Topological robotics: motion planning in projective spaces

In this paper we study one of the most elementary problems of the topological robotics: rotation of a line, which is fixed by a revolving joint at a base point: one wants to bring the line from its initial position A to a final position B by a continuous motion in the space. The final goal is to construct a motion planning algorithm which will perform this task once the initial position A and the final position B are presented. This problem becomes hard when the dimension of the space is large. Any such motion planning algorithm must have instabilities, i.e. the motion of the system will be discontinuous as a function of A and B. These instabilities are caused by topological reasons. A general approach to study instabilities of robot motion was suggested recently in papers [7, 8]. With any path-connected topological space X one associates in [7, 8] a number TC(X), called the topological complexity of X . This number is of fundamental importance for the motion planning problem: TC(X) determines character of instabilities which have all motion planning algorithms in X . The motion planning problem of moving a line inR reduces to a topological problem of calculating the topological complexity of the real projective space TC(RP), which we tackle in this paper. We compute the number TC(RP) for all n ≤ 23 (see the table in section 6). This will probably be useful for applications. Surprisingly, the problem of finding a general formula for TC(RP) turns out to be quit difficult. One of the main results of this paper claims that for n 6= 1, 3, 7, the number TC(RP) coincides with the smallest integer k such that the projective space RP admits an immersion into R. This means that our problem of finding TC(RP) is equivalent to the immersion problem for real projective spaces. The latter is a famous, classical, well researched, topological problem, see [5, 10] for a survey. A general answer to the immersion