Algebraic Input-output Angle Equation Derivation Algorithm for the Six Distinct Angle Pairings in Arbitrary Planar 4R Linkages

This paper describes a generalised algorithm that can be applied to any single degree of freedom parallel kinematic chain to determine the algebraic polynomial that represents the input-output equation relating any pair of distinct angles between any pair of links in the kinematic chain. There are six such algebraic polynomials for an arbitrary four-bar linkage. The algorithm consists of assigning standard Denavit-Hartenberg coordinate systems and parameters to the open kinematic chain. The open chain is conceptually closed by equating the forward kinematic transformation that maps coordinates of points in the "end-effector" coordinate system to the relatively non-moving base coordinate system to the identity matrix. The resulting transformation is mapped to Study soma coordinates wherein the twist and joint angles have been converted to tangent half-angle parameters. Elimination theory is then applied to the soma coordinates revealing a single algebraic polynomial in terms of the link lengths and the desired angle pair. Example applications are discussed for continuous approximate synthesis, mobility classification, and the design parameter space.