Feedback Stabilization of a Class of Unstable Nonholonomic Systems

[+] Author and Article Information
Bill Goodwine

Aerospace and Mechanical engineering, University of Notre Dame, Notre Dame, IN 46556

Miloš Žefran

Electrical Engineering and Computer Science, The University of Illinois at Chicago, Chicago, IL 60607

J. Dyn. Sys., Meas., Control 124(1), 221-230 (Oct 11, 2000) (10 pages) doi:10.1115/1.1433479 History: Received October 11, 2000
Copyright © 2002 by ASME
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Stépán,  G., 1991, “Chaotic motion of wheels,” Veh. Syst. Dyn., 20, pp. 341–351.
Schlippe,  B., and Dietrich,  R., 1941, “Shimmying of a pneumatic wheel,” Lilienthal-Gesellschaft für Luftfahrtforschung, 140, pp. 125–160. Translated for the AAF in 1947 by Meyer & Company.
Kalker,  J. J., 1990, “Wheel-rail rolling contact,” Wear, 144, pp. 243–261.
F. Böhm and M. Kollatz, 1989, “Some Theoretical Models for Computation of Tire Nonuniformities,” Vol. 12.124 of Fortschrittberichte VDI, VDI Verlag.
Plaut,  R. H., 1996, “Rocking instability of a pulled suitcase with two wheels,” Acta Mech., 117, pp. 165–179.
Barta,  G., and Stépán,  G., 1995, “Untersuchung quasi-periodischer schwingungen von mit reifen versehenen radern,” Z. Angew. Math. Mech., 75, pp. S77–S78.
Goodwine,  Bill, and Stépán,  Gábor, 2000, “Controlling unstable rolling phenomena,” J. Vib. Control, 6, pp. 137–158.
Bill Goodwine and Gábor Stépán, 1996, “Stabilization of the classical shim-mying wheel,” Proceedings of the 2nd European Nonlinear Oscillations Conference.
H. B. Pacejka, 1988, “Modelling of the pneumatic tyre and its impact on vehicle dynamic behaviour,” Research report no. i72B, TU Delft.
M. Žefran and J. W. Burdick, 1998, “Stabilization of systems with changing dynamics,” Hybrid Systems: Computation and Control, LNCS 1386, Springer.
P. Peleties and R. A. DeCarlo 1991, “Asymptotic stability of m-switched systems using lyapunov like functions,” Proceedings of the American Control Conference, pp. 1679–1684.
Michel,  A. N., and Hu,  B., 1999, “Towards a stability theory of general hybrid dynamical systems,” Automatica, 35, No. 3, pp. 371–384.
Branicky,  Michael S., 1998, “Multipe lyapunov functions and other analysis tools for switched and hybrid systems,” IEEE Trans. Autom. Control, 43, No. 4, pp. 475–482.
C. A. Yfoulis, A. Muir, B. B. O. L. Pettit, and P. E. Wellstead, 1998, “Stabilization of orthogonal piecewise linear Lyapunov-like functions,” Proceedings of the IEEE Conference on Decision and Control, pp. 1476–1481.
Johansson,  M., and Rantzer,  A., 1998, “Computation of piecewise quadratic lyapunov functions for hybrid systems,” IEEE Trans. Autom. Control, 43, No. 4, pp. 555–559.
S. Petterson and B. Lennartson, 1996, “Stability and robustness of hybrid systems,” Proceedings of the 35th Conference on Decision and Control, pp. 1202–1207, Kobe, Japan.
H. Nijmeijer and J. M. Schumacher, 1999, “Four decades of mathematical system theory,” J. W. Polderman and H. L. Trentelman, eds., The Mathematics of Systems and Control: From Intelligent Control to Behavioral Systems, pp. 73–83. Univ. of Groningen.
Filippov,  A. F., 1964, “Differential equations with discontinuous right-hand sides,” AMS Translations Ser. 2, 42, pp. 199–231.
V. I. Utkin, 1978, Sliding modes and their application in variable structure systems. Mir Publishers, Moscow.
Eduardo D. Sontag, 1999, “Stability and stabilization: discontinuities and the effect of disturbances,” Nonlinear analysis, differential equations and control (Montreal, QC, 1998), pp. 551–598, Kluwer Acad. Publ., Dordrecht.
Brogliato,  B., Niculescu,  S., and Monteiro-Marques,  M., 2000, “On tracking control of a class of complementary-slackness hybrid mechanical systems,” Syst. Control Lett., 39, No. 4, pp. 255–166.
Mills,  J. K., and Lokhorst,  D. M., 1993, “Control of robotic manipulators during general task execution: A discontinuous control approach,” Int. J. Robot. Res., 2, pp. 146–163.
Pagilla,  P. R., and Tomizuka,  M., 1997, “Contact transition control of nonlinear mechanical systems subject to unilateral constraint,” ASME J. Dyn. Syst., Meas., Control, 119, pp. 749–759, Dec.
Tarn,  T. J., Wu,  Y., Xi,  N., and Isidori,  A., 1996, “Force regulation and contact transition control,” IEEE Control Syst. Mag., 16, pp. 32–40, Feb.
W. Kohn, A. Nerode, J. B. Remmel, and Xiolin Ge, 1994, “Multiple agent hybrid control: carrier manifolds and chattering approximations to optimal control,” Proceedings of the 33rd IEEE Conference on Decision and Control, pp. 4221–4227, Lake Buena Vista, FL.
A. Puri, 1995, “Theory of hybrid systems and discrete event systems,” PhD thesis, U. C. Berkeley.
A. Deshpande and P. Varaiya, 1995, “Viable control of hybrid systems,” LNCS 999, pp. 128–147. Springer-Verlag.
J. Lygeros, D. N. Godbole, and S. S. Sastry, 1996, “A game theoretic approach to hybrid system design,” LNCS 1066, pp. 1–12. Springer-Verlag.
J. O. Moody and P. J. Antsaklis, 1996, “Supervisory control of petri nets with uncontrollable/unobservable transitions,” Proceedings of the 35th IEEE Conference on Decision and Control, Kobe, Japan.
Sontag,  E. D., 1981, “Nonlinear regulation: The piecewise linear approach,” IEEE Trans. Autom. Control, AC-26, pp. 346–358.
A. Hassibi and S. P. Boyd, 1998, “Quadratic stabilization and control of piecewise-linear systems,” in Proceedings of American Control Conference, Vol. 6, pp. 3659–64, Philadelphia, PA, June.
K. X. He and M. D. Lemmon, 1998, “Lyapunov stability of continuous-valued systems under the supervision of discrete-event transition systems,” Hybrid Systems: Computation and control, LNCS 1386, pp. 175–189. Springer.
Kolmanovsky,  I., and McClamroch,  H. N., 1996, “Hybrid feedback laws for a class of cascade nonlinear control systems,” IEEE Trans. Autom. Control, 41, No. 9, pp. 1271–1282.
Burridge,  R. R., Rizzi,  A. A., and Koditschek,  D. E., 1999, “Sequential composition of dynamically dexterous robot behaviors,” Int. J. Robot. Res., 18, No. 6, pp. 534–555.
E. Sontag and H. J. Sussmann, 1995, “Nonsmooth control-lyapunov functions,” Proceedings of the 34th IEEE Conference on Decision and Control, New Orleans, LA.
G. A. Lafferriere, 1994, “Discontinuous stabilizing feedback using partially defined Lyapunov functions.” Proceedings of the 33rd IEEE Conference on Decision and Control, Orlando, FL.
J. Malmborg, B. M. Bernhardsson, and K. J. Åström, 1996, “A stabilizing switching scheme for multi-controller systems,” 13th IFAC World Congress, San Francisco, CA.
W. Hahn, 1967, Stability of motion, Springer-Verlag.
Hassan K. Khalil, 1992, Nonlinear systems, Macmillan, New York.
E. D. Sontag, 1990, Mathematical Control Theory, Springer-Verlag.


Grahic Jump Location
A linearizing controller applied in (a) rolling (left) and (b) sliding (right)
Grahic Jump Location
Discrete layer in control of the shimmying wheel
Grahic Jump Location
A typical simulation run
Grahic Jump Location
A modified controller guarantees decreasing of V3
Grahic Jump Location
Uncontrolled Memory effect tire
Grahic Jump Location
Controlled memory effect tire




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