Rendezvous Guidance Trajectories via Multiple-Subarc Sequential Gradient-Restoration Algorithm

We consider the three-dimensional rendezvous between a target spacecraft in a circular orbit and a chaser spacecraft with an initial separation distance and an initial separation velocity. We assume that the chaser spacecraft has variable mass and that its trajectory is governed by three controls, one determining the thrust magnitude and two determining the thrust direction. We employ the Clohessy–Wiltshire equations, describing the relative motion of the chaser vis-à-vis the target, and the multiple-subarc sequential gradient-restoration algorithm to produce first optimal trajectories and then guidance trajectories for the following problems: P1—minimum time, fuel free; P2—minimum fuel, time free; P3—minimum time, fuel given; P4—minimum fuel, time given; and P5—minimum $\text{time}\times \text{fuel}$ , time and fuel free. Clearly, P1 and P2 are basic problems, while P3, P4, and P5 are compromise problems. Problem P1 leads to a two-subarc solution including a max-thrust subarc followed by another max-thrust subarc. Problem P2 leads to a four-subarc solution including two coasting subarcs alternating with two max-thrust subarcs. Problems P5 leads to a three-subarc solution including two max-thrust subarcs alternating with one coasting subarc. Problems P3 and P4 include P1, P2, and P5 as particular cases and lead to two-, three-, or four-subarcs solutions depending on the prescribed value of fuel or time. For all problems, the thrust magnitude control is saturated at one of its extreme values: in optimization studies, we determine the best thrust direction controls; in guidance studies, we force the thrust direction controls to be constant in each subarc and determine the best thrust direction parameters. Of course, the time lengths of all the subarcs must also be determined. The computational results show that, for Problems P1–P5, the performance index of the multiple-subarc guidance trajectory approximates well the performance index of the multiple-subarc optimal trajectory: the pairwise relative differences in performance index are less than $1∕100$ in all cases. To sum up, the produced guidance trajectories are highly efficient and yet quite simple in implementation.