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Deployable antenna kinematics using tensegrity structure design

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Deployable antenna kinematics using tensegrity structure design
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viii, 103 leaves : ill. ; 29 cm.

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Thesis (Ph. D.)--University of Florida, 2000.
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Includes bibliographical references (leaves 98-102).
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Vita.
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by Byron Franklin Knight.

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Full Text
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 Ian 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.
IV


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iv
ABSTRACT vii
CHAPTERS
1 BACKGROUND 1
Space Antenna Basis 1
Antenna Requirements 2
Improvement Assumptions 3
2 INTRODUCTION 5
Tensegrity Overview 5
Related Research 7
Related Designs 8
Related Patents 10
3 STUDY REQUIREMENTS 13
Stability Criterion 13
Stowage Approach 13
Deployment Approach 13
Mechanism Issues 15
4 BASIC GEOMETRY FOR THE 6-6 TENSEGRITY APPLICATION 16
Points, Planes, Lines, and Screws 17
The Linear Complex 19
The Hyperboloid of One Sheet 22
Regulus Plticker Coordinates 24
Singularity Condition of the Octahedron 26
Other Forms of Quadric Surfaces 28
v


5 PARALLEL PLATFORM RESULTS 31
3-3 Solution 31
4-4 Solution 39
6 6-6 DESIGN 42
6-6 Introduction 42
Sketch 42
Evaluating the Jacobian 45
Optimization Solution 46
Variable Screw Motion on the Z-Axis 48
Special Tensegrity Motions 55
7 DEPLOYMENT AND MECHANICS 57
Strut Design 57
Strut/Tie Interaction 63
Deployment Scheme 65
Previous Related Work 66
Alabama Deployment Study 68
Deployment Stability Issues 69
8 STOWAGE DESIGN 75
Minimized Strut Length 76
3-3 Optimization 76
4-4 Optimization 84
6-6 Optimization 86
9 CONCLUSIONS 90
Applying Tensegrity Design Principles 91
Antenna Point Design 95
Patent Disclosure 97
Future Work 97
REFERENCES 98
BIOGRAPHICAL SKETCH 103
vi


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 mesh-surface reflectors have resembled folded umbrellas,
with incremental redesigns utilized to save packaging size. These systems are typically
over-constrained 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
vn


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 3-3 (octahedron) and 4-4
(square anti-prism) structures, the 6-6 (hexagonal anti-prism) 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
antenna-feed 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 NASAs
Tracking Data Relay Satellite. In recent years, parabolic, mesh-surface, 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 self-deploying 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, over-driven, 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 1-1.
Figure 1-1. 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
Frequency systems (greater than 20GHz) in low earth orbit (LEO), the antenna aperture
need only be a few meters in diameter. But for an L-band, 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 Snelsons 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 2-1 is a four-vertex, 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 2-1. A Simple Tetrahedron and Tripod Frame
The Octahedron (6-vertices, 12-members, and 8-faces) is the basis for this research to
apply tensegrity to deployable antenna structures. Figure 2-2 presents the simple structure
5


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 six-leg parallel platform. It is from this mathematics that the new designs are
derived.
Figure 2-2. 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
energy-efficient 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


7
proposes a family of residential, commercial, and industrial structures that are both
esthetically 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 long-term 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 Boeings (Delta)
rocket services.
The significance of this study is that U.S. manufacturers are meeting their goals for
short-term research (achieving program performance), but have greatly neglected the
long-term 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 (6-struts and 4-
vertices) is the basis for this approach. From that, the authors propose a series of
octahedral cells (12-struts and 6-vertices) to build the adaptive structure (Figures 2-3 and
2-4). The conclusion is that from well-defined 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].


9
Figure 2-3. Octahedral Truss Notation
Figure 2-4. 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.
Warnaar developed criteria for deployable-foldable 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 three-dimensional volume. The author also presented a
tutorial series [Wamaar and Chew, 1990 (a & b)]. This series of algorithms presents a
mathematical representation for the folded (three-dimensional 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 14-meter diameter, three-ring optical truss, was
designed for space observation missions (Figure 2-5). A design study was performed [Wu
and Lake, 1996] using the Taguchi methods to define key parameters for a Pareto-optimal
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
200-kg 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
Lower surface
Figure 2-5. Three-ring 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 three-dimensional
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 three-dimensional 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 3-1 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 6-6 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. T his
approach does have excellent merit for deployable arrays, as he presents in the paper.
13


14
Figure 3-1. Deployed and Stowed Radial Rib Antenna Model
The tensegrity 6-6 antenna structure would utilize a deployment scheme whereby the
lowest energy state for the structure is in a tensegrity position. Figure 3-2 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 3-2. 6-6 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
3-3 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
Angle-Unconstrained
Revolute Joint
Figure 3-3. The Struts Are Only Constrained in Translation


CHAPTER 4.
BASIC GEOMETRY FOR THE 6-6 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 4-1 shows the rotation of the
6-6 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 4-1. A 6-6 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.
16


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 non-linear 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 of pre-stressed 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 of tensegrity 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 co-axial 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
by


18
r = x i + y j + zk
(4-1)
X Y Z
Referencing Hunt f 19901, these coordinates can be written x = , y = , z = - .
W W W
This expresses the point in terms of the homogeneous coordinates (W;X, Y,Z). A point
X Y Z
is completely specified by the three independent ratios , , and therefore there are
WWW
an ooJ points in three space.
Similarly, the equation for a plane can be expressed in the form
D + Ax + By + Cz = 0
or in terms of the homogeneous point coordinates by
Dw + Ax + By + Cz = 0
(4-2)
(4-3)
The homogeneous coordinates for a plane are (D; A,B.C) and a plane is completely
specified by three independent ratios
ABC
DDD
Therefore, there are an oc> planes in
three space. It is well known that in three space the plane and the point are dual.
Using Grassmanns [Meserve, 1983] determinant principles the six homogeneous
coordinates for a line, which is the join of two points (xj ,yj ,z\) and (x2, y 2 > z2 )> can
be obtained by counting the 2x2 determinants of the 2x4 array.
1 x, yi zx
1 *2 y2 z2
(4-4)
L =
P =
1 X]
1 x2
yi zi
y2 z21
M =
Q =
1 yi
i y2
zi xi
z2 x2
N =
R =
1 Zj
1 z2
xi yi
x2 y2
(4-5)


19
The six homogeneous coordinates (L,M,N;P,Q,R) or (s; Sq ) are superabundant by 2
since they must satisfy the following relationships.
S-S = L2+M2+N2 = d2 (4-6)
where d is the distance between the two points and,
S S0 = LP + MQ + NR = 0 (4-7)
which is the orthogonality condition. Briefly, as mentioned, the vector equation for a line
is given by r x S = Sq Clearly, S and Sq are orthogonal since S-So = S- fxS = O.A
line is completely specified by four independent ratios. Therefore, these are an oo^ 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 Sq *= 0, and the pitch
, ~ j, LP + MQ + NR 11 ii 5 -i
is defined by h = It follows that there are an oo screws in three space.
L2 +M2 +N2
By applying Balls 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 (oc' of lines); (ii) a linear


20
2 1
congruence (oo of lines); or (iii) a ruled surface called a cylindroid (oo 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
2
of a screw with pitch h on the z-axis. For such an infinitesimal twist, a system of oo
coaxial helices of equal pitch is defined. Every point on the body lies on a helix, with the
3 2
velocity vector tangential to the helix at that point. Such a system of oo tangents to oo
coaxial helices is called a helicoidal velocity field.
A
z C0,h
Va=h co
cox a
Va=h co
coxb
Figure 4-2. Two equal-pitched helices


21
In Figure 4-2, 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 z-axis. After one complete
revolution, the points have moved to A and B\ with AA,=BB=27rh. Both advance
along the z-axis a distance h0 for a rotation 0. Now, the instantaneous tangential
velocities are Vta = to x a and Ytb = co x b. Further, Va=hco and Vta=o) x a. The ratio
IYa|/|Yta| = h/a = tan a, or h=a tan a. Similarly, h/b = tan P, or h=b tan p.

I z
Figure 4-3. A Pencil of Lines in the Polar Plane a Through the Pole A
Further, Figure 4-3 (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 (oo1) through the pole A. Clearly, as a point A moves on the helix, an oo2
lines is generated. If we now count oo1 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 oo-'


22
lines, which comprises the linear complex. All such lines are reciprocal to the screw of
pitch h on the z-axis. The result with respect to anti-prism tensegrity structures will be
shown in (4-26) and (4-27) and it is clear by (4-28) 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 4-4). The surface is represented
by the equation
2 2 ?
a2 b2 c2
(4-8)
which is a standard three-dimensional geometry equation. This equation can be factored
into the form
( x z V
+ -
Va cj
X z
Va cj
( y
V b
V
i-y
A
(4-9)
and can become an alternate form
fx
z'
Ml
-+-
Va
C)
l b J
ii y]
(x z\
i
l
l bj
Va cj
(4-10)
Similarly,
(x
z)
L y]
-+-
1 + J
Va
C)
V bj
U)
N |
I
X I
V bj
U c)
(4-11)
The equations can be manipulated to form:


23
z
-+-
=p
(
i+-
and
=P
1 X
I
In
Va C J
l bj
l bj
Va cj
(4-12)
z
Figure 4-4. 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 4-10. Similarly, any point on
the surface, which is generated by the line equation, also satisfies the equations in 4-12 as
they are derived from 4-10. The system of lines, which is described by 4-12, 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 r\
fx z']
- + -
Va c)
= T|
V by
and
f y ^
V by
= r|
^ x z ^
fa c)
(4-13)


24
The lines that correspond to q constitute a second regulus, which is complementary to
the original regulus and also lies on the surface of the hyperboloid.
Regulus PlUcker Coordinates
Using Plcker Coordinates [Bottema and Roth, 1979], three equations describe a line:
S (L, M, N) and S0 (P, Q, R)
Ny Mz = P
Lz- Nx = Q (4-14)
Mx Ly = R
Expanding 4-12, the equations become
p abe bcx + p acy abz = 0
and (4-15)
abe p bcx acy + pabz = 0
The Plcker 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,
p abc be p ac ab
(4
_ abc -pbc -ac pab_
Therefore,
P = ab2c2
Q = a2bc2
P
1
P
1
R = a2b2c
-1
-P
P
-1
-1
P
= ab2c2(l -p2)
= -2a2bc2p
= a2b2c(l + p2)
(4-17)
and


25
L = a^ be
P
-1
-1
P
M = ab2c
-1
-1
P
-P
N abc2
-1
P
-P
-1
= -a2bc(l p2)
- 2ab2cp
= abc2(l + p2)
(4-18)
This set of coordinates is homogeneous, and we can divide through by the common factor
abc. Further, we have in ray coordinates:
L = -a(l p2) P = bc(l p2)
M = 2bp Q = -2acp
N = c(l + p2) R = ab(l + p2)
By using the same method for developing the Pllicker coordinates and the
homogeneous ray coordinates, the r) equations are developed with 4-13.
pabc-bcx-pacy-abz=0
and
abc-r|bcx+acy+r|abz=0
and
pabc -be -pac -ab
_ abc -pbc ac pab
to form the Pliicker coordinates
P = ab2c2
Q = a2bc2
R = a2b2d
p -1
1 -p
p -r\
1 1
p -1
1 p
= ab2c2(l-p2)
= 2a2bc2p
= a2b2c(l+p2)
(4-19)
(4-20)
(4-21)
(4-22)
and


26
L = abc
-ft -1
1 ft
= a2bc(l-r)2)
-1 -1
= 2ab2cn
M=ab2c
ft -ft
N = abc?
-1 -r
-ft 1
i =-ab (4-23)
yielding, after dividing by the common factor abc, the ray coordinates:
L = a(l-r|2) P = bc(l-rf)
(4-24)
M = 2br| Q = 2acr|
N = -c(l + r|2) R = ab(l + rj2)
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 (q) 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 3-3 parallel platform and the octahedron will
be developed. Figure 4-5 is a plan view of the octahedron (3-3 platform) with the upper
det J|
platform in a central position for which the quality index, X =
det J
= 1 [Lee et al.
m
1998]. When the upper platform is rotated through 90 about the normal z-axis the
octahedron is in a singularity. Figure 4-6 illustrates the singularity for (j) = 90 when X 0
since det J I = 0 The rank of Jis therefore 5 or less. It is not immediately obvious from
the figure why the six connecting legs are in a singularity position.


27
Figure 4-5. Octahedron (3-3) Platform in Central Position
Figure 4-6. Octahedron Rotated 90 into Tensegrity
Flowever, this illustrates a plan view of the octahedron with the moving platform
ABC rotated through ((> = 90 to the position A'B'C'. As defined by Lee et al. [1998],
the coordinates of A'B'C' are


28
x'A =rcos(90 + 30)
x'B = rcos(90 + 30)
x'c = rsin(90)
where r = ¡=
V3
y'A = r sin(90 + 30)
y'B = -r sin(90 + 30)
y'c = rcos(90)
(4-25)
By applying the Grassmann principles presented in (4-4), at (j) = 90, the k components
for the six legs are N¡ = z and Rj = ab where i=l, 2, ...6. The Pliicker coordinates of
all six legs can be expressed in the form
S J =
ok
Li Mi z; Pi Qi
6
(4-26)
Therefore, a screw of pitch h on the z-axis is reciprocal to all six legs and the coordinates
for this screw are
ST=[0, 0, 1; 0 0 hj (4-27)
For these equations,
, ab ab
hz + = 0 or h = (4-28)
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:


29
2 "> 2
Xy 77
r H H 1
a2 b2 c2
(4-29)
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-T. 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 4-7) is described by the equation
x2 l l 1
a2 b2 c2
(4-30)
Figure 4-7. 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.
x2 y2
f
f k2)
1+--2
V c )
b2
1+ 2
l c J
=l,z=k
(4-31)
The traverse axis is y = 0, z = k, for all values of k The lengths of the semi-axes are
! k U
1 y jb.
cz
1 + -. They are smallest for k = 0, namely a and b, and increase
c2
without limit as Ik increases. The hyperbola is not composite for any real value of k.


CHAPTER 5.
PARALLEL PLATLORM RESULTS
3-3 Solution
Previous University of Llorida CIMAR research [Lee et al. 1998] on the subject of 3-
3 parallel platforms, Figure 5-1 is the basis work for this research. Their study addressed
the optimal metrics for a stable parallel platform.
The octahedral manipulator is a 3-3 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 leg-lines; 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 in-parallel 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 S-P-S, where S denotes a ball and socket joint, and P denotes a
prismatic, or sliding kinematic pair. The terminology 3-3 is introduced to indicate the
number of connection points in the base and top platforms. Clearly, for a 3-3 device.
31


32
there are 3 connecting points in the base, and in the top platforms as shown in Figure 5-1.
A 6-6 device would have 6 connecting points in the top and base platforms.
Figure 5-1. 3-3 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 (z-axis) 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 Pliicker 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 (A,), which is defined as the ratio of the J
determinant to the maximum value.
The significance between this 3-3 manipulator research and tensegrity is the
assumption that there is a correlation between the stability of a 6-strut platform and a 3-
strut, 3-tie 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 (A,) is the ratio of the determinant of J to the
maximum possible value of the determinant of J. The dimensionless quality index is
defined by
Idet J|
X =
Idet J
I im
(5-1)
In later chapters, this same approach applied here for the J matrix of the 3-3 platform
will be used for calculating that of the 6-6 tensegrity structure. For the later case the lines
of the connecting points are defined by a 6x12 matrix and will require additional
mathematic manipulation. In this case, a 6x6 matrix defines the lines of the 3-3 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
a
2
a
2^3
0
, B
a
2
2V3
0
, C
0
a
7J
0
( b
b
X
(n b
X
b
b
X
J
273
-h
y
> Eb
0 ~r
l 73
-h
J
, Ec
273
-h
J
(5-2)
(5-3)
The Grassmann method for calculating the Pliicker coordinates is now applied to the
3-3 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.


34
Si-
s2 =
S3-
s4 =
s5 =
S =
-b a + b
ah ah ab
2 2V3 h
2V3 2 2V3.
a 2b-a
ah ah ab
2 2V3
2V3 2 2V3
a a-2b
ah ah ab
2 2V3 h
2V3 2 2V3
a b a + b ah
2V3 ~h;
b b-2a
2 2V3
b 2a b
2
h;
-h;
2V3
ah 0 ab
ah ab
~2
2V3
(5-4)
ah
0
2^3
ab
2V3 V3 2V3
which yields the matrix for this system of
T 1
det J =
&
S, S2 S3 s4 s5 s6
(5-5)
The normalization divisor is the same for each leg (they are the same length),
therefore, l = V L2 + M2 + N2 = ab + b2 + 3h:) and the expansion of the
determinant yields
3V3aVh3
det J =
f a2 -ab + b2 2
\3
3
+ h"
(5-6)
Dividing above and below by yields
3\/3aV
det J -
a2 ab + b2
\3
3h
+ h
(5-7)
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


35
equating to zero to obtain a maximum value for det J yields the following expression for
h.
h = hm =Ji(a2-ab + b2
(5-8)
If we now select values for a and b, (5-7) yields the value hm for det J to be a maximum.
detJl =
lm
27aV
321a2 -ab + b2 J2
(5-9)
Further, we now determine the ratio y=b/a to yield a maximum absolute value
det J| Substituting b= ya in Equation 5-7 yields
detJ -
lm
m 3 3 3
27a y a
3 3
y a
27aJ
32(a^-ya^+yV)2 ^¡3 32
3 1
f \
vY T 2
(5-10)
To get the absolute maximum value of this determinant, the derivative with respect to y is
taken which yields:
il--'
y2 1 ~y)
b 0
y = = 2
a
= 0
(5-11)
Substituting this result in (5-8) gives:
= 1
(5-12)
This work shows some similarity to the values to be derived for the 6-6 platform. The
original quality index equation reduces to a function of (platform height) / (platform
height at the maximum index).


36
A, =
f
8
v
h
\3
U +
f
\
(5-13)
The resulting quality index plots for this 3-3 structure are found in Figures 5-2
through 5-6. In Figure 5-2, 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 plots 45 locations.
Figure 5-2. Coplanar translation of Platform from Central Location: Contours of Quality
Index


37
Figure 5-3. Rotation of Platform About Z-axis
Figure 5-4. Rotation of Platform About X-axis


38
Figure 5-5. Rotation of Platform About Y-axis
As expected, rotations about the z-axis 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 y-axes over 20. This could be valuable for antenna
repointing without using an antenna gimbal.
Figure 5-6 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.


39
Figure 5-6. Quality Index as a Function of the Height Ratio
4-4 Solution
The 4-4 parallel platform (Figure 5-7) is a square anti-prism. The calculations of the
4-4 quality index are similar to those for the 3-3 platform; however, because the 4-4 line
coordinates yield a 6x8 matrix, the determinant cannot be calculated directly and we
introduce JJ [Knight, 1998], the product of the matrix and its transpose. As with the 3-3
platform, X is defined as the ratio of the Jacobian determinant to the maximized J
determinant.
iy
i
Figure 5-7. The 4-4 Parallel Platform (plan view)


40
X =
det JJ 1
det JmJm
(5-14)
T ? 9 ?
From the Cauchy-Binet theorem, it can be shown that det J J = Af + A? +... + An.
Each A is the determinant of a 6x6 submatrix of the 6x8 matrix. It is clear that (5-14)
reduces to (5-1) for the 6x6 matrix. This method can be used for any 6xn matrix. As with
the 3-3 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 h
\ B^
J
0 h
V
, c
0 h
\ D-^
J
0 h
V
( b b ^
(b b >
fb b 'l
( b b ^
, F
, G
- 0
, H
l 2 2 J
U 2 J
U 2 J
l 2 2 )
(5-15)
Therefore, the Jacobian matrix is
b
b
\2a b
b
b
- y¡2a + b
yTa b
-V2a +
2
2
2
2
2
2
2
2
- V2a + b -
- yfla + b
b
\2a b
\2a b
b
b
b
2
2
2
2
2
2
2
2
T 1
h
h
h
h
h
h
h
h
i6
bh
bh
bh
bh
bh
bh
bh
bh
2
2
2
2
2
2
2
2
bh
bh
bh
bh
bh
bh
bh
bh
2
2
2
2
2
2
2
2
V2ab
\2ab
\2ab
Jlab
\2ab
v2ab
V2ab
V2ab
4
4
4
4
4
4
4
4
It follows that yjd
2t JJT S
given by
(5-16)
Vd
et J JT =
32^aVh3
(a2 -V2ab + b2 +2h2f
(5-17)


41
By following the same procedure as used for the 3-3 parallel platform, the key 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 =j^(a2-V2ab + b2) (5-18)
Again, as presented in the 3-3 analysis, this maximum value for h is included in
(5-17) to provide the maximum determinant.
I 2a2b2
VdetJmjJ,= y (5-19)
(a2-V2ab + b:^
To determine the ratio y=b/a for the maximum expression for (5-19), b=ya is substituted.
The numerator and denominator are also both divided by y3a3.
det Jm Jm
2a-
. V2 1
1_ +~2
y y2)
3
^2
(5-20)
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 = = V2
a
(5-21)


CHAPTER 6.
6-6 DESIGN
6-6 Introduction
The 6-6 in-parallel platform (a hexagonal anti-prism) 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 4-4 platform, the Cauchy-Binet theorem is used to
determine the index. Once the mathematics is determined, further attention will be
applied to antenna design.
Sketch
Figure 6-1 presents the 6-6 in-parallel 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 6-1. A 6-6 Parallel Platform (Hexagonal Anti-Prism)
42


43
A plan view of the 6-6 parallel (redundant) platform is shown in Figure 6-2. 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 x-y-z coordinate
system is located in the base with the x-y plane in the base plane. Hence, the base
coordinates are
Gc
Figure 6-2. A Plan View for the 6-6 Parallel Platform (Hexagonal Anti-Prism)
b
2
0
Gc [0 b 0]
(6-1)


44
D
y3b b
O
G,
V3b _b
2 2
Gf[0 -b o]
(6-2)
The coordinates for the top platform vertices at the central position are (6-3) where h
is the height of the top platform above the base.
A [a Oh] B
D[-a 0 h] E
a V3a
h
c
i
i
1 fa
$31
nr
i
2
2
2 2
a %/3a
It
u
a V3a
2
2
11
r
11
2 2
(6-3)
Applying Grassmanns method (see Chapter 4) to obtain the line coordinates yields
the following 12 arrays.
S,[Ga A]:
S3[Gb B]:
S5[Gc C]:
S7[Gd D]
S9[Ge E]:
V3b
~Y~
a
V3b
2
a
2
0 b O'
a V3a
h
2 2
0
2
0 h
b
2
J3a
S2[Gb A]:
S4[Gc B]:
1^*0
2 2
1
0 h
1 0 b 0
1 a V3a ,
2
S11 [Gf F]:
1 -
1
1
1
2
'1 0
1 i -
%/3b b o
2 2
-a Oh
v^3b b
2
a \/3a
0
h
S6[Gd C]:
S8[Ge D]:
Sio[Gf E]:
2
V3b
T"
a
~2
V3b
2
-a
b
2
V3a
2
b
~2
0 h
0
h
0
(6-4)
1 0
1 --
-b 0
SI2[GA F]:
V3b
2
a
2
-b 0
%/3a ,
0
2


45
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.
t = ~.
2\l
J3U
a b
V
+ b2+4h2
(6-5)
Evaluating the Jacobian
The J matrix, comprised of the line coordinates for the twelve legs, is a 6x12 array.
2a-Vib 2a
-Vib a-Vib
a
-a
i
p
+
cr
b
-b
Via b
V3a -2b
Via-2b
Via-b
2h
2h
2h
2h
2h
2h
-bh
bh
bh
2bh
2bh
bh
-VJbh
Vibh -Vibh
0
0
Vibh
ab
-ab
ab
-ab
ab
-ab
-2a + Vib -
2a + Vib
a + Vib
-a
a
a Vib
-b
b
Via + b
- Via + 2b
- Via + 2b
- Via + b
2h
2h
2h
2h
2h
2h
bh
-bh
-bh
-2bh
2h
-bh
Vibh
Vibh
Vibh
0
-2bh
-Vibh
ab
-ab
ab
- ab
ab
-ab


46
X
J is, therefore, the transpose (a 12x6 matrix).
2a-V3b
b
2h
-bh
-V3bh
ab
2a-V3b
-b
2h
bh
-V3bh
- ab
a V^3b
V3a b
2h
bh
-V3bh
ab
a
V3a-2b
2h
2bh
0
-ab
-a
V3a 2b
2h
2bh
0
ab
-a + V3b
V3a b
2h
bh
V3bh
-ab
- 2a + V3b
-b
2h
bh
V3bh
ab
- 2a + V3b
b
2h
-bh
x/3bh
-ab
-a + V3b
- V3a + b
2h
-bh
^3bh
ab
- a
-V3a + 2b
2h
- 2bh
0
-ab
a
-v/3a + 2b
2h
- 2bh
0
ab
a -V3b
- V3a + b
2h
-bh
-V3bh
-ab
Optimization Solution
(6-7)
Lee et al. [1998] developed the optimization method for the 3-3 and 4-4 platforms.
The method for calculating the optimization value for the 6-6 J matrix (non-symmetric) is
an extension of the 4-4 platform solution. The quality index X is given by
VdetJJ1
y/det Jm Jm
(6-8)
For this example, Vdet JJT is calculated.
Vd
et JJT =54
a3b3h3
a2-x/3ab + b2+h2f
(6-9)


47
As with the 4-4 parallel platform calculation, the maximum height (h) must be found. To
find this expression, the numerator and denominator of (6-9) 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 = Va2-V3ab + b2 (6-10)
As with the 4-4 analysis, this maximum value for h is included in (6-9) to provide the
maximum determinant.
det JmJm
54
a3b3
a2-V3ab + b2)2
(6-11)
This yields the k value (quality index) as a function of a and b.
k =
yd
et JJ
8h3 a2
V3ab + b2)2
detJmJm (a2-V3ab + b2+h2)3
(6-12)
This index (k) is a value between zero (0) and one (1), which represents the stability of
the structure.
As with the 4-4 structure, the ratio y=b/a, which represents the parameter ratio at the
maximum quality index, is determined by substituting for b=ya.
det JmJm
54
3 3 3
aya
8
a2 V3aya + (ya)2 j2
(6-13)
3 3
Again, the numerator and denominator are both divided by y a
det JmJm
54
1 _V3
U2 y
+i
(6-14)


48
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 6-6 platform.
d_
dy
J__V3
y2 y
+ 1
3
2 3
~~ 2
1 V3 1
-2 V3
~T + ~2
Y Y
= 0
(6-15)
The vanishing of the first bracket of the right side of the equation yields imaginary
solution, whilst the second bracket yields
Y
_2__ b
V3 a
(6-16)
h = -i and b = 2a
V3
S
(6-17)
Variable Screw Motion on the Z-Axis
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((()z)
(6-18)
Ya = rsin((j>z)
(6-19)
1 V3
XB = rcos((|)z + 60) = r( cos (6-20)
1 V3
Yg = rsin((t)z + 60) = r(sin (6-21)
i s
Xq = rcos(<|)z +120) = -r(-cos (6-22)
1 3
YC =rsin( (6-23)


49
Xp = rcos((])z + 180) = -rcos(j);
Yd = rsin((|)z +180) = r sin (()z
1
V3 .
X£ = rcos((j)z + 240 ) = -r(-cos(j)z sinz)
1 V3
Yp = r sin(4> z + 240) = -r( sin 4> z + -y cos (j)z)
1
73 .
XF = rcos((j)z +300 ) = r(cos(j)z + sin(|)z)
1 V3
Yp = rcos((J)z + 300) = r(sin (j)z cos<|)z)
(6-24)
(6-25)
(6-26)
(6-27)
(6-28)
(6-29)
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 + Xp = 0 YA + YB + Yc + YD + YE + Yp = 0 (6-30)
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.
, 0
2 2
S2 [G b A]:
! 73b
2
^ 0
2
Li XA Ya hj
1
X
>
<
S,[Ga a]:
(6-31)


50
1
V3b
b o'
2
i
o
o
1
s3[gb
B]:
2
s4[gc
B]:
1
xB
Yb h
_1 XB Yb hj
rl
0
b 0]
} V3b b
2 2
s5[gc
C]:
S[GD
C]:
1
Xc Yc hj
.1 Xc Yc
1
V3b
b
-Jib b
2 2
S7[GD
D]:
2
2
0
s8[ge
D]:
1
xD
Yd
h
.1 xD Yd
1
V3b
b
0
'1 0 -b 0"
s9[ge
E]:
2
2
S i o [G F
E]:
1
xE
ye
h
[l XE Ye hj
0
h
Sn[GF F]:
1
o
i
cr
o
i
SI2[GA F]:
1 -b 0
2 2
1 Xp Yp h
1 XF Yp h
The Plcker coordinates are defined by the 2x2 determinants of these 2x4 arrays.
Sj =
sT =
sj =
A
X
B
;T
vV
( b^
1Ya+t
1
bh
V3bh
2 ,
2
2
Ab^
ya-E
2y
1
bh
V3bh
2 )
b 2
2
V3b^
' bj
Yb"2
Y
bh
Sbh
2 J
I
2 "
2
(Yb b)
h; bh
0
-bXB]
(YC b)
h; bh
0
- bXc]
(6-32)
(6-33)
(6-34)
(6-35)
(6-36)
(6-37)
(6-38)
(6-39)
(6-40)
(6-41)


Xr +
x/3b
Y,
C
XD +
x/3b
yd-x
XD +
V3b
yd+-
Xp +
V3b
Yp + -
[XE (Ye + b) h; -bh
[XF (YF+b) h; -bh
Xp-
V3b
Yp +
bh
x/3bh
2
2
bh
x/3bh
2
2
bh
V3bh
2
2
bh
x/3bh
2
2
bXE]
bXp]
bh
x/3bh
2
2
^(V3YC+XC)
^3YD+XD)
^(V3Yd-Xd)
^(73Ye-Xe)
^(V3Ye + Xf)
(6-42)
(6-43)
(6-44)
(6-45)
(6-46)
(6-47)
(6-48)


52
This yields the transpose of the Jacobian matrix.
jt =
XA -
XA -
xB-
2
V3b
V3b
X
B
Xc
'xc + ^
V J
f
XD + 7
V 1 J
'xD+^'
2
V3b^
Xp +
V3b
XE
XF
YA+b)
2)
Vb]
2)
bA
yb-
J
i
(Yb b)
(Yc b)
M)
Y bl
\Yd ~2)
V]
2)
bA
XE +
2y
(YE + b)
(YF + b)
rv 42b)
XF
Yp +
2
v 2
l 2)
h
h
h
h
h
h
h
h
h
h
h
h
bh
V3bh
2
2
bh
x/3bh
2
2
bh
V3bh
2
2
bh
0
bh
0
bh
V3bh
2
2
bh
V3bh
2
2
bh
V3bh
2
2
bh
V3bh
2
2
-bh
0
-bh
0
bh
x/3bh
2
2
(V3Ya+Xa)
(V3Ya-Xa)
b(73YB-XB)
~ bXB
-bXc
-b(V3Yc+Xc)
_ (V3 Y d + X D)
-^(V3Yd-Xd)
-b(V3YE-XE)
bXE
bXF
(V3YF + XF)
(6-49)
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.
Lo + Mo + No
(v V3b)
z
f b^
XA r-
+
YA +7T
2
v z ;
l 2)
+ IT
Xa ~ V3bXA + + Ya + bYA + ^- + h2
(6-50)


53
i i 3b ? 2 i* o
r cos (|)z-v3brcos())z + + r sin (|)z + brsincj)z + + h
b2 .2
= r2 + br(sin = a2 + ab(sin ,1
2
(6-51)
Similarly, lengths of the even numbered legs are equal.
a'
ab(sin (6-52)
Lee et al. [1998] used the following notation to describe the screw motion.
M = J*TSD
(6-53)
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, A0, etc.). To
normalize the leg coordinates, each value is divided by the instantaneous leg lengths.
AT
A s|c'T' A /N* A
6f¡ =S; 5D = 5D
ti
(6-54)
Calculating the summation of the individual coordinates shows that all the values
are zero except for N and R.
T I I I I I V ^b V3b V3b yflb
Lt + l3 + l5 + L7 + L9 + Lh Xa + XB + Xq + Xd + + Xg + + Xp
V3b
- rcosq)z 1- r
f
V3 .
-cos<|)z --sin4>z
v2 2 j
x/3b
1 i V3 ,
cos(b7 + simp..
2 2
- rcosij),
(6-55)
Sb
+ r
1 i V3 ,
cos x/3b
+ + r
1 a >/3 .
cos = 0


54
L. L. L L
Mj + M 3 + M g + M7 + M9 + M|j = Ya + + Yg ~ + Y^ b + Yd + Yp H + Yp + b
= rsin 1 1 V3 .
COS(pz + -^-SHl y
' b f
r
9
1 V3 ,
C0S(tz sm<(>2
- b r sin 2
b
r
2
f1 A ^ .
cos x2 2 j
^ b r
-b + r
2
1 A ^ .
-cosq)z sin (j)z
+ b = 0
(6-56)
N] + N 3 + N5 + Ny + Ng + N ] j 6h
bh bh
bh bh
P| + P7 + P^ + P7 + Pq + Pi 1 1 1- bh H 1bh 0
11
(6-57)
(6-58)
~ ^ ^ ^ ^ ^ V^bll ^bb ^bb ^bb n n /A
Qj + Q3 + Q5 + Q7 + Q9 + Qi 1 b 0 H 1 b 0 0 (6-59)
R|+R3 + R5+R7+R9+Rn = j(V3YA+XA)+^(V3YB-XB)-bXc-b(x/3YD+XD)
b(^YE-XE)+bX,
br
T
f
(x/s Sin (J)z + COS(()z )+ ~
r R 3
-^-sin y
+br
br
2
cos(()z + sin 4>z + (V3sinz)
2 2 y 2
V
^ y3 3 ^
sin <()z + cos V 2 2 J
1 A Y 3 I
cosq)z sin (pz
Y
+ br
V
cos <))z sin Vs ^
^ sin 4>z H cos (j)z
= 3br(V3 sin 4>z + cos The second pair of legs sum similarly.
(6-60)
L2 + L4 + L6 + L8-bLio + Li2 0
M 2 -b M 4 + M ^ + M g + M j Q + M j 2 = 0
?2 + ^4 + ^6 + ^8 + PlO + Pl2 = 0
Q2+Q4+Q6+Q8+Ql0+Ql2 = b
(6-61)
N 2 "b Nj + + Ng + Njq + N12 6h
R2 + R4 + R5 + Rg + R]q + Rj2 = jbr(VV sin (|)z + cos (6-62)


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.
61n51
o'o
(N ] + N 3 + N 5 + N 7 + N 9 + N ] j )5z + (R| + R 3 + R 5 + R 7 + Rg + R] 1 )S 2
= 6z5z + 3br(V3 sin §z + cos
(6-63)
IqSIq = z8z + ^ V3 sin<|)z + cos(j)z]5(j).
(6-64)
The even leg calculation yields a similar result.
L5L =z8z +
e e 2
J3 sin 4>z cos (6-65)
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.
z8z = -
br
2
y¡3 sin(()z + cos(j)z 5(j)
(6-66)
The pitch is defined by the ratio of linear z change to rotation about the z-axis.
8z
5z
(6-67)
This yields the pitch equation.
8z br [ r- '
p = = [v3 sin 4>z +cos(pz
5(j>
2z
(6-68)
The subsequent integration yields the z calculation. This proves that the odd numbered
struts can be commanded to yield a pitch motion (z and 0z motions are coupled).


56
J z5z = -
z0
br
2
I
o
V3sin(j)z +cos(j)z 3 = Zq br[V3(cos (6-69)
(6-70)
Equation (6-70) can be modified (a=r and zo=0) to define the square of the platform
height.
z2 = ab|sin(j)z V3 (cos (})z +l)} (6-71)
Therefore, the platform height (z) is the root of (6-71).
z = Vabjsin(j)z -V3(cos(j>z +l)}2 (6-72)
This result shows that for a given twist about the z-axis (<|>z), there is a corresponding
displacement along the z-axis, defined by a finite screw (p=z/ Figure 6-3. The Pitch Relationship


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 six-degree 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
semi-precision, mesh reflector is presented.
Strut Desiun
In order to deploy the struts from a stowed position, the end points of the stowage-to-
deployment plan must be defined. Figure 7-1 presents a nominal 15-meter (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=60). The strut lengths are shown increasing for simplicity, but an actual design would
57


58
show the upper surface approaching the lower surface as the struts rotated to the
tensegrity position.
Figure 7-1. 6-6 Structure Rotated from a=0 to a=60 (Tensegrity)
Figure 7-2. Dimensions for Model Tensegrity Antenna


59
Based on these design assumptions the structure (Figure 7-2) would have the values
found in Table 7-1.
Table 7-1. Deployable Tensegrity Design Values
Design Parameter
NOMINAL 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 7-3, 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.


60
00
o
E
ir>
E
0.5 m.
E
Os
E
ir)
Figure 7-3. 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 7-4 shows a simple hinge design, which could have an over-center 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 7-5 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 7-6 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), LGarde (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 7-4. Folding Hinge Design


62
Figure 7-6. 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.


63
Table 7-2. Strut Deployment Trade Study
Strut
Deployment
Design
Advantages
Disadvantages
Folding
Design history
Design relevance to other
industries
Moderate deployment forces
Potential stiffness non-
linearities
Potential hinge surface galling
Locking hardware required
Sliding
Minimal deployment forces
Positive locking position
Potential bending stiffness non-
linearities
Limited design history
Potential contact surfaces
galling
Telescoping
Compact packaging
Minimal stiffness non-
linearities
Requires interference fittings at
deployment
Potential contact surface galling
Large deployment forces
Inflatables
Very compact packaging
Near homogeneous deployed
structure
Advanced materials
application
Requires deployment pump and
tubing
Weight savings limited
Expensive
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 deployment.
Stowed Ties: By simply folding the ties along the struts (Figure 7-7), 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.


64
Figure 7-7. 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 7-8). This added hardware
(potentially one motor per strut) increases complexity, weight, and therefore cost.
A trade study for these approaches is presented below.


65
Table 7-3. Strut/Tie Trade Study
Strut/Tie Interaction
Design
Advantages
Disadvantages
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, snag-free design
REQUIRES ADDITIONAL
HARDWARE
Reel Ties (elastic)
Stiffness constant
adjustments
COMPLEX DESIGN AND
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 cord-ties must be stowed to
ensure deployment success. This issue will be addressed further in Chapter 8.
Deployment Scheme
Figure 7-9 presents a potential deployment scheme. The requirements for this
operation are primarily low shock load and continuous motion. Despite the inherent self
deploying nature of tensegrity 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.


66
Deployment and Surface Adjustment
Figure 7-9. 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.
Motros work [1996] has predominantly focused on the stability of tensegrity
structures, including force density, non-linear analysis and morphology. Despite his clear


67
focus on the engineering aspects of tensegrity, 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 3-3, octahedron tensegrity is an excellent example of a spatial
structure. He has developed multiple tensegrity structure designs, which solve some of
the toughest curved-surface 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 cable-strut systems as an
extension of tensegrity. Reciprocal prisms (RP) and crystal-cell pyramidal (CP) grids,
which technically exclude tensegrity systems, are the basis for his space frame
applications. He developed a hierarchy of feasible cable-strut 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 double-layer tensegrity grids [Wang and Liu, 1996], His work in the
feasibility of these new applications is very important to space structure development.
Dr. S. Pellegrinos 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 [Calladme 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 semi-solid 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, gas-filled
shock absorbers, would allow a self-deploying 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.
Cx + Kx = 0 (7-1)
Deployment Stability Issues
The calculations for the 3-3 design, which were presented in Chapter 3 (Parallel
Platform Results), suggest that there is a singularity at the tensegrity position. Figure 7-10
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 Kenners 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.


70
Figure 7-10. Octahedron Configurations
To improve the design and stability of the tensegrity structure, while not affecting the
self-deployability, another set of ties is added between the vertex of the base and the
opposite vertex of the platform (Figure 7-11).


Figure 7-11. Redundant 3-3 Structure
This results in four ties at the end of each strut, versus the three in the original design.
Again, the angles (|) and a represent the works of CIMAR and Kenner, respectively.
Figure 7-12 presents the rotations from the Central Position, through the Aligned and
Tensegrity Positions, to the Crossover Position, where the struts intersect.


72
Central Position
<))=0o a=-60
F E
Aligned Position
Tensegrity Position (singularity)
4>=90 a=30
Crossover Position (interference)
(j)=120 a=60
Figure 7-12. Redundant Octahedron Configurations
The mathematics to calculate this 3-3+ structure is similar that for the 4-4 and 6-6
structures, in that the Cauchy-Binet theorem is employed. Because there are now nine (9)
connections between the platform and the base, the resultant J is a 6x9 matrix.
J =
S8
S9
(7-2)


73
Therefore, J is a 9x6 matrix.
rT
51
52
53
54
55
56
57
58
s9
(7-3)
As shown in Chapter 5, the quality index is calculated using the determinant of the
combined matrices (det JJ ). 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 7-13. The quality index remains relatively constant as the platform rotates
through 120, 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 3-3 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 3-3 design, A,=0 at a=30, 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.


74
Redundant Octahedron
Figure 7-13. 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 8-1). With this
method, a 15-meter diameter antenna would have a stowed package volume of
approximately 4-meter height and 4-meter diameter. This extra fold along the spar length
greatly increases the material content, complexity of the structure, and touch labor to
assemble the system.
Figure 8-1. 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 6-6 quality index (Central Position), the height h was equal to
75


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, 15-meter 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 (6-6 Design), the diameter of the
deployed surface (2a for the 6-6 design), and the strut length (1). The purpose of this
analysis is to design a stable structure while minimizing the strut length for the 15-meter
antenna. The 6-6 design is the basis for the deployable design. Table 8-1 presents the
geometric relationships for the three candidate structures (3-3, 4-4. and 6-6).
Table 8-1. The Three Tensegrity Structure Designs Considered
Design
# of Struts
# of Ties (total)
Tip-to-Tip Diameter
3-3
3
9
a
4-4
4
12
V2a
6-6
6
18
2a
3-3 Optimization
The tensegrity position for the 3-3 structure, as defined in Chapter 7, is at (|)=90o and
a=30. 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 3-3 structure, the Central Position is defined as (|)=0o or a=-60. As presented
in Chapter 5, the determinant of the J matrix and the determinant of the maximum of this
matrix (Jm) are |j|
3V3aVlr
( 2 i i 2 h3
a -ab + b ?
+ h
and Jm =
27aV
, respectively.
321a2 -ab + b2^
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
l [ 2 2) d
hm = J-\a -ab + b /, found by taking the partial derivative and setting it equal to
, lJl
zero (a calculus inflection point). Calculating the quality index, X = p-1- yields:
r m
X=-
th3(a2-ab+b2]2
^a2-ab+b2 2^
+h
V
(8-1)
7


78
As the lim (A.), which means that the base reduces to a point, the Equation 1 reduces to
b=>0
X =
8^3
f
a h
h
v 3h
\3
a )
(8-2)
Rooney et al. [1999] refers to this design as the tensegrity pyramid'.
As a first-design, the ratio a/h=l is chosen. This further reduces the equation to
fi+l)
J
9
f4l
v3 J
,3,
(8-3)
which is an acceptable quality index (optimum is A.=T.O). But to define a class of
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
X =
8v^3
' 1 '3
3y
+ Y
and taking the derivative
dy
(8-4)
of the denominator, the maximum values for the quality
index (the denominator equals zero) is found at y =
plot of the quality index (A.) vs. the ratio values (y =
1
73
a
a 0.58. Figure 8-2 presents the
At this value of y, the quality
index has a relative value of 1.0.


79
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 structures 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 off-axis moment (see Figure 8-3). Additionally, it is
impractical to connect an antenna structure at a point, as moment loads would approach
infinity.


80
Figure 8-3. Reduction of the Base to Zero
Based on these observations, a compromised geometry is necessary. To this end, the
base should be minimized, and the y = ratio chosen for the maximized quality index.
a
Table 8-2 presents the results of three choices of Base Planar Tie length (b) with
a a
maximized quality index. Figures 8-4, 8-5, and 8-6 present the curves for the b= ,
2 4
and cases, respectively.


81
Table 8-2. Quality Index for b= and Cases
2 4 8
b
J
hm
r m|
X
y at ^max
a
2
32
3a/3
1 +h T
,4h a2>
- = 0.5a
2
^ 0.2a3
32
\
1
1 f
+ y
,4r J
0.50
a
4
256
3y¡3
13 + h '
48h + a2,
\3
^la 0.52a
4V3
27a 0.02a3
416V13
72
13V39
f 13 "l
+ Y
V 48y J
3
0.52
a
8
3^3
^ 0.55a
8
3^a 0.002a3
608VI9
19V9
0.54
2048
'57 + h V
,192h + a2J
64
f 57 ^
l192y ,
3
f
b
V
Figure 8-4. X vs. y


82


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. Stern [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 3-3 design, thereby
improving the quality index. For the 4-4 and 6-6 structures, the index approaches 1.0 for
virtually any position. Figure 8-7 presents the design for the 3-3 structure. In this case,
the A. varies only 25% from 0.75 to 1.0 (as shown in Figure 7-13).
Figure 8-7. Reduction of the Base to Zero (Redundant Octahedron)
a) b0; c) b=0


84
4-4 Optimization
As presented in Chapter 4, the Jacobian (J) for the 4-4 structure is a 6x8 matrix, and
an understanding of the Cauchy-Binet Theorem aids in obtaining the quality index. As
previously presented, the numerator for the quality index (A.) reduces to
Vd
et JJ 1 =
I
32V2a-VhJ
(a2 -V2ab + b2 + 2h2f
- The denominator represents the maximum
possible vale for the numerator was found by using h=0. This value is
det JmJm
2aV
(a- 4l ab + b2 ) 2
. The height (h), which is used to find the
denominator, is hm = y-^(a2 V2ab + b ). Again, following the work in Chapter 4, the
quality index is therefore.
16v/2h3(a2-^ab+b2)2
(8-5)
(a2-V2ab + b2+2h2f
As the lim (l) this reduces to
b=>0
, 16V2a3h3
= (a2 + 2h2J
By using y = , the equation reduces further to
a
16^2 h3 16V2 I6V2
( h2^
3
( a
3
0
a + 2 h
+ 2
2y + -
l a )
a;
l y)
(8-6)
(8-7)


85
with a maximum X at y =
1
Ti
0.71. Figure 8-8 plots X vs. y.
Figure 8-8. X vs. y
y =
ay
for the Square Anti-prism


86
3 3 3
Similarly, the equations for b equals - and are presented in Table 8-3. Figure
2 4 8
8-9 presents the first A. vs. y plot. The second and third cases are similar, but it is obvious
that the y value at A.max changes significantly between a/2 and a/8.
Table 8-3. y at b= and
2 4 8
b
X (y=h/a)
y st Amax
a
2
/
16^2
V
\ 3/
5 1 V2
4 V2 j
T5-1V
,4 V2 ,
2
V y
1
72
0.52
(1 1
2y + -
l y
5 1 V
v4 V2
\3
/
a
4
16V^2
V
17 1 >
16 2V2 y
3
2
'17 1 ^
16 2V2
2
V 2
72
0.60
fo 1
l y
'17 1 v
,16 2a/2a
a
8
/
I6V2
V
65 1 "
64 4V2>
3
2
'65 1 N
64 4V2
2
1/
72
0.65
^ 1
2y + -
^ y
65 1 'j
,64 4,/2,
\3
)
6-6 Optimization
The 6-6 tensegrity design is the basis for this new class of deployable antenna
structures. The calculations are similar to those for the 4-4 to solve the 6x12 J matrix.
The numerator for X, taken from Chapter 4, is
54aVh3
(a2 --\/3 ab + b2 + h2 J3
. The
denominator, which is found by using h equals zero is


87
/det 1
54a3b3
\laei J m J m "
^ x Ilia 11 VulUv la 11 y yd V d dU 1 L/ J
8(a2 -V3ab + b2} 2
quality index is therefore,
The
A 8h3(a2 V3ab + b2f2
"(a2-V3ab + b2+h2)3
(8-8)
As the lim (A.) this reduces to
b=>0
X =
8 a3h3
(a 2 + h 2 J
(8-9)
By using y = , the equation reduces further to
a
A,=
81r
8
h2^
a +
v a j
3 rn u\3 ( ix3
a h
h
vh ay
1
y+
V V
(8-10)
with a maximum A. at y = 1. Figure 8-10 plots A. vs. y.
j -j
Similarly, the equations for b equals and are presented in Table 8-4. Figure
2 4 8
8-11 presents the A, vs. y plot for the case. The second and third cases are similar.
Again, the y at A.max values vary greatly as b is reduced from a/2 to a/8. Keeping the work
of Stern [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.


88
Quality Index
Y
Figure 8-10. X vs. y
a;
for the Hexagonal Anti-prism
Quality Index (a/2)
Y
(
Figure 8-11. X vs. y
\
for the Hexagonal Anti-prism


89
Table 8-4. A and y for b= and
2 4 8
b
A. (y=h/a)
y at A-max
a
2
8
'i VY
v4" 2 J
3
2
'5 vy
l4"2,
* 0.62
( 1
Y +
l y
'5 V3Y
l4 2 JJ
3
a
4
8
'i7_VT
16 4
\ V
3
2
r 17 vy
[l6 4 /
1
V 2
* 0.79
( i
y +
l v
'l7 VY
[l6~ 4 J
y
>
a
8
8
^65 vr
,64' 8 ,
3
V2
'65 VY
[m'T
\ 2
0.89
1
y +
l y
'65 VY
,64' 8 J
\3
)


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, Pliicker, 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
90


91
deploy (preferably self deploy) and stow to allow placement in the space environment.
After development of the 3-3, 4-4, and 6-6 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 6-6 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 mathematic analysis of the quality
index for these augmented 3-3 and 4-4 structures showed a marked improvement in the
indices. For the 6-6 design, the basic tensegrity design with 12 platform/base connections


92
(Figure 9-la) is augmented to a total of 18 (Figure 9-lb), 24 (Figure 9-2a), 30 (Figure 9-
2b), and 36 (Figure 9-3).
(a)
(b)
Figure 9-1. Hexagonal Anti-prism Designs
(a) Basic Tensegrity Design (12 platform/base connections); (b) Augmented Tensegrity
Design (18)
Figure 9-2. Augmented 6-6 Hexagonal Anti-prism Designs
(a) Augmented Tensegrity Design (24); (b) Augmented Tensegrity Design (30)


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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 UNIVER ITY OF FLORIDA 2000

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Copyright 2000 B y Byron Franklin Knight

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For this work I thank Mary my friend my partner and my wife

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ACKNOWLEDGMENTS This research has been a labor of love beginning with m y 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 Ian 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 e x tensive knowledge of mathematics To m y Committee Chairman Dr. Joseph Duffy I give my heartfelt thanks. You have taught me that to grow the developments of the 21 s t Century we need the wisdom and dedication of the Renaissance. Sir y ou are an English Gentleman m y teacher and m y 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 l V

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TABLE OF CONTENTS ACKNOWLEDGMENT lV ABSTRACT ........... ......... .... .................................................. .. .................... ....... .. ... . .. ..... vii CHAPTERS BACKGROUND ...... .. .. ................................................................................... .. ..... .. .. .. 1 Space Antenna Basis ...................................................................................................... l Antenna Requirements ......... ...... ... ....... .. .............. .. .. ............ ... ............................ .... .. 2 Improvement Assumptions .. ................ .............. .. ...... ........ .... .. ...... .......................... 3 2 INTRODUCTION ................................. .......................... ............................. .... ... ... ..... 5 Tensegrity Overview ...................................................................................................... 5 Related Research ...... .................. .. ...... ... .. .. ....... .... ... .. ........ .. .... ...... ....... .. ..... .... .. .... ... .. 7 Related Designs ............................................................................................................. 8 Related Patents ... ...... ................................................................................................... 10 3 STUDY REQUIREMENTS ........................................................................................ 13 Stability Criterion ......... ... .......... ....... ...... ............................... .... .. .............................. 13 Stowage Approach ....... .. ........ ........... ....................... ... ..... .. ......... ........ ........................ 13 Deployment Approach ...... ........................ ..... ... .. .. ................................................ ... 13 Mechanism Issues .......... ... .. .. .............................. .. .. .. ....... .. ....................... .. ..... ..... .... 15 4 BASIC GEOMETRY FOR THE 6-6 TENSEGRITY APPLICATION ....... ... ............ 16 Points Planes Lines and Screws .................. ..... ............. ...... ......... ..... ............. .......... 17 The Linear Complex ............. ...................... ....... ........... ................. ... .... .... ..... ............ 19 The Hyperboloid of One Sheet .................................................................................... 22 Regulus Plucker Coordinate ...... ................. .. ... ..... ... ... .. ............................................ 24 ingularity Condition of th Octahedron ..................................................................... 26 Other Form of Quadric urfa e ................ ........ ..... ... .... ... ...... ........... ................. ... .. 28

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5 PARALLEL PL ATF ORM RES UL TS .. .. .. .. ... . ............ .. ........................ .... . .. .. ... 31 3 3 Solution ..... ...... .. .. ... ... ... ...... ......... ......... .. ...... .. .. ............................ ... .. .. ............. 3 1 4-4 Solution ................................. .. .. ....... ... .......... .... ... ... .... .. . ................... .. .. .. .. .... 39 6 6-6 DESIGN .... .. .. ... . .. ..................... .... . .. .. . ... .. .. .. .. .. ... .. . .. ...... ... .. .. ... .. ... .. .. ........ 42 6-6 Introduction ............ ... . .... .. .. ... .. .. .. ... ............ ...... .. .. .. ................... ....... ... ...... 42 Sketch ..... .. .. ... .. .. ... .. ......... .. ... ................ ..... ... .. ...... .. .. .. ... .. .. .... ... .. ....... .. .. ... .... .. 42 Eva luating the Jacobian .. ....... ... .. .. .. .. ...... ... ............... .. ................................... . .... .45 Optimization Solution .... ....... ....... .. ... . ... ................... .. .. ....... ... .. .. .. ... . . ....... ...... .46 Variable Screw Motion on the Z-Axis ........ .................... . ... ........... .. .. ... .................. .48 Special Tensegrity Motions .. ... .. ... .. ...... ..... .......... ..... .. ........................................... 55 7 DEPLOYMENT AND MECHANICS ..... .. . ............................ .... ... .. ....... .... ...... .. .. .. 57 Strut Design ....................................... .... ..... .... ...... ..... .. ... ..... ... .. .. . ............ . . ..... . .. 57 Strut / Tie Interaction ......... ...... .. ..... .. ....... . .... .. .. .. ........ .. ............... .. ... ... .. ......... .... ...... 63 Deplo y ment Scheme .. ......... .... .. .. ... .. ... ... .... ... .. . ...... .. . ......... .... .. . .. .. ......... .. .......... 65 Previous Related Work ............ ... .. ..... ........ ... .. ...... .. . .. .. .. ... .. .. .. .. ... .... ....... .... ........... 66 A labama Deplo y ment Study .. ... ... . .. .......... ... .. .. .. .................. ............ ............. .. . .. .. 68 Deplo y ment Stability Issues ........ .. ... ...... .. ... .. ..... .. .. .. ... .. . .. ... .. ......... ...... .. ...... .. 69 8 STOWAGE DESIGN .... .. .............. .. .. .......... .... ... ..... .. ... ... .. .. ... .... ......... .... .. .. ..... . 75 Minimized Strut Length .... .. .. ... .. .. ...... .. ... .. .. .. .. .. ...... ...... ............ .. ........ .. .. ............. 7 6 3-3 Optimization .. ... ... ...... .. ... ... .. .. .. ... .. .. ... .. .. .... ............. ....... . .. ........ .. ... .. .. ... ..... 76 4-4 Optimization .... .. .. .. ... .. .. .. .. .. .. .. ... .. ...... .. .. .. .. .. .... ... .. ...... ............. .......... .. .. ... .. 84 6-6 Optimization . .. .. .. .. .. .................................... .. .. ... .. ..... .. .. .. .. ... ................... ... .. ... 86 9 CONCLUSIONS ... .......... .. ........... ........... ...... .. .. .... .. .. .. ....... .. ... ... ................. ......... 90 Applying Tensegrity Design Principles ... .. .. ...... ... .. .. .... ........................................... 91 Antenna Point Design ........... . ... .............. ... ........ .. .. .......... .. ........ .. ..... ... ................ 95 Patent Disclosur e . ..................... .. ... ..... .. .. .. ..... .... .. .. ..... ............ ... ... ......... ........... ... 97 Future Work ........... .. ...... ... . ... .. .................. . ........ .. ....... .. ... ...................... ... .. .. ......... 97 REFERENCES ................................. ................. .. .. .. .. ............... .. ... ... ....... .... ....... ........... 98 BIOGRAPHICAL SKETCH .. .. ...... .. .. ..... ... .. .. .. .. ............................ .. .. ... .. .. ........ .......... 103 Vl

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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 Philosoph y DEPLOY ABLE 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 mesh-surface reflectors have resembled folded umbrellas with incremental redesigns utilized to save packaging size. These systems are typically over-constrained 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 thi increm ntal approach. This work applies an approach called ten egrity to d playable antenna d lopm nt. Kenneth nel on a tudent of R Buel min t r Full r in nt d t n g rit tru tur 111 1948. Such structure use a minimum numb r of c mpr ion m mb r ( trut ) ; tabilit is maintained u ing tension m mber (ti ) Th n v It intr du d in thi rk i that II

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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 3-3 (octahedron) and 4-4 (square anti-prism) structures the 6-6 (hexagonal anti-prism) 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 antenna feed 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 Vlll

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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 mesh-surface 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 Snel on (stud nt of R. Buckminster Fuller) in 1948 [Connelly and Black 1998]. The name tensegrity is deriv d from the words Tensile and Integrity, and was originally developed for architectural sculptures. The advantage of this type of design is that there i a minimum of compression tubes (herein referred to a strut ) ; the tability of th t d

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2 through the use of tension members (ties). Specifically this work addresses the new application for self-deploying 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 over-driven 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

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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 1-1. Figure 1-1. 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 payload Improvement Assumptions The basis for this research is the assumption that th stowed density for deplo able antennas can be greatly increa ed while maintaining the reliability that th pac community ha enjoyed in the pa t. Failur of the tructure i unacc ptabl but if th

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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 Frequency systems (greater than 20GHz) in low earth orbit (LEO) the antenna aperture need only be a few meters in diameter. But for an L-band 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.

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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 2-1 is a four-vertex 6member 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 2-1. A imple Tetrahedron and Tripod Frame Th Octah dron (6-v rtic 12-memb r and 8-fac ) i th ba i fi r thi r apply t n grity to depl yabl ant nna structur 1 g ur 2-2 pr nt th impl tru tur 5

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6 and tensegrity application (rotated about the center with alternate struts replaced b y 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 six-leg parallel platform. It is from this mathematics that the new designs are deri v ed. Figure 2-2. 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 energy-efficient 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

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7 proposes a family of residential commercial and industrial structures that are both esthetically 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 long-term 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 short-term research (achieving program performance) but have greatly neglected the long-term goals which has traditionally been funded by the government. The top candidate technologies include tructural elements materials and tructur for ctr rn devices and large deployable antenna ( > 25 meter diam ter) Whil th r ha meter subsystems developed to meet GEO sy tern requir m nt durin g th 1990 th large deployable requirem nt ha y et to b addr d r d ev lop d hi r a r h w ill addr son po sibl oluti n t buildin g uch a ub y t m

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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 ( 6-struts and 4vertices) is the basis for this approach. From that the authors propose a series of octahedral cells (12-struts and 6-vertices) to build the adaptive structure (Figures 2-3 and 2-4). The conclusion is that from well-defined 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].

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9 z Figure 2-3 Octahedral Truss Notation Cell 2 Cell I Figure 2-4. Long Chain Octahedron VGT The most complex issue in developing a reliable deployable structur d ign i the packaging of a light weight subsystem in as small a volume a po ibl hil that the deployed structure meets the sy tern requir ments and mi n p rforman Warnaar developed criteria for deployable-foldabl tru tructur [Warna a r l 9 2 ]. H

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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 three-dimensional volume The author also presented a tutorial series [W arnaar and Chew 1990 ( a & b)]. This series of algori thrns presents a mathematical representation for the folded (three-dimensional volume in a two dimensional area) truss This work aids in determining the various combinations for folded truss design. NASA Langley Research Center has extensi v e experience in developing truss structures for space. One application a 14-meter diameter three-ring optical truss was designed for space observation missions (Figure 2-5). A design study was performed [Wu and Lake 1996] using the Taguchi methods to define key parameters for a Pareto-optimal 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 anal y ses 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 200-kg 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

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11 rather than specific antenna designs. Some of these patents address new approaches that have not been seen in publication. Upper surface Lower surface Figure 2-5. Three-ring 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 three-dimensional 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 three-dimensional framework. Other inventors continued work in expandable network to meet th n d of International Space Station. Natori [ 1985] u ed beam and triangular plat s to form a tetrahedral unit. These units formed a lin ar truss; his work included both joint and hinge details and the stowage / deployment kinematics. Kitamura and Y ama hiro [ 1990]

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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.

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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 3-1 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 6-6 system i to manipulate the l g joining th hub to the antenna to create a tensegrity structure Onoda suggests a sliding hing to achi deployment but such a package still requires a large height for th tow d tru tur hi approach does have excellent m rit ford plo y abl array a h pr nt in th p p r.

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14 Figure 3-1. Deployed and Stowed Radial Rib Antenna Model The tensegrity 6-6 antenna structure would utilize a deployment scheme whereby the lowest energy state for the structure is in a tensegrity position. Figure 3-2 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 3-2. 6-6 Tensegrity Platform

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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 3-3 shows this concept using a deployment mechanism for the ties ; spherical joints would be necessary to ensure that there are only translational constraints. Angle-Unconstrained Revolute Joint Elastic Ties Deployed from the Strut (3 each) Figure 3-3. The Struts Are Only Constrained in Translation

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CHAPTER 4. BASIC GEOMETRY FOR THE 6-6 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 4-1 shows the rotation of the 6-6 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 4-1 A 6-6 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. 16

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17 Defocus or the cupping" of the structure must be corrected once the subsystem i s 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 non-linear 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 of pre-stressed 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 of tensegrity 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 co-axial 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 point planes lines and Screw Theory is presented. Points, Planes, Lin and Screws The vector equation for a point can b expr s d int rm of th e art tan c rdinat by

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18 r=xi+yj+zk (4-1) Referencing Hunt [1990] these coordinates can be written x = ~ y = :!_, z = .!:_. w w w This expresses the point in terms of the homogeneous coordinates (W; X Y Z) A point is completely specified by the three independent ratios ~,:!_,.!:_and therefore there are WWW an 00 3 points in three space. Similarly the equation for a plane can be expressed in the form D +Ax+ By+ Cz = 0 or in terms of the homogeneous point coordinates by Dw + Ax + By+ Cz = 0 (4-2) (4-3) The homogeneous coordinates for a plane are (D ; A B C) and a plane is completely specified by three independent ratios ( A B C) Therefore there are an oo 3 planes in D D D 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 1 Y I z1) and (x 2 y 2 z2) can be obtained by counting the 2x2 determinants of the 2x4 array. [: X 1 Y i Z]] X2 Y2 Z2 (4 -4 ) 1 X1 1 Yi 1 Z1 L= M= N= 1 X2 1 Y2 1 Z2 (4-5) P= Y I Zj Q= ZJ X1 R= X1 Yi Y2 Z2 Z2 X2 X2 Y2

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19 Th homogeneous coordinates (L M N ; P Q R) or (s; So) are superabundant by 2 since they must satisfy the following relationships. (4-6) where d is the distance between the two points and S So =LP+ MQ + NR = 0 (4-7) which is the orthogonality condition. Briefly as mentioned the vector equation for a line is given by r x S = S 0 Clearly Sand So are orthogonal since S So = S r x S = 0. A line is completely specified by four independent ratios. Therefore these are an oo 4 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 -:t O and the pitch LP+MQ+NR 5 1s defined by h = ------. 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 de ign 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 octah dral manipulators. Instant mobility of the deployable ten egrity antenna structur i of much interest within th design community. Thi instant mobility is cau d by th Lin ea r Dependence of Lines. This occurs when the connecting line of a tructur b m linearly dependent. They can belong to (i) a linear c mpl ( oo f !in (ii ) a !in a r

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20 congruence ( e x} of lines); or (iii) a ruled surface called a cylindroid ( 00 1 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 hon the z-axis. For such an infinitesimal twist a system of 00 2 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 of 00 3 tangents to 00 2 coaxial helices is called a helicoidal velocity field. ... z co h Figure 4-2 Two equal-pitched helices

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21 In Figure 4-2 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 Bare taken on the respective radii and both cylinders are on the same z -axis After one complete revolution the points have moved to A' and B' with AA'=BB'=2nh. Both advance along the z-axis a distance h0 for a rotation 0 Now the instantaneous tangential velocities are Vr a = w x and Vtb = w x Q. Further V a= hw and Vr a=w x ~The ratio I Val /l Vral = h/a = tan a or h=a tan a. Similarly h/b = tan ~ or h = b tan ~.. I Z Figure 4-3. A Pencil of Lines in the Polar Plane a Through the Pole A Further Figure 4-3 (see [Hunt 1990]) illustrates a pole A through which a h e li x 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 plan a contain a pencil of lines ( 00 1 ) through the pole A. Clearly as a point A move on th h Ii an 00 2 lines is generated. If we now count 00 1 cone ntric h lie of pitch h and con id r th totality of the 00 2 lines generated at each polar plan on a in g l h li ill g n ra t 00 3

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22 lines which comprises the linear complex. All such lines are reciprocal to the screw of pitch h on the z-axis The result with respect to anti-prism tensegrity structures will be shown in ( 4-26) and ( 4-27) and it is clear by ( 4-28) 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 4-4). The surface is represented by the equation 2 2 2 X Y Z + =1 a 2 b2 c2 (4-8) which is a standard three-dimensional geometry equation. This equation can be factored into the form (4-9) and can become an alternate form ( 4-10) Similarly (4-11) The equations can be manipulated to form :

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23 ( 4-12) z y X Figure 4-4. 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 4-10. imilarly any point on the surface, which is generated by the line equation also satisfies the equations in 4-12 a they are derived from 4-10. The system of line which is described by 4-12 her p i a parameter is called a regulu of lines on thi hyperboloid. Any indi idual !in f th regulu is called a generator. A similar set of quation can b er at d fi r th alu 17 ( 4-1 )

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24 The lines that correspond tori constitute a second regulus which is complementary to the original regulus and also lies on the surface of the hyperboloid. Regulus Plucker Coordinates Using Plucker Coordinates [Bottema and Roth 1979] three equations describe a line: S (L M N) and S 0 (P Q R) Ny-Mz=P Lz-Nx =Q Mx-Ly = R Expanding 4-12 the equations become pabc bcx + pacyabz = 0 and abc p bcx acy + pabz = 0 ( 4-14) ( 4-15) The Plucker 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 be pac -ab] abc p be ac pab Therefore Q = a 2 be 2 p p = -2a 2 be 2 p 1 -1 R = a 2 b 2 c p 1 = a 2 b 2 c(l + p 2 ) 1 p and ( 4-16) ( 4-17)

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L = a 2 be p 1 = -a 2 bc(l p 2 ) -1 p 2 -1 -1 2 M = ab c = 2ab cp p -p N=abc 2 -l p =abc 2 (l+p 2 ) -p -1 25 (4-18) This set of coordinates is homogeneous and we can divide through by the common factor abc. Further we have in ray coordinates: L=-a(l-p 2 ) M=2bp N = c(l + p 2 ) P = bc(l p 2 ) Q = -2acp R = ab(l + p 2 ) By using the same method for developing the Plucker coordinates and the homogeneous ray coordinates the ri equations are developed with 4-13. riabc-bcx-riacy-abZ=O and abc-ri bcx+acy+riabZ= 0 and [ riabc abc to form the Plucker coordinates P=ab2c21'l -1 =ab2c2(1-ri2) 1 Q=a2bc2 -11 =2a2bc217 R = a 2b21~ 1 = a 2b2c(l +172) and ( 4-19) ( 4-20) ( 4-21) (4-22)

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2 -1 M=ab ri ? -1 N=abc -ri =2ab 2 cri =-ab2-(1 +ri 2 ) 1 26 yielding, after dividing by the common factor abc the ray coordinates: L = a (l -ri 2 ) P = be (1 ri 2 ) M = 2bri Q = 2acri N = -c (1 + ri 2 ) R = ab (1 + ri 2 ) (4-23) ( 4-24 ) 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 3-3 parallel platform and the octahedron will be developed Figure 4-5 is a plan view of the octahedron (3-3 platform) with the upper i det J i platform in a central position for which the quality index A = i i = 1 [Lee et al. detJm 1998]. When the upper platform is rotated through 90 about the normal z -axis the octahedron is in a singularity. Figure 4-6 illustrates the singularity for = 90 when A= 0 since i det J i = 0 The rank of J is therefore 5 or less. It is not immediately obvious from the figure why the six connecting legs are in a singularity position.

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27 Figure 4-5. Octahedron (3-3) Platform in Central Position Figure 4-6. Octahedron Rotated 90 into Tensegrity However, this illustrates a plan view of the octahedron with the moving platform ABC rotated through = 90 to the position A 'B 'C'. As defin db Lee t al. [ 1998], the coordinates of A 'B'C' are

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28 x:,\ = r cos(90 + 30) YA = r sin(90 + 30) X13 = rcos(90 + 30) Y B = -rsin(90 + 30) Xe= rsin(90) Ye= rcos(90) a wherer = (4-25 ) ,.. B y appl y ing the Grassmann principles presented in (4-4) at = 90 the k components for the six legs are Ni = z and R i =_!_ab where i = 1 2 ... 6. The Plucker coordinates of 6 all six legs can be expressed in the form S = L T [ I I M I ~] (4-26) Therefore a screw of pitch h on the z-axis is reciprocal to all six legs and the coordinates for this screw are For these equations ab hz+ = 0 or 6 h =ab 6z ( 4-27) (4-28) 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 b y the equation:

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x2 y2 z2 + + =l a2 b2 c2 29 ( 4-29) 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 ~. 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 = O 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 x2 y2 z2 =l a2 b 2 c 2 ( 4-30 ) z X

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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 h y perbola. ( 4-31 ) The traverse axis is y = 0 z = k for all values of k The lengths of the semi-axes are ~ 2 a b 1 + They are smallest for k = 0 namely a and b and increase c2 without limit as l k l increases. The hyperbola is not composite for any real value of k.

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CHAPTER 5 PARALLEL PLATFORM RES UL TS 3-3 Solution Previous University of Florida CIMAR research [Lee et al. 1998] on the subject of 33 parallel platforms Figure 5-1 is the basis work for this research Their study addressed the optimal metrics for a stable parallel platform The octahedral manipulator is a 3-3 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 leg-lines ; 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 in-parallel 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 S-P-S where S denote a ball and sock t joint and P den t a prismatic or sliding kinematic pair. The terminology 3-3 is introduc e d to indicat th number of connection points in the base and top platform l ea rl for a 3 d 1c 31

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32 there are 3 com1ecting points in the base and in the top platforms as shown in Figure 5-1 A 6-6 device would have 6 connecting points in the top and base platforms. Figure 5-1. 3-3 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 (z-axis) 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 Plucker 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 (A), which is defined as the ratio of the J determinant to the maximum value. The significance between this 3-3 manipulator research and tensegrity is the assumption that there is a correlation between the stability of a 6-strut platform and a 3strut 3-tie tensegrity structure. If true this would greatly improve the stability prediction possibilities for deployable antennas based on tensegrity. As described in the abstract

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33 paragraph above the quality index (A) is the ratio of the determinant of J to the maximum possible value of the determinant of J. The dimensionless quality index is defined by A = _/ d e tJ _/ l det J i m (5-1) In later chapters this same approach applied here for the J matrix of the 3-3 platform will be used for calculating that of the 6-6 tensegrity structure For the later case the lines of the connecting points are defined by a 6x 12 matrix and will require additional mathematic manipulation. In this case a 6x6 matrix defines the lines of the 3-3 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 0 } B(a a (5-2) b --h} EB ( 0 b b -2 3 -h ) (5-3) The Grassmann method for calculating the Plucker coordinates is now applied to the 3-3 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 ana describing the connecting lines

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34 [ a-b a+b h ah ah ab] S1= -2 2 [ a 2b-a -h ; ah ah ab] S2 = -2 2 [ a a-2b h ah ah ab] S3 = -2 2 [ a-b a+b ah ah ab] S4= -h; 2 2 [b b-2a h ah 0 ab] S5 = 2 [b 2a-b -h; ah ab] s6 = 2 0 which yields the matrix for this system of The normalization divisor is the same for each leg (they are the same length) therefore = )L' + M + N ab+ b + 3h 2 ) and the expansion of the determinant yields Dividing above and below by h 3 yields l det J I = ,, ( a 2 -ab+ b 2 J.:i 4 ---+h 3h (5-4) (5-5) (5-6) (5-7) 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

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35 equating to zero to obtain a maximum value for det J yields the following expression for h (5-8) If we now select values for a and b (5-7) yields the value hm for det J to be a maximum. (5-9) Further we now determine the ratio y=b/a to yield a maximum absolute value l detJ I m Substituting b= ya in Equation 5-7 yields (5-10) To get the absolute maximum value of this determinant the derivative with respect toy is taken which yields: 1 (1-3-J = 0 y2 y b y = = 2 a Substituting this result in (5-8) gives: h = 1 a (5-11) (5-12) This work shows some similarity to the valu s to b d rived for th 6-6 platform Th original quality index equation reduces to a function of (platform height) I (platform height at the maximum index)

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36 (5-13) The resulting quality index plots for this 3-3 structure are found in Figures 5-2 through 5-6. In Figure 5-2 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 45 locations. y 2 --.-----------+----------, X -2 --+-------r----+-----,--------1 -2 -1 0 2 Figure 5-2 Coplanar translation of Platform from Central Location: Contours of Quality Index

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1 2 1 0 >< Cl) 0 8 ti :,... 0 6 (1J :::i 0 0 4 0 2 0 0 9 0 1 2 1 0 X QJ tJ 0 8 >, 0 6 ro ::::, a 0 4 0 2 0 0 6 0 37 6 0 3 0 0 3 0 Ro ta tio n 0 Figure 5-3. Rotation of Platform About Z-axis 4 0 2 0 0 Rotation 0 2 0 Figure 5-4. Rotation of Platform About X a x i 6 0 9 0 4 0 6 0

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1 2 1 0 X (I) "O 0 8 >, 0 6 ro :::J a 0 4 0 2 0 0 6 0 4 0 2 0 38 0 Rotation 0 2 0 Figure 5-5. Rotation of Platform About Y-axis 4 0 6 0 As expected rotations about the z axis 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 xand y-axes over 20. This could be valuable for antenna repainting without using an antenna gimbal. Figure 5-6 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

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1 200 1 1000 r---0 800 ,< 0 600 0.400 0 200 +---39 h / hm Figure 5-6. Quality Index as a Function of the Height Ratio 4-4 Solution The 4-4 parallel platform (Figure 57) is a square anti-prism. The calculations of the 4-4 quality index are similar to those for the 3-3 platform; however because the 4-4 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 3-3 platform A is defined as the ratio of the Jacobian determinant to the maximized J determinant. E i Y I I --X F Figur 57 Th 4-4 Parall I Platform (plan )

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40 (5-14) From the Cauchy-Binet theorem it can be shown that detJ J T = b.f + b.~ + ... + b.~ Each b. is the determinant of a 6x6 submatrix of the 6x8 matrix. It is clear that (5-14) reduces to (5-1) for the 6x6 matrix. This method can be used for any 6xn matrix. As with the 3-3 platform, the determinant is calculated. As shown in the figure, the value for the side of the platform (moving plane) is a. Similarly; bis 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(o .fia h} B( ~a 0 h} c(o .fia h} n(~a 0 h} ---2 2 (5-15) E(-~ b o} F(~ b o} o(~ b o} H(-~ b o: 2 2 2 2 Therefore, the Jacobian matrix is b b 2a-b b b 2a-b + b 2 2 2 2 2 2 2 2a+ b + b b 2a-b b b 2 2 2 2 2 2 2 J= 1 h h h h h h h h (5-16) e 6 bh bh bh bh bh bh bh bh 2 2 2 2 2 2 2 2 bh bh bh bh bh bh bh bh 2 2 2 2 2 2 2 2 2ab 2ab ------4 4 4 4 4 4 4 4 It follows that .J detJJ T is given by 3 3 3 = 32-vL.a b h (a 2 + b 2 + 2h 2 ) (5-17)

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41 By following the same procedure as used for the 3-3 parallel platform the key 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 b y h 3 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. (5-18) Again, as presented in the 3-3 analysis, this maximum value for h is included in (5-17) to provide the maximum determinant. (5 -19 ) To determine the ratio y=b/a for the maximum expression for (5-19), b=ya is substituted The numerator and denominator are also both divided by y3a 3 (5-20) To get the maximum value of this determinant the deri va tive with respect toy is taken. This yields the ratio between a b and h. a (5-21)

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CHAPTER 6. 6-6 DESIGN 6-6 Introduction The 6-6 in-parallel platform (a hexagonal anti-prism) 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 4-4 platform, the Cauchy-Binet theorem is used to determine the index. Once the mathematics is determined further attention will be applied to antenna design. Sketch Figure 6-1 presents the 6-6 in-parallel 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 6-1. A 6-6 Parallel Platform (Hexagonal Anti-Prism) 42

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43 A plan view of the 6-6 parallel (redundant) platform is shown in Figure 6-2. Double lines depict the base and top platform outlines. Heavy lines depict the connector s. The base coordinates are G A through G F; the platform coordinates are A through F The first segment is S 1 connecting points G A (base) and A (platform) ; the last segment is S 1 2 connecting points G A and F. The base coordinates are all fixed and the x-y-z coordinate system is located in the base with the x-y plane in the base plane. Hence the base coordinates are G e Top Platform a Figure 6-2. A Plan View for the 6-6 Parallel Platform (Hexa g onal Anti-Pri s m ) b 2 b 2 o] Gc[o b o] (6 -1 )

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44 (6-2) The coordinates for the top platform vertices at the central position are (6-3) where h is the height of the top platform above the base. A[a O h] B[ 1 h] c[1 ~a h] o[a o h] E[-1 h] F[ 1 ~a h] (6-3) Applying Grassmann's method (see Chapter 4) to obtain the line coordinates yields the following 12 arrays. 1 S5[Gc C]r S7[Go o] [: J3b 2 a J3b 2 a 2 b 2 Fa 2 0 b a Fa 2 2 b 2 -a 0 h 1 b 0 2 2 1 a ha h 2 2 b hol Fa 2 2 a 0 b a Fa 2 1 b 0 1 Ss [GE o]: [: S1o[Gr E] [: 1 2 2 a ha h 2 J3b 2 -a 0 a 2 J3b 2 a 2 -b Fa 2 b 2 0 3a h 2 (6 -4 )

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45 ounting the 2x2 det erm inant s (see C hapt er 4) y i e ld s th e [L M, N; P Q R] lin e coordinates for each of the twelve legs The normali ze d line coordinates were found b y di vi ding the calculated value by the nominal lengths of the legs for the central po s ition (6 -5 ) Evaluating the Jacobian The J matri x, comprised of the line coordinates for the twel ve le gs i s a 6x 1 2 array. a -a -a+ 3b b b -2 b 2h 2h 2h 2h 2h 2h 2'2 e 2 bh bh bh 2bh 2bh bh 0 0 ab -a b ab -a b ab -a b (6 -6 ) -2a + -a a b b + b + 2b + 2b + b 2h 2 h 2h 2h 2h 2 h bh bh bh -2 bh 2 h bh 0 -2b h 3 bh ab -a b ab -a b ab -a b

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46 JT is therefore the transpose (a 12x6 matrix). 2a3b b 2h -bh ab -b 2h bh -ab 2h bh ab a 2h 2bh 0 -ab -a 2h 2bh 0 ab 1 2h bh -ab 2 12 .e 12 (6-7) -b 2h bh ab -2a + b 2h -bh -ab 2h -bh ab -a + 2b 2h -2bh 0 -ab a + 2b 2h -2bh 0 ab 2h -bh -ab Optimization Solution Lee et al. [ I 998] developed the optimization method for the 3-3 and 4-4 platforms. The method for calculating the optimization value for the 6-6 J matrix (non-symmetric) is an extension of the 4-4 platform solution. The quality index A is given by For this example det J J T is calculated. 3 3 3 det JJ T = 54 a b h (a 2 + b 2 + h 2 ) (6-8) (6-9)

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47 As with the 4-4 parallel platform calculation the maximum height (h) must be found. To find this expression, the numerator and denominator of (6-9) 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. (6-10) As with the 4-4 analysis this maximum value for his included in (6-9) to provide the maximum determinant. 3 3 J JT = 54 a b et m m 8 3 (a 2 b 2 F This yields the A value ( quality index) as a function of a and b. 3 A= = 8h 3 (a 2 b 2 p (a 2 + b 2 + h 2 f (6-11) (6-12) This index (A) is a value between zero (0) and one (1) which represents the stability of the structure. As with the 4-4 structure, the ratio y=b/a which represents the parameter ratio at the maximum quality index is determined by substituting for b=ya. 3 3 3 Jd J 1 r = 54 a y a et m m 3 8 (a 2 +(ya)2 P (6-13) Again the numerator and denominator are both divided by y3a 3 (6 -14)

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48 By differentiating the denominator w ith respect toy the maximum and minimum va lues are determined. This yields the solution for the most stable geometry for the 6-6 platform. The vanishing of the first bracket of the right side of the equation yields imaginary solution whilst the second bracket yields 2 b y= = a h = and b = 2a Variable Screw Motion on the Z-Axis (6 -15 ) (6 -16 ) (6-17) 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. (6 -18 ) (6 -19 ) 0 1 XB = rcos(~z + 60 ) = r( cos~ 2 -sm~ 2 ) 2 2 (6-20) (6-21) (6 22) (6 23)

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49 X 0 = rcos(~ 2 + 180) = -rcos~ 2 (6-24) Y 0 = rsin(~ 2 + 180) = -rsin~ 2 (6-25) XE= rcos(~ 2 + 240) = -r(_!_cos~ 2 sin~ 2 ) 2 2 (6-26) (6-27) (6-28) (6-29) 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. (6-30) 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 Plucker line coordinates are calculated using the Grassmann principle by counting the 2 x 2 determinants of each of the 2 x 4 arrays. :] (6-31)

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50 B]: [: b :l B]: [: 0 b :1 S3[Gs S4[Ge 2 2 (6-32) Xs YB XB Ya c] [: 0 b :1 c] [: b :l Ss[Ge sdGo 2 2 (6-33) Xe Ye Xe Ye o] [: b :l o] [: b :l S7[Go Sg[GE 2 2 2 2 (6-34) Xo Yo Xo Yo E]t b :l E]: [: 0 -b 01 S9[GE S1o[GF 2 2 (6-35) XE YE XE YE h F] [: 0 -b 01 F] [: b :l S1 i(GF S12[GA 2 2 (6-36) XF YF h XF YF The Pl ticker coordinates are defined by the 2x2 determinants of these 2x4 arrays. 'T [( ( b) h bh A+ XA )j S1 = XA 2 YA +2 (6-37) 2 2 2 'T [[ ( b) h bh -XA )j (6-38) S2 = XA 2 YA -2 -' 2 2 T [ [ J ( b) bh -xB)j (6-39) S3 = XB 2 YB -2 h; -2 2 A T [ S4 = Xs (Ys b) h bh 0 -bX 8 ] (6-40) A T [ S5 = Xe (Ye b) h bh 0 bXe] (6-41)

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51 SJ = [ [Xe+ ~b J (Ye-%) h bh 3bh +Xe)] -(6-42) 2 2 T [[ ( b) h bh +Xo)] S7 = Xo+ 2 Yo-2 -(6-43) 2 2 l = [ [ x O + ~b J (Yo+%) h bh -Xo )] -(6 -44 ) 2 2 T [[ ( bJ h bh -xE)] S9 = XE + 2 YE +2 -(6-45) 2 2 T [ S10 = XE (YE+b) h; -bh 0 bXE] (6 -46 ) T [ S11 = XF (Y F + b) h ; -bh 0 bXF] (6-47) 'T [[ ( b J h bh + XF l] S12 = XF2 YF+2 -(6 -48 ) 2 2

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52 This yields the transpose of the Jacobian matrix. ( XA ~b] (YA+%] h bh 3bh +XA) -2 2 2 ( XA ~b] (YA-%] h bh -XA) -2 2 2 ( Xs~b] ( Ys -%] h bh -Xs) -2 2 2 Xs (Ys b) h bh 0 -bX 8 Xe (Ye b) h bh 0 -bXe (Xe+ ~b] (Ye-%] h bh +Xe) JT = 2 2 2 (6-49) [ Xo + ~b] (Yo-%] h bh 2 2 2 [ Xo + ~b] (Yo+%] h bh -XD) 2 2 2 (XE+ ~b] (YE+%] h bh 3bh _!?_( 3YE -XE) 2 2 2 XE (YE+ b) h -bh 0 bXE XF (YF + b) h -bh 0 bXF ( XF ~b] ( YF + %] h bh + XF) -2 2 2 The first three of the six Plucker coordinates define the length of the leg. The odd numbered legs for this structure are the same length. I f o =[L 2 +M 2 +N 2 ] 2 0 0 0 I l ( r ( b r 2 r = XA 2 + YA +2 +h I 2 3b 2 b 2 l 2 2 y = XA + 4 + YA+ bYA + 4 +h (6-50)

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53 I = r r 2 cos 2 ~z 3brcos ~z + 3 2 + r 2 sin 2 ~z + br sin ~z + b: + h 2 r I = [r 2 + br(sin ~z cos~ 2 )+ b 2 + h 2 ]2 I = [a 2 + ab(sin ~z cos~ 2 )+ b 2 + h 2 ] 2 Similarly lengths of the even numbered legs are equal. 1 = [a 2 ab(sin~ 2 + cos~z)+ b 2 + h 2 ] 2 Lee et al. [1998] used the following notation to describe the screw motion (6-51) (6-52) (6 -53) This notation describes an incremental change in leg length as a product of the normalized line coordinates (J T) and the platform incremental change (~x, ~8 etc.). To normalize the leg coordinates each value is divided by the instantaneous leg lengths. T "* T A S A 8 f =S 8D= 1 8D I I f I (6-54) Calculating the summation of the individual coordinates shows that all the values are zero except for N and R. 3 b +-+ 2 = rco s "' --+ r cos"' --s in"' --r cos"' + in"' rco .+. ( I J ( I 3 J 't' z 2 2 't' z 2 't' z 2 2 't' z 2 't' L 't' z + -r cos"' sin"' + + r cos"' + s in"' = 0 ( I J ( I 3 J 2 2 't' z 2 't' z 2 2 't' z 2 't' z

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54 b b b b M1 + M3 +Ms+ M7 + M9 + M, I= YA+ + Ys + Ye -b+ Yo+ YE+ + YF + b 2 2 2 2 b ( 1 .J3. J b ( I .J3 J b = rsm + +r cos + sin -r cos -sin -rs1n z 2 2 z 2 z 2 2 z 2 z z -~ r ( _!_cosJ. + .fj sin J. J + + r ( _!_cosJ. .fj sin J. J + b = 0 2 2 'l' z 2 'l'z 2 2 'l'z 2 'l' z N 1 + N 3 + N s + N 7 + N 9 + N 1 1 = 6h bh bh bh bh P1 + p,., + Ps + P7 + P9+P11 = + + bh + + -bh = 0 j 2 2 2 2 Q1 +Q,., +Qs +Q7 +Q9 +Q11 = ------+O+--+--+O = 0 j 2 2 2 2 +br(_!_cos 2 +isin 2 ) + +cos 2 ) 2 2 2 br [( ,1,. 3 ,1,. J ( I ,1,. ,1,. J ] b ( I ,1,. ,1,. J 2 2Stn 'l' z + 2COS'l' z 2COS'l'z 2Stn 'l' z + r 2sm 'l'z + 2COS'l'z = 3br(FJ sin 2 + cos 2 ) The second pair of legs sum similarly. L2 + L4 + L6 + Lg + L10 + L12 = 0 M2 +M 4 +M6 +Mg +M10 +M12 =0 P2 + P4 + P6 +Pg+ Pio+ P12 = 0 Q2 + Q4 + Q6 + Qg + Q10 + Q12 = 0 N 2 + N 4 + N6 +Ng+ N10 + N12 = 6h R 2 +R 4 +R 6 +Rg +R 10 +R 12 +cos~ 2 ) (6-56) (6 -57 ) (6 -58 ) (6-59) (6-60) (6 -61 ) (6-62)

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55 dding 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. 610810 =(N1 +N3 +N5 +N7 +N9 +N11)8z+(R1 +R 3 + Rs + R7 + R9 + R11)8~ 2 = 6zoz + sin ~z + cos~ 2 ~~ z ( 6 63 ) (6-64) The even leg calculation yields a similar result. (6-65) 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. The pitch is defined by the ratio of linear z change to rotation about the z-axis 8z p = 8~z This yields the pitch equation. 8z br [ r::; ] p = = -v3 Slll~z + COS~z 8~ 2 2z (6-66) (6-67) (6-68) The subsequent integration yields the z calculation. Thi proves that the odd numb r d struts can be commanded to yield a pitch motion (z and 0z motion ar coupled)

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56 2 br z [ lf zoz = f sin~ 2 + cos~ 2 p~ 2 zo 2 0 (6-69) (670) Equation (6-70) can be modified (a=r and z 0 =0) to define the square of the platform height. Therefore, the platform height (z) is the root of (6-71). 1 z = +1)} 2 (671) (672) This result shows that for a given twist about the z-axis ( ~z) there is a corresponding displacement along the z-axis defined by a finite screw (p=z/~z) as shown. z2 0 5 0 I -0 5 -1 -1 5 -2 -2 5 -3 -3 5 -4 Figure 6-3. The Pitch Relationship

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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 six-degree 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 semi-precision mesh reflector is presented. Strut Design In order to deploy the struts from a stowed position the end points of the stowage-to deployment plan must be defined. Figure 7-1 present a nominal 15-m t r (tip to tip) deploy d surface with six struts. This first p ition i con ider d th tartin p iti n (a = O) according to Kenner (1976). The ub qu nt sk tch s ho r ( a = 60). The strut lengths are hown incr a ing fi r implicit but an a tu a l d i n uld 57

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58 show the upper surface approaching the lower surface as the struts rotated to the tensegrity position. Figure 7-1. 6-6 Structure Rotated from a=0 to a=60 (Tensegrity) 14 m I 7 m. il Figure 7-2. Dimensions for Model Tensegrity Antenna

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59 Based on these design assumptions the structure (Figure 7-2) would have the v alues found in Table 7-1. Table 7-1. Deployable Tensegrity Design Values Design Parameter NOMINAL 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 7-3 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.

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Taurus (Orbital) 60 a 00 0 a If) 0.5 m. Extended Delta (Boeing) 2.7 m. a Figure 7-3. 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 7-4 shows a simple hinge design which could have an over-center 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 7-5 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

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61 efficiency increase this could become a viable option. Figure 7-6 pr e sents thi s configuration which would encourage tapered diameter struts which impro e the specific stiffness of a complete system Inflatable Struts: A very different approach but one that has been gaining fa or with the space structures design community is inflatable s pars The l e ader s 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 w eight of the deployment system is comparable to the three mechanical deplo y ment 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. Fi g ur 7-4. oldin g Hin g D 1 g n

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62 Figure 7-5. Sliding Coupling Design '-+-1 1 ...__ ---=-----=-----=-----=-----=-----=----=:!..... 1 1 Figure 7-6. 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.

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63 Table 7-2. Strut Deployment Trade Study Strut Deployment Advantages Disadvantages Design Folding Design history Potential stiffness nonDesign relevance to other linearities industries Potential hinge surface galling Moderate deployment forces Locking hardware required Sliding Minimal deployment forces Potential bending stiffness nonPositive locking position linearities Limited design history Potential contact surfaces galling Telescoping Compact packaging Requires interference fittings at Minimal stiffness nondeployment 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 deployment. Stowed Ties: By simply folding the ties along the struts (Figure 7-7) th can b released by force restraints which are highly sensitive and as the load r a h a predetermined value will release the ties Elastic ties would a th reel to take up the slack but the disadvantage i xtrem load in th prior to deployment. This could be requir d for month

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64 Figure 7-7. 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 7-8) This added hardware (potentially one motor per strut) increases complexity weight and therefore cost. Figure 7-8 Reel Ties A trade study for these approaches is presented below

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65 Tabl 7-3. trut / Tie Trade Study Strut/Tie Interaction Advantages Disadvantages Design tow d Ties ( cord) High stiffness Can only be used for Minimal Creep planar ties due to elasticity needs tow d Ties (elastic) Ease of stowage HIGH STOWAGE LOADS Reel Ties ( cord) Clean snag-free 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 cord ties must be stowed to ensure deployment success. This issue will be addressed further in Chapter 8. Deployment Scheme Figure 7-9 presents a potential deployment scheme. The requirements for thi operation are primarily low shock load and continuous motion. Despite the inher nt If deploying nature of tensegrity structures they cannot be allowed to spring into po ition for fear of introducing high shock and vibration loading into th th ha deployed changing tension in th ti and therefore po ition of th trut an a lt r surface accuracy.

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66 I Extensio o)l seuarati i> I Release > I Deployment and Surface Adjustment I Figure 7 9. 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 non-linear analysis and morphology Despite his clear

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67 focus on the engineering aspects of tensegrity 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 3-3 octahedron tensegrity is an excellent example of a spatial structure. He has developed multiple tensegrity structure designs, which solve some of the toughest curved-surface 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 cable-strut systems as an extension of tensegrity. Reciprocal prisms (RP) and crystal-cell pyramidal (CP) grids which technically exclude tensegrity systems are the basis for his space frame applications He developed a hierarchy of feasible cable-strut 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 double-layer 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 cone rn with these kinematic systems and recent system developments have required even higher precision from much lighter structures. By developing the mathematics for cab! constrained nodes You [ 1997] has be n abl to very accurate I y model th po i ti n f mesh antenna surfaces including proven experimental results. Studies in th anal i f mechanisms [Calladine and Pellegrino 1991 ] folding concepts for fle ibl but olid surface reflectors [Tibbalds et al. 1998] and shape control bas don tr anal

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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 semi-solid 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. (l 999a)] 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 gas-filled shock absorbers would allow a self-deploying 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

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69 al o ugg ted that other forms of deployment control should be con idered due to the high cost of space qualification for these subsystems. The viable options presented included : spiral prings pneumatic cylinders and compression springs ince 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 (l)] suggesting that a controlled sequence could be determined to ensure safe low transient force deployment. Cx +Kx = 0 (7-1) Deployment Stability Issues The calculations for the 3-3 design which were presented in Chapter 3 (Parallel Platform Results) suggest that there is a singularity at the tensegrity position. Figure 7-10 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 O at the Central Position and increases as the platform rotates counterclockwise The angle a is equal to O 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 Align e d Position are not tabl a th tie are in compression. The Tensegrity Po ition is a stable critical point. Thi ugg that the design has in tantaneous mobility and any minor perturbation to the tructur hil not necessarily causing instability would provide ufficient energy too cillat th antenna enough to d grade antenna performanc

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70 E Central Position F E Aligned Position F ~=0 a=-60 ~=60 a=0 E F E F Tensegrity Position (singularity) Crossover Position (interference) ~=90 a=J0O ~=120 a=60 Figure 7-10. Octahedron Configurations To improve the design and stability of the tensegrity structure while not affecting the self-deployability another set of ties is added between the vertex of the base and the opposite vertex of the platform (Figure 7-11).

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71 y X E F Figure 7-11. Redundant 3-3 Structure This results in four ties at the end of each strut versus the three in the original design. Again, the angles~ and a represent the works of CIMAR and Kenner respectively. Figure 7-12 presents the rotations from the Central Position through the Aligned and Tensegrity Positions, to the Crossover Position where the struts intersect.

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72 E F E F Central Position Aligned Position ~=0 a=-60 ~=60 a=0 E F E F Tensegrit y Position (singularity ) Crossover Position (interference) ~ = 90 a=30 ~=120 a=60 Figure 7-12 Redundant Octahedron Configurations The mathematics to calculate this 3-3+ structure is similar that for the 4-4 and 6-6 structures in that the Cauchy-Binet theorem is employed. Because there are now nine (9 ) connections between the platform and the base the resultant J is a 6 x 9 matrix. ( 7-2 )

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73 Therefore JT is a 9x6 matrix. S1 S 2 S3 S4 JT = Ss (7-3) s6 S7 s& S9 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 7-13. The quality index remains relatively constant as the platform rotates through 120 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 3-3 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 3-3 design A. = 0 at a = 30 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 deplo y abl e antenna i couple to the end and midpoints of the strut exten s ion of these tructural m mb r could alter the surface of the antenna ther b y p rforming arious or simultan ou mission tasks. If this were true the same ant nna r fl ctor could b u d to communicat with more than on location.

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74 Redundant Octahedron 1 0 -1 / A -2 -3 I \ j Statdard Octahedron \ j I \ -300 -200 -100 0 \ 100 Cl 30 0 200 300 Figure 7-13. Quality Index vs. Rotation About the Vertical Axis

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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 8-1 ). With this method a 15-meter diameter antenna would have a stowed package volume of approximately 4-meter height and 4-meter diameter. This extra fold along the spar length greatly increases the material content complexity of the structure and touch labor to assemble the system. I 15 m I ~ 4m I I M UI ~ 4 m Figure 8-1 Current Deployable Antenna Design This chapter addresses the final goal for this research: a study of th ten egrit structure parameters This approach will incr a e the efficiency of th to d packag by maximizing the use of the spars for the ant nna and not ju t the tructur In hapt r 6 at the maximum 6-6 quality index (Central Po ition) the h ight h a qual t 75

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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 15-meter 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 ( 6-6 Design) the diameter of the deployed surface (2a for the 6-6 design) and the strut length (I). The purpose of this analysis is to design a stable structure while minimizing the strut length for the 15-meter antenna. The 6-6 design is the basis for the deployable design. Table 8-1 presents the geometric relationships for the three candidate structures (3-3 4-4 and 6-6). Table 8-1. The Three Tensegrity Structure Designs Considered Design # of Struts # of Ties (total) Tip-to-Tip Diameter ,., ,., 3 9 a .)_) 4-4 4 12 -fia 6-6 6 18 2a 3-3 Optimization The tensegrity position for the 3-3 structure as defined in Chapter 7 is at ~ =90 and a=30 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

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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 3 -3 structure the Central Position is defined as ~=0 or a = -60 As presented in Chapter 5 the determinant of the J matrix and the determinant of the maximum of this 27a 3 b 3 matrix (J m) are I l l = 3 and 1 1 m I = 3 respectively. 4 (a 2 -a 3 b+b 2 J ( 2 2\::l + h 2 32 a ab+ b }' 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 2 ab+ b 2 ) found by taking the partial derivative .!!__ and setting it equal to 3 ah 1 1 1 zero (a calculus inflection point) Calculating the quality index A= l l m I yields: 3 &13h 3 a2 -ab+ b 2 2 /l.,= ---'----...L...{ a2 -ab+ b2 ]3 ---+h2 3 (8-1)

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78 As the lim (A) which means that the base reduces to a point, the Equation 1 reduces to b O A=---9(~ + ~)3 (8-2) 3h a Rooney et al. [1999] refers to this design as the tensegrity pyramid ". As a first-design the ratio a/h=l is chosen This further reduces the equation to (8-3) which is an acceptable quality index (optimum is A=l.O). 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 (8-4) and taking the derivative !_ of the denominator the maximum values for the quality &y index (the denominator equals zero) is found at y = "'0.58. Figure 8-2 presents the plot of the quality index (A) vs. the ratio values ( y = h ). At this value of y the quality a index has a relative value of 1 0

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79 Quality Inde x 1 50 ~---------------------~ 1.00 -0 .SO -1---7'~~~ ... --,;.; 1-~ "':/>JI-1 50 ~---------------------~ y Figure 8-2. A. vs. r(: = I J 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=O), 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 off-axis moment (see Figure 8-3). Add itionall y it is impractical to connect an antenna structure at a point as moment loads wo uld approach infinity.

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80 b
O b=O Figure 8-3 Reduction of the Base to Zero Based on these observations a compromised geometry is necessary To this end the base should be minimized and the y = h ratio chosen for the maximized quality index a Table 8-2 presents the results of three choices of Base Planar Tie length (b) with maximized quality index. Figures 8-4 8-5 and 8-6 present the curves for the b= .! .! 2 4 a and cases respectively. 8

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81 Ta bl e 8-2. Quality Index for b = ~ ~ and Cases 2 4 8 b 1 1 1 hm 1 1 m l A y at Am ax 3 3 1 a ,., a 3 Ur +rJ 2 ( I h J = 0.5a 0.2a 0.50 32 + 2 32 4h a 2 a 3 ( 13 r 256( E_ +J:_ r a~ 0.52a 2 7a 0.02a3 4 4 3 72 +y 0 52 48h a 2 48y 0.55a 0.002a 3 a 2048( _I!_ + _!:_ r 64 ( 57 + r 8 8 0.54 19 2 h a 2 1 92y y Qualit y Inde x (a/2) 1.50 1 .00 -----0 50 -1-1 00 1 .50 y

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82 Qualit y Index (a/4) 1.50 .. ----1 I 1.00 I 0 50 ,-< 0 00 l;:)I;:) -0 SO -1 00 -1.5 0 ------I I y Figure 8-5. A vs. r( b = J Quality Index (a/8) 1 50 I ----~-. I i 0.50 ,-< 0 00 l;:)I;:) o SIO 1 00 -1 50 y Figure 8-6 A vs. r( b = f J

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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. Stern [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 3-3 design thereby improving the quality index For the 4-4 and 6-6 structures the index approaches 1 0 for virtually any position. Figure 87 presents the design for the 3-3 structure. In this case the 'A, varies only 25% from 0.75 to 1.0 (as shown in Figure 7-13). (a) (b) ( c ) Figure 8-7. Reduction of the Base to Zero (Redundant Octah dron) a) b < a ; b) b => 0 ; c) b = 0

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84 4-4 Optimization As presented in Chapter 4 the Jacobian (J) for the 4-4 structure is a 6x8 matrix and an understanding of the Cauchy-Binet Theorem aids in obtaining the quality index. As previously presented the numerator for the quality index (A) reduces to a 3 b 3 h 3 det JJ T = ( J The denominator represents the maximum a 2 ab+ b 2 + 2h 2 possible vale for the numerator was found by using h=O. This value is 3 3 -J detJ m J = 2 a b The height (h ) which is used to find the (a 2 ab + b 2 )3 2 denominator is hm = 2 ab + b 2 ) Again following the work in Chapter 4 the quality index is therefore 3 A= ab+ b 2 ) 2 (a 2 ab+ b 2 + 2h 2 ) As the lim (A) this reduces to b O A=----(a 2 + 2h 2 ) By using y = h the equation reduces further to a A= h 3 = = [ a+
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85 with a maximum A at y = Jz "'0.71. Figure 8-8 plots A vs. y Quality Inde x 1 .50 -I SO ~-----------------------~ y Figure 8-8. A vs. y ( y =:) for the Square Anti-prism Quality Inde x (a/2) 4 00 3 00 2 00 +---1.0 0 -+---------,< 0 0 0 r-;;.;;;;;i;;;;;.;:;;::::::: .---,-,-,,-, -2 00 -3 00 -4 00 y '-----~-Figure 8-9. A v --r( b = ~) for the q uare Anti-prism

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86 Similarly the equations for b equals ~ and are presented in Table 8-3 Figure 2 4 8 8-9 presents the first A vs. y plot. The second and third cases are similar, but it is obvious that they value at Amax changes significantly between a/2 and a/8. b a 2 a 4 a 8 a a a Table 8-3."' at b=and I 2 4' 8 A (y=h/a) y at Amax ( 5 1 J 3 ~ 4 1 (5 1 J 4 0.52 l l J 2 16 ( 2y + _!_ (!_2 l JJ3 y 16 _ l J 2 64 ( 2 y + _!_(65 _ l JJ 3 y 64 2 ''/ 65 1 -64 2 '6-6 Optimization 0 60 0.65 The 6-6 tensegrity design is the basis for this new class of deployable antenna structures. The calculations are similar to those for the 4-4 to solve the 6x 12 J matrix ~-54a 3 b 3 h 3 The numerator for A taken from Chapter 4 is
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87 T 54a 3 b 3 detJ m J m = ------3 1 This h value is h m = a 2 ab+ b 2 The quality index is therefore 'A=8h 3 (a 2 3ab+b 2 }3 2 (a 2 3 ab + b 2 + h 2 ) As the lim (A) this reduces to b O 8 a 3 h 3 A=---(a 2 + h 2) By using y = h the equation reduces further to a 8h 3 8 8 'A= (a+ h:r = (~ +~r = (y+~J3 with a maximum A at y = 1. Figure 8-10 plots A vs. y. (8-8) (8-9) (8-10) Similarly the equations for b equals ~ and are presented in Table 8-4. Figure 2 4 8 8-11 presents the A vs. y plot for the~ case. The second and third cases are similar 2 Again they at Amax values vary greatly as b is reduced from a/2 to a/8 K ping the ork of Stern [1999] in mind to minimize the ti force b=a/4 i cho en a a c mpr m1 Using this chosen ratio h/a=0.79 b/a=0.25 and therefore b/h=0.32

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88 ---------, Quality Index I 1.50 1.00 0 50 ,< 0 00 -0.5-0 t>-~ I\~ ,.? 0~ 1.00 1.50 y Figure 8-10. A vs y ( y = : ) for the Hexagonal Anti-prism Quality Index (a/2) 2 50 2 00 -+--------1.50 -+---------------1----"" c--------------t 1.00 ________________ ,_ ____ __,__ _____ ------t 0 50 +--------------1 --------=--~ ----l ,< 0 00 +--,~~~~-~-~-~~-1.50 -+------------1--------------------t -2 00 -+------------"' '-----------------------1 -2 50 y Figure 8-11. A vs. y ( b = ~) for the Hexagonal Anti-prism

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89 a a a Table 8-4. A and y for b = and 2' 4 8 b "A (y=h/a) Y at Amax 3 s[~r a [~r ~062 2 (r+H~JJ sU 2 a 16 4 ( _12 2 0 79 4 (r+H:~ '7JJ3 16 4 s( 65 a 64 8 ( 65 J 2 0.89 8 (r+H::~Jr 64 8

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CHAPTER 9 CONCLUSIONS The requirements process introduced in Chapter I 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 90

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91 deploy (preferably self deploy) and stow to allow placement in the space environment. fter de elopment of the 3-3 4-4 and 6-6 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 6-6 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 tie above th ba ic tensegrity design (two ties from each base vertex). A mathematic analysis of th qualit index for these augmented 3-3 and 4-4 structure howed a marked impro em nt in th indice For the 6-6 design th ba ic t n grit d 1gn ith 12 platfi rm a nn ti n

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92 (Figure 9-1 a) is augmented to a total of 18 (Figure 9-1 b ) 24 (Figure 9-2a) 30 (Figure 92b) and 36 (Figure 9-3) (a) (b) Figure 9-1. Hexagonal Anti -pri sm Designs (a) Basic Tensegrity Design (12 platform/base connections); (b) Augmented Tensegrity Design (18) (a) (b) Figure 9-2 Augmented 6-6 Hexagonal Anti-prism Designs (a) Augmented Tensegrity Design (24) ; (b) Augmented Tensegrity Design (30)

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93 Figure 9-3. Augmented 6-6 Hexagonal Anti-prism Design with the Maximum (36) Number of Platform / Base Connections As presented in Chapter 7 the augmented 3-3 index (1,,, 9 ) only varies between 75 and 1.0 (Figure 9-4) with three minimum potential energy position located symmetrically about the Central Position (basic platform design position). Note that only one additional tie per vertex is required increasing the number of platform / base connections from 6 to 9 The quality index ()1, 16 ) for the 4-4 with four ties between each base vertex and the corresponding platform vertices the value varies less than 5% between the maximum and minimum. Again the number of minimum potential energy nodes is equal to the numb r of sides in the geometry and these nodes are symmetrically placed about the Central Position. The point design wi ll use one set of additional ties between the base and th platform as shown in Figure 9-1 (b ). This will implify the calculation but till impr the stability of the structure above that of for th ba ic t ns grity d ign

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94 1 0 /----..... / "' I \ (a) I \ 0 9 I '( ,1, 9 I \ 0 8 I \ / "' ;,..., 0 7 0 6 0 5 0.4 0 C: 0 3 0 2 ( d ) ( d ) :::::i 0. 0 1 (1) X 0 0 -180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 Rotation Angle (degree) Figure 9-4. A Comparison of the A. vs. Curves for the Basic 0--6) and Augmented (11.9) Tensegrity Designs for the Octahedron <"'< X (I) -0 E 1 0 L ____ -----<' /416_ 0 9 i 0 8 0 7 0 6 I ___ __ _______ _.... 0 5 ro 0.4 :::::i a 0 3 0 2 J ,,-1 Te n segr it y 0 1 (d) (d) -180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 Rotation Angle (degree) Figure 9-5. A Comparison of the A. vs. Curves for the Basic (As) and Augmented (11.16) Tensegrity Designs for the Square Anti-Prism

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95 Antenna Point Design As presented in Chapter 7 a 15-meter tip-to-tip design meets the needs of current systems. For this research this dimension is increased to 16-meters providing sufficient structure to suspend a 15-meter diameter parabolic surface within Chapter 8 has shown that the ratios between a b and h can be varied to improve the efficiency and utility of the spars while ensuring the stability ( quality index) is sufficient to form a usable structure The design that can meet the requirements of the space community while improving the subsystem efficiency is presented in Figure 9-6. To simplify the design these parameter ratios were altered slightly to h/a=0.50 b/a=0.25 and b/h=0.50 This design maintains the necessary b/a ratio while only altering the h value slightl y to flatten the cup of the antenna. This applies to the f/d ratio to be addressed later. This chosen design is only one of a family of choices which include various numbers of extra ties a/b and h/a ratios. The assumption is to avoid the instantaneous mobility issue for tensegrity-class structures ; an additional set of ties would be included. Nominally this would be one set connecting each base vertex with its corresponding platform vertex above. Figure 9-7 presents the baseline design. Note that the reflector surface suspended within the strut framework could accommodate a focus to diameter (f/d) ratio of approximately 0.3 This value is typical for deployable antennas in ser v ice toda y. A noted in the figure the distance from the parabola vertex to the edge of th e antenna structure is nominally 2-meters. For al 5-meter diameter parabolic reflector smfac th focal point (for an f/d = 0 3) would be a total of 3.75-meters abo v e the parabola e rt ex. Therefore the focal point is located at 1.75-met e rs a bo th e d g o f th e tructur hi

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96 significant because additional structure may be needed to place the feed electronics for this antenna. What changes a hexagonal anti-prism to the truncated structure on which the 6-6 tensegrity structure is based in a hat" structure above and below the platform and base. This extra structure above what is being considered the edge of the antenna structure could be used for this feed support structure. ~3~ (2b) Figure 9 6. The Proposed 6-6 Hexagonal Anti-prism Deployable Tensegrity Antenna Design Focal + ------~ .... / J2J4m "_ ...... / Structure Envelop Figure 97. The Relationship Between the Antenna Structure Envelope and the Focal Point Location

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97 Patent Disclosure Harris Corporation Melbourne Florida, in conjunction with the University of Florida Gainesville, Florida, has filed a patent disclosure on the application of tensegrity structures to self-deploying space antennas The concept of using tensegrity of structures even space structures has been suggested in technical literature over the last two decades. The specifics for applying tensegrity the concept of self-deployment and the mathematics which proves its value to the design community is the basis for this patent. This new approach has the potential to radically change the deployable structures market place reducing cost weight and complexity therefore improving the subsystem efficiency. Future Work Although current deployable antenna design approaches meet most of the goals necessary for space flight the cost and development time are still much too great. This research which addresses the application of tensegrity structures to these subsystems has proven both stability and possible special motions (screw theory based) which would meet the mission needs Future work in this area would address these special motions particularly as they affect the reflector surface which is suspended between the struts. Additionally a trade between strut deployment schemes relative to subsystem stiffness should be performed to benefit from the advances over the last decade in inflatable structures. Of all the subsystems necessary for space missions deployable antennas are potentially the least package-efficient.

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REFERENCES Ball R S (1998) "A Treatise on the Theory of Screws ," Cambridge University Press London Bottema 0 ., Roth B. (1979) "Theoretical Kinematics Dover Press New York Calladine C. Pellegrino S ., (1991 ) First-Order Infinitesimal Mechanisms ," International Journal of Solid Structures Vol. 27 No. 4 pp. 505-515. Connelly R ., and Black A. (1998) Mathematics and Tensegrity ," American Scientist Vol. 86 pp. 142-151. Duffy J. Knight B. Crane C ., Rooney J ., (1998) An Investigation of Some Special Motions of an Octahedron Manipulator Using Screw Theory ," Proceedings of the 1998 Florida Conference on Recent Advances in Robotics Duffy J. Rooney J. Knight B. Crane C. (June 1998) An Analysis of the Deployment of Tensegrity Structures Using Screw Theory ," Proceedings of Sixth International Symposium on Advances in Robot Kinematics: Analysis and Control Strobl Austria Kluwer Academic Publishers. Fuller R B. (1960) "Tensegrity ," Portfolio and Art News Annual No. 4 the Art Foundation Press Inc. New York. Fuller R.B ., (1975) Synergetics: Explorations in the Geometry of Thinking ," Macmillan Publishing New York. Gabriel J.F. (1977) Beyond the Cube: the Architecture of Space ," John Wiley and Sons Inc. New York. Hamlin G. J ., Sanderson A. C. (1998) TETROBOT A Modular Approach to Reconfigurable Parallel Robotics Kluwer Academic Publishers. Hunt K. H. ( 1990) Kinematic Geometry of Mechanisms ," Oxford Engineering Sciences Series7 Oxford University Press New York. Jessop C.M. (1969) A Treatise on the Line Complex ," Chelsea Publishing Company New York NY (1 s t publication in Cambridge UK 1903) Kaplan R. Schultz J. L. (October 23 1975) Deployable Reflector Structure ," Grumman Aerospace Corporation United States Patent Number: 4030102. 98

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99 Kawaguchi, K. Hangai Y., Pellegrino S. Furuya H. (December 1996) Shape and Stress Control Analysis of Prestressed Truss Structures ," Journal of Reinforced Plastics and Composites Vol. 15 pp. 1226 1236. Kenner H. (January 1976) Geodesic Math and How to Use It ," University of California Press Berkeley Los Angeles. Kitamura T. Yamashiro K. (August 31 1990) Extendable Mast ", Japan Aircraft Mfg. Co ., Ltd. United States Patent Number : 5085018. Knight B. (June 1998) An Analysis of Special Redundant Motions for a Square Platform ," CIMAR Paper Krishnapillai A. (1988) Deployable Structures ," United States Patent Number: 5167100. Larson W J. and Wertz J. R. (1992) Space Mission Analysis and Design ," Microcosm Inc. Los Angeles Lee, J. Duffy, J. and Hunt K.H. (1998) "A Practical Quality Index based on the Octahedral Manipulator ," The International Journal of Robotics Research Vol. 17 No. 10 pp. 1081-90. Marcus M. and Mine H (1965) "Introduction to Linear Algreba The Cauchy-Binet Theorem The Macmillan Company New York. Meserve B., (1983) Fundamental Concepts of Geometry ," Dover Press New York. Motro R. Editor (1992) Special Issue of the International Journal of Space Structures: Tensegrity Systems ," Multi-Science Publishing Co. Ltd. Vol. 7 No. 2. Motro R. (1996) "Structural Morphology of Tensegrity Systems ," International Journal of Space Structures Multi-Science Publishing Co. Ltd ., Vol. 11 Nos I & 2, pp 233-240. Natori M. (1985) Jointed Extendible Truss Beam ," Japan Aircraft Mfg. Co. Ltd. United States Patent Number: 4655022. Nelson R.A. (1983) Biaxial Scissors Fold Post Tensioned Structure ," LTV Aerospac and Defense Co. United States Patent Number: 4539786. Newman D. D. (1998) Classification and Optimization of Actuators for Space Deployment Mechanisms ," M.S.M.E. Thesis University of Alabama Tuscaloosa Onoda J. (May 3 1985) Deployable Trnss Structure ," Japan Aircraft Mfg Co., Ltd., United States Patent Number: 4745725

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BIOGRAPHICAL SKETCH During his 18-year career Mr. Knight has worked for Harris (Melbourne FL 198285 ; 1990-present) Unisys (Reston VA, 1986-88) and Lockheed (Kennedy Space Center FL 1985-86) developing electro-mechanical systems. In 1998 he had the great honor of serving as a Technology Fellow to the National Reconnaissance Office, managing a communications dissemination project. This achievement-focused organization contributed greatly to this research guiding the requirements and designs necessary to meet the space mission needs for future space systems. Currently Mr. Knight is leading an R&D effort to study the applicability of tensegrity structures to space antenna systems. Mr. Knight s post-secondary education began with math and physics undergraduate work at Florida State University which led to a Bachelor of Science in Mechanical Engineering (1982) from the University of Michigan (Ann Arbor). He was a Research Assistant to Dr. George S. Springer a national authority in composite structures. He returned to earn a Master of Science in Mechanical Engineering (1990) from North Carolina State University (Raleigh). This research under Dr. Thomas A. Dow focused on precision manufacturing design and control including assembly of the country s first nanometer-accurate testbed 103

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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy Graduate Research Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope ~nd quality, as a dissertation for the degree of Doctor of Philosophy. Carl D Crane III Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality as a dissertation for the degree of Doctor of Philosophy. Ali A. H Seireg Ebaugh Professor of Mechanical Engineering I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality as a dissertation for the degree of Doctor of Philosophy

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I certify that I have read this study and that in my opinion it conforms to acceptable tandards of scholarly presentation and is fully adequate in scope and quality, a a dis ertation for the degree of Doctor of Philosophy. Gloria~ Wiens Associate Professor of Mechanical Engineering This dissertation was submitted to the Graduate Faculty of the College of Engineering and to the Graduate School and was accepted as partial fulfillmerrt of the requirements for the degree of Doctor of Philosophy. May 2000 f .. Ohanian Dean, College of Engineering Winfred M. Phillips Dean, Graduate School

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