Discrete Kinematic Geometry and Saddle Synthesis of Spatial Linkages

The matrix representation of the discrete spatial movement of a rigid body is given at first. The spatial discrete kinematic geometry is indicated by applying the saddle point programming twice. Firstly, a discrete point-trajectory, traced by a point of a rigid body, is globally compared with the constraint surfaces of spatial linkages, a spherical surface and a cylindrical surface, both the saddle spherical surface fitting and the saddle cylindrical surface fitting are designated as the mathematical models of the saddle point programming. The normal fitting errors of the two saddle surface fittings are discussed for five, six, seven and multiply separated positions of a moving body. Secondly, the saddle sphere points or the saddle cylinder points are optimally located by the saddle point programming and certainly exist on the moving body. The line-trajectories, traced by the lines of a moving body, likes the that of point-trajectory for the saddle point programming, are also globally compared with the constraint ruled surfaces of spatial linkages, a constant ruled surface and its two degenerated ruled surfaces, which are viewed as the saddle spherical image curve fitting and the saddle striction curve fitting (on a cylindrical surface). Three mathematical models of the three saddle constant ruled surface fittings are presented by means of the saddle point programming, whose normal fitting errors are discussed. There exist the saddle constant axis lines and their degenerated lines on the moving body. The mathematical model of the discrete kinematic synthesis of spatial four-bar linkages RCCC, RRSS, RRSC, are established, whose initial points are provided by saddle sphere points, saddle cylinder points, saddle constant axis lines and their degenerated lines on the moving body, and three numeral examples indicate the solving process of the mathematical models.