Workspace evaluation of manipulators through finite-partition of SE(3)

Workspace analysis and optimization are important in a manipulator design. As the complete workspace of a 6-DOF manipulator is embedded into a 6-D space, it is difficult to quantify and qualify it. Most 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 R3R3. It is known that the SO(3) is homeomorphic to a solid ball D3 with antipodal points identified while the geometry of R3R3 can be regarded as a cuboid. The complete 6-D workspace SE(3) is at the first time parametrically and proportionally partitioned into a number of elements with uniform convergence based on its geometry. As a result, a basis volume element of SE  (3) is formed by the product of a basis volume element of R3R3 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 becomes the simple summation of the basis volume elements of SE(3). Two new global performance indices, i.e., workspace volume ratio (Wr) and global condition index (GCI), are defined over the complete 6-D workspace. A newly proposed 3RP PS parallel manipulator is optimized based on this finite-partition approach. As a result, the optimal dimensions for maximal workspace are obtained, and the optimal performance points in the workspace are identified.