AUTONOMOUS REGULATION OF SPACECRAFT MOTION ABOUT UNSTABLE PERIODIC ORBITS USING IMPULSIVE MANEUVERS

Transcription

1 AAS AUTONOMOUS REGULATION OF SPACECRAFT MOTION ABOUT UNSTABLE PERIODIC ORBITS USING IMPULSIVE MANEUVERS Hiroaki Fukuawa This paper presents a control method for autonomous regulation of spacecraft motion about unstable periodic orbits b occasional impulsive maneuvers. In this method, the impulsive maneuver to be applied at each prescribed time is obtained b modulating the difference between the actual state and the reference state of the motion at the time. The method is based on that of Generalied Sampled-data Hold Function (GSHF) control which is known to be useful in stabiliing motion about unstable periodic orbits using continuous maneuvers. A formula for impulsive maneuvers is obtained so that the impulsive maneuvers mimic the efficac of a stabiliing GSHF controller. The method is applied to a halo orbit spacecraft in the Hill s circular three-bod problem as an illustration. INTRODUCTION In recent ears, increasing attention has been being drawn to utiliation of unstable periodic orbits for space flight (See Refs. 1 5). Unstable periodic orbits in the three-bod problem not onl are often useful per se for their unique locations, but also pla a central role in new tpes of transfers. In order for a spacecraft to fl an unstable periodic orbit in practice, however, it is essential to constantl regulate its motion about the orbit b active control maneuvers, for otherwise even an infinitesimal deviation from the orbit will lead to eventual escape from the vicinit of the orbit. The question is how such regulation should be performed. The purpose of this paper is to present a control method to autonomousl regulate the spacecraft motion about a given arbitrar unstable periodic orbit. As being tpical in space flight, control maneuvers here are given in terms of impulsive ones. An impulsive maneuver can be thought of a jump in velocit, customaril understood b V, which is in practice approimatel realied b velocit change of the amount in a ver short time interval. We derive a formula which provides the impulsive maneuver V to be applied at each prescribed time for each difference between the actual state and the reference state of the spacecraft motion at the time. Appling the impulsive maneuvers provided b this formula frequentl enough, the spacecraft motion is epected to be stabilied about the reference periodic orbit and converge to it asmptoticall. Control of spacecraft motion about unstable periodic orbits, especiall with the application to halo orbits, is not new. Nonetheless, a sharp distinction can be drawn between our control method presented in this paper and currentl dominant control methods used Engineer, Estech Corp., 89-1 Yamashita-cho, Naka-ku, Yokohama , Japan; hiroaki.fukuawa@estech.co.jp

2 in the past unstable periodic orbit missions such as Soho and Genesis (See Refs. 6,7). In those missions the spacecrafts flew in the vicinit of unstable halo orbits in the sun-earth sstem. There, each individual station-keeping control maneuver was designed off-line during the operation, i.e. the control is non-autonomous. Although it was proven to be effective in those missions, the non-autonomous station-keeping strateg is computationall and operationall epensive and is limited to the case where the instabilit time scale is relativel large so that not ver frequent maneuvers are required. Our control method is, on the other hand, an autonomous one. It is computationall efficient, operationall inepensive, and readil applicable to the case where frequent maneuvers are required. Thus, this method is promising for future unstable periodic orbit missions to which the currentl available non-autonomous station-keeping methods are not ver suitable. Possible eamples of such missions are one such that the spacecraft is to fl an orbit whose instabilit time scale is ver small, such as libration point orbits in a planet-moon sstem, and one such that man spacecrafts are to be deploed in unstable orbits and the must be controlled at the same time. Availabilit of autonomous control methods will certainl increase the possibilit of future space flight applications. There are in fact several previous works presenting control methods for autonomous regulation or stabiliation of spacecraft motion about unstable periodic orbits (See Refs. 8 11). A major difference between this work and those previous works is that while this work uses impulsive maneuvers for regulation, the previous works use continuous maneuvers. Some advantages of using impulsive maneuvers are that the are often easier to be implemented in actual space flight, and that the leave us free-flight intervals for accurate trajector determination between maneuvers. To our knowledge there is no other work in the current literature presenting a general method for autonomous regulation of motion about unstable periodic orbits using impulsive maneuvers. The presented method is based on that of Generalied Sampled-data Hold Function (GSHF) control (See Refs. 12, 13). Reference 11 showed that GSHF control can be used to stabilie the spacecraft motion about unstable periodic orbits. The idea behind GSHF control is ver simple: sample the output of the sstem at the beginning of each period and, during the net period, generate the continuous control input b modulating the sampled output using a periodic hold function. Although formall GSHF control onl feeds back sampled output, practicall it can be implemented as continuous output feedback control to improve robustness. An advantage of GSHF control is that it makes possible for us to use well-developed linear time-invariant control techniques for stabiliation of linear periodic sstems. We obtain the formula for impulsive maneuvers in two steps. First, we design a GSHF controller that stabilies the linearied spacecraft motion about a target unstable periodic orbit. Net, we divide the period of the periodic orbit into N subintervals and assume that the impulsive maneuvers are applied at the N nodes over the period. The formula is then molded so that the impulsive maneuvers mimic the efficac of the GSHF controller. The remainder of the paper is as follows. In the net section, we formall state the problem considered. Subsequentl, we summarie basic results and a design technique for GSHF control that stabilies the spacecraft motion about unstable periodic orbits. Then, we derive a formula for impulsive maneuvers based on the GSHF control. Finall, we verif the efficac of our method using an unstable L 2 halo orbit in the Hill s circular three-bod problem.

3 PROBLEM STATEMENT Free flight motion of a spacecraft can be in general described b the sstem of differential equations { ẋ = v, (1) v = g(, v), where (t) R 3 is the position vector and v(t) R 3 is the velocit vector in some arbitrar coordinate frame. Let (t) [(t) T, v(t) T ] T and assume that (1) has an orbitall unstable a periodic solution (t) = φ(t) with period T, i.e., φ(t + T) = φ(t), t. (2) Let C be the path of the periodic solution φ(t), t, which is a simple closed curve in the state space. The curve C is called a periodic orbit of (1). Since φ(t) is an orbitall unstable solution, a solution of (1) starting arbitraril close to but not on C escapes the vicinit of C after some time. Suppose it is desired that the spacecraft sta close to the periodic orbit C. Because of orbital instabilit of C, it is required for the spacecraft to appl control maneuvers from time to time to cancel deviations from C in order to sta close to C. In this paper we consider intermittent impulsive maneuvers to control the spacecraft. Impulsive maneuvers can be thought of jumps onl in velocit. Define v(t) lim ǫ {v(t + ǫ) v(t)}. (3) In a free motion, v(t) = for all t. However, if an impulsive maneuver is applied at time τ, then v(τ). In practice, an impulsive maneuver is approimatel realied b velocit change in a ver short time interval. Let {t i } be a given arbitrar sequence of time, where at each time t i, i =, 1, 2,..., an impulsive maneuver is to be applied. The goal of this paper is to obtain a rule, or a formula, h(, ) such that the sequence of impulsive maneuvers v(t i ) = h((t i ), t i ), i =, 1, 2,..., (4) stabilies the motion of (1) about the periodic orbit C. GSHF CONTROL OF LINEAR PERIODIC SYSTEMS If the sstem (1) is linearied about the unstable periodic solution φ(t), the linearied sstem is an unstable linear periodic sstem. If a controller is designed to achieve stabiliation of this linear periodic sstem, then the controller achieves local stabiliation of (1) about φ(t). The purpose of this section is to describe the method of Generalied Sampleddata Hold Function (GSHF) control which can be easil designed to stabilie linear periodic sstems. The GSHF control described in this section will serve as a basis in obtaining a formula for impulsive maneuvers in the net section. a For the formal definition of orbital stabilit of solutions of differential equations, see for eample Ref. 14.

4 GSHF Basics Consider the sstem (1) with an added control input { ẋ = v, v = g(, v) + u, (5) where u(t) R 3 is the control input. Note that the control input enters the sstem as an acceleration term. Linearie (5) about the periodic solution φ(t) to obtain the linear periodic sstem δ = A(t)δ + Bu, (6) where δ(t) = (t) φ(t) and A(t + T) = A(t), t, (7) [ I A(t) = g g v ], B = =φ(t) [ I ]. (8) Let Φ(t, t ) be the open-loop state transition matri of (6), which can be obtained b solving B (7) and (9), it can be shown that Φ(t, t ) = A(t)Φ(t, t ), Φ(t, t ) = I. (9) Φ(t + T, t + T) = Φ(t, t ), t, t. (1) Generalied Sampled-data Hold Function (GSHF) control has the form u(t) = H(t)δ(kT), t [kt, (k + 1)T), k =, 1, 2,..., (11) H(t + T) = H(t), t. (12) Application of the Variation of Constants (VOC) formula (See for eample Ref. 15) to the linear sstem (6) gives (k+1)t δ((k + 1)T) = Φ((k + 1)T, kt)δ(kt) + Φ((k + 1)T, τ)bu(τ)dτ, (13) kt kt+σ δ(kt + σ) = Φ(kT + σ, kt)δ(kt) + Φ(kT + σ, τ)bu(τ)dτ, σ [, T), (14) kt for k =, 1, 2,.... Substituting (11) in (13) and (14), and using the periodicit (7), (1) and (12), we obtain δ((k + 1)T) = Ψ(T, )δ(kt), (15) for k =, 1, 2,..., where δ(kt + σ) = Ψ(σ, )δ(kt), σ [, T), (16) t Ψ(t, t ) Φ(t, t ) + Φ(t, τ)bh(τ)dτ. t (17)

5 Equation (15) governs the state transition of the linear closed-loop sstem (6), (11) at discrete moments t = kt, k =, 1, 2,.... It can be shown that the linear closed-loop sstem is asmptoticall stable if and onl if all eigenvalues of the so called closed-loop monodrom matri Ψ(T, ) have moduli strictl less than one (See Ref. 12). Now, let H(t) in (11) have the form H(t) = F(t)K, (18) then Ψ(T, ) can be written as where F(t + T) = F(t), t, K = constant, (19) Ψ(T, ) = Φ(T, ) + W(T, )K, (2) W(t, t ) t t Φ(t, τ)bf(τ)dτ. (21) The form of (2) suggests that, in this case, the eigenvalues of Ψ(T, ) are arbitraril assignable b choice of the constant matri K if, and onl if, the pair ( Φ(T, ), W(T, ) ) is controllable. Note that the sies of matrices F(t) and K are et to be specified as long as F(t)K is 3 6. Also note that W(t, t ) can be obtained b solving Ẇ(t, t ) = A(t)W(t, t ) + BF(t), W(t, t ) =. (22) In practice, GSHF control in the original form (11) is not advisable because, since the sstem evolves as open-loop between sampling moments, it ields poor disturbance rejection and poor robustness with respect to model errors. But, b (16), we have δ(kt) = Ψ(σ, ) 1 δ(kt + σ), σ [, T), k =, 1, 2,..., (23) and substitution of this in (11) ields u(kt + σ) = H(σ)Ψ(σ, ) 1 δ(kt + σ), σ [, T), k =, 1, 2,.... (24) Hence, GSHF control (11) can be implemented as a continuous feedback control as (24) whenever Ψ(σ, ) is nonsingular. It is preferable in practice that GSHF control be implemented as a continuous feedback control in the form of (24), especiall when it is applied to nonlinear sstems as in the present case. Hold Function Design When F(t) in (18) is specified, all we have to do to obtain a GSHF control law that stabilies the sstem (6) is to choose K such that Ψ(T, ) in (2) is Schur, i.e., all eigenvalues have moduli strictl less than one. Such a K can be obtained b Linear Quadratic Regulator design (See for eample Ref. 16) as follows. Consider the quadratic performance inde J = { δ(t) T Qδ(t) + u(t) T Ru(t) } dt, (25)

6 where Q and R are the smmetric positive definite weighting matrices. When GSHF control (11), (18) is applied, this continuous-time quadratic performance inde can be transformed into the discrete-time quadratic performance inde with J = Q = R = V = δ(kt) T (Q + K T R K + 2V K)δ(kT), (26) k= T T T The matri K that minimies J in (26) is given b Φ(τ, ) T QΦ(τ, )dτ, (27) W(τ, ) T QW(τ, ) + F(τ) T RF(τ)dτ, (28) Φ(τ, ) T QW(τ, )dτ. (29) K = (W(T, ) T SW(T, ) + R ) 1 ( ) W(T, ) T SΦ(T, ) + V T, (3) where S is the positive semi-definite solution of the Algebraic Riccati Equation Φ(T, ) T SΦ(T, ) S + Q (Φ(T, ) T SW(T, ) + V )( W(T, ) T SW(T, ) + R ) 1 ( ) W(T, ) T SΦ(T, ) + V T =. If the pair (Φ(T, ), W(T, )) is stabiliable, R >, Q V R 1 V T, and (Q V R 1 V T, Φ(T, ) W(T, )R 1 V T ) has no unobservable mode on the unit circle, then it is guaranteed that K in (3) makes Ψ(T, ) in (2) Schur, or asmptoticall stable. Practicall, the weighting matrices Q, R, and V ma be directl chosen instead of Q, R, and V. Choice of the periodic gain matri F(t) is not as eas. The matri F(t) should be chosen considering the inter-sampling behavior of the closed-loop sstem governed b (16). Preferabl, F(t) should be chosen so that an induced norm Ψ(t) p is as small as possible for t [, T] for good performance. However, designing F(t) as such is not a trivial problem because Ψ(t) depends on both F(t) and K, while K is chosen after F(t) is specified. In our eperience, however, the choice F(t) = B T Φ(T, t) T, t [, T), (32) seems to work ver well in man cases. Note that with this F(t), W(T, ) = T coincides with the reachabilit grammian from t = to t = T. (31) Φ(T, τ)bb T Φ(T, τ) T dτ (33)

7 AUTONOMOUS IMPULSIVE MANEUVERS Suppose that we have designed a GSHF controller u(t) = H(t)δ(kT), t [kt, (k + 1)T), k =, 1, 2,..., (34) H(t + T) = H(t), t, (35) that stabilies the linear sstem (6) and achieves satisfactor performance when it is applied to the nonlinear sstem (5). The goal of this section is to obtain a formula for impulsive maneuvers that approimatel achieve the efficac of (34). and define Divide the time interval [, T) into N arbitrar subintervals [σ j, σ j+1 ), j =, 1,...,N 1, σ =, σ N = T, (36) (see Figure 1). An application of the VOC formula to (6) gives t k j kt + σ j (37) t k δ(t k j+1) = Φ(t k j+1, t k j)δ(t k j+1 j) + Φ(t k j+1, τ)bu(τ)dτ, (38) t k j for j =, 1,...,N 1 and k =, 1,.... Substituting (34) in (38) and using the periodicit, we obtain [ ] t k δ(t k j+1) = Φ(t k j+1, t k j)δ(t k j+1 j) + Φ(t k j+1, τ)bh(τ)dτ δ(kt) t k j [ ] σj+1 = Φ(σ j+1, σ j )δ(t k j) + Φ(σ j+1, τ)bh(τ)dτ δ(kt). (39) Now, we consider the impulsive control { u ū k j (t) = (constant), if t [tk j, tk j + ǫ);, if t [t k j + ǫ, tk j+1 ), (4) σ j Figure 1: Subintervals of the period

8 where ǫ is an infinitesimal time. Let δ (t) denote the solution of (6) when the control (4) is applied, then, b the VOC formula, t k δ (t k j+1) = Φ(t k j+1, t k j)δ (t k j+1 j) + Φ(t k j+1, τ)bu (τ)dτ, (41) t k j for j =, 1,...,N 1 and k =, 1,.... Substituting (4) in (41) and using the periodicit, we see that δ (t k j+1) = Φ(t k j+1, t k j)δ (t k j) + B formall ǫ, we obtain t k j +ǫ t k j Φ(t k j+1, τ)bū k jdτ = Φ(σ j+1, σ j )δ (t k j) + Φ(σ j+1, σ j + ǫ) σj +ǫ σ j Φ(σ j + ǫ, τ)bū k jdτ. (42) δ (t k j+1) = Φ(σ k j+1, σ k j )δ (t k j) + Φ(σ j+1, σ j )Bū k jǫ. (43) Since it is desired that the impulsive control (4) have the efficac of the continuous GSHF control (34), we require δ (t k j ) = δ(tk j ) for j =, 1,...,N 1 and k =, 1,.... This leads to, b (39) and (43), [ ] σj+1 Φ(σ j+1, σ j )Bū k jǫ = Φ(σ j+1, τ)bh(τ)dτ δ(kt). (44) σ j Therefore, the impulses ū k j ǫ that achieve the efficac of the GSHF control (34) at time tk j are obtained b solving (44). However, the equation (44) is over-determined because the number of unknowns ū k jǫ is three, while the number of equations is si. Therefore, an eact solution of (44) does not eist in general. Nevertheless, the best approimate solution in the least square sense can be obtained as [ ] σj+1 ū k jǫ = [Φ(σ j+1, σ j )B] + Φ(σ j+1, τ)bh(τ)dτ δ(kt), (45) σ j where { } + indicates Moore-Penrose inverse. Since both matrices Φ(σ j+1, σ j ) and B have full ranks, we have [Φ(σ j+1, σ j )B] + = [ B T Φ T (σ j+1, σ j )Φ(σ j+1, σ j )B ] 1 B T Φ T (σ j+1, σ j ). (46) B the wa, using VOC formula again, we have B formall ǫ, we obtain δ (t k j + ǫ) = Φ(t k j + ǫ, t k j)δ (t k j) + t k j +ǫ t k j Φ(t k j + ǫ, τ)bū k jdτ. (47) δ (t k j + ǫ) = δ (t k j) + Bū k jǫ. (48)

9 This implies, b the definition of impulsive maneuvers (3), v(t k j) = ū k jǫ. (49) Therefore, b (45) and (49) we obtain [ ] σj+1 v(t k j) = [Φ(σ j+1, σ j )B] + Φ(σ j+1, τ)bh(τ)dτ δ(kt). (5) Furthermore, since, b (16), σ j δ(kt) = Ψ 1 (σ j, )δ(kt + σ j ) = Ψ 1 (σ j, )δ(t k j), (51) we obtain [ ] σj+1 v(t k j) = [Φ(σ j+1, σ j )B] + Φ(σ j+1, τ)bh(τ)dτ Ψ 1 (σ j, )δ(t k j). (52) If H(t) has the form as in (18), then (52) can be written as σ j v(t k j) = [Φ(σ j+1, σ j )B] + W(σ j+1, σ j )K [Φ(σ j, ) + W(σ j, )K] 1 δ(t k j). (53) The formulae (52) and (53) provide impulsive maneuvers which mimic continuous feedback implementation of GSHF control. Once the matrices Φ(σ j+1, σ j ) and W(σ j+1, σ j ) are computed and stored over one period, the impulsive maneuver v(t k j ) to be applied at time t k j = kt + σ j is readil obtained b (53) for each deviation δ(t k j ) = (tk j ) φ(tk j ) at that time. Since we have made an approimation in (45) in obtaining the formula (52) or (53), the stabilit of the controlled trajectories with the maneuvers should be checked b simulations. Also, maneuvers should be applied frequentl enough in order to cancel nonlinear deviations before the grow up too much. The number of maneuvers over one period, N, ma be determined b trial and error so that satisfactor performance is achieved. So far, we have assumed that the timing of maneuvers t k j, j =, 1,...,N 1, k =, 1,..., is predetermined and fied. However, it is often desired to change the timing of maneuvers during operation depending on situations. For eample, while relativel frequent maneuvers are required when the deviation from the reference periodic solution φ(t) is large because in this case errors due to neglected nonlinearit tend to grow fast, relativel less frequent maneuvers are necessar when the deviation is small. In some cases, when the deviation is so small that it is indistinguishable from navigation errors, it ma be desirable to postpone a maneuver until the deviation becomes distinguishabl large. Actuall, we can easil generalie the result to adopt fleible timing maneuvers. Replace σ j and σ j+1 in (53) b σ and σ + λ, respectivel, then we obtain v(kt + σ) = [Φ(σ + λ, σ)b] + W(σ + λ, σ)k [Φ(σ, ) + W(σ, )K] 1 δ(kt + σ). (54) Using (54), it is possible to obtain the maneuver at time kt +σ for an σ. The value of λ is chosen according to when the net maneuver is epected to be applied. Furthermore, it is a ver hand propert of GSHF control that it allows much fleibilit when to sample the

10 state for feedback and our maneuver rule inherit this propert. Indeed, using the propert in (16), (54) can be written as v(kt +σ) = [Φ(σ + λ, σ)b] + W(σ+λ, σ)k [Φ(σ ξ,) + W(σ ξ,)k] 1 δ(kt +σ ξ). (55) for ξ < σ. Equation (55) allows us to obtain the impulsive maneuver to be applied at time kt + σ from the state at time kt + σ ξ. This generalied formula is useful when it is required to obtain the impulsive maneuver ahead of time or when [Ψ(σ, ) + W(σ, )K] in (54) is singular or close to singular. APPLICATION TO HALO ORBIT SPACECRAFTS In this section, as an eample application of our method, we consider autonomous regulation of a spacecraft along an unstable halo orbit in a general planet-satellite sstem. The model we use here for the spacecraft s motion is that of the Hill s circular three bod problem which is ver general (See Refs ). The basic Hill s assumptions require a near circular orbit for the planetar satellite and a sufficientl small relative mass ratio of the satellite to the planet. These conditions are satisfied for almost all planetar satellites in the solar sstem, eceptions being Earth s moon and Pluto s moon, Charon, for which the mass ratios are not small (Ref. 2). The Hill s equations of motion for the spacecraft is given b ẍ 2ωẏ = ÿ + 2ωẋ = = V, (56) V, (57) V, (58) V (,, ) = µ r ω2 (3 2 2 ), r = , (59) where [,, ] T describes the position of the spacecraft in a rotating Cartesian frame such that the origin coincides with the center of the satellite, the -ais lies along the planetsatellite line pointing outward, the -ais points along the motion of the satellite about the planet, and the -ais completes the right-handed triad, ω is the satellite s rotation rate about the planet assuming circular motion, and µ is the satellite s mass parameter. In order to avoid using specific values for ω and µ, we can remove those parameters from the equations b scaling time b 1/ω and length b (µ/ω 2 ) 1/3. The resulting dimensionless equations are ẍ 2ẏ = ÿ + 2ẋ = = U, (6) U, (61) U, (62) U(,, ) = 1 r (32 2 ), r = (63)

11 We shall use these dimensionless equations in the following. In order to get some sense of numbers that appear in the following, however, we note that a unit time, a unit length, and a unit speed in the dimensionless scale approimatel correspond to 58.1 das, km, and 43m/s in the sun-earth scale, respectivel. Let [,, ] T and v [ẋ,ẏ, ż] T, then the sstem (6)-(62) can be written in the form of (1) as where ẋ = v, (64) v = U + 2Jv, (65) J = 1 1. (66) The sstem has families of three dimensional periodic orbits known as halo orbits near colinear libration points. Some halo orbits are unstable. One of such unstable halo orbits near the L 2 libration point is shown in Figure 2. This halo orbit is obtained with the initial condition below. () = ẋ() = () =. ẏ() = (67) () = ż() = Note that specification of all the digits in the above data is required because of the severe instabilit of the orbit. The period of this halo orbit is T In the sun-earth scale this halo orbit corresponds to the one used in Genesis mission (See Refs. 9, 21). Linearie the sstem about the periodic solution to obtain [ ] [ δ δ δv = A δv [ I A = 2J 2 U 2 ], (68) ], (69) where the matri 2 U/ 2 is evaluated along the periodic solution. The characteristic multipliers of the periodic orbit, i.e., the eigenvalues of the open-loop monodrom matri Φ(T, ) which can be computed b integrating over one period, are Φ(t, ) = AΦ(t, ), Φ(, ) = I, (7) 1563, 1/1563,.98 ±.197i, 1., 1., and the first eigenvalue indicates that the halo orbit is in fact a ver unstable periodic orbit. We shall now consider stabiliing the spacecraft motion about this halo orbit. First, we shall design a GSHF controller that stabilies the spacecraft motion about the halo orbit using the results described in the previous section. We choose F(t) as in (32)

12 (a) - projection (b) - projection (c) - projection Figure 2: The nominal halo orbit and snthesie K as in (3)-(31) with the choice of weighting matrices Q = I, R = 1 2 I, and V =. The eigenvalues of the resulting closed-loop monodrom matri Ψ(T, ) are.333,.49 ±.18i, , ,.54, indicating asmptotic stabilit. Figure 3 shows a simulated controlled trajector of the spacecraft with this GSHF control implemented as continuous feedback. The initial state for the simulation was set at the point where () is shifted b.463 (about 1 5 km in the sun-earth scale) from the initial state for the periodic solution in (67). The following series of simulations are obtained with the same initial state for comparison and the simulation time is 5T. It can be seen from the simulation result that the spacecraft motion is successfull stabilied about the halo orbit b this GSHF control (a) - projection (b) - projection (c) - projection Figure 3: Controlled trajector of the halo orbit spacecraft b GSHF control Net, we consider impulsive maneuvers to achieve stabiliation of the spacecraft motion about the halo orbit following the results in the last section. We divide [, T) into N equall spaced subintervals as in (36) with σ j = jt/n and the impulsive maneuvers are to be applied at time kt + jt/n, j =, 1,...,N, k =, 1,.... Based on the GSHF controller design {F(t), K} above, we immediatel obtain a maneuver rule b (53). We simulate

13 controlled trajectories of the spacecraft with the impulsive maneuvers for three different numbers of N, N = 1, N = 4, and N = 2, and the results are shown in Figures 4, 5, and 6, respectivel. In all three cases the spacecraft motion is successfull stabilied about the halo orbit b these impulsive maneuvers. It can be seen b comparing the simulation results that the control performance is better and closer to that of the continuous feedback GSHF control as the frequenc of maneuvers is higher. Although, the performance deteriorates as the frequenc of maneuvers becomes less, we find that in the present eample the spacecraft motion can be stabilied b onl two impulsive maneuvers per period (a) - projection (b) - projection (c) - projection Figure 4: Controlled trajector of the halo orbit spacecraft b impulsive maneuvers of equal intervals for N = 1; Ten impulsive maneuvers are applied per period (a) - projection (b) - projection (c) - projection Figure 5: Controlled trajector of the halo orbit spacecraft b impulsive maneuvers of equal intervals for N = 4; Four impulsive maneuvers are applied per period The maneuver rule was obtained so that the impulsive maneuvers mimic the GSHF control. In order to illustrate this point, Figure 7 and 8 show the state transition of the linearied spacecraft motion with the GSHF control and the impulsive maneuvers, respectivel, for over the first period. It is seen that the state transition in both cases, especiall that of the position, match ver well. This illustrates that the impulsive maneuvers indeed mimic the GSHF control. Finall, Figure 9 shows the magnitude and the timing of each impulsive maneuver applied in the above simulations for the three different values of N. The plot shows that the total maneuver cost tend to be less when N is larger. This is

14 (a) - projection (b) - projection (c) - projection Figure 6: Controlled trajector of the halo orbit spacecraft b impulsive maneuvers of equal intervals for N = 2; Two impulsive maneuvers are applied per period.4.2 δ δ δ δv δv δv time (a) Position time (b) Velocit Figure 7: State transition of the linearied spacecraft motion with GSHF control over the first period.4.2 δ δ δ δv δv δv time (a) Position time (b) Velocit Figure 8: State transition of the linearied spacecraft motion with impulsive maneuvers of equal intervals for N = 1 over the first period

15 because the more frequentl the maneuvers are applied, the tighter the regulation is and the quicker the spacecraft converges to the reference halo orbit N=1 N=4 N=2 v time Figure 9: Magnitude and timing of the impulsive maneuvers; a unit speed in the dimensionless scale is about 43m/s in the sun-earth scale CONCLUSIONS AND REMARKS In this paper, we presented a method for autonomous regulation of spacecraft motion about unstable periodic orbits using occasional impulsive maneuvers. A formula for impulsive maneuvers was obtained so that the mimic Generalied Sampled-data Hold Function (GSHF) control which can be easil designed to stabilie the motion about a reference periodic orbit through monodrom matri assignment. A design technique for GSHF control was summaried briefl. The formula produces the impulsive maneuver to be applied at each time b modulating the difference between the actual state and the reference state at the time. Although timing and frequenc of impulsive maneuvers can be arbitraril chosen, the should be chosen b trial and error so that satisfactor control performance is achieved. The more frequentl the maneuvers are applied, the more similar the effect of the maneuvers are to that of continuous GSHF control on which the maneuver rule is based and in general the better it is in performance and control efficienc. As advanced implementation, it is also possible to change the timing of maneuvers during operation. Although not treated in this paper, we note that it is possible to obtain a maneuver rule based on Event-driven GSHF control to achieve more robust stabiliation with respect to parametric uncertainties (See Ref. 11). The presented method was illustrated b regulation of spacecraft motion about a halo orbit in the Hill s circular three-bod problem and the simulation results show ver well the effectiveness of the method. REFERENCES [1] G. Góme, J. Llibre, R. Martíne, and C. Simó, Dnamics and Mission Design Near Libration Points Vol. I. World Scientific Monograph Series, World Scientific Publishing, 21. [2] G. Góme, A. Jorba, C. Simó, and J. Masdemont, Dnamics and Mission Design Near

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