Fast Direction-Conditioned Reachability for Motion Prediction Under Model Uncertainty

To avoid collisions, a robot must repeatedly predict where nearby agents may move, usually with an imperfect model of their dynamics. Reachable sets provide such predictions, but computing them when the system matrices themselves are uncertain can become computationally expensive and conservative for frequent replanning. Moreover, a planner often needs to know only how far an agent can move in one particular direction, for example toward the robot, rather than the complete reachable set. We propose a direction-conditioned reachability method for linear systems with uncertain state and input matrices. Given a query direction $d$, the method selects one admissible model $(A^\star,B^\star)$ whose reachable set extends nearly as far along $d$ as the reachable set of the entire uncertain model family, and then computes the reachable set of only this model with a standard reachability solver. On an uncertain linearized bicycle model, the complete selection-and-computation pipeline is about three times faster than computing the reachable set of the full uncertain family in the CORA toolbox, while its extent along $d$ is within $5\%$ of the full family's in the reported directions. We also use the method in a closed-loop multi-vehicle simulation in which the robot queries, at each replanning step, how far each nearby vehicle can move toward it, and replans to avoid the resulting sets.