| The geometric, algebraic, and gradient methods of planar kinematic synthesis are examined and reviewed to define the general procedure of the kinematic synthesis of linkages problem. For a given dyad in the prescribed positions, the link lengths and joint loci can be calculated using the geometric, algebraic or gradient methods. Here, the gradient methods show that the Burmester curve and other related planar entities are actually equipotential curves.; to generate the equipotential surfaces of the joint loci using a single parameter.; However, in general spatial problems, a single parameter cannot be used because the screw axes do not intersect. Because of this, a dual angle described by a dual number is needed to describe the motion along the axes of the screws. Although numerical solutions to the general spatial synthesis problem were developed in the 1960s, few works examined the geometry of the spatial C-C and spherical R-R dyads in an attempt to generate the spatial equivalent of Burmester curves and other related entities.; The present work enables extending the gradient method to the spherical R-R and spatial C-C dyads by using the dihedral angle as a scalar parameter. The solution surfaces for the rotation were generated, and superposition of solutions yield solution loci for problems with more precision positions specified. Further, by dualizing the dihedral angle, a formula giving the distance relationships between the screws was developed.; During the research, it was discovered that the geometric relations, defining the scalars used in the spatial gradient methods, could also be used to develop spatial graphical methods for generating solution sets. This procedure is briefly demonstrated for special cases. |
