Four bugs on a rectangle

The problem of four bugs in cyclic pursuit starting from a 2-by-1 rectangle is considered, and asymptotic formulas are derived to describe the motion. In contrast to the famous case of four bugs on a square, here the trajectories quickly freeze to essentially one dimension. After the first rotation about the centre point, the scale of the configuration has shrunk by a factor of 10^427907250, and this number is then exponentiated four more times with each successive cycle. Relations to Knuth’s double-arrow notation and level-index arithmetic are discussed

Cyclic Pursuit on Compact Manifolds

We study a form of cyclic pursuit on Riemannian manifolds with positive injectivity radius. We conjecture that on a compact manifold, the piecewise geodesic loop formed by connecting consecutive pursuit agents either collapses in finite time or converges to a closed geodesic. The main result is that this conjecture is valid for nonpositively curved compact manifolds.Comment: Typos and minor details fixe

Broadcast Guidance of Multi-Agent Systems

We consider the emergent behavior of a group of mobile agents guided by an exogenous broadcast signal. The agents’ dynamics is modelled by single integrators and they are assumed oblivious to their own position, however they share a common orientation (i.e. they have compasses). The broadcast control, a desired velocity vector, is detected by arbitrary subgroups of agents,that upon receipt of the guidance signal become "ad-hoc" leaders. The control signal and the set of leaders are assumed to be constant over some considerable intervals in time. A system without "ad-hoc" leaders is referred to as autonomous. The autonomous rule of motion is identical for all agents and is a gathering process ensuring a cohesive group. The agents that become leaders upon receipt of the exogenous control add the detected broadcast velocity to the velocity vector dictated by the autonomous rule of motion. This paradigm was considered in conjunction with several models of cohesive dynamics, linear and non-linear, with fixed inter-agent interaction topology, as well as systems with neighborhood based topology determined by the inter-agent distances. The autonomous dynamics of the models considered provides cohesion to the swarm, while, upon detection of a broadcast velocity vector, the leaders guide the group of agents in the direction of the control. For each local cohesion interaction model we analyse the effect of the broadcast velocity and of the set of leaders on the emergent behavior of the system. We show that in all cases considered the swarm moves in the direction of the broadcast velocity signal with speed set by the number of agents receiving the control and in a constellation determined by the model and the subset of "ad-hoc" leaders. All results are illustrated by simulations

Cyklisk jakt och flykt i planet

Let n bugs constitute the corners of an n-sided polygon. If the bugs cyclically pursue each other, the positions of the bugs will satisfy a system of ordinary differential equations, which we study. We examine the system for different n, but focus on the case n=3. When n=3, the bugs form a triangle. In this case, the solution will converge to some point. We study how the convergence occur. Ignoring translation, rotation and scaling, the triangle converges to a line. Further, we also consider when the three bugs escape from each other. If we again ignore rotation, translation and scaling, the triangle converges to an equilateral triangle. Finally, most theory in this thesis is already known, but we present a new proof for the convergence when three bugs pursuit each other.Vi har n=3 insekter: S1, S2 och S3 placerade i ett plan. Insekterna kommer då, oavsett hur de placeras i planet, kunna ses som hörnen i en triangel. Vi låter sedan insekterna jaga varandra cykliskt. Hur ser vägen ut som insekterna tar? Redan 1877 formulerade Edouard Lucas denna fråga och sedan dess har problemet studerats och även kompletterats med nya frågor av flera forskare. Till exempel kan man fråga sig om alla insekter kommer att kollidera samtidigt eller inte. Dessutom kan antalet insekter ökas. I den här uppsatsen kommer vi framförallt att fokusera på när insekterna bildar en (ickedegenererad) triangel. I det fallet kommer alla insekterna att krocka samtidigt, även om en insekt är långt ifrån de två andra, så att triangeln som bildas är oliksidig. Om antalet insekter är större är det inte säkert att alla krockar samtidigt. Låt nu antalet insekter vara tre. Insekterna kommer alltså att krocka samtidigt, men hur rör de sig i förhållande till varandra fram tills kollisionen? Det kan visas att om insekternas startpositioner inte bildar en liksidig triangel, så kommer insekternas positioner gå mot att ligga på en linje. För ett större antal insekter verkar det som att så länge n<7 går insekterna mot att ligga på en linje, vilket överensstämmer med fallet ovan med tre insekter. Däremot om n är större än eller lika med 7 tycks insekterna konvergera mot en regelbunden polygon innan kollisionen sker. Som nämndes tidigare, är det inte säkert att alla insekter krockar samtidigt om antalet insekter är fler än tre. Om vi har fyra insekter kommer inte alla att krocka samtidigt om insekternas startpositioner bildar en konkav fyrhörning. Däremot kommer de kollidera samtidigt om fyrhörningen som insekterna utgör är konvex. Om antalet insekter är fler än fyra, är det vanligaste att alla insekter krockar samtidigt. Vidare, kan insekterna istället fly från varandra. Även här fokuserar vi på n=3. I det fallet kommer triangeln som insekterna bildar att expandera obegränsat men vinklarna kommer konvergera mot pi/3. Triangeln går alltså mot att bli liksidig

Juegos de evasión

The goals of the work are twofold. The first one is about the background of mathematics of pursuit and evasion, models that are known as Chases and Escapes games. Five problems are solved using clasic techniques from Mathematic Analysis, and a sixth one of a different nature where its solution is described from a probabilistic point of view. The second one, of divulgative character and realized in colaboration with an Architecture student, consists in the design of a real park where the problems mentioned before can be brought into practise. The park, which is supposed to be placed in Seville, becomes as an idea to bring closer the world of mathematics to people from the city throughout a pleasant experience, showing that mathematics attached to a logical reasoning are a powerfull tool.Universidad de Sevilla. Grado en Matemática

Coordination of Multiple Autonomous Vehicles with Directed Communication Graphs

A considerable amount of research has been conducted on coordination control of vehicle formations in the last few decades due to its possible applications in numerous areas, such as patrolling, search and rescue, vehicle platoons, etc. Information flow between vehicles is vital for the formation to function properly. Controllability and stability of the formation depends on the type of information flow that is employed between vehicles. This work focuses on using a type of directed communication graph, called the ring graph, for coordination control of vehicle platoons. A detailed study of literature is undertaken to clearly understand the limitations of the various types of undirected graphs in terms of stability and scalability. The use of ring graph in vehicle platoons is investigated. The problems associated with the ring graph in its basic form related to communication sensor range and the scalability of ring graph are investigated. Methods to create a ring graph topology that requires minimal communication distance over a given formation are also investigated. A survey of literature was undertaken to investigate the formulation of the Traveling Salesman Problem (TSP), constraints involved in TSP, methods to solve TSP including exact and approximate algorithms. The problem of finding a ring graph for a given vehicle formation is addressed by formulating this problem as a special case of the TSP. Properties such as stability and scalability of the formation with a controller based on the ring graph are studied. Lack of scalability of the controller that is based on the ring graph is discussed. Alternate ring graph topologies which address the practical issues such as communication range of transmitters are proposed. Simulations which show that the alternate ring graphs have similar properties as the basic ring graph are carried out. Controller based on multiple ring graphs on a vehicle platoon and its advantages over the single ring topologies are studied. An algorithm based on the branch and bound method is proposed to solve the modified TSP and to generate a ring graph for the given vehicle platoons, or two- and three-dimensional vehicle formations. An experimental platform consisting of multiple homogeneous autonomous robots is developed. A coordination controller using ring graph is implemented on the experimental platform and results for a number of platoon initial condition scenarios are presented.Mechanical & Aerospace Engineerin

Cyclic Pursuit: Symmetry, Reduction and Nonlinear Dynamics

In this dissertation, we explore the use of pursuit interactions as a building block for collective behavior, primarily in the context of constant bearing (CB) cyclic pursuit. Pursuit phenomena are observed throughout the natural environment and also play an important role in technological contexts, such as missile-aircraft encounters and interactions between unmanned vehicles. While pursuit is typically regarded as adversarial, we demonstrate that pursuit interactions within a cyclic pursuit framework give rise to seemingly coordinated group maneuvers. We model a system of agents (e.g. birds, vehicles) as particles tracing out curves in the plane, and illustrate reduction to the shape space of relative positions and velocities. Introducing the CB pursuit strategy and associated pursuit law, we consider the case for which agent i pursues agent i+1 (modulo n) with the CB pursuit law. After deriving closed-loop cyclic pursuit dynamics, we demonstrate asymptotic convergence to an invariant submanifold (corresponding to each agent attaining the CB pursuit strategy), and proceed by analysis of the reduced dynamics restricted to the submanifold. For the general setting, we derive existence conditions for relative equilibria (circling and rectilinear) as well as for system trajectories which preserve the shape of the collective (up to similarity), which we refer to as pure shape equilibria. For two illustrative low-dimensional cases, we provide a more comprehensive analysis, deriving explicit trajectory solutions for the two-particle "mutual pursuit" case, and detailing the stability properties of three-particle relative equilibria and pure shape equilibria. For the three-particle case, we show that a particular choice of CB pursuit parameters gives rise to remarkable almost-periodic trajectories in the physical space. We also extend our study to consider CB pursuit in three dimensions, deriving a feedback law for executing the CB pursuit strategy, and providing a detailed analysis of the two-particle mutual pursuit case. We complete the work by considering evasive strategies to counter the motion camouflage (MC) pursuit law. After demonstrating that a stochastically steering evader is unable to thwart the MC pursuit strategy, we propose a (deterministic) feedback law for the evader and demonstrate the existence of circling equilibria for the closed-loop pursuer-evader dynamics

Toward a Framework for Systematically Categorizing Future UAS Threat Space

Title from PDF of title page, viewed September 21, 2022Dissertation advisor: Travis FieldsVitaIncludes bibliographical references (pages 241-270)Dissertation (Ph.D.)--Department of Civil and Mechanical Engineering, Department of Electrical and Computer Engineering. University of Missouri--Kansas City, 2021The development of unmanned aerial vehicles (UAVs) is occurring as fast or faster than any other innovation throughout the course of human history. Building an effective means of defending against threats posed by malicious applications of novel technology is imperative in the current global landscape. Gone are the days where the enemy and the threat it poses are well defined and understood. Defensive technologies have to be modular and able to adapt to a threat technology space which is likely to recycle several times over during the course of a single defense system acquisition cycle. This manuscript wrestles with understanding the unique threat posed by UAVs and related technologies. A thorough taxonomy of the problem is given including projections for how the defining characteristics of the problem are likely to change and grow in the near future. Next, a discussion of the importance of tactics related to the problem of a rapidly changing threat space is provided. A discussion of case studies related to lessons learned from military acquisition programs and pivotal technological innovations in the course of history are given. Multiple measures of success are proposed which are designed to allow for meaningful comparisons and honest evaluations of capabilities. These measures are designed to facilitate discussions by providing a common, and comprehensible language that accounts for the vast complexity of the problem space without getting bogged down by the details. Lastly, predictions for the future threat space comprising UAVs is given. The contributions of this work are thus threefold. Firstly, an analytic framework is presented including a detailed parameterization of the problem as well as various solution techniques borrowed from a variety of fields. Secondly, measures of success are presented which attempt to compare the effectiveness of various systems by converting to expected values in terms of effective range, or extending the popular concept of kill chain and collapsing effectiveness into units of time. A novel technique for measuring effectiveness is presented whereby effectiveness is composed of various individual probabilities. Probabilities and associated distributions can be combined according to the rules of joint probabilities and distributions and allows performance against a probabilistic threat to be measured succinctly and effectively. The third contribution concerns predictions made with respect to the UAS threat space in the future. These predictions are designed to allow for defensive systems to be developed with a high expected effectiveness against current and future threats. Essentially this work comprises a first attempt toward developing a complete framework related to engagement and mission level modeling of a generic defensive system (or combination of systems) in the face of current and future threats presented by UAS.Introduction -- Literature review -- War gaming -- Measures of success -- Conclusion