Charles W. Wampler

The calibration index and taxonomy for robot kinematic calibration methods

The major approaches toward kinematic calibration are unified by considering an end-point measurement system as forming a joint and closing the kinematic loop. A calibration index is in troduced, based on the mobility equation, that considers sensed and unsensed joints and single and multiple loops and ex presses the surplus of measurements over degrees of freedom at each pose. Past work using open-loop calibration, closed-loop calibration, and screw axis measurement is classified according to this calibration index. Numerical issues are surveyed, in cluding task variable scaling, parameter variable scaling, rank determination, pose selection, and input noise handling.

On the inverse kinematics of redundant manipulators

Many conventional nonredundant manipulators have singu lar configurations, near which some small motions of the end-effector require excessive and physically unrealizable joint speeds. Consequently, the usable workspace of the ma nipulator is effectively reduced. It has been proposed that high joint speeds could be avoided by introducing redundant joints and using an appropriate kinematic inversion algo rithm. For a very general class of kinematic inversion algo rithms, the theorems of this paper state some fundamental relations between the properties of the algorithm and its ability to resolve such problems. These results have practical implications in the design of controllers for redundant ma nipulators, especially when real-time sensory input is used to modify the manipulators trajectory.

Forward displacement analysis of general six-in-parallel SPS (Stewart) platform manipulators using soma coordinates

Abstract General six-in-parallel SPS platform manipulators are constructed of six telescoping legs, each connecting a stationary base platform to a moving platform via spherical joints. These are often termed “generalized Stewaart platforms”. given the legnths of teh six legs, teh forward displacement problem is to find the location of the end platform relative to the base platform. It was first demonstrated numerically that the problem may in general have at most 40 nonsingular solutions and this bound has been verified using several different mathematical arguments. The problem is reformulated in this report using a classical representation of rigid-body displacements: Studys soma coordinates, or equivalently, dual quaternions. This provides a much simpler analytical proof of the upper bound of 40. Moreover, the simple form of the equations may be useful in further studies of the problem.

Numerical Decomposition of the Solution Sets of Polynomial Systems into Irreducible Components

In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on its multiplicity. The bound is sharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroes of a finite number of polynomials.

Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems

Many polynomial systems have solution sets comprised of multiple irreducible components, possibly of different dimensions. A fundamental problem of numerical algebraic geometry is to decompose such a solution set, using floating-point numerical processes, into its components. Prior work has shown how to generate sets of generic points guaranteed to include points from every component. Furthermore, we have shown how monodromy can be used to efficiently predict the partition of these points by membership in the components. However, confirmation of this prediction required an expensive procedure of sampling each component to find an interpolating polynomial that vanishes on it. This paper proves theoretically and demonstrates in practice that linear traces suffice for this verification step, which gives great improvement in both computational speed and numerical stability. Moreover, in the case that one may still wish to compute an interpolating polynomial, we show how to do so more efficiently by building a structured grid of samples, using divided differences, and applying symmetric functions. Several test problems illustrate the effectiveness of the new methods.

Inverse kinematic functions for redundant manipulators

A general approach to the kinematic control of redundant manipulators using inverse kinematic functions is presented. The principal objective is to find an inverse kinematic function and a workspace such that the ratio of joint speed to workspace speed is bounded. The definition of such feasible workspaces leads naturally to optimization problems aimed at reducing this bound. The application of these results to the inverse kinematics of a redundant wrist illustrates both the theoretical importance of the approach and its immediate practicality. However, for general redundant manipulators, the development of algorithms for constructing feasible workspaces and the refinement of the optimization problem and its solution remain open questions ripe for further research.

Using Monodromy to Decompose Solution Sets of Polynomial Systems into Irreducible Components

To decompose solution sets of polynomial systems into irreducible components, homotopy continuation methods generate the action of a natural monodromy group which partially classifles generic points onto their respective irreducible components. As illustrated by the performance on several test examples, this new method achieves a great increase in speed and accuracy, as well as improved numerical conditioning of the multivariate interpolation problem. 2000 Mathematics Subject Classiflcation. Primary 65H10; Secondary 13P05, 14Q99, 68W30.

Adaptive Multiprecision Path Tracking

This article treats numerical methods for tracking an implicitly defined path. The numerical precision required to successfully track such a path is difficult to predict a priori, and indeed it may change dramatically through the course of the path. In current practice, one must either choose a conservatively large numerical precision at the outset or rerun paths multiple times in successively higher precision until success is achieved. To avoid unnecessary computational cost, it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path. We present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates. We then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations.

Advances in Polynomial Continuation for Solving Problems in Kinematics

For many mechanical systems, including nearly all robotic manipulators, the set of possible configurations that the links may assume can be described by a system of polynomial equations. Thus, solving such systems is central to many problems in analyzing the motion of a mechanism or in designing a mechanism to achieve a desired motion. This paper describes techniques, based on polynomial continuation, for numerically solving such systems. Whereas in the past, these techniques were focused on finding isolated roots, we now address the treatment of systems having higher-dimensional solution sets. Special attention is given to cases of exceptional mechanisms, which have a higher degree of freedom of motion than predicted by their mobility. In fact, such mechanisms often have several disjoint assembly modes, and the degree of freedom of motion is not necessarily the same in each mode. Our algorithms identify all such assembly modes, determine their dimension and degree, and give sample points on each.

Some facts concerning the inverse kinematics of redundant manipulators

Many conventional nonredundant manipulators have singular configurations, near which some small motions of the end-effector require excessive and physically unrealizable joint speeds. Consequently, the usable workspace of the manipulator is effectively reduced. It has been proposed that high joint speeds could be avoided by introducing redundant joints and using an appropriate kinematic inversion algorithm. For a very general class of kinematic inversion algorithms, the theorems of this paper state some fundamental relations between the properties of the algorithm and its ability to resolve these problems. These results have practical implications in the design of controllers for redundant manipulators, especially when real-time sensory input is used to modify the manipulators trajectory.