For redundant robotic manipulators, we study the design problem of Jacobian inverse kinematics algorithms of desired performance. A specific instance of the problem is addressed, namely the optimal approximation of the Jacobian pseudo-inverse algorithm by the extended Jacobian algorithm. The approximation error functional is derived for the coordinate-free representation of the manipulator’s kinematics. A variational formulation of the problem is employed, and the approximation error is minimized by means of the Ritz method. The optimal extended Jacobian algorithm is designed for the 7 degrees of freedom (dof) POLYCRANK manipulator. It is concluded that the coordinate-free kinematics representation results in more accurate approximation than the coordinate expression of the kinematics.