Coordinated Attitude Control of Spacecraft Formation without Angular Velocity Feedback: A Decentralized Approach

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1 AIAA Guidance, Navigation, and Control Conference - 3 August 29, Chicago, Illinois AIAA Coordinated Attitude Control of Spacecraft Formation without Angular Velocity Feedback: A Decentralized Approach A. R. Mehrabian, S. Tafazoli, and K. Khorasani In this paper, motivated by recent developments of velocity-free spacecraft (SC) attitude control techniques and behavioral-based SC formation control a decentralized control algorithm for attitude coordination of SC formation without angular velocity feedback is presented. Asymptotic stability of the SC formation is guaranteed by using Lyapunov analysis and LaSalle s theorem. The advantage of our proposed algorithm is that it requires limited information exchange among the SC in the formation (only attitude information exchange is necessary). Additionally, the proposed algorithm can be extremely useful when angular velocity information of the SC in the formation in not available due to sensor failures or communication constraints. Unlike other popular methods in the robotics area which tend to assume simple dynamics such as linear systems and single or double integrator dynamic models, in this paper, the full nonlinear attitude dynamics of the SC is considered to track fast time-varying reference trajectories. Nomenclature e Euler vector θ Euler angle, rad q The unit-quaternion q Vector part of the quaternion q 4 Scalar part of the quaternion R Rotation matrix [] Cross product matrix Q(q) Quaternion product matrix ω Spacecraft angular velocity vector, rad/sec E(q) Matrix based on quaternion u External torque (control effort), N-m F B Body frame F I Inertial frame J Spacecraft moments of inertial matrix, kg-m 2 δq Quaternion error δω Angular velocity error, rad/sec q jn Quaternion error between jth and nth spacecraft ω jn Angular velocity error between jth and nth spacecraft, rad/sec A Constant matrix B Constant matrix This research is supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) Strategic Projects. PhD Candidate, Department of Electrical and Computer Engineering, Faculty of Engineering and Computer Science, Concordia University, 455 de Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G M8. armehrabian@ieee.org Adjunct Assistant Professor, Department of Electrical and Computer Engineering, Faculty of Engineering and Computer Science, Concordia University, 455 de Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G M8. Also with the Canadian Space Agency (CSA), Saint-Hubert, QC, Canada J3Y 8Y9. stafazoli@videotron.ca Professor, Department of Electrical and Computer Engineering, Faculty of Engineering and Computer Science, Concordia University, 455 de Maisonneuve Blvd. West, Montreal, Quebec, Canada H3G M8. kash@ece.concordia.ca Copyright 29 by A. R. Mehrabian, S. Tafazoli, and K. Khorasani. American Published Institute by the ofamerican Aeronautics Institute andof Astronautics Aeronautics and Astronautics, Inc., with permission. of5

2 P Constant matrix Q Constant matrix [] First time derivative, sec - [] Second time derivative, sec -2 Controller s constant proportional gain Controller s constant derivative gain Identity matrix m Number of spacecraft in the formation λ p λ d I. INTRODUCTION Distributed spacecraft (SC) formation flying has received considerable attention in the passed few years - 2. This new technology allows SC to autonomously react to each other s orbit changes quickly and more efficiently. It enables the collection of different types of scientific data unavailable from a single satellite, such as stereo views or simultaneously collecting data of the same ground scene at different angles. In addition, rather than flying all the instruments on one costly satellite, by using this technology it allows scientists to obtain unique measurements by combining data from several satellites. SC formation maintenance requires control of relative distances among two or more SC. In some applications, e.g. deep space interferometry, in addition to maintaining a specific distance, the SC are required to maintain a specified relative attitude during formation maneuvers. 2 Four general approaches are available for SC formation control, namely multi-input multi-output (MIMO), leader-follower (LF), virtual structure (VS), and behavioral. Roughly speaking in the LF approach, there is a leader agent and a set of follower(s). The problem is how to design a controller to ensure that the follower(s) indeed track the leader. The following approaches are typical control methods that are employed within this structure, namely: () proportional/derivative (PD) control; 2 (2) feedback linearization design approach; 3 (3) Sliding mode control; 7 and (4) LQR and H control. 8 In the VS approach, the SC behave as rigid bodies embedded in a larger, virtual rigid body. The overall motion of the virtual structure as specified by positions and orientations of the SC within it are used to generate reference trajectories for each individual SC to track using its own controllers. In the behavioral approach, the control laws that govern the motion of each spacecraft are derived by, weighting the importance of several desired behaviors including formation keeping and goal seeking. The advantages of behavioral approach are: 2 () it is a distributed (not centralized) strategy; (2) it is very robust (fault-tolerant); (3) it requires a low information exchange; and (4) it also provides more flexibility in control algorithm design. The attitude control of a SC without angular velocity feedback has been studied by several researchers. Specifically, based on passivity arguments, Tsiotras 3 extended the results reported in Ref. 4 for SC attitude stabilization using Modified-Rodrigues-Parameters (MRP). Later Akella 5 extended the results that are reported in Ref. 3 and developed a tracking attitude controller for a SC. Recently, Tayebi 6 proposed a quaternion-based dynamic output controller for the attitude tracking problem of a rigid SC without using the angular velocity information. Two advantages can be sought by using these methods, namely: () numerical differentiation difficulties due to the noise-corrupted attitude signals can be avoided and; (2) they are backed up with a rigorous stability analysis. Design of decentralized attitude control for formation flying SC without requiring angular velocity will considerably reduce the communication requirements and eliminates the need for angular rate measurement. It also improves the robustness of the SC formation. It is shown in a recent study on SC on-orbit failures 7 that mechanical gyroscopes, which are used for measuring the SC angular velocity, are the components with the highest failure rates in the Attitude and Orbit Control subsystem (AOCS) of SC (see Fig. 9 in Ref. 7). Therefore, development of an efficient velocity-free control method for SC formation flying will considerably increase the chance of survival and completion of these important missions. A decentralized attitude control law for SC formation without requiring angular velocity has appeared in Ref. 2. In this paper, based on passivity analysis, the authors presented a decentralized behavioral control architecture for SC in the ring coordination architecture. However, the feedback control gains need to be adjusted for different initial SC attitudes. In addition, the final angular velocity of the SC formation is 2of5

3 assumed to be zero in the reported analysis. More recently, the authors in Ref. 8 and Ref. 9 extended the results reported in Ref. 6 to design a decentralized attitude controller for a rigid SC in formation based on LF and behavioral approaches, respectively. In the present paper, inspired from Ref. we propose a decentralized control law that is based on behavioral approach for coordinated attitude control of SC formation using passivity arguments. We extend the results reported in Ref. 4 and Ref. 5 to design a controller without requiring angular velocity information. We assume that the final angular velocity of SC formation is non-zero. However, the control algorithm for the case when the final angular velocity of SC is zero, which is desirable in deep space interferometry missions, is shown to be easily derived from our developed analysis. The proposed control strategy is model independent in the sense that it does not require knowledge of the SC moments of inertia for the latter case. We conduct rigorous stability analysis for development of our proposed control algorithm. In contrast to existing methods in the robotics research community, 2, 2 which tends to assume rather simple dynamics such as linear systems and single or double integrator dynamic models, our proposed strategy in this paper primarily deals with complex dynamical systems consisting of nonlinear SC attitude dynamics that are controlled to track time-varying reference trajectories. II. SPACECRAFT FORMATION STRUCTURE Generally, there are three types of SC formation, namely: 22 () trailing formations (e.g. Earth Observing- satellite program); (2) cluster formations (e.g. European Space Agency s Cluster Mission); and (3) constellation formations (e.g. GPS satellites). Usually, these formations are made up of several smallsats which weight less than 2 kg. In this paper, we consider a formation formed with m SC. All the SC in the formation are 3-axis stabilized and fully actuated. This enables the SC to change their attitude from any initial attitude to any desired attitude rather quickly. We only consider the attitude dynamics of the SC, so that the developed algorithm can be applied to any SC formation where attitude synchronization is required. Our main objective is to develop a decentralized control algorithm that enables the SC to first align their attitudes within the formation, followed by the SC formation tracking a desired attitude specified for all the SC. This implies that the desired attitude assigned by the supervisor (ground station operator), is available to all the SC in the formation (there is no leader in the formation). In the development of the control strategy, it will be assumed that the SC have no information about their own and other SC angular velocity and only attitude information is available for exchange and feedback. This could occur due to either rate sensor failure 7 or communication constraints among the SC. A control algorithm is proposed below that relies only on the availability of attitude information for attitude alignment of the SC in the formation. Our analysis shows that stability of the formation can be guaranteed for different information flow architectures among the SC in the formation. In order to show the stability of the SC formation, first we consider a fully connected formation (aka tight formation topology) as shown in Fig.. According to our derived analytical results, the stability of the formation can be guaranteed by using minimal information exchange among the SC as long as the flow is bi-directional and the SC graph is connected. 23 It should be noted that in the tight formation topology, although the information flow is centralized (attitude information of all the SC are available to all SC in the formation), the controller is decentralized as each controller does not require the decision from the other controllers. III. SPACECRAFT ATTITUDE DYNAMICS AND KINEMATICS We review the spacecraft (SC) attitude dynamics and kinematics in the section. Unit quaternion is used to present the attitude of the SC. The unit quaternion, for the jth SC is defined as: [ ] [ ] e q j = j sin( θj 2 ) q cos( θj 2 ) = j () q j,4 where e j =(e j,,e j,2,e j,3 )istheeuleraxis,θ j is the Euler angle, q j is the vector part, and q j,4 is the scalar part of the quaternion, subject to the constraint:. q 2 j,4 + q T j q j = 3of5

4 SC SC 2 SC n SC 3 SC 4 SC m Figure. The schematic of a tight coordination architecture. A rotation matrix is a 3 3 matrix that represents the attitude of one reference frame with respect to another. For example, the rotation matrix denoted by R(q jn ) represents the rotation from frame F n to F j. The rotation matrix is related to the quaternion through: 24 R(q j )=(q 2 j,4 q T j q j) +2 q j q T j 2q j,4 q j (2) where ω represents the cross product matrix, i.e.: ω 3 ω 2 ω = ω 3 ω (3) ω 2 ω Single subscript q j represents the attitude of the corresponding reference frame F j with respect to the inertial frame F I. Similarly, double subscripts q jn represents the attitude of the corresponding reference frame F n with respect to frame F j. In general, +q jn and q jn both represent the same rotation matrix, and this sign ambiguity can be resolved by using the kinematic equation below: 24 q jn = 2 [ ω jn ωjn T ω jn ] q jn (4) 2 E(q jn)ω jn (5) where ω jn =[ω jn,, ω jn,2, ω jn,3 ] T is the angular velocity vector of the jth SC with respect to the nth SC, whichisdefinedbelow: ω jn = ω j R(q jn )ω n (6) where ω j is the angular velocity vector of the body frame of the jth SC, Fj B, with respect to the inertial frame F I. In addition, the matrix E(q) isgivenas: The inverse of the quaternion is defined as: q 4 q 3 q 2 q E(q) = 3 q 4 q q 2 q q 4 q q 2 q 3 q =[ q q 2 q 3 q 4 ] T (8) (7) 4of5

5 The following equations corresponding to the relative states of the jth and the nth SC will be used subsequently: R(q jn )=R T (q nj ) (9) q nj = q jn = R(q nj ) q jn () The individual SC in the formation is modeled as a rigid body. The equations of motion for a rigid body in absence of any external disturbance torque is governed by: J j ω j = u j ω j J j ω j () where J j is the moment of inertia matrix of the jth SC expressed in the body frame, Fj B,andu j is the external torque expressed in Fj B. The formulation given above can be extended to the SC with momentum exchange actuators, e.g. control moment gyroscopes (CMGs), similar to the method that was presented in Ref. 24 or the variable speed CMGs (VSCMGs), similar to the method that was presented in Ref. 5. IV. SPACECRAFT ATTITUDE ERROR DYNAMICS For a SC in a formation we define two error measures. These measures are the station-keeping and formation-keeping attitude state errors. The station-keeping error is defined as the attitude state error of an individual SC with respect to its absolute desired attitude state. The station-keeping error, δq j, is defined as: δq j = Q(q j )q j (2) where q j is the desired attitude of the SC formation and the matrix Q(q) is defined as: q 4 q 3 q 2 q q Q(q) = 3 q 4 q q 2 q 2 q q 4 q 3 q q 2 q 3 q 4 The station-keeping angular velocity error, δω j, is defined as: δω j = ω j R(δq j )ω j (3) where ωj is the absolute desired angular velocity vector expressed in the absolute desired reference frame. The first derivative of the station-keeping angular velocity error is obtained as: 5 δ ω j = ω j R(δq j ) ω j + ω j R(δq j)ω j (4) We can now state the governing equations for the attitude error δq j and the angular velocity error δω j as follows: δ q j = 2 E(δq j)δω j (5) J j δ ω j = u j ω j J ( j ω j + J j R(δqj ) ω j + ω j R(δq j)ωj ) (6) If we consider that the desired angular velocity of the SC in the formation is zero, i.e. ω j = ω j =, (6) simplifies to: which implies that in this case we have: J j δ ω j = u j ω j J j ω j (7) δ ω j ω j Formation-keeping error, for the jth SC is the attitude state error of the jth SC with respect to the other SC in the formation. The relative attitude error between the jth and the nth SC is defined as: q jn = Q(q n )q j = Q(δq (8) n )δq j 5of5

6 The time derivative of q jn is defined in (5). The relative angular velocity vector of the jth SC with respect to the nth SC, ω jn, that is defined in (6) can be re-written in terms of δω j and δω k,namely: ω jn = δω j R(q jn )δω n (9) Our main goal in this paper is to design a decentralized controller for each SC which commands the actuators in order guarantee coordinated SC attitude and angular velocity alignment, i.e. q j q n (or equivalently, q jn ) and ω jn. This objective is designated as the formation-keeping behavior. Furthermore, we have to ensure that the designed controllers guarantee that each SC attitude converges to the commanded attitude, i.e. δq j andδω j. This objective is designated as the station-keeping behavior. The constraint that we impose here is that there are no information exchanges regarding angular velocity of the SC in the formation. V. SINGLE SPACECRAFT ATTITUDE CONTROL WITHOUT VELOCITY FEEDBACK In this section, we first introduce an attitude control law for a single SC. The synchronized attitude control of SC formation will be considered in the next section. The following theorem is presented first to guarantee stability of a single SC without angular velocity feedback. Theorem : Consider the jth SC error kinematics (5) and the dynamics (6) and let the control input, u j, be computed by using the following dynamic controller: { u j = δ q j 2E T (δq j )B T P(Az j + Bδq j )+J j R(δq j ) ω j +Υ j J jυ j ż j = Az j + Bδq j (2) where A R 4 4 is Hurwitz and B is given by: [ ] B = (2) and where denotes an identity matrix, and matrix P is a symmetric, positive-definite solution of the following Lyapunov equation: A T P + PA = Q (22) for any 4 4 symmetric positive-definite matrix Q. Furthermore, we define Υ j = R(δq j ) δω j. Then the closed-loop system (5), (6) and (2) is globally asymptotically stable. Proof: Consider the following positive-definite radially unbounded Lyapunov function candidate: V j = 2 δωt j J j δω j + 2 δ qt j δ q j + 2 (δq j,4 ) 2 +2(Az j + Bδq j ) T P(Az j + Bδq j ) (23) The time derivative of this function along the trajectories of the closed-loop system (5), (6), and (2) can be computed as follows: d dt V j =δωj T J jδ ω j + δ q T j δ q j +(δq j,4 )δ q j,4 +2( z T j Pż j + ż T j P z j) =δωj T [ uj ω j J ( j ω j + J j R(δqj ) ω j + ω j R(δq j)ωj )] + δω T j δ q j +2 [ (Aż j + Bδ q j ) T Pż j + ż T j P(Aż j + Bδ q j ) ] =δωj T [ δ q j 2E T (δq j )B T Pż j + J j R(δq j ) ω j +Υ j J jυ j ω j J j ω j ( + J j R(δqj ) ω j + ω j R(δq j)ωj ) ]+δω T j δ q j +2 [ (ż T j AT + δ q T j BT )Pż j + ż T j P(Aż j + Bδ q j ) ] =δωj T ( Υ j J j Υ j ω j J j ω j + J j ω j R(δq j)ωj ) 2δωj T E T (δq j )B T Pż j +2 [ ż T j A T Pż j + δ q T j B T Pż j + ż T j PAż j + ż T ] j PBδ q j (24) In view of (3) and the fact that ω ω =,itcanbeshownthat: 5 δω T j ( Υ j J jυ j ω j J j ω j + J j ω j R(δq j)ω j ) = 6of5

7 Thus, we have: V j = 2δωj T E T (δq j )B T Pż j +2ż T j (A T P + PA)ż j +4δ q T j B T Pż j (25) = 2ż T j Qż j 2δωj T E T (δq j )B T Pż j +4δ q T j B T Pż j Equation (5) can be re-written as δωj T ET (δq j )=2δ q T j. Therefore, from (25) we obtain: V j = 2ż T j Qż j (26) which is a negative semi-definite function. Note that since V j is radially unbounded all the solutions remain bounded. Consider now the set H = {(δω j,δ q j, z j ): V =}. OnH we have z j = ż j =, which from (2) implies that δ q j =. This, from (5) implies that δω j =sincee(δq j ) is nonsingular for all δq j.theseresultscan be used along with (3), (6) and (2) to demonstrate that δ q j =. SinceA is Hurwitz, (2) also implies that z j =. Thus, the largest invariant set in H is the set H = {(δω j,δ q j, z j ) H: δω j = δ q j = z j =}. By invoking LaSalle s invariance theorem 25 and noting that V is radially unbounded, the closed-loop system is globally asymptotically stable. This completes the proof of the theorem. We are now in a position to state the following result. Corollary : Consider the case when the desired angular velocity is zero, i.e. ω j = ω j =. Then the following control law can be used to stabilize the SC dynamics (7): u j = δ q j 2E T (δq j )B T P(Az j + Bδq j ) (27) where z j is defined in (2). Proof: This follows directly from Theorem, and can be shown by replacing δω j with ω j in the Lyapunov function (23), and its time derivative (24). VI. COORDINATED SPACECRAFT ATTITUDE CONTROL WITHOUT VELOCITY FEEDBACK The design of synchronized SC formation control reorientation is presented in this section by assuming that no angular velocity measurement is available for each SC and for information exchange among the SC. The developed controller is based on the sum of the control action for the station-keeping behavior and the control action for the formation-keeping behavior. A. Station-Keeping Controller Inspired from (2), the station-keeping control law for the jth SC is defined according to: { u s j = λp j δ q j 2λ d j ET (δq j )B T P(Az j + Bδq j )+J j R(δq j ) ω j +Υ j J jυ j ż j = Az j + Bδq j (28) where superscript s stands for the station-keeping behavior, λ p j >, and λd j parameters. The matrices A and B are defined in the previous section. > are the controller constant B. Formation-Keeping Controller The control law for the formation-keeping behavior for the jth SC, u f j, is defined as: { u f j = m n= λpjn q jn m ( n= λd jn E T (q jn )B T P(Az jn + Bq jn ) R(q jn )E T (q nj )B T P(Az nj + Bq nj ) ) ż jn = Az jn + Bq jn n =, 2,...,m, j n (29) where m is the number of the SC in formation. In addition we have: λ p jn = λp nj, λd jn = λd nj. (3) Through the constraints (3) we assume that the communication flows among the SC are bi-directional. Different coordination architectures can be built by proper selection of the formation flow gains λ p jn and λd jn. For example, selecting λ p 2 = λp 2 = λd 2 = λd 2 = eliminates the connection between the SC and the in the formation. We are now in the position to present our main results. 7of5

8 VII. MAIN RESULTS The main result of this paper is presented in the following theorem. Theorem 2: Considerm SC in a formation where the dynamic equations of the jth SC is given by (6). The decentralized control law for the jth SC obtained by adding the station-keeping behavior (28) and the formation-keeping behavior (29) given as: u j =u s j + u f j = λ p j δ q j 2λ d j ET (δq j )B T P(Az j + Bδq j )+J j R(δq j ) ω j +Υ j J jυ j λ p jn q jn n= (3) ( E T (q jn )B T P(Az jn + Bq jn ) R(q jn )E T (q nj )B T P(Az nj + Bq nj ) ) n= λ d jn where z j, z nj, j,n =, 2,...,m, j n are defined in (28) and (29) will guarantee that the closed-loop system signals remain all bounded and the SC formation is globally asymptotically stable. Proof: Consider the following positive-definite radially unbounded Lyapunov function candidate: V j = 2 δωt j J j δω j + 2 λp j δ qt j δ q j + 2 λp j (δq j,4 ) 2 +2λ d j (Az j + Bδq j ) T P(Az j + Bδq j ) [ + λ p T jn ( q jn q jn +( q jn,4 ) 2) + λ d jn(az jn + Bδq jn ) T P(Az jn + Bδq jn ) ] (32) n= The time derivative of V j along the trajectories of the closed-loop system is governed by: V j =δωj T J j δ ω j + λ p j δ qt j δ q j + λ p j (δq j,4 ) δ q j,4 +2λ d j ( z T j Pż j + ż T j P z j ) + λ p 2 jn qt jnω jn + λ d T jn ( z jn Pż jn + ż T ) (33) jnp z jn n= n= n= From (25), the above equation simplifies to: [ V j = 2λ d j żt j Qż m j δωj T λ p jn q jn + λ d ( jn E T (q jn )B T Pż jn R(q jn )E T (q nj )B T ) ] Pż nj + 2 λ p jn qt jnω jn + n= n= λ d jn n= ( z T jn Pż jn + ż T jnp z jn ) In order to show the stability of the SC formation, we consider the sum of the component Lyapunov functions. The composite Lyapunov function is given by: V = 2λ d j żt j Qż j δωj T ( λ p jn q jn + λ d ( jn E T (q jn )B T Pż jn R(q jn )E T (q nj )B T )) Pż nj + 2 j= j= n= j= n= λ p jn qt jn ω jn + j= n= j= n= λ d T jn ( z jn Pż jn + ż T jn P z jn) Using (6), () and (3) we can show that: λ p 2 jn qt jnω jn = λ p jn δωt j q jn (36) j= n= (34) (35) Furthermore, according to (3) and noting the fact that q T R( q) = q T we obtain: λ d T jn ( z jn Pż jn + ż T jn P z ) m jn = λ d jnżt jn Qż jn j= n= + j= n= j= n= δω T j ( λ d jn ( E T (q jn )B T Pż jn R(q jn )E T (q nj )B T Pż nj )) (37) 8of5

9 Consequently, from (35)-(37) we obtain: V = 2λ d j żt j Qż j λ d jnżt jn Qż jn (38) j= j= n= Consider now the set H = {(δω j,δ q j, z j,ω jn, q jn, z jn ): V =}. OnH we have ż j =, which from (28) implies that δ q j =. This from (5) implies that δω j =. It can be further shown that by using (3), (6) and (2) one gets δ q j = δ q n = ; thus from (8) we have q jn =. SinceA is Hurwitz, (2) also implies that z j =. Similarly, ż jn = results in having q jn = from (29) and from (5) we have ω jn =. Therefore, the largest invariant set in H is the set Ĥ = {(δω j,δ q j, z j,ω jn, q jn, z jn ) H: δω j = δ q j = z j = ω jn = q jn = z jn =}. By invoking LaSalle s invariance theorem 25 and since V is radially unbounded, it follows that the system is globally asymptotically stable. This completes the proof of the theorem. To demonstrate stability preservation of the SC formation with minimal communication exchange achieved by using the above control laws, we consider the following typical example. Consider a SC formation that is formed from three SC (m = 3) with only two bi-directional connections (see Fig. 2). For this system with minimal connections we have: λ p j, λd j >,j =, 2, 3 λ p 23 = λp 32 = λd 23 = λ d 32 =, λ p 2 = λp 2,λp 3 = λp 3,λd 2 = λd 2,λd 3 = λd 3 Based on the parameters that are given above the Lyapunov function (32) is a positive-definite radially unbounded function. According to (38), the time derivative of the Lyapunov function along with the trajectories of the system is given by: V = 3 2λ d j żt j Qż j λ d 2 (żt 2 Qż 2 + ż T 2 Qż 2) λ d 3 (żt 3 Qż 3 + ż T 3 Qż 3) (4) j= First note that all the terms in the equation above are quadratic so it is a negative semi-definite function. Having the first term in (4) equal to zero implies δω j = δ q j = z j =, which according to (8) yields q jn =. Semi-negativeness of the second term in (4) and knowing the fact that A is Hurwitz implies z 2 = z 2 = ω 2 = ω 2 =. Similarly, noting semi-negativeness of the third term in the above equation we can conclude z 3 = z 3 = ω 3 = ω 3 =. From (6) and knowing the fact that the rotation matrix for the unit quaternion attitude representation (2) is nonsingular, we conclude that ω 32 = ω 23 =. Obviously, z 32, z 23 are not defined for this system. Therefore, using LaSalle s invariance theorem 25 and given that V is radially unbounded, it follows that the system is globally asymptotically stable. The global asymptotic stability of the SC formation having a larger number of SC in the formation when not all SC are directly connect to each other although the SC formation graph is connected, can be shown in a similar manner. We now have the following Corollary. (39) SC Figure 2. Spacecraft formation having three members with minimal connection (information exchange). Corollary 2: Consider the case when the desired SC formation angular velocity is zero, i.e. ω j = ω j =. Then the following control law can be used to stabilize the SC formation dynamics: u j =u s j + u f j = λ p j δ q j 2λ d j E T (δq j )B T Pż j n= λ p jn q jn n= (4) λ d ( jn E T (q jn )B T P(Az jn + Bq nj ) R(q jn )E T (q nj )B T P(Az nj + Bq jn ) ) 9of5

10 Proof: It follows directly from Theorem 2, and can be shown by replacing δω j with ω j in the Lyapunov function (32) and its time derivative (33). Remark : Examples of formation architectures that can be constructed with four SC with guaranteed stability by using our proposed control laws are shown in Fig. 3. The ring topology in shown in Fig. 3(a). In Fig. 3(b), a LF-like topology is shown where the attitude of all SC in the formation is available to SC only while the desired attitude is available to all the SC in the formation. A Minimal formation topology where the formation is constructed by requiring minimum information flows is shown in Fig. 3(c). SC SC SC (a) (b) (c) Figure 3. Examples of different formation architectures that can be constructed by using our proposed control laws with guaranteed asymptotic stability: (a) ring topology; (b) LF-like topology; (c) minimal connection topology. Remark 2: Other approaches for designing velocity-free controllers for the SC formation may be considered. For example, one may try to use an observer to estimate the velocity similar to the one that is introduced in Ref. 26. However, a separation principle-like property considered in this paper is conjectured but not shown formally. Thus, the closed-loop stability is not analyzed. 4 In addition, the observer-based controller introduced in Ref. 27 requires knowledge of the SC moment of inertia matrix and the reaction wheels (RW) angular velocities (if it is used in the AOCS subsystem), which makes the control law quite complex. The results obtained in Ref. 28 on the velocity-free quaternion-based tracking controller cannot be implemented here either since only local stability of the closed-loop system was shown to be guaranteed. VIII. SIMULATIONS The performance of our proposed control law is demonstrated in simulations in this section. For the purpose of simulations, we consider four SC in the formation with different initial attitudes and different moments of inertia. The attitude of the SC in the formation is available to all the SC in the formation. However, the SC have no information regarding their own and other SC angular velocities and moments of inertia. The final formation angular velocity is assumed to be zero in the simulations. The controller parameters are selected as: λ p j =., λd j =., λp jn =.2, λd jn =.6, for j, n =, 2, 3, 4andj n (42) which represents a tight formation topology. We select higher gains for the formation-keeping behavior when compared to the station-keeping behavior in order to emphasize on the requirement of formation-keeping. In addition, the matrices A and Q are chosen as: A = 4 4, Q = Based on the matrices selected above, the solution to the Lyapunov equation (22) is found as: P = The initial attitude error of the four SC in the formation are given by: δq () = [,,.8,.6] T,δq 2 () = [,,, ] T, δq 3 () = [,.2,,.9798] T,δq 4 () = [.9,,,.4359] T of 5

11 The moments of inertia of the four SC are given below: J = diag([, 5, 5]), J 2 = diag([8, 3, 2]), J 3 = diag([2, 7, 2]), J 4 = diag([2,, 2]), Note that the initial conditions for the linear dynamical systems z j, z jn, j,n =, 2,...,m, j n are chosen randomly between zero and one. Using the SC and the controller parameters as indicated above, the time response of the SC in the formation are obtained. The error quaternions are shown in Fig. 4 during the first 4 seconds of the maneuver SC δ q δ q δ q 3 δ q Figure 4. Error quaternions for the tight formation with four SC during the first 4 seconds. As can be seen from this figure, the SC in the formation first align their attitudes. Subsequently, they try to reduce the formation attitude error (station-keeping behavior is dominant in this period). This was expected according to the selected weights of the station-keeping and the formation-keeping behaviors in (42). It is interesting to note that due to the priority of the formation-keeping behavior, the second SC which has initial station-keeping error of zero moves away from its initial attitude and goes towards other SC in the formation. This behavior makes the formation-keeping error zero, although increases the station-keeping error. Approximately after 5 seconds, the formation-keeping maneuver ends (implying that the four SC are all in the same attitude and the formation attitude error is very close to zero). Consequently, the SC in the formation try to reduce the formation attitude error as time evolves. In order to demonstrate convergence of the SC formation to the desired attitude, the time response of the system is shown for 8 seconds in Fig. 5, which clearly shows the convergence of the SC states. The control efforts for the SC in the formation are shown in Fig. 6 for the first 4 seconds in three different channels. It is clear that the control efforts magnitude are reasonable and converge to zero as the SC align their attitude and the formation attitude error decreases. These results confirm that our analytical conclusions obtained earlier are substantiated. It is interesting to note that in the simulations convergence of the closed-loop system states appears to be exponential which is an advantage of our designed controller. In order to demonstrate the performance and stability of our proposed control laws when links among the SC are disconnected two scenarios are considered. In the first scenario, it is assumed that the second and the third SC are no longer connected to each other, while all the other links are present. In this case we have, λ p 23 = λp 32 = λd 23 = λd 32 =. In the second scenario, we assume that the first and the fourth SC as well as the second and the third SC are not connected. This essentially realizes the ring topology as shown of 5

12 SC δ q δ q δ q 3 δ q Figure 5. Error quaternions for the tight formation with four SC during the first 8 seconds. u [N.m] Control effort vs. time SC u 2 [N.m] u 3 [N.m] Figure 6. Control effort for the four SC in the tight formation during the first 4 seconds. 2 of 5

13 .6.4 SC.2 δ q δ q δ q 3 δ q Figure 7. Error quaternions for the formation with four SC with no connection between the second and the third SC during the first 4 seconds SC δ q δ q δ q 3 δ q Figure 8. Error quaternions for the ring formation with four SC during the first 4 seconds. 3 of 5

14 in Fig. 3(a). The time responses of the SC in the formation are shown in Fig. 7 and Fig. 8, respectively. In simulations, it can be observed that the response of the SC in the formation does not change significantly by loosing connections between SC and the formation-keeping error, in both cases, converge to zero almost at the same time as it converged in the tight formation architecture (around 2 seconds). However, we have noticed rapid responses and early misalignment among the SC that are not connected to each other in the latter case (refer to the first seconds in Fig. 8 and compare it with the first seconds of Fig. 4). The significance is that transient delays are introduced in the attitude alignment for a few seconds at the beginning of the maneuver which is not surprising. IX. CONCLUSIONS This paper presents a decentralized approachfor coordinated attitude maneuver for SC formation without requiring and feeding back angular velocity information. The proposed algorithm is specially useful when the information exchanges among the SC in a formation is limited, i.e. when only attitude position information exchanges are possible among the SC. The proposed algorithm can also be used when one or more of the SC rate gyroscopes, which are commonly used for measuring SC angular velocity, has failed and the SC angular velocity information is no longer available. The global asymptotic stability of the SC formation by using our proposed control algorithm is guaranteed through Lyapunov analysis and LaSalle s theorem. The stability analysis is also shown to assist one in determining the minimum of information exchanges among the SC while guaranteeing the asymptotic stability of the formation. It is also important to emphasize that in our proposed method, the desired (final) angular velocity of the SC formation does not need to be zero. We have shown that a simplified version of the proposed control algorithm can be employed when the desired angular velocity of the SC formation is zero (as desired in deep space interferometry missions). In this case, the proposed control strategy is model independent in that it does not require knowledge of the SC moments of inertia matrices. References D.P. Scharf, F.Y. Hadaegh, and S.R. Ploen, A Survey of Spacecraft Formation Flying Guidance and Control (part II): Control, in Amer. Contr. Conf., P.K.C. Wang and F.Y. Hadaegh, Coordination and Control of Multiple Microspacecraft Moving in Formation, J. Astro. Sci., vol. 44, no. 3, pp , M. Mesbahi, and F.Y. Hadaegh, Formation Flying Control of Multiple Spacecraft via Graphs, Matrix Inequalities and Switching, J. Guid., Contr., and Dyn., vol. 24, no. 2, pp , 2. 4 V. Kapilal, A.G. Sparks, J.M. Buffington, and D.P. Qiguo Yan, Spacecraft Formation Flying: Dynamics and Control, in Amer.Contr.Conf., S.-J. Chung, U. Ahsun, and J.-J. E. Slotine, Application of Synchronization to Formation Flying Spacecraft: Lagrangian Approach, J. Guid., Contr., and Dyn., vol. 32, no. 2, 29, in press. 6 W. Ren, Distributed Attitude Alignment in Spacecraft Formation Flying, Int. J. Adaptive Contr. and Signal Processing, vol. 2, pp. 95 3, J.B. Cruz, Jr.S.V. Drakunov, and M.A. Sikora, Leader-Follower Strategy Via a Sliding Mode Approach, J. Optimization Theo. Appl., vol. 88, no. 2, pp , S.R. Stann, R.K. Yedavalli, and A.G. Sparks, Design of a LQR Controller of Reduced Inputs for Multiple Spacecraft Formation Flying, in Amer. Contr. Conf., pp , 2. 9 W. Ren, and R.W. Beard, Decentralized Scheme for Spacecraft Formation Flying via the Virtual Structure Approach, J. Guid., Contr., and Dynam., vol. 27, no., pp , 24. J. Lawton, R.W. Beard, and F.Y. Hadaegh, Elementary Attitude Farmation Maneuvers via Leader-Following and Behavior-Based Control, in AlAA Guid. Navig., Contr. Conf., pp , 2. M. VanDyke, and C.D. Hall, Decentralized Coordinated Attitude Control Within a Formation of Spacecraft, J. Guid., Contr., and Dynam., vol. 29, no. 5, pp. 9, J.R. Lawton, and R.W. Beard, Synchronized Multiple Spacecraft Rotations, Automatica, vol. 38, no. 8, pp , P. Tsiotras, Further Passivity Results for the Attitude Control Problem, IEEE Trans. Auto. Contr., vol.43,no., pp , F. Lizarralde, and J.T. Wen, Attitude Control Without Angular Velocity Measurements: A Passivity Approach, IEEE Trans. Auto. Contr., vol. 4, no. 3, pp , M. R. Akella, Rigid Body Attitude Tracking without Velocity Feedback, Systems and Control Letters, vol. 42, pp , 2. 6 A. Tayebi, Unit Quaternion-Based Output Feedback for the Attitude Tracking Problem, IEEE Trans. Auto. Contr., vol. 53, no. 6, pp , of 5

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