Nonlinear Dynamics

, Volume 85, Issue 3, pp 1533–1546 | Cite as

Gait optimization and energetics of ballistic walking for an underactuated biped with knees

  • Jae-Sung Moon
  • Joonbum Bae
Original Paper


In this paper, we study gait optimization of ballistic walking in order to understand the natural dynamics of an underactuated biped with knees. We also propose applications for our understandings. Our optimization problem is solved by fixing energy levels, and then, we attempt to explain how optimal gaits are formed by examining the role of each joint in speeding up. In addition, we explain some natural characteristics of walking. Based on the results, we propose a new cost function to generate various walking gaits, including the optimum. Finally, we evaluate and discuss the energy efficiency of our ballistic walker and other bipedal walkers including humans.


Ballistic walking Passive dynamic walking Gait optimization Energetics  Natural dynamics Biped with knees 



This work was supported by the 2010 Research Fund (1.100036.01) of UNIST (Ulsan National Institute of Science and Technology), the 2012 Creativity and Innovation Research Fund (1.120047.01) of UNIST, and the 2015 Research Fund (1.150033.01) of UNIST.

Supplementary material

11071_2016_2777_MOESM1_ESM.mp4 (206 kb)
Supplementary material 1 (mp4 206 KB)
11071_2016_2777_MOESM2_ESM.mp4 (208 kb)
Supplementary material 2 (mp4 208 KB)


  1. 1.
    Adolfsson, J., Dankowicz, H., Nordmark, A.: 3D passive walkers: finding periodic gaits in the presence of discontinuities. Nonlinear Dyn. 24(2), 205–229 (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ahmadi, M., Buehler, M.: The ARL monopod II running robot: control and energetics. In: Proceedings of IEEE International Conference on Robotics Automation, vol. 3, pp. 1689–1694. Detroit, MI (1999)Google Scholar
  3. 3.
    Ahmadi, M., Buehler, M.: Controlled passive dynamic running experiments with the ARL-monopod II. IEEE Trans. Robot. 22(5), 974–986 (2006)CrossRefGoogle Scholar
  4. 4.
    Alexander, R.M.: Walking made simple. Science 308(5718), 58–59 (2005)Google Scholar
  5. 5.
    Aoustin, Y., Formal’skii, A.M.: 3d walking biped: optimal swing of the arms. Multibody Syst. Dyn. 32(1), 55–66 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Aoustin, Y., Tlalolini Romero, D., Chevallereau, C., Aubin, S.: Impulsive control for a thirteen-link biped. In: Proceedings of International Workshop Advanced Motion Control, pp. 439–444. Istanbul, Turkey (2006)Google Scholar
  7. 7.
    Asano, F., Luo, Z.W., Yamakita, M.: Biped gait generation and control based on a unified property of passive dynamic walking. IEEE Trans. Robot. 21(4), 754–762 (2005)CrossRefGoogle Scholar
  8. 8.
    Beletskii, V.V., Chudinov, P.S.: Parametric optimization in the problem of bipedal locomotion. Izvestiya AN SSSR Mekhanika Tverdogo Tela 12(1), 25–35 (1977)Google Scholar
  9. 9.
    Bhounsule, P.A., Cortell, J., Grewal, A., Hendriksen, B., Karssen, J.G.D., Ruina, A.: Low-bandwidth reflex-based control for lower power walking: 65 km on a single battery charge. Int. J. Robot. Res. 33(10), 1305–1321 (2014)CrossRefGoogle Scholar
  10. 10.
    Channon, P.H., Hopkins, S.H., Pham, D.T.: Derivation of optimal walking motions for a bipedal walking robot. Robotica 10(2), 165–172 (1992)CrossRefGoogle Scholar
  11. 11.
    Chevallereau, C., Aoustin, Y.: Optimal reference trajectories for walking and running of a biped robot. Robotica 19(5), 557–569 (2001)CrossRefGoogle Scholar
  12. 12.
    Choi, J.H., Grizzle, J.W.: Feedback control of an underactuated planar bipedal robot with impulsive foot action. Robotica 23(5), 567–580 (2005)CrossRefGoogle Scholar
  13. 13.
    Collins, S.H., Ruina, A., Tedrake, R., Wisse, M.: Efficient bipedal robots based on passive-dynamic walkers. Science 307(5712), 1082–1085 (2005)CrossRefGoogle Scholar
  14. 14.
    Collins, S.H., Wisse, M., Ruina, A.: A three-dimensional passive-dynamic walking robot with two legs and knees. Int. J. Robot. Res. 20(7), 607–615 (2001)CrossRefGoogle Scholar
  15. 15.
    Formal’skii, A.M.: Motion of anthropomorphic mechanism under impulsive control. In: Proceedings of Institute of Mechanics, Lomonosov Moscow State University, pp. 17–34 (1978) (in Russian) Google Scholar
  16. 16.
    Formal’skii, A.M.: Ballistic walking design via impulsive control. J. Aerosp. Eng. 23(2), 129–138 (2010)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Gabrielli, G., von Kármán, T.: What price speed? Mech. Eng. 72(10), 775–781 (1950)Google Scholar
  18. 18.
    Garcia, M., Chatterjee, A., Ruina, A.: Efficiency, speed, and scaling of two-dimensional passive-dynamic walking. Dynam. Stabil. Syst. 15(2), 75–99 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Garcia, M., Chatterjee, A., Ruina, A., Coleman, M.: The simplest walking model: stability, complexity, and scaling. ASME J. Biomech. Eng. 120(2), 281–288 (1998)CrossRefGoogle Scholar
  20. 20.
    Goswami, A., Thuilot, B., Espiau, B.: A study of the passive gait of a compass-like biped robot: symmetry and chaos. Int. J. Robot. Res. 17(12), 1282–1301 (1998)CrossRefGoogle Scholar
  21. 21.
    Greenwood, D.T.: Principles of Dynamics, 2nd edn. Prentice Hall, Englewood Cliffs, NJ (1988)Google Scholar
  22. 22.
    Grizzle, J.W., Abba, G., Plestan, F.: Asymptotically stable walking for biped robots: analysis via systems with impulse effects. IEEE Trans. Automat. Contr. 46(1), 51–64 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Hobbelen, D.G.E., Wisse, M.: Controlling the walking speed in limit cycle walking. Int. J. Robot. Res. 27(9), 989–1005 (2008)CrossRefGoogle Scholar
  24. 24.
    Hurmuzlu, Y., Chang, T.H.: Rigid body collisions of a special class of planar kinematic chains. IEEE Trans. Syst. Man Cybern. 22(5), 964–971 (1992)Google Scholar
  25. 25.
    Hurmuzlu, Y., Marghitu, D.B.: Rigid body collisions of planar kinematic chains with multiple contact points. Int. J. Robot. Res. 13(1), 82–92 (1994)CrossRefGoogle Scholar
  26. 26.
    Hurmuzlu, Y., Moskowitz, G.: The role of impact in the stability of bipedal locomotion. Dynam. Stabil. Syst. 1(3), 217–234 (1986)CrossRefzbMATHGoogle Scholar
  27. 27.
    Kuo, A.D.: Stabilization of lateral motion in passive dynamic walking. Int. J. Robot. Res. 18(9), 917–930 (1999)CrossRefGoogle Scholar
  28. 28.
    Margaria, R.: Positive and negative work performances and their efficiencies in human locomotion. Int. Z. angew. Physiol. Einschl. Arbeitsphysiol. 25(4), 339–351 (1968)Google Scholar
  29. 29.
    Margaria, R.: Biomechanics and Energetics of Muscular Exercise. Clarendon Press, Oxford, U.K. (1976)Google Scholar
  30. 30.
    McGeer, T.: Passive dynamic walking. Int. J. Robot. Res. 9(2), 62–82 (1990)CrossRefGoogle Scholar
  31. 31.
    McGeer, T.: Dynamics and control of bipedal locomotion. J. Theor. Biol. 163(3), 277–314 (1993)CrossRefGoogle Scholar
  32. 32.
    Mochon, S., McMahon, T.A.: Ballistic walking. J. Biomech. 13(1), 49–57 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Mochon, S., McMahon, T.A.: Ballistic walking: an improved model. Math. Biosci. 52(3–4), 241–260 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Moon, J.S., Lee, S.M., Bae, J., Youm, Y.: Analysis of period-1 passive limit cycles for flexible walking of a biped with knees and point feet. Robotica (2015). doi: 10.1017/S0263574715000144
  35. 35.
    Moon, J.S., Spong, M.W.: Classification of periodic and chaotic passive limit cycles for a compass-gait biped with gait asymmetries. Robotica 29(7), 967–974 (2011)CrossRefGoogle Scholar
  36. 36.
    Moon, J.S., Stipanović, D.M., Spong, M.W.: Gait generation and stabilization for nearly passive dynamic walking using auto-distributed impulses. Asian J. Control (2015). doi: 10.1002/asjc.1206
  37. 37.
    Parker, T.S., Chua, L.O.: Practical Numerical Algorithms for Chaotic Systems. Springer, New York, NY (1989)CrossRefzbMATHGoogle Scholar
  38. 38.
    Pratt, J.E.: Exploiting inherent robustness and natural dynamics in the control of bipedal walking robots. Ph.D. Thesis, Massachusetts Institute of Technology, Cambridge, MA (2000)Google Scholar
  39. 39.
    Roussel, L., Canudas-de-Wit, C., Goswami, A.: Generation of energy optimal complete gait cycles for biped robots. In: Proceedings of IEEE International Conference on Robotics Automation, pp. 2036–2041. Leuven, Belgium (1998)Google Scholar
  40. 40.
    Schiehlen, W.: Energy-optimal design of walking machines. Multibody Syst. Dyn. 13(1), 129–141 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Spong, M.W., Holm, J.K., Lee, D.: Passivity-based control of bipedal locomotion. IEEE Robot. Autom. Mag. 14(2), 30–40 (2007)CrossRefGoogle Scholar
  42. 42.
    Spong, M.W., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control. Wiley, Hoboken, NJ (2006)Google Scholar
  43. 43.
    Sreenath, K., Park, H.W., Poulakakis, I., Grizzle, J.W.: A compliant hybrid zero dynamics controller for stable, efficient and fast bipedal walking on MABEL. Int. J. Robot. Res. 30(9), 1170–1193 (2011)CrossRefGoogle Scholar
  44. 44.
    Srinivasan, M., Ruina, A.: Computer optimization of a minimal biped model discovers walking and running. Nature 439, 72–75 (2006)CrossRefGoogle Scholar
  45. 45.
    Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Addison-Wesley, Reading, Mass (1994)Google Scholar
  46. 46.
    Tedrake, R., Zhang, T.W., Fong, M.f., Seung, H.S.: Actuating a simple 3D passive dynamic walker. In: Proceedings of IEEE International Conference on Robotics Automation, pp. 4656–4661. New Orleans, LA (2004)Google Scholar
  47. 47.
    Vukobratović, M., Borovac, B.: Zero-moment point—thirty five years of its life. Int. J. Humanoid Robotics 1, 157–173 (2004)CrossRefGoogle Scholar
  48. 48.
    Vukobratović, M., Juričić, D.: Contribution to the synthesis of biped gait. In: Proceedings of IFAC Symposium on Technical and Biological Problem on Control. Erevan, USSR (1968)Google Scholar
  49. 49.
    Westervelt, E.R., Grizzle, J.W., Koditschek, D.E.: Hybrid zero dynamics of planar biped walkers. IEEE Trans. Automat. Contr. 48(1), 42–56 (2003)Google Scholar
  50. 50.
    Wisse, M., Schwab, A.L., van der Linde, R.Q.: A 3D passive dynamic biped with yaw and roll compensation. Robotica 19(3), 275–284 (2001)CrossRefGoogle Scholar
  51. 51.
    Wittenburg, J.: Dynamics of System of Rigid Bodies. Teubner, Stuttgart (1977)CrossRefzbMATHGoogle Scholar
  52. 52.
    Yamakita, M., Asano, F.: Extended passive velocity field control with variable velocity fields for a kneed biped. Adv. Robot. 15(2), 139–168 (2001)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.School of Mechanical and Nuclear EngineeringUlsan National Institute of Science and TechnologyUlsanKorea
  2. 2.Department of Mechanical Engineering, and School of Mechanical and Nuclear EngineeringUlsan National Institute of Science and TechnologyUlsanKorea

Personalised recommendations