What, you say, are you talking about? Robots? Have you completely lost it? Not at all, all shall soon become clear. I was looking over in the arXiv today at recent math.AT goings-on when I saw a pile of papers by Michael Farber, who I met at the very cold but quite enjoyable Prospects in Mathematics conference in Durham last winter. He found out that I went in for algebraic topology and gave me some survey papers of his on the field of the application of topology to robot motion planning.

How do you see the relevance of topology to motion-planning? Suppose you have a robotic arm with a few elbows on it. Then the position can be set by specifying some parameters (say, the angle and rotation) at each elbow, and the set of all possible positions is the configuration space, . Let be the space of all continuous paths in , and send a path to the ordered pair of points given by the initial and final points of the path. This will now be a (Serre) fibration, so suddenly we’re doing algebraic topology. Now a motion planning algorithm would take an initial and final point and give you a path between them. This is just a section of this fibration! Thus a “continuous” motion planning algorithm is just a continous section. It is fairly easy to see that such an algorithm exists iff is contractible (you can build a retraction to a point), so generally doesn’t exist. Thus the interesting thing is to see how discontinuous they need to be.

Farber introduces a numerical invariant called the topological complexity which has a few equivalent (for nice enough spaces) definitions, one of which is “the smallest number of disjoint sets that cover the space and on which the section is continuous” along with some technicalities (the sets should be Euclidean Neighbourhood Retracts). There are then bounds for this number in terms of the topological properties of X, but in particular a lower bound in terms of the cohomology, in the form of the zero-divisors-cup-length of the cohomology of the space. Pretty cool stuff.

Now of course you’d like to know the (co)homology of some configuration spaces. That’s exactly what you’ll find in Farber/Schütz “Homology of planar polygon spaces”.