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.

Figures (5)

• Figure E1: Multiple target, multiple destination estimates given very noisy data
• Figure E2: Given sparse observations, proportional guidance using Kalman filtering misses on evenly matched target and pursuer entities due to lag caused by lack of a nonlinear dynamics constraint in the filter
• Figure E3: Reachable regions $\mathcal{R}$ at terminal time for the two pursuer deployment scenarios targeting perpendicular contact. Left: The unrestricted scenario where the initial pursuer configuration is unconstrained. Right: The restricted scenario where initial headings are limited to east or west. The green dot denotes the launch base station, and the light green regions highlight spatiotemporal coordinates where perpendicular contact is feasible according to the value function $u$.
• Figure E4: Optimal rendezvous points and pursuer trajectories for unrestricted (left) and restricted (right) scenarios. The green lines represent the optimal rendezvous paths derived from the value function $u$, targeting the stochastic trajectory ensembles (orange tubes) and their mean paths (red). The terminal points are marked by cyan circles of radius $R$, representing the optimized spatiotemporal coordinates for successful rendezvous and parameter disambiguation.
• Figure E5: In the same slow-pursuer scenario, follower dynamics using a Kalman filter utterly fails due to lack of planning