satellite.
As seen, the base satellite tracking controller asymptotically stabilizes when tracking the
reference trajectory of the target satellite. As expected, due to the previous coarse align-
ment, in the case of the target satellite tracking controller, a fast convergence of payload
directions occurs.
117
Figure 6.11 shows the trajectories of the Euler angles with respect to each orbit reference
frame. Initially, the Euler angles of the satellites are zero. In other words, the payload,
body and orbit frame are aligned. During the pointing and tracking maneuvers, the pitch
angle changes the most while the roll angle is altered half as dramatically and the yaw
angle is only slightly changed.
0 5 10 15 200
0.05
0.1||σ
||
0 5 10 15 200
0.05
0.1
||δω|
|
0 5 10 15 200
0.05
0.1
t (seconds)
||s||
BaseTarget
Figure 6.10: P-P relative tracking control simulation (||σ||, ||δω||, ||s||)
0 5 10 15 20−10
−5
0
5
10
15
20
t (seconds)
Eul
er a
ngle
s (d
eg)
Base
Base
Base
Target
Target
Target
PitchRollYaw
Figure 6.11: Time history of Euler angles
118
6.8 Evaluation of Satellite Relative Tracking Controls
Using the sliding mode scheme, we have developed two types of relative tracking control:
Body-to-Body and Payload-to-Payload. The difference between the two types of control is
related to the definition of the MRP vectors.
The MRP vector defined by a rotation matrix develops B-B relative tracking control system.
In this type of control system, the base satellite tracking controller causes the payload frame
of the base satellite to track the reference frame, which is defined by the movements of
the target satellite. The two factors that affect the movements are the relative position
and velocity vectors as seen by the base satellite. While the base satellite is tracking the
reference frame defined by the target satellite, the target satellite tracking controller causes
the payload frame of the target satellite to track the payload frame of the base satellite.
Thus, during the tracking maneuvers, the two payload frames are aligned. This relative
tracking control can be a robust control technique for synchronizing the payload frame of
the base satellite with that of the target satellite. However, when the links between two
payloads is the only concern, the other two non-payload axes, in the orthogonal frame, can
be involved in unnecessary maneuvers. On the other hand, the synchronized maneuvers of
the two non-payload axes can be used as a beneficiary orientation for other objectives.
The MRP vector defined by unit direction vectors develops P-P relative tracking control
system. For this control type, the base satellite tracking controller points and tracks the
payload toward the line of sight of the target satellite as seen by the base satellite. In
the target satellite, the tracking controller causes the payload of the target satellite to
simultaneously track the payload of the base satellite. During this maneuver, a principal
Euler axis, perpendicular to the plane established by the two satellites’ payloads, works
as a rotation axis. This plane represents an optimal trajectory for a payload to align with
a desired payload. This trajectory gives a great advantage of a fast convergence rate and
less control effort for satellite-to-satellite links. However, the two non-payload axes are
unconstrained unlike the case in B-B relative tracking control. When the synchronization
119
of the two non-payload axes is not a concern, P-P relative tracking control will be more
appropriate than B-B relative tracking control.
Figures 6.12 and 6.13 illustrate the tracking errors and control torques of B-B and P-P
relative tracking controls. For the numerical simulations, the parameter values are the
same as Tables 6.2 and 6.3, and the initial orientation of the target satellite payload is
coarsely aligned with the payload frame of the base satellite. Thus, the tracking errors of
the target satellite tracking controllers are small and P-P relative tracking control shows
slightly faster convergence than the B-B relative tracking control, as seen in Fig 6.12. In
the cases of the base satellite tracking controller, the tracking error of B-B relative tracking
control is relatively large and shows a slow convergence rate. This results are due to the
fact that the non-payload axes in the payload frame perform the maneuver to align with
the reference frame defined by the movements of the target satellite.
In general, the maneuvers of satellite tracking problems require a large control torques at
the beginning of the operation. A less control effort is a critical issue when designing the
control system because the magnitude of the actuator is limited. Figure 6.13 shows the
comparison of control torques between two relative tracking controls. The initial control
torques of B-B relative tracking control are about 16 Nm and 6 Nm for the base and
target satellite tracking controllers, respectively, whereas P-P relative tracking control are
the lower initial control torques of about 8 Nm and 5 Nm for the base and target satellite
tracking controllers. In the cases of P-P relative tracking control, the initial control inputs
are dramatically decreased to nearly zero during the first 1 sec, while the B-B relative
tracking control still requires additional control efforts after 1 sec.
Consequently, for the tracking maneuvers of satellite-to-satellite links, P-P relative tracking
control is more appropriate than B-B relative tracking control in terms of the convergence
rate and control effort.
120
0 5 10 15 200
20
40
60
80
100
120
140
160
t (seconds)
Tra
ckin
g er
ror
(deg
)
BaseTarget
Body to Body Relative Control
Payload to Payload Relative Control
Figure 6.12: Comparison of the tracking errors
0 2 4 6 80
2
4
6
8
10
12
14
16
t (seconds)
Mag
nitu
de o
f con
trol
vec
tor
BaseTarget
Body to Body Relative Control
Payload to Payload Relative Control
Figure 6.13: Comparison of the control torques
121
6.9 Conclusions
In Chapter 6, we developed relative tracking control systems for satellite-to-satellite links.
For the links between satellites, the reference trajectory representing the relative move-
ments of satellites is required for tracking, and obtained from the exact solutions of the
GROM model. With this reference trajectory, we used a sliding mode control technique
to make the control system robust. The resulting control system is only limited by the
practical constraints of control torques regardless of the size of the modeling uncertainties.
Two types of relative tracking control systems are developed with different MRPs defi-
nitions determined by a rotation matrix and unit direction vector. In the case of B-B
relative tracking control, the payload frames of two satellites are simultaneously aligned.
This control system shows a slow convergence rate and more control torque for satellite-
to-satellite links. On the contrary, P-P relative tracking control is only concerned with the
alignment of the two payloads instead of the payload frames. This system provides fast
convergence rates and less control efforts compared to the B-B relative tracking control
system. Consequently, P-P relative tracking control systems are more appropriate when
dealing with the links of satellite payloads, than B-B relative tracking control systems.
Chapter 7
Conclusions and Recommendations
This chapter summarizes the conclusions of the dissertation and suggests the proposals for
future work based on the contributions.
7.1 Conclusions
Dynamics and control problems of large-scale relative motion are a complex task, compared
to the problems associated with small-scale relative motion. Thus, the problems involving
large-scale relative motion commonly rely on numerical integrations of the equations of
motion. This dissertation proposes the following analytic solutions for the analysis and
design of satellite relative motion problems.
First, we developed an exact and efficient analytic solution of satellite relative motion in
spherical coordinates, using a direct geometrical approach. With the resulting solutions, we
also derived linearized equations of motion for small-scale relative motion. The linearized
equations provide geometrical insight useful in the design of cross-track formations. The
validity of the proposed solutions is evaluated with existing analytic solutions in terms of
modeling accuracy and efficiency.
122
123
Second, the derived relative positions in Chapter 3 were converted into the general para-
metric equations of cycloids and trochoids. Using the relationship between the general
equations of the parametric curves and the derived parametric relative equations, new ob-
servations for relative motion geometry are found. One of the new findings states that the
relative motion dynamics of circular orbit cases in polar views are exactly the same as the
mathematical models of cycloids and trochoids. We also found that the number of petals
or cusps specifying parametric relative orbits can be identified as the number of vertical
tracks of a target satellite as seen by a base satellite. Furthermore, we conclude that rel-
ative orbit frequency γ and relative inclination iR
are involved in defining the parametric
relative orbits.
Third, we developed the PC theory to create repeating space tracks of target satellites as
seen by a base satellite. In this theory, the rotating reference frame uses a base satellite
orbit. When dealing with a base satellite orbit that is circular, we can distribute an infinite
number of the target satellite orbits on the base satellite orbit plane, using a real number
system for node spacing. When considering an elliptical orbit of a base satellite, we can
distribute the target satellites with a single orbit plane. The PC theory consists of satellite
phasing rules to obtain the orbit element set and closed form formulae to describe the
repeating space tracks. The satellite phasing rules provide the orbit element sets for the
following types: repeating relative orbits in the ECI and ECI′ frames, repeating ground
track orbits in the ECEF frame, and repeating space tracks with a single orbit. The
evaluation of the PC theory illustrated better performances in comparison to the existing
FC theory in terms of node spacing and constellation design process.
Fourth, we proposed relative tracking control systems using a sliding mode scheme for
satellite-to-satellite links. For the tracking problem, the analytic solutions in Chapter 3
are used to derive the relative angular velocity and acceleration representing the reference
trajectory. Two types of relative tracking controls were developed with different MRPs
definitions: Body-to-Body and Payload-to-Payload. In the numerical simulations, the
tracking control systems were examined and evaluated in terms of convergence rate and
124
control torque for appropriateness in practical applications of inter-satellite links.
The analytic solutions and tools proposed in the overall dissertation will be highly valuable
in mission analysis and design for relative motion systems involving not only a single base
and target satellite but also systems involving multiple target satellites.
7.2 Recommendations
In this section, we recommend future works to extend the findings and results of this
dissertation. Although there may exist many potential applications for the results, the
following specific suggestions represent several feasible future work.
A relative orbit design tool including a visualization tool should be developed in terms of
relative orbit frequency γ, relative inclination iR, and eccentricity e. The design tool will
allow a designer to easily determine what kind of pattern one satellite will produce as seen
from another satellite. This design tool will be important for understanding how to design
and point payloads and instruments for inter-satellite links. Furthermore, the relative orbit
design tool will provide the analysis and design for repeating ground tracks with respect
to the Earth, in terms of relative orbit frequency γ, inclination i, and eccentricity e.
Based on the relative orbit design tool, the PC theory can be extended to many potential
applications of various space missions, in particular to multiple satellite systems involving
inter-satellite links. A PC design and analysis tool should be made with three-dimensional
visualization graphics for the demonstrations of repeating relative orbits and repeating
ground tracks building on existing commercial software. Finally, the PC design and analysis
tool along with the relative orbit design tool will allow engineers and scientists to design
and analyze complex dynamics problems of satellite relative motion.
As discussed in the control portion of the dissertation, the Body-Body and Payload-Payload
relative tracking control systems each have advantages and disadvantages resulting from
their respective orientations. In B-B relative tracking control, the control system involves
125
unnecessary maneuvers of two non-payload axes when linking two payloads. In Payload-
Payload relative tracking control, the two non-payload axes are unconstrained. Thus, a
multi-axis target tracking control system can be developed that combines both a P-P and
a B-B relative tracking control system. This system begins with a tracking maneuver to
link payloads using a P-P relative tracking control system. The secondary maneuver uses a
B-B relative tracking control system and allows an arbitrary non-payload axis to be aligned
with a desired pointing direction, for example when sun tracking is required.
Appendix A
Spherical Geometry and Spherical
Coordinate System
Let us consider an object X and an arbitrary point Y on the sphere that has the north
pole Z and the origin O at the center in Fig A.1. The object X always moves on the sphere
keeping a spherical triangle 4XY Z. To describe the relationship between angles and sides
of 4XY Z, the following spherical triangle laws can be used.
The spherical law of sines states that
sinA
sin a=
sinB
sin b=
sinC
sin c(A.1)
and the spherical law of cosines for angles are that
cosA = − cosB cosC + sinB sinC cos a (A.2a)
cosB = − cosA cosC + sinA sinC cos b (A.2b)
cosC = − cosA cosB + sinA sinB cos c (A.2c)
126
127
Figure A.1: Spherical triangles and spherical coordinates on the sphere
The spherical law of cosines for sides are given by
cos a = cos b cos c+ sin b sin c cosA (A.3a)
cos b = cos a cos c+ sin a sin c cosB (A.3b)
cos c = cos a cos b+ sin a sinB cosC (A.3c)
Next, we determine the position vector of X on the sphere. The position X in the spherical
coordinates can be described with the three coordinates of the radial distance r, the azimuth
angle α, and the elevation angle δ, as shown in Fig A.1. The azimuth angle α is measured
from the reference axis x and δ is the elevation angle from the local horizon to the object.
The position vector in the spherical coordinates is transformed to the coordinates in the
rectangular system (x, y, z):
x = r cosα cos δ (A.4a)
y = r sinα cos δ (A.4b)
z = r sin δ (A.4c)
Appendix B
Unit Sphere Approach
The relative position on the unit sphere is given by [10]
∆x
∆y
∆z
= [RBRT
T− I]
1
0
0
(B.1)
where ∆x, ∆y, ∆z, are the radial, in-track, and cross-track relative position of target
satellite on the unit sphere. The direction cosine matrix, Rj , of the base and target
satellite is written by
Rj =
cuj cΩj − cij suj sΩj cij cΩj suj + cuj sΩj sij suj
−suj cΩj − cij cuj sΩj cij cuj cΩj − suj sΩj sij cuj
sij sΩj −sij cΩj cij
(B.2)
where j refers to B and T , and the letters s and c are abbreviations for sine and cosine,
respectively.
128
129
The relative positions of unit sphere approach can be expanded as
∆x = −1 + c2(0.5iB)c2(0.5i
T)c(u
T− u
B+ Ω
T− Ω
B)
+s2(0.5iB)s2(0.5i
T)c(u
T− u
B− Ω
T+ Ω
B)
+s2(0.5iB)c2(0.5i
T)c(u
T+ u
B+ Ω
T− Ω
B)
+c2(0.5iB)s2(0.5i
T)c(u
T+ u
B− Ω
T+ Ω
B)
+0.5siBsi
T
[
c(uT− u
B) − c(u
T+ u
B)]
(B.3a)
∆y = c2(0.5iB)c2(0.5i
T)s(u
T− u
B+ Ω
T− Ω
B)
+s2(0.5iB)s2(0.5i
T)s(u
T− u
B− Ω
T+ Ω
B)
−s2(0.5iB)c2(0.5i
T)s(u
T+ u
B+ Ω
T− Ω
B)
−c2(0.5iB)s2(0.5i
T)s(u
T+ u
B− Ω
T+ Ω
B)
+0.5siBsi
T
[
s(uT− u
B) + s(u
T+ u
B)]
(B.3b)
∆z = −siBs(Ω
T− Ω
B)cu
T−[
siBci
Tc(Ω
T− Ω
B) − ci
Bsi
T
]
suT
(B.3c)
The relative velocities of unit sphere approach are
∆x = −c2(0.5iB)c2(0.5i
T)s(u
T− u
B+ Ω
T− Ω
B)(ν
T− ν
B)
−s2(0.5iB)s2(0.5i
T)s(u
T− u
B− Ω
T+ Ω
B)(ν
T− ν
B)
−s2(0.5iB)c2(0.5i
T)s(u
T+ u
B+ Ω
T− Ω
B)(ν
T+ ν
B)
−c2(0.5iB)s2(0.5i
T)s(u
T+ u
B− Ω
T+ Ω
B)(ν
T+ ν
B)
−0.5siBsi
T
[
s(uT− u
B)(ν
T− ν
B) − s(u
T+ u
B)(ν
T+ ν
B)]
(B.4a)
∆y = c2(0.5iB)c2(0.5i
T)c(u
T− u
B+ Ω
T− Ω
B)(ν
T− ν
B)
+s2(0.5iB)s2(0.5i
T)c(u
T− u
B− Ω
T+ Ω
B)(ν
T− ν
B)
−s2(0.5iB)c2(0.5i
T)c(u
T+ u
B+ Ω
T− Ω
B)(ν
T+ ν
B)
−c2(0.5iB)s2(0.5i
T)c(u
T+ u
B− Ω
T+ Ω
B)(ν
T+ ν
B)
+0.5siBsi
T
[
c(uT− u
B)(ν
T− ν
B) + c(u
T+ u
B)(ν
T+ ν
B)]
(B.4b)
∆z = siBs(Ω
T− Ω
B)su
Tν
T−[
siBci
Tc(Ω
T− Ω
B) − ci
Bsi
T
]
cuTν
T(B.4c)
130
The actual relative motion between the two satellites is written by
x = rT(1 + ∆x) − r
B(B.5a)
y = rT∆y (B.5b)
z = rT∆z (B.5c)
The relative velocity vectors of the target satellite are given by
x = rT(1 + ∆x) + r
T∆x− r
B(B.6a)
y = rT∆y + r
T∆y (B.6b)
z = rT∆z + r
T∆z (B.6c)
Appendix C
Numerical Design Processes of FCs
and PCs
In this appendix, we examine the numerical design processes of PCs and FCs for the three
types of repeating space tracks: repeating ground track orbits in the ECI frame, repeating
relative orbits in the ECI′ frame, and repeating relative orbits in the ECI frame.
Repeating Ground Track Orbit in the ECI Frame
Given:
- ECEF frame (ω⊕ = 7.292115 × 10−5rad/sec)
- Orbital elements of 1st satellite in the ECI frame:
a1=20270km, e1 = 0.01, i1=15, Ω1 = 0, ω1 = 0, M10 = 0
- The number of target satellites: N = 3
- 36 evenly spaced distribution of Ωk k = 1, 2, 3
131
132
Find:
- Find the orbital elements of the satellites for repeating ground track orbit.
1) Flower Constellations
The phasing rules of the FC theory are written in the following form [73]:
Ωk = 2πFn
Fd
(k − 1), Mk0 = 2πFnNp + FdFh
FdNd
(1 − k), k = 1, 2, ..., N (C.1)
where Fn, Fd, and Fh are phasing related parameters. For the given example problem,
the design parameters are chosen as the values shown in Table C.1. Using the phasing
rules in Eq. (C.1), we obtain the orbit element set of the ascending nodes and initial mean
anomalies as follows:
Ω2 = 2π1
10(2 − 1) = 36, M20 = 2π
1 × 3 + 10 × 0
10 × 1(1 − 2) = −108 (C.2a)
Ω3 = 2π1
10(3 − 1) = 72, M30 = 2π
1 × 3 + 10 × 0
10 × 1(1 − 3) = −216 (C.2b)
Table C.1: Design parameters of FCs
Rotating frame 1st satellite Phasing parameters
Np = 3, Nd = 1
e1 = 0.01 N = 3
ECEF frame i1 = 15 Fn = 1
Ω1 = 0 Fd = 10
ω1 = 0 Fh = 0
M10 = 0
133
2) Parametric Constellations
The phasing rules of the PC theory for a repeating ground track orbit are expressed in the
following forms:
Ωk = Ω1 + θΩ(k − 1), Mk0 = −γΩk, k = 1, 2, ..., N (C.3)
Table C.2 shows the design parameters for the example problem.
Table C.2: Design parameters of PCs
Rotating frame 1st satellite Phasing parameters
γ = 3
ω⊕ → a e1 = 0.01
i = 0 i1 = 15 N = 3
Ω = 0 Ω1 = 0 θΩ = 36 (β = 0.1)
M0 = 0 ω1 = 0
M10 = 0
Using the phasing rules in Eq. (C.3), we derive the element set of the ascending nodes and
initial mean anomalies:
Ω2 = 0 + 36(2 − 1) = 36, M20 = −3 × 36 = −108 (C.4a)
Ω3 = 0 + 36(3 − 1) = 72, M30 = −3 × 72 = −216 (C.4b)
The orbit element set of ascending nodes and initial mean anomalies of PCs in Eq. (C.4)
is the same as the values of FCs in Eq. (C.2).
Repeating Relative Orbit in the ECI′ Frame
Given:
- Base satellite circular orbit:
aB=8000km, i
B=15, Ω
B= 30, M
B0= 0
134
- Relative orbital elements of 1st target satellite in the ECI′ frame:
a1=8000km, e1 = 0.01, i1=15, Ω1 = 0, ω1 = 0, M10 = 0
- The number of target satellites: N = 3
- 36 evenly spaced distribution of Ωk k = 1, 2, 3
Find:
- Find the orbital elements of the target satellites for repeating relative orbit.
1) Flower Constellations
In the ECI′ frame, we have defined the relative orbit elements [a, e, ik, Ωk, ωk, Mk0]. Ac-
cording to the FCs theory, the values of Ωk and Mk0 in the ECI′ frame are distributed
using the following sequence:
Ωk = 2πFn
Fd
(k − 1), Mk0 = 2πFnNp + FdFh
FdNd
(1 − k), k = 1, 2, ..., N (C.5)
where Fn, Fd, and Fh are phasing related parameters.
a. Design parameters
In the given example problem, the relative ascending node, Ωk, is distributed as 36 evenly
spaced values, and all of the satellites have the same ak, ek, ik, and ωk in the ECI′ frame.
The design parameters for the example problem are shown in Table C.3.
b. Orbital parameters of the base and target satellites
Table C.4 shows the orbital parameters of three target satellites for the given example
problem.
135
Table C.3: Design parameters of FCs
Base satellite 1st target satellite Phasing parameters
Np = Nd = 1
aB
= 8000km e1 = 0.01 N = 3
iB
= 15 i1 = 15 Fn = 1
ΩB
= 30 Ω1 = 0 Fd = 10
MB0
= 0 ω1 = 0 Fh = 0
M10 = 0
Table C.4: Orbital parameters of the base and target satellites
Satellites a(km) e i (deg) Ω(deg) ω(deg) M0(deg)
Base 8000 0 15 30 0 0
Target 1 8000 0.01 i1 = 15 Ω1 = 0 ω1 = 0 M10 = 0
Target 2 8000 0.01 i2 = 15 Ω2 = 36 ω2 = 0 M20 = −36
Target 3 8000 0.01 i3 = 15 Ω3 = 72 ω3 = 0 M30 = −72
c. OEECI′ =⇒ (r, v)PQW
In this section we convert the relative orbital elements into the PQW position and velocity
vectors.
Step 1: Determine the corresponding true anomalies of the initial mean anomalies using
Newton’s method. Table C.5 shows the resulting true anomalies.
Step 2: Find the PQW position and velocity vectors using the true anomalies in Table C.5.
rPQW =
p cos ν
1+e cos ν
p sin ν
1+e cos ν
0
, vPQW =
−√
µ
psin ν
√
µ
p(e+ cos ν)
0
(C.6)
136
Table C.5: True anomalies of initial mean anomalies
k Mk0(deg) νk0(deg)
1 0 → 0
2 -36 → -36.6804
3 -72 → -73.0940
where p = a(1− e2) = 7999.2. The resulting PQW position and velocity vectors are shown
in Table C.6.
Table C.6: PQW position and velocity vectors
Component r1(km) r2(km) r3(km) v1(km/sec) v2(km/sec) v3(km/sec)
x 7920.0 6364.2 2319.4 0 4.2167 6.7540
y 0 -4740.3 -7631.3 7.1296 5.7318 2.1234
z 0 0 0 0 0 0
d. (r, v)PQW =⇒ (r, v)ECI′
The PQW position and velocity vectors in Table C.6 are rotated with the following rela-
tions.
rECI′ = R313 rPQW = R3(−Ωk)R1(−i)R3(−ωk) rPQW (C.7a)
vECI′ = R313 vPQW = R3(−Ωk)R1(−i)R3(−ωk) vPQW (C.7b)
The resulting position and velocity vectors, (r, v)ECI′, in the ECI′ frame are shown in
Table C.7.
137
Table C.7: Position and velocity vectors in the ECI′ frame
Component r1(km) r2(km) r3(km) v1(km/sec) v2(km/sec) v3(km/sec)
x 7920.0 7840.1 7727.2 0 0.1571 0.1365
y 0 36.4 -71.9 6.8867 6.9576 7.0572
z 0 -1226.9 -1975.1 1.8453 1.4835 0.5496
e. (r, v)ECI′ =⇒ (r, v)ECI
Using the base satellite orbital elements, the position and velocity vectors in the ECI′ frame
are transformed into the position and velocity vectors in the ECI frame. The rotation
matrix is given by
rECI = R3(−Ω1)R1(−i1)R3(−M10) rECI′ (C.8a)
vECI = R3(−Ω1)R1(−i1)R3(−M10) rECI′ (C.8b)
Table C.8 shows the resulting position and velocity vectors of the target satellites in the
ECI frame.
Table C.8: Position and velocity vectors in the ECI frame
Component r1(km) r2(km) r3(km) v1(km/sec) v2(km/sec) v3(km/sec)
x 6858.9 6613.3 6471.1 -3.0872 -3.0332 -3.2191
y 3960.0 4225.5 4246.2 5.3472 5.5662 5.8485
z 0 -1175.6 -1926.4 3.5648 3.2337 2.3574
f. (r, v)ECI =⇒ OEECI
This section converts the position and velocity vectors (r, v)ECI into the orbital elements
in the ECI frame. The following example shows the case of the second target satellite.
138
The position and velocity vectors of the second target satellite (k = 2) are
~rECI = 6613.3I + 4225.5J − 1175.6K (C.9a)
~vECI = −3.0322I + 5.5662J + 3.2337K (C.9b)
Step 1: Begin by finding the angular momentum:
~h = ~r × ~v = 20208I − 17821J + 49624K (C.10)
The magnitude of ~h: |~h|= 56467
Step 2: Find the node vector ~n using a cross product:
~n = K ×~h = 17821I + 20208J (C.11)
The magnitude of ~n: |~n|= 26943
Step 3: Find the eccentricity vector ~e:
~e =1
µ(~v ×~h) − ~r
r
= 0.0042I + 0.0090J + 0.0015K (C.12)
The magnitude of ~e: |~e|= 0.01
Step 4: Find the inclination, i2:
cos i2 =hK
|~h|=
49624
56467= 0.8788
i2 = cos−1(0.8788) = 28.4998 (C.13)
Step 5: Find the ascending node, Ω2:
cos Ω2 =nI
|~n| =17821
26943= 0.6614
Ω2 = cos−1(0.6614) = 48.5920 (C.14)
Step 6: Find the argument of perigee, ω2:
cosω2 =~n · ~e|~n||~e| = 0.9478
ω2 = cos−1(0.9478) = 18.5920 (C.15)
139
Step 7: Find the initial true anomaly, ν20:
cos ν20 =~e · ~r|~e||~r| = 0.8019
ν20 = cos−1(0.8019) = −36.6804 (C.16)
Thus,
E20 = cos−1( e2 + cos ν20
1 + e2 cos ν20
)
= −36.3395 (C.17)
Step 7: Finally, the initial mean anomaly, M20:
M20 = E20 − e2 sinE20 = −36.0 (C.18)
In the same manner, we can compute the orbital elements of the first and third target
satellites. The resulting orbital elements are shown in Table C.9.
Table C.9: Resulting orbital elements of target satellites (FCs)
Satellites a(km) e i (deg) Ω(deg) ω(deg) M0(deg)
Target 1 8000 0.01 30 30 0 0
Target 2 8000 0.01 28.4998 48.5920 18.5920 -36.0
Target 3 8000 0.01 24.1731 66.9494 36.9494 -72.0
2) Parametric Constellations
The satellite phasing rules in terms of the relative orbital elements defined in the ECI′
frame are given by
Ωk = Ω1 + θΩ(k − 1), k = 1, 2, ..., N (C.19a)
ik = cos−1(
cos iB
cos ik − sin iB
sin ik cos(MB0
+ Ωk))
(C.19b)
Ωk = ΩB
+ sin−1(sin(M
B0+ Ωk) sin ik
sin ik
)
(C.19c)
ωk = φk + ωk (C.19d)
Mk0 = −γΩk + φk − ωk (C.19e)
140
Design parameters and computation of orbital elements
For the given example problem, the design parameters of PCs are chosen as the values
shown in Table C.10.
Table C.10: Design parameters of PCs
Base satellite 1st target satellite Phasing parameters
γ = 1
aB
= 8000km e1 = 0.01
iB
= 15 i1 = 15 N = 3
ΩB
= 30 Ω1 = 0 θΩ = 36 (β = 0.1)
MB0
= 0 ω1 = 0
M10 = 0
Using the satellite phasing rules in Eq. (C.19), we compute the orbital elements in the ECI
frame as follows (k = 2):
Ω2 = 0 + 36(2 − 1) (C.20a)
i2 = cos−1(
cos 15 cos 15 − sin 15 sin 15 cos(0 + 36))
= 28.4998 (C.20b)
Ω2 = 30 + sin−1(sin(0 + 36) sin 15
sin 28.4998
)
= 48.5920 (C.20c)
ω2 = tan−1[sin(48.5920 − 30) sin 15 sin 28.4997
cos 15 − cos 28.4997 cos 15
]
+0
= 18.5920 (C.20d)
Mk0 = −36 + 18.5920 − 18.5920
= −36 (C.20e)
In the same manner, we can compute the orbital elements of the first and third target
satellites, as shown in Table C.11. The resulting orbital elements of PCs in Table C.11 are
141
the same as the orbital elements of FCs in Table C.9.
Table C.11: Resulting orbital elements of target satellites (PCs)
Satellites a(km) e i (deg) Ω(deg) ω(deg) M0(deg)
Target 1 8000 0.01 30 30 0 0
Target 2 8000 0.01 28.4998 48.5920 18.5920 -36.0
Target 3 8000 0.01 24.1731 66.9494 36.9494 -72.0
Repeating Relative Orbit in the ECI frame
Given:
- Base satellite circular orbit:
aB=8000km, i
B=15, Ω
B= 30, M
B0= 0
- Orbital elements of 1st target satellite in the ECI frame:
a1=8000km, e1 = 0.01, i1=30, Ω1 = 30, ω1 = 0, M10 = 0
- The number of target satellites: N = 3
- 36 evenly spaced distribution of Ωk k = 1, 2, 3
Find:
- Find the orbital elements of the target satellites for repeating relative orbit.
1) Flower Constellations
For the given problem of the repeating relative orbit in the ECI frame, the phasing rules
of existing FCs are unable to directly use in order to obtain orbit element sets. To apply
142
the phasing rules to the given problem, the orbit elements of 1st target satellite must be
transformed into the relative orbit elements using a geometrical approach. The existing
FCs then obtain the desired orbit element sets through the same design procedures in the
ECI′ frame. In this case, the existing FCs are involved with more complicated constellation
design process, compared to the design process in the ECI′ frame.
2) Parametric Constellations
The satellite phasing rules in terms of orbital elements are given by
Ωk = Ω1 + θΩ(k − 1), k = 1, 2, ..., N (C.21a)
ik = cos−1( cos i
R√
1 − sin2 iB
sin2 ∆Ωk
)
+ tan−1(tan iB
cos ∆Ωk) (C.21b)
ωk = φk + ωR
(C.21c)
Mk0 = γ(MB0
− φ1(k)) − ωR
(C.21d)
Design parameters and computation of orbital elements
For the given example problem, the design parameters are shown in Table C.12.
Table C.12: Design parameters of PCs
Base satellite 1st target satellite Phasing parameters
γ = 1
aB
= 8000km e1 = 0.01
iB
= 15 i1 = 30 N = 3
ΩB
= 30 Ω1 = 30 θΩ = 36 (β = 0.1)
MB0
= 0 ω1 = 0
M10 = 0
Using the satellite phasing rules in Eq. (C.21), we can directly obtain the orbital elements
143
as follows (k = 2):
Ω2 = 30 + 36 = 66 (C.22a)
i2 = cos−1( cos 15√
1 − sin2 15 sin2(66 − 30)
)
+ tan−1(tan 15 cos(66 − 30))
= 24.4622 (C.22b)
ω3 = tan−1[sin(66 − 30) sin 15 sin 24.4622
cos 15 − cos 24.4622 cos 15
]
+0
= 36 (C.22c)
M20 = (0 − tan−1[sin(66 − 30) sin 15 sin 24.4622
− cos 24.4622 + cos 15 cos 15
]
) − 0
= 289.8787 (C.22d)
In the same manner, we can compute the orbital elements of the first and third target
satellites, as shown in Table C.13.
Table C.13: Resulting orbital elements of target satellites (PCs)
Satellites a(km) e i (deg) Ω(deg) ω(deg) M0(deg)
Target 1 8000 0.01 30 30 0 0
Target 2 8000 0.01 24.4622 66 36 289.8787
Target 3 8000 0.01 9.4667 102 72 217.1840
For the design of the repeating relative orbit in the ECI frame, the satellite phasing rules
of PCs are direct solutions to obtain the desired orbit elements, while FCs requires many
additional design steps.
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