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Attitude control of spinning satellites using environmental forces Pande, Kailash Chandra 1973-12-31

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ATTITUDE CONTROL OF SPINNING SATELLITES USING ENVIRONMENTAL FORCES by KAILASH CHANDRA PANDE B.Sc. Eng. (Hons.), Banaras Hindu University, 1966 M.Sc, University of Saskatchewan, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 19 7 3 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that publication, in part or in whole, or the copying of this thesis for financial gain shall not be allowed without my written permission. KAILASH CHANDRA PANDE Department of Mechanical Engineering The University of British Columbia, Vancouver 8, Canada Date ABSTRACT The feasibility of utilizing the environmental forces for three-axis librational damping and attitude control of spinning satellites is investigated in detail. An appreciation of the environmental influence is first gained through a librational dynamics study of spinning, axisymmetric, cylindrical satellites in the solar radiation pressure field. The highly nonlinear, nonautonomous, coupled equations of motion are analyzed approximately using the method of variation of parameters. The closed form solution proves to be quite useful in locating periodic solutions and resonance characteristics of the system. A numerical parametric analysis, involving large amplitude motion, establishes the effect of the radiation pressure to be substantial and destabilizing. Next, a possibility of utilizing this adverse influence to advantage through judiciously located rotatable control surfaces is explored. A controller configuration for a dual-spin spacecraft is analyzed first. The govern ing equations, in the absence of a known exact solution, are solved numerically to evaluate the effect of system parameters on the performance of the control system. The available control moments are found to be sufficient to iii compensate for the rotor spin decay, thus dispensing with the necessity of energy sources maintaining the spin rate. The controller is able to damp extremely severe disturbances in a fraction of an orbit and is capable of imparting arbitrary orientations to a satellite, thus permitting it to undertake diverse missions. The development of an efficient yet structurally simple controller configuration is then considered. A logical approach for solar controller design is proposed which suggests a four-plate configuration. Its performance in conjunction with a bang-bang control law is studied in detail. The utilization of maximum available control moments leads to a substantial improvement of the damping characteristics. Attention is then focussed on using the earth's magnetic field interaction with onboard dipoles for attitude control. Magnetic torquing, however, is unable to provide first order pitch control in near equatorial orbital planes. The shortcoming is overcome by hybridizing the concepts of magnetic and solar control. Two magnetic controller models, employing a single rotatable dipole or two fixed dipoles, are proposed in conjunction with a solar pitch controller. The system performance is evaluated for a wide range of system parameters and initial conditions. Although high spin rates lend considerable gyroscopic stiffness to the spacecraft, the controllers continue to be quite effective even in the absence of any spin. Even with extremely severe disturbances, damping times of the order of a few orbital degrees are attainable. As before, the concept enables a satellite to change the desired attitude in orbit. The effectiveness of the controllers at high altitudes having been established, the next logical step was to extend the analysis to near-earth satellites in free molecular environment. A hybrid control system, using the solar pressure at high altitudes and the aerodynamic forces near perigee, is proposed. The influence of important system parameters on the bang-bang operation of the controller is analyzed. The concept appears to be quite effective in damping the satellite librations. Both the orbit normal and the local vertical orientations of the axis of symmetry of the satellite are attainable. However, for arbitrary pointing of the symmetry axis, small limit cycle oscillation about the desired final orientation results. Finally, the time-optimal control, through solar radiation pressure, of an unsymmetrical satellite executing planar pitch librations is examined analytically. The switching criterion, synthesized for the linear case, is found to be quite accurate even when the system is subjected to large disturbances. V Throughout, the semi-passive character of the system promises an increased life-span for a satellite. vi TABLE OF CONTENTS Chapter Page 1. INTRODUCTION . . 1 1.1 Preliminary Remarks 1 1.2 Literature Review 2 1.3 Purpose and Scope of the Investigation 12 2. LIBRATIONAL DYNAMICS OF SPINNING AXISYMMETRIC SATELLITES IN PRESENCE OF SOLAR RADIATION PRESSURE . 15 2.1 Formulation of the Problem 16 2.2 Analytical Results 23 2.2.1 Approximate analytical solution 2 3 2.2.2 Periodic solutions of the system 30 (a) High frequency oscillations 31 (b) Low frequency oscillations 32 (c) Solar pressure excited oscillations 36 (d) Accuracy of the analytical solution 38 (e) Stability of periodic solutions 38 2.2.3 Resonance 41 2.3 Numerical Results 5 vii Chapter Page 2.3.1 Significant system parameters . . 4 5 2.3.2 System plots 47 2.3.3 Design plots 50 2.4 Concluding Remarks 2 3. ATTITUDE CONTROL USING SOLAR RADIATION PRESSURE 54 3.1 Feasibility of Solar Pressure Control. . 55 3.1.1 Equations of motion 55 3.1.2 Controller configuration .... 59 3.1.3 Control strategy 65 3.1.4 Results and discussion 68 (a) Nutation damping 69 (b) Rotor spin decay 75 (c) Attitude control 78 (d) Illustrative example .... 79 3.2 Improvement of Controller Design .... 81 3.2.1 Equations of motion 83 3.2.2 Development of controller models 83 3.2.3 Control strategy 90 3.2.4 Results and discussion 96 (a) Libration damping in circular orbits 9(b) Libration damping in elliptic orbits 99 (c) Attitude control 102 (d) Illustrative example . . . . 104 viii Chapter Page 3.3 Concluding Remarks 105 4. MAGNETIC-SOLAR HYBRID ATTITUDE CONTROL . . . 108 4.1 Formulation of the Problem 109 4.1.1 Equations of motion 109 4.1.2 Magnetic roll-yaw control .... m 4.1.3 Solar pitch control 119 4.2 Results and Discussion 121 4.2.1 Nutation damping 124.2.2 High spin rates and spin decay 129 4.2.3 Attitude control 131 4.2.4 Illustrative example ...... 133 4.3 Concluding Remarks . 135 5. AERODYNAMIC-SOLAR HYBRID ATTITUDE CONTROL . . 136 5.1 Formulation of the Problem 137 5.1.1 Equations of motion 137 5.1.2 Controller configuration and generalized forces 137 5.2 Control Strategy 142 5.2.1 High altitude 143 5.2.2 Low altitude 145 5.3 Results and Discussion 148 5.3.1 Libration damping 149 5.3.2 Attitude control 154 5.4 Concluding Remarks 156 ix Chapter Page 6. TIME-OPTIMAL PITCH CONTROL USING SOLAR RADIATION PRESSURE . . . 158 6.1 Formulation of the Problem 159 6.2 Time-Optimal Synthesis 163 6.3 Results and Discussion 167 6.4 Concluding Remarks .......... 171 7. CLOSING COMMENTS 172 7.1 Summary of the Conclusions 172 7.2 Recommendations for Future Work .... 174 BIBLIOGRAPHY 176 APPENDIX I 184 APPENDIX II 5 X LIST OF TABLES Table Page 4.1 Response With High Spin Rates 130 xi LIST OF FIGURES Figure Page 1.1 Schematic diagram of the proposed plan of study 14 2.1 Geometry of motion of spinning satellite in the solar pressure environment 17 2.2 Evaluation of generalized forces due to ,solar radiation pressure: (a) curved surface 19 (b) flat ends2.3 Frequency and initial conditions for high frequency periodic motion 33 2.4 Frequency and initial conditions for low frequency periodic motion 35 2.5 Initial conditions leading to solar pressure excited periodic motion 37 2.6 Typical periodic solutions of the system 39 2.7 Resonance conditions in system parameter space 4 3 2.8 Typical responses under resonant conditions 4 2.9 Typical unstable responses demonstrating the significance of system parameters ... 46 2.10 System plots showing the coning angle and average nodding frequency as affected by: (a) inertia parameter 48 (b) spin parameter .  8 (c) orbit eccentricity 48 (d) orbit inclinationxii Figure Page 2.11 Typical stability charts showing adverse influence of solar radiation pressure: (a) e = 0 51 (b) e = 0.13.1 Geometry of motion of dual-spin satellite in the solar pressure environment 56 3.2 Solar controller configuration 62 3.3 Radiation force on a plate element 63 3.4 Influence of the solar controller gains VK'^i on. the response 70 3.5 Damped response as affected by satellite inertia and spin parameters 71 3.6 Effect of solar parameters and solar aspect angle on the damped response 73 3.7 Response plot showing limit cycle oscillations in eccentric orbits and their removal through the modified control function 74 3.8 Response as affected by: (a,b) orbital inclination from the ecliptic 76 (c) rotor spin decay 7(d) modified control function 76 3.9 Effectiveness of the solar controller in achieving arbitrary orientations of the satellite 80 3.10 Projected controller performance in achieving nutation damping and attitude control of INTELSAT IV and Anik satellites 82 3.11 Development of solar controller con figurations 85 3.12 Proposed four-plate solar controller model  9 3.13 Quadrant constraint for plate rotation 6 94 l xiii Figure Page 3.14 Optimization plots for the solar controller gain m 97 3.15 (a) Typical optimum response in circular orbits 100 (b) Limit cycle oscillation in elliptic orbits(c) Variation of limit cycle amplitude with system parameters 100 3.16 Effectiveness of the four-plate model in achieving arbitrary orientations of the satellite 103 4.1 Geometry of motion of dual-spin satellite in the earth's magnetic field 110 4.2 Magnetic-solar controller configurations . . 115 4.3 Optimization plots for the magnetic-solar controller gain m 122 4.4 Damped response and time history of con trols subsequent to initial impulsive disturbance 127 4.5 Damped response and time history of con trols subsequent to initial position disturbance . . . . 128 4.6 Effectiveness of the magnetic-solar controller in imparting arbitrary orientations to the satellite 132 5.1 Aerodynamic-solar hybrid controller configuration 139 5.2 Optimization plots for the aerodynamic-solar controller gain m 150 5.3 Typical responses showing the effectiveness of the aerodynamic-solar controller at different orbital positions 153 xiv Figure Page 5.4 Effectiveness of the aerodynamic-solar controller in imparting arbitrary orientations to the satellite 155 6.1 Geometry of motion of unsymmetrical satellite in the solar pressure environment 160 6.2 (a) Variation of IQ, I with ? 162 1 ijrmax (b) Phase plane portrait of the system . . . 162 6.3 (a) Variation of transient amplitude Ix,(G)I with initial condition 1 1 1 max x2(0) 168 (b) Variation of switching time 0 and final time 9f with initial condition x2(0) . . . t 168 6.4 System response to impulsive disturbance . . 169 ACKNOWLEDGEMENT The author wishes to express his gratitude to Dr. V.J. Modi for the guidance given throughout the preparation of the thesis. His encouragement, help and patience have been invaluable. The investigation reported in this thesis was supported by the National Research Council of Canada Grant No. A-2181. xvi LIST OF SYMBOLS A,A^ area of single control plate, i=l,2,3,4 B geomagnetic induction vector B^,B_.,Bk; components of B/Dm along the x,y,z and Bxn'Byn'Bzn] Xn'Yn'zn axes' respectively C,C\ solar parameter, i=1, 2,3,4 C,Ci 2ppoR\iei/y, 1=1,2,3,4 Cm magnetic parameter 3 25 D geomagnetic dipole moment, M /R ; M =8.1 x 10 m e e 3 gauss-cm E(9) (1+e) 3/(l+ecos6) 4 G solar aspect ratio I inertia parameter of axisymmetric (I^=Iz) satellite, I /I x y I ,1 ,1 principal moments of inertia of the satellite I _»I „ moment of inertia of the platform and the rotor about the axis of symmetry of the satellite, respectively J platform inertia fraction, I /I xp/ x J(6) { (1+e) 2 (l+2ecos0+e2)/(l+ecos9) 4} [{R (1+e)/ P n (1+ecosG)-R }/(R -R )] e ' p e 3 1/2 viscous damping parameter, Kd(Rp/u) / /I K K-. viscous damping coefficient XVI1 inertia parameter of unsymmetrical satellite, (i -I )/i ' z y x center of force pericenter total generalized force, l=a , & ,y, \ ,\\) generalized force due to solar radiation pressure, i=a,3,Y/^ generalized force due to magnetic interaction, generalized force due to aerodynamic forces, i=3,Y,A distance between the satellite center of mass and the center of force radius of the earth distance between the pericenter and the center of force center of mass of the satellite switching function, i=&,y,\ kinetic energy gravitational potential energy polarity velocity of the satellite center of mass orbit eccentricity distance between the satellite center of mass and the hinge point T, Figure 3.11a xvi ii 2 dipole strength ha,hg constants of motion h perigee altitude p i,i inclinations of the orbital plane from the m ecliptic and the equatorial planes, respectively i,j,k unit vectors along x,y and z axes, respectively £,r length and radius of cylindrical satellite, respectively m bang-bang control system gain m mass of the satellite s ne,nr rates of the earth's rotation and regression of the line of nodes, respectively n (3K.COS2IJJ )1//2, -TT/4<IJ; <TT/4 p,p^,P2 unit vector along dipole orientation, p=Pii+Pj:+Pkk -5 2 PQ solar radiation pressure, 4.65 x 10 dynes/cm r position vector of area element from the center of mass of the satellite t time u unit vector in the direction of the sun, u. i+u^ j+u, k I 3J k u^ coscj) (sinycosBcosn+sinBsinn) + sin(J>{cosi (sinycosB sinn-sinBcosn)- sinicosBcosy} u_. cos<})cosYcosri+sin(j) (cosicosysinn+sinisinY) XXX cos<j> (sinysinBcosn-cosBsinn) +sintj>{ cosi (siny sinBsinn+cosBcosn)-sinisinBcosy} control vector unit vector in the direction of atmospheric flow relative to the satellite center of mass, v. i+v . j+v, k 1 j k 2 1/2 {sin8+e(sinBcos6-cosBsinysin6)}/(l+2ecos0+e ) 2 1/2 -ecosysinB/(l+2ecos0+e ) 2 1/2 -{cosB+e(cosBcos0+sinBsinysin9)}/(l+2ecos0+e ) ' inertial coordinate system with x normal to 1 n the orbital plane and yn along the line of nodes referred to the equatorial plane inertial coordinate system with x' normal to the orbital plane and y* along the pericenter rotating orbital coordinate system with XQ normal to the orbital plane and yQ along the local vertical intermediate body coordinates resulting from rotations y and 3 about z and y, axes, o 11 respectively platform-fixed coordinate system coning angle of the axis of symmetry, cos ^(cosBcosy) state-transition matrix (Equation 6.8) angles between the vernal equinox and the lines of nodes referred to the ecliptic and the equatorial planes, respectively XX a,Q,y,X attitude angles a1_'a2 constants in Chapter 2 and Appendices I and II; longitude and latitude of the plate support arm in the xyz-reference, respectively (Figure 3.11a) Bc,Yc/^c position control parameters Sf,Yf/^£ final desired attitude 3m longitude of the plane containing both the geographic and geomagnetic polar axes from the vernal equinox (Figure 4.1) 6,6^ control plate rotation, i=l,2,3,4 e distance between the center of pressure and the center of mass of the satellite (Chapter 2); distance between the center of pressure of the control plate and the hinge point T of the support arm (Chapter 3) e. distance between the center of pressure of control plate and the satellite center of mass, i=l,2,3,4 em geomagnetic polar axis declination, 11.4° C n+i|j-tan ^ (tancjicosi) n,n (u)+9) and (w +9) , respectively m m 1 ' r 9 orbital angle angular position of unit vectors p and p^ m S f from the y axis (Figure 4.2) switching time and final time, respectively XXI characteristic multipliers, i=l,2,3,4 u gravitational constant y^,v^ gains of linear controller with saturation constraints angle of incidence, ? = cos "'"(u'n) ; 1 m i=l,2,3,4 C • angle between the direction of relative a ax flow and plate normal, %, = cos ^"(vn); 3. i=l,2,3 p,x reflectivity and transmissibility of satellite surface or control plate, respectively p ,p atmospheric density and its value at the a ap 2 perigee altitude h^, respectively a spin parameter damping time cf) solar aspect angle, angle between the sun-line and the line of nodes referred to the ecliptic plane rm mm 4> planar pitch attitude of unsymmetrical satellite \\i nominal pitch attitude of unsymmetrical satellite o),^ arguments of the perigee referred to the ecliptic and the equatorial planes, respectively xxii ID average nodding frequency of the satellite axis of symmetry ^•^,^2 frequencies of the high and low frequency periodic motion, respectively Dots and primes indicate differentiation with respect to t and 8 respectively. The subscript o indicates initial condition. 1 1. INTRODUCTION 1.1 Preliminary Remarks Success of a vast majority of space missions depends on the ability of a spacecraft to point accurately in the desired direction. Even a correctly positioned satellite tends to deviate in time from its preferred orien tation due to environmental influences, such as, micrometeor-ite impacts, solar radiation pressure, aerodynamic forces, gravitational and magnetic field interactions, etc. Fortunately, several methods of attitude control are available which damp the resulting undesirable librations. These procedures may be broadly classified as active, passive and semi-passive (or semi-active). Active stabilization procedures involve mass expulsion schemes and/or components requiring a large amount of energy, an expensive commodity aboard a spacecraft, leading to increased weight and space requirements with a reduced satellite life-span. The main advantage of the technique is its ability to achieve a specified orientation with almost any desired degree of accuracy. Stabilization techniques requiring no power consumption are termed passive. This is generally achieved by designing satellites with physical characteristics which 2 interact with the environmental forces in a manner so as to attain a specific equilibrium position. Spin stabilization presents an alternative that relies on the inherent tendency of a spinning body to maintain its attitude in space. The pointing accuracies attained through passive methods, however, are limited and deteriorate due to the influence of environmental forces. The semi-passive methods attempt to utilize the environmental forces, through the introduction of appropriate controllers, and thereby achieve attitude control. The possibility of attaining high pointing accuracies with low power consumption promises an increased satellite life-span. The design of suitable controller configurations, however, requires a thorough understanding of the system dynamics under the influence of the environmental force used for control. The development and analysis of several semi-passive control systems, with particular reference to spinning satellites, forms the main objective of this thesis. 1.2 Literature Review Spinning bodies have received,in the past decade, considerable attention owing to their particular stability properties. For rigid axisymmetric bodies under the 3 influence of gravity forces and with the axis of spin perpendicular to the orbital plane, Thomson''" (1962) presented a stability criterion using linearized analysis while 2 Pringle (1964) investigated motion in the large employing the Hamiltonian as a Lyapunov function. Asymmetry was taken 3 into account by Kane and Shippy (1963) applying the Floquet 4 theory. The same method was used later by Kane and Barba (1966) to deal with motion in the small for arbitrary eccentricity. Wallace and Meirovitch^ (1967) studied the same problem by an asymptotic analysis in conjunction with 6-9 Lyapunov's direct method. Neilson and Modi (1968-72) gave insight into the problem of stability in the large by making use of the integral manifold concept. According to classical mechanics, the stable rotational motion of a rigid body in absence of external forces is possible only if the axis of rotation is a prin cipal axis of least or greatest inertia. If the body is not rigid and energy is dissipated by the cyclic forces acting on it while under nutation, then only the motion about the axis of maximum inertia is stable. It turns out that for slowly spinning rigid satellite, the internally dissipated energy is such as to overcome the stabilizing influence of gravity and the system ends up in a state of tumbling about the axis of maximum moment of inertia: the classical example is that of Explorer 1^. 4 The constraint of "major axis spin rule" was sub sequently removed by the introduction of the dual-spin concept which allows two sections to nominally rotate about a common axis at different rates relative to inertial space. An early paper by Roberson11 (1958) had anticipated that torques generated by a disc rotating about an axis fixed in a rigid body could substantially affect its motion. However, the first fundamental contribution to the feasibility of 12 dual-spin stabilization was by Landon and Stewart (1964): an energy-sink method indicated no constraints on inertias when energy dissipation took place on the slowly-rotating 13 part of the system. Iorillo , a year later, extended this concept to the case where energy dissipation occurs on both bodies. 14 Likms (1967) developed, for a specific configura tion involving an axisymmetric rotor and an asymmetric body containing a ball-in-tube damper constrained to move parallel to the rotor axis, an accurate stability criterion based 15 on the Routh analysis. Mingori (1969) took a more general approach which involved two dissipative sections. The Floquet analysis stressed the sensitivity of the system behaviour to the relative effectiveness of the sources of 16 energy dissipation. Pringle (1969) extended his theorems on Lyapunov stability to the case of dual-spin spacecraft and gave a rigorous proof of the "maximum moment of inertia 17 spin-axis" rule. Cloutier (1968) investigated the 5 stability and performance of a nutation damper consisting of mass shifting perpendicular to the spin-axis: again, it led to no restrictions on inertia ratios or damper size when dissipation occurs on a despun platform. In another 18 paper (1969), the author extended the analysis to a damper involving two degrees of freedom in a plane perpen dicular to the spin axis. An approximate solution was derived for the nutation angle and its decay was optimized 19 in terms of system parameters. Sen (1970) studied a four mass nutation damper whose design constraints were not as 20 severe as those of Likins. Vigneron (1971) applied the method of averaging to obtain a closed-form first approxi mation solution for a dual-spin system containing both 21 platform and rotor mounted dampers. Bainum et al. (1970) conducted a stability and performance analysis of the dual-spin Small Astronomy Satellite (SAS-A) and found that asymmetry noticeably deteriorates the performance of the nutation 22 damping system. In a subsequent paper (1972), the authors included the effect of damping in the momentum wheel by permitting the plane of the wheel to flex with two degrees of freedom with respect to the hub. The analysis established stability criteria for the SAS-A satellite. Although passive methods for attenuating nutation are generally reliable and conceptually simple, their effectiveness may be limited. A device containing an actively controlled mass, capable of attaining any nutation 6 23 angles, was studied by Kane and Scher (1969). Mingori 24 et al. (1971) analyzed both semi-passive and active nutation dampers for dual-spin spacecraft, the former in volving single axis control moment gyros (CMG's) whose rotational motion relative to the spacecraft was restrained passively by a spring and dashpot. The active damper was realized by controlling the CMG's in accordance with the information from a rectilinearaccelerometer. Both devices were found to be capable of reducing nutation several times faster than passive dampers of equal mass. Studies which take the gravity torque into account are rare but offer, nevertheless, valuable results. Of 25 particular interest is the conclusion by Kane and Mingori (1965) that the stability of undamped axisymmetric dual-spin satellites is equivalent to that of rigid spinning bodies. 2 6 White and Likins (1969) extended the research to slightly asymmetric system by making use of asymptotic expansions 27 2 8 and resonance lines. The efforts of Roberson et al. ' (1966, 1969) and Yu29 (1969) should also be noted who analyzed the equilibrium positions of a single rigid body containing a symmetric, constant speed, fixed axis rotor, also called a gyrostat, in presence of gravity forces. The concept of dual-spin spacecraft gained sufficient recognition by 1967 to be considered seriously as a design alternative"^. The first flight data for a prolate dual-spin satellite (Tacsat 1), however, became publicly available 7 31 only recently (1970). It was found that the spacecraft did not maintain the nominal inertial attitude but executed stable limit cycle in a nearby state of free precession. None of the previous analyses employing linear models for the energy dissipation mechanisms could explain this 32 anomaly. In a recent paper, Likins et al. (1971) found, via the 'energy-sink' method, the limit cycle oscillation to be a consequence of nonlinearities in the damping forces. On the other hand, the possibility of constant or variable amplitude limit cycles due to nonlinear restoring forces 33 has been indicated by Mmgon et al. (1972) . It should be emphasized that although several authors have recognized the importance of environmental forces, they were ignored in the analyses of spinning space craft discussed above. Although extensive volume of literature exists on the librational dynamics of gravity stabilized satellites34'35,et al" , serious efforts at analyzing the influence of environmental forces and exploiting them for 3 6 attitude control are relatively recent. Roberson (1958) 37 gave a general outline of the problem and Wiggins (19 64) presented estimates of the relative magnitudes of these forces. Clancy and Mitchell38 (1964) and Modi et al.39,40 (1971) investigated the influence of solar radiation pressure on the attitude motion of satellites. The effect of the atmosphere on satellite librations was the subject of 8 study by Debra41 (1959) , Schrello42 (1961), Garber43 (1963) , 44 Meirovitch and Wallace (1966), et al. The environmental effect, in general, was found to be detrimental to the satellite performance. On the other hand, the environmental forces offer an exciting possibility of trajectory and attitude control through the introduction of a carefully designed controller. As no mass expulsion or active gyros requiring large power consumption are involved, these schemes are essentially semi-passive and hence promise an increased satellite life span. The use of solar radiation pressure for propulsion 45 within the solar system was first proposed by Garwin (19 58). 46 Sohn (1959) suggested a specific configuration using plates of large surface areas to orient the satellite with respect 47 to the sun. Galitskaya and Kiselev (1965) studied, qualitatively, the principle of libration control of space 48 probes about three axes. Mallach (1966) presented a system for solar damping of gravity oriented satellites and gave a simplified analysis using average torques. Modi and 49 Flanagan (1971) examined the planar attitude control of a gravity gradient system in an ecliptic orbit using the 50 solar pressure as a damping torque. Modi and Tschann (1971) extended the analysis by the introduction of a displacement and velocity sensitive controller enabling the satellite to attain any desired orientation. Modi and 9 51 Kumar (1973) further generalized the concept for the case of three degree of freedom motion. Their analysis demon strated the feasibility of achieving general three-axis librational damping and attitude control. Literature on the use of solar pressure for attitude control of spinning vehicles appears to be rather 52 limited. Ule (1963) was the first to consider aligning the spin axis along the sun-satellite line employing a corner mirror array fixed to the spacecarft. Similar devices 53 54 were further explored by Peterson (1966) , Colombo (1966) , 55 56 Falcovitz (1966), et al. Crocker (1970) considered the same problem using spacecraft-fixed and spring-mounted paddles. An adequate nutation damper was assumed so that the angular momentum vector remained close to the spin axis. A possibility of using the solar pressure for general three-axis librational damping and attitude control of spinning satellites remains virtually unexplored. It would Joe appropriate to mention here the experi-57 ment aboard Mariner IV spacecraft , conducted on depletion of the attitude control gas, to align the roll axis along the sun-line using passive solar radiation control. Each of the four solar panels was provided with a rotatable solar pressure vane for this purpose. Unfortunately, one of the vanes proved to be inoperative during a major portion of the mission. However, subsequent reactivation of the vane enabled the solar pressure control system, in con-junction with active gyros, to maintain the spacecraft attitude within 1° of the sun-line. Generation of control torques through the inter action of onboard electromagnetic dipoles and the earth's magnetic field appears to be particularly attractive as the system reliability is enhanced by the elimination of moving parts. Libration damping of gravity oriented satellites 5 8 was considered by Alper and O'Neill (1967) who proposed a 59 passive hysteresis damper. Bainum and Mackison (1968), on the other hand, considered three mutually perpendicular electromagnets controlled according to the sample and hold concept. Although time constants of approximately one to two orbits for roll-yaw damping were achieved, the inadequacy of the system for pitch control in equatorial orbits, where the geomagnetic field is nearly parallel to the orbit normal, became apparent. The problem of maintaining the spin axis of a satellite perpendicular to the orbital plane has been a 6 0 subject of considerable study by Vrablik et al. (1965), Sonnabend61 (1967) and many others. Fischell62 (1966) considered the possibility of using magnetic control for 6 3 regulating the spin rate of a satellite. Wheeler (1967) investigated the use of a single dipole along the spin axis for both attitude control and nutation damping. The analysis, however, assumes the desired final orientation 11 to be inertially fixed and the spin rate constant during 64 the control maneuver. Sorensen (1971) applied the Kalman filter technique to estimate pointing errors for a system with limited attitude determination capabilities and developed the minimum energy control law using these estimates. In a recent paper, Shigehara6^ (1972) studied a control law, based on the asymptotic stability criterion, for the spin-axis and the spin rate using dipoles along the axis and perpendicular to it, respectively. The first operational magnetically controlled satellites, 6 6 the TIROS wheels, were discussed by Hecht and Manger (1964) 6 7 and Lindorfer and Muhlfelder (1966). The use of aerodynamic forces for attitude control of satellites in near-earth orbits was a subject of several early discussions by Wall68 (1959), Schrello69 (1961), et al. In practice, however, it has been used successfully only for the pitch control of C0SM0S-149, with the other 70 degrees of freedom governed by gyroscopic forces 71 Ravindran (1971) optimized, through linearization, a set of controller flaps for a satellite in a circular orbit. 72 Modi and Shrivastava (1971), on the other hand, have proposed several schemes of semi-passive aerodynamic control lers. Their nonlinear analysis showed the system to be effective in damping severe disturbances in a fraction of an orbit. The performance of the controller appeared promising even in elliptic orbits where the corrective 12 moments are available only over a portion of the trajec-73 tory. In a subsequent paper (1973), the authors optimized the performance of the controller in both circular and elliptic orbits, using the damping time and the steady state pointing error as the respective criteria. 1.3 Purpose and Scope of the Investigation From the foregoing, it is evident that the influence of environmental forces on the attitude motion of spinning satellites and their utilization for attitude control has received little attention in the past. On the other hand, the importance of such a study becomes apparent when one recognizes the fact that the majority of the communications, applied technology and scientific satellites are indeed spin stabilized. The thesis aims at filling this gap by systematically analyzing environmental effects and exploit ing them to advantage over a wide range of operational altitudes. The influence of solar radiation pressure, con stituting the dominant environmental force at high altitudes, on the librational motion is examined first. Both analytical and numerical techniques are employed to study the system response. Stability of the periodic solutions is ascertained using the Floquet theory. Numerical results establish regions of nontumbling motion in the system parameter space. 13 Next, the possibility of using solar radiation pressure for general three-axis libration damping and attitude control of a dual-spin system is explored. The results establish the effectiveness of the concept. Effort is then directed towards devising a solar controller model that is structurally simple and operationally efficient. This is followed by an investigation of attitude damping and control utilizing the earth's magnetic field. A comparative study of two controller models is conducted. For the reason pointed out earlier, the magnetic controllers fail to provide first order pitch control in synchronous orbits. To compensate for this, the magnetic controllers are hybridized with a solar pitch controller. Attitude control of near-earth satellites in elliptic trajectories, normally preferred to minimize degeneration of the orbit due to atmospheric drag, is considered next. A hybrid control system, utilizing the aerodynamic forces at low altitudes and solar radiation pressure when beyond the atmosphere, is proposed. Finally, the problem of time-optimal pitch control of satellites using the radiation pressure is examined analytically. Figure 1.1 schematically presents the plan of study. Librational dynamics in presence of solar radiation pressure 1 L ATTITUDE CONTROL OF SPINNING SATELLITES USING ENVIRONMENTAL FORCES ± Attitude control using solar radiation pressure Attitude control using the earth's magnetic field and solar radiation pressure Analytical study • Periodic solutions • Resonance Numerical study • System response • Libra tional stability Feasibility study Nutation damping Attitude control r Attitude control using aerodynamic forces and solar radiation pressure Improvement of controller design • Libration damping •Attitude control Time-optimal pitch control using solar radiation pressure Feasibility of magnetic-solar hybrid control •Nutation damping •Attitude control Feasibility of aerodynamic-solar hybrid control Libration damping Attitude control Optimal synthesis | • System response Figure 1.1 Schematic diagram of the proposed plan of study i—• 15 2. LIBRATIONAL DYNAMICS OF SPINNING AXISYMMETRIC SATELLITES IN PRESENCE OF SOLAR RADIATION PRESSURE This chapter investigates the attitude dynamics of axisymmetric, cylindrical, spinning satellites under the influence of solar radiation pressure and gravity gradient torques. The equations of motion of the system are obtained first using the classical Lagrangian formulation followed by an evaluation of the generalized forces due to the radiation pressure. As the nonlinear, nonautonomous, coupled equations of motion do not possess a known closed-form solution, an approximate study is undertaken using Butenin's extension of 74 the method of slowly varying parameters . The approximate analytical solution proves to be an excellent tool in locating periodic solutions of the system, whose importance in the attitude dynamics study of satellites has been well emphasized 9 7 5 V 6 77 ' ' . The Floquet theory is employed to examine the variational stability of the periodic solutions. The possibility of resonant oscillations of the system in presence of the solar torque is also investigated. The quasi-linear analytical method, however, fails to describe the large amplitude motion of the system. To this end, the governing equations are analyzed numerically 16 and the librational response is studied as a function of the system parameters. The available information is con densed in the form of design plots, which clearly emphasize the importance of the solar parameter characterizing the radiation pressure torque, and should prove useful during the design of an attitude control system. 2.1 Formulation of the Problem Figure 2.1 shows an axisymmetric cylindrical satellite with the center of mass S moving in a Keplerian orbit about the center of force 0. The spatial orientation of the axis of symmetry of the satellite is completely specified by two successive rotations y and B, referred to as roll and yaw, respectively, which define the attitude of the satellite principal axes x,y,z with respect to the inertial reference frame x',y',z'. The satellite spins in the x,y,z reference with angular velocity a. In terms of these modified Eulerian rotations, the expressions for the potential and kinetic energies to 0(1/R ) are obtained as: U g -ym /R - y{(I /2R3)(I-l)/I}(1-3 sin2ycos2B) (2.1) T 2 + (B-8siny)2 + (ycosB+SsinBcosy)2) (2.2) Figure 2.1 Geometry of motion of spinning satellite in the solar pressure environment 18 Neglecting orbital perturbations due to the 7 8 79 librational motion ' , the classical Lagrangian formulation yields the governing equations of motion in the roll, yaw and spin degrees of freedom as: (d/dtj{-I(a-ysinft+QcosBcosy)sin3+(Ycos3+8sin3cosY)cos3) +1 (a--ysin3+9cos3cosY) 8cos3siny+ (3-8sinY) 8cosy + (Ycos3 + 8sin3cosY)8sin3sinY+3(y/R3) (1-1) * 2 sinycosYcos 3 = Q^/I (2.3a) (d/dt)(3-esiny)+(Ycos3+6sin3cosY){I +(Ysin3-8cos3cosY)>-3(y/R3)(I " Qe/Zv (d/dt)(a-Ysin3+8cos3cosY) = Q /I Ot X where Q^(i=Yf3,a) represent the generalized forces due to solar radiation pressure. Consider an area element dA, of the curved surface of the satellite, of length dx, angular width d0 and located at an axial distance x from the center of mass S such that the surface normal n makes an angle 0 with the y axis (Figure 2.2a). The force acting on the area element due to solar radiation pressure is given by, (a-ysin3+8cos3cosy) 2 -1)sin ysin3cos3 (2.3b) (2.3c) 19 Figure 2.2 Evaluation of generalized forces due to solar radiation pressure: (a) curved surface; (b) flat ends 20 dF = -podA|cos£|{(l-T)u+ps> (2.4) with the resulting moment about the satellite center of mass as M = C rr X dF (2.5) A U where the integration extends over the entire curved surface "seen" by the sun. The limits of integration for x are -(V2-e) to (l/2+z) and those for 9 are tan-1 (-u./u.) to J K {ir'+tan ^ (-u-/u, ) } , which correspond to £=TT/2 . 1 K-Evaluation of the integral in Equation (2.5), after considerable algebraic manipulations, leads to M = {- (TT/2) p£r2(1-T-p) u.u.+2 (l-T+p/3) p r£eu, / 2+ 2}j O 1 .K OJs!U> Ui J K + {(TT/2)P £r2(l-T-p)u.u.-2 (l-T+p/3)p rku./ 2^_ 2}k (2.6) ' *o in ' *0 1 u.+u, J J j k The expression for the moment due to the flat ends of the cylindrical satellite (Figure 2.2b) is obtained in a similar manner, ME = { (7T/2)poS.r2 (l-T-p)uiuk+Trpor2e (l-T-p)uk|ui| }j -{ (7T/2)po5,r2(l-T-p)uiUj+Trpor2£(l-T-p)Uj|ui| }k (2.7) The total moment due to solar radiation pressure is thus given by 21 M = Mc + ^ = 2(l-T + p/3)por£euk[/u2+u2+(7rr/2£) { (1-T-p)/ j k (l-T+p/3)}|u±|]j-2(l-T+p/3)por£eujx [/ 2+u2 +(TTr/2A) {• (1-T-p)/(l-T + p/3) } | u± | ]k (2.8) j k The application of the principle of virtual work yields the generalized forces in the Y»3 and a degrees of freedom as: Q =-2 (l-T + p/3) p r&eu.[/ 2, 2 +(iTr/2£)x Y o i 1 u .+u, 7 J D k {(1-T-p)/(l-T+p/3)}|u±|]cos3 (2.9a) QB = 2 (l-T + p/3) p rfceuk [/ 2 2 +{T\r/2l)x j k {(l-T-p)/(l-T+p/3)}|Ui|] Q = 0 a The generalized force in the a degree of freedom being zero, a first integral of motion defining the satellite spin rate a is furnished by Equation (2.3c), (2.9b) (2.9c) • • • a-Ysin3+6cos3cosY = h (2.10) As ha is a measure of the spin rate, a dimension less spin parameter c, defined as a = (a/9)I = (h /9) I6=6=Y=0 a -1 (2.11) 9=0 may be used to eliminate the cyclic coordinate a. Changing the independent variable from t to B, through the Keplerian orbital relations R = hg/y(l+ecos6) (2.12a) 9 = hQ/R2 (2.12band making use of the spin parameter a, the governing equations of motion in the roll and yaw degrees of freedom (Equations 2.3a and b, respectively) transform to: Y" -23,Y'tan3+23'cosY-I(a+lj{(1+e)/(l+ecos9)}2x (3'-sinY)sec8+{3(I-l)/(l+ecos8)-1}x sinYcosY-{2esin9/(l+ecos9)} (Y1+cosYtanB) =-{ (1+e) 3/(l+ecos9) 4}Cu. {/^2~^2 +G|ui|}secB (2.13a) 3 uj uk x B" -Y'COSY+[I(a+1){(1+e)/(1+ecosG)}2 +(Y'sinB-cosBcosY)](Y'cosB+cosYsinB) -{3(I-l)/(l+ecos9)}sin2Ysin$cosB 23 -{2esin9/(l+ecos9)}(B'-siny) = {(1+e)3/(l+ecos6)4}Cu, {/ 2, 2 +G|u. |} (2.13b) uj+uk 1 where the solar parameter C and the solar aspect ratio G are defined as: C = (2R3/uIy)pQrile (l-T+p/3) G = (7rr/2£) (l-T-p)/(l-T+p/3) (2.14) 2.2 Analytical Results 2.2.1 Approximate analytical solution In absence of a known, exact, closed form solution to such a complex system, it was decided to analyze the problem approximately using Butenin's extension of the method of variation of parameters. The case of w=c()=0 is analyzed here which leads to a considerable reduction in algebra with out affecting the physics of the problem. Replacing the trigonometric functions of the dependent variables by their series expansions, ignoring fourth and higher order terms in B,Y, and their derivatives 2 as well as terms of o(e ), Equations (2.13) take the form: Y" +n2Y-^1B'=2Ce-C(l+3e)cos9+2Cecos20+f(Y,Y',3,3',9) (2.15a) 3" +n23+&2Y' = -C(l+3e)sin0+2Cesin29+g(Y,Y' ,3,3' ,9) (2.15b) where n2 = 3I-4+I(a+1)(l+2e) n2 = I (a+1) (l+2e)-1 £1 = l2 = I(CT+1) (1+2e)-2 (2.16) and the nonlinear functions f and g are defined as f = C(l+3e-4ecos9)[Y2cos6/2+32sin0cos9/2 +Y2cos39/2+Y3sin9cos29-G|Ycos8+3sin9|cos9] -2I(a+l) (3'-Y) ecos9+3(1-1)Yecos0+2(Y' + 3)esin9 +1 (a+l)(l+2e-2ecos9) (Y3+33' 32-3Y32)/6 + 233'Y'+3'Y2+(2/3) { (31-4)-3 (I-l)ecos9>Y3 + (33/3-3Y2/2)2esin9 g = C (l+3e-4ecos9) [Y3cos9 + 32sin39/2+Y2sin9cos29/2 +Y3sin29cos9-G|Ycos0+3sin9|sin9]+2I(a+1) * 25 x(Y'+S)ecose+2(61-Y)esinB+I(a+1) (l+2e-2ecos9)(3Y'32+3y23+33) -2Y'32-3Y2-Y'23-Y'Y2-233/3+3(1-1)(1-ecosG)y2B+Y3esin9/3 (2.17) The solution for the corresponding linear system (i.e., f=g=0) is given by Y = asin(k,9+3,)+bsin(k„9+39)+A +A.cos9+A„cos29 11 A 1 ° 1 * (2.18a) 3 = a, acos (k, 9 + 3,)+a bcos (k.,9+3.,)+B, sin0+B_sin28 1 112 i 2. i 2. (2.18b) where a, b, 3-|_'32 are constants to be determined from initial conditions and the characteristic frequencies k^ and k2 (k^>k2) are the roots of the equation k4-(n2+n2+Ji1£2)k2+n2n2 = 0 (2.19) The constants a., A. and B. are defined as: ii l a. = (k2-n?) A,k. , i = 1,2 l l 1 ' 1 l A = 2Ce/n2 o 1 A1 = C(l+3e) (l-n2-5,1)/(l-k2) (1-k2) A2 = 2Ce(n2+2il1-4)/(4-k2) (4-k2) 26 B1 = C(l+3e) (l-n2-£2)/(l-k2) (1-k2,) B2 = 2Ce(n2+2^2-4)/(4-k2)(4-k2) (2.20) A solution of the similar form is sought for the nonlinear system, however, allowing the amplitude and phase to be functions of 0, i.e., Y = a(0)sin(k10+31 (0))+b(0)sin(k„0+6„(0))+A 11 2 2 O +A1cos8+A2cos20 (2.21a) 8 = a1a(8) cos (kjj^O + B-L (0) )+a2b (0) cos (k20 + 82 (0) ) +B1sin0+B2sin20 (2.21b) As the four unknown functions defining the variable amplitude and phase cannot be determined from four initial conditions alone, the solution in the present form is over-specified. Hence four constraint relations must be obtained. Keeping the first derivatives of y and 8 to be the same as that of the linear system gives two of the constraint relations: a' sinijj1+b' sinip2+a8'1cosip1+b82cosiJ;2 = 0 (2.22a) a^a ' cosi^+o^b' cosij;2-a^aB^sini^-a2b82sinip2 = 0 (2.22b) 27 where ^ = k1e+B1(e), y2 = k2e+B2(e) Equations of motion (2.15) in conjunction with the assumed solution(2.21) yield the other two constraint relations as: k^'cos^1+k2bIcos^2-k1a6^sin^1-k2b82sin^2 = f* (2.23a) alkla'S^n^l+a2^2^'S^n^2+alkla^lCOS^l+a2k2b^2COS^2 = (2.23b) * * where f , g are the modified nonlinear functions. Solving Equations (2.22) and (2.23) simultaneously for a', b', B| and B2 yields: a' = -{^1g*sin^1-(ii2/a1) f*cos^1}/(k2-k2) b' = {iiig*sin^2-(£2/a2) f*cos^2}/(k2-k2) 31 = -{£1g*cos^1+(^2/a1) f*sini|J1}/{a(k2-k2) } 32 = {ii1g*cos^2+(£2/a2) f*sinij;2}/{b(k2-k2) } (2.24) Equations (2.24) represent an exact transformation of the two second order equations of motion (2.15) into four 28 first order differential equations. For small amplitude * * motion, f and g are small. Consequently, a, b, 3-^ and 32 are slowly varying parameters. Using their average values over one period gives: a' = -U/8TT3 (k2-k2) } (2TT r2TT 0 J 0 2TT {£^g sin^ -(Jl2/a1)f cosiJJ1}d^1d^2de b' = {l/8TT3(k2-k2) } 2TT 2TT 2^ {£^g sin^ -(A2/a2)f cosi^} dij^d^de 3j_ = -{l/8Ti3a(k2-k2) } /•2TT r2v 0 0 {5--^g COSIJJ^ + (£2/a1)f sin^ldi^dij^de 32 = {l/8Tr3b(k2-k2) } 2T\ 2TT 0 J 2TT 7C {i^g cos^2 + (£2/a2)f sin^d^dij^dQ (2.25) Solving Equations (2.25) for a, b, 3-^ and 32, and substituting in Equations (2.21), the solution takes the form: 29 Y = asin (o3ie+c1)+bsin (co26+c2)+Ao+A1cose+A2cos29 (2.26a) 6 = a1acos (w10+c1)+a2bcos (to2e+c2)+B1sin6+B2sin2e (2.26b) where a = [{(6;+a2a)2(Y0-Ao-A1-A2)-B1-2B2)/(a1a);L-a2ca2)}2 ,2 1/2 + {(a0Y'-a)„6 )/(a,a>.,-a,a>.I) } ] (2.27a) 2. O 2 O X ^X b = [{ (B^+a1o)1(Yo-Ao-A1-A2)-B1-2B2)/(a1co1-a2a)2) }2 2 I/2 + {(a1Y^-a31Bo)/(a1u)2-a2a)l)} ] (2>2?b) c1 = tan 1[{ (3^+a2w2(Yo-Ao-A1-A2)-B1-2B2)/(a10)1-a2a)2) }/ * (a2Yo~a)2^o) /(a1a)2_a2tJJl) ^ (2.27c) c2 = tan-1 [{ (B(^+a1co1 (YQ-Ao-A1-A2)-B1-2B2)/(a1o)1-a2co2) }/ { (a1Y(!)-(jJ1B0)/(a1co2-a2w1) }] (2.27d) The frequencies and (JO2 are represented by rather lengthy functions (Appendix I) of librational amplitudes and system parameters, 30 U). f (a,b,I,a,e,C,G) CO- = f2 (a,b,I,a,e,C,G) (2.28a) (2.28b) 2.2.2 Periodic solutions of the system The approximate, closed-form solution shows the system response to be characterized by three distinct components: response at 'high' frequency <JO^ , 'low' frequency a)2 and the orbital frequency. The resulting motion would, in general, be non-periodic except in the special situation when the frequencies oo^ and u>2 assume rational values for non-zero C, or, the ratio W2./(JJ2 is a rational number with C = 0. A search extending over a reasonable range of system parameters and initial conditions showed such frequency combinations to be rare indeed. On the other hand, the solution indicates that the system would execute periodic motion subject to initial conditions which excite only one of the three frequencies. Various relationships exist in the initial condition space for which the resulting motion is periodic. The high and low frequency periodic librations in the absence of solar radiation pressure (C = 0) are discussed first, followed by the solar pressure excited periodic oscillations of orbital frequency. 31 (a) High frequency oscillations (C = 0) In the absence of solar radiation pressure, A^ = B^ = 0. Hence, the system would execute high frequency periodic motion for b = 0. This is satisfied by two sets of initial conditions. The oscillations of type I result with Y = 3' = 0 o o Y^ = (0)1/a1)8o (2.29a) The corresponding amplitude and frequency of motion are obtained from Equations (2.27a) and (2.28a), respec tively, as: a = 80/a1 Wl = kl + fi (^0/ai_' I'a'e) (2.29b) The other set of initial conditions, leading to high frequency oscillations of type II, is readily found to be, Y' = B = 0 O Q B' = -a1u),Y (2.30a) o 1 1 o with the amplitude and the frequency of motion, 32 u>1 = 'k1 + f1(Y0/If(?,e) (2. 30b) Relations (2.29) and (2.30), governing the initial conditions for high frequency periodic oscillations and the resulting frequency (oscillations per orbit), are plotted in Figure 2.3 for typical values of satellite parameters. The system behaves as a hard-spring oscillator showing an increase in the frequency with amplitude. Note that the change in satellite's configuration from spherical (I = 1) to disc-like (I = 2) results in a corresponding increase in frequency. An increase in orbital eccentricity also has similar effect. (b) Low frequency oscillations (C = 0) initial conditions leading to a = 0. As in the case of the high frequency oscillations, two distinct relationships between the initial states are found to yield periodic motions of low frequency. The low frequency periodic oscillations result from Periodic motion of type I is obtained with, Y = 0 ' o (2.31a) C = o e = o stable a = 2 e = 0.05 it stable ©unstable 0.2 _ 0.4 0 0.2 0.4 Figure 2.3 Frequency and initial conditions for high fre quency periodic motion 34 with amplitude and frequency as b = 6D/a2 u2 = k2 + f2(6o/a2,I,a,e) (2.31b) while that of type II is governed by the conditions, Y' = 8 = 0 o o 6' = -a„u„Y (2.32a) o 2 2 o leading to b = ^o w2 = k2 + r2(YQ,I,a,e) (2.32b) Relations (2.31) and (2.32) are presented in Figure 2.4 for typical values of the system parameters. The variation of the frequency is found to be relatively small. The slight decrease in its value with increasing amplitude indicates a soft-spring type of nonlinear effect over the range of initial conditions considered. Influence of the inertia parameter follows essentially the same trend as before. On the other hand, an increase in eccentricity tends to reduce the frequency of motion. 35 C = o e = o • stable a = 2  = 0.05 0 0.2 p 0.4 0 0.2 y 0.4 Figure 2.4 Frequency and initial conditions for low fre quency periodic motion (c) Solar pressure excited oscillations (C ^ 0) For the solar pressure excited periodic solutions, having the same frequency as that of the orbital motion, a = 0, b = 0. Substituting these conditions in Equations (2.27a and b) results in the initial state, Y = A + A, + A„ o o 1 2 Y. , o 8o - 0 Figure 2.5 shows the variation of the initial conditions YQ and 3^ with the satellite spin parameter for both circular and noncircular orbital motion. The influence of the solar parameter C is to raise the initial conditions for periodic motion. The initial state appears to be highly sensitive to the spin rate for slowly spinning satellites, however, it asymptotically approaches a constant value with increasing spin parameter. In general, the effect of eccentricity is to increase the magnitude of the initial conditions except at the lower end of the spin parameter spectrum. 37 Figure 2.5 Initial conditions leading to solar pressure excited periodic motion 38 (d) Accuracy of the analytical solution To assess the accuracy of this analytical pro cedure in predicting the periodic solutions, the governing equations of motion (2.13) were integrated numerically with initial conditions derived from the analytical solution. Typical responses are presented in Figure 2.6. In circular orbits, the librational response is observed to be periodic with insignificant error. The frequency of oscillations is also predicted very accurately, thus demonstrating the effectiveness of the approximate, closed-form analysis. The method continues to predict the periodic motions quite accurately even in elliptic orbits. The accuracy, however, was found to deteriorate with increasing e. Results showed the amplitude and frequency of the numerical ly generated response to be within five percent of their analytically predicted values for e <_ 0.1, the normal range of interest. (e) Stability of periodic solutions The stability of periodic solutions can be studied using variational analysis. The variational equations are obtained by letting Y = Y + Y 1 'p 'v 39 25 0 -25 o = 2 Y P e = o 1 = 1.5 C=o Y = B'=oB=o.38 e = o.o5 _ Y0 = 1.6486 \A ;'\A rf /'• \ 1 r i ' J* i\ ' /' I '11 \\ \k \ '"x f \ \ ' */ \ 1 v' W \ J CO Y0=1.8027 \ '«/ \ I'M /'A ~ 1 \ \ .VI ' V \ i 1 \ ' '/ \ \ •11 1 I \ i J \ . 'J \ / M ' M / M / \ I * / • i > /1 i , / • i * / • i 1 : i i M I./ • i• 1 1 i. ( . I . / • i * / > i > i 'I./ i. / • • • / \: ''/•'/ '. 1.' / i \i / 1 l.' / ! 1.' / 1 . 1.1 i it I 1 l; / 1 l; / > LI 1 1 f 1 V f I f t 'if III 'Il 'if 1 'If ilf >4f 'If i '\ / i i\ / • '\ / 'Ml > ' \J V/ ' \ J " V y 10 1 25 e = o - . R= 0.4353 /\ro A i / X i J 1 = 1.5 C = 0 = P0=0 Yo=0.3 e = o.05 3 = 0.4165 -' 0 *•X / * /I '1 V 1 /V \ 1 I '\ 1 1 *\ I 1 / I 1 V • r\ ' I t / Y 1 0 t \ 1 / 1 \ 1 / ' \ I / 1 \ i' / i I 1 / ' \ 1 / ' \ i 1 ' \ i / i \ 1 / 1 \ 1 / i 1 1 / 1 1 1 / i 1 / ' \ * / "~ \ l / ' \ i / \ % I ' I i / 1 1 / \ ' » / V i / * \ i / \ i / 1 \ i / \ • i / 1  i 1 \ i / 1 \ 1 / MA * \ » / * \ * \ i \ , / i \ i \ i / *i \ , / i \ i \ \i 'A\ /V \ 1 / 1 \ —• \ 1 /1\ 1 / ' \ \ t / '\ 1 I ' \ \ 1 / '\ I / • \ \ 1 / 1\ XJ ' \ 1 \ 1 / 1 \ •' \ ' / \ 1 \ ' / 1 \ \ * f 1 I \ • / 1 I I/ 1 25 \ A ! \ A i i \ t * \ / • ^ \ » \ I V 1 s / (b) V / » 1 ^ ^-^ i » i \ /• * vy i > 25 e = o Y0=-0.20 = 2 C=0.3 G= co= 0 = i = Y0'= arbitrary = p = o e = o.o5 f0=- 0.2307 l'= -0.3157" 0 B= -0.2 75 •o /" •» / I \ ^—*. ' ' \ 0 / A • / \ -//\\// • / . \ >/ * y / \_/« < \ / A \ X i \ \ '/ ''\ • / 7\ / * \ '  i \ f/ / \ \/ / y / • \y / / \ 1 / 1 \ • i • / * \ L ' / * \ " \ 1 / 1 \ \ 1 / 1\ * / ' \ \ X / 1 \ \ \f 1 \ \ 1 'v J *  V 25 i i (c) A./ 1 \_/ 1 0 1 2 3 0 1 2 3 9- orbits Figure 2.6 Typical periodic solutions of the system where yp, Bp represent a periodic solution and y , Bv are small perturbations. Substituting in Equations (2.13) and linearizing with respect to Yv and Bv leads to: y" = F-y + F_y' + F_B + F.B' v 1 v 2 v 3 v 4 v B" = G,Y + G_y1 + G-B + G.B' (2.34) v 1 v 2 v 3 v 4 v where the details of lengthy functions Fi * Fi(YP'3P'9)' 1 = x'2'3'4 G± = Gi(Yp,Bp,e), i = 1,2,3,4 (2.35) are given in Appendix II. For circular orbital motion, the functions F^ and G^ assume the form F^ = F^(yp,3p), G^ = G^(Yp,8p) and hence have periodicity of the solution. The Floquet theory can thus be applied to investigate the stability of the varia tional system (2.34). The stability criteria for distinct roots can be expressed as: |AI| <_ 1 , i = 1,2,3,4; stable any of the \X±\ > 1 , i = 1,2,3,4; unstable (2.36) 41 For satellite librations in noncircular orbits, the functions F^ and are periodic only if the solution period is a rational multiple of the orbital period. The Floquet theory may again be used to assess the variational stability of these solutions. Figures 2.3 - 2.5 also show the results of this analysis. The stable motion at smaller values of the initial conditions is, of course, anticipated. What is of particular significance is the possibility of periodic motions of large amplitudes. The amount of computational effort involved for stability analysis in an eccentric trajectory is enormous. Large values of the common period of the libra tional and orbital motion lead to extended integration limiting the stability investigation to isolated points (Figure 2.3). Of course, the solar pressure excited motion of the orbital period does not present this problem (Figure 2.5). The ability of the approximate closed form solution to predict nonlinear character of the system is indeed promising. 2.2.3 Resonance It is of particular relevance to recognize here several possibilities of solar pressure excited resonance (Equations 2.15). Equivalence of or k2 to the frequency of one of the forcing terms implies existence of certain combinations of the satellite inertia parameter I, spin parameter a and the orbital eccentricity e which would lead to unbounded motion. In circular orbits, the condition and/or k2=l results in the resonance conditions, 1 = 1 or, I (a + 1) -2 = 0 (2.37) The conditions for resonance in elliptic orbits are obtained, for one of the frequencies k^, assuming the value 1 or 2, as: 1 = 1 or, I(a + 1) (1 + 2e)-2 = 0 or, 2I(a+l) (l+2e)-(I+l)± f/l2-18I+33 = 0 (2. 38) Figure 2.7 shows resonance conditions (2.37) and (2.38) in the system parameter space with typical responses presented in Figure 2.8. The large amplitude beat phenomenon in some cases indicates near resonant conditions. The value of C is purposely taken here to be small to emphasize the destabilizing effect of the radiation pressure. Larger values of the solar parameter as often observed in practice (C - 1.5 for Anik, 2 for the CTS) would magnify the amplitude build-up. This clearly indicates the need for avoiding such 43 44 C = o.i G=co=cp = o i = arbitrary y-- Y„'= P0= P> o Y° p° e = 0 |=1.5 Q = 0.3333 0 2 4 6 8 10 0-orbits Figure 2.8 Typical responses under resonant conditions 45 critical combinations of system parameters for a safe satellite design. 2.3 Numerical Results The quasi-linear character of the analytical soluti limits its usefulness to the study of small amplitude motion. As anticipated, with large amplitudes, its accuracy was found to deteriorate due to the increased effect of system nonlinearities. A parametric study of the system was, there fore, carried out by numerically integrating the governing equations of motion. The Adams-Bashforth predictor-corrector quadrature with the Runge-Kutta starter was used, in con junction with a step size of 3°, which gave results of sufficient accuracy without involving excessive computational effort. 2.3.1 Significant system parameters The significance of the inertia parameter I, the spin parameter a and the orbital eccentricity e in the librational dynamics of spin stabilized satellites cannot be overemphasized. Sensitivity of the system response to these parameters is vividly demonstrated in Figure 2.9. It shows the variation of $, the angular deviation of the axis of symmetry of the satellite from the orbit normal, against 9, the position of the satellite in an orbit. It is apparent that a judicious choice of parameter values is essential to avoid tumbling motion ($>u/2). 46 Figure 2.9 Typical unstable responses demonstrating the, significance of system parameters 47 Of particular interest is the disturbing influence of solar radiation pressure. Note that the value of C as small as 0.5, which would physically correspond to e = 0.1 ft for INTELSAT IV category of satellites at synchronous altitude, causes the satellite to tumble over. The critical values of eccentricity, intertia and spin parameters would only accentuate this behaviour. Of course, in actual practice, a higher spin rate and/or active control system would counter this tendency. Nevertheless, the analysis clearly brings out the fact that the solar parameter C is of the same importance as I, a and e in the design of the satellite attitude control system. 2.3.2 System plots In order to better understand the solar pressure excited dynamical behaviour of the satellite, the system parameters were varied over the desired range and the librational response observed. To isolate and emphasize the influence of the radiation pressure no other disturbances in the form of initial conditions were introduced. The resulting information was condensed in the form of system plots. Figure 2.10a shows the effect of the satellite inertia parameter I on the coning amplitude $ and the c max average "nodding" frequency of the axis of symmetry, con, expressed as oscillations per orbit, for different values of 48 G = 0.5 CJ=0 0 = 45° Y0 = = P0 = P0' = 0 C = 0.05 C =0.1 C = o.2 (a) a = 2 e = o i =45 JW -0^0 5" C=0 ~ (b) T 1 = 1.5 C = o -3 -1 1 = 45 1=1.5 Q = 2 I e = 0 o. 2 o e Figure 2.10 System plots showing the- coning angle and average nodding frequency as affected by: (a) inertia parameter; (b) spin parameter; (c) orbit eccentricity; (d) orbit inclination the solar parameter C. It may be observed that a satellite, when the solar pressure effects are neglected, remains in the equilibrium position ($ = o). However, with non zero C (say, C = 0.05), the librational motion is excited which increases in amplitude with increasing C. It is of interest to recognize the presence of a critical value of 1=1 leading to large amplitude motion finally resulting in instability. This amplitude build-up was found to occur even for very small values of the solar parameter C, thus confirming the resonant behaviour at these critical combin ations of the system parameters predicted earlier by the analytical method. On the other hand, the average nodding frequency of the axis of symmetry appears to be relatively unaffected by changes in the inertia or solar paremeter. The effect of the spin parameter a on the libra tional behaviour is indicated in Figure 2.10b. Here again, critical values of the spin parameter exist for which the satellite tumbles over. A large value of the spin parameter, in general, leads to smaller coning angles as anticipated. Figure 2.10c shows the influence of the orbit eccentricity on the attitude motion. In general, higher values of the orbit eccentricity result in larger amplitude motion. Unlike the effect of the inertia and the spin parameters, no resonant behaviour is noticed for the typical values of system parameters considered here. 50 The influence of the significant orbital para meters, such as i, 4>, OJ and the solar aspect ratio G, on the satellite performance was also investigated. The amplitude of oscillation was found to reduce gradually with an increase in the orbital inclination from the ecliptic (Figure 2.10d). Changes in the solar aspect angle cf>, which depends upon the location of the line of nodes and the appar ent position of the sun, did not affect the amplitude of librations and their frequency. The influence of the perigee position co and the solar aspect ratio G was also found to be insignificant. 2.3.3 Design plots From design considerations, it would be desirable to assess the magnitude of the solar pressure torque that a satellite can withstand without exceeding the permissible bound of libration as governed by the mission requirements. This bound then would establish a criterion for stability. Here, the stability limit is purposely taken as a large value of $ = TT/2 to emphasize the vulnerability of the satellite's performance to the solar pressure torque. Figure 2.11a shows a typical stability chart for librational motion in a circular orbit with the radiation pressure as the only excitation. The equations of motion (2.13) were integrated over 15-20 orbits for a range of values of satellite inertia and spin parameters. The result-Figure 2.11 Typical stability charts showing adverse influence of solar radiation pressure: (a) e = 0; (b) e = 0.1 52 ing information about the maximum amplitude of the coning angle $ was then condensed in the form of stability max J plots in the I-a space. The analysis shows that in addition to the main stable region for high inertia parameter values, there also exist small isolated stable areas. However, there are substantial unstable regions even for positive spin or large inertia parameters. It is observed that the stable areas reduce drastically, as expected, with an increase in the value of the solar parameter. The effect of the orbital eccentricity on the stability of librational motion is presented in Figure 2.11b. An increase in orbital eccentricity further enhances the destabilizing influence of the solar pressure. The effect appears to be more pronounced for satellites with the retrograde spin. 2.4 Concluding Remarks The important features of the analysis and the conclusions based on them may be summarized as follows: (i) The approximate analytical solution developed using the method of slowly varying parameters proves to be an excellent tool in establishing the periodic solutions of the system. The closed form character of the solution provides con siderable insight into the system behaviour. 53 (ii) The analytical method predicts the system response and frequency quite accurately in circular orbits. Even for noncircular orbital motion (e <_ 0.1), the errors are confined to less than 5%. (iii) The influence of the solar pressure on the satellite librations, in general, is adverse. The analysis shows, however, that the system can execute stable periodic motions of con siderable magnitude in the solar pressure field under suitable initial conditions. (iv) There exist combinations of system parameters for which large amplitude oscillations result, even in the presence of a very small solar torque, due to resonant interaction, (v) The solar parameter affects stability of the motion substantially and hence merits equal consideration with the satellite inertia and spin parameters and the orbital eccentricity. 54 3. ATTITUDE CONTROL USING SOLAR RADIATION PRESSURE The analysis of the last chapter clearly estab lishes the substantial adverse influence of solar radiation pressure on satellite librations. On the other hand, the findings of earlier investigations of gravity oriented 49-51 systems suggest that the radiation force can provide effective damping torques to maintain a satellite in a desired attitude. However, as pointed out in the literature review, available analyses of solar pressure control of spinning satellites are only of a preliminary nature. This is unfortunate because, in many space applications, a satellite with a directional sensor has a preferred orientation which is normally achieved by mounting the device on a stabilized platform aboard the spinning satellite. .The spin, through a gyroscopic moment, provides stability while the platform, despun by control moments, tracks a given object in space. The concept of solar pressure control provides an exciting possibility of stabilizing the entire system through a semipassive approach.. This chapter investigates the feasibility of the general three-axis nutation damping and attitude control of spinning satellites using a solar controller sensitive to angular displacement and velocity errors. The analysis 55 is kept quite general to accommodate eccentric orbits and arbitrary inclinations of the orbital plane with respect to the ecliptic. The nonlinear, nonautonomous, coupled equations of motion are analyzed numerically and the influence of system parameters on the response studied. In the latter part of the chapter, a logical approach for controller design is developed. Analytical solutions for the control variables are obtained which suggest reduced software requirements. Several examples using representative satellites illustrate the effectiveness of the control system and help gain an appreciation as to the controller size required for a desired performance. 3.1 Feasibility of Solar Pressure Control 3.1.1 Equations of motion Figure 3.1 shows an axisymmetric (I = 1^) satellite with the center of mass S moving in a Keplerian orbit about the center of force 0. The satellite consists of a central body I, spinning at a constant average angular velocity, connected to a stabilized platform II through a viscous damper effective in axial rotation. The spatial orien tation of the axis of symmetry, as stated before, is specified by two successive rotations y and 8, referred to as roll and yaw, respectively, which define the attitude of the satellite principal axes x,y,z with respect to the inertial Figure 3.1 Geometry of motion of dual-spin satellite in the solar pressure environment reference frame x', y', z1. The rotor and the platform spin in the x,y,z reference with angular velocities a and X, respectively. In terms of these modified Eulerian rotations, the expressions for the potential and kinetic energies to 0(1/R ) are obtained as: U = -ym /R-y{(I /2R3)(I-1)/I}(l»3sin2Ycos28) (3.1) y o 2i. T = (mg/2)(R2+r262)+(Ix/2I){(1-J)I(a-ysin3+8cos8cosY)2 + JI(A-YsinB+0cos3cosY)2+(B-8sinY)2 • 2 + (YcosB+6sin(3cosY) > (3.2) The Rayleigh dissipation function is given by F = (l/2)I<d(X-a)2 (3.3) Using the Lagrangian formulation, the equation of motion in the a degree of freedom becomes, (d/dt) {Ivv. (a-YsinB+6cosBcosY) > (a-X) = (3.4) As the rotor is considered to spin at a constant average spin rate (i.e., no spin decay), the usual assumption in the analysis of dual-spin spacecraft, 58 Qa = Kd(a-X) (3.5) This is equivalent to assuming an energy source countering the bearing drag on the rotor. The first integral a-Ysin3+6cos3cosy = ha (3.6) may now be used to eliminate the cyclic coordinate a through the spin parameter a defined as, (a/9) j =(h /6) 'e=3=Y=o 9=0-1 (3-7) Neglecting orbital perturbations due to librational 78 79 motion ' and making use of the spin parameter, the Lagrangian formulation yields the governing equations of motion in the roll, yaw and pitch degrees of freedom. Changing the independent variable from t to 0 through the Keplerian orbital relations, they finally take the form: Y" -2 3' (Y'tan3-cosY)-(3'-sinY)sec3[(l-J)I(a+l) x { (l+e)/(l+ecos9)} 2+JI(A 1-y'sin3+cos3cosy)] +{3(1-1)/(1+ecose)-l}sinYcosY-(2esin9/(l+ecos9)} x (Y'+cosYtan3) =Qy (3.8a) 59 8" -Y'cosY-{2esin9/(l+ecos9)}(81-sinY)+(Y'cos8+cosYsin8) x [ (1-J)I(a+1){(1+e)/(l+ecos9)}2+JI(A'-Y'sin8+cos8cosy) +(Y'sin3-cos8cosY)]-3{(I-l)/(l+ecos0)}sin2Ysin8cos8 = QD p (3.8b) A" -Y" sin8-{2esin0/(l+ecos9)}(A'-Y'sin8+cos8cosY) -3IY'cos3-Ylcos8sinY-3'cosYsin8+(K/JI) x { (1+e) 3y/2/(l+ecos9) 2} [A 1 -y1 sin8+cos8cosY- (a+1) x { (l+e)/(l+ecos9)}2] = QA (3.8c) where Q^(i = Y*8,A) represent the generalized forces due to solar radiation pressure. 3.1.2 Controller configuration A controller, in general, consists of light, rigid, highly reflective plates (membranes) suitably mounted on the platform to be stabilized. The control moments resulting from the solar radiation force on the plates may be varied by changing any one of the following: (i) the distance between the center of pressure and the satellite center of mass by translating the plate support; 60 (ii) the area of the membrane through wrapping or unfurling portions of it; (iii) the projected area of the plate as 'seen' by the sun by rotating the plate. Practical considerations make the last alternative the most attractive, especially, when servomotors can be located within the controlled environment of the spacecraft. In order that the three degrees of freedom of the system, namely the roll Yr the yaw 6 and the platform pitch X, be controlled independently, it is necessary to provide at least three independent plate rotations 6^, S2, and 63. Various controller configurations were studied which, in general, yield expressions for the generalized forces of the form: QY = QY(5lf 62, 63) Q3 = QB(6lf 62, 63) QA = QA(51# 62, 63) (3.9) where the functions in the right hand side are transcendental. As the rotations 6^, an<^ ^3 are real, the simultaneous maxima |Ct I „ . |QQL„v and IQ^I for which Y max p max 1 A max the set of Equations (3.9) possesses a real solution, represent the physical limit on the generalized forces that a particular controller configuration can yield. The problem of determining the maximum values of Q^, Q^, and Q^, which would satisfy the above criterion is, in general, a complex one. Any attempt to simplify the problem, through the choice of controllers yielding completely decoupled moments, would lead to increased hardware complexities. One is, therefore, forced to compromise by selecting a controller configuration resulting in a partial uncoupling of the . Figure 3.2 shows the schematic diagram of the proposed semi-passive controller. It consists of five sets df plates Pi(i = 1,2,3) and P^(j = 1,2) with their axes mounted on the platform. The plates are permitted rotations 6^(i = 1,2,3) about these axes. The angles 6^(i = 1, 2) are measured from the yz-plane and 6^ is measured from a platform-fixed reference line at an angle A from the y axis. At a given instant, the set P^, controlling the A motion, operates in conjunction with the sets P^(i = 1»2) or Pj (j = 1,2) or P^Pj (i j) i which provide corrective torques in the y and 8 degrees of freedom. The determination of the moments due to solar radiation pressure is somewhat involved. Figure 3.3 shows a plate in an arbitrary orientation with respect to the sun. The force on an elemental plate area dA is given by Figure 3.2 Solar controller configuration 63 Normal Reflected (i-T)fDdA|cos£ Incident u Plate element P^dA|cos£ | Figure 3.3 Radiation force on a plate element 64 dF = -p dA|cos£ | {(l-T)u+ps} (3.10) with the resulting moment about the satellite center of mass as M = f r X dF (3.11) A Expressing the angle of incidence £ and the unit vectors u and s as functions of the attitude, angles, the solar aspect angle (f> and the plate rotations 6^, and evaluating the integral in Equation (3.11) yield the desired expression for M. The application of the principle of virtual work finally leads to the generalized forces Q^. The expressions are rather lengthy, however, ignoring the terms of order (1-T-p)/2p compared to unity, which is justi fiable for surfaces of high reflectivity, results in a considerable similification: = -E(0) [U1C11 cos ? ^ cos 5 1sin61sinA +^^2 I cos ^ ^ COS?2S^"n^2COS^-' sec^ (3.12a) = -E (9) [U1C11cos K 11cos K 1sin61cosA - U2C2 | cos E, 2 | cos £ 2sinS2sinA] (3.12b) Qx = -E(6) (C3/J) |cos £3|cos ?3 (3.12c) where Ui = +1 for Pif -1 for P^; i = 1,2 cos ^ = u. cos6. + (u . sinA-u, cosA) sin6. X 1 X 1 K X cos ?2= uicos62+(UjCosA+uksinA)sin62 (3.13) cos C3=-UjSin(63+A)+ukcos(63+A) The solar parameters, C^, are defined as Ci = (4pP0Rp/yIy)Aiei' 1 = 1'2 C3 = (4ppoR3/yIx)A3£3 (3.14) 3.1.3 Control strategy The generalized forces are controlled in a velocity and position sensitive manner according to the relations: Qy = -y YY'-vy(Y-Yc) (3.15a) QB = ~yB6'"v3(6"6c) (3.15bQA = ~yAA'"VA(A"Ac) (3 .15c) where the system gains , are chosen according to some suitable criterion, such as, the least time of damping or the maximum permissible displacement during a nutation cycle. The position control parameters y , Bc, and X^, however, are functions of the desired final orientation Yff Bf, and X^ and are obtained from the equilibrium consideration of the controlled system (Equations 3.8). For nutation damping in a circular orbit, this results in Yc = Bc = 0, Ac = -(l/vA)Ka/JI (3.16) In principle, the plate rotations can be obtained by substituting Equations (3.15) into Equations (3.12) and solving for 6^. This, of course, implies specification of the signs of and U^, i.e., a combination of the plate sets to be operated. However, there are still several mathematical problems as the system of Equations (3.12) may not possess a real solution. A trial with different sign combinations of and is thus necessary. Fortun ately, for many applications, |u.| << |u.|, | u,| and X 1 j K executes small oscillations in the neighbourhood of nominal pitch attitudes X^ = 0, TT/2 , ir or 3TT/2 . In such situations the required signs of U-^ and may be determined analytically. Furthermore, the controller may be unable to provide the corrective moments demanded by the system (Equations 3.15) at all times due to its physical limita tions. It is, therefore, necessary to introduce saturation constraints on the control moments Q.. Hence the real 1 solution for 6. has to be determined consistent with this x constraint. being a function of 6^ only, its maximum attainable value can easily be found, But 0^, Qg are functions of both 6^ and hence it is necessary to specify a rational criterion for the controller operation. In the present analysis, based on physical considerations, it is taken to be the maximum of the total 2 2 2 1/2 transverse torque, (Q^cos 3+Q^) . This occurs at 6 = Tr/2+tan~1[ (3/2) (u.sinA-u, cosA)/u.±{ (9/4) (u.sinA X T K 1 "J lO, I = E(6) (C-/J) (u2+u.2) 1 A 1 max 3' j k (3.17) -UVCOSA)2/u2+2}1/2] K 1 (3.18a) 3 and 6 +u, sinA) 2/u2+2}1/2] k x (3.18b) 68 Substitution for 6^(i = 1,2) from Equations (3.18) into Equations (3.12) yields the desired maximum values of and Qg. Appropriate signs of and are to be introduced which yield signs of Q and Qa consistent J ^ ymax 3max with those governed by Equations (3.15). The control procedure may now be summarized as follows: (i) sense the roll, yaw and pitch angles and rates, orbital position and the apparent position of the sun; (ii) compute the control moments demanded by the system using Equations (3.15); (iii) evaluate the maximum attainable moments using Equations (3.17, 3.18, 3.12a and b); (iv) compare the moment demand with the attainable values. If the demand exceeds the maximum available, set it equal to the latter; (v) determine the plates to be operated for roll-yaw control (through and U2) and the rotations 6^, $2 and 6^ from Equations (3.12). 3.1.4 Results and discussion The response of the system was studied by numeric ally integrating the equations of motion (3.8) along with the control relations (3.15). Again the Adams-Bashforth predictor-corrector quadrature with the Runge-Kutta starter was used in conjunction with a step size of 3°. The impor tant system parameters were varied gradually over the range of interest and the controller performance evaluated both in circular and elliptic orbits. In general, the system is exposed to extremely severe disturbances, much higher than it is likely to encounter in the normal operation, to evaluate the controller's performance under adverse conditions. (a) Nutation damping Figure 3.4 summarizes the influence of the controller gains u^, on the response. In general, an increase in y^ provides an overdamped character to the system (Figures 3.4a and b) while a corresponding increase in for a given y^ results in oscillations suggesting a reduction in damping (Figures 3.4a and c). The existence of an optimum choice of controller gains for a set of given system parameters is thus apparent. This is indicated in Figure 3.4d. The effects of the satellite inertia parameter I and the spin parameter a, presented in Figure 3.5 suggest that short, disc-like satellites (I = 1.5) withstand and damp a given disturbance relatively better than long, slender satellites (I = 0.5). It is of interest to point out that here the value of J, representing the ratio of the axial inertias of the platform and the satellite, is taken as 0.5. The analyses with J varying from 0.25 to 0.7 5 showed the system response to remain virtually unaffected. This is 70 20 10 0 1 r-* - h = 5 : \ V. =20 • 1 1 i t i i ii i i~'/\ • jr\ \ ' 1 V •• 1 1 / \ V; = 25 , 1 , I f 1 11^ \ 1 \ K x ^-A. »-•,.. ^, \*^-.'' j \ / \ / -(C) \ / \./ 0 0.25 0.50 0 0.25 0.50 0.75 Orbit Figure 3.4 Influence of the solar controller gains y., v. on the response 71 e = o K=o.i }A.=io Yf=p=Af=o Y° i,CJ=0 C|=4 V|=2S Y=P0=X=o P° J = 0.5 <f>=60° 0.25 0.50 0.25 0 Orbit Figure 3.5 Damped response as affected by satellite inertia and spin parameters 72 understandable as large J makes it more amenable to con troller corrections. The effect of the spin parameter a was also found to be relatively insignificant. This is to be expected as the natural stiffness of the system, to which the spin parameter contributes, is largely provided by the controller gain (Figures 3.5c and d). Figure 3.6 shows the influence of the solar parameters and aspect angle <j> on the controller per formance. The limitation imposed by on the maximum allowable control moments is clearly reflected in the response plots (Figures 3.6a and b). Note that an increase in results in an overall improvement of the transient performance of the system. However, there is a restriction as to the maximum attainable values for as imposed by the control plate areas and their moment arms. The relative position of the sun (cf>) affects the response in the y and 3 degrees of freedom through the corres ponding change in the available control moment. Thus roll and yaw attain different relative amplitudes in an orbit, how ever, their damping time remains essentially unaffected (Figures 3.6c and d). The performance of the controller in an eccentric orbit remains essentially the same except for a steady state limit cycle appearing in the A degree of freedom due to the periodic forcing function dependent on e (Equation 3.8c). This is indicated in Figures 3.7a and b, where the 73 e = o i,co=o rli = io Yf= Pf = Xf-o - 0 Y 1 =i J = 0.5 Vj=25 Y0 = P0=X0=o - P° a = i K= o.i Y> P0'-X>i " 30 15h 0h V-(j> = 60° 10 0 C;= 6 (b) \ / I 37.5 25.0 12.5 (j) = 0 oh 25.0 12.5 <j)=90° JL 0.25 0.50 0.75 0 Orbit 0.25 0-50 Figure 3.6 Effect of solar parameters and solar aspect angle on the damped response 74 Figure 3.7 Response plot showing limit cycle oscillations in eccentric orbits and their removal through the modified control function roll and yaw motions damp out quite quickly but the platform pitch persists as a limit cycle. The sustained oscillation of the platform would, in general, be highly undesirable. Fortunately, it can be eliminated using a modified control relation for sensitive to the eccentricity induced disturbance, Xc = (l/vx)[-2esin6/(l+ecos6)+{(K/JI)(1+e)3/2/(l+ecos6)2}x U-(a+l) (1+e) 2/(l+ecos9) 2}] (3.15c)' Figures 3.7c and d illustrate the effectiveness of the modified control relation in eliminating the limit cycle. The information presented so far pertains to orbital motion in the plane of the ecliptic (i = 0). Of course, depending upon the mission, the orbital plane would be at an angle to the ecliptic. A systematic study showed the effect of i to be confined to local changes in response character without significantly altering the overall con trol performance. The plots in Figures 3.8a and b substantiate this conclusion. (b) Rotor spin decay The present analysis considers the rotor (body I) to have a constant average spin rate. Apparently, this would be achieved through some active energy source. However, the influence of possible rotor spin decay on the librational 76 e = o J =0.5 V. = 25 vrxr° Y" l=i Cj=4 U)= 0 \=G=x0=o P° a = i U.= 10 • 1 X'=|3'=X>' x° 1 i K=o.i Cf>=60° —i i 22.5 0 1 = 23.5 ' / 1 / i = 50° \ i \ 15.0 / f _ / 1 / \ \ i \ \ i 1 Pi \ / \ 7.5 / \ \ - \ \ K * * A —,1 1 a\ 1 J \ 1 1 \ 1 \ * i \ \ \ 0 \ ^ v J \ V 1 \ * II x.* Mb) \ • 1 r i=o 4>=70° a=o |' i \ K=o.5 i \ i i (/ \ ! \ K=o.5 it i i ii * i i . i .. i i » i * i » 7/'\ \ * * rv \ \ / . (c) ,i \ i -(d) ,V/ i 15 10 0 -10 0 0.25 Figure 3.8 0.50 Orbit 0.75 Response as affected by: (a,b) orbital inclin ation from the ecliptic; (c) rotor spin decay (d) modified control function response of the system would be appropriate to explore. Although a still remains a cyclic coordinate, and hence the first integral is available, it is not possible to eliminate the rotor spin degree of freedom from the rest of the equations of motion. Thus one is faced with the solution of a 7th order system (as against the 6th order in the previous case). As can be anticipated, the roll and yaw motions damp out as before, coupling with the a degree of freedom being weak. However, the platform pitch angle A tends to drift away from its preferred orientation A^ (Figure 3.8c). This does not reflect in any way a limitation on the capability of the controller but failure on our part to exploit it fully. The validity of this observation becomes quite apparent when one examines the modified pitch equation in the presence of dissipation, A" -Y" sinB-{2esine/(l+ecos8)}(A'-Y'sinB+cosBcosY) -B'Y'cosB-Y'cosBsiny-B'cosYsinB+(K/JI)x {(l+e)3/2/(l+ecose)2}(A,-a') = Qx (3.8c)' Note the dynamic coupling between the rotor and the platform. On the other hand, the control relation for (Equation 3.15c) does not involve a' explicitly. Thus the controller's potential to account for the rotor spin decay is not utilized. The situation can easily be 78 corrected by modifying the control function as indicated, QA = -yAA,-a(6)a,-vA(X-A ) (3.15c)" where a(6) = (K/JI)(1+e)3/2/(l+ecos0)2 Ac = -(l/vA)(2esin6)/(1+ecosG) A typical system response using this modified control function is presented in Figure 3.8d. Implication of this analysis is rather far reaching. It is no longer necessary to maintain the rotor spin rate from attitude dynamics considerations as the controller is able to.provide sufficient torque irrespective of the spin rate. The system thus has a truly semi-passive character promising an increased satellite life-span. (c) Attitude control The controller provides an interesting possibility of changing the satellite's preferred orientation in orbit. This is accomplished by using the position control parameters Yc, Bc and A^ in accordance with the desired equilibrium configuration y^, 6^, and A^. As an illustration, for a static equilibrium of the satellite in a circular orbit, Equations (3.8) in conjunction with Equations (3.15) lead to 79 Yc = (1/v )[(sinyf/cosBf){(l-J)I(a+l) +JIcosBfcosYf:}+(3I-4) sinY^cosy^] +Y (3.19a) (3.19b) Xc = (Vvx)-[(K/JI){cosSfcosYf-a-l}]+Xf (3.19c) Note that in this particular case of e = 0, the position control parameters are fixed once the final orientation is specified. On the other hand, for the case of an eccentric orbit, the parameters depend, in addition, on the satellite position in the orbit. Figure 3.9 illustrates the versatility of the semi-passive controller in achieving arbitrary orientations in space, for both circular and elliptic orbits. This suggests an exciting possibility for a space vehicle to extend its range of applicability and undertake diverse missions. (d) Illustrative example of the concept through a preliminary attitude dynamics study of the two well-known satellites, INTELSAT IV and Anik, when provided with the proposed controller. Appropriate It was decided to demonstrate the effectiveness 80 Figure 3.9 Effectiveness of the solar controller in achieving arbitrary orientations of the satellite 81 geometrical and inertia properties were assigned. As seen before, it is no longer necessary to spin a satellite for attitude control. However, nominal value of a = 1 was taken to account for possible spin introduced from other consider ations, e.g., temperature control. The solar parameter value of C. - 2 for INTELSAT IV can be obtained with control 2 plate areas A^ = 3 ft and moment arms ^ = 5 ft. Similarly, 2 - 5 for Anik may be achieved with plate areas A^ = 1.2 ft and moment arms = 5 ft. Subjecting the satellites to a disturbance equivalent to that imparted by micrometeorite impacts over 24 hrs, which represents an enormous magnifi-8 0 cation of the real situation, gives -0.05 < y', 8', A' ^ — o o o £0.05. It is apparent (Figure 3.10) that the controller is able to damp such a severe disturbance quite effectively with the maximum deviation from the preferred orientation of less than 0.25°. The figure also shows the controller's effectiveness in achieving specified spatial orientations. 3.2 Improvement of Controller Design Potential of the concept having been established, attention is next directed at rendering it more efficient and structurally more attractive. As the solar controller was found to be quite effective even in the absence of any spin, the case of a nonspinning satellite is now considered.. 82 e = o J = o.5 jij=io Y0=(3=A=o 0=1 K=0.5 Vi = 25 Y0=g=X=0.05 i =23.5° <f> = 60° CO = 0 1--Y -p° 0.2 INTELSAT IV 1=0.7 C:=2 Yf=Pf = A=o H 75 -75 _Yf=10° ($=-45° Af=45°J 0.2 h 0.1 0 Anik l = i C; = 5 Yf=Pf=Af=o 75 -75 l_Yf=20° (3 = 60° A=-50°_ 1/8 1/4 3/8 0 Orbit 1/8 1/4 Figure 3.10 Projected controller performance in achieving nutation damping and attitude control of INTELSAT IV and Anik satellites 83 3.2.1 Equations of motion The governing equations of motion follow directly from Equations (3.8) on substituting J = 1, K = 0: Y" -28* (Y'tan3-cosy)-I(A'-y'sinB+cosBcosy)(B'-siny)secB +{3(1-1)/(l+ecos9)-l}sinYcosY-{2esin6/(l+ecos8)}x (Y'+cosytanB) = Qy (3.20a) B" -Y'cosY-{2esin9/(l+ecose)}(B'-sinyJ + d(A'-y'sinB +COSBCOSY)+(Y1sinB-cosBcosy)}(Y'cosB+cosysinB) -3{ (I-l)/(l+ecos9) }sin2YsinBcosB = QD (3.20b) P A" -Y" sinB-{2esin9/(l+ecos9)}(A'-Y'sinB+cosBcosy) -B'Y1cosB-Y'cosBsiny-B'cosysinB = Q, (3.20c) 3.2.2 Development of controller models Controllers capable of providing general libration al damping and three-axis attitude control require the three degrees of freedom of the system (y, 3 and A) to be controlled independently. A versatile solar controller configuration would thus result in generalized forces as functions of at least three control variables, 6.. Of l various configurations satisfying this requirement, only those permitting relatively easier solution for the control variables with a given set of would be practically feasible. 84 At the outset, it appears instructive to study the control moments generated by a single plate P with its support arm gimballed at the point T on the axis of symmetry of the satellite (Figure 3.11a). The arm supporting the plate is allowed rotations and a2 while the plate can turn about it through an angles measured from the plane TSW. Following the procedure discussed earlier in detail, the generalized forces due to solar radiation pressure can be written as: QY = -CE (0) |cos?|cos?{(h/e+sina2) (sina^cosS+cosa^sina^sinS) 2 +cosa,cos a_sin6}/(I cosB) (3.21a) j. z y = -CE (0) |cos?|cos?{(h/e) (cosa^cosS-sina^sino^sinfi) +cosa,sina0cos6-sina„sinS}/I (3.21b) 12 2 y Q, = CE(0) |cos?[cos? cosa_cos6/I (3.21c) A c. X where cos? = -u^coso^sinfi+Uj(sina^cos<5 + cosa^sina2sin6) -u, (cosa,cos6-sina,sina„sinS) (3.22) k 1 12 As is evident, each of the 0^ depends on the variables o^, and 6. The transcendental nature of these functions makes it difficult to establish the bounds Figure 3.11 Development of solar controller configurations 86 IQ I , |Q„! and lQ-,1 within which Equations (3.21) 1 y max' 1 8'max 1 A 1 max ^ possess real solutions for a^, a2 and 6. It is, therefore, necessary to consider modifications which would reduce these equations to a more amenable form. A logical approach seems to lie in devising con figurations where some of the three variables become con stants. The best choice, from a practical viewpoint, would be to remove the gimbal and fix the axis of the plate to the satellite body, thereby rendering and a2 constants. This reduction in the number of control variables, of course, would have to be compensated by a corresponding increase in the number of plates rotating about body-fixed axes. Numerous controller configurations were considered to this end, how ever, for conciseness, only a few are discussed here. Figure 3.11b shows an arrangement of three plates P^(i = 1,2,3) rotatable about three mutually perpendicular body-fixed axes. The rotations 6^(i = 1,2) are measured from the yz-plane while 6^ is measured from the xy-plane. The evaluation of the generalized forces resulting from this configuration may be carried out either from the first principles, or, by appropriate substitutions for a^, a2 and 6 for each plate into Equations (3.21). These are found to be: 87 = E (0) {(^ |cos£11cos?1cos61cosA-C2|cos£2|cos?2cos62sinA +C3|cos?3|cos53sin63>/(I cos6) (3.2 3a) = E(0){-C1|cos?1|cos^1cos61sinA-C2|cos?2|cos52cos62cosA +C3|cos£3|cos£3cosS3}/Iy (3.23b) = E (0){C1|cos?1|cos51sin61+C2|cos?2|cos?2sin62}/Ix (3.23c) where cos£^ = u^cos6^+(u_.sinA-u^cosA) sinS^ cos^ = u^cos52+(UjCosA+u^sinA) sin62 cos£3 =-UjSin63+ukcos63 (3.24) It may be observed that the dependence of the generalized forces Qi(i = y,3,A) on the control variables 6^(i = 1,2,3) is quite involved and the system of Equations (3.23) does not appear to lend itself to an analytical treatment. Figures 3.11c and d show alternative arrangements, An analysis of the generalized force expressions, which are essentially of the same nature as Equations (3.23), showed them to present similar difficulties. 88 Following the approach discussed in preceding sections, the controller configuration shown in Figure 3.12 is finally arrived at. The performance of this model is investigated in detail as both the hardware and software requirements are relatively simple to implement. It consists of four plates P^(i = 1,2,3,4) with their support arms forming a crisscross. The plates and P2 rotate about the axis of symmetry of the satellite, the rotations 6^ and 62 being measured from the xy-plane. The plates P^ and P^ rotate about arms, lying in the plane perpendicular to the axis of symmetry, with the rotations 63 and 6^ measured from the yz-plane. The moments generated by this arrangement are found to be: = E(8) [C^|cosC11cos?1sin61-C2|cos?2|cos£2sin62 +l{C3 | cos£3 I cos^cosS 3~C;j | cos£4 | cos^cos6^}cosX] sec 3 (3.25a) Qg = E (9) [C]_ I cos^ I cos?1cos(S1-C2 | cos?2 I cos52cosS2 +1{-C^I cos?^Icos?3Cos63+C4I cos£4|cosE^cosfi^sinA] (3.25b) = E(6){C3|cos53|cos?3sin6 3+C4|cos?4|cos?4sin64} (3.25c) where 90 cos?. = -u .sinS . +u. cos6 . , i = 1.2 cos?_ = u.cosS_+(u.sinA-u, cosA)sin6_ cos?. = u. cos6 .-(u . sinA-u. cos A) sinS . (3.26) 4 I 4 ] k 4 and the solar parameters are defined as, C. = (2pp R3/yI )A.e., i = 1,2 l co p y l l = (2ppoR3/yIx)Aiei, i = 3,4 (3.27) Although Equations (3.25) indicate a rather complex dependence of on the control variables, it is possible to obtain relatively simple analytical solutions for (i = 1,2,3,4) through a judicious control strategy. 3.2.3 Control strategy An inspection of Equations (3.25) indicates that the plates and P2 do not produce any moment in the pitch (A) degree of freedom, hence this control will have to be accomplished through plates P^, P^. Ideally, one would like their operation to be free of coupling roll-yaw moments. Fortunately, this can be achieved through a simple control law: 91 •off or •off , TT/2, (3.28) the option being dicated by the sign required of Q^. 6^ = 'off* implies its rendering the corresponding cos£^ = 0, i.e., physically, the sun-line would lie in the plane of the plate. A pure pitch moment thus produced has the magnitude Q, = E(9)C_ „ (u.sinX-u. cosA) 2 (3.29) where C., = C„ = C0 „ is assumed for convenience. the orientation of the axis of symmetry of the satellite, plates P^ and maY be operated simultaneously or one at a time with the other 'off. The latter mode of operation, resulting in greater transverse torques, is considered here. The generalized forces in the roll and yaw degrees of freedom now take the form: Q = ±E (9) C . I -u . sin6 . +u, cos6 . I (-u . sin6 . +u, cos6 . ) sin6 . secB y i' j l k l1 i l k l l (3.30a) QQ = ±E ( 8) C . I-u . sin6 .+u. cos6 . I (-u . sin6 .+u, cos6 .) cos6 . 3 • il lk I 1 3 • i k i I (3.30b) 3,4 In order to provide the roll-yaw moments controlling where the plus and minus signs correspond to plates and ?2' respectively. Note that the moments Q and are coupled through the plate rotation angle 6^. The next step would be to maximize the available moments consistent with the control requirements of appropriate sign. There are several aspects to this problem. One approach is to maximize the total transverse torque, 2 2 2 1/2 (Q^cos B+Qg) . Unfortunately, the resulting critical <5^ (orienting the control plate so as to maximize cos^) yields only two of the four possible combinations of the roll and yaw moment directions: Qy, Qg: ++,+-,-+, An alternate procedure would be to maximize |Q | and ||, in turn, with respect to 6^. the maximum value of |Q^| = Q™ is found to occur at 6im = ^/2+tan 1[{-3u.±(9u2+8u2)1/2}/2uk] (3.31a) with the corresponding |Q^| = being given by Equation (3.30b) with 6. =6. . Similarly, the maximum of |QR| = Qm occurs at 6im = tan~1[{3uk±(9u2+8u2)1/2}/2uj] (3.31b) I l c with the corresponding \Q^\ = Q given by Equation (3.30a). The proper sign of the quantity within the square-root is determined from the coupling constraint imposed by Equations (3.30), i.e., tan6± = Q^cosB/Qg (3.32) A consideration of Equations (3.31a and b) indicates that in each case, the two values of <5^ , corresponding to the plus and the minus signs, lie in adjacent quadrants. Hence, one of them always satisfies the constraint relation (3.32). Thus use of the proper sign in the evaluation of 6. im (Equations 3.31) in conjunction with an appropriate choice of the plate or P2 (Equations 3.30) enables the controller to provide all possible sign combinations of Q^ and Qg. Of m c cm the two admissible sets of moments, (Q^, Qg) and (Q^, Qg), the one resulting in a greater magnitude of the total torque to control the axis of symmetry of the satellite, i.e., (Q2cos2B+Qg)1/2, is used. For example, let the required control moments be Q^> 0 and Qg< 0. As cosB is positive, tan 6^< 0- Thus 6^ must lie either in the second or the fourth quadrant (Figure 3.13). Let the two values of as given by Equation (3.31a) be (6. ) and (6. ) as shown. Note that they im a^ im a.2 have to be in adjacent quadrants as pointed out before. Let the 6^m corresponding to Equation (3.31b) be in the third and the fourth quadrants. Consistent with our require-94 Figure 3.13 Quadrant constraint for plate rotation 6. 95 merits, the admissible values are (6. ) and (6. ). . Of im a2 im b1 these, the one resulting in a higher value of the total transverse torque is used in the analysis. The control moments thus determined are applied in a bang-bang fashion in conjunction with linear displace ment and velocity sensitive switching functions, Qi = ~Qis9n s± (3.33) where Si = i' + m(i-if), i = Y,B,A (3.34) 2 2 2 2 Qy = Q™, QB = for (C-J cos2B+Qg )>(Q° cos2B+Qm ) Qy = QY, Qg-Qg for (Q'yl cosZB+Q^ ) < (Q° cos^B+Qm ) and m represents the system gain, chosen according to some suitable criterion. The control procedure may now be summarized as follows: (i) sense the roll, yaw and pitch angles and rates, orbital position and the solar aspect angle; (ii) determine the signs required of according to the switching criterion (Equations 3.33); (iii) during pitch control, use (6^ = TT/2, <5^ = 'off or (63 = 'off, 64 = TT/2) for Q^/ (UjSinA-u^cos X) ^ 0, respectively (Equations 3.25c and 3.26). (iv) for roll-yaw control, compute the four values of 6^ = 6^ from Equations (3.31a and b) and of the two satisfying the quadrant constraint (3.32), choose the 6^ producing the greater resultant transverse torque. Substitute into Equation (3.30a or b) and select the plate to be operated through the sign required in the right hand side. 3.2.4 Results and discussion As before, the response of the system is studied by numerically integrating the equations of motion (3.20) along with the control relations (3.33). Much smaller inte gration steps were required due to the bang-bang nature of the control law. A step size of 0.1° gave results of sufficient accuracy for the solar parameters 5. For cases with larger values of C^, the step size had to be correspondingly decreased. The controller performance was evaluated in both circular and elliptic orbits and the influence of important system parameters investigated. (a) Libration damping in circular orbits Figure 3.14 summarizes the performance of the controller in damping the librational motion in circular orbits in the form of optimization plots for the controller gain m. It shows the variation of the damping time x,, 97 e = 0 CO = 0 0=45° i = 2 3.5° Y,= P,= A,= o Yf=ft=Af=o J = Y,P,X i 1 i i i i I 0 5 10 15 20 25 30 m Figure 3.14 Optimization plots for the solar controller gain m 98 defined here as the time taken for all the three libration angles to settle within 0.1° of the final orientation, with the gain m for different combinations of system parameters and initial conditions. The plots, in general, indicate the existence of an optimum value of the system gain result ing in the least time of damping. The influence of the satellite inertia parameter I on the performance of the controller is indicated by a comparison of curves (a) and (b). Despite the drastic change in the satellite mass distribution from pencil-like (I = 0.1) to spherical (1=1), only a small variation in the optimum value of m and the general damping behaviour is observed. It may be pointed out here that curves (a) and (b) were obtained using a relatively small value of the solar para meters = 2. For larger values of C^, the effect of I was hardly perceivable. The effectiveness of the control system in libration damping of satellites with a wide variety of mass distributions is thus apparent. The effect of the solar parameters C\ , characterizing the magnitude of the solar torques, is observed by comparing curves (a and c) and (b and d). An increase in not only results in a substantial reduction in the damping time but renders the controller performance relatively insensitive to the system gain m as well. A comparison of curves (d and f) and (c and e) shows the effect of the initial disturbances. An increase 99 in the initial conditions results in longer damping times as anticipated. In addition, for smaller values of C\, the system performance shows a strong dependence on the controller gain m (curves d and f). The influence of the apparent position of the sun, indicated by the solar aspect angle cj>, and the perigee argument w on the system behaviour was also studied. The results, however, showed local changes in the response character only, leaving the damping performance essentially unaffected. Typical optimal responses for a satellite in an equatorial orbit are presented in Figure 3.15a. Even when subjected to severe impulsive disturbances in the roll, yaw and pitch degrees of freedom simultaneously, the controller is able to damp the satellite librations in a few degrees of orbital travel with the amplitudes remaining quite small. (b) Libration damping in elliptic orbits The effectiveness of the controller in damping the roll-yaw (y, 3) oscillations of the axis of symmetry of the satellite remained virtually unchanged in eccentric orbits. The pitch (A) degree of freedom, however, executes a steady-state limit cycle (Figure 3.15b). These oscillations result due to the presence of the periodic forcing function dependent on e (Equation 3.2 0c) and the inability of the controller to generate sufficient torque to counter it at all times. 100 Figure 3.15 (a) Typical optimum response in circular orbits; (b) Limit cycle oscillation in elliptic orbits; (c) Variation of limit cycle amplitude with system parameters 101 From design considerations, it is of interest to investigate the limit cycle amplitude as a function of system parameters. Once the roll-yaw oscillations are completely damped, Equations (3.20a and b) are identically satisfied. Letting Y=Yi=3=Bi=0 and substituting from Equations (3.29 and 3.33) into Equation (3.20c), the governing equation for the pitch oscillations reduces to A" -{2esin8/(l+ecos6) } A * = -C-. -E(6)x 3 i 4 2 (u.sinA-u. cosA) sgn S,+2esin0/(l+ecos6) (3.20c)1 J K A The limit cycle oscillation being the steady-state solution of this equation, its amplitude may be expressed as A = f(C_ . ,e,m, A,-,i,(d,<t)) (3.36) max v3,4f The largest amplitudes would result when the maximum magnitude of the eccentricity induced disturbance and the minimum (zero) magnitude of the pitch control torque occur simultaneously. The latter would physically correspond to the projection of the sun-line on the yz-plane becoming coincident with the axis about which plates and P^ rotate. Note that such a situation would arise only twice a year for an earth satellite. Considering small e, the condition for worst limit cycle is given by 102 tancf) = (tanA^sinoo-cosoo) seci/ (tanA^cosoo+sinoo) (3.37) Figure 3.15c shows the variation of the pitch limit cycle amplitude with the orbital eccentricity for a range of values of the solar parameter C, . under the most adverse situation. Even here, only moderate values of 4 are required to limit the amplitude to a generally acceptable value. (c) Attitude control The ability of the controller in imparting arbi trary orientations to the satellite and thus enabling it to undertake diverse missions appears interesting to explore. Figure 3.16a shows, for both circular and elliptic orbits, the effectiveness of the control system in providing an arbitrary pitch attitude to the satellite while the axis of symmetry remains normal to the orbital plane. The ability to align the symmetry axis with the local vertical direction and simultaneously attain a desired pitch attitude is indicated in Figures 3.16b and c. Note the excessive over shoots with a larger value of the system gain which suggest the use of small m for a smooth transition between widely differing attitudes. Thus with the present control system, an antenna aboard the spacecraft is able to scan substantial regions of the sky as arbitrary pitch attitudes are attainable 103 0 10 12 Orbits Figure 3.16 Effectiveness of the four-plate model in achieving arbitrary orientations of the satellite 104 with the axis of symmetry either along the orbit normal or the local vertical. The possibility of stabilizing the satellite axis of symmetry at other orientations was also investigated. The analysis suggests that the controller is able to achieve this but only at the cost of higher values of C. . 1 (d) Illustrative example In order to obtain preliminary estimates of the control plate areas and moment arms required, the proposed Canadian Communications Technology Satellite (CTS) and Anik, are considered. The latter is assumed to be non-spinning which represents an adverse situation. An impulsive disturbance of 0.1 is applied in all the three degrees of freedom which is in excess of that imparted by micrometeorite impacts over 24 hours. As the inertia parameter (I - 0.1 for CTS, 1 for Anik) did not affect the performance signifi cantly, most curves in Figure 3.14 (especially curves c, d and e) are representative of the librational behaviour of both satellites considered here. The solar parameter value of C^ - 5 can be obtained for CTS using plate areas 2 2 A^ 2 = ±2 rt r ^ = 1.2 ft with the moment arms = 10 ft. Similarly, C^ - 20 for Anik may be achieved with each of 2 the plates having an area of 5 ft and moment arm 10 ft. As indicated by curves (d and c) in Figure 3.14 these moderate control plate areas are sufficient to damp the satellite 105 librations within a few degrees of orbital travel. The librational amplitudes also remained within a fraction of a degree. The plots presented in Figure 3.16 indicate the effectiveness of the controller in providing attitude control to the CTS. Finally, a comment concerning earth shadow, which would render the solar controllers ineffective, is appropriate here. For a geostationary orbit, the influence of shadow is confined to quarter of the satellite's lifespan, and even here, only during 5% of the orbital period. The results showed the controller performance to remain virtually unaffected. 3.3 Concluding Remarks The significant conclusions based on the analysis may be summarized as follows: (i) The feasibility of a semi-passive controller using solar radiation pressure for three-axis nutation damping and attitude control of spinning satellites is clearly demonstrated, (ii) The system is capable of damping extremely severe disturbances in a fraction of an orbit. The time of damping can be further reduced by an optimum choice of the system gains, (iii) With the use of the general controller configura tion a directional device aboard the stabilized platform may be earth-oriented, space-oriented 106 or made to track a specified celestial object through a proper choice of the position control parameters. Even with the simplified model, scanning of a substantial region of the sky is possible. The solar pressure controllers thus impart versatility to a space vehicle in under taking diverse missions, (iv) The effectiveness of the system remains unaffected, .even during a spin decay, with a proper choice of the modified control function. Thus it is no longer necessary to maintain a constant spin rate by compensating for the dissipated energy, (v) A logical approach for evolving a three-axis solar controller model suitable for practical implemen tation is presented. The proposed four-plate controller model appears to be quite attractive due to its simplicity of design and the associated software. (vi) The use of bang-bang control relations substantially improves the system performance. For solar parameter values attainable in practice, a wide range of controller gain yield near optimum performance. Satellites with different mass distributions exhibit essentially the same damping characteristics. 107 (vii) As the controllers do not require any mass expulsion scheme or active gyros involving large power consumption, the solar pressure control system is essentially semi-active. This promises as increased lifespan. 108 4. MAGNETIC-SOLAR HYBRID ATTITUDE CONTROL The earth's magnetic field presents an interesting possibility of generating control moments by interaction with onboard dipoles. Magnetic torquing appears to be particularly attractive as it offers increased reliability through the elimination of moving parts. The shortcoming 5 9 of the method, as pointed out by Bainum and Mackison and others, lies in its inability to produce sufficient pitch control torques in equatorial orbits where the geomagnetic field is nearly parallel to the orbit normal. On the other hand, the usefulness of placing satellites in equatorial orbits cannot be overemphasized, e.g., the communications satellites in the geostationary orbit. A need for evolving a system which retains the simplicity of the magnetic concept and yet able to provide pitch control, is thus apparent. Having established the effectiveness of the solar pressure controller, it is proposed here to utilize it for damping pitch oscillations. This chapter explores the feasibility of the three-axis nutation damping and attitude control of a dual-spin satellite using a magnetic-solar hybrid control system. Two magnetic controller models are considered. A bang-bang control law with linear displacement and velocity sensitive switching functions is employed and analytical 109 solutions for the control variables are obtained. The performance of the control system is evaluated numerically and the results are presented as functions of system parameters. An illustrative example towards the end establishes the effectiveness of the system. 4.1 Formulation of the Problem 4.1.1 Equations of motion The geometry of motion (Figure 4.1) of the satellite being the same as the dual-spin spacecraft considered in Chapter 3, Equations (3.8) apply where (i = y,$,X) now represent the total generalized forces due to the magnetic and solar controllers. For convenience, the governing equations of motion are cited again: Y11 -23' (Y'tanB-cosy)-(3*-siny) sec 3 [(l-J)I(a+l) x { (1+e)/(1+ecosG)}2+JI(X'-y'sin3+cos3cosy)] +{3(1-1)/(l+ecos0)-l}sinycosY-{2esine/(l+ecos6)} x (Y'+cosytanB) = Q„ (4.1a) 3" -Y'cosY-{2esin0/(l+ecos6)}(31-siny)+(y'cosB+cosysinB) x [ (1-J)I(a+1){(1+e)/(l+ecos9)}2+JI(X'-y'sin3+cos3cosy) 110 Figure 4.1 Geometry of motion of dual-spin satellite in the earth's magnetic field Ill +(Y'sinB-cos3cosY)]-3{(I-l)/(1+ecose)} sin2YsinBcosB= (4.1b) A" -Y" sinB-{2esin6/(l+ecos6)}(A'-Y'sinB+cosBcosYi-B'Y'cosB -Y'cosBsinY-B'cosYsinB+(K/JI){(1+e)3^2/(l+ecos6)2} x [Al-Y,sinB+cosBcosY-(a+l) { (1+e)/.(1+ecosG) }2] = (4.1c) 4.1.2 Magnetic roll-yaw control Consider a single dipole onboard the platform with - 2 an arbitrary orientation p, strength h^ and polarity U. The moment generated by interaction with the earth's magnetic field is then given by, M = P X B = Uh2p X B (4.2) Expressing the unit vector p and the geomagnetic induction vector B in terms of their components along the xyz-axes and using the principle of virtual work, the generalized forces in the roll, yaw and pitch degrees of freedom can be written as: Q = UC (p.B.-p.B.)secB/(l+ecos9) (4.3a) Y m i j j i 112 QB = uSn(pkBi~piBk)/(1+ecose) QAm= U<Cn/JI) (PjB^B^/d+ecose) (4.3b) (4.3c) where the magnetic parameter C , characterizing the magnitude of the magnetically generated moments, is defined as C = h M /yl m m e y (4.4) The body components of B/D are given by the relations m • — B . l B . 3 Bk cosBcosy cosBsinycosnm cosBsinysinnm tsinBsinn m -sinBcosn m -siny cosycosr^ cosysinn m sinBcosy sinBsinycosn sinBsinysinn m ' m -cosBsinn m +cosBcosn m B xn B yn B zn (4.5) The geomagnetic induction components along the orbit normal, ascending node and perpendicular to the line 81 of nodes in the orbital plane are well established . For an earth-centered canted dipole model, they are: B, =• -cosi cose + sini sintj) sine xn mm m m m (4.6a) 113 Byn = (3/2) sinimsin2nrncosem+(l/2) [cos<j>m+ (3/2) x { (1+cosi ) cos (2n-<j) )+(l-cosi )cos(2n + cj) )}]sine m mm m mm m (4.6b) Bzn = (1/2) sinim (l-3cos2nm) cosem+(1/2) [cosimsincj)m +(3/2){(1+cosi )sin(2n )+(l-cosi )x m mm m sin(2n +6)}]sinem (4.6c) m m m rm It may be pointed out here that the angle <$> varies due to the earth's rotation and the regression of the line of nodes, according to <j>m = (ne+nr) . Its governing equation with the orbital angle as the independent variable is, <t>^ = (ne+nr) (R3/y)1/2(l+e)3/2/(l+ecos0)2 (4.7) which has the solution (for e < 1) *m = cj)mo+{(ne+nr) (Rp/^)1/2 d+e) 3/2/d~e2) } x [-esin9/(l+ecos9)+{2/(l-e2)1^2}tan-1 { (1-e) 1/2tan(6/2)/(l+e) 1/2} I (4.8) 114 where the appropriate quadrant for the arctan function is to be introduced. The present analysis ignores the nodal regression as a dynamic effect which is equivalent to assuming the earth to be gravitationally spherical. For controlling satellite nutations in an equatorial orbit with the nominal position of the spin axis along the orbit normal, it can be easily shown that the 2 2 2 1/2 maximum transverse torque, (Q cos 3+QQ) , results when y p p^ = 0. Accordingly, two magnetic controller models with dipoles in the satellite's transverse plane are considered here (Figure 4.2). Model A consists of a single dipole rotatable about the x axis in the platform fixed reference x ,y , z . p v p,jrp' p Physically, it would correspond to a single electromagnet rotating about the axis of symmetry or two (or more) fixed electromagnets with variable currents. The moments generated are given by, QY = ~ucmPjBisec3/d+ecose) (4.9a) Qg = UCmpkBi/(l+ecos9) (4.9b) QAm = u(Cm/JI)(PjBk-pkBj)/(l+ecose) (4.9c) where 115 Figure 4.2 Magnetic-solar controller configurations 116 Pj = cos(6m+A), pk = sin(0m+X) (4.9d) o A constant dipole level h , leading to a constant value of the magnetic parameter C , is assumed in the present analysis. The components of the total transverse torque are controlled according to the relations: sgn(Q^cosB) = -sgn (4.10a) sgn QQ = -sgn SD (4.10b) P P Q cosB/Qg = S /Sp (4.10c) where the switching functions S^(i = y,B) are defined as Si = i'+ m(i-if), i = y,3 (4.11) and m represents the system gain. This results in a simple control law for the dipole angle 8^ (Figure 4.2), )m = -A-tan_1(S3/SY) (4.12) Note that no polarity reversals are required if the angle 6^ is permitted any values in the range 0 to 2TT . However, it would be desirable to restrict 0 to the range m -TT/2 to TT/2 (for the gimballed electromagnet) and permit polarity reversals, which leads to the controls: 117 3m = ~tan_1{(S3+SYtanA)/(SY-SgtanX)} (4.13a) U = sgn{(S^cosX-S^sinX)/B± > (4.13b) where the principal value of the arctan function is to be admitted. Model B consists of two mutually perpendicular dipoles with orientations p^ and p2 fixed in the platform and allowed polarity reversals (Figure 4.2). Substituting for the unit vectors p^ and p2 in Equations (4.3), the magnetic control moments become, Q = -(U p +U„p„.)C B.secB/(l+ecos0) (4.14a) Q8 = (Ulplk+U2p2k)CmBi/(1+ecos9) (4.14b) QXm = {Ul(pljBk-plkBj)+U2(p2jBk-p2kBj)}x (Cm/JI)/(1+ecose) (4.14c) where pij = p2k = cos(em+X), plk = -p2j = sin(em+X) (4.14d) 118 Although the bang-bang controller can no longer distribute the total transverse moment between the roll and yaw degrees of freedom in proportion to the demands governed by the switching functions, appropriate signs of Q and (Equations 4.10a and b) may be achieved with the polarity controls: for I tan(9 + X)|<1, U, = sgn(S /B.cos(9 +X)} 1 m 1 1 3 Y i m U0 = -sgn{S„/B.cos(9 + A)} (4.15a) 2 pi m for|tan(9M+A)|>1, = -sgnCS^/B^in (9M+X) } U0 = sgn{S /B.sin(0 +X)} (4.15b) 2 3 Y i m Note that 9^ defining the locations of the dipoles with respect to the platform-fixed axes is a constant in this case. For the final pitch orientation X^ = 0 (the axis y pointing towards the earth), minimum cancellation of the torques due to the two dipoles occurs when 0^ = 0. Further more, for most applications requiring —TT/4 < X < TT/4, the controls reduce to: U1 = sgn(S /B.^, U2 =- sgn (Sg/B^ (4.16) 4.1.3 Solar pitch control The platform pitch control is accomplished using a solar pressure controller consisting of plates PP which are allowed a rotation 6 about the axis of symmetry of the satellite (Figure 4.2). For highly reflective plates, the pure pitch moment resulting from solar radiation pressure is (Equation 3.12c), QXs = -C{(1+e)3/(l+ecose)4}|-UjSin(6+A) +ukcos(S+A)|{-UjSin(6+A)+ukcos(6+A)} (4.17) where C = (4ppoR3/uIxp)AE (4.18) Using the control relation Q, = -|Q, I sgn S, (4.19) As 1 As'max ^ A with SA = A'+m(A-Af), (4.20) the control law for the plate rotation 6 becomes, for u_.>0, 6 = tan~1(uk/uj)-A-(7r/2)sgn Sx for u_.<0, <5 = Tr+tan-1 (uk/Uj)-A-(TT/2) sgn SA 120 for Uj = 0, 6 = cos_1{-sgn(ukSA)} (4.21) where the principal value of the arctan function is to be introduced. The total generalized force in the pitch degree of freedom is then given by, QA=QAm+QAs <4'22> It may be pointed out here that the components of the unit vector in the direction of the sun involve orbital parameters Q, i and ui referred to the ecliptic plane while the formulation of the magnetic controller requires the same angles referred to the earth's equatorial plane. The relations governing these parameters for an arbitrary orbital plane are readily obtained as: sinQ, sini = sin^sini (4.23a) m m cosfi cosw -sinfi sino) cosi = cos^cosoj-sinfisincocosi mm m m m (4.23b) cosfi sinu) +sin^ cosw cosi = cosfisinco+sinficosucosi m m m m m (4.23c) 121 4.2 Results and discussion The response of the system was studied by numeric ally integrating the equations of motion (4.1) along with the control relations governing the magnetic and solar moments, i.e., Equations (4.9, 4.13, 4.21 and 4.22) and Equations (4.14, 4.15, 4.21 and 4.22) for models A and B, respectively. The Adams-Bashforth predictor-corrector quadrature with the Runge-Kutta starter was used with a step size of 0.1°. The important system parameters were varied gradually over the range of interest and the controller performance evaluated both in circular and elliptic orbits. The amount of information thus generated is rather extensive; however, for conciseness, only the typical results sufficient to establish trends are presented here. 4.2.1 Nutation damping Figure 4.3 summarizes the performance of the proposed magnetic-solar hybrid controller in damping the nutational motion of the satellite spin axis and the pitch oscillation of the platform. It shows the variation of the damping time with the controller gain m for a variety of combinations of the system parameters and initial conditions. Here the damping time is defined as the time required for all the three degrees of freedom; namely, the roll y, yaw 8 and the platform pitch A, to settle within 122 e = 0 i =o J = i,K = o m for o = o J = Y,p,X <P 0 = 45 i = 23.5° J = 0.5, K= o.oi for o to C = 4 0 = io° r\=n-o mo m com=co =30° if=0 30 20 10 0 30 Model A Cm=i,l = i,a=o,j = o.i C , = 2,1 = 0.1-1,0 = 0-10, j = o.i (C) Cm=4,| = i,a = o,j = o.i (A) 20 10 C=o.5,| = i,a = o,j = o.i Cm=i,l=o.i,a=o,j=o.i Model B Cm=i,l=i,a=io,j = o.i Cm= 2,1 = 1 C=i,l = i,a = o,j = o.i Cm=2,| = ifa = ofj =o.i ± ± ( B) o 10 40 20 30 m Figure 4.3 Optimization plots for the magnetic-solar controller gain m 50 123 0.1° of the desired final orientation. The responses of both magnetic controller models A and B are presented, which, in general, indicate the existence of an optimum value of the system gain m leading to the minimum time of damping, and the sensitivity of the optimum to the system parameters and initial conditions. The influence of the satellite inertia and spin parameters on the attitude dynamics of passively stabilized satellites is well recognized. Their effect on the nutation damping performance of the present controlled system, however, appears to be negligible. This is indicated by curve (a) in Figure 4.3A where the results for pencil-like to spherical mass distributions (I = 0.1 to 1) and non-spinning to moderate spin rates (a = 0 to 10) were virtually indistinguishable. The banding together of curves (a,b and f) in Figure 4.3B reflects similar system behaviour. The insensitivity of the transient response to I and moderate values of a is understandable as these parameters, con tributing restoring forces to the system, are now largely provided for by the controller gain m. The performance under high spin rates is investigated in a later section. The effect of the magnetic parameter C , character izing the magnitude of the magnetic control torques, is indicated by a comparison of curves (b,a and c) in Figure 4.3A for the case of the single rotatable dipole. As anticipated, increasing the value of C leads to a corres-ponding reduction in the damping time T^. Larger values of the magnetic parameter (curves a and c) show a response pattern that is relatively insensitive to the controller gain m, thus indicating a large range of values of the latter to provide near-optimum damping. It may, however, be pointed out that the maximum attainable value of C is * m subject to constraints imposed by the electromagnet weight and power requirement. Curves (c,a and d) indicate similar effect of the magnetic parameter for the model with two fixed transverse dipoles (Figure 4.3B). A comment concerning the influence of the solar parameter would be appropriate here. An increased value of C led to a reduced damping time for the pitch (A) degree of freedom. However, even with a value of C = 2, the pitch motion, in general, damped out faster than the roll-yaw motion, except for the case of large viscous drags on the platform (large K, a). A comparison of curves (a and d) in Figure 4.3A and curves (d and e) in Figure 4.3B shows the effect of initial conditions on the performance of controller models A and B, respectively. In both cases, an increased impulsive disturbance not only leads to an anticipated increase in the damping time but also renders the response quite sensitive to the system gain m. With larger values of C , however, the latter effect was found to be less m pronounced. 125 It is of interest to compare the performance of the two magnetic controller models in damping the attitude motion. A suitable criterion for the comparison is the same total electromagnet weight and power consumption which requires the value of the magnetic parameter Cm for model A to be twice the value for model B. Comparisons may be made of curves (b, a and c) in Figure 4.3A with curves (c, a and d) in Figure 4.3B, respectively, towards this end. The results clearly indicate a better performance of the single rotatable dipole model A. This is explained by the ability of this model to distribute the total magnetic torque between the roll and yaw degrees of freedom in proportion to their demands, determined here by the switch ing functions S and Sg, respectively. The single rotating dipole model A may be obtained by using a rotating electromagnet, which, however, would involve additional weight and power requirement for the turning mechanism and a reduction in the system reliability due to physical movement of the electromagnet. Alternately, two fixed electromagnets with variable currents may be employed. Unfortunately, this would require each electro magnet to be the same size as the single rotating one in order to maintain the same resultant dipole strength as the latter, thus doubling the total weight. The choice between the physical arrangements leading to model A or B would be governed by such mission oriented factors as the system 126 reliability, associated hardware and software, and performance requirements. Typical damped responses of a satellite subjected to an extremely severe impulsive disturbance are shown in Figure 4.4 for both controller models. The time history of the controls is also included. Figure 4.4a shows the response in a circular orbit with a nominal value of a = 1, which may be required from such considerations, as tempera ture, control. The damped attitude motion of a nonspinning satellite in an eccentric orbit is shown in Figure 4.4b, which indicates a slight increase in the maximum amplitude and the damping time. The effectiveness of the controllers in damping such a severe impulsive disturbance in a few degrees of the satellite's orbital travel is thus apparent. The amplitudes are also limited to a few degrees. The controls, in general, require rather infrequent switching until the corresponding amplitudes become very small and chatter initiates. This may, however, be prevented by the inclusion of suitable deadbands in the control relations which would depend on the pointing accuracies required. The effectiveness of the control system in capturing a satellite from initial roll, yaw and pitch errors is presented in Figure 4.5. Both circular and eccentric orbit as well as nonspinning and moderate spin rate cases are considered. The initial error of 20° in 127 1 = 1 I - 0 m m = i o X = rl=A=0 Y 0 = 45 i =23.5° Am=n = 0 Y0=(30=X = 0.3 C = 2 0=10° w =wr30° Y=B=X=0 - indicates initiation of chatter e = o a = l J = 0.5 K =o.oi 1 1 1 1 1 v Model A \ Model B V\ C =2 \ \ C = l 3 i-l 25 50 0 25 0 -degrees 50 75 Figure 4.4 Damped response and time history of controls subsequent to initial impulsive disturbance 128 1=1 'rrT0 m=5 Y0 = P0 = A0=20° Y° r-2fl°I=io-50nm=n = 0- *t~-*'z° — -P° 0-2 Cp - 1 0 m o m wm=Lu=30t Yf=Pf=Xf=o X° indicates initiation of chatter u. 180 0 6 - 20- (b) 1 60 120 0 60 9-degrees 120 180 Figure 4.5 Damped response and time history of controls subsequent to initial position disturbance 129 each degree of freedom is corrected in approximately one-third of an orbit with model A and half an orbit with model B. The performance may be improved further through an optimum choice of the controller gain. 4.2.2 High spin rates and spin decay The discussion so far pertains to satellites which are either nonspinning or have moderate spin rates. The effectiveness of the control system (model B) for satellites with high spin rates is indicated by the response data presented in Table 1, which shows the coning amplitude of the spin axis, $max* and the damping time x^. The coning amplitude $max reached in the absence of the controller is also presented. For low spin rates, the control system, in addition to providing quick nutation damping, helps keep the amplitudes low. However, for high spin rates, it essentially acts as a damper. A comparison of the controller performance with m = 10 and m = 30 shows its effect on the amplitude to be slight but the damping time is affected appreciably, up to spin parameter values as high as a = 200. These observations substantiate the earlier conclusion regard ing the controller gain lending stiffness to the system. For still higher spin rates (a > 200), however, the influence of m is negligible indicating the dominance of the gyroscopic restoring forces. 130 TABLE 4.1 Response With High Spin Rates e = 0 I = 1.2 i=0 ft = ft = 0 Y=B=A=0 m m o o o J = 0.5 <j> = 45° i = 23.5° u> = u = 30° Y1 = B1 = 0.3 moo K = 0 C = 4 cj> = 10° A' = 0 = B. = A = 0 mo o f f f C = 2, m m = 10 C = 2, m m = 30 C =0 m a $° max Ta max Td max 0 1. 86 28.20 1. 86 29.50 39.88 10 1.78 28.26 1.77 30.86 6.61 25 1. 61 26.04 1. 53 34. 56 2.99 35 1.39 30.96 1. 34 43.56 2.19 50 1.12 24. 22 1.10 25.50 1.56 75 0.84 21. 84 0.84 35.56 1.05 100 0.67 19. 44 0.67 33. 88 0.79 200 0.37 15.54 0. 37 22.26 0. 40 300 0. 25 11.82 0.25 13. 20 0. 27 400 0.19 9. 70 0.19 9.78 0.20 500 0.16 6.68 0.16 6.66 0.16 600 0.13 4. 50 0.13 4. 52 0.13 131 The analysis so far considered the rotor (body I) to have a constant average spin rate. Apparently this would be achieved through some active energy source compen sating for rotor spin decay due to bearing losses. Even in the absence of such energy supply, the analysis continues to be valid provided the spin parameter o remains sensibly invariant over a time interval of the order of the damping times attained here. This is of considerable value as any spin decay is likely to occur very slowly indeed. Thus the data presented in Table 1 can also be used to predict the long range performance of a spinning satellite. The controller's effectiveness in achieving quick nutation damping even in the absence of any spin promises an increased satellite life-span. 4.2.3 Attitude control The control system offers the exciting possibility of stabilizing the satellite along any arbitrary orientation in space, thus enabling it to undertake diverse missions. This may be achieved through the parameters y^, 3^ and A^, defining the final desired orientation, which are incorporated in the switching functions S^, and s^, respectively. Figure 4.6 shows the ability of the controller in achieving a variety of spatial orientations, the time taken being well within an orbit. Note that a fairly small value of the gain m used here leads to a smooth transition between widely 132 1 = 1.2 'm^0 m = I = P»=x=o - Y 0 = 45° 1=23.5° n =n = m ° Y; =P:=X'=O..-—|f c J = 4 = 1,K 0 =10° m o = o for o - o m ; j = 0.5 30° K=o.oi for a *o A° 80 4 0 0 40 0 -40 40 0 -40 Model A Cm = 4 Model B Cm = 2 y'' e = o a = o / Y=90° B=o X = 20 _ / T / f / / / / y'' e = o a = o /Y=90° p = o /• ' ' 0 X=40 f e=o a=o " e = 0 o-0 Y =30 B=-30* X,= -'o° YF=30 pF=-20 <"— x"'°° N e = o.2 a = i " .-"""* Y =30° P = -30° X = -io° e = 0.2 a = i /•^"{^-**° PF=30° ' 0 / X^-io \ 0 0.5 0 0.5 1.0 Orbits Figure 4.6 Effectiveness of the magnetic-solar controller in imparting arbitrary orientations to the satellite 133 different attitudes. On the other hand, larger values of m were found to result in an undesirable overshoot of the final orientation. It may be pointed out further that except for final orientations along equilibrium points of the system, the controller must at all times provide corrective torques to counter the gravity gradient and gyroscopic moments. Moderate values of the magnetic parameter taken here, C =4 for model A and C = 2 for m m model B, are found to be sufficient for both nonspinning and a nominal spin rate in circular as well as eccentric orbits (Figure 4.6). 4.2.4 Illustrative example It appears interesting to evaluate the performance of the magnetic-solar controller through a preliminary attitude dynamics study of two well-known satellites, the proposed Canadian Communications Technology Satellite (CTS) and Anik, when provided with the proposed control system. For convenience the latter may be considered as nonspinning which only represents an adverse situation. Values of the magnetic parameter cm = 4 for model A and 2 for model B are attainable with total dipole levels of approximately 2 200 and 20 ampere-meter for the CTS and Anik, respectively. 2 A control plate area A = 0.5 ft and moment arm e = 5 ft yield a solar parameter value of C - 2. On the. other hand, pitch control may be achieved using a reaction wheel of 134 capacity = 0.1 lb-ft-sec/day which is equivalent to C = 2. An impulsive disturbance of 0.1 is applied in all the three degrees of freedom simultaneously which is in excess of that imparted by micrometeorite impacts over 24 hrs, and thus represents an enormous magnification of the real situation. As the inertia parameter (I - 0.1 for CTS, 1 for Anik) did not affect the performance significantly, most of the results presented in this chapter are representative of these satellites. It is apparent (Figure 4.3) that the controllers are able to damp out such a severe disturbance in about 3° and 5° of the orbit with models A and B, respec tively. The maximum deviation from the orbit normal attitude also remained less than 0.2°. Any arbitrary orientations may be imparted to the satellites in well within an orbital period (Figure 4.6). Finally, it may be mentioned that the comment con cerning the earth shadow made earlier also applies here. The results showed the controller performance to remain virtually unaffected. Furthermore, the analysis ignores, dynamics due to relative motion of the gimballed electro magnet (model A) and the solar control plates as well as the shadowing of the latter by the satellite. 135 4.3 Concluding Remarks The conclusions based on the analysis may be summarized as follows: (i) The analysis clearly establishes the potential of a magnetic-solar hybrid system for nutation damping and attitude control of satellites, (ii) The concept permits interchangeability of the solar controller and a variable speed pitch momentum wheel, thus effectively providing a gyromagnetic control system, (iii) The ability of the system in damping extremely severe disturbances in a few degrees of the orbit makes it quite suitable for applications, such as, communications satellites, (iv) Even with rotor spin decay, the system continues to function effectively which promises an increased satellite life-span. (v) It is possible for a satellite to attain any arbitrary orientation in space, both in circular and elliptic orbits, thus widening the scope of its mission. 136 5. AERODYNAMIC-SOLAR HYBRID ATTITUDE CONTROL The analyses in the earlier chapters apply primarily to satellites in high altitude orbits since the influence of the earth's atmosphere was ignored. However, several space applications, such as, weather forecasting, earth resources exploration, military scouting, etc., depend on high resolution photography and hence necessarily negotiate near-earth trajectories. Unfortunately, interaction with free molecular reaction forces reduces their effective life-span leading to expensive periodic replacement. One possible solution would be to employ an elliptic trajectory making a space vehicle to spend major portion of its orbital period above the atmosphere and dip into it only when actively engaged in its mission. This presents an interesting situation where aerodynamic forces may be used to advantage for attitude control, possibly in conjunction with solar radiation pressure. The present chapter explores the feasibility of such a hybrid control system. The governing equations of motion, together with a bang-bang control law, are analyzed numerically and the influence of the important system parameters on the performance is evaluated. 5.1 Formulation of the Problem 5.1.1 Equations of motion The case of a nonspinning satellite is considered here, which, in view of the analysis in Chapter 4, represents an adverse situation. Equations (3.20) therefore apply where Q^(i = Y,B,A.) now represent the total generalized forces due to the solar pressure and the aerodynamic forces: y" -26* (y1 tanB-cosy)-I(A'-y1sinB+cosBcosy) (B'-siny)secB + {3 (1-1)/(l+ecos6)-l}sinYcosy-{2esin0/(l+ecos6)} x (y'+cosytanB) = Qy (5.1a) B" -Y'cosy-{2esine/(l+ecose)}(B'-siny)+{ I(X'-y1sinB+cosBcosy) +(y1sinB-cosBcosy)}(y1cosB+cosysinB)-3{(1-1)/(l+ecos9)} x . 2 sm ysxnBcosB = (5.1b) X" -y" sinB-{2esin9/(l+ecos6)}(X'-y'sinB+cosBcosy) -B'Y1cos3_Y'cosBsiny-B1cosysinB = (5.1c) 5.1.2 Controller configuration and generalized forces The controller configuration studied for establish ing the feasibility of solar pressure control is considered 138 for the hybrid aerodynamic-solar control as well (Figure 5.1). The generalized forces due to Solar radiation pressure were found earlier (Equations 3.12) as: Q^s = -E (0) [LT1C11 cos?11 cos?1sin61sinA + U2C2 I cos?2 I cos?2s:'-n^2COS^''sec^ (5.2a) Qgs = -E(0) [U1C11 cos?11 cos?1sin61cosA - U2 C2 I cos?2 I cos?2sin(52sin^ J (5.2b) Q, = -E(0)C_ cos?Jcos?_ (5.2c) AS J J -J where U. = +1 for P., -1 for P!; i = 1,2 1 1 1 cos?, = u. cos S, + (u . s in A—u, cosA)sin5. 1 1 1 j k 1 cos?2 = u^cos62+(UjCosA+u^sinA)sin62 cos?^ = -UjSin(6^+A)tu^cos(6^+A) (5.3) The solar parameters, C\ , are given by, ci = (4pP0Rp/Mly )Aiei' 1 = 1,2 X i Figure 5.1 Aerodynamic-solar hybrid controller configuration 140 C3 = (4pPoRp/yIx)A3£3 (5'4) The resultant free molecular force on an elemental 8 2 area, considering specular reflection, may be expressed as , dF = (1/2)p V2dA|cos£ |{(C -CTcosec£ |cos£ |)v d a. a. U Lt 3. 3. + (CTcosec? sgncos? )n} (5.5) ij a a Evaluating the total moment about the satellite center of mass due to the aerodynamic forces and using the principle of virtual work leads to: Qya = J(0)[UlCallCos5all{vj-(CL/CD)cosec?al(vjlCOS?all -sin61sinAsgncos?a-])}+U2Ca2 | cosCa2 I ^vj~ ^L^D^ cosec^a2 (Vj | cos?a2 | -sin62cosAsgncosCa2) ^-ca3 I cos^a3 ^ ^Cl/CD^ cosec?^^|cos?a3|}v^cos(6^+A)]sec3 (5.6a) Sa = -J(9) [UlCallcos?all{vk-(CL/CD)cosec?al(vklcos?all tsinS^cosAsgncos^) }+u2ca2 I cos^a2 I ^vk~ ^Cl/CD^ cosec^a2 (vk | cos?a2 | -sin62sinAsgncos?a2) }~Ca3 | cosCa3 | (1- (C^/C^) x cosec?a3|cosCa3|}v^sin(63+A)] (5.6b) 141 QXa = J(e)Ca3[lCOS?a3l+(CL/CD)sin?a3lcOS?a3 (5'6c) where U. = +1 for P., -1 for P!; i = 1,2 l ii cos? , = v. cosS, + (v . sinA-v. cosA) sin6, al I 1 j k 1 cos?a2 = v^cos62+(vjcosA+v^sinA)sin62 cos? _ = -v . sin ( 6 +A)+v, cos (6 + A) (5.7) The aerodynamic parameters, are defined as C . = (cr p R2/I )A.e., i = 1,2 ai DKap p/ y' I i' ' Ca3 = <CDpaEPp/Ix,A3e3 (5'8) The variation of the atmospheric density with altitude, incor porated in the definition of the function J(0), is modelled according to the relation p = p {(R-R )/(R -R )}n (5.9) a ap e p e where n varies in the range -5 to -7 depending on the altitude^3. The total generalized forces due to the radiation pressure and the aerodynamic forces may be expressed as functions of the control variables U\ (i=l,2) and S^(i=l,2,3) in the form: 142 QY = QYS(Ul'U2'6l'62,+QYa{Ul'U2'6l'62'63)  Q3 = QBs(Ul'U2'6l'62)+Q3a(Ul'U2'6l'62'63) °A = QAs(63)+QAa(63) In view of Equations (5.2 and 5.6), the dependence of the moments on the control variables is quite complicated. However, it is possible to obtain relatively simple analytical solutions for the plate rotations 6^ through a judicious control strategy. 5.2 Control Strategy The magnitudes of the control moments are constrained by the control surface areas and moment arms (through the solar parameters C. and the aerodynamic parameters C .). In l ai order to utilize the maximum moments available from the controller, the following bang-bang control law is employed here: Q. = -|Q.I sgn S. (5.11) l 1 l'max ^ l where the switching functions S. are defined as Si = i'+m(i-if), i = Y,3,A (5.12) (5.10a) (5.10b) (5.10c) and m represents the system gain. The determination of the simultaneous maxima, |Q.I , that the controller can provide, is rather involved 11 max c The problem may be simplified considerably by recognizing that the solar pressure and the aerodynamic forces represent the respective dominant influences at high and low altitudes Only over a small portion of an elliptic trajectory are the two effects comparable in magnitude. The plate rotations 6^, producing the maximum moments, may thus be obtained in accordance with the solar pressure at high altitudes and aerodynamic forces at low altitudes. The switch-over point can be determined by comparing the magnitudes of the two forces. 5.2.1 High altitude Maximization of the solar pitch moment, Q, , leads AS to the following control law for 6_: for u. > 0, 6_ = tan (u, /u .) - A-(TT/2) sgn S, J JS K J A for Uj < 0, 63s = TT+tan-1 (uk/Uj)-A-(TT/2) sgn for u. = 0, 63s = cos "'"{sgn (u^S ^) }-A (5.13) where the principal value of the arctah function is to be admitted. The solar roll-yaw control moments Q and Q„ J ys 3s being coupled through the rotations 6^ and 6^, the total 144 2 2 2 1/2 transverse torque, (Q cos $+QR ) ' , is maximized which occurs at: 6, = ff/2+tan-1[ (3/2) (u.sinA-u, cosA)/u.±{ (9/4) x (u^sinX-Uj^cosA) 2/u2+2}1/2] (5.14a) + for { (u . sinA-u, cosA)/u-} > 0, J K 1 and 6 = u/2+tan"1[(3/2) (u .cosA+u, sinA)/u.±{(9/4) (u.cosA ^& J K 1 J +uksinA)2/u2+2}1/2] (5.14b) ± for {(u.cosA+u, sinA)/u.} > 0. ~\ K 1 In general, all sign combinations of Q and QR y S P S are available through an appropriate choice of the control plates to be operated. Occasionally, due to the time-varying nature of the components u^, u_. , u^, the desired signs of Q and Q„ may not be available. In such situations, the ys |3s J roll-yaw control plates are to be turned 'off, which may be done either by making the corresponding 6^ = 0 or, by aligning the plates parallel to the incident radiation, i.e., choosing 6^ so as to render the corresponding cos?^ = 0. The control moments at high altitude thus become, 145 QY = QYs(Ul'U2'6ls'62s)+QYa(Ul'U2'6la'62s'63s)  QB = Q0s(Ul'U2'6la'62s,+Q3a(Ul'U2'6ls'62s'63s> QX = Qxs(63s,+QXa(63s) 5.2.2 Low altitude In a manner similar to the solar control at high altitude, the moments due to the aerodynamic forces may be maximized at low altitude. The exact determination of the critical 6^ being more complicated in this case, simplifying assumptions such as | v^| , |v_.|<<|vk| and CL/CD<< 1 are employed. These approximations, however, extend only as to the determination of the maximizing 6^, the subsequent evaluation of the moments being exact. The control law for the rotation 6^, maximizing the aerodynamic pitch control torque Q, , is found to be: A a f or v . > 0, 6_ = tan ^ (V. /v .) -X+ (TT/2) sgn S, j 3a k y ^ X for v. < 0, 6 _ = iT+tan (v, /v .)-A+ (TT/2) sgn S, ] 3a k j X for v. = 0, '6_ = cos 1{-sgn (v, S, ) }-X (5.16) J o a K A where the principal value of the arctan function is to be introduced. (5.15a) (5.15b) (5.15c) 146 The aerodynamic roll-yaw moment is maximized by the rotations 6^ and 62 given by, S, = TT/2-tan "*"{v. / (v . sinA-v, cosA) } (5.17a) la l j k <5 = TT/2-tan_1{v. / (v .cosA+v, sinA) } (5.17b) ^a I j K An investigation of Equations (5.6) indicates that, in highly eccentric orbits, the drag component of the aero dynamic force governs the directions of both the roll and the yaw moments. As a result, all sign combinations of C) and QD , in general, are not available. This presents an ya pa option as to controlling either the roll or the yaw moment and retaining the associated torques in the other degree of freedom. Recognizing that \Qa | >> | Q | for small yaw angles pa y a B, a large proportion of the total transverse torque may be utilized by selecting the control plates according to the sign required of Qg. A preliminary investigation of the system performance, however, revealed the effect of the coupled roll moment Q to be generally adverse. The roll-yaw control law in the aerodynamic region is, therefore, modified to exercise control action only over those portions of the trajectory where the associated roll moment Qya is also of the correct sign, the controller being switched 'off other wise. The latter may be accomplished by making all the roll-yaw control plates parallel to each other or to the flow 147 direction (by choosing 6^ so as to render the corresponding cos? • = 0). ai The control moments at low altitude thus take the form: Qy ' Qys^l'^^la'^a^V^l'^'^a'^a'Sa) (5'18a) QB = QBs(Ul'U2'6la'62a,+QBa(Ul'U2'6la'62a'63a) (5-18b) QX = QXs"<63a,+QXa(63a) (5-18c) The control procedure may be summarized as follows: (i) sense the roll, yaw and pitch angles and rates, orbital position and the solar aspect angle. Estimate the atmospheric density, (ii) determine the switch-over point by comparing the magnitudes of the solar and aerodynamic parameters. It may change due to variations in the atmospheric density. (iii) for pitch control, provide rotation 6^ determined from Equations (5.13) and (5.16) at high and low altitudes, respectively, (iv) for roll-yaw control at high altitude, compute 6^(i = 1,2) from Equations (5.14) and provide these rotations to the sets of plates resulting in the signs of Qva and QRc governed by Equation 148 (5.11). If the proper signs are not available, turn the roll-yaw control plates 'off1. At low altitude, compute 6^(i = 1,2) from Equations (5.17) and determine the plate sets yielding maximum Q^a of the proper sign. If the associated Q is of the correct sign, provide these rotations; otherwise, turn the roll-yaw control plates 'off. 5.3 Results and Discussion The response of the proposed hybrid control system was studied by numerically integrating the equations of motion (5.1) along with the appropriate control relations (Equations 5.15 or 5.18). The Adams-Bashforth predictor-corrector quadrature with the Runge-Kutta starter was used. A step size of 0.1° at high altitudes and 0.02° at low altitudes gave results of sufficient accuracy. The important system parameters were varied gradually over the range of interest and the controller performance evaluated. For conservative estimate of the controller's performance, it was purposely subjected to severe disturbances. It should be pointed out here that the atmospheric density depends, in addition to the altitude, on several classes of solar and geophysical phenomena. In the present analysis, density variations due to the latter over a few orbits of the satellite are ignored and a reference atmos phere corresponding to an exospheric temperature of 1250°K is 149 84 -14 3 considered . The value of p = 0.74 x 10 gm/cm at cl the perigee altitude h = 250 miles and the drag coefficient P = 2.2 yield the ratio C ./C. - 50. The switch-over D J ai 1 point is found to be at an altitude of approximately 500 miles. 5.3.1 Libration Damping The performance of the controller in damping the librational motion of the satellite is summarized in Figure 5.2 in the form of optimization plots for the controller gain m. The damping time x^, defined as the time taken for all the three libration angles to settle within 1° of the desired orientation, is presented as a function of the system gain. Various combinations of the important system parameters and initial conditions are considered. The plots, in general, indicate the existence of an optimum value of the system gain resulting in the minimum settling time. The influence of the satellite inertia parameter I on the controller performance is indicated by a comparison of curves (a) and (b). A reduction in the damping time with an increased value of I is apparent. Curves (f), (c) and (g) exhibit a similar trend for the case of a larger C^(=l). The zero settling time in curve (g) simply implies that, with this set of parameters, none of the attitude angles exceeded 1° in amplitude. The advantage of disc-like satellite mass distribution (I > 1) is thus obvious. 150 e =o.i 0 = 45° CL/CD = o.i hp=25o mi. Y0=P0=X=o CO = 0 i = 23.5° Cgj/C^so j=Y,p,X Yf = pf=Af=o 0 5 10 15 20 m Figure 5.2 Optimization plots for the aerodynamic-solar controller gain m 151 The effect of the solar parameters C\ and the aerodynamic parameters C ^, which are directly related for a given plate size and moment arm, is shown by curves (a) and (c), and (b) and (g). Increasing the value of C\ leads to a substantial reduction in the damping time. This, of course, can be anticipated as C\ characterize the magnitudes of the control moments available. Their maximum attainable values in practice, however, are constrained due to considera tions such as launch, deployment and operation. It is interesting to determine the physical size of the controller yielding a given C. (or C .). For example, consider a x ax satellite with the mass properties of the Canadian communi cations satellite Anik (I - 1). A value of C. =1 (C . = 50) x ax is found to be attainable with plate sizes of about 5' x 5' and moment arms = 10 ft for a perigee altitude h^ = 250 miles. A comparison of curves (c), (d) and (e) indicates the system performance as affected by the initial impulsive disturbance. As expected, an increased disturbance implies larger damping times. In addition, this results in a left ward shift of the optimum gain m, suggesting the use of a smaller system gain for quick damping of large impulsive disturbances. The optimum gain being dependent on initial conditions, a value of m would have to be selected that promises reasonably good libration damping rates for all initial conditions that the satellite is likely to encounter 152 in its normal operation. This does not appear to be difficult as for y' = B1 = A' < 0.2, which represents an 'o o o — extremely severe disturbance, a large range of values of the system gain m yield near-optimum performance. Typical damped responses of the satellite are presented in Figure 5.3, with severe initial disturbances applied at different positions of an elliptic trajectory. Figure 5.3a shows the damping of a disturbance encountered at the pericenter. The large aerodynamic yaw and pitch moments restrict the corresponding amplitudes to a negligible value. On the other hand, the satellite executes a small roll oscillation resulting from the relatively smaller roll control torques and occasional loss of roll-yaw control at low altitude. Note the small hump in yaw due to a reduction of control moments in the neighbourhood of the switch-over point (0 = 67°). Figure 5.3b shows the response to a disturbance occuring just before the switch-over point. The satellite traverses a short distance through the aero dynamic region and most of the damping occurs under the influence of solar pressure torques. Figure 5.3c presents the response to a disturbance applied at the apocenter where solar radiation pressure has the greatest influence. As anticipated, quick libration damping results with the attitude errors settling within 1° in about 30° of the satellite's orbital travel. The same disturbance, applied shortly before the satellite is 153 l = i 0 = 45° Cj=o.5 hp = 250 mi. Y0=P0=X=o Y° e=o.i i=23.5° CL/CD=o.i Y^P^X^0-2 P° LO=0 m=5 Cai/Ci=50 Yf=Pf=XfQ X° 40 80 120 60 100 140 180 3.0 \-1.5 \-1 1™ 6 r * # i * * i i • i i eo=i8o° v'\ - 3 / # / / i i •9 = 270° • 0 0 /V \ .... .1 1 (C) v (d) i i 180 220 260 300 270 9-degrees o 90 180 Figure 5.3 Typical responses showing the effectiveness of the aerodynamic-solar controller at different orbital positions about to re-enter the aerodynamic region, leads to the response indicated in Figure 5.3d. The loss of continuous roll-yaw control in the aerodynamic region is reflected by the relatively larger (-7°) roll amplitude and the longer damping time. 5.3.2 "Attitude control At times, missions involving diverse objectives may require a satellite to change its preferred orientation in orb The controller's ability to impart any desired orientation to the satellite is explored here. Figure 5.4a shows the effectiveness of the control system in providing arbitrary pitch attitudes with the axis of symmetry of the satellite along the orbit normal. The ability to align the symmetry axis with the local vertical direction and simultaneously attain a desired pitch attitude is indicated in Figure 5.4b for a slender satellite (I = 0.1). For satellites with large I, the controller was able to accomplish the same, but only at the cost of higher values of as it must now overcome the gravity torques in addition to the inertia of the satellite. Note that the attitude angle 3 represents the planar oscillation in the local vertical configuration. The steady state motion noticeable for the case of e = 0.2, resulting from the eccentricity induced disturbance (Equation 5.1b), indicates the need of a larger value of C.. for its elimination 155 C,= i i=23.5° CL/CD=0.1 Y=P=A=° 'o "0 o 0 Y CA) =0 m = 0.5 Cai/Cj=50 Y = B = X =o.i o ro o — p° 0=45° hP=2 50 mi. 40 \- e = 0.1 I =2 Yf = PF=0 X = 45 e = 0.2 1=2 Yf=Pf=o XF=30° 120 60 e = o.i I =0.1 YF=9 0° p=o X=20° 30 /"'yzo0 PF=-30° \ Xf = io° ' Yf=o PF=30° Xf=-io° •30 0 e =0.1 I =1 (C) e =0.2 I =1 0 Orbits Figure 5.4 Effectiveness of the aerodynamic-solar controller in imparting arbitrary orientations to the satellite 156 Figure 5.4c presents results for stabilization along arbitrarily chosen values for all the three degrees of freedom. Finally, a comment concerning the earth's shadow, which would render the controller ineffective in the solar pressure mode, is appropriate here. It is apparent that its effect would be negligible in near-polar orbits. On the other hand, the influence would be maximum for orbits in the plane of the ecliptic. Even in the latter case, the effect of shadow would be of little consequence if the apparent position of the sun is in the neighbourhood of the apocenter as now the control in the shadow region is primarily accomplished aerodynamically. Hence, depending upon the mission, a judicious selection of the location of the line of nodes, orbital inclination from the ecliptic and the perigee argument could effectively minimize the influence of the earth's shadow. 5.4 Concluding Remarks The significant conclusions based on the analysis may be summarized as follows: (i) The feasibility of aerodynamic-solar hybrid control of near-earth satellites in elliptical orbits is established. (ii) The control system is capable of damping extremely severe disturbances in a fraction of an orbit 157 with the maximum amplitudes during the process limited to a few degrees, (iii) The controller permits the spacecraft to undertake diverse missions through stabilization along arbitrary attitudes. The pointing accuracies appear to be sufficient for many applications of near-earth satellites, (iv) The system is essentially semi-active which promises an increased satellite life-span. 158 6. TIME-OPTIMAL PITCH CONTROL USING SOLAR RADIATION PRESSURE The investigations presented in the preceding chapters clearly establish the possibility of utilizing the environ mental forces to achieve general three-axis librational damping and attitude control. With the control systems offering increased satellite life-times through their semi-passive character, it seems logical to direct efforts at improving their performance. This involves two aspects, namely, physical controller design and efficient control laws. The former having been stressed in the earlier chapters, attention here is directed at the possibility of using optimal control laws. As the energy required to turn the control plates is low and could be generated easily through the use of solar cells, the performance index need only include the damping time which is of prime concern. Time-optimal control of multi-degree of freedom systems, such as the coupled roll-yaw-pitch motions of a satellite, can generally be achieved only through enormous software complexities since the solution of a two point boundary value problem is involved. This is why switching criteria that are simple functions of the state variables were considered in the earlier analyses. On the other hand, a single degree of freedom system may lend itself to an analytical synthesis of the time-optimal 159 switching criterion. This is significant as, if successful, it not only could be applied to several situations of practical importance (platform pitch control of a spinning satellite or pure pitch control of a gravity gradient system 70 as in the case of COSMOS-149 ) but may also suggest switching laws that are likely to be efficient in controlling the general motion. This chapter investigates the development of the time-optimal control law for the planar pitch motion of a satellite. The utilization of solar radiation pressure for attitude control in a circular orbit is considered. The analysis leads to a useful relationship between the magnitude of the disturbance, control plate areas and moment arms, and the corresponding minimum damping time. 6.1 Formulation of the Problem Figure 6.1 shows an unsymmetrical satellite executing planar pitch libration \\i, with the center of mass S moving in a circular orbit about the center of force 0. The governing equation of motion is well-known, ijj" + 3Kisim|;cosi|j = (6.1) where represents the generalized force due to solar radiation pressure. Figure 6.1 Geometry of motion of unsymmetrical satellite in the solar pressure environment The solar pressure controller consists of two highly reflective control plates and P2 which are permitted rotations 6^ and 62, respectively, in the orbital plane. The center of pressure of each plate is taken to lie on the satellite y axis (could be anywhere on the yz-plane) so as to yield a pure pitch moment. The moment generated by the controller is Qip = ±Ci' sin(6i+?) I sin (6i+?) cos6i, i = 1,2 (6.2) (+ for P , - for P2) where the solar parameter is defined as C. = (2pp R3/yI )A. e . (l-sin2(j)sin2i) (6.3) I ^o p x 1 1 Through a judicious choice of the plate to be operated (P^ or ?2) » in accordance with the angle (6^+£), may be controlled in sign. The magnitude of the control moment, ||, varies with both the angle £ and the control variable 6.. Its maximum with respect to 6. occurs at i i 6im = tan-1[(-3/2)tanc±{(9/4)tan2?+2}1/2] (6.4) where the ± signs apply for tan C £ 0, respectively. The variation of |Q,[ with L is shown in Figure 6.2a where 1 \p 1 max ^ 162 163 = C2 = C is assumed for convenience. The system is able to provide a value |Q,| = (2/3/3)C at all times. The governing equation of motion (6.1) may thus be presented as 4>" + SK^sin^cosiJj = u(9) (6.5) with - (2/3/3) C-3Kisini|)ecos^e < u(0);< (2/3/3) C-^K^ini^cos^ A symmetrical band on the control, |u(9)|<_C*= (2/3/3) C- | 3KisiniJ;ecosiJJe| , (6.6) is considered here for convenience. Its effect is only to yield a slightly conservative bound on the control either on the plus or the minus side depending on the nominal attitude . re 6.2 Time-Optimal Synthesis Using the state variables x^ = i> and x2 = i>' , and linearizing about the nominal attitude ^ = i1e> the system (6.5) can be expressed in the form, (6.7) 164 where A = - • 0 1 0 2 0 B = -n 1 and u(6) I< C Taking the initial time 8Q = 0, the state-transition matrix $(0) is obtained as $(9) = (6.8) cosnG (l/n)sinn0 -nsinn0 cosnS The solution for the system of Equation (6.7) then becomes, x(0) = $(8)x(0) + $ (0--T)Bu(T)dT 0 (6.9) A control U(T) is sought which will bring the system state from x(0) to x(8f) = 0 in minimum 8f. Substituting the initial and the final states in Equation (6.9) results in <H-x)Bu(T)dT = -x(0) (6.10) The solution for u(8) bringing the system state to * rest in minimum Gf is well-known to be u(8) = ±C , with the number of switches depending upon the initial state of 8 5 the system . Considering initial states that can be driven to rest in a single switch, the control takes the form 165 where u(9)=K1, 0 £ 9 < 0g u(9) = K2, 0g < 9 < Q Kl> = |K2> = C*' (6.11) Substituting for u(9) in Equation (6.10) leads to: K1 = n{-nx1(0)(sinn9f-sinn0s)+x2(0)(cosn9f-cosn0s)}/A (6.12a) K2 = n{nx1(0)sinn9g+x2(0)(l-cosn9g)}/A (6.12b) where A = sin(n0„-n0 )-sinnGJ-+sinn9 f s f s The proper signs of and K2 are best obtained from the phase plane portrait of the optimally controlled system (Figure 6.2b). The trajectories are circular arcs * 2 * with centers at x^ = ±C /n , x2 = 0 for u = ±C , respectively. The switching boundary is composed of semi-circles passing through the origin. For any initial condition x(0), the system state moves on the switching boundary for 0g <_ 0 <_ 0^ as shown in the figure, It is apparent that all initial states lying within the region ABCDA of the phase diagram can be driven to the origin with a single switch of the control. For optimal response, the control u(0) assumes the value * u = -C if the system state lies above the switching boundary * and u = +C if it is below the switching boundary. Equations (6.12) may now be solved to obtain the switching time 0 and the final time 0_ for a given initial s f 3 condition. This yields the open-loop realization of the control in the form u = u(0). On the other hand, use of the switching boundary yields the feedback realization u = u (x) , which makes the system self-correcting to slight deviations of the state vector. Of particular interest are impulsive disturbances which a satellite is likely to encounter through, say, micrometeorite impacts. The phase portrait immediately yields the maximum impulsive disturbance from which the satellite can be brought to rest in a single switch of the control and the error amplitude during the process as: x~ (0) 2 max = 2/2 (6.13a) x, (0) 1 max {l+x2(0)}1/2-l (6.13b) 167 where the normalized state variables x^ and i<2 are defined as 3^(6) = n2x1(9)/C* (6.13c) x2 (0) = n x2(0)/C* (6.13d) The variation of the normalized error amplitude with the initial impulsive disturbance is shown in Figure 6.3a. The switching and the final times obtained by solving Equations (6.12) are also presented as functions of the impulsive disturbance (Figure 6.3b). 6.3 Results and Discussion In order to ascertain the applicability of the optimal control law synthesized from linear theory to the actual nonlinear system, the response of both the linearized and nonlinear equations governing the motion was evaluated. The two systems were subjected to the same disturbance and control. Figures (6.4a and b) show response plots indicating the effect of the inertia parameter to be negligible for the pitch attitude nominally along the local vertical. As anticipated, the open-loop and the feedback response of the linear system are identical. On the other hand, the open-loop control system is unable to bring the nonlinear system to rest exactly. The feedback system, however, accomplishes 168 0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 6.3 (a) Variation of transient amplitude |x, (8)| with initial condition xo(0); (b) Variation of switching time 0S and final time 9^ with initial condition x„(0) 169 Open loop , feedback (linear system) Open loop ( nonlinear system ) Feedback (nonlinear system) C = 10 x2(o) -0.5 1 T f\ ib =o / \ / \ K.= 0.5 1 1 / \ K.= i.o 1 \ {b) -1 1 i i + =30C K =0.5 15 (O K = i.o 30 0 15 6 - deg rees 30 45 Figure 6.4 System response to impulsive disturbance 17 this, the nonlinear system state approaching the origin asymptotically using a number of switches of the control. I order to avoid any relay-chatter, it appears advisable to use only a single switch for the actual nonlinear system as well and employ a passive device to damp the small residual motion in the neighbourhood of the origin. Figures 6.4c and d present the system response for a satellite stabilized in an arbitrary pitch attitude. Note that the gravity gradient torque now represents a destabilizing effect, which the controller must neutralize in addition to countering the disturbance. The longer dampi time required with a higher value of the inertia parameter * clearly reflects a greater reduction in the value of C for increased K.. 1 The system response may now be projected for the pitch control of the CTS. At synchronous altitude, the 2 value of C =10 corresponds to A. - 2.5 ft and e. = 10 ft. x c i I When subjected to an extremely severe impulsive disturbance of ^(O) = 0.5, a damping time of the order of 20° of the orbit is attained (Figure 6.4). A disturbance ^(O) = 0.1 on the other hand would be damped out in approximately 4°. (Figure 6.3b). The system thus appears promising. It should be pointed out here that the constraint |[£(2/3/3)C represents the most adverse situation as || may attain a value as large as C during certain orbital positions (Figure 6.2a). The performance of the controller, 171 therefore, would always exceed the responses presented here (Figure 6.4). 6.4 Concluding Remarks (i) The analysis clearly demonstrates the feasibility of the time-optimal pitch control of satellites using the solar pressure, (ii) The controller is capable of damping extremely severe disturbances in a few degrees of the satellite's orbital travel. The transient amplitude is also small, (iii) The controller is able to provide nominal control at an orientation which is not an equilibrium position of the uncontrolled system, (iv) The optimal control strategy, developed for the linearized system, may be applied effectively to the actual nonlinear system. 172 7.. CLOSING COMMENTS 7.1 Summary of the Conclusions The significant conclusions based on the pre ceding investigation may be summarized as follows: (i) The solar radiation pressure, normally neglected in the analysis of spinning satellites, can affect the librational performance substantially. It merits the same consideration as the inertia properties, spin rate and eccentricity during the design of an attitude control system, (ii) The environmental forces can be used quite effectively to provide three-axis libration damping and attitude control of spinning space craft. (iii) As substantial control moments are available even with the use of moderate controller sizes, it does not appear necessary to spin a satellite from attitude control considerations. Of course, the presence of spin would improve the nutation damping performance, (iv) A logical procedure is established for the development of an effective solar pressure con trol system. This should prove useful in evolving a suitable controller depending upon the mission requirements. 173 (v) The magnetic roll-yaw controller, in conjunction with the solar pitch controller, provides an efficient three-axis control system, (vi) The hydrid aerodynamic-solar system offers effective control of near-earth satellites in elliptic trajectories which promise an increased life-span. (vii) Utilization of the maximum available control moments, through bang-bang operation, leads to smaller damping times compared to the linear control law with saturation constraints. The near-optimum performance resulting for a wide range of system parameters and initial distur bances is particularly attractive, (viii) Approximate analytical techniques can be used quite effectively during preliminary stages of satellite design. For small amplitude motion, usually the case of interest, they can predict the libration amplitude and frequency with considerable accuracy, (ix) The attitude control systems analyzed here are semi-passive, as they do not involve any mass expulsion schemes and/or active gyros requiring large power consumption. This promises an increased satellite life-span. 174 7.2 Recommendations for Future Work The investigation reported here suggests several topics for future exploration. Only some of the important problems are mentioned here: (i) The feasibility of using environmental forces to advantage having been established here, it appears logical to direct efforts at improving the efficiency of these systems. The design approach suggested in Chapter 3 could be applied to devise alternative controller configurations. An extension of the approach in Chapter 6 to the synthesis of optimal or suboptimal control laws for the coupled motion is likely to improve the control performance, (ii) A detailed hardware oriented study would permit precise comparison of the methods proposed here with the currently used active control systems. Such analyses should also include any inertia variations and reaction forces arising from con trol system operation, (iii) With ever increasing energy requirements, the use of large solar panels by the future generation of satellites is likely to be indispensable. Substantial energy savings may result if the semi-175 passive attitude control systems utilizing the environmental forces could be extended to flexible satellite configurations. The dynamical problem is going to be more involved. Hence, more sophisticated controllers and control strategies would be required as the disturbances, such as, panel vibration, differential thermal heating, etc., are of the continuous type. The problem appears to be quite useful as well as challenging, (iv) The possibility of employing the environmental forces for orbital transfer and trajectory control appears interesting. As the concept promises considerable energy saving and reduction in payload (no mass expulsion is required), it could be of immense value in interplanetary travel as well where long time durations are involved. It presents an exciting possibility of achieving controlled variations in the orbital parameters of a space-vehicle. Optimization of system performance in the foregoing represents a vast challenging area that has remained virtually unexplored. 176 BIBLIOGRAPHY 1. Thomson, W.T., "Spin Stabilization of Attitude against Gravity Torques," The Journal of the Astronautical  Sciences, Vol. 9, No. 1, January 1962, pp. 31-33. 2. Pringle, R., Jr., "Bounds on the Librations of a Symmet rical Satellite," AIAA Journal, Vol. 2, No. 5, May 1964, pp. 908-912. 3.. 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Mingori, D.L., Tseng, G.T.fand Likins, P.W., "Constant and Variable Amplitude Limit Cycles in Dual-Spin Spacecraft," Journal of Spacecraft and Rockets, Vol. 9, No. 11, November 1972, pp. 825-830. 179 34. Proceedings of the Symposium on Gravity Gradient Attitude Stabilization, The Aerospace Corp., December 1968. 35. Shrivastava, S.K. , Tschann, C.,and Modi, V.J., "Librational Dynamics of Earth Orbiting Satellites -A Brief Review," Proceedings XIV Congress of  Theoretical and Applied Mechanics, Indian Society of Theoretical and Applied Mechanics, Kharagpur, December 1969, pp. 284-306. 36. Roberson, R.E., "Attitude Control of a Satellite Vehicle - An Outline of the Problem," Proceedings  of the Vlllth International Astronautical Congress, Wein-Springer-Verlag, Berlin, 1958, pp. 317-319. 37. Wiggins, L.E., "Relative Magnitudes of the Space-Environment Torques on a Satellite," AIAA Journal, Vol. 2, No. 4, April 1964, pp. 770-771. 38. Clancy, T.F. and Mitchell, T.P., "Effects of Radiation Forces on the Attitude of an Artificial Earth Satellite," AIAA Journal, Vol. 2, No. 3, March 1964, pp. 517-524. 39. Modi, V.J. and Flanagan, R.C, "Effect of Environmental Forces on the Attitude Dynamics of Gravity Oriented Satellites: Part I - High Altitude Orbits," Aeronautical  Journal, Royal Aeronautical Society, Vo. 75, November 1971, pp. 783-793. 40. Modi, V.J. and Kumar, K., "Librational Dynamics of Gravity Oriented Satellites under the Influence of Solar Radiation Pressure," Proceedings of the  International Symposium on Computer-Aided Engineering, University of Waterloo, May 1971, pp. 359-377. 41. Debra, D.B., "The Effect of Atmospheric Forces on Satellite Attitude," The Journal of Aeronautical Sciences, Vol. 6, 1959, pp. 40-45. 42. Schrello, D.M., "Aerodynamic Influence on Satellite Librations," ARS Journal, Vol. 31, No. 3, March 1961, pp. 442-444. 43. Garber, T.B., "Influence of Constant Disturbance Torques on the Motion of Gravity Gradient Stabilized Satellites," AIAA Journal, Vol. 1, No. 4, 1963, pp. 968-969. 180 44. Meirovitch, L. and Wallace, F.B., Jr., "On the Effect of Aerodynamic and Gravitational Torques on the Attitude Stability of Satellites," AIAA Journal, Vol. 4, No. 12, December 1966, pp. 2196-2202. 45. Garwin, R.L., "Solar Sailing - A Practical Method of Propulsion Within the Solar System," Jet Propulsion, Vol. 28, No. 3, March 1958, pp. 188-190. 46. Sohn, R.L., "Attitude Stabilization by Means of Solar Radiation Pressure," ARS Journal, Vol. 29, No. 5, 1959, pp. 371-373. 47. Galitskaya, E.B. and Kiselev, M.I., "Radiation Control of the Orientation of Space Probes," Cosmic Research, Vol. 3, No. 3, May-June 1965, pp. 298-301. 48. Mallach, E.G., "Solar Pressure Damping of the Libra tions of a Gravity Oriented Satellite," AIAA Student Journal, Vol. 4, No. 4, December 1966, pp. 143-147. 49. Modi, V.J. and Flanagan, R.C., "Librational Damping of a Gravity Oriented System using Solar Radiation Pressure," Aeronautical Journal, Royal Aeronautical Society, Vol. 75, No. 728, August 1971, pp. 560-564. 50. Modi, V.J. and Tschann, C., "On the Attitude and Librational Control of a Satellite using Solar Radiation Pressure," Astronautical Research 1970, Proceedings of  the XXI Congress of the International Astronautical  Federation, Editor-in-chief: L.G. Napolitano, North-Holland Publishing Co., Amsterdam, 1971, pp. 84-100. 51. Modi, V.J. and Kumar, K., "Coupled Librational Dynamics and Attitude Control of Satellites in Presence of Solar Radiation Pressure," Astronautical Research 1971, Proceedings of the XXII Congress of the International  Astronautical Federation, Editor-in-chief: L.G. Napolitano, D. Reidel Publishing Co., Dordrecht, Holland, 1973, pp. 37-52. 52. Ule, L.A., "Orientation of Spinning Satellites by Radiation Pressure," AIAA Journal, Vol. 1, No. 7, July 1963, pp. 1575-1578. 53. Peterson, C.A., "Use of Thermal Re-Radiative Effects in Spacecraft Attitude Control," CSR-TR-6 6-3, May 1966, Massachusetts Institute of Technology Center for Space Research, Cambridge, Mass. 181 54. Colombo, G., "Passive Stabilization of a Sunblazer Probe by Means of Radiation Pressure Torque," CSR-TR-66-5, 1966, Massachusetts Institute of Technology Center for Space Research, Cambridge, Mass. 55. Falcovitz, J., "Attitude Control of a Spinning Sun-Orbiting Spacecraft by Means of a Grated Solar Soil," CSR-TR-66-17, December 1966, Massachusetts Institute of Technology Center for Space Research, Cambridge, Mass . 56. Crocker, M.C. II, "Attitude Control of a Sun-Pointing Spinning Spacecraft by Means of Solar Radiation Pressure," Journal of Spacecraft and Rockets," Vol. 7, No. 3, March 1970, pp. 357-359. 57. Scull, J.R., "Mariner IV Revisited, or the Tale of the Ancient Mariner," presented at the 20th Congress of the International Astronautical Federation, Argentina, October 1969. 58. Alper, J.R. and O'Neill, J.P., "A New Passive Hysteresis Damping Technique for Stabilizing Gravity-Oriented Satellites," Journal of Spacecraft and Rockets, Vol. 4, No. 12, December 1967, pp. 1617-1622. 59. Bainum, P.M. and Mackison, D.L., "Gravity-Gradient Stabilization of Synchronous Orbiting Satellites," Journal of the British Interplanetary Society, Vol. 21, 1968, pp. 341-369. 60. Vrablik, E.A., Black, W.L., and Travis, L.J., "LES-4 Spin Axis Orientation System," TN 1965-48, 1965, Lincoln Lab., Massachusetts Institute of Technology, Cambridge, Mass. 61. Sonnabend, D., "A Magnetic Control System for an Earth Pointing Satellite," Proceedings of the Symposium on  Attitude Stabilization and Control of Dual-Spin Space craft, Rept. RT-0158(3307-01)-16, November 1967, Aerospace Corp., El Segundo, Calif., pp. 121-144. 62. Fischell, R.E., "Spin Control for Earth Satellites," Peaceful Uses of Automation in Outer Space, edited by J.A. Aseltine, Plenum Press, New York, 1966, pp. 211-218. 63. Wheeler, P.C., "Spinning Spacecraft Attitude Control via the Environmental Magnetic Field," Journal of  Spacecraft and Rockets, Vol. 4, No. 12, December 1967, pp. 1631-1637. 182 64. Sorensen, J.A., "A Magnetic Attitude Control System for an Axisymmetric Spinning Spacecraft," Journal of Space  craft and Rockets, Vol. 8, No. 5, May 1971, pp. 441-448. 65. Shigehara, M., "Geomagnetic Attitude Control of an Axisymmetric Spinning Satellite," Journal of Spacecraft  and Rockets, Vol. 9, No. 6, June 1972, pp. 391-398. 66. Hecht, E. and Manger, W.P., "Magnetic Attitude Control of the TIROS Satellites," Torques and Attitude Sensing  in Earth Satellites, edited by S. Fred Singer, Academic Press, New York, 1964, pp. 127-135. 67. Lindorfer, W. and Muhlfelder, L., "Attitude and Spin Control for TIROS Wheel," AIAA/JACC Guidance and  Control Conference, Seattle, Wash., 1966, pp. 448-461. 68. Wall, J.K., "The Feasibility of Aerodynamic Attitude Stabilization of a Satellite Vehicle," presented at ARS Controllable Satellite Conference, M.I.T., April 1959. 69. Schrello, D.M., "Passive Aerodynamic Attitude Stabiliz ation of Near Earth Satellites, Vol. I, Libration due to Combined Aerodynamic and Gravitational Torques," WADD Tech. Report 61-133, Vol. I, July 1961. 70. Sarychev, V.A., "Aerodynamic Stabilization System of the Satellites," Proceedings of the International  Colloquium on Attitude Changes and Stabilization of  Satellite, Paris, October 1968, pp. 177-179. 71. Ravindran, R., "Optimal Aerodynamic Attitude Stabiliz ation of Near Earth Satellites, Ph.D. Thesis, University of Toronto, April 1971. 72. Modi, V.J. and Shrivastava, S.K., "On Librations of Gravity-Oriented Satellites in Elliptic Orbits through Atmosphere," AIAA Journal, Vol. 9, No. 11, November 1971, pp. 2208-2216. 73. Modi, V.J. and Shrivastava, S.K., "On the Optimized Performance of a Semi-Passive Aerodynamic Controller," AIAA Journal, Vol. 11, No. 8, August 1973, pp. 1080-1085. 74. Butenin, N.V., Elements of the Theory of Nonlinear  Oscillations, Blaisdell Publishing Company, New York, 1965, pp. 201-217. 183 75. Modi, V.J. and Brereton, R.C, "Periodic Solutions Associated with the Gravity-Gradient-Oriented System. Part I: Analytical and Numerical Determination," AIAA Journal, Vol. 7, No. 7, July 1969, pp. 1217-1225. 76. Modi, V.J. and Brereton, R.C, "Periodic Solutions Associated with the Gravity-Gradient-Oriented System. Part II: Stability Analysis," AIAA Journal, Vol. 7, No. 8, August 1969, pp. 1465-1468. 77. Minorsky, N., Nonlinear Oscillations, D. Van Norstrand Company Inc., Princeton, 1962, pp. 127-133, 390-415. 78. Moran, J.P., "Effects of Plane Librations on the Orbital Motion of a Dumbbell Satellite," ARS Journal, Vol. 31, No. 8, August 1961, pp. 1089-1096. 79. Yu, E.Y., "Long-Term Coupling Effects Between the Librational and Orbital Motions of a Satellite," AIAA Journal, Vol. 2, No. 3, March 1964, pp. 553-555. 80. Ehricke, K.A., Principles of Guided Missile Design, D. Van Nostrand Co. Inc., New Jersey, 1960, pp. 241-256. 81. RCA, "RCA Flywheel Stabilized, Magnetically Torqued Attitude Control System for Meteorological Satellites, CR-232, 1965, NASA. . 82. Schaaf, S.A. and Chambre, P.L., Flow of Rarefied Gases, Princeton University Press, 1961, pp. 17-24. 83. Jensen, J., Townsend, G. , Kork, K., and Kraft, D., Design Guide to Orbital Flight, McGraw Hill, New York, 1962, pp. 179-264. 84. Jacchia, L.G., "Static Diffusion Models of the Upper Atmosphere with Empirical Temperature Profiles," Smithsonian Contributions to Astrophysics, Vol. 8, No. 9, 1965, Smithsonian Institution, Washington, D.C, pp. 215-257. 85. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mischenko, E.F., The Mathematical Theory of Optimal  Processes, Pergamon Press Ltd., New York, 1964, pp. 26-34. 184 APPENDIX I The expressions for the frequencies and w2 are given by, o>1= f1 (a,b,I,a,e,C,G) = F(a,b,a1,a2,k1,k2, 2,1,£2,C,G,A1,B1) = ^-{1/ (k2-k2) } £c (l+3e) { (5A1+7B1-4G) l±a2+ (7A1+B1~4G) %2 }/ 16a1+£1[{I(a+1)(l+2e)-4}{k1(3a2a2+2a2b2+2B2)+4k2a1a2b2}/ 16+{I (a+1) (l+2e)+2 (31-4) }a;L (a2+4b2+4A2)/32+{I(a+1) (l+2e) -4}x a1(a2a2+2a2b2+2B2)/16-{3a1k2a2/8+a1k2b2/4+a1A2/4+a2k1k2b2/2} -k1(a2+b2+A2)/4] + (£2/a1) [{l(a+l) (l+2e)+4 (31-4)}x (a2+2b2+2A2)/16-{l(a+1)(l+2e)}a1k1(a2a2+a2b2+B2)/8 - [I(a+1) (l+2e)}(a2a2+4a2b2+4B2)/32-a2k2a2/8 - a1a2k1k2b2/2+a1k1A1B1/2-a2k2b2/2-a1k1(3a2+2b2+2A2)/8]J w_ = F(b,a,a0,ot,,k0,k,rZ0,C,GrA1,B,) 185 APPENDIX II The functions and G^ (i=l,2,3,4) are given £-1(a+1){(l+e)/(l+ecos9)}2cosyp-{3(I-l)/(l+ecos9) -DcosB (cos2Yp-sin2Yp) - [{2esin6/ (l+ecos9) }sinBp -23pcos6p]sinYp-{(1+e)3/(l+ecos9)4}Ccos9 x 2 [{(sinYpSinBpcos0-cos3pSin9)sinBp-sinYpcos9}cos Ypcos9/ 2 2 1/2 { (sinYpsin3pCOs9-cosBpSin9) + (cosYpcos9) } 2 2 1/2 {(sinYpSin3pCOS0-cosBpSin9) +(cosYpcos0) } sinYp -G j sinYpcos6pCos0+sin3pSin9|sinYp]J secBp [{2esin9/(l+ecos9)}cosBp+23psinBp]secBp [{3 (I-l)/(l+ecos9)}sin3psinYpcosYp+{2esin9/(l+ecos9)} x (cos3pCOSYp-YpSin3p)+23pYpCOs3p+{Yp +(23p-sinYp)x 186 x cosYp}singp-{(1+e)3/(l+ecos6)4}Ccos9(sinYpsinBp x cos0-cosBpsin0)(sinYpcosBpcos0+sinBpsin0)cosYp/((sinyp x 2 2 1/2 sin3pcos0-cosBpsin0) +(cosYpcos0) } ' ]secBp [I(a+1){(l+e)/(l+ecos0)}2+2YpsinBp-2cosBpcosYp]secBp I(a+1){(l+e)/(l+ecos0)}2sinBpsinYp+ 2{3 (I-l) / (l+ecos0) -1} sinBpcosBpsiriYpcosYp 2 2 -{2esin0/(l+ecos0)}cosYp-(cos Bp~sin Bp+l)YpsinYp +{(1+e)3/(l+ecos0)4}Ccos0[{(sinYpsinBpcos0-cosBpsin0)x sinB^-sinY^cos©} (sinY^sinBcCOsO-cosBpSin©) COSYP/ 2 2 1/2 {(sinYpsinBpcos0-cosBpsin0) +(cosYpcos0) } 2 2 1/2 +cosYpsinBp{(sinYpsinBpcos0-cosBpsin0) +(cosYpcos0) } +GcosYpsinBp|sinYpcosBpcos0+sinBpsin0|] 187 [-I(a+D { (l+e)/(l+ecos0) }2+2YpsinBp]cosB! 2 2 + (cos Bp-sin Bp+DcosYp I(a+1){(l+e)/(l+ecos6)}2(Ypsin8p-cosBpcosYp) + [{3 (I-l)/(l+ecos0)}sin2Yp+cos2Yp-Yp2] (cos2Bp-sin2Bp) -4YpSinBpcosBpcosYp+{(1+e)3/(l+ecos0)4}C x 2 [(sinYpSinBpCos0-cosBpSin6) (sinYpcos3pCOS0+sinBpsin0)/ 2 2 1/2 {(sinYpSinBpCOse-cosBpSine) +(cosYpcos0) } 2 +(sinYpCosBpcos0+sinBpSin0){(sinYpsinBpcos0-cosBpsin0) 2 1/2 +(cosYpcos8) } ' +G(sinYpCOsBpCOS0+sin3pSin0) x |sinYpcosBpCOS0+sinBpSin0|] 2esin0/(l+ecos0) 

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