Plenary Talks

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30 Years at GM with Charles Wampler: From Topological Robotics to Mathematical Fashion Design, Danny Baker, General Motors
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This talk will briefly summarize aspects of Charles Wampler’s work over the last 30 years that many people attending the conference may not be aware of. I will try to at least touch on important contributions that Charles has made to robotics at GM, but I will focus mostly on some of his other work, things that I have a more detailed knowledge of.

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Kinematic mapping - recent results
and applications, Manfred Husty, University of Innsbruck
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Various mathematical formulations are used to describe the kinematics of mechanisms and
robots. The mathematical modelling is the basis for kinematic analysis and synthesis, e.g.
the computation of motion capabilities, singularities, workspaces, operation modes, velocities
and accelerations on one hand and to obtain design parameters on the other. Vector/matrix
formulation with trigonometric functions is the most favoured approach in the engineering
research community to solve these tasks. A less well known, but nevertheless very successful
approach relies on an algebraic formulation via the point model of Euclidean displacements in
the kinematic image space. In this approach, mechanism constraints are described with algebraic
(polynomial) equations and these sets of equations pertaining to some given mechanism or robot,
are solved with the powerful tools of algebraic and numerical algebraic geometry.

Within the talk some new results concerning the kinematics of lower dimensional parallel
manipulators, such as direct kinematics, workspace, operation and assembly modes and some
aspects of the kinematics of cable robots will be discussed. In detail the following topics will be addressed: methods to establish the sets of equations – the canonical equations, image space transformations; direct kinematics of n < 6-cable manipulators in crane configuration; inverse kinematic mapping and motion interpolation; Jacobian, singularities and operation modes;
examples.

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New Results for Anderson Acceleration: Theory and Applications, Tim Kelley, North Carolina State University
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Anderson Acceleration was designed to accelerate Picard iteration in the context of electronic structure computations in Physics. In this talk we will review our results on convergence, in particular our recent work on the cases in which he fixed point map is corrupted with errors. We consider uniformly bounded errors and stochastic errors with infinite tails. We prove local improvement results which describe the performance of the iteration up to the point where the accuracy of the function evaluation causes the iteration to stagnate. We will also discuss the EDIIS variation of the method, which is used in quantum chemistry applications.

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The Design of Linkage Systems to Trace Specified Curves,
Mike McCarthy, University of California, Irvine
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This talk examines design techniques for linkages to draw specified curves. We consider the design of straight-line linkages, the synthesis of four-bar and six-bar linkages to pass a coupler curve through a given set of points, and Kempe’s Universality Theorem that states for any plane algebraic curve there is a linkage that draws the curve. We show that mechanical Fourier synthesis can be used to draw approximations to a wide range of plane curves. And finally, we present a linkage system that draws trigonometric Bezier curves that can be configured to sign your name.

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How and Why Polynomial Continuation Came to General Motors: A Reminiscence 1980-1988,
Alexander Morgan, General Motors (retired)
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Within a few years of 1980, problems from three different engineering departments at GM Research—Computer Science, Engine Research, and Power Systems—which had come up in three entirely different applications were brought to the Mathematics Department to see if we could help solve them. Each problem reduced to finding all solutions to a system of polynomial equations.

Although the engineers were aware of many methods for solving many kinds of systems of equations, there was at that time no standard practical method for finding all solutions to systems of polynomial equations. The favored approach for several of these engineering groups—reducing the system to a single polynomial—can be particularly challenging to implement, even if it works “in theory.” A specific example will be given that illustrates this point.

The polynomial continuation method we eventually used was available in the recent mathematics literature. (The earliest paper I can find is from 1977.) But the engineers in these departments would never have found the method they needed in this literature, because:
the outlets used by the mathematicians (books and journals) were unlikely to be looked at by the engineers, who would tend to look in “their own” literature. For the same reason, successful mathematical methods for one area of engineering might not be discovered by engineers in a different area; the language the engineers used to describe their problems was radically different from the language used in the mathematics literature: polynomial systems, homotopies, continuation, ideals, functional differential equations, fixed points, degree theory, etc.;
The mathematics used to justify the methods—concerning either piecewise linear or smooth manifolds in Euclidean space—was unfamiliar to these engineers; the ultimate effectiveness of a method described in a paper is difficult to judge for a particular application. “Small” issues can make a method very challenging to apply or worthless.

The story here, then, is how these barriers were overcome at General Motors leading to three successful applications in three different departments and also helping to inspire the broadly useful and recognized area of applied algebraic geometry.

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Historical Survey of Numerical Algebraic Geometry, Andrew Sommese, University of Notre Dame
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In the middle 80's, the scope and power of continuation for computing solutions of polynomial systems was greatly increased by recognition, in research centered at General Motors Research Laboratories, of the algebraic geometry that underlay the systems of polynomials arising in practice. The catalyst for this development was the wealth of polynomial systems arising from applications at General Motors. Such polynomial systems displayed remarkable properties. Generally, they were quite sparse and highly structured. They were also surprisingly clean, e.g., when parameters were included, they typically had small integers as coefficients. Lines and other positive dimensional subsets of solution sets were regularly encountered in the computation of solutions of the polynomial systems. In an organic development, the view of these positive-dimensional sets as annoying and useless objects evolved to a realization the sets are interesting, ubiquitous, and useful. Numerical algebraic geometry grew out of the development of effective numerical algorithms to compute and manipulate the positive dimensional solution sets of polynomial systems.

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Algebraic Geometry for Projection Kinematic Analysis of DNA Origami Nano-Mechanisms, Haijun Su, Ohio State University
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Kinematic analysis and synthesis has been a central part of machine design and robotics. The key step of these kinematics problems is essentially solving a system of polynomial equations. Recently we have pioneered in applying kinematic principles to the design of tiny (tens of nm) mechanisms using the emerging DNA origami nanotechnology. We have demonstrated this technology by designing and fabricating a series of nanoscale classic kinematic joints and kinematic mechanisms. Currently, design validation analysis of these DNA origami mechanisms relies on measurement of projected geometric parameters, measured from Transmission Electron Microscopy (TEM) or Atomic Force Microscopy (AFM) images. In this talk, I will present a new method called “projection kinematics” for the determination of 3D configurations of DNA Origami Mechanisms (DOM) based on a single 2D projected image. We first derive classical kinematic equations of these DOMs and then project all parameters onto a projection plane. These projected parameters must satisfy a set of algebraic equations that we solve using elimination method or polynomial homotopy continuation methods. The solutions from these equations give projected configurations of a DOM in space. If redundant measurements are available, they can be exploited to eliminate ambiguous solutions.

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Research highlights associated with the characterization of battery materials for
electrified vehicle applications, Mark Verbrugge, General Motors
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After providing a brief overview of (i) General Motors Research and Development and (ii) progress in electric vehicle technology and existing opportunities, I will delve into mathematical models for the characterization of active materials used to construct battery systems of high energy density (Wh/L) and specific energy (Wh/kg). The scope of the talk will cover thermodynamics, electrode (charge transfer) kinetics, irreversible thermodynamics associated with transport phenomena (with an emphasis on solid-state diffusion), and cell modeling.