needed to accelerate the maneuver process and reduce the response time This control approach can also be applied. to the synchronous attitude rotation of multiple spacecraft about single multiple given axes which is useful in the. continuous observation of a planetary surface using cameras attached to a number of spacecraft 7. 2 Modeling of Spacecraft Formation Flying, The dynamics of spacecraft formation flying has been studied by many researchers For the leader spacecraft runs. in an elliptical orbit the relative motion between the leader and the follower spacecraft is governed by the following. nonlinear equation 4 5,mf q C q N q Rl ul Fd uf 1, where mf is the mass of the follower spacecraft the relative position vector q R3 x t y t z t. and are the orbit angular velocity and acceleration Rl is the distance from the Earth to the leader space. craft uf R3 ufx ufy ufz and ul R3 ulx uly ulz are the control vectors for the follower and the. leader spacecraft Fd R3 Fdx Fdy Fdz T is the constant disturbance difference vector the Coriolis like matrix. C R3 3 and the nonlinear term N q Rl ul R3 are,mf K x y z Rl x mf 2 x mf y ulx. C 2mf 1 0 0 N mf K x y z Rl y Rl R 2,l mf 2 y mf x uly 2. mf K x y z Rl z ulz,with K x y z Rl x2 Rl y 2 z 2 2. Following a similar procedure as that in 4 the following linear parameterized equation for Eq 1 can be obtained. mf p C q N q Rl ul Fd W p q q Rl ul uf 3, where p R3 px py pz T is a dummy variable W R3 5 is the regressor matrix that is composed of known. functions R4 mf Fdx Fdy Fdz is the system s constant parameter vector and t is defined as its. estimate vector and W can be explicitly defined as follows. px 2 y K x y z Rl 2 x y ulx 1 0 0,py 2 x K x y z Rl y Rl R 2. l x 2 y uly 0 1 0,pz K x y z Rl z ulz 0 0 1,3 Adaptive Synchronization Controller. The implementation of formation flying depends on accurate relative position control In this paper we first consider a. leader follower formation configuration to develop the controller Then we apply the controller to the case of multiple. spacecraft formation flying, By defining qd t R3 xd t yd t zd t T as the desired relative position trajectory and assuming its first. two time derivatives are bounded the position tracking error e t R3 becomes. e t qd t q t 5,3 1 Generalized Synchronization Error. Synchronization error is used to identify the performance of the synchronization controller i e how one trajectory. converges with respect to each other There are various ways to choose the synchronization error In this paper we. propose the following synchronization error t which is a linear combination of position tracking error e t. where 1 2 n T Rn 1 T Rn n is a generalized synchronization transformation matrix. By choosing different matrix T we can form different synchronization errors In our investigation we choose the. following synchronization transformation matrix, From Eqs 6 7 we know that if e t 0 and t 0 can be realized at the same time ei t i 1 2 3 will. go to zero at the same rate Therefore the control objective becomes to achieve e t 0 and t 0 as t in. the presence of unknown parameters,3 2 Controller Development. For controller design a coupled position error e t e 1 e 2 e n T Rn which contains both the position. tracking error e t and the synchronization error t is further introduced 9. e t e t BTT d 8, where B diag is a positive coupling gain matrix the corresponding coupled velocity error is e t. e t BTT t and the detailed coupled position error is. e 1 t e1 t 2 1 2 n d,e 2 t e2 t 2 2 3 1 d,e n t en t 2 n n 1 1 d. It can be seen from Eq 9 that the synchronization error i t appears in e i t as 2 i t and i t in e i 1 t. and e i 1 t In this way the coupled position errors are driven in opposite directions by i t which contributes to. the elimination of the synchronization error i t, The coupled filtered tracking error r t Rn is defined as 4 9. r t e t e t 10, with the constant diagonal positive definite control gain matrix Rn n. Then the controller is designed to contain an adaptation on line estimation law for unknown parameter and. feedback terms,uf t W t Kr t Ks TT t 11, where K Rn n Ks Rn n are two constant diagonal positive definite control gain matrices and the estimated. parameter t is subject to the following adaptation law. with the constant diagonal positive definite adaptation gain matrix R4 4. Therefore the closed loop dynamics for the parameter estimation error vector t b. Moreover the dummy variable p in Eq 3 has the following expression. p q d e BTT 14, Theorem 1 The proposed adaptive synchronization controller Eqs 10 11 12 guarantees the global asymptotic. convergences to zero of both the position tracking error e t and the position synchronization error t i e. lim e t t 0 15, Proof Define the following positive definite Lyapunov function. e 1 rT mf r 1,V r e 1 T K s 1,e T 1 TT d B Ks TT d 16. 2 2 2 2 0 0,and its derivative with respect to time t is. e rT mf r e T Ks TT d B Ks TT 17,After some mathematical manipulations we can get. e rT Kr TT T BKs TT T Ks 0, Following the standard process as that in 10 all signals in the adaptive synchronization controller and system. can be proved to be bounded during the closed loop operation. From Eq 18 we have r t L2 TT t L2 and t L2 Hence lim r t 0 and lim t 0 can be. obtained according to Corollary 1 1 in 10 Furthermore we can conclude that lim e t e t 0 using Lemma. 1 6 in 10 When t 0 one can get e1 t e2 t en t 0 by considering lim e t 0 and the form. of the synchronization matrix in Eq 7 Also from Eq 18 we know that V 0 only if e t 0 Therefore. lim e t 0 can be concluded using LaSalle s theorem 11 Thus we finally reach. lim e t t 0,4 Simulation Results,4 1 Leader Follower Spacecraft Pair. A leader follower formation flying configuration is considered in this section The leader spacecraft runs in an. elliptical orbit with orbital elements semi major axis a 42241 km eccentricity e 0 2 and mean motion n. 7 2722 10 5 rad s The masses of the leader and the follower spacecraft are ml 1550 kg and mf 410. kg Fd 1 025 6 248 2 415 T 10 5 N The desired relative position trajectory is chosen to be xd t. 100 sin 4 t 1 0 exp 0 05t3 m yd t 100 cos 4 t 1 0 exp 0 05t3 m zd t 0 m and the initial conditions. are q 0 30 0 b, 200 T m q 0 0 0 0 T m s 0 diag 0 8 0 7 0 7 0 8 The control and. adaptation gains are K diag 0 13 0 12 0 09 Ks diag 0 03 0 03 0 03 diag 0 04 0 04 0 04. diag 900 28 28 9 10 5 and B diag 8 0 8 0 8 0 10 4, Figure 1 shows the simulation results of adaptive tracking control of formation flying without synchronization. strategy Figure 2 gives the corresponding simulation results with synchronization strategy For the purpose of. comparison the 2 norms of e t and t after 5 hours are calculated for all simulations and listed in Table 1 It. can be seen from the results that although the position tracking error vector e t 0 can be achieved by using the. adaptive controller without synchronization strategy the differences between the position tracking errors of all axes. are large i e the synchronization errors are large However with the proposed adaptive synchronization controller. the synchronization performance can be observably improved Take X axis as an example the 2 norms of the position. tracking error and the synchronization error are 341 5 m and 1442 8 m respectively without synchronization strategy. With synchronization strategy the corresponding 2 norms have become 940 8 m and 150 7 m The synchronization. error has been remarkably reduced Moreover Table 1 shows the control efforts needed for performing these control. strategies The results show that more fuel consumption is needed for using synchronization controller For example. to maneuver and maintain the X axis relative position in 30 hours with the adaptive controller a fuel consumption. of 382 1 N s is needed However 742 6 N s is necessary for using the synchronization strategy. 4 2 Multiple Spacecraft in Formation, In this section we assume four spacecraft are requested to maneuver from their initial relative positions Ri0 i. 1 2 3 4 to the final positions Rif along the following trajectory and to form a circular formation. Rdi t Ri0 Rif Ri0 1 exp Ci t3 19,where C1 0 01 C2 0 02 C3 0 03 C4 0 04. For this case we can apply the synchronization strategy in two ways internal and external The internal synchro. nization error t is the synchronization error between different axes of one spacecraft This is the same as that. in the leader follower configuration The external synchronization error E t however denotes the synchronization. error between a given axis of all spacecraft Therefore the total coupled position error becomes. e t e t B T d A TE T E d 20, where m denotes the spacecraft number n is the number of axes of one spacecraft e eT1 eT2 eTm T e. e 1 T e 2 T e m T T T1 T2 Tm T E ET1 ET2 ETn T Rmn 1 ei ei1 ei2 ein T. ei ei1 ei2 ein T i i1 i2 in T Rn 1 Ei 1i 2i mi T Rm 1 B. diag BT1 BT2 BTm A diag AT1 AT2 ATm Rmn mn are two diagonal synchronization gain ma. trices Bi i1 i2 in T Ai i1 i2 in T T diag T 1 T 2 T m Rmn mn is. the internal synchronization transformation matrix with T i Rn n T diag T 1 T 2 T n Rmn mn. is the external synchronization transformation matrix with T i Rm m and another transformation matrix TE is. 1 for i 1 n j j 1 m i i 1 2 m,0 for others j 1 2 n. Figure 3 gives the simulation results using internal synchronization strategy only Figure 4 gives the results with. both internal and external synchronization strategies Table 2 shows the parameters and control gains for MSFF. simulation Other gains are kept the same as those in the leader follower configuration It can be seen from these. simulation results that the synchronization errors of these four spacecraft about any given axis X Y and Z have. been remarkably reduced by applying the external synchronization strategy. 5 Conclusions, This paper presents the development of an adaptive nonlinear synchronization controller for Multiple Spacecraft. Formation Flying MSFF With this controller both the position tracking errors and the position synchronization. errors can be guaranteed to globally converge to zero even in the presence of uncertain parameters Different from the. previous adaptive controllers for formation flying this controller can achieve synchronized motion among multiple. axes of one spacecraft and or any given axis of multiple spacecraft while realizing the convergences of position. tracking errors Simulations are conducted on the leader follower configuration and multiple spacecraft formation. flying to verify the effectiveness of the proposed controller Future work under consideration includes 1 adaptive. synchronization control for MSFF with time varying parameters 2 adaptive synchronization control for MSFF with. synchronization between translational orbital motion and attitude motion. References, 1 Vassar R H and Sherwood R B 1985 Formationkeeping for a pair of satellites in a circular orbit Journal of. Guidance Control and Dynamics 8 2 pp 235 242, 2 Carpenter J R 2002 Decentralized control of satellite formations International Journal of Robust and Nonlinear. Control 12 2 3 pp 141 161, 3 Vadali S R Vaddi S S and Alfriend K T 2002 An intelligent control concept for formation flying satellites. International Journal of Robust and Nonlinear Control 12 2 3 pp 97 115. 4 de Queiroz M S Kapila V and Yan Q 2000 Adaptive nonlinear control of multiple spacecraft formation flying. Journal of Guidance Control and Dynamics 23 3 pp 385 390. 5 Wong H and Kapila V 2003 Adaptive learning control based periodic trajectory tracking for spacecraft formations. In Proceedings of the 42nd IEEE Conference on Decision and Control pp 3597 3602. 6 Kang W Sparks A and Banda S 2001 Coordinated control of multisatellite systems Journal of Guidance Control. and Dynamics 24 2 pp 360 368, 7 Wang P K C Hadaegh F Y and Lau K 1999 Synchronized formation rotation and attitude control of multiple. free flying spacecraft Journal of Guidance Control and Dynamics 22 1 pp 28 35. 8 Koren Y 1980 Cross coupled biaxial computer controls for manufacturing systems ASME Journal of Dynamic. Systems Measurement and Control 102 2 pp 256 272, 9 Sun D 2003 Position synchronization of multiple motion axes with adaptive coupling control Automatica 39 6. pp 997 1005, 10 Dawson D M Hu J and Burg T C 1998 Nonlinear Control of Electric Machinery Marcel Dekker Inc. 11 Khalil H 1996 Nonlinear Systems 2nd ed Prentice Hall. Table 1 Performance evaluation wihtout with synchronization strategy. Error Control Without With,ex 2 m 78 8 940 8,ey 2 m 341 5 940 8. ez 2 m 1540 1 887 3,x 2 m 419 2 88 1,y 2 m 1442 8 150 7. R z 2 m 1576 7 160 3,R uxf N s 382 1 742 6,R uyf N s 498 9 527 9. uzf N s 115 7 424 0, Table 2 Parameters of multiple spacecraft in formation flying. Parameter Value i 1 2 3 4,mfi kg 410 500 600 660,Fdi 10 5 N 1 025 6 248 2 415 1 9106 1 960 1 517. 1 925 4 850 2 455 2 250 6 850 3 156,Ri0 m 150 10 20 10 130 20. 140 10 20 30 160 20,R i0 m 0 0 0,Rif m 100 0 0 0 100 0 100 0 0 0 100 0. R if m 0 0 0,0 0 7 0 85 1 15 1 3,Bi 10 3 8 0 8 0 8 0. Ai 10 3 8 0 8 0 8 0,x tracking error,y tracking error. 150 z tracking error,position tracking error m,y m 100 100 x m 0 5 10 15 20 25 30. 200 200 time hr,x synchronization error x axis,y synchronization error y axis. Adaptive Synchronization Control of Multiple Spacecraft Formation 2 Modeling of Spacecraft Formation Flying The dynamics of spacecraft formation ying has been

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