The minimum torque norm control scheme for redundant manipulators has long been known to exhibit instabilities. In the paper an analysis is made of a class of acceleration-level redundancy resolution schemes that includes this and other well-known schemes. It is proved that divergence of the joint velocity norm to infinity in finite time is possible, and in fact does occur for self-motions of mechanisms with one degree of redundancy under almost all such schemes in the class, i.e., generically. The schemes that do not diverge in finite time are associated with conserved quantities and include minimum acceleration norm and dynamically-consistent redundancy resolution schemes. It is also proved that exponential dissipation of nullspace velocity cannot eliminate the divergence.

The minimum torque norm control scheme for redundant manipulators has long been known to exhibit instabilities. In this paper, an analysis is made of a class of acceleration-level redundancy resolution schemes that includes the minimum torque, minimum acceleration, and other well known schemes. It is proved that divergence of joint velocity norm to infinity in finite time is possible, and in fact does occur for self motions of mechanisms with one degree of redundancy under almost all such schemes in the class. The schemes that do not diverge in finite time are associated with conserved quantities and include minimum acceleration norm and dynamically consistent redundancy resolution schemes. Besides the divergence, all these schemes can exhibit local instabilities in the form of abrupt increases in joint velocities and torques. Simulations are presented that illustrate the types of divergence and instability. The details of the analysis should be useful to designers seeking to modify these resolution schemes. For example, it is proved that linear dissipation or nullspace velocity cannot stabilize the algorithms.

Resolved acceleration motion control using the pseudoinverse or weighted pseudoinverse of the Jacobian matrix is a well-known algorithm for control of redundant robotic manipulators. Several authors have observed instabilities in simulations using this algorithm. In this paper the cause of the instability is determined, and the corresponding growth of joint velocities and acceleration is characterized in terms of the smallest singular value of the Jacobian matrix of the kinematic function, and a nearly-conserved quantity analogous to angular momentum. The analysis is supported by simulations. Based on the analysis, a stabilizing modification to the control scheme is derived that does not rely on the simple addition of a kinematic component, but rather addresses the cause of the instability directly. Simulations show that the new control algorithm stabilizes the motion in agreement with the analysis.

Solutions of the inverse kinematic problem for redundant manipulators are studied for workspace paths that pass near, but not through, kinematic singularities. Widely-used algorithms such as pseudo-inverse velocity control may compute solution paths that have low joint velocities but very high joint accelerations, which may not be physically desirable or even feasible. In this paper we show the existence of solutions with low joint space acceleration, which use the redundancy of the mechanism to track the workspace trajectory by means of a more natural joint motion. We derive a redundancy resolution algorithm at the velocity level that is computationally undemanding and tracks the desirable low-acceleration solution path. Simulations illustrating the feasibility of the proposed algorithm are presented.

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.