Distortion Bounds of Subdivision Models for SO(3)

In the subdivision approach to robot path planning, we need to subdivide the configuration space of a robot into nice cells to perform various computations. For a rigid spatial robot, this configuration space is $SE(3)=\mathbb{R}^3\times SO(3)$. The subdivision of $\mathbb{R}^3$ is standard but so far, there are no global subdivision schemes for $SO(3)$. We recently introduced a representation for $SO(3)$ suitable for subdivision. This paper investigates the distortion of the natural metric on $SO(3)$ caused by our representation. The proper framework for this study lies in the Riemannian geometry of $SO(3)$, enabling us to obtain sharp distortion bounds.

Theory and Explicit Design of a Path Planner for an SE(3) Robot

We consider path planning for a rigid spatial robot with 6 degrees of freedom (6 DOFs), moving amidst polyhedral obstacles. A correct, complete and practical path planner for such a robot has never been achieved, although this is widely recognized as a key challenge in robotics. This paper provides a complete "explicit" design, down to explicit geometric primitives that are easily implementable. Our design is within an algorithmic framework for path planners, called Soft Subdivision Search (SSS). The framework is based on the twin foundations of $\epsilon$-exactness and soft predicates, which are critical for rigorous numerical implementations. The practicality of SSS has been previously demonstrated for various robots including 5-DOF spatial robots. In this paper, we solve several significant technical challenges for SE(3) robots: (1) We first ensure the correct theory by proving a general form of the Fundamental Theorem of the SSS theory. We prove this within an axiomatic framework, thus making it easy for future applications of this theory. (2) One component of $SE(3) = R^3 \times SO(3)$ is the non-Euclidean, non-orientable space SO(3). We design a novel topologically correct data structure for SO(3). Using the concept of subdivision charts and atlases for SO(3), we can now carry out subdivision of SO(3). (3) The geometric problem of collision detection takes place in $R^3$, via the footprint map. Unlike sampling-based approaches, we must reason with the notion of footprints of configuration boxes, which is much harder to characterize. Exploiting the theory of soft predicates, we design suitable approximate footprints which, when combined with the highly effective feature-set technique, lead to soft predicates. (4) Finally, we make the underlying geometric computation "explicit", i.e., avoiding a general solver of polynomial systems, in order to allow a direct implementation.

Proceedings of the 1st Workshop on Robotics Challenges and Vision

Proceedings of the 1st Workshop on Robotics Challenges and Vision (RCV2013)