Guidance Trajectories for Spacecraft Rendezvous

In a previous paper of Miele et al. (J. Optim. Theory Appl. 132(1), 2007), we employed the single-subarc sequential gradient-restoration algorithm to optimize the three-dimensional rendezvous between a target spacecraft in a planar circular orbit and a chaser spacecraft with an initial separation distance and separation velocity. The achieved continuous solutions are characterized by two, three, or four subarcs depending on the performance index (time, fuel) and the constraints. In this paper, based on the solutions in Miele et al. (J. Optim. Theory Appl. 132(1), 2007), we employ the multiple-subarc sequential gradient-restoration algorithm to produce pieced guidance trajectories implementable in real time via constant control components. In other words, in this investigation, we force the controls to behave as parameters in each subarc. With the above understanding, we investigate four problems: (P1) minimum time, fuel free; (P2) minimum fuel, time free; (P3) minimum time, fuel given; (P4) minimum fuel, time given. Problem P1 results in a two-subarc solution, each subarc with constant controls: a max-thrust accelerating subarc followed by a max-thrust braking subarc. Problem P2 results in a four-subarc solution, each subarc with constant controls: an initial coasting subarc, followed by a max-thrust braking subarc, followed by another coasting subarc, followed by another max-thrust braking subarc. Problems P3 and P4 result in two, three, or four-subarc solutions depending on the performance index and the constraints, albeit with constant controls in each subarc. For each of the problems studied, the performance index of the multiple-subarc pieced guidance trajectory approximates well the performance index of the single-subarc continuous optimal trajectory of Miele et al. (J. Optim. Theory Appl. 132(1), 2007) as well as the performance index of the multiple-subarc pieced optimal trajectory: the pairwise relative differences in performance index are less than 1/100. This research was supported by NSF under Grant CMS-0218878.