EQUULEUS Trajectory Design

Abstract

This paper presents the trajectory design for EQUilibriUm Lunar-Earth point 6U Spacecraft (EQUULEUS), which aims to demonstrate orbit control capability of CubeSats in the cislunar space. The mission plans to observe the far side of the Moon from an Earth-Moon L2 (EML2) libration point orbit. The EQUULEUS trajectory design needs to react to uncertainties of mission design parameters such as the launch conditions, errors, and thrust levels. The main challenge is to quickly design science orbits at EML2 and low-energy transfers from the post-deployment trajectory to the science orbits within the CubeSat’s limited propulsion capabilities. To overcome this challenge, we develop a systematic trajectory design approach that 1) designs over 13,000 EML2 quasi-halo orbits in a full-ephemeris model with a statistical stationkeeping cost evaluation, and 2) identifies families of low-energy transfers to the science orbits using lunar flybys and solar perturbations. The approach is successfully applied for the trajectory design of EQUULEUS.

Introduction

EQUilibriUm Lunar-Earth point 6U Spacecraft (EQUULEUS) will be the first CubeSat mission that explores a libration point orbit at the Earth-Moon L2 (EML2) [9, 18]. The launch opportunity is provided by NASA’s Artemis 1 (formerly known as Exploration Mission 1: EM-1) as one of 13 piggyback CubeSats onboard the Space Launch System rocket. EQUULEUS is developed by JAXA and the University of Tokyo and aims to demonstrate orbit control capabilities by a CubeSat for future lunar missions. In order to reach the EML2 region within the CubeSat’s limited propulsion capabilities, EQUULEUS will take advantage of lunar flybys (LFBs) and solar gravity to follow a low-energy transfer.

Many studies investigated a class of low-energy transfers. Conley [11] theoretically investigated low-energy transfers to the Moon based on the phase-space structure of collinear Lagrange points in the restricted three-body problem. Belbruno and Miller [4] found a low-energy transfer to the Moon, which requires less Δv compared with the high-energy Hohmann-type direct transfer. The mechanism of the low-energy lunar transfer of [4] is based on the use of the solar tidal force [29], which accelerates a trajectory, pumps up its perigee, and reduces \(v_{\infty }\) with respect to the Moon if an apogee is placed in the 2nd or 4th quadrant around the Earth. These transfers are well explained in the dynamical system theory context, as they shadow invariant manifolds of unstable orbits [14, 23, 31, 37]. In past missions, Hiten [40] and Gravity Recovery and Interior Laboratory (GRAIL) [10, 36] demonstrated low-energy transfers to the Moon; Acceleration, Reconnection, Turbulence and Electrodynamics of the Moon’s Interaction with the Sun (ARTEMIS) [1] reached the EML2 libration point orbit exploiting LFBs and solar perturbation. These past investigations and missions provide insight into the dynamical structure of low-energy transfers to the Moon as well as how the theories have been applied to flight missions.

EQUULEUS also plans to perform three types of scientific observations in the cislunar space [18], one of which will observe the far side of the Moon from an EML2 libration point orbit. EML2 halo orbits are chosen as science orbit candidates to provide constant visibility from the Earth and satisfy science requirements. It necessitates us to have stationkeeping operations due to the instability of a halo orbit.

Theoretical investigation on halo orbits is often based on the circular restricted three-body problem (CR3BP) (e.g., [5, 26]). Since the CR3BP is an autonomous Hamiltonian system, a constant energy integral called the Jacobi constant exists in its associated rotating frame, which parameterizes a variety of periodic orbits including halo orbits [5, 21, 26]. While the CR3BP provides insights into dynamical structures in the three-body regime, the real-world dynamics are time-varying and not autonomous systems anymore; instead, natural dynamical substitutes of CR3BP halos in the non-autonomous systems are quasi-halo orbits [7, 12, 21]. Conversion from CR3BP halos to quasi-halos requires numerical techniques to make them fit the time-varying dynamical system. For instance, Guzzetti et al [24] used an interactive catalog for quick orbit design, and [12] proposed a modified orbit refinement scheme using the Lagrangian formalism. As for the stationkeeping operations on libration point orbits, previously proposed strategies include unstable mode canceling (the Floquet mode approach) [22] and the target point method [27]. These strategies were successfully applied to stationkeeping planning within realistic constraints [15, 16, 28, 32, 33].

Comparing with these past missions and researches, EQUULEUS has two critical mission constraints that make its mission analysis challenging. The first constraint is the CubeSat’s limited Δv budget and thrust levels due to the small bus. The second one is the constrained initial condition (IC): a piggyback payload does not have any control over the launch conditions, and the ICs can be subject to last minute changes due to the primary payload’s final trajectory. Such constraints are typically shared by secondary payload CubeSats; for example, the Near Earth Asteroid (NEA) Scout mission, which is another Artemis 1 secondary payload, carried out contingency trajectory design and analysis [34].

To overcome the challenge, we develop a systematic trajectory design approach consisting of transfer and science orbit phases. The basic idea of the approach is to patch the two phases and then optimize the discontinuous trajectories to yield continuous, almost ballistic trajectories. Each of the phases addresses the challenging tasks in the following two paragraphs.

The transfer trajectory design addresses three main tasks. The first task is to provide a robust and flexible framework of the transfer design. Due to the constrained IC, which targets a low-altitude LFB followed by a ballistic escape from the Earth-Moon system, EQUULEUS must change the first LFB condition with its thruster to remain within the Earth-Moon system. Given the potential future changes of the ICs, the transfer design needs to be capable of dealing with various ICs. The second task is to minimize Δv1 (the first burn before the first LFB) as much as possible. The current operation plan has only 6 days to complete the check-out operations, Δv1 execution, orbit determination (OD), and trajectory correction maneuvers (TCMs) before the first LFB [9]. Considering the limited thrust capability of \(\sim 3\) mN, the operations before the first LFB will be tight, and therefore smaller Δv1 is desirable. The third task is to produce almost ballistic transfer trajectories, since the Δv budget is limited to 77 m/s for the whole mission, including TCMs and clean-up maneuvers (CUMs) before and after LFBs, stationkeeping Δv on the science orbit, and attitude control maneuvers.

The science orbit design addresses two primary tasks. The first task is to design a huge number of science orbits in a full-ephemeris model. Since the transfer phase using the lunisolar gravity perturbations can lead to a variety of orbit insertion (OI) conditions, this phase needs to design many orbits in order to find as many promising pairs as possible when patched with the transfer phase. The second task is to evaluate the stationkeeping cost for each of the orbits, since the stationkeeping cost may occupy a large fraction of the limited Δv budget. To obtain reliable cost evaluation results, the statistical stationkeeping cost must be quantified under possible stochastic errors such as navigation and maneuver execution errors.

The two phases are patched by propagating their trajectory legs forward and backward in time. Our current approach patches the legs at first apogees after the first LFB. We note that our previous method patched them at perilunes [30, 41, 42]. However, the current approach has two advantages over the previous method. 1) The current method enables easier designs of transfer trajectories using one LFB because it is difficult to extract good initial guesses from forward-propagated states at first LFBs. On the other hand, substantial differences arise after the first LFB due to the sensitivity of trajectories near the Moon. Note that a greater number of LFBs result in larger navigation Δv and more challenging operations. 2) It is easier to predict Δv to patch forward and backward legs before optimization as detailed in “Transfer Trajectory Design Results”. In our previous method, the sensitivity of perilune states due to the lunar gravity makes the Δv prediction difficult.

The rest of this paper is structured as follows. After providing some backgrounds in “Background”, an overview of the trajectory design approach is illustrated in “Trajectory Design Approach”. Sections “Transfer Phase” and “Science Orbit Phase” present our approach to the design of transfer trajectories and science orbits, respectively. Section “Results” shows and discusses the trajectory design results. Section “Conclusions” concludes this paper. Note that, at the time of writing, the actual launch states are not known yet; the results in this paper use one of the notional ICs provided by NASA for the launch on 2018 OCT 07.Footnote 1 Our approach has been demonstrated also with other notional ICs provided by NASA (e.g., [3, 13]).

Background

Mission Overview

EQUULEUS is a 6U CubeSat jointly developed by the University of Tokyo and JAXA, with the following mission objectives: 1) demonstration of the orbit control capability in the cislunar space via the CubeSat’s limited propulsion capabilities and 2) scientific observations in the cislunar space.

The propulsion system, which is newly developed for EQUULEUS, uses water as the propellant [2]. Under the current design, its expected capabilities are characterized as 3.3 mN of thrust magnitude, 70 sec of Isp, and 77 m/s of Δv budget for the whole mission. The Δv budget is derived from the 1.22 kg of water propellant planed to be onboard.

EQUULEUS has three scientific instruments [18]: Plasmaspheric Helium ion Observation by Enhanced New Imager in eXtreme ultraviolet (PHOENIX), an extreme UV imager to observe the Earth plasmasphere; Cis-Lunar Object detector within THermal insulation (CLOTH), dust detectors to assess the solid object distribution in the cislunar space; and DEtection camera for Lunar impact PHenomena IN 6U Spacecraft (DELPHINUS), a pair of cameras to detect meteors impacting on the far side of the Moon. Figure 1 shows a conceptual figure of the EQUULEUS mission. DELPHINUS will perform its observations from an EML2 libration point orbit. Among several types of libration point orbits, EML2 halo orbits are chosen as candidate science orbits to provide constant visibility from the Earth with wide varieties of the lunar far side observation geometries.

Fig. 1
figure1

A conceptual figure of the EQUULEUS spacecraft observing the lunar far side

CR3BP

The initial guesses of the EQUULEUS science orbits are generated based on the circular restricted three-body problem (CR3BP). The CR3BP models dynamics of a massless spacecraft under the point-mass gravity attractions of two primaries m1 and m2 (mEarth and mMoon in our case) that move on circular orbits around their barycenter (BC) [39]. The CR3BP dynamics are described in a rotating frame centered at the Earth-Moon BC, where the x-axis is aligned with the Earth-Moon line, the z-axis is aligned with the orbit normal, and the y-axis forms the right-hand system. This paper calls the rotating frame Earth-Moon Hill frame. The CR3BP dynamics are expressed in the rotating frame as follows:

$$ \ddot{x}-2\dot{y}=\frac{\partial U}{\partial x},\quad \ddot{y}+2\dot{x}=\frac{\partial U}{\partial y},\quad \ddot{z}=\frac{\partial U}{\partial z}. $$
(1)

U is the pseudo-potential function defined as:

$$ U = \frac{1}{2}(x^{2}+y^{2})+\frac{1-\mu}{d}+\frac{\mu}{r}, $$
(2)

where \(d=\sqrt {(x+\mu )^{2}+y^{2}+z^{2}} \) and \(r= \sqrt {(x-1+\mu )^{2}+y^{2}+z^{2}}\). Note that the system is nondimensionalized by the scaling parameters

$$ \begin{array}{@{}rcl@{}} l_{\text{unit}}& =& l_{\text{EM}}\qquad\qquad\qquad\qquad\qquad(\mathrm{distance\ unit}),\\ t_{\text{unit}} &=& \sqrt{l_{\text{EM}}^{3}/(\mu_{\text{Earth}}+\mu_{\text{Moon}})}\quad ~\quad\quad(\mathrm{time\ unit}), \end{array} $$
(3)

so that the dynamics are independent from time (instead, the scaling parameters are time-varying). lEM is the Earth-Moon distance. μEarth and μMoon are the gravitational constants of the Earth and Moon, respectively. The Jacobi constant is computed by \(J_{C} = 2U - \dot {x}^{2}- \dot {y}^{2}- \dot {z}^{2}\). Since the equations of motion are symmetric about the x-z plane, an orbit is periodic if and only if the two x-z plane crossings in one revolution are both perpendicular to the x-z plane.

Target point method

The present paper evaluates the statistical stationkeeping cost based on the target point method [27]. The target point method provides an analytical expression of an optimal stationkeeping Δv that minimizes the weighted sum of Δv and future position deviations from the reference orbit. The analytical expression of Δv aids in reducing the computational complexity of our stationkeeping cost evaluation procedure as detailed in “Stationkeeping analysis”.

The target point method is briefly reviewed in the following. Let M and \( {m}_{i}\in \mathbb {R}^{3} \) (i = 1,2,...,M) represent the number of future targeting points and the position deviation at the i-th targeting point, respectively. Then the optimal control problem for the targeting is formulated as:

$$ \begin{array}{@{}rcl@{}} \min_{\Delta {v}_{\text{plan}}} {\Delta}{v}_{\text{plan}}^{\top} {\Delta}{v}_{\text{plan}}+\sum\limits_{i=1}^{M}{{m}_{i}^{\top} R_{i}{m}_{i}}, \end{array} $$
(4)

where \({\Delta } {v}_{\text {plan}}\in \mathbb {R}^{3}\) is a stationkeeping control input; \(R_{i}\in \mathbb {R}^{3\times 3} \) is a weight matrix for the i-th target point. Let \({\Phi }_{\tau _{2},\tau _{1}}\in \mathbb {R}^{6\times 6}\) represent a State Transition Matrix (STM) from time t = τ1 to τ2 along a reference trajectory. The STM, \({\Phi }_{\tau _{2},\tau _{1}}\), can be expressed with 3 × 3 sub-block matrices \(A_{\tau _{2}, \tau _{1}}, B_{\tau _{2}, \tau _{1}}, C_{\tau _{2}, \tau _{1}}\), and \(D_{\tau _{2}, \tau _{1}}\) as

$$ \begin{array}{@{}rcl@{}} {\Phi}_{\tau_{2}, \tau_{1}}= \left[\begin{array}{ll} A_{\tau_{2}, \tau_{1}} & B_{\tau_{2}, \tau_{1}} \\ C_{\tau_{2}, \tau_{1}} & D_{\tau_{2}, \tau_{1}} \end{array}\right]. \end{array} $$
(5)

Let p(tc) and e(tc) be the estimates of the position and velocity deviations, respectively, from the reference at time t = tc (e.g., OD cutoff time). The future position deviation mi after a Δv execution at t = tDV can be approximately expressed as:

$$ \begin{array}{@{}rcl@{}} {m}_{i}= B_{t_{i}, t_{c}}{e}(t_{c})+B_{t_{i}, t_{\text{DV}}}{\Delta}{v}_{\text{plan}}+A_{t_{i}, t_{c}}{p}(t_{c}). \end{array} $$
(6)

Then we have the analytical solution of Eq. 4 as [27]:

$$ \begin{array}{@{}rcl@{}} &&{\Delta}{v}_{\text{plan}} = \sum\limits_{i=1}^{M} \alpha_{i} {e}(t_{c}) + {\upbeta}_{i} {p}(t_{c}), \end{array} $$
(7)
$$ \begin{array}{@{}rcl@{}} &&\qquad\ \ \ \begin{array}{ll} \alpha_{i} &= -[I_{3} + B_{t_{i},t_{\text{DV}}}^{\top} R_{i} B_{t_{i}, t_{\text{DV}}}]^{-1}\cdot B_{t_{i}, t_{\text{DV}}}^{\top} R_{i} B_{t_{i}, t_{c}},\\ {\upbeta}_{i} &= -[I_{3} + B_{t_{i},t_{\text{DV}}}^{\top} R_{i} B_{t_{i}, t_{\text{DV}}}]^{-1}\cdot B_{t_{i}, t_{\text{DV}}}^{\top} R_{i} A_{t_{i}, t_{c}}. \end{array} \end{array} $$
(8)

Trajectory optimization software

A trajectory optimization software jTOP is used in optimization processes of our analysis, including the design of transfer trajectories and quasi-halo orbits. jTOP optimizes trajectories via the direct multiple-shooting method using the SNOPT optimization package [20] and covers from low-fidelity to high-fidelity dynamics and both impulsive and finite thrust maneuvers. The capability successfully contributed to the trajectory design of PROCYON [8], the first deep-space micro spacecraft [17].

Trajectory Design Approach

Figure 2 shows a flowchart of our trajectory design approach. The approach patches two legs at apogees in the Apogee patching process, which are propagated forward (Forward propagation process) and backward (Backward propagation process) in time from the separation and quasi-halo orbits around the EML2 (computed in the Halo generation process), respectively, and optimizes them to minimize total Δv in the Optimization process.

Fig. 2
figure2

A flowchart of the trajectory design approach for EQUULEUS

The following two sections describe the design of the transfer trajectory and science orbit phases. Section “Transfer Phase” highlights each process in Fig. 2 (except for the Halo generation process), and “Science Orbit Phase” addresses the Halo generation process, which presents design and stationkeeping analysis of the science orbits.

Transfer Phase

Forward Propagation Process

The forward propagation process propagates trajectories forward in time from the separation for 210 days including gravitational effects of the Sun, Earth, and Moon as point masses, and saves the epochs and the states of every apogee pass. After one day from the separation, Δv1 is applied to change the first LFB condition. The computation is a three-dimensional grid search for Δv1 in terms of the magnitude, right ascension 𝜃 and declination ψ with respect to the ecliptic.

Table 1 summarizes the search set and corresponding number of grid points in this analysis.

Table 1 Grid search parameters for forward propagation

Figure 3 shows an example of a trajectory propagated forward in time after Δv1 = 2 m/s. The first lunar flyby causes a high out-of-plane component at the apogee, where most of the backward legs from quasi-halo orbits (see “Backward propagation process” cannot reach due to the associated manifold flows. We confirm that other possible combinations of 𝜃 and ψ with Δv1 = 2 m/s also result in highly inclined orbits. On the other hand, larger Δv1 results in a variety of orbits after the first LFB, including less inclined trajectories that experience closer encounters with the backward legs. Note that the results of the forward propagation strongly depend on the launch condition.

Fig. 3
figure3

An example of a trajectory (blue) propagated forward in time with Δv1= 2 m/s, right ascension 𝜃 = 250, and declination ψ = 0 in the Earth-centered J2000 frame. The arrows indicate the directions of motion

Figure 4 shows values of (a) x, (b) y, and (c) z at apogees in the Earth-centered J2000 frame in terms of the epoch as the result of the grid search. The forward and backward legs (see “Backward propagation process” and “Apogee patching process”) are patched at first apogees of the forward legs to simplify the analysis. According to Fig. 4, ranges of time and positions of the first apogees are found to be

$$ \begin{array}{@{}rcl@{}} &&\text{2018~NOV~09} \leq \text{epoch~(UTC)} \leq \text{2018~DEC~31}, \\ &&-12.5\times10^{5} \leq x \leq -5\times10^{5} \ \text[km], \\ &&-3.8\times10^{5} \leq y \leq 16\times10^{5} \ \text[km], \\ &&0 \leq z \leq 12\times10^{5} \ \text[km]. \end{array} $$
(9)

Note that the strong nonlinearity of the lunisolar gravity perturbations scatters the collective states at the first apogee. The data of the backward legs satisfying Eq. (9) is extracted to reduce the number of possible combinations, but it is also possible to patch at other apogees to produce more families of solutions if necessary.

Fig. 4
figure4

ax, by, cz in terms of the epoch at apogees of trajectories propagated forward in time as the result of the grid search. The states at the first apogee are scattered due to the strong nonlinearity of the lunisolar gravity perturbations

Backward propagation process

The backward propagation process propagates trajectories backward in time for 240 days from Earth-Moon quasi-halo orbits including gravitational effects of the Sun, Earth, and Moon as point masses, and saves the epochs and the states of every apogee pass. In order to escape in the fastest way, we apply small Δv along the minimum stretching directions of 1,000 equally spaced nodes on candidate quasi-halo orbits.

The minimum stretching directions are defined as in Shadden et al [38] and minimize the 2-norm of the final variation at t = tf for a given initial perturbation δx0:

$$ {\left\Vert\delta {x}_{\mathrm f}\right\Vert}_{2} = \sqrt{ \delta {x}_{0}^{\top} {\Phi}_{t_{\mathrm{f}},t_{0}}^{\top} {\Phi}_{t_{\mathrm{f}},t_{0}} \delta {x}_{0} }, $$
(10)

where \({\Phi }_{t_{\mathrm {f}},t_{0}}^{\top } {\Phi }_{t_{\mathrm {f}},t_{0}}\) is the Cauchy-Green tensor. Based on Eq. 10, the minimum stretching direction corresponds to the eigenvector associated to the minimum eigenvalue of the Cauchy-Green tensor, which was used in the computation of Lagrangian coherent structures [19, 25] and an analysis of the low-energy gravitational capture of BepiColombo to Mercury [6]. The trajectories can be regarded as an extension of stable manifolds in autonomous systems to non-autonomous systems.

Figure 5 shows sample trajectories propagated backward in time by applying small Δv(= 0.1 m/s) to minimum stretching directions from one quasi-halo orbit including gravitational effects of the Sun, Earth, and Moon as point masses. EQUULEUS aims to get on one of these time-dependent analogs of stable manifolds [19] to be captured into a quasi-halo orbit with small insertion Δv.

Fig. 5
figure5

Sample trajectories propagated backward in time from a quasi-halo orbit around the EML2 in the Moon-centered, Earth-Moon line fixed rotating frame

Apogee patching process

The apogee patching process patches forward (“Forward Propagation Process”) and backward (“Backward propagation process”) legs at first apogees. We only use the data at apogees of the backward propagation that satisfy Eq. (9). A knn-search algorithm (MATLAB®;’s knnsearch function) is used to find k-nearest neighbors in terms of time and position at apogees, and extract pairs such that differences of time and distance between apogees are less than prescribed tolerances (TOLE.t and TOLE.rmag). If TOLE.t and TOLE.rmag are small enough, the magnitude of the difference of velocities of forward and backward pairs at apogees (discontinuity Δv) may be a good criterion to measure the quality of initial guesses, which is ideally equal to |Δvtotal| − |Δv1| after optimization (Δvtotal: total Δv for a transfer).

Figure 6 shows forward (Δv1 = 6 m/s, blue dots) and backward (green dots) states at apogees, displayed with those satisfying tolerances = 0.025 days and = 25,000 km (gray circles for forward and red crosses for backward) in the position and velocity planes. Due to the strict tolerance in terms of the position TOLE.rmag, gray circles and red crosses almost overlap in the position planes, whereas the differences in the velocities correspond to Δv required to patch forward and backward legs.

Fig. 6
figure6

Forward (Δv1= 6 m/s, blue dots) and backward (green dots) states at apogees, displayed with those satisfying the tolerances TOLE.t= 0.025 days and TOLE.rmag= 25,000 km (gray circles for forward and red crosses for backward) in ax-y, bx-z, cvx-vy, dvx-vz planes in the Earth-centered J2000 frame

Optimization process

The optimization process optimizes the initial guesses satisfying tolerances TOLE.t and TOLE.rmag (gray and red pairs in Fig. 6) to obtain continuous trajectories of minimal total impulsive Δv by jTOP using the direct multiple shooting scheme. The optimization process includes gravitational effects of the Sun, Jupiter, Earth, and Moon as point masses, where arcs are propagated forward and backward in time from each phase, and additional phases are introduced in sensitive regions such as flybys to improve the convergence. Feasibility and optimality tolerances for SNOPT are set to be 10− 8. The inclusion of solar radiation pressure is out of the scope of this analysis because it does not substantially affect the values of Δv nor the proposed trajectory design approach.

Science Orbit Phase

Science Orbit Design

Figure 7 shows a flowchart of our approach to the science orbit design. The approach consists of three processes: 1) CR3BP halo orbit design, 2) Quasi-halo orbit design, and 3) Continuation. Each process is highlighted in the following subsections.

Fig. 7
figure7

A flow chart of the science orbit design

CR3BP Halo Orbit Design

The CR3BP halo orbit design process designs nondimensional CR3BP halo orbits of various Jacobi constants. A bisection method is used to compute the periodic halos, imposing the two x-z plane crossings of an orbit to be perpendicular to the x-z plane under the system Eq. (1). The computation produces periodic orbits and their periods τHalo (both nondimensional). Figure 8I shows an example of a nondimensional halo orbit from the CR3BP.

Fig. 8
figure8

Quasi-halo orbit design procedure with example orbits. (I): A nondimensional CR3BP halo orbit. (II): A dimensionalized CR3BP halo, realized by propagating dimensionalized nodes of a CR3BP halo in the ephemeris model (also an initial guess of the optimization process). (III): An optimized quasi-halo orbit in the ephemeris model. (IV): An epoch-shifted quasi-halo orbit (also an initial guess of the continuation process). Each trajectory is shown in the Moon-centered, Earth-Moon line fixed rotating frame. The magenta and green arcs respectively represent forward and backward propagations

Quasi-Halo Orbit Design

The quasi-halo orbit design process designs quasi-halo orbits in a full-ephemeris model, which includes gravitational effects of the Sun, Earth, Moon, Mercury, Venus, Mars BC, Jupiter BC, Saturn BC, Neptune BC, and Pluto BC (point masses). A CR3BP halo is used as the initial guess. An orbit is represented by a set of stacked nodes, where each node contains its epoch and state (position and velocity in the Moon-centered inertial frame) and durations of forward and backward propagations that connect the node with the next nodes. It consists of two subprocesses: Dimensionalization process and Optimization process.

(a) Dimensionalization process

The process 2.(a) dimensionalizes a CR3BP halo orbit to produce the initial guess for the full-ephemeris optimization. The dimensionalization units are the same as the scaling parameters in Eq. 3. Note that, since the Earth-Moon distance lEM varies depending on the time, lunit must be taken at the epoch we want to dimensionalize the node. tunit also varies according to the variation of lunit.

The dimensionalization process is as follows. Defining t0 as the initial epoch of a quasi-halo orbit, the first node is realized by dimensionalizing the apolune state using lunit and tunit at t0. Let TQH be the duration of a quasi-halo orbit to be designed in the ephemeris model; TQH = 180 days is chosen in our mission design to prepare long enough science orbits. Equally spaced epochs are defined at tj = t0 + τHalotunitj/2, (j = 1,2, ..., n), which places nodes around apo- and peri-lunes in the dimensionalized system. The maximum index of j (i.e., n) is chosen so that (tnt0) is about TQH. Then each node is dimensionalized by lunit and tunit at its associated epoch. The dimensionalized nodes are converted to the Moon-centered inertial frame from the Earth-Moon Hill frame (defined in “CR3BP”).

The stacked, dimensionalized nodes do not realize a continuous orbit in the ephemeris model anymore. If one propagates these nodes forward and backward in time for τHalotunit/2 in the ephemeris model, it yields discontinuities at the ends of the propagated arcs due to perturbations that are unmodeled in the CR3BP. An example of the dimensionalized halo shown in Fig. 8II displays noticeable gaps between the ends of the propagated arcs.

(b) Optimization process

The process 2.(b) takes the stacked, dimensionalized nodes as the initial guess and yields a continuous, ballistic quasi-halo orbit (Fig. 8III) by minimizing the total of the discontinuity Δv down to zero in the full-ephemeris model. The optimization is performed by jTOP with a tolerance of 10− 8 for SNOPT (both feasibility and optimality).

Two kinds of constraints are imposed in the optimization. One (C-1) is on all of the apolune nodes, and the other (C-2) is on the first and final nodes of the trajectory to be optimized. C-2 can be either on apo- or peri-lunes depending on the number of nodes of a quasi-halo orbit. The first node is constrained by both C-1 and C-2. C-1 constraints the apolune nodes to be y = 0 and \(z\in [z_{\min \limits }, z_{\max \limits }]\) (almost constant z-amplitude); and C-2 imposes the first and final nodes to be at the perilunes with y = 0 and \(x\in [x_{\min \limits }, x_{\max \limits }]\) (perpendicular crossing with respect to the x-z plane). Different values of \(z_{\min \limits }\) and \(z_{\max \limits }\) (C-1) and \(x_{\min \limits }\) and \(x_{\max \limits }\) (C-2) are chosen depending on the families of halos as listed in Table 2.

Table 2 Constraints in the quasi-halo optimization process

Examples of quasi-halo families after the optimization process are shown in Fig. 9. They include 8 families: northern and southern families with four different values of the Jacobi constants JC, which are labeled 1-8 in the ascending order in terms of the JC values with “north-first” convention. These orbits are selected to provide various levels of the orbit energy of the science orbits (represented by the Jacobi constants) in order to allow a wide variety of OI conditions from the transfers. It is also possible to design more families of quasi-halo orbits to help the patching process find more pairs if necessary.

Fig. 9
figure9

Designed quasi-halo families shown in the Moon-centered, Earth-Moon line fixed rotating frame. The magenta and green arcs respectively represent forward and backward propagation

The properties of the designed quasi-halo families are summarized in Table 3, which includes the Jacobi constants of the original CR3BP halos; north or south classifications; (pseudo-)orbital periods, which are calculated from the differences between the epochs of the first and third nodes (the first and second apolunes); durations of the trajectories; and scaled linear instabilities. The scaled linear instabilities are logarithms of the maximum eigenvalues of the Cauchy-Green tensor (i.e., \(\ln {(\lambda _{\mathrm max}({\Phi }^{\mathrm T}{\Phi }))} \)) scaled by that of the family 1, where the STMs (Φ) are integrated for 50 days from the initial epochs of orbits. The integration duration is chosen in order to evaluate the instabilities over long enough trajectories while avoiding overflows in the STM computation. The quantities of the period and linear instability of each family are represented by those of a quasi-halo orbit with the initial epoch 2019 May 31. The linear instability analysis implies that the families 1 and 2 are least unstable and families 7 and 8 are most unstable among the 8 families.

Table 3 Properties of designed quasi-halo families

Continuation

The continuation process computes families of quasi-halo orbits in order to allow the patching process to have a wide variety of OI epochs. The process produces a series of orbits of different initial epochs for each family; the range of the initial epochs is from 2018 SEP 01 to 2019 OCT 31 with six-hour intervals. The six-hour interval is chosen to make the time resolution high enough compared to the time scale of the dynamics (e.g., quasi-halo period \(\sim 7\)-14 days, the Moon revolution around the Earth \(\sim 27\) days). If necessary, it is also possible to design more orbits for each family to 1) adapt to further postponement of the launch by widening the epoch range or 2) find more patching solutions by increasing the resolution.

This process consists of Epoch-shifting process (3.(a) in Fig. 7) and Optimization process (3.(b)), and repeats them using the output of 3.(b) as a reference to the next epoch-shifting process.

(a) Epoch-shifting process

Taking the output of either 2.(b) or 3.(b) as a reference trajectory, the epoch-shifting process generates an initial guess of the next quasi-halo orbit at a slightly different epoch from the reference’s.

First, convert the coordinate frame of the reference from the Moon-centered inertial to the Moon-centered, Earth-Moon line fixed rotating frame. Then a small time shift (6 hours in our study) is applied to each epoch of all nodes of the reference, without any changes in their positions and velocities. A set of the epoch-shifted nodes is re-converted into the inertial frame (the epoch-shifted initial guess).

While the epoch-shifted nodes no longer represent a continuous quasi-halo orbit due to the time-dependent nature of the dynamical system, the discontinuities are expected to be smaller than those of a dimensionalized CR3BP halo. Figure 8IV shows an example of the propagated arcs from the epoch-shifted nodes, which confirms smaller discontinuities than those of the dimensionalized CR3BP halo in Fig. 8II. In other words, in terms of the discontinuities, this process yields better initial guesses to design epoch-shifted quasi-halo orbits than naively constructing the guesses from the CR3BP solutions.

We emphasize that the epoch-shifting process is performed in the Moon-centered rotating frame in order to reduce the influence of the eccentricity of the lunar orbit around the Earth.

(b) Optimization process

The process 3.(b) takes the epoch-shifted nodes as the initial guess and yields a continuous, ballistic quasi-halo orbit in the same manner as in 2.(b). The smaller discontinuities of the initial guess contribute to better convergence in the optimization process; in our study, the continuation process yields in total 13,362 candidate science orbits (1,704 for each family) in the full-ephemeris model.

Stationkeeping analysis

To evaluate the stationkeeping costs of the candidate science orbits, we perform a Monte-Carlo analysis with N trials. The stationkeeping analysis models stochastic errors of OI, \( {\epsilon }_{\text {OI}} \sim \mathcal {N}( {0}_{6}, P_{\text {OI}}) \), OD, \( {\epsilon }_{\text {OD}} \sim \mathcal {N}( {0}_{6}, P_{\text {OD}})\), and maneuver execution, \( {\epsilon }_{\text {DV}} \sim \mathcal {N}( {0}_{3}, P_{\text {DV}})\), where \(\mathcal {N}( {0},P)\) represents a multivariate Gaussian distribution with zero mean and covariance of P; POI, POD, and PDV are the covariance matrices of the OI, OD, and maneuver execution errors, respectively. Note that 𝜖DV represents the percentage error of the planned maneuver magnitude. Then the perturbed states and Δv are expressed as

$$ \begin{array}{@{}rcl@{}} {x}(t_{\text{OI}}) &=& {x}_{\text{true}}(t_{\text{OI}}) + {\epsilon}_{\text{OI}},\\ {x}(t_{\text{OD}}) &=& {x}_{\text{true}}(t_{\text{OD}}) + {\epsilon}_{\text{OD}},\\ {\Delta}{v} &=& {\Delta} {v}_{\text{plan}}(I_{3}+{\epsilon}_{\text{DV}}). \end{array} $$
(11)

Figure 10 schematically shows a sequence of stationkeeping operations from an OI to the first maneuver execution. An OI occurs at time t = tOI (“△” in Fig. 10), after which a sequence of maneuvers is executed with time interval ΔtDV (“◇”). Before each of the maneuvers, an OD is initiated (“△”) and terminated (“∘”) Δtc before the maneuver execution epoch (OD cutoff time). Following this convention, the duration of the i-th OD operation is ΔtDV −Δtc. Each stationkeeping maneuver targets two downstream points Δt1t2 after an OD’s initiation (“\(\square \)” in Fig. 10); hence our analysis uses M = 2 in Eq. 7. Recall that each maneuver is computed according to the weighting matrices Ri (i = 1,2 in our analysis). After each maneuver execution, the next OD process initiates to prepare for the next maneuver. To represent these parameters compactly, let

$$ \begin{array}{@{}rcl@{}} {\Lambda}=\{R_{i}, {\Delta} t_{\text{DV}}, {\Delta} t_{\mathrm{c}}, {\Delta} t_{i}\}, \end{array} $$
(12)

be the set of the design parameters in the Monte-Carlo simulation.

Fig. 10
figure10

Schematic figure of stationkeeping simulation procedure. Shown in the Moon centered, Earth-Moon line fixed rotating frame, projected onto the x-y plane

Table 4 summarizes parameters assumed in the stationkeeping analysis, which includes covariances of navigation and execution errors and OD, cutoff, and target point time. Note that three cases of Δv interval time are tested (ΔtDV = 1,2,3 weeks) to assess the trade-off relationship between the maneuver execution frequency and stationkeeping cost.

Table 4 Stationkeeping analysis parameters

Algorithm

Since the Δv cost depends on the design parameters in Λ, identifying the optimal parameter set Λ is critical to appropriately evaluate the stationkeeping cost. To identify Λ, the grid search method is used to explore the parameter space of Ri and ΔtDV. A weighting matrix is parameterized by a scalar ri as Ri = riI3. The grid search parameters are summarized in Table 5, which implies that the number of parameter sets to be explored is L = 15 × 15 × 3 = 675. Recalling that N trials are simulated for a set of Monte-Carlo simulations, each grid search procedure requires LN stationkeeping simulation trials for a science orbit. N = 10,000 is used in our analysis. Given we have 13,362 science orbits, the total stationkeeping simulations to be performed are \(13,362\times L\times N \sim 9.0\times 10^{10} \).

Table 5 Grid search parameters for the stationkeeping analysis

The overall procedure to identify Λ and its associated cost is described in Algorithm 1, where k = 1,2,...,L represents the index of parameter sets to be explored; the function (explained in detail later) performs a Monte-Carlo analysis for a given parameter set Λk; \(\bar {\mathcal {V}}(k)\) and \( \sigma _{\mathcal {V}}(k)\) represent the mean and standard deviation of the stationkeeping costs of N trials for Λk. This algorithm returns the optimal expected value of the cost \(\bar {\mathcal {V}}^{*} \) and its standard deviation \(\sigma _{\mathcal {V}}^{*} \) with the optimal parameter set Λ.

figurea

In light of the discussion above, developing an efficient algorithm is critical to carry out the procedure in Algorithm 1 in a reasonable amount of time. The function efficiently performs a set of Monte-Carlo simulations taking advantage of matrix operations. The next paragraph details .

figureb

Main computational complexities of the stationkeeping cost evaluation procedure are due to the orbit propagation and the computation of optimal stationkeeping Δvs. Our approach converts these tasks to matrix operations to reduce the computational load. STMs are used to propagate the spacecraft’s deviation from a reference orbit. This approximation is reasonable for our preliminary analysis because the spacecraft should remain in the vicinity of its nominal path at all times. As for computing stationkeeping Δvs, Eq. 7 gives the analytical solution of Eq. 4. Stacking the state vectors of all N trials to form a matrix enables us to compute the optimal Δvs for each trial at once. The MCSK algorithm is described in Algorithm 2, where δxtrue,δxestvplan, and ΔvDV represent the true and estimated values of the state deviation and values of planned and executed Δv, respectively; tend,tc,tDV, and ti are the final time of the reference trajectory, OD cutoff time, maneuver execution time, and target point time, respectively; \(\mathcal {E}_{\text {OI}}\triangleq [ {\epsilon }_{\text {OI}_{1}}, {\epsilon }_{\text {OI}_{2}},\ ...\ , {\epsilon }_{\text {OI}_{N}}]\), \(\mathcal {E}_{\text {OD}}\triangleq [ {\epsilon }_{\text {OD}_{1}}, {\epsilon }_{\text {OD}_{2}},\ ...\ , {\epsilon }_{\text {OD}_{N}}]\), and \(\mathcal {E}_{\text {DV}}\triangleq [ {\epsilon }_{\text {DV}_{1}}, {\epsilon }_{\text {DV}_{2}},\ ...\ , {\epsilon }_{\text {DV}_{N}}]\) are stacked random vectors representing OI, OD, and execution errors. These random vectors can be generated by a standard Gaussian random number generator; for example, our analysis generates \(\mathcal {E}_{\text {OI}}\) as \(\mathcal {E}_{\text {OI}}\gets \sqrt {P_{\text {OI}}}\ \mathtt {randn}(6,N) \) on MATLAB®;, where \(\sqrt {A}\) for a matrix A satisfies \(\sqrt {A}\sqrt {A}^{\top }=A \). This function performs N trials of Monte-Carlo simulations in one run and returns an array \(\mathcal {V}\in \mathbb {R}^{1\times N} \) that contains the Δv costs for each of N trials.

Results

The proposed trajectory design approach is applied to the EQUULEUS mission design. Sections “Transfer Trajectory Design Results” and “Science orbit design results” present the results of the transfer and science phases, respectively.

Transfer Trajectory Design Results

Figure 11 shows the results of optimizing initial guesses of discontinuity Δv < 100 m/s. In order to avoid falling into the same local minima, we only optimize one initial guess in each family, where families are classified in terms of dates of LFBs and time of flight (TOF). Figure 11a shows the values of TOF and Δvtotal, which reveals several families of solutions and trade-off between TOF and Δvtotal. Figure 11b shows Δv1 and Δvtotal, which indicates that the initial guess of Δv1 (6 m/s, the dotted line) is not much changed via the optimization process. Figure 11c shows the discontinuity Δv at apogees before optimization and |Δvtotal|−|Δv1| after optimization. There is a near-linear relationship (the dotted line) between these values for most of the solutions. Thus, in most cases, the sum of Δv1 and discontinuity Δv at apogees provides good approximations of Δvtotal for optimal solutions, which would help us filter initial guesses to be optimized (we could eliminate initial guesses of too large Δv1 + Δvapogee).

Fig. 11
figure11

a TOF and Δvtotal of optimal solutions. b Δv1 and Δvtotal of optimal solutions. c Discontinuity Δv at apogees before optimization and Δvtotal − Δv1 after optimization

Figure 12a shows a transfer trajectory of the optimal solution with Δvtotal = 11.5 m/s, Δv1 = 6.4 m/s, TOF= 199 days in the Earth-centered, Sun-Earth line fixed rotating frame. The trajectory reaches a science orbit after one LFB only, which would save Δv for trajectory correction maneuvers (TCMs) and relax the complexity of the operation. The important events of this transfer are summarized in Table 6.

Fig. 12
figure12

Optimal transfer trajectories with (a) Δvtotal = 11.5 m/s, Δv1 = 6.4 m/s, TOF= 199 days, and (b) Δvtotal = 13.7 m/s, Δv1 = 5.7 m/s, TOF= 207 days in the Earth-centered, Sun-Earth line fixed rotating frame. The diamonds represent locations of Δv, where the final maneuver, Δv4 in these cases, inserts the spacecraft from the transfer trajectory into the science orbit, and the magenta and green arcs represent forward and backward propagation legs of the multiple shooting scheme of jTOP, respectively

Table 6 Important events of the transfer in Fig. 12a

Figure 12b shows a transfer trajectory of the optimal solution with Δvtotal = 13.7 m/s, Δv1 = 5.7 m/s, TOF= 207 days in the Earth-centered, Sun-Earth line fixed rotating frame. This solution uses multiple LFBs but results in smaller Δv1. Whereas having multiple LFBs contributes to reducing Δv1, the trade-off between the number of LFBs (Δv for TCMs and operation complexity) and the magnitude of Δv1 would need to be taken into account for the selection of the baseline transfer trajectory. Table 7 summarizes the important events of the transfer in Fig. 12b.

Table 7 Important events of the transfer in Fig. 12b

Science orbit design results

As mentioned in “Science Orbit Phase”, the science orbit design process yields over 13,000 quasi-halo orbits in the full-ephemeris model and evaluates their stationkeeping cost by Monte-Carlo simulations. The total CPU time to prepare all of the 13,000+ science orbits was around 16 days on a desktop machine with Intel®; CoreTM i7-6900K processor. Among the designed orbits, this section highlights a portion of the candidate science orbits that meet the criteria of the patching process with transfer trajectories (see “Transfer Phase” for detail). The patching process downselected 13 quasi-halo orbits from the large pool of full-ephemeris trajectories as science orbit candidates. Table 8 summarizes the properties of the selected orbits with the result of the stationkeeping cost analysis.

Table 8 Candidate science orbits with the stationkeeping costs

The table confirms two general trends in terms of the stationkeeping Δv cost. First, it shows that the further the x −position of an orbit apolune is from the Moon, the larger its stationkeeping cost becomes, which is consistent with the instability analysis in “Science Orbit Phase”. Second, the stationkeeping cost is larger for less frequent stationkeeping control opportunities, which makes it infeasible to maintain the orbits within the Δv budget in some cases, as seen in the stationkeeping cost results of ID-2,4,5, etc; it reveals the trade-off relationship between the frequency of operation opportunities and the stationkeeping Δv cost. Since very frequent operation opportunities cannot be expected for CubeSat missions in general, the trade-off relationship is an important factor of practical mission design considerations. On the other hand, the stationkeeping costs for ID-3, 6, and 10 show that the costs with Δv/2weeks are smaller than those with Δv/1week. Potential reasons may include that the frequency of Δv/1week is too high compared to their orbital periods or Δv locations are not optimal in terms of the geometries. The optimal spacing of stationkeeping executions would depend on operation parameters as well as the selection of halo families. Exploring the optimal spacing is out of the scope of this article because our primary objective is to provide a systematic trajectory design approach in complex dynamical systems. Optimal spacing of stationkeeping on unstable orbits is discussed in Renault and Scheeres [35] with a simplified model.

The current baseline trajectory is determined based on the combined results of the transfer and science phases. Recall that the overall Δv budget is 77 m/s. From the stationkeeping cost point of view, the transfer-science pairs of ID-3, 6, and 10 are selected as good candidates (quasi-halo families of 1 and 2). Among the three pairs, we select the pair of ID-3 as the current baseline since it reaches the science orbit with the smallest Δv and TOF. Figure 13 shows a close-up trajectory describing the transfer to the science orbit insertion of the current baseline solution (ID-3) in the Moon centered, Earth-Moon line fixed rotating frame.

Fig. 13
figure13

Baseline science orbit shown in the Moon centered, Earth-Moon line fixed rotating frame. OI epoch: 2019 APR 07 11:05

Conclusions

This paper presented our trajectory design approach for the EQUULEUS mission, which consists of two phases: transfer trajectory and science orbit phases. We developed a systematic approach to 1) compute families of low-energy Earth-Moon transfers using lunisolar gravity forces with the constrained launch condition, and 2) design a large number of quasi-halo orbits with the stationkeeping cost evaluation, both in the high-fidelity ephemeris model. The application of this two-stage approach for the mission analysis of EQUULEUS delivered a baseline trajectory consisting of a 10.2 m/s low-energy transfer that injects the spacecraft in a low-cost quasi-halo orbit (7.4 m/s annual stationkeeping) after 184 days. The systematic approach would benefit mission designers to efficiently design complex low-energy transfers to cheap science orbits within the chaotic dynamics of the Earth-Moon system.

Notes

  1. 1.

    As of September 2019, a new set of ICs for the launch in June 2020 is also available provided by NASA.

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Acknowledgements

Part of the work described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. We thank the former and current members of the EQUULEUS mission analysis team for their support: Chit Hong Yam, Daniel Garcia, Tomohiro Yamaguchi, Quentin Verspieren, Yuki Kayama, Stefano Bonasera, Mattia Pugliatti, Yosuke Kawabata, Takuya Chikazawa, Kento Ichinomiya, Tomoya Kitade, Masahiro Fujiwara, and Diogene A. Dei Tos.

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Oguri, K., Oshima, K., Campagnola, S. et al. EQUULEUS Trajectory Design. J Astronaut Sci 67, 950–976 (2020). https://doi.org/10.1007/s40295-019-00206-y

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Keywords

  • EQUULEUS
  • Low-energy transfer
  • Halo orbit
  • Earth-Moon Lagrange point
  • Multi-body dynamics
  • Optimization
  • CubeSat