Research Papers

Passive Dynamic Biped Walking—Part I: Development and Validation of an Advanced Model

[+] Author and Article Information
Christine Q. Wu

Department of Mechanical Engineering,
University of Manitoba,
Winnipeg, MB R3T 5V6, Canada

Angle between the legs at the instance of heel strike.

Software developed by Quanser.

The sliding velocity is less than the Stribeck velocity.

Does not switch between impact and motion equations like the traditional impact-based passive walking model.

Contributed by the Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received December 11, 2012; final manuscript received February 28, 2013; published online April 19, 2013. Assoc. Editor: Parviz Nikravesh.

J. Comput. Nonlinear Dynam 8(4), 041007 (Apr 19, 2013) (10 pages) Paper No: CND-12-1219; doi: 10.1115/1.4023934 History: Received December 11, 2012; Revised February 28, 2013

Passive dynamic walking is a manner of walking developed, partially or in whole, by the energy provided by gravity. Studying passive dynamic walking provides insight into human walking and is an invaluable tool for designing energy-efficient biped robots. The objective of this research was to develop a continuous mathematical model of passive dynamic walking, in which the Hunt–Crossley contact model, and the LuGre friction model were used to represent the normal and tangential ground reactions continuously. A physical passive walker was built to validate the proposed mathematical model. A traditional impact-based passive walking model was also used as a reference to demonstrate the advancement of the proposed passive dynamic walking model. The simulated gait of the proposed model matched the gait of the physical passive walker exceptionally well, both in trend and magnitude.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.


Mochon, S., and McMahon, T.1980, “Ballistic Walking: An Improved Model,” Math. Biosci., 52(3), pp. 241–260. [CrossRef]
McGeer, T., 1990, “Passive Dynamic Walking,” Int. J. Robot. Res., 9(2), pp. 62–82. [CrossRef]
McGeer, T., 1990, “Passive Walking With Knees,” Proc. of the 1990 IEEE International Conference on Robotics and Automation, Cincinnati, OH, pp. 1640–1645.
Goswami, A., Espiau, B., and Keramane, A., 1997, “Limit Cycles in a Passive Compass Gait Biped and Passivity-Mimicking Control Laws,” Auton. Rob., 4(3), pp. 273–286. [CrossRef]
Goswami, A., Thuilot, B., and Espiau, B., 1998, “A Study of the Passive Gait of a Compass Like Biped Robot: Symmetry and Chaos,” Int. J. Robo. Res., 17(12), pp. 1282–1301. [CrossRef]
Garcia, M., Chatterjee, A., Ruina, A., and Coleman, M., 1998, “The Simplest Walking Model: Stability, Complexity, and Scaling,” ASME J. Biomech. Eng., 120(2), pp. 281–288. [CrossRef]
Ikemata, Y., Yasuhara, K., Sano, A., and Fujimoto, H., 2008, “A Study of the Leg–Swing Motion of Passive Walking,” Proc. of the 2008 IEEE International Conference on Robotics and Automation, Pasadena, CA, pp. 1588–1593.
Borzova, E., and Hurmuzlu, Y., 2004, “Passively Walking Five-Link Robot,” Automatica, 40(4), pp. 621–629. [CrossRef]
Hurmuzlu, Y., and Chang, T.-H., 1992, “Rigid Body Collisions of a Special Class of Planar Kinematic Chains,” IEEE Trans. Syst. Man Cybern., 2(5), pp. 964–971. [CrossRef]
Mu, X., and Wu, Q., 2006, “On Impact Dynamics and Contact Events for Biped Robots Via Impact Effects,” IEEE Trans. Syst. Man Cybern., Part B: Cybern., 36(6), pp. 1364–1372. [CrossRef]
Wu, Q., Sekhavat, P., Sepehri, N., and Peles, S., 2005, “On Design of Continuous Lyapunov's Feedback Control,” J. Franklin Inst., 342(6), pp. 702–723. [CrossRef]
Gilardi, G., and Sharf, I., 2002, “Literature Survey of Contact Dynamics Modeling,” Mech. Mach. Theory, 37(10), pp. 1213–1239. [CrossRef]
Olsson, H., Åström, K., Canudas de Wit, C., Gäfvert, M., and Lischinsky, P., 1998, “Friction Models and Friction Compensation,” Eur. J. Control, 4(3), pp. 176–195. Available at: http://www.gipsa-lab.grenoble-inp.fr/~carlos.canudas-de-wit/publications/friction/dynamic_friction_EJC_98.pdf
Boos, M., and McPhee, J., 2012, “Volumetric Modeling and Experimental Validation of Normal Contact Dynamic Forces,” ASME J. Comput. Nonlinear Dyn., 8(2), p. 021006. [CrossRef]
Qi, F., Wang, T., and Li, J., 2010, “The Elastic Contact Influences on Passive Walking Gaits,” Robotica, 29, pp. 787–796. [CrossRef]
Jafarian, M., 2010, “Variable Stiffness for Robust and Energy Efficient 2D Bipedal Locomotion,” M.Sc. thesis, University of Twente, Enschede, Netherlands.
Piiroinen, P., Dankowicz, H., and Nordmark, A., 2003, “Breaking Symmetries and Constraints: Transitions From 2D to 3D in Passive Walkers,” Multibody Syst. Dyn., 10, pp. 147–176. [CrossRef]
David, A., and Bruneau, O., 2011, “Bipedal Walking Gait Generation Based on the Sequential Method of Analytical Potential (SMAP),” Multibody Syst. Dyn., 26, pp. 367–395. [CrossRef]
Font-Llagunes, J., Barjau, A., Pàmies-Vilà, R., and Kövecses, J., 2012, “Dynamic Analysis of Impact in Swing-Through Crutch Gait Using Impulsive and Continuous Contact Models,” Multibody Syst. Dyn., 28, pp. 257–282. [CrossRef]
Machado, M., Moreira, P., Flores, P., and Lankarani, H., 2012, “Compliant Contact Force Models in Multibody Dynamics: Evolution of the Hertz Contact Theory,” Mech. Mach. Theory, 53, pp. 99–121. [CrossRef]
Wu, Q., and Sabet, N., 2004, “An Experimental Study of Passive Dynamic Walking,” Robotica, 22, pp. 251–262. [CrossRef]
Wu, Q., and Chen, J., 2010, “Effects of Ramp Angle and Mass Distribution on Passive Dynamic Gait– An Experimental Study,” Int. J. Human. Robot., 7(1), pp. 55–72. [CrossRef]
Kazi, R., Koop, D., and Wu, C. Q., 2011, “Experimental Study on Passive Dynamic Bipedal Walking: Comparing Test Platforms and Effects of Parameter Changes on Gait Patterns,” Proc. of the ASME 4th Annual Dynamic Systems and Control Conference, Arlington, VA.
Ning, L., Junfeng, L., and Tianshu, W., 2009, “The Effects of Parameter Variation on the Gaits of Passive Walking Models: Simulations and Experiments,” Robotica, 27(4), pp. 511–528. [CrossRef]
Hunt, K., and Crossley, F., 1975, “Coefficient of Restitution Interpreted as Damping in Vibroimpact,” J. Appl. Mech., 42(2), pp. 440–445. [CrossRef]
Hertz, H., 2006, “On the Contact of Elastic Solids,” Miscellaneous Papers, MacMillan, London, Chap. 5, pp. 146–162.
Canudas de Wit, C., Olsson, H., Åström, K., and Lischinsky, P., 1995, “A New Model for Control Systems With Friction,” IEEE Trans. Autom. Control, 40(3), pp. 419–425. [CrossRef]
Flores, P., and Ambrósio, J., 2010, “On the Contact Detection for Contact-Impact Analysis in Multibody Systems,” Multibody Syst. Dyn., 25(1), pp. 103–122. [CrossRef]
Shampine, L., and Reichelt, M., 1997, The Matlab ODE Suite, MathWorks, Inc., Natick, MA.
Koop, D., 2010, “Passive Dynamic Bipedal Walking: Ramp-Treadmill Comparison and Gait Variation Due to Parameter Change,” B.Sc. thesis, University of Manitoba, Winnipeg, MB.
Olsson, H., 1996, “Control Systems With Friction,” Ph.D. thesis, Lund Institute of Technology, Lund, Sweden.
Lankarani, H., and Nikravesh, P., 1994, “Continuous Contact Force Models for Impact Analysis in Multibody Systems,” Nonlinear Dyn., 5(2), pp. 193–207. Available at : http://link.springer.com/article/10.1007%2FBF00045676?LI=true#.


Grahic Jump Location
Fig. 2

Schematics of interpenetration of two bodies

Grahic Jump Location
Fig. 3

Visualization of the LuGre model

Grahic Jump Location
Fig. 4

Transition to double support phase when Leg 1 is the stance leg

Grahic Jump Location
Fig. 5

Transition to single support phase to single support on Leg 2

Grahic Jump Location
Fig. 9

Example of data from HM2L

Grahic Jump Location
Fig. 6

Determining the interpenetration value for Leg i

Grahic Jump Location
Fig. 7

Picture of HM2L with features labeled

Grahic Jump Location
Fig. 8

Photos of HM2L on the ramp

Grahic Jump Location
Fig. 10

Leg angle phase portrait. (a) Proposed model. (b) Impact-based model.

Grahic Jump Location
Fig. 11

Normal and friction force versus time

Grahic Jump Location
Fig. 12

Friction state observer

Grahic Jump Location
Fig. 13

Comparison between experimental walker, proposed model, and impact-based model. (a) Step length. (b) Step period. (c) Average hip velocity.

Grahic Jump Location
Fig. 14

Comparison of the inner leg angle (α) of the simulation and experimental gait



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In