Differentiation-Free Taylor Approximation of Finite Motion in Closed Loop Kinematics

Higher-order derivatives of kinematic mappings give insight into the motion characteristics of complex mechanisms. Screw theory and its associated Lie group theory have been used to find these derivatives of loop closure equations up to an arbitrary order. However this has not been extended to the higher-order derivatives of finite motion as given by the inverse or forward kinematic model of closed loop mechanisms. In this paper, a recursive algorithm is presented, consisting solely of matrix multiplications, which is capable of giving these higher-order derivatives of kinematic models of closed loop linkages. It depends on a simplified representation of the higher-order derivatives of an open chain. From these higher-order derivatives a Taylor expansion of a finite motion becomes available. The evaluation of this method on a Taylor approximation (up to 5th order) of the inverse kinematic model of a 5-bar mechanism shows a good approximation in a large part of workspace around the evaluation point.