Nonlinear Dynamics

, Volume 70, Issue 3, pp 2089–2094

Stability of quadruped robots’ trajectories subjected to discrete perturbations

Original Paper


In this paper, we study the stability of a mathematical model for trajectory generation of a qua-druped robot. We consider that each movement is composed of two types of primitives: rhythmic and discrete. The discrete primitive is inserted as a perturbation of the purely rhythmic movement. The two primitives are modeled by nonlinear dynamical systems. We adapt the theory developed by Golubitsky et al. in (Physica D 115: 56–72, 1998; Buono and Golubitsky in J. Math. Biol. 42:291–326, 2001) for quadrupeds gaits. We conclude that if the discrete part is inserted in all limbs, with equal values, and as an offset of the rhythmic part, the obtained gait is stable and has the same spatial and spatiotemporal symmetry groups as the purely rhythmic gait, differing only on the value of the offset.


Quadruped locomotion Modular trajectory Symmetry Central pattern generators H/K theorem 


  1. 1.
    Bizzi, E., d’Avella, A., Saltiel, P., Trensch, M.: Modular organization of spinal motor systems. The Neuroscientist 8(5), 437–442 (2002) CrossRefGoogle Scholar
  2. 2.
    Bullock, D., Grossberg, S.: The VITE model: a neural command circuit for generating arm and articulator trajectories. In: Kelso, J., Mandell, A., Shlesinger, M. (eds.) Dynamic Patterns in Complex Systems, pp. 206–305. World Scientific, Singapore (1988) Google Scholar
  3. 3.
    Buono, P.L., Golubitsky, M.: Models of central pattern generators for quadruped locomotion I. Primary gaits. J. Math. Biol. 42, 291–326 (2001) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Collins, J.J., Stewart, I.: Coupled nonlinear oscillators and the symmetries of animal gaits. J. Nonlinear Sci. 3, 349–392 (1993) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Degallier, S., Santos, C.P., Righetti, L., Ijspeert, A.: Movement generation using dynamical systems: a drumming humanoid robot. In: Humanoid’s06 IEEE-RAS International Conference on Humanoid Robots, Genova, Italy (2006) Google Scholar
  6. 6.
    Golubitsky, M., Stewart, I.: The Symmetry Perspective. Progress in Mathematics, vol. 200. Birkhauser, Basel (2002) MATHCrossRefGoogle Scholar
  7. 7.
    Golubitsky, M., Stewart, I., Buono, P.L., Collins, J.J.: A modular network for legged locomotion. Physica D 115, 56–72 (1998) MathSciNetMATHGoogle Scholar
  8. 8.
    Golubitsky, M., Stewart, I., Buono, P.L., Collins, J.J.: Symmetry in locomotor central pattern generators and animal gaits. Nature 401, 693–695 (1999) CrossRefGoogle Scholar
  9. 9.
    Grillner, S., Wallén, P., Saitoh, K., Kozlov, A., Robertson, B.: Neural bases of goal-directed locomotion in vertebrates—an overview. Brains Res. Rev. 57, 2–12 (2008) CrossRefGoogle Scholar
  10. 10.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, 3rd edn., pp. 150–154. Springer, New York (1997) Google Scholar
  11. 11.
    Matos, V., Santos, C.P., Pinto, C.M.A.: A brainstem-like modulation approach for gait transition in a quadruped robot. In: Proceedings of the 2009 IEEE/RSJ International Conference on Intelligent RObots and Systems, IROS 2009, St Louis, MO, USA, October (2009) Google Scholar
  12. 12.
    Marsden, J., McCracken, M.: Hopf Bifurcation and Its Applications. Springer, New York (1976) MATHCrossRefGoogle Scholar
  13. 13.
    Pearson, K.G.: Common principles of motor control in vertebrates and invertebrates. Annu. Rev. Neurosci. 16, 265–297 (1993) CrossRefGoogle Scholar
  14. 14.
    Pinto, C.M.A., Golubitsky, M.: Central pattern generators for bipedal locomotion. J. Math. Biol. 53, 474–489 (2006) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Pinto, C.M.A., Rocha, D., Santos, C.P.: Hexapod robots: new CPG model for generation of trajectories. J. Numer. Anal. Ind. Appl. Math. 7(1–2), 15–26 (2012). ISSN 1790–8140 Google Scholar
  16. 16.
    Pinto, C.M.A., Rocha, D., Santos, C.P.: Biped robots: effects of small perturbations on the generation of modular trajectories. J. Numer. Anal. Ind. Appl. Math. 7(1–2), 27–37 (2012). ISSN 1790–8140 Google Scholar
  17. 17.
    Pinto, C.M.A., Santos, C.P., Rocha, D., Matos, V.: New developments on online generation of trajectories in quadruped robots. J. Numer. Anal. Ind. Appl. Math. 7(1–2), 39–57 (2012). ISSN 1790–8140 Google Scholar
  18. 18.
    Pinto, C.M.A., Santos, C.P., Rocha, D., Matos, V.: Impact of discrete corrections in a modular approach for trajectory generation in quadruped robots. In: AIP Conf. Proc. Numerical Analysis and Applied Mathematics ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics, Halkidiki, Greece, vol. 1389(1), pp. 509–513 (2011) Google Scholar
  19. 19.
    Pinto, C.M.A., Tenreiro Machado, J.A.: Fractional central pattern generators for bipedal locomotion. Nonlinear Dyn. 62(1), 27–37 (2010) MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Taga, G., Yamaguchi, Y., Shimizu, H.: Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment. Biol. Cybern. 65, 147–169 (1991) MATHCrossRefGoogle Scholar
  21. 21.
    Tenore, F., Etienne-Cummings, R., Lewis, M.A.: Entrainment of silicon central pattern generators for legged locomotory control. In: Thrun, S., Saul, L., Scholkopf, B. (eds.) Proc. of Neural Information Processing Systems, vol. 16. MIT Press, Cambridge (2004) Google Scholar
  22. 22.
    Tira-thompson, E.: Digital servo calibration and modeling (2008). doi:

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Centro de Matemática da Universidade do PortoPortoPortugal
  2. 2.Instituto Superior de Engenharia do PortoPortoPortugal

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