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Modeling and Control of Electrodynamic Tethers - an Energy and Topology Approach Downloaded from orbit.dtu.dk on: Oct 04, 2021 Modeling and Control of Electrodynamic Tethers - an Energy and Topology Approach Larsen, Martin Birkelund Publication date: 2010 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Larsen, M. B. (2010). Modeling and Control of Electrodynamic Tethers - an Energy and Topology Approach. Technical University of Denmark. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. Users may download and print one copy of any publication from the public portal for the purpose of private study or research. You may not further distribute the material or use it for any profit-making activity or commercial gain You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Martin Birkelund Larsen Modeling and control of Electrodynamic Tethers – an Energy and Topology Approach PhD thesis, May, 2010 Modeling and Control of Electrodynamic Tethers – an Energy and Topology Approach Martin Birkelund Larsen Technical University of Denmark Kongens Lyngby 2010 Technical University of Denmark Department of Electrical Engineering Automation and Control (AUT) Elektrovej, building 326 DK-2800 Kgs. Lyngby Denmark Tel: (+45)45253576 Fax: (+45)45881295 E-mail: [email protected] ISBN 978-87-92465-25-2 Summary A space tether is a cable used to connect spacecrafts in an orbiting structure. If an electrical current is lead through the tether, it can be utilized to provide propulsion for the spacecraft. In this case the cable is referred to as an electro- dynamic tether. The system utilizes the magnetic field of the Earth for creating a Lorentz force along the tether which occur when a current carrying wire oper- ates in a magnetic field. The use of electrodynamic tethers are interesting since they operate solely on electrical energy, which can be provided by solar panels of the spacecrafts. In this way the amount of propellant a spacecraft need to bring from Earth can be reduced. In this thesis the modeling and control of electrodynamic tethers are investi- gated, both when a single tether is used to connect two spacecrafts, and when the tethers are used i more general formations of spacecrafts. One of the main challenges when using electrodynamic tethers is that the force created along the tether is based on an external uncontrollable condition, namely the magnetic field. Even whit a known model of the magnetic field, limitations to the creation of the Lorentz force still exists, since the force can only be generated perpendic- ular to the instantaneous magnetic field. Furthermore, the control problem is complicated by the time variations in the magnetic field. This thesis solves these problems by utilizing an energy-based system description and a passivity-based control design. An advantage of the energy-based approach is that the stability of the system can easily be investigated, based on the energy flow in the system. Systems of several spacecrafts connected by tethers has many applications, for example in connection with space telescopes and space stations. Tethered for- mations are advantageous, compared to formations of free-flying spacecrafts, since a predetermined geometry of spacecrafts is easily maintained. This thesis investigates the use of electrodynamic tethers for such tethered satellite forma- ii tions with focus on the modeling and control aspects. One can think of many different structures for solving tasks in space, and separate derivations of the dynamical equations can be cumbersome. It can therefore be advantageous to be able to model a formation independent of its topology, i.e. the way tethers and satellites are interconnected. The thesis treats a class of formations in a generic framework, using graph theory to describe the topology of the forma- tions. The framework can be used both to deduce the equations of motion for the attitude motion of the formation and for control design regarding the same motion. The main part of the thesis consists of five scientific papers which have been submitted for international journals and conferences during the PhD project. Resum´e Et rumkabel (space tether) er et kabel, der forbinder rumfartøjer i en formation. Ledes en elektrisk strøm gennem dette kabel, kan det anvendes til at p˚avirke fremdriften af rumfartøjet. I dette tilfælde benævnes kablet elektrodynamisk (electrodynamic tether). Systemet udnytter jordens magnetfelt til at danne en Lorentzkraft distribueret langs kablet. Denne kraft opst˚ar ved en vekselvirkning mellem det strømførende kabel og magnetfeltet. Brugen af elektrodynamiske kabler er interessant, da de kan fungere alene ved tilførslen af elektrisk energi, som kan leveres af solpaneler p˚arumfartøjerne i formationen. P˚adenne m˚ade kan mængden af brændstof, der skal medbringes fra jorden, reduceres. I denne afhandling er modelleringen og styringen af elektrodynamiske kabler undersøgt, b˚ade i forbindelse med et enkelt kabel der forbinder to rumfartøjer og i forbindelse med formationer af flere rumfartøjer. Et af hovedproblemerne ved brugen af elektrodynamiske kabler er, at kraften der dannes langs kablet, er afhængig af en ekstern størrelse, nemlig jordens magnetfelt. Selv med en kendt model af jordens magnetfelt, er dannelsen af Lorentzkraften underlagt restriktioner, da den kun kan dannes ortogonalt p˚aden aktuelle feltvektor. Reg- uleringsproblemet bliver ydermere vanskeliggjort af tidsvariationerne i jordens magnetfelt. I denne afhandling er disse problemer løst ved anvendelsen af en energibaseret systembeskrivelse samt en passivitetsbaseret reguleringsmetode. Metoden har blandt andet den fordel at stabiliteten af systemet let kan un- dersøges gennem energistrømmen i systemet. Flere rumfartøjer forbundne med kabler har flere anvendelsesomr˚ader, blandt andet i forbindelse med rumteleskoper og rumstationer. Forbundne formationer har den fordel, frem for formationer af frie rumfartøjer, at en fast struktur let kan opretholdes. Denne afhandling undersøger brugen af elektrodynamiske kabler i s˚adanne forbundne formationer, med fokus p˚amodellering og styring. Man kan forstille sig mange forskellige formationer til løsning af opgaver i rummet, og det iv kan være tidskrævende at finde de dynamiske ligninger for alle formationerne separat. Det er derfor fordelagtigt at kunne modellere formationerne uafhængigt af deres topologi dvs. hvordan kabler og rumfartøjer er forbundne. Afhandlingen undersøger en gruppe formationer med en fælles beskrivelse, hvor formationernes topologi er udtrykt ved hjælp af grafteori. Beskrivelsen kan bruges b˚ade til at finde bevægelsesligningerne for formationen, samt til at designe regulatorer. Hoveddelen af nærværende afhandling best˚ar af fem artikler indsendt til inter- nationale tidsskrifter og konferencer i løber af ph.d. projektet. Preface This thesis is written based on the research conducted during my PhD project at the Technical University of Denmark, Department of Electrical Engineering. The project was carried out from May 2007 to May 2010 founded by a DTU scholarship. The supervisor of the project was Professor Mogens Blanke. The thesis constitutes a collection of articles which have been submitted for conferences and journals during the project. Since the articles represent work carried out over a three year period, a consistent nomenclature have not been possible. This thesis treats the use of electrodynamic tethers in connection with single tether systems and tethered satellite formations. For clarity the thesis is divided according to these research areas. This division can, in some contexts, appear enforced due to the gradual transition between the areas. The main focus of the thesis is on the modeling and control aspects of such tethered satellite systems. Martin Birkelund Larsen Copenhagen, May 2010 vi Papers included in the thesis [A] Larsen, M. B., Blanke M. Control by damping injection of electrodynamic tether system in an inclined orbit. In proceedings of American Control Conference ACC’09, 2009, 4824-4829, St. Louis, USA. Published [B] Larsen, M. B., Blanke M. Stabilization of periodic solutions in a tethered satellite system by damping injection. In proceedings of European Control Conference ECC’09, 2009, 2169-2174, Budapest, Hungary. Published [C] Larsen, M. B., Blanke M. Passivity-based control of a rigid electrodynamic tether. Journal of Guidance, Control, and Dynamics, 2010. Submitted [D] Larsen, M. B., Smith, R. S., Blanke M. Modeling of tethered satellite formations using graph theory. Acta Astronautica, 2010. Submitted [E] Larsen, M. B., Smith, R. S., Blanke M. Decentralized control of tethered satellite formations using electrodynamic tethers. Journal of Guidance, Control, and Dynamics, 2010. Submitted The following paper was published during the PhD project, but not included in this thesis. The paper was based on my MSc project and related to the work in this thesis. Larsen, M. B., Blanke M. Nonlinear control of electrodynamic tether in • equatorial or somewhat inclined orbits. In proceedings of Mediterranean Conference on Control & Automation MED ’07, 2007, 1-6, Athens, Greece. Published viii Acknowledgements
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