Abstract
The aim of this note is to examine the conditions of stability of a simple robotic task: we consider a one degree-of-freedom (dof) robot that collides with a spring-like environment with stiffness k, the goal being to stabilize the system in contact with the environment. We study conditions on the feedback gains that guarantee quadratic Lyapunov stability of the task with a well-conditioned solution to the Lyapunov equation. It is shown that when the environment's stiffness k grows unbounded, those conditions yield unbounded values of the gains. Motivated by the stability analysis of the impact Poincaré map in the perfectly rigid case ( \(k = + \infty\)), we propose an analysis that is independent of k. It enables us to conclude on global asymptotic convergence of the system's state towards the equilibrium point. This work can also be seen as the study of stability of a contact (force control) phase, taking into account the unilateral feature of the constraint.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
An, C. H., and Hollerbach, J. M., “Dynamic stability issues in force control of manipulators,” IEEE Int. Conf. on Robotics and Automation, Raleigh, NC, pp. 890-896, 1987.
Branicky, M. S., “Stability of switched and hybrid systems,” in Proc. IEEE Conf. Decision and Control, Lake Buena Vista, 1994.
Brogliato, B., Niculescu, S. I., and Orhant, P., “On the control of finite-dimensional mechanical systems with unilateral constraints,” IEEE Transactions on Automatic Control, vol. 42(2), pp. 200-215, 1997.
Lozano, R., and Brogliato, B., “Stability of interconnected passive systems,” European Control Conference, Brussels, 1997.
Colgate, J. E., and Hogan, N., “Robust control of dynamically interacting systems,” Int. J. of Control, vol. 48(1), pp. 65-88, 1988.
Corless, M., and Leitmann, G., “Continuous state feedback guaranteing unifrom ultimate boundedness for uncertain dynamical systems,” IEEE Trans. on Automatic Control, vol. 26(5), 1981.
Hogan, N., “On the stability of manipulators performing contact tasks,” IEEE J. of Robotics and Automation, vol. 4, pp. 677-686, 1988.
Kazerooni, H., “Contact instability of the direct drive robot when constrained by a rigid environment,” IEEE Trans. on Automatic Control, vol. 35, pp. 710-714, 1990.
Leela, S., “Stability of measure differential equations,” Pacific J. of Mathematics, vol. 55(2), pp. 489-498, 1974.
Marth, G. T., Tarn, T. J., and Bejczy, A. K., “Stable transition phase for robot arm control,” in Proc. IEEE Int. Conf. on Robotics and Automation, Atlanta, pp. 355-362, 1993.
Mills, J. K., and Lokhorst, D. M., “Stability and control of robotic manipulators during contact/noncontact task transition,” IEEE Trans. on Robotics and Automation, vol. 9(3), pp. 335-345, June 1993.
Mills, J. K., and Lokhorst, D.M., “Control of robotic manipulators during general task execution: A discontinuous control approach,” Int. Journal of Robotics Research, vol. 12(2), pp. 146-163, 1993.
Moreau, J. J., “Unilateral contact and dry friction in finite freedom dynamics,” in Nonsmooth Mechanics and Applications, J.J. Moreau and P.D. Panagiotopoulos, editors, CISM Courses and Lectures, no. 302, Springer-Verlag, pp. 1-82, 1988.
Moreau, J. J., “Some numerical methods in multibody dynamics: Application to granular materials,” Second European Solid Mechanics Conf., Genoa, Italy, 1994, European J. of Mechanics, A/Solids, vol. 13(6), pp. 93- 116.
Paoli, L., and Schatzman, M., “Mouvement à un nombre fini de degrés de liberté avec contraintes unilatérales: cas avec perte d'énergie,” Mathematical Modelling and Numerical Analysis (Modèlisation Mathématique et Analyse Numérique ), vol. 27(6), pp. 673-717, 1993.
Schmaedeke, W. W., “Optimal control theory for nonlinear vector differential equations containing measures,” SIAM J. of Control, ser. A, vol. 3(2), pp. 231-280, 1965.
Slotine, J. J., and Li, W., “Adaptive manipulator control: A case study,” IEEE Trans. on Automatic Control, vol. 33, pp. 995-1003, 1988.
Tornambé, A., “Global regulation of planar robot arm striking a surface,” IEEE Transactions on Automatic Control, vol. 41, 1996.
Tornambé, A., “Modeling and controling two-degree-of freedom impacts,” in Third IEEE Mediterranean Conference on Control and Automation, Limassol, Cyprus, July 1995. See also IEE Proceedings Control Applications, vol. 143(1), pp. 85-90, 1996.
Vidyasagar, M., Nonlinear Systems Analysis, Prentice-Hall, second edition, 1993.
Volpe, R., and Khosla, P., “A theoretical and experimental investigation of impact control for manipulators,” Int. Journal of Robotics Research, vol. 12(4), pp. 351-365, 1993.
Vol'pert, A. I., and Hudjaev, S. I., Analysis in Classes of Discontinuous Functions and Equations of Mathematical Physics, Martinus Nijhoff Publisher: Dordrecht, 1985.
Wang, Y., “Dynamic modeling and stability analysis of mechanical systems with time-varying topologies,” ASME J. of Mechanical Design, vol. 115, pp. 808-816, 1993.
Whitney, D. E., “Force feedback control of manipulator fine motions,” ASME J. of Dynamic Systems, Meas. and Control, vol. 99, pp. 91-97, 1977.
Youcef-Toumi, K., and Gutz, D. A., “Impact and force control: Modeling and experiments,” ASME J. of Dynamic Systems, Meas. and Control, vol. 116, pp. 89-98, 1994.
Yoshikawa, T., “Dynamic hybrid position/force control of robot manipulators-Description of hand constraints and calculation of joint driving force,” IEEE Trans. on Robotics and Automation, vol. 3(5), pp. 386- 392, 1987.
Brogliato, B., Nonsmooth Impact Mechanics. Models, Dynamics and Control, Springer Verlag, LNCIS 220, 1996.
Rights and permissions
About this article
Cite this article
Brogliato, B., Orhant, P. Contact Stability Analysis of a One Degree-of-Freedom Robot. Dynamics and Control 8, 37–53 (1998). https://doi.org/10.1023/A:1008226913003
Issue Date:
DOI: https://doi.org/10.1023/A:1008226913003