THE JACOBIAN MATRIX FOR A FLEXIBLE MANIPULATOR

This article develops the exact first-order endpoint Jacobian matrix f or a general n-degrees of freedom tree-like robot with flexible links. The Jacobian is developed in terms of the joint axes, the link deform ations, and the relative position vectors using cross products. To hav e the correct first-order endpoint Jacobian matrix, the second-order k inematics is used to describe a flexible link. Using two local Jacobia n matrices enables to write the endpoint Jacobian sub-matrix associate d to a flexible link, which is similar to the column of the Jacobian a ssociated to a joint. An example with a one-link flexible arm rotating in a vertical plane illustrates the usefulness of the endpoint Jacobi an in calculating the torque required to apply an endpoint force and t he link deformation resulting from this force. An experimental verific ation proves the validity of the developed Jacobian and suggests that using only first-order kinematics results in serious errors in the pre diction of the beam's curvatures and deformations. (C) 1995 John Wiley & Sons, Inc.