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Description: The motion planning problem consists of generating a state trajectory, and associated input, that connects the given initial state set and the final state set while satisfying the dynamics of the system as well as a given safety criterion, which is critical in many applications in Robotics. Search graph has been widely used to model the process of searching for the solution to the motion planning problem, which employs vertices to represent the states of the systems and edges to represent the state trajectory that connects the states represented by their endpoint vertices. Various algorithms, such as breadth-first search algorithm and rapid exploring random tree (RRT) algorithm, can be used to construct the search graph and to search for the solution to the motion planning problem. While the search graph-based algorithm has been widely used throughout the motion planning problems for the purely continuous-time systems and purely discrete-time systems, fewer efforts are devoted to the motion planning problems for hybrid systems.
Hybrid systems are systems that contain both continuous-time dynamics, called flows, and discrete-time dynamics called jumps. The classic motion planning algorithms do not consider both continuous and discrete dynamics, which leads to the incompatibility for motion planning for hybrid systems.
Motivated by this deficiency, we propose to study motion planning algorithms for hybrid systems. In this report, we will present the need for the motion planning algorithm for hybrid systems and provide some preliminary results. In particular, we firstly define the motion planning problem for a hybrid system and propose a forward/backward propagation algorithm to solve it.
Next, we present our hyEQ RRT algorithm and some theoretical results about its probabilistic completeness property. Finally, we present our proposal about the proofs of the completeness of the proposed algorithms, methods to solve the optimal motion planning problem, and the control design to tracking the generated motion plan.