Novel Coordinate Transformation and Robust Cooperative Formation Control for Swarms of Spacecraft

Transcription

1 4th International Conerence on Spacecrat Formation Flying Missions an echnologies May St. Hubert Quebec Canaa Novel Coorinate ransormation an Robust Cooperative Formation Control or Swarms o Spacecrat Insu Chang 1 Soon-Jo Chung 1 Fre Y. Haaegh 2 1 University o Illinois at Urbana-Champaign Urbana USA; 2 Jet Propulsion Laboratory Caliornia Institute o echnology Pasaena USA Abstract his paper presents a new coorinate transormation metho or controlling a large number o spacecrat moving in elliptical orbits. A new coorinate transormation metho or phase synchronization o spacecrat in relative elliptical orbits is introuce to eectively maintain esire ormation patterns. he propose controller which employs both the aaptive graph Laplacian matrix an the istance-base connectivity rule synchronizes the relative motions o spacecrat with a guarantee property o robustness. A complex time-varying network topology constructe by the propose controller relaxes the stanar requirement o consensus stability even permitting stabilization on an arbitrary unbalance graph. A challenging example o reconiguring swarms o spacecrat shows the reliability o the coorinate transormation metho an the eectiveness o the propose phase synchronization controller. 1. Introuction Spacecrat ormation lying (SFF) has been extensively stuie ue to its potential applications in uture space science missions such as stellar intererometry. Despite the beneits o SFF there exist unique challenges in guiance navigation an control o SFF [1]. his paper ocuses on a more challenging application o swarms o ormation lying spacecrat [2]. he sheer number o spacecrat (1s) involve in spacecrat swarms signiicantly complicates the ormation control problem. he obective o this paper is to introuce a new coorinate transormation metho as well as a novel ormation controller or swarms o spacecrat. here have been a variety o stuies on SFF. Surveys o SFF guiance an control are given by [3 4]. Various architectures or SFF were introuce in [5]. A leaer-ollower system has been the most popular metho [6] an several types o ecentralize coorinate-base control have also been extensively stuie [7 8]. Various linear an nonlinear control methos have been propose (e.g. integrator backstepping control [9] passivity-base control [1] output eeback [11] aaptive control [12]). Base on [13] a combine controller or attitue an orbital motions o SFF was introuce in [8] which utilizes phase synchronization in circular motions. Exact nonlinear ynamics an cooperative control or SFF base on the aaptive Graph Laplacians an istance-base connectivity were introuce in [14] uner the assumption that each spacecrat knows its esire traectory explicitly.

2 We can summarize the main contributions o the paper as ollows. First the propose coorinate transormation metho an the phase angle shit metho acilitate phase angle shits in any elliptical orbits in three-imensional (3D) space so that the motions o spacecrat in elliptical orbits can be escribe by the combination o circular an sinusoial motions in a new coorinate system. Secon the propose phase synchronization controller which is proven to be exponentially stabilizing the spacecrat traectories actively attempts to in time-varying tracking an iusive coupling gains by means o the aaptive graph Laplacians an the istance-base connectivity metho [14]. his way a large number o spacecrat in a swarm o spacecrat can be eectively synchronize. he phase synchronization controller eliminates the nee or having a balance graph (i.e. the number o inputs or coupling is the same as that o outputs) or synchronization [15]. We investigate the eectiveness o the propose methos by using a swarm o spacecrat rotating an reconiguring in multiple concentric relative orbits. he organization o this paper is as ollows: In Section 3 the problem statement is presente. In Section 4 the new coorinate transormation metho an the phase angle shit metho are introuce. he propose phase synchronization control law is given in Section 5. In Section 6 results o simulation are iscusse. Finally concluing remarks are given in Section Problem Statement Suppose that there is a swarm o spacecrat consisting o spacecrat whose iniviual ynamic moel ( th agent ( )) is given by the Euler-Lagrangian (EL) ormulation M ( q ) q C ( q q ) q G ( q ) τ D ( q q ) (1) where an enote the states the inertia matrix the Coriolis/centriugal orces the gravitational orce the control input an the non-conservative orces respectively. Note that is chosen such that ( ) is skew-symmetric which plays a critical role in stability proos in this paper. Note that the exact nonlinear ynamic moels or chie an eputy spacecrat which inclue J perturbation an atmospheric rag as isturbances can be oun in [17]. 2 We want to esign a ormation control an phase synchronization controller such that the spacecrat in a swarm can maintain their positions with boune errors with respect to the esire traectories in multiple concentric ellipses. In orer to construct the esire traectories it is assume that there is a pre-esignate orbit leaer spacecrat which can be either a physical spacecrat or a virtual one. he orbit leaer spacecrat is use or more systematic hanling o the network (e.g. ormation keeping reconiguration etc.). Figure 1 shows two rames (the Earth Centere Inertial (ECI) an

3 the Local Vertical an Local Horizontal (LVLH) rames) an a swarm o spacecrat moving in multiple concentric ellipses. Fig. 1 Coorinate systems an spacecrat swarm in multiple concentric orbits 4. Perioic Relative Orbits (PROs) In this paper we consier perioic motions or the esire traectory. he perioic motions speciically in the LVLH rame are calle PROs. he PROs can be obtaine rom the solutions o the HCW equation [16]: x ( t) x / ns (3x 2 y ) c 4x 2 y / n nt nt y ( t) 2{ x / nc (3x 2 y / n) s } 3 nt / 2(4x 2 y / n) ( y 2 x / n) nt nt z ( t) z c z / ns nt nt (3) where q [ ] x y z enotes the relative motion o the esire traectory an x y an z enote the initial values o x y z that can be chosen arbitrarily. he timevarying variable n / a is the mean motion o the chie spacecrat. 3 In this paper we assume that PROs are concentric with ( ) the center o the LVLH rame. hereore the conitions or PROs without a rit in the LVLH rame are y 2nx an x ny /2. By substituting the two contions into Eq. (3) the perioic solutions without a rit in the LVLH rame can be expresse in a compact orm as x ( t) a s y ( t) 2 a c z ( t) z s (4) e e e e max z where ae ( x / n) (3x 2 y / n) e e nt e tan ((3nx 2 y) / x ) 2 2 zmax z ( z / n) z z an 1 z tan ( nz / z) respectively. Note that the solutions escribe elliptic orbits with the relative semimaor axis ( 2a e ) which is twice longer than the relative semiminor axis ( a e ). he solutions in Eq. (5) will be use as esire traectories o the orbit leaer spacecrat.

4 5. New Coorinate ransormation an Phase Angle Shit in Elliptical Orbits We iscuss a metho to erive esire traectories or ollower (eputy) spacecrat rom a single pre-etermine esire traectory (i.e. an orbit leaer spacecrat) in a particular elliptical orbit. We assume that the esire traectory can be obtaine by using communications among neighbors in the network. Fig. 2 Deinition o an the Relationship between the intermeiate an the new rames 5.1 Coorinate ransormation in Elliptical Orbits Although we have the esire traectory in Eq. (4) the angle rotation using rotation matrices is iicult in ellipses. hereore by transorming the current rame to another the elliptic orbit can be expresse by a combination o simple orms. In orer to in the transormation matrix we consier two consecutive coorinate transormations o the rame. he irst transormation is to rotate the original rame to an intermeiate rame such that the motion is expresse in a 2D motion i.e. the orbit is locate in the intermeiate x - y plane (see Fig. 2). In this case we shoul in positions or the maximum an minimum istances rom the origin. he istance between the origin an the agent can be expresse rom Eq. (4) as l a s a c z s (5) ( ) e ( ) 4 e e ( ) max ( ) e where nt () with the time-varying angular rate nt (). Dierentiating Eq. (5) with respect to yiels the extrema o the istance. l / s 9a z 6 a z c / (2 l) (2 2 ) e max e max (2( e )).5tan (3 a s z s ) / ( 3 a c z c ) e (2 e ) max (2 ) (2 ) max (2 ) z e e (6) Hence l( ) has extrema or an 3 / 2. hat is obtaining the positions o lmin l( ) an lmax l(3 / 2 ) which are the x ˆ N an yˆ N axes ater the normalization yiels the irst rotational matrix R as: n

5 max ( ) 2 max ( ) max max ( ) 1 l aes l aec e l z s e R n lmin aec( ) 2l e min aes( ) lmin zmax c ( ) e (7) l 2 minlmax 2a ezmax c( ) aezmax s( ) 2a e e z e N he secon rotation is relate to ining an angle ( ) with respect to the x axis such x that the motion in the new x - y plane (i.e. x ' - y ' plane) becomes circular. One can notice rom Fig. 2 that l min becomes a raius o the circle in the new rame. hereore can be expresse as Figure 2 illustrates the einition o an the geometrical relationship between the two rames. he secon transormation matrix is eine as (8) [ ] (9) Hence the coorinate transormation rom the original rame to the new rame can be oun by combining Eqs. (7) an (9) as where q ' [ x' y' z ' ] an q [ x y z ]. 5.2 Phase Angle Shit Metho in Elliptical Orbits he propose phase angle shit metho enables spacecrat to shit their positions in elliptical orbits in 3D space simultaneously with only one phase shit angle. From the einition o an ellipse in Eq. (4) the irection o is opposite to that o i the ollower spacecrat are assume to ollow their orbit leaer spacecrat. With this inormation we consier motions in the - plane in avance. he angle rotation or the - plane can be expresse as (1) x c( 1) s( 1) lmin c( ) x xy (( 1) ) (11) y s( 1) c( 1) lmin s( ) y For the phase angle shit in the z axis we eine an auxiliary variable Z as 2 2 Z : z max c( where ) z max lmax lmin. hen the angle rotation or z an Z can be eine as z c( 1) s( 1) zmax s( ) z z (( 1) ) (12) Z s( 1) c( 1) zmax c( ) Z

6 Hence combining Eqs. (11) an (12) yiels the propose phase angle shit or the th agent as xy (( 1) ) 2 q : q q (13) 2 z (( 1) ) Z zmax c( ( 1) ) : 1 where 2 is the 2 2 zero matrix an 1 (( 1) ) enotes the phase angle shit matrix with ( 1) rom the leaer agent in the new rame. he time erivative o the esire traectory q '' is use requently in this paper: (14) where q '' [ ' ' ' ' ] x y z Z with Z ' z ' z ' / Z '. Notice that the auxiliary variable Z ' in Eq. (13) is purely use to apply the phase angle shit to. hus the auxiliary variable can be remove rom the state vector ater the rotation. 6. Robust Formation Control or Phase Synchronization in Swarms o Spacecrat In this section we introuce the aaptive phase synchronization controller. We begin with a transormation o the composite state variables with R an Moiication o States an Composite Variables In Eqs. (1) an (13) we transorme the esire traectory in the original rame ( ) to those in the new rame ( an ). For the transormation o in Eq. (1) is use i.e. q R q (15) ' From Section 5.1 the value o can be obtaine which plays an important role when ining the auxiliary Z in q'' [ x' y' z ' Z ' ] :[ q ' Z ' ] which can be oun as ' Z ' z ' c R q cot( ) (16) max ( ) 3 q an R 3 is the thir row vector o R. where z ' max R 3 / s( ) From the einitions o q ' q composite variables ( s ' is written as ollows: '' q ' an q '' in Eqs. (1) (13) (15) (16) the moiie s '' ) are obtaine as well. he moiie composite variable s'' q'' q'' Λ''( q'' q '' ) (17) 1 1

7 where is a positive iagonal matrix q'' [( Rq ) Z' ] an q'' [( Rq '' ) Z ' ]. he phase angle shit matrix is eine in Eq. (13). he 1 composite variable s ' is irectly obtaine rom the irst three elements in s ''. 6.2 Phase Synchronization Controller Given the ynamic moels in Eq. (1) an the esire traectories in Eq. (4) we shoul transorm the original rame to the new rame by using the coorinate transormation in Eq. (1) such that we can use the phase angle shit metho in Eq. (13). By letmultiplying Eq. (1) by R an q R Rq R q ' the ynamic moels escribe in 1 the new rame are written as R.. ( ) 3 1 ( ) 3 1 ( ) ( ) ( t). 1 3 ( ( ) ) R M q R 33 R C q q R 1 3 ( ( ) ) G 33 Z D R G R q R D R q R q q q Z R τ R M q R Z R C q q R (18) where ( RM ( ) q R ) an 33 ( ( ) R C q q R ) enote the (33) elements o 33 RM ( q ) R an R C ( q q ) R an G an Z ' D are eine by using the similar proceure in Eq. (16) as Z ' G R G cot( ) D R D cot( ) (19) Z ' 3 Z ' 3 In orer to construct the propose phase synchronization controller the active parameter aaptation is introuce or the ynamic moel in Eq. (1) or the purpose o tuning the tracking an iusive coupling gains. he propose ormation control an phase synchronization controller or Eq. (1) incluing the nominal communication with the l th an m th agents can be written as:.. ( ) ( ) 3 1 ( ) R τ t R M q R R C q q 3 1 ( ) ( ) R R D R q R q R G R q q Z 1 3 ( ( ) ) r q 33 ˆ 13 r R M q R D G Z Z k ( ) ( 1s k 2 lsl msm W s ) c e (2).... Y ( ) ( ) q q q r q r b R G R q k 1 k2( l l m m) ( ) YZ e s G s s W s c b Z where k1 2k2 or p 3 [13] ρ [ 1 2 ] p s'' { 1s '' 1 2s '' 2 ps '' p} i (( i) ) an e'' q'' q ''. A constant is a esign parameter an q'' r [ q' r Z ' r ] where q' : [ ' ' ' ] r x r y r z r R q with r q r q Λ( q q ) an Z' R q cot( ). r 3 r Detaile inormation or the gain aaptation ( ) an parameter estimation ( ) can be oun in [17]. Furthermore the nonlinear stability proos are erive in [17] which also shows that the synchronization error is smaller than the tracking error.

8 7. Numerical Valiations In this section we evaluate the eectiveness o the propose aaptive phase synchronization controller by simulating a reconiguration o a swarm o spacecrat (S/C). he center o the LVLH rame is the position o the chie S/C escribe by the Proposition 1 in [14]. We can escribe the motions o all S/C in the swarm by using Proposition 2 in [14]. In the simulation we want to control the motions o 18 S/C in two orbits which are concentric ellipses in the same orbital plane. he initial positions o the S/C are ranomly chosen with a uniorm istribution with a range o.5km x y z.5km No initial velocity is assigne to all o the S/C. It is assume that 6 S/C are place in the inner orbit an the rest o them are in the outer orbit. For simulations the physical parameters are chosen as ollows: m 1 kg 2 A 1m CD 2.. he altitue o the chie S/C is 4km an the initial classical orbital elements are [ a e i ] [ R E 4 45eg 3eg eg 1eg] where R enotes E the raius o the Earth. he unmoele isturbance Δ ist incluing system's 5 2 uncertainties is set as Δ ist / m 1 km/s. he initial conitions o the esire traectory or the orbit leaer S/C are eine as: 5 x 1km y 1km z.5km z 1 km/s. he values o x an y can be oun by using the constraints in Eq. (4). For the controller in Eq.(2) the esign parameters are set as ollows: k 1 3 k2 1 1 Λ iag(.1.1.1). For the gain aaptation law [17] Σ iag(11 1) ck max 1 1. Initial values o the aaptive c iusive gains c are set to 's. For the communication no more than 5 connections with neighbors are allowe. he parameter estimation law is not use or simplicity an clarity o the perormance o the propose phase synchronization controller. Figure 3 shows the traectory motions or the S/C in 3D space uring 9 s controlle by the propose phase synchronization controller. Note that ater reaching their orbits S/C ollow their esire traectories without a rit. he irst igure in Fig. 4 shows the traectories o the 18 S/C in the swarm. Note that only x (i.e. the irst axis in the new rame ( x - y - z )) was chosen because x y an z have very similar results. In the igure the traectories are converging to each other regarless o their initial positions beore approaching the esire traectory iniviually. he contraction rate is obtaine as 2.2. he estimate value o the maximum state error is q.5m or iniviual S/C. he secon igure in Fig. 4 shows the gains variations or the 7th S/C (the irst one in the outer orbit) base on the gain aaptation law. he gains o not have values uring 15 sec ue to the zero initial conitions. Note that the 7th S/C has gains only or its neighbors in the same orbit (8th an 18th S/C) an those in the inner orbit (1st an 2n S/C).

9 In orer to evaluate the perormance we compare the propose synchronization controller an the nominal gain-base controller with properly tune gains. For the simulations it is assume that all S/C are initially locate at the origin. he irst igure in Fig. 5 shows the convergent time with respect to the size o the orbits (the number enotes the ratio to the orbit eine previously in this section.) with the same amount o uel or reconigurations. As the orbit size gets bigger the propose synchronization controller makes the S/C converge aster which is ue to the gain aaptation law. On the other han base on the same convergent time the propose synchronization controller uses less uel consumption ( V ) or the reconiguration than the nominal gainbase controller (the secon igure in Fig. 5). hrough the simulation results we showe that the propose phase synchronization controller more eiciently synchronize the motions o the spacecrat uring the reconigurations o the swarm. Fig. 3 raectories o the 18 S/C or 9s 8. Conclusions Fig. 4 raectories x ater transorming them to the position o x (upper igure) an aapte iusive gains or the 7th S/C Fig. 5 Comparison o the perormance: convergent time (upper igure) an uel consumption (lower igure) We presente a novel coorinate transormation metho an a new phase synchronization control strategy or swarms o spacecrat constructe by the networke Euler-Lagrangian (EL) systems moving in elliptical orbits. he propose coorinate transormation metho an phase angle shit metho acilitate the phase angle rotation in ellipses thereby playing a crucial role in the phase synchronization control law. he propose phase synchronization controller which guarantees smaller error boun robustly synchronize the motions o spacecrat with relaxe sensing topology or synchronization stability in the swarm. Acknowlegements he research was carrie out in part at the Jet Propulsion Laboratory Caliornia Institute o echnology uner a contract with the National Aeronautics an Space Aministration. 211 Caliornia Institute o echnology. Government sponsorship acknowlege. he irst author was supporte by the University o Illinois at Urbana- Champaign. he authors grateully acknowlege Dr. Lars Blackmore an Dr. Behcet Acikmese at JPL/NASA an Daniel Morgan at UIUC or their constructive iscussion.

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