Seven pictures from Applied Algebra and Geometry: Picture #2
The Society for Industrial and Applied Mathematics, SIAM, has recently released a journal of Applied Algebra and Geometry called SIAGA. The poster for this journal (see here) features seven pictures. In my last blog post, I wrote about the first picture. Here, I describe the mathematics represented by the second picture.
In the first section of this post “The Context”, I’ll set the mathematical scene. In the second section “The Picture” I’ll talk about the specifics of this particular image, representing Robotics.
Robotics-the design of mechanical machines to perform complex tasks-is a burgeoning field of study. The range and precision of robots’ motion are limited by the mechanics of their constituent parts, which traditionally consist of rigid pieces connected by joints. The interplay between robotics, algebra, and geometry arises naturally from the kinematics of these pieces working together. In fact, numerical algebraic geometry largely arose from these applications. For an introduction, see Chapter 6 of the undergraduate textbook “Ideals, Varieties and Algorithms” by Cox, Little and O’Shea (2007).
A rigid part floating unconstrained in three-dimensional space has six degrees of freedom, but the joints of a mechanism restrict its motion. For most joints used in robotics, polynomial equations can describe these restrictions. With multiple pieces working together, one task is to find the location of a selected terminating part. For example, for fixed locations and angles of your shoulder, elbow, wrist and carpometacarpal joint, what is the location of your thumb? This is known as forward kinematics. Furthermore, given a desired location of your thumb, what are the possible angles of your arm and hand that would achieve that location. This is inverse kinematics.
The numerical algebraic geometry tool of homotopy continuation can help answer these questions. In homotopy continuation, we first solve an easier but related set of polynomial equations. Then we deform the easier system to the more challenging problem of interest via a homotopy map. We use the solutions of the easier problem to obtain those of the harder problem by numerically tracking the paths of the original solutions as they deform under the homotopy. For more, see “Numerically Solving Polynomial Systems with Bertini” by Bates, Hauenstein, Sommese, and Wampler (SIAM Books, 2013).
The image depicts a special combination of rigid pieces and joints called a Griffis-Duffy Type I Platform. It consists of two equilateral triangles, one fixed at the base and the other held above it by six rigid legs. Each leg connects a vertex of one triangle to a midpoint of an edge on the other. Although the lengths of each leg are fixed, the angles at each joint are free to move.
The geometry of this problem yields a system of polynomial equations that describes its kinematics: if we fix the point shown on top of the upper triangle, the collection of positions it can reach is the red curve, which is an algebraic set of degree 40.
The picture was created by Charles Wampler of General Motors and Douglas Arnold
of the University of Minnesota. It appeared on the poster for the IMA Thematic Year on Applications of Algebraic Geometry in 2006-07, in which significant progress was made connecting the use of algebraic geometry tools to industrial and applied mathematics.