Spatial Generalizations of Planar Point-Angle and Path Generation Problems

This paper deals with the spatial generalizations of two classical planar synthesis problems: the point-angle and path generation problems. The two planar synthesis problems involve the guidance of a point through specified positions by using planar four-bar linkages. In spatial generalizations, we are concerned with the guidance of a line by using spatial 4C linkages. The equivalent screw triangle is employed to derive the synthesis equations of the spatial 4C linkage. The synthesis of the RCCC linkage is achieved by constraining the translational motion in the driving C joint. Our results show that the synthesis of the 4C linkage for line guidance yields the same maximum number of positions as the planar 4-bar linkage for point guidance. The maximum number of positions of the path generation of a line is nine, while that of the lineangle problem is five. In addition to presenting the spatial generalizations of planar synthesis problems, the results provided in this paper can be used to design spatial four-bar linkages to match line specifications, in which only an infinitely-extended line, such as a laser beam, is of interest.