In this paper we present a passivity based control for spacecraft formation flying. We derive both attitude and position control. The controls do not require velocity information. They only require information about the position and attitude of the given robot and two of its neighbors. Furthermore, our approach has the advantage that we provide convergence results and establish bounds on the formation error

Beard, Randal W.

Hadaegh, Fred Y.

Lawton, Jonathan

English

DigitalCommons@USU

I I I I I I I I I I I I I I I I I I I SPACECRAFT FORMATION FLYING VIA DYNAMIC COMPENSATION Jonathan Lawton Randal W. Beard Electrical and Computer Engineering Department Brigham Young University Fred Y. Hadaegh Jet Propulsion Laboratory California Institute of Technology Abstract In this paper we present a passivity based control for spacecraft formation flying. We derive both attitude and position control. The coutrols do not require velocity information. They only require information about the position and attitude of the given robot and two of its neighbors. FUrthermore, our approach has the advantage that we provide convergence results and establish bounds on the formation error. Nomenclature The inertial position of the ithspacecraft. formation with respect to the inertial frame. ifF The desired position of the ith ·spacecraft in the formation frame. ri The distance to go for each spacecraft i.e. ri = ri - rp - rdiF ii; The formation error between the ith and jt11spacecraft, i.e. ri- r; is The Orientation of the it11spacecraft with respect to the F frame. q.;; The relative orientation of the ith spacecraft with respect to the jt11spacecraft. The indices are defined mod N i.e. rN+l = r1 and ro =rN. 1 Introduction The inertial velocity of the it11spacecraft. The multiple agent formation problem has ap-The unit quaternion representing plication to areas in both robotics and aerospace. attitude of the itA spacecraft with In [Decou, 1991a] and [Decou, 1991 b) a free-flying respect to the inertial frame. multiple spacecraft interferometer is proposed. The vector part of q;. For this application a group of spacecraft fiy in a The scaJ.ar part of q;. rigid formation within very fine tolerances. In avi-The angular velocity of the ith ation [Blake and Multhopp, 1998) ftying multiple spacecraft with respect aircraft in formation can reduce the induced drag to the inertial frame. on each airplane. This can result in less power The control force applied to the ith expended and thus fuel savings. spacecraft. In addition there are many applications of for-The control torque applied to the ith mation control to robotics. Many robots working spacecraft. together can push a box [Kube and Zhang, 1996), The mass of the· it" spacecraft. move large awkward objects [Dickson et al., 1996), The inertia of the it11spacecraft. move a large number of small objects ef-The inertial final position of the fectively [Vidal et ·al., 1996), or sweep out a formation. given area to obtain information about the The unit quatemion representing terrain [Rao et al., 1996) or the environment desired attitude of the [Kurabayashi et al., 1996, Anderson et al., 1996]. 1 Specifically we will address the problem of maintaining multiple spacecraft in rigid forma-tion while maintaining alignment among the spacecraft to within a fine tolerance. Many of the previous approaches to formation con-trol implement leader-follower hierarchal control [Wang, 1991, Wang and Hadaegh, 1996]. In these approaches some agents are designated as lead-ers. The leaders establish the motion for the for-mation and the remaining agents (followers) track the leader's motion to a fixed separation distance. The advantage of this control is that it implements feedback control on each agent and there exists convergence results for these control algorithms [Wang, 1991, Wang and Radaegb, 1996]. The dif-ficulty with implementing these approaches is the need to have position, velocity and acceleration information of one or more of the the neighbor-ing agents. Furthermore the leader-follower ap-proach ·provides for many points of failure. H a given agent is impaired or fails completely there exists no feedback between the impaired agent and its leaders in the hierarchal chain. The impaired agent will simply be left behind. One approach that adds feedback between all of the agents in the formation is the virtual structure approach [Tan and Lewis, 1997]. In this approach each agent checks its position within the given for-mation, then it runs an open loop control to cor-rect for any formation errors. This process is then iteratively repeated. Other alternatives to hierarchal control are biological-model based control [Sekiyama and Fukuda, 1996, Mitsumoto et al., 1996], behavior based con-trol [Balch and Arkin, 1998) template based control[Beard and Hadaegh, 1998]. These controls all require velocity and acceleration information about neighboring agents or they do not guarantee that strict bounds on the relative distance between agents are meet. The approach that. we present here is to implement a passivity based control [Lizarralde and Wen, 1996, Tsiotras, 1998, Ortega et al., 1995] on each spacecraft. The control only requires information about its own position and attitude and the position and atti-tude of two neighboring spacecraft. No velocity or acceleration information is required. The paper is organi'Pd as follows: • Section 2 reviews the equations of motion of the problem and introduces the notion of for-mation control. 2 • Section 3 presents the position and attitude controls for each spacecraft. • Section 4 gives some simulations. • Section 5 presents the conclusions of the pa-per. 2 Problem Statement Consider a fleet of N spacecraft. The motion of each spacecraft is governed by a translational and a rotational set of equations. The translational equations are given by double integratordynamics ~i;=f[. (1) For the rotational equations we will use a unit quatemion attitude representation. A useful re-view of properties of the unit quatemion are given in the Appendix. In terms of the unit quaternion the rotational equations are given by Jiwi = -wf J;w; + -r1 Qs = ~E(Cii)w;, where the cross product operator is de&ned by (2) and E( q) is given in the appendix. The formation structure is defined by • r F, the desired inertial position of the forma-tion center. • QF, the desired inertial attitude of each space-craft. • rfF, the desired position of each spacecraft with respect to the formation center. The formation control problem is to derive con-trols 'Tf and fi such that 1. lfi;ll-+ 0 2. p(Qi,'IF)-+ 0 while 3. E~tllii,i+t(t)ll < Et 4. E~l p(q;(t),qi+l(t)) < E2, I I I I I •• I I 1: I I I I I I I I I I I I I I I ... I_ I-I I I I I I I I I I I where the metric p(qh'IJ) is de6ned in the Ap. pendix. Furthermore we wish to implement this con-trol in such a way that the coordination between spacecraft is closed loop. In other words if the ithspacecraft is moving too far away from the other spacecraft it will slow down. Second we wish to implement this control with as little communica-tion as possible. Specifically, we will only require that each spacecraft have infOrmation about its own position and that of two other spacecraft. To implement this control we will use a passive con-trol law . 3 The Main Result Do to difficulty in communicating velocity in-. formation-from· one spacecraft to another, we will iniplemem a passive control. Theorem 3.1 and Theorem 3.2 will present a control that is moti-vated by that giveit in [Lizarralde and Wen, 1996, TSiOtras, 1998] First we present results for the translational problem.. Theorem.S.l Given that _ 1. ·A Hunmtz, fl. B full rank, 3. P> 0 such that ATP+PA= -Q, 4- K>O 5. c> 0 6. a>O then the control r; = -aKri - Kri,i-1 - Kii,i+1 - C'lli (3) r;(o) =rio (4) vi(O) = 0 (5) Zi = Azi + Bri (6) 'Yi = BTPA.xi + BTPBii (7) xi(O) = -A-1 Bri(O), N N E rf.i+1 (t)Kri,i+1 (t) · ~ E ( ri,i+l (O)Kr;,i+l (0) i=1 ' i=l + aii(O)TKri(O)) (8) 3 Proof: Consider the function N V 1 "'< -TK- -T K = 2 L...., ari ri + ri,i+1 ri,i+1 i=l + v'[~vi + c:i:f ~zi)· (9) Dift"erentiating we find that N V = E vf (aKri + Kii,i+1 + Ki;,i-1 + li i=1 cBTp• ) Ci ·TQ· + z· --z. z· ' 2 ' s N = E vf (Kii + Kii,i+l + Kii,i-1 + !i i=1 Consider the set n = {ii,i+t,i;, Vi,ZiiV = 0}. Let fi be the largest invariant set in n. We have that Zi = 0. It follows that Vt ~ 0, Xi = Xio a constant. From equation (6) and the fact that · B is full rank we know that Vt ~ 0 ri = riD a constant. Therefore we have that Yi = 0. Since 1/i = BT Pzi, tli = 0. Since vi = 0 it follows that in fi, ff = 0. On this set we get from (3) that ri,i-1 + ri,i+l + ari = o. (tO) Since riJ = ii - i; then we may write equation (10) as a matrix equation +a -1 0 0 -1 -1 2+a -1 0 0 ®l·l!) =0. 0 -1 2+o 0 0 -1 0 0 -1 2+o (11} The matrix in equation (11) is nonsingular Va > 0. Therefore ii = 0 in ll. Hence by LaSalle's Invariant set theorem ri = 0. The bound (8) fol-lows from the fact that V :5 0 and Vi (O) = 0 and Zi(O) = 0. • Note from equation (9), that by choosing a small places low emphasis on ii or little impor-tance on getting to the final destination relative to maintaining formation. By choosing a small a most of the control effort goes toward maintaining formation. Theorem 3.2 presents an analogous control law for maintaining formation alignment. ··--... , ___ ._ Theorem 3.2 Given that 1. A HunDitz, e. B full nmk, 9. P > 0 such that ATP+ PA = -Q, 4- k>O 5. c>O 6. a>O then the control -r~ = -akVec(q;qj;.)- kVec('li'li+t) -kVec(Qiqi'_1)- CJJi (12) w;(O) = 0 'li(O) =Clio z; = Az; + BV ec( CliqF) (13) y; = BTP.Az; + BTPBVec(Qiqp), (14) asymptotically stoblizes qsq} and N ... N Ep(q;(t),fi+t(t)) s Ep(q;(O),q;+t(O)) i=l i=t N +a Ep(q;(O),qF(O)) (15) i=l Proof: Given the Lyapunov function candi-date the proof is analogous to the proof of Theorem 3.1. • The next section applies this control to the three spacecraft formation problem. We provide simu-lations for the translational control problem. 4 Simulations Consider the three spacecraft problem with • r'fF = r1 (0) = [5, 0, O]T e ~F = r2(0) = (0,5,Q]T • rfF = rs (0) = [0, 0, 5]T • r'J;. = [10, 10, 10]T 4 ~~r;: . : : . I 0 10 20 30 -40 50 Ill 70 80 90 100 FUrthermore, suppose that each spacecraft is ini-tially in perfect formation i.e. it2 = ~s = 0. Given the control parameters K = B = P = -A = Is, c = 1 and a = .1. Figure 1 plots the quantities it, it2, and its versus time. Observe that the quatity it slowly decays to zero. Thus each spacecraft slowly converges to its final posi-tion. Furthermore, since the quanties it and r13 remain small, we know that the formation is begin keep during the entire maneuver. Now consider the same problem, but with a small random position error added on to each spacecraft. Thus it2(0) :F r23(0) :F 0. Given the control parameters -A= B =Is, K =lOis, P = 5I3, c = 1 and a = .01 Figure 2 plots r1, it2 and r23 versus time for the case where we start with some initial position error. Again note that the error it slowly decays to zero. This indicates that spacecraft one is converging to it proper final formation position. We also see that the formation keeping is improving with time since i12, its -+ 0. For both of these examples similar plots for the second and third spacecraft are obtained. 5 Conclusion The main result of this paper was the devel-opment of a passivity based formation control. This control rigidly moves a group of N spacecraft from one position to another while maintaining spacecraft formation and alignment. This control guarantees both formation keeping and spacecraft I I I I I I I I I ,, I I I I I I I I I I I I I I I I I I I I I I I • , I I I I ~b 0 10 2D 3) <40 ~~it:: 0 ;g; : : : : : I 0 10 2D 3D 40 Figure 2: Plot of llr12ll, llr23ll and flit II alignment to within a given bound. The advantage of using a passivity based control is that the con-trol does not depend on velocity information. In formation problems it is difficult enough to com-municate positions between agents. It is nearly impossible to communicate accurate velocity~ formation. Thus this approach has the advantage of being more easily implemented than previous controls. The set of unit quaternions is de6ned by q E {p E FIPT p = 1 }. The quaternion is generally partioned into a vector and a scaler component i.e. q= [~' where q E B 3 and ij E B. The conjugate of q is We define the following quaternion matrix o~ erators (16) The composition of two unit quaternions is closed 5 under the operation '11'12 = Ht(CI1)'12 = H2('12)Cb· (17) For this paper we will define the following metric on the set of unit quaten;lions p(Ql,Q2) = 2(1- qf q,). (18) Since Cb, and Q2 are both unit vectors, p(Q1,Q2) > 0 for Q1 :# '12· For Cb = Q2 p(Q1,Q2) = 0. Acknowledgement The researcl:l was partially supported by the Jet Propulsion Laboratory, California Institute of Technology under contract #96-1245, and by the NASA Rocky Mountain Space Grant Consortium. References (Anderson et al., 1996] Anderson, M., Mckay, M., and Richardson, B. (1996). Multirobot atJto.. mated indor floor cbaracterization team. 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Spacecraft Formation Flying Via Dynamic Compensation

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Spacecraft Formation Flying Via Dynamic Compensation

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