HyRRT-Connect: Bidirectional Motion Planning for Hybrid Dynamical Systems

TL;DR

HyRRT-Connect addresses motion planning for hybrid dynamical systems by introducing a bidirectional search in the hybrid time domain, leveraging a backward-in-time hybrid system, trajectory reversal, and concatenation to form a valid forward plan. A reconstruction procedure is used to smooth potential discontinuities along the flow when forward and backward partial plans are combined, and exact jump connections provide another path to reduce discontinuities. The paper proves that reversal and concatenation preserve hybrid dynamics under mild assumptions and demonstrates substantial computational gains over prior approaches on an actuated bouncing ball and a high-dimensional walking robot. The work advances scalable, provably sound planning for hybrid systems and enables efficient, robust trajectories for legged locomotion and other hybrid-control tasks.

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 that the motion plan 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. effectively eliminating the discontinuity. The proposed algorithm is applied to an actuated bouncing ball system and a walking robot example to highlight its computational improvement.

Figures (9)

• Figure 1: A motion plan to Problem \ref{['problem:motionplanning']}. The green square denote the initial state set. The green star denotes the final state set. The blue region denotes the flow set. The red region denotes the jump set. The solid blue lines denote flow and the dotted red lines denote jumps in the motion plan.
• Figure 2: The actuated bouncing ball system in Example \ref{['example:bouncingball']}.
• Figure 3: The biped system in Example \ref{['example:biped']}. The angle vector $\theta$ contains the planted leg angle $\theta_{p}$, the swing leg angle $\theta_{s}$, and the torso angle $\theta_{t}$. The velocity vector $\omega$ contains the planted leg angular velocity $\omega_{p}$, the swing leg angular velocity $\omega_{s}$, and the torso angular velocity $\omega_{t}$. The input $u$ is the input torque, where $u_{p}$ is the torque applied on the planted leg from the ankle, $u_{s}$ is the torque applied on the swing leg from the hip, and $u_{t}$ is the torque applied on the torso from the hip.
• Figure 4: Examples demonstrating the trajectories of reversal and concatenation operations.
• Figure 5: Motion plans for actuated bouncing ball example solved by HyRRT-Connect, where precise connection during the flow is achieved.

Theorems & Definitions (55)

• Definition 2.1: (Hybrid time domain)
• Definition 2.2: (Hybrid input)
• Definition 2.3: (Hybrid arc)
• Definition 2.4: (Solution pair to a hybrid system)
• Definition 2.5: (Concatenation operation)
• Example 3.1: (Actuated bouncing ball system)
• Example 3.2: (Walking robot)
• Definition 4.1: (Backward-in-time hybrid system)
• Example 4.2: (Actuated bouncing ball system in Example \ref{['example:bouncingball']}, revisited)
• Remark 4.3