Underactuated Robot Manipulators

Project Objectives and Current Work

Underactuated robots are those robots with both passive and active joints. For example, when a joint motor in a fully-actuated robot fails, that joint becomes passive. It is important in such cases to design control techniques to still control the robot, if possible. Another example is a hyper-redundant snake-like robot with several degrees-of-freedom. In this case one would like to see if control is possible when not all joints are actuated. Space applications may also benefit from the study of underactuated robots, for a free-floating satellite can be modelled as a 6 degrees-of-freedom passive mechanism. The overall system (satellite + manipulator) can be considered as a combination of actuated and passive subsystems, i.e., as an underactuated one. Humans commonly utilize underactuation to perform tasks more precisely or more easily. For example, the martial art star Bruce Lee can project the tip of his favorite weapon, a numbchuck, on a precise point in space to produce a tremendous impact force. The numbchuck is composed of 3 hardwood segments connected by 2 passive joints, and the overall combination of the numbchuck with Bruce Lee's arm is an underactuated system.

The issues that we are interested in investigating include:

• characterization of the dynamic coupling responsible for driving the passive joints through the movement of the active ones;
• rational utilization of the dynamic coupling to control the system with maximized performance;
• stabilization of the system around an equilibrium point, irrespective of the number of actuators available;
• control of underactuated systems with substantial modelling uncertainties and disturbances;
• optimal control of the system when each passive joint is equipped with an on/off brake;
• conditions for controllability of the passive joints through application of torques at the active ones; and the relationship between dynamic coupling and controllability.

We have been studying the above issues since 1994, and have accomplished substantial results. We have defined a dynamic coupling index to measure the amount of dynamic coupling between the actuated and the passive joints of an underactuated manipulator. We have developed a robust control scheme to stabilize all joints of an underactuated manipulator to an equilibrium point. We have showed, based on results from differential geometry, that a non-zero coupling index implies controllability of the passive joints via application of torques at the actuators.

We built two underactuated manipulators, one with 2 links and 1 passive joint, and another with 3 links and 2 passive joints (shown below). We use the brakes in our control method to eliminate the nonholonomic constraints in the dynamic equations associated with the passive joints. Both robots are locally controllable anywhere inside their workspaces, since their coupling indices are both non-zero at any given configuration. Control of all joints is then possible with only one actuator.

Figure 1: Two-link underactuated manipulator with one passive joint.

Figure 2: Three-link underactuated manipulator with two passive joints.

Future research will include:

• extension of the work to parallel chain mechanism;
• extension of the work to underpowered systems;
• investigation of systems with a free-floating base;
• study of gymnast motion with underactuation;
• planning motion profiles for optimization.

Project Researchers

• Marcel Bergerman
• H. Ben Brown
• Christopher Lee
• Bin Liang
• Yangsheng Xu

Project Publications

• Bergerman, M.; Xu, Y. "Planning collision-free motions for underactuated manipulators in constrained configuration space." IEEE International Conference on Robotics and Automation, Albuquerque, NM, EUA, April 1997.
• Liang, B.; Xu, Y.; Bergerman, M. "Dynamically equivalent manipulator for space manipulator system: Part 1." IEEE International Conference on Robotics and Automation, Albuquerque, NM, EUA, April 1997.
• Bergerman, M.; Xu, Y. "Robust joint and Cartesian control of underactuated manipulators." Transactions of the ASME, Journal of Dynamic Systems, Measurement, and Control, vol. 118, no. 3, Sep. 1996, pp. 557-565. (Abstract or Postscript) Also available as a CMU technical report.
• Liang, B.; Xu, Y.; Bergerman, M. ``Mapping a space manipulator to a dynamically equivalent manipulator." Technical Report CMU-RI-TR-96-33, Carnegie Mellon University, Sep. 1996.
• Bergerman, M.; Xu, Y. "Optimal control sequence for underactuated manipulators." Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, USA, April 1996, pp. 3714-3719. (Abstract or Postscript)
• Bergerman, M.; Lee, C.; Xu, Y. "A dynamic coupling index for underactuated manipulators." Journal of Robotic Systems, vol. 12, no. 10, October 1995, pp. 693-707.x (Abstract or Postscript) Also available as a CMU technical report.
• Bergerman, M.; Lee, C.; Xu, Y. "Dynamic coupling of underactuated manipulators." Proceedings of the 4th IEEE Conference on Control Applications, Albany, USA, September 1995, pp. 500-505. (Abstract or Postscript)
• Bergerman, M.; Lee, C.; Xu, Y. "Experimental study of an underactuated manipulator." Proceedings of the 1995 IEEE/RSJ International Conference on Intelligent Robots and Systems, Pittsburgh, USA, August 1995, vol. 2, pp. 317-322. (Abstract or Postscript)Also available as a CMU technical report.
• Bergerman, M.; Xu, Y. "Robust control of underactuated manipulators: analysis and implementation." Proceedings of the 1994 IEEE International Conference on Systems, Man and Cybernetics, San Antonio, USA, October 1994, pp. 925-930. (Abstract or Postscript)
• Lee, C.; Xu, Y. "Actuability of underactuated manipulators." Technical Report, The Robotics Institute, Carnegie Mellon University, CMU-RI-TR-94-13, 1994.