A Line Geometric Foundation for Finite Screw Systems Associated With Spatial Linkages

This paper investigates the line varieties corresponding to finite screw systems associated with spatial linkages. This research is based on the correspondence between a screw and a linear complex, and a screw system corresponds to the intersection of the linear complexes. In finite kinematics, two screw systems associated with the finite motions of the revolute-revolute (R-R) and prismatic-revolute (P-R) open chains have been discovered. These two screw systems also led to the discovery of the screw systems associated with the finite motions of the spatial 4R, spatial RPRP, and other overconstrained linkages. By using the intersection operation of linear complexes, this paper finds the linear reguli corresponding to the finite motions of R-R and P-R chains. Then we utilize the sum operation of the linear reguli corresponding to the R-R and P-R chains to obtain the hyperbolic linear congruences corresponding to the finite motions of the spatial 4R and RPRP linkages. The result presented here serves as a line geometric foundation for finite screw systems associated with spatial linkages. In addition, with CAD drawings, this paper enables the visualization of the obtained line varieties and their operations.