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The concept of exceptional point (EP) is demonstrated experimentally in the case of a simple mechanical system consisting of two linearized coupled pendulums. Exceptional points correspond to specific values of the system parameters that yield defective eigenvalues. These spectral singularities which are typical of non-Hermitian system means that both the eigenvalues and their associated eigenvectors coalesce. The existence of an EP requires an adequate parameterization of the dynamical system. For this aim, the experimental device has been designed with two controllable parameters which are the length of one pendulum and a viscous-like damping which is produced via electromagnetic induction. Thanks to the observation of the free response of the coupled pendulums, most EP properties are experimentally investigated, showing good agreements with theoretical considerations. In contrast with many studies on EPs, mainly in the field of physics, the novelty of the present work is that controllable parameters are restricted to be real-valued, and this requires the use of adequate search algorithms. Furthermore, it offers the possibility of exploiting the existence of EPs in time-domain dynamic problems.

In the context of autonomous vehicles, one of the most crucial tasks is to estimate the risk of the undertaken action. While navigating in complex urban environments, the Bayesian occupancy grid is one of the most popular types of maps, where the information of occupancy is stored as the probability of collision. Although widely used, this kind of representation is not well suited for risk assessment: because of its discrete nature, the probability of collision becomes dependent on the tessellation size. Therefore, risk assessments on Bayesian occupancy grids cannot yield risks with meaningful physical units. In this article, we propose an alternative framework called Dynamic Lambda-Field that is able to assess generic physical risks in dynamic environments without being dependent on the tessellation size. Using our framework, we are able to plan safe trajectories where the risk function can be adjusted depending on the scenario. We validate our approach with quantitative experiments, showing the convergence speed of the grid and that the framework is suitable for real-world scenarios., Comment: 2021 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works

In the context of autonomous robots, one of the most important tasks is to prevent potential damage to the robot during navigation. For this purpose, it is often assumed that one must deal with known probabilistic obstacles, then compute the probability of collision with each obstacle. However, in complex scenarios or unstructured environments, it might be difficult to detect such obstacles. In these cases, a metric map is used, where each position stores the information of occupancy. The most common type of metric map is the Bayesian occupancy map. However, this type of map is not well suited for computing risk assessments for continuous paths due to its discrete nature. Hence, we introduce a novel type of map called the Lambda Field, which is specially designed for risk assessment. We first propose a way to compute such a map and the expectation of a generic risk over a path. Then, we demonstrate the benefits of our generic formulation with a use case defining the risk as the expected collision force over a path. Using this risk definition and the Lambda Field, we show that our framework is capable of doing classical path planning while having a physical-based metric. Furthermore, the Lambda Field gives a natural way to deal with unstructured environments, such as tall grass. Where standard environment representations would always generate trajectories going around such obstacles, our framework allows the robot to go through the grass while being aware of the risk taken., Comment: Published in the Special Issue "Frontiers in Mobile Robot Navigation" of Sensors. https://www.mdpi.com/1424-8220/21/22/7562

A numerical algorithm is proposed to explore in a systematic way the trajectories of the eigenvalues of non-Hermitian matrices in the parametric space and exploit this in order to find the locations of defective eigenvalues in the complex plane. These non-Hermitian degeneracies also called exceptional points (EP) have raised considerable attention in the scientific community as these can have a great impact in a variety of physical problems. The method requires the computation of successive derivatives of two selected eigenvalues with respect to the parameter so that, after recombination, regular functions can be constructed. This algebraic manipulation permits the localization of exceptional points (EP), using standard root-finding algorithms and the computation of the associated Puiseux series up to an arbitrary order. This representation, which is associated with the topological structure of Riemann surfaces allows to efficiently approximate the selected pair in a certain neighbourhood of the EP. Practical applications dealing with guided acoustic waves propagating in straight ducts with absorbing walls and in periodic guiding structures are given to illustrate the versatility of the proposed method and its ability to handle large size matrices arising from finite element discretization techniques. The fact that EPs are associated with optimal dissipative treatments in the sense that they should provide best modal attenuation is also discussed., Comment: Fix typo in Eq. (17), add EasterEig library url, few other enhancements

In a context of autonomous robots, one of the most important task is to ensure the safety of the robot and its surrounding. Most of the time, the risk of navigation is simply said to be the probability of collision. This notion of risk is not well defined in the literature, especially when dealing with occupancy grids. The Bayesian occupancy grid is the most used method to deal with complex environments. However, this is not fitted to compute the risk along a path by its discrete nature, hence giving poor results. In this article, we present a new way to store the occupancy of the environment that allows the computation of risk for a given path. We then define the risk as the force of collision that would occur for a given obstacle. Using this framework, we are able to generate navigation paths ensuring the safety of the robot., Comment: 2019 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes,creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works