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Efficient walking with optimization for a planar biped walker with a torso by hip actuators and springs

Published online by Cambridge University Press:  27 August 2010

Terumasa Narukawa*
Affiliation:
Center for Education and Research of Symbiotic, Safe and Secure System Design, Keio University, 4-1-1 Hiyoshi, Kohoku-ku, Yokohama 223-8526, Japan
Masaki Takahashi
Affiliation:
Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
Kazuo Yoshida
Affiliation:
Department of System Design Engineering, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan
*
*Corresponding author. E-mail: narukawa@2003.jukuin.keio.ac.jp

Summary

This paper focuses on the use of passive dynamics to achieve efficient walking with simple mechanisms. A torso is added to a biped walker; and hip actuators, instead of ankle actuators, are used. A numerical approach is used to find the optimal control trajectories. A comparison between the cost functions of simple feedback control and optimal control is presented. Next, springs are added to the biped walking model at the hip joints to demonstrate the advantage of hip springs in terms of energy cost and ground conditions. The comparison between the torque costs with and without hip springs indicates that hip springs reduce the torque cost, particularly at a high walking speed.

Type
Articles
Copyright
Copyright © Cambridge University Press 2010

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References

1.Collins, S., Ruina, A., Tedrake, R. and Wisse, M., “Efficient bipedal robots based on passive-dynamic walkers,” Science 307, 10821085 (2005).CrossRefGoogle ScholarPubMed
2.McGeer, T., “Passive dynamic walking,” Int. J. Robot. Res. 9 (2), 6282 (1990).CrossRefGoogle Scholar
3.McMahon, T. A., “Mechanics of locomotion,” Int. J. Robot. Res. 3 (2), 428 (1984).CrossRefGoogle Scholar
4.Goswami, A., Espiau, B. and Keramane, A., “Limit Cycles and Their Stability in a Passive Bipedal Gait,” Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minnesota, USA (1996) pp. 246251.Google Scholar
5.Garcia, M., Chatterjee, A., Ruina, A. and Coleman, M., “The simplest walking model: Stability, complexity and scaling,” J. Biomech. Eng.-Trans. ASME 120 (2), 281288 (1998).CrossRefGoogle ScholarPubMed
6.McGeer, T., “Stability and control of two-dimensional biped walking,” Tech. Rep. CSS-IS TR 88-01. (Simon Fraser University Burnaby, BC, Canada, 1988).Google Scholar
7.Asano, F., Yamakita, M., Kamamichi, N. and Luo, Z.-W., “A novel gait generation for biped walking robots based on mechanical energy constraint,” IEEE Trans. Robot. Autom. 20 (3), 565573 (2004).CrossRefGoogle Scholar
8.Narukawa, T., Takahashi, M. and Yoshida, K., “Level-Ground Walk Based on Passive Dynamic Walking for a Biped Robot with Torso,” IEEE International Conference on Robotics and Automation, Roma, Italy (2007) pp. 32243229.Google Scholar
9.Narukawa, T., Takahashi, M. and Yoshida, K., “Numerical simulations of level-ground walking based on passive walk for planar biped robots with torso by hip actuators,” JSME J. Syst. Des. Dyn. 2 (2), 463474 (2008).Google Scholar
10.Chatterjee, A. and Garcia, M., “Small slope implies low speed for McGeer's passive walking machines,” Dyn. Stab. Syst. 15 (2), 139157 (2000).CrossRefGoogle Scholar
11.Gomes, M. and Ruina, A., “A Passive Dynamic Walking Model that Walks on Level Ground,” The 27th Annual Meeting For the American Society of Biomechanics, Ohio, USA (2003).Google Scholar
12.Gomes, M. and Ruina, A., “A walking model with no energy cost,” (2005) (in revision). Available online: http://ruina.tam.cornell.edu/.Google Scholar
13.Duindam, V. and Stramigioli, S., “Optimization of Mass and Stiffness Distribution for Efficient Bipedal Walking,” Proceedings of the International Symposium on Nonlinear Theory and Its Applications, Bruges, Belgium (2005).Google Scholar
14.Duindam, V., Port-Based Modeling and Control for Efficient Bipedal Walking Robots Ph.D. Thesis (Enschede, The Netherlands: The University of Twente, 2006).Google Scholar
15.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
16.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
17.Garcia, M., Stability, Scaling, and Chaos in Passive-Dynamic Gait Models Ph.D. Thesis (Cornell University, 1999).Google Scholar
18.Anderson, F. C. and Pandy, M. G., “Dynamic optimization of human walking,” J. Biomech. Eng. 123, 381390 (2001).CrossRefGoogle ScholarPubMed
19.Chevallereau, C. and Aoustin, Y., “Optimal reference trajectories for walking and running of a biped robot,” Robotica 19, 557569 (2001).CrossRefGoogle Scholar
20.Chow, C. K. and Jacobson, D. H., “Studies of human locomotion via optimal programming,” Math. Biosci. 10, 239306 (1971).CrossRefGoogle Scholar
21.Hardt, M., Kreutz-Delgado, K. and Helton, J. W., “Optimal Biped Walking with a Complete Dynamical Model,” Proceedings of the 38th IEEE Conference on Decision and Control, Arizona, USA (1999) pp. 29993004.Google Scholar
22.Srinivasan, M. and Ruina, A., “Computer optimization of a minimal biped model discovers walking and running,” Nature 439, 7275 (2006).CrossRefGoogle ScholarPubMed
23.Ono, K. and Liu, R. Q., “Optimal biped walking locomotion solved by trajectory planning method,” J. Dyn. Syst. Meas. Control-Trans. ASME 124 (4), 554565 (2002).CrossRefGoogle Scholar
24.Roussel, L., Canudas-De-Wit, C. and Goswami, A., “Generation of Energy Optimal Complete Gait Cycles for Biped Robots,” Proceedings of the 1998 IEEE International Conference on Robotics and Automation, Leuven, Belgium (1998) pp. 20362041.Google Scholar
25.Betts, J. T., “Survey of numerical methods for trajectory optimization,” J. Guid. Control Dyn. 21 (2), 193207 (1998).CrossRefGoogle Scholar
26.Hardt, M. and von Stryk, O., “Dynamic modeling in the simulation, optimization, and control of bipedal and quadrupedal robots,” Z. Angew. Math. Mech. 83 (10), 648662 (2003).CrossRefGoogle Scholar
27.Mombaur, K. D., Bock, H. G., Schloder, J. P. and Longman, R. W., “Open-loop stable solutions of periodic optimal control problems in robotics,” Zamm-Z. Angew. Math. Mech. 85, 499515 (2005).CrossRefGoogle Scholar
28.Mombaur, K. D., Bock, H. G., Schloder, J. P. and Longman, R. W., “Self-stabilizing somersaults,” IEEE Trans. Robot. 21 (6), 11481157 (2005).CrossRefGoogle Scholar
29.Tomlab Optimization Inc., PROPT - Matlab Optimal Control Software. Available online: http://tomdyn.com/.Google Scholar
30.The MathWorks, MATLAB and Simulink. Available online: http://www.mathworks.com/.Google Scholar
31.Gill, P. E., Murray, W. and Saunders, M. A., “SNOPT: An SQP algorithm for large-scale constrained optimization,” SIAM J. Optim. 12 (4), 9791006 (2002).CrossRefGoogle Scholar