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Some flower-like gaits in the snakeboard’s locomotion

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Abstract

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.

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Abbreviations

∇:

affine connection

φψ, φφ :

snakeboard’s phases of the shape variables

λi :

Lagrange multipliers

ψ, ϕ:

snakeboard’s body coordinates

ωj i :

constraint one-forms

ω, ωψ, ωϕ :

snakeboard’s frequencies of the shape variables

a, aψ, aϕ :

snakeboard’s amplitudes of the shape variables

A :

mecanical connection

D :

constraint distribution

F i :

generalized forces

g :

an element of the Lie group, SE(3)

g :

the corresponding Lie algebra of g

G :

the fiber

HQ :

horizontal space of Q

I −1 :

locked inertia tensor

J, J r, J w :

snakeboard’s body, rotor and wheel inertias

k o, k a :

the ratio of frequencies and amplitudes of ϕ to ψ respectively

L :

Lagrangian

M :

base manifold

M ij :

metric tensor in base space

p a :

generalized momenta

q i :

generalized coordinates

Q :

configuration manifold

r :

shape coordinates of the locomotive system

S :

constrained fiber distribution

TQ :

tangent space of Q

VQ :

Vertical space of Q

x, y, θ:

snakeboard’s center of mass position

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Authors and Affiliations

  1. Department of Mechanical Engineering, School of Engineering, Shiraz University, Shiraz, 71348–13668, Iran

    A. R. Asnafi & M. Mahzoon

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  1. A. R. Asnafi
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  2. M. Mahzoon
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Correspondence to A. R. Asnafi or M. Mahzoon.

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Asnafi, A.R., Mahzoon, M. Some flower-like gaits in the snakeboard’s locomotion. Nonlinear Dyn 48, 77–89 (2007). https://doi.org/10.1007/s11071-006-9053-9

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