AIAA Guiance, Navigation, an Control Conference an Exhibit 5-8 August, Monterey, California AIAA -9 VIRTUAL STRUCTURE BASED SPACECRAT ORMATION CONTROL WITH ORMATION EEDBACK Wei Ren Ranal W. Bear Department of Electrical an Computer Engineering Brigham Young University Provo, UT 8 fweiren, bearg@ee.byu.eu Abstract ormation control for multiple vehicles has become an active research area in recent years. Generally there are three approaches to this problem, namely leaerfollowing, behavioral, an virtual structure approaches. In this paper, formation control ieas for multiple spacecraft using virtual structure approach are presente. If there is no formation feeback from spacecraft to the virtual structure, the spacecraft will get out of formation when the virtual structure moves too fast for the spacecraft to track or the total system must sacrifice convergence spee in orer to keep the spacecraft in formation. The spacecraft may also get out of formation when the system is affecte by internal or external isturbances. To overcome these rawbacks, a novel way of introucing formation feeback from spacecraft to the virtual structure is illustrate in etail. An application of these ieas to multiple spacecraft interferometers in eep space is given. Introuction ormation control for multiple vehicles has become an active research area in recent years. Applications in this area inclue the coorination of multiple robots, UAVs, satellites, aircraft, an spacecraft. 5 Many papers have been publishe to eal with ifferent control strategies, schemes, an applications of multiple vehicle control. While the applications are ifferent, the funamental approaches to formation control are similar: the common theme being the coorination of multiple vehicles to accomplish an objective. Generally there are three approaches to multi-vehicle coorination reporte in the literature, namely leaerfollowing, behavioral, an virtual structure approaches. In the leaer following approach,,, one of the agents is esignate as the leaer, with the rest of the agents esignate as followers. The leaer tracks a pre-efine trajectory, an the followers track a transforme version of the leaer s states. The avantage of leaer following is that group behavior is irecte by specifying the behavior of a single quantity: the leaer. The isavantage is that there is no explicit feeback to the formation. Another isavantage is that the leaer is a single point of failure for the formation. In the remainer of the paper, we use the following efinition: 7 The group feeback from vehicles to the formation is referre to as formation feeback. In the behavioral approach, 8 several esire behaviors are prescribe for each agent. The basic iea is to make the control action of each agent a weighte average of the control for each behavior. Possible behaviors inclue collision avoiance, obstacle avoiance, goal seeking, an formation keeping. The avantage of the behavioral approach is that it is natural to erive control strategies when agents have multiple competing objectives. In aition, there is explicit feeback to the formation since each agent reacts accoring to the position of its neighbors. Another avantage is that the behavioral approach lens itself naturally to a ecentralize implementation. The isavantage is that the group behavior cannot be explicitly efine, rather the group behavior is sai to emerge. In aition, it is ifficult to analyze the behavioral approach mathematically an guarantee its group stability. In the virtual structure approach,, 7, the entire formation is treate as a single structure. The virtual structure can evolve as a rigi boy in a given irection with some given orientation an maintain a rigi geometric relationship among multiple vehicles. The avantage of the virtual structure approach is that it is fairly easy to prescribe a coorinate behavior of the group. The isavantage is that requiring the formation to act as a virtual structure limits the class of potential applications of this approach. Another isavantage is that its current evelopment lens itself to a centralize control implementation. In the case of the application of synthesizing multiple Copyright by the, Inc. All rights reserve.
spacecraft interferometers in eep space, it is esirable to have a constellation of spacecraft act as a single rigi boy in orer to image stars in eep space. As a result, it is suitable to choose the virtual structure approach to accomplish formation maneuvers. In the remainer of the paper we use the term formation an virtual structure interchangeably. In general, there is a ilemma when there is no feeback applie from spacecraft to the virtual structure. On the one han, if the virtual structure evolves too fast, the spacecraft cannot track their esire trajectories accurately an they will get out of formation. On the other han, the virtual structure might be slowe own sufficiently to allow the spacecraft to track their trajectories accurately. However, this results in unreasonably slow formation ynamics. Also, when performing formation maneuvers, the total system is often isturbe by internal or external factors. or example, some spacecraft may fail for a perio of time ue to mechanical or electrical malfunctions or may isintegrate from the formation ue to external isturbances in eep space. If there is no formation feeback from spacecraft to the formation, the faile or isintegrate spacecraft will be left behin while the other spacecraft still keep moving towars their final goals, an the entire system cannot ajust to maintain formation. ormation feeback from spacecraft to the virtual structure provies a goo compromise between formation keeping an convergence spee as well as improve group stability an robustness. In Ref. the authors introuce a coorination architecture for spacecraft formation control which subsumes leaer-following, behavioral, an virtual structure approaches to the multi-agent coorination problem. In Ref. formation maneuvers are easily prescribe an group stability is guarantee, but formation feeback is not inclue. In Ref. 7 formation feeback is use for the coorinate control problem for multiple robots. This paper is aime as a further evelopment for Ref. an Ref. 7. The main contribution of this paper is to propose a novel iea of introucing formation feeback from spacecraft to the virtual structure an apply this iea to the spacecraft interferometry problem so that formation keeping is guarantee throughout the maneuver an the total system robustness is improve. The ecentralize control implementation of the virtual structure approach nees to be explore in the future. The outline of this paper is as follows. In section we introuce spacecraft ynamics. In section we escribe virtual structure equations of motion for spacecraft. In section we present formation control strategies with formation feeback. In section 5 we illustrate simulation results for spacecraft formation control. By comparing the results with an without formation feeback, we emonstrate the superiority of the system with formation feeback over the one without formation feeback. Spacecraft Dynamics In this paper each spacecraft is moele as a rigi boy, with r i, v i, q i an! i representing the position, velocity, unit quaternion, an angular velocity of the ith spacecraft, where r i, v i, an! i are vectors an q i is a unit quaternion use to represent the attitue of a rigi boy. We will represent r i, v i, an! i in terms of their components in the inertial frame C O. or simplicity, we use the same symbol to enote a vector an its corresponing coorinate frame representation in the remainer of the paper. Euler s theorem for rigi boy rotations states that the general isplacement of a rigi boy with one point fixe is a rotation about some axis. Let z represent a unit vector in the irection of rotation, calle the eigenaxis, an let represent the angle of rotation about z, calle the Euler angle. The unit quaternion representing this rotation is given by q = [z T sin(=); cos(=)] T = [~q T ; q] T, where ~q is a vector with its components represente in the given coorinate frame an q is a scalar. It is easy to see that q an q represent the same attitue. To simplify our iscussion in the remainer of the paper, we assume q. Given a vector p, the corresponing cross-prouct operation p is efine as p = p p p p p p where p =[p ; p ; p ] T in terms of its components in the given coorinate frame. If we let C O O be a rotation matrix that represents the orientation of the frame C O with respect to C O, then r O = C O Or O, where r O an r O represent vector r in terms of its components in the frame C O an C O separately. The relationship between the unit quaternion q represente in the frame C O an the rotation matrix C O O is efine as 5 ; C O O =(q )I +~q ~q T q~q : The relationship between two rotation matrices C O O an C OO is given as C OO = C T O O =(q )I +~q ~q T +q~q : The multiplication of two quaternions is given by the formula q a q b = Q(q b )q a, where Q(q b ) = q b I ~q b ~q b ~q T : Let q b q be the inverse of the quaternion q given by q ~q ~q b = =. Suppose that q q the unit quaternions q an q represent the actual attitue
an the esire attitue of a rigi boy respectively, then the attitue error is given by q e = q q ~qe = : q e The translational ynamics for the spacecraft are C C C _r i =v i M i _v i =f i ; () where M i is the mass of the ith spacecraft, an f i is the control force. The rotational ynamics for the spacecraft are _q i =! i ~q i + q i! i _q i =! i ~q i () C O given by igure : Coorinate frame geometry. C J i _! i =! i J i! i + i ; where J i is the moment of inertia of the ith spacecraft, an i is the control torque on the ith spacecraft. Virtual Structure Equations Of Motion or Spacecraft In the virtual structure approach, we treat the entire formation as a rigi boy with place-holers fixe in the formation to represent the esire position an attitue for each spacecraft. As the virtual structure evolves in time, the place-holers trace out trajectories for each spacecraft to track. The relative orientation of each place-holer within the formation is fixe with respect to each other. 7 The control is erive in four steps: first, the esire ynamics of the virtual structure are efine, secon, the motion of the virtual structure is translate into the esire motion for each spacecraft, thir, tracking controls for each spacecraft are erive, an finally, formation feeback is introuce from each spacecraft to the virtual structure. The coorinate frame geometry is shown in igure. rame C O is an inertial frame. Since the formation can be thought of as a rigi boy with inertial position r, velocity v, attitue q, an angular velocity!, we efine the formation reference frame C locate at r with an orientation given by q with respect to the inertial frame C O. We also have one reference frame C i imbee in each spacecraft to represent the configuration of each spacecraft. Each spacecraft can be represente either by position r i, velocity v i, unit quaternion attitue q i, an angular velocity! i with respect to the inertial frame C O or by r i, v i, q i, an! i with respect to the formation reference frame C. Virtual structure equations of motion for each placeholer, that is, the esire motion for each spacecraft are r i (t) =r (t) +C O (t)r i (t) v i (t) =v (t) +C O (t)v i (t) +! (t) (C O (t)r i (t)) () q i (t) =q (t)q i (t)! i (t) =! (t) +C O (t)! i (t); where C O (t) is the rotation matrix of the frame C O with respect to C, an superscript above a vector means a esire state for each spacecraft. C O is given as C O = C T O =(q )I +~q ~q T +q ~q : Group maneuvers that preserve formation shape can be achieve as a succession of elementary formation maneuvers. Therefore, we will introuce virtual structure equations of motion for spacecraft via elementary formation maneuvers. The elementary formation maneuvers inclue translations, rotations, an expansions/contractions. Let (t) = [ (t); (t); (t)] T with its components represent the expansion/contraction rates along each formation reference frame axis. An expansion/contraction matrix is efine as (t) =iag((t)), which is a iagonal matrix. Generally all parameters in () can vary with time. However, if we are concerne with formation maneuvers that preserve the overall formation shape, ri, v, i qi, an! i are constant. To realize elementary formation maneuvers, we can vary r an v to translate the formation, vary q (q can be transforme to the rotation matrix C O.) an! to rotate the formation, an replace ri in the first an secon equations in () by (t)r i an replace v i in the secon equation in () by _(t)r i to expan or contract the formation. Arbitrary formation maneuvers can be realize by varying r (t), v (t), q (t),! (t), (t), an (t) _ simultaneously. In the case of preserving the overall formation shape uring
formation maneuvers, the equations of motion are given as r i (t) =r (t) +C O (t)(t)r i v i (t) =v (t) +C O (t) _ (t)r i q i (t) =q (t)q i! i (t) =! (t): +! (t) (C O (t)(t)r i ) () Note that! i is zero since q i is constant. The erivatives of the esire states are given by _r i (t) =_r (t) + _ C O (t)(t)r i + C O (t) _ (t)r i _v i (t) =_v (t) + _ C O (t) _ (t)r i + C O (t) (t)r i + _! (t) (C O (t)(t)r i ) (5) +! (t) ( _ C O (t)(t)r i + C O _ (t)r i ) _q i (t) =_q (t)q i _! i (t) =_! (t): rom (), we can see that if the velocity of the formation is zero, that is, v (t) =,! (t) =, an (t) _ =, then the esire velocity of each spacecraft is zero, that is, vi (t) =an! i (t) =. Also, if both the velocity an acceleration of the formation are zero, that is, v (t) =,! (t) =, (t) _ =, _v (t) =, _! (t) =, an (t) =, then both the esire velocity an acceleration of each spacecraft are zero, that is, vi (t) =,! i (t) =, _v i (t) =, an _! i (t) =. ormation Control Strategies With ormation eeback Let X i = [r T i ; vt i ; qt i ;!T i ]T an X i = [ri T ; v T i ; q T i ;! T i ] T represent the states an esire states for the ith spacecraft with respect to the inertial frame C O repectively. Let X i = [ri T ; vt i ; qt i ;!T i ]T T ;v i T ;q i T ;! i T ] T an Xi = [ri represent the states an esire states for the ith spacecraft with respect to the formation frame C respectively. Let X = [r T ;vt ;qt ;!T ;T ; _ T ]T an X = [r T ;vt ;qt ;!T ;T ; _ T ]T represent respectively the states an esire states for the virtual structure with respect to the inertial frame C O. We know that Xi is constant since we want to preserve the formation shape uring the group maneuvers. The aim of the formation maneuver is to evolve X (t) to X (t) while guaranteeing that X i(t) tracks Xi (t). Accoringly, a formation maneuver is efine as follows: A formation maneuver is asymptotically achieve if X (t)! X (t) an X i(t)! Xi (t) as t!. The control law for each spacecraft without formation feeback is given by the following lemma. Lemma. Let X = [r T ; v T ; q T ;! T ] T, X = [r T ; v T ; q T ;! T ] T, an let B @ _r _v _q J _! C A = B @ where (!) = If _v K r (r r ) K v (v v ) (!)q! J! + K q ~q e K! (!! )!!! T an q e = q q.. K r = K T r >, K v = K T v >, K q = K T q >, K! = K T! >,. r L [; ) \ L [; ),. _! +! L [; ) \ L [; ), then X X! as t!. C A ; () Proof: see Ref. an Ref.. In the case of the control law for the virtual structure, we nee to a two equations to () since an _ are use to represent the expansion/contraction rate of the formation, an pairs (K r ; K v ), (k q ; K! ), (K ; K _ ) correspon to translation, rotation, an expansion/contraction gains for the formation respectively. We also assume that q an! satisfy the rotational ynamics. Note that in the simple case when X is constant, the rotational ynamics is satisfie obviously. The control law for the virtual structure is given as follows. Lemma. Let _r _v _q C B B @ _! _ = C B A @ _v _v K r(r r ) K v(v v ) (! )q _! + k q~q e K! (!! ) _ K ( ) K _( _ ; C A _ ) (7) where q e = q q. If K r, K v, K!, K, an K _ are symmetric positive efinite matrices, an k q is a positive scalar, then X X! as t!. Proof: We can rewrite the secon equation in (7) as r r = K r(r r ) K v(_r _r ): Let ~r = r r, then ~r = K r ~r K v _ ~r. Since K r an K v are positive efinite, it is obvious to see that
r r! an v v! asymptotically as t!. Rewriting the thir equation in (7) as _~q = (q!! ~q ) _q =!T ~q : (8) Base on the assumption above, this equation is also true for q, so we can get _~q = (q!! ~q ) _q =!T ~q : (9) Let ~q = q q an ~! =!!. rom (8) an (9), we know that _~q = (q!! ~q q! +! ~q ) _~q = (!T ~q! T ~q ): () Let V = ~q T ~q, V = ~!T ~!, an consier the Lyapunov function caniate: V = k q V + V : Differentiating V, we get _V = ~q T _ ~q = ~! T (q ~q ~q ~q q ~q ): After some manipulation, we also know that q e = q q q = ~q + ~q ~q +q ~q ~q T ~q + q q which means that ~q e = q ~q +~q ~q +q ~q. Thus, _V = ~! T ~q e. Rewriting the fourth equation in (7) as ~! _ = k q ~q e K! ~!, an ifferentiating V, we can arrive at V _ = ~! T ~! _ = ~! T (k q~q e K! ~!). Therefore, V _ = k _ q V + _V = ~! T K! ~!. Let =f(~q ; ~! )j V _ =g, an be the largest invariant set containe in.on, V _ =, which implies that ~! since K! is positive efinite. When we plug! =! into the fourth equation in (7), we can show that ~q e = [; ; ] T, which implies that q = q. Therefore, by LaSalle s invariance principle, q q! an!!! asymptotically as t!. Thus X X! as t!. Therefore, X (t)! X (t) an X i(t)! Xi (t) as t!, an formation maneuvers without formation feeback are asymptotically achieve. ; or a secon orer system s + k s + k =,if we efine rise time t r an amping ratio, then natural frequency! n is approximately :8=t r. Therefore, if we let k =! n = (:8=t r) an k =! n = (:8=t r ), the transient specifications for the system are satisfie. We can efine K r, k q, K accoring to k, an efine K v, K!, K _ accoring to k. rom Lemma. an., we can see that the performe maneuver will be achieve an the spacecraft will track their esire states in the en. However, how well the spacecraft will preserve the formation shape uring the maneuver is not guarantee by this control law. or example, the errors r i (t) ri (t) an q i (t) qi (t) for the ith spacecraft may be large uring the maneuver, that is, the spacecraft may get out of the esire formation. If the virtual structure moves too fast, the spacecraft coul fall far behin their esire positions. If the virtual structure moves too slowly, the maneuver cannot be achieve within a short time. Therefore, we introuce formation feeback from the spacecraft to the virtual structure to overcome these rawbacks. We will introuce nonlinear gains in the control law for the virtual structure. Let X = [X T ;XT ; ;XT N ]T an X = [X T ;X T ; ;X T N ]T, where N is the number of spacecraft in the formation. The performance measure is efine as X X. We woul like to esign the nonlinear gains to meet the following requirements. When the spacecraft are out of the esire formation, that is, X X is large, the virtual structure will slow own or stop, allowing the spacecraft to regain formation. When the spacecraft are maintaining formation, that is, X X is small, the virtual structure will keep moving towar its final goal. A caniate for such gains can be efine as = K+K X X, where K = K T > is the gain when there is no formation feeback, an K = K T > is the formation gain which weights the performance measure X X.We can see that X X! )! K X X!)!: We can use nonlinear gains v,!, an _ to replace K v, K!, an K _ in (7), where nonlinear gains are efine as follows. v =K v + K X X! =K! + K X X () _ =K _ + K X X : Of course, we can use ifferent K an rise times for pairs (K r ; v ), (k q ; q ), an (K ; _ ) to change the weights of translation, rotation, an expansion/contraction effects. As a result, nonlinear gains slow 5
own or spee up the virtual structure base on how far out of the esire formation the spacecraft are. The control law for the virtual structure with formation feeback is given as follows. Lemma. Let B @ _r _v _q _! _ = C B A @ _v _v K r(r r ) v(v v ) (! )q _! + k q~q e! (!! ) _ K ( ) _ ( _ _ ) ; C A () where q e = q q. If v,!, an _ are given by (), then X X! as t!. Proof: We can follow the same proceure as Lemma. except that we use nonlinear gains v,!, an _ to replace the linear gains K v, K!, an K respectively _ everywhere in the proof. Since v,!, an _ are positive efinite, we can show that X X! as t!. Combine with the control law for each spacecraft, we know that X (t)! X (t) an X i(t)! Xi (t) as t!. ormation maneuvers with formation feeback are asymptotically achieve. When X (t) is specifie for the virtual structure, X (t) will track X (t) accoring to the control law for the virtual structure with formation feeback. If the formation moves too fast, X X will increase. As a result of the formation feeback, the virtual structure will slow own for the spacecraft to track their esire states, that is, to keep the formation. Thus X X will ecrease corresponingly, an the formation can keep moving towar its goal with a reasonable spee. As this couple proceure procees with time, the formation maneuver will be asymptotically achieve. 5 Simulation Results In this section we will consier a group of three spacecraft each with mass given 5 Kg. The esire original positions of the three spacecraft are given by r = [8; ; ]T, r = [; 8; ]T, r = [; ; 8]T meters an the esire original attitues are given by q = q = q = [; ; ; ]T with respect to the formation frame C. We suppose that the three spacecraft start from rest with some initial errors. The three spacecraft will perform a formation maneuver of a combination of translation, rotation, an expansion. The formation will start from rest with inertial position r () = [; ; ] T an inertial attitue q () = [; ; ; ] T to the esire position r = [; ; ]T an esire attitue q = [ut sin(=); cos(=)] T, where u = [= p ; = p ; = p ] T, an expan.5 times the original size. r i r i (m) abs( r i r i+ r i+ r i+ ) (m).5.5 (a) spacecraft # spacecraft # 8 x (c) i= i= 8 q i.5..5..5..5 q i+ (b) spacecraft # spacecraft # 8 8 x () i= i= 8 igure : Position an attitue errors without formation feeback (convergence time: 9 sec). In simulation, we will plot absolute position an attitue errors as well as relative position an attitue errors for each spacecraft to the time when the system converges. When X X + X X < :, we say that the system has converge. Absolute position error is represente by absolute ifference between actual position an esire position for each spacecraft. Absolute attitue error is represente by absolute ifference between actual attitue an esire attitue for each spacecraft. Since the formation shape is an equilateral triangle an the three spacecraft keep the same attitue in the formation, we use absolute ifference between lengths of the sies in the equilateral triangle to represent the relative position error an absolute ifference between the attitue of each spacecraft to represent the relative attitue error. If the formation is preserve exactly, the relative position an attitue errors shoul be zero. In this section, we use a subscript i ( i ) which is efine moulo to represent the states for the ith spacecraft. or each figure in this section, in part (a), we plot absolute position errors represente by r i ri. In part (b), we plot absolute attitue errors represente by q i qi. In part (c), we plot relative position errors represente byjkr i r i+ k kr i+ r i+ kj. In part (), we plot relative attitue errors represente by kq i q i+ k. Note that sometimes some curves may coincie with each other. igure shows the formation maneuver without formation feeback. igure shows the formation maneu-
r i r i (m).5..5..5. (a) spacecraft # spacecraft # q i....8.. (b) spacecraft # spacecraft # r i r i (m) 8 (a) spacecraft # spacecraft # q i..5... (b) spacecraft # spacecraft #.5.. 5 5 5 5 8 8 abs( r i r i+ r i+ r i+ ) (m) x (c) i= i= 5 5 q i+ 8 x () i= i= 5 5 abs( r i r i+ r i+ r i+ ) (m) 7 5 (c) i= i= 8 q i+.7..5.... () i= i= 8 igure : Position an attitue errors with formation feeback (convergence time: 5 sec). ver with formation feeback. By comparing each part of igure an igure, we can see that the maximum absolute an relative errors of the system without formation feeback is larger than that of the one with formation feeback. Also the system without formation feeback converges faster than the one with formation feeback when we choose the same rise time for both of them. When we ecrease the rise time in igure to let the system converge faster, the corresponing errors will increase significantly. Similarly, we can also increase the rise time to ecrease the errors, but the system will converge more slowly. In igure, since the system has formation feeback, we can choose smaller rise time than that in igure to let the system converge faster while the errors are still maintaine within a reasonable range. In igure an 5, we simulate the formation maneuver results when spacecraft # fails from 5th to th secon with an without formation feeback respectively. Since there is no formation feeback in igure, the virtual structure keeps moving towar its final goal even if one of the spacecraft fails for some time. As a result, spacecraft # cannot track its esire states satisfactorily, an the system has very large absolute an relative errors uring the perio when spacecraft # fails. In fact, in this case the spacecraft are out of formation for some time. However, in igure 5, since there is formation feeback, the virtual structure slows own to preserve the formation when one of the spacecraft fails for a perio of time. As a result, the system in igure 5 has much smaller absolute an relative errors than the one in igure. The formation is preserve much better than that in igure even if spacecraft # fails for 5 secons. Within the same range of error, the system with formation feeback can choose smaller rise time, an thus igure : Position an attitue errors without formation feeback when spacecraft # fails for 5 secons (convergence time: 9 sec). converge faster than the one without formation feeback. Within the same range of convergence spee, the system with formation feeback will have smaller errors than the one without formation feeback. We know that absolute an relative errors will ecrease when the formation gain K increases, but the convergence spee will ecrease corresponingly. At the same time, when the formation gain K ecreases, the system will converge faster, but the absolute an relative errors will increase corresponingly. We also know by simulation that it is har to choose goo rise time beforehan to achieve a goo performance in the system without formation feeback. However, a wie range of rise times work well in the system with formation feeback. Conclusion In this paper we have investigate a novel iea of introucing formation feeback uner the scheme of virtual structures through a etaile application of this iea to the problem of synthesizing multiple spacecraft in eep space. Introucing formation feeback from spacecraft to the formation has several avantages. irst, the system can achieve a goo performance in improving convergence spee an ecreasing maneuver errors. Secon, formation feeback as a sense of group stability an robustness to the whole system. Thir, formation feeback improves the robustness with respect to choosing gains for ifferent spacecraft. inally, formation feeback makes formation keeping more robust to synchronization issues an the variability on each spacecraft. 7
r i r i (m) abs( r i r i+ r i+ r i+ ) (m).5.... (a) spacecraft # spacecraft # 5 5.7..5.... (c) i= i= 5 5 q i q i+.5..5..5..5 (b) spacecraft # spacecraft # 5 5.... () i= i= 5 5 igure 5: Position an attitue errors with formation feeback when spacecraft # fails for 5 secons (convergence time: 5 sec). Acknowlegements This work was fune by Jet Propulsion Laboratory, California Institute of Technology uner grant #9-5. References [] P. K. C. Wang, Navigation strategies for multiple autonomous mobile robots moving in formation, Journal of Robotic Systems, vol. 8, no., pp. 77 95, 99. Control an Dynamics, vol., pp. 8 5, January 999. [7] B. Young, R. Bear, an J. Kelsey, A control scheme for improving multi-vehicle formation maneuvers, in Proceeings of the American Control Conference, (Arlington, VA), June. [8] J. Lawton, R. Bear, an. Haaegh, Elementary attitue formation maneuvers via behavior-base control, in Proceeings of the AIAA Guiance, Navigation an Control Conference, (Denver, CO), August. AIAA Paper No. AIAA--. [9] T. Balch an R. C. Arkin, Behavior-base formation control for multirobot teams, IEEE Transactions on Robotics an Automation, vol., pp. 9 99, December 998. [] J. Lawton, B. Young, an R. Bear, A ecentralize approach to elementary formation maneuvers, in IEEE International Conference on Robotics an Automation, (San ransico, CA), April. [] R. W. Bear, J. Lawton, an. Y. Haaegh, A feeback architecture for formation control, IEEE Transactions on Control Systems Technology, vol. 9, pp. 777 79, November. [] P. C. Hughes, Spacecraft Attitue Dynamics. John Wiley & Sons, 98. [] J. T.-Y. Wen an K. Kreutz-Delgao, The attitue control problem, IEEE Transactions on Automatic Control, vol., pp. 8, October 99. [] M. A. Lewis an K.-H. Tan, High precision formation control of mobile robots using virtual structures, Autonomous Robots, vol., pp. 87, 997. [] B. Wie, H. Weiss, an A. Apapostathis, Quaternion feeback regulator for spacecraft eigenaxis rotations, AIAA Journal of Guiance an Control, vol., pp. 75 8, May 989. [] P. K. C. Wang an. Y. Haaegh, Coorination an control of multiple microspacecraft moving in formation, The Journal of the Astronautical Sciences, vol., no., pp. 5 55, 99. [5]. Y. Haaegh, W.-M. Lu, an P. K. C. Wang, Aaptive control of formation flying spacecraft for interferometry, in IAC, IAC, 998. [] P. Wang,. Haaegh, an K. Lau, Synchronize formation rotation an attitue control of multiple free-flying spacecraft, AIAA Journal of Guiance, 8