an:05177991
Zbl 1138.70316
Gauthier, J. P.; Zakalyukin, V. M.
Robot motion planning: a wild case
EN
Proc. Steklov Inst. Math. 250, 56-69 (2005); translation from Tr. Mat. Inst. Steklova 250, 64-78 (2005).
2005
a
70E60 53C17 70G45 93C10
Summary: A basic problem in robotics is a constructive motion planning problem: given an arbitrary (nonadmissible) trajectory \(\Gamma\) of a robot, find an admissible \(\varepsilon\)-approximation (in the sub-Riemannian (SR) sense) \(\gamma(\varepsilon)\) of \(\Gamma\) that has the minimal sub-Riemannian length. Then, the (asymptotic behavior of the) sub-Riemannian length \(L(\gamma (\varepsilon))\) is called the metric complexity of \(\Gamma\) (in the sense of Jean). We have solved this problem in the case of an SR metric of corank 3 at most. For coranks greater than 3, the problem becomes much more complicated. The first really critical case is the 4-10 case (a four-dimensional distribution in \(\mathbb {R}^{10}\). Here, we address this critical case. We give partial but constructive results that generalize, in a sense, the results of our previous papers.
For the entire collection see [Zbl 1116.37001].