Transverse linearization for controlled mechanical systems with several passive degrees of freedom

IEEE Transactions on Automatic Control

Volumn 55, Issue 4, 2010, Pages 893-906

  Shiriaev, Anton S ^a,b   Freidovich, Leonid B ^a   Gusev, Sergei V ^c  

^a UMEÅ UNIVERSITY   (Sweden)

^b NORWEGIAN UNIVERSITY OF SCIENCE AND TECHNOLOGY   (Norway)

^c ST PETERSBURG STATE UNIVERSITY   (Russian Federation)

Author keywords

Moving Poincar section; Orbital stability; Synchronisation of mechanical system; Transverse linearization; Underactuated mechanical systems; Virtual holonomic constraints

Indexed keywords

MECHANICAL SYSTEMS; ORBITAL STABILITY; POINCARE; SYNCHRONISATION; TRANSVERSE LINEARIZATION; UNDERACTUATED MECHANICAL SYSTEMS; VIRTUAL HOLONOMIC CONSTRAINTS;

ELECTRIC DRIVES; LINEARIZATION; MECHANICAL ENGINEERING; PENDULUMS; SYSTEM STABILITY; VIBRATIONS (MECHANICAL);

MECHANICS;

EID: 77950823133     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2010.2042000     Document Type: Article

References (36)

1. J. Hauser and A. Banaszuk, "Feedback linearization of transverse dynamics for periodic orbits," Syst. Control Lett., vol.26, pp. 95-105, 1995.
2. L. T. Aguilar, I. Boiko, L. Fridman, and R. Iriarte, "Generating self-excited oscillations via two-relay controller," IEEE Trans. Autom. Control, vol.54, no.2, pp. 416-420, Feb. 2009.
3. A. A. Andronov and A. Vitt, "On Lyapunov stability," Exper. Theor. Phys., vol.3, pp. 373-374, 1933.
4. B. Brogliato, R. Ortega, and R. Lozano, "Global tracking controllers for flexible-joint manipulators-A comparative study," Automatica, vol.31, no.7, pp. 941-956, 1995.
5. B. Brogliato and A. Rio, "On the control of complementary-slackness juggling mechanical systems," IEEE Trans. Autom. Control, vol.45, no.2, pp. 235-246, Feb. 2000.
6. L. B. Freidovich, A. S. Shiriaev, F. Gordillo, F. Gómez-Estern, and J. Aracil, "Partial-energy-shaping control for orbital stabilization of high frequency oscillations of the furuta pendulum," IEEE Trans. Control Syst. Technol., vol.17, no.4, pp. 853-858, Jul. 2009.
7. L. B. Freidovich, A. S. Shiriaev, and I. R. Manchester, "Stability analysis and control design for an underactuated walking robot via computation of a transverse linearization," in Proc. 17th IFAC World Congress, Seoul, Korea, 2008, pp. 10166-10171.
8. J. W. Grizzle, G. Abba, and F. Plestan, "Asymptotically stable walking for biped robots: Analysis via systems with impulse effects," IEEE Trans. Autom. Control, vol.46, no.1, pp. 51-64, Jan. 2001.
9. J. W. Grizzle, C. H. Moog, and C. Chevallereau, "Nonlinear control of mechanical systems with an unactuated cyclic variable," IEEE Trans. Autom. Control, vol.50, no.5, pp. 559-576, May 2005.
10. S. V. Gusev, A. S. Shiriaev, and L. B. Freidovich, "LMI approach for solving periodic matrix riccati equation," in Proc. 3rd IFAC Workshop Periodic Control Syst., St. Petersburg, Russia, 2007 [Online]. Available: http://www.lib.physcon.ru/download/p1428.pdf
11. J. K. Hale, Ordinary Differential Equations. Melbourne, FL: Krieger, 1980.
12. P. Hartman, Ordinary Differential Equations. New York: Wiley, 1964.
13. J. Hauser and C. C. Chung, "Converse Lyapunov functions for exponentially stable periodic orbits," Syst. Control Lett., vol.23, no.1, pp. 27-34, 1994.
14. P. X. La Hera, L. B. Freidovich, U. Mettin, and A. S. Shiriaev, "Newapproach for swinging up the furuta pendulum: Theory and experiments," Mechatronics, vol.19, no.8, pp. 1240-1250, 2009.
15. I. A. Hiskens, "Stability of hybrid system limit cycles: Application to the compass gait biped robot," in Proc. 40th IEEE Conf. Decision Control, Orlando, FL, 2001, pp. 774-779.
16. H. K. Khalil, Nonlinear systems, 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 2002.
17. G. A. Leonov, "Generalization of the andronov-vitt theorem," Regular Chaotic Dyn., vol.11, no.2, pp. 281-289, 2006.
18. G. Liu, I. Mareels, and D. Nešić, "Way point tracking of a spherical pendulum inverted," in Proc. IEEE Int. Conf. Control Autom., Guangzhou, China, 2007, pp. 3145-3150.
19. K. Lynch and C. Black, "Control of underactuated manipulation by real-time nonlinear optimization," in Robotics Research, J. Hollerbach and D. Koditschek, Eds. London, U.K.: Springer-Verlag, 2000, pp. 51-58.
20. S. Nair and N. E. Leonard, "Stable synchronization of mechanical system networks," SIAM J. Control Optim., vol.47, no.2, pp. 661-683, 2008.
21. S. Nakaura, Y. Kawaida, T. Matsumoto, and M. Sampei, "Enduring rotatory motion control of devil stick," in Proc. 6th IFAC Nonlin. Control Syst. Symp., Stuttgart, Germany, 2004, pp. 1073-1078.
22. C. Nielsen and M. Maggiore, "On local transverse feedback linearization," SIAM J. Control Optim., vol.7, no.5, pp. 2227-2250, 2008.
23. R. Ortega, A. Loria, P. J. Nicklasson, and H. Sira-Ramirez, Passivity- Based Control of Euler-Lagrange Systems. Berlin, Germany: Springer-Verlag, 1998.
24. H. Poincaré, Oeuvres Comleés. Paris, France: Gauthier-Villars, 1916-1954, vol.11.
25. R. Ronsse, P. Lefevre, and R. Sepulchre, "Sensorless stabilization of bounce juggling," IEEE Trans. Robotics, vol.22, no.1, pp. 147-159, Jan. 2006.
26. N. Rouche and J. Mawhin, Ordinary Differential Equations. Stability and Periodic Solutions. London, U.K.: Pitman Publishing, Ltd., 1980.
27. W. Rugh, Linear System Theory, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 1996.
28. A. S. Shiriaev, L. B. Freidovich, and I. R. Manchester, "Can we make a robot ballerina perform a pirouette? orbital stabilization of periodic motions of underactuated mechanical systems," Annu. Rev. Control, vol.32, no.2, pp. 200-211, 2008.
29. A. S. Shiriaev, J. W. Perram, and C. Canudas-de-Wit, "Constructive tool for orbital stabilization of underactuated nonlinear systems:Virtual constraints approach," IEEE Trans. Autom. Control, vol.50, no.8, pp. 1164-1176, Aug. 2005.
30. A. S. Shiriaev, J. W. Perram, A. Robertsson, and A. Sandberg, "Periodic motion planning for virtually constrained Euler-Lagrange systems," Syst. Control Lett., vol.55, pp. 900-907, 2006.
31. G. Song and M. Zefran, "Stabilization of hybrid periodic orbits with application to bipedal walking," in Proc. Amer. Control Conf., Minneapolis, MN, 2006, pp. 2504-2509.
32. M. W. Spong, "Partial feedback linearization of underactuated mechanical systems," in Proc. Int. Conf. Intell. Robots Syst.,Munich, Germany, 1994, pp. 314-321.
33. M. W. Spong, K. Khorasani, and P. Kokotovic, "An integral manifold approach to the feedback control of flexible joint robots," IEEE J. Robot. Autom., vol.RA-3, no.4, pp. 291-300, Apr. 1987.
34. M. Urabe, Nonlinear Autonomous Oscillations. New York: Academic Press, 1967.
35. E. R. Westervelt, J. W. Grizzle, C. Chevallereau, J. H. Choi, and B. Morris, Feedback Control of Dynamic Bipedal Robot Locomotion. Orlando, FL: CRC Press, 2007.
36. T. Yoshizawa, Stability theory by Liapunov's Second Method. Tokyo, Japan: The Mathematical Society of Japan, 1966.