Forward-Integration Riccati-Based Feedback Control for Spacecraft Rendezvous Maneuvers on Elliptic Orbits

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1 5st IEEE Cnference n Decisin and Cntrl December -3,. Maui, Hawaii, USA Frward-Integratin Riccati-Based Feedback Cntrl fr Spacecraft Rendezvus Maneuvers n Elliptic Avishai Weiss, Ilya Klmanvsky, Mrgan Baldwin, R. Sctt Erwin, and Dennis S. Bernstein Abstract We apply the frward-integrating Riccati-based feedback cntrller, which has been develped in ur previus wrk fr stabilizatin f time-varying systems, t a maneuvering spacecraft in an elliptic rbit arund the Earth. We simulate rendezvus maneuvers n Mlniya and Tundra rbits. We demnstrate that the cntrller perfrms well under thrust cnstraints, in the case where the spacecraft can thrust in nly the rbital tangential directin, in the case where the thrusters may perate nly intermittently due t faults r pwer availability, with thrust directin errrs, and, finally, in an utput feedback cnfiguratin where nly relative psitin measurements are available. I. INTRODUCTION Traditinally, relative mtin maneuvers are perfrmed using pen-lp planning techniques []. Ad hc maneuver crrectins may be emplyed t cmpensate fr errrs inherent in pen-lp cntrl. Eamples f relative mtin maneuvers include rendezvus, dcking, debris avidance, and frmatin flying. In particular, literature n spacecraft rendezvus cntrl prblems is abundant, see e.g., [], [] and references therein. Recently, mre interest has been emerging in clsed-lp maneuvering, especially fr missins that invlve frmatin flying r autmated rendezvus, dcking, and primity peratins. The XSS- [4] spacecraft has been develped by the Air Frce Research Labratry as a platfrm fr demnstrating relative mtin capabilities. In this paper, we address a class f relative mtin cntrl prblems fr spacecraft n elliptic rbits. Elliptic rbits are used t deply a variety f spacecraft fr cmmunicatins and planet/star bservatin purpses. Fr instance, Mlniya and Tundra rbits hst cmmunicatin satellites launched frm predminately nrthern latitudes [8]. The linearized equatins f mtin fr a spacecraft n an elliptic rbit are time-varying, thus impeding their treatment using feedback cntrl techniques that assume time-invariant plant mdels. As such, we emply a recently develped frward-integrating Riccati (FIR) cntrller fr time-varying systems [5] in rder t stabilize the spacecraft t a desired rbital psitin. Unlike the backwards-in-time Riccati cntrller arising in ptimal cntrl thery, the FIR cntrller can achieve stabilizatin f a linear time-varying system This wrk was supprted by ASEE thrugh the Air Frce Summer Faculty Fellw Prgram (SFFP). A. Weiss, I. Klmanvsky, and D.S. Bernstein are with the Department f Aerspace Engineering, University f Michigan, Ann Arbr, MI 489 USA. M. Baldwin and R.S. Erwin are with the Space Vehicles Directrate, Air Frce Research Labratry, Albuquerque, NM. withut requiring future knwledge f time-varying mdel parameters. Our cnclusins, based n etensive simulatins n a nnlinear mdel that includes perturbatin frces, are that the FIR cntrller stabilizes spacecraft relative mtin dynamics n elliptic rbits, and is rbust t many errr surces, including severe thrust magnitude and directin deviatins, and even intermittent thrust availability. The paper is rganized as fllws. In sectin II, we present a linear time-varying mdel fr spacecraft relative mtin n elliptic rbits. Sectin III describes the FIR cntrller. In sectin IV we give simulatin results that highlight the effectiveness f the FIR cntrller fr spacecraft relative mtin cntrl. Finally, cncluding remarks, including future wrk, are given in sectin V. II. SPACECRAFT MODEL In traditinal relative mtin prblems, an appraching spacecraft is maneuvered clse t a target spacecraft in a nminal rbit. The target spacecraft is assumed t be at the rigin f Hill s frame [3]. The equatins f mtin fr an appraching spacecraft arund a target spacecraft depend nnlinearly n the rbital radius. Fr elliptic rbits f arbitrary eccentricity, the linearizatin f these equatins is described by the linear time-varying equatins [6], F m c = ẍ R F y m c = ÿ + µ R 3(t) + h ẏ, (t) µ h R 3(t) R 4(t) h R 4(t) + (v (t) R (t)) h R 4 (t) y (v (t) R (t)) h R 4 (t) h + R ẋ, (t) F z = z + µ m c R 3(t) z, () where, y and z are (relative) crdinates f the spacecraft in Hill s frame, F,F y,f z are cmpnents f the eternal frce vectr (ecluding gravity) acting n the spacecraft, h is the specific relative angular mmentum, R (t) is the nminal time-varying rbital radius, and v (t) is the nminal time-varying rbital velcity. Equatin () assumes that the target spacecraft mtin is in an ideal Keplerian rbit; if its mtin is affected by perturbatins, F,F y,f z have t be mdified t accunt fr these perturbatins []. In the subsequent develpment, we assume that F,F y,f z y //$3. IEEE 75

2 are thrust frces that are generated by the FIR cntrller; these frces can be realized via n-bard thruster n-ff time allcatin and attitude cntrl system cmmands []. III. FORWARD-INTEGRATING RICCATI CONTROLLER In [5], we analyzed a Riccati-based cntrller fr stabilizing a class f linear time-varying systems. Unlike standard, backwards-integrating Riccati-based cntrllers f finite hrizn ptimal cntrl thery, the apprach f [5] integrates a Riccati equatin frward in time. As such, the cntrller des nt require advance knwledge f the system dynamics, and thus is applicable t rendezvus maneuvers n elliptic rbits, where the nminal rbital psitin and velcity are updated via grund-based measurements but are nt knwn with high precisin in advance due t unmdeled disturbances. The frward-integrating Riccati-based (FIR) cntrller assumes a linear system mdel and takes the frm ẋ(t) =A(t)(t)+B(t)u(t), () u(t) = R BT (t)p f (t)(t), (3) where P f (t) is the slutin t the frward-in-time cntrl Riccati differential equatin P f (t) =A T (t)p f (t)+p f (t)a(t) P f (t)b(t)r BT (t)p f (t)+r, (4) with the initial-time bundary cnditin P f (t ). Since (4) is integrated frward in time, advance knwledge f A(t) and B(t) is nt required. In [5] it is shwn that, if the clsed-lp dynamics matri is symmetric, then the FIR cntrller is asympttically stabilizing. We als shwed, using averaging thery, that, in the case f peridically time-varying systems and under suitable assumptins, there eists a perid belw which the dynamics f the clsed-lp system are asympttically stable. In ther wrds, clsed-lp stability is guaranteed fr systems with time-varying dynamics f sufficiently high frequency. Nte that tuning the FIR cntrller is similar t tuning LQR, namely, by adjusting the relative weighting matrices R and R. In this paper, we apply the FIR cntrller (3)-(4) t the spacecraft rendezvus prblem n elliptic rbits. We shw thrugh etensive numerical eperiments that the cntrller is stabilizing and has gd perfrmance and rbustness. IV. NUMERICAL STUDIES In the fllwing simulatins we cnsider spacecraft in bth Mlniya and Tundra rbits, highly elliptical gesynchrnus rbits with high inclinatin used by cmmunicatin satellites [8]. See Fig. fr a plt f a Mlniya rbit. The rbital elements [7] used fr the Mlniya rbit are given by (a, e, i,,!, ) = (6559 km,.7448, 63.7, 6.346, 8.646, ), and fr the Tundra rbit we use (a, e, i,,!, ) = z (km) Fig. : Mlniya Orbit. The sphere represents the Earth. (464 km,.3, 63.7, 6.346, 8.646, ). These rbital elements give an initial psitin R () and velcity v (t) fr the target spacecraft with which we wish t rendezvus. Nte that, since we let =, we start at rbital perigee. The mass f the chaser spacecraft is m c = 4 kg, and the parameters f the FIR cntrller (3)-(4) are R =.I 6, R =, and P f () = I 6. These values were tuned t give apprpriate nminal respnse time and reasnable thrust usage ver a set f typical maneuvers that the spacecraft is epected t eecute. We test the cntrller in a high fidelity nnlinear simulatin that includes bth J and air drag perturbatins based n the Harris-Priester mdel [9]. The cntrller has n knwledge f these perturbatins althugh we assume that accurate psitin and velcity infrmatin are available at the current time instant. A. Multiple Initial Cnditins We use the FIR cntrller (3)-(4) fr varius chaser spacecraft initial cnditins n the Mlniya rbit, where the bjective is t rendezvus the chaser spacecraft with the target spacecraft. Fig. a shws a 3D plt fr the initial cnditins [ () y() z()] = ±[5 5 5] km, while Fig. b shws a prjectin nt the rbital plane fr varius ther initial cnditins in bth v-bar and r-bar appraches. These simulatin results are based n a nnlinear mdel with J and air drag effects; they demnstrate that the cntrller is stabilizing even fr large deviatins in the initial cnditins. Nte that, fr pen-lp maneuver planning, the applicability f () is generally limited t 5-km maneuvers. Stabilizing maneuvers were als btained when the nminal rbital psitin is nt at the perigee; the perigee lcatin is mst challenging n an elliptic rbit due t faster mtin and larger influence f disturbances such as air drag. All subsequent simulatins are perfrmed at perigee. B. Thrust saturatin Let u ma = N be the maimum thrust magnitude. If the cntrller specifies a thrust cmmand with nrm greater 4 753

3 5 4 5 z (km) (a) 3D relative mtin plt (a) Orbital plane prjectin (b) Orbital plane prjectin. Fig. : (a) 3D relative mtin plt fr initial cnditins near perigee n a Mlniya rbit; (b) Orbital plane prjectin fr multiple initial cnditins near perigee n a Mlniya rbit (b) Thrust vectr cmpnents Fig. 3: Rendezvus maneuver perfrmed at perigee n a Mlniya rbit with N saturated thrust. (a) Orbital plane prjectin; (b) Thrust vectr cmpnents. than u ma, we let u(t) u sat (t) =u ma u(t). (5) We use the FIR cntrller (3)-(4) fr the rendezvus maneuver, where the bjective is t bring the chaser spacecraft frm the initial psitin [ () y() z()] = [5 5 5] km, with zer initial relative velcity, t rest at the desired final psitin, [ y z] = [ ]. Fig. 3a shws the maneuver prjected nt the rbital plane fr the Mlniya rbit. Fig. 3b gives the cmpnents f the thrust vectr. Nte that the thrust is saturated t N. The spacecraft is able t rendezvus with the target within.5 rbits. Fig. 4 shws the same plts fr the Tundra rbit. All subsequent simulatins are perfrmed with thrust saturatin. C. Thrust aligned with the ram directin We nw cnsider the case where the spacecraft thrusts in nly the tangential (ram) directin (±y ais in Hill s frame). This case is practically relevant if the spacecraft rientatin cannt be changed in rder t pint its thruster. We use the FIR cntrller (3)-(4) fr the rendezvus maneuver, where the bjective is t bring the chaser spacecraft (a) Orbital plane prjectin (b) Thrust vectr cmpnents Fig. 4: Rendezvus maneuver perfrmed at perigee n a Tundra rbit with N saturated thrust. (a) Orbital plane prjectin; (b) Thrust vectr cmpnents. 754

4 (a) Orbital plane prjectin (a) Orbital plane prjectin (b) Thrust vectr cmpnents Fig. 5: Rendezvus maneuver perfrmed at perigee n a Mlniya rbit with N saturated thrust aligned with the ram directin. (a) Orbital plane prjectin; (b) Thrust vectr cmpnents (b) Thrust vectr cmpnents Fig. 6: Rendezvus maneuver perfrmed at perigee n a Tundra rbit with N saturated thrust aligned with the ram directin. (a) Orbital plane prjectin; (b) Thrust vectr cmpnents. frm the initial psitin [ () y() z()] = [5 5 ] km, with zer initial relative velcity, t rest at the desired final psitin, [ y z] = [ ]. Let u ma = N. Fig. 5a shws the maneuver prjected nt the rbital plane fr the Mlniya rbit. Fig. 5b gives the cmpnents f the thrust vectr. Nte that nly the tangential thrust is used and that it is saturated t N. The spacecraft is able t rendezvus with the target within.5 rbits. Fig. 6 shws the same plts fr the Tundra rbit. Finally, we d nt cnsider radial-nly thrust since the spacecraft dynamics are uncntrllable in this case, even fr circular rbits. D. Intermittent Thrust Availability and Thrust Directin Errrs We nw highlight the rbustness f the FIR cntrller t intermittent thrust availability and thrust-directin errrs. We assume that the thrusters are able t perate fr minutes every 3 minutes in rder t simulate the situatin where ccasinal burns are used t rendezvus with the target. Additinally, we assume that the attitude cntrller is nt capable f crrectly pinting the thruster in the desired directin, s that the requested thrust vectr is rtated by arund a randm bdy-fied vectr. The chaser spacecraft is initially at [ () y() z()] = [5 5 5] km, with zer initial relative velcity, and the bjective is t bring it t rest at the desired final psitin [ y z] = [ ]. Let u ma = N. Fig. 7a shws the maneuver prjected nt the rbital plane fr the Mlniya rbit. Fig. 7b gives the cmpnents f the thrust vectr. Nte that the thrust is saturated t N and fires nly every 3 minutes. The spacecraft is able t rendezvus at the target within.5 rbits. Fig. 8 shws the same plts fr the Tundra rbit. E. Output Feedback We nw cnsider the case where the full-state measurement is nt available. In particular, we assume that we d nt have measurements f the relative velcity, that is C(t) R 3 6 = I

5 (a) Orbital plane prjectin (a) Orbital plane prjectin (b) Thrust vectr cmpnents Fig. 7: Rendezvus maneuver perfrmed at perigee n a Mlniya rbit with N saturated thrust that is available fr nly minutes every 3 minutes and is rtated by degrees abut a randm bdy vectr. (a) Orbital plane prjectin; (b) Thrust vectr cmpnents. (b) Thrust vectr cmpnents Fig. 8: Rendezvus maneuver perfrmed at perigee n a Tundra rbit with N saturated thrust that is available fr nly minutes every 3 minutes and is rtated by degrees abut a randm bdy vectr. (a) Orbital plane prjectin; (b) Thrust vectr cmpnents. We cnsider the bserver-based dynamic cmpensatr ˆ(t) =A(t)ˆ(t)+B(t)u(t) + F (t) y(t) C(t)ˆ(t), (6) u(t) = R BT (t)p f (t)ˆ(t), (7) where F (t) =Q(t)C T (t)v, and Q(t) is prduced using the estimatr Riccati equatin Q(t) =A(t)Q(t)+Q(t)A T (t) Q(t)C T (t)v C(t)Q(t)+V. (8) We let V = I 6, and V = 5 in rder t slw dwn the cnvergence f the estimated states t enhance the visibility f the simulatin. The chaser spacecraft is initially at [ () y() z()] = [5 5 5] km, with zer initial relative velcity, and the bjective is t bring it t rest at the desired final psitin [ y z] = [ ]. Let u ma = N. Fig. 9a shws the maneuver prjected nt the rbital plane fr the Mlniya rbit. Fig. 9c gives the cmpnents f the thrust vectr. The thrust is saturated t N. Fig. 9e shws the relative velcity states and estimates. The estimated states cnverge t the true state values and the spacecraft rendezvus with the target within.5 rbits. Fig. 9(b),(d),(f) shw the same plts fr the Tundra rbit. V. CONCLUSION We have shwn that the FIR cntrller is stabilizing fr rendezvus maneuvers n elliptic rbits with large initial cnditins and in the presence f thrust limitatins, including bth saturatin and intermittent thrust availability. This has been demnstrated with simulatins n a high-fidelity nnlinear mdel with J and air drag perturbatins. The FIR cntrller is advantageus fr general linear timevarying systems, des nt require future knwledge f mdel parameters, is tuned similarly t cnventinal LQR, and has sme stability guarantees presented in [5]. Future wrk includes etending the theretical stability guarantees beynd the results given in [5] and including state cnstraints []. 756

6 (a) Orbital plane prjectin (b) Orbital plane prjectin (c) Thrust vectr cmpnents (d) Thrust vectr cmpnents Relative Velcity (km/sec).5 Relative Velcity (km/sec) (e) Relative velcity cmpnents and estimated states (f) Relative velcity cmpnents and estimated states Fig. 9: Output feedback rendezvus maneuver perfrmed at perigee n Mlniya (left) and Tundra (right) rbits with -N saturated thrust. Only relative psitin data is assumed t be available. (a),(b) Orbital plane prjectin; (c),(d) Thrust vectr cmpnents; (e),(f) Relative velcity cmpnents and estimated states. REFERENCES [] W. Fehse. Autmated Rendezvus and Dcking f Spacecraft. Cambridge Aerspace Series, 3. [] K.T. Alfriend, S.R. Vadali, P. Gurfil, J.P. Hw and L.S. Breger, Spacecraft Frmatin Flying, Elsevier Astrdynamics Series,. [3] G. W. Hill, Researches in the Lunar Thery, American Jurnal f Mathematics, vl., n., 878, pp. 56. [4] XSS- micr satellite, [5] A. Weiss, I. Klmanvsky, and D. S. Bernstein, Frward-Integratin Riccati-Based Output-Feedback Cntrl f Linear Time-Varying Systems Prc. Amer. Cnf. Cntr., Mntreal, Canada, June, pp [6] H. D. Curtis, Orbital mechanics fr engineering students, Butterwrth- Heinemann, 5. [7] B. Wie. Space Vehicle Dynamics and Cntrl. AIAA Educatin Series, 8. [8] P. Frtescue, G. Swinerd and J. Stark, Spacecraft Systems Engineering, Wiley, 4th Editin,. [9] O. Mntenbruck and E. Gill, Satellite : Mdels, Methds, Applicatins, Springer,. [] A. Weiss, I. Klmanvsky, M. Baldwin, and R. S. Erwin, Mdel Predictive Cntrl f Three Dimensinal Spacecraft Relative Mtin Prc. Amer. Cnf. Cntr., Mntreal, Canada, June, pp