Efficient geometric algorithms for robot kinematic design

This article addresses the problem of designing a robotic mechanism such that it end-effector frame comes closest to reaching a set of desired goal frames. We formulate this as an optimization problem, in which the kinematic parameters are selected to minimize the total distance between the end-effector frame and each goal frame. The objective function is defined in terms of a class of distance metrics on the rigid body motions that are invariant with respect to choice of fixed reference frame. A main contribution of this article is an explicit expression for the gradient of this objective function with respect to the kinematic parameters. With this analytic gradient, efficient optimization algorithms can now be developed for the design of general spatial mechanisms. Our design methodology is illustrated with an example involving the base positioning of two cooperating robots