A novel method of controlling robotic manipulators near singularities is introduced. In traditional rate control, a local first-order approximation of the robot kinematics is used, which may fail at a singularity. However, if the approximation is extended to the second order, it becomes possible to solve for the joint rates. The proposed method uses a multidimensional Newton-Raphson technique to solve the resulting set of coupled, quadratic equations. The method is presented in conjunction with a description of rate control in the complex domain, which allows for end-effector locations outside the manipulators workspace. Two examples employing quadratic rate control are presented for a 6R robot in a singular configuration, and one example applying the method to a closed trajectory is shown.>

A method for using complex numbers in the kinematic solution of robotic manipulators is presented. The inverse kinematics algorithm of a manipulator produces complex joint angles if it is given an end effector location outside its workspace. Although complex joint angles cannot be implemented physically, they can be converted to real numbers through a suitable mapping, and the robot approximates the configuration required to produce the desired location within the limits of its workspace. It is shown that for certain manipulators, the real-part mapping produces a minimization of the end effector location error. Examples of the technique using 2R and 3R manipulators are presented.>