Abstract
We consider a class of mechanical systems with an arbitrary number of passive (nonactuated) degrees of freedom, which are subject to a set of nonholonomic constraints. We assume that the challenging problem of motion planning is solved giving rise to a feasible desired periodic trajectory. Our goal is either to analyze orbital stability of this trajectory with a given time-independent feedback control law or to design a stabilizing controller. We extend our previous work done for mechanical systems without nonholonomic constraints. The main contribution is an analytical method for computing coefficients of a linear reduced-order control system, solutions of which approximate dynamics that is transversal to the preplanned trajectory. This linear system is shown to be useful for stability analysis and for design of feedback controllers orbitally, exponentially stabilizing forced periodic motions in nonholonomic mechanical systems.We illustrate our approach on a standard benchmark example.
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References
Arnold, V.I., Kozlov, V.V., Neishtadt, A.I.: Mathematical Aspects of Classical and Celestial Mechanics, vol. 3. Springer, Berlin (1988)
Banaszuk, A., Hauser, J.: Feedback linearization of transverse dynamics for periodic orbits. Systems and Control Letters 26, 95–105 (1995)
Bloch, A., Baillieul, J., Crouch, P., Marsden, J.: Nonholonomic Mechanics and Control. Springer, New York (2003)
Bloch, A.M., Reyhanoglu, M., McClamroch, N.H.: Control and stabilization of nonholonomic dynamic systems. IEEE Trans. Automatic Control 37(11), 1746–1757 (1992)
Freidovich, L., Shiriaev, A., Manchester, I.: Stability analysis and control design for an un-deractuated walking robot via computation of a transverse linearization. In: Proc. 17th IFAC World Congress, Seoul, Korea, pp. 10,166–10,171 (2008)
Freidovich, L., Shiriaev, A.: Transverse linearization for mechanical systems with passive links, impulse effects, and friction forces. In: Proc. 48th IEEE Conf. Decision and Control (CDC 2009) / the 28th Chinese Control Conference, Shanghai, China, pp. 6490–6495 (2009)
Freidovich, L.B., Robertsson, A., Shiriaev, A.S., Johansson, R.: Periodic motions of the Pen-dubot via virtual holonomic constraints: Theory and experiments. Automatica 44(3), 785–791 (2008)
Hale, J.: Ordinary Differential Equations, Krieger, Malabar (1980)
Hussein, I.I., Bloch, A.M.: Optimal control of underactuated nonholonomic mechanical systems. IEEE Trans. Automatic Control 53(3), 668–682 (2008)
Leonov, G.: Generalization of the Andronov-Vitt theorem. Regular and Chaotic Dynamics 11(2), 281–289 (2006)
Neimark, J.I., Fufaev, F.A.: Dynamics of Nonholonomic Systems. A.M.S. Translations of Mathematical Monographs, vol. 33, Providence, RI (1972)
Poincaré, H.: Oeuvres completes, vol. 11. Gauthier-Villars, Paris (1916-1954)
Shiriaev, A., Freidovich, L.: Computing a transverse linearization for mechanical systems with two and more passive degrees of freedom. IEEE Trans. Automatic Control 54(12), 2882–2888 (2009)
Shiriaev, A., Freidovich, L., Gusev, S.: Transverse linearization for controlled mechanical systems with several passive degrees of freedom. IEEE Trans. Automatic Control 55(4), 893–906 (2010)
Shiriaev, A., Freidovich, L., Manchester, I.: Can we make a robot ballerina perform a pirou-ette? Orbital stabilization of periodic motions of underactuated mechanical systems. Annual Reviews in Control 32(2), 200–211 (2008)
Shiriaev, A., Perram, J., Canudas-de-Wit, C.: Constructive tool for orbital stabilization of un-deractuated nonlinear systems: Virtual constraints approach. IEEE Trans. Automatic Control 50(8), 1164–1176 (2005)
Shiriaev, A., Perram, J., Robertsson, A., Sandberg, A.: Periodic motion planning for virtually constrained Euler-Lagrange systems. Systems and Control Letters 55, 900–907 (2006)
Urabe, M.: Nonlinear Autonomous Oscillations. Academic Press, New York (1967)
Westervelt, E.R., Grizzle, J.W., Chevallereau, C., Choi, J.H., Morris, B.: Feedback Control of Dynamic Bipedal Robot Locomotion. CRC Press, Taylor and Francis Group, Boca Raton, FL (2007)
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Freidovich, L.B., Shiriaev, A.S. (2012). Transverse Linearization for Underactuated Nonholonomic Mechanical Systems with Application to Orbital Stabilization. In: Johansson, R., Rantzer, A. (eds) Distributed Decision Making and Control. Lecture Notes in Control and Information Sciences, vol 417. Springer, London. https://doi.org/10.1007/978-1-4471-2265-4_11
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DOI: https://doi.org/10.1007/978-1-4471-2265-4_11
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