A General Method for Determining Algebraic Input-Output Equations for Planar and Spherical 4R Linkages

Abstract

A new and completely general method for determining the algebraic input-output (IO) equations for planar and spherical 4R linkages is presented in this paper. First, the forward kinematics transformation matrix of an arbitrary planar or spherical open 4R kinematic chain is computed in terms of its Denavit-Hartenberg parameters, where the link twist and joint angles are converted to their tangent half-angle parameters. This transformation matrix is mapped to its corresponding eight Study coordinates. The serial kinematic chain is conceptually closed by equating the forward kinematics transformation to the identity matrix. Equating the two corresponding Study arrays yields four equations in terms of the four revolute joint angle parameters. Gröbner bases are then used to eliminate the two intermediate joint angle parameters leaving an algebraic polynomial in terms of the input and output joint angle parameters and the four twist angle or link length parameters. In the limit, as the sphere radius becomes infinite and the link twist angle parameters are expressed as ratios of arc length and sphere radius in the general spherical algebraic IO equation, the only terms that remain are those in the planar 4R IO equation.