Randomized path planning on manifolds based on higher-dimensional continuation

Despite the significant advances in path planning methods, highly constrained problems are still challenging. In some situations, the presence of constraints defines a configuration space that is a non-parametrizable manifold embedded in a high-dimensional ambient space. In these cases, the use of sampling-based path planners is cumbersome since samples in the ambient space have low probability to lay on the configuration space manifold. In this paper, we present a new path planning algorithm specially tailored for highly constrained systems. The proposed planner builds on recently developed tools for higher-dimensional continuation, which provide numerical procedures to describe an implicitly defined manifold using a set of local charts. We propose to extend these methods focusing the generation of charts on the path between the two configurations to connect and randomizing the process to find alternative paths in the presence of obstacles. The advantage of this planner comes from the fact that it directly operates into the configuration space and not into the higher-dimensional ambient space, as most of the existing methods do.