Efficient Jacobian inversion for the control of simple robot manipulators

Symbolic inversion of the Jacobian matrix for spherical wrist arms is investigated. It is shown that, taking advantage of the simple geometry of these arms, the closed-form solution of the system Q = J-1X, representing a transformation from task space to joint space, can be obtained very efficiently. The solutions for PUMA, Stanford, and a six-revolute-joint coplanar arm, along with all singular points, are presented. The solution for each joint variable is found as an explicit function of the singular points which provides a better insight into the effect of different singular points on the motion and force exertion of each individual joint. For the above arms, the computation cost of the solution is on the same order as the cost of forward kinematic solution and it is significantly reduced if forward kinematic solution is already obtained. A comparison with previous methods shows that this method is the most efficient to date.