A Computationally Efficient Planar Rigid Body Distance Metric

A means of assessing the quality of a given rigid body configuration relative to a desired position and orientation is developed. Here, the assessment is based on the desired positions of all particles that constitute the body. This measure of quality has previously been shown to meet the mathematical requirements of a metric. This metric, however, has been largely dismissed in practical application due to the difficulty in performing its calculation. This paper describes procedures to efficiently calculate this metric for planar positioning problems. A method of obtaining an analytical formulation for planar displacement for any polygonal body is presented. An approximation function for the metric is also presented. This approximation function is very easily calculated and provides an intuitive appreciation of the metric values. Calculation times for the metric for standard polygonal body geometries is on the order of 100 microseconds on a laptop computer. Calculation times for the approximation function for standard polygonal bodies is on the order of microseconds