The jacodian matrix for a flexble manipulator

This article develops the exact first-order endpoint Jacobian matrix for a general n-degrees of freedom tree-like robot with flexible links. The Jacobian is developed in terms of the joint axes, the link deformations, and the relative position vectors using cross products. To have the correct first-order endpoint Jacobian matrix, the second-order kinematics is used to describe a flexible link. Using two local Jacobian matrices enables to write the endpoint Jacobian sub-matrix associated to a flexible link, which is similar to the column of the Jacobian associated to a joint. An example with a one-link flexible arm rotating in a vertical plane illustrates the usefulness of the endpoint Jacobian in calculating the torque required to apply an endpoint force and the link deformation resulting from this force. An experimental verification proves the validity of the developed Jacobian and suggests that using only first-order kinematics results in serious errors in the prediction of the beam's curvatures and deformations © 1995 John Wiley & Sons, Inc.