Soft Subdivision Planner for a Rod

In this paper we consider the motion planning problem to find a path for a rod (also known as a line segment or ladder) amidst polyhedral obstacles in 3D. The configuration space of a rod has 5 degrees of freedom (DOFs) and may be identified with R × S. Devising theoretically-sound but practical algorithms for motion planning of robots with more than 4 DOFs is a major challenge. To our knowledge, there is no explicit exact planning algorithm for a rod. Even if one exists, exact implementation seems out of the question. We also believe that exact algorithms have little practical value in robot motion planning because of the inherent inaccuracies in all real-world robots and environments. So we want ε-approximations, but we must be careful not to use the ε parameter as if it is an exact parameter (e.g., replacing a point p by an ε-ball centered at p with sharp boundaries). In our recent work [3, 5, 6, 1], we propose the notion of a resolution-exact algorithm where ε is used in a “soft” manner. Specifically, we propose a subdivision framework based on the twin foundations of ε-exactness and soft predicates. In this paper we develop an ε-exact planner for a rod in 3D under this framework. Technical Background and Setting We consider a robot R0 which is a rod (closed line segment) of length l0 > 0 in R . The two endpoints of the rod are denoted B and A, and called (respectively) the base and apex of R0. The standard placement of the rod is defined to be the line segment [(0, 0, 0), (0, 0, l0)] where (0, 0, 0) and (0, 0, l0) correspond to the placements of B and A, respectively. As a rigid body, the rod has 5 DOFs. We let its configuration space be Cspace = R 3 ×S and a typical configuration is a pair (q, α) where q = (x, y, z) ∈ R and α ∈ S is a (rod) orientation. This configuration defines a transformation of R whereby the base and apex of the robot are respectively transformed to the points B[q, α] := q and A[q, α] := q + l0α. The space S is unsuitable for subdivision. So we introduce a parametrized model Ŝ2 which is the boundary of the cube [−1, 1] ⊆ R. In general, if X and