A Geometric Theory for Synthesis and Analysis of Sub-6 DoF Serial Manipulator Subchains

Motivated by the work of Herve and his coworkers, this paper presents a rigorous and precise geometric theory for the synthesis and analysis of sub-6 DoF serial manipulator subchains. First, we review the basic properties of the Special Euclidean group SE(3), Lie subgroups and submanifolds of SE(3). With low dimensional subgroups and submanifolds providing models for the so called primitive generators, the high dimensional subgroups and regular submanifolds provide models for the set of desired end-effector motions. Two important classes of regular submanifolds of SE(3) are studied in detail. Then, starting from a given list of primitive generators, we give a rigorous definition of the synthesis problem for a serial manipulator subchain, and develop a general procedure for solving the synthesis problem when the set of desired end-effector motions is a Lie subgroup or a regular submanifold.