Attitude Regulation About a Fixed Rotation Axis

Transcription

1 AIAA Journal of Guidance, Control, & Dynamics Revised Submission, December, 22 Attitude Regulation About a Fixed Rotation Axis Jonathan Lawton Raytheon Systems Inc. Tucson, Arizona Randal W. Beard Electrical and Computer Engineering Department Brigham Young University, Provo, Utah 8462 Abstract This paper considers the attitude control problem of rotating a rigid body from its current attitude to a desired attitude such that the instantaneous rotation axis is aligned with an externally defined axis of rotation. The key idea is to factor the attitude quaternion into rotations parallel and perpendicular to the desired rotation axis. State feedback and output feedback control strategies are designed to minimize the perpendicular component. Simulation results are provided to illustrate the salient features of the approach. Corresponding author: beard@ee.byu.edu. 1

2 Nomenclature q Unit quaternion representation of the attitude of the spacecraft. q d Unit quaternion representation of the desired attitude of the spacecraft. e = q d q, Unit quaternion representation of the attitude error. ˆp Vector part of the unit quaternion p. p Scalar part of the unit quaternion p. q = ( ˆq T, q ) T, the conjugate of q. 1 = (,,, 1) T, the unit quaternion identity. ω Angular velocity of the spacecraft. ω = (ω T, ) T. J Inertia matrix of the spacecraft. τ Torque applied to the spacecraft. u Desired axis of rotation. e The component of e that is parallel to u. e The component ( ) of e that is perpendicular to u. θ = 2 tan 1 u T ê e., ω = θu, the angular velocity associated with e. ω The angular velocity associated with e. 1 Introduction Rigid body attitude regulation is a problem that has generated much interest. A great deal of work has been done in this area, including attitude tracking, 1 inverse optimal stabilization, 2 approximate solutions to the optimal attitude control problem 3 and H suboptimal control. 4, 5 These results assume that the spacecraft angular velocity is known. A good approximation of the spacecraft angular velocity is often not available. Passivity-based control has been derived to regulate the attitude of a spacecraft 6, 7 without velocity information. Sometimes the path taken to the final orientation is as important as is the final orientation. For example, a problem that has attracted attention lately is the spacecraft formation control problem. 8 The attitude formation control problem described in Ref. 8, 9 requires that the spacecraft constellation perform a maneuver such that the attitude of the individual spacecraft remain synchronized. To accomplish this, the spacecraft must rotate about axes of rotation that are parallel to each other. 2

3 This type of maneuver leads to an interesting spacecraft attitude control problem. If the spacecraft deviates from the desired axis of rotation, then the objective of the control law is to re-align the axis of rotation, with the other spacecraft, as well as synchronize the angle of rotation along that direction. Attitude synchronization has been discussed in Ref. 8. The objective of this paper is to present a model independent attitude control law, that facilitates the types of formation maneuvers discussed above, i.e., to rotate a rigid body, as closely as possible, about an externally assigned axis of rotation. Our objective is in contrast to the well known problem of eigenaxis rotation where the goal is to rotate through the shortest possible angle. If 1, 11 the spacecraft if performing a rest-to-rest maneuver, and it deviates from the original eigenaxis, the control objective is still to maneuver the spacecraft to its final attitude through the shortest possible angle. Therefore, unlike the problem addressed in this paper, the eigenaxis may change in time, due to disturbances or model uncertainty. Current strategies for eigenaxis rotations use a combination of feedback control and feed-forward control terms. Wie, Weiss and Arapostathis 12 used feedback on the quaternion error combined with feed-forward of the gyroscopic term to implement eigenaxis rotations via quaternion regulation. These regulation results were extended to the tracking problem by Weiss. 13 The tracking results are also model dependent and require cancellation of the gyroscopic term. Some effort has been made to derive a model independent approach to quaternion regulation via eigenaxis rotations. Cristi, Burl and Russo 14 developed adaptive control strategies to approximate the control found in Ref. 12. Output adaptive control is used in Ref. 15 to approximate an inertia matrix. Our approach it to consider a rest-to-rest maneuver and to factor the required rotation into a rotation about u, where u is an arbitrary unit vector fixed in the inertial frame, followed by a rotation about an axis perpendicular to u. The control law then damps out the rotation perpendicular to u. A paper that is related to our approach is Ref. 16 which addresses the problems of spin stabilization about an arbitrary axis defined in the body frame. However, Ref. 16 differs from our approach in several aspects. First, the rotation axis in Ref. 16 is fixed in the body frame, in contrast to our approach where u is fixed in the inertial frame. Second, in Ref. 16, the rotation axis must be known a priori, as the control strategy depends explicitly on the rotation axis. On the other hand, in our paper u is an input to the controller, which is designed independent of u. Finally, the control strategy 3

4 in Ref. 16 is model dependent in that it requires knowledge of J, where our result is independent of J. The paper is organized as follows. In Section 2, the mathematical notation and models used in the paper will be introduced. In addition, a precise statement of the rotation axis attitude control problem will be given. In Section 3 we present two control strategies for rotation axis attitude control. The first is a state feedback result that assumes both attitude and angular velocity information. The second result removes the assumption that angular velocity is available. In addition, some heuristic rules for tuning the proposed strategies are given. In Section 4 we present simulation results that demonstrate our technique. In Section 5 we offer some conclusions. 2 Background and Problem Statement This section provides brief background material on unit quaternions. Complete discussions on the use of unit quaternions for attitude representation can be found in Refs. 17, 18. A unit quaternion is a four dimensional unit vector. A rotation of φ radians about a unit vector z is represented by the unit quaternion p = p 1 p 2 p 3 p ( ) ( ( ˆp = sin φ ) = 2 z p cos ( ) φ where ˆp is the vector part of the quaternion and p is the scalar part of the quaternion. The conjugate of a unit quaternion, which represents a rotation of φ about z is given by ( ) ˆp p =. Quaternion multiplication is defined by the equation ( ) qˆp + p pq = ˆq + ˆp ˆq q p ˆq T. ˆp The identity quaternion is given by 1 = (,,, 1) T were pp = p p = 1. On occasion we will have need to refer to the scalar and vector parts of the 4 p 2 ),

5 product of two or more quaternions. In that case we will use the notation (pq) and pq refer to the scalar and vector parts of pq respectively. To simplify the equations in several places we will use the notation ( ) ω ω =. If the attitude of the spacecraft is represented by q, then the equations of motion for the spacecraft are given by q = 1 q ω 2 (1) J ω = ω Jω + τ, (2) where ω is the angular velocity, J is the inertia matrix and τ is the applied torque. If q represents the attitude of the spacecraft, and q d represents its desired attitude, then the error quaternion, which represents the attitude error between q and q d is given by e = q d q. This paper considers the attitude control problem of rotating a rigid body from its current attitude q to a desired attitude q d, where we would like the instantaneous axis of rotation to align as closely as possible, with an externally defined desired axis of rotation that is fixed in the inertial frame. Let the desired axis of rotation be represented by the unit vector u. A block diagram of the proposed approach is shown in Figure 1, were both q d and u are defined externally. 3 Rotation Axis Control In this section we present the main results of the paper. Section 3.1 shows how to factor a unit quaternion into two unit quaternions that represent a rotation about a fixed inertial axis u, and a rotation about an axis perpendicular to u. Section 3.2 derives a state feedback control law that minimizes the motion perpendicular to u. Section 3.3 offers intuition about how the gains can be selected for a desired response. Finally, an output feedback result is presented in Section

6 3.1 Quaternion Factorization Let e = q d q be the error quaternion and let u be an arbitrary desired axis of rotation fixed in the inertial frame. We define e as the unit quaternion that represents the portion of the rotation of e that is parallel to u, i.e., ( ( sin θ ) ) e (u) = 2 u cos ( ) θ (3) where (êt ) θ = 2 tan 1 u. (4) e Similarly, we can define the unit quaternion perpendicular to the desired axis of rotation as ( ( cos θ ) e (u) = e e = 2 ê e sin ( ( θ 2) u sin θ ) 2) u ê cos ( ) θ e + sin ( θ. (5) 2 2) êt u Lemma 3.1 Let e be the unit quaternion representing attitude error and let u be an arbitrarily defined desired axis of rotation, and let e (u) and e (u) be defined as in Equations (3) and (5), respectively, then the following statements hold. 1. e = e (u)e (u). 2. ê T u =. 3. ê T ê =. 4. The angular velocity ω = θu satisfies the rigid body kinematic equation 5. The angular velocity 2 ė = 1 2 e ω. ω = ω e ω e, satisfies the rigid body kinematic equation ė = 1 2 e ω. Proof: The proof of this Lemma is based on standard quaternion manipulation. Full details appear in Ref

7 3.2 State Feedback Rotation Axis Control In this section we will consider the control strategy given by τ = k 1 ê d 1 ω k 2 ê d 2 (I uu T )ω, (6) where the first two term are similar to PD control using quaternion feedback, 1 and the second two terms feedback the portion of the error quaternion that is perpendicular to u. Theorem 3.1 Consider the spacecraft whose dynamics are given by Equations (1) and (2). Let u be an arbitrary unit vector representing the desired axis of rotation in the inertial frame, let e be given by Equation (5), and let the input torque be given by Equation (6). 1. If the constants k 1, k 2, d 1, and d 2 are positive and k 1 k 2, then ê(t) and ω(t) asymptotically. 2. In addition, if k 1 e() k 2 e () ωt ()Jω() < 2k 1 + 2k 2 then e(t) 1 and ω(t) asymptotically. 3. If condition 1 holds and the initial state satisfies e() 1, ω() =, e () 1 δ, δ2 + (k 1 /k 2 ) e() 1 2 ɛ, (7) then (q, ω) (1, ) asymptotically, and e (t) 1 ɛ for all t. The first statement of the theorem guarantees regulation of the attitude error to zero. The second statement excludes the case of regulation of the error to 1. The third statement guarantees that if the component of the initial attitude error that is perpendicular to u is bounded by δ, that throughout the maneuver it remains bounded by ɛ, where ɛ can be made arbitrarily close to δ by an appropriate choice of k 1 and k 2. Proof: 7

8 (1) Consider the Lyapunov function candidate V = k 1 e k 2 e ωt Jω, which is zero if and only if e = 1 and ω =. Differentiating V we obtain, V = 2k 1 (e 1) T ė + 2k 2 (e 1)ė + ω T J ω. Noting that ė = q d q = 1 2 qd q ω = 1 e ω and using Lemma 3.1 and Equation (2) we 2 obtain V = k 1 (e 1) T e ω + 2k 2 (e 1)e ω + ω T τ. Observing that we get ( ) T ( ) ê (e 1) T e ω + ω ê e ω = e 1 ê T ω = ω T ê, V = k 1 ω T ê + k 2 ω T ê + ω T τ. (8) Using the definition of ω, it is straight forward to show that ω T ê = ω T ê. Therefore V can be written as V = k 1 ω T ê + k 2 ω T ê + ω T τ = ω T (τ + k 1 ê + k 2 ê ) = ω T ( d 1 ω d 2 (I 3 uu T )ω ) = ω T Dω, (9) where D = (d 1 I 3 + d 2 (I 3 uu T )), and the third line was obtained from Equation (6). Noting that D is symmetric and positive definite, we see that V. Let Ω = {(e, ω) V = } and let Ω be the largest invariant subset in Ω. On Ω, ω(t) and τ (t), therefore from Equations (2) and (6) we get that k 1 ê + k 2 ê =. (1) Since ê = e ê + e ê + ê ê, 8

9 substituting into Equation (1) gives k 1 e ê + (k 2 + k 1 e )ê + k 1 ê ê =. (11) If we multiply Equation (11) by ê T, êt and (ê ê ) T, respectively we obtain the following relations: e ê 2 =, (12) (k 2 + k 1 e ) ê 2 =, (13) ê ê 2 =. (14) Noting from Lemma 3.1(3) that ê T ê =, Equation (14) indicates that ê and ê are simultaneously parallel and perpendicular. Therefore, either ê or ê must be zero. If ê = then e = ±1 and from Equation (12) ê =. On the other hand, if ê = then e = ±1 and provided that k 1 k 2 Equation (13) implies that ê =. Therefore ê = ê =, which implies that e = ±1 and e = ±1, or equivalently e = e e = ±1. Therefore Ω = {(±1, )} and LaSalle s Invariance principle ensures that ê(t) and ω(t) asymptotically. (2) Suppose that e = 1. If follows that e e = 1, or e = e. Therefore ê = ê, and e = e. Since ê and ê are simultaneously equal and perpendicular they must both equal zero. From Equation (4), ê = implies that θ =, which implies from Equation (5) that e = e = 1. Thus we can see that e = e = 1. Therefore, when e = 1 and ω = the Lyapunov function is equal to 2k 1 + 2k 2. Since V if V (e(), ω()) < 2k 1 + 2k 2, the attitude state can never reach the state (e, ω) = ( 1, ), which implies that the attitude will converge to (1, ) since this is the only remaining element of Ω. 9

10 (3) Note that since ω() = and q() 1 that condition 2 holds and thus (q, ω) (1, ) asymptotically. From Lyapunov theory we get k 2 e (t) 1 2 k 1 e(t) k 2 e (t) ω(t)t Jω(t) = V (t) V () from which we obtain = k 1 e() k 2 e () ω()t Jω() = k 1 e() k 2 e () 1 2 (since ω() = ) k 1 e() k 2 δ 2, e (t) 1 δ 2 + (k 1 /k 2 ) e() 1 2 ɛ. 3.3 Gain Selection Heuristics The objective of this section is to offer some simple heuristics for choosing the gains k 1, d 1, k 2, and d 2. We begin by combining Equations (2) and (6) to obtain J ω + ω Jω + d 1 ω + d 2 (I 3 uu T )ω + k 1 ê + k 2 ê =. (15) Projecting this equation onto the desired axis of rotation by multiplying both sides of Equation (15) by u T, gives u T J ω + u T (ω Jω) + d 1 u T ω + k 1 u T ê =. (16) Note that k 2 and d 2, the gains associated with off axis motion, do not appear in Equation (16). Assuming that the off-axis motion is small we can approximate ê and ω as ê sin(θ/2)u and ω θu. Substituting these approximations into Equation (16) gives u T Ju θ + d 1 θ + k1 sin(θ/2) =. (17) For small angles sin(θ/2) θ/2 which allows us to write Equation (17) as θ + d 1 u T Ju θ + k 1 θ =, (18) 2u T Ju 1

11 which is a second order linear system. The gains k 1 and d 1 can be selected to give the desired linear response for small angles near the axis of rotation. Selection of k 2 should be made based on Equation (7). Given ɛ, δ < ɛ, k 1, and e() 1, k 2 is chosen such that k 2 k 1 e() 1 ɛ 2 δ 2. The gain d 2 adds damping to the motion of e. While the selection of d 2 appears arbitrary, note however, from Equation (6) that the matrix d 1 I 3 + d 2 (I 3 uu T ) multiplies ω. Therefore if d 2 >> d 1, the condition number of this matrix becomes large, and small, possibly numeric, variations in ω may cause large swings in τ. Based on simulation, we have found that 1d 1 d 2 1d 1 results in adequate damping. 3.4 Output Feedback Rotation Axis Control The control law given in Equation (6) requires knowledge of both e and ω. This section uses the passivity ideas presented in Refs. 6, 7 to remove the dependence on ω. Theorem 3.2 Consider the spacecraft whose dynamics are given by Equations (1) and (2). Let u be an arbitrary unit vector representing the desired axis of rotation fixed in the inertial frame, let e be given by Equation (5), and let the input torque be given by the following dynamic system α = Aα + Be, y = B T P Aα + B T P Be, τ = k 1 ê k 2 ê ê y. (19) 1. If k 1 >, k 2 >, k 1 k 2, A IR 4 4 is Hurwitz, B IR 4 4 is full rank, and P is the positive definite solution to A T P + P A = Q, where Q IR 4 4 is negative definite, then ê(t) and ω(t) asymptotically. 2. In addition, if α() = A 1 Be() and k 1 e() k 2 e () ωt ()Jω() < 2k 1 + 2k 2 then e(t) 1 and ω(t) asymptotically. 11

12 3. If condition 1 holds, and the initial conditions satisfy q() 1 α() = A 1 Be(), ω() =, e () 1 δ, δ2 + (k 1 /k 2 ) e() 1 2 ɛ, then e (t) 1 ɛ for all t. Proof: Note that the filter equation can be expressed as α = Aα + Be y = B T P Aα + B T P Be ᾱ = Aᾱ + Bė, y = B T P ᾱ, (2) where ᾱ = α. Consider the Lyapunov function candidate V = k 1 e k 2 e ωt Jω + ᾱ T P ᾱ. Differentiating V and substituting in Equation (2) and using the same steps as in the previous section we get: V = ω T (τ + k 1 ê + k 2 ê ) ᾱ T Qᾱ + 2ᾱ T P Bė = ω T (τ + k 1 ê + k 2 ê ) ᾱ T Qᾱ + y T (eω) = ω T (τ + k 1 ê + k 2 ê ) ᾱ T Qᾱ + (y eω) = ω T (τ + k 1 ê + k 2 ê ) ᾱ T Qᾱ + ω T ê y = ω T (τ + k 1 ê + k 2 ê + ê y) ᾱ T Qᾱ = ᾱ T Qᾱ. 12

13 Let Ω be the largest invariant subset of Ω = {(e, ω, ᾱ) V = }. On Ω, ᾱ(t). This also implies that ᾱ(t). From Equation (2), on this set, ė(t). Since ω = 2e ė and τ = J ω + ω Jω, it also follows that ω(t) τ (t). Furthermore, since y = B T P ᾱ, on Ω, y(t). Therefore k 1 ê + k 2 ê =. The arguments for the rest of the proof are similar to the proof of Theorem Simulation Results In this section we present simulation results for a spacecraft whose model is given by Equations (1) and (2) where J = Simulation results for four different controllers will be presented. The first controller is the state feedback, rotation axis controller given in Equation (6) where k 1 = 115, k 2 = 115, d 1 = 736 and d 2 = 736. The second controller is the output feedback, rotation axis controller given in Equation ((19) ) where u (u k 1 = 92, k 2 = 115, A = 1I 4, B = 54I 4, and Q = 17I 4 1 T ). For comparison purposes we will also present simulation result for the model dependent eigenaxis controller presented in Ref. 12 and given by τ = ω Ĵω k1ĵê d1ĵω, where k 1 =.1, d 1 =.64, and Ĵ is an approximation to J to within 1%. In addition, we will compare our results to the quaternion PD control suggested in Ref. 1 which is given by where k 1 = 1 and d 1 = 7. τ = k 1 ê d 1 ω, 13

14 4.1 Example 1. The first maneuver that we will present is a rest-to-rest eigenaxis maneuver where the initial conditions of the spacecraft are given by q() = sin ( ) 1 4π 18 cos ( ), ω() = 4π 18 and the desired attitude is given 1 q d = sin(). cos() In other words, the desired rest-to-rest maneuver is a 4 degree rotation about the axis (1,, ) T. The desired axis of rotation is also given by u = (1,, ) T. Figure 2 shows the simulation results. Note that the control gains for each of the four controllers have been tuned to give similar response for this particular maneuver. The top-left subplot shows the convergence of the combined state metrics e 1 + ω. The top-right subplot shows the norm of the torque τ. Note that each control strategy requires roughly the same amount of torque for roughly the same convergence rate. The bottom-left subplot plots e 1 and therefore shows the deviation of the attitude from the desired axis of rotation, which in this case is the eigenaxis at time zero. Note that since neither the model dependent eigenaxis control strategy nor the PD control strategy have feedback on e small deviations from the desired axis of rotation are not compensated for. The bottom-right subplot shows a graph of ê(t) ê(t) u. ê(t) ê(t) Note that is the instantaneous rotation axis of the spacecraft with respect to the desired attitude. Therefore the bottom-right subplot indicates the deviation of the instantaneous rotation axis from the desired axis of rotation. Figure 2 shows that the instantaneous rotation axis may deviate substantially from the original desired eigenaxis. 14

15 4.2 Example 2. For the second example, consider the same rest-to-rest maneuver presented in Example 1, but where the desired rotation axis is given by cos ( ) 3π 18 u = sin ( ) 3π. 18 In other words, the desired rotation axis is 3 degrees off of the eigenaxis at time zero. Figure 3 shows the simulation results for this case. Note first that the motion of the spacecraft perpendicular to u is driven to zero quickly by the proposed controllers. However, movement about a desired rotation axis as opposed to the natural eigenaxis comes at the price of control effort. The control torque is four time greater for the rotation axis controllers, than the eigenaxis and PD control strategies. In example 1, the desired rotation axis was the natural eigenaxis and therefore the output of the four controllers was essentially the same. In this example, the rotation axis control strategies attempt to rotate the spacecraft to an orientation such that the remaining maneuver is aligned with the desired rotation axis. The alignment with the desired rotation axis requires extra bandwidth. 5 Conclusions In this paper we have presented a model independent attitude control law that is designed to align the instantaneous rotation axis of a spacecraft about an externally defined desired axis of rotation. Both state feedback and output feedback strategies were presented. The most obvious application of this approach is attitude formation control where a group of spacecraft must rotate about the same axis of rotation. The two simulation examples illustrate that the approach in fact regulates the instantaneous rotation axis to the desired axis of rotation. In addition, the examples illustrate that additional control effort is required to effect these types of maneuvers. Acknowledgment This research was supported by the Jet Propulsion Laboratory, California Institute of Technology under contract #

16 References 1 Wen, J. T.-Y. and Kreutz-Delgado, K., The Attitude Control Problem, IEEE Transactions on Automatic Control, Vol. 36, No. 1, October 1991, pp Krstić, M. and Tsiotras, P., Inverse Optimal Stabilization of a Rigid Spacecraft, IEEE Transactions on Automatic Control, Vol. 44, No. 5, May 1999, pp Lawton, J., Beard, R. W., and McLain, T., Successive Galerkin Approximation Of Nonlinear Optimal Attitude Control, Proceedings of the American Control Conference, San Diego, CA, June 1999, pp Kang, W., Nonlinear H Control and Its Application to Rigid Spacecraft, IEEE Transactions on Automatic Control, Vol. 4, No. 7, July 1995, pp Dalsmo, M. and Egeland, O., State Feedback H -Suboptimal Control of a Rigid Spacecraft, IEEE Transactions on Automatic Control, Vol. 42, No. 8, August 1997, pp Lizarralde, F. and Wen, J., Attitude Control Without Angular Velocity Measurement: A Passivity Approach, IEEE Transactions on Automatic Control, Vol. 41, No. 3, March 1996, pp Tsiotras, P., Further Passivity Results For The Attitude Control Problem, IEEE Transactions on Automatic Control, Vol. 43, No. 11, 1998, pp Lawton, J. and Beard, R. W., Synchronized Multiple Spacecraft Rotations, Automatica, Vol. 38, No. 8, August 22, pp Lawton, J., Beard, R., and Hadaegh, F., Elementary Attitude Formation Maneuvers via Behavior-Based Control, Proceedings of the AIAA Guidance, Navigation and Control Conference, Denver, CO, August 2, AIAA Paper No. AIAA Bilimoria, K. D. and Wie, B., Time-Optimal Three-Axis Reorientation of a Rigid Spacecraft, AIAA Journal of Guidance, Control, and Dynamics, Vol. 16, No. 3, May-June 1993, pp

17 11 Steyn, W. H., Near-Minimum-Time Eigenaxis Rotation Maneuvers Using Reaction Wheels, AIAA Journal of Guidance, Control and Dynamics, Vol. 18, No. 5, September-October 1995, pp Wie, B., Weiss, H., and Apapostathis, A., Quaternion Feedback Regulator for Spacecraft Eigenaxis Rotations, AIAA Journal of Guidance and Control, Vol. 12, No. 3, May 1989, pp Weiss, H., Quaternion-Based Rate/Attitude Tracking System with Application to Gimbal Attitude Control, AIAA Journal of Guidance, Control, and Dynamics, Vol. 16, No. 4, July-August 1993, pp Cristi, R., Burl, J., and Russo, N., Adaptive Quaternion Feedback Regulation for Eigenaxis Rotations, Journal of Guidance, Control, and Dynamics, Vol. 17, No. 6, November-December 1994, pp Costic, B. T., Dawson, D. M., de Queiroz, M. S., and Kapila, V., Quaternion-Based Adaptive Attitude Tracking Controller Without Velocity Measurements, AIAA Journal of Guidance, Control and Dynamics, Vol. 24, No. 6, November December 21, pp Zhao, R. and Posbergh, T. A., Stabilization of a Rotating Rigid Body by the Energy Momentum Method, Proceedings of the IEEE Conference on Decision and Control, Tucson, Arizona, December 1992, pp Wertz, J. R., editor, Spacecraft Attitude Determination and Control, Kluwer Academic Publishers, Shuster, M. D., A Survey of Attitude Representations, The Journal of the Astronautical Sciences, Vol. 41, No. 4, October December 1993, pp Lawton, J., Multiple Spacecraft Elementary Formation Maneuvers, Ph.D. thesis, Brigham Young University, Provo, UT 8462, 2. 17

18 q u d e Rotation Axis Controller τ Spacecraft ω q Figure 1: Proposed block diagram of the closed-loop system. 18

19 e 1 + ω States e 1 + ω. State Feedback Output feedback Eigenaxis with J PD without J τ Control Torque τ..1 1 e time d 5 x 1 3Error perpendicular to u : e time norm diff time e/ e u d time Figure 2: Rest-to-rest maneuver about the eigenaxis. 19

20 e 1 + ω States e 1 + ω. State Feedback Output feedback Eigenaxis with J PD without J τ Control Torque τ time time.2 Error perpendicular to u d : e 1..7 e/ e u d e 1.1 norm diff time time Figure 3: Rest-to-rest maneuver where the desired rotation axis is 3 degrees off of the eigenaxis at time zero. 2