Decentralized Guidance and Control for Spacecraft Formation Flying Using Virtual Target Configuration

Transcription

1 Decentralized Guidance and Control for Spacecraft Formation Flying Using Virtual Target Configuration Daero Lee 1, Sasi Prabhaaran Viswanathan 2, Lee Holguin 2, Amit K. Sanyal 4 and Eric A. Butcher 5 Abstract This paper proposes a novel guidance and control scheme for decentralized spacecraft formation flying on SE(3) via virtual target configuration. The configuration space for a spacecraft modeled as a rigid body is the Lie group SE(3), which is the set of positions and orientations of the spacecraft moving in three-dimensional Euclidean space. A combined guidance and feedbac control law in continuous time for the full (translational and rotational) motion of a rigid body under the influence of external forces and torques is designed. The guidance scheme for each spacecraft in the formation is based on the relative motion states between itself and the virtual target, and aims to reach a final relative configuration according to the desired formation. The relative configuration is described in terms of the exponential coordinates on SE(3). The control law then uses inversion of the dynamics equations of motion to obtain the desired trajectory given by the guidance scheme, in order to achieve the desired formation. Thus, each spacecraft can correct and update its relative configuration with respect to the virtual target in an autonomous manner. Local asymptotic stability of the desired trajectory in the nonlinear state space is demonstrated analytically. Numerical simulation results are presented to show the capability to perform a decentralized formation flying for the equilateral triangle formation. 1. INTRODUCTION Spacecraft formation flying (SFF) is a technology applicable for many space missions such as monitoring of the Earth and its surrounding atmosphere, geodesy, deep space imaging and exploration, and in-orbit servicing and maintenance of spacecraft. The formation of multiple spacecraft has several benefits including increased feasibility, improved accuracy, system robustness, flexibility, reduced cost, and reconfigurability 1]-3]. SFF cluster of spacecraft also offers a graceful degradation of performance for a spacecraft failure. If a single spacecraft malfunctions due to system failure, the remaining spacecraft may continue to perform the mission by maing up for its fault 3],4]. Therefore, SFF technology has seen a lot of research interest during the last few decades. In general, SFF control can be categorized into two types according to how the control decisions are made 2],4]-6]. A centralized control is a type of coordinated control where a leader spacecraft wors as manager to offer local control actions for the distributed systems 6],7]. On the other hand, decentralized control lets each spacecraft mae its own control decisions according to its desired objectives 2],4],8]- 12]. A decentralized control has two primary benefits over a 1 Daero Lee, Post Doctoral Researcher, Department of Mechanical and Aerospace Engineering MAE, New Mexico State University NMSU, Las Cruces, NM 883. daerolee@nmsu.edu 2 Sasi Prabhaaran & Lee Holguin, Grad Students, MAE, NMSU. {sashi, holguinl}@nmsu.edu 4 Amit Sanyal & 5 Eric Butcher, Assistant & Associate Professors, MAE, NMSU. {asanyal, eab}@nmsu.edu centralized control: fault-tolerance and simpler control law. Any problems with the leader in a centralized control may lead to the disruption of the entire system whereas the failure of the local control agent in a decentralized control scheme does not necessarily collapse the entire system because it is confined to the failed local control agent. Decentralized control is able to have relatively simple control law because the whole controller design is divided into smaller control agents. Furthermore, a decentralized control offers mild degradation of the system performance and reliability in the failure of local control agent 4],9]. In this paper, we propose a new decentralized guidance and control of multiple autonomous spacecraft, which are modeled as rigid bodies, to create a formation with reference to a virtual target. The virtual target has a specified motion nown to all spacecraft in the formation, and therefore a central supervisor is not required for formation eeping. The configuration space for each spacecraft modeled as a rigid body is the Lie group SE(3)13]-16], which is the set of positions and orientations of the spacecraft moving in three-dimensional Euclidean space. General treatments of mechanical systems whose configuration spaces are Lie groups are given in the references 17]-18]. The group SE(3) is the semi-direct product of b R 3 and the group of rigid body orientations of SO(3), i.e, SE(3)R 3 SO(3). The guidance scheme uses the reference states that consist of relative configuration and velocities (angular velocity and translational velocity) about the virtual target to each spacecraft. The relative configuration and velocities of each spacecraft are represented in exponential coordinates using exponential mapping 16],18]. The guidance scheme is able to generate the reference states autonomously from any given initial state and correct the reference states such that corresponding exponential states decay exponentially. The feedbac control that subsumes the control force and the control torque is derived using dynamic inversion. The feedbac tracing control of multiple spacecraft are required to maneuver large ranges of motion in 3-D Euclidean space and maintain the desired formation while aligning attitudes for a virtual target configuration. The feedbac controller has no discontinuity in the phase change. Moreover, the feedbac controller is proven to be asymptotically stable via Lyapunov stability analysis. The objective of this study is to propose a novel decentralized guidance and control scheme for spacecraft formation flying that enables a group of spacecraft to trac the reference trajectories and maintain a desired formation while synchronizing each spacecraft s configuration (position and

2 attitude) with respect to a virtual target configuration. The equilateral triangle formation about the virtual target in a highly eccentric orbit is selected for the SFF simulation scenario. The simulation results show the capability and effectiveness of the proposed decentralized guidance and control for spacecraft formation flying on SE(3). The remainder of this paper is organized as follows. In section 2, rigid body dynamics models of the target and the multiple spacecraft are described. Section 3 describes decentralized guidance and control scheme on SE(3). Section 4 presents analytical stability analysis. Section 5 presents numerical simulations. Finally, section 6 presents a concluding discussion of results obtained in this paper. 2. RIGID BODY DYNAMICS MODELS The dynamics of the virtual target spacecraft and the dynamics of the spacecraft desired to be in formation are described. All spacecraft are assumed to be rigid bodies and be in a central gravitational field in the Earth s orbital environment. The virtual target spacecraft is assumed to move in unconstrained natural motion in this environment. The spacecraft are required to achieve the desired position and then maintain it at a certain distance from the virtual target such that they can maintain an equilateral triangle formation about the virtual target. Each spacecraft body frame is configured to coincide with the target body frame. The dynamics models used here do not include effects due to the Earth oblateness (J2), atmospheric drag or solar pressure. A. Target States The virtual target is modeled as a rigid body that orbits around the Earth in a central gravity field. The configuration space of the virtual target is the special Euclidean group SE(3) which is the set of all translational and rotational motions of a rigid body. A subscript( ) is used to specify the virtual target states and parameters. The virtual target attitude is represented by the rotation matrix R SO(3) that transforms a body-fixed frame to the Earth-centered inertial (ECI) frame. The virtual target position is expressed by the inertial position vector b R 3 that begins from the origin of the ECI frame to the center of the mass of the target. Translational and angular velocities of the target are expressed by the vector ν R 3 and Ω R 3, respectively which are represented in the target-fixed frame. The inematics of the target are written as follows: ḃ R ν, Ṙ R (Ω ) (1) where the operator ( ) : R 3 so(3) is the cross-product operator defined by v v 1 v 2 v 3 v 2 v 3 v 1 v 3 v 2 v 1 Here, so(3) is the Lie algebra of SO(3), alternately the linear space of 3 3 sew-symmetric matrices. Let the target mass be denoted m and its moment of inertia J. The dynamics of the target object is given by: m ν m ν Ω + F g (b,r ) (2) J Ω J Ω Ω + M g(b,r ) (3) where Fg,Mg R 3 denote the gravity force and gravity gradient moment on the target, respectively, as given by Fg ( ( m µ ) 15 µ(p T ) J p) ( µ ) p + b 3 2 b 7 p 3 b 5 J p (4) ( ) µ Mg 3 b 5 (p J p) (5) p (R ) T b (6) and µ is the gravitational parameter of the Earth, which is assumed to be much larger than the target and chaser in this application. The state space of the target is TSE(3) SE(3) se(3) and the motion states of the target are given by (b,r,ν,ω ) SE(3) se(3). The configuration of the virtual target can also be represented on SE(3) as the following 4 4 matrix: g R b ] SE(3). 1 We also denote the vector of body velocities of the virtual target by ] ξ Ω R 6. b Thereafter, the inematics equation (1) can be expressed as follows: ġ g (ξ ), where (ξ ) (Ω ) b ] se(3). (7) We express the mass and inertia properties assigned to the virtual target, and the vector of gravity forces and moments, as follows: ] ] ϕg M g R 6, and I J m R 6 6. I F g We represent the adjoint and co-adjoint actions of SE(3) on se(3), which are defined in the usual sense as that between a Lie group and its corresponding Lie algebra, as linear operators 19]. The adjoint action is given by ] R Ad g b, s.t. Ad R R g X (gx g 1 ), (8) where ( ) 1 : se(3) R 6 is the inverse of the vector space isomorphism ( ) : R 6 se(3). The adjoint representation of se(3) can also be represented in matrix form as ad ζ Θ β Θ ] where ζ Θ β ]. (9) Therefore, the co-adjoint operator can be expressed in matrix form as ad ζ (ad ζ ) T Θ β ] Θ. (1)

3 Using the co-adjoint operator, the dynamics equations (2)-(3) can be expressed in the compact form: I ξ ad ξ I ξ + ϕ g, (11) which can be used to generate the state trajectory of the virtual target. B. Spacecraft Dynamics The configuration of the spacecraft is given by the position vector from the origin of the geocentric inertial frame to the center of mass of the spacecraft (denoted by b R 3 ), and the attitude given by the rotation matrix from a bodyfixed coordinate frame fixed to the geocentric inertial frame (denoted R SO(3)). The inematics for the spacecraft taes the same form as the inematics for the target: b R ν, R R (Ω ). (12) 3. GUIDANCE AND CONTROL SCHEME ON SE(3) A. Setting up a Formation Let the configuration of the formation be given by (h 1 f,h2 f,...,hn f ) SE(3)n, where h f denotes the fixed relative configuration of the th spacecraft to the virtual target. The h f should provide appropriate inter-spacecraft separations. Let (g,ξ ) SE(3) R 6 TSE(3) denote the states (configuration and velocities) of the th spacecraft. In this study, the number of the spacecraft is selected as three to maintain an equilateral triangle formation about the virtual target. The initial configuration of three spacecraft and the virtual target for an equilateral triangle formation flying in the virtual target body frame is illustrated in Fig. 1. The number of the spacecraft can be extended to produce a different formation. The inematic equation (12) can be expressed as follows: ġ g (ξ ), where (ξ ) (Ω ) ν ], (13) ν R 3 is the translational velocity of the spacecraft and Ω R 3 is the angular velocity of the th spacecraft, both vectors being expressed in the spacecraft s body frame. The state space of the spacecraft s motion states is also SE(3) se(3). Let φ c : SE(3) se(3) R 3 denote the feedbac control force acting on the spacecraft, and let τ c : SE(3) se(3) R 3 denote the feedbac control torque acting on the spacecraft. The dynamics equations of motion for the spacecraft are therefore given as follows: m ν m ν Ω + φ c (b,r,ν,ω ) + F g (b,r ) (14) J Ω J Ω Ω + τ c (b,r,ν,ω ) + M g (b,r ) (15) where F g,m g R 3 denote the gravity force and gravity gradient moment, respectively, on the spacecraft. The gravity force and moment on the spacecraft have the same functional form as the gravity force and moment on the target, given by equations (4)-(5). Note that unlie the dynamics models of terrestrial unmanned vehicles (as in 2]) or spacecraft in (nearly) circular orbits (as in 21]), for spacecraft moving in general orbits or trajectories, the gravity forces and moments vary with the location (inertial position vector b ) of the spacecraft. The dynamics equations (14)-(15) can then be expressed in the compact form: where ϕg M g F g I ξ ad ξ I ξ + ϕ e + ϕ c (16) ] ] ] R 6, ϕc τ c φc R 6, and I J m R 6 6. I where, ϕ c R 6 is the vector of control inputs (torque and force) and ϕ g R 6 are the external torque and force due to gravity on this spacecraft. Fig. 1. Initial configuration for an equilateral triangle formation in the virtual target body-fixed frame. Given the virtual target trajectory generated by equations (7)-(11), the relative configuration of the th spacecraft to the virtual target is h (t) ( g (t) ) 1 g (t) for t t (17) Let us define η (t) se(3) by ( η (t) ) log ( (h f ) 1 h (t) ) (18) The relative velocities of the target to this spacecraft are obtained by taing a time derivative of both sides of (18) and substituting equations (13), (7) and (17), which gives ξ ξ Ad (h ) 1ξ, (19) where ξ is the relative velocity of the virtual target with respect to the th spacecraft in its body coordinate frame. Let us denote the quantities h h (t ), η η (t ), ξ ξ (t ) and ξ ξ (t ) at initial time t, and assume that all these quantities are nown; these are nown if the initial

4 state of the th spacecraft is nown. Then η η (t ) can be obtained from equation (19) and 22] as follows: η G(η ) ξ. (2) In 22], we obtain the following expansion for G(X), for X R 6, in terms of the adjoint operator on se(3): G(X) I ad X + A(θ)ad 2 X + B(θ)ad 4 X, (21) where θ 2 A(θ) θ cot(θ/2) θ 2 8 csc2 (θ/2), and θ 2 B(θ) θ cot(θ/2) θ 2 8 csc2 (θ/2). Here θ X R is the norm of X R R 3, which is the vector of the first three components of the exponential coordinate vector X R 6, and corresponds to the exponential coordinates for rotational motion. The spacecraft formation control scheme given here is based on a desired evolution of the exponential coordinates η giving the relative configuration of the th spacecraft with respect to the virtual target. Suppose this desired evolution is given by η P η Kη, (22) where P and K are positive definite 6 6 matrices. The second-order differential equation (22) can be made to produce critically damped response for η. The gains P and K are chosen to be of the form P 2ζ ω I 6 6, K ω 2 I 6 6, (23) where ω is the natural frequency and ζ is the damping ratio. With η (t) given by equation (22) and initial conditions η and η, one obtains the spacecraft s states from equations (17)-(19) as where g (t) g (t)h (t) (24) ξ (t) G 1 (η (t)) η (t) + Ad (h (t)) 1ξ (t) (25) h (t) h f exp( (η (t)) ). If we assume that the principal angles corresponding to relative orientation of the spacecraft with respect to the virtual target are not too large, then G ( η ) I and one can approximate ξ as ξ (t) η (t) + Ad (h (t)) 1ξ (t). (26) Thereafter, one can approximate the accelerations on the spacecraft in its body frame as ξ (t) η (t) ad η (t) Ad (h (t)) 1ξ (t) + Ad (h (t)) 1 ξ (t). (27) Equations (24)-(27) can be used to re-write the relative dynamics in terms of (h, η ) as follows: ḣ h ( η ), (28) I η ad ξ I ξ + ϕ g + ϕ c + I (ad ξ Ad (h ) 1ξ Ad (h ) 1 ξ ) (29) Substituting equation (22) into equation (29) which is assumed exact, the feedbac control law used to stabilize the relative motion to the desired formation is obtained as ϕc I ( P η Kη ad ξ Ad (h ) 1ξ + Ad (h ) 1 ξ ) ad ξ I ξ ϕ g. (3) The dynamics of the feedbac system in exponential coordinates is then given by η G(η ) ( ξ Ad (h ) 1ξ ) (31) ξ P η Kη ad ξ Ad (h ) 1ξ + Ad (h ) 1 ξ (32) where the feedbac of the velocities ξ and the exponential coordinates η give the relative configuration between the spacecraft and the virtual target is used, along with the virtual target s velocities and accelerations. The above equations can be implemented with a standard integration scheme lie Euler s method or Runge-Kutta methods, since η and ξ are vectors in R 6. Thereafter, the relative configuration is obtained from h h f exp( (η ) ) (33) and the absolute configuration of the th spacecraft is g g h g h f exp( (η ) ). (34) Note that G(η ) I in the limit as η approaches the zero vector. Therefore, the approximation in equation (29) is accurate for small relative configuration. B. Stability Analysis The stability of the feedbac system consisting of equations (28)-(3) is analyzed here. Theorem 3.1: The close-loop system given by equations (28)-(3) is asymptotically stable at (h, ξ ) (h f,). Proof: The feedbac control law in equation (3) is substituted into equation (29) to give I η I ( P η Kη ), (35) with equality holding when η. Therefore, in a neighborhood of the origin, the exponential coordinate vector η is exponentially stable. Since η corresponds to h h f, and since ξ G(η ) η, this leads to the desired result. 4. NUMERICAL SIMULATION RESULTS This section presents numerical simulation results for all spacecraft maneuvers obtained by applying the decentralized guidance and control in section 3 to achieve and maintain an equilateral triangle formation. The mass and moment of inertia of the target and every spacecraft were identically selected from realistic micro spacecraft values. The mass of each spacecraft is : m 56.7 g and the moment of inertia diag ] g m 2. The virtual target was selected as the Molniya orbit whose eccentricity is high to add complexity to the guidance and control tass. Spacecraft 1 is assumed to be initially aligned with the virtual target frame. On the other hand, the spacecraft 2 and

5 S1 S2 S3 Relative Translational Values Relative position (m) Relative velocity (m/s) ] T ] T ] T ] T ] T ] T TABLE I INITIAL TRANSLATIONAL ERRORS OF THE CHASERS Relative Rotational Values Attitude error (deg) Relative angular velocity (rad/s) S ] T S ] T S ] T TABLE II INITIAL ROTATIONAL ERRORS OF THE CHASERS Fig. 2. Demonstration of an equilateral triangle formation in the virtual target body-fixed frame. 3 are required to rotate the principal rotation angles by 3 and 45 degrees about the virtual target body-fixed frame, respectively. Table 1 shows the initial relative positions and velocities of spacecraft 1, 2 and 3, respectively. S1, S2 and S3 denote spacecraft 1, spacecraft 2 and spacecraft 3, respectively. Table 2 shows the initial principal rotation angle differences between each spacecraft and the virtual target and relative angular velocities of spacecraft 1, 2 and 3 about the virtual target, respectively. The natural frequency ω and the damping ration ζ in equation (23) are selected by.83 and 1.2, respectively. The selected scenario in this study consists of an approach maneuver and a station eeping maneuver while they attempt to coincide with the virtual target body frame in one hour and then maintain the attitude alignment. The desired relative distance between each spacecraft is set to be 1 m. We use step size of h.1 sec for one orbital period in our numerical integration scheme. Fig. 2 shows that an equilateral triangle formation flying results in the target body-fixed frame is demonstrated for one period of the Molniya orbit (12 hours). Every spacecraft could almost achieve the desired positions from the given positions while they attempted to align their body frames with the virtual target body frame in about 18 seconds.then, they could maintain both the equilateral triangle formation and the attitude alignment. Equilateral triangle formation flying results are also plotted in the Clohessy-Wiltshire (CW) 23] frame in Fig. 3. The rotated trajectory was caused by the fact that the virtual target attitude was not fixed not fixed in the target LVLH frame. Fig. 4 shows the Earthgravity force and moment of the spacecraft 3. The Earthgravity force and moment of each spacecraft increase as the body comes closer to the Earth. Fig. 5 shows the norms of position tracing errors about the desired positions. After approach phase finished, the norms of the position tracing errors converged to less than.6 m. However, the increases in the norms of the position tracing errors in the bottom plot in Fig. 5 was caused by large variation in the Earth gravity Fig. 3. Demonstration of an equilateral triangle formation in the virtual target LVLH frame. Fig. 4. Earth-gravity force and moment.

6 force. Figure 6 shows the norms of the attitude tracing errors whose principal rotation angles were initially rotated, 3 and 45 degrees, respectively. Spacecraft 2 and 3 could coincide with the target body-fixed frame in 18 seconds with the norms of attitude tracing errors.1 degrees. Fig. 7 shows that the relative distances between each spacecraft that converge to 1 m in about 1 seconds and they are then maintained with error less than.2 m. This figure result obviously demonstrates that the desired relative distance between each spacecraft are surely satisfied. Fig. 7. Relative distances between each chaser. command that is about 4.6 N among them because the largest rotational maneuver was simultaneously enforced. Fig. 5. The norms of position tracing errors. Fig. 8. Norms of control forces. Fig. 6. Time plots of the norm of tracing error in attitude. The increasing magnitudes of control forces and the control torques in the bottom plots in Fig. 8 and Fig. 9 were caused by the large variation in the Earth-gravity force and moment in the perigee. We also note that the norms of control force and torque converge to lower positive limit during approach and attitude alignment maneuvers and then they maintained fairly constant values to counteract with nonzero force and moment. In Fig. 8, the norms of control forces of all spacecraft are plotted. Spacecraft 3 required the largest control force Fig. 9. Norms of control torques.

7 5. CONCLUSION This study proposed a new and novel a decentralized guidance and control scheme for spacecraft formation flying using a virtual target configuration. An equilateral triangle formation flying scenario was successfully fulfilled for one period of the Molniya orbit. Each spacecraft has a capability to perform its own mission autonomously about the virtual target that always maintains a normal woring condition. The feedbac control includes the embedded guidance that produces the relative configuration using exponentially decaying exponential coordinates. It is practically stable in tracing a desired trajectory in the nonlinear state of rigid body motion on SE(3). It can also produce moderate control input to achieve two different phases that consist of approach maneuver and station-eeping maneuver while attempting to achieve attitude alignment with the virtual target body-fixed frame. A stability analysis on the nonlinear state space shows the asymptotic stability of the desired trajectory with this control scheme. Numerical simulation results demonstrate that the proposed guidance and control scheme can be successfully applied for a decentralized spacecraft formation flying and achieve the excellent tracing performance. REFERENCES 1] R. W. Beard, Jonathan Lawton and Fred Y. Hadaegh, A Coordination Architecture for Spacecraft Formation Control, IEEE Transactions on Control Systems Technology, Vol. 9, No. 6, pp ] W. Ren and Randal W. Beard, Decentralized Scheme for Spacecraft Formation Flying via the Virtual Structure Approach, Journal of Guidance, Control and Dynamics, Vol. 27, No. 1, 24, pp ] C. Sabol, Rich Burns, and Craig A. McLaughlin, Satellite Formation Flying Design and Evolution, Journal of Spacecraft and Rocets, Vol. 38, No. 2, 21, pp ] M. C. VanDye. and C. D. Hall, Decentralized Coordinated Attitude Control Within a Formation of Spacecraft, Journal of Guidance, Control and Dynamics, Vol. 29, No. 5, 26, pp ] J. Bae and Y. Kim, Design of Optimal Controllers for Spacecraft Formation Flying Based on the Decentralized Approach, International Journal of Aeronautical and Space Sciences, Vol. 1, No. 1, 29, pp ] C. De La Cruz and R. Carelli, Dynamic Modeling and Centralized Formation Control of Mobile Robots, IEEE Industrial Electronics, IECON, 32nd Annual Conference, 26, pp ] S. Keshmiri and S. Payandeh, A Centralized Framewor to Multi-robots Formation Control: Theory and Application, CARE@AI 9/CARE@IAT 1 Proceedings of the CARE@AI 29 and CARE@IAT 21 international conference on Collaborative agents - research and development, pp ] A. Zou and K. Dev Kumar, Distributed Attitude Coordination Control for Spacecraft Formation Flying, IEEE Transactions on Aerospace and Electronics Systems, Vol. 48, No. 2, 212, pp ] J. Russell Carpenter, Decentralized Control of Satellite Formations, International Journal of Robust and Nonlinear Control, Vol. 12, Issue 2-3, 22, pp ] J. R. T. Lawton, Randal W. Beard, and Brett J. Young, A Decentralized Approach to Formation Maneuvers, IEEE Transactions on Robotics and Automation, Vol. 19, No. 6, 23, pp ] J. Erdonga, J. Xiaoleib, S. Zhaoweia, Robust decentralized attitude coordination control of spacecraft formation, System and Control Letters, Vol. 57, Issue. 7, 28, pp ] Y. Lang and Ho Lee, Decentralized Formation Control and Obstacle Avoidance for Multiple Robots with Nonholonomic Constraints, Proceedings of the 26 American Control Conference, Minneapolis, Minnesota, 26, pp ] A. K. Sanyal, N. Nordvist, and M. Chyba, An Almost Global Tracing Control Scheme for Maneuverable Autonomous Vehicles and its Discretization, IEEE Transactions on Automatic Control, Vol. 56, No. 2, 211, pp ] A. K. Sanyal, N. Nordvist, and M. Chyba, A Lie Group Variational Integrator for Rigid Body Motion in SE(3) with Applications to Underwater Vehicle Dynamics, IEEE Conference on Decision and Contro, Atlanta, GA, 21, pp ] A. Sanyal, L. Holguin, S. Prabhaaran Viswanathan, Guidance and Control for Spacecraft Autonomous Chasing and Close Proximity Maneuvers, 7th IFAC Symposium on Robust Control Design Robust Control Design, Vol. 7, Part 1, Denmar Aalborg Congress and Culture Centre, Denmar, ] T. Lee, M. Leo, and N. Harris McClamroch, Dynamics of Connected Rigid Bodies in a Perfect Fluid, American Control Conference, St. Louis, MO, 29, pp ] J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry, Springer, 2nd Edition, ] F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, Springer, New Yor, ] V. S. Varadarajan, Lie Groups, Lie Algebras and Their Representations, Springer, New Yor, ] A. K. Sanyal, N. Nordvist and M. Chyba, An Almost Global Tracing Control Scheme for Maneuverable Autonomous Vehicles and its Descretization, IEEE Transactions on Automatic Control, Vol. 56, No. 2, 211, pp ] A. K. Sanyal, T. Lee, M. Leo, and N. H. McClamroch, Global Optimal Attitude Estimation using Uncertainty Ellipsoids, Systems and Control Letters, Vol. 57, No. 3, 28, pp ] F. Bullo and R. M. Murray, Proportional derivative (PD) control on the Euclidean group, European Control Conference, Vol. 2, Rome, Italy, 1995, pp ] A. K. Sanyal and M. Chyba, Robust Feedbac Tracing of Autonomous Underwater Vehicles with Disturbance Rejection, In Proceeding American Control Conference, St. Louis, MO, 29, pp ] W. H. Clohessy and R. H. Wiltshire, Terminal Guidance System for Spacecraft Rendezvous, Journal of the Aerospace Sciences, Vol. 27, 196, pp ] S. Segal, and P. Gurfil, Effect of Kinematic Rotation-Translation Coupling on Relative Spacecraft Translational Dynamics, Journal of Guidance, Control and Dynamics, Vol. 32, No. 3, May- June 29, pp ] K. T. Alfriend, P. Gurfil, S. Rao Vadali, Jonathan How, Louis Breger, Spacecraft Formation Flying: Dynamics, Control and Navigation, 21 Elsevier Ltd., 1st ed, 21.