Motion Planning for Hybrid Dynamical Systems: Framework, Algorithm Template, and a Sampling-based Approach

TL;DR

This work addresses motion planning for systems with both continuous and discrete dynamics by formulating a general hybrid framework and introducing HyRRT, a sampling-based planner that alternates between flow and jump extensions. The method relies on a forward propagation template, complete hybrid-input libraries, and continuous/discrete simulators to construct a search tree whose edges encode solution pairs $(\phi,\upsilon)$. A key contribution is the inflation-based probabilistic completeness theory, which shows that HyRRT succeeds with probability approaching one as the number of iterations grows, even without positive clearance in the original system, by working on an inflated hybrid system $\mathcal{H}_{\delta}$. The approach is validated on actuated bouncing ball and walking robot models, and a HyRRT software tool is provided to reproduce the results and apply the framework to other hybrid systems in robotics.

Abstract

This paper focuses on the motion planning problem for the systems exhibiting both continuous and discrete behaviors, which we refer to as hybrid dynamical systems. Firstly, the motion planning problem for hybrid systems is formulated using the hybrid equation framework, which is general to capture most hybrid systems. Secondly, a propagation algorithm template is proposed that describes a general framework to solve the motion planning problem for hybrid systems. Thirdly, a rapidly-exploring random trees (RRT) implementation of the proposed algorithm template is designed to solve the motion planning problem for hybrid systems. At each iteration, the proposed algorithm, called HyRRT, randomly picks a state sample and extends the search tree by flow or jump, which is also chosen randomly when both regimes are possible. Through a definition of concatenation of functions defined on hybrid time domains, we show that HyRRT is probabilistically complete, namely, the probability of failing to find a motion plan approaches zero as the number of iterations of the algorithm increases. This property is guaranteed under mild conditions on the data defining the motion plan, which include a relaxation of the usual positive clearance assumption imposed in the literature of classical systems. The motion plan is computed through the solution of two optimization problems, one associated with the flow and the other with the jumps of the system. The proposed algorithm is applied to an actuated bouncing ball system and a walking robot system so as to highlight its generality and computational features.

Figures (8)

• 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: The association between states/solution pairs and the vertices/edges in the search tree. The blue region denotes $X_{0}$, the green region denotes $X_{f}$, and the black region denotes $X_{u}$. The dots and lines between dots in Figure \ref{['fig:searchgraph_searchgraph']} denote the vertices and edges associated with the states and solution pairs in Figure \ref{['fig:searchgraph_statespace']}. The path $p = (v_{1}, v_{2}, v_{3}, v_{cur}, v_{new})$ in the search graph in Figure \ref{['fig:searchgraph_searchgraph']} represents the solution pair $\tilde{\psi}_{p} = \overline{\psi}_{e_{1}}|\overline{\psi}_{e_{2}}|\overline{\psi}_{e_{3}}|\psi_{new}$ in Figure \ref{['fig:searchgraph_statespace']}.
• Figure 5: The concatenation $\phi$ of trajectory $\phi_{2}$ to trajectory $\phi_{1}$.

Theorems & Definitions (100)

• Definition 2.1: (Hybrid time domain)
• Definition 2.2: (Hybrid input)
• Definition 2.3: (Hybrid arc)
• Definition 2.4: (Types of hybrid arcs)
• Definition 2.5: (Solution pair to a hybrid system)
• Definition 2.6: ($(\tau, \epsilon)$-closeness of hybrid arcs)
• Example 3.1: (Actuated bouncing ball system)
• Example 3.2: (Walking robot)
• Definition 4.1
• Remark 4.2