Daniel A. Brake

The Complete Solution of Alt–Burmester Synthesis Problems for Four-Bar Linkages

Precision-point synthesis problems for design of four-bar linkages have typically been formulated using two approaches. The exclusive use of path-points is known as “path synthesis”, whereas the use of poses, i.e. path-points with orientation, is called “rigid-body guidance” or the “Burmester problem”. We consider the family of “Alt-Burmester” synthesis problems, in which some combination of path-points and poses are specified, with the extreme cases corresponding to the typical approaches. The Alt-Burmester problems that have, in general, a finite number of solutions include Burmester’s original five-pose problem and also Alt’s problem for nine path-points. The elimination of one path-point increases the dimension of the solution set by one, while the elimination of a pose increases it by two. Using techniques from numerical algebraic geometry, we tabulate the dimension and degree of all problems in this Alt-Burmester family, and provide more details concerning all the zeroand one-dimensional cases.

Bertini_real: Software for One- and Two-Dimensional Real Algebraic Sets

Bertini_real is a command line program for numerically decomposing the real portion of a one- or two-dimensional complex irreducible algebraic set in any reasonable number of variables. Using numerical homotopy continuation to solve a series of polynomial systems via regeneration from a witness set, a set of real vertices is computed, along with connection information and associated homotopy functions. The challenge of embedded singular curves is overcome using isosingular deflation. This decomposition captures the topological information and can be used for further computation and refinement.

On Computing a Cell Decomposition of a Real Surface Containing Infinitely Many Singularities

Numerical algorithms for decomposing the real points of a complex curve or surface in any number of variables have been developed and implemented in the new software package Bertini_real. These algorithms use homotopy continuation to produce a cell decomposition. The previously existing algorithm for surfaces is restricted to the “almost smooth” case, i.e., the given surface must contain only finitely many singular points. We describe the use of isosingular deflation to remove this almost smooth condition and describe an implementation of deflation via Bertini with MATLAB.

Intrinsic localized modes in two-dimensional vibrations of crystalline pillars and their application for sensing

We present a complete analysis on the possibility of exciting and observing the intrinsic localized modes (ILMs) in a crystalline linear array of nano pillars. We discuss the nano-fabrication techniques for these arrays and visualization procedures to observe the real-time dynamics. As a consequence, we extend previous models to the study of two dimensional vibrations to be consistent with these restrictions. For these pillars, the elastic properties and hence the dynamics depend on the pillars shape and the orientation of the crystal axes. We show that ILMs do form in the system, but their stability, defect pinning, and reaction to friction strongly depend on the crystals properties, with the optimal dynamics only achieved in a rather small region of the parameter space. We also demonstrate fabrication techniques for these pillars and discuss the applications of these pillar arrays to sensing.

Algorithm 976: Bertini_real: Numerical Decomposition of Real Algebraic Curves and Surfaces

Bertini_real is a compiled command line program for numerically decomposing the real portion of a positive-dimensional complex component of an algebraic set. The software uses homotopy continuation to solve a series of systems via regeneration from a witness set to compute a cell decomposition. The implemented decomposition algorithms are similar to the well-known cylindrical algebraic decomposition (CAD) first established by Collins in that they produce a set of connected cells. In contrast to the CAD, Bertini_real produces cells with midpoints connected to boundary points by homotopies, which can easily be numerically tracked. Furthermore, the implemented decomposition for surfaces naturally yields a triangulation. This CAD-like decomposition captures the topological information and permits further computation on the real sets, such as sampling, visualization, and three-dimensional printing.

Decomposing Solution Sets of Polynomial Systems Using Derivatives

A core computation in numerical algebraic geometry is the decomposition of the solution set of a system of polynomial equations into irreducible components, called the numerical irreducible decomposition. One approach to validate a decomposition is what has come to be known as the “trace test.” This test, described by Sommese, Verschelde, and Wampler in 2002, relies upon path tracking and hence could be called the “tracking trace test.” We present a new approach which replaces path tracking with local computations involving derivatives, called a “local trace test.” We conclude by demonstrating this local approach with examples from kinematics and tensor decomposition.

Numerical Local Irreducible Decomposition

Globally, the solution set of a system of polynomial equations with complex coefficients can be decomposed into irreducible components. Using numerical algebraic geometry, each irreducible component is represented using a witness set thereby yielding a numerical irreducible decomposition of the solution set. Locally, the irreducible decomposition can be refined to produce a local irreducible decomposition. We define local witness sets and describe a numerical algebraic geometric approach for computing a numerical local irreducible decomposition for polynomial systems. Several examples are presented.

Homotopies for Connected Components of Algebraic Sets with Application to Computing Critical Sets

Given a polynomial system f, this article provides a general construction for homotopies that yield at least one point of each connected component on the set of solutions of \(f = 0\). This algorithmic approach is then used to compute a superset of the isolated points in the image of an algebraic set which arises in many applications, such as computing critical sets used in the decomposition of real algebraic sets. An example is presented which demonstrates the efficiency of this approach.

ILLUSTRATION OF NUMERICAL ALGEBRAIC METHODS FOR WORKSPACE ESTIMATION OF COOPERATING ROBOTS AFTER JOINT FAILURE

We consider the estimation of the post-failure workspace of two two-link serial robots where the free-swinging failure of one robot’s last joint is handled by having the functional robot grasp the final link of the broken robot. We present an algorithm for finding the optimal placement of such synergistic robot arms and the optimal grasp point on the final link of the broken robot. To determine whether a point in the pre-failure workspace of the broken robot remains in the post-failure workspace, we solve an inverse kinematics problem in the form of a polynomial system. Homotopy continuation provides an efficient means for solving such polynomial systems, so we use the software package Bertini. Finally, Monte-Carlo methods are used to estimate the post-failure workspace.

Workspace Multiplicity and Fault Tolerance of Cooperating Robots

Cooperating robotic systems, especially in the context of fault-tolerance of complex robotic mechanisms, is an important question for theoretical and applied studies. In this paper, we focus on one measure of fault tolerance in robots, namely, the multiplicity of the configurations for reaching a particular point in the workspace, which is difficult to measure using traditional methods. As a particular example, we consider the case of a free-swinging failure of a robotic arm that is handled by having a cooperating functional robot grasp the link adjacent to the failed joint. We present an efficient method to compute the multiplicity measure of the workspace, based on the tools from numerical algebraic geometry, applied to the inverse kinematics problem re-cast in the form of a polynomial system. To emphasize the difference between our methods and more traditional approaches, we compute the measure of workspace based on the multiplicity of configurations, and optimize placement of synergistic robot arms and the optimal grasp point on each link of the broken robot based on this measure.