Development of a Velocity Metric for Rigid-Body Planar Motion

Two motions of motion quality have been developed for planar motion. They are the best motion measure and the velocity metric . The best motion measure identifies the best motion for a given displacement. The velocity metric quantifies the discrepancy between two planar motions for the same rigid body. The best motion measure compares the motion of each particle on the body to an ideal , but usually unobtainable, motion. This ideal motion moves each particle from its current position to its desired position on a straight-line path. Although the ideal motion is not a valid rigid body motion, this does not preclude its use as a reference standard in evaluating valid rigid body motions. It is shown that the best motion measure can be reduced to the product of two components: the average distance from a point in the plane to the body, and a term based on configuration parameters. The optimal instantaneous planar motion for general rigid bodies in translation and rotation is characterized. This optimal motion is defined as the geodesic motion of a frame located at the geometric center of the body. In other words, the geometric center of the body will move in a straight-line path to its desired position, while the body rotates about the axis perpendicular to the plane. The optimal angular velocity is a function of the discrepancy between the current configuration and the desired configuration. A velocity measure used to evaluate how closely one motion is to another is also defined. This second measure is shown to meet the mathematical requirements of a metric. For a single particle, the velocity metric is the norm of the difference in translational velocity. It is shown that for most cases, this metric can be characterized as the average distance from a point in space to the body. This point is the instantaneous center of the motion associated with the discrepancy between the two motions. Because the measures described above depend on body geometry, a method to analytically evaluate any polygonal body based on polygonal decomposition into simpler bodies is provided. Results for example planar positioning problems are presented