Hybrid Differential Dynamic Programming in the Circular Restricted Three-Body Problem

Particle Swarm Optimization-Based Co-State Initialization for Low-Thrust Minimum-Fuel Trajectory Optimization

“…As expected from the previous discussion, Trajectory α is the most fuel-efficient, requiring just 35.34 kg of fuel, while Trajectory γ requires nearly double the fuel mass (i.e., 61.27 kg), and Trajectory β is the most fuel expensive (at 81.28 kg). It should be noted that the fuel-optimal trajectory discovered through the application of the proposed method (i.e., Trajectory α) agrees with results published by Aziz et al [22]. A slightly lower fuel requirement is found here, which is explained by the fact that the Hybrid Differential Dynamic Programming (HDDP) algorithm applied by Aziz et al enforced a constant direction of thrust over each integration step and can lag in x (km) x (km) x (km) switching thrust on or off [22], whereas indirect methods allow thrust to be applied in the optimal direction of the primer vector at every point along the trajectory and enforce throttling on or off at the optimal times.…”

“…Note that the mass unit (which is a unique value for a given scenario) is defined as the initial mass of the spacecraft m i . The chosen scenarios provide problems of moderate difficulty to benchmark our approach while allowing for validation and comparison to previous works [21,22,34,35,36,37].…”

“…For the L 2 halo orbit to L 1 halo orbit transfer problem, the spacecraft was chosen to have an initial mass m i = 2000 kg, a maximum thrust T max = 1.5 N, and a specific impulse I sp = 2000 s, as in reference [22]. The transfer duration was fixed to 12.7 days and the terminal physical boundary conditions for the TPBVP, identifying the initial state of the spacecraft on an L 2 halo orbit and the target point on an L 1 halo orbit, were defined as [22] These correspond to initial and target halo orbits with periods of 14.2 and 11.2 days respectively.…”

“…Herein, the proposed methodology is presented in the framework of the Circular Restricted Three Body Problem (CR3BP), and important considerations in regard to solution optimality and computational performance are discussed. Case studies, based on previous works [21,22], are presented which shed light on the promising performance of the proposed methodology while allowing for its validation. Guidance on the selection of the particle swarm size, a heuristic parameter of the optimization algorithm which has a significant effect on co-state initialization performance, is also provided.…”

Game-Theoretic Flexible-Final-Time Differential Dynamic Programming Using Gaussian Quadrature

“…Trajectory optimization is a key enabler to improve the level of autonomy of an aerial vehicle [1][2][3]. Differential dynamic programming (DDP) has become a widespread method to solve nonlinear trajectory optimization problems due to its well-established problem formulation framework and fast convergence characteristics in recent years [4][5][6].…”