Rapidly exponentially stabilizing control lyapunov functions and hybrid zero dynamics

IEEE Transactions on Automatic Control

Volumn 59, Issue 4, 2014, Pages 876-891

  Ames, Aaron D ^a   Galloway, Kevin ^b   Sreenath, Koushil ^c   Grizzle, Jessy W ^d  

^a Texas AandM University   (United States)

^b UNITED STATES NAVAL ACADEMY   (United States)

^c CARNEGIE MELLON UNIVERSITY   (United States)

^d UNIVERSITY OF MICHIGAN   (United States)

Author keywords

Bipeds; Lyapunov functions; nonlinear control; robotics

Indexed keywords

BIPED LOCOMOTION; CHAOS THEORY; COMPUTER SIMULATION; LYAPUNOV FUNCTIONS; LYAPUNOV METHODS; ROBOTICS;

BIPEDAL WALKING ROBOT; BIPEDS; CONTROL LYAPUNOV FUNCTION; EXPONENTIAL CONVERGENCE; INPUT OUTPUT LINEARIZATION; NON LINEAR CONTROL; STABILIZING CONTROLLERS; SYSTEM WITH IMPULSE EFFECTS;

DYNAMICS;

EID: 84897417600     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2014.2299335     Document Type: Article

References (56)

1. A. D. Ames, "First steps toward automatically generating bipedal robotic walking from human data," in Robotic Motion and Control, ser. LNICS. New York: Springer, 2012, vol. 422, pp. 89-116.
2. A. D. Ames, "First steps toward underactuated human-inspired bipedal robotic walking," in Proc. IEEE Int. Conf. Robot. Autom., 2012, pp. 1011-1017.
3. A. D. Ames, E. A. Cousineau, and M. J. Powell, "Dynamically stable robotic walking with NAO via human-inspired hybrid zero dynamics," in Proc. 15th ACM Conf. Hybrid Syst.: Computat. Control, 2012, pp. 135-144.
4. A. D. Ames,K. Galloway, and J. W. Grizzle, "Control Lyapunov functions and hybrid zero dynamics," in Proc. 51st IEEE Conf. Decision Control, 2012, pp. 6837-6842.
5. Z. Artstein, "Stabilization with relaxed controls," Nonlinear Anal. Theory Methods Appl., vol. 7, no. 11, pp. 1163-1173, 1983.
6. D. D. Bainov and P. S. Simeonov, Systems with Impulse Effects: Stability, Theory and Applications. Chichester, U. K.: Ellis Horwood Limited, 1989.
7. A. Banaszuk and J. Hauser, "Feedback linearization of transverse dynamics for periodic orbits," in Proc. 33rd IEEE Conf. Decision Control, Dec. 1994, vol. 2, pp. 1639-1644.
8. S. P. Bhat and D. S. Bernstein, "Continuous finite-time stabilization of the translational and rotational double integrators," IEEE Trans. Autom. Control, vol. 43, no. 5, pp. 678-682, 1998. (Pubitemid 128736491)
9. F. Blanchini and S. Miani, Set-TheoreticMethods in Control. Boston, MA: Birkhäuser, 2008.
10. C. Chevallereau, G. Abba, Y. Aoustin, F. Plestan, E. R. Westervelt, C. Canudas-de-Wit, and J. W. Grizzle, "RABBIT: A testbed for advanced control theory," IEEE Control Syst. Mag., vol. 23, no. 5, pp. 57-79, Oct. 2003.
11. S. Diop, J. W. Grizzle, P. E. Moraal, and A. Stefanopoulou, "Interpolation and numerical differentiation for observer design," in Proc. Amer. Control Conf., 1994, pp. 1329-1333.
12. R. A. Freeman and P. V. Kokotovic, Robust Nonlinear Control Design. Boston, MA: Birkhäuser, 1996.
13. K. Galloway, K. Sreenath, A. Ames, and J. Grizzle,Walking with Control Lyapunov Functions (2012). Youtube Video [Online]. Available: http://youtu. be/onOd7xWbGAk.
14. K. Galloway, K. Sreenath, A. D. Ames, and J. W. Grizzle, Torque saturation in bipedal robotic walking through control Lyapunov function based quadratic programs 2013, arXiv:1302. 7314 (cs. SY).
15. R. Goebel, R. Sanfelice, and A. Teel, "Hybrid dynamical systems," IEEE Control Syst. Mag., vol. 29, no. 2, pp. 28-93, Apr. 2009.
16. 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. (Pubitemid 32191020)
17. J. W. Grizzle, C. Chevallereau, A. D. Ames, and R. W. Sinnet, "3D bipedal robotic walking: models, feedback control, and open problems," in Proc. 8th IFAC Symp. Nonlin. Control Syst., Bologna, Italy, 2010, pp. 505-532.
18. J. W. Grizzle, C. H. Moog, and C. Chevallereau, "Nonlinear control of mechanical systems with an unactuated cyclic variable," IEEE Trans. Autom. Control, vol. 30, no. 5, pp. 559-576, May 2005. (Pubitemid 40784429)
19. F. Grognard and C. C. d. Wit, "Design of orbitally stable zero dynamics for a class of nonlinear systems," Syst. Control Lett., vol. 51, pp. 89-103, 2004.
20. W. M. Haddad, V. S. Chellaboina, and S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control. Princeton, NJ: Princeton Univ. Press, 2006.
21. J. Hauser and C. C. Chung, "Converse Lyapunov functions for exponentially stable periodic orbits," Syst. Control Lett., vol. 23, pp. 27-34, 1994.
22. A. Isidori, Nonlinear Control Systems: An Introduction, 3 ed. Berlin, Germany: Springer-Verlag, 1995.
23. A. Isidori and C. Moog, C. Byrnes and A. Kurzhanski, Eds., "On the nonlinear equivalent of the notion of transmission zeros," in Proc. IIASA Conf.: Modeling Adaptive Control, Berlin, Germany, 1988, pp. 146-157.
24. A. Jadbabaie, J. Yu, and J. Hauser, "Stabilizing receding horizon control of nonlinear systems: a control Lyapunov function approach," in Proc. IEEE Amer. Control Conf., 1999, vol. 3, pp. 1535-1539. (Pubitemid 32078684)
25. Z. Jiang and Y. Wang, "A converse Lyapunov theorem for discretetime systems with disturbances," Syst. Control Lett., vol. 45, no. 1, pp. 49-58, 2002. (Pubitemid 33107170)
26. H. K. Khalil, Nonlinear Systems, 3 ed. Upper Saddle River, NJ: Prentice-Hall, 2002.
27. M. Kristic, I. Kanellakopoulos, and P. Kokotovic, "Nonlinear and adaptive control design," in Adaptive and Learning Syst. for Signal Processing, Commun. and Control. New York: Wiley, 1995.
28. B. Morris and J. W. Grizzle, "Hybrid invariant manifolds in systems with impulse effects with application to periodic locomotion in bipedal robots," IEEE Trans. Autom. Control, vol. 54, no. 8, pp. 1751-1764, Aug. 2009.
29. B. Morris and J. W. Grizzle, "A restricted Poincaré map for determining exponentially stable periodic orbits in systems with impulse effects: Application to bipedal robots," in Proc. 44th IEEE Conf. Decision Control, Seville, Spain, 2005, pp. 4199-4206. (Pubitemid 46297629)
30. B. Morris, E. R. Westervelt, C. Chevallereau, G. Buche, and J. W. Grizzle, "Achieving bipedal running with RABBIT: Six steps toward infinity," in Fast Motions Symposium on Biomechanics and Robotics, ser. Lecture Notes in Control and Inform. Sciences. Heidelberg, Germany: Springer-Verlag, 2006, pp. 277-297.
31. A. Papachristodoulou and S. Prajna, "On the construction of Lyapunov functions using the sum of squares decomposition," in Proc. 41st IEEE Conf. Decision Control, 2002, vol. 3, pp. 3482-3487.
32. H.-W. Park, K. Sreenath, J. W. Hurst, and J. W. Grizzle, "Identification of a bipedal robot with a compliant drivetrain: Parameter estimation for control design," Control Syst. Mag., vol. 31, no. 2, pp. 63-88, Apr. 2011.
33. I. R. Petersen and B. R. Barmish, "Control effort considerations in the stabilization of uncertain dynamical systems," Syst. Control Lett., vol. 9, pp. 417-422, 1987. (Pubitemid 18531449)
34. M. Reyhanoglu, A. v. d. Schaft, N. H. McClamroch, and I. Kolmanovsky, "Dynamics and control of a class of underactuated mechanical systems," IEEE Trans. Autom. Control, vol. 44, no. 9, pp. 1663-1671, 1999.
35. W. Rugh, Linear System Theory. Englewood Cliffs, NJ: Prentice Hall, 1996.
36. R. G. Sanfelice, "Control Lyapunov functions and stabilizability of compact sets for hybrid systems," in Proc. 50th IEEE Conf. Decision Control, 2011, pp. 7404-7409.
37. S. Sastry, Nonlinear Systems: Analysis, Stability and Control. New York: Springer-Verlag, 1999.
38. R. Sepulchre, M. Jankovic, and P. Kokotovic, Constructive Nonlinear Control Communications and Control Engineering. London, U. K.: Springer Verlag, 1997.
39. A. Shiriaev, J. W. Perram, and C. C.-D. 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. (Pubitemid 41410114)
40. A. S. Shiriaev, L. B. Freidovich, and S. V. Gusev, "Transverse linearization for controlled mechanical systems with several passive degrees of freedom," IEEE Trans. Autom. Control, vol. 55, no. 4, pp. 893-906, Apr. 2010.
41. G. Song and M. Zefran, "Underactuated dynamic three-dimensional bipedal walking," in Proc. IEEE Int. Conf. Robot. Autom., Orlando, FL, May 2006, pp. 854-859. (Pubitemid 44940179)
42. E. Sontag, "A Lyapunov-like stabilization of asymptotic controllability," SIAM J. Control Optim., vol. 21, no. 3, pp. 462-471, 1983.
43. E. Sontag, "A 'universal' contruction of Artstein's theorem on nonlinear stabilization," Syst. Control Lett., vol. 13, pp. 117-123, 1989. (Pubitemid 20721269)
44. E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems. New York: Springer, 1998, vol. 6.
45. E. D. Sontag and Y. Wang, "On characterizations of the input-to-state stability property," Syst. Control Lett., vol. 24, no. 5, pp. 351-359, 1995.
46. E. D. Sontag and Y. Wang, "New characterizations of input-to-state stability," IEEE Trans. Autom. Control, vol. 41, no. 9, pp. 1283-1294, 1996. (Pubitemid 126768503)
47. M. W. Spong, "Energy based control of a class of underactuated mechanical systems," in Proc. IFAC World Congress, San Francisco, CA, 1996, pp. 431-435.
48. K. Sreenath, H. W. Park, I. Poulakakis, and J. W. Grizzle, "ACompliant Hybrid Zero Dynamics Controller for Stable, Efficient and Fast Bipedal Walking onMABEL," Int. J. Robot. Res. vol. 30, pp. 1-, 2011 [Online]. Available: http://www-personal. umich. edu/~parkhw/IJRR2010. pdf
49. D. Sun, "A further result on an implicit function theorem for locally Lipschitz functions," Oper. Res. Lett., vol. 28, no. 4, pp. 193-198, 2001. (Pubitemid 32610342)
50. W. Tan and A. Packard, "Stability region analysis using polynomial and composite polynomial Lyapunov functions and sum-of-squares programming," IEEE Trans. Autom. Control, vol. 53, no. 2, pp. 565-571, 2008. (Pubitemid 351493893)
51. E. Wendel and A. D. Ames, "Rank deficiency and superstability of hybrid systems," Nonlinear Anal.:Hybrid Syst., vol. 6, pp. 787-805, 2012.
52. E. Westervelt, J. W. Grizzle, and D. E. Koditschek, "Hybrid zero dynamics of planar biped walkers," IEEE Trans. Autom. Control, vol. 48, no. 1, pp. 42-56, Jan. 2003.
53. E. R. Westervelt, G. Buche, and J. W. Grizzle, "Experimental validation of a framework for the design of controllers that induce stable walking in planar bipeds," Int. J. Robot. Res., vol. 24, no. 6, pp. 559-582, Jun. 2004.
54. E. R. Westervelt, J. W. Grizzle, C. Chevallereau, J. H. Choi, and B. Morris, Feedback Control of Dynamic Bipedal Robot Locomotion. Boca Raton, FL: CRC Press, 2007.
55. H. Ye,A. N. Michel, and L. Hou, "Stability theory for hybrid dynamical systems," IEEE Trans. Autom. Control, vol. 43, no. 4, pp. 461-474, Apr. 1998. (Pubitemid 128737370)
56. W. Zhang, A. Abate, J. Hu, and M. Vitus, "Exponential stabilization of discrete-time switched linear systems," Automatica, vol. 45, no. 11, pp. 2526-2536, 2009.