Singularity-consistent parameterization of robot motion and control

The inverse kinematics problem is formulated as a parameterized autonomous dynamical system problem, and respective analysis is carried out It is show n that a singular point of work space can be mapped either as a critical or a noncritical point of the autonomous system, depending on the direction o f approach to the singular point. Making use of the noncritical mapping, a closed-loop kinematic controller with asymptotic stability and velocity lim its along degenerate singular or near-singular paths is designed. The autho rs introduce a specific type of motion along the reference path, the so-cal led natural motion. This type of motion is obtained in a straightforward ma nner from the autonomous dynamical system and always satisfies the motion c onstraint at a singular point. In the vicinity of the singular point, natur al motion slows down the end-effector speed and keeps the joint velocity bo unded. Thus, no special trajectory replanning will be required. In addition , the singular manifold can be crossed, if necessary. Further on, it is sho wn that natural motion constitutes an integrable motion component. The rema ining, nonintegrable motion component is shown to be helpful in solving a p roblem related to the critical point mapping of the autonomous system. The authors design a singularity-consistent resolved acceleration controller wh ich they then apply to singular or near-singular trajectory tracking under torque limits. Finally, the authors compare the main features of the singul arity-consistent method and the damped-least-squares method. It is shown th at both methods introduce a so-called algorithmic error in the vicinity of a singular point. The direction of this error is, however different in each method. This is shown to play an important role for system stability.