Controllability of a 3-DOF manipulator with a passive joint under a nonholonomic constraint

Controllability of a manipulator with a passive joint which has neither an actuator nor a holding brake is investigated. The manipulator has 3 degrees of freedom in a horizontal plane and the third joint is passive. The dynamic constraint on the third link is 2nd-order nonholonomic. The controllability is proved by constructing examples of the input trajectories from arbitrary initial slates to arbitrary desired slates. The proof is intuitively understandable, and the construction of the input directly leads to the trajectory planning. Simulations show that the manipulator can reach the desired position and velocity by the constructed input.