Finite-Partition of SE(3) and its Applications on Workspace Optimization of Parallel Manipulators

Workspace analysis and optimization are important in a manipulator design. As the complete workspace of a 6-DOF manipulator is embedded into a 6-dimensional space, it is difficult to quantify and qualify it. Most of the literatures only considered the 3-D sub workspaces of the complete 6-D workspace. In this paper, a finite-partition approach of the special Euclidean group SE(3) is proposed based on the topology properties of SE(3), which is the product of special orthogonal group SO(3) and Ropf³. It is known that the SO(3) is homeomorphic to a solid ball D³ with antipodal points identified while the geometry of Ropf³ can be regarded as a cuboid. Furthermore, the solid ball and the cuboid can be parametrically and proportionally partitioned into a number of elements. Therefore, a basis volume element of SE(3) is the product of a basis volume element of Ropf³ and a basis volume element of SO(3), which is the product of a basis volume element of D³ and its associated integration measure. By this way, the integration of the complete 6-D workspace volume become the simple summation of the basis volume elements of SE(3). Two global performance indices, i.e., workspace volume ratio (W_r) and global condition index (GCI)., are defined over the complete 6-D workspace. An optimization algorithm is developed for a 3RPlowbarPS parallel manipulator to illustrate the effectiveness of the finite-partition approach. As a result, the workspace optimization method is valid although it is computationally intensive