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-imensional 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 R^3. It is known that the SO(3) is homeomorphic to a solid ball D^3 with antipodal points identified while the geometry of R^3 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 R^3 and a basis volume element of SO(3), which is the product of a basis volume element of D^3 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 3 RPPS 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.