Abstract
This paper study the stabilization of mechanical system with impulse effects around a hybrid limit cycle, the proposed control approach is based on LaSalle’s invariance principle for hybrid systems and Layounov constraint based method. Theorem 2 shows necessary and sufficient condition of the existence and the uniqueness of the developed controller which leads to a system of partial differential equations (PDE) whose solutions are the kinetic and potential energy of smooth Lyapunov function, furthermore Theorem 3 gave an alternative existence condition which states that the largest positively invariant set should be nowhere dense and closed and it is none other than the hybrid limit cycle itself.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J. W. Grizzle, G. Abba, and F. Plestan, “Asymptotically stable walking for biped robots: analysis via systems with impulse effects,” IEEE Trans. on Automatic Control, vol. 46, no. 1, pp. 51–64, Jan. 2001. [click]
T. McGeer, “Passive dynamic walking,” Int. J. Robotics Research, vol. 9, no. 2, pp. 62–82, 1990.
R. W. Brockett, “Asymptotic stability and feedback stabilization,” R. W. Brockett, R. S. Millman, H. J. Sussmann (Eds), Differential Geometric Control Theory, vol. 27, no 1, pp. 181–191, Birkhaüser, Boston, 1983.
Z. P. Jiang, “Controlling underactuated mechanical systems: a review and open problems,” J. Levine, P. Mullhaupt, (Eds), Advances in the Theory of Control, Signals and Systems with Physical Modeling,pp. 77–88, Springer Berlin Heidelberg, 2011.
E. R. Westervelt, J. W. Grizzle, and D. E. Koditschek, “Hybrid zero dynamics of planar biped walkers,” IEEE Trans. on Automatic Control, vol. 48, no 1, pp. 42–56, 2003. [click]
K. A. Hamed, B. G. Buss, and J. W. Grizzle, “Continuous-time controllers for stabilizing periodic orbits of hybrid systems: Application to an underactuated 3D bipedal robot,” Proc. of 53rd IEEE Conference on Decision and Control, pp. 1507–1513, December 2014.
A. Shiriaev, L. Freidovich, and I. Manchester, “Can we make a robot ballerina perform a pirouette? orbital stabilization of periodic motions of underactuated mechanical systems,” Annual Reviews in Control, vol. 32, no 2, pp. 200–211, 2008.
A. Shiriaev, J. W. Perram, and C. Canudas-de-Wit, “Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach,” IEEE Trans. on Automatic Control, vol. 50, no 8, pp. 1164–1176, 2005. [click]
J. K. Holm and M. W. Spong, “Kinetic energy shaping for gait regulation of underactuated bipeds,” Proc. of IEEE Multi-conference on Systems and Control, pp. 1232–1238, 2008.
R. W. Sinnet and A. D. Ames, “Energy shaping of hybrid systems via control Lyapunov functions,” Proc. of 2015 American Control Conference (ACC), pp. 5992–5997, 2015.
M. W. Spong and F. Bullo, “Controlled symmetries and passive walking,” IEEE Trans. on Automatic Control, vol. 50, no. 7, pp. 1025–1031, July, 2005. [click]
R. Naldi and R. G. Sanfelice, “Passivity-based control for hybrid systems with applications to mechanical systems exhibiting impacts,” Automatica, vol. 49, no 5, pp.1104–1116, 2013. [click]
M. W. Spong, “Passivity based control of the compass gait biped,” Proc. of IFAC TriennialWorld Congress, pp. 19–24, 1999.
A. Bloch, N. Leonard, and J. Marsden, “Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem,” IEEE Trans. on Automatic Control, vol. 45, no 12, pp. 2253–2270, 2000. [click]
A. Bloch, N. Leonard, and J. Marsden, “Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping,” IEEE Trans. on Automatic Control, vol. 46, no 10, pp. 1556–1571, 2001. [click]
D. E. Chang, “The Method of Controlled Lagrangians: Energy Plus Force Shaping,” SIAM J. Control and Optimization, vol. 48, no 8, pp. 4821–4845, 2010. [click]
R. Ortega, A. J. Van der Sahft, B. Maschke, and G. Escobar, “Interconnection and damping assignment passivity based control of port-controlled Hamiltonian systems,” Automatica, vol. 38, no 4, pp. 585–596, 2002. [click]
D. E. Chang, A. Bloch, N. E. Leonard, J. Marsden, and C. A. Woolsy,“The equivalence of controlled Lagrangians and IDA-passivity based control,” ESAIM Controle Optim. Calc. Var, vol. 8, pp. 393–422, 2002. [click]
D. Auckly, L. Kapitansky, and W. White, “Control of nonlinear underactuated systems,” Comm. Pure and Applied Mathematics, vol 53, no 3, pp.354–369, 2002.
B. Gharsifard, A. D. Lewis, and A. R. Mansouri, “A geometric framework for stabilization by energy shaping: sufficient conditions for existence of solutions,” Communications in Information and Systems, vol. 8, no 4, pp. 353–398, 2008.
S. Grillo, J. Marsden, S. Nair, “Lyapunov constraints and global asymptotic stabilization,” Journal of geometric mechanics AIMS, vol 3, no 2, pp.145–196, June 2011. [click]
C. M. Marle, “Kinematic and geometric constraints, servomechanism and control of mechanical systems,” Rend. Sem. Mat. Univ. Pol. Torino, vol.54, no 4, pp. 353–364, 1996.
S. Grillo, L. Salomone, and M. Zuccalli, “On the relationship between the energy shaping and the Lyapunov constraint based methods.,” arXiv preprint arXiv:, 1601.03975, 2016.
M. Chaalal and N. Achour, “Stabilization of a Class of Mechanical Systems with Impulse Effects by Lyapunov Constraints,” Proc. of IEEE 20th International Conference on Methods and Models in Automation and Robotics, pp. 335–340, 2015.
J. K. Hale, Ordinary Differential Equations, Robert E. Krieger Publishing Company, New York, 1980.
F. Bullo and A. Lewis, Geometric Control of Mechanical Systems, Springer-Verlag, New York, 2005.
P. de Leon and R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland 1989.
B. Brogliato, Nonsmooth Mechanics. Models, Dynamics and Control, 3rd edition, Springer International Publishing, 2016.
F. Bullo and M. Zefran, “Modeling and Controllability for a Class of Hybrid Mechanical systems,” IEEE Trans. on Robotics and Automation, vol, 18, no. 4, pp. 563–573, 2002. [click]
R. I. Leine and N. van de Wouw, Stability and Convergence of Mechanical Systems with Unilateral Constraints, vol. 36, Springer-Verlag, Berlin Heidelberg New-York, 2008.
J. Hauser and C. C. Chung, “Lyapunov functions for exponentially stable periodic orbits,” Systems and Control Letters, vol. 23, no 1, pp. 27–34, 1994.
H. K Khalil, Nonlinear Systems, 3rd edition, Prentice hall, New Jersey, 2002.
V. Chellaboina, S. P. Bhat, and W. M. Haddad, “An invariance principle for nonlinear hybrid and impulsive dynamical systems. nonlinear analysis,” Nonlinear Analysis: Theory, Methods and Applications, vol. 53, no 3, pp. 527–550, 2003. [click]
W. M. Haddad, V. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity and Control, Princeton Univ. Press, Princeton, NJ. 2006.
S. N. Simic, S. Sastry, K. H. Johansson, and J. Lygeros, “Hybrid limit cycles and hybrid Poincaré-Bendixson,” Proc. of IFAC 15th Triennial World Congress, vol. 35, no1, pp. 197–202, 2002.
M. Posa, M. Tobenkin, and R. Tedrake, “Stability analysis and control of rigid-body systems with impacts and friction,” IEEE Trans. on Automatic Control, vol. 61, no 6, p. 1423–1437, 2016. [click]
L. A. Steen and J. A. Seebach, Counterexamples in Topology, vol. 18, Springer-Verlag, New York, 1978.
S. Grillo, L. Salomone, and M. Zuccalli, “On the asymptotic stabilizability of underactuated systems with two degrees of freedom and the Lyapunov constraint based method,” arXiv preprint arXiv:, 1604.08475, 2016.
A, Goswami, B. Espiau, and A. Keramane, “Limit cycles in a passive compass gait biped and passivity-mimicking control laws,” Autonomous Robots, vol. 4, no 3, pp. 273–286, 1997. [click]
A. Goswami, B. Thuilot, and B. Espiau, “A study of the passive gait of a compass-like biped robot symmetry and chaos,” The International Journal of Robotics Research, vol. 17, no 12, pp. 1282–1301, 1998.
Author information
Authors and Affiliations
Corresponding author
Additional information
Recommended by Associate Editor Shihua Li under the direction of Editor Hyun-Seok Yang.
Mohammed Chaalal received his B.S. degree in Electrical Engineering from Guelma University, Algeria in 2008. The Msc degree in Automatic Control and Robotics from Houari Bumediene University of Sciences and Technology (USTHB), Algiers, Algeria in 2010. He is currently pursuing, Ph.D. studies at the faculty of Electronics and Computer Science, at the USTHB. His research interests include control theory, geometric control, geometric mechanics and its application to robotic systems.
Noura Achour is a full Professor at University of USTHB at Algiers. She received a degree in electronics engineering from the ENP (Ecole Nationale Polytechnique) at Algiers in 1983, followed by a Magister degree in process control in 1992 and a Ph.D. degree in motion planning in 2004 from the USTHB University. She is Director of Research at LRPE (Laboratoire de Robotique, Parallélisme et Electroénergétique) at USTHB. Her main research activities are in the areas of control systems, robotics, autonomous robots, artificial intelligence and motion planning.
Rights and permissions
About this article
Cite this article
Chaalal, M., Achour, N. Stabilizing periodic orbits of a class of mechanical systems with impulse effects: A Lyapunov constraint approach. Int. J. Control Autom. Syst. 15, 2213–2221 (2017). https://doi.org/10.1007/s12555-016-0387-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12555-016-0387-x