Nonlinear Dynamics

, Volume 95, Issue 3, pp 2307–2324 | Cite as

The wheeled three-link snake model: singularities in nonholonomic constraints and stick–slip hybrid dynamics induced by Coulomb friction

  • Tal Yona
  • Yizhar OrEmail author
Original Paper


The wheeled three-link snake model is a well-known example of an underactuated robotic system whose motion can be kinematically controlled by periodic changes of its internal shape, coupled with nonholonomic constraints. A known problem of this model is the existence of kinematic singularities at symmetric configurations where the three constraints become linearly dependent. Another critical assumption of this model is that the constraints of zero lateral slippage always hold, which requires large friction at the ground contact. This assumption breaks down when the inputs’ actuation frequency becomes too large, or when passing through singular configurations where the constraint forces grow unbounded. In this work, we extend the kinematic model by allowing for wheels slippage when the constraint forces reach an upper bound imposed by Coulomb friction. Using numerical simulations, we analyze the system’s hybrid dynamics governed by stick–slip transitions at the three wheels. We study the influence of actuation frequency on evolution of stick–slip periodic solutions which induce reversal in direction of net motion, and also show the existence of optimal frequencies that maximize the net displacement per cycle or mean translational speed. In addition, we show that passing through kinematic singularities is overcome by stick–slip transitions which keep the constraint forces and body velocity at finite bounded values. The analysis proves that in some cases, simple kinematic models of underactuated robotic locomotion should be augmented by the system’s hybrid dynamics which accounts for realistic frictional bounds on contact forces.


Under-actuated robots Nonholonomic systems Robotic locomotion Friction Stick-slip transitions Hybrid dynamics 


Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

Supplementary material

Supplementary material 1 (mp4 13496 KB)


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Faculty of Mechanical EngineeringTechnion - Israel Institute of TechnologyHaifaIsrael

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