Spacecraft Formation Flying Near Sun-Earth L2 Lagrange Point: Trajectory Generation and Adaptive Output Feedback Control

2005 American Control Conference ThB04.5 June 8-10, 2005. Portland, OR, USA

Spacecraft Formation Flying near Sun-Earth L2 Lagrange Point: Trajectory Generation and Adaptive Output Feedback Control

Hong Wong and Vikram Kapila Mechanical, Aerospace, and Manufacturing Engineering, Polytechnic University, Brooklyn, NY 11201

Abstract— In this paper, we present a trajectory gen- Spacecraft trajectory designs for single spacecraft eration and an adaptive, output feedback control design missions near the Sun-Earth Lagrange points include methodology to facilitate spacecraft formation flying near Lyapunov and Halo orbits [16], [18], [19]. These periodic the Sun-Earth L2 Lagrange point. Specifically, we create trajectories have the characteristic that spacecraft do a spacecraft formation by placing a leader spacecraft on a not require fuel to stay on these orbits. Thus, these desired Halo orbit and a follower spacecraft on a desired periodic trajectories are well suited as locations for a quasi-periodic orbit surrounding the Halo orbit. We develop leader spacecraft and a formation of follower spacecraft the nonlinear dynamics of the follower spacecraft relative can be placed in the vicinity of these trajectories. to the leader spacecraft, wherein the leader spacecraft is Current literature for formation design near the Sun- assumedtobeonadesiredHaloorbittrajectory.Inaddition, Earth L2 Lagrange point is scarce, with the exception of we design a formation maintenance controller such that the [9], [13], [17]. In [13], reference trajectories for follower follower spacecraft tracks a desired trajectory. Specifically, spacecraft are computed using classical orbital elements, we design an adaptive, output feedback position tracking resulting in bounded orbits around the leader spacecraft controller, which provides a filtered velocity measurement on a periodic orbit. In [17], feedback control is utilized to and an adaptive compensation for the unknown mass of the produce reference trajectories for follower spacecraft. In follower spacecraft. The proposed control law is simulated addition, [9] provides a method of generating reference for the case of the leader and follower spacecraft pair and is trajectories for follower spacecraft using a numerical shown to yield semi-global, asymptotic convergence of the method, where the resulting trajectories are quasi- relative position tracking errors. periodic. Current approaches for spacecraft control near the I. Introduction L2 Lagrange point require position and velocity sensors for feedback control purposes [7], [8], [11], [12], [14]. Equilibrium positions in the restricted three body However, exploiting the nonlinear, adaptive, output problem (RTBP) of the Sun-Earth system, known as the feedback control design methodologies of [3], [6], [15] to Lagrange points, have been exploited as key locations control spacecraft near the L2 Lagrange point eliminates for space-based astronomical observation stations [1], the need for velocity sensors, thus reducing the cost and [10]. As seen in Figure 1(a), Lagrange points L1,L2, mass of the spacecraft. and L3 are collinear with the Sun and Earth while L4 In this paper, we develop a leader-follower spacecraft and L5 each combined with the Sun and Earth yields formation, where the leader spacecraft is on a periodic, an equilateral triangle. A primary benefit of operating Halo orbit around the L2 Lagrange point in the Sun- observation stations in the vicinity of the Lagrange Earth system and the follower spacecraft is to track a points is that spacecraft near these points obtain nearly desired relative trajectory. Specifically, in Section II, we an unobstructed view of the galaxy, unhindered by the develop the dynamics of the follower spacecraft relative atmospheric and geomagnetic forces. to the leader spacecraft. Next, in Section III, we design a Spacecraft formation flying (SFF) has the potential to desired quasi-periodic relative trajectory for the follower enhance space-based imaging/interferometry missions spacecraft in the spirit of [9]. In contrast to [9], our by distributing mission tasks (usually conducted by trajectory design exploits the analytical properties of a monolithic spacecraft) to many small spacecraft. the quasi-periodic relative trajectories to characterize Incorporating this technology into future space mis- spacecraft formations using a parameter set. In Section sions near the Sun-Earth Lagrange points can enlarge IV, we formulate a trajectory tracking control problem. the sensing aperture and increase versatility of future In Section V, we develop an adaptive, output feedback observation platforms. However, effective utilization of control algorithm to enable the follower spacecraft to this new technology requires proper design of spacecraft track this desired quasi-periodic relative trajectory. formations and for each spacecraft in the formation In Section VI, we provide illustrative simulations to to be precisely controlled to maintain a meaningful demonstrate the efficacy of the proposed trajectory gen- baseline. eration and control design schemes. Finally, in Section VII, we give some concluding remarks. Research supported in part by the National Aeronautics and Space Administration–Goddard Space Flight Center II. System Model under Grant NGT5-151 and the NASA/New York Space In this section, we develop a nonlinear model charac- Grant Consortium under Grant 39555-6519. terizing the position dynamics of the follower spacecraft 0-7803-9098-9/05/$25.00 ©2005 AACC 2411 relative to the leader spacecraft near the L2 Lagrange spacecraft dynamics of (1), with the control thrust u point in the Sun-Earth system. Referring to Figure 1, set to zero. First, the Poincaré-Lindstedt method is we assume that the Earth and the Sun rotate in a used to find a high order analytic approximation to circular orbit around the Sun-Earth system barycenter a periodic trajectory in the neighborhood of the L2 (center of mass) with a constant angular speed ω. Lagrange point. Next, the initial conditions, based on In addition, we attach an inertial coordinate system the Poincaré-Lindstedt method, are used as an initial {X, Y, Z} to the Sun-Earth system barycenter and a seed in a numerical algorithm to find a better set of {x ,y ,z } rotating, right-handed coordinate frame L2 L2 L2 initial conditions leading to a periodic trajectory. This x to the L2 Lagrange point with the L2 -axis pointing numerical algorithm applies a Taylor series expansion z along the direction from the Sun to the Earth, the L2 - to the spacecraft states with respect to the initial axis pointing along the orbital angular momentum of conditions and time and truncates higher order terms, y the Sun-Earth system, and the L2 -axis being mutually such that for Halo orbits the result is a set of 3 linear x z perpendicular to the L2 and L2 axes, and pointing in equations with 4 unknown variables. Families of orbits the direction that completes the right-handed coordi- can be characterized by fixing one of the unknown nate frame. variables so that the result gives an equal number of equations to unknowns. Solving the aforementioned A. Dynamics of a Spacecraft Relative to the L2 La- linear matrix equation and using the result to update grange Point the previous set of initial conditions, we obtain a new initial condition guess. In order to describe the dynamics of a spacecraft The spacecraft dynamics are then propagated using formation near the L2 Lagrange point, we must first the new updated set of initial conditions to verify describe the dynamics of a spacecraft relative to the L2 T trajectory periodicity. If the trajectory is sufficiently q t xyz ∈ R3 Lagrange point. To do so, let ( ) = [ ] close to being periodic, then the initial conditions denote the position vector from the spacecraft to the {x ,y ,z } can be used for further simulation, else the above L2 Lagrange point, expressed in the L2 L2 L2 numerical algorithm is used to solve for a new set of R t ∈ R3 coordinate frame. In addition, let S→s( ) and initial conditions. Since the collinear Lagrange points R t ∈ R3 E→s( ) denote the position vectors from the are inherently unstable [19], long-term propagation of Sun and Earth, respectively, to the spacecraft. Finally, spacecraft dynamics using the initial conditions ob- R R R let L2 , E,and S denote the distances between tained in the above manner is futile. However, by the Sun-Earth system barycenter and the L2 Lagrange exploiting the symmetry property of Halo orbits (see point, the Earth, and the Sun, respectively. Then, below), we can artificially obtain a periodic orbit by the mathematical model describing the position of a computing trajectory information during half of a pe- spacecraft relative to the L2 Lagrange point is given by riod and reusing this trajectory data throughout other [21] simulations. mq¨+ Cq˙ + N(q, s) = u, (1) Halo orbits are classified as periodic trajectories that {x ,z } × are symmetric with respect to the L2 L2 plane (i.e., where m is the mass of the spacecraft, C ∈ R3 3 is y 0 −10 L2 = 0), and are not confined to be in the orbital C mω 100 plane of the Sun and Earth. Halo orbits have the a Coriolis-like matrix defined as = 2 000, 3 distinguishing characteristic that their projections on N ∈ R is a nonlinear term consisting of gravitational {y ,z } the L2 L2 plane are curves that resemble a Halo. effects and inertial forces T In this paper, we let q (t)=[x y z ] ∈ R3 denote ⎡ µ (x+R +R ) µ (x+R −R ) ⎤ H H H H S L2 S E L2 E − ω2 x R R 3 + R 3 ( + L2 ) the position vector from a point on a Halo orbit to ⎢ S→s E→s ⎥ the L Lagrange point, expressed in the {x ,y ,z } ⎢ ⎥ 2 L2 L2 L2 µS y µE y 2 coordinate frame. An initial seed for the numerical N m⎢ 3 + 3 − ω y ⎥, = ⎣ RS→s RE→s ⎦ algorithm of [19] consists of a spacecraft starting on {x ,z } y µS z µE z the L2 L2 plane with a nonzero initial L2 and 3 + 3 T RS→s RE→s z q x z L2 velocity (i.e., H(0) = [ H(0) 0 H(0)] and T 3 q y z x and u(t) ∈ R is the thrust control input to the ˙H(0) = [0 ˙H(0) ˙d(0)] ). Updates to the initial L2 y spacecraft. Furthermore, the constants µE and µS in the position and L2 velocity contribute to finding a closed N µ GM µ GM z definition of are defined as E = E and S = S, periodic trajectory. In addition, the initial L2 position respectively, where G is the universal gravitational determines the size of the Halo orbit. Figure 1(b) shows constant, ME is the mass of the Earth, and MS is the a typical Halo orbit trajectory around the L2 Lagrange mass of the Sun. point. In this paper, we use Halo orbits as the reference B. Halo Orbit Trajectory trajectory for the leader spacecraft. The control design In this subsection, we describe a method to gen- framework of [21] can be employed to ensure that the spacecraft dynamics of (1) tracks a Halo orbit reference erate thrust-free, periodic trajectories around the L2 Lagrange point in the form of Halo orbits. We present a trajectory. In a subsequent subsection, we will describe succinct overview of a numerical algorithm to generate the dynamics of the follower spacecraft relative to the leader spacecraft on the Halo orbit. Finally, we denote these periodic trajectories. Additional details on the R t ∈ R3 R t ∈ R3 generation of these periodic trajectories can be found S→H( ) and E→H( ) as the position in [16], [18], [19]. vectors from the Sun and the Earth, respectively, to One numerical method [19] of generating thrust- the Halo orbit. Remark 2.1: The Halo orbit trajectory satisfies the free periodic orbits around the L2 Lagrange point in the Sun-Earth system involves finding a proper set of spacecraft dynamics of (1) under the condition that the position and velocity initial conditions to propagate the spacecraft control input is zero. Moreover, we express 2412 the leader spacecraft dynamics on the Halo orbit as III. Spacecraft Formation Design mq¨ + Cq˙ + N(q , H) = 0. (2) In this section, we exploit [9] to develop a method of H H H designing reference trajectories for the follower space- We note that the Halo orbit is a periodic trajectory craft relative to the leader spacecraft on the Halo orbit with a frequency denoted as ωH. trajectory. Specifically, we present a method of designing quasi-periodic orbits around a nominal Halo orbit. C. Follower Spacecraft Dynamics These quasi-periodic orbits will be used as the desired trajectories for the follower spacecraft. Furthermore, In this subsection, we describe the dynamics of the we will exploit special characteristics of these quasi- follower spacecraft relative to the leader spacecraft periodic orbits to parameterize spacecraft formations tracking a no-thrust, periodic Halo orbit trajectory q about the leader spacecraft on the Halo orbit. H without deviating from this orbit for all time. To We begin by expressing the relative position dynamics describe the dynamics of the follower spacecraft, we 3 of (5) in a state-space form, i.e., let x1(t) ∈ R be express the position vector of the follower spacecraft rel- 3 defined as x1 qf and x2(t) ∈ R be defined as x2 q˙f . ative to the L2 Lagrange point in the coordinate frame = = T Then (5) can be written as {x ,y ,z } q t x y z ∈ R3 L2 L2 L2 as fL ( )= fL fL fL . 2 2 2 2 x x In addition, we denote R → (t) ∈ R3 and R → (t) ∈ ˙ 1 2 S sf E sf X˙ = = − , (7) R3 f x −m 1 C x N x , as the position vectors from the Sun and Earth, ˙ 2 f ( f 2 + f ( 1 sf )) respectively, to the follower spacecraft. Using (1), the T X t xT xT ∈ R6 follower spacecraft dynamics relative to the L2 Lagrange where f ( ) = 1 2 and we assume that point can be expressed as uf =0,∀t ≥ 0. Next, we linearize the nonlinear terms on the right hand side of (7), in the neighborhood of m q C q N q , u , f ¨fL2 + f ˙fL2 + fL2 ( fL2 sf )= f (3) Xf =0,toobtain m C ∈ where f is the mass of the follower spacecraft, f X˙ f = AXf , (8) R3×3 C m ω is a Coriolis-like matrix defined as f = 2 f × 0 −10 where A(t) ∈ R6 6, defined as 100 3 · N ∈ R 03 I3 000, fL2 is a nonlinear term consisting A −1 dNf (x1,sf ) −1 = −m x =0 −m Cf ,isatimevarying of gravitational effects and inertial forces defined as f dx1 1 f N mf N q , u t ∈ R3 matrix with elements that are periodic with time. fL2 = m ( fL2 sf ), and f ( ) is the thrust control Note that 03 denotes the 3 × 3 zero matrix, I3 denotes input to the follower spacecraft. dNf (x1,sf ) × x Next, we define the relative position between the the 3 3 identity matrix, and dx1 1=0 denotes q t ∈ R3 q q × N x , follower and the leader spacecraft f ( ) as f = fL2 the 3 3 Jacobian matrix of f ( 1 sf ) evaluated at −q . To obtain the dynamics of the follower spacecraft x1 = 0. The period of oscillation of A is the same H A relative to the leader spacecraft, we differentiate qf with as the period of the nominal Halo orbit, i.e., is respect to time twice and multiply both sides of the periodic with a frequency ωH. Furthermore, the time A resulting equation by mf to produce dependence of characterizes the dynamics resulting from the linearization of (7) as a nonautonomous, m q m q − m q . A f ¨f = f ¨fL2 f ¨H (4) linear differential equation with a periodic matrix. Consequently, we employ Floquet theory [4] to q Next, we solve for ¨H in (2), multiply the resulting transform (8) into an autonomous, linear differential m equation by f , and substitute the result into (4) to equation so as to facilitate an explicit solution of (8). yield We begin by introducing the notion of a fundamental matrix [4] of (8) denoted as ϕ(t) ∈ R6×6.Next,we mf q¨f + Cf q˙f + Nf (qf , sf )=uf , (5) denote the Halo orbit period as TH. Using Floquet 3 theory, we utilize the transformation where (3) has been used. Note that Nf ∈ R is a non- N N q , − N q , −1 linear term defined as f = fL2 ( fL2 sf ) H( H H), Xf = PYf ,Yf = P Xf , (9) N ∈ R3 N mf N q , where H is defined as H = m ( H H). 6 where Yf (t) ∈ R is a vector composed of the trans- Remark 2.2: The Coriolis matrix Cf satisfies the × T 3 formed state X and P (t) ∈ R6 6 is a matrix with skew-symmetric property of x Cf x =0,∀x ∈ R . f Remark 2.3: The left-hand side of (5) produces an elements that are periodic with time [4], to transform the nonautonomous differential equation of (8) into affine parameterization mf q¨f + Cf q˙f + Nf (qf , sf )= Y q , q ,q , m m (¨f ˙f f sf ) f ,where f is the unknown, constant Y˙ = BY , (10) mass of the follower spacecraft and Y (·) ∈ R3 is a f f regression matrix defined as where B ∈ R6×6 is a constant matrix. Following [4], the B ϕ T T matrix can be computed using and H as follows 1 − Y [Y1 Y2 Y3] , (6) B ϕ 1 ϕ T = = TH log (0) ( H) , where the log function Y ,Y ,Y ∈ R Y x denotes the logarithm of a matrix. Furthermore, the where 1 2 3 are defined as 1 = ¨f µ x x R R µ x x R −R P matrix can be computed using ϕ and B as follows 2 S( f + H+ L2 + S) E( f + H+ L2 E) −Bt −2ωy˙f − ω xf + R 3 + R 3 P t ϕ t e P S→sf E→sf ( )= ( ) . Note that the matrix is nonsingular µ x R R µ x R −R S( H+ L2 + S) E( H+ L2 E) ∀t ∈ R, such that the transformation of (9) is unique − 3 − 3 , Y y¨ +2ωx˙ RS→H RE→H 2 = f f [5]. −ω2y µS(yf +yH) µE(yf +yH) − µS yH − µE yH f + R 3 + R 3 R 3 R 3 , The autonomous, linear differential equation of (10) is S→sf E→sf S→H E→H µS(zf +zH) µE(zf +zH) µS zH µE zH equivalent to (8) in the transformed set of coordinates. and Y3 z¨f + R 3 + R 3 − R 3 − R 3 , B = S→sf E→sf S→H E→H Furthermore, the eigenvalues of the matrix are de- respectively. noted as the characteristic exponents [5], which describe 2413 the stability characteristics of any trajectory that is Remark 3.1: To facilitate subsequent illustrative ex- sufficiently near the nominal Halo orbit. It is observed amples, we approximate the Halo orbit and the P in [17] that direct computation of the eigenvalues of matrix using Fourier series approximations. Since both B results in a pair of hyperbolic eigenvalues, a pair of qH and P are periodic with the same period, the zero eigenvalues, and a pair of nonzero, pure, imaginary resulting Fourier series approximations are convergent eigenvalues. We denote the pair of hyperbolic eigenval- to the actual forms of qH and P . To compute the time λ λ q P ues as h1 and h2 and the frequency corresponding to derivatives of H and , we analytically differentiate the nonzero, pure, imaginary eigenvalues as ωQ.Next, the Fourier series approximations with respect to time. q we perform a coordinate transformation of the form Thus, it follows that... df and its time derivatives, viz q q q X˙ X¨ ., ˙df ,¨df ,and df or equivalently f and f , Yf = TZf, (11) are computed using qH, P ,andZf ,andtheirtime 6 where Zf (t) ∈ R is a vector composed of the trans- derivatives, i.e., Y T ∈ R6×6 formed state f and is a time indepen- X˙ PTZ˙ PTZ˙ , X¨ PTZ¨ PT˙ Z˙ PTZ¨ , dent, linear transformation matrix, which transforms f = f + f f = f +2 f + f (14) the B matrix into a modal matrix form given by where (13) has been used. 01 0101 Ω=diag , −λ λ λ λ , −ω2 . 00 h1 h2 ( h1 + h2 ) Q 0 IV. Trajectory Tracking Problem Formulation Then, (10) is transformed into Z˙ =ΩZ . In this section, we formulate a control design problem f f q Now it is trivial to obtain the following solution for such that the follower spacecraft relative position f q Zf analytically tracks a desired relative position trajectory df , i.e., lim q (t) − q (t) = 0. The effectiveness of this control T t→∞ f df Z = [Zf1 Zf2 Zf3 Zf4 Zf5 Zf6 ] ,Z (t) ∈ R,i=1,...,6, f fi objective is quantified through the definition of a (12) e t ∈ R3 Z Z Z t, Z Z ,Z position tracking error ( ) as where f1 = f1 (0) + f2 (0) f2 = f2 (0) f3 = −λh Zf (0)+Zf (0) λ t λh Zf (0)−Zf (0) λ t 2 3 4 e h1 1 3 4 e h2 ,Z e q − q . λ −λ + λ −λ f4 = f df (15) h1 h2 h1 h2 = −λh Zf (0)+Zf (0) λ t λh Zf (0)−Zf (0) λ t 2 3 4 λ e h1 1 3 4 λ e h2 The goal is to construct a control algorithm that λ −λ h1 + λ −λ h2 , h1 h2 h1 h2 Z D ω t φ Z − Dω ω t φ obtains the aforementioned tracking result in the pres- f5 = cos( Q + ), and f6 = Q sin( Q + ), ence of the unknown constant follower spacecraft mass Z i ,..., ith fi (0), =1 6, denotes the initial condition m . We assume that the velocity measurements of the Z D, φ ∈ R f of the vector f ,and are parameters that follower spacecraft relative to the leader spacecraft on a characterize size, location, and shape of the relative nominal Halo orbit are not available for feedback, i.e., trajectory around the nominal Halo orbit. Eq. (12) q˙ is unknown. Z f reveals that the general solution of f may not be To facilitate the control development, we assume periodic for arbitrary initial conditions. However, by that the desired trajectory q and its first three time Z df properly choosing the initial condition f (0) the terms derivatives are bounded functions of time. Next, we corresponding to the pair of zero eigenvalues and the define the follower spacecraft mass estimation error hyperbolic eigenvalues that produce unstable and/or m˜ f (t) ∈ R as asymptotically stable motion can be eliminated, thus Z m m − m , resulting in periodic motion for f . The remaining ˜ f = ˆ f f (16) periodic terms in (12) allow the trajectory designer Z D φ wherem ˆ f (t) ∈ R is the follower spacecraft mass freedom to choose the parameters f1 (0), ,and to satisfy mission specifications. estimate. To compute the follower spacecraft trajectory relative V. Adaptive Output Feedback Position Tracking to the nominal Halo orbit requires transformation from Controller Zf −→ Xf in the form of X PTZ , In this section, we design a desired compensation f = f (13) adaptation control law (DCAL) [3] that asymptoti- where (9) and (11) have been used. Note that the P cally tracks a pre-specified follower spacecraft rela- matrix is composed of elements which are periodic with tive position trajectory, despite the unknown constant follower spacecraft mass m and the lack of follower respect to time, with frequency ωH, whereas the solution f of Z is composed of elements which are periodic with spacecraft relative velocity measurements. In order f to state the main result of this section, we define respect to time, with frequency ωQ. Consequently, the X auxiliary error variables ϑ(t) ∈ R9 and r(t) ∈ R10 solution of f is a trajectory with two frequency com- T T ponents ωH and ωQ. It is observed that the frequencies ϑ eT eT ηT r eT eT ηT m as = f and = f ˜ f , ωQ and ωH are linearly independent, i.e., the condition a ω a ω ai ∈ Z i , Z respectively. In addition, we define positive con- 1 Q + 2 H =0, , =12, where is the 1 − λ λ kη λ { ,m , 1} set of integers, holds only for ai =0,i =1, 2(see[2] stants 1, 2,and as 1 = 2 min 1 f Γ , 1 − λ { ,m , 1} kη m k − − for details on linearly independent frequencies). Such 2 = 2 max 1 f Γ ,and = f ( 1) 1, re- a trajectory containing linearly independent frequency spectively. Finally, we define a new regression matrix Y · ∈ R3 Y · Y q , q ,q , components is termed as a quasi-periodic trajectory (see d( ) as d( ) = (¨df ˙df df sdf ), where the [2] for details on quasi-periodic functions). Thus, the Xf linear parameterization of Remark 2.3 has been used. Y R t ∈ R3 trajectory has the characteristic of being quasi-periodic. Note that in the definition of d, S→sd ( ) and X f Finally, we utilize f as the desired trajectory of the R → (t) ∈ R3 are denoted as the position vectors q t ∈ R3 E sdf follower spacecraft relative to the Halo orbit df ( ) , from the Sun and Earth, respectively, to the desired qT qT T X i.e., df ˙df = f . trajectory of the follower spacecraft. 2414 A. Velocity Filter Design C. Stability Analysis To account for the lack of follower spacecraft relative In this subsection, we present the main theorem to velocity measurements viz.,˙qf , a filtered velocity error ensure semi-global, asymptotic stability of the position 3 signal ef (t) ∈ R is produced using a filter. The tracking error. following design is based on the framework of [3]. The Theorem 5.1: Let k ∈ R be a constant, positive, filter is constructed using the position tracking error e control gain and Γ ∈ R be a positive constant. Then, as an input, as shown below the adaptive, output feedback control law consisting of (17), (18), and ef = −ke + p, (17) uf = Ydmˆ f + kef − e, (26) t k> p t ∈ R3 T T where 0 is a positive, constant filter gain, ( ) mˆ =ˆm (0) − Γ Y (σ)(e(σ)+ef (σ)) dσ − ΓY e is a pseudo-velocity tracking error generated using f f d d 0 t dY T σ p − k p k2 e, p ke . Y T e d ( )e σ dσ, ˙ = ( +1) +( +1) (0) = (0) (18) +Γ d (0) (0) + Γ dσ ( ) (27) 0 To obtain the closed-loop dynamics of ef ,wetake ensures semi-global asymptotic convergence of the po- the time derivative of the filtered velocity error signal sition tracking errors as delineated by lim e(t)=0,ifk ef and replace the dynamics ofp ˙(t) from (18) to obtain t→∞ 1 2 λ2 is selected such that kη > ρ r(0) where ρ(·) e −kη − e e, 4 λ1 ˙f = f + (19) is a nondecreasing function. Proof. We begin by substituting (26) into (24) to η t ∈ R3 where ( ) is an auxiliary tracking error variable obtain the closed-loop dynamics for η defined as m η Y m ke − e − C η − m k − η X . η e e e . f ˙ = d ˜ f + f f f ( 1) + (28) = +˙+ f (20) Next, differentiating (16) with respect to time and using In addition, rearranging the definition of η gives the (27) and (20), we obtain the closed-loop dynamics for closed-loop error dynamics fore ˙(t) the spacecraft mass estimation error m˙ m˙ − Y T σ η. e˙ = η − e − ef . (21) ˜ f = ˆ f = Γ d ( ) (29) We define a positive-definite, candidate Lyapunov func- 1 T 1 T 1 T 1 − B. Open-Loop Auxiliary Tracking Error Dynamics V e ef e e m η η 1m2 tion as = 2 f + 2 + 2 f + 2 Γ ˜ f . Applying In this subsection, we develop the open-loop dynamics Rayleigh-Ritz’s theorem on V results in η of the auxiliary tracking error variable .Webegin λ ϑ2 ≤ λ r2 ≤ V ≤ λ r2. by differentiating η of (20) with respect to time, 1 1 2 (30) multiplying both sides of the resulting equation by mf , V e e Next, differentiating with respect to time and sub- substituting for ˙f from (19) and ˙ from (21), and stituting the closed-loop dynamics of (19), (21), (28), rearranging terms to yield and (29) into the result, we obtain m η m q − m q − m k − η − m e , ˙ T T T T f ˙ = f ¨f f ¨df f ( 1) 2 f f (22) V = −ef ef − e e − mf (k − 1)η η + η X , (31) where the definition of (15) has been used. Next, we where the skew-symmetry property of Remark 2.2 has been used. In addition, utilizing the upper bound on X substitute mf q¨f from (5) into (22) and rearrange terms to obtain to upper bound (31), we get V˙ ≤−ϑ2 − k η2 ρ ϑ ϑη, m η −m q − C q u − C e − N q , η + ( ) (32) f ˙ = f ¨df f ˙df + f f ˙ f ( f sf ) kη −mf (k − 1)η − 2mf ef , (23) where the definition of has been used. Bounding the last two terms on the right hand side of (32) by q e q where ˙f has been replaced by ˙ +˙df .Weaddand completing the squares results in N q , subtract f ( df sdf ) to and from the right hand side of ρ2 (ϑ) (23) and substitutee ˙ from (21) to write the open-loop V˙ ≤− 1 − ϑ2. (33) dynamics of η as follows 4kη 2 ρ (ϑ) kη kη > mf η˙ = −Ydmf + uf − Cf η − mf (k − 1)η + X , (24) Note that if is chosen such that 4 ,then V˙ is negative semidefinite, i.e., Y where the definition of the desired regression matrix d 2 has been used and X (t) ∈ R3 is defined as V˙ ≤−βϑ , (34) where β is some positive constant defined as β 1 X C e e N q , −N q , − m e . 2 = = f ( + f )+ f( df sdf ) f( f sf ) 2 f f (25) − ρ (ϑ) 4kη . Utilizing (30) yields a sufficient condition for Remark 5.1: For X defined in (25), a boundedness (34) as follows condition is required such that a stability result can be ⎛ ⎞ formulated. Using the mean value theorem, as in [15], V t 2 1 2 ⎝ ( )⎠ X X ≤ ρ ϑ ϑ V˙ ≤−βϑ ,kη > ρ . we can bound as follows ( ) ,where λ (35) ρ(·) is some nondecreasing function. 4 1 2415 V V˙ q Since is a non-negative function and is a negative To show the resulting trajectories of df ,givendiffer- V Z D φ semi-definite function, is a non-increasing function. ent numerical values for parameters f1 (0), ,and , V ∈L V r t ≤ V r < ∞ q Z Thus, ∞ as described by ( ( )) ( (0)) , we simulated df using a parameter set: f1 (0) = 0, t ≥ 0. Using (30) and V (r(t)) ≤ V (r(0)), we obtain a D =0.0001, and φ = 0 rad. By computing the sufficient condition for (35) eigenvalues of the B matrix, we determined ωQ = 6.286301816644046 × 10−5 1 . Figure 2(a) shows the 1 λ day V˙ ≤−βϑ2,k> ρ2 2 r . quasi-periodic trajectory relative to the nominal Halo η (0) (36) π 4 λ1 orbit for parameter values of φ =0,φ = ,and φ π φ 4 = 2 . Figure 2(a) illustrates that changes in denote From V ∈L∞, we know that ef ,e,η,m˜ f ∈L∞.Since e, e ,η ∈L e ∈L changes in the initial position of the spacecraft along f ∞, it follows from (20) that ˙ ∞; hence, a given quasi-periodic trajectory. Next, we simulated q q due to the bound of df ,˙df , we can use (15), (19), q using a parameter set: Z (0) = 0, D =0.0002, q , q , e , η ∈L df f1 (20), (25), and (28) to conclude that f ˙f ˙f ˙ ∞. and φ = 0 rad. Figure 2(b) shows the desired quasi- Similar signal chasing arguments can now be employed periodic trajectory relative to the nominal Halo orbit. to show that all other signals in the closed-loop system Note that the parameter D determines the size and remain bounded. Using (34), it can be easily shown shape of the desired quasi-periodic trajectory relative that e, ef ,η ∈L .Sincee, ef ,η ∈L∞, using Barbalat’s q 2 to the nominal Halo orbit. We also simulated df using a Lemma [6], we conclude that lim e(t),ef (t),η(t)=0. Z . D φ t→∞ parameter set: f1 (0) = 0 0001, =0,and =0rad. Thus, the result of Theorem 5.1 follows. For this parameter set, Figure 3(a) shows a periodic trajectory relative to the nominal Halo orbit with the ω q VI. Simulation Results same period as H. Finally, we simulated df using a Z . D . φ In this section, we present illustrative examples that parameter set: f1 (0) = 0 0001, =00001, and =0 incorporate the algorithms presented in Sections III rad. For this parameter set, Figure 3(b) shows the quasi- and V. Specifically, we provide details on computing periodic trajectory relative to the nominal Halo orbit. the quasi-periodic trajectories described in Section III. Next, we provide a simulation of the follower spacecraft B. Adaptive Output Feedback Control of the Follower relative dynamics (5), utilizing the control laws of (17), Spacecraft (18), (26), and (27) so that the follower spacecraft tracks The control law consisting of (17), (18), (26), a desired quasi-periodic trajectory relative to a nominal and (27) was simulated for the follower spacecraft Halo orbit. dynamics relative to the leader spacecraft on a In all simulations, we employ the Sun-Earth sys- nominal Halo orbit (5). When tracking desired tem circular orbit parameters [19], [20]: G =6.671 3 quasi-periodic trajectories, we initialized the follower ×10−11 m , ω =2.73774795629 × 10−3 rad , M = kg·s2 day S spacecraft with the set of initial conditions given 30 24 as qf (0) = [−2.61921376240742 − 2.57780484325713 1.9891 × 10 kg, ME =5.974 ×10 kg, 1 AU = 5 . × 8 R . −0.13648677396294] × 10 km and q˙f (0) = 1 496 10 km, and L2 =1010033599267463 AU, − . . − . where 1 AU stands for 1 Astronomical Unit denoting the [ 0 1469110370264 42 1353617291110 0 1469092330256] ×102 km distance between the Sun and the Earth. Furthermore, day. The control and adaptation gains are we consider that the follower spacecraft has a mass obtained through trial and error in order to obtain of mf = 1000kg. Finally, the distances RS and RE M good performance for the tracking error response. The R E × R can be computed as S = M +M 1AU and E = following resulting gains were used in this simulation M E S k . . × 5 S =4497 and Γ = 9 3 10 . In addition, the follower M M × 1AU, respectively. E+ S spacecraft mass parameter estimate was initialized to mˆ f (0) = 600 kg. A simulation of the follower spacecraft A. Quasi-Periodic Trajectory Generation tracking the desired quasi-periodic trajectory of Figure Applying the numerical algorithm presented in 2(a) is performed. The trajectory qf is shown in Figures Subsection II-B results in a family of initial con- 4(a) and 4(b). Figure 5 shows the position tracking ditions for the Halo orbit from which we have error e and the pseudo-velocity error ef .Thecontrol q selected the following initial condition H(0) = input uf is shown in Figure 6(a). Finally, the follower 5 [−2.61921376240742 0 −0.13648677396294] × 10 km spacecraft mass estimatem ˆ f is shown in Figure 6(b). andq ˙ (0) = [0 4.21353617291110 0] ×103 km .In H day addition, the Halo orbit period is determined to be VII. Conclusion 3 TH =1.135225027876099 × 10 day. Figure 1(b) shows In this paper, we designed desired quasi-periodic tra- the Halo orbit relative to the L2 Lagrange point and jectories for the follower spacecraft relative to the leader {x ,y } {x ,z } its projections onto the L2 L2 , L2 L2 ,and spacecraft on the Halo orbit. The size, location, and {y ,z } L2 L2 planes. We utilized 25 terms of a Fourier shape of these trajectories were characterized by a pa- series to approximate the Halo orbit trajectory qH. rameter set. Illustrative simulations were performed to The fundamental matrix ϕ describedinSectionIIIis show these parameter characteristics. Next, a Lyapunov numerically computed using A(t) as followsϕ ˙ = A(t)ϕ, design was used to develop an adaptive, output feed- −Bt ϕ(0) = I6, ∀t ∈ [0,TH]. Thus, using P (t)=ϕ(t)e , P back controller, which yielded semi-global, asymptotic is numerically computed ∀t ∈ [0,TH]. Next, we compute convergence of the relative position tracking errors. The a Fourier series approximation of P , where we retain 25 control law required only position error measurements terms of the series approximation. This is used with the while estimating velocity error measurements through Z q analytic expression for f to compute df and its time a filtering scheme. Simulation results were presented to derivatives analytically from (14). show good trajectory tracking. 2416 References [1] http://nssdc.gsfc.nasa.gov/space/isee.html, website of Inter- national Sun-Earth Explorers Project Information. [2] H. Bohr, Almost Periodic Functions. Springer, Berlin, 1933. [3] T. Burg, D. Dawson, J. Hu, and M. de Queiroz, “An Adaptive Partial State Feedback Controller for RLED Robot Manipulators,” IEEE Transaction on Automatic Control, Vol. 41, No. 7, pp. 1024–1031, 1996. [4]C.T.Chen,Linear System Theory and Design.Oxford University Press, Oxford, NY, 1999. [5] E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw Hill, New York, NY, 1955. [6] D. M. Dawson, J. Hu, and T. C. Burg, Nonlinear Control of Electric Machinery. Marcel Dekker, New York, NY, 1998. [7] P. Di Giamberardino and S. Monaco, “Nonlinear Regulation (a)

6 4 in Halo Orbits Control Design,” Proceedings of Conference x 10 x 10 1 2

on Decision and Control, Tuscan, AZ, pp. 536–541, 1992. 1.5 0.5 [8] G. Gomez, J. Llibre, R. Martnez, J. Rodrguez-Canabal, and 1 0.5 C. Simo, “On the Optimal Station-Keeping Control of Halo 0 Orbits,” Acta Astronautica, Vol. 15, No. 6, pp. 391–397, 1987. y (km) z (km) 0 −0.5 −0.5 [9] G. Gomez, J. Masdemont, C. Simo, “Lissajous Orbits around −1

−1 −1.5 Halo Orbits,” AAS/AIAA Space Flight Mechanics Meeting, −3 −2 −1 0 1 2 −1 −0.5 0 0.5 1 5 6 AAS Paper 97-106, 1997. x (km) x 10 y (km) x 10 4 x 10 4 x 10 2 [10] K. C. Howell, B. T. Barden, R. S. Wilson, and M. W. Lo, Actual 1.5 L “Trajectory Design using a Dynamical Systems approach 2 2 1

with Application to Genesis,” Proceedings of the AAS/AIAA 0 0.5 z (km) Astrodynamics Specialist Conference, Sun Valley, ID, AAS z (km) 0 −2 −0.5 Paper 97–709, 1997. 1 2 −1 6 0 0 x 10 5 [11] K. C. Howell and S. C. Gordon, “Orbit Determination Error −2 x 10 −1.5 y (km) −1 −4 x (km) −3 −2 −1 0 1 2 5 Analysis and a Station Keeping Strategy for Sun Earth L1 x (km) x 10 Libration Point Orbits,” Journal of Astronautical Sciences, (b) Vol. 42, pp. 207–228, 1994. [12] K. C. Howell and H. J. Pernicka, “Station-Keeping Method Fig. 1. (a) Sun-Earth system schematic diagram and (b) for Libration Point Trajectories,” Journal of Guidance and Halo orbit trajectory of the leader spacecraft relative to the L2 Control, Vol. 16, pp. 151–159, 1993. Lagrange point [13] F. Y. Hsiao and D. J. Scheeres, “Design of Spacecraft Forma- tion Orbits Relative to a Stabilized Trajectory,” AAS/AIAA Space Flight Mechanics Meeting, AAS Paper 03-175, 2003. Relative Trajectory Initial Condition for φ = 0 [14] T. M. Keeter, Station-Keeping Strategies for Libration Point Initial Condition for φ = π/4 Initial Condition for φ = π/2 Orbits: Target Point and Floquet Mode Approaches, Master’s 200 thesis, School of Aeronautics and Astronautics, Purdue Uni- 150 versity, West Lafayette, IN, 1994. 100 [15] M. S. de Queiroz, D. Dawson, T. Burg, “Position/Force Con- 50 0

trol of Robot Manipulators without Velocity/Force Measure- z (km) ments,” International Journal of Robotics and Automation, −50 Vol. 12, pp. 1–14, 1997. −100 −150 [16] D. L. Richardson, “Analytic Construction of Periodic Orbits −200 about the Collinear Points,” Celestial Mechanics,Vol.22, 2 1 1 pp. 241-253, 1980. 0.5 4 0 x 10 0 [17] D. J. Scheeres, F. Y. Hsiao, and N. X. Vinh, “Stabilizing 4 −1 x 10 −0.5 Motion Relative to an Unstable Orbit: Applications to −2 y (km) −1 Spacecraft Formation Flight,” Journal of Guidance, Control, x (km) and Dynamics, Vol. 26, No. 1, pp. 62–73, 2003. (a)

[18] V. Szebehely, Theory of Orbits. Academic Press, New York, Relative Trajectory NY, 1967. Initial Condition

[19] R. Thurman and P. A. Worfolk, The Geometry of Halo Orbits 300 in the Circular Restricted Three-Body Problem,Geometry 200 Center Research Report GCG95, University of Minnesota, 1996. 100 [20] D. A. Vallado, Fundamentals of Astrodynamics and Appli- 0 z (km) cations. McGraw Hill, New York, NY, 1997. −100

[21] H. Wong and V. Kapila, “Adaptive Nonlinear Control of −200 Spacecraft Near Sun-Earth L2 Lagrange Point,” Proceedings −300 of the American Control Conference, Denver, CO, pp. 1116– 4 2 1.5 1121, 2003. 1 4 0 0.5 x 10 0 4 −2 −0.5 x 10 −1 −4 y (km) −1.5 x (km) (b) Fig. 2. Trajectory of the follower spacecraft relative to the nominal Halo orbit using: (a) Zf (0) = 0,D =0.0001,φ =0,φ = π π 1 4 ,andφ = 2 and (b) Zf1 (0) = 0,D =0.0002,φ=0

2417 Relative Trajectory 4 Initial Condition x 10 2

1 300 0

200 Error x (km) −1

−2 100 0 500 1000 1500 2000 2500 3000 4 x 10 2 0 1 z (km)

−100 0

Error y (km) −1 −200 −2 0 500 1000 1500 2000 2500 3000 4 −300 x 10 4 2

2 2 1 1 4 0 0 x 10 0 4

−2 x 10 Error z (km) −1 −1 −4 y (km) −2 −2 x (km) 0 500 1000 1500 2000 2500 3000 Time (days) (a) (a) Relative Trajectory Initial Condition 400 200

300 0 x (km/day) f

e −200 200 −400 0 500 1000 1500 2000 2500 3000 100 400 0 200 z (km)

−100 0 y (km/day) f

e −200 −200 −400 0 500 1000 1500 2000 2500 3000 −300 4 400

2 2 200 1 4 0 0 x 10 0

4 z (km/day) f

−2 x 10 e −200 −1 −4 y (km) −2 −400 x (km) 0 500 1000 1500 2000 2500 3000 Time (days) (b) (b) Fig. 3. Trajectory of the follower spacecraft relative to the Fig. 5. (a) Position tracking error and (b) ﬁltered velocity error nominal Halo orbit using: (a) Zf (0) = 0.0001, D =0,φ =0 1 signal and (b) Zf1 (0) = 0.0001, D =0.0001, φ =0

−6 Actual trajectory x 10 Initial condition 2

1 150 0

100 Control x (N) −1

−2 0 500 1000 1500 2000 2500 3000 50 −6 x 10 2 0 1 z (km)

−50 0

Control y (N) −1 −100 −2 0 500 1000 1500 2000 2500 3000 −6 −150 x 10 2 2 1 2 1 0 1.5 0 4 −1 1 x 10 0.5 −2 4 0 x 10 Control z (N) −1 −3 −0.5 −4 −2 y (km) −1 x (km) 0 500 1000 1500 2000 2500 3000 time (days) (a) (a) 630

200 620 150

100 610 50

0 z (km) −50 600

−100

−150 590 Mass parameter estimate (kg) −200 2

1 1 580 0.5 4 0 x 10 0 4 −1 x 10 −0.5 −2 570 y (km) −1 x (km) 0 500 1000 1500 2000 2500 3000 time(days) (b) (b) Fig. 4. Trajectory of the follower spacecraft relative to the Fig. 6. Follower spacecraft (a) control input and (b) mass nominal Halo orbit using: (a) normal view and (b) zoomed in parameter estimate view

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