Sampling-based motion planning with differential constraints

摘要

Since differential constraints which restrict admissible velocities and accelerations of robotic systems are ignored in path planning, solutions for kinodynamic and non-holonomic planning problems from classical methods could be either inexecutable or inefficient. Motion planning with differential constraints (MPD), which directly considers differential constraints, provides a promising direction to calculate reliable and efficient solutions. A large amount of recent efforts have been devoted to various sampling-based MPD algorithms, which iteratively build search graphs using sampled states and controls. This thesis addresses several issues in analysis and design of these algorithms. Firstly, resolution completeness of path planning is extended to MPD and the first quantitative conditions are provided. The analysis is based on the relationship between the reachability graph, which is an intrinsic graph representation of a given problem, and the search graph, which is built by the algorithm. Because of sampling and other complications, there exist mismatches between these two graphs. If a solution exists in the reachability graph, resolution complete algorithms must construct a solution path encoding the solution or its approximation in the search graph in finite time. Secondly, planners are improved with symmetry-based gap reduction algorithms to solve their gap problem, which dramatically increases time to return a high quality solution trajectory whose final state is in a small neighborhood of a goal state. The improved planners quickly obtain high quality solutions by minimizing gaps in solution path candidates, which is greatly accelerated using symmetries of robotic systems to avoid numerical integration. Finally, a heuristic is designed to solve metric sensitivity of RRT-based planners, which means that RRT-based methods have difficulties in escaping local minima when the given metric provides a poor approximation of the cost-to-go. Instead of designing a metric, the heuristic is obtained by collecting collision information online and assigning a real value to each node in the search graph. A node with a higher value means that the number of trajectories from the node that have been detected in collision is larger. Local minima are more likely to be avoided when nodes with smaller values are given higher probability to be extended.