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In this paper, we present a new approach for solving a broad class of energy-optimal trajectory problems in relative motion using the recently developed Theory of Functional Connections (TFC). A total of four problem cases are considered and solved, i.e. rendezvous and intercept with fixed and free final time. Each problem is constrained and formulated using an indirect approach which casts the optimal trajectory problem as a system of linear or nonlinear two-point boundary value problems for the fixed and free final time cases, respectively. Using TFC, we convert each two-point boundary value problem into an unconstrained problem by analytically embedding the boundary constraints into a "constrained expression." The latter includes a free-function that is expanded using Chebyshev polynomials with unknown coefficients. Regardless of the values of the unknown coefficients, the boundary constraints are satisfied and simple optimization schemes can be employed to numerically solve the problem (e.g. linear and nonlinear least-square methods). To validate the proposed approach, the TFC solutions are compared with solutions obtained via an analytical based method as well as direct and indirect numerical methods. In general, the proposed technique produces solutions to machine level accuracy. Additionally, for the cases tested, it is reported that computational run-time within the MATLAB implementation is lower than 28 and 300 ms for the fixed and free final time problems respectively. Consequently, the proposed methodology is potentially suitable for on-board generation of optimal trajectories in real-time. • Energy-optimal problems in relative motion are solved via TFC. • Boundary constraints are analytical embedded reducing the search space. • Functional interpolation is performed with linear and non-linear least-squares. • TFC is compared with other state-of-the-art methods. • TFC is shown to solve all problems with a loss accuracy on the order of 10–17 and with a computational runtime between 28 ms (linear problems) and 300 ms (non-linear problems). [ABSTRACT FROM AUTHOR]

Abstract An interplanetary trajectory optimization of the multi-asteroid rendezvous is investigated for a solar sail-based spacecraft. The optimal control problem is formulated using the calculus of variations and Pontryagin's maximum principle and the first order necessary conditions for optimality of the problem are derived. To achieve the optimal solution, the indirect method transforms the optimal control problem into a multiple point boundary value problem and solves it via shooting method, and the pseudospectral method discretizes the state and control variables and solves the NLP problem. In this paper, a numerical method combining the global optimization algorithm and a nonlinear equation solver is proposed to get the optimal solution of the indirect method. Additionally, costates are estimated to validate the optimality of the solution obtained via the pseudospectral method. Finally, numerical examples are given to verify the effectiveness of the proposed methods. The results indicate that the optimal solutions to the problems with two and three targets can be solved using the proposed indirect method algorithm. As the number of visiting targets is more than three, the direct method will be more efficient. Highlights • Necessary condition for optimality of multi-asteroid rendezvous problem is derived. • A shooting method is developed to solve the multiple point boundary value problem. • A hybrid method combining the indirect method and the pseudospectral method is developed. [ABSTRACT FROM AUTHOR]