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Dynamics and control of an orbiting space platform based tethered satellite system Lakshmanan, Prem Kumar 1989

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D Y N A M I C S A N D C O N T R O L O F A N ORBITING S P A C E P L A T F O R M B A S E D T E T H E R E D S A T E L L I T E S Y S T E M P r e m K u m a r Lakshmanan B. Tech. (Hons), Banaras Hindu University, 1982 M.A.Sc, University of British Columbia, 1985 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T F O R T H E D E G R E E O F D O C T O R O F PHILOSOPHY in The Faculty of Graduate Studies Department of Mechanical Engineering We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F BRITISH C O L U M B I A October 1989 © P r e m Kumar Lakshmanan, 1989 In p resen t i ng this thesis in partial fu l f i lment of the requ i remen ts fo r an a d v a n c e d d e g r e e at the Un ivers i ty of Brit ish C o l u m b i a , I agree that t he Library shal l m a k e it f reely avai lable fo r re fe rence and s tudy. I fur ther agree that p e r m i s s i o n fo r ex tens ive c o p y i n g of this thes is fo r scho lar ly p u r p o s e s may be g ran ted by the h e a d o f m y d e p a r t m e n t o r by his o r her representa t ives . It is u n d e r s t o o d that c o p y i n g o r pub l i ca t i on of this thesis for f inancia l ga in shal l no t b e a l l o w e d w i t hou t m y wr i t ten p e r m i s s i o n . D e p a r t m e n t of MELC^^A-^. £* 0>A,£te ^ T h e Univers i ty of Brit ish C o l u m b i a V a n c o u v e r , C a n a d a Da te @cfo£e*r S^oP.J^S? DE-6 (2/88) A B S T R A C T A relatively general mathematical model is proposed for studying the coupled attitude dynamics of space platform supported tethered subsatellite systems ac-counting for offset of the tether attachment point. The offset is treated as a function of time subject to constraints. General energy expressions allowing for flexibility of the tether as well as the platform are derived. The governing equations account for: (i) three-dimensional librational motion of the platform; (ii) inplane and out-of-plane libration of the tether of finite mass and con-nected to the platform with an offset; (iii) time dependent variations on the attachment point of the tether; (iv) generalized force contributions due to various active controllers; (v) orbits of arbitrary eccentricities; (vi) deployment and retrieval of the tether from the space platform. The second order coupled, nonlinear, nonautonomous, differential equations are linearized about a quasi-static equilibrium position. After nondimensionalizing with respect to the orbital rate and characteristic dimensions of the structure, they are collated into matrix form and integrated numerically. A n extensive response analysis is carried out over a range of system parameters, operational maneuvers and orbit eccentricity to assess complex interactions involved and help evolve suitable control strategies. Two control schemes, tether tension modulation and thruster control, are ex-tended to the case of an offset of the tether attachment point. It is shown that a linear control strategy is sufficient to control the tether inplane as well as out-of-plane librations in the presence of an out-of-plane offset. ii A new approach to control of platform based tethered satellite systems is pro-posed that utilizes motion of the offset to control the coupled system dynamics. The scheme involves specification of offset accelerations based on feedback of sys-tem states and feedforward of offset states. Controllability of the linearized equa-tion is established numerically and relative merits of the three control strategies assessed. Results indicate that the controllers are effective even in the presence of severe disturbances during all three mission phases of deployment, stationkeeping and retrieval. During stationkeeping, the tension control procedure demands larger energy for shorter tethers. Damping characteristics of the thruster control are in-deed superior but at the expense of the energy. The offset control has a tendency to dynamically isolate the tethered subsatellite from the space platform. From energy consideration, it proved to be the best, particularly at shorter tether lengths. How-ever, due to offset constraint in a practical situation, its effectiveness diminishes with an increase in the tether length and becomes virtually ineffective for a tether length over 1 km. During retrieval, hybrid control strategies utilizing tension or thruster control at the onset of retrieval, with offset control at shorter tether lengths proved to be quite energy efficient. For space application, the thruster-ofFset hybrid con-trol strategy appears to be quite promising both in terms of system dynamics and energy demand. iii T A B L E O F C O N T E N T S A B S T R A C T ii LIST O F FIGURES viii LIST O F T A B L E xvi LIST O F S Y M B O L S xvii 1. I N T R O D U C T I O N 1 1.1 Preliminary Remarks 1 1.1 Background 4 1.1 Scope of the Present Investigation 7 2 F O R M U L A T I O N O F T H E P R O B L E M 10 2.1 System Description 10 2.2 Kinematics of the System 11 2.2.1 Reference frames and domains 11 2.2.2 Position vectors 15 2.2.3 Deformation vectors 15 2.2.4 Constraints 19 2.3 Librational Generalized Coordinates 22 2.4 Kinetic Energy 26 2.5 Potential Energy 29 iv 2.6 Strain Energy . 30 2.7 Equations of Motion 32 3. S Y S T E M D Y N A M I C S 37 3.1 Retrieval Scheme 37 3.2 Results and Discussions 39 3.2.1 Platform offset 39 3.2.2 Tether parameters 47 3.2.3 Payload mass 49 3.2.4 Platform inertia 49 3.2.5 Orbit eccentricity 54 3.2.6 Tether retrieval 54 4. C O N T R O L O F T H E T E T H E R E D S A T E L L I T E S Y S T E M 58 4.1 Preliminary Remarks 58 4.2 State Space Representation of the Mathematical Model . . . . 58 4.2.1 Equilibrium configuration 60 4.3 Control Strategies 62 4.3.1 Tension control 64 4.3.2 Thruster control 65 4.3.3 Offset control 67 4.4 Results and Discussion 72 4.4.1 Reference case (stationkeeping at 100 m) 72 v 4.4.2 Orbit eccentricity 83 4.4.3 Platform inertias 87 4.4.4 Tether length 91 4.4.5 Tether mass 104 4.4.6 Payload mass 112 4.4.7 Deployment 119 4.4.8 Tether retrieval 123 4.4.9 Hybrid control strategies 139 5. E X P E R I M E N T A L V A L I D A T I O N O F T H E O F F S E T C O N T R O L S T R A T E G Y 146 5.1 Introduction 146 5.2 Theoretical Representation of the Ground-based Tethered System 152 5.3 System Description 154 5.3.1 Controller 154 5.3.2 Actuator 157 5.3.3 Sensors 159 5.4 Sampled Data and Finite Difference Discretization 162 5.5 Selection of Sample Step-Size 164 5.5.1 Process dynamics criterion 165 5.5.2 Control algorithm criterion 165 5.6 Software 167 vi 5.7 Results and Discussion 170 C O N C L U D I N G R E M A R K S 186 R E F E R E N C E S 190 A P P E N D I C E S 195 A . Equations of Motion for the Rigid System in an Arbitrary Orbit 195 B. System Equations in Matrix Form 203 C . Matrices Related to Optimal Control 213 D. Computer-Translator Interface 215 E . Step-Motors and Translator Modules 216 F . Sensors Technical Information 221 vii LIST O F FIGURES 1-1 A schematic diagram of the Space Station based tethered microgravity facility. 3 1-2 Tethered dumbbell satellite illustrating the working principle 4 1- 3 A schematic diagram of the plan of study 9 2- 1 A schematic diagram of the platform based tethered satellite system showing the boom-trolley arrangement to vary the offset vector dp. Various forces acting on the system due to control are also indicated. Ma, Mp, and represent nondimensionalized momentum wheel torques while Ta, T 7 , and indicate the thruster forces. Dx>p, DViP, and DZ)P are the three dimensional time dependent offset components. . 11 2-2 The space platform based tethered satellite configuration. Vectorial description of the kinematics is established through position vectors R{ (t = p, r, s, t), offset vectors dp, dB, and the tether line vector Ijt. Body frames a;,-,j/,-,,z,- are attached to each component of the system 13 2-3 Deformation of the tether as measured in the tether fixed frame xt,yt,zt-PP' represents the tether deformation with components ut, v*, and wt that are functions of the axial coordinate yt and independent time variable t 17 2-4 Transverse deformation of the platform as measured in the platform fixed frame xp,yp,zp. PP' represents the transformation of an arbitrary point on the platform due to deformation. The deformation u p is a function of the space coordinates yp, zp, and time t 19 viii 2-5 Description of orientations of the platform, subsatellite and tether relative to the orbital coordinate frame using Eulerian rotations. . . . 24 2- 6 Position vector I\ of an arbitrary mass element in the inertial space. Note, its relation to the orbital and body frames 27 3- 1 Mission profile for TSS-1 to be launched in early 1991 40 3-2 Dynamical response of the platform supported tethered satellite system in the reference configuration with the offsets set to zero 43 3-3 Effect of inplane and out-of-plane offsets on the coupled response of the tether and platform: (a) DVtP = Dz>p = 0 44 (b) DXtP = DZtP = 0 45 (c) Ac,p = Dy,p = 0 46 3-4 Effect of increasing tether the mass (fivefold) on the response of the platform based tethered satellite system 48 3-5 Effect of increasing the payload mass on the coupled space platform-tether dynamics 50 3-6 Effect of varying platform inertias on the coupled dynamics: (a) Jyy,p = JZZ,P = 10- 1 8 x 1 ° 7 kgm 2 52 (b) IyVtP = 4.08 x 107 kgm 2 and Izz>p = 16.33 x 107 kgm 2 53 3-7 Effect of orbital eccentricity on the dynamics of the space platform based tethered satellite system 55 3-8 System response during retrieval. Note, even with such a small retrieval rate and initial disturbance, the platform based tethered subsatellite system quickly becomes unstable 57 ix 4-1 Closed loop pole placement for tension control of the tethered satellite system 65 4-2 Closed loop pole placement for thruster control of the tethered satellite system 66 4-3 Block diagram of the offset controlled platform based tethered satellite system 69 4-4 Closed loop pole allocation for the offset controlled system 70 4-5 Three dimensional view showing motion of the offset during retrieval of the tether from 100 m to 10 m in presence of a large disturbance. . . . 71 4-6 Performance of the system in the reference stationkeeping configuration: (a) time histories of platform and tether response during the tension control 74 (b) variation of the tether tension and controller effort during the tension control 75 (c) time histories of platform and tether response during the thruster control 77 (d) variation of control effort during the thruster based strategy. . . . 79 (e) platform and tether response during the offset strategy. 80 (f) time histories of the tether attachment point, associated forces and platform moments during the offset control strategy 82 4-7 Response of the system in the stationkeeping reference configuration while negotiating an elliptic trajectory of e = 0.05: (a) tension control 84 (b) thruster control 85 (c) offset control 86 x 4-8 Effect of platform inertia on the response of a tethered satellite system: (a) tension control 88 (b) thruster control 89 (c) offset control 90 4-9 Effectiveness of the control strategies for a tether length of 1 km: (a) system response with the tension control strategy 93 (b) control effort time histories during the tension modulation strategy. 94 (c) system response with the thruster control strategy 95 (d) control effort time histories during the thruster control strategy. . . 96 4-10 Effect of tether length on the performance of the offset control strategy: (a) system response, 1 km tether 98 (b) control effort, 1 km tether 99 (c) system response, 750 m tether 100 (d) system response, 500 m tether 101 (e) system response, 250 m tether 102 4-11 Effect of increasing the tether mass to 10 kg on the performance of the various control strategies: (a) system response during the tension modulation approach 106 (b) control effort time histories during the tension modulation approach. 107 (c) system response during the thruster control 108 (d) control effort time histories for the thruster case 109 (e) system response during the offset control approach 110 (f) offset and control effort time histories I l l 4-12 Effect of increasing the payload mass on effectiveness of the three control strategies: xi (a) system response during the tension modulation approach 113 (b) control effort time histories during the tension modulation approach. 114 (c) system response during the thruster control 115 (d) control effort time histories for the thruster case 116 (e) system response during the offset control approach 117 (f) offset and control effort time histories 118 4-13 Effect of deployment on the system response during: (a) the tension control strategy. 120 (b) the thruster control strategy 121 (c) the offset control strategy 122 4-14 System response during retrieval from 100 m to 10 m in the presence of the tension control strategy: (a) retrieval time of 1 orbit 125 (b) retrieval time of 0.68 orbit 126 (c) retrieval time of 0.37 orbit 127 4-15 System response during from 100 m to 10 m with the thruster control strategy: (a) retrieval time of 1 orbit 129 (b) retrieval time of 0.68 orbit 130 (c) retrieval time of 0.37 orbit 131 4-16 System response during from 100 m to 10 m with the offset control strategy: (a) retrieval time of 1 orbit 132 (b) retrieval time of 0.68 orbit 133 (c) retrieval time of 0.37 orbit 134 xii 4-17 Response of the system during exponential retrieval of the tether from 1000 m to 10 m in 0.37 orbit: (a) tension control strategy 136 (b) thruster control 137 (c) offset control. Note, the tether initial condition is now reduced to at = 7t = 1 ° . Platform initial conditions remain unchanged 138 4-18a System response with the tension modulation procedure followed by the offset control during exponential retrieval from 1000 m to 10 m in 0.37 orbit 140 4-18b Controller time history for tension-offset hybrid control 141 4-19a Thruster-offset hybrid controlled system response during exponential retrieval from 1000 m to 10 m in 0.37 orbit 143 4- 19b Controller effort time history for the thruster-offset hybrid control. . 144 5- 1 Photograph of the test-rig constructed to validate offset control strategy: (a) aluminum frame; (b) inplane motor; (c) carriage for out-of-plane motion; (d) wooden stand; (e) linear bearings; (f) tethered payload 149 5-2 Digital hardware used in the experiment: (a) translator modules, deployment and retrieval; (b) translator module, offset motions; (c) power supply; (d) function generator . 150 5-3 Carriage reel and sensor mechanism: (a) potentiometer on mounting bracket; (b) moveable aluminum leaf mechanism with slot for tether; (c) retrieval motor and reel housing; (d) tether; (e) inplane traverse with linear bearings; (f) payload 151 5-4 Schematic diagram of the offset control test rig 155 xiii 5-5 Block diagram representation of the ground based offset control experiment 156 5-6 The offset positioning mechanism showing the trapezoidal frame and linear bearings that allow the carriage to be located anywhere within the motion envelope 158 5-7 Tether attitude motion sensor with potentiometers as transducers. . 161 5-8 Flowchart for the offset acceleration based control algorithm showing initialziation and control phases 168 5-9 Plots showing comparison between numerical and experimental response results for uncontrolled and controlled conditions of the system during the stationkeeping phase. The system is subjected to a large inplane disturbance of at (0) = - 1 5 ° 172 5-10 Typical response data for the system, in the stationkeeping phase, subjected to an out-of-plane disturbance of 'yt(O) = —15° 174 5 - l la Experimental data for the tether stationkeeping at 2.25m with a large initial disturbance of 15° in both the inplane as well as the out-of-plane directions 176 5 - l lb Carriage position during controlled response of the system, in the stationkeeping mode, subjected to a large initial disturbance of 15° in both the planes 177 5-12a Experimentally obtained response for the 1.5 m tethered system in the stationkeeping mode. Note, the system is subjected to the same disturbance as in Fig. 5-11 178 5-12b Offset trajectory during the controlled response of the 1.5 m tethered subjected to an initial disturbance of at(0) — 7t(0) = 15° 179 xiv 5-13a Response of the tethered system to an initial displacement of 15° in both the planes. The tether was stationkeeping at 0.75m 180 5-13b Carriage trajectory during the controlled stationkeeping response of the 0.75 m tether. A n initial displacement of 15° was applied in both the planes 181 5-14 System response during uniform deployment from 0.75 m to 2.25 m in 9 s. The initial disturbance of 15° was applied in both the inplane and out-of-plane directions 182 5-15 Experimentally obtained response plots showing effectivenss of the offset control strategy for two different retrieval rates: (a) L = -0.08 m s 184 (b) L = -0.17 m s 185 xv LIST O F T A B L E S 2- 1 Eulerian rotation sequence for the ith frame starting from the orbital frame orientation 23 3- 1 Reference configuration parameter values for dynamics simulation. . . 37 3- 2 Parametric representation of the exponential retrieval maneuver. . . . 38 4- 1 Equilibrium configuration for a platform based tether (1000 m) with 20 m offset 61 4-2 Reference configuration parameters for controlled response simulation. 73 4- 3 Controller effort 144 5- 1 Variation of the controller performance with tether length 175 xvi LIST O F S Y M B O L S c constant of exponential retrieval djtP,djtP nominal offset values and their corresponding deviations, re-spectively, along orthogonal directions; j = x,y,z drrii differential mass element on the tth body dVi differential volume element on the tth body e eccentricity fc Nyquist frequency fe 2esin6/l + ecosO fep 1/1 + ecos6 h orbit altitude ha angular momentum per unit mass of the system ii,ji,ki unit vectors of tth body frame / stretched length of tether / / final value of I after deployment or retrieval li initial value of T before deployment or retrieval lrej reference tether length (same as the initial length of the tether) lXi, lVi, lz< direction .cosines of the tth frame T instantaneous nominal unstretched tether length / deviation of the tether length from its nominal value m mass of the entire system xvii mp mass of the space platform mr mass of the reel mechanism and undeployed tether mrat mr + mg + mt — m - mp m8 mass of the subsatellite met ma + mt m8t2 m8 + mt mass of the tether n number of generalized coordinates {?} n X 1 vectors of generalized coordinates, velocities, and accel-erations, respectively r number of inputs ?i position vector of a mass element dm,- on the ith body; • = P, r,M t independent time variable t0 initial time tx tQ + At t-i t0 — At At discrete time step size {u} r x 1 vector of controller inputs U{, V{, Wi platform and tether deformations expressed as separable func-tions of space and time; i = p, t Xo,y0,z0 orbital reference frame Xi,yi,Z{ reference frame attached to the tth body; t = p,r,s,t xviii [A] system characteristic matrix [B] control influence matrix [B] n x r matrix formed by coefficients of the r controller inputs [C] n X n gyroscopic matrix consisting of coefficients of the gener-alized velocities {q} C a i , C n . cosine(a,) and cosine(7t), respectively {D}, { £ ) } , { £ > } vector of controlled offsets and their derivatives Di,p> D{tP nondimensionalized nominal and controlled offsets, respectively; * = x,y,z —- — 2 — 2 Dxx,p Dy}p + Dz,p — — 2 — 2 Dyy,p Dx,p + Dz,p DZZiP DyiP + DXfP Dxy,p Dx.pDy.p Dyz.P DyiPDZtP Dzx,p Dz,pDx,p E Young's modulus Fi i = p,r,s,t frame attached to the tth body; t = p, r, s, t Fx,Fy,Fz forces required to move offsets [G] optimal control gain matrix GM gravitational constant Ixyp,Iy,p>Iz,p platform inertias about xp, yp and zp axes, respectively [/] identity matrix J quadratic regulator performance index xix \K\ nxn stiffness matrix made up of coefficients of the generalized displacements {q} L l/lref L l/lref [M] nxn mass matrix composed of coefficients of the generalized acceleration {q} Kat > Kat optimal gains for tether inplane state feedback K^t, optimal gains for tether out-of-plane state feedback Kf)z piKfo gains for inplane offset feedforward Kfs J Kk gains for out-of-plane offset feedforward D*>p £>x,p MP)8 ratio of the platform mass to subsatellite mass Mt>B ratio of the tether mass to subsatellite mass M Q , M / 3 , M 7 momentum wheel torques about xp, yp and zp directions, re-spectively, nondimensionalized with respect to malref62 {P} vector of terms not associated with generalized coordinates or control inputs. It arises due to nonzero equilibrium position caused by the offset [Q] state penalty matrix Qi equilibrium tether tension force Qap generalized force associated with ap [R] control penalty matrix Rc position vector of system centre of mass from the inertial frame Ri position vector of the »th domain relative to the inertial frame; i = P,r,s,t xx \S] solution of the Ricatti matrix equation S a i , S l i sine(at) and sine(7,), respectively T kinetic energy of the system Ti kinetic energy of the tth body [Tij] transformations matrix relating a-,-,t/i,z» and Xj,yj,Zj Ta,T^,Ti thruster forces along at, 7t and yt directions, respectively, nondimensionalized with respect to mel*ej02 Up potential energy of the system due to gravity Upi potential energy of the tth body due to gravity UB strain energy of the system due to deformation Usi strain energy of the tth body due to deformation V{ volume of the tth domain; t = p,t ap> Pp, lp pitch, yaw and roll of the platform, respectively <*a»A»»7a pitch, yaw and roll of the subsatellite, respectively Q-tilt tether inplane and out-of-plane rotations, respectively e strain in a differential element v Poisson's ratio Pt mass per unit length of tether pv mass per unit area of platform Pi vector measured in the tth frame w x», Wyi,vzi scalar components of the w,-, the angular velocity of the tth body a stress in a differential element xxi true anomaly of the orbit position vector of elemental mass rfm» on the tth body relative to the inertial frame xxii 1. I N T R O D U C T I O N 1.1 Preliminary Remarks Advent of the Space Shuttle and the proposed U.S. Space Station have presented a wide range of possibilities for space exploration and exploitation. One approach to this end is the concept of Tethered Satellite System (TSS). The innovative idea is attributed to the Russian scientist Tsiolkovsky who explored the possibility of using an extremely high tower, extending from the equator to the geosynchronous altitude and beyond, as a means for escaping the Earth's gravity. In his paper[l], aptly called "Day Dreams of Heaven and Earth", he discussed changes in the grav-ity force an individual would experience as he climbed up the tower thus uncovering one of the principal novelties of the concept that has now grown to become the TSS. As far as the actual application is concerned, interest in the system was initially associated with the retrieval of a stranded astronaut by throwing a buoy on a tether from the rescue vehicle and reeling in the tether. The first space based experiments with tether were carried out during the two Gemini missions [2]. The manned space vehicle Gemini XI was tethered to an Agena vehicle and a rotating configuration maintained through the Gemini thruster reaction control system. Later, the Gemini XII flight demonstrated feasibility of a gravity gradient stabilized tethered configu-ration. Since then, possible applications of the system have grown to cover a broad spectrum: (i) sophisticated scientific experiments aimed at gravity gradient, magnetic, ionospheric, aerothermodynamic and radio astronomy experiments; (ii) use of the tethered system as a flying wind tunnel; (iii) deployment of payloads into new orbits and retrieval of satellites for 1 servicing; (iv) provision of a desired controlled microgravity environment for scientific experiments and space manufacturing; (v) generation of electricity (electrodynamic tether); (vi) power and cargo transfer; and many others. N A S A has shown considerable interest in exploring some of the possibilities and has planned, in collaboration with the Italian Space Agency (ASI), an experiment involving deployment, stationkeeping and retrieval of a 20 km long electrodynamic tether. During deployment, a reel mechanism releases the tether to send the subsatellite to a desired altitude. The length of the tether is held fixed at this point during which various mission activities may be carried out. This phase is called stationkeeping. Once the mission is completed, the subsatellite is reeled back (retrieval phase). The first mission, TSS-1, has been approved and is scheduled for launch in early 1991. The system will consist of a U.S. built deployer and an Italian built subsatellite. In general, such a tethered payload may be deployed from the Space Shuttle, the proposed Space Station (Freedom) or any orbiting platform. Fig. 1-1 shows schematically a typical configuration of the space station based tethered satellite system as outlined in a N A S A Technical Memorandum [3]. The fundamental principle governing tether dynamics can be illustrated by a simple two-body tethered system as shown in Fig. 1-2. This "dumbbell" satellite orbits the earth such that its centre of mass follows the prescribed orbital trajectory. The top mass experiences a larger centrifugal force than the gravitational force, being in an orbit higher than that of the centre of mass. The reverse occurs at the lower mass causing a resultant moment to act on the system such that it oscillates about the local vertical like a pendulum with the tether maintained taut under 2 9080 Kg 3.8 x 10- 4 g's CONTAMINATION—FREE AND ISOLATION LEVEL 1Km SPACE c£» PROCESSING 4 | FACILITY LJ 100m \ 90800 Kg \ 3.8x10-5 g's 0 9'« ("ZERO G") LEVEL Figure 1-1 A schematic diagram of the Space Station based tethered micro-gravity facility. 3 Mass Figure 1-2 Tethered dumbbell satellite illustrating the working principle. 4 tension. Now, as the length of the tether decreases during retrieval, the conservation of angular momentum dictates the swinging oscillations of the tether to amplify. In conjunction with a space platform supported tethered satellite system, the swinging motion could increase to the point that the tether wraps itself around the space platform. Furthermore, the presence of any offset between the tether attachment point and the platform's centre of mass may impose an additional moment on the platform. Thus the dynamics of the platform becomes intrinsically coupled with those of the tether and the subsatellite. 1.2 Background The vast potential of tether connected orbiting systems has led to numerous investigations aimed at their dynamics and control during deployment, operational (stationkeeping) and retrieval phases. In its utmost generality the problem is quite challenging as the system dynamics is governed by a set of ordinary and partial nonlinear, nonautonomous and coupled differential equations that account for: (i) three dimensional rigid body dynamics (librational motion) as well as elastic response of the end bodies; (ii) swinging inplane and out-of-plane motions of the tether, of finite mass and elasticity, with longitudinal and transverse vibrations superposed on them; (iii) offsets of the tether attachment points with the problem further compli-cated by their specified variations in case of the offset control strategy; (iv) external environmental forces due to aerodynamic drag in a rotating atmosphere, solar radiation pressure, thermal stresses, etc. (v) control forces provided by tether tension, thrusters, etc., and their com-5 binations. Over the years, investigators have attempted to obtain some insight into the complex dynamics of the system using a variety of models which have been sum-marized in two review papers by Misra and Modi [4,5]. In general, the studies show that the dynamics of the system during deployment is stable, however, the retrieval dynamics is basically unstable. The system involves a negative damping approxi-mately proportional to V/lrej, where f and lrej are the instantaneous unstretched and reference lengths, respectively, and prime denotes differentiation with respect to the true anomaly 6. This suggests a need for an active control strategy, partic-ularly to limit inplane (at) and out-of-plane (7*) swing motions of the tether. The pioneering contribution that may help realize this objective is due to Rupp[6] who introduced a tension control law for the system. Librational motion in the orbital plane was analyzed and the growth of pitch oscillations during the retrieval phase noted. The system was further studied in detail by Baker et al. [7] taking into ac-count the three dimensional character of the dynamics as well as the aerodynamic drag in a rotating atmosphere. Several more sophisticated models have followed since [8-16], however, one of the major conclusions of all the analyses remains es-sentially the same: even when the various tension control schemes are used, a large amplitude motion can result under certain conditions, particularly during retrieval, which may not be acceptable. Tension is directly dependent on the gravity gradient, i.e., on the tether length. Hence, at shorter lengths during retrieval, there exists a possibility of the tether be-coming slack rendering the tension control strategy ineffective. This can be avoided by providing thrusters in the three orthogonal directions at the subsatellite end. The strategy has been shown to be quite effective, however, use of thrusters in proxim-6 ity of a space platform is undesirable due to the danger of plume impingement and safety considerations. 1.3 Scope of the Present Investigation With this as background, a model is proposed here that studies coupled Vibra-tional dynamics and control of a tethered satellite system supported by an orbiting platform (e.g., the Space Station). To gain better insight of the system dynamics, the model focuses on the key factors associated with the complex dynamics of such a spacecraft operating in the absence of environmental forces. The study is initiated with a relatively general formulation that accounts for the three dimensional librational motion of a plate-type space platform and a tether connected subsatellite of an arbitrary inertia distribution. The tether is treated as a continuum with a finite mass whose point of attachment is offset from the center of mass of the platform and is capable of being moved subject to reasonable constraints based on the platform geometry and available technology of telerobotics. General energy expressions accounting for flexibility of the tether as well as the space platform are derived (Chapter 2 ) . This is followed by the response analysis for various combinations of the system parameters, operational maneuvers and orbit eccentricity (Chapter 3). As pointed out earlier, control of the system, especially during retrieval, presents a challenging task. The coupling introduced by the offset causes the en-tire system to have drastically different dynamic characteristics. The two major schemes that have been successfully utilized to control two-body tethered systems, namely the tether tension modulation and active thruster control, are extended to the offset case and their relative merit assessed. Recognizing that there are cer-7 tain situations where conventional actuators cannot be used, a new approach that utilizes the offset motion itself to control the coupled system dynamics is studied in detail and its performance compared with the tension and thruster strategies. Hybrid control strategies during retrieval, using the offset procedure in conjunction with the tension or thruster approach with optimal switching, are also developed as they appeared to offer considerable promise (Chapter 4). Finally, as a logical culmination to the study, the proposed offset control strategy is validated using a ground-based real-time tethered satellite model. Effectiveness of the offset control is tested during stationkeeping, deployment, and retrieval phases with the system subjected to demanding disturbances (Chapter 5). Chapter 6 presents concluding remarks based on the study and offers sug-gestions for future study which are likely to be profitable. It is not intended here to amass a vast volume of data necessary for design. Emphasis throughout is on the development of a methodology, appreciation of the physical concepts involved, and validation of an approach to control a two-body tethered system as offered by judicious manipulation of the attachment point. Figure 1-3 summarizes the plan of study. 8 D Y N A M I C S & C O N T R O L O F T H E S P A C E P L A T F O R M B A S E D T E T H E R E D S A T E L L I T E S Y S T E M D E F I N E P H Y S I C A L P R O B L E M ( K I N E M A T I C S )  D Y N A M I C S C O N T R O L L E R D E S I G N L a g r a n g e F o r m u l a t i o n S e l e c t p l a u s i b l e c o n t r o l v a r i a b l e s a n d o b t a i n t h e i r g e n e r a l i z e d f o r c e s A u t o n o m o u s C o n s e r v a t i v e ( c i r c u l a r o r b i t s ) N o n - A u t o n o m o u s C o n s e r v a t i v e ( e l l i p t i c o r b i t s ) N o n - A u t o n o m o u s N o n - C o n s e r v a t i v e ( d e p l o y m e n t / r e t r i e v a l ) T e t h e r T e n s i o n + M o m e n t u m W h e e l s B M a t r i x M i c r o - T h r u s t e r s + M o m e n t u m W h e e l s T e t h e r O f f s e t M o t i o n + M o m e n t u m W h e e l s C o m b i n e d S y s t e m D y n a m i c s A M a t r i x B M a t r i x S t a t e v a r i a b l e f e e d b a c k w i t h o p t i m a l g a i n s e l e c t i o n T e t h e r E f f e c t s Tether length Tether mass E n d B o d y E f f e c t s Payload Mass Space Platform Inertias D e p l o y / R e t r i e v e Exponential Proliles O r b i t a l E f f e c t s Eccentricity Altitude T e n s i o n c o n t r o l l e d r e s p o n s e T h r u s t e r c o n t r o l l e d r e s p o n s e O f f s e t c o n t r o l l e d r e s p o n s e V A L I D A T I O N O F O F F S E T C O N T R O L S T R A T E G Y Figure 1-3 A schematic diagram of the plan of study. 2. F O R M U L A T I O N O F T H E P R O B L E M 2.1 System Description This chapter describes a relatively general approach to study the dynamics of a space platform based tethered satellite system. In the beginning, a kinematical description of the system consisting of a space platform, tether and subsatellite is obtained, without any restriction on the offsets of the points of attachment at the platform and the subsatellite ends. This is followed by the introduction of constraints involved to obtain kinematical relations between the tether, platform, subsatellite and centre of mass of the system. Next, an Eulerian description of the platform, subsatellite and tether orientation is introduced. This information is uti-lized to obtain the kinetic, potential and strain energies of the system. Finally, the energy is processed according to the Lagrange formulation procedure to obtain the governing equations of motion. Even a numerical solution of the highly nonlinear, nonautonomous, and coupled differential equations using a high speed computer presented a formidable challenge. A balance must be struck between an elaborate detailed model that is not conducive to mathematical analysis and the one that re-tains essential physical features of the system and permits development of suitable control strategies. The system consists of a large space platform, considered as a free-free plate, in an arbitrary orbit around the Earth (Fig. 2-1). It is connected by a thin, flexible cable (tether) to a rigid subsatellite. The tether is capable of being deployed, retrieved or maintained at a fixed length from the space platform on the end of a boom-trolley arrangement. This allows the point of attachment of the tether to move relative to the centre of mass of the platform. The point of attachment of the 10 Figure 2-1 A schematic diagram of the platform based tethered satellite sys-tem showing the boom-trolley arrangement to vary the offset vector dp. Various forces acting on the system due to control are also indi-cated. M a , Mp, and M 7 represent nondimensionalized momentum wheel torques while Ta, T 7 , and indicate the thruster forces. A e , P > DVtp, and DZ)P are the three dimensional time dependent offset components. 11 tether at the subsatellite end is also permitted such freedom of motion, although a similar boom-trolley arrangement is not shown explicitly in the figure. The platform based momentum wheels provide control torques about the platform principal axes as shown. In addition, one may attempt to control the system by manipulating the tether tension, orthogonal set of thrusters (Ta, T 7 , and T^) at the subsatellite, or the tether attachment offset with respect to the system centre of mass. Their application in suitable combinations may also present interesting possibilities for control. 2.2 Kinematics of the System 2.2.1 Reference frames and domains Four distinct domains can be clearly established for the system described in Fig. 2-2. Domain 'p' represents the flexible plate or space platform. The frame Fp (xp,yp,Zp) is attached to the centre of mass of the undeformed platform with its axes along the principal directions of the undeformed plate. Vectors described relative to this rotating reference frame are distinguished by the subscript 'p'. Mass of the platform is considered constant (rhp = 0). Domain's ' consists of the rigid subsatellite attached at the end of the tether. The frame F8 (xg,ys,z8) is used to describe its orientation in space. The origin of this frame is at the centre of mass of the subsatellite with the axes oriented along the principal directions of the subsatellite. Vectors measured relative to this rotating frame carry subscript 's'. Mass of the rigid subsatellite is taken to be constant (m8 = 0). The tether is considered to be made up of two domains. The domain 'r' consists of the rigid undeployed tether that is wrapped around the reel (enlarged view in 12 CO T R O L L E Y / R E E L INERTIAL F R A M E Figure 2-2 SUBSATELLITE —* R E E L RADIUS 6r E N L A R G E D , R E L O C A T E D VIEW OF T R O L L E Y / R E E L Vt . ^ The space platform based tethered satellite configuration. VectoriaUescription of the kinematics is estab-lished through position vectors J2,- (i = p,r,M), offset vectors dp, da, and the tether line vector ljt. Body frames £,-,t/,-,2i are attached to each component of the system. Fig. 2-2). The frame Fr (xr,yr,zr) is attached to the center of mass of the domain and its axes are oriented along the principal directions of the domain. Vectors described relative to this rotating frame are distinguished by the subscript V . The domain is considered to be rigid. As the tether is deployed or retrieved, the mass of the trolley changes. The deployed portion of the tether is flexible and belongs to domain't'. The frame Ft (xt,yt,zt) has its origin at the point of attachment of the tether at the reel end. The t/t-axis is along the undeformed direction of the tether, that is, the direction of the tether in absence of transverse vibrations. Note, the frame xt,yt,zt is not attached to the centre of mass of the deployed tether. Vectors described —* relative to this rotating frame are characterized by the subscript *t\ 6r represents position vector of the tether frame xt,yt,zt relative to the reel frame xr,yr,zr. It accounts for the finite dimension of the reel mechanism. In practice, however, it is expected that this vector will be smaller compared to others involved in describing kinematics of the system and can therefore be considered negligible. The mass of the deployed portion of the tether varies with time according to the deployment or retrieval maneuver. The total mass composed of the deployed (domain't') and undeployed (domain 'r') portions of the tether is a constant (rht = — rh r). As a part of the analysis, two more frames are required to complete the kine-matic picture: an inertial frame attached to the centre of the Earth; and the orbital frame F0 {x0,y0,z0). The orbital frame is so oriented that the y0-axis is along the local vertical (the line that joins the Earth's centre with the system centre of mass); the a-0-axis along the orbit normal (the line normal to the plane of the orbit); and the 20-axis completes the right-handed triad. The orientations of all the body fixed frames were established relative to the orbital frame as functions of time. The true anomaly 6 of the system is the angle swept by the line joining the 14 instantaneous centre of mass of the system, C , and the Earth's centre in the plane of the orbit. 2.2.2 Position vectors With the reference frames selected, the vectors describing the system can be defined easily. The centre of mass of the system, C, followed a Keplerian orbit. This is based on the assumption that the orbital motion is unaffected by attitude motion of the satellite[l7,18,19]. In Fig. 2-2, Rp and Re are the position vectors of the origins of frames Fp and FB relative to C, respectively. Rr is the position vector of the center of mass of the domain V (the trolley) relative to C. Rt is the position vector of an arbitrary point on the tether relative to C. The mass of a given domain is denoted by mt- (t = p, s, r, t). Similarly, position vector of a differential mass element in domain ' i ' with respect to the corresponding body frame is denoted as (* = Pi si r> *)• F ° r example, Rp is the position vector to the center of mass of the space platform, of mass mp, from the system centre of mass C. A differential mass element dmp on the platform is located by the position vector rp as shown in Fig. 2-2. 2.2.3 Deformation vectors The tether and platform are the flexible components of the system. Their deformations are functions of space and time and are denoted as u, , v,-, and tu, along Xi, y,-, and z,- (i = p, i) directions, respectively. The quantities u,-, v,-, W{ can be expressed as separable functions of space and time using admissible functions having the following properties: (i) they are linearly independent; 15 (ii) they satisfy kinematic boundary conditions. Figure 2-3 shows a typical deformed configuration of the tether. The line EF represents a vector between the points of attachment of the tether to the platform and the subsatellite. Its magnitude is the stretched length of the tether. In other words: _ A EF = l(t)jt = {/ + / + vt(l + l,t)}jt. (2.1) —* Here: jt = unit basis vector along the j/t-axis; / = stretched length of the tether; / = commanded or nominal unstretched length of the tether; / = variation in tether length treated as a generalized coordinate for the purpose of control. Since the tether diameter is very small (of the order of a few millimetres), defor-mations depend essentially on the axial coordinate only. Thus, the point P{0, yt, 0} in the undeformed configuration transforms (dashed line) to P'{ut(yt, i), vt(yt,t) + yt, u>t{yt,t)} on the deformed tether. The tether deformations can be expressed as: Mvt,t) = ^2 An{t)<t>n{yt) ; (2.2a) n=l Myut) = Bn(t)4>n(yt); (2.26) n=l Mvut) = J2 cnV)4>n{vt); (2.2c) n=l where: <f>n{yt), ipn(yt) = admissible functions of the tether axial coordinate; iVi = number of admissible functions used to represent 16 Figure 2-3 Deformation of the tether as measured in the tether fixed frame xt,yt,zt. PP' represents the tether defor-mation with components u*, vt, and Wt that are functions of the axial coordinate yt and independent time variable t. tether transverse vibration; N2 = number of admissible functions used to represent tether longitudinal vibration. The kinematic boundary conditions dictate that transverse deformations at the supported ends be zero, that is, u t(0, t) = wt{0, t) = ut{l, t) = wt(l, t) = 0 . (2.3) The longitudinal deformations have only one geometric boundary condition at yt = 0 end given by vt(0, t) = 0. At the other end a dynamic boundary condition relating the stretch and static tension in the tether line can be obtained [20]. The admissible functions <t>n{yt) and V'n(yt) were obtained from reference [21]. So far as the platform is concerned, following assumptions are made with respect to its deformations: (i) the platform thickness is small compared to its other dimensions (typically 100 m width, 120 m length); (ii) the mass and stiffness properties of the platform are considered uniformly dis-tributed throughout; (iii) elastic displacements in the plane of the platform are considered negligible com-pared to those normal to the plane. (iv) transverse deformations of the platform are considered small; (v) the platform is unconstrained and hence can be treated as a uniform free-free plate in space. As shown in Fig. 2-4, a point P{0, y p , zp} on the undeformed platform transforms to the point P'{up(yp, zp, t), yp, zp} in the deformed configuration. Thus, as before, 18 p'{up{yP,zP,t),yp,zp} Figure 2-4 Transverse deformation of the platform as measured in the platform fixed frame xp,yp,zp. PP' represents the transformation of an arbitrary point on the platform due to deformation. The deformation up is a function of the space coordinates yp, zp, and time t. plate deformations can be written as: "3 n=l 0: p 0. (2.4) where r}n{yp,zp) satisfy the geometric boundary conditions for a free-free plate and Nz equals number of admissible functions used to represent the platform's trans-verse vibration. Suitable candidates for admissible functions for a free-free plate were obtained from references [22,23]. 2.2.4 Constraints The nature of the tethered satellite system involves unique constraint relation-ships between the various components. These constraints help eliminate some of the vectors involved in the kinematic description of the overall system. It follows from vector geometry in Fig. 2-2 that: A further constraint equation is obtained from the geometry of motion about the centre of mass C. By definition, the first moment of mass about C must be zero, R8 = Rp + dp + Ijt — de ; (2.5) Rr = Rp + dp ; (2.6) Rt = Rp + dp + rt. (2.7) (2.8) 20 —• —» —» —* —* The vectors Rp and RB can now be expressed in terms of dp, d8, fp, ft and Ijt. By definition of the centre of mass for a rigid body, fm r» dm,- = 0 (t = r, s). Using this property and substituting constraints Eqs. (2.5)-(2.7) into Eq. (2.8), Rp and R8 can be obtained as: Rp = — — \(m8 + mt + mr)dp + ma(ljt - dB) m L + / fpdmp+ / ftdmt ; (2.9) J tnp J mt and R e = — — mpdp — (mp + mt + mr)ht + (mp + mt + mr)dp + fp dmp + / ft dmt . (2.10) J tnp J mf It should be noted that f rp dmp ^ 0 as the platform may deform with time. . —* —* — • fp, rt and ljt are functions of u,-, v,-, and Wi (t = p, t) and hence, Rp and i? a are —# —* functions of dp, da, I and the deformations. —* —# It is useful to obtain the time derivatives of Rp and R8 as they are needed later in the formulation. In what follows, a derivative relative to a moving noninertial, reference frame Fi (i = p, r, s, t) is denoted as ( The relationship between the time derivative of a vector A measured relative to the inertial and noninertial frames can be shown to be dA fdA\ where Wi is the angular velocity of frame F< relative to the inertial reference. The mass per unit length (pt) of the tether as well as the mass per unit area (pp) of the platform are assumed to be uniform. Thus: f fm j dmt = I pt dyt; J mt JO 21 f fPA fpB / dmp = / / pp dypdzp . J mp JO Jo Using the Leibnitz rule for differentiation within an integral, the time derivatives of Rp and R8 can be written as: R„ — 1 m L Ra — ~ (m r + mt + rnB){(dp)Fp + wp x dp} + mB{(i + vt)jt + + vt){ut x jt)} - m8{(d8)Fe + CJ8X d8} /PA fPB Qf J0 {(-dr)Fp + * x r p } p p d y p d z p /"'(*) drt ~ i + J0 { ( ~ d t } p t + Q t X r ^ p t d y t + + V t ^ P t l j t J ; ^ [ - m p { ( 4 ) F p + Up x dp} - (trip + mt + m r){(/ + t)t)j't + (* + v t)(w t x j*t)} + (rrip + mt + mr){(d8)F8 + uj8 x <fa} J0 ^~cH^pP + Q p X fp}pp d y p d Z p fl(t) dr. -1 + J0 ^ ~ d t + * ^  RFYT + ^ + W T ^ T / J ' J • (2.11) (2.12) 2.3 Librational Generalized Coordinates The generalized coordinates associated with deformations having been identi-fied, the next logical step would be to introduce generalized coordinates defining librational motion (rigid body rotation) of the system. This permits identification of frames in terms of Eulerian rotations. The orientation of each frame Fi (t = p, r, 5 , t) is obtained relative to the orbital frame through three modified Eulerian rotations starting from the orbital frame, 22 which was established in Section 2.1. The rotation sequence is as follows: Table 2-1 Eulerian rotation sequence for the ith frame starting from the orbital frame orientation. Old Frame Angle Turned Axis New Frame Fo = xo,yo,zo ai z i , y i,2i* xi,yi,zi Pi yi a-2,y2,Z2* li Z2 Xi,yi,Zi = Fi * xi,yi>zi and Z2,y2,22 are used to denote intermediate frames in each case. It is also illustrated schematically in Fig. 2-5, where the intermediate frames are not labeled for clarity. For the case of the tether, two rotations at and 7$ are sufficient to describe its orientation in space. This means the second rotation Pt is set to zero and frame X i , y i , z i and x^,y2,Z2 collapse to a single frame. The rotational transformations can be described using orthogonal matrices. For example: Pi = [T IO] PO , (2.13) where: Thus where: pi = vector measured in frame x i , y i , 2 i ; Po = vector measured in frame xo,yo,zo ; [T"IOJ = transformation matrix relating x i , y i , z i and xo,yo>zo Pi= [ r i 2 ] [ r 2 1 ] [ r 1 0 ] / 5 b , (2.14) N - Cn s1{ 0 - 5 7 , C 7 , 0 0 0 1 23 Figure 2-5 Description of the orientations of the platform, subsatellite and tether relative to the orbital coordinate frame using Eulerian rotations. M - CPi 0 -Sfr 0 1 0 Sfc 0 Cfa M - 1 0 0 0 Cai Scci 0 -Son Con This gives angular velocity of the frame Fi as = TI2 + NN{5] + N{;} ' {e + aJCPiCn + PiSn -{e + adCPiSn + PiCii [0 + a t)S/? t- + 7," (2.15) where 6 = orbital rate. It is also necessary to obtain direction cosines (/,•) of the frame F{ relative to the local vertical as they are needed in evaluation of the gravitational potential energy described in the next section: li = {Ui, lvv /«,•} = {»'« • 3o, 3i' jo, h • Jo} = [iii] [T2I] [T10] 0 1 0 SctiSpiCn + CctiSn = •{ -SctiSPiSn + CatCn -SaiCPi (2.16) 25 2.4 Kinetic energy The kinetic energy, T, of the entire system consists of four parts, each corre-sponding to a separate domain: T= ]T Tit (2.17) i=p,r,a,t = J2 \ I t-Tidrm, (2.18) i=p,r,8,t Jmi —• where I\ = velocity of the differential mass element <fmt-. As shown in Fig. 2-6, fi = Rc + Ri + fi; (2.19) fi = tic + hi + (?i)F. + (uJi x ft) . (2.20) Substituting from Eq. (2.20) into Eq. (2.18), kinetic energy for the tth domain becomes, Ti = ^-[kc • kc] + • hi] + ±aTim + Um. (*) Fj * (*) F i d m i + ^ c ' [mi^ + Im (*) F i d m i + Ui x J fi <fm,j + Ri - (fij F dmi + w,-x ^ ri dm tj + wt- • J7,. (2.21) Here: [/,] = inertia matrix of the tth domain relative to frame Fi; Hi = angular momentum of the tth domain relative to frame Fi, / (fi x fi) dm,-. Jmi 26 Figure 2-6 Position vector ft- of an arbitrary mass element in the inertial space. Note, its relation to the orbital and body frames. The total kinetic energy of the system can thus be written as T = -[mRc • Re] + -mRp • Rp + -(m8 + mt + mr)dp - dp £t £t & + - m a d s • d a + - 2^ (tDt- [/,•]£,• + wt- • Hi) i=p,r,s,t + (m8 + mt + mr)dp + ma{ ^{h't) ~ d8 J + ^ - [ r { ( § ) r , + ' ( i ) + - - ^ ^ Here: 1 -—mRc • iE c = kinetic energy due to orbital motion; At 1 -—mRp ' Rp = contribution due to shift in the center of mass; 1. - 1 —(m8 + m* + m r ) d p • d p + —m8de • d8 = contribution due to offset motion; 1 tD^ [/]£,• + u;,- • Hi = kinetic energy due to rotational motion; i=p,r,8,t — [ [ (-zr~) ' (~ir~) Ppdxjpdzp = kinetic energy due to platform 2 Jo Jo \ at / Fp \ at / FP deformation; 5/„ *' (£f)***"+1/„"" '(^) • ' 0 " * • 1 ( d . -*. d . -* .~\ + « m * i TvOt) • -rvOtJT kinetic energy due to change in tether length 2 I at dt > 28 and deformations. The rest of the terms are due to coupling between these effects. 2.5 Potential Energy The gravitational potential energy for an element of mass dm,- can be written as GM , , . dUG. = - -r—j dmi • (2.23) Using binomial expansion with truncation of the series after second degree terms, the potential energy for the tth domain can be written as GMrrn UGi = where: G = universal gravitational constant; M = mass of the Earth; A p = Rp + tp ; A a = Ra + Te ; A t = Rp + dp + ft; A r = Rp + dp + fr . Hence the gravitational potential energy for the entire system can be written as GMm GM R c + 2 ^ | " [ ( m P + m * + m r ) { ^ - J ? p - 3 f f ° - ^ ) 2 } + (m t + mr) {2Rp • dp + dp • dp - 6{j0 • Rp) (j0 • dp) - 3(7o • 4 ) 2 } + ™*{RS • Rs - 3(j 0 • R e ) 2 } + 2 { ^ A fpppdypdzp} • {tip - 3(; 0 • RP)]o} + 2 { J q { ) Wt dyt] • [tip + dp- 3(jQ -tip + jo- <Tp)Jo} 29 - £ {tr[/,] + 3/T[/t]/;}]. (2.24) i=p,r,e,t Here the first term represents potential energy of the system treated as a point mass. The rest of the terms are due to the system's finite dimensions. 2.6 Strain Energy The strain energy (U8) of the system is due to deformation of the flexible com-ponents, that is, the tether and space platform, Ue = U8p + U8t, (2.25) where: U8p = strain energy due to transverse deformation of the platform; U8t = strain energy due to longitudinal and transverse deformation of the deployed tether. Now, Uai = i / at dVi (i = p,t); where: a = stress in a differential element of volume dV{; e = strain in a differential element of volume dV{; Vi = volume of the ith domain. For a thin plate with the following assumptions: (i) points lying on a normal to the middle surface remain on the normal after deformation; (ii) the middle surface remains strain free; (iii) deformations due to shear stresses are neglected; 30 u -if Eti \(d2u*>Y i r a 2"M2 where: E = Young's Modulus; v = Poisson's ratio; Vp = volume of the space platform; tp = thickness of the space platform. The strain energy for the tether is based on the theory of vibrating strings [21]. In order to account for interactions between longitudinal and transverse modes [20], transverse displacement terms up to the second order are retained in the strain expression, dyt 2 I V dyt' \ ayt ) ) The corresponding strain energy expression from Eq. (2.25) becomes — W . , w [ £ + i { ( s ) , + ( S ) , } ] , * » (2-"> where A = area of cross-section of the tether. Using Eqs.(2.2), (2.4), (2.26) and (2.27), the strain energy of the entire system is given by (2.26) 31 2.7 Equations of Motion Using the Lagrangian procedure, the governing equations of motion can be obtained from dAdq) dq + dq ~ Q g ' where: q = a,-, /?,-, 7,-, 7, Aa, Bb, Ca, Dc; i = p,r,s,t; a = 1, 2, . . . ,Ni ; 6 = 1,2, . . . , J V 2 ; c = l , 2, . . . , J V 8 ; f/ = Ug + ^ . As can be expected, the complex character of the system leads to highly non-linear, nonautonomous and coupled equations of motion which are not suitable for development of appropriate control strategies. The following assumptions rendered the equations more tractable retaining essential characteristics of the system: (i) The energy associated with vibrations of the tether and platform are considered negligible compared to contribution from the librational motion. This results in ignoring of the deformational dynamics. 32 (ii) The centre of mass of the system is considered coincident with the centre of mass of the space platform. The platform mass accounts for over 99% of the entire system mass, thus clearly justifying this assumption. (iii) The offset of the tether attachment point at the subsatellite end is considered negligible compared to other kinematical lengths (e.g., platform dimensions, tether length, tether attachment point offset, etc.). These assumptions together with linearization lead to a coupled, nonautonomous set of second order differential equations governing the system dynamics. These equations account for: (i) three dimensional librations of the space platform and subsatellite negotiating an arbitrary trajectory; (ii) swinging inplane and out-of-plane motions of the tether of finite mass; (iii) three dimensional offset of the tether attachment point form the space platform's centre of mass; (iv) effect of controlled variations of the offset attachment point; (v) effect of deployment and retrieval of the tether. Note, the equations permit investigation of control strategies using tether ten-sion, thruster offset, etc., and their hybrid combinations. They have not been reported in literature. Note that tension and thruster control procedures have lim-itations at shorter tether lengths. There is a possibility of the tether becoming slack rendering the control ineffective. Use of thrusters in vicinity of the platform is undesirable from safety and plume impingement considerations. As an example, the platform and tether pitch equations are presented here to ap-preciate formidable character of the system even after the asssumptions mentioned before: 33 ap - Platform Pitch Equation a p [j* ) P + rrirstiSyj, + <%p)] + ap[-mtldytP - m8tJdyiP - ^-l8dZyP - mBt2(l8 + W)dZjP ^ • { 3 m r 8 t ( J J p - rj^p) + met22ldyiP + 3(IZiP - Iy>p)} , + rhtldX)P + 7P [2m mratdXiPdz>p - m8tTedZ)P + m8t2l82dy>p + - r 8 t ( J J p d^p met22 — Pp [mratdx.pdy.p] — Pp ^2rnr8t8dXtPdZtP^ + (3P ^2mr8t8dXiPdZji + matTdXiP - m8t2T82dX)P^ + -^3 - { -3m r a t J X ( P f i y ) P - m at 22lJ Z ( P}j - \ rat0dXtPdytP^ + 7 p ^ 2mrat9dXtPdVtP + ^-T0dXiP + mat: + mBtT0dx>p + ^^{3mratdXiPdZiP}^ + dt m8t2ldytP^ + dt ^ tetldy.p + m8t22l0dZtP + a f [~^ ~^ !/>P + ~2~^*,p + meJdy,p + m8t2 i8tWdz.v — m8t2W dv,v + —^-m8t2ldViX)\ — I m8tdZiP \ + I\mt6dytP + m8t0dytP r r7 t^ ** \~2^y'p + met2idy,P + m + m8tUdZtP - 8t2W2dytP + -^-rn8t2TdytP^ • — TTl' 1 - r + m8t20dytP + — + 2(18 + 10)1  8t2Vdy>p  2dZtP\ + I ]m8t2VdyiP + —v&y>p + mt2Vdy,p + m8ttrdz>p + -^-{m8t22dZiP + mtdz<p}^ — dy>p ^ mrBtdz>p + dy>p ^2mrBt0dytP^ + dVtP ]2mrBt0dyiP + m8t2T0 + m8t2l0 + -^-{Smr8tdz>p}^ + dZ)P [mr8tdy)P^ + dz>p^2mrBt0dz>p^ + dZ)P^2mr8t0dZtP + m8t2T02 - r h j - m8tJ ^•{Smr8tdytP} + mat2/}] + 0 Ix,p + " W ( 4 , P + ^ , P ) + mat2/rfy)Pj + 8 ^ —ldytP + m8tldy>p + m at 22Jd Z I P}J = Qap ; - m8tldZiP + -^3-{3m r atdy ) Pd* l P + : mBt2{ldyiP + WdZ)P}] + rhtldz>p (2.29a) at - Tether Pitch Equation o=p [m«t2^ y,p] + "p [-2m 8t 2/fi Z ) P] + a v + Pp \-m8t2ldXtP^ + Pp ^ m a t 2rfi Z ) P —p3 - m a t 2 / a ri ) P j + % [2mat2/a f X l Pj T J GM T J I -m8t2ldytP + -j~p-rn8t2ldy)P^ + &t[matsP] + at[2ma l 2/T] + at [mBt2ldyiP + ^ { 3 ( 1 + ^ i l ) ! LM, 34 + 2m a t 2W y ) P}] + 1 [2mat2J\ + 1 [mat2{-dz<p + jjfdz.p} + 2(1 + LMt,8)V] + by)P[2mat2l] + bz,P[{l + LMt>a)J\ + Dz>p[mat2{^-1- T}] + 2mat2Vl + mat2{^fldZ)P - ldz>p} = TJ. (2.296) Of course, in general, there are 8 equations governing the system's controlled motion (9 for tension control). As can be clearly seen, the nominal offsets dj,p(j = x, y, z) strongly couple all the generalized coordinates. If they are set to zero, the platform motion would decouple completely from the tether inplane and out-of-plane motions. A A A Terms containing dj)P, djtP, djiP represent the effect of variations in the offsets from their corresponding nominal values. That is, —* djtP = scalar j-component of vector dp, A = dJtP + dj>p . —* Since da has been neglected, the subsatellite motions (cca, (3a, 7 a ) decouple and consequently can be studied separately. In the present study the focus is on the platform-tether interaction dynamics and control, hence the subsatellite equations are not involved. For the classical Keplerian orbit, R c = GM(l + ecos6) ' ( 2 ' 3 0 ) where: hB = angular momentum per unit mass of the system = R2J; e = orbit eccentricity. 35 Equation (2.30) permits use of the true anomaly (0) as an independent variable instead of time by substituting: d _ • d di = ed0] dl?~ \d$*~ l + ecos0d0~) ' The equations, nondimensionalized with respect to melfef02, are presented in Ap-pendix A . The highly coupled, nonautonomous, second order differential equations are not amenable to any known closed-form solution. Hence one is forced to resort to a nu-merical integration scheme to study the system response and control dynamics. The numerical scheme (IMSL:DGEAR) is adopted which utilizes an Adams predictor-corrector (multivalue) algorithm. Accuracy of this scheme is relatively high as a variable order of up to twelve could be used. The Jacobian of the set of differential equations is evaluated numerically by a finite difference approach. 36 3. S Y S T E M D Y N A M I C S Appreciation of the system dynamics often aids in the development of an ap-propriate control strategy. Before development and evaluation of various control strategies, it was considered desirable to study dynamics of the platform based tethered subsatellite system. The governing nonautonomous coupled equations were numerically integrated to assess parametric effects of the platform inertia, tether and subsatellite masses, retrieval rate, and orbit eccentricity. More importantly, influence of the offset in each direction {Dx>p, DVtP, and DZ)P) was thoroughly in-vestigated. The key parameter values used in the simulation are summarized below: Table 3-1 Reference configuration parameter values for dynamics simulation. Orbital Altitude 500 km Eccentricity 0, 0.01, 0.05 Platform dimensions 100 m x 120 m Platform mass 100,000 kg Platform inertias: Iz,p 20.33 x 107 kgm 2 8.33 x 107 kgm 2 Iz,p 12.00 x 107 kgm 2 Subsatellite mass 100 kg Tether mass 20 kg Tether length 1000 m Maximum offset in each direction 20 m Retrieval rate (max.) 0.17 m/s Initial disturbance (max.) 10° 3.1 Retrieval Scheme Before proceeding with the parametric analysis of the system response, a remark on the retrieval scheme used would be appropriate. For exponential retrieval, the 37 rate is proportional to the length of the tether: l{t) = l r e f e c t ; f = clrefect = cl; I=c2lrefect = c2l. The time required for retrieval to be completed in m orbits is given by 2irm (3.1a) (3.16) (3.1c) t = 6 (3.1d) Here 8, for an arbitrary orbit, represents average angular rate. Let: /,• = initial tether length = lref ; / / = final tether length . FromEq. (3.1a), or £ = JLln{ 0 2nm (lref J The various combinations of retrieval parameters used in the parametric analysis are listed below: Table 3-2 Parametric representation of exponential retrieval maneuvers. Retrieval Time (Orbits) 1 0.68 0.37 c/e for lref = 100 m - 0.37 - 0.54 - 0.99 / / = 10 m c/B for lref = 1000 m / / = 10m - 0.73 - 1.08 - 1.98 38 Note, the maximum retrieval rate corresponding to c/6 = -1.98 s - 1 was 2.3 m/s, while the minimum retrieval rate used was 0.04 m/s. The TSS-1 misssion requires a maximum retrieval rate of around 0.7 m/s [24] as shown in Fig. 3-1. 3.2 Results and Discussions 3.2.1 Platform Offset As pointed out earlier, presence of the tether attachment point offset from the platform center of mass leads to a strong coupling between the platform and tether librations. This means that swinging motions of the tether (at, 7t) affect the platform rotations (ap, (5p, and 7P). To illustrate this point, consider the tether equations for the special case of stationkeeping (L = L = 0): at — Tether Inplane Swing (i + ^ ) D » W - »d + - a + H^JJM? + 2(1 + ^ 4 ) ^ + (1 + + 3{(1 + *%*-)D„ 2 ' " r 3 / * 2 + ( i + ^ ) } « , + 2 ( 1 + ^ ) L ' + 2 ( 1 + ~^)b;,t + (l + '^K, = T.L ; 7t — Tether Out-of-Plane Swing - (1 + ^ ) X > . ^ - (1 + ^)D„rPP + (1 + + (1 + + (1 + ^H' + {3(1 + ^ ) D „ P + 4(1 + ^ ) £ } 7 . - (1 + LM,,.)D.J. - (1 + - ( l + ^ ) i D ; ' , p - ( l + ^ ) L D „ p = -T,I ; 39 Pre-Deploy Quiescent !° _ M ai o tn o A L — Deviation from Nominal Length - (1 + LMt,8)DZyPa'p' - 2(1 + LMt<8)DyiPa'p + 3(1 + LMtl8)DZiPap + 2(1 + LMt>8)DX)Pl3'p + (1 + LMt,e)DXtPip' - 3(1 + LMtj8)Dx>plp - 2(1 + - (1 + LMti8)DXiplt + (l + L M t i f l ) L " - z{lMt,8{Dy>p + |) + (1 + + (1 + LMt,8)Dlp - 3(1 + LMt>8)Dy,p - 2(1 + LMt,8)D'z>p - T t = QT . When the out-of-plane offset DXfP is set to zero, the inplane degrees of freedom (ap, at, and L) completely deouple from the out-of-plane degrees of freedom (/?p, 7 P , and 7t) in the linear case. This decoupling is illustrated in the tether equations given below: at : Tether Inplane Swing i 1 + — ^ — ) D v , P a P ~ 2(1 + — 2 — ) D * , P a p + I1 + ~~3—i at + 3 U 1 + — ^ — ) D v , v , LMt «M / LMt,,}, , LMt e \ A / / LAft a . m -+ (1 + —j^)}* + 2(1 + —j^W + 2(1 + -^±)D'y>p + (1 + -^-)D% = TaL ; 7t • Tether Out-of-Plane Swing - ( i + ^ ) D „ r p ' p > - ( i + i ^ ) ^ P + ( i + ^ ) 5 „ , P y ; + (i + ^ ) * W f c t a + + (3d + ^ )J> . „ + 4(i + ^ ) £ } 7 l - ( i + ^ ) l 6 i ' , p - ( i + ^ i i ) l D , , p = - r 7 l : A L : Deviation from Nominal Length - (1 + LMt,8)Dz,pa'p' - 2(1 + LMt,e)DViPa'p + 3(1 + LMt>8)DZtPap - 2(1 + + (1 + LMt,8)L" ~ s{LMt>8(Dv,P + + ^Y1) }L + (1 + lMt,8)blp 41 - 3(1 + LMt>s)Dy>p - 2(1 + LMttB)D'z>p = -Tt + Qj Figure 3-2 shows response of the system to a disturbance as specified on the diagram. Note, both the platform and the tether are subjected to an initial distur-bance but the resulting response does not show any coupling (as expected) between the platform and tether dynamics. In order to study effect of the out-of-plane offset, DX)P was now assigned a value of 20 m leaving the inplane components, DVtP and Dz>p of the offset vector dp, equal to zero. Figure 3-3 (a) shows response of the system in this configuration. The platform was given an initial displacement of 1° in roll (7P), yaw (/?p), and pitch (ap), while the tether was subjected to a 10° displacement in both inplane (ctt) and out-of-plane (7*) directions as before. In practice, this would be considered a rather large disturbance. However, it was purposely chosen to help identify parametric and coupling effects. Furthermore, system damping (structural, viscous, etc.) was neglected to help focus on the parametric interactions. Clearly, the platform roll (7P) is strongly coupled with the tether out-of-plane swing (7t). Note the transfer of energy between the two modes as indicated by their relative amplitudes. Of interest is the beat-type phenomenon suggesting a small deviation between the two frequencies. The beat period is around 7 orbits. The platform pitch attains a maximum amplitude of 2° while platform roll amplitudes does not exceed 1°, the initial condition. Similarly, effect of an inplane offset along the local vertical DViP was isolated by setting DX)P = DZiP = 0 with the same initial conditions (Fig. 3-3b). Amplitude of the platform roll now increased to 1.5° showing a coupling between these two degrees of freedom. As expected, the platform pitch and yaw motions are not affected by the offset along the local vertical: their amplitudes remained confined 42 Initial Conditions Satellite Parameters a l ° ( 0 ) = 7 t < > (0) = 10 0 V(o)=/sp°(o)=rp0(o)=i0 Tether Parameters p t = 2xl0~ 2 kg /m Stationkeeping at 1000m DX ( P= D y p = D 2 ( p = 0m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 2.0 Time, orbits Figure 3-2 Dynamical response of the platform supported tethered satellite system in the reference configuration with the offsets set to zero. 43 Initial Conditions Satellite Parameters « t O ( 0 ) = 7 t ° ( 0 ) = 10 o < ( 0 ) = / S p ° ( 0 ) = r p » = l ° Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 1000m D x p=20m,Dy p=Dz p=0m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 10.0 Time, orbits Figure 3-3 Effect of inplane and out-of-plane offsets on the coupled response of the tether and platform: (a) DVyP = Dz>p = 0. 44 Initial Conditions Satellite Parameters a t ° ( 0 ) = 7 t ° ( 0 ) = 10 0 « p ° (0 )= /V (0 )=7 p ° (0 )= l O Tether Parameters p t = 2 x l 0 " 2 kg /m Stationkeeping at 1000m Dyp=20m,Dxp=Dzp=0m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 2.0-r Time, orbits Figure 3-3 Effect of inplane and out-of-plane offsets on the coupled response of the tether and platform: (b) Dx,p = Dz>p = 0. 45 Initial Conditions Satellite Parameters a l o ( 0 ) = 7 l ° ( 0 ) = 1 0 ° V(0)=/V(0)=7pO(0)=lO Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 1000m Dzp=20m,Dxp=Dyp=0m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 2.0 20.0 Figure 3-3 Effect of inplane and out-of-plane offsets on the coupled response of the tether and platform: (c) Dx>p = DViP = 0. 46 to the 1° initial condition. Effect of an inplane offset along the local horizontal Dz>p is shown in Fig. 3-3(c). The other offset components are set to zero and the identical disturbance is used as before. However, now the platform yaw (/?p) is found to be coupled to the tether inplane motion (oct). This indicates that the inplane offset Dz>p leads to a forced motion of the platform in yaw, induced by the tether's out-of-plane swing. The frequency of the tether out-of-plane motion (7t) was not close to that of the plaform yaw motion and hence a low frequency beat response (over 25 orbits) was observed between the two degrees of freedom (not shown). The maximum platform yaw amplitude reached is 1.5°, while platform pitch and roll motions do not exceed the initial disturbance of 1°. 3.2.2 Tether Parameters Figure 3-4 shows effect of increasing the tether mass on the system response. To facilitate comparison, the system configuration is chosen to have a purely out-of-plane offset, i.e., DXiP = 20m; DVjP = Dz>p = 0. The tether mass is increased by a factor of five to 100 kg. Comparing with Fig. 3-3(a), it is easy to notice that the trends remain essentially the same. However, now the beat frequency is a little higher (period « 5 orbits). Amplitude of the platform roll (7p) is found to be close to 10° . From Eq. (A.3) for platform roll, it can be clearly seen that a larger tether mass (i.e., larger Mt)B) increases the coupling between the platform and tether dynamics. 47 Initial Conditions Satellite Parameters a t o (0) = 7 t ° ( 0 ) = 10 o « p ° ( 0 ) = / ? p o ( 0 ) = 7 p o ( 0 ) = l o Tether Parameters p t = l x l O " 1 kg /m Stationkeeping at 1000m DXjp=20m,Dy>p=D2_p=0m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 20.0-1 Time, orbits Figure 3-4 Effect of increasing the tether mass (fivefold) on the response of the platform based tethered satellite system. 48 3.2.3 Payload Mass Figure 3-5 shows response of the system with a subsatellite five times massive compared to that considered in the reference configuration (Fig. 3-2). From the physics of the problem, one would expect the interaction between the space platform and tether to be stronger as the inertial effects due to the swinging tether and payload become larger. This is clearly indicated by the larger platform amplitude in pitch ( « 15° ) . The platform roll motion is coupled to the tether out-of-plane motion as before, and a beat response is present again. However, now the beat frequency is much higher with a period of less than 2.5 orbits compared to that observed in Fig. 3-3(a). The platform pitch is also seen to be coupled more strongly with an amplitude close to 8° . A slight beat response in the tether inplane motion can also be discerned. Thus an increase in the payload mass leads to a stronger coupling between the platform and tether dynamics. 3.2.4 Platform Inertia The nominal dimensions of the homogeneous platform are PA = 100 m and PB = 120 m. This results in an aspect ratio PB/PA ° f 1-2. The largest platform inertia is given by Ixx,p = J£(PA + H) = 20.33 x 107 kg m 2 . In order to study effect of the platform inertia, two aspect ratios were considered: (a) PB/PA = U a n d (b) PB/PA = 2; keeping IXX)P constant. The dimensions of the platform for an aspect ratio of 1, are PA = 110.5 m, PB = 110.5 m. Similarly, for an aspect ratio of 2, PA = 70 m, PB = 140 m. 49 Initial Conditions Satellite Parameters « l ° ( 0 ) = 7 l ° ( 0 ) = 10 o « p ° ( 0 ) = / S p o ( 0 ) = 7 p o ( 0 ) = l o Tether Parameters p t = 2xl0~ 2 kg /m Stationkeeping at 1000m D x p=20m,Dy p=Dz p=0m m p = 100,000kg m s = 500 kg Orbit Parameters e = 0 , h = 500Km 20.0 0 2 4 6 8 Time, orbits Figure 3-5 Effect of increasing the payload mass on the coupled space platform-tether dynamics. 50 Case (a): IyVtP = Izz>p = 10.18 x 107 kgm 2 (Square Platform) Figure 3-6(a) shows response of a square platform to a purely out-of-plane offset configuration [DXtP = 20 m, Dy>p = Dz>p = 0) and with initial conditions as before. Comparing with Fig. 3-3(a), the response appears to be quite similar. However, the pitch shows frequency modulations. Note, there are two distinct platform frequencies. The higher frequency corresponds to that of the inplane tether motion. The low frequency is around 4 cycles/orbit. This corresponds to the pitch frequency of the platform without any tether coupling. The offset terms DXXtP, Dyy>p, and DZZ)P (Eqs. A.1 - A.3) modify the overall inertia of the platform. They introduce contribution of the mass of the payload and the tether concentrated at a point 20 m from the centre of mass of the space platform. The pitch (ap) frequency is governed by a combination of the platform inertia and that contributed by the tether and payload via the offset. Case (b): Iyy>p = 4.08 x 107 kgm 2 , IZZtP = 16.33 x 107 kgm 2 (Rectangular Platform) Figure 3-6(b) presents response of the elongated space platform. As expected, the platform pitch interaction with the tether inplane motion is indeed increased. A beat phenomenon between the two is observed with a period of slightly less than 3 orbits. It should be recognized that the out-of-plane (roll) motions of the platform and tether remain, however the amplitudes are distinctly lower than those observed in Case (a). This is due to the larger roll inertia (IZZlP). The beat period for the out-of-plane motion (ft) was observed to be 4 orbits. As seen in Figs. 3-3(a) and 3-6(a), the platform and tether roll motions are coupled and energy transfer between the two is clearly discernible. The frequency of the beat response depends on the platform inertia as evident from Eq. (A.3). 51 Initial Conditions Satellite Parameters « t ° ( 0 ) = 7 t o (0) = 10 o « P ° ( 0 ) = / ? p o ( 0 ) = 7 p o ( 0 ) = l o Tether Parameters p t = 2xl0~ 2 kg /m Stationkeeping at 1000m Dx>p=20m,DyiP=DZiP=0m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km PA = 110.5 m, PB = 110.5 m Time, orbits Figure 3-6 Effect of varying platform inertias on the coupled dynamics: ( a ) 7 v y , P = 7«,p = 10-18 x1 0 7 kSm2-52 Initial Conditions Satellite Parameters a l ° ( 0 ) = 7 t ° ( 0 ) = 10 o < ( 0 ) = / 3 p o ( 0 ) = 7 p ° ( 0 ) = l O Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 1000m 5xp=20m,Dyp=Dzp=0m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km PA = 70 m, Ps = 140 m Time, orbits Figure 3-6 Effect of varying platform inertias on the coupled dynamics: (b) IyyiP = 4.08 x 10 7 kgm 2 and IZZjP = 16.33 x 10 7 kgm 2. 53 3.2.5 Orbit Eccentricity Effect of orbit eccentricity is to introduce a periodic forcing function on the system. In general, as can be expected, the effect is observed mainly in the inplane degrees of freedom (ap, at). Here, the tethered satellite system is subjected only to the eccentricity induced excitation with a 20 m offset in each orthogonal direction, i.e., DjiP — 20 m (j = x, y, z). Figure 3-7 shows typical response plots for e = 0.01 and 0.05. It is of interest to observe that even with this small departure from the circular trajectory, the platform deviates from the equilibrium configuration by a significant amount. As expected, the higher the eccentricity, the larger the ampli-tudes of response. Both the pitch responses show similar trends and frequencies are the same. However, in general, an eccentricity induced disturbance has more effect on the space platform response. For e = 0.05, the maximum platform amplitude is around 5° while the tether deviation from the equilibrium is only 0.9°. 3.2.6 Tether retrieval Finally, dynamical response of the system during exponential retrieval of the tether from 100 m to 10 m in 0.37 orbit is studied. It should be noted that this is a relatively slow retrieval rate with a maximum speed of 0.11 m/s. A small initial dis-turbance of 1° in the tether inplane and out-of-plane directions is introduced. The disturbance is purposely chosen to be small to emphasize influence of the retrieval maneuver. It is apparent (Fig. 3-8) that the tether amplitude in pitch quickly grows to around 15° even before the retrieval maneuver is completed. Note, the platform 54 Initial Conditions Satellite Parameters a t ° ( 0 ) = 7 t ° ( 0 ) = 0 ° « p o ( 0 ) = / S p o ( 0 ) = 7 p o ( 0 ) = 0 ° Tether Parameters p t = 2xl0~ 2 kg /m Stationkeeping at 1000m Dx,p=Dyp=Dzp= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e=0.01,0.05, h = 500Km 10.0 -1.0 LEGEND e = 0.05 Time, orbits Figure 3-7 Effect of orbital eccentricity on the dynamics of the space platform based tethered satellite system. 55 response also suggests instability, particularly in yaw. It should be recognized that the presence of the offsets also represents a large disturbance, however, the system during the stationkeeping phase was stable (Fig. 3-3(a)). This emphasizes the fact that such interaction dynamics can not be neglected in a mathematical model to obtain realistic results. Furthermore, it is obvious that some form of control will be required during retrieval to counter destabilizing influence even in the presence of a very small disturbance. 56 Initial Conditions Satellite Parameters a t ° ( 0 ) = 7 t ° ( 0 ) = l ° « P ° ( 0 ) = / 3 p < > ( 0 ) = 7 p o ( 0 ) = 0 O Tether Parameters p t = 2xl0~ 2 kg /m Retrieval 100 m to 10m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Time, orbits Figure 3-8 System response during retrieval. Note, even with such a small retrieval rate and initial disturbance, the platform based tethered subsatellite system quickly becomes unstable. 57 4 . C O N T R O L O F T H E T E T H E R E D S A T E L L I T E S Y S T E M 4.1 Preliminary Remarks This chapter presents development of the optimal control strategies for the tethered satellite system. The concepts of using tether tension and thruster as means for control are extended to the case of a space platform based tethered satellite with a finite offset of the attachment point. The system dynamics with moving offsets is successfully utilized to develop a new method of control referred to as the "offset control". In the beginning, the equations of motion are presented in the standard state space matrix form. The controllability of the system for each method of control - tension, thruster and offset - is established numerically using Householder's transformation. Optimal controller gains for the state variable feedback are established by solving the steady state algebraic Ricatti equation. The performance index, defined by controller energy and deviation of the state from the equilibrium point, is minimized. The matrix equations are solved numerically to obtain controlled response of the system as well as the effort required. 4.2 State Space Representation of the Mathematical Model The linearized dynamical equations presented in Appendix A can be written in a matrix form as W W + \C}{q} + \K}{q} = [£]{«} + {P} , (4.1) where: {?}) {?}> { £ } = n x 1 vectors of generalized coordinates, velocities, and accelerations, respectively; 58 [M] n x n mass matrix made up of coefficients of the generalized acceleration {q}; [C] = n x n gyroscopic matrix consisting of coefficients of the generalized velocities {q}] [K] = n x n stiffness matrix made up of coefficients of the generalized displacements {q}\ {u} = r x 1 vector of controller inputs; [J5] = n x r matrix of coefficients of the r controller inputs; {P} = vector of terms not associated with generalized Details of the above matrices are presented in Appendix B. The generalized accel-erations {q} can be obtained from Eq. (4.1) by premultiplying it with [ M ] - 1 and rearranging the terms, {q} = -[M]- 1[C]{g> - [M^IKM + [M]-1[J9]{u} + [M] - 1 {P } . (4.2) Let { £ } be a 2n x 1 state vector consisting of generalized displacements and veloc-coordinates or control inputs. It arises due to nonzero equilibrium position caused by the offsets. ities, (4.3) i.e., 59 Using Eqs. (4.2) and (4.3), {2} can be written as {2} = [A]{x} + [B]{u} + {P} , where: {P} (4.4) [A] = [0] nxn -[M)-i[K] nxn [I] nxn -[M]-l[C] nxn = system characteristic matrix; [B] = .1 [0] n x r [M)-*[B] nx r [0] n x 1 = control influence matrix; M-1[P] n x 1 null matrix; [0] [I] — nxn identity matrix. 4.2.1 Equilibrium configuration The equilibrium configuration of the system is not at {2} = 0. This is because of the offsets which cause the platform to rotate until the moments about the system center of mass are balanced. Let {2} = {xeq} be the nonzero equilibrium configuration of the system. Then, from Eq. (4.4), {0} = \A]{xeq} + {P} 60 i.e., {xeq} = -{A^iP} . (4.5) It should be realized that, strictly speaking, there is no equilibrium state. Eq. (4.4) represents a linear time varying system, where [A] and {P} are functions of time. They change due to deployment/retrieval maneuvers and eccentricity effects. The underlying assumption is that during each time-step of integration, the system has a quasistatic equilibrium configuration obtainable from Eq. (4.5). In general, {xeq} is a function of the platform, tether and subsatellite inertias, tether length and offset dimensions. For a space platform based tethered satellite system having parameter values as shown in Table 3-1 and with DXtP = Dy>p = DZtP = 20 m, the equilibrium configuration is as shown below: Table 4-1 Equilibrium configuration for a platform based tether (1000 m) with 20 m offset. EQUILIBRIUM C O N F I G U R A T I O N ap 5.38494° PP 0.07706° 1P 1.75577° a* 0 It 2.57596° x 10~ 3 L 0 In general, state of a system can be considered to have the form {x} = {xeq} + {x} , (4.6) where {x} represents deviation from the equilibrium state. Noting that {xeq} = 0 based on the quasistatic assumption, Eq. (4.4) gives {x} = [A}({x} + {xeq}) + \B]{u} + {P}. 61 Using Eq . (4.5), the classical matrix state equation for motion about the quasistatic equilibrium configuration is obtained as {*} = [A]{x} + [ £ ] { u > . (4.7) 4.3 Control Strategies As pointed out earlier, control of the system, especially during retrieval, presents a challenging task. The coupling introduced by the offset (DXtP, Dy>p, DZiP) cause the system to have significantly different dynamic characteristics. The tether oscil-lations in and out of the orbital plane perturb the roll, yaw and pitch motions of the platform, and vice versa. Furthermore, it may be pointed out that the conventional design procedures applicable to single-input single-output systems for pole place-ment by gain selection cannot be used here to obtain a unique control law. In order to develop a control strategy consistent with proper speed (bandwidth), damping characteristics and the available power limitation, one has to use an optimal control theory [25]. The control procedure adopted is of the classical L Q R type with the performance index [26] for minimum tracking error and energy given by r°° J = / ({x}T[Q]{x} + {u}T[R}{u})dt. Jo Here [Q] and [R] are the positive definite state and control penalty matrices, re-spectively. The optimal control {u} that minimizes this criterion is given by {«} = -[R-l}[B)T[S]{x} = -\G]{x} , where [S] is the solution to the steady state Ricatti matrix equation -[S}[A]-[A)T[S] + [S)[B][R]-l[Bf[S]-[Q] = [0] . 62 The requirements for a unique control law are: • The pair [A, B] is controllable. • All system states must be observable. The system states were assumed available either through direct measurement or by design of a suitable observer. Substituting the feedback law into the system equation Eq. (4.7) yields {*} = [A]{x} - {B][G]{x} representing response of the closed loop system. The characteristic equation for the system is given by det\s[I) - ([A] - [B)[G])} = 0 . The closed loop poles of the system are therefore given by the eigenvalues of [A] — \B][G] [27]. The control law design consists of choosing appropriate state penalties ([Q] matrix) and control penalties ([R] matrix) to obtain desired location of the poles. For a real system, one would expect the poles to be either purely real or complex conjugates. Controllability of the system was established by the numerical technique that utilizes the Householder's transformation [28]. In Eq. (4.7), [A] and [B] are the system characteristic and control influence matrices, of dimensions n x n and n x r, respectively. For example, in the case of thruster control, n = 12 and r = 6. The system is controllable if and only if the rank of [C] = n. Here [C] is the 12 x 72 matrix given by [C}= [B\AB\A2B\---An-lB]. It is not easy to check linear dependence of the columns of such a large order ma-trix, particularly in the presence of numerical round-off error. Instead the matrix is triangularized using Householder's approach that transforms [C]T to the upper 63 triangular [C]T as shown in the schematic diagram below, If there are any zeroes in the diagonal of the upper triangular submatrix of [C] T , then the (n x n) submatrix is singular, [C] is not of rank n and the system is not controllable. Various combi-nations of tether length and offset were tried and for all the cases the system was found to be controllable. 4.3.1 Tension control It is apparent that instantaneous length of the tether affects its tension and librational dynamics through coupling (Eqs. A-l,-•-,A-6). Hence the tether ten-sion can be utilized to control the system. In the present case, its effectiveness is assessed in the presence of platform rotations as well as the tether offset during stationkeeping and retrieval phases. The tether equations indicate that the out-of-plane motion is unaffected by linear tension modulations [29]. However Eq. (A.5), that takes into account effects of the out-of-plane offset (Z) I i P), showed that the system is controllable and a linear strategy can be used effectively. The station is controlled by momentum wheels. Ma,Mp and M 7 represent the nondimension-alized momentum wheel torques (Fig. 2-1). Here, {u} is a 4 x 1 control vector consisting of M a , M 0 , M 7 and Te , where Te represents tension in the tether line 64 nondimensionalized with respect to m a / r c / 0 2 , i.e., (Ma ' { Te ) Poles for the case of a 1000 m tether with 20 m offsets in the three orthogonal directions and attached to a platform of 100,000 kg mass carrying a tethered payload of 100 kg are shown in Fig. 4-1. Note, the closed loop poles of the tension controlled system are quite close to the imaginary axis leading to a poor speed of response and low damping rate. 3 co X < C O c C D E 2 1 0 -1 -2 -3 H I I a o POLES -2 Real Axis -1 Figure 4-1 Closed loop pole placement for tension control of the tethered satel-lite system. 4.3.2 Thruster control Effectiveness of thrusters during retrieval of the tether has been demonstrated by several authors [30,31]. However, in these investigations librational dynamics of the station and offset of the tether attachment point were ignored. In the present 65 case, to control tether librations, a scheme that utilizes a set of orthogonal thrusters is used. The nondimensionalized thruster force acts along the tether line while Ta and T 7 are directed normal to the tether line in the orbital plane and along the orbit normal, respectively. Now {u} is a 6 X 1 control vector, {«> Mp M 7 Ta I T, ) The analysis showed that the closed loop poles for the thruster control strategy can be placed very far to the left of the imaginary axis. This was avoided by increasing the control penalties as thruster forces tended to be the highest in magnitude among the different controller efforts. The poles shown in Fig. 4-2 are for the same case as that for the tension control to facilitate comparison. As expected, the thruster control has better damping characteristics compared to the tension control strategy. The platform rotational frequencies are lower as the poles associated with them have smaller imaginary components. 3 to CO c CO E 2 • 1 • 0 -1 • -2 -3 • i n n i B n B • I i •3 2 Real Axis 1 0 B POLES Figure 4-2 Closed loop pole placement for thruster control of the tethered satellite system. 66 4.3.3 Offset control As pointed out before, it is apparent from the libration equations that the offset A perturbations denoted by dy ) P and their derivatives affect all the degrees of freedom. In order to control the system one may choose any set of three input variables, e.g., the offset positions, velocities or accelerations. If one selected Dx,p, Dy>p, and DZiP as the offset control variables (Ma, Mp and M 7 would make up rest of the control variables), then f D*M ) {D} = {D}(t) = { DytP(t) } . I t>.A*) > From the feedback law, {£>} = - \G]{x} , where {2} is the state vector. In order to evaluate the offset velocity {D(t)} and acceleration {D(t)}, one would have to differentiate the state vector twice. On the other hand, choosing the three offset accelerations (DXiP, DVtP, DZ)P ) as control inputs, and integrating them to obtain the offset velocities and positions, presented a number of advantages: • The linearized closed loop equations indicated that the three offset accelerations are the only set of variables directly influencing all the degrees of freedom. The offset velocities have no effect on the tether out-of-plane swing, while the offset positions do not influence the tether inplane motion when the equations are 67 linearized. • From the numerical standpoint, integration schemes are more accurate than differentiation algorithms. • From the practical standpoint, a relationship between the motor torques (pro-portional to accelerations) and the motor input voltage is direct and hence a servo-loop is not necessary. • A n offset position specification scheme may result in velocities and accelerations that are impossible to implement through conventional motors. On the other hand, the maximum offset accelerations required for control could be easily lowered by adjusting elements of the penalty matrix [R]. Accordingly, the control vector {u}, of order 6 x 1, is ( Ma ^ M~ u = < D b I D x,p y,p z,p It should be noted that the offset velocities and positions become specified quanti-ties in this case and must be accounted for appropriately in the equations during numerical simulations. Furthermore, they must be restricted and returned to rest at the end of the control, preferably at the starting position. This was achieved through the offset state feedforward. The system control block diagram can be represented as indicated in Fig. 4-3. The offset control technique is similar to the act of balancing a rod on the palm of one's hand. As can be expected, for a given angular disturbance, the motion required at the point of attachment of the tether would grow proportional to the 68 r = 0 - A r M 0 U = ^ J9 x,p y,p x = Ax + Bu SYSTEM MODEL ACCOUNTING FOR TIME DEPENDENT OFFSETS STATE VARIABLE FEEDBACK O F F S E T POSITION AND VELOCITY FEEDFORWARD Figure 4-3 Block diagram of the offset controlled platform based tethered satellite system. length of the tether. The closed loop pole allocation is shown in Fig. 4-4. Note, as the complex poles have large real parts, the damping characteristics are favorable. Figure 4-5 represents a three dimensional view of the offset motion during a typical control maneuver. The tether is undergoing retrieval from 100 m to 10 m after being subjected to a large initial disturbance of 10° in both inplane (at) and out-of-plane (7t) directions. It should be noted that scales in the x and z directions are larger relative to that in the y direction (local vertical). The platform motions are not shown for clarity. The solid dot represents the tethered payload. The motion of the point of attachment on the boom-trolley traces the curve at the lower end of the tether. CO X < CO c O) CO E -2 • B H3 Q D B B POLES 4 -3 -2 -1 Real Axis Figure 4-4 Closed loop pole allocation for the offset controlled system. 70 Figure 4-5 Three dimensional view showing motion of the offset during retrieval of the tether from 100 m to 10 m in presence of a large disturbance. 4.4 Results and Discussion The first order differential equations (Eq. 4.7) were integrated numerically with respect to the true anomaly (6). In the simulation, for noncircular orbits as well as during deployment and retrieval maneuvers, the equilibrium configuration changes quasi-statically. Under these circumstances, the optimal gains were updated at each time-step (every 0.1 radian of the orbit). Solution of the steady state Ricatti equation requires determination of eigenvalues of a matrix of order 2n where n is the order of the system. A typical run took 30 minutes on an Apollo Domain 3000 workstation which operates at a process speed of 3.8 MIPS (Millions Instructions Per Second). Simulations were carried out over a range of system parameters involving plat-form dimensions, ratio of the platform to tether mass, tether length, offset and initial conditions. Obviously, an enormous amount of data was amassed. Only a typical set of response data with demanding platform inertias, severe initial condi-tions (10° in each degree of freedom) and rapid retrieval rates are presented here to assess the controllers' performance and establish their relative merits. The control effort requirements for the cases considered are summarized in Table 4.3 at the end of the section. When relevant to the discussion, the control effort time histories are also presented to gain better insight into the controller's performance. 4.4.1 Reference case (stationkeeping at 100 m) In order to facilitate comparison, a reference case was established with a typical set of parameter values summarized in the table below: 72 Table 4-2 Reference configuration parameters for controlled response simula-tions. Orbital Altitude 500 km Eccentricity 0 Platform dimensions 100 m x 120 m Platform mass 100,000 kg Platform inertias: II,P 20.33 x 107 kgm 2 8.33 x 107 kgm 2 Iz,p 12.00 x 107 kgm 2 Subsatellite mass 100 kg Tether mass 2 kg Tether length 100 m Maximum offset in each direction 20 m Initial disturbance 10° Figure 4-6(a) and 4-6(b) illustrate effectiveness of the tension control strategy for the system in the reference configuration. As can be seen in Fig. 4-6(a), the tension control in conjunction with the momentum wheels stabilizes the platform and tether motions in slightly over 4 orbits. The platform motion (ap, /?p, 7P) has a maximum amplitude of 10° corresponding to the initial conditions. The maximum tether motion (at, it) is also confined to the 10° initial conditions. As expected, the tether inplane [at) motion is damped much faster compared to the out-of-plane (7t) swing which is affected only through coupling provided by a 20 m offset of the tether attachment in the three orthogonal directions. The length of the tether changes in accordance with the controlled tension demanded. These fluctuations are represented by the generalized coordinate L. The unstretched commanded (reference) length L remained constant at 100 m during 73 Initial Conditions Satellite Parameters c , t ° ( 0 ) = 7 t ° ( 0 ) = 1 0 ° < ( 0 ) = / 3 p o ( 0 ) = 7 p ° ( 0 H 0 o Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 100m D x,p=D y p=D 2 p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 20.0 o Time, orbits Figure 4-6 Performance of the system in the reference stationkeeping config-uration: (a) time histories of platform and tether response during the tension control. 74 Initial Conditions Satellite Parameters a t O(0) = 7 l O (0) = 1 0 o <(0)=/V(0)=7 p°(0H0 o Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 100m D v =D~ =D~ = 20m x.p y.p z,p m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 0.2 -12.5 -f-1 1 1 r- 1 T 0 2 4 6 8 Time, orbits Figure 4-6 Performance of the system in the reference stationkeeping config-uration: (b) variation of the tether tension and controller effort during the tension control. 75 the stationkeeping maneuver. As shown in Fig. 4-6(a), the length decreases sharply following the release of the displaced tether. Within 1 orbit, the length fluctutations are reduced significantly and the complete control is achieved in less than four orbits. Considering the severity of the disturbance, this is indeed an impressive performance. Time history of the tension force (Te) and momentum wheel torques (Ma, Mp, M 7 ) corresponding to this case are depicted in Fig. 4-6(b). As can be expected, when the length decreases sharply, the tension reaches a maximum, of around 0.2 N . The tension fluctuations are much smaller following this peak and settle to an equilibrium tension of 0.08 N. Note, the equilibrium tension is a function of the tether length, tether mass, payload mass and altitude of the orbit. The corresponding momentum wheel torques reach a maximum value of around 21 Nm. The largest moment is required for the roll correction ( M 7 ) . The inplane requirements for the tension force was found to be 169.3 Ns. The integrated mo-mentum wheel energy requirement was 480.6 J. Figures 4-6(c) and 4-6(d) illustrate effectiveness of the thruster control strategy for the same reference configuration. The thruster control together with momen-tum wheels is capable of stabilizing the tether and platform in around 3.5 orbits. Although the tether swing motion starts out with 10° in pitch and roll, it ap-proaches the equilibrium orientation in essentially a pure convergence fashion, with little overshoot, suggesting increased effectiveness of the controller in this mode. This is further illustrated by damping to a lower amplitude (< 3°) within the first oscillation. Figure 4-6(d) shows the corresponding thruster forces as well as momentum wheel torques. The maximum thrust required is around 0.008 N in the inplane 76 I n i t i a l C o n d i t i o n s Satellite Parameters a t ° ( 0 ) = 7 t ' ! ( 0 ) = 1 0 o a p °(0)=,3 p °(0)=7 p o (0)=10 o Tether Parameters p t = 2 x l 0 - 2 k g / m Stationkeeping at 100m D x . p = D y i P = D Z i P = 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m - 5 . 0 - T 0 1 2 3 4 5 6 7 Time, orbits Figure 4-6 Performance of the system in the reference stationkeeping config-uration: (c) time histories of platform and tether response during the thruster control. 77 direction (Ta). The peak thrust values along the tether line (T^) and in the out-of-plane direction (T 7) are around 0.007 N and -0.005 N, respectively. It should be emphasized that these relative magnitudes are functions of the tether as well as platform motions. In other words, it is the entire state vector {x} that governs the control vector {u} for a particular set of optimal gains [G]. As shown in Fig. 4-6(d), the maximum momentum torque is around 24 Nm which is higher compared to that in the tension controlled case (Fig. 4-6b). As before, the largest moment is required for the platform roll correction ( M 7 ) . The total thruster based impulse requirement is 87.1 Ns while the corresponding momentum wheel energy demand is 503.2 J. Figures 4-6(e) and 4-6(f) illustrate effectiveness of the offset control strategy in stabilizing the reference configuration during stationkeeping. Once again, the platform and tether motions remain within the initial disturbance of 10° in each of the degrees of freedom. The offset control strategy in conjunction with momentum wheel torques is able to stabilize the platform as well as the tether motions in around 4 orbits (Fig. 4-6e). The tether oscillations in both pitch and roll (at, It) are found to decay uniformly. Note, the damping effectiveness is essentially the same for both inplane and out-of-plane tether motions. As pointed out earlier, the controlled offset motions Dj,p (j = x, y, z), were re-stricted to a maximum of 20 m in each direction. These motions were superimposed on the nominal offsets DjtP. As shown in Fig. 4-6(f), the offset excursions are within the 20 m constraint with Dx>p having a maximum deviation of close to 18 m. Fx, Fy, Fz denote the corresponding forces required at the boom-trolley end to implement the demanded offset motion. It is of interest to note that the maximum offset force requirement is only 0.0011 N in the out-of-plane direction. The inplane force has a peak value of -0.001 N. It may be emphasized that the offset velocity requirements 78 I n i t i a l C o n d i t i o n s Satellite Parameters « t ° ( 0 ) = 7 l O ( 0 ) = 1 0 o « p O (0)=/? p °(0)=7 p °(0)=10 o Tether Parameters p t = 2 x l 0 ~ 2 k g / m Stationkeeping at 100m D x . p =D y p =D 2 p = 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m ^ 0.010 E 40.0 0 1 2 3 4 5 6 7 Time, orbits Figure 4-6 Performance of the system in the reference stationkeeping config-uration: (d) variation of control effort during the thruster based strategy. 79 Initial Conditions Satellite Parameters V ( 0 ) = 7 l o ( 0 ) = 1 0 ° «pO(0)=PpO(0)=7p°(0)=10o Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 100m Dx.p=Dy>p=D2p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km Time, orbits Figure 4-6 Performance of the system in a reference stationkeeping configura-tion: (e) platform and tether response during the offset strategy. 80 are relatively small involving motions that take place in around half an orbit. This resulted in very small accelerations and forces. The momentum wheel requirements are also relatively lower in the case of offset control.The maximum roll correction torque ( M 7 ) is around 17 Nm. The lower momentum wheel torques suggests favourable influence of the offset motion on the control of the space platform. This is apparent from the terms containing _DyiP (j = x, y, z) in Eqs.(A.l) - (A.3). In other words, the linear regulator control moves the tether attachment point optimally in such a way as to provide a controlling influence on all system states including those associated with the platform motion. From the response of the system in the reference configuration with three dif-ferent control strategies, it is clear that in terms of effectiveness in stabilizing tether motions quickly, thruster control has the best performance. However, the thruster control has the drawback of being an expensive option to use in space as the fuel required has to be carried with the satellite. Furthermore, as pointed out before, thruster operation raises the question of safety, particularly in the vicinity of the platform, and the possible danger of plume impingement. On the other hand, the offset strategy requires relatively little energy and is reasonably efficient in control-ling both the tether as well as the platform motions. From physical consideration, it promises to be more effective at shorter tether lengths, however, this aspect needs to be confirmed. The key attractive feature of the tension control strategy is its simplicity and ease of implementation. Note, only one moving part, namely the reel, is required for deployment and retrieval, irrespective of the control strategy used. As can be expected, it is not easy to make a comparisibn of the three con-trol strategies. Each strategy varies in terms of the number and type of control 81 E Initial Conditions Satellite Parameters a l ° ( 0 ) = 7 l ° ( 0 ) = 1 0 o «PO (0)=/3 p °(0)=7 p °(0)=10 o Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 100m Dx.p=Dy.p=Dzp= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km Time, orbits Figure 4-6 Performance of the system in the reference stationkeeping configu-ration: (f) time histories of the tether attachment point, associated forces and platform moments during the offset control strategy. 82 variables. In other words, the order of the control vector {u} and its contents are different. This means that the penalty matrices \R] and [Q] are different (Appendix C). Hence, to assess their relative merit, it seems appropriate to test performance over a range of system parameters. This is discussed in some length in the following subsection. 4.4.2 Orbit eccentricity A n orbit eccentricity introduces a time dependent forcing function in the pitch degrees of freedom (ap,at). This perturbation is continuous and the controller effort is constantly required to stabilize the system. Figures 4-7 study effect of the eccentricity on the space platform based tethered satellite system in the reference configuration. Figure 4-7(a) shows the response during tension control for e = 0.05. As can be expected, the pitch oscillations are not stabilized completely and a residual platform oscillation of 0.5° remains even after 6 orbits. The same is true for the tether inplane motion [at). Out-of-plane motion of the tether is also observed due to coupling introduced by the length control. These motions in turn induce a small variation in the tether length L suggesting application of the tension control. Fig. 4-7(b) shows the corresponding response during thruster control. Once again a residual platform pitch oscillation of 0.5° is observed even after 6 orbits. However, the tether oscillations are, relatively, very small (< 0 .01°) . In case of the offset control (Fig. 4-7c), the platform pitch (ap) settles to the steady state limit cycle oscillations with an amplitude of 0 .5° , however, now the tether pitch motion is confined to around 0.03° after 4 orbits. It is clear that all the three strategies are equally effective in controlling the system even when the orbit is slightly eccentric. Their control performance with 83 I n i t i a l C o n d i t i o n s Satellite Parameters c , t ° (0 ) = 7 l ° ( 0 ) = 1 0 ° <(0 )= /V (0 )=7 p ° (0H0° Tether Parameters p t = 2 x l 0 " 2 k g / m Stationkeeping at 100m Dx.p=Dy.p=D2p= 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 .05, h = 5 0 0 K m 10.0 o " OH 0.0 •10.0 10.0 0 . 0 -E ft 7 _R LEGEND o n s LEGEND o o 7t 4 6 Time, orbits Figure 4-7 Response of the system in the stationkeeping reference configura-tion while negotiating an elliptic trajectory of e = 0.05: (a) tension control. 84 I n i t i a l C o n d i t i o n s Satellite Parameters « t O ( 0 ) = 7 t ° ( 0 ) = 1 0 o « p C (0 )= / ? p O (0 )=7 p ° (0 )=10 o Tether Parameters pt = 2 x l 0 " 2 k g / m Stationkeeping at 100 m DXlP=Dy.p=Dz>p= 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 .05 , h = 5 0 0 K m Time, orbits Figure 4-7 Response of the system in the stationkeeping reference configura-tion while negotiating an elliptic trajectory of e = 0.05: (b) thruster control. 85 Initial Conditions Satellite Parameters « t O ( 0 ) = 7 t ° ( 0 ) = 10 o «p O (0)=/? p O (0)= 7 p °(0)=10 O Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 100m Dx.p=Dy p=D2 p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0.05, h=500Km -IO.O H , , , 1 1 0 2 4 6 8 10 Time, orbits Figure 4-7 Response of the system in the stationkeeping reference configura-tion while negotiating an elliptic trajectory of e = 0.05: (c) offset control. 86 respect to the platform motion is similar. On the other hand, thruster and offset control strategies control the tether motion better suggesting their suitability in situations where the stationkeeping accuracy of high order is required. Scientific experiments and manufacturing in the microgravity environment would be one such situation. The offset control does appear attractive because of its relatively small energy demand. 4.4.3 Platform inertia Effect of the platform inertia on the controller's performance was studied by changing its aspect ratio to 2. This corresponds to the platform dimensions of: PA = 70 m; P B = 140 m with inertias: IXX>P = 20.33 x 107 kgm 2 ; I Y Y < P = 4.08 x 107 kgm 2 ; and IZz,p = 16.33 x 107 kgm 2 . Note, now the roll platform inertia is larger than that for the reference case while the yaw platform inertia is smaller. This would suggest that the roll platform motion will take longer to stabilize while the corresponding yaw motion will be damped faster compared to the response behaviour observed in the reference case. The tension controlled response to the same disturbance as before is shown in Fig. 4-8(a). As expected, the platform roll motion requires a longer time to control. Due to coupling, the platform pitch and yaw degrees of freedom are also affected and attain the steady state only after around seven orbits. The tether librations follow the trend similar to that observed in the reference case (Fig. 4-6a) and motions were damped in around 6 orbits. Note, the inplane tether motion is less than that in the reference case. This may be attributed to persistent platform oscillations storing a part of the system energy. The variation of L shows essentially similar trend as before. The thruster controlled response is presented in Fig. 4-8(b). Once again, the 87 10.0 Initial Conditions o / ~ \ , ^ o < V ( 0 ) = 7 t ° ( 0 ) = 1 0 '(0)=ft a (0)=7 p °(0)=10° p ° ( o ) = V - ° Tether Parameters pt = 2xl0~ 2 kg /m Stationkeeping at 100 m Satellite Parameters Dx,p=Dy p=Dz p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 h = 500Km 4 6 Time, orbits Figure 4-8 Effect of platform inertia on the response of a tethered satellite system: (a) tension control. 88 Initial Conditions Satellite Parameters a t ° ( 0 ) = 7 t o ( 0 ) = 1 0 ° «P ° ( 0 )= /V ( 0 ) = 7 p ° ( 0 ) = 1 0 O Tether Parameters p t = 2xl0~ 2 kg /m Stationkeeping at 100 m D =D =D = 20m x,p y.p z,p m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km PA = 70 m, PB = 140 m 10.0-i - i o . o 4 15.0 -r 10.0-Time, orbits Figure 4-8 Effect of platform inertia on the response of a tethered satellite system: (b) thruster control. 89 Initial Conditions Satellite Parameters « t ° ( 0 ) = 7 t ° ( 0 ) = 1 0 o « p °(0)=(3 p o (0)=7 p °(0)=10 o Tether Parameters pt = 2xl0~ 2 kg/m Stationkeeping at 100m D =D =D =• 20m x,p y,p z,p m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km PA = 70 m, PB = 140 m Time, orbits Figure 4-8 Effect of platform inertia on the control of a tethered satellite sys-tem: (c) offset control. 90 platform pitch and roll take around 7 orbits to be stabilized, while the yaw returns to equilibrium quicker than that in the reference case. Tether motions were damped in around 5 orbits. Characteristic of the thruster control, as observed earlier, the tether response is rather small and is damped in around 5 orbits. The offset control (see Fig. 4-8c) continues to be effective as before but the time taken to return the system to equilibrium is a little longer (around 7 orbits). The results illustrate a characteristic feature of the offset strategy where the controller attempts to stabilize the platform motion as well as tether motions in an optimal fashion so as to minimize the momentum wheel torques. Thus, offset controlled re-sponse always shows finite tether motion until the platform oscillations are damped. This tendency can be reduced by increasing the state penalties associated with the platform response thus increasing the contribution of the momentum wheel torques in stabilization of the space platform. 4.4.4 Tether length In an attempt to assess the effect of tether length, simulations were carried out for 1 km long tether, i.e., the tether was ten times longer than that considered in the reference case. A longer tether leads to larger inplane and out-of-plane inertias (indicated by the terms containing L in Eqs. A.4 and A.5). This means that energy imparted to the system through a disturbance (initial conditions) is higher, even though the angular displacement is still 10° . On the other hand, as the tether length increases, the equilibrium tension in the tether also increases promising an improvement in effectiveness of the tension control scheme. From Eqs. (A.4) and (A.5) it is clear that the generalized moments associated with the thruster forces are proportional to the tether length L. This would suggest an improvement in 91 the thruster controller's effectiveness with the tether length. On the other hand; the offset control strategy relies on moving the point of attachment of the tether to "balance" the tether and the payload. Thus, intuitively it is apparent that with an increase in the tether length, the attachment point will have to move a larger amount to stabilize the tether. The translational displacement at the payload end of a 1000 m tether, for an inplane angular displacement of 10° can be calculated as (1000 x 10 X 7r/180) = 175 m. This means that during the uncontrolled ± 1 0 ° oscillations of the tether, the payload excursions would cover a distance of 350 m. Ideally one would like to damp the motion critically, which would require the offset to move around 175 m so that the point of attachment of the tether is always directly under the payload and the tether is aligned with the local vertical. This ideal motion requirement of the offset may be referred to as the critical displacement A of offset motion Dcr. Dcr = tether displacement (rad) x L (m) However, in the present study, the offset is restricted to 20 m and hence the motion of the tether is no longer critically damped. It is clear that as the tether length is A reduced, Dcr decreases and the offset control becomes more effective. Figure 4-9(a) shows the response of the system under tension control with a tether length of 1000 m. It is apparent that although the energy of the initial disturbance is raised by two orders of magnitude (inertia of the tether is increased by 102), the severe disturbance is arrested in around 4 orbits. The response trends of the platform and the tether are quite similar to those corresponding to the reference case. The system oscillations are confined to the 10° initial displacement. It is A interesting to note that L, the length generalized coordinate, exhibits a relatively large change (around 53 m) reflecting variation in the controlled tension T e . As 92 Initial Conditions Satellite Parameters a t ° ( 0 ) = 7 t o ( 0 ) = 10 o «pO(0)=/V(0)=7p O(0)=10° Tether Parameters p t = 2xl0~ 2 kg /m Stationkeeping at 1000m Dx.p=Dy,p=Dz = 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 20.0 -10.0 1100.0 900.0 LEGEND ex. o 0 2 4 6 Time, orbits Figure 4-9 Effectiveness of the control strategies for a tether length of 1 km: (a) system response with the tension control strategy. 93 Initial Conditions Satellite Parameters a l o (0) = 7 l o ( 0 ) = 10 o V(o)=/sp°(o)=Tp°(o)=io° Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 1000m D x,p=D y p=D z p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 0.95 Time, orbits Figure 4-9 Effectiveness of the control strategies for a tether length of 1 km: (b) control effort time histories during the tension modulation strat-egy. 94 I n i t i a l C o n d i t i o n s Satellite Parameters a t ° ( 0 ) = 7 l o ( 0 ) = 1 0 ° a p o ( 0 ) = / ? p o ( 0 ) = 7 p ° ( 0 ) = 1 0 ° Tether Parameters p t = 2 x l 0 ~ 2 k g / m Stationkeeping at 1000m D x . p = D y , p = D 2 > p = 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m Time, orbits Figure 4-9 Effectiveness of the control strategies for a tether length of 1 km: (c) system response with the thruster control strategy. 95 Initial Conditions Satellite Parameters «t O ( 0 ) = 7 t ° ( 0 ) = 10 o apo(0)=^po(0)=7p'>(0)=10o Tether Parameters p t = 2xl0~ 2 kg /m Stationkeeping at 1000m D x,p=Dy p=D 2 p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km Z O.lO-i -0.10-1 , , , , h 0 0.5 1 1.5 2 2.5 3 E 40.0-T 0 1 2 3 4 5 6 7 Time, orbits Figure 4-9 Effectiveness of the control strategies for a tether length of 1 km: (d) control effort time histories during the thruster control strategy. 96 expected the equilibrium tension in the tether was higher at 0.8 N with maximum tension reaching 0.9 N (Fig. 4-9b). The peak demand on the momentum wheel torque was found to be around 26 Nm and, as before, is associated with the roll degree of freedom. The control impulse contributed by the tension variations was 190.2 Ns while momentum wheel energy requirement was found to be 518.2 J. The corresponding response with the thruster control strategy is shown in Figure 4-9(c). Indeed the controller continues to be quite effective in damping the large disturbance in a relatively short time (around 4 orbits). In fact, the tether motion is virtually negligible after 2 orbits. Figure 4-9(d) shows the corresponding thruster force and momentum wheel torque time histories. Note, the peak thruster forces are required in the pitch and roll directions with Ta = 0.06 N and T 7 = —0.06 N. The higher rotational energy of the tether due to an increase in inertia is reflected in the larger magnitudes (10 fold) of the thurster forces. One may wonder why the forces do not increase proportionally (to 102). The explanation lies in the fact that now the effective thruster moment arm is 10 times larger. The momentum wheel torques are also higher as in the tension controlled case for the same reasons. The maximum torque is around 34 Nm, again required for the roll correction. The thruster force impulse requirement amounted to 166.6 Ns which is quite close to the tension control case. The energy demand on the momentum wheels was found to be 556.1 J, significantly larger compared to the reference case of 100 m. The offset control strategy, being the focus of this thesis, was studied in more detail for four different tether lengths of 1000 m, 750 m, 500 m and 250 m (in addition to 100 m for the reference case). The offset controlled response for a 1000 m long tether is depicted in Fig. 4-10(a). As explained earlier, the 20 m constraint imposed on the offset motion restricts the damping characteristics of the 97 I n i t i a l C o n d i t i o n s Satellite Parameters a t ° ( 0 ) = 7 t ° ( 0 ) = 10 o V(0)=/?p o<0)=7p°(0H0o Tether Parameters p t = 2 x l 0 ~ 2 k g / m Stationkeeping at 1000m Dx.p=Dy p=Dz p= 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m 10.0 -10.0 4 10.0-r Time, orbits Figure 4-10 Effect of tether length on the performance of the offset control strategy: (a) system response, 1 km tether. 98 E I n i t i a l C o n d i t i o n s Satellite Parameters a l o ( 0 ) = 7 l ° ( 0 ) = 1 0 o V (0 )= /? p ° (0 )= 7 p ° (0 )=10° Tether Parameters p t = 2 x l 0 ~ 2 k g / m Stationkeeping at 1000m D v =DV =17, = 20m x.p y.p z,p m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m 0 5 10 15 20 Time, orbits Figure 4-10 Effect of tether length on the performance of the offset control strategy: (b) control effort, 1 km tether. 99 Initial Conditions Satellite Parameters a l o (0) = 7 l o ( 0 ) = 10 o ap°(O)=0p°(O)=7po(O)=lO° Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 750m Dx.p=Dy,p=DZiP= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 10.0 Time, orbits Figure 4-10 Effect of tether length on the performance of the offset control strategy: (c) system response, 750 m tether. 100 I n i t i a l C o n d i t i o n s Satellite Parameters a t ° ( 0 ) = 7 t ° ( 0 ) = 1 0 ° V(O)=0pc(O)=7po(O)=lOo Tether Parameters p t = 2 x l 0 ~ 2 k g / m Stationkeeping at 500m D x . P =D y > p =D 2 p = 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m 10.0 -IO.O H , , , , , — i — | 1 0 2.5 5 7.5 10 12.5 15 17.5 Time, orbits Figure 4-10 Effect of tether length on the performance of the offset control strategy: (d) system response, 500 m tether. 101 Initial Conditions Satellite Parameters a t o (0) = 7 l o ( 0 ) = 1 0 ° <(o)=/3p0(o)=rp°(o)=io0 Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 250m Dx.P=Dy,p=D2p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km Time, orbits Figure 4-10 Effect of tether length on the performance of the offset control strategy: (e) system response, 250 m tether. 102 controller. It becomes particularly evident at larger tether lengths. It should be noted that the allowable motion is only 0.114 Dcr which gives some appreciation as to the severity of the constraint. Even then, the platform motion is reduced to less than 1° in amplitude in around 4 orbits with residual oscillations due to coupling with the tether dynamics. On the other hand, the tether oscillations were damped gradually to 3° in 15 orbits and attained steady state only after 28 orbits (not shown). Figure 4-10(b), shows time histories of the tether attachment point and the control efforts. The tether being long, the offset excursions approach the maximum permissible limit of 20 m initially but reduce to 2 m in 15 orbits. However, the associated control forces still remain extremely small in magnitude (0.007 N maximum). The demand on the platform based control momentum gyros showed a maximum of 17 Nm which is comparable to that observed in the reference case for a 100 m tether. This suggests that the offset controller continues to damp the platform motion in an optimal fashion. The impulse requirement for the offset forces was 42.3 Ns while the momentum wheels demanded 460.3 J . Figures 4-10(c) to Figure 4-10(e) study systematically the influence of reducing the tether length. A marked improvement in performance of the offset control with reduction in the tether length is quite apparent. Note, for the case of stationkeeping with a 250 m tether, the disturbance is damped in around 7 orbits (4 orbits for the reference case of 100 m tether, Fig. 4-6e ), thus confirming potential of this form of control strategy, particularly at shorter tether lengths. This study on the effect of tether length brings to light characteristic features of the three control strategies. Effectiveness of the tension and thruster control pro-cedures improve with longer tether lengths. The thruster control tends to consume a comparable amount of energy as the tension modulation strategy as the tether 103 length increases. The severely constrained offset strategy is much more energy ef-ficient at a cost of speed of damping. Of course at a longer tether length, there is an increase in demand on the control moment gyros due to larger inertia of the deployed tether. Even here, the the offset controlled system resulted in a smaller increase in the momentum wheel energy demand (6.2 J) compared to that for the tension control (37.6 J) or the thruster control (52.9 J). This indicates that the offset control has an ability to isolate influence of the inertial effects of the tether on the platform. 4.4.5 Tether mass The mass of the tether is changed by varying its linear density (pt). In the reference configuration it was taken as 2 x 10~2 kg/m. This corresponds to the mission requirements for an electrodynamic tether [24]. The linear density was now increased 5 fold (pt = 1 0 - 1 kg/m) to assess the effect of a more massive tether on the controllers' performance. A higher linear density causes the tether inertia to increase, thereby increasing the energy of rotation of the tether for the same initial condition. Figure 4-11(a) shows system response during tension control. Although the platform and tether response trends are similar to that for the reference case, there are some distinct differences in magnitudes. At a given instant, the tether inplane displacement (at) is distinctly larger indicating the effect of higher tether inertia. On the other hand, the controlled length variations, L, are smaller (max-imum about 5 m) as a smaller change in length is required to cause an equivalent change in tension. As expected, Fig. 4-11(b), shows the equilibrium tension level to be higher (0.11 N) compared to that for the reference case (0.08 N). Note, the required tension variations as well as the momentum wheel torques are now higher 104 (0.29 N and 40 N m maximum, respectively). These resulted in the tension force impulse requirement of of 181.4 Ns while the momentum wheel energy demand was 542.1 J. Figure 4-11(c) depicts the thruster controlled response of the system under similar initial conditions. The platform motion continues to be stabilized within 4 orbits as for the reference configuration, however, the controller effort in terms of thruster force as well as momentum wheel torques are larger to account for the higher energy stored in the more massive tether (Fig. 4-1 Id). The pitch and roll demand the maximum controlling thrust of around 0.04 N . The demand on the momentum wheels is also higher than that observed in the reference case. Note, the roll moment (M 7 ) required peak value of 36 Nm. The thruster force impulse requirement was 153.2 Ns while the momentum wheel energy demand was 560.7 J. The offset control not only remains effective but shows an improvement in per-formance (Fig. 4-lle). Note, in Eq. (A.4) and (A.5), the coefficients of J 9 " p and b'xtP a r e functions of Mt<e, the ratio of tether mass to subsatellite mass. A higher values of pt results in an increase in this ratio and hence the offset controller be-comes more effective. The corresponding offset force and momentum wheel torque time histories are shown in Fig. 4-11(f). The impulse requirement was found to be 36.9 Ns, with the momentum wheel energy demand of 451.3 J. A n increase in the offset force related impulse requirement may be attributed to the higher rotational energy of the tether. Thus with an increase in the tether mass, all three strategies show an increase in energy demand. The offset control effectiveness improves, however, the energy required for control also increases almost two-fold. On the other hand, the tension controller shows a smaller increase in energy utilization with a small loss in effec-tiveness. The thruster control shows a marginal improvement in performance with 105 Initial Conditions Satellite Parameters c l ° ( 0 ) = 7 l < > (0) = 10 0 V(O)=0 p°(O)= 7 p°(O)=lO° Tether Parameters p t = l x l O - 1 kg /m Stationkeeping at 100 m D x.p=Dy p=D 2 p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 10.0 o ~ o . o -•10.0 10.0 0.0-+ -10.0 110.0 100.0-90.0 2 3 4 5 Time, orbits Figure 4-11 Effect of increasing the tether mass to 10 kg on the performance of the various control strategies: (a) system response during the tension modulation approach. 7 106 I n i t i a l C o n d i t i o n s Satellite Parameters a l ° ( 0 ) = 7 l < , ( 0 ) = 1 0 ° V(0)=/?p o(0)=7p o(0)=10o Tether Parameters p t = l x l O " 1 k g / m Stationkeeping at 100 m D x , p =D y p =D 2 p = 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m -40 .0 H 1 1 1 1 1 1 — i 0 1 2 3 4 5 6 7 Time, orbits Figure 4-11 Effect of increasing the tether mass to 10 kg on the performance of the various control strategies: (b) control effort time histories during the tension modulation approach. 107 I n i t i a l C o n d i t i o n s Satellite Parameters a i ° ( 0 ) = 7 t ' , ( 0 ) = 1 0 O « p O ( 0 ) = / ? p O ( 0 ) = 7 p ° ( 0 ) = 1 0 o Tether Parameters p t = l x l O " 1 k g / m Stationkeeping at 100 m D v =DV =L\ = 20m x.p y.p z,p m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m Figure 4-11 Effect of increasing the tether mass to 10 kg on the performance of the various control strategies: (c) system response during the thruster control. 108 I n i t i a l C o n d i t i o n s Satellite Parameters « t ° ( 0 ) = 7 t O ( 0 ) = 1 0 ° «p O (0)=/? p °(0)=7 p o (0)=10 O Tether Parameters p t = l x l O - 1 k g / m Stationkeeping at 100 m D x .p=D y p=20m,D 2 p=20m m p = 1 0 0 , 0 0 0 k g m B = 100 ke Orbit Parameters e = 0 , h = 5 0 0 K m B 40.0 Time, orbits Figure 4-11 Effect of increasing the tether mass to 10 kg on the performance of the various control strategies: (d) control effort time histories for the thruster case. 109 I n i t i a l C o n d i t i o n s Satellite Parameters a t ° ( 0 ) = 7 t ° ( 0 ) = 1 0 ° c , p ° (0 )=p p ° (0 )= 7 p ° (0 )=10° Tether Parameters p t = l x l O " 1 k g / m Stationkeeping at 100 m D =D =D = 20m x,p y.p z,p '-win m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m -IO.O H , , 1 , 1 — — i 1 0 1 2 3 4 5 6 7 Time, orbits Figure 4-11 Effect of increasing the tether mass to 10 kg on the performance of the various control strategies: (e) system response during the offset control approach. 110 B I n i t i a l C o n d i t i o n s Satellite Parameters « l ° ( 0 ) = 7 l ° ( 0 ) = 1 0 o c . p t , (0 )= /? p t > (0 )= 7 p o (0 )=10° Tether Parameters pt = l x l O " 1 k g / m Stationkeeping at 100 m D x , p =D y p=D 2 p= 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m Time, orbits Figure 4-11 Effect of increasing the tether mass to 10 kg on the performance of the various control strategies: (f) offset and control effort time histories. I l l an increase in energy utilization. 4.4.6 Payload mass In order to study the effect of payload mass on the controllers' performance, a 500 kg subsatellite was considered. The effect of an increased payload (from 100 to 500 kg) is to increase the tether tension without affecting its mass. Hence the results here should show some similarity with those obtained in the previous section. Figure 4-12 presents response results and control effort time histories for the three control strategies under study. The expected similarity is quite evident although there are local variations. For the tension control case (Figs. 4-12a and 4-12b), this is understandable as now the equilibrium tension is higher (0.4 N against 0.11 N in the previous case) although inertia of the tether-payload subassembly is also increased. These affect the performance in opposite fashion, however, the overall effect is to marginally increase the impulse requirement to 190.1 Ns and the momentum wheel energy demand to 546.7 J (compared to 181.4 Ns and 542.1 J , respectively for the tether mass increase case discussed in section 4.4.5). The thruster and offset control strategies (Figures 4-12c and 4-12f), do not show any significant changes in trends except that now the energy demand is slightly increased ( compared to the case in section 4.4.5). Of course, with respect to the reference case, the tension control strategy shows an improvement in performance as expected, however the tether tension has virtu-ally no effect on the thruster controlled performance. In fact, it increases the energy consumption due to a larger payload (the thruster force impulse requirement is in-creased to 163.9 Ns while momentum wheel energy demand is 569.6 J compared to 159.2 Ns and 560.7 J, respectively, for the previous tether mass increase case). 112 I n i t i a l C o n d i t i o n s Satellite Parameters a t o ( 0 ) = 7 t o ( 0 ) = 1 0 o « p ° ( 0 ) = , 5 p ° ( 0 ) = r p ° ( 0 H 0 o Tether Parameters p t = 2 x l O ~ 2 k g / m Stationkeeping at 100 m D X i P =D y p =D z p = 20m m p = 1 0 0 , 0 0 0 k g m s = 500 k g Orbit Parameters e = 0 , h = 5 0 0 K m 20.0 o 0 . 0 -«—3 + -10.0 120.0 100.0 80.0 i 4 LEGEND a. __7_P. LEGEND o o 7t 2 3 Time, orbits Figure 4-12 Effect of increasing the payload mass on effectiveness of the three control strategies: (a) system response during the tension modula-tion approach. 6 113 Initial Conditions Satellite Parameters c, t o(0) = 7 l ° ( 0 ) = 10 o a p ° (0 )= /? p ° (0 )= 7 p ° (0 )=10° Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 100 m D x . p = D y i P = D 2 > p = 20m m p = 100,000kg m s = 500 kg Orbit Parameters e = 0 , h = 500Km Time, orbits Figure 4-12 Effect of increasing the payload mass on effectiveness of the three control strategies: (b) control effort time histories during the ten-sion modulation approach. 114 Initial Conditions Satellite Parameters « t ° ( 0 ) = 7iO(0) = 10 o a p ° (0 )= ,3 p ° (0 )=7 p ° (0 )=10 o Tether Parameters p t = 2xl0~ 2 kg /m Stationkeeping at 100 m Dx,p=Dy>p=Dzp= 20m m p = 100,000kg m s = 500 kg Orbit Parameters e = 0 , h = 500Km 0 1 2 3 4 5 6 7 Time, orbits Figure 4-12 Effect of increasing the payload mass on effectiveness of the three control strategies: (c) system response during the thruster control. 115 Initial Conditions Satellite Parameters a l ° ( 0 ) = 7 l ° ( 0 ) = 10 o « p ° (0 )= / ? p o (0 )=7 p ° (0 )=10 o Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 100 m Dx.p=DyiP=20m^z p=20m m p = 100,000kg m s = 500 kg Orbit Parameters e=0 , h=500Km Z 0.08 £ 40.0 0 1 2 3 4 5 6 7 Time, orbits Figure 4-12 Effect of increasing the payload mass on effectiveness of the three control strategies: (d) control effort time histories for the thruster case. 116 Initial Conditions Satellite Parameters « l ° ( 0 ) = 7 t ° ( 0 ) = 10 o « p °(0)=/3 p < > (0)= 7 p °(0)=10 c > Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 100m D x , p = D y p = D z p = 20m m p = 100,000kg m s = 500 kg Orbit Parameters e = 0 , h = 500Km 20.0 Time, orbits Figure 4-12 Effect of increasing the payload mass on effectiveness of the three control strategies: (e) system response during the offset control approach. 117 G Initial Conditions Satellite Parameters « l O ( 0 ) = 7 l ° ( 0 ) = 10 o apO(0)=/3po(0h7po(0)=10o Tether Parameters p t = 2xl0~ 2 kg/m Stationkeeping at 100m D x ,p=D y p =T3 z p = 20m m p = 100,000kg m s = 500 kg Orbit Parameters e = 0 , h = 500Km Time, orbits Figure 4-12 Effect of increasing the payload mass on effectiveness of the three control strategies: (f) offset and control effort time histories. 118 In Eqs. (A.4) and (A.5), the terms associated with offset acceleration control are [(1 + LMtt8)L\b'lp and [- ( l + LMtts/2)L}b'^p. It should be noted that these terms are nondimensionalized with respect to payload mass and can be written as l(ma + mtL)/m8L]D"p and [—{ma + mtL/2)/meL]D"p. Clearly, increasing payload mass would be beneficial to the offset control performance. A comparison with the response results for the reference configuration is presented earlier (Fig. 4-6e). An improvement in performance is indeed noticeable. 4.4.7 Deployment The nondimensionalized commanded or nominal length of the tether L is a specified coordinate. Its values range from 0 to 1, representing complete retrieval and fully deployed states of the tether, respectively. L' is a nonzero quantity during deployment and retrieval, in Eqs. (A.4) and (A.5). It appears in the coefficients of tether pitch and roll rates (aj and 7J). During deployment L' has a positive value and makes these coefficients positive resulting in a damping of the corresponding motions. The exponential deployment maneuver was conducted with a starting length of 10 m, and the system subjected to the same initial conditions, to a final length of 1000 m in 0.37 orbit. Fig. 4-13(a), shows deployment of the tether from from 0 around 1000 m where the variations due to L are superimposed over the exponentially increasing L. As expected, the tether inplane motion is stabilized rather quickly in around 2 orbits due to the damping effect of deployment. The out-of-plane motion (7$) takes a little longer to return to the equilibrium state as it is affected only through coupling. The platform motions required 4.5 orbits to stabilize. Figure 4-13(b) shows the thruster controlled response during deployment with 119 o ~ OH Initial Conditions Satellite Parameters a t ° ( 0 ) = r t ° ( 0 ) = 1 0 ° ° p ° ( 0 ) = ^ p o ( 0 ) = 7 p o ( 0 ) = 1 0 o Tether Parameters p t = 2 x l 0 ~ 2 k g / m Deployment 10m to 1000m Dx.p=Dy>p=D2fP= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e=0 , h=500Km 10.0 Deployment completed in 0.37 orbit o.o-10.0 10.0 o.o--10.0 2000.0 2 3 4 5 Time, orbits Figure 4-13 Effect of deployment on the system response during: (a) the tension control strategy. 120 I n i t i a l C o n d i t i o n s Satellite Parameters a l ° ( 0 ) = r l o ( 0 ) = 1 0 ° V ( 0 ) = / 3 p ° ( 0 ) = 7 p ° ( 0 ) = 1 0 ° Tether Parameters p t = 2 x l 0 ~ 2 k g / m Deployment 10m to 1000m D =D =D = 20m x,p y,p z,p m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m Deployment completed in 0.37 orbit Time, orbits Figure 4-13 Effect of deployment on the system response during: (b) the thruster control strategy. 121 Initial Conditions Satellite Parameters a t ° ( 0 ) = 7 l ° ( 0 ) = 1 0 o « P ° ( 0 ) = / S p o ( 0 ) = 7 p o ( 0 ) = 1 0 o Tether Parameters p t = 2 x l 0 ~ 2 k g / m Deployment 10m to 1000m D x,p=Dy p=D 2 p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 10.0 Deployment completed in 0.37 orbit OH o ~ 03- 0.0 a -10.0 20.0 10.0-0.0 10.0 1000.0-LEGEND a. ft y ° 2 4 6 8 10 12 14 Time, orbits Figure 4-13 Effect of deployment on the system response during: (c) the offset control strategy. 122 the identical conditions. The thruster control distinctly shows better performance damping the tether motion in less than 2 orbits. Even the platform motion is arrested a little faster (4 orbits compared to 4.5 orbits for the tension control case). Results of the offset control are presented in Fig. 4-13(c). The offset constraint of 20 m restricts motion of the tether attachment point to 0.114 Dcr (for a 1000 m tether). Although the platform motion is damped quite quickly as in the case of the other two control procedures, the tether motion persists for a significantly longer time (around 13 orbits). A coupling between the platform and tether continued to perturb the former until tether stabilization was accomplished. Comparing the three control strategies, the application of thruster control appears to be most effective in stabilizing the deploying tether when the initial disturbance is of the order of 10° or higher. However, for smaller disturbances, which may be more realistic as the initial tether length is only 10 m, the offset control may prove to be equally suitable. 4.4.8 Tether retrieval The coefficients of a't and in Eqs. (A.4) and (A.5), contain V , the derivative of nominal tether length (nondimensionalized with respect to lref). During retrieval L' is negative, making the coefficients of the tether pitch and roll rates negative. This corresponds to a system with negative damping, i.e., the tether motion gradually increases with time. Physically this is apparent. A reduction in the tether length leads to a corresponding reduction in moments of inertia of the system, hence the principle of conservation of angular momentum requires the disturbance to amplify. Thus control during retrieval represents the most challenging aspect of the tethered 123 satellite dynamics. L' can be varied in two ways: (a) changing the time of retrieval for a given length, or (b) changing the length to be retrieved in a specified time. To facilitate comparison, the system in the reference configuration was simulated in an expo-nential retrieval maneuver where the tether length is reduced from 100 m to 10 m in 1 orbit. Next, the same retrieval maneuver was conducted with shorter retrieval times of 0.68 orbit and 0.37 orbit. Finally, the retrieval was conducted with an initial tether length of 1000 m to a final length of 10 m. Figure 4-14 studies effectiveness of the tension control strategy during the three different retrieval rates. The tension controlled response for exponential retrieval of a 100 m tether in the reference configuration, to a final length of 10 m in 1 orbit is presented in Figure 4-14(a). The controlled length variations L are significant: around 10 m. It is of particular interest to note that the inherently unstable char-acter of the retrieval maneuver leads to a peak displacement of around -16° in at although initial disturbance is 10° in all the degrees fo freedom. The out-of-plane 7t motion is also excited by the maneuver. This can be attributed to the term T({ [2(1 + L M t , a / 2 L ' Z ] in Eq. (A.5). At the end of the retrieval, the tether length is quite small (10 m) leading to a small value of tension (0.01 N) and the controller's effectiveness diminishes. It takes around 4 orbits to damp the platform motion and approximately 6 orbits to return the tether to its equilibrium state. For a faster retrieval rate with the maneuver completed in 0.68 orbit (Fig. 4-14b), the peak at increases to -19° . Even the out-of-plane swing reaches around -18° . Once the destablilizing influence of the retrieval ends (upon completion of the maneuver) the controller asserts its effectiveness. The controlled variations of the tether length (L) follow essentially the same trend as before. 124 Initial Conditions Satellite Parameters a t ° ( 0 ) = 7 l ° ( 0 ) = 1 0 ° «p°(0)=/3po(0)=V(0)=10o Tether Parameters p t = 2 x l 0 " 2 kg/m Retrieval 100m to 10m Dx.p=Dy p=D2p= 20m m p = 100,000kg rn s = 100 kg Orbit Parameters e = 0 , h = 500Km -40.0-lO.O-i s Time, orbits Figure 4-14 System response during retrieval from 100 m to 10 m in the presence of the tension control strategy: (a) retrieval time of 1 orbit. 125 Initial Conditions Satellite Parameters a l ° ( 0 ) = 7 l < > (0)=10 0 <(0 )=P p ° (0)=7 p O (0H0 o Tether Parameters p t = 2 x l 0 " 2 kg /m Retrieval 100m to 10m DX i P=D y p=D z p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 0 1 2 3 4 5 6 7 Time, orbits Figure 4-14 System response during retrieval from 100 m to 10 m in the presence of the tension control strategy: (b) retrieval time of 0.68 orbit. 126 Initial Conditions Satellite Parameters ° t < > ( 0 ) = 7 l O ( 0 ) = 1 0 o a p o ( 0 ) = / 3 p ° ( 0 ) = 7 p ° ( 0 H 0 o Tether Parameters p t = 2xl0~ 2 kg/m Retrieval 100m to 10m Dx.P=Dy>p=Dzp= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km Retrieval completed in 0.37 orbit Time, orbits Figure 4-14 System response during retrieval from 100 m to 10 m in the presence of the tension control strategy: (c) retrieval time of 0.37 orbit. 127 The response results show essentially the same trends at a still faster retrieval completed in 0.37 orbit (Fig. 4-14c). The peak tether pitch displacement, at , is -26° . This is what one would have expected intuitively. The faster retrieval rate represents a larger disturbing influence on the system. From the controlled variations in tether length L superimposed on the nominal tether length L, it is apparent that the deployed tether length can be at times close to a couple of metres. This is a critcial situation in tension control of the tethered satellite system. Effect of the similar retrieval sequences was studied for case of thruster control. Figure 4-15(a) shows the response for a retrieval time of 1 orbit. Now the peak at response does not exceed the 10° initial condition. The out-of-plane tether roll response (74) is controlled rather effectively.The platform dynamics is also effectively controlled in around 3 orbits. When retrieval time is decreased to 0.68 orbit (Fig. 4-15b), at slightly exceeds the initial condition of 10° ( « 11° ) , however, the control continues to remain quite effective. The system returns to equilibrium in little over three orbits. With a further increase in the retrieval rate (maneuver completed in 0.37 orbit, Fig. 4-15c), the peak response of the tether inplane motion shows a further rise (at, peak = 14°) , however, the general trends remain the same and the controller's effectiveness is apparent. Unfortunately, as mentioned earlier, safety considerations may not permit use of thrusters in the vicinity of the platform. Effectiveness of the offset controller is assessed during the same maneuvers in Figs. 4-16. As before, the peak tether responses continue to increase with the retrieval rate, however, now the tether roll motion proves to be more susceptible and reaches a value of 25° for the retrieval completed in 0.37 orbit (Fig. 4-16c). It is 128 I n i t i a l C o n d i t i o n s a t ° ( 0 ) = 7 l ° ( 0 ) = 1 0 ° ap o(0)= /3p o(0)=7 p c ,(0)=10o  Tether Parameters p t = 2 x l 0 ~ 2 k g / m Retrieval 100m to 10m Satellite Parameters D x , p=D y , p=D 2 i p= 20m Orbit e = 0 1 0 0 , 0 0 0 k g 100 k g Parameters h = 5 0 0 K m 10.0 Retrieval completed in 1 orbit a o ~ a 00-- 5 .0 100.0-1 E Figure 4-15 0 1 2 3 4 5 6 7 Time, orbits System response during from 100 m to 10 m with the thruster control strategy: (a) retrieval time of 1 orbit. 129 In iii a ] Conditions Satellite Parameters a t o (0) = 7 l o ( 0 ) = 1 0 ° a p ° ( 0 ) = ^ p o ( 0 ) = 7 p o ( 0 ) = 1 0 ° Tether Parameters p t = 2xl0~ 2 kg/m Retrieval 100m to 10m D * . p = D y i P = D 2 i P = 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km _j , , ! j , , 0 1 2 3 4 5 6 7 Time, orbits Figure 4-15 System response during from 100 m to 10 m with the thruster control strategy: (b) retrieval time of 0.68 orbit. 130 I n i t i a l C o n d i t i o n s Satellite Parameters « t O ( 0 ) = 7 t ° ( 0 ) = 1 0 O < ( 0 ) = / ^ ( 0 ) = 7 p ° ( 0 H 0 O Tether Parameters p t = 2 x l 0 ~ 2 k g / m Retrieval 100m to 10m D x,P=D y > p=D 2 > p= 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m u u j ! ( j j j [ 0 1 2 3 4 5 6 7 Time, orbits Figure 4-15 System response during from 100 m to 10 m with the thruster control strategy: (c) retrieval time of 0.37 orbit. 131 Initial Conditions Satellite Parameters a t ° ( 0 ) = 7 t ° ( 0 ) = 1 0 ° apo(O)=0p°(O)=7p°(O)=lOo Tether Parameters p t = 2xl0~ 2 kg /m Retrieval 100m to 10m D x,p=Dy p=D 2 p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km 10.0 Retrieval completed in 1 orbit Figure 4-16 3 4 5 Time, orbits System response during from 100 m to 10 m with the offset control strategy: (a) retrieval time of 1 orbit. 132 Initial Conditions Satellite Parameters a t o(0) = 7 t o(0)=10° « p O(0)=/S p°(0)=7 p°(0)=10 o Tether Parameters p t = 2xl0~ 2 kg/m Retrieval 100m to 10m Dx,p=Dy>p=D2iP= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 500Km Retrieval completed in 0.68 orbit LEGEND o.o H 1 1 1 1 1 1 0 1 2 3 4 5 6 7 Time, orbits Figure 4-16 System response during from 100 m to 10 m with the offset control strategy: (b) retrieval time of 0.68 orbit. 133 I n i t i a l C o n d i t i o n s Satellite Parameters « l ° ( 0 ) = 7 t ° ( 0 ) = 1 0 O «pO(0)=/3 p O(0)=7 p°(0)=10 o Tether Parameters p t = 2 x l 0 ~ 2 k g / m Retrieval 100m to 10m D x .p=D y p =D z p = 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m 10.0 o 0 . 0 --10.0-1 50.0 -50 .0 100.0 £ 0.0 2 3 4 Time, orbits T 5 LEGEND a. ft 7 .p.. LEGEND o a t o 7t 7 Figure 4-16 System response during from 100 m to 10 m with the offset control strategy: (c) retrieval time of 0.37 orbit. 134 gratifying to note that the offset control remains effective even at a relatively high retrieval rate and damps the system dynamics within 4 orbits. It was decided to increase the retrieval rate further to correspond to that for the TSS-1 mission requirements by exponentially retrieving a tether with a longer starting length of 1 km in 0.37 orbit. Effectiveness of the three control strategies were studied as before. Figure 4-17(a) shows the system response during the tension control. The tether pitch response now reaches a maximum of 34° . The coupling between at and ap due to the nominal offsets causes the latter to deviate to 26° as well. The corresponding controller length variations (L) were as high as 20 m. The controller is quite effective and all the motions are stabilized within 5 orbits. The system response for identical conditions and retrieval rate employing the thruster control procedure is depicted in Fig. 4-17(b). The trends are essentially the same as those observed earlier (Fig. 4-15c). The faster retrieval seems to affect the platform dynamics more, which now takes around five orbits to stabilize (as against 3 orbits in Fig. 4-15c). In the case of the offset controller, the severe restriction limiting offset to within 0.114 Dcr curtailed its ability to control such a large retrieval induced excitation. However, with the tether initial disturbance reduced to 1° and using a retrieval time of 1 orbit (platform disturbance same as before), the controller was successful in damping the resulting motion. Note, the peak tether response reaches a maximum of 26° yet the system is stabilized within 2 orbits after completion of the retrieval maneuver. Platform motion remains small, however, it persists for a longer time. These results indicate that although the offset control performed quite well in most situations, it was limited by the 20 m constraint imposed on the displacement of the point of tether attachment. If the retrieval rate is modest and the tether 135 I n i t i a l C o n d i t i o n s Satellite Parameters a t ° ( 0 ) = 7 t ° ( 0 ) = 1 0 O V(o)=V(o)=7p0(o)=io0 Tether Parameters pt = 2 x l 0 ~ 2 k g / m Retrieval 1000m to 10m D x . p =D y p =D 2 p = 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m Retrieval completed in 0.37 orbit 2 4 6 Time, orbits Figure 4-17 Response of the system during exponential retrieval of the tether from 1000 m to 10 m in 0.37 orbit: (a) tension control strategy. 136 I n i t i a l C o n d i t i o n s Satellite Parameters « l ° ( 0 ) = 7 l ° ( 0 ) = 1 0 o «pO(O)=0p°(O)=7po(O)=lOo Tether Parameters p t = 2 x l 0 ~ 2 k g / m Retrieval 1000m to 10m D x > p =D y p =D 2 p = 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e = 0 , h = 5 0 0 K m 1000.0 a i i i i I 1 1 1 0 1 2 3 4 5 6 7 Time, orbits Figure 4-17 Response of the system during exponential retrieval of the tether from 1000 m to 10 m in 0.37 orbit: (b) thruster control. 137 I n i t i a l C o n d i t i o n s Satellite Parameters « t ° ( 0 ) = 7 t O ( 0 ) = l O V (0)=/3 p o (0)=7 p °(0)=10 o Tether Parameters P t = 2 x l ( T 2 k g / m Retrieval 1000m to 10m D x . p =D y p =D 2 p = 20m m p = 1 0 0 , 0 0 0 k g m s = 100 k g Orbit Parameters e=0 , h = 5 0 0 K m Time, orbits Figure 4-17 Response of the system during exponential retrieval of the tether from 1000 m to 10 m in 0.37 orbit: (c) offset control. Note, the tether initial condition is now reduced to at = 7 t = 1 ° . Platform initial conditions remain unchanged. 138 length is relatively small (i.e., Dcr is smaller), the performance improves dramat-ically and it results in the most energy efficient controller. This feature may be exploited to advantage using a hybrid control scheme that utilizes the thruster or the tension control at the onset of retrieval when L and V are relatively large fol-lowed by the offset control when they become acceptably small. Performance of the system during the hybrid control is discussed in the following section. 4.4.9 Hybrid control strategies Two hybrid control strategies were developed for the case of retrieval from 1000 m to 10 m in 0.37 orbit. They employed the following combinations: (a) the tension modulation strategy followed by the offset control; (b) the thruster control procedure followed by the offset control strategy. The switch over point from tension to offset control strategy was based on energy consideration. As the offset control at shorter tether length demands the least energy among the three damping procedures, one would like to arrange for switch relatively early. The switch over point was established systematically through this least energy criterion. The optimal switching point was obtained by iteratively searching along the tether length till energy for the combined control was minimized. Figure 4-18(a) shows response for a combination of the tension control followed by the offset control. The hybrid control strategy is indeed quite effective. The maximum tether response is confined to 10° and both the platform and tether motions are damped in less than three orbits. Figure 4-18(b) shows the corresponding controller effort required. The tether tension reaches a maximum value of 0.23 N . Initially, the controlled offset remains motionless until the switching occurs at L = 201 m. Now the offset control takes 139 a <^ o " 02. Initial Conditions Satellite Parameters « t ° ( 0 ) = 7 t ° ( 0 ) = 1 0 ° «p 0(o)=f5p°(o)=rp 0(o)=io 0 Tether Parameters p t = 2 x l 0 ~ 2 k g / m Retrieval 1000m to 10m Dx.p=Dy>p=D2p= 20m m p = 100,000kg m s = 100 kg Orbit Parameters e = 0 , h = 5 0 0 K m 2Q Q Retrieval completed in 0.37 orbit -12.5 2000.0-1000.0 0.0 I 0 3 Time, orbits T 5 Figure 4-18a System response with the tension modulation procedure followed by the offset control during exponential retrieval from 1000 m to 10 m in 0.37 orbit. 140 Initial Conditions Satellite Parameters a t O(0) = 7t°(0) = 10 o <(O)=0pO(O)=7p°(O)=lOo Tether Parameters pt = 2xl0~ 2 kg/m Retrieval 1000m to 10m nx.p=5y.p=D2p= 20m m p = 100,000kg ms = 100 kg Orbit Parameters e=0 , h=500Km 6 20 .0 - : o.o -20.0 0 1000.0 201 10.0 \ /-Hybrid Control , \ / Switching at 201m. • v./ LEGEND I 0.3 - 0.2 0.1 0.0 0.2 0.4 Time, orbits 0.6 0.8 • I I 1 1 1 1 LEGEND A • \ \ K /• -..V, » » Up • ' i ' i : \ 0) Figure 4-18b Controller time history for tension-offset hybrid control. 141 over. The peak demand on the momentum wheel output is only 24 Nm for the yaw correction. A finite non-zero offset remains at the end of the maneuver because at the time of switching, L may not necessarily be zero (see Eq. A.1-A.6). Figure 4-19(a) shows response of the system during thruster-offset hybrid control strategy. Even here the maximum tether pitch is limited to only 13°. Note, the out-of-plane tether motion is effectively damped in a near optimal fashion. The entire system is stabilized in less than 3 orbits. The corresponding demand on the controller effort is presented in Fig. 4-19(b). A brief burst in thruster firing continues until L = 269 m at which point the offset control commences. It is of interest to recognize that the peak momentum wheel torque is significantly lower than the previous case (11 Nm compared to 24 Nm) and the system is stabilized in around 3 orbits. To get better appreciation as to the controller demand over the range of system parameters studied during deployment stationkeeping and retrieval, the impulse and energy values are presented together in Table 4-3. At the outset, the energy efficient character of the offset control in all situations (except the retrieval from 1000 m —• 10 m) is apparent. It is successful and energy efficient even at 1000 m in the station-keeping mode. However, stationkeeping being the most important phase when the payload (subsatellite) is pursuing its intended mission, time taken to damp a dis-turbance may become the prime consideration. This may tip the balance in favour of the thruster control at a cost of higher energy consumption. The offset strat-egy is well suited for deployment, however, for retrieval, the thruster-offset hybrid control procedure appears to be ideal. It is better in terms of dynamical response, damping time and energy demand. Furthermore, the switch over point being at a tether length of 269 m the possible problems of safety and plume impingement are 142 Initial Conditions Satellite Parameters « t ° ( 0 ) = 7 t O ( 0 ) = 1 0 o «pO(0)=/3pO(0)=7p°(0)=10O Tether Parameters p t = 2 x l 0 " 2 k g / m Retrieval 1000m to 10m Dx.p=Dyp=Dzp= 20m m p = 100,000kg m s = 500 kg Orbit Parameters e = 0 , h = 5 0 0 K m Retrieval completed in 0.37 orbit 12.5-Time, orbits Figure 4-19a Thruster-offset hybrid controlled system response during exponen-tial retrieval from 1000 m to 10 m in 0.37 orbit. 143 Initial Condit ions a l ° ( 0 ) = 7 l o ( 0 ) = 1 0 ° « p O ( 0 ) = ^ O ( 0 ) = 7 p ° ( 0 ) = 1 0 O  Tether Parameters p t = 2 x l 0 ~ 2 k g / m Retrieval 1000m to 10m Satellite Parameters D * . P = D y . P = D z . P = 20m m p = 100 ,000kg m s = 100 k g Orbit Parameters e = 0 20 .0-N x* -20.0 E 20.0 10.0-o.o --10.0 1000.0 E 500.0-0.0 h = 5 0 0 K m 0 2 3 Time, orbits 1 LEGEND H /-Hybrid Control Switching at 269m. 1 Jr.. . . J i . . 0.125 - 0.000 -0.125 Figure 4-19b Controller effort time history for the thruster-offset hybrid control. 144 Table 4-3 Controler efort Case Configuration Tension Thruster Ofset Stationkeping Reference Case 169.3 Ns, 480.6 J 87.1Ns, 503.2 J 2.9Ns, 44.1 J PB/PA = 2 174.3 Ns, 502.7 J 9.2 Ns, 525.4 J 26.1Ns, 467.4 J 100 m 190.2 Ns, 518.2 J 16.6Ns, 56.1 J 42.3Ns, 450.3 J Pt = lx 10_1 kg/m 181.4 Ns, 542.1 J 59.2Ns, 560.7 J 56.9Ns, 451.3 J me = 50  kg 190.1 Ns, 546.7 J 163.9Ns, 569.6 J 57.1Ns, 469.1 J Deployment 10 m -f 100 m 0.37 orbit 184.1 Ns, 513.5 J 19.7Ns, 560.8 J 41.4Ns, 450.1 J Retrieval 10 m -> 10 m 1 orbit 1656.9 Ns, 546.5 J 143.3Ns, 573.9 J 85.0Ns, 469.3 J 0.68 orbit 1701.1 Ns, 55.8 J 1510.5Ns, 594.7 J 95.7Ns, 475.6 J 0.37 orbit 1781.9 Ns, 581.4 J 583.4Ns, 609.8 J 55.3 Ns, 490.8 J Retrieval 100 m -» 10 m 0.37orbit6501.1 Ns, 641.1 J 6076.1Ns, 649.2 J 10621.1Ns, 509.3 J* HybridControl Retrieval 100 m -> 10 m 0.37 orbit tension/ofset 3903.5 Ns, 312.9 J 503.7 Ns, 208.4 J 0.37 orbit thruster/ofset 2623.1 Ns, 280.6 J 525.5Ns, 218.3 J * Retrieval in one orbit with the initial tether disturbance reduced to a*(0) = 7*(0) = 1°. 5 . E X P E R I M E N T A L V A L I D A T I O N O F T H E O F F S E T C O N T R O L S T R A T E G Y 5.1 Introduction In Chapter 4, it was seen that a mathematical model of the offset controller performed well for the conditions it was tested in. It proved feasible even though constrained to a maximum offset motion of only 20m in each direction. As men-tioned earlier, these constraints were introduced keeping in mind the present-day spacecraft and available technology in the field of telerobotics. The next logical step is to demonstrate validity of the control strategy through a simple ground-based experiment. This chapter describes such an experiment to stabilize the tether using the digital computer control of the point of attachment. Based on the available literature in the field, it is clear that this is the first time such an experiment has been carried out. The results of the experiment clearly indicate that the control algorithm is feasible and robust. Ever since the launch of the first satellite in 1957, there have been only a few space based experiments in the general area of satellite dynamics and control, and none pertaining to the tether system. As pointed out earlier, in late 1966, two Gem-ini missions (XI and XII) were used to establish tethered satellite systems in space with the Atlas-Agena D spent stage [2]. A rotating cable counterweight configu-ration as well as the gravity gradient stable equilibrium orientation were attained. However, these can hardly be called controlled experiments. Since 1980, a joint U.S.-Japan space project T . P . E . (Tethered Payload Experiment) used a sounding rocket based tether to conduct a series of three tests [32]. The purpose of the ex-periments was to gather technical and scientific data supporting the electrodynamic 146 tethered satellite mission. Several configurations were tested where, in each case, a daughter payload was deployed, tests carried out (lasting only about 5 minutes) and the payload released. The tether, 418 m in length and 0.66 mm diameter, was made of stainless steel and coated with teflon. The maximum deployment rate was 1.48 m/s. The results provided an evidence for plasma discharge. More recently, a N A S A funded project K. I .T .E . (Kinetic Isolation Tethered Experiment, [33]) is in progress at Stanford University where the objective is to control orientation of an instrumented tethered platform (subsatellite) by utilizing tether tension to apply moments through a moveable offset. With this as background, an experiment was planned to simulate essential dy-namics of the tethered satellite system. The objective of the proposed experiment is to test the physical model with a real time implementation of the offset control strat-egy. The success of such an experiment would help validate the control algorithm for actual prototype development. Some of the issues crucial to an experimental project of this nature are as follows: (i) The physical system should portray, as closely as possible, the most significant dynamical effects encountered in space. (ii) It was clear that the tether inplane and out-of-plane swing motions are the most dominant characteristics of the system. Furthermore, as seen in the dynamics of the tethered satellite system (Chapter 3), perturbations of the space platform by the tether motion are quite small justifying the use of a fixed frame to model the space platform for the purpose of testing the control strategy. The experiment consists of a test-rig from which is suspended an object at the end of a nylon cable (representing the tether and payload) in a stable equilibrium position. The point of attachment of the tether could be moved in a horizontal 147 plane by means of electrical motors. The motors respond to commands generated by a digital computer in accordance with the offset control algorithm. Informa-tion regarding the angles that the tether made with two orthogonal planes is fed back electronically to the computer so that the closed loop digital control can be implemented. Figure 5-1 shows the actual test rig with a tether supported payload. The step-motors move the carriage, and hence the tether attachment point, in a horizontal plane fairly accurately through commands from a digital computer. Figure 5-2 shows the electronic hardware assoicated with digital control of the offset acceler-ation; while a close-up view of the carriage with the tether reel and mechanisms associated with sensing of the tether angles are indicated in Fig. 5-3. Note, the carriage also carries a third motor which is used to reel in or out the tether to simulate retrieval and deployment operations, respectively. In this experimental model, the tether and payload are physical systems that behave according to the laws of nature and are characterized by continuous time representations. The computer, however, can only process digital data obtained at finite intervals of time limited by its execution speed. The tether angles must be sampled at intervals of time to convert the continuous physical phenomenon into discrete time information. These results are then transformed electronically into digital data which the computer can manipulate. Thus, at any given instant of time, the model dynamics will generate digital, analog discrete time, and ana-log continuous time signals. Such systems are broadly classified as sampled data systems. In digital control theory, such a system is analyzed by converting its continuous time representation into the corresponding discrete time analog. This facilitates introduction of a step-size into the system dynamics. A suitable step-148 Figure 5 -1 Photograph of the test-rig constructed to validate offset control strategy: (a) aluminum frame; (b) inplane motor; (c) carriage for out-of-plane motion; (d) wooden stand; (e) linear bearings; (f) tethered payload. Figure 5-2 Digital hardware used in the experiment: (a) translator module deployment and retrieval; (b) translator module, offset motions; (c) power supply; (d) function generator. 150 Figure 5-3 Carriage reel and sensor mechanism: (a) potentiometer on mount-ing bracket; (b) moveable aluminum leaf mechanism with slot for tether; (c) retrieval motor and reel housing; (d) tether; (e) inplane traverse with linear bearings; (f) payload. 151 size must then be selected which preserves dynamics of the physical system while providing the computer with a realistic processing time. The first step to this end is to reduce the general mathematical model developed for space platform based tethered satellite system in orbit to the realm of the ground-based laboratory experiment. 5.2 Theoretical Representation of the Ground-based Tether By suitably introducing mathematical constraints corresponding to laboratory conditions, one can reduce the more general equations (Eqs. A . l - A.6) derived earlier. In particular, the following constraints are introduced to arrrive at the laboratory model with offset control during the stationkeeping phase: Dy,P = by>p = by>p = 0; ap = ap = ttp = 0; pp = 0p = 0p = 0 ; 1P = 1P = % = 0 ; Mt,s = 0; A A A L = L = L = 0 ; Dj,p = 0 (j = x,y,z); (5.1) which reduces the tether equations to: a'/ + 3a t = -b'^p ; (5.2a) -ti + 4rtt = D'lP + Dz,P. (5.2b) These equations are not the same as the linearized equations of motion of a pen-dulum suspended from a fixed support on the Earth and free to move in both the 152 planes. One would expect the two equations to be symmetric which is not the case here. This apparent discrepancy can best be understood by looking at the original tether equations before substitution of the gravity gradient terms as functions of 0, the orbital rate. The second term in Eq. (5.2b) is derived from r. A i f . L . - o ' o GM, MteL.-« it For a circular orbit, GM -2 R* Nondimensionalizing with respect to L202 and noting the constraints in Eq. (5.1) reduce the above expression to 4^. For a body in orbit about the Earth, the net centrifugal force, which exactly balances the gravitational force at its center of mass, cannot be neglected. However, for a ground-based system, the centrifugal force due to the Earth's rotation is rel-atively small and hence omitted here. Neglecting the centrifugal force contribution in Eq. (5.2) and substituting 3GM/R* = g/L gives: a'l + | a t = -t)'jip ; (5.3a) and -# + jpt = -f>",p\ (5.36) the classical simple pendulum equations. Note, the last term in Eq. (5.2b) is the redundant term of the expression (1 + ^ Mt)L02bx>p appearing in Eq. (A.5). It represents the moment due to orbital centrifugal force in presence of an out-of-plane offset. Similarly, for the ground-based deployment or retrieval phase the reduced governing equations are: a'l + 2L'a't + | a t = -bz<p - L'D'Z p ; (5.4a) 153 X,p 5 (5.46) with the corresponding control relations as: bXtP = Krft + Kntn + KbxbXtP + K5xpbx x,p • z,p J (5.4d) (5.4c) 5.3 System Description Figure 5-4 is a schematic diagram of the test rig used to experimentally validate the offset control strategy in real time. The test apparatus is made of three main parts: the controller; the actuator; and the sensor. Figure 5-5 is a block diagram of the offset control system showing the relationship between the various subsystems and the plant (tethered system) to be controlled. 5.3.1 Controller The controller was an IBM X T compatible computer running at a clock speed of 8 MHz. At this speed the time for one loop is 79 ms. A typical loop consists of a complete cycle of sensing, calculation of corrective control effort and dispatching of actuation signals to move the tether attachment point. Neglecting the gravity gradient effect and recognizing planar motion of the attachment point, the governing equations for the ground-based system as well as the control algorithm were shown to reduce to Eqs. (5.4a - 5.4d). The control algorithm establishes the required offset (motor) accelerations and then employs a two-step integration process to obtain the corresponding velocities 154 INPLANE MOTOR 5.5 cm dia. SPROCKET OUT OF PLANE MOTOR Step motor drivers accessed through parallel port Potentiometers sensors accessed through game port /TETHERED SATELLITE \ OFFSET CONTROL ALGORITHM PERSONAL COMPUTER BASED CONTROL 7 Figure 5-4 Schematic diagram of the offset control test rig. 155 T R A N S L A T O R S (STEP MOTOR CONTROL) S T E P M O T O R S (ACTUATOR DRIVES OFFSET CONTROL) REFERENCE INPUT = 0 DIGITAL C O M P U T E R C O N T R O L R A T E E S T I M A T I O N DRIFT C O R R E C T I O N S A M P L I N G T IME C O N T R O L P L A N T (PHYSICAL MODEL OF TETHERED SATELLITE) TETHER ANGLES 1 Z E R O ORDER S A M P L E A N D HOLD G A M E B O A R D P O T E N T I O M E T E R S (ANALOG TO DIGITAL CONVERTER) (SENSORS) Figure 5 - 5 Block diagram representation of the ground based offset control experiment. and positions. This information is sent to the actuators (stepper-motors) through the parallel port connection on the computer. Details of the port pin connections are presented in Appendix D. 5.3.2 Actuator The actuator system can be visualized as a very large x,y-plotter where, instead of a pen moving across a plotter bed, a carriage representing the tether attachment point traverses a horizontal plane. The motion of the carriage was controlled by a pair of step-motors thus providing the tether attachment point inplane and out-of-plane freedom of motion. The carriage carried a reel mechanism, driven by a motor, to deploy and retrieve the tethered payload. All the three motors were controlled by commands generated by a digital computer that utilized the strategy as explained above. From the mechanical hardware point of view, 3x2.5x0.25 in. aluminum alloy sections were used to construct the horizontal trapezoidal platform, 2.075x1.812 m, as shown in Figure 5-6. The inplane traverse and carriage were made from a 1/4 in. aluminum plate and supported by a system of linear bearings thus resulting in their relatively smooth motion. The entire structure (with motors) was clamped to a wooden stand at a height of 3.2 m from the ground. The transfer of torque from the motors to the inplane traverse and the carriage was through a system of pulleys and polythene coated steel chains. A comment concerning the step-motor drive would be appropriate here. Step-motors are electromagnetic incremental drives which respond in fixed angular steps to input pulses (or digital commands). They are particularly useful in situations where digital drives are required in an open-loop configuration [34,35]. In the case 157 1.743 m ALUMINUM FRAME BEARING SUPPORT OFFSET MOTION ENVELOPE (1.64 mx 1.57 m ) INPLANE TRAVERSE WELDED SUPPORT \SSSV>»SS\S\SSS\NSSSSSSSS\\ 1.812 m STEEL LINEAR BEARINGS (ALL HATCHED AREAS) 0.17m | 0.25 m OUT OF PLANE TRAVERSE (CARRIAGE) 2.075 m Figure 5-6 The offset positioning mechanism showing the trapezoidal frame and linear bearings that allow the carriage to be located anywhere within the motion envelope. 158 of the offset controller, the use of step-motors is not really necessary as acceleration is the specified quantity. However, step-motors considerably reduced the sensor requirements of the controller because of their open-loop digital character. This meant that the displacement of the step-motor was always directly related to the input signal (assuming no steps were missed) and position feedback of the carriage was not required. As mentioned before, three step motors were utilized in the ex-periment. Two were used for the offset positioning (inplane and out-of-plane) while the third was for the tether deployment and retrieval. All the three-step motors were of the permanent magnet type. The position accuarcy was within 3 to 5% of the desired value. Details of the motors and translator modules are given in Appendix E . 5.3.3 Sensors The role of sensors is to provide information concerning the state of the system. Thus choice of sensors is an integral part of the control system design. There are several options available: shaft encoders; strain gauges; variable inductance transducers; piezoelectric transducers; potentiometers; etc. Potentiometers offered several positive features: (i) high voltage low impedance output signals requiring no amplification; (ii) impedance can be varied simply by changing the resistance and supply voltage; (iii) favourable vibration disturbance rejection characteristics, an important aspect for the present experiment; (iv) recently developed resistive film potentiometers can provide infinitesimal reso-lutions limited only by the signal to noise ratio and mechanical properties; (v) relatively inexpensive. 159 In the present case, a pair of potentiometers, made of high resistance con-ductive plastic with resistance proportional to length, served as sensors. Further technical information related to these potentiometers may be found in Appendix F. As the length is varied mechanically according to the displacement, the voltage varies continuously and proportionally. The potentiometer is an analog displace-ment transducer. This means that the signal has to be sampled at a suitable rate and converted to a digital signal using an analog-to-digital converter. In conjunction with the personal computer, this requirement is most conveniently met by utilizing a game-board. The game-board stores a digital signal in the form of an 8 bit word. This means that the largest decimal value is given by 2 8 — 1 = 255. The digital values range from 0 to 256 which corresponds to a change in resistance of 100 kJ7. The potentiometers had a maximum resistance of 500 kfl over 265° giving effective resolution of the sensor as — x 100 x — = 0.74° . 500 256 It should be noted that the sensitivity can be improved quite readily if desired by changing the maximum potentiometer resistance and/or the supply voltage. A n important step in the sensor development was the design of a suitable mech-anism, to move the potentiometers as a joystick would, to detect the tether's swing-ing motion. The mechanism had to perform without interfering with the deploy-ment/retrieval maneuvers and yet maintain the unavoidable friction at a very small level. The design consisted of a pair of light, slotted aluminum metal leaves as shown in Fig. 5-7. The metal strips are semicircular hence the moment arm is uniform for all tether angles. The tether passed through slots in the two strips per-mitting detection of the swinging motion in either plane, even during deployment 160 POTENTIOMETER | MOUNTING BRACKET ALUMINUM MOUNTING PLATE INPLANE AXIS 500 KQ LINEAR POTENTIOMETERS 0.3125 cm dia. HOLE POTENTIOMETER SHAFT ALUMINUM LEAF MECHANISM TO TRANSFER TETHER SWING MOTIONS TO POTENTIOMETER SHAFT MOTIONS 0.25 m Figure 5-6 Tether attitude motion sensor with potentiometers as transducers. and retrieval. 5.4 Sampled Data and Finite Difference Discretization As pointed out before, Eq. (5.4) represents the continuous time model for the system. However, the experimental test facility employs a digital computer con-trolled sampled-data system. In order to determine the optimal Ricatti gains, it is necesssary to model the system in discrete time. Since the equations for the ground-based model are symmetric, the finite difference conversion is identical for the two degrees of freedom and hence only the inplane equation discretization procedure is discussed here. Let: t0 = initial time ; At = time step ; ti = t 0 + At; t_i = t0 — At. Furthermore, let subscripts o, 1, -1 refer to values of variables at time t0, t 1 } and t _ i , respectively. If A i is chosen sufficiently small (choice of A t is discussed in the following section) the following relations can be defined: Similarly, dat ~dt d2at dt2 t=tc = at t = tc = ott t=u atl - at_t ti - t-i 2At (5.5) at t=to+At - at t=to- At At 162 _ A t A t A t 0^-20^ + a t . ! " (Alp ' ( 5 - 6 ) On substituting these relations in Eq. (5.4a), it is easy to solve for at1 as, cttl = 2a t o - a * , ! - y{At)2ato - bz,p{At)2 . (5.7) Li Moreover, the controlled offset accelerations presented in Eqs. (5.4c) and (5.4d) must be discretized to correspond to the digital computer output. Substituting from Eq. (5.7) into Eq. (5.4c) gives: bz>Po = KM (a'12A7"l) + Katato + K h z b Z t P + K3zpbz,p . (5.8) One can define derivatives of DZtP in terms of discrete time values as before: D*,PO f ^ p . (5.10) Using Eqs. (5.9) and (5.10) to eliminate DZ)P from Eq. (5.8) leads to expression for finite difference time-stepping as, _ 1 {2 + K£>z(At)2}bZ)t * ' P 1 (1 - K3At/2) Vx" 1 " J > , l P i — J ~ z > * > ° -{l + KbzAtl2)bz>p_x + KatAt/2{Dz,Pl - ^ , p _ x ) + Kat{At)2DZjPo\ . (5.11) In order to initiate the finite difference procedure, the initial conditions have to be set-up in a suitable form. From Eqs. (5.5) and (5.6), one can express atl in 163 terms of a t c a n c * derivatives as atl = i ( A t ) 2 a t o + atoAt + ato . (5.12) From Eq. (5.8), at t = tot &t„ + | < * t o = -bZtPo . (5.13) Substituting for &t0 from Eq. (5.13) into Eq. (5.12) gives the expression for the first time-step in terms of the initial conditions, a h = \{At)2{-bz<Po - j2-ato) + a t o A t + <*u . (5.14) * *re/ Similarly, expressing DZfPl in terms of Dz>Po and its derivatives using Eqs. (5.9) and (5.10) leads to an expression for the first time-step as Dz,Pl = \{At)2bZtPo + (At)bZ)Po + Dz>Po , (5.15) or in terms of initial conditions, Dz,Pl = \{At)2 [K^dcto + Katato + p£* ) P o + K ^ D ^ ] + (At)bZiPo+DZtPo. (5.16) 5.5 Selection of Sample Step-Size A choice of the sampling rate is crucial to the performance of a computer con-trolled data system. At has to be sufficiently large so that the control algorithm processing and data transfer can be completed during the time for each control-step. At the same time, A t should be small enough to meet the system dynamics (response speed) requirements. 164 5.5.1 Process dynamics criterion Shannon's sampling theorem [36] states that the maximum meaningful fre-quency of a sampled data system is the Nyquist frequency fc which is given by half the sampling rate, fe = 1/2At. In order to control the process, the frequency constant of the sensor signal should stay within the Nyquist frequency. From the numerical simulation, one can obtain the highest frequency of the process for the shortest tether length of 0.75 m as: Shortest Process Period = 1.7373 s; Highest Process Frequency = 0.5756 Hz. According to Shannon's theorem, fc should be greater than 0.5756 Hz, i.e., the sam-pling period At required to represent the tether swing motion accurately should be less than 0.869 s. 5.5.2 Control algorithm criterion The control algorithm consists of two parts: determination of the required offset acceleration based on optimally obtained feedback gains; and modulation of the accelerations using feedforward of the offset states. The first step involves simple multiplications while the latter utilizes finite difference integration of the offset accelerations. Obviously, accuracy of the control strategy will be affected if the finite difference integrations do not yield accurate estimate of the actual offset states. The accuracy as well as convergence of the numerical integration scheme is influenced by the step-size which, in turn, is governed by the real time implementation of one algorithm loop. Levy and Kroll [37] have obtained the criterion for choosing the 165 finite difference step-size. For convergent stable solutions, it requires T At< - , where T= most significant period of the system. In practice, however, to adequately follow the response and retain good accuracy, Levy and Kroll suggested At < ZL. Using the value of the shortest process period indicated before, this gives At < 86.87 ms. The criterion suggests that the control algorithm imposes a severe constraint on the step-size. In other words, the time of execution of one control loop should be minimized. To achieve this, a number of measures had to be taken during the software development for the control strategy: • The program was written in C language which yields the fastest executable code among compiled languages for a personal computer. • The entire C program was called as a subroutine from the BASIC main pro-gram to allow utilization of powerful built-in functions (in BASIC) that would otherwise have to be programmed in the assembler code separately. • Once the software development was complete, the compiled C code was opti-mized to increase the execution speed. • The information on constants, loop time, potentiometer calibration and initial carriage position was introduced before the control loop was entered. • Al l data collected regarding tether angles and carriage positions were stored in real time in arrays during controlled operation. The data was then dumped into data files on magnetic media when the control was complete. With these programming features in place, the control algorithm loop time was 166 measured to be 79 ms on an I B M X T compatible computer running at a clock speed of 8 MHz. This value is within the control algorithm criterion constraint of 86.87 ms. 5.6 Software It would be useful to make a brief comment pertaining to the control computer code. The entire software was developed as a C language subroutine of a B A -SIC language main programme. This allowed the use of built-in BASIC functions like STICK (which accesses the game port) without having to reprogram them in assembler code. The C language has several advantages: • It has a versatile programming environment which combines convenience of a higher-level language like F O R T R A N with very close control of the memory, as in the case of the assembly language. • It can generate a very fast code especially when the optimization option of the compiler is used. • It provides superior control of peripherals and hardware through the computer. • It is highly portable, i.e., it can be easily adapted to different machines. Tech-nical information on adapting C language to I B M X T compatible machines is available[38,39,40]. Details of the controller algorithm are depicted as a flowchart in Fig. 5-8. The program was written to be completely interactive and assumes no technical back-ground on the part of the user. It essentially has two phases of operation: (a) ini-tialize; and (b) control. During the initialize phase, user-computer interaction is maximum while computer-peripheral interaction is minimum. In the second phase of control, the opposite holds true. During phase (a), the speed and time of response 167 INITIALIZATION Read In Optimal Feedback/Feed Forward Gains Set Timing For Loop Set Potentiometer Zero Values Enter Deployment/Retrieval Step Size Read In Carriage Position LOOPING CONTROL Accept Retrieval/Deployment Commands From Keyboard SENSE TETHER STATE Read Tether Angle Values From Potentiometers Calculate Angular Velocities IMPLEMENT CONTROL STRATEGY Obtain Offset Accelerations in Inplane and Out of Plane Directions Integrate To Obtain Offset Velocities And Positions I COMMAND ACTUATORS Send Pulses To Motors While Checking For Run Off Correct For Drift in Potentiometer Values Figure 5 - 8 Flowchart for the offset acceleration based control algorithm show-ing initialziation and control phases. 168 are not crucial; while in phase (b) these factors are critical. To begin with, phase (a) is entered. It prompts the user for the name of the file containing the optimal gains for tether state feedback and offset state feedforward. Next, the routine establishes the loop time At. This value is used for all numerical differentiations and integrations that occur as part of the offset control strategy. Due to dead-zone effects, the sensors may not be exactly at their zero positions. The readings from the sensors (potentiometers) during phase (a) are taken as their zero values. The routine then prompts the user for a deployment/retrieval step-size. This would determine the speed of deployment/retrieval during the control phase. The initial carriage position is taken as the zero offset point. Finally, the program prompts the user for an output file name if the data is to be stored. This completes phase (a). After initialization, the control phase is entered. At the beginning of each loop, the routine checks for keyboard input to decide if deployment, retrieval or station-keeping mode of operation is to be performed. Next, the tether state is sensed by accessing the game board* which, in turn, is interfaced with the sensors. The sen-sors produce an analog signal which is digitally encoded at the game-board. These values are converted into radians by applying experimentally determined calibration constants. The corresponding angular rates (a* and it) are then estimated by direct differentiation of angle values (at and 7*). Once the tether state is established, the control strategy of Eqs. (5.4c) and (5.4d) is applied giving the offset accelerations required to stabilize the disturbed tether. The offset positions are simply the alge-braic sum of pulses that have been sent from the computer to the actuators at any given instant. These accelerations are integrated twice using the finite difference * A computer board that transforms analog signals at the game port into a digi-tally encoded signal. 169 approach to obtain the corresponding offset positions and velocities, i.e., the offset states in pulses and pulses per second, respectively. The number of pulses required are then converted into bit patterns (strings of O's and l's) and sent as actuator commands from the parallel port (on the back of the computer). The offset state is also fed forward to be used in the next control-step. While this is in progress, the routine performs a number of error checks to assess if the system is behaving as planned. It determines if: (i) the pulses supplied to the motors would drive the carriage off the test rig (this never occured in practice because of the self-centering nature of the control law); (ii) the sensors were drifting; (iii) an emergency controller exit had been actuated for safety reasons. Finally, the sensor and actuator readings are stored on the computer hard disk providing a permanent record of the experiment. This procedure in the control phase is repeated until the tethered payload is brought to rest. 5.7 Results and Discussion The experimental validation of the offset control strategy provided considerable insight into the actual performance and ultimate feasibility of the, approach. The concept of controlling the motion of a tethered object by specification of the ac-celeration of the point of attachment, with feedback of the state and feedforward of the offset, proved to be viable. As is usually the case in such circumstances, a large amount of response results were collected through a series of carefully planned experiments. Only a small sample of response related data is recorded here. Before turning to the results it may be pointed out that: (i) Damping effects cannot be eliminated completely from a real system. 170 The sources of energy dissipation are: aerodynamic drag; material damp-ing in the tether, frame and carriage; and Coulomb friction in the linear bearings and potentiometers. (ii) Electronic noise is always present in a real system due to interference from the environment and power fluctuations. (iii) The initial conditions cannot be matched exactly since, during the ex-periment, the tether was released by hand. (iv) Nonlinearities in the real system may modify the response behaviour from that predicted by the mathematical model. The major nonlinear contribution is due to backlash in the sensor. A slight clearance had to be provided for the tether to pass through slots in the leaf mechanism (Fig. 5.6). This introduced a small dead-zone as well as backlash error in the region where the tether can swing without the leaf mechanism moving. However, the emphasis here is not on correlation between numerical and exper-imental data but on assessing validity of the offset control strategy. The system was purposely subjected to a very large disturbance of 15° (at and/or it) to assess the controller's effectiveness under a demanding situation. In practice, an external excitation would seldom result in motions larger than a few degrees. The initial condition, though large, is within the sensor's dynamic range of 26 .5° . The plots show time histories of the tether response as well as inplane (DZ,P) a n d out-of-plane (DXIP) motions of the carriage. As a first step in understanding the system behavior, it was subjected to a purely inplane disturbance of at(0) = —15°. Figure 5-9 compares numerical and experi-mental response results during the stationkeeping phase. The uncontrolled response 171 I n i t i a l C o n d i t i o n s Svstem Parameters m t = 8 x l 0 " 4 k g m s = 0.11 k g L(0) = 2 . 2 5 m a t °(0) =-15°, 7 t ° ( 0 ) = 0 ° t = 0 LEGEND al theory 0( t expt. OH N 0.2 0.0 0.2 0 10 C O N T R O L L E D 20 30 Time, s LEGEND o , a t theory CKf expt. ;A A A / 1 LEGEND A D z p theory A D z ( p . . . . e x P t -40 50 Figure 5-9 Plots showing comparison between numerical and experimental re-sponse results for uncontrolled and controlled conditions of the sys-tem during the stationkeeping phase. The system is subjected to a large inplane disturbance of at(0) = —15°. 172 is fairly well predicted by the numerical simulations, however, frequency attenua-tion observed during the experiment is slightly lower. This may be attributed to nonlinearities of the real system and small differences in the initial condition as pointed out before. Comparison of results during the control phase continues to show good correlation. Note, the offset motion is confined to the plane of the dis-turbance because of the uncoupled character of the system, and is less than ± 2 0 cm even for the large initial disturbance of 0^ (0) = —15°. The fact that such a severe disturbance is damped in less than 17 s is indeed encouraging. Figure 5-10 shows response of the system during the stationkeeping phase but now subjected to an out-of-plane disturbance of the same magnitude as before. The general character of both the uncontrolled and controlled motions are simi-lar to those observed in Fig. 5-9, except for some minor differences. The higher rate of amplitude reduction suggests larger frictional damping in the out-of-plane direction. Of interest is a slight inplane oscillation during the uncontrolled and controlled phases. It was difficult to provide pure out-of-plane initial condition due to air drafts and other environmental disturbances. Although the disturbance is essentially damped in around 17 s as before, a small amplitude high frequency in-plane motion persists perhaps due to noise in the sensors. Motion of the point of attachment of the tether shows essentially the same trend as before. Figures 5-11, 5-12, and 5-13 present response of the system, during stationkeep-ing, for three different tether lengths (2.25 m, 1.5 m, and 0.75 m). Motion of the point of attachment is also presented in each case using two distinct formats. The initial displacement of 15° was applied in both the planes and controlled as well as uncontrolled modes examined. At the outset it is apparent that the frequency of tether swing in both inplane and out-of-plane directions are exactly the same, 173 I n i t i a l C o n d i t i o n s Svstem Parameters m t = 8 x 1 0 - 4 kg m s = 0.11 k g 1(0) = 2 . 2 5 m a t°(0) = 0° , 7 t ° ( 0 ) =-15° 1 = 0 20.0 0.0 -20.0 0.25 C O N T R O L L E D 20 30 Time, s LEGEND o o 7t Figure 5-10 Typical response data for the system, in the stationkeeping phase, subjected to an out-of-plane disturbance of 7*(0) = - 1 5 ° . 174 however, damping in the out-of-plane direction is higher as indicated by the larger decrement in amplitude per cycle. The controlled response presented here indicates that the reduced mathematical model, which resulted in a set of uncoupled differen-tial equations, is accurate. Furthermore, it clearly demonstrates that the principle of superposition is applicable to the system and therefore a linearized analysis can be justified. Note the controller is able to damp the motion in around 20 s (Fig. 5-1 la) with the peak offset displacement less than 0.2 m (Fig. 5-llb). As can be expected, effectiveness of the control improves at the shorter lengths (Fig. 5-12, 5-13) and the same disturbance is damped in less than 5 s for the shortest tether of 0.75 m tested. Note, an important feature of the control strategy is its self-centering nature. The final position of the carriage is always close to the origin. A small error (1 — 5%) in the terminal condition is attributed to the sensor backlash. Thus the offset con-trol strategy appears to be ideally suited to relatively short tethered systems where the performance in terms of damping time, offset motion, and positioning error is particularly attractive as indicated below: Table 5 -1 Variation of controller performance with tether length. Tether Damping Maximum Offset Motion, m Final Positioning Length, m Time, s Inplane Out-of-plane Error, % 2.25 17 0.15 0.2 5 1.5 14 0.15 0.17 1 0.75 5 0.12 0.05 0.05 Figure 5-14 shows response of the system during deployment, essentially a stable maneuver unless the deployment rate is beyond the critical value. Here the tether payload is deployed from 0.75 m to 2.25 m in 9 s, with the initial disturbance of 15° in each plane as before. The damped motion, even during the uncontrolled phase 175 I n i t i a l C o n d i t i o n s Svstem Parameters m t = 8 x l 0 " 4 k g m s = 0.11 k g L(0) = 2 . 2 5 m a t°(0) = 15°, 7 t ° ( 0 ) = 15° • L = 0 Figure 5 - l l a Experimental data for the tether stationkeeping at 2.25m with a large initial disturbance of 15° in both the inplane as well as the out-of-plane directions. 176 Initial Conditions Svstem Parameters m t = 8x l0~ 4 kg m s = 0.11 kg 1(0) = 2.25m a t °(0) = 15°, 7 t ° ( 0 ) = 15° 1 = 0 B i 0 . 3 n x" 0.2-0.1-~ i 1 1 0.1 0.2 0.3 A D o.i z,p,m —0.2 Figure 5 - l l b Carriage position during controlled response of the system, in the stationkeeping mode, subjected to a large initial disturbance of 15° in both the planes. 177 Initial Conditions System Parameters m t = 8xl0~ 4 kg m s = 0.11 kg 1(0) = 1.5m a t°(0) = 15°, 7 t ° ( 0 ) = 15° 1 = 0 20.0 -20.0 25.0 -25.0 0.2 o.o H 0.2 U N C O N T R O L L E D wvwwv LEGEND o a t 1 1 1 . C O N T R O L L E D LEGEND o a t o LEGEND A D x , p A 0 10 15 Time, s 20 25 Figure 5-12a Experimentally obtained response for the 1.5 m tethered system in the stationkeeping mode. Note, the system is subjected to the same disturbance as in Fig. 5-11. 178 Initial Conditions System Parameters m t = 8 x l 0 " 4 k g m s = 0.11 kg 1(0) = 1.5m ? l ° ( 0 ) = 15°, 7 t ° ( 0 ) = 15° 1 = 0 z,p' Figure 5-12b Offset trajectory during the controlled response of the 1.5 m teth-ered subjected to an initial disturbance of at(0) = 7t(u) = 15°. 179 Initial Conditions System Parameters m t = 8xl0~ 4 kg m s = 0.11 kg L(0) = 0.75m a t°(0) = 15°, 7 t ° ( 0 ) = 15° L = 0 25.0 -25.0 x" 0.2-i 0.0--0.2 0 6 8 Time, s 10 C O N T R O L L E D LEGEND o " t 1 1 1 1 1 1 LEGEND A A .....D,,R.... 12 14 Figure 5-13a Response of the tethered system to an initial displacement of 15° in both the planes. The tether was stationkeeping at 0.75m. 180 Initial Conditions System Parameters m t = 8 x l 0 " 4 k g m s = 0.11 kg 1(0) = 0.75m ? l ° ( 0 ) = 15°, 7 t ° ( 0 ) = 15° 1 = 0 0.10 - i x -0.15 J Figure 5-13b Carriage trajectory during the controlled stationkeeping response of the 0.75 m tether. A n initial displacement of 15° was applied in both the planes. 181 Initial Conditions System Parameters m t = 8xl0~ 4 kg m s = 0.11 kg L(0) = 0.75m a t°(0) = 15°, 7 t ° ( 0 ) = 15° t = 0.17m/s 25.0 0.0-•25.0 0.2 0 . 0 -S C O N T R O L L E D LEGEND o "t o 1 1 1 LEGEND A D * , P A .....D.?,P.... 1 I I Time, s Figure 5-14 System response during uniform deployment from 0.75 m to 2.25 m in 9 s. The initial disturbance of 15° was applied in both the inplane and out-of-plane directions. 182 suggests stability due to increased inertia. With the offset control activated, the disturbance is damped continuously during deployment and the motion in both the degrees of freedom is arrested almost completely by the end of deployment. Of course, retrieval would represent the most demanding condition for assessing the controller's effectiveness as now the system is inherently unstable. A higher retrieval rate would only accentuate the instability and test the controller's perfor-mance under an extremely difficult situation. To that end, retrieval maneuvers were tested for two different rates of 0.08 m/s and 0.17 m/s. In each case, the tether length was reduced uniformly from 2.25 m to 0.75 m. At this point, the retrieval motor was stopped and the stationkeeping control activated. Figure 5-15a shows uncontrolled as well as offset controlled response of the sys-tem during the slower retrieval rate (0.08 m/s) with a large initial disturbance in both the planes as before. As can be expected, with the tether length becoming smaller, inertia of the system is lowered causing tether oscillations to grows to con-serve angular momentum. During the uncontrolled operation, the system response grows to a maximum amplitude of 38° as the retrieval ends. Subsequently, the fric-tional damping causes the amplitudes to drop in both the planes. With the offset controller in operation, the response shows a damped motion throughout. Thus the controller is able to overcome the destabilizing effect of the retrieval quite effectively. Note, the tether librations are completely damped in around 3 s after completion of the retrieval. The controller performance continues to be remarkable even at a rapid retrieval rate (Fig. 5-15b) 183 Initial Conditions System Parameters m t = 8xl0~ 4 kg m s = 0.11 kg 1(0) = 2.25m a t°(0) = 15°, 7 t ° ( 0 ) = 15° • L = - O . O B m / s -50.0 20.0 0.0--20.0 0.2 0.0-B 0 10 U N C O N T R O L L E D LEGEND o 7t u C O N T R O L L E D LEGEND o 7t i I I LEGEND A A ....P.z .R.. . . i i i 20 30 Time, s 40 Figure 5-15 Experimentally obtained response plots showing effectivenss of the offset control strategy for two different retrieval rates: (a) L = -0.08 m s. 184 Initial Conditions System Parameters m t = 8xl0~ 4 kg m s = 0.11 kg 1(0) . = 2.25m a t°(0) = 15°, 7 t ° ( 0 ) = 15° I = -0 .17m/s -50.0 20.0 o.o--20.0 0.2 0.0-a x 0.2 3.0 C O N T R O L L E D LEGEND o o 7t 1 i i LEGEND A D x , P _ 1 r 1 r 1 * \ V A ....?.Z|R.... 1 i i 20 30 Time, s Figure 5-15 Experimentally obtained response plots showing effectivenss of the offset control strategy for two different retrieval lengths: (b) L = -0.17 m s. 185 6. C O N C L U D I N G R E M A R K S Using a relatively general dynamical formulation, the motion of an orbiting platform supported tethered subsatellite system was studied. Based on the analysis, the following comments can be made: (i) Even the rigid body system dynamics is governed by extremely lengthy, nonlinear, nonautonomous and coupled equations of motion. Simplifica-tion introduced by linearization about a quasi-static equilibrium point leaves the situation formidable but tractable. (ii) Rigid body rotations of the platform and tether are strongly coupled. A A A As shown in Chapter 2, each of the offets DXtP, Dy>p, and DZtP, had a unique coupling influence between the space platform and tether motions. A Of particular importance is the out-of-plane offset DXtP, that couples the inplane and out-of-plane degrees of freedom. In essence, the tether behaves as a "dynamical energy transmission line" between the space platform and subsatellite. It provides unique constraints on the motions of the two end bodies. (iii) Retrieval of a tether is dynamically an unstable maneuver. In the pres-ence of offsets, it could lead to undesirable oscillations and eventual insta-bility of not only of the tether but also the space platform itself. Hence, consideration of the offset effects on the system dynamics is essential in any realistic study. Control of the space platform based tethered satellite system was achieved through three control laws involving tension modulation, thruster mounted at the end of the tether (subsatellite end), and motion of the tether attachment point at 186 the space platform end (Chapter 4). It was shown that a linear tension control law, which takes advantage of the coupling introduced by the offset, is sufficient to damp the tether motion. All the three control strategies were shown to be effective even in the presence of a large disturbance. Based on the simulation response results, following conclusions can be made: (i) For short tethers (100 m), the energy demanded by the tension control strategy is the highest while the offset control presents a possibility of considerable saving in fuel. Thruster control provides the best damping characteristics in terms of quick return to the equilibrium state. (ii) For a longer tether (1 km), the thruster and tension controller demands are comparable. However, the offset controller still remains energy effi-cient though its effectiveness is reduced by the 20 m constraint on the point of attachment motion. (iii) During retrieval of a long tether, in the presence of relatively severe disturbances and the aforementioned constraint, the damping time with the offset controller becomes quite large. In fact at critical combinations of the tether length, retrieval rate and external disturbances, the offset control becomes ineffective. On the other hand, the thruster and tension control strategies continue to perform relatively well. This fact led to the development of hybrid control strategies involving thrusters or tension modulations at the onset of the retrieval followed by the offset control when the subsatellite is close to the platform. Results indicate that the hybrid control strategies are quite promising both in terms of response dynamics and energy demand. (iv) The ground based experimental validation of the offset control strategy 187 provided considerable insight into the actual performance and ultimate feasibility of the approach. The concept of controlling the motion of a tethered object by specification of the acceleration at the point of attachment, with feedback of the system state and feedforward of the offset is shown to be viable. In terms of simplicity and ease of implementation, the tension controller has obvious advantages. The reel mechanism is a part of the tethered satellite system irrespective of the control procedure selected. Adding a tension controller to such a system would be relatively simple, however, its main drawback is the possible problem of tether slackness at shorter lengths due to reduced equilibrium tension. The offset controller involves more moving parts, however, its ability to isolate the space platform and tether dynamics makes it attractive for tether supported sensitive microgravity based payloads. Also, it is ideally suited for supporting a docking platform, for the Shuttle serving the Space Station. Thruster control represents a relatively reliable and effective control mechanism. However, it demands extra fuel onboard. Furthermore, there are safety related issues associated with firing of thrusters close to the space platform. With reference to the proposed space station Freedom, all the participating agencies have shown interest in tethered supported facilities aimed at a variety of missions. Recognizing highly flexible character of the station, the problem of dy-namics and control of the space station based tethered facility attains the challenge of a higher order than that encountered or studied so far. Several researchers have initiated investigations in that area, however, eventually one will have to make the models more sophisticated to account for station, tether and payload flexibility. Influence of free molecular reaction forces, solar radiation induced heating and elec-188 tromagnetic forces for conducting tethers will have to be incorporated realistically. This class of problems have remained virtually untouched so far. A new class of problems involving tether supported three, four or more bodies is also receiving some attention lately. The objectives are quite varied ranging from simultaneous sounding of the environment at several altitudes to communication antenna, payload transfer and microgravity control. This represents a fertile field of considerable practical significance and promise. Orbital transfer using a tether depends strongly on the attitude dynamics of the tethered system. A parametric study aimed at interactions between the orbital dynamics and swing motions to optimize the transfer would lay a sound foundation for further evolution of that subject. Our present knowledge in the area consists of isolated studies with specific idealized configurations of limited value. Eventually on-orbit demonstration of the control algorithms would have to be planned so as to check their validity and to modify them appropriately. The pro-posed TSS-1 mission is a welcome step in that direction. However, there is con-siderable scope for ground based and smaller space based experiments focused at specific aspects pertaining to material properties, damping, deployment mechanism, sensor-actuator design and operation, reduction of observed information and state estimation, control strategies, etc. Ground based and inflight verification of existing dynamical models and control schemes is fundamental to the success of a variety of visionary concepts involving application of tethers in space. 189 R E F E R E N C E S [1] Tsiolkovsky, K . E . , "Speculations between Earth and Sky," izd-vo AN-SSSR, Sci-ence Fiction Works, Moscow, 1859, p. 35 (reprinted 1959). [2] Lang, D.L . , and Nolting, R.K. , "Operations with Tethered Space Vehicles," NASA SP-138, Gemini Summary Conference, February, 1967. [3] Von Tiesenhausen, G . et. al. "The Roles of Tethers On Space Stations", NASA TM-86519, October, 1985. [4] Misra, A . K . , and Modi, V . J . , "Dynamics and Control of Tether Connected Two-Body Systems - A Brief Review," Invited Address, SSrd Congress of the In-ternational Astronautical Federation, Paris, France, September 1982, Paper No. IAF-82-316; also Space 2000, Editor L . G . Napolitano, A I A A Publisher, 1983, pp. 473-514. [5] Misra, A . K . , and Modi, V . J . , " A Survey on the Dynamics and Control of Teth-ered Satellite Systems," NAS A/AIAA/PSN International Conference on Tethers, Arlington, V A , September 1986, Paper No. AAS-86-246; also Advances in the Astronautical Sciences, Editors: P . M . Bainum et al., American Astronautical Society, Vol. 62, pp. 667-719. [6] Rupp, C . C , " A Tether Tension Control Law for Tethered Subsatellites Deployed Along Local Vertical," NASA T M X-64963, September 1975. [7] Baker, P.W., et al, "Tethered Subsatellite Study," NASA T M X73314, March 1976. [8] Kulla, P., "Dynamics of Tethered Satellites," Proceedings of the Symposium on Dynamics and Control of Non-Rigid Spacecraft, Frascati, Italy, May 1976, pp. 349-354. 190 [9] Kalaghan, P.N., et al., "Study of the Dynamics of a Tethered Satellite System (Skyhook)," Final Report, Contract NAS8-32199, Smithsonian Institution, As-trophysical Observatory, Cambridge, Massachusetts, March 1978. [10] Modi, V . J . , and Misra, A . K . , "On the Deployment Dynamics of Tether Con-nected Two-Body Systems," Acta Astronautica, Vol. 6, No. 9, 1979, pp. 1183-1197. [11] Misra, A . K . , and Modi, V . J . , " A General Dynamical Model for the Space Shuttle Based Tethered Subsatellite System," Advances in the Astronautica! Sciences, American Astronautical Society, Vol. 40, Part II, 1979, pp. 537-557. [12] Bainum, P . M . , and Kumar V . K . , "Optimal Control of the Shuttle-Tethered Sys-tem," Acta Astronautica, Vol. 7, No. 12, 1980, pp. 1333-1348. [13] X u , D . M . , Misra A . K . and Modi, V . J . , "Three Dimensional Control of the Shut-tle Supported Tethered Satellite Systems During Deployment and Retrieval," Proceedings of the 3rd VPI&SU/AIAA Symposium on Dynamics and Control of Large Flexible Spacecraft, Blacksburg, Virginia, Editor: L. Meirovitch, June 1981, pp. 453-469. [14] X u , D . M . , Misra A . K . , and Modi, V . J . , "On Vibration Control of Tethered Satel-lite Systems," NASA/JPL Workshop on Applications of Distributed Systems The-ory to the Control of Large Space Structures, Pasadena, California, July 1982., pp. 317-327. [15] Modi, V . J . , Gung, C.F . , and Misra, A . K . , "Effect of Damping on the Control Dynamics of the Space Shuttle Based Tethered System," The Journal of the Astronautical Sciences, Vol. 21, No. 1, 1983, pp. 135-149. [16] Fan R. and Bainum, P .M. , "The Dynamics and Control of a Space Platform Con-nected to a Tethered Subsatellite," NASA/AIAA/PSN International Conference 191 on Tethers, Arlington, V A , September 1986. [17] Moran, J.P., 'Effects of Planar Librations on the Orbital Motio of a Dumbbell Satellite,'' ARS Journal, Vol. 31, No. 8, August 1961, pp. 1089-1096. [18] Yu, E.Y. , "Long-term Coupling Effects Between the Librational and Orbital Motions of a Satellite," AIAA Journal, Vol. 2, No. 3, March 1964, pp. 553-555. [19] Misra, A . K . , and Modi, V . J . , "The Influence of Satellite Flexibility on Orbital Motion," Celestial Mechanics, Vol. 17, 1978, pp. 145-165. [20] X u , D . M . , "Dynamics and Control of the Shuttle Supported Tethered Subsatellite Systems," Ph.D. dissertation, Department of Mechanical Engineering, McGill University, Canada, 1984. [21] Nayfeh, A . H . , and Mook, D . T . , Nonlinear Oscillation, John Wiley and Sons, New York, 1979, pp. 485-500. [22] Gorman, D.J . , "Free Vibration Analysis of Rectangular Plates," Elsevier North Holland, Inc., New York, 1982, pp. 147-161. [23] Nowacki, W., "Dynamics of Elastic Systems," Chapman & Hall Ltd., 1963, Lon-don, pp. 198-259. [24] Penzo, P.A., and Ammann, P.W. (Editors), "Tethers in Space Handbook," N A S A Office of Space Flight, 2nd Ed. , May 1989. [25] Friedland, B., Control System Design - An Introduction to State Space Methods, McGraw-Hill Series in Electrical Engineering, 1986, pp. 337-350. [26] Kuo, B .C . , Automatic Control Systems, 3rd Edition, Prentice Hall, Inc., 1975, pp. 599-611. [27] Franklin, G.F. , Powell, J .D. , and Emami-Naeini, A . , Feedback Control of Dy-namic Systems, Addison-Wesley Publishing Co., Inc., 1986, pp. 324-327. 192 [28] Bierman, J . G . , "Factorization Methods for Discrete Sequential Estimation," Math-ematics in Science and Engineering, Academic Press, New York, 1977, Vol. 128, pp. 62-€3. [29] Modi, V . J . , Gung, C.F . , Misra, A . K . , and Xu, D . M . , "On the Control of the Space Shuttle Based Tethered Systems," 82nd Congress of the International As-tronautical Federation, Rome, Italy, September 1981, Paper No. IAF-81-316. [30] Banerjee, A . K . , and Kane, T.R. , "Tethered Satellite Retrieval with Thruster Augmented Control," Journal of Guidance, Control, and Dynamics, Vol. 7, No. 1, 1984, pp. 45-50. [31] X u , D . M . , Misra, A . K . , and Modi, V . J . , "Thruster Augmented Active Control of a Tethered Satellite During its Retrieval," Journal of Guidance, Control, and Dynamics, Vol. 9, No. 6, November-December 1986, pp. 663-672. [32] Sasaki, S., et al., "Results from a Series of Tethered Rocket Experiments," Jour-nal of Spacecraft, Vol. 24, No. 5, pp. 444-453. [33] Kline-Schroder, R., and Powell, J .D., "Recent Laboratory Results of the K I T E Attitude Dynamics Simulator," Proceedings of the Second International Confer-ence on Tethers in Space, Venice, Italy, October, 1987, pp. 61-66. [34] Kenjo, T . , Stepping Motors and their Microprocessor Control, Clarendon Press, Oxford, England, 1984. [35] Acarnley, P.P., Stepping Motors: A Guide to Modern Theory and Practice, Peter Peregrinus Ltd., Stevenage, England, 1982. [36] DeSilva, C.W. , Control Sensors and Actuators, Prentice-Hall, Englewood Cliffs, N.J. , 1989. [37] Levy, S., and Kroll, W.D. , "Errors Introduced by Finite Space and Time In-crements in Dynamic Response Computations," Proceedings of the First U.S. 193 National Congress of Applied Mechanics, pp. 1-8, 1951. [38] Hohenstein, C.L . , Computer Peripherals for Minicomputers, Microcomputers and Personal Computers, McGraw-Hill Book Co., 1980. [39] Money, S.A., Microprocessors in Instrumentation and Control, Collins Profes-sional & Technical Books, 1985. [40] Lafore, R., Microsoft C Programming for the IBM, Howard W. Sams & Co., Hayden Books, 1987. [41] Superior Electric Co., "Slo-Syn Micro Series Step Motors," Catalog M S C 1086, U.S.A. 194 Appendix A : Equations of Motion for the Rigid System in an Arbitrary Orbit The kinetic and potential energy expressions were obtained for the flexible plat-form and the tether (Eqs. 2.22, 2.24, and 2.28). Recognizing that the corresponding equations of motion will be too formidable for the system control study, the lin-earized equations for the rigid system in an arbitrary orbit are presented here. Platform Pitch aP"[l*x,P + (1 + Mt>a)DXXtP\ + ctp'[lXXiP + (1 + Mtt8)DXXtP\ (/e) + ap[-(L'Mtte{L'DyjP) - (1 + LMt,8){L"Dy,p) - (1 + LMtt8)L'BytPfe - (1 + —^LfeDZiP + (1 + —J±)LDy>p + /e p{s(l + Mt)8)(D2Z>p - DyViP) + 2(1 + ^ l ) L D y > p + 3 ( / „ , p - Iyy>p)}] + /?p" [ - ( l + Mtt8)DxytP\ + /?p'[-2(l + Mt,B)DXZiP - (1 + Mt)8)Dxy>pfe + /3P[-2(1 + Mt,8)feDXZiP + (L'Mt>8)L'Dx>p+ (1 + LMt)B)L"DX)P + (1 + LMt,8)LDXtPh - (1 + ^±)LDXiP + /.p{-3(l + Mt,8)Dxy>p - 2(1 + ^h±)LDXt9}] + V ' [ " ( l + MttB)DXStP\ + 7P'[2(1 + Mt)8)Dxy>p - (1 + Mt,B)DXZtPfe} + 7p[2(l + Mt,8)feDxy,p + 2(1 + lMt,B)L'Dx>p + (1 + ^2l)LDXtPfe + fep{Hl + Mt>8)DXZiP}} W'[(i + ^)IA,, P] + at' [2(1 + LMt,B)L'Dy>p + 2(1 + ^ 1 ) X / > . , P + (1 + ^ )I£ V ) P/ e] 195 + at [(L'Mt,8)L'Dy>p + (1 + LMt,s)L"DytP + (1 + LMt,8)L'Dy<pfe + 2(1 + LMt,8)L'Dz,p + (1 + ^ i ) L D Z i P / e - ( l + ^ ) Z P , , p + / e p { ( l + ^ ) L ^ p } ] + it" o + V + 7*[0j + L " [ - ( l + L M M ) D * i P ] + L'[2(l + LMt,8)Dy>p - 2LfMti8DZiP - (1 + LMt,8)Dz>pfe + £ [ ( 1 + ^ ) A , , p / e + ^ Dy,pfe + 2 ( L ' M M ) ^ i P + (1 + LMt,8)DZtP + / e p J 2 ( l + ^ i ) P 2 ) P + L M t , a P Z ) P j ] + ^ ' I p[o]+^ l P[o]+z > x ) P[o] + £ > ; ' | P [ - ( 1 + M t , a ) ^ ) P ] + £>i,p[2(l + Mt,8)DVtP - (1 + Mf,a)A*,P/e] + £ > y , p [2(1 + Mt)8)Dy>pfe + (1 + ^ ^ ) I / e + 2(1 + LMt,8)L' + fep{z{l + Mt,8)Dz,p)} + b'J>p [(1 + Mt ( a)5 V i P] + D'ZiP [2(1 + M t , a ) ^ i P + (1 + Mtt8)Dy,pfe] + D, ) P [2(1 + Mt,8)DZ)Pfe + (1 + ^ y ^ ) L - ( L ' M t ) 8 ) L ' - (1 + LMt,8)L" - (1 + LMt,8)L'fe + / e p {3(l + M ^ p y . p + 2(1+^1)1}] + /e [M/*«,P + (1 + M t , a ) £ > X X ) P + (1 + ^ ^ ) L D V ) p + 2(1 + LMt,8)L'Dy,p - (L'Mt,8)L'DZiP - (1 + LMt,8)L"Dz,p -(l + LMt,s)L'Dz,pfe+{l + ^^)LDZtP + fep{z(l + L)DyZtP + 2(1 + ^±)LDZtP} = -TtDz>p + TaDytP + Ma . {A.l) 196 Platform Yaw ap" [-(1 + Mtt8)DxyjP\ + ap' [ 2 ( 1 + Mtt8)Dxz>p - (1 + Mt}8)DxytPfe] + ap [(1 + LMt>B)DX)PL" + (1 + LMtt8)DXtPL'fe + (L'Mt>e)L'DXtP - (1 + ^ ) I 5 I > P + / E P { - 3 ( l + M t ) 8)5 x y, p - 2(1 + ™h±)LDXt9}\ + PP" [hy,P + (1 + Mt,a)DyVlP] + fip' [lyy>p + (1 + Mt>8)Dyy,p] fe + /? p [ - ( l + Mt>8)feDyz>p - (1 + ^±)LDZ)Pfe - 2(1 + ^±)L'DZ>P + {Ixx,p - / « , P ) + (1 + Mtt8)(DXXtP - D„tPj\ + V [ " I 1 + Wt,.)^»»,p] + V [( Jxx,p " IVy,p - J«,p) + (1 + Mt>8){Dxx>p - Dyy>p - Dzz>p) + (1 + LMt,.)L'(DMtP - Dy>p) - (1 + Mt>8)Dy2tPfe] + lp[fe{(hx,p ~ Iyy,p) + (1 + M t , a ) ( / > „ , „ - Dyy,p) } + (1 + LMt>8)L"{Dz,p - DVtP) + (l + LMt>8)L'{DZtP - DVtP)fe + (L'Mt,8)L'(Dz,p - Dy,p) + 2(1 + LMt,e)L'Dy,p + (i + ±Mhl)LDytPfe - (1 + JUi,.)/W] + « , " [ - ( ! + ^ H ) I D . „ ] + « / [ - 2(l + ^ ) X 'D x , p - - (1 + ^ ) L 5 x , p / e ] + a t [ - ( l + LMtt8)L"DXiP - (1 + LMtt8)L'DXiPfe - {L'Mt,8)L'DXtP + { 1 + ^ ) L D x , p - f e p ( l + ^ ) L D x , p } + ^[-(l + ^ ) L D z , p ] + V [ - 2 ( 1 + LMt)8)L'DZiP - (1 + ^ l ) Z D Z i P / e ] + l t [ - ( l + LMtt8)L"DZtP - {L'Mt>e)L'DZtP - f e p ( l + lMht)LDx%p\ 197 + L"[o] + 2'[-2(1 + lMt,.)Dt,p] + LMt>a)DX)Pfe - 2 L ' M t , . £ x > p ] + &lp [(1 + Mt>a)DZtP] + b'x>p [(1 + Mt>a)DZiPfe + I>x,P[-(l + Mt,a)DViPfe - (1 + ^ Y±)lfe + (1 + Mt,a)DZtP - 2(1 + LMt>a)L'] + D ; ' ) P [ O] + D; i p[-2(1 + Mt,a)DX)P] + by,p - (1 + Mt,a)DXiPfe] + b'lP [-(i + M f,e)5x,p] + b'Z)P [-(i + M M ) A c , P / e ] + bz,p [(i + M M ) £ X , P ] + h [-(1 + M f i 8 ) P x y , p - (1 + ^ ) L D X t P ] - 2(1 + LMt,a)L'Dx>p + (1 + M t , a ) £ x * i P = T 7 J D * ) P - T a P X | P + Mp . (A.2) Platform Roll -(1 + Mt)B)DXZtP\ + < [ - 2 ( l + Mt,a)DxytP - (1 + M ^ A c z . p / e ] + a p [2 ( l + LMtt8)L'Dx>p + (1 + ^ ^ ) L D x > p f e + / e p {3(l + M t ) 8 ) 5 I 2 ) P } ] + / 9 p " [ - ( l + M , , f l ) ^ ) P ] + /V [(J„, P + / y y > P - /*,, P) + (1 + M « , . ) { S „ l P + A/V,P - Dxx>p} - ( l + M , > a ) £ y * , p / e ] + Pp [fc{hz,p + (1 + Mt,a)DZZtP} - (1 + Mt>a)Dyz>p + 2(1 + LMttB)L'DyiP + (1 + ^ t ) L D y > p f e + lp"[lzz,p + (1 + M « f . ) 5 „ > p ] + 7 p ' [ ( / « , p + (1 + M t . O^w . p J / e ] + 7 p [ ( l + Mt,a)DyZiPfe - (1 + LMt,B)L"Dy,p + (1 + LMt)B)L'Dy,pfe + (IXX,p - Iyy,p) + (1 + M t , . ) ( 0 « l P " A/V,p) + (l + ^ ^ ) L D y > p - {L'Mtta)L'DyiP + / e p { s ( l + M t > a ) ( P x x > p - D y y ,p) + 3( / X X i P - J y y , p ) 198 + 2{l + ^)LDytP}} W'[o] W[-2(1 + ^ )Z/>,,P] + at [-2(1 + LMt,e)L'DStP - (1 + ^ y ^ ) L 5 I ) P / e ] + lt"[(l + ^ ) L D y , p ] + V[2(l + LMt,8)L'Dy>p + (1 + ^ i ) Z A / , r / e ] + 7t (1 + LMt,8)L"DytP + (1 + LMtt8)L'Dy>pfe + [L'Mty8)L'DytP + /eP(l + ^ ) L D y > p ] + L"[(l + LMt>8)Dx,p] + L'[2{L'Mt)8)DX)P + (1 + LMt,8)Dx>pf^ + 2[-(l + lMtt8)DXiP + L"Mt,8DXtP + {L'Mt,8)Dx,pfe -2/ e p(l + LMM)5X ) P] + ^ 'p[-(1 + ^ ) ^ , P ] + £>;iP[-(l + Mt i8)5y,p/c] + £>x,P[-(l + Mti8)(feDZiP - Dy,p) - (1 + ^ ^ ) L + (1 + LMt>8)L" + (1 + LMt>8)L'fe + (L'Mt>8)L' - /ep{s(l + MM)_Dtf)P + 2(1 + ^±)L}] + b^p [(1 + Mt,8)Dx,p] + D'ytp [(1 + Mt,8)DXtPfe + Dy,p[-{1 + Mtt8)Dx>p - /ep{3(l + Mt>8)DX)P} + b'lP[o] + bz>p[-2(l + Mt,8)DXiP] + Dz,p[-(1 + Mt,8)feDXtP] + fe[-(l + Mt,s)D + (1 + LMtt8)L"DXtP + (1 + LMtt8)L'DXiPfe + (L'Mt>8)L'DXtP LMi - (1 + M M)£ S ! / ) P - (1 + ^^)LDX, 2 .p + / e p [-3(1 + Mtt8)DxyiP - 2(1 + ±MM-)LDX,P] = - T ^ p + TtDx>p + M,. {A.3) 199 Tether Pitch OLr (l + ^ M)Ifl„] + [-2(1 + ^ ) W , „ + (1 + ^ ) L B y , p f . + «P[-(I + ^ ) l / A , - (i + ^ ) ^ ) P + M i + ^ ) W > , „ + /V'[-(i + + 2 + & 2 )LDXtPfe] + PP[(l + ^ ) L D x , p - f e p ( l + LM^)LDX,P] + 7p"[o] + V [ 2 ( l + ^ ) ^ , P ] + 7P[(1 + ^ ) X / e ^ x , p ] + a t"[(l + + a / [ 2 ( l + + (1 + ^ ) L V e ] 3 + a t [ ( l + ^ ) { L P Z ( P / e + L5J/>p} + /ep{3(l + )^2/ + 2(1 + )LDy,p}] + 1t"[0\ + V [ 0 j +7t[°J + L » [ 0 ] + L ' [ 2 ( 1 + ^ ) L ] + L[(l + LMttB)feDyiP + (1 + ^ ^ ) { - A , ) P + / e p 5 z , p } + 2(1 + ^ ^ ) f e l + 2(1 + LMt,B)L' + £ » p [ o ] + £)'I)P[o] + i)I)Pfo LMt + ^ [ o ] + ^ > p[2(i + f^djx] + i>,„[(i + f^d)/. l ] + j&Jp[(1 + LMttB)L] + b'2tP[(1 + ^ ) L f e ] + £>*,P[(1 + ^ ) { f e P L - L}] + fe [(1 LMtiB - 2 LMt,i + 2(1 + XMt 2 I M + — r 1 ^ ) ! / + (1 + M ) Z / X + ( 1 + ^ h L ) { h p l D 2 , p - LDZ,P) = TaL (AA) 200 Tether Roll ap" 0\+ap' 0 +a p | 0 + /jp[-/.p(i + ^ )Zfl., p] 2 + 7 P"[( l + ^ ) L D y > p ] + 7 / [ ( l + ^ ) ^ , P / « + 7 P /ep(l + )^y,p] + a t "[0j + at'[oJ +at[Oj + V [d + ^ ) I 2 ] + V[2(l + ^ )L>L + (1 + ^ ) Z 2 / e ] + 7t[(l + 2 + (1 + ^ ) I 5 V > P { 1 + 2/ep} + I/" [o] + L' [o] + L [-(1 + LM M ) / e p 5 I ( P ] + i^'p[o] + ^ t . p [ ° ] + ^ . p [ ° ] + £>:',„ [o]+ft,p[o]+^lP[o] " ( 1 + = -T^L •)LDt,f (A.5) Tether Tension ftp"[-(l + LMt,B)Dz,p] + a p ' [ - 2 ( l + LMt>8)DV)P - (1 + L M 4 , . ) D , l P / e ] + a p [(1 + LMt,8){-feDy>p + D z > p + 2/e pA*,P)] + fip" [o] + /?„' [2(1 + lMt,.)5x,p] + £ P [(1 + ZAft,a)/e + 7 p " [(1 + LMt>8)DXtP] + l p ' [(1 + LMt>e)DXtPfe] + l p [(1 + L M t > 8 ) { - D x > p - 2fepDXtP)] 201 + at" [o] + at' [-2(1 + = ^ ) L ] + at [(1 + LMt,B){f.Dy,p - D2>p + fepDz>p) + It" [o] + it' [o] + 7t [-(1 + LMt,s)fePDXiP] + L" [(1 + LMt,s]\ + V [l'Mt,s + (1 + LMt,.)fe] + L[L"Mt,8 + {L'Mt,e)fe - Mt,8feDZiP - Mt,8Dy>p - 2M t , a / e p D ! / ( P - ( l + L M f | 8 ) ( l + 2/ e p)j + KP[°) + ^ , P[o] + bx>P[o] + D'y,P + D'yiP [(1 + XMt,a)/e] + A / . P [~(1 + •t'-Wt,«)(l + 2/ep)] + £) p^[o] + £;,p[-2(l + IM M)] + £,,p[-(l + LMt,8)fe} + fe[-{l + LMt,9)DMtP] + L'Mi L'+{1 + LMt,8)L" + (1 + LMt,.)L'fe it + (1 + LMt . JHVp " 2 /ep5 y ) p) + (1 + - 2L/ep) (A.6) Subsatellite's Equations of Motion Ct"lxx,a + a'8 [lxx,8fe] + 0t8 [3{Izz,e ~ Iyy,,] + Ixx.sfe = 0 ', (A.7o) P'JlvVyB + P'e [-^yy,«/e] + 7?a [-^ xx.a — ^zz.aj + 7a [-Txx.a — Iyy,8 ~ Izz,e^ + le[(IxXls-Iyy,8)fe]=0; (A.76) /?8 [-Jzz.a — -^ I/!/,* ~~ -^xx.aj Pe [jzz.a/ej + 7a [-^ zz.aj + 7a [^zz,a/ej + 7a \{Ixx,s ~ Iyy,.) + 3/ ep(/xx,a - Iyy,s)] = 0 . {A.7c) 202 Appendix B: System Equations in Matrix Form In Eqs. (A.l) - (A.6), the terms containing the generalized coordinates as well as their velocities and accelerations can be collected to obtain a set of matrix equations in the following form: \M}{g} + [C]{q} + \K){q} = [B]{u} + {P} . The following properties of these matrices are noteworthy: in the absence of de-ployment, retrieval and orbit eccentricity, [M] and [K] are real symmetric matrices, while [C] is real skew symmetric. With nonautonomous effects, [K] and [C] lose their symmetric and skew symmetric character. The elements of these matrices are presented below: Mass Matrix [M] M M M M M Mi M i M M M M ) = J*X>P + (I + M , . ) A B * , P ; 1.2) = -(l + Mt,a)DxytP; 1.3) = -(l + Mt,a)Dxz>p; 1.4) = {l + LMt)8/2)LDy>p; 1.5) = 0 ; 1.6) = -{l + LMtt8)Dz>p; 2,1) = -{l + Mt,8)Dxy>p', ^ ) = I ^ y t P + ( l + Mti8)Dyy>p; 2.3) = - ( l + M t > B) JD„Z ) P ; 2.4) = -{l + LMtt8/2)LDx>p; 203 M ( 2 , 5 ) = - (1 + LMt,a/2)LDz>p ; M(2,6) = 0 ; M ( 3 , l ) = -(1 + Af 4 i«)5„,p; M(3,2) = -{l + Mt,8)Dyz>p', M(3,3) = J j z > p + (1 + Mt,B)Dzz>p ; M ( 3 , 4 ) = 0 ; M(3,5) = (1 + LMt,a/2)LDy,p ; M(3,6) = (1 + LMt>a)DXtP ; M(4, l ) = ( l + £ M M / 2 ) L 5 y ) P ; M(4,2) = -(1 + LMtta/2)LDXtP ; M ( 4 , 3 ) = 0 ; M(4,4) = ( l + I M t , 8 / 3 ) L 2 ; M(4,5) = 0 ; M(4,6) = 0 ; M ( 5 , l ) = 0 ; M ( 5 , 2 ) = - (1 + LMttB/2)LDZtP ; M(5,3) = (1 + LMt)a/2)LDy>p ; M(5,4) = 0 ; M ( 5 , 5 ) = (1 + I M M / 3 ) L 2 ; M(5,6) = 0 ; M ( 6 , l ) = - ( l + L M M ) A s ) P ; 204 M(6,1) = 0; M(6,1) = (1 + XMt,a)£> M(6, l ) = 0; M ( 6 , l ) = 0; M ( 6 , l ) = (1 + LMt,s) • Gyroscopic Matrix [C] C ( l , l ) = {lL,P + (1 + LMt,e)Dxx>p)fe; C(l,2) = -2(1 + LMt,a)DXZtP - (1 + LMt,8)Dxy>pfe; C(l,3) = 2(1 + LMt,e)DxytP - (1 + LMt,e)DX2,pfe ; C(l ,4) = 2(1 + LMt)8)L'DytP + 2(1 + LMt>e/2)LDy>p + {l + LMt>a/2)LDytPfe] C(l ,5) = 0; C ( l , 6) = 2(1 + LMt,8)DytP + 2L'Mtj8DZtP - (1 + LMtt8/2)Dz>pfe; C(2,1) = 2(1 + ZM t | . )A«,p - (1 + LMt>8)Dxy>pfe ; C(2,2) = ( j j ^ + (1 + LMi)8)DyytP)h 5 C(2,3) = (/*x,p - / y 2 y,p - & , P ) + (1 + LMt,8)(DXXtP - A/V,P - A«,P) + (1 + LMt,8)L'{Dz,p - P y ,p) - (1 + LMt,8)DXy,pfe; C(2,4) = -2(1 + LMt,8/2)L'DXtP - LMtl.L'/2DX,P - (1 + LMt,8/2)LDx>pfe ; C(2,5) = -2(1 + LMt,8)L'DZtP - (1 + LMt>8/2)LDz,pfe ; C(2,6) = -2(1 + LMt,8)DX)P; 205 C(3,1) = -2(1 + LMt>8)Dxy,p - (1 + LMt,8)Dxz,pfe ; C(3,2) = [Izz,p + Iyy,p ~ Ixx,p) + (1 + LMt>a){Dzz,p + Dyy,p ~ Dxx,p) - (1 + LMt,a)Dxy>pfe; C(3,3) = (i2,lP + (1 + LMt>a)DZz,p)fe; C(3,4) = -2(1 + LMtta/2)LDXtP ; C(3,5) = 2(1 + LMttB)L'DytP + (1 + LMt>a/2)LDy)Pfe ; C(3,6) = 2L'Mt>eDXtP + (1 + lMt,a)Dx>pU \ C(4,1) = -2(1 + LMt,a/2)LDZ)P + (1 + LMt,a/2)LDy>pfe ; C(4,2) = - (1 + LMt>a/2)LDx>pfe; C(4,3) = 2(1 + LMtia/2)LDXtP ; C(4,4) = 2(1 + LMt,a/2)LL' + (1 + LMt,8/3)L2fe ; C(4,5) = 0; C(4 ,6 )=2( l + X M t ) J / 2 ) L ; C(5, l ) = 0 (7(5,2) = -(1 + LMt,a/2)LDz,pfe; C(5,3) = (1 + LMt,a/2)LDylPfe ; C(5,4) = 0; C(5,5) = 2(1 + LMtt8/2)LL' + (l + LMt,8/3)L2fe ; C ( 5 , 6 ) = 0 ; C(6,1) = -2(1 + LMt,e)Dy)P - (1 + LMt>a)DZtPU ; C(6,2) = 2(1 + L M t ) 8 ) D I i P ; C(6,3) = (1 + LMt>a)DXtPfe; C(6,4) = -2(1 + LMty8/2)L ; 206 C(6,5) = 0; C(6,6) = LMt,8' + (1 + LMt,s)fe . Stiffness Matrix [if] i f (1, 1) = - (1 + LMt,8)L"Dy,p - (L'Mt,8)L'Dy>p - (1 + LMt)8)L'Dy>pfe - (1 + LMt,a/2)LDZiPfe + (1 + LMt,8/2)LDVtP + / e p{3(l + Mt,8){D2ZiP - Dyy,p) + 2(1 + LMt,8/2)LDy>p + 3(I2ZZtP - I2yytP)} ; if(1,2) = -2(1 + Mt)8)DX2tPfe (L'Mtt8)L'DXiP + (1 + LMt>8)L"DXtP - (1 + LMtte)L'DXtPfe - (1 + LMti8/2)LDXtP + / e p { -3 ( l + M t ) 8 ) D x y ) P - 2(1 + LMt>8/2)LDXiP} ; i f (1, 3) = 2(1 + M t t 8 ) D x y t P f e + 2(1 + LMtt8)L'DXtP + (1 + LMt,8/2)LDXtPfe + / e p{3(l + M t > a ) D I Z ( p } ; If (1,4) = (1 + LMtt8)L"DytP + (L'Mt,8)L'Dy,p + (1 + LMt,8)L'DyiPfe + (1 + LMt,8)L'DZiP + (1 + LMt,8/2)LDZtPfe - (1 + LMt,8/2)LDy,p + fep{(l + LMt,8/2)LDy,p}; i f (1,5) = 0 ; i f (1,6) = (1 + LMt,8)DyiPfe + 2(L'Mt,8)Dy,p + (1 + LMtt8)DZtPfe + / e p{2(l + LMt)8)DZ)P} ; i f (2,1) = (1 +. LMt,8)L"DXtP + (1 + LMt>8)l'Dx>pfe 207 (L'Mt,.)L'Bx,p - (1 + LMttB/2LDXtP + /ep{-3(l + Mt,8)DxytP - 2(1 + LMt>e/2)LDXtP} ; K{2,2) = -(1 + Mt,B)DyZ)Pfe - (1 + LMt,B/2LDZtPfe - 2(1 + LMt,B/2)L'Dz>p + (4 ) P - 42z,P) + (1 + JWt,.)(l>««,p - Dzz>p); JT(2,3) = / e { ( /L,p - J?„,P) + (1 + M t > a ) (D X I , p - 5 y y > p ) } + (1 + LMt,B)L"(DZtP - DVtP) + (1 + LMt,B)L'(Dz>p - DVtP)fe + (L'Mt)B)L'(Dz>p - DyiP) + 2(1 + LMt,B)L'DyiP + (1 + LMt,B/2)LBytPfe - (1 + MttB)Dyz,p ; K(2,4) = -(1 + LMt)B)L"Dx>p - (1 + LMt<B)L'Dx>pfe - {L'Mt,a)L'DXtP + (1 + LMt,B/2)LDXtP - fep{l + LMtiB/2)ZBx>p; K{2,5) = -(1 + LMttB)L"DZtP - {L'n)L'Dz,p - / e p(l + LMt,8/2)LDZ)P ; #(2,6) = -(1 + LMt,B)DXtPfe - 2L'Mt,BD11A ; X(3,1) = 2(1 + LMttB)L'DXtP + (1 + LMt,sl2)LDx>pfe + /ep{3(l + Mt(8) Ac*,p} ; #(3,2) = /e{/*2,p + (1 + Mt>.)5MtP} - (1 + MM)5y*,P + 2(1 + LMt,B)L'Dy>p + (1 + LMttB/2)LDy>pfe ; K(3,3) = (1 + MtjB)Dyz>pfe - (1 + LMt>a)L"Dy>p - {L'MttB)L'Dy<p + (1 + LMt,a)L'Dy>pfe + (4 2x,P " ij y > p) + (1 + M t , a ) (£ x x ,p - 5 y t , , P ) 208 + (1 + LMt,8/2)LDy,p + / e p{3(l + Mt,8)(DXXiP - DyyiP) + 3(I2XXiP - I2y>p) + 2(1 + LMt>8/2)LDy,p} ; i f (3,4) = -2(1 + LMt,8)L'Dx<p - (1 + LMti8/2)LDx>pfe; i f (3,5) = (1 + LMt,8)L"Dy>p + (1 + LMt>8)L'by,pfe + {L'Mtj8)L'DyiP + fep{l + LMt>s/2)LDy>p ; i f (3,6) = L"Mtt8DXtP + L'Mtt8DXiPfe - (1 + LMt,8)DXtP - 2 / e p ( l + LMtt8)DXtP ; i f (4,1) = - (1 + LMt>8/2)LDz>pfe - (1 + LMt>8/2)LDy>p + / e p ( l + LMt,8/2)lDytP ; i f (4,2) = (1 + LMtt8/2)LDx>p - / e p ( l + LMt>8/2)lDX)P ; if(4,3) = (l + L M t , a / 2 ) X D I ) P / e ; i f (4,4) = (1 + LMt)8/2)LDZ)Pfe + (1 + LMti8l2)L'Dz>p + / c p{3(l + LMt>8/3)L2 + 2(1 + LMtt8/2)LDyiP} ; if(4,5) = 0; i f (4,6) = (1 + LMti8)Dv>pfe + 2(1 + L M t > a ) L ' + 2(1 + LMt>8/2)feL + (1 + LMt,8){-Dz>p + fepDZtP} ; #(5,1) = 0 ; i f (5,2) = - / e p ( l + LMt,8/2)LDZtP ; i f (5,3) = / e p ( l + LMt>8/2)LDy)P ; i f ( 5 , 4 ) = 0 ; i f (5,5) = (1 + LMt,8/2)feLDz,p + (1 + L M t ( 8 / 3 ) L 2 { l + 3/ep} 209 + (1 + LMt,./2)LDVtP{l + 2fep} ; #(5,6) = - (1 + LMt,s)fepDXtP ; K{6,1) = - (1 + LMt,e)Dy,pfe + (1 + LMt,e){-DZ)P + 2fepDz,p ; #(6,2) = {l + LMt,8)Dx,pfe; K{6,3) = (1 + LMt,8){-DXtP - 2/epDS)p) ; X(6,4) = (1 + LMt>e)DytPfe + (1 + l M t l , ) ( - 5 f l P + fepDZiP); if(6,5) = {1 + lMt,B)DttP fep j tf(6,6) = L"Mtt. + L'MttJe - MttJeDZtP - MttaDyiP - 2MttJepDyiP - (1 + LMt,s) (1 + 2 / e p ) . Controller Coefficient Matrix [ £ ] Only the nonzero elements of [£ ] are listed below: Tension Control £ ( 1 , 1 ) = 1; £ ( 2 , 2 ) = 1; 5(3,3) = 1; £ ( 6 , 4 ) = 1. Thruster Control £ ( i , i ) = 1; £ ( 1 , 5 ) = {l + Mtt.)DMtP't B{l,6) = -{l + Mt)8)Dy)P; £ ( 2 , 2 ) = 1; 210 £ ( 2 , 4 ) = - ( l + MM)P*,P; £ ( 2 , 6 ) = (1 + Mt,B)bXtP ; £ ( 3 , 3 ) = 1; £ ( 3 , 4 ) = {l + Mt,.)Dv,P; £ ( 3 , 5 ) = - ( l + M t ) a)£*,P; £ ( 4 , 6 ) = {l + Mt,s)L; £ ( 5 , 3 ) = - ( l + Mt,e)DXtP; £ ( 6 , l ) = - ( l + M t )8)£v,p-, £ ( 6 , 2 ) = (1 + Mt,a)bXtP ; Offset Control £ ( 1 , 1 ) = i ; £ ( l , 4 ) = £ y , P ; £ ( i , 6 ) = -bz<p\ £ ( 2 , 2 ) = 1; £ ( 2 , 4 ) = - £ x , P ; £ ( 2 , 5 ) = ; £ ( 3 , 3 ) = 1; £ ( 3 , 5 ) = -by>p; £ ( 3 , 6 ) = D I > P ; £ ( 4 , l ) = £ y , P - , £ ( 4 , 2 ) = -DX)P; £ ( 4 , 4 ) = L ; 211 B{5,2)=Dz>p; B(5,Z) = -bViP; B(5,5)=X; B{6, l) = -DStP; £ ( 6 , 3 ) = /> I ( P; £ ( 6 , 6 ) =L. Nonzero Equilibrium Matrix \P) P ( l , 1) = (1 + LMtj2)LDz<p + / e p J 3 ( l + Mt<e)DyZiP + 2(1 + LMt>a/2)LDZiP] ; P(2,l) = (l + M t , . ) £ P(3,1) = - (1 + Mt>a)Dxy>p - (1 + LMt,e/2)LDXiP + / e p { - 3 ( l + Mtt8)DxytP - 2(1 + LMt,./2)lDx,p} ; P(4,1) = (1 + L M t , . / 2 ) L / ) , , p { / e p - l } ; P(5,1) = - (1 + LMt,a/2)LDx>p ; P(6,1) = (1 + L M t i , ) f i y , p { - l - 2/e p} + (1 + LMt>a/2)l[-l - 2fep] . 212 Appendix C : Matrices Related to Optimal Control The state and control penalty matrices [Q] and [R] used in the quadratic per-formance index for optimal control (section 4.3), are presented here for each of the control strategies. Since cross coupling weights were not used, only diagonal elements Ra and Qa are shown here: Tension Control IQ) = 500 500 500 200 200 100 500 500 500 200 200 100 [R] = 100 100 100 2000. Thruster Control '25 25 25 10 [Q} = 10 25 25 25 10 10 213 200 200 [R} = 400 400 400. Offset Control 300 [Q) = 300 300 100 100 300 300 300 100 100 [R} = 300 300 300 10 10 10 The feedforward matrices [V] and [P] in the offset control are diagonal and can be represented as: and where [V] = V[I]; [P] = P[J]; 7 = 3 x 3 identity matrix. For a 100 m tether: V = 14, P = 5. For a 1000 m tether: V = 25, P = 9. 214 Appendix D: Computer-Translator Interface Table D . l Parallel Port Pin Configurations (25 Pin D-Type Con-nector) PIN N O . SIGNAL 1 Ground 2 Inplane Translator (Clockwise) 3 Inplane Translator (Counterclockwise) 4 Out-of-plane Translator (Clockwise) 5 Out-of-plane Translator (Counterclockwise) 6 Retrieval Translator 7 Deployment Translator 8 Inplane Translator Ground 9 Out-of-plane Translator Ground 10 Retrieval/Deployment Translator Ground 11-25 Common Ground Table D .2 Game Port Pin Configurations (15 Pin D-Type Connec-tor) PIN N O . SIGNAL 1 Inplane Potentiometer 5V D C 2 On/Off 3 Inplane Potentiometer Ground 4,5 Cable Shield Ground 6 Out-of-plane Potentiometer Ground 7 On/Off 8 Out-of-plane Potentiometer Ground 9-15 Common Ground 215 Appendix E : Step-Motors and Translator Modules The experimental setup used three step-motors, all of the permanent magnet type. Here, the rotor is a permanent magnet, while the stator consists of a stack of teeth with several pairs of field windings. The windings can be switched on and off in sequence to produce electromagnetic pole-pairs that cause the rotor to move in increments. The sequential switching of the windings is accomplished electronically by the translator modules described later. A n in-depth discussion on step-motors and their working principles is well documented[34,35]. Relevant mass, geometry, and other system parameters used in the analysis are listed below: Mass of carrige ( m c a r r i a g e ) = 70.43 kg Mass of inplane traverse (mtransverae) = 2.23 kg Mass of inplane motor (m^) = 3.52 kg Mass of out-of-plane motor (mout) = 1.48 kg Mass of retrieval motor (mret) = 0.57 kg Mass of pendulum {mpend) = 0.1108 kg Reel diameter (dreei) = 2 x 10~2 m Pulley diameter (dpuuey) = 4.4 x 10~2 m Shortest tether length (L) = 0.75 m Maximum expected motor speed = 400 pulse/s (obtained from theoretical calculations) Friction force (estimated) = 0.23 kg Motor Torque Required .Assuming the most efficient case, the inplane or out-of-plane transverse ac-celeration (a) would be the same as the maximum tangential acceleration of the 216 pendulum, a = sin 8max = 6.54 rad/s 2 L In the case of the retrieval motor, for a maximum retrieval rate of 0.17 m/s within I s , 0.17 , , , a = : r ^ r = 8.5rad/V 1 x 0.02 ' The inertia load information is as follows: Inplane traverse load = m c a r r i a g e + mtraverae + mout + mret +mpend = 4.82 kg; In-plane traverse load inertia = 4.82 x (^P""^) 2 = 3.11 x 10~ 3 kgm 2 ; Out-of-plane traverse load = m c a r r i a g e + mret + mpend = 1.11 kg; Out-of-plane traverse load inertia = 7.17 x 1 0 - 4 kgm 2 ; Retrieval load = mpend = 0.1108 kg; Retrieval load inertia = 1.11 x 10~5 kgm 2 ; Traverse friction inertia = 0.23 x ( dP u 2" ey )2 = 1 . 4 8 x 10~4 kgm 2 ; Retrieval friction inertia = 0.23 x {^f1)2 = 2.3 x 1 0 - 5 kgm 2 . Now the motor torque (T) can be calculated from T = (Load inertia + Friction inertia + Rotor inertia) a . The rotor inertia data were supplied by the manufacturer [41]: Retrieval motor rotor inertia = 0.12 x 1 0 - 4 kgm 2 ; Inplane motor rotor inertia = 1.87 x 1 0 - 4 kgm 2 ; Inplane motor torque requirement = 2.25 Ncm; Out-of-plane motor torque requirement = 0.6 Ncm; Retrieval motor torque requirement = 0.03 Ncm. 217 The torque-speed characteristics for the motors are indicated in Fig. E-1..E-3. At the rated maximum expected speed (400 pps), the torque available is as follows: Inplane motor = 2.4 Ncm; Out-of-plane motor = 1.0 Ncm; Retrieval motor = 0.3 Ncm. The retrieval motor selection was purposely selected conservatively to allow the use of larger reels and payloads if required. 400 (282.4) 320 (226.0) 240 (169.4) 160 (113.0) 80 (56.5) TYPICAL PERFORMANCE CHARACTERISTICS [>> A / 1 \ L, OWER-•oroue 3.3A, 21 (V DC N \ 40 32 24 t 16 g O CL 8 1000 2000 3000 4000 5000 6000 7OO0 8000 9000 1OO00 SPEED (1.8* STEPS PER SECOND) Figure E - l Typical performance characteristics of the M093-FD14 step-motor used to drive the inplane traverse. 150 (105.9) 120 (84.7) 90 (63.5) 60 (42.4) 30 (21.2) TYPICAL PERFORMANCE CHARACTERISTICS \ / • • • • / -POWER -2A, 2( VDC / -TORQU : 30 24 18 12 & O a 6 1000 2000 3OO0 4OO0 5000 6000 7000 8000 9000 10000 S P E E D ( 1 . 8 * S T E P S P E R S E C O N D ) Figure E-2 Typical performance characteristics of the M091-FD09 step-motor used to drive the carriage in the out-of-plane direction. 218 Figure E-3 Typical performance characteristics of the M061-FD08 step-motor used for deployment and retrieval of the tethered payload. Translator modules The translator module has logic circuitry to interpret a pulse train and "trans-late" it into the corresponding switching sequence for stator field windings (on/off/ reverse state for each phase of the stator). The translator also has a solid-state switching circuitry (using gates, latches, triggers, etc.) to direct field currents to the appropriate phase windings according to the specified switching state. Control signals within the translator are of the order of 10 mA, whereas the phase exci-tation requires a current of several amperes. Therefore, the control signals must be amplified by switching amplifiers for phase excitation. The power to operate the translator and phase excitation amplifiers comes from a D C source. The en-tire unit, consisting of a translator, amplifiers and a power supply, is termed as 'motor-drive system'. Since two of the motors were high torque type, series re-sistor translators were used. The additional distinctive features of the translators selected were: (1) an integrated chip PMM8713 for stepping phase control, which simplified circuit configuration and allowed excitation modes to be changed easily; 219 (2) a built-in power-down circuit with 15 KHz overpass switching that chopped the motor excitation current to prevent energy loss and overheating of the motors. Translator Characteristics Input power voltage Input power current Maximum pulse rate Standing by current chopper frequency Operating temperature Size D C 24V to D C 30V. 6A/phase maximum 10,000pps with ideal stepping motor 15 KHz minimum -10°C to 50°C 135 mm (w) x 100mm (h) x 30mm (d), (including heatsink) 220 Appendix F : Sensor Technical Information The tether attitude sensors described in Section 5.3.3, employed a pair of poten-tiometers as analog displacement transducers. Any high quality, low friction poten-tiometer could be used for this purpose. In this case, a pair of 1/2 W, 500K linear conductive plastic potentiometers, manufactured by Clarostat Company, U.S.A. , was selected. According to the manufacturers, the potentiometers were manufac-tured in a controlled environment to ensure a reliable, homogeneous, and strain free resistance element. Potentiometer Specifications Resistance Tolerance ± 1 0 % up to 500K Effective Rotation 265° Dielectric Strength 750 V A C for 60 SEC A T M Working Voltage 350 V D C across terminals Dynamic Noise 1.5% total resistance Linearity (Independent) ± 5 % Voltage Coefficient 0.008%/V (max.) Rotational Life 50,000 cycles Power Rating 0.5 W at 70°C Torque Range 1.4x 10" 3 to 3.18 x 10" 2 Nm Weight 8.16 x 10" 3 kg Functional Output Linear 221 LIST OF PUBLICATIONS 1. Modi V.J. and Lakshmanan, P.K., "A Near Optimum Strategy for Semi-Passive Attitude Control of Large Communications Satellites," AIAA/AAS, Astrodynamics Conference, Seattle, Washing-ton, U.S.A., August 1984, paper no. AIAA-84-2007. 2. Lakshmanan P.K., Modi V.J. and Misra A.K., "Dynamics of the Tethered Satellite System," proceedings of the 11th Canadian Congress of Applied Mechanics, Edmonton, Canada, June 1987, Vol. 1, paper A-94. 3. Lakshmanan P.K., Modi V.J. and Misra A.K., "Dynamics and Control of the Tethered Satellite System in the Presence of Offsets," Acta Astronautica, Vol. 15, No. 12, pp. 1053-1057, 1987. 4. Lakshmanan P.K., Modi V.J. and Misra A.K., "Dynamics and Control of the Tethered Satellite System in the Presence of Offsets," - Proceedings of the Sixth VPI&SU/AIAA Symposium on Dynamics and Control of Large Struc-tures, June 29 - July 1, 1987. - Proceedings of AAS/AIAA Astrodynamics Specialist Conference, Kalispell, Montana, August 10 - 13, 1987, paper AAS-87-434. 5. Lakshmanan P.K., Modi V.J. and Misra A.K., "Linear Regulator Control of the Tethered Satellite System in the Presence of Offsets," IMACS/IFACS-DPS Symposium, Hiroshima, Japan, October 6th-9th, 1987. 6. Lakshmanan P.K., Modi V.J. and Misra A.K., "Dynamics and Control of the Space Station Based Tethered Payload," Proceedings of NASA/AIAA/PSN Second International Conference on Tethers in Space, October 5th-8th, 1987, Venice, Italy. 7. Lakshmanan P.K., Modi V.J. and Misra A.K., "Dynamics and Control of the Tethered Satellite System in the Presence of Offsets," Acta Astronautica, Vol 19, No. 2, pp. 145-160, 1989. 8. Lakshmanan P.K., Modi V.J. and Misra A.K., "Space Station Based Tethered Payload: Con-trol Strategies and Their Relative Merit," Proceedings of the Third International Conference on Tethers in Space, 17-19 May 1989. 9. Lakshmanan P.K., Modi V.J. and Misra A.K., "Control of an Orbiting Platform Supported Teth-ered Satellite System," Proceedings of the IFAC Symposium, Japan, July 1989 (in press). 10. Lakshmanan P.K., Modi V.J. and Misra A.K., "Dynamics and Control of an Orbiting Tethered System," Proceedings of the 28th Annual Conference of the Society of Instrument and Control Engineers (SICE), Matsuyama, Japan, 1989. 11. Modi V.J., Lakshmanan P.K., Misra A.K., Picha R.J., Chan S., and Vakil S., "Experimental Demonstration of Offset Control With an Application to the Tethered Payload," Proceedings of the 12th Canadian Congress of Applied Mechanics, Carleton, Canada, June 1989. 

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