Curvature theory for a two-degree-of-freedom planar linkage

This paper shows that the method of kinematic coefficients can be applied in a straightforward manner to the kinematic analysis of mechanisms with more than one input. For illustrative purposes, the paper presents an example of a two-input linkage or two-degree-of-freedom linkage; namely, the well-known planar five-bar linkage. The two inputs are the side links of the five-bar linkage which are assumed to be cranks. Since the linkage is operated by two driving cranks independently then the linkage can produce a wide variety of motions for the two coupler links. The kinematic coefficients are partial derivatives of the two coupler links with respect to the two input crank angles and separate the geometric effects of the mechanism from the operating speeds. As such they provide geometric insight into the kinematic analysis of a mechanism. A practical application of the five-bar linkage is to position the end-effector of an industrial robotic manipulator, for example, the General Electric model P50 robotic manipulator. The paper then presents closed-form expressions for the radius of curvature and the center of curvature of an arbitrary coupler curve during the complete operating cycle of the linkage. The analytical equations that are developed in the paper can be incorporated, in a straightforward manner, into a spreadsheet that is oriented towards the path curvature of a multi-degree-of-freedom linkage. The author hopes that, based on the results presented here, a variety of useful tools for the kinematic design of planar multi-degree-of-freedom mechanisms will be developed for planar curve generation.