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HyRRT-Connect: A Bidirectional Rapidly-Exploring Random Trees Motion Planning Algorithm for Hybrid Systems

Nan Wang, Ricardo G. Sanfelice

TL;DR

HyRRT-Connect tackles motion planning for hybrid systems by running bidirectional searches on forward and backward hybrid-time trees and stitching their trajectories. It introduces a backward-in-time hybrid system to enable reverse propagation, and a reconstruction procedure to smooth potential flow discontinuities, with optional exact jump connections to avoid reconstruction when possible. The approach is validated on an actuated bouncing ball and a walking robot, showing substantial speedups over prior methods and robust convergence as the tolerance parameter $\delta$ decreases. This work advances practical planning for hybrid dynamics by integrating forward/backward synthesis, reconstruction, and jump-based connections to yield reliable motion plans efficiently.

Abstract

This paper proposes a bidirectional rapidly-exploring random trees (RRT) algorithm to solve the motion planning problem for hybrid systems. The proposed algorithm, called HyRRT-Connect, propagates in both forward and backward directions in hybrid time until an overlap between the forward and backward propagation results is detected. Then, HyRRT-Connect constructs a motion plan through the reversal and concatenation of functions defined on hybrid time domains, ensuring the motion plan thoroughly satisfies the given hybrid dynamics. To address the potential discontinuity along the flow caused by tolerating some distance between the forward and backward partial motion plans, we reconstruct the backward partial motion plan by a forward-in-hybrid-time simulation from the final state of the forward partial motion plan. By applying the reversed input of the backward partial motion plan, the reconstruction process effectively eliminates the discontinuity and ensures that as the tolerance distance decreases to zero, the distance between the endpoint of the reconstructed motion plan and the final state set approaches zero. The proposed algorithm is applied to an actuated bouncing ball example and a walking robot example so as to highlight its generality and computational improvement.

HyRRT-Connect: A Bidirectional Rapidly-Exploring Random Trees Motion Planning Algorithm for Hybrid Systems

TL;DR

HyRRT-Connect tackles motion planning for hybrid systems by running bidirectional searches on forward and backward hybrid-time trees and stitching their trajectories. It introduces a backward-in-time hybrid system to enable reverse propagation, and a reconstruction procedure to smooth potential flow discontinuities, with optional exact jump connections to avoid reconstruction when possible. The approach is validated on an actuated bouncing ball and a walking robot, showing substantial speedups over prior methods and robust convergence as the tolerance parameter decreases. This work advances practical planning for hybrid dynamics by integrating forward/backward synthesis, reconstruction, and jump-based connections to yield reliable motion plans efficiently.

Abstract

This paper proposes a bidirectional rapidly-exploring random trees (RRT) algorithm to solve the motion planning problem for hybrid systems. The proposed algorithm, called HyRRT-Connect, propagates in both forward and backward directions in hybrid time until an overlap between the forward and backward propagation results is detected. Then, HyRRT-Connect constructs a motion plan through the reversal and concatenation of functions defined on hybrid time domains, ensuring the motion plan thoroughly satisfies the given hybrid dynamics. To address the potential discontinuity along the flow caused by tolerating some distance between the forward and backward partial motion plans, we reconstruct the backward partial motion plan by a forward-in-hybrid-time simulation from the final state of the forward partial motion plan. By applying the reversed input of the backward partial motion plan, the reconstruction process effectively eliminates the discontinuity and ensures that as the tolerance distance decreases to zero, the distance between the endpoint of the reconstructed motion plan and the final state set approaches zero. The proposed algorithm is applied to an actuated bouncing ball example and a walking robot example so as to highlight its generality and computational improvement.
Paper Structure (17 sections, 5 theorems, 8 equations, 2 figures, 2 algorithms)

This paper contains 17 sections, 5 theorems, 8 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Notation and Preliminaries
  2. Notation
  3. Preliminaries
  4. Problem Statement and Applications
  5. Algorithm Description
  6. Overview
  7. Backward-in-time Hybrid System
  8. Construction of Motion Plans
  9. HyRRT-Connect Algorithm
  10. Motion Plan Identification and Reconstruction
  11. Same State Associated with Vertices in ${\mathcal{T}}^{\mathrm{fw}}$ and ${\mathcal{T}}^{\mathrm{bw}}$
  12. Reconstruction Process
  13. Convergence of ${\phi}^{\mathrm{r}}$ to $X_{f}$
  14. Connecting Forward and Backward Search Trees via Jump
  15. Software Tool and Simulation Results
  16. ...and 2 more sections

Key Result

Proposition 2

Given two solution pairs $\psi_{1} = (\phi_{1}, u_{1})$ and $\psi_{2} = (\phi_{2}, u_{2})$ to a hybrid system ${\mathcal{H}}^{\mathrm{fw}}$, their concatenation $\psi = (\phi, u) = (\phi_{1}|\phi_{2}, u_{1}|u_{2})$, denoted $\psi = \psi_{1}|\psi_{2}$, is a solution pair to ${\mathcal{H}}^{\mathrm{fw

Figures (2)

  • Figure 1: Motion plans for the actuated bouncing ball example.
  • Figure 2: Selected states of the partial motion plans for the walking robot system.

Theorems & Definitions (16)

  • Definition 1: Solution pair to a hybrid system
  • Definition 2: Concatenation operation
  • Example 1: Actuated bouncing ball system
  • Example 2: Walking robot
  • Definition 3: Backward-in-time hybrid system
  • Proposition 2
  • Remark 3
  • Definition 4: Reversal of a solution pair
  • Proposition 4
  • Remark 6
  • ...and 6 more