Trusted Polytopic Action Sets for Fast Planning in Underactuated Systems

Autonomous motion planning under unknown nonlinear dynamics requires learning system properties while navigating toward a target. In this work, we develop a hierarchical planning-control framework that enables online motion synthesis with limited prior system knowledge. The state space is partitioned into polytopes and approximates the unknown nonlinear system using a piecewise-affine (PWA) model. The local affine models are identified once the agent enters the corresponding polytopes. To reduce computational complexity, we introduce a non-uniform adaptive state space partition strategy that refines the partition only in task-relevant regions. The resulting PWA system is abstracted into a directed weighted graph, whose edge existence is incrementally verified using reach control theory and predictive reachability conditions. Certified edges are weighted using provable time-to-reach bounds, while uncertain edges are assigned information-theoretic weights to guide exploration. The graph is updated online as new data becomes available, and high-level planning is performed by graph search, while low-level affine feedback controllers are synthesized to execute the plan. Furthermore, the conditions of classical reach control theory are often difficult to satisfy in underactuated settings. We therefore introduce relaxed reachability conditions to extend the framework to such systems. Simulations demonstrate effective exploration-exploitation trade-offs with formal reachability guarantees.

Agile quadruped navigation in dynamic environments demands trajectories that are both dynamically feasible and provably collision-free. Most real-time planners achieve tractability through a convex reduction that requires simplified kinematic models, discarding motion that the robot can readily execute, and certify safety only at discretization points, leaving open possible inter-sample collisions with moving obstacles. We present a real-time motion planning framework that retains the fully-actuated $SE(2)$ model of a quadruped, with its anisotropic, heading-coupled velocity bounds, and accepts the resulting nonconvexity. By parameterizing trajectories as composite Bernstein polynomials, we exploit the convex hull property to certify collision avoidance and the input bounds over the entirety of each planning horizon. Solved at online replanning rates, the optimization runs in closed loop, commanding body velocities to the locomotion controller. We validate the approach in simulation against ablations and external baselines, and in hardware experiments on a quadruped navigating among unknown moving obstacles, recovering both lateral and heading-coupled motion that simplified models exclude.

Motion planning algorithms compute control sequences that drive autonomous robots to goal regions while avoiding unsafe states. Existing methods, from sampling-based planning to deep reinforcement learning, typically provide task-completion guarantees only with respect to a nominal model or simulator, which may be invalidated when the true dynamics are unknown or difficult to model accurately. This letter addresses this limitation for systems with unknown dynamics and an available approximate nominal model, contributing a planner-agnostic constraint-tightening procedure that equips existing planners with a probabilistic task-completion guarantee on the true system. We leverage conformal prediction to provide a probabilistic bound on the nominal-to-true trajectory deviation over a distribution of planning problems. We tighten the planning constraints using that bound, and show that solving the tightened problem under the nominal model is a sufficient condition for solving the original problem on the true system with a prescribed probability. We validate the theoretical guarantees empirically and demonstrate substantially improved task completion relative to nominal-model planning.