Abstract
Currently, passive robots are designed following a trial and error process in which the existence of a stable walking cycle for a given passive robot’s model is analyzed using Poincaré maps. The standard stability analysis procedure suffers from discretization aliasing, and it is not able to deal with complex passive models. In this paper a methodology that allows finding conditions on the robot’s parameters of a given passive model in order to obtain a stable walking cycle is proposed. The proposed methodology overcomes the aliasing problem that arises when Poincaré sections are discretized. Basically, it implements a search process that allows finding stable subspaces in the parameters’ space (i.e., regions with parameters’ combinations that produce stable walking cycles), by simulating the robot dynamics for different parameters’ combinations. After initial conditions are randomly selected, the robot’s dynamics is modeled step by step, and in the Poincaré section the existence of a walking cycle is verified. The methodology includes the definition of a search algorithm for exploring the parameters’ space, a method for the partition of the space in hypercubes and their efficient management using proper data structures, and the use of so-called design value functions that quantify the feasibility of the resulting parameters. Among the main characteristics of the proposed methodology are being robot independent (it can be used with any passive robot model, regardless of its complexity), and robust (stable subspaces incorporate a stability margin value that deals with differences between the robot’s model and its physical realization). The methodology is validated in the design process of a complex semi-passive robot that includes trunk, knees, and non-punctual feet. The robot also considers the use of actuators, controllers and batteries for its actuation.
Similar content being viewed by others
References
Coleman, M.J.: A stability study of a three-dimensional passive-dynamic model of human gait. Ph.D. Thesis, Cornell University (1998)
Collins, S.H., Wisse, M., Ruina, A.: A three-dimensional passive-dynamic walking robot with two legs and knees. Int. J. Rob. Res. 20(7), 607–615 (2001)
Collins, S., Ruina, A., Tedrake, R., Wisse, M.: Efficient bipedal robots based on passive-dynamic walkers. Science 307(5712), 1082–1085 (2005)
Garcia, M.: Stability, scaling, and chaos in passive-dynamic gait models. Ph.d. Thesis, Sibley School of Mechanical and Aerospace Engineering, Cornell University (1999)
Goswami, A., Espiau, B., Keramane, A.: Limit cycles and their stability in a passive bipedal gait. In: Proceedings of the IEEE 1996 International Conference on Robotics and Automation, vol. 1, pp. 246–251. Minneapolis, USA (1996)
Goswami, A., Goswami, A., Thuilot, B., Thuilot, B., Espiau, B., Espiau, B.: Compass-like Biped Robot—Part i: Stability and bifurcation of passive gaits. Tech. Rep. 2996, INRIA: Institut National de Recherche en Informatique et en Automatique (1996)
Grizzle, J., Plestan, F., Abba, G.: Poincare’s method for systems with impulse effects: application to mechanical biped locomotion. In: Proceedings of the 38th IEEE Conference on Decision and Control, vol. 4, pp. 3869–3876 (1999)
Grizzle, J., 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)
Grizzle, J., Abba, G., Plestan, F.: Correction to “asymptotically stable walking for biped robots: analysis via systems with impulse effects”. IEEE Trans. Automat. Contr. 46(3), 513–513 (2001)
Haruna, M., Ogino, M., Hosoda, K., Asada, M.: Yet another humanoid walking—passive dynamic walking with torso under simple control. In: Proceedings of the 2001 IEEE International Conference on Intelligent Robots and Systems, vol. 1, pp. 259–264 (2001)
Hsu Chen, V.: Passive dynamic walking with knees: a point foot model. Master Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (2007)
Kajita, S., Espiau, B.: Springer Handbook of Robotics, 1st edn., chap. 16, pp. 361–389. Springer (2008)
Kuo, A.D.: Stabilization of lateral motion in passive dynamic walking. Int. J. Rob. Res. 18(9), 917–930 (1999)
McGeer, T.: Passive dynamic walking. Tech. Rep. CSS-ISS TR 88-02, Simon Fraser University, Burnaby, British Columbia, Canada (1988)
McGeer, T.: Passive bipedal running. In: Proceedings of the Royal Society of London, Series B, Biological Sciences, vol. 240, pp. 107–134 (1990)
McGeer, T.: Passive dynamic walking. Int. J. Rob. Res. 9(2), 62–82 (1990)
McGeer, T.: Passive walking with knees. In: Proceedings of the 1990 IEEE International Conference on Robotics and Automation, vol. 3, pp. 1640–1645 (1990)
McGeer, T.: Principles of Walking and Running. Advances in Comparative and Environmental Physiology, vol. 11, chap. 4. Springer-Verlag (1992)
McGeer, T.: Dynamics and control of bipedal locomotion. J. Theor. Biol. 163, 277–314 (1993)
McGeer, T.: Passive dynamic biped catalogue, 1991. In: Proceedings of the 2nd International Symposium on Experimental Robotics II, pp. 465–490. Springer-Verlag, London, UK (1993)
Morimoto, J., Atkeson, C.: Learning biped locomotion. IEEE Robot. Autom. Mag. 14(2), 41–51 (2007)
Morris, B., Grizzle, J.: A restricted Poincaré map for determining exponentially stable periodic orbits in systems with impulse effects: application to bipedal robots. In: 44th IEEE Conference on Decision and Control and 2005 European Control Conference. CDC-ECC ’05, pp. 4199–4206 (2005)
Morris, B., Grizzle, J.: Hybrid invariance in bipedal robots with series compliant actuators. In: 45th IEEE Conference on Decision and Control, pp. 4793–4800 (2006)
Morris, B., Grizzle, J.: Hybrid invariant manifolds in systems with impulse effects with application to periodic locomotion in bipedal robots. IEEE Trans. Automat. Contr. 54(8), 1751–1764 (2009)
Ohta, H., Yamakita, M., Furuta, K.: From passive to active dynamic walking. In: Proceedings of the 38th IEEE Conference on Decision and Control, vol. 4, pp. 3883–3885 (1999)
Paul, C., Yokoi, H., Matsushita, K.: Design and control of humanoid robot locomotion with passive legs and upper body actuation. In: Proceedings of the International Symposium on Robotics. Paris, France (2003)
Pratt, J.: Exploiting inherent robustness and natural dynamics in the control of bipedal walking robots. Ph.d. Thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (2000)
Tedrake, R.: Actuating a simple 3d passive dynamic walker. In: Proceedings of the IEEE International Conference on Robotics and Automation, pp. 4656–4661 (2004)
Westervelt, E., Grizzle, J., Koditschek, D.: Hybrid zero dynamics of planar biped walkers. IEEE Trans. Automat. Contr. 48(1), 42–56 (2003)
Westervelt, E.R., Grizzle, J.W., Chevallereau, C., Choi, J.H., Morris, B.: Feedback Control of Dynamic Bipedal Robot Locomotion. CRC Press, Boca Raton, FL (2007)
Wisse, M.: Essentials of dynamic walking—analysis and design of two-legged robots. Ph.d. Thesis, Faculty of Mechanical Engineering, Technische Universiteit Delft (2004)
Wisse, M.: Three additions to passive dynamic walking; actuation, an upper body, and 3d stability. In: Proceedings of the 2004 IEEE International Conference on Humanoid Robots, vol. 1, pp. 113–132 (2004)
Wisse, M., Feliksdal, G., Van Frankkenhuyzen, J., Moyer, B.: Passive-based walking robot. IEEE Robot. Autom. Mag. 14(2), 52–62 (2007)
Wisse, M., Hobbelen, D., Rotteveel, R., Anderson, S., Zeglin, G.: Ankle springs instead of arc-shaped feet for passive dynamic walkers. In: Proceedings of the 2006 IEEE International Conference on Humanoid Robots, pp. 110–116 (2006)
Yamasaki, F., Hosoda, K., Asada, M.: An energy consumption based control for humanoid walking. In: Proceedings of the 2002 IEEE International Conference on Intelligent Robots and Systems, vol. 3, pp. 2473–2477 (2002)
Yamasaki, F., Endo, K., Asada, M., Kitano, H.: A control method for humanoid biped walking with limited torque. In: RoboCup 2001: Robot Soccer World Cup V, pp. 60–70. Springer-Verlag, London, UK (2002)
Yamasaki, F., Endo, K., Asada, M., Kitano, H.: An energy-efficient walking for a low-cost humanoid robot pino. AI Mag. 23(1), 60–61 (2002)
Yamasaki, F., Endo, K., Kitano, H., Asada, M.: Acquisition of humanoid walking motion using genetic algorithm-considering characteristics of servo modules. In: Proceedings of the 2002 IEEE International Conference on Robotics and Automation, vol. 3, pp. 3123–3128 (2002)
Zhao, M., Zhang, J., Dong, H., Liu, Y., Li, L., Su, X.: Humanoid robot gait generation based on limit cycle stability. In: RoboCup 2008: Robot Soccer World Cup XII, pp. 403–413. Springer-Verlag, Berlin, Heidelberg (2009)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Vallejos, P., Ruiz-del-Solar, J. & Swett, F. A New Methodology for the Design of Passive Biped Robots: Determining Conditions on the Robot’s Parameters for the Existence of Stable Walking Cycles. J Intell Robot Syst 63, 503–523 (2011). https://doi.org/10.1007/s10846-010-9524-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10846-010-9524-6