Symbolical singular value decomposition for a 7-DOF manipulator and its application to robot control

The method for symbolical singular value decomposition (SVD) based on Jacobian decomposition is broadened to cover simple redundant manipulators. The redundant robot should be decomposed into its nonredundant part and an actually redundant subrobot. The Jacobians should be expressed in some intermediate coordinate frames in order to obtain the simplest symbolical expressions. The Jacobian for the nonredundant part should be decomposed into several submatrices of the order 1×1, 2×2 or 3×3. In the case of 3×3 matrices, it is not always possible to obtain the symbolical expressions for SVD. The redundant subrobot Jacobian should also be further decomposed into submatrices of the order m_s× n_s, with m_s being less or equal to 2. By deriving the symbolical damped least-squares solution, the numerical complexity is reduced about 15 times as compared to the numerical SVD of the Jacobian submatrices of the same order. Simulations at the kinematic control level have shown very low position error and limited joint velocities