The extended Jacobian method is a popular approach for controlling a kinematically redundant arm which allows one to resolve redundancy by locally optimizing an objective function and to gain repeatability for a cyclic end effector trajectory. It is a special case of a family of methods called constraint function methods. It has been found that the occurrence of algorithmic singularities can cause severe difficulties and the advantages of the methods such as repeatability might no longer exist. The purpose of this paper is to study the characteristics of algorithmic singularities, especially those of corank 1. A result of the authors on kinematic singularities is used to obtain a sufficient condition for the existence of solution paths at algorithmic singularities of the constrained function method. The phenomenon of branch repeatability is shown to occur at an algorithmic singularity. We also show that the extended Jacobian method cannot successfully optimize the objective function beyond the singularity without loss of continuity of the joint derivative. Examples are given to demonstrate the use of our theoretical results.

Resolved motion rate control, which uses a linear approximation to the kinematic function to find joint velocities, can fail at kinematic singularities. Pohl and Lipkin (1991) proposed rate control using a quadratic approximation to the full kinematic function, but this algorithm has disadvantages. Here we propose a hybrid algorithm that works at singularities, is computationally efficient, and finds all physically meaningful solution trajectories. Furthermore, it is shown that this algorithm finds smoother solutions than, the damped least squares algorithm for paths passing through singular points.

A new kinematic algorithm for the control of mechanisms is described, which produces smoother and more accurate control near kinematic singularities than the usual damped least-squares method. The new method predicts the occurrence of singularities, evaluates the singular terms of the inverse kinematic function using a nonsingular expression, and smoothly interpolates between the singular and nonsingular forms. The desingularized expression makes use of the rates of change of the singular vectors and values of the Jacobian matrix; formulas for these derivatives are presented. Simulations of the control of two and six joint serial manipulators moving through and near singular configurations are given to illustrate the method.

Resolved motion rate control is an algorithm for solving the path-tracking problem in robotic control which can fail at singular points of the kinematic function. The questions of existence and smoothness of solutions to the path tracking problem at singular points have not heretofore been addressed. In this paper we find a new second-order condition which, when satisfied, ensures the existence of a solution path with continuous, bounded joint rates. The condition is related to the curvature of the path at the singular value. We prove that a modification of the usual resolved motion rate control algorithm can successfully compute this solution path. As an application, we give a sufficient condition for the existence of self-motion for redundant manipulators at singular points. We derive a simple formula for the rate of recovery of the manipulability measure near the singularity. Several realistic examples are presented, for which we compute exact solutions to typical path tracking problems passing through singular points.

Resolved motion rate control is an algorithm for solving the path-tracking problem in robotic control which can fail at singular points of the kinematic function. In this paper we find a new second-order condition which, when satisfied, ensures the existence of a solution path with continuous, bounded joint rates. The condition is related to the curvature of the path at the singular value. As an application, we give a sufficient condition for the existence of self-motion for redundant manipulators at singular points. We derive a simple formula for the rate of recovery of the manipulability measure along the path at the singularity, and prove that the usual resolved motion rate control algorithm can be modified to compute these solution paths. Several realistic simulations are presented.

The goal of this paper is to study the behavior of general redundant manipulators near singular configurations, and investigate the escapability of redundant manipulators at a singularity. Escapability means that the manipulator can actually reconfigure itself from a singular posture to a nonsingular posture via self-motion. Criteria for the classifications of escapable and inescapable singularities are established. Examples are given to demonstrate the use of these criteria.