Numerical analysis of displacements in spatial mechanisms with ball joints

The ball joint, often referred to as a spherical or ‘S’ joint is modeled using dual-number coordinate-transformation matrices. The joint consists of concave and convex spherical surfaces engaged to prevent translations but allowing three degrees of freedom, all of which are rotations. Derivative-operator matrices to be used in the Fischer–Paul adaptation of the Uicker–Denavit–Hartenberg numerical scheme for displacement analysis of spatial mechanisms are developed. The generalized slider-crank (CSSP) mechanism is presented as an example featuring ball joints where coordinate-transformation matrices modeling links with ball joints are used in a concatenation with analogous matrices modeling links with revolute, prismatic or cylindrical joints to analyze the displacements.