The Synthesis of Planar RR and Spatial CC Chains and the Equation of a Triangle

Abstract

This paper formulates the planar and spatial versions of an equation that determines one vertex of a triangle in terms of the other two vertices and their interior angles. The fact that a slight modification of Sandor and Erdman’s standard form equation for the design of RR chains yields this planar triangle equation is the basis for identifying the equivalent equation for a spatial triangle as the standard form equation for CC chains. The simultaneous solution of two of the planar equations yields an analytical expression of Burmester’s relationship between the fixed pivot of an RR chain and the relative position poles of its floating link. A similar solution of simultaneous spatial triangle equations yields Roth’s generalization of this insight, specifically, the fixed axis of a CC chain views two relative screw axes in one-half the dual crank rotation angle. These results provide the foundation for generalizing planar linkage synthesis techniques based on complex numbers to the synthesis of spatial linkages.