Dynamics and Control of Relative Motion in an Unstable Orbit*

AOO-39779

AIAA Paper 2000-4135

DYNAMICS AND CONTROL OF RELATIVE MOTION IN AN UNSTABLE ORBIT*

D.J. ScheeresWd N.X. Vinh* Departmen Aerospacf o t e Engineering The University of Michigan Ann Arbor, MI 48109-2140

Abstract - modified as a function of the applied control law. We show tha entirn a t e clas sucf so h control lawn sca We study the relative dynamics and control of two be defined and their stability analyzed as a time pe- spacecraft orbiting in close proximity in an unsta- riodic linear system e fueTh l . expenditur sucr efo h ble halo orbit. The understanding of a pair of such control laws is often quite small, but scales with the spacecraft is immediately extendable to the under- mean distance spacecraftbetweeo tw e nth . standing of a constellation of such spacecraft. This A fundamental problem with using center mani- issun a currenf s ei o t interest given plan spacer sfo - folds of an unstable orbit for relative motion is the based interferometric telescopes and the attractive- difficult f computino y g non-linea r eve(o r n linear) nes placinf o s g such system orbitn si s abou Le jth t orbits that remai centee th n ni r manifold cure -Th . libration pointe papeTh .r focusee specifith n so - t ren(numericaar t e statth f d analyticaleo an l s i ) cation of control laws to stabilize the relative mo- given in Harden and Howell (1998) and Gomez et al. tion, and analyzes the relative dynamics and fuel (1998). From these analyses we see that extremely consumptio spacecrafte th f no brieA . f discussiof no precise computation (numerica r analyticalo l - re s )i relative th e orbit determinatio alss ni o given. quired to maintain an orbit in the center manifold unstabln oa f e halo orbit. Small deviations froe mth Introduction center manifold quickly caus e trajectorth e - di o yt This paper studies the stabilization of translational verge alon unstable gth e manifol hale th of d o orbit . motion relative to an unstable periodic orbit in the In this pape propose w r approacn a e h whic- had Hill problem (whicconsideree b n h ca subse a s da t dresses the problem from a different direction, cast- e restricteoth f d 3-body problem). Result f thio s s ing the problem as a trajectory control problem from study wil relevane b dynamice l th o t tcontrod san l the start. Using this methodology we change the fo- oconstellatioa f f spacecrafno unstabln a n i t- eor computatioe th cu f so n from remainin centee th n gi r bital environment suc founs ha d nea Earth-Sue th r n manifold to continually thrusting to remove compo- libration points. It will also shed light onto the prac- nent f motioe hyperbolio s th n ni c manifoldse Th . tical contro computatiod an l singla f no e spacecraft basic controo effect w t la l thi - s derivei sap d dan trajectory over long time spans. plie orbito dt s abou e unstablth t e halo orbite W . We investigate the application of feedback con- conclude with some examples showin e controgth l trol law stabilizo st relative th e e trajectore th f o y la t wwora non-linean ki r situations commend an , t spacecraft in the sense of Lyapunov. Thus, the rela- levee on th controf lo l acceleration neede othed dan r tive trajectory may still contain oscillatory motions, practical issue implementationf so . which in this context can be interpreted as stable motion centee th n i sr manifol periodie th f do c orbit Model of Spacecraft Motion For computing the non-linear motion of the space- "Copyright ©2000 The American Institute of Aeronautic Hile s th l e equationus craf e w t motionf o s , whice har Astronauticd an s rightl IncAl . s reserved. t Assistant Professor of Aerospace Engineering; Senior simplifiea de 3-bod forth f mo y problem applicable Member AIAA; email: [email protected] motioe th spacecrafa o f t no t neae Earthth r- ac , * Professo f Aerospaco r e Engineering, Emeritus counting for the perturbation of the sun (Marchal,

. three 64)p Th .e dimensional motio s governeni d equationse byth :

x — (1)

y + (2)

= — - z (3)

where u is the mean motion of the central, attracting body abou perturbine th t g body (assuming circular motion), and /j. is the gravitational attraction of the centra lsituatior bodyou n I thachoos.e e b n w to t i e/ of Earth and u> to correspond to an orbital period of one year. The x axis lies along the line from the 1.5»*06 perturbing body to the central body and remains X(km) fixe thin di s rotating frameaxiy e s th ,point s along directioe th travecentraf e no th f o l l body aboue th t Figure 1: Halo orbit trajectory projected into the x-y perturbing body, and the z axis is normal to the plane. ecliptic. These equations are non-integrable and ex- hibit a range of complex solutions, analogous to the restricted 3-body problem. and henc a seculae r increas thein ei r relative distance. Rather, the second spacecraft is assumed to For our investigation we consider a periodic orbit solutio thesf no e equations hala , o more orbib o et t non-periodia bn ei c vicinit e orbi th firse n th i tt f yo spacecraft. specific, which is a fully 3-dimensional orbit centered about the libration points. Figures 1-3 show this particular halo orbit projected intmaie oth n coor- Linearized Dynamics dinate planes e initiaTh . l condition f thio s s peri- e firsTh t spacecraf a trajector s ha t y defines a d odic orbit are chosen to be similar to the Genesis R(t V, 0 0;R ) , R(t+T R(£))= perio,e wherth s di eT halo orbit during its main mission phase (Howell et of motion. Naturally velocite th ,spacecrafte th f yo , al. 1997). The period of this orbit is 178.9 days. V(i;R0, V0), is also periodic with period T. The orbie unstables Th i t , wit hsingla e hyperbolic sta- second spacecraft has its own trajectory defined with unstabld an e bl e manifold with characteristic expo- position and velocity r(i;r0,v0) and v(t;r0,v0). If nen4.75= a t 10~7x 7characteristia /( s c timf eo the distance betweespacecrafo tw e nth assumes i t d 24.3 days) e orbi stablo Th t.tw alses oscillaoha - to be relatively small, then the trajectory of the tion modes, one with period equal to the periodic second spacecraft can be approximated using lin- orbit (as all periodic orbits have) and the other with earized equations of motion. Define x = [r, v] and period slightly greater than one orbital period . The V] .e , differencth n[R = eX between these We assume that the first spacecraft follows this pe- state s <5xsi , whic s assumehi e "smallb o dt " rela- riodic orbit. We do not discuss the navigation (i.e. , dynamicderivee e statar e Th tivx th . 5 ed o X et f so orbit determinatio trajectord nan y control f thio ) s as: spacecraft, but assume that it is kept on track and is always relatively close to the periodic orbit. Some = x-X (4) practical aspect navigatinf so g suc hspacecrafa e ar t = x-X (5) discussed in Scheeres et al. (1999). = F(X + - F(X) (6) Relative Motion of the Spacecraft (7) In our ideal problem, we assume that the first A(t)S-x. (8) spacecraft is following a periodic halo orbit and the dF A(t) second spacecraf a neighborin n o s i t g trajectory. 9X (9) Since we desire the two spacecraft to maintain a close A(t+T) A(t) (10) relative distanc eaco et h other assume w , e thae th t second spacecraft is not on a neighboring periodic where dynamicF(Xe th s )i s function corresponding orbit, as this would imply a different orbit period Equationo t s 1-3 solutioe Th . relativr nfo e motion

1.50*06 -800000-600000-400000-200000 0 200000 400000 600000 SOOOOO X(km) Y(ton)

Figur : Hale2 o orbit trajectory projectez d x- inte oth Figure 3: Halo orbit trajectory projected into the y-z plane. plane.

can then be expressed as: lie alon hyperbolie gth c manifold periodie th f so - cor bit. A spacecraft in such an orbit will naturally cir- <5x = $(t, (11) culate abou periodie tth c orbi quasi-periodia n ti - cor (12) bit, its trajectory winding onto a torus that encloses the periodic orbit in 3-dimensional space. Such a relative orbital configuratio attractivs ni , dependeas - ,t0) = A(t)$(t,t0) (13) initiae th inn glo conditions spacecrafte giveth o nt , (14) , *o) = I an ensemble of spacecraft distributed along such ini- where $ is the state transition matrix, computed tial conditions may serve a useful purpose as a con- abou periodie th t c orbitinitiae <5xth d s i an 0, l offset stellation of spacecraft (Harden and Howell 1998). of the spacecraft from the periodic orbit. Unfortunately, the practical implementation of such famila orbitf yo extremels i y difficult, requiring pre- Long-Term Relative Motion cision placement and control of the spacecraft at all case f long-terth eo r Fo m motion, i.e., spacecraft points along the trajectory, as the center manifold is motion on the order of the orbit period or longer, the itsel unstabln a f e object. state transition matriwrittee producb e n th xs nca a t of two matrices by application of Floquet's theorem Short-Term Relative Motion (Cesari, pp. 55 - 59): For practical consideration insteay ma e dsw con- cern ourselves with the relative dynamics of the (t o)£> $(t,t0) = P(t-t0)e ~* (15) spacecraft over a time much less than the orbital period. While, in principle, the description of rela- periodiwhera s i eP cs i matri D periof d xo an dT tive motion give Equation i hold5 n1 s true t doei , s a constant matrix which has, as its eigenvalues, the not give us a direct indication of relative motion over characteristic exponent e periodith f o s c orbit over shorter time periods. one orbital period. As noted earlier, our periodic represenWn eca state th t e transition matrie th f xo halo orbit has one pair of hyperbolic characteris- periodic orbit ove perio e produce ron th s da mapf to - c exponent ti circulatioo tw d an s n frequenciese on , pings over much shorter time intervals withie th n orbie equath slightle to t lperio on d ydan longer. period of motion: presence Duth unstable o th et f eo e manifold- un , T controlled relative motion will rapidly move away (16) fro vicinite mperiodie th th f yo c orbit. wis e spacecrafe Iw fh th maintaio t t nlong-tera m trajectory in the neighborhood of the periodic orbit Wher mappine eth g ove tima r e interva repres i t lA - sented as $(tj + At, ti) and conforms to the equation: we mus centee t th plac n i r t emanifoli perie th f -do odic orbit, define space th orbitf s eo da t s thano o d t 6t,ti) (17)

0 < 6t < At < T (18) t eaca h poine show ar timen i tFigurd n ni an , . e4 We see that for our base periodic orbit the structure For At small enough we can expand A(t) in a Taylor is consistent, having a pair of hyperbolic roots and Series expansion, yielding: two pair of oscillatory roots at each moment in time. A(ti + 6t) = A(ti) + A'(ti}St (19) The generalization of this result to the family of halo t beeorbitno ns performesha yetf Figuro n I .s da e5 r problemou r fine Fo w ,d tha force th t e partia- ma l we sho characteristie wth c exponent numere th f so - trix A(t) does not vary strongly over time, mean- ically computed map r decreasin , t^sfo )3>(ti At + g ing that the time interval At can be chosen small . valueComparisoAt f so e characteristith f no - cex enoug ensuro ht e that ||A(t;)| ||4'(ti)At||> | . With ponents wit exponente hth s derive t eacda h instant this restriction, we find that the state transition ma- of time clearly shows thae actuath t l characteristic trix differential equation can be approximated over exponents converge to the "instantaneous" charac- short time intervals as: teristic exponent t shrinksA s sa . (*i + St, ti) = A(ti)$(ti+6t,ti) +... (20) CtwartwWte Exponent givin a gtim e invariant syste t mleadina g order. Thus solutio e expressee th ,b n nca : das (21) I + A(ti)5t + ... (22) where, sinc time eth e interva choses i lsmalle b o nt , we neglect higher orderexponentiae th f o s l expansion. The relative motion of the spacecraft can then be characterized ove a shorr t perioe th f tim o dy b e 300 1000 1500 2000 2500 3000 eigenvalue eigenvectord san exponentiae th f so l map, Tim«(houra) defined fro equatione mth : Figure 4: Instantaneous characteristic exponents and u A 3>= u (23) e periodith frequencie f o e c th orbi f o s t oveperioe on r d of motion. or equivalently: (AJ-$)0 = u (24) eigenvectore th eigenvalue whers th i s i u eA d e.an Using Equatio reducee b thi2 n n2 : s ca d to

(A-l)0 .I-A(ti) = u (25) St For a time invariant system the eigenvalue of the state transition matrix Equation i , A s equa,i 4 n2 l to e^\ where 7 is the characteristic exponent of

systeme th . Thusst eigenvalue th , Equatiof eo n ca 5 n2 approximatee b : das A-l 7 lim (26) (5t->0 St Figure 5: Characteristic exponents of the unstable hy- From this sequenc f approximationeo e thase te w s perbolic mode of the periodic orbit computed for the the relative motion of the S/C over a short time state transition maps (t0 -I- iAt,t - l)At)i ( = t 0+ A , span centere t understootime da b en Uca anay db - T/N, orbie equaT th to t lperiod exponente Th . e sar lyzin eigenvaluee gth eigenvectord san matrie th f so x computed for increasing values of N. It is clear that A(ti). Such an analysis resembles the analysis of an becomeN s a s large (an t smalldA ) tha exponente th t s equilibrium point carrie t eaca t h dou tim . eti converg instantaneoue th o et s valuemape th f .so Given the base periodic orbit, the characteristic exponents and frequencies of A(ti) can be computed derivinn I controwilr these e gw ou us l ew "shortlla -

term" dynamics to guide our thinking. It is a fairly vectospacecrafe th f o r projectes i t d int stable oth e natural approach, sinc precursoea spacecrafa r fo r t and unstable manifolds, and that this projection is to diverge from the periodic orbit over a longer time multiplie gaia y ndb constan acceleration a d an t n span will be for it to diverge from the periodic orbit of this magnitud s applieei d alon e stablgth d an e ove a shorter r time span . n thinThusca f o ke w , unstable manifolds, respectively. Thus, the control relative motion alon unstable gth e manifolda e b o st acceleration which we apply is: precursor to motion along the unstable manifold of e fulth l orbit s definea , Floquey b d t theory. Note, - = Tc u_u 5r (27) tha fule th tl orbistily unstablee b lma t , e eveth f ni where crG is the gain constant and 6r is the mea-

instantaneous map is stable at each time step. We sured relativ2 e position vector betweespaceo tw e -nth wilexamplee se l thif so s later. craft (i.e. offsee th , t betwee periodie nth c orbid an t spacecraft)d 2n e notth e W e. thaeigenvaluee th t s Stabilizing the Relative Motion ±er(ti) eigenvectord an s u±(i; well-definee ar ) - pe d A contro formulates i w la l d whic stabilizn hca e eth riodic functions of time, with period equal to the relative motio spacecrafte th f no firse W t. derive eth base periodic orbit. control by focusing on stabilizing the "short-term" Given this specific control law we can reformulate dynamics, as described previously. We then show the relative dynamics of our system, taking advan- that this "local stabilization" can also stabilize the tage of the Lagrangian structure of the equations of long-term relative motio spacecrafte th f no . motion. Denot e linearizeeth d equation motiof so n for relative motion as: Local Stabilization <5r "- 2u JSr'0 - V = dr Neglecting the larger issue of long-term stability rr (28) for now focue w stabilizin,n o s relative gth e motion 0 1 0 between the two spacecraft over short time spans. J = -100 (29) Ther e manar e y different approachee sb than ca t 000 taken here, depending on what form we wish the where u is the rotation rate of the coordinate frame resulting motio constellatioe th takeo r nt Fo . - nap and (ti)V is the second partial of the force ma-

plications we are considering it would be most ben- trix evaluateTT d along the periodic orbit, making it eficial if the relative control between the spacecraft time-varyinga , periodic matri wells xa . Thene th , would allo stablr wfo e oscillations ,foune sucar s dha eigenvalue eigenvectord san conr ou -n i e s us tha e w t in the center manifold of an orbit. If the control law trol schem computee ear d from: givn ca e such solutions, the have nw e considerable 2 freedo chooso mt changd ean e relativeth e motion [a ! :p 2wcr J - VTT] u± = 0 (30) of the spacecraft throughout the orbit. Specifically, if the control law reduces the relative motion to os- Implementin controe gth l acceleratio Equatiof no n cillatory motion, the nvarieta f motionyo e b n ca s 27 into Equation 28 has the effect of modifying our implemente propey b d r choice intiath f lo e condi- potential force contribution lineae th o rst system: tions. Additionally, the application of one or two <5r 2u>J5r'"— impulsive maneuver excitn sca edifferena t aspecf o t -{V; -CT2G[u u£+u_u!;]}(Jr = 0 (31) relative th e motio changd nan dynamice eth e th f so r + constellation. Technically wise w choos,o h t conea - This control law maintains the Hamiltonian nature trol law which will establish Lyapunov stability of e problemoth f d providean , s local stabilite th f yi relative th e motion. gain constan s chosei G t n sufficiently e largeth n I . We propos specifiea c contro whicw prola n l hca - Appendix we show that this control law can always vide Lyapunov stability of our local, short time mo- stabiliz etime-invarianta , hyperbolic unstabl- de e2 sufficientla tio r nfo y large gain constant- im o T . gree of freedom Hamiltonian system, and give a plement this contro givea t a ln tim eevaluate tw j e heuristic justification for scaling our gain constant the local eigenstructure of the matrix A(ti) and find characteristie byth c exponent squared. e characteristith c exponent e hyperbolith f so - cmo e contro Th jusw la tl consist a "reshaping f so f o " tion, ±a(ti), and find the eigenvectors that define the local force structure by proper application of e stabl th d unstablan e e manifold f thio s s motion, thrusting. We shall see that since the magnitude u±(ti), where the + denotes the unstable manifold of these force rathes i s r vicinite smale th th n i lf yo —e th denote d an stable sth e manifold stabilizo T . e libration point e amounsth acceleratiof o t n needed motioe th thee nw n assume tha relative tth e position to implement this control is also relatively small and

could be provided by a low-thrust propulsion system minimum computation then consist integrae th f so - in many cases. tion of the periodic orbit (6 equations) plus the in- tegratio modifiee th f no d state transition matri6 x(3 Long-Term Stability equations) integratioe Th . performes nwa d usinga Equatio generalize1 n3 e relativth o t s e motiof no variable step size RK78 algorithm, with error con- our spacecraft over longer time spane wells th sa s a , trol applied to the entire set of 42 equations. As spectrum of the dynamics, Equation 30, is well de- the gain is increased, it is the numerical integration fine t eacda stabilithe momenth s a yd e typan tth f eo of $ that controls the step size choice, and hence it local motion doe t changsno e ove entire th r e orbit. is most convenient to carry along the computation Thus, the eigenvalues ±cr and eigenvectors u± are e periodi spectruoe th f th s d it c an f m orbito t i s a , all well defined functions of time. local motion will then be evaluated at the appropri- ate time for inclusion in the 3? computation. This is Computation of Stability Stabilization of the rela- found to be simpler than pre-computing and storing tive motion over short time spans is nice, but does the periodic orbit and its local stability characteri- t necessarilno y guarantee thae motioe th tth f o n zation. spacecraft wile stablb l e over longer time spans. stabilite closee Th th f yo d loop syste then me ca n b Technical conditions for which such a time periodic evaluated by computing the eigenvalues, p, of this stable systeb discussee y ear m ma greaten di r detail map: in Cesari (1963). However, these mathematical conditions, even if definite, will in general not be easily = 0 (33) appliee currenth o dt t system wher have ew e fully general motion in three dimensions defined numer- As detaile Marchan di 171-173p (p l eigenvaluee )th s ically and a linear map that changes significantly of this map must occur in complex conjugate pairs with time. The stability of the system can be evalu- and in inverse pairs, meaning that for every complex ated, however applicatioy b , Floquef no t theord yan eigenvalue fj,, its conjugate p, must also be an eigen- numerical integration. value, and that for every eigenvalue with magnitude Again, since the eigenvalues and eigenvectors used different than 1, |/x| ^ 1, there must also be an eigen- in the feedback control are functions only of the pe- value of magnitude 1 / |/i|. Stability of this system oc- riodic orbit, they are in turn periodic, making the cur seigenvaluee wheth l nal s have unit magnitude, linear dynamical system defined in Equation 31 time i.e., reside on the unit circle in the complex plane periodic. Also, sinc e controth e l force onle ar sa y havd fore an eth m p.e= ±tff . Conversely unstabln a , e functio principln i f positione o n ar d e derivablan , e map will either have a pair of multipliers along the fro mpotentiaa l function thae e controse th te w , l rea lfore axisth m f o , ±(p, l/p) r wilo , l hav efoura - law doe t changno s e conservativth e e structurf eo som f compleeo x multiplier e unith tf o f s of tha e li t

dynamicse th . Thus characteristie th , c exponentf so circle and have the form pe(\/p)e. Either y ±ie

lineae oveperiop th e ron ma d will follo generae wth l of these modes will lea exponentiallo dt ±ie y unstable rules for conservative, Hamiltonian systems. relative motion between the spacecraft. The algorithm which we apply is given, in brief, followss a periodie Th . accompanyincs it orbi d an t g For the case of a completely stable map, there are state transition matrix, modified by introduction of three pairs of eigenvalues, each pair characterized by stabilizatioe th n control s numericalli , y integrated the angle 0$, i = 1,2,3. This angle denotes the total over one period of motion. At each time step the winding angle of the mode about the periodic orbit, spectru opee th nf m o loo p syste mcomputes i d dan modulus 27r. Taken together, they define a quasi- fed into Equation 31. The gain constant G is a free periodic orbit with three seperate "average" periods parameter- thain s specifiee i t e starth th f o t a d oveperiodie on r generae c th orbit r Fo l. character- tegration. The modified, time-varying matrix A(t) izatio f thesno e oscillation n considemodeca e w s r used in the state transition matrix computation is the angles to be functions of time along the orbit, specified as: Oi(t), with their final values computed from the monodromy map being Qi = mod(0i(!T),27r). Unfortu- 0 nately , extraco therdirect o n y s ei wa tt &i(t) from A(t) = (32) V - cr2G [u+u+ + u_ul] the numerically computed monodromy map. This rr iten a f interestms o i , however e frequencth s a , f yo resultine Th g state transition matri denotes xi s da these stable motions will change wit gaie hth n con- (t0 + T, to). The orbit frequency, meaning tha spacecrafa t t having

motion along this mode can encircle the periodic or- motion stable over the entire periodic orbit. How- t severabi l times ove perioe base on th r ef do orbit . ever note w , e that stabilit locae th f ly o motio n does Thus, denning and computing these mean frequen- t guaranteno stabilite eth monodrome th f yo - yma f practicao cie s i s l interese relativth r fo te motion. trix. And factn fini ,e w d, tha monodrome th t p yma Ideally, we compute them as: does not become stable until the gain G reaches a valu 2.19f eo . (34) n general I s stabli r e finp w fo ed, ma thae th t gains greater than this value. There are exceptions (35) correspondin resonanceo gt s betwee oscillatioe nth n We find that the number Ni can be estimated from modes and with the periodic orbit period. However, the time history of the spectrum of the state transi- in analog Mathiee th o yt u equation gaie - th n be s a , tion matrix countiny 2 and —1/P2- The next interval of instability is 6.10 < modee th 5.9G , f whero 2< s o havtw e ea -/ Zi(t)dt (36) Jo resonance, one having twice the period of the other, and resulting in a complex set of four multipliers. where the frequency w,(t) is defined as an instanta- We note tha magnitude th t multipliere th f eo n si neous characteristic frequenc e lineath f ro y matrix these intervals are "small", the maximum multiplier A(t), which has been stabilized by our application in the above intervals being less than 1.1 over one of local control. We find that reasonable estimates orbit period (corresponding to a characteristic time e meaoth f n frequenc e stablth f o ye modee th f o s greater tha years)n5 . Thi especialls i s y smals a l relative motion can be found from this quadrature. compare originae th o dt l instabilit periodie th f yo c Improvements to this approach will be investigated orbit whiccharacteristia s hha c days 3 tim2 n f I e.o futuree inth . many cases these instabilities would not appreciably thif o shampee relativus e th r e control. Such inter- Stability as a function of Gain As formulated, val instabilitf so expectede yar genera n i d an ,l cor- there is only one parameter for the controller, the respond to the eigenvalues of the monodromy map gain constant G. We see that, as this constant intersectin unie th t n circlgo e (see - Marchade a r fo l is increased, the nature of the linear dynamics tailed discussion of possible stability transitions). As changes r valuewheG Fo e monodromf . o th sn y noted in Marchal, the presence of oscillatory, linear matri stables xi l e relativspacal th , f eo e motions stability does not imply non-linear stability of the e founb y computinb n d ca e characteristith g c relative orbits. However, we are assured that any frequencies and eigenvectors of the matrix. Motion non-linear instabilities wilt grono l w exponentially, under this map will in general lie on quasi-periodic but will only grow, at most, as a polynomial in time. tori surrounding the periodic orbit - somewhat For practical considerations, this implies that such similar to motion in the center manifold but with instabilities should be controllable by standard nav- additional frequencie chooso t s e froeliminad man - igation practices over time. tion of instability concerns. To give insight into As the gain of the control loop increases, we find these possible modes of motion we evaluate the that the mean oscillation frequencies of the system characteristic e monodromth f o s ys i matri G s xa defined earlier, u>i, change as well. For our system, increased. thesf o e e on mean frequencies remains relatively con- For G < 1.17 the control law has insufficient au- stant and is consistently less than the periodic orbit thorit stabilizo yt locae eth l motio t everna y time frequency (27T/T). This oscillation mode is associ- along the periodic motion, meaning that there are ated most strongly with "out-of-plane" motion along time intervals where the spectrum of A(t) is not com systeme -z th axi relatef d e initiae so th an th , o dt l pletely stable. We note that the monodromy matrix stable oscillation uncontrollee modth f eo d relative is also found to be unstable for this situation. Only motion. While the control loop does influence this for value greateG f so r than instantaneou e 1.1th 7s i s stable oscillation mode, it never causes it to stray

far from its original value. Relative Trajectory ——— Conversely e meath , n frequenc e otheo th f tw ro y Trajector— —— y oscillation modes increase generaln i ,squar e th s a ,e root of the gain G. One of these modes origi- nates from the original open loop oscillation of period T, while the other is created when the closed loop control removes the hyperbolic manifolds. By specifying the control gain G the "natural dynam- relative th ics f o " e motiomodifiede b n nca , causing the relative motion between the spacecraft to follow different paths characteriziny B . e oscillagth - tion plane f theso s e motions over tims possii t ei - ble to distribute a constellation of spacecraft so that they, as a group, are oriented toward a specific direc-

tionr largFo .e value thesG f so e mean frequencies 600000 800000 te*O6 1.2e*06 1.4

Stabilizing non-linear motion The linear stability tem. In Figure 12 we show a plot of the motion rel- r halou or controe oorbith fo f w t la lhavin g bee- nes ative to the periodic orbit for a case with an initial tablished numerically, it is of interest to verify that it position offset of 5 x 10s km and control gain of 10, can actually stabiliz non-lineaea r trajectory about projecte z coordinat d an d y int e eoth planes. Also a periodic orbit. To verify this, we apply the control shown is the linear solution to Equation 31, started acceleration given in Equation 27 to the full equa- with the same value of offset but obviously follow- tion motiof so spacecrafa f no t moving accordino gt ing a different trajectory as compared to motion in Equations 1-3. The initial conditions of this mo- the "real" system. This give indication sa n thar ou t tio denotee nar offsen a s dta fro initiae mth l condi- control law remains robust when the relative motion periodie tionth f o s cinitiae orbith t a lt timed an , longeo isn r well defined wit lineae hth r equationf so relative th e position betwee trajectore ne th th d yan motion. periodic orbit, needee controlth r dfo computes i , d as the vector difference between the two solutions at Control Implementation each momen timen i t . This formulatio entirels i n y Having shown tha r controtou stabilizew la l s rela- consistent, as the computed difference is equivalent tive motion and is robust when applied to fully non- measuree th o t d difference spacebetweeo tw e -nth linear motion, it is necessary to compute the thrust craft, and the local stability computations are based needed to properly apply the control. In our com- on the periodic orbit, which can in principle be computations we keep track of the total "AV" expended puted on-board the controlled spacecraft. by performin quadraturee gth : To push the control law to a somewhat extreme condition gave spacecrafe w , eth initian a t l position \T \dt (37) offset witfro m e periodik mhth s 10 cx orbi1 f o t ~ L c 100d e resultinan Th .gain0 1 f so g trajectoriee ar s actuae Th l time history instantaneoue th f o s thrust shown in Figures 6-11. Each of these trajectories is and accumulated AV change gaie initiad th n an s sa l integrated ovedays0 r40 , whic mors hi - e pe tha o ntw riods of the periodic orbit. We note that our control conditions are varied. An estimate of the necessary law is able to maintain the stable relative motion thrus rela e instangivee ty th b leve -an y n nb t a lca t over this time frame. It is significant to note that tion: computatioe th periodie th f no c orbit itself over this 2cr2GR (38) lengt timf ho non-trivials ei strengte th o t f e ho du , instabilitys it . where R is the amplitude of the relative motion of We also note that the control law is being applied the spacecraft in the x-y plane, a ~ 4.7 x 10~7/s iregiona n where non-linearities dominat syse eth - is the characteristic exponent of our periodic orbit,

Relative Trajector— y Relative Trajectory ——— Periodic Trajectory — Periodic Trajectory ———

1e+06 1.1e+06 1.26*06 1.39406 t.4e+O6 1.5O+06 1.6e*06 1.7&+06 1.8e+O6 •800000 -600000 -400000 -200000 0 200000 400000 600000 800000 X(km)

Figure 7: Controlled and halo orbit trajectory projected Figure 8: Controlled and halo orbit trajectory projected z planex- int e o.th Contro applieds i l0 gai1 f no , with into the y-z plane. Control gain of 10 is applied, with an initian a . l positiokm s 10 n x offse1 f o t initial position offset of 1 x 10s km. e controth s i l G gain d an . thae e Thuse th te w s bital environments nominaa s A . l cas considee ew r thrust will scal QA4GRs ea nm/s2. formation flying spacecraft in an unstable halo orbit To apply a gain of 100 (oscillation periods of 15 and about the Earth-Sun libration point. The relative days1 3 ) wit hrelativa e constellation siz f 100eo 0 dynamic f theso s e spacecraf e considerear t d along km requires a thrust of 44 micrometers/s2. For a wit practicae hth l computatio relativf no e orbitd san 10spacecraftg 0k , this translates int oa thrus f o t the measurement and dynamics of relative orbit un- 4.4 milli-Newtons low-thrusa r Fo . enginn io t e with certainty between the spacecraft. We find a family of a characteristic velocity of 30 km/s over a 2 year simple control laws that stabilize the relative motion mission, the resulting mass fuel fraction used is on appead vere an b yo rt robust cosfeasibile d an tTh . - the order of 5%. itimplementinf yo g these control law addressee sar d To successfully apply the control law we must mea- and found to be reasonable. sure the relative position along the directions of sta- Acknowledgements ble and unstable motion. Radial distance and radial The work described here was funded by the TMOD rate can be measured with a ranger and Doppler Technology Progra grana my b t Propul t Je fro e mth - extractor (we assume near-continuous communica- sion Laboratory, California Institut Technologf eo y tion between the spacecraft in support of their sci- whic undes hi r contract wit Nationae hth l Aeronau- entific mission). If multiple receivers are placed on ticSpacd san e Administration. spacecrafte th relative th , e orientatio spacee th f no - determinee b crafn ca t d and processiny ,b - in e gth Appendix ertially measured attitude relative th , e positiof no We first consider stabilizatio simplea f no , 1DOF, the spacecraft can be determined at each moment. hyperbolic unstable motion e generiTh . c forr mfo Scheeren I . foun(1999s al t wa se d t i ) tha orbie th t t this cas: eis uncertaint singla f yo e spacecraf unstabln a n i t e halo orbit tende minimizee b o dt d alon locae gth l stable f-<70 = 2r (39) and unstable manifolds of the trajectory. A similar characteristie wherth s i ea c exponen unstae th f o t- result for the relative orbit determination between systeme bl thin I . s cas intuitiveln ea y obvious stabi- two spacecraft would support the potential feasibil- lization is to apply a position feedback acceleration r proposeou f o y dit approac trajectoro ht y control. of the form — Kr, yielding the modified system: Studie thif so s effect wil futuremade e b lth n ei . = 0 (40) Conclusions Now obvioun a , s conditio stabilitr cr> nfo 2 ,K s yi In this paper we address issues concerned with the or rewritin

control of formation flying spacecraft in unstable or- comes G 2 > 1. This simple result motivates us to

Relative Trajector— —— y Relative Trajector— —— y (Periodic Trajector— —— y Periodic Trajectory ———

600000 800000 1e*06 t.2e*06 1.to*0$ 1.6e»06 1.8O+06 2e*O6 2.2e*06 10+06 1.10+06 1.2&+O6 1.3e+06 1.4e*O6 1.5e*06 1.6e*06 1.7e+06 1.SO+06 X{krn) X(km)

Figure 9: Controlled and halo orbit trajectory projected Figure 10: Controlled and halo orbit trajectory pro- into the x-y plane. Control gain of 100 is applied, with jected into the x-z plane. Control gain of 100 is applied, an initia. l positiokm 0 1 nx offse1 f o t wit initian ha l. positiokm 0 1 n x offse1 f o t

scal r feedbaceou ksquare e loca th gaith y f lnb eo Vxy T 2o;cr (47) characteristic exponent. " Next consider application of our local stabilization The gain matrix that is added to the left hand side control scheme to an unstable hyperbolic point in a of Equatio (correspondin1 n4 Equatioo gt : is ) n32 rotating 2DOF frame opee Th n. loop dynamice sar specifie lineae th y rdb equation: 8x" 0 1 5x' Sy" -1 0 6y' (48)

v Vv V 5x xy XX = 0 (41) Vvxy V*yy Reforming the characteristic equation for this case characteristie Th c equatio thir nfo s syste: mis yields: 4 22 A[4o; + A ] V(420 n + - IXVi V V - V?yy)= x- y A4+ k,2-V«-VUA2 Fo singla r y hyperbolic unstabl- as e systen ca e mw 2 sume, in general, that u — xxy v VVyy — V — 0 (49) 2 V V -Vv < 0 (43) where 'xx'yy xy The hyperbolic characteristic exponents of the sys- (50) tem are computed from:

(44) (51)

~ V** - ~ 4 (VXIVyy - vxy = (52) which is guaranteed to be positive if the inequality in Equation 43 holds. The corresponding eigenvectors To guarantee linear stability of this system requires will be of the form: that three condition mete sb :

u± = (45) 0 (53) 2 0 > y VxxV (54V - yy )y > 0 u± = (46) 2 2 [v (55 4 0 - Vix Vyy- )]V- yy> y]

Relative Trajectory Non-linear solution • Periodic Trajectory Linear solution -

•800000-600000-400000-200000 0 200000 4000OO 600000 800000 0 Y(km) Y(km)

Figur : Controlle11 e hald dan o orbit trajectory pro- Figure 12: The non-linear and linear trajectory compu- jecte z planed y- int e o.th Contro applieds i 0 l gai10 f ,no tations relativ periodie th o et c orbit, projected inte oth wit initialn ha . positiokm s 10 n x offse1 f o t y-z plane. Control gain of 10 is applied, with an initial position offset of 5 x 105 km. We note that Condition 53 can be rewritten as:

2 each momen timf dependo d t ean estise onlth -n yo V- xx V- yy 0 > G (56) mat relativf eo e position betwee spacecrafe nth d an t which can be satisfied for large enough G. Next, an estimate of the position of the spacecraft on the expanding Condition 54 results in the form: nominal periodic orbit.

2 References [(l + ul) {u +Vxx + Vyy - Harden, B.T., K.C. Howell, "Fundamental + (l + 4) (1 + «i) > 0(57) Motions Near Collinear Libration Points and Their Transitions, : 361-37846 S "JA , 1998. chosee b n Againthane ca largse G te w ,e enougo ht Cesari, L., Asymptotic Behavior and Stability guarantee stability. Finally note w , e that Condition Problems in Ordinary Differential Equations, 55 can be rewritten as 2nd Ed., Academic Press, 1963. Gomez . MasdemontJ , G. , . SimoC , , "Quasihalo Orbits Associated with Libration Points," 0 (58) JA : 135-176S46 , 1998. Howell, K.C., B.T. Barden, M.W. Lo, "Application which inspectionn o , , reduce e sufficienth o st t con- of Dynamical Systems Theor Trajectoro yt y dition Desig Libratioa r nfo n Point Mission", - 178 1 , 16 1997JA : S45 . - Vxx - Vyy 0 (59) Marchal, C., The Three-Body Problem, Elsevier, 1990. or equivalently Scheeres, D.J. . Han,D . Hou,Y , "The Influencf eo Unstable Manifolds Orbin o t Uncertainty,"

V- V xx - yy 0 (60) submitted to JGCD, 1999. for which G can always be chosen large enough to stabilize. Should this condition be true, then Con- dition 56 follows trivially. Thus we note that our proposed alwayn controca w s lla stabiliz hypere eth - bolic unstable equilibrium poin largr fo t e enoug. hG The attractiveness of this controller is that it has a natural specification for our time-varying system at

202