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Dynamics and Control of Relative Motion in an Unstable Orbit*

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Dynamics and Control of Relative Motion in an Unstable Orbit* (c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)1 Sponsoring Organization. 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, 192 (c)2000 American Institut Aeronauticf eo Astronautics& r Publisheso d with Permissio Author(sf no ) and/or Author(s)1 Sponsoring Organization. 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 dis- tance. 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 193 (c)2000 American Institut Aeronauticf eo Astronautics& r Publisheso d with Permissio Author(sf no ) and/or Author(s)1 Sponsoring Organization. 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 = $(t, (11) culate abou periodie tth c orbi quasi-periodia n ti - cor (12) bit, its trajectory winding onto a torus that encloses the periodic orbit in 3-dimensional space. Such a rel- ative orbital configuratio attractivs ni , dependeas - ,t0) = A(t)$(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).
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