Inverse kinematic problem near singularities for simple manipulators: symbolical damped least-squares solution

The application of damped least-squares in solving the inverse kinematic problem near singularities requires numerically expensive singular value decomposition (SVD) of the Jacobian matrix and introduces some position error. Here the damped least-squares solution is obtained by dividing the Jacobian matrix into several submatrices of the order 1*1 or 2*2 and deriving a symbolic SVD for these submatrices. This is possible for simple manipulators where the inverse Jacobian can be obtained in analytical form. The SVD for the trivial 1*1 submatrices are also trivial, while for 2*2 matrices it can be easily derived in symbolic form. Simulations carried out at the kinematic control level for the Stanford manipulator and the PUMA-600 robot show that very good tracking of the specified trajectories may be achieved. Position error outside the trajectory is reduced to minimum, while the joint velocities are limited.