Structural synthesis of two-dof differential screw mechanism by the multi-body kinematics

A novel structural synthesis of a mechanism based on the multibody kinematics is proposed in this paper. Instead of the degree-of-freedom analysis by the Grübler’s equation, numbers of the generalized coordinates and those of the constraint equations are introduced for the number synthesis of the mechanism. Sufficient and independent constraints which satisfy the number synthesis are described for the structural synthesis. The Jacobian matrix is derived by the derivative of the constraint equations with respect to time. By using the generalized coordinates, the proposed synthesis is widely applied for the lower degree analysis such as the three-dof planar mechanism, as well as the six-dof spatial mechanism without modifying the primary calculation method of the multibody kinematics. The lower degree analysis removes the redundant coordinates and over-constrained conditions, thus improves the complicated calculation, such as analyzing the planar mechanism using six-dof full spatial analysis. The proposed method is applied to the synthesis of the two-dof differential screw mechanism. The differential screw is comprised of two screw-nut mechanisms, each of them has two mechanical elements. The differential screw mechanisms are divided into two types; The coaxial serial-type differential screw mechanism in which one of the end of each non-driven mechanical element is directory connected to each other, and the parallel-type in which the non-driven mechanical element is connected via mechanical pairs. The rotational pair and the sliding pair are treated as the zero lead and the infinity lead of the screw pair as well as the general helix screw pair. A total of eight generalized coordinates, six kinematic constraints and two driving constraints are set for the number analysis. By evaluating the singularity of the Jacobian matrix from the generalized input velocity and that of the output element, forty combinations of the parallel-type differential screw mechanisms and twenty-four combinations of the serial-type are derived.