Computer modelling and geometric construction for four-point synthesis of 4R spherical linkages

In this paper, computer modelling and geometric construction of Burmester curve for synthesis of spherical mechanisms is presented. Rigid shell guidance in four specified positions on a sphere is performed by a 4R spherical linkage. The synthesis of such a linkage requires obtaining the associated Burmester curves on the reference sphere. Based on the Burmester theory and beside the computational and modelling abilities of the symbolic mathematical software namely Maple, an accurate as well as fast procedure for geometric construction of Burmester curve is developed. In the first part, the concepts of orientation and position on the sphere, pole of the motion and the related parameters are extended for modelling by the Maple. In the second part, the concepts of complementary axis quadrilateral and its imaginary motion, the center and circle axis cones and Burmester curve are derived. In the final part, using the prepared procedure and through a numerical example, a 4R spherical linkage for guiding an antenna to meet four specified postures in a three-dimensional working space is synthesized.