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 = Â¡=

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 NÂ¡ = 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 ^Â¡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

- yÂ¡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 .

XÂ£ = 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

6fÂ¡ =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 Â§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

yÂ¡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

3yÂ¡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)