Spherical 4R Linkage Algebraic $$\boldsymbol{v_i}$$-$$\boldsymbol{v_j}$$ Input-Output Equations

Abstract

AbstractIn this paper the algebraic polynomial equations relating the relative orientations between the six distinct pairs of rigid links in an arbitrary spherical 4R mechanism are derived. First, the forward kinematics transformation matrix of an arbitrary spherical open 4R kinematic chain is computed in terms of its Denavit-Hartenberg parameters, where all angles are converted to their tangent half-angle parameters. This transformation matrix is mapped to its corresponding four non-zero Study soma coordinates. The serial kinematic chain is conceptually closed by equating the forward kinematics transformation to the identity matrix. Gröbner bases and resultants are then used to eliminate the two intermediate joint angle parameters leaving an algebraic polynomial in terms of the selected input and output (IO) joint angle parameters and the four twist angle or link length parameters. This yields six independent algebraic IO Equations. Their utility is demonstrated with two function generator continuous approximate synthesis examples. KeywordsSpherical four-bar linkage\(v_i\)-\(v_j\) algebraic input-output equationsContinuous approximate synthesis