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Published in:
2018 | OriginalPaper | Chapter
Complete Nonholonomy of the Rolling Ellipsoid - A Constructive Proof
Authors : F. Rüppel, F. Silva Leite, R. C. Rodrigues
Published in: Modeling, Dynamics, Optimization and Bioeconomics III
Publisher: Springer International Publishing
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Abstract
We present a constructive proof of the complete nonholonomy of the rolling ellipsoid. The rolling motions are assumed to be over the affine tangent space at a point of the n-ellipsoid and both manifolds are considered embedded in \(\mathbb {R}^{n+1}\), equipped with a metric that results from a convenient deformation of the Euclidean metric. The deformation is defined through a positive definite matrix D whose eigenvalues are the semi-axis of the ellipsoid. The rolling motion has the usual constraints of non-slip and non-twist. Showing that the rolling ellipsoid is a complete nonholonomic system reduces to showing that one can move between two arbitrary admissible configurations by rolling without slipping and without twisting. We exhibit piecewise linear paths on the affine tangent space along which the ellipsoid rolls in order to perform the forbidden motions, twists and slips.