Hostname: page-component-848d4c4894-r5zm4 Total loading time: 0 Render date: 2024-06-21T12:32:12.420Z Has data issue: false hasContentIssue false

Analysis of period-1 passive limit cycles for flexible walking of a biped with knees and point feet

Published online by Cambridge University Press:  13 March 2015

Jae-Sung Moon*
School of Mechanical and Nuclear Engineering, Ulsan National Institute of Science and Technology, Ulsan, 689-798, Korea. E-mails:,
Seong-Min Lee
School of Mechanical and Nuclear Engineering, Ulsan National Institute of Science and Technology, Ulsan, 689-798, Korea. E-mails:,
Joonbum Bae
School of Mechanical and Nuclear Engineering, Ulsan National Institute of Science and Technology, Ulsan, 689-798, Korea. E-mails:, Department of Mechanical Engineering, Ulsan National Institute of Science and Technology, Ulsan, 689-798, Korea. E-mail:
Youngil Youm
School of Mechanical and Nuclear Engineering, Ulsan National Institute of Science and Technology, Ulsan, 689-798, Korea. E-mails:,
*Corresponding author. E-mail:


In this paper, we investigate dynamic walking as a convergence to the system's own limit cycles, not to artificially generated trajectories, which is one of the lessons in the concept of passive dynamic walking. For flexible walking, gait transitions can be performed by moving from one limit cycle to another one, and thus, the flexibility depends on the range in which limit cycles exist. To design a bipedal walker based on this approach, we explore period-1 passive limit cycles formed by natural dynamics and analyze them. We use a biped model with knees and point feet to perform numerical simulations by changing the center of mass locations of the legs. As a result, we obtain mass distributions for the maximum flexibility, which can be attained from very limited location sets. We discuss the effect of parameter variations on passive dynamic walking and how to improve robot design by analyzing walking performance. Finally, we present a practical application to a real bipedal walker, designed to exhibit more flexible walking based on this study.

Copyright © Cambridge University Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


1. Ames, A. D., “Human-inspired control of bipedal walking robots,” IEEE Trans. Autom. Control 59 (5), 11151130 (2014).CrossRefGoogle Scholar
2. Aoi, S. and Tsuchiya, K., “Bifurcation and chaos of a simple walking model driven by a rhythmic signal,” Int. J. Non-Linear Mech. 41 (3), 438446 (2006).CrossRefGoogle Scholar
3. Asano, F. and Kawamoto, J., “Modeling and analysis of passive viscoelastic-legged rimless wheel that generates measurable period of double-limb support,” Multibody Syst. Dyn. 31 (2), 111126 (2014).CrossRefGoogle Scholar
4. Asano, F. and Yamakita, M., “Virtual gravity and coupling control for robotic gait synthesis,” IEEE Trans. Syst. Man Cybern. A 31 (6), 737745 (2001).CrossRefGoogle Scholar
5. Chevallereau, C., Abba, G., Aoustin, Y., Plestan, F., Westervelt, E. R., Canudas-de-Wit, C. and Grizzle, J. W., “RABBIT: A testbed for advanced control theory,” IEEE Control Syst. Mag. 23 (5), 5779 (2003).Google Scholar
6. Collins, S. H., Ruina, A., Tedrake, R. and Wisse, M., “Efficient bipedal robots based on passive-dynamic walkers,” Science 307 (5712), 10821085 (2005).CrossRefGoogle ScholarPubMed
7. Dempster, W. T. and Gaughran, G. R. L., “Properties of body segments based on size and weight,” Am. J. Anat. 120 (1), 3354 (1967).CrossRefGoogle Scholar
8. Garcia, M., Chatterjee, A., Ruina, A. and Coleman, M., “The simplest walking model: Stability, complexity, and scaling,” ASME J. Biomech. Eng. 120 (2), 281288 (1998).CrossRefGoogle ScholarPubMed
9. Goswami, A., Thuilot, B. and Espiau, B., “A study of the passive gait of a compass-like biped robot: Symmetry and chaos,” Int. J. Robot. Res. 17 (12), 12821301 (1998).CrossRefGoogle Scholar
10. Grizzle, J. W., Hurst, J., Morris, B., Park, H.-W. and Sreenath, K., “MABEL, a New Robotic Bipedal Walker and Runner,” Proceedings of the American Control Conference, St. Louis, MO (2009) pp. 2030–2036.Google Scholar
11. Grizzle, J. W., Abba, G. and Plestan, F., “Asymptotically stable walking for biped robots: analysis via systems with impulse effects,” IEEE Trans. Autom. Control 46 (1), 5164 (2001).CrossRefGoogle Scholar
12. Hirose, M. and Kenichi, O., “Honda humanoid robots development,” Phil. Trans. R. Soc. A 365 (1850), 1119 (2007).CrossRefGoogle Scholar
13. Hobbelen, D. G. E. and Wisse, M., “Controlling the walking speed in limit cycle walking,” Int. J. Robot. Res. 27 (9), 9891005 (2008).CrossRefGoogle Scholar
14. Hobbelen, D. G. E. and Wisse, M., Limit Cycle Walking. Humanoid Robots: Human-like Machines (Hackel, M., ed.) (Vienna, Austria: I-Tech Education and Publishing, 2007).Google Scholar
15. Howell, G. W. and Baillieul, J., “Simple Controllable Walking Mechanisms which Exhibit Bifurcations,” Proceedings of the IEEE Conference Decision Control, Tampa, FL (1998) pp. 3027–3032.Google Scholar
16. Hurmuzlu, Y. and Moskowitz, G., “The role of impact in the stability of bipedal locomotion,” Dyn. Stabil. Syst. 1 (3), 217234 (1986).Google Scholar
17. Hurmuzlu, Y. and Marghitu, D. B., “Rigid body collisions of planar kinematic chains with multiple contact points,” Int. J. Robot. Res. 13 (1), 8292 (1994).CrossRefGoogle Scholar
18. Hurst, J. W. and Rizzi, A. A., “Series compliance for an efficient running gait,” IEEE Robot. Autom. Mag. 15 (3), 4251 (2008).CrossRefGoogle Scholar
19. Kaneko, K., Kanehiro, F., Morisawa, M., Akachi, K., Miyamori, G., Hayashi, A. and Kanehira, N., “Humanoid Robot HRP-4 - Humanoid Robotics Platform with Lightweight and Slim Body,” Proceedings of the IEEE/RSJ International Conference on Intelligent Robots Systems, San Francisco, CA (Sep. 2011) pp. 4400–4407.CrossRefGoogle Scholar
20. Karssen, J. G. D. and Wisse, M., “Running with improved disturbance rejection by using non-linear leg springs,” Int. J. Robot. Res. 30 (13), 15851595 (2011).CrossRefGoogle Scholar
21. Kim, J.-Y., Park, I.-W. and Oh, J.-H., “Experimental realization of dynamic walking of the biped humanoid robot KHR-2 using zero moment point feedback and inertial measurement,” Adv. Robot. 20 (6), 707736 (2006).CrossRefGoogle Scholar
22. McGeer, T., “Passive dynamic walking,” Int. J. Robot. Res. 9 (2), 6282 (1990).CrossRefGoogle Scholar
23. Moon, J.-S., “Stability Analysis and Control for Bipedal Locomotion using Energy Methods,” Ph.D. Dissertation (Urbana, IL: University of Illinois at Urbana-Champaign, 2011).Google Scholar
24. Moon, J.-S. and Spong, M. W., “Classification of periodic and chaotic passive limit cycles for a compass-gait biped with gait asymmetries,” Robotica 29 (7), 967974 (2011).CrossRefGoogle Scholar
25. Moon, J.-S., Stipanović, D. M. and Spong, M. W., “Gait generation and stabilization for nearly passive dynamic walking using auto-distributed impulses,” submitted, 2013.Google Scholar
26. Narukawa, T., Masaki, T. and Yoshida, K., “Biped Locomotion on Level Ground by Torso and Swing-Leg Control based on Passive-Dynamic Walking,” Proceedings IEEE/RSJ International Conference on Intelligent Robots Systems, Edmonton, Canada (Aug. 2005) pp. 4009–4014.CrossRefGoogle Scholar
27. Park, I.-W., Kim, J.-Y., Lee, J. and Oh, J.-H., “Mechanical design of the humanoid robot platform, HUBO,” Adv. Robot. 21 (11), 13051322 (2007).CrossRefGoogle Scholar
28. Poulakakis, I. and Grizzle, J. W., “The spring loaded inverted pendulum as the hybrid zero dynamics of an asymmetric hopper,” IEEE Trans. Autom. Control 54 (8), 17791793 (2009).CrossRefGoogle Scholar
29. Pratt, G. and Justin, M., “The DARPA robotics challenge,” IEEE Robot. Autom. Mag. 20 (2), 1012 (2013).CrossRefGoogle Scholar
30. Spong, M. W., Hutchinson, S. and Vidyasagar, M., Robot Modeling and Control (Wiley, Hoboken, NJ, 2006).Google Scholar
31. Sreenath, K., Park, H.-W., Poulakakis, I. and Grizzle, J. W., “A compliant hybrid zero dynamics controller for stable, efficient and fast bipedal walking on MABEL,” Int. J. Robot. Res. 30 (9), 11701193 (2011).CrossRefGoogle Scholar
32. Strogatz, S. H., Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering (Addison-Wesley Pub., Reading, MA, 1994).Google Scholar
33. van Oijen, T. P., Karssen, J. G. D. and Wisse, M., “The effect of center of mass offset on the disturbance rejection of running robots,” Int. J. Humanoid Robotics 10 (2), 122 (2013).CrossRefGoogle Scholar
34. Vukobratović, M. and Borovac, B., “Zero-moment point – thirty five years of its life,” Int. J. Humanoid Robot. 1, 157173 (2004).CrossRefGoogle Scholar
35. Vukobratović, M. and Juričić, D., “Contribution to the Synthesis of Biped Gait,” Proceedings IFAC Symposium Technical and Biological Problem on Control, Erevan, USSR (1968).CrossRefGoogle Scholar
36. Vukobratović, M. and Stepanenko, Y., “On the stability of anthropomorphic systems,” Math. Biosci. 15, 137 (1972).CrossRefGoogle Scholar
37. Yamakita, M. and Asano, F., “Extended passive velocity field control with variable velocity fields for a kneed biped,” Adv. Robot. 15 (2), 139168 (2001).CrossRefGoogle Scholar