Parallel High-fidelity Trajectory Optimization with Application to CubeSat Deployment in an Earth-moon Halo Orbit Hongru Chen1) Mickaël Gastineau1) Daniel Hestroffer1) and Vishnu Viswanathan1) 1) Observatoire de Paris, IMCCE, 75014 Paris, France (Received 24th April 2018) The application of CubeSats to interplanetary missions is currently being exploited. However, trajectories of interplanetary CubeSats are greatly constrained by the very limited propulsion capacity. Electric propulsion systems that enjoy a high Isp as well as a high Δv budget is promising to CubeSats. However, the low thrust of electric systems may not be applicable for some trajectories needing for quick response. In addition to propulsion, the ephemeris and gravity of celestial bodies can exert positive or negative influence on CubeSat trajectories, which results in significant Δv saving or loss to the CubeSat. CubeSat trajectories should be designed as well as verified in a high-fidelity model accommodating the ephemeris and gravity of celestial bodies, and specifications of the CubeSat and the propulsion system. A parallel high-fidelity trajectory optimization tool, PHITO, is presented in this paper. How PHITO is applied to trajectory analysis and propulsion system sizing is discussed through a case study of the phasing trajectory of an Earth-moon halo orbit. The result of Δv reveals the effect of the full-ephemeris n-body dynamics. Key words: Trajectory Optimization, Interplanetary CubeSats, Parallel Computation 1. Introduction applicable for the trajectories requiring quick maneuvers. The application of CubeSats to interplanetary space In addition to propulsion, the gravity of celestial missions is currently exploited. Compared to Earth- bodies exerts positive or negative influence on the orbiting CubeSats, interplanetary CubeSats generally trajectory, depending on the mission date. As need to perform orbit change and deployment. For propellant cost is crucial to a CubeSat, it is necessary instance, the EM-1 mission has selected 13 CubeSats to design and analyze CubeSat trajectories in the high- intended for a wide range of mission objectives fidelity model accommodating the ephemeris and including landing on the moon, lunar observation, gravity of celestial bodies, and the specification of the solar sail demonstration, and asteroid flyby(1. The 13 propulsion system. CubeSats will be released at different mid-points, and This paper presents a Parallel HIgh-fidelity head to different destinations. By deploying CubeSats, Trajectory Optimization tool (PHITO). Same as other collaborative operations, such as inter-satellite developed trajectory optimization tools, such as ranging, which is useful for autonomous navigation, MALTO(3, GALLOP(4, and jTOP(5, the present tool and interferometric measurement, can be performed. adopts techniques of direct optimization and multiple For instance, the AIM mission is planed to rendezvous shooting. The tool is programmed to be compatible with the binary asteroid 65803 Didymos and then with parallel computing environment. To be specific, deploy a lander and two or more CubeSats to perform the propagation of trajectory states and the partial inter-satellite network(2. There will be many novel derivatives of each trajectory segment is implemented exploration manners enabled by deployed CubeSats. in parallel, which greatly saves the running time. The However, upon release, CubeSats can only rely on the tool is applied to the low-thrust phasing trajectory of a small propulsion system on board to reach desired CubeSat to an Earth-moon L2 halo orbit. Preliminary destinations. design in a simplified model shows this trajectory The propulsion capacity of a CubeSat is greatly imposes relatively high requirement of the thrust. The restrained by its limited mass and volume budget, thus trajectories as well as required Δv are solved for the trajectory design of CubeSat is subject to strict different mission starting dates. The feasibility of a constraint compared to general interplanetary certain type of propulsion system and the influence of spacecraft. Electric propulsion systems that generally the ephemeris on Δv are thus revealed. come with high specific impulse, Isp, are promising for interplanetary CubeSats. A high Isp suggests that the CubeSat can enjoy a high Δv budget for a certain mass budget. One drawback of the electric propulsion system is the generally low thrust, which may not be Proceedings of iCubeSat 2018, the 7th Interplanetary CubeSat Workshop, Paris, France 1 2. Structure of PHITO final states [xf, mf]. In addition, to facilitate optimization, the propagator also integrates the partial derivatives of [xf, mf] with respect to [xi, mi, FT], 2.1. High-fidelity Propagation which consists of another 7 x 9 = 63 elements. The time derivatives of the partial derivatives are To have high-fidelity trajectories, the propagator summarized in Appendix A. takes into account thrusting force, solar radiation 2.2. Optimization pressure, the ephemeris and gravity field of the celestial bodies, such as the Sun, the Earth, the moon, Jupiter, etc. The thrusting force is modeled as a 2.2.1. Multiple shooting continuous force vector, FT, acting on the trajectory when the thruster is turned on. As the optimization is The optimization is programmed based on direct based on direct optimization and multiple shooting optimization method and multiple shooting method. methods (see next section), where trajectory states and Both methods discretize trajectory states and controls. controls are discretized, FT is made constant The multiple shooting method is illustrated in Fig. 1. during the given period of propagation. The The whole trajectory is divided into N segments. The - equations of motion written in the form of state of the starting node of each segment, xk , is set free for optimization. For trajectory continuity, the x f () x ordinary differential equation, , are, optimizer tunes the initial state and control, in order for the final state to match the initial state of the next segment. Although the number of free variables and constraints is increased, the sensitivity to initial (1) guesses, however, is reduced, and thus the robustness of the optimization is improved. The interested reader where n is the number of celestial bodies taken into is referred to Ref. 6) and 7). account, mi denotes the gravitational parameter of the i-th celestial body, rji denotes the relative position from the j-th celestial body to the i-th celestial body, riSat denotes that from the i-th celestial body to the CubeSat, vjsat the relative velocity of CubeSat to the j- th celestial body, FSRP the solar radiation pressure, and m the mass of the CubeSat. In Eq. (1), the j-th celestial Figure 1. Concept of multiple shooting body is chose to be the center of the coordinate. The choice of the center depends on the trajectory 2.2.2. Objectives, variables, and constraints scenario. For instance, if the trajectory is around the moon, then the moon is suitable for the central body. The objective of the trajectory optimization is to If the trajectory is mostly heliocentric, the Sun or solar minimize the propellant loss, Δm. Free variables - system barycenter can be the center. In case of solar generally include the states of the starting node, xk , system barycenter, the second term of the second row and the control, FT,k, of each segment. Trajectory of the left-hand side is zero. FSRP is expressed as, states meet equality constraints at the initial and final boundaries and the match points. FT is constrained by (2) the maximum thrust of the thruster, namely, ||FT|| ≤ FTmax. The optimizer tunes free variables such that Δm where Cr and A are the reflectivity coefficient and the is minimized and all constraints are met. The used exposed area of the CubeSat, respectively, AU is the optimizer is the Sparse Nolinear OPtimization Tool, astronomical unit, and P is the solar power at 1 AU. SNOPT(8, (9. These parameters are considered constant by the current propagator. Positions of celestial bodies are 2.2.3. Gradients obtained from an ephemeris tool. The propellant loss The optimizer needs gradients of constraints and with time due to thrusting is computed by Eq. (3), objective with respect to states and controls to improve the feasibility and optimality of the solution (3) at each iteration. As mentioned in the previous section, partial derivatives with respect to initial states where is the mass loss rate, Isp is the specific and controls are propagated along with trajectory impulse, and g0 is the standard gravity on the Earth. states. Using chained rule, gradients of constraints and The input of the propagator consists of initial objective with respect to initial states and controls can trajectory states xi, including position and velocity, be easily derived. The obtained gradients are fed to initial mass mi of the CubeSat, and FT. The propagator the optimizer. In this way, the accuracy, speed, and integrates Eq. (1) and Eq. (3). The output includes Proceedings of iCubeSat 2018, the 7th Interplanetary CubeSat Workshop, Paris, France 2 robustness of the optimization process are greatly improved. 2.3. Parallel implementation and ephemeris tool Since the propagation and evaluation of gradients of each segment is uncoupled, running time can be greatly saved by propagating of trajectory segments and the partial derivatives in parallel. However, for calling ephemeris during propagation, the most widely-used ephemeris tool SPICE is not compatible with parallel computing environment. To implement the parallel algorithm in a parallel environment, the ephemeris tool CALCEPH(10 that is developed by IMCCE and designed to be compatible with parallel computing environment is used. The current PHITO calls the propagator using the MATLAB parallel routine PARFOR. The parallel implementation of the optimization is thus verified. 3. Case Study: Halo Orbit Phasing Trajectory 3.1. Problem statement PHITO is applied to the optimization of a halo orbit phasing trajectory to demonstrate its use for trajectory analysis. A previous work proposes a mission concept Fig. of providing the positioning service for landers or 2.