April 2004

Adaptive Nonlinear Control of Spacecraft Near Sun-Earth L2 Lagrange Point by

Hong Wong and Vikram Kapila Department of Mechanical, Aerospace, and Manufacturing Engineering Polytechnic University, Brooklyn, NY 11201 [hwong01@utopia, vkapila@duke].poly.edu

Abstract In this paper, an adaptive control algorithm is presented to enable a spacecraft to track desired trajectories near the L2 Lagrange point in the Sun-Earth system. An adaptive full-state feedback control law, designed using a Lyapunov-type analysis, exhibits globally asymptotic position tracking in the presence of the unknown spacecraft mass. The control law is simulated using a Lyapunov and a Halo orbit in the vicinity of the L2 Lagrange point as the periodic reference trajectory.

Key Words: Adaptive control, trajectory tracking, Lagrange points, Lyapunov orbit, Halo orbit Running Title: Spacecraft Control in Lyapunov and Halo Orbits

1.

Introduction Space-based missions near the Lagrange points of the Sun-Earth system have recently

received significant attention for their strategic benefits in long-term astronomical observations. The Lagrange points are defined as equilibrium positions in the restricted three body problem (RTBP), see Figure 1 for details. The first three Lagrange points in the RTBP (labeled as L1 , L2 , and L3 ) are points that are collinear with the two primary masses. The last two Lagrange points in the RTBP (labeled as L4 and L5 ) are equilibrium points such that each of these points combined with the two primary masses yields an equilateral triangle. Each of the five equilibrium positions can host a spacecraft for an indefinite amount of time. However, in the Sun-Earth system, the collinear Lagrange points are unstable equilibrium points (i.e., the linearization of the spacecraft dynamics on any of the collinear Lagrange points result in a Jacobian matrix that has unstable eigenvalues) thus necessitating stationkeeping control. The Lagrange points L4 and L5 are locally stable (i.e., the linearization of the spacecraft dynamics on either of these Lagrange points result in a Jacobian matrix that has two pairs of distinct, complex conjugate, purely imaginary eigenvalues and two real, negative eigenvalues). Furthermore, periodic and quasi-periodic orbits neighboring the collinear Lagrange points can host an arbitrary number of spacecraft for long time durations using station-keeping control. For precise details on the stability of Lagrange points, see Ref. 1. One of the first space missions to utilize the L1 Lagrange point was the International SunEarth Explorer-3,2 which collected data on solar wind, cosmic-rays, plasma waves, etc., with negligible disturbances from the Earth. The mission duration lasted less than four-years and opened the path for future space exploration to the Sun-Earth Lagrange points. Currently, the Genesis mission resides about the L1 Lagrange point. Genesis mission objectives include obtaining precise solar isotopic measurements and obtaining highly sensitive measurements of solar wind, as described in Ref. 3.

1

Future space missions, such as Ref. 4, that intend to utilize the L2 Lagrange point as the location for deep-space observations and/or interstellar communication have the advantage that solar influences on the spacecraft are minimal and space observations can be conducted on a frequent basis. In contrast, spacecraft that are to perform the same types of missions in either Sun-synchronous or Low Earth Orbits about the Earth are not suitable because these orbits expose the spacecraft to harsh physical conditions (e.g., gravitational and/or atmospheric disturbances, space debris, etc.). Current spacecraft control designs that facilitate the trajectory tracking of periodic or quasi-periodic orbits about the collinear Lagrange points are classified as impulsive and continuous methods. The dominant literature in impulsive spacecraft control near the collinear Lagrange points consists of the Floquet mode approach, wherein tailoring control inputs cancel instability effects for a spacecraft orbiting the Lagrange points,5 – 7 and the target mode approach, wherein impulsive control inputs are generated using the error feedback and by minimizing a cost function.8 – 10 However, the Floquet mode approach suffers from large transient errors that arise from tailoring control input, which “tries” to cancel instability. Additionally, the resulting control design for the target mode approach does not guarantee the spacecraft tracking error to be globally stable for the periodic or quasi-periodic orbit. Alternatively, a nonlinear control algorithm has been presented in Ref. 11 that utilizes the nonlinear dynamics of a spacecraft relative to the L2 Lagrange point to track a quasihalo orbit. Ref. 11 accounts for the eccentricity of the Sun-Earth system by treating the elliptical Sun-Earth motion as perturbations from an ideal circular Sun-Earth motion and by canceling these perturbations as a disturbance force. However, uncertainty in the spacecraft mass has not been accounted for in this approach. In this paper, an adaptive control algorithm is presented to enable a spacecraft to track desired trajectories near the L2 Lagrange point in the Sun-Earth system. First, in Section 2, the mathematical model for a spacecraft relative to the L2 Lagrange point in the Sun-Earth 2

system is described. Next, a trajectory tracking control problem is formulated in Section 3. In Section 4, using a Lyapunov-based approach, a full state feedback control law and a parameter update algorithm are designed, which facilitate the tracking of given reference trajectories in the presence of unknown spacecraft mass. Illustrative simulations are included in Section 5 to demonstrate the efficacy of the proposed controller. Finally, some concluding remarks are given in Section 6.

2.

System Model In this section, we present the nonlinear model characterizing the position dynamics of a

spacecraft relative to the Lagrange point L2 in the Sun-Earth system. We assume that the Earth and the Sun rotate in a circular orbit about the Sun-Earth system barycenter (center of mass). In addition, we consider that an inertial coordinate system {X, Y, Z} is attached to the Sun-Earth system barycenter. Finally, we assume that a rotating right-handed coordinate frame {xb , yb, zb } is attached to the Sun-Earth system barycenter with the xb -axis pointing along the direction from the Sun to the Earth, the zb -axis pointing along the orbital angular momentum of the Sun-Earth system, and the yb -axis being mutually perpendicular to the xb and zb axes, and pointing in the direction that completes the right-handed coordinate frame. The equation of motion of a spacecraft is derived by a direct application of Newton’s second law to the spacecraft, such that the only external forces influencing the spacecraft motion are the gravitational forces from the Sun and the Earth, thus yielding maI = −GmMS

RS→s RE→s − GmME , 3 RS→s  RE→s 3

(1)

where m is the mass of the spacecraft, aI (t) ∈ R3 is the inertial acceleration vector of the spacecraft, G is the universal gravitational constant, MS is the mass of the Sun, RS→s (t) ∈ R3 is the position vector from the Sun to the spacecraft, ME is the mass of the Earth, and 3

RE→s (t) ∈ R3 is the position vector from the Earth to the spacecraft. Next, the inertial acceleration of the spacecraft is expressed in the rotating coordinate frame {xb , yb , zb } as follows (see Ref. 12 for details) aI = ab + Ω × (Ω × rb ) + 2Ω × vb ,

(2)

where we are assuming that the Sun-Earth system is rotating at a constant rate about its barycenter, ab (t) ∈ R3 is the acceleration vector of the spacecraft relative to the coordinate frame {xb , yb, zb }, Ω is the angular velocity vector of the Sun-Earth system, rb (t) ∈ R3 is the position vector of the spacecraft relative to the coordinate frame {xb , yb , zb }, and vb ∈ R3 is the velocity vector of the spacecraft relative to the coordinate frame {xb , yb , zb }. In addition, rb , vb , ab , and Ω, can be expressed in the coordinate frame {xb , yb, zb }, respectively as ˆ rb = x¯ˆı + yˆ  + z k,

ˆ vb = x¯˙ ˆı + yˆ ˙  + z˙ k,

ˆ ¨¯ˆı + y¨ˆ + z¨k, ab = x

ˆ Ω = ω k,

(3)

where {¯ x(t), y(t), z(t)} ∈ R3 are the components of the position vector of the spacecraft expressed in the coordinate frame {xb , yb , zb } and ω is the constant angular speed of the Sun-Earth system. Note that ˆı, ˆ, and kˆ denote the unit vectors in the xb , yb , and zb directions, respectively. Furthermore, RS→s and RE→s can be expressed in the coordinate frame {xb , yb , zb }, respectively as ˆ x + RS ) ˆı + yˆ  + z k, RS→s = (¯

(4)

ˆ x − RE ) ˆı + yˆ  + z k, RE→s = (¯

(5)

where RS is the distance from the origin of the inertial coordinate frame to the Sun and RE is the distance from the origin of the inertial coordinate frame to the Earth. Substituting (2) into (1), using (3)–(5), and dividing by the spacecraft mass m, produces the mathematical model describing the position of the spacecraft relative to the Sun-Earth 4

system barycenter expressed in the rotating coordinate frame {xb , yb, zb } as follows1 µS (¯ x + RS ) µE (¯ x − RE ) − , 3 RS→s  RE→s 3 µS y µE y − , y¨ + 2ω x¯˙ − ω 2 y = − 3 RS→s  RE→s 3 µE z µS z − , z¨ = − 3 RS→s  RE→s 3

¨¯ − 2ω y˙ − ω 2 x¯ = − x

(6)

 where µS  = GMS and µE = GME .

Remark 2.1. The collinear Lagrange points relative to the Sun-Earth system barycenter are obtained by setting the y and z position components and all velocity and acceleration components in (6) to zero. This yields a set of quintic polynomial equations for x¯ which needs to be solved independently to obtain each collinear Lagrange point (see Ref. 1 for further details). The collinear Lagrange points, as shown in Figure 1, are ordered as L1 , L2 , and L3 , where L1 is located between the Earth and the Sun, L2 is located on the far side of the Earth in the positive xb direction, and L3 is located on the far side of the Sun in the negative xb direction. In addition, we denote the distance between the Sun-Earth system barycenter and the Lagrange point L2 as RL2 . Next, we perform a translational coordinate transformation of the form x¯ = x + RL2 to transform the coordinate frame {xb , yb, zb }

x ¯=x+RL2

−→

{xL2 , yL2 , zL2 }. Now, (6) can be expressed

in the new coordinate frame {xL2 , yL2 , zL2 } as µS (x + RL2 + RS ) µE(x + RL2 − RE ) − , RS→s 3 RE→s 3 µS y µE y − , y¨ + 2ω x˙ − ω 2 y = − RS→s 3 RE→s 3 µE z µS z − , z¨ = − RS→s 3 RE→s 3

x¨ − 2ω y˙ − ω 2 (x + RL2 ) = −

(7)

where RS→s and RE→s can be expressed in the coordinate frame {xL2 , yL2 , zL2 }, as ˆ  + z k, RS→s = (x + RL2 + RS ) ˆı + yˆ

(8)

ˆ  + z k. RE→s = (x + RL2 − RE ) ˆı + yˆ

(9)

5

After pre-multiplying the nonlinear dynamics of (7) with the spacecraft mass m and inserting the thrusting control forces for the spacecraft, the nonlinear position dynamics of the spacecraft relative to the L2 Lagrange point can be arranged in the following compact form m¨ q + C q˙ + N(q) = u,

(10)

where q(t) ∈ R3 is defined as q

 =

[x y

z]T ,

(11)

C is a Coriolis-like matrix defined as 

C

 =



0 −1 0   2mω  1 0 0  , 0 0 0

(12)

N is a nonlinear term consisting of gravitational effects and inertial forces

N

 µS (x+RL +RS ) µ (x+RL2 −RE ) 2 + E RE→s 3 R  3 S→s    µS y Ey + RµE→s − ω 2y =m  3  RS→s 3 µS z RS→s 3

+

− ω 2 (x + RL2 )

µE z RE→s 3

   , 

(13)

and u(t) ∈ R3 is the thrust control input to the spacecraft. The following remarks further facilitate the subsequent control design and stability analysis. Remark 2.2. The Coriolis matrix C satisfies the skew symmetric property of xT Cx = 0,

∀x ∈ R3 .

(14)

Remark 2.3. The left-hand side of (10) produces an affine parameterization mξ1 + Cξ2 + N(q) = Y (ξ1 , ξ2 , q)m,

(15)

where ξj (t) ∈ R3 , for j = {1, 2}, are dummy variables with components ξjx , ξjy , and ξjz , m is the unknown, constant mass of the spacecraft, and Y ∈ R3 is a regression matrix, composed

6

of known functions, defined as 

Y

3.

 =

   

µS (x+RL2 +RS ) µ (x+RL2 −RE ) + E RE→s RS→s 3 3 Sy Ey 2ωξ2x − ω 2 y + RµS→s + RµE→s 3 3 µS z µE z ξ1z + RS→s 3 + RE→s 3

ξ1x − 2ωξ2y − ω 2 (x + RL2 ) + ξ1y +

   . 

(16)

Problem Formulation In this section, we formulate a control design problem such that the spacecraft position

q tracks a desired position trajectory qd ∈ R3 , i.e., lim q(t) = qd (t). The effectiveness of this t→∞

control objective is quantified through the definition of a position tracking error e(t) ∈ R3 as e

 =

q − qd .

(17)

The goal is to construct a control algorithm that obtains the aforementioned tracking result in the presence of the unknown constant spacecraft mass m. We assume that the position and velocity measurements (i.e., q and q) ˙ of the spacecraft relative to the L2 Lagrange point are available for feedback. To facilitate the control development, we assume that the desired trajectory qd and its first two time derivatives are bounded functions of time. Next, we define the spacecraft mass estimation error m(t) ˜ ∈ R as m ˜

 =

m ˆ − m,

(18)

where m(t) ˆ ∈ R is the spacecraft mass estimate.

4.

Adaptive Position Tracking Controller In this section, we design an adaptive feedback control law that asymptotically tracks a

pre-specified spacecraft position trajectory, in the presence of the unknown constant spacecraft mass m. In order to state the main result of this section, we define the following 7

notation. A filter tracking error variable r(t) ∈ R3 is defined as r

 =

e˙ + αe,

(19)

where α ∈ R3×3 is a constant, diagonal, positive-definite, control gain matrix, an augmented error variable η(t) ∈ R6 is defined as 

η

 =

r e



,

(20)

and a positive constant λ is defined as λ = min {λmin {K} , λmin {Kp α}} ,

(21)

where λmin{·} denotes the minimum eigenvalue of a matrix and K, Kp ∈ R3×3 are constant, diagonal, positive-definite matrices. Additionally, we solve for e˙ in (19) to produce e˙ = r − αe.

(22)

Finally, we define a new regression matrix Yd (·) ∈ R3 as Yd (·)

 =

Y (¨ qd − αe, ˙ q˙d − αe, q),

(23)

where (15) has been used with ξ1 = q¨d − αe˙ and ξ2 = q˙d − αe, in the definition of (16). Theorem 4.1. Let K, Kp ∈ R3×3 be constant, diagonal, positive-definite matrices and Γ ∈ R be a positive constant. Then, the adaptive control law ˆ − Kp e − Kr, u = Yd m m ˆ˙ = −ΓYdT r,

(24) (25)

ensures global asymptotic convergence of the position and velocity tracking errors as delineated by ˙ = 0. lim e(t), e(t)

t→∞

8

(26)

Proof. We begin by rewriting the spacecraft position dynamics (10) in terms of the filtered tracking error variable (19). To this end, differentiating (19) with respect to time, multiplying both sides of the resulting equation by m, using e¨ = q¨ − q¨d from (17), and rearranging terms yields ˙ + m¨ q. mr˙ = −m(¨ qd − αe)

(27)

Substituting for m¨ q from (10) in (27), we obtain ˙ − C q˙ − N(q) + u. mr˙ = −m(¨ qd − αe)

(28)

Next, we expand (19) by noting that e˙ = q˙ − q˙d . Then solving for q, ˙ substituting the result into (28), and rearranging terms yields ˙ − C(q˙d − αe) − N(q) − Cr + u, mr˙ = −m(¨ qd − αe) = −Yd m + u − Cr,

(29)

where the definition of (23) has been used. Eq. (29) characterizes the open-loop dynamics of r. Now, substituting (24) into (29) results in the following closed-loop dynamics for r ˜ − Kp e − Kr − Cr, mr˙ = Yd m

(30)

where the definition of (18) has been used. Finally, note that differentiating (18) with respect to time and using (25), produces the closed-loop dynamics for the spacecraft mass estimation error m ˜˙ = −ΓYdT r.

(31)

Now, we utilize the error systems of (30) and (31) along with the positive-definite, candidate Lyapunov function defined by 1 T 1 T 1 2 m ˜ , V (t)  = mr r + e Kp e + 2 2 2Γ

9

(32)

to prove the above stability result for the position and velocity tracking errors. Specifically, differentiating (32) with respect to time yields 1 ˙ V˙ (t) = r T mr˙ + eT Kp e˙ + m ˜ m. ˜ Γ

(33)

Now, the substitution of the closed-loop dynamics of (22) and (30) into (33) results in



1 ˙ ˜ , ˜ YdT r + m V˙ (t) = −r T Kr − eT Kp αe + m Γ

(34)

where the property of (14) has been used. Finally, substituting (31) into the parenthetical term of (34), we obtain V˙ (t) = −r T Kr − eT Kp αe ≤ −λη2 ≤ 0,

(35)

where the definitions of (20) and (21) have been used. Since V is a non-negative function and V˙ is a negative semi-definite function, V is a non-increasing function. Thus V (t) ∈ L∞ as described by V (r(t), e(t), m(t)) ˜ ≤ V (r(0), e(0), m(0)), ˜

t ≥ 0.

(36)

Using standard signal chasing arguments, all signals in the closed-loop system can now be shown to be bounded. Using (22) and (30) along with the boundedness of all signals in the closed-loop system, we now conclude that η˙ ∈ L∞ . Solving the differential inequality of (35) results in V (0) − V (∞) ≥ λ

 ∞ 0

η(t)2 dt.

Since V (t) is bounded, t ≥ 0, we conclude that η(t) ∈ L∞

(37)

L2 , t ≥ 0. Finally, using

Barbalat’s Lemma,13, 14 we conclude that lim η(t) = 0.

t→∞

(38)

Using the definition of r and η in (19) and (20), respectively, the limit statement of (38), and Lemma 1.6 of Ref. 13, yield the result of (26). 10

Remark 4.1. Similar to the traditional adaptive control designs in Ref. 14, the stability result of Theorem 4.1 neither requires nor guarantees the convergence of m(t) ˆ to m. However, Eqs. (25) and (38) with the signal chasing arguments used in the proof of Theorem 4.1 yield the convergence of m(t) ˆ to some constant value.

5.

Simulation Results In this section, the efficacy of the adaptive control methodology is demonstrated for the

desired trajectories of the spacecraft that are generated using the ideal, thrust-free, periodic trajectories around the L2 Lagrange point in the form of Lyapunov and Halo orbits. A succinct overview of a numerical algorithm to generate the thrust-free periodic trajectories around the L2 Lagrange point is given below. Additional details on the generation of these periodic trajectories can be found in Refs. 15–19. One numerical method,19 of generating thrust-free periodic orbits around the L2 Lagrange point in the Sun-Earth system involves finding a proper set of position and velocity initial conditions to propagate the spacecraft dynamics of (7). First, the Poincar´e-Lindstedt method is used to find a high order analytic approximation to a periodic trajectory in the neighborhood of the L2 Lagrange point. Next the initial conditions, based on the Poincar´eLindstedt method, are used as an initial seed in a numerical algorithm to find a better set of initial conditions leading to a periodic trajectory. This numerical algorithm applies a Taylor series expansion to the spacecraft states with respect to the initial conditions and time and truncates higher order terms, such that the result is a set of linear equations (Lyapunov orbits: 2 equations, Halo orbits: 3 equations) for a set of unknown variables (Lyapunov orbits: 3 variables, Halo orbits: 4 variables). Families of orbits can be characterized by fixing one of the unknown variables so that the result gives an equal number of equations to unknowns (Lyapunov orbits: 2 equations, 2 unknowns, Halo orbits: 3 equations, 3 unknowns). Solving the aforementioned linear matrix equation and using the result to update the previous set

11

of initial conditions provides a new initial condition guess. The spacecraft dynamics are then propagated using the new updated set of initial conditions to verify trajectory periodicity. If the trajectory is sufficiently close to being periodic, then the initial conditions can be used for further simulation, else the above numerical algorithm is used to solve for a new set of initial conditions. Since the collinear Lagrange points are inherently unstable, long-term propagation of spacecraft dynamics using the initial conditions obtained in the above manner is futile. However, we can artificially obtain close to a periodic orbit by computing trajectory information during half of a period and reusing this trajectory data throughout other simulations. Lyapunov orbits are classified as periodic trajectories set in the orbital plane {xL2 , yL2 } (i.e., zL2 = 0). An initial seed for the numerical algorithm of Ref. 19 consists of a spacecraft starting on the xL2 axis with a nonzero initial yL2 velocity (i.e., qd (0) = [xd (0) 0 0]T and q˙d (0) = [0 y˙ d(0) 0]T ). Updates to the initial yL2 velocity contributes to finding a closed periodic trajectory in the orbital plane. In addition, the initial xL2 position determines the size of the Lyapunov orbit. Halo orbits are classified as periodic trajectories that are symmetric with respect to the {xL2 , zL2 } plane, (i.e., yL2 = 0). Unlike Lyapunov orbits, Halo orbits are periodic trajectories which are not confined to be in the orbital plane of the Sun and Earth. Halo orbits have the distinguishing characteristic that its projection on the {yL2 , zL2 } plane is a curve that resembles a Halo. An initial seed for the numerical algorithm of Ref. 19 consists of a spacecraft starting on the {xL2 , zL2 } plane with a nonzero initial yL2 and zL2 velocity (i.e., qd (0) = [xd (0) 0 zd (0)]T and q˙d (0) = [0 y˙d (0) z˙d (0)]T ). Updates to the initial xL2 position and yL2 velocity contributes to finding a closed periodic trajectory. In addition, the initial zL2 position determines the size of the Halo orbit. Applying the numerical algorithm presented in Ref. 19 results in a set of initial conditions for a desired Lyapunov orbit given as qdL (0) = [−1.501026450400631 0 0] × 103 km, 12

q˙dL (0) = [0 2.692047849889458 0] × 10 km . Furthermore, applying the numerical algorithm day in Ref. 19 results in a set of initial conditions for a desired Halo orbit given as qdH (0) = [1.67698167739895 0 −0.49947314521862] × 105 km,

q˙dH (0) = [0 − 3.39920850756425 0]

km . ×103 day When tracking Lyapunov orbits, we initialize the spacecraft with the set of initial condi





˙ = 0.7q˙dLx (0) 1.3q˙dLy (0) q˙dLz (0) . tions given as q(0) = 1.3qdLx (0) 0.7qdLy (0) qdLz (0) , q(0) When tracking Halo orbits, we initialize the spacecraft with the set of initial conditions given





˙ = 0.7q˙dHx (0) 1.3q˙dHy (0) q˙dHz (0) . as q(0) = 1.3qdHx (0) 0.7qdHy (0) qdHz (0) , q(0) The adaptive control law of (24) and (25) has been simulated for the nonlinear dynamics of (10) such that the spacecraft outputs are q(t) and q(t), ˙ with the Sun-Earth system circular m3 , orbit parameters19, 12 : G = 6.671 × 10−11 kg ·s2 ×1024 kg,

1 AU = 1.496 × 108 km,

MS = 1.9891 × 1030 kg,

ME = 5.974

RL2 = 1.010033599267463 AU, where 1AU stands for

1 Astronomical Unit denoting the distance between the Sun and the Earth and we consider that the spacecraft has a mass of m = 1000kg. In addition, the distances RS and RE can be computed as RS =

ME ME +MS

× 1AU and RE =

MS ME +MS

× 1AU .

The control and adaptation gains, in the control law of (24) and (25), are obtained through trial and error in order to obtain good performance for the tracking error response. The following resulting gains were used in this simulation Kp = diag (1, 1, 1) × 7.49 × 10−3 , K = diag (1, 1, 1) × 5.47 × 103 , α = diag (1, 1, 1) × 8.49 × 10−1 , and Γ = 88.9. In addition, the spacecraft mass parameter estimate was initialized to m(0) ˆ = 900 kg. A simulation of a spacecraft tracking the desired Lyapunov orbit is performed. The actual trajectory q is shown in Figure 2. Figures 3 and 4 show the tracking error e and velocity tracking error e, ˙ respectively. The control input u is shown in Figure 5. Finally, the spacecraft mass estimate m ˆ is shown in Figure 6. Next, a simulation of a spacecraft tracking the desired Halo orbit is performed. The actual trajectory q is shown in Figure 7. Figures 8 and 9 show the tracking error e and velocity tracking error e, ˙ respectively. The control input 13

u is shown in Figure 10. Finally, the spacecraft mass estimate m ˆ is shown in Figure 11.

6.

Conclusion In this paper, we designed an adaptive control algorithm for the position dynamics of a

spacecraft to enable it to perform trajectory tracking relative to the L2 Lagrange point in the Sun-Earth system. A Lyapunov-type design was used to construct a full state feedback control law and parameter estimates which facilitate the tracking of periodic reference trajectories with global asymptotic convergence. Simulation results were given to illustrate the efficacy of the control design.

7.

Acknowledgments Research supported in part by the National Aeronautics and Space Administration–

Goddard Space Flight Center under Grant NAG5-11365 and the NASA/New York Space Grant Consortium under Grant 39555-6519.

14

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Thurman, R. and Worfolk, P. A., The Geometry of Halo Orbits in the Circular Re-

stricted Three-Body Problem, Geometry Center Research Report GCG95, University of Minnesota, 1996.

Figure Captions Figure 1: Sun-Earth system schematic diagram Figure 2: Lyapunov orbit: trajectory of the spacecraft relative to the L2 Lagrange point Figure 3: Lyapunov orbit: position tracking error Figure 4: Lyapunov orbit: velocity tracking error Figure 5: Lyapunov orbit: control input to spacecraft Figure 6: Lyapunov orbit: mass parameter estimate of spacecraft Figure 7: Halo orbit: trajectory of the spacecraft relative to the L2 Lagrange point Figure 8: Halo orbit: position tracking error Figure 9: Halo orbit: velocity tracking error Figure 10: Halo orbit: control input to spacecraft Figure 11: Halo orbit: mass parameter estimate of spacecraft

Figure 1: Sun-Earth system schematic diagram

5000

4000

Actual L2

3000

Initial position Desired

2000

y (km)

1000

0

−1000

−2000

−3000

−4000

−5000 −2500

−2000

−1500

−1000

−500

0 x (km)

500

1000

1500

2000

2500

Figure 2: Lyapunov orbit: trajectory of the spacecraft relative to the L2 Lagrange point

19

100 Position error x (km)

0 −100 −200 −300 −400 −500 −600

0

200

400

600

800

1000

1200

1400

0

200

400

600

800 time (days)

1000

1200

1400

Position error y (km)

600

400

200

0

−200

Figure 3: Lyapunov orbit: position tracking error

Velocity error x (km/day)

10

5

0

−5

−10

0

200

400

600

800

1000

1200

1400

0

200

400

600

800 time (days)

1000

1200

1400

Velocity error y (km/day)

10

5

0

−5

−10

Figure 4: Lyapunov orbit: velocity tracking error

20

−7

Control x (N)

15

x 10

10

5

0 0

200

400

600

800

1000

1200

1400

200

400

600

800 time (days)

1000

1200

1400

−7

x 10

Control y (N)

0

−5

−10

−15

0

Figure 5: Lyapunov orbit: control input to spacecraft

900 899.99

Mass parameter estimate (kg)

899.98 899.97 899.96 899.95 899.94 899.93 899.92 899.91 899.9

0

200

400

600

800 time(days)

1000

1200

1400

Figure 6: Lyapunov orbit: mass parameter estimate of spacecraft

21

6

1

4

x 10

6

x 10

4 2 z (km)

y (km)

0.5

0

0 −2 −4

−0.5

−6 −1 −4

−2

0 x (km)

2

−8 −1

4

−0.5

5

x 10

0 y (km)

0.5

1 6

x 10

4

6

x 10

Actual L

4

x 10 5

Initial position Desired

2 z (km)

z (km)

2

4

0 −5

−4

1 −10 −5

0 0

5

x 10

5

−1

0 −2

6

x 10

y (km)

−6 −8 −4

−2

x (km)

0 x (km)

2

4 5

x 10

Figure 7: Halo orbit: trajectory of the spacecraft relative to the L2 Lagrange point

Position error x (km)

4

10 5 0

Position error y (km)

−5 1

0 4 x 10

200

400

600

800

1000

1200

0 4 x 10

200

400

600

800

1000

1200

0

200

400

600 time (days)

800

1000

1200

0 −1 −2

Position error z (km)

x 10

1 0 −1 −2

Figure 8: Halo orbit: position tracking error

22

−400

0

200

400

600

800

1000

1200

0

200

400

600

800

1000

1200

0

200

400

600 time (days)

800

1000

1200

200 0 −200 −400 200 0 −200 −400

Figure 9: Halo orbit: velocity tracking error

−5

1 Control x (N)

Velocity error x (km/day)

−200

x 10

0 −1 −2 5

Control y (N)

Velocity error y (km/day)

0

0 −6 x 10

200

400

600

800

1000

1200

0 −6 x 10

200

400

600

800

1000

1200

0

200

400

600 time (days)

800

1000

1200

0 −5 −10 4

Control z (N)

Velocity error z (km/day)

200

2 0 −2

Figure 10: Halo orbit: control input to spacecraft

23

902

901

Mass parameter estimate (kg)

900

899

898

897

896

895

894

0

200

400

600 time(days)

800

1000

1200

Figure 11: Halo orbit: mass parameter estimate of spacecraft

24