Trajectory optimization method via direct control history update in computational guidance

Computational guidance demands highly precise and efficient trajectory optimization. The model predictive static programming (MPSP) methods and their variants are commonly-used approaches for this problem. However, if the iterative number and discrete points are limited, the solutions may become suboptimal. Motivated by these approaches, a discrete trajectory optimization method is proposed via direct control history update for fixed terminal state constraints. Additionally, a generalized trajectory optimization method is proposed by a low-dimensional weight matrix within a continuous-time architecture, which eliminates the need for initial system discretization. Finally, the equivalence of the two methods is proven. The proposed method utilizes a Lagrange multiplier with a symbolic solution to directly update the control history, thereby enhancing both computational accuracy and efficiency. Although the proposed method is motivated by the current MPSP method, it solves the Lagrange multiplier for the actual dynamic equations rather than for the deviation dynamic equations used in the MPSP method. Consequently, these two methods are radically different. Applied to a near-space vehicle, the trajectory generated by the proposed method is regarded as optimal. Meanwhile, the optimization time is decreased by at least several times when compared to the MPSP method under same level accuracy.