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 suitably selecting two involutive vector fields. This then leads to a constructive sufficient condition for the periodicity of the joints based on the usage of both the 1-form defining the output manifold and complementary integrable 1-forms. The results are illustrated on a five-link rotary redundant robot (5R robot).