Tracking manipulator trajectories with ordinary singularities: a null-space based approach

This paper discusses the problem of tracking manipulator end-link trajectories passing through ordinary singularities. It is based on a method developed by J. Kieffer (1992) who suggested to parametrize the end-link trajectory by a parameter which is not time, and to consider this parameter as a dependent variable. We show that this is equivalent to a kinematically redundant mechanism comprising the original nonredundant kinematic chain augmented by one more virtual link. The trajectory is actually mapped onto the joint space augmented by the parameter variable. While the new mechanism performs only a "self-motion" tracking the projected trajectory through regular points and ordinary singularities, the end-link of the real mechanism tracks the desired trajectory. Our solution is based on effective null-space methods which have been so far widely applied to kinematically redundant manipulators. Asymptotic convergence of the tracking error to zero and stability is guaranteed through a conventional closed-loop approach.