A mathematical framework for rigid contact detection between quadric and superquadric surfaces |
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Authors: | Daniel S Lopes Miguel T Silva Jorge A Ambrósio Paulo Flores |
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Affiliation: | (1) Data Storage Institute, (A*STAR) Agency for Science, Technology and Research, DSI Building, 5 Engineering Drive 1, Singapore, 117608, Singapore |
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Abstract: | The calculation of the minimum distance between surfaces plays an important role in computational mechanics, namely, in the
study of constrained multibody systems where contact forces take part. In this paper, a general rigid contact detection methodology
for non-conformal bodies, described by ellipsoidal and superellipsoidal surfaces, is presented. The mathematical framework
relies on simple algebraic and differential geometry, vector calculus, and on the C2 continuous implicit representations of the surfaces. The proposed methodology establishes a set of collinear and orthogonal
constraints between vectors defining the contacting surfaces that, allied with loci constraints, which are specific to the type of surface being used, formulate the contact problem. This set of non-linear
equations is solved numerically with the Newton–Raphson method with Jacobian matrices calculated analytically. The method
outputs the coordinates of the pair of points with common normal vector directions and, consequently, the minimum distance
between both surfaces. Contrary to other contact detection methodologies, the proposed mathematical framework does not rely
on polygonal-based geometries neither on complex non-linear optimization formulations. Furthermore, the methodology is extendable
to other surfaces that are (strictly) convex, interact in a non-conformal fashion, present an implicit representation, and
that are at least C2 continuous. Two distinct methods for calculating the tangent and binormal vectors to the implicit surfaces are introduced:
(i) a method based on the Householder reflection matrix; and (ii) a method based on a square plate rotation mechanism. The
first provides a base of three orthogonal vectors, in which one of them is collinear to the surface normal. For the latter,
it is shown that, by means of an analogy to the referred mechanism, at least two non-collinear vectors to the normal vector
can be determined. Complementarily, several mathematical and computational aspects, regarding the rigid contact detection
methodology, are described. The proposed methodology is applied to several case tests involving the contact between different
(super) ellipsoidal contact pairs. Numerical results show that the implemented methodology is highly efficient and accurate
for ellipsoids and superellipsoids. |
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