Department of Aeronautics and Astronautics Stanford University Stanford, California THE CONTROL AND USE OF LIBRATION-POINT SATELLITES by Robert W. Farquhar SUDAAR NO. 350 July 1968 Prepared for the National Aeronautics and Space Administration Under Grant NsG 133-61 '3 t ABSTRACT This study is primarily concerned with satellite station-keeping in the vicinity of the unstable collinear libration points, L1 and L2. Station-keeping problems at other libration points, including an "isosce- les-triangle point," are also treated, but to a lesser degree. A detailed analysis of the translation-control problem for a satel- lite in the vicinity of a collinear point is presented. Simple linear feedback control laws are formulated, and stability conditions are deter- mined for both constant and periodic coefficient systems. It was found that stability could be achieved with a single-axis control that used only range and range-rate measurements. The station-keeping cost with this control is given as a function of the measurement noise. If Earth- based measurements are available, this cost is very Pow. Other transla- tion-control analyses of this study include (1) solar-sail control at the Earth-Moon collinear points, (2) a limit-cycle analysis for an on-off :b control system, and (3) a method for stabilizing the position of the mass center of a cable-connected satellite at a collinear point by simply changing the length of the cable with an internal device. b, For certain applications, a satellite is required to follow a small quasi-periodic orbit around a collinear libration point. Analytical estimates for the corrections to this orbit due to the effects of non- linearity, eccentricity, and perturbations are derived. Several possible libration-point missions are proposed. It is sug- gested that libration points could be utilized for (1) lunar far-side communications, (2) lunar and interplanetary transportation systems, (3) deep-space optical communications, (4) monitoring solar-induced phenomena in and beyond the Earth's magnetosphere, and (5) low-frequency radio astronomy. iii ACKNOWLEDGMENTS The author wishes to express his gratitude to his advisor, Prof. John V. Breakwell, for his inspirational guidance and continual availa- bility throughout the course of this research. He also wishes to thank Ahmed Kame1 for checking some of the mathematics. The conscientious efforts of Marge Holmgren and Bill Terluin during the preparation of this report are appreciated. The support of the Electronics Research Center of NASA in this research is gratefully acknowledged. This work is dedicated to Bonnie, Pamela, Robin, and Patricia. iv Q CONTENTS .Page 3 INTRODUCTION ........................... 1 A . Problem Statement ..................... 1 B . Previous Contributions ................... 2 C . Contributions of This Research ............... 3 Chapter I . PRELIMINARY DEVELOPMENTS ............... 5 A . The Restricted Three-Body Problem ............. 5 . 1 . Equations of Motion .................. 5 2 . Jacobian Constant and the Surfaces of Zero Relative Velocity ........................ 9 3 . Libration Points .................... 12 B . Equations of Motion in the Vicinity of Libration Points . 15 1 . Collinear Points .................... 15 2 . Equilateral-Triangle Points .............. 20 3 . Isosceles-Triangle Points ............... 24 4 . Stability of the Linearized Equations of Motion in the Elliptic Case ..................... 33 3 C . Numerical Data for Some of the Collinear Points in the Solar System ........................ 33 1 . Mass-Ratio Dependent Quantities ............ 34 2 . Conversion Factors for Normalized Units ........ 35 3 of of . Comparison L2 Distance and Extent Planetary Shadow., ....................... 36 Chapter I1 . PERTURBATIONS AND NOMINAL PATH CONTROL ....... 45 A . Additional Accelerations Acting on a Libration-Point Satellite ......................... 45 1 . Gravitational Perturbations .............. 45 2 . Solar Radiation Pressure ................ 45 3 . Thrust Control ..................... 46 B . Control about Nominal Path ................. 46 1 . Basic Strategy ..................... 47 2 . Periodic Orbits .................... 48 3 . Higher-Order Corrections for a Periodic Orbit ..... 49 V H P c CONTENTS (Cont) .Page a . Response to a Periodic Input ............ 50 b . Nonlinear Correction ................ 51 c . Eccentricity Correction .............. 52 d . Effect of Gravitational Perturbations and Solar Radiation Pressure ................. 54 4 . Linearized Equations of Motion Relative to a Nominal Path .......................... 55 C . Examples of Gravitational Perturbations .......... 57 1 . Solar Perturbation near the Earth-Moon Collinear Points ......................... 57 a . De Pontecoulant's Expressions for the Motion of theMoon ...................... 58 b . Derivation of Direct Solar Effect ......... 59 c . Cancellation of First-Order Terms at Collinear Point ....................... 61 d . Periodic Orbit Correction ............. 62 2 . Effect of the Moon at the Sun-Earth Collinear Points . 65 3 . Some Other Interesting Examples ............ 70 a . Jupiter's Effect at the Sun-Earth Isosceles-Triangle Points ....................... 70 b . Solar Effect at the Earth-Moon Equilateral-Tsiangle Points ....................... 71 Chapter I11 . LINEAR FEEDBACK CONTROL .............. 73 A . Collinear Points ...................... 73 1 . Routh Stability Conditions ............... 74 2 . Root Loci and Closed-Loop Response ........... 76 3 . Floquet Stability Investigations ............ 84 a . Eccentricity Effect ................ 87 b . Motion Relative to a Periodic Orbit ........ 91 B . Equilateral-Triangle Points ................ 92 1 . Routh Stability Conditions ............... 96 2 . Root Loci and Closed-Loop Response ........... 97 Chapter IV . STATION-KEEPING ...................101 A . Average Control-Acceleration Requirements .........101 vi . CONTENTS (Cont) 1 . Cost Estimates for Noise Inputs .......... 101 a . Optimization ................. ...103 b . Estimation of Noise Level ........... ...106 . c . Costs at the Earth-Moon Collinear Points ...... 107 2 . Cost for Sinusoidal Control Acceleration with Noise ...108 3 . Cost for a Constant Displacement ......... ...110 B . Solar-Sail Control at the Earth-Moon Collinear Points . ...111 1 . Basic Concepts .................. ...113 2 . Sail Variations with Radial-Axis Control ........ 117 3 . Examples ..................... ...118 Chapter V . ON-OFF CONTROL SYSTEM .................120 A . Limit Cycles: Exact Analysis ................120 1 . Application of Harmonic Method .............121 2 . Closed-Form Solution ..................127 B . Stability of Limit Cycles ..................131 1 . Tsypkin’s Method .................... 131 2 Results for Special Case = K) . (e2 - e 1 .........132 3 . Procedure for General Case (e2 .el f n) ........135 C . Solar-Sail Control at the Sun-Earth L1 Point ........139 D . Limit Cycles: Approximate Analysis .............140 1 . Accuracy of Single-Axis Approximation ..........140 2 . Phase-Plane Method ...................143 3 Sample Calculation at the Earth-Moon L Point 146 . 2 ...... Chapter VI . STABLE CABLE ..................... 148 A . Stabilization Procedure ...................148 1 . One-Dimensional Analysis ................150 a . Shifting Equilibrium Point .............150 b . Cable Control .................... 152 2 . Three-Dimensional Analysis ............... 154 B . Other Considerations .................... 163 1 . Cable Extension Limitations ...............163 vii CONTENTS (Con t 2 . Structural Comments .................. 169 Chapter VI1 . APPLICATIONS ................... 171 A . Supporting Role for Lunar and Planetary Missions ..... 171 1 . Utilization of the Earth-Moon Collinear Points in Future Lunar Operations ................ 171 a . Lunar Communications ............... 171 (i) Lissajou-Orbit Concept ........... 171 (a) Occultation Geometry ......... 172 (6) Phase-Jump Control .......... 175 (ii) Halo-Orbit Concept ............. 180 (iii) Hummingbird Concept ............ 181 (iv) Comparison of Different Techniques ..... 182 (v) Concluding Remarks ............. 182 3 b . Rendezvous Technique ............... 183 2 . An Interplanetary Transportation System with Terminals at the Sun-Planet Collinear Points .......... 186 3 3 . Deep-Space Communications using a Relay Satellite at an Earth-Moon Equilateral-Triangle Point ....... 188 B . Scientific Usefulness ................... 189 1 . A Multiple-Satellite System for Monitoring Solar-Induced Phenomena In and Beyond the Earth's Magnetosphere ... 189 2 . Low-Frequency Radio Astronomy from the Earth-Moon L2 Point ....................... 190 CONCLUSIONS AND RECOMMENDATIONS ................. 192 Appendix A . DERIVATION OF EQ . (4-18) .............. 193 Appendix B . ECCENTRICITY CORRECTION FOR THE Z-AXIS OSCILLATION AT THE EARTH-MOON L2 POINT ............ 197 REFERENCES ........................... 198 viii NOTATION AND UNITS Q Not at ion A brief list of symbols is given here. Unlisted symbols are defined in the chapter where they appear. AAA oscillation amplitude along coordinate axis x' y' z defined by Eq. (1-44) BL defined by Eq. (1-45) cL defined by Eq. (1-46) DL + Fc' Fey' Fez) control acceleration K acceleration magnitude -+ Nt), (Px, Pys Pz) perturbing acceleration e orbital eccentricity m (1) mass (2) in Section C-1 of Chapter 11, it is the ratio of the mean motions of the Earth and the Moon (m = 0.07480133) (3) in Chapter V, it is the fraction of the total time that the