Guidance and Control--II [1 ed.] 9781600862687, 9781563478628

Citation preview

Purchased from American Institute of Aeronautics and Astronautics

Guidance and Control—II

Purchased from American Institute of Aeronautics and Astronautics

Progress in ASTRONAUTICS and AERONAUTICS (a continuation of Progress in Astronautics and Rocketry)

A series of volumes sponsored by American institute of Aeronautics and Astronautics 1290

Avenue of the Americas, New York, New York 10019

Progress Series Editor Martin Summerfield Princeton University, Princeton, New Jersey

Titles in the Series Volume 1. SOLID PROPELLANT ROCKET RESEARCH. 1960 Editor: MARTIN SUMMERFIELD, Princeton University, Princeton, New Jersey Volume 2. LIQUID ROCKETS AND PROPELLANTS. 1960 Editors: LOREN E. BOLLINGER, The Ohio State University, Columbus, Ohio; MARTIN GOLDSMITH, The RAND Corporation, Santa Monica, California; AND ALEXIS W. LEMMON JR., Battelle Memorial Institute, Columbus, Ohio Volume 3. ENERGY CONVERSION FOR SPACE POWER. 1961 Editor: NATHAN W. SNYDER, Institute for Defense Analyses, Washington, D. C. Volume 4. SPACE POWER SYSTEMS. 1961 Editor: NATHAN W. SNYDER, Institute for Defense Analyses, Washington, D. C. Volume 5. ELECTROSTATIC PROPULSION. 1961 Editors: DAVID B. LANGMUIR, Space Technology Laboratories, Inc., Canoga Park, California; ERNST STUHLINGER, NASA George C. Marshall Space Flight Center, Huntsville, Alabama; AND J. M. SELLEN JR., Space Technology Laboratories, Inc., Canoga Park, California Volume 6. DETONATION AND TWO-PHASE FLOW. 1962 Editors: S. S. PENNER, California Institute of Technology, Pasadena, California; AND F. A. WILLIAMS, Harvard University, Cambridge, Massachusetts Volume 7. HYPERSONIC FLOW RESEARCH. 1962 Editor: FREDERICK R. RIDDELL, Avco Corporation, Wilmington, Massachusetts Volume 8. GUIDANCE AND CONTROL. 1962 Editors: ROBERT E. ROBERSON, Consultant, Fullerton, California; AND JAMES S. FARRIOR, Lockheed Missiles and Space Company, Sunnyvale, California

ACADEMIC PRESS • NEW YORK AND LONDON

Purchased from American Institute of Aeronautics and Astronautics

Progress in ASTRONAUTICS id AERONAUTICS (a continuation of Progress in Astronautics and Rocketry)

A series of volumes sponsored by American Institute of Aeronautics and Astronautics 1290 Avenue of the Americas, New York, New York 10019

Progress Series Editor

Martin Summerfield Princeton University, Princeton, New Jersey

Titles in the Series Volume 9. ELECTRIC PROPULSION DEVELOPMENT. 1963

Editor: ERNST STUHLINGER, NASA George C. Marshall Space Flight Center, Huntsville, Alabama Volume 10. TECHNOLOGY OF LUNAR EXPLORATION. 1963

Editors: CLIFFORD I. CUMMINGS AND HAROLD R. LAWRENCE, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California

Volume 11. POWER SYSTEMS FOR SPACE FLIGHT. 1963

Editors: MORRIS A. ZIPKIN AND RUSSELL N. EDWARDS, Space Power and Propulsion Section, Missiles and Space Division, General Electric Company, Cincinnati, Ohio Volume 12. IONIZATION IN HIGH-TEMPERATURE GASES. 1963

Editor: KURT E. SHULER, National Bureau of Standards, Washing ton, D.C. Associate Editor: JOHN B. FENN, Princeton University, Princeton, New Jersey Volume 13. GUIDANCE AND CONTROL —II. 1964

Editors: ROBERT C. LANGFORD, General Precision Inc., Little Falls, New Jersey; and CHARLES J. MUNDO, Institute of Naval Studies, Cambridge, Massachusetts (Other volumes are planned)

ACADEMIC PRESS • NEW YORK AND LONDON

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

and Control Edited by

Robert C. Langford Aerospace Research Center, General Precision Inc., Little Falls, New Jersey

Charles J. Mundo Institute of Naval Studies, Center for Naval Analyses, Cambridge, Massachusetts

A Selection of Technical Papers based mainly on the American Institute of Aeronautics and Astronautics Guidance and Control Conference held at Cambridge, Massachusetts August 12-14, 1963

ACADEMIC PRESS • NEW

YORK • LONDON • 1964

Purchased from American Institute of Aeronautics and Astronautics

COPYRIGHT© 1964 BY ACADEMIC PRESS INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS ACADEMIC PRESS INC.

Ill FIFTH AVENUE NEW YORK, NEW YORK 10003

United Kingdom Edition Published by

ACADEMIC PRESS INC. (LONDON) LTD.

BERKELEY SQUARE HOUSE, LONDON W. 1

Library of Congress Catalog Card Number: 62 - 13119

PRINTED IN THE UNITED STATES OF AMERICA

Purchased from American Institute of Aeronautics and Astronautics

THE GUIDANCE AND CONTROL COMMITTEE OF THE AMERICAN INSTITUTE OF AERONAUTICS AND ASTRONAUTICS August 1963

Robert C. Langford, Chairman General Precision Inc., Little Falls, N. J.

Malcolm J. Abzug Douglas Aircraft Company, Santa Monica, Calif. Robert M. L. Baker Jr. Lockheed-California Company, Los Angeles, Calif. Arthur E. Bryson Jr. Harvard University, Cambridge, Mass. Robert H. Cannon Jr. Stanford University, Stanford, Calif. J. A. Ce stone Department of the Navy, Washington, D. C.

David Chandler North American Aviation, Inc., Downey, Calif. R. A. Duffy U. S. Air Force, Ballistic Systems Division, Los Angeles, Calif. R. C. Duncan Department of Defense, Washington, D. C.

Alan Greene Raytheon Company, Bedford, Mass. Albert J. Kelley NASA Headquarters, Washington, D. C. Cornelius T. Leondes University of California, Los Angeles, Calif. Howard F. Matthews NASA Ames Research Center, Moffett Field, Calif. David L. Mellen Minneapolis-Honeywell Regulator Company, Minneapolis, Minn. Charles J. Mundo Institute for Defense Analyses, Cambridge, Mass. vii

Purchased from American Institute of Aeronautics and Astronautics

Robert E. Roberson Consultant, Fullerton, Calif. Robert L. Roderick Hughes Aircraft Company, Culver City, Calif. Charles W. Sarture Aerospace Corporation, Los Angeles, Calif. Stanley Schmidt Philco/We stern Development Laboratories, Palo Alto, Calif. Thomas H. Thompson Bellcomm, Inc., Washington, D. C. H. Philip Whitaker Massachusetts Institute of Technology, Cambridge, Mass.

viii

Purchased from American Institute of Aeronautics and Astronautics

PREFACE

The present volume is intended both as a report on, and an authoritative reference guide to, the present status of the closely connected fields of Guidance and Control, particularly as related to missiles and space vehicles. Like its predecessor in the series, Volume 8 on "Guidance and Control," edited by R. E. Roberson and J. S. Farrior, this volume had its genesis in a Specialist Conference that was organized by the American Institute of Aeronautics and Astronautics and that was held at the Massachusetts Institute of Technology in August 1963. Although in no sense a proceedings of this Conference, the present volume uses the same subdivisions of technology, i. e., Attitude Control, Inertia! Guidance, Onboard Techniques, Manned Control, Deep Space Guidance and Navigation, Rendezvous, and Landing. To set the stage and to orient those readers who may not have had exposure to the more rapidly growing areas, the Editors have invited certain introductory review articles. They will assist the reader's entry into the subject and into the literature, and in addition they contain information basic to the more complex situations covered in the succeeding papers. In the selection of papers the guiding rule has been the exposure of diverse lines of inquiry, with particular attention given to those current lines of research which appear to be most promising. A somewhat intriguing aspect of the Conference was the partial eclipse of "control" by "guidance" as demonstrated by the greater vigor of the latter line of research. In this sense, also, this volume follows the same pattern as the Conference. Approximately equal treatment is given to passive and active methods of attitude control of satellites. In the passive method, the use of the gradient of the earth's gravity field for stabilization is an effective demonstration of the use of minute forces. At interplanetary distances, another method that uses solar radiation pressure for a Mars Mariner flight makes an even better demonstration. In the treatment of active systems, the reader's attention is drawn specifically to an integral pulse frequency control technique of novel form which is based on the emission of a control impulse each time the integral of the error exceeds the preset amount. Lack of space in this volume and in the Conference prevented a full disclosure of methods of attitude control now being tried. However, several papers are included covering the problem of reorienting the spin axis of a satellite without introducing wobble into the final motion. ix

Purchased from American Institute of Aeronautics and Astronautics

Inertia! guidance has been with us now for several decades, and regrettably improvements are appearing in continually smaller quanta. The ingenuity of present workers in the field is represented by such diverse developments as drift reductions in platforms using standard components, the exploitation of highly sophisticated components such as electrostatic gyroscopes to give improved performance, and finally the combination of multiple sensor inputs in statistical form. An intriguing paper is the one directed to rigid-body dynamics which delineates the key relationships that describe coupled rotary actions in platform suspension systems. It is thought that further development of this analysis may illuminate the resonant excitations of elements of a gyro-stabilized platform which often have resulted in the degradation during a test program of the tested unit beyond the specification limits imposed by the operational problem. The vagaries of the human pilot in the man-machine control loop have received much attention. Perhaps the bewildering variety of ways in which he may behave have caused vehicle designers to be cautious in accepting mathematical analysis of man's role in these control loops. There appear to be intriguing indications that successful pilots operate over spectra that are more predictable and that are amenable to analysis. Other major treatments of the role of the pilot are presented in terms of models for human action and in terms of the predictive possibilities of a human with and without past visible stimuli. The use of external observables to update inertial references is a subject of great practical importance. Their use to correct platform drift and perform various corrections is covered in several papers using radar sensing, star vectors, sun line vectors, and horizon sensing of earth and near-earth planets and moon. The combination of these various frequencies of the electromagnetic spectrum is further exploited in extensions of Kalman f s work to secure increased accuracies from combinations of sensors.

Although it is inevitable that this volume will not even partially cover the fields of guidance and control, an attempt was made to produce a sequel to the Roberson and Farrior edition and to set the stage for the establishment of a series on guidance and control to keep the reader informed of the important contributions in this expanding field. Recognition should be given to the AIAA Guidance and Control Technical Committee (Chairman, Robert C. Langford) that sponsored the Conference at the Massachusetts Institute of Technology. Charles J. Mundo, as Technical Chairman, was responsible for the over-all concept of the Conference and its accomplishments. Special recognition should be given to C. J. Mundo, C. Gould, A. H. Greene, G. Kilpatrick, M. Riess, E. Walner, W. Wrigley, L. F. Gilchrist, J. C. Houbolt, H. Bramson, H. H. Haglund, R. H. Cannon Jr. , and F. P. Morrison, who performed the task of evaluation and solicitation of papers. x

Purchased from American Institute of Aeronautics and Astronautics

Finally, the Editors wish to express their appreciation to Martin Summerfield, Series Editor; Leil Tanenholz, General Precision, Inc.; Ruth F. Bryans, Managing Editor, AIAA Scientific Publications, and members of her staff for help in producing this volume and in bridging the gap from Conference to copy with constructive comment.

Robert C. Langford General Precision, Inc. Charles J. Mundo Institute of Naval Studies April 1964

XI

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

CONTENTS The Guidance and Control Committee. . . . . . . . . . . . . . . . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. Active and Passive Attitude Control for Space Vehicles An Introduction to Section on Attitude Control . . . . . . . . . . . . . Robert H. Cannon Jr. and Daniel B. DeBra Current Status of Progress in Attitude Control . . . . . . . . . . . . E. I. Ergin A System for Passive Gravity-Gradient Stabilization of Earth Satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robert E. Fischell and Frederick F. Mobley Theoretical and Practical Aspects of Solar Pressure Attitude Control for Interplanetary S p a c e c r a f t . . . . . . . . . . . James D. Acord and John C. Nicklas Control Moment Gyro Gravity Stabilization. . . . . . . . . . . . . . . E. D. Scott Applications of Gyro stabilizers to Satellite Attitude Control. . . . J. E. De Lisle, E. G. Ogletree, and B. M. Hildebrant Integral Pulse Frequency On-Off Control . . . . . . . . . . . . . . . . R. L. Farrenkopf, A. E. Sabroff, and P. C. Wheeler A Time Optimal Attitude Control System Designed to Overcome Certain Sensor Imperfections . . . . . . . . . . . . . . M. J. Abzug, L. W. Martin, and F. T. Ohgi Two-Pulse Attitude Control of an Asymmetric Spinning Satellite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. C. Wheeler Bank Angle Control System for a Spinning Satellite . . . . . . . . . H. Patapoff Several Linear Stabilization and Reorientation Control System Configurations for a Rotating, Manned Orbital Space Station . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mitchell Loebel xiii

vii ix

3 7

37

73 103 149

185

231

261 289

313

Purchased from American Institute of Aeronautics and Astronautics

II. Inertia! Guidance for Space Flight Drift of a Stable Platform Caused by Gyro Rotor Unbalance . . . . Myron Kayton Stellar Alignment of a Gyroscopic ally Stabilized Platform During F r e e - F a l l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saul Moskowitz Navigation and Guidance Systems Employing a Gimballess IMU. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Robert M. Bumstead and Wallace E. Vander Velde Rigid Body Dynamics - Stable Platform Isolation Systems. . . . . Melvin Feinberg Interrogation of Spherical-Rotor Free Gyros . . . . . . . . . . . . . Keith D. Graham III.

341

359

391 421 439

Onboard Techniques for Interplanetary Flight

Background and Requirements on Radar Sensors for Spacecraft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Richard J. Taylor and Jack H. Kalish Stellar Techniques for Midcourse Navigation Guidance . . . . . . . K. C. Kochi and H. L. Dibble Infrared Horizon Sensor Techniques for Lunar and Planetary Approaches . . . . . . . . . . . . . . . . . . . . . . . . Gerald Falbel and Robert W. Astheimer

Novel Solar System Compasses for Interplanetary Flight. . . . . . Ralph Deutsch IV. Manned Control of Space Vehicles Pilot-Vehicle Control System Analysis . . . . . . . . . . . . . . . . . Duane T. McRuer and Dunstan Graham A Survey of the Development of Models for the Human Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. I. Elkind Some Predictive Characteristics of the Human C o n t r o l l e r . . . . . T. B. Sheridan, M. H. Merel, J. G. Kreifeldt, and W. R. Ferrell V. Deep Space Guidance and Navigation Discussion of Guidance Policies for Multiple-Impulse Correction of the Trajectory of a S p a c e c r a f t . . . . . . . . . . . . D. W. Curkendall and C. G. Pfeiffer

A Class of Unified Explicit Methods for Steering Throttle able and Fixed-Thrust Rockets . . . . . . . . . . . . . . . . . . . . . . . . George W. Cherry xiv

467 495

551

583

603

623

645

667

689

Purchased from American Institute of Aeronautics and Astronautics

New Investigations in the Field of Error Propagation . . . . . . . . Wayne H. Tempelman Theory of Error Compensation in Astro-Inertial Guidance Systems for Low-Thrust Space Missions . . . . . . . . . . . . . . Hermann M. Dusek

Statistical Filtering of Space Navigation Measurements . . . . . . James E. Potter and Robert G. Stern

727

755

775

VI. Rendezvous

Survey of Rendezvous Progress . . . . . . . . . . . . . . . . . . . . . . Arthur W. Vogeley and Roy F. Brissenden Fuel Optimization in Orbital Rendezvous . . . . . . . . . . . . . . . . A. A. Goldstein, A. H. Greene, and A. T. Johnson Guidance System Complexity vs Fuel Consumption for Midcourse Correction of Rendezvous . . . . . . . . . . . . . . . . . Robert M. O'Brien Application of PontryaginTs Maximum Principle to the Lunar Orbit Rendezvous Problem . . . . . . . . . . . . . . . . . . . . . . . E. S. Armstrong and J. H. Suddath Guidance Dynamics for the Terminal Phase of Rendezvous . . . . Carl Grubin

805 823

845

865

887

VII. Re-Entry and Landing A Look at the Re-Entry Problem . . . . . . . . . . . . . . . . . . . . . Donald L. Baradell

911

A Simple Re-Entry Guidance System . . . . . . . . . . . . . . . . . . . John A. Love and L. W. Neustadt A Minimum Fuel Vertical Touchdown Lunar Landing Guidance Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. A. Hall, R. G. Dietrich, and K. E. Tiernan

931

Contributors to Volume 13 . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

965

995

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

AN INTRODUCTION TO SECTION ON ATTITUDE CONTROL Robert H. Cannon, Jr.* Stanford University, Stanford, Calif. and

Daniel B. DeBra** Lockheed Missiles and Space Company, Sunnyvale, Calif. In this section a comprehensive survey paper is followed by papers on passive control methods, active control systems and control of rapidly spinning vehicles. The survey paper, by E. I. Ergin of Space Technology Laboratories, begins by describing the kinds and spectrum of control requirements with which the designer of an attitude control system may be faced as a result of the variety of spacecraft tasks encountered and contemplated. Next, a survey is presented of the numerous conceptual approaches and the devices which have been developed to implement them. Comparisons are given between several possible techniques for meeting a given requirement. The accompanying diversity of analytical design requirements is described. Finally, a critical comparison of the several possible methods for testing attitude control systems is presented. Under Passive Attitude Control, papers discuss methods of utilizing the gradient in the earth's gravity field to achieve satisfactory stabilization, and the use of solar pressure for the same purpose in interplanetary missions. In a paper on augmented gravity-gradient stabilization, R. E. Fischell and F. F. Mobley, of the Applied Physics Laboratory, Johns Hopkins University, present flight test data obtained from a recent flight in which a boom and coil-spring Presented at the AIAA Guidance and Control Conference, Cambridge, Mass., August 12-14, 1963. * Professor of Aeronautics and Astronautics. *# Supervisor, Dynamics and Control Analysis.

Purchased from American Institute of Aeronautics and Astronautics

R. H. CANNON JR. AND D. B. DeBRA

damper, developed by the Applied Physics Laboratory, was used with magnetic hysteresis to stabilize passively a gravitygradient oriented satellite. For vehicles initially spin stabilized, the satellite is first de-tumbled using magnetic hysteresis rods and a yo-yo de-spin mechanism. Then with the satellite aligned to the earth's magnetic field by an electromagnet the boom is extended while the satellite is temporarily oriented along the local vertical. The boom extends 100 ft and the coil spring extends an additional 30 ft from the end of the boom. As the satellite swings back and forth about the local vertical the mass on the end of the coil spring extends and contracts the spring, which is specially designed to dissipate energy by mechanical hysteresis.

Another paper on passive or semi-passive stabilization is that of J. D. Acord and J. C. Nicklas of the Jet Propulsion Laboratory. Their scheme, which will be used for the Mariner Mars space vehicle, uses solar pressure to maintain the vehicle always oriented toward the direction of the sun: the center of solar pressure of the vehicle is arranged to be behind its mass center so that nominally the vehicle would have very slow undamped oscillations about a position along the radius vector from the sun. Damping is introduced into the system by means of an ingenious thermal arrangement. If the vehicle has an angular error with respect to the sun line, a baffle arrangement (radiation valve) admits heat from the sun warming a bimetal strip which connects solar paddles to the vehicle. The thermal time constant of the bimetal strip is of the order of 40 min, which is long enough so that the resulting change in solar paddle orientation takes place with sufficient phase shift with respect to the vehicle motion to provide damping.

An active pneumatic system is used for coarse control. The passive system operates within the pneumatic system deadband. The solar pressure equilibrium orientation is assured of lying within this deadband by allowing the solar paddles which are mounted on a "dry hinge" to move as the gas valves are fired, thus adjusting the equilibrium position toward the center of the deadband. In the papers by E. D. Scott, of Lockheed, and by J. E. DeLisle, B. M. Hildebrant, and E. G. Ogletree, of the Massachusetts Institute of Technology Instrumentation Laboratory, very comprehensive discussions are given of the utilization of single-axis gyros with viscous restraint to augment control and to provide damping of the natural oscillations of a gravity stabilized vehicle. The gyros enable a vehicle to reach and maintain stable equilibrium in the local-vertical orientation. DeLisle, Hildebrant, and Ogletree describe the motions and the selection of design parameters for systems of two gyros with 4

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

output axes along the roll axis and with the momenta symmetrically arranged about the orbital angular velocity vector. Their criterion for optimization is to minimize the integral of the error squared. The method enables them to get results in closed form. In their presentation at the August 1963, AIAA Specialist Meeting the authors showed a motion picture produced by an IBM 7090 digital computer under the direction of E. E. Zajac of the Bell Telephone Laboratories. It depicted the large-angle and capture-mode behavior with excellent clarity.

Scott presents an independent derivation of the governing relations and introduces additional gyro arrangements. Of four two-gyro arrangements, he shows two to be effective, pointing out that their relative performance suits one of them to missions requiring tight roll control and the other to missions requiring tight yaw control. Two single-gyro configurations are also discussed. Scott describes methods of solution, including the use of Monte Carlo optimization techniques in selecting system parameters for a particular configuration. Finally, he presents extensive design tables based on numerous optimization solutions that he has obtained. The group of papers on active attitude control begins with the work of A. E. Sabroff, R. L. Farrenkopf and P. C. Wheeler and follows with a paper by M. J. Abzug and F. T. Ohgi. The paper by Sabroff, Farrenkopf, and Wheeler, of the Space Technology Laboratories, describes a new type of on-off attitude control known as integral pulse frequency control, which emits a standard control impulse each time the integral of the error changes by a pre-set amount. Several powerful analytical techniques are used to establish a fundamental understanding of the stability of such systems, and their superiority in certain cases is demonstrated. Simulation studies, including noise, verify and augment the analytical results. The paper by M. J. Abzug, L. W. the Douglas Aircraft Company extends and impulse-optimal attitude control zone type imperfections exist in the sensors.

Martin, and F. T. Ohgi of the theory of time-optimal to the case where deadattitude and attitude-rate

The three papers on rapidly spinning vehicles are addressed to the problem of reorienting the spin axis of the satellite without introducing wobble in the final motion.

Purchased from American Institute of Aeronautics and Astronautics

R. H. CANNON JR. AND D. B. DeBRA

P. C. Wheeler discusses the application of two-pulse control to the general class of vehicles including those with three distinct principal moments of inertia. The design angle between two sets of jets is given as a function of vehicle shape parameters. This angle allows each jet to be fired at the same inertial location so the change in angular momentum is added while the wobble effects are cancelled.

H. Patapoff discusses an application of this method to a symmetrical satellite which is required to keep its spin axis normal to the orbit plane. The effects of restricted sensing of only roll errors, misalignments, and control on spin speed are discussed.

In the last paper in this section M. Loebel studies two control laws for control of a spinning manned space station by gas jets on a two-degree-of-freedom torqued gyro. He shows the law using the body axis components of the vehicle angular velocity (body rates) and their integral allows one to control the spin axis of the space station about a nonprincipal geometrical axis of the vehicle - a desirable condition for docking space ferrys.

Purchased from American Institute of Aeronautics and Astronautics

CURRENT STATUS OF PROGRESS IN .ATTITUDE CONTROL

E. I. Ergln* TRW Space Technology Laboratories, Inc. Redondo Beach, Calif, Abstract Past attempts in establishing the state of the art in spacecraft attitude control have concentrated on cataloguing various satellites in terms of their mission objectives and design features, or on describing specific hardware developments. The discussions in this paper are developed along the lines of "what is common" rather than "what is different" with the idea that seemingly -unrelated efforts in this new technology can be put in their proper perspective, and that some unresolved problems can be defined. To achieve this objective, first the control system requirements (from a performance rather than a mission point of view) and the implications of such requirements in terms of control system design are discussed. Second, specific attitude control approaches and the associated performance or implementation problems, or both, are considered in "Conceptual Design," and some of the theoretical problems are discussed in "Analytical Design." Finally, the question of "Ground Test Techniques" for space vehicles is treated qualitatively since this subject has received relatively little attention in the literature in spite of its technological importance. I. Introduction Several recent survey papers^>2 have classified in detail the past and the present generation of satellites and space vehicles, their mission objectives, and their important design features. OthersS have discussed the recent advances in the field primarily from the hardware point of view. Because of the large number of possible control configurations, the current status of control system design cannot be established by simple generalizations. Nevertheless, some important Presented at the AIAA Guidance and Control Conference, Cambridge, Mass., August 12 - 1^, •^Member of the Technical Staff.

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

considerations and problems can be logically explored in relation to specific phases of development such as conceptual design (Sec. II), analytical studies (Sec. Ill), system development and manufacturing, and integration and test (Sec. IV). Development and manufacturing phases will not be considered here because of their specialized nature, and the integration and test discussion will be primarily concerned with the present status of ground test techniques for performance evaluation of attitude control systems (ACS).

The function of attitude control in space vehicles is to orient one or more spacecraft axes in prescribed reference directions in the presence of perturbations. The control system design problem associatid with this function is to establish a system configuration which meets the mission requirements and constraints in performance, while optimizing the overall system in terms of established criteria. The final control configuration will thus be a function of performance requirements, the expected perturbations, and the optimization criteria. Since the optimization criteria, performance requirements, and the perturbations determine the feasibility and the complexity of particular control configurations, their influence on system design will first be briefly examined in the following paragraphs. 1. System Optimization

Optimization determines the "best" system configuration on the basis of selected criteria and within the constraints of the particular mission. However, selection of valid criteria is just as important as the optimization process. At present, the most significant design optimization criteria for ACS are minimum weight and power, and maximum reliability. Optimization on these bases is a formidable, if not impossible, analytical problem. Although specific elements (such as reaction wheels, or coil sizes for magnetic control) can be optimized in terms of weight , the development of a control system concept within the optimization criteria depends on the experience and the ingenuity of the designer. The above criteria are usually manifested in the selection of control concepts utilizing a minimum number of components (particularly electromechanical devices) and a maximum reliance on passive or switching elements, or both. Hence we have the increased use of "bang-bang" controllers for reaction 8

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

wheels as well as gas jets, since switching elements use less average power and have fewer components than proportional devices. Similarly, much effort has been expended on the design of systems using the environmental fields (such as gravity gradient and radiation pressure) to minimize the control system complexity, increase reliability, and reduce weight and power. Even though some analytical work has been done on minimum time or impulse controllers, or both, these efforts remain primarily as academic problems since they treat single-axis control system performance only in a significant transient state (large attitude errors) and totally neglect important mechanization problems. Furthermore, in focusing attention toward the transient behavior of the ACS, two important factors are neglected: l) for most space missions, the significant transient occurs very infrequently compared to quasisteady-state operation (limit-cycling for exarrrple) where attitude errors remain small; and 2) when such significant transient states exist, the simplifying assunrptions allowing the treatment of the control system on a single-axis basis can no longer be justified because of the major contribution to transient behavior made by kinematic and dynamic crosscoupling between the three control axes*

2. Performance Requirements and Perturbations Performance requirements establish the direction and the accuracy of the attitude orientation and are dependent on the mission as well as on other subsystems such as power, communication, temperature control, etc.5 It should be noted that these prescribed directions can be considered stationary if they rotate at rates much below the basic attitude control or perturbation frequencies. Thus, the orbital rate can in general be neglected in the design of a reaction-wheel, gasjet control system for earth satellites. On the other hand, the evaluation of disturbance-torque effects to establish the momentum storage or impulse requirements, or both, must consider the rotation of the local vertical (orbit rate) since significant disturbance-torque perturbations are at orbit frequencies. Similarly, an ACS design for gravity-gradient stabilization must again take into account the orbit motion inasmuch as the control system dynamics will have frequencies of the order of orbit rate. The control system performance, and hence its design, will be influenced by the nature and the magnitude of the expected perturbations. These perturbations may be associated either

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

with torque disturbances (environmental or nonenvlronmental) or the angular motion of the reference frame. The first category includes the effects of guidance engine firing, start and stop of internal rotating devices (such as tape recorders), and the environmental torques due to gravitational and magnetic fields, solar radiation, and aerodynamic pressure. The reference-frame motion category consists of perturbations associated with the closed-loop guidance of space vehicles for the purposes of rendezvous, orbital docking, interception, or lunar landing. The attitude-controlreference direction in these instances is coupled directly into the translational motion of the spacecraft and is therefore dependent on the overall guidance-control performance. The control-system dynamic performance requirements are defined explicitly or implicitly in terms of accuracy, speed of response, and the dynamic range of the torque-generating devices. These requirements, in conjunction with the expected perturbations, determine both the overall control system complexity and the complexity of specific elements, such as sensors and actuators. Since these requirements form the basis for comparison of various attitude control concepts, the implications of each performance requirement will be briefly discussed.

5:* Accuracy: Primarily, the accuracy requirement provides constraints on the sensor design in terms of "near null" characteristics as well as null sensitivity and stability. In addition, it establishes the small signal requirements (such as granularity, linearity, threshold effects, etc.) on the actuators. Generally, in the absence of the other two performance requirements, the actuator and sensor requirements can be met as long as the sensor receives sufficient signal power to provide a reasonable S/N ratio. Signal power should be plentiful for a sun sensor, for example. Aside from attitude control aspects, high-accuracy requirements (scientific satellites such as Orbiting Astronomical Observatory (OAO) and Advanced Orbiting Solar Observatory (AOSO) have attitude-orient at ion accuracy requirements in the range of 0.1 to 5 arc-sec) result in significant overall spacecraft design constraints. These constraints may be associated with the initial sensor alignment or temperature and structural control, or both, to provide acceptable thermal gradients and structural deflections (both dynamic and thermoelastic) at the spacecraft-sensor-experiment interfaces.

10

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

—* Response: Response requirement establishes the transient character of the attitude motion in the presence either of significant attitude perturbations or attitude reference rotations (commanded maneuvers). The three important considerations in determining control-system response requirements are related to the initial stabilization, disturbance torques, and the commanded maneuvers. Following separation from the boost vehicle, the control system must stabilize the spacecraft to its nominal orientation within a specified time for power, temperature control, communication, or guidance purposes. In general, significant initial conditions in terms of -unbounded attitude errors and bounded, but large, attitude rates may exist at the start of the initial stabilization (so-called acquisition) process. Nonenvironmental disturbances, such as these resulting from thrusters used for guidance or those from the start and stop of internal rotating devices, (e. g., tape recorders) which generate step-torque disturbances, must be compensated for by the control system before attitude errors exceed the established accuracy limits. Conversely, environmental torques caused by radiation pressure, gravity gradient, aerodynamic, and magnetic field disturbances are low-frequency perturbations compared to basic attitude control frequencies (except during reentry), and do not significantly influence the system response requirements unless the same environmental sources are used for control torque generation. Special spacecraft maneuvers may be associated with either the relatively rapid rotation of the reference frame (such as in rendezvous-docking or landing maneuvers where the attitude reference is strongly influenced by guidance corrections), or with operational requirements for special scientific experiments (the AOSO spacecraft, for example, must scan an area of 5- by 5-arc-min in a 5-^in period to an accuracy of 5 arc-sec within the specified scan pattern. The usual design approach taken to meet the fast-response requirements aurplifies the effects of nonlinearities, particularly near the required null operating regions. For example, an increase in the actuator (or controller) dynaMc range requirements would influence the near-null nonlinearities. Thus, the multimode controllers become necessary when the system has a relatively fast-response, low-accuracy control mode (e. g., acquisition) coupled with a slow-response, higher-accuracy control mode (normal mode). In cases of 11

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

extreme accuracy requirements (such as in OAO), the multimode controller has more "levels" of control authority (two sets of reaction wheels, magnetic torquers, and gas jet torquers) and is correspondingly more complex.

The general acceptance of momentum control devices (e. g., reaction wheels) to compensate only for cyclic disturbance torques can lead to faulty design decisions* The most important function of momentum control devices may be to perform orientation maneuvers rather than merely to compensate for cyclic disturbances. £. Dynamic range: Dynamic range requirement defines the control torque range necessary to attain the specified control capability at various phases of the mission. This requirement is, of course, related not only to the response requirements of the control system but also to the particular sensor-controller configuration. In general, however, the control torque requirements fall into three categories: 1) Overcoming the environmental disturbance torques; 2) Maneuvering the spacecraft (such as acquisition, and other re-orientation maneuvers); and 3) Overcoming major disturbances (for example, due to a guidance engine firing).

There are usually orders-of-magnitude differences between the torque requirements in the three categories. OGO, for example, has control accelerations of about 6 x 10"^ deg/sec^ for acquisition, 5 x 10~3 deg/sec^ for orbit maneuvers (posteclipse turn), and only about 10~5 deg/sec^ to perform during normal operating modes. Attitude control systems that depend totally on environmental sources for control torque generation (such as gravity gradient or radiation pressure) possess significant limitations in terms of maneuvering or acquisition capabilities, or both. This limitation may necessitate auxiliary controlling elements to perform these functions. II.

Conceptual Design

It was already pointed out that the selection of a particular control concept is dependent on control system requirements in terms of pointing accuracy, control-torque dynamic range, speed of response, and kinematic constraints (maneuvering requirements). The control systems may then be characterized by their capabilities in being able to satisfy most-relaxed to most-stringent performance requirements. Such a characterization also results in a classification that is closely related to the inherent complexity, and hence to weight, power requirements, and reliability of the control 12

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

systems. However, in many instances, relaxed performance requirements from the attitude control system imply increased performance penalties in the other subsystems, such as power, temperature control, communications, etc. Hence only over-all spacecraft system tradeoffs can determine the relative merits of the performance penalties and establish reasonable control system requirements. Generally the accepted categorization of the control concepts, both in terms of their relative simplicity and their performance limitations is passive, semipassive, and active controllers, each containing a relatively broad variety of configurations.

1. Passive Control a. Spin-stabilization techniques: The simplest passive control system may be obtained by spin stabilization of the spacecraft when only one spacecraft axis must be oriented in some inertia! direction with relatively low accuracy. Early spacecraft and satellites, such as Explorer and Pioneer, contained no control equipment except spin-up rockets. Clearly, the system operation depended totally on having sufficient spin momentum so that expected external torques for the duration of the mission would not precess the spin axis outside the required orientation accuracy limits. Spin stabilization is still an attractive control technique for missions where low-accuracy inertia! orientation of only one spacecraft axis is sufficient to meet the particular system requirements. Recent studies^ have shown that, for special orbits and spinaxis orientation requirements, uncontrolled spin stabilization for equatorial or near-polar orbits (orbit regression also negligible) is feasible even for 1- to 3-y** duration orbits. The effects of disturbances such as earthfs magnetic field, gravity gradient, and radiation pressure produce predictably low precession angles when the spin axis is normal to the orbit plane. Fortunately for the control engineer, not many missions can admit a totally passive, spin-stabilized control system. In general, there are requirements either for changing the spin-axis orientation several times during the flight (for example, in interplanetary probe) or for controlling its orientation to a reference which is only quasiinertia! (for example, a reference direction normal to the orbit plane precesses in inertia! space because of orbit regression, for nonpolar orbits).

b. Control using environmental fields; Some widely discussed and partially implemented passive control techniques 13

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

make use of the environmental fields to generate control torques. Most common of these is, of course, the gravity gradient control. Certain missions can be conceived where the utilization of solar-radiation pressure, earth's magnetic field, and even aerodynamic control surfaces may provide passive stabilization. There are, however, some significant operational limitations and design considerations in the utilization of environmental fields for passive attitude control: 1) The system will have an extremely low speed of response (of the order of hours), and a limited acceleration capability (small control torques)*

2) The spacecraft will, in general, be limited to having a "fixed" nominal (equilibrium) orientation established by the field used to generate control moments. There is little or no flexibility for changing arbitrarily the nominal spacecraft orientation during the mission. In addition, the control requirements will dictate to a great extent the mass distribution and the shape factors for the spacecraft. That is, the vehicle will be designed to satisfy the control requirements.

3) The spacecraft control authority and the orientation accuracy limitations will cause it to be sensitive to predictable or unpredictable perturbations. This implies that, for example, for a gravity-gradient-stabilized satellite, the disturbing moments due to other external forces (such as radiation pressure, magnetic field, or aerodynamic torques) or internal perturbations (such as tape recorders) must be accurately predicted and properly compensated for. k) The spacecraft will, in general, require some active control mechanism for initial stabilization. Because of the limited control authority, the spacecraft must initially be placed near its equilibrium orientation with initially low angular momentum.. Gravity-gradient-stabilized satellites, for example, can stabilize only if the vehicle's initial angular rates are less than one or two times the orbital rate. TRMC satellite' accomplishes the initial stabilization task by utilizing despin magnets (removing energy from the system through hysteresis-loop losses) and an electromagnet turned on and off by ground command. This active control requirement for initial stabilization is very often neglected in the discussion of the passive control systems. 5) Generally, a mechanism for introducing damping into the spacecraft motion must be provided. 14

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

In spite of their performance limitations, passive control systems possess the very important advantage of simplicity* Gravity-gradient control is already a flight-proven technique for relatively low altitude (100- to 200-mile) orbits. Some •unique and apparently •workable schemes for attitude cottrol "by solar sails have also been proposed, if not yet applied, for interplanetary flights" & . Acord and Nicklas, in the latter reference, propose the utilization of the thermoelastic behaviour of bimetallic strips to introduce damping into the motion, thus eliminating the need for an active damping device. .£» Design problems; Certain problems, however, associated with the conceptual design of passive control systems require further consideration. 1. Damping: The question of damping is a critical one both in terms of what can be provided and what is necessary. In most cases, the damping requirements defined in terms of transient response of coupled or uncoupled axes are not sufficient for design purposes, since the system will eventually damp out any reasonable initial conditions whether within one orbit or three orbits. A more important criterion for requirements should be based on the effects of perturbations. Environmental disturbances as well as orbit perturbations (the orbit eccentricity, for example) will introduce forcing terms at or near the natural frequencies of the satellite's ACS. Some of these effects have been discussed in the literature-^ for undamped systems but have not been directed

towards establishing criteria by which the damping requirements can be developed in terms of desired orientation accuracy, orbit parameters, and satellite physical parameters. In addition, much work is needed in damping materials, which must provide damping in the presence of vehicle rates of the order of 10"^ to 10~5 rad/sec after many months of orbital operation. 2. Initial stabilization; The question of initial stabilization needs to be explored further. The unique stabilization technique utilized for TRAAC satellite, for example, may not work well for an equatorial satellite because the earthfs magnetic field is approximately normal to the equatorial plane. A much heavier reliance on ground command links as an in-line control function for initial stabilization must be considered. Even though such reliance is not conceptually unattractive, the problems of ground track, visibility time, and computation and command periods require detailed 15

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

system studies. 3. Disturbances: The question of disturbances is important, particularly with increased spacecraft size and orbit altitude (for gravity-gradient-controlled satellites). Because of decreased control authority with increased orbit altitudes (torque per unit angular deviation is proportional to the square of orbit rate) the outgassing of materials and the variation in surface emissivities and reflectivities may cause serious control difficulties. The same considerations hold for passive solar-radiation control systems relative to tolerances in the physical character of the solar sails. Hence procedures must be developed for properly evaluating these sigiificant disturbance effects in the expected spacecraft operating environment. h. Three-axis control: Three-axis control has not been fully resolved for passively stabilized spacecraft. The control about the local vertical axis depends totally on orbit kinematics for gravity-gradient stability. For solarradiation pressure, however, the control about the sun line remains a significant, though not fully explored, problem. Possibly, angular rates about the vehicle axis parallel to the sun line may cause system instability for certain control parameters if closed-loop control about this axis is lacking. 5. Ground test evaluation: Finally, effective ground test evaluation of control system performance is as important as it is difficult to achieve and is particularly true for the damping mechanisms that are nominally to work in a zero-g environment. The spring-mass damper used on TRAAC was tested by means of ingenious torsional oscillators with natural periods of the order of 1.5 hr. Testing of other possible damping configurations (possibly fluid or magnetic dampers with a properly controlled environment) is a difficult task at best, particularly if natural periods of about 6 to 12 hr are desired (future communication satellites, for example, may operate at orbits with periods of 6, 12, or 2k hr). Hence the development of ground test techniques for passively controlled satellites is neither well understood nor defined. 2. Semipassive Control Semipassive control systems provide both improved performance in terms of response time and possibly more control authority over the passive controllers and generally require more relaxed kinematic and dynamic constraints. Spinning satellites with active controllers (either pneumatic or 16

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

magnetic) and gravity-gradient or solar-radiation pressure controlled satellites with, active damping mechanisms (such as gyros or reaction-wheel dampers) fall into this category. a. Gravity-gradient control with active damping mechanisms: A great deal of work has been done and documented on the use of single-degree-of-freedom floated gyros as dampers and moment-urn storage devices in spacecraft attitude control systems-^ ""^5. The gyros act as momentum storage devices and provide stability because of gyroscopic torques due to body angular rates, independently of the gravitational field. The gravity gradient can, of course, be utilized as an external torque to remove momentum from the system. This is an important distinction relative to passive gravity-gradient control where the gravitational torque directly provides the control moments, An excellent discussion of control system response with gyro controls is given in Ref. 16. Even though the use of two reaction wheels as damping devices for gravitygradient control has not been reported in the open literature, their application has been studied in connection with highaltitude, gravity-gradient-stabilized satellites, and design procedures have been developed. The major drawback to the use of reaction wheels is the additional necessity for an earth sensor while gyros combine both sensing and controlling functions. Since the gyroscopic torques are due to angular rates of the order of orbit rate, the gyro perturbation torques (torques that cause drift) must be extremely small. This requirement, although it can be met with the state-ofthe-art gyros, necessitates the use of inertial quality gyros with drift rates less than 1 deg/hr. The initial stabilization is again a critical performance problem with actively deorrped, gravity-gradient systems. Researchers have very often overlooked the fact that a gyrodamped system, for example, may be capable of capturing from initial rates about equivalent to those with passive dampers, while a reaction wheel damper may capture from initial rates that are 5 to 10 times as high. It is interesting to note that the so-called "acquisition limitation" is a function of the saturation behavior of the controlling element (output angle freedom for the gyro, wheel speed saturation of the reaction wheel). To get compatible acquisition capability equivalent to reaction wheels, the gyro output angle freedom would have to be increased to the 50- to 80-deg range while maintaining the low drift rates during normal operation.

b. Spin stabilization with active controllers: Active control of spinning satellites has been extensively treated in 17

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

the literature for specific mission requirements^-? "19. The primary application thus far has "been for scientific and communication satellites even though extension of this concept to photographic and possibly surveillance missions is under active consideration. In addition to the inherent gyroscopic stability, the spin motion itself can "be utilized to advantage for simple information processing either for attitude reference or for generating the data required for the

Probably the most complex, active, spin- stabilized ACS to date is utilized on OSO I spacecraft, -which performed satisfactorily during its initial flight test. The gyroscopic stability was obtained by spinning a platform on which a nonspinning scientific experiment package was mounted through two gimbals, one of them having an infinite degree of freedom (about the spin axis). Note that this does not differ from having one large constant -speed reaction wheel internal to the spacecraft with fine spin-speed modulation, jet thrusters, and a single gimbal system for the experiments. It can be classified as either active or semipassive, the former term probably being more accurate. Unlike other spin -stabilized satellites, it does provide a pointing accuracy for the experiments in the order of arc -minutes.

In the communication satellite field, examples are more plentiful: Relay I, Telstar, and Syncom are all spinstabilized with capability to precess the spin momentum to keep it in its proper inertial or orbital orientation within relatively relaxed accuracy requirements. Depending on orbit inclination and altitude, either magnetic (Relay I, Transit) or pneumatic control systems ( Syncom) can be employed to provide the precessional torques. Some important design problems are associated with both surveillance type or scientific satellites requiring precise spin-axis orientation, and design considerations associated with manned orbital space stations can be included in this category. Consider the problem of precisely orienting a body-fixed axis (nominally along the maximum principal moment of inertia) along a specified inertial direction. Because of uncertainties or changes in the location of the true principal moment-of -inertia axes, the body-fixed nominal axes (an optical axis, for example) will precess around the spin axis at an angle corresponding to the misalignment angle and at the spin frequency. This "wobbling" motion of the nominal axis can significantly limit the quality of experiments or, in the case of a manned orbital space station, complicate the 18

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

rendezvous and docking maneuvers^. Also, the inertia distribution within the spacecraft may be time-varying in some

random fashion (as a result of crew movement, for example).

The control problem is now shifted from that of orienting the spin momentum in a prescribed inertial direction to orienting precisely a body-fixed axis nominally close to the spin axis in a prescribed inertia! orientation. Assuming that a fully stabilized configuration is not a desirable alternative (for example, the spin may be necessary to generate an artificial gravitational field for the crew), several possible alternatives are discussed below: 1) OSO I approach where one portion of the vehicle is nonspinning but gimballed to the main spinning body through relatively vulnerable bearings. 2) Momentum control approach where the angular velocity components measured along the transverse moment of inertia axes are utilized to drive reaction wheels for wobble control, and the gimballing of the main "spin reaction wheel" to effectively shift the vehicle moment-urn from the actual principal moment-of-inertia axis to the nominal spin axis. This approach possesses some significant operational difficulties. First, the transverse react ion-wheel momentum will have to be transferred between the two wheels at a frequency corresponding to spin frequency with corresponding power consumption; and second, the spin-control-wheel gimbal system will have to be accurately timed to the spin speed to achieve stable performance. Other mechanizations are, of course, possible for the momentum control approach (such as a single wheel with two gimbals and speed control) but they all require precise tuning of the control system as a function of spin speed and Involve significant power consumption. 3) Inertia control approach where the position of movable inertias within the vehicle are controlled until the nominal and the actual spin axes coincide. Even though this approach may look attractive conceptually, the development of the control laws in terms of measurable attitude position and rate information does not appear simple, and the possible implementation problems cannot be predicted at this stage. It is clear that much analytical work is necessary before realistic design comparisons can be made and tradeoffs established between various approaches to the control problem posed above for spinning satellites.

19

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

Active Control

Active attitude control systems provide the greatest design flexibility for the controlled orientation of a spacecraft in the presence of significant perturbations and maneuvers. Their control capability relative to accuracy and dynamic response is limited only by the sensors or the system complexity, or both (including weight and power requirements) that can reasonably be tolerated. Basic control concepts utilizing external control moments or moment-urn storage devices, or both, have been treated extensively in the Iiterature'3j2\ Design flexibility of these control concepts is clear from an examination of spacecraft currently developed or under development. The OGO control system, for example, provides three-axis, earth-sun orientation control for satellites in either low-altitude circular orbits (150 naut miles) or highly elliptical, cislunar orbits (perigee of 100 naut miles and apogee of 60,000 naut miles) or any intervening orbit. OAO provides for control accuracies of the order of 0.1 arc-sec, while AOSO can provide not only accurate control but also accurately programmed scan. The only thing common to all these control systems is the provision for momentum storage (reaction wheels) and the provision for generating external control moments (pneumatic jets for OGO and AOSO, and pneumatic jet and magnetic control for AOA. At the extremes are the relatively simple pneumatic control system for Mariner spacecraft, and the relatively complex control and display systems for manner spacecraft, such as for Project Apollo and LEM (Lunar Excursion Module).

Problems associated with the control and guidance of manner spacecraft such as for Project Apollo are both too complex and philosophical for the scope of this article. The questions of the allotment of control functions (manual versus automatic), the questions of redundancy and backup modes, and the problems of abort are extremely mission-sensitive and concept-dependent. However, one interesting aspect to the manner spacecraft systems is the increased emphasis on quasilinear controllers. Manned spacecraft require the application of control moments that are a linear function of the pilot's stick position unlike most unmanned spacecraft where the emphasis has been on bang-bang controllers because of simplicity and reliability. As a result, increasing attention is being paid to quasilinear operation of bang-bang valves by pulse frequency or pulse width modulation, or both, and the related control-system design problems. Even though pulse modulation.techniques are not new, their application to spacecraft attitude control systems is relatively novel'5. 20

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

a. Momentum storage techniques; Momentum storage techniques for attitude control of unmanned space vehicles may vary from the unique multigyro configuration^ to the more conventional reaction wheels. Aside from the "bearing reliability arguments associated with fixed-speed devices (gyros) versus variablespeed devices (reaction wheels) or the power considerations, the reaction wheels offer a greater design flexibility particularly for vehicles with large maneuvering requirements. They have provided the primary control torques for Nimbus, OGO, OAO, and AOSO, all presently under development* The primary application for these devices has been in the area of control of periodic momenta either for maneuvering or for cyclic disturbance-torque compensation. The momentum storage devices, however, can function either as gyroscopic stabilizers or provide information on the total momentum state of the whole spacecraft as inputs into other controllers. The most obvious application of the former concept can readily be seen in OSO I and can be extended to other similar applications. It can also be shown, for example, that a constantspeed reaction wheel with the spin axis normal to the orbit plane (pitch) significantly improves the yaw (local vertical) "stiffness" of a gravity-gradient-stabilized satellite reducing the magnitude of uncontrolled yaw oscillations. The effect of the wheel, in this case is, of course, to increase the dynamical coupling between the yaw and the roll channels and, hence,utilize the gravity gradient for more effective control of yaw. The latter concept, that of utilizing the momentum storage devices as momentum-state-information sources can be found, for example, in Nimbus and OAO. These satellites utilize reaction-wheel-speed information for activating the pneumatic jets to change the momentum state of the spacecraft. On the other hand, OGO pneumatic jets are activated directly by the sensor error information, thus operating, in principle, independently of the momentum state of the spacecraft. Control laws for the wheel-drive systems for these three spacecraft also differ significantly. Both OAO and Nimbus utilize the sensor output with a lead-lag network for stability augmentation to drive the reaction wheels through a linear amplifier. OGO reaction wheels are driven in a bang-bang manner with full voltage of proper sign being applied whenever the output of the lead-lag filter exceeds the threshold level. Thus it is possible to achieve increased reliability by replacing the linear amplifier with a threshold detector and a switch^. However, the high control precision required from the OAO control system necessitates a much more careful evaluation of the effects of the control laws as well as of the drive system 21

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

nonlinear!ties on the system performance. On the other hand, the dynamical and kinematic nonlinearities (such as those associated "with derived rate information or intermode gyroscopic coupling, or "both) can, in general, be neglected in the design of even2 the precise control systems utilizing reaction wheels °>29 for at least near-null operation. This latter consideration raises the question of necessity for research into the reaction-sphere type of momentum storage devices, since their only performance advantage lies in the elimination of gyroscopic cross-coupling terms. But unless significant gains can be made in the -weight and power requirements with no increase in overall system complexity for reaction spheres over reaction -wheels, their utility as a momentum control device will remain limited.

b. External-moment control techniques: External moments necessary for spacecraft attitude control have to date been generated primarily by pneumatic systems (Ranger, Mariner, OGO, OAO) even though the OAO control system includes provisions for momentum dumping (desaturation of momentum storage devices) by utilizing the earth's magnetic field. Relatively short-term missions (several months for Mariner) with limited reorientation maneuvers and low-attitude accuracy requirements (a few degrees) are well suited for considering direct control methods (Ranger and Mariner spacecraft depend totally on mass expulsion for attitude control). The major limitation in the exclusive use of mass expulsion (such as pneumatic systems) for attitude control is the weight penalty associated both with relatively low specific impulse (about 60 Ib-sec/lb for nitrogen) and large tankage weight to contain the highpressure gas (about 1.5 times the weight of the gas). More efficient mass-expulsion devides such as vapor jet, exploding wire, and ion engines are in developmental stages and appear to have significant limitations in terms of thrust capability, duty cycle, or dynamic range. Nevertheless, it is feasible to consider, even with present mass-expulsion systems, the elimination of momentum storage requirements for low-accuracy control of satellites either inertially oriented or having limited maneuvering requirements. It is interesting to note that the momentum added to the AOSO satellite from the algebraic addition of external disturbances (including gravity gradient, solar-radiation pressure, and magnetic and aerodynamic torques) is about 0.1 ft-lb-sec per orbit while each 5-a^c-min by 5 arc-min scan would ideally require (if done without the reaction wheels) about ko ft-lb-sec. If a bang-bang mass expulsion system was to be designed to meet the present jitter and jitter-rate requirements (l arc-sec and 0.5 arc-sec, respectively) the impulse required for limit cycle operation 22

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

would be about k ft-Xb-sec per orbit. However, if the jitter amplitude (hence the accuracy) were to be relaxed to ^0 arcsec, the impulse required for limit cycle operation would decrease to 0.1 ft-lb-sec per orbit. Unfortunately the dynamic range of the torque to meet both the maneuvering requirements and the jitter specifications would have to be about 100 to 1, suggesting possibly multilevel thrusters. The above example illustrates both the importance of maneuvers in determining the desirability of momentum storage and the feasibility of mass-expulsion techniques with either the relaxation of control requirements or improvements in specific impulse. The use of the earth's magnetic field for direct control or momentum dumping has received serious consideration, and, of course, has been implemented on OAO. The magnetic torquing concept is attractive from both the possible savings in propulsion weight (expelled mass) and the probable improvements in reliability through elimination of the electromechanical valves and actuators. However, some limitations exist in its application to satellite attitude control aside from the variations of the earth!s magnetic field strength with altitude:

1) Since the earth's magnetic field vector, as measured in satellite coordinates, is time-varying as a function of orbit parameters and the satellite orientation, it is necessary to provide on-board devices (magnetometers) to measure the components of this vector in spacecraft coordinates in addition, of course, to normal attitude sensors. .Alternatively, magnetometers may be eliminated by utilizing ephemeris information and computing the field vector at the particular longitude and latitude of the satellite and commanding correction at appropriate times. Such an approach suffers from the magnetic field uncertainties at high altitudes (above 5&00 naut miles) and the necessity of in-line ground command link for control. 2) The torque applied to the vehicle is the vector crossproduct of the spacecraft generated magnetic moment and the earth's magnetic field vector. Hence, there is always a particular direction (along the field vector) along which no torque can be generated. Consequently, it is not possible to generate arbitrary magnitude torques about all three vehicle control axes at the same instant. If the direction of the field vector remains stationary in the orbit coordinates, magnetic control cannot be used, such as 24-hr equatorial orbits. Magnetic control for orbits near the magnetic equator (about 11-deg inclination with respect to geographic equator) may prove difficult since the orbit plane component of the field 23

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

vector remains small and may be significantly influenced by magnetic field disturbances.

Fortunately, for orbits with reasonable inclination (about 30 deg and more) the control concept is an attractive one. Since the torques about the three vehicle axes are not independent (only two can be specified arbitrarily), the magnetic control has been applied for momentum control rather than attitude (orientation) control, thus requiring momentum storage devices* The required magnetic moment can be generated by passing current through three mutually perpendicular coils fixed in the spacecraft. The control laws were originally evolved^OjSl ^-o app^y a torque continuously so as to reduce the angular momentum in a plane perpendicular to the instantaneous field vector. Since the field vector is not inertially fixed (for inclined orbits) as measured in orbit coordinates, it is thus possible to reduce the total angular momentum of the spacecraft. This mechanization, however, not only requires the computation of the vector cross-product of the momentum vector and the field vectdr (six analog multiplications), but also the generation of currents in proportion to the components of the cross-product (hence, a linear current amplifier). Specific hardware design and optimization problems for such control laws are reported-^. Later studies significantly simplified the mechanization problem by developing a control law based on simple logic, switching, but still variable, current coils^* This control law basically allows the reduction of momentum along one vehicle axis whenever the field vector is in proper orientation to affect momentum removal about that particular axis. Later studies further simplified the mechanization problem by utilizing fixed current coils, variable torquing periods, and simple field-component threshold detectors instead of magnetometers. Even though the scheme is not as efficient in terms of power utilization, it offers a great deal of mechanization simplicity.

III.

.Analytical Design Problems

The design of techniques for linear control systems is both well developed and understood and requires no further exploration in connection with spacecraft attitude control problems. Fortunately, appropriately linearized dynamical equations allow in-the-small investigation (small error regions) of the performance of the ACS and introduce physical insight for the problem at hand. Unfortunately, however, many of the significant control system design problems cannot be linearized because of the dynamic and kinematic character of the equations of motion, or because of the nonlinearity of the

24

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

controller itself. Analytical design difficulties can, then, "broadly be categorized as "being either the result of dynamic and kinematic nonlinearities In the equations of motion, or nonlinear!ties in the control loop. Even though the two categories are obviously not independent, each by itself presents a formidable task to the analyst.

1. Kinematic Nonlinearities The most critical problem in the first category is associated with initial stabilization of the spacecraft to the desired attitude orientation. During the act of separation of a spacecraft from its rocket booster, an arbitrary but bounded amount of angular impulse is added to the spacecraft. For spin-stabilized vehicles spun up before separation, the spin momentum can usually absorb this impulse without significant change in the total angular vector, thus providing a reasonably well-defined initial orientation region for subsequent control corrections (Syncom II, Relay I, Telstar, etc.). The problem of correcting large, initial-orientation errors for spinning satellites is discussed in Refs. 18 and 21. The addition of momentum to a nonspinning spacecraft such as OGO, AOSO, and OAO will cause the vehicles to tumble at bounded rates (depending on the separation impulse), but they will have unpredictable initial orientations. At other times during the mission, the spacecraft may lose reference because of a temporary failure and have to reacquire its nominal orientation. Since small-angle approximations are not valid, the system equations involve kinematic transformations relating the orientation of the vehicle axes to the nominal reference frame. This kinematic nonlinear!ty is much more significant than the Buler dynamic coupling terms (due to unequal principal moments of inertia) for most acquisition situations in determining the performance of the acquisition system, since generally sufficient control authority exists to overcome the dynamic coupling torques. One important exception to the statement of sufficient control authority is in control systems utilizing environmental fields (such as gravity gradient) to generate control moments. Some significant contributions have been made (if small in number) to the analytical study of in-the-large stability of the ACS. DeBra has investigated in detail the stability boundaries of a satellite in a gravitational field for various inertia distributions and orbit parameters^. Even though much of this work was done by numerical analysis (digital computation), the results provide an insight to the problem. Several investigators have considered the problem of 25

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

stabilizing a spacecraft from arbitrary initial conditions to an inertial reference (more precisely, the sun)35. Windeknecht proved the in-the-large stability of the system analytically by utilizing Lyapunov stability criteria and evaluated the influence of control system parameters (rate and position gains) on stability and convergence rate* The study assumed the availability of proportional torquers (proportional jets) and rate gyros* The attitude information was provided by sun sensors, which gave the direction cosines of the sun vector in body coordinates. We cannot yet predict the practical implications of these studies or the possible outcome of future efforts in this area. Nevertheless, such studies shed some light on one of the least understood and formalized design problems in the spacecraft attitude control field. Present-generation, nonspinning satellites and spacecraft utilize a variety of acquisition techniques and procedures* The TRAAC acquisition technique is to use the earth*s magnetic field for initial despin and for later orientation. OGO and OAO first acquire the sun (easiest reference to locate) and then search for earth and star references, respectively. Nimbus goes directly into earth acquisition from an initial orientation close to local vertical. Both OGO and OAO use a rate gyro to control rates about one body axis during sun acquisition, and in earth or star search modes. Nimbus originally had three rate gyros for acquisition and presently has none. Ranger and Mariner spacecraft acquire the sun and use rate gyros in all axes to stabilize the vehicle.

Unfortunately, attitude sensors (such as horizon, sun, or star sensors) provide limited information either due to field of view or the kinematic restrictions (for example, the sunsensor outputs are independent of the rotation of the spacecraft about the sun line), and furthermore, body rates must be readily extracted from this information for damping when the attitude errors are large. For small attitude errors, the differential of the sensor-output derived rate usually approximates actual body rates. The nature of the information obtained by the derived-rate technique is, of course, a function of the sensor output* Consider, for example, the sun acquisition problem, where direction cosines a , a , and az define the orientation of the sun vector with respect to x, y, and z body axes. If wx, wy, and wz are the components of angular velocity about the three body axes respectively and Jf the z axis is to be oriented towards the sun (that is a^ and a are the measured error signals) then

26

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

Thus, the differentiation of the direction cosine a , for example, instead of giving (-w ), as required for damping, may have any value including ^(-HW ) if a is (-l)(the sun to the ""back" of the spacecraft). During OG8 acquisition studies, it was found that the saturation of the sun-sensor outputs at about 35 deg was unexpectedly desirable since an increase in the linear range degraded the system stability instead of improving it. Of course, the saturation of the sun signal effectively caused £ix and eL to be zero for large error angles, thus providing no rate information instead of false rate information. The problem is further complicated with earth sensors, since the outputs of horizon scanners are highly nonlinear for large attitude errors, depending on the particular scanning and information processing technique. At times, this factor strains the best physical intuition for finding appropriate control lavs. 2. Control System Nonlinearities

The second class of analytical design problems is associated with control system nonlinearities . In this case, it is assumed that the spacecraft is near its nominal orientation and that usual approximations to linearize the dynamic and kinematic equations are valid; that is, the system can be analyzed in a single-axis basis. The only system nonlinearity is usually in the generation of the control moments. It should be noted that the so-called single-axis analysis is very often extended to large -attitude -error regions to include the effects of sensor nonlinearities (saturation, for example, in the analysis. In many cases, such extensions may be justified to provide an insight to the problem. On the other hand, restraint should be exercised in arriving at specific conclusions as a result of these over-extended assumptions. Again, the assumptions cannot be justified by mere consideration of Euler dynamic terms since kinematic nonlinearities in general may be more significant.

Because of reliability considerations, a great deal of emphasis has been placed on the use of fixed -thrust, on-off controllers in spacecraft attitude control systems. The concept of on-off control has extended from pneumatic (or massexpulsion) systems to reaction wheel control^ and magnetic momentum-dumping schemes. Specifically, the pneumatic control 27

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

systems have operated almost exclusively on the bang-bang principle "by turning on a gas jet whenever the attitude error signal (properly lead-compensated for stability) exceeds a threshold level, and turned off when the signal falls back within the dead zone. In the absence of external disturbances, such controllers operate in a limit cycle with rates dependent on the compensation-time constants, delays in the thruster loop, and the hysteresis in the on-off switch. The torque levels for bang-bang controllers are determined primarily either by initial stabilization requirements (OGO, OAO) or by the expected external disturbances due to guidance rocket firing (Mariner, Ranger) or both. During the normal operating modes, however, these torques are usually several orders of magnitude greater than they need be. Since the impulse expenditure in limit cycling is proportional to the square of the limit cycle rates for a given dead band, several investigators have studied possible mechanizations to reduce the limit cycle rates. The fundamental limitation, of course, is the minimum possible on-time for the valves themselves. Hence, efforts have been directed toward achieving control in the limit cycle regions with minimum valve-on times.

The so-called logically controlled pulse schemesS^ provide a reasonable but very limited approach to the problem, since the system behavior away from the limit cycle regions cannot be well-defined and generally results in relatively complex system configuration37 based on arbitrary constraints and optimization criteria. Ths most promising alternative to logically controlled pulses lies in the utilization of pulse modulation techniques where the control system commands fixedanrplitude pulses of either varying width and constant frequency (pulse width) or varying frequency and constant width (pulse frequency modulation). Thus, the following comments on the analytical design of nonlinear controllers will be limited to the bang-bang and pulse modulation techniques. .Analytical tools for determining the limit cycle behavior of bang-bang systems for spacecraft applications are well documented^. Additionally, much work has been done in the field of contactor servos primarily for optimum control (minimum time for example), even though the latter techniques have not found any significant applications in spacecraft attitude control. Two basic problems associated with optimum control are the definition of the optimization criteria and the problem of mechanization. The minimum-time criteria establishes the control laws to force the system from any initial state to its final desired state (a single point in phase space) in the least time. It does not, for example, take into account 28

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

the behavior of the system near the desired final state (limit cycle region) due to system imperfections and measurement limitations, which may "be a far more critical performance factor than the time to converge. Unfortunately, the minimum-impulse (hence, minimum-fuel) criteria for a singleaxis control system provides trivial solutions unless constraints are established on time-to-converge as well as on the maximum effort (maximum available acceleration). A general criterion for performance optimization (other than qualitative) cannot even be defined for most attitude control applications, let alone formulated. Particularly, weight, power, and reliability factors normally overpower other optimization considerations, allowing the system analyst limited design choices to optimize performance.

The approach in the development of pulse modulated controllers, however, has been somewhat different. The investigators have selected certain attractive (either because of mechanization ease or because of some intuitive arguments) control laws and developed the techniques for the analysis of their performance in terms of the system parameters. In spite of controller nonlinearities, it has been possible to prove analytically the stability of the system, the convergence rate, and the limit cycle behavior for certain pulse modulated controllers. The most recent contribution to pulse modulation techniques for attitude control is described in Ref. 25. The so-called integral pulse frequency controller has not only distinct performance advantages (quasilinear operation, and small limit-cycle impulse utilization and noise rejection) but also is a system for which complete, simple, design procedures have been developed after a lengthy and complex analytical process. Thus though it may not be the best control law relative to certain performance criteria, it is well-understood. Then as similar investigations yield results for different control laws,it will be possible to select the best mission-oriented control system. IV. Spacecraft Testing 1. Testing Objectives Realistic performance testing of spacecraft attitude control systems has been the subject of considerable discussion, study, and experimentation. The major goals of these tests are: the evaluation of the control system performance in its various modes of operation; the utilization of the test results as a measure of performance acceptability; and, possibly, the evaluation of the effects of various failure modes 29

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

on spacecraft performance. The implied assumption is that the expected dynamic as veil as the ambient environment can reasonably be duplicated in ground testing to provide useful performance information.

First, consider the type of information that can "be obtained if there were complete flexibility in the design of a ground performance test facility: l) detection of engineering errors in translating the system design into an integrated control system (such as polarity errors); 2) component malfunctions such as faulty sensor operation; 3) effects of signal interference and noise; h) functional integrity of the control system in providing logical operation and mode switching; and 5) control system performance in terms of stability, impulse consumption, limit cycle operation, and response to perturbations. 2.

Present Test Approaches

Next consider, in the order of their complexity, the present test approaches that have been devised to evaluate functional integrity and performance of a control system:

1) Static end-to-end tests in integrated spacecraft or the control system by itself* In these tests, the response of the control system to various sensor stimuli can be measured and compared with the predicted results. It satisfies 2), 3); and *0 completely, l) only partially since under special circumstances, polarity errors may not be detected; and, of course, 5) not at all.

2) Single-axis dynamic tests where one-control-channel operation can be examined. Although this test satisfies only 5) partially and none of the others, it may give significant

design information during the development of the control system.

3) Three-axis dynamic tests of a scaled-down control system on an air-bearing table. This test satisfies all test objectives except 5) (and possibly 3); since in most cases realistic impulse consumption and limit cycle operation cannot be obtained due to the scaling of the control system components such as pneumatic jets and reaction wheels). h) Three-axis dynamic tests of full-scale, integrated spacecraft on an air-bearing table. This test would satisfy ail objectives on the assumption that it can be implemented reasonably and made to work, (Since the ACS must provide the 30

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

control torques for properly orienting the air-bearing table, the effects of environmental disturbances (particularly the gravitational torques due to mass unbalances) may become intolerable.) 3. Approaches for Dynamic Testing of Integrated Spacecraft It should be noted that the static end-to-end and three-axis scaled-down dynamic tests -would satisfy all test objectives except the system impulse consumption* This exception, however, has little significance since the expected disturbances in space environment cannot be reasonably simulated in any air-bearing test setup* The one significant shortcoming to this approach is that it does not provide for the dynamic testing of an integrated spacecraft. Whether the dynamic testing of an integrated spacecraft, in addition to static and scaled-down tests of a control system, is worth the additional facility and test costs is. a difficult question and not yet answered by quantitative arguments* Assuming that such tests are desirable or necessary, several alternatives to the air-bearing testing can be envisioned. First, it is clear that the major problem associated •with full-scale airbearing tests can be eliminated if the ACS does not have to supply the control torques for moving the test bed. Two possible test approaches can then be formulated: a) Fixed-bed testing: In this case, the spacecraft remains fixed at all times and the stimuli for the sensors can be servoed to satisfy the proper geometric relations between the spacecraft and the stimuli (such as the sun and the earth). The spacecraft dynamic equations, together with necessary kinematic relationships, are programmed in a computer. The computer measures the control action (for example, the speed of the reaction wheels or the gas-jet firings) and solves the dynamic and kinematic equations to determine the effects of the control action on relative orientation of the spacecraft and the stimuli, and drives the stimuli to their proper position with respect to the spacecraft. The motion of the stimuli with respect to the spacecraft-mounted sensors will, of course, generate error signals for the control system, and thus the loop is closed. It should be noted that the computer can include all necessary disturbance torques in the dynamical equations with little difficulty. Note also that in such a test setup it is possible to test not only gravitygradient stabilization systems with active damping but systems employing magnetic torquing. Its first drawback, however, is in the lack of visual observation of the spacecraft going through its various modes of operation. In addition, the 31

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGIN

stimuli drive mechanisms are likely to be complex to accommodate, for example, eccentric orbits (variable earth size) and variable earth-sun-star geometry, and to compensate for the parallax effects of the stimuli with respect to the sensors. This results from the fact that stimuli-sensor distance is not large enough compared to the spacecraft dimensions as is the case, of course, in orbital operations. To eliminate parallax, collimated beams in the order of spacecraft dimensions can be obtained for weak -point sources, such as the stars, by standard optical techniques but present a significant problem for the sun and the earth. Hence the parallax compensation for the latter sources can best be achieved by moving the stimuli. In general, the actual spacecraft telemetry channels can be utilized to measure the control action for computing the stimuli drive equations. b) Servoed-bed testing: Another alternative method to full-scale air-bearing tests is to utilize a secondary control system to drive the test bed on which the spacecraft is mounted. This concept is very similar to fixed-bed testing except that the computer generates drive equations for the test-bed servo system rather than for the stimuli. It is important to note that the parallax problem still exists, and it may be necessary to provide limited stimuli motion to compensate for it. In this case, the test bed need not be supported by an air-bearing system and, in fact, may be easier implemented if a gimballed bed is used. Because of the necessity of stimuli motion to compensate for parallax, such a test facility, like all the others, cannot be considered a "general" test facility, but must be tailored to the specific configuration of the spacecraft to be tested. Nevertheless, it does offer a greater flexibility in test design. It is difficult to assess the merits of these alternatives in terms of cost, test flexibility, and the design tradeoffs •without detailed studies. It would, however, be interesting to classify and document the information obtained from dynamic tests of the spacecraft control systems to date that could not have been obtained by static or scaled-down tests, or both, and thus provide reasonably quantitative arguments for full-scale dynamic tests.

The test requirements and objectives for a manned spacecraft mission such as for Project Apollo are, of course, significantly different. Each phase of the mission (Little Joe flights, orbital rendezvous, and lunar flight) can be considered as a test for the following phase, thus providing multilevel inflight test capabilities. There are extensive 32

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

dynamic ground test requirements as veil, particularly for the simulation of lunar landing conditions. Definition of an overall test philosophy, and the design and developmental problems associated with the ground test facilities have not "been yet documented in the open literature and thus cannot "be commented upon. References

Space Technology Lab., "Space log " (June 1963)* o

Dzilvelis, A., "Satellite attitude control systems," Astronaut Aerospace Eng. I, 78-82 (March 1963). ^ Haeussermann, W., "Recent advances in attitude control of space vehicles," ARS J. 32, 188-195 (1962). k

Tuchyner, H. J., "Optimization of a mercury reaction flywheel system for three axis attitude control," AIAA Summer Meeting, Preprint 63-213 (June 1963).

Whitford, R. K., "Design of attitude control systems for earth satellites," Space Technology Lab. Rept.23l3-0001-RU-000 (June 1961). Patapoff, H., "Attitude drift of a spin-stabilized satellite due to the earthTs magnetic and gravitational fields," XlVth Intern. Astronaut. Congr., Paris, France (October 1963).

7

Fishell, R. E. and Mobley, F. M., "A system for passive gravity-gradient stabilization of earth satellites," AIAA Guidance and Control Conference Paper 63-326, Mass. Inst. Tech. (August 1963). o

Sohn, R. L., "Attitude stabilization by means of solar radiation pressure," ARS J. 29, 371-373 (1959). Q

Acord, J. D. and Mcklas, J. C., "Theoretical and practical aspects of solar pressure attitude control for interplanetary spacecraft," Jet Propulsion Lab. TR 32-^67 (June 1963).

33

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERG1N

Magnus, K., "Rotational motions of satellites in a radial gravity field/' Center of Research Eng. Sci . Univ. Kansas (June 1963).

Roberson, R. E., "Gyroscopic control of satellites using integrating gyros," Proc. First Symp. Rockets and Astronaut, Tokyo, Japan (1959). 12

Burt, E. G. C., "On the attitude control of earth satellites," Eighth Anglo-American Aeronautical Conference, London (September 1961). ^ White, J. S. and Hansen, Q. M., "Study of a satellite attitude control system using integrating gyros as torque sources," NASA TDN-1073 (September 1961). Scott, E. D., "Control moment gyro gravity stabilization," AIAA Guidance and Control Conference Paper 63-32^, Mass. Inst. Tech. (August 1963). 1

^ DeLisle, J. E., Ogletree, E. G. and Hildebrant, B. M., "Applications of gyrostabilizers to satellite attitude control," AIAA Guidance and Control Conference Paper 63-325* Mass. Inst. Tech. (August 1963).

Cannon, R. H., "Basic response relations for space vehicle attitude control using gyros," Joint Automatic Control Conference, Minnesota (June 1963)• 17 Le Compte, G. W., Bland, J. G., "Simply mechanized attitude control for spinning vehicles," AIAA Guidance and Control Conference, Mass. Inst. Tech. (August 1963). 18

Windeknecht, T. G., "A simple system for sun orientation of a spinning satellite," ARS Preprint 6l-20*l~l898, Joint IAS-ARS Meeting (June 196l). "^ Reid, H. J., Garner, H. D., "Miniguide - A simplified attitude control for spin-stabilized vehicles," AIAA Summer Meeting, Los Angeles, No. 63-210 (June 1963).

34

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

20

Wheeler, P. C., "Two-pulse attitude control of an asymmetric spinning satellite," AIAA Guidance and Control Conference Paper 63-338, Mass. Inst. Tech. (August 1963). 21 Patapoff, H., "Bank angle control system for a spinning satellite," AIAA Guidance and Control Conference Paper 63-339, Mass. Inst. Tech. (August 1963). 22

Loebel, M., "Stabilization and control of a rotating manned space station - A total system concept," AIAA Guidance and Control Conference Paper 63-3^0, Mass. Inst. Tech. (August 1963). po J

Ergin, E., Norum, V, and Windeknecht, T,, "Techniques for analysis of nonlinear attitude control systems for space vehicles," Aeronaut. Systems Div., ASD-TDR-62-208, Vols. I-IV (June 1962). 2^ De Bra, D. B., Cannon, R. H., "Momentum vector considerations in wheel - set satellite control system design," ARS Guidance and Control Conference, Stanford Univ. (August 1961). 25

^ Farrenkopf, R., Sabroff, A. and Wheeler, P., "Integral pulse frequency on-off control," AIAA Guidance and Control Conference Paper 63-328, Mass. Inst. Tech. (August 1963). Lopez, A. and Ratcliff, J., "Results of studies on a twin gyro attitude control system for space vehicles," AIAA Guidance and Control Conference Paper, Mass. Inst. Tech. (August

1963). 27

Reeves, E., Otten, D. and Martin, L., "Design of an attitude control system for an earth-sun oriented satellite," Intern. Conf. Satellite Commun., London (December 1962); also Space Tech. Lab. Rept. 2313-6003-RU-OOO. White, J. S. and Hansen, Q. M., "Study of systems using inertia wheels for precise attitude control of a satellite," NASA TN D-691 (April 1961).

29 Roberson, R. E., "Dynamical model for fine pointing attitude control of the orbiting astronomical observatory," Joint Automatic Control Conf., Minnesota (June 1963). 35

Purchased from American Institute of Aeronautics and Astronautics

E. I. ERGiN

White, J. S., Shigemoto, F. H. and Bourguin, K., "Satellite attitude control utilizing the earth's magnetic field/1 NASA TN D-1068 (August 1961). •* Buckingham, A. G., "A nev method of attitude control utilizing the earth's magnetic field for long life space vehicles," ARS Guidance, Control, and Navigation Conference, Stanford Univ. (August 1961). og J

Braumiller, J., et al, "Magnetic torquers for space vehicle control," Aeronaut. System Div. ASD-TDR-63-7^ (January 1963). oo JJ

McElvain, R. J., "Satellite angular momentum removal utilizing earth's magnetic field," Am. Astronaut. Soc., Goddard Memorial Symposium 62-53 (March 1962). ok

De Bra, D. B., "The large attitude motions and stability, due to gravity of a satellite vith passive damping in an orbit of arbitrary eccentricity about an oblate body," Stanford Univ., SUDAER 126 (May 1962). 0

35 Windeknecht, T. G., "On design of in-the-large stability in spacecraft attitude control," Space Tech. Lab. Kept, 93H-6001-KUOOO (July 1962). 3 Gaylord, R. J. and Keller, W. N., "Attitude control system using logically controlled pulses," ARS Guidance and Control Conference, Stanford Univ. (August 1961).

37 Abzug, M. J., Martin, L. W. and Ohgi, F. T., "A time optimal attitude control system designed to overcome certain sensor imperfections," AIAA Guidance and Control Conference Paper 63-329, Mass. Inst. Tech. (August 1963).

36

Purchased from American Institute of Aeronautics and Astronautics

A SYSTEM FOR PASSIVE GRAVITY-GRADIENT STABILIZATION OF EARTH SATELLITES Robert E. Fischell* and Frederick F. Mobley/ The Johns Hopkins University, Applied Physics Laboratory, Silver Spring, Md. Abstract The possibility of using the earth's gravity field for vertical stabilization of near-earth satellites has intrigued theoreticians for a number of years. The very small stabilizing torque available, and the lack of a natural damping mechanism have been recognized as the major problems. The satellite 1963 22A was designed to solve these problems. This satellite was launched into a polar orbit at a ^00 naut mile mean altitude with an eccentricity of 0.0032 in mid-1963. The initial spin of the satellite was removed by mechanical and magnetic despin devices. An electromagnet was then energized within the satellite to cause it to be aligned along the local magnetic field direction. Twelve hours later when the satellite passed over the north magnetic pole it was vertical with the correct side facing earthward. A 100 ft boom was then extended from the satellite and the magnetic dipole was turned off. The satellite then librated with an initial peak amplitude of approximately ^5°« Both in-plane and cross-plane oscillations were observed. A very weak spring with a mass at its end was then deployed from the end of the boom. The libration motions of the satellite caused the spring to extend and contract and thereby absorb energy due to mechanical hysteresis in a cadmium coating on the spring wire. Damping of the satellite librations was also accomplished by means of magnetic hysteresis rods which were previously used to remove the spin of the satellite and to damp its oscillations about the local magnetic field direction. Within a week after boom deployment the peak angle of oscillation of the satellite was damped below 10°. Presented as Preprint 63-326 at the AIAA Guidance and Control Conference, Cambridge, Mass., August 12-1^4-, 1963^Principal Staff Physicist. /Senior Engineer. 37

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

Vertical stabilization of a satellite is achievable by means of the earth's gravity-gradient. It is desirable and achievable to obtain gravity-gradient stabilization by employing only passive control and damping devices.

There are numerous advantages for satellites that are aligned vertically with the same side continually facing the earth. Probably the greatest advantage is that a directional satellite antenna can be utilized to enhance the signal strength of radio transmission both to and from the orbiting satellite. In a similar manner, improved optical tracking of a brilliant flashing light, such as used on the AJMA satellite, can be obtained by directing the light only in a downward direction. The power gain (compared to an isotropic radiator) that can be realized by means of a gravity stabilized satellite which sends all its radiation to cover the earth from horizon-to-horizon is shown in Fig. 1. At the altitude of a synchronous satellite (2^ hr orbital period) gravity stabilization is almost a requirement for efficient communication to and from the satellite. Gravity-gradient stabilization offers many advantages for earth observations by means of cameras on an orbiting spacecraft. For example, the quantity of usable pictures produced from a meteorological or surveillance satellite can be considerably increased when the cameras are always directed toward the earth's surface.

Several scientific experiments to study corpuscular and electromagnetic radiation are more profitably performed on a vertically oriented spacecraft. A less obvious advantage is that gravity stabilization can be employed to improve the operation of solar cell power generating systems. With a system of gravity stabilization, it is possible to design the satellite so that the projected area of the solar cells is inversely proportional to the fraction of an orbital period in sunlight.1 This assures that the average electrical power generation is constant irrespective of the fraction of the time that the satellite is in the sun. This same principle of a change in projected area depending on the percentage of time the satellite is illuminated by the sun can be applied to enhance the thermal design of a satellite.-1Theory of Gravity-Gradient Attitude Stabilization

The principle on which gravity-gradient attitude stabiliza38

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

tion is "based is quite simple. A considerable nuniber of theoretical papers have been -written on the subject. Putting these ideas into practice is, however, a most difficult engineering problem. Assuming a satellite with cylindrical symmetry about its Z axis, with moments of inertia Ix and Iz, the gravity- gradient torque is given by

(I - I) sin 2 0

T = where uo and

(dynes-cm)

(l)

= orbit angular rate (rad/sec), 9 = angle between the satellite's Z axis and the local vertical.

Since the satellite has cylindrical mass symmetry, Ix = I . To develop a substantial torque it is necessary that I be very much greater than Iz. We also see that the gravitygradient torque is less effective for satellites at very high altitudes where the orbital period is very great and therefore 2 U)Q is very small. The natural period of oscillation (libration period) of a gravity stabilized satellite is given by

»0

(sec)

(2)

(sec)

(3)

in the plane of the orbit; and by

TI = ——— . n tt)O Y! - IZ7/IX 7

in the plane perpendicular to the orbit. For a satellite having orbital period of 100 min and having Ix » Iz we find that Til = 57.8 min and Tj_ = 50.0 min. These very long libration periods, when combined with the trivial torques that are available, make damping of the satellite oscillations a most difficult problem.

Procedure for Achieving Stabilization For an earth satellite to achieve passive gravity- gradient stabilization it is necessary to follow certain procedures. 39

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND F. F. MOBLEY

These procedures will of course differ somewhat for various satellite missions, "but some problems common to all will "be discussed herein. It should be presumed that the long extension, or boom, that is required to alter the mass distribution of the satellite will be extended after the satellite is in orbit.

The first thing that must be accomplished is to remove virtually all the spin that may have been imparted to the satellite during the launch procedure. A device which rapidly removes the spin energy of a satellite is the so-called "yoyo" consisting of two weights attached to cables which are wrapped around the satellite.5 When the weights are released they spin out from the satellite causing a tension in the cables which results in a retarding torque on the satellite. This device has been successfully employed on three Applied Physics Laboratory satellites as well as on several Tiros satellites. To guarantee the very low angular rates that are required for erecting a comparatively weak extendible boom, one can employ magnetic hysteresis rods.° By rotating in the earth's magnetic field these rods remove the spin energy of the satellite because of their magnetic hysteresis loss. Magnetic damping has been successfully employed for removing the spin energy of the ANNA and other satellites. The 13 satellite employed both "yo-yo" and magnetic despin devices to eliminate unwanted spin.5 For the first seven days, magnetic rods mounted perpendicular to the satellite's spin axis created a retarding torque by magnetic hysteresis loss reducing the spin rate from 2.80 to 2.60 rps. On the seventh day the Tfyo-yo" despin weights were deployed reducing the spin rate from 2.6 rps to approximately 0.08 rps. The magnetic rods reduced the remaining spin to approximately two revolutions per orbit (rpo) in a period of less than 10 days. This final spin rate of less than 0.001 rps is quite slow enough to allow the deployment of an extendible boom. The next procedure is to align the satellite vertically with the correct side facing downward.. This can be accomplished by energizing an electromagnet rigidly attached to the satellite. The direction of the resulting magnetic dipole moment is along the satellitef s Z (symmetry) axis. It can be shown that the satellite will then align its Z axis along the local magnetic field direction.f The magnetic hysteresis rods that were used to remove the spin energy of the satellite will also damp the oscillations of the satellite about the local magnetic field direction.7 A magnetically stabilized satellite over the earth's magnetic pole will be stabilized along the local vertical (which is the direction of the local magnetic field) with a particular, pre-determined face of the satellite 40

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

directed downward. The tumbling rate of the satellite at this time will be 1.5 rpo. 7 The satellite is now in a most advantageous condition for capture into gravity- gradient attitude stabilization . The boom would then be erected and the electromagnet turned off by radio command from a ground station. The satellite will then have its tumbling angular rate reduced by the ratio of the satellite's moment of inertia after the erection of the boom compared to the moment of inertia before. For a typical satellite design, the moment of inertia might be increased by a factor of 100, resulting in a decrease in the satellite's tumbling rate to 0.015 rpo, which is essentially stopped in inertial space. In order to be vertically stabilized the satellite must then achieve a tumbling rate in inertial space of 1.0 rpo. Immediately after the boom is erected the satellite continues in its orbital motion with its Z axis essentially fixed in inertial space. As the satellite moves away from the magnetic pole, a gravity- gradient torque will act upon it tending to align the Z axis along the local vertical direction. The angle with the local vertical will continue to increase until the gravity- gradient torque causes the satellite to develop an angular rate of 1.0 rpo. The satellite angle with the vertical will then decrease as the gravitygradient torque continues to act, resulting in a planar libration motion of the satellite. This is illustrated in Fig. 2. The maximum angle developed with the local vertical is of great interest. If this angle is less than 90°, capture of the satellite into gravity- gradient attitude stabilization will result. The angle can be calculated rather simply by equating the angular kinetic energy that the satellite must develop (to achieve an angular rate of 1.0 rpo) to the work done by the gravity- gradient torque as the satellite moves

out to that maximum angle.

The satellite angular kinetic energy when it achieves an angular rate of 1.0 rpo is given by

K.E. = I Ix ci£

(ergs)

(1*)

The work done on the satellite by the gravity- gradient torque is given by

W = J m Td9

(ergs)

41

(5)

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

where

0. = initial angle off the vertical,

and

0 = the maximum, angle with the local vertical direction to which the satellite will swing.

Taking the expression for T from Eq. (l) gives W =

tt

(Ix - Iz)

sin 20d0

(6)

9

i

W=

I ^o

(l

x "Iz) (cOS 20i "COS 2V

Equating (h) and (7) and solving for 0 yields the result

m

arc cos

cos 20. - •=2

3 IX - I,

(8)

For 0. = 0, 0 = 35.36° which is well below 90° and the satellite will therefore be captured into the vertical stabilization condition. Furthermore, the maximum angle to which the satellite will librate is independent of the orbital period (and therefore independent of the satellite altitude). For Ix » Iz , the maximum angle is also independent of the values of I and I . When the satellite is not exactly aligned along the local field direction when the boom is erected, the satellite will swing out to a larger angle with respect to the local vertical direction. The result of 0 as a function of 0. i for Ix » Iz is shown in Fig. 3- It is apparent that 0m increases as the absolute value of 0. increases; i.e., for

positive or negative initial deviation angles off the vertical, the maximum angle 0 will increase. The limiting angle for capture, 7 0 = 90°, 7 occurs at 0. = 5^-°m i The previous discussion describes the capture process for a satellite whose orbit takes it over one of the earth's magnetic poles. This would include orbits with inclinations between 7^-° ancl 90° f°r capture over the north magnetic pole and between 68° and 90° for capture over the south magnetic pole. This capture process can be accomplished using the forementioned method for satellites with a considerably lower inclination, but oscillations perpendicular to the orbital

42

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

plane will result. These oscillations would then have to "be damped out just as the initial oscillation in the plane of the orbit would "be damped out. The technique of initial alignment along the earth's magnetic field is quite practical for orbital inclinations as low as 30°. At 30° N latitude, (and 8U° W longitude) a magnetically stabilized satellite would only be 27° off the local vertical direction. After the boom is extended, a satellite captured at this point would have an initial cross-orbit oscillation of 27° as well as the initial oscillation in the plane of the orbit of 35.36°. An alternate scheme to that described above would be to allow the satellite's Z axis to drift in a random manner and merely wait to erect the boom by a command from some station when the Z axis was observed to be within some acceptable angle (e.g. 30°) with respect to the local vertical. Another possibility would be to use a system of reaction wheels or gas jets to obtain the initial orientation that is required for capture. The next procedure, damping of the satellite librations, is undoubtedly the most difficult that must be accomplished to achieve gravity-gradient stabilization. Damping is required to remove initial librations and to reduce the effect of perturbing torques and impulses. Several methods have been suggested for this purpose."^9 There are undoubtedly numerous possibilities for achieving damping. A most promising method, the use of an energy dissipating spring, was suggested by R. R. Newton of the Applied Physics Laboratory. For the satellite 1961 a7]2, damping of the libration motion was to be accomplished by this ultra-weak spring fastened to the end of a long boom.10 As the satellite oscillates about the local vertical, the radial force in the direction of the boom varies due to the difference in gravity-gradient force as a function of the angle 9 plus an additional force contribution due to the ~?

term originating from the librational motion. This varying force causes the spring to move in and out. The libration energy is then absorbed by mechanical hysteresis in the spring. Fig. k is an artist's concept of the satellite 1961 oT]2 in orbit with the boom and spring extended. The spring with its associated end mass, is held rigidly to the end of the boom during the launch and magnetic stabilization phases by means of a block of subliming material. After the boom is erected this material sublimes gradually. First to be released is the 43

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

mass that is attached to the end of the spring. The gravitygradient force then acts on this mass and tends to pull the spring out. The spring is released from the subliming material one coil at a time. Although the spring had zero length in its equilibrium position in a zero g field, the gravitygradient force acting on the end mass was to extend the spring to a nominal length of approximately ^0 ft. Due to a mechanical problem the "boom failed to extend and gravity stabilization was not achieved with this satellite. -LO

Analysis of Libration Damping R. R. Newton and J. L. Vanderslice of the Applied Physics Laboratory, and B. Paul of the Bell Telephone Laboratories^1""-^ have analyzed the motion of a boom and spring system. The theoretical results of these investigators indicate that the spring provides considerable damping of the librations for the gravity- gradient stabilized satellite.

Reference 11 provides a theoretical analysis of the motions in the orbital plane of a satellite with damping spring and end mass. A viscous damping approximation is used for the damping effect of the spring. When the satellite motions are completely damped, it is shown in Ref . 11 that the equilibrium length of the spring due to the gravity- gradient force is given by: 3Lm m2

o

k - 3m CD

where

L = length of the boom, k = spring constant of the damping spring,

and

m = mass at the end of the spring.

2 It is obvious from Eq. (9) that k must be greater than 3m ^ or else the spring would continue to extend indefinitely (assuming k remains constant).

The results of Ref. 11 indicate that there are three normal modes of oscillation as illustrated in Fig. 5. In all cases when the libration angle is maximum, the spring is at its equilibrium length. Mode 1 provides the lowest frequency. This mode is characterized by having the angles 9 and \|r inphase, with & (the spring length) decreasing at the moment 9 is a maximum ( 9m) . In mode 2, 9 and i|r are out-of -phase and I

44

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

is increasing at the time 0 = 9 . Mode 3 is characterized by

m

9 and ^ out-of-phase with JL decreasing when 9 = 9 .

For I » I , the frequency of each mode varies as a x z function of spring constant, orbital period, boom length, and mass at the end of the spring. A summary of these theoretical results follows: /" P Mode 1: oo-, varies from 0 to oo /3 as k/mco varies from 3 to °°.

Since k/muo = oo corresponds to no spring, only one mode of oscillation would remain, and the frequency predicted here coincides with the well-known libration frequency of a rigid "dumbbell" shaped satellite. r P 2 Mode 2: varies from co v 7 to °° as k/moaJD varies from 3 to O

Mode 3:

CJD~ varies from oo through a minimum back to oo as k/ moo varies from 3 to oo. The minimum is approximately

Vs Minimizing the time to damp the most poorly damped modes leads to optimum parameter values of

mL I

mco

1.3

= 27

(10)

(11)

Letting y = the fraction of peak spring energy dissipated in each cycle of spring motion, the theoretical time to damp to 1/e ' of the initial value of 9 m is about l.h days, for Y = 0.50, for a satellite with orbital period of 100 min.

J. L. Vanderslice has developed a comprehensive numerical analysis of the dynamics of this problem (Ref. 13). His equations of motion are accurate to the second order in the dependent variables, and include seven degrees of freedom covering combined in-plane and cross-plane oscillations, rotation about the symmetry axis, and spring wind-up. A

45

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

viscous model was used for the mechanical hysteresis of the damping spring, and a simple mathematical model was used for the magnetic hysteresis action. The analysis is primarily f-} concerned with the special case mL /I = 2 and Y = 0.15. x Major conclusions with respect to design parameters are: 1) The damping spring constant should he chosen so that

k/mu) lies between 6 and 20. ' o

2) A spring damping constant y = 0.15 or greater is desirable, "but even 0.08 is adequate. 3)

For a 100 ft "boom length and I

o = 500 slug-ft , an end

mass of 3 l^s "was found adequate. Small changes from this value showed no appreciable effect.

U) Magnetic hysteresis rods are essential for good damping of cross-orbit librations. Rods should be placed in a plane in the satellite normal to the boom axis in two sets at right angles. The volume of each set can be in the range 5 to 35 cm.3. Rod magnetic properties should be comparable to AEM Vf50 permalloy. Fig. 6 is illustrative of typical damping curves that are obtained from the theory developed by J. L. Vanderslice. These curves show the extent to which damping can be achieved. They also show the relative effectiveness of the spring and magnetic rods for damping the librations of the satellite in the orbital plane and perpendicular to that plane.

Effect of Perturbing Torques It has been shown that a satellite having IX » IZ and with

effective damping spring and hysteresis rods will stabilize vertically along the direction of the earth's gravity gradient. To assure that this is the case for a practical satellite design one must examine all possible torques that tend to perturb the satellite off its vertical position. The principal perturbing torques are 1) magnetic, 2) solar radiation pressure, and 3) aerodynamic. Magnetic interactions are a principal perturbing torque for satellite altitudes below 1,000 miles; solar radiation pressure would be the most significant disturbing effect for a satellite in a synchronous orbit; the effect of aerodynamic drag is relatively unimportant for a satellite altitude above 500 miles.

46

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

The magnetic torque is a result of the interaction of the earth's magnetic field with any permanent or induced dipole moment from permeable material in the satellite. When hysteresis rods are employed for damping they are usually the largest source of magnetic dipole moment. The torque resulting from the interaction with the earth's magnetic field is given "by T = M H sin 9

(dyne-cm)

(12)

•where M = satellite's magnetic dipole moment (unit-pole cm), H = earth's magnetic intensity at the satellite (oersted), and cp = angle between the earth's magnetic field and the magnetic dipole of the satellite. The gravity stabilization experiment was performed on the satellite 1963 22A. (henceforth to be referred to as 22A). Fig. 7 is an artist's concept of this satellite, in orbit, with boom and spring extended. It has been calculated that the magnetic perturbing torque on the satellite 22A (altitude = tOO naut miles) results in less than 1.0° displacement of the satellite from the local vertical direction. Since the earth's magnetic field intensity varies as the inverse cube of the distance from the center of the earth, orbits considerably higher than tOO miles altitude will produce a significantly smaller magnetic perturbing torque. For a gravity stabilized satellite with an asymmetrical distribution of area about the center of mass there will be a net perturbing torque due to solar radiation pressure. The torque resulting from solar radiation pressure can be readily computed by taking a summation of the solar radiation pressure moments about the center of mass of the spacecraft. For the satellite 22A, this torque will cause approximately 0.3° displacement of the satellite from the local vertical direction. Although the radiation pressure torque is small for the comparatively short boom used for the 22A experiment, the use of much larger booms to stabilize a satellite at higher altitudes would produce appreciable solar radiation pressure torques. This perturbation can be made negligible by designing the spacecraft so that the sum of the solar radiation torques about the center of mass is essentially zero. 47

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

Aerodynamic drag can disturb the vertical orientation for satellites at low orbital altitudes. The torque due to aerodynamic drag is given "by T

a = ±1 «>A Ai PV'di

where

to)

(C.,). = drag coefficient of the i

A. = projected area of the i

surface,

surface,

p = atmospheric density at satellite altitude, v = orbital velocity,

and

d. = moment arm of the i

surface.

For the 22A satellite configuration, the aerodynamic torque for an altitude of kOO miles would result in an approximate deviation off the vertical of 0.9°. At an altitude of 300 miles the displacement off the vertical would "be approximately 10°. This effect can toe reduced "by proper design of the spacecraft. Thermal Distortion The side of the "boom that is illuminated reaches a higher temperature than the opposite side. This produces a thermal "bending of the "boom. When the "boom axis is normal to the sun line, there will "be a deflection of the "boom ends (and therefore the satellite) off the vertical direction. This angle is approximately 5° for the 22A satellite when the boom axis is normal to the sun line. Though thermal "bending is not an external perturbing torque, it is the principle effect in causing the 22A satellite to deviate off the local vertical direction.

In addition to a steady-state deflection, there is a rapid motion imparted to the boom when it leaves the earth's shadow and is illuminated with sunlight. This effect can cause the boom and end masses to vibrate at its natural frequencies. Orbit Eccentricity

An additional deviation from vertical stabilization will be observed when the gravity-gradient stabilized satellite is in an eccentric orbit. J. L. Vanderslice of the Applied Physics Laboratory has shown that, for a well damped satellite as just described, the maximum deviation off the vertical is given by 48

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

the eccentricity expressed in radians. For example, an eccentricity of 0.02 would cause a maximum deviation off the vertical of 1.15°. Design of the Satellite 1963 22A The pertinent physical characteristics of the satellite 22A are as follows :

number of magnetic rods = l\magnetic rod dimensions = 120 cm "by 0.280 cm dia

total rod volume

=29.6 cm 2

electromagnet residual strength < 10 pole- cm P I = 1 =12 slug-ft (prior to boom extension) Q

1 = 1 = 2100 slug-ft (immediately after boom extension) 1 = 1 = 720 slug-ft

O

(after deployment of the end mass)

I = 22 slug-ft2 boom length = 100 ft

damping spring constant = 2.lU x 10"

Ib/ft

zero force spring length = ij in. equilibrium spring length= 20 ft.

spring dissipation per cycle (y)

_n " °'5°

weight of end mass P mL /I

= 3«85 Ibs

km(JJ2

= 18

=1.65 (after deployment of end mass)

Magnetic Damping System: The satellite 22A employed four magnetic damping rods which were aligned perpendicular to the symmetry axis of the satellite. The rod material was permalloy with a nominal composition of ^7*5^ nickel, 52.5$ iron. The rods were annealed in 49

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

a dry hydrogen atmosphere to develop the highest possible value of initial permeability. The purpose of these rods was threefold: l) to remove the spin of the satellite about its symmetry axis, 2) to damp oscillations about the local magnetic field direction during the magnetic stabilization phase, and 3) to damp the crossorbit oscillations of the gravity stabilized satellite. Magnetic Stabilization System: The magnetic stabilization system consisted of an electromagnet in addition to the damping rods previously described. The electromagnet was fabricated with a core of unannealed iron. When energized by radio command the electromagnet develops a dipole moment of 2.5 x ICr" pole-cm. With the electromagnet turned on and the satellite spin rate less than 0.1 rpm, the satellite will align itself along the local magnetic field direction. When the command is given for the electromagnet current to be turned off, a capacitor mounted across the electromagnet terminals in the satellite causes an alternating slowly decaying current in the windings. This causes the iron core in the electromagnet to become demagnetized. The residual dipole moment of the electromagnet is less than 100 pole-cm. Since the iron is unannealed it does not develop a significant induced dipole moment in the low magnetic field strength experienced by the orbiting satellite. Extendible Boom: The extendible boom used for this satellite was built by The De Havilland Aircraft of Canada. Fig. 8 is a photograph of this unit. The boom is of the "self-erecting" type, i.e., it is deployed by the energy stored in the wound tape. The tape material is beryllium copper, 2 mils thick and the tape width (when laid flat) is 2 in. When deployed the tape forms a cylinder of 0.55 in. in diameter and 100 ft long. The weight of the boom tape is 1.35 lt>sJ "the weight of the entire unit is 3.2 Ibs.

The deployment of the boom is initiated by a radio command to the satellite. This causes the detonation of an explosive squib which unlatches the boom mechanism allowing it to deploy.

50

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

Design of the Damping Spring: The fa"brication of a damping spring and the design of a simple "but reliable means of deploying it in orMt present a difficult engineering problem. The satellite 22A employed a helical spring with an equilibrium, length under zero force of lj in. The diameter of the helical spring is 7.6 in. The spring consists of 70 turns of 0.008 in. dia beryllium copper wire. Since beryllium copper is an "excellent" spring material, it does not provide sufficient hysteresis loss. To obtain good damping, a 0.0008-in.-thick layer of the mechanically soft material cadmium was electrolytically deposited on the outer surface of the beryllium copper wire. A 0.0002 in. coating of gold was electrolytically deposited on the outer surface of the cadmium to prevent the cadmium from subliming in the hard vacuum of space. When completely fabricated, ^ annealed and coated, the spring had a constant of 2.1^4- x 10 pounds per foot. A torsion pendulum was used to measure the damping characteristics of a short sample of the spring wire. Air damping was eliminated by operating the pendulum in a vacuum chamber at pressure less than 10~^ mm Hg.^ The period of the torsional pendulum was set at 55 ^in "to closely correspond to the natural period of libration for the nominal orbit of the 22A satellite. By this method it was determined that the spring had a damping coefficient of 50$; i.e., 50/0 of the maximum energy stored in the spring was dissipated on each oscillation. This compares with an energy loss of less than l/o per cycle for the uncoated beryllium copper wire.

To prevent any tangling or other damage to this ultra-weak spring during handling and launching operations it was necessary to encapsulate the spring in a solid subliming material. The material selected for this purpose was a compound of the "benzene family known as Mphenyl. Biphenyl was also used to hold the spring end mass during the erection of the boom. After the boom was extended the biphenyl was to sublime away; first releasing the end mass, then allowing one coil of the spring to extend at a time. Complete deployment of the spring from the subliming material was to "be accomplished within a period of one day.

Attitude Detection System:

To determine if the gravity stabilization experiment is successful it is necessary to instrument the satellite with an attitude detection system. To accomplish the attitude measurements a system of magnetic aspect sensing and solar aspect 51

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

sensing was employed. The magnetic measurements were accomplished "by means of a three-axis vector magnetometer. The resolution of the magnetometers was "better than ±J-°. Sun sensors were mounted on the satellite to determine orientation relative to a line from the sun. Five sun sensors, each with angular resolution of approximately ±1.0°, were mounted to determine solar aspect relative to the +Z, +X, -X, +Y, and -Y directions of the satellite. The object of this system was to determine the satellite attitude in inertial space. The displacement of the satellite from the vertical both in-plane and cross-plane can then "be determined. The analysis of magnetometer and sun sensor data to determine angle with the vertical was performed "by J. ¥. Teener and H. D. Black of the Applied Physics Laboratory. Spring End Mass:

A flashing light was used as the end mass for the spring in order to obtain additional information on the performance of the gravity stabilization system. Separate solar cells were provided to power this unit. The flash repetition rate was slightly less than once per minute. The intensity of the light as measured at the satellite provides information as to the extension length of the spring. The system was also capable of determining if the flashing light was within a 5% 10°, or 20° half-cone angle with respect to the +Z axis of the satellite.

Experimental Results

The satellite 1963 22A was launched into a near polar orbit. Some of the orbital parameters are as follows:

perigee apogee inclination eccentricity orbital period

= = = = =

389-2 naut miles il^.5 naut miles 90.01° 0.0032 99-71 min

The small value for the eccentricity was highly desirable for a gravity stabilization experiment. When fully stabilized, with the initial libration motion damped, the eccentricity of the orbit would provide deviations off the vertical of only 0.18°. 52

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

Satellite Despin: Immediately "before separation from the rocket, the satellite had a spin rate of approximately 200 rpm. Several minutes later a ny°-y°Tr despin (as described in Ref. 5) ™"as used to reduce the spin rate to 1.02 rpm. The magnetic damping rods then reduced the spin rate to essentially zero in 59 to. Fig. 9 is a curve of spin rate as a function of time for this satellite. At 69 hr after launch time the spin rate about the Z axis was less than one revolution in 120 min. Magnetic Stabilization: The electromagnet was turned on 69 hr after injection into orbit. The satellite was nearly 90° off the local magnetic field direction when the electromagnet was turned on. As shown in Fig. 10, the satellite then proceeded to oscillate about the local.magnetic field direction with a natural period of 15 min, it-- sec, which is within jfo of the theoretical value for this magnetic field intensity, dipole moment and satellite moment of inertia. Twelve hours later the magnetic rods had damped this oscillation to a peak value of 15°. This damping agrees very closely with the theoretical value determined by the method of Ref. 7Boom Deployment and Gravity Capture: Special precautions were taken to assure that the command for boom extension would be given when the satellite was in the proper orientation, i.e., boom away from the earth, and as near vertical as possible to minimize the amplitude of oscillations. The three-axis vector magnetometers aboard the satellite provided a convenient method of detormining the satellite orientation relative to the earth magnetic field. At high latitude the magnetic f j.eld approaches the vertical and can be used to indicate satellite orientation. Fig. 11 shows the variation of this angle with time during the pass when the boom was extended. During the pass the magnetometer outputs were read and converted into an angle of the satellitef s symmetry axis off the magnetic field direction. This curve is also shown in Fig. 11. Making a "worst case" assumption, the angles were added to get a net angle of the symmetry axis from the local vertical direction (also plotted in Fig. 11). As the satellite was observed to be approaching the vertical the command for boom extension was given. At that time the satellite was within 1.0° of the local magnetic 53

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

field direction, and within 12° of the local vertical.

Fig. 12 shows the telemetry record of the "boom extension process. Upon transmission of the command, the TM voltage changed from -.19 v to -.10 v to indicate release of the latch, then changed to +.31 v indicating initial extension of the "boom. Thereafter there is a cyclic change in voltage from .31 to .11 v for each revolution of the spool upon which the iDOom tape was wrapped. A total of 28^ revolutions were observed in an elapsed time of 77 sec, indicating a complete "boom extension of 100 ft. Fig. 13 shows the "boom extension length vs time during this process as determined from the telemetry record. Up to a"bout 25 sec the "boom extension was slightly faster than that observed in ground tests; probably because in the ground tests there was additional friction from support equipment. Beyond 25 sec the rate was slower than anticipated. The entire extension was completed after 77 sec as compared to an expected time of U8 sec. Fig. 1^ shows the spool rotation rate vs time. After 25 sec the spool rotation rate dropped to about 2 cps and became erratic indicating unexpected increased friction in the mechanism. In spite of this resistance to the spool motion, full extension of the boom was achieved. Damping Spring Deployment: On the satellite pass immediately following that in which the boom was extended, the telemetry system indicated that the biphenyl "lock" had not released; therefore, the spring deployment had not begun. This was as expected. On a pass 10 hr later the biphenyl lock was observed to have been released, and signals from the flashing light system were received, indicating that the damping spring had started to deploy. The amplitude of the flashing light signals was initially quite

low indicating one of two possible situations, viz., either the spring was deployed over 100 ft, or the reflector was tipped beyond ^5° from the satellite and the satellite was receiving a much attenuated signal from the edges of the flash beam. During the next several days flashing light signals were received regularly with increasing strength indicating an approach of the spring to its equilibrium extension of 20 ft from the end of the boom. A typical light flash observed by telemetry is shown in Fig. 15Libration Damping:

It is well recognized that damping of the libration motions is the critical problem in achieving passive gravity-gradient stabilization. From the limited field of vision of a single

54

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

telemetry receiving station at the Applied Physics Laboratory one can only observe the satellite during 10 to 1^4- min of its 100 min orbit during h to 5 passes a day. As previously described, solar aspect detectors and vector magnetometers in the satellite provide data from which the satellite attitude in inertial space can be determined. By doing this many times and observing consistent approach to the vertical and maintenance of that orientation, effective damping can be inferred. Fig. l6 shows the average deviation of the satellite from the vertical as a function of time after boom erection. Fig. IT shows the in-plane and cross-plane deviation off the vertical for several times after the boom was erected. From Figs. l6 and 17 one can definitely discern that the satellite librations have damped down from an initial value of approximately ^4-5° to a peak oscillation of something less than 10°.

The damping of the satellite librations has been complicated by a comparatively high frequency oscillation of the satellite that has been observed on every pass when the satellite passed from the earth's shadow into sunlight. A probable cause of this oscillation is the extremely rapid heating of the boom tape on the side of the satellite facing the sun which causes an impulse to be imparted to the boom resulting in the oscillation of the boom and satellite. Fig. 18 illustrates this effect as observed 10 days after the boom was erected. Since the satellite is massive compared to the boom it is believed that the point where the boom joins the satellite is a node for the wave along the boom, causing maximum angular deviations of the satellite at that point. Since beryllium copper is an extremely low hysteresis loss material, this motion does not damp out in an orbital period. It is probable that the steady state thermal bending of the boom, in addition to the dynamic excitation discussed above, is keeping the satellite from more closely approaching the vertical.

Studies at the Applied Physics Laboratory (Ref. 15) have shown that the effect of thermal bending can be drastically reduced if the boom is plated with a highly reflective material so that virtually none of the energy from the sun is absorbed by the boom. It has also been determined experimentally that additional overlap of the boom tape significantly reduces the thermal bending. By a combination of silver plating and increasing the boom tape overlap, it was experimentally determined that the thermal bending decreased by a factor of at least 20. This should reduce both the 55

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

steady state and dynamic thermal "bending problems observed with the 22A satellite to a point where they produce a deviation of less than 1° off the vertical.

Conclusions The satellite 1963 22A was successfully captured into a condition of passive gravity-gradient stabilization. All elements of the stabilization system, including magnetic despin, magnetic orientation devices, the extendible boom, the damping spring, and the attitude detection system performed satisfactorily. An unexpected high frequency oscillation of the "boom and satellite system was observed which was most probably a dynamic effect resulting from thermal bending. This has caused the satellite to stabilize with a maximum deviation off the vertical of approximately 10°. The directional antennas of the 22A satellite will operate properly when the satellite is stabilized within 20° of the local vertical at all times. This objective has been successfully accomplished by the satellited passive gravity-gradient attitude stabilization system.

References Fischell, R. E., "Magnetic and gravity attitude stabilization of earth satellites," The Johns Hopkins University, Applied Physics Laboratory, Rept. CM-996 (May 1961). 2 Roberson, R. E., "Gravitational torque on a satellite

vehicle," J. Franklin Inst. 265, 13-22 (January 1958). 3

Klemperer, W. B., and Baker, R. N., "Satellite librationsj' Astronaut. Acta 3, 16-27 (1957). Nidley, R. A., "Gravitational torque on a satellite of arbitrary shape," ARS J. jJO, 203-20^- (February 1960). Kershner, R. B., and Newton, R. R., "Attitude control of artificial satellites," Space Astrophysics (McGraw Hill Book Co., Inc., New York, 1961), Chap. 1^.

Fischell, R. E., "Magnetic damping of the angular motions of earth satellites," ARS J. _31, 1210-1217 (l96l). v1

Fischell, R. E., "Passive magnetic attitude control for earth satellites," Am. Astronaut. Soc., Paper 62-8 (January 1962). 56

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

Kamm, L. J., " 'Vertistat1 — An improved satellite orientation device/' ARS J. 32, 911-913 (June 1962). Q

^ Anom., "Damping spring for gravity- stabilized satellites/1 APL Tech. Digest 2, pp. 20-21 (November-December 1962). 10

Fischell, R. E., "The TEIAAC Satellite/1 APL Tech. Digest 1, J, 2-9 (January-February 1962) . Newton, R. R., "Damping of a gravitationally stabilized satellite," The Johns Hopkins University, Applied Physics Laboratory Rept. TG-W (April 1963). 12

Paul, B., "Planar librations of an extensible dumbbell satellite," AIAA J. 1, ^Ll-Ul8 (February 1963). Vanderslice, J. L., "Dynamic analysis of gravity- gradient satellite with passive damping/1 The Johns Hopkins University, Applied Physics Laboratory Rept. TG 502 (June 1963). Smola, J. F., Schrantz, P. R., and Tossman, B.:E., "The torsional damping capacity of a thin wire and its application to the damping of a librating satellite," The Johns Hopkins University Applied Physics Laboratory Rept. CM-1039 (August 1963) ^ Private Communication with Mr. J. E. Phenix of the Applied Physics Laboratory (July 19^3) •

57

Purchased from American Institute of Aeronautics and Astronautics 28 r-

po m

n
60

Q

O DO

-75

-60

-45

-30

-15

0

15

30

45

INITIAL ANGLE OFF THE VERTICAL, B\ ( Degrees)

Fig. 3 Maximum angle off the vertical as a function of initial angle.

60

75

Purchased from American Institute of Aeronautics and Astronautics

O

n n O 8

Fig. Ij- Design configuration of Satellite 1961 c*T]2.

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

LOCAL VERTICAL

ORBITAL MOTION

MODE 1

MODE 2

MODE 3

Fig. 5 Normal modes of li~bration in the orbital plane.

62

Purchased from American Institute of Aeronautics and Astronautics

V = VOLUME OF HYSTERESIS RODS ( cm*) SPRING DAMPING CONSTANT - 15%

IN-PLANE LIBRATION (PEAK ANGLE)

O c

o

O m

o n O

05

CO

>CROSS-PLANE LIBRATION / (PEAK ANGLE)

TIME ( D a y s )

Fig. 6 Theoretical damping of satellite li"brations for a satellite -with damping spring and hysteresis.

-H 70

O

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

Fig.

F. F. MOBLEY

7 Artist's concept of 1963 22A in orbit.

64

Purchased from American Institute of Aeronautics and Astronautics

Fig. 8 100 ft extendible boom unit. 1.0 OB-

CL

£ o*

OA

0.2 •*»

1 10

90 40 TIME AFTER

50 LAUNCH

60 (HOURS)

70

*M

1 80

90

Fig. 9 Spin rate as a function of time after launch Satellite 1963 22A 65

100

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

ANGLE BETWEEN EARTH FIELD AND SATELLITE AXIS.

START PASS DAY 170 1045 UT

BOOM EXTENDED ELECTROMAGNET OFF

START PASS DAY 169,2157 UT.

ELECTROMAGNET TURNED ON

Fig.

10

Magnetic stabilization of 1963 22A path of the

projection of the earth magnetic field in the satellite X-Y plane.

66

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

4On

383634-

323028-

SATELLITE ANGLE WITH VERTICAL

2624-

22-

MAGNETIC FIELD ANGLE WITH VERTICAL

2018161412-

SATELLITE ANGLE WITH

10-

MAGNETIC FIELD

8 6 4

IO47

46

49

IO5O

51

52

53

54

55

56

57

58

59

MOO

TIME (HRS.- MIN., UT)

Fig. 11

Gravity capture of the Satellite 1963 22A.

67

Purchased from American Institute of Aeronautics and Astronautics

R. E. FISCHELL AND

F. F. MOBLEY

TM VOLTAGE

TOTAL SPOOL

REVOLUTIONS'284

SPOOL REVOLUTIONS

END OF — EXTENSION

(.31 VOLTS-NOMINAL)

.20(.11 VOLTS --M NOMINAL)

.10-

0-.10-

-LATCH RELEASE (-.10 VOLTS NOMINAL) -PRE-EXTENSION (-.19 VOLTS NOMINAL)

-.20'

TOTAL TIME FOR-* EXTENSION «77SEC.

1057:47.7 UT FINISH 1056130.7 UT START

Fig.

12

Telemetry record of boom extension Satellite 1963 22A 19 June 1963 (Day No. 170).

10080-

60BOOM EXTENSION -FT40-

20-

10

Fig.

20

30 40 50 60 TIME AFTER COMMAND -SECS-

13 Boom extension vs

68

time.

70

80

CO

s

H« 0

s*

CQ

CD O O

H CD

CD

d-

h3

vn

H

•

Cfe

O

-no

o^

P

HD

CO HO P

CD X c+ CD ^

OJ O

H- O d- 4 CD ty H' H d-

H i

CD HH ^

d-

CO O

CD

O O

H-

CK5

6-

01-

O

SPOOL

REVOLUTION

RATE

-CPS-

I

O

n O

O

n

6

O

Purchased from American Institute of Aeronautics and Astronautics

o

H-

c+

ch 0)

sr

cn P

c

o z

i

§

A318OW 'J 'd QNV n3HDSIJ '3 'H

A V E R A G E ANGLE OFF THE VERTICAL (degrees)

02,

Purchased from American Institute of Aeronautics and Astronautics

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—I

-A

IH-ORBIT ANGLE

•X CROSS-ORBIT ANGLE

3 -30

< 1.

Fig. 17

~

i

Libration motion at several times after "boom erection.

2°

40 TIME (seconds)

Fig. 18 High frequency oscillations of the satellite.

71

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

THEORETICAL AND PRACTICAL ASPECTS OF SOLAR PRESSURE ATTITUDE CONTROL FOR INTERPLANETARY SPACECRAFT James D. Acord* and John C. Nicklas* Jet Propulsion Laboratory, California Institute of Technology, Pasadena,

Calif.

Abstract This paper first briefly covers the theory of photon momentum transfer to the exposed surfaces and gives an engineering derivation of the essential equations and parameters for dealing with the resulting forces in a practical manner. Various configurations of spacecraft equipment and control surfaces are examined as to magnitudes and directions of solar pressure torques and the necessary conditions for control. A particular spacecraft configuration is chosen for further examination and analysis to conveniently illustrate the principles involved. Block diagrams and transfer functions are given for several control and damping loop configurations. Some of the major system integration considerations are discussed, with particular emphasis on a minimum-interface combination of the solar p r e s s u r e control system with an impulsive mass expulsion system. System parameter optimization by graphical parametric analysis is illustrated using the combined impulsive-solar p r e s s u r e system as an example. Finally, some of the special considerations and peculiarities involved in mechanizing such a low-torque, low-response, control system are discussed. A practical, simple, mechanization is presented to illustrate the salient points. Presented as Preprint 63-327 at the AIAA Guidance and Control Conference, Cambridge, M a s s . , August 12-14, 1963. This paper presents the results of one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, under Contract No. NAS 7-100, sponsored by NASA. -^Guidance and Control Project Engineer, Mariner Mars Spacecraft ^Engineering Group Supervisor, Advanced Systems Development Group 73

Purchased from American Institute of Aeronautics and Astronautics

J. D. ACORD AND J. C. NICKLAS

Introduction

In the literature there are many r e f e r e n c e s to the effects of the solar photon p r e s s u r e field on attitude control and flight path of space vehicles. 1, 2, 3 Although the f o r c e s are small their neglect can cause trajectory e r r o r s of several thousand miles on a planetary mission, and if the spacecraft is improperly balanced, significant increases in attitudecontrol gas consumption may result. On the basis of flight telemetry data from the Mariner II Venus mission, the spacecraft experienced solar pressure imbalance torques of at least 10 to 30 dyne centimeters per axis. These torques alone accounted for about half of the total gas consumed. In theory it is quite simple to design a passively stable spacecraft. It is necessary only to make certain that the centroid of solar p r e s s u r e on the craft is "down Sun" from the center of mass. 4, 5, 6 The action is analogous to that of a sea anchor that holds the bow of a boat into the wind during a storm. Unfortunately such a configuration is not sufficient for control. In mechanizing a system utilizing a solar p r e s sure control, the major problems fall into two areas: 1) Such a system has no inherent damping and would thus continuously oscillate about its neutral position with an amplitude depending on initial angular position and rate. Methods which have been proposed to obtain damping include some form of viscous friction working against a secondary inertia or the use of a rate feedback signal. Although theoretically sound, the practical implementation of such systems at natural frequencies on the order of several hours per cycle (typical values) presents formidable problems. 2) The neutral position or center of oscillation of the passive system does not usually lie at the null position of the primary attitude control system. Prelaunch adjustment is manifestly unfeasible due to the extremely minute f o r c e s involved. The method of solar p r e s s u r e utilization to be discussed p r e sents a practical solution to these problems. Derivation of Solar P r e s s u r e Forces

Consider a surface of area A whose normal vector is at an angle \\t radians to a collimated radiation field of power density I watts/m^ as shown in Fig. 1. Further assume that the surface reflects a fraction p, 0 < p < 1, of the incoming energy and absorbs a fraction 1-p. Of the amount reflected, consider that a fraction s, 0 < s < 1, is reflected specularly, and a fraction 1 -s is reflected diffusely or with uniform 74

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

intensity as seen from the hemisphere about the lighted side of the surface.

The incoming photons may be considered to have an equivalent mass in accordance with the famed electromagnetic m a s s - e n e r g y equivalence relationships E - me 2 . From this the photon equivalent momentum P may be expressed as P = E/c. The force developed as a result of photon momentum t r a n s f e r _ a t a surface is entirely defined by the vector relationship F = d/dt P.

The area A intercepts a beam of radiation with cross section A COSLJJ. If that beam were entirely absorbed at the surface, p = 0, then the momentum t r a n s f e r r e d to the surface in unit time d/dt P is the total momentum of photons contained in a volume of cross section A cosiji and length c. From the relation P = E/c, if P is now taken as Pv, the photon momentum in the volume of space cA cos^j, then E represents the photon energy in the same volume or E V = IA cosijj, the power density multiplied by the cross-sectional area. The power density I is the energy per unit time through unit area. Summing up for this case,

dt

= Pv = E , =*

The direction of the resulting force is the same as the direction of incoming radiation since all photons are completely absorbed. A fundamental quantity to be used extensively in subsequent analysis may now be defined: P£ = I/c is the radiation p r e s s u r e (force per unit area) acting on a nonreflective surface normal to the incoming radiation. Expressing P£ in convenient units,

Pf = 3.33 x 10~ 4 I

(1)

where P£ is now in dynes/m^ and I is in watts/m . Near Earth with I = 1.4 x 103 w/m^, the foregoing equation yields Pf = 0. 47 dynes/m 2 . As shown in the diagram in Fig. 1, the momentum of the photons intercepted by the surface will be changed in various ways depending on the surface conditions. From the foregoing discussion, if p ^ 0, the force developed on the surface would be PfA cosijfU]; where Uj is a unit vector in the direction of incoming radiation. The component of radiation force Pa due only to absorved photons may then be calculated from the relation 75

Purchased from American Institute of Aeronautics and Astronautics

J. D. ACORD AND J. C. NICKLAS

Fa = (1 - p) P f A cos^Uj

(2)

Of the fraction p of photons reflected from the surface, a subfraction sp is reflected specularly as from a mirror with the reflected beam and the incoming beam making equal angles with the normal to the surface. In this case the normal component of momentum in the reflected ray is equal and opposite to that in the incoming ray for the sp subfraction, and the tangential component is unchanged. Applying the law of momentum conservation at the surface, the component of force F g due only to specularly reflected photons is given by F

s

= 2spP,A cos^U x

n

(3)

where Un is a unit normal vector to the surface. The remaining subfraction (1 - s)p of the reflected photons is assumed to be diffusely reflected, The incoming photon momentum may be considered as stopped at the surface and subsequently reradiated uniformly into the hemisphere containing -U n . The force due to stopping the incoming photons F-QJ may be calculated from Eq. (2) by changing the expression for the photon momentum subfraction from 1 - p to (1 - s)p:

FD]. = (1 - s)p P f A cos^Uj

(4)

The outgoing momentum of diffusely reflected photons may be calculated by integrating over the hemisphere and making use of the symmetry about the normal to the surface. It is apparent that the tangential components of outgoing momentum will be cancelled due to symmetry leaving only the normal components to integrate over the hemisphere. The result is that the effective momentum transfer to the surface due to outgoing diffusely reflected photons is just two -thirds what it would have been if the same nurnber of photons were reflected along the normal. The force Fj)R due to diffuse reflection of the (1 - s)p subfraction is given by

(1 - s ) p 2 P A c o s ^ U

(5)

Resolving the forces along the direction of incident radiation into normal and tangential components and gathering terms, the total force vector F is

76

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

F =

-^-U - s) cosijj + (1 + sp) cos i (1 - sp) P-Acos^ sim|jUT

where U,-p is a unit vector^along the intersection of the surface with the plane containing Uj and U n and directed as shown in Fig. 1.

Application to the Attitude Control Problem

In a system using forces generated by solar radiation p r e s s u r e , special attention must be given to the static stability of the system and the manner in which damping can be introduced. A system will be said to be statically stable when the center of mass of the spacecraft lies between the Sun and the point of application of the resultant solar p r e s s u r e f o r c e (Fig. 2). This point of application will be called the center 'of pressure. In this case whenever the spacecraft rotates away from the stable or null position, a restoring or stabilizing force occurs. If the center of pressure lies between the Sun and the center of mass, solar p r e s s u r e will generate a destabilizing force whenever the spacecraft moves away from the null position. This is a statically unstable system. There are several ways in which a system can be made statically stable. In one technique, a reflective, vane like appendage is used as a solar stabilizer. The vane is mounted to the spacecraft on the side away from the Sun so that when the spacecraft has the proper attitude, the Sun' s rays are parallel to the vane, and no forces are generated (Fig. 3). When the spacecraft and hence the vane are at some angle with respect to the Sun's rays, a restoring force will be developed tending to return the spacecraft to the null position. In another method reflective panels have one or two degrees of freedom with respect to the spacecraft (Fig. 4). These panels are mounted in such a position that, with respect to the Sun, the resulting solar p r e s s u r e force acts behind the center of mass of the spacecraft. For a statically stable system, the stable position occurs when the center of pressure, the spacecraft center of mass, and the Sun are colinear. This will be called the zero or null position of the spacecraft. Any rotating of the spacecraft about its center of mass is denoted by 0 g . If for any reason the spacecraft is disturbed from the null position, it will oscillate about this position with an amplitude depending 77

Purchased from American Institute of Aeronautics and Astronautics

J. D. ACORD AND J. C. NICKLAS

upon the values of 9 S and 6 S at the time that the disturbance disappears. The equation of motion of the spacecraft is given by J 0 + FZ0 = 0 s s

(7) x '

where J s = moment of inertia of the spacecraft about its center of mass F = resultant solar p r e s s u r e force Z = distance between center of p r e s s u r e and center of mass. Since Eq. (7) contains no damping term, any control system using solar p r e s s u r e forces must be mechanized to include a means of damping.

Various techniques of obtaining damping have been suggested. In one method, a fluid is used to dissipate energy by means of viscous friction. Another proposed technique utilizes a rotor passing through a magnetic field to generate damping f o r c e s . More conventional means such as employing a rate measuring device on the spacecraft and actively controlling the vane position have been suggested. However, all these methods of obtaining damping have the same disadvantage: they would be very difficult to mechanize. In the technique proposed in this paper, this disadvantage is eliminated. It is only necessary that the spacecraft be statically stable. If this is the case, plots of position and velocity time are sinusoids with the velocity leading the position by 90° as shown in Fig. 5. The most effective damping occurs when a force is always in opposition to the velocity. This f o r c e F^ is shown as a sine wave in Fig. 5, and it lags 0 by 90°. One is thus led to the conclusion that damping can be obtained by using a f o r c e which lags the position by 90°. Rather than this intuitive approach, the terms of servo analysis can be used to describe the damping requirements. In the usual system the damping force is proportional to the negative of the velocity signal, i. e. , to a signal leading 0 by 90° and opposite in sign to it. This is negative velocity feedback. In a steady-state condition for sinusoidal oscillations, the signal can also be described as lagging 0 by 90° and of the same sign. This is the same as a signal proportional to F^ just described. If F^ lags 0 by less than 90°, there are times when F^ is not a retarding f o r c e but is causing the velocity to increase (Fig. 6). In this figure, the shaded areas represent the impulse that would be expended in retarding the spacecraft, and the unshaded areas under the F^ curve show the impulse that would increase the velocity. As long as the shaded area 78

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

exceeds the unshaded area, a net damping impulse remains, and the oscillations will decay to zero. It can be seen that some damping will be obtained whenever F^ lags 9 by an angle greater than zero but less than 180°. Also apparent from this discussion is that the optimum case arises when the lag angle is 90° as shown in Fig. 5.

System Integration

A quick look at some typical parameters associated with practical spacecraft and control-surface dimensions will demonstrate that, despite mathematical similarity, solar p r e s s u r e control systems require an almost entirely new technology. More will be said of this later. The item of primary interest at this point is that the extremely low t o r ques and slow response speeds of such systems make them useful mainly for long-term cruise control on interplanetary flights. The more dynamic flight phases such as initial attitude acquisition and attitude maneuvering generally require faster-acting, higher-torque systems such as conservative momentum interchange systems using torqued flywheels or gyros or nonconservative systems using hot or cold gas jets. To illustrate the aspects of system integration involved, a conservative flywheel system will be briefly considered followed by a more detailed study involving an impulsive coldgas jet system. ? If a momentum conservative primary attitude control system is used alone, the angular momentum of the total spacecraft must remain constant. Thus, during the initial stabilization process, any angular momentum removed from the body of the spacecraft must be stored on the flywheels. In addition, any external bias torques due to magnetic field interaction, gravitational gradients, outgassing of equipment or radiation torque imbalances will result in continuous acceleration of the flywheels and eventual saturation or centrifugal destruction of them. A set of solar-pressure control surfaces properly adjusted can accomplish two important tasks in such a system with quite simple control logic. The control surfaces should be continuously adjusted so that the resulting s o l a r - p r e s s u r e torque about each spacecraft axis is proportional to some function of the flywheel velocity and its integral about that axis. If the intentional imbalance torque so derived is in a direction to force reduction of the wheel velocity through the primary control system, a nearly ideal situation results. The accumulated wheel momentum is first slowly worked off against the intentional solar-pressure imbalance. The imbalance is reduced as the wheel speed is reduced until a steady-state condition is

79

Purchased from American Institute of Aeronautics and Astronautics

J. D. ACORD AND J. C. NICKLAS

reached with the wheel running at a low constant speed and the control surfaces set to balance out all external torques. Such a cooperative system has many advantages. It requires no mode-switching with attendant reliability problems; it can be a linear system that is capable of very tight angular accuracy; and it uses high-power, high-response control only when required by the control task. In considering solar-pressure control as a cooperative element with an impulsive, cold gas attitude control system., it is convenient to first define the total system design goals and from these develop the guidelines for integration. Probably the most important system design goals are the following 1) Maximum mass efficiency. This is the ratio of total control angular impulse actually required by the spacecraft mission profile to the total system weight, including fuel, allocated to perform that control task. The ratio may be considered as the product of three sub-ratios: specific impulse of fuel, mass fraction of system or ratio of fuel weight to total system weight, and the control system, design efficiency. The first two are generally familiar terras, and their product may be defined as a total system specific impulse. The third is calculated by first summing the angular impulse required to overcome all disturbances, intentional or otherwise, for the complete mission, and dividing by the linear impulse actually carried to p e r f o r m the total task. If the gas jet system is used alone, its design efficiency may be improved by such things as: a) Increasing jet lever arm to increase the ratio of angular impulse to linear impulse. b) Reducing turning rates during attitude reorientations to minimize acceleration and deceleration time. c) Increasing limit cycle size and decreasing limit cycle velocity. These have an effect only if the spacecraft is torque balanced precisely enough so that an actual limit cycle occurs. 7 Experience has shown this to be virtually impossible without some means of adjusting the center of mass or the center of radiation p r e s s u r e or both in flight. d) Reducing contingency r e s e r v e s . Obviously the ideal system would not require any expenditure of impulse to maintain either a fixed position or a constant velocity with respect to the inertial r e f e r e n c e frame. 2) Minimum electrical peak power and total energy. Electrical power requirements for attitude control may be divided into two parts. The f i r s t is a preferably small constant demand to operate sensors and standby signal processing 80

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

electronic circuitry. This is minimized by the usual circuit design techniques. The second is the incremental power required when the jet valves or other intermittent loads are

actuated.

The peak power levels of these loads are of con-

cern as well as their duty cycle or total duration which is related to the design efficiency defined above. 3) Maximum reliability. Reliability may be defined in many ways but the definition best suited to the present discussion is "performing the attitude control task for the total mission. " In addition to its dependence on the usual items of component types, counts, load factors, and equipment operating time, the reliability of an attitude control system is critically dependent on the number of actuations of intermittently operating components, such as gas valves,

relays, etc.

The task of system integration is to define the interfaces between the primary gas jet system and the cooperating

but secondary solar-pressure system so as to maximize the

degree of design goal achievement. The solar-pressure system must be regarded as secondary because it cannot perform

the total attitude control task alone whereas the gas jet system can. A properly integrated solar-pressure system can significantly contribute to design goals 1 and 3 of the f o r e -

going list while causing little or no degradation of goal 2. The following is a list of system interface guidelines and reasons for them based on the foregoing considerations:

1)

The number and complexity of electrical inter-

faces between the primary control system and the solarp r e s s u r e system should be reduced to an absolute minimum and should be designed to favor the reliability of the primary

system. 2) The most important duty of the solar-pressure control system should be to balance out all external torques

on the spacecraft. Minimum external torque results in near minimum gas consumption and number of jet actuation cycles, thus contributing to all three design goals.

3) The two control systems should be cooperative, not exclusive, in their operation. Otherwise one system

would have to be shut down during operation of the other to avoid fighting. This implies that the null or balance point of

the solar-pressure system must be adjustable to fall within

the jet system deadband.

The overall system proposed in this paper will now be examined with special reference to the aforementioned guidelines. The system is based on primary control by a minimum

impulse cold nitrogen gas jet system? similar to that used on 81

Purchased from American Institute of Aeronautics and Astronautics

J. D. ACORD AND J. C. NICKLAS

Mariner II. To the basic system is added an array of four solar-pressure control surfaces and their actuators and control electronics. Both the gas jets and control vanes are located at the tips of the solar panels to maximize lever arm length in accordance with design goal 1. Each gas jet-control actuator assembly represented schematically in Fig. 7, is a self-contained unit requiring only a dc power source, a gas supply line, and gas jet command signals to operate. The gas jets on opposing panels operate as f o r c e couples while the solar-pressure vanes operate differentially about a preset erected position. The innermost or torque balancing actuation of the vanes is driven by a stepping motor and gear train with integral electronics. The control logic is arranged so that each gas jet actuation will command one step of the vane actuator so as to increase solar pressure torque in the same direction as that of the particular gas jet impulse. Any torque bias, resulting in a preponderance of jet pulses in a single direction to combat it, will thus cause the solar vanes to move so as to reduce the bias. The end result of such a procedure is a configuration that is balanced with the solarp r e s s u r e null point within the jet system deadband. Power for this adaptive balancing system is derived from the p r i mary control system supply through suitable isolation with the stepping pulse energy stored on capacitors in the vane control electronics Control signals are taken from the adjacent gas valves through isolation. The primary system is thus made almost completely invulnerable to any solarp r e s s u r e system failure. Limit switches for vane travel are provided to minimize torque imbalance resulting from a runaway failure mode in the vane system. As the spacecraft approaches a balanced condition, characterized by less frequent jet pulses and greater penetration of the deadband as shown in the phase plane plots of Fig. 8, the thermal mechanical damping actuator begins to be effective. This is a phase-lagged positive position feedback system as described earlier and is mechanized by a spacecraft position sensitive solar energy valve controlling the heat input to a bimetal strip with large thermal inertia. Again referring to Fig. 8, the end result will be zero velocity condition at a stable point somewhere within the jet deadband. Unless disturbed, the jets will not be required to operate, but they have not been turned off and are ready for instant action in case of disturbance. System Analysis A method of choosing the system parameters can be illustrated with the aid of the schematic shown in Fig. 9. Here 82

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

the center of pressure of the spacecraft without the vanes attached lies between the center of mass and the Sun. As previously discussed this is a statically unstable system. Thus the purpose of attaching vanes is threefold: 1) to make the spacecraft system statically stable; 2) to align the stable position of the spacecraft with the null of the primary attitude control system; and 3) to supply retarding forces to damp out any oscillations that may occur. In order to establish the vane area and the desired vane dynamics, it is necessary to determine the torque gains of the system. Torque gain is used here to mean the magnitude of the value of the ratio of applied torque to spacecraft angular position 9 g . There are several gains to be considered. First is that of the spacecraft without any vanes attached and which will be denoted K^. Thus for small values of 0 S , T

d

= K

d9s

where, in this example, T^ is a destabilizing torque. The value for K^ is dependent upon the force Fa due to the absorbed radiation and for a symmetric spacecraft is found from the expression

Kd = F a Z

(9)

Here Z is the distance between the center of p r e s s u r e of the spacecraft without vanes and its center of mass. If, for the moment, the vanes are assumed to be rigidly attached to the spacecraft, a restoring torque T s caused by the vanes occurs when the spacecraft moves away from the null position. Thus, T

s

= -K 0 s s

( 10) v '

where K s is the stabilizing gain. As has been previously discussed, in order to damp out any oscillation it is necessary to move the vanes in such a manner that the vane torques lag the spacecraft position. From some nominal or zero position the vanes can be moved to generate this retarding torque. Letting the position gain of the vanes be Kv, the change in vane position A9 V is

(ii) 83

Purchased from American Institute of Aeronautics and Astronautics

J. D. ACORD AND J. C. NICKLAS

where Tv is the vane time constant. is described by

The torque of the vanes

(12) where K^ is the torque gain of the vanes. The block diagram for this system (assuming that the Sun is inertially fixed) is shown in Fig. 10. In the part representing the spacecraft dynamics, J is the spacecraft moment of inertia about its center of mass. Because of the inability to align the center of mass exactly with the geometric center and because of variations in reflectivity over the exposed surface, the stable position of the spacecraft may not be the desired orientation with respect to the Sun. However, in this example it will be assumed that any bias torques have been cancelled in the manner previously mentioned, and the effects of these bias torques on K^ will be discussed later. The reference position 0j^ is nominally zero. The values of the system constants are chosen by considering the closed loop expression for

6 9

K K^ v T

s

R

(TvS + 1)(JS 2 -

Letting

K

K 2

"

K

vKT s - Kd

the foregoing expression becomes 0 K, K u> 3 _s_ _ ____________1 7L n____________ 9

R

3 S3 + K nicon S2 + con2S + K,u (l - K ^£ ) i n

84

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

The roots of the characteristic equation can be found for various values of KI and K£, and the plot shown in Fig. 11 can be constructed. Since the roots turn out to be a complex pair and a real root, both the percentage overshoot and damping ratio are important. By means of Fig. 11, the selection of desired values for damping ratio and overshoot establish values for K^ and K£.

The task that remains is to interpret the values of the physical and system constants from the desired values of K^ and K^. This is accomplished by considering the expressions for the torques acting on the system. Denoting the absorbed and reflected forces on the left and right vanes by F^L* FRL, F*AR' and Fj^, respectively, letting the area of each vane be A, assuming that the vanes are specularly reflective, i. e. , s = 1, and referring to Fig. 9 the forces are F

AL

= P A

f ^ - P>

COS 9

( VOL - 9 s )

= 2P£AP = P f A(l - p) cos(9 VOR FRR = 2P f Ap Here QyOL. anc^ ^VOR a r e the respective angles of the left and right vanes with the solar panels. The torque T V due to the vanes is then T

v = ^ F RL sin ( 9 VOL - 9 s >

+

^R F RR sin ' 8 VOR

+

V

The meaning of -^L, -^R, LL, and L/R is best described by Fig. 9: 9

VOR

9

= 9

VO

+ A9

VOL - °VO "

v

A9

and

6 = A9 85

v

+ 9

s

v

Purchased from American Institute of Aeronautics and Astronautics

J. D. ACORD AND J. C. NICKLAS

Performing the indicated substitutions and putting all variables in terms of the system parameters gives an expression for a torque-to-area ratio. Thus in functional notation, T

-^ = g (p f , P, h, z, w, e s , evo, 6)

(is)

In this expression, the vane width w may or may not be dependent on the value of A. In the discussion that follows, it will be assumed that w is independent of the vane area. However, if this is not the case, the results can be modified by finding T

and using its value to calculate an appropriate correction factor. The validity of the new results obtained will depend upon the accuracy of the original estimate of w and the magnitude of the partial derivative. The value for 9yQ is determined by assuming 9 S = 0 and finding the value for which maximizes the expression

•-86

A curve of T V /A as a function of 6 can then be plotted. this example P

In

= 0 . 4 7 dynes/m 2

P = 0. 8

h = 224 cm Z = 30. 5 cm

w = 61 cm For these values @VO ~ 37. 5° and the resulting curve for T V /A is shown in Fig. 12< The slope of this curve is -K-p/A. If bias torques are acting on the spacecraft initially, the steady -state position of the vanes will be at some angle A9 V away from 9yo- Since the gain 86

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

T

K _s_

" A

~

8-j! A

80

s

is dependent upon A6 V , it is necessary to determine K S /A as a function of A9 V . The results of this computation are shown in Fig. 13.

Sufficient information is now available to determine the values of the physical and system constants. The approach can best be explained by using the curves obtained in the example just given. Assume that a minimum damping ratio of 0. 15 and a maximum overshoot of 30% is desired. Then from Fig. 11, K L = 1. 03 Assume also that the maximum expected bias torque is 75 dyne -cm and that the destabilizing gain K^ at the spacecraft is K^ = 2. 5 dyne-cm/deg. It must now be decided what the degradation in gain K s will be acceptable as the vanes move to cancel the bias torque. Assume this to be 10%. Then from Fig. 13, a 10% degradation of K g allows a A6 V of 11°, and

6.0 x I P " 4 dy?e " Cm < < 6 . 6 7x 2 T ~ A — cm

1 0 -4

-deg

dyne-cm 2 , cm

-deg

Using Fig. 1Z, the fact that a maximum bias torque is 75 dyne -cm, and (A9 v ) max = 11°, the area A of each vane is A = 7500 cm^. This is approximately 8 ft^ and is the minimum area that can be used. The range of values for K s is then

4

5

dyne -cm < deg

K

s

< 5

Q

'

dyne -cm deg

Assume

K

s

= 4. 7 5 , =/T, 05 + 00 S/C STRUCTURE /

Fig.

BIMETAL ACTUATOR AT TEMPERATURE T

17 Combined sensor-actuator for solar pressure vanes.

101

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

CONTROL MOMENT GYRO GRAVITY STABILIZATION E. D. Scott* Lockheed Missiles and Space Co., Sunnyvale, Calif. Abstract This article deals with the control of gravity-stabilized vehicles, using control moment gyros. The attitude control devices used to dampen vehicle librations are single-degreeof-freedom integrating gyros called control moment gyros (CMG's). The CMG's control the vehicle to an orbital reference established by the earth-to-satellite radius vector and the orbital angular momentum vector. Design considerations presented include the effect of orbital eccentricity, the attitude offsets produced by torques which are constant in a body reference, limitations a.nd variations of values of CMG parameters such as angular momentum, damping, gimbal axis spring restraint, the orientation of the CMG within the vehicle, and the misalignment of vehicle principal axes to the vehicle reference axes. The CMG control system is optimized, using montecarlo techniques, and the results of the optimization are shown. The optimization technique is discussed and its utility illustrated. Linearization techniques and their limitations are given, and an example is presented showing where linearization can produce serious errors in system analysis. This limitation can be overcome by using partially nonlinear and linear methods. This approach is developed and illustrated. Nomen clature t 1^ Iy !„ w

h Y

= principal moment of inertia about the roll axis, X = 1, principal moment of inertia about the pitch axis, Y (normalized) = principal moment of inertia about the yaw axis, Z = 1, average orbit rate (normalized) = h . = hp, angular momentum of each gyro = 90° - a - 90°-a , Vee angle of system

Pl'€;i/*2'P2'€2' alls™ent angles of CMG^ CMG2 T = maximum time constant of the system Presented as Preprint 63-32^ at the AIAA Guidance and Control Conference, Cambridge, Mass., August 12 - 14, 1963. •^•Research Specialist tNomenclature relates to Table 2.

103

Purchased from American Institute of Aeronautics and Astronautics

E. D. scon DF SD

= minimum damping factor = standard deviation of T -where the 3 sigma tolerance on h^ h2, C^, C2, KX and K2 is 20$ C-|>Cp = viscious damping constant for CMG-,, CMGp K

1'K2 = gyr° s£ring constant for CMGL, CMGp d> s = roll offset for Mx = 1 Q = pitch offset for M. = 1 s j j, = yaw offset for M = 1 'S

Z

= yav offset for w

ai = •*-

= vav

offset for oo^ = 1 = max roll oscillation for M = 1 sin t. Also max yaw oscillation \\> X = X = max pitch oscillation for M = 1 sin t

*

y

= max yaw oscillation for M = 1 sin t. Also ^ =9 z max pitch oscillation foryM = 1 sin t Y z = max yaw oscillation for M = 1 sin t. Also max roll Z oscillation 0 = 0 = max pitch oscillation for e = 0.01 = max yaw oscillation for e = 0.01 offset due to a

1^ increase in h

offset due to a

ip^

1# increase in h = yaw offset due to a 1° increase in #

^2 (j^p^

= yaw offset due to a 1° increase in «2 = roll offset due to a 1° error in BT Alsocjjft. = (jjg 2 roll offset due to a 1° error in p^ -1 ^cy = yaw offset due to a 1° error in a H The moon is gravity-stabilized to the earth by virtue of its differing moments of inertia, this being the reason why one side of the moon continually faces the earth. Gravity stabilization can be considered by using a simple model. This model is an approximation and is good only for small angles. Consider the following:

Kepler's Laws state that a satellite's center of mass will/ trace out an ellipse in a plane containing the central force field. The key point is this, that the center of mass traces out the ellipse. The gravity field created by the central force field is actually curvilinear, varying in strength with the radius, and therefore the center of gravity of the satellite does not coincide with its center of mass. Since the centrifugal force acts through the center of mass and the gravity force acts through the center of gravity, a torque couple is generated when the center of gravity and center of mass do not coincide with the radius vector. The mass-center of gravity separation is quite small. However, it is sufficient to stabilize an orbiting satellite. To illustrate the distance between the center of mass and center of gravity, consider a 3K Ib satellite with moments of inertia 3>000,

105

Purchased from American Institute of Aeronautics and Astronautics

E. D. SCOTT

3,000, 300 slug ft2 orbiting at 500 naut miles. The center of mass to center of gravity separation is 0.00004 in.

A more accurate representation of the gravity gradient torque can be formed by evaluating the gravity forces throughout the satellite. > The equation of the gravity torque in dyadic-^-3 form is

— 3k A = A rg = ~^=rr x I • r r

f_ x (1;

where fg is the gravity torque, k is the earth's gravity field constant -which is equal to the universal gravity field constant times the mass of the earth (k = 1.^07645 x 10+1^ ft3/ sec2), r is the magnitude of the radius vector, r is the unit radius vector and I is the dyadic representation of the satellited inertia tensor.

Various systems have used gravity stabilization successfully, for example, the TRMC 5 satellite and the Agena vehicle on a number of the programs for which it is the satellite. The main reason why gravity stabilization is successful is that the orbital torque environment is small compared to gravity restoring torques. The gravity stabilization system cannot be used at low altitudes because the aerodynamic torques become quite significant as the altitude is reduced. Gravity stabilization typically performs best at 500 naut miles, is marginal at 300 naut miles and, is completely useless at 150 naut miles. To illustrate the torque environment for a satellite, consider the following:

Satellite characteristics: Cross-sectional area, 100 ft2 Center of mass to center of pressure = 1 ft Weight, 3,000 Ib Moments of inertia, 3,000, 3,000, 300 slug ft2 Daring the ascent phase: Aerodynamic torques, 1 ft-lb Vehicle maneuvering, 10 ft-lb Vehicle thrusting, 100 ft-lb Initial venting, 10 ft-lb

Venting after one day, 0.0001 ft-lb On orbit torque environment for 300-naut mile orbit: Aerodynamic torques, 0.001 ft-lb Solar Torques, 0.0001 ft-lb Magnetic torques, 0.00005 ft-lb (oscillatory) Eccentricity equal 0.01, 0.00007 ft-lb (oscillatory) Gravity torques for a 1-degree error, 0.0002 ft-lb 106

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

rate of change of angular momentum) "which is directly proportional to the vehicle rate and so generates damping. The CMG may "be thought of as two devices in one, as a sensor and an actuator. As a gyro it senses the vehicleT s rates and as a large angular momentum device it produces vehicle control torques. In a sense, this is somewhat different from the normal attitude control philosophy. Normally, the control engineer has an attitude sensor which is separate from the actuator. The attitude sensors often are the attitude gyros of the satellite and in a pneumatic attitude control system the gas jets are the actuators. One point should "be stressed at this time. The CMG is actually a control device and must not be used as a sensor per se. In fact the control moment gyro used for damping the roll axis of a vehicle might be at its null position for a constant roll offset and the control moment gyro used for damping the yaw axis of a vehicle might be at null for a constant yawoffset. The control moment gyro is quite different from an attitude gyro. First of all, the control moment gyro has a large angular momentum in comparison to that of an attitude gyro. For a satellite orbiting at 500 naut miles with moments of inertia 3,000, 3,000, 300 slug ft2, the angular momentum of the CMG would be on the order of 10° gm-cm^/sec, while the attitude gyro may range somewhere between ICr- to 1C)5 gm-cm^/sec. The control moment gyro may have a spring restraint as indicated

by ECL- (2):

0
€ .

The scaling of the system from the normalized parameters to the actual satellite system parameters is as follows: Normalized

Actual

G

i

K.

K.I =

M

y M z

V =" M 1 = oo _ z o ZT z

OJ

ojdl1

-i

dl

T Time Constant All Angles

= OJ

o

OJ__

dl

T~ = T/COQ Unchanged

127

Purchased from American Institute of Aeronautics and Astronautics

E. D. SCOTT

As an illustration of the design procedure, consider a satellite with the folio-wing characteristics,

3^

= 3,000 ft-lb-sec2

Iy

= 3,000 ft-lb-sec 2

Iz

= 300 ft-lb-sec 2

Altitude = 500 run then = 1.0 = 0.1 o

co

= l.Olij. x 10"J r/sec

co I

= 3-0^2 ft-lb-sec

coQ2I

= 3.085 x 103 ft-in

•which produces a system Normalized h

l

=

K-L

= 1-553

K2

= 1.19T

a

= -26.87°

^

= -153.13°

P! = P2 €

T

2

= e

3.69 x 10"3 ft-lb/r

i=

0

= 90° = 3.03 128

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

Normalized

Actual

DF = 0.253

0.253

= 20.38 s 6 = 21.22 s

6.6° for 10~3 ft-lb

^

9.9° for 10~3 ft-lb

s

6.88° for 10"3 ft-lb

=27.80

6X = 21.22

6.6° for 10"3 ft-lb

6,4, = 1^0. 90 ?

12.6° for 10"3 ft-lb

d; y

= 22.05

6.80° for 10"3 ft-lb

^

= 36.00

11.1° for 10"3 ft-lb

y «y

6ec = 2.223

2.22° e = 0.01

ipTec = 1.191 ^

1.19° 'e = 0.01

The sensitivities are scaled 1:1.

The last feature of the CMG system is the region of static stability. 5 Figure 1^ shows the region of stability for torques constant in the body axes for a horizontal V-CMG system optimized for IX/IY = 1*° an-d IZ/^Y = °-3- A point of interest is that the stability is one-sided in pitch and yaw, being very strong for positive z axis torques and weak for negative torques. The figure is only an approximation since only the on-axis values were obtained, the remaining portions were sketched in. References DeBra, D.B., "The large attitude motions and stability, due to gravity of a satellite with passive damping in an orbit of arbitrary eccentricity about an oblate body/' Ph.D. Dissertation Submitted to Stanford University (May 1962). P Debra, D. and Delp, R.H., "Satellite stability and natural

frequencies in a circular orbit," The Astronaut Sci., VIII (1962). ———

, E.G.C., "On the attitude control of earth satellites," Great Britain RAE Space-TN-3, January 1962 (paper read at the Eighth Anglo-American Aeronautical Conference in London, September 1961). 129

Purchased from American Institute of Aeronautics and Astronautics

E. D. SCOTT

4

Doolin, B.F., "The application of matrix methods to coordinate transformations occurring in systems studies involving large motions of aircraft," NACA TN 3968 (May 1957)•

•^Garber, T.B., "Influence of constant disturbing torques on the motion of gravity-gradient stabilized satellites," AIAA

Journal^ 968 (April 1963).

Goldstein, H., Classical Mechanics (Addison-Wesley, Reading, Mass.j March 1956), Y Kay, M.G., "Attitude control of satellites using integrating gyros," M.I.T. Inst. Lab. Reprt. T-208 (Conference) (June 1959)o

Klass, P.J., "New gyro technique orients satellite," Avia-

tion Week, 76, No. 1, 68-73 (1962). 9Plummer, H.C., An Introductory Treatise on Lfynamical

Astronomy (Dover Publications, New York, I960)y pp. 22, 24. Roberson, R.E., "Gravitational torque on a satellite vehicle," J. Franklin Inst., 265 (1958).

Kouth, E.M., "The advanced part of a treatise on the dynamics of a system of rigid bodies," (Dover, Publications, New York, 1955), Chap. XII. 12

Stearns, E.V.B., Navigation and Guidance in Space (Prentice-Hall, New York, 1963\ pp. 71-85.

^Wilson, E.B. (Gibbs, J.W.), Vector Analysis (Dover Publications, New York, 1960), ]k DeLisle, J.E., Hildebrant, B.M., and Ogletree, E.G., "Quarterly summary report for U.S. Air Force Contract AF o4(6Vf)-769," M.I.T. Inst. Lab. Repts. R-357A, B, C, D (1962). ^Hildebrant, B.M., Lombardo, J., and Petranic, T., "Analytic techniques applied to satellite gyro-stabilizers," M.I.T. Inst. Lab. Rept. R-398 (February 1963).

130

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

Table I Reference to transformations

DIRECTION

COSINE

EUL£R

A X I S Of

ANGLE

ROTATION

Zr

ORBITAL

REFERENCE

xr,

Yr'

%

AXES ^————————

(/)

BODY

c< X, Y, Z

PRINCIPAL

ocz- 2 1 2,n| l,n

2K|z o

2K

l*2,n= z l,n 6

^i=s3n z?£>n n B^-Bgn *%n n+1

l,n = sgn z 2,n n -sgn Z-, n+1 ^ ^Q

5

These state equations are obtained by relating the filter output, x(t), to the controller output, m(t), for t ... Since an impulse of known polarity was fired at t , it is possible to determine x(t .) and x(t .,) in terms or =t - t An ~n+l n' x(t)is then substituted into Eq. (l) and A evaluated. Substitution of this time increment into the x ana x equations yields state equations for the filter output, and it is a simple matter to relate these equations to attitude error and error rate to obtain Eqs. (5) and (6). Table 1 is necessary since the polarity of the right hand side of Eq. (l) is unknown a priori. Ambiguity arises in solving for A since it need merely satisfy a quadratic equation. Table ln lists those conditions which will insure that the least possible positive root is obtained. This is the only solution with physical significance. Thus given the system state at the n control pulse, (z., z ) the state equations together with Table 1 specify 2,n+l) thereby providing a complete solution to the response of the system.

Equations (11) and (12) possess a particularly simple transformation characteristic. The state solution plane is merely partitioned into two regions in each of which similar state equations hold. This is illustrated in Fig. 2. Here DI is the set of elements in D lying above P plus the elements of P in D in the fourth quadrant. D is the domain in D below P plus the elements of P in D in ^he second quadrant. A state "solution plane" can be obtained from Fig. 2 by plotting a locus of vs. for fixed values of z., "2,n These loci are identical parabolas symmetric with respect to the z axis. An illustration is provided in Fig. 3 for 194

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

K = 0.5.

The system state z is a point in the plane. The parabola that threads z determines z., - . z £ D, implies a decrease n l,n-fl n 1 ^ of one unit in z_ to determine z^ - . z v£ D«_ implies an n 2 increase of one unit. The new states determine z - "which can n-t-1

"be used to obtain z . n~i~2

A typical trajectory is illustrated

in Fig. 3 ^here the crosses represent specific states and the arrows establish their order. It is interesting to observe from the state Eqs. (ll) and (12) that a single dimensionless parameter K, determines every performance characteristic of the normalized system. Eqs. (11) and (12) along with Table 1 form the starting point for the analyses to follow. In-the-Large Behavior Practical Stability

An often-applied approach to establishing a suitable V(z )

function is to consider a linear first approximation > " > > " . In terms of the block diagram of Fig. 1, the first approximation used here is defined to be that linear system which has the same average controller output when the input is constant. With this constraint Eq.. (l) can be evaluated directly to give A

. - t = A/x n = tn+1 n 'n

The average controller output over the interval A = (nI/A)xn

3 ) v( J/1

is (111)

Thus the linear system gain corresponding to the integral pulse frequency controller is jjtl/A. A necessary and sufficient Lyapunov function for this linear system (for K>0.25) is given by fl The present analysis is limited to the case of conjugate complex roots of the linear first approximation although the approach can be easily extended to the case where 0(r ) condition (iv) of the theorem is satisfied. Evaluation of the boundary of the set which defines the set U is considered for the case vhere K = 0-5- The shaded area or Fig. 4 pictures the set U . The inexact use of functional notation is preferred in order to retain the familiar Av notation. ThusAv(z ) = Av(r y ft ) through the transformation of variables given by Eq. (19). 197

Purchased from American Institute of Aeronautics and Astronautics

FARRENKOPF, SABROFF, AND WHEELER

Condition (v) of the theorem is automatically satisfied since V vas chosen as described in the paragraph following the theorem. To demonstrate the sufficiency of the so-defined V , condition (vi) of the theorem is examined by considering

(23) Z

Z

+ KZ

" l,n+l 2,n+l

2,n+l

for z in U . In the domain D, , utilizing the state Eqs. (18) and the transformation Eq.. (19), Eq.. (23) can be written, for Z 2,n=°'as

for K = 0.5» In like manner, applying the transformation of Eq. (21) gives, for LJzy n /^ )/V(r , )^ ) and AV(r ,0 )/V(r ,0 )^ incorporation of Eqs. (29) and (30) and

taking the limit as r -*> oo , leads to the result

K Since Eq. (3!) is in the form of Eq. (26), it establishes the greatest lover bound on the in-the-large convergence rate of the IFF system. Thus V

f =

1/K

(K > 0.25)

( 3 6 )

Comparison of Eqs. (36) and (33) shows that the IFF system and its linear first approximation have identical convergence characteristics for large r . This is of considerable significance since it implies that the transient performance characteristics of the two systems are equivalent in the large, and in view of the relationship between the IFF system and the first approximation, would be expected to be similar elsewhere.

In order to demonstrate this fact, it is necessary to establish for all 'n W and r0 u the largest value of rn that violates Eq. (26) for various particular values of ^, termed ^ • By this approach a closed boundary enclosing z = 0 will be established in the state plane such that for z noutside the enclosed region,

AvUJ

An

(37)

Torr-Tp

For the case of K = 0.5> three values of ^ were chosen for investigation: ^ =1.0, 1.5 and 1-75- Fig. 6 illustrates the boundary and Pits associated region for each value of >7 . Notice that even for ^ = 1.75 a relatively small subset of the domain D will not satisfy Eq. (37)* It is interesting to note that the degradation of the transient convergence rate occurs near the partition line F. Outside this region, the transient response of the IFF system and its linear first approximation would be expected to be similar. These results have been verified by an analog computer simulation and are presented subsequently.

On this basis a reasonable transient design procedure for the IFF system is available through its linear first approximation. In terms of the normalized second order linear system the relative damping ratio and undamped natural frequency are given by V2

2(K)

V -T72 ~

(K)

201

(38)

Purchased from American Institute of Aeronautics and Astronautics

FARRENKOPF, SABROFF, AND WHEELER

and for large r these parameters//also define the transient response of the IFF system. Ultimate State Considerations

It has been mentioned that the IFF controlled system will not converge to a single point in the state space, but can, at best, go into some limit cycle. From a practical standpoint, only those limit cycles with a low number of control pulses per cycle are of interest since the others generally imply a high fuel rate expenditure. Thus it is desirable to show that the system will converge to a two pulse limit cycle a large proportion of the time. The existence, stability and uniqueness of this particular ultimate state will now be considered, and higher order limit cycle sets are discussed.

Existence Two pulse limit cycles necessarily contain one point above Assuming (z- , z ) to lie above the partition line, it follows that 1,n £,n

and one below the partition line in Fig. 2.

J

Z

_ rZ

V2

-1

2

l,n+2 - [ l,n-H - 2K(z^n-l) J

( 3 9 ) V2

For the limit cycle condition z

l,n+2 = Z2,n

The solution of these equations is

r > 5(r ) i s sufficiently large for practical systems. 202

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

Not all possible solutions of Eqs. (^3) and (kk) can "be accepted since not all imply the assumed state locations •with respect to the partition line. Utilizing Table 1, it can be demonstrated that two pulse limit cycle existence is guaranteed if

( -

s) K < z2n < i - (f ( -

in "which case z1j. • n and z,JL« n"r JL are given by Eqs. (^-3)

an
1 the effect of four-pulse limit cycles results in J being greater than A Design Procedure

In any design problem there are certain specifications to be met as well as basic limitations due to implementation considerations. For most spacecraft attitude control problems the most basic specifications deal with steady-state attitude accuracy and the maintenance of control (perhaps with reduced accuracy) when the vehicle is subject to non- environmental disturbance torques such as those arising from guidance or station keeping thrust sources. Additional requirements may be imposed by the characteristics of a particular mission; for example, the system may be required to recover from some specified attitude error within a prescribed time. Finally, within the framework of the explicit design specifications, it is desirable to maintain the lowest possible rate of impulse consumption. Actually the crossing may occur for K-cl.O (but greater than 2/3) since the assumptions concerning the periods of the elements of S»-. and S^ and the non existence of S^ may no longer be valid. Of course the analytical extension to this case is trivial in concept.

212

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL— II

In implementing the integral pulse frequency control system two constraints are of particular significance. The time constant T, which is usually derived by a lead net-work approximation, is limited in magnitude by the size of available components; and by the need for simplicity. The pulse width, h, has a lover limit which depends partially upon the method of propulsion. For spacecraft control purposes, then, it is desirable to consider the manner in "which the basic specifications and implementation constraints affect the selection of the system parameters. If 9 is the required accuracy during unperturbed limit cycle operation, the desirability of maintaining a low rate of impulse usage indicates that (from Eq. (56))///

KT = 20^

( 6 9 )

Furthermore, the available control acceleration (jj./h) must at all times exceed the disturbance acceleration, X^* To avoid excessive impulse consumption during limit cycle operation, ji must be as small as possible. Thus the constraint

H > hXD(t)

(TO)

-where h is made as small as is feasible. Generally, some safety factor (T ( & > 0) will be provided, so that

n = (i+o-)h(XD)max

(Ti)

Having thus specified h, i^and the product KT, selection of K will complete the design. As mentioned above, K will be selected to satisfy secondary system requirements. It is useful to express the remaining performance measures of the previous section in terms of 9 , p., h and K. From Eq. (68), the

777

In actual application one may not be overjoyed with the numbers which appear. For example, in a spacecraft attitude control application where the desired limit cycle amplitude (9 ) is 1.5°, the control acceleration (jjt/h) is 1.0 deg/sec2 and the pulse width (h) is 20 msec, in order to obtain a K = 0.5 Eq. (69) would require a T equal to ^00 sec. This is a result of not incorporating a dead zone in the present controller; an addition which is trivially simple from an application viewpoint.

213

Purchased from American Institute of Aeronautics and Astronautics

FARRENKOPF, SABROFF, AND WHEELER

rate of impulse consumption is

(72) The offset resulting from a constant disturbance is :

The unsaturated transient response is described "by

(75) The region of unsaturated operation is defined by

9 -f T 0 | « / K U h

( 7 6 )

If two or more of the above are of interest, a compromise may be necessary in the selection of K.// /For instance, an increase in K to reduce 9 will have the undesirable effect of increasing the impulse consumption rate, J. On the other hand, a decrease in K to reduce the impulse requirement will require an increase in T and, moreover, will increase the sensitivity of the system to sensor noise. P2 = -Cit + C2 c

( 1 0 )

PS = 0; PS = s

Thus, H becomes

H = C!X2 + (-Cit + C2)u + C3

( 1 1 )

The optimum u which minimizes x3(t) is the u which maximizes the function H. In this case L u = Y" (s^n P2)

236

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

This restriction will keep the product of p2u always positive. From the adjoint equations, p2 is seen to be a linear function of the form P2 = -Cjt + C2

( 1 3 )

Then the sign of p2 may be one of the following four forms: 1) always positive, 2) always negative, 3) initially positive, then negative, and h) initially negative, then positive. These four forms of p2 lead to four different control torque applications. The system equations are solved for these four cases of torque application.

In the first case, the sign of the control torque is always positive until the system terminal conditions are satisfied at which time the control torque goes to zero. The system equations are integrated.

_ ut2

•^1 """

o cL

^-2

o,

"^*1

o

X 2 = Ut + X2

At t = T, these equations become 0 = 2Si + X2°T + Xl° (15)

0 = uT + x2

The convergence time T is then x ° T = - -jjj-

(16)

and since T>0 and u>0, x2° 2(AeI/L ), the C thrust termination of the last cr£ £ cycle occurs outside of ±Ae region and the system drifts to the opposite side of the phase plane to start another series of rapid convergence trajectories. Thus 21 A* is the limiting L value of k(tcr - tc , ).

* \ Selection of System Constants

C

The sufficient convergence condition, Eq. (33)$ is stated in terms of specific times on a trajectory. It is desirable for design purposes to eliminate this time dependence and to find a relationship among the system parameters that defines this sufficient condition. The constant k can be related to other system parameters as follows. The last rapid convergence trajectory starts with some attitude rate e whose magnitude is smaller than |Ae|, and with an attitude error e . If the system is assumed to be free of errors, the time of control signal sign change can be computed. Starting from e and e both of sign opposite to L , the trajectory crosses the e axis at the time

245

Purchased from American Institute of Aeronautics and Astronautics

ABZUG, MARTIN, AND OHGI

.

I "n L ^

(310

——mm

with e =

n

n

o

)2 =

(35)

The signal sign change occurs at I

= £'= fo . 2 2

/« *2

(36)

for the case of no external torque, and, by definition e = 61. The time required to reach this point from e e = 0 is t

cr

*

_

t

cdo

*

= [ - ( - e L

c

2L C

n

-e )

°

(37)

Now, Eq. (33) states that for convergence k(t

cr,.

- tcd ) < 21 Ae

v-

= 2t

m

(33)

where t is the duration of the minimum-impulse or m * 21 Ae

(38)

k
0Q

= initial angular displacements of vehicle

, A 1

=

I x

-I y I

y

I

Xx

"I

^Z

A2

=

6 56Z

= gyro torquer gimbal displacements about Y, Z axes

y

References Hazeltine, W.R., "Passive damping of wobbling satellites: general stability theory and example", Journal of the Aerospace Sciences, Vol. 29, No. 5, pp. 543-549. o

Hazeltine, W.R., "Nutation damping rates for a spinning satellite", Aerospace Engineering, Vol. 21, No. 3, pp. 10-17. Grout, P.D.,"The motion of a spinning body equipped with a dynamic damper containing viscous friction", Report IDP 1389, U.S. Naval Ordnance Test Station, China Lake, California (August 22, 1961).

^ Carrier, G.F., "The theory of the annular damper", Report GM-TM-0165-00265, Space Technology Laboratories, Los Angeles, California (July 21, 1958). 5 Carrier, G.F. and Miles, J.W., "On the annular damper for a freely precessing gyroscope", Report TN-59-0000-00230, Space Technology Laboratories, Los Angeles, California (January 29, 1959). 6

Den Hartog, J.P., Mechanical Vibrations (McGraw - Hill Book Co., New York, 1956), 4th ed., pp. 87-102. 7 Adams, J.J., "Study of an active control system for a spinning body", NASA TN D-905, Langley Research Center, Langley Field, Virginia (June 196l),

8 Kurzhals, P.R. and Adams, J.J., "Dynamics and stabilization of the rotating space station", Astronautics, Vol. 7, No. 9, pp. 25-29. " Grubin, C., "Generalized two - impulse scheme for Reorienting a spin stabilized vehicle", Progress in Astronautics and

331

Purchased from American Institute of Aeronautics and Astronautics

M. LOEBEL

Aeronautics - Guidance and Control (Academic Press, New York,

1962), Vol. 8, pp. 649-668.

1° Wheeler, P.C., "Two - pulse attitude control of an asymmetric spinning satellite11, AIAA Preprint 63-338 (August 1963). 11 Progress Report - Manned Orbital Space Station Technical Summary, Report SID 63-36-1, Space and Information Systems Division, North American Aviation, Inc., Downey, California (February 1963). 12 Goldstein, H., Classical Mechanics (Addison - Wesley Publishing Co., Inc., Cambridge, Mass., 1950), page 134. 1^ Etkin, B., Dynamics of Flight (John Wiley and Sons, Inc., Toronto, Canada, 1958), Appendix A, pp. 442,443.

332

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

333

o

60

o

O

CO CO

§ d o

CO

CO JJ

CO

o CO ex. CO

4J •H

a)

60

Pn

•H

Purchased from American Institute of Aeronautics and Astronautics H A X

^BODY

00 CO

o m CD m

BODY

Y

BODY ROTATION SEQUENCE: $, Q,

Fig. 2 Coordinate system transformation.

Fig. 3 Precession wheel gimballing.

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—I

IMAG

-- 0.4 [y = lz = 9.1 X 106 SLUG-FT2 Po= 1/3 RAD/SEC. A =0.835 H = 5 0 X 103$LUG-FT 2 /SEC

Po -- 0.3

REAL -0.08

-0.1

-0.06 -0.04

-0.02 --0.01

-0.02 -0.03 -0.2 K2=l

-*> -0.3

————————

Fig. 4

K, = 30

Root locus - gyro stabilization system.

335

Purchased from American Institute of Aeronautics and Astronautics

M. LOEBEL

IMAG.

l y = lz = 9 . 1 X 106 SLUG-FT2

A =0.835

--0.31

Po = 1/3 RAD/SEC H = 50,000 SLUG-FT2/SEC INDICATED VALUES OF K 3 ARE TO BE MULTIPLIED BY TO 6

--0.30

K, = 30

K, = 10

REAL

1.0

---0.02

Ki = 10

*i-»r\ 0.3

Fig. 5

K3=0.2

0.1

0.03 .._, -0.33 -Po 0.05 0.01 -0.34

Root locus - gyro/jet reorientation control system (primary mode).

336

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

20 REAL

-0.6 ——I—

-0.5

-0.4

l y = lz = 9.1 X 106 SLUG-FT 2 p = 1/3 RAD/SEC \ = 0.835 INDICATED VALUES OF K's ARE TO BE MULTIPLED BY 106

ACCELERATIONS - — — — — RATES

Fig0 6

Root locus - all jet reorientation control system (secondary mode).

337

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

This page intentionally left blank

Purchased from American Institute of Aeronautics and Astronautics

DRIFT OF A STABLE PLATFORM CAUSED BY GYRO ROTOR UNBALANCE Myron Kayton* Litton Industries, Woodland Hills, Calif* Abstract This paper discusses several sources of drift in a gyrostabilized platform, caused by static radial unbalance of the rotors or by axial runout in the rotor bearings* Some of the drift results from unbalances in two or more gyro rotors; some results from unbalance in only one rotor. All vibrationinduced drifts vary with spin frequency and platform gimbal angles* Methods of controlling these drifts are discussed, and an example is evaluated numerically* The effect is not serious if the gyro rotors are well-balanced* Nomenclature x,y,z

= unit vectors fixed in the stable element (figure 1 and figure 2)

k]_

=

k2

= principal angular stiffness of the stable element about the y axis

k3

= vertical stiffness of the stable element

9^

=

rotation of the rigid stable element about the x axis

@2

=

rotation of the rigid stable element about the y axis

principal angular stiffness of the stable element about the x axis

Presented as Preprint 63-305 at the AIAA Guidance and Control Conference, Cambridge, Mass., August 12-14, 1963. ^Senior Research Engineer, Guidance and Control Systems Division*

341

Purchased from American Institute of Aeronautics and Astronautics

M. KAYTON

A

= azimuth or heading angle of the platform relative to the stable element [positive azimuth is clockwise from above (figure 2)]

nip

= mass of the stable element including gyros, accelerometers, servo-motors, gears, etc*

I

= polar moment of inertia of the stable element about x or y

"2 2

=

co~3 2L

r

VS

=

k /m 3 p = center-to-center distance between the spin axes of the two gyros

=

radial unbalance of the top gyro rotor

0 r^

= axial unbalance of the top gyro rotor

SQ

= radial unbalance of the bottom gyro rotor

s,

= axial unbalance of the bottom gyro rotor

0

= phase angle between top and bottom gyro rotors

0..

= phase angle between the radial and axial unbalance

0

s

o>s

= frequency of vibration excitation [angular velocity of rotor or retainer-ring]

mQ

= rotor mass

mQgr

=

H

= angular momentum of the gyro

F r Fs

= vibration excitation force on the top gyro

of the top gyro

=

phase angle between the radial and axial unbalance of the bottom gyro

rotating or axially vibrating unbalance, dyne-cm

vibration excitation force on the bottom gyro 342

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

T

= vibration excitation torque on the stable element

a"

=

a" B

= acceleration of the bottom gyro

ft » ft t ft x y z

=

acceleration of the top gyro

instantaneous drift rates of the platform around x f y and z f due to self-excited vibration

?1 9 H * 0 £ y z

=

ft

= mean drift rate due to simple inter-rotor coupling [typical term of ftx and f t !

ft ? »U,

= mean drift rate due to anisoelastic coupling [typical term of ft and £7 ] x y

mean values of Q t 0 f 0 respectively x y z

Discussion

It is commonly believed that the presence of a static unbalance in the rotor of a gyro causes no net drift since the unbalance torque averages out to zero over many cycles of rotor rotation. This paper shows that if several gyros are mounted on a platform and if each gyro has a statically unbalanced rotor or has axial runout in the rotor bearings, the gyros can excite themselves and each other in such a manner as to cause significant rectified drift. In physical terms the unbalanced rotating forces excite the stable element in linear and angular vibration. Since all the rotors normally operate from the same frequency source and turn in phase, the rotating unbalances are in phase with the vibration reaction forces. This causes rectified drift. Some characteristics of this drift are:

1) Mathematically, the rectified drift rate is composed of three types of terms, one of which is relatively independent of heading and two of which are proportional to the sine and cosine of twice the heading. The latter two vanish if the platform gimbals are isoelastic at all headings*

2) The drift rate due to inter-rotor coupling is different each time the system is started because of the random 343

Purchased from American Institute of Aeronautics and Astronautics

M. KAYTON

phase in which the wheels lock into synchronism. Singlerotor coupling does not depend on rotor-to-rotor phase, No phase angle exists that would make the drift rate vanish at all headings«

3) The magnitude of these drifts increases as the size of platforms is reduced.

*O This effect exists with approximately 50% more severity in platforms using single-degree-of-freedom (SDF) gyros than those using two-degree-of-freedom (TDF) gyros because of the presence of an additional unbalanced rotor*

5) In-phase radial-axial vibrations exist at rotor speed and retainer speed in all ball bearings. Experimental evidence at the authors' firm has confirmed the existence of inter-rotor and anisoelastic coupled drift rates when badly balanced rotors are used on a stable platform. These drift rates can be reduced by the following methods; 1) Improved rotor balancing, including static radial balance and axial runout. In the example considered in this paper, rotors must be balanced to 24 dyne-cm to achieve 0.2 deg/hr of rectified drift. Bearing manufacturers must take care to control radial and axial runout, 2) Preventing the two rotors from spinning in phase. This can be done by interrupting power or by driving the wheels at different frequencies. This procedure will not eliminate single-rotor coupling, which is caused by radialaxial interactions within each gyro. It is not necessary to put constraints on the gimbal structure to reduce these effects; adequate static balancing of rotors should be sufficient. Making the gimbal structure isoelastic does not remove all of the inter-rotor coupling. Additional damping in the gimbal structure will prevent the amplification of inter-rotor coupling if structural resonances are near the spin or retainer frequency. Damping will also reduce the fluctuation of inter-rotor drift as the platform gimbal angles change. Vibration-induced drift will be least affected by platform heading if the spin frequency is outside the range of the primary platform resonances. The importance of vibration control in the design of gimbal systems should not be underemphasized,

344

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

Analysis

Figure 1 shows the stable element of a stable platform. This type of construction, using two TDF gyros or three SDF gyros, is representative of the gimbal configuration used by many manufacturers of aircraft inertial equipment« In the example discussed here, the platform is stabilized with TDF gyros. No additional generality would result from considering SDF gyros. The top gyro controls azimuth about its outer gimbal axis z and pitch about its inner gimbal axis y. The bottom gyro controls azimuth about its outer gimbal axis z and roll about its inner gimbal axis x. The stable element is assumed to be rigid but capable of linear vertical vibrations and of angular vibrations about the horizontal axes due to servo compliance and to structural and bearing compliance in the gimbals. The dynamics of vibration in each principal axis are assumed to be second order.

The mass center of the top and bottom gyro wheels are at "r = r (z cos to t + x sin to t) + y r . cos (to t + ) + y sin (to t + ) J s o s o 1

(i)

cos (to t + 4> S

Both rotors are synchronous. Only a first harmonic axial wander is considered since little generality is added by taking the Fourier series for an arbitrary axial motion of the mass center. The mechanical excitation forces caused by the unbalanced wheels are Top gyro

F

r

= m r to (z cos to t + x sin to t) o o s s s

2 •f m r .to cos (to t -f d> J y o 1 s s 1 345

Purchased from American Institute of Aeronautics and Astronautics

M. KAYTON

Bottom gyro F

s

= m s to Cz cos x(to t -I- Ycb ) + yy sin (to t + Y )] o o s s o' s o' 4- m

s .to

2 ^

o i s

(2

x cos (to t 4- cb + Tcb ) Y

s

o

2

These forces cause a torque on the stable element

TF = LS x ( F r . FS) T

F

=

m

oWs

L [s

=

"^o^s

L [r

o

8in

X

T

F

o

(w s t

Sinu

+

*0) - r! c°s ( u s t + ^ t ) ]

st " Sl

COS ( W t +

s

(3)

^ o ^ *2 )]

Thus the equations that describe the three degrees of freedom of the stable element, including the restoring forces are

- 8 j (k1 cos 2 A + k2 sin2 A)

^ 2

y

F

- e ^ < k < sin2A + v 2

A

2

^ p i

cos 2 A)

k -k -I- 6, -^ —— sin 2A = I 6 0 1 2 p 2 +F

rz

sin 2A = I

2

sz

(4)

- k0 z = m z 3 p

These equations can be written as ?

2 m u),s L

?

e* 1 + w^ e - wl. e, = A 1 C 2 - r

^I

P

cos (to t -f )] 346

[s o sin (tost +o ) (5)

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

~ P 9; + UL 9. - CO^ 8

2

T

3

2

C

1

m =

2

CO Ij °T — [r

I P

cos (CO t + 0 s o

o

sin CO t - s.

s

i

+ 00)] 2

... (5)

(cont) m CO^ 2 z*+ C00 z = ——2—2 [ r cos CO t + s cos (CO t + 0 ) ] 3 m o s o s o P

where k

cos

2

A+k

COA = A

sin

2

A

I k

1

sin

^ ~

2

A + k

,

1 o^" "2 sin C 21

Ct

cos

2

A

2A

P

UL = k./m 3 3 p

The solutions for the steady-state forced motion are 4 m CO z = + ————-——r— [ (r

,£ \

°

+ s

o

cos 0 )

o

cos CO t - s sin 0 sin CO t] s o o s

4 CO L

m

5 E T° A —— E1 A P

cos

CO st + B sin COs t]

4 T CO L

m

—Lf —t± A—— ^ " C cos ^ St + D sin COS t] P

347

Purchased from American Institute of Aeronautics and Astronautics

M. KAYTON

where (CO

2

-D

2

- CO ) (r S

cos 0

1

1

- s

O

sin 0 ) O

2 + CO s cos (0 + 0 ) C 1 2 o

(cont)

B = (CO2 - 0>2) (r. sin 0. + s cos 0 ) ±3 s 1 1 o o + W

2

CCro+

S

l

sin

(02+0o)l

2

C = (COA - 60 ) s. cos (00 + 0 ) A s 1 2 o

+ 00_ ( r n cos 0. - s sin 0 ) C 1 1 o o

D = (or - to?)Cr •"•

S

O

+ s. sin (0, + 0 )] I

L,

Q

2 + CO (r sin 0. + s cos 0 ) o;^ these types of drift are

1

^-— s

larger than those caused by simple coupling. They exist in the presence of radial or axial unbalances in one rotor alone or combinations of radial and axial unbalances in one or both rotors.

The mechanism of excitation of anisoelastic drift is for the axial or radial unbalance to excite all three modes of vibration at a nonprincipal heading. The other unbalances rectify the resulting vibration. The behavior of these two types «f terms with frequency is shown in Figure 3b. The azimuth drift rates have a much simpler form involving only simple coupling and sin 2A anisoelastic coupling. This is because of the assumption that xy motions can occur only 353

Purchased from American Institute of Aeronautics and Astronautics

M. KAYTON

in rotation and not in translation* A more general analysis is not warranted since many of the other assumptions (such as that of a simple mass-spring, second-order structure) are gross simplifications. It is believed that this analysis is sufficient to illustrate the problem, estimate the magnitude of the resulting drift, and engineer a cure. A more complete analysis would have to include pitch and roll angle changes as well as heading changes.

An analogous rectification mechanism occurs in platforms stabilized by means of SDF gyros except that the unbalances and phase angles of the third rotor must be included. As a result, unbalance-induced drift can be fifty percent larger in such platforms than in platforms using TDF gyros. Thus it is apparent that if the spin speed is far removed from all resonances5 and and if r « r « s «* s f, the mag^ 0 1 0 l :s are the rectified drifts

n=

™ gr ar in ™ ju T2 m o o 2H I P

to

2 r s g

+

/((0z

1

Z

- (0z \)

r

g

(18)

For a typical platform gm-cm 2 /sec

H = 2 X 10 2L = 12 in.

I to

P s

=

30.5 cm

= 150 lb-in.2 = 4.4 X 105 gm-cm 2 * = 400 cps

2 1

2 2

tO . - U)_

= 800-ir rad/sec tO

fi = 3.5 X 10"4 (m gr) 2 deg/hr

(19)

If the rectified drift is to be less than 0.2 deg/hr, the rotors must be balanced to 24 dyne-cm of static unbalance.

The estimate assumes that the spin speed is above all resonances. The presence of nearby resonances will tend to raise the drift rates shown in Eq. (18). 354

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

Experimental evidence within the author's firm has shown the existence of all three types of drift rates when badly

balanced rotors are used to stabilize a platform. Suitable balancing of rotors reduces these effects, as predicted. The effects of unbalance in the retainer rings of the spin axis ball bearings are exactly the same as the effects of rotor unbalance. The inter-rotor effects at retainer speeds are less significant than at rotor speed because, unlike the rotors, the retainers do not rotate in synchronism (due to slippage and axial motion of the balls in the races). Hence this drift will vary at the beat frequencies of the retainers. The easiest cure for inter-rotor drift, and for its variation with changing gimbal angles, is to-reduce wheel unbalance. A more sophisticated cure for all forms of inter-rotor coupling is to prevent the rotors from locking into phase; perhaps by interrupting power, or by using different spin speeds for the two gyros. This procedure does not reduce single-rotor coupling due to r 2 , s^ r r O r s^. Except for the addition of gimbal damping, structural changes are not warranted to reduce these effects. Added damping of the gimbal structure would reduce the magnification of vibration-induced drift if the spin speed lay near a gimbal resonance and will reduce its dependence on heading. The use of isoelastic platform gimbals would not reduce simple inter-rotor coupling. The resonances of the stable element should be far above the spin and retainer frequencies (or far below, as a second choice) to minimize simple inter-rotor coupling. Anisoelastic inter-rotor coupling is reduced by improving rotor balance, by making the gimbals as isoelastic as possible in all positions, by adding damping, and by keeping gimbal resonances away from the spin and retainer speeds.

Bibliography 1 Fellows, W . E . , "The Performance of Gyroscopes in a Vibration Environment," ASME Paper Number 60-WA-246 (Pec. 1960). 2 Stewart, R . M . , "Some Effects of Vibration and Rotation on the Drift of Gyroscopic Instruments," ARS J. 29* 22-28 (1959).

3 Vaccaro, R.J. and D.D. Martin, "Investigation of Steady-State Anisoelastic Torques in Gimbal Systems Under Vibration," ASME Paper Number 58-A-250 (Dec. 1958).

355

Purchased from American Institute of Aeronautics and Astronautics

M. KAYTON

GYRO SENSITIVE AXES fy

GYRO ROTOR.

AND v|j

TOP GYRO

STABLE ELEMENT

GIMBAL AXIS

ACCELEROMETERS I

GYRO SENSITIVE AXES 0

Fig* 1

and

BOTTOM GYRO

Typical stable element.

356

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—I

ROLL GIMBAL

PITCH GIMBAL STABLE ELEMENT

Fig.

2

Platform at an azimuth A,

357

Purchased from American Institute of Aeronautics and Astronautics

M. KAYTON

2HI p

.2 2 , 2 (w - w, X) (w -

2

AMPLITUDE

FREQUENCY

SIMPLE INTER-ROTOR COUPLING

ANISOELASTIC INTER-ROTOR COUPLING AMPLITUDE

(m r L) o o

2

2 2 4 (w, - u>_ ) (co ) sin 2A 1 '

SPIN FREQUENCY

Fig« 3

Variation of inter-rotor coupling with platform heading and frequency.

358

Purchased from American Institute of Aeronautics and Astronautics

STELLAR ALIGNMENT OP A GYROSCOP1CALLY STABILIZED PLATFORM DURING FREE-FALL Saul Moskowitz-xKollsman Instrument Cort>., Elmhurst, N. Y. Abstract The design of instrumentation for space navigation is contingent upon the meaningful definition of the mission, proper choice of navigation or guidance concept and the optimum integration of the human operator. A resulting equipment configuration will fall within the "Spectrum of Navigation Instrumentation," lying somewhere between the extremes of completely manual and fully automatic operation. Certain of the possible configurations which meet the operational requirements of rendezvous, orbital transfer, lunar landing, and atmospheric re-entry include an inertial (gyroscopically stabilized) platform as a basic navigational component. The operational in-flight alignment of such a platform by celestial means is the subject of this paper. The requirements of each operation are defined, and the theoretical aspects of each given solution are investigated, leading to a realistic evaluation of computational requirements and the problems of mechanization. The approaches considered are based upon the use of a four gimbal, all attitude platform along with various feasible combinations of manual sights, an automatic celestial tracker, a moderate accuracy or a high accuracy horizon seeker (depending upon availability) and a space sextant. In addition to conventional approaches, the use of an integrated system incorporating the unique Celestially-Locked Astro-Inertial Measurement Unit is described. Digital Presented as Preprint 63-306 at the AIAA Guidance and Control Conference, Cambridge, Mass., August 12-

iU, 1963.

•^Project Engineer for Space Navigation.

359

Purchased from American Institute of Aeronautics and Astronautics

S. MOSKOWITZ

computational techniques are weighed against analog methods. Computer loading is discussed. Schematic representations and moding diagrams for the different approaches are included, as are basic accuracy considerations. The principal purpose of this paper is an objective engineering evaluation of the employment of a gyroscopically stabilized platform during extended periods of free-fall rather than an advocation of its unquestioned use under all circumstances of space flight.

1.

Introduction

We are today at the threshold of an historical transition in manned space flight. Man's essentially passive function in space travel is about to be replaced by his active participation in this operation. Spacecraft "guidance" is about to be superceded by space navigation. It is in this context that the problem of instrumentation must be considered and solved. This paper presents an analysis of one aspect of navigational instrumentation for space flight, that which involves the use of a gyroscopically stabilized inertial reference platform. (For a discussion of the overall problem see Ref.l.) In order to determine the instrumentation requirements (or range of feasible instrumentation) for a given operation, one must consider the mission itself and the operational constraints imposed upon the mission. In all cases, the limiting constraints imposed on any space navigation system are the characteristics of the available observables. A property of a terrestrial, celestial, solar, or orbiting body must be measurable before it can provide useful information. These observables must be defined before specific instrumentation or data reduction techniques can be set forth. The foreseeable space missions can be considered in terms of the following basic groupings: Earthorbital flight; lunar flight (circumnavigation or landing); and deep-space flight (circumnavigation of Mars or Venus).

These classifications categorize the total mission. It must be recognized though that each of these missions may have aspects or phases common to all. 360

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

These phases then dictate the type, speed, and accuracy of the required observations and computations.

The phase classifications indicated below prove particularly useful in the definition of the system requirements and specifications of the responsive system characteristics: 1) boost-to-injection (Earth or Moon take-off; 2) early midcourse flight; 3) late midcourse flight; ij.) near-body (orbital) flight; 5) variation of orbital trajectory: 6) rendezvous with target in orbit: and ?/ landing (including atmospheric re-entry for the Earth). Operational considerations indicate that there is a possible place for the use of a gyroscopically stabilized inertial reference platform in the navigational functions required for phases 1, ij., 5, 6, and 7 in the foregoing. Phase 1 needs little further comment. It is the classical problem of missile boost guidance. The inertial platform is ground aligned and its accelerometer outputs are used to derive cut-off information. It is for phases ij., 5, 6, and 7 that it is necessary to consider in-flight alignment of the platform. Near-Body Flight During near-body or orbital flight, a local vertical reference can be of value for purposes of navigation to future points by celestial sightings as well as preparation for the initiation of the operations of orbital transfer and landing. Note though, that if stellar sightings are used for alignment of the platform (instead of a horizon seeker, for instance) it is not meaningful to use such a derived vertical for purposes of celestial navigation. To do so would be to be caught in meaningless circular operations.

Variation of Orbital Trajectory A number of general exploratory as well as defensive military operations will require that the astronaut alter his trajectory to meet changing situations, initiate rendezvous operations with targets in either Earth orbits or lunar orbits, or prepare for re-entry and landing at a safe base. 361

Purchased from American Institute of Aeronautics and Astronautics

S. MOSKOWITZ

Thrust application will have to be referenced to a chosen coordinate frame which can be retained by a stabilized platform* Rendezvous with Target in Orbit

The normal rendezvous operation will be required for refueling, the construction or repair of orbiting space stations, and the return to the primary vehicle after a lunar landing. Most rendezvous guidance schemes advanced to date are based on the reduction of target line-of-sight angular rates to zero and then target closure by a simultaneous reduction of range and range-rate* The angular rates of the line-of-sight must be measured against a fixed, but arbitrary, reference. This reference can be an inertial platform. However, the platform need not be aligned to any particular orientation. Landing (with Atmospheric Re-Entry )

Self-contained navigation systems must include the capability for a lunar landing (where applicable) and the safe return to an Earth base. Thus, in addition to considering the aerodynamic characteristics of the spacecraft, it is necessary to obtain data relating to its present and predicted motion. Attitude control is of great importance and must, at some point, be based upon the determination of the local vertical. The gyroscopically stabilized platform provides a means for "remembering" the instantaneous local vertical. Although various mission phases have been discussed above, the detailed examination of a single operation, that of alignment of the platform to a local Earth vertical during free-fall, provides sufficient insight into all such related problems that it alone is discussed in this paper. The following sections contain a theoretical description of the problem, realistic evaluation of computational requirements, and a suggested mechanization for a complete system. Block diagrams illustrating the various modes of operation are included within the text. It is important to remember that the formulation presented is based upon a general consideration of the problem and so is open to modification based upon the special requirements of, or constraints imposed upon a 362

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

particular application. 2.

Phases of Operation

Alignment of a gyroscopically stabilized platform to the local Earth vertical by means of celestial data may be considered in terms both of equipment configuration and phases of operation. The possible equipment configurations are: 1) inertial platform, automatic star tracker; 2) inertial platform, moderate accuracy horizon seeker, automatic star tracker: 3) inertial platform, high accuracy horizon seeker, automatic star tracker: Ij.) inertial platform, moderate accuracy horizon seeker, sextant; 5) inertial platform, sextant, automatic star tracker; 6) inertial platform, sextant; 7) inertial platform, sextant, high accuracy horizon seeker; 8) inertial platform, sextant, moderate accuracy horizon seeker; and 9) inertial platform, sextant, automatic star tracker, moderate accuracy horizon seeker. Figure 1 presents the schematic combination of all of the previously listed components. All of these instruments are mounted upon a common rigid base so that pointing information derived by each can be related to the others through inter-gimbal transformations. A i[.-gimbal inertial platform (capable of complete attitude freedom) is indicated. Both the horizon sensor and star tracker are in level, cross-level gimballing configurations. In the above listing, the horizon sensor was indicated with either high or low accuracy. Ideally, a high accuracy instrument is preferable. To date, however, only limited accuracies have been demonstrated in flyable hardware. Realistically, therefore, it may be necessary to base an overall system upon the availability of only moderately accurate horizon sensors. Figure 1 also shows a pictorial representation of a space sextant with both visul and automatic readouts. The gimbal suspension shown effectively isolates the sextant from the spacecraft's attitude and angular motion. Therefore, the instrument may be employed for data acquisition even under conditions of low, constant, arbitrary tumble rates. The sextant can be mounted behind either a flat window or astrodome, the astrodome providing wider field capabilities. With appropriate isolation 363

Purchased from American Institute of Aeronautics and Astronautics

S. MOSKOWITZ

bellows it can even be open to the space environment. The space sextant shown in the overall schematic is based upon a design put forth by the Kollsman Instrument Corporation during the early part of 1961. Figure 2 shows this instrument in greater detail. (Although some aspects of later versions of this instrument are somewhat different from the illustrated unit, it is nontheless representative of a type of sextant currently under consideration.)

The three phases of operation are: A) coarse alignment; B) search and acquisition; and C) fine alignment. The possible combinations of items 1 through 9 with A through C are shown in Pig. 3* The choice between the various approaches will be determined to a great extent by the configuration of the spacecraft (due to possible obscuration of portions of the heavens) and the porper functioning of each component and subsystem. The total composite of instrumentation as shown in Pig. 1 supplies a redundancy which becomes evident from an examination of Fig/ 3. Thus depending upon his actual situation, the astronaut will have to resort to one or another of the possible approaches. 3.

Initial Data and Orbital Parameters

The geometry of Earth-orbital flight is shown pictorially as Fig* ij.. The spacecraft is in a closed elliptical orbit about the Earth. The track of the spacecraft's nadir on the Earth's sphere (or similarly the intersection of the Earth's sphere and the orbital plane) traces out an inertially fixed great circle which is the projection of the orbit. The track of the spacecraft's zeniths traces out a great circle on the celestial sphere. Upon the celestial sphere (essentially at infinity) lie the reference navigation stars.

Based upon computational considerations it is desirable to store reference star data in terms of the components (in a fixed inertia! space) of normalized pointing vectors to the stars. Thus, each of the required reference stars is characterized by the vector (in matrix notation), 364

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

^TT)

(1)

s(l)

=

For practical purposes the reference space for the storage of stellar data is a North-First Point of Aries oriented frame. Transformation of the ¥Ti7 into the coordinate frame defined by the orbital plane and the point of perigee is accomplished by the means of the matrix A,

s f = As

(2)

where s is the column fmatrix of the pointing vector in reference space, s the column matrix in orbital plane space and A is the square matrix /An

A13

A =

A22

(3)

A32

These nine components can be transmitted to the vehicle from the ground tracking station complex or computed in the vehicle. In either case, A is formed from the product of three rotational transformations, , 6, and ty( o), in that order. The first, j£, corresponds to a rotation about the polar axis which brings the First Point of Aries axis into the plane containing the normal to the orbital plane and the polar axis. The second, &9 represents the angle of inclination between the polar axis and the normal. The third, llf(o), corresDonds to a rotation about the normal to the orbital plane which brings the transformed First Point of Aries axis into the point of perigee. Thus the matrix A is given by \ A =

/cos0 -sin6 0\

in0 0

cos0 0 0

365

1

/i o 0

\

(10

Purchased from American Institute of Aeronautics and Astronautics

S. MOSKOWITZ

If, then, the position of the vehicle in the

orbital plane is defined by the parameters llf( t), the orbital angle (true anomaly), and r(t), the length of the radius vector to the vehicle, the transformation of the stellar pointing vector components into the local Earth level-orbital plane space is accomplished by means of the matrix (^"(t) ), /I

0

0

0

cosT/f(t) sin^(t)

(5)

cos T/f(t)}

Depending upon whether the platform is continuously torqued to local Earth vertical or it is held to a

perigee point orientation and the local vertical is obtained analytically, the transformation given by (5) will or will not be applied to the components of f

s . In either caseT^(t) must be computed as a function of time.

The computation of "^(t) can be based upon some set

of the constants of the orbit. If the onboard computations are to be performed by a computer function in a DDA mode, then the appropriate constants of the orbit can be r(O), the initial value of the radius vector, E, the total energy of the system, and i, the angular momentum of the system. Then solution of the following pair of equations will yield the desired function.

Instead of the foregoing approach, a concept developed from the technology of special purpose analog computation can be employed. The appropriate differential equations can be solved in parametric form for 1/T(t) as a function of t, time, £, the eccentricity \ of the orbit, and T, the period of the

orbit. This functional relationship is shown graphically as Pig. 5>. Actually shown is the difference between the true anomaly for a circular 366

Purchased from American Institute of Aeronautics and Astronautics

GUIDANCE AND CONTROL—II

orbit and an orbit of eccentricity, £ . Note that by using T normalized time, the family of curves is dependent only upon £ and correspondingly independent of orbital size. Ij..

Initial Alignment of the Platform

Initial, or coarse, alignment can be achieved either by aligning the spacecraft with the orbital plane and uncaging to this attitude, or by slaving the platform to a gimballed horizon seeker after the spacecraft has been oriented in azimuth, or by the simple expedient of using handset attitude inputs. For the second, and most complicated alternate, the problem becomes that of developing those error signals for the torquing of the gyros which will result in the platform assuming the desired orientation. The horizon seeker vertical is denoted by the vector /0\

(8)

v 5 \v/

If the platform is to be continuously precessed into a local vertical orientation, equation (8) will be transformed from horizon seeker space, through vehicle space, into platform space to yield the vector

.A

/

/

e2 \ 3" /

tt

(9)

3 I

If the platform is to be kept fixed in inertial space and the solution for the vertical to be analytical, then v f f must also be rotated through -TJf(t) to yield vield the the vector components corrmonents / v

2

, t

t v 3-;

\

367

ei'\ e 2'

do)

Purchased from American Institute of Aeronautics and Astronautics

S. MOSKOWITZ

Initial alignment is accomplished by driving either & T I and 6pf ff or I T £ ! ^° zero* Consider the former alternative . - Horizon seeker cross-level axis - Horizon seeker level axis - Platform outer roll axis P-^ - Platform pitch axis

/?2 - Platform inner roll axis p\ - Platform azimuth axis

Using the foregoing notation (based upon Pig. 1), the transformation of v into V T T is given by (11)

Each transformation corresponding to an Eulerian rotation is represented in matrix notation below:

1

0

0

1

-sin p 2

0

1

1

0

sin/