26 research outputs found
Cut time in sub-Riemannian problem on Engel group
The left-invariant sub-Riemannian problem on the Engel group is considered.
The problem gives the nilpotent approximation to generic nonholonomic systems
in four-dimensional space with two-dimensional control, for instance to a
system which describes motion of mobile robot with a trailer. The global
optimality of extremal trajectories is studied via geometric control theory.
The global diffeomorphic structure of the exponential mapping is described. As
a consequence, the cut time is proved to be equal to the first Maxwell time
corresponding to discrete symmetries of the exponential mapping
Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry
We consider a series of optimal control problems with 2-dimensional control
lying in an arbitrary convex compact set . The considered problems are
well studied for the case when is a unit disc, but barely studied for
arbitrary . We derive extremals to these problems in general case by
using machinery of convex trigonometry, which allows us to do this identically
and independently on the shape of . The paper describes geodesics in
(i) the Finsler problem on the Lobachevsky hyperbolic plane; (ii)
left-invariant sub-Finsler problems on all unimodular 3D Lie groups (SU(2),
SL(2), SE(2), SH(2)); (iii) the problem of rolling ball on a plane with
distance function given by ; (iv) a series of "yacht problems"
generalizing Euler's elastic problem, Markov-Dubins problem, Reeds-Shepp
problem and a new sub-Riemannian problem on SE(2); and (v) the plane dynamic
motion problem.Comment: 50 pages, 56 figure