Abstract
In this paper, we consider hybrid models of mechanical systems undergoing impacts, Lagrangian hybrid systems, and study their periodic orbits in the presence of Zeno behavior—an infinite number of impacts occurring in finite time. The main result of this paper is explicit conditions under which the existence of stable periodic orbits for a Lagrangian hybrid system with perfectly plastic impacts implies the existence of periodic orbits in the same system with non-plastic impacts. Such periodic orbits contain phases of constrained and unconstrained motion, and the transition between them necessarily involves Zeno behavior. The result is practically useful for a wide range of unilaterally constrained mechanical systems under cyclic motion, as demonstrated through the example of a double pendulum with a mechanical stop.
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References
Ames, A.D., Sastry, S.: Hybrid Routhian reduction of Lagrangian hybrid systems. In: Proc. American Control Conference, pp. 2640–2645 (2006)
Ames, A.D., Zheng, H., Gregg, R.D., Sastry, S.: Is there life after Zeno? Taking executions past the breaking (Zeno) point. In: Proc. American Control Conference, pp. 2652–2657 (2006)
Di Bernardo, M., Garofalo, F., Iannelli, L., Vasca, F.: Bifurcations in piecewise-smooth feedback systems. International Journal of Control 75, 1243–1259 (2002)
Bourgeot, J.M., Brogliato, B.: Tracking control of complementarity Lagrangian systems. International Journal of Bifurcation and Chaos 15(6), 1839–1866 (2005)
Brogliato, B.: Nonsmooth Mechanics. Springer, Heidelberg (1999)
Camlibel, M.K., Schumacher, J.M.: On the Zeno behavior of linear complementarity systems. In: Proc. IEEE Conf. on Decision and Control, pp. 346–351 (2001)
Chevallereau, C., Westervelt, E.R., Grizzle, J.W.: Asymptotically stable running for a five-link, four-actuator, planar bipedal robot. International Journal of Robotics Research 24(6), 431–464 (2005)
Collins, S.H., Wisse, M., Ruina, A.: A 3-D passive dynamic walking robot with two legs and knees. International Journal of Robotics Research 20, 607–615 (2001)
Grizzle, J.W., Abba, G., Plestan, F.: Asymptotically stable walking for biped robots: Analysis via systems with impulse effects. IEEE Trans. on Automatic Control 46(1), 51–64 (2001)
Heymann, M., Lin, F., Meyer, G., Resmerita, S.: Analysis of Zeno behaviors in a class of hybrid systems. IEEE Trans. on Automatic Control 50(3), 376–384 (2005)
Hirsch, M.W.: Differential Topology. Springer, Heidelberg (1980)
Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations of Non-smooth Mechanical Systems. Springer, Heidelberg (2004)
McGeer, T.: Passive walking with knees. In: Proc. IEEE Int. Conf. on Robotics and Automation, vol. 3, pp. 1640–1645 (1990)
Miller, B.M., Bentsman, J.: Generalized solutions in systems with active unilateral constraints. Nonlinear Analysis: Hybrid Systems 1, 510–526 (2007)
Morris, B., Grizzle, J.W.: A restricted Poincaré map for determining exponentially stable periodic orbits in systems with impulse effects: Application to bipedal robots. In: Proc. IEEE Conf. on Decision and Control and European Control Conf., pp. 4199–4206 (2005)
Murray, R.M., Li, Z., Sastry, S.: A Mathematical Introduction to Robotic Manipulation. CRC Press, Boca Raton (1993)
Or, Y., Ames, A.D.: Stability of Zeno equlibria in Lagrangian hybrid systems. In: Proc. IEEE Conf. on Decision and Control, pp. 2770–2775 (2008)
Or, Y., Ames, A.D.: A formal approach to completing Lagrangian hybrid system models. Submitted to ACC 2009 (2009), www.cds.caltech.edu/~izi/publications.htm
Or, Y., Ames, A.D.: Existence of periodic orbits with Zeno behavior in completed Lagrangian hybrid systems. Technical Report (2009), www.cds.caltech.edu/~izi/publications.htm
Pfeiffer, F., Glocker, C.: Multibody Dynamics with Unilateral Contacts. John Wiley and Sons, New York (1996)
Pogromsky, A.Y., Heemels, W.P.M.H., Nijmeijer, H.: On solution concepts and well-posedness of linear relay systems. Automatica 39(12), 2139–2147 (2003)
Pratt, J., Pratt, G.A.: Exploiting natural dynamics in the control of a planar bipedal walking robot. In: Proc. 36th Annual Allerton Conf. on Communications, Control and Computing, pp. 739–748 (1998)
Shen, J., Pang, J.-S.: Linear complementarity systems: Zeno states. SIAM Journal on Control and Optimization 44(3), 1040–1066 (2005)
Stronge, W.J.: Impact Mechanics. Cambridge University Press, Cambridge (2004)
van der Schaft, A., Schumacher, H.: An Introduction to Hybrid Dynamical Systems. Lecture Notes in Control and Information Sci. Springer, Heidelberg (2000)
van der Schaft, A.J., Schumacher, J.M.: The complementary-slackness class of hybrid systems. Math. of Control, Signals, and Systems 9(3), 266–301 (1996)
Zhang, J., Johansson, K.H., Lygeros, J., Sastry, S.: Zeno hybrid systems. Int. J. Robust and Nonlinear Control 11(2), 435–451 (2001)
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Or, Y., Ames, A.D. (2009). Existence of Periodic Orbits with Zeno Behavior in Completed Lagrangian Hybrid Systems. In: Majumdar, R., Tabuada, P. (eds) Hybrid Systems: Computation and Control. HSCC 2009. Lecture Notes in Computer Science, vol 5469. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00602-9_21
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DOI: https://doi.org/10.1007/978-3-642-00602-9_21
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