Expedited Award for Novel Research: Geometric Representations of Functional Constraints Using Clifford Algebras

制造部件中的功能约束可确保互连零件的正确相对运动。运动学的最新结果表明，射影三空间的Clifford代数可以方便地将这些函数约束表示为二次代数曲面。Clifford代数是一个具有乘积运算的向量空间，它包含关于基础空间的度量性质的信息。在三维情况下，Clifford代数也表示空间刚性位移组。在这项研究中，研究人员将推导出六个空间函数约束的流形，展示该理论如何将两个“孔中钉”约束的平面匹配表示为双曲面的交点。这项研究的目的是将这些结果推广到一种形式主义，即几何地规定两部分的相对空间位置和方向的界限。这一几何规定将有助于装配验证和公差验证。

Functional constraints in a manufactured assembly assure the proper relative movement of interconnected parts. Recent results in kinematics show that the Clifford Algebra of projective three space provides a convenient representation of these functional constraints as quadratic algebraic surfaces. A Clifford Algebra is a vector space with a product operation that contains information about the metric properties of the underlying space. In the case of three dimensions the Clifford Algebra also represents the group of spatial rigid displacements. In this research the investigator will derive the manifolds for six spatial functional constraints, show how this theory represents the planar mating of two "peg-in-hole" constraints as the intersection of hyperboloids. The goal of the research is to generalize these results to a formalism that prescribes geometrically the bounds of relative spatial position and orientation of two parts. This geometric prescription will facilitate assembly verification and tolerance verification.