Exponential map in the generalized Dido problem

Abstract.The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained. Mathematics Subject IntroductionProblems of sub-Riemannian geometry have been actively studied by geometric control methods, see books [2,3,7,10]. One of the central and hard questions in this domain is a description of cut and conjugate loci. Detailed results on the local structure of conjugate and cut loci were obtained in the 3-dimensional contact case [1,6]. Global results are restricted to symmetric low-dimensional cases, primarily for left-invariant problems on Lie groups (the Heisenberg group [5,22], the growth vector (n, n(n + 1)/2) [9,11,12], the groups SO(3), SU(2), SL(2) and the Lens Spaces [4]).The paper continues this direction of research: we start to study the left-invariant sub-Riemannian problem on the group of motions of a plane SE(2). This problem has important applications in robotics [8] and vision [13]. On the other hand, this is the simplest sub-Riemannian problem where the conjugate and cut loci differ one from another in the neighborhood of the initial point.The main result of the work is an upper bound on the cut time t cut given in Theorem 5.4: we show that for any sub-Riemannian geodesic on SE(2) there holds the estimate t cut ≤ t, where t is a certain function defined on the cotangent space at the identity. In a forthcoming paper [21] we prove that in fact t cut = t. The bound on the cut time is obtained via the study of discrete symmetries of the problem and the corresponding Maxwell points -points where two distinct sub-Riemannian geodesics of the same length intersect one another.This work has the following structure. In Section 1 we state the problem and discuss existence of solutions. In Section 2 we apply Pontryagin Maximum Principle to the problem. The Hamiltonian system for normal extremals is triangular, and the vertical subsystem is the equation of mathematical pendulum. In Section 3Keywords and phrases. Optimal control, sub-Riemannian geometry, differential-geometric methods, left-invariant problem, Lie group, Pontryagin Maximum Principle, symmetries, exponential mapping, Maxwell stratum. * The second author is partially supported by Russian Foundation for Basic Research, and we endow the cotangent space at the identity with special elliptic coordinates induced by the flow of the pendulum, and integrate the normal Hamiltonian system in these coordinates. Sub-Riemannian geodesics are parameterized by Jacobi's functions. In Section 4 we construct a discrete group of symmetries of the exponential mapping by continuation of reflections in the phase cylinder of the pendulum. In the main Section 5 we obtain an explicit description of Maxwell strata corresponding to the group of discrete symmetries, and prove the upper ...

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“…The general construction of elliptic coordinates was developed in [15,16,19], here they are adapted to the problem under consideration.…”

Section: Exponential Mappingmentioning

confidence: 99%

“…In the main Section 5 we obtain an explicit description of Maxwell strata corresponding to the group of discrete symmetries, and prove the upper bound on cut time. This approach was already successfully applied to the analysis of several invariant optimal control problems on Lie groups [15][16][17][18][19][20]. …”

mentioning

confidence: 99%

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Моисеев¹,

Sachkov²

2009

ESAIM: COCV

114

149

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“…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]). The sub-Riemannian problem on SE(2) is the first problem in this series, where a complete solution was obtained (local and global optimality, cut time and cut locus, optimal synthesis).…”

Section: Resultsmentioning

confidence: 99%

“…, corresponding to discrete symmetries ε i . The following upper bound is the main result of work [9]: 13) where t(λ) = min(t 1 ε i (λ)) is the first Maxwell time corresponding to the group of symmetries G. We recall the explicit definition of the function t(λ) below in equations (2.20)-(2.24).…”

Section: Introductionmentioning

confidence: 99%

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

Sachkov¹

2009

ESAIM: COCV

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Abstract.The left-invariant sub-Riemannian problem on the group of motions (rototranslations) of a plane SE(2) is studied. Local and global optimality of extremal trajectories is characterized. Lower and upper bounds on the first conjugate time are proved. The cut time is shown to be equal to the first Maxwell time corresponding to the group of discrete symmetries of the exponential mapping. Optimal synthesis on an open dense subset of the state space is described.Mathematics Subject Classification. 49J15, 93B29, 93C10, 53C17, 22E30.

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“…. , 8 as a frame of left-invariant vector fields on . Consider the left-invariant sub-Riemannian structure ( , Δ, ) defined by 1 , 2 as an orthonormal frame: Δ = span( 1 ( ), 2 ( )), ( , ) = .…”

Section: Letmentioning

confidence: 99%

On the free Carnot (2, 3, 5, 8) group

Gauthier¹,

Sachkov²

2015

PSTA

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Abstract. We consider the free nilpotent Lie algebra with 2 generators, of step 4, and the corresponding connected simply connected Lie group , with the aim to study the left-invariant sub-Riemannian structure on defined by the generators of as an orthonormal frame.We compute two vector field models of by polynomial vector fields in R 8 , and find an infinitesimal symmetry of the sub-Riemannian structure. Further, we compute explicitly the product rule in and the right-invariant frame on .

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Exponential map in the generalized Dido problem

Cited by 51 publications

References 32 publications

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

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

On the free Carnot (2, 3, 5, 8) group

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