Skip to main content
SpringerLink
Log in

Hopping on Even Ground and Up Stairs with a Single Articulated Leg

  • Published:
Journal of Intelligent and Robotic Systems Aims and scope Submit manuscript

Abstract

This paper presents a method for generating gaits for a one-legged articulated hopping robot. A static optimization procedure produces the initial joint velocities for the flight phase, using the principle of conservation of angular momentum and assuming (nearly) passive flight. Two novel objective functions for this static optimization enable one to choose different gaits by simply changing a few parameters. A dynamic optimization procedure yields a solution for the flight trajectory that minimizes control effort. The stance phase (when the foot is touching the ground) becomes a standard two point boundary value problem, also solved with a dynamic optimization procedure. During the stance phase, the physical joint limitations, ground reaction forces, and the trajectory of the zero-moment point all constrain the solution. After these single-phase optimizations, a complete-cycle optimization procedure, incorporating both flight and stance phases, further reduces the control effort and balances the motion phases. In simulation, the leg hops on even ground and up stairs, exhibiting energy-efficient and intuitively satisfying gaits.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Srinivasan, M., Ruina, A.: Computer optimization of a minimal biped model discovers walking and running. Nature 439, 72–75 (2006)

    Article  Google Scholar 

  2. Alexander, R. McN.: A model of bipedal locomotion on compliant legs. Philos. Trans. R. Soc. Lond. B Biol. Sci. 338(1284), 189–198 (1992)

    Article  Google Scholar 

  3. Vukobratović, M., Borovac, B.: Zero-moment point—thirty five years of its life. Int. J. Humanoid Robotics 1(1), 157–173 (2004)

    Article  Google Scholar 

  4. Sardain, P., Bessonnet G.: Forces acting on a biped robot: center of pressure - zero moment point. IEEE Trans. Syst. Man Cybern. Part A Syst. Humans 34(5), 630–637 (2004)

    Article  Google Scholar 

  5. Popovic, M.B., Goswami, A., Herr, H.: Ground reference points in legged locomotion: definitions, biological trajectories and control implications. Int. J. Rob. Res. 24(12), 1013–1032 (2005)

    Article  Google Scholar 

  6. Kondak, K., Hommel, G.: Control of online computation of stable movement for biped robots. In: Proceedings of the IEEE/RSJ Intl. Conf. Intelligent Robots and Systems, pp. 874–879. Las Vegas (2003)

  7. Westervelt, E.: Toward a coherent framework for the control of planar biped locomotion. Ph.D. Dissertation, University of Michigan at Ann Arbor, June (2003)

  8. Raibert, M.H.: Legged Robots That Balance. The MIT Press (1986)

  9. Gregorio, P., Ahmadi, M., Buehler, M.: Design, control, and energetics of an electrically actuated legged robot. IEEE Trans. Syst. Man Cybern. 27B(4), 626–634 (1997)

    Google Scholar 

  10. Ahmadi, M., Buehler, M.: Stable control of a simulated one-legged running robot with hip and leg compliance. IEEE Trans. Robot. Autom. 13(1), 96–103 (1997)

    Article  Google Scholar 

  11. Brown, B., Zeglin, G.: The bow leg hopping robot. In: Intl. Conf. Robotics and Automation, pp. 781–786. Leuven, Belgium (1998)

    Google Scholar 

  12. De Man, H., Lefeber, D., Vermeulen, J.: Control on irregular terrain of a hopping robot with one articulated leg. In: ICAR Workshop II: New Approaches on Dynamic Walking and Climbing Machines, pp. 72–76. Monterey, CA (1997)

  13. Vermeulen, J.: Trajectory generation for planar hopping and walking robots: an objective parameter and angular momentum approach. Ph.D. Dissertation, Vrije Universiteit Brussel, Belgium (2004)

  14. Ikeda, T., Iwatani, Y., Suse, K., Mita, T.: Analysis and design of running robots in touchdown phase. In: Proceedings Intl. Conf. Control and Automation, pp. 496–501. Hawaii (1999)

  15. Hyon, S., Emura, T., Mita, T.: Dynamics-based control of a one-legged hopping robot. J. Syst. Control Eng. 217(2), 83–98 (2003)

    Google Scholar 

  16. Spong, M.W., Hutchinson, S., Vidyasagar, M.: Robot Modeling and Control. John Wiley & Sons (2006)

  17. Cabodevila, G., Abba, G.: Quasi optimal gait for a biped robot using genetic algorithm. In: Proceedings of the IEEE Intl. Conf. Systems, Man and Cybernetics, Computational Cybernetics and Simulations, pp. 3960–3965. Orlando, FL (1997)

  18. Channon, P.H., Hopkins, S.J., Pham, D.H.: Simulation and optimal walking motions for a bipedal robot. Math. Comput. Model. 14, 463–467 (1990)

    Article  Google Scholar 

  19. Channon, P.H., Hopkins, S.J., Pham, D.H.: Derivation of optimal walking motions for a bipedal walking robot. Robotica 10, 165–172 (1992)

    Article  Google Scholar 

  20. Chevallereau, C., Aoustin, Y.: Optimal reference trajectories for walking and running of a biped robot. Robotica 19, 557–569 (2001)

    Article  Google Scholar 

  21. Hardt, M.W.: Multibody Dynamical Algorithms, Numerical Optimal Control, with Detailed Studies in the Control of Jet Engine Compressors and Biped Walking. Ph.D. Dissertation, UCSD (1999)

  22. Yen, V., Nagurka, M.L.: Suboptimal trajectory planning of a five-link human locomotion model. In: Stein, J.L. (ed.) Biomechanics of Normal and Prosthetic Gait, BED-vol. 4 and DSC vol. 7, pp. 17–22. ASME Winter Annual Meeting, Boston, MA (1987)

  23. Azevedo, C., Poignet, P., Espiau, B.: Moving horizon control for biped robots without reference trajectory. In: Proceedings Intl. Conf. Robotics and Automoation, vol. 3, pp. 2762–2767. Washington, DC (2002)

  24. Rostami, M., Bessonnet, G.: Sagittal gait of a biped robot during the single support phase. Part 2: optimal motion. Robotica 19, 241–253 (2001)

    Google Scholar 

  25. Bessonnet, G., Chesse, S., Sardain, P.: Optimal gait synthesis of a seven-link planar biped. Int. J. Rob. Res. 23(10–11), 1059–1073 (2004)

    Article  Google Scholar 

  26. Roussel, L., Canudas de Wit, C., Goswami, A.: Comparative study of method for energy-optimal gait eneration for biped robots. In: Proceedings Intl. Conf. Informatics and Control, pp. 1205–1212. St. Petersburg (1997)

  27. Becerra, V.M.: Solving optimal control problems with state constraints using nonlinear programming and simulation tools. IEEE Trans. Educ. 47(3), 377–384 (2004)

    Article  Google Scholar 

  28. Meghdari, A., Aryanpour, M.: Dynamic modeling and analysis of the human jumping process. J. Intell. Robot. Syst. 37(1), 97–115 (2003)

    Article  Google Scholar 

  29. Fujimoto, Y.: Trajectory generation of biped running robot with minimum energy consumption. In: Proc. of the IEEE International Conference on Robotics and Automation, pp. 3803–3808 (2004)

  30. Collins, S., Ruina, A., Tedrake, R., Wisse, M.: Efficient bipedal robots based on passive-dynamic walkers. Science 307(5712), 1082–1085 (2005)

    Article  Google Scholar 

  31. Lesbastard, V., Aoustin, Y., Plestan, Y.: Observer-based control of a walking biped robot without orientation measurement. Robotica 24, 384–400 (2006)

    Google Scholar 

  32. Hass, J., Herrmann, J.M., Geisel T.: Optimal mass distribution for passivity-based bipedal robots. Int. J. Rob. Res. 25, 11, 1087–1098 (2006)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, University of Calgary, Calgary, Alberta, Canada, T2N 1N4

    Q. Guo & C. J. B. Macnab

  2. Department of Mechanical Engineering, University of Calgary, Calgary, Alberta, Canada, T2N 1N4

    J. K. Pieper

Authors
  1. Q. Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. J. B. Macnab
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. K. Pieper
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to C. J. B. Macnab.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Guo, Q., Macnab, C.J.B. & Pieper, J.K. Hopping on Even Ground and Up Stairs with a Single Articulated Leg. J Intell Robot Syst 53, 331–358 (2008). https://doi.org/10.1007/s10846-008-9244-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10846-008-9244-3

Keywords

Search

Navigation