Nonlinear attitude control of a tether-connected multi-satellite in three-dimensional space

Transcription

1 IEEE ransations on Aerospae and Eletroni Systems Vol. 46 o. 4 pp Otober onlinear attitude ontrol of a tether-onneted multi-satellite in three-dimensional spae Insu Chang Student ember IEEE Sang-Young Park and Kyu-Hong Choi Abstrat he objetive of the urrent researh is to analyze the attitude dynamis and ontrol of a tethered satellite formation flying the tethered units are modeled as extended rigid bodies. he three-inline array system is used in this study and the general formulation of the equations of motion of the system is obtained through a Lagrangian approah. In this researh attitude motions of the tethered satellite system are analyzed in a three-dimensional free-spae system to omplement previous works. he State-Dependent Riati Equation (SDRE ontroller is used to regulate the attitude errors. he stability region for the SDRE-ontrolled tethered satellite system is also estimated using a numerial method to show globally asymptoti stability for the ontrol method. Centralized and deentralized approahes are applied to the dynami system to ompare the performane of ontrolling the attitude motion. he SDRE ontroller performs well in both the entralized and deentralized approahes for the attitude ontrol of tethered satellites in formation flying. Index erms Centralized and deentralized approah onlinear attitude ontrol SDRE ontroller ethered formation flying S I. IRODUCIO AELLIE tethered formation flying is a system onsisting of satellites onneted by tethers to implement a unified spae mission. he onept of ontrol for tethered formation flying has attrated onsiderable attention in the last thirty years beause it has several advantages suh as the signifiant redution of fuel onsumption the improvement of ontrol auray ompatibility with various spae missions et. [] [3]. Beause of these benefits the tethered formation flying system has been applied to various spae missions suh as PF anusript reeived April 7 8; revised January 4 9. his work was supported by the Korean Siene and Engineering Foundation (KOSEF through the ational Researh Laboratory Program funded by the inistry of Siene and ehnology (o. 68-6j-8. Insu Chang is with Astrodynamis and Control Lab. Department of Astronomy Yonsei University Seoul -749 Republi of Korea ( ihang@al.yonsei.a.kr. Sang-Young Park is with Astrodynamis and Control Lab. Department of Astronomy Yonsei University Seoul -749 Republi of Korea (orresponding author to provide phone: ; fax: ; spark@galaxy.yonsei.a.kr. Kyu-Hong Choi is with Astrodynamis and Control Lab. Department of Astronomy Yonsei University Seoul -749 Republi of Korea ( khhoi@galaxy.yonsei.a.kr. [4] DARWI [5] SPECS [6] et. For tethered formation flying both the relative positions and the attitudes of the individual satellites need to be aurately ontrolled to point toward the same inertial target [7]. Attitude dynamis and ontrol for tethered satellite systems have reeived onsiderable attention in many papers [8] []. However these studies onsidered a gravity-gradient Earth-orbiting tether while the attitude ontrol of tethered platforms in tether formations has not been studied. enon and Bombardelli [7] investigated the possibility of utilizing a passive ontrol sheme for attitude ontrol of individual satellites. However the passive ontrol methods have limitations in ontrolling dynami systems espeially in settling time and regulating the errors of the systems [3]. Chung et al. [4] studied the attitude motion in tethered formation flying by introduing ontration theory. hey also evaluated the ontrol algorithm by applying it to a hardware system. However they desribed the attitude motions in two-dimensional spae beause the out-of-plane attitude motion is assumed to be deoupled from the in-plane attitude motion. When the moment of inertia with respet to eah axis is not the same the out-of-plane attitude motion is atually oupled with the in-plane attitude motion. Hene attitude motions in three- dimensional spae should be studied. It is very hallenging to formulate dynami systems and ontrol the systems in three-dimensional spae. he main objetive of the urrent study is to analyze the dynamis and ontrol of a tethered satellite formation the tethered units are modeled as extended rigid bodies. In this study the tethered formation onsists of three satellites that form the shape of an inline array in three-dimensional spae. In the urrent researh the equations of motion are formulated for satellite attitude motions of tethered formation flying in three-dimensional free spae. he general formulation of the governing equations of motions for the tethered satellite system is very omplex; hene it is desirable to apply the proedure of the Lagrangian formulation to derive the equations of motion for suh a omplex dynami system. For the analysis the tethers are assumed to have negligible mass and to be inextensible and rigid. here are two main approahes to handling the multi-satellite systems: entralized and deentralized approahes. Centralized and deentralized approahes for

2 attitude motions an be represented for the tethered system. By means of deomposing the design of the global ontroller into smaller ontrol agents the ontrol an be relatively simple and very robust [5]. In the urrent study the two approahes are ompared in order to determine the better approah of ontrolling the tethered attitude motion. For ontrolling the attitude motions of tethered formation flying a nonlinear ontroller should be applied sine linear ontrollers are not suitable beause of the high nonlinearity of the tethered satellite system. One of ontributions of the urrent researh is to introdue the State-Dependent Riati Equation (SDRE tehnique for nonlinear attitude ontrol problems in the tethered satellite formation flying system. he SDRE tehnique whih is one of nonlinear sub-optimal and robust ontrol methods is relatively easy to handle and is systemati [6] so that it has been applied to various lasses of nonlinear systems. However the drawbak of the SDRE tehnique is the diffiulty it presents in estimating the stability region beause the losed-loop form for SDRE-ontrolled systems is typially not known [7]. In the urrent researh the stability region for the SDRE-ontrolled systems is estimated using a numerial method. In numerial simulations the robustness of the ontrol method is investigated in the presene of external disturbanes. he simulation results for the attitude ontrol demonstrate the effetiveness of the SDRE nonlinear ontroller. he urrent study addresses multiple hallenges. First nonlinear attitude dynamis are formulated for tethered satellite systems in a formation flying in three-dimensional free spae. Seond a nonlinear SDRE ontroller is applied for the first time to the tethered formation flying system for nonlinear attitude motion. hird the stability region for the SDRE-ontrolled attitude system is also estimated to guarantee its asymptoti stability. Lastly using the SDRE ontroller the entralized and deentralized approahes for tethered satellite formation flying are ompared to determine whih is more effetive. he urrent paper onsists of three main setions and onlusions. In Setion II the attitude dynamis of tethered formation flying in three dimensions are introdued and the equations of motion for a single-tethered system and for a three-inline array system are formulated. In Setion III the attitude ontrol systems of tethered formation flying are onstruted. he entralized and deentralized approahes are used to establish the attitude ontrol systems. he SDRE ontroller is applied as a nonlinear ontroller for the attitude ontrol systems. he stability region of the SDRE ontroller is also estimated for the tethered attitude ontrol systems. In Setion IV numerial simulations and results are presented and analyzed. he onlusions are disussed in Setion V. II. REPRESEAIO OF HE EHERED AIUDE OIOS A. Equations of otion for a Single-ethered System Before deriving the attitude motion of the three-inline array system the equations of motion of a single-tethered system are presented in order to apply the equations to the three-inline array system whih will be presented in the next sub-setion. Fig. shows the attitude motion of the single-tethered system in the three-dimensional spae. Here the x-y-z oordinate denotes a free spae. Fig. (a depits the motion of the satellite and Fig. (b shows the motion projeted in the x-y plane l and r are the tether length and the length from the enter of mass of the satellite to the end-point of the tether respetively. oreover and denote the angle between the x-axis and the diretion of the projeted tether into the x-y plane the angle between the diretion of the projeted tether into the x-y plane and the diretion of the enter of mass of the satellite from the end-point of the tether and the angle between the diretion of the tether and the x-y plane respetively. represents the angle between the diretion of the tether and the diretion of the enter of mass of the satellite from the end-point of the tether and desribes the rotational angle about the axis whose diretion is from the end-point to the enter of mass of the satellite respetively. It is assumed that the tether is set to zero mass and is inextensible; that is the length of the tether is onstant and l l. he governing equations of motion for the attitude motion of the tethered satellite system an be derived using a Lagrangian formulation of Eq. ( [8] beause of the omplexity of the dynami system. L( qq ( qq U( q d L L i dt q i qi L and U denote the Lagrangian kineti energy and potential energy respetively. i q i q and q are external torque for the ith independent variable on the body the independent variable the vetor of the independent variables and the time derivative of the independent variable vetor respetively. In the satellite attitude motion the independent variables are and. Here it is assumed that the potential term in Lagrange s equations is negleted ( U ( q beause of the assumption that the tethered satellite system operates in a very weak gravity field suh as Lagrange libration points [9]. he derivation of the kineti energies is shown in Appendix A. By solving Lagrange s equations the governing equations of motion an be expressed as follows: ( q q ( q q q ( q and. (

3 3 Here matries and are the moment of inertia matrix and the frition oeffiient matrix at hinge respetively. Eah element in the matries and is listed in Appendix B. Here prime is used to distinguish the matries and whih are used in Eq. (3 and are modified from and respetively. oreover subsript is used to distinguish the matrix here from the matrix in the total matrix for the whole dynami model disussed in the next sub-setion. ote that diretion an be easily ontrolled beause the term is deoupled from other variables. he ontrol regarding the angle is not an issue and an be easily inluded when it is independently alulated. herefore the term related to the angle is exluded in the urrent researh. Omitting the term (i.e. in the governing equations Eq. ( an be rewritten as follows: ( q q ( q q q ( q and ote that unlike the statement made by Chung et al. [4] the in-plane ( and out-of-plane ( motions are highly oupled (see eah element of and Appendix B. herefore the attitude motions of the tethered satellite system should be desribed in the three-dimensional free spae. B. Equations of otion for the hree-inline Array System he objetive of this sub-setion is to desribe the attitude motion of the three-inline array system in three dimensional free spae (Fig.. Here the x -y -z oordinate denotes the desired rotating oordinate with respet to the enter of mass of the enter satellite. he x and y -axes are in the rotating plane and the z -axis is normal to the plane. ost of the array systems need to be rotated so that the tethers are taut and straight []. Fig. (a illustrates the attitude motion of the three-inline array system in three-dimensional free spae and Fig. (b depits the motion whih is projeted in the x -y plane. he oordinates with subsripts and denote the shifted referene oordinate of the enter satellite (x -y -z to measure the angles ( and and. In order to simplify the expressions of the dynami motions angles ( and and are set to measure from the x-axis and x-y plane respetively (right image in Fig. [4]. ote that the mass size and moment of inertia of eah satellite in the tethered satellite system are assumed to be idential. he governing equations of motion for the three-inline array system are derived by using the Lagrangian formulation as used in the single-tethered system. Here the independent variables are and. and denote the angle between the x-axis and the diretion of the end-point of the tether with respet to the enter of the enter satellite and the angle of the end-point of the tether and the x-y plane respetively. he other variables were desribed in the previous sub-setion. ote that the independent variables with subsripts and refer to the variables for the first and seond satellites respetively. hen the governing equations of motion for the three-inline array beome qq ( qqq ( ( q and he elements of denote the ontrol torques for eah diretion. oreover the elements of the matries and are listed in Appendix B. Remark: he equation of motion for a spinning angle (suh as of the enter satellite along the line between two tether juntion points an be expressed as I x. his means that this equation is deoupled with other angles. herefore ontrolling this angle is exluded in the urrent researh for the same reason as ontrolling in II-A. C. Extension to ethered ultiple Satellite Systems he struture for three-inline tethered systems an be easily extended to multiple satellite systems. hat is if new satellites are added to the urrent three-inline array the same struture for and ( th new satellite in Eq. (3 will be added to 3

4 4 and in Eq. (4. he first elements of and * will be * plaed in 4( 4( elements of to and in Eq. (4. Furthermore states ( q and ontrol torques ( in Eq. (4 an be expressed as q q q qn 3 n. herefore the dimension of and in Eq. (4 for the tethered (n+-satellite array system (inluding the enter satellite is 4n. III. COSRUCIO OF HE COROL SYSE A. Centralized and Deentralized Approahes in the Control System here are two main methods of ontrolling satellites in formation: the entralized and deentralized approahes. Eah approah has its own benefits and drawbaks []. In the entralized approah the leader satellite alulates the ontrol values and then transmits the information to follower satellites. By using this method the satellites an generate proper ontrol values so that energy suh as the eletriity and fuel of the satellites an be saved. However this method an plae a high omputational burden on the enter satellite. In order to apply the entralized approah to the dynami system Eq. (4 is used to onstrut the global ontrol design. ote that the ontroller algorithm is onstruted using the SDRE tehnique desribed in the next subsetion. Hene state-spae equations are needed. he state-spae equations for the whole dynami system an be derived from equations of motion suh as Eq. (4. By multiplying obtained: q q in both sides of Eq. (4 Eq. (5 an be (5 hen the state-spae equation for the entralized approah beomes x A ( x xb ( x u (6 t t At ( x Bt ( x x E q q and u. E is an identity matrix. In the deentralized approah the global ontrol design is deomposed into small loal ontrol agents. he attitude motions of the first and seond satellites an be separated from the whole dynami motion so that the three satellites an be desribed independently. hat is in the matries and in Eq. (4 the equations of motion of the first and seond satellites an be divided into independent equations beause they are deoupled exept for the terms related to the enter satellite. Hene the equations of motion for the whole tethered satellite system an be divided into three equations of motion for eah satellite. q q q q q q q q q q q q : for the enter satellite : for the first satellite : for the seond satellite q q q q q q In order to apply the SDRE ontrollers to the ontrol algorithms using the deentralized approah the state-spae equation for eah satellite should be derived separately. he state-spae equations for the loal ontrol agents (the first and seond satellites for the dynami system an be derived from equations of motion suh as in Eq. (7. By multiplying and in both sides of the three equations in Eq. (7 respetively the dynami equations an be expressed as follows: q q q q q q q q q q q q Here on the right sides of both equations q (7 q q an be regarded as disturbanes beause states are not related to the system s states. For the same reason q q and q q an be regarded as disturbanes from the seond and third equations in Eq. (8. hus these terms are finally omitted from the equations of motion and are inluded as disturbane torques (8 4

5 5 when numerial simulations are performed. hus the state-spae equations of the deentralized approah for the three satellites an be written as x A( x x B( x u : for the enter satellite x A( x xb( x u : for the first satellite (9 x A ( x x B ( x u : for the seond satellite x q q x q q x q q u u u A( x A( x E E 44 A( x E B( x B( x and B( x Remark : If (n+ satellites are used for the tethered formation system as mentioned in the previous hapter then the equations of motions for eah satellite an be straightforwardly expanded from and in setion II-C extended to q q q q : for the enter satellite q q q q : for the first satellite q q q q : for the seond satellite (7' q q q q : for the third satellite q q q q : for the n satellite. n n n n n n n th hen the extended state-spae equations (Eq. (9 for eah satellite in multiple tethered systems an be expressed as x A( x x B( x u : for the enter satellite x A( x xb( x u : for the first satellite x A( x x B( x u: for the seond satellite x 3 A3( x3 x3 B3( x3 u3 : for the third satellite (9' x A ( x x B ( x u : for the n satellite. n n n n n n n th B. SDRE Controller for the ethered Formation Satellites In order to regulate the state errors a suitable ontroller is required. In the ase of a tethered satellite system linear ontrollers are not suitable beause of the high nonlinearity of the tethered satellite system. Even with small state errors the linear ontrollers annot make the dynami system s errors onverge to zero. herefore nonlinear ontrollers are needed. oreover if possible robustness and optimality properties should be applied to the dynami system to guarantee the fail-safety and energy or time saved. here are many nonlinear ontrol methods. In the urrent researh the SDRE tehnique [6] is applied to the dynami systems. he SDRE ontroller is a nonlinear sub-optimal and robust ontrol method. Beause the ontrollers onstrut the dynami model matries diretly from the nonlinear dynami equations this ontrol method uses few assumptions for onstruting the ontrollers. oreover beause the method of proessing the SDRE ontrollers follows that of the LQR ontrollers onstruting the SDRE ontrollers is relatively easier than onstruting other nonlinear ontrollers. Due to this benefit the SDRE ontrollers have been applied to the satellite systems [] [5]. Like in LQR ontrollers state-spae equations suh as Eq. (6 or Eq. (9 are needed to apply the SDRE ontrollers to the dynami systems. Unlike LQR ontrollers whih use a onstant or time-varying dynami model matrix ( A and B or A ( t and B ( t however the SDRE ontrollers update the dynami model matrix ( Ax ( and the input oupling matrix ( Bx ( in the state-spae equation in every given time step. hus the dynami model matrix in the SDRE ontrollers an reflet real dynami motion. Hene the SDRE ontrollers an determine the proper ontrol values for the nonlinear dynami motions. From the state-spae equations that are desribed as Eq. (9 the initial state values x ( t x ( t and x( t are given and x n ( t R ( n x t R ( n x t R u m ( t R ( m u t R and ( m u t R are assumed. he performane index to be minimized is J i i i i t i i i xqx uru dt ( i denotes the index of satellite in the three-inline array system. Q i and R i are a real symmetri positive semi-definite matrix and a real symmetri positive definite matrix respetively. he optimal ontroller for attitude ontrol is given by u K ( x x. ( i i i i Ki( x i is the optimal feedbak gain matrix for the ith satellite suh that K x R x B x P x ( i( i i ( i i ( i i( i. Pi is the unique positive-definite solution to the following algebrai Riati equation for the ith satellite: A ( x P( x P( x A ( x P( x B R ( x B P( x Q i i i i i i i i i i i i i i i i i (3 When using the same method the SDRE ontroller of the entralized approah an be established for the three-inline array system. C. Estimation of the Stability Region for the SDRE Controlled System Beause of the diffiulty of finding the losed-loop form for the SDRE ontrolled system it is hard to hek the stability of 5

6 6 the SDRE ontrolled dynami system. Beause the urrent researh mainly studies the deentralized approah the stability regions for the deentralized approah ase are estimated. In fat the stability regions for the SDRE ontrolled entralized three-inline array an be heked using the same method. In the urrent study the method derived by Brai et al. [7] is used to estimate the stability regions for the SDRE ontroller of the three-inline array system. In order to use Brai et al. s method the linearized state-spae equations from the state-dependent state-spae equations are required. Hene by using the Jaobian method the nonlinear state-spae equations suh as Eq. (9 must be linearized. In order to simplify the alulation Eq. (4 an be linearized in advane. hen the model matries and are linearized to the following form: ( q ( q u (4 ( and ( are the matries whih are linearized from and respetively. he omponents of ( and ( are listed below: ( ( 44 ( 88 mr Iz ( 3 ( 7 ( 5 ( 9 mr mrl ( 33 ( 77 m( rl Iz ( 34 ( 78 mr mrl I z ( ( 66 mr Iy ( 4 ( 8 ( 6 ( mr ( 55 ( 99 m( rl Iy ( 56 ( 9 mr mrl I y. ote that some omponents not listed have zero values. herefore the linearized state-spae equation (Eq. (5 an be easily obtained from Eq. (4. Equation (5 an be also derived by diretly linearizing the nonlinear Eq. (6. he two results are the same for the partiular problem treated in this paper beause the SDRE ontroller is a regulator. he state-spae equations (Eq. (6 for eah satellite an be obtained by separating from Eq. (5 beause of the deoupling of the first and seond satellites in the equation. x A( xb( u (5 x A( x B( u : for the enter satellite x A( xb( u : for the first satellite (6 x A( x B( u : for the seond satellite A( A( E A( E E A( ( B( ( B( E ( B( and ( B( he values of the matrix Q are needed to find the solution ( P for the Lyapunov equation [6] defined as follows: PA A P Q. (7 A denotes a linearized dynami model matrix suh as A A A and A. Here if Q is positive definite the origin whih has the dynami model matrix ( A is asymptotially stable. hen the solutions of the Lyapunov equations defined as Eq. (8 an be obtained from Eq. (7. APPA Q AP PA Q AP PA Q (8 In the urrent researh Q Q and Q are equal to the weighting matrix used in the SDRE ontrollers. he values of Q * matries in Eq. (9 were hosen based on the ontrol riteria: the angular errors should be onverged to. rad within 3 se. Q diag 5 5 Q diag Q diag (9 From the solutions of Eq. (8 the Lyapunov funtions and time derivatives of the Lyapunov funtions an be derived as below: V ( x x Px V ( x V ( x V ( x x Px x Pf( x V ( x x Px x Pf( x V ( x x P x x Pf( x xpx xpx ( ( x f( x A( x x B( x u Ax ( Bx ( Kx ( x. he area satisfying the onditions that Eq. ( is positive and Eq. ( is negative belongs to the stability region aording to Lyapunov loal stability [7]. Fig. 3 shows the simulation results for the stability regions for the SDRE ontroller of the three-inline array satellite systems. Unfortunately beause of the dimension problem not all states an be depited in three dimensions. Hene some states are set to speifi values and the other states are plotted. Fig. 3 shows one of the stability regions under the assumption that some state variables are fixed. Fig. 3(a desribes the stability region of the enter satellite under the assumption that.rad/s. Fig. 3(b shows the stability regions for the first satellite under the assumption that magnitudes of all angular veloities are equal to.rad/s. ote that the grids in the graphs denote the 6

7 7 asymptoti stability regions. In Fig. 3(a the dynami system is guaranteed asymptoti stability in all state regions. In Fig. 3(b however the first satellite an be unstable in some state regions (un-grid regions. Hene the ontrol logi should be modified. By replaing and modifying omponents in the matries ( Ax ( and Bx ( the ontrollability and stability onditions an be hanged. For all the state regions all satellite systems must be stable for the SDRE ontroller. ote that the state-dependeny of the input oupling matrix ( Bx ( an affet the stability ondition of the SDRE ontrolled system. So the input oupling matrix an be adjusted to alulate the ontrol values that make all the satellite systems stable. In other words the ontrol values that orrespond to the adjusted input oupling matrix an make the feedbak system stable. In this researh the input oupling matrix is linearized to remove the unstable regions in Fig 3(b. he linearized input oupling matrix ( B ( is used to alulate the P( x matrix from the Riati equation of PxA(x ( Ax ( Px ( PxBRx ( ( ( B ( Px ( Q. hen the ontrols an be provided as u( t Kxx ( Rx ( B( Pxx. ( he linearized input oupling matrix is used to estimate the ontrol values that serve as feedbak to the original nonlinear dynamis x f( x A( x xb( x u( A( x B( x K( x x. Hene the harateristis of the original system dynamis an be kept. Fig. 4 shows the results of the stability regions using a linearized B matrix. It is verified that these angles in the stability regions over from 9deg to 9deg. hese are the reasonable regions for the satellite attitude motions. As seen in Fig. 4 all of the verified regions turned out to be globally asymptotially stable. he environment model does not math the ontroller model beause the environment model has the term of worst-ase disturbanes as mentioned in setion IV. he globally asymptoti stability is shown under the environment model with worst-ase disturbanes. he stability of the seond satellite is also verified by using the same proedure. herefore the SDRE ontrolled three-inline array system using the deentralized approah is guaranteed to be globally asymptotially stable. When using the same method it an be shown that the dynami system using the entralized approah of the SDRE ontrolled three-inline array system is also globally asymptotially stable. IV. UERICAL SIULAIOS AD RESULS For numerial simulations it is assumed that all state information an be measured perfetly. here are two primary types of simulations: one is for various initial errors and the other is for various tether lengths. In the numerial simulations the SDRE algorithm is used for both the entralized and deentralized approahes and the performanes of the two approahes are thus ompared. For the main parameters of the satellites in the tethered satellite system used in the simulations it is assumed that the mass the length from the enter of mass of the satellite to the attahment point of the tether and the moment of inertia are equal to m 3kg r.75m and I diag( kgm respetively. oreover it is assumed that there are disturbane torques in eah axis. he total magnitude of the disturbane torques applied to the axes in eah satellite is set to -5 m as the worst-ase and its diretion is hosen randomly using sine funtions. he disturbane torques ating on the satellites for simulation are as -5 follows: [.6sin( t/3.8sin( t/ 47] m for the enter satellite -5 [.5sin( pt / 34.73sin( pt /5.4sin( pt / 67.68sin( t / 3] m for the first satellite and 5 [.sin( t/ 4.66sin( t/ 7.3sin( t/ 76.75sin( t/ 59] m for the seond satellite respetively. he magnitude of the 4 potential at the L point is about 7.4 times that of the altitude of 6 km on the Earth (less than -5 m [8]. oreover beause of the distane effet the disturbane torque is muh smaller than that on the Earth due to the gravity gradient at the L point. hus we set -5 m as the magnitude of the worst-ase disturbane torques in order to model a true environment inluding atuator misalignment gravity gradient torque solar radiation torque ignoring potential energy in the equations of motion the oupling between axis et. Under the worst-ase environment the stability and performane of the system are numerially evaluated. In the first type simulation the tether length is set to a onstant value ( l m. In this simulation the performanes of the entralized and the deentralized approahes are ompared. he SDRE ontrollers are well-onduted in the simulation of both the entralized and the deentralized approahes. hree tests are performed under different initial state errors; the state errors in the first test are relatively small values and those in the third are relatively large. Detailed information for the initial state errors is as follows. he initial state errors for the first test are [ ] [. -.] rad/s [ ] [. -.] rad [ ] [ ] rad/s [ ] [ ] rad [ ] [.. -..] rad/s [ ] [ ] rad. he initial state errors for the seond test are [ ] [.5.4] rad/s [ ] [. -.3]rad [ ] [ ] rad/s [ ] [ ] rad [ ] [ ] rad/s [ ] [ ] rad. he initial state errors for the third test are [ ] [.6 -.7] rad/s [ ] [-.5.4]rad [ ] [ ] rad/s [ ] [ ] rad [ ] [ ] rad/s [ ] [ ] rad. Fig. 5 shows the results of the omparisons of the entralized and deentralized approahes. he graphs are the results for the third test. Graphs in Fig. 5(a show the results for the enter satellite. All angular errors and angular rate errors are 7

8 8 onverged to zeros in 3s. Graphs in Fig. 5(b show the results for the first satellite. he ontrols for body angles suh as are onverged quiker than the tether angles suh as and. he reason for this is that in the ase of the regulation of the tether angles ( and the satellites have to make a ontrol fore using the satellite s orbit thrusters. hat is these tether angles are ontrolled by translation motions of eah satellite. In real situations there may be physial limitations that prevent these motions suh as maximum ontrol values from thruster and reation wheel assembly. he last graph in Fig. 5(b shows the results of the magnitude of ontrol torque used to regulate the state errors. ote that the magnitudes of the ontrol torques are almost 3 m in the early stage of the test. However beause the tether length is set to m the real ontrol fores required to make the ontrol torques are lower than 3. he first and the seond tests yield very similar phenomena. he entralized and deentralized approahes use global and loal ontrollers respetively. he two approahes give almost the same results even though the ontrol systems are different. oreover the SDRE ontrollers are well-proessed in these simulation results. ables and show the results for the total magnitude of the angular momentum for eah axis. he results are obtained by integrating the magnitude of ontrol torques applied to the dynami system from the initial time to the time when all angular errors are onverged to -5 rad. In the results the ontrol system of the deentralized approah spends ~7 % more total angular momentum (sum of angular momentum for eah axis than that of the entralized approah to regulate the errors. Hene the deentralized approah is less effetive in regulating the state errors in the early stages of most test results. hat is the ontrol system of the deentralized approah spends more time and requires more total angular momentum to regulate the state errors than the ontrol system of the entralized approah. he reason for this phenomenon is that the ontrol system of the deentralized approah regards the state errors of the enter satellite as disturbane errors. Consequently the ontroller of the deentralized approah annot generate suitable ontrol torque in the early stage. On the other hand the entralized approah an minimize ontrol values beause the ontroller inludes all the states of the whole dynami system. In the sense of the omputational load however the entralized approah (global ontrol system needs more time to ompute the ontrol torques beause the sizes of the dynami model matrix and input oupling matrix are.5 times larger than those of the matries for the deentralized approah (loal ontrol system. In the seond type simulation the tether length is varied as l m. Under the large initial state errors the two approahes are applied to the dynami system independently as the tether length varies. he SDRE ontrollers are well-proessed in the simulations both of the entralized and deentralized approahes; that is all state errors onverge to zero. ables 3 and 4 show the total angular momentum values for the entralized and deentralized approahes respetively. Regardless of the tether length the deentralized approah usually requires a more total angular momentum than the entralized approah. If the soure for ontrol fores is thrust the deentralized approah generally needs more fuel. V. COCLUSIOS In the urrent researh ontrol systems for tethered satellite formation flying are investigated. he equations of motion for satellite attitude in three-dimensional free spae are formulated. o ontrol attitude errors the SDRE ontroller is for the first time utilized as a nonlinear ontrol of the tethered formation flying. he stability for the SDRE ontrolled system is also estimated using a numerial method. he SDRE ontrolled formation system is guaranteed to be globally asymptotially stable. o develop the attitude ontrol systems for the tethered formation flying the entralized and deentralized approahes are employed by utilizing the SDRE ontroller and their performanes are ompared. he deentralized approah an redue the satellite s omputational burden; however it requires relatively more ontrol fores. he entralized approah an redue ontrol efforts but it an also inrease the omputational burden to evaluate ontrol values. herefore a hybrid approah that ombines the two approahes may be preferable for the tethered formation mission in order to redue the omputational load as well as fuel onsumption. APPEDIX A Derivation of the kineti energies; A. One Satellite Case Coordinate of the enter of mass rc ( r r r3 r los os ros( os( r lsin sin ros( sin( r3 lsin rsin( Veloity of the enter of mass an be obtained by time derivatives of the oordinates of the enter of mass vc ( v v v3 v l os os lsin os l os sin r( sin( os( r( os( sin( v l os sin lsin sin l os os r( sin( sin( r( os( os( v l sin los r( os( 3 he kineti energy of the tethered satellite (rigid body an be obtained by summing the translational energy and rotational energy for the enter of mass. t m v. C Rotational kineti energy an be obtained as r IG I z Iy I x. 8

9 9 Hene the total kineti energy an be expressed as. t r B. hree-inline Array Case Coordinate of the first satellite s enter of mass r C ( r r r3 r ros oslos os ros( os( r ros sinlos sin ros( sin( r3 rsin lsin rsin( Veloity of the first satellite s enter of mass an be obtained by differentiating the oordinate of the first satellite s enter of mass v C ( v v v3 v r sin osr os sinl sin os l os sin r( sin( os( r( os( sin( v r sin sinr os osl sin sin l os os r( sin( sin( r( os( os( v r os l os r( os( 3 mrl os osmr os os( Coordinate of the seond satellite s enter of mass 6 mr sin sin( os( mr os os( r C ( r r r3 7 mrlsin os sin( mr sin os( sin( r ros oslos os ros( os( 8 mr sin os( sin( r ros sinlos sin ros( sin( 9 mrlsin sin os( mr sin sin( os( r3 rsin lsin rsin( mrl os os mr os os( Veloity of the seond satellite s enter of mass an be obtained mr sin sin( os( mr os os( by differentiating the oordinate of the seond satellite s enter 33 ml os mr os ( mrl os os( os I z of mass 34 mr os ( mrlos os( os Iz v C ( v v v3 35 mrlsin os( sinmrl ossin( sin v r sin osr os sinl sin os l 36 mrlos sin( sin os sin 44 mr os ( Iz r( sin( os( r( os( sin( 45 mrlsin os( sin v r sin sinr os osl sin sin l os os 55 mrlsinsin( osmrlosos( r( sin( sin( r( os( os( ml mr I y v3 r os l os r( os( 56 mr mrlsin sin( os mrl osos( I y Coordinate of the enter satellite s enter of mass r 66 mr Iy C (. 77 ml os mr os ( mrl os os( os I he kineti energy of the tethered satellites an be obtained by summing the translational energy for enter of mass and 78 mr os ( mrlos os( os Iz rotational energy for the enter of mass for three satellites. 79 mrlsinos( sin mrlossin( sin t t t t m vc m vc 7 mrlossin( sin 88 mr os ( Iz r r r r Iz Iy 89 mrlsin os( sin 99 mrlsinsin( os mrlosos( Iz Iy I z I y ml mr I y otal energy an be obtained by summing the kineti energy for 9 mr mrlsin sin( os mrlos os( Iy translational and rotational energies. mr Iy Ix. t r APPEDIX B Eah element in the matries and in Eq. ( and in Eq. (3 and and in Eq. (4 are listed as follows: mr os I z 3 mrlosos os( mr os os( os( 4 mr os os( os( 5 mrlossin sin( mr ossin( sin( 6 mr ossin( sin( 7 mrl os osos( mr os os( os( 8 mr os os( os( 9 mrlossin sin( mr ossin( sin( mr ossin( sin( mr I y 3 mrlsin os sin( mr sin os( sin( 4 mr sin os( sin( mrlsin sin os( mr sin sin( os( 5 z 9

10 mr os sin mrl os os sin( mr ( os sin( 3 os( mr ( os os( sin( mr ( os sin( os( mr ( 4 os os( sin( mrl os sin os( mr ( os os( 5 sin( mrl os os sin( mr ( os os( sin( 6 7 mrl os os sin( mr ( os sin( os( mr ( os os( sin( mr ( os sin( os( mr ( 8 os os( sin( 9 mrl os sin os( mr ( os os( sin( mrl os os sin( mr ( os os( sin( mr os sin mrl sin os os( mr ( sin sin( 3 sin( mr ( sin os( os( mr ( sin sin( sin( mr ( 4 5 sin os( os( mrl mr mrl os sin sin sin sin( ( sin os( os( mrl sin os os( mr ( os sin( mr ( sin os( os( mr ( 6 os sin( mrl sin os os( mr ( sin sin( 7 sin( mr ( sin os( os( mr ( sin sin( sin( mr ( 8 sin os( os( 9 mrl mrl mr os sin sin sin sin( ( sin os( os( mrl sin os os( mr ( os sin( mr ( sin os( os( mr ( os sin( mrl sin os os( mr os os( 3 sin( mrl os ossin( mr sin os( os( mrl os os sin( mr os os( sin( 3 mr ( os( sin( mrl( os 33 sin( os mr ( os( sin( mrl( os 34 sin( os mrl( osos( sin ml os sin mrl sin os( os 35 mrl( os os( sin 36 mr os os( sin( mr sin os( 4 os( mr os os( sin( 4 mr ( os( sin( mrl os 43 os( sin 44 mr ( os( sin( mrl os os( sin mrl sin os( os 45 mrl mrl mr 5 os sin os( sin sin sin( 5 os sin( os( mr sin sin( sin( mrl os sinos( mr os sin( os( mrl sin os mr sin os( mrl os sin( os mrl( sin os( os 53 ml ossin mr ( os( sin( mrl( sin sin( sin mrl( sin 54 os( os mr ( os( sin( mrl os sin( os mrl( sin os( os mrl( ossin( mrl( sin os( os mrl( sin sin( sin mrl( os sin( mr os sin( os( mr sin sin( 6 sin( mr os sin( os( mr sin os( 6 mrl os sin( os mr ( os( sin( mr ( os( sin( mrl os sin( os mrl sin sin( sin 65 mrl sinos( mrl sin os os( mr os os( 7 sin( mrl os os sin( mr sin os( os( mrl os os sin( mr os os( sin( 7 mr ( os( sin( mrl( os 77 sin( os 78 mr ( os( sin( mrl( os sin( os mrl( osos( sin ml os sin mrl sin os( os 79 mrl( os os( sin 7 mr mr sin os( 8 os os( sin( os( mr os os( sin( 8 mr ( os( sin( mrl os 87 os( sin mr ( os( sin( mrl os os( sin mrlsinos( os

11 mrl mrl mr 9 os sin os( sin sin sin( os sin( os( mr sin sin( sin( mrl os sin os( mr os sin( os( 9 mrl sin os mr sin os( 97 mrl ossin( os mrl( sinos( os ml ossin mr ( os( sin( 98 mrl( sinsin( sin mrl( sin os( os mr ( os( sin( 99 mrl os sin( os mrl( sin os( os mrl( ossin( 9 mrl( sin os( os mrl( sin sin( sin mrl( ossin( mr os sin( os( mr sin sin( sin( mr os sin( os( mr sin os( 7 mrl os sin( os mr ( os( sin( 8 mr ( os( sin( 9 mrl os sin( os mrl sin sin( sin mrl sin os( REFERECES [] O. ori and S. atunaga Formation and attitude ontrol for rotational tethered satellite lusters Journal of Spaeraft and Rokets Vol. 44 o. 7 pp.. [] K. akaya. Iai O. ori and S. atunaga On formation deployment for spinning tethered formation flying and experimental demonstration Proeedings of the 8 th International Symposium on Spae Flight Dynamis unih Germany Otober 4 ESA 548. [3]. Sabatini and G. B. Palmerini Dynamis of a 3D rotating tethered formation flying faing the Earth IEEE Aerospae Conferene Big Sky ontana arh 7. [4] P. R. Lawson he terrestrial plant finder IEEE Aerospae Conferene Big Sky ontana arh pp [5] C. V.. Fridlund DARWI he infrared spae interferometer Proeedings of the Conferene Darwin and Astronomy he Infrared Spae Interferometer Stokholm Sweden ovember 999 pp. 8. [6] D. Leisawitz J. C. ather S. H. oseley Jr. and X. Zhang he submillimeter probe of the evolution of osmi struture Astrophysis and Spae Siene Vol pp [7] C. enon and C. Bombardelli Self-stabilising attitude ontrol for spinning tethered formations Ata Astronautia Vol. 6 7 pp [8] L. G. Lemke J. D. Powell and X. He Attitude ontrol of tethered spaeraft he Journal of Astronautial Sienes Vol. 35 o. 987 pp [9] S. Bergamashi and F. Bonon Coupling of tether lateral vibration and subsatellite attitude motion Journal of Guidane Control and Dynamis Vol. 5 o pp [] S. Pradhan B. J. odi and A. K. isra ether-platform oupled ontrol Ata Astronautia Vol pp [] V. J. odi G. Gilardi and A. K. isra Attitude ontrol of spae platform based tethered satellite system Journal of Aerospae Engineering Vol. o. 998 pp [] P. Williams. Watanabe C. Blanksby P. rivailo and H. A. Fujii Libration ontrol of flexible tethers using eletromagneti fores and movable attahment Journal of Guidane Control and Dynamis Vol. 7 o. 5 4 pp [3] E. Garia C. S. Webb and. J. Duke Passive and ative ontrol of a omplex flexible struture using reation mass atuators Journal of Vibration and Aoustis Vol. 7 o. 995 pp. 6. [4] S. J. Chung J. J. E. Slotine and D. W. iller onlinear model redution and deentralized ontrol of tethered formation flight Journal of Guidane Control and Dynamis Vol. 3 o. 7 pp [5] E. J. Davison he deentralized stabilization and ontrol of a lass of unknown non-linear time-varying systems Automatia Vol. 974 pp [6] J. R. Cloutier State-dependent Riati equation tehniques: an overview Proeedings of the Amerian Control Conferene Albuquerque ew exio June 997 pp [7] A. Brai. Innoenti and L. Pollini Estimation of the region of attration for state-dependent Riati equation ontrollers Journal of Guidane Control and Dynamis Vol. 9 o. 6 6 pp [8] B. Wie Spae Vehile Dynamis and Control AIAA Eduation Series AIAA Virginia 998 pp [9] D. A. Vallado Fundamentals of Astrodynamis and Appliations 3rd edition iroosm Press California 7 pp [] K. D. Kumar and. Yasaka Rotating formation flying of three satellites using tethers Journal of Spaeraft and Rokets Vol. 4 o. 6 4 pp [] I. Chang S. Y. Park and K. H. Choi A deentralized attitude ontrol for satellite formation flying via the state-dependent Riati equation tehnique AAS/AIAA Astrodynamis Speialist Conferene akina Island ihigan August 7 AAS [] D. K. Parrish and D. B. Ridgely Attitude ontrol of a satellite using the SDRE method Proeedings of the Amerian Control Conferene Albuquerque ew exio June 997 pp [3] D.. Stansbery and J. R. Cloutier Position and attitude ontrol of a spaeraft using the state-dependent Riati equation tehnique Proeedings of the Amerian Control Conferene Chiago June pp [4] W. Luo and Y. C. Chu Attitude ontrol using the SDRE tehnique 7 th International Conferene on Control Automation Robotis and Vision Singapore Deember pp [5]. R. D. ayeri and. irshams Gyrosat attitude ontrol using nonlinear SDDRE method 4 th IAA Symposium on Small Satellites for Earth Observation Berlin Germany 3. [6] H. K. Khalil onlinear Systems 3rd edition Prentie Hall Upper Saddle River ew Jersey pp [7] J. J. E. Slotine and W. Li Applied onlinear Control Prentie Hall Englewood Cliffs ew Jersey 99 pp [8] J. R. Wertz and W. J. Larson Spaeraft ission And Design 3rd edition iroosm Press California 999 pp

12 (a Attitude motion in three-dimensional free spae (b Projeted attitude motion in the x-y plane Fig.. Illustration of the single-tethered system

13 3 (a Attitude motion in three-dimensional free spae (b Projeted attitude motion in the x-y plane Fig.. Illustration of the three-inline array system 3

14 4 (a For enter satellite (fixed (b For the first and seond satellites (fixed angular rates Fig. 3. Stability regions of the SDRE ontrolled three-inline array system for the deentralized approah using state-dependent B(x matrix 4

15 5 (a For enter satellite (fixed (b For the first and seond satellites (fixed angular rates Fig. 4. Stability regions of the SDRE ontrolled three-inline array system for the deentralized approah using linearized B matrix 5

16 6 (a Angular errors and angular rate errors for the enter satellite (b Angular errors angular rate errors and ontrol torques for the first satellite Fig. 5. Comparisons of the entralized and deentralized approahes in large initial errors 6

17 7 ABLE I AGULAR OEUS ( ms OF EACH AXIS I HE CERALIZED APPROACH FOR DIFFERE IIIAL SAE ERRORS State Center Satellite First Satellite Seond Satellite Error -axis -axis -axis -axis -axis -axis -axis -axis -axis -axis st test nd test rd test ABLE II AGULAR OEUS ( ms OF EACH AXIS I HE DECERALIZED APPROACH FOR DIFFERE IIIAL SAE ERRORS State Center Satellite First Satellite Seond Satellite Error -axis -axis -axis -axis -axis -axis -axis -axis -axis -axis st test nd test rd test ether length(l ABLE III AGULAR OEUS ( ms OF EACH AXIS I HE CERALIZED APPROACH WIH RESPEC O EHER LEGHS Center Satellite First Satellite Seond Satellite -axis -axis -axis -axis -axis -axis -axis -axis -axis -axis 5 m m m m m ether length(l ABLE IV AGULAR OEUS ( ms OF EACH AXIS I HE DECERALIZED APPROACH WIH RESPEC O EHER LEGHS Center Satellite First Satellite Seond Satellite -axis -axis -axis -axis -axis -axis -axis -axis -axis -axis 5 m m m m m