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

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo></mml:math> , and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo></mml:math> 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo></mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo></mml:math> is associated with the location of the centroid of the planar object and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>z</mml:mi> <mml:mo>=</mml:mo> <mml:mtext>sin</mml:mtext> <mml:mspace></mml:mspace> <mml:mi>θ</mml:mi></mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>θ</mml:mi></mml:math> 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.