Functional constraints as algebraic manifolds in a Clifford algebra

The characterization of the relative position of parts in an assembly can be viewed as a generalized robot kinematics problem in which the set of acceptable assemblies is the workspace of the system. The authors show how the Clifford algebra of projective space provides algebraic manifolds characterizing this allowable movement in an assembly. They examine six basic constraints that represent the mating of cylindrical, spherical, and polyhedral features: pure rotation about a line, pure rotation about a point, pure translation along a line, contact of a point with a plane, contact of a plane with a point, and contact of a line with a line. The authors derive parametric and algebraic formulas for the six primitive functional constraints. To illustrate how this theory gives a geometric form to functional constraints, the hyperboloid that defines the planar peg-in-hole constraint is derived. It is shown that the constraint that two pegs fit into two holes is the intersection of two hyperboloids