Multibody System Dynamics

, Volume 26, Issue 3, pp 213–243 | Cite as

Adaptive control of straight worms without derivative measurement

  • Carsten Behn


In this paper we consider an adaptive control problem of finite DoF worm-like locomotion systems (WLLS) which contact the ground with Coulomb dry friction. Using a rough mathematical friction law the system is shown to belong to a system class that allows adaptive control. Gaits from the kinematic theory can be tracked by means of adaptive controllers. For this we introduce two different adaptive controllers for λ-tracking and focus on that one which is not based on the derivative of the output. We pay attention to the analysis of such systems and present some theoretical control investigations including proofs. Numerical simulations of tracking different reference signals under arbitrary choice of the system parameters demonstrate and illustrate that the introduced simple adaptive controllers work successfully and effectively. Current experiments are aimed at the justification of theoretical results.


Worm-like locomotion Adaptive tracking control Relative degree two Asymmetric Coulomb friction 


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  1. 1.
    Armstrong-Hélouvry, B., Dupont, P., Canudas de Wit, C.: A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 30(7), 1083–1138 (1994) MATHCrossRefGoogle Scholar
  2. 2.
    Awrejcewicz, J., Olejnik, P.: Analysis of dynamic systems with various friction laws. Appl. Mech. Rev. 58, 389–411 (2005) CrossRefGoogle Scholar
  3. 3.
    Barbălat, I.: Systèmes d’équations différentielles d’oscillations non linéaires. Rev. Math. Pures Appl., Bucharest IV, 267–270 (1959) Google Scholar
  4. 4.
    Behn, C.: Ein Beitrag zur adaptiven Regelung technischer Systeme nach biologischem Vorbild. Cuvillier, Göttingen (2005) Google Scholar
  5. 5.
    Behn, C., Zimmermann, K.: Biologically inspired locomotion systems and adaptive control. In: Proceedings ECCOMAS Thematic Conference Multibody Dynamics, Madrid, Spain, 21–24 June 2005 Google Scholar
  6. 6.
    Behn, C., Zimmermann, K.: Adaptive λ-tracking for locomotion systems. Robot. Auton. Syst. 54, 529–545 (2006) CrossRefGoogle Scholar
  7. 7.
    Liu, W., Menciassi, A., Scapellato, S., Dario, P., Chen, Y.: A biomimetic sensor for a crawling minirobot. Robot. Auton. Syst. 54, 513–528 (2006) CrossRefGoogle Scholar
  8. 8.
    Dragan, V., Halanay, A.: Stabilization of Linear Systems. Birkhäuser, Boston (1999) MATHGoogle Scholar
  9. 9.
    Hahn, W.: Stability of Motion. Springer, Berlin (1967) MATHGoogle Scholar
  10. 10.
    Hardy, G.H., Littlewood, J.E., Polya, G.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1994) Google Scholar
  11. 11.
    Ilchmann, A.: Non-identifier-based adaptive control of dynamical systems: a survey. IMA J. Math. Control Inf. 8, 321–366 (1991) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Kurzweil, J.: Ordinary Differential Equations—Introduction to the Theory of Ordinary Differential Equations in the Real Domain. Elsevier Science, Amsterdam (1986) MATHGoogle Scholar
  13. 13.
    Meier, P., Dietrich, J., Oberthür, S., Preuß, R., Voges, D., Zimmermann, K.: Development of a peristaltically actuated device for the minimal invasive surgery with a haptic sensor array. In: Micro- and Nanostructures, pp. 66–89. Shaker, Aachen (2004) Google Scholar
  14. 14.
    Miller, G.: The motion dynamics of snakes and worms. Comput. Graph. 22, 169–173 (1988) CrossRefGoogle Scholar
  15. 15.
    Miller, D.E., Davison, E.J.: An adaptive controller which provides an arbitrarily good transient and steady-state response. IEEE Trans. Autom. Control 36, 68–81 (1991) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Nakamura, T., Kato, T., Iwanaga, T., Muranaka, Y.: Peristaltic crawling robot based on the locomotion mechanism of earthworms. In: Proceedings 4th IFAC-Symposium on Mechatronic Systems, Heidelberg, Germany, 12–14 September, 2006 Google Scholar
  17. 17.
    Natanson, I.P.: Theorie der Funktionen einer reellen Veränderlichen, 5th edn. Akademie-Verlag, Berlin (1981) MATHGoogle Scholar
  18. 18.
    Olsson, H., Åström, K.J., Canudas de Wit, C., Gräfert, M., Lischinsky, P.: Friction models and friction compensation. Eur. J. Control 4, 176–195 (1998) MATHGoogle Scholar
  19. 19.
    Ostrowski, J.P., Burdick, J.W., Lewis, A.D., Murray, R.M.: The mechanics of undulatory locomotion: the mixed kinematic and dynamic case. In: Proceedings IEEE International Conference on Robotics and Automation, Nagoya, Japan (1995) Google Scholar
  20. 20.
    Preuß, R., Stubenrauch, M.: Miniaturization of fully compliant, worm-like motion systems. In: Blickhan, R. (ed.) Motion Systems—Collected Short Papers, pp. 58–62. Shaker, Aachen (2001) Google Scholar
  21. 21.
    Sontag, E.D.: Mathematical Control Theory, 2nd edn. Springer, New York (1998) MATHGoogle Scholar
  22. 22.
    Steigenberger, J.: On a class of biomorphic motion systems. Preprint No. M12/99, Faculty of Mathematics and Natural Sciences, TU Ilmenau (1999) Google Scholar
  23. 23.
    Steigenberger, J.: On nonholonomic systems and biomorphic motion. In: Proceedings 44. Internationales Wissenschaftliches Kolloquium (IWK), Ilmenau, Germany, 20–23 September 1999, pp. 216–219 (1999) Google Scholar
  24. 24.
    Steigenberger, J.: Modeling artificial worms. Preprint No. M02/04, Faculty of Mathematics and Natural Sciences, TU Ilmenau (2004) Google Scholar
  25. 25.
    Steigenberger, J.: Some theory towards a stringent definition of “locomotion”. Multibody Syst. Dyn. (2011). doi: 10.1007/s11044-011-9245-z MATHGoogle Scholar
  26. 26.
    Steigenberger, J., Behn, C., Zimmermann, K., Abaza, K.: Worm-like locomotion: theory, control and application. In: Proceedings of the 3rd International Symposium on Adaptive Motion in Animals and Machines (AMAM), Ilmenau, Germany, 25–30 September, 2005 Google Scholar
  27. 27.
    Walter, W.: Analysis 2, 4., durchgesehene und ergänzte Auflage. Springer, Berlin (1995) Google Scholar
  28. 28.
    Ye, X.: Universal λ-tracking for nonlinearly-perturbed systems without restrictions on the relative degree. Automatica 35, 109–119 (1999) MATHCrossRefGoogle Scholar
  29. 29.
    Zimmermann, K., Zeidis, I.: Ein mathematisches Modell für peristaltische Bewegung als Grundlage für das Desgin wurmartiger Mikroroboter. In: Proceedings 44. Internationales Wissenschaftliches Kolloquium (IWK), Ilmenau, Germany, 20–23 September 1999, pp. 220–227 (1999) Google Scholar
  30. 30.
    Zimmermann, K., Zeidis, I.: Mathematical models and prototypes of worm-like motion systems using magnetic materials. In: Theory and Practice of Robots and Manipulators, Proc. of the 15th CISM IFToMM Symposium (RoManSy 15), Montreal, Canada (2004) Google Scholar
  31. 31.
    Zimmermann, K., Preuß, R., Lysenko, V.: Lokomotions- und Manipulationssysteme für die Mikrorobotik auf der Basis nachgiebiger Strukturen. In: Proceedings Robotik 2000, Berlin, Germany, 29–30 June 2000, pp. 491–496 (2000) Google Scholar
  32. 32.
    Zimmermann, K., Zeidis, I., Behn, C.: Mechanics of Terrestrial Locomotion—With a Focus on Non-Pedal Motion Systems. Springer, Berlin (2009) MATHGoogle Scholar
  33. 33.
    Zimmermann, K., Zeidis, I., Lysenko, V.: An approach to the modelling of peristaltic motion using continuum mechanics—first steps in technical realization. In: Blickhan, R. (ed.) Motion Systems—Collected Short Papers, pp. 53–57. Shaker, Aachen (2001) Google Scholar
  34. 34.
    Zimmermann, K., Zeidis, I., Naletova, V., Turkov, V.: Waves on the surface of a magnetic fluid layer in a travelling magnetic field. J. Magn. Magn. Mater. 268(1–2), 227–231 (2004) CrossRefGoogle Scholar

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© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Technical MechanicsIlmenau University of TechnologyIlmenauGermany

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