Analysis of 3D Passive Walking Including Turning Motions for the Finite-width Rimless Wheel

Document Type: Research Paper


1 Research assistant in control laboratory

2 Associate professor, School of Mechanical Engineering, University of Tehran, Iran

3 PhD student, School of Mechanical Engineering, University of Tehran, Iran.


The focus of studies in the field of passive walking has often been on straight walking, while less attention has been paid to the field of turning motions. In this paper, the passive motions of a finite width rimless wheel as the simplest 3D model of passive biped walkers was investigated with a focus on turning motions. For this purpose, the hybrid model of the system consisting of continuous and discontinuous phases of motion was derived with respect to a vertical fixed frame that was independent of the surface profile. A Poincaré map corresponding to a step is one of the common methods used for the determination of periodic motions (limit cycles) and their specifications. In this study, it was emphasized that the Poincaré map has only one fixed point, indicating only one stable periodic motion that is parallel to the steepest slope surface. It is also shown that if the wheel is released from an orientation other than the steepest slope, the wheel turns towards the slope surface and eventually, its motion continues on the only existing stable limit cycle (passive limited turning). The effect of variation among some parameters of the initial conditions on rotational behaviour and its convergence were investigated.


[1]. McGeer, T., 1990, Passive dynamic walking,
The International Journal of Robotics Reseach,
9(2), pp. 62-82.
[2]. Coleman, M.J. 1998, A stability study of a
three-dimensional passive dynamic model of
human gait, Cornell University, PhD thesis.
[3]. Coleman, M.J., Chatterjee, A. and Ruina, A.,
1997, Motions of a rimless spoked wheel: a
simple three-dimensional system with impacts,
Dynamics and Stability of Systems, 12(3), pp.
[4]. Smith, A. C. and Berkemeier, M. D., 1998, The
motion of a finite-width rimless wheel in 3D, in
Robotics and Automation Proceedings IEEE
International Conference, Leuven, Belgium.
[5]. Sabaapour, M.R., Hairi-Yazdi, M.R. and
Beigzadeh, B., 2014, Towards passive turning in
biped walkers, Procedia Technology, 12, pp. 98-
[6]. Sabaapour, M. R., Hairi-Yazdi, M. R., and
Beigzadeh, B., 2014, Passive turning motion of
3D rimless wheel: novel periodic gaits for
bipedal curved walking, Submitted to Advanced
[7]. Spong, M. W., and Bullo, F., 2002, Controlled
symmetries and passive walking, in in Proc.
15th Triennial World Congress..
[8]. Goswami, A., Espiau, B., and Keramane, A.,
1997, Limit cycles in a passive compass gait
biped and passivity-mimicking control laws,
Autonomous Robots, 4(3), pp. 273-286.
[9]. Shih, C., Grizzle, J.W. and Chevallereau, C.,
2009, Asymptotically stable walking and
steering of a 3D bipedal robot with passive point
feet, IEEE Transactions on Robotics.