Motion synthesis of mechanisms using constraint manifolds in image space

Kinematic mappings, quaternion algebra, and constraint manifolds in the algebraic image space are applied to the problems of the dimensional synthesis of mechanisms. Dimensions of a mechanism are determined such that a tracer frame fixed on the coupler will pass through or at least as close as possible to the desired positions and orientations in the physical space as the input link rotates about its fixed joint. First, using kinematic mappings, the desired positions and orientations of the tracer frame of the mechanism can be mapped onto points in a hyperspace in which the motion of the tracer frame can be represented by a curve. Second, using quaternion algebra, the structure equations representing the transformations from the reference frame to the tracer frame via each leg, each crank-coupler dyad of the mechanism, form the constraint manifolds of the mechanism. Finally, the problem of dimensional synthesis thus becomes one of finding a curve, generated by the intersection of constraint manifolds and fulfilling the constraint equations of kinematic mappings, which passes through or near the desired image points. The dimensions of the mechanism are found by using total least square algorithms to minimize the normal distance between all the desired image points and image curve of the tracer frame. Using this approach, the synthesis problems of all three types of mechanisms, planar, spherical, and spatial, can be formulated similarly. It provides a straightforward tool for general motion synthesis problems. The theory is illustrated by numerical examples of planar and spherical mechanisms