Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T06:01:38.795Z Has data issue: false hasContentIssue false

Alternative proofs of four stability properties of rigid-link manipulators under PID position control

Published online by Cambridge University Press:  19 April 2012

Ryo Kikuuwe*
Affiliation:
Department of Mechanical Engineering, Kyushu University, Fukuoka, Japan
*
*Corresponding author. E-mail: [email protected]

Summary

This paper presents new proofs of four stability properties (semiglobal strict passivity, semiglobal asymptotic stability, semiglobal input-to-state stability, and semiglobal uniform ultimate boundedness with an arbitrarily reducible ultimate bound) of a rigid-link manipulator under proportional-integral-derivative (PID) position control. The proofs employ a strict Lyapunov function and a novel parameterization to provide four inequality conditions for the stability properties. In those inequalities, arithmetic operations on physical quantities are physically consistent if the joints are all revolute or all prismatic. A gain selection procedure is presented by which the ultimate bounds of velocity error, position error, and its integral can be independently designed.

Type
Articles
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alvarez-Ramirez, J., Kelly, R. and Cervantes, I., “Semiglobal stability of saturated linear PID control for robot manipulators,” Automatica 39 (6), 989995 (2003).CrossRefGoogle Scholar
2.Alvarez-Ramirez, J., Santibáñez, V. and Campa, R., “Stability of robot manipulators under saturated PID compensation,” IEEE Trans. Control Syst. Technol. 16 (6), 13331341 (2008).CrossRefGoogle Scholar
3.Arimoto, S., Control Theory of Nonlinear Mechanical Systems: A Passivity-based and Circuit-Theoretic Approach (Oxford University Press, Oxford, 1996).CrossRefGoogle Scholar
4.Arimoto, S. and Miyazaki, F., “Stability and Robustness of PID Feedback Control for Robot Manipulators of Sensory Capability,” In: Robotics Research, First International Symposium (Brady, M. and Paul, R., eds.) (MIT Press, Cambridge, MA, 1984) pp. 783799.Google Scholar
5.Cervantes, I. and Alvarez-Ramirez, J., “On the PID tracking control of robot manipulators,” Syst. Control Lett. 42, 3746 (2001).CrossRefGoogle Scholar
6.Chaillet, A., Loria, A. and Kelly, R., “Robustness of PID-controlled manipulators vis á vis actuator dynamics and external disturbances,” Eur. J. Control 13 (6), 563576 (2007).CrossRefGoogle Scholar
7.Choi, Y. and Chung, W. K., PID Trajectory Tracking Control for Mechanical Systems (Springer, Berlin, 2004).CrossRefGoogle Scholar
8.Hernández-Guzmán, V. M., Santibáñez, V. and Silva-Ortigoza, R., “A new tuning procedure for PID control of rigid robots,” Adv. Robot. 22 (9), 10071023 (2008).CrossRefGoogle Scholar
9.Kelly, R., “A tuning procedure for stable PID control of robot manipulators,” Robotica 13 (2), 141148 (1995).CrossRefGoogle Scholar
10.Kelly, R., Santibáñez, V. and Loría, A., Control of Robot Manipulators in Joint Space (Springer, Berlin, 2005).Google Scholar
11.Khalil, H. K., Nonlinear Systems, 3rd ed. (Prentice Hall, Upper Saddle River, NJ, 2002).Google Scholar
12.Loria, A., Lefeber, E. and Nijmeijer, H., “Global asymptotic stability of robot manipulators with linear PID and PI2D control,” Stab. Control: Theory Appl. 3 (2), 138149 (2000).Google Scholar
13.Meza, J. L., Santibáñez, V. and Campa, R., “An estimate of the domain of attraction for the PID regulator of manipulators,” Int. J. Robot. Autom. 22 (3), 187195 (2007).Google Scholar
14.Mulero-Martínez, J. I., “Uniform bounds of the Coriolis/centripetal matrix of serial robot manipulators,” IEEE Trans. Robot. 23 (5), 10831089 (2007).CrossRefGoogle Scholar
15.Pervozvanski, A. A. and Freidovich, L. B., “Robust stabilization of robotic manipulators by PID controllers,” Dyn. Control 9 (3), 203222 (1999).CrossRefGoogle Scholar
16.Ortega, R., Loria, A. and Kelly, R., “A semiglobally stable output feedback PI2D regulator for robot manipulators,” IEEE Trans. Autom. Control 40 (8), 14321436 (1995).CrossRefGoogle Scholar
17.Qu, Z. and Dorsey, J. F., “Robust PID control of robots,” Int. J. Robot. Autom. 6 (4), 228235 (1991).Google Scholar
18.Rocco, P., “Stability of PID control for industrial robot arms,” IEEE Trans. Robot. Autom. 12 (4), 606614 (1996).CrossRefGoogle Scholar
19.Wen, J. T. and Murphy, S. H., “PID control for robot manipulators,” CIRSSE Document #54, Rensselaer Polytechnic Institute (1990). Available at http://ntrs.nasa.gov/.Google Scholar