Convex Obstacles Research Articles

Local Path Planning for Dual-Arm Mobile Robot Shuttling within Truss

A new local path planning method for dual-arm mobile robot shuttling within the truss is presented. Like the probabilistic roadmaps method, this method proceeds in two stages: preprocessing stage and path planning stage. In preprocessing stage, the workspace is divided into a set of non-overlapping cubical cells. The nodes in the free workspace are stored in a matrix. In path planning stage, three query strategies are adopted to search the path from start point to goal point. Take use of vertex query strategy, the smooth path can be acquired in a fraction of a second. The algorithm is simple, and is applicable to any static environment with convex obstacles.

Optimal Transport with Branching Distance Costs and the Obstacle Problem

We address the Monge problem in metric spaces with a geodesic distance: $(X,d)$ is a Polish space and $d_N$ is a geodesic Borel distance which makes $(X,d_N)$ a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along a family of geodesics. We introduce three assumptions on a given $d_{N}$-monotone transference plan $\pi$ which imply, respectively, strong consistency of disintegration, continuity of the conditional probabilities of the first marginal, and a regularity property for the geometry of chain of transport rays. We show that this regularity is sufficient for the construction of a transport map with the same transport cost of $\pi$. We apply these results to the Monge problem in $\mathbb{R}^{d}$ with smooth, convex, and compact obstacle obtaining the existence of an optimal map, provided the first marginal is absolutely continuous with respect to the d-dimensional Lebesgue measure.

Simultaneous study of the diffraction by a 2D-convex obstacle through boundary layer method and microlocal analysis

In this paper we consider the high frequency diffraction of electromagnetic waves by a strictly convex obstacle. We restrict ourselves to the case of an obstacle in 2D and to the diffraction of a TE or TM wave. The aim of this paper is to present simultaneously with the same notations, unknowns and special functions the boundary layer analysis of this problem with stretched variables using the intuition of the solutions and the microlocal analysis way of construction of an adapted parametrix. We concentrate on the case of diffraction, i.e. studying the representation of a creeping wave on the boundary, which occurs in the case of the study of a glancing point. We use both methods to derive the exact solution near a diffractive point, obtaining results in terms of suitable Airy functions.

Homotopical complexity of 2D billiard orbits

Traditionally, rotation numbers for toroidal billiard flows are defined as the limiting vectors of average displacements per time on trajectory segments. Naturally, these creatures live in the (commutative) vector space ℝn, if the toroidal billiard is given on the flatn-torus. The billiard trajectories, being curves, often getting very close to closed loops, quite naturally define elements of the fundamental group of the billiard table. The simplest non-trivial fundamental group obtained this way belongs to the classical Sinai billiard, i.e. the billiard flow on the 2-torus with a single, strictly convex obstacle (with smooth boundary) removed. This fundamental group is known to be the groupF2freely generated by two elements, which is a heavily noncommutative, hyperbolic group in Gromov’s sense. We define the homotopical rotation number and the homotopical rotation set for this model, and provide lower and upper estimates for the latter one, along with checking the validity of classically expected properties, like the density (in the homotopical rotation set) of the homotopical rotation numbers of periodic orbits.The natural habitat for these objects is the infinite cone erected upon the Cantor set Ends (F2) of all ŋds" of the hyperbolic groupF2. An element of Ends (F2) describes the direction in (the Cayley graph of) the groupF2in which the considered trajectory escapes to infinity, whereas the height functiont(t≧ 0) of the cone gives us the average speed at which this escape takes place.The main results of this paper claim that the orbits can only escape to infinity at a speed not exceeding\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt 2 $ \end{document}, and any directione∈ Ends (F2) for the escape is feasible with any prescribed speeds, 0 ≦s≦\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\sqrt 2 $ \end{document}/2. This means that the radial upper and lower bounds for the rotation setRare actually pretty close to each other.

A cell decomposition approach to visibility-based pursuit evasion among obstacles

In this paper, we address the problem of surveillance in an environment with obstacles. We consider the problem in which a mobile observer attempts to maintain visual contact with a target as it moves through an environment containing obstacles. This surveillance problem is a variation of traditional pursuit–evasion games, with the additional condition that the pursuer immediately loses the game if at any time it loses sight of the evader. We analyze this tracking problem as a game of kind. We use the method of explicit policy to compute guaranteed strategies for surveillance for the observer in an environment containing a single corner. These strategies depend on the initial positions of the observer and the target in the workspace. Based on these strategies a partition of the visibility polygon of the players is constructed. The partitions have been constructed for varying speeds of the observer and the target. Using these partitions we provide a sufficient condition for escape of a target in a general environment containing polygonal obstacles. Moreover, for a given initial target position, we provide a polynomial-time algorithm that constructs a convex polygonal region that provides an upper-bound for the set of initial observer positions from which it does not lose the game. We extend our results to the case of arbitrary convex obstacles with differentiable boundaries. We also present a sufficient condition for tracking and provide a lower-bound on the region around the initial position of the target from which the observer can track the target. Finally, we provide an upper bound on the area of the region in which the outcome of the game is unknown.

Generalized Strichartz estimates on perturbed wave equation and applications on Strauss conjecture

In this paper we prove a general Strichartz estimate for certain perturbed wave equations, and here we can drop the nontrapping hypothesis and handle trapping obstacles with some loss of derivatives for data in the local energy decay estimates. We then give the obstacle version of the sharp life span for semilinear wave equations when $n=3,p<p_c$, by using the real interpolation method, and by getting corresponding finite time Strichartz estimates (see Section 3). Finally, as another application, we get the Strauss conjecture for semilinear wave equations with several convex obstacles when $n=3,4$ (see Section 4).

A Model Reduction Algorithm for Solving Multiple Scattering Problems Using Iterative Methods

In this paper we propose a new method for solving high-frequency scattering problems by multiple objects. A model reduction algorithm based on the macro basis function (MBF) method is used to find an approximate solution within a subspace spanned by the solutions of several single scattering subproblems. Different iterative methods for generating the MBFs are compared. The whole process relies on a finite element approach and is applied to convex obstacle scattering.

Optimal transport with convex obstacle

We consider the Monge transportation problem when the cost is the squared geodesic distance around a convex obstacle. We show that there exists at least one—and in general infinitely many—optimal transport maps.

Gauge equivalence and inverse scattering for long-range magnetic potentials

For Schrodinger operators with long-range magnetic vector potentials and short range electric scalar potentials in an exterior domain Ω in Rn with n ⩾ 2, we show that there is a one-to-one correspondence between the gauge equivalent classes of Hamiltonians and those of S-matrices if Ω is exterior to a bounded convex obstacle.

Scattering for the critical and localised semilinear wave equation

In this paper, we prove a scattering theorem for the critical wave equation outside convex obstacle. The proof relies on generalized Strichartz estimates.

A fully discrete Galerkin method for high frequency exterior acoustic scattering in three dimensions

Standard Galerkin discretization techniques (with locally- or globally-supported basis functions) for boundary integral equations are inefficient for high frequency three dimensional exterior scattering simulations because they require a fixed number of unknowns per wavelength in each dimension, leading to large CPU time and memory requirements to set up the dense Galerkin matrix, with each entry requiring evaluation of multi-dimensional highly oscillatory integrals. In this work, using globally-supported basis functions, we describe an efficient fully discrete Galerkin surface integral equation algorithm for simulating high frequency acoustic scattering by three dimensional convex obstacles that includes a powerful integration scheme for evaluation of four dimensional Galerkin integrals with high-order accuracy. Such high-order order accuracy for various practically relevant frequencies ( k ∈ [1, 100,000]) substantially improves on approximations based on standard asymptotic techniques. We demonstrate the efficiency of our algorithm for spherical and non-spherical convex scattering for several wavenumbers 1 ⩽ k ⩽ 100,000 for low to high order prescribed tolerance. Our fully discrete algorithm requires only mild growth in the number of unknowns and CPU time as the frequency increases.

Analytic continuation of the resolvent of the Laplacian and the dynamical zeta function

Abstract Let s 0 0 be the abscissa of absolute convergence of the dynamical zeta function Z ( s ) for several disjoint strictly convex compact obstacles K i ⊂ R N , i = 1 , … , κ 0 , κ 0 ⩾ 3 and let R χ ( z ) = χ ( − Δ D − z 2 ) −1 χ , χ ∈ C 0 ∞ ( R N ) , be the cut-off resolvent of the Dirichlet Laplacian − Δ D in Ω = R N ∖ ⋃ i = 1 κ 0 K i ¯ . We prove that there exists σ 2 s 0 such that Z ( s ) is analytic for R ( s ) ⩾ σ 2 and the cut-off resolvent R χ ( z ) has an analytic continuation for ℑ ( z ) − i σ 2 , | R ( z ) | ⩾ C . To cite this article: V. Petkov, L. Stoyanov, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

An Amplitude Finite Element Formulation for Multiple-Scattering by a Collection of Convex Obstacles

We present a multiple-scattering solver for nonconvex geometries obtained as the union of a finite number of convex obstacles. The algorithm is a finite element reformulation of a high-frequency integral equation technique proposed previously. It is based on an iterative solution of the scattering problem, where each iteration leads to the resolution of a single scattering problem in terms of a slowly oscillatory amplitude.

On the Schrödinger equation outside strictly convex obstacles

We prove sharp Strichartz estimates for the semiclassical Schrodinger equation on a compact Riemannian manifold with a smooth, strictly geodesically concave boundary. We deduce classical Strichartz estimates for the Schrodinger equation outside a strictly convex obstacle, local existence for the H1-critical (quintic) Schrodinger equation, and scattering for the subcritical Schrodinger equation in three dimensions.

Obstacle Numbers of Graphs

An obstacle representation of a graph G is a drawing of G in the plane with straight-line edges, together with a set of polygons (respectively, convex polygons) called obstacles, such that an edge exists in G if and only if it does not intersect an obstacle. The obstacle number (convex obstacle number) of G is the smallest number of obstacles (convex obstacles) in any obstacle representation of G. In this paper, we identify families of graphs with obstacle number 1 and construct graphs with arbitrarily large obstacle number (convex obstacle number). We prove that a graph has an obstacle representation with a single convex k-gon if and only if it is a circular arc graph with clique covering number at most k in which no two arcs cover the host circle. We also prove independently that a graph has an obstacle representation with a single segment obstacle if and only if it is the complement of an interval bigraph.

Analysis of multiple scattering iterations for high-frequency scattering problems. II: The three-dimensional scalar case

In this paper, we continue our analysis of the treatment of multiple scattering effects within a recently proposed methodology, based on integral-equations, for the numerical solution of scattering problems at high frequencies. In more detail, here we extend the two-dimensional results in part I of this work to fully three-dimensional geometries. As in the former case, our concern here is the determination of the rate of convergence of the multiple-scattering iterations for a collection of three-dimensional convex obstacles that are inherent in the aforementioned high-frequency schemes. To this end, we follow a similar strategy to that we devised in part I: first, we recast the (iterated, Neumann) multiple-scattering series in the form of a sum of periodic orbits (of increasing period) corresponding to multiple reflections that periodically bounce off a series of scattering sub-structures; then, we proceed to derive a high-frequency recurrence that relates the normal derivatives of the fields induced on these structures as the waves reflect periodically; and, finally, we analyze this recurrence to provide an explicit rate of convergence associated with each orbit. While the procedure is analogous to its two-dimensional counterpart, the actual analysis is significantly more involved and, perhaps more interestingly, it uncovers new phenomena that cannot be distinguished in two-dimensional configurations (e.g. the further dependence of the convergence rate on the relative orientation of interacting structures). As in the two-dimensional case, and beyond their intrinsic interest, we also explain here how the results of our analysis can be used to accelerate the convergence of the multiple-scattering series and, thus, to provide significant savings in computational times.

Analysis of multiple scattering iterations for high-frequency scattering problems. I: the two-dimensional case

We present an analysis of a recently proposed integral-equation method for the solution of high-frequency electromagnetic and acoustic scattering problems that delivers error-controllable solutions in frequency-independent computational times. Within single scattering configurations the method is based on the use of an appropriate ansatz for the unknown surface densities and on suitable extensions of the method of stationary phase. The extension to multiple-scattering configurations, in turn, is attained through consideration of an iterative (Neumann) series that successively accounts for further geometrical wave reflections. As we show, for a collection of two-dimensional (cylindrical) convex obstacles, this series can be rearranged into a sum of periodic orbits (of increasing period), each corresponding to contributions arising from waves that reflect off a fixed subset of scatterers when these are transversed sequentially in a periodic manner. Here, we analyze the properties of these periodic orbits in the high-frequency regime, by deriving precise asymptotic expansions for the “currents” (i.e. the normal derivative of the fields) that they induce on the surface of the obstacles. As we demonstrate these expansions can be used to provide accurate estimates of the rate at which their magnitude decreases with increasing number of reflections, which defines the overall rate of convergence of the multiple-scattering series. Moreover, we show that the detailed asymptotic knowledge of these currents can be used to accelerate this convergence and, thus, to reduce the number of iterations necessary to attain a prescribed accuracy.

Magnebike: A magnetic wheeled robot with high mobility for inspecting complex‐shaped structures

AbstractThis paper describes the Magnebike robot, a compact robot with two magnetic wheels in a motorbike arrangement, which is intended for inspecting the inner casing of ferromagnetic pipes with complex‐shaped structures. The locomotion concept is based on an adapted magnetic wheel unit integrating two lateral lever arms. These arms allow for slight lifting off the wheel in order to locally decrease the magnetic attraction force when passing concave edges, as well as laterally stabilizing the wheel unit. The robot has the main advantage of being compact (180 × 130 × 220 mm) and mechanically simple: it features only five active degrees of freedom (two driven wheels each equipped with an active lifter stabilizer and one steering unit). The paper presents in detail design and implementation issues that are specific to magnetic wheeled robots. Low‐level control functionalities are addressed because they are necessary to control the active system. The paper also focuses on characterizing and analyzing the implemented robot. The high mobility is shown through experimental results: the robot not only can climb vertical walls and follow circumferential paths inside pipe structures but it is also able to pass complex combinations of 90‐deg convex and concave ferromagnetic obstacles with almost any inclination regarding gravity. It requires only limited space to maneuver because turning on the spot around the rear wheel is possible. This high mobility enables the robot to access any location in the specified environment. Finally the paper analyzes the maximum payload for different types of environment complexities because this is a key feature for climbing robots and provides a security factor about the risk of falling and slipping. © 2009 Wiley Periodicals, Inc.

On the complexity of umbra and penumbra

Computing shadow boundaries is a difficult problem in the case of non-point light sources. A point is in the umbra if it does not see any part of any light source; it is in full light if it sees entirely all the light sources; otherwise, it is in the penumbra. While the common boundary of the penumbra and the full light is well understood, less is known about the boundary of the umbra. In this paper we prove various bounds on the complexity of the umbra and the penumbra cast on a fixed plane by a segment or convex polygonal light source in the presence of convex polygonal or polyhedral obstacles in R 3 . In particular, we show that a single segment light source may cast on a plane, in the presence of two disjoint triangles, four connected components of umbra and that two fat convex and disjoint obstacles of total complexity n can give rise to as many as Ω ( n ) connected components of umbra. In a scene consisting of a segment light source and k disjoint convex polyhedra of total complexity n, we prove an Ω ( n k 2 + k 4 ) lower bound on the maximum number of connected components of the umbra and a O ( n k 3 ) upper bound on its complexity; if the obstacles may intersect, we only prove an upper bound of O ( n 2 k 2 ) . We also prove that, in the presence of k convex polyhedra of total complexity n, some of which are light sources, the umbra cast on a plane may have in the worst case Ω ( n 2 k 3 + n k 5 ) connected components and has complexity O ( n 3 k 3 ) (the polyhedra are supposed pairwise disjoint for lower bounds and possibly intersecting for the upper bounds). These are the first bounds on the size of the umbra in terms of both k and n. These results prove that the umbra, which is bounded by arcs of conics, is intrinsically much more intricate than the boundary between full light and penumbra which is bounded by line segments and whose worst-case complexity is, as we show, in Ω ( n k + k 4 ) and O ( n k α ( k ) + k 4 ) ; moreover, if there are only O ( 1 ) light sources of total complexity m, then the complexity is in Ω ( n α ( k ) + k m + k 2 ) and O ( n α ( k ) + k m α ( k ) + k 2 ) .

Sojourn times, manifolds with infinite cylindrical ends, and an inverse problem for planar waveguides

We prove that two particular entries in the scattering matrix for the Dirichlet Laplacian on ℝ × (−γ, γ) \( \mathcal{O} \) determine an analytic strictly convex obstacle \( \mathcal{O} \). With an additional symmetry assumption, one entry suffices. Part of the proof is an integral identity involving an entry in the scattering matrix and a distribution related to the fundamental solution of the wave equation. This identity holds for general manifolds with infinite cylindrical ends. A consequence of this is a relationship between the singularities of the Fourier transform of an entry in the scattering matrix and the sojourn times of certain geodesics.