Solving for manipulator joint rates in singular positions

Computing manipulator joint velocities and accelerations, given the end-effector Cartesian rates (velocities and accelerations), is complicated by the singular positions the manipulator can assume along a trajectory. At such configurations the Jacobi matrix that relates Cartesian end-effector rates to joint rates cannot be inverted. Geometric approaches in solving for joint rates yield expressions that are indeterminate at such configurations. It is shown that exact solutions for the joint rates can be obtained in most singularity situations if the system of differential kinematic relations is properly partitioned. A solution methodology is developed and illustrated by an example