3 First and Second Order Kine - matic Analysis

A kinematic singularity occurs when the Jacobian matrix associated with the forward kinematic map loses rank. At a singular connguration, the end-eeector velocities are restricted to a subspace of the task space velocities. Also, rate control schemes that require the inversion of the Jacobian matrix fail at such singulari-ties. However, even at a singular connguration, if the kinematic map is generic the end-eeector can achieve a desired end-eeector acceleration. The second order kinematic equations are derived for a serial chain manipulator using screw theory to demonstrate this. These equations are then used to develop a scheme to determine the joint rates from the desired end-eeector velocity and acceleration at singularities. The theory and the algorithm are illustrated using simple examples and computer simulations.