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Advances in the Theory of Control, Signals and Systems with Physical Modeling pp 77–88Cite as

Controlling Underactuated Mechanical Systems: A Review and Open Problems

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Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 407))

Abstract

This chapter provides a short review on the popular yet still very important area of controlling underactuated mechanical systems. New solutions to the simultaneous stabilization and tracking problem are proposed for nonholonomic mobile robots using state and output feedback. Some open problems are discussed with a unique objective to solicit fundamentally novel techniques for the further development of modern nonlinear control theory.

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References

  1. Acosta, J.A., Ortega, R., Astolfi, A., Mahindrakar, A.D.: Interconnection and damping assignment passivity-based control of mechanical systems with underactuation degree one. IEEE Trans. Automat. Contr. 50, 1936–1955 (2005)

    Article  MathSciNet  Google Scholar 

  2. Astolfi, A.: Discontinuous control of nonholonomic systems. Systems and Control Letters 27, 37–45 (1997)

    Article  MathSciNet  Google Scholar 

  3. Bloch, A., Baillieul, J., Crouch, P.E., Marsden, J.E.: Nonholonomic Mechanics and Control. Springer, NY (2003)

    Book  MATH  Google Scholar 

  4. Brockett, R.W.: Asymptotic stability and feedback stabilization. In: Brockett, R.W., Millam, R.S., Sussmann, H.J. (eds.) Differential Geometric Control Theory, pp. 181–191. Birkhäuser, Boston (1983)

    Google Scholar 

  5. Buccieri, D., Perritaz, D., Mullhaupt, P., Jiang, Z.P., Bonvin, D.: Velocity-scheduling control for a unicycle mobile robot: theory and experiments. IEEE Trans. Robotics and Automation 25, 451–458 (2009)

    Google Scholar 

  6. Coron, J.M.: On the stabilization of some nonlinear control systems: results, tools, and applications. In: Clarke, F.H., Stern, R.J. (eds.) Nonlinear Analysis, Differential Equations and Control, pp. 307–367. Kluwer Academic Publishers, Dordrecht (1999)

    Google Scholar 

  7. Dixon, W.E., Dawson, D.M., Zergeroglu, E., Behal, A.: Nonlinear Control of Wheeled Mobile Robots. Springer, New York (2001)

    MATH  Google Scholar 

  8. Do, K.D., Jiang, Z.P., Pan, J.: Universal controllers for stabilization and tracking of underactuated ships. Systems & Control Letters 47, 299–317 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  9. Do, K.D., Jiang, Z.P., Pan, J.: Simultaneous tracking and stabilization of mobile robots: an adaptive approach. IEEE Trans. Automatic Control 49, 1147–1152 (2004)

    Article  MathSciNet  Google Scholar 

  10. Do, K.D., Jiang, Z.P., Pan, J.: A global output-feedback controller for simultaneous tracking and stabilization of unicycle-type mobile robots. IEEE Trans. on Robotics and Automation 20, 589–594 (2004)

    Article  Google Scholar 

  11. Do, K.D., Pan, J., Jiang, Z.P.: Robust and adaptive path following for underactuated autonomous underwater vehicles. Ocean Engineering 31, 1967–1997 (2004)

    Article  Google Scholar 

  12. Do, K.D., Jiang, Z.P., Pan, J.: Robust adaptive path following of underactuated ships. Automatica 40, 929–944 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Egerstedt, M., Hu, X.: A hybrid control approach to action coordination for mobile robots. Automatica 38, 125–130 (2002)

    Article  MATH  Google Scholar 

  14. Egerstedt, M., Hu, X., Stotsky, A.: Control of mobile platforms using a virtual vehicle approach. IEEE Transaction on Automatic Control 46, 1777–1782 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Fliess, M., Lévine, J., Martin, P., Rouchon, P.: Flatness and defect of nonlinear systems: Introductory theory and examples. Int. J. Control 61, 1327–1361 (1995)

    Article  MATH  Google Scholar 

  16. Fliess, M., Lévine, J., Martin, P., Rouchon, P.: A Lie-Backlund approach to equivalence and flatness of nonlinear systems. IEEE Trans. Autom. Control 44, 922–937 (1999)

    Article  MATH  Google Scholar 

  17. Fontaine, I., Lozano, R.: Non-linear Control for Underactuated Mechanical Systems. Springer, London (2001)

    Google Scholar 

  18. Fossen, T.I.: Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cybernetics (2002) ISBN 82-92356-00-2

    Google Scholar 

  19. Jiang, Z.P.: Lyapunov design of global state and output feedback trackers for nonholonomic control systems. Int. J. of Control 73, 744–761 (2000)

    Article  MATH  Google Scholar 

  20. Jiang, Z.P.: Global tracking control of underactuated ships by Lyapunov’s direct method. Automatica 38, 301–309 (2002)

    Article  MATH  Google Scholar 

  21. Jiang, Z.P., Nijmeijer, H.: Tracking control of mobile robots: a case study in backstepping. Automatica 33, 1393–1399 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  22. Jiang, Z.P., Nijmeijer, H.: A recursive technique for tracking control of nonholonomic systems in chained form. IEEE Trans. Automatic Control 44, 265–279 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jiang, Z.P., Lefeber, E., Nijmeijer, H.: Saturated stabilization and tracking of a nonholonomic mobile robot. Syst. & Control Lett. 42, 327–332 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Jiang, Z.P., Lin, Y., Wang, Y.: Stabilization of nonlinear time-varying systems: A control Lyapunov function approach. J. Syst. Sci. & Complexity 22, 683–696 (2009)

    Article  MathSciNet  Google Scholar 

  25. Kolmanovsky, I., McClamroch, N.H.: Developments in nonholonomic control problems. IEEE Control Syst. Mag. 15, 20–36 (1995)

    Article  Google Scholar 

  26. Krstic, M., Kanellakopoulos, K., Kokotovic, P.V.: Nonlinear and Adaptive Control Design. John Wiley & Sons, New York (1995)

    Google Scholar 

  27. Lee, T.C., Jiang, Z.P.: A generalization of Krasovskii-LaSalle theorem for nonlinear time-varying systems: converse results and applications. IEEE Trans. Automatic Control 50, 1147–1163 (2005)

    Article  MathSciNet  Google Scholar 

  28. Lee, T.C., Jiang, Z.P.: Uniform asymptotic stability of nonlinear switched systems with an application to mobile robots. IEEE Trans. Automatic Control 53, 1235–1252 (2008)

    Article  MathSciNet  Google Scholar 

  29. Lee, T.C., Song, K.T., Lee, C.H., Teng, C.C.: Tracking control of unicycle-modeled mobile robots using a saturation feedback controller. IEEE Trans. Control Systems Technology 9, 305–318 (2001)

    Article  Google Scholar 

  30. Levine, J.: Analysis and Control of Nonlinear Systems: A Flatness-based Approach. Springer, Heidelberg (2009)

    MATH  Google Scholar 

  31. Lizarraga, D.A.: Obstructions to the existence of universal stabilizers for smooth control systems. Math. Control, Signals, Syst. 16, 255–277 (2003)

    MathSciNet  Google Scholar 

  32. Malisoff, M., Mazenc, F.: Constructions of Strict Lyapunov Functions. Springer, London (2009)

    Book  MATH  Google Scholar 

  33. Morin, P., Samson, C.: Control of nonholonomic mobile robots based on the transverse function approach. IEEE Trans. Robotics 25, 1058–1073 (2009)

    Article  Google Scholar 

  34. Murray, R.M., Sastry, S.S.: Nonholonomic motion planning: Steering using sinusoids. IEEE Trans. Autom. Control 38, 700–716 (2003)

    Article  MathSciNet  Google Scholar 

  35. Olfati-Saber, R.: Nonlinear control of underactuated mechanical systems with application to robotics and aerospace vehicles. Ph.D. Thesis, Massachusetts Institute of Technology (2001)

    Google Scholar 

  36. Ortega, R., Spong, M., Gomez-Estern, F., Blankenstein, G.: Stabilization of a class of underactuated mechanical systems via interconnection and damping assignment. IEEE Trans. Automat. Contr. 47, 1218–1233 (2002)

    Article  MathSciNet  Google Scholar 

  37. Qu, Z., Wang, J., Plaisted, C.E., Hull, R.A.: Global-stabilizing near-optimal control design for nonholonomic chained systems. IEEE Trans. Automat. Control 51, 1440–1456 (2006)

    Article  MathSciNet  Google Scholar 

  38. Refsnes, J.E., Sorensen, A.J., Pettersen, K.Y.: Model-based output feedback control of slender-body underactuated AUVs: Theory and experiments. IEEE Trans. Control Systems Technology 16, 930–946 (2008)

    Article  Google Scholar 

  39. Reyhanoglu, M., van der Schaft, A., McClamroch, N.H., Kolmanovsky, I.: Dynamics and control of a class of underactuated mechanical systems. IEEE Trans. Automat. Contr. 44, 1663–1672 (1999)

    Article  MATH  Google Scholar 

  40. Samson, C.: Velocity and torque feedback control of a nonholonomic cart. In: Canudas de wit, C., et al. (eds.) Advanced Robot Control, pp. 125–151 (1991)

    Google Scholar 

  41. Sontag, E.D., Sussmann, H.: Remarks on continuous feedback. In: Proc. IEEE Conf. Decision and Control, Albuquerque, pp. 916–921 (1980)

    Google Scholar 

  42. Sontag, E.D.: Input to state stability: Basic concepts and results. In: Nistri, P., Stefani, G. (eds.) Nonlinear and Optimal Control Theory, pp. 163–220. Springer, Berlin (2007)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Polytechnic Institute of New York University, Six Metrotech Center, Brooklyn, NY, 11201, U.S.A.

    Zhong-Ping Jiang

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  1. Zhong-Ping Jiang
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Editors and Affiliations

  1. CAS, Unité Mathématiques et Systémes, MINES-ParisTech, 35, rue Saint-Honoré, 77300, Fontainebleau, France

    Jean Lévine

  2. Laboratoire d’ Automatique, Faculté des Sciences de l’Ingénieur Station 9, CH-1015, Lausanne, Switzerland

    Philippe Müllhaupt

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Jiang, ZP. (2010). Controlling Underactuated Mechanical Systems: A Review and Open Problems. In: Lévine, J., Müllhaupt, P. (eds) Advances in the Theory of Control, Signals and Systems with Physical Modeling. Lecture Notes in Control and Information Sciences, vol 407. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16135-3_7

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  • DOI: https://doi.org/10.1007/978-3-642-16135-3_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-16134-6

  • Online ISBN: 978-3-642-16135-3

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