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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 R+ and a discrete group of reflections Z2 × Z2 × Z2. The cut locus admits a stratification with 6 three-dimensional strata, 12 two-dimensional strata, and 2 one-dimensional strata. Three-dimensional 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. [ABSTRACT FROM AUTHOR]

This work studies a number of approaches to solving the motion planning problem for a mobile robot with a trailer. Different control models of car-like robots are considered from the differential-geometric point of view. The same models can also be used for controlling a mobile robot with a trailer. However, in cases where the position of the trailer is of importance, i.e., when it is moving backward, a more complex approach should be applied. At the end of the article, such an approach, based on recent works in sub-Riemannian geometry, is described. It is applied to the problem of reparking a trailer and implemented in the algorithm for parking a mobile robot with a trailer. [ABSTRACT FROM AUTHOR]

The left-invariant sub-Riemannian problem on the Engel group is considered. The problem gives the nilpotent approximation to generic rank two sub-Riemannian problems on four-dimensional manifolds. 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. [ABSTRACT FROM AUTHOR]

We study the geometric structures associated with curvature radii of curves with values on a Riemannian manifold (M, g). We show the existence of sub-Riemannian manifolds naturally associated with the curvature radii and we investigate their properties. In the particular case of surfaces these sub-Riemannian structures are of Engel type. The main character of our construction is a pair of global vector fields f 1 , f 2 , which encodes intrinsic information on the geometry of (M, g). [ABSTRACT FROM AUTHOR]

Abnormal trajectories are of particular interest for sub-Riemannian geometry, because the most complicated singularities of the sub-Riemannian metric are located just near such trajectories. Important open questions in sub-Riemannian geometry are to establish whether the abnormal length minimizers are smooth and to describe the set filled with abnormal trajectories starting from a fixed point. For example, the Sard conjecture in sub-Riemannian geometry states that this set has measure zero. In this paper, we consider this and other related properties of such a set for the left-invariant sub-Riemannian problem with growth vector . We also study the global and local optimality of abnormal trajectories and obtain their explicit parametrization. [ABSTRACT FROM AUTHOR]

The structure of the intersection of the sub-Riemannian sphere on the Cartan group with a 3-dimensional invariant manifold of main symmetries is described. [ABSTRACT FROM AUTHOR]

The space of -jets of a real function of one real variable admits the structure of Carnot group type. As such, admits a submetry (sub-Riemannian submersion) onto the Euclidean plane. Horizontal lifts of Euclidean lines (which are the left translates of horizontal one-parameter subgroups) are thus globally minimizing geodesics on . All -geodesics, minimizing or not, are constructed from degree polynomials in according to [7-8, 9], reviewed here. The constant polynomials correspond to the horizontal lifts of lines. Which other polynomials yield globally minimizers and what do these minimizers look like? We give a partial answer. Our methods include constructing an intermediate three-dimensional "magnetic" sub-Riemannian space lying between the jet space and the plane, solving a Hamilton – Jacobi (eikonal) equations on this space, and analyzing period asymptotics associated to period degenerations arising from two-parameter families of these polynomials. Along the way, we conjecture the independence of the cut time of any geodesic on jet space from the starting location on that geodesic. [ABSTRACT FROM AUTHOR]

We describe all Carnot algebras with growth vector , their normal forms, an invariant that distinguishes them, and a basis change that reduces such an algebra to a normal form. For every normal form, we calculate the Casimir functions and symplectic foliations on the Lie coalgebra. We describe the invariant and the normal forms of left-invariant -distributions. We also obtain a classification of all left-invariant sub-Riemannian structures on -Carnot groups up to isometry and present models of these structures. [ABSTRACT FROM AUTHOR]

We consider left-invariant optimal control problems on connected Lie groups. The Pontryagin maximum principle gives necessary optimality conditions. Namely, the extremal trajectories are the projections of trajectories of the corresponding Hamiltonian system on the cotangent bundle of the Lie group. The Maxwell points (i.e., the points where two different extremal trajectories meet each other) play a key role in the study of optimality of extremal trajectories. The reason is that an extremal trajectory cannot be optimal after a Maxwell point. We introduce a general construction for Maxwell points depending on the algebraic structure of the Lie group. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3 and 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3-step nilpotent Lie groups, we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of time-optimal problems on 3-step free Carnot groups, we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group. [ABSTRACT FROM AUTHOR]

The structure of the intersection of the sub-Riemannian sphere on the Engel group with a two-dimensional invariant set of discrete symmetries is described: regularity, analytic properties, exp-log category, Whitney stratification, multiplicity of points, characterization in terms of abnormal trajectories, conjugate points and Maxwell points, and explicit expressions for the sub-Riemannian distance to singular points. [ABSTRACT FROM AUTHOR]

We describe all Carnot algebras with growth vector (2, 3, 5, 6), their normal forms, an invariant that separates them, and a change of basis that transforms such an algebra into a normal form. For each normal form, Casimir functions and symplectic foliations on the Lie coalgebra are computed. An invariant and normal forms of left-invariant (2, 3, 5, 6)-distributions are described. A classification, up to isometries, of all left-invariant sub-Riemannian structures on (2, 3, 5, 6)-Carnot groups is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we present a novel model of the primary visual cortex (V1) based on orientation, frequency, and phase selective behavior of V1 simple cells. We start from the first-level mechanisms of visual perception, receptive profiles. The model interprets V1 as a fiber bundle over the two-dimensional retinal plane by introducing orientation, frequency, and phase as intrinsic variables. Each receptive profile on the fiber is mathematically interpreted as rotated, frequency modulated, and phase shifted Gabor function. We start from the Gabor function and show that it induces in a natural way the model geometry and the associated horizontal connectivity modeling of the neural connectivity patterns in V1. We provide an image enhancement algorithm employing the model framework. The algorithm is capable of exploiting not only orientation but also frequency and phase information existing intrinsically in a two-dimensional input image. We provide the experimental results corresponding to the enhancement algorithm. [ABSTRACT FROM AUTHOR]

In the setting of step 2 sub-Finsler Carnot groups with strictly convex norms, we prove that all infinite geodesics are lines. It follows that for any other homogeneous distance, all geodesics are lines exactly when the induced norm on the horizontal space is strictly convex. As a further consequence, we show that all isometric embeddings between such homogeneous groups are affine. The core of the proof is an asymptotic study of the extremals given by the Pontryagin Maximum Principle. [ABSTRACT FROM AUTHOR]

In the setting of step 2 sub-Finsler Carnot groups with strictly convex norms, we prove that all infinite geodesics are lines. It follows that for any other homogeneous distance, all geodesics are lines exactly when the induced norm on the horizontal space is strictly convex. As a further consequence, we show that all isometric embeddings between such homogeneous groups are affine. The core of the proof is an asymptotic study of the extremals given by the Pontryagin Maximum Principle. [ABSTRACT FROM AUTHOR]

We present a PDE-based approach for finding optimal paths for the Reeds-Shepp car. In our model we minimize a (data-driven) functional involving both curvature and length penalization, with several generalizations. Our approach encompasses the two- and three-dimensional variants of this model, state-dependent costs, and moreover, the possibility of removing the reverse gear of the vehicle. We prove both global and local controllability results of the models. Via eikonal equations on the manifold Rd×Sd-1 we compute distance maps w.r.t. highly anisotropic Finsler metrics, which approximate the singular (quasi)-distances underlying the model. This is achieved using a fast-marching (FM) method, building on Mirebeau (Numer Math 126(3):515-557, 2013; SIAM J Numer Anal 52(4):1573-1599, 2014). The FM method is based on specific discretization stencils which are adapted to the preferred directions of the Finsler metric and obey a generalized acuteness property. The shortest paths can be found with a gradient descent method on the distance map, which we formalize in a theorem. We justify the use of our approximating metrics by proving convergence results. Our curve optimization model in Rd×Sd-1 with data-driven cost allows to extract complex tubular structures from medical images, e.g., crossings, and incomplete data due to occlusions or low contrast. Our work extends the results of Sanguinetti et al. (Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications LNCS 9423, 2015) on numerical sub-Riemannian eikonal equations and the Reeds-Shepp car to 3D, with comparisons to exact solutions by Duits et al. (J Dyn Control Syst 22(4):771-805, 2016). Numerical experiments show the high potential of our method in two applications: vessel tracking in retinal images for the case d=2 and brain connectivity measures from diffusion-weighted MRI data for the case d=3, extending the work of Bekkers et al. (SIAM J Imaging Sci 8(4):2740-2770, 2015). We demonstrate how the new model without reverse gear better handles bifurcations. [ABSTRACT FROM AUTHOR]

For the sub-Riemannian problem on the group of motions of Euclidean space we present explicit formulas for extremal controls in the special case where one of the initial momenta is fixed. [ABSTRACT FROM AUTHOR]

The notion of curvature discussed in this paper is a far-going generalization of the Riemannian sectional curvature. It was first introduced by Agrachev et al. ([2015]), and it is defined for a wide class of optimal control problems: a unified framework including geometric structures such as Riemannian, sub-Riemannian, Finsler, and sub-Finsler structures. In this work, we study the generalized sectional curvature of Carnot groups with rank-two distributions. In particular, we consider the Cartan group and Carnot groups with horizontal distribution of Goursat-type. In these Carnot groups, we characterize ample and equiregular geodesics. For Carnot groups with horizontal Goursat distribution, we show that their generalized sectional curvatures depend only on the Engel part of the distribution. This family of Carnot groups contains naturally the three-dimensional Heisenberg group, as well as the Engel group. Moreover, we also show that in the Engel and Cartan groups, there exist initial covectors for which there is an infinite discrete set of times at which the corresponding ample geodesics are not equiregular. [ABSTRACT FROM AUTHOR]

Global optimality analysis in sub-Riemannian problem on the Lie group SH(2) is considered. We cutout open dense domains in the preimage and in the image of the exponential mapping based on the description of Maxwell strata. We then prove that the exponential mapping restricted to these domains is a diffeomorphism. Based on the proof of diffeomorphism, the cut time, i.e., time of loss of global optimality, is computed on SH(2). We also consider the global structure of the exponential mapping and obtain an explicit description of cut locus and optimal synthesis. [ABSTRACT FROM AUTHOR]

This paper is concerned with two systems from sub-Riemannian geometry. One of them is defined by a Carnot group with three generatrices and growth vector (3, 6, 14), the other is defined by two generatrices and growth vector (2, 3, 5, 8). Using a Poincaré map, the nonintegrability of the above systems in the general case is shown. In addition, particular cases are presented in which there exist additional first integrals. [ABSTRACT FROM AUTHOR]

We study local and global optimality of geodesics in the left invariant sub-Riemannian problem on the Lie group SH(2). We obtain the complete description of the Maxwell points corresponding to the discrete symmetries of the vertical subsystem of the Hamiltonian system. An effective upper bound on the cut time is obtained in terms of the first Maxwell times. We study the local optimality of extremal trajectories and prove the lower and upper bounds on the first conjugate times. We also obtain the generic time interval for the n-th conjugate time which is important in the study of sub-Riemannian wavefront. Based on our results of n-th conjugate time and n-th Maxwell time, we prove a generalization of Rolle's theorem that between any two consecutive Maxwell points, there is exactly one conjugate point along any geodesic. [ABSTRACT FROM AUTHOR]

We consider the problem P of minimizing $\int \limits _{0}^{L} \sqrt {\xi ^{2} + \kappa ^{2}(s)} \, \mathrm {d}s$ for a curve x in $\mathbb {R}^{3}$ with fixed boundary points and directions. Here, the total length L≥0 is free, s denotes the arclength parameter, κ denotes the absolute curvature of x, and ξ>0 is constant. We lift problem P on $\mathbb {R}^{3}$ to a sub-Riemannian problem P on SE(3)/({ 0}×SO(2)). Here, for admissible boundary conditions, the spatial projections of sub-Riemannian geodesics do not exhibit cusps and they solve problem P . We apply the Pontryagin Maximum Principle (PMP) and prove Liouville integrability of the Hamiltonian system. We derive explicit analytic formulas for such sub-Riemannian geodesics, relying on the co-adjoint orbit structure, an underlying Cartan connection, and the matrix representation of SE(3) arising in the Cartan-matrix. These formulas allow us to extract geometrical properties of the sub-Riemannian geodesics with cuspless projection, such as planarity conditions, explicit bounds on their torsion, and their symmetries. Furthermore, they allow us to parameterize all admissible boundary conditions reachable by geodesics with cuspless spatial projection. Such projections lay in the upper half space. We prove this for most cases, and the rest is checked numerically. Finally, we employ the formulas to numerically solve the boundary value problem, and visualize the set of admissible boundary conditions. [ABSTRACT FROM AUTHOR]

We consider the left-invariant sub-Riemannian and Riemannian structures on the Heisenberg groups. A classification of these structures was found previously. In the present paper, we find (for each normalized structure) the isometry group, the exponential map, the totally geodesic subgroups, and the conjugate locus. Finally, we determine the minimizing geodesics from identity to any given endpoint. (Several of these points have been covered, to varying degrees, by other authors.) [ABSTRACT FROM AUTHOR]

Under consideration is the sub-Riemannian geodesic flow for the Goursat distribution. We find the level surfaces of the first integrals that are in involution and study the trajectories in the phase space whose projections to the horizontal plane are closed curves. [ABSTRACT FROM AUTHOR]

We present a new flexible wavefront propagation algorithm for the boundary value problem for sub- Riemannian (SR) geodesics in the roto-translation group SE(2) = R2 × S1 with a metric tensor depending on a smooth external cost C : SE(2) → [δ, 1], δ > 0, computed from image data. The method consists of a first step where an SR-distance map is computed as a viscosity solution of a Hamilton-Jacobi-Bellman system derived via Pontryagin's maximum principle (PMP). Subsequent backward integration, again relying on PMP, gives the SR-geodesics. For C = 1 we show that our method produces the global minimizers. Comparison with exact solutions shows a remarkable accuracy of the SR-spheres and the SR-geodesics. We present numerical computations of Maxwell points and cusp points, which we again verify for the uniform cost case C = 1. Regarding image analysis applications, trackings of elongated structures in retinal and synthetic images show that our line tracking generically deals with crossings. We show the benefits of including the SR-geometry. [ABSTRACT FROM AUTHOR]

We consider the sub-Riemannian length minimization problem on the group of motions of pseudo-Euclidean plane that form the special hyperbolic group SH(2). The system comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional. We apply the Pontryagin maximum principle to obtain the extremal control input and the sub-Riemannian geodesics. A change of coordinates transforms the vertical subsystem of the normal Hamiltonian system into the mathematical pendulum. In suitable elliptic coordinates, the vertical and the horizontal subsystems are integrated such that the resulting extremal trajectories are parametrized by the Jacobi elliptic functions. Qualitative analysis reveals that the projections of normal extremal trajectories on the xy-plane have cusps and inflection points. The vertical subsystem being a generalized pendulum admits reflection symmetries that are used to obtain a characterization of the Maxwell strata. [ABSTRACT FROM AUTHOR]

This paper presents a semidiscrete alternative to the theory of neurogeometry of vision, due to Citti, Petitot, and Sarti. We propose a new ingredient, namely, working on the group of translations and discrete rotations SE(2,N). The theoretical side of our study relates the stochastic nature of the problem with the Moore group structure of SE(2,N). Harmonic analysis over this group leads to very simple finite dimensional reductions. We then apply these ideas to the inpainting problem which is reduced to the integration of a completely parallelizable finite set of Mathieu-type diffusions (indexed by the dual of SE(2,N) in place of the points of the Fourier plane, which is a drastic reduction). The integration of the the Mathieu equations can be performed by standard numerical methods for elliptic diffusions and leads to a very simple and efficient class of inpainting algorithms. We illustrate the performances of the method on a series of deeply corrupted images. [ABSTRACT FROM AUTHOR]

We consider a natural extension of the Petitot–Citti–Sarti model of the primary visual cortex. In the extended model, the curvature of contours is taken into account. The occluded contours are completed via sub-Riemannian geodesics in the four-dimensional space M of positions, orientations, and curvatures. Here, M = R 2 × SO (2) × R models the configuration space of neurons of the visual cortex. We study the problem of sub-Riemannian geodesics on M via methods of geometric control theory. We prove complete controllability of the system and the existence of optimal controls. By application of the Pontryagin maximum principle, we derive a Hamiltonian system that describes the geodesics. We obtain the explicit parametrization of abnormal extremals. In the normal case, we provide three functionally independent first integrals. Numerical simulations indicate the existence of one more first integral that results in Liouville integrability of the system. [ABSTRACT FROM AUTHOR]