Multi-phase trajectory optimization of aerospace vehicle using sequential penalized convex relaxation

This paper presents a novel methodology for solving the multi-phase trajectories of aerospace vehicles in the framework of convex optimization. As a result of the inherent non-smooth, non-linear, and staged features of the problem, the application of convex optimization is confronted by three categories of tough non-convex terms, non-convex functions, phase linkage constraints, and free endpoints. To overcome these difficulties, a combination of a time-projection approach, and a sequential relaxation and penalization method is developed in this paper. Firstly, free singularity points from linkage constraints and endpoints are eliminated by projecting the time history onto a normalized time interval. Subsequently, the resulting fixed-time problem is equivalently converted in to a pre-semi-definite form and sequentially relaxed as an approximated convex optimization problem. Furthermore, to increase the convergence of the proposed method, a penalization term is included to control the search direction toward the true solution. Analyses are performed to ensure that the convergence and robustness of the proposed algorithm is well guaranteed. A numerical comparison of the results with those from the state-of-art pseudo-spectral nonlinear programming solver GPOPS suggests that the algorithm outputs similar trajectories, but in only one tenth of the computation time.