Affiliations Adjunct Professor, Department of Applied & Computational Mathematics & Statistics, University of Notre Dame, Notre Dame, IN Technical Fellow, General Motors Research and Development Center, Warren, MI

Research Statement

I primarily study the kinematics of robots and mechanisms. This field is concerned with collections of rigid bodies with geometric constraints between them. Examples of such constraints are rotational hinges, prismatic (linearly sliding) joints, and spherical (ball-and-socket) joints. Mathematical models based on ideal joints and rigid bodies closely approximate the motion of many practical devices, ranging from steering and suspension systems on vehicles to multi-limbed robots.

The most common geometric constraints involve entities that are algebraic (points, lines, planes, cylinders, and spheres), and since squared distances are algebraic also, the mathematical models are algebraic. Thus, the questions to be answered fall within the domain of algebraic geometry. In the late 19th century and early 20th century, these questions were actively pursued in mathematical circles, and such well-knowns as Cayley, Chebychev, Kempe, Schönflies, Study, and Sylvester made significant contributions. Subsequently, the main thread of algebraic geometry moved to a higher level of abstraction and the study of kinematics became mainly the province of engineers. Cross-fertilization between the fields resumed in the late 20th century as fast computers and the bloom of robotics inspired engineers to ask new and difficult questions that have once again drawn the attention of applied mathematicians.

To answer questions from kinematics, as well as algebraic questions from other disciplines such as chemistry and computer graphics, one needs to describe and manipulate the solution sets of systems of polynomial equations. One of several computational techniques for addressing these systems is polynomial continuation. In 1995, Andrew Sommese (Notre Dame) and I coined the term numerical algebraic geometry to describe a new class of algorithms to deal with positive-dimensional solution sets, built on top of existing techniques of polynomial continuation for finding isolated solutions. Past work with Andrew and Jan Verschelde (UIC) includes algorithms to compute irreducible decompositions, membership tests, and the intersection of algebraic varieties. Currently, I'm working with Andrew Sommese, Daniel Bates, and Jon Hauenstein on extensions to these methods and on the software package, Bertini (see below). Among the newer developments, our regeneration methods, which solve systems equation by equation, look particularly promising for solving large, sparse systems.

Work at General Motors

In addition to the mathematical work described in my research publications, I earn my keep, so to speak, by helping GM develop and deploy applications of robotics and related technologies in our manufacturing facilities. An important thrust has been the use of programmable tooling in the production of automotive bodies, allowing flexible production of multiple body styles on the same equipment and faster introduction of new models. A top concern is making this equipment accurate, so that our vehicles are of the highest quality, which effort is connected to my research in robot calibration.

Most recently, I am a member of the GM/NASA team that developed Robonaut2, a.k.a. R2, a humanoid robotic torso. With two 7-DOF (degrees of freedom) arms and highly dexterous hands, it has the capacity to closely mimic human motion. Replete with over 350 motion and force sensors, the robot is an outstanding platform for work in autonomous dexterous manipulation and human interaction. While NASA intends to send a robonaut into space to work alongside of astronauts or as part of a precursor mission to prepare a site for a manned mission, GM has hopes of someday using it in manufacturing. Spin-off technologies may find their way into our products, such as in vehicle safety systems.

Robonaut2 is the product of a NASA and General Motors robotics partnership.

Robonaut2 is designed to work safely with humans.

Read

On April 14, 2010, we announced that Robonaut 2 will go to the International Space Station on the last space shuttle flight (STS-133 mission), currently scheduled for November. Initially, R2 will be installed in the station's Destiny laboratory. The video below gives more information on R2's mission and the preparations that have been made for operation in space.

Polynomial Continuation and Kinematics

For more about the connection between kinematics and polynomials, along with an introduction to polynomial continuation and numerical algebraic geometry, view the slide show (45 min. with audio) recorded at my talk to the SIAM Annual Meeting, Pittsburgh, July 13, 2010. Or you can view a PDF of just the slides.

Publications

• Publication list in PDF format.
• Selected preprints and links.
• Our book:

[Click to enlarge]

The Numerical Solution of Systems of Polynomials Arising in Engineering and Science by Andrew J. Sommese and Charles W. Wampler, II World Scientific, 2005

The book is available from World Scientific and internet booksellers.
Here is a list of errata.
The homotopy continuation codes that accompany the book can be downloaded at: HomLab. The codes provided include the general purpose solver called HomLab (a suite of Matlab m-files) along with m-file codes to be used in working the exercises at the end of each chapter.

Software for polynomial continuation

Our software
• Bertini
  by Daniel Bates, Jon Hauenstein, Andrew Sommese, and Charles Wampler, is a C program for solving polynomial systems.
  Key features:
  • Finds isolated solutions by total degree or multihomogeneous degree homotopies.
  • Implements the latest method, called "regeneration," which efficiently finds isolated solutions by introducing the equations one-by-one.
  • Finds positive dimensional solution sets and breaks them into irreducible components.
  • Has adaptive multiprecision arithmetic for maintaining accuracy in larger problems.
  • Endgames for fast, accurate treatment of singular roots.
  • Simple input file format.
  • Provides for construction of user-defined homotopies, such as parameter homotopies.
  • Supports parallel computing.
• HomLab
  by Charles Wampler, is a suite of MatLab routines for learning about polynomial continuation. Although created for use with the book by Sommese and Wampler, HomLab is a general-purpose solver, fast enough for moderately-sized systems. If you are concerned about speed, numerical accuracy, and user-friendliness, try Bertini. If you want to learn the techniques of polynomial continuation from the inside, HomLab is your entry point.
  
Other people's software
• PHC
  is a code written in Ada, by Jan Verschelde.
  Key features:
  • Treatment of isolated solutions includes polyhedral homotopy (also known as the BKK approach, mixed volume, or polytope method).
  • Treatment of positive-dimensional solutions includes irreducible decomposition and diagonal homotopy.
  • The PHC pages also include a large collection of interesting examples.
• HOM4PS-2.0
  by T.Y. Li, T.L. Lee, and C.H. Tsai. Code for polyhedral homotopy (serial and parallel versions).
  Key features:
  • Very fast polyhedral homotopy method.

Journal Links

I am on the editorial boards of the following:

• Mechanism and Machine Theory
• International Journal of Robotics Research

Other journals of particular interest:

• ASME Journal of Mechanical Design
• ASME Journal of Mechanism and Robotics
• IEEE Transactions on Robotics (T-RO)
• SIAM Journal of Numerical Analysis

Other Links

SIAM Activity Group (SIAG) on Algebraic Geometry.
IEEE Robotics and Automation Society

Contact Info

General Motors R&D Center, MC 480-106-359, 30500 Mound Road, Warren, MI 48092, USA
charles.w.wampler(at)gm.com

Maintained by Charles Wampler/ charles.w.wampler(at)gm.com /revised June 4, 2011