It is well known that R. Ball's screw theory was based on the Lie algebra (and its dual) of the special Euclidean group SE(3), a branch of modern mathematics called Lie group theory developed after his time. A major limitation of the screw theory in mechanism synthesis is when certain basic machine elements, e.g., the parallelogram joint in the Delta robot and the end-effector of a mechanism, e.g., a five-axis machine and the limbs of many PKMs lack a group structure.

In this talk, I will outline an approach to use the more complete Lie group theory of SE(3), including its subgroups, submanifolds and homogeneous spaces, for modeling complex motions of spatial mechanisms and for systematic synthesis of serial and parallel mechanisms. I will also introduce a new type of kinematic structure called quotient kinematics machines (QKMs) and discuss their systematic synthesis.