But suppose your arm had six or seven joints, with sections of varying lengths. Or suppose you had four or five or six arms. Could you straighten them all out -- or just move them from one arrangement to another -- without having them bump into each other or get tangled up? What if someone else rearranged them when you weren't looking?
Now you can appreciate some of the issues facing the designers and programmers of industrial robots. A group of mathematicians and computer scientists will try to throw some light on what turns out to be a curiously complex mathematical challenge during a symposium at the annual meeting of the American Association for the Advancement of Science, Feb. 14-19 at the Sheraton Boston and Boston Marriott Copley Place hotels. The symposium, "Robot Arm Manipulation: Geometric Challenges," will be held at 3 p.m. Saturday, Feb. 16.
"It's one part of the application of the theory of rigid structures to problems that come up in engineering and biology," said Robert Connelly, professor of mathematics at Cornell University, Ithaca, N.Y., who organized the symposium.
In addition to robot arms, the mathematics applies to such problems as the unfolding of antennas on satellites and the folding of protein chains, which can be seen as a straightening problem in reverse. "The symposium would be a success if one person said 'I can use these results to understand a problem,' " Connelly says
Connelly's specialty is geometry. All the other speakers in the symposium are computer scientists working in the relatively new field of computational geometry, in which computers manipulate mathematical representations of geometric objects. When it comes to jointed arms, mathematicians deal with abstract versions they call "polygonal arcs," which are geometric figures made up of straight line segments connected to other straight line segments at their ends. It has become a sort of game for one mathematician to draw a figure and for others to try to prove that it can be straightened out without one part bumping into another.
Connelly and Erik Demaine, assistant professor of electrical engineering and computer science at the Massachusetts Institute of Technology, and GŁnter Rote of the Free University of Berlin have made a small breakthrough, proving mathematically that any arc in a plane can be straightened. They also have developed an algorithm that straightens an arc -- or, sometimes, an arm. Simply, the algorithm says that the distance between all points of the arc must constantly increase.
The proof also applies to the "convexifying" of polygons: a closed shape made up of jointed lines may be compressed, folded and distorted into what looks like a maze, but the mathematical proof shows that any such figure can be unfolded to an open polygon, provided none of the lines cross.
The proof will be published in a forthcoming special issue of the journal Discrete and ComputationalGeometry.
Unfortunately, Connelly says, it is still possible to tangle up an arm in three dimensions in such a way that it can't be straightened, and multiple arms are a whole new set of problems.
In addition to Connelly, speakers in the symposium include Demaine, who is expected to offer some updates on the recent work; Sue Whitesides, McGill University associate professor of computer science, discussing practical applications of the new algorithm; and Ileana Streinu, associate professor of computer science and chair of the Department of Computer Science at Smith College, with an alternative algorithm.
Related World Wide Web sites: The following sites provide additional information on this news release. Some might not be part of the Cornell University community, and Cornell has no control over their content or availability.
o Robert Connelly's home page, with links to examples: http://www.math.cornell.edu/~connelly/
o Erik Demaine's page on the problem: http://db.uwaterloo.ca/~eddemain/linkage/
o AAAS annual meeting information: http://xserver.aaas.org/meetings/
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