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

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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.