Functional Constraints as Algebraic Manifolds in a Clifford Algebra

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 for these functional constraints. 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 a subset of the Clifford algebra also represents the group of spatial rigid displacements as a six-dimensional manifold in an eight-dimensional space R8. Thus, the set of relative positions of two parts are mapped to a manifold in R8. This manifold is a geometric representation of the functional constraint. In this paper we derive the manifolds for six spatial functional constraints that define quadric algebraic manifolds. We then show how this theory can be used in the special case of the planar mating of two “peg-in-hole” constraints.