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Attitude control of spinning satellites using environmental forces 1973

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ATTITUDE CONTROL OF SPINNING SATELLITES USING ENVIRONMENTAL FORCES by KAILASH CHANDRA PANDE B.Sc. Eng. (Hons.), Banaras Hindu U n i v e r s i t y , 1966 M.Sc, U n i v e r s i t y of Saskatchewan, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mechanical E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA November, 19 7 3 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia , I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e fo r reference and s tudy . I fu r ther agree tha t pe rmiss ion for ex t ens ive copying of t h i s t h e s i s fo r s c h o l a r l y purposes may be granted by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t ha t p u b l i c a t i o n , i n p a r t or i n whole , or the copying o f t h i s t h e s i s fo r f i n a n c i a l g a i n s h a l l not be a l lowed wi thou t my w r i t t e n p e r m i s s i o n . KAILASH CHANDRA PANDE Department of Mechanica l Eng inee r ing The U n i v e r s i t y o f B r i t i s h Columbia , Vancouver 8 , Canada Date ABSTRACT The f e a s i b i l i t y of u t i l i z i n g the environmental forces for three-axis l i b r a t i o n a l damping and attitude control of spinning s a t e l l i t e s i s investigated i n d e t a i l . An appreciation of the environmental influence i s f i r s t gained through a l i b r a t i o n a l dynamics study of spinning, axisymmetric, c y l i n d r i c a l s a t e l l i t e s i n the solar radiation pressure f i e l d . The highly nonlinear, nonautonomous, coupled equations of motion are analyzed approximately using the method of va r i a t i o n of parameters. The closed form solution proves to be quite useful i n locating periodic solutions and resonance c h a r a c t e r i s t i c s of the system. A numerical parametric analysis, involving large amplitude motion, establishes the e f f e c t of the radiation pressure to be substantial and d e s t a b i l i z i n g . Next, a p o s s i b i l i t y of u t i l i z i n g t h i s adverse influence to advantage through jud i c i o u s l y located rotatable control surfaces i s explored. A c o n t r o l l e r configuration for a dual-spin spacecraft i s analyzed f i r s t . The govern- ing equations, i n the absence of a known exact solution, are solved numerically to evaluate the e f f e c t of system parameters on the performance of the control system. The available control moments are found to be s u f f i c i e n t to i i i compensate for the rotor spin decay, thus dispensing with the necessity of energy sources maintaining the spin rate. The c o n t r o l l e r i s able to damp extremely severe disturbances in a f r a c t i o n of an o r b i t and i s capable of imparting arbi t r a r y orientations to a s a t e l l i t e , thus permitting i t to undertake diverse missions. The development of an e f f i c i e n t yet s t r u c t u r a l l y simple c o n t r o l l e r configuration i s then considered. A l o g i c a l approach for solar c o n t r o l l e r design i s proposed which suggests a four-plate configuration. Its performance i n conjunction with a bang-bang control law i s studied i n d e t a i l . The u t i l i z a t i o n of maximum available control moments leads to a substantial improvement of the damping c h a r a c t e r i s t i c s . Attention i s then focussed on using the earth's magnetic f i e l d i nteraction with onboard dipoles for attitude control. Magnetic torquing, however, i s unable to provide f i r s t order p i t c h control i n near equatorial o r b i t a l planes. The shortcoming i s overcome by hybridizing the concepts of magnetic and solar control. Two magnetic c o n t r o l l e r models, employing a single rotatable dipole or two fixed dipoles, are proposed i n conjunction with a solar p i t c h c o n t r o l l e r . The system performance i s evaluated for a wide range of system parameters and i n i t i a l conditions. Although high spin rates lend considerable gyroscopic s t i f f n e s s to the spacecraft, the c o n t r o l l e r s continue to be quite e f f e c t i v e even i n the absence of any spin. Even with extremely severe disturbances, damping times of the order of a few o r b i t a l degrees are attainable. As before, the concept enables a s a t e l l i t e to change the desired attitude i n o r b i t . The effectiveness of the c o n t r o l l e r s at high altitudes having been established, the next l o g i c a l step was to extend the analysis to near-earth s a t e l l i t e s i n free molecular environment. A hybrid control system, using the solar pressure at high a l t i t u d e s and the aerodynamic forces near perigee, i s proposed. The influence of important system parameters on the bang-bang operation of the c o n t r o l l e r i s analyzed. The concept appears to be quite e f f e c t i v e i n damping the s a t e l l i t e l i b r a t i o n s . Both the o r b i t normal and the l o c a l v e r t i c a l orientations of the axis of symmetry of the s a t e l l i t e are attainable. However, for a r b i t r a r y pointing of the symmetry axis, small l i m i t cycle o s c i l l a t i o n about the desired f i n a l orientation r e s u l t s . F i n a l l y , the time-optimal control, through solar radiation pressure, of an unsymmetrical s a t e l l i t e executing planar p i t c h l i b r a t i o n s i s examined a n a l y t i c a l l y . The switching c r i t e r i o n , synthesized for the l i n e a r case, i s found to be quite accurate even when the system i s subjected to large disturbances. V Throughout, the semi-passive ch a r a c t e r of the system promises an increased l i f e - s p a n f o r a s a t e l l i t e . v i 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 A n a l y t i c a l Results 23 2.2.1 Approximate a n a l y t i c a l solution 2 3 2.2.2 Periodic solutions of the system 30 (a) High frequency o s c i l l a t i o n s 31 (b) Low frequency o s c i l l a t i o n s 32 (c) Solar pressure excited o s c i l l a t i o n s 36 (d) Accuracy of the a n a l y t i c a l solution 38 (e) S t a b i l i t y of periodic solutions 38 2.2.3 Resonance 41 2.3 Numerical Results 45 v i i Chapter Page 2.3.1 S i g n i f i c a n t system parameters . . 4 5 2.3.2 System plots 47 2.3.3 Design plots 50 2.4 Concluding Remarks 52 3. ATTITUDE CONTROL USING SOLAR RADIATION PRESSURE 54 3.1 F e a s i b i l i t y 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) I l l u s t r a t i v e example . . . . 79 3.2 Improvement of Controller Design . . . . 81 3.2.1 Equations of motion 83 3.2.2 Development of c o n t r o l l e r models 83 3.2.3 Control strategy 90 3.2.4 Results and discussion 96 (a) L i b r a t i o n damping i n c i r c u l a r orbits 96 (b) L i b r a t i o n damping in e l l i p t i c o r bits 99 (c) Attitude control 102 (d) I l l u s t r a t i v e example . . . . 104 v i i i 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 121 4.2.2 High spin rates and spin decay 129 4.2.3 Attitude control 131 4.2.4 I l l u s t r a t i v e 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 a l t i t u d e 143 5.2.2 Low al t i t u d e 145 5.3 Results and Discussion 148 5.3.1 L i b r a t i o n damping 149 5.3.2 Attitude control 154 5.4 Concluding Remarks 156 i x 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 185 X LIST OF TABLES Table Page 4.1 Response With High Spin Rates 130 x i LIST OF FIGURES Figure Page 1.1 Schematic diagram of the proposed plan of study 14 2.1 Geometry of motion of spinning s a t e l l i t e i n the solar pressure environment 17 2.2 Evaluation of generalized forces due to ,solar radiation pressure: (a) curved surface 19 (b) f l a t ends 19 2.3 Frequency and i n i t i a l conditions for high frequency periodic motion 33 2.4 Frequency and i n i t i a l conditions for low frequency periodic motion 35 2.5 I n i t i a l conditions leading to solar pressure excited periodic motion 37 2.6 Typical periodic solutions of the system 39 2.7 Resonance conditions i n system parameter space 4 3 2.8 Typical responses under resonant conditions 44 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) i n e r t i a parameter 48 (b) spin parameter . 4 8 (c) o r b i t e c c e n t r i c i t y 48 (d) o r b i t i n c l i n a t i o n 48 x i i F igure Page 2.11 T y p i c a l s t a b i l i t y charts showing adverse i n f l u e n c e of s o l a r r a d i a t i o n pressure: (a) e = 0 51 (b) e = 0.1 51 3.1 Geometry of motion of d u a l - s p i n s a t e l l i t e i n the s o l a r pressure environment 56 3.2 S o l a r c o n t r o l l e r c o n f i g u r a t i o n 62 3.3 Ra d i a t i o n force on a p l a t e element 63 3.4 Influence of the s o l a r c o n t r o l l e r gains V K ' ^ i on. the response 70 3.5 Damped response as a f f e c t e d by s a t e l l i t e i n e r t i a and s p i n parameters 71 3.6 E f f e c t of s o l a r parameters and s o l a r aspect angle on the damped response 73 3.7 Response p l o t showing l i m i t c y c l e o s c i l l a t i o n s i n e c c e n t r i c o r b i t s and t h e i r removal through the modified c o n t r o l f u n c t i o n 74 3.8 Response as a f f e c t e d by: (a,b) o r b i t a l i n c l i n a t i o n from the e c l i p t i c 76 (c) r o t o r s p i n decay 76 (d) modified c o n t r o l f u n c t i o n 76 3.9 E f f e c t i v e n e s s of the s o l a r c o n t r o l l e r i n achieving a r b i t r a r y o r i e n t a t i o n s of the s a t e l l i t e 80 3.10 P r o j e c t e d c o n t r o l l e r performance i n achieving n u t a t i o n damping and a t t i t u d e c o n t r o l of INTELSAT IV and Anik s a t e l l i t e s 82 3.11 Development of s o l a r c o n t r o l l e r con- f i g u r a t i o n s 85 3.12 Proposed f o u r - p l a t e s o l a r c o n t r o l l e r model 8 9 3.13 Quadrant c o n s t r a i n t f o r p l a t e r o t a t i o n 6 94 l x i i i Figure Page 3.14 Optimization plots for the solar c o n t r o l l e r gain m 97 3.15 (a) Typical optimum response i n c i r c u l a r orbits 100 (b) Limit cycle o s c i l l a t i o n i n e l l i p t i c o r bits 100 (c) Variation of l i m i t cycle amplitude with system parameters 100 3.16 Effectiveness of the four-plate model i n achieving a r b i t r a r y orientations of the s a t e l l i t e 103 4.1 Geometry of motion of dual-spin s a t e l l i t e i n the earth's magnetic f i e l d 110 4.2 Magnetic-solar c o n t r o l l e r configurations . . 115 4.3 Optimization plots for the magnetic-solar c o n t r o l l e r gain m 122 4.4 Damped response and time history of con- t r o l s subsequent to i n i t i a l impulsive disturbance 127 4.5 Damped response and time history of con- t r o l s subsequent to i n i t i a l p o s i t i o n disturbance . . . . 128 4.6 Effectiveness of the magnetic-solar co n t r o l l e r i n imparting a r b i t r a r y orientations to the s a t e l l i t e 132 5.1 Aerodynamic-solar hybrid c o n t r o l l e r configuration 139 5.2 Optimization plots for the aerodynamic- solar c o n t r o l l e r gain m 150 5.3 Typical responses showing the effectiveness of the aerodynamic-solar c o n t r o l l e r at d i f f e r e n t o r b i t a l positions 153 xiv Figure Page 5.4 Effectiveness of the aerodynamic-solar c o n t r o l l e r i n imparting a r b i t r a r y orientations to the s a t e l l i t e 155 6.1 Geometry of motion of unsymmetrical s a t e l l i t e i n the solar pressure environment 160 6.2 (a) Variation of IQ, I with ? 162 1 i j r m a x (b) Phase plane p o r t r a i t of the system . . . 162 6.3 (a) Variation of transient amplitude I x , ( G ) I with i n i t i a l condition 1 1 1 max x 2(0) 168 (b) Var i a t i o n of switching time 0 and f i n a l time 9 f with i n i t i a l condition x 2(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 i n thi s 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 B x n ' B y n ' B z n ] Xn' Yn' zn a x e s ' respectively C,C\ solar parameter, i=1, 2,3,4 C,C i 2 p p o R \ i e i / y , 1=1,2,3,4 C m 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 r a t i o I i n e r t i a parameter of axisymmetric (I^=I z) s a t e l l i t e , I /I x y I ,1 ,1 p r i n c i p a l moments of i n e r t i a of the s a t e l l i t e I _»I „ moment of i n e r t i a of the platform and the rotor about the axis of symmetry of the s a t e l l i t e , respectively J platform i n e r t i a f r a c t i o n , I /I xp / x J(6) { (1+e) 2 (l+2ecos0+e 2)/(l+ecos9) 4} [{R (1+e)/ P n (1+ecosG)-R }/(R -R )] e ' p e 3 1/2 viscous damping parameter, K d(Rp /u) / /I K K-. viscous damping c o e f f i c i e n t XVI1 i n e r t i a parameter of unsymmetrical s a t e l l i t e , ( i -I ) / i ' z y x center of force pericenter t o t a l generalized force, l=a , & ,y, \ ,\\) generalized force due to solar radiation pressure, i=a,3,Y/^ generalized force due to magnetic i n t e r a c t i o n , generalized force due to aerodynamic forces, i=3,Y ,A distance between the s a t e l l i t e 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 s a t e l l i t e switching function, i=&,y,\ k i n e t i c energy g r a v i t a t i o n a l p o t e n t i a l energy p o l a r i t y v e l o c i t y of the s a t e l l i t e center of mass or b i t e c c e n t r i c i t y distance between the s a t e l l i t e center of mass and the hinge point T, Figure 3.11a x v i i i 2 d i p o l e s t r e n g t h h a,hg constants o f motion h p e r i g e e a l t i t u d e p i , i i n c l i n a t i o n s o f the o r b i t a l plane from the m e c l i p t i c and the e q u a t o r i a l planes, r e s p e c t i v e l y i , j , k u n i t v e c t o r s along x,y and z axes, r e s p e c t i v e l y £,r l e n g t h and r a d i u s of c y l i n d r i c a l s a t e l l i t e , r e s p e c t i v e l y m bang-bang c o n t r o l system g a i n m mass of the s a t e l l i t e s n e , n r r a t e s of the e a r t h ' s r o t a t i o n and r e g r e s s i o n of the l i n e of nodes, r e s p e c t i v e l y n (3K . C O S2 I J J ) 1 / / 2 , -TT/4<IJ; <TT/4 p,p^,P 2 u n i t v e c t o r along d i p o l e o r i e n t a t i o n , p=P ii+ Pj:+P kk -5 2 P Q s o l a r r a d i a t i o n p r e s s u r e , 4.65 x 10 dynes/cm r p o s i t i o n v e c t o r o f area element from the center of mass of the s a t e l l i t e t time u u n i t v e c t o r i n the d i r e c t i o n o f the sun, u. i+u^ j+u, k I 3J k u^ coscj) (sinycosBcosn+sinBsinn) + sin(J>{cosi (sinycosB s i n n - s i n B c o s n ) - s i n i c o s B c o s y } u_. cos<})cosYcosri+sin(j) ( c o s i c o s y s i n n + s i n i s i n Y ) X X X cos<j> (sinysinBcosn-cosBsinn) +sintj>{ c o s i (siny sinBsinn+cosBcosn)-sinisinBcosy} c o n t r o l vector u n i t v ector i n the d i r e c t i o n of atmospheric flow r e l a t i v e to the s a t e l l i t e center of mass, v. i+v . j+v, k 1 j k 2 1/2 {sin8+e ( s i n B c o s 6-cosBsinysin6)}/(l+2ecos0+e ) 2 1/2 -ecosysinB/(l+2ecos0+e ) 2 1/2 -{cosB+e(cosBcos0+sinBsinysin9)}/(l+2ecos0+e ) ' i n e r t i a l coordinate system w i t h x normal to 1 n the o r b i t a l plane and y n along the l i n e of nodes r e f e r r e d to the e q u a t o r i a l plane i n e r t i a l coordinate system w i t h x' normal to the o r b i t a l plane and y* along the p e r i c e n t e r r o t a t i n g o r b i t a l coordinate system w i t h X Q normal to the o r b i t a l plane and y Q along the l o c a l v e r t i c a l intermediate body coordinates r e s u l t i n g from r o t a t i o n s y and 3 about z and y, axes, o 11 r e s p e c t i v e l y p l a t f o r m - f i x e d coordinate system coning angle of the a x i s of symmetry, cos ^(cosBcosy) s t a t e - t r a n s i t i o n matrix (Equation 6.8) angles between the v e r n a l equinox and the l i n e s of nodes r e f e r r e d t o the e c l i p t i c and the e q u a t o r i a l planes, r e s p e c t i v e l y XX a,Q,y,X a t t i t u d e angles a1_'a2 constants i n Chapter 2 and Appendices I and I I ; l o n g i t u d e and l a t i t u d e of the p l a t e support arm i n the x y z - r e f e r e n c e , r e s p e c t i v e l y (Figure 3.11a) B c,Y c/^ c p o s i t i o n c o n t r o l parameters S f , Y f / ^ £ f i n a l d e s i r e d a t t i t u d e 3m l o n g i t u d e of the plane c o n t a i n i n g both the geographic and geomagnetic p o l a r axes from the v e r n a l equinox (Figure 4.1) 6,6^ c o n t r o l p l a t e r o t a t i o n , i=l,2,3,4 e d i s t a n c e between the c e n t e r of pressure and the c e n t e r of mass of the s a t e l l i t e (Chapter 2); d i s t a n c e between the c e n t e r of p r e s s u r e o f the c o n t r o l p l a t e and the hinge p o i n t T of the support arm (Chapter 3) e. d i s t a n c e between the c e n t e r of p r e s s u r e o f c o n t r o l p l a t e and the s a t e l l i t e c e n t e r of mass, i=l,2,3,4 e m geomagnetic p o l a r a x i s d e c l i n a t i o n , 11.4° C n+i|j-tan ^ (tancjicosi) n,n (u)+9) and (w +9) , r e s p e c t i v e l y m m 1 ' r 9 o r b i t a l angle angular p o s i t i o n of u n i t v e c t o r s p and p^ m S f from the y a x i s (Figure 4.2) s w i t c h i n g time and f i n a l time, r e s p e c t i v e l y X X I c h a r a c t e r i s t i c m u l t i p l i e r s , i=l,2,3,4 u g r a v i t a t i o n a l constant y^,v^ gains of l i n e a r c o n t r o l l e r w i t h s a t u r a t i o n c o n s t r a i n t s angle of i n c i d e n c e , ? = cos "'"(u'n) ; 1 m i=l,2,3,4 C • angle between the d i r e c t i o n of r e l a t i v e a ax flow and p l a t e normal, %, = cos ^" ( v n ) ; 3. i=l,2,3 p , x r e f l e c t i v i t y and t r a n s m i s s i b i l i t y of s a t e l l i t e surface or c o n t r o l p l a t e , r e s p e c t i v e l y p , p atmospheric d e n s i t y and i t s value at the a ap 2 perigee a l t i t u d e h^, r e s p e c t i v e l y a s p i n parameter damping time cf) s o l a r aspect angle, angle between the sun- l i n e and the l i n e of nodes r e f e r r e d to the e c l i p t i c plane rm m m 4> planar p i t c h a t t i t u d e of unsymmetrical s a t e l l i t e \\i nominal p i t c h a t t i t u d e of unsymmetrical s a t e l l i t e o),^ arguments of the perigee r e f e r r e d to the e c l i p t i c and the e q u a t o r i a l planes, r e s p e c t i v e l y x x i i ID average nodding frequency of the s a t e l l i t e a x i s of symmetry ^•^,^2 frequencies of the high and low frequency p e r i o d i c motion, r e s p e c t i v e l y Dots and primes i n d i c a t e d i f f e r e n t i a t i o n w i t h respect to t and 8 r e s p e c t i v e l y . The s u b s c r i p t o i n d i c a t e s i n i t i a l c o n d i t i o n . 1 1. INTRODUCTION 1.1 Preliminary Remarks Success of a vast majority of space missions depends on the a b i l i t y of a spacecraft to point accurately in the desired d i r e c t i o n . Even a c o r r e c t l y positioned s a t e l l i t e tends to deviate i n time from i t s preferred orien- tat i o n due to environmental influences, such as, micrometeor- i t e impacts, solar radiation pressure, aerodynamic forces, g r a v i t a t i o n a l and magnetic f i e l d interactions, etc. Fortunately, several methods of attitude control are available which damp the r e s u l t i n g undesirable l i b r a t i o n s . These procedures may be broadly c l a s s i f i e d as active, passive and semi-passive (or semi-active). Active s t a b i l i z a t i o n 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 s a t e l l i t e l i f e - s p a n . The main advantage of the technique i s i t s a b i l i t y to achieve a s p e c i f i e d orientation with almost any desired degree of accuracy. S t a b i l i z a t i o n techniques requiring no power consumption are termed passive. This i s generally achieved by designing s a t e l l i t e s with physical c h a r a c t e r i s t i c s which 2 interact with the environmental forces i n a manner so as to a t t a i n a s p e c i f i c equilibrium p o s i t i o n . Spin s t a b i l i z a t i o n presents an alternative that r e l i e s on the inherent tendency of a spinning body to maintain i t s attitude i n 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 u t i l i z e the environmental forces, through the introduction of appropriate c o n t r o l l e r s , and thereby achieve attitude control. The p o s s i b i l i t y of attaining high pointing accuracies with low power consumption promises an increased s a t e l l i t e l i f e - s p a n . The design of suitable c o n t r o l l e r 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 p a r t i c u l a r reference to spinning s a t e l l i t e s , forms the main objective of this thesis. 1.2 Literature Review Spinning bodies have received,in the past decade, considerable attention owing to t h e i r p a r t i c u l a r s t a b i l i t y properties. For r i g i d axisymmetric bodies under the 3 influence of gravity forces and with the axis of spin perpendicular to the o r b i t a l plane, Thomson''" (1962) presented a s t a b i l i t y c r i t e r i o n using l i n e a r i z e d analysis while 2 Pringle (1964) investigated motion i n 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 l a t e r by Kane and Barba (1966) to deal with motion i n the small for a r b i t r a r y e c c e n t r i c i t y . Wallace and Meirovitch^ (1967) studied the same problem by an asymptotic analysis i n conjunction with 6-9 Lyapunov's d i r e c t method. Neilson and Modi (1968-72) gave insight into the problem of s t a b i l i t y i n the large by making use of the in t e g r a l manifold concept. According to c l a s s i c a l mechanics, the stable r o t a t i o n a l motion of a r i g i d body i n absence of external forces i s possible only i f the axis of rotation i s a p r i n - c i p a l axis of least or greatest i n e r t i a . I f the body i s not r i g i d and energy i s dissipated by the c y c l i c forces acting on i t while under nutation, then only the motion about the axis of maximum i n e r t i a i s stable. I t turns out that for slowly spinning r i g i d s a t e l l i t e , the i n t e r n a l l y dissipated energy i s such as to overcome the s t a b i l i z i n g influence of gravity and the system ends up i n a state of tumbling about the axis of maximum moment of i n e r t i a : the c l a s s i c a l example i s 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 d i f f e r e n t rates r e l a t i v e to i n e r t i a l space. An early paper by Roberson 1 1 (1958) had anticipated that torques generated by a disc rotating about an axis fixed i n a r i g i d body could s u b s t a n t i a l l y a f f e c t i t s motion. However, the f i r s t fundamental contribution to the f e a s i b i l i t y of 12 dual-spin s t a b i l i z a t i o n was by Landon and Stewart (1964): an energy-sink method indicated no constraints on i n e r t i a s when energy d i s s i p a t i o n took place on the slowly-rotating 13 part of the system. I o r i l l o , a year l a t e r , extended this concept to the case where energy d i s s i p a t i o n occurs on both bodies. 14 L i k m s (1967) developed, for a s p e c i f i c configura- tion involving an axisymmetric rotor and an asymmetric body containing a ball-in-tube damper constrained to move p a r a l l e l to the rotor axis, an accurate s t a b i l i t y c r i t e r i o n based 15 on the Routh analysis. Mingori (1969) took a more general approach which involved two d i s s i p a t i v e sections. The Floquet analysis stressed the s e n s i t i v i t y of the system behaviour to the r e l a t i v e effectiveness of the sources of 16 energy d i s s i p a t i o n . Pringle (1969) extended his theorems on Lyapunov s t a b i l i t y to the case of dual-spin spacecraft and gave a rigorous proof of the "maximum moment of i n e r t i a 17 spin-axis" r u l e . C l o u t i e r (1968) investigated the 5 s t a b i l i t y and performance of a nutation damper consisting of mass s h i f t i n g perpendicular to the spin-axis: again, i t led to no r e s t r i c t i o n s on i n e r t i a r a t i o s or damper siz e when d i s s i p a t i o n occurs on a despun platform. In another 18 paper (1969), the author extended the analysis to a damper involving two degrees of freedom i n a plane perpen- dicul a r to the spin axis. An approximate solution was derived for the nutation angle and i t s 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 L i k i n s . Vigneron (1971) applied the method of averaging to obtain a closed-form f i r s t approxi- mation solution for a dual-spin system containing both 21 platform and rotor mounted dampers. Bainum et a l . (1970) conducted a s t a b i l i t y and performance analysis of the dual- spin Small Astronomy S a t e l l i t e (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 e f f e c t of damping i n the momentum wheel by permitting the plane of the wheel to f l e x with two degrees of freedom with respect to the hub. The analysis established s t a b i l i t y c r i t e r i a for the SAS-A s a t e l l i t e . Although passive methods for attenuating nutation are generally r e l i a b l e and conceptually simple, t h e i r effectiveness may be l i m i t e d . A device containing an ac t i v e l y controlled mass, capable of attaining any nutation 6 23 angles, was studied by Kane and Scher (1969). Mingori 24 et a l . (1971) analyzed both semi-passive and active nutation dampers for dual-spin spacecraft, the former i n - volving single axis control moment gyros (CMG's) whose rot a t i o n a l motion r e l a t i v e to the spacecraft was restrained passively by a spring and dashpot. The active damper was rea l i z e d by c o n t r o l l i n g the CMG's i n 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 o f f e r , nevertheless, valuable r e s u l t s . Of 25 pa r t i c u l a r i n t e r e s t i s the conclusion by Kane and Mingori (1965) that the s t a b i l i t y of undamped axisymmetric dual-spin s a t e l l i t e s i s equivalent to that of r i g i d spinning bodies. 2 6 White and Liki n s (1969) extended the research to s l i g h t l y asymmetric system by making use of asymptotic expansions 27 2 8 and resonance l i n e s . The e f f o r t s of Roberson et a l . ' (1966, 1969) and Y u 2 9 (1969) should also be noted who analyzed the equilibrium positions of a single r i g i d body containing a symmetric, constant speed, fixed axis rotor, also c a l l e d a gyrostat, i n presence of gravity forces. The concept of dual-spin spacecraft gained s u f f i c i e n t recognition by 1967 to be considered seriously as a design alternative"^. The f i r s t f l i g h t data for a prolate dual-spin s a t e l l i t e (Tacsat 1), however, became p u b l i c l y available 7 31 only recently (1970). I t was found that the spacecraft did not maintain the nominal i n e r t i a l attitude but executed stable l i m i t cycle i n a nearby state of free precession. None of the previous analyses employing l i n e a r models for the energy d i s s i p a t i o n mechanisms could explain t h i s 32 anomaly. In a recent paper, L i k i n s et a l . (1971) found, v i a the 'energy-sink' method, the l i m i t cycle o s c i l l a t i o n to be a consequence of n o n l i n e a r i t i e s i n the damping forces. On the other hand, the p o s s i b i l i t y of constant or variable amplitude l i m i t cycles due to nonlinear restoring forces 33 has been indicated by Mmgon et a l . (1972) . I t should be emphasized that although several authors have recognized the importance of environmental forces, they were ignored i n the analyses of spinning space- c r a f t discussed above. Although extensive volume of l i t e r a t u r e exists on the l i b r a t i o n a l dynamics of gravity s t a b i l i z e d s a t e l l i t e s 3 4 ' 3 5 , e t a l " , serious e f f o r t s at analyzing the influence of environmental forces and e x p l o i t i n g them for 3 6 attitude control are r e l a t i v e l y recent. Roberson (1958) 37 gave a general outline of the problem and Wiggins (19 64) presented estimates of the r e l a t i v e magnitudes of these forces. Clancy and M i t c h e l l 3 8 (1964) and Modi et a l . 3 9 , 4 0 (1971) investigated the influence of solar r a d i a t i o n pressure on the attitude motion of s a t e l l i t e s . The e f f e c t of the atmosphere on s a t e l l i t e l i b r a t i o n s was the subject of 8 study by Debra 4 1 (1959) , S c h r e l l o 4 2 (1961), Garber 4 3 (1963) , 44 Meirovitch and Wallace (1966), et a l . The environmental e f f e c t , i n general, was found to be detrimental to the s a t e l l i t e performance. On the other hand, the environmental forces o f f e r an e x c i t i n g p o s s i b i l i t y of trajectory and attitude control through the introduction of a c a r e f u l l y designed c o n t r o l l e r . As no mass expulsion or active gyros requiring large power consumption are involved, these schemes are e s s e n t i a l l y semi-passive and hence promise an increased s a t e l l i t e l i f e - span. The use of solar radiation pressure for propulsion 45 within the solar system was f i r s t proposed by Garwin (19 58). 46 Sohn (1959) suggested a s p e c i f i c configuration using plates of large surface areas to orient the s a t e l l i t e with respect 47 to the sun. Galitskaya and Kiselev (1965) studied, q u a l i t a t i v e l y , the p r i n c i p l e of l i b r a t i o n control of space 48 probes about three axes. Mallach (1966) presented a system for solar damping of gravity oriented s a t e l l i t e s and gave a s i m p l i f i e d analysis using average torques. Modi and 49 Flanagan (1971) examined the planar attitude control of a gravity gradient system i n an e c l i p t i c o r b i t using the 50 solar pressure as a damping torque. Modi and Tschann (1971) extended the analysis by the introduction of a displacement and v e l o c i t y sensitive c o n t r o l l e r enabling the s a t e l l i t e to a t t a i n 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 f e a s i b i l i t y of achieving general three-axis l i b r a t i o n a l damping and attitude control. L i t e r a t u r e on the use of solar pressure for attitude control of spinning vehicles appears to be rather 52 limited. Ule (1963) was the f i r s t to consider aligning the spin axis along the s u n - s a t e l l i t e l i n e 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 a l . 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 p o s s i b i l i t y of using the solar pressure for general three- axis l i b r a t i o n a l damping and attitude control of spinning s a t e l l i t e s remains v i r t u a l l y unexplored. I t would Joe appropriate to mention here the experi- 57 ment aboard Mariner IV spacecraft , conducted on depletion of the attitude control gas, to ali g n the r o l l 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 t h i s purpose. Unfortunately, one of the vanes proved to be inoperative during a major portion of the mission. However, subsequent rea c t i v a t i o n of the vane enabled the solar pressure control system, i n con- junction with active gyros, to maintain the spacecraft attitude within 1° of the sun-line. Generation of control torques through the i n t e r - action of onboard electromagnetic dipoles and the earth's magnetic f i e l d appears to be p a r t i c u l a r l y a t t r a c t i v e as the system r e l i a b i l i t y i s enhanced by the elimination of moving parts. L i b r a t i o n damping of gravity oriented s a t e l l i t e s 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 orb i t s for roll-yaw damping were achieved, the inadequacy of the system for pitch control i n equatorial o r b i t s , where the geomagnetic f i e l d i s nearly p a r a l l e l to the o r b i t normal, became apparent. The problem of maintaining the spin axis of a s a t e l l i t e perpendicular to the o r b i t a l plane has been a 6 0 subject of considerable study by Vrablik et a l . (1965), Sonnabend 6 1 (1967) and many others. F i s c h e l l 6 2 (1966) considered the p o s s i b i l i t y of using magnetic control for 6 3 regulating the spin rate of a s a t e l l i t e . 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 f i n a l orientation 11 to be i n e r t i a l l y fixed and the spin rate constant during 64 the control maneuver. Sorensen (1971) applied the Kalman f i l t e r technique to estimate pointing errors for a system with limited attitude determination c a p a b i l i t i e s and developed the minimum energy control law using these estimates. In a recent paper, Shigehara 6^ (1972) studied a control law, based on the asymptotic s t a b i l i t y c r i t e r i o n , for the spin-axis and the spin rate using dipoles along the axis and perpendicular to i t , respectively. The f i r s t operational magnetically controlled s a t e l l i t e s , 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 s a t e l l i t e s i n near-earth or b i t s was a subject of several early discussions by W a l l 6 8 (1959), S c h r e l l o 6 9 (1961), et a l . In practice, however, i t has been used successfully only for the p i t c h control of C0SM0S-149, with the other 70 degrees of freedom governed by gyroscopic forces 71 Ravindran (1971) optimized, through l i n e a r i z a t i o n , a set of c o n t r o l l e r flaps for a s a t e l l i t e i n a c i r c u l a r o r b i t . 72 Modi and Shrivastava (1971), on the other hand, have proposed several schemes of semi-passive aerodynamic c o n t r o l - l e r s . Their nonlinear analysis showed the system to be e f f e c t i v e i n damping severe disturbances i n a f r a c t i o n of an o r b i t . The performance of the c o n t r o l l e r appeared promising even i n e l l i p t i c o r b i t s where the corrective 12 moments are available only over a portion of the t r a j e c - 73 tory. In a subsequent paper (1973), the authors optimized the performance of the c o n t r o l l e r i n both c i r c u l a r and e l l i p t i c o r b i t s , using the damping time and the steady state pointing error as the respective c r i t e r i a . 1.3 Purpose and Scope of the Investigation From the foregoing, i t i s evident that the influence of environmental forces on the attitude motion of spinning s a t e l l i t e s and t h e i r u t i l i z a t i o n for attitude control has received l i t t l e attention i n 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 s c i e n t i f i c s a t e l l i t e s are indeed spin s t a b i l i z e d . The thesis aims at f i l l i n g t h i s gap by systematically analyzing environmental e f f e c t s and e x p l o i t - ing them to advantage over a wide range of operational a l t i t u d e s . The influence of solar radiation pressure, con- s t i t u t i n g the dominant environmental force at high a l t i t u d e s , on the l i b r a t i o n a l motion i s examined f i r s t . Both a n a l y t i c a l and numerical techniques are employed to study the system response. S t a b i l i t y of the periodic solutions i s ascertained using the Floquet theory. Numerical results e s t a b l i s h regions of nontumbling motion i n the system parameter space. 13 Next, the p o s s i b i l i t y of using solar radiation pressure for general three-axis l i b r a t i o n damping and attitude control of a dual-spin system i s explored. The results e s t a b l i s h the effectiveness of the concept. E f f o r t i s then directed towards devising a solar c o n t r o l l e r model that i s s t r u c t u r a l l y simple and operationally e f f i c i e n t . This i s followed by an investigation of attitude damping and control u t i l i z i n g the earth's magnetic f i e l d . A comparative study of two c o n t r o l l e r models i s conducted. For the reason pointed out e a r l i e r , the magnetic c o n t r o l l e r s f a i l to provide f i r s t order p i t c h control i n synchronous o r b i t s . To compensate for t h i s , the magnetic c o n t r o l l e r s are hybridized with a solar pitch c o n t r o l l e r . Attitude control of near-earth s a t e l l i t e s in e l l i p t i c t r a j e c t o r i e s , normally preferred to minimize degeneration of the o r b i t due to atmospheric drag, i s considered next. A hybrid control system, u t i l i z i n g the aerodynamic forces at low a l t i t u d e s and solar r a d i a t i o n pressure when beyond the atmosphere, i s proposed. F i n a l l y , the problem of time-optimal p i t c h control of s a t e l l i t e s using the radiation pressure i s examined a n a l y t i c a l l y . Figure 1.1 schematically presents the plan of study. L i b r a t i o n a l dynamics i n presence of solar rad i a t i o n pressure 1 L ATTITUDE CONTROL OF SPINNING SATELLITES USING ENVIRONMENTAL FORCES ± Attitude control using solar r a d i a t i o n pressure Attitude control using the earth's magnetic f i e l d and solar r a d i a t i o n pressure A n a l y t i c a l study • Periodic solutions • Resonance Numerical study • System response • Libra- t i o n a l s t a b i l i t y F e a s i b i l i t y study Nutation damping Attitude control r Attitude control using aerodynamic forces and solar radiation pressure Improvement of c o n t r o l l e r design • L i b r a t i o n damping •Attitude control Time-optimal p i t c h control using solar r a d i a t i o n pressure F e a s i b i l i t y of magnetic-solar hybrid control •Nutation damping •Attitude control F e a s i b i l i t y 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, c y l i n d r i c a l , spinning s a t e l l i t e s under the influence of solar r a d i a t i o n pressure and gravity gradient torques. The equations of motion of the system are obtained f i r s t using the c l a s s i c a l 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 i s undertaken using Butenin's extension of 74 the method of slowly varying parameters . The approximate a n a l y t i c a l solution proves to be an excellent tool i n locating periodic solutions of the system, whose importance i n the attitude dynamics study of s a t e l l i t e s has been well emphasized 9 7 5 V 6 77 ' ' . The Floquet theory i s employed to examine the v a r i a t i o n a l s t a b i l i t y of the periodic solutions. The p o s s i b i l i t y of resonant o s c i l l a t i o n s of the system i n presence of the solar torque i s also investigated. The quasi-linear a n a l y t i c a l method, however, f a i l s to describe the large amplitude motion of the system. To this end, the governing equations are analyzed numerically 16 and t h e l i b r a t i o n a l r e s p o n s e i s s t u d i e d a s a f u n c t i o n o f t h e s y s t e m p a r a m e t e r s . The a v a i l a b l e i n f o r m a t i o n i s c o n - d e n s e d i n t h e f o r m o f d e s i g n p l o t s , w h i c h c l e a r l y e m p h a s i z e t h e i m p o r t a n c e o f t h e s o l a r p a r a m e t e r c h a r a c t e r i z i n g t h e r a d i a t i o n p r e s s u r e t o r q u e , a n d s h o u l d p r o v e u s e f u l d u r i n g t h e d e s i g n o f an a t t i t u d e c o n t r o l s y s t e m . 2.1 F o r m u l a t i o n o f t h e P r o b l e m F i g u r e 2 .1 shows an a x i s y m m e t r i c c y l i n d r i c a l s a t e l l i t e w i t h t h e c e n t e r o f mass S m o v i n g i n a K e p l e r i a n o r b i t a b o u t t h e c e n t e r o f f o r c e 0 . The s p a t i a l o r i e n t a t i o n o f t h e a x i s o f s y m m e t r y o f t h e s a t e l l i t e i s c o m p l e t e l y s p e c i f i e d b y t w o s u c c e s s i v e r o t a t i o n s y a nd B, r e f e r r e d t o as r o l l a n d y a w , r e s p e c t i v e l y , w h i c h d e f i n e t h e a t t i t u d e o f t h e s a t e l l i t e p r i n c i p a l a x e s x , y , z w i t h r e s p e c t t o t h e i n e r t i a l r e f e r e n c e f r a m e x ' , y ' , z ' . The s a t e l l i t e s p i n s i n t h e x , y , z r e f e r e n c e w i t h a n g u l a r v e l o c i t y a . I n t e r m s o f t h e s e m o d i f i e d E u l e r i a n r o t a t i o n s , t h e e x p r e s s i o n s f o r t h e p o t e n t i a l a n d k i n e t i c e n e r g i e s t o 0 ( 1 / R ) a r e o b t a i n e d a s : U g - ym /R - y { ( I / 2 R 3 ) ( I - l ) / I } ( 1 - 3 s i n 2 y c o s 2 B ) ( 2 . 1 ) T 2 + ( B - 8 s i n y ) 2 + (ycosB+SsinBcosy) 2) ( 2 . 2 ) Figure 2.1 Geometry of motion of spinning s a t e l l i t e i n the solar pressure environment 18 Neglecting o r b i t a l perturbations due to the 7 8 79 l i b r a t i o n a l motion ' , the c l a s s i c a l Lagrangian formulation yie l d s the governing equations of motion i n the r o l l , 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/R 3) (1-1) * 2 sinycosYcos 3 = Q^/I (2.3a) (d/dt)(3-esiny)+(Ycos3+6sin3cosY){I +(Ysin3-8cos3cosY)>-3(y/R 3)(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 s a t e l l i t e , of length dx, angular width d0 and located at an a x i a l 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 i s given by, (a-ysin3+8cos3cosy) 2 - 1 ) s i n ysin3cos3 (2.3b) (2.3c) 19 Figure 2.2 E v a l u a t i o n of g e n e r a l i z e d forces due to s o l a r r a d i a t i o n pressure: (a) curved surface; (b) f l a t ends 20 dF = -podA|cos£|{(l-T)u+ps> (2.4) with the re s u l t i n g moment about the s a t e l l i t e 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 l i m i t s of integration for x are -(V2-e) to (l/2+z) and those for 9 are t a n - 1 (-u./u.) to J K {ir'+tan ^ (-u-/u, ) } , which correspond to £=TT/2 . 1 K- Evaluation of the i n t e g r a l i n Equation (2.5), after considerable algebraic manipulations, leads to M = {- (TT/2) p£r 2(1-T-p) u.u.+2 (l-T+p/3) p r£eu, / 2+ 2}j O 1 .K O J s ! U > Ui J K + { ( T T/2)P £r 2(l-T-p)u.u.-2 (l-T+p/3)p r k u . / 2^_ 2}k (2.6) ' *o i n ' * 0 1 u.+u, J J j k The expression for the moment due to the f l a t ends of the c y l i n d r i c a l s a t e l l i t e (Figure 2.2b) i s obtained i n a simi l a r manner, ME = { ( 7 T/2)p oS.r 2 ( l - T - p ) u i u k + T r p o r 2 e ( l - T - p ) u k | u i | }j - { ( 7 T/2)p o5,r 2(l-T-p)u iUj+Trp or 2£(l-T-p)Uj|u i| }k (2.7) The t o t a l moment due to solar radiation pressure i s thus given by 21 M = M c + ^ = 2 ( l - T + p/3)p or£eu k[/ u2 + u2+(7 r r/2£) { ( 1 - T - p ) / j k ( l - T + p / 3 ) } | u ± | ] j - 2 ( l - T + p / 3 ) p o r £ e u j x [/ 2 + u2 +(TTr/2A) {• ( 1 - T - p ) / ( l - T + p/3) } | u ± | ]k (2.8) j k The application of the p r i n c i p l e of v i r t u a l work yield s the generalized forces i n the Y»3 and a degrees of freedom as: Q =-2 ( l - T + p/3) p r & e u . [ / 2, 2 + ( i T r / 2 £ ) x Y o i 1 u .+u, 7 J D k { ( 1 - T - p ) / ( l - T + p / 3 ) } | u ± | ] c o s 3 (2.9a) Q B = 2 ( l - T + p/3) p rfceu k [/ 2 2 +{T\r/2l)x j k { ( l - T - p ) / ( l - T + p / 3 ) } | U i | ] Q = 0 a The generalized force i n the a degree of freedom being zero, a f i r s t i n t e g r a l of motion defining the s a t e l l i t e spin rate a i s furnished by Equation (2.3c), (2.9b) (2.9c) • • • a-Ysin3+6cos3cosY = h (2.10) As h a i s a measure of the s p i n r a t e , a dimension- l e s s s p i n parameter c, d e f i n e d as a = (a/9)I = (h /9) I6=6=Y=0 a -1 (2.11) 9=0 may be used to e l i m i n a t e the c y c l i c coordinate a. Changing the independent v a r i a b l e from t to B, through the K e p l e r i a n o r b i t a l r e l a t i o n s R = hg/y(l+ecos6) (2.12a) 9 = h Q/R 2 (2.12b) and making use of the s p i n parameter a, the governing equations of motion i n the r o l l and yaw degrees of freedom (Equations 2.3a and b, r e s p e c t i v e l y ) transform t o : 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)} (Y 1+cosYtanB) =-{ (1+e) 3/(l+ecos9) 4}Cu. {/^2~^2 +G|u i|}secB (2.13a) 3 u j u k x B" -Y ' C O S Y + [ I(a+1){(1+e)/(1+ecosG)} 2 + ( Y'sinB-cosBcosY)](Y'cosB+cosYsinB) - { 3 ( I - l ) / ( l + e c o s 9 ) } s i n 2 Y s i n $ c o s B 23 -{2esin9/(l+ecos9)} (B'-siny) = {(1+e) 3/(l+ecos6) 4}Cu, {/ 2, 2 +G|u. |} (2.13b) u j + u k 1 where the solar parameter C and the solar aspect r a t i o G are defined as: C = (2R 3/uI y)p Qrile ( l - T + p / 3 ) G = (7rr/2£) ( l - T - p ) / ( l - T + p / 3 ) (2.14) 2.2 A n a l y t i c a l Results 2.2.1 Approximate a n a l y t i c a l solution In absence of a known, exact, closed form solution to such a complex system, i t was decided to analyze the problem approximately using Butenin's extension of the method of v a r i a t i o n of parameters. The case of w=c()=0 i s analyzed here which leads to a considerable reduction i n algebra with- out a f f e c t i n g the physics of the problem. Replacing the trigonometric functions of the dependent variables by t h e i r series expansions, ignoring fourth and higher order terms i n B,Y, and t h e i r derivatives 2 as well as terms of o(e ), Equations (2.13) take the form: Y " +n 2 Y-^ 1 B'=2Ce-C(l+3e)cos9+2Cecos20+f(Y,Y',3,3',9) (2.15a) 3" + n 2 3 + & 2 Y ' = -C(l+3e)sin0+2Cesin29+g (Y ,Y ' ,3,3' ,9) (2.15b) where n 2 = 3I-4+I(a+1)(l+2e) n 2 = I (a+1) (l+2e)-1 £1 = l2 = I ( C T + 1 ) ( 1 +2e)-2 (2.16) and the nonlinear functions f and g are defined as f = C(l+3e-4ecos9)[Y 2cos6/2+3 2sin0cos9/2 +Y 2cos 39/2+Y3sin9cos 29-G|Ycos8+3sin9|cos9] -2I(a+l) (3'-Y) ecos9+3(1-1)Yecos0+2(Y' + 3)esin9 +1 (a+l)(l+2e-2ecos9) (Y 3+33' 3 2-3Y3 2)/6 + 233'Y'+3'Y2+(2/3) { (31-4)-3 (I-l)ecos9>Y 3 + (3 3/3-3Y 2/2)2esin9 g = C (l+3e-4ecos9) [Y3cos9 + 3 2sin 39/2+Y 2sin9cos 29/2 +Y3sin 29cos9-G|Ycos0+3sin9|sin9]+2I(a+1) * 25 x(Y'+S )ecose+2(6 1 -Y)esinB+I(a+1) (l+2e-2ecos9)(3Y'3 2+3y 23+3 3) -2Y'3 2-3Y 2-Y' 23-Y'Y 2-23 3/3+3(1-1)(1-ecosG)y 2B+Y 3esin9/3 (2.17) The solution for the corresponding l i n e a r system ( i . e . , f=g=0) i s given by Y = asin(k,9+3,)+bsin(k„9+3 9)+A +A.cos9+A„cos29 1 1 A 1 ° 1 * (2.18a) 3 = a, acos (k, 9 + 3,)+a bcos (k.,9+3.,)+B, sin0+B_sin28 1 1 1 2 i 2. i 2. (2.18b) where a, b, 3-|_'32 a r e constants to be determined from i n i t i a l conditions and the c h a r a c t e r i s t i c frequencies k^ and k 2 (k^>k2) are the roots of the equation k 4-(n 2+n 2+Ji 1£ 2)k 2+n 2n 2 = 0 (2.19) The constants a., A. and B. are defined as: i i l a. = (k 2-n?) A,k. , i = 1,2 l l 1 ' 1 l A = 2Ce/n 2 o 1 A1 = C(l+3e) ( l - n 2 - 5 , 1 ) / ( l - k 2 ) (1-k 2) A 2 = 2Ce(n 2+2il 1-4)/(4-k 2) (4-k 2) 26 B 1 = C(l+3e) (l-n 2-£ 2)/(l-k 2) (1-k2,) B 2 = 2Ce(n 2+2^ 2-4)/(4-k 2)(4-k 2) (2.20) A solution of the si m i l a r form i s sought for the nonlinear system, however, allowing the amplitude and phase to be functions of 0, i . e . , Y = a(0)sin(k 10+3 1 (0))+b(0)sin(k„0+6„(0))+A 1 1 2 2 O +A 1cos8+A 2cos20 (2.21a) 8 = a 1a(8) cos (kjĵ O + B-L (0) )+a 2b (0) cos (k 20 + 8 2 (0) ) +B 1sin0+B 2sin20 (2.21b) As the four unknown functions defining the variable amplitude and phase cannot be determined from four i n i t i a l conditions alone, the solution i n the present form i s over- sp e c i f i e d . Hence four constraint relations must be obtained. Keeping the f i r s t derivatives of y and 8 to be the same as that of the l i n e a r system gives two of the constraint r e l a t i o n s : a' sinijj1+b' sinip2+a8'1cosip1+b82cosiJ;2 = 0 (2.22a) a^a ' cosi^+o^b' cosij; 2-a^aB^sini^-a 2b8 2sinip 2 = 0 (2.22b) 27 where ^ = k 1e+B 1(e), y2 = k 2e+B 2(e) Equations of motion (2.15) i n conjunction with the assumed solution(2.21) y i e l d the other two constraint relations as: k ^ ' c o s ^ 1 + k 2 b I c o s ^ 2 - k 1 a 6 ^ s i n ^ 1 - k 2 b 8 2 s i n ^ 2 = f* (2.23a) a l k l a ' S ^ n ^ l + a 2 ^ 2 ^ ' S ^ n ^ 2 + a l k l a ^ l C O S ^ l + a 2 k 2 b ^ 2 C O S ^ 2 = (2.23b) * * where f , g are the modified nonlinear functions. Solving Equations (2.22) and (2.23) simultaneously for a', b', B| and B 2 y i e l d s : a' = - { ^ 1 g * s i n ^ 1 - ( i i 2 / a 1 ) f*cos^ 1}/(k 2-k 2) b' = { i i i g * s i n ^ 2 - ( £ 2 / a 2 ) f*cos^ 2}/(k 2-k 2) 31 = -{ £ 1g* co s^ 1+(^ 2/a 1) f*sini|J 1}/{a(k 2-k 2) } 3 2 = {ii 1g*cos^ 2+(£ 2/a 2) f*sinij; 2}/{b(k 2-k 2) } (2.24) Equations (2.24) represent an exact transformation of the two second order equations of motion (2.15) into four 28 f i r s t order d i f f e r e n t i a l equations. For small amplitude * * motion, f and g are small. Consequently, a, b, 3-̂  and 3 2 are slowly varying parameters. Using t h e i r average values over one period gives: a ' = - U / 8 T T 3 ( k 2 - k 2 ) } (2TT r2TT 0 J 0 2TT {£^g s i n ^ - ( J l 2/a 1)f cosiJJ 1 }d^ 1 d^ 2 de b' = { l / 8 T T 3 ( k 2 - k 2 ) } 2TT 2TT 2^ {£^g s i n ^ - ( A 2 / a 2 ) f cosi^} d i j ^d^de 3j_ = - { l / 8 T i 3 a ( k 2 - k 2 ) } /•2TT r2v 0 0 {5--̂ g COSIJJ^ + (£ 2/a 1)f s i n ^ l d i ^ d i j ^ d e 3 2 = { l / 8 T r 3 b ( k 2 - k 2 ) } 2T\ 2TT 0 J 2TT 7C { i^g cos^ 2 + (£ 2/a 2)f s i n ^ d ^ d i j ^ d Q (2.25) Solving Equations (2.25) for a, b, 3-̂  and 3 2 , and substituting i n Equations (2.21), the solution takes the form: 29 Y = asin (o3 ie+c 1)+bsin (co 26+c 2)+A o+A 1cose+A 2cos29 (2.26a) 6 = a 1acos (w 10+c 1)+a 2bcos ( to 2e+c 2)+B 1sin6+B 2sin2e (2.26b) where a = [{(6;+a 2a) 2(Y 0-A o-A 1-A 2)-B 1-2B 2)/(a 1a ) ; L-a 2ca 2)} 2 ,2 1/2 + {(a0Y'-a)„6 )/(a,a>.,-a,a>.I) } ] (2.27a) 2. O 2 O X ^ X b = [{ (B^+a 1 o) 1(Y o-A o-A 1-A 2)-B 1-2B 2 ) / (a 1 co 1 -a 2 a) 2 ) }2 2 I/2 + { (a 1Y^-a3 1 B o ) / (a 1 u) 2 -a 2 a ) l ) } ] ( 2 > 2 ? b ) c 1 = tan 1[{ (3^+a 2w 2(Y o-A o-A 1-A 2)-B 1-2B 2)/(a 1 0) 1-a 2a) 2) }/ * ( a2 Yo~ a )2^o ) / ( a 1 a ) 2 _ a 2 t J J l ) ^ (2.27c) c 2 = t a n - 1 [{ (B(^+a1co1 (Y Q-A o-A 1-A 2)-B 1-2B 2) / (a 1 o) 1 -a 2 co 2 ) }/ { (a 1Y (! )-(jJ 1 B 0 ) / (a 1 co 2 -a 2 w 1 ) }] (2.27d) The frequencies and (JO 2 are represented by rather lengthy functions (Appendix I) of l i b r a t i o n a l amplitudes and system parameters, 30 U). f (a,b,I,a,e,C,G) C O - = f 2 (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 d i s t i n c t components: response at 'high' frequency <JÔ  , 'low' frequency a) 2 and the o r b i t a l frequency. The r e s u l t i n g motion would, in general, be non-periodic except i n the sp e c i a l s i t u a t i o n when the frequencies oô  and u>2 assume r a t i o n a l values for non-zero C, or, the r a t i o W2./(JJ2 i s a r a t i o n a l number with C = 0. A search extending over a reasonable range of system parameters and i n i t i a l 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 i n i t i a l conditions which excite only one of the three frequencies. Various relationships e x i s t i n the i n i t i a l condition space for which the r e s u l t i n g motion i s periodic. The high and low frequency periodic l i b r a t i o n s i n the absence of solar radiation pressure (C = 0) are discussed f i r s t , followed by the solar pressure excited periodic o s c i l l a t i o n s of o r b i t a l frequency. 31 (a) High frequency o s c i l l a t i o n s (C = 0) In the absence of solar ra d i a t i o n pressure, A^ = B^ = 0. Hence, the system would execute high frequency periodic motion for b = 0. This i s s a t i s f i e d by two sets of i n i t i a l conditions. The o s c i l l a t i o n s of type I r e s u l t with Y = 3' = 0 o o Y^ = ( 0 ) 1 / a 1 ) 8 o (2.29a) The corresponding amplitude and frequency of motion are obtained from Equations (2.27a) and (2.28a), respec- t i v e l y , as: a = 8 0 / a 1 W l = k l + f i ( ^ 0 / a i _ ' I' a' e) (2.29b) The other set of i n i t i a l conditions, leading to high frequency o s c i l l a t i o n s of type I I , i s readi l y found to be, Y' = B = 0 O Q B ' = -a 1u ) ,Y (2.30a) o 1 1 o with the amplitude and the frequency of motion, 32 u>1 = 'k1 + f 1(Y 0 / I f(?,e) (2. 30b) Relations (2.29) and (2.30), governing the i n i t i a l conditions for high frequency periodic o s c i l l a t i o n s and the res u l t i n g frequency ( o s c i l l a t i o n s per o r b i t ) , are plotted i n Figure 2.3 for t y p i c a l values of s a t e l l i t e parameters. The system behaves as a hard-spring o s c i l l a t o r showing an increase i n the frequency with amplitude. Note that the change i n s a t e l l i t e ' s configuration from spherical (I = 1) to d i s c - l i k e (I = 2) results i n a corresponding increase i n frequency. An increase i n o r b i t a l e c c e n t r i c i t y also has s i m i l a r e f f e c t . (b) Low frequency o s c i l l a t i o n s (C = 0) i n i t i a l conditions leading to a = 0. As i n the case of the high frequency o s c i l l a t i o n s , two d i s t i n c t relationships between the i n i t i a l states are found to y i e l d periodic motions of low frequency. The low frequency periodic o s c i l l a t i o n s r e s u l t from Periodic motion of type I i s obtained with, Y = 0 ' o (2.31a) C = o e = o s table a = 2 e = 0.05 it s table ©unstable 0.2 _ 0.4 0 0.2 0.4 Figure 2.3 Frequency and i n i t i a l conditions for high f r e - quency periodic motion 34 with amplitude and frequency as b = 6 D/a 2 u 2 = k 2 + f 2 ( 6 o / a 2 , I , a , e ) (2.31b) while that of type II i s governed by the conditions, Y' = 8 = 0 o o 6' = -a„u„Y (2.32a) o 2 2 o leading to b = ^o w 2 = k 2 + r 2(Y Q,I,a,e) (2.32b) Relations (2.31) and (2.32) are presented i n Figure 2.4 for t y p i c a l values of the system parameters. The v a r i a t i o n of the frequency i s found to be r e l a t i v e l y small. The s l i g h t decrease i n i t s value with increasing amplitude indicates a soft-spring type of nonlinear e f f e c t over the range of i n i t i a l conditions considered. Influence of the i n e r t i a parameter follows e s s e n t i a l l y the same trend as before. On the other hand, an increase i n e c c e n t r i c i t y tends to reduce the frequency of motion. 35 C = o e = o • s tab le a = 2 e = 0.05 0 0.2 p 0.4 0 0.2 y 0.4 Figure 2.4 Frequency and i n i t i a l c o n d i t i o n s f o r low f r e - quency p e r i o d i c motion (c) Solar pressure excited o s c i l l a t i o n s (C ^ 0) For the solar pressure excited periodic solutions, having the same frequency as that of the o r b i t a l motion, a = 0, b = 0. Substituting these conditions i n Equations (2.27a and b) results i n the i n i t i a l state, Y = A + A, + A„ o o 1 2 Y . , o 8 o - 0 Figure 2.5 shows the v a r i a t i o n of the i n i t i a l conditions Y Q and 3^ with the s a t e l l i t e spin parameter for both c i r c u l a r and noncircular o r b i t a l motion. The influence of the solar parameter C i s to r a i s e the i n i t i a l conditions for periodic motion. The i n i t i a l state appears to be highly sensitive to the spin rate for slowly spinning s a t e l l i t e s , however, i t asymptotically approaches a constant value with increasing spin parameter. In general, the e f f e c t of e c c e n t r i c i t y i s to increase the magnitude of the i n i t i a l conditions except at the lower end of the spin parameter spectrum. 37 F i g u r e 2.5 I n i t i a l c o n d i t i o n s l e a d i n g t o s o l a r p r e s s u r e e x c i t e d p e r i o d i c m o t i o n 38 (d) Accuracy of the a n a l y t i c a l solution To assess the accuracy of t h i s a n a l y t i c a l pro- cedure i n predicting the periodic solutions, the governing equations of motion (2.13) were integrated numerically with i n i t i a l conditions derived from the a n a l y t i c a l solution. Typical responses are presented i n Figure 2.6. In c i r c u l a r o r b i t s , the l i b r a t i o n a l response i s observed to be periodic with i n s i g n i f i c a n t error. The frequency of o s c i l l a t i o n s i s 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 i n e l l i p t i c o r b i t s . 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 f i v e percent of t h e i r a n a l y t i c a l l y predicted values for e <_ 0.1, the normal range of i n t e r e s t . (e) S t a b i l i t y of periodic solutions The s t a b i l i t y of periodic solutions can be studied using v a r i a t i o n a l analysis. The v a r i a t i o n a l equations are obtained by l e t t i n g 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 ' 1 1 \ \ \k \ '"x f \ \ ' */ \ 1 v' W \ J CO Y 0=1.8027 \ '«/ \ I 'M / ' A ~ 1 \ \ . V I ' V \ i 1 \ ' '/ \ \ • 1 1 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; / > L I 1 1 f 1 V f I f t ' i f I I I ' I l ' i f 1 ' I f i l f > 4 f ' I f i ' \ / i i \ / • '\ / ' M l > ' \J V/ ' \ J " V y 1 0 1 25 e = o - . R= 0.4353 /\ro A i / X i J 1 = 1.5 C = 0 = P0=0 Y o=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 / M A * \ » / * \ * \ 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 Y 0 =-0.20 = 2 C = 0 . 3 G= co= 0 = i = Y0'= arbi t rary = p = o e = o .o5 f 0 =- 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- o rb i ts Figure 2.6 T y p i c a l p e r i o d i c s o l u t i o n s of the system where y p , B p represent a periodic solution and y , B v are small perturbations. Substituting i n Equations (2.13) and l i n e a r i z i n g with respect to Y v and B v leads to: y " = F - y + F_ y ' + F_B + F.B' v 1 v 2 v 3 v 4 v B" = G , Y + G _ y 1 + G-B + G .B' (2.34) v 1 v 2 v 3 v 4 v where the d e t a i l s of lengthy functions F i * F i ( Y P ' 3 P ' 9 ) ' 1 = x ' 2 ' 3 ' 4 G± = G i ( Y p , B p , e ) , i = 1,2,3,4 (2.35) are given i n Appendix I I . For c i r c u l a r o r b i t a l motion, the functions F^ and G ^ assume the form F^ = F ^ ( y p , 3 p ) , G ^ = G ^ ( Y p , 8 p ) and hence have p e r i o d i c i t y of the solution. The Floquet theory can thus be applied to investigate the s t a b i l i t y of the v a r i a - t i o n a l system (2.34). The s t a b i l i t y c r i t e r i a for d i s t i n c t roots can be expressed as: | A I | <_ 1 , i = 1,2,3,4; stable any of the \X±\ > 1 , i = 1,2,3,4; unstable (2.36) 41 For s a t e l l i t e l i b r a t i o n s i n noncircular o r b i t s , the functions F^ and are periodic only i f the solution period i s a r a t i o n a l multiple of the o r b i t a l period. The Floquet theory may again be used to assess the v a r i a t i o n a l s t a b i l i t y of these solutions. Figures 2.3 - 2.5 also show the results of th i s analysis. The stable motion at smaller values of the i n i t i a l conditions i s , of course, anticipated. What i s of p a r t i c u l a r significance i s the p o s s i b i l i t y of periodic motions of large amplitudes. The amount of computational e f f o r t involved for s t a b i l i t y analysis i n an eccentric t r a j e c t o r y i s enormous. Large values of the common period of the l i b r a - t i o n a l and o r b i t a l motion lead to extended integration l i m i t i n g the s t a b i l i t y investigation to is o l a t e d points (Figure 2.3). Of course, the solar pressure excited motion of the o r b i t a l period does not present this problem (Figure 2.5). The a b i l i t y of the approximate closed form solution to predict nonlinear character of the system i s indeed promising. 2.2.3 Resonance It i s of p a r t i c u l a r relevance to recognize here several p o s s i b i l i t i e s of solar pressure excited resonance (Equations 2.15). Equivalence of or k 2 to the frequency of one of the forcing terms implies existence of certain combinations of the s a t e l l i t e i n e r t i a parameter I, spin parameter a and the o r b i t a l e c c e n t r i c i t y e which would lead to unbounded motion. In c i r c u l a r o r b i t s , the condition and/or k 2=l results i n the resonance conditions, 1 = 1 or, I (a + 1) -2 = 0 (2.37) The conditions for resonance i n e l l i p t i c o r b i t s 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/l 2-18I+33 = 0 (2. 38) Figure 2.7 shows resonance conditions (2.37) and (2.38) i n the system parameter space with t y p i c a l responses presented i n Figure 2.8. The large amplitude beat phenomenon in some cases indicates near resonant conditions. The value of C i s purposely taken here to be small to emphasize the d e s t a b i l i z i n g e f f e c t of the radiation pressure. Larger values of the solar parameter as often observed i n practice (C - 1.5 for Anik, 2 for the CTS) would magnify the amplitude build-up. This c l e a r l y indicates the need for avoiding such 43 44 C = o.i G = c o = c p = o i = a r b i t r a r y y-- Y„'= P0= P > o Y° p° e = 0 |=1.5 Q = 0.3333 0 2 4 6 8 10 0 - o r b i t s Figure 2.8 Typical responses under resonant conditions 45 c r i t i c a l combinations of system parameters for a safe s a t e l l i t e design. 2.3 Numerical Results The quasi-linear character of the a n a l y t i c a l s o l u t i l i m i t s i t s usefulness to the study of small amplitude motion. As anticipated, with large amplitudes, i t s accuracy was found to deteriorate due to the increased e f f e c t of system n o n l i n e a r i t i e s . A parametric study of the system was, there- fore, c a r r i e d out by numerically integrating the governing equations of motion. The Adams-Bashforth predictor-corrector quadrature with the Runge-Kutta s t a r t e r was used, i n con- junction with a step size of 3°, which gave res u l t s of s u f f i c i e n t accuracy without involving excessive computational e f f o r t . 2.3.1 S i g n i f i c a n t system parameters The significance of the i n e r t i a parameter I, the spin parameter a and the o r b i t a l e c c e n t r i c i t y e i n the l i b r a t i o n a l dynamics of spin s t a b i l i z e d s a t e l l i t e s cannot be overemphasized. S e n s i t i v i t y of the system response to these parameters i s v i v i d l y demonstrated i n Figure 2.9. It shows the v a r i a t i o n of $, the angular deviation of the axis of symmetry of the s a t e l l i t e from the o r b i t normal, against 9, the p o s i t i o n of the s a t e l l i t e i n an o r b i t . I t i s apparent that a judicious choice of parameter values i s e s s e n t i a l to avoid tumbling motion ($>u/2). 46 Figure 2.9 T y p i c a l unstable responses demonstrating the, s i g n i f i c a n c e of system parameters 47 Of p a r t i c u l a r i n t e r e s t i s the disturbing influence of solar radiation pressure. Note that the value of C as small as 0.5, which would ph y s i c a l l y correspond to e = 0.1 f t for INTELSAT IV category of s a t e l l i t e s at synchronous a l t i t u d e , causes the s a t e l l i t e to tumble over. The c r i t i c a l values of e c c e n t r i c i t y , i n t e r t i a and spin parameters would only accentuate t h i s behaviour. Of course, i n actual practice, a higher spin rate and/or active control system would counter this tendency. Nevertheless, the analysis c l e a r l y brings out the fact that the solar parameter C i s of the same importance as I, a and e i n the design of the s a t e l l i t e attitude control system. 2.3.2 System plots In order to better understand the solar pressure excited dynamical behaviour of the s a t e l l i t e , the system parameters were varied over the desired range and the l i b r a t i o n a l response observed. To i s o l a t e and emphasize the influence of the radiation pressure no other disturbances i n the form of i n i t i a l conditions were introduced. The r e s u l t i n g information was condensed i n the form of system plo t s . Figure 2.10a shows the e f f e c t of the s a t e l l i t e i n e r t i a parameter I on the coning amplitude $ and the c max average "nodding" frequency of the axis of symmetry, con, expressed as o s c i l l a t i o n s per o r b i t , for d i f f e r e n t 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 -0̂ 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) i n e r t i a parameter; (b) spin parameter; (c) o r b i t e c c e n t r i c i t y ; (d) o r b i t i n c l i n a t i o n the solar parameter C. I t may be observed that a s a t e l l i t e , when the solar pressure e f f e c t s are neglected, remains i n the equilibrium p o s i t i o n ($ = o). However, with non- zero C (say, C = 0.05), the l i b r a t i o n a l motion i s excited which increases i n amplitude with increasing C. I t i s of in t e r e s t to recognize the presence of a c r i t i c a l value of 1 = 1 leading to large amplitude motion f i n a l l y r e s u l t i n g i n i n s t a b i l i t y . 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 c r i t i c a l combin- ations of the system parameters predicted e a r l i e r by the a n a l y t i c a l method. On the other hand, the average nodding frequency of the axis of symmetry appears to be r e l a t i v e l y unaffected by changes i n the i n e r t i a or solar paremeter. The e f f e c t of the spin parameter a on the l i b r a - t i o n a l behaviour i s indicated i n Figure 2.10b. Here again, c r i t i c a l values of the spin parameter e x i s t for which the s a t e l l i t e tumbles over. A large value of the spin parameter, i n general, leads to smaller coning angles as anticipated. Figure 2.10c shows the influence of the o r b i t e c c e n t r i c i t y on the attitude motion. In general, higher values of the o r b i t e c c e n t r i c i t y r e s u l t i n larger amplitude motion. Unlike the e f f e c t of the i n e r t i a and the spin parameters, no resonant behaviour i s noticed for the t y p i c a l values of system parameters considered here. 50 The influence of the s i g n i f i c a n t o r b i t a l para- meters, such as i , 4>, OJ and the solar aspect r a t i o G, on the s a t e l l i t e performance was also investigated. The amplitude of o s c i l l a t i o n was found to reduce gradually with an increase i n the o r b i t a l i n c l i n a t i o n from the e c l i p t i c (Figure 2.10d). Changes i n the solar aspect angle cf>, which depends upon the location of the l i n e of nodes and the appar- ent position of the sun, did not a f f e c t the amplitude of l i b r a t i o n s and t h e i r frequency. The influence of the perigee position co and the solar aspect r a t i o G was also found to be i n s i g n i f i c a n t . 2.3.3 Design plots From design considerations, i t would be desirable to assess the magnitude of the solar pressure torque that a s a t e l l i t e can withstand without exceeding the permissible bound of l i b r a t i o n as governed by the mission requirements. This bound then would e s t a b l i s h a c r i t e r i o n for s t a b i l i t y . Here, the s t a b i l i t y l i m i t i s purposely taken as a large value of $ = T T/2 to emphasize the v u l n e r a b i l i t y of the s a t e l l i t e ' s performance to the solar pressure torque. Figure 2.11a shows a t y p i c a l s t a b i l i t y chart for l i b r a t i o n a l motion i n a c i r c u l a r o r b i t with the radiat i o n pressure as the only e x c i t a t i o n . The equations of motion (2.13) were integrated over 15-20 o r b i t s for a range of values of s a t e l l i t e i n e r t i a and spin parameters. The r e s u l t - Figure 2.11 T y p i c a l s t a b i l i t y charts showing adverse i n f l u e n c e of s o l a r r a d i a t i o n pressure: (a) e = 0; (b) e = 0.1 52 ing information about the maximum amplitude of the coning angle $ was then condensed i n the form of s t a b i l i t y max J plots i n the I-a space. The analysis shows that i n addition to the main stable region for high i n e r t i a parameter values, there also e x i s t small i s o l a t e d stable areas. However, there are substantial unstable regions even for p o s i t i v e spin or large i n e r t i a parameters. I t i s observed that the stable areas reduce d r a s t i c a l l y , as expected, with an increase i n the value of the solar parameter. The e f f e c t of the o r b i t a l e c c e n t r i c i t y on the s t a b i l i t y of l i b r a t i o n a l motion i s presented i n Figure 2.11b. An increase i n o r b i t a l e c c e n t r i c i t y further enhances the d e s t a b i l i z i n g influence of the solar pressure. The e f f e c t appears to be more pronounced for s a t e l l i t e s 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 a n a l y t i c a l 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 ( i i ) The a n a l y t i c a l method predicts the system response and frequency quite accurately i n c i r c u l a r o r b i t s . Even for noncircular o r b i t a l motion (e <_ 0.1), the errors are confined to less than 5%. ( i i i ) The influence of the solar pressure on the s a t e l l i t e l i b r a t i o n s , i n general, i s adverse. The analysis shows, however, that the system can execute stable periodic motions of con- siderable magnitude i n the solar pressure f i e l d under suitable i n i t i a l conditions. (iv) There e x i s t combinations of system parameters for which large amplitude o s c i l l a t i o n s r e s u l t , even i n the presence of a very small solar torque, due to resonant in t e r a c t i o n , (v) The solar parameter affects s t a b i l i t y of the motion subs t a n t i a l l y and hence merits equal consideration with the s a t e l l i t e i n e r t i a and spin parameters and the o r b i t a l e c c e n t r i c i t y . 54 3. ATTITUDE CONTROL USING SOLAR RADIATION PRESSURE The analysis of the l a s t chapter c l e a r l y estab- l i s h e s the substantial adverse influence of solar r a d i a t i o n pressure on s a t e l l i t e l i b r a t i o n s . On the other hand, the findings of e a r l i e r investigations of gravity oriented 49-51 systems suggest that the radiation force can provide e f f e c t i v e damping torques to maintain a s a t e l l i t e i n a desired attitude. However, as pointed out i n the l i t e r a t u r e review, available analyses of solar pressure control of spinning s a t e l l i t e s are only of a preliminary nature. This i s unfortunate because, in many space applications, a s a t e l l i t e with a d i r e c t i o n a l sensor has a preferred orientation which i s normally achieved by mounting the device on a s t a b i l i z e d platform aboard the spinning s a t e l l i t e . .The spin, through a gyroscopic moment, provides s t a b i l i t y while the platform, despun by control moments, tracks a given object in space. The concept of solar pressure control provides an exciting p o s s i b i l i t y of s t a b i l i z i n g the entire system through a semipassive approach.. This chapter investigates the f e a s i b i l i t y of the general three-axis nutation damping and attitude control of spinning s a t e l l i t e s using a solar c o n t r o l l e r sensitive to angular displacement and v e l o c i t y errors. The analysis 55 i s kept quite general to accommodate eccentric o r b i t s and a r b i t r a r y i n c l i n a t i o n s of the o r b i t a l plane with respect to the e c l i p t i c . The nonlinear, nonautonomous, coupled equations of motion are analyzed numerically and the influence of system parameters on the response studied. In the l a t t e r part of the chapter, a l o g i c a l approach for c o n t r o l l e r design i s developed. A n a l y t i c a l solutions for the control variables are obtained which suggest reduced software requirements. Several examples using representative s a t e l l i t e s i l l u s t r a t e the effectiveness of the control system and help gain an appreciation as to the c o n t r o l l e r size required for a desired performance. 3.1 F e a s i b i l i t y of Solar Pressure Control 3.1.1 Equations of motion Figure 3.1 shows an axisymmetric (I = 1^) s a t e l l i t e with the center of mass S moving i n a Keplerian o r b i t about the center of force 0. The s a t e l l i t e consists of a central body I, spinning at a constant average angular v e l o c i t y , connected to a s t a b i l i z e d platform II through a viscous damper e f f e c t i v e i n a x i a l rotation. The s p a t i a l o r ien- tation of the axis of symmetry, as stated before, i s s p e c i f i e d by two successive rotations y and 8, referred to as r o l l and yaw, respectively, which define the attitude of the s a t e l l i t e p r i n c i p a l axes x,y,z with respect to the i n e r t i a l F i g u r e 3.1 Geometry of motion of d u a l - s p i n s a t e l l i t e i n the s o l a r p r e s s u r e environment reference frame x', y', z 1 . The rotor and the platform spin in the x,y,z reference with angular v e l o c i t i e s a and X, respectively. In terms of these modified Eulerian rotations, the expressions for the pot e n t i a l and k i n e t i c energies to 0(1/R ) are obtained as: U = -ym /R-y{(I /2R 3)(I-1)/I}(l»3sin 2Ycos 28) (3.1) y o 2i. T = (m g/2)(R 2+r 26 2)+(I x/2I){(1-J)I(a-ysin3+8cos8cosY) 2 + JI(A-YsinB+0cos3cosY) 2+(B-8sinY) 2 • 2 + (YcosB+6sin(3cosY) > (3.2) The Rayleigh d i s s i p a t i o n function i s 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) {I v v. (a-YsinB+6cosBcosY) > (a-X) = (3.4) As the rotor i s considered to spin at a constant average spin rate ( i . e . , no spin decay), the usual assumption i n the analysis of dual-spin spacecraft, 58 Q a = K d(a-X) (3.5) T h i s i s e q u i v a l e n t to assuming an energy source c o u n t e r i n g the b e a r i n g drag on the r o t o r . The f i r s t i n t e g r a l a-Ysin3+6cos3cosy = h a (3.6) may now be used to e l i m i n a t e the c y c l i c c o o r d i n a t e a through the s p i n parameter a d e f i n e d as, (a/9) j =(h /6) 'e=3=Y=o 9 = 0 - 1 ( 3 - 7 ) N e g l e c t i n g o r b i t a l p e r t u r b a t i o n s due to l i b r a t i o n a l 78 79 motion ' and making use o f the s p i n parameter, the Lagrangian f o r m u l a t i o n y i e l d s the governing equations of motion i n the r o l l , yaw and p i t c h degrees of freedom. Changing the independent v a r i a b l e from t t o 0 through the K e p l e r i a n o r b i t a l r e l a t i o n s , they f i n a l l y take the form: Y" -2 3' ( Y ' t a n 3 - c o s Y ) - ( 3 ' - s i n Y ) s e c 3 [ ( 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) = Q y (3.8a) 59 8" -Y'cosY-{2esin9/(l+ecos9)}(8 1-sinY)+(Y'cos8+cosYsin8) x [ (1 - J ) I(a+1){(1+e)/(l+ecos9) } 2 + J I(A'-Y'sin8+cos8cosy) + ( Y ' s i n 3 - c o s 8 c o s Y ) ] - 3 { ( I - l ) / ( l + e c o s 0 ) } s i n 2 Y s i n 8 c o s 8 = Q D p (3.8b) A " - Y " sin8-{2esin0/(l+ecos9)}(A'-Y'sin8+cos8cosY) -3 IY'cos3-Y lcos8sinY-3'cosYsin8+ ( K / J I ) x { (1+e) 3 y / 2 / ( l + e c o s 9 ) 2 } [A 1 -y1 sin8+cos8cosY- (a+1) x { ( l + e ) / ( l + e c o s 9 ) } 2 ] = Q A (3.8c) where Q ^ ( i = Y*8,A) r e p r e s e n t the g e n e r a l i z e d f o r c e s due to s o l a r r a d i a t i o n p r e s s u r e . 3.1.2 C o n t r o l l e r c o n f i g u r a t i o n A c o n t r o l l e r , i n g e n e r a l , c o n s i s t s o f l i g h t , r i g i d , h i g h l y r e f l e c t i v e p l a t e s (membranes) s u i t a b l y mounted on the p l a t f o r m t o be s t a b i l i z e d . The c o n t r o l moments r e s u l t i n g from the s o l a r r a d i a t i o n f o r c e on the p l a t e s may be v a r i e d by changing any one of the f o l l o w i n g : (i) the d i s t a n c e between the c e n t e r of p r e s s u r e and the s a t e l l i t e c e n t e r of mass by t r a n s l a t i n g the p l a t e support; 60 ( i i ) the area of the membrane through wrapping or unfurling portions of i t ; ( i i i ) the projected area of the plate as 'seen' by the sun by rotating the plate. P r a c t i c a l considerations make the l a s t alternative the most a t t r a c t i v e , e s p e c i a l l y , 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 r o l l Yr the yaw 6 and the platform p i t c h X, be controlled independently, i t i s necessary to provide at least three independent plate rotations 6^, S 2, and 63. Various c o n t r o l l e r configurations were studied which, i n general, y i e l d expressions for the generalized forces of the form: Q Y = Q Y ( 5 l f 6 2 , 6 3 ) Q 3 = Q B ( 6 l f 6 2 , 63) Q A = Q A ( 5 1 # 6 2 , 63) (3.9) where the functions i n the ri g h t hand side are transcendental. As the rotations 6^, a n < ^ 3̂ a r e r e a l , the simultaneous maxima |Ct I „ . | Q Q L „ v and I Q ^ I for which Y max p max 1 A max the set of Equations (3.9) possesses a r e a l solution, represent the physical l i m i t on the generalized forces that a p a r t i c u l a r c o n t r o l l e r configuration can y i e l d . The problem of determining the maximum values of Q^, Q^, and Q^, which would s a t i s f y the above c r i t e r i o n i s , i n general, a complex one. Any attempt to simplify the problem, through the choice of co n t r o l l e r s y i e l d i n g completely decoupled moments, would lead to increased hardware complexities. One i s , therefore, forced to compromise by selecting a c o n t r o l l e r configuration r e s u l t i n g i n a p a r t i a l uncoupling of the . Figure 3.2 shows the schematic diagram of the proposed semi-passive c o n t r o l l e r . I t consists of f i v e sets df plates P i ( i = 1,2,3) and P^(j = 1,2) with t h e i r 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 ^ i s measured from a platform-fixed reference l i n e at an angle A from the y axis. At a given instant, the set P^, c o n t r o l l i n g the A motion, operates i n 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 i s somewhat involved. Figure 3.3 shows a plate i n an ar b i t r a r y orientation with respect to the sun. The force on an elemental plate area dA i s given by F i g u r e 3.2 Solar c o n t r o l l e r c o n f i g u r a t i o n 63 Normal Reflected ( i -T ) fDdA|cos£ Incident u Plate element P ^ d A | c o s £ | Figure 3.3 Radiation force on a p l a t e element 64 dF = -p dA|cos£ | {(l - T)u+ps} (3.10) with the re s u l t i n g moment about the s a t e l l i t e 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 in t e g r a l i n Equation (3.11) y i e l d the desired expression for M. The application of the p r i n c i p l e of v i r t u a l work f i n a l l y leads to the generalized forces Q^. The expressions are rather lengthy, however, ignoring the terms of order ( 1 - T - p ) / 2 p compared to unity, which i s j u s t i - f i a b l e for surfaces of high r e f l e c t i v i t y , r e s u l t s i n a considerable s i m i l i f i c a t i o n : = -E(0) [U1C11 cos ? ̂  cos 5 1sin6 1sinA +^^2 I c o s ^ ̂  C O S?2 S^" n^2 C O S^-' s e c ^ (3.12a) = -E (9) [U 1C 11cos K 11cos K 1sin6 1cosA - U 2C 2 | cos E, 2 | cos £ 2 s i n S 2 s i n A ] (3.12b) Qx = -E(6) (C 3/J) |cos £ 3|cos ? 3 (3.12c) where U i = +1 for P i f -1 for P^; i = 1,2 cos ^ = u. cos6. + (u . sinA-u, cosA) sin6. X 1 X 1 K X cos ? 2= u icos6 2+(UjCosA+u ksinA)sin6 2 (3.13) cos C 3 = -UjSin(6 3+A)+u kcos(6 3+A) The solar parameters, C^, are defined as C i = ( 4 p P 0 R p / y I y ) A i e i ' 1 = 1 ' 2 C 3 = ( 4 p p o R 3 / y I x ) A 3 £ 3 (3.14) 3.1.3 Control strategy The generalized forces are controlled i n a v e l o c i t y and position sensitive manner according to the re l a t i o n s : Q y = -y YY'-v y(Y-Y c) (3.15a) QB = ~ y B 6 ' " v 3 ( 6 " 6 c ) (3.15b) QA = ~ y A A ' " V A ( A " A c ) (3 .15c) where the system gains , are chosen according to some suitable c r i t e r i o n , such as, the least time of damping or the maximum permissible displacement during a nutation cycle. The position control parameters y , B c, and X^, however, are functions of the desired f i n a l orientation Yff Bf, and X^ and are obtained from the equilibrium consideration of the controlled system (Equations 3 . 8 ) . For nutation damping i n a c i r c u l a r o r b i t , t h i s results i n Y c = B c = 0, A c = - ( l / v A ) K a / J I (3.16) In p r i n c i p l e , the plate rotations can be obtained by substituting Equations (3.15) into Equations (3.12) and solving for 6 ^ . This, of course, implies s p e c i f i c a t i o n of the signs of and U^, i . e . , a combination of the plate sets to be operated. However, there are s t i l l several mathematical problems as the system of Equations (3.12) may not possess a r e a l solution. A t r i a l with d i f f e r e n t sign combinations of and i s thus necessary. Fortun- ately, for many applications, |u.| << |u.|, | u,| and X 1 j K executes small o s c i l l a t i o n s i n 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 a n a l y t i c a l l y . Furthermore, the c o n t r o l l e r may be unable to provide the corrective moments demanded by the system (Equations 3.15) at a l l times due to i t s physical limita- tions. It i s , 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 6̂  only, i t s maximum attainable value can easily be found, But 0^, Qg are functions of both 6̂  and hence i t is necessary to specify a rational criterion for the controller operation. In the present analysis, based on physical considerations, i t is taken to be the maximum of the total 2 2 2 1/2 transverse torque, (Q^cos 3+Q̂ ) . 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) -U VCOSA) 2/u 2+2} 1 / 2] K 1 (3.18a) 3 and 6 +u, sinA) 2/u 2+2} 1 / 2] k x (3.18b) 68 Substitution for 6^(i = 1,2) from Equations (3.18) into Equations (3.12) y i e l d s the desired maximum values of and Qg. Appropriate signs of and are to be introduced which y i e l d 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 r o l l , yaw and pit c h angles and rates, o r b i t a l p o sition and the apparent position of the sun; (i i ) compute the control moments demanded by the system using Equations (3.15); ( i i i ) evaluate the maximum attainable moments using Equations (3.17, 3.18, 3.12a and b); (iv) compare the moment demand with the attainable values. I f the demand exceeds the maximum available, set i t equal to the l a t t e r ; (v) determine the plates to be operated for r o l l - 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- a l l y 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 s t a r t e r was used i n conjunction with a step size of 3°. The impor- tant system parameters were varied gradually over the range of i n t e r e s t and the c o n t r o l l e r performance evaluated both i n c i r c u l a r and e l l i p t i c o r b i t s . In general, the system i s exposed to extremely severe disturbances, much higher than i t i s l i k e l y to encounter in the normal operation, to evaluate the c o n t r o l l e r ' s performance under adverse conditions. (a) Nutation damping Figure 3.4 summarizes the influence of the c o n t r o l l e r gains u^, on the response. In general, an increase i n y^ provides an overdamped character to the system (Figures 3.4a and b) while a corresponding increase i n for a given y^ results i n o s c i l l a t i o n s suggesting a reduction i n damping (Figures 3.4a and c ) . The existence of an optimum choice of c o n t r o l l e r gains for a set of given system parameters i s thus apparent. This i s indicated i n Figure 3.4d. The e f f e c t s of the s a t e l l i t e i n e r t i a parameter I and the spin parameter a , presented i n Figure 3.5 suggest that short, d i s c - l i k e s a t e l l i t e s (I = 1.5) withstand and damp a given disturbance r e l a t i v e l y better than long, slender s a t e l l i t e s (I = 0.5). I t i s of int e r e s t to point out that here the value of J, representing the r a t i o of the a x i a l i n e r t i a s of the platform and the s a t e l l i t e , i s taken as 0.5. The analyses with J varying from 0.25 to 0.7 5 showed the system response to remain v i r t u a l l y unaffected. This i s 70 20 10 0 1 r- * - h = 5 : \ V. =20 • 1 1 i t i i i i i i i i ~'/\ • jr\ \ ' 1 V • 1 1 / \ V ; = 25 , 1 , I f 1 1 1 1^ \ 1 \ K x ^ - A . »-•,.. ^ , \*^-.'' j \ / \ / - (C) \ / \./ 0 0.25 0.50 0 0.25 0.50 0.75 Orb i t Figure 3.4 Influence of the solar c o n t r o l l e r 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 | = 2 S 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 s a t e l l i t e i n e r t i a and spin parameters 72 understandable as large J makes i t more amenable to con- t r o l l e r corrections. The e f f e c t of the spin parameter a was also found to be r e l a t i v e l y i n s i g n i f i c a n t . This i s to be expected as the natural s t i f f n e s s of the system, to which the spin parameter contributes, i s largely provided by the c o n t r o l l e r gain (Figures 3.5c and d). Figure 3.6 shows the influence of the solar parameters and aspect angle <j> on the c o n t r o l l e r per- formance. The l i m i t a t i o n imposed by on the maximum allowable control moments i s c l e a r l y r e f l e c t e d i n the response plots (Figures 3.6a and b). Note that an increase in r esults i n an o v e r a l l improvement of the transient performance of the system. However, there i s a r e s t r i c t i o n as to the maximum attainable values for as imposed by the control plate areas and t h e i r moment arms. The r e l a t i v e position of the sun (cf>) affects the response i n the y and 3 degrees of freedom through the corres- ponding change i n the available control moment. Thus r o l l and yaw a t t a i n d i f f e r e n t r e l a t i v e amplitudes i n an o r b i t , how- ever, t h e i r damping time remains e s s e n t i a l l y unaffected (Figures 3.6c and d). The performance of the c o n t r o l l e r i n an eccentric o r b i t remains e s s e n t i a l l y the same except for a steady state l i m i t cycle appearing i n the A degree of freedom due to the periodic forcing function dependent on e (Equation 3.8c). This i s indicated i n Figures 3.7a and b, where the 73 e = o i,co=o r l i = io Yf= Pf = Xf-o - 0 Y 1 =i J = 0.5 V j = 2 5 Y0 = P0=X0=o - P ° a = i K= o.i Y> P0'-X>i " 30 15h 0 h V - (j> = 6 0 ° 10 0 C;= 6 (b) \ / I 37.5 25.0 12.5 (j) = 0 oh 25.0 12.5 < j ) = 9 0 ° JL 0.25 0.50 0.75 0 Orbit 0.25 0-50 Figure 3.6 E f f e c t of s o l a r parameters and s o l a r aspect angle on the damped response 74 Figure 3.7 Response p l o t showing l i m i t c y c l e o s c i l l a t i o n s i n e c c e n t r i c o r b i t s and t h e i r removal through the modified c o n t r o l f u n c t i o n r o l l and yaw motions damp out quite quickly but the platform p i t c h p e r s i s t s as a l i m i t cycle. The sustained o s c i l l a t i o n of the platform would, i n general, be highly undesirable. Fortunately, i t can be eliminated using a modified control r e l a t i o n for sensitive to the e c c e n t r i c i t y induced disturbance, X c = (l/v x)[-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 i l l u s t r a t e the effectiveness of the modified control r e l a t i o n i n eliminating the l i m i t cycle. The information presented so far pertains to o r b i t a l motion i n the plane of the e c l i p t i c ( i = 0). Of course, depending upon the mission, the o r b i t a l plane would be at an angle to the e c l i p t i c . A systematic study showed the e f f e c t of i to be confined to l o c a l changes i n response character without s i g n i f i c a n t l y a l t e r i n g the o v e r a l l con- t r o l performance. The plots i n Figures 3.8a and b substantiate t h i s conclusion. (b) Rotor spin decay The present analysis considers the rotor (body I) to have a constant average spin rate. Apparently, t h i s would be achieved through some active energy source. However, the influence of possible rotor spin decay on the l i b r a t i o n a l 76 e = o J =0.5 V. = 25 vr xr° 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 P i \ / \ 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 ( i / \ ! \ K=o.5 i t i i i i i * i i . i . i . i i » i * i » 7/' \ \ * * r v \ \ / . (c) , i \ i -(d) , V / i 15 10 0 -10 0 0.25 Figure 3.8 0.50 Orbit 0.75 Response as a f f e c t e d by: (a,b) o r b i t a l i n c l i n - a t i o n from the e c l i p t i c ; (c) r o t o r s p i n decay (d) modified c o n t r o l f u n c t i o n response of the system would be appropriate to explore. Although a s t i l l remains a c y c l i c coordinate, and hence the f i r s t i n t e g r a l i s available, i t i s not possible to eliminate the rotor spin degree of freedom from the rest of the equations of motion. Thus one i s faced with the solution of a 7th order system (as against the 6th order in the previous case). As can be anticipated, the r o l l and yaw motions damp out as before, coupling with the a degree of freedom being weak. However, the platform p i t c h angle A tends to d r i f t away from i t s preferred orientation A^ (Figure 3.8c). This does not r e f l e c t i n any way a l i m i t a t i o n on the c a p a b i l i t y of the c o n t r o l l e r but f a i l u r e on our part to exploit i t f u l l y . The v a l i d i t y of t h i s observation becomes quite apparent when one examines the modified pitch equation i n the presence of d i s s i p a t i o n , 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 + e c o s e ) 2 } ( A , - a ' ) = Qx (3.8c)' Note the dynamic coupling between the rotor and the platform. On the other hand, the control r e l a t i o n for (Equation 3.15c) does not involve a' e x p l i c i t l y . Thus the c o n t r o l l e r ' s potential to account for the rotor spin decay i s not u t i l i z e d . The s i t u a t i o n can e a s i l y be 78 corrected by modifying the control function as indicated, Q A = -y AA ,-a(6)a ,-v A(X-A ) (3.15c)" where a(6) = (K/JI)(1+e) 3 / 2/(l+ecos0) 2 A c = -(l/v A)(2esin6)/(1+ecosG) A t y p i c a l system response using t h i s modified control function i s presented i n Figure 3.8d. Implication of t h i s analysis i s rather far reaching. I t i s no longer necessary to maintain the rotor spin rate from attitude dynamics considerations as the c o n t r o l l e r i s able to.provide s u f f i c i e n t torque i r r e s p e c t i v e of the spin rate. The system thus has a t r u l y semi-passive character promising an increased s a t e l l i t e l i f e - s p a n . (c) Attitude control The c o n t r o l l e r provides an i n t e r e s t i n g p o s s i b i l i t y of changing the s a t e l l i t e ' s preferred orientation i n o r b i t . This i s accomplished by using the position control parameters Y c, B c and A^ i n accordance with the desired equilibrium configuration y^, 6^, and A^. As an i l l u s t r a t i o n , for a s t a t i c equilibrium of the s a t e l l i t e i n a c i r c u l a r o r b i t , Equations (3.8) i n conjunction with Equations (3.15) lead to 79 Y c = (1/v ) [ ( s i n y f / c o s B f ) { ( l - J ) I ( a + l ) +JIcosBfcosYf:}+(3I-4) sinY^cosy^] +Y (3.19a) (3.19b) X c = ( V v x)-[(K/JI){cosS fcosY f - a-l}]+X f (3.19c) Note that i n t h i s p a r t i c u l a r case of e = 0, the po s i t i o n control parameters are fixed once the f i n a l orientation i s sp e c i f i e d . On the other hand, for the case of an eccentric o r b i t , the parameters depend, i n addition, on the s a t e l l i t e p o s ition in the o r b i t . Figure 3.9 i l l u s t r a t e s the v e r s a t i l i t y of the semi-passive c o n t r o l l e r i n achieving ar b i t r a r y orientations i n space, for both c i r c u l a r and e l l i p t i c o r b i t s . This suggests an e x c i t i n g p o s s i b i l i t y for a space vehicle to extend i t s range of a p p l i c a b i l i t y and undertake diverse missions. (d) I l l u s t r a t i v e example of the concept through a preliminary attitude dynamics study of the two well-known s a t e l l i t e s , INTELSAT IV and Anik, when provided with the proposed c o n t r o l l e r . Appropriate It was decided to demonstrate the effectiveness 80 Figure 3.9 Effectiveness of the solar c o n t r o l l e r in achieving a r b i t r a r y orientations of the s a t e l l i t e 81 geometrical and i n e r t i a properties were assigned. As seen before, i t i s no longer necessary to spin a s a t e l l i t e 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 f t and moment arms ^ = 5 f t . S i m i l a r l y , 2 - 5 for Anik may be achieved with plate areas A^ = 1.2 f t and moment arms = 5 f t . Subjecting the s a t e l l i t e s to a disturbance equivalent to that imparted by micrometeorite impacts over 24 hrs, which represents an enormous magnifi-8 0 cation of the re a l s i t u a t i o n , gives -0.05 < y', 8', A' ^ — o o o £0 . 0 5 . I t i s apparent (Figure 3.10) that the c o n t r o l l e r i s able to damp such a severe disturbance quite e f f e c t i v e l y with the maximum deviation from the preferred orientation of less than 0.25°. The figure also shows the co n t r o l l e r ' s effectiveness i n achieving s p e c i f i e d s p a t i a l orientations. 3.2 Improvement of Controller Design Potential of the concept having been established, attention i s next directed at rendering i t more e f f i c i e n t and s t r u c t u r a l l y more a t t r a c t i v e . As the solar c o n t r o l l e r was found to be quite e f f e c t i v e even i n the absence of any spin, the case of a nonspinning s a t e l l i t e i s now considered.. 82 e = o J = o . 5 ji j=io Y0=(3=A=o 0 = 1 K=0.5 V i = 25 Y0=g=X=0.05 i = 2 3 . 5 ° <f> = 60° CO = 0 1- -Y -p° 0.2 INTELSAT IV 1=0.7 C :=2 Yf=Pf = A=o H 75 -75 _Y f=10° ( $ = - 4 5 ° 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 c o n t r o l l e r performance i n achieving nutation damping and attitude control of INTELSAT IV and Anik s a t e l l i t e s 83 3.2.1 Equations of motion The governing equations of motion follow d i r e c t l y 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) = Q y (3.20a) B" -Y'cosY-{2esin9/(l+ecose)}(B'-sinyJ + d(A'-y'sinB +COSBCOSY)+(Y 1sinB-cosBcosy)}(Y'cosB+cosysinB) -3{ (I-l)/(l+ecos9) }sin 2YsinBcosB = Q D (3.20b) P A" - Y " sinB-{2esin9/(l+ecos9)}(A'-Y'sinB+cosBcosy) -B'Y 1cosB-Y'cosBsiny-B'cosysinB = Q, (3.20c) 3.2.2 Development of c o n t r o l l e r models Controllers capable of providing general l i b r a t i o n - a l damping and three-axis attitude control require the three degrees of freedom of the system (y , 3 and A) to be controlled independently. A v e r s a t i l e solar c o n t r o l l e r configuration would thus r e s u l t i n generalized forces as functions of at least three control variables, 6.. Of l various configurations s a t i s f y i n g t h i s requirement, only those permitting r e l a t i v e l y easier solution for the control variables with a given set of would be p r a c t i c a l l y f e a s i b l e . 84 At the outset, i t appears i n s t r u c t i v e to study the control moments generated by a single plate P with i t s support arm gimballed at the point T on the axis of symmetry of the s a t e l l i t e (Figure 3.11a). The arm supporting the plate i s allowed rotations and a 2 while the plate can turn about i t through an angles measured from the plane TSW. Following the procedure discussed e a r l i e r i n d e t a i l , the generalized forces due to solar radiation pressure can be written as: Q Y = -CE (0) |cos?|cos?{(h/e+sina 2) (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,sina 0cos6-sina„sinS}/I (3.21b) 1 2 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^sina 2sin6) -u, (cosa,cos6-sina,sina„sinS) (3.22) k 1 1 2 As i s evident, each of the 0^ depends on the variables o^, and 6. The transcendental nature of these functions makes i t d i f f i c u l t to esta b l i s h the bounds Figure 3.11 Development of s o l a r c o n t r o l l e r c o n f i g u r a t i o n s 86 I Q I , | Q „ ! and lQ-,1 within which Equations (3.21) 1 y max' 1 8'max 1 A 1 max ^ possess r e a l solutions for a^, a 2 and 6. It i s , therefore, necessary to consider modifications which would reduce these equations to a more amenable form. A l o g i c a l approach seems to l i e i n devising con- figurations where some of the three variables become con- stants. The best choice, from a p r a c t i c a l viewpoint, would be to remove the gimbal and f i x the axis of the plate to the s a t e l l i t e body, thereby rendering and a 2 constants. This reduction i n the number of control variables, of course, would have to be compensated by a corresponding increase i n the number of plates rotating about body-fixed axes. Numerous con t r o l l e r configurations were considered to t h i s 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 6̂  i s measured from the xy-plane. The evaluation of the generalized forces r e s u l t i n g from thi s configuration may be ca r r i e d out either from the f i r s t p r i n c i p l e s , or, by appropriate substitutions for a^, a 2 and 6 for each plate into Equations (3.21). These are found to be: 8 7 = E (0) {(^ |cos£ 1 1cos? 1cos6 1cosA-C 2|cos£ 2|cos? 2cos6 2sinA +C 3|cos? 3| c o s 5 3sin6 3>/(I cos6) (3.2 3a) = E(0){-C 1|cos? 1|cos^ 1cos6 1sinA-C 2|cos? 2| c o s 5 2cos6 2cosA +C 3|cos£ 3|cos£ 3cosS 3}/I y (3.23b) = E ( 0 ) { C 1 | c o s ? 1 | c o s 5 1 s i n 6 1 + C 2 | c o s ? 2 | c o s ? 2 s i n 6 2 } / I x (3.23c) where cos£^ = u^cos6^+(u_.sinA-u^cosA) sinS^ c o s ^ = u^cos5 2+(UjCosA+u^sinA) s i n 6 2 cos£ 3 =-UjSin6 3+u kcos6 3 (3.24) I t may be observed that the dependence of the generalized forces Q i ( i = y,3,A) on the control variables 6^(i = 1,2,3) i s quite involved and the system of Equations (3.23) does not appear to lend i t s e l f to an a n a l y t i c a l treatment. Figures 3.11c and d show alternative arrangements, An analysis of the generalized force expressions, which are e s s e n t i a l l y of the same nature as Equations (3.23), showed them to present si m i l a r d i f f i c u l t i e s . 88 Following the approach discussed i n preceding sections, the c o n t r o l l e r configuration shown i n Figure 3.12 i s f i n a l l y arrived at. The performance of th i s model i s investigated i n d e t a i l as both the hardware and software requirements are r e l a t i v e l y simple to implement. I t consists of four plates P ^ ( i = 1,2,3,4) with t h e i r support arms forming a c r i s s c r o s s . The plates and P 2 rotate about the axis of symmetry of the s a t e l l i t e , the rotations 6̂  and 62 being measured from the xy-plane. The plates P^ and P^ rotate about arms, l y i n g i n the plane perpendicular to the axis of symmetry, with the rotations 63 and 6^ measured from the yz-plane. The moments generated by t h i s arrangement are found to be: = E(8) [C^|cosC11cos?1sin61-C2|cos?2|cos£2sin62 +l{C 3 | cos £ 3 I cos^cosS 3~C;j | cos£4 | cos^cos6^}cosX] sec 3 (3.25a) Q g = E (9) [C]_ I c o s ^ I cos? 1cos(S 1-C 2 | cos? 2 I cos5 2cosS 2 +1{-C^I cos?^Icos?3Cos63+C4I cos£4|cosE^cosfi^sinA] (3.25b) = E ( 6 ) { C 3 | c o s 5 3 | c o s ? 3 s i n 6 3 + C 4 | c o s ? 4 | c o s ? 4 s i n 6 4 } (3.25c) where  9 0 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 R 3/yI )A.e., i = 1,2 l co p y l l = ( 2 p p o R 3 / y I x ) A i e i , i = 3,4 (3.27) Although Equations (3.25) indicate a rather complex dependence of on the control variables, i t i s possible to obtain r e l a t i v e l y simple a n a l y t i c a l 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 P 2 do not produce any moment i n the pi t c h (A) degree of freedom, hence th i s control w i l l have to be accomplished through plates P^, P^. Ideally, one would l i k e t h e i r operation to be free of coupling roll-yaw moments. Fortunately, t h i s can be achieved through a simple control law: 91 • o f f or • o f f , T T / 2 , ( 3 . 2 8 ) the option being dicated by the sign required of Q^. 6̂  = 'off* implies i t s rendering the corresponding cos£^ = 0, i . e . , p h y s i c a l l y , the sun-line would l i e in the plane of the plate. A pure pi t c h moment thus produced has the magnitude Q, = E(9)C_ „ (u.sinX-u. cosA) 2 (3.29) where C., = C„ = C 0 „ i s assumed for convenience. the orientation of the axis of symmetry of the s a t e l l i t e , plates P^ and maY be operated simultaneously or one at a time with the other ' o f f . The latter mode of operation, resulting in greater transverse torques, is considered here. The generalized forces in the r o l l 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 l 1 i l k l l (3.30a) QQ = ±E ( 8) C . I-u . sin6 .+u. cos6 . I (-u . sin6 .+u, cos6 .) cos6 . 3 • i l l k I 1 3 • i k i I (3.30b) 3,4 In order to provide the roll-yaw moments c o n t r o l l i n g 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 thi s problem. One approach i s to maximize the t o t a l transverse torque, 2 2 2 1/2 (Q^cos B+Qg) . Unfortunately, the res u l t i n g c r i t i c a l <5̂ (orienting the control plate so as to maximize c o s ^ ) yi e l d s only two of the four possible combinations of the r o l l and yaw moment directions: Q y, Q g: ++,+-,-+, An alternate procedure would be to maximize |Q | and | | , i n turn, with respect to 6 ^ . the maximum value of |Q̂ | = Q™ i s found to occur at 6im = ^ / 2 + t a n 1[{-3u.±(9u 2+8u 2) 1 / 2}/2u k] (3.31a) with the corresponding |Q̂ | = being given by Equation (3.30b) with 6 . = 6 . . S i m i l a r l y , the maximum of |QR| = Q m occurs at 6 i m = tan~ 1[{3u k±(9u 2+8u 2) 1 / 2}/2u j] (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 i s 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 i n each case, the two values of <5̂  , corresponding to the plus and the minus signs, l i e i n adjacent quadrants. Hence, one of them always s a t i s f i e s the constraint r e l a t i o n (3.32). Thus use of the proper sign i n the evaluation of 6. im (Equations 3.31) i n conjunction with an appropriate choice of the plate or P 2 (Equations 3.30) enables the c o n t r o l l e r to provide a l l possible sign combinations of Q^ and Qg. Of m c c m the two admissible sets of moments, (Q^, Qg) and (Q^, Qg), the one r e s u l t i n g i n a greater magnitude of the t o t a l torque to control the axis of symmetry of the s a t e l l i t e , i . e . , (Q 2cos 2B+Qg) 1 / 2 , i s used. For example, l e t the required control moments be Q̂ > 0 and Qg< 0. As cosB i s p o s i t i v e , tan 6̂ < 0- Thus 6̂ must l i e either i n 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 i n adjacent quadrants as pointed out before. Let the 6^m corresponding to Equation (3.31b) be i n the t h i r d and the fourth quadrants. Consistent with our require- 94 Figure 3.13 Quadrant c o n s t r a i n t f o r p l a t e r o t a t i o n 6. 95 merits, the admissible values are (6. ) and (6. ). . Of im a 2 im b 1 these, the one re s u l t i n g i n a higher value of the t o t a l transverse torque i s used i n the analysis. The control moments thus determined are applied i n a bang-bang fashion i n conjunction with l i n e a r displace- ment and v e l o c i t y s e n s i t i v e switching functions, Q i = ~ Q i s 9 n s ± (3.33) where S i = i ' + m ( i - i f ) , i = Y , B , A (3.34) 2 2 2 2 Q y = Q ™ , Q B = for (C-J cos 2B+Qg ) > ( Q ° cos 2 B+Q m ) Qy = Q Y, Qg-Qg for (Q'yl cos Z B+Q^ ) < ( Q ° cos^B+Q m ) and m represents the system gain, chosen according to some suitable c r i t e r i o n . The control procedure may now be summarized as follows: (i) sense the r o l l , yaw and p i t c h angles and rates, o r b i t a l position and the solar aspect angle; ( i i ) determine the signs required of according to the switching c r i t e r i o n (Equations 3.33); ( i i i ) during pitch control, use (6^ = TT/2, <5̂  = ' o f f or (6 3 = ' o f f , 6 4 = TT/2) for Q^/ ( U j S i n A - u ^ c o s X) ^ 0, respectively (Equations 3.25c and 3.26). (iv) for roll-yaw control, compute the four values of 6^ = 6̂  from Equations (3.31a and b) and of the two s a t i s f y i n g the quadrant constraint (3.32), choose the 6̂  producing the greater resultant transverse torque. Substitute into Equation (3.30a or b) and select the plate to be operated through the sign required i n the r i g h t hand side. 3.2.4 Results and discussion As before, the response of the system i s studied by numerically integrating the equations of motion (3.20) along with the control r e l a t i o n s (3.33). Much smaller i n t e - gration steps were required due to the bang-bang nature of the control law. A step size of 0.1° gave results of s u f f i c i e n t accuracy for the solar parameters 5. For cases with larger values of C^, the step size had to be correspondingly decreased. The c o n t r o l l e r performance was evaluated i n both c i r c u l a r and e l l i p t i c o r b i t s and the influence of important system parameters investigated. (a) L i b r a t i o n damping i n c i r c u l a r o r b i t s Figure 3.14 summarizes the performance of the c o n t r o l l e r i n damping the l i b r a t i o n a l motion i n c i r c u l a r orbits i n the form of optimization plots for the c o n t r o l l e r gain m. I t shows the v a r i a t i o n of the damping time x,, 97 e = 0 CO = 0 0 = 4 5 ° i = 2 3.5° Y,= P,= A,= o Y f=ft=A f=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 c o n t r o l l e r gain m 98 defined here as the time taken for a l l the three l i b r a t i o n angles to s e t t l e within 0.1° of the f i n a l orientation, with the gain m for d i f f e r e n t combinations of system parameters and i n i t i a l conditions. The p l o t s , i n general, indicate the existence of an optimum value of the system gain r e s u l t - ing i n the least time of damping. The influence of the s a t e l l i t e i n e r t i a parameter I on the performance of the c o n t r o l l e r i s indicated by a comparison of curves (a) and (b). Despite the d r a s t i c change in the s a t e l l i t e mass d i s t r i b u t i o n from p e n c i l - l i k e (I = 0.1) to spherical ( 1 = 1 ) , only a small v a r i a t i o n i n the optimum value of m and the general damping behaviour i s observed. It may be pointed out here that curves (a) and (b) were obtained using a r e l a t i v e l y small value of the solar para- meters = 2. For larger values of C^, the e f f e c t of I was hardly perceivable. The effectiveness of the control system in l i b r a t i o n damping of s a t e l l i t e s with a wide variety of mass d i s t r i b u t i o n s i s thus apparent. The e f f e c t of the solar parameters C\ , characterizing the magnitude of the solar torques, i s observed by comparing curves (a and c) and (b and d). An increase i n not only results i n a substantial reduction i n the damping time but renders the c o n t r o l l e r performance r e l a t i v e l y i n s e n s i t i v e to the system gain m as well. A comparison of curves (d and f) and (c and e) shows the e f f e c t of the i n i t i a l disturbances. An increase 99 i n the i n i t i a l conditions r e s u l t s i n longer damping times as anticipated. In addition, for smaller values of C\, the system performance shows a strong dependence on the con t r o l l e r 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 re s u l t s , however, showed l o c a l changes i n the response character only, leaving the damping performance e s s e n t i a l l y unaffected. Typical optimal responses for a s a t e l l i t e i n an equatorial o r b i t are presented i n Figure 3.15a. Even when subjected to severe impulsive disturbances i n the r o l l , yaw and pitch degrees of freedom simultaneously, the c o n t r o l l e r i s able to damp the s a t e l l i t e l i b r a t i o n s i n a few degrees of o r b i t a l t r a v e l with the amplitudes remaining quite small. (b) L i b r a t i o n damping i n e l l i p t i c o r b i t s The effectiveness of the c o n t r o l l e r i n damping the roll-yaw ( y , 3) o s c i l l a t i o n s of the axis of symmetry of the s a t e l l i t e remained v i r t u a l l y unchanged i n eccentric o r b i t s . The p i t c h ( A ) degree of freedom, however, executes a steady- state l i m i t cycle (Figure 3.15b). These o s c i l l a t i o n s r e s u l t due to the presence of the periodic forcing function dependent on e (Equation 3.2 0c) and the i n a b i l i t y of the con t r o l l e r to generate s u f f i c i e n t torque to counter i t at a l l times. 100 Figure 3.15 (a) T y p i c a l optimum response i n c i r c u l a r o r b i t s ; (b) L i m i t c y c l e o s c i l l a t i o n i n e l l i p t i c o r b i t s ; (c) V a r i a t i o n of l i m i t c y c l e amplitude w i t h system parameters 101 From design considerations, i t i s of in t e r e s t to investigate the l i m i t cycle amplitude as a function of system parameters. Once the roll-yaw o s c i l l a t i o n s are completely damped, Equations (3.20a and b) are i d e n t i c a l l y s a t i s f i e d . Letting Y = Y i = 3 = B i = 0 and substituting from Equations (3.29 and 3.33) into Equation (3.20c), the governing equation for the pit c h o s c i l l a t i o n s 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 l i m i t cycle o s c i l l a t i o n being the steady-state solution of t h i s equation, i t s amplitude may be expressed as A = f(C_ . ,e,m, A,-,i,(d,<t)) (3.36) max v 3 , 4 f The largest amplitudes would r e s u l t when the maximum magnitude of the e c c e n t r i c i t y induced disturbance and the minimum (zero) magnitude of the p i t c h control torque occur simultaneously. The l a t t e r would ph y s i c a l l y 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 s i t u a t i o n would arise only twice a year for an earth s a t e l l i t e . Considering small e, the condition for worst l i m i t cycle i s given by 102 tancf) = (tanA^sinoo-cosoo) s e c i / (tanA^cosoo+sinoo) (3.37) Figure 3.15c shows the v a r i a t i o n of the p i t c h l i m i t cycle amplitude with the o r b i t a l e c c e n t r i c i t y for a range of values of the solar parameter C, . under the most adverse s i t u a t i o n . Even here, only moderate values of 4 are required to l i m i t the amplitude to a generally acceptable value. (c) Attitude control The a b i l i t y of the c o n t r o l l e r i n imparting a r b i - trary orientations to the s a t e l l i t e and thus enabling i t to undertake diverse missions appears i n t e r e s t i n g to explore. Figure 3.16a shows, for both c i r c u l a r and e l l i p t i c o r b i t s , the effectiveness of the control system in providing an arbitrary p i t c h attitude to the s a t e l l i t e while the axis of symmetry remains normal to the o r b i t a l plane. The a b i l i t y to al i g n the symmetry axis with the l o c a l v e r t i c a l d i r e c t i o n and simultaneously a t t a i n a desired p i t c h attitude i s indicated i n 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 t r a n s i t i o n between widely d i f f e r i n g attitudes. Thus with the present control system, an antenna aboard the spacecraft i s able to scan substantial regions of the sky as a r b i t r a r y p i t c h attitudes are attainable 103 0 1 0 1 2 Orb i t s Figure 3.16 Effectiveness of the four-plate model in achieving a r b i t r a r y orientations of the s a t e l l i t e 104 with the axis of symmetry either along the o r b i t normal or the l o c a l v e r t i c a l . The p o s s i b i l i t y of s t a b i l i z i n g the s a t e l l i t e axis of symmetry at other orientations was also investigated. The analysis suggests that the c o n t r o l l e r i s able to achieve t h i s but only at the cost of higher values of C. . 1 (d) I l l u s t r a t i v e example In order to obtain preliminary estimates of the control plate areas and moment arms required, the proposed Canadian Communications Technology S a t e l l i t e (CTS) and Anik, are considered. The l a t t e r i s assumed to be non- spinning which represents an adverse s i t u a t i o n . An impulsive disturbance of 0.1 i s applied i n a l l the three degrees of freedom which i s i n excess of that imparted by micrometeorite impacts over 24 hours. As the i n e r t i a parameter (I - 0.1 for CTS, 1 for Anik) did not a f f e c t the performance s i g n i f i - cantly, most curves i n Figure 3.14 (especially curves c, d and e) are representative of the l i b r a t i o n a l behaviour of both s a t e l l i t e s considered here. The solar parameter value of C^ - 5 can be obtained for CTS using plate areas 2 2 A^ 2 = ± 2 r t r ^ = 1.2 f t with the moment arms = 10 f t . S i m i l a r l y , C^ - 20 for Anik may be achieved with each of 2 the plates having an area of 5 f t and moment arm 10 f t . As indicated by curves (d and c) i n Figure 3.14 these moderate control plate areas are s u f f i c i e n t to damp the s a t e l l i t e 105 l i b r a t i o n s within a few degrees of o r b i t a l t r a v e l . The l i b r a t i o n a l amplitudes also remained within a f r a c t i o n of a degree. The plots presented i n Figure 3.16 indicate the effectiveness of the c o n t r o l l e r i n providing attitude control to the CTS. F i n a l l y , a comment concerning earth shadow, which would render the solar c o n t r o l l e r s i n e f f e c t i v e , i s appropriate here. For a geostationary o r b i t , the influence of shadow i s confined to quarter of the s a t e l l i t e ' s l i f e s p a n , and even here, only during 5% of the o r b i t a l period. The results showed the c o n t r o l l e r performance to remain v i r t u a l l y unaffected. 3.3 Concluding Remarks The s i g n i f i c a n t conclusions based on the analysis may be summarized as follows: (i) The f e a s i b i l i t y of a semi-passive c o n t r o l l e r using solar radiation pressure for three-axis nutation damping and attitude control of spinning s a t e l l i t e s i s c l e a r l y demonstrated, ( i i ) The system i s capable of damping extremely severe disturbances i n a f r a c t i o n of an o r b i t . The time of damping can be further reduced by an optimum choice of the system gains, ( i i i ) With the use of the general c o n t r o l l e r configura- tion a d i r e c t i o n a l device aboard the s t a b i l i z e d platform may be earth-oriented, space-oriented 106 or made to track a s p e c i f i e d c e l e s t i a l object through a proper choice of the pos i t i o n control parameters. Even with the s i m p l i f i e d model, scanning of a substantial region of the sky i s possible. The solar pressure c o n t r o l l e r s thus impart v e r s a t i l i t y to a space vehicle i n 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 i t i s no longer necessary to maintain a constant spin rate by compensating for the dissipated energy, (v) A l o g i c a l approach for evolving a three-axis solar c o n t r o l l e r model suitable for p r a c t i c a l implemen- tatio n i s presented. The proposed four-plate c o n t r o l l e r model appears to be quite a t t r a c t i v e due to i t s s i m p l i c i t y of design and the associated software. (vi) The use of bang-bang control relations s u b s t a n t i a l l y improves the system performance. For solar parameter values attainable i n practice, a wide range of c o n t r o l l e r gain y i e l d near optimum performance. S a t e l l i t e s with d i f f e r e n t mass di s t r i b u t i o n s exhibit e s s e n t i a l l y the same damping c h a r a c t e r i s t i c s . 107 ( v i i ) As the c o n t r o l l e r s do not re q u i r e any mass expulsion scheme or a c t i v e gyros i n v o l v i n g l a r g e power consumption, the s o l a r pressure c o n t r o l system i s e s s e n t i a l l y semi-active. This promises as increased l i f e s p a n . 108 4. MAGNETIC-SOLAR HYBRID ATTITUDE CONTROL The earth's magnetic f i e l d presents an in t e r e s t i n g p o s s i b i l i t y of generating control moments by inte r a c t i o n with onboard dipoles. Magnetic torquing appears to be p a r t i c u l a r l y a t t r a c t i v e as i t offers increased r e l i a b i l i t y through the elimination of moving parts. The shortcoming 5 9 of the method, as pointed out by Bainum and Mackison and others, l i e s i n i t s i n a b i l i t y to produce s u f f i c i e n t pitch control torques i n equatorial o r b i t s where the geomagnetic f i e l d i s nearly p a r a l l e l to the o r b i t normal. On the other hand, the usefulness of placing s a t e l l i t e s i n equatorial orbits cannot be overemphasized, e.g., the communications s a t e l l i t e s i n the geostationary o r b i t . A need for evolving a system which retains the si m p l i c i t y of the magnetic concept and yet able to provide pit c h control, i s thus apparent. Having established the effectiveness of the solar pressure c o n t r o l l e r , i t i s proposed here to u t i l i z e i t for damping pitch o s c i l l a t i o n s . This chapter explores the f e a s i b i l i t y of the three-axis nutation damping and attitude control of a dual-spin s a t e l l i t e using a magnetic-solar hybrid control system. Two magnetic c o n t r o l l e r models are considered. A bang-bang control law with l i n e a r displacement and v e l o c i t y sensitive switching functions i s employed and a n a l y t i c a l 109 solutions for the control variables are obtained. The performance of the control system i s evaluated numerically and the res u l t s are presented as functions of system parameters. An i l l u s t r a t i v e 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 s a t e l l i t e being the same as the dual-spin spacecraft considered i n Chapter 3, Equations (3.8) apply where ( i = y,$,X) now represent the t o t a l generalized forces due to the magnetic and solar c o n t r o l l e r s . For convenience, the governing equations of motion are c i t e d again: Y 1 1 -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)}(3 1-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 d u a l - s p i n s a t e l l i t e i n the earth's magnetic f i e l d I l l +(Y'sinB-cos3cosY)]-3{(I-l)/(1+ecose)} sin 2YsinBcosB= (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 [A l-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 a r b i t r a r y orientation p, strength h^ and p o l a r i t y U. The moment generated by in t e r a c t i o n with the earth's magnetic f i e l d i s then given by, M = P X B = Uh 2p X B (4.2) Expressing the unit vector p and the geomagnetic induction vector B i n terms of t h e i r components along the xyz-axes and using the p r i n c i p l e of v i r t u a l work, the generalized forces i n the r o l l , 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 = u S n ( p k B i ~ p i B k ) / ( 1 + e c o s e ) QAm= U< Cn/ J I ) ( P j B ^ B ^ / d + e c o s e ) (4.3b) (4.3c) where the magnetic parameter C , characterizing the magnitude of the magnetically generated moments, i s defined as C = h M / y l m m e y (4.4) The body components of B / D are given by the r e l a t i o n s m • — B . l B . 3 Bk cosBcosy c o s B s i n y c o s n m c o s B s i n y s i n n m t s i n B s i n n m -sinBcosn m - s i n y cosycosr^ cosysinn m sinBcosy sinBsinycosn s i n B s i n y s i n n m ' m -cosBsinn m +cosBcosn m B xn B yn B zn (4.5) The geomagnetic induction components along the o r b i t normal, ascending node and perpendicular to the l i n e 81 of nodes i n the o r b i t a l plane are well established . For an earth-centered canted dipole model, they are: B, =• - c o s i cose + s i n i sintj) s ine xn m m m m m (4.6a) 113 Byn = ( 3 / 2 ) s i n i m s i n 2 n r n c o s e m + ( l / 2 ) [cos<j>m+ (3/2) x { (1+cosi ) cos (2n-<j) )+(l-cosi )cos(2n + cj) )}]sine m mm m m m m (4.6b) B z n = (1/2) s i n i m (l-3cos2n m) cose m+(1/2) [cosi msincj) m +(3/2){(1+cosi )sin(2n )+(l-cosi )x m m m m sin(2n + 6 ) } ] s i n e m (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 l i n e of nodes, according to <j>m = (n e+n r) . Its governing equation with the o r b i t a l angle as the independent variable i s , <t>̂  = (n e+n r) ( R 3 / y ) 1 / 2 ( l + e ) 3 / 2 / ( l + e c o s 0 ) 2 (4.7) which has the solution (for e < 1) *m = c j )mo + { ( ne + nr ) ( R p / ^ ) 1 / 2 d+e) 3 / 2 / d ~ e 2 ) } x [-esin9/(l+ecos9)+{2/(l-e 2) 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 i s to be introduced. The present analysis ignores the nodal regression as a dynamic e f f e c t which i s equivalent to assuming the earth to be g r a v i t a t i o n a l l y spherical. For c o n t r o l l i n g s a t e l l i t e nutations i n an equatorial o r b i t with the nominal p o s i t i o n of the spin axis along the o r b i t normal, i t can be e a s i l y shown that the 2 2 2 1/2 maximum transverse torque, (Q cos 3+QQ) , re s u l t s when y p p^ = 0. Accordingly, two magnetic c o n t r o l l e r models with dipoles i n the s a t e l l i t e ' s transverse plane are considered here (Figure 4.2). Model A consists of a single dipole rotatable about the x axis i n the platform fixed reference x ,y , z . p v p , j rp' p Physically, i t would correspond to a single electromagnet rotating about the axis of symmetry or two (or more) fix e d electromagnets with variable currents. The moments generated are given by, QY = ~ u c m P j B i s e c 3 / d + e c o s e ) (4.9a) Q g = UC mp kB i/(l+ecos9) (4.9b) QAm = u(C m/JI)(PjB k-p kBj)/(l+ecose) (4.9c) where 115 Figure 4.2 Magnetic-solar c o n t r o l l e r configurations 116 Pj = cos(6 m+A), p k = sin(0 m+X) (4.9d) o A constant dipole l e v e l h , leading to a constant value of the magnetic parameter C , i s assumed i n the present analysis. The components of the t o t a l transverse torque are controlled according to the r e l a t i o n s : sgn(Q^cosB) = -sgn (4.10a) sgn Q Q = -sgn S D (4.10b) P P Q cosB/Qg = S /S p (4.10c) where the switching functions S ^ ( i = y,B) are defined as S i = i'+ m ( i - i f ) , i = y,3 (4.11) and m represents the system gain. This results i n a simple control law for the dipole angle 8^ (Figure 4.2), ) m = -A-tan _ 1(S 3/S Y) (4.12) Note that no p o l a r i t y reversals are required i f the angle 6^ i s permitted any values i n the range 0 to 2TT . However, i t would be desirable to r e s t r i c t 0 to the range m - T T / 2 to T T/2 (for the gimballed electromagnet) and permit p o l a r i t y reversals, which leads to the controls: 117 3m = ~ t a n _ 1 { ( S 3 + S Y t a n A ) / ( S Y - S g t a n X ) } (4.13a) U = sgn{(S^cosX-S^sinX)/B ± > (4.13b) where the p r i n c i p a l value of the arctan function i s to be admitted. Model B consists of two mutually perpendicular dipoles with orientations p^ and p 2 fixed i n the platform and allowed p o l a r i t y reversals (Figure 4.2). Substituting for the unit vectors p^ and p 2 i n Equations (4.3), the magnetic control moments become, Q = -(U p +U„p„.)C B.secB/(l+ecos0) (4.14a) Q 8 = ( U l p l k + U 2 p 2 k ) C m B i / ( 1 + e c o s 9 ) (4.14b) QXm = { U l ( p l j B k - p l k B j ) + U 2 ( p 2 j B k - p 2 k B j ) } x (C m/JI)/(1+ecose) (4.14c) where p i j = p2k = c o s ( e m + X ) , p l k = - p 2 j = s i n ( e m + X ) (4.14d) 118 Although the bang-bang c o n t r o l l e r can no longer d i s t r i b u t e the t o t a l transverse moment between the r o l l and yaw degrees of freedom i n proportion to the demands governed by the switching functions, appropriate signs of Q and (Equations 4.10a and b) may be achieved with the p o l a r i t y controls: for I tan(9 + X)|<1, U, = sgn(S /B.cos(9 +X)} 1 m 1 1 3 Y i m U 0 = -sgn{S„/B.cos(9 + A) } (4.15a) 2 p i m for|tan(9 M+A)|>1, = -sgnCS^/B^in (9 M+X) } U 0 = 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 i s a constant i n this case. For the f i n a l p itch 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 < T T / 4 , the controls reduce to: U 1 = sgn(S /B.^, U 2 =- sgn (Sg/B^ (4.16) 4.1.3 Solar pitch control The platform pitch control i s accomplished using a solar pressure c o n t r o l l e r consisting of plates PP which are allowed a rotation 6 about the axis of symmetry of the s a t e l l i t e (Figure 4.2). For highly r e f l e c t i v e plates, the pure pi t c h moment r e s u l t i n g from solar radiation pressure i s (Equation 3.12c), Q X s = -C{(1+e) 3/(l+ecose) 4}|-UjSin(6+A) +u kcos(S+A)|{-UjSin (6+A)+u kcos (6+A)} (4.17) where C = ( 4 p p o R 3 / u I x p ) A E (4.18) Using the control r e l a t i o n Q, = -|Q, I sgn S, (4.19) As 1 As'max ^ A with S A = A'+m(A-Af), (4.20) the control law for the plate rotation 6 becomes, for u_.>0, 6 = t a n ~ 1 ( u k / u j ) - A - ( 7 r / 2 ) s g n S x for u_.<0, <5 = T r + t a n - 1 (u k/Uj)-A-(TT/2) sgn S A 120 for U j = 0, 6 = cos _ 1{-sgn(u kS A)} (4.21) where the p r i n c i p a l value of the arctan function i s to be introduced. The t o t a l generalized force i n the pitch degree of freedom i s then given by, Q A = Q A m + Q A s <4'22> It may be pointed out here that the components of the unit vector i n the d i r e c t i o n of the sun involve o r b i t a l parameters Q, i and ui referred to the e c l i p t i c plane while the formulation of the magnetic c o n t r o l l e r requires the same angles referred to the earth's equatorial plane. The relations governing these parameters for an a r b i t r a r y o r b i t a l plane are rea d i l y obtained as: sinQ, s i n i = s i n ^ s i n i (4.23a) m m cosfi cosw -sinfi sino) cosi = cos^cosoj-sinfisincocosi m m m m m (4.23b) cosfi sinu) +sin^ cosw cosi = cosfisinco+sinficosucosi m m m m m (4.23c) 121 4.2 R e s u l t s and d i s c u s s i o n The response o f the system was s t u d i e d by numeric- a l l y i n t e g r a t i n g the equations of motion (4.1) along w i t h the c o n t r o l r e l a t i o n s governing the magnetic and s o l a r moments, i . e . , Equations (4.9, 4.13, 4.21 and 4.22) and Equations (4.14, 4.15, 4.21 and 4.22) f o r models A and B, r e s p e c t i v e l y . The Adams-Bashforth p r e d i c t o r - c o r r e c t o r quadrature w i t h the Runge-Kutta s t a r t e r was used w i t h a step s i z e of 0.1°. The important system parameters were v a r i e d g r a d u a l l y over the range of i n t e r e s t and the c o n t r o l l e r performance e v a l u a t e d both i n c i r c u l a r and e l l i p t i c o r b i t s . The amount of i n f o r m a t i o n thus generated i s r a t h e r e x t e n s i v e ; however, f o r c o n c i s e n e s s , o n l y the t y p i c a l r e s u l t s s u f f i c i e n t to e s t a b l i s h trends are presented here. 4.2.1 N u t a t i o n damping F i g u r e 4.3 summarizes the performance of the proposed m a g n e t i c - s o l a r h y b r i d c o n t r o l l e r i n damping the n u t a t i o n a l motion of the s a t e l l i t e s p i n a x i s and the p i t c h o s c i l l a t i o n of the p l a t f o r m . I t shows the v a r i a t i o n of the damping time w i t h the c o n t r o l l e r g a i n m f o r a v a r i e t y of combinations of the system parameters and i n i t i a l c o n d i t i o n s . Here the damping time i s d e f i n e d as the time r e q u i r e d f o r a l l the three degrees of freedom; namely, the r o l l y, yaw 8 and the p l a t f o r m p i t c h A , to s e t t l e w i t h i n 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° i f = 0 30 20 10 0 30 Model A C m =i,l = i,a=o,j = o.i C , = 2,1 = 0 . 1 - 1 , 0 = 0 - 1 0 , 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,| = i f a = o fj =o.i ± ± ( B) o 10 40 20 30 m Figure 4.3 Optimization plots for the magnetic-solar c o n t r o l l e r gain m 50 123 0.1° of the desired f i n a l orientation. The responses of both magnetic c o n t r o l l e r models A and B are presented, which, i n general, indicate the existence of an optimum value of the system gain m leading to the minimum time of damping, and the s e n s i t i v i t y of the optimum to the system parameters and i n i t i a l conditions. The influence of the s a t e l l i t e i n e r t i a and spin parameters on the attitude dynamics of passively s t a b i l i z e d s a t e l l i t e s i s well recognized. Their e f f e c t on the nutation damping performance of the present controlled system, however, appears to be n e g l i g i b l e . This i s indicated by curve (a) i n Figure 4.3A where the re s u l t s for p e n c i l - l i k e to spherical mass d i s t r i b u t i o n s (I = 0.1 to 1) and non- spinning to moderate spin rates (a = 0 to 10) were v i r t u a l l y indistinguishable. The banding together of curves (a,b and f) i n Figure 4.3B r e f l e c t s s i m i l a r system behaviour. The i n s e n s i t i v i t y of the transient response to I and moderate values of a i s understandable as these parameters, con- t r i b u t i n g restoring forces to the system, are now largely provided for by the c o n t r o l l e r gain m. The performance under high spin rates i s investigated i n a l a t e r section. The e f f e c t of the magnetic parameter C , character- i z i n g the magnitude of the magnetic control torques, i s indicated by a comparison of curves (b,a and c) i n Figure 4.3A for the case of the single rotatable dipole. As anticipated, increasing the value of C leads to a corres- ponding reduction i n the damping time T^. Larger values of the magnetic parameter (curves a and c) show a response pattern that i s r e l a t i v e l y i n s e n s i t i v e to the c o n t r o l l e r gain m, thus in d i c a t i n g a large range of values of the l a t t e r to provide near-optimum damping. I t may, however, be pointed out that the maximum attainable value of C i s * m subject to constraints imposed by the electromagnet weight and power requirement. Curves (c,a and d) indicate s i m i l a r e f f e c t 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 pi t c h (A) degree of freedom. However, even with a value of C = 2, the p i t c h motion, i n 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) i n Figure 4.3A and curves (d and e) i n Figure 4.3B shows the e f f e c t of i n i t i a l conditions on the performance of c o n t r o l l e r models A and B, respectively. In both cases, an increased impulsive disturbance not only leads to an anticipated increase i n the damping time but also renders the response quite sensitive to the system gain m. With larger values of C , however, the l a t t e r e f f e c t was found to be less m pronounced. 125 I t i s of in t e r e s t to compare the performance of the two magnetic c o n t r o l l e r models i n damping the attitude motion. A suitable c r i t e r i o n for the comparison i s the same t o t a l electromagnet weight and power consumption which requires the value of the magnetic parameter C m for model A to be twice the value for model B. Comparisons may be made of curves (b, a and c) i n Figure 4.3A with curves (c, a and d) i n Figure 4.3B, respectively, towards t h i s end. The results c l e a r l y indicate a better performance of the single rotatable dipole model A. This i s explained by the a b i l i t y of t h i s model to d i s t r i b u t e the t o t a l magnetic torque between the r o l l and yaw degrees of freedom i n proportion to t h e i r 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 i n the system r e l i a b i l i t y due to physical movement of the electromagnet. Alternately, two fixed electromagnets with variable currents may be employed. Unfortunately, t h i s would require each e l e c t r o - magnet to be the same size as the single rotating one i n order to maintain the same resultant dipole strength as the l a t t e r , thus doubling the t o t a l 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 r e l i a b i l i t y , associated hardware and software, and performance requirements. Typical damped responses of a s a t e l l i t e subjected to an extremely severe impulsive disturbance are shown in Figure 4.4 for both c o n t r o l l e r models. The time history of the controls i s also included. Figure 4.4a shows the response i n a c i r c u l a r o r b i t 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 s a t e l l i t e i n an eccentric o r b i t i s shown i n Figure 4.4b, which indicates a s l i g h t increase i n the maximum amplitude and the damping time. The effectiveness of the c o n t r o l l e r s in damping such a severe impulsive disturbance i n a few degrees of the s a t e l l i t e ' s o r b i t a l t r a v e l i s thus apparent. The amplitudes are also limited to a few degrees. The controls, i n general, require rather infrequent switching u n t i l the corresponding amplitudes become very small and chatter i n i t i a t e s . This may, however, be prevented by the inclusion of suitable deadbands i n the control relations which would depend on the pointing accuracies required. The effectiveness of the control system i n capturing a s a t e l l i t e from i n i t i a l r o l l , yaw and pit c h errors i s presented i n Figure 4.5. Both c i r c u l a r and eccentric o r b i t as well as nonspinning and moderate spin rate cases are considered. The i n i t i a l error of 20° i n 127 1 = 1 I - 0 m m = i o X = rl=A=0 Y 0 = 45 i =23.5° A m = n = 0 Y0=(30=X = 0.3 C = 2 0=10° w = w r 3 0 ° Y=B=X=0 - i n d i c a t e s i n i t i a t i o n of c h a t t e r e = o a = l J = 0.5 K = o . o i 1 1 1 1 1 v M o d e l A \ M o d e l B V \ C =2 \ \ C = l 3 i - l 25 50 0 25 0 - d e g r e e s 50 75 Figure 4.4 Damped response and time h i s t o r y of c o n t r o l s subsequent to i n i t i a l impulsive disturbance 128 1 = 1 ' r r T 0 m = 5 Y0 = P0 = A0=20° Y° r - 2 f l ° I = i o - 5 0 n m = n = 0 - * t ~ - * ' z ° — - P ° 0 - 2 Cp - 1 0 m o m w m = L u = 3 0 t 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 h i s t o r y of c o n t r o l s subsequent to i n i t i a l p o s i t i o n disturbance 129 each degree o f freedom i s c o r r e c t e d i n approximately one- t h i r d o f an o r b i t w i t h model A and h a l f an o r b i t w i t h model B. The performance may be improved f u r t h e r through an optimum ch o i c e o f the c o n t r o l l e r g a i n . 4.2.2 High s p i n r a t e s and s p i n decay The d i s c u s s i o n so f a r p e r t a i n s t o s a t e l l i t e s which are e i t h e r nonspinning or have moderate s p i n r a t e s . The e f f e c t i v e n e s s o f the c o n t r o l system (model B) f o r s a t e l l i t e s w ith h i g h s p i n r a t e s i s i n d i c a t e d by the response data presented i n Table 1, which shows the coning amplitude o f the s p i n a x i s , $ m a x * and the damping time x^. The coning amplitude $ m a x reached i n the absence of the c o n t r o l l e r i s a l s o p r e sented. For low s p i n r a t e s , the c o n t r o l system, i n a d d i t i o n to p r o v i d i n g quick n u t a t i o n damping, helps keep the amplitudes low. However, f o r h i g h s p i n r a t e s , i t e s s e n t i a l l y a c t s as a damper. A comparison of the c o n t r o l l e r performance w i t h m = 10 and m = 30 shows i t s e f f e c t on the amplitude to be s l i g h t but the damping time i s a f f e c t e d a p p r e c i a b l y , up t o s p i n parameter values as h i g h as a = 200. These o b s e r v a t i o n s s u b s t a n t i a t e the e a r l i e r c o n c l u s i o n r e g a r d - i n g the c o n t r o l l e r g a i n l e n d i n g s t i f f n e s s t o the system. For s t i l l h i g h e r s p i n r a t e s (a > 200), however, the i n f l u e n c e of m i s n e g l i g i b l e i n d i c a t i n g the dominance o f the g y r o s c o p i c r e s t o r i n g f o r c e s . 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° Y 1 = B 1 = 0.3 m o o 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 T d 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 a n a l y s i s so f a r cons ide red the r o t o r (body I) to have a constant average s p i n r a t e . Apparen t ly t h i s would be achieved through some a c t i v e energy source compen- s a t i n g fo r r o t o r s p i n decay due to bea r ing l o s s e s . Even i n the absence o f such energy supp ly , the a n a l y s i s cont inues to be v a l i d p rov ided the s p i n parameter o remains s e n s i b l y i n v a r i a n t over a time i n t e r v a l of the order o f the damping times a t t a i n e d here . T h i s i s o f c o n s i d e r a b l e va lue as any s p i n decay i s l i k e l y to occur very s l o w l y indeed . Thus the data presented i n Table 1 can a l s o be used to p r e d i c t the long range performance o f a s p i n n i n g s a t e l l i t e . The c o n t r o l l e r ' s e f f e c t i v e n e s s i n a c h i e v i n g qu ick n u t a t i o n damping even i n the absence o f any s p i n promises an i nc rea sed s a t e l l i t e l i f e - s p a n . 4 .2 .3 A t t i t u d e c o n t r o l The c o n t r o l system o f f e r s the e x c i t i n g p o s s i b i l i t y o f s t a b i l i z i n g the s a t e l l i t e a long any a r b i t r a r y o r i e n t a t i o n i n space, thus enab l ing i t t o undertake d i v e r s e m i s s i o n s . Th i s may be achieved through the parameters y ^ , 3^ and A^, d e f i n i n g the f i n a l d e s i r e d o r i e n t a t i o n , which are i nco rpo ra t ed i n the s w i t c h i n g func t ions S^, and s^, r e s p e c t i v e l y . F i g u r e 4.6 shows the a b i l i t y of the c o n t r o l l e r i n a c h i e v i n g a v a r i e t y of s p a t i a l o r i e n t a t i o n s , the time taken be ing w e l l w i t h i n an o r b i t . Note tha t a f a i r l y s m a l l va lue o f the ga in m used here leads to a smooth t r a n s i t i o n between w i d e l y 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 Mode l A C m = 4 M o d e l B C m = 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 = - i o ° e = 0.2 a = i / • ^ " { ^ - * * ° PF=30° ' 0 / X ^ - i o \ 0 0.5 0 0.5 1.0 O r b i t s Figure 4.6 E f f e c t i v e n e s s of the magnetic-solar c o n t r o l l e r i n imparting a r b i t r a r y o r i e n t a t i o n s to the s a t e l l i t e 133 d i f f e r e n t attitudes. On the other hand, larger values of m were found to r e s u l t i n an undesirable overshoot of the f i n a l orientation. I t may be pointed out further that except for f i n a l orientations along equilibrium points of the system, the co n t r o l l e r must at a l l 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 s u f f i c i e n t for both nonspinning and a nominal spin rate i n c i r c u l a r as well as eccentric orbits (Figure 4.6). 4.2.4 I l l u s t r a t i v e example I t appears i n t e r e s t i n g to evaluate the performance of the magnetic-solar c o n t r o l l e r through a preliminary attitude dynamics study of two well-known s a t e l l i t e s , the proposed Canadian Communications Technology S a t e l l i t e (CTS) and Anik, when provided with the proposed control system. For convenience the l a t t e r may be considered as nonspinning which only represents an adverse s i t u a t i o n . Values of the magnetic parameter c m = 4 for model A and 2 for model B are attainable with t o t a l dipole le v e l s of approximately 2 200 and 20 ampere-meter for the CTS and Anik, respectively. 2 A control plate area A = 0.5 f t and moment arm e = 5 f t y i e l d 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 i s equivalent to C = 2. An impulsive disturbance of 0.1 i s applied i n a l l the three degrees of freedom simultaneously which i s i n excess of that imparted by micrometeorite impacts over 24 hrs, and thus represents an enormous magnification of the r e a l s i t u a t i o n . As the i n e r t i a parameter (I - 0.1 for CTS, 1 for Anik) did not a f f e c t the performance s i g n i f i c a n t l y , most of the r e s u l t s presented i n thi s chapter are representative of these s a t e l l i t e s . I t i s apparent (Figure 4.3) that the cont r o l l e r s are able to damp out such a severe disturbance i n about 3° and 5° of the o r b i t with models A and B, respec- t i v e l y . The maximum deviation from the o r b i t normal attitude also remained less than 0.2°. Any ar b i t r a r y orientations may be imparted to the s a t e l l i t e s i n well within an o r b i t a l period (Figure 4.6). F i n a l l y , i t may be mentioned that the comment con- cerning the earth shadow made e a r l i e r also applies here. The results showed the c o n t r o l l e r performance to remain v i r t u a l l y unaffected. Furthermore, the analysis ignores, dynamics due to r e l a t i v e motion of the gimballed e l e c t r o - magnet (model A) and the solar control plates as well as the shadowing of the l a t t e r by the s a t e l l i t e . 135 4.3 Concluding Remarks The conclusions based on the analysis may be summarized as follows: (i) The analysis c l e a r l y establishes the pote n t i a l of a magnetic-solar hybrid system for nutation damping and attitude control of s a t e l l i t e s , ( i i ) The concept permits interchangeability of the solar c o n t r o l l e r and a variable speed p i t c h momentum wheel, thus e f f e c t i v e l y providing a gyromagnetic control system, ( i i i ) The a b i l i t y of the system i n damping extremely severe disturbances i n a few degrees of the o r b i t makes i t quite suitable for applications, such as, communications s a t e l l i t e s , (iv) Even with rotor spin decay, the system continues to function e f f e c t i v e l y which promises an increased s a t e l l i t e l i f e - s p a n . (v) I t i s possible for a s a t e l l i t e to a t t a i n any arb i t r a r y orientation i n space, both i n c i r c u l a r and e l l i p t i c o r b i t s , thus widening the scope of i t s mission. 136 5. AERODYNAMIC-SOLAR HYBRID ATTITUDE CONTROL The analyses i n the e a r l i e r chapters apply primarily to s a t e l l i t e s i n high a l t i t u d e o r b i t s since the influence of the earth's atmosphere was ignored. However, several space applications, such as, weather forecasting, earth resources exploration, m i l i t a r y scouting, etc., depend on high resolution photography and hence necessarily negotiate near-earth t r a j e c t o r i e s . Unfortunately, i n t e r a c t i o n with free molecular reaction forces reduces t h e i r e f f e c t i v e l i f e - s p a n leading to expensive periodic replacement. One possible solution would be to employ an e l l i p t i c trajectory making a space vehicle to spend major portion of i t s o r b i t a l period above the atmosphere and dip into i t only when a c t i v e l y engaged i n i t s mission. This presents an in t e r e s t i n g s i t u a t i o n where aerodynamic forces may be used to advantage for attitude control, possibly i n conjunction with solar radiation pressure. The present chapter explores the f e a s i b i l i t y 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 i s evaluated. 5.1 Formulation of the Problem 5.1.1 Equations of motion The case of a nonspinning s a t e l l i t e i s considered here, which, i n view of the analysis i n Chapter 4, represents an adverse s i t u a t i o n . Equations (3.20) therefore apply where Q^(i = Y,B,A.) now represent the t o t a l generalized forces due to the solar pressure and the aerodynamic forces: y" -26* (y 1 tanB-cosy)-I(A'-y 1sinB+cosBcosy) (B'-siny)secB + {3 (1-1)/(l+ecos6)-l}sinYcosy-{2esin0/(l+ecos6)} x (y'+cosytanB) = Q y (5.1a) B" -Y'cosy-{2esine/(l+ecose)}(B'-siny)+{ I(X'-y 1sinB+cosBcosy) +(y 1sinB-cosBcosy)}(y 1cosB+cosysinB)-3{(1-1)/(l+ecos9)} x . 2 s m ysxnBcosB = (5.1b) X" -y" sinB-{2esin9/(l+ecos6)}(X'-y'sinB+cosBcosy) -B'Y 1cos3 _Y'cosBsiny-B 1cosysinB = (5.1c) 5.1.2 Controller configuration and generalized forces The c o n t r o l l e r configuration studied for e s t a b l i s h - ing the f e a s i b i l i t y of solar pressure control i s considered 138 f o r t h e h y b r i d a e r o d y n a m i c - s o l a r c o n t r o l a s w e l l ( F i g u r e 5 . 1 ) . T h e g e n e r a l i z e d f o r c e s due t o S o l a r r a d i a t i o n p r e s s u r e w e r e f o u n d e a r l i e r ( E q u a t i o n s 3 . 1 2 ) a s : Q ŝ = - E (0) [L T 1 C 1 1 c o s ? 1 1 c o s ? 1 s i n 6 1 s i n A + U2C2 I cos?2 I c o s?2 s :'- n^2 C O S^'' s e c ^ ( 5 . 2 a ) Qgs = - E(0) [U 1 C 1 1 c o s ? 1 1 c o s ? 1 s i n 6 1 c o s A - U 2 C 2 I c o s ? 2 I c o s ? 2 s i n ( 5 2 s i n ^ J ( 5 . 2 b ) Q, = - E ( 0 ) C _ c o s ? J c o s ? _ (5.2c) A S J J -J where U. = +1 f o r P., -1 f o r P!; i = 1,2 1 1 1 cos?, = u. cos S , + (u . s i n 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 s o l a r parameters, C \ , are given b y , c i = ( 4 p P 0 R p / M l y ) A i e i ' 1 = 1 , 2 X i Figure 5.1 Aerodynamic-solar hybrid c o n t r o l l e r configuration 140 C 3 = ( 4 p P o R p / y I x ) A 3 £ 3 ( 5 ' 4 ) The r e s u l t a n t free molecular force on an elemental 8 2 area, c o n s i d e r i n g specular r e f l e c t i o n , may be expressed as , dF = (1/2)p V 2dA|cos£ |{(C -C Tcosec£ |cos£ |)v d a. a. U Lt 3. 3. + (C Tcosec? sgncos? )n} (5.5) ij a a E v a l u a t i n g the t o t a l moment about the s a t e l l i t e center of mass due to the aerodynamic forces and using the p r i n c i p l e of v i r t u a l work leads t o : Q y a = J ( 0 ) [ U l C a l l C o s 5 a l l { v j - ( C L / C D ) c o s e c ? a l ( v j l C O S ? a l l -sin6 1sinAsgncos? a-])}+U 2C a 2 | c o s C a 2 I ^ v j ~ ^ L ^ D ^ c o s e c ^ a 2 ( V j | c o s ? a 2 | -sin6 2cosAsgncosC a 2) ̂ - c a 3 I c o s ^ a 3 ^ ^ C l / C D ^ cosec?^^|cos? a 3|}v^cos(6^+A)]sec3 (5.6a) S a = - J ( 9 ) [ U l C a l l c o s ? a l l { v k - ( C L / C D ) c o s e c ? a l ( v k l c o s ? a l l t s i n S ^ c o s A s g n c o s ^ ) } + u 2 c a 2 I c o s ^ a 2 I ̂ vk~ ^ C l / C D ^ c o s e c ^ a 2 (v k | c o s ? a 2 | -sin6 2sinAsgncos? a 2) }~Ca3 | c o s C a 3 | (1- (C^/C^) x c o s e c ? a 3 | c o s C a 3 | } v ^ s i n ( 6 3 + A ) ] (5.6b) 141 QXa = J ( e ) C a 3 [ l C O S ? a 3 l + ( C L / C D ) s i n ? a 3 l c O S ? a 3 ( 5 ' 6 c ) where U. = +1 for P., -1 for P!; i = 1,2 l i i cos? , = v. cosS, + (v . sinA-v. cosA) sin6, a l I 1 j k 1 cos? a2 = v^cos62 +( vj c o sA+v^sinA)sin62 cos? _ = -v . s i n ( 6 +A)+v, cos (6 + A) (5.7) The aerodynamic parameters, are defined as C . = (cr p R 2/I )A.e., i = 1,2 a i DKap p/ y' I i ' ' Ca3 = < CD pa E P p / I x , A3 e3 ( 5 ' 8 ) The v a r i a t i o n of the atmospheric density with a l t i t u d e , incor- porated i n the d e f i n i t i o n of the function J ( 0 ) , i s modelled according to the r e l a t i o n p = p {(R-R )/(R -R ) } n (5.9) a ap e p e where n varies i n the range -5 to -7 depending on the a l t i t u d e ^ 3 . The t o t a l 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 = Q Y S ( U l ' U 2 ' 6 l ' 6 2 , + Q Y a { U l ' U 2 ' 6 l ' 6 2 ' 6 3 ) Q3 = Q B s ( U l ' U 2 ' 6 l ' 6 2 ) + Q 3 a ( U l ' U 2 ' 6 l ' 6 2 ' 6 3 ) °A = Q A s ( 6 3 ) + Q A a ( 6 3 ) In view of Equations (5.2 and 5.6), the dependence of the moments on the control variables i s quite complicated. However, i t i s possible to obtain r e l a t i v e l y simple a n a l y t i c a l 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 a i order to u t i l i z e the maximum moments available from the co n t r o l l e r , the following bang-bang control law i s employed here: Q. = -|Q.I sgn S. (5.11) l 1 l'max ^ l where the switching functions S. are defined as S i = i ' + m ( i - i f ) , i = Y,3,A (5.12) (5.10a) (5.10b) (5.10c) and m represents the system gain. The d e t e r m i n a t i o n o f t h e s i m u l t a n e o u s m a x i m a , |Q.I , t h a t t h e c o n t r o l l e r c a n p r o v i d e , i s r a t h e r i n v o l v e d 1 1 max c The p r o b l e m may be s i m p l i f i e d c o n s i d e r a b l y b y r e c o g n i z i n g t h a t t h e s o l a r p r e s s u r e a n d t h e a e r o d y n a m i c f o r c e s r e p r e s e n t t h e r e s p e c t i v e d o m i n a n t i n f l u e n c e s a t h i g h a n d l o w a l t i t u d e s O n l y o v e r a s m a l l p o r t i o n o f a n e l l i p t i c t r a j e c t o r y a r e t h e two e f f e c t s c o m p a r a b l e i n m a g n i t u d e . The p l a t e r o t a t i o n s 6^, p r o d u c i n g t h e maximum m o m e n t s , may t h u s be o b t a i n e d i n a c c o r d a n c e w i t h t h e s o l a r p r e s s u r e a t h i g h a l t i t u d e s a n d a e r o d y n a m i c f o r c e s a t l o w a l t i t u d e s . The s w i t c h - o v e r p o i n t c a n be d e t e r m i n e d b y c o m p a r i n g t h e m a g n i t u d e s o f t h e t w o f o r c e s . 5 . 2 . 1 H i g h a l t i t u d e M a x i m i z a t i o n o f t h e s o l a r p i t c h moment , Q, , l e a d s AS t o t h e f o l l o w i n g c o n t r o l l a w f o r 6_: f o r u . > 0 , 6_ = t a n (u, / u .) - A-(TT/2) s g n S, J JS K J A f o r U j < 0 , 6 3 s = T T + t a n - 1 ( u k/Uj )-A- ( T T / 2 ) s g n f o r u . = 0 , 6 3 s = c o s "'"{sgn (u^S )̂ }-A ( 5 . 1 3 ) w h e r e t h e p r i n c i p a l v a l u e o f t h e a r c t a h f u n c t i o n i s t o be a d m i t t e d . The s o l a r r o l l - y a w c o n t r o l moments Q and Q„ J ys 3s b e i n g c o u p l e d t h r o u g h t h e r o t a t i o n s 6^ a n d 6^, t h e t o t a l 144 2 2 2 1/2 transverse torque, (Q cos $+QR ) ' , i s maximized which occurs at: 6, = ff/2+tan-1[ (3/2) (u.sinA-u, cosA)/u.±{ (9/4) x (u^sinX-Uj^cosA) 2/u 2+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 +u ksinA) 2/u 2+2} 1 / 2] (5.14b) ± for {(u.cosA+u, sinA)/u.} > 0. ~\ K 1 In general, a l l sign combinations of Q and Q R 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 s i t u a t i o n s , the ys |3s J roll-yaw control plates are to be turned ' o f f , which may be done either by making the corresponding 6^ = 0 or, by aligning the plates p a r a l l e l to the incident radiation, i . e . , choosing 6^ so as to render the corresponding cos?^ = 0. The control moments at high a l t i t u d e thus become, 145 QY = Q Y s ( U l ' U 2 ' 6 l s ' 6 2 s ) + Q Y a ( U l ' U 2 ' 6 l a ' 6 2 s ' 6 3 s ) QB = Q 0 s ( U l ' U 2 ' 6 l a ' 6 2 s , + Q 3 a ( U l ' U 2 ' 6 l s ' 6 2 s ' 6 3 s > Q X = Q x s ( 6 3 s , + Q X a ( 6 3 s ) 5.2.2 Low a l t i t u d e In a manner s i m i l a r to the solar control at high a l t i t u d e , the moments due to the aerodynamic forces may be maximized at low a l t i t u d e . The exact determination of the c r i t i c a l 6^ being more complicated i n this case, simplifying assumptions such as | v^| , |v_.|<<|vk| and C L/C D<< 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 pi t c h control torque Q, , i s 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 p r i n c i p a l value of the arctan function i s to be introduced. (5.15a) (5.15b) (5.15c) 146 The aerodynamic roll-yaw moment i s maximized by the rotations 6^ and 6 2 given by, S, = TT/2-tan "*"{v. / (v . sinA-v, cosA) } (5.17a) l a l j k <5 = T T/2-tan _ 1{v. / (v .cosA+v, sinA) } (5.17b) ^a I j K An investigation of Equations (5.6) indicates that, i n highly eccentric o r b i t s , the drag component of the aero- dynamic force governs the directions of both the r o l l and the yaw moments. As a r e s u l t , a l l sign combinations of C) and QD , i n general, are not available. This presents an ya pa option as to c o n t r o l l i n g either the r o l l or the yaw moment and retaining the associated torques i n the other degree of freedom. Recognizing that \Qa | >> | Q | for small yaw angles pa y a B, a large proportion of the t o t a l transverse torque may be u t i l i z e d by selecting the control plates according to the sign required of Qg. A preliminary investigation of the system performance, however, revealed the e f f e c t of the coupled r o l l moment Q to be generally adverse. The roll-yaw control law i n the aerodynamic region i s , therefore, modified to exercise control action only over those portions of the trajectory where the associated r o l l moment Qya i s also of the correct sign, the c o n t r o l l e r being switched ' o f f other- wise. The l a t t e r may be accomplished by making a l l the r o l l - yaw control plates p a r a l l e l to each other or to the flow 147 d i r e c t i o n (by choosing 6^ so as to render the corresponding cos? • = 0). a i The control moments at low a l t i t u d e thus take the form: Q y ' Q y s ^ l ' ^ ^ l a ' ^ a ^ V ^ l ' ^ ' ^ a ' ^ a ' S a ) ( 5 ' 1 8 a ) Q B = Q B s ( U l ' U 2 ' 6 l a ' 6 2 a , + Q B a ( U l ' U 2 ' 6 l a ' 6 2 a ' 6 3 a ) ( 5 - 1 8 b ) Q X = QXs"< 63a , + QXa ( 63a ) ( 5 - 1 8 c ) The control procedure may be summarized as follows: (i) sense the r o l l , yaw and p i t c h angles and rates, o r b i t a l position and the solar aspect angle. Estimate the atmospheric density, ( i i ) determine the switch-over point by comparing the magnitudes of the solar and aerodynamic parameters. It may change due to variations i n the atmospheric density. ( i i i ) for pitch control, provide rotation 6̂  determined from Equations (5.13) and (5.16) at high and low a l t i t u d e s , respectively, (iv) for roll-yaw control at high a l t i t u d e , compute 6^(i = 1,2) from Equations (5.14) and provide these rotations to the sets of plates r e s u l t i n g i n the signs of Q v a and Q R c governed by Equation 148 (5.11). I f the proper signs are not available, turn the roll-yaw control plates ' o f f 1 . At low a l t i t u d e , compute 6 ^ ( i = 1,2) from Equations (5.17) and determine the plate sets y i e l d i n g maximum Q^a of the proper sign. I f the associated Q i s of the correct sign, provide these rotations; otherwise, turn the roll-yaw control plates ' o f f . 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 s t a r t e r was used. A step size of 0.1° at high alt i t u d e s and 0.02° at low a l t i t u d e s gave results of s u f f i c i e n t accuracy. The important system parameters were varied gradually over the range of i n t e r e s t and the c o n t r o l l e r performance evaluated. For conservative estimate of the co n t r o l l e r ' s performance, i t was purposely subjected to severe disturbances. I t should be pointed out here that the atmospheric density depends, i n addition to the a l t i t u d e , on several classes of solar and geophysical phenomena. In the present analysis, density variations due to the l a t t e r over a few o r b i t s of the s a t e l l i t e are ignored and a reference atmos- phere corresponding to an exospheric temperature of 1250°K i s 149 84 -14 3 considered . The value of p = 0.74 x 10 gm/cm at cl the perigee a l t i t u d e h = 250 miles and the drag c o e f f i c i e n t P = 2.2 y i e l d the r a t i o C ./C. - 50. The switch-over D J a i 1 point i s found to be at an a l t i t u d e of approximately 500 miles. 5.3.1 L i b r a t i o n Damping The performance of the c o n t r o l l e r i n damping the l i b r a t i o n a l motion of the s a t e l l i t e i s summarized i n Figure 5.2 i n the form of optimization plots for the c o n t r o l l e r gain m. The damping time x^, defined as the time taken for a l l the three l i b r a t i o n angles to s e t t l e within 1° of the desired orientation, i s presented as a function of the system gain. Various combinations of the important system parameters and i n i t i a l conditions are considered. The p l o t s , i n general, indicate the existence of an optimum value of the system gain r e s u l t i n g i n the minimum s e t t l i n g time. The influence of the s a t e l l i t e i n e r t i a parameter I on the c o n t r o l l e r performance i s indicated by a comparison of curves (a) and (b). A reduction i n the damping time with an increased value of I i s apparent. Curves ( f ) , (c) and (g) exhibit a si m i l a r trend for the case of a larger C^(=l). The zero s e t t l i n g time i n curve (g) simply implies that, with this set of parameters, none of the attitude angles exceeded 1° i n amplitude. The advantage of d i s c - l i k e s a t e l l i t e mass d i s t r i b u t i o n (I > 1) i s thus obvious. 150 e =o.i 0 = 4 5 ° C L / C D = o.i hp=25o mi. Y0=P0=X=o CO = 0 i = 23 . 5 ° C g j / C ^ s o j=Y,p,X Y f = pf=Af=o 0 5 10 15 20 m Figure 5.2 Optimi z a t i o n p l o t s f o r the aerodynamic-solar c o n t r o l l e r gain m 151 The e f f e c t of the solar parameters C\ and the aerodynamic parameters C ^, which are d i r e c t l y related for a given plate s i z e and moment arm, i s shown by curves (a) and (c), and (b) and (g). Increasing the value of C\ leads to a substantial reduction i n the damping time. This, of course, can be anticipated as C\ characterize the magnitudes of the control moments available. Their maximum attainable values i n practice, however, are constrained due to considera- tions such as launch, deployment and operation. I t i s inte r e s t i n g to determine the physical size of the c o n t r o l l e r y i e l d i n g a given C. (or C . ) . For example, consider a x ax s a t e l l i t e with the mass properties of the Canadian communi- cations s a t e l l i t e Anik (I - 1). A value of C. = 1 (C . = 50) x ax i s found to be attainable with plate sizes of about 5' x 5' and moment arms = 10 f t for a perigee a l t i t u d e h^ = 250 miles. A comparison of curves (c), (d) and (e) indicates the system performance as affected by the i n i t i a l impulsive disturbance. As expected, an increased disturbance implies larger damping times. In addition, t h i s results i n a l e f t - ward s h i f t 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 i n i t i a l conditions, a value of m would have to be selected that promises reasonably good l i b r a t i o n damping rates for a l l i n i t i a l conditions that the s a t e l l i t e i s l i k e l y to encounter 152 i n i t s normal operation. This does not appear to be d i f f i c u l t 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 y i e l d near-optimum performance. Typical damped responses of the s a t e l l i t e are presented i n Figure 5.3, with severe i n i t i a l disturbances applied at d i f f e r e n t positions of an e l l i p t i c t r ajectory. Figure 5.3a shows the damping of a disturbance encountered at the pericenter. The large aerodynamic yaw and p i t c h moments r e s t r i c t the corresponding amplitudes to a n e g l i g i b l e value. On the other hand, the s a t e l l i t e executes a small r o l l o s c i l l a t i o n r e s u l t i n g from the r e l a t i v e l y smaller r o l l control torques and occasional loss of roll-yaw control at low a l t i t u d e . Note the small hump i n yaw due to a reduction of control moments i n 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 s a t e l l i t e 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 l i b r a t i o n damping results with the attitude errors s e t t l i n g within 1° i n about 30° of the s a t e l l i t e ' s o r b i t a l t r a v e l . The same disturbance, applied shortly before the s a t e l l i t e i s 153 l = i 0 = 45° Cj=o.5 h p = 250 mi. Y0=P0=X=o Y° e=o.i i=23.5° C L /C D =o.i Y ^ P ^ X ^ 0 - 2 P° LO=0 m=5 Cai/Ci=50 Yf=P f=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 - d e g r e e s o 90 180 Figure 5.3 Typical responses showing the effectiveness of the aerodynamic-solar c o n t r o l l e r at d i f f e r e n t o r b i t a l positions about to re-enter the aerodynamic region, leads to the response indicated i n Figure 5.3d. The loss of continuous roll-yaw control i n the aerodynamic region i s r e f l e c t e d by the r e l a t i v e l y larger (-7°) r o l l amplitude and the longer damping time. 5.3.2 "Attitude control At times, missions involving diverse objectives may require a s a t e l l i t e to change i t s preferred orientation i n orb The c o n t r o l l e r ' s a b i l i t y to impart any desired orientation to the s a t e l l i t e i s explored here. Figure 5.4a shows the effectiveness of the control system i n providing a r b i t r a r y pitch attitudes with the axis of symmetry of the s a t e l l i t e along the o r b i t normal. The a b i l i t y to a l i g n the symmetry axis with the l o c a l v e r t i c a l d i r e c t i o n and simultaneously a t t a i n a desired pitch attitude i s indicated i n Figure 5.4b for a slender s a t e l l i t e (I = 0.1). For s a t e l l i t e s with large I, the c o n t r o l l e r was able to accomplish the same, but only at the cost of higher values of as i t must now overcome the gravity torques i n addition to the i n e r t i a of the s a t e l l i t e . Note that the attitude angle 3 represents the planar o s c i l l a t i o n i n the l o c a l v e r t i c a l configuration. The steady state motion noticeable for the case of e = 0.2, r e s u l t i n g from the e c c e n t r i c i t y induced disturbance (Equation 5.1b), indicates the need of a larger value of C.. for i t s elimination 155 C,= i i = 2 3 . 5 ° C L / C D = 0 . 1 Y=P=A=° 'o "0 o 0 Y CA) = 0 m = 0 . 5 C a i / C j=50 Y = B = X =o.i o ro o — p° 0 = 4 5 ° h P = 2 5 0 mi. 4 0 \ - e = 0.1 I = 2 Yf = P F = 0 X = 4 5 e = 0.2 1=2 Y f=P f=o X F = 3 0 ° 1 2 0 6 0 e = o.i I = 0 . 1 Y F = 9 0 ° p=o X = 2 0 ° 3 0 /"'yzo0 P F = - 3 0 ° \ X f = io° ' Yf=o P F = 3 0 ° X f = - i o ° • 3 0 0 e =0.1 I =1 ( C ) e =0.2 I = 1 0 Orbi ts Figure 5.4 Effectiveness of the aerodynamic-solar c o n t r o l l e r i n imparting a r b i t r a r y orientations to the s a t e l l i t e 156 Figure 5.4c presents r e s u l t s for s t a b i l i z a t i o n along a r b i t r a r i l y chosen values for a l l the three degrees of freedom. F i n a l l y , a comment concerning the earth's shadow, which would render the c o n t r o l l e r i n e f f e c t i v e in the solar pressure mode, i s appropriate here. It i s apparent that i t s e f f e c t would be n e g l i g i b l e i n near-polar o r b i t s . On the other hand, the influence would be maximum for o r b i t s i n the plane of the e c l i p t i c . Even i n the l a t t e r case, the e f f e c t of shadow would be of l i t t l e consequence i f the apparent position of the sun i s i n the neighbourhood of the apocenter as now the control in the shadow region i s primarily accomplished aerodynamically. Hence, depending upon the mission, a judicious selection of the location of the l i n e of nodes, o r b i t a l i n c l i n a t i o n from the e c l i p t i c and the perigee argument could e f f e c t i v e l y minimize the influence of the earth's shadow. 5.4 Concluding Remarks The s i g n i f i c a n t conclusions based on the analysis may be summarized as follows: (i) The f e a s i b i l i t y of aerodynamic-solar hybrid control of near-earth s a t e l l i t e s i n e l l i p t i c a l o r b i t s i s established. ( i i ) The control system i s capable of damping extremely severe disturbances in a f r a c t i o n of an o r b i t 157 w i t h the maximum amplitudes during the process l i m i t e d to a few degrees, ( i i i ) The c o n t r o l l e r permits the space c r a f t to undertake d i v e r s e missions through s t a b i l i z a t i o n along a r b i t r a r y a t t i t u d e s . The p o i n t i n g a c c u r a c i e s appear to be s u f f i c i e n t f o r many a p p l i c a t i o n s of near-earth s a t e l l i t e s , (iv) The system i s e s s e n t i a l l y semi-active which promises an increased s a t e l l i t e l i f e - s p a n . 158 6. TIME-OPTIMAL PITCH CONTROL USING SOLAR RADIATION PRESSURE The i n v e s t i g a t i o n s presented i n the p r e c e d i n g chapters c l e a r l y e s t a b l i s h the p o s s i b i l i t y of u t i l i z i n g the e n v i r o n - mental f o r c e s to achieve g e n e r a l t h r e e - a x i s l i b r a t i o n a l damping and a t t i t u d e c o n t r o l . With the c o n t r o l systems o f f e r i n g i n c r e a s e d s a t e l l i t e l i f e - t i m e s through t h e i r semi-passive c h a r a c t e r , i t seems l o g i c a l to d i r e c t e f f o r t s a t improving t h e i r performance. T h i s i n v o l v e s two a s p e c t s , namely, p h y s i c a l c o n t r o l l e r d esign and e f f i c i e n t c o n t r o l laws. The former having been s t r e s s e d i n the e a r l i e r c h apters, a t t e n t i o n here i s d i r e c t e d a t the p o s s i b i l i t y of u s i n g o p timal c o n t r o l laws. As the energy r e q u i r e d to t u r n the c o n t r o l p l a t e s i s low and c o u l d be generated e a s i l y through the use of s o l a r c e l l s , the performance index need o n l y i n c l u d e the damping time which i s of prime concern. Time-optimal c o n t r o l of multi-degree of freedom systems, such as the coupled r o l l - yaw-pitch motions of a s a t e l l i t e , can g e n e r a l l y be achieved o n l y through enormous software c o m p l e x i t i e s s i n c e the s o l u t i o n of a two p o i n t boundary value problem i s i n v o l v e d . T h i s i s why s w i t c h i n g c r i t e r i a t h a t are simple f u n c t i o n s of the s t a t e v a r i a b l e s were co n s i d e r e d i n the e a r l i e r a n a l y s e s . On the other hand, a s i n g l e degree of freedom system may lend i t s e l f to an a n a l y t i c a l s y n t h e s i s of the time-optimal 159 switching c r i t e r i o n . This i s s i g n i f i c a n t as, i f successful, i t not only could be applied to several situations of p r a c t i c a l importance (platform p i t c h control of a spinning s a t e l l i t e or pure pi t c h control of a gravity gradient system 70 as i n the case of COSMOS-149 ) but may also suggest switching laws that are l i k e l y to be e f f i c i e n t i n c o n t r o l l i n g the general motion. This chapter investigates the development of the time-optimal control law for the planar p i t c h motion of a s a t e l l i t e . The u t i l i z a t i o n of solar radiation pressure for attitude control i n a c i r c u l a r o r b i t i s considered. The analysis leads to a useful rel a t i o n s h i p 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 s a t e l l i t e executing planar pitch l i b r a t i o n \\i, with the center of mass S moving in a c i r c u l a r o r b i t about the center of force 0. The governing equation of motion i s well-known, i j j " + 3K isim | ; cos i | j = (6.1) where represents the generalized force due to solar radiation pressure. Figure 6.1 Geometry of motion of unsymmetrical s a t e l l i t e i n the solar pressure environment The solar pressure c o n t r o l l e r consists of two highly r e f l e c t i v e control plates and P 2 which are permitted rotations 6 ^ and 6 2 , respectively, in the o r b i t a l plane. The center of pressure of each plate i s taken to l i e on the s a t e l l i t e y axis (could be anywhere on the yz-plane) so as to y i e l d a pure p i t c h moment. The moment generated by the c o n t r o l l e r i s Qip = ± C i ' s i n ( 6 i + ? ) I sin (6i+?) cos6 i, i = 1,2 (6.2) (+ for P , - for P 2) where the solar parameter i s defined as C. = (2pp R 3/yI )A. e . (l - s i n 2 ( j ) s i n 2 i ) (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 6 i m = tan - 1[(-3/2)tanc±{(9/4)tan 2?+2} 1 / 2] (6.4) where the ± signs apply for tan C £ 0, respectively. The vari a t i o n of |Q,[ with L i s shown i n Figure 6.2a where 1 \p 1 max ^ 162 163 = C 2 = C i s assumed for convenience. The system i s able to provide a value |Q,| = (2/3/3)C at a l l 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-3K isini|) ecos^ e < u(0);< (2/3/3) C - ^ K ^ i n i ^ c o s ^ A symmetrical band on the control, |u(9)|<_C*= (2/3/3) C- | 3K isiniJ; ecosiJJ e| , (6.6) i s considered here for convenience. Its e f f e c t i s only to y i e l d a s l i g h t l y 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 x 2 = i>' , and l i n e a r i z i n g 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 i n i t i a l time 8 Q = 0, the s t a t e - t r a n s i t i o n matrix $(0) i s 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 ) i s sought which w i l l bring the system state from x(0) to x ( 8 f ) = 0 i n minimum 8 f. Substituting the i n i t i a l and the f i n a l states in Equation (6.9) re s u l t s i n <H-x)Bu(T)dT = -x(0) (6.10) The solution for u(8) bringing the system state to * rest i n minimum G f i s well-known to be u(8) = ±C , with the number of switches depending upon the i n i t i a l state of 8 5 the system . Considering i n i t i a l states that can be driven to rest in a single switch, the control takes the form 165 where u ( 9 ) = K 1 , 0 £ 9 < 0 g u(9) = K2, 0 g < 9 < Q Kl> = |K2> = C*' (6.11) Substituting for u(9) in Equation (6.10) leads to: K 1 = n{-nx 1(0)(sinn9 f-sinn0 s)+x 2(0)(cosn9 f-cosn0 s)}/A (6.12a) K 2 = n{nx 1(0)sinn9 g+x 2(0)(l-cosn9 g)}/A (6.12b) where A = sin(n0„-n0 )-sinnGJ-+sinn9 f s f s The proper signs of and K 2 are best obtained from the phase plane p o r t r a i t of the optimally controlled system (Figure 6.2b). The t r a j e c t o r i e s are c i r c u l a r arcs * 2 * with centers at x^ = ±C /n , x 2 = 0 for u = ±C , respectively. The switching boundary i s composed of semi-circles passing through the o r i g i n . For any i n i t i a l condition x(0), the system state moves on the switching boundary for 0 g <_ 0 <_ 0^ as shown i n the figure, It i s apparent that a l l i n i t i a l states l y i n g within the region ABCDA of the phase diagram can be driven to the o r i g i n with a single switch of the control. For optimal response, the control u(0) assumes the value * u = -C i f the system state l i e s above the switching boundary * and u = +C i f i t i s below the switching boundary. Equations (6.12) may now be solved to obtain the switching time 0 and the f i n a l time 0_ for a given i n i t i a l s f 3 condition. This y i e l d s the open-loop r e a l i z a t i o n of the control in the form u = u(0). On the other hand, use of the switching boundary y i e l d s the feedback r e a l i z a t i o n u = u (x) , which makes the system s e l f - c o r r e c t i n g to s l i g h t deviations of the state vector. Of p a r t i c u l a r i n t e r e s t are impulsive disturbances which a s a t e l l i t e i s l i k e l y to encounter through, say, micrometeorite impacts. The phase p o r t r a i t immediately y i e l d s the maximum impulsive disturbance from which the s a t e l l i t e can be brought to rest i n 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 + x 2 ( 0 ) } 1 / 2 - l (6.13b) 167 where the normalized state variables x^ and i<2 are defined as 3^(6) = n 2x 1(9)/C* (6.13c) x 2 (0) = n x 2(0)/C* (6.13d) The v a r i a t i o n of the normalized error amplitude with the i n i t i a l impulsive disturbance i s shown i n Figure 6.3a. The switching and the f i n a l 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 a p p l i c a b i l i t y of the optimal control law synthesized from l i n e a r theory to the actual nonlinear system, the response of both the l i n e a r i z e d 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 e f f e c t of the i n e r t i a parameter to be n e g l i g i b l e for the pitch attitude nominally along the l o c a l v e r t i c a l . As anticipated, the open-loop and the feedback response of the li n e a r system are i d e n t i c a l . On the other hand, the open- loop control system i s 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 i n i t i a l condition x o ( 0 ) ; (b) Variation of switching time 0 S and f i n a l time 9̂  with i n i t i a l condition x „ ( 0 ) 169 Open loop , f e e d b a c k ( l inear s y s t e m ) O p e n loop ( non l inear s y s t e m ) F e e d b a c k ( n o n l i n e a r s y s t e m ) C = 10 x 2 (o ) -0.5 1 T f \ ib =o / \ / \ K.= 0.5 1 1 / \ K.= i.o 1 \ {b) -1 1 i i + =30 C K =0.5 15 ( O K = i.o 30 0 15 6 - deg r e e s 30 45 Figure 6.4 System response to impulsive disturbance 17 t h i s , the nonlinear system state approaching the o r i g i n asymptotically using a number of switches of the control. I order to avoid any relay-chatter, i t 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 i n the neighbourhood of the o r i g i n . Figures 6.4c and d present the system response for a s a t e l l i t e s t a b i l i z e d i n an ar b i t r a r y p i t c h a t t i t u d e . Note that the gravity gradient torque now represents a d e s t a b i l i z i n g e f f e c t , which the c o n t r o l l e r must neutralize in addition to countering the disturbance. The longer dampi time required with a higher value of the i n e r t i a parameter * c l e a r l y r e f l e c t s a greater reduction i n the value of C for increased K.. 1 The system response may now be projected for the pitc h control of the CTS. At synchronous a l t i t u d e , the 2 value of C =10 corresponds to A. - 2.5 f t and e. = 10 f t . 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 or b i t i s attained (Figure 6.4). A disturbance ^(O) = 0.1 on the other hand would be damped out i n approximately 4°. (Figure 6.3b). The system thus appears promising. I t should be pointed out here that the constraint |[£(2/3/3)C represents the most adverse s i t u a t i o n as | | may at t a i n a value as large as C during cer t a i n o r b i t a l positions (Figure 6.2a). The performance of the c o n t r o l l e r , 171 therefore, would always exceed the responses presented here (Figure 6 . 4 ) . 6.4 Concluding Remarks (i) The analysis c l e a r l y demonstrates the f e a s i b i l i t y of the time-optimal p i t c h control of s a t e l l i t e s using the solar pressure, ( i i ) The c o n t r o l l e r i s capable of damping extremely severe disturbances i n a few degrees of the s a t e l l i t e ' s o r b i t a l t r a v e l . The transient amplitude i s also small, ( i i i ) The c o n t r o l l e r i s able to provide nominal control at an orientation which i s not an equilibrium position of the uncontrolled system, (iv) The optimal control strategy, developed for the l i n e a r i z e d system, may be applied e f f e c t i v e l y to the actual nonlinear system. 172 7.. CLOSING COMMENTS 7.1 Summary of the Conclusions The s i g n i f i c a n t conclusions based on the pre- ceding investigation may be summarized as follows: (i) The solar radiation pressure, normally neglected in the analysis of spinning s a t e l l i t e s , can af f e c t the l i b r a t i o n a l performance su b s t a n t i a l l y . It merits the same consideration as the i n e r t i a properties, spin rate and e c c e n t r i c i t y during the design of an attitude control system, ( i i ) The environmental forces can be used quite e f f e c t i v e l y to provide three-axis l i b r a t i o n damping and attitude control of spinning space- c r a f t . ( i i i ) As substantial control moments are available even with the use of moderate c o n t r o l l e r sizes, i t does not appear necessary to spin a s a t e l l i t e from attitude control considerations. Of course, the presence of spin would improve the nutation damping performance, (iv) A l o g i c a l procedure i s established for the development of an e f f e c t i v e solar pressure con- t r o l system. This should prove useful i n evolving a suitable c o n t r o l l e r depending upon the mission requirements. 1 7 3 (v) The magnetic roll-yaw c o n t r o l l e r , i n conjunction with the solar p i t c h c o n t r o l l e r , provides an e f f i c i e n t three-axis control system, (vi) The hydrid aerodynamic-solar system o f f e r s e f f e c t i v e control of near-earth s a t e l l i t e s in e l l i p t i c t r a j e c t o r i e s which promise an increased l i f e - s p a n . (vii) U t i l i z a t i o n of the maximum available control moments, through bang-bang operation, leads to smaller damping times compared to the l i n e a r control law with saturation constraints. The near-optimum performance r e s u l t i n g for a wide range of system parameters and i n i t i a l d i s t u r - bances i s p a r t i c u l a r l y a t t r a c t i v e , ( v i i i ) Approximate a n a l y t i c a l techniques can be used quite e f f e c t i v e l y during preliminary stages of s a t e l l i t e design. For small amplitude motion, usually the case of in t e r e s t , they can predict the l i b r a t i o n 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 s a t e l l i t e l i f e - s p a n . 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 f e a s i b i l i t y of using environmental forces to advantage having been established here, i t appears l o g i c a l to d i r e c t e f f o r t s at improving the e f f i c i e n c y of these systems. The design approach suggested i n Chapter 3 could be applied to devise alt e r n a t i v e c o n t r o l l e r configurations. An extension of the approach i n Chapter 6 to the synthesis of optimal or suboptimal control laws for the coupled motion i s l i k e l y to improve the control performance, ( i i ) 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 i n e r t i a variations and reaction forces a r i s i n g from con- t r o l system operation, ( i i i ) With ever increasing energy requirements, the use of large solar panels by the future generation of s a t e l l i t e s i s l i k e l y to be indispensable. Substantial energy savings may r e s u l t i f the semi- 175 passive attitude control systems u t i l i z i n g the environmental forces could be extended to f l e x i b l e s a t e l l i t e configurations. The dynamical problem i s going to be more involved. Hence, more sophisticated c o n t r o l l e r s and control strategies would be required as the disturbances, such as, panel vibration, d i f f e r e n t i a l thermal heating, etc., are of the continuous type. The problem appears to be quite useful as well as challenging, (iv) The p o s s i b i l i t y of employing the environmental forces for o r b i t a l transfer and trajectory control appears inte r e s t i n g . As the concept promises considerable energy saving and reduction i n payload (no mass expulsion i s required), i t could be of immense value i n interplanetary t r a v e l as well where long time durations are involved. It presents an exciting p o s s i b i l i t y of achieving controlled variations in the o r b i t a l parameters of a space- vehicle. Optimization of system performance i n the foregoing represents a vast challenging area that has remained v i r t u a l l y unexplored. 176 BIBLIOGRAPHY 1. Thomson, W.T., "Spin S t a b i l i z a t i o n of Attitude against Gravity Torques," The Journal of the Astronautical Sciences, Vol. 9, No. 1, January 1962, pp. 31-33. 2. Pringle, R., J r . , "Bounds on the Librations of a Symmet- r i c a l S a t e l l i t e , " AIAA Journal, Vol. 2, No. 5, May 1964, pp. 908-912. 3.. Kane, T.R. and Shippy, D.J., "Attitude S t a b i l i t y of a Spinning Unsymmetrical S a t e l l i t e in a C i r c u l a r Orbit," The Journal of the Astronautical Sciences, Vol. 10, No. 4, Winter 1963, pp. 114-119. 4. Kane, T.R. and Barba, P.M., "Attitude S t a b i l i t y of a Spinning S a t e l l i t e i n an E l l i p t i c Orbit," Journal of Applied Mechanics, Vol. 33, June 1966, pp. 402-405. 5. Wallace, F.B., J r . , and Meirovitch, L., "Attitude I n s t a b i l i t y Regions of a Spinning Symmetric S a t e l l i t e i n an E l l i p t i c Orbit," AIAA Journal, Vol. 5, No. 9, September 1967, pp. 1642-1650. 6. Neilson, J.E., "On the Attitude Dynamics of Slowly- Spinning Axisymmetric S a t e l l i t e s under the Influence of Gravity-Gradient Torques," Ph.D. Thesis, University of B r i t i s h Columbia, November 1968. 7. Modi, V.J. and Neilson, J.E. "Attitude Dynamics of Slowly-Spinning Axisymmetric S a t e l l i t e s under the Influence of Gravity-Gradient Torques," Proceedings of the xxth international Astronautical Congress, Pergamon Press Ltd., Oxford, 1972, pp. 563-596. 8. Modi, V.J. and Neilson, J.E. "Roll Dynamics of a Spinning Axisymmetric S a t e l l i t e i n an E l l i p t i c Orbit," Journal of the Royal Aeronautical Society, Vol. 72, No. 696, December 1968, pp. 1061-1065. 9. Modi, V.J. and Neilson, J.E. "On the Periodic Solutions of Slowly-Spinning Gravity-Gradient Systems," C e l e s t i a l Mechanics - An International Journal of Space Dynamics, Vol. 5, No. 2, 1972, pp. 126-143. 10. L i k i n s , P.W., "Effects of Energy Dissipation on the Free-Body Motions of Spacecrafts," No. 32-860, Jet Propulsion Laboratory Technical Report, 1966. 177 11. Roberson, R.E., "Torques on a S a t e l l i t e Vehicle from Internal Moving Parts," Journal of Applied Mechanics, Vol. 25, 1958, pp. 196-200. 12. Landon, V.D. and Stewart, B., "Nutational S t a b i l i t y of an Axisymmetric Body Containing a Rotor," Journal of Spacecraft and Rockets, Vol. 1, No. 6, December 1964, pp. 682-684. 13. I o r i l l o , A.J., "Nutation Damping Dynamics of Axisymmetric Rotor S t a b i l i z e d S a t e l l i t e s , " American Society of Mechanical Engineers, Winter Meeting, November 1965. 14. L i k i n s , P.W., "Attitude S t a b i l i t y C r i t e r i a for Dual- Spin Spacecraft," Journal of Spacecraft and Rockets, Vol. 4, No. 12, December 1967, pp. 1638-1643. 15. Mingori, D.L., "Effects of Energy Dissipation on the Attitude S t a b i l i t y of Dual-Spin S a t e l l i t e s , " AIAA Journal, Vol. 7, No. 1, January 1969, pp. 20-27. 16. Pringle, R., J r . , " S t a b i l i t y of the Force-Free Motions of a Dual-Spin Spacecraft," AIAA Journal, Vol. 7, No. 6, June 1969, pp. 1054-1063. 17. C l o u t i e r , G.J., "Stable Rotation States of Dual-Spin Spacecraft," Journal of Spacecraft and Rockets, Vol. 5, No. 4, A p r i l 1968, pp. 490-92. 18. Cloutier, G.J., "Nutation Damper Design P r i n c i p l e s for Dual-Spin Spacecraft," The Journal of the Astronautical Sciences, Vol. 16, No. 2, March-April 1969, pp. 79-87. 19. Sen, A.K., " S t a b i l i t y of a Dual-Spin S a t e l l i t e with a Four-Mass Nutation Damper," AIAA Journal, Vol. 8, No. 4, A p r i l 1970, pp. 822-823. 20. Vigneron, F.R., " S t a b i l i t y of a Dual-Spin S a t e l l i t e with Two Dampers," Journal of Spacecraft and Rockets, Vol. 8, No. 4, A p r i l 1971, pp. 386-389. 21. Bainum, P.M., Fuechsel, P.G.7and Mackison, D.L. "Motion and S t a b i l i t y of a Dual-Spin S a t e l l i t e with Nutation Damping," Journal of Spacecraft and Rockets, Vol. 7, No. 6, June 1970, pp. 690-696. 22. Bainum, P.M., Fuechsel, P.G.,and Fedor, J.V., " S t a b i l i t y of a Dual-Spin Spacecraft with a F l e x i b l e Momentum Wheel," Journal of Spacecraft and Rockets, Vol. 9, No. 9, September 1972, pp. 640-646. 178 23. Kane, T.R. and Scher, M.P., "A Method of Active Control Based on Energy Considerations," Journal of Spacecraft and Rockets, Vol. 6, No. 5, May,1969, pp. 633-636. 24. Mingori, D.L., Harrison, J.A.,and Tseng, G.T., "Semipassive and Active Nutation Dampers for Dual- Spin Spacecraft," Journal of Spacecraft and Rockets, Vol. 8, No. 9, May 1971, pp. 448-455. 25. Kane, T.R. and Mingori, D.L., "Eff e c t of a Rotor on the Attitude S t a b i l i t y of a S a t e l l i t e in a C i r c u l a r Orbit," AIAA Journal, Vol. 3, No. 5, May 1965, pp. 936-940. 26. White, E.W. and L i k i n s , P.W., "The Influence of Gravity Torque on Dual-Spin S a t e l l i t e Attitude S t a b i l i t y , " The Journal of the Astronautical Sciences, Vol. 16, No. 1, January-February 1969, pp. 32-37. 27. Roberson, R.E. and Hooker, W.W., "Gravitational E q u i l i b r i a of a Rigid Body Containing Symmetric Rotors," Proceedings of the x v i j t h International Astronautical Congress, Gordon and Breach Inc., New York, 1967, pp. 203-210. 28. Longman, R.W. and Roberson, R.E., "General Solutions for the E q u i l i b r i a of Orbiting Gyrostats Subject to Gravitational Torques," The Journal of the Astronautical Sciences, Vol. 16, No. 2, March-April 1969, pp. 49-58. 29. Yu, E.Y., "Attitude S t a b i l i t y of an Orbiting Vehicle Containing a Gyrostat," Journal of Spacecraft and Rockets, Vol. 6, No. 8, August 1969, pp. 948-951. 30. Proceedings of the Symposium on Attitude S t a b i l i z a t i o n and Control of Dual-Spin Spacecraft, TR 0158(3307-01) -16, November 1967, Aerospace Corp., E l Segundo, C a l i f . , pp. 73-109. 31. Johnson, C.R., "Tacsat 1 Nutation Dynamics," AIAA Paper 70-455, Los Angeles, C a l i f . , 1970. 32. L i k i n s , P.W., Tseng, G.T., and Mingori, D.L., "Stable Limit Cycles due to Nonlinear Damping i n Dual-Spin Spacecraft," Journal of Spacecraft and Rockets, Vol. 8, No. 6, June 1971, pp. 568-574. 33. Mingori, D.L., Tseng, G.T.fand L i k i n s , P.W., "Constant and Variable Amplitude Limit Cycles i n 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 S t a b i l i z a t i o n , The Aerospace Corp., December 1968. 35. Shrivastava, S.K. , Tschann, C.,and Modi, V.J., " L i b r a t i o n a l Dynamics of Earth Orbiting S a t e l l i t e s - 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 S a t e l l i t e Vehicle - An Outline of the Problem," Proceedings of the V l l l t h International Astronautical Congress, Wein-Springer-Verlag, B e r l i n , 1958, pp. 317-319. 37. Wiggins, L.E., "Relative Magnitudes of the Space- Environment Torques on a S a t e l l i t e , " AIAA Journal, Vol. 2, No. 4, A p r i l 1964, pp. 770-771. 38. Clancy, T.F. and M i t c h e l l , T.P., "Effects of Radiation Forces on the Attitude of an A r t i f i c i a l Earth S a t e l l i t e , " 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 S a t e l l i t e s : Part I - High A l t i t u d e Orbits," Aeronautical Journal, Royal Aeronautical Society, Vo. 75, November 1971, pp. 783-793. 40. Modi, V.J. and Kumar, K., " L i b r a t i o n a l Dynamics of Gravity Oriented S a t e l l i t e s 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 E f f e c t of Atmospheric Forces on S a t e l l i t e Attitude," The Journal of Aeronautical Sciences, Vol. 6, 1959, pp. 40-45. 42. Schrello, D.M., "Aerodynamic Influence on S a t e l l i t e L ibrations," 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 S t a b i l i z e d S a t e l l i t e s , " AIAA Journal, Vol. 1, No. 4, 1963, pp. 968-969. 180 44. Meirovitch, L. and Wallace, F.B., J r . , "On the E f f e c t of Aerodynamic and Gravitational Torques on the Attitude S t a b i l i t y of S a t e l l i t e s , " AIAA Journal, Vol. 4, No. 12, December 1966, pp. 2196-2202. 45. Garwin, R.L., "Solar S a i l i n g - A P r a c t i c a l Method of Propulsion Within the Solar System," Jet Propulsion, Vol. 28, No. 3, March 1958, pp. 188-190. 46. Sohn, R.L., "Attitude S t a b i l i z a t i o n 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 L i b r a - tions of a Gravity Oriented S a t e l l i t e , " AIAA Student Journal, Vol. 4, No. 4, December 1966, pp. 143-147. 49. Modi, V.J. and Flanagan, R.C., " L i b r a t i o n a l 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 L i b r a t i o n a l Control of a S a t e l l i t e using Solar Radiation Pressure," Astronautical Research 1970, Proceedings of the XXI Congress of the International Astronautical Federation, E d i t o r - i n - c h i e f : L.G. Napolitano, North- Holland Publishing Co., Amsterdam, 1971, pp. 84-100. 51. Modi, V.J. and Kumar, K., "Coupled L i b r a t i o n a l Dynamics and Attitude Control of S a t e l l i t e s i n Presence of Solar Radiation Pressure," Astronautical Research 1971, Proceedings of the XXII Congress of the International Astronautical Federation, E d i t o r - i n - c h i e f : L.G. Napolitano, D. Reidel Publishing Co., Dordrecht, Holland, 1973, pp. 37-52. 52. Ule, L.A., "Orientation of Spinning S a t e l l i t e s by Radiation Pressure," AIAA Journal, Vol. 1, No. 7, July 1963, pp. 1575-1578. 53. Peterson, C.A., "Use of Thermal Re-Radiative E f f e c t s i n Spacecraft Attitude Control," CSR-TR-6 6-3, May 1966, Massachusetts I n s t i t u t e of Technology Center for Space Research, Cambridge, Mass. 181 54. Colombo, G., "Passive S t a b i l i z a t i o n of a Sunblazer Probe by Means of Radiation Pressure Torque," CSR-TR- 66-5, 1966, Massachusetts I n s t i t u t e of Technology Center for Space Research, Cambridge, Mass. 55. F a l c o v i t z , J . , "Attitude Control of a Spinning Sun- Orbiting Spacecraft by Means of a Grated Solar S o i l , " CSR-TR-66-17, December 1966, Massachusetts Inst i t u t e of Technology Center for Space Research, Cambridge, Mass . 56. Crocker, M.C. I I , "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. S c u l l , 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 S t a b i l i z i n g Gravity-Oriented S a t e l l i t e s , " Journal of Spacecraft and Rockets, Vol. 4, No. 12, December 1967, pp. 1617-1622. 59. Bainum, P.M. and Mackison, D.L., "Gravity-Gradient S t a b i l i z a t i o n of Synchronous Orbiting S a t e l l i t e s , " Journal of the B r i t i s h 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 S a t e l l i t e , " Proceedings of the Symposium on Attitude S t a b i l i z a t i o n and Control of Dual-Spin Space- c r a f t , Rept. RT-0158(3307-01)-16, November 1967, Aerospace Corp., E l Segundo, C a l i f . , pp. 121-144. 62. F i s c h e l l , R.E., "Spin Control for Earth S a t e l l i t e s , " Peaceful Uses of Automation i n 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 F i e l d , " 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- c r a f t and Rockets, Vol. 8, No. 5, May 1971, pp. 441-448. 65. Shigehara, M., "Geomagnetic Attitude Control of an Axisymmetric Spinning S a t e l l i t e , " 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 S a t e l l i t e s , " Torques and Attitude Sensing in Earth S a t e l l i t e s , 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 F e a s i b i l i t y of Aerodynamic Attitude S t a b i l i z a t i o n of a S a t e l l i t e Vehicle," presented at ARS Controllable S a t e l l i t e Conference, M.I.T., A p r i l 1959. 69. Schrello, D.M., "Passive Aerodynamic Attitude S t a b i l i z - ation of Near Earth S a t e l l i t e s , Vol. I, L i b r a t i o n due to Combined Aerodynamic and Gravitational Torques," WADD Tech. Report 61-133, Vol. I, July 1961. 70. Sarychev, V.A., "Aerodynamic S t a b i l i z a t i o n System of the S a t e l l i t e s , " Proceedings of the International Colloquium on Attitude Changes and S t a b i l i z a t i o n of S a t e l l i t e , Paris, October 1968, pp. 177-179. 71. Ravindran, R., "Optimal Aerodynamic Attitude S t a b i l i z - ation of Near Earth S a t e l l i t e s , Ph.D. Thesis, University of Toronto, A p r i l 1971. 72. Modi, V.J. and Shrivastava, S.K., "On Librations of Gravity-Oriented S a t e l l i t e s i n E l l i p t i c 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 O s c i l l a t i o n s , B l a i s d e l l Publishing Company, New York, 1965, pp. 201-217. 183 75. Modi, V . J . and B r e r e t o n , R.C, " P e r i o d i c S o l u t i o n s A s s o c i a t e d w i t h the G r a v i t y - G r a d i e n t - O r i e n t e d System. P a r t I : A n a l y t i c a l and Numerical Determination," AIAA J o u r n a l , V o l . 7, No. 7, J u l y 1969, pp. 1217- 1225. 76. Modi, V . J . and B r e r e t o n , R.C, " P e r i o d i c S o l u t i o n s A s s o c i a t e d w i t h the G r a v i t y - G r a d i e n t - O r i e n t e d System. P a r t I I : S t a b i l i t y A n a l y s i s , " AIAA J o u r n a l , V o l . 7, No. 8, August 1969, pp. 1465-1468. 77. Minorsky, N., N o n l i n e a r O s c i l l a t i o n s , D. Van Norstrand Company Inc., P r i n c e t o n , 1962, pp. 127-133, 390-415. 78. Moran, J.P., " E f f e c t s of Plane L i b r a t i o n s on the O r b i t a l Motion of a Dumbbell S a t e l l i t e , " ARS J o u r n a l , V o l . 31, No. 8, August 1961, pp. 1089-1096. 79. Yu, E.Y., "Long-Term C o u p l i n g E f f e c t s Between the L i b r a t i o n a l and O r b i t a l Motions of a S a t e l l i t e , " AIAA J o u r n a l , V o l . 2, No. 3, March 1964, pp. 553-555. 80. E h r i c k e , K.A., P r i n c i p l e s of Guided M i s s i l e Design, D. Van Nostrand Co. Inc., New J e r s e y , 1960, pp. 241- 256. 81. RCA, "RCA Flywheel S t a b i l i z e d , M a g n e t i c a l l y Torqued A t t i t u d e C o n t r o l System f o r M e t e o r o l o g i c a l S a t e l l i t e s , CR-232, 1965, NASA. . 82. Schaaf, S.A. and Chambre, P.L., Flow of R a r e f i e d Gases, P r i n c e t o n U n i v e r s i t y P r e s s , 1961, pp. 17-24. 83. Jensen, J . , Townsend, G. , Kork, K., and K r a f t , D., Design Guide to O r b i t a l F l i g h t , McGraw H i l l , New York, 1962, pp. 179-264. 84. J a c c h i a , L.G., " S t a t i c D i f f u s i o n Models of the Upper Atmosphere wi t h E m p i r i c a l Temperature P r o f i l e s , " Smithsonian C o n t r i b u t i o n s t o A s t r o p h y s i c s , V o l . 8, No. 9, 1965, Smithsonian I n s t i t u t i o n , Washington, D.C, pp. 215-257. 85. P o n t r y a g i n , L.S., B o l t y a n s k i i , V.G., Gamkrelidze, R.V., and Mischenko, E.F., The Mathematical Theory of Optimal Processes, Pergamon Press L t d . , 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,a 1,a 2,k 1,k 2, 2, 1,£ 2,C,G,A 1,B 1) = ^ - { 1 / (k 2-k 2) } £ c (l+3e) { (5A1+7B1-4G) l±a2+ (7A1+B1~4G) % 2 }/ 16a 1+£ 1[{I(a+1)(l+2e)-4}{k 1(3a 2a 2+2a 2b 2+2B 2)+4k 2a 1a 2b 2}/ 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} -k 1(a 2+b 2+A 2)/4] + (£ 2/a 1) [{l(a+l) (l+2e)+4 (31-4)}x (a 2+2b 2+2A 2)/16-{l(a+1)(l+2e)}a 1k 1(a 2a 2+a 2b 2+B 2)/8 - [I(a+1) (l+2e)}(a 2a 2+4a 2b 2+4B 2)/32-a 2k 2a 2/8 - a 1a 2k 1k 2b 2/2+a 1k 1A 1B 1/2-a 2k 2b 2/2-a 1k 1(3a 2+2b 2+2A 2)/8]J w_ = F(b,a,a 0,ot,,k 0,k, rZ 0,C,G rA 1,B,) 185 APPENDIX II The functions and G^ (i=l,2,3,4) are given £-1(a+1){(l+e)/(l+ecos9)} 2cosy p-{3(I-l)/(l+ecos9) -DcosB ( c o s 2 Y p - s i n 2 Y p ) - [{2esin6/ (l+ecos9) }sinB p -23 pcos6 p]sinY p-{(1+e) 3/(l+ecos9) 4}Ccos9 x 2 [{(sinYpSinB pcos0-cos3pSin9)sinBp-sinY pcos9}cos Y pcos9/ 2 2 1/2 { (sinY psin3pCOs9-cosBpSin9) + (cosY pcos9) } 2 2 1/2 {(sinYpSin3pCOS0-cosBpSin9) +(cosY pcos0) } s i n Y p -G j sinY pcos6pCos0+sin3pSin9|sinY p]J secB p [{2esin9/(l+ecos9)}cosBp+23 psinBp]secB p [{3 (I-l)/(l+ecos9)}sin3 psinY pcosYp+{2esin9/(l+ecos9)} x (cos3pCOSYp-YpSin3p)+23pYpCOs3p+{Yp +(23p-sin Y p)x 186 x cosY p}sing p-{(1+e) 3/(l+ecos6) 4}Ccos9(sinY psinB p x cos0-cosB psin0)(sinY pcosB pcos0+sinB psin0)cosY p/((siny p x 2 2 1/2 sin3 pcos0-cosB psin0) +(cosY pcos0) } ' ]secB p [I (a+1){(l+e)/(l+ecos0)} 2+2Y psinB p-2cosB pcosY p]secB p I (a+1){(l+e)/(l+ecos0)} 2sinB psinY p+ 2{3 (I-l) / (l+ecos0) -1} sinB pcosB psiriY pcosY p 2 2 -{2esin0/(l+ecos0)}cosY p-(cos B p ~ s i n B p+l)Y psinY p +{(1+e) 3/(l+ecos0) 4}Ccos0[{(sinY psinB pcos0-cosB psin0)x sinB^-sinY^cos©} (sinY^sinBcCOsO-cosBpSin©) C O S Y P / 2 2 1/2 {(sinY psinB pcos0-cosB psin0) +(cosY pcos0) } 2 2 1/2 +cosY psinB p{(sinY psinB pcos0-cosB psin0) +(cosY pcos0) } +GcosY psinB p|sinY pcosB pcos0+sinB psin0|] 187 [-I(a+D { (l+e)/(l+ecos0) } 2+2Y psinB p]cosB! 2 2 + (cos Bp-sin Bp+DcosYp I(a+1){(l+e)/(l+ecos6)} 2(Y psin8 p-cosB pcosY p) + [{3 (I-l)/(l+ecos0)}sin 2Yp+cos 2Y p-Yp 2] ( c o s 2 B p - s i n 2 B p ) -4YpSinB pcosB pcosYp+{(1+e) 3/(l+ecos0) 4}C x 2 [(sinYpSinBpCos0-cosBpSin6) (sinY pcos3pCOS0+sinB psin0)/ 2 2 1/2 {(sinYpSinBpCOse-cosBpSine) +(cosY pcos0) } 2 +(sinYpCosB pcos0+sinBpSin0){(sinY psinB pcos0-cosB psin0) 2 1/2 +(cosY pcos8) } ' +G(sinYpCOsBpCOS0+sin3pSin0) x |sinY pcosBpCOS0+sinBpSin0|] 2esin0/(l+ecos0)

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