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Dynamically Feasible Path Planning in Cluttered Environments via Reachable Bezier Polytopes

Noel Csomay-Shanklin, William D. Compton, Aaron D. Ames

TL;DR

This work proposes a layered control architecture that efficiently produces collision free and dynamically feasible paths for nonlinear control systems, and demonstrates the framework on the tasks of 3D hopping in a cluttered environment.

Abstract

The deployment of robotic systems in real world environments requires the ability to quickly produce paths through cluttered, non-convex spaces. These planned trajectories must be both kinematically feasible (i.e., collision free) and dynamically feasible (i.e., satisfy the underlying system dynamics), necessitating a consideration of both the free space and the dynamics of the robot in the path planning phase. In this work, we explore the application of reachable Bezier polytopes as an efficient tool for generating trajectories satisfying both kinematic and dynamic requirements. Furthermore, we demonstrate that by offloading specific computation tasks to the GPU, such an algorithm can meet tight real time requirements. We propose a layered control architecture that efficiently produces collision free and dynamically feasible paths for nonlinear control systems, and demonstrate the framework on the tasks of 3D hopping in a cluttered environment.

Dynamically Feasible Path Planning in Cluttered Environments via Reachable Bezier Polytopes

TL;DR

This work proposes a layered control architecture that efficiently produces collision free and dynamically feasible paths for nonlinear control systems, and demonstrates the framework on the tasks of 3D hopping in a cluttered environment.

Abstract

The deployment of robotic systems in real world environments requires the ability to quickly produce paths through cluttered, non-convex spaces. These planned trajectories must be both kinematically feasible (i.e., collision free) and dynamically feasible (i.e., satisfy the underlying system dynamics), necessitating a consideration of both the free space and the dynamics of the robot in the path planning phase. In this work, we explore the application of reachable Bezier polytopes as an efficient tool for generating trajectories satisfying both kinematic and dynamic requirements. Furthermore, we demonstrate that by offloading specific computation tasks to the GPU, such an algorithm can meet tight real time requirements. We propose a layered control architecture that efficiently produces collision free and dynamically feasible paths for nonlinear control systems, and demonstrate the framework on the tasks of 3D hopping in a cluttered environment.

Paper Structure

This paper contains 12 sections, 3 theorems, 15 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Reduced Order Models and Problem Statement
  4. Bézier Curves and Graphs
  5. Kinodynamic Bézier Graphs
  6. Build Graph
  7. Cut Graph
  8. Find Path
  9. Refine the Path
  10. Results
  11. Simulation and Hardware
  12. Conclusion

Key Result

Lemma 1

Given two points $\mathbf{x} _0, \mathbf{x} _T \in \mathbb{R}^n$, there exists a unique matrix $\mathbf{D}$ such that $\mathbf{P} \bm{ \mathbf{P} } = \mathbf{D} $ are the control points of a Bézier curve $\mathbf{x} _d(\cdot)$ satisfying $\mathbf{x} _d(0) = \mathbf{x} _0$ and $\mathbf{x} _d(T

Figures (6)

  • Figure 2: The proposed framework performing path planning around segmented obstacles. A coarse dynamically feasible path is solved for on a graph with Bézier polynomial edges, and is further refined with MPC.
  • Figure 3: The path planning framework presented in Algorithm \ref{['algo:BezGraph']}. From left to right: a) A Bézier graph is constructed, b) it is cut based on the present obstacles, c) a path is solved, and d) the path is refined with MPC.
  • Figure 4: The heuristic employed to check if the Bézier curve is collision free. First, obstacle membership is checked. If not satisfied, a single proposed separating hyperplane is checked. Finally, the Quadratic Program \ref{['eqn:feas_check']} is solved.
  • Figure 5: 500 randomly generated obstacles with graph replanning at 10 Hz and \ref{['eqn:mpc_cost']} at 200 Hz. Cyan indicates the Bézier graph solve $\mathbf{v} _k^*$ and blue the MPC Bézier solution $\mathbf{x} _d(\cdot)$.
  • Figure 6: A long-horizon maze with 600 cells, 300 obstacles and 50,000 edges. The graph cut is solved at 10 Hz, and graph solve and MPC both run at 200 Hz. Blue represents the graph solve, and red the closed loop system behavior.
  • ...and 1 more figures

Theorems & Definitions (5)

  • Definition 1
  • Lemma 1: future_acc
  • Theorem 1: future_acc
  • Theorem 2
  • proof