Hierarchical Motion Planning and Control under Unknown Nonlinear Dynamics via Predicted Reachability

Abstract

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.

Figures (7)

• Figure 1: A pipeline of the proposed motion planning and control framework.
• Figure 2: Schematic illustration of the maximal and minimal control input range $u_{l'j}$.
• Figure 3: A 2D example illustrates uniform and non-uniform state space partitioning. The blue dot represents the current state and the red dot represents the target state. (Left) Uniform partitioning. (Right) Non-uniform partitioning.
• Figure 4: Simulation results for a fully actuated mobile robot operating under unknown dynamics. Each column corresponds to a distinct time instant during the simulation. The top row illustrates the robot’s trajectory within the simulated environment, while the bottom row depicts the associated reachability analysis and prediction, nonuniform adaptive state space partitioning, and high-level planning performed on the reachability-guaranteed graph. Blue arrows indicate facets that are provably reachable, red arrows denote the absence of feasible transitions, and the absence of an arrow represents uncertainty in edge existence. The initial tile is highlighted in yellow and the target tile is highlighted in green. The green dot and blue dot denote the current tile and the next tile on the predicted graph respectively. The red line shows the optimal path computed at each discrete planning step.
• Figure 5: Simulation results for an under-actuated mobile robot operating under unknown dynamics. Each column corresponds to a distinct time during the simulation. The top row illustrates the robot's trajectory (marked as orange line) in the simulated environment. The bottom row depicts the associated reachability analysis and prediction, nonuniform adaptive state space partitioning, and high-level planning performed on the reachability-guaranteed graph. Blue arrows indicate facets that are reachable. The target tile is marked by a red star. The red line represents the sequence of polytopes selected by the high-level planner, and the green dot denotes the robot's current state.

Theorems & Definitions (10)

• Definition 1
• Definition 2
• Proposition 1
• Proposition 2
• Remark 1
• Remark 2
• Proposition 3
• Remark 3
• Proposition 4
• Remark 4