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

Cyclic pursuit without coordinates: Convergence to regular polygon formations

Abstract-We study a multi-agent cyclic pursuit model where each of the identical agents moves like a Dubins car and maintains a fixed heading angle with respect to the next agent. We establish that stationary shapes for this system are regular polygons. We derive a sufficient condition for local convergence to such regular polygon formations, which takes the form of an inequality connecting the angles of the regular polygon with the heading angle of the agents. A block-circulant structure of the system's linearization matrix in suitable coordinates facilitates and elucidates our analysis. Our results are complementary to the conditions for rendezvous obtained in earlier work [Yu et al., IEEE Trans. Autom. Contr., Feb. 2012]

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

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