Study of the Ballistic Capabilities of a Tethered Satellite System
by
Jennifer Marie Grant
A thesis submitted to the Graduate Faculty of
Auburn University
in partial fulfillment of the
requirements for the Degree of
Master of Science
Auburn, Alabama
August 4, 2012
Copyright 2012 by Jennifer Marie Grant
Approved by
David Cicci, Chair, Professor of Aerospace Engineering
Andrew Sinclair, Associate Professor of Aerospace Engineering
George Flowers, Professor of Mechanical Engineering and Dean of the Graduate School
ii
Abstract
This thesis is an investigation into the ballistic capabilities of a Tethered Satellite System
(TSS) when the subsatellite is released from the system. This topic is of particular interest
because TSS could be potentially used as threats, or weapons. There is a need to determine the
velocity change or angular velocity that is required to cause a subsatellite to enter an impact
trajectory. Once the subsatellite enters an impact trajectory, the ground range covered by the
new trajectory and the time to impact are determined. A simple dumbbell model is used to
represent the TSS in a dynamical simulation. Changes to the velocity of the system were
introduced at release point in the orbit in order to cause the subsatellite to enter an impact
trajectory toward the Earth after release from the TSS. The parameters of the TSS that affect the
impact trajectories are the altitude, tether length, and release point. A comparison is then done
for changes in these parameters in order to determine the maximum and minimum ground range
and time to impact for the various cases studied. An analytical solution is also developed to
determine the maximum and minimum ranges when given a range of changes in velocity and to
find the angular velocity and velocity change necessary for a given a set of initial conditions and
desired impact trajectories.
iii
Acknowledgments
I would like to thank my advisor, Dr. Cicci, for helping me to get through graduate
school and complete this thesis. I would also like to thank Dr. Cicci and Dr. Gross for their
support during the more stressful times of my college education. Without their support I might
have given up due to all the ups and downs in life.
I would also like to thank my father, Richard K. Grant, for his continued support and his
encouragement for me to pursue an engineering degree. I will always remember those late night
calls where he is trying to help me solve homework problems, telling me that I can complete my
thesis, and telling me to remember to relax and say a little prayer. I also appreciate the support
my stepmother, Patty Caldwell Grant, gave me after she married my father.
Finally, I would like to dedicate this thesis to my mother, Maria T. Grant. She never got
to see her daughter graduate with a Bachelor?s degree in Aerospace Engineering and go after a
Master?s Degree in Aerospace Engineering; however, without her I would never have considered
going for a Masters. Thanks to her I have followed my dreams and now they have come true.
iv
Table of Contents
Abstract ....................................................................................................................................... ii
Acknowledgments ....................................................................................................................... iii
List of Tables ............................................................................................................................. vii
List of Figures .............................................................................................................................. x
Nomenclature ............................................................................................................................. xv
1. Introduction ............................................................................................................................. 1
1.1 Historical Background .............................................................................................. 1
1.2 TSS Applications ...................................................................................................... 5
1.3 Previous Research ..................................................................................................... 7
1.4 Problem Description ............................................................................................... 13
2. Relevant Theory Used in TSS Analysis ................................................................................ 16
2.1 Tethered Satellite System Dynamics ...................................................................... 16
2.1.1 Equations of Motion ................................................................................ 21
2.2 Attitude Dynamics Using Euler Parameters ........................................................... 23
2.3 Rigid Body Dynamics ............................................................................................. 26
2.4 Orbital Elements of a Trajectory ............................................................................. 29
2.5 Numerical Integration Using a 4th Order Runge Kutta ........................................... 35
2.6 Analytical Approach ............................................................................................... 37
3: Model and Problem Formulation .......................................................................................... 39
v
3.1 Problem Setup of the TSS ..................................................................................... 40
3.2 Changes Applied to the System to Create Impact of SubSatellite ........................ 50
3.3 Variations to the Problem Setup ............................................................................. 51
3.4 Analytical Solutions ................................................................................................ 53
3.4.1 Equations to Find the Range and Time to Impact .................................... 55
3.4.2 Algorithm to Find the Range and Time to Impact ................................... 59
3.4.3 Equations to Find the Velocity Change and Angular Velocity.................. 64
3.4.4 Algorithm to Find the Velocity Change and Angular Velocity ............... 66
4: Results for Analytical Solutions .......................................................................................... 68
4.1 Analytical Results for the Range and Time to Impact .............................................. 68
4.1.1 Case A Results ........................................................................................... 69
4.1.2 Case B Results ........................................................................................... 72
4.1.3 Case C Results ........................................................................................... 75
4.1.4 Case D Results ........................................................................................... 78
4.2 Analytical Results for the Impulsive Velocity Change and the Angular Velocity . 80
4.2.1 Ground Range of 1500 km and ............................................ 81
4.2.2 Ground Range of 3000 km and ............................................ 87
5: Results for Numerical Integration Simulation ..................................................................... 95
5.1 Results for First Configuration or Case 1 ................................................................ 96
5.1.1 Impulsive Velocity Change of 1.0 km/s in the Positive Y  Direction ...... 96
5.1.2 Impulsive Velocity Change of 1.0 km/s in the Positive Z  Direction ..... 104
5.1.3 Impulsive Velocity Change of 3.0 km/s in the Negative Z  Direction ... 113
5.1.4 Comparison between Velocity Changes for First Configuration ............. 121
vi
5.2 Results for Second Configuration or Case 2 ........................................................... 122
5.2.1 Impulsive Velocity Change of 1.0 km/s in the Positive Z Direction ...... 123
5.2.2 Impulsive Velocity Change of 1.5 km/s in the Negative Z  Direction ... 132
5.2.3 Impulsive Velocity Change of 1.0 km/s in the Positive Y  Direction .... 140
5.2.4 Comparison between Velocity Changes for Second Configuration ........ 149
5.3 Comparison between the Analytical Results and the Simulation Results ............ 150
6: Conclusions and Future Work ........................................................................................... 155
References ............................................................................................................................... 157
Appendix A: Sample Results from Numerical Integration ..................................................... 161
Appendix B: Sample Results from Analytical Solutions ........................................................ 164
vii
List of Tables
Table 3.1: Subsatellite Position Components for Cases A  D .................................................. 54
Table 4.1: Minimum Impulsive Velocity Change Needed for Case A ....................................... 70
Table 4.2: Maximum Ground Range and Time to Impact for Case A ....................................... 71
Table 4.3: Minimum Ground Range and Time to Impact for Case A ........................................ 71
Table 4.4: Minimum Impulsive Velocity Change Needed for Case B ....................................... 74
Table 4.5: Maximum Ground Range and Time to Impact for Case B ........................................ 74
Table 4.6: Minimum Ground Range and Time to Impact for Case B ........................................ 75
Table 4.7: Minimum Impulsive Velocity Change Needed for Case C ....................................... 77
Table 4.8: Maximum Ground Range and Time to Impact for Case C ........................................ 77
Table 4.9: Minimum Ground Range and Time to Impact for Case C ........................................ 77
Table 4.10: Minimum Impulsive Velocity Change Needed for Case D ..................................... 79
Table 4.11: Maximum Ground Range and Time to Impact for Case D ..................................... 80
Table 4.12: Minimum Ground Range and Time to Impact for Case D ...................................... 80
Table 4.13: Impulsive Velocity Change and Angular Velocity for Case A and B at Three
Altitudes When R = 1500 km and ?imp = 270? ........................................................ 83
Table 4.14: Impulsive Velocity Change and Angular Velocity for Case A and B at Three Tether
Lengths When R = 1500 km and ?imp = 270? ......................................................... 85
Table 4.15: Impulsive Velocity Change and Angular Velocity for Case C and D at Three
Altitudes When R = 1500 km and ?imp = 270? ........................................................ 86
Table 4.16: Impulsive Velocity Change and Angular Velocity for Case C and D at Three Tether
Lengths When R = 1500 km and ?imp = 270? ......................................................... 87
viii
Table 4.17: Impulsive Velocity Change and Angular Velocity for Case A and B at Three
Altitudes When R = 3000 km and ?imp = 270? ........................................................ 89
Table 4.18: Impulsive Velocity Change and Angular Velocity for Case A and B at Three Tether
Lengths When R = 3000 km and ?imp = 270? ......................................................... 91
Table 4.19: Impulsive Velocity Change and Angular Velocity for Case C and D at Three
Altitudes When R = 3000 km and ?imp = 270? ........................................................ 92
Table 4.20: ImpulsiveVelocity Change and Angular Velocity for Case C and D at Three Tether
Lengths When R = 3000 km and ?imp = 270? ......................................................... 94
Table 5.1: Impulsive Velocity Changes for Case 1 .................................................................... 96
Table 5.2: Range and Impact Time for km/s at Three Altitudes for Case 1 ............ 97
Table 5.3: Range and Impact Time for km/s at Three Tether Lengths
for Case 1 ................................................................................................................. 101
Table 5.4: Maximum and Minimum Values as Increases for Case 1 ........................... 104
Table 5.5: Range and Impact Time for km/s at Three Altitudes for Case 1 .......... 106
Table 5.6: Range and Impact for km/s at Three Tether Lengths for Case 1 .......... 109
Table 5.7: Maximum and Minimum Values as Increases for Case 1 ........................... 113
Table 5.8: Range and Impact Time for km/s at Three Altitudes Case 1 ............. 114
Table 5.9: Range and Impact Time for km/s at Three Tether Lengths
for Case 1 ............................................................................................................... 118
Table 5.10: Maximum and Minimum Values as Increases for Case 1 ......................... 121
Table 5.11: Impulsive Velocity Changes for Case 2 ................................................................ 123
Table 5.12: Range and Impact Time for km/s at Three Altitudes for Case 2 ........ 125
Table 5.13: Range and Impact Time for km/s at Three Tether Lengths
for Case 2 ............................................................................................................... 128
Table 5.14: Maximum and Minimum Values as Increases for Case 2 ......................... 132
Table 5.15: Range and Impact Time for km/s at Three Altitudes for Case 2 ..... 133
ix
Table 5.16: Range and Impact Time for km/s at Three Tether Lengths
for Case 2 ............................................................................................................... 137
Table 5.17: Maximum and Minimum Values as Increases for Case 2 ......................... 140
Table 5.18: Range and Impact Time for km/s at Three Altitudes for Case 2 ........ 141
Table 5.19: Range and Impact Time for km/s at Three Tether Lengths
for Case 2 ............................................................................................................... 145
Table 5.20: Maximum and Minimum Values as Increases for Case 2 ......................... 148
Table 5.21: Results for Configuration 1 and Case B with km/s ............................ 150
Table 5.22: Comparison of Results for Configuration 1 and Case B with km/s ... 151
Table 5.23: Results for Configuration 1 and Case B with km/s ............................ 151
Table 5.24: Comparison of Results for Configuration 1 and Case B with km/s ... 152
Table 5.25: Results for Configuration 2 and Case A with km/s ............................ 152
Table 5.26: Comparison of Results for Configuration 2 and Case A with km/s ... 152
Table 5.27: Results for Configuration 2 and Case A with km/s ............................ 153
Table 5.28: Comparison of Results for Configuration 2 and Case A with km/s ... 153
Table A.1: Simulation Results as Altitude Increases for Case 1 at km/s .............. 161
Table A.2: Simulation Results as Tether Length Increases for Case 1 at km/s ..... 162
Table A.3: Simulation Results as Increases for Case 1................................................. 163
Table B.1: Analytical Results as Altitude Increases for Case A When R = 1500 km and
........................................................................................................... 164
Table B.2: Analytical Results as Tether Length Increases for Case A When R = 1500 km and
........................................................................................................... 165
Table B.3: Analytical Results as Increases for Case A When H = 500 km
and = 4 km .......................................................................................................... 166
x
List of Figures
Figure 2.1: Tethered Satellite System Model ............................................................................ 17
Figure 2.2: Geocentric Equatorial Coordinate System .............................................................. 19
Figure 2.3: Body Fixed Coordinate System for a Negative ? ................................................... 21
Figure 2.4: Orbital Elements of a Trajectory ............................................................................. 30
Figure 3.1: Initial Setup of the TSS for a Negative ? ............................................................... 40
Figure 3.2: Analytical Cases ...................................................................................................... 54
Figure 4.1: Ground Range vs. for Case A ....................................................................... 69
Figure 4.2: Time to Impact vs. for Case A...................................................................... 70
Figure 4.3: Ground Range vs. for Case B ....................................................................... 73
Figure 4.4: Time to Impact vs. for Case B ...................................................................... 73
Figure 4.5: Ground Range vs. for Case C ....................................................................... 76
Figure 4.6: Time to Impact vs. for Case C ...................................................................... 76
Figure 4.7: Ground Range vs. for Case D ....................................................................... 78
Figure 4.8: Time to Impact vs. for Case D...................................................................... 79
Figure 4.9: Velocity Change vs. Altitude for Case A and B; R = 1500 km and
?imp = 240? .............................................................................................................. 82
Figure 4.10: Angular Velocity vs. Altitude for Case A and B; R = 1500 km and
?imp = 240? .............................................................................................................. 82
Figure 4.11: Velocity Change vs. Tether Length for Case A and B; R = 1500 km and
?imp = 240? .............................................................................................................. 84
xi
Figure 4.12: Angular Velocity vs. Tether Length for Case A and B; R = 1500 km and
?imp = 240? .............................................................................................................. 84
Figure 4.13: Velocity Change vs. Altitude for Case C and D; R = 1500 km and
?imp = 240? .............................................................................................................. 86
Figure 4.14: Velocity Change vs. Tether Length for Case C and D; R = 1500 km and
?imp = 240? .............................................................................................................. 87
Figure 4.15: Velocity Change vs. Altitude for Case A and B; R = 3000 km and
?imp = 240? .............................................................................................................. 88
Figure 4.16: Angular Velocity vs. Altitude for Case A and B; R = 3000 km and
?imp = 240? .............................................................................................................. 88
Figure 4.17: Velocity Change vs. Tether Length for Case A and B; R = 3000 km and
?imp = 240? .............................................................................................................. 90
Figure 4.18: Angular Velocity vs. Tether Length for Case A and B; R = 3000 km and
?imp = 240? .............................................................................................................. 90
Figure 4.19: Velocity Change vs. Altitude for Case C and D; R = 3000 km and
?imp = 240? .............................................................................................................. 92
Figure 4.20: Velocity Change vs. Tether Length for Case C and D; R = 3000 km and
?imp = 240? .............................................................................................................. 93
Figure 5.1: Ground Range vs. Altitude for km/s in Case 1 ..................................... 96
Figure 5.2: Time to Impact vs. Altitude for km/s in Case 1 .................................... 97
Figure 5.3: Trajectory at 500 km Altitude for km/s in Case 1 ................................. 98
Figure 5.4: Trajectory at 1000 km Altitude for km/s in Case 1 .............................. 99
Figure 5.5: Trajectory at 1500 km Altitude for km/s in Case 1 ............................... 99
Figure 5.6: Ground Range vs. Tether Length for km/s in Case 1 .......................... 100
Figure 5.7: Time to Impact vs. Tether Length for km/s in Case 1 ........................ 100
Figure 5.8: Trajectory at 55 km Tether Length for km/s in Case 1 ....................... 102
Figure 5.9: Trajectory at 100 km Tether Length for km/s in Case 1 ..................... 102
xii
Figure 5.10: Ground Range vs. for Case 1 .................................................................... 103
Figure 5.11: Time to Impact vs. for Case 1 .................................................................. 103
Figure 5.12: Ground Range vs. Altitude for km/s in Case 1 ................................. 105
Figure 5.13: Time to Impact vs. Altitude for km/s in Case 1 ................................ 105
Figure 5.14: Trajectory at 500 km Altitude for km/s in Case 1 ............................. 106
Figure 5.15: Trajectory at 650 km Altitude for km/s in Case 1 ............................. 107
Figure 5.16: Trajectory at 850 km Altitude for km/s in Case 1 ............................. 107
Figure 5.17: Ground Range vs. Tether Length for km/s in Case 1 ........................ 108
Figure 5.18: Time to Impact vs. Tether Length for km/s in Case 1 ....................... 109
Figure 5.19: Trajectory at 55 km Tether Length for km/s in Case 1 ..................... 110
Figure 5.20: Trajectory at 100 km Tether Length for km/s in Case 1 ................... 110
Figure 5.21: Ground Range vs. for Case 1 .................................................................... 111
Figure 5.22: Time to Impact vs. for Case 1 ................................................................... 111
Figure 5.23: Lofted Trajectory at km/s for Case 1 ................................................ 112
Figure 5.24: Ground Range vs. Altitude for km/s in Case 1 .............................. 114
Figure 5.25: Time to Impact vs. Altitude for km/s in Case 1 ............................. 114
Figure 5.26: Trajectory at 500 km Altitude for km/s in Case 1 .......................... 115
Figure 5.27: Trajectory at 1000 km Altitude for km/s in Case 1 ........................ 116
Figure 5.28: Trajectory at 1500 km Altitude for km/s in Case 1 ........................ 116
Figure 5.29: Ground Range vs. Tether Length for km/s in Case 1 ..................... 117
Figure 5.30: Time to Impact vs. Tether Length for km/s in Case 1 .................... 117
Figure 5.31: Trajectory at 55 km Tether Length for km/s in Case 1 .................. 118
Figure 5.32: Trajectory at 100 km Tether Length for km/s in Case 1 ................ 119
xiii
Figure 5.33: Ground Range vs. for Case 1 .................................................................... 120
Figure 5.34: Time to Impact vs. for Case 1 ................................................................... 120
Figure 5.35: Ground Range vs. Altitude for km/s in Case 2 ................................. 124
Figure 5.36: Time to Impact vs. Altitude for km/s in Case 2 ................................ 124
Figure 5.37: Trajectory at 500 km Altitude for km/s in Case 2 ............................. 127
Figure 5.38: Trajectory at 650 km Altitude for km/s in Case 2 ............................. 126
Figure 5.39: Trajectory at 850 km Altitude for km/s in Case 2 ............................. 126
Figure 5.40: Ground Range vs. Tether Length for km/s in Case 2 ........................ 127
Figure 5.41: Time to Impact vs. Tether Length for km/s in Case 2 ....................... 128
Figure 5.42: Trajectory at 55 km Tether Length for km/s in Case 2 ..................... 129
Figure 5.43: Trajectory at 100 km Tether Length for km/s in Case 2 ................... 129
Figure 5.44: Ground Range vs. for Case 2 .................................................................... 130
Figure 5.45: Time to Impact vs. for Case 2 ................................................................... 130
Figure 5.46: Lofted Trajectory at km/s for Case 2 ................................................ 131
Figure 5.47: Ground Range vs. Altitude for km/s in Case 2 .............................. 132
Figure 5.48: Time to Impact vs. Altitude for km/s in Case 2 ............................. 133
Figure 5.49: Trajectory at 500 km Altitude for km/s in Case 2 .......................... 134
Figure 5.50: Trajectory at 1000 km Altitude for km/s in Case 2 ........................ 134
Figure 5.51: Trajectory at 1300 km Altitude for km/s in Case 2 ........................ 135
Figure 5.52: Ground Range vs. Tether Length for km/s in Case 2 ..................... 136
Figure 5.53: Time to Impact vs. Tether Length for km/s in Case 2 .................... 136
Figure 5.54: Trajectory at 55 km Tether Length for km/s in Case 2 .................. 137
Figure 5.55: Trajectory at 100 km Tether Length for km/s in Case 2 ................ 138
xiv
Figure 5.56: Ground Range vs. for Case 2 .................................................................... 139
Figure 5.57: Time to Impact vs. for Case 2 ................................................................... 139
Figure 5.58: Ground Range vs. Altitude for km/s in Case 2 ................................. 140
Figure 5.59: Time to Impact vs. Altitude for km/s in Case 2 ................................ 141
Figure 5.60: Trajectory at 500 km Altitude for km/s in Case 2 ............................ 142
Figure 5.61: Trajectory at 1000 km Altitude for km/s in Case 2 .......................... 142
Figure 5.62: Trajectory at 1500 km Altitude for km/s in Case 2 .......................... 143
Figure 5.63: Ground Range vs. Tether Length for km/s in Case 2 ....................... 144
Figure 5.64: Time to Impact vs. Tether Length for km/s in Case 2 ...................... 144
Figure 5.65: Trajectory at 55 km Tether Length for km/s in Case 2 ..................... 146
Figure 5.66: Trajectory at 100 km Tether Length for km/s in Case 2 ................... 146
Figure 5.67: Ground Range vs. for Case 2 .................................................................... 147
Figure 5.68: Time to Impact vs. for Case 2 .................................................................. 148
xv
Nomenclature
ATEx Advanced Tether Experiment
a Semimajor axis
First value in the principal axis vector
Second value in the principal axis vector
Third value in the principal axis vector
? Acceleration vector, Principal axis
? Acceleration vector of the subsatellite
? Acceleration vector of the center of mass
C Direction cosine matrix
Direction cosine matrix element from the ith row and jth column
E Energy, Eccentric anomaly
e Eccentricity
ex Xcomponent of the eccentricity vector
ey Ycomponent of the eccentricity vector
ez Zcomponent of the eccentricity vector
? Eccentricity vector
? Force Vector
? Gravitational Force
Function for the fourth order Runge Kutta
xvi
G Universal gravitational constant
H Altitude
? Angular momentum vector
h Magnitude of the angular momentum vector
hi Xcomponent of the angular momentum vector
hj Ycomponent of the angular momentum vector
hk Zcomponent of the angular momentum vector
? Unit vector for the X axis in the geocentric equatorial coordinate system
? Unit vector for the X2 axis in the body fixed coordinate system
i Inclination
? Unit vector for the Y axis in the geocentric equatorial coordinate system
? Unit vector for the Y2 axis in the body fixed coordinate system
? Unit vector for the Z axis in the geocentric equatorial coordinate system
? Unit vector for the Z2 axis in the body fixed coordinate system
First step in the fourth order Runge Kutta
Second step in the fourth order Runge Kutta
Third step in the fourth order Runge Kutta
Fourth step in the fourth order Runge Kutta
Magnitude of the tether length vector
? Tether length vector, position of relative to
Magnitude of the position vector of relative to the center of mass of the system
? Position vector of relative to the center of mass of the system
Magnitude of the position vector of relative to the center of mass of the system
xvii
? Position vector of relative to the center of mass of the system
M total mass of a TSS; Mean anomaly
ME Mass of the Earth
M2 Secondary mass, usually of a planet in a twobody problem
Subsatellite mass
Main satellite mass
NORAD North American Aerospace Defense Command
N Magnitude of the node vector
Nj Ycomponent of the node vector
? Node vector
n Mean motion
Mean motion of the center of mass
PMG Plasma Motor Generator
ProSEDS Propulsive Small Expendable Deployer System
R Ground range covered by the subsatellite after release
Radius of the Earth
Position Magnitude of the center of mass
Xcomponent of the position of the center of mass of the TSS
Ycomponent of the position of the center of mass of the TSS
Zcomponent of the position of the center of mass of the TSS
? Position vector of the center of mass
?? First derivative of the position vector of the center of mass
?? Second derivative of the position vector of the center of mass
xviii
r Magnitude of the position vector
rp Position magnitude at perigee
rx X component of the position vector
ry Y component of the position vector
rz Z component of the position vector
? Position vector
Position magnitude of the subsatellite
Xcomponent of the position of the subsatellite
Ycomponent of the position of the subsatellite
Zcomponent of the position of the subsatellite
? Position vector of the subsatellite
Position magnitude of the main satellite
? Position vector of the main satellite
SEDS Small Expendable Deployment System
TiPS Tether Physics and Survivability Experiment
TSS Tethered Satellite System
Time at subsatellite impact
Time to impact from release
Final time
Initial time
Time at subsatellite release
V Magnitude of the velocity vector
? Velocity vector
xix
Velocity magnitude of the center of mass
Vx X component of the velocity vector
Vy Y component of the velocity vector
Vz Z component of the velocity vector
? Velocity vector of the center of mass
Circular velocity of the center of mass
Velocity of the center of mass in the Xdirection
Velocity of the center of mass in the Ydirection
Velocity of the center of mass in the Zdirection
Velocity magnitude of the subsatellite
? Velocity vector of the subsatellite
? Velocity vector of the subsatellite at release
Velocity magnitude of the subsatellite in the Xdirection
Velocity magnitude of the subsatellite in the Ydirection
Velocity magnitude of the subsatellite in the Zdirection
? ? Relative velocity of with respect to the center of mass
Circular velocity of the subsatellite
Velocity magnitude of the main satellite
? Velocity vector of the main satellite
? ? Relative velocity of with respect to the center of mass
Circular velocity of the main satellite
? State Matrix
?? First derivative of the state matrix
xx
X, Y, Z Axes for the geocentric equatorial coordinate system
X2, Y2, Z2 Axes for the body fixed coordinate system.
Xposition of the subsatellite
? Xacceleration of the subsatellite
Yposition of the subsatellite
? Yacceleration of the subsatellite
Zposition of the subsatellite
? Zacceleration of the subsatellite
Xposition of the center of mass
? Xacceleration of the center of mass
Yposition of the center of mass
? Yacceleration of the center of mass
Zposition of the center of mass
? Zacceleration of the center of mass
Time step
Velocity change of the center of mass in the Xdirection
Velocity change of the center of mass in the Ydirection
Velocity change of the center of mass in the Zdirection
? Angular acceleration vector
Fourth Euler parameter
First Euler parameter
Second Euler parameter
Third Euler parameter
xxi
? First derivative of the fourth Euler parameter
? First derivative of the first Euler parameter
? First derivative of the second Euler parameter
? First derivative of the third Euler parameter
? Angle between the position and velocity vector
? True anomaly
True anomaly at subsatellite impact
True anomaly at subsatellite release
? Gravitational parameter
? True anomaly
? Rotation angle, Principal angle
? Longitude of the ascending node
? Argument of periapsis
? Angular velocity vector
Xcomponent of the Angular Velocity
Ycomponent of the Angular Velocity
Zcomponent of the Angular Velocity
1
Chapter 1: Introduction
This research is a conceptual study to evaluate the possible ballistic capabilities of a
satellite that is launched from a space based platform. While any space based platform could be
used, the platform chosen for this study is a Tethered Satellite System (TSS). A TSS is used as
the platform because different release point configurations can be created having different
velocities. The subsatellite of the TSS will be the object that is launched or released from the
system. The subsatellite will be released from the TSS after a velocity change is placed on the
subsatellite. This velocity change could be the result of an impulsive velocity change maneuver
done by the subsatellite or it could be the result of an increase in the angular velocity of the
TSS. The ballistic capabilities that will be studied are the ground range and time to impact after
the subsatellite is released from the system. Different release point configurations and changes
in the TSS parameters will be analyzed to determine the options in the available ground ranges
and times to impact. This chapter will discuss the history and applications of a TSS, the previous
work done on TSS, and a more detailed problem description for this study.
1.1 Historical Background
A Tethered Satellite System (TSS) is made up of two satellites connected by a cord or
tether. The two end bodies are called the main satellite and the subsatellite. The subsatellite is
in a lower orbit than the main satellite and is deployed from the main satellite using the tether.
The main satellite can be a larger satellite than the subsatellite or even a shuttletype orbiter.
The idea of a TSS was first introduced in 1895 by a Russian scientist by the name of Tsiolkoskii,
2
who presented the concept of a ?space tower? or a space elevator. Tsiolkoskii?s idea of a space
tower was to have a system that would be attached to the surface of the Earth with the other end
being in a geosynchronous orbit about the Earth. The objective of the space tower was to allow
an object to travel up the tower and be released once it reached the top. At the top of the tower
the object would be released into a geosynchronous orbit. The idea of a space tether is attributed
to Tsiolkovskii because of his proposal for the space tower.7
In 1965 the idea of a TSS was put to the test by NASA during the Gemini XI and Gemini
XII missions. In Gemini XI, the subsatellite was connected to the upper stage and both the sub
satellite and the upper stage swung around the center of mass of the system and generated an
artificial gravity.7 The Gemini XII mission was designed to test the gravitygradient stabilization
of a TSS.17 Both the Gemini XI and Gemini XII missions verified the assumptions and
calculations done by scientists thereby, proving that tethered satellites could be useful. Issac
Artsutanov, was the first person to propose a nonsynchronous sky hook. This sky hook was a
rotation tethered system that would approach Earth?s surface with a zero relative velocity at the
perigee of the tethered system?s orbit.3
Despite the missions and the ideas presented in the 1960s the ?Era of TSS? did not begin
until the 1970s when Mario Grossi and Giuseppe Colombo proposed their idea for a skyhook to
NASA and the Italian Space Agency.7 Grossi was from the Smithsonian Astrophysical
Observatory and Colombo was working at the University of Padua in Italy. The skyhook
proposal involved a small satellite that could be deployed from a shuttle orbiter along a twenty to
one hundred kilometer tether. The small satellite could then help with exploration and data
collection of the Earth?s upper atmosphere.18 This proposal by Grossi and Colombo generated
interest in TSS for NASA and the Italian Space Agency.
3
The interest generated by the proposal by Grossi and Colombo resulted in NASA and the
Italian Space Agency formally agreeing to collaborate on TSS projects. While several short
tethered experiments were conducted on the shuttle during the 1980s, it was not until 1992 that
the first major joint project between the two agencies was launched.18 The satellite was called
the TSS1 and was launched from the Atlantis orbiter. The TSS1 was the first test of a TSS in
space.7 The purpose of the TSS1 mission was to study the electrodynamic properties of the
Earth?s ionosphere. This required that the tether be made of an electrically conducting
material.18 However, during the mission the reeling mechanism for the tether and subsatellite
malfunctioned. This caused the tether to stop its deployment. In other words the tether did not
deploy from the shuttle to its full length.7 Despite this malfunction the TSS1 mission proved
that a satellite could be deployed and controlled. The mission also showed that a TSS could be
easily controlled and showed that the system was more stable than predicted.5
The Small Expendable Deployment System 1 (SEDS1) and the Plasma Motor Generator
(PMG) missions were conducted during 1993. The SEDS1 successfully demonstrated the
downward freereeling deployment of the subsatellite in a TSS. The PMG?s mission was to
investigate the possibility of using an electrically conducting tether to generate power that could
be used to create thrust for the TSS. This was the first example of propulsion for a system that
did not require any propellant. A year later the SEDS2 was deployed and remained in orbit for
five days until space debris or a micrometeorite severed the tether.7 The purpose for deploying
the SEDS2 was to study long term tether dynamics.18 Even though the tether was severed, the
SEDS2 mission collected useful data.
In 1995 and 1996 three more missions were launched to further study the dynamics and
applications of TSS. These missions were the OEDIPUSC, the TSS1R, and the Tether Physics
4
and Survivability Experiment (TiPS). The OEDIPUSC system was made up of two rocket
payloads and was launched in 1995. After the system reached the apogee of the TSS orbit, the
tether was intentionally cut in order to study a postapogee trajectory.18 The TSS1R and TiPS
systems were launched in 1996. The TSS1R is named because it is the relaunch of the TSS1
mission. During this launch the tether was not severed and the mission was able to prove that a
tether could be used to create a high voltage charge as it passes through the Earth?s ionosphere.7
The TiPS satellite was launched by the Naval Research Laboratory.18 The system consisted of
two small satellites rather than a shuttle orbiter and satellite combination. The TiPS satellite
remained in space for several years and provided enough data to further study tether satellite
dynamics and orbital motion.7
The last TSS mission launched in the 1990s was the Advanced Tether Experiment
(ATEx). The Naval Research Laboratory launched the ATEx system with a goal to demonstrate
TSS stability, control, and attitude determination of the two end bodies. The plan behind the
mission was to deploy the subsatellite and tether out to 6 km over the course of three and a half
days; however, after only eighteen minutes of deployment the ATEx system was jettisoned from
the main satellite. The reason behind the jettison of the ATEx was because the tether angle
sensor detected that an outoflimits condition had been reached and automatically jettisoned the
system. It is believed that the condition was detected due to a thermal expansion of the tether.7,18
Very few tether missions have been launched in the 21st century. Some missions were
proposed but later cancelled. One successful launch was the PICOSat in 2001. The PICOSat
consisted of a small TSS that had a mass less than a kilogram. In 2002 the Propulsive Small
Expendable Deployer System (ProSEDS) was proposed. This system was a propulsive
experiment that was designed to test the ability of a TSS to draw energy from the magnetic field
5
of the Earth and then use that energy as an electric thruster to raise or lower the TSS?s orbit.7
Work was done on the ProSEDS until NASA cancelled the project due to the fact that key
parameters in the performance of the TSS could not be met.27
1.2 TSS Applications
TSS applications can be broken into three categories: space transportation, electrically
conducting tethers, and atmospheric exploration. The first application that will be discussed is
the space transportation category, which focuses on the movement of the satellite system. A TSS
can be used to put a satellite into a higher or lower orbit than the original orbit without the use of
boosters. The system can also be used to return a discarded payload to the Earth and help during
the docking process of a shuttle to a space station. These applications can be done because
energy and momentum exchange occurs between the main satellite and the subsatellite during
the deployment of the subsatellite. This same momentum and energy exchange can also help
with controlling the attitude of the main satellite.18 A TSS could be used to perform aero
braking when approaching a planet. Aerobreaking occurs when the subsatellite is deployed
into the planet?s atmosphere where atmospheric drag would affect the satellite. When
aerodynamic drag is exerted on the subsatellite, the system will theoretically decrease to a
capture speed from the approach speed before deployment. This decrease in the satellite speed is
caused by the tension in the tether instead of using a propulsive maneuver to slow down the
satellite.7
Since the space transportation application for TSS deals with movement of all or part of
the system, fuel savings and artificial gravity could be a result.18 The fuel savings are beneficial
for obvious reasons. The benefit from artificial gravity may not be as obvious. Artificial gravity
is created by the spin rate of a spacecraft rotating about its center of mass. However, in order for
6
a shuttle orbiter to generate enough artificial gravity the required spin rate would be too high for
the crew of the shuttle to withstand. When a TSS is used to create an artificial gravity, the spin
rate of the system is at an acceptable level for the crew.7
The second category of TSS applications is electrically conducting tethers, which allows
for the transportation of energy. In this case the energy is in the form of electricity that is
generated from the Earth?s magnetic field. The electricity will produce a current along the tether
which will create a force that can move the TSS into a higher or lower orbit. In other words the
force generated by the electric current can be used instead of propellant in a propulsive maneuver
to change the trajectory of the TSS. If the tether can conduct electricity, it can also be used as an
antenna for the system, which can transmit electromagnetic waves in the direction that the tether
is pointed.18 There are two main benefits to using a conducting tether. The first one is reduced
cost. If the conducting tether can provide a force to move the satellite system, the amount of fuel
needed for the mission is reduced and the fuel cost will be reduced as well. The second benefit is
that there is a reduced risk of a chemical propellant contaminating any external part of the TSS.7
While fuel will always be needed on the TSS, a conducting tether can reduce the amount the
system carries; thereby, decreasing the risk and cost of the mission.
The final category of tether applications is atmospheric exploration. In atmospheric
exploration an instrument is deployed into the Earth?s atmosphere from a large spacecraft, like
the shuttle orbiter. A TSS is a great system for atmospheric exploration because the tether can be
extended into areas of the upper atmosphere that are too ?rarified? for the aircraft and too dense
for shuttle orbiters or other large spacecraft. One use of tethers for atmospheric exploration is
the gathering of physical, meteorological, and environmental data over a long course of time.
The tethers could also be used for surveillance and measurements of the Earth?s magnetic and
7
electric fields. Perhaps the most useful application for tethers in atmospheric exploration is the
lowering of a model into the atmosphere for highspeed and high altitude aircraft. Wind tunnels
are normally used to determine the characteristics of an aircraft design; however, high altitudes
and high speeds are harder to replicate inside a wind tunnel. Overall, the tether can be used to
collect data for a variety of experiments in atmospheric exploration.18
Atmospheric exploration is an area of great interest, especially when it comes to how the
sun interacts with the Earth?s atmosphere. The peak area for this interaction occurs anywhere at
or below a four hundred kilometer altitude above the Earth. This region contains the maximum
gradient in electron content and temperature in the Earth?s atmosphere. The higher areas of this
segment are hard to reach with an aircraft and difficult to maintain for long periods of time for a
large spacecraft. By extending a tether into this region long term data can be collected. It?s
important to note that once the scientific community has an understanding on how to use TSS to
study the Earth?s atmosphere this application can be extended to other planets.7
1.3 Previous Research
TSS research has mainly focused on the orbit determination and the identification of TSS
throughout the years. Some researchers have studied the dynamics and attitude of tethered
satellites and there has been some research done that looks at the impact trajectory of the sub
satellite and the new orbital elements of the trajectory for the subsatellite once it has been
released from the system; however, there has not been an effort, to this writer?s knowledge, to
determine what can cause an impact trajectory and what will lead to a maximum range with the
best time to impact for the subsatellite. That being said this section will still discuss some
previous research that has helped to formulate the problem and model development for this
research as well as for research studying impacting trajectories.
8
In 1983 Hoots, Roehrich, and Szebehely published a study that focused on TSS analysis
and the effect of the tether on both satellites. The authors discussed the effect of the tether on the
speed of the satellites in the system. The tether causes the subsatellite to have a lower velocity
than its expected circular velocity without the tether and the main satellite actually has a higher
velocity than its circular velocity without the tether. Hoots, Roehrich, and Szebehely also
determined the conditions for impact after the subsatellite is released from the TSS and define
how to modify the gravitational constant in order to make the Keplerian acceleration valid while
the tether remains intact. The authors also pointed out the problem of detecting a TSS system
and what it does to detection programs, such as the one used by North American Aerospace
Defense Command, NORAD.13
Several papers have also been published concerning the modeling, orbit determination,
and quick identification of TSS. Qualls and Cicci24 published a paper in 2007 that studied the
modifications of classic preliminary orbit determination methods to distinguish between a
tethered satellite and an untethered satellite. It was determined that the Herrick ? Gibbs Method
and the true anomaly iteration technique provided the most accurate results in determining if the
object in question was tethered or not.24 Rossi, Cicci and Cochran Jr.25 published a paper in
2004 that discussed an analytical study of the dynamics of a TSS to determine the conditions for
periodic motions about the equilibrium state of the system. It was determined that when only the
gravitational and oblateness forces were placed on the system that the periodic motion of the
TSS depended on the physical characteristics of the tether; however, if the aerodynamic drag was
placed on the system, the drag must be bounded in order to create a periodic motion.25 A paper
was published by Lovell, Cho, Cochran Jr., and Cicci19 in 2003, that examined the factors that
caused the bodies of a TSS to behave differently and how each of the factors creates a
9
discrepancy in the determination of a satellite reentry. The authors determined that the tether
force and the librational motion of the system were key factors that caused a tethered satellite to
appear to be on a re ? entry course.19 In 2002, Cicci, Cochran Jr, Qualls, and Lovell9 published a
paper that provided a methodology to perform an identification, orbit determination, and orbit
prediction of a TSS. For the identification of a TSS preliminary orbit determination methods
were used and ridge ? type estimation methods were used for the orbit determination of the TSS.
The authors used a TSS dynamic model for the long term orbit prediction of the TSS.9 Another
paper would be published by Cicci, Qualls, and Lovell in 2001.8 This paper discussed the use of
ridge ? type estimation methods to identify a TSS when a ?small arc? of observational data was
available.8 The final paper to be discussed by this group of authors was published in 1998 by
Cho, Cochran, Jr., and Cicci.6 In this paper a perturbed two ? body model was used for the TSS
during orbit determination. The orbit determination was done using only the observations from
one of the satellites in the system. The characteristics of the perturbed motion on the TSS was
studied using this data.6
Another key author in TSS studies is A. K. Misra. Misra21 published a paper in 2008
titled ?Dynamics and Control of Tethered Satellite Systems.? In this paper Misra studies the
nonlinear roll and pitch motions on a TSS. Misra found that the aerodynamic drag produced
librational motion on the TSS and that electrodynamic forces would affect the pitch and roll of
the TSS; therefore, the drag and electrodyanmic forces would change the stability of the TSS.21
Modi, Gilardi, and Misra22 published a paper in 1998 that studied the attitude control of a TSS.
The authors used an Nth order lagrangian formulation to study the attitude control of space
platform based TSS and used a Liapunov method for the control of the system. The Liapunov
control was found to be effective in stabilizing the TSS.22
10
Curtis Hilton Stanley wrote his Master?s thesis at the University of Colorado at Colorado
Springs about apparent impacting trajectories, identification, and orbit determination of TSS in
2010. Stanley defines an apparent trajectory as being the ?trajectory that an end mass would
follow if not tethered.? If the apparent trajectory would intersect the Earth, then it becomes an
apparent impact trajectory. He looked at circular and elliptical orbits with and without librational
motion to determine the apparent impacting trajectory and used batch filters to determine the
trajectory of the satellite and identify the subsatellite as being a part of a TSS. Stanley?s
research focused on determining the minimum tether length needed to create an apparent
impacting trajectory and simulating a ballistic missile trajectory along with the apparent
impacting trajectory to form a comparison between the two. Once the comparison was
determined he could use that knowledge to have a program identify an object as being a ballistic
missile or a satellite attached to a TSS. Stanley never looked into an actual impacting trajectory
and the range that the end mass could obtain once released from a TSS.28
In 1994 Naigang, Dun, Yuhua, and Naiming23 discussed the calculation of the orbital
elements for the subsatellite after it is released from the TSS. The purpose behind this
discussion is the momentum transfer between the two satellites that could place a payload in a
higher orbit about the Earth. The orbital elements are calculated from position and velocity
vectors, which are determined using the pitch angle, roll angle, magnitude of the position, the
true anomaly, and their derivatives. The authors then compare the orbital elements before
release to those after the release of the satellite. They found that the change in the right
ascension and the inclination angle of the subsatellite after release is small because of the small
roll rate present in their examples. The authors also note that the biggest change is found in the
semimajor axis.23
11
Nammi Jo Choe and Thomas Alan Lovell did research into the orbit determination and
detection of TSS and state estimation of TSS, respectively. Choe looked at a two satellite TSS
and a three satellite TSS in 2003. She also developed a basic model for the TSS by assuming a
massless tether and representing the end satellites as point masses. An algorithm was developed
by Choe to detect the TSS systems. Once the TSS was detected, Choe used an estimation
process to determine the orbit of the system over the orbital period of the systems. The process
developed by Choe allowed for the detection, state estimation, and calculation of tether
parameters for a two and three body TSS.7 Lovell used a ?batchtype differential corrections
filtering scheme? or a gradient based algorithm and a genetic algorithm to determine the state
estimation of a TSS. He then compared the two techniques in order to determine which one had
better accuracy, speed, and robustness of the two algorithms. His research showed that the
genetic algorithm could give an accurate solution over short periods of data. Since long term
filters such as the gradient based method depend on the initial approximation, Lovell suggested
that a hybrid approach be used. This hybrid approach would use the genetic algorithm to
determine the initial approximation and the gradient based method would be used to solve for the
estimation of the state.18
Research addressing TSS dynamics includes the study of perturbed motion, constrained
dynamics, attitude and control, librational motion, and stability of the TSS. Sungki Cho5
analyzed the perturbed motion of TSS and used filters to identify the perturbed motion of the
satellite system in 1999. The perturbed motion of one detected satellite could be used to identify
the satellite as part of a TSS. Cho used a least squares batch filter in order to identify the satellite
as a member of a TSS and to determine the motion of the system.5 In 2001 Peter Beda used
Lagrange?s equations to define the equations of motion for the TSS. He then put a constraint on
12
the system to make sure that the tether length remained constant. The equations of motion and
the control constraint were then used in a numerical simulation to determine the attitude of a
TSS.2 Aaron Schutte and Brian Dooley26 developed a control law for the constrained motion of
a TSS. This control law or constraint equations could then be used to simplify the modeling and
simulation of a TSS.26
An extended rigid body model of a TSS was used by Insu Chang, SangYoung Park, and
KyuHong Choi4 in 2010 to study the attitude and control of a TSS. In order to simulate this, the
authors used a statedependent Riccati equation to model the nonlinear attitude control of a TSS.
A numerical integration was used to determine the stability of the system using the Riccati
equation. The TSS was found to be asymptotically stable when using the Riccati equation to
model the nonlinearities in the system.4 Joshua Ellis and Christopher Hall11 also studied the
stability of a TSS by looking at the system?s out of plane librations. Equations of motion were
developed by Ellis and Hall to model the inplane and outofplane librations of the TSS. The
eigenvalues for the librational motions were then used to determine the stable and unstable areas
of motion for a TSS.11
In 2008 Kosei Ishimura and Ken Higuchi15 wrote a paper that described the coupling
between the pitch, axial vibration, and orbital motion of TSS. The authors looked at the effects
of these motions on the eigenvalues and eigenvectors. Ishimura and Higuchi also looked into the
influence of the mass ratio and natural frequency on the characteristics of the system, specifically
the eigenvalues. The natural frequency was found to have the greatest influence on the coupling
between the pitch, axial vibration, and orbital motion while the mass ratio was found to influence
the pitch motion and found to affect the value for the eigenvalues and eigenvectors.15 Also in
2008, Hao Wen, Dongping P. Jin, and Haiyan Y. Hu29 discussed the relative advances in the
13
dynamics and control of TSS and provided a comprehensive look at all the advances in TSS
dynamics and control. Some of the topics discussed are controls for tether deployment and
retrieval of the subsatellite, further studies into producing artificial gravity with a tethered
system, reentry and rendezvous missions assisted by a tether, and stability of a TSS. Some of
the studies discussed by Wen, Jin, and Hu focused on a two satellite tethered system, while
others looked at three or more satellites tethered together.29
The majority of the research conducted concerning TSS focused on constraints and
attitude and control and dynamics of tethered systems. Only a few studies have discussed impact
trajectories. Stanley?s28 focus was on the apparent impacting trajectory of a subsatellite without
being cut from the TSS. He compared this apparent impacting trajectory with a typical ballistic
missile trajectory. This information was then used to identify a tethered satellite, when only one
end of the tethered satellite was detected. After the subsatellite was identified as part of a TSS,
Stanley focused on the orbit determination of a TSS. Naigang, Dun, Yuhua, and Naiming23
focused on the changes in the orbital elements after the subsatellite was released from a TSS
with librational motion. All the information found in the research stated above was used to help
formulate the best model and equations of motion for this study.
1.4 Problem Description
The purpose of this study is to determine the possible ballistic capabilities of a satellite
launched from a space based platform. For the purposes of this study a TSS was chosen for the
launching platform because it can be used to release the satellite with different release
configurations. The subsatellite of the TSS will be released from the system after a velocity
change is placed on the subsatellite. There are two methods to create such a velocity change on
the subsatellite: an impulsive velocity change maneuver and an increase in the angular velocity
14
of the system. The impulsive velocity change is achieved when the subsatellite produces a
thrust in a certain direction. The resulting velocity change from the thrust maneuver will affect
the velocity of the subsatellite and may result in an impact trajectory after the subsatellite is
released from the TSS. For the increase in the angular velocity of the system, the subsatellite
and the main satellite will produce a thrust that would cause the TSS to rotate faster. This
increase in the rotation of the system will affect the velocity of both satellites and may cause the
subsatellite to impact the Earth after it is released from the system. If an impact does occur, the
range covered by the subsatellite?s impacting trajectory and the time it takes for the subsatellite
to impact the Earth after launch will be calculated. Range is used to describe the ground range
covered by the subsatellite after it is released from the TSS. The range and time to impact are
the ballistic capabilities that will be analyzed in this study.
Changes in the parameters of the space based platform, or the TSS, are also analyzed to
determine how they might affect the capabilities of the subsatellite after release. The
parameters that are investigated are the altitude of the main satellite and the tether length because
both of these parameters will affect the launching position of the subsatellite relative to the
Earth. For each combination of parameters an impulsive velocity change maneuver is placed on
the subsatellite and it is released from the system. If the subsatellite impacts the Earth, the
range and time to impact of the impacting trajectory are calculated in order to determine the
impact of the changing on the ballistic capabilities of the subsatellite.
This research takes a threat analysis point of view for a satellite launched from a space
based platform. Using the TSS as the platform, the subsatellite will be placed in different
release configurations with various altitudes, tether lengths, and velocity changes placed on the
subsatellite before it is released. For each velocity change evaluated, a corresponding angular
15
velocity of the TSS is calculated which would produce such a velocity change. The ballistic
capabilities for different cases are then compared in order to determine a range of ballistic
capabilities for each configuration.
16
Chapter 2: Relevant Theory Used in TSS Analysis
There are many ways to model a TSS with different coordinate systems and assumptions.
Basic effects caused by the tether on the end bodies will be discussed before developing a
detailed model for the TSS. After the detailed model is defined, the equations of motion and
perturbation effects will be introduced. Since rotational motion is included in the model, attitude
control parameters will be used to assist with the transformations between the body coordinate
system and the inertial coordinate system. Orbital elements will be described since the elements
will be used to determine if the trajectory of the subsatellite after release will impact the Earth.
Finally, discussions of rigid body dynamics, the integration method used to model the TSS, and
orbital mechanics will be presented. In the discussion that follows, it will be assumed that the
center of mass of the TSS will move in a circular orbit.
2.1 Tethered Satellite System Dynamics
The tether in the system exerts a force on both the main satellite and the subsatellite.
This force leads to a change in the velocities of both of the satellites. Depending on the position
of the mass, the tether will either increase or decrease the velocity of each satellite to a point
where it is greater than or less than the satellite?s untethered circular velocity. For this
discussion the setup for the TSS is shown in Figure 2.1. The main satellite generally has a mass
equal to or larger than the subsatellite. Hoots, Roehrich, and Szebehely13 discussed the effects
of the tether on the masses and developed equations for the velocity of both masses and the
center of mass in their study. The first step was to determine the location of the center of mass.
17
Once the position of the center of mass was determined, the equation for circular velocity was
then used to find the circular velocity for the center of mass.13
Figure 2.1: Tethered Satellite System Model
For a circular orbit, the position vector of the center of mass relative to the center of the Earth
and the circular velocity of the center of mass are calculated using equation (1) and equation (2),
respectively. 13
? ? ?
?
18
The variables and are the mass of the subsatellite and the main satellite, respectively.
The magnitude of the position vectors of the subsatellite and the main satellite are and , ? is
the gravitational parameter, and the magnitude of the position vector of the center of mass is .
In order to find the velocities of the end masses the mean motion of the center of mass
must first be calculated. Equation (3) defines the mean motion as a function of the gravitational
constant and the position of the center of mass.13
?
Once the mean motion is calculated, the velocities of the end masses can be found as a function
of the mean motion and the position of each mass. Equation (4) and Equation (5) define the
equations used to obtain the velocity of the main satellite and the subsatellite, respectively.13
Equation (3) can then be substituted into equations (4) and (5) to give equations (6) and (7) for
the velocities of the main satellite and the subsatellite, repectively.13
?
?
When the satellites are not connected by a tether, both the main satellite and subsatellite
have a circular velocity that is defined similar to the circular velocity of the center of mass given
19
by equation (2). The actual velocities of the satellites when connected with the tether and the
unconnected circular velocity of the satellites can then be compared. For the situation shown in
Figure 2.1, Hoots, Roehrich, and Szebehely13 determined that the actual velocity of the main
satellite when tethered is larger than the circular velocity of the main satellite when it is not
tethered. The subsatellite?s actual velocity when tethered was found to be smaller than the
circular velocity the subsatellite has when not tethered. This means that when the two masses
are put together in a TSS formation the main satellite?s mass will speed up, while the mass of the
subsatellite will be slowed down relative to their untethered circular velocities.13
In order to model a TSS a set of coordinate systems needs to be defined before the
equations of motion can be determined. An inertial coordinate system will be used to describe
the motion of the system, and will be selected to be the geocentric equatorial coordinate system.
The geocentric  equatorial coordinate, Earth ? centered Inertial, or ECI system is depicted in
Figure 2.2.
Figure 2.2: Geocentric Equatorial Coordinate System
20
The origin of the geocentric equatorial system is fixed at the center of the Earth. The positive X
direction points in the vernal equinox direction. The positive Z axis goes through the North Pole
and the positive Y axis is ninety degrees from the positive X and Z axes, such that the XY plane
lies in the Earth?s equatorial plane. ?, ?, and ? are the unit vectors for the X, Y, and Z axes,
respectively.1 The TSS is also depicted in Figure 2.2. The end masses are considered to be point
masses and the tether is assumed to be a massless, rigid tether. The position vector for the main
satellite and the subsatellite are ? and ? , respectively, while the position vector, ? , represents
the position of the center of mass of the TSS. The fourth vector, ? , in Figure 2.2 is the tether
length vector or the position of relative to .
Since the end masses are naturally rotating about the center of mass in a counter 
clockwise rotation about the positive X ? axis due to the gravity gradient affect, a body
coordinate system will be needed in order to determine the position of the end masses relative to
the center of mass of the system. This position will then be transformed into the inertial
coordinate system. Figure 2.3 shows the body coordinate system for the TSS. The fixed
coordinate system now has an origin located at the center of mass of the TSS. The X2, Y2, and
Z2 coordinate system is the body coordinate system after a rotation about the X axis. The TSS
rotates about the center of mass; therefore, the body coordinate system will rotate about the X
axis. This means that the X2 axis will line up with the X axis and an angle will be created
between the Y and Y2 axes and the Z and Z2 axes. The unit vectors for the body fixed coordinate
system are ? , ? , and ? , which correspond to the X2, Y2, and Z2 axes, respectively. The rotation
angle, ?, represents the angle between the fixed coordinate system and the body coordinate
system caused by the angular velocity. The two vectors in Figure 2.3 represent the distances
21
from the center of mass to the end bodies. The position vector of relative to the center of
mass of the system is ? . The vector, ? , is the position vector of relative to the center of
mass. Both vectors are expressed in body components. The rotation angle can be used to
express the vectors in the geocentric equatorial system by creating a direction cosine matrix and
using attitude dynamics. This methodology will be discussed after the introduction of the
equations of motion.
Figure 2.3: Body Fixed Coordinate System for a Negative ?
2.1.1 Equations of Motion
As stated earlier the end satellites will be modeled as two point masses and the tether will
be modeled as a massless, rigid tether. It was also mentioned earlier that due to the tether force
only the center of mass of the system can be modeled using Keplarian motion. Therefore, the
22
TSS will be considered as a rigid body in this study. In this section the equations of motion for
the center of mass of the system will be determined using twobody forces. The position and
velocity of the two end masses will be determined using rigid body dynamics.
The equations of motion of a spacecraft relative to the Earth are wellknown and can be
written as
? ??
where
Equation (8) is also known as the Keplarian acceleration of the system. Since equation (8) is a
vector, it has ?, ?, and ? components. This means that there are a total of three scalar equations
that can be used to describe the motion of the system. These final equations of motion are
expressed in equations (10) through (12).
?
?
?
These equations can be used to find the velocity and position components of the center of mass
of the TSS using numerical integration; however, the position and velocity of the end masses
have yet to be determined. In order to find the position of the end masses, the attitude dynamics
of the TSS must be considered.
23
2.2 Attitude Dynamics Using Euler Parameters
According to Hughes14, spacecraft attitude dynamics is an applied science that aims to
understand and predict spacecraft orientation and how the orientation evolves over time. The
part of attitude dynamics that this research utilizes is the determination of equations to describe
rotational motion and the differential equations that govern the motion equations. There are
many ways to represent the attitude dynamics of a spacecraft. In this text, the Euler parameters
are chosen because they have no singularities at any orientation; however, the disadvantage of
the Euler parameters is that they lack uniqueness. In other words, a positive Euler parameter can
describe the same attitude orientation as a negative Euler parameter. The Euler parameters are
then placed into a direction cosine matrix, which describes the rotation of the system and allows
for a transformation between two coordinate systems.14 In this case the direction cosine matrix
will allow for the transformation of the body coordinate system into the fixed geocentric
equatorial coordinate system.
Euler?s theorem states that any rigid body orientation can be achieved by a single rotation
about the principal axis, ?, through a principal angle, ?. The principal axis and angle can be
determined by examining the eigenvalues and eigenvectors of the direction cosine matrix. The
principal axis is a vector made up of three components. If the principal axis and angle are
known, the Euler parameters can be found. Equations (13) through (16) define the four Euler
parameters in terms of the principal axis and angle.14
( )
( )
24
( )
( )
The fourth Euler parameter is labeled using the symbol .14
The direction cosine matrix, C, is a 3x3 matrix and can be expressed in terms of the four
Euler parameters.
[
]
If the Euler parameters are not known but a direction cosine matrix is known, the Euler
parameters can be found using the elements from the direction cosine matrix. This must be done
in a series of steps. The first step is to calculate the squared values of each of the Euler
parameters.
[ ]
[ ]
[ ]
[ ]
Where,
[ ]
25
The second step is to determine the largest squared value from equation (18) through (22). Once
that is determined, take the square root of the largest value in order to find one of the Euler
parameters. Finally, the remaining three Euler parameters can be determined using the three of
the following equations.14
For example, the value for was found in step two. In order to find the other three Euler
parameters equation (21) can be used to find , equation (28) can be sued to find , and
equation (27) can be used to find .14
The Euler parameters will change with time and the introduction of an angular velocity.
In order to do this kinematics are introduced into the rotational motion equations. The motion of
the Euler parameters can be described using the first derivative of the parameters. The first
derivative of the Euler parameters is a function of the angular velocity and the original Euler
parameters. The first order derivatives are expressed in matrix format in equation (29).14
26
[
?
?
?
? ]
[
][
]
The , , and variables are the X, Y, and Zcomponents of the angular velocity.14 Now
that the rotational motion equations have been defined, the position and velocity of the end
masses in the TSS can be determined using rigid body dynamics.
2.3 Rigid Body Dynamics
Since the TSS will be modeled as a rigid body, the center of mass is the only point on the
TSS that can be modeled using Keplarian motion. This means that the position and velocity of
the main satellite and the subsatellite must be calculated using rigid body dynamics. According
to Ginsberg12 the motion of a rigid body is a ?superposition of a translation and a pure rotation.?
For the translational motion all points on the rigid body follow the movement of an arbitrary
point on the body.12 The arbitrary point for the TSS is the center of mass of the system. In
Figure 2.3, the X, Y, Z coordinate system is placed at the center of mass of the TSS because
when there is no angular velocity on the system the body coordinate system will line up with the
geocentric equatorial system.
Another important note is that the position vectors between the center of mass and the
two end masses will have constant components relative to the moving reference frame.12 The
moving reference frame in this case is the body fixed coordinate system; however, the final result
needs to be expressed in terms of the geocentric equatorial coordinate system. The position of
the center of mass is already expressed in terms of the geocentric equatorial coordinate system;
therefore, all that is left to do is to convert the position vectors between the center of mass and
27
the two end masses into geocentric equatorial components. This is done by multiplying the
direction cosine matrix by the vector of the distance between the center of mass and the end
masses.
? [ ] ?
? [ ] ?
Equations (30) and (31) give the position vector from the center of mass to the subsatellite and
to the main satellite, respectively, in geocentric equatorial coordinates. The next two equations
express the position of the subsatellite and the main satellite in geocentric equatorial coordinates
using the relative position equation.20
? ? ?
? ? ?
At this point, the disadvantage to using Euler parameters needs to be considered. Recall that the
Euler parameters have a lack of uniqueness and the negative Euler parameters could describe the
same position as positive Euler parameters. The sign on the position vectors from the center of
mass to the end masses must be checked to make sure that the addition and subtraction of the
distances matches up with the model.
Now that the position has been found for the end masses, the velocity of the subsatellite
and the main satellite needs to be found using rigid body dynamics. Ginsberg12 and Meriam20
define the velocity as the velocity of an arbitrary point plus the relative velocity between the
point where the velocity is to be determined and the arbitrary point.12,20 The arbitrary point
28
remains the center of mass of the TSS. Equation (34) and (35) defines the velocity of the sub
satellite and the main satellite, respectively.12,20
? ? ? ?
? ? ? ?
The second term on the right hand side of both equations is the relative velocity between the two
end masses and the center of mass. The relative velocity is defined as the angular velocity vector
crossed with the position vector between both points.12,20
? ? ? ?
? ? ? ?
The angular velocity is described in geocentric equatorial coordinates and the position vectors
that will be used are those defined in equations (30) and (31). Equation (36) can be substituted
into equation (34) to get the velocity of the subsatellite. The velocity of the main satellite can
be found by substituting equation (37) into equation (35).
? ? ? ?
? ? ? ?
Equations (32), (33), (38), and (39) are the rigid body equations used to find the position and
velocity of the end masses of the TSS.
The movement of the center of mass of the TSS and the end masses differ slightly. The
center of mass will move in a circular orbit. Rossi25 states that if the TSS is modeled as a solid
body it will rotate with a constant angular velocity in a counter ? clockwise direction. If the TSS
29
moves only in an orbital plane, the system will have a radial configuration or it will form a
straight line that is tangent to the circular orbit of the center of mass. The points of the TSS, the
center of mass and the two end masses, must be collinear because the system is being treated as a
rigid solid body.25 If there is no librational motion, the center of mass and both the end mass of
the TSS will move in concentric circles with the lower mass always being the subsatellite.19
Kaplan16 states that the lower mass of the TSS will always remain in the subsatellite position
because of the gravity gradient on the system. The gravity gradient will create a spin
stabilization that will keep the system at a local vertical; therefore, the subsatellite will always
be below the main.16 The natural motion of the TSS is a rotation of the system in a counter ?
clockwise direction with the lower satellite always remaining the subsatellite because of the
gravity gradient.
2.4 Orbital Elements of a Trajectory
At the time of the subsatellite?s release, the orbital elements of the subsatellite?s impact
trajectory will be calculated. The six orbital elements that describe an orbit are the semimajor
axis (a), the eccentricity (e), the inclination (i), the longitude of the ascending node (?), the
argument of periapsis (?), and the true anomaly (?). Figure 2.4 labels some of the orbital
elements of an orbit.
30
Figure 2.4: Orbital Elements of a Trajectory10
The semimajor axis defines the size of the orbit and the eccentricity defines the shape of a conic
orbit.1 The other four orbital elements are the inclination, the longitude of the ascending node,
the argument of perigee or periapsis, and the true anomaly. The inclination is the angle between
the ? geocentric equatorial unit vector and the angular momentum vector, ?. The longitude of
the ascending node is measured counter clockwise about the Zaxis from the Xaxis to the
ascending node on the orbit. The argument of periapsis is the angle between the ascending node
and the periapsis or perigee point of the orbit. The argument of periapsis is measured in the
plane of the satellite?s orbit in the direction of travel of the satellite. The true anomaly, ?, is
measure from the perigee point to the position vector of the satellite. The true anomaly is
measured in the direction of travel of the satellite.1 Each of these four parameters are shown in
the orbit depicted in Figure 2.4.10
31
The orbital elements can be determined when the position and the velocity vector are
known for a satellite. The eccentricity can be found by calculating the eccentricity vector.
? (
) ?
? ? ?
The magnitude of the eccentricity vector can then be found to get the eccentricity. Equation (41)
is used to calculate the magnitude of the eccentricity vector.10
?
The second orbital element that can be found is the semimajor axis. In order to find the semi
major axis the angular momentum must be calculated.
? ? ?
The magnitude of the angular momentum vector can then be calculated using the same formula
that was used to find the eccentricity vector.
Equation (43) can then be rearranged to find the semimajor axis in terms of the eccentricity, the
angular momentum, and the gravitational parameter.10
32
The inclination can then be found by using the vector properties for an angle between two
vectors. The two vectors that will be used are the angular momentum vector and the unit vector
for the Z axis.10
? ?  ?
The magnitude of a unit vector is always one and the angular momentum vector dotted with the
unit vector will lead to the Zcomponent of the angular momentum vector.
Equation (46) can then be arranged to find the inclination angle.10
( )
The inclination will always be less than or equal to one hundred and eighty degrees.
The longitude of the ascending node can be found by forming the node vector, ?. The
node vector is a constant vector that points in the direction of the ascending node. Equation (48)
can be used to find the node vector.10
? ? ? ? ?
The node vector consists of the X and Y components of the angular momentum. The magnitude
of the node vector can be found in the same way that the magnitude of the eccentricity vector
was found. The longitude of the ascending node is the angle between the X unit vector and the
node vector.10
? ?  ?
33
Equation (51) can be used find the longitude of the ascending node; however, the quadrant of the
longitude of the ascending node must be determined. In order to find the quadrant for the
longitude of the ascending node the Ycomponent for the node vector must be looked at. The test
to determine the quadrant of the longitude of the ascending node is described in equation (52).10
For the case when the Ycomponent of the node vector is less than zero, the longitude of the
ascending node must be recalculated. The new longitude of the ascending node can be found in
equation (53).10
The argument of periapsis is the angle between the node vector and the eccentricity
vector. The same process to find the longitude of the ascending node and the inclination are used
to find the argument of periapsis.10
(
? ?
)
The quadrant for the argument of periapsis must also be determined. This time the variable that
will be looked at to determine the quadrant is the Zcomponent of the eccentricity vector. The
test is described in Equation (55).10
34
For the case when the Zcomponent of the eccentricity vector is negative, the argument of
periapsis must be subtracted from three hundred and sixty degrees. This will result in an angle
greater than one hundred and eighty degrees.10
The final orbital element to be found is the true anomaly. The true anomaly is the angle
between the eccentricity vector and the position vector. This means that the true anomaly can be
found using the same method that was used for the previous three orbital elements. Equation
(57) gives the true anomaly as a function of the eccentricity vector and position vector, as well as
their magnitudes.10
( ? ? )
The quadrant for the true anomaly must be determined as well. The test involves the result from
taking the dot product of the position vector with the velocity vector.10
? ?
? ?
If the dot product is found to be negative, the true anomaly must be recalculated in order to be
greater than one hundred and eighty degrees. The new true anomaly for a negative dot product is
given in equation (59).10
35
The six orbital elements found in this section can be used to describe the trajectory of the
subsatellite once it is released from the TSS. Some of the elements can be used to determine if
the subsatellite will impact the Earth without having to propagate the trajectory of the sub
satellite forward in time. The elements that are used to determine impact are the semimajor axis
and the eccentricity vector. These two orbital elements can be used to calculate the distance at
the perigee point of the trajectory. The perigee point on a trajectory is the lowest point of the
satellite?s orbit.10
If the perigee point is found to be less than the radius of the Earth, then the satellite will impact
the Earth. Being able to determine if the subsatellite will impact before propagating the motion
forward in time will save computational time. The goal of this study is to determine the impact
capabilities of the impact trajectories after release and how different changes affect the time to
impact and range; therefore, if the satellite does not impact the Earth, the propagation of the
motion of the satellite does not need to be done.
2.5 Numerical Integration Using a 4th Order Runge Kutta
Numerical integration is used to solve ordinary differential equations. This feature
allows a state matrix consisting of position, velocity, and constants to be propagated forward in
time. Numerical integration methods use a Taylor series expansion to solve the ordinary
differential equations. The Euler method approximates the solution to the ordinary differential
equations using the firstorder term from the Taylor series. The first and second terms of the
Taylor series are used for the improved Euler method. The Runge Kutta method used for this
research approximates the higher derivatives using finitedifference expressions. This means
36
that the higher derivatives do not have to be calculated from the original ordinary differential
equation that is given.17
The approximation for the Runge Kutta is done by calculating data through a series of
time steps from an initial time, to, to a final time, tf. The order of the Runge Kutta method is
determined by the number of steps used to estimate the state vector after a time step. For this
research a fourth order Runge Kutta is used because it is the most common version of the Runge
Kutta method. Equations (61) through (64) list the four steps used for the fourth order Runge
Kutta method.17
[ ? ]
[ ? ]
[ ? ]
[ ? ]
The function, , is used to find the first derivative of the state vector using the state vector, ?, at
the initial time. After the first step, step two and three are calculated after half of the time step,
?t, has passed. The fourth step is calculated at the initial time plus the time step. After each
time step a new state vector is calculated.17
? ?
The four step process is continued until the final time is reached and a final state vector is
calculated. A Runge Kutta method can be implemented in MATLAB using a WHILE loop or a
37
for loop to get from the initial time to the final time using any time step. Typical values for the
final time are the orbital period of the satellite and the time for a single day. The setup of the
state vector and the terms of the first derivative of the state vector are discussed in chapter three.
2.6 Analytical Approach
In order to establish an analytical approach to the TSS problem more orbital mechanics
equations must be introduced. The energy of an orbit remains constant and can be calculated by
using the semimajor axis or the position and velocity magnitudes of the satellite. Both ways to
calculate the energy of an orbit are shown in equation (66).10
The trajectory equation is used to find the position of a satellite when the semimajor axis, the
eccentricity, and the true anomaly of the satellite are known as given in equation (67).
The energy equation and the trajectory equation will be used later in chapter 3 to find an
analytical solution to the range and time to impact problem for the subsatellite.
The next equation set will be used to find the time to impact for the analytical solutions.
Kepler?s Equation gives the mean anomaly in terms of the eccentric anomaly, E, and the
eccentricity.10
The mean anomaly can be expressed in terms of the mean motion, n, and a difference in time.10
38
The difference in time stands for the time at the eccentric anomaly minus the time at perigee.
Equations (68) and (69) can be defined at two different points, a point one and a point two, and
combined into the differenced Kepler?s equation.10
( )
( )
Where,
?
[( )
?
( )]
The difference in time expressed in equation (71) is the time from release to impact. Point one is
the place in the orbit where the subsatellite is released from the TSS and point two is the place
in the impact trajectory of the subsatellite where the subsatellite impacts the Earth. Equations
(71), (72), and (73) can be used to find the time to impact. Equation (73) is the same for points
one and two. The only value that changes is the true anomaly. The first true anomaly
corresponds to the true anomaly at release and the second true anomaly corresponds to the true
anomaly at impact.
39
Chapter 3: Model and Problem Formulation
In order to investigate this problem, a simplified model is developed for the TSS and the
initial velocity for the center of mass of the TSS is determined. A numerical integration method
is used to propagate the motion of the center of mass of the system using Keplerian motion. The
position and velocity of the end masses is then determined relative to the center of mass. Once
the system successfully travels in its baseline orbit, changes into the system can be introduced.
There are three main changes that are applied to the system: a velocity change on the sub
satellite, a change in the altitude of the TSS, a change in tether length of the TSS, and a change
in the release point location. The change in velocity is needed in order to cause the subsatellite
to enter an impact trajectory, while the tether length and altitude are changed in order to
determine their effect on the range and time to impact. As stated before, the velocity change
placed on the subsatellite will be a result of an impulsive velocity change maneuver or a change
to the angular velocity of the system. The tether is cut after the velocity change is implemented
and the trajectory of the released mass is calculated to determine if the trajectory of the sub
satellite impacts the Earth after release. If the subsatellite impacts the Earth, then the time to
impact and the range from release to impact are calculated. This information is then used to
determine how the changes in the system affect the ballistic capabilities of the subsatellite after
release. An analytical solution is also developed to examine situations where the subsatellite is
directly above, below, to the right, and to the left of the main satellite. At each of these
orientations the analytical solution can be used to determine the maximum and minimum ranges
and times to impact when given a range of velocity changes. A second set of analytical solutions
40
are also developed in order to determine an equivalent angular velocity that can be done in place
of an impulsive velocity change. The setup and algorithms for the simulation and analytical
solutions are discussed in this chapter.
3.1 Problem Setup of the TSS
Before numerical integration of the equations of motion can be performed and before an
impact of the subsatellite can be evaluated an initial position and velocity must be determined.
The initial setup of the TSS needs to be performed in such a way that the position and velocity
of the center of mass and the end masses can be easily determined. The initial setup for the TSS
is shown in Figure 3.1.
Figure 3.1: Initial Setup of the TSS for a Negative ?
41
The center of mass of the TSS is initially placed along the Z axis so that the position of each of
the end masses can be determined from the altitude, TSS orientation, and tether length. The
direction of travel of the TSS is counterclockwise about the origin of the geocentric equatorial
coordinate system and about the positive X  axis. The orbit for the center of mass is assumed to
be circular in shape.
In order to determine the positions of the end masses and the center of mass the altitude,
tether length, and the size of the masses must be chosen. It?s also important to keep in mind that
the tether itself is modeled as a massless, rigid tether. A 10:1 ratio between the main satellite and
the subsatellite was selected; therefore, the mass of the main satellite was assigned a value of
1000 kg and the subsatellite was assigned a mass of 100 kg. The altitude measures the height of
the main satellite above the Earth. The subsatellite is then ?lowered down? from the main
satellite until the tether is completely extended. For the baseline case the altitude was 500 km
above the Earth?s surface with a tether length of 4 km. The angular velocity of the system is set
to zero. Changes to the baseline values will be made in order to determine the effects of the
altitude and tether length on the range and time to impact for an impact trajectory of the sub
satellite. The masses of the two satellites will always remain constant.
With the baseline values determined, the position of the end masses and the position and
velocity of the center of mass can be determined. In order to determine the positions of the TSS,
the system is assumed to be completely aligned with the Z axis. In other words the rotation angle
is zero. Since the TSS is aligned with the Z axis of the geocentric equatorial system, the
positions of the end masses can be determined by adding and subtracting values without having
to worry about the other two axes directions. The main satellite position can be calculated by
adding the radius of the Earth to the altitude. The tether length can then be subtracted from the
42
position of the main satellite to get the position of the subsatellite. The calculations for the
position of the main satellite and for the subsatellite are shown in equations (74) and (75),
respectively.
Both of the positions are initially in the ? direction. The position and velocity of the center of
mass are then calculated using equations (1) and (2). The position for the center of mass is along
the ? direction. The velocity of the center of mass that is calculated is the circular velocity and
is tangent to the circular orbit. Since the TSS moves in a counterclockwise direction, the initial
velocity of the center of mass is in the negative ? direction.
With the position of the center of mass determined, the length between the center of mass
and the two end masses can be calculated. For this initial case the direction of the lengths
between the center of mass and the two end masses is in the ? direction.
The next step after determining the distances between the end masses and the center of mass is to
find the four Euler parameters with the rotation angle in place. In order to find the four Euler
parameters a direction cosine matrix must be generated. The direction cosine matrix for this
instant in time and with a rotation angle can be calculated using the rotation transformation about
an X axis.12
43
[
] 9
The process described in chapter 2 to find the Euler parameters from a given direction cosine
matrix using equations (18) through (28) are used to find the Euler parameters for the initial set
up of the TSS. As the direction cosine matrix changes the direction of the distance between the
end masses and the center of mass will change; however, the magnitude of the distances will
remain constant. In order to determine the new directions the direction cosine matrix must be
premultiplied by the magnitude of the two lengths. This process is shown in equations (30) and
(31). The vector components are then added or subtracted from the position of the center of
mass to determine the new position of the end masses. This process is shown in equations (32)
and (33).
Before the numerical integration is started the velocity for the main satellite and sub
satellite must be calculated. Equations (38) and (39) are used to find the velocity vector for the
subsatellite and the main satellite, respectively. The determination of the velocity vector and the
determination of the position vector for the end masses will be done at each time step until the
subsatellite is released. This is done in order to keep track of the position and velocity vector
for the subsatellite at release. Once the position and the velocity vectors have been calculated
the numerical integration can be started.
In order to do a numerical integration, the initial and final time and an initial state vector
are needed. The initial time for the system is set to zero and a final time is set to one day.
Giving a day for the motion of the TSS, the release of the subsatellite, and the motion of the
subsatellite allows for the determination of the time to impact and the range of an impact
44
trajectory. The time step used for the numerical integration is five seconds; however, the time
step is decreased later when the tether length is increased. Before the release of the subsatellite,
the state vector will contain 15 terms. The terms include the position and velocity components
of the center of mass, the true anomaly, the four Euler parameters, the angular velocity, and the
rotation angle.
?
[
? ?
?
]
The function in the numerical integration calculates the first derivative of the state vector.
The derivative of the state vector is calculated in four steps as described in equations (61)
through (64). In order to get the derivative of the state vector the derivatives of the individual
elements must be taken. The first derivative of the state vector contains the velocity of the center
of mass, the acceleration of the center of mass, the first derivative of the true anomaly, the first
derivative of the four Euler parameters, the angular acceleration, and the derivative of the
rotation angle. Equation (80) gives the first derivative of the state vector.
??
[
? ?
?
?
?
?
?
?
? ]
45
The velocity of the center of mass is taken from the original state vector. The first derivative of
the velocity of the center of mass is equal to the acceleration of the center of mass. Since the
acceleration of the center of mass was not in the original state vector, the acceleration must be
calculated using equations (10) through (12). The derivative of the true anomaly must now be
calculated.
Lovell?s procedure18 is used to determine the first derivative of the true anomaly using
the position and velocity of the center of mass. The first step is to calculate the magnitude of the
angular momentum in geocentric coordinates and polar coordinates. Equation (81) gives the
magnitude of the angular momentum for geocentric coordinates.18
 ? ? ?
Lovell then finds the magnitude of the angular momentum in terms of the body fixed coordinate
system expressed in polar coordinates. Equation (82) gives the magnitude for the angular
momentum using the body fixed coordinates assigned earlier in chapter two.18
 ? ?  ? ? ? ? ?  ?
The angular momentum of an orbit remains constant and the magnitude will always be the same
value no matter what coordinate system is used to express the components. The next step is to
set equation (81) and (82) equal to each other and solve for the first derivative of the true
anomaly. The solution is given below.
? ?
46
The X, Y, and Z components from the position vector and the velocity vector can be taken from
any position or velocity vector. For this case the components will come from the position and
velocity vector of the center of mass, so that the true anomaly of the system can be tracked.
The first derivatives of the Euler parameters, the angular acceleration, and the first
derivative of the rotation angle now have to be determined. The derivatives of the four Euler
parameters are calculated using equation (29). The angular acceleration and angular velocity for
all cases is kept at zero. The derivative of the rotation angle is equal to the angular velocity. The
equivalent angular velocity needed to replace the impulsive velocity change for impact is in the
negative or positive X2 ? direction of the body coordinate system, which is parallel to the X ?
direction of the ECI coordinate system. Since the study is focused on the inplane motion of the
subsatellite, rotation of the TSS about the Y2 and Z2 axes is zero. This means that the Y2 and Z2
components of the equivalent angular velocity will be zero.
This process is continued within a loop until the point that the subsatellite is released.
Prior to release an impulsive velocity change is done by the subsatellite. After the velocity
change, the subsatellite is released from the system at the desired configuration and the release
point position and velocity of the subsatellite are calculated as described below.
? ? ?
The term on the left hand side of equation (84) stands for the velocity of the subsatellite at
release. An equivalent angular velocity that could cause impact can then be calculated by setting
equation (84) equal to equation (38).
? ? ? ?
47
The angular velocity in the X  direction can then be solved for using equation (85). There are
two ways to calculate the angular velocity in the X  Direction. If the velocity of the subsatellite
changes in the Ydirection, equation (86) is used; and equation (87) is used to find the angular
velocity when the velocity changes in the Zdirection.
Equations (86) and (87) can then be solved for the velocity change of the subsatellite. The
velocity change is expressed in vector form in equation (88).
? ? ?
From equation (88), it can be seen that the velocity change of the subsatellite is related to the
angular velocity of the system. This is why an impulsive velocity change or a change in the
angular velocity can lead to the same ballistic capabilities of the subsatellite.
Before a new numerical integration is done for the subsatellite only, the orbital elements
of the subsatellite?s impact trajectory are calculated in order to determine if the subsatellite will
impact the earth. Since the position and velocity vectors of the subsatellite have been calculated
the orbital elements can be calculated using the process discussed in section 2.4. Equation (60)
is then used to calculate the perigee position of the subsatellite along its impact trajectory. If the
perigee position of the subsatellite is greater than the radius of the Earth, the subsatellite will
not impact the earth and the program breaks out of the loop to begin a new test from the
beginning. This is done in order to save computational time. When the perigee position is found
48
to be less than the radius of the Earth, a new state vector is created in order to model the motion
of the subsatellite after release from the system.
The state vector for the subsatellite after release is smaller than the state vector needed
for the TSS. The Euler parameters are no longer needed to model the subsatellite, since the sub
satellite is considered to be a point mass. There is also no rotation angle for the subsatellite
because all vectors can now be expressed in terms of the geocentric coordinates only. The new
state vector that will be used in a 4th order Runge Kutta is expressed in equation (89).
? [ ? ?
]
The first derivative of the state vector is then calculated for the numerical integration.
?? [ ? ?
]
The velocity vector is taken from the original state vector. The acceleration of the subsatellite
can be calculated using the same equations as before by replacing the position and acceleration
center of mass values with the position and acceleration of the subsatellite.
?
?
?
The state vector for the subsatellite is then propagated forward in time using the
numerical integration. This is done until impact occurs or a full day goes by. After a full day
49
goes by, the simulation stops because a day to reach an impact point is too long of a travel time
for a threat. When the magnitude of the position vector is equal to or less than the radius of the
Earth, impact of the subsatellite will occur. When this happens, the program breaks out of the
loop and completes the numerical integration.
Before the range and time to impact can be calculated, the true anomaly at impact and the
true anomaly at release must be calculated using the eccentricity and the semimajor axis from
the impact trajectory. The first step is to take the trajectory equation in equation (67) and solve
for the true anomaly.
(
)
The magnitude for the position of the subsatellite in equation (94) is dependent upon the true
anomaly that is trying to be found. If equation (94) is used to find the true anomaly at impact
than the position of the subsatellite is equal to the radius of the Earth.
(
)
Equation (95) may give a true anomaly at impact that is less than the true anomaly at release.
Since impact occurs after the true anomaly at release, the true anomaly at impact must be greater
than the true anomaly at release. This change can be done by following the test in equation (96).
With the true anomaly at impact in the correct quadrant, the range and time to impact can be
calculated using the following equations.
50
( )
The angle for the arc length in equation (97) is the absolute value of the true anomaly at impact
minus the true anomaly at release. The time to impact subtracts the time at release from the final
time when impact occurs. The impact of the subsatellite can take longer than a day depending
on the type of changes that are placed on the system. Usually an impact that takes longer than a
day means that the impact trajectory of the subsatellite is hyperbolic. If the impact takes longer
than a day, the program continues to run until the end of the day but will not calculate the range
or time to impact because the subsatellite has not impacted yet. For the range and time to
impact to be calculated the total time for the numerical integration to run must be increased.
With this test complete the process can be continued with new altitude and tether length values.
The changes placed on the system to create an impact trajectory will be discussed in the next
section.
3.2 Changes Applied to the System to Create Impact of SubSatellite
For this research two configurations are chosen for the subsatellite upon release during
the numerical simulation. These configurations were chosen to model possible real world
scenarios for releasing the subsatellite from the TSS. The first configuration of the subsatellite
at release will have the subsatellite located below the main satellite with a rotation angle of 0.5
radians. The second configuration will place the subsatellite above the main satellite using a
rotation angle of about 3.6416 radians. The rotation angle is depicted in figure 3.1 and
measures the position of the subsatellite from the vertical Z ? axis. The rotation angle also
51
keeps the subsatellite in a leading position. More points can be chosen and the program can be
easily changed to reflect the new configuration or configurations of the subsatellite at release.
Simply releasing the subsatellite from the TSS by cutting the tether does not always
create an impact trajectory. In order to create impact trajectories an impulsive velocity change of
the subsatellite or a change in the angular velocity of the system must be introduced prior to the
time of release. The change in velocity of the subsatellite is increased or decreased in the Z
direction or the Y direction. All changes in velocity are tested within the program because a
velocity change in the positive Y direction may cause the impact of the subsatellite to occur at a
small velocity change, while a large change in velocity in the negative Z direction may be
needed in order to cause an impact. In order to get the change in the subsatellite velocity to
occur, a thrust force is generated by the subsatellite. The thrust force is not modeled in the
program because the research is not concerned with how the velocity change occurs, but on what
is the minimum velocity change needed to cause impact and how the velocity change will affect
the range and time to impact for the subsatellite system. It is also important to note that a
change in velocity still may not cause the subsatellite to impact. The test for impact is done
because of the fact that an impact still may not occur even with a large velocity change. If a
change in velocity is not done on the subsatellite, a change in the angular velocity can be placed
on the system prior to the time of release in order to create an impact. This angular velocity is
produced by the main satellite and subsatellite generating a thrust that would cause the TSS to
rotate about the center of mass. Once again, the thrust changes are not modeled.
The velocity of the subsatellite can also be changed by introducing a velocity change in
the X direction; however, this case is not investigated because the motion of the TSS is restricted
to the YZ plane. Introducing a change in the X direction will cause out of plane librational
52
motion that will need to be accounted for. While the out of plane librational motion can be
introduced into the program, it will severely complicate the program and the setup or model of
the TSS. One of the goals of this research is to investigate the impact of the changes in altitude,
tether length, and angular velocity using a simplified model for the TSS. For this reason velocity
changes in the X direction on the subsatellite are not considered in this research.
3.3 Variations to the Problem Setup
The changes in the velocity of the subsatellite discussed in the previous section will help
to lead to an impact and will affect the range and time to impact for the subsatellite. The main
properties of the TSS that will be changed are the altitude and the tether length of the system.
When a change in the velocity of the subsatellite leads to an impact, changes in the altitude and
tether length will be analyzed in order to determine if these changes lead to an increase or a
decrease in the range and time to impact.
The altitude that is being changed is the altitude for the main satellite. In this case the
tether length of the TSS is kept constant at four kilometers. The range of the altitude goes from
five hundred kilometers to fifteen hundred kilometers. The minimum value of five hundred
kilometers was chosen because the drag force due to the Earth?s atmosphere does not have to be
taken into account at this point. As the altitude of a satellite decreases and gets closer to the
Earth, the atmospheric drag must be taken into account. The altitude is increased in increments
of fifty kilometers. At each altitude the program is run as discussed in section 3.1 in order to
determine the range and time to impact for the subsatellite after release. The increase in altitude
should lead to an increase in the range and time to impact because the subsatellite is located at a
higher position at release.
53
The range of values used for the tether length is one kilometer to one hundred kilometers.
Three kilometers are used as the increment change from one kilometer to one hundred
kilometers. For this case the altitude is held constant at a value of five hundred kilometers. The
tether length can either increase or decrease the range and time to impact of the subsatellite
depending on the location of the subsatellite relative to the main satellite at the point of release.
For example, if the subsatellite is located above the main satellite, an increase in the tether
length should lead to an increase in the range and time to impact. On the other hand, if the sub
satellite is located below the main satellite, an increase in the tether length should decrease the
range and time to impact of the subsatellite.
3.4 Analytical Solutions
This section discusses the formulation of equations and algorithms for the analytical
solution to the impact problem of a subsatellite after it is released from a TSS. The first
analytical solution that is determined can be used to find the range and time to impact given
initial conditions for the TSS. The second analytical solution solves the problem backwards. In
other words, given the range, true anomaly at impact, and some other initial conditions find the
change in the angular velocity of the TSS or the impulsive velocity change needed in order to
achieve the desired range at impact. The reason behind the analytical approach is to force the
subsatellite to be directly below, above, to the right, or to the left of the main satellite. The set
up for the four cases are depicted in figure 3.2, which is located before the next section.
The position of the center of mass, the main satellite, and the subsatellite must be
determined for each case. Each position component can be described in terms of the radius of
the Earth, the altitude of the main satellite, and the tether length. The position of the main
satellite will always be in the positive Z ? direction and equal to the radius of the Earth plus the
54
altitude. Table 3.1 gives the position of the subsatellite for each of the four cases. Case A, B, C,
and D correspond to when the subsatellite is located above, below, to the right, and to the left of
the main satellite, respectively. In table 3.1 the components of the subsatellite position are given
for each case. The Xcomponent of the subsatellite position vector is always equal to zero
because the orbital motion is kept in the YZ plane.
Table 3.1 Subsatellite Position Components for Cases A  D
Case XComponent (km) YComponent (km) ZComponent (km)
A 0 0 Re + H + Lt
B 0 0 Re + H  Lt
C 0 Lt Re + H
D 0  Lt Re + H
Figure 3.2 Analytical Cases
55
3.4.1 Equations to Find the Range and Time to Impact
For this set of analytical equations the goal is to find expressions for the range and time
to impact. It is assumed that the altitude of the main satellite, the length of the tether, the angular
velocity of the TSS, and the impulsive velocity change of the subsatellite are known. The
angular velocity can have three components in the X2, Y2, and Z2 ? directions; however, the Y2
and Z2 ? components of the angular velocity are equal to zero because the trajectory of the TSS
remains in the YZ plane. For all cases an impulsive velocity change is placed on the sub
satellite; therefore, the angular velocity in the X2 or X direction will also be zero . Since the only
known values are the altitude, tether length, angular velocity, and the velocity change of the sub
satellite, the range and time to impact must be functions of these four known values for the given
mass ratio.
In order to find the range the true anomaly at impact and the true anomaly at release,
equation (97) must be expressed in terms of the four known variables. Since the radius of the
Earth is a constant and the magnitude of the position vector is a function of the altitude, the
tether length, and the radius of the Earth, the semimajor axis and the eccentricity of the
impacting trajectory must now be expressed in terms of the four known values. The energy
equation, or equation (66), can be used to find the semimajor axis.
Equation (43) can then be used to solve for the eccentricity.
?
56
Equation (99) can then be substituted into equation (100).
?
Equation (102) expresses the angular momentum of the subsatellite?s impact trajectory.
? ?  ? ?
The point of release for the subsatellite is located at the point near the apogee of the impact
trajectory, where the flight path angle is zero. The angular momentum can now be expressed in
terms of the magnitudes of the position and velocity vectors of the subsatellite after release.
The velocity of the subsatellite can be written as
? ? ? ? ?
The velocity of the center of mass can be found by using equation (2). The angular velocity only
has a single vector component.
? ?
The velocity vector for the subsatellite can be written as
? [ ] ? [ ( )] ?
The final step is to calculate the magnitude of the subsatellite velocity at release. A change in
velocity can then be added to the velocity of the subsatellite to get the velocity at release, as
57
shown in equation (84). The magnitude of the subsatellite velocity can then be found by taking
the square root of the components squared.
Now that all the terms have been found for the true anomaly the range equation can be
defined. The true anomaly at release is assumed to be one hundred and eighty degrees.
Equations (99), (101), and (102) are substituted into equation (95) to get the true anomaly at the
impact point. The true anomaly at the impact point is then substituted into equation (97). The
range formula given in equation (107) is now a function of the altitude, tether length, angular
velocity, and velocity change of the subsatellite.
( (
) )
Where,
?
In order to get the time to impact for the impact trajectory, the time at release and the
time at impact must be calculated. The time at release can then be found by taking half of the
orbital period, given by
?
58
Since the true anomaly at impact is known, it can be substituted into equation (73) to get the
eccentric anomaly at impact. The time at impact can then be found by using equations (68) and
(69), giving
?
The initial time in equation (111) is equal to zero because it is the time at perigee.
?
This same method can be used to find the time at release if the true anomaly at release is not
equal to one hundred and eighty degrees. The true anomaly at release would be used in equation
(73) to find the eccentric anomaly at release.
The time to impact can now be found by subtracting the time at impact from the time at
release, as
Another method can be used to find the position of the subsatellite if the center of mass
position and velocity vector, and the rotational angle are known in addition to the other values.
The position vector between the center of mass and the subsatellite is defined in equation (114).
? ? ?
Substituting into equation (32) gives
59
? ( ) ? ?
which has a magnitude of
?
For the case when the angular momentum of the subsatellite is
? ( ? ?) ( ? ?)
Equation (117) can then be simplified to
? ( ) ?
Having the magnitude of
?
Equation (116) and (119) can then be used in the second method to find the range and time to
impact, when the center of mass position and velocity vector, and the rotational angle are known.
3.4.2 Algorithm to Find the Range and Time to Impact
Since all the equations are defined for the analytical solution to find the range and time to
impact, an algorithm can be developed for use in a computer program, such as MATLAB, or for
use in a hand calculation of the results. Remember that the known values for this algorithm are
the altitude of the main satellite, the tether length, the angular velocity, and the impulsive
velocity change of the subsatellite. The algorithm is listed in step format beginning on the next
page.
60
Step 1: Calculate the position of the subsatellite at release, position of the main satellite, and the
position of the center of mass. The position of the subsatellite is calculated using table 3.1 and
the position of the center of mass is calculated using
? ? ?
Step 2: Calculate the velocity of the center of mass using
?
Step 3: Calculate the velocity of the subsatellite at release using
? [ ] ? [ ( )] ?
? ? ?
Step 4: Calculate the semimajor axis using
Step 5: Calculate the angular momentum using
Step 6: Calculate the eccentricity using
?
61
Step 7: Calculate the perigee position using
and test for impact using equation (119).
Step 8: Calculate true anomaly at impact using
(
)
And assign the true anomaly at release value using equation (121).
The true anomaly at impact must then be placed in the correct quadrant using
Step 9: Calculate the range using
( )
Note: Step eight and nine can be combined into one step by using
( (
) )
Where,
62
?
Step 10: Calculate the time at release using
?
Step 11: Calculating the eccentric anomaly at impact using
[( )
?
( )]
Step 12: Calculating the time at impact using
?
If the true anomaly at release is much less than or much greater than one hundred and eight
degrees use equation (122) to find the time at impact.
Step 13: Calculate the time to impact. If the true anomaly at impact was found to already be
greater than the true anomaly at release use
63
If the true anomaly at impact was changed in step 8, use equation (123) to calculate the time to
impact. This eliminates the issue caused by a negative eccentric anomaly.
If the true anomaly at release is much less than or much greater than one hundred and eight
degrees use equations (122) and (113) to find the time to impact.
If the center of mass position and velocity vectors and the rotational angle are known in
addition to the other values, some steps in the algorithm will change. The changes to the
algorithm are listed below.
Step 1: Calculate the length between the center of mass and the subsatellite using
)
Then calculate the position of the subsatellite at release using
? ( ) ? ?
Step two can then be omitted.
Step 5: Calculate the angular momentum using equation (119).
?
Step 8: Calculate the true anomaly at release using
(
)
64
The final steps remain the same for this set of given values.
3.4.3 Equations to Find the Velocity Change and Angular Velocity
In this section equations are developed to find the impulsive velocity change of the sub
satellite and the equivalent angular velocity of the TSS needed to cause an impact if an impulsive
velocity change is not to be placed on the system given a set of known values. Five known
values are needed for each solution. Four of the known values remain the same for each
solution. These are the range, the true anomaly at impact, the altitude, and the tether length. The
fifth known value depends upon what is being solved for. If the velocity change of the sub
satellite is to be determined, the angular velocity of the TSS must be known. When the change
in the angular velocity of the TSS is being solved for, the velocity change of the subsatellite
caused by the angular velocity must be known. Since the known values have changed the
equations found in section 3.4.1 need to be rearranged to find the velocity change and the change
angular velocity.
With this analytical solution the assumption that the point of release is located at the
apogee of the impact trajectory no longer holds. This means that the true anomaly at release
must be determined. This is done by first getting rid of the absolute value in equation (97) by
multiply the left hand side by a negative one and then solving for the true anomaly at the release
point for the subsatellite.
65
The next step is to use the true anomaly at impact to find the semimajor axis of the impacting
trajectory in terms of the eccentricity of the impacting trajectory. This is done by solving for the
semimajor axis in equation (99).
Equation (126) is then substituted into equation (94).
( ( )
)
Equation (127) is then used to solve for the eccentricity of the impact trajectory.
Two different equations must be developed for finding the impulsive velocity change of
the subsatellite and for finding the change in angular velocity of the TSS. In order to find the
velocity change of the subsatellite when given an angular velocity, equation (38) must be
substituted into equation (84).
? ? ? ? ? ? ?
Since the angular velocity of the TSS is assumed to be zero when an impulsive velocity change
is applied, the cross product in equation (129) is dropped and the velocity change can be solved
for by using equations (130) and (131).
66
The equivalent angular velocity for a velocity change of the subsatellite can be found using
equations (86) and (87) depending upon the direction of the velocity change.
3.4.4 Algorithm to Find the Velocity Change and Angular Velocity
The algorithm to find the impulsive velocity change of the subsatellite and the algorithm
to find the change in the angular velocity of the TSS to cause an impact of the subsatellite given
a set of known parameters vary slightly. The first algorithm that will be looked at is the
algorithm to solve for the impulsive velocity change of the subsatellite.
Step 1 ? 6: Same as the steps discussed in section 3.4.2.
Step 7: Calculate the velocity change using equation (130) or equation (131).
If the velocity change calculated is reasonable, then the algorithm can be stopped. If the velocity
change is not reasonable, the altitude or tether length must be increases for decreased to reach a
reasonable value.
The second algorithm that will be looked at is the algorithm to solve for the equivalent
angular velocity change of the subsatellite.
Step 1 ? 6: Same as the steps discussed in section 3.4.2.
Step 7: Calculate the equivalent angular velocity using equation (86) or equation (87).
67
If the angular velocity change calculated is reasonable, then the algorithm can be stopped. If the
angular velocity change is not reasonable, the altitude or tether length must be increases for
decreased to reach a reasonable value. The effects of the change in altitude and tether length on
the angular velocity and on the velocity change of the subsatellite are shown in chapter four.
68
Chapter 4: Results for Analytical Solutions
The cases that will be considered for this chapter are cases A through D, which were
described in chapter three, section four. Case A and B are when the subsatellite is located
directly above and directly below the main satellite, while Case C and D are when the sub
satellite is located immediately to the right, in the direction opposite the velocity vector, and
immediately to the left, in the same direction as the velocity vector, of the main satellite. The
first section of this chapter will discuss the ground range and the time to impact found by using
the algorithm discussed in chapter three, section 3.2. The second section discusses the results
when the algorithm discussed in chapter three, section 3.4 is used to find the impulsive velocity
change and the change in the angular velocity of the TSS.
4.1 Analytical Results for the Range and Time to Impact
For this case the altitude, tether length and the impulsive velocity change are known and
are used to calculate the ground range and time to impact. For the results presented here, the
altitude and tether length are kept constant, while the impulsive velocity change is increased
from zero kilometers per second to six kilometers per second. Three different setups for the
altitude and the tether length are evaluated. The three setups are a five hundred kilometer
altitude and a four kilometer tether length, an eight hundred kilometer altitude and a four
kilometer tether length, and a five hundred kilometer tether length and a forty kilometer tether.
These three setups are used in order to investigate the results obtained by increasing the altitude
and increasing the tether length.
69
4.1.1 Case A Results
The position of the TSS at the point of release was shown in figure 3.2. In this situation
the velocity of the subsatellite is directed solely in the negative Y ? direction and it is assumed
any impulsive velocity change that takes place must be done in the positive Y ? direction to
create impact. The trends for the ground range and the time to impact as the magnitude of the
impulsive velocity change increases hold for all three setups.
Figure 4.1: Ground Range vs. +?V1y for Case A
0
2000
4000
6000
8000
10000
12000
14000
16000
0 1 2 3 4 5 6 7
Gr
ou
nd
Ran
ge
, R
(km
)
Change in Velocity in the Y  Direction, Delta Vy (km/s)
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
70
Figure 4.2: Time to Impact vs. +?V1y Increases for Case A
A minimum impulsive velocity change in the positive Y ? direction is required in order to
create an impact trajectory for each of the three setups. The minimum impulsive velocity
needed to create an impact point changes when the altitude is increased to eight hundred
kilometers; however, a change in the tether length does not result in a change to the minimum
impulsive velocity change needed for an impact trajectory. The minimum impulsive velocity
change values needed for the three setups are listed in table 4.1.
Table 4.1: Minimum Impulsive Velocity Change Needed for Case A
Altitude (km) Tether Length (km)
Minimum Velocity Change needed in
+ Y  direction (km/s)
500 4 0.2
800 4 0.3
500 40 0.2
The maximum ground range and time to impact occur at the minimum impulsive velocity
change, while the minimum ground range and time to impact occur at the maximum impulsive
0
500
1000
1500
2000
2500
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Change in Velocity in the Y  Direction, Delta Vy (km/s)
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
71
velocity change. Depending upon the altitude and the tether length the value for the ground
range and time to impact could increase or decrease. For these three cases an increase in the
tether length to a value of forty kilometers, results in the largest increase in the ground range and
the time to impact. The increase in the altitude to eight hundred kilometers results in a small
decrease in the ground range and time to impact. These effects are shown in figures 4.1 and 4.2.
The next two tables list the actual maximum and minimum values for the ground range and time
to impact for each of the setups. The maximum values occur at the minimum impulsive
velocity change placed on the subsatellite for the three cases listed.
Table 4.2 Maximum Ground Range and Time to Impact for Case A
Altitude
(km)
Tether Length
(km)
Ground Range
(km)
Time to Impact
(s)
500 4 13065.3 1793.5
800 4 13220.3 1899.3
500 40 15165.6 2077.0
The maximum value for the ground range and the time to impact occur when the tether length
increases to forty kilometers and the velocity change is 0.2 kilometers per second in the positive
Y ? direction. The increase in altitude leads to an increase in the ground range and the time to
impact, but an increased velocity change is needed in order to create an impact point at the
higher altitude. This results in the smaller increase to the ground range and time to impact.
Table 4.3: Minimum Ground Range and Time to Impact for Case A
Altitude
(km)
Tether Length
(km)
Ground Range
(km)
Time to Impact
(s)
500 4 549.9 349.8
800 4 636.5 456.4
500 40 570.1 363.8
72
The minimum values for the ground range and time to impact occur when the maximum
impulsive velocity change is placed on the subsatellite. For Cases A through D this maximum
velocity change is six kilometers per second in the positive Y ? direction. As the velocity change
increases the ground range and the time to impact are decreased. The true minimum value
comes from the first setup, where the altitude is five hundred kilometers and the tether length is
four kilometers. This time the increase in altitude results in a higher minimum value for the
ground range and time to impact. The third setup serves as a midpoint for the minimum values
that can be obtained.
4.1.2 Case B Results
For case B, the subsatellite is located below the main satellite. The center of mass, main
satellite, and subsatellite have a velocity solely in the negative Y ? direction. In order to cause
the subsatellite to enter an impact trajectory, the impulsive velocity change of the subsatellite
must be in the positive Y ? direction. The ground range and time to impact decrease as the
impulsive velocity change in the positive Y ? direction is increase. The decrease in the ground
range and the time to impact as the impulsive velocity change increases are shown in figures 4.3
and 4.4 on the next page.
73
Figure 4.3: Ground Range vs. +?V1y for Case B
Figure 4.4: Time to Impact vs. +?V1y for Case B
The minimum impulsive velocity change in the positive Y ? direction needed for the subsatellite
to have an impact trajectory for each of the three setups is listed in table 4.4.
0
2000
4000
6000
8000
10000
12000
14000
0 1 2 3 4 5 6 7
Gr
ou
nd
Ran
ge
, R
(km
)
Change in Velocity in the Y  Direction, Delta Vy (km/s)
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Change in Velocity in the Y  Direction, Delta Vy (km/s)
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
74
Table 4.4: Minimum Impulsive Velocity Change Needed for Case B
Altitude (km) Tether Length (km)
Minimum Velocity Change needed in
+ Y  direction (km/s)
500 4 0.2
800 4 0.3
500 40 0.2
The maximum and minimum ground range and time to impact occur at the minimum impulsive
velocity change and the maximum impulsive velocity change of six kilometers per second,
respectively. An increase in the altitude leads to a larger increase in the ground range and time
to impact, than an increase in the tether length.
Table 4.5: Maximum Ground Range and Time to Impact for Case B
Altitude
(km)
Tether Length
(km)
Ground Range
(km)
Time to Impact
(s)
500 4 12696.5 1743.2
800 4 12980.0 1865.2
500 40 11279.4 1548.9
The maximum ground range and time to impact occur when the altitude is increased because as
the altitude increases the subsatellite moves further away from the surface of the Earth. The
lowest maximum value occurs when the tether length is increased, because the subsatellite gets
closer to the surface of the Earth as the tether increases. If the altitude is increased further, the
ground range and time to impact should also increase; however, if the tether length increases
instead, the ground range and time to impact will decrease.
75
Table 4.6: Minimum Ground Range and Time to Impact for Case B
Altitude
(km)
Tether Length
(km)
Ground Range
(km)
Time to Impact
(s)
500 4 545.3 346.7
800 4 633.1 453.7
500 40 524.3 332.4
The minimum values for the ground range and the time to impact are affected by the increase in
the altitude and tether length in the same way that the maximum values are affected. The largest
ground range and time to impact happens when the altitude increases, while the true minimum
occurs when the tether length is increased. The ground range and time to impact calculated for
case B in all setups is less than the ground range and time to impact calculated in case A.
4.1.3 Case C Results
The subsatellite is located immediately to the right of the main satellite. In this position
the subsatellite?s velocity is only in the negative Y ? direction; therefore, an impulsive velocity
change in the positive Y ? direction is needed in order to create an impact for the subsatellite.
As the impulsive velocity change in the positive Y ? direction is increased, the ground range and
the time to impact decrease. The trends mentioned above can be seen in figures 4.5 and 4.6 on
the next page.
76
Figure 4.5: Ground Range vs. +?V1y for Case C
Figure 4.6: Time to Impact vs. +?V1y for Case C
The minimum impulsive velocity change needed to create the first impact trajectory for each set
up is listed on the next page in table 4.7.
0
2000
4000
6000
8000
10000
12000
14000
16000
0 1 2 3 4 5 6 7
Gr
ou
nd
Ran
ge
, R
(km
)
Change in Velocity in the Y  Direction, Delta Vy (km/s)
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Change in Velocity in the Y  Direction, Delta Vy (km/s)
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
77
Table 4.7: Minimum Impulsive Velocity Change Needed for Case C
Altitude (km) Tether Length (km)
Minimum Velocity Change needed
in + Y  direction (km/s)
500 4 0.2
800 4 0.3
500 40 0.2
The maximum ground range and time to impact will occur at the minimum impulsive velocity
changes listed in table 4.7 for each setup. When the impulsive velocity change reaches the
maximum value of six kilometers per second, the ground range and time to impact will be at
their minimum values. The maximum and minimum values are listed for the ground range and
time to impact below in table 4.8 and table 4.9.
Table 4.8: Maximum Ground Range and Time to Impact for Case C
Altitude
(km)
Tether Length
(km)
Ground Range
(km)
Time to Impact
(s)
500 4 12945.5 1758.1
800 4 13140.0 1875.6
500 40 13535.9 1667.9
Table 4.9: Minimum Ground Range and Time to Impact for Case C
Altitude
(km)
Tether Length
(km)
Ground Range
(km)
Time to Impact
(s)
500 4 547.8 348.2
800 4 635.0 455.0
500 40 549.4 347.2
The largest maximum value that is obtained for case C occurs when the tether is increased,
because an increase in the tether length places the subsatellite further away from the impact
78
point. The largest minimum value for case C occurs when the altitude is increased because the
altitude places the subsatellite in a higher orbit.
4.1.4 Case D Results
The position of the subsatellite for case D is directly to the left main satellite. In this set
up the velocity of the main satellite, the center of mass, and the subsatellite is only in the
negative Y ? direction; therefore, the impulsive velocity change needs to be in the positive Y ?
direction in order to cause the subsatellite to enter an impact trajectory. The ground range and
time to impact for all three setups decreases as the impulsive velocity change in the positive Y ?
direction increases. The change in ground range is shown below, while the change in the time to
impact is shown in figure 4.8 on the next page.
Figure 4.7: Ground Range vs. +?V1y for Case D
0
2000
4000
6000
8000
10000
12000
14000
16000
0 1 2 3 4 5 6 7
Gr
ou
nd
Ran
ge
, R
(km
)
Change in Velocity in the Z  Direction, Delta Vz (km/s)
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
79
Figure 4.8: Time to Impact vs. +?V1y for Case D
The minimum impulsive velocity change needed to send the subsatellite on an impact trajectory
follows the same trends that are in the previous cases.
Table 4.10 Minimum Impulsive Velocity Change Needed for Case D
Altitude (km) Tether Length (km)
Minimum Velocity Change needed in
+ Y  direction (km/s)
500 4 0.2
800 4 0.3
500 40 0.2
The maximum and minimum values for the ground range and the time to impact are listed in
tables 4.11 and 4.12, respectively. The maximum ground range and time to impact occur when
the minimum impulsive velocity changes listed in table 4.10 are placed on the subsatellite. The
minimum ground range and time to impact occur when the impulsive velocity change is equal to
six kilometers per second.
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Change in Velocity in the Z  Direction, Delta Vz (km/s)
H = 500 km Lt = 4 km
H = 800 km Lt = 4 km
H = 500 km Lt = 40 km
80
Table 4.11: Maximum Ground Range and Time to Impact for Case D
Altitude
(km)
Tether Length
(km)
Ground Range
(km)
Time to Impact
(s)
500 4 12945.5 1758.1
800 4 13140.2 1875.6
500 40 13535.1 1667.9
Table 4.12: Minimum Ground Range and Time to Impact for Case D
Altitude
(km)
Tether Length
(km)
Ground Range
(km)
Time to Impact
(s)
500 4 547.8 348.2
800 4 635.0 455.0
500 40 549.4 347.2
The values for the maximum and minimum ground range and time to impact are the same as the
values in case C. The only difference is the position vector of the subsatellite in both cases;
therefore, the trajectories are only shifted over because of the subsatellite?s orientation with the
main satellite.
4.2 Analytical Results for the Impulsive Velocity Change and the Angular Velocity
In this section it is assumed that the altitude, tether length, ground range, and true
anomaly at impact are known. The algorithm discussed in chapter three, section 3.4 is used to
get the results in this section. Two scenarios are looked at to use the analytical results to find the
impulsive velocity change and change in angular velocity of the TSS at different altitudes and
tether lengths. The first scenario sets the ground range at fifteen hundred kilometers with a true
anomaly at impact of two hundred and forty degrees or 4.1888 radians. The second scenario sets
the ground range at three thousand kilometers with a true anomaly at impact of two hundred and
forty degrees. As each scenario is looked at, case A and B are placed on the same plot because
81
in both situations the velocity of the subsatellite and the impulsive velocity change are along the
Y ? axis, while case C and D are placed on the same plot. For all cases the velocity of the sub
satellite is in the negative Y ? direction and the impulsive velocity change is in the positive Y ?
direction.
4.2.1 Ground Range of 1500 km and
The impulsive velocity change and the equivalent change in angular velocity of the TSS
were first found for case A and case B. The altitude starts at five hundred kilometers and is then
increased to fifteen hundred kilometers in increments of one hundred kilometers. As the altitude
increases the impulsive velocity change and the change in angular velocity of the TSS needed to
reach a ground range of fifteen hundred kilometers decreases in magnitude. The impulsive
velocity change for case A and B are slightly different as the altitude increases and are in the
same direction. For case A the change in angular velocity must be in the negative X ? direction
to cause a clockwise rotation of the TSS to create the same velocity change. The change in
angular velocity must be in the positive direction for case B to cause a counterclockwise
rotation. The plots of the impulsive velocity change and the change in angular velocity as a
function of changing altitude are shown on the next page and are followed by a table that gives
specific values for the velocity change and the angular velocity at three different altitudes.
82
Figure 4.9 Velocity Change vs. Altitude for Case A and B; R = 1500 km and ?imp = 240?
Figure 4.10 Angular Velocity vs. Altitude for Case A and B; R = 1500 km and ?imp = 240?
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 500 1000 1500 2000
Ch
an
ge
in
Ve
loc
ity
in t
he
Y
 D
ire
ction
,
De
lta
Vy
(km
/s)
Altitude, H (km)
Case A
Case B
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0 500 1000 1500 2000
An
gu
lar
Ve
loc
ity,
Om
eg
a_
x (r
ad
s/s)
Altitude, H (km)
Case A
Case B
83
Table 4.13 Impulsive Velocity Change and Angular Velocity at Three Altitudes for Case A and
B When R = 1500 km and ?imp = 240?
Case A Case B
Altitude
(km)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
500 0.6048 0.1663 0.5981 0.1645
1000 0.5012 0.1378 0.5025 0.1382
1500 0.0681 0.0187 0.0731 0.0201
From table 4.13 it can be seen that the values for the impulsive velocity change and the
magnitude of the change in angular velocity differ slightly, because of the placement of the sub
satellite with respect to the main satellite. For both cases as the altitude increases the required
impulsive velocity change and change in angular velocity of the system is decreased.
The next change that is placed on case A and B is an increase in the tether length. The
tether length starts at four kilometers and is increased to one hundred and fourteen kilometers in
increments of ten kilometers. As the tether length increases the impulsive velocity change of the
subsatellite is decreased for case B, while the impulsive velocity change increases for case A.
The magnitude of the equivalent angular velocity needed to reach the same ground range and
true anomaly at impact decreases as the tether length increases. These trends are shown in
figures 4.11 and 4.12. The plots are then followed by a table that lists specific impulsive velocity
changes and angular velocities that are needed to achieve the desired ground range and impact
point.
84
Figure 4.11 Velocity Change vs. Tether Length for Case A and B; R = 1500 km and ?imp = 270?
Figure 4.12 Angular Velocity vs. Tether Length for Case A and B; R = 1500 km and ?imp = 240?
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 20 40 60 80 100 120
Ch
ange i
n Vel
oc
ity
in
th
e Y
 D
irec
tion
,
De
lta
Vy
(km
/s)
Tether Length, Lt (km)
Case A
Case B
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100 120
An
gu
lar
Ve
loc
ity,
Om
eg
a_
x (r
ad
s/s)
Tether Length, Lt (km)
Case A
Case B
85
Table 4.14: Impulsive Velocity Change and Angular velocity at Three Tether Lengths for Case A
and B When R = 1500 km and ?imp = 240?
Case A Case B
Tether
Length
(km)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
4 0.6048 0.1663 0.5981 0.1645
54 0.6431 0.0131 0.5515 0.0112
114 0.6804 0.0066 0.4856 0.0047
The impulsive velocity changes increase for case A as the tether length increases because the
subsatellite is located above the main satellite. The impulsive velocity change decreases for
case B because the subsatellite is located below the main satellite. The change in angular
velocity of the TSS needed to create the same ground range and impact point in case A and B
decrease because the increase in tether length is a larger value than the velocity change.
Now the effect of increasing altitude and tether length will be investigated for cases C
and D. The change in angular velocity of the TSS for both cases was found to be zero because an
impulsive change in velocity in the positive Y ? direction on the subsatellite will not lead to a
rotation of the TSS. A change in velocity in the positive or negative Z ? direction will lead to a
rotation. For a ground range of fifteen hundred kilometers and a true anomaly of two hundred
and forty degrees, the increase in altitude will results in a decrease of the magnitude of the
velocity change.
86
Figure 4.13 Velocity Change vs. Altitude for Case C and D; R = 1500 km and ?imp = 240?
Table 4.15: Impulsive Velocity Change and Angular Velocity at Three Altitudes for Case C and
D When R = 1500 km and ?imp = 270?
Case C Case D
Altitude
(km)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
500 0.6015 N/A 0.6015 N/A
1000 0.5019 N/A 0.5019 N/A
1500 0.0706 N/A 0.0706 N/A
At each altitude the magnitude of the impulsive velocity change is the same for each case. This
similarity occurs because the subsatellite in case C and D has the same altitude as the main
satellite.
The increasing tether length for case C and D leads to an increase in the magnitude of the
impulsive velocity change. The change in angular velocity is still equal to zero because of the
reasons explained for the changing altitude. Three values for impulsive velocity change and
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 500 1000 1500 2000
Ch
an
ge
in
Ve
loc
ity
in t
he
Z
Dir
ec
tion
,
De
lta
Vz (
km
/s)
Altitude, H (km)
Case C
Case D
87
change in angular velocity are shown in table 4.16 for both cases at an altitude of five hundred
kilometers, one thousand kilometers, and fifteen hundred kilometers.
Figure 4.14 Velocity Change vs. Tether Length for Case C and D; R = 1500 km and ?imp = 240?
Table 4.16: Impulsive Velocity Change and Angular Velocity at Three Tether Lengths for Case
C and D When R = 1500 km and ?imp = 240?
Case C Case D
Tether
Length
(km)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
4 0.6015 N/A 0.6015 N/A
54 0.6017 N/A 0.6017 N/A
114 0.6023 N/A 0.6023 N/A
4.2.2 Ground Range of 3000 km and
The same increase in altitude and tether length are placed on the four cases; however, the
desired ground range has been doubled in order to see if there are any changes to the impulsive
velocity change and change in angular velocity trends by increasing the altitude and tether
0.6014
0.6015
0.6016
0.6017
0.6018
0.6019
0.602
0.6021
0.6022
0.6023
0.6024
0 20 40 60 80 100 120
Ch
an
ge
in
Ve
loc
ity
in t
he
Y
Dir
ec
tion
,
De
lta
Vy
(km
/s)
Tether Length, Lt (km)
Case C
Case D
88
length. As in the previous section, the first two cases that will be looked at are case A and B and
the first change that will be investigated is the increase in the altitude of the main satellite. The
increase in altitude leads to an increase in the magnitude of the impulsive velocity change and
change in angular velocity for both cases. The impulsive velocity change and change in angular
velocity of the TSS as a function of altitude are shown in figures 4.15 and 4.16.
Figure 4.15 Velocity Change vs. Altitude for Case A and B; R = 3000 km and ?imp = 240?
Figure 4.16 Angular Velocity vs. Altitude for Case A and B; R = 3000 km and ?imp = 240?
0
0.2
0.4
0.6
0.8
1
1.2
0 500 1000 1500 2000
Ch
an
ge
in
Ve
loc
ity
in t
he
Y

Dir
ec
tion
, D
elta
Vy
(k
m/
s)
Altitude, H (km)
Case A
Case B
0.4
0.3
0.2
0.1
0
0.1
0.2
0.3
0.4
0 500 1000 1500 2000
An
gu
lar
Vel
oc
ity
, Ome
ga_
x
(rad
s/s)
Altitude, H (km)
Case A
Case B
89
Table 4.17: Impulsive Velocity Change and Angular Velocity at Three Altitudes for Case A and
B When R = 3000 km and ?imp = 240?
Case A Case B
Altitude
(km)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
500 0.5968 0.1641 0.5857 0.1611
1000 0.9200 0.2530 0.9128 0.2510
1500 1.0687 0.2939 1.0641 0.2926
With this ground range the increase in altitude leads to an increase in the impulsive velocity
change because a greater velocity change is needed in order to create an impact at a higher
altitude and to achieve a greater ground range. As the velocity change increases, the change in
angular velocity increases as well because a larger equivalent angular velocity is needed to create
the same impact point. The angular velocity for case A is negative because the angular velocity
of the system needs to be in the negative X ? direction in order to create a clockwise rotation.
The angular velocity is positive in order to create a counterclockwise rotation for case B.
The increase in tether length will also affect the impulsive velocity change and the
change in angular velocity for case A and B. The impulsive velocity change placed on the sub
satellite at release decreases with the increasing tether length for case B; however, the impulsive
velocity change increases with increase tether length in case A. The increase in tether length
also causes the change in angular velocity of the TSS to decrease in magnitude. This decrease in
angular velocity is a result of the increase in the distance between the center of mass and the sub
satellite. As the distance increases the amount of rotation or angular velocity that is required to
place the TSS in the proper orientation to cause the subsatellite to enter an impact trajectory
decreases. The impulsive velocity change and change in angular velocity as a function of
increasing tether length are shown in figures 4.17 and 4.18.
90
Figure 4.17 Velocity Change vs. Tether Length Case A and B; R = 3000 km and ?imp = 240?
Figure 4.18 Angular Velocity vs. Tether Length for Case A and B; R = 3000 km and ?imp = 240?
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 20 40 60 80 100 120
Ch
ange i
n Vel
oc
ity
in
th
e Y
 D
irec
tion
,
De
lta
Vy
(km
/s)
Tether Length, Lt (km)
Case A
Case B
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0 20 40 60 80 100 120
An
gu
lar
Ve
loc
ity,
Om
eg
a_
x (r
ad
s/s)
Tether Length, Lt (km)
Case A
Case B
91
Table 4.18: Impulsive Velocity Change and Angular Velocity at Three Tether Lengths for Case
A and B When R = 3000 km and ?imp = 240?
Case A Case B
Tether
Length
(km)
Velocity Change
(km/s)
Angular Velocity
(rads/s)
Velocity Change
(km/s)
Angular Velocity
(rads/s)
4 0.5968 0.1641 0.5857 0.1611
54 0.6642 0.0135 0.5149 0.0105
114 0.7414 0.0072 0.4260 0.0041
The impulsive velocity change for case A increases with increasing tether length because the
subsatellite is placed in a higher orbit away from the Earth. The impulsive velocity change for
case B decreases with increasing tether length because the subsatellite is lowered into an orbit
closer to the Earth. The change in angular velocity of the TSS needed for impact decreases with
increasing tether length. The magnitudes of the angular velocities differ between the two cases
because of the different impulsive velocity changes. The angular velocity in case A is in the
negative X ? direction because a clockwise rotation is needed for impact.
The increase in altitude and increase in tether length will now be applied to case C and D.
From the previous case in section 4.2.1, it is expected that the results for case C and D for each
change should be equal in magnitude and direction. The first parameter that will be increased on
case C and D is the altitude. As the altitude increases the impulsive velocity change required for
the subsatellite to reach a ground range of three thousand kilometers at a true anomaly at impact
of two hundred and forty degrees increase as well. The change in angular velocity remains at
zero because the impulsive velocity change direction in the positive Y ? direction does not create
a rotation on the system. Figure 4.19 shows the impulsive velocity change as a function of the
92
increase in altitude, and is followed by a table that list specific impulsive velocity change and
change in angular velocity values for three altitudes.
Figure 4.19 Velocity Change vs. Altitude for Case C and D; R = 3000 km and ?imp = 240?
Table 4.19: Impulsive Velocity Change and Angular Velocity at Three Altitudes for Case C and
D When R = 3000 km and ?imp = 240?
Case C Case D
Altitude
(km)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
500 0.5912 N/A 0.5912 N/A
1000 0.9164 N/A 0.9164 N/A
1500 1.0664 N/A 1.0664 N/A
As expected the magnitudes and directions of the impulsive velocity change are equal between
the two cases. This is a result of the placement of the subsatellite in the same orbit height as the
main satellite at release. The larger ground range causes the impulsive velocity change to
increase in order to reach the desired impact.
0
0.2
0.4
0.6
0.8
1
1.2
0 500 1000 1500 2000
Ch
an
ge
in
Ve
loc
ity
in t
he
Y
Dir
ec
tion
,
De
lta
Vy
(km
/s)
Altitude, H (km)
Case C
Case D
93
The last change that will be looked at is the increase in the tether length for case C and D.
As the tether length increases the impulsive velocity change increases. The change in angular
velocity of the TSS is once again zero because of the direction of the velocity change. The
increase in the impulsive velocity change as a function of the tether length is shown below.
Table 4.20 lists the values for the impulsive velocity change and the change in angular velocity
at a tether length of four kilometers, fiftyfour kilometers, and one hundred and fourteen
kilometers. The table can be found on the next page.
Figure 4.20 Velocity Change vs. Tether Length for Case C and D; R = 3000 km and ?imp = 240?
0.591
0.5912
0.5914
0.5916
0.5918
0.592
0.5922
0.5924
0.5926
0.5928
0 20 40 60 80 100 120
Ch
an
ge
in
Ve
loc
ity
in t
he
YD
ire
ction
,
De
lta
Vy
(km
/s)
Tether Length, Lt (km)
Case C
Case D
94
Table 4.20: Impulsive Velocity Change and Angular Velocity at Three Tether Lengths for case C
and D When R = 3000 km and ?imp = 240?
Case C Case D
Tether
Length
(km)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
Velocity Change
(km/s)
Angular
Velocity (rads/s)
4 0.5912 N/A 0.5912 N/A
54 0.5916 N/A 0.5916 N/A
114 0.5926 N/A 0.5926 N/A
The impulsive velocity change for both case C and D have the same magnitudes and directions
because the subsatellite for both cases is located at the same orbit height as the main satellite.
95
Chapter 5: Results for Numerical Integration Simulation
The process discussed in section 3.1 is now implemented with the changes discussed in
sections two and three in chapter three. Three different impulsive velocity changes were chosen
for the two different release configurations. For each configuration and impulsive velocity
change the effect of changes in altitude and tether length were investigated. The first
configuration corresponds to when the subsatellite is below the main satellite. The second
configuration corresponds to when the subsatellite is above the main satellite. For each change
in altitude and tether length three values are chosen to discuss in detail after the general trend is
introduced. For the change in altitude the tether length is kept constant at four kilometers. The
altitude is kept at five hundred kilometers when the tether length is changed. A range of
impulsive velocity changes are then looked at to determine the minimum impulsive velocity
change needed to impact and the change in angular velocity of the TSS needed to impact. For
the range of impulsive velocity changes the altitude is kept at five hundred kilometers and the
tether length is kept at four kilometers. For all cases and release points the rotation angle is 0.5
radians because it is a good midrange oscillation that will keep the subsatellite leading the
system. Some sample results for the numerical integration simulation can be found in Appendix
A.
96
5.1 Results for First Configuration or Case 1
Three values were chosen for the impulsive velocity change of the subsatellite as shown
in table 5.1. An impulsive velocity change in the negative Ydirection was not chosen because
the velocity change would only increase the subsatellite velocity and an impact trajectory would
not be created.
Table 5.1: Impulsive Velocity Changes for Case 1
Velocity Change (km/s)
1.0
N/A
1.0
3.0
5.1.1 Impulsive Velocity Change of 1.0 km/s in the Positive YDirection
The first change that will be looked at is the change in altitude. Remember that the
altitude defines the height above the Earth for the main satellite and the subsatellite is a distance
away from the main satellite equal to the tether length.
Figure 5.1 Ground Range vs. Altitude for ?V1y = 1.0 km/s in Case 1
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 500 1000 1500 2000
Gr
ou
nd
Ran
ge
, R
(km
)
Altitude, H (km)
97
For the first configuration an increase in the altitude of the main satellite leads to an increase in
the range. This makes sense because as the altitude is increased the subsatellite is in a higher
orbit and it travels for a longer time before impact is reached. The time increase can be seen in
Figure 5.2. As the orbit of the subsatellite gets higher the ground range covered by the sub
satellite in its impact trajectory will increase as well.
Figure 5.2 Time to Impact vs. Altitude for ?V1y = 1.0 km/s in Case 1
An altitude of five hundred kilometers, one thousand kilometers, and one thousand five hundred
kilometers are looked at in more detail. Table 5.2 lists the ground range covered by the sub
satellite after release and the time to impact.
Table 5.2 Range and Impact Time for km/s at three Altitudes for Configuration 1
Altitude (km) Ground Range (km) Time to Impact (s)
500 4505.7 700
1000 6377.7 1055
1500 7822.9 1370
0
200
400
600
800
1000
1200
1400
1600
0 200 400 600 800 1000 1200 1400 1600
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Altitude, H (km)
98
The lowest ground range and time to impact happen at an altitude of five hundred kilometers,
while the maximum for each happens at an altitude of one thousand five hundred kilometers. To
achieve maximum ground range covered by the subsatellite with a constant tether length and
impulsive velocity change the altitude should be increased. If a minimum ground range is
needed the altitude should be decreased. The next three figures show the trajectory for the sub
satellite after release at the three altitude values and illustrate how the range and time to impact
increase as the altitude increases.
Figure 5.3 Trajectory at 500 km Altitude for ?V1y = 1.0 km/s in Case 1
99
Figure 5.4 Trajectory at 1000 km Altitude for ?V1y = 1.0 km/s in Case 1
Figure 5.5 Trajectory at 1500 km Altitude for ?V1y = 1.0 km/s in Case 1
100
Increasing the tether length also affects the time to impact and the ground range covered
by the subsatellite after release. As the tether length increases the ground range and the time to
impact decrease as well. This is due to the fact that the subsatellite is located below the main
satellite.
Figure 5.6 Ground Range vs. Tether Length for ?V1y = 1.0 km/s in Case 1
Figure 5.7 Time to Impact vs. Tether Length for ?V1y = 1.0 km/s in Case 1
4050
4100
4150
4200
4250
4300
4350
4400
4450
4500
4550
0 20 40 60 80 100 120
Gr
ou
nd
Ran
ge
, R
(km
)
Tether Length, Lt (km)
630
640
650
660
670
680
690
700
710
0 20 40 60 80 100 120
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Tether Length, Lt (km)
101
A tether length of four kilometers, fiftyfive kilometers, and one hundred kilometers are
examined in more detail. The ground range and time to impact for these three cases can be
found in table 5.3.
Table 5.3: Range and Impact Time for km/s at Three Tether Lengths for Case 1
Tether Length
(km) Ground Range (km) Time to Impact (s)
4 4505.7 699
55 4297.5 665
100 4110.8 635
The maximum range and time to impact occur when the tether length is shorter. If a
shorter ground range or time to impact is desired, an increase in tether length can achieve the
desired results instead of decreasing the altitude. Figure 5.3 shows the impact trajectory of the
subsatellite at an altitude of five hundred kilometers and a tether length of four kilometers. The
next two figures show the impact trajectories of the subsatellite for the other two tether lengths.
The changes in the impact trajectory of the subsatellite after release are small. The subsatellite
travels along a similar impact trajectory path for each tether length; however, as the tether
increases the point of impact moves closer to the point of release. This movement of the impact
point results in a decrease of the ground range and time to impact of the subsatellite.
102
Figure 5.8 Trajectory at 55 km Tether Length for ?V1y = 1.0 km/s for Case 1
Figure 5.9 Trajectory at 100 km Tether Length for ?V1y = 1.0 km/s for Case 1
103
Finally, the impulsive velocity change of the subsatellite is increased while the altitude
and tether length are kept constant at five hundred kilometers and four kilometers, respectively.
The effect of the impulsive velocity change increase on the ground range and time to impact can
be determined by inspecting figures 5.10 and 5.11.
Figure 5.10 Ground Range vs. +?V1y for Case 1
Figure 5.11 Time to Impact vs. +?V1y for Case 1
0
2000
4000
6000
8000
10000
12000
14000
0 1 2 3 4 5 6 7
Gr
ou
nd
Ran
ge
, R
(km
)
Change in Velocity in the YDirection, Delta Vy (km/s)
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Change in Velocity in the YDirection, Delta Vy (km/s)
104
As the impulsive velocity change is increased the ground range and the time to impact
decreases because the larger the impulsive velocity change is the smaller the impact orbit will be.
In other words the point of impact will move closer to the release point. The minimum
impulsive velocity change needed to cause an impact was 0.2 km/s in the positive Ydirection.
The maximum and minimum values for the ground range and time to impact are shown in table
5.4.
Table 5.4 Maximum and Minimum Values as Increases for Case 1
Velocity
Change
(km/s)
Ground Range
(km)
Time to Impact
(s)
0.2 12746.8 1755
3 1924.3 430
6 545.6 350
While an impulsive velocity change of six kilometers achieves the shortest ground range and
time to impact, the fuel needed to achieve this impulsive velocity change and impact values will
be greater than the fuel needed to achieve impact at the lowest impulsive velocity change. It is
important to remember that there are tradeoffs when selecting an impulsive velocity change.
5.1.2 Impulsive Velocity Change of 1.0 km/s in the Positive Z ? Direction
For this impulsive velocity change, an increase in the altitude will also lead to an increase
in the ground range covered by the subsatellite at release and the time to impact. This matches
with the results found in the previous section. The increase of the ground range and time to
impact can be seen in figures 5.12 and 5.13. The maximum and minimum values for the ground
range and time to impact are listed in table 5.5.
105
Figure 5.12 Ground Range vs. Altitude for ?V1z = 1.0 km/s in Case 1
Figure 5.13 Time to impact vs. Altitude for ?V1z = 1.0 km/s in Case 1
The impulsive velocity change placed on the subsatellite does not result in an impact at higher
altitudes; however, if the impulsive velocity change for the subsatellite is increased an impact
can occur. For this velocity change an altitude of five hundred kilometers, six hundred and fifty
kilometers, and eight hundred and fifty kilometers will be looked at in more detail.
23000
24000
25000
26000
27000
28000
29000
0 200 400 600 800 1000
Gr
ou
nd
Ran
ge
, R
(km
)
Altitude, H (km)
0
1000
2000
3000
4000
5000
6000
0 200 400 600 800 1000
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Altitude, H (km)
106
Table 5.5 Range and Impact Time for km/s at Three Altitudes for Case 1
Altitude (km)
Ground Range
(km) Time to Impact (s)
500 24015.7 3920
650 25498.6 4235
850 28753.9 4825
The ground range and time to impact are larger than the values in the previous section in table
4.2 because the subsatellite is lofted up higher in its impact trajectory. Another way to phrase it
is that the angle of the subsatellite?s impact trajectory at release is greater than the angle for the
impact trajectory at release when the velocity change in the positive Y ? Direction was done. As
the altitude is increased the impact trajectory of the subsatellite becomes longer and the point of
impact moves further around the Earth. The impact trajectories for the subsatellite at an altitude
of five hundred kilometers, six hundred and fifty kilometers, and eight hundred and fifty
kilometers can be seen in figures 5.14, 5.15, and 5.16.
Figure 5.14 Trajectory at 500 km Altitude for ?V1z = 1.0 km/s in Case 1
107
Figure 5.15 Trajectory at 650 km Altitude for ?V1z = 1.0 km/s in Case 1
5.16 Trajectory at 850 km Altitude for ?V1z = 1.0 km/s in Case 1
An increasing tether length for an impulsive velocity change in the positive Z ? direction
has the same impact on the ground range and time to impact as it did when the impulsive
108
velocity change was in the positive Y ? direction. Both values decrease as the tether length
increases; therefore, the maximum ground range and time to impact will occur at the smallest
tether length and the minimum ground and time to impact will occur at the largest tether length
value. The decrease in ground range and time to impact can be seen in figures 5.17 and 5.18. A
list of ground ranges and times to impact are given in table 5.6.
Figure 5.17 Ground Range vs. Tether Length for ?V1z = 1.0 km/s in Case 1
21500
22000
22500
23000
23500
24000
24500
0 20 40 60 80 100 120
Gr
ou
nd
Ran
ge
, R
(km
)
Tether Length, Lt (km)
109
Figure 5.18 Time to Impact vs. Tether Length for ?V1z = 1.0 km/s in Case 1
Table 5.6 Range and Impact Time for km/s at Three Tether Lengths for Case 1
Tether Length (km)
Ground Range
(km) Time to Impact (s)
4 24015.7 3920
55 22804.3 3700
100 21821.4 3520
The ground range and time to impact for an impulsive velocity change in the positive Z ?
direction is greater than the ground range and time to impact for an impulsive velocity change in
the positive Y ? direction for an increasing tether length because of the lofted trajectory. Figure
5.14 shows the impact trajectory for an altitude of 500 km and a tether length of 4 km. The other
two impact trajectories are shown in figures 5.19 and 5.20.
3450
3500
3550
3600
3650
3700
3750
3800
3850
3900
3950
4000
0 20 40 60 80 100 120
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Tether Length, Lt (km)
110
Figure 5.19 Trajectory at 55 km Tether Length for ?V1z = 1.0 km/s in Case 1
Figure 5.20 Trajectory at 100 km Tether Length for ?V1z = 1.0 km/s in Case 1
Figures 5.21 to 5.22 show the decrease of the ground range and the increase of the time to
impact as the impulsive velocity change is increased.
111
Figure 5.21 Ground Range vs. +?V1z for Case 1
Figure 5.22 Time to Impact vs. +?V1z for Case 1
The increase of the impulsive velocity change in the positive Z ? direction also leads to a
decrease in the ground range of the subsatellite; however, the time to impact starts to increase
after a three kilometer per second impulsive velocity change. This happens because the sub
satellite is getting lofted higher but is impacting at a closer point to the release point. Figure 5.23
0
5000
10000
15000
20000
25000
30000
35000
0 1 2 3 4 5 6 7
Gr
ou
nd
Ran
ge
, R
(km
)
Change in Velocity in the Z  Direction, Delta Vz (km/s)
0
5000
10000
15000
20000
25000
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Change in Velocity in the Z  Direction, Delta Vz (km/s)
112
shows the impact trajectory for an impulsive velocity change of three kilometers per second.
The subsatellite is lofted higher after it is released from the TSS with an impulsive velocity
change of three kilometers per second in the positive Z ? direction than it is when the impulsive
velocity change is one kilometer per second.
Figure 5.23 Lofted Trajectory at ?V1z = 3.0 km/s for Case 1
The minimum impulsive velocity change needed to cause impact was 0.6 km/s in the
positive Ydirection. The maximum and minimum values for the ground range and time to
impact are shown in table 5.7 on the next page.
113
Table 5.7 Maximum and Minimum Values as Increases for Case 1
Velocity
Change
(km/s)
Ground Range
(km)
Time to Impact
(s)
0.6 28675.1 4270
3 21282.8 5600
6 20654.0 22965
For an impulsive velocity change in the positive Z ? direction a smaller ground range can be
achieved by increasing the impulsive velocity change; however, the time to impact will increase
as the impulsive velocity change becomes greater than three kilometers per second.
5.1.3 Impulsive Velocity Change of 3.0 km/s in the Negative Z ? Direction
The third impulsive velocity change investigated in this section is higher than the
previous two sections because a larger impulsive velocity change in the negative Z ? Direction
was needed in order to create impacts at higher altitudes. During this altitude range the ground
range covered by the subsatellite during its impact trajectory and the time to impact increases
because of reasons explained in the previous two sections. Figures 5.24 and 5.25 show the
increasing trend of the ground range and time to impact as the altitude of the subsatellite
increases. The values for the ground range and time to impact for three altitudes are given in
table 5.8.
114
Figure 5.24 Ground Range vs. Altitude for ?V1z = 3.0 km/s in Case 1
Figure 5.25 Time to Impact vs. Altitude for ?V1z = 3.0 km/s in Case 1
Table 5.8 Range and Impact Time for km/s at Three Altitudes for Case 1
Altitude (km)
Ground Range
(km) Time to Impact (s)
500 21297.8 170
1000 22536.7 345
1500 23795.4 530
21000
21500
22000
22500
23000
23500
24000
0 200 400 600 800 1000 1200 1400 1600
Gr
ou
nd
Ran
ge
, R
(km
)
Altitude, H (km)
0
100
200
300
400
500
600
0 200 400 600 800 1000 1200 1400 1600
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Altitude, H (km)
115
The ground range and time to impact are lower than the ground range and time to impact that
were calculated in the previous section. The time to impact is significantly smaller because of
the steep descent of the impact trajectory caused by the large impulsive velocity change in the
negative Z  direction. The large impulsive change in velocity in the negative Z ? direction,
causes the subsatellite to enter a very small orbit that will send the subsatellite into the Earth
quicker than the other impulsive velocity changes. The impact trajectories can be seen in the next
three figures.
Figure 5.26 Trajectory at 500 km Altitude for ?V1z = 3.0 km/s in Case 1
116
Figure 5.27 Trajectory at 1000 km Altitude for ?V1z = 3.0 km/s in Case 1
Figure 5.28 Trajectory at 1500 km Altitude for ?V1z = 3.0 km/s in Case 1
An impact is found for all values of the tether length from one kilometer to one hundred
kilometers. As the tether length is increased for this impulsive change in velocity the ground
117
range and the time to impact decrease. These two trends can be seen in Figures 5.29 and 5.30.
Table 5.9 gives specific values for three tether lengths.
Figure 5.29 Ground Range vs. Tether Length for ?V1z = 3.0 km/s in Case 1
Figure 5.30 Time to Impact vs. Tether Length for ?V1z = 3.0 km/s in Case 1
20850
20900
20950
21000
21050
21100
21150
21200
21250
21300
21350
0 20 40 60 80 100 120
Gr
ou
nd
Ran
ge
, R
(km
)
Tether Length, Lt (km)
135
140
145
150
155
160
165
170
0 20 40 60 80 100 120
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Tether Length, Lt (km)
118
Table 5.9 Range and Impact Time for km/s at Three Tether Lengths for Case 1
Tether Length (km)
Ground Range
(km) Time to Impact (s)
4 21297.8 167
55 21080.4 152.5
100 20884.8 140
The decrease in ground range and time to impact are caused by the impact trajectory of the sub
satellite after release. The impact point of the subsatellite slowly moves closer to the release
point; however, the impact trajectory becomes steeper as the tether length increases. This
increase in the steepness of the impact trajectory leads to the decrease in time that is seen in
figure 5.30. The impact trajectory of the subsatellite with a four kilometer tether length is
shown in figure 5.26. The impact trajectory for the subsatellite with a fifty ? five kilometer
tether length and a one hundred kilometer tether length are shown in figure 5.31 and 5.32.
Figure 5.31 Trajectory at 55 km Tether Length for ?V1z = 3.0 km/s in Case 1
119
Figure 5.32 Trajectory at 100 km Tether Length for ?V1z = 3.0 km/s in Case 1
When the impulsive velocity change placed on the subsatellite is increased in the
negative Z ? direction, the ground range and the time to impact are decreased. This decrease
occurs because the impulsive velocity change increase results in a steeper descent toward the
impact point and the movement of the impact point is toward the point of release. Figures 5.33
and 5.34 show ground range and time to impact as a function of the impulsive velocity change
increase and are located on the next page.
120
Figure 5.33 Ground Range vs. ??V1z for Case 1
Figure 5.34 Time to Impact vs. ??V1z for Case 1
The minimum impulsive velocity change required for an impact trajectory at a five hundred
kilometer altitude and a tether length of four kilometers is 0.6 kilometers per second in the
negative Z ? direction. This impulsive velocity change is increased until it reaches 6.0
0
5000
10000
15000
20000
25000
30000
35000
7 6 5 4 3 2 1 0
Gr
ou
nd
Ran
ge
, R
(km
)
Change in Velocity in Z  Direction, Delta Vz (km/s)
0
200
400
600
800
1000
1200
1400
7 6 5 4 3 2 1 0
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Change in Velocity in Z  Direction, Delta Vz (km/s)
121
kilometers per second. The maximum and minimum values for the ground range and time to
impact are shown in table 5.10.
Table 5.10: Maximum and Minimum Values as ??V1z Increases for Case 1
Velocity
Change
(km/s)
Ground Range
(km)
Time to
Impact (s)
0.6 28929.6 1160
3 21297.8 170
6 20667.6 85
5.1.4 Comparison between Velocity Changes for First Configuration
For the first configuration point a small impulsive velocity change in the positive Y and Z
? directions leads to an impact trajectory; however, a minimum value of 0.6 kilometers per
second in the negative Z ? direction is needed to create the first impact point when the altitude is
at five hundred kilometers and the tether length is at four kilometers. If the altitude is increased,
the minimum impulsive velocity change needed in the positive Z ? direction has to increase as
well in order to achieve an impact at an altitude greater than eight hundred and fifty kilometers.
The other two impulsive velocity changes do not require a further increase in order to achieve an
impact trajectory for the altitudes tested in the simulation.
The impulsive velocity changes in the Y and Z ? directions responded to changes in
altitude and tether length in the same way. The range and time to impact increased for an
increasing altitude and decreased for an increasing tether length. The tether length decreased the
range in both cases by less than two hundred kilometers. An increase in the altitude for an
impulsive velocity change in the positive Z ? direction lead to a greater increase in the range
when compared to the increase found in the range for an impulsive velocity change in the
122
positive Y ? direction; therefore, if the objective is to have the longest range increasing the
altitude and having an impulsive velocity change in the positive Z ? direction will lead to the
maximum range that can be achieved for this configuration. The minimum range can be
achieved by increasing the tether length and implementing an impulsive velocity change in the
positive Y ? direction.
Another way to decrease the range in all three examples is to increase the impulsive
velocity change. As the impulsive velocity change gets higher the range and time to impact
decrease for a change in the positive Y ? direction and for a change in the negative Z ? direction.
For the case when the impulsive velocity change is increased in the positive Z ? direction, the
range decreases but the time to impact starts to increase again after an impulsive velocity change
of three kilometers per second. This occurs because the impact trajectory of the subsatellite
becomes lofted. As the subsatellite is lofted into a higher orbit, the time it takes to impact the
Earth will increase but the range may remain the same or slightly decrease. For this attempt in
decreasing range, it is better to increase the impulsive velocity change in the positive Y ?
direction or in the negative Z ? direction; however, having an impulsive velocity change of six
kilometers in either direction will require a lot more fuel than changing the tether length or
altitude of the TSS.
5.2 Results for Second Configuration or Case 2
For the second configuration, the subsatellite is located above the main satellite. This is
similar to case A for the analytical solutions. The three impulsive velocity changes that were
applied to the subsatellite in the second configuration are shown in table 5.11 on the next page.
123
Table 5.11: Impulsive Velocity Changes for Case 2
Velocity Change (km/s)
1.0
1.5
N/A
1.0
An impulsive velocity change in the positive Y ? direction is not investigated because the
velocity of the subsatellite is in the positive Y ? direction; therefore, an increase in the velocity
in the positive Y ? direction will only increase the size of the subsatellite trajectory after release
and its perigee point, which will not result in an impact of the subsatellite.
5.2.1 Impulsive Velocity Change of 1.0 km/s in the Positive Z ? Direction
For an increase in the altitude, the ground range covered by the impact trajectory and the
time to impact increases; however, at altitudes greater than eight hundred and fifty kilometers the
subsatellite does not impact the Earth. This means that a greater impulsive velocity change in
the positive Z ? direction is needed in order to create an impact trajectory. The ground range and
time to impact as a function of altitude are shown in figures 5.35 and 5.36. The figures are then
followed by a table that lists the ground range and time to impact for an altitude of five hundred
kilometers, six hundred and fifty kilometers, and eight hundred and fifty kilometers.
124
Figure 4.35 Ground Range vs. Altitude for ?V1z = 1.0 km/s in Case 2
Figure 4.36 Time to Impact vs. Altitude for ?V1z = 1.0 km/s in Case 2
0
5000
10000
15000
20000
25000
30000
35000
0 200 400 600 800 1000 1200 1400 1600
Gr
ou
nd
Ran
ge
, R
(km
)
Altitude, H (km)
0
1000
2000
3000
4000
5000
6000
0 200 400 600 800 1000 1200 1400 1600
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Altitude, H (km)
125
Table 5.12 Ground Range and Impact Time for ?V1z = 1.0 km/s at Three Altitudes for Case 2
Altitude (km)
Ground Range
(km) Time to Impact (s)
500 24210.6 3955
650 25730.4 4275
850 29736.1 4955
The impact trajectory of the subsatellite is lofted until a maximum apogee is reached and then it
begins its return toward the Earth. The increase in the ground range is caused by the movement
of the impact point further away from the release point as the altitude increases. The increase in
the time to impact is caused by the fact that as the altitude increases the area of the impact
trajectory that is lofted increases as well. The impact trajectories for a five hundred kilometer
altitude, a six hundred and fifty kilometer altitude, and an eight hundred and fifty kilometer
altitude are shown below.
Figure 5.37 Trajectory at 500 km Altitude for ?V1z = 1.0 km/s in Case 2
126
Figure 5.38 Trajectory at 650 km Altitude for ?V1z = 1.0 km/s in Case 2
Figure 5.39 Trajectory at 850 km Altitude for ?V1z = 1.0 km/s in Case 2
127
The next change that is looked at for this impulsive velocity change is an increase in the
tether length. The time to impact and the ground range increase as the tether length becomes
longer. This means that the minimum range that can be achieved will occur at the shortest tether
length, while the maximum range will occur at the longest length that the tether can reach. The
increasing trends for the range and time to impact are shown in figures 5.40 and 5.41.
Figure 5.40 Ground Range vs. Tether Length for ?V1z = 1.0 km/s in Case 2
24000
24500
25000
25500
26000
26500
27000
0 20 40 60 80 100 120
Gr
ou
nd
Ran
ge
, R
(km
)
Tether Length, Lt (km)
128
Figure 5.41 Time to Impact vs. Tether Length for ?V1z = 1.0 km/s in Case 2
A list of ground ranges and times to impact for a tether length of four kilometers, fiftyfive
kilometers, and one hundred kilometers is given in table 5.29.
Table 5.13 Range and Impact Time for ?V1z = 1.0 km/s at Three Tether Lengths for Case 3
Tether Length (km)
Ground Range
(km) Time to Impact (s)
4 24210.6 3955
55 25495.2 4180
100 26800.6 4400
The increase in tether length causes the ground range to increase because the point of impact
moves further away from the point of release and the height of the subsatellite is increasing.
This same change in the impact point also causes an increase in the time to impact because if the
impact point moves further away from the release point the time to get to an impact will increase.
The impact trajectory at a four kilometer tether length is shown in figure 5.37. The other two
impact trajectories are shown in the figures below.
3900
4000
4100
4200
4300
4400
4500
0 20 40 60 80 100 120
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Tether Length, Lt (km)
129
Figure 5.42 Trajectory at 55 km Tether Length for ?V1z = 1.0 km/s in Case 2
Figure 5.43 Trajectory at 100 km Tether Length for ?V1z = 1.0 km/s in Case 2
The last change that is looked at in order to cause an impact trajectory is the increase of
the magnitude for the impulsive velocity change while keeping the altitude and tether length
130
constant at five hundred kilometers and four kilometers. The effect of the increase in the
impulsive velocity change on the ground range and time to impact are shown in figures 5.44 to
5.45.
Figure 5.44 Ground Range vs. +?V1z for Case 2
Figure 5.45 Time to Impact vs. +?V1z for Case 2
0
5000
10000
15000
20000
25000
30000
0 2 4 6 8
Gr
ou
nd
Ran
ge
, R
(km
)
Change in Velocity in the Z  direction, Delta Vz (km/s)
0
5000
10000
15000
20000
25000
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Change in Velocity in the Z  direction, Delta Vz (km/s)
131
The increase in the time to impact is caused by the lofted trajectory after the subsatellite release.
Figure 5.46 shows the lofted trajectory of the subsatellite when an impulsive velocity change of
three kilometers per second in the positive Z ? direction is placed on the system.
Figure 5.46 Lofted Trajectory at ?V1z = 1.0 km/s for Case 2
The ground range decreases because the impact point moves closer to the release point. The
minimum impulsive velocity change in the positive Z ? direction that was needed to cause an
impact at a five hundred kilometer altitude and a tether length of four kilometers was 0.7
kilometers per second. The ground range and time to impact achieved during the minimum
impulsive velocity change, a velocity change of 3.0 kilometers per second, and a velocity change
of 6.0 kilometers per second are listed in table 5.14
132
Table 5.14 Maximum and Minimum Values as +?V1z Increases for Case 3
Velocity
Change
(km/s)
Ground Range
(km)
Time to Impact
(s)
0.7 26743.1 4090
3 21344.7 5625
6 20690.8 23140
5.2.2 Impulsive Velocity Change of 1.5 km/s in the Negative Z ? Direction
For an impulsive velocity change of 1.5 kilometers per second in the negative Z ?
direction an increase in the altitude leads to an increase in the range and time to impact; however,
no impact trajectory is created at altitudes greater than thirteen hundred kilometers. A minimum
impulsive velocity change of three kilometers per second in the negative Z ? direction is required
to have an impact at all altitudes that are tested. The increase in the range and time to impact as
the altitude is increased for a velocity change of 1.5 kilometers per second our shown in figures
5.47 and 5.48.
Figure 5.47 Ground Range vs. Altitude for ?V1z = 1.5 km/s in Case 2
0
5000
10000
15000
20000
25000
30000
35000
0 200 400 600 800 1000 1200 1400
Gr
ou
nd
Ran
ge
, R
(km
)
Altitude, H (km)
133
Figure 5.48 Time to Impact vs. Altitude for ?V1z = 1.5 km/s in Case 2
Since an impact trajectory does not occur at altitudes greater than thirteen hundred
kilometers, the results for an impact trajectory at five hundred kilometers, one thousand
kilometers, and thirteen hundred kilometers will be examined in more detail. A list of the ground
range and time to impact achieved at these three altitude values are given in table 5.15.
Table 5.15 Range and Impact Time for ?V1z = 1.5 km/s at Three Altitudes for Case 2
Altitude (km)
Ground Range
(km) Time to Impact (s)
500 22683.8 345
650 25665.0 760
800 28841.2 1180
The small time to impact is caused by the steep impact trajectory of the subsatellite after release.
The impact trajectories of the subsatellite at the three different altitudes are shown in figures
5.49 to 5.51.
0
200
400
600
800
1000
1200
1400
0 200 400 600 800 1000 1200 1400
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Altitude, H (km)
134
Figure 5.49 Trajectory at 500 km Altitude for ?V1z = 1.5 km/s in Case 2
Figure 5.50 Trajectory at 1000 km Altitude for ?V1z = 1.5 km/s in Case 2
135
Figure 5.51 Trajectory at 1300 km Altitude for ?V1z = 1.5 km/s in Case 2
After inspecting the impact trajectory figures, it can be deduced that as the altitude increases the
point of impact of the subsatellite moves further along the surface of the Earth. This movement
of the impact point results in an increase in the ground range. The increase of the time to impact
is a direct result of the increase of the altitude of the TSS.
Even though an impact trajectory cannot be obtained for all altitudes, an impact trajectory
can still be found for all the tested tether lengths. The trends for an impulsive velocity change of
1.5 kilometers per second in the negative Z ? direction are shown in figures 5.52 and 5.53.
136
Figure 5.52 Ground Range vs. Tether Length for ?V1z = 1.5 km/s in Case 2
Figure 5.53 Time to Impact vs. Tether Length for ?V1z = 1.5 km/s in Case 2
Since the ground range and time to impact increase as the tether length increases, the maximum
range and time to impact will result at the maximum tether length. The ground range, time to
impact, and impact trajectories will be looked at in more detail for the same tether length values
used in previous sections.
22600
22700
22800
22900
23000
23100
23200
23300
23400
23500
23600
0 20 40 60 80 100 120
Gr
ou
nd
Ran
ge
, R
(km
)
Tether Length, Lt (km)
340
345
350
355
360
365
370
375
380
385
390
0 20 40 60 80 100 120
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Tether Length, Lt (km)
137
Table 5.16 Range and Impact Time for ?V1z = 1.5 km/s at Three Tether Lengths for Case 2
Tether Length (km)
Ground Range
(km) Time to Impact (s)
4 22683.8 344.5
55 23144.0 366.5
100 23540.6 385.5
The cause behind the increase in ground range and time to impact as the tether length increases
can be found by examining the impact trajectories at each tether length. The impact trajectory
for the four kilometer tether length and five hundred kilometer case is shown in figure 5.49. The
other two impact trajectories also have an altitude of five hundred kilometers. The impact
trajectories for these other two tether length values are shown below.
Figure 5.54 Trajectory at 55 km Tether Length for ?V1z = 1.5 km/s in Case 2
138
Figure 5.55 Trajectory at 100 km Tether Length for ?V1z = 1.5 km/s in Case 2
The increase in range is caused by the movement of the impact point further along the Earth?s
surface, while the time to impact is increased because the orbit is at a higher position. As the
position or height of the orbit increases so will the time to impact.
The final change that is looked at in this section is an increase in the impulsive velocity
change. The altitude and tether length are kept constant at five hundred kilometers and four
kilometers, respectively. As the magnitude of the impulsive velocity change is increased in the
negative Z ? direction, the ground range and time to impact decrease. The decrease in the
ground range and the time to impact can be seen in figures 5.56 through 5.57.
139
Figure 5.56 Ground Range vs. ??V1z for Case 2
Figure 5.57 Time to Impact vs. ??V1z for Case 2
The decrease in the ground range and the time to impact follows the trends displayed in
the previous sections for different release points and impulsive velocity changes. In order to
create an impact trajectory at this configuration a minimum impulsive velocity change value of
0
5000
10000
15000
20000
25000
30000
7 6 5 4 3 2 1 0
Gr
ou
nd
Ran
ge
, R
(km
)
Change in Velocity in the Z  Direction, Delta Vz (km/s)
0
100
200
300
400
500
600
700
800
900
1000
7 6 5 4 3 2 1 0
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Change in Velocity in the Z  Direction, Delta Vz (km/s)
140
0.7 kilometers per second in the negative Z ? direction is needed. Table 5.17 list values for the
ground range and time to impact for three impulsive velocity changes.
Table 5.17 Maximum and Minimum Values as ??V1z Increases for Case 2
Velocity
Change
(km/s)
Ground Range
(km)
Time to Impact
(s)
0.7 26663.4 860
3 21329.7 170
6 20677.2 85
5.2.3 Impulsive Velocity Change of 1.0 km/s in the Positive Y ? Direction
The ground range and time to impact increase as the altitude is increased and the tether
length is kept constant at four kilometers with an impulsive velocity change of one kilometer per
second in the positive Y ? direction. The results for the ground range and time to impact as the
altitude increases are shown below.
Figure 5.58 Ground Range vs. Altitude for ?V1y = 1.0 km/s in Case 2
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
0 500 1000 1500 2000
Gr
ou
nd
Ran
ge
, R
(km
)
Altitude, H (km)
141
Figure 5.59 Time to Impact vs. Altitude for ?V1y = 1.0 km/s in Case 2
Altitudes of five hundred kilometers, one thousand kilometers, and fifteen hundred kilometers
are examined in more detail. The values for the ground range and time to impact at these
altitudes are listed in table 5.18.
Table 5.18: Range and Impact Time for ?V1y = 1.0 km/s at Three Altitudes for Case 2
Altitude (km)
Ground Range
(km) Time to Impact (s)
500 4544.8 705
1000 6412.0 1060
1500 7856.0 1375
The range and time to impact achieved with this impulsive velocity change are less than the
ground range and time to impact calculated for different altitudes for the other two impulsive
velocity changes. This is a result of the type of impact trajectory that is created after the velocity
change is put into place and the subsatellite is released from the TSS. The impact trajectory of
the subsatellite at each altitude is shown on the next two pages.
0
200
400
600
800
1000
1200
1400
1600
0 200 400 600 800 1000 1200 1400 1600
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Altitude, H (km)
142
Figure 5.60 Trajectory at 500 km Altitude for ?V1y = 1.0 km/s in Case 2
Figure 5.61 Trajectory at 1000 km Altitude for ?V1y = 1.0 km/s in Case 2
143
Figure 5.62 Trajectory at 1500 km Altitude for ?V1y = 1.0 km/s in Case 2
From the impact trajectories depicted in figure 5.60 through 5.62, it can be concluded that the
ground range increases because as the altitude increases the release point occurs earlier in the
TSS orbit and the impact point occurs further on the surface of the Earth. The increase in the
time to impact happens because the impact trajectory is placed on a higher orbit that requires
more time to travel around in order to get to the impact point.
An increase in the tether length of the TSS affects the ground range and time to impact of
the impact trajectory after the subsatellite is released from the system. For a constant impulsive
velocity change of one kilometer per second in the positive Y ? direction, an increase in tether
length leads to an increase in the ground range and time to impact.
144
Figure 5.63 Ground Range vs. Tether Length for ?V1y = 1.0 km/s in Case 2
Figure 5.64 Time to Impact vs. Tether Length for ?V1y = 1.0 km/s in Case 2
The list of the ground range and the time to impact at three tether lengths are listed in
table 5.19 on the next page.
4400
4500
4600
4700
4800
4900
5000
5100
5200
0 20 40 60 80 100 120
Gr
ou
nd
Ran
ge
, R
(km
)
Tether Length, Lt (km)
690
700
710
720
730
740
750
760
0 20 40 60 80 100 120
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Tether Length, Lt (km)
145
Table 5.19 Range and Impact Time for ?V1y = 1.0 km/s at Three Tether Lengths for Case 2
Tether Length (km)
Ground Range
(km) Time to Impact (s)
4 4544.8 703
55 4839.0 729
100 5103.0 752
The ground range and time to impact are increased because an increase in the tether length
causes the subsatellite to be in a slightly higher orbit, which in turn causes the impact trajectory
to start from a higher point and move the impact point further along the Earth?s surface. In order
to see if this is true the impact trajectories must be examined. The impact trajectory of the sub
satellite at a four kilometer tether length is shown in figure 5.60. The impact trajectories for the
other two tether lengths are shown below. As the tether length increases the height in the impact
trajectories are increased. This increase is what results in the increase in the time to impact
shown in table 5.37. The plot of the impact trajectories also show that the impact point of the
subsatellite travels forward along the surface of the Earth. The movement of the impact point is
not significantly large; therefore, a small increase in the ground range results from the increase of
the tether length.
146
Figure 5.65 Trajectory at 55 km Tether Length for ?V1y = 1.0 km/s in Case 2
Figure 5.66 Trajectory at 100 km Tether Length for ?V1y = 1.0 km/s in Case 2
147
An increase of the magnitude of the impulsive velocity change in the positive Y ?
direction will also lead to a change in the ground range and time to impact. The increase in the
impulsive velocity change results in a decrease of the ground range and the time to impact. This
happens because the impact trajectory of the subsatellite after release decreases in size as the
impulsive velocity change increases; therefore, the subsatellite will impact the Earth sooner at
higher impulsive velocity changes. The effect of the impulsive velocity change on the ground
range and time to impact values are shown in figures 5.67 through figures 5.68.
Figure 5.67 Ground Range vs. +?V1y for Case 2
0
2000
4000
6000
8000
10000
12000
14000
0 1 2 3 4 5 6 7
Gr
ou
nd
Ran
ge
, R
(km
)
Change in Velocity in the Y  Direction, Delta Vy (km/s)
148
Figure 5.68 Time to Impact vs. +?V1y for Case 2
The minimum impulsive velocity change in the positive Y ? direction that is needed to create an
impact trajectory for the second configuration is 0.2 kilometers per second. Note that this is the
minimum impulsive velocity change needed when the altitude of the main satellite is five
hundred kilometers and the tether length is kept at four kilometers. A maximum impulsive
velocity change of six kilometers per second in the positive Y ? direction is place on the system;
therefore, the minimum time to impact and the minimum ground range will occur at this
maximum impulsive velocity change. Values of the ground range and time to impact are listed
in the table below.
Table 5.20: Maximum and Minimum Values at +?V1y Increases for Case 2
Velocity
Change
(km/s)
Ground Range
(km)
Time to Impact
(s)
0.2 13058.2 1785
3 1938.1 435
6 549.4 350
0
200
400
600
800
1000
1200
1400
1600
1800
2000
0 1 2 3 4 5 6 7
Tim
e t
o Im
pac
t, t_i
mp
ac
t (s)
Change in Velocity in the Y  Direction, Delta Vy (km/s)
149
5.2.4 Comparison between Velocity Changes For Second Configuration
The impulsive velocity changes that are placed on the subsatellite result in different
values for the ground range and the time to impact. For the second configuration an impulsive
velocity change placed in the positive Z ? direction will result in the maximum ground range and
time to impact for the subsatellite. These values for ground range and time to impact can be
further increased by increasing the altitude and the tether length; however, a larger impulsive
velocity change in the positive Z ? direction will be needed to cause an impact trajectory at
altitudes above eight hundred and fifty kilometers. For the case of an impulsive velocity change
in the positive Z ? direction, the tether length can be increased in order to increase the ground
range without needing an increase in the magnitude of the impulsive velocity change. In order to
save on fuel, the increase in tether length is the better way to achieve a longer ground range for
this impulsive velocity change direction.
An impulsive velocity change in the positive Y ? direction results in the minimum ground
range and time to impact for the second configuration. The ground ranges achieved with this
impulsive velocity change can be increased by placing the TSS in a higher altitude or by
increases the tether length. The increase in altitude leads to a larger increase for the ground
range, while the increase in tether length has a smaller effect on the ground range and time to
impact. In this case, the method used to increase the ground range depends upon how big of an
increase is needed in order to reach the desired impact point.
As at any release point, the magnitude of the impulsive velocity change can be increased
in order to achieve a lower ground range. The time to impact will also decrease with the ground
range when the impulsive velocity change is in the negative Z ? direction or the positive Y ?
direction. When an impulsive velocity change is placed in the positive Z ? direction, the time to
150
impact will increase because the subsatellite is lofted into a higher orbit upon release. In other
words, one has to consider the tradeoff of an increase in the time to impact in order to achieve a
shorter ground range. Overall, impact trajectories will result at these impulsive velocity changes
for the second configuration; however, which impulsive velocity change to be used depends
upon if a maximum or minimum ground range is desired.
5.3 Comparison between the Analytical Results and the Simulation Results
In order to compare the analytical approach and the simulation results the algorithm to
find the ground range and time to impact for the analytical calculation is used. The first
configuration from the simulation is compared to case B from the analytical solution because
both have the subsatellite below the main satellite. Case A from the analytical solution is then
compared to the second configuration from the simulation because both have the subsatellite
located above the main satellite. For each comparison an impulsive velocity change of one
kilometer per second and six kilometers per second in the positive Y ? direction will be placed
on the subsatellite.
The first comparison will be done using the first release configuration. The impulsive
velocity change that is used for the simulation and the analytical calculations is one kilometer per
second in the positive Y ? direction. The ground range and time to impact calculated by the
simulation and the analytical approach are listed in table 5.21.
Table 5.21: Results for Configuration 1 and Case B with ?V1y = 1.0 km/s
Ground Range (km) Time to Impact (s)
Simulation 4.5057232E+03 700
Analytical 4.4980330E+03 696.98
151
The percent error and the difference between the simulation value and the analytical value for the
ground range and time to impact are calculated and listed below.
Table 5.22 Comparison of Results for Configuration 1 and Case B with ?V1y = 1.0 km/s
Ground Range (km) Time to Impact (s)
% Error 0.17068 0.43108
Difference 7.69028 3.01755
The percent error between the two ground ranges is less than one percent and the actual
difference is about 7.7 kilometers. The percent error for the time to impact is also less than one
percent but the actual difference between the two times is about 3.02 seconds with the time to
impact from the simulation being the greater value.
The next comparison will be done at the second configuration with an impulsive velocity
change of 6.0 kilometers per second in the positive Y ? direction. The calculated values for the
ground range and time to impact for both methods are listed below.
Table 5.23: Results for Configuration 1 and Case B with ?V1y = 6.0 km/s
Ground Range (km) Time to Impact (s)
Simulation 5.4562965E+02 350
Analytical 5.4529027E+02 346.72
The percent error and the direct difference between the ground ranges and times to impact are
given in table 5.24 on the next page.
152
Table 5.24 Comparison of Results for Configuration 1 and Case B with ?V1y = 6.0 km/s
Ground Range (km) Time to Impact (s)
% Error 0.062201 0.936190
Difference 0.339384 3.276665
The percent error for the ground range and the time to impact are less than one. The difference
between the ground ranges is 0.34 kilometers and the difference between the times is about 3.28
seconds. So far the analytical solution is giving similar results to those from the simulation.
The third comparison is done with the second configuration and an impulsive velocity
change placed on the subsatellite of one kilometer per second in the positive Y ? direction. The
simulation results and the analytical approach results are listed below in table 5.25.
Table 5.25: Results for Configuration 2 and Case A with ?V1y = 1.0 km/s
Ground Range (km) Time to Impact (s)
Simulation 4.5448026E+03 705
Analytical 4.5458193E+03 704.69
The analytical result for the ground range is greater than the ground range obtained from the
simulation results. The time to impact calculated during the simulation is greater than the time to
impact calculated by the analytical approach. The actual difference between the simulation
values and the analytical values and the percent error are listed below.
Table 5.26: Comparison of Results for Configuration 2 and Case A with ?V1y = 1.0 km/s
Ground Range (km) Time to Impact (s)
% Error 0.02237 0.04446
Difference 1.01666 0.31343
153
The percent errors between the ground ranges and the times to impact are less than one percent.
This is the same trend that has been seen in the past two comparisons. The difference between
the ground range determined by the simulation and the ground range calculated using the
analytical approach is about one kilometer. The difference between the times to impact is about
0.313 seconds.
The final comparison will be done with the second configuration and an impulsive
velocity change of 6.0 kilometers per second in the positive Y ? direction.
Table 5.27: Results for Configuration 2 and Case A with ?V1y = 6.0 km/s
Ground Range (km) Time to Impact (s)
Simulation 5.4939447E+02 350
Analytical 5.4986409E+02 349.86
The ground range calculated by the analytical approach is slightly larger than the ground range
calculated by the simulation. The percent error and the actual difference between the two values
are presented in table 5.28.
Table 5.28: Comparison of Results for Configuration 2 and Case A with ?V1y = 6.0 km/s
Ground Range (km) Time to Impact (s)
% Error 0.08548 0.04084
Difference 0.46964 0.14295
The percent error between the two ground ranges and the percent error between the two times to
impact is less than one percent. The difference between the two ground ranges was about 0.45
kilometers. The time to impact calculated by the simulation is greater than the time to impact
that was calculated using the simulation approach by almost 0.14 seconds.
154
More comparisons are not shown in this write up because the results have a percent error
of less than one percent. The maximum difference between the times was found to be about 3.3
seconds, while the maximum difference for the ground ranges was 7.7 kilometers. The errors
between the simulation and the analytical approach are caused by the rotation angle that is
introduced into the simulation in order to keep the subsatellite in a leading position. Overall the
analytical approach provides a good estimate of what the time to impact and ground range will
be after the subsatellite release. A simulation should still be run in order to determine the
impact trajectory after the subsatellite is released and to place the impulsive velocity change in
different directions on the subsatellite.
155
Chapter 6: Conclusions and Future Work
The altitude, tether length, and the impulsive velocity change have been varied to
determine the type of ground range and time to impact that can be achieved. The effects of the
altitude, tether length, and impulsive velocity change on the ground range and time to impact
were investigated using a numerical simulation approach and an analytical approach. The
increase in altitude caused the ground range and time to impact to increase for all configurations
and impulsive velocity changes. If the subsatellite is located below the main satellite, an
increase in tether length will cause the ground range and time to impact to decrease; however, if
the subsatellite is located above the main satellite, an increase in the tether length will result in
an increase in the ground range and time to impact. An increase in the impulsive velocity change
on the subsatellite will always cause the ground range and the time to impact to decrease;
however, there are a few occasions where the time to impact will increase because of the lofted
impact trajectory. The changes in altitude and tether length are the better methods to increase or
decrease the ground range and time to impact because an increase in an impulsive velocity
change will result in a greater fuel usage. The analytical solutions were found to follow these
exact same trends for a subsatellite located above or below the main satellite. For the cases
when a subsatellite had the same altitude as the main satellite, the increase in altitude resulted in
an increase in the ground range and time to impact, while an increase in the tether length resulted
in an increase to the ground range and a decrease to the time to impact.
156
The final part of this research was to compare the analytical solution to the numerical
simulation by comparing the ground range and the time to impact. The percent error between the
ground ranges and times to impact for the analytical solution and the numerical solution was
found to always be less than one percent. The percent errors and differences between the two
approaches are caused by the fact that a rotation angle was introduced into the simulation in
order to model the fact that the subsatellite is leading the main satellite. The analytical approach
can be used to provide an initial estimate of the ground range and time to impact; however, the
numerical simulation should be used in order to track the trajectory of the subsatellite before
and after its release from the TSS.
In order to improve the ground range and time to impact calculated in the numerical
simulation a more detailed model of the TSS should be put into place for future research. The
effect of the Earth?s oblateness and the aerodynamic drag on the satellites can be added to the
simulation. The aerodynamic drag will impact the movement of the subsatellite as it gets closer
to the Earth?s surface. The inclusion of the gravitational gradient in the model will keep the TSS
in a natural orbital motion where the subsatellite will remain in a lower orbit than the main
satellite at all times. By adding the above three conditions to the model that is being used in the
numerical simulation, the movement of the TSS will be model more accurately and the
simulation will provide a more accurate result for the position of the subsatellite at release. If
the effects of the Earth?s oblateness and the effects of the aerodynamic drag are placed on the
subsatellite after release, a more accurate model for the movement of the subsatellite after
release will be in place and more precise ground ranges and times to impact can be calculated.
157
References
1. Bate, Roger R., Mueller, Donald D., White, Jerry E., Fundamentals of Astrodynamics,
Dover Publications, Inc., New York, 1971.
2. Beda, Peter B., ?Anticipatory Computing in the Attitude Control of Satellites,? Paper
American Institute of Physics 0735400121. 2001.
3. Beletsky, V. V., and Levin, E.V., Advances in the Astronautical Sciences: Dynamics
of Space Tether Systems, Vol. 83, AAs, Sand Diego, CA, 1993, p. 7 ? 9.
4. Chang, Insu, Park, SangYoung, and Choi, KyuHong, ?Nonlinear Attitude Control of a
TetherConnected MultiSatellite in ThreeDimensional Space,? Paper IEEE
Transactions on Aerospace and Electronic Systems, Vol. 46, No. 4, 2010.
5. Cho, Sungki, ?Analysis of the Orbital Motion of a General Tethered Satellite System,?
Ph.D. Dissertation, Dept. of Aerospace Engineering, Auburn University, Auburn,
AL, June 1999, p. 1 ? 3, p. 10 ? 23.
6. Cho, S., Cochran, Jr., J. E., and Cicci, D. A., ?Identification and Orbit Determination of
Tethered Satellite Systems.?Applied Mathematics and Computation, Vol. 117,
pp. 301 ? 312, 2001.
158
7. Choe, Nammi J., ?Detection and Orbit Determination of Tethered Satellite Systems,?
Ph.D. Dissertation, Dept. of Aerospace Engineering, Auburn University, Auburn,
Al, August 2003, p. 1 ? 8, p. 20 ? 24.
8. Cicci, D. A., Qualls, C., and Lovell, T. A., ?A look at Tethered Satellite Identification Using
Ridge ? Type Estimation Methods,? Applied Mathematics and Computation, Vol. 119
pp. 297 ? 316, 2001.
9. Cicci, D. A., Cochran, Jr., J. E., Qualls, C., and Lovell, T. A.,?Quick  Look Identification
and Orbit Determination of a Tethered Satellite,?The Journal of Astronautiacl Sciences,
Vol. 50, No. 3, pp. 339 ? 353, July ? September 2002.
10. Curtis, Howard D., Orbital Mechanics for Engineering Students, Elsevier, New York, 2005.
11. Ellis, Joshua R., and Hall, Christopher D. ?Outofplane Librations of Spinning Tethered
Satellite Systems,?Celestial Mechanics and Dynamical Astronomy, Vol. 106, No. 1,
p. 39 ? 67, 2009.
12. Ginsberg, Jerry, Engineering Dynamics, Cambridge University Press, New York, New
York, 2008.
13. Hoots, F. R., Roehrich, R. L., and Szebehely, V. G., ?Space Shuttle Tethered Satellite
Analysis,? Technical Report #83 ? 5, Directorate of Astrodynamics, Peterson
Air Force Base, CO, August 1983.
14. Hughes, Peter C., Spacecraft Attitude Dynamics, Dover Publications, Inc., Mineola, New
York, 1984.
159
15. Ishimura, Kosei, and Higuchi, Ken, ?Coupling among Pitch Motion, Axial Vibration, and
Orbital Motion of Large Space Structures,? Journal of Aerospace Engineering, Vol.
21, No. 2, April 1, 2008.
16. Kaplan, M. H., Modern Spacecraft Dynamics and Control. Ed. 1st, Wiley, John and Sons
Incorporated, New Jersey, 1976, pp. 199.
17. Kulakowski, Bohdan T., Gardner, J. F., Shearer, J. L., Dynamic Modeling and Control of
Engineering Systems, Cambridge University Press, New York, 2007. pg. 126 ? 129.
18. Lovell, Thomas A., ?State Estimation of Tethered Satellites Using Conventional and
Intelligent SystemsBased Paradigms,? Ph.D. Dissertation, Dept. of Aerospace
Engineering, Auburn University, Auburn, AL, December 2001, p. 1 ? 7, p. 16 ? 38.
19. Lovell, T. A., Cho, S., Cochran, Jr., J. E., and Cicci, D. A., ?A Study of the ReEntry Orbit
Discrepancy Involving Tethered Satellites,? Acta Astronautica, Vol. 53, Issue 1,
pp. 21 ? 33, 2003.
20. Meriam, J. L., and Kraige, L. G., Engineering Mechanics: Dynamics, 6th Ed., John Wiley
and Sons, Inc., New Jersey, 2007.
21. Misra, A.K., ?Dynamics and Control of Tethered Satellite Systems,? Acta Astronautica,
Vol. 63, pp. 1169 ? 1177, 2008.
22. Modi, V. J., Gilardi, G., Misra, A.K., ?Attitude Control of Space Platform Based Tethered
Satellite Systems.?Journal of Aerospace Engineering, Vol. 11, No. 2, pp. 24 ? 31, 1998.
160
23. Naigang, Cui, Dun, Liu, Yuhua, Liu, and Naiming Qi, ?The Calculation of the Orbital
Elements of a Tethered Satellite System after Release,? Paper AIAA 943746CP, 1994.
24. Qualls, C. and Cicci, D. A, ?Preliminary Orbit Determination of a Tethered Satellite,?
Applied Mathematics and Computation, Vol. 188, pp. 462 ? 471, 2007.
25. Rossi, E. V., Cicci, D. A., and Cochran Jr., J. E., ?Existence of Periodic Motions of a Tether
Trailing Satellite,?Applied mathematics and Computation. Vol. 155, pp. 269 ? 281, 2004.
26. Schutte, Aaron D., and Dooley, Brian A., ?Constrained Motion of Tethered Satellites,?
Journal of Aerospace Engineering, Vol. 18, No. 4, October 2005.
27. Space Dynamics Laboratory, ?ProSEDS,? Space Dynamics Laboratory Webpage,
http://www.sdl.usu.edu/programs/proseds
28. Stanley, Curtis H., ?Apparent Impacting Trajectories, Identification, and Orbit
Determination of Tethered Satellite Systems,? Master?s Thesis, University of Colorado,
Colorado Springs, CO, 2010.
29. Wen, Hao, Jin, Dongping P., and Hu, Haiyan Y., ?Advances in Dynamics and Control of
Tethered Satellite Systems,?Acta Mechanica Sinica, Vol. 24, p. 229 ? 241, 2008.
161
Appendix A: Sample Results from Numerical Integration
Table A.1: Simulation Results as Altitude Increases for Case 1 at ?V1y = 1.0 km/s
Altitude
(km) Time to Impact (s) Ground Range (km)
500 700 4505.723
550 740 4726.313
600 775 4937.022
650 815 5139.085
700 850 5333.504
750 885 5521.106
800 920 5702.580
850 955 5878.512
900 990 6049.402
950 1020 6215.682
1000 1055 6377.728
1050 1085 6535.871
1100 1120 6690.403
1150 1150 6841.584
1200 1180 6989.646
1250 1215 7134.797
1300 1245 7277.226
1350 1275 7417.105
1400 1305 7554.589
1450 1340 7689.820
1500 1370 7822.930
162
Table A.2: Simulation Results as Tether Length Increases for Case 1 at ?V1y = 1.0 km/s
Tether
Length
(km) Time to Impact (s) Ground Range (km)
1 701 4517.880
4 699 4505.723
7 697 4493.558
10 695 4481.383
13 693 4469.199
16 691 4457.006
19 689 4444.803
22 687 4432.590
25 685 4420.367
28 683 4408.133
31 681 4395.889
34 679 4383.634
37 677 4371.369
40 675 4359.092
43 673 4346.803
46 671 4334.502
49 669 4322.190
52 667 4309.865
55 665 4297.528
58 663 4285.178
61 661 4272.815
64 659 4260.439
67 657 4248.049
70 655 4235.645
73 653 4223.227
76 651 4210.795
79 649 4198.348
82 647 4185.886
85 645 4173.409
88 643 4160.917
91 641 4148.408
94 639 4135.884
97 637 4123.343
100 635 4110.786
163
Table A.3: Simulation Results as +?V1y Increases for Case 1
Velocity
Change
(km/s) Time to Impact (s) Ground Range (km)
0
0.1
0.2 1755 12746.853
0.3 1335 9553.004
0.4 1130 7953.634
0.5 1000 6929.881
0.6 905 6197.195
0.7 840 5636.828
0.8 785 5188.739
0.9 740 4818.745
1 700 4505.723
1.1 670 4235.805
1.2 640 3999.452
1.3 620 3789.851
1.4 595 3601.989
1.5 580 3432.081
1.6 560 3277.210
1.7 545 3135.081
1.8 530 3003.864
1.9 520 2882.078
2 510 2768.509
2.1 500 2662.151
2.2 490 2562.160
2.3 480 2467.828
2.4 470 2378.548
2.5 465 2293.802
2.6 455 2213.141
2.7 450 2136.178
2.8 440 2062.572
2.9 435 1992.024
3 430 1924.271
164
Appendix B: Sample Results from Analytical Solutions
Table B.1: Analytical Results as Altitude Increases for Case A When R = 1500 km and ?imp =
240?
Altitude (km)
Velocity Change in the Y
(km/s) Angular Velocity (rads/s)
500 0.60484 0.16633
600 0.62511 0.17191
700 0.62185 0.17101
800 0.59824 0.16452
900 0.55718 0.15322
1000 0.50120 0.13783
1100 0.43258 0.11896
1200 0.35332 0.09716
1300 0.26517 0.07292
1400 0.16966 0.04666
1500 0.06812 0.01873
165
Table B.2: Analytical Results as Tether Length Increases for Case A When R = 1500 km and
?imp = 240?
Tether Length (km)
Velocity Change in the Y
(km/s) Angular Velocity (rads/s)
4 0.60484 0.16633
14 0.61305 0.04817
24 0.62098 0.02846
34 0.62864 0.02034
44 0.63602 0.01590
54 0.64314 0.01310
64 0.64999 0.01117
74 0.65658 0.00976
84 0.66292 0.00868
94 0.66900 0.00783
104 0.67483 0.00714
114 0.68041 0.00657
166
Table B.3: Analytical Results as ?V1y Increases for Case A When H = 500 km and Lt = 4 km
Velocity
Change
(km/s) Ground Range (km) Time to Impact (s)
0
0.1
0.2 13065.345 1793.531
0.3 9700.880 1355.315
0.4 8053.841 1142.390
0.5 7007.363 1008.651
0.6 6261.278 914.513
0.7 5692.005 843.645
0.8 5237.520 787.848
0.9 4862.674 742.480
1 4545.819 704.687
1.1 4272.780 672.600
1.2 4033.819 644.941
1.3 3821.997 620.798
1.4 3632.213 599.504
1.5 3460.618 580.556
1.6 3304.248 563.568
1.7 3160.775 548.237
1.8 3028.342 534.322
1.9 2905.447 521.628
2 2790.860 509.997
2.1 2683.561 499.298
2.2 2582.698 489.421
2.3 2487.551 480.274
2.4 2397.508 471.778
2.5 2312.045 463.867
2.6 2230.707 456.484
2.7 2153.102 449.579
2.8 2078.887 443.108
2.9 2007.758 437.034
3 1939.451 431.323