Rochester Institute of Technology RIT Scholar Works Theses 1-2006 Dynamics and control of Tethered Satellite System Royce L. Abel Follow this and additional works at: https://scholarworks.rit.edu/theses Recommended Citation Abel, Royce L., "Dynamics and control of Tethered Satellite System" (2006). Thesis. Rochester Institute of Technology. Accessed from This Thesis is brought to you for free and open access by RIT Scholar Works. It has been accepted for inclusion in Theses by an authorized administrator of RIT Scholar Works. For more information, please contact [email protected]. Dynamics and Control of Tethered Satellite System By Royce L. Abel A thesis submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in Mechanical Engineering Approved by: Dr. Josef Torok Edward Hensel for Josef Torok Rochester Institute of Technology (Thesis Advisor) Dr. Agamemnon Crassidis Agamemnon Crassidis Rochester Institute of Technology Dr. AlanNye Alan Nye Rochester Institute of Technology Dr. Edward Hensel Edward Hensel Rochester Institute of Technology Department of Mechanical Engineering College of Engineering Rochester Institute of Technology Rochester, NY 14623 Jan uary 2006 Permission Granted: I, Royce L. Abel, Hereby grant permission to the Wallace Memorial Library of the Rochester Institute of Technology to reproduce my thesis entitled Dynamics and Control of Tethered Satellite Systems in whole or part. Any reproduction will not be for commercial use or profit. Royce L. Abel Royce L. Abel i Abstract Tethered Satellite Systems (TSS) has been an ongoing project around the world for many decades. This study examined the dynamics and controls of tethered satellites. The equations of motion were developed and simulated with no control and small ec centric orbits. A control law was derived by associating the Hamiltonian of the system to a Lyapunov Function. This control law was then simulated to demonstrate ability and robustness. The tethered system can be retrieved and deployed in a small num ber of orbits, and return from a perturbation quickly. Examination of a non thruster controlled system resulted in asymptotic convergence in retrieval and station keeping, but the inplane angle, orbital plane, converged during deployment. Time to complete the phases was similar to out of plane thruster controlled. The maximum eccentricities were found for retrieval and deployment when the system started at the perihelion of the orbit with a lOORm tether. Drag was added to the model because it effects the dynamics when the subsatellite is very close to earth. Most of the effects are seen in the inplane angle. With aerodynamic drag the simulation showed that offsets would result in the inplane angle. At fully deployed the inplane angle was small. A Lya punov controller for tether satellites works very well under ideal conditions. Once aerodynamics are included into the model, offsets and oscillations occur because of the new forces that unbalance the system compared to the model without drag. Acknowledgements: I would like to thank my parents for supporting me during my time at RIT. I would also like to thank my thesis advisor, Dr. Torok, for his time, guidance, and input on my thesis. His excellent personality and teaching style made my education and stay at RIT much more fulfilling. There are many people that also supported in editing my thesis. I would like to thank them as well: Linda Abel, Richard Abel, Bill Watkins, Richele Watkins, Erin Long, Erin Canfield, and Tulin Dadali. Without their assistance grammar checking this thesis would have been tedious. Lastly, I would like to thank the members of my review board for helping me finish. Especially Dr. Crassidis. Without his assistance this thesis would never have been finished. 111 Contents 1 Background Information 3 1.1 Nomenclature 3 1.2 Tethered Satellites Overview 6 1.3 Orbital Mechanics 7 1.4 Nonlinear Stability 11 2 Literature Review 13 3 Dynamics of Tethered Satellites 21 3.1 Equations of Motion of Tethered Satellites 21 3.1.1 Assumptions 21 3.1.2 Development of the Equations of Motion 23 3.2 Analysis of EOM without control 29 3.2.1 Station Keeping Phase 31 3.2.2 Deployment Phase 33 3.2.3 Retrieval Phase 35 4 Lyapunov Based Feedback Control of Tethered Satellites With and Without Aerodynamics 37 4.1 Conservative System 37 4.1.1 Formulation of Control Algorithm 37 of System with 4.1.2 Simulation Conservative Feedback Control .... 43 4.1.3 Simulation of Eccentric Motion 51 4.2 Non-Conservative System (Aerodynamic Effects) 65 4.2.1 Modification of Equations of Motion based on Nonconservative System 65 4.2.2 Simulation of Non-Conservative Systems with Feedback Control 66 5 Conclusions & Recommendations 70 Appendices 77 Chapter 1 Background Information 1.1 Nomenclature Symbol Explanation ma Mass of Subsatellite mb Mass of Basesatellite G Gravitational Constant Me Mass of Earth f True Anomaly e Eccentricity P Focal Parameter for elliptical orbit a Semimajor axis UJo Angular velocity of basesatellite Ft Perturbation Forces Wi Thrust Forces s Length of Tether P Density of tether f Unit vector to tension E Modulus of Elasticity T^o, Tj, Tether tension points 1 Unstretched tether length t Time R Origin to point on Tether T Tension Vi Velocity of end-body 0 In-Plane motion <f> Out-Of-Plane motion Ri Location of Satellite A: Sub and B: Base Ui Velocity of the tether leaving satellite Non-Conservative Parameters: c Nondimensional Aerodynamic Drag Aa Subsatellite Drag Area 2.01 Cds Subsatellite Drag Coefficient 2.0 Cdt Tether Drag Coefficient 2.0 "'orbit Orbital Altitude 220000 m h0 Scaling Parameter 10800 m "ref Reference Altitude for Air Density 120000 kg/m2 Pref Reference Air Density 4.47e-9 P Air Density variable k Initial Length 1000m I'm Desirable Final Length 100000m m Mass of the subsatellite 500 Kg 1.2 Tethered Satellites Overview The space tether concept started during the late 1800's and then progressed in the 60's causing NASA to give direction to the field in the 70's. NASA examined the progress up to that point and studied the feasibility of ideas and gave direction to the study of tethered systems, especially tethered satellites. Some applications consid ered in the 70's are: orbiting antenna, shuttle-borne tethered satellite, electrodynamic powered tether, and possibly a space station tether system. The result of NASA's in volvement in the field was the creation of a single short term goal for tether satellite researchers: achieve 100km deployment with the creation of newer materials, control laws, and all the supporting subsystems. The control laws and dynamics of tethered satellites is the primary subject of this thesis. The basic tethered satellite is composed of three parts: the basesatellite, tether, and subsatellite. The basesatellite contains the subsatellite and tether until deployment. In some cases the basesatellite can be considered a satellite, or it could be a shuttle, space station, or planet/moon. The tether simply connects the two satellites. Currently the tether is generally made from composite material consisting mostly of aluminum and kevlar. The subsatellite is released from the basesatellite towards an attracting body. The attracting body will be Earth in this study. There are three dynamic phases of a tethered satellite system: elongation, station- keeping, and retracting phase. Of the three phases the elongation phase is generally not discussed in literature because it is naturally stable and needs very little control. When the subsatellite is released it is attracted by Earth. The only way the subsatellite can increase the natural rate of elongation is with some sort of propulsion. The sta- tionkeeping phase and retraction phase need active control for stability, especially when atmospheric effects are taken into account. When there are no assumptions, the dynamics become overly difficult because the dynamics are governed by a set of ordinary and partial nonlinear, non-autonomous and coupled differential equations. These conditions create a list of problems to consider.[1] Three-dimensional rigid body dynamics (librational motion) of the station and subsatellite In-plane and out-of-plane pendulum type motions of the tether of finite mass Offset of the tether attachment point from the space station's center of mass as well as controlled variations of the off-set Transverse vibrations of the tether External forces Due to these conditions, certain assumptions are required that are discussed in a later chapter. The control laws that are found in literature are numerous. Previous literature [1] discusses various types of control laws including tension, thruster, and offset control. The thesis goal is to consider these control laws with a focus on newer control theories that may help improve upon past attempts. Thruster control and non-thruster based control will be examined. The thruster control laws intuitively should have better performance than other types, but also include disadvantages: fuel and interaction between thruster and basesatellite, causing possible object damage.[2] 1.3 Orbital Mechanics In this introduction of orbital mechanics, only the two-body problem will be ex amined pertaining to this research. More information on the general n-body object dfsderivation is contained in reference [3]. The two body problem discusses the inter action between two masses without any other body in the universe. These bodies will be spherical and points in space as shown in Figure 1.1. As can be shown from Newton's Second Law the following forces of attraction can be found: MrM = Tl5 (1.1) T Figure 1.1: Two Body Problem Located in Inertial Space and GMmr mr. = - (1.2) then, GMm2: Mtutm (1.3) and GM2mr MmrL = (1.4) r2 r after subtracting and cancelling out like terms, the above equations become: G{M + m) - = rm rM t, r (1.5) with, r = rm - rM (1.6) or in standard form, tpO *=^ <l-8> with /x = G(M + m).