On Achieving Periodic Joint-Motion for Redundant Robots

The consequence of the loss of involutivity of a specific set of vector fields on the periodicity of the joint motion is examined for redundant robots. An output task, defined as a one dimensional periodic closed curve embedded in a two dimensional working surface, is realized through the computation of joint velocities in the configuration space. Depending on the manner in which the joint velocity is computed from the end-effector velocity, the resulting joint motion can become unpredictable and of a chaotical nature, even though the end-effector movement is periodic and predictable. The paper proposes an improvement over classical pseudo-inverse computation of the joint motion by first suitably selecting two involutive vector fields (used as a basis for parameterization) in the tangent bundle of the output manifold. It also presents a sufficient condition for the periodicity of all the joint configuration based on the involutivity of two vector fields in the tangent bundle of the joint space. The results are illustrated on a five-link rotary redundant robot (5R robot).