Paper:

# Arc Feet Effects on Stability Based on a Simple Oscillator-Driven Walking Model

## Shinya Aoi^{*}, Yuuki Sato^{**}, and Kazuo Tsuchiya^{*}

^{*} Dept. of Aeronautics and Astronautics, Graduate School of Engineering, Kyoto University, Yoshida-honmachi, Sakyo-ku, Kyoto 606-8501, Japan

^{**} Dept. of Aeronautics and Astronautics, Graduate School of Engineering, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan

Lower-extremity movement in bipedal walking is characterized by a foot-rolling motion that includes heel-strike and toe-off. We investigated the dynamical influence of this movement on walking stability using a simple walking model that has a circular arc at the end of each leg. The leg is driven by a rhythmic signal from an internal oscillator to generate walking. We focused on stability characteristics due to the arc foot based on (1) the stability region for parameters such as mass distribution and walking speed, in which the circular arc radius is optimal when it is almost the same length as the leg to maximize the stable region and (2) the rate of convergence to stable walking, which is maximized by a circular arc radius of zero. These two conflicting results imply that the optimal radius of a circular arc for local stability is a trade-off between the two criteria, reflecting a dynamic feature of bipedal walking that should be considered in biped robot design.

*J. Robot. Mechatron.*, Vol.20, No.5, pp. 709-718, 2008.

- [1] P. G. Adamczyk, S. H. Collins, and A. D. Kuo, “The advantages of a rolling foot in human walking,” J. Exp. Biol., 209, pp. 3953-3963, 2006.
- [2] A. D. Hansen and D. S. Childress, “Effects of adding weight to the torso on roll-over characteristics of walking,” J. Rehabil. Res. Dev., 42(3), pp. 381-390, 2005.
- [3] A. D. Hansen, D. S. Childress, and E. H. Knox, “Roll-over shapes of human locomotor systems: effects of walking speed,” Clin. Biomech., 19, pp. 407-414, 2004.
- [4] R. M. Alexander, “Simple models of human motion,” Appl. Mech. Rev., 48, pp. 461-469, 1995.
- [5] R. M. Alexander, “Walking made simple,” Science, 308(5718), pp. 58-59, 2005.
- [6] A. D. Kuo, “Energetics of actively powered locomotion using the simplest walking model,” J. Biomech. Eng., 124, pp. 113-120, 2002.
- [7] T. McGeer, “Passive dynamic walking,” Int. J. Robot. Res., 9, pp. 68-82, 1990.
- [8] M. Coleman and A. Ruina, “An uncontrolled toy that can walk but cannot stand still,” Phys. Rev. Lett., 80(16), pp. 3658-3661, 1998.
- [9] S. H. Collins, M. Wisse, and A. Ruina, “A three-dimensional passive-dynamic walking robot with two legs and knees,” Int. J. Robot. Res., 20(7), pp. 607-615, 2001.
- [10] S. H. Collins, A. L. Ruina, R. Tedrake, and M. Wisse, “Efficient bipedal robots based on passive-dynamic walkers,” Science, 307, pp. 1082-1085, 2005.
- [11] F. Asano and Z. Luo, “On energy-efficient and high-speed dynamic biped locomotion with semicircular feet,” Proc. IEEE/RSJ Int. Conf. on Intell. Robots Syst., pp. 5901-5906, 2006.
- [12] R. Tedrake, T. W. Zhang, M. Fong, and H. S. Seung, “Actuating a simple 3D passive dynamic walker,” Proc. IEEE Int. Conf. on Robot. Autom., pp. 4656-4661, 2004.
- [13] M. Wisse, A. L. Schwab, R. Q. van der Linde, and F. C. T. van der Helm, “How to keep from falling forward: elementary swing leg action for passive dynamic walkers,” IEEE Trans. Robotics, 21(3), pp. 393-401, 2005.
- [14] S. Aoi and K. Tsuchiya, “Self-stability of a simple walking model driven by a rhythmic signal,” Nonlin. Dyn., 48(1-2), pp. 1-16, 2007.
- [15] M. Wisse, D.G.E. Hobbelen, R. J. J. Rotteveel, S. O. Anderson, and G.J. Zeglin, “Ankle springs instead of arc-shaped feet for passive dynamic walkers,” Proc. IEEE-RAS Int. Conf. on Humanoid Robots, pp. 110-116, 2006.
- [16] S. Aoi and K. Tsuchiya, “Locomotion control of a biped robot using nonlinear oscillators,” Auton. Robots, 19(3), pp. 219-232, 2005.
- [17] S. Aoi and K. Tsuchiya, “Bifurcation and chaos of a simple walking model driven by a rhythmic signal,” Int. J. Non-Linear Mech., 41(3), pp. 438-446, 2006.
- [18] S. Aoi and K. Tsuchiya, “Stability analysis of a simple walking model driven by an oscillator with a phase reset using sensory feedback,” IEEE Trans. Robotics, 22(2), pp. 391-397, 2006.
- [19] S. Aoi and K. Tsuchiya, “Adaptive behavior in turning of an oscillator-driven biped robot,” Auton. Robots, 23(1), pp. 37-57, 2007.
- [20] S. Mochon and T. A. McMahon, “Ballistic walking,” J. Biomech., 13(1), pp. 49-57, 1980.
- [21] J.W. Grizzle, G. Abba, and F. Plestan, “Asymptotically stable walking for biped robots: Analysis via systems with impulse effects,” IEEE Trans. Automat. Contr., 46(1), pp. 51-64, 2001.
- [22] M. Coleman, A. Chatterjee, and A. Ruina, “Motions of a rimless spoked wheel: A simple three-dimensional system with impacts,” Dyn. Stab. Syst., 12(3), pp. 139-160, 1997.
- [23] F. Asano and Z. Luo, “The effect of semicircular feet on energy dissipation by heel-strike in dynamic biped locomotion,” Proc. IEEE Int. Conf. on Robot. Autom., pp. 3976-3981, 2007.
- [24] M. Garcia, A. Chatterjee, A. Ruina, and M. Coleman, “The simplest walking model: Stability, complexity, and scaling,” ASME J. Biomech. Eng., 120(2), pp. 281-288, 1998.
- [25] M. Noguchi and K. Hirata, “Effect of round foot shape on stability of passive walking -From linearized Poincaré map viewpoint-,” Proc. 8th SICE System Integration Division Annual Conference, pp. 263-264, 2007. (in Japanese).
- [26] Y. Sugimoto and K. Osuka, “Stability analysis of passive-dynamicwalking focusing on the inner structure of poincaré map,” Proc. IEEE Int. Conf. on Advanced Robotics, pp. 236-241, 2005.

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