Nonlinear Dynamics

, Volume 48, Issue 1–2, pp 77–89 | Cite as

Some flower-like gaits in the snakeboard’s locomotion

  • A. R. Asnafi
  • M. Mahzoon
Original Article


The snakeboard is a modified version of the skateboard in which the front and back pairs of wheels can pivot freely about a vertical axis (see Fig. 1). The rider can generate motion by coupling a turning of his/her feet which lie on wheel platforms with an appropriate twisting of his/her body without kicking off the ground.

The snakeboard was first presented in details by [Lewis et al., in Proceedings of the 1994 IEEE International Conference on Robotics and Automation, San Diego, CA, USA, May 1994, pp. 2391–2400]. In literature, it also has been studied as a prototype of the symmetrical nonholonomic locomotion systems. Geometrical modeling, finding the gaits, presenting the controllability ideas and designing desired trajectories are the subjects that can be found in the literature of the snakeboard.

In this paper, we present some symmetric sensitive flower-like gaits for the snakeboard by suitable tuning of the input parameters. The highly symmetric patterns generated by these gaits, besides their inherent beauty, sensitivity to parameter variations and coherency; exemplify the rich information content of the underlying nonlinear system.


Geometric mechanics Nonholonomic systems Robotic locomotion Snakeboard Symmetric patterns Underactuated systems 


affine connection

φψ, φφ

snakeboard’s phases of the shape variables


Lagrange multipliers

ψ, ϕ

snakeboard’s body coordinates


constraint one-forms

ω, ωψ, ωϕ

snakeboard’s frequencies of the shape variables

a, aψ, aϕ

snakeboard’s amplitudes of the shape variables


mecanical connection


constraint distribution


generalized forces


an element of the Lie group, SE(3)


the corresponding Lie algebra of g


the fiber


horizontal space of Q


locked inertia tensor

J, Jr, Jw

snakeboard’s body, rotor and wheel inertias

ko, ka

the ratio of frequencies and amplitudes of ϕ to ψ respectively




base manifold


metric tensor in base space


generalized momenta


generalized coordinates


configuration manifold


shape coordinates of the locomotive system


constrained fiber distribution


tangent space of Q


Vertical space of Q

x, y, θ

snakeboard’s center of mass position


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, School of EngineeringShiraz UniversityShirazIran

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