EasyChair Preprint. A Backstepping Sliding Mode Controller Design for Spacecraft Formation Flying

Transcription

1 EasyChair Preprint 06 A Backstepping Siding Mode Controer Design for Spacecraft Formation Fying Amin Imani, Mohsen Bahrami and Jafar Roshanian EasyChair preprints are intended for rapid dissemination of research resuts and are integrated with the rest of EasyChair. Apri 9, 08

2 IAA-AAS-DyCoSS4-- A BACKSEPPING SLIDING MODE CONROLLER DESIGN FOR SPACECRAF FORMAION FLYING Amin Imani, * Mohsen Bahrami, and Jafar Roshanian In this paper, a backstepping siding mode controer is deveoped for tracking contro of spacecraft formation fying on eiptica orbits. he controer is designed in accordance with the noninear mode of reative motion, and combines the advantages of backstepping and siding mode contro techniques. After appying the backstepping method to incorporate the tracking errors and Lyapunov functions, a siding mode controer is deveoped to guarantee the Lyapunov stabiity, handing a noninearities, robustness against uncertainties and tracking the desired trajectory. It is supposed that the eader and foower spacecraft are in a ow Earth orbit whie J perturbation and atmospheric drag are considered as externa disturbances. he performance of the proposed controer in tracking the desired formation is compared to a siding mode controer. Simuation resuts confirm the effectiveness of the proposed controer. INRODUCION Spacecraft formation fying (SFF) has increasingy attracted attention during the ast few decades. Using this method, a arge and expensive spacecraft can be repaced with a number of smaer, ess expensive and cooperative spacecraft which work as an integrated unit and fufi the purpose of the mission. Besides the simper design and faster aunching process, the main advantage of this system ies in reiabiity and fexibiity which makes nove and innovative appications in space and the Earth science missions incuding observation of the Earth and its atmosphere, geodesy, deep space imaging with high resoution, in-orbit servicing and spacecraft maintenance. A common method for impementation of SFF is Leader/Foower architecture. Based on this method, one spacecraft is controed as a eader in a reference orbit whie other spacecraft -as foowersadjust their positions reative to the eader and track the desired reative trajectory. In this study, attention is confined to contro probem of SFF on eiptica orbits. A robust and precise noninear controer is necessary for spacecraft formation process, because it has noninear coupe dynamics subjected to externa disturbances. * Ph.D. Candidate, Schoo of Mechanica Engineering, Iran University of Science and echnoogy (IUS), ehran , Iran. E-mai: aimani@mecheng.iust.ac.ir Professor, Department of Mechanica Engineering, Amirkabir University of echnoogy, ehran 5875, Iran. E-mai: mbahrami@aut.ac.ir Professor, Department of Aerospace Engineering, K.N. oosi University of echnoogy, ehran , Iran. E- mai: roshanian@kntu.ac.ir

3 It is we known that siding mode contro (SMC) is a robust method to contro noninear systems which has satisfactory performance to keep the systems insensitive to the uncertainties and disturbances. An effective robust method for sateite contro using a siding mode controer was presented based on Hi s equations. Parameters incuding the equivaent damping ratio, bandwidth and thrust were aso estimated to minimize the fue cost. Hui et a. designed a ow-eve SMC for SFF based on noninear reative dynamics of circuar reference orbit in another study 3. hey used the controer to contro eader, foower and entire-formation maneuvering in ow-earth orbits. Second order siding mode controer was aso appied to contro of spacecraft reative transation by Pukdeboon 4. In another research, Hui and Li presented a termina SMC to reduce fue consumption, track desired trajectory and accompish formation reconfiguration in a proper time 5. An adaptive termina siding mode controer was designed for spacecraft formation fying by Wang and Sun, in which Leader/Foower architecture was considered and the convergence of the desired trajectory to a neighborhood in finite time was proved anayticay 6. In some studies, adaptive SMC was used to contro the reative position and attitude of spacecraft in a formation 7,8. Bae and Kim designed an adaptive controer based on neura network to compensate the modeing error, externa disturbance and noninearities to improve the performance of the siding mode controer 7. Backstepping controer as a noninear controer is a recursive process to contro noninear systems. It is based on Lyapunov theory and can be designed with a systematic strategy. he system is initiay represented in sma subsystems, then by a recursive approach virtua Lyapunov-based contro signas are determined for each subsystems. Actua contro aw can be obtained from the ast step 9. So, the backstepping technique is a fexibe approach in designing controers for noninear systems. Backstepping method was used to contro spacecraft formation fying in some researches,0-. Adaptive backstepping controer was designed for sateite formation with mass uncertainty and thruster error in some studies,. In another study, Kristiansen et a. deveoped an integrator backstepping approach based on noninear mode of reative motion. In the present work, siding mode contro and backstepping technique are combined to design a robust backstepping siding mode controer (BSMC) for spacecraft formation contro on eiptica orbits. After appying the backstepping method to incorporate the tracking errors and Lyapunov functions, a siding mode controer is deveoped to guarantee the Lyapunov stabiity, handing a system noninearities, robustness against uncertainties and disturbances and tracking the desired formation. In this paper, it is assumed that the spacecraft move in ow Earth orbits and are subjected to the perturbing effects of J and atmospheric drag. he performance of the proposed controer in tracking the desired trajectory is compared to a conventiona siding mode controer. SPACECRAF RELAIVE MOION DYNAMICS In this section, dynamic mode of reative motion for SFF is specified. It is assumed that each spacecraft is a point mass. As shown in Figure, C X, Y, Z is the inertia coordinate system and r and r f are the position vectors of eader and foower spacecraft, respectivey. he coordinate system C { x, y, z} is a moving frame ocated on the eader s center of mass. Herein, y axis is aong the direction of r () t, x is aong the direction of eader veocity vector and norma to y, and z axis competes the right-handed C coordinate frame.

4 Figure. Inertia and Moving Coordinate Systems he dynamics of the eader and foower in the inertia reference frame can be written as r r d u 3 r () r r d u f 3 f f f r f () 3 where r r, = km s is the constant of the Earth gravity, u is contro input vector and d is the vector of externa disturbance. It is supposed that the foower spacecraft shoud be controed; the eader is subjected to perturbations, and it moves in an uncontroed baistic trajectory; therefore u 0. Position vector of the foower reative to the eader is defined as rf r x y z. he noninear reative dynamics of the foower with respect to the moving frame is described as 3 C ( ) F (,,, r, N, n) D u (3) f where D d d (4) f D is the differentia perturbation subjecting to the formation and, 0 0 ( N ) x y C ( ) 0 0, F (,,, r, N, n) ( N ) y ( N n ) r x N z and (5) 3

5 a e e N ( ) sin,, n 3 3 r r r a( e ), r 3 x ( r ) ecos y z in which, ae,, are the anguar veocity, the orbita semi-major axis, the eccentricity and the true anomay of the eader spacecraft, respectivey. CONROLLER DESIGN Siding Mode Contro Siding mode controer as a robust controer is an appropriate technique for noninear systems contro. It provides a systematic approach to the probem of maintaining stabiity and consistent performance in the face of modeing inaccuracies 4. Because J disturbance and atmospheric drag are dominant perturbations in ow-earth orbits, we consider them as externa disturbances in this study. hey have bounded vaues based on spacecraft atitude, so for D in (6) we have i where the positive constant q i is considered as 5,6 he siding surface is chosen as D q i,,3 (7) i i 7 q i 0 km s (8) (6) s e e (9) 3 where e d and e d. d, d R are the desired reative position and veocity of the foower. s [ s, s, s3 ] and diag (,, 3 ) are siding surfaces and sopes of siding surfaces, respectivey. According to the siding mode contro methodoogy 4, the contro input aw is determined as u C ( ) F ( ) e k sgn( s ) (0) f d where k diag ( k, k, k 3) is the gain matrix determined based on the amount of uncertainties, disturbance and reaching time to siding surface. Sign function vector sgn( s) is a coumn matrix of sign functions s s s s () sgn( ) sgn( ),sgn( ),sgn( ) 3 o guarantee stabiity, Lyapunov second method is used; V s s is considered as Lyapunov function and k i is determined in such a way that V 0. hus by choosing ki qi, the cosed-oop system wi be gobay asymptoticay stabe. In practice, sign functions in Equation (0) can be repaced with continuous saturation functions to eiminate chattering phenomenon, then the controer aw becomes uf C ( ) F ( ) d e k sat ( s,) () 4

6 where sat( s, ) sat( s, ), sat( s, ), sat( s, ) and saturation function sat( s, ) is 3 where is a positive scaar constant. Backstepping Siding Mode Contro sat ( s, ) s s (3) i i i In order to utiize the benefits offered by the siding mode and backstepping controer, these two techniques are combined to deveop a backstepping siding mode controer for spacecraft formation. First, state-space form of the noninear dynamic (3) is considered X, X X : racking error can be defined as X X X C ( ) X F ( ) D uf d i (4) X X (5) so X is considered as virtua contro for first subsyestem. It can be determined from the first Lyapunov function as foows where 0 ; accordingy V is Virtua contro input is seected as V Z Z (6) V Z Z Z X (7) d X K Z (8) d where K is a positive diagona matrix. So V becomes V Z K Z (9) which means that the first subsystem is asymptiticay stabe. he second step of the controer design foows by defining the new variabe Z X. So we have Siding surface is defined as Z X X d K Z Z K Z Z C ( ) X F ( ) D uf (0 s Z Z () where diag (,, 3) are the sopes of siding surfaces. Second Lyapunov function is considered as foows Differentiating () gives V V s s () 5

7 V Z Z s s Z Z K Z s Z Z Z Z Z K Z s C ( ) X F ( ) D u Z K Z f Backstepping siding mode contro aw is designed as (3) uf C ( ) X F ( ) Z K Z K s K 3 sgn s (4) where K and K 3 are positive diagona matrices. hus Equation (3) is written as We know that Di qi, so we have V Z Z sgn Z K Z s K s s D K s 3 (5) s D K sgn( s ) s ( q K ) 3 3 i i 3i i thus by choosing K3i qi, the time derivative of Lyapunov function is simpified to (6) Note that Equation (7) can be rewritten as where V Z Z Z K Z s K s (7) V Z QZ (8) K K K I Q K I K and K K 4 I 0 matrix Q is positive definite; herefore Z and Z wi converge to zero, and the asymptotica stabiity of the proposed backstepping siding mode contro system is guaranteed. Z Z Z. According to the Schur s emma 7, if In order to overcome the chattering phenomenon, saturation function (3) is considered instead of sign function in the proposed contro input (4). SIMULAION RESULS AND DISCUSSION he noninear mode of spacecraft formation (3) is used for simuation purposes. For contro part, it is assumed that and are measurabe and avaiabe. he perturbation due to the Earth s obateness with respect to the inertia frame is given as 5 (9) D J 5Z X 3X 7 5 r r JR e 5Z Y 3Y 7 5 r r 3 5Z 9Z 7 5 r r (30) 6

8 where J , Re km is the mean equatoria radius of the Earth. his perturbation can be transformed to the moving frame by using coordinate transformation. Atmospheric drag is aso given as 6 D CDS ˆ drag v v (3) m where m is the mass of spacecraft, C D is the drag coefficient, S is the effective surface, is the oca atmosphere density, v is the reative veocity of the spacecraft with respect to the atmosphere and ˆv is the reated unit vector. Constant parameters are assumed as foows: m 00kg C S 0.5m D Figure. hree-dimensiona rajectory of the Foower Using BSMC he most significant density variation is due to the height above the surface. Locay, this variation might be accounted for by the barometric formua 6 h h0 ( h) 0 exp H where 0 is the density at the reference height h 0, and H is the scaing height at h 0. he reference vaues are seected as 8 h km kg m H 7.835km he initia orbita eements of the eader are supposed as: a km e 0. i (3) 7

9 Figure 3. ota racking Error he desired reative trajectory of the foower is a circuar formation with a radius of km in x- y pane 9. he center of the desired formation is ocated on (0000,0,0)m. he period of the foower movement entirey ies on the eader spacecraft anguar veocity around the Earth and is obtained as 6300s. he initia reative errors of the foower in the moving frame are chosen as ( ex e y ez ) ( )(m) ( ex e y ez ) (0. 0)(m / s) he parameters of the both controers are considered beow diag(,,), k (4 0 ) diag(,,) (60 ) diag(,,), K (60 ) diag(,,), K diag(,,), K (0 ) diag(,,) (33) 5 and 0 for both controers. 8

10 Figure4. racking Error aong x-, y- and z- Axis Figure5. Reative Veocity Error aong x-, y- and z- Axis According to the seected gains, stabiity condition for both controers is fufied. Figure shows a three-dimensiona view of the foower motion reative to the eader using the proposed 9

11 backstepping siding mode controer input. System s response in reducing the tracking errors during one period of movement using both controers have been presented in Figure 3 and Figure 4. Reative veocity error has been aso demonstrated in Figure 5. Figure 6. Contro Inputs of the Foower Using BSMC and SMC As shown in the figures, the backstepping siding mode controer has more accurate response in tracking the desired formation. Figure 6 shows contro inputs, and fue cost has been demonstrated in Figure 7. Fue cost can be obtained as t u u u (34) 0 x y z V dt abe. Comparison of Both Controers at the End of One Orbit Controer e ( m ) V( ms) BSMC SMC he performance of both controers at the end of one orbit have been compared in abe, in which e is tota tracking error and V is fue cost. Based on the resuts, the proposed backstepping siding mode controer has a superior performance in tracking the desired trajectory and aso has ower fue cost compared with the conventiona siding mode controer. 0

12 CONCLUSION Figure7. Fue Cost of the Foower Using BSMC and SMC In this study, a backstepping siding mode controer was designed for spacecraft formation contro on eiptica orbits. he controer design was based on the noninear mode of spacecraft formation whie perturbation of J and atmospheric drag were considered as externa disturbances. Using Lyapunov second theory, the stabiity of the cosed-oop system was guaranteed. In comparison the performance of the proposed backstepping siding mode controer to a conventiona siding mode controer, simuation resuts confirmed more efficient and superior performance of the proposed controer in tracking the desired formation and fue cost. REFERENCES R. Kristiansen and P.J. Nickasson, Spacecraft Formation Fying: A Review and New Resuts on State Feedback Contro. Acta Astronautica. Vo. 65, No., 009, pp H.H. Yeh, E. Neson and A. Sparks, Noninear racking Contro for Sateite Formations. AIAA J. Guidance Contro Dyn. Vo. 5, No., 00, pp L. Hui, L. Junfeng and H. Baoying, Siding Mode Contro for Low-hrust Earth-Orbiting Spacecraft Formation Maneuvering. J. Aerospace Sci. echno. Vo. 0, No. 7, 006, pp C. Pukdeboon, Second-Order Siding Mode Controers for Spacecraft Reative ransation. App. Math. Sci. Vo. 6, No. 00, 0, pp L. Hui and J. Li, ermina Siding Mode Contro for Spacecraft Formation Fying. IEEE rans. Aerospace Eectron. Sys. Vo. 45, No. 3, 009, pp J. Wang and Z. Sun, 6-DOF Robust Adaptive ermina Siding Mode Contro for Spacecraft Formation Fying. Acta Astronautica. Vo. 73, 0, pp

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