Robot Motion Planning: A Wild Case

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.