Sensor-based Planning for a Rod-shaped Robot in Three Dimensions: Piecewise Retracts of R3 × S2

Abstract

We present a new roadmap for a rod-shaped robot operating in a three-dimensional workspace, whose configuration space is diffeomorphic to R3 × S2. This roadmap is called the rod hierarchical generalized Voronoi graph (rod-HGVG) and can be used to find a path between any two points in an unknown configuration space using only the sensor data. More importantly, the rod-HGVG serves as a basis for an algorithm to explore an unknown configuration space without explicitly constructing it. Once the rod-HGVG is constructed, the planner can use it to plan a path between any two configurations. One of the challenges in defining the roadmap revolves around a homotopy theory result, which asserts that there cannot be a one-dimensional deformation retract of a non-contractible space with dimension greater than two. Instead, we define an exact cellular decomposition on the free configuration space and a deformation retract in each cell (each cell is contractible). Next, we “connect” the deformation retracts of each of the cells using a roadmap of the workspace. We call this roadmap a piecewise retract because it comprises many deformation retracts. Exploiting the fact that the rod-HGVG is defined in terms of workspace distance measurements, we prescribe an incremental procedure to construct the rod-HGVG that uses the distance information that can be obtained from conventional range sensors.