Sinusoidal control and limit cycle analysis of the dissipative Chaplygin sleigh

Original Paper


In this paper, we investigate the behavior of an underactuated mixed-dynamic nonholonomic system, a Chaplygin sleigh, subjected to viscous dissipation and sinusoidal forcing. The viscous dissipation is in the allowable directions of motion and preserves the nonholonomic constraint. The inclusion of such dissipative effects produces limit cycle oscillations in a reduced velocity space. We find analytical approximations to such limit cycles and use these to determine sinusoidal inputs to control the speed of the sleigh. We further show small changes to the sinusoidal input can steer the sleigh to any desired direction. Invariant structures like limit cycles can be expected to be seen in the dynamics of other nonholonomic systems when the effects of viscous dissipation are included. The findings we report here are therefore applicable to a broad class of both terrestrial and aquatic locomotion systems with nonholonomic constraints.


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This paper is based upon work supported by the National Science Foundation under Grant Number CMMI 1563315.

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Conflict of interest

The authors have no conflict of interest to report.


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringClemson UniversityClemsonUSA

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