Rendezvous Planning from Sparse Observations of Optimally Controlled Targets

Abstract: We develop a probabilistic framework for \emph{rendezvous planning}: given sparse, noisy observations of a fast-moving target, plan rendezvous spatiotemporal coordinates for a set of significantly slower seeking agents. The unknown target trajectory is estimated under uncertain dynamics using a filtering approach that combines a kernel-based maximum a posteriori estimation with Gaussian process correction, producing a mixture over trajectory hypotheses. This estimate is used to select spatiotemporal rendezvous points that maximize the probability of successful rendezvous. Points are chosen sequentially by greedily minimizing failure probability in the current belief space, which is updated after each step by conditioning on unsuccessful rendezvous attempts. We show that the failure-conditioned update correctly captures the posterior belief for subsequent decisions, ensuring that each step in the greedy sequence is informed by a statistically consistent representation of the remaining search space, and derive the corresponding Bayesian updates incorporating temporal correlations intrinsic to the trajectory model. This result provides a systematic framework for planning under uncertainty in applications of autonomous rendezvous such as unmanned aerial vehicle refueling, spacecraft servicing, autonomous surface vessel operations, search and rescue missions, and missile defense. In each, the motion of the target entity can be modeled using a system of differential equations undergoing optimal control for a chosen objective, in our example case Hamilton--Jacobi--Bellman solutions for minimum arrival time of a Dubins car with uncertain turning radius and destination.