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Motion Planning and Control with Unknown Nonlinear Dynamics through Predicted Reachability

Zhiquan Zhang, Gokul Puthumanaillam, Manav Vora, Melkior Ornik

TL;DR

This work tackles motion planning under unknown nonlinear dynamics by partitioning the state space into polytopes and approximating each region with a piecewise-affine model. It builds a predictive reachability graph where edges are incrementally validated via affine system identification and reachability analysis, and edges to uncertain regions receive heuristic weights to balance exploration and exploitation. The core mechanism combines facet reachability-based controller synthesis with a graph-search-based planner (via Dijkstra) to steer the agent online toward a target polytope. The approach is demonstrated in simulation on a mobile robot with unknown terrain dynamics, showing that the predictive graph can anticipate reachability in unexplored regions and guide feasible trajectories using online affine controllers.

Abstract

Autonomous motion planning under unknown nonlinear dynamics presents significant challenges. An agent needs to continuously explore the system dynamics to acquire its properties, such as reachability, in order to guide system navigation adaptively. In this paper, we propose a hybrid planning-control framework designed to compute a feasible trajectory toward a target. Our approach involves partitioning the state space and approximating the system by a piecewise affine (PWA) system with constrained control inputs. By abstracting the PWA system into a directed weighted graph, we incrementally update the existence of its edges via affine system identification and reach control theory, introducing a predictive reachability condition by exploiting prior information of the unknown dynamics. Heuristic weights are assigned to edges based on whether their existence is certain or remains indeterminate. Consequently, we propose a framework that adaptively collects and analyzes data during mission execution, continually updates the predictive graph, and synthesizes a controller online based on the graph search outcomes. We demonstrate the efficacy of our approach through simulation scenarios involving a mobile robot operating in unknown terrains, with its unknown dynamics abstracted as a single integrator model.

Motion Planning and Control with Unknown Nonlinear Dynamics through Predicted Reachability

TL;DR

This work tackles motion planning under unknown nonlinear dynamics by partitioning the state space into polytopes and approximating each region with a piecewise-affine model. It builds a predictive reachability graph where edges are incrementally validated via affine system identification and reachability analysis, and edges to uncertain regions receive heuristic weights to balance exploration and exploitation. The core mechanism combines facet reachability-based controller synthesis with a graph-search-based planner (via Dijkstra) to steer the agent online toward a target polytope. The approach is demonstrated in simulation on a mobile robot with unknown terrain dynamics, showing that the predictive graph can anticipate reachability in unexplored regions and guide feasible trajectories using online affine controllers.

Abstract

Autonomous motion planning under unknown nonlinear dynamics presents significant challenges. An agent needs to continuously explore the system dynamics to acquire its properties, such as reachability, in order to guide system navigation adaptively. In this paper, we propose a hybrid planning-control framework designed to compute a feasible trajectory toward a target. Our approach involves partitioning the state space and approximating the system by a piecewise affine (PWA) system with constrained control inputs. By abstracting the PWA system into a directed weighted graph, we incrementally update the existence of its edges via affine system identification and reach control theory, introducing a predictive reachability condition by exploiting prior information of the unknown dynamics. Heuristic weights are assigned to edges based on whether their existence is certain or remains indeterminate. Consequently, we propose a framework that adaptively collects and analyzes data during mission execution, continually updates the predictive graph, and synthesizes a controller online based on the graph search outcomes. We demonstrate the efficacy of our approach through simulation scenarios involving a mobile robot operating in unknown terrains, with its unknown dynamics abstracted as a single integrator model.

Paper Structure

This paper contains 15 sections, 2 theorems, 23 equations, 2 figures, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES AND NOTATIONS
  3. Transition Systems and Directed Weighted Graphs
  4. Polytopes
  5. Notations and Terminology
  6. PROBLEM FORMULATION
  7. PROPOSED APPROACH
  8. Facet Reachability and Controller Synthesis
  9. Predictive Reachability-Guaranteed Graph
  10. Motion Planning Framework
  11. EXPERIMENTAL RESULTS
  12. CONCLUSION AND FUTURE WORK
  13. Affine Dynamic Difference Bounds
  14. Affine System Identification
  15. Proof of Proposition 1

Key Result

Proposition 1

Consider $2^m$ sets of inequalities formulated by narrow ineq. where $\varepsilon_A$, $\varepsilon_B$, $\varepsilon_c$ are bounds for dynamics difference between $(\bar{A}_l, \bar{B}_l, \bar{c}_l)$ and $(\bar{A}_{l'}, \bar{B}_{l'}, \bar{c}_{l'})$ are estimated as described in Appendix A. The vectors $\bar{B}_{l'}^-$ and $\bar{B}_{l'}^+$ are defined by: If at least one of these sets is feasible,

Figures (2)

  • Figure 1: The initial tile is marked yellow while the target tile is marked green. (left) The ground truth of possible transitions between polytopes. The dynamics are linearized at the center of each grid cell according to \ref{['piecewise affine']}. (center) The exploration and navigation process. The sequence of the bright blue dots represents the result of graph planning. Blue arrows denote guaranteed facet reachability (i.e., the confirmed presence of edges until the mission is completed), red arrows indicate the absence of an edge and the lack of an arrow denotes uncertainty regarding edge existence. (right) The trajectory generated by the controller synthesized by \ref{['eq: controller']}.
  • Figure 2: The trajectory of the mobile robot. The green and purple markers denote the initial and target locations respectively.

Theorems & Definitions (4)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Proposition 2