On the Control of Robotic Manipulators at Positions of Singularity

摘要

Robotic manipulator control usually results in the end-effector tracking a position profile, either in terms of Cartesian coordinates or in terms of joint angles. The most common control schemes for robotic manipulators are the methods of resolved motion and resolved rates proposed by Lu. The (Jacobian) transformation between the joint angles, joint rates, and joint accelerations and the end-effector's position, velocity, and acceleration must be determined and, for control, the inverse of the transformation is also required. There exist positions of the end-effector position such that the transformation is singular, that is, the angular quantities are not unique and the angular position either cannot be extracted from the linear position or is nonexistent. Some of these singularities are physical and not the result of the mathematical coordinates chosen. In any case, both the mathematical and physical singularities must be dealt with in order to control the robot motion. In this paper, an algorithm is proposed to determine the null space of the robotic system by computing the gradient of the determinant of the Jacobian. The zero elements of the gradient represent redundant joints and the null space of the gradient represents the linear combinations of the joint values which would cause the system to remain in a singular state. The algorithm is very intuitive and simple to implement.