Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane

Abstract

Abstract.The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE (2) , extremal trajectories were defined, their local and global optimality were studied. In this paper the global structure of the exponential mapping is described. On this basis an explicit characterization of the cut locus and Maxwell set is obtained. The optimal synthesis is constructed.Mathematics Subject Classification. 49J15, 93B29, 93C10, 53C17, 22E30.

Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

“…The solution to the sub-Riemannian problem on SE(2) obtained in the previous paper [9], this one, and the subsequent paper [19] is based on a detailed study of the action of the discrete group of symmetries of the exponential mapping. This techniques was already partially developed in the study of related optimal control problems (nilpotent sub-Riemannian problem with the growth vector (2, 3, 5) [13][14][15][16] and Euler's elastic problem [17,18]).…”

“…It will play a crucial role in the subsequent analysis of the global structure of the exponential mapping in Section 3 and in the subsequent work [19].…”

“…Figure 12 shows the sphere S π/2 in rectifying coordinates (R 1 , R 2 , θ). Figure 13 represents the same sphere with a cut-out opening the singularities of the sphere: the sphere intersects itself at the local components of the cut locus Cut loc described in [19]. Figure 14 shows embedding of the same sphere to the solid torus.…”

“…In the subsequent work [19] we complete our study of problem (1.1)-(1.5). There we describe explicitly the Maxwell strata and cut locus, and characterize the optimal synthesis in this problem.…”

Cut Locus and Optimal Synthesis in Sub-Riemannian Problem on the Lie Group SH(2)

“…In this paper we extend the global optimality analysis similar to [9]. We decompose the image M = SH (2) and the preimage of the exponential mapping into open dense sets based on the Maxwell strata and conjugate loci and prove that the exponential mapping between these sets is a diffeomorphism.…”