Efficient Control with an Order (n) Recursive Inversion of the Jacobian for an n-Link Serial Manipulator

Abstract

Given desired task space trajectories, the robot control problem is to choose the joint torques, T, so that the actual trajectories track the desired ones. If joint-based control algorithms are used, the transformation from task space to joint space requires an inversion of the Jacobian matrix, J. Typically this requires order (n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) operations where n is the number of links in a serial manipulator. In this paper we will show how a technique developed by Rodriguez [15] can be applied to reduce the order (n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) inversion of the Jacobian to an order (n) inversion of the product J* J by formulating the n-link robot equation as a spatially recursive algorithm in the form of a filtering and smoothing problem. It is shown that with a proper model, the inverse Jacobian problem is equivalent to solving the forward dynamics problem for the same model. The algorithm is modified to yield the least squares solution if the Jacobian matrix is singular.