Geometric Algorithms for Part Orienting and Probing

Abstract

In this thesis, detailed solutions are presented to several problems dealing with geometric shape and orientation of an object in the field of robotics and automation. We first have considered a general model for shape variations that allows variation along the entire boundary of an object, both in two and three-dimensional space. Based on this model of shape variation, we have studied the problem of orienting planar parts. We have focused on orienting a planar imperfect part with pushing by placing a single friction-less line like jaw in different orientations. We then have shown how the family of possible parts defined by this model can be simultaneously oriented into the smallest possible interval of final orientations after applying a bounded predetermined number of pushes. We have considered the same model and investigated the locus of the center of mass for parts with shape variation. We have considered parts with uniform mass distribution and bounded the location of the center of mass for both two and three-dimensional cases. Pose statistics is a fundamental topic related to part orienting. We have considered a family of 3D objects whose initial pose is uniformly random. We assumed that the object falls onto a flat surface in presence of gravity under quasi-static conditions. We defined a type of geometric eccentricity and showed that 3D eccentric objects with high probability rest at a pose, which is close to a specific plane or specific line. In this work, we also investigated a novel type of geometric probing. The goal is to interactively determine geometric shape and orientation of an unknown objects by using special measurements. We have defined a type of proximity probing which returns the distance to the boundary of the object in question. This work has concentrated on the case where the object is a convex polygon P in the plane. The goal is to find an upper bound on the number of measurements required to exactly determine P. We have proposed an algorithm that the number of requiring probes is linear in the number of vertices. Furthermore, our method is computationally very efficient, requiring only constant computation time per probe, for a total linear time complexity. We also considered the same task of using these proximity probes to identify P, but from a finite set of convex polygons. We presented an algorithm achieving this and bound the number of probes.