sub-Riemannian Problem Research Articles

Numerical Solution of a Left-Invariant Sub-Riemannian Problem on the Group $\mathrm{SO}(3)$

We consider a left-invariant sub-Riemannian problem on the Lie group of rotations of a three-dimensional space. We find the cut locus numerically, in fact we construct the optimal synthesis numerically, i. e., the shortest arcs. The software package nutopy designed for the numerical solution of optimal control problems is used. With the help of this package we investigate sub-Riemannian geodesics, conjugate points, Maxwell points and diffeomorphic domains of the exponential map. We describe some operating features of this software package.

Time-Optimal Problem in the Roto-Translation Group with Admissible Control in a Circular Sector

We study a time-optimal problem in the roto-translation group with admissible control in a circular sector. The problem reveals the trajectories of a car model that can move forward on a plane and turn with a given minimum turning radius. Our work generalizes the sub-Riemannian problem by adding a restriction on the velocity vector to lie in a circular sector. The sub-Riemannian problem is given by a special case when the sector is the full disc. The trajectories of the system are applicable in image processing to detect salient lines. We study the local and global controllability of the system and the existence of a solution for given arbitrary boundary conditions. In a general case of the sector opening angle, the system is globally but not small-time locally controllable. We show that when the angle is obtuse, a solution exists for any boundary conditions, and when the angle is reflex, a solution does not exist for some boundary conditions. We apply the Pontryagin maximum principle and derive a Hamiltonian system for extremals. Analyzing a phase portrait of the Hamiltonian system, we introduce the rectified coordinates and obtain an explicit expression for the extremals in Jacobi elliptic functions. We show that abnormal extremals are of circular type, and they correspond to motions of a car along circular arcs of minimal possible radius. The normal extremals in a general case are given by concatenation of segments of sub-Riemannian geodesics in SE2 and arcs of circular extremals. We show that, in a general case, the vertical (momentum) part of the extremals is periodic. We partially study the optimality of the extremals and provide estimates for the cut time in terms of the period of the vertical part.

Abnormal Extremals in the Sub-Riemannian Problem with Growth Vector $(2,3,5,8,14)$

We study the left-invariant sub-Riemannian problem on the free nilpotent Lie group of rank $2$ and step $5$. We describe some abnormal trajectories and some properties of the set filled by nice abnormal trajectories starting at the identity of the group.

Abnormal extremals in the sub-Riemannian problem for a general model of a robot with a trailer

A symmetric mathematical model of a wheeled robot with a trailer is considered for various types of coupling between the robot and the trailer. It is shown that for fixed coupling parameters and fixed initial position of the robot with trailer there are two symmetric abnormal extremals. In motion along these trajectories the robot and the trailer traverse normal extremal trajectories for the sub-Riemannian problem on the group of motions of the plane; the coupling point always draws an inflectional elastica or a straight line. Bibliography: 33 titles.

New second-order optimality conditions in sub-riemannian geometry

We study the geometry of the second-order expansion of the extended end-point map for the sub-Riemannian geodesic problem. Translating the geometric reality into equations we derive new second-order necessary optimality conditions in sub-Riemannian Geometry. In particular, we find an ODE for velocity of an abnormal sub-Riemannian geodesics. It allows to divide abnormal minimizers into two classes, which we propose to call 2-normal and 2-abnormal extremals. In the 2-normal case the above ODE completely determines the velocity of a curve, while in the 2-abnormal case the velocity is undetermined at some, or at all points. With some enhancement of the presented results it should be possible to prove the regularity of all 2-normal extremals (the 2-abnormal case seems to require study of higher-order conditions) thus making a step towards solving the problem of smoothness of sub-Riemannian abnormal geodesics. As a by-product we present a new derivation of Goh conditions. We also prove that the assumptions weaker than these used in [Boarotto, Monti, Palmurella, 2020] to derive third-order Goh conditions, imply piece-wise-C2 regularity of an abnormal extremal. Marty me i Zuzi

Abnormal extremals in the sub-Riemannian problem for a general model of a robot with a trailer

Рассматривается симметричная математическая модель колесного робота с прицепом с различными видами сцепки робота и прицепа. Исследуется соответствующая субриманова задача в анормальном случае принципа максимума Понтрягина. Доказано, что при фиксированных параметрах сцепки и начальном положении робота с прицепом существуют две симметричные анормальные экстремали. При движении вдоль этих траекторий робот и прицеп следуют вдоль нормальных экстремальных траекторий для субримановой задачи на группе движений плоскости, при этом точка сцепки всегда описывает инфлексионную эластику либо прямую. Библиография: 33 названия.

Cut time in the sub-Riemannian problem on the Cartan group

We study the sub-Riemannian structure determined by a left-invariant distribution of rank 2 on a step 3 Carnot group of dimension 5. We prove the conjectured cut times of Yu. Sachkov for the sub-Riemannian Cartan problem. Along the proof, we obtain a comparison with the known cut times in the sub-Riemannian Engel group, and a sufficient (generic) condition for the uniqueness of the length minimizer between two points. Hence we reduce the optimal synthesis to solving a certain system of equations in elliptic functions.

Correction to: Conjugate Time in the Sub-Riemannian Problem on the Cartan Group

A correction to this paper has been published: https://doi.org/10.1007/s10883-021-09569-8 .

Conjugate Time in the Sub-Riemannian Problem on the Cartan Group

The Cartan group is the free nilpotent Lie group of rank 2 and step 3. We consider the left-invariant sub-Riemannian problem on the Cartan group defined by an inner product in the first layer of its Lie algebra. This problem gives a nilpotent approximation of an arbitrary sub-Riemannian problem with the growth vector (2,3,5). In previous works, we described a group of symmetries of the sub-Riemannian problem on the Cartan group, and the corresponding Maxwell time — the first time when symmetric geodesics intersect one another. It is known that geodesics are not globally optimal after the Maxwell time. In this work, we study local optimality of geodesics on the Cartan group. We prove that the first conjugate time along a geodesic is not less than the Maxwell time corresponding to the group of symmetries. We characterize geodesics for which the first conjugate time is equal to the first Maxwell time. Moreover, we describe continuity of the first conjugate time near infinite values.

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 Ω. The considered problems are well studied for the case when Ω is a unit disc, but barely studied for arbitrary Ω. 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 Ω. 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 Ω; (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.

The structure of abnormal extremals in a sub-Riemannian problem with growth vector

A left-invariant sub-Riemannian problem on a free nilpotent Lie group of step 4 with two generators is considered. The structure of abnormal extremals is described. These extremals are shown to define an abnormal foliation of the annihilator of the square of the distribution, which is given by the intersections of this annihilator with the leaves of the symplectic foliation of the Lie coalgebra. The question of abnormal trajectories being strictly/nonstrictly abnormal is investigated, their projection onto a plane of the distribution are described, estimates for the corank are given and examples of nonsmooth trajectories are constructed. Bibliography: 14 titles.

An instability criterion for volume-preserving area-stationary surfaces with singular curves in sub-Riemannian 3-space forms

We study stable surfaces, i.e., second order minima of the area for variations of fixed volume, in sub-Riemannian space forms of dimension 3. We prove a stability inequality and provide sufficient conditions ensuring instability of volume-preserving area-stationary $$C^2$$ surfaces with a non-empty singular set of curves. Combined with previous results, this allows to describe any complete, orientable, embedded and stable $$C^2$$ surface $$\Sigma $$ in the Heisenberg group $${\mathbb {H}}^1$$ and the sub-Riemannian sphere $${\mathbb {S}}^3$$ of constant curvature 1. In $${\mathbb {H}}^1$$ we conclude that $$\Sigma $$ is a Euclidean plane, a Pansu sphere or congruent to the hyperbolic paraboloid $$t=xy$$ . In $${\mathbb {S}}^3$$ we deduce that $$\Sigma $$ is one of the Pansu spherical surfaces discovered in Hurtado and Rosales (Math Ann 340(3):675–708, 2008). As a consequence, such spheres are the unique $$C^2$$ solutions to the sub-Riemannian isoperimetric problem in $${\mathbb {S}}^3$$ .

Asymptotics of Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space

Asymptotics of Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space

Dynamic interpolation for obstacle avoidance on Riemannian manifolds

ABSTRACT This work is devoted to studying dynamic interpolation for obstacle avoidance. This is a problem that consists of minimising a suitable energy functional among a set of admissible curves subject to some interpolation conditions. The given energy functional depends on velocity, covariant acceleration and on artificial potential functions used for avoiding obstacles. We derive first-order necessary conditions for optimality in the proposed problem; that is, given interpolation and boundary conditions we find the set of differential equations describing the evolution of a curve that satisfies the prescribed boundary values, interpolates the given points and is an extremal for the energy functional. We study the problem in different settings including a general one on a Riemannian manifold and a more specific one on a Lie group endowed with a left-invariant metric. We also consider a sub-Riemannian problem.

Symmetries and Parameterization of Abnormal Extremals in the Sub-Riemannian Problem with the Growth Vector (2, 3, 5, 8)

Symmetries and Parameterization of Abnormal Extremals in the Sub-Riemannian Problem with the Growth Vector (2, 3, 5, 8)

Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater

One of the main approaches to investigating sub-Riemannian problems is Mitchell's theorem on nilpotent approximation, which reduces the analysis of a neighbourhood of a regular point to the analysis of the left-invariant sub-Riemannian problem on the corresponding Carnot group. Usually, the in-depth investigation of sub-Riemannian shortest paths is based on integrating the Hamiltonian system of Pontryagin's maximum principle explicitly. We give new formulae for sub-Riemannian geodesics on a Carnot group with growth vector and prove that left-invariant sub-Riemannian problems on free Carnot groups of step 4 or greater are Liouville nonintegrable. Bibliography: 30 titles.

Cut Locus in the Sub-Riemannian Problem on Engel Group

This paper considers the left-invariant sub-Riemannian problem on the Engel group. The stratification of the cut locus and the structure of optimal synthesis are described.

Maxwell Strata and Cut Locus in the Sub-Riemannian Problem on the Engel Group

We consider the nilpotent left-invariant sub-Riemannian structure on the Engel group. This structure gives a fundamental local approximation of a generic rank 2 sub-Riemannian structure on a 4-manifold near a generic point (in particular, of the kinematic models of a car with a trailer). On the other hand, this is the simplest sub-Riemannian structure of step three. We describe the global structure of the cut locus (the set of points where geodesics lose their global optimality), the Maxwell set (the set of points that admit more than one minimizer), and the intersection of the cut locus with the caustic (the set of conjugate points along all geodesics). The group of symmetries of the cut locus is described: it is generated by a one-parameter group of dilations $\mathbb{R}_+$ and a discrete group of reflections $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimen-sional strata of the cut locus are Maxwell strata of multiplicity 2 (for each point there are 2 minimizers). Two-dimensional strata of the cut locus consist of conjugate points. Finally, one-dimensional strata are Maxwell strata of infinite multiplicity, they consist of conjugate points as well. Projections of sub-Riemannian geodesics to the 2-dimensional plane of the distribution are Euler elasticae. For each point of the cut locus, we describe the Euler elasticae corresponding to minimizers coming to this point. Finally, we describe the structure of the optimal synthesis, i.e., the set of minimizers for each terminal point in the Engel group.

Extremal Controls in the Sub-Riemannian Problem on the Group of Motions of Euclidean Space

For the sub-Riemannian problem on the group of motions of Euclidean space we present explicit formulas for extremal controls in a special case, when one of the initial momenta is fixed.

Symmetric Riemannian Problem on the Group of Proper Isometries of Hyperbolic Plane

We consider the Lie group PSL(2) (the group of orientation preserving isometries of the hyperbolic plane) and a left-invariant Riemannian metric on this group with two equal eigenvalues that correspond to space-like eigenvectors (with respect to the Killing form). For such metrics we find a parametrization of geodesics, the conjugate time, the cut time and the cut locus. The injectivity radius is computed. We show that the cut time and the cut locus in such Riemannian problem converge to the cut time and the cut locus in the corresponding sub-Riemannian problem as the third eigenvalue of the metric tends to infinity. Similar results are also obtained for SL(2).