Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

An approach to dynamics and control of flexible space structure Chen, Yuan 1993

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
ubc_1993_spring_chen_yuan.pdf [ 3.03MB ]
[if-you-see-this-DO-NOT-CLICK]
Metadata
JSON: 1.0080915.json
JSON-LD: 1.0080915+ld.json
RDF/XML (Pretty): 1.0080915.xml
RDF/JSON: 1.0080915+rdf.json
Turtle: 1.0080915+rdf-turtle.txt
N-Triples: 1.0080915+rdf-ntriples.txt
Original Record: 1.0080915 +original-record.json
Full Text
1.0080915.txt
Citation
1.0080915.ris

Full Text

AN APPROACH TO DYNAMICS AND CONTROL OF FLEXIBLE SPACE STRUCTURE By YUAN CHEN B. A. Sc. (Electrical Engineering), Shanghai Jiao Tong University,1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 1993 C) YUAN CHEN, 1993 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. (Signature)  Department of ILl^cLoj n eo The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract A relatively general formulation for studying dynamics and control of a large class of systems, characterized by interconnected rigid bodies, beam and plate type structural members forming a tree topology with two level of branching, is developed using the Lagrangian procedure. The governing equations are discretized using two fundamentally different approachs of system and component modes synthesis. Versatility of the formu- lation is demonstrated, through its application to two systems of contemporary interest, in the presence of nonlinear control: (i) Space station based, two-arm mobile, flexible manipulator; (ii) NASA proposed configuration involving a flexible slewing arm abode a flexible truss with a rigid antenna. Relative merit of the two discretization procedures is assessed over a range of system parameters through comparison of the controlled and uncontrolled responses. Results suggest that the component mode synthesis, though relatively easy to implement, can lead to inaccurate response unless the boundary con- ditions are modeled precisely. Unfortunately, this is seldom possible, particularly with complex systems of current interest. On the other hand, discretization through system modes, though precise, would require frequent updating leading to an increase in the computational time. The investigation represents an original contribution of far-reaching consequence to the field. Such a planned approach to assess discretization methodologies with reference to space based system has not be reported in open literature. Table of Contents Abstract List of Figures List of Tables^ viii List of Symbols ix Acknowledgement^ xiii 1 INTRODUCTION 1 1.1 Preliminary Remarks ^ 1 1.2 Background to the Problem ^ 2 1.3 Scope of the Present Investigation 6 2 KINEMATICS OF THE SYSTEM 8 2.1 Coordinate System ^ 10 2.2 Position of Spacecraft in Space ^ 13 2.3 Shift in the Center of Mass 16 2.4 Elastic and Thermal Deformations ^ 17 2.4.1^Background ^ 17 2.4.2^Deformation Expression for Beam-type Substructure ^ 19 2.4.3^Thermal Deformation^ 20 2.4.4^Transverse vibration 21 2.5 Rotation Matrices  ^22 3 KINETICS OF THE SYSTEM^ 24 3.1 Kinetic Energy ^  24 3.2 Potential Energy  26 3.3 Equations of Motion ^  27 4 AN APPROACH TO CONTROL^ 31 4.1 Feedback Linearization Technique (FLT) ^  31 4.2 Control Implementation Procedures  35 5 COMPUTATIONAL CONSIDERATIONS^ 38 5.1 Program Structure of the Component Modes Method (CMM) ^ 38 5.2 Program Structure for the System Modes Method (SMM)^ 40 6 RESULTS AND DISCUSSION^ 45 6.1 Two Link Mobile Servicing System  45 6.2 NASA's Cotrol-Structure Interaction Model ^  59 7 CONCLUDING REMARKS AND RECOMMENDATIONS FOR FU- TURE WORK ^ 64 Bibliography^ 66 A DETAILS OF Tsys, In, AND Hn,^ 70 B INPUT FILES FOR THE CMM 79 C INPUT DATA FILES FOR THE SMM^ 81 iv List of Figures 1.1 One of the earlier configurations of the proposed space stationFreedom .^3 1.2 Schematic diagrams of the four milestone Configurations of the evolv- ing space station Freedom: (a) First Milestone Configuration (FMC) (b) Man Tended Configuration (MTC) (c) Permanently Manned Configura- tion (PMC) (d) Assembly Complete Configuration (ACC)   4 2.3 A system of interconnected bodies, forming two levels of branching, con- sidered for study. It can represent a vast variety of systems including the proposed space station ^9 2.4 Body fixed coordinate system Fp(Xp, Yp, Zp) and the solar radiation inci- dent angles.  ^11 2.5 Coordinate systems used to identify position of a mass element undergoing rigid body as well as vibrational motions and thermal deformations. . . . 12 2.6 Orbital parameters for a spacecraft defining position of its center of mass. 14 2.7 Modified Eulerian rotations specifying arbitrary orientation of the system in space ^15 4.8 Block diagram of a feedback linearized control system ^ 34 4.9 Block diagram of the Quasi-closed loop control system  37 5.10 Program flow-chart for the CMM simulation ^  39 5.11 Flow-chart showing the numerical integration procedure using the CMM^41 5.12 Program flow-chart for the system modes determination. ^ 42 5.13 Program flow-chart for the SMM simulation ^  43 6.14 Geometry of the two-link manipulator supported by a rigid space platform. 47 6.15 Effect of increasing the amplitude as well as speed of the slewing maneuver, in the plane of the orbit, with the MSS located at the center of the station: (a) 45° in 10 min; (b) 90° in 10 min; (c) 180° in 10 min.   50 6.16 Effect of the slewing speed on the system response with the manipulator at the center of the main truss. The slewing maneuver of 180°(inplane) is completed in: (a) 10 min; (b) 5 min; (c) 2 min.   51 6.17 Response plots showing the effect of the manipulator's location: (a) Ma- nipulator at the center of the station; (b) manipulator at the tip of the station. The slewing rate is 180° in 10 min.   52 6.18 Response plots showing the effect of the manipulator's location during the slewing maneuver of 180° in 5 min. (a) manipulator at the center of the station (b) manipulator at the tip of the station   53 6.19 Controlled response using the FLT applied to the librational degrees of freedom with the manipulator at the center of the station: slewing ma- neuver of 45° in 10 minutes.   54 6.20 Controlled response using the FLT applied to the librational degrees of freedom with the manipulator at the center of the station: slewing ma- neuver of 90° in 10 minutes  55 6.21 Controlled response using the FLT applied to the librational degrees of freedom with the manipulator at the center of the station: slewing ma- neuver of 180° in 10 minutes.   56 6.22 System response in the presence of FLT control with the manipulator located at the tip of the station: slewing maneuver of 180° in 10 minutes. 57 vi 6.23 System response in the presence of FLT control with the manipulator located at the tip of the station: slewing maneuver of 1800 in 5 minutes.^58 6.24 Geometry of the model proposed by NASA for the Control-Structure- Interaction (CSI) study.   61 6.25 Response of the NASA's CSI model during a 180° maneuver in 10 minutes: (a) discretization using the CMM; (b) discretization using the SMM. The manipulator is located at the center of the truss.   62 6.26 Effect of the manipulator location on the system response during the 180° slewing maneuver in 10 minutes: (a) discretization using the CMM; (b) discretization using the SMM. The manipulator is situated at the tip of the truss.   63 vii List of Tables 6.1 Numerical values used in simulation of the two-link manipulator ^ 46 6.2 Numerical values used during simulation of the CSI configuration dynamics 60 viii List of Symbols ab^ beam radius di,j position vectors from 0 c to Oi and Oi to 0, respectively &Tic, drn„ dmij^elemental mass in body Bc, Bi, and Bi,j, respectively orbit inclination with respect to the ecliptic plane kb^ thermal conductivity of an appendage direction cosines of R,,, with respect to Xi,, Yp, Zp axes 1b^ beam length thermal reference length of the beam lc, /i, li,j^ length of bodies Bc, 132 and B2 ,3, respectively mb beam mass per unit length mc, mi, rn2,3^mass of bodies Bc, Bi and B2,3, respectively number of 132,3 bodies attached to body Bi vector representing flexible and rigid generalized coordinates (4r)d ^ ^ vector representing the desired value of the rigid generalized coordinates solar radiation intensity; W/m2 time b V 7 w^ transverse vibration of a beam in its Y and Z directions, respectively C2`, Cfj^ transformation matrices defining orientation of relative to Fc, respectively ix ricm EIZZ Ccm Cf -Fecni —f Ccni position vector from C2 to the instantaneous cenetr of mass of the spacecraft position vector from 0, to the center of mass of the undeformed spacecraft center of mass of the undeformed and deformed configurations of the spacecraft, respectively flexural rigidity of a beam about its Y and Z axes, respectively angular momentum of the spacecraft with respect to the Xc,Y , Z, axes, respectively inertia matrix of spacecraft with respect to the Z, axes, respectively Ixr, I,izz^principal inertia of body Bk about Xk,Yk, Zk axes, respectively; k = c,i or i, total mass of spacecraft N, Nj^ total number of B, and .132,,3 bodies, respectively 0,, 0i, 0,,j origins of the coordinate axes for bodies Bc , BR: and B2,3 respectively (2f,(2,^control effort vectors for flexible and rigid coordinates, respectively position vector from the center of force to the instantaneous center of mass of the spacecraft Rc, Hz, R. ^ ^position vectors of the mass elements dm,, dm, and dm,,i, respectively, as measured from the center of force potential energy of spacecraft; Ue Ug unit matrix ET,^ strain energy of the spacecraft Ug gravitational potential energy of the spacecraft X0, Yo, Zo^ inertial coordinate system located at the center of the earth X p , Yp , Z^ coordinate axes with origin at Cf and parallel to xc, yo, oz,, respectively X8, Y3, Z,^ orbital frame with X, in the direction of the orbit-normal, Y. along the local vertical, and Z, towards the local horizontal 8c, 8i, 6i,i^ vectors representing transverse vibration of the mass elements dm,,dmi, dmi,j, respectively rotation about the local horizontal axis, Z1 , of the intermediate frame Xi, A^ rotation about the local horizontal axis, Y2, of the intermediate frame X2, Y2, Z2 gravitational constant matrix representing slewing motion of the body .132,3 9^ true anomaly 03, On, slew angle and maximum slew angle,respectively longitude of the ascending node Pc, Pi, Pi,i^ vectors denoting positions of dm,, dmi and dmij, respectively, in the undeformed configuration of the spacecraft 'is^ slewing period Tc, Ti, vectors denoting thermal" xi deformations of dm, , dmi and dmi,j, respectively argument of the perigee point librational velocity vector with respect to Xp, Yp, axes rotation about the orbit normal, X, Abrivations cm : center of mass CMM : Component Modes Method FLT : Feedback Linearization Technique MSS : Mobile Servicing System SMM : System Modes Method Dot (.) and (') represent differentiations with respect to time t and true anomaly 0, respectively. xii Acknowledgement I would like to thank Prof. V. J. Modi for his guidance throughout my M.A.Sc program and the preparation of the thesis as well as many other academic and nonacademic aspects. I would like to express my appreciation to Dr. A. Ng, Dr. A. Suleman and Dr. H. Mah for their help in understanding the previous related work. I would like to thank my wife Chao-ying for her unreserved support and understand- ing. Without it the completion of the thesis would have been impossible. This project was suported by the Natural Sciences and Engineering Research Council of Canada, Grant No. A-2181; and Networks of Centers of Excellence Program, Grant No. IRIS/C-8, 5-55380. Both the grants were held by Dr. Modi. Chapter 1 INTRODUCTION 1.1 Preliminary Remarks The Unite States, together with Canada, Japan and the European Space Agency, has committed itself to the establishment of a space station by the turn of this century. It will be used for scientific exploration, satellite launch and maintenance, manufacture of prod- ucts in the favourable microgravity environment, the Earth-oriented applied technologies and numerous other applications. The primary design requirement for the space station is to provide a versatile, ex- pandable, permanent, manned facility for the tasks mentioned above. It will contain laboratories for a wide range of fundamental investigations. Furthermore, the space sta- tion will serve as a platform for satellite launch and repair; as well as assembly of space structures which may be too large, in terms of size and weight, to transport by the space shuttle or other launch vehicles available today. One of the space station configurations considered by NASA is shown in Figure ( 1.1). It had a 150m long main truss (power boom), aligned with the orbit normal with eight photovoltaic arrays parallel to the local vertical, each extending to thirty-three meters and together generating 75 kW of power. The gimbaled solar array blankets provide power at any relative alignment of the space station with respect to the Sun, and heat rejection is achieved by nonrotating radiators. Habitation, laboratory and logistics mod- ules are located near the geometric center of the power-boom. The geometry of the space 1 Chapter 1. INTRODUCTION^ 2 station has already gone through several modifications and, at present, its configuration is undergoing further revision. However, one thing is clear: the space station will be a gigantic and highly flexible structure. It will be the largest platform (more than 100m in length) ever assembled in space. Such a gigantic, massive structure, however, cannot be carried in its entirety to the operational orbit. It will have to be constructed in space by integration of hardware delivered by a number of flights of the space shuttle. For instance, the the first space shuttle flight will deliver four truss bays of the power-boom, two of which are outboard of the articulating alpha joint, which allows the photovoltaic solar arrays to track the Sun. The hardware delivered will also include a radiator, two Reaction Control System (RCS) modules, fuel storage tanks for flight control and reboost, and limited avionic and communication equipment. Once assembled, it will be a fully functional spacecraft, awaiting the second shuttle flight to progress to the next stage of the assembly sequence. NASA has identified, for such an evolving structure, four milestone configurations as shown in Figure ( 1.2). Given the large size of this orbiting system and the expected growth from the initial configuration, the structural flexibility will be a key parameter governing the space station dynamics and control. The presence of environmental and operational disturbances will only add to the complexity of the problem. Hence, thorough understanding of interactions between librational and flexible body dynamics is of importance, for the appropriate control system design, to attain the desired performance. 1.2 Background to the Problem Historically, experiments with scale models have been routinely carried out to obtain useful information for the prototype design. One is, therefore, tempted to use similar SPACE STATION 'FREEDOM' Figure 1.1: One of the earlier configurations of the proposed space station Freedom PV Array V*4 PV Radiator Station Radiator Laboratory Module Stinger/Resistojet (b) PV Array Laboratory Module Logistic Module Habitation Module Japanese Module ESA Module PV Radiator Station Radiator Stinger/Resistojet (d) PV Array Laboratory Module Logistic Module Habitation Module PV Radiator Station Radiator Stinger/Resistojet (c) Chapter 1. INTRODUCTION^ 4 Figure 1.2: Schematic diagrams of the four milestone Configurations of the evolving space station Freedom: (a) First Milestone Configuration (FMC) (b) Man Tended Con- figuration (MTC) (c) Permanently Manned Configuration (PMC) (d) Assembly Complete Configuration (ACC). Chapter 1. INTRODUCTION^ 5 approach for the design of space-based structures. In fact, facilities attempting to simu- late some aspects of space environment have proved useful in testing of relatively small and essentially rigid spacecraft. However, ground-based experiments with large flexi- ble space structures have been found to be of limited value due to practical difficulty in simulating environmental effects such as gravity gradient, magnetic, free molecular, microgravity, solar radiation, etc. This has led to increased dependence on numerical methods, particularly with larger, extremely elastic and complex space configurations. A general formulation applicable to a large class of systems is always attractive. Once the governing equations of motion are established and the associated integration program is operational, it becomes a powerful versatile tool. Over the past three decades, Modi et al. as well as number of other investigators have attempted to obtain relatively general formulations for progressively complex systems to study their dynamical behaviour [1]- [8]. this has also helped in development of several linear as well as nonlinear control strategies. Of course, a number of other investigators have also approached this class of problems in a variety of different ways [9]. This has led to the generation of a vast body of literature which has been reviewed quite effectively by Modi [10] [11], Ng [12], Suleman [13], and Mah [14]. In the above mentioned developments, focus has been on the systems characterized by a large number of interconnected flexible bodies forming a tree topology. In general, dynamics of such systems is governed by a set of 'hybrid', nonlinear, nonautonomous and coupled equations of motion. Here 'hybrid' refers to the set containing both partial (elastic motion) as well as ordinary (rigid body motion) differential equations. To help obtain useful information with relative ease, it is conventional to discretize the partial differential equations into an ordinary set. This is achieved by representing the elastic deflections through a series of time dependent generalized coordinates and spatially vary- ing admissible functions, ideally satisfying geometric and natural boundary conditions. Chapter 1. INTRODUCTION^ 6 There are two fundamentally different choices for the admissible functions: component modes, which try to simulate the system behaviour through synthesis of the local com- ponent behavior; and system modes which attempt to accomplish the same but through the consideration of more global behaviour. Although component modes and their varied forms of synthesis have been used in practice quite widely, as shown by Suleman [13], through a simple example, there is a serious doubt as to the validity of the approach. The major problem is associated with the difficulty in satisfying complex boundary conditions in multibody systems. As future space structures are going to be highly flexible and can only be represented as interconnected multibody systems, the choice of modal functions for discretization has become an issue of enormous importance. This is understandable as validity of the dynamical response and control results depend on the accuracy of the discretization approach. 1.3 Scope of the Present Investigation With this as background, the thesis aims at analyzing, understanding and hopefully arriv- ing at better appreciation of the problem of discretization through a systematic study at a fundamental level. To begin with kinematics and kinetics of a flexible, multibody system with two levels of branching is described and the governing equations of motion obtained using the Lagrangian procedure. A computer code capable of accommodating discretiza- tion through both Component Mode Method (CMM) and System Mode Method(SMM) is developed as an extension of the contributions by Ng [12], Suleman [13] and Mah [14]. Two systems of contemporary interest are considered to asses comparative response: (i) two-link Mobile Flexible Manipulator(MFM) operating on a space platform; (ii) configuration proposed by the Control-Structure Interaction (CSI) program at NASA Langley Research Center [15]. Chapter 1. INTRODUCTION^ 7 Performance of the above two configurations is studied over a range of important system parameters to have some appreciation of the conditions leading to unacceptable performance. Finally, a control strategy, based on the Feedback Linearization Technique (FLT) accounting for the complete nonlinear dynamics of the system, is developed and its effectiveness assessed for the libration/vibration control of the above two configurations using both the discretization procedures. The thesis ends with concluding comments and recommendations for future work in this area. Such a comparative study of discretization procedure, as applied to multibody systems with tree topology and in the presence of nonlinear control, has never been reported before. Chapter 2 KINEMATICS OF THE SYSTEM The system model is identified first and reference coordinate systems explained. This is followed by a kinematic study aimed at establishing position and velocity of an ar- bitrary element of the system under consideration in terms of specified and generalized coordinates. To help study a wide variety of systems, a relatively general model of interconnected beam and plate type members, interconnected to form a tree topology, is considered in Figure ( 2.3). The system is in an arbitrary orbit about the center force, which is also the origin of the inertial reference from F0 comprised of coordinates X0, Yo, Z0. Cf is the instantaneous center of mass of the system. It consists of the central body B, to which are connected structural members Bi referred to as B1, B2, • • • , BN. This may be looked upon as the first level of branching from the trunk of a tree (main body). In turn each Bi body (i = 1, 2, • • • , N) is connected to Bi,3 bodies (j = 1, 2, • • ,ni), the second level of branching. Thus the number of bodies forming the system are 1 N E n3. j=1 The number of members N and ni as well as their positions are kept arbitrary. Fur- thermore the members are free to undergo arbitrary translational and rotational maneu- vers at the joints. This permits simulation of a vast variety of systems of contemporary and future interest. For instance, to simulate the proposed space station Freedom (present configuration which is undergoing changes), the central body Bc may simulate the main truss with the modules, stinger, radiators, photovoltaic arrays and manipulator repre- sented by Bi and B2 bodies as shown inFigure ( 2.3) 8 -N Flexible Body B Flexible Body B 1 *\ Orbit N Flexible \\ Body B1,n1 \\\ Flexible Body B2 Central Body Bc Yo / 0 R cm Flexible Body BN Flexible Body B11 Chapter 2. KINEMATICS OF THE SYSTEM^ 9 Perigee Figure 2.3: A system of interconnected bodies, forming two levels of branching, considered for study. It can represent a vast variety of systems including the proposed space station. Chapter 2. KINEMATICS OF THE SYSTEM^ 10 2.1 Coordinate System Consider the system model as described before. Let X0, Yo, Zo be the inertial coordinate reference located at the center of the earth. The centers of mass of undeformed and deformed configurations of the system are located at C2 and c, respectively. There is a body coordinate system attached to each member of the model which is helpful in defining relative motion between the members. Thus, reference frame 11 is attached to Bc at an arbitrary point O. Frame Fi, with origin at 02, is attached to bodies B, at the connecting point between body Bi and B. In addition, for defining attitude and solar radiation incidence angles, a reference frame is located at Cf such that the axes Xi,, Yp and Zp are parallel to Xc, Y and Zc, respectively as shown in Figure ( 2.4). Now, an arbitrary mass element dmi on body Bi can be reached through a direct path from 0, via Oi. Oc, in turn, is located with respect to the instantaneous center of mass Cf and the inertial reference frame Fo. Thus, the motion of dmi, caused by librational and vibrational motions of Bc and Bi, can be expressed in terms of the inertial coordinate system. Similarly, frame Fi,i is attached to body Bi,i and has its origin at the joint between Bi and Bi The relative position of Oi with respect to 0, is denoted by the vector di, while c12 ,3 defines the position of O relative to O. The location of the elemental mass of the central body, dmc, relative to Oc is defined by pc +77-, +Sc. Here pc indicates the undeformed position of the element; Tc, the thermal deformation; and finally Sc expresses the transverse vibration of the element. Similarly, T-i and S. define the location of the elemental mass dm„ in body B,, relative to Oi . For the elemental mass dm,,i of body B,,j, its position relative to 0,,j is defined by and Si J. The coordinate systems are shown in Figure (2.5). Orientation of the coordinate axes Xi, Y„ Zi and X, Y ,3, Z relative to Xc, Y, Z, is defined by the matrices CT and CZ,pi,j, respectively such that Chapter 2. KINEMATICS OF THE SYSTEM^ 11 Figure 2.4: Body fixed coordinate system Fp(Xp, Yp, Zp) and the solar radiation incident angles. Chapter 2. KINEMATICS OF THE SYSTEM^ 12 Figure 2.5: Coordinate systems used to identify position of a mass element undergoing rigid body as well as vibrational motions and thermal deformations. Chapter 2. KINEMATICS OF THE SYSTEM^ 13 = TL, = CiciLi =^ (2.1) where^is the matrix denoting the motion of body B2 ,3 relative to body Bi; 774 (k = p, c,i, or i , j) is the unit column vector in the corresponding coordinate axes. For instance, = {;:cjc, n4criT. It should be noted that the thermal deformation and transverse vibra- tion of B, and Bi result in the time varying characteristics of C. and C, respectively. 2.2 Position of Spacecraft in Space Consider an arbitrary spacecraft in orbit, as shown in Figure ( 2.6), with its instanta- neous center of mass at Cf . At any instant, the position of Cf is determined by the orbital elements p,i,w, c, R77, and O. Here 8 is the longitude of the ascending node; i, the inclination of orbit with respect to the ecliptic plane; co, the argument of the perigee point; c, the eccentricity of the orbit; R, the distance from the center of the earth to Cf ; and 9 , the true anomaly of the orbit. In general, p, co and c are fixed while and 19 are considered, approximately, functions of time. As the spacecraft has finite dimensions, i.e. it has mass as well as inertia, it is also free to undergo librational motion about its center of mass (in addition to the orbital motion). Let X3, Y5, Z, represent the orbital frame with coordinates aligned along the orbit normal , local vertical, and local horizontal, respectively. Any spatial orientation of Xp, Yp, Zp with respect to X8, Y3, Z, can be described by three modified Eulerian rotations in the following sequence: a pitch motion, 7,b, about the X5-axis giving the first set of intermediate axes X1, Y, Z1; a roll motion, 0, about the Z1-axis generating the second set of intermediate axes X2, Y2, Z2; and finally a yaw motion, A, about the Y2-axis yielding Xp, n and Zr,, as shown in Figure ( 2.7). Chapter 2. KINEMATICS OF THE SYSTEM^ 14 Figure 2.6: Orbital parameters for a spacecraft defining position of its center of mass. Z , Z2 Z s xs x 1 Local Horizontal Yl z^Local Vertical N Chapter 2. KINEMATICS OF THE SYSTEM^ 15 Figure 2.7: Modified Eulerian rotations specifying arbitrary orientation of the system in space. Chapter 2. KINEMATICS OF THE SYSTEM^ 16 From the system geometry and Eulerian rotations, the librational velocity vector is given by c7) = [--4. sin A + (e -F1j)) cos cos Arip + R — (è -c.b) sin Orip -F[cos A + (à +1.k)cos Osin A]rcp,^ (2.2) where 0 represents the orbital rate of the spacecraft. 2.3 Shift in the Center of Mass Instantaneous position of the center of mass Cf serves as an important reference point in identifying position of the system in orbit. However, it is affected the system flexibility as well as translational and slewing motions of the appendages. Hence, determination of its position, at a given instant, is important [16] [17]. In Figure (2.5), C2 and Cf represent the centers of mass of the system during un- deformed and deformed conditions, respectively. The vector Cf , denotes the position of Cf relative to Ct. Thus it represents the shift in the instantaneous center of mass of the spacecraft due to its deformation. This information is necessary in evaluation of the kinetic and potential energies of the system. The vector from the origin of inertial coordinates, the center of the earth, to the mass elements dmc, dmi and dmij are denoted by Rc,Ri,Ri,j, respectively. These vectors can be written as: R,,i — Cf —^+ To, + "Tc + 8c;^ (2.3) Ri^Rcm — Cf^+ +^+ + Si); (2.4) = R — Cf^+ d +^ Tij ^ (2.5) Chapter 2. KINEMATICS OF THE SYSTEM^ 17 Taking moment about the center of force gives 1 R = — {I Rcdmc+ E^Ridmi E fidmi,j1}.^(2.6)cm M^ j=1 mid Substituting Eqs.(2.3)-(2.5) into Eq.(2.6) results in 1 Can= where cicm =^+ of finc c^c^c i=1 (7)- +77-- + )dm + E {f^+ Cf(Pi^Si)dmi ni + E f + C:di,j  ^ , J=1 mid (2.7) Ccm -  = position vector to Ci, the center of mass of the undeformed spacecraft, with respect to Oc; —Ccnif = position vector to Cf, relative to Ci; ni = number of Bi,i bodies attached to body Bi; N = number of Bi bodies; M = total mass. 2.4 Elastic and Thermal Deformations 2.4.1 Background In recent years, greater emphasis has been placed on the design of high-speed, lightweight, precision mechanical systems. These systems, in general, incorporate various types of Chapter 2. KINEMATICS OF THE SYSTEM^ 18 driving, sensing, and controlling devices working together to achieve specified perfor- mance requirements under different loading conditions. In many of these applications, systems cannot be treated as collection of rigid bodies; i.e. flexible character of the system must be accounted for. In such cases, a mechanical system can be modeled as a collection of both rigid and flexible bodies, or an entirely flexible system depending on the situation. The flexible members may be represented as beams, plates, shells, membranes or their combinations. The design and performance analysis of such systems through dynamic simulation can be achieved provided the de- formation effect is incorporated in the mathematical model. The motion of a rigid body in the multibody system can be described by six co- ordinates defining its translation and rotation. They lead to a set of six independent second-order differential equations of motion. The exact configuration of a deformable body, however, can be identified only by infinite number of coordinates. Dynamics of such continuous systems leads to a set of partial differential equations of motion which are both time- and space-dependent. To get meaningful information about the complex system behaviour, it is convenient to convert the mathematical representation into a set of ordinary differential equations by specifying deformations in terms of admissible functions of space and time dependent generalized coordinates. Admissible functions should satisfy as many natural and geometric bound- ary conditions as possible, and often structural modes are used to that end. However, the choice of structural modes which will converge to the right answer is still a subject of considerable controversy particularly for complex, highly flexible systems with ill-defined boundary conditions. Broadly speaking, the choice of modes can be classified into two categories leading to two different methods for studying system dynamics and control (i) modes defining motion of the entire system vibrating in unision, normally obtained Chapter 2. KINEMATICS OF THE SYSTEM^ 19 through finite element approach [13], referred to here as the System Mode Method (SMM); (ii) simpler, individual component modes appropriately synthesized to represent the sys- tem behaviour [12] called the Component Mode Method (CMM). In the SMM, the structure is first subdivided into finite elements with degrees of freedom at the nodes. Using the local degrees of freedom as generalized coordinates, the mass and stiffness matrices of the element can be derived readily. Applying the boundary conditions for the system and compatibility requirements between adjacent elements, the system mass and stiffness matrices can be assembled from the corresponding matrices of the elements. The system modes can be evaluated numerically by using the conventional finite element method. In the CMM, the system's flexural motion is represented in terms of the components' dynamics. The first step is to obtain the series of admissible functions, by solving the eigenvalue problem for each component, representing its elastic deformation. Next, the admissible functions are assembled so that the individual member dynamics becomes homogeneous constituent of the total system response. 2.4.2 Deformation Expression for Beam-type Substructure The governing equation for transverse vibration of a thermally fixed beam is given by (942 52mb^522 Eh, ^+ m ^t  Mb = 0,^(2.8)ax4 ax2^ao with the appropriate boundary conditions. As an example, for a cantilever beam fixed at x = 0 and free at x = lb the boundary conditions are: Chapter 2. KINEMATICS OF THE SYSTEM^ 20 ^Wb -49C4ib = 0,^at x = 0;^(2.9)ax a2wb^a3wb a Alb EIb^ .114 = E ^ t 0 at x = lb.^(2.10)ax2 ax3^ax Here E/b is the bending stiffness of the beam; mb, the mass per unit length; and /14 the thermal bending moment given by .11/4 =EatT(x,y, z)zdA; .Area (2.11) where T(x, y, z) is the difference between the ambient temperature and the temperature at a point on the substructure with coordinate (x, y, z); and af is the thermal expansion coefficient of the beam material. The integral is over the cross-sectional area of the substructure. In general, a closed-form solution for the system is not available. The problem is over- come by assuming the thermal and elastic deformations to be uncoupled, i.e. the solution for thermal displacements can be obtained independent of the elastic displacements. 2.4.3 Thermal Deformation The effect of thermal deformation on transient dynamics of large space structures can be significant, however, the associated analysis, in general, would be quite formidable. Fortunately, as shown by Modi and Brereton[1] the time constant of the heat balance equation is quite small, i.e. the system attains the deformed state almost instantaneously. The steady state solution shows that the shape function for a thermally flexed free-free beam is give by: 8 = —ln[cos (1)] cos^ (2.12) Chapter 2. KINEMATICS OF THE SYSTEM^ 21 8 = —in icos lt, (\n )lcos z; (2.13) where [s^al ^ + zicbc-t (qsas )3/4 (8^eb .)1 aaqq, 7rebab)^4 — Eb ) 2ab^kbbb (2.14) Here: Si,; biz = deflections in the transverse Y and Z directions, respectively; Oy; q = solar radiation incidence angles with respect to appendage reference Y and Z directions, respectively; the distance along the appendage axis from the center point; /g = the thermal reference length of the appendage. It is a function of the solar radia- tion intensity(q,), Stefa-Boltzman constant(crt), appendage dimensions (ab, bb), and appendage physical properties (as, at, eb, kb). Ng and Modi [18] showed that the solution to Eqs.(2.12) and (2.13) can be approxi- mated with an error of 3% for niit < 0.6: 2 ay^1 (77 0— — —̂ s y; it,^2^lt,) (2.15) 2 Sz^1 (ri —) cos Os.^ (2.16) /t,^2^/g Thus, for instance, the thermal deformation of a beam-type body Bi can be simulated as 2^x• -= {0, --() COS^(-12 COS Oz,i}T.2 1:^"1 2 /7 (2.17) Chapter 2. KINEMATICS OF THE SYSTEM^ 22 2.4.4 Transverse vibration There are numerous possibilities for selection of the admissible functions. Consider, for example, the body .13c, a free-free beam, the admissible functions may be given by: Or(x) = cosh (f3r32-) + cos (T) — -yr [sinh (f3r5r) + sin (r- /31th^ib^lb (2.18) r =1,2,•--; where Or/b is the solution of the equation cosh (3r) COS (13r) —1 = 0;^ (2.19) and -yr is given by r^sin (,8r)^sinh (13r) (2.20)7^— COS (f3r) + cosh (Or) • For B. and Bi j treated as cantilever beams: Or(x) = cosh (Prl — cos (Or^— -yr {sinh (f3r1 — sin (or-x— tb (2.21) r = 1,2,— •; where Or is the solution of the equation cosh (Or) cos (Or) + 1 = 0;^ (2.22) and -yris given by = sin Pr) sinh (3")) „yr COS (Or) — cosh (0*) (2.23) 2.5 Rotation Matrices Matrix CT appearing in Eq.(2.1) denotes orientation of the frame Fi relative to the frame F. Two rotation sequences are needed to determine C. The first one, Cic'r, defines Chapter 2. KINEMATICS OF THE SYSTEM^ 23 the rigid body orientation of Fi with respect to Fc; while the second one, CT's, specifies rotation of the frame Fi relative to Fl due to elastic and thermal deformations of the body B. The Eulerian rotations have the following sequence: rotation 07.' about the Xcaxis, followed by Our about 1''-axis; and finally Orz about Z'-axis; i.e. cos In — sin Orz sin g cos g o 0 cos Byr 0 - sin 19; 1 0 0 0 0 1 0 0 COS Or - sin 8; 1 — sin g 0 cos OrY _ 0 sin 8: cos Br. cos 9; cos erz sin g sin g cos erz — cos 9: sin trz cos In cos trz sin 9; + sin 8; sin Brz cos 9; sin 19; sin 8; sin g sin g + cos g cos erz cos In sin 9; sin Brz — sin 8; cos Orz — sin Of sin en cos 8; cos in cos 0; (2.24 ) Similarly c:" = cos 9t cos 0:: sin 01 sin^cos 9-zf — cos O sin 19:7ze cos 9 cos 0-zf sin 9-;` + sin 0.1 sin 19-I cos 19:: sin 81 sin 01 sin q sin 0:: + cos 0-1 cos t91 cos 9 sin 19L` sin 19-zf — sin 8-1 cos 0-zf — sin 19f^sin q cos eifi^ cos 81 cos Oil;  (2.25) Finally, = c7,f x (2.26) Chapter 3 KINETICS OF THE SYSTEM With the kinematics of the system established, the next logical step is to obtain the kinetic and potential energies of the system model under consideration. Application of the Lagrangian principle will then provide the equations of motion, one for each generalized coordinate, governing the system dynamics. 3.1 Kinetic Energy The kinetic energy, T, of the system is given by 1^ ni T = — Rclicdmc E [f Rdmi E I Rijjdmi,]} , 2^nic^ i=1 (3.27) where Rc,^and Ri are obtained by differentiating Eqs.(2.3)—(2.5) with respect to time: =^cm—af —Ci +.t-c-1-6.c-Fox(—Cf —C:,„+pcd-T-c-1-8c); (3.28) =^- R1,3 = -Fc7) x [—Cf — Ci + + C^+^82)];^ (3.29) +(azikti,i +^+^+^+^+ si,J) x [-Cc _ cii +^CZidi CZikti,;(A^T-i,i 82,3)].^(3.30) 24 Chapter 3. KINETICS OF THE SYSTEM^ 25 Substituting Eqs.(3.28-3.30) into Eq.(3.27), the kinetic energy expression can written in the form T = Tort) + [T,„ Th T jr Tt^+ Th,jr Th,t Th,v T,r,t Tint, Tt,e] 1 +—ZiiT Y sc7.; 4.4771181,8, (3.31) 2 Or 1 , T = Torb Tay, + —C7i' 1580 + H sys ,^ (3.32) 2^Y where (47 is the librational velocity vector; Ins, the inertia matrix; Hsu, , the angular imomeutum with respect to the I', frame ; orLy.0 ,the kinetic energy due to pure ro-2 tation ; and C.-vT1/88 , the kinetic energy due to coupling between the rotational motion, transverse vibraion, and thermal deformation. Tsys represents the kinetic energy contri- butions due to various other effects contributed by flexibility, shift in the center of mass and assoiated coupling terms. The subscripts involved are as follows: orb = orbital motion; cm = centre of mass motion; h = hinge position between body Bc and Bi or between body Bi and Bi,j; jr = joint rotation due to elastic and thermal deformation; t = thermal defomation; v = transverse vibration. Chapter 3. KINETICS OF THE SYSTEM^ 26 For instance, Tt,,, refers to the contribution of kinetic energy due to the rate of thermal deformation and transverse vibration velocity. Details of the kinetic energy expression are given in Appendix A. isy, and Hsys represent the inertia and angular momentum vectors of the system, respectively. They are time dependent and can be given by: Lys^Ian + + 1r + +^+ Ih,t +^+ + (3.33)  /Lys -,-- Hem+ H h + H^Ht+ He+ 1 kir + 1 h,,. H h,t H +H,.,t^H,.,„^Ht,„,. (3.34) Here subspript r denotes contribution from the rigid body components. The details of Lys and Hn, are also given in Appendix A. 3.2 Potential Energy The potential energy , U, of the system has two sources: gravitational potential energy, Ug; and strain energy, Ue, due to transverse vibration and thermal deformation, U = Ue Ug.^ (3.35) The potential energy due to the gravity gradient is given by = { f dmc rn dmi^dmijil ^ Rc^i=1^i^j=1 mij 14J Substituting the expressions for Rc, Ri and Rij from Eqs.(2.3)-(2.5) and neglecting the terms of order 1/R4 and higher, Ug can be written as (3.36) M Pe U g = ficm ^ 2R3 trans[Ie Y e] +  314 -IT I 1 2R3^ssY (3.37) h2 R= /4(1^c cos 8)' (3.41) Chapter 3. KINETICS OF THE SYSTEM^ 27 where tte is the gravitational constant and 7 reprensents the direction cosine vector of ltm with respect to the Xp, Yp, zp axis, 1= (cos 7/) sin 4. cos A + sin sin A)i, + cos '17b cos g — c ^(cos "1k sin 4' sin ). — sing') cos AP-cc.^ (3. 38) The strain energy expression for a beam is Ue,beam = 1 I EIy a2 wb) 2 ETI a2 vb 2 vzz  ^dlb ,^(3.39)—2^dx2^ax2 where EIvy and EIzz are the bending stiffnesses of the beam about the Y and Z axes, respectively. 3.3 Equations of Motion Using the Lagrangian procedure, the equation of motion can be obtained from d aT aT au — —aq + —aq Fq,^ (3.40) where q and Fq are the generalized coordinates and generalized forces , respectively. Normally, the effect of librational and vibrational motions on the orbital trajectory is small. Hence , the orbit can be expressed by the classical Keplerian relations: R9 = h; (3.42) where h is the angular momentum per unit mass of the system; ice is the gravitational constant; and E is the eccentricity of the orbit. The general form of the equations of motion can be written as Chapter 3. KINETICS OF THE SYSTEM Mr^Mr, f of, 01/ Ail Cb CA KA QA 28 171, Cg, Kg, Qq, Mf,r^ mf / / qn, Cg,„ Kg„.. Qq., (3.43) Or M(q)re C(q, q', 0) K(q, 0) = Q(0),^(3.44) where primes denote defferentiation with respect to the true normaly 0. Note, n, rep- resents the number of vibrational degrees of freedom. Hence, the total number of gen- eralized coordinates Ng equals the three librational coordinates IP, q5, A plus nv. M is a nonsingular asymmetric matrix of dimension Ng x Ng. The entries in M come from the second order terms of dId0(OT I aqi). C is a Ng x 1 vector. The terms here are derived from two sources: Coriolis contribution of dId0(37115q9; and from 871/aq' . K, also a Ng x 1 vector, denotes the stiffness of the system. It is composed of the terms from ÔU/öq.Q, the generalized force vector of dimension Ng x 1, is evaluated using the virtual work principle. Nonlinear entries in M together with nonlinear and time varying compo- nents of C, K and Q result in a set of coupled, nonlinear and nonautonomous equations of motion. As pointed out before, the equations of motion are applicable to a wide variety of systems because of the relatively general character of the model. They range from com- munication satellites to the evolving space station Freedom and robotic manipulators, Chapter 3. KINETICS OF THE SYSTEM^ 29 to mention a few configurations of current interest. More important features of the formulation may be summarized as follows: (i) It is applicable to an arbitrary number of beam, plate and rigid body members, in any desired orbit, interconnected to form an open branch-type topology. (ii) Joints are provided with translational and rotational degrees of freedom. For ex- ample, in the Space Station configuration, the solar panels can undergo predefined slewing and translational maneuvers with respect to the central body, to track the Sun for optimal power production. (iii) The formulation accounts for the effects of transient system inertias, shift in the center of mass, geometric nonlinearities, shear deformations and rotary inertias. It also considers solar radiation induced thermal effects. Thus it is possible to study the complex system dynamics involving interactions between librational motion, transverse vibrations and thermal deformations. (vi) Environmental forces (aerodynamic, magnetic, etc.) and operational disturbances (Orbiter docking, EVA activity, etc.) can be incorporated readily through general- ized forces and initial conditions. (v) The governing equations are programmed in a modular fashion to isolate the effects of slewing , librational dynamics, flexibility, orbital parameters, etc. (vi) The equations are amenable to discretization using both component as well as system modes thus facilitating comparison between their relative merit. (vii) The equations of motion can be cast, quite readily, into a form suitable for both linear and nonlinear control studies. Chapter 4 AN APPROACH TO CONTROL A system is usually designed according to prescribed specifications to achieve a desired level of performance. Besides performing the required tasks efficiently, it must also exhibit several desirable properties such as stability, quick response, disturbance rejection, etc. For flexible space based systems, governed by highly nonlinear, nonautonomous and coupled equations of motion, this is possible only in the presence of active control. Misawa and Hedrick [19] have reviewed the pertinent literature at some length. As can be expected, the control strategy can vary rather widely. There are literally thousands of control algorithms written to meet a variety of situations and performance requirements. More recent approaches include extended linerization [20],[21]; feedback linearization [22]-[24] or, in simplified cases, the inverse control [25] and several others [26]- [32]. In the present study, the Feedback Linearization Technique (FLT), which completely accounts for the system's nonlinear dynamics, is used to control large, flexible, multibody structures in space. 4.1 Feedback Linearization Technique (FLT) Feedback linearization is an approach for the nonlinear control system design . The basic idea is to algebraically transform the nonlinear system dynamics into an equivalent un- coupled canonical form (partly or fully ) so that linear control techniques can be applied. 30 Chapter 4. AN APPROACH TO CONTROL^ 31 It also can be used as a model-simplifying method in the development of robust or adap- tive nonlinear controllers. In its simplest form, the FLT amounts to cancellation of the nonlinearities so that the closed-loop dynamics is in a linear form. It is applicable to a class of nonlinear systems described by the so-called companion form, or the controlla- bility canonical form. A system is said to be in the companion form if its dynamics is represented by x(n) = f(x)+ b(x)u,^ (4.45) where u is the scalar control input; x is the scalar output of interest with = [x,±, • • • ,x(n)1T as the state vector; and f(x), a nonlinear function of the states . Note, in the state-space representation, Eq.(4.45) can be written as  xl X2 di^ (4.46) xn^f(x) b(x)u Using the control input (assuming b to be non-zero) for the systems which can be expressed in the controllability canonical form, U =^— f].^ (4.47) One can cancel the nonlinearities and obtain the simple input-outout relation (multiple- integrator form ) Xn-1 Xn (4.48) with the control law Chapter 4. AN APPROACH TO CONTROL^ 32 V = -- k1 X1^ —kn_1xn-1,^ ( 4. 49 ) where the ki's are chosen so that Pn + kn-iP12-1 +2 .^+ki is a stable polynomial. At the same time, one can place the poles at the desired locations, i.e. reaching the pole- placement target. This leads to an exponentially stable dynamics, (4.50) which implies that x(t) —> 0. For the tasks involving tracking of a desired output xd(x), the control law (n)^ (n-1)U = Xd — kie — k—, • • • ,^, (4.51) with e = x(t)— xd(t) as the tracking error, leads to the exponentially convergent tracking . Note, similar results would be obtained if the scalar x was replaced by a vector and the scalar b by an invertible square matrix. When the nonlinear dynamics is not in the controllability canonical form, one may have to use transformations to arrive the form before using the above mentioned feedback linearization approach. At times, only partial linearization of the original dynamics is possible. Consider a system as = f (x , u). (4.52) Let us apply the technique referred to as input-state linearization. There are two steps involved. First, one finds a state transformation Z = w(x) and a input transforma- tion U g (x , v) so that the nonlinear system dynamics is transformed into that of an equivalent linear system. Next, one may use a standard linear technique (such as the Chapter 4. AN APPROACH TO CONTROL^ 33 —IP- 0 l̂ib • " V- KT Z U=g (x, v) x=f (x, u) Linearization Loop Pole Placement Loop Z—w (x) Figure 4.8: Block diagram of a feedback linearized control system pole placement) to design the equivalent input v. The block diagram for the procedure is presented in Figure (4.8). Followings remarks can be made about the method: (i) The result is valid in a large region of the state space except for the singularities. (ii) The technique involves combination of the state transformation as well as the input transformation, with the state feedback in both. Thus, it is a linearization by feedback and hence the designation Feedback Linearization Technique. This is fundamentally different from the Jacobian linearization for a small range operation which is the basis for linear control. (iii) In order to implement the control law, the new state components (Z) must be available. They must be physically meaningful, measurable and amenable to de- termination from Z=w(x) . (iv) In general, one relies on the system model for both the controller design and the Chapter 4. AN APPROACH TO CONTROL^ 34 computation . If there is an uncertainty in the model, it will lead to errors in the computation of both the new state Z and the control input U. (v) Tracking control can also be considered. However, the desired motion then needs to be expressed in terms of the complete new state vector. Complex computations may be involved in transforming the desired motion specifications ( of physical output variables ) into specifications in terms of the new state. 4.2 Control Implementation Procedures As shown in Chapter 3 , dynamics of a flexible spacecraft with and 4f corresponding to librational and vibrational generalized coordinates, respectively, is given by [ Mr,r Mr,1 { r + {Fr^{Qr Mf,r Mf,f^f^Ff^Q f (4.53) Here Mr,,. is a 3 x 3 matrix for the librational degrees of freedom; Mrj is a 3 x (Ng — 3) matrix, representing coupling between the rigid and flexible generalized coordinates; Mf,r = Mr,f; and Mfj is a (Nq — 3) x (Ng — 3) submatrix for the flexible degrees of freedom. Fr and Qr are 3 x 1 vectors representing the first and second order coupling terms and the generalized force for the rigid part of the system, respectively. Similarly, Ff and Q f are (Ng —3) x 1 vectors corresponding to the first and second order coupling terms and generatized force for the flexible part of the system, respectively. Assuming only the generalized coordinates of the librational degrees of freedom to be observable, the control force Q f is not applicable and hence set to zero. The objective is to determine the control input Q,. such that the closed-loop system is linearized and has desired pole-positions. Consistent with the assumption: Chapter 4. AN APPROACH TO CONTROL^ 35 mfrrir+ mrf4f + Fr = Qr; ^ (4.54) Mfr^Mf f^F f^; (4.55) which can be solved for "4, and^as: ^Mir F = Qr;^ (4.56) -74f^MI) f; (4.57) where: M = Mrr MrfAlf; Mfr; ^ (4.58) F = FR— Mr.fMhieF f• (4.59) Fortunately, the system has the controllability canonical form. A suitable choice for Q,. would be ^C2r = 71177+P;^ (4.60) with "r) = (47.)d + Ko[(r)d — 4r] + ki2R4r)d 4r]- ^(4.61) Now the controlled equations of motion become: = "I); ^ (4.62) f _^mfru^F f.^ (4.63 ) Chapter 4. AN APPROACH TO CONTROL^ 36 Here: Q,. , M(Md + F + M(Kv6 + IC136); and e = (q7.)d— qr is the error between desired output and the system output . The controller is composed of two parts: the primary controller Qra, = M(Md + F; and the secondary controler: QT,s = M(kve + kpe). The quasi-closed loop control block diagram is shown in Figure (4.9). Note, the controller solves the tracking, stability and linearization problems at the same time. As shown in Chapter 6, the control strategy improves the system behaviour significantly. Chapter 4. AN APPROACH TO CONTROL^ 37 Figure 4.9: Block diagram of the Quasi-closed loop control system Chapter 5 COMPUTATIONAL CONSIDERATIONS As pointed out before, formulation of the problem for dynamical and control studies of a multibody system, comprised of rigid bodies, beam and plate type members is a challenging task. The governing equations were found to be extremely lengthy (even in the matrix form), highly nonlinear, nonautonomous, and coupled. Equally formidable is the development of an efficient numerical code for their integration. This chapter briefly indicates approach used to this end. Modular nature to help isolate influence of various system parameters as well as user friendliness were the two characteristic features which guided the process. Depending on the discretization process used, two separate programs were required. 5.1 Program Structure of the Component Modes Method (CMM) The overall program is composed of several stages or phases. The first one is the in- put phase, followed by numerical integration, output part and completion test. It is schematically shown in Figure (5.10). The input phase includes: (i) orbital parameters p,i ,(4.; and eccentricity e; (ii) program control parameters, simulation period, numerical integration tolerance, ini- tial step-size, and output data; (iii) initial conditions of the state variables. 38 Chapter 5. COMPUTATIONAL CONSIDERATIONS^ 39 Figure 5.10: Program flow-chart for the CMM simulation Chapter 5. COMPUTATIONAL CONSIDERATIONS^ 40 Two data files used are referred to as MODEL and MODE. They are described in the Appendix B. The method DGEAR was chosen to perform the numerical integration of the equations of motion. This is a mature method. In most cases, it is robust. Also it can automatically adjust the integration step-size and select appropriate iteration procedure. FCN block calculates the right function values, e.g. value of the the state vector at the time of T + AT. Here AT is the time step-size used by the DGEAR at the instant. Output and completion test-blocks are determined by the input data file. They are used to output the data in a proper format and stop the simulation at a chosen time, respectively. The numerical integration flowchart is presented in Figure (5.11). 5.2 Program Structure for the System Modes Method (SMM) The first step is to obtain system modes using the finite element method. Thus the program has two major parts: the finite element module; and the dynamic analysis module. They are presented in Figure (5.12) and (5.13), respectively. Updating of the system modes for spacecraft with time dependent geometries, such as rotating solar panels or slewing robotic arms, cannot be overemphasized. System modes reflect elastic character of the entire spacecraft, hence their variation with the geometry can affect the system response significantly. There is an important decision to be made: how often to update the system modes? Obviously one has to strike a balance between the available computational resources, need for real-time information and accuracy. Long updating time may lead to discontinuity in response, which may not be acceptable. Of course, by frequent updating one can avoid such problem, however, only at a considerable computational cost. The input I block in Figure (5.12) introduces material properties of the structure Chapter 5. COMPUTATIONAL CONSIDERATIONS^ 41 Figure 5.11: Flow-chart showing the numerical integration procedure using the CMM Chapter 5. COMPUTATIONAL CONSIDERATIONS^ 42 Figure 5.12: Program flow-chart for the system modes determination. Chapter 5. COMPUTATIONAL CONSIDERATIONS^ 43 Figure 5.13: Program flow-chart for the SMM simulation Chapter 5. COMPUTATIONAL CONSIDERATIONS^ 44 necessary for calculation of the system modes. Input II, in Figure (5.13), enters the simulation period, numerical integration and program control parameters, mode update information, etc. Details of the input files are presented in the Appendix C. Chapter 6 RESULTS AND DISCUSSION 6.1 Two Link Mobile Servicing System With the formulation applicable to a class of multibody systems in hand and the associ- ated computer code operational, the next logical step is to illustrate its effectiveness. To that end two different configurations of contemporary interest are considered: (i) a two-arm, mobile, flexible manipulator operating on a rigid platform; (ii) NASA proposed system for Control-Structure-Interaction(CSI) study. There are three basic objectives: (i) potential to undertake parametric studies if desired; (ii) effectiveness of the FLT for control accounting for the complete nonlinear system dynamics; (iii) relative merit of the system and component modes discretization. It is not intended here to generate a vast body of results for each of the systems under consideration. Of course the code can provide the information as desired by spacecraft designers. The objective is primarily to show the potential of this powerful versatile tool. The proposed space station Freedom, expected to be operational by the turn of the century, will support a Mobile Servicing System (MSS). It is essentially a two link manip- ulator on a mobile base which can traverse along the main truss of the station treated as a 45 Chapter 6. RESULTS AND DISCUSSION^ 46 Table 6.1: Numerical values used in simulation of the two-link mani ulator M (kg) L (m) w(rad/s) .Ix(kgm2) Iyy(kgm2) Izz(kgm2) Center Body 240,120 115.35 rigid 8 x 105 2.67 x 108 2.67 x 108 Upper Link 1,800 7.5 0.167 101 33,750 33,750 Lower Link 1,800 7.5 0.167 101 33,750 33,750 free-free beam. The system is expected to be the workhorse for the station's construction, maintenance, operation, and future evolution. The objective of the simulation here is to assess: (a) effect of the more important system parameters on the response; (b) effectiveness of the nonlinear feedback linearization control. The system geometry is shown in Figure (6.14) with the numerical values used in the simulation presented in Table (6.1). The slewing maneuver of the manipulator is taken to be a sine-ramp function thus resulting in zero velocity and acceleration at the beginning and end of the operation: Os = { Om • (r/T8) — (0,,/27) sin(277-/7-8), r; < Tm Om^ T > rn, . ^(6. 64) Here: 08, slew angle at an instant 0 (true anomaly); 19,, maximum slew angle; and -r8, duration of the slew. To help isolate the effect, only the lower arm was considered to undergo the specified slewing motion in the plane of the orbit. Also only the first cantilever mode is used to represent the link flexibility. The space station is taken to be rigid although the formulation and the program accounts for its flexibility. Figure (6.15) shows the effect of increasing the magnitude as well as speed of the slewing maneuver. Three cases are considered corresponding to the slewing magnitude of 45°, 90° and 180° completed in 10 minutes with the manipulator located at the center of the station. Note, the inplane maneuver excites only the pitch librational motion, Chapter 6. RESULTS AND DISCUSSION^ 47 Figure 6.14: Geometry of the two-link manipulator supported by a rigid space platform. Chapter 6. RESULTS AND DISCUSSION^ 48 i.e. the out-of plane librations, roll and yaw, remain zero. The response follows the expected trend. With the increase in amplitude and speed of the maneuver, from 450 at 4.5°/min to 1800 at 18°/min, the peak pitch response increases from 0.01° to 0.035°. Corresponding vibratory response at the tip of the lower arm (611) in the 1714 direction is also shown for the three cases. Note., the maximum 611 increases significantly, from 0.3mm to 1.2mm . The effect of speed is isolated in the response results given in Figure (6.16). Now, the fixed inplane maneuver of 180° is completed at increasingly faster rates (18° /min, 36°/min, and 90°/min). As can be expected, the librational response is modulated at the vibra- tional frequency with the pick çb exceeding 0.050. Note, the tip response at the lower arm exceeds 10cm for fastest maneuver considered ! This clearly suggests that active control would be necessary during large , fast maneuvers to maintain the response within an acceptable level. Figure (6.17) assesses the effect of manipulator location on the system response for a 180° maneuver completed in 10 minutes. Two cases are considered : (a) manipulator located at the center of the station as before (same as Case (a) in Figure (6.16)); (b) manipulator located near the tip of the station. Note, besides the librational and manip- ulator tip response, deflection time history of the upper arm end (81) is also plotted. As can be expected, several observations of interest can be made: (i) As expected, the pitch amplitude increases significantly due to a large reaction mo- ment about the center of mass. In the present case it increases almost by a factor of 10 ! (ii) The inplane deflection 81 of the upper arm is large than that of the lower arm 611. (iii) Effect of the MSS location on the vibratory response of its arms, though present, is relatively small. Chapter 6. RESULTS AND DISCUSSION^ 49 Figure (6.18) studies the effect MSS location during a faster maneuver of 1800 in 5 minutes (twice the previous slewing rate). Except for the frequency modulation in the pitch, the response character remains essentially the same as in Figure (6.17). Next the attention was turned to the system response with the librational control using the FLT. Typical results are given for the influence of the slewing speed with the manipulator located at the center of the system in Figures (6.19)-(6.21) and at the tip of the station Figures (6.22)-(6.23). Note, the librational motion is controlled, as expected, quite well depending upon the choice of gains. The vibratory motion being uncontrolled continues to persist as no damping is considered. There are several options available to control the manipulator vibrations using the conventional proportional control strategies, linear or nonlinear, with or without damping. These options are not pursued here as the main objective is to assess applicability of the FLT to this class of system. The extension to a hybrid control strategy with the FLT applied to the librational degrees and proportional (or other conventional control) control procedure applied to the vibrational degrees is rather routine. 0.025 V °0.000 -0.025 0.025 0.000 -0.025 0.050^0.100 (b) 0.000 0.150^0.200 0.250 0.000 0.050 0.100 0.150 0.200 0.250 -0.00100 -0.00000 0.00100 -0.00000 -0.00100 0.000 0.00100 0.050 0.100 0.150 0.200 0.250 (a) 0.050^0.100 0.150^0.200 0.250 0.025 0.000 -0.025 0.000 ^ 0.050 ^ 0.100 ^ 0.150 ^ 0.200 ^ 0.250 0.000^0.050 0.100^0.150 0.200^0.250 -0.00000 _0.00100 - 8^(m) Chapter 6. RESULTS AND DISCUSSION^ 50 TIME, Obit TIME, Obit Figure 6.15: Effect of increasing the amplitude as well as speed of the slewing maneuver, in the plane of the orbit, with the MSS located at the center of the station: (a) 45° in 10 min; (b) 90° in 10 min; (c) 180° in 10 min. 0.050 0.025 °-0. 000 -0.025 -0.050 0.000 0.050 0.100 0.150 0.200 0.250 0.050 0.025 0 -0.000 -0.025 -0.050 0.000 0.050 0.100 0.150 0.200 0.250 0.025 iv 0 -0.000 0.050 -0.025 -0.050 0.000 0.050 0.100 0.150 0.200 0.250 -0.1%0000 ^ .000 0.050 0.100^0.150 0.200^0.250 0.100 0.050 0.000 -0.050 0.100 0.2500.050 0.2000.150 0.100 0.050 0.000 -0.050 (m) TIME, Orbit Chapter 6. RESULTS AND DISCUSSION ^ 51 TIME, Orbit Figure 6.16: Effect of the slewing speed on the system response with the manipulator at the center of the main truss. The slewing maneuver of 1800(inplane) is completed in: (a) 10 min; (b) 5 min; (c) 2 min. 0.100 TIME, Orbit -0.0075 0.000 0.00200 ^ 0.000^0.050 0.150^0.200 52 0.2500.000 0.050 0.100 0.150 TIME, Orbit 0.200 0.250 -0.0050 0.000 0.00150 -0.0025 -0.025 0.000 0.0050 0.050^0.100 0.050 0.100 0.150^0.200 0.150 0.200 0.250 0.250 0.00100 80.00050 (TO 1,1 -0.00000 -0.00050 -0.00100 0.0025 (m) -0.0000 0.025 -0.000 0.00100 8 1,1 (m) -0.00000 -0.00100 1 II 0.000 0.1000.050 0.2000.150 0.250 0.2000.050 0.1500.100 0.250 Chapter 6. RESULTS AND DISCUSSION (a) 0.0025 0.0000 8 1 (m) -0.0025 -0.0050 -0.00 -0.10 -0.20 r 0 1 -0.30 -0.40 -0.50 -0.60 (b) Figure 6.17: Response plots showing the effect of the manipulator's location: (a) Manip- ulator at the center of the station; (b) manipulator at the tip of the station. The slewing rate is 1800 in 10 min. Chapter 6. RESULTS AND DISCUSSION 53 0.025 N' ° .000 -0.025 0.1500.100 0.2000.000 0.050 0.250 (a) 0.025 8 (r.) 0.000 -0.025 -0.050^0.100 0.150^0.200 0.2500.000 0.0100 - 0.0050 - 0.0000 -0.0050 -0.0100 - -0.01y000 ^ 0.050^0.100 0.150^0.200 0.250 -0.20 o N' -0.40 -0.00 -0.10 -0.50 -0.60 0.000 0.050 0.100 0.150 0.200 0.250 0.050 0.025 8 (m) 0.000 -0.025 - -0.0%0000^ 0.100^0.150 0.200^0.250 (b) 0.01000 0.00500 0.00000 -0.00500 -0.01000 -0.0159,0000 0.050^0.100 0.150^0.200 0.250 TIME, Orbit TIME, Orbit Figure 6.18: Response plots showing the effect of the manipulator's location during the slewing maneuver of 1800 in 5 min. (a) manipulator at the center of the station (b) manipulator at the tip of the station. 8 i (m) _0.0000 0.0010 0.0005 -0.0005 0.150 0.2000.100 0.2500.000 0.050 8^,m, -0.0000 0.0010 0.0005 -0.0005 0.050 0.100 0.2000.150 0.2500.000 TIME, Orbit -° .if000 0.50 0.25 V °-0.00 -0.25 0.2000.050 0.100 0.150 0.250 TIME, Orbit Chapter 6. RESULTS AND DISCUSSION^ 54 Controlled ^ Uncontrolled Figure 6.19: Controlled response using the FLT applied to the librational degrees of freedom with the manipulator at the center of the station: slewing maneuver of 45° in 10 minutes. .000 0.150 0.200 0.2500.1000.050 5 (.) -0.00000 0.00200 0.00100 -0.00100 0.100 0.2000.000 0.050 0.150 0.250 0.0010 0.0005 5^(In) -0.0000 -0.0005 0.2000.100 0.150 0.2500.000 0.050 0.0200 0.0100 w 0 -0.0000 -0.0100 0.000 0.050 0.100 0.150 0.200 0.250 8 i (m) _0.00000 0.00200 0.00100 -0.00100 0.000 0.050 0.100 0.150 0.200 0.250 Controlled TIME, Orbit Uncontrolled 0.50 0.25 w 0 -0.00 -0.25 TIME, Orbit Chapter 6. RESULTS AND DISCUSSION^ 55 Figure 6.20: Controlled response using the FLT applied to the librational degrees of freedom with the manipulator at the center of the station: slewing maneuver of 900 in 10 minutes. 0.50 0.25 v 0 -0.00 -0.25 -0.P000 0.050 0.100 0.150 0.200 0.250 -0.025 0.000 0.0050 0.0025 5 (m) -0.0000 -0.0025 -0.0050 0.000 0.00200 0.00100 5 1,1 (m) -0.00000 -0.00100 iir 0 -0.000 0.025 0.050 0.050 0.100 0.100 0.150 0.150 0.200 0.200 0.250 0.250 Chapter 6. RESULTS AND DISCUSSION^ 56 Controlled Uncontrolled  0.0050 0.0025 5 1 (m) -0.0000 -0.0025 -0.0050 0.000 ^ 0.050^0.100 ^ 0.150^0.200 ^ 0.250 0.00200 0.00100 5 1,1 (m) -0.00000 -0.00100 0.000 ^ 0.050 ^ 0.100 ^ 0.150 ^ 0.200 ^ 0.250 ^ 0.000 ^ 0.050 ^ 0.100 ^ 0.150 ^ 0.200 ^ 0.250 ^TIME, Orbit TIME, Orbit Figure 6.21: Controlled response using the FLT applied to the librational degrees of freedom with the manipulator at the center of the station: slewing maneuver of 1800 in 10 minutes. Chapter 6. RESULTS AND DISCUSSION 57 0.000 0.0030 0.0020 0.0010 -0.0000 -0.0010 -0.0020 11 1 11 m 1,1 0.050^0.100^0.150^0.00 0.000 0.050 0.100 0.150 0.200 0.2500.000 0.050 0.100 0.150 0.200 0.250 NI -0.00 -0.25 0.50 0.25 -0.5&000 ^ 0.050 0.100 0.150 0.200 0.250 0.0100 0.0050 8 1 (m) -0.0000 -0.0050 -0.00 -0.10 -0.20 y -0.30 -0.40 -0.50 -0.60 0.000 0.050 0.100 0.150 0.200 0.250 8 1 (m) -0.0000 0.0100 0.0050 -0.0050 UncontrolledControlled 0.00300 0.00200 0.00100 -0.00000 -0.00100 -0.00200 0.1000.250 0.250 8 1,1 (m) 0.000^0.050 0.150^0.00 TIME, Orbit TIME, Orbit Figure 6.22: System response in the presence of FLT control with the manipulator located at the tip of the station: slewing maneuver of 180° in 10 minutes. Chapter 6. RESULTS AND DISCUSSION 58 Controlled Uncontrolled -0.60 0.000 0.050 0.100 0.1500.050 0.150 0.2000.100 0.200 0.2500.250 0.50 0.25 0 -0.00 -0.25 -0.10.000 -0.00 -0.10 -0.20 NI 0-0.30 -0.40 -0.50 0.050^0.100 0.150^0.200 0.2500.250 .000 -0.0V ^ .000 8^(m) 0.150^0.2000.050^0.100 0.2500.050^0.100 0.150^0.200 0.01000 0.00500 -0.00000 -0.00500 -0.01000 0.0000.250 0.050^0.100^0.150^0.200 0.025 8 1 (m) 0.000 -0.025 0.025 8 (m) 0.000 -0.025 0.0100 0.0050 -0.0000 -0.0050 -0.0100 - 0.000 TIME, Orbit TIME, Orbit Figure 6.23: System response in the presence of FLT control with the manipulator located at the tip of the station: slewing maneuver of 1800 in 5 minutes. Chapter 6. RESULTS AND DISCUSSION^ 59 6.2 NASA's Cotrol-Structure Interaction Model As pointed out before, with increase in size of the space structures, flexibility effects have progressively become more important in ensuring successful completion of a given mission. The frequency spectrum for a flexible space based system often shows extremely low fundamental frequency with closely spaced and/or overlapping higher modes. This raises the possibility of the control system bandwidth interfering with the system dynam- ics making understanding of the Control-Structure-Interaction(CSI) problems and their resolution extremely important. This has led to the establishment of a major project by NASA, involving ground as well as space based experiments, to better appreciate the CSI oriented issues [9]. The CSI project has established an evolutionary structural model for dynamics and control study as shown in Figure (6.24). It consists of a uniform beam supporting a one arm flexible manipulator and a fixed rigid antenna. The manipulator can be located at any desired position on the beam. Two such positions are shown in the figure: manipulator located at the center or the tip of the beam. Objective is to control the manipulator's tip position during slewing maneuvers. In the present study, this is attempted using the FLT with the flexibility discretization accomplished using system as well as component modes. Thus the response results would help assess: (i) effectiveness of the control strategy using the FLT; (ii) comparative response using the two discretization procedures. The numerical values used in the simulation are presented in Table (6.2) The manipulator, initially aligned with the main truss, executes a slewing maneuver of 108° in 10 minutes when located at: (i) the midpoint of the space station (beam); (ii) the tip of the truss, i.e. 57.5 m from the center. Corresponding results are given in Figures (6.25) and (6.26), respectively. Chapter 6. RESULTS AND DISCUSSION ^ 60 Table 6.2: Numerical values used during simulation of the CSI configuration dynamics M (kg) L (m) f(Hz) Izx(kgm2) Iyy(kgm2) Izz(kgm2) Main truss 18,000 115 0.193 1.9733 x 107 1.01 x 104 1.9733 x 107 Link 3,200 15 0.015 1,805 25.500 25.500 Tip Point Mass 1,000 - - - - The results shows that the nonlinear control using the FLT continues to be quite effective. Both the manipulator and the truss tip motions are rather small. Furthermore, the general trend of the response time history is essentially the same except for local details. The main difference seems to be the chattering response with the SMM which is attributed to the updating of the system mode (fundamental) during the slewing maneuver. In the present case the new system mode was calculated every 2.78 minutes (100 min./36 steps), to save the computational effort. Of course, by more frequent updating, the chattering can be eliminated. Chapter 6. RESULTS AND DISCUSSION^ 61 Figure 6.24:^Geometry of the model proposed by NASA for the Con- trol-Structure-Interaction (CSI) study. 1.0000E-4 5.0000E-5 -4.3656E-11 -5.0000E-5 0.000 0.050 0.100 0.150 0.200 Chapter 6. RESULTS AND DISCUSSION^ 62 2.5000E-7 3.2685E-13 -2.5000E-7 -5.0000E-7 -7.5000E-7 -1.0000E-60.000 5.0000E-6 4.0927E-12 -5.0000E-6 -1.0000E-5 8 (m) 0.050^0.100^0.150^0.200^0.250 -1.5000E-5 • c^• '^• '^'^•0.000^0.050^0.100^0.150^0.200 0.00100 0.00050 -0.00000 -0.00050 8 ( ) -0.00100 0.000 0.050 0.100 0.150 0.200 0.250 (a) (b) Figure 6.25: Response of the NASA's CSI model during a 1800 maneuver in 10 minutes: (a) discretization using the CMM; (b) discretization using the SMM. The manipulator is located at the center of the truss. 0.000^0.050^0.100^0.150 0.00020 0.00010 0.00000 -0.00010 -0.00020 6.156^6.00 (b) 0.200 Chapter 6. RESULTS AND DISCUSSION^ 63 0.000 0.050 0.100 0.150 0.200 0.250 0.0010 0.0005 0.0000 -0.0005 -0.0010 0.000 0.050 0.100 0.150 0.200 0.250 ( a ) 1.0000E-6 5.0000E-7 -2.6290E-13 -5.0000E-7 2.0000E-5 -3.6380E-12 -2.0000E-5 Figure 6.26: Effect of the manipulator location on the system response during the 1800 slewing maneuver in 10 minutes: (a) discretization using the CMM; (b) discretization using the SMM. The manipulator is situated at the tip of the truss. Chapter 7 CONCLUDING REMARKS AND RECOMMENDATIONS FOR FUTURE WORK Advent of the computer revolution has made it possible for dynamicists and control engi- neers to analyze complex space based systems. Dynamics of a flexible orbiting platform, supporting a mobile flexible manipulator, represents a system never envisaged by pio- neers of the classical mechanics including Newton, Euler, Lagrange and Hamilton. Yet, their elegant methodologies help us cast such formidable systems into into mathematical models, and computers assist in their analyses. The present thesis has attempted to tackle a class of such challenging problems of considerable practical importance. More important aspects of the study and associated results may be summarized as follows: (i) An approach to a relatively general formulation for studying dynamics and control of systems, characterized by interconnected flexible bodies, has been explained. (ii) A computer code for the above model, leading to extremely lengthy, highly nonlinear, nonautonomous and coupled equations of motion, has been developed. (iii) The formulation together with the operational integration program represent pow- erful tools for design and parametric evaluation of the system performance. (iv) Application of the above mentioned development to a specific system of the space station based flexible manipulator, in the presence of nonlinear FLT control, sug- gests versatility of the computer code. Results show that large and rapid maneuvers 64 Chapter 7. CONCLUDING REMARKS AND RECOMMENDATIONS FOR FUTURE WORK65 can lead to unacceptable response. The information is fundamental to the design of the system as well as the controller. (v) The study using the NASA's control-structure interaction model shows the response trends to be essentially the same with the component or the system mode dis- cretization. The differences are primarily local in character, at least for the case considered. The study presents a first step in approaching even the preliminary level of under- standing for the dynamics and control of complex flexible systems. The thesis has es- tablished a methodology, however, systematic parametric studies are necessary with a variety of configurations, of practical importance, to develop a data bank with perfor- mance and design charts. This would demand considerable time, effort and computer cost, but should lead to important design tools so urgently needed for the next genera- tion of spacecraft. It will also help in planning of the proposed space based experiments aimed at dynamics and control of flexible structures, analysis of results when such exper- iments are conducted, with the assessment and improvement of the models and control algorithms. Bibliography [1] Modi,V.J., and Brereton,R.C., " Planar Librational Stability of a Flexible Satel- lites",A/AA Journal, Vol.6, No.3, March 1968, pp.511-517. [2] Modi, V.J., and Flanagan, R.C., " Effect of Environmental Forces on the Atti- tude Dynamics of Gravity Oriented Satellites, Part I: High Attitude Orbits", The Aeronautical Journal of the Royal Aeronautical Society, Vol.75, November 1971, pp. 783-793. [3] Modi, V.J., and Flanagan, R.C., "Effect of Environmental Forces on the Attitude Dynamics of Gravity Oriented Satellites, Part II: Intermediate Altitude Orbits Ac- counting for Earth Radiation", The Aeronautical Journal of the Royal Aeronautical Society, Vol.75, December 1971, pp.846-849. [4] Flanagan, R.C., and Modi, V.J., " Effect of Environmental Forces on the Attitude Dynamics of Gravity Oriented Satellites, Part III: Close Earth Orbits Accounting for Aerodynamic Forces", The Aeronautical Journal of the Royal Aeronautrical Society, Vol.76, January 1972, pp.34-40. [5] Modi, V.J., and Flanagan, R.C., " Librational Damping of a Gravity Oriented Sys- tem Using Solar Radiation Pressure", The Aeronautical Journal of the Royal Aero- nautical Society, Vol.75, August 1971, pp.560-564. [6] Modi, V.J., and Kumar,K., " Librational Dynamics of a Satellite with Thermally Flexed Appendages", The Journal of the Astronautical Sciences, Vol.25, No.1, January-March 1977, pp.3-20. [7] Ibrahim, A.M., Mathematical Modelling of Flexible Multibody Dynamics with Ap- plications to Orbiting Systems, Ph.D. Thesis, The University of British Columbia, April 1988. [8] Chan,J.K., Dynamics and Control of an Orbiting Space Platform Based Mobile Flex- ible Manipulator, M.A.Sc. Thesis, The University of British Columbia, Vancouver, Canada, April 1990. [9] Special Issue on Computer Methods in Flexible Multibody Dynamics, The Interna- tional Journal of Numerical Methods in Engineering, Vol.32, No.8, December 1991. [10] Modi,V.J., "Attitude Dynamics of Satellites with Flexible Appendages- A Brief Re- view", Journal of Spacecraft and Rocket, Vol.58, No. 11, November 1974, pp.743-751. 66 Bibliography^ 67 [11] Modi,V.J., "Spacecraft attitude Dynamics : Evolution and Current Challenges", Invited Address to the NATO/AGARD Symposium on the Space Vehicle Flight Mechanics, Luxembourg, Novermber 1989; also Proceedings of the Symposium, AGARD-CP-489, pp.15-1 to 15-26; also Acta Astronautica, Vol.21 No.10, 1990, pp.689-718. [12] Ng,A.C., Dynamics and Control of Orbiting Flexible Systems:A Formulation with Applications, Ph.D.Thesis, The University of British Columbia, Vancouver, Canada, April 1992. [13] Suleman,A., Dynamics and Control of Evolving Space Platforms: an Approach with Application, Ph.D. Thesis, The University of British Columbia, Vancouver,Canada, Augest 1992. [14] Mah,H.W., On the Nonlinear Dynamics of a Space Platform Based Mobile Flexible Manipulator, Ph.D. Thesis, The University of British Columbia, Vancouver, Canada. October 1992. [15] NASA Langley Material on the Space Station Freedom as a Transportation Node and the NASA CSI Program, NASA Langley Space Station Office and CSI Program Office. [16] Mah, H.W., and Modi, V.J., " A Relatively General Formulation for Studying Dy- namics of the Space Station Based MRMS with Applications", AIAA 26th Aerospace Sciences Meeting, January 1988 Reno,Nevada,U.S.A., paper No.AIAA-88-0674. [17] Modi, V.J.,and Suleman,A., " An Approach to Dynamics of Flexible Orbiting Sys- tems with Application to the Proposed Space Station", 42nd Congress of the Inter- national Astronautical Federation, October 1991, Montreal, Canada, Paper No.IAF- 91-293. [18] Ng, A.C, and Modi, V.J., "Dynamics of Gravity Oriented satellites with Thermally Flexed Appendages", AAS/AIAA Astrodynamics Specialist Conference, Kalispell, Montana,U.S.A., 1987, Paper No. AAS-87-432. [19] Misawa, E.A., and Hedrick, J.K., "Nonlinear Observers-A State-of-the-Art Survey", ASME Journal of Dynamic Systems, Measurement, and Control, Vol.111, September 1989, pp.344-352. [20] Baumann, W.T., and Rugh, W.J., " Feedback Control of Nonlinear Systems By Extended Linearization", IEEE Journal of Automatic Control, Vol.31, No.1, January 1986, pp.40-46. Bibliography^ 68 [21] Baumann, W.T., "Feedback Control of Multi-input Nonlinear Systems by Extended Linearization", IEEE Journal of Automatic Control, Vol.33, No.2, February 1988, pp.193-197. [22] Wang, D., and Vidyasagar, M., " Control of a Class of Manipulators with a Single Flexible Link-Part I: Feedback Linearization", ASME Journal of Dynamic Systems, Measurement, and Control, Vol.11, December 1991, pp.655-661. [23] Wang, D., and Vidyasagar, M., " Control of a Class of Manipulators with a Sin- gle Flexible Link-Part II: Observer-Controller Stabilization", ASME Journal of Dy- namic Systems, Measurement, and Control, Vol. 11, December 1991, pp.662-668. [24] Modi, V.J., Karray, F., and Chan, J.K., " On the Control of a Class of Flexible Manipulators Using Feedback Linearization Approach", .42nd Congress of the In- ternational Astronautical Federation, October 1991, Montreal, Canada, Paper No. IAF-91-324. [25] Karray,F., Modi, V.J., and Chan, J.K., " Inverse Control of Flexible Orbiting Ma- nipulators", Proceedings of the American Control Conference, Boston. Mass. U.S.A., June 1991, Editor: A.G. Ulsoy, pp. 1909-1912. [26] Sharan, A.M., Jain, J., and Kalra, P., "Efficient Methods for Solving Dynamic Prob- lems of Flexible Manipulators", ASME Journal of Dynamic Systems, Measurement, and Control, Vol.114, March 1992, pp.78-88. [27] Tosunoglu, S., Lin, S.H., and Tesar, D., "Accessibility and Controllability of Flexible Robotic Manipulators", ASME Journal of Dynamic Systems, Measurement, and Control, Vol.114, March 1992, pp.50-58. [28] Fuliu,V., Rattan, K.S, and Brown Jr.,H.B., "Modeling and Control of Single-Link Flexible Arms with Lumped Masses", ASME Journal of Dynamic Systems, Mea- surement, and Control, Vol.114, March 1992, pp.59-69. [29] Spong, M.W., and Vidyasagar, M., " Robust Linear Compensator Design for Non- linear Robotic Control", Proceedings of the IEEE Conference on Robotics and Au- tomation, St. Louis, Missouri,U.S.A., March 1985, pp.954-959. [30] Spong, M.W., and Vidyasagar, M., " Robust Nonlinear Control of Robot Manip- ulator", Proceedings of the 24th IEEE Conference on Decision and Control, Fort Lauderdale, Florida,U.S.A., December 1985, pp. 1767-1772. [31] De Silva, C.W., "Trajectory Design for Robotic Manipulator in Space Application", J.Guidance, Control and Dynamics, Vol.14, No.3, 1991, pp.670-674. Bibliography^ 69 [32] Xu, J., Bainum, P.M., and Li, F., "Reaction Rejection Techniques for Control of Space-Robotic Structures", IMACS/SICE International Symposium on Robotics, Mechantronics and Manufacturing Systems, Kobe, Japan, September 1992. Appendix A DETAILS OF Lys, Ins AND H The details of the components that make up T of Eq.(3.31) are as follows: 1^•^• Torb = — M R,„ • Rcni; 2 1 = --MC, • C„; 2 1 N Th E2 i=1 [I;1 -2d.dm• + E f (d. i + C7dij) • (di + C7dii)dmij ;mi^2^2 j=-1 mij Tjr 67(pi +^Si) • dic(Pi Ti Si)dmi + E p {6Tdii +^+^+ + sii)} j=1 mi.a {67dii (67,pai,./^T-ii 8ii)} dmi,ii 1^•^•^1^• Tt — I 'Tc • T-cdmc + — 2 2 {.1 (cT.T ) • (cT2)dm, 2 mc mi 70 Appendix A. DETAILS OF Tsy,, I ^1151,8 71 ni + E^(CZillio';" ij) • (CZi No+ ij)drnio ; j=1 Mij 1 NT = —1•Scdnt, + — E^i) • (CfS. i)dmi2 .17.,^2 is=1 ni + E f^• (azip,i,jeii)dmi,i j=1 ni • c—^—Thor — E Idi • fci^Ti + 61i)drnt. E^+ i=1^mi j=-1 {Of cli; +^+ CZpai ,j )(Ti; +^+ S2i ildm2 ,j 1 ni Th,t E + E f^+ cTdii) • (Cf,ilLi,.frii)dmi,ii=1 frni ni =^• (Cf8i)dmi E f (di + cfdij) • (cfjp,i,isii)dmi i=1 j=1 mij Tir,t = E Ef^+ + Si)(67-i)dmi+ mi {6 L ^7dij  ^ Sij)} j=1 mij •(CZjiii,j7fij)drni^; Appendix A. DETAILS OF Tn,, Ins AND I I n,^ 72 Tjr,t, = E [f + + si)(676-..i)dmi+ i=i mi Ef fafdii +^+^+ + Sij)} ni j=1 ; =r ^I• Scdmc+ E [f (Cf'*i) (C)dmi  i=i mi ni + E f (czipi,J7-ii) • ( CZipilSij)dmi,j . 1 rni The system inertia, Ins, given by Eq.(3.33) is the sum of the following components: 'cm —m- recni^— UcmCY,,i ; = E I {di • diU — didi}dmi+ EJ {(di+ CNi,;) • ( di +i=1 mi^ j=1 —(di Cfc/i,j)(di Cfdadmiji =^{-,ac • Tcu — Tc-gcldinc Appendix A. DETAILS OF Tn., In, AND H.y.^ 73 N „ + E 1 j {(c) • (CTA)U (C7Ti)(Cfiii)}dmi i=1 mi ni r ^+ E^{(cf,ith,3T)z,) • (cf,3itigt.i)uj=1 rn'd ; It = f {7-, • T,U Tcrc}dMc + E {fm {(c7T-i) • (C)U — (C7T-i)(CT-iildmi^i=1^ i ni + E f {(czjiL,,,,Tij) • (cfjp,,,t-ii)u —(Cf,j14,iTi3)(Cf,314,3"Tii)}dm.,i1; ^Iv = f {6, •^— 6,S,}dm, mc + E [f {(cfsi) • (C76i)U — (Cf6i)(Cf6iildmi mi ni + E I^• (cz,/,,,,xxj=1 Appendix A. DETAILS OF Tn,, In, AND Hsy,^ 74 Ih,r = E f2di • (cf)u - di(Cfpi) — (Cf-)dildmi i=1 mi + E f {2(di C7dii) (CZjp,i,jp-ii)U — (di + j=1 CTdii) }dmi ,j1 ; = E [f {2di • (C)U — di(CT-f-i) — + EJ {2(di C7dij) • (CZitti,iTii)U — (di + Cfdij)(CfjpidTii) —(CZiktidrii)(di C7dij)}dmi Ih,t, = E {f {2di • (Cf8j)U —4(CfSi)— (C:8i)dildmi i=1 mi + E {2(di + cTdii) • (c,,,Aijaii)u - (di + j=1 mid C7dii)}dmij ] {25c .7-cu — pc.rc — Tcpc}dmc mc + E {2(cfAi) • (c)u - (Cfpi)(Cf7r-i) — (CTi-i)(Caii)}dmi i=i mi Appendix A. DETAILS OF Tsy,, I ^Hn, 75 ni + E f^• (cf,paz,ifii)u -mj —(6 3 ,3T=2)(cT,J kiz,3 1)i, )}dm,31 = f {2Tc • ScU --Pc& — 6cPc}drricmc + E {f {2(C) • (Cf8i)U — (C)(CfSi) — (Cf5i)(CT)i)}dm,ii=1 mi ni + E^(C27,iiti,iTzi)(CC:pai,36zi) —( G7,3/.1/43623 )(67,,A2 ,315, )1drni,3] ; h,,, = f {2T-, • 8,U —^— rn„ + E {f {2(cf,T-i) • (CT8i)U — (CTT-i)(Cf8j) — (C78i)(CfT-i)}dmi i=i mi ni + E {2(czip,i,,,r-ii) • (c,)u - J=1 mid -^)(CZ jiLidrij )1^. Here, U is a unit matrix. Appendix A. DETAILS OF Tsy,, Ins AND Hsy,^ 76 The components of Hn,, the system angular momentum vector in Eq.(3.34) , are as given below: ni Hh = E [I (di x di)dmi E / (di + CNi;) x (di + CT.dij)dmid i=1 rni^ j=1 Mij = E [f Cf(Ti+ 77-^Si) x^Si)dmi ni + E Icifdii+ ̂ ii)} rnij x { Ofdii^+^ri;^; Ht = fmc(Tic X't)dnic E (9777--i x (C)dmi mi ni + E f^x^; mij Hy = Inic(Sc X .8c)dmc E [j (c78, x (C)dmi ni + E (czjizi,j8i; x j=1 mi,i Appendix A. DETAILS OF T55, In, AND Hn,^ 77 H ki, E^x 6T(Toi^+ 8i)dmi i=1 ni E I^x {67c/i; (dZitti,j^+ Sii)}dmi,j x^+ Cfclij)}dmid ;niII kr = E I (qpi x di)drni + E f {Cf,paigij i=1^ j=1 mij H kt = E [f (cT,Ti x + 1, x i=1 mi +E I^x (C/ij=1 H kt, = E (c:si x + x CTSi)dmi i=1 mi ni + EJ {cjaii x (di + j=1 +(di + Ci7clij) x (CLILi,j8. ii)}dmi,;]; 11,,t = I (I), X i-c)dmc + E [f (qpi x cfT-i)dmi mc i=i mi Appendix A. DETAILS OF Tns, Ins AND H58^ 78 ni +E f x j=1 mid f (T) c X Sc)dnic E [f^x CfSi)dmimc^i=1 Htm = f^x 8c -F3c x c)thnc E [1^x^C78i xmc i=i mi ni +Ef^x j=1 X Appendix B INPUT FILES FOR THE CMM Input Data File for the Model: * MSS data based on Julius's two arms model as of June 20, 1990 * Read in Ni, Nj * 1 1 (Ni * * Modeb -- Modes for beam; Modepx -- Modes for plate in X * Modepy — Modes for plate in Y 1 1 1 (Modeb, Modepx, Modepy * * Nqc - central body g.c. * 0 (Nqc 133.4d0 1538d0 .0001d0 .00d0 .0d0 .3d0 (Rmi, RLi, 'WLi, TLi 2629.59d0 .0030d00 1.d0 0.0d0 .00d0 .00d0 (Rloi, Rh -.5d0 -.5d0 0.0d0^(Ai * * Nqi(1) - 1st body g.c. * 2^(Nqi 1.d0^1.d0 .0001d0 .10d0 .453562d2 .3d0 (Rmc, RLc, WLc, TLc 3333d0 .7511d-3^1.d0 0.0d0^.00d0 .00d0 (Rloc, Ric 0.d0^0.0d0^.0d0 (Bi=.5 or .0665 0.d0^0.0d0^.5d0 (Ac * * Nqj(1) - 1st j body g.c. * 1^2^(Nqj 1.d0^1.d0 .0001d0 .10d0 .453562d2 .3d0 (Rmi, RLi, WLi, TLi 3333d0 .7511d-3^1.d0 0.0d0^.00d0 .00d0 atIoi, RR 1.0d0^.060^.0d0 (Bi .060^.060^+.5d0 (Al 79 Appendix B. INPUT FILES FOR THE CMM^ 80 Input Data File for the Mode: * MAIN19D This is MSS model case * long -longitude; inc - inclination; * argu - argument; Ecc - eccentricity 1.570796 0.d0 1.570796 0.d0 * TOL - tolerance; HE - initial step size 1.d-6 1.d-6 * METH - method; MITER - * INDEX - 1^0^1 * Nor - No. of Orbit; Npto - No. of points/Orbit * Nstop - Termination of output; Nout -- Termination of vib. output 1 1800 450 450 * Ncm - shift in cm; Nshade - Thermal Effect; Nengy - Evaluate energy * Nang - use of alt. w; Nslew - Axis of slew 1^0^0^1^3 * Shadei - Shadow angle; Tslew - Period of slew * Aslew - angle of slew 110.d0 36.d0 -180.d0 * Ncon -- nonlinear control 0 * (Gp(i), Gv(i)) i=1 to 3 0.64 1.6 0.64 1.6 0.64 1.6 * Y(iq), iq=1 to 7 0.0000d0 0.0000d0 0.0000d0 0.d0 0.d0 0.d0 0.d0 * Y(Nq-t-iq), iq=1 to 7 0.000d0 0.000d0 0.0000d0 0.d0 0.d0 0.d0 0.d0 Appendix C INPUT DATA FILES FOR THE SMM Data File for the INPUT I: PREP7 /OUTPUT,t1a_out /COM 27TH DECEMBER 1990 /TITLE, MODAL ANALYSIS OF SPACE STATION (FEL - FIRST ELEMENT LAUNCH) KAN,2 KAY,2,40 KAY,3 /COM DEFINE ELEMENT TYPES, 63=PLATE ELEMENT /COM^4=BEAM ELEMENT /COM 21=GENERALIZED MASS ET,1,4 ET,2,21 /COM /COM DEFINE MATERIAL PROPERTIES FOR CENTRAL TRUSS(BEAM) EX,1,1.5E9 NUXY,1,0.333 R,1,3.464,1,1,1.86,1.86 DENS,1,45 /COM DEFINE MATERIAL PROPERTIES FOR MANIPULATOR (BEAM) EX,2,1.E7 NUXY,2,0.333 R,2,3.464,1,1,1.86,1.86 DENS,2,61.66 IOM DEFINE MASS PROPERTIES FOR MSS PAYLOAD R,3,1000,1000,1000,1,1,1 81 Appendix C. INPUT DATA FILES FOR THE SMM^ 82 /COM^ ######## /COM DEFINE NODES FOR MAIN TRUSS N,1,0.,57.5,0. N,21,0.,-57.5,0. FILL /COM DEFINE NODES FOR MANIPULATOR N,22,-0.1307,56.0,0. N,31,-1.307,42.56,0. FILL /COM NLIST,ALL /COM ###########################W############### /COM DEFINE ELEMENTS FOR MAIN TRUSS TYPE,1 REAL,1 MAT,1 E,1,2 E,2,3 E,3,4 E,4,5 E,5,6 E,6,7 E,7,8 E,8,9 E,9,10 E,10,11 E,11,12 E,12,13 E,13,14 E,14,15 E,15,16 E,16,17 E,17,18 E,18,19 E,19,20 E,20,21 Appendix C. INPUT DATA FILES FOR THE SMM^ 83 /COM DEFINE ELEMENTS FOR MANIPULATOR TYPE,1 MAT,2 REAL,2 E,1,22 E,22,23 E,23,24 E,24,25 E,25,26 E,26,27 E,27,28 E,28,29 E,29,30 E,30,31 /COM DEFINE ELEMENT FOR MANIPULATOR PAYLOAD TYPE,2 REAL,3 E,31 /COM ########################################### WAVES M,31,ALL tota1,12 /SHOW,x11 /VIEW,1,1,1,1 ENUM EPLOT ITER,I,1 AFWRITE FINISH /EXEC /INPUT,27 FINISH Appendix C. INPUT DATA FILES FOR THE SMM^ 84 Data File for the INPUT II: ##############################################*################# # *SPACECRAFT CONFIGURATION DATA* # (FIRST MILESTONE CONFIGURATION) # INPUT FILE TO FMC SPACE STATION # PROGRAM NAMES: FEL GF.F => FLT NONLINEAR CONTROL # ################################################################ IMATXC => INERTIA MATRIX FOR CENTRAL BODY ^ 19733353.13d0^0.D0^0.DO 0.0D0^10103.48d0^0.D0 0.01)0^0.DO^19733353.13d0 KEPLERIAN ORBITAL PARAMETERS CCMAT => INITIAL 1-2-3 ORIENTATION OF SPACECECRAFT 0.D0 90.D0 O.DO ECCENT => ECCENTRICITY OF ORBIT 0.0D0 HEIGHT => HEIGHT AT PERIGEE (KM) 4.132 PERIOD => PERIOD OF ORBIT (MINUTES) 1.D2 SLEWING MANEUVRE PARAMETERS FOR APPENDAGES ROTSLW => 1-2-3 MANEUVRE ANGLES FOR APPENDAGES #1 0.01)0 0.60D0 (THETAO,THETAF IN ORBITS) #1 0.0D0 0.00D0 (ALPHAO,ALPHAF IN ORBITS) #1 0.01)0 0.0D0 (BETAO,BETAF) #1 0.0D0 0.60130 (GAMMAO,GAMMAF) TRANSLATION MANEUVRE PARAMETERS FOR APPENDAGES TRASLW => TRANSLATION VECTORS FOR APPENDAGES #1 0.0D0 0.0130 (TRANSO,TRANSF) #1 0.0D0 0.0D0 (XO.XF) #1 0.0130 0.0D0 (YO,YF) #1 0.0D0 0.0D0 (Z022) Appendix C. INPUT DATA FILES FOR THE SMM^ 85 9'0%%%%%%9:9%%%%9'0%%%%%%%%9'0%%%%%%%%%%%%%%%%%%%%%%%%%9'0%%%%e709'0%%%%% STRUCTURAL PROPERTIES FOR APPENDAGES IMATXI => INERTIA MATRIX FOR APPENDAGES APPENDAGE #1 1805.74D0^O.DO^O.DO ^ 0.0D0^255000.00D0^O.DO 0.0D0^O.DO^255000.00d0 RHOLNI => FIRST MOMENT OF AREA Bc O.DO^0.0D0^0.0D0 #1 39000.0D0^0.0D0^0.0D0 DIO => INITIAL HINGE POSITION VECTOR FOR APPENDAGES #1 ODO^0.D0^0.D0 ENIASS => MASS PROPERTIES Bc 17905.5D0 #1 4200.D0 ROT1NI => 1-2-3 INITIAL RIGID ORIENTATION OF APPENDAGES #1 O.DO^O.DO^O.DO IMSL => IMSL:DGEAR PARAMETERS HE => 1.D-8 TOL => 1.D-6 METH => 2 MITER => 0 INDEX => 1 SIMUL => SIMULATION RUN TIME NOR => 0.20D0 NPTO => 10.d3 INICO INTTIAL CONDMONS (IN DEGREES FOR ATTITUDE MOTION) DLSPO => O.DO^0.D0^040^0.0d0^0.d0^0.d0^0.d0^0.d0^0.130 0.D0 0.D0 0.D0 0.D0 VELO => 0.D0^0.130^0.130^0.130^0.D0^0.D0^0.130^0.D0^0.D0 0.130 0.D0 0.D0 0.130 Appendix C. INPUT DATA FILES FOR THE SMM^ 86 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% MODAL => MODAL EIGENVECTOR FILENAMES STEP! => MODAL EIGENVECTOR FILENAMES mass0701 mass0801 mass0901 mass1001 mass1101 STEP2 => MODAL EIGENVECTOR FILENAMES mass0702 mass0802 mass0902 mass 1002 mass1102 STEP35 => MODAL EIGENVECTOR FILENAMES mass0'735 mass0835 mass0935 mass1035 mass1135 STEP36 => MODAL EIGENVECTOR FILENAMES mass0736 mass0836 mass0936 mass1036 mass 1136 STEP37 => MODAL EIGENVECTOR FILENAMES mass0'737 mass0837 mass0937 mass 1037 mass1137 Appendix C. INPUT DATA FILES FOR THE SMM^ 87 OPTION => SIMULATION AND OUTPUT OPTIONS ICTRL =>^1 (0=UNCONTROLLED, 1=CONTROLLED ) IQUASI =>^1 (1=OPEN LOOP • 2=CLOSE LOOP) IGFOFT =>^0 {1=INCLUDE G.F , 0= NO G.F. ) ISOLAR =>^1 (0DONT PRINT, 1= PRINT ) IPVRAD =>^0 (0-=DON'T PRINT, 1= PRINT ) IRCS =>^0 (0=DONT PRINT, 1= PRINT ) ISTING =>^0 (0=PRINT , 1= PRINT ) ITRUSS =>^1 (0=DON'T PRINT, 1= PRINT ) IAERO => 0 (1=INCLUDE AERO, 0= DO NOT ) IUPDAT =>^1 EOF

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
Japan 6 0
United States 6 1
China 2 1
City Views Downloads
Tokyo 6 0
Ashburn 2 0
Beijing 2 0
Washington 2 0
Redmond 1 0
Irvine 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080915/manifest

Comment

Related Items

Admin Tools

To re-ingest this item use button below, on average re-ingesting will take 5 minutes per item.

Reingest

To clear this item from the cache, please use the button below;

Clear Item cache