Strategy for Optimal, Long-Term Stationkeeping of Libration Point Orbits in the Earth-Moon System

Thomas A. Pavlak∗ and Kathleen C. Howell† Purdue University, West Lafayette, IN, 47907-2045, USA

In an eﬀort to design low-cost maneuvers that reliably maintain unstable libration point orbits in the Earth-Moon system for long durations, an existing long-term stationkeeping strategy is augmented to compute locally optimal maneuvers that satisfy end-of-mission constraints downstream. This approach reduces stationkeeping costs for planar and three- dimensional orbits in dynamical systems of varying degrees of ﬁdelity and demonstrates the correlation between optimal maneuver direction and the stable mode observed during ARTEMIS mission operations. An optimally-constrained multiple shooting strategy is also introduced that is capable of computing near optimal maintenance maneuvers without formal optimization software.

I. Introduction

Most orbits in the vicinity of collinear libration points are inherently unstable and, consequently, stationkeeping strategies are a critical component of mission design and operations in these chaotic dynamical regions. Stationkeeping is particularly important for libration point missions in the Earth-Moon system since fast time scales require that orbit maintenance maneuvers be implemented approximately once per week. Assuming that acceptable orbit determination solutions require 3-4 days to obtain, stationkeeping ∆V planning activities must be quick, efficient, and effective. Furthermore, the duration of a libration point mission is often dictated by the remaining propellant so a key capability is maintenance maneuvers that are low-cost. Thus, to accommodate a likely increase in future operations in the vicinity of the Earth-Moon libration points, fast, reliable algorithms capable of rapidly computing low-cost stationkeeping maneuvers, with little or no human interaction, are critical. The libration point stationkeeping problem has been explored in both the Sun-Earth and Earth-Moon systems by numerous researchers. The majority of these orbit maintenance methods can be classified as “short-term” approaches given that the goal for each individual maneuver is maintenance of the spacecraft within the vicinity of the libration point for only the immediate future. Optimal short-term strategies, some incorporating Floquet theory to eliminate the unstable component of spacecraft error, have been explored extensively by Farquhar,1,2 Breakwell et al.,3 and others.4–6 Globally optimal maneuvers have been sought via a global search stationkeeping algorithm designed for the Sun-Earth L2 James Webb Space Telescope by Janes and Beckman.7 During operation of the first Earth-Moon libration point mission, ARTEMIS,8 – depicted in Figure1 – the L1 and L2 quasi-periodic orbits were maintained over 1-2 revolutions at a time using direct optimization.9 Post-processing of maneuver data revealed that the ∆V direction aligned closely with the associated stable mode for virtually all of the approximately 60 stationkeeping maneuvers that were successfully implemented.10 A number of “long-term” stationkeeping approaches, specifically concerned with ensuring that the spacecraft meets specific end-of-mission constraints, have also been developed, but are generally non-optimal. A method employed by Grebow et al.12 and Folta et al.13 targets a rigidly maintained baseline trajectory. The long-term stationkeeping strategy presented previously by Pavlak and Howell14 is flexible and able to

∗Ph.D. Candidate, School of Aeronautics and Astronautics, Purdue University, Armstrong Hall of Engineering, 701 W. Stadium Ave, West Lafayette, Indiana 47907-2045, Student Member AIAA. †Hsu Lo Professor of Aeronautical and Astronautical Engineering, School of Aeronautics and Astronautics, Purdue Univer- sity, Armstrong Hall of Engineering, 701 W. Stadium Ave, West Lafayette, Indiana 47907-2045; Fellow AAS; Associate Fellow AIAA.

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American Institute of Aeronautics and Astronautics Figure 1. ARTEMIS P1 Earth-Moon Trajectory11

maintain libration point orbits with a variety of mission requirements using multiple shooting. The algorithm employs stationkeeping maneuvers and targets many revolutions ahead – in contrast to the more limited 1-2 revolutions that serve as the basis in some short-term strategies – and, thus, is capable of maintaining trajectories for an arbitrary number of revolutions and/or meeting end-of-mission constraints such as a specific set of lunar arrival conditions. The analysis in this investigation is motivated by a need for flexible libration point orbit stationkeeping methods that are capable of reliably delivering a spacecraft to a set of desired end-of-mission conditions, final targets that may occur many revolutions downstream, and accomplish the goal for minimum cost in terms of ∆V . Furthermore, the empirical results generated from analysis of the ARTEMIS trajectory maneuvers currently represent only a single data set, so the observed alignment of the ∆V and stable mode directions warrants further exploration. The current investigation first seeks to augment the long-term stationkeeping scheme discussed by Pavlak and Howell14 with direct optimization to yield a flexible method for computing low-cost orbit maintenance maneuvers that is well-suited for automation. The optimal long-term strategy is also employed to further explore the empirical results extracted from the previous ARTEMIS stationkeeping analysis. Stationkeeping costs are reduced over results obtained via non-optimal long-term stationkeeping in the circular restricted three-body and higher-fidelity models. It is also demonstrated that knowledge obtained from ARTEMIS mission stationkeeping operations can be applied to develop algorithms to rapidly compute “near-optimal” stationkeeping maneuvers using only existing differential corrections strategies that do not require commercial optimization software.

II. System Models

Dynamical models of multiple levels of fidelity are utilized in this analysis. The circular restricted three-body problem (CR3BP) serves as the baseline model and is used for preliminary trajectory design and stationkeeping activities. A higher-fidelity ephemeris model incorporating lunar eccentricity and solar gravity – both of which can be significant when operating in the vicinity of the Earth-Moon libration point orbits – is included as well.

II.A. Circular Restricted Three-Body Problem The circular restricted three-body problem represents a dynamical model that governs the motion of a “massless” spacecraft under the simultaneous inﬂuence of two gravitational bodies – in this case, the Earth and the Moon. The system is further simpliﬁed by assuming that the orbits of the Earth and Moon are both circular and coplanar; each body rotates about a common barycenter. It is often convenient to formulate the governing equations in the CR3BP with respect to a reference frame centered at the Earth-Moon barycenter;

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American Institute of Aeronautics and Astronautics the rotational rate of the frame is equal to the orbital rate of the bodies. The rotating positive x-axis is aligned with the Earth-Moon line, the z-axis is defined to be orthogonal to the orbital plane of the gravitating bodies, and the y-axis completes the right-handed triad. The nondimensional mass parameter, µ, is defined as m µ = M (1) mE + mM and effectively represents the relative influence of the smaller body, i.e., the Moon, within the three-body system. The mass of the Earth and the Moon are denoted mE and mM , respectively. Spacecraft state information is described by a Cartesian six-dimesional state vector, h iT x = x, y, z, x,˙ y,˙ z˙ (2)

in which positions and velocities are expressed in the barycentered rotating frame. Note that bold symbols are used to represent vector quantities. Motion within the circular-restricted three-body problem is governed by the set of nondimensional second-order scalar diﬀerential equations,

(1 µ)(x + µ) µ (x 1 + µ) x¨ 2y ˙ x = − 3 − 3 (3) − − − d1 − d2 (1 µ) y µy y¨ + 2x ˙ y = −3 3 (4) − − d1 − d2 (1 µ) z µz z¨ = −3 3 (5) − d1 − d2

with the scalar Earth-spacecraft and Moon-spacecraft distances – represented by d1 and d2, respectively – expressed as follows, q 2 2 2 d1 = (x + µ) + y + z (6) q d = (x 1 + µ)2 + y2 + z2 (7) 2 − Trajectories are computed in the CR3BP by numerically integrating the equations of motion written in ﬁrst-order vector form, i.e., x˙ = f (t, x) (8) where x is the six-dimensional state vector described previously.

II.B. Libration Point Orbit Stability The equations of motion in the circular restricted three-body problem are autonomous when expressed in terms of a rotating frame and admit ﬁve well-known equilibrium solutions, i.e., the libration points. The three collinear libration points that lie along the Earth-Moon line are dynamically unstable as are many of the periodic orbits in their vicinity. The monodromy matrix, M, associated with a periodic libration point orbit is computed by integrating the state transition matrix (STM), Φ (t, ti), for one orbital period, P , i.e.,

M = φ (ti + P, ti) (9)

Since the monodromy matrix corresponds to a discrete linear mapping, the orbital stability is assessed by examining the eigenvalues of M relative to a unit circle. The eigenvalues, λi, exist in reciprocal pairs and are interpreted as follows:

λi 1 - Stable eigenvalue | | ≤ λi 1 - Unstable eigenvalue | | ≥ λi = 1 - Corresponds to orbit periodicity

This classiﬁcation is further illustrated in Figure2. Stable and unstable manifolds exist as higher dimensional surfaces governing ﬂow into and out of unstable periodic libration point orbits. Local manifold information is obtained from the eigenvectors, ν, of a

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American Institute of Aeronautics and Astronautics Periodicity

Neutrally Stable Stable Unstable

Figure 2. Example Eigenvalue Diagram for Unstable Periodic Orbit

monodromy matrix computed at a desired fixed point along the periodic orbit. The local stable and unstable manifolds – νs and νu, respectively – associated with a fixed point on an L2 Lyapunov orbit are projected onto configuration space and depicted in Figure 3(a). Note that the stable and unstable subspaces extend in both “+” and “ ” directions. Similarly, the local manifold directions associated with various points along the orbit appear− in Figure 3(b).

(a) (b)

Figure 3. Local L2 Lyapunov Manifold Directions

II.C. Ephemeris Model Higher-ﬁdelity ephemeris models are utilized to more accurately predict stationkeeping ∆V costs during actual mission operations. This analysis incorporates the eﬀects of lunar eccentricity and solar gravity via a Moon-Earth-Sun gravity model in which the relative positioning of the bodies are dictated by the JPL DE405 ephemerides. Ephemeris motion of a spacecraft, “i”, relative to a central body, “q”, is governed by the second-order N-body relative equations of motion,

n ! µq X rij rqj r¨qi = rqi + µj (10) −r3 r3 − r3 qi j=1 ij qj j=6 i,q

where rqj describes the position of a perturbing body “j” with respect to the central body and is obtained directly from DE405. The vector rij represents the position of each perturbing body relative to the spacecraft and the gravitational parameter of each body is denoted µk. Ephemeris trajectories are computed by

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American Institute of Aeronautics and Astronautics rewriting Equation (10) in ﬁrst-order form and numerically integrating in a Moon-centered, Earth J2000 reference frame. Like the CR3BP equations of motion, the higher-ﬁdelity equations of motion are also nondimesionalized by the characteristic length, l∗, and time, t∗, associated with the Earth-Moon system. These constants are listed in Table1 along with the gravitational parameters of the Moon, Earth, and Sun and the lunar radius, rM .

Table 1. Problem Constants Parameter Value Units l∗ 385,692.50 km t∗ 377,084.152667038 s 3 2 µM 4902.80058214776 km /s 3 2 µE 398,600.432896939 km /s 3 2 µS 132,712,440,017.987 km /s

rM 1738.2 km

III. Numerical Methods

The dynamically sensitive nature of most collinear libration point orbits prohibit the computation of such orbits for multiple revolutions using conventional single shooting algorithms. Thus, a long-term stationkeeping strategy requires a multiple shooting approach to distribute trajectory sensitivities and to successfully target mission constraints far into the future. The multiple shooting scheme is incorporated into existing commercial direct optimization software to compute locally optimal stationkeeping maneuvers.

III.A. Multiple Shooting In this analysis, multiple shooting is utilized to compute trajectories containing many libration point revolutions and, in some cases, transfers between libration point orbits and/or to the lunar vicinity. Multiple shooting is initially implemented to compute a continuous reference solution and, within the context of the long-term stationkeeping algorithm, to ultimately compute maneuvers to target a speciﬁc set of end- of-mission constraints using the reference as an initial guess. To formulate the algorithm, the path is ﬁrst discretized into n patch points associated with n 1 trajectory segments as illustrated in Figure4. A dis- − t t Tn-1 xn xn-1 xt T 2 1 … xn, τn xn-1, τn-1 x2, τ2 x1, τ1

(a) Initial Guess

Tn-1 t t t xn xn-1 x2 T1 x , τ … xn-1, τn-1 n n x2, τ2

x1, τ1

(b) Converged

Figure 4. Multiple Shooting Schematic continuous initial guess is depicted in Figure 4(a) while a converged solution satisfying a set of user-deﬁned constraints appears in Figure 4(b). The vectors, xi are the six-dimensional states associated with the patch t points and the symbols xi represent the ﬁnal numerically integrated state along each segment. The epoch at each patch point and the integration time corresponding to each trajectory segment are labeled, Ti and τi, respectively.

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American Institute of Aeronautics and Astronautics The initial guess is differentially corrected to satisfy the desired trajectory requirements by first estab- lishing a vector of design variables, X. When computing CR3BP solutions in this analysis, X, is defined such that it contains the full state at each path point as well as the integration times of all segments along the path, i.e.,   x1  .   .       xn  X =   (11)  T1     .   .  Tn−1 and has length 7n 1. Trajectories in the circular restricted three-body problem are time-invariant, however, solutions in the ephemeris− model are time-dependent and, thus, it is necessary to augment the variable vector with the epochs, τi, associated with the interior patch points as well. Any user-defined trajectory constraints are included in a vector of equality constraints of the form F (X) = 0. With multiple shooting, constraints are typically included to enforce continuity in position and velocity between the end of each integrated segment and the next patch point, i.e., t x xi = 0 (12) i − in addition to any additional constraints at the beginning or end of the trajectory or along the path. Note that, to perform ephemeris corrections, it is also necessary to ensure that the trajectory is continuous in time at each patch point. Gradient information is obtained from the Jacobian matrix,

∂F (X) DF (X) = (13) ∂X and, since the problem is typically deﬁned with more variables than constraints, iterations to adjust the variable vector proceed using the minimum norm update equation, i.e.,

T h T i−1 Xj+1 = Xj DF Xj DF Xj DF Xj F Xj (14) − to generate a converged solution, X∗. The multiple shooting algorithm is implemented in this analysis to compute continuous reference solutions, transition trajectories from the CR3BP to higher-ﬁdelity ephemeris models, and is also an integral component of the long-term stationkeeping strategy.

III.B. Long-Term Stationkeeping Strategy The long-term stationkeeping strategy is described previously by Pavlak and Howell14 and is not fundamentally altered in this investigation. The underlying goal is the computation of stationkeeping maneuvers that ensure a spacecraft satisfies a set of end-of-mission constraints far into the future without requiring that the spacecraft strictly adhere to a pre-defined baseline trajectory. In this sense, the stationkeeping strategy is based in a trajectory planning approach.15 The long-term method does, however, require a reference trajectory that serves as an initial guess for the maneuver planning algorithm. A multiple-revolution reference trajectory is first computed using an orbit discretized into patch points and a multiple shooting scheme is then applied. Collinear libration point orbits typically depart after only 1-2 revolutions when numerically integrated, but utilizing multiple trajectory segments enables libration point orbits to be computed for an arbitrary number of revolutions. If desired, invariant manifolds can also be incorporated into the reference trajectory design to achieve low-cost transfers between libration point orbits or to the lunar vicinity. The long-term stationkeeping strategy is fundamentally a targeting problem and a schematic is depicted in Figure5. The algorithm originates at the initial state along the reference solution and the trajectory is propagated to the location of the first stationkeeping maneuver, denoted with a red dot. The maneuver then targets a set of end-of-mission constraints using multiple shooting and utilizing the remaining portion of the reference trajectory as an initial guess. Randomly distributed navigation/modeling and maneuver execution errors are superimposed on the planned maneuver and the trajectory is propagated forward to the location of the next stationkeeping ∆V maneuver. This process is summarized as follows:

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American Institute of Aeronautics and Astronautics Figure 5. Long-Term Stationkeeping Example

1. Obtain reference solution in desired dynamical model 2. Apply simulated navigation errors to initial state 3. Integrate from initial point to ﬁrst stationkeeping maneuver location 4. Compute maneuver using remaining leg of reference solution as initial guess 5. Apply simulated navigation and maneuver execution errors 6. Integrate to next stationkeeping maneuver location 7. Repeat steps 4-6 until ﬁnal stationkeeping maneuver is implemented 8. Apply simulated navigation and maneuver execution errors 9. Integrate to end-of-mission condition (lunar arrival, etc.) These steps are repeated for the required number of stationkeeping maneuvers. Note that the reference solution is updated following the computation of each maneuver. Since each long-term simulation is stochastic in nature, actual mission stationkeeping costs are approximated utilizing Monte Carlo simulations.

III.C. Direct Optimization During mission operations, orbit maintenance costs can significantly impact mission duration and/or mass budgets and, consequently, minimizing stationkeeping ∆V expenditure is often a critical mission objective. And, while multiple shooting is a robust method for computing solutions near an initial guess, it may not, in general, yield optimal results. In this analysis, however, the long-term stationkeeping algorithm is augmented with direct optimization to compute maintenance maneuvers that are locally optimal with respect to ∆V . This is achieved by first defining an objective function, J (X), given by J (X) = ∆V 2 (15) || || for each stationkeeping maneuver, ∆V . Note that, for a single maneuver, this formulation is equivalent to the objective J (X) = ∆V whose derivative becomes singular for ∆V = 0. Equation (15) is implemented because it offers the distinct|| || advantage of not experiencing singular derivatives when one of the maneuvers vanishes, i.e., approaches zero.15 The optimal solution is also required to satisfy a set of inequality constraints, g (X) 0, as well as equality constraints, h (X) = 0. The constraints include the traditional continuity constraints≤ enforced in the multiple shooting algorithm in addition to any other user-defined constraints. Thus, the optimization problem for each stationkeeping maneuver is formally stated, Minimize J (X) = ∆V 2 || || Subject to: g (X) 0 ≤ h (X) = 0

Ti 0 ≥

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American Institute of Aeronautics and Astronautics where the variables associated with the trajectory segment integration time, Ti, are also explicitly required to be non-negative. In this analysis, this optimization problem is solved using sequential quadratic programming (SQP) via the MATLAB function fmincon because the problem is nonlinear and SQP is able to intrinsically incorporate inequality and equality constraints as well as bounds on the design variables. While not required, analytical gradients of the objective function and constraints are supplied to improve convergence. For this optimal orbit maintenance problem, the long-term stationkeeping algorithm is executed as discussed in Section III.B, with the substitution of SQP in place of a minimum norm update to compute each maneuver.

IV. Optimal Long-Term Stationkeeping Results

The long-term stationkeeping strategy is robust, flexible, and can be integrated within the context of a direct optimization problem in a straightforward manner. The augmented maneuver planning method is first explored in the circular restricted three-body problem to validate and explore empirical results obtained during an ARTEMIS post-mission analysis. The optimal long-term stationkeeping strategy is also applied to a variety of trajectories in a higher-fidelity ephemeris model. Lastly, lessons learned from previous ARTEMIS stationkeeping activities are used to compute optimal maneuvers without the use of gradient-based optimization software.

IV.A. Optimal Long-Term Stationkeeping in the CR3BP The ARTEMIS mission is comprised of two spacecraft – P1 and P2 – and was the ﬁrst mission to successfully orbit in the vicinity of an Earth-Moon libration point. The two spacecraft were maintained in L1 and L2 quasi-periodic orbits using a short-term stationkeeping approach that utilized direct optimization to compute stationkeeping maneuvers that maintain the spacecraft 1-2 revolutions downstream by bounding the x- velocity at a series of x-axis crossings.13 Post-processing the maneuver data revealed that the ∆V directions + − aligned closely with the stable mode direction, i.e., νs or νs , for virtually all of the 60+ stationkeeping maneuvers.9, 10 Figure6 demonstrates the alignment between the ∆ V vector and the stable mode direction for one of the maneuvers actually implemented by the ARTEMIS P1 spacecraft during mission operations. One revolution of the ARTEMIS P1 L2 quasi-periodic trajectory is projected into the xy-plane and appears along with the local stable (blue) and unstable (red) manifolds and ∆V direction (black) at a particular stationkeeping maneuver location. This is a signiﬁcant result considering that the optimizer determined

−0.2 Earth, Moon 6 −0.4 Earth 4 −0.6 −0.8 2 km) km) −1 4 4 0 10 Moon L2 10 −1.2 × × ( −2 ( y y −1.4 ∆V −4 ∆V −1.6 −1.8 −6 −2 0 5 10 7.5 8 8.5 x ( 104 km) x ( 104 km) × × (a) (b)

Figure 6. Alignment between ∆V Direction and the Stable Eigenvector

these maneuver directions using only a gradient-based search and did not possess any a priori knowledge of the orbit stability characteristics. Yet, the optimizer eﬀectively sought to return the spacecraft to the

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American Institute of Aeronautics and Astronautics vicinity of the reference path. Employing the optimal long-term stationkeeping strategy, this analysis ﬁrst seeks to demonstrate that the alignment of the stationkeeping ∆V vectors with the stable mode directions is not unique to the ARTEMIS mission. To examine the stationkeeping history, consider a reference L2 planar Lyapunov and three-dimensional halo orbit in the circular restricted three-body problem that can both be maintained for 12 revolutions, that is, approximately six months. The reference L2 Lyapunov and halo orbits are depicted in Figures 7(a) and 7(b), respectively. The goal of this initial investigation is conﬁrmation that, consistent

(a) L2 Lyapunov Orbit (b) L2 Halo Orbit

Figure 7. Periodic CR3BP Orbits

with the ARTEMIS spacecraft trajectories, the locally-optimal long-term stationkeeping maneuvers align with the stable modes of the associated orbits. To ensure that the ∆V directions are not biased by the speciﬁc stationkeeping constraints incorporated for the ARTEMIS trajectories, three constraint strategies are applied to the Lyapunov and halo orbits in the CR3BP in the form of three sample cases: Case 1. Using extra revolutions to achieve desired behavior (minimal endpoint constraint) Case 2. Bounding x-position at the ﬁnal xz-plane crossing Case 3. Departing the orbit via an unstable manifold and targeting a desired set of lunar arrival conditions

For Case 1, a 16-revolution reference solution for both the planar L2 Lyapunov and 3-D halo orbits is generated via multiple shooting. Then, each long-term stationkeeping maneuver targets four revolutions farther downstream than the spacecraft is actually intended to travel. The only required end-point constraint is that the spacecraft return to the xz-plane after 16 revolutions, i.e., yf = 0. Case 1 is intended to represent the least constrained example. Case 2 employs a 12-revolution reference trajectory and stipulates that, at the final xz-plane crossing, the x-position of the spacecraft lies within the bounds [α, β], i.e., α xf β. The Case 2 constraints are illustrated in Figure8. For this analysis, the bounds are defined to≤ be 100≤ km on either side of the final x-position of the 12-revolution reference orbit, thus, β α = 200 km. The reference path for the final sample constraint, Case− 3, completes 12 revolutions of the libration point, departs the orbit via an unstable manifold trajectory, and arrives in the lunar vicinity at a periapse altitude of 1500 km. The reference trajectories representing the L2 Lyapunov trajectory and the 3-D halo orbit for this example are depicted in Figure 9(a) and 9(b), respectively. The orbit maintenance ∆V maneuvers are computed via the long-term stationkeeping strategy using lunar periapse altitude as the only end-of-mission constraint. Optimal stationkeeping costs are computed for each of the three cases, representing three types of constraints, using the long-term stationkeeping strategy outlined in Section III.B. The optimal stationkeeping costs are compared to results computed with a non-optimal long-term stationkeeping strategy, i.e., utilizing multiple shooting and a minimum norm update instead of direct optimization, and a 16-revolution reference consistent with constraint Case 1. For each case, the mean stationkeeping cost to maintain the orbit for 12 revolutions – requiring 23 stationkeeping maneuvers – is computed using 500 Monte Carlo trials incorporating conservative 1 σ navigation/modeling errors of 1 km and 1 cm/s, respectively, and 1 σ maneuver − −

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American Institute of Aeronautics and Astronautics Figure 8. Long-Term Stationkeeping Example

(a) L2 Lyapunov Orbit (b) L2 Halo Orbit

Figure 9. Reference Solutions Incorporating Departure Along Unstable Manifold execution errors of 1%. The alignment between the stationkeeping ∆V vectors and the stable mode directions is assessed by computing the angle, θ, between the vectors for each maneuver. The deﬁnition of the angle θ is illustrated in Figure 10.

Figure 10. Angle, θ, Between ∆V Vector and Stable Eigenvector

Optimal and non-optimal variations of the long-term stationkeeping strategy are successfully employed to maintain both the L2 Lyapunov and the halo orbit for 12 revolutions using the three sample end-of-mission constraints. A ∆V cost comparison is summarized in Table2 and a number of trends are readily apparent. As anticipated, the optimal long-term stationkeeping strategy reduces the cost associated with maintaining

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American Institute of Aeronautics and Astronautics Table 2. CR3BP Optimal Stationkeeping Cost Comparison

Orbit Case Mean ∆Vtot (m/s) Mean θ (deg) Median θ (deg) Non-optimal 10.64 39.89 57.28 Optimal, Case 1 8.03 5.84 1.59 L2 Lyapunov Optimal, Case 2 8.06 9.08 1.73 Optimal, Case 3 8.05 7.99 2.50 Non-optimal 10.48 35.92 43.96 Optimal – Case 1 8.30 15.84 3.88 L2 Halo Optimal – Case 2 8.49 7.93 0.13 Optimal – Case 3 8.46 3.73 0.12

the orbits in the CR3BP – by approximately 24% for the L2 Lyapunov orbit and 19-21% for the L2 halo orbit. It is also clear that the reduction in ∆V is correlated with a reduction in the angle between the ∆V vector and the stable mode direction, θ. Consistent with the ARTEMIS spacecraft trajectories and the corresponding maneuver history, the optimal maneuvers computed using the long-term stationkeeping strategy do align closely with the stable mode direction, particularly when the median angle is considered. The discrepancy between the mean and median angles is likely the result of the direct optimization scheme converging on locally optimal solutions that, in some cases, are far from the stable mode direction. Of course, these outliers skew the mean value. Also notable, in both CR3BP orbits, the Case 1 constraints result in the lowest stationkeeping costs, followed by Case 3 and Case 2. This is due to the fact that Case 1 requires the least restrictive orbit constraints while Case 2 enforces the most rigid requirements. The fundamental results, however, conﬁrm that the optimal long-term stationkeeping strategy is capable of reliably computing maneuvers satisfying a variety of end-of-mission constraints and that the optimal ∆V directions align closely with the stable mode direction as empirically demonstrated in the ARTEMIS stationkeeping results.

IV.B. Optimal Long-Term Stationkeeping in the Ephemeris Model The optimal long-term stationkeeping strategy is an inherently flexible formulation and is easily adapted to maintaining orbits in higher-fidelity models. In this analysis, a Moon-Earth-Sun point mass model incorporating DE405 ephemerides is utilized. The method is applied to higher-fidelity analogues of the L2 Lyapunov trajectory and the 3-D halo orbit previously introduced, but is applicable to more complex trajectories as well.

IV.B.1. Application to an L2 Lyapunov Trajectory and an L2 3-D Halo Orbit Similar to the evaluation of the long-term stationkeeping strategy in terms of a circular restricted three- body model in Section IV.A, the method is now implemented to maintain an L2 Lyapunov trajectory and a 3-D halo orbit in an ephemeris model for 12 revolutions. For purposes of demonstration, only the sample constraint in Case 1 is assessed employing a 16-revolution reference solution. Using the 16-revolution planar Lyapunov and 3-D halo reference solutions in the CR3BP as initial guesses, the higher-ﬁdelity reference solutions are computed using a Moon-Earth-Sun point mass ephemeris model and an epoch associated with January 1, 2020 00:00:00 UTC. The converged quasi-periodic L2 Lyapunov and halo ephemeris reference solutions appear in Figure 11. To simulate actual mission stationkeeping costs, 500 optimal and non-optimal Monte Carlo trials are again completed incorporating the same navigation/modeling and maneuver execution errors implemented in the CR3BP trials. For both the ephemeris Lyapunov and halo orbits, the optimal long-term stationkeeping algorithm is again successful in reducing the cost to maintain the trajectories for 12 revolutions as illustrated in Table3. The optimal scheme reduces the orbit maintenance cost by 21% for the ephemeris L2 Lyapunov orbit and by 13% for the ephemeris L2 halo orbit. The overall reduction in ∆V cost in the ephemeris model compared to the results generated in the circular restricted three-body problem can most likely be attributed to a favorable lunar and/or solar phasing associated with the arbitrarily selected simulation epoch.

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American Institute of Aeronautics and Astronautics (a) L2 Lyapunov Orbit (b) L2 Halo Orbit

Figure 11. Quasi-periodic Ephemeris Orbits

Table 3. Ephemeris Optimal Stationkeeping Cost Comparison

Orbit Case Avg ∆Vtot (m/s) Non-optimal 9.60 L2 Lyapunov Optimal 7.63 Non-optimal 9.25 L2 Halo Optimal 8.06

IV.C. Application to the ARTEMIS Mission Trajectories The optimal long-term stationkeeping algorithm is readily applicable to the ephemeris ARTEMIS Earth- Moon libration point orbit trajectories as well. ARTEMIS reference solution design is brieﬂy discussed. Sensitivity of the stationkeeping ∆V cost to simulated navigation/modeling error magnitude is also explored.

IV.C.1. ARTEMIS Reference Trajectory Design The ARTEMIS spacecraft trajectories are considerably more complex than the ephemeris orbits examined in Section IV.B.1, but the underlying principles are the same. The ARTEMIS P1 and P2 spacecraft began in elliptical Earth orbits as part of the THEMIS mission.16 A series of lunar flybys were used to set up a final lunar flyby to transfer the spacecraft into the Sun-Earth dynamical regime.17 Shadowing Sun-Earth and Earth-Moon invariant manifolds, the ARTEMIS P1 and P2 spacecraft were eventually delivered to Earth-Moon L2 quasi-periodic orbit insertion conditions in August 2010 and October 2010, respectively. The present analysis considers only the Earth-Moon libration point orbit phase of the ARTEMIS mission. The construction of the relatively complex reference trajectories, similar to those employed by ARTEMIS, is explained in detail by Pavlak and Howell,14 but the basic trajectory elements include:

1. L2 orbit insertion

2. Some number of L2 revolutions

3. Transfer to L1 via an L2 unstable manifold and an L1 stable manifold

4. Some number of L1 revolutions

5. Transfer to the lunar vicinity via an L1 unstable manifold The individual components are initially designed separately in the circular restricted three-body problem. Planar Lyapunov orbits are “stacked” at L1 and L2 to achieve the desired “loiter” duration in the vicinity of each libration point. Transfers between the L2 and L1 orbits and from L1 to the lunar vicinity are designed using invariant manifold segments. The ARTEMIS P1 and P2 trajectories are ﬁrst converged in the CR3BP with multiple shooting and are then transitioned to the higher-ﬁdelity model to produce the ephemeris reference solutions depicted in Figure 12.

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American Institute of Aeronautics and Astronautics 5 5 L1 L1 Moon Initial Moon State Initial km) km)

4 State 4 0 0 10 10 L2

× L2 × ( ( z z −5 −5

−5 5 −5 5 0 0 0 0 5 −5 5 −5 10 4 10 4 4 y (× 10 km) 4 y (× 10 km) x (× 10 km) x (× 10 km) (a) ARTEMIS P1 (b) ARTEMIS P2

Figure 12. ARTEMIS Baseline Trajectories

IV.C.2. ARTEMIS Optimal Long-Term Stationkeeping The versatility of the long-term stationkeeping strategy, both with and without the direct optimization option, is explored by applying the algorithm to the ARTEMIS P1 and P2 ephemeris trajectories. The method is implemented as in the previous example. However, in comparison to the ephemeris L2 Lyapunov and halo reference solutions from Section IV.B.1, the ARTEMIS trajectories are highly constrained: 1. Fixed initial position and epoch 2. Final lunar periapse altitude speciﬁed 3. Fixed ﬁnal epoch at lunar arrival

The initial position and epoch are specified by an L2 insertion state consistent with the actual orbit determination data and a requirement that the spacecraft must arrive at a lunar periapse altitude of 1500 km at a precise epoch stipulated by the science requirements. The optimal long-term stationkeeping strategy is implemented in Monte Carlo simulations comprised of 500 trials for the ARTEMIS trajectories. But, unlike results for the previous sample L2 Lyapunov and halo orbits, no reduction in stationkeeping cost is achieved when compared with the non-optimal method. For the ARTEMIS P1 spacecraft, both the non-optimal and optimal methods yield average total stationkeeping costs of 14.40 m/s. Similarly, the average total non-optimal and optimal ∆V costs to maintain the ARTEMIS P2 spacecraft are both equal to 13.18 m/s. While these results warrant further exploration, the initial interpretation is that the highly constrained nature of the ARTEMIS long-term stationkeeping problem – and the fixed time of flight, in particular – results in both the traditional multiple shooting algorithm and the direct optimizer converging on very similar, feasible local solutions. However, it is likely that these stationkeeping costs can be reduced if the locations of the maintenance maneuvers, themselves, are optimized. Stationkeeping maneuver location design is an ongoing research effort.

IV.C.3. ARTEMIS Stationkeeping Cost Error Sensitivity Analysis The average total stationkeeping costs obtained via Monte Carlo simulation for the ARTEMIS P1 and P2 trajectories are equivalent to per-maneuver ∆V costs of 45 cm/s and 49 cm/s, respectively. While these costs are consistent with previous Earth-Moon stationkeeping results obtained using similar error levels,12, 13 the costs are higher than those implemented during the actual ARTEMIS spacecraft operations.9 This discrepancy originates in the fact that the 1 σ position and velocity errors of 1 km and 1 cm/s, respectively, as employed in this analysis represent conservative,− pre-mission estimates that are considerably higher than those observed during the ARTEMIS mission.18 To assess the sensitivity of the ARTEMIS stationkeeping cost to these simulated navigation and modeling errors, the ARTEMIS stationkeeping simulations conducted in Section IV.C.2 are repeated incorporating 1/5 and 1/10 of the original position and velocity error levels. The maneuver execution error is ﬁxed at a 1 σ level of 1% for each simulation. The computed stationkeeping ∆V costs that result from each 500-trial Monte− Carlo simulation appear in Table4. The table illustrates a

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American Institute of Aeronautics and Astronautics Table 4. ARTEMIS Stationkeeping Cost Error Sensitivity

Orbit 1 σ Position Error (km) 1 σ Velocity Error (cm/s) Avg ∆Vtot (m/s) − − 1.00 1.00 14.40 ARTEMIS P1 0.20 0.20 2.92 0.10 0.10 1.49 1.00 1.00 13.18 ARTEMIS P2 0.20 0.20 2.64 0.10 0.10 1.32

relationship between position and velocity error and stationkeeping cost that is nearly linear. Thus, reducing the 1 σ errors by a factor of 10 – a level more representative of the errors observed during ARTEMIS ﬂight operations− – reduces the average total stationkeeping cost by approximately an order of magnitude to the per-maneuver ∆V cost levels consistent with the actual values for the ARTEMIS P1 and P2 spacecraft. While the linear relationship between the error and stationkeeping costs is not unexpected, these results are of interest because they demonstrate that the long-term stationkeeping method is capable of computing stationkeeping costs that are consistent with actual libration point mission operations in this regime.

IV.D. Optimally-Constrained Multiple Shooting – Preliminary Results A recent modiﬁcation to the long-term stationkeeping strategy enables the computation of near-optimal orbit maintenance maneuvers without the use of formal optimization procedures. As mentioned previously, the actual ARTEMIS trajectories and associated maneuvers empirically demonstrate a correlation between the directions associated with optimal libration point orbit stationkeeping maneuvers and the local stable mode direction. The new approach, termed “optimally-constrained multiple shooting,” incorporates insight from the ARTEMIS mission libration point orbit operations into the existing non-optimal long-term stationkeeping strategy. Speciﬁcally, the multiple shooting algorithm that is employed to compute each stationkeeping maneuver is augmented with an additional constraint to ensure that each ∆V is computed such that it aligns exactly with the associated stable mode direction, i.e.,

∆Vˆ = ν+ (16) ± s for each maneuver, ∆V . Note that the stable mode direction can be computed in the same dynamical model as the reference trajectory. Equation (16) is expressed as a single constraint in the multiple shooting algorithm with the mathematical expression

2 ∆Vˆ ν+ 1 = 0 (17) · s − The original non-optimal long-term stationkeeping strategy is implemented but employs the new optimally- constrained multiple shooting algorithm as the core. A caveat to the optimally-constrained multiple shooting strategy is necessary since, in a small number of cases, the algorithm may not be able to compute a ∆V that meets all conditions, i.e., satisﬁes trajectory continuity constraints and is aligned with the stable mode direction. In these cases, the additional stable mode alignment constraint is not enforced in the multiple shooting algorithm and a non-optimal maneuver is computed as accomplished previously. To demonstrate the updated scheme, the L2 Lyapunov and halo orbits are maintained for 12 revolutions in the CR3BP using the 16-revolution reference solutions constraints consistent with Case 1. Stationkeeping maneuvers are still executed at all xz-plane crossings. The results of the Monte Carlo simulations incorporating optimally-constrained multiple shooting appear in Table5 with results obtained via the non-optimal and direct optimization methods included for comparison. For both the L2 Lyapunov planar trajectory and the 3-D halo orbit, the optimally-constrained multiple shooting approach produces mean total stationkeeping costs that are consistent and slightly lower than those produced using commercial direct optimization software. It is also important to note that the constraint requiring that the ∆V vector be aligned with the stable mode direction yields convergence in nearly all simulations. Of the 11,500 maneuvers performed in each 500-trial Monte Carlo simulation, 98.9% of L2 Lyapunov stationkeeping maneuvers and 100% of L2 halo maneuvers align with their respective stable mode directions. It is still necessary to verify the results

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American Institute of Aeronautics and Astronautics Table 5. CR3BP Optimally-Constrained Multiple Shooting Stationkeeping Cost Comparison

Orbit Case Avg ∆Vtot (m/s) Avg θ (deg) Median θ (deg) Non-optimal 10.64 39.89 57.28 L2 Lyapunov Direct Optimization 8.03 5.84 1.59 Optimally-Constrained M.S. 8.01 0.93 4.81e-4 Non-optimal 10.48 35.92 43.96 L2 Halo Direction Optimization 8.30 15.84 3.88 Optimally-Constrained M.S. 8.22 4.26e-4 4.22e-4

obtained via the optimally-constrained multiple shooting approach in an ephemeris model, but similar trends are expected. Nevertheless, while the current results are only preliminary, augmenting the long-term stationkeeping algorithm with optimally-constrained multiple shooting represents a promising alternative strategy that is capable of reliably designing low-cost stationkeeping maneuvers for both planar and three-dimensional libration point orbits in a chosen dynamical model without requiring any formal direct optimization software.

V. Conclusion

The long-term stationkeeping strategy supplies a robust, reliable approach for maintaining spacecraft in unstable orbits with minimal human interaction. In this current investigation, the long-term stationkeeping algorithm is expanded to compute locally optimal ∆V maneuvers in both the circular restricted three-body and higher-ﬁdelity ephemeris models. As anticipated, reductions in total mission stationkeeping costs are achieved by incorporating a direct optimizer into the existing algorithm. Both L2 Lyapunov and halo orbits are maintained in multiple dynamical models and reductions in ∆V cost between 12-25% are observed compared to the non-optimal approach. Optimal and non-optimal stationkeeping costs are equal for the tightly constrained ARTEMIS P1 and P2 trajectories. An optimally-constrained multiple shooting approach is developed that rapidly produces near-optimal maneuvers without requiring additional direct optimization software. The use of reference trajectories in the long-term stationkeeping strategy results in rapid, reliable convergence that eﬃciently renders the automated computation of low-cost orbit maintenance maneuvers in a robust algorithm that is readily applicable to a wide variety of libration point orbit mission scenarios.

Acknowledgments

The authors wish to thank David Folta and Mark Woodard for their assistance with research eﬀorts at Goddard Space Flight Center. This work was supported at NASA Goddard and at Purdue University by the NASA Space Technology Research Fellowship (NSTRF) under NASA Grant No. NNX11AM85H.

References

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