Adjoint Jacobian closed-loop kinematic control of robots

We propose a new technique for closed-loop kinematic control of nonredundant robotic mechanisms, based on the adjoint matrix of the kinematic Jacobian. Using the Lyapunov direct method, we show that the adjoint Jacobian approach guarantees asymptotic stability at regular points, around singularities, and at so-called instantaneous self-motion singularities. The new property, as compared to previous approaches, is that direction of motion can be precisely controlled at those points. To guarantee the asymptotic stability around any singularity and at instantaneous self-motion singularities, the desired (scalar) end-effector velocity is appropriately modified, and in the same time, restriction on the joint velocity norm according to a user-specified value is imposed. In the vicinity of a singularity an error in the position along the desired path is tolerated, which however, does not lead to deviation from the path.