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

This article deals with the computational kinematic geometry of bounded planar objects rather than that of infinitely large moving spaces. In this article, we present a new formulation for kinematic convexity based on an average-distance minimizing motion sweep of a bounded planar object. The resulting single-degree-of-freedom (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=sinθ with θ being the angle of rotation. For three poses, a 2DOF 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.