Regular and Chaotic Dynamics

, Volume 13, Issue 4, pp 267–282 | Cite as

Steering by transient destabilization in piecewise-holonomic models of legged locomotion

Nonholonomic Mechanics


We study turning strategies in low-dimensional models of legged locomotion in the horizontal plane. Since the constraints due to foot placement switch from stride to stride, these models are piecewise-holonomic, and this can cause stride-to-stride changes in angular momentum and in the ratio of rotational to translational kinetic energy. Using phase plane analyses and parameter studies based on experimental observations of insects, we investigate how these changes can be harnessed to produce rapid turns, and compare the results with dynamical cockroach data. Qualitative similarities between the model and insect data suggest general strategies that could be implemented in legged robots.

Key words

biomechanics hybrid dynamical system insect locomotion passive stability piecewise holonomy robotics turning transient instability 

MSC2000 numbers

37J15 37J25 70E18 70H03 37J60 92B05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Dickinson, M.H., Farley, C.T., Full, R.J., Koehl, M.A.R., Kram, R., and Lehman, S., How Animals Move: An Integrative View, Science, 2000, vol. 288, pp. 100–106.CrossRefGoogle Scholar
  2. 2.
    Saranli, U., Schwind, W.J., and Koditschek, D.E., Rhex: A Simple and Highly Mobile Hexapod Robot, Int. J. Robotics Research, 2001, vol. 20, no. 7, pp. 616–631.CrossRefGoogle Scholar
  3. 3.
    Full, R.J. and Tu, M.S., Mechanics of Six-Legged Runners, J. Exp. Biol., 1990, vol. 148, pp. 129–146.Google Scholar
  4. 4.
    Full, R.J. and Tu, M.S., Mechanics of a Rapid Running Insect: Two-, Four-, and Six-Legged Locomotion, J. Exp. Biol., 1991, vol. 158, pp. 215–231.Google Scholar
  5. 5.
    Ting, L.H., Blickhan, R., and Full, R.J., Dynamic and Static Stability in Hexapedal Runners, J. Exp. Biol., 1994, vol. 197, pp. 251–269.Google Scholar
  6. 6.
    Full, R.J., Yamauchi, A., and Jindrich, D.L., Maximum Single Leg Force Production: Cockroaches Righting on Photoelastic Gelatin, J. Exp. Biol., 1995, vol. 198, pp. 2441–2452.Google Scholar
  7. 7.
    Schmitt, J. and Holmes, P., Mechanical Models for Insect Locomotion: Dynamics and Stability in the Horizontal Plane — Theory, Biol. Cybern., 2000, vol. 83, no. 6, pp. 501–515.MATHCrossRefGoogle Scholar
  8. 8.
    Schmitt, J. and Holmes, P., Mechanical Models for Insect Locomotion: Stability and Parameter Studies, Phys. D, 2001, vol. 156, nos. 1–2, pp. 139–168.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Schmitt, J., Garcia, M., Razo, C., Holmes, P., and Full, R.J., Dynamics and Stability of Legged Locomotion in the Horizontal Plane: A Test Case Using Insects, Biol. Cybern., 2002, vol. 86, no. 5, pp. 343–353.MATHCrossRefGoogle Scholar
  10. 10.
    Seipel, J.E., Holmes, P., and Full, R.J., Dynamics and Stability of Insect Locomotion: A Hexapedal Model for Horizontal Plane Motions, Biol. Cybern., 2004, vol. 91, no. 2, pp. 76–90.MATHCrossRefGoogle Scholar
  11. 11.
    Kukillaya, R. and Holmes, P., A Hexapedal Jointed-Leg Model for Insect Locomotion in the Horizontal Plane, Biol. Cybern., 2007, vol. 97, nos. 5–6, pp. 379–395.CrossRefMathSciNetGoogle Scholar
  12. 12.
    Jindrich, D. and Full, R.J., Dynamic Stabilization of Rapid Hexapedal Locomotion, J. Exp. Biol., 2002, vol. 205, pp. 2803–2823.Google Scholar
  13. 13.
    Goldstein, H., Classical Mechanics, New York: Addison-Wesley, 1922.Google Scholar
  14. 14.
    Neĭmark, J.I. and Fufaev, N.A., Dynamics of Nonholonomic Systems, Providence, RI: Amer. Math. Soc., 1972.MATHGoogle Scholar
  15. 15.
    Ruina, A., Non-holonomic Stability Aspects of Piecewise Holonomic Systems, Rep. Math. Phys., 1998, vol. 42, no. 1/2, pp. 91–100.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Coleman, M.J. and Holmes, P., Motions and Stability of a Piecewise Holonomic System: The Discrete Chaplygin Sleigh, Regul. Chaotic Dyn., 1999, vol. 4, no. 2, pp. 1–23.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Holmes, P., Full, R.J., Koditschek, D., and Guckenheimer, J. The Dynamics of Legged Locomotion: Models, Analyses, and Challenges, SIAM Rev., 2006, vol. 48, no. 2, pp. 207–304.MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Zill, S.N. and Moran, D.T., The Exoskeleton and Insect Proprioception III. Activity of Tibial Campaniform Sensilla during Walking in the American Cockroach, Periplaneta Americana, J. Exp. Biol., 1981, vol. 94, pp. 57–75.Google Scholar
  19. 19.
    Schmitt, J. and Holmes, P., Mechanical Models for Insect Locomotion: Dynamics and Stability in the Horizontal Plane — Application, Biol. Cybern., 2000, vol. 83, no. 6, pp. 517–527.MATHCrossRefGoogle Scholar
  20. 20.
    Camhi, J.M. and Johnson, E.N., High-Frequency Steering Maneuvers Mediated by Tactile Cues: Antennal Wall-Following in the Cockroach, J. Exp. Biol., 1999, vol. 202, pp. 631–643.Google Scholar
  21. 21.
    Cowan, N.J., Lee, J., and Full, R.J., Task-Level Control of Rapid Wall Following in the American Cockroach, J. Exp. Biol., 2006, vol. 209, pp. 1617–1629.CrossRefGoogle Scholar
  22. 22.
    Lee, J., Lamperski, A., Schmitt, J., and Cowan, N. J., Task-Level Control of the Lateral Leg Spring Model of Cockroach Locomotion, in Fast Motions in Biomechanics and Robotics: Optimization and Feedback Control. Lecture Notes in Control and Information Sciences, Diehl, M. and Mombaur, K., Eds, Heidelberg: Springer-Verlag, 2006, no. 340, pp. 167–188.Google Scholar
  23. 23.
    Lee, J., Sponberg, S.N., Loh, O.Y., Lamperski, A.G., Full, R.J., and Cowan, N.J., Templates and Anchors for Antenna-Based Wall Following in Cockroaches, IEEE Trans. Robotics, 2008, vol. 24, no. 1, pp. 130–143.CrossRefGoogle Scholar
  24. 24.
    Jindrich, D.L. and Full, R.J., Many-Legged Maneuverability: Dynamics of Turning in Hexapods, J. Exp. Biol., 1999, vol. 202, pp. 1603–1623.Google Scholar
  25. 25.
    Back, A., Guckenheimer, J., and Myers, M., A Dynamical Simulation Facility for Hybrid Systems, in Lecture Notes in Computer Science, Berlin: Springer-Verlag, 1993, vol. 736, pp. 255–267.Google Scholar
  26. 26.
    Guckenheimer, J. and Johnson, S., Planar Hybrid Systems, in Lecture Notes in Computer Science, Berlin: Springer-Verlag, 1995, no. 999, pp. 202–225.Google Scholar
  27. 27.
    Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, New York: Springer-Verlag, 1983.MATHGoogle Scholar

Copyright information

© MAIK Nauka 2008

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringPrinceton UniversityPrincetonUSA
  2. 2.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA

Personalised recommendations