Research Papers

Passive Dynamic Biped Walking—Part II: Stability Analysis of the Passive Dynamic Gait

[+] Author and Article Information
Christine Q. Wu

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

Difference between the next estimate and current estimate.

Normalized with the leg length, total mass, and average hip velocity.

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 March 1, 2013; published online March 26, 2013. Assoc. Editor: Parviz Nikravesh.

J. Comput. Nonlinear Dynam 8(4), 041008 (Mar 26, 2013) (10 pages) Paper No: CND-12-1220; doi: 10.1115/1.4023933 History: Received December 11, 2012; Revised March 01, 2013

Passive dynamic walking is an excellent tool for evaluating biped stability measures, due to its simplicity, but an understanding of the stability, in the classical definition, is required. The focus of this paper is on analyzing the stability of the passive dynamic gait. The stability of the passive walking model, validated in Part I, was analyzed with Lyapunov exponents, and the geometry of the basin of attraction was determined. A novel method was created to determine the 2D projection of the basin of attraction of the model. Using the insights gained from the stability analysis, the relation between the angular momentum and the stability of gait was examined. The angular momentum of the passive walker was not found to correlate to the stability of the gait.

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


Vukobratovic, M., and Borovac, B., 2004, “Zero-Moment Point – Thirty-Five Years of Its Life,” Int. J. Human. Robot., 1(1), pp. 157–173. [CrossRef]
Goswami, A., 1999, “Postural Stability of Biped Robots and the Foot-Rotation Indicator (FRI) Point,” Int. J. Robot. Res., 18(6), pp. 523–533. [CrossRef]
Herr, H., and Popovic, M., 2008, “Angular Momentum in Human Walking,” J. Exp. Biol., 211, pp. 467–481. [CrossRef] [PubMed]
Silverman, A., Wilken, J., Sinitski, E., and Neptune, R., 2012, “Whole-Body Angular Momentum in Incline and Decline Walking,” J. Biomech., 45, pp. 965–971. [CrossRef] [PubMed]
Popovic, M., Hofmann, A., and Herr, H., 2004, “Angular Momentum Regulation During Human Walking: Biomechanics and Control,” Proc. of the 2004 IEEE International Conference on Robotics and Automation, New Orleans, LA, Vol. 3, pp. 2405–2411.
Goswami, A., and Kallem, V., 2004, “Rate of Change of Angular Momentum and Balance Maintenance of Biped Robots,” Proc. of the 2004 IEEE International Conference on Robotics and Automation, New Orleans, LA, pp. 3785–3790.
Popovic, M., Hofmann, A., and Herr, H., 2004, “Zero Spin Angular Momentum Control: Definition and Applicability,” Proc. of the 2004 IEEE RAS/RSJ International Conference on Humanoid Robots, Los Angeles, CA, Vol. 1, pp. 478–493.
Popovic, M., Goswami, A., and Herr, H., 2005, “Ground Reference Points in Legged Locomotion: Definitions, Biological Trajectories and Control Implications,” Int. J. Robot. Res., 24(12), pp. 1013–1032. [CrossRef]
Wight, D., Kubica, E., and Wang, D., 2008, “Introduction of the Foot Placement Estimator: A Dynamic Measure of Balance for Bipedal Robotics,” ASME J. Comput. Nonlinear Dyn., 3(1), p. 011009. [CrossRef]
Millard, M., McPhee, J., and Kubica, E., 2012, “Foot Placement and Balance in 3D,” ASME J. Comput. Nonlinear Dyn., 7(2), p. 021015. [CrossRef]
Schwab, A., and Wisse, M., 2001, “Basin of Attraction of the Simplest Walking Model,” Proc. of the ASME Design Engineering Technical Conferences , pp. 531–539.
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]
Zhang, P., Tian, Y., and Liu, Z., 2009, “Bisection Method for Evaluation of Attraction Region of Passive Dynamic Walking,” Proc. of the 4th International Conference on Autonomous Robots and Agents, Wellington, New Zealand, pp. 692–697.
Wolf, A., Swift, J., Swinney, H., and Vastano, J., 1985, “Determining Lyapunov Exponents From a Time Series,” Phys. D, 16, pp. 285–317. [CrossRef]
Sobczyk, M., Perondi, E., and Cunha, M. A., 2010, “A Continuous Approximation of the LuGre Friction Model,” ABCM Symposium Series in Mechatronics, Vol. 4, ABCM, Ouro Preto, pp. 218–228.
Jarvis, R., 1973, “On the Identification of the Convex Hull of a Finite Set of Points in the Plane,” Inf. Process. Lett., 2, pp. 18–21. [CrossRef]
Dempster, W., 1955, Space Requirements of the Seated Operator: Geometrical, Kinematic, and Mechanical Aspects of the Body, With Special Reference to the Limbs, WADC, Ohio, pp. 55–159.
Collins, S., Wisse, M., and Ruina, A., 2001, “A Three Dimensional Passive-Dynamic Walking Robot With Two Legs and Knee,” Int. J. Robot. Res., 20(7), pp. 607–615. [CrossRef]
Borzova, E., and Hurmuzlu, Y., 2004, “Passively Walking Five-Link Robot,” Automatica, 40(4), pp. 621–629. [CrossRef]


Grahic Jump Location
Fig. 2

Evolution of an initial (hyper)sphere of initial conditions

Grahic Jump Location
Fig. 3

Effect of kv on Sv

Grahic Jump Location
Fig. 4

Points of interest (POI) labeled on the θ-θ· phase portrait

Grahic Jump Location
Fig. 5

Offset correction visualization

Grahic Jump Location
Fig. 6

Finding the first point on the BoA edge

Grahic Jump Location
Fig. 7

Finding the first edge on the BoA

Grahic Jump Location
Fig. 8

Without the reverse protocol

Grahic Jump Location
Fig. 9

Relative error between the current Lyapunov exponent λ(t) to the final Lyapunov exponent λ

Grahic Jump Location
Fig. 10

BoA edge algorithm validation case: b/l = 0.35, POI 1

Grahic Jump Location
Fig. 11

BoA for point of interest one

Grahic Jump Location
Fig. 12

Visualization of the swing leg BoA abnormality

Grahic Jump Location
Fig. 13

Area of the BoA swing leg

Grahic Jump Location
Fig. 14

Area of the BoA stance leg

Grahic Jump Location
Fig. 15

Torque and angular momentum versus the center of mass

Grahic Jump Location
Fig. 16

Normalized torque and angular momentum versus the center of mass

Grahic Jump Location
Fig. 17

Angular momentum of the walker about the center of mass over one step

Grahic Jump Location
Fig. 18

Ground reaction torque over one step

Grahic Jump Location
Fig. 19

BoA for point of interest one

Grahic Jump Location
Fig. 20

BoA for point of interest two

Grahic Jump Location
Fig. 21

BoA for point of interest three

Grahic Jump Location
Fig. 22

BoA for point of interest four

Grahic Jump Location
Fig. 23

BoA for point of interest five

Grahic Jump Location
Fig. 24

BoA for point of interest six

Grahic Jump Location
Fig. 25

BoA for point of interest seven

Grahic Jump Location
Fig. 26

BoA for point of interest eight




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