Curve Shortening and the Rendezvous Problem for Mobile Autonomous Robots

If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular point. This curve evolution is called Euclidean curve shortening and the result is known as the Gage-Hamilton-Grayson Theorem. Motivated by the rendezvous problem for mobile autonomous robots, we address the problem of creating a polygon shortening flow. A linear scheme is proposed that exhibits several analogues to Euclidean curve shortening: The polygon shrinks to an elliptical point, convex polygons remain convex, and the perimeter of the polygon is monotonically decreasing.Comment: 15 pages, 18 figure

Dirichlet sigma models and mean curvature flow

The mean curvature flow describes the parabolic deformation of embedded branes in Riemannian geometry driven by their extrinsic mean curvature vector, which is typically associated to surface tension forces. It is the gradient flow of the area functional, and, as such, it is naturally identified with the boundary renormalization group equation of Dirichlet sigma models away from conformality, to lowest order in perturbation theory. D-branes appear as fixed points of this flow having conformally invariant boundary conditions. Simple running solutions include the paper-clip and the hair-pin (or grim-reaper) models on the plane, as well as scaling solutions associated to rational (p, q) closed curves and the decay of two intersecting lines. Stability analysis is performed in several cases while searching for transitions among different brane configurations. The combination of Ricci with the mean curvature flow is examined in detail together with several explicit examples of deforming curves on curved backgrounds. Some general aspects of the mean curvature flow in higher dimensional ambient spaces are also discussed and obtain consistent truncations to lower dimensional systems. Selected physical applications are mentioned in the text, including tachyon condensation in open string theory and the resistive diffusion of force-free fields in magneto-hydrodynamics.Comment: 77 pages, 21 figure

Estratégia de controle cooperativo baseado em consenso para um grupo multi-veículos

Tese (doutorado) - Universidade Federal de Santa Catarina, Centro Tecnológico, Programa de Pós-Graduação em Engenharia de Automação e Sistemas, Florianópolis, 2013Resumo: Nesta tese é apresentada uma proposta de controle cooperativo baseada em consenso para resolver o problema de rendezvous para um grupo multi-veículos. Como caráter essencial de cooperação os veículos, estes compartilham informação acerca do conhecimento individual, o qual está sujeito a falhas na comunicação com origem diversa, desde faltas nos dispositivos de comunicação até perdas de pacote. A estratégia de controle cooperativo proposta consiste no desenvolvimento de leis de controle descentralizadas para que cada veículo determine sua trajetória de consenso até o ponto de encontro, que a priori é desconhecido para alguns integrantes da equipe. Para tal, utiliza-se uma abordagem baseada em controle preditivo, o que permite a inclusão de requisitos de resposta, bem como de restrições, de modo a manter o caráter de cooperação entre os veículos do grupo. De forma complementar à proposta, adicionam-se à formulação das trajetórias de consenso, restrições de conectividade entre veículos, restrições de ângulo de sensoriamento em relação ao ponto de encontro e à formação de cobertura sobre o ponto de referência. Trata-se ainda neste trabalho de protocolos de comunicação com intuito de melhorar a pontualidade na troca periódica de informações, melhorando a convergência da tarefa de consenso. Abstract: In this thesis, a cooperative control strategy for teams of multiple autonomous vehicles to solve a basic coordination problem, called rendezvous problem, is presented. The vehicles share information about individual knowledge in order to cooperate. It is considered that communication failures can be occurs due to packet losses and device faults. In the cooperative proposal, a decentralized control laws are developed to drive all the vehicles to a reference position by performing trajectories with consensus. The rendezvous point is a priori unknown to some group members. The calculus of the consensus trajectories is based on receding horizon control, which allows to include response requirements and constraints for maintaining cooperation between the vehicles of the group. Additionally, it is added to the consensus trajectories formulation: connectivity constraints between vehicles, optimization of the sensing angle relative to the rendezvous point and coverage formation on the reference point. Finally, in this work, communication protocolsare studied for improving the timeliness in the information exchange, which improves the convergence of consensus task

Curve shortening and the rendezvous problem for mobile autonomous robots”, preprint

If a smooth, closed, and embedded curve is deformed along its normal vector field at a rate proportional to its curvature, it shrinks to a circular point. This curve evolution is called Euclidean curve shortening and the result is known as the Gage-Hamilton-Grayson Theorem. Motivated by the rendezvous problem for mobile autonomous robots, we address the problem of creating a polygon shortening flow. A linear scheme is proposed that exhibits several analogues to Euclidean curve shortening: The polygon shrinks to an elliptical point, convex polygons remain convex, and the perimeter of the polygon is monotonically decreasing. Index Terms Distributed control, curve shortening, mobile autonomous robots

Consensus in multi-agent systems with time-delays

Different consensus problems in multi-agent systems have been addressed in this thesis. They represent improvements with respect to the state of the art. In the first part of the thesis in luding Chapters 2, 3, and 4, the state of the art of the representation and stability analysis of consensus problems, time-delay systems, and sampled-data systems have been presented. Novel contributions have been illustrated in Chapters 5-8. Particularly, in Chapter 5 we reported the results of Zareh et al. (2013b), where we investigated the consensus problem for networks of agents with double integrator dynamics affected by time-delay in their coupling. We provided a stability result based on the Lyapunov-Krasovskii functional method and a numerical proc edure based on an LMI condition which depends only on the algebraic connectivity of the considered network topologies, thus reducing greatly the computational complexity of the procedure. Obviously, this result implies the existence of a minimum dwell time such that the proposed consensus protocol is stable for slow swit things between network topologies with suffient algebraic connectivity. Future work will involve actually computing such a dwell time by adopting a multiple Lyapunov function method and evaluating the worst case sider only delayed relative measurements instead of delayed absolute values of the neighbors' state variables. The results of Zareh et al. (2013a) were addressed in Chapter 6, in which a on- tinuous time version of a consensus on the average protocol for arbitrary strongly connected directed graphs is proposed and its convergence properties with respect to time delays in the local state update are characterized. The convergenc e properties of this algorithm depend upon a tuning parameter that an be made arbitrary small to prove stability of the networked system. Simulations have been presented to corroborate the theoretical results and show that the existenc e of a small time delay an a tually improve the algorithm performance. Future work will include an extension of the mathematical characterization of the proposed algorithm to consider possibly heterogeneous or time-varying delays. In Chapter 7 we proposed a PD-like consensus algorithm for a second-order multi- agent system where, at non-periodic sampling times, agents transmit to their neighbors information about their position and veloc ity, while each agent has a perfect knowledge of its own state at any time instant. Conditions have been given to prove onsensus to a ommon xed point, based on LMIs verification. Moreover, we also show how it is possible to evaluate an upper bound on the de ay rate of exponential convergence of stable modes. In Chapter 8, mainly based on our paper Zareh et al. (2014b), we considered the same problem as in Chapter 7. The main contribution consists in proving consensus to a common fixed point, based on LMIs verification, under the assumption that the network topology is not known and the only information is an upper bound on the connectivity. Two are the main directions of our future research in this framework. First, we want to compute analytically an upper bound on the value of the second largest eigenvalue of the weighted adjacency matrix that guarantees consensus, as a function of the other design parameters. Second, we plan to study the case where agents do not have a perfect knowledge of their own state