## Controllability of a 3-DOF Manipulator with a Passive Joint under a Nonholonomic Constraint

### Hirohiko Arai

#### Proceedings of 1996 IEEE International Conference on Robotics and Automation (ICRA'96), pp.3707-3713, Minneapolis, Minnesota, April 1996.

#### [Abstract]

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 states to arbitrary desired states. 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.

#### [Keywords]

manipulator, passive joint, dynamics, nonholonomic constraint, controllability

#### [Contributions and Applications]

This paper proves controllability of a 3-DOF manipulator with a passive joint which has neither an actuator nor a brake. This system is under a 2nd-order nonholonomic constraint, which is represented as a non-integrable differential equation including the accelerations. There have not been studies which proved the global controllability of such a mechanical system. The proof in this paper is easy to understand, and the constructed input can be directly used as the desired trajectory in the control stage. The result shows the possibility of positioning without using a brake in this type of manipulator. This method allows the control of more joints than actuators, and reduces the weight, energy consumption, and cost of the manipulator. It will be also effective in fault-tolerant control of a manipulator. Extension of this approach will enable 6-DOF manipulation of an object in non-gravitational space with only 3-DOF translational forces.