Kevin A. O'Neil

Divergence of linear acceleration-based redundancy resolution schemes

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.

Removing singularities of resolved motion rate control of mechanisms, including self-motion

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.

Statistics of two-dimensional point vortices and high-energy vortex states

Results from the theory ofU-statistics are used to characterize the microcanonical partition function of theN-vortex system in a rectangular region for largeN, under various boundary conditions, and for neutral, asymptotically neutral, and nonneutral systems. Numerical estimates show that the limiting distribution is well matched in the region of major probability forN larger than 20. Implications for the thermodynamic limit are discussed. Vortex clustering is quantitatively studied via the average interaction energy between negative and positive vortices. Vortex states for which clustering is generic (in a statistical sense) are shown to result from two modeling processes: the approximation of a continuous inviscid fluid by point vortex configurations; and the modeling of the evolution of a continuous fluid at high Reynolds number by point vortex configurations, with the viscosity represented by the annihilation of close positive-negative vortex pairs. This last process, with the vortex dynamics replaced by a random walk, reproduces quite well the late-time features seen in spectral integration of the 2d Navier-Stokes equation.

Instability of pseudoinverse acceleration control of redundant mechanisms

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.

On the existence and the manipulability recovery rate of self-motion at manipulator singularities

This article investigates the escapability of redundant manipu lators at a singularity. Escapability means that the manipulator can reconfigure itself from a singular posture to a nonsingular posture via self-motion. Criteria for the classifications of es capable and inescapable singularities are established based on multivariable calculus theorems. For the general case, we give necessary conditions for the escapability of singular configura tions. For singular configurations with only one lost degree of freedom in the workspace, sufficient conditions for the escapa bility are provided. We then show how the arm can escape the singularity at the maximum rate if these sufficient conditions are satisfied. Our method gives the best self-motion among the set of all feasible self-motions, in the sense that the manipula bility measure of the arm increases most rapidly for the chosen self-motion. It is also shown that at a singular configuration with more than one lost degree of freedom in the workspace, the initial rate of recovery of manipulability is very slow, re gardless of the possibility of self-motion. Examples are given to demonstrate the use of these criteria.

Symmetric configurations of vortices

Abstract Stationary and uniformly translating configurations of point vortices are examined when the vortices are arranged in symmetric rings, lines, and lattices. Algebraic equations for the reduced systems are found, and the solutions counted using algebraic geometry.

On the Limiting Distribution of Pair-Summable Potential Functions in Many-Particle Systems

For systems with finite phase space volume, the density of states can be viewed as a multiple of the probability density of the energy, when the phase space variables are independent uniformly distributed random variables. We show that the distribution of a random variable proportional to the sum of pairwise interactions of independent identically distributed random variables converges to a limiting distribution as the number of variables goes to infinity, when the interaction satisfies certain homogeneity requirements. The moments of this distribution are simple combinations of cyclic integrals of the potential function. The existence of this limit gives information about the structure function of some systems in statistical mechanics having pair-summable interactions, even in the absence of a thermodynamic limit. The result is applied to several examples, including systems of two-dimensional point vortices.

Escapability of singular configuration for redundant manipulators via self-motion

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.

On the existence and characteristics of solution paths at algorithmic singularities [kinematically redundant arms]

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.

Continuous parametric families of stationary and translating periodic point vortex configurations

The number of periodic arrangements of point vortices - point vortex streets - in two-dimensional fluid flow that are stationary is known to be finite for a generic choice of vortex circulations. When all circulations are the same in absolute value, however, stationary vortex street configurations have been associated with the zeros of certain trigonometric polynomials containing free complex parameters. The presence of these parameters may prove useful in constructing point vortex models of shear layers and wakes. In this paper it is shown that such a continuum of stationary configurations exists in a much wider class of point vortex street systems. The circulations may take on many values, not just two, providing increased flexibility in the modelling context. A simple method for computing these configurations is derived. The effects of symmetries on the solution polynomials are described, and illustrated with examples. In addition, novel translating vortex street configurations are found having arbitrary translation velocity and containing free parameters for vortex circulations ±1 and also for vortex circulations +1, -2.