The Definition, Determination, and Characterization of Acceleration Sets for Spatial Manipulators

Abstract

In this article, the approach developed by the authors for sys tematically studying the acceleration capability and properties of the end effector of a planar manipulator is extended to the general, serial, spatial manipulator possessing three degrees of freedom. The acceleration of the end effector at a given config uration of the manipulator is a linear function of the actuator torques and a (nonlinear) quadratic function of the joint ve locities. By decomposing the functional relationships between the inputs (actuator torques and joint velocities) and the output (acceleration of the end effector) into two fundamental map pings—a linear mapping between the actuator torques and the acceleration space of the end effector and a quadratic (nonlin ear) mapping between the joint velocities and the acceleration space of the end effector—and by deriving the properties of these two mappings, it is possible to determine the properties of all acceleration sets that are the images of the appropri ate input sets under the two fundamental mappings. A central feature of this article is the determination of the properties of the quadratic mapping, which then makes it possible to obtain analytic expressions for most acceleration properties of interest. We show that a fundamental way of studying these quadratic mappings is in terms of the mapping of (input) line congru ences into (output) line congruences. The article concludes with the application of the analytical results to the computa tion of the various acceleration properties of an actual spatial manipulator.