Abstract This paper presents a screw theory based method for determining the set of possible second-order relative motions between two 3D rigid objects which are initially in a finite number of unilateral point contacts. Using adjoint transformation for twist and twist-derivative, the motion space allowed by each contact is referred to a common coordinate system. Thereafter, a set of quadratic inequalities in twist and twist-derivative coordinates are solved for obtaining the resultant allowable motions. We show that a minimum of three frictionless contacts are sufficient for curvature based form-closure. Also, it is shown exactly that a regular tetrahedron can be in second-order form-closure with four identical spheres in point contact.

中文翻译：

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多点接触中物体的运动空间分析，应用到形式闭合和运动学对

摘要 本文提出了一种基于螺旋理论的方法，用于确定最初处于有限数量的单边点接触中的两个 3D 刚体之间可能的二阶相对运动集。对扭曲和扭曲导数使用伴随变换，每个接触允许的运动空间被称为公共坐标系。此后，求解一组扭曲和扭曲导数坐标中的二次不等式以获得最终的允许运动。我们表明，对于基于曲率的形式闭合，最少三个无摩擦接触就足够了。此外，它准确地表明了一个正四面体可以在具有四个相同球体点接触的二阶闭包中。