SATELLITE ATTITUDE MOTION MODELS 

FOR CAPTURE AND RETRIEVAL 

INVESTIGATIONS 

John E. Cochran, Jr. and Brian S. Lahr 

Aerospace Engineering Department 

Auburn University, Alabama 36849-3501 
SATELLITE ATTI'CUUE MOTION MODELS 

FOR CAPTURE AND RETRIEVAL 

INVESTIGATIONS 

John E. Cochran, Jr. and Brian S. Lahr 

Aerospace Engineering Department 

Auburn University, Alabama 36849-3501 

for 

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 

George C. Marshall Space flight Center 

under 

Contract NASB-36470 

Administered through 

Engineering Experiment Station 

Auburn University, Alabama 36849 

October 1986 
FOREWORD 

This report was prepared under Contract NAS8-36470 with the 

National Aeronautics and Space Administration, George C. Marshall Space 

Flight Center, Marshall Space Flight Center, Alabama. Mr. Harry 

Buchanan was the Contracting Officer's Technical Representative. 

He provided valuable advice and support. 

The contributions of this effort of Dr. Krishna Kumar in the forms 

of analyses and helpful discussions were sincerely appreciated. During 

the course of this work, Mr. Daryl Thaxton contributed in several areas. 

He was especially effective in revising the codes for analytical models. 

His contributions are gratefully acknowledged. Regarding the 

preparation of this report, Mr. Jack Landreth's fine work on the 

illustrations is acknowledged. To Mrs. Marjorie McGee the authors owe 

special thanks for her expert typing of the manuscript. 
TABLE OF CONTENTS 

Page 

LISTOFTABLES . . . iv 

LISTOFFIGURES .e . . V 

LISTOF SYMBOLS . . X 

I. INTRODUCTION I 

1.0 Background . . . . . . . . . . . . . . . . . . . 1 

1.1 The Capture and Retrieval Problem . . . . . . . . 2 

1.2 Scope of This Investigation . . . . . 3 

11. CATEGORIZATION OF SATELLITES 6 

2.0 Rationale . . . . . . . . . . . . . . . . . . . . 6 

2.1 Categories . . . . . . . . . . . . . . . . . 6 

111. ANALYTICAL MODELS FOR TORQUE-FREE ROTATIONAL MOTION 

OFSATELLITES e. 10 

3.0 Introduction . . . . . . . . . . . . . . . 10 

3.1 Single-Rigid-Body . . . . . . . 10 

3.2 Dual-Spin . . . . . . . . . . . . . . . . . . . . 12 

3.3 Multi-Rotor . . . . . . . . . . . . . . . . . . . 12 

TV. SATELLITE MODEL FOR NUMERICAL SIMULATIONS 16 

4.0 Introduction . . . . . . . . . . . . . . . . . . 16 

4.1 Satellite Physical Model . . . . . . . . . . 17 

4.2 Equations of Motion . . . . . . . . . . . 24 

V. ENVIRONMENTAL TORQUES 27 

5.0 Introduction . . . . . . . . . . . . 27 

5.1 Gravity-Gradient Torques . . 29 

5.2 Aerodynamic Torques . . . . . . 31 

ii 
TABLE OF CONTENTS - Continued 

Page 

VI. SHORT-TERM APPLICABILIlY OF TORQUE-FREE 

ANALYTICAL MODEL 38 

6.0 Introduction . . . . . . . . . . . . . 38 

6.1 Applicability of Analytical Solutions . . . . 39 

6.2 Low Angular Momentum Examples . 59 

VII. PREDICTION OF LONG-TERM MOTION . . . . . . . . . . . . . 108 

7.0 Rationale . . . . . . . . . . . . . . . . . . . . 108 

7.1 Averaging Method . . . . . . . . . . . . . . . 109 

7.2 Equations of Motion . . . . . . . . 112 

7.3 Results . . . . . . . . . . . . . . . . . . . . . 116 

VIII. CONCLUSIONS AND RECOMMENDATIONS 128 

8.1 Conclusions . . . . . . . . . . . . . . . . . . . 128 

8.2 Recommendations . . . . . . . . . . 129 

REFERENCES 131 

APPENDICES........................... 134 

A. SINGLE-RIGID-BODY SATELLITE MATHEMATICAL MODEL . . . 135 

B. DUAL-SPIN SATELLITE MATHEMATICAL MODEL . 140 

C. MULTI-ROTOR SATELLITE MATHEMATICAL MODEL. e . a 149 

D. GRAVITY-GRADIENT AND AERODYNAMIC TORQUE 

MATHEMATICAL MODELS 158 

iii 
LIST OF TABLES 

1. Data for Numerical Verification of Torque-Free Analytical 

Simulation of Attitude Motion of Satellites with Asymmetric Internal Angular Momenta . . . . . . 

2. Data for Torque-Free Analytic Simulation of Attitude 

Motion of Satellites with Asymmetric Internal Angular 

Momenta.. . . . . . . . . . . . . . . . . . . . . . 3. Hubble Space Telescope (HST) Numerical Simulation, 

Satellite Physical Data . . . . . . . . . . . . . . 

4. Hubble Space Telescope (HST) Numerical Simulation Data . . 

5. LANDSAT and AXAF Numerical Simulation, Satellite 

Physical Data . . . . .. . . . . . . . . . . . . 

6. LANDSAT and AXAF Numerical Simulation Data . . 7. Data for Semi-Analytical Example 1 . . . 

40 

41 

60 

61 

62 

63 

118 

8. Data for Semi-Analytical Example 2 (CRRES) . . . . . . 123 

iv 
LIST OF FIGURES 

1. Block Diagram of Categories of Uncontrolled Satellites . 7 

2. Single-Rigid-Body Satellite Model. . . . . . . . . . 11 

3. Dual-Spin Satellite Model . . . . . . . . . 13 

4. Multi-Rotor Satellite Model . . . 15 

5. Generic Satellite Physical Model . . . . 18 

6. Inertial, Orbital Plane, and Orbiting Coordinate Systems 19 

7. Orbital Plane and Angular Momentum Coordinate Systems 20 

8. Angular Momentum and Body-Fixed Coordinate Systems . 21 

9. Local Vertical and Body-Fixed Coordinate Systems 23 

10. Angle of Attack and Sideslip Angles . . . . . . . . . . 34 

11. Diagram of Aerodynamic Cylinder Model . . . . . . 35 

12 a. Multi-Rotor Numerical Verification Case I, Precession 

Angle Time-History . . . . . . . . 42 

12 b. Multi-Rotor Numerical Verification Case I, Nutation Angle 

Time-History . . . . .. . . . . . . . . . . . . . . . . 43 

12 c. Multi-Rotor Numerical Verification Case I, Angle of Proper 

Rotation Time-History . . . . . . . . 44 

13 a. Multi-Rotor Numerical Verification Case 11, Precession 

Angle Time-History . . . . . . . . . . . . . . . . . . . . 45 

13 b. Multi-Rotor Numerical Verification Case 11, Nutation Angle 

Time-History . . . . . I . . . . . . . . . . . . . 46 

13 c. Multi-Rotor Numerical Verification Case 11, Angle of Proper 

Rotation Time-History . . . . . . 47 

14 a. Multi-Rotor Numerical Verification Case 111, Precession 

Angle Time-History . . I . . . . . . . . . 48 

V 
14 b. Multi-Rotor Numerical Verification Case 111, Nutation Angle 

Time-History... ..................... 49 

14 c. Multi-Rotor Numerical Verification Case 111, Angle of Proper 

Rotation Time-History ................... 50 

15 a. Multi-Rotor Numerical Verification Case IV, Precession Angle 

Time-History ........................ 51 

15 b. Multi-Rotor Numerical Verification Case IV, Nutation Angle 

Time-History ........................ 52 

15 c. Multi-Rotor Numerical Verification Case IV, Angle of Proper 

Rotation Time-History ................... 53 

16 a. Fidelity of Multi-Rotor Results in the Presence of 

Gravity-Gradient Torque, HAM Case, Nutation Angle 

Time-History ........................ 55 

16 b. Fidelity of Multi-Rotor Results in the Presence of 

Gravity-Gradient Torque, HAM Case, Angle of Proper 

Rotation Time History ................... 56 

17 a. Fidelity of Multi-Rotor Results in the Presence of 

Gravity-Gradient Torque, LAM Case, Nutation Angle 

Time-History ........................ 57 

17 b. Fidelity of Multi-Rotor Results in the Presence of 

Gravity-Gradient Torque, LAM Case, Angle of Proper 

Rotation Time-History ................... 58 

18 a. Time-Histories of Euler-Parameters for HST, 

Case I (250 km. Orbital Altitude) ............. 66 

18 b. Time-Histories of Roll, Pitch, and Yaw Angles for HST, 

Case I (250 km. Orbital Altitude) ............. 67 

18 c. Time-Histories of Angular Momentum Components for HST, 

Case I (250 km. Orbital Altitude) . . . . . 68 

18 d. Time-Histories of Aerodynamic Torque Components for HST, 

Case I (250 km. Altitude) ................. 69 

18 e. Time-Histories of Gravity-Gradient Torque Components for 

HST, Case I (250 km. Orbital Altitude) . . 70 

19 a. Time-Histories of Euler-Parameters for HST, 

CASE I1 (800 km. Orbital Altitude) . . 72 

19 b. Time-Histories of Roll, Pitch, and Yaw Angles for HST, 

Case I1 (800 km. Orbital Altitude) ............. 73 

vi 
19 c. Time-Histories of Angular Momentum Components for HST, 

Case IT (800 km. Orbital Altitude) ............. 74 

19 d. Time-Histories of Angular Momentum Components for HST, 

Case I1 (800 km. Orbital Altitude) ............. 75 

19 e. Time-Histories of Gravity-Gradient Torque Components for 

HST, Case I1 (800 km. Orbital Altitude) . . . . 76 

20 a. Time-Histories of Euler-Parameters for HST, 

Case I11 (184 km. x 316 km. Orbital Altitude) . . 78 

20 b. Time-Histories of Roll, Pitch, and Yaw Angles for HST, 

Case I11 (184 km. x 316 km. Orbital Altitude) . . . . 79 

20 c. Time-Histories of Angular Momentum Components for HST, 

Case I11 (184 km. x 316 km. Orbital Altitude) . . 80 

20 d. Time-Histories of Aerodynamic Torque Components for HST, 

Case 111 (184 km. x 316 km. Orbital Altitude) . . 81 

20 e. Time-Histories of Gravity-Gradient Torque Components for 

HST, Case I11 (184 km. x 316 km. Orbital Altitude) . 82 

21 a. Time-Histories of Euler-Parameters for HST, 

Case IV (184 km. x 316 km. Orbital Altitude without 

Environmental Torques) ................... 84 

21 b. Time-Histories of Roll, Pitch, and Yaw Angles for HST, 

Case IV (184 km. x 316 km. Orbibal Altitude without 

Environmental Torques) ................... 85 

21 c. Time-Histories of Angular Momentum Components for HST, 

Case IV (184 km. x 316 km. Orbital Altitude without 

Environmental Torques) ................... 86 

22 a. Time-Histories of Euler-Parameters for LANDSAT, 

Case I (352 km. Orbital Altitude, Gravity-Gradient Torque 

Only) ........................... 89 

22 b. Time-Histories of Roll, Pitch, and Yaw Angles for LANDSAT, 

Case I (352 km. Orbital Altitude, Gravity-Gradient 

Torque Only), ........................ 90 

22 c. Time-Histories of Angular Momentum Components for LANDSAT, 

Case I (352 km. Orbital Altitude, Gravity-Gradient 

Torque Only) ......................... 91 

22 d. Time-Histories of Gravity-Gradient Torque Components for 

. LANDSAT, Case I (352 km. Orbital Altitude, GravityGradient Torque Only) ................... 92 

vii 
23 a. Time-Histories of Euler Parameters for LANDSAT, 

Case I1 (705 km. Orbital Altitude, Gravity-Gradient 

Torque Only) ........................ 93 

23 b. Time-Histories of Roll, Pitch, and Yaw Angles for LANDSAT, 

Case I1 (705 km. Orbital altitude, Gravity-Gradient 

Torque Only) ........................ 94 

23 c. Time-Histories of Angular Momentum Components for LANDSAT, 

Case I1 (705 km. Orbital Altitude, Gravity-Gradient 

Torque Only) ........................ 95 

23 d. Time-Histories of Gravity-Gradient Torque Components for 

LANDSAT, Case I1 (705 km. Orbital Altitude, GravityGradient Torque Only) ................... 96 

24 a. Time-Histories of Euler-Parameters for AXAF, 

Case I (400 km. Orbital Altitude) ............. 98 

24 b. Time-Histories of Roll, Pitch, and Yaw Angles for AXAF, 

Case I (400 km. Orbital Altitude) ............. 99 

24 c. Time-Histories of Angular Momentum Components for AXAF, 

Case I (400 km. Orbital Altitude) ............. 100 

24 d. Time-Histories of Aerodynamic Torque Components for AXAF, 

Case I (400 km. Orbital Altitude) ............. 101 

24 e. Time-Histories of Gravity-Gradient Torque Components for 

AXAF, Case I (400 km. Orbital Altitude) . . . . . 102 

25 a. Time-Histories of Euler-Parameters for AXAF, 

Case I1 (400 km. Orbital Altitude, No Environmental 

Torques) .......................... 105 

25 b. Time-Histories of Roll, Pitch, and Yaw Angles for AXAF, 

Case I1 (400 km. Orbital Altitude, No Environmental 

Torques) .......................... 106 

25 c. Time-Histories of Angular Momentum Components for AXAF, 

Case I1 (400 km. Orbital Altitude, No Environmental 

Torques) .......................... 107 

26. Semi-Analytic Example 1, Short-Term Motion, Stage 1. 

Time-Histories (a) Nutation Angle; (b) Angle.of Proper 

Rotation; (c) Precession Angle . . . . 119 

27. Semi-Analytic Example 1, Intermediate Motion, Stage 2. 

- 

(a) ; (b) ; (c) YH ................... 120 

IH 

viii 
28. Semi-Analytic Example 1, Long-Term Motion, Stage 3. = = % vs. YH .........................121 

29. Comparison of Direct Integration Results and SemiAnalytic Results for Semi-Analytic Example 1 ........ 122 

30. Semi-Analytic Example 2 (CRRES), Short-Term Motion, 

Time-Histories, Stage 1. (a) Nutation Angle; 

(b) Angle of Proper Rotation; (c) Precession Angle . . . 124 

31. Semi-Analytic Example 2 (CRRES), Intermediate Motion, 

Stage 2. Time Histories (a) $; (b) ; (c) yH ...... 125 

yH 

32. Semi-Analytic Example 2 (CRRES), Long-Term Motion, I P 

Stage 3. vs. \yH ..................... 126 

B.l Angular Momentum and Body-Fixed Coordinate Systems 

for Appendices B and C ................... 143 

ix 
LIST OF SYMBOLS 

English Symbols 

cj3, j=1,2,3 

D 

dSj , j=1,2,3 

" 

;;R'cR 

H 

Principal centroidal moments of inertia about the 

satellite's xb-, yb-, zb-axes, respectively. 

a = H sine 

b = H cos0 

Center of mass of the satellite model; also geometric 

centroid of right-circular cylinder. 

Center of mass of the rotor. 

"Free molecular flow" accommodation constant, 2.2 

Direction cosines between the satellite's body-fixed axes 

and the earth's polar axis. 

Direction cosines of the position vector of the 

satellite's center of mass in the satellite's body-fixed 

coordinate system. 

Moment of inertia of the rotor about its symmetry axis 

Differential surface elements of the right-circular 

cylinder used in deriving the aerodynamic torque. 

True anomaly. 

Periodic function of 5 and y used to define $. 

Periodic functions of - x and y used to define i. 

Magnitude of total rotational angular momentum. 

Total angular momentum vector of the satellite expressed 

in the satellite's body-fixed coordinate system. 

Total angular momentum vector of the satellite expressed 

in the angular momentum coordinate system. 

Angular momentum vector of the rotor about the rotor's 

center of mass expressed in the r-basis. 

X 
h - 

z 

IR 

Lbr 

M 

pa 

P 

yH 

pJI 

R 

R - 

Re 

Internal angular momentum vector of the satellite 

expressed in the satellite's body-fixed coordinate 

system. 

Xb-, yb', and zb-components, respectively, of h. - 

Inertia dyadic of the satellite. 

Moment of inertia of the rotor about the rotor's z-axis. 

Moments of inertia about the satellite's Xb', Yb', and 

zb-axes, respectively, from inertia dyadic g. - 

Products of inertia found in the inertia dyadic 2 - 

Orbital inclination. 

Inertia matrix of the rotor about CR in the b-basis. 

Constant used in the expansion of the gravity-gradient 

torque expression for the first spherical harmonic. 

Centroidal inertia matrix of the complete satellite less 

the contribution by the rotor about its axis of rotation 

with respect to the satellite. 

Transformation matrix from the r-basis to the b-basis. 

Mean anomaly. 

Angular momentum of the rotor about its axis of rotation 

with respect to the satellites. 

Momentum conjugate to the angle 4. 

Momentum conjugate to the angle YH 

Momentum conjugate to the angle +. 

Components of angular velocity about the Xb', yb-, 

zb-axes . Radius of the satellite's orbit. 

Position vector of the satellite's center of mass in the 

satellite's body-fixed coordinate system. 

Equatorial radius of the earth. 

xi 
r , j=1,2,3 -j 

Position vectors from the centroid of the right-circular 

cylinder to the surface element dSj, j=1,2,3, 

respectively. 

Sj, j=1,2,3 Surfaces of the right-circular cylinder. 

T - 

T 

-a 

T 

73 

External torque vector. 

Aerodynamic torque vector. 

xb-, yb-, and z -components, respectively of Ta. b 

Gravity-gradient torque vector. 

xb-, yb-, and z -components, respectively of T . b -a 

External torque vector expressed in the angular momentum 

coordinate system . Xb', yb-, and zb-components, respectively, of the 

vector - T. 

%-, yH-, and z -components, respectively, of the 

vector T . H T ,T ,T 

XH 'H 'H 

- 

Ta 

t 

Torque on the rotor about its axis of relative rotation. 

Time 

U Sum of the argument of periapse and true anomaly. 

u ,v ,w Xb', yb-, and zb-components of 1. 

sj, j=1,2 Function of and used to make the differential 

equations for simpler than those for x. - 

- V Total velocity vector of an element on the satellite's 

surface. 

Translational velocity vector of the satellite's center 

of mass in the satellite's body-fixed coordinate system. 

Function of H and 

equations for 

used to make the differential Xj' - 

simpler than those for y. 

xii 
- X 

- Y 

Greek Symbols 

a 

Vector (column matrix) containing the "slow" variables, 

used in the semi-analytic method. 

Vector (column matrix) containing the "fast" variables, 

used in the semi-analytic method. 

Angle of relative rotation between the satellite, 

"carrier body ," or platform and rotor. 

aa Aerodynamic angle of attack. 

- B Euler parameter vector, - B = (f31 B2 f33)T. 

Ba Aerodynamic sideslip angle. 

Bj, j=0,1,2,3 Euler parameters. 

E Small parameter used in perturbation analysis. 

OH Angle between the orbital plane and the XH yH-plane. 

0 Classical nutation angle. 

ej, j=1,2,3 

x Latitude of satellite center of mass. 

Roll, pitch, and yaw angles, respectively. 

- x Angular velocity of the body-fixed coordinate 

system with respect to the angular momentum system. 

= (3 -986x105 --L) km2 ' 'E sec 

Gravitational constant 

P Atmospheric density. 

4 Classical angle of proper rotation. 

YH Angle used to locate xH-axis with respect to orbital 

plane. 

R 

r 

- 

QO 

Classical precession angle. 

"Node," the longitude of the ascending node; also spin 

rate of rotor relative to platform of dual-spin 

satellite. 

\ 

Angular velocity vector of the rotor in the r-basis. 

Initial angular rate of rotor. 

xiii 
W Argument of periapse. 

- w Angular velocity of the satellite in the satellite's 

body-fixed coordinate system. 

Angular velocity of the angular momentum coordinate %I 

sys tem. 

- w Angular velocity of the satellite in the r-basis. 

r 

xH-, yH- and z -components, respectively, of LOO. axH wyH @ZH H 

ax ,w ,wz xb-, yb-, and zb-components, respectively, of goo 

0 yo 0 

w Initial y. -0 

Coordinate Svstems 

CxbYbzb 

CPXPY PZP 

CRXRYR~R 

EXYZ 

Unit Vectors 

a , j=1,2,3 

b , j=1,2,3 

c , j=1,2,3 

e , j=1,2,3 

A 

-j 

-j 

-j 

-j 

:,?,E 

A 

A 

A 

ALIA 

A 

Satellite body-fixed coordinate system. 

Angular momentum coordinate system. 

Dual-spin satellite platform-fixed coordinate system. 

Dual-spin satellite rotor-fixed coordinate system. 

Local vertical coordinate system. 

Relative wind coordinate system. 

Orbiting coordinate system. 

Earth-centered inertial coordinate system. 

Orbital plane coordinate system. 

Angular momentum system unit vector triad. 

Satellite body-fixed system unit vector triad. 

Orbital plane coordinate system unit vector triad. 

Orbiting coordinate system 

Inertial coordinate system 

Unit vectors normal to the 

xiv 

unit vector triad. 

unit vector triad. 

surfaces SI, S2, and S3. 
Other Nomenclature 

(*I Time rate of change of the elements of ( ). 

Averaged value(s) of the elements of ( ). 

Tilde matrix formed from the elements of ( ). 

Double averaged value(s) of the elements of ( ). 

Time rate of change of the components of the vector 

quantity (-). 

Symbols used in the appendices are defined as they are introduced. 

xv 
I INTRODUCTION 

1.0 Background 

To date, successful satellite capture and recovery/repair 

operations have been executed by Space Shuttle astronauts using the 

Remote Manipulating Arm and the Manned Maneuvering Unit .I B~ 

both of these valuable tools are very much range-limited. Furthermore, 

attempts to capture uncontrolled satellites within the range of the 

Orbiter can lead to the dangerous exposure of the Space Shuttle crew 

members and/or the Orbiter to a possibly violently gyrating, 

uncontrolled satellite. For both of these reasons most future capture 

and recovery missions will likely be the domain of remotely piloted 

Orbital Maneuvering Vehicles (OMVs) .3 *4 

However , 

The potential usefulness of OMVs to affect the capture and 

retrieval of satellites should be significantly increased by adaptable, 

yet accurate, simulations of the target satellite's attitude motions. 

Two primary mathematical models are needed for such simulation 

capability. First, a long-term model is needed for predicting a 

satellite's attitude motion as much as six months or more in advance. 

Second, a "short-term" model adaptable and rigorous enough for use in 

simulators to train the OW "pilots" for specific satellite recoveries, 

is needed. 

A1 t hough the 

center of mass is 

problem of modeling a spacecraft's motion about its 

one which arose because of space flight, the study of 

1 
2 

spacecraft attitude dynamics is basically a contemporary application of 

classical theoretical mechanics. The analytical approach to solving the 

differential equations of rotational motion of a satellite is typified 

by Jacobi's classical solution for the free rotational motion of a 

triaxial rigid body in terms of elliptic integrals and functions. 

Although analytical solutions have been obtained for only relatively 

simple models of satellites, the more recent wide availability of 

digital computers has made it possible to use straightforward numerical 

integration of differential equations of motion for models composed of 

rigid and flexible bodies, subjected to various torques and forces. 

1.1 The Capture and Retrieval Problem 

A time period of several months may elapse between the failure of a 

satellite (for example, a control system failure, power system failure, 

maneuvering propellant depletion, or other non-catastrophic type 

problems which might precipitate the need for capture and retrieval) and 

the first possible opportunity, or "window ," to capture and retrieve it. 

Hence, an apparently necessary capability is that of accurately 

predicting the evolution of the attitude motion of a disabled satellite 

several months into the future, for the purposes of capture feasibility 

analyses and mission planning. In addition to long-term prediction 

capability, short-term attitude motion simulation capability is needed 

to train remote operators. Since all satellites which may be the 

objects of capture and retrieval missions will most certainly not fit 

into a single category of satellite or attitude motion, simulation 

capability must include more than just a single-axis-of-rotation, 

single-rigid-body model. The satellite model should be rigorous enough 
3 

to include the presence of spinning rotors or reaction wheels 

distributed symmetrically, or asymmetrically. The flexibility 

of the satellite might also be included. 

Another problem involves the quantity and quality of data on the 

states of the satellite's orbital and attitude motions. These, of 

course, are functions of the satellite's "mode of failure." There are 

three general modes, or scenarios, of "failure" which may precipitate a 

recovery. The first scenario involves a satellite which has not lost 

its telemetry, but is uncontrollable. In this case, ground controllers 

would, through the satellite's telemetry data, have at least partial 

attitude motion information. The second scenario involves a satellite 

which is both uncontrolled and has lost its telemetry capability. Such 

a satellite's state was possibly known prior to failure, but additional 

information can only be obtained through remote observation, if the 

satellite is observable. The third scenario involves space debris which 

are, of course, neither controllable nor capable of telemetry. Thus, 

the debris' rotational states are unknown except for estimates based on 

any observations which might have been possible. Therefore, in the 

capture of debris, any of a wide variety of rotational states may be 

encountered. 

- 1.2 Scope of 'his Investigation 

The primary purpose of this research is to provide mathematical 

models which may be used in investigation of various aspects of the 

remote capture and retrieval of uncontrolled satellites. Emphasis has 

been placed on analytical models; however, to verify analytical 

solutions, numerical integration must be used. Also, for satellites of 
4 

of certain types, numerical integration may be the only practical or 

perhaps the only possible method of solution. 

for analytical and numerical work, uncontrolled satellites were 

categorized using criteria based on (1) orbital motions, (2) external 

angular momenta, (3) internal angular momenta, (4) physical 

characteristics and (5) the stability of their equilibrium states. 

Chapter 11 deals specifically with categorization. 

First, to provide a basis 

Second, several analytical solutions for the attitude motions of 

satellite models were compiled, checked, corrected in some minor 

respects and their short-term prediction capabilities were investigated. 

These models should be useful in studying the attitude behavior of 

satellites which have considerable angular momentum, and in "driving" 

short-term simulation, during which human operators try to capture 

satellites. Single-rigid-body, dual-spin and multi-rotor configurations 

are treated. The models are described in Chapter 111 and Appendices A, 

B and C of this report. Copies of computer codes for evaluating the 

solutions are being supplied separately. 

Third, as indicated above, to verify the analytical models and to 

see how the "true" motion of a satellite which is acted upon by 

environmental torques differs from its corresponding torque-free motion, 

a numerical simulation code was developed. This code contains a 

relatively general satellite model and models for gravity-gradient and 

aerodynamic torques. The spacecraft physical model for the code and the 

equations of motion are given in Chapter IV. The two environmental 

torque models are described later in Chapter V and Appendix D. 
5 

Fourth, the use of torque-free analytical solutions to represent 

satellite attitude motion is considered in Chapter VI. Analytical 

results and numerical results, including gravity-gradient and 

aerodynamic torque effects for satellites are presented. 

Fifth, there are cases in which analytical solutions are not known 

for the unperturbed attitude motion of a satellite, but available 

integrals of the motion and extensive numerical results indicate that 

the motion will in a great many cases be almost periodic. To model the 

motion in such cases, a semi-analytic method was developed. This method 

is based on concepts from the Generalized Method of Averaging (GMA), but 

its application is primarily numerical. The method is described in 

Chapter VII. It has been used to predict satellite attitude motion over 

long periods of time. Results obtained using the semi-analytic method 

and direct numerical integration are presented in Chapter VII. 

Finally, Chapter VI11 contains conclusions and recommendations. 

It should be noted here that this effort did not include an 

investigation of the effects of internal dissipation of ,energy on 

satellite motion. This was considered to be beyond the scope of this 

effort. ,. 
11. CATEGORIZATION OF SATELLITES 

2.0 Rationale 

Artificial earth satellites can be categorized according to (1) the 

orbital motion of their centers of mass, (2) their gross (external) 

rotational motion, (3) their internal motion, (4) their individual 

physical characteristics and (5) the stability of their equilibrium 

statese5 The motion of a satellitels center of mass about the earth is 

of importance in determining whether the satellite may be reached with 

the means available. If the satellite is accessible, the orbital 

characteristics are needed in modeling the effects of environmental 

torques. The initial rotational motion of the satellite about its 

center of mass as well as its internal angular momentum determine the 

satellite's initial rotational kinetic energy and total rotational 

angular momentum. Using these and satellite physical data, one can 

predict the general principal characteristics of the uncontrolled 

attitude motion which may evolve. Examples of this uncontrolled 

attitude motion include continuous, steady rotation and librating 

motion. 

,. 2.1 Categories 

The block diagram shown in Fig. 1 depicts categories of 

uncontrolled satellites based upon orbital criteria.5 The first two 

categories, LEO (Low Earth Orbit) and HE0 (High Earth Orbit), are based 

6 
7 

I aa > 

Iu 

c 

I- - 

m 

a 

000 I 

wwoo 

JIZW 

. 

m 

aJ 

U 

d 

4 

d 

aJ 

U 

a 

m 

a 

al 

d 

d 

0 

b 

U 

C 

0 

U 

s 

w 

0 

0 

bo 

aJ 

U 

a 

u 

w 

0 

0 

0 

d 

m 

d 

. 

M 

d 

E 
a 

upon the perigee altitude of the satellite's orbit. A LEO is considered 

one which has a perigee altitude of less than 600 km. For those periods 

of time during which the satellite operates in the LEO altitude range, 

it is subjected to significant aerodynamic torques as well as other 

environmental torques. For that portion of the satellite's orbit in 

which it moves in the HE0 altitude range, aerodynamic torques become 

much less significant than gravity-gradient torques 

The next two categories, NCO (Near Circular Orbit) and EQ (Elliptic 

Orbit), determine the satellite's orbital motion with respect to 

altitude and time. An elliptic orbit of significant eccentricity 

subjects the satellite to substantial periodic changes in the 

environmental torques which affect the satellite's attitude motion. On 

the other hand, for a satellite moving in a circular, or near circular, 

orbit, the magnitudes of the environmental torques to which it is 

subjected remain in a narrow range. 

Uncontrolled satellites also can be placed into several 

subcategories based on their rotational motion about their respective 

centers of mass. These subcategories include High Kinetic Energy (HKE), 

Low Kinetic Energy (LKE), High Angular Momentum.(HAM), and Low Angular 

Momentum (LAM). The HKE satellites possess rotational kinetic energy 

which is orders of magnitude larger than the maximum possible potential 

energy due to gravity-gradient torque acting upon the satellite. This 

high kinetic energy state could be due to a high "gross" rate of 

rotation; that is, a high rotational rate of the entire body, and/or due 

to large amounts of internal angular momentum and internal rotational 

kinetic energy. Satellites in the HAM subcategory possess rotational 
9 

angular momentum, - H, due either to gross or internal relative motion of 

sufficient magnitude that the ratio of the magnitude of the applied 

torque to the magnitude of - H is small. 

and HKE are not mutually exclusive. In point of fact, they overlap a 

great deal. 

Obviously, the subcategories HAM 

Other subcategories are defined to deal more specifically with the 

inertia characteristics of the individual satellites. Two of these 

subcategories include Asymmetric Inertia (AI) ellipsoid, and Symmetric 

Inertia (SI) ellipsoid. Still other subcategories may be based upon the 

initial orientation of the satellite’s body-fixed axes with respect to 

the radius vector between the center of the earth and the satellite’s 

center of mass. This orientation is of central importance in the 

determination of the torque on the satellite which in turn are modeled 

in stability analyses. The stability of its initial motion has a direct 

bearing upon the later motion of a satellites under the effects of 

aerodynamic and/or gravity-gradient torques. Those subcategories, 

associated with aerodynamic torques include AS, ANS, AUS (Aerodynamically Stable, Neutrally Stable, and Unstable initial states, respectively). 

Similarly, for Gravity Gradient stability, we have the subcategories 

GGS, GGNS, and GGUS.S 

static stability rather than dynamic stability. 

In all these the type of stability considered is 

Finally, satellites can be categorized according to their construction and their components. Such categories relate directly to the 

choice of the physical model and will approximate a satellite best. 

Examples are dual-spin satellites and satellites which contain reaction 

wheels for attitude control. 
111. ANALYTICAL MODELS FOR TORQUE-FREE 

ROTATIONAL MOTION OF SATELLITES 

3 .O Introduction 

The kinds of satellites to be considered in this investigation as 

typical candidates for capture and retrieval include single-rigid-body 

and dual-spin satellites as well as satellites which contain multiple 

spinning rotors (reaction wheels). In this Chapter three torque-free 

analytical models for these three types of satellites are discussed. 

Such models serve at least three purposes. First, if its rotational 

angular momentum is not too small, over short time periods such as that 

required for capture, the actual attitude motion of a satellite is 

essentially the same as its torque-free motion. Second, to predict 

satellite motion over longer periods of time, perturbation theories for 

satellite attitude motion,6-1° analogous to those for satellite orbital 

motion may be developed on the basis of analytical solutions for 

torque-free motion. Third, analytical solutions provide a means for 

checking the accuracy of numerical integration procedures. 

3.1 Single-Rigid-Body Satellite Model 

As the name implies, the single-rigid-body model consists of a 

single, rigid, asymmetric body. Using this single-rigid-body model, and 

general perturbation methods 

Cochran, 

Beletski ,9 Crenshaw and Fitzpatrick,6 

and Liu and Fitzpatrick , * have developed theories to predict 

10 
11 

i” 

Fig. 2. Single-Rigid-Body Satellite Model. 
12 

the attitude motion in the presence of gravity-gradient torques. If 

the satellite modeled (see Fig. 2) is rotating rapidly about its center 

of mass, then its short-term motion is essentially classical "freeEulerian" motion. This "free-Eulerian" motion may be described by 

an exact analytical solution which involves elliptic functions and 

integrals .I1 ,12 

model. 

Appendix A contains the details of this mathematical 

3.2 Dual-Spin Satellite Model 

Satellites which contain a single spinning rotor and a nominally 

despun platform, or "carrier" body , are called "dual-spin" satellites 

(see Fig. 3). More specifically, the dual-spin satellite can be modeled 

as a system of two rigid bodies coupled together in such a manner that a 

rotation about an axis fixed in both bodies is the only relative degree 

of freedom. A general, exact analytical solution for the torque-free 

attitude motion of an arbitrary dual-spin satellite does not exist. 

However, Cochran, Shu and Rew,13 building on previous work (Ref 12, 

page 37) have obtained a complete, exact, analytical solution for the 

case in which: (1) one body is axisymmetric; (2) the axisymmetric body 

rotates relative to the other about the former's axis of symmetry; (3) 

the other body is asymmetric; (4) the axis about which relative rotation 

occurs is the axis of major, or minor, moment of inertia of the asymmetric body; (5) the relative rotation is either free (no internal torque), 

or the relative spin rate is constant. A summary of the mathematical 

model of Ref. 13 is presented in Appendix B for completeness. 
13 

/ 

Fig. 3. Dual-Spin Satellite Model. 
14 

3.3 Multi-Rotor Satellite Model 

An exact analytical solution is also available for the torque-free 

attitude motion of a particular model of a satellite which contains 

spinning rotors; i .e., "momentum wheels ," or "reaction wheels ." In this 

solution, also due to Cochran and Shu,14 the satellite physical model 

(see Fig. 4) is that of a gyrostat, consisting of two axisymmetric, 

constant-speed rotors, or momentum wheels, and a "carrier" rigid body 

which has a mass distribution such that the centroidal inertia ellipsoid 

of the system of rigid bodies is axisymmetric.14 The model is general 

enough to represent any axisymmetric satellite that contains an 

arbitrary number of axisymmetric rotating components which together 

produce a resultant relative, or "internal," angular momentum vector 

that is not parallel to a principal axis of the system. As noted, the 

solution does, however, require constant internal angular momentum and 

an axisymmetric system. The solution is given in Appendix C. 

- 

- 
15 

Fig. 4. Multi-Rotor Satellite Hodel. 
TV. SATELLITE MODEL FOR NUMERICAL SIMULATIONS 

4.0 Introduction 

Many satellites contain variable-speed spinning rotors or other 

moving parts. Should an exact analytical solution for the torque-free 

attitude motion of such a satellite exist, it will undoubtedly be much 

more complicated than the single-rigid-body analytical solution 

discussed in Chapter 3 and given in Appendices A, B and C. Thus, for 

complex models, a practical alternative is to use numerical integration 

techniques to obtain even the torque-free, or "unperturbed" solutions .15 

Furthermore, to determine the motion of a satellite which is dynamically 

complex and/or is exposed to significant perturbing torques (for 

example, a LAM satellite in LEO), numerical integration of the equations 

of motion is the only approach generally applicable. In this Chapter, a 

physical model of a satellite is described. The corresponding 

mathematical model presented following the physical model has been 

incorporated in a digital simulation code along with models of 

gravity-gradient and aerodynamic torques. This code utilizes a 

fourth-order, Runge-Kutta, numerical integration algorithm to produce 

attitude motion time-histories. 

The purposes of the simulation code are: (1) to verify the 

correctness of the analytical solutions; (2) to determine the conditions 

under which torque-free solutions can be considered reasonably valid; 

16 
17 

and (3) to produce attitude motion time-histories for cases in which 

analytical solutions are not known. 

4.1 Satellite Physical Model 

The "general" satellite model (see Fig. 5) consists of two rigid 

bodies, each of arbitrary mass distribution, which are coupled together 

so that the smaller body, or rotor, can be rotated with respect to the 

main body of the satellite, only about an axis which is fixed in the 

satellite and the rotor, and passes through the center of mass of the 

rotor. This physical model can be used to represent single-rigid-body 

and dual-spin satellites and also satellites which possess internal 

angular momentum due to constant-speed multiple, spinning components. 

Since the equations of motion for the general model are to be integrated 

numerically, environmental torques of some complexity may be included. 

Because the description of the attitude motions of satellites 

requires the use of reference frames and coordinate systems, the 

coordinate systems in Figs. 6, 7, and 8 are introduced. They are all 

dextral, orthogonal systems. The nonrotating EXYZ (inertial) 

coordinate system, depicted in Fig. 6, has its origin at the center of 

the earth and is considered inertially fixed. The unit vector triad 

(I,J,K) is fixed to the EXYZ system. The coordinate system, Ex y z 

0 0 0' 

is called the "orbital plane system" and is defined by requiring that 

the %-axis lie along the line of nodes of the satellite's orbit, 

AAA 

--- 

directed toward the ascending node, and that the zo-axis be 

perpendicular to the orbital plane, collinear with the orbital areal 

velocity vector. AAA 

The unit vector triad (c~,E~,~~) is fixed to the - 
18 

yb / 

I 

Fig. 5. Generic Satellite Physical Model. 
19 

/ i 

Fig. 6. Inertial, Orbital Plane, and Orbiting Coordinate Systems. 
20 

I 

Fig. 7. Orbital Plane and Angular Momentum Coordinate Systems. 
21 

\ 

Fig. 8- Angular Momentum and Body-Fixed Coordinate Systems. 
Exoyozo 

has its 

Fig. 6. 

22 

coordinate system. An "orbiting coordinate system" Cgqs which 

origin at the center of mass of the satellite is also shown in 

The 6-axis of this coordinate system is collinear with - R, the 

position vector of C. The q-axis lies in the orbital plane and is 

directed in the sense of increasing true anomaly, f. The unit vector 

triad The orientation of the orbiting 

frame is defined by using the standard angles: Q, the longitude of the 

ascending node; i, the inclination; w, the argument of periapse; and f, 

the true anomaly. The notation, u=&f, is used for convenience. 

Ann (sl,e2,c3) is attached to Cgns. 

For cases in which the satellite's rotational angular momentum is 

large (HAM) compared to the magnitude of the external torque, the 

orientation of the satellite with respect to the inertial system is 

specified by first introducing the "angular momentum coordinate system,*' 

CXHYHZII, which has its zH-axis collinear with - H, the total angular 

momentum vector of the satellite due to rotation about its center of 

mass (see Fig. 7). The angle \YH is used to locate the xH-axis which 

lies along the intersection of the orbital plane and the plane which 

passes through C and is perpendicular to - H. The angle between these 

two planes is %. 

CXHYHZH system. Finally, the orientation of the satellite-fixed, 

centroidal, principal system Cxbybzb with respect to CXHYHZH system is 

defined by using the Euler angles JI, 0, and + (as shown in Fig. 8) in a 

3-1-3 rotation sequence. The angles JI, 0, and 41 are identified as the 

classical angles of precession, nutation, and proper rotation, 

respectively. 12 

Ann 

A unit vector triad (~t,a_~,a_~) is associated with the 

The unit vector triad (bl ¶kZ ,k3) is 

attached to CXbYbZb 
23 

yb 

X 

Fig. 9. Local Vertical and Body-Fixed Coordinate Systems. 
24 

Alternatively, for the LAM cases in which the satellite's angular 

momentum vector is small in magnitude compared to that of the external 

torque, a "local vertical" system Cxvyvzv is used which has its zv-axis 

directed parallel to the negative e-axis of the Ceqr system and its 

xv-axis parallel to the orbital velocity vector. 

satellite body-fixed system CxbYbZb with respect to the Cxvyvzv system 

is defined by the angles 83, 82, and 01 (as shown in Fig. 9) in a 

3-2-1 rotation sequence. 

pitch, and yaw angles, respectively. 

The orientation of the 

Angles €I1, e2, and e., correspond to roll, 

4.2 Eauations of Motion 

The equations governing the attitude dynamics of the "general*' 

satellite model depicted in Fig. 5, may be derived using H, the angular 

momentum of the satellite, expressed in the satellite's body-fixed 

components; Pa, the angular momentum of the rotor about the axis of 

rotation of the rotor with respect to the satellite; a, the angle of 

relative rotation; and four Euler parameters, Bj, j=0,1,2,3. 

- 

Let ,K denote the centroidal inertia matrix of the complete 

satellite, less the contribution by the rotor about the axis of rotation 

of the rotor with respect to the satellite. Then, the total angular 

momentum can be expressed as 

_H=y_o +h, (4.2 .l) 

where - w is the absolute angular velocity of the satellite in the 

satellite's body-fixed components (b-basis) and where - h, the internal 

angular momentum vector, is 
25 

(4.2.2) 

where J - is the inertia matrix of the rotor about CR, the center of mass 

of the rotor, in the b-basis; hris the transformation matrix from the 

r-basis (a coordinate system fixed in the rotor with its 3-axis along 

the axis of rotation of the rotor with respect to the satellite) to the 

b-basis; and D is the moment of inertia of the rotor about the rotor 

axis of rotation. 

The differential equation for the matrix, €I, is 

.. - fi = H -- g-L(lJ-h_) + T , 

where 

(4 -2.3) 

(4.2.4) 

I 

and - T is the external torque expressed in the satellite's body-fixed 

components. The momentum Pa varies according to 

(4.2.5) 

where -%/c,is the angular momentum vector of the rotor about its own 

r 

center of mass, expressed in the r-basis; w and are the angular 

velocities of the satellite and the rotor, respectively, also expressed 

- 

r r 

in the r-basis; and T, is the torque on the rotor about the axis of 

relative rotation. 
I 

and 

26 

The Euler parameters Bj, j=0,1,2,3, vary with time according to 

(4.2.7) 

where 4 = ( B1 4 (see, for example, Ref. 17, pp. 17-18 and 26). 

The direction cosine matrix, which defines the orientation of the 

Cx,ybzb system with respect to the inertial coordinate system EXYZ may 

be expressed in terms of B , j=lS2,3, as follows: 

&I 

J 
5.0 Introduction 

V. TORQUES 

The environmental torques which act on artificial satellites are 

very small compared to, for example, the aerodynamic torques which act 

on aircraft. However, when even small torques act over significant 

periods of time, large changes in attitude motion can occur. As 

discussed in the chapter on "Categorization," if the time period of 

concern is relatively short, and if the magnitude of the satellite's 

rotational kinetic energy is relatively large compared to the work done 

on the satellite by its environment, the change in a satellite's 

attitude will occur very slowly. A very important factor governing the 

effects of environmental torques on the attitude motion of satellites is 

the ratio of ITI, - the magnitude of the perturbing torque to IHI, - the 

magnitude of the satellite's total angular momentum. When the ratio 

1T1/1H1 -- is very small, say fl(10-4), as it would be for a HAM satellite, 

the short-term effects may be considered "minute." However, over long 

periods of time they can significantly perturb a HAM satellite's 

attitude motion. 

for a LAM satellite in LEO, the satellite's attitude motion will be very 

strongly perturbed, even in the short term. In some cases, tumbling 

motion which exhibits no well-defined pattern may occur. Under other 

circumstances a LAM satellite will tend to oscillate "about a position 

of relative equilibrium."6 s9 

When lTl/1H1 -- is much larger, sayel), as it might be 

27 
28 

Generally, environmental torques fall into two main categories. 

The first category consists of environmental torques which are derivable 

from a potential function. The most studied member of this category is 

the gravity-gradient torque; however portions of the "aerodynamic" 

torque (due to interactions of the atmosphere and a satellite) may also 

be derivable from a potential function. These torques are therefore 

"conservative." However, when the motion of the center of mass of the 

satellite is specified, there is generally no integral of the motion 

corresponding to conservation of energy, but in the special case of a 

circular orbit, a Jacobi-type integral does exist if only "conservative" 

torques are present. The other category of torques is that of 

"dissipative ," or "damping ," torques. These nonconservative torques 

cause secular changes in the rotational kinetic energy of the satellite 

and its attitude motion. The more significant of these are: the 

dissipative parts of the aerodynamic torques; the torques due to the 

solar radiation pressure--direct from the sun, as well as reflected by 

the earth and its atmosphere; and torques resulting from the interaction 

of conducting parts of satellites with the earth's magnetic field. The 

most dominant in orbits of altitude up to 300 km are the aerodynamic 

torques .6,9 9 16 

Many of the perturbation techniques employed in the treatment of 

Satellite orbital motion can be employed analogously to study satellite 

attitude motion in the presence of perturbing environmental torques. 

Any further discussion, or reference, to environmental torques in this 

report will center on the two most dominant, gravity-gradient and 

aerodynamic. 
29 

5.1 Gravity-Gradient Torques 

According to Beletskii ,9 the primary effect of gravity-gradient 

torques "amounts to secular precession of the angular momentum vector 

around the normal to the orbital plane." He goes on to say that 

"periodic nutations of the angular momentum vector (with a period 

comparable to the satellite's orbital period) are superimposed on this 

secular precession."9 

Gravity-gradient torques are due to the fact that the earth's 

gravitational field is not uniform. This non-uniformity in both the 

magnitude and direction of the field over distances even as small as the 

dimensions of an orbiting artificial satellite result in a "gravitygradient" potential over the satellite body. This gravity-gradient 

potential over the satellite results in a gravity-gradient torque about 

the satellite's center of mass. This gravity-gradient torque was first 

considered in the context of celestial mechanics in 1749 by d'Alembert 

and Euler, and later in 1780 by Lagrange.l7~l8 

A number of factors are involved in determining the perturbation 

effect of gravity-gradient torque upon the satellite's attitude motion. 

If one assumes that the satellite is a single-rigid-body orbiting a 

spherical primary in a circular orbit, the problem is greatly 

simplified. With these assumptions, the gravity-gradient torque becomes 

a function of the distance from the center of mass of the satellite to 

the center of the earth; the values of the principal moments of inertia; 

and the orientation of the satellite's body-fixed axes with respect to 

the radius vector between the center of the earth and the satellite's 

center of mass. Using the above assumptions, the gravity-gradient 
30 

torques on an asymmetric satellite in a Newtonian gravitational field 

can be expressed in the form9 sr6 I ,l9 

r(C-B) '23'331 

(5.1.1) 

where R is the magnitude of R, the position vector 

the earth to the satellite's center of mass; A, B, 

- from the center of 

and C are the 

principal centroidal moments of inertia; and the cj3, j=1,2,3, are the 

direction cosines of - R in the satellite's body-fixed coordinate system. 

If the above simplifying assumptions are not made the expression for the 

gravity-gradient torque is much more complex. For example, if one 

assumes an "oblate earth," the expression for the xb-component of the 

gravity-gradient torque has the form" 

3' J R2 

E e (C-B)[5(1-7 sin2X)c23c33 2R 

'E T = (C-B) c c 23 33 i- gX 

(5.1.2) 

where J2 = 1.083~10'3, X is the latitude of the center of mass of the 

satellite, Re is the equatorial radius of the earth, and ~ej3, j=1,2,3, 

are the direction cosines between the 

polar axis. Note that the additional 

Eq. (5.1.2) diminish at a rate of R-5 

diminish at a rate of R-3. For small 

due to the asphericity of the primary 

body-fixed axis and the earth's 

gravity-gradient torque terms in 

while the terms in Eq. (5.1.1) 

satellites, the additional terms 

a can be neglected as they normally 
31 

result in less than a 1% difference, even for satellites in "low earth 

orbits."L7 However, if one were to consider a larger satellite, such as 

the proposed Space Station, then the higher order terms would take on 

greater significance .I I 

Another concern is that of more than one celestial body. In the 

earth, moon and sun system one must consider the gravity-gradient 

effects of the other celestial bodies in much the same manner that 

one considers their perturbing effects upon orbital trajectories. For 

near earth orbits, one may assume that the satellite is within the 

"sphere of influence" of the earth, and include only the perturbing 

effects of the earth. To illustrate this point, consider that for a 

geosynchronous orbit the gravity-gradient torque due to the moon is less 

than 0.0023% of that due to the earth, and the gravity-gradient torque 

at geosynchronous altitudes due to the sun is less than 0.0011% of that 

due to the earth. * 

To summarize, for the purposes of the simulations used in this 

investigation, the following assumptions were found to give a 

sufficiently accurate representation of gravity-gradient torques on the 

satellites modeled: (1) the satellite mass distribution is that of a 

single, tri-inertial (or asymmetric), rigid body; (2) its center of mass 

is moving in a two-body orbit about a spherical primary; (3) the 

greatest dimension of the satellite is much smaller than the radius of 

the orbit of its center of mass; and (4) the orbital plane may be 

rotating in a prescribed manner. 
32 

5.2 Aerodynamic Torques 

As noted in the introduction to this chapter on environmental 

torques, there are a number of dissipative torques. The most dominant 

of these in the LEO altitude range are the aerodynamic torques. 

Although well outside the domain of "atmospheric vehicles," artificial 

earth satellites are by no means free of the effects of the atmosphere. 

This is demonstrated by the constant attention which those who track 

satellites pay combating the problem of orbital decay due to atmospheric 

drag. The treatment of aerodynamics at orbital altitudes as opposed to 

lower altitudes is different. At orbital altitudes the "mean free 

paths" between molecules are on the order of a kilometer or more, much 

larger than the dimensions of LEO satellites. This means that one need 

not consider collisions between molecules approaching the satellite's 

surface with those leaving the surface. Thus, one can use a 

" f ree-molecular flow model" to approximate the "aerodynamics" of 

satellites. This treatment greatly simplifies the determination of the 

aerodynamic torques acting upon a ~atellite.~ sL I 

There are four primary aerodynamic effects on satellites moving in 

the free-molecular flow regime. The first effect is similar to the 

"Weather Cock" effect which occurs in the lower portions of the 

atmosphere. This effect is due to the fact that the center of pressure 

and the center of mass of the satellite do not coincide. As a result of 

this, there is a restoring torque which tends to stabilize the satellite 

in the direction of the resultant of the orbital velocity of the 

satellite and the local velocity of the atmosphere due to the earth's 

rotation. The second effect is due to the angular velocity of the 
33 

satellite about its own center of mass. The torques developed due to 

this effect are known as the "spin damping dissipative torq~es."6,~ 

Since even for high rates of satellite rotation the linear velocity due 

to rotation is much smaller than the satellite's orbital velocity, these 

torques can be considered to vary linearly with angular velocity. The 

last two effects are small by comparison and are usually ignored. These 

are pressure gradient and the molecular thermal velocity effects. The 

pressure gradient effect is due to the density differential which exists 

over the satellite surface (being higher on the portion of the satellite 

facing the earth). 

generally neglected because it is much smaller than the orbital velocity 

of the satellite.9 

The contribution of molecular thermal velocity is 

For the purposes of this research effort, only the first two 

aerodynamic effects were considered to be significant. The model chosen 

to simulate the satellite in a free-molecular flow regime is a rightcircular cylinder, a shape which is common among satellites. The 

coordinate axes chosen, as seen in Fig. 9, are very similar to those 

used by Etkin20 to describe the motion of atmospheric vehicles. 

that the axis of symmetry has been designated the xA-axis in a 

CXA~AZA system. For the purposes of this work, it is assumed that 

the axes of CAXAYAZA are aligned with the axes of CXbYbzb, but that C 

and CA do not coincide. The CAX~Y~Z~, "relative wind" system, has its 

Note 

xw-axis along the orbital velocity vector and its +-axis lies in the 

orbit plane and points (nominally) toward the center of the earth. 

The angles, aa and &, between the relative wind system and the 

body-fixed system are the angle of attack and the sideslip angle, 

respectively, of the satellite (see Fig. 10). For the purpose of 
34 

Fig. 10. Angle of Attack and Sideslip Angles. 
35 

Figrn 11. Diagram of Aerodynamic Cylinder Model. 
36 

integrating the force per unit area, the cylinder is divided into three 

surfaces, Si, S2 and S3 (see Fig. 11). Position vectors, L~, z2, and 

from the centroid of the cylinder to the surface elements dS1, dS2, 

and dS3, respectively, are used in deriving an expression for the torque 

g3 

about CA. The aerodynamic torque on the cylinder may be written as 

follows : 

(5.2.1) 

where p is atmospheric density, cd is an accommodation constant, 

21, g2, and n 

A A 6 

are the normal unit vectors normal to each respective -3 

surface, ,V = yc + ,w x r , j=1,2, or 3, where 

velocity of the satellite's center of mass,* and ,w = (p q r)T. 

is the translational 

-j 

The 

results of the derivation of the aerodynamic torques on the right 

circular cylinder are given in Appendix D. 

It must be realized that any model of the aerodynamic torque on a 

satellite in orbital space is an approximation. Two factors which limit 

the accuracy of the modeling are variations in atmospheric density, and 

"shadowing," or "blanketing," of portions of the body by its other 

portions, for example, solar panels.lls21 Since the purpose of the 

sirrulations contained in this thesis is to determine qualitatively the 

*Note that strictly speaking the velocity 

However, this would only modify the results by 

j=1,2,3, slightly. 

should be used. 

changing the gj, 
37 

effects of environmental torques, a rather simple exponential 

atmospheric density model was adopted. Obviously, this model does not 

take into account such sources of variations in density as the diurnal 

cycles, orbital inclination (latitude variations at equal altitudes), 

and solar activity. Also, although a right-circular cylinder is a good 

approximation to the shape of many satellites, for satellites which have 

complex geometries (such as those with large sun seeking solar arrays) 

shadowing is a major problem. Such arrays can, at certain satellite 

attitudes, blanket large portions of the satellite surface resulting in 

torques greatly different from those on an unshadowed However, 

the modeling of these complex satellite arrays for all possible aspects 

is extremely difficult and is beyond the scope of this effort. 
VI. SHORT-TERM APPLICABILITY OF TORQUE-FREE 

ANALYTICAL MODEL 

6.0 Introduction 

When a closed-form analytical solution exists for the torque-free 

motion of a particular satellite, it is generally the more efficient 

solution for short-term attitude motion simulations. Numerical 

solutions, although generally available, tend to require much more 

computer time than torque-free analytical solutions, especially in the 

case of satellites with high rates of rotation which necessitate the use 

of small integration step-sizes. However, when one attempts to 

incorporate the perturbing effects of environmental torques into the 

analytical approach, the complexity of the analytical solution increases 

significantly. In some cases, specifically, those in which the 

magnitude of the total angular momentum is much larger than the 

magnitude of the environmental torques acting upon the satellite, the 

torque-free analytical solutions are very accurate over short time 

periods. In other cases, for example, those in which the magnitude of 

the environmental torque is greater than the magnitude of the total 

angular momentum of the satellite, a numerical approach incorporating 

environmental torques is usually necessary. 

The purposes of this chapter are to investigate the fidelity and 

applicability of the analytical solutions discussed in Section 3.3 and 

to present some results of numerical simulations of LAM satellite attitude motion. First, to verify their fidelity, the analytical solutions 

38 
39 

are compared with results from the numerical simulation code. Second, 

to determine the applicability of the torque-free analytical solutions, 

they are compared with numerical simulation results obtained using the 

same initial conditions, but including gravity-gradient torques. This 

comparison illustrates how, at lower angular momentum levels (as 

compared to the magnitude of the environmental torque), the fidelity of 

the torque-free analytical solutions degenerates rapidly, indicating the 

need to use either an analytical or a numerical solution which 

incorporates environmental torques. Third, to show that there are cases 

for which the analytical solutions cannot be expected to provide 

accurate results, some examples of simulated attitude time-histories of 

actual satellites in the LAM category are presented. 

6.1 Applicability of Analytical Solutions 

To verify the analytical solutions, the numerical simulation code 

was run with the data shown in Table 1. The data used in the analytical 

solutions is given in Table 2. Note that the moments of inertia in 

Table 1 differ slightly from those in Table 2. This is due to the fact 

that the numerical code uses a single movable rotor to simulate the two 

rotors of the analytical model. In each, the system is axisymmetric. 

As verification of the analytical solution, the results for the four 

cases are shown in Figs. 12, 13, 14, and 15. The variable "TAU" is a 

"no~idimensional" time, defined as TAU = Ht/A. The numerical results are 

shown as triangles superimposed on the corresponding analytical 

time-histories (continuous curves) of the precession, nutation, and 

proper rotation angles, respectively. The first three cases (shown in 

Figures 12, 13, and 14) are for an oblate satellite with various initial 

angular velocities. The fourth case (Fig. 15) is for a rapidly spinning 
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43 

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Fig. 12 b. Multi-Rotor Numerical Verification Case I, Nutation Angle 

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44 

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Fig. 13 a. Multi-Rotor Numerical Verificatiop Case 11, Precession 

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Fig. 15 C. Multi-Rotor Numerical Verification Case IV, Angle of Proper 

Rotation Time-History. 
54 

prolate, dual-spin satellite with a misaligned rotor. As can be seen, 

the numerical and analytical results are, within the numerical 

precession of the calculations, the same. 

The numerical simulation code was modified to demonstrate the 

fidelity of the analytical solution for HAM cases in the presence of 

environmental torques. The modifications to the code included the 

incorporation of gravity-gradient torques, and a routine which would 

allow the sampling of the time-histories of the three attitude angles at 

a time many nutation periods in the future. The computer program was 

used to integrate the equations of motion over the intervening period, 

but output to data files for plotting was limited to the desired sample. 

The case which was run used the data from Case I11 in Table 1. The 

attitude motion of the satellite as perturbed by gravity-gradient torque 

was simulated for a time equal to a quarter of an orbit (1342.6 sec for 

an orbital altitude of 250 km). As can be seen in Fig. 16, the torquefree analytical time histories and the projected numerical solutions 

agree in amplitude very well. There is a "phase shift" which is probably due to to the fact that the nutation period was not exactly equal 

to the number used to determine the beginning of the second sample time. 

To determine the magnitude of rotational angular momentum for which 

there would be significant disagreement between the two solutions, the 

initial angular momentum was successively reduced. To obtain the 

results shown in Fig. 17, the initial angular velocity and internal 

angular momentum components were each divided by 120. The results were 

chosen for illustrative purposes because they show a large difference in 

the amplitudes of the solutions, but the attitude motion structure has 
55 

I I I I I I I 

-Analytical 

+Numerical 

I I I 

2: 5 5' 7: 5 1 

TAU 

Fig. 16 a. Fidelity of Multi-Rotor Results in the Presence of 

Gravity-Gradient Torque, HAM Case, Nutation Angle 

Time-History. 
56 

51 I I I I I I L I 

-Analytical 

+Numerical 

I 

I I I 

2: 8 8' 7: s I 

TAU 

Fig. 16 b. Fidelity of Multi-Rotor Results in the Presence of 

Gravity-Gradient Torque, HAM Case, Angle of Proper 

Rotation Time History. 

, 
57 

Fig. 17 a. Fidelity of Multi-Rotor Results in the Presence of 

Gravity-Gradient Torque, LAM Case, Nutation Angle 

Time-History . 
A 

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1: 8 8' 7: 8 1 

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Fig. 17 b. Fidelity of Multi-Rotor Results in the Presence of 

Gravity-Gradient Torque, LAM Case, Angle of Proper 

Rotation Time-History . 
59 

not yet broken down. That is, the three angles of precession, nutation, 

and proper rotation are still appropriate for describing the motion of 

the satellite. As was noted previously in Section 4.1, at low angular 

momentum levels, roll, pitch, and yaw angles are more appropriate for 

describing the satellite's attitude motion because then $, e, and Q are 

not well defined. The amplitudes of the LAM (torqued) numerical results 

are significantly greater than the HAM torque-free analytical results. 

One could infer from these that the torque-free analytical solutions 

would be appropriate for short-term HAM satellite attitude motion 

simulations, and that the (torqued) numerical simulations more 

appropriate for short-term LAM satellite attitude motion simulations. 

6.2 Low Angular Momentum Examples 

The capability of the numerical simulation code when applied to low 

angular momentum (LAM) satellite simulations, was demonstrated by making 

eight simulation runs using data for three actual satellites. Four 

simulation runs were made using data for the Hubble Space Telescope 

(HST). Then, using the data from the Earth Resources Satellite 

(LANDSAT) and the Advanced X-Ray Astrophysics Facility (AXAF), two 

computer runs were made for each of the two satellites. The satellite 

physical data for the HST satellite simulations can be found in Table 3. 

The corresponding orbital data can be found in Table 4. The data used 

for the LANDSAT and AXAF simulation runs found in Tables 5 and 6 is 

presented in the same manner. 

All of the results from the simulations are in the form of 

time-histories of selected parameters over five orbital periods. 

These parameters are: (1) the Euler parameters, Bo, 61, 62, and 63; 
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(2) orientation of the satellite with respect to its local vertical 

coordinate system in terms of roll, pitch, and yaw angles; (3) the 

body-fixed components of the satellite's total angular momentum vector; 

(4) where applicable, the body-fixed components of the aerodynamic 

torque, and (5) again, where applicable, the components of the gravitygradient torque, also with respect to the satellite's body-fixed axes. 

The four HST cases run include circular LEO, circular HEO, and 

elliptic LEO cases, all with environmental torques, and an elliptic LEO 

case without environmental torques. The results for the HST are shown 

in Figures 18, 19, 20, and 21. The LANDSAT cases are a circular LEO 

case and a circular HE0 case. 

torques. The two AXAF cases run are both circular LEO, one with 

environmental torques and one without. The results of the LANDSAT 

simulations are found in Figures 22 and 23. The results of the AXAF 

simulations are found in Figures 24 and 25. A more detailed analysis of 

the simulation results is made in the following paragraphs. 

6.2.1 HST Simulations 

Both include only gravity-gradient 

In the first HST case, as with other HST cases, the satellite is 

initially "pitching down," or rotating, about its negative yb-axis at 

orbital rate. The satellite also has some internal angular momentum due 

to a reaction wheel. spinning about its xb-axis. 

begin with a 45" pitch angle with respect to the local horizontal. 

All of the HST cases 

HST Case I 

The first figure for the HST (see Fig. 18 a) shows the 

time-histories of the Euler parameters, henceforth referred to as the 

"Beta plots." The individual Euler parameters oscillate in two modes; a 
65 

"low" frequency mode with a comparatively large amplitude and a "high" 

frequency mode with a much smaller amplitude. The time-histories of the 

roll, pitch, and yaw angles are presented in Fig. 18 b. Note that the 

pitch angle begins at 45" and oscillates at a relatively high frequency 

at progressively lower amplitude due to aerodynamic damping. After five 

orbits, the amplitude of the oscillation is approximately half its 

initial value. The roll angle shows relatively steady oscillations at 

low frequency, approximately equal to the orbital frequency and a 

smaller amplitude high-frequency mode. The yaw angle shows the same 

high frequency oscillation with fairly regular pulse-like changes in 

amplitude. 

The third figure (see Fig. 18 c) shows plots of the angular 

momentum components in which the damping of the component about the 

yb-axis is commensurate with the damping in pitch. The angular momentum 

component about the xb-axis remains nearly constant, and is due to the 

internal angular momentum and the lack of environmental torques about 

that axis. 

it has relatively small amplitude pulses. 

The xb-component is similar to the yaw oscillations in that 

The variations in the components of aerodynamic torque, shown in 

Fig. 18 d, are proportional to the corresponding angular momentum plots 

in all three components. In order to give the satellite model a margin 

of static stability, the center of gravity was placed 1/10 of the length 

of the satellite forward along the xb-axis from center of pressure (the 

centroid of the right-circular cylinder model). This was also done for 

the AXAF satellite. 
66 

I I I I I I I c 

Fig. 18 a. Time-Histories of Euler-Parameters for HST, 

Case I (250 km. Orbital kltitude). 
67 

I I I I I I I 

Fig. 18 b. Time-Histories of Roll, Pitch, and Yaw Angles for HST, 

Case I (250 km. Orbital Altitude). 
68 

0 *. 

rn v I I I I I I I 

x 

I 

1 1 1 t 4b eb 1lO q ' 230 260 I TIE (& 010 

Fig. 18 c. Time-Histories of Angular Momentum Components for HST, 

Case I (250 km. Orbital Altitude). 
69 

0 

I I I I I I I I 

Fig. 18 d. Time-Histories of Aerodynamic Torque Components for HST, 

Case I (250 km. Altitude). 
70 

. I I I I I I I c 

Fig. 18 e. Time-Histories of Gravity-Gradient Torque Components for 

HST, Case I (250 km. Orbital Altitude). 
71 

The fifth, and last, set of plots for this case is given in 

Fig. 18 e. These are the time-histories of the components of the 

gravity-gradient torque. Here again the variations in the three 

components are proportional to those of the corresponding components of 

the vectors shown in the previous three figures. Notice, however, that 

the maximum amplitude of the oscillations in the gravity-gradient torque 

components is two orders of magnitude smaller than that for the 

aerodynamic torque. 

HST Case I1 

Results for the second HST case were obtained using all of the same 

initial conditions as the first case except of orbital altitude. The 

first HST case was a circular orbit with an altitude of 250 km. In the 

second case, the orbit model was a circular orbit with an altitude of 

800 km. The Beta plots for this case (see Fig. 19 a) show somewhat 

similar low frequency/high amplitude oscillations of the parameter, but 

the high frequency/low amplitude oscillations have disappeared. The 

attitude angles (see Fig. 19 c) are very different from that for the 

first case. The pitch angle shows oscillations of similar amplitude 

initially, but does not exhibit the same aerodynamic damping. Also, the 

frequency of the oscillation in pitch is much lower for this case. The 

roll and yaw angles both oscillate at lower frequencies but at greater 

amp-.itudes than in the previous case. Also, note that roll angle 

oscillation is changing secularly in the negative direction. The 

angular momentum plot (see Fig. 19 c) shows that the Xb-component 

remains constant. Note that the maximum amplitude the oscillations is 

an order of magnitude less than the first case. This is due to the 
72 

Fig. 19 a. Time-Histories of Euler-Parameters for HST, 

CASE I1 (800 la. Orbital Altitude). 
73 

0 

- I I I I I I I c 

0 

1 1 1 1 I Yo- rb a0 ti0 20-9 240 280 , TIE (i"PC) 010 

Fig. 19 b. Time-Histories of Roll, Pitch, and Yaw Angles for HST, 

Case I1 (800 km. Orbital Altitude). 
74 

I 

I I I rb 80 ti0 1bo fop 2Lo ti0 1 , TIME (SEC) 010 

Fig. 19 c. Time-Histories of Angular Momentum Components for HST, 

Case I1 (800 km. Orbital Altitude). 
75 

0 

0, I I I I I I I 

- 

- 

Fig. 19 d. Time-Histories of Angular Momentum Components for HST, 

Case I1 (800 km. Orbital Altitude). 
76 

c 

0 

CI 

c 

n S 

z 

W 

v) 

0 

c 

L 

I I I I I I I 

'1 

Fig. 19 e. Time-Histories of Gravity-Gradient Torque Components for 

HST, Case I1 (800 km. Orbital Altitude). 
77 

lower orbital rate and low level of excitation by aerodynamic torque. 

The yb-component is varying secularly in the positive sense while 

oscillating slowly with amplitude which decreases and then increases. 

The zb-component exhibits what appear to be long-period variations. 

While changing in this manner, the zb-component is also oscillating much 

like the yb-component. The aerodynamic torque components (see Fig. 19 

d) are several orders of magnitude smaller than they were in the first 

case; in fact, they are smaller than the gravity-gradient torque 

components (see Fig. 19 e). This is due to the fact that the model of 

the density of the atmosphere decreases exponentially with altitude, 

whereas the gravity-gradient torque is proportional to the inverse cube 

of orbital radius. 

HST Case 111 

The third HST case is a low, slightly elliptic orbit (e = .Ol). As 

a result of the ellipticity, the orbital altitude ranges from a perigee 

altitude of 184 km to an apogee altitude of 316 km. The Beta plots in 

Fig. 20 a exhibit two frequency modes seen in Case I. But the 

amplitudes of the lower frequency/high amplitude modes show some 

parameters with growing amplitudes and others with diminishing 

amplitudes. The attitude plot for this case (see Fig. 20 b) shows that 

the roll angle oscillates in two frequency modes as in the first case, 

but it is also changing secularly in a negative sense. The pitch angle 

time history contains a high frequency oscillation (as in the first 

case) and is decreasing in amplitude as time progresses, but there are 

amplitude pulses which initially coincide with apogee passage. These 

pulses most likely result from the lower level of pitch damping due to 
78 

I I 1 I 1 

'0' 10 bo 120 204 2k 2bo 

TIE (8C) *lo 

Fig. 20 a. Time-Histories of Euler-Parameters for HST, 

Case I11 (184 km. x 316 km. Orbital Altitude). 
79 

I I I I I I I 

0 1 

0 

1 1 1 1 1 t! .2Lo 280 10 80 li0 TIE (Arc) 010 *O4 

Fig. 20 b. Time-Histories of Roll, Pitch, and Yaw Angles for HST, 

Case 111 (184 km. x 316 km. Orbital Altitude). 
80 

0 

z I I I I I I I 

Fig. 20 C. Time-Histories of Angular Momentum Components for HST, 

Case I11 (184 km. x 316 km. Orbital Altitude). 
81 

01 

0- I I I I I I I 

"1 z 

Fig. 20 d. Time-Histories of Aerodynamic Torque Components for HST, 

Case I11 (184 km. x 316 km. Orbital Altitude). 
82 

Te, 

- . ...,. 

Fig. 20 e. Time-Histories of Gravity-Gradient Torque Components for 

HST, Case I11 (184 km. x 316 km. Orbital Altitude). 
83 

lower atmospheric density at the apogee altitude. 

oscillations have approximately the same frequency as the oscillation in 

pitch. However, the yaw angle oscillations begin with zero amplitude 

and then increase. Initially, the yaw amplitude pulses coincide with 

the pitch amplitude pulses in frequency. 

amplitude of the yaw angle pulses grows larger than the diminishing 

pitch angle pulses, it also becomes out of phase. 

zb-components of angular momentum in Fig. 20 c behave very similarly to 

the pitch and yaw angles, respectively, of the attitude plot. The 

xb-component of angular momentum remains nearly constant due to the 

internal angular momentum of the satellite and the fact that there is no 

aerodynamic torque (Fig. 20 e) about the xb-axis. The yb-component of 

aerodynamic torque decreases in amplitude with periodic pulses which 

coincide with perigee passage. The zb-component of aerodynamic torque 

remains at a relatively low amplitude with pulses also coincident with 

the perigee yb-component pulses. 

zb-components of the gravity-gradient torque pulse not so much as a 

result of the changing orbital altitude as with the changing of 

orientation due to the rolling motion of the satellite. That is, when 

the yb-axis is pointed more towards the center of the earth, the 

zb-component of gravity-gradient torque increases in magnitude. 

comerse is also true. 

HST Case IV 

The yaw angle 

As time progresses and the 

The yb' and 

The amplitude of the yb-and 

The 

This case has the same initial conditions as Case 111. However, in 

this simulation the environmental torque subroutines were "disconnected" 

so that the motion is torque-free. The Beta plots (Fig. 21 a) show only 
84 

I I I I I I I 

I 

Fig. 21 a. Time-Histories of Euler-Parameters for HST, 

Case IV (184 km. x 316 km. Orbital Altitude without 

Environmental Torques). 

a 
85 

Fig. 21 b. Time-Histories of Roll, Pitch, and Yaw Angles for HST, 

Case IV (184 km. x 316 km. Orbital Altitude without 

Environmental Torques). 
86 

i' Yx 

Fig. 21 c. Time-Histories of Angular Momentum Components for HST, 

Case IV (184 km. x 316 km. Orbital Altitude without 

Environmental Torques). 
87 

the expected low frequency/high amplitude oscillations. The attitude 

niigtes (Fig. 21 b) show the same secular variation in roll angle and 

almost the same for low frequency oscillation as in Case 111, but the 

high frequency oscillztion has disappeared. 

frequency/low amplitude oscillation and slight secular departure in the 

negative direction. 

a frequency similar to that of pitch angle. The yaw angle does not 

appear to change secularly. The angular momentum (Fig. 21 c) has a 

constant xb-component, and the yb-and zb-components exhibit singlefrequency, low-amplitude oscillations, but are out of phase with each 

other. The low amplitude is due to the low angular rates which are 

The pitch angle has a low 

The yaw angle shows a slightly higher amplitude and 

unexcited by environmental torques. 

6.2.2 LANDSAT Simulations 

The LANDSAT attitude motion simulations differ from the HST simulations in a few respects. First, the satellite physical data indicates 

that the satellite's moments of inertia are an order of magnitude 

smaller than the HST. Second, the LANDSAT model possesses no internal 

angular momentum as did the HST. Third, the initial orientation of 

LANDSAT model with respect to the local vertical differs from that of 

the HST. The HST simulations began with an initial pitch angle of 45", 

but the LANDSAT simulations begin with a 10" pitch angle and a -80" yaw 

angle. Fourth, the initial angular rate of the HST was a negative pitch 

about the yb-axis at orbital rate, while the initial angular rate of the 

LANDSAT is ncoslOo about the xb-axis and nsinl0" about the zb-axis, 

where n is the orbital mean motion for a given orbital radius. Fifth, 
88 

the orbits which are used for the LANDSAT simulations are both inclined 

at 98.2" as compared to the 28.5O inclination used for the HST cases. 

The first LANDSAT case is for a circular orbit with an orbital 

altitude of 352 km. The second LANDSAT case is also for a circular 

orbit, but the orbital altitude is 705 km. In both of the LANDSAT 

cases, only gravity-gradient torques were simulated. 

Case I 

In the 352 km orbital altitude case, the Beta plots (Fig. 22 a) are 

irregular oscillations which show strong orbital coupling. 

oscillations are most similar to the HST Case IV Beta plot (Fig. 21 a), 

but are somewhat more irregular. The attitude angle plots (Fig. 22 b) 

shows that roll angle increases secularly with some irregular 

"oscillations." The pitch angle time-history also shows similar 

irregular "oscillations ," but no secular departure. The yaw angle 

time-history contains higher amplitude irregularities and is also not 

changing secularly. The angular momentum plot (Fig. 22 c) indicates 

that this case is a much lower angular momentum case than the HST cases. 

Note that the scale used in the plots is two orders of magnitude smaller 

than that for the previous HST angular momentum component plots. The 

xb-component remains nearly constant and the yb-and zb-components have 

low frequency/low amplitude oscillations and no apparent secular change. 

The gravity-gradient torque component plot (Fig. 22 c) shows that the 

gravity-gradient torque is an order of magnitude less than it was for 

the HST cases. This is due to the fact that the LANDSAT satellite is 

much smaller than the HST. 

These 
89 

4b sb 1 io 1 bo 2bp 2io 2b0 8hO 

TIME (SEC) 816 

Fig. 22 a. Time-Histories of Euler-Parameters for LANDSAT, 

Case I (352 km. Orbital Altitude, Gravity-Gradient Torque 

Only). 
90 

I I I I I I I 

4b sb rlo 1 bo 2k 2bo 1 

TIE (SEC) 

Fig. 22 b. Time-Histories of Roll, Pitch, and Yaw Angles for LANDSAT, 

Case I (352 km. Orbital Altitude, Gravity-Gradient 

Torque Only). 

I 
91 

*! I I I I I I I m 

P 

Fig. 22 c. Time-Histories of Angular Momentum Components for LANDSAT, 

Case I (352 km. Orbital Altitude, Gravity-Gradient 

Torque Only). 
92 

I I I I I I I 

- 

... 

- 

- 

I 

... 

... 

1 1 

4b d li0 2q 240 2h a20 

TIE (;&) *Ib 

Fig. 22 d. Time-Histories of Gravity-Gradient Torque Components for 

LANDSAT, Case I (352 km. Orbital Altitude, GravityGradient Torque Only) - 
93 

\I 

4b 120 1 bo 2q 2'rO 280 

TIME (SEC) a10 

Fig. 23 a. Time-Histories of Euler Parameters for LANDSAT, 

Case I1 (705 km. Orbital Altitude, Gravity-Gradient 

Torque Only). 
94 

0 

t- I I I I I I I 

Fig. 23 b. Time-Histories of Roll, Pitch, and Yaw Angles for JANDSAT, 

Case I1 (705 km. Orbital altitude, Gravity-Gradient 

Torque Only). 
95 

Fig. 23 c. Time-Histories of Angular Momentum Components for LANDSAT, 

Case I1 (705 km. Orbital Altitude, Gravity-Gradient 

Torque Only). 
96 

c 

b 

0 

.I 

n P 

e - 20 

I I I I I I I 

Fig. 23 d. Time-Histories of Gravity-Gradient Torque Components for 

LANDSAT, Case I1 (705 km. Orbital Altitude, GravityGradient Torque Only). 
97 

Case I1 

The results of the second case are nearly identical to that of the 

first case. The Beta plot (Fig. 23 a) are almost identical to that for 

the first case. The attitude angles for both cases (Figs. 22 b and 23 

b) are also almost identical. The angular momentum component plots 

(Figs. 22 c and 23 c) are slightly different. The orbital mean motion 

in this case is very slightly lower due to the fact that the orbital 

altitude is 705 km. As a result of the slightly lower angular velocity 

magnitude, the angular momentum components have slightly lower 

amplitudes. Again in the gravity-gradient torque plot (Fig. 23 d), the 

amplitudes are slightly smaller due to the higher orbital altitude. 

6.2.3 AXAF Simulations 

This is the final satellite considered in the numerical 

simulations. The AXAF satellite is more similar to the HST than the 

LANDSAT. Geometrically, the sizes of the AXAF and the HST are nearly 

identical. However, the moments of inertia of the AXAF are slightly 

more than twice as large as those of the HST. The AXAF model includes 

no internal angular momentum. The initial orieycation of the satellite 

model has a 45' pitch angle with respect to the local horizontal as does 

the HST. Also, the AXAF model has a 45' yaw in the negative sense. The 

initial angular rates of the AXAF satellite model are ~.OX~O'~ rad/sec 

about the xb-axis and ~.OX~O'~ rad/sec about the yb-axis. The orbital 

inclination is the same as for the HST simulations, 28.5'. Both of the 

AXAF simulations assume a circular orbit with an altitude of 400 km. In 

the first case, both of the environmental torques were simulated. In 

the second case, no environmental torques were included. 
98 

V W” 

b 

Fig. 24 a. Time-Histories of Euler-Parameters for AXAF, 

Case I (400 km. Orbital Altitude). 
99 

Fig. 24 b. Time-Histories of Roll, Pitch, and Yaw Angles for AXAF, 

Case I (400 km. Orbital Altitude). 
100 

Fig. 24 c. Time-Histories of Angular Momentum Components for AXAF, 

Case I (400 km. Orbital Altitude). 
101 

I I I I I I I I 

eb I ob0 ob0 ah0 4b 1ho TIME (!&] 010 20! 

Fig. 24 d. Time-Histories of Aerodynamic Torque Components for AXAF, 

Case I (400 km. Orbital Altitude). 
I I I I I I I c 

X 

Fig. 24 e. Time-Histories of Gravity-Gradient Torque Components for 

AM, Case I (400 km. Orbital Altitude). 
103 

Case I 

The Beta plots (Fig. 24 a) show the reappearance of the two 

frequency mode oscillations seen in the first HST (LEO) case. Here, the 

high frequency/low amplitude oscillations have a lower frequency and are 

more irregular than those in the first HST case. This is due to the 

pitch and yaw angles appear to be almost periodic with no secular 

variations. They behave much like the pitch angle in the second HST 

case (see Fig. 19 b). The components of angular momentum (Fig. 24 c) 

also behave in a manner which suggests that this case represents a 

"transition" between the first and second HST cases. Note that the 

maximum amplitude of the angular momentum is an order of magnitude 

greater in this case. There is no apparent damping effect due to 

aerodynamic torque on any of the components. Note that the xb-component 

is not constant as it has been in previous cases. The reason for this 

can be seen in the gravity-gradient torque component plot (Fig. 24 e). 

The xb-component of gravity-gradient torque is not constant because 

there is a large difference between the moments of inertia Iyy and Izz. 

The aerodynamic torque components (Fig. 24 d) are essentially 

proportional to the angular momentum components. Note that the maximum 

amplitude of aerodynamic torque lies somewhere between those for the 

~ __~ - 

high €requency/low amplitude oscillations have a lower frequency and are 

more irregular than those in the first HST case. This is due to the 

intermediate altitude of this case (400 km), which is between the first 

HST case (250 km) and the second HST case (800 km) where the high 

frequency oscillations have disappeared. 

behaves much differently than in previous cases. The roll angle 

increases to approximately 650°, then decreases to around 200O. The 

The roll angle in Fig. 24 b 
104 

first and second HST cases. Also, note that the maximum amplitude of 

the gravity-gradient torques is approximately twice that of the first 

HST case. This is due, as we stated previously, to the fact that there 

are greater differences in the values of the three principal moments of 

inertia. 

Case 11 

This final numerical simulation case has the AXAF model with 

initial conditions identical to the first AXAF case. The differences, 

between the two are a result of the absence of environmental torques in 

s, 

the second case. 

The Beta plots (Fig. 25a) show only the single low frequency/high 

amplitude oscillations. The attitude angle plots (Fig. 25 b) show that 

all three angles "oscillate" at a relatively low frequencies with 

significant irregularities. The final plot of this section, the angular 

momentum plots (Fig. 25 c), are similar to the corresponding plots for 

the second HST case. That is, the xb-component remains constant, while 

the other two exhibit smooth oscillations of the same frequency and 

amplitude and are slightly out of phase. 

6.2.4 Summary of Low Angular Momentum Examples 

The results of the eight cases given in this section indicate that 

the time-histories of some of the variables used to describe the 

attitude motion do not exhibit well-defined patterns. In some cases, 

aerodynamic and gravity-gradient torques perturb the attitude motion of 

low angular momentum satellites to such an extent that existing 

torque-free analytical solutions are not valid for any significant 

period of time. 
105 

/ 

Fig. 25 a. Time-Histories of Euler-Parameters for AXAF, 

Case I1 (400 km. Orbital Altitude, No Environmental 

Torques). 
106 

0 

0 I I I I I I I 

a 

0 

h. 

m 

0 

0; 

J 

Fig. 25 b. Time-Histories of Roll, Pitch, and Yaw Angles for AXAF, 

Case I1 (400 km. Orbital Altitude, No Environmental 

Torques). 
107 

0 * - I I I I I I I 

L 

Fig- 25 c. Time-Histories of Angular Momentum Components for AXAF, 

Case I1 (400 km. Orbital Altitude, No Environmental 

Torques). 
VI1 . PREDICTION OF LONG-TERM MOTION 

7.0 Rationale 

As was stated previously, there will likely be a time lapse of 

several months between a satellite's "failure" and a recovery attempt. 

Therefore, a need exists for an accurate long-term attitude motion 

simulation for mission feasibility studies and planning. This chapter 

focuses on this need. As with the short-term attitude motion 

simulations, there are two approaches to the problem of perturbed 

long-term attitude motion. These are the General Perturbation Methods 

(GPM), and the Special Perturbation Methods (SPM). To apply the usual 

GPM, one needs an analytical solution to the unperturbed problem. 

Examples of unperturbed solutions include two-body orbital motion and 

"free-Eulerian" rotational motion. Tractable analytical expressions for 

perturbations are also required. The SPM most commonly applied are that 

of numerical methods. Examples of numerical perturbation methods 

applied to orbital motion include the work of Cowell and En~ke.2~ 

Applications to attitude motion include the work of Kraige and 

Junkins .25 The short-term motion simulation, described in Chapter IV, 

also uses numerical perturbation methods. 

Both GPM and SPM have shortcomings which hamper their application 

to attitude dynamics problems. The application of the GPM is difficult 

or perhaps impossible if the satellite is dynamically complex. On 

the other hand, the numerical approach, although generally applicable, 

108 
109 

often requires large amounts of computer time, especially when higher 

angular rates necessitate the use of very small integration step sizes. 

For these reasons, a "hybrid" method was developed and labeled the 

"Semi-Analytic method." 

averaging concepts from GPM, and numerical integration techniques from 

SPM to obtain "averaged equations of motion" which govern the long-term 

attitude motion of certain classes of satellites. 

This method incorporates the use of analytical 

r. 

7.1 Averaging Method 

In order to apply the General Method of Averaging to the equations 

of motion to obtain the "averaged equations of motion," new variables 

must be introduced. These variables are elements of - x and - y, where 

x - = (H HcosOH YH), (7.1.1) 

(7.1.2) 

where M is the mean anomaly. 

and the elements of - y are the "fast" variables. 

The elements of - x are the "slow" variables 

The time derivatives of 

- - 

and 

x and Y take the forms, 

- = E f& y) (7.1.3) 

(7.1.4) 

where go.z, y) is the "unperturbed" t-.ne derivative of y, E fl(x, y) 

and E gl(x, y) are the "perturbations" due to external torques on the 

satellite, and E is the usual small parameter introduced for 

perturbation analyses. 
110 

Here, it will be assumed that the functions of fl(x, y) and 

gj(x, y), j = 0,1, are periodic with period 2rr in the elements of y. 

Since the elements of - y are Euler angles, this will be the case for a 

variety of torques. 

A new set of variables (f, - E) is defined by 

and 

(7.1.5) 

(7 -1.6) 

where the functions ,u and v j = 1,2, are to be chosen to make the 

differential equations for - x and 

Equations (7.1.5) and (7.1.6) may be differentiated with respect to time 

j -j' 

simpler than those for 5 and y. 

to get 

0 

0- 

- x=x+c; -1 +&,+ ... 

and 

i = E + E 4, + E2 p, + ... 

(7 -1 -7) 

(7 a1 -8) 

The respective right-hand sides of Equations (7 4.3) and (7.1.7), and 

(7.1.4) with (7.1.8) may be equated to provide 

0 

- (7.1.9) 

- 

x + E - -1 + €2 i2 + ... - E fl(5, y) 

and 

where the arguments of ,u and v have been dropped for brevity. j -j 

The aim of the averagingmethod is to obtain averaged equations of 

motion which do not contain the "fast variables." It is assumed that 
111 

the & and 

powers of E in Eqs. (7.1.5) and (7.1.6) are grouped, then the 

are @(1) (or smaller), and the coefficients of equal -j -j 

averaged equations of motion may be obtained in the forms, 

and 

(7.1.11) 

(7 -1.12) 

where an overbar over a function indicates its "average" value. This 

requires that the "periodic perturbations" be chosen such that 

(7 -1 J3) 

and 

where ( )o indicates evaluation of the function ( ) at E = 0. The 

function g1 can be found by putting the zeroth-order solutions for 

x and into fl and integrating with respect to time. Then, v may be 

found by solving Equation (7.1.14), which is a linear differential 

equation if the zeroth-order solutions for x - and 

"average" is defined using the zeroth-order solutions for 

a f:ee, single-body satellite, the angles 

- 

- -1 

are used. Here, 

and y. For 

and 5 are ordinarily 

monotonic, while terms like sinOHsin$ have zero averages. Other terms, 

for example, cos9, usually have non-zero torque-free average values. 
112 

In torque-free motion, when the satellite contains spinning rotors 4 may 

be bounded. A further explanation of this behavior is given in Ref. 15. 

7.2 Equations of Motion 

The equations of motion for this semi-analytic method were derived 

from expressions for the angles YH, OH, Q, &, 8 and !HI, - the magnitude 

of the total angular momentum vector. The angles were described in 

Section 4. 

angular momentum coordinate system with respect to the orbital plane 

coordinate system (see Fig. 7). The angles I), 8, and Q describe the 

short-term motion of the satellite's body-fixed coordinate system with 

respect to the angular momentum coordinate system (see Fig. 8). In 

The angles YH and OH describe the long-term motion of the 

vector notation, the total rotational angular momentum vector can be 

written as 

-- H = i*g+ ,h , (7.2.1) 

where 2 - is the principal, centroidal inertia dyadic, matrix, y is the 

angular velocity vector, and - h is the internal angular momentum vector. 

In matrix notation, this expression may be written as 

(Hx H H )T = I g + (hx h h )T (7 e2.2) YZ - YZ 

An expression for the angular velocity vector can be derived from 

Eq. (7.2.2). It is found that 

-- 0 = 2-1 (E - h) . (7.2.3) 

The time rate of change of the angular momentum vector, H, written in 

vector form is 

- 
113 

where 

In matrix notation, the external torque expression is 

where 

and 

H 

0 -H 

Z Y 

0 -H 

Z X 

0 

HX 

-H 

Y 

H can -3 writ - en in the s tellit 's b 

(7 e2 *5) 

(7.206) 

(7 e2 -7) 

dy-fixed system, or in the angular 

momentum system. In the satellite system, it is written 

... fi=H_w+x. (7.2.8) -- 

Alternatively, by substituting in Equation (7.2.3), one finds 

In the angular momentum system, 

where H+ = (0 0 H)T, H = /HI and xH = (T T T~ IT. The 

"H H 

- 

(7.2.9) 

angillar velocity of the angular momentum system, -%, includes the 

secular effects of the regression of the line of nodes of the 

satellite's orbit. Explicitly, 
114 

=w 

[i si SYH 

w 

"H 

YH 

w 

-H z 

+ 

4 

8I-I 

i sinOH 

Y C0SOH 

H 

. 

H 

1 

The relationship between ,w and % is ,w = % + A, where 

i 

(7 e2 ell) 

(7 e2 -12) 

is the angular velocity of the satellite with respect to the angular 

momentum system. 

The equations of motion for the short-term variables include 

expressions for A, 6, 4 and 6. 

define 0, the nutation angle, where a and b are defined as 

The parameters a and b are used to 

a E H sine 

and 

b E H cos0 . 

(7.2 13a) 

(7 -2.13b) 

The components of the angular momentum vector can then be found from the 

relations 

Hx = a sin$, 

H = a cos$, Y 

(7.2.14a) 

(7.2 J4b) 
115 

and 

H, = b. (7.2.14~) 

The time derivatives of a and b, which may be integrated to find the 

time-history of the nutation angle, are 

= b{[a sin4 - hx) cos$]/A - [(a cos$ - hy) sin$]/B> 

+ Tx sin$ + T cos 

Y (7.2.15) 

- (h /B) a sin$ + TZ9 (7.2.16) Y 

where A, B, and C are the principal centroidal moments of inertia, 

and where T, is the zb-component of external torque. From Eqs. 

(7.2.14), (7.2.15) and (7.2.16), one may obtain the following equations 

for 4 and $: 

& = (b-h,)/C-b[B sin2$ + A cos2$]/AB 

+ (b/a)(A h cos+ + B hx sin+)/AB Y 

+ (Tx/a) cos4 - (Ty/a)sin$ 

b = H[B sin2$ + A cos2$]/AB - {[Ah 

(7.2.17) 

and 

cos4 + Bhx sin$l/AB 

Y 

(7.2.18) 

The time derivatives of the long-term variables YH, OH, and H are 

?H = - d sin i cosyH cotOH - fi cos i + (T /H)sinOH (7.2.19) 

"H 

$ = - d sin i sin\YH - T /H 

YH 

(7.2.20) 
116 

and 

8=TZ , 

n 

(7 a2.21) 

where Tx , T 

angular momentum system. 

, and TZ are the components of external torque in the 

H ’H H 

7.3 Results 

To demonstrate the capability of the semi-analytic simulation 

method, two examples are given. The first example15 involves an 

axisymmetric satellite which contains two axisymmetric rotors on the 

satellite’s xb-and yb-axes, respectively. The second example concerns 

the Combined Release and Radiation Effects Satellite (CRRES).26 In this 

section, a brief description of the computer code designed to use the 

semi-analytic method is first given. Next, the results of the two 

examples are presented and finally, a summary of these results is made. 

For descriptive purposes the semi-analytic computer code is 

broken down into four parts: the initialization process; the short-term 

stage, or Stage 1; the intermediate stage, or Stage 2; and the long-term 

stage., or Stage 3. 

In the initialization process, 1, - the principal centroidal inertia 

matrix; w the initial angular velocity vector; h, the internal 

angular momentum vector; and the initial values for the angles $, 

and GH, are entered. 

vector, - H, are then used to find the initial values of the angles 

-0 ’ 

yH 

From f, go, and 5, the total angular momentum 

0 and 9. The initial gravity-gradient torque is also calculated to find 

the initial values for Stage 1. 
117 

The six equations of motion for the Stage 1 integration are those 

the momentum conjugate to $, where P 

$ = H; pQ9 the for: Q; 6; YH; P 

momentum conjugate to Q, where P = b; and P , the momentum conjugate 

$' 

Q IH 

to YH, where P = H COSO~. The expressions for 4, &, i " H, i, and 

. yH 

OH can be found in the previous section. 

determined either by using an analytical solution, as in the first 

The nutation period is 

example, or numerically, as in the second example. The six equations of 

motion are then integrated over one nutation period to find average 

values of P P P and $, which are to be used as initial conditions 

for the "averaged equations of motion" for Stage 2. This completes the 

0' 4' $H, 

short-term stage, or Stage 1. 

The intermediate stage, or Stage 2, uses these "averaged equations 

of motion." These four averaged equations of motion are integrated 

numerically, over an orbital period. The integration step size is much 

larger (two orders of magnitude) than the step size used in Stage 1. 

The results are doubly averaged values of P 

averaged again over the orbital period to obtain as the initial values 

and $H which are used then 

$H 

- 

for the "doubley averaged equations of motion" for and TH. 

IH 

In the final stage, or Stage 3, the two doubly averaged equations 

of motion are integrated over a specified "long" period of time using 

integration step sizes on the order of one-half day. 

- Exrinple 1 

Data for the first example is given in Table 7. Using the 

analytical solution, a nutation period of 4.6 seconds was obtained and 

the mean time rates of change of Q and $ were found to be approximately 

1.367 radlsec and 2.132 radlsec, respectively. Plots of I$, 0, and $ are 
118 

shown in Figure 26 for the Stage 1 portion of the semi-analytical 

solution process. The mean value of e during the short-term motion is 

approximately 6.267 deg. For Stage 2, the numerically determined motion 

in s, 

long-term motion is shown in Fig. 28 in the form of a OH vs. YH plot. 

The analytical solution to the averaged equations yields the same 

- , and YH is shown in Figure 27. Results for Stage 3, or 

yH - - 

informat ion. 

A = B = 400 kg-m2 

C = 200 kg-m2 

440) = 0 

YH(0) = 0 

hx = 20 kg-m2/s 

hy = 0 

h, = 150 kgm2/s e=O 

o~(0) = 80 deg 

R = 6778.27 km 

%(O) = 0.1 rad/s 

~(0) = 0.001 rad/s 

~~(0) = 3.5 rad/s 

i = 28.5 deg 

fi = -6 deglday 

n = 3.998 deglmin 

As a partial check on the accuracy of the semi-analytical theory, 

the full equations of motion were integrated numerically using a step 

size of 0.2 seconds. Results using a fourth-order Runge-Kutta algorithm 

on a Harris 800 minicomputer over 29,164 seconds of CPU time are 

compared with the semi-analytical results in Fig. 29. The small initial 

difference in YH and jH is due to using the average value from Stage 2 

of the semi-analytical approach. The CPU time for the application of 

- 

the semi-analytic theory is around 12 seconds. Clearly, good accuracy 
119 

8 

r : # 4: I 

TIN ISEC) 

1: 8 a' 4: 8 

(b) 

TI* (SEC) 

Fig. 26. Semi-Analytic Example 1, Short-Term Motion, Stage 1. 

Time-Histories (a) Nutation Angle; (b) Angle of Proper 

Rotation; (c) Precession Angle. 
120 

a 

2b 4b 06 

TIE (SEC) *lo2 

(a) 

c 

n 

(9 

w 

CI 

Y 

a 

i 

. 

ro 8b 

TIHE (S&) *l(Tz 

Fig. 27. Semi-Analytic Example 1, Intermediate Motion, Stage 2. 
12 1 

W' 

.) 

r3 

3 

a e 

Y 

li 0= 

0 

. 

c 

c 

Fig. 28. Semi-Analytic Example 1, Long-Term Motion, Stage 3. 
'0 122 

Fig. 29. 

- DIRECT NUMERICAL 

0 SEMI-ANALYTIC 

Comparison of Direct Integration Results and SemiAnalytic Results for Semi-Analytic Example 1. 
123 

can be maintained with tremendous savings of computer time using the 

semi-analytic theory. 

As a second example, long-term attitude motion of the CRRES 

satellite was simulated using the semi-analytic method. The data for 

this simulation is presented in Table 8. Using a numerical solution, a 

nutation period of 3.9 seconds was determined. Plots of 9, 0, and 6 are 

shown in Fig. 30 for the Stage 1 portion of the semi-analytic process. 

The mean value of 8 during the short-term motion is approximately 4.58 

degrees. 

and YH is shown in Fig. 31. 

are shown in Fig. 32 in the form of a GH vs YH plot. 

- 

"H' and 

For Stage 2, the numerically determined motion in G, 

- 

Results from Stage 3, for long-term motion, 

- - 

-I 

Table 8. Data for Semi-Analytic -- Example 2 (CRRES) 

A = 2263.13 kgm2 Y(0) = 0 

B = 1917.5 kg-m2 \YH(o) = 0 

C = 3719.65 kg-m2 %(O) = 5 deg 

h, = 0. R = 7378.27 km 

hy = 0. 

h, = 0. 

%(O) = 0.15 rad/s 

q(0) = 0. 

~~(0) = 1.0472 rad/s 

e=O 

i = 28.5 deg 

h = -6 deg/day 

n = 3.4246 deg/min 
124 

Fig. 30. Semi-Analytic Example 2 (CRRES), Short-Term Motion, 

Time-Histories, Stage 1. (a) Nutation Angle; 

(b) Angle of Proper Rotation; (c) Precession Angle. 
125 

I 20 0 

TIME (&C) @It2 

Y 

Fig. 31. Semi-Analytic Example 2 (CRRES), Intermediate Motion, 

Stage 2. Time Histories (a) $; (b) ; (c) TH 

*H 
126 

Fig. 32. Semi-Analytic Example 2 (CRRES), Long-Term Motion, 0 = Stage 3. OH vs. YH 
127 

The CPU time required for the second example was 9.22 seconds. The 

integration time step size was 0.1 sec for Stage 1, 20 sec for Stage 2 

and 1.2 day for Stage 3. 

A semi-analytic method for predicting the long-term attitude motion 

of uncontrolled satellites has been described. The method can be used 

effectively when the square of the rotational angular momentum of the 

satellite under consideration is much larger than the maximum magnitude 

of the torque multiplied by the largest moment of inertia of the 

satellite. The method is applicable in both cases in which the 

unperturbed solution is known and in cases for which it must be 

determined numerically. The problem of determining the attitude motion 

of a satellite with constant internal angular momentum and subjected to 

gravity-gradient torque was used in the first example. 

determining the attitude motion of a satellite which has no internal 

angular momentum, but is spinning quite rapidly and is subjected to 

gravity-gradient torque was considered in the second example. The speed 

with which solutions can be obtained using the method, relative to that 

of straightforward numerical integration is of the order of 2430 to 1. 

Such ttme savings are of considerable significance. 

The problem of 
VIII. CONCLUSIONS AND RECOMMENDATIONS 

8.1 Conclusions 

Much of the emphasis of this investigation has been placed on 

analytical solutions for the attitude motion of uncontrolled satellites. 

Analytical solutions to the torque-free equations of rotational motion 

for three satellite models were verified using numerical integration. 

The majority of this verification was done using a digital simulation 

code which contains a rather general dynamic model of a two-body 

satellite. The applicability of one of the torque-free analytical 

solutions for the attitude motion of satellites with asymmetric internal 

angular momentum was tested using a model for the gravity-gradient 

torque. It was determined that for high angular momentum cases the 

fidelity of the torque-free analytical solution was very good. However, 

for cases in which the satellite is modeled as slowly tumbling in the 

presence of environmental torques, the torque-free analytical solution 

does not accurately model the motion. Numerical results were obtained 

using physical characteristics of actual satellites for cases of low 

angular momentum in the presence of environmental torques. 

The investigation of long-term attitude motion emphasized the 

semi-analytic method. Good results with a substantial savings in CPU 

time were obtained using this method. It can be utilized in cases where 

the unperturbed analytical solution is known, and also those in which 

the unperturbed solution must be determined numerically. 

128 
129 

8.2 Recommendations 

It is recommended that for short-term modeling of satellite 

attitude motion, for use in capture and retrieval simulators, the 

"driver" for dynamically simple, high angular satellites should be a 

"torque-free," closed-form analytical solution if available. This 

recommendation is based upon the "computational-speed advantage" of 

closed-form, analytical solutions. If, on the other hand, the satellite 

in question has such complex dynamics that an analytical solution is not 

available and/or is a low angular momentum satellite which is moving in 

the presence of significant environmental torques, a perturbed 

analytical solution or a numerical solution is required. Numerical 

integration of the attitude equations becomes more efficient in the low 

angular momentum regime because larger integration step sizes may be 

employed at lower angular rates. 

No analytical solutions for short-term prediction of the attitude 

motion of low angular momentum satellites are included in this report. 

It is recommended that such solutions be developed. These solutions 

would be very useful in satellite capture and retrieval operations. 

The recommendation for long-term prediction of satellite attitude 

motion is to use a semi-analytic method. Such a method should provide 

results which are sufficiently accurate for the purposes of predicting 

the state of the satellite's attitude motion sufficiently far in advance 

for capture and retrieval mission planning. 

The semi-analytic method presented herein does not address the 

problem of internal energy dissipation. Therefore, it is recommended 
130 

that "energy-sink" models be incorporated in the present semi-analytic 

method. 

Finally, for low earth orbit cases, due account must be taken of 

the aerodynamic torque. 

aerodynamics of each target in low earth orbit be modeled. It is 

therefore recommended that methods be developed to produce quickly 

attitude time histories of fairly general satellites perturbed by 

aerodynamic torques. 

To do this precisely would require that the 
REFERENCES 

1. Covault, C., "Shift Urged in Shuttle Landing Site," Aviation Week & 

Space Technology, April 23, 1984, pp. 18-23. 

2. Marsh, A. K., "Satellite Retrieval Succeeds Despite Equipment 

Problem," Aviation Week & Space Technology, November 19, 1984, pp. 

16-19. 

3. Covault, C., "Orbiter Crew Restores Solar Max," Aviation Week & 

Space Technology, April 16, 1984, pp. 18-28. 

4. Lennox, H., et a1 . , "Teleoperator Maneuvering System, *' Preliminary 

Definition Study, George C. Marshall Space flight Center, Alabama, 

June 1983. 

5. Cochran, John E., Jr., and Lahr, B. S., "Satellite Recovery-- 

Attitude Dynamics of the Targets," AASIAIAA Astrodynamics 

Specialists Conference, Vail, Colorado, August 12-15, 1985, Paper 

AAS 85-324. 

6. Crenshaw, J. W., and Fitzpatrick, P. M., "Gravity Effects on the 

Rotational Motion of a Uniaxial Artificial Satellite," AIAA 

Journal, Vol. 6, No. 11, pp. 2140-2145. 

- 

7. Cochran, J. E., Jr., "On the Effects of Gravity-Gradient Torque on 

the Rotational Motion of a Triaxial Artificial Satellite in a 

Precessing Elliptic Orbit," Celestial Mechanics, Vol. 6, No. 4, 

1972, pp. 127-150. 

8. Liu, J. J. F., "General Study of the Rotational Motion of an 

Orbiting Rigid Body About Its Center of Mass Under the Influence of 

Conservative and Nonconservative Torques," Dissertation, Auburn 

University, Alabama, December 1971. 

9. Beletskii, V. V., "Motion of an Artificial Satellite About Its 

Center of Mass," NASA TT F-429, U.S. Department of Commerce, 

Springfield, Virginia, 1966. 

10. Liu, J. J. F., and Fitzpatrick, P. M., "Poisson Equations of 

Rotational Motion for a Rigid Triaxial Body with Applications to a 

tumbling Artificial Satellite, Celestial Mechanics, Vol. 12, No. 4, 

December 1975, pp. 463-487. 

11. Jacobi, C. G. J., Journal fir Mathematiks, XXXIX, 1944, pp. 293f. 

131 
132 

12. Leimanis, E., The General Problem of Motion of Coupled Rigid Bodies 

about a Fixed Point, Springer-Verlag, New York, Inc., New York, 

1965, pp. 12f. 

13. Cochran, J. E., Jr., Shu, P. H., and Rew, S. D., "Attitude Motion 

of Asymmetric Dual-Spin Spacecraft," Journal of Guidance and 

Control, Vol. 5, No. 1, January-February 1982, pp. 37-42. 

14. Cochran, J. E., and Shu, P. H., "Attitude Motion of Spacecraft with 

Skewed Internal Angular Momenta," The Journal of the Astronautical 

Sciences, Vol. XXXI, No. 2, April-June, 1983, pp. 203-215. 

15. Lahr, B. S., Cochran, J. E., Jr., and Kumar, K., *'Semi-Analytical , 

Theory for Long-Term Rotational Motion of Uncontrolled Satellites," 

AIAA Paper No. 86-2272, AIM/AAS Astrodynamics Conference, 

Williamsburg, Virginia, August 18-20, 1986. 

16. Shrivasta, S. K., and Modi, V. J., "Satellite Attitude Dynamics and 

Control in the Presence of Environmental Torques--A Brief Survey," 

Journal of Guidance and Control, Vol. 6, No. 6, November-December 

1983, pp. 461-471. 

17. Hughes, P. C., Spacecraft Attitude Dynamics, John Wiley & Sons, New 

York, 1986, pp. 232-272. 

18. NASA: "Spacecraft Gravitational Torques," NASA SP-8024, May 1969. 

19. Cochran, J. E., Jr., "On the Effects of Gravity-Gradient Torque on 

the Rotational Motion of a Triaxial Artificial Satellite in a 

Precessing Elliptic Orbit," Dissertation, University of Texas, 

Austin, Texas, 1970. 

20. Etkin, B., Dynamics of Atmospheric Flight, Wiley, New York, 1972, 

pp. 89. 

21. Evans, W. J., "Aerodynamics and Radiation Disturbance Torques on 

Satellites Having Complex Geometry," The Journal of Astronautical 

Sciences, Vol. IX, No. 4, 1962, pp. 93-99. 

22. Stanley, M. G., "Satellite Data for Uncontrolled Motion Study," 

Memorandum, George C. Marshall Space Flight Center, Marshall Space 

Flight Center, Alabama, April 9, 1985. 

23. Doughorty, H., et al., "Space Telescope, Observatory in Space ," 

AIM Paper No. 80-1784, Presented at the AIAA Guidance and Control 

Conference, Danvers, Massachusetts, August 11-13, 1980. 

24. Danby, J. M. A., Fundamentals of Celestial Mechanics, The Macmillan 

Company, New York, 1962, pp. 234-238. 
133 

25. Kraige, L. G., and Junkins, J. L., "Perturbation Formulations for 

Satellite Attitude Dynamics ," Celestial Mechanics , - 13 , 1976 , pp. 

39-64. 

26. Henrickson, C., et al., "NASA/MSFC Combined Release and Radiation 

Effects Satellites (CRRES), Delta Critical Design Review," Ball 

Aerospace Systems Division under Contract NAS8-34025, April 2-5, 

1985. 

27. Cochran, J. E., Jr., and Speakman, N. O., "Rotational Motion of a 

Free Body Induced by Mass Redistribution," Journal of Spacecraft 

and Rockets, Vol. 12, No. 2, February 1975, pp. 89-95. 

28. Byrd, P. F., and Friedman, M. D., Handbook of Elliptic Integrals 

for Engineers and Scientists, Springer-Verlag, New York, 1971. 
APPENDICES 

134 
APPENDIX A 

SINGLE-RIGID-BODY MATHEMATICAL MODEL 

135 
136 

A.l Introduction 

Many satellites may be considered to be "single bodies" which are 

fairly rigid. The simplest physical model for such a satellite which 

takes into account the possibility of mass asymmetries is an asymmetric, 

or "tri-inertial," rigid body. Although the solution for the torquefree rotational motion of such a body is well knownLL because it forms 

the basis for perturbed motion analyses and, principally, because it 

does provide a good approximation to the motion of many satellites over 

short time periods, a summary of the solution is included in this 

report. This summary closely follows that given in Ref. 27. 

A.2 Mathematical Model 

Let the principal, centroidal moments of inertia of the body be A, 

B and C where either OB>C or C>B>A. Let the motion of principal axes - - 

Xb, Yb and Zb, be measured with respect to a (fixed) rotational angular 

momentum coordinate system which has its zH-axis collinear with H, the - 

angular momentum of the body about its center of mass C. Let the 

principal axes components of w, the angular velocity of the body, be 

9, %, and oZ. 

- 

Also, let 6, 8 and + denote Euler angles such that 

w = 6 - 6 sine 

X 

(A.2 -1) 
137 

Note that the Euler angle sequence used here is a 3-2-1 sequence. 

The equations of rotational motion for the body are 

Also, the 

or 

A 

j’ 

where ,b 

A;, + (C-B)w w 

Y= 

B; + (A-C)wz wx 

C;, + (B-A)wx w 

Y 

angular momentum 

H = Amx il + Bw 

Y 

A 

Y - 

A 

- H = H(-sin€I - bl + 

-0 

=o 

-0 

can be expressed as either 

A A 

i2 + cwz b3 3 

(A.2.2) 

(A.2.3a) 

j=1,2,3, are unit vectors attached to the Cx y z system. bbb 

Equations (A.2.1) have two immediate integrals, 

Aw2 + Bw2 + Cw: = 2 (A.2.4b) 

X Y 

and 

A29 + B2 + C2$ = H2 , (A.2 e4b) “y 

where is the rotational kinetic energy. These integrals may be used 

in Eqs. (A.2.1) to obtain2] the equation, 

= X2(1 - k2 sin2E), (A.2.5) 

where the new variable 5 E sin’l (wy/Q) ,Q 

e = B(C-B), A2 = [(C-B)(H2 - 2 Ar)J/(ABC) and 

k2 = [(B-A)(2Cy -H2)]/[(C-B)(H2 - 2 AT)]. 

It follows from Eq. (A.2.5) that 

- (d/e)*’2 , d = 2 C f -H2, 

sing = sn u (A.2 -6) 
138 

where sn u is one of the Jacobian elliptic functions, u = At - v, and 

v is the value of u at t=O. Hence, 

yr=Qsnu (A.2.7) 

For C>B>A, the integrals (A.2.4) amd (A.2.5) and Eq. (A.2.7) 

may be used to get 

ox = P cn u (A.2 -8) 

and 

oz =Rdnu, (A.2 -9) 

where P2 = (2C3- H2)/[A(C-A)], R2 = (H2 - 2Ag)/[C(C-A)] and cn u and 

dn u are Jacobian elliptic functions. 

By using Eqs. (A.2.3) and defining p = AP/H, q = BQ/H and 

r = CR/H, the following relationships may be obtained from Eqs. (A.2.7) 

through (A.2.9): 

-sin0 = p cn u , 

cos0 sin4 = q sn u (A.2 .lo) 

and 

cos9 cos4 = r dn u . The angle of precession can be obtained from 

which is derivable from Eqs. (A.2.1) and (A.2.2). By using 

tan+ = (B/C)(q/r)(sn u/dn u) in Eq. (A.2.11), it is found that 

t 

I# = $to + (H/C)[l + g2 sn2 u/(l + u2 sn2u)]dt , 

where 

g2 = c2 (C-A)/A 

(A.2 ell) 

(A.2.12) 
139 

and 

02 = [A(C-B)]/[C(B-A)] k2 

The integral in (A.2.12) is a form of elliptic integrals of the 

third kind and may be evaluated using methods described in Ref. 28. 
APPENDIX B 

DUAL-SPIN SATELLITE MATHEMATICAL MODEL 

140 
141 

R.l Introduction 

A "dual-spin'' satellite is usually a satellite which is composed of 

a nonrotating, or "despun," "platform" and a rapidly rotating "rotor." 

More generally, a dual-spin satellite is one which has two parts which 

spin at different rates. The latter description is used for the 

purposes of this appendix. Additional requirements on the model of a 

dual-spin satellite adopted here are explained in the following, 1 

A solution to the equations of torque-free attitude motion of a 

particular dual-spin satellite physical model is given in this appendix. 

It is taken from Ref. 13. The physical model consists of a rigid, 

axisymmetric body and a rigid asymmetric body (see Fig. 3). Either can 

represent the platform (rotor). The axisymmetric body rotates with 

respect to the asymmetric body about its axis of symmetry which is 

parallel to one of the axes of extrema1 moment of inertia of the asymmetric body. For the solution given, the torque between the rotor and 

platform is assumed to be zero. However, the form of the solution is 

the same if the relative spin rate of the rotor is constant. 

B.2 Equations of Motion 

Let A*, B and C represent the principal, centroidal moments of 

inertia of the satellite about the Xb-, yb' and zb-axes, respectively, 

which are fixed in the asymmetric body. Also, let B1 denote the moment 

of inertia of the axisymmetric body about its symmetry axis, which is 

assumed to be parallel to the xb-axis. Furthermore, let A = A* - B1. 

The components of the rotational angular momentum are 

H = Aox + Pa (B .2. la) 

H = Bo (B -2.1 b) 

X 

Y Y 
142 

and 

HZ = Coz , (B -2 .IC) 

where %, w, and wz are the components of angular velocity of the 

asymmetric body and 

P = B1(ox + Sl) Ci 

(B -2 .2) 

is the angular momentum of the axisymmetric body about its symmetry 

axis. 

In Eq. (B.2.2), Sl is the x-axis component of the angular velocity 

of the axisymmetric body with respect to the other body. 

Because the motion is torque-free, the angular momentum system 

CXHYHZH (see Fig. B-1) is fixed and the Euler angles I#, 8 and + are 

defined so that 

H, = 

- HY - 

H, = 

and 

where H = (Hx2 

C, the xH-axis 

H case , (B .2.3a) 

H sine sin+ (B e2 -3b) 

H sine COS+ , (B e2 -3~) 

+ Hy2 + Hz2)”2. 

is aligned with ,H -and the Euler angle sequence is 1-2-1. 

Note that in this appendix and Appendix 

The equations of motion and the kinematic equations for $ and a, 

the angle of relative rotation of the axisymmetric body, are 

fi Y = - { [(A-C)/(AC)]Hx + Pa/A}HZ , (B e2 -4b) 
143 

H 

Fig. B. 1 Angular Xomentum and Body-Fixed Coordinate Systems for 

Appendices B and C. 
144 

P =o 

a 

(B -2.4d) 

and 

= H(H~~/B + H~~/c)(H~-H X 2) 

B.3 Solutions for $, 8, I$ and a 

Equations (B .2.4) have the "immediate" integrals' 

~2 = ~~2 + H 2 + H,Z, constant , Y 

P = constant 

a 

and 

2~ = (H~-P~)~/A + H 2/B + HZ2/C, constant. 

Y 

(B .2.4e) 

(B .2.4f) 

(B .3 .l) 

(B -3 -2) 

These three integrals may be used to write 

f (H ) = H 2/I(B(A-C)]/[A(B-C>]) = [Hx + C Pa/(A-C)I2 

0 2x Y 

and 

+ B P /(A-B)I2, (B.3.4b) 

a 

+ Hz2 [A(B-C)[/[C(A-B)I - [H, 

0 0 

where a subscript o denotes an initial value. 

Hence, H>. and H, are functions of H,, and $2 may be written in the 

form, 

fi2, [ (A-C) (A-B)/(ABC) If2(Hx)f3(Hx) (B e3 -5) X 
145 

The solut€on to Eq. (B.3.5) depends upon whether the roots of the 

ccirliitlo~is f2 = 0 ;iiid F7 = 0 are real or complex. For physically 

reiillzable motion, the roots of E2 = 0 must be real. The roots of 

f3 = 0 may, however, be either real or complex. 

possible. 

Case 1. Four Real Roots, a>b>c>d 

Hence, two cases are 

In this case, 

Hx = (D1 + D2 sn2u)/(D3 + D4 sn2u), (B a3 -6) 

where sn u is a 

k= 

Jacobian elliptic function of modulus 

[ (a-b) (c-d) 1 / [ (a-c) (b-d) I } (B e3 -7) 

and argument, 

u=Xt-uo, (B.3.8) 

in which 

(B a3 -9) x = { [(A-B)(A-C)/(BC)]1/2[(a-c)(b-d)] I /2 }/2A . 

The forms of the Dj, j=1,2,3,4, depend upon H, . It turns out that 

0 

Hx 

D1 = 

D1 = d(a-c) , D2 = a(c-d) , D3 = a-c and D4 = c-d. 

Case 2. Two Real Roots 

is never between b and c for real motion. For b < H, - < a, 

0 

- 

0 

a(b-d), D2 = d(a-b), D3 = b-d and D4 = a-b. For d < H, - < c, 

0 

- 

Tf only the roots of f2 = 0 are real, the solution for H, has the 

form, 

(B .3 .lo) 
146 

Here, cn u is a Jacobian elliptic function of modulus 

k = {[(a-b)2 - (a* - B*I2]/(4 a*8*)}1/2, (Be3 all) 

where a and b are the real roots (a>b) and a* and B* are the real and 

imaginary parts, respectively, of the complex root c. In the argument, 

u = At + uo, 

X = [ (A-C)(A-B)/(A2BC) J 1/2(a*f3*)1/2 (B.3 -12) 

The constants Cj, j=1,2,3,4, are defined as C1 = aB* + ba*, 

C2 = ba* - ag*, C3 = a* + B* and C4 = a* - B*. 

Solutions to the equations for $, +, and a may be obtained for each 

of the two cases above. For Case 1, the solution for $ is 

where 

2 U 

J, = 1 b. 1 du/(l-aj2 sn2u) + 

j-0 u 

0 

(B e3 J3) 

(B .3.14) 

> = [2AT-(H+P )2]/(2A) c2 a 

c1 = H/A 

= D4/(D4H f D2) (B.3.15) 4 ,6 

- [2AT-(H-Pa)2]/(2A) ; c c3 - 

5 97 

a. = 0 

C = (D3D2 - DID4)/[(D3H f D1)(D4H f D2)J 

oi2 = - (D4H - D2)/(D3H - D1) 

22 = - (D4H + D2)/(D3H + D1) 

(B -3 -16) 
147 

The integrals which appear in the solution for Q are elliptic integrals 

oE the third kind (see Ref. 28, pp. 232-237), 

For Case 2, the solution for J, is 

2 U 

6 = d. I du/(l - Bj2 cn u) + J,o , j=o J u 

0 

(B a3.17) 

where the dj and $j have the same forms as bj and aj but Ck replaces % 

in the definitions. Also, the modulus and parameter X have the required 

forms. The integrals in Eq. (B.3.17) are also elliptic Integrals of the 

third kind (see Ref. 28, pp. 215 and 232-237). 

The solution for 9 has the form, 

2 U 

for case 1. The constants pj are defined as 

PO = [Pa/A - c3c6 + c2c4]/X , 

PI = c3c7/x 

P2 = c2c5/X . 

and 

(B .3.18) 

(B.3.19) 

For Case 2, the solution for I$ is analogous to the solution for Q 

in that case. 

Solutions for a are similar to those for I$ and 0, but are simpler 

because only one elliptic integral is involved. For Case 1, 

a = a (u - uo) + al du/(l - a32 sn2u) 3 

0 

(B a3.20) 

a 
148 

where 

(B -3.21) 

and 

a32 = - D4/D3 

The solution for a for Case 2 can be obtained in the manner described 

above for J, and 4. 
APPENDIX C 

MULTI-ROTOR SATELLITE MATHEMATICAL MODEL 

149 
150 

C .1 Introduction 

The model presented in Appendix B cannot be used to represent a 

satellite which contains a part (or parts) which is rotating about an 

axis which is not parallel to a principal axis of the carrier body. 

model which can be used to represent such a satellite, if the system is 

A 

axisymmetric, is presented in this appendix. In fact, the satellite 

model can be used to represent a satellite which contains several 

constant-speed, rigid "rotors ." Hence, the label, "Multi-Rotor 

Sat e 11 it e. " 

The physical model is depicted in Fig. 4. It consists of a 

torque-free carrier body and one, or more, constant-speed, axisymmetric, 

rigid rotors. These rotors are arranged so that their resultant angular 

momentum due to rotation relative to the carrier body (internal angular 

momentum) is not parallel to any axis, except possibly the yb-axis of 

the system CXbYbZb fixed in the carrier body. The xb-axis of the system 

is an axis of symmetry. Hence, the yb-axis can always be chosen such 

that the zb-component of the internal angular momentum is zero. 

angular momentum system CXHYHZH is fixed because there are no external 

The 

torques . 

Equations of Motion 

The mathematical model presented here is based, for the most part, 

on Ref. 14. 

respectively, of the total angular momentum about C. Also, let hx and 

hy denote the xb-and Yb-components, respectively, of the internal 

angular momentum and let A and C be the principal centroidal moments of 

Let Hx, Hy and H, denote the Xb', yb-and zb-components, 
151 

€nert€a of the system of carrier body and rotors, with A about the 

xb-axis. Then, the equations of rotational motion may be put into the 

forms, 14 

fi =Hh/C, (C .2. la) 

X ZY 

and 

II = {[(A-C)/(AC)]Hx + hx/A}Hy - H h /C 

Z XY 

(C .2. IC) 

The corresponding kinematic equations are 

5, = (H/C)[1 - h H /(H2-Hx2)] (C.2.2b) 

YY 

Here, Q and J, are the angle of proper rotation and the precession angle 

of the body, respectively (see Fig. B.l in previous appendix). 

The nutation angle 0 is given by 

e = cos-' (Hx/H) , 

where H = (HX2 + Hy2 + Hz2) 112 . (C.2.3) 

The following expressions for the angular velocity components 

wx, 9, and w, may also be required at times: 

ox = (Hx- hx>/A (C .2.4a) 

wz = HZ/C (C -2 -4~) 

Furthermore, the angles 0, + and J, can be used to write 

Hx = H cos0 (C .2.5a) 

H = H sine sin+ (C.2.5b) 

Y 
152 

and 

HZ = H sin0 cos$ , (C.2.5~) 

Equations (C.2.1) admit from "first" and "second" integrals 

H = constant and 

(H -hx)2/A + (H -h )2/C + HZ2/C = 2T, constant (C -2.6) 

X YY 

Equation (C.2.6) may be rewritten (by using -- H*H = constant) in the form, 

H = aHx + 2bHx + c , (C.2.7) Y 

where 

a = (C-A)/(2Ahy) , 

b = - hxC/(2Ahy) 

and 

c = [(C/A)hx2 + h + H2 - 2CT]/(2hy). Y 

(C .2.8a) 

(C .2.8b) 

(C .2.8c) 

Equation (C.2.7) represents a family of parabolic cylinders in angular 

momentum space. The terminus of the vector H describes, on the sphere 

H2 = H + H 

cylinder with the sphere. 

- 

+ Hz2, a curve which is an intersection of a parabolic X Y 

These integrals may be used in Eq. (C.2.1a) to get the equation 

where 

2(Hx) = a2[-Hx4 + c3 Hx3 + c2 Hx2 + C1 Hx CO1 

HZ 

c3 

c2 Y X 

= - 4 hxC/(A-C) 

= - 4(A2h 2 + C2h 2)/(A-C)2 

+ 2[Chx2 + Ah + A(H2 - 2CT)]/(A-C) 

Y 

(C .2 .lo) 

(C .2 .lla) 

e (C.2.11b) 
153 

= 4[C2h 3 + ACh h + AChx(H2 - 2CT)]/(A-C)2 (C .2 .llc) c1 X XY 

c = A2{4h 2H2 - [(C/A)hx2 + h + H2 - 2CT]2)/(A-C)2. 

0 Y Y 

(C.2.11d) 

C.3 Solutions for 8, 4 and 

The general solution to Eq. (C.2.9) can be expressed in terms of 

elliptic functions of the time. The form of the solution depends on the 

number of real roots of the quartic equation, HZ2(H,) = 0. 

Case 1. Four Real Roots 

In this case, if the roots are H, , j=1,2,3,4, and H, < Hx for 

j j k 

k> j , then 

Hx = (D1 + D2 sn2u)/(D3 + D4 sn2u) , (C e3.1) 

where sn u is Jacobi's sine amplitude function and the Dj are determined 

by H, , the initial value of H,. 

e 

0 

0 

If H 

D1 =H (H, -H ); D3=H -H 

x1 2 x4 x2 x4 

D2=H (H, -H );D4=H -H 

1 x2 x1 

If Hx2 H, > H, , then 

0 3 

D1=H (H, -H ); D3=H -H 

1 x3 x1 

(C.3.2) 

(C.3.3) 

D2 = Hx (H, - H ); D4 = Hx - H 

12 x3 2 
154 

X Finally, if Hx - > H 

3 0 

D1=H (Hx -H );D3=H -H 

x4 1 x3 x1 x3 

(C .3.4) 

D2 =H (Hx -H ); D4=H -H , x1 3 x4 x3 ,4 

The argument of the elliptic functions is 

u = At + uo , 

where 

~2 = [ahy/(2C)][(H - H )(Hx - H 11 

,2 x4 1 x3 

(C .3.5) 

(C a3 -6) 

(C.3.7) 

By using the solution for H, in Eq. (C.2.2a), and considerable 

algebraic manipulation, one may put the equation for C$ into the form, 

3 U 

(C .3.8) 

(C .3.9) 
(C.3.10) 

The integrals in Eq. (C.3.18) are elliptic integrals of the third 

kind. A set of subroutines has been developed to evaluate these. 

By using a similar procedure, it may be shown that 

du 3 U 

+ = 1 Qj 1 - a1 snzu ++, 9 

j =O U 

0 

where 

(C.3.11) 

(C -3.12) 
156 

Case 2. Two Real Roots 

When the quartic equation has two real roots, HX1 > Hx2, and two 

complex roots, y f 61, the solution for H, has the form, 

HX = (C1 + C2 cn u)/(C3 + C4 cn u), (C -3 -13) 

where, if, 

r = [(H~ - y>2 + 621172 (C .3.14a) 

1 

and 

then 

(C.3.14b) 

C1=Hxs+H r 

1 x2 

C2 = Hx r - Hs, 

C3=r+s 

and 

C4=r-s. 

The modulus of cn u is given by 

As in Case 1, u has the form, 

u = At +uo . 

But, for this case, 

A = ah [r~l~/~/C. Y 

(C .3.15a) 

(C -3 15b) 

(C a3 -15~) 

(C e3 J5d) 

(C.3.16) 

((2.3.17) 

e 

(C -3.18) 
157 

Regarding the forms of the solutions for I$ and $, the only way in 

which this case differs from Case 1 is that the Cj are used in place of 

the Dj and the elliptic function cn u replaces snh. 

integrals are still of the third kind (see Ref. 28, pp. 215, 232-237). 

The elliptic 
APPENDIX D 

GRAVITY-GRADIENT AND AERODYNAMIC TORQUE MATHEMATICAL MODELS 

158 
159 

D.l Gravity-Gradient Torque Mathematical Models 

Two mathematical models were derived during this investigation to 

model the gravity-gradient torque experienced by a satellite orbiting 

the earth. The first model was used for the short-term numerical 

simulation program. The second was used for the long-term semi-analytic 

simulation program. 

Short-Term Mathematical Model 

A $1- > 

-g 

The unit vector e describes the attitude relationship between the 

satellite's body-fixed axes and the zv-axis of the local vertical 

system. Recall that the +-axis always points toward the center of the 

earth. Thus, the third column of the LbV - transformation matrix provides 

the unit vector of interest in the matrix form, 

(D.1.1) 

This expression for e may then be used to complete the expression, 

-g 

(D.1.2) 

where p~ is the gravitational constant of the earth, R is the distance 

from the center of the earth to the center of mass of the satellite and 

- I is the matrix of the satellite. 

Long-Term Mathematical Model 

The model of the perturbing effects of gravity-gradient torque on 

the satellite attitude motion used in the semi-analytic simulation is 

based on a rigid axisymmetric body with centroidal moments of inertia 

A=BfC. The components of the gravity-gradient torque can be derived by 

* 
160 

taking partial derivatives of the expression for gravity-gradient 

potential, 

V = [3pE/(2R3)](C-A) COS~X, (De1 -3) g 

where the angle x is the angle between the position vector of the 

satellite's center of mass and the body-fixed zb-axis (the axis of 

symmetry). 

defined in the body of the report, the cos2x may be expressed as 

follows: 

In terms of the angles 8, OH, $, and p = PH - u, which are 

D.2 Aerodynamic Torque Mathematical Model 

The results of the derivation of the aerodynamic torques on the 

right-circular cylinder are as follows. The total aerodynamic torque 

acting on the right circular cylinder is a sum of the torques 
161 

T , T , and T contributed by the surfaces S1, S2, and S3, 

respectively (as shown in Fig. 11). That is 

-al -a 2 -a3 

T =T +Ta +T 

-a -al 2 -a3 

The torque contributed by the front endcap, Si, is 

T = (Ta T T IT 

Z 

al 

Y 

al 

X 

1 1 -a 

where 

-'II - p cd rc4 [2up - Wr - V~J, 

a T 

a, L 

X 

(D -2 -1) 

(D -2.2) 

(D .2.3) 

where rc is the radius of the right circular cylinder, U, V, W are the 

xb-, yb-, and zb-components of the "relative wind," where 

U = IV I cosa cos0 

'C a a' 

and r are the components of angular velocity about Xb', yb', and 

V = Ixcl sinBa, and W = Iycl sinaa cosBa, and p, q, 

zb-axes, respectively, and x1 is the distance from the centroid of the 

cylinder to center of the front end cap along the xb-axis. The 

aerolynamic torque contributed by the sides of the cylinder, surface 

S2, is 

Ta IT 

z 

2 

= (Ta2 T a2 

X Y 

-a2 

(D .2.4) 
162 

where 

L 

X 

T =-- 1 r {W[+ U rc L - 4- (d12 - d22)I 2"d c 

Y 

a2 

)(rc2 L + d13 + d23) W2 

J (VZ+WZ ) 

+ 

(rc2 L 

- $ U rc(d12-d22)] - r[5 2vw 

J (VZ+WZ) 

+ d13+d23>1}, 

and 

T =-- 1 r (V[J(VZ+WZ) (d12-d22) - 3 rC L] a3 2"d c L 

z 

V2 )(r2 L + d13 + d23> 

C 

+ r[$(J-g + 

J (VZ+WZ) 

- - ll U rc (d12-d22>>1} 

4 (D.2.5) 

Here, L is the length of the cylinder, dl is the distance along the 

xb-axis between the front endcap and the center of gravity, and d2 is 

the distance from the rear endcap to the center of gravity. Finally, 

the aerodynamic torque contributed by the rear endcap, S3, is 
163 

(U.2.6) 

where 

and 

T =-TPC~ IT r 2 u[2vx2 -(rc2 + 2x2*)rI . (D.2.7) 

z 

a3 

Here, x2 is the distance from the center of the rear endcap to the 

centroid of the cylinder.