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DEPLOYABLE ANTENNA KINEMATICS USING TENSEGRITY STRUCTURE DESIGN By BYRON FRANKLIN KNIGHT A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2000 Copyright 2000 By Byron Franklin Knight For this work I thank Mary, my friend, my partner, and my wife. ACKNOWLEDGMENTS This research has been a labor of love, beginning with my first job as a "new grad" building deployable antenna tooling in 1982. There have been numerous mentors along this path who have assisted me to gain the knowledge and drive to attack such a difficult problem. I thank Gerry Perkins, Doug Worth, and Jerry Cantrell for giving me that first job and allowing me to indulge my interests. I thank Dr. Bobby Boan and Joe Cabrera for guiding me through necessary original growth that allowed this knowledge to blossom. I thank lan Stem for his enthusiasm, energy, and creativity. Most of all, I thank my associate, Ms. Geri Robb, for trusting me, guiding me, and protecting me. I wish to acknowledge my family; we truly are the lowest paid group per degree on this earth, but we are rich in each other. I thank my parents, George and Mary, and their brood: Dewey, ML, Ally, Mary, Mo, Karen, Tracy, George M., and Little Byron. I thank the Kennedys for letting me join their clan. I thank my committee, Drs. C. Crane, A. Seireg, R. Selfridge, and G. Wiens for their assistance toward this work. I also thank Dr. Joseph Rooney of the Open University in England for his generous assistance and extensive knowledge of mathematics. To my Committee Chairman, Dr. Joseph Duffy, I give my heartfelt thanks. You have taught me that to grow the developments of the 21st Century we need the wisdom and dedication of the Renaissance. Sir, you are an English Gentleman, my teacher and my mentor. I shall not forget this gift you give me. More than teaching me engineering, you taught me the proper way for a gentleman to toil at his labor of love. TABLE OF CONTENTS page ACKNOW LEDGG ENTS ........................................................................................... iv ABSTRACT...................................................................................................................... vii CHAPTERS 1 BACKGROUND ..................................................................................................... 1 Space Antenna Basis................................................................................................ 1 Antenna Requirements............................................................................................ 2 Improvement Assumptions................................................................................. 3 2 INTRODUCTION ................................................................................................... 5 Tensegrity Overvie ................................................................................................ 5 Related Research...................................................................................................... 7 Related Designs ....................................................................................................... 8 Related Patents ......................................................................................................... 10 3 STUDY REQUIREM ENTS ........................................................................................ 13 Stability Criterion...................................................................................................13 Stowage Approach .................................................................................................. 13 Deployment Approach..............................................................................................13 M mechanism Issues .................................................................................................. 15 4 BASIC GEOMETRY FOR THE 66 TENSEGRITY APPLICATION......................16 Points, Planes, Lines, and Screws................................................. ............................ 17 The Linear Complex .............................................................................................. 19 The Hyperboloid of One Sheet ...............................................................................22 Regulus Pliicker Coordinates .................................................................................... 24 Singularity Condition of the Octahedron.....................................................................26 Other Forms of Quadric Surfaces ...............................................................................28 5 PARALLEL PLATFORM RESULTS .............................................................................31 33 Solution............................................................................................................ 31 44 Solution............................................................................................................39 6 66 D ESIGN ...............................................................................................................42 66 Introduction ...........................................................................................................42 Sketch..........................................................................................................................42 Evaluating the Jacobian .......................................................................................... 45 Optim ization Solution..................................................................................................46 V ariable Screw M otion on the ZA xis ....................................................................48 Special Tensegrity M options .........................................................................................55 7 D EPLO Y M EN T A N D M EC HAN ICS ........................................... .................................57 Strut D design .................................................................................................................57 Strut/Tie Interaction.....................................................................................................63 D eploym ent Schem e ....................................................................................................65 Previous Related W ork .............................................................................................66 A labam a D eploym ent Study....................................................................................... 68 D eploym ent Stability Issues ........................................................................................69 8 STO W A G E D ESIGN .................................................................................................75 M inim ized Strut Length.................................... ...................... ..................................76 33 O ptim ization ..........................................................................................................76 44 Optim ization ........................................................... ........................................84 66 Optim ization ..........................................................................................................86 9 CON CLU SION S..........................................................................................................90 A applying Tensegrity Design Principles......................................................................91 A antenna Point D design ..................................................................................................95 Patent D disclosure .........................................................................................................97 Future W ork .................................................................................................................97 REFEREN C ES ..................................................................................................................98 BIO G RA PH ICA L SKETCH ..........................................................................................103 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy DEPLOYABLE ANTENNA KINEMATICS USING TENSEGRITY STRUCTURE DESIGN By Byron Franklin Knight May 2000 Chairman: Dr. Joseph Duffy Major Department: Mechanical Engineering With vast changes in spacecraft development over the last decade, a new, cheaper approach was needed for deployable kinematic systems such as parabolic antenna reflectors. Historically, these meshsurface reflectors have resembled folded umbrellas, with incremental redesigns utilized to save packaging size. These systems are typically overconstrained designs, the assumption being that high reliability necessary for space operations requires this level of conservatism. But with the rapid commercialization of space, smaller launch platforms and satellite buses have demanded much higher efficiency from all space equipment than can be achieved through this incremental approach. This work applies an approach called tensegrity to deployable antenna development. Kenneth Snelson, a student of R. Buckminster Fuller, invented tensegrity structures in 1948. Such structures use a minimum number of compression members (struts); stability is maintained using tension members (ties). The novelty introduced in this work is that the ties are elastic, allowing the ties to extend or contract, and in this way changing the surface of the antenna. Previously, the University of Florida developed an approach to quantify the stability and motion of parallel manipulators. This approach was applied to deployable, tensegrity, antenna structures. Based on the kinematic analyses for the 33 octahedronn) and 44 (square antiprism) structures, the 66 (hexagonal antiprism) analysis was completed which establishes usable structural parameters. The primary objective for this work was to prove the stability of this class of deployable structures, and their potential application to space structures. The secondary objective is to define special motions for tensegrity antennas, to meet the subsystem design requirements, such as addressing multiple antennafeed locations. This work combines the historical experiences of the artist (Snelson), the mathematician (Ball), and the space systems engineer (Wertz) to develop a new, practical design approach. This kinematic analysis of tensegrity structures blends these differences to provide the design community with a new approach to lightweight, robust, adaptive structures with the high reliability that space demands. Additionally, by applying Screw Theory, a tensegrity structure antenna can be commanded to move along a screw axis, and therefore meeting the requirement to address multiple feed locations. Viii CHAPTER 1. BACKGROUND Space Antenna Basis The field of deployable space structures has matured significantly in the past decade. What once was a difficult art form to master has been perfected by numerous companies, including TRW, Hughes. and Harris. The significance of this maturity has been the reliable deployment of various antenna systems for spacecraft similar to NASA's Tracking Data Relay Satellite. In recent years, parabolic, meshsurface, reflector development has been joined by phased arrays (flat panel structures with electronically steered beams). Both of these designs are critical to commercial and defense space programs. An era has begun where commercial spacecraft production has greatly exceeded military/civil applications. This new era requires structural systems with the proven reliability and performance of the past and reduced cost. This dissertation addresses one new approach to deployable antenna design utilizing a kinematic approach known as tensegrity, developed by Kenneth Snelson (student of R. Buckminster Fuller) in 1948 [Connelly and Black, 1998]. The name tensegrity is derived from the words Tensile and Integrity, and was originally developed for architectural sculptures. The advantage of this type of design is that there is a minimum of compression tubes (herein referred to as struts); the stability of the system is created 2 through the use of tension members (ties). Specifically, this work addresses the new application for selfdeploying structures. Antenna Requirements James R. Wertz of Microcosm, Inc., a leading spacecraft designer, defines a system's requirements through a process of identifying broad objectives, reasonably achievable goals, and cost constraints [Larson and Wertz, 1992]. Space missions vary greatly, and the requirements, goals, and costs associated with each task also vary greatly, but one constraint is ever present: "space is expensive". The rationale behind this study of new deployable techniques is related to the potential cost savings to be gained. The mission objective for a large, deployable space antenna is to provide reliable radio frequency (RF) energy reflection to an electronic collector (feed) located at the focus of the parabolic surface. The current state of deployable parabolic space antenna design is based on a segmented construction, much like an umbrella. Radial ribs are connected to a central hub with a mechanical advantaged linear actuator to drive the segments into a locked, overdriven, position. Other approaches have been proposed utilizing hoop tensioners (TRW) and mechanical memory surface materials (Hughes), but as of this publication, these alternative approaches have not flown in space. To meet this objective, an analysis of mathematics and electrical engineering yields three parameters: defocus, mispointing, and surface roughness. For receiving antennas, defocus is the error in the reflector surface that makes the energy paint an area, rather than converge on the focal point. Mispointing is the misplacement of the converged energy to a position other than the designed focal point. Surface roughness, or the approximation to a theoretical parabolic surface, defines the reflector's ability to reflect 3 and collect a given band of RF energy. Higher band reflectors require a more accurate surface that better approximates the theoretical parabola. Similarly for transmitting antennas, defocus generates divergent rays of energy (rather than parallel) from the reflector surface; mispointing directs these waves in the wrong direction. Defocus (focal area vice point) and mispointing (focus located in the wrong position) are illustrated in Figure 11. ^1 Figure 11. Defocus and Mispointing on a Parabolic Antenna In recent years, numerous Department of Defense organizations have solicited for new approaches to deployable antenna structures. The Air Force Research Laboratories (AFRL) are interested in solutions to aid with their Space Based Laser and Radar programs. Specifically, they have requested new solutions to building precision deployable structures to support the optical and radar payloads. Improvement Assumptions The basis for this research is the assumption that the stowed density for deployable antennas can be greatly increased, while maintaining the reliability that the space community has enjoyed in the past. Failure of these structures is unacceptable, but if the 4 stowed volume is reduced (therefore an increase in density for a given weight), launch services could be applied much more efficiently. The implementation of multiple vehicle launch platforms (i.e. Iridium built by Motorola) has presented a new case where the launch efficiency is a function of the stowed spacecraft package, and not the weight of the electronic bus. For Extremely High Frequenc) systems (greater than 20GHz) in low earth orbit (LEO), the antenna aperture need only be a few meters in diameter. But for an Lband, geosyncronous (GEO) satellite (i.e. AceS built by Lockheed Martin), the antenna aperture diameter is 15 meters. And to reach GEO, less weight and payload drag must be achieved to ensure a more efficient ascent into the orbit. Currently, these systems stow within the rocket launchers much like folded inverted umbrellas. This greatly limits the stowage efficiency, greatly increasing the launcher shroud canister height. This research addresses a concept to improve this efficiency. CHAPTER 2. INTRODUCTION Tensegrity Overview Pugh [1976] simplified Snelson's work in tensegrity structures. He began with a basic description of the attractions and forces in nature that govern everyday life. From there he described the applications in history of tensile and compressive members in buildings and ships to achieve a balance between these forces to achieve the necessary structures for commerce and living. The introduction of Platonic Solids presents the simplicity and art of tensile/compressive structures. The Tetrahedron in Figure 21 is a fourvertex, 6 member structure. Framing the interior with a strut (tetrapod) system and connecting the vertices with ties can create the tensegrity. The ties must, of course, always be in tension. Figure 21. A Simple Tetrahedron and Tripod Frame The Octahedron (6vertices, 12members, and 8faces) is the basis for this research to apply tensegrity to deployable antenna structures. Figure 22 presents the simple structure 6 and tensegrity application (rotated about the center, with alternate struts replaced by ties). From this simple structure, we have been able to create a class of deployable structures using platform kinematic geometry. It is apparent that the tensegrity application resembles a sixleg parallel platform. It is from this mathematics that the new designs are derived. Figure 22. The Simple, Rotated, and Tensegrity Structure Octahedron The work of Architect Peter Pearce [1990] studies the nature of structures and the discovery of the Platonic Solids. Plato was able to determine the nature of structures, and the structure of nature (a duality), through observing naturally occurring systems such as spider webs. Building on this work, Pearce was able to document other natural phenomena (soap bubbles, Dragonfly wings, and cracked mud) to establish energy minimization during state change. The assumption here is that nature uses the most energyefficient method. From these assumptions and an understanding of stress and strain in structural members (columns and beams), he was able to present a unique solution for simple, durable, high strength structures. From these conclusions, he proposes a family of residential, commercial, and industrial structures that are both aesthetically pleasing and functional. Related Research The most comprehensive study of the technology needs for future space systems to be published in the last decade was released by the International Technology Research Institute [WTEC, 1998]. This NSF/NASA sponsored research "commissioned a panel of U.S. satellite engineers and scientists to study international satellite R&D projects to evaluate the longterm presence of the United States in this industry." A prior study was undertaken in 1992 to establish that there was significant activity in Europe and Asia that rivaled that of the U.S., and benchmarked this R&D to U.S. capability. The later study added market, regulatory, and policy issues in addition to the technology developments. The conclusion was that while the U.S. holds a commanding lead in the space marketplace, there is continual gaining by both continents. This is evident in space launch, where Ariane Space has nearly achieved the capabilities of Boeing's (Delta) rocket services. The significance of this study is that U.S. manufacturers are meeting their goals for shortterm research (achieving program performance), but have greatly neglected the longterm goals, which has traditionally been funded by the government. The top candidate technologies include structural elements, materials and structures for electronic devices, and large deployable antennas (>25 meters diameter). While there have been 14 meter subsystems developed to meet GEO system requirements during the 1990s. the large deployable requirement has yet to be addressed or developed. This research will address one possible solution to building such a subsystem. 8 Related Designs Tetrobots [Hamlin and Sanderson, 1998] have been developed in the last few years as a new approach to modular design. This approach utilizes a system of hardware components, algorithms, and software to build various robotic structures to meet multiple design needs. These structures are similar to tensegrity in that they are based on Platonic Solids (tetrahedral and octahedral modules), but all the connections are made with truss members. Tensegrity utilizes only the necessary struts (compression members) and ties (tensile members) to maintain stability. Adaptive trusses have been applied to the field of deployable structures, providing the greatest stiffness and strength for a given weight of any articulated structure or mechanism [Tidwell et al. 1990]. The use of the tetrahedron geometry (6struts and 4 vertices) is the basis for this approach. From that, the authors propose a series of octahedral cells (12struts and 6vertices) to build the adaptive structure (Figures 23 and 24). The conclusion is that from welldefined forward analyses (position, velocity and acceleration), this adaptive truss would be useful for deployed structures to remove position or motion errors caused by manufacturing, temperature change, stress, or external force [Wada et al. 1991]. Figure 23. Octahedral Truss Notation Cell n Cell 2 y Cell 1 Figure 24. Long Chain Octahedron VGT The most complex issue in developing a reliable deployable structure design is the packaging of a light weight subsystem in as small a volume as possible, while ensuring that the deployed structure meets the system requirements and mission performance. Wamaar developed criteria for deployablefoldable truss structures [Warnaar 1992]. He 10 addressed the issues of conceptual design, storage space, structural mass, structural integrity, and deployment. This work simplifies the concepts related to a stowed two dimensional area deploying to a threedimensional volume. The author also presented a tutorial series [Warnaar and Chew, 1990 (a & b)]. This series of algorithms presents a mathematical representation for the folded (threedimensional volume in a two dimensional area) truss. This work aids in determining the various combinations for folded truss design. NASA Langley Research Center has extensive experience in developing truss structures for space. One application, a 14meter diameter, threering optical truss, was designed for space observation missions (Figure 25). A design study was performed [Wu and Lake, 1996] using the Taguchi methods to define key parameters for a Paretooptimal design: maximum structural frequency, minimum mass, and the maximum frequency to mass ratio. Tetrahedral cells were used for the structure between two precision surfaces. 31 analyses were performed on 19,683 possible designs with an average frequency to mass ratio between 0.11 and 0.13 Hz/kg. This results in an impressive 22 to 26 Hz for a 200kg structure. Related Patents The field of deployable space structures has proven to be both technically challenging and financially lucrative during the last few decades. Such applications as large parabolic antennas require extensive experience and tooling to develop, but this is a key component in the growing personal communications market. The patents on deployable space structures have typically focused on the deployment of general truss network designs, 11 rather than specific antenna designs. Some of these patents address new approaches that have not been seen in publication. Upper surface Ring 3 Lower surface Figure 25. Threering Tetrahedral Truss Platform The work of Kaplan and Schultz [1975], and, Waters and Waters [1987] specifically applies strut and tie construction to the problem of deployable antennas, but the majority of patents address trusses and the issues associated with their deployment and minimal stowage volume. Nelson [1983] provides a detailed design for a threedimensional rectangular volume based on an octahedron. His solution to deployment uses a series of ties within the truss network. Details of the joints and hinges are also included. When networked with other octahedral subsets, a compact stow package could be expanded into a rigid threedimensional framework. Other inventors continued work in expandable networks to meet the needs of International Space Station. Natori [1985] used beams and triangular plates to form a tetrahedral unit. These units formed a linear truss; his work included both joint and hinge details and the stowage/deployment kinematics. Kitamura and Yamashiro [1990] 12 presented a design based on triangular plates, hinged cross members, and ties to build expanding masts from very small packages. Onoda [1985, 1986, 1987a, 1987b, 1990] patented numerous examples of collapsible/deployable square truss units using struts and ties. Some suggested applications included box section, curved frames for building solar reflectors or antennas. Onoda et al. [1996] published results. Rhodes and Hedgepeth [1986] patented a much more practical design that used no ties, but employed hinges to build a rectangular box from a tube stowage volume. Krishnapillai [1988] and Skelton [1995] most closely approximate the research presented herein, employing the concepts of radial struts and strut/tie combinations, respectively. The combination of these approaches could provide the necessary design to deploy a small package to a radial backup surface, as with a deployable antenna. CHAPTER 3. STUDY REQUIREMENTS Stability Criterion The primary assumption for this research is that improved stability will provide a superior deployable structure. Applying a tensegrity approach, the secondary assumption is a resultant lower system development cost. The development of this new approach to antenna systems, assuming these criteria, will provide a usable deployable product with greatly reduced component count, assembly schedule, and final cost, but with equal stability and system characteristics to the currently popular radial rib antenna system. From this assumption, increased stowage density will be realized. Stowage Approach Figure 31 shows a deployed and stowed antenna package, utilizing a central hub design. Most current deployable antenna designs use this approach. For a single fold system, the height of the stowed package is over one half of the deployed diameter. The approach taken in this research is to employ Tensegrity Structural Design to increase the stowed package density. Deployment Approach The deployable approach for this 66 system is to manipulate the legs joining the hub to the antenna, to create a tensegrity structure. Onoda suggests a sliding hinge to achieve deployment, but such a package still requires a large height for the stowed structure. This approach does have excellent merit for deployable arrays, as he presents in the paper. Figure 31. Deployed and Stowed Radial Rib Antenna Model The tensegrity 66 antenna structure would utilize a deployment scheme whereby the lowest energy state for the structure is in a tensegrity position. Figure 32 shows this position, with the broken lines representing the ties (tension) and the solid lines representing the struts (compression). Clearly, equilibrium of this structure requires that the tie forces sum to match the compression forces at the end of each strut. Figure 32. 66 Tensegrity Platform 15 Mechanism Issues Rooney et al. [1999] developed a concept for deploying struts and ties using a "reel" design, thereby allowing the ties to stow within the struts. This simple, yet durable approach solves the problem of variable length ties for special antenna designs, such as those with multiple feed centers (focal points on the parabolic antenna surface). Figure 33 shows this concept, using a deployment mechanism for the ties; spherical joints would be necessary to ensure that there are only translational constraints. Elastic Ties Deployed ,!from the Strut (3 each) Tube AngleUnconstrained Revolute Joint Figure 33. The Struts Are Only Constrained in Translation CHAPTER 4. BASIC GEOMETRY FOR THE 66 TENSEGRITY APPLICATION The application of tensegrity structures to the field of deployable antenna design is a significant departure from currently accepted practices. Not only must this new structure meet the system parameters previously described, but there also must be a process to validate the performance reliability and repeatability. Figure 41 shows the rotation of the 66 structures through tensegrity. Tensegrity occurs when all struts are in compression, and all ties are in tension. When describing a stable structure, the struts cannot be in tension because they only interface with tensile members (ties). Figure 41. A 66 Structure Rotated through Tensegrity As presented in Chapter 1, the accepted subsystem mechanical requirements applied to deployable parabolic antennas are defocus, mispointing, and surface roughness. 17 Defocus, or the "cupping" of the structure, must be corrected once the subsystem is deployed to correct any energy spreading which occurs. A correctly shaped parabolic antenna surface may not direct the radio frequency (RF) energy in the correct direction (to the right focal point). This is known as mispointing. Practically, antenna design requires that the theoretical focal "point" be a "plane", due to energy management issues of RF transmitter/receivers. The surface accuracy is a coupled effect, which is influenced by the nonlinear stiffness (displacement is not linear with respect to the applied force), structural time constant, and general stability of the backup reflector structure and facing antenna mesh surface. Positioning and control of this mesh surface defines the antenna's "accuracy". Pellegrino (The University of Cambridge) has developed applicable tools for calculating the motions ofprestressed nodes by actuating flexible ties [You, 1997]. In order to address adequately these three design parameters, the stability of this subsystem must be assured. During his career, Hunt [1990] has addressed line geometry, the linear dependence of lines, the linear complex, and the hyperboloid. All of these studies have direct application in the case oftensegrity structures. This linear dependence relates to the stability of the structure. For this to occur, the two sets of lines on the tensegrity structure, the struts and ties, must lie on coaxial hyperboloids of one sheet. This builds the case to explain how such a structure in tensegrity can be stable yet at a singularity, having instantaneous mobility. To explain this, an introduction into points, planes, lines, and Screw Theory is presented. Points, Planes, Lines, and Screws The vector equation for a point can be expressed in terms of the Cartesian coordinates r =xi+yj+zk (41) Referencing Hunt [1990], these coordinates can be written x = , y = , z =  This expresses the point in terms of the homogeneous coordinates (W;X,Y,Z). A point XYZ is completely specified by the three independent ratios and therefore there are W W W an oo points in three space. Similarly, the equation for a plane can be expressed in the form D + Ax + By + Cz = 0 (42) or in terms of the homogeneous point coordinates by Dw + Ax + By + Cz = 0 (43) The homogeneous coordinates for a plane are (D;A,B, C) and a plane is completely specified by three independent ratios B Therefore, there are an oo planes in three space. It is well known that in three space the plane and the point are dual. Using Grassmann's [Meserve, 1983] determinant principles the six homogeneous coordinates for a line, which is the join of two points (x ,yl, Zl) and (x 2, Y 2, 2), can be obtained by counting the 2x2 determinants of the 2x4 array. [1 Xl Yi ZI (44) 1 X2 Y2 Z2 L= 1 xl M= 1 yl N= z2 1 x2 1 y2 1 z2 (45) = Q=Z x R= x2 Y2 Y2 z2 z2 x2 x2 Y2 19 The six homogeneous coordinates (L, M, N; P, Q, R) or (S;So) are superabundant by 2 since they must satisfy the following relationships. S S = L +M' +N2 =d2 (46) where d is the distance between the two points and, S S =LP +MQ +NR = (47) which is the orthogonality condition. Briefly, as mentioned, the vector equation for a line is given by x S = So. Clearly, Sand S0 are orthogonal since S So = S ix S = 0. A line is completely specified by four independent ratios. Therefore, these are an o4 lines in three space. Ball [1998, p.48] defines a screw by, "A screw is a straight line with which a definite linear magnitude termed the pitch is associated". For a screw, S SO # 0, and the pitch * i LP+MQ+NR 5  is defined by h = L MQ + NR It follows that there are an oo screws in three space. L2 +M2+N2 By applying Ball's Screw Theory, the mathematics are developed to show that this class of tensegrity structures can follow a screw. This is very applicable in antenna design to allow a subsystem to direct energy to multiple feed centers. The Linear Complex Many models have been developed for the geometry and mobility of octahedral manipulators. Instant mobility of the deployable, tensegrity, antenna structure is of much interest within the design community. This instant mobility is caused by the Linear Dependence of Lines. This occurs when the connecting lines of a structure become linearly dependent. They can belong to (i) a linear complex (oo3 of lines); (ii) a linear 20 congruence (cc2 of lines); or (iii) a ruled surface called a cylindroid (oo0 of lines). The linear complex has been investigated by, for example, Jessop [1903]. Of interest here is the linear complex described by Hunt [1990], which will be described shortly. Before proceeding, it is useful to note that the resultant of a pair of forces, which lie on a pair of skew lines, lies on the cylindroid. The resultant is a wrench, which is simply a line on the cylindroid with an associated pitch h. The resultant is only a pure force when a pair of forces intersects in a finite point or at infinity (i.e. they are parallel). Hunt [1990] describes a linear complex obtained by considering an infinitesimal twist of a screw with pitch h on the zaxis. For such an infinitesimal twist, a system of oo2 coaxial helices of equal pitch is defined. Every point on the body lies on a helix, with the velocity vector tangential to the helix at that point. Such a system ofoo3 tangents to 002 coaxial helices is called a helicoidal velocity field. A z 0),h \ $3V,=h CO b SVtIb= cx b Figure 42. Two equalpitched helices 21 In Figure 42, two helices are defined, one lying on a circular cylinder of radius a, and the other on a coaxial circular cylinder of radius b. Two points A and B are taken on the respective radii and both cylinders are on the same zaxis. After one complete revolution, the points have moved to A' and B', with AA'=BB'=2nh. Both advance along the zaxis a distance hO for a rotation 0. Now, the instantaneous tangential velocities are Vta = 0) x a and Vtb = o x b. Further, Va=ho) and Yta=) x a. The ratio IYaI/ltal = h/a = tan a, or h=a tan a. Similarly, h/b = tan P, or h=b tan P. A 1 Figure 43. A Pencil of Lines in the Polar Plane a Through the Pole A Further, Figure 43 (see [Hunt, 1990]) illustrates a pole A through which a helix passes together with a polar plane a. The pencil of lines in a which pass through A are normal to the helix (i.e. the vector through A tangent to the helix). The plane a contains a pencil of lines (oo') through the pole A. Clearly, as a point A moves on the helix, an oo2 lines is generated. If we now count oo' concentric helices of pitch h, and consider the totality of the oo2 lines generated at each polar plane on a single helix, we will generate o3 22 lines, which comprises the linear complex. All such lines are reciprocal to the screw of pitch h on the zaxis. The result with respect to antiprism tensegrity structures will be shown in (426) and (427) and it is clear by (428) that the pitch h is given by ab/6z. The Hyperboloid of One Sheet Snyder and Sisam [1914] developed the mathematics to describe a hyperbola of rotation, known as the hyperboloid of one sheet (Figure 44). The surface is represented by the equation 2 2 2 x y =2 1 (48) a2 b2 c which is a standard threedimensional geometry equation. This equation can be factored into the form x = 1+ (49) acac b b x + z)(x z) =(I +b)(l b (49) and can become an alternate form (a + c) b) =x p (410) y xz I b+ a c Similarly, (x z Y a+ ( )rl (411) The equations can be manipulated to form: X Z 1+Y and (1 =  a cj bJ bj ,a cj Figure 44. A Ruled Hyperboloid of One Sheet These formulae describe the intersection of two planes, which is a line. Therefore, for every value of p there is a pair of plane equations. Every point on the line lies on the surface of the hyperboloid since the line coordinates satisfy 410. Similarly, any point on the surface, which is generated by the line equation, also satisfies the equations in 412 as they are derived from 410. The system of lines, which is described by 412, where p is a parameter, is called a regulus of lines on this hyperboloid. Any individual line of the regulus is called a generator. A similar set of equations can be created for the value rn x+ = r(1 and (l+) = n( ) (413) ac b b ac (412) The lines that correspond to il constitute a second regulus, which is complementary to the original regulus and also lies on the surface of the hyperboloid. Regulus Pliicker Coordinates Using Plicker Coordinates [Bottema and Roth, 1979], three equations describe a line: S (L, M, N) and So (P, Q, R) Ny Mz= P Lz Nx =Q (414) MxLy=R Expanding 412, the equations become p abc bcx + p acy abz = 0 and (415) abc p bcx acy + pabz = 0 The Pliicker axis coordinates for the line in the p regulus are obtained by counting the 2x2 determinants of the 2x4 arrays, which are built from these equations. pabc bc pac ab (416) abc pbc ac pab Therefore, P = ab2c2 1 = ab2 c2(1p2) 1 p Q= a2bc2 P P = 2a2bc2p (417) R= a2b2c p =a2b2c(l+p 2) 1 p and L=a2bc = a2bc(lp2) 1 p M= ab2c = 2ab2cp (418) p p N =abc2 1 =abc2(l+p2) p 1 This set of coordinates is homogeneous, and we can divide through by the common factor abc. Further, we have in ray coordinates: L=a(1p2) P=bc(1p2) M = 2bp Q= 2acp (419) N =c(1+ p2) R = ab(1+ p2) By using the same method for developing the Pliicker coordinates and the homogeneous ray coordinates, the ri equations are developed with 413. rlabcbcxrlacyabz= 0 and (420) abcrbcx+acy+T abz=0 and Trabc bc ilac ab] abc nrbc ac laab (4 to form the Pliicker coordinates P=ab2c2 1 =ab2c2(lr2) Q=a2bc211 =2a2bc2 (422) R= a2b2 1 1a2b2cl+2 I T1 and 26 L= a2b =a2bc(1n2 1 1 M=ab2c =2ab2c (423) I1 n N=ab? =ab?(l+112) yielding, after dividing by the common factor abc, the ray coordinates: L=a(112) P=bc(1T2) M = 2bl Q = 2acil (424) N=c(l+,q2) R=ab(l+r12) This series of calculations shows that the lines of the tensegrity structure lie on a hyperboloid of one sheet, either in the "forward" (p) or the reverse (ri) directions. The next section addresses the linear dependence inherent in the lines of a hyperboloid of one sheet and therefore the effect on the stability of the tensegrity structure. Singularity Condition of the Octahedron In Chapter 5, a comparison between a 33 parallel platform and the octahedron will be developed. Figure 45 is a plan view of the octahedron (33 platform) with the upper _det JL platform in a central position for which the quality index, X = 1 [Lee et al. Idet Jm 1998]. When the upper platform is rotated through 900 about the normal zaxis the octahedron is in a singularity. Figure 46 illustrates the singularity for < = 90 when X = 0 since Idet JJ = 0. The rank ofJ is therefore 5 or less. It is not immediately obvious from the figure why the six connecting legs are in a singularity position.  X Ec b EA Figure 45. Octahedron (33) Platform in Central Position EB Ec EA Figure 46. Octahedron Rotated 900 into Tensegrity However, this illustrates a plan view of the octahedron with the moving platform ABC rotated through < = 900 to the position A'B'C'. As defined by Lee et al. [1998], the coordinates of A'B'C' are 28 xA =rcos(900+ 300) y' =rsin(900+300) x3 =r cos(900 + 300) y' =rsin(90 + 30) x' = rsin(900) y' = rcos(900) (425) a where r = 43 By applying the Grassmann principles presented in (44), at 4 = 900, the k components for the six legs are Ni = z and Ri = ab where i=l, 2,...6. The Plicker coordinates of 6 all six legs can be expressed in the form ^TF ab1 Si= Li Mi z; Pi Qi (426) Therefore, a screw of pitch h on the zaxis is reciprocal to all six legs and the coordinates for this screw are S =[0, 0, 1; 0 0 h (427) For these equations, ab ab hz + = 0 or h = (428) 6 6z It follows from the previous section that all six legs lie on a linear complex and that the platform can move instantaneously on a screw of pitch h. This suggests that the tensegrity structure is in a singularity and therefore has instantaneous mobility. Other Forms of Quadric Surfaces The locus of an equation of the second degree in x, y, and z is called a quadric surface. The family that includes the hyperboloid of one sheet includes the ellipsoid, described by the equation: 2 + Z=1 (429) a2 b2 c2 The surface is symmetrical about the origin because only second powers of the variables (x, y, and z) appear in the equation. Sections of the ellipsoid can be developed, as presented by Snyder and Sisam [1914], including imaginary sections where the coefficients become V. If the coefficients are a=b>c then the ellipsoid is a surface of revolution about the minor axis. If the coefficients are a>b=c then it is a surface of revolution about the major axis. If a=b=c then the surface is a sphere. If a=b=c=0 the surface is a point. Although it is not relevant to this tensegrity structure analysis, the hyperboloid of two sheets (Figure 47) is described by the equation x2y22 =1 (430) a2 b2 c2 ,Z Figure 47. Hyperboloid of Two Sheets 30 Snyder and Sisam [1914] state, "It is symmetric as to each of the coordinate planes, the coordinate axes, and the origin. The plane z = k intersects the surface in the hyperbola." 2 y2 =1,zk 2 k2 (431) a2 k b2I The traverse axis is y = 0, z = k, for all values of k. The lengths of the semiaxes are k2 1 k2 a 1+ ,b 1+ . They are smallest for k = 0, namely a and b, and increase without limit as increases. The hyperbole is not composite for any real value ofk. without limit as Jkl increases. The hyperbola is not composite for any real value of k. CHAPTER 5. PARALLEL PLATFORM RESULTS 33 Solution Previous University of Florida CIMAR research [Lee et al. 1998] on the subject of 3 3 parallel platforms, Figure 51 is the basis work for this research. Their study addressed the optimal metrics for a stable parallel platform. The octahedral manipulator is a "33" device that is fully in parallel. It has a linear actuator on each of its six legs. The legs connect an equilateral platform triangle to a similar base triangle in a zigzag pattern between vertices. Our proposed quality index takes a maximum value of 1 at a central symmetrical configuration that is shown to correspond to the maximum value of the determinant of the 6x6 Jacobian matrix of the manipulator. This matrix is none other than that of the normalized line coordinates of the six leglines; for its determinant to be a maximum, the platform triangle is found to be half of the size of the base triangle, and the perpendicular distance between the platform and the base is equal to the side of the platform triangle. The term inparallel was first coined by Hunt [1990] to classify platform devices where all the connectors (legs) have the same kinematic structure. A common kinematic structure is designated by SPS, where S denotes a ball and socket joint, and P denotes a prismatic, or sliding kinematic pair. The terminology 33 is introduced to indicate the number of connection points in the base and top platforms. Clearly, for a 33 device. 32 there are 3 connecting points in the base, and in the top platforms as shown in Figure 51. A 66 device would have 6 connecting points in the top and base platforms. EBy Ec b EA Figure 51. 33 Parallel Platform (plan view) The parameter a defines the side of the platform (the moving surface); parameter b defines the side of the base; and parameter h defines the vertical (zaxis) distance between the platform and the base. The assumption that "more stable" is defined as being further away from a singularity. For a singularity, the determinant (det J) of the Jacobian matrix (J), the columns of which are the Plticker coordinates of the lines connecting the platform and the base, is zero. The most stable position occurs when det J is a maximum. These calculations create the "quality index" (k), which is defined as the ratio of the J determinant to the maximum value. The significance between this 33 manipulator research and tensegrity is the assumption that there is a correlation between the stability of a 6strut platform and a 3 strut, 3tie tensegrity structure. If true, this would greatly improve the stability prediction possibilities for deployable antennas based on tensegrity. As described in the abstract 33 paragraph above, the quality index (X) is the ratio of the determinant of J to the maximum possible value of the determinant ofJ. The dimensionless quality index is defined by =det J1 = detJm (51) detJ In later chapters, this same approach applied here for the J matrix of the 33 platform will be used for calculating that of the 66 tensegrity structure. For the later case the lines of the connecting points are defined by a 6x12 matrix and will require additional mathematics manipulation. In this case, a 6x6 matrix defines the lines of the 33 platform, and the determinant is easily calculated. The matrix values are normalized through dividing by the nominal leg length, to remove any specific design biases. The centroid of the triangle is considered to be the coordinate (0,0). From that basis, the coordinates for the upper and lower platforms are A 20 B 0 C 0 0 (52) (2 2, 3 (b 2b2F3 b b E A (h bE~ b h Ebh C h (53) The Grassmann method for calculating the Plicker coordinates is now applied to the 33 design, as described in Chapter 4. Briefly, the coordinates for a line that joins a pair of points can easily be obtained by counting the 2x2 determinants of the 2x4 array describing the connecting lines. S1  " 2 S2  2 0 a S3 =  2 S  S2 =[ b S b S 2 ah 24 ah h;23 2 /3 2 3 2ba 2i  a2b 2 3 b a+b 23 b2a 2b h 2 3 2ab 2, ah ah h; ah ab 22 Sah ab 3 2 2,F3 ah ab 3 2 2fJ ah 2 ab 0 2,3 ab 2 2J3 which yields the matrix for this system of detJ= S1 e6 84 $5 The normalization divisor is the same for each leg (they are the same length), therefore, e = dL+2 + + N2 (a (a2 ab+b2 +3h2) V3 and the expansion of the determinant yields det J = 4 3 a 3b3h3 a2 _ab+b 2 2 3 Dividing above and below by h3 yields Idet J = 33a3b3 (57) The key to calculating the maximum value for the quality index is to find the maximum height, h. Differentiating the denominator of the determinant with respect to h, and (56) ab 24i ah h; 2,3 (54) (55) 35 equating to zero to obtain a maximum value for det J yields the following expression for h. h = h = F 2 ab+b2 (58) If we now select values for a and b, (57) yields the value hm for det J to be a maximum. 27a3b3 det J = Idetm 3 (59) 32(a2 ab+b2 Further, we now determine the ratio y=b/a to yield a maximum absolute value fdet J m Substituting b= ya in Equation 57 yields 1 d 27a3 3a3 y3 3 27a3 detJm ~ 3 (510) 32(a2ya2 +2a2) y3a3 32 2 + To get the absolute maximum value of this determinant, the derivative with respect to y is taken which yields: 1 0 2o l Y(511) b 7 ==2 a Substituting this result in (58) gives: h =1 (512) a This work shows some similarity to the values to be derived for the 66 platform. The original quality index equation reduces to a function of (platform height) / (platform height at the maximum index). 8(h 8hm 1+(h hm (513) The resulting quality index plots for this 33 structure are found in Figures 52 through 56. In Figure 52, the quality index varies about the geometric center of the structure, with usable working area (index greater than 0.8) within half of the base dimension (b). It is interesting to note that these are not circles, but slightly flattened at the plot's 450 locations. Y 2 0.6 0.8 0 1.0 2 2 1 0 1 2 Figure 52. Coplanar translation of Platform from Central Location: Contours of Quality Index 1 .2 1 .0 0.8 0.6 0 .4 0.2 0.0 60 30 0 30 Rota tio n Figure 53. Rotation of Platform About Zaxis 60 40 20 0 20 Ro ta tio n 9 Figure 54. Rotation of Platform About Xaxis 1 .2 1 .0  V 0.8 0 .6 0 .2 0 .0 60 40 20 0 20 40 60 R ota tio n 0 Figure 55. Rotation of Platform About Yaxis As expected, rotations about the zaxis yield values approaching zero, where the singularity occurs. What is unique is that there are workable quality indices when the structure is rotated about the x and yaxes over 200. This could be valuable for antenna repointing without using an antenna gimbal. Figure 56 presents the change in quality index due to the height of the platform relative to the maximum value. Obviously, the greatest value (1.0) occurs when these values are equal. From this it is apparent that a working envelope of 40% (+/20% about the maximum) is achievable. Again, this discovery is helpful in the design on working antenna systems to address multiple feed centers. 1.200 1.000  0.800 < 0.600 0.400 0.200 0.000  0 . h/hm Figure 56. Quality Index as a Function of the Height Ratio 44 Solution The 44 parallel platform (Figure 57) is a square antiprism. The calculations of the 44 quality index are similar to those for the 33 platform; however, because the 44 line coordinates yield a 6x8 matrix, the determinant cannot be calculated directly and we introduce JJT [Knight, 1998], the product of the matrix and its transpose. As with the 33 platform, X is defined as the ratio of the Jacobian determinant to the maximized J determinant. Figure 57. The 44 Parallel Platform (plan view) b / 40 SdetJJT = t JT (514) det Jm From the CauchyBinet theorem, it can be shown that det J. jT = A + 2+ A. A1 ... n Each A is the determinant of a 6x6 submatrix of the 6x8 matrix. It is clear that (514) reduces to (51) for the 6x6 matrix. This method can be used for any 6xn matrix. As with the 33 platform, the determinant is calculated. As shown in the figure, the value for the side of the platform (moving plane) is a. Similarly; b is the value for the base side. The distance between the upper surface and the base surface is h. The definition of the line coordinate endpoints is A 0 a h, B 1a 0 h, C 0 a h, D a 0 h; 2 2 2 2 ( )(515) E b b 0, F b b 0 Gb b 0 H( b 0 ( 2 2 2 2 2 2 2 Therefore, the Jacobian matrix is b b V2ab b b 2a+b 52ab I2a+b 2 2 2 2 2 2 2 2 V a+b a+b b /2a b /2ab b b b 2 2 2 2 2 2 2 2 h h h h hh h h h J=(516) 6 bh bh bh bh bh bh bh bh (516) 2 2 2 2 2 2 2 bh bh bh bh bh bh bh bh 2 2 2 2 2 2 2 2 V2ab V2ab iab J2ab 2ab V2ab V2ab &2ab 4 4 4 4 4 4 4 4 It follows that Vdet jjT is given by / J T 32dfa3b3h Vdet.J = 32 ab3b+h3 (517) (a2 /2ab+b2 + 2h2 By following the same procedure as used for the 33 parallel platform, the ke3 to calculating the maximum value for the quality index is to find the maximum height, h. To find this expression, the numerator and denominator are both divided by h3, to ensure that h is only found in the denominator. Differentiating the denominator with respect to h, and equating this value to zero provides the maximum expression. h= hm (= a2 ab+b2 (518) Again, as presented in the 33 analysis, this maximum value for h is included in (517) to provide the maximum determinant. detJJ 2a3b3 (519) (a2 vab+ b2( To determine the ratio 'y=b/a for the maximum expression for (519), b=ya is substituted. The numerator and denominator are also both divided by ya3. T 2a3 detmJm 3 (520) To get the maximum value of this determinant, the derivative with respect to y is taken. This yields the ratio between a, b, and h. Y = b = (521) a CHAPTER 6. 66 DESIGN 66 Introduction The 66 inparallel platform (a hexagonal antiprism) is the basis for this new deployable antenna design. Using the previously derived mathematics, similar quality index values are developed. This defines the stability of the structure once it is in an equilibrium position. As with the 44 platform, the CauchyBinet theorem is used to determine the index. Once the mathematics is determined, further attention will be applied to antenna design. Sketch Figure 61 presents the 66 inparallel platform. This is a highly redundant parallel platform with 12 legs for 6 degrees of freedom, but can also be manipulated to define an antenna subsystem by applying tensegrity structure design. This approach will be presented in a later chapter. Figure 61. A 66 Parallel Platform (Hexagonal AntiPrism) 43 A plan view of the 66 parallel (redundant) platform is shown in Figure 62. Double lines depict the base and top platform outlines. Heavy lines depict the connectors. The base coordinates are GA through GF; the platform coordinates are A through F. The first segment is Si connecting points GA (base) and A (platform); the last segment is S12 connecting points GA and F. The base coordinates are all fixed and the xyz coordinate system is located in the base with the xy plane in the base plane. Hence, the base coordinates are Go G S6 C B S3 S7 S2 D Top Platform Ss S,1 S9 E a F S12 Slo SO y b zx GF Figure 62. A Plan View for the 66 Parallel Platform (Hexagonal AntiPrism) GA 0 GB 0 GC[0 b 0] (61) 2 2 12 2 0 GE 3b I 1 2 b 2 0o GF[0 b 0] (62) The coordinates for the top platform vertices at the central position are (63) where h is the height of the top platform above the base. A[a 0 h] B S D[a 0 h] E a 2 3a h 2 23 2 h2 2 h] F[ \3a 2 2 Applying Grassmann's method (see Chapter 4) to obtain the line coordinates yields the following 12 arrays. SI[GA A]: 1 S3[GB B]: 1 SS Gc C]: S7GD D: 1 S9[GE E: 1 SII[GF F]: L b 0 2 0 h b 0 2 /3a h 2 3b 2 a 23b 2 a 2 0 b 0] a 3a 2 2 2 a S3b 2 a 2 b 2 0 h b 2 _/3a 2 S2 GB A]: k 1 0 S4[GC BI: a 2 S6[GD C]: Sg[GE DI: 3b 2 a b 23a 2 1 3b 2 a 2 2 Sa 0 1 0 SIo[GF E] : a h 2 1 3b S12[GA F]: 2 a 2 b 2 (63) (64) b 2 43a 2 b 2 0 b X3a 2 b 2 13a 2 GD 3 2 Counting the 2x2 determinants (see Chapter 4) yields the [L, M, N; P, Q, R] line coordinates for each of the twelve legs. The normalized line coordinates were found by dividing the calculated value by the nominal lengths of the legs for the central position. = 4 aFb +b2+4h2 (65) 2 2 Evaluating the Jacobian The J matrix, comprised of the line coordinates for the twelve legs, is a 6x12 array. 2av3b 2aVb al3b a a a+ 3b b b V3ab Via2b V3a2b Viab 1 2h 2h 2h 2h 2h 2h 212 12 bh bh bh 2bh 2bh bh 3bh 3bh i3bh 0 0 V3bh ab ab ab ab ab ab (66) 2a+43b 2a+\3b a+V3b a a aV3b b b V3a+b 3a+2b V3a+2b ,3a+b 2h 2h 2h 2h 2h 2h bh bh bh 2bh 2h bh V3bh Vibh V/3bh 0 2bh 3bh ab ab ab ab ab ab 46 JT is, therefore, the transpose (a 12x6 matrix). 2aV3b 2a V3b aV 3b a a a+ 3b  2a + 3b 2a+V3b a+V3b a a a J3b b b Va b V3a2b 3a 2b V3ab b b V3a+b .3a+2b 3a + 2b  3a + b ODtimization Solution Lee et al. [1998] developed the optimization method for the 33 and 44 platforms. The method for calculating the optimization value for the 66 J matrix (nonsymmetric) is an extension of the 44 platform solution. The quality index X is given by I = (68) For this example, /det jjT is calculated. T a3b3 h3 Vdet JJ = 54 a3 a2 ab + b2 + h2 (69) 1 212 12 bh bh bh 2bh 2bh bh bh bh bh 2bh 2bh bh  /bh  /bh  3bh 0 0 3bh bbh V/3bh V3bh 0 0  /3bh ab  ab ab ab ab ab ab  ab ab ab ab  ab (67) 47 As with the 44 parallel platform calculation, the maximum height (h) must be found. To find this expression, the numerator and denominator of (69) are both divided by h3, to ensure that h is only found in the denominator. Then, differentiating with respect to h and equating to zero provides the maximum expression. h=hm = a2 3ab+b2 (610) As with the 44 analysis, this maximum value for h is included in (69) to provide the maximum determinant. de mT 54 a3b VdetJm =a ab 3 (611) V2 ab + b2 This yields the X value (quality index) as a function of a and b. = detT 8h3(a2 ab+b2) (612) FdetJmJT (a2 3ab+b2 +h2 This index (X) is a value between zero (0) and one (1), which represents the stability of the structure. As with the 44 structure, the ratio y=b/a, which represents the parameter ratio at the maximum quality index, is determined by substituting for b=ya. , ,T 54 a3 3a3 /det JmJ = 3 (613) 8 (a2 a + (ya)2 Again, the numerator and denominator are both divided by y3a.  d 54 a3 8detJmJ =J 3 (614) (72 7 By differentiating the denominator with respect to y, the maximum and minimum values are determined. This yields the solution for the most stable geometry for the 66 platform. 3 1 2 1 J23 2 i2 + =0F (615) By y2 y 2 3 2 The vanishing of the first bracket of the right side of the equation yields imaginary solution, whilst the second bracket yields 2 b  J a (616) a 2a h= and b= (617) Variable Screw Motion on the ZAxis Duffy et al. [1998] presented a study of special motions for an octahedron using screw theory. The moving platform remains parallel to the base and moves on a screw of variable pitch (p). The screw axis is along the Z direction. XA =rcos(oz) (618) YA = rsin(oz) (619) XB = rcos(z + 600) = r( cos sin4z) (620) 2 2 YB = rsin(Oz +600)= r(sin 0 +cos z) (621) 2 2 1 .J3 XC rcos(Qz + 1200)= r( cosI z + sino4) (622) S=sin(r(sin (623) YC =r sin(O, + 1200) = r( sin Oz , cos ), (623) 2 2 XD =rcos(oz +1800)=rcos z (624) YD = rsin( z + 1800) = rsin0 z (625) 1 XE = rcos( z + 2400) = r(cos z sin c) (626) 2 2 1 J? YE = rsin(z + 2400) =r( sinoz + cos ~) (627) 2 2 XF = rcos(1z + 300)= r(cosoz +sin z) (628) 2 2 Yp = rcos(Oz + 3000) =r(sinz  cos z) (629) 2 2 It is important to recognize that simply actuating the struts by giving each the same incremental increase or decrease in length can produce the motion. Continuity requires that the sum of the coordinates (about the circle defined) sums to zero. XA+XB+XC+XD+XE+XF=O YA YB+YC+YD+YE+YF=O (630) Similar to previous octahedron and square platform papers, the radius from the center of the structure to the platform coordinates is equal to the length of the platform side (r =a). Using the base and platform coordinates previously defined, the Pliicker line coordinates are calculated using the Grassmann principle by counting the 2 x 2 determinants of each of the 2 x 4 arrays. SF3b b 5 b b S[GA A] 2 2 S2[GB A]: 2 2 (631) 1 XA YA h XA YA h S3[GB B]: 1 S5Gc c C]: 1 Xc S7[GD D: S9[GE E]: S :[1 SII[GF F]: ,3b b 0 0 2 2 XB YB h_ b 01 YC h  b 0 01 2 2 XD YD h 3b b 0 2 2 XE YE h 0 b 0 XF YF h S4 [GC B]: S6[GD C]1 1 Ss[GE D]: Sio[GF E]: S12[GA ]: 1 0 b 0 XB YB h f3b 2 XC 3bb b 2 2 XD YD h 0 b 0 XE YE h J3b 2 XF b 2 YF The Plicker coordinates are defined by the 2x2 determinants of these 2x4 arrays. l= XA 2 (YA+ 2 bh [3bh 2 2 T b VTb S2 = PXA 2 sT [ S3 = XB (YA (YB bh J3bh h; 2 2 bh V3bh h; 2 2 2 b1 2 b (3YA XA) S(3YYB XB)] 2 S4 =XB S5 = [XC (YB b) h; bh 0 bXBI (Ycb) h; bh 0 bXc] (632) (633) (634) (635) (636) b /( 2 +XA ) (637) (638) (639) (640) (641) YC b) 2 c 2 +b) (YD 2 bh r3bh h; 2 2 bh V3bh h; 2 2 b (3Yc + Xc) 2) (b ,3Y + X D 2 bh V3bh 2 2 b (3Y 2 b (YE +b 2 2 (YE + b) (YF + b) h; bh bh h; 2 0 bXE] JVbh 2 b (YE XE 2 h; bh 0 bXF] f Eb) (F + 2 bh h; 2 V3bh 2 b (3YpF +XF) 2 T = XC b ) S6 = Xc + 2 (642) (643) (YD XD) (644) (645) A T S1 = [XF "T S12 = XF (646) (647) (648) S = S7 = XD Sg = X[D + A T S9 = XE + .V This yields the transpose of the Jacobian matrix. XA 3b) fb + I b h b b) XA X i) YA + h 2 2 f 3b f b XB b (YB)h h XB (YBb) h (YC b) YC ) 2 , 2 (YD SYET + (YE + b) (YF + b) YFp+ I b h h h h h h h h bh 2 bh 2 bh 2 bh bh bh 2 bh 2 bh 2 bh 2  bh  bh bh I bbh 2 V3bh 2 V3bh 2 0 ,Sbh 2 ,Fbh 2 ,rbh 2 , bh 2 0 0 ,fbh h, b(2/YA + XA) b (YA +XA) b (YB, XB) 2 2 bXB bXC b 3( 2 b ,3D 2 b (/3YD 2 b 23Y, 2 +Xc) +XD) XD) XE) bXE bXF S(V3Y +Xi) } The first three of the six Pliicker coordinates define the length of the leg. The odd numbered legs for this structure are the same length. o =L 2M + N 2 2 = XA b + YA+( +h2 [X2 3b 2 b2 = X FbXA+ + YA+bYA + + 4 4 (650) T = (649) b) +2 2 [ ) 7 / 3b2 ]71 = r2 cos2 zVbrcosz + 3b +r2 sin2 Oz +brsinz + +h22 4 4 = r2 +br(sinz 3cos z)+b2 +h2 2 = [a2 + ab(sin (z 3cos z)+b2 + h2J (651) Similarly, lengths of the even numbered legs are equal. e = a2 ab(sin(z + cos z)+b2 +h2] (652) Lee et al. [1998] used the following notation to describe the screw motion. 6e J*T (653) This notation describes an incremental change in leg length as a product of the normalized line coordinates (J*T) and the platform incremental change (Ax, AO, etc.). To normalize the leg coordinates, each value is divided by the instantaneous leg lengths. ^T i=S*T = SiA6 (654) ti Calculating the summation of the individual coordinates shows that all the values are zero except for N and R. _Sb J. b j, b ,Jb Ll+L3+L5+L7+L9+Ll =XA +XB +XC+XD+ +XE+ +XF 2 2 2 2 Jib ( Ab r o . =rcosz b 3 +r cos z sin (z  rcos z +sin((zrcosz, (655) +r cosz 3sinz ++r cosz + sin z =0 2 2 2 2 2 2 b b b b Ml +M3+M5+M7 +M9+M11 =YA + YB + YCb+YD + YE + + YF +b 2 2 2 2 *n b V3A+{c. b_ b  =rsin +z+ +r( cos +zz+sin +z r 1cos +z sin 2 brsin^z (656) ++rcos z sin z +b=0 2 2 2 NI+N3+N5+N7+N9+N11 =6h P1 +P3 +P5 +P7 +P9 +P11 bh bh bh bh = + + bh + + bh = 0 2 2 2 2 (657) (658) (659) S^bh fbh AJ3bh 3bh Ql+Q3 +Q5 +Q7 +Q9 +Q11= +0+ + F+0=0 2 2 2 2 RI +R3+R5 +R7+R9+R = 3YA +XA)+ (V3BXB)bXC /YD +X 2 2 2 b (.YE XE)+bXF 2 br/ ) br f[ N =br sin z + cos z)+ sin 2 2 2 3 +cos ,z 2 1 s Cos csmi( z 2 2 3 br +b{ coslz +snz +,i(v3siniz+cosz)J br f3 23 1 flO, V3 11 1 smin +cos cosz smz +b sinz + cosOz 2 2 2 2 2 2 2 = 3br( sin z + cos z) The second pair of legs sum similarly. L2 +L4 +L6 +L8 +L10 +L12 = 0 M2 +M4 +M6 +M8 +M10 +M12 =0 P2 +P4 P6 +P8 +10 +P2 =0 Q2 +Q4 +Q6 +Q8 +Q +Q12 =0 N2 +N4 +N6 +N8 +N10 +N12 =6h R2 +R4 +R6 +R8 +R10 +R12 =3br(J/3sin z, +cos z) (660) (661) (662) b f1 3 br( cosz + sin z 2 2 2 55 Adding the first, third, fifth, seventh, ninth, and eleventh rows of the matrix and substituting the expressions for the coordinates yields the necessary expression. Note that z replaces h in this calculation. 610o10 =(N +N3+N5+N7 +N9+Nll)8z+(Ri+R3 +R5+R7R +R9 +Rll)5 z t (663) = 6z8z + 3br(j3 sin < z + cos z )5z 10610 = zz + .3sin z +cos z zz (664) 2 The even leg calculation yields a similar result. le81e = zSz + [3 sin z cos Oz J z (665) Special Tensegrity Motions Using the assumption that the even numbered legs are struts (2, 4, 6, 8, 10, and 12 have no longitudinal displacement) then the equation reduces to a function of rotation and translation. zz = 3 sin z + cos z 15z (666) 2 The pitch is defined by the ratio of linear z change to rotation about the zaxis. 8z P = z (667) This yields the pitch equation. = br[ 3 sin z + cosz ] (668) 8P0z 2z The subsequent integration yields the z calculation. This proves that the odd numbered struts can be commanded to yield a pitch motion (z and Oz motions are coupled). z br +Z jz z=[ sin z ++cosjz zp (669) z 2 0 z2 = z br{3(cos z +1)sin z (670) Equation (670) can be modified (a=r and zo=0) to define the square of the platform height. z2 = ab{in z 3(cos z +1)} (671) Therefore, the platform height (z) is the root of (671). z = rab in z 3(cos z +1)}2 (672) This result shows that for a given twist about the zaxis (Qz), there is a corresponding displacement along the zaxis, defined by a finite screw (p=z/Q4), as shown. Figure 63. The Pitch Relationship 0.5 1 0  0.5 a 1  1.5 2 2.5  3 3.5 4 CHAPTER 7. DEPLOYMENT AND MECHANICS While this research addresses the theory for a new class of deployable antenna structures, there remains significant work in defining the mechanics of such a subsystem. There does appear to be a potential reduction in mechanical component count as compared to current systems. This chapter addresses a potential deployment scheme, the mechanics necessary to achieve the motion, and some potential mechanisms to support these motions. Paramount to this design study is the combination of struts and ties. Waters and Waters [1987] suggested that there should be twelve (12) struts and twelve (12) ties for his hyperboloidal antenna model. This research suggests that there need only be six struts to define a sixdegree of freedom structure. First, the struts are defined, including various approaches to deployment. Second, the strut/tie length and stiffness ratios are addressed. Third, a useful approach to deploying a semiprecision, mesh reflector is presented. Strut Design In order to deploy the struts from a stowed position, the end points of the stowageto deployment plan must be defined. Figure 71 presents a nominal 15meter (tip to tip) deployed surface with six struts. This first position is considered the starting position (a=0) according to Kenner (1976). The subsequent sketches show rotation to tensegrity (a=600). The strut lengths are shown increasing for simplicity, but an actual design would show the upper surface approaching the lower surface as the struts rotated to the tensegrity position. LI=OO Figure 71. 66 Structure Rotated from a=00 to a=600 (Tensegrity) 15 m. 916 . 19 m. Figure 72. Dimensions for Model Tensegrity Antenna 59 Based on these design assumptions the structure (Figure 72) would have the values found in Table 71. Table 71. Deployable Tensegrity Design Values NOMINAL VALUE Design Parameter NOMNAL VALUE Tip to Tip Diameter 15 meters Deployed Height 14 meters Planar Ties (top and bottom) 7 meters Tension Ties (upper to lower) 16 meters Struts (upper to lower) 19 meters Based on this model, it is clear that this structure would require a stowage space approximately 20 meters in length and an isosceles triangle three times the diameter of the struts. For a conventional 75 mm tube design, the total stowage volume would be a 20 m. long x 0.25 m. diameter. This is unacceptable for spacecraft design, as the trend in launch vehicle design is toward smaller systems, with correspondingly smaller fairings. In Figure 73, the nominal dimensions are presented for the Taurus and Delta launch vehicle. It is obvious from these sketches that a 20m x .25m antenna could not fit in even the 7.2m x 2.7m Extended Delta fairing. Design experience shows that the center of gravity for the spacecraft should be maintained at the centerline of the launch vehicle; therefore the usable height could be reduced to 5.3m x 2.7m. Clearly, a method for deploying the struts must be developed. The following examples are suggested for solving this design issue. 0.5 m. /0.5 m. e 2m. Extended 2 Delta e V" (Boeing) Taurus (Orbital) 2.7m. Figure 73. Taurus and Delta Launch Vehicle Fairings * Folding Hinge Struts: Numerous antenna systems have been developed in the last 30 years that utilize folding struts. They usually require some drive motion to deploy, including a latching mechanism at the end of the deployment travel. Figure 74 shows a simple hinge design, which could have an overcenter locking mechanism. * Sliding Coupling Struts: Similar to the folding design, sliding struts could be used, with a locking mechanism at the end of travel. Typically there is less force necessary to latch these struts, as it would take significant force to return them to the sliding configuration. Figure 75 shows this configuration, with a large angle sliding surface to lock the surface into place. Springs could be used to hold the mechanism in position. * Telescoping Struts: Due to excessive weight and drive force required telescoping struts have not been applied to deployable space applications. As motor cost and 61 efficiency increase, this could become a viable option. Figure 76 presents this configuration, which would encourage tapered diameter struts, which improve the specific stiffness of a complete system. * Inflatable Struts: A very different approach, but one that has been gaining favor with the space structures design community, is inflatable spars. The leaders in the field are ILC Dover (DE), L'Garde (CA), and SPS (AL). This approach can minimize the stowed spar volume, but analysis has shown that the size and weight of the deployment system is comparable to the three mechanical deployment schemes. The deployment requires a charge of gas energy, which requires a space qualified pump and tubing. One patented approach uses a UV hardening polymer that creates a solid structure once the inflatable is deployed. Another uses humidity evacuation technology to harden the tube. In all cases, structural integrity on orbit cannot be maintained merely by gas pressure; a solid structure must be provided. Figure 74. Folding Hinge Design L~FI Figure 75. Sliding Coupling Design Figure 76. Telescoping Design The greatest advantage to inflatables is that once the struts are deployed, they are almost uniform in cross sectional area and material properties. The mechanical approaches presented above introduce stiffness discontinuities at a minimum, and non linear load responses as the worst case. A trade study of these approaches is presented below. [[ P 63 Table 72. Strut Deployment Trade Study Strut Deployment Advantages Disadvantages Design Folding Design history Potential stiffness non Design relevance to other linearities industries Potential hinge surface galling Moderate deployment forces Locking hard\\are required Sliding Minimal deployment forces Potential bending stiffness non Positive locking position linearities Limited design history Potential contact surfaces galling Telescoping Compact packaging Requires interference fittings at Minimal stiffness non deployment linearities Potential contact surface galling Large deployment forces Inflatables Very compact packaging Requires deployment pump and Near homogeneous deployed tubing structure Weight savings limited Advanced materials Expensive application Strut/Tie Interaction The key to maintaining control over the surface once the antenna is deployed, as well as modifying the surface direction and accuracy, is the strut/tie interaction. Two approaches have been studied to manage the ties during deploy ment. Stowed Ties: By simply folding the ties along the struts (Figure 77), they can be released by force restraints, which are highly sensitive and as the loads reach a predetermined value, will release the ties. Elastic ties would save the need for a reel to take up the slack, but the disadvantage is extreme loads in the tension ties prior to deployment. This could be required for months. Figure 77. Stowed Ties *Reel Ties: Whether or not the ties are elastic, a reel could be used to take up the slack, changing the forces in the structure (Figure 78). This added hardware (potentially one motor per strut) increases complexity, weight, and therefore cost. Figure 78. Reel Ties A trade study for these approaches is presented below. 65 Table 73. Strut/Tie Trade Study Strut/Tie Interaction ie ne n Advantages Disadvantages Design Stowed Ties (cord) High stiffness Can only be used for Minimal Creep planar ties due to elasticity needs Stowed Ties (elastic) Ease of stowage HIGH STOWAGE LOADS Reel Ties (cord) Clean, snagfree design REQUIRES ADDITIONAL HARDWARE Reel Ties (elastic) Stiffness constant COMPLEX DESIGN AND adjustments POTENTIAL STIFFNESS CREEP One design issue, which is critical to the mission success of this type of subsystem, is snag prevention. Since these antennas are deployed remotely, any potential snag could degrade or destroy the reflector surface. By using elastic ties, which are under prestress, they are less likely to catch on deploying struts. Similarly, the cordties must be stowed to ensure deployment success. This issue will be addressed further in Chapter 8. Deployment Scheme Figure 79 presents a potential deployment scheme. The requirements for this operation are primarily low shock load and continuous motion. Despite the inherent self deploying nature oftensegrity structures, they cannot be allowed to "spring" into position for fear of introducing high shock and vibration loading into the system. Once the system has deployed, changing tension in the ties, and therefore position of the struts, can alter surface accuracy. ,xtensionge atio I Release > SDeployment and Surface Adjustment Figure 79. Deployment Scheme Previous Related Work During the 1990s, tensegrity structures became increasingly applicable to space structure design, including space frames, precision mechanisms, and deployables. The leading names in this new field have been Motro (France), Wang (China), Pellegrino (England), and Skelton (United States). Motro [1992] edited a special edition of the International Journal of Space Structures, which was dedicated to tensegrity. Kenneth Snelson wrote an introductory letter for this edition describing his invention, Fuller's contribution to its development, and the synergy between art and engineering. Motro's work [1996] has predominantly focused on the stability of tensegrity structures, including force density, nonlinear analysis and morphology. Despite his clear 67 focus on the engineering aspects oftensegrity, he has an excellent grasp of the artistic applications for this work. There is a clear development of stable, strut/tie structures from rectilinear (one dimensional), planar (two dimensional), through to spatial (three dimensional). The 33, octahedron tensegrity is an excellent example of a spatial structure. He has developed multiple tensegrity structure designs, which solve some of the toughest curvedsurface problems for space structures. This class of structure requires extremely lightweight with excellent geometric stability and deployability. Wang [1998 a & b] has performed some of the best work on cablestrut systems as an extension oftensegrity. Reciprocal prisms (RP) and crystalcell pyramidal (CP) grids, which technically exclude tensegrity systems, are the basis for his space frame applications. He developed a hierarchy of feasible cablestrut systems that include his new discoveries and tensegrity. Starting with triangular RP and CP simplexes, square, pentagonal, and hexagonal systems are developed to build cable domes, ring beams [Wang, 1998c], and doublelayer tensegrity grids [Wang and Liu, 1996]. His work in the feasibility of these new applications is very important to space structure development. Dr. S. Pellegrino's staff at the University of Cambridge has focused on the application of tensegrity to deployable space structures. Precision is of great concern with these kinematic systems, and recent system developments have required even higher precision from much lighter structures. By developing the mathematics for cable constrained nodes. You [1997] has been able to very accurately model the position of mesh antenna surfaces, including proven experimental results. Studies in the analysis of mechanisms [Calladine and Pellegrino, 1991], folding concepts for flexible but solid surface reflectors [Tibbalds et al. 1998], and shape control based on stress analysis 68 [Kawaguchi et al. 1996] have all greatly contributed to the state of the art. Infinitesimal mechanism analysis has led to prestressing conditions, which are critical to understanding deployable tensegrity structures. Their work with semisolid antenna reflectors has solved some of the fundamental problems associated with deploying these delicate systems. Launch capacity (size and weight) has continually reduced in recent years, requiring multiple folding systems to provide larger and larger structures. Obviously, once these structures are deployed and in operation, the surface must be maintained to meet performance requirements. Pellegrino has led the community in predictive models for using stress profiles (and node position control) to ensure reflector surface positioning is maintained. Skelton and Sultan [1997] has seen the control of tensegrity structures as a new class of smart structures. This work has been applied to deployable telescope design [Sultan et al. (1999a)], where precision is orders of magnitude tougher than deployable antennas. He has also been instrumental in the development of integrated design [Sultan and Skelton, 1997] and reduction of prestress [Sultan et al. (1999b)], which are critical to solving position correction and dynamic control issues. Alabama Deployment Study The University of Alabama provided a deployment study for Harris Aerospace that suggested some alternative approaches to deployment. One such approach, gasfilled shock absorbers, would allow a selfdeploying system like this tensegrity structure, to maintain a controlled deployment sequence. This study found that, based on the current design practices deployable space structures, the highest scoring actuator was the motor and lead screw combination. This is the most common scheme employed today. Alabama 69 also suggested that other forms of deployment control should be considered due to the high cost of space qualification for these subsystems. The viable options presented included: spiral springs, pneumatic cylinders, and compression springs. Since the tensegrity design provides the spring energy, a pneumatic design might be of use. The proportional velocity law governed this passive type design (damper). The energy equation is first order from stowage to deployment [Equation (1)], suggesting that a controlled sequence could be determined to ensure safe, low transient force deployment. Ci +Kx = 0 (71) Deployment Stability Issues The calculations for the 33 design, which were presented in Chapter 3 (Parallel Platform Results), suggest that there is a singularity at the tensegrity position. Figure 710 presents a sequence from the Central Position, through the Aligned Position and the Tensegrity Position to the Crossover Position, where the struts intersect. The angle is equal to 0 at the Central Position and increases as the platform rotates counterclockwise. The angle a is equal to 0 in the aligned position. The former value is consistent with the CIMAR calculations. The later value is consistent with Kenner's works. For the tensegrity design, the Central and Aligned Positions are not stable, as the ties are in compression. The Tensegrity Position is a stable critical point. This suggests that the design has instantaneous mobility, and any minor perturbation to the structure, while not necessarily causing instability, would provide sufficient energy to oscillate the antenna enough to degrade antenna performance. E F E F Central Position E Aligned Position =00 a=60 0=600 6a=00 B A E F E F Tensegrity Position (singularity) Crossover Position (interference) 0=900 a=300 <=1200 a=600 Figure 710. Octahedron Configurations To improve the design and stability of the tensegrity structure, while not affecting the selfdeployability, another set of ties is added between the vertex of the base and the opposite vertex of the platform (Figure 711). x E b Figure 711. Redundant 33 Structure This results in four ties at the end of each strut, versus the three in the original design. Again, the angles (j and a represent the works of CIMAR and Kenner, respectively. Figure 712 presents the rotations from the Central Position, through the Aligned and Tensegrity Positions, to the Crossover Position, where the struts intersect. Central Position 0=00 a=600 Tensegrity Position (singularity) =900 a=300 Aligned Position 6=600 a=0o Crossover Position (interference) 4=1200 a=600 Figure 712. Redundant Octahedron Configurations The mathematics to calculate this "33+" structure is similar that for the 44 and 66 structures, in that the CauchyBinet theorem is employed. Because there are now nine (9) connections between the platform and the base, the resultant J is a 6x9 matrix. J=[, S2 3 S4 S5 6 7 S8 S9] (72) Therefore, J is a 9x6 matrix. Si S2 S3 S4 JT= 5 (73) S6 S7 Sg .59. As shown in Chapter 5, the quality index is calculated using the determinant of the combined matrices (det JJT). The ratios for a, b, and h, which represent the maximum quality index ratios, were also calculated. The significance of this design change is shown in the Figure 713. The quality index remains relatively constant as the platform rotates through 1200, varying a total of 25%, from a minimum of.75, to a maximum of 1.0. This amount of variation is negligible, as compared to the standard 33 design, and suggests that the fourth tie creates redundancy, avoiding the singularity at tensegrity. The structure is stable and practical. Note that for the standard 33 design, K=0 at a=300, as predicted by the calculations in Chapter 5. Further, there is a suggestion here that the articulation of a single strut could provide necessary antenna surface motions. Since the reflector surface for a deployable antenna is couple to the ends and midpoints of the struts, extension of these structural members could alter the surface of the antenna, thereby performing various or simultaneous mission tasks. If this were true, the same antenna reflector could be used to communicate with more than one location. 300 200 100 Redundant Octahedron StaAdard Octahedron I I 0 \100 200 300 Figure 713. Quality Index vs. Rotation About the Vertical Axis CHAPTER 8. STOWAGE DESIGN An efficient (minimized) stowage volume is an equally important requirement to the deployment and antenna functions previously presented. Typically, antennas are designed with extra folds along the length of the struts to reduce the launch vehicle shroud height requirement. For a standard "hub and spoke" design deployable antenna, an extra fold can be included at the midpoint of the spar (see section view in Figure 81). With this method, a 15meter diameter antenna would have a stowed package volume of approximately 4meter height and 4meter diameter. This extra fold along the spar length greatly increases the material content, complexity of the structure, and touch labor to assemble the system. 15m 4m hub spoke 4m Figure 81. Current Deployable Antenna Design This chapter addresses the final goal for this research: a study of the tensegrity structure parameters. This approach will increase the efficiency of the stowed package, by maximizing the use of the spars for the antenna, and not just the structure. In Chapter 6, at the maximum 66 quality index (Central Position), the height h was equal to 76 approximately 0.6 times a. The base dimension b was equal to approximately 1.2 times a. Modifying the a/b and a/h ratios would reduce the length of the spars. This would improve the efficiency of the structure by maximizing the deployed structure (tip to tip) diameter for a minimized strut length. Minimized Strut Length As presented in Chapter 7, the typical launch vehicle (Extended Delta Class) shroud could not accommodate the baseline, 15meter diameter deployed tensegrity antenna, wherein the strut length is 19 meters. The following is a mathematical trade analysis between the size of the base (b) as defined in Chapter 5 (66 Design), the diameter of the deployed surface (2a for the 66 design), and the strut length (1). The purpose of this analysis is to design a stable structure while minimizing the strut length for the 15meter antenna. The 66 design is the basis for the deployable design. Table 81 presents the geometric relationships for the three candidate structures (33, 44, and 66). Table 81. The Three Tensegrity Structure Designs Considered Design # of Struts # of Ties (total) TiptoTip Diameter 33 3 9 a 44 4 12 52a 66 6 18 2a 33 Optimization The tensegrity position for the 33 structure, as defined in Chapter 7, is at =900 and a=300. Despite any changes in the a, b, or h values, tensegrity structures maintain the same rotation angle relative to the Central Position (Chapter 7). This characteristic of 77 tensegrity, related to the static force balance in each strut. This position is uniquely in a singularity at this equilibrium position. Unfortunately, the quality index approaches zero at the tensegrity position. This is known as a "stable critical point", which means that the structure has instantaneous mobility (i.e. small forces can produce motion), but because the energy is at a minimum in this position, the structure is stable. The quality index is zero because the determinant (det J) becomes zero. To determine this mathematic trade, the Central Position will be analyzed and the results hypothesized for the tensegrity structures. For the 33 structure, the Central Position is defined as 0=00 or a=600. As presented in Chapter 5, the determinant of the J matrix and the determinant of the maximum of this 3. ~ a3b3h3 27a3 b3 matrix (Jm) are J = 3 aa3b3h3 and Jm = 27ab respectively. 4 a2 ab + b2 + h2 32(a2 ab+b2) 3 Jm is a simplification of the J matrix with a substitution of the maximum height (hm) values. This geometry corresponds to the maximized quality index. The value is hm = (a ab + b ), found by taking the partial derivative and setting it equal to 3 Oh zero (a calculus inflection point). Calculating the quality index, X = yields: IJmI 8h3 (a2 ab+b2 3 ( 2 ,2 3 (81) 78 As the lim (k), which means that the base reduces to a point, the Equation 1 reduces to b= 0 9 a hF3 (82) 9( a +h 3h a Rooney et al. [1999] refers to this design as the "tensegrity pyramid". As a firstdesign, the ratio a/h=l is chosen. This further reduces the equation to 3 8 = 0.65 9(+1) 9( 4 3 (83) 3 3 which is an acceptable quality index (optimum is X=1.0). But to define a class of h structures with acceptable Quality Indices, a new value y is introduced. This value, y =  a or h = ya, represents the ratio of the side of the platform relative to the height of the structure. This changes the equation to 9(1 )3 (84) a and taking the derivative of the denominator, the maximum values for the quality 1 index (the denominator equals zero) is found at y = 0.58. Figure 82 presents the h plot of the quality index (k) vs. the ratio values (y = ). At this value ofy, the quality a index has a relative value of 1.0. Quality Index 1.50 1.00 0.50 X 0.00 0.50 1.00 1.50 Figure 82. X vs.y =1 Although there appears to be a mathematic benefit to designing a deployable platform, such as a tensegrity structure, with a base width of zero (hence a point) there are practical engineering limitations. The most obvious one is that the lines of the ties and the struts approach each other. This reduces the structure's stability to zero. As the ties that define the base approach zero length (b=0), the ties that define the platform cease to be in tension. This is due to the connecting ties becoming collinear with the struts, and therefore ceasing to create an offaxis moment (see Figure 83). Additionally, it is impractical to connect an antenna structure at a point, as moment loads would approach infinity. b0 b=0 Figure 83. Reduction of the Base to Zero Based on these observations, a compromised geometry is necessary. To this end, the h base should be minimized, and the y = ratio chosen for the maximized quality index. a Table 82 presents the results of three choices of Base Planar Tie length (b) with a a maximized quality index. Figures 84, 85, and 86 present the curves for the b= , , 2 4 and a and cases, respectively. 8 a Table 82. Quality Index for b= , 2 a a , and Cases 4 8 Quality Index (a/2) 1.50  1.00 0.50 . 0.00 0.50 1.00 .. 1.50 J....... Figure 84. X vs. b J[ hm JmI X yat max a 3_ 3 1 a a 333a3 1 2 (1 h = 0.5a 3a 0.2a3 1 0.50 32 + 2 32 +Y 4h a2 4y 3132 a 3V/3 3 313 4 13 h 3 4 a3a 0.52a 27a 0.02a3 13 0 2564 + 42 416l 02a3 48y 52 3xf3 19J19 a 3,f3 ,3 49a 3,/a 3 3 S 8 57 h a 0.55a 0.002a3 f 57 0.5 8 2048 + 8 6089 64 + 54 192h a2 192y ~ ~~ I S0" \' \ \ i y( b=) I, 1 Is i r' Sbfi s__~ N. \* . Figure 85. X vs. b = Quality Index (a/8) S  a  o o v v Figure 86. X vs. y b =  8 82 Quality Index (a/4) . .. .. . 1.50 1.00 0.50 C 0.00 0.50 1.00 1.50  1.50 1.00 0.50 0 000 ~ _ I I I I I ~~~~ 83 The conclusion drawn by this analysis of the base size is that there is no appreciable improvement by making the base larger or smaller. That is, by using just the stability of the structure (quality index) as the decision criterion. Stem [1999] developed a series of equations to describe the forces in the ties as the platform (a) and base (b) dimensions are varied. Simply put, the ratio of a/b changes linearly with the force in the ties. In other words, if the base dimension is reduced by 50%, the force in the base ties increases by 50%. Based on this research, it would be impractical to reduce the base dimension to a/8, as the forces would increase an order of magnitude. Therefore, the ratio a/4 was chosen because it reduces the strut lengths, provides a sufficient base dimension to attach the antenna, and still does not increase the tie forces too greatly. As presented in Chapter 7, additional ties can be included in the 33 design, thereby improving the quality index. For the 44 and 66 structures, the index approaches 1.0 for virtually any position. Figure 87 presents the design for the 33 structure. In this case, the X varies only 25% from 0.75 to 1.0 (as shown in Figure 713). (a) (b) (c) Figure 87. Reduction of the Base to Zero (Redundant Octahedron) a) b0; c) b=0 84 44 Optimization As presented in Chapter 4, the Jacobian (J) for the 44 structure is a 6x8 matrix, and an understanding of the CauchyBinet Theorem aids in obtaining the quality index. As previously presented, the numerator for the quality index (k) reduces to i 32_a b h VdetJJT = 322a3b3. The denominator represents the maximum (a2 2i ab+b2 +2h22) possible vale for the numerator was found by using h=0. This value is det JmJT 2a3b3 The height (h), which is used to find the (a2 /ab + b2)2 denominator, is hm = !j(a2 2ab + b2). Again, following the work in Chapter 4, the quality index is therefore, 3 16h3 (a2 ab+b2 (85) (a2 Vab+ b2 +2h2 As the lim (.) this reduces to b= 0 16,2 a3h3 S= (86) (a2 +2h2 h By using y = , the equation reduces further to a 2 h3 a h 1I(87) a+2 +2 2y + a a y with a maximum X at y = 0.71. Figure 88 plots X vs. y. Quality Index Quality Index Figure 88. X vs. y = ) for the Square Antiprism Quai Index (a2) Qualit) Index (a/2) 4.00 3.00 2.00 1.00 0< 0.00 2.00 3.00 4.00 V c* 8* \ j \ * Figure 89. X vs. y b = a for the Square Antiprism 86 aa a Similarly, the equations for b equals , and are presented in Table 83. Figure 2 4 8 89 presents the first X vs. y plot. The second and third cases are similar, but it is obvious that the y value at Xmax changes significantly between a/2 and a/8. as a Table 83. y at b= and  2 4 8 66 Optimization The 66 tensegrity design is the basis for this new class of deployable antenna structures. The calculations are similar to those for the 44 to solve the 6x 12 J matrix. The numerator for X, taken from Chapter 4, is vdet JJ =54abh The (a2 r3ab+b2 +h2) denominator, which is found by using h equals zero is 87 /detJm J = 54ab3 This h value is hm =(a2 3ab+b2. The 8(a2 ab + b2 quality index is therefore, 8h3 2 3 ( ab + b2 (88) (a2 ab+b2 +h2 As the lim (X) this reduces to b=:>0 8a3h S= 8ah (89) (a2 +h 2 h By using y = , the equation reduces further to a 8h3 8 8 h2 3 a h3 ( 3 (810) a+J Clh (j +j with a maximum X at y = 1. Figure 810 plots X vs. y. a a a Similarly, the equations for b equals and are presented in Table 84. Figure 2 4 8 811 presents the X vs. y plot for the case. The second and third cases are similar. 2 Again, the y at Xmax values vary greatly as b is reduced from a/2 to a/8. Keeping the work of Stem [1999] in mind to minimize the tie forces, b=a/4 is chosen as a compromise. Using this chosen ratio, h/a=0.79, b/a=0.25, and therefore, b/h=0.32. Quality Index 1.50 Figure 810. vs.y = for the Hexagonal Sa Antiprism Quality Index (a/2)   i  Figure 811. X vs. y b= ) for the Hexagonal Antiprism \  2.50 2.00 1.50 1.00 0.50 < 0.00 rv 1.50 2.00 2.50 a Table 84. X and for b= , 2 a a , and  4 8 b X (y=h/a) y at Xmax 5 /3" 8 5  a 4 2 f5 /32 oY 2 2 1(5 43 20 y+  y4 2 (17 1( 3 Y2 a 16 4 17 /31Y2 0.79 4 1 17 "F3 6 41 166 4 8 65 45 32 a 64 8 ( "65 23. 8 1 65 V33 64 8 y+ I, A 64 8 __ CHAPTER 9. CONCLUSIONS The requirements process introduced in Chapter 1 comes from a history defined by predictive engineering and unfortunate system failures. The participating hardware development companies have been greatly aided over the years by the work of analyst such as James R. Wertz. This process is based on problem definition and end vision, with a activity definition to reach the end goals. Space structures in general, and precision subsystems such as deployable antennas in particular, have become mired in this predictive process. The critical need for these subsystems has driven the development process to be extremely conservative, building larger, heavier, and stronger structures than are necessary to meet the mission requirements. This work has applied the theories of some of the greatest minds in mathematics (Ball, PlUcker, etc.) and engineering (Kenner, Hunt, etc.) to the simple and elegant architectural designs of Snelson and Fuller. The premise for embarking on this work was that architecture, by definition, leans more toward art than engineering, but combines form with function. Pearce (1990) accurately presented the theory whereby nature abhors inefficiency, requiring everything from dragonfly wings to cracked mud to find a minimal potential energy. It is this confidence in the efficiency of nature and its obvious tie with architecture which defines this work. In Chapter 3, a geometrical stability criterion measured by the quality index was introduced as defining an acceptable design. Within this stability, the structure should 91 deploy (preferably self deploy) and stow to allow placement in the space environment. After development of the 33, 44, and 66 parallel structures, this theory was applied to the tensegrity position. It is most interesting to note that this position happens to occur when the quality index is zero. This is known as a "stable critical point" in Chaos Theory. In this position, the structure has instantaneous mobility, whereby small perturbations can create small deflections of the antenna. Adding extra connecting ties between the "platform" and the "base" nullifies the instant mobility and provides a very stable structure. Further analysis proved that the antenna surface of this class of structures can be commanded to move on a screw whose axis is perpendicular to the surface. This happens to be a useful function for antenna surfaces, allowing them to address various feed centers (located at the focal points of the parabola). Applying Tensegrity Design Principles The idea for applying tensegrity design to deployable antennas has been suggested numerous times over the last two decades, but this work has addressed the mathematics necessary to prove its stability and therefore its applicability. The 66 structure has been chosen to provide enough radial spars on which to "hang" the reflective surface of the antenna. Again, possible advantages and disadvantages of the instantaneous mobility issue at the tensegrity position warrant further investigation. An improvement was presented for these designs with additional ties above the basic tensegrity design (two ties from each base vertex). A mathematics analysis of the quality index for these augmented 33 and 44 structures showed a marked improvement in the indices. For the 66 design, the basic tensegrity design with 12 platform/base connections (Figure 9la) is augmented to a total of 18 (Figure 91b), 24 (Figure 92a), 30 (Figure 9 2b), and 36 (Figure 93). Figure 91. Hexagonal Antiprism Designs (a) Basic Tensegrity Design (12 platform/base connections); (b) Augmented Tensegrity Design (18) (a) (b) Figure 92. Augmented 66 Hexagonal Antiprism Designs (a) Augmented Tensegrity Design (24); (b) Augmented Tensegrity Design (30) 
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