Kinematic Convex Combinations of Multiple Poses of a Bounded Planar Object Based on an Average-Distance Minimizing Motion Sweep

Abstract

Convex combination of points is a fundamental operation in computational geometry. By considering rigid-body displacements as points in the image spaces of planar quaternions, quaternions and dual quaternions, respectively, the notion of convexity in Euclidean three-space has been extended to kinematic convexity in 𝑆⁢𝐸⁡(2),𝑆⁢𝑂⁡(3), and 𝑆⁢𝐸⁡(3) in the context of computational kinematic geometry. This paper deals with computational kinematic geometry of bounded planar objects rather than that of infinitely large moving spaces. In this paper, we present a new formulation for kinematic convexity based on an average-distance minimizing motion sweep of a bounded planar object. The resulting 1-DOF motion sweep between two planar poses is represented as a convex combination in the configuration space defined by (𝑥,𝑦,𝑧) where (𝑥,𝑦) is associated with the location of the centroid of the planar object and 𝑧 =sin ⁡𝜃 with 𝜃 being the angle of rotation. For three poses, a 2-DOF motion sweep is developed that not only minimizes the combined average squared distances but also attains a convex-combination representation so that existing algorithms for convex hull of points can be readily applied to the construction and analysis of kinematic convex hulls. This results in a new type of convex hull for planar kinematics such that its boundaries are defined by the average-distance minimizing sweeps of the bounded planar object.