Towards finding the shortest-paths for 3D rigid bodies
Abstract
In this work; we analyze and present an algorithm to find shortest-paths for generic rigid bodies. We derived the necessary conditions for optimality using Lagrange multipliers; and compared it to the conditions derived from Pontraygin's Maximum Principle. We derived the equations of the necessary conditions using geometric Jacobian; drawing inspiration from the similarity between the rigid-body systems and the arm-like systems. In the previous work, the analysis focused on finding shortest-paths to reach goals in positions only. This work extends the analysis to find the shortest-path to reach a goal with complete configuration in 3D. We show that the algorithm is resolution complete even when the orientations are included. To overcome the complexity of 3D orientations; we describe the system using three points in the robot frame; and show that this parameter system is redundant but can derive the same necessary conditions as those derived using the minimum parameters (configuration). We used a 3D Dubins system to demonstrate the correctness of the analysis and the algorithm.