We consider path planning for a rigid spatial robot moving amidst polyhedral obstacles. Our robot is either a rod or a ring. Being axially-symmetric, their configuration space is R^3 x S^2 with 5 degrees of freedom (DOF). Correct, complete and practical path planning for such robots is a long standing challenge in robotics. While the rod is one of the most widely studied spatial robots in path planning, the ring seems to be new, and a rare example of a non-simply-connected robot. This work provides rigorous and complete algorithms for these robots with theoretical guarantees. We implemented the algorithms in our open-source Core Library. Experiments show that they are practical, achieving near real-time performance. We compared our planner to state-of-the-art sampling planners in OMPL [Sucan et al., 2012]. Our subdivision path planner is based on the twin foundations of epsilon-exactness and soft predicates. Correct implementation is relatively easy. The technical innovations include subdivision atlases for S^2, introduction of Sigma_2 representations for footprints, and extensions of our feature-based technique for "opening up the blackbox of collision detection".

The concept of resolution-exact path planning is a theoretically sound alternative to the standard exact algorithms, and provides much stronger guarantees than probabilistic or sampling algorithms. It opens the way for the introduction of soft predicates in the context of subdivision algorithm. Taking a leaf from the great success of the Probabilistic Road Map (PRM) framework, we formulate an analogous framework for subdivision, called Soft Subdivision Search (SSS). In this video, we illustrate the SSS framework for a trio of simple planar robots: disc, triangle and 2-links. These robots have, respectively, 2, 3 and 4 degrees of freedom. Our 2-link robot can also avoid self-crossing. These algorithms operate in realtime and are relatively easy to implement.