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Dynamics and control of multibody tethered systems using an order-N formulation Kalantzis, Spiros 1996

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D Y N A M I C S A N D C O N T R O L OF M U L T I B O D Y T E T H E R E D SYSTEMS USING A N O R D E R - N F O R M U L A T I O N S P I R O S K A L A N T Z I S B. Eng. (Honours), McGill University, Montreal, Canada, 1994 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E 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 B R I T I S H C O L U M B I A September 1996 © Spiros Kalantzis, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT The equations of motion for a multibody tethered satellite system in three di-mensional Keplerian orbit are derived. The model considers a multi-satellite system connected in series by flexible tethers. Both tethers and subsatellites are free to un-dergo three dimensional attitude motion, together with deployment and retrieval as well as longitudinal and transverse vibration for the tether. The elastic deformations of the tethers are discretized using the assumed mode method. The tether attachment points to the subsatellites are kept arbitrary and time varying. The model is also ca-pable of simulating the response of the entire system spinning about an arbitrary axis, as in the case of OEDIPUS-A/C which spins about the nominal tether length, or the proposed BICEPS mission where the system cartwheels about the orbit nor-mal. The governing equations of motion are derived using a non-recursive order(N) Lagrangian procedure which significantly reduces the computational cost associated with the inversion of the mass matrix, an important consideration for multi-satellite systems. Also, a symbolic integration and coding package is used to evaluate modal integrals thus avoiding their costly on-line numerical evaluation. Next, versatility of the formulation is illustrated through its application to two different tethered satellite systems of contemporary interest. Finally, a thruster and momentum-wheel based attitude controller is developed using the Feedback Lin-earization Technique, in conjunction with an offset (tether attachment point) control strategy for the suppression of the tether's vibratory motion using the optimal Lin-ear Quadratic Gaussian-Loop Transfer Recovery method. Both the controllers are successful in stabilizing the system over a range of mission profiles. ii T A B L E OF C O N T E N T S A B S T R A C T . .• i i T A B L E O F C O N T E N T S i i i L I S T O F S Y M B O L S v i i L I S T O F F I G U R E S x i A C K N O W L E D G E M E N T xv 1. I N T R O D U C T I O N 1 1.1 Prel iminary Remarks 1 1.2 Brief Review of the Relevant Literature 7 1.2.1 Mul t ibody O(N) formulation 7 1.2.2 Issues of tether modelling . . . 10 1.2.3 Att i tude and vibration control 11 1.3 Scope of the Investigation 12 2. F O R M U L A T I O N O F T H E P R O B L E M 14 2.1 Kinematics 14 2.1.1 Prel iminary definitions and the itfl position vector . . . 14 2.1.2 Tether flexibility discretization 14 2.1.3 Rotat ion angles and transformations . 17 2.1.4 Inertial velocity of the ith link 18 2.1.5 Cyl indr ica l orbital coordinates 19 2.1.6 Tether deployment and retrieval profile 21 i i i 2.2 Kinetics and System Energy 22 2.2.1 Kinet ic energy 22 2.2.2 Simplification for rigid links 23 2.2.3 Gravitat ional potential energy 25 2.2.4 Strain energy 26 2.2.5 Tether energy dissipation 27 2.3 O(N) Form of the Equations of Mot ion 28 2.3.1 Lagrange equations of motion 28 2.3.2 Generalized coordinates and position transformation . . 29 2.3.3 Velocity transformations 31 2.3.4 Cyl indrical coordinate modification 33 2.3.5 Mass matrix inversion 34 2.3.6 Specification of the offset position 35 2.4 Generalized Control Forces 36 2.4.1 Preliminary remarks 36 2.4.2 Generalized thruster forces 37 2.4.3 Generalized momentum gyro torques 39 3. C O M P U T E R I M P L E M E N T A T I O N 42 3.1 Preliminary Remarks 42 3.2 Numerical Implementation 43 3.2.1 Integration routine 43 3.2.2 Program structure 43 3.3 Verification of the Code 46 iv 3.3.1 Energy conservation . 4 6 3.3.2 Comparison with available data 50 4. D Y N A M I C S I M U L A T I O N 53 4.1 Prel iminary Remarks 53 4.2 Parameter and Response Variable Definitions 53 4.3 Stationkeeping Profile 56 4.4 Tether Deployment 66 4.5 Tether Retrieval 71 4.6 Five-Body Tethered System 71 4.7 B I C E P S Configuration 80 4.8 O E D I P U S Spinning Configuration . . . 88 5. A T T I T U D E A N D V I B R A T I O N C O N T R O L 93 5.1 Att i tude Control 93 5.1.1 Prel iminary remarks 93 5.1.2 Controller design using Feedback Linearization Technique 94 5.1.3 Simulation results 96 5.2 Control of Tether's Elastic Vibrations 110 5.2.1 Prel iminary remarks . 110 5.2.2 System linearization and state-space realization . . . . 1 1 0 5.2.3 Linear Quadratic Gaussian control with Loop Transfer Recovery 114 5.2.4 Simulation results . 121 v 6. C O N C L U D I N G R E M A R K S . 1 2 4 6.1 Summary of Results . 124 6.2 Recommendations for Future Study 127 B I B L I O G R A P H Y 128 A P P E N D I C E S I. T E N S O R R E P R E S E N T A T I O N O F T H E E Q U A T I O N S O F M O T I O N 133 1.1 Prel iminary Remarks 133 1.2 Mathematical Background 133 1.3 Forcing Function 135 II. R E D U C E D E Q U A T I O N S O F M O T I O N 137 II. 1 Preliminary Remarks 137 II.2 Derivation of the Lagrangian Equations of Mot ion 137 v i LIST OF SYMBOLS A / j intermediate length over which the deployment/retrieval profile, for the itfl l ink, is sinusoidal A Lagrange multipliers EU,QU longitudinal state and measurement noise covariance matrices, respectively matrix containing mode shape functions of the \ t h flexible link pitch, roll and yaw angles of the i * ' 1 link time-varying modal coordinate for the i * ' 1 flexible link link strain and stress, respectively damping factor of the } t h attitude actuator set of attitude angles (fjj = {c*i, structural damping coefficient for the \ t h l ink, EjJEj true anomaly Earth's gravitational constant, GMejB?e density of the ith link fundamental frequency of the link longitudinal tether vibra-tion A{ tether's cross-sectional area Af,Bf,Cj,Df state-space representation of flexible subsystem Dj inertial position vector of frame Fj Dj magnitude of Dj —* —* Df). transformation matrix relating Dj and Ds. D r i inplane radial distance of the first l ink DSi {Drv6i, DZ1 }T for i = 1 and {DX{, Dy{, £> 2 .} T for 1 m Pi w 0 i . v i i Dx- transformation matrix relating D j and Ds-DXi, Dyi, Dz- Cartesian components of D{ DZl out-of-plane position component of first link E{ Young's elastic modulus for the ith l ink Ej^ contribution from structural damping to the augmented complex modulus E* F system's conservative force vector FQ inertial reference frame Fi \ t h link body-fixed reference frame n x n identity matrix alternate form of inertia matrix for the i * ' 1 l ink K A Z $i(k + dXi+1) Kei \ t h link kinetic energy M(q, t) system's coupled mass matrix Mma., Mma., Mmy- control moments in the pitch, roll and yaw directions, respec-tively for the \ t h link Mr, fr rigid mass matrix and force vector, respectively Mred; fred-. Qred mass matrix, force vector and generalized coordinate vector for the reduced model, respectively Mt block diagonal decoupled mass matrix symmetric, decoupled mass matrix N total number of links 0{N) order-N Pi(9i) column matrix [T^gi,Tp.g^T^gi) —* Q nonconservative generalized force vector Qu actuator coupling matrix or longitudinal L Q R state weighting matrix v i i i Rdm- inertial position of the \ t h link mass element drrii RP transformation matrix relating qt and tf Ru longitudinal L Q R input weighting matrix Rv,Rn, Rd transformation matrices relating qt and q S(f, M\dc) generalized acceleration vector of coupled system, q, for the full nonlinear, flexible system th uiirv l u w u u u maniA th T j i l ink rotation matrix Tta.,Tto control thrust in the pitch and roll direction for the im l ink, respectively \rQi maximum deployment/retrieval velocity of the \ t h l ink Vei l ink strain energy Vgi link gravitational potential energy Rayleigh dissipation function arising from structural damping in the ith link di position vector to the frame F{ from the frame F{_i • dc desired offset acceleration vector (i*' 1 l ink offset position) drrii infinitesimal mass element of the \ t h l ink dx^dy^dZi Cartesian components of d{ along the local vertical, local hori-zontal and orbit normal directions, respectively / F-Q —* gi r igid and flexible position vectors of drrii, fj + <E»^<5^  i,j,k unit vectors { 1 , 0 , 0 } r , {0,1,0}-^ and {0,0, l } r , respectively li length of the \ t h l ink raj mass of the \ t h link nfj total number of flexible modes for the i * ' 1 l ink, nXi + nyi + nZi nq total number of generalized coordinates per link, nfj + 7 nqq system's total number of generalized coordinates, Nnq ix number of flexible modes in the longitudinal, inplane and out-of-plane transverse directions, respectively for the i ^ link number of attitude control actuators { # , - . . , $ } T set of generalized coordinates for the itfl l ink which accounts for interactions with adjacent links {q?v---,QtN}T set of coordinates for the independent \ t h link (not connected to adjacent links) rigid position of dm, in the frame Fj position of centre of mass of the itfl l ink relative to the frame Fj i + $DJi desired settling time of the j * ' 1 attitude actuator actuator force vector for entire system flexible deformation of the link along the Xj, yi and Zj direc-tions, respectively control input for flexible subsystem Cartesian components of fj actual and estimated state of flexible subsystem, respectively output vector of flexible subsystem LIST OF FIGURES 1-1 A schematic diagram of the space platform based N-body tethered satellite system 2 1-2 Associated forces for the dumbbell satellite configuration 3 1- 3 Some applications of the tethered satellite systems: (a) multiple communication satellites at a fixed geostationary location; (b) retrieval maneuver; (c) proposed B I C E P S configuration 6 2- 1 Vector components of the i * ' 1 and (i- l )* ' 1 chain links 15 2-2 Vector components in cylindrical orbital coordinates 20 2-3 Inertial position of subsatellite thruster forces 38 2- 4 Coupled force representation of a momentum-wheel on a rigid body. . . 40 3- 1 Flowchart showing the computer program structure 45 3-2 Kinet ic and potential.energy transfer for the three-body platform based tethered satellite system: (a) variation of kinetic and potential energy; (b) percent change in total energy of system 48 3-3 Kinet ic and potential energy transfer for the five-body platform based tethered satellite system: (a) variation of kinetic and potential energy; (b) percent change in total energy of system 49 3-4 Simulation results for the platform based three-body tethered system originally presented in Ref.[43] 51 3- 5 Simulation results for the platform based three-body tethered system obtained using the present computer program 52 4- 1 Schematic diagram showing the generalized coordinates used to describe the system dynamics 55 4-2 Stationkeeping dynamics of the three-body STSS configuration without offset: (a) attitude response; (b) vibration response 57 4-3 Stationkeeping dynamics of the three-body STSS configuration xi with offset along the local vertical: (a) attitude response; (b) vibration response 60 4-4 Stationkeeping dynamics of the three-body STSS configuration with offset along the local horizontal: (a) attitude response; (b) vibration response 62 4-5 Stationkeeping dynamics of the three-body STSS configuration with offset along the orbit normal: (a) attitude response; (b) vibration response 64 4-6 Stationkeeping dynamics of the three-body STSS configuration wi th offset along the local horizontal, local vertical and orbit normal: (a) attitude response; (b) vibration response 67 4-7 Deployment dynamics of the three-body STSS configuration without offset: (a) attitude response; (b) vibration response 69 4-8 Deployment dynamics of the three-body STSS configuration with offset along the local vertical: (a) attitude response; (b) vibration response 72 4-9 Retrieval dynamics of the three-body STSS configuration with offset along the local vertical and orbit normal: (a) attitude response; (b) vibration response 74 4-10 Schematic diagram of the five-body system used in the numerical example. 77 4-11 Stationkeeping dynamics of the five-body STSS configuration without offset: (a) attitude response; (b) vibration response 78 4-12 Deployment dynamics of the five-body STSS configuration wi th offset along the local vertical: (a) attitude response; (b) vibration response 81 4-13 Stationkeeping dynamics of the three-body B I C E P S configuration x i i with offset along the local vertical: (a) attitude response; (b) vibration response 84 4-14 Cartwheeling dynamics with deployment for the three-body B I C E P S with offset along the local vertical: (a) attitude response; (b) vibration response 86 4-15 Spin dynamics (7 = l ° / s ) of the three-body O E D I P U S configuration with offset along the local vertical: (a) attitude response; (b) vibration response 89 4- 16 Spin dynamics (7 = 10° / s ) of the three-body O E D I P U S configuration wi th offset along the local vertical: (a) attitude response; (b) vibration response 91 5- 1 Controlled dynamics of the three-body STSS during stationkeeping, using the nonlinear, rigid F L T controller with offset along the local horizontal, local vertical and orbit normal: (a) attitude and vibration response; (b) control actuator time histories. 98 5-2 Controlled dynamics of the three-body STSS during stationkeeping, using the nonlinear, flexible F L T controller with offset along the local horizontal, local vertical and orbit normal: (a) attitude and vibration response; (b) control actuator time histories. 101 5-3 Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear, rigid F L T controller with offset along the local vertical: (a) attitude and vibration response; (b) control actuator time histories. 103 5-4 Deployment dynamics of the three-body STSS, using the nonlinear, rigid F L T controller with offset along the local vertical: (a) attitude and vibration response; (b) control actuator time histories. 105 5-5 Retrieval dynamics of the three-body STSS, using the non-linear, rigid F L T controller with offset along the local vertical and orbit normal: (a) attitude and vibration response; (b) control actuator time histories. 108 x i i i 5-6 Block diagram for the L Q G / L T R estimator based compensator. . . . 115 5-7 Singular values for the L Q G and L Q G / L T R compensator compared to target return ratio: (a) longitudinal design; (b) transverse design 118 5-8 Controlled dynamics of the three-body STSS during stationkeeping, using the nonlinear, rigid F L T attitude controller and L Q G / L T R offset vibration controller: (a) attitude and libration controller response; (b) vibration and offset response 122 xiv A C K N O W L E D G E M E N T Firs t and foremost, I would like to express the most genuine gratitude to my supervisor, Prof. V i n o d J . M o d i , whose unending guidance and encouragement proved most invaluable during the course of my studies. I am also deeply indebted to Dr . Satyabrata Pradhan, for his limitless patience and technical advice from which this work would not have been possible. I would also like to extend a word of appreciation towards Prof. A r u n K . Mis ra ( M c G i l l University) for ini t iat ing my interest in space dynamics and control. Also, I would like to thank my collegues, Dr . Sandeep Munshi , M r . Mathieu Caron and M r . Yuan Chen as well as Dr . Mae Seto, M r . Gary L i m and M r . Mark Chu for their words of advice and helpful suggestions that heavily contributed to all aspects of my studies. Final ly, I would like to thank all my family and friends here in Vancouver and back home in Montreal whose unending support made the past two years the most memorable of my life. The research project was supported by the Natural Sciences and Engineering/ Research Council 's P G S - A Scholarship held by the author as well as by N S E R C grant A-2181 held by Prof. V . J . M o d i . xv 1. INTRODUCTION 1.1 Prel iminary Remarks W i t h the ever changing demands of the world's population, one often wonders about the commitment to the space program. On the other hand, the importance of worldwide communications, global environmental monitoring as well as long-term habitation in space have demanded more ambitious and versatile satellites systems. It is doubtful that the Russian scientist Tsiolkovsky[l], when he first proposed the uti l izat ion of the Earth's gravity-gradient environment in the last century, would have envisioned the current and proposed applications of tethered satellite systems. A tethered satellite system consists of two or more subsatellites connected to each other by long, thin cables or tethers which are in tension (Figure 1-1). The subsatellite, which can also include the shuttle and space station, are in orbit together. The first proposed use of tethers in space was associated with the rescue of stranded astronauts by throwing a buoy on a tether, and reeling it to the rescue vehicle. Prel iminary studies of such systems led to the discovery of the inherent instability during the tether retrieval[2]. Nevertheless, the bir th of tethered systems occured during the Gemini X I and X I I missions[3] in 1966 when a short tether (10-30m) was used to generate the first artificial gravity environment in space (0.00015g) by cartwheeling (spinning) the system about its center of mass. The mission also demonstrated an important use of tethers, for gravity gradient stabilization. The force analysis of a simplified model illustrates this point. Consider the dumbbell satellite configuration orbiting about the centre of Ear th as shown in Figure 1-2. Satellite 1 and satellite 2 are connected to each other by a tether with 1 Space Station (Satellite 1 ) 2 Satellite N Figure 1-1 A schematic diagram of the space platform based N-body tethered satellite system. 2 Figure 1-2 Associated forces for the dumbbell satellite configuration. 3 the whole system free to librate about the system's centre of mass, C M . The two major forces acting on each body are due to the centrifugal and gravitational effects. Since body 1 is further away from the earth, its gravitational force, Fg^y is smaller than that of body 2, i.e. Fgi < Fg^. However its centrifugal force, FCl, is greater then the centrifugal force experienced by the second satellite (FCl > FC2). Moreover, for body 1, Fg^ < Fc^ in contrast to Fg2 > FC2 for body 2. Thus, resolving the force components along the tether, there is an evident resultant tension force, Ff, present in the tether. Similarly, adding the normal components of the two forces vectorially results in a force, FR, which restores the system to its stable equil ibrium along the local vertical. These tension and gravity-gradient restoring forces are the two most important features of tethered systems. Several milestone missions have flown in the recent past. The U S A / J a p a n project, T P E (Tethered Payload Experiment) [4], was one of the first to be launched using a sounding rocket to conduct environmental studies. The results of the T P E provided support for the N A S A / I t a l y shuttle based tethered satellite system, referred to as the TSS-1 mission[5]. This experiment aimed to study the electrodynamic effects of a shuttle borne satellite system, with a conductive tether connection, where an electric current was induced in the tether as it swept through the Earth 's magnetic field. Unfortunetely the two attempts, in August 1992 and February 1996, resulted in only partial success. The former suffered from a spool failure resulting in a tether deployment of only 256m of a planned 20km. During the latter attempt, a break in the tether's insulation resulted in arcing leading to tether rupture. Nevertheless, the information gathered by these attempts st i l l provided the engineers and scientists invaluable information on the dynamic behaviour of this complex system. Canada, also paved the way with novel designs of tethered system, aimed primari ly at the study of the ionosphere and Northern Lights (Aurora Borealis). 4 The O E D I P U S A and C (Observation of Electrified Distributions in the Ionospheric P lasma - a Unique Strategy) missions[6] launched from a sounding rocket, in January 1989 and November 1995 respectively, provided insight into the complex dynamical behaviour of two comparable mass satellites, connected by a 1 km long spinning tether. Final ly, two experiments were conducted by N A S A , called S E D S I and II (Small Expendable Deployable System) [7], which hold the current record of 20 km long tether. In each of these missions, a small probe (26 kg) was deployed from the second stage of a D E L T A II rocket thus succesfully demonstrating the feasibility of long tether deployment. Retrieval of the tether, which is significantly more difficult, has not yet been achieved. Several proposed applications are also currently under study. These include the study of earth's upper atmosphere using probes lowered from the shuttle in a low earth orbit ( L E O ) , to the deployment and retrieval of satellites for servicing. Even more promising application concerns the generation of power for the proposed International Space Station using conductive tethers. The Canadian Space Agency (CSA) has also proposed a dual-mass satellite system B I C E P S (Bl-static Canadian Experiment on Plasmas in Space) to make simultaneous measurements at different locations in the environment for correlation studies[8]. A unique feature of the B I C E P S mission is the deployment of the tether aided by the angular momentum of the system, which is ini t ia l ly cartwheeling about its orbit normal (Figure 1-3). In addition to the three-body configuration, multi-tethered systems have been proposed for monitoring Earth's environment in the global project, monitored by N A S A , entitled Mission to Planet Ear th ( M T P E ) . The proposed mission aims to study pollution control through the understanding of the dynamic interactions be-5 Retrieval of Satellite for Servicing \ / < — Satellite Figure 1-3 Some applications of the tethered satellite systems: (a) multiple com-munication satellites at a fixed geostationary location; (b) retrieval maneuver; (c) proposed BICEPS configuration. 6 tween the Earth's atmosphere, biosphere, hydrosphere and cryosphere. This wi l l be accomplished with multiple tethered instrumentation payloads simultaneously sound-ing at different altitudes for spatial correlations. Such mult ibody systems are consid-ered to be the next stage in the evolution of tethered satellites. Micro-gravity payload modules suspended from the proposed space station for long-term experiments as well as communications satellites with increased line-of-sight capability represent just two of the numerous possible applications under consideration. 1.2 Brief Review of the Relevant Literature The design of multi-payload systems wi l l require extensive dynamic analysis and parametric study before the final mission configuration can be chosen. This wi l l include a review of the fundamental dynamics of the simple two-body tethered systems and how its dynamics can be extended to those of multi-body tethered system in a chain or, more generally, a tree topology. 1.2.1 Multibody 0(N) formulation Many studies of two-body systems have been reported. One of the earliest contribution is by Rupp[9] who was the first to study planar dynamics of the sim-plified Shuttle based Tethered Satellite System (STSS). His pioneering work led to the discovery of pitch oscillation growth during the retrieval phase. Later, Baker[10] advanced the investigation to the third dimension in addition to adding atmospheric effects to the system. A more complete dynamical model was later developed and analyzed by M o d i and Misra[ l l ,12] which included deployment and tether flexibility. A comprehensive survey of important developments and milestones in the area have been compiled and reviewed by Mis ra and Modi[13]. The major conclusions based on the literature may be summarized as follows: • the stationkeeping phase is stable; 7 • deployment can be unstable if the rate exceeds a crit ical speed; • retrieval of the tether is inherently unstable; • transverse vibrations can grow due to the Coriolis force induced during de-ployment and retrieval. Misra , Amier and Modi[14] were one of the first to extend these results to the three-satellite, double pendulum, case. This simple model, which includes a variable length tether, was sufficient to uncover the multiple equilibrium configurations of the system. Later, the work was extended to include out-of-plane motion and a preliminary reel-rate control law to regulate the librational motion[15]. Kesmir i and Misra[16] developed a general formulation for N-body tethered systems, based on the Lagrangian principle where three-dimensional motion and flexibility are accounted for. However, from their work, it is clear that as the number of payload or bodies increases, the computational cost to solve the forward dynamics also increases dramatically. Tradit ional methods of inverting the mass matrix, M, in the equation M'L?+F = Q to solve for the acceleration vector proved to be computationally costly, being of the order for practical simulation algorithms. O(N^) refers to a mult ibody formula-tion algorithm whose computational cost is proportional to the cube of the number of bodies, N, used. It is therefore clear that more efficient solution strategies have to be considered. Many improved algorithms have been proposed to reduce the number of com-putational steps required for solving for the acceleration vector. Rosenthal[17] pro-posed a recursive formulation based on the triangularization of the equations of mo-tion to obtain an 0(N2) formulation. He also demonstrates the use of a symbolic manipulation program to reduce the final number of computations. A recursive L a -8 grangian formulation proposed by Book[18] also points out the relative inefficiency of conventional formulations and proceeds to derive an algorithm which is also 0(N2). A recursive formulation is one where the equations of motion are calculated in order, from one end of the system to the other. It usually involves one or more "passes" along the links to calculate the acceleration of each link (forward pass), and the constraint forces (backward or inverse pass). Non-recursive algorithms express the equations of motion for each link independent of the other. Although the coupling terms st i l l need to be included, the algorithm is, in general, more amenable to parallel computing. A st i l l more efficient dynamics formulation is an O(N) algorithm. Here the computational effort is directly proportional to the number of bodies or links used. Several types of such algorithms have been developed over the years. The one based on the Newton-Euler approach has recursive equations and is used extensively in the field of multi-joint robotics[19,20]. It should be emphasized that efficient computation of the forward and inverse dynamics is imperative if any on-line (real-time) control of the joint is to be achieved. Hollerbach[21], proposed a recursive O(N) formulation based on Lagrange's equations of motion. However, his derivation was primari ly fo-cused on the inverse dynamics and did not improve the computational efficiency of the forward dynamics (acceleration vector). Other recursive algorithms include meth-ods based on the principle of vir tual work[22] and Kane's equations of motion[23]. Keat[24] proposed a method based on a velocity transformation that eliminated the appearance of constraint forces and was recursive in nature. O n the other hand, K u r d i l a , Menon and Sunkel[25] proposed a non-recursive algorithm based on the Range-Space method, which employs element-by-element approach used in modern finite-element procedures. Authors also demonstrated the potential for parallel com-putation of their non-recursive formulation. The introduction of a Spatial Operator Factorization[26,27,28], which utilizes an analogy between mult ibody robot dynamics and linear filtering and smoothing theory, to efficiently invert the system's mass ma-9 t r ix is another approach to a recursive algorithm. Their results were further extended to include the case where joints follow user-specified profiles[29]. More recently, Prad-han et al. [30] proposed a non-recursive Lagrangian algorithm for factorizing the mass matrix leading to an O(N) formulation of the forward dynamics of the system. A n efficient O(N) algorithm where the number of bodies, N , in the system varies on-line has been developed by Banerjee[31]. 1.2.2 Issues of tether modelling The importance of including tether flexibility has been demonstrated by several researchers in the past. However, there are several methods available for modelling tether flexibility. Each of these methods has relative advantages and limitations wi th regards to the computational efficiency. A continuum model, where flexible motion is discretized using the assumed mode method, has been proposed[32,33] and succesfully implemented. In the majority of cases, even one flexible mode is sufficient to capture the dynamical behaviour of the tether[34]. K i m and Vadali[35] proposed a so-called bead model or lumped-mass approach that discretizes the tether using point masses along its length. However, in order to accurately portray the motion of the tether, a significantly high number of beads are needed thus increasing the number of computation steps. Consequently, this has led to the development of a hybrid between the last two approaches[16], i.e. interpolation between the beads using a continuum approach thus requiring less number of beads. Finally, the tether flexibility can also be modelled using a finite-element approach[36] which is more suitable for transient response analysis. In the present study, the continuum approach is adopted due to its simplicity and proven reliability in accurately conveying the vibratory response. In the past, the modal integrals arising from the discretization process have been evaluted numerically. In general, this leads to unnecessary effort by the com-puter. Today, these integrals can be evaluated analytically using a symbolic in-10 tegration package, e.g. Maple V . Subsequently, they can be coded in F O R T R A N directly by the package which could also result in a significant reduction in debugging time. More importantly, there is considerable improvement in the computational effi-ciency [16], especially during deployment and retrieval where the modal integral must be evaluated at each time-step. 1.2.3 Attitude and vibration control In view of the conclusions arrived at by some of the researchers mentioned above, it is clear that an appropriate control strategy is needed to regulate the dy-namics of the system, i.e. attitude and tether vibration. First , the attitude motion of the entire system must be controlled. This, of course, would be essential in the case of a satellite system intended for scientific experiments such as the micro-gravity facili-ties aboard the proposed Internation Space Station. Vibra t ion of the flexible members wi l l have to be checked if they affect the integrity of the on-board instrumentation. Thruster as well as momentum-wheel approach[34] have been considered to regulate the rigid-body motion of the end-satellite, as well as the swinging motion of the tether. The procedure is particularly attractive due to its effectiveness over a wide range of tether lengths as well as ease of implementation. Other methods in-clude tension and length rate control which regulate the tether's tension and nominal unstretched length, respectively[9,15]. It is usually implemented at the deployment spool of the tether. More recently, an offset strategy involving time dependent mo-tion of the tether attachment point to the platform has been proposed[37,38]. It overcomes the problem of plume impingement created by the thruster control and the ineffectiveness of tension control at shorter lengths. However, the effectiveness of such a controller can become limited with an exceedingly long tether due to a prac-tical l imi t on the permissible offset motion. In response to these two control issues, a hybrid thruster/offset scheme has been proposed to combine the best features of the 11 two methods[39]. In addition to attitude control, these schemes can be used to attenuate the flexible response of the tether. Tension and length rate control[12] as well as thruster based algorithms[33,40] have been proposed to this end. M o d i , et al. [41] have suc-cessfully demonstrated the effectiveness of an offset strategy to damp undesired v i -brations. Passive energy dissipative devices, e.g. viscous dampers, are also another viable solution to the problem. The development of various control laws to implement the above mentioned strategies has also recieved much attention. A n eigenstructure assignment in conduc-tion wi th an offset controller for vibration attenuation and momemtum wheels for platform libration control has been developed[42], in addition to the controller design from a graph theoretic approach[41]. Also, non-linear feedback methods, such as the Feedback Linearization Technique ( F L T ) , that are more suitable for controlling highly nonlinear, non-autonomous, coupled systems have also been considered[43]. It is important to point out that several linear controllers, including the clas-sic state feedback Linear Quadratic Regulator ( L Q R ) , have received considerable attention[39,44]. Moreover, robust methods such as the Linear Quadratic Guas-s ian /Loop Transfer Recovery ( L Q G / L T R ) method have also been developed and im-plemented on tethered systems[43]. A more complete review of these algorithms as well as others applied to tethered system has been presented by Pradhan[43]. 1.3 Scope of the Investigation The objective of the thesis is to develop and implement a versatile as well as computationally efficient formulation algorithm applicable to a large class of tethered satellite systems. The distinctive features of the model can be summarized as follows: 12 • TV-body, 0(N) tethered satellite system in a chain-type topology; • the system is free to negotiate a general Keplerian orbit and permitted to undergo three dimensional inplane and out-of-plane l ibrtational motion; • r igid bodies constituting the system are free to execute general rotational motion; • three dimensional flexibility present in the tether which is discretized using the continuum assumed-mode method, with an arbitrary number of flexible modes; • energy dissipation through structural damping is included; • capability to model various mission configurations and maneuvers including the tether spin about an arbitrary axis; • user-defined, time dependent deployment and retrieval profiles for the tether as well as the tether attachment point (offset); • attitude control algorithm for the tether and rigid bodies, using thrusters and momentum-wheels, based on the Feedback Linearization Technique; • the algorithm for suppression of tether vibration using an active offset (tether attachment point) control strategy based on the optimal linear control law ( L Q G / L T - R ) . To begin with , in Chapter 2, kinematics and kinetics of the system are derived using the O(N) formulation methodology, developed by Pradhan, et al. [30]. Chap-ter 3 discusses issues related to the development of the simulation program and its validation. This is followed by a detailed parametric study of several mission pro-files, simulated as particular cases of the versatile formulation, in Chapter 4. Next, the attitude and vibration controllers are designed and their effectiveness assessed in Chapter 5. The thesis ends with concluding remarks and suggestions for future work. 13 2. F O R M U L A T I O N OF T H E P R O B L E M 2.1 Kinematics 2.1.1 Preliminary definitions and the i ^ position vector The derivation of the equations of motion begins with the definition of the inertial position of an arbitrary link of the mult ibody system. Let the i ^ link of the system be free to translate and rotate in 3-D space. From Figure 2-1, the position vector Rdm^ to the mass element drrn on the i ^ link, wi th reference to the inertial frame FQ , can be written as Here D{ represents the inertial position of the \ t h body fixed frame Fj relative to FQ; ri = [xi,yi,Zi]T is the rigid position vector of drrii wi th respect to F J ; and f/(fi) is the flexible deformation at fj also with respect to the frame Fj. Bo th these vectors are relative to Fj . Note, in the present model, tethers (i even) are considered flexible and X{ corresponds to the nominal position along the unstretched tether while \a and ZJ are, by definition, equal to zero. On the other hand, the rigid bodies referred to as satellites (i odd), including the platform, are taken to be rigid, i.e. f?(fj) = 0. T j is the rotation matrix used to express body-fixed vectors with reference to the inertial frame. 2.1.2 Tether flexibility discretization The tether flexibility is discretized, with an arbitrary but finite number of Di + Ti(?i + f?(?i)). (2.1) 14 15 modes in each direction, using the assumed-mode method as Ui \ \ nvi . wi I I i = 1 nzi • (2.2) where nXi, nyi and n 2 ^ are the number of modes used in the X{, m and Z{ directions, respectively. For the ] t h longitudinal mode, the admissible mode shape function is taken as / , . \ 2 j - l Hi(xi,ii)=[j:) , (2.3) where l{ is the i ^ tether length[32]. In the case of inplane and out-of-plane transverse deflections, the admissible functions are &yi(xi,li) = &Zi{xuk) = v ^ s i n JKXj k (2.4) where y/2 is added as a normalizing factor. In this case both the longitudinal and transverse mode shapes satisfy the geometric boundary conditions for a simple string in axial and transverse vibration. Eq.(2.2) is recast into a compact matrix form as ff(?i) = $i(xi,li)5i(t), (2.5) where $i(x{, l{) is the matrix of the tether mode shapes defined as 0 0 ^i(xiJi) = ^ X ; • • • ^XA 0 0 0 0 G 5R 3 x n f* (2.6) and nfj = nx- + nyi + nZi. Note, $J(XJ,/J) is a function of the spatial coordinate x\ and not of yi or ZJ. However, it is also a function of the tether length parameter, li, 16 which is time-varying during deployment and retrieval. For this reason, care must be exercised when differentiating $J(XJ,/J) with respect to time, such that d d d k) = Jt®i(Xi' li) = ^ 7 $ t ( ^ i . k)Xi + gf$i(xi> li)k- (2-7) Since x\ represents the nominal rate of change of the position of the elemental mass drrii, it is equal to /j (deployment/retrieval rate). Let t ing ( 3 d \ dx~ + dT)^Xi,li^ ( 2 ' 8 ) one arrives at $i(xi,li) = $Di(xi,li)ii. (2.9) The time varying generalized coordinate vector, 5{ in Eq.(2.5), is composed of the longitudinal and transverse components, i.e. /<M*)\ G s f t n f i x l , (2.10) where 5X^, 5y- and Sz^ are nx- x 1, ny. x 1 and nz- x 1 size vectors and correspond to $ X { , and $ Z { , respectively. 2.1.3 Rotation angles and transformations The matrix T j in Eq.(2.1) represents a rotation transformation from the body fixed frame Fj to the inertial frame FQ, i.e. FQ = T j F j . It is defined by the 3-2-1 ordered sequence of elementary rotations[46]: 1. P i tch , F[ = Cf ( a j )F 0 ; 2. R o l l , F / ' = C f (/3j)F/ = C7f (/3j)Cf ( a j )F 0 ; 3. Yaw, F , = C7( 7 i )F / ' = C7(7l)cf\^)Cf(ai)FQ = Q F 0 ; 17 where: c f (A) = -Sai Cai 0 0 and C?(7i ) = 0 -s0i] 0 1 0 0 cpi\ " i 0 0  0 0 (2.11) (2.12) (2.13) Here Eq.(2.11) corresponds to a rotation of ctj (pitch) about the inertial axis ZQ (orbit normal) of frame FQ resulting in a new intermediate frame F ' . It is then followed by a roll rotation, fa, about the axis y\ of F- giving a second intermediate frame F-'. Final ly, a spin rotation 7J about the axis z" of F-' is given resulting in the body fixed frame Fj for the i < / l link. Since all the three rotation matrices are orthogonal, it follows that FQ = C~1Fi = Cj F, and hence C CPi Cai ~ CH Sai + SrYi SPi Sai 5 7 i S&i + Cli SPi Cc*i CPi Sai C1i Cai + 5 7 i SPi ~ S1{ Ca{ + C 7 • Sp Sai (2.14) SliCPi CliCPi Here Cx and Sx are abbreviations for cos(x) and sin(x), respectively. 2.1.4 Inertial velocity of the ith link —* Lett ing pj = fj - f $j(5j and differentiating Eq.(2.1) with respect to time, gives ,m,- D j + T j f j + T^Si + T j £ . + T j $ j 5 j (2.15) Each of the terms appearing in Eq.(2.15) must be addressed individually. Since fj represents the rigid body motion within the link, it only has meaning for the deployable tether; and since yi = Zj = 0 for al l tether links (i even), fj = Ifi where 1 is the unit vector {1, 0, 0}T. For the case of rigid links (i odd), fj = 0. 18 Evaluating the derivative of each angle in the rotation matrix gives T^i9i = Taiai9i + Tp.pigi + T^fagi (2.16) where Tx = J j I V Collecting the scaler angular velocity terms and defining the set Vi = {°^> A) 7i}^\ Eq.(2.16) can be rewritten as T i f t = Pi{9i)fji (2.17) where Pi(gi) is a column matrix defined by Pi(9i)Vi = [Taigi^p^i.T^gilffi. (2.18) Inserting Eq.(2.9) and Eq.(2.17) into Eq.(2.15), and defining Sj = 2 + leads to 2.1.5 Cylindrical orbital coordinates The Di term in Eq.(2.19) describes the orbital motion of the \ t h l ink and is composed of the three Cartesian coordinates DXi, A ^ , DZi. However, over the cycle of one orbit, DXi and Dyi vary dramatically, by around the order of Earth's radius. This must be avoided since large variations in the coordinates can cause severe truncation errors during their numerical integration. For this reason, it is more convenient to express the Cartesian components in terms of more stationary variables. This is readily accomplished using cylindrical coordinates. However, it is only necessary to transform the first l ink's orbital components, i.e. L>i, since only D\ is a generalized coordinates. From Figure 2-2, it is apparent (2.19) that (2.20) 19 Figure 2-2 Vector components in cylindrical orbital coordinates. 20 Eq.(2.20) can be rewritten in matrix form as cos(0i) 0 0 s i n ( 0 i ) 0 0 0 0 1 Dri h (2.21) = DTlDSv where Dri is the inplane radial distance; 9\, the true anomaly; and DZl, the out-of-plane distance normal to the orbital plane. The total derivative of D\ wi th respect to time gives, in column matrix form, 1 1 1 1 J (2.22) = D D l D 3 v where [cos(^i) -Dnsm(8i) 0 ] (2.23) For all remaining links, i.e. i = 2 ... N, cos 0 A - j S i n ^ i )  sin(^i) D r i c o s ( 0 i ) 0 0 0 1 DTl = D D i = I d 3 , i = 2...N, (2.24) where is the n x n identity matrix and 2...N. (2.25) 2.1.6 Tether deployment and retrieval profile The deployment of the tether is a critical maneuver for this class of systems. Cr i t i ca l deployment rates would cause the system to librate out of control. Normally, a smooth, sinusoidal profile is chosen (S-curve). However, one would also like a long tether extending to ten, twenty or even a hundred kilometers in a reasonable time, say 3-4 orbits. This is usually difficult or impossible to accomplish solely wi th one S-curve profile. Often, a composite S-curve-Steady-S-curve profile is adopted. Thus, 21 the deployment scheme for the \ t h tether can be summarized as: k = "Tr { l - cos (^r(ti - to,-)) 1, \ <U< h.; 2 r V A V * , k = Vov tii<ti<t2i\ k = IT j 1 - C ° S {lkitl " ^ } ' ^ < *i < ^ (2.26) where to-, tt. are the ini t ia l and final times of the deployment or retrieval maneuver, respectively. A t j = t\. - tQ. — tt. — t2- and the steady deployment velocity VQ. is calculated based on the continuity of l{ and l{ at the specified intermediate times t\{ and to.. For the case of and A/i = /i(ti.) - ^(t 0 -) = - ^ ( t 2 i ) , (2.27) 2Ali + li(tf.) -h(tQ.) V = lUli  %l^lL (2.28) 1 tf.-tQ. v y A t , = (2.29) 2.2 Kinetics and System Energy 2.2.1 Kinetic energy The kinetic energy of the ith link is given by Kei = lJm Rdrnfi^Ami. (2.30) Setting Eq.(2.19) in a matrix-vector form, gives the relation 4 , ^ ^ PM) T i $ 4 TiSi]i[t (2.31) 22 where m \ k J x l (2.32) and nq = -nfj + 7 is the number of generalized coordinates in each link. Inserting Eq.(2.31) into Eq.(2.30) and integrating, the kinetic energy for the \ t h l ink can be rewritten as 1 .T (2.33) where Mti is the link's symmetric mass matrix, Mt% = m i D D , T D D . DD.TPi{fgidmi) DD.TTi J ^dm, D n / T , / Sidm, rn sym , fP?(9i)Pi(9i)dmi J P[{g^T^idm, J P / ' ( £ ) T ^ m sym sym J ^f^idm, f ^f^dmi sym sym sym J sfsidmi £ Sfttiqxnq (2.34) and mi is the mass of the Ith link. The kinetic energy for the entire system, i.e. N bodies, can now be stated as 1 . • T • 1 • T Ke = ^ J2 %MtA = o « Mtqt, 2 ^ *-i i=l (2.35) where qt = {qj^,^, • • -,qfN}T a n d Mt is a block diagonal, decoupled mass matrix of the system with Mf- on its diagonal, (2.36) Mt = 0 0 0 Mt2 0 0 . 0 . MH . 0 0 0 0 0 0 • MtN 2.2.2 Simplification for rigid links The mass matrix given by Eq.(2.34) simplifies considerably for the case of rigid 23 links and is given as Mti m, D D lT D D l DD^Piiffidmi) 0 sym 0 0 fP?(fi)Pi(?i)dmi 0 0 0 0 Id nfj 0 m~ , i = 1 , 3 ,5 , . . . , N. (2.37) Moreover, when mult iplying the matrices in the (1,1) block term of Eq.(2.37), one gets D D l 1 D D l = 1 0 0 0 D n 2 0 (2.38) 0 0 1 which is a diagonal matrix. However, more importantly, i t is independent of 6\ and hence is always non-singular ensuring that is always invertible. In the (1,2) block term of M ^ , it is apparent that f fidmi is simply the definition of the centre of mass of the \ t h rigid link and hence given by / To drrii — miff (2.39) where fcmi is the position vector of the centre of mass of link i . For the case of the (2,2) block term, a little more attention is required. Expanding the integrand using Eq.(2.18), where <?j = fj, gives fj Ta{ Tafi fj T Q . Tp.fi f- T Q . T 7 i f j sym sym Now, since each of the terms of Eq.(2.40) is a scaler or a 1 x 1 matrix, it is therefore equal to its trace, i.e. sum of its diagonal elements, thus 'tr(ffTaiTTaifi) tr(ffTaiTTp.fi) tr(ff Ta^T^fi)' fjT^Tp.n rfTp.^fi (2.40) P/(rj)Pj(fi) = syrri trifjTpTTp.fi) t r ^ T ^ T ^ sym sym tr(ff T 7 . T r 7 . f j) J (2.41) Using two properties of the trace function[47], namely: tr(ABC) = tr(BCA) = tr{CAB); (2.42) 24 and j tr(A) =tr(^J A^j; (2.43) where A, B and C are real matrices, it becomes quite clear that the (2,2) block term of Eq.(2.37) simplifies to / P?(fi)Pi{?i)dmi = tr(TaiTTaiImi) tT{TaiTTp.Imi) t r (T Q . r T 7 . 7 m . ) tr{TaTTpImi) tr(TpTT7Jmi) sym sym sym (2.44) where Im- is an alternate form of the inertia tensor of the \ t h link which, when expanded, is Im« = I nfi dm %• i sym sym sym J x^dmi J X{y{dmi / X{Z{dm. sym J yi2dmi f yiZidm, sym J z^dmi J xVi -L VH sym ijxxi + lyy, IZZJ)/2 (2.45) Note, this is in terms of moments of inertia and products of inertia of the \ t f l rigid link. A similar simplification can be made for the flexible case where T{ is replaced with Qi resulting in a new expression for Im.. 2.2.3 Gravitational potential energy The gravitational potential energy of the ith link, due to the central force law, can be written as V9i f dm, f -A* / — = -A* / •>™<i \RdmA dm; inn \L>i + T j ^ | ' (2.46) where /z = 3.986 x 105 km 3/s 2 (Earth's gravitational constant). Expanding binomially and retaining up to third order terms gives Vgi ~ ~ D T 2D? [J gfgidrrii + 2/3f T ; j gidm{ —DT D2 i ^ l I I Df Tt I grffdmiTiDi (2-< 7) 25 where D{ = \D{\ is the magnitude of the inertial position vector to the frame F{. The integrals in the above expression are functions of the flexible mode shapes and can be evaluated through symbolic manipulation using an existing package, such as Maple V . In turn, Maple V can translate the evaluated integrals into F O R T R A N and store the code in a file. Furthermore, for the case of rigid bodies, it can be shown quite readily that j gjgidrrii = j rjfidrrn = [IXXi + Iyy{ + IZZi)/2. (2.48) Similarly, the other integrals in Eq.(2.34) can be integrated symbolically. 2.2.4 Strain energy When deriving the elastic strain energy of a simple string in tension, the assumptions of high tension and low amplitude of transverse vibrations are generally valid. Consequently, the first order approximation of the strain-displacement relation is generally acceptable. However, for orbiting tethers in a weak gravitational-gradient field, neglecting higher order terms can result in poorly modelled tether dynamics. Geometric nonlinearities are responsible for the foreshortening effects present in the tether. These cannot be neglected because they account for the heaving motion along the longitudinal axis due to lateral vibrations. The magnitude of the longitudinal oscillations diminish as the tether becomes shorter[35]. W i t h this as background, the tether strain, which is derived from the theory of elastic vibrations[32,48], is given by (2.49) where Uj, V{ and W{ are the flexible deformations along the Xj, and Z{ direction respectively, and are obtained from Eq.(2.5). The square bracketed term in Eq.(2.49) represents the geometric nonlinearity and is accounted for in the analysis. 26 The total strain energy of a flexible link is given by Ve% = I f CiEidVl (2.50) Substituting the stress-strain relationship, 0{ = E(£i, the strain energy is now given by Vei=l-EiAi J l \ l d x h (2.51) where E^ is the tether's Young's modulus and A{ is the cross-sectional area. The tether is assumed to be uniform thus E{A{. which is the flexural stiffness of the tether, is assumed to be constant. Eq.(2.51) is evaluated symbolically for arbitrary number of modes. 2.2.5 Tether energy dissipation The evaluation of the energy dissipation due to tether deformations remains a problem not well understood even today. Here, this complex phenomenon is rep-resented through a simplified structural damping model[32]. In addition, the system exhibits hysteresis, which also must be considered. This is accomplished using an augmented complex Young's modulus EX^Ei+jEi., (2.52) where Ej^ is the contribution from the structural damping and j = The augmented stress-strain relation is now given as a* = E*£i = Ei(l+jrldi)ei, (2.53) where 77^. = Ej./E^ <C 1 is defined as the structural damping coefficient determined experimentally [49]. 27 If £i is a harmonic function, with frequency UJQ.-, then jet = (2.54) wo,-% Substituting into Eq.(2.53) and rearranging the terms gives e* + I — I £i = o-i + ad., (2.55) where ad. and a, are the stress with and without structural damping, respectively. Now the Rayleigh dissipation function[50] for the ith tether can be expressed as / (ii) Jo _ 1 EjAiVdi rU (2-56^ 2 w 0 .  The strain rate, ii, is the time derivative of Eq.(2.49). The generalized external force due to damping can now be written as Qd = {Q^, • • • ,QdN}T where dWd. ^ = - i p ! = 2 ' 4 ( 2 . 5 7 ) = 0, i = 1,3, ...,N. 2.3 0(N) Form of the Equations of Mot ion 2.3.1 Lagrange equations of motion W i t h the kinetic energy expression defined and the potential energy of the whole system given by Pe = ]Cn=l(^ 5i+^ ej); the equations of motion can be obtained quite readily using the Lagrangian principle where q is the set of generalized coordinates, to be defined in a later section. Substituting Eqs.(2.35,2.47,2.51 and 2.57) into Eq.(2.58) leads to the familiar matr ix form of the non-linear, non-autonomous, coupled equations of motion for the 28 system, M{q, t)t+ F(q, 'q, t) = Q(q, 'q, t), (2.59) where M(q, t) is the nonlinear symmetric mass matrix; and F(q, q, t) is the forcing function, which can be written in matrix form as Q(<f> Qi 0 is the vector of non-conservative generalized forces, including control in-puts, acting on the system. A detailed expansion of Eq.(2.60) in tensor notation is developed in Appendix I. Numerical solution of Eq.(2.59) in terms of q would require inversion of the mass matrix. However, M is a full matrix of size nqq x nqq, where nqq = Nnq is the total number of generalized coordinates. Therefore, direct inversion of M would lead to a large number of computation steps, of the order nqq^ or higher. The objective here is to develop an order-N, O(N), algorithm for obtaining M _ 1 that minimizes the computational effort. This is accomplished through the following transformations ini t ia l ly developed for the stationkeeping case by Pradhan et al. [30]. 2.3.2 Generalized coordinates and position transformation During the derivation of the energy expressions for the system, the focus was on the decoupled system, i.e. each link was considered independent of the others. Thus each link's energy expression is also uncoupled. However, the constraints forces between two links must be incorporated in the final equations of motion. Let e Unqxl (2.61) m be the set of coupled coordinates of the \ t h l ink such that q = {qT, q2 , • • •, 0%}T i s the 29 system's generalized coordinates. Let qt- be the set of auxiliary decoupled coordinates, defined by Eq.(2.32), such that qt = {qj^, qj^,..., Qtj^}1 • The only difference between qtj and q, is the presence of D{ against d,. is the position of F{ wi th respect to the inertial frame FQ, whereas d{ is defined as the offset position of F{ relative to and projected on the frame. It can be expressed as d{ = {dXi,dyi,dZi}'^. For the special case of link 1, D\ = d\.. From Figure 2-1, can written as Di = A - l + T i _ i { J i _ i * + di) + T V i S i - i f t - i + dXi)8i-!. (2.62) Denoting KAj = $j(Zj + dx- ,), Eq.(2.62) can be rewritten in summation form as i &i = H ( T i - 1 ^ ' + T i - l K A i - l % - l + T j - l ^ - l ) • ( 2 - 6 3 ) 3 = 1 Introducing the index substitution k — j — 1 into Eq.(2.63), i-1 D{ = ^2 (Tfc_i<4 + TkKAk6k + Tkilk) + T^di (2.64) k=l since d\ = D\, TQ = 1^ and IQ, DQ, SQ are null vectors. Recasting Eq.(2.64) in matr ix form leads to i-1 Qt, Jfe=l where: and For the entire system, T fc-1 0 TkKAk 0 0 0 0 0 0 0 0 0 0 0 0 -r T i - i 0 0 0" 0 0 0 0 0 0 G . 0 0 0 1. Qt = RpQ, G *R n < ? x l . (2.65) (2.66) G Unq x l (2.67) (2.68) 30 where RP R 1 0 0 R{ R 2 0 R \ RP2 R 3 0 0 0 R \ RP . . . R P N _ 1 RN J (2.69) Here, RP is the lower block triangular matrix which relates qt and q. 2.3.3 Velocity transformations The two sets of velocity coordinates qti and qi are related by the following transformations: (i) First Transformation Differentiating Eq.(2.62) with respect to time gives the inertial velocity of the \ t h l ink as D{ = A - l + Ti_iK + + K A i _ i ^ _ i } + T j _ i ^ + Ti-ii + T i - i K A i - i ^ + T j - i K A ^ i ^ i . (2.70) (2.71) ( Defining K A D ; = d/ddXi+1{KAi}, K A L ; = d / d / ; { K A ; } and hi = di+1 +lfi +KA^i, results in K A ^ i = K A D j . i o ^ + K A L i _ i i _ i = KADi_iiTdi + K A L j _ i ^ _ i . Inserting Eq.(2.71) into Eq.(2.70) and rearranging gives = A _ ! + T i . ^ / d g + K A D ^ x ^ i ^ } ^ + Pi-^hi-iH-i + T i _ i K A i _ i 5 ; _ i + T i_ i{2 + K A L j _ i 5 j _ i } 4 _ i . (2.72) Using Eq.(2.72) to define recursively - D j _ i and applying the index substitution, /c = j — 1, as in the case of the position transformation, it follows that Di = ^ { T ^ i K D ^ ! ^ + Pk(hk)ffk + T f c K A f c ^ + TfcKLfc4} Jfc=l (2.73) + T j _ i K D j _ i < i . ii 31 where K D j _ i = + K A D ^ i ^ - i r 7 , and K L j = i + K A L j 5 j . Finally, by following a procedure similar to that used for the position transformation, it can be shown that for the entire system ?t = Rvil, (2-74) where Rv is given by the lower block diagonal matrix Here: Rv Rf 0 0 R\ R$ 0 R\ Rl RJ 0 0 0 T3V nN-l R •N e Mnqqxl. Ri = T i _ i K D i _ i TiKAi T j K L j 0 0 0 0 •o 0 0 . 0 0 0 0 0 e K n < ? x l ; RS T i-1 0 0 o-0 0 0 0 0 0 0 0 0 1. G Mnq x l (2.75) (2.76) (2.77) (ii) Second Transformation The second transformation is simply Eq.(2.72) set into matrix form. This leads to the expression for qt. as (2.78) where Id Pi'hi) T j K A j T j K L j 0 0 0 0 0 0 0 0 0 0 0 0 G $lnq x l Thus, for the entire system, (2.79) £ = Rnjt + Rd'q = Vdnqq - RnrlRdi (2.80) 32 where: and - 0 0 0 RI 0 0 Rn = 0 0 . 0 0 'Rf 0 0 0 r>d K2 0 Rd = 0 0 Rl 0 0 0 RN-1 0 G 5R n w x l; 0 0 0 e 5 R N ^ X L . (2.81) (2.82) 2.3.4 Cylindrical coordinate modification Because of the use of cylindrical coorfmates for the first body, it is necessary to slightly modify the above matrices to account for this coordinate system. For the first link, d\ — D\ = D ^ D S L , thus Qti = 01 => DDTigi, (2.83) where D D T i Eq.(2.69) now takes the form DTL 0 0 0 / d 3 0 0 0 ID 0 0 nfj 0 RP = P f D T T i R2 0 P f D T T i Rp2 R3 iJfDTTi i?^ . . . x l 0 0 0 RPN-I  RN (2.84) (2.85) In the velocity transformation, a similar modification is necessary since D\ = DD^DS^. Mak ing the substitution, it can be shown that Eq.(2.76) and Eq.(2.79) now 33 takes the form, for i = 1, T)V _ pn _ r t j — rt^ — 0 0 0 0 0 0 0 0 0 0 0 0 (2.86) The rest of the matrices for the remaining links, i.e. i = 2 ... N, are unchanged. 2.3.5 Mass matrix inversion Returning to the expression for kinetic energy, substitution of Eq.(2.74) into Eq.(2.35) gives the first of two expressions for kinetic energy as Ke = -q RvT MtRv (2.87) Introduction of Eq.(2.80) into Eq.(2.35), results in the second expression for Ke as 1:X 2q (hnqq-RnrlRd Mt (idnqq - RnrlRd (2.88) Thus there are two factorizations for the system's coupled mass matrix given as M = RV MtRv; M (hnqq - RnrlRd Mt ( i d n q q - R n r 1 R d (2.89) (2.90) Inverting Eq.(2.90) r-1 _ ( r>d\~^ M "  = (Ra) \ l d n q q - Rn)Mf1 \(Rd)~l(Idnqq - Rn) (2.91) Since both Rd and Mt are block diagonal matrices, their inverse is simply the inverse 34 of each block on the diagonal, i.e. Mr1 h 0 0 0 0 0 0 0 0 0 0 0 0 tN (2.92) Each block has the dimension nq x nq and is independent of the number of bodies in the system, thus the inversion of M is only linearly dependent on N, i.e. an O(N) operation. 2.3:6 Specification of the offset position The derivation of the equations of motion using the O(N) formulation requires that the offset coordinate d{ be treated as a generalized coordinate. However, this offset may be constrained or controlled to follow a prescribed motion dictated either by the designer or the controller. This is achieved through the introduction of Lagrange multipliers[51]. To begin with, the multipliers are assigned to all the 'd ' equations. Thus, letting f = F - Q, Eq.(2.59) becomes Mq + f = P C A , (2.93) where A G K 3 - 7 V x l is the vector of Lagrange multipliers and Pc G s R n ? 9 x 3 A r is the permutation matrix assigning the appropriate Aj to its corresponding d\ equation. Inverting M and pre-multiplying both sides by PcT gives pcTM-\pc = ( dc + P°TM-^f), (2.94) where dc is the desired offset acceleration vector. Note, both dc and PcTM lf are known. Thus, the solution for A has the form A pcTM-lpc - l dc + PC1 M (2.95) 35 Now, substituting Eq.(2.95) into Eq.(2.93) and rearranging the terms gives "q = S(f, M\l) = -M~lf + M~lPc which is the new constrained vector equation of motion with the offset specified by dc. Note that pre-multiplying M~l by P c T provides the rows of M _ 1 corresponding to the d equations. Similarly post-multiplying by Pc leads the columns of the matrix corresponding to the d equations. 2.4 Generalized Control Forces 2.4.1 Preliminary remarks The treatment of non-conservative external forces acting on the system is considered next. In reality, the system is subjected to a wide variety of environmental forces including atmospheric drag, solar radiation pressure, Earth 's magnetic field, as well as dissipative devices to mention a few. However, in the present model, only the effect of active control thrusters and momemtum-wheels is considered. These generalized forces wi l l be used to control the attitude motion of the system. In the derivation of the equations of motion using the Lagrangian procedure, the contribution of each force must be properly distributed among all the generalized coordinates. This is acheived using the relation nu ^5 Qk=E Pern ' ^ (2-97) m=l ° Q k where Qk and qk are the kth elements of Q and <f, respectively[51]. Fem is the mth external force acting on the system at the location Rm from the inertial frame, and nu is the total number of actuators. Derivation of the generalized forces due to thrusters and momentum wheels is given in the next two subsections. pcTM-lpc (dc + PcTM- Vj, (2-96) 36 2.4.2 Generalized thruster forces One set of thrusters is placed on each subsatellite, i.e. al l the rigid bodies, except the first (platform), as shown in Figure 2-3. They are capable of firing in the three orthogonal directions. From the schematic diagram, the inertial position of the thruster located on body 3 is given as Defining R3 = di + Tid2 + T2(hi + KA2<52) + T 3 f c m 3 = d\ + dfY + T 3 f c m 3 . df = Tjdj+i + Tj+1{lj+ii + KAj+i5j+i), the thruster position for body 5 is given as (2.98) (2.99) R5 = di+ dfx + d / 3 + T5fcm, Thus, in general, the inertial position for the thruster on the \ t h body can be given (2.100) as i - 2 k=l i1 cm; (2.101) Let the column matrix [Q\, i iT 9 R-K=rR% dqj lAni WjRi ARi (2.102) where i = 3, 5 , . . . , N and j — 1, 2 , . . . , i. The vector derivative of R, wi th respect to the scaler components qk is stored in the kth column of Thus, for the case of three bodies, Eq.(2.97) for thrusters becomes '*1 Q Q3 ' 3 -1 T 3 T t 2 , (2.103) 37 Satellite 3 — > -» cm, D1=d1 — > a — > 'o Figure 2-3 Inertial position of subsatellite thruster forces. 38 where ft{ = {0,Tta.,Tt^}T is the thrust vector for the \ t h l ink (tether i-1). Here Ttn and Tta are the thrust forces along the inplane m and out-of-plane Z{ directions, i Pi respectively. For the case of 5-bodies, Q ~Q>1 0 0 ( T ^ 2 ) (2.104) The result for seven or more bodies follows the pattern established by Eq.(2.104). 2.4.3 Generalized momentum gyro torques When deriving the generalized moments arising from the momemtum-wheels on each rigid body (satellite) including the platform, it is easier to view the torques as coupled forces. From Figure 2-4, it is apparent that, for the ith l ink, the generalized forces constituting the couple are (2.105) where: Fei = Faij acting at e*i = eX{i\ Fe2 = -Fp.k acting at e 2 = C a ^ l F e 3 = F7ik acting at e 3 = ey.j. Expanding Eq.(2.105), it becomes clear that 3 Qk = ^ F e i i=l d dqk (2.106) which is independent of Ri- Thus the moments Mma. = 2FaieXi, Mm^ = 2Fp.eXi and Mmi. = 2FlieVi. Transforming the coordinates to the body fixed frame using the rotation matrix, the generalized force due to the momentum-wheels on link i can 39 Figure 2-4 Coupled force representation of a momentum-wheel on a rigid body. 40 be written as d d d Qi = {Td} • ^ { T i t } M m a . - {Tik} • —{T^Mmp. + { T ^ } • { T J j M ^ Qi*Mmi, where: Mm{ = {Mma., M m / ? , Mmi.} T . <3 { T J } . ^ { T , ? } - { T i f c } • ^ { T i £ } { T ^ } - ^ { T z j } (2.107) (2.108) Note, combining the thruster forces and momentum-wheel torques, a compact expression for the generalized force vector, for 5 bodies, can be written as Q Q ? T 3 0 Q ? T 5 0 0 0 0 0 <33T3 Q¥ Q53T5 0 0 0 0 Q4T5 0 0 0 0 Q55T5 / M M I \ Ti *2 m 3 Quu. (2.109) The pattern of Eq.(2.109) is retained for the case of N links. 41 3 . C O M P U T E R I M P L E M E N T A T I O N 3.1 Prel iminary Remarks The governing equations of motion for the TV-body tethered system were de-rived in the last chapter using an efficient O(N) factorization method. These equa-tions are highly nonlinear, nonautonomous and coupled. In order to predict and appreciate the character of the system response under a wide variety of disturbances, it is necessary to solve this system of differential equations. Although, the closed-form solutions to various simplified models can be obtained using well-known analytic methods, the complete nonlinear response of the coupled system can only be obtained numerically. However, the numerical solution of these complicated equations is not a straight-forward task. To begin with, the equations are stiff[52], be. they have attitude and vibration response frequencies, separated by about an order of magnitude or more, which makes their numerical integration suceptible to accuracy errors, i f a properly designed integration routine is not used. Secondly, the factorization algorithm that ensures efficient inversion of the mass matrix, as well as the time-varying offset and tether length, significantly increase the size of the resulting simulation code, to well over 10,000 lines. This raises computer memory issues which must be addressed in order to properly design a single program that is capable of simulating a wide range of configurations and mission profiles. It is further desired that the program be modular in character to accomodate design variations with ease and efficiency. This chapter begins with a discussion on issues of numerical and symbolic in-42 tegration. This is followed by an introduction to the program structure. Final ly , the validity of the computer program is established using two methods, namely the con-servation of total energy; and comparison of the response results for simple particular cases reported in the literature. 3.2 Numerical Implementation 3.2.1 Integration routine A computer program coded in F O R T R A N was written to integrate the equa-tions of motion of the system. The widely available routine used to integrate the dif-ferential equations, D G E A R [ 5 3 ] , is based on Gear's predictor-corrector method[54]. It is well suited to stiff differential equations, as i t provides automatic step-size adjust-ment and error-checking capabilities at each iteration cycle. These features provide superior performance over traditional 4*^ order Runge-Kut ta methods. Like most other routines, the method requires that the differential equations be transformed into a first order, state space, representation. This is easily accomplished by letting thus X - ^ ' ^ ' f ^ ) =?<•{*•*). (3 -2 ) where S(f,M\dc) is defined by Eq.(2.96). Eq.(3.2) represents the new set of 2nqq first order, differential equations of motion. 3.2.2 Program structure A flow chart representing the computer program's structure is presented in Figure 3-1. The M A I N program is responsible for calling the integration routine. It also coordinates the flow of information throughout the program. The first subroutine 43 called is INIT which initializes the system variables such as the constant parameters (mass, density, inertia, etc.), and the ini t ial conditions, I .C., (attitude angles, orbital motion, tether elastic deformation, etc.) from user-defined data files. Furthermore, all the necessary parameters required by the integration subroutine D G E A R (step-size, tolerance) are provided by INIT. The results of the simulation, i.e. time history of the generalized coordinates, are then passed on to the O U T P U T subroutine, which sorts the information into different output files to be subsequently read in by the auxiliary plott ing package. The majority of the simulation running time is spent in the F C N subroutine which is called repeatedly by D G E A R . F C N generates the fv(X,i) vector defined in Eq.(3.2). It is composed of several subroutines which are responsible for the com-putation of all the time-varying matrices, vectors and scaler variables, derived in Chapter 2, that comprise the governing equations of motion for the entire system. These subroutines have been carefully constructed to ensure maximum efficiency in computational performance as well as memory allocation. Two important aspects in the programming of F C N should be outlined at this point. As mentioned earlier, some of the integrals defined in Chapter 2 are notably more difficult to evaluate manually. Traditionally, they have been integrated numer-ically on-line (during the simulation). In some cases, particularly stationkeeping, the computational cost is quite reasonable since the modal integrals need only be evalu-ated once. However, for the case of deployment or retrieval, where the mode shape functions vary, they have to be integrated at each time-step. This has considerable repercussions on the total simulation time. Thus, in this program, the integrals are evaluated symbolically using M A P L E V[55]. Once generated, they are translated into F O R T R A N source code and stored in ' I N C L U D E ' files to be read by the compiler. Al though, these files can be lengthy (1000 lines or more) for a large number of flexible 44 INIT DGEAR PARAM. I.C. & SYS. PARAM. MAIN < * DGEAR < FCN OUTPUT STATE ACTUATOR FORCES MOMENTS & THRUST FORCES VIBRATION r OFFS DYNA ET MICS Figure 3-1 Flowchart showing the computer program structure. 45 modes, they st i l l require less time to compute compared to their on-line evaluation. For the case of deployment, the performance improvement was found be as high as 10 times for certain cases. Secondly, the O(N) algorithm presented in the last section involves matrices that contained mostly zero terms (RP, Rv and M±). The sparse structure of these ma-trices can be used advantageously to improve the computational efficiency by avoiding multiplications of zero elements. The result is a quick evaluation of fv(X,t). More-over, there is a considerable saving of computer memory as zero elements are no longer stored. Final ly , the C O N T R O L subroutine, which is part of F C N is designed to cal-culate the appropriate control forces, torques and offset dynamics which are required to stabilize the librational ( A T T I T U D E ) and vibrational ( V I B R A T I O N ) motion of the system. 3.3 Verification of the Code The simulation program was checked for its validity using two methods. The first verified the conservation of total energy in the absence of dissipation. The second approach was a direct comparison of the planar response generated by the simulation program with those available in the literature. A s numerical results for the flexible, three-dimensional, A^-body model are not available, one is forced to be content with the validation using a few simplified cases. 3.3.1 Energy conservation The configuration considered here is that of a 3-body tethered system with the following parameters for the platform, subsatellite and tether: 46 h = 1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760 kg • m 2 , platform inertia; kg • m 2 , end-satellite inertia; 200 0 0 0 400 0 0 0 400 m\ = 90,000 kg, mass of the space station platform; m2 = 500 kg, mass of the end-satellite; EfAt = 61,645 N , tether elastic stiffness; pt — 4.9 kg /km, tether linear density. The tether length is taken to remain constant at 10 km (stationkeeping case). A n in i t ia l disturbance in pitch and roll of 2° and 1°, respectively is given to both the rigid bodies and the flexible tether. In addition, the tether is ini t ia l ly deflected by 4.5 m in the longitudinal direction at its end and 0.5 m in both the inplane and out-of-plane directions in the first mode. The tether attachment point offset at the platform (satellite 1) end is 1 m in all the three directions, i.e. d2 = {1,1, l}Tm. The attachment point at the subsatellite (satellite 2) end is 1 m in the x% direction, i.e. fcm3 = {1,0, 0}T, The variation in the kinetic and potential energies is presented for the above-mentioned case in Figure 3-2(a). As seen from the plot, there is a continuous mutual exchange between the kinetic and potential energies, however the variation in the total energy remains zero (Figure 3-2b). It should be noted that after 1 orbit, the variation of both kinetic and potential energy does not return to 0. This is due to the attitude and elastic motion of the system which shifts the centre of mass of the tethered system in a non-Keplerian orbit. Similar results are presented for the 5-body case, i.e. a double pendulum (Fig-ure 3-3). Here the system is composed of a platform and two subsatellites connected 47 STSS configuration: 3-Body a 1(0)=a 2(0)=a t(0)=2° P1(0)=p2(0)=p,(0)=1° 5x(0)=4.5m, 8y(0)=52(0)=0.5m dx(0)=dy(0)=d2(0)=1m Stationkeeping: lt=10km Variation of Kinetic and Potential Energy (a) Time (Orbits) Percent Variation in Total Energy O.OEOh -5.0E-12 -1.0E-11 I . . i i 0 1 2 (b) Time (Orbits) Figure 3-2 Kinetic and potential energy transfer for the three-body platform based tethered satellite system: (a) variation of kinetic and potential energy; (b) percent change in total energy of system. 48 STSS configuration: 5-Body a 1(0)=a 2(0)=a t 1(0)=2° a 3(0)=a t 2(0)=2.5° P1(0)=p2(0)=p3(0)=pt1(0)=pt2(0)=1< 5x(0)=5m; 8y(0)=5z(0)=0.5m dx(0)=dy(0)=dz(0)=0 Stationkeeping: 1^ =1,2=1 Okm (a) 2.0E6 1.0E6r-0.0E0 -1.0E6 -2.0E6 Variation of Kinetic and Potential Energy - / 1 I I I 1 I I I I / AP e \ / ~_ i i 1 i i i i 0.0E0 UJ -5.0E-12 U J < Time (Orbits) Percent Variation in Total Energy -1.0E-11h (b) Time (Orbits) Figure 3 -3 Kinet ic and potential energy transfer for the five-body platform based tethered satellite system: (a) variation of kinetic and potential en-ergy; (b) percent change in total energy of system. 49 in sequence by two tethers. The two subsatellites have the same mass and inertia, i.e. 7713 = 7 7 i 2 a n d I3 = I21 a n d are separated by identical tether segments, each 10 k m in length. The platform has the same properties as before, however, there is no tether attachment point offset present in the system (d, = fcmi = 0). In al l the cases considered, energy was found to be conserved. However, it should be noted that for the cases where the tether structural damping, deployment, retrieval, attitude and vibration control are present, energy is no longer conserved since al l these features add or subtract energy from the system. 3.3.2 Comparison with available data The alternative approach used to validate the numerical program involves comparison of system responses with those presented in the literature. A sample case considered here is the planar system whose parameters are the same as those given in Section 3.3.1. A n ini t ia l pitch disturbance of 2° is imparted to both the platform and the tether together with an ini t ial tether deformation of 0.8 m and 0.01 m in the longitudinal and transverse directions, respectively in the same manner as before. The tether length is held constant at 5 km and it is assumed to be connected to the centre of mass of the rigid platform and subsatellite. The response time histories from Ref.[43] are compared with those obtained using the present program in Figures 3-4 and 3-5. Here ap and at represent the platform and tether pitch, respectively; and B i , C i are the tether's longitudinal and transverse generalized coordinates, respectively. A comparison of Figures 3-4 and 3-5 clearly shows a remarkable correlation between the two sets of results, in both the amplitude and frequency. The same trend persisted even with several other comparisons. Thus, a considerable level of confidence is provided in the simulation program as a tool to explore the dynamics and control of flexible, multibody tethered systems. 50 a p(0) = 2°; 0,(0) = 2C 6,(0) = 0.8 m 0,(0) = 0.01 m Stationkeeping, L = 5 km D p y = 0; D p 2 = 0 a; o.o c. -o.oo -0 .01 Figure 3 - 4 Simulation results for the platform based three-body tethered system originally presented in Ref.[43]. 51 STSS configuration: 3-Body a1(0)=a2(0)=at(0)=2° p1(0)=(32(0)=(3t(0)=0 8x(0)=0.8m, 5y(0)=0.01m, 5Z(0)=0 dx(0)=dy(0)=dz(0)=0 Stationkeeping: l,=5km Satellite 1 Pitch Angle Tether Pitch Angle i . . , . -i i- i i , i , i i i d 1 2 0 1 2 Time (Orbits) Time (Orbits) Longitudinal Vibration ^MAAAAAAA/WVWWWWVVVW* : V , , , • i , • , . . i , I , , i . J 0.0 0.5 1.0 1.5 2.0 Time (Orbits) Transverse Vibration Time (Orbits) Figure 3-5 Simulation results for the platform based three-body tethered system obtained using the present computer program. 52 4. D Y N A M I C SIMULATION 4.1 Prel iminary Remarks Understanding the dynamics of a system is critical to its design and devel-opment for engineering application. Due to obvious limitations imposed by flight tests and simulation of environmental effects in ground based facilities, space based systems are routinely designed through the use of numerical models. This Chapter studies the dynamical response of two different tethered systems during deployment, retrieval and stationkeeping phases. In the first case, the Space platform based Teth-ered Satellite System (STSS), which consists of a large mass (platform) connected to a relatively smaller mass (subsatellite) with a long flexible tether, is considered. The other system is the O E D I P U S / B I C E P S configuration involving two comparable mass satellites interconnected by a flexible tether. In addition, the mission requirement of spin about an arbitrary axis, the tether axis for O E D I P U S - A / C and cartwheeling, or spin about the orbit normal, for the proposed B I C E P S mission, is accounted for. A s mentioned in Chapter 2, the tether flexibility is modelled using the as-sumed mode discretization method. Although the simulation program can account for an arbitrary number of vibrational modes in each direction, only the first mode is considered in this study as it accounts for most of the strain energy[34] and hence dominates the vibratory motion. The parametric study considers single as well as double pendulum type systems with offset of the tether attachment points. The system's stability is also discussed. 4.2 Parameter and Response Variable Definitions The system parameters used in the simulation, unless otherwise stated, are 53 selected as follows for the STSS configuration: 1091430 -8135 328108 -8135 8646050 27116 328108 27116 8286760 h = h = 200 0 0 0 400 0 0 0 400 • m\ = 90,000 kg (mass of the space platform); kg • m 2 (platform inertia); J kg • m 2 (subsatellite inertia); • 777,2 = 500 kg (mass of the subsatellite); • EtAt = 61,645 N (tether stiffness); • Pt = 4.9 k g / k m (tether density); • Vd — 0.5% (tether structural damping coefficient); • fcm^ = { l ; 0 , 0 } r m (tether attachment point at the subsatellite). The response variables are defined as follows: • ai,0\: satellite 1 (platform) pitch and roll angles, respectively; • CK2, / ? 2 : satellite 2 (subsatellite) pitch and roll angles, respectively; • at, fit'- tether pitch and roll angles, respectively; • If. tether length; • d = {dx, dy, dz}T-. tether attachment position relative to satellite 1 (platform); —* • S = {8X, 8y, Sz}: tether flexible generalized coordinates in the longitudinal x, inplane transverse y, and out-of-plane transverse z directions, respectively. The attitude angles and are measured with respect to the Local Ve r t i c a l -Loca l Horizontal ( L V L H ) frame. A schematic diagram illustrating these variable is presented in Figure 4-1. The system is taken to be in a nominal circular orbit, at an altitude of 289 km, with an orbital period of 90.3 minutes. 54 Satellite 1 (Platform) Y a w Figure 4-1 Schematic diagram showing the generalized coordinates used to de-scribe the system dynamics. 55 4.3 Stationkeeping Profile To facilitate comparison of the simulation results, a reference case, based on the three-body STSS system with zero offset (d = 0, r c m 3 = 0) and a tether length of It = 20 km, is considered first (Figure 4-2). The system is subjected to an ini t ial disturbance in pitch and roll of 2° and 1° , respectively, to all the three bodies. In addition, an ini t ia l longitudinal deflection of 14 m from the tether's unstretched position is given, together with a transverse inplane and out-of-plane deflection of 1 m at the tether's mid-length. From the attitude response given in Figure 4-2(a), it is apparent that the three bodies oscillate about their respective equilibrium positions. Since coupling between the individual rigid-body dynamics is absent (zero offset), the corresponding librational frequencies are unaffected. Satellite 1 (Platform) displays amplitude modulation arising from the products of inertia. The result is a slight increase of amplitude in the pitch direction and a significantly larger decrease in amplitude in the roll angle. Figure 4-2(b) clearly shows coupling of the tether's attitude motion with its flexibility dynamics. The tether vibrates in the axial direction about its equilibrium position (at approximately 13.8 m) with two vibration frequencies, namely 0.12 Hz and 3.1 x 1 0 - 4 Hz. The former is the first longitudinal flexible mode which is dissi-pated through the structural damping while the latter arises from the coupling with the pitch motion of the tether. Similarly, in the transverse direction, there are two frequencies of oscillations arising from the flexible mode and its coupling wi th the attitude motion. As expected, there is no apparent dissipation in the transverse flex-ible motion. This is attributed to the weak coupling between the longitudinal and transverse modes of vibration since the transverse strain in only a second order effect relative to the longitudinal strain where the dissipation mechanism is in effect. Thus, there is very litt le transfer of energy between the transverse and longitudinal modes resulting in a very long decay time for the transverse vibrations. 56 STSS configuration: 3-Body a1(0)=a2(0)=a,(0)=2° (31(0)=p2(0)=p,(0)=1° 8x(0)=14m,8y(0)=8z(0)=1m dx(0)=dy(0)=dz(0)=0 Stationkeeping: lt=20km Satellite 1 Pitch Angle Satellite 1 Roll Angle 0 1 2 3 4 5 0 1 2 3 4 5 Time (Orbits) Time (Orbits) Satellite 2 Pitch Angle Satellite 2 Roll Angle (a) Time (Orbits) Time (Orbits) Figure 4-2 Stationkeeping dynamics of the three-body STSS configuration with-out offset: (a) attitude response. 57 STSS configuration: 3-Body a1(0)=cx2(0)=cx,(0)=2° P1(0)=p2(0)=p,(0)=1° 5x(0)=14m,8y(0)=82(0)=1m dx(0)=dy(0)=dz(0)=0 Stationkeeping: lt=20km Tether Longitudinal Vibration 16| ' ' ' ' 1 ' 1 ' ' 1 ' ' 1 ' 1 ' r-Time (Orbits) Tether In-Plane Transverse Vibration i— 1 — 1 — 1 — 1 —r r ' ' ' I I 1 1 1 ' I 1 1 1 1 I 1 • ' ' ^ • • I I J 0 1 2 3 4 5 Time (Orbits) Tether Out-of-Plane Transverse Vibration (b) Time (Orbits) Figure 4-2 Stationkeeping dynamics of the three-body STSS configuration with-out offset: (b) vibration response. 58 Introduction of the attachment point offset significantly alters the response of the attitude motion for the rigid bodies. W i t h a i m offset along the local vertical at the platform (dx = 1 m) and fcm^ = {1,0,0} m, coupling between the tether and end-bodies is established by providing a lever arm from which the tether is able to exert a torque that affects the rotational motion of the rigid end-bodies (Figure 4-3). Bo th the platform and subsatellite now oscillate at the same frequency as the tether, whose motion remains unaltered. In the case of the platform, there is also an amplitude modulation due to its non-zero products of inertia, as explained before. However, the elastic vibration response of the tether remains essentially unaffected by the coupling. Providing an offset along the local horizontal direction (dy = 1 m) results in a more dramatic effect on the pitch and roll response of the platform, as shown in Figure 4-4(a). Now, the platform oscillates about its new equilibrium position of -90° . The roll motion is also significantly disturbed resulting in an increase to over 10° in amplitude. On the other hand, the rigid body dynamics of the tether and the subsatellite, as well as the tether flexible motion, remain the same (Figure 4-4b). Figure 4-5 presents the response when the offset of 1 m, at the platform end, is along the z direction, i.e. normal to the orbital plane. The result is a larger amplitude pitch oscillation about the reference equilibrium position, as shown in Figure 4-5(a), while the roll equilibrium is now shifted to approximately 90° . Note, there is little change in the attitude motion of the tether and end-satellites, however, there is a noticeable change in the out-of-plane transverse vibration of the tether (Figure 4-5b) which may be due to the large amplitude platform roll dynamics. Final ly, by setting a i m offset in all the three direction simultaneously, the equil ibrium position of the platform in the pitch and roll angle is altered by approxi-mately 30° (Figure 4-6a). However, the response of the system follows essentially the 59 STSS configuration: 3-Body a1(0)=cx2(0)=cxt(0)=2° p1(0)=(32(0)=pt(0)=1° 8x(0)=14m,8y(0)=8z(0)=1m dx(0)=1m, dy(0)=dz(0)=0 Stationkeeping: lt=20km Satellite 1 Pitch Angle Satellite 1 Roll Angle Time (Orbits) Time (Orbits) Tether Pitch Angle Tether Roll Angle L • . . . I . . . . I • . . • < • • • » • • • • -I h . . , , I . . , . I . . , . I . . . . I . . i . H 0 1 2 3 4 5 0 1 2 3 4 5 Time (Orbits) Time (Orbits) Satellite 2 Pitch Angle Satellite 2 Roll Angle (a) Time (Orbits) Time (Orbits) Figure 4-3 Stationkeeping dynamics of the three-body STSS configuration with offset along the local vertical: (a) attitude response. 60 STSS configuration: 3-Body a 1(0)=a 2(0)=a t(0)=2° (31(0)=p2(0)=pt(0)=1° 5x(0)=14m,5y(0)=5z(0)=1m dx(0)=1m,dy(0)=dz(0)=0 Stationkeeping: lt=20km Tether Longitudinal Vibration 1 6 i — i — i — i — i — i — i — i — ' — i — i — ' — ' — ' — ' — i — ' — < -Time (Orbits) Tether In-Plane Transverse Vibration y • i i , i . , i i i . . i i i i i i i i i i i 0 1 2 3 4 5 Time (Orbits) Tether Out-of-Plane Transverse Vibration r 1 1 1 . 1 1 . 1 ' 1 ' 1 ' ' 1 1 1 ' ' 1 1 ' ' 1 q h i . . . I i i I i i i i I i i i i 3 0 1 2 3 4 5 (b) Time (Orbits) Figure 4-3 Stationkeeping dynamics of the three-body STSS configuration with offset along the local vertical: (b) vibration response. 61 STSS configuration: 3-Body a1(0)=cx2(0)=o,(0)=2° (31(0)=p2(0)=p,(0)=1° 8x(0)=14m,8y(0)=8z(0)=1m dx(0)=dz(0)=0, dy(0)=1m Stationkeeping: lt=20km Satellite 1 Pitch Angle i ' ' 1 ' i Satellite 1 Roll Angle 1 2 3 4 Time (Orbits) Tether Pitch Angle 1 2 3 4 Time (Orbits) Tether Roll Angle CD edT 1 2 3 Time (Orbits) Satellite 2 Pitch Angle 1 2 3 4 Time (Orbits) Satellite 2 Roll Angle CD C O . (a) Figure 4-1 2 3 4 Time (Orbits) 1 2 3 4 Time (Orbits) 4 Stationkeeping dynamics of the three-body STSS configuration with offset along the local horizontal: (a) attitude response. 62 STSS configuration: 3-Body a1(0)=cx2(0)=ot(0)=2° p1(0)=p2(0)=pt(0)=1° 8x(0)=14m,8y(0)=8z(0)=1m dx(0)=dz(0)=0, dy(0)=1m Stationkeeping: lt=20km Tether Longitudinal Vibration 16| 1 > 1 • 1 ' ' ' ' 1 ' ' ' ' 1 ' 0 1 2 3 4 5 Time (Orbits) Tether In-Plane Transverse Vibration i — 1 — 1 — 1 — 1 — i — ' — < — 1 — 1 — i — ' — 1 — 1 — V . . , , I . . , . I • • • . I i i i I i i 1 1 3 0 1 2 3 4 5 Time (Orbits) Tether Out-of-Plane Transverse Vibration r 1 , , , 1 1 1 1 1 1 . 1 1 1 1 > ' • ' | ' ' ' ' q V . . . , i , , , , 1 , , , . 1 . 1 . . 1 1 ' — . 1 — 3 0 1 2 3 4 5 (b) Time (Orbits) Figure 4-4 Stationkeeping dynamics of the three-body STSS configuration with offset along the local horizontal: (b) vibration response. 63 STSS configuration: 3-Body a1(0)=o2(0)=at(0)=2° (31(0)=p2(0)=pt(0)=1° 8x(0)=14m,8y(0)=8z(0)=1m dx(0)=dy(0)=0, dz(0)=1m Stationkeeping: lt=20km Satellite 1 Pitch Angle Satellite 1 Roll Angle 2 3 4 Time (Orbits) Tether Pitch Angle 1 2 3 4 Time (Orbits) Tether Roll Angle 1 2 3 4 Time (Orbits) Satellite 2 Pitch Angle 2 3 4 Time (Orbits) Satellite 2 Roll Angle (a) Figure 4-1 2 3 4 Time (Orbits) 2 3 4 Time (Orbits) 5 Stationkeeping dynamics of the three-body STSS configuration with offset along the orbit normal: (a) attitude response. 64 STSS configuration: 3-Body a1(0)=cx2(0)=ot(0)=2° . P1(0)=p2(0)=(3,(0)=1° 8x(0)=14m,8y(0)=82(0)=1m dx(0)=dy(0)=0, d2(0)=1m Stationkeeping: lt=20km Tether Longitudinal Vibration 1 6 i — r — i — ' — i — i — ' — i — i — ' — i — ' — i — 1 — 1 — i — ' — < — < — < — r Time (Orbits) Tether In-Plane Transverse Vibration (b) Time (Orbits) Figure 4-5 Stationkeeping dynamics of the three-body STSS configuration with offset along the orbit normal: (b) vibration response. 65 same trend with only minor perturbations in the transverse elastic vibratory response of the tether (Figure 4-6b). 4.4 Tether Deployment Deployment of the tether from an ini t ial length of 200 m to 20 km is explored next. Here, the deployment length profile is critical to ensure the system's stability. It is not desirable to deploy the tether too quickly since that can render the tether slack. Hence, the deployment strategy detailed in Section 2.1.6 is adopted. The tether is deployed over 3.5 orbits with the sinusoidal acceleration and deceleration occurring over a 4 km length ( A ^ 2 ) with the remaining deployment at a constant velocity of V0 = 1.46 m/s . Initially, the tether is stretched only slightly, S = {1.304 x 1 0 _ 3 , 0 , 0} m, with al l the other states of the system, i.e. pitch, roll and transverse displacements remaining zero. The response for the zero offset case is presented in Figure 4-7. For the platform, there is no longer any coupling with the tether, hence it oscillates about the equilibrium orientation determined by its inertia matrix. However, there is st i l l coupling between the subsatellite and the tether since rcm^ — {1,0,0} m resulting in complete domination of the subsatellite dynamics by the tether. Consequently, the pitch motion ini t ial ly grows by almost 50° but eventually subsides as the tether deployment rate decreases. It is interesting to note that the out-of-plane motion is not affected by the deployment. This is because the Coriolis force, which is responsible for the tether pitch motion, does not have a component in the z direction (as it does in the y direction), since it is always perpendicular to the orbital rotation (Q) and the deployment rate (It), i.e. Q, x It. Similarly, there is an increase in the amplitude of the transverse elastic oscil-lations of the tether, again due to the Coriolis effect (Figure 4-7b). Moreover, as the 66 STSS configuration: 3-Body a1(0)=a2(0)=cxt(0)=2° P1(0)=p2(0)=pt(0)=1° 8x(0)=14m,5y(0)=52(0)=1m dx(0)=dy(0)=d2(0)=1m Stationkeeping: lt=20km Satellite 1 Pitch Angle Satellite 1 Roll Angle i , . . . i . . . . i . . . . i . . . . i . . . . i i . . . i , . , . i , . . . i , , , . i 0 1 2 3 4 5 0 1 2 3 4 5 Time (Orbits) Time (Orbits) Tether Pitch Angle Tether Roll Angle l . . . . i . . . . i . . . . i . . . . i . . . . -I h i i i i I i i i i l- i i i i l i i i i I i i- i i H 0 1 2 3 4 5 0 1 2 3 4 5 Time (Orbits) Time (Orbits) Satellite 2 Pitch Angle Satellite 2 Roll Angle (a) Time (Orbits) Time (Orbits) Figure 4-6 Stationkeeping dynamics of the three-body STSS configuration with offset along the local horizontal, local vertical and orbit normal: (a) attitude response. 67 STSS configuration: 3-Body a1(0)=cc2(0)=a,(0)=2° p1(0)=p2(0)=(3t(0)=1° 5x(0)=14m,6y(0)=52(0)=1m dx(0)=dy(0)=d2(0)=1m Stationkeeping: lt=20km Tether Longitudinal Vibration 1 6 1 — i — i — i — ' — i — 1 — 1 — ' — ' — i — ' — 1 — 1 — 1 — i — 1 — 1 — 1 — 1 — r 0 1 2 3 4 5 Time (Orbits) Tether In-Plane Transverse Vibration (b) Time (Orbits) Figure 4-6 Stationkeeping dynamics of the three-body STSS configuration with offset along the local horizontal, local vertical and orbit normal: (b) vibration response. 68 STSS configuration: 3-Body a,(0)=a2(0)=ot(0)=0 P1(0)=p2(0)=pt(0)=0 8X(0)=1.304 x 10'3m, 8y(0)=8z(0)=0 dx(0)=dy(0)=dz(0)=0 Deployment: l.=0.2km to 20km in 3.5 Orbits Tether Length Profile E Satellite 1 Pitch Angle 1 2 3 4 Time (Orbits) Satellite 1 Roll Angle — i — i — i — i — i — i — i — i — [ — i — i — i — i — | — 1 2 3 4 Time (Orbits) Tether Pitch Angle 1 2 3 4 Time (Orbits) Tether Roll Angle CD % o.o ax i i—i—i—I—i—i—i—r-_ 1 _ L 1 2 3 4 Time (Orbits) Satellite 2 Pitch Angle 1 2 3 4 Time (Orbits) Satellite 2 Roll Angle -50 (a) 1 2 3 4 Time (Orbits) 1 2 3 4 Time (Orbits) Figure 4-7 Deployment dynamics of the three-body STSS configuration without offset: (a) attitude response. 69 STSS configuration: 3-Body a1(0)=a2(0)=at(0)=0 p1(0)=p2(0)=(3t(0)=0 8X(0)=1.304x 103m, 8y(0)=52(0)=0 dx(0)=dy(0)=d2(0)=0 Deployment: l,=0.2km to 20km in 3.5 Orbits Tether Longitudinal Vibration to -0.2 (b) Figure 4-7 2 3 Time (Orbits) Tether In-Plane Transverse Vibration 0 1 , . , . 1 •, . . . 1 . . . . 1 . . 1 2 3 4 Time (Orbits) Tether Out-of-Plane Transverse Vibration 5 : 1 , 1 , 1 1 1 i i | i i . . | ^ _ i , , , , i , , . . i . . . . i L , ...-I : 1 2 , 3 4 5 Time (Orbits) Deployment dynamics of the three-body STSS configuration without offset: (b) vibration response. 70 tether elongates, its longitudinal static equilibrium position also changes due to an increase in the gravity-gradient tether tension. It may be pointed out that, as in the case of attitude motion, there are no out-of-plane tether vibrations induced. When the same deployment conditions are applied to the case of 1 m offset in the x direction (Figure 4-8), coupling effects are re-introduced. The platform pitch exceeds -100° during the peak deployment acceleration. However, as the tether pitch motion subsides, the platform pitch response returns to a smaller amplitude oscillation about its nominal equilibrium. On the other hand, the platform roll motion grows to over 20° in amplitude in the terminal stages of deployment. Longitudinal and transverse vibrations of the tether remain virtually unchanged from the zero offset case except for minute out-of-plane perturbations due to the platform librations in roll. 4.5 Tether Retrieval The system response for the case of the tether retrieval from 20 km to 200 m, with an offset of 1 m in the x and z directions at the platform end, is presented in Figure 4-9. Here the Coriolis force induced during retrieval renders the tether unstable (Figure 4-9a). Consequently, the pitch motion of the platform is also destabilized through coupling. The z offset provides an additional moment arm that shifts the roll equilibrium to 35° , however, this degree of freedom is not destabilized. From Figure 4-9(b), it is apparent that the transverse mode of the tether vibration is also disturbed producing a high frequency response in the final stages of retrieval. This disturbance is responsible for the slight increase in the roll motion for both the tether and subsatellite. 4.6 Five-Body Tethered System A five-body chain link system is considered next. To facilitate comparison with 71 STSS configuration: 3-Body a,(0)=a2(0)=at(0)=0 P1(0)=P2(0)=P,(0)=0 5X(0)=1.304 x 10'3m, 8y(0)=8z(0)=0 dx(0)=1m,dy(0)=d2(0)=0 Deployment: lt=0.2km to 20km in 3.5 Orbits Tether Length Profile E Satellite 1 Pitch Angle 1 2 3 Time (Orbits) Satellite 1 Roll Angle ' • • • . i • • • • i 1 2 3 4 Time (Orbits) Tether Pitch Angle 1 2 3 Time (Orbits) Tether Roll Angle i . . . . i . . . . i . . . . i J I I i _ 1 2 3 4 Time (Orbits) Satellite 2 Pitch Angle 1 2 3 4 Time (Orbits) Satellite 2 Roll Angle 3E-3h D ) T J ^ 0 E 0 eg CQ. -3E-3 (a) 1 2 3 4 Time (Orbits) 1 2 3 4 Time (Orbits) Figure 4-8 Deployment dynamics of the three-body STSS configuration with offset along the local vertical: (a) attitude response. 7 2 STSS configuration: 3-Body a1(0)=a2(0)=at(0)=0 p1(0)=P2(0)=(3t(0)=0 Sx(0)=1.304 x 10"3m, 5y(0)=5z(0)=0 dx(0)=1m,dy(0)=dz(0)=0 Deployment: l,=0.2km to 20km in 3.5 Orbits Tether Longitudinal Vibration Time (Orbits) Tether In-Plane Transverse Vibration n 1 1 1 f I . . . . I . . . . I _j I I I I I 1 1 1 1 1 2 3 4 5 Time (Orbits) Tether Out-of-Plane Transverse Vibration J 1 I , I 1 I I , I , , I I I I , I L _ _ . I I d 1 2 3 4 5 Time (Orbits) Deployment dynamics of the. three-body STSS configuration with offset along the local vertical: (b) vibration response. (b) Figure 4-8 73 STSS configuration: 3-Body a1(0)=a2(0)=at(0)=0 p\(0)=P2(0)=(3t(0)=0 Sx(0)=14m,8y(0)=82(0)=0 dy(0)=0,dx(0)=d2(0)=1m Retrieval: l,=20km to 0.2km in 3.5 Orbits Satellite 1 Pitch Angle (a) Time (Orbits) Figure 4-9 Retrieval dynamics of the along the local vertical ar Tether Length Profile i i i i i i i i i i I 0 1 2 3 Time (Orbits) Satellite 1 Roll Angle Time (Orbits) Tether Roll Angle b_. i i . I i , , . I i , i L _ J 0 1 2 3 Time (Orbits) Satellite 2 Roll Angle _ — | r i i | 1 i i i——j i 1 r — i — : 0.4- -0 1 2 3 Time (Orbits) three-body STSS configuration with offset d orbit normal: (a) attitude response. 74 STSS configuration: 3-Body a1(0)=a2(0)=ocl(0)=0 (31(0)=f32(0)=pt(0)=0 8x(0)=14m,8y(0)=6z(0)=0 dy(0)=0,dx(0)=dz(0)=1m Retrieval: l.=20km to 0.2km in 3.5 Orbits Tether Longitudinal Vibration E 10 to Time (Orbits) Tether In-Plane Transverse Vibration 1 2 Time (Orbits) Tether Out-of-Plane Transverse Vibration (b) Time (Orbits) Figure 4-9 Retrieval dynamics of the three-body STSS configuration with offset along the local vertical and orbit normal: (b) vibration response. 75 the three-body system dynamics studied earlier, the chain is extended through the addition of two bodies, a tether and a subsatellite, with the same physical properties as before (Section 4.2). Thus the five-body system consists of a platform (satellite 1), tether 1, subsatellite 1 (satellite 2), tether 2 and subsatellite 2 (satellite 3), as shown in Figure 4-10. The offset of tether 1 at the platform-end is denoted by d2 while that of tether 2 to subsatellite 1 as 04. In this numerical example fcm^ = fcm^ = 0, i.e. tethers 1 and 2 are attached to the centre of mass of subsatellites 1 and 2, respectively. The system response for the case where the tether attachment points coincide wi th the centres of mass of the rigid bodies, i.e. zero offset (d2 = d\ — 0), is presented in Figure 4-11. Here o>t\ and at2 represent the pitch motion of tethers 1 and 2, respectively, whereas 0:3 is the pitch motions of satellite 3. A similar convention is adopted for the rol l angle 0. A s before, the zero offset eliminates the coupling between the tethers and satellites such that they are now free to librate about their static equil ibrium positions. However, their is s t i l l mutual coupling between the two tethers. This coupling is present regardless of the offset position since each tether is capable of transferring a force to the other. The coupling is clearly evident in the pitch response of the two tethers (Figure 4 - l l a ) . O n the other hand, the roll motion appears uncoupled as the in i t ia l conditions in rol l for the two tethers are identical. Thus the motion is in phase and there is no transfer of energy through coupling. A s expected, during the elastic response, the tethers vibrate about their static equil ibrium positions and mutually interact (Figure 4 - l l b ) . However, due to the variation of tension along the tether (x direction), they do not have the same longitudinal equilibrium. Note, relatively large amplitude transverse vibrations are present, particularly for tether 1, suggesting strong coupling effects with the longitudinal oscillations. 76 Figure 4-10 Schematic diagram of the five-body system used in the numerical example. 77 STSS configuration: 5-Body a1(0)=(x2(0)=at1(0)=2° a3(0)=a12(0)=2.5° p1(0)=r32(0)=p3(0)=pt1(0)=pt2(0)=r 82(0)=84(0)={5,0.5)0.5}m d2(0)=d4(0)={0,0,0} Stationkeeping: lt1=lu=10km Pitch: Roll: Satellite 1 Pitch Angle Satellite 1 Roll Angle o 1 2 Time (Orbits) Tether 1 Pitch and Roll Angle 0 1 2 Time (Orbits) Tether 2 Pitch and Roll Angle Time (Orbits) Satellite 2 Pitch and Roll Angle Time (Orbits) Satellite 3 Pitch and Roll Angle Time (Orbits) Time (Orbits) Figure 4-11 Stationkeeping dynamics of the five-body STSS configuration with-out offset: (a) attitude response. 78 STSS configuration: 5-Body a1(0)=a2(0)=oct1(0)=2° a3(0)=at2(0)=2.5o p1(0)=p2(0)=(33(0)=pt1(0)=pt2(0)=10 62(0)=54(0)={5,0.5,0.5}m d2(0)=d4(0)={0,0,0} Stationkeeping: lt1=1^ =1 Okm Tether 1 Longitudinal Vibration Tether 2 Longitudinal Vibration i i 1 i 1 r 1 1 1 1 (b) Time (Orbits) Time (Orbits) Figure 4-11 Stationkeeping dynamics of the five-body STSS configuration with-out offset: (b) vibration response. 79 The system dynamics during deployment of both the tethers in the double-pendulum configuration is presented in Figure 4-12. Deployment of each tether takes place from 200 m to 20 km in 3.5 orbits. The pitch motion of the system is similar to that observed during the three-body case. The Coriolis effect causes the tethers to pitch through a large angle ( « 50° ) which in turn disturbs the platform and subsatellite due to the presence of offset along the local vertical. On the other hand, the roll response for both the tethers is damped to zero as the tether deploys, in confirmation with the principle of conservation of angular momentum, whereas the platform response in roll increases to around 20° . Subsatellite 2 remains virtually unaffected by the other links, since the tether is attached to its centre of mass thus eliminating coupling. Finally, the flexibility response of the two tethers, presented in Figure 4-12(b), shows similarity with the three-body case. 4.7 BICEPS Configuration The mission profile of the Bl-static Canadian Experiment on Plasmas in Space (BICEPS) is simulated next. It is presently under consideration by the Canadian Space Agency. To be launched by the Black Brant 2000/500 rocket, it would in-volve interesting maneuvers of the payload before it acquires the final operational configuration. As shown in Figure 1-3, at launch the payload (two satellites) with an undeployed tether is spinning about the orbit normal (phase 1). The internal dampers next change the motion to a flat spin (phase 2) which, in turn, provides momentum for deployment (phase 3). When fully deployed, the one kilometre long tether will connect two identical satellites carrying instrumentation, video cameras and transmitters as payload. The system parameters, for the BICEPS configuration, are summarized below: [5.9 0 0 1 • I\ = I2 = 0 36.6 0 kg • m 2 (satellite inertias) [ 0 0 39.2_ • mi '= 777,2 = 200 kg (mass of the satellites) 80 STSS configuration: 5-Body a,(0)=cx2(0)=al1(0)=2° a3(0)=c (^0)=:2.5o P1(0)=P2(0)=p3(0)=f3t1(0)=Pt2(0)=r 52(0)=84(0)={1.304x 10'3,0,0}m d2(0)=d4(0)={1,G,0}m Deployment: lt1=le=0.2km to 20km in 3.5 Orbits Pitch: Roll: Satellite 1 Pitch Angle Time (Orbits) Satellite 2 Pitch and Roll Angle T 1—1 1 1 I I I I | I I I 1 I | I I <~r Tether Length Profile i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i i . . . . i . . . . i . . . . i i 0 1 2 3 4 5 Time (Orbits) Satellite 1 Roll Angle 0 1 2 3 4 5 Time (Orbits) Tether 1 & 2 Roll Angle L ' i ' ' | ' ' ' ' I ' ' ' ' I 1 ' ' ' I 1 1 ' ' -I -Ih I- • I . . . I i i i i I i i i i 1 0 1 2 3 4 5 Time (Orbits) Satellite 3 Pitch and Roll Angle i- ' ' ' ' i ' 1 ' 1 i 1 1 1 1 i 1 1 1 1 i 1 1 1 1 i L . . . . i . . . . i i i i i 1 i i — i — i I i i—i—LJ 0 1 2 3 4 5 Time (Orbits) Figure 4-12 Deployment dynamics of the five-body S T S S configuration with off-set along the local vertical: (a) attitude response. 81 STSS configuration: 5-Body a1(0)=a2(0)=oct1(0)=2° a3(0)=at2(0)=2.5° P1(0)=P2(0)=P3(0)=Pt1(0)=pt2(0)=1o 82(0)=54(0)={1.304 x 10-3,0,0}m d2(0)=d4(0)={1,0,0}m Deployment: lt1=l,2=0.2km to 20km in 3.5 Orbits Tether 1 Longitudinal Vibration Tether 2 Longitudinal Vibration Time (Orbits) Time (Orbits) Tether 1 Z Tranverse Vibration Tether 2 Z Tranverse Vibration Figure 4-12 Deployment dynamics of the five-body STSS configuration with off-set along the local vertical: (b) vibration response. 82 • EtAt = 61,645 N (tether stiffness) • pt = 3.0 k g / k m (tether density) • It = 1 km (tether length) • rid = 1-0% (tether structural damping coefficient) The system is in a circular orbit at a 289 km altitude. Offset of the tether attachment point to the satellites, at both ends, is taken to be 0.78 m in the x direction. The response of the system in the stationkeeping phase, for a prescribed set of in i t ia l conditions, is given by Figure 4-13. As in the case of the STSS configuration, there is strong coupling between the tether and the satellites' rigid body dynamics causing the latter to follow the attitude of the tether. However, because of the smaller inertias of the payloads, there are noticeable high frequency modulations arising from the tether flexibility. Response of the tether in the elastic degrees of freedom is presented in Figure 4-13(b). Note, the longitudinal vibrations decay quite rapidly (< 0.5 orbits) due to the structural damping, however, it has vir tual ly no effect on the transverse oscillations. The unique mission requirement of B I C E P S is the proposed use of its ini t ia l angular momentum in the cartwheeling mode to aid in the deployment of the tethered system. The maneuver is considered next. The response of the system during this maneuver wi th an ini t ia l cartwheeling rate of 5 ° / s is illustrated in Figure 4-14(a). The in i t ia l tether length is taken to be 10 m and is deployed to 1 km. There is an in i t ia l increase in the pitch motion of the tether, however, due to the conservation of angular momentum, the cartwheeling rate decreases proportionally to the square of the change in tether deployment rate. The result is a rapid drop in the cartwheeling rate unti l the system stops rotating entirely and simply oscillates about its new equilibrium point. Consequently, through coupling, the end-bodies follow the same trend. The roll response also subsides once the cartwheeling motion ceases. However, 83 BICEPS configuration: 3-Body a1(0)=o2(0)=at(0)=2° P1(0)=P2(0)=pt(0)=1° 8X(0)=0.01 m, 6y(0)=52(0)=0.1 m dx(0)=0.78m, dy(0)=d2(0)=0 Stationkeeping: l,=1 km Satellite 1 Pitch Angle Satellite 1 Roll Angle - i , 1 1 1 1 i - i q r ' ' 1 1 ' -> ' I- . . . . I . . . . i t i i i i I H 0 1 2 0 1 2 Time (Orbits) Time (Orbits) Tether Pitch Angle Tether Roll Angle F — 1 — 1 — ' — 1 — i — 1 — 1 — ' — 1 — = i F — 1 — ' — ' — 1 — i — 1 — 1 — 1 — 1 — q (a) Time (Orbits) Time (Orbits) Figure 4-13 Stationkeeping dynamics of the three-body BICEPS configuration with offset along the local vertical: (a) attitude response. 84 BICEPS configuration: 3-Body a1(0)=o2(0)=at(0)=2° Pl(0)=p2(0)=(3t(0)=1° 8X(0)=0.01 m, 8y(0)=82(0)=0.1 m dx(0)=0.78m, dy(0)=d2(0)=0 Stationkeeping: lt=1km Tether Longitudinal Vibration 1E-2[ « = 5 E - 3 | 0.00 0.02 Time (Orbits) Tether In-Plane Transverse Vibration Time (Orbits) Tether Out-of-Plane Transverse Vibration Time (Orbits) Figure 4-13 Stationkeeping dynamics of the three-body BICEPS configuration with offset along the local vertical: (b) vibration response. 85 BICEPS configuration: 3-Body a1(0)=o2(0)=at(0)=2° a1(0)=a2(0)=at(0)=57s Pl(0)=P2(0)=(3t(0)=1° 8X(0)=1.15 x 10"3m, 8y(0)=82(0)=0 dx(0)=0.78m, dy(0)=d2(0)=0 Cartwheeling Deployment: lt=10m to 1km in 1 Orbit Tether Length Profile E 2000 cu T3 2000 co CD 2000 C D CD T J Satellite 1 Pitch Angle 1 2 Time (Orbits) Satellite 1 Roll Angle cn CD 33, cdl Time (Orbits) Tether Pitch Angle Time (Orbits) Tether Roll Angle Time (Orbits) Satellite 2 Pitch Angle 1 2 Time (Orbits) Satellite 2 Roll Angle (a) Time (Orbits) Time (Orbits) Figure 4-14 Cartwheeling dynamics with deployment for the three-body BICEPS with offset along the local vertical: (a) attitude response. 86 BICEPS configuration: 3-Body a1(0)=a2(0)=at(0)=2° a, (0)=a2(0)=a,(0)=57s P1(0)=P2(0)=f3,(0)=1° 8X(0)=1.15x10'3m, 8y(0)=8z(0)=0 dx(0)=0.78m, dy(0)=d2(0)=0 Cartwheeling Deployment: l.=1 Om to 1 km in 1 Orbit .5E-3h co 0E0 0.25 0.00 E, to" -0.25 0.02 -0.02 (b) Figure 4-14 Tether Longitudinal Vibration Time (Orbits) Tether In-Plane Transverse Vibration Time (Orbits) Tether Out-of-Plane Transverse Vibration Time (Orbits) Cartwheeling dynamics with deployment for the three-body BICEPS with offset along the local vertical: (b) vibration response. 87 as the deployment maneuver is completed, the satellites continue to exhibit a small amplitude, low frequency roll response with small period perturbations, induced by the tether elastic oscillations, superposed on it (Figure 4-14b). 4.8 OEDIPUS Spinning Configuration The mission entitled Observation of Electrified Distributions of Ionospheric Plasmas - A Unique Strategy ( O E D I P U S ) also consists of a 1 k m tethered system with two similar end-masses. It was first flown in a suborbital flight in 1989 ( O E D I P U S A ) followed by a recent study in November 1995 ( O E D I P U S - C ) in which the University of Br i t i sh Columbia was one of the participants. Dynamically, O E D I P U S represents a unique system never encountered before. Launched by a Black Brant rocket developed at Br is to l Aerospace L t d . , the system, i.e. the end-bodies together wi th the tether, spins about the longitudinal axis of the tether to achieve stabilized alignment wi th Earth 's magnetic field. The response of the system undergoing a spin rate of j = l ° / s is presented in Figure 4-15 using the same parameters as those outlined for the B I C E P S con-figuration in Section 4.7. Again , there is strong coupling between the three bodies (two satellites and tether), due to the nonzero offset. The spin motion introduces an additional frequency component in the tether's elastic response which is transferred to the l ibrational motion of the satellites. A s the spin rate increases to 1 0 ° / s (Figure 4-16), the amplitude and frequency of the perturbations also increase, however, the general character of the response remains essentially the same. 88 OEDIPUS configuration: 3-Body a1(0)=a2(0)=at(0)=2° i(0)^[2(0H(0)=Y/s p1(0)=p2(0)=p,(0)=r 8X(0)=1.2 x 10"3m, 8y(0)=82(0)=0.1m dx(0)=0.78m, dy(0)=d2(0)=0 Stationkeeping: l,=1 km Satellite 1 Pitch Angle Satellite 1 Roll Angle — i 1 1 1 a r 1 1 1 r (a) Time (Orbits) Time (Orbits) Figure 4-15 Spin dynamics (7 = l°/s) of the three-body OEDIPUS configuration with offset along the local vertical: (a) attitude response. 89 OEDIPUS configuration: 3-Body a1(0)=a2(0)=cct(0)=2° Y1(0)=Y2(0)=Yt(0)=l7s (31(0)=p2(0)=pt(0)=1° 8X(0)=1.2 x 103m, 8y(0)=5z(0)=0.1 m dx(0)=0.78m, dy(0)=d2(0)=0 Stationkeeping: lt=1 km 1E-2 to OEO E 0.0 to Tether Longitudinal Vibration 0.05 Time (Orbits) Tether In-Plane Transverse Vibration Time (Orbits) Tether Out-of-Plane Transverse Vibration Time (Orbits) Figure 4-15 Spin dynamics (7 = l ° / s ) of the three-body O E D I P U S configuration with offset along the local vertical: (b) vibration response. 90 OEDIPUS configuration: 3-Body a1(0)=a2(0)=a,(0)=2° Yi(0)=Y2(0)=rt(0)=107s 81(0)=p2(0)=pt(0)=1° 8X(0)=1.2 x 10'3m, 8y(0)=82(0)=0.1m dx(0)=0.78m, dy(0)=d2(0)=0 Stationkeeping: lt=1km Satellite 1 Pitch Angle Satellite 1 Roll Angle (a) Time (Orbits) Time (Orbits) Figure 4-16 Spin dynamics (7.= 10°/s) of the three-body OEDIPUS configura tion with offset along the local vertical: (a) attitude response. 91 OEDIPUS configuration: 3-Body a1(0)=a2(0)=ot(0)=2° Yi(0)H(0)4(0)«107s P1(0)=P2(0)=pt(0)=1° 8X(0)=1.2 x 103m, 8y(0)=82(0)=0.1m dx(0)=0.78m, dy(0)=dz(0)=0 Stationkeeping: lt=1 km Tether Longitudinal Vibration Figure 4-16 Spin dynamics (7 = 10°/s) of the three-body OEDIPUS configura-tion with offset along the local vertical: (b) vibration response. 92 5. ATTITUDE AND VIBRATION CONTROL 5.1 Att itude Control 5.1.1 Preliminary remarks The instability in the pitch and roll motions during the retrieval of the tether, the large amplitude librations during its deployment and the marginal stability during the stationkeeping phase suggest that some form of active control is necessary to satisfy the mission requirements. This section focuses on the design of an attitude controller with the objective to regulate the librational dynamics of the system. As discussed in Chapter 1, a number of methods are available to accomplish this objective. This includes a wide variety of linear and nonlinear control strategies which can be applied in conjuction with tension, thrusters, offset, momentum-wheels, etc. and their hybrid combinations. One may consider a finite number of Linear T ime Invariant (LTI) controllers, scheduled discretely over different system configurations i.e. gain scheduling[43]. This would be relevant during deployment and retrieval of the tether where the configuration is changing with time. A n alternative may be to use a Linear T ime Varying ( L T V ) model in conjuction with an adaptive control scheme where on-line parametric identification may be used to advantage[56]. The options are vir tually limitless. O f course, the choice of control algorithms is governed by several important factors: effective for time-varying configurations, computationally efficient for real-time implementation, and simple in character. Here the thruster/momentum-wheel system in conjuction with the Feedback Linearization Technique ( F L T ) is chosen as it accounts for the complete nonlinear dynamics of the system and promises to have good overall performance over a wide range of tether lengths. It is well suited for 93 highly time-varying systems whose dynamics can be modelled accurately, as in the present case. The proposed control method utilizes the thrusters located on each rigid satel-lite, excluding the first one (platform), to regulate the pitch and rol l motion of the tether to which it is attached, i.e. the thrusters located on satellite 2 (link 3) reg-ulates the attitude motion of tether 1 (link 2). O n the other hand, the l ibrational motion of the rigid bodies (platform and subsatellite) is controlled using a set of three momentum wheels placed mutually perpendicular to each other. The F L T method is based on the transformation of the nonlinear, time-varying governing equations into a L T I system using a nonlinear time-varying feedback. De-tailed mathematical background to the method and associated design procedure are discussed by several authors[43,57,58]. The resulting L T I system can be regulated using any of the numerous linear control algorithms available in the literature. In the present study, a simple P D controller is adopted arid is found to have good perfor-mance. The choice of an F L T control scheme satisfies one of the criteria mentioned earlier, namely: valid over a wide range of tether lengths. The question of compu-tational efficiency of the method wi l l have to be addressed. In order to implement this controller in real-time, the computation of the system's inverse dynamics must be executed quickly. Hence a simpler model, that performs well, is desirable. Here the model based on the rigid system with nonlinear equations of motion is chosen. Of course, its validity is assessed using the original nonlinear system that accounts for the tether flexibility. 5.1.2 Controller design using Feedback Linearization Technique The control model used is based on the rigid system with the governing equa-94 tions of motion given by Mr'qr + fr = Quu, (5.1) where the left hand side represents inertia and other forces while the right hand side is the generalized external force due to thrusters and momentum-wheels and is given by Eq.(2.109). Lett ing, ~fr = fr~ QuU (5.2) and substituting in Eq.(2.96), 'ir = s(fr,Mr\dc). (5.3) Expanding Eq.(5.3), it can be shown that "qr = S(fr, Mr\dc) - S(QU, Mr\6)u (5-4) —* = Fr - Qru, where S(QU, Mr\0) is the column matrix given by S(Qu,Mr\6) = [s(Qu(:,l),Mr\0), 5 (Q u ( : , 2 ) ,M r | 0 ) , . . . , S(QU{:, nu), M r | 0 ) ] (5.5) and Qu{'-,i) is the i ^ column of Qu. Extract ing only the controlled equations from Eq.(5.4), i.e. the attitude equations, one has Ire = Frc ~~ Qrcu (5.6) — vrci where vrc is the new control input required to regulate the decoupled linear system. A t this point, a simple P D controller can be applied, i.e. Vrc = qrcd +Kv{Qrcd-Qrc) + Kp(Qrcd~ Qrc)- (5.7) Eq.(5.7) represents the secondary controller with Eq.(5.4) as the primary controller. Kp and Kv are the proportional and derivative gain matrices with qrcdi Qrcd an(^ qrcd 95 as the desired acceleration, velocity and position vector for the attitude angles of each controlled body, respectively. Solving for u from Eq.(5.6) leads to u Qrc {Frc ~ vrcj • (5.8) 5 .1.3 Simulation results The F L T controller is implemented on the three-body STSS tethered system. The choice of proportional and derivative matrix gains is based on a desired settling time of ts. — 0.5 orbit and a damping factor of = 0.7 for both the pitch and roll actuators in each body. Given O and ts., it can be shown that -In ) . 0 5 V / T ^ r (5.9) and hence: kn • —10, V (5.10) where kp. and kVj are the diagonal elements of the gain matrices Kp and Kv in Eq.(5.7), respectively[59]. Note, as each body-fixed frame is referred directly to the inertial frame FQ, the nominal pitch equilibrium angle is zero only when measured from the L V L H frame (OJJ — 0\). However, it is 6\ when referred to FQ. Hence, the desired position vector qrC(l is set equal to 0\(t), i.e. the time-varying orbital angle, for the pitch motion; and zero for the roll and yaw rotations, such that qrcd 37Vxl (5.11) 96 The desired velocity and acceleration are then given as: e U 3 N x l ; (5.12) E K 3 7 V x l . (5.13) When the system is in a circular orbit, qrc^ = 0. Figure 5-1 presents the controlled response of the STSS system, defined in Chapter 4, in the stationkeeping phase with offset d2 = {1,1,1}T m and If = 20 km. A s mentioned earlier, the F L T controller is based on the nonlinear, rigid model. Note, the pitch and roll angles of the rigid bodies as well as of the tether are now attenuated in less than 1 orbit (Figure 5-la). Consequently, the longitudinal vibration response of the tether is free of coupling arising from the librational modes of the tether. This leaves only the vibration modes which are eventually damped through structural damping. The pitch and roll dynamics of the platform require relatively higher control moments, Ma\ and Mpi, respectively (Figure 5-lb) , since they have to overcome the extra moments generated by the offset of the tether attachment point. Furthermore, the platform demands an additional moment to maintain its orientation at zero pitch and roll angle, since it is not the nominal equilibrium position. The non-zero static equilibrium of the platform arises due to its products of inertia. On the other hand, the tether pitch and roll dynamics require only a small ini t ia l control force of about ± 1 N , which eventually diminishes to zero once the system is 97 and o 0 Qrcj — o o 0 0 Oiit) 0 0 1 N \ N 1° I o STSS configuration: 3-Body a1(0)=a2(0)=at(0)=2< P1(0)=p2(0)=pt(0)=1 8x(0)=14m,8y(0)=8z(0)=1m dx(0)=dy(0)=dz(0)=1m Controlled Stationkeeping: lt=20km Tether Pitch and Roll Angles Sat. 1 Pitch and Roll Angles 0.5 1.0 1.5 Time (Orbits) Sat. 2 Pitch and Roll Angles 2.0 CO CD T J , cdT cT - — ' — ' 1 1 1 ! 1 1 I - " " 1" Pitch -A Roll -A -1 • • ' -1 Time (Orbits) Tether Y Tranverse Vibration 1 2 Time (Orbits) Tether Z Tranverse Vibration > to N to Time (Orbits) Time (Orbits) Longitudinal Vibration 14.5h (a) Time (Orbits) Figure 5-1 Controlled dynamics of the three-body STSS during stationkeeping, using the nonlinear, rigid FLT controller with offset along the local horizontal, local vertical and orbit normal: (a) attitude and vibration response. 98 1 STSS configuration: 3-Body a1(0)=ct2(0)=cct(0)=20 P1(0)=p2(0)=pt(0)=1° 8x(0)=14m,8y(0)=8z(0)=1m dx(0)=dy(0)=d2(0)=1m Controlled Stationkeeping: lt=20km Sat. 1 Pitch Control Moment Sat. 1 Roll Control Moment Figure 5-1 Controlled dynamics of the three-body STSS during stationkeeping, using the nonlinear, rigid FLT controller with offset along the local horizontal, local vertical and orbit normal: (b) control actuator time histories. 99 stabilized. Similarly for the end-body, a very small control torque is required to attenuate the pitch and roll response. If the complete nonlinear, flexible dynamics model is used in the feedback loop, the response performance was found to be virtually identical with minor differences, as shown in Figure 5-2. For example, now the pitch and roll response of the platform is exactly damped to zero with no steady-state error as against the minor, almost imperceptible deviation for the case of the controller based.on the rigid model (Figure 5-la). In addition, the rigid controller introduces additional damping in the transverse mode of vibration where none is present, when the full flexible controller is used. This is due to a high frequency component in the at motion that slowly decays the transverse motion through coupling. As expected, the full nonlinear, flexible controller, which now accounts the elastic degrees of freedom in the model, introduces larger fluctuations in the control requirement for each actuator except at the subsatellite, which is not coupled to the tether since f c m 3 = 0. The high frequency variations in the pitch and roll control moments at the platform are due to longitudinal oscillations of the tether and the associated changes in the tension. Despite neglecting the flexible terms, the overall controlled performance of the system remains quite good. Hence, the feedback of the flexible motion is not considered in subsequent analysis. When the tether offset at the platform is restricted to only 1 m along the x direction, a similar response is obtained for the system. However, in this case the steady-state error in the platform's rigid body motion is much smaller (Figure 5-3a). In addition, from Figure 5-3(b), the platform's control requirement is significantly reduced. Figure 5-4 presents the controlled response of the STSS deploying a tether 100 STSS configuration: 3-Body a1(0)=a2(0)=oct(0)=2° p1(0)=P2(0)=p,(0)=1° 8x(0)=14m,5y(0)=82(0)=1m dx(0)=dy(0)=dz(0)=1m Controlled Stationkeeping: l,=20km Tether Pitch and Roll Angles c n T J , cdT Sat. 1 Pitch and Roll Angles 1 r 0.0 0.5 1.0 1.5 Time (Orbits) Sat. 2 Pitch and Roll Angles c n CD cdT a" o T J CQ. - - I" ' 1 1 1 I 1 1 1 1 P i t c h : R o l l • 1 -0 1 2 Time (Orbits) Tether Y Tranverse Vibration 0 1 2 Time (Orbits) Tether Z Tranverse Vibration E, > 10 N CO Time (Orbits) Time (Orbits) Longitudinal Vibration 14.5h (a) Time (Orbits) Figure 5-2 Controlled dynamics of the three-body STSS during stationkeeping, using the nonlinear, flexible FLT controller with offset along the local horizontal, local vertical and orbit normal: (a) attitude and vibration response. 101 STSS configuration: 3-Body a1(0)=a2(0)=cxt(0)=2° 31(0)=p2(0)=p,(0)=1° 8x(0)=14m,8y(0)=8z(0)=1m dx(0)=dy(0)=d2(0)=1m Controlled Stationkeeping: lt=20km r , , . . i , i , , i t i — i — i — i — i — i — i — i — . — J 0 1 2 0 1 2 Time (Orbits) Time (Orbits) Tether Pitch Control Thrust Tether Roll Control Thruster (b) Time (Orbits) Time (Orbits) Figure 5-2 Controlled dynamics of the three-body STSS during stationkeeping, using the nonlinear, flexible FLT controller with offset along the local horizontal, local vertical and orbit normal: (b) control actuator time histories. 102 STSS configuration: 3-Body a1(0)=o2(0)=a,(0)=2° P1(0)=p2(0)=(3t(0)=1° 8x(0)=14m,5y(0)=6z(0)=1m dx(0)=1m,dy(0)=dz(0)=0 Controlled Stationkeeping: lt=20km Tether Pitch and Roll Angles Sat. 1 Pitch and Roll Angles 0.5 1.0 1.5 Time (Orbits) Sat. 2 Pitch and Roll Angles CD eg CQ. 0 1 2 Time (Orbits) Tether Y Tranverse Vibration 1 2 Time (Orbits) Tether Z Tranverse Vibration > CO Time (Orbits) Time (Orbits) Longitudinal Vibration 14.5h (a) Time (Orbits) Figure 5-3 Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear, rigid FLT controller with offset along the local vertical: (a) attitude and vibration response. 103 STSS configuration: 3-Body a1(0)=o2(0)=al(0)=2o P1(0)=p2(0)=pt(0)=1° 8x(0)=14m,8y(0)=8z(0)=1m dx(0)=1m,dy(0)=dz(0)=0 Controlled Stationkeeping: lt=20km Sat. 1 Pitch Control Moment 11 1 ' ' 1 r Sat. 1 Roll Control Moment 0 1 2 Time (Orbits) Tether Pitch Control Thrust 0 1 2 Time (Orbits) Sat. 2 Pitch Control Moment 0 1 2 Time (Orbits) Tether Roll Control Thruster -0.7 o 1 Time (Orbits'* Sat. 2 Roll Control Moment 1E-5 0E0F Time (Orbits) Controlled dynamics of the three-body STSS during stationkeeping using the nonlinear, rigid FLT controller, with offset along the local vertical: (b) control actuator time histories. 104 STSS configuration: 3-Body a1(0)=a2(0)=a,(0)=2° Pl(0)=p2(0)=pt(0)=1° 6X(0)=1.304 x 103m, 8y(0)=82(0)=0 dx(0)=1m,dy(0)=d2(0)=0 Controlled Deployment: l.=0.2km to 20km in 3.5 Orbits Tether Pitch and Roll Angles Sat. 1 Pitch and Roll Angles o 1 2 Time (Orbits) Tether Y Tranverse Vibration 0 1 2 Time (Orbits) Sat. 2 Pitch and Roll Angles CD CD CM CO. Time (Orbits) Tether Z Tranverse Vibration to 15.0 10.0 1 2 3 0 1 2 Time (Orbits) Time (Orbits) Longitudinal Vibration to (a) ,* 5.0 2 3 Time (Orbits) Figure 5 - 4 Deployment dynamics of the three-body STSS, using the nonlinear, rigid FLT controller with offset along the local vertical: (a) attitude and vibration response. 105 STSS configuration: 3-Body a1(0)=oc2(0)=al(0)=20 P1(0)=p2(0)=pt(0)=1° 8X(0)=1.304 x 10"3m, 5y(0)=52(0)=0 dx(0)=1m,dy(0)=dz(0)=0 Controlled Deployment: l.=0.2km to 20km in 3.5 Orbits Tether Length Profile i ' 1 2 0 h E 2*L Sat. 1 Pitch Control Moment t — r — i — i — | — i — i — i — i — | — r — i — i — i — | — r — 0 . 5 1 2 3 Time (Orbits) Sat. 1 Roll Control Moment ! i i " t | t — t — i — [ i i — r -1 2 Time (Orbits) Tether Pitch Control Thrust 1 2 3 Time (Orbits) Tether Roll Control Thruster 2 E - 5 1 2 3 Time (Orbits) Sat. 2 Pitch Control Moment 1 2 Time (Orbits) Sat. 2 Roll Control Moment 1 E - 5 E O E O Time (Orbits) Deployment dynamics of the three-body STSS, using the nonlinear, rigid FLT controller with offset along the local vertical: (b) control actuator time histories. 106 from 0.2 km to 20 km in 3.5 orbits with a i m offset along the tether length. A s before, the system's attitude motion is well regulated by the controller. However, the control cost increases significantly for the pitch motion of the platform and tether due to the Coriolis force induced by the deployment maneuver (Figure 5-4b). Initially, there is a sinusoidal increase in the pitch control requirement for both the tether and platform, as the tether accelerates to its constant velocity, VQ. Then the control requirement remains constant for the platform at about 60 N m as opposed to the tether, where the thrust demand increases linearly. Finally, when the tether decelerates, the actuators' control input reduces sinusoidally back to zero. The rest of the control inputs remain essentially the same as those in the stationkeeping case. In fact, the tether pitch control moment is significantly less since the tether is short during the ini t ia l stages of control. However, the inplane thruster requirement, Tat, acts as a disturbance and causes 6y to nearly double from the uncontrolled value (Figure 4-8a). O n the other hand, the tether has no out-of-plane deflection. Final ly , the effectiveness of the F L T controller during the crit ical maneuver of retrieval from 20 km to 0.2 k m in 3.5 orbits is assessed in Figure 5-5. A s in the earlier cases, the system response in pitch and roll is acceptable however, the controller is unable to suppress the high frequency elastic oscillations induced in the tether by the retrieval. O f course, this is expected as there is no active control of the elastic degrees of freedom. However, the control of the tether vibrations is discussed, in detail, in Section 5.2. The pitch control requirement follows a similar trend, in magnitude, as in the case of deployment with only minor differences due to the addition of offset in the out-of-plane direction ( d 2 = { 1 , 0 , 1 } ^ m). This is also responsible for the higher roll moment requirement for the platform control (Figure 5-5b). 107 STSS configuration: 3-Body a1(0)={X2(0)=a,(0)=2° P1(0)=p2(0)=pt(0)=1° 5x(0)=14m,8y(0)=52(0)=0 dx(0)=1m,dy(0)=dz(0)=0 Controlled Retrieval: l.=20km to 0.2km in 3.5 Orbits Tether Pitch and Roll Angles Sat. 1 Pitch and Roll Angles 0 1 2 Time (Orbits) Sat. 2 Pitch and Roll Angles CO CD cdT CO CD T J , CM CO. 1 2 Time (Orbits) Tether Y Tranverse Vibration Time (Orbits) Tether Z Tranverse Vibration 15.0 10.0 1 2 3 0 1 2 Time (Orbits) Time (Orbits) Longitudinal Vibration CO (a) x 5.0 2 3 Time (Orbits) Figure 5 -5 Retrieval dynamics of the three-body STSS, using the non-linear, rigid FLT controller with offset along the local vertical and orbit normal: (a) attitude and vibration response. 108 STSS configuration: 3-Body a1(0)=ct2(0)=at(0)=2° p1(0)=p2(0)=(3,(0)=1° 5x(0)=14m,5y(0)=82(0)=0 dx(0)=1m,dy(0)=dz(0)=0 Controlled Retrieval: l.=20km to 0.2km in 3.5 Orbits Tether Length Profile E 1 2 Time (Orbits) Tether Pitch Control Thrust 2E-5 1 2 Time (Orbits) Sat. 2 Pitch Control Moment OEOft 1 2 3 Time (Orbits) Sat. 1 Roll Control Moment 1 2 3 Time (Orbits) Tether Roll Control Thruster 1 2 Time (Orbits) Sat. 2 Roll Control Moment 1E-5 E z 2 0E0 (b) Time (Orbits) Time (Orbits) Figure 5-5 Retrieval dynamics of the three-body STSS, using the non-linear, rigid FLT controller with offset along the local vertical and orbit normal: (b) control actuator time histories. 109 5.2 Control of Tether's Elastic Vibrations 5.2.1 Prel iminary remarks The requirement of a precisely controlled micro-gravity environment as well as the accurate release of satellites into their final orbit demands additional control of the tether's vibratory motion, in addition to its attitude regulation. To that end, an active vibration suppression strategy is designed and implemented in this section. The strategy adopted is based on offset control, i.e. time dependent variation of the tether's attachment point at the platform (satellite 1). A l l the three degrees of freedom of the offset motion are used to control both the longitudinal as well as the inplane and out-of-plane transverse modes of vibration. In practice, the offset controller can be implemented through the motion of a dedicated manipulator or a robotic arm supporting the tether, which in turn is supported by the platform. The focus here is on the control of elastic deformations during stationkeeping (i.e. fully deployed tether, fixed length) as it represents the phase when the mission objectives are carried out. This section begins with the linearization of the system's equations of motion for the reduced three body stationkeeping case. This is followed by the design of the optimal control algorithm, Linear Quadratic Gaussian-Loop Transfer Recovery ( L Q G / L T R ) , based on the reduced model and its implementation on the full nonlinear model. Final ly, the system's response in the presence of offset control is presented which tends to substantiate its effectiveness. 5.2.2 System linearization and state-space realization The design of the offset controller begins with the linearization of the equations of motion about the system's equilibrium position. Linearization of extremely lengthy (even in matrix form), highly nonlinear, nonautonomous and coupled equations of 110 motion presents a challenging problem. This is further complicated by the fact the pitch angle ot{ is not referred to the L V L H frame. Hence the equilibrium pitch angle is not a constant, but is equal to the instantaneous orbital angle 9\(t). Two methods are available to resolve this problem. One may use the non-stationary equations of motion in their present form and derive a controller based on the Linear T ime Varying ( L T V ) system. Alternatively, one can design a Linear T ime Invariant (LTI) controller based on a new set of reduced governing equations representing the motion of the three body tethered system. A l -though several studies pertaining to L T V systems have been reported[56,60] the latter approach is chosen because of its simplicity. Furthermore, the L T I controller design is carried out completely off-line and thus the procedure is computationally more efficient. The reduced model is derived using the Lagrangian approach with the gener-alized coordinates given by where <y.{ and are the pitch and roll angles of link i relative to the L V L H frame, respectively. 5X, 8y and 5Z are the generalized coordinates associated with the flexible tether deformations in the longitudinal, inplane and out-of-plane transverse direc-tions, respectively. Only the first mode of vibration is considered in the analysis. The nonlinear, nonautonomous and coupled equations of motion for the teth-ered system can now be written as Qred = [<Xl,Pl,<X2,P2,fix,Sy,5z,a3,02>], (5.14) M, redQred + fr — 0> (5.15) where Mred and frec[ are the reduced mass matrix and forcing term of the system 111 respectively. They are functions of qred and qred in addition to the time varying offset position, d2 and its time derivatives d2 and d2. A detailed derivation of Eq.(5.15) is given in Appendix II. A n additional consequence of referring the pitch motion to a local frame is that the new reduced equations are now independent of the orbital angle 9\\ and under the further assumption of the system negotiating a circular orbit, 91 remains constant. The nonlinear reduced equations can now be linearized about their static equi-l ibr ium position. For all the generalized coordinates, the equilibrium position is zero wi th the exception of 5X which has a non-zero equilibrium, 5XQ. The offset position, d2) is linearized about its ini t ial position, d2^. Linearizing Eq.(5.15) and recasting into matr ix form gives Msqred + CsQred + KsQred + Md^2 + Cdd2 + Kdd2 + fs = 0, (5.16) where Ms, Cs, Ks, Md, Cd, Kd and / s are constant matrices. Mul t ip ly ing throughout by Ms 1 gives qr = -Ms 1Csqr - Ms lKsqr - Ms~lCdd22 - Ms-"Kdd2 - Ms~"Mdd2 - Mg~1 fs. r-1 (5.17) Defining ud = d2 and v = {qT,d^}T, Eq.(5.17) can be rewritten as Pu t t ing v = + -M~lCs -M~'Cd -1/ 0 -Mg~lMd Id 0 v + -M~lKs -M~lKd 0 0 -M-lfs 0 MCv + MKv + Mlud +Fs. *- '4 ) - ( ;> . the L T I equations of motion can be recast into the state-space form as (5.18) (5.19) x =Ax' + Bud + Fd, (5.20) 112 where: A = MC MK Id12 o 24x24. and Let B = \ ^ U 5 R 2 4 X 3 -Fd = I FJ ) e f t 2 4 * 1 — _ i _ -5? (5.21) (5.22) (5.23) (5.24) where x is the perturbation vector from the constant equilibrium state vector xeq. Substituting Eq.(5.24) into Eq.(5.20) gives the linear, perturbation state equation modelling the reduced tethered system as •J x — x — Ax + Bud + (Fd + Axeq). (5.25) It can be shown that for ideal control[60], Fd + Axeq = 0 thus giving the familiar state-space equation S=Ax + Bud. (5.26) The selection of the output vector completes the state space realization of the system. The output vector consists of the longitudinal deformation from the equil ibrium position, SXQ, of the tether at xt = h\ the slope of the tether due to the transverse deformation at xt = 0; and the offset position d2 from its in i t ia l position d2Q. Thus, the output vector y is given by K(o)sx V dx ~ dXQ dy ~ dyQ d z - d z0 J Jy dx da XQ vo \ d z - dZQ ) dy dyQ (5.27) 113 where < ( 0 ) = d/dxi[$v{xi)]Xi=o and < ( 0 ) = d/dxi[4>w(xi)]Xi=0. 5.2.3 Linear Quadratic Gaussian control with Loop Transfer Recovery W i t h the linear state space model defined (Eqs.5.26,5.27), the design of the controller can commence. The algorithm chosen is the Linear Quadratic Gaussian ( L Q G ) estimator based optimal controller[60,61]. The L Q G is a widely used optimal controller with its theoretical background well developped by many authors over the last 25 years. It involves of the design of the Kalman-Bucy Fi l ter ( K B F ) which provides an estimate of the states x, and a Linear Quadratic Regulator ( L Q R ) , which is separately designed assuming all the states x are known. Both the L Q R and K B F designs independently have good robustness properties, i.e. retain good performance when disturbances, due to model uncertainty, are included. However the combined L Q R and K B F designs, i.e. the L Q G design may have poor stability margins in the presence of model uncertainties. This l imitat ion has led to the development of an L Q G design procedure that improves the performance of the compensator, by recovering the full state feedback robustness properties at the plant input or output (Figure 5-6). This procedure is known as the Linear Quadratic Guassian-Loop Transfer Recovery ( L Q G / L T R ) control algorithm. A detailed development of its theory is given in Ref.[61] and hence, is not repeated here for conciseness. The design of the L Q G / L T R controller involves the repeated solution of com-plicated matrix equations too tedious to be executed by hand. Fortunenately, the en-tire algorithm is available in the Robust Control Toolbox[62] of the popular software package M A T L A B . The input matrices required for the function are the following: (i) state space matrices A,B,C and D (D = 0 for this system); (ii) state and measurement noise covariance matrices E and 0 , respectively; (iii) state and input weighting matrix Q and R, respectively. 114 u MODEL UNCERTAINTY SYSTEM PLANT y u LQR CONTROLLER X KBF ESTIMATOR u y LQG COMPENSATOR PLANT X = f v ( X %t ) COMPENSATOR x = A k x - B k y ud = C k * f y Figure 5-6 Block diagram for the LQG/LTR estimator based compensator. 115 A s mentionned earlier, the main objective of this offset controller is to regu-—* late the tether vibration described by the 5 equations. However, from the response of the uncontrolled system presented in Chapter 4, it is clear that there is a large difference between the magnitude of the librational and vibrational frequencies. This separation of frequencies allow for the separate design of the vibration and attitude —* —* controllers. Thus, only the flexible subsystem, composed of the S and d equations, is required in the offset controller design. Similarly, there is also a wide separation of frequencies between the longitudinal and transverse modes of vibrations permitt ing —* the decoupling of the flexible subsystem into a set of longitudinal (Sx and dx) and transverse (5y, Sz, dy and dz) subsystems. The appended offset system, d, must also be included since it acts as the control actuator. However, it is important to note that the inplane and out-of-plane transverse modes can not be decoupled because their oscillation frequencies are of the same order. The two flexible subsystems are summarized below: (i) Longitudinal Vibra t ion Subsystem This is defined by %u — Auxu + Buudu\ (5.28) where xu = {Sx,dX2,5x,dX2}T; u d u = dX2 and, from Eq.(5.27), 0 0 $u(lt) 0 0 0 0 1 '0 0 1 0" 0 0 0 1 (5.29) Au and Bu are the rows and columns of A and B, respectively and correspond to the components of xu. The L Q R state weighting matrix Qu is taken as Qu = •1 0 0 o -0 1 0 0 0 0 1 0 .0 0 0 10. (5.30) 116 and the input weighting matrix Ru = 1^. The state noise covariance matr ix is selected as 1 0 0 0 0 1 0 0 0 0 4 0 LO 0 0 1 while the measurement noise covariance matrix is taken as (5.31) i o 0 15 (5.32) Given the above mentionned matrices, the design of the L Q G / L T R compen-sator is computed using M A T L A B with the following 2 commands: (i) kfu = l q r c ( ^ ( i , C „ , d i a g m x ( E u , e u ) ) / ; (ii) [Aku>Bku>Cku>Dku\ =^ry{Au,Bu,Cu,Du,kfu,Qu,Ru,r); where r is a scaler. The first command is used to compute the Ka lman filter gain matrix kfu. Once kfu is known, the second command is invoked returning the state space representation of the L Q G / L T R dynamic compensator to: (5.33) xu — Akuxu Bkuyu; where xu is the state estimate vector oixu. The function ' l t ry ' performs loop transfer recovery at the system output, i.e. the return ratio at the output approaches that of the K B F loop given by Cu(sld - Au)Kfu[61}. This is achieved by choosing a sufficently large scaler value, r , in ' l t ry ' such that the singular values of the return ratio approach those of the target design. For the longitudinal controller design, r — 5 x 10 5 . Figure 5-7(a) compares the singular values of the recovered compensator design and the non-recovered L Q G compensator design with respect to the target. Unfortunetaly , perfect recovery is not possible, especially at higher frequencies, 117 100 co > CO -50 -100 10 Longitudinal Compensator Singular Values i I I I N I I — i — i i i i n i i — i — i i i i ini— i—i i i i i i i i — i — i i M i I I Target r = 5x10 5 r = 0 (LQG) i i i i I I i ii i i i 1 1 1 1 i i i i i 1 1 1 1 ii i I I i i i i 11111 -3 10 -2 (a) 10"1 10° Frequency (rad/s) 101 10" Transverse Compensator Singular Values 100 50 X3 > 0 CO -50 •100 10 -i—i i 111II i 111 ni 1—i I I I I I I I 1—i I I I N I I 1—i i M 111 Target r = 50 r = 0 (LQG) m i i i i i i n i i i i I I I I I I I 1—i I I I I I I I 1—i i 111 n -3 10 -2 (b) 10"1 10° Frequency (rad/s) 101 10" Figure 5-7 Singular values for the LQG and LQG/LTR compensator compared to target return ratio: (a) longitudinal design; (b) transverse design. 118 because the system is non-minimal, i.e. it has transmission zeros with positive real parts[63]. (ii) Transverse Vibra t ion Subsystem Here: Xy — AyXy + ByU(lv; Dv = Cvxv; with xv = {6y, 8Z, dV2, dZ2, Sy, 6Z, dV2, dZ2}T] udv = {dy2,dZ2}T and (5.34) Cy 0 0 0 0 %(0) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 h 0 0 0 0 0 0 < ( 0 ) o 0 0 1 0 0 0 1 0 0 0 \ / 2 ? r h 0 0 0 0 1 0 0 1 (5.35) Av and Bv are the rows and columns of the A and B corresponding to the components O f Xy. The state L Q R weighting matrix Qv is given as Qv 10 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 10 4 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0. 0 0 1 0 0 0 0 0 0 0 0 1 (5.36) — Ry =w The state noise covariance matrix is taken "1 0 0 0 0 0 0 0 ' 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 9 x 10 4 0 .0 0 0 0 0 0 0 9 x 10 4 (5.37) 119 while the measurement noise covariance matrix is 1 0 0 0 0 1 0 0 0 0 5 x 10 4 0 LO 0 0 5 x 10 4 (5.38) As in the development of the longitudinal controller, the transverse offset con-troller can be designed using the same two commands ('lqrc' and ' l try') wi th the input matrices corresponding to the transverse subsystem with r = 50. The result is the transverse compensator system given by: (5.39) where xv is the state estimates of xv. The singular values of the target transfer function is compared with those achieved by the L Q G and L Q G / L T R in Figure 5-7(b). It is apparent that the recovery is not as good as that for the longitudinal case, again due to the non-minimal system. Moreover, the low value of r indicates that only a lit t le recovery in the transverse subsystem is possible. This suggests that the robustness of the L Q G design is almost the maximum that can be achieved under these conditions. Denoting f y = { f j , f j } r , xf = {x^,x^}T and ud = {udu, ^ } T , the longi-tudinal compensator can be combined to give: xf = $d = Aku 0 0 Akv\ Cku o Cu 0 Bku 0 B Xf Xf = Ckxf, 0 kv Hf = Akxf - Bkyf; (5.40) Vu Vv 0 Cv Xf = CfXf. Defining a permutation matrix Pf such that Xj = PfX, where X is the state vector of the original nonlinear system given in Eq.(3.2), the compensator and full nonlinear 120 system equations can be combined as: X (5.41) Ckxf. A block diagram representation of Eq.(5.41) is shown in Figure 5-6. 5.2.4 Simulation results The dynamical response for the stationkeeping STSS with a tether length of 20 km and ini t ia l offset position of 1 m in each direction is presented in Figure 5-8 when both the F L T attitude controller and the L Q G / L T R offset controller are activated. As expected, the offset controller is successful in quickly damping the elastic vibrations in the longitudinal and transverse direction (Figure 5-8b). However, from Figure 5-8(a), it is clear that the presence of offset control requires a larger control moment to regulate the attitude of the platform. This is due to the additional torque created by the larger moment arm, around 2 m and 1.25 m in the inplane and out-of-plane directions, respectively, introduced by the offset control. In addition, the control moment for the platform is modulated by the tether's transverse vibration through offset coupling. However, this does not significantly affect the l ibrational motion of the tether whose thruster force remains relatively unchanged from the uncontrolled vibration case. Final ly , it can be concluded that the tether elastic vibration suppression though offset control in conjunction with thruster and momentum-wheel attitude control presents a viable strategy for regulating the dynamics tethered satellite sys-tems. 121 STSS configuration: 3-Body a1(0)=a2(0)=at(0)=2° P1(0)=p2(0)=pt(0)=1° 8x(0)=14m,6y(0)=52(0)=1m dx(0)=dy(0)=dz(0)=1m Stationkeeping: I, = 20km LQG/LTR Vibration Control Sat. 1 Pitch and Roll Angles Tether Pitch and Roll Angles — , . : 1 1 1 1 1 1 I 1 1 < 1 ~ ^ ' 1 ^ l _ Time (Orbits) Time (Orbits) Sat. 1 Roll Control Moment Tether Roll Control Thrust (a) Time (Orbits) Time (Orbits) Figure 5-8 Controlled dynamics of the three-body STSS during stationkeeping, using the nonlinear, rigid FLT attitude controller and LQG/LTR offset vibration controller: (a) attitude and libration controller re-sponse. 122 STSS configuration: 3-Body a1(0)=a2(0)=a,(0)=2° p1(0)=(32(0)=Q,(0)=1° ox(0)=14m,Sy(0)=5z(0)=1m dx(0)=dy(0)=dz(0)=1m Stationkeeping: lt = 20km LQG/LTR Vibration Control Tether X Offset Position Time (Orbits) Tether Y Tranverse Vibration 2.0r Time (Orbits) Tether Y Offset Position t o 1 2 Time (Orbits) Tether Z Tranverse Vibration Time (Orbits) Tether Z Offset Position (b) Figure 5-8 Time (Orbits) Time (Orbits) Controlled dynamics of the three-body STSS during stationkeeping, using the nonlinear, rigid FLT attitude controller and LQG/LTR offset vibration controller: (b) vibration and offset response. 123 6 . C O N C L U D I N G R E M A R K S 6.1 Summary of Results The thesis has developed a rather general dynamics formulation for a multi-body tethered system undergoing three-dimensional motion. The system is composed of multiple rigid bodies connected in a chain configuration by long flexible tethers. The tethers, which are free to undergo libration as well as elastic vibrations in three dimensions, are also capable of deployment, retrieval and constant length stationkeep-ing modes of operation. Two types of actuators are located on the rigid satellites: thrusters and momentum-wheels. The governing equations of motion are developed using a new Order(N) ap-proach that factorizes the mass matrix of the system such that it can be inverted efficiently. The derivation of the differential equations is generalized to account for an arbitrary number of rigid bodies. The equations were then coded in FORTRAN, for their numerical integration, with the aid of a symbolic manipulation package that algebraically evaluated the integrals involving the modes shapes functions used to dis-cretize the flexible tether motion. The simulation program was then used to assess the uncontrolled dynamical behaviour of the system under the influence of several system parameters including offset at the tether attachment point, stationkeeping, deploy-ment and retrieval of the tether. The study covered the three-body and five-body geometries, recently flown OEDIPUS system, and the proposed BICEPS configura-tion. It represents innovation at every phase: a general three dimensional formulation for multibody tethered systems; an order-N algorithm for efficient computation; and application to systems of contemporary interest. Two types of controllers, one for the attitude motion and the other for the 124 flexible vibratory motion, are developed using the thrusters and momentum-wheels for the former and the variable offset position for the latter. These controllers are used to regulate the motion of the system under various disturbances. The attitude controller is developed using the nonlinear Feedback Linearization Technique ( F L T ) and is based on the nonlinear but rigid model of the system. O n the other hand, the separation of the longitudinal and transverse frequencies from those of the attitude response allows for the development of a linear optimal offset controller using the robust Linear Quadratic Gaussian-Loop Transfer Recovery ( L Q G / L T R ) method. The effectiveness of the two controllers is assessed through their subsequent application to the original nonlinear flexible model. More important original contributions of the thesis which have not been re-ported in the literature include the following: (i) the model accounts for the motion of a multibody chain-type system under-going librational and, in the case of tethers, elastic vibrational motion in all the three directions; (ii) the dynamics formulation is based on a recursive O(N) Lagrangian algorithm that efficiently computes the system's generalized acceleration vector; (iii) the development of an attitude controller, based on the Feedback Lineariza-tion Technique, for a multibody system using thrusters and momentum-wheels located on each rigid body; (iv) the design of a three degree of freedom offset controller to regulate elastic v i -brations of the tether using the Linear Quadratic Gaussian and Loop Transfer Recovery method ( L Q G / L T R ) ; (v) substantiation of the formulation and control strategies through the applica-tion to a wide variety of systems thus demonstrating its versatility. 125 The emphasis throughout has been on the development of a methodology to study a large class of tethered systems efficiently. It was not intended to compile an extensive amount of data concerning the dynamical behaviour through a planned variation of system parameters. O f course, a designer can easily employ the user-friendly program to acquire such information. Rather, the objective was to establish trends based on the parameters which are likely to have more significant effect on the system dynamics, both uncontrolled as well as controlled. Based on the results obtained, the following general remarks can be made: (i) The presence of the platform products of inertia modulates the attitude re-sponse of the platform and gives rise to non-zero equilibrium pitch and roll angles. (ii) Offset of the tether attachment point significantly affects the equil ibrium ori-entation of the system as well as its dynamics through coupling. W i t h a rel-atively small subsatellite, the tether dynamics dominate the system response with high frequency elastic vibrations modulating the librational motion. On the other hand, the effect of the platform dynamics on the tether response is negligible. As can be expected, the platform dynamics is significantly affected by the offset along the local horizontal and the orbit normal. (iii) For a three-body system, deployment of the tether can destabilize the platform in pitch as in the case of nonzero offset. However, the roll motion remains undisturbed by the Coriolis force. Moreover, deployment can also render the tether slack if it proceeds beyond a critical speed. (iv) Uncontrolled retrieval of the tether is always unstable in pitch. It also leads to high frequency elastic vibrations in the tether. (v) The five-body tethered system exhibits dynamical characteristics similar to those observed for the three-body case. As can be expected, now there are additional coupling effects due to two extra bodies, a rigid subsatellite and a 126 flexible tether. (vi) Cartwheeling motion of the B I C E P S configuration can also be used to ad-vantage in the deployment of the tether. However, exceeding a crit ical ini t ia l cartwheeling rate can result in a large tension in the tether which eventually causes the satellite to bounce back rendering the tether slack. (vii) Spinning the tether about its nominal length, as in the case of O E D I P U S , introduces high frequency transverse vibrations in the tether which, in turn, affect the dynamics of the satellites. (viii) The F L T based controller is quite successful in regulating the attitude mo-tion of both the tether and rigid satellites, in a short period of time, during stationkeeping, deployment as well as retrieval phases. 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M . , First Leaves: A Tutorial Introduction to Maple V, Springer-Verlag, New York, U . S . A . , 1992. [56] Tsakalis, K . S., and Ioannou, P. A . , Linear Time-Varying Systems: Control and Adaptation, Prentice-Hall Inc., Englewood Cliffs, New Jersey, U . S . A . , 1993, pp. 148-180. [57] Su, R . , " O n the Linear Equivalents of Nonlinear Systems," Systems and Control Letter, V o l . 2, No. 1, 1982, pp. 48-52. [58] M o d i , V . J . , Karray, F . , and Chan, J . K . , "On the Control of a Class of Flexible Manipulators Using Feedback Linearization Approach," 4.2nd Congress of the International Astronautical Federation, October 1991, Montreal , Canada, Paper No. IAF-91-324. 131 [59] K u o , B . C , Automatic Control Systems, Prentice-Hall Inc., Englewood Cliffs, New Jersey, U . S . A . , 1987, pp. 314-326. [60] Athans, M . , "The Role and Use of the Stochastic Linear-Quadrat ic-Guassian Problem in Control System Design," IEEE Transactions on Automatic Control, V o l . A C - 1 6 , No. 6, December 1971, pp. 529-552. [61] Maciejowski, J . M . , Multivariable Feedback Design, Addison-Wesley Publ ishing Company, Wokingham, England, 1989, Chapters 1-5. [62] Chiang, R . Y . , and Safonov, M . G , Robust Control Toolbox User's Guide, The M a t h Works Inc., Natick, Mass., U . S . A . , 1992, pp. 2.72-2.74. [63] Stein, G , and Athans, M . , "The L Q G / L T R Procedure for Mult ivariable Feedback Control Design," IEEE Transactions on Automatic Control, V o l . A C - 3 2 , No. 2, February 1987, pp. 105-114. [64] Borisenko, A . I., and Tarapov, I. E . , Vector and Tensor Analysis with Applica-tions, Dover Publications, Inc., New York, U . S . A . , 1979. [65] Lass, H . , Vector and Tensor Analysis, M c G r a w - H i l l Book Company, Inc., New York, U . S . A . , 1950. 132 A P P E N D I X I: TENSOR R E P R E S E N T A T I O N OF T H E EQUATIONS OF M O T I O N 1.1 Prel iminary Remarks The derivation of the equations of motion for the multibody tethered system involves the product of several matrices as well as the derivative of matrices and vectors with respect to other vectors. In order to concisely code these relationships in FORTRAN, the matrix equations of motion are expressed in tensor notation: Tensor mathematics in general is an extremely powerful tool which can be used to solve many complex problems in physics and engineering[64,65]. However, only a few preliminary results of Cartesian tensor analysis are required here. They are summarized below. 1.2 Mathematical Background The representation of matrices and vectors in tensor notation simply involves the use of indices referring to the specific elements of the matrix entity. For example: v = vk (k = l...N); (I.l) A = Aij (i = l...N,j = l...M); where vk is the kth element of vector v, and A{j is the element on the \ t h row and 3th column of matrix A. It is clear from Eq.(I.l) that exactly one index is required to completely define an entire vector whereas two independent indices are required for matrices. For this reason, vectors and matrices are known as first-order and second-order tensors, respectively. Scaler variables are known as zeroth-order tensors since, in this case, an index is not required. 133 Matrix operations are expressed in tensor notation similar to the way they are programmed using a computer language, such as FORTRAN or C. They are expressed in a summation notation known as Einstein notation. For example, the product of a matrix with a vector is given as w = Av (1.2) or, in tensor notation, W i = ^2^2Aij(vkSjk) 4-1 <L3> j where 8jk is the Kronecker delta. Dropping the summation symbol, the matrix-vector product can be expressed compactly as Wi = AijVj = Aimvm. (1.4) Here, j is a dummy index since it appears twice in a single term and hence can be replaced by any other index. Note that since the resulting product is a vector, only one index is required, namely i that appears exactly once on both sides of the equation. Similarly E = aAB + bCD (1.5) or E{j — aAimBmj + bCimDmj ~ aBmjAim "I" bDmjCim, (1.6) where A, B, C, D and E are second-order tensors (matrices) and a and b are zeroth-order tensors (scalers). In addition, once an expression is in tensor form, it can be treated similar to a scaler and the terms can be rearranged. The transpose of a matrix is also easily described using tensors. One simply switches the position of the indices. For example, let w = ATv then, in tensor-notation Wi = AjiVj. (1.7) 134 The real power of tensor notation is in its unambiguous expression of terms containing partial derivatives of scalers, vectors or matrices with respect to other vectors. For example, the Jacobian matrix of y = f(x) can be easily expressed in tensor notation as di-dx-j-'1* ( L 8 ) —* which is a second-order tensor as expected. If f(x) = Ax, where A is constant, then f - * a .) according to the current definition. If C = A(x)B(x) then the time derivative of C is given by C = A{x)B{x) + A(x)B{x) (1.10) or, in tensor notation, Cij = AimBmj + AimBmj (1.11) = XhiAimfcBmj + AimBmj^k), which is more clear than its equivalent vector form. Note that A^^ and Bmjk are third-order tensors that can be readily handled in F O R T R A N . 1.3 Forcing Function The forcing function of the equations of motion as given by Eq.(2.60) is - . u . 1-SdM- dqtdPe * , 9 f i m F(q, q,t) = Mq - -q —q + —— - Qd (2.60) This expression can be converted into tensor form to facilitate its implementation into F O R T R A N source code. Consider the first term Mq. From Eq.(2.89), the mass matrix M is given by T M — Rv MtRv which, in tensor form is Mtj = RZiM^R^j. (1.12) 135 Taking the time derivative of M gives Mij = qs{Rvni^MtnrnRvmj + RvniMtnTn^Rvmj + RvniMtnmRvmjs). (1.13) The second term on the right hand side in Eq.(2.60) is given by \-JPdMu 1 . „, . / T x 2Q ~cWq = 2Qs s r ' k Q r ' ^ ^ Expanding Msrk leads to M sr,/c = Rns,kMtnmRmr + RnsMtnmkRmr + RnsMtnmRmr,k- (L15) Finally, the potential energy term in Eq.(2.60) is also expanded into tensor form to give dPe = dqt dPe dq dq dqt . = dq^QP^ and since ^ = Rp(q)q, then Now, inserting Eqs.(I.13-I.17) into Eq.(2.60) and rearranging, the tensor form of the forcing term can how be stated as Fk(q^q^)=(RPsk + R P n J : q n ) ^ - - Q d k + QsQr | (^Rlk,s ~ 2 M t n m R m r + (^RnkMtnm,s ~ 2~RnsMtnm>k^ R m r (^RnkRmr,s ~ ~2RnsRmr,k^j ' where Fk(q, q, t) represents the kth component of F. (1.18) 136 A P P E N D I X II: R E D U C E D EQUATIONS OF M O T I O N II. 1 Prel iminary Remarks The reduced model used for the design of the vibration controller is represented by two rigid end-bodies, capable of three-dimensional attitude motion, interconnected with a fixed length tether. The flexible tether is discretized using the assumed-mode method with only the fundamental mode of vibration considered to represent longitudinal as well as in-plane and out-of-plane transverse deformations. In this model, tether structural damping is neglected. Finally, the system is restricted to a nominal circular orbit. II .2 Derivation of the Lagrangian Equations of Mot ion Derivation of the equations of motion for the reduced system follows a similar procedure to that for the full case presented in Chapter 2. However, in this model, the straightforward derivation of the energy expressions for the entire system is under-taken, ignoring the Order(N) approach discussed previously. Furthermore, in order to make the final governing equations stationary, the attitude motion is now referred to the local LVLH frame. This is accomplished by simply defining the following new attitude angles: c*i = on + 6\, Pi = Pi, (HI) 7t = 0, where and are the pitch and roll angles, respectively. Note that spin is not considered in the reduced model hence 7J = 0. Substituting Eq.(II.l) into Eq.(2.14), 137 gives the new expression for the rotation matrix as T- ' CpiSa^ Cai+61 S 0 . S a . + e i -S 0 c, Pi (II.2) W i t h the new rotation matrix defined, the inertial velocity of the elemental mass drrii o n the ^ link can now be expressed, using Eq.(2.15), as (2.15) Since the tether length is assumed to be fixed, T j f j and Tj$j<^ of Eq.(2.15) are identically zero. Also, defining (II.3) Vi = I Pi i , Rdm- f ° r the reduced system is given by where and Rdm, =Di + Pi(gi)m + T/^iSi Pi(9i)m= [Tai9i T0.gi T^] fj/ (II.4) (II.5) (II.6) D1 = D D l D S v 52 = 31 + P{(d2)m +T[d2, D3 = D2 + {l2 + dX2}Pi{i)m + T'2i6x. Note that $ j for i = 2 is defined in Section 2.1.2 whereas for i = 1 and 3, it is the null matrix. Since only one mode is used for each flexible degree of freedom, 5i = [6Xi,6yi,6Zi]T for i = 2. The kinetic energy of the system can now be written as i=l Z i = l Jmi Rdm;Rdm;dmi (UJ) 138 Expanding K&i using Eq.(II.4), KH = \ \miD%D{ + 2DiP,i{ j gidmi)fji' + 2D{ T / /.<M™^ +tf J P^(9i)P'i{9i)d^l + 2mT j PFmTi'^dm^ (II.8) where the integrals are evaluated using the procedure similar to that described in Chapter 2. The gravitational and strain energy expressions are given by Eq.(2.47) and Eq.(2.51), respectively, using the newly defined rotation matrix T/ in place of TV Substituting the kinetic and potential energy expressions in d ( 8Ke\ d(Ke - Pe) , i • i Q ^ 0 (II.9) d t \dqred/ d(ired with Qred = [ al> / ? l > a 2 , / ? 2 A 2 A 2 ' < ^ 2 ' Q : 3 ! / ? 3 ] (5-14) and d2 regarded as a time-varying parameter, the nonlinear, non-autonomous equa-tions of motion for the reduced system can finally be expressed as MredQred + fred = ®- (5-15) 139 

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