zbMATH — the first resource for mathematics

Robot motion planning: a wild case. (English. Russian original) Zbl 1138.70316
Proc. Steklov Inst. Math. 250, 56-69 (2005); translation from Tr. Mat. Inst. Steklova 250, 64-78 (2005).
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].

70E60 Robot dynamics and control of rigid bodies
53C17 Sub-Riemannian geometry
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
93C10 Nonlinear systems in control theory
Full Text: MNR