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Kinematic Analysis of a Planar Tensegrity Mechanism with Pre-Stressed Springs

Material Information

Title:
Kinematic Analysis of a Planar Tensegrity Mechanism with Pre-Stressed Springs
Creator:
Vikas, Vishesh
Place of Publication:
[Gainesville, Fla.]
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (45 p.)

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering
Mechanical and Aerospace Engineering
Committee Chair:
Crane, Carl D.
Committee Members:
Dixon, Warren E.
Schueller, John K.
Graduation Date:
8/9/2008

Subjects

Subjects / Keywords:
Coordinate systems ( jstor )
Degrees of polynomials ( jstor )
Determinants ( jstor )
Kinematics ( jstor )
Matrices ( jstor )
Mechanical springs ( jstor )
Polynomials ( jstor )
Rigid structures ( jstor )
Tensegrity structures ( jstor )
Wrenches ( jstor )
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
kinematic, pre, static, tensegrity
Genre:
Electronic Thesis or Dissertation
born-digital ( sobekcm )
Mechanical Engineering thesis, M.S.

Notes

Abstract:
This thesis presents the equilibrium analysis of a planar tensegrity mechanism. The device consists of a base and top platform that are connected in parallel by one connector leg (whose length can be controlled via a prismatic joint) and two spring elements whose linear spring constants and free lengths are known. The thesis presents three cases: 1) the spring free lengths are both zero, 2) one of the spring free lengths is zero and the other is nonzero, and 3) both free lengths are nonzero. The purpose of the thesis is to show the enormous increase in complexity that results from nonzero free lengths. It is shown that six equilibrium configurations exist for Case 1, twenty equilibrium configurations exist for Case 2, and no more than sixty two configurations exist for Case 3. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (M.S.)--University of Florida, 2008.
Local:
Adviser: Crane, Carl D.
Statement of Responsibility:
by Vishesh Vikas.

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5.2 Numerical Example

Values of the given parameters were selected to be the same as in the previous

example (Sec 3.2) with the exception of the free lengths of spring 1 and 2 which are

now set to Lol = 5.1 m, LO2 = 6.6309 m. Equations 5-6, 5-7, 5-8, 5-9 are calculated

numerically. Next, the Sylvester Matrix (SBFL) WaS obtained by operations explained in

Section (5.1). Finally, Equation 5-15 is solved for xl. Observations for solutions of xl were

Circular points at infinity 26 of the solutions were equal to fi. It must be the case
that the 104 degree polynomial can be divided by (1 + xf )13. It 1S 1100 Surprising as
H1, H3, 1I, 3~ are equal to (1 + x ).

Complex Solutions 38 of the remaining 78 solutions were complex.

Real, 'extraneous Solutions 16 of the solutions were real but did not satisfy
Equations 5-2, 5-3, 5-4, 5-5.

Real, 'relevant Solutions 24 of the solutions were real and satisfied Equations 5-2,
5-3, 5-4, 5-5. For two of these solutions (Case 3 and Case 6) fl, f2 arT ZeoO.









BIOGRAPHICAL SKETCH

Vishesh Vikas was born on the 31st of May, 1983 in New Delhi, India. He attended

his high school at Delhi Public School RKE Purant, Delhi. He relieved his Bachelor in

Technology in Mechanical Engineering from Indian Institute of Technology, Guwahati in

May of 2005. After that, he worked at MAIA, INRIA Lorraine(LORIA), France. In 2007,

he joined the Center for Intelligent Machines and Robotics(CIMAR) at the University of

Florida, completing his Masters of Science degree in Mechanical Engineering in August of

2008. Upon completion of his MS, Vishesh will pursue PhD in Department of Aerospace

and Mechanical Engineering at University of Florida.









REFERENCES


[1] Fuller R. S to *i,. 1.. The Geometry of Ti,.: 1.,:I MacMillan Publishing Co., Inc.,
New York, 1975.

[2] Edmondson A. A Fuller Ex~lphonetion: The S to *i,.1..i~ Geometry of R. Buckminster
Fuller. Birkhauser, Boston, 1987.

[:3] Tobie R.S. A report on an inquiry into the existence, formation and representation of
tensile structures. Master of industrial design thesis, Pratt Institute, New York, 1976.

[4] Pugh A. An Introduction to T. 0 i ;; University of California Press, 1976.

[5] Duffy J., Yin J. and Crane C. An analysis for the design of self-deploi- .1.1.' tensegrity
and reinforced tensegrity prisms with elastic ties. International Jourmal of Robotics
and Automation. Special I~ssue on C'omplicence and C'omplicent Alechanism~s, 17, 2002.

[6] Roth B. and Whiteley W. Tensegrity frameworks. In American Mathematical Society,
editor, Transactions of the American M~athematical S .. .:. It; page 419446, 1981.

[7] Ingher D. http://www.childrenshospital.org/research/nhrtnert~tl Havard
Medical School.

[8] Hanaor A. Aspects of design of double 1.,-< c tensegrity domes. Journal of Sp~ace
Structures, 7(2):101-11:3, 1992.

[9] Hanaor A. Geometrically rigid double-] n,-< c tensegrity grids. Journal of Sp~ace
Structures, 9(4):227-2:38, 1994.

[10] Motro R. Tensegrity systems: the state of the art. Journal of Sp~ace Structures, 7(2):
75-83, 1992.

[11] Juan, S.H. and Mirats Tur, J.M. Tensegrity frameworks: Static analysis review.
Alechanista and Iafechine The ..<;; 2007. in press.

[12] Fu F. Structural behavior and design methods of tensegrity domes. Journal of
C'onstructional Steel Research, 61(1):25-35, 2005.

[1:3] Helton J., Adhikari R., Pinaud J., Skelton, R. and C'I I.. W. An introduction to the
mechanics of tensegrity structures. In IEEE, editor, Proceedings of the 40th IEEE
conference on Decision and control, page 42544258, 2001.

[14] S. Levin. The tensegrity-truss as a model for spine mechanics: Biotensegrity. Journal
of Alechanic~s in M~edicine and B..~~I J..;,i 2(:3&4)::375-388.

[15] Furuya H. Concept of deploi- .1.1., tensegrity structures in space applications. Journal
of Sp~ace Structures, 7(2):14:3151, 1992.










CHAPTER 6
CONCLUSION

The purpose of this thesis was to show the significant increase in complexity that re-

sults when springs with nonzero free lengths are incorporated in pre-stressed mechanisms.

It has been shown that six equilibrium configurations exist for the case of a simple planar

niechanisni with two springs where both springs have zero free lengths. Twenty equilib-

riunt configurations were found for the case where one of the springs had a nonzero free

length. For the case where both springs had nonzero free lengths, seventy eight solutions

sets were obtained once the circular points at infinity were disregarded. Sixteen of these

seventy eight, did not satisfy the equation set which means that the presented elimination

technique introduced extraneous roots. The remaining sixty two solutions satisfied the

equations, but two solutions in the numerical example resulted in cases where the lines

along the three legs did not intersect which is puzzling. Additional work needs to be done

before this simple case is fully understood. The approach presented here does however

bound the dimension of the solution. The goal is to extend this work to spatial devices in

order to develop a thorough understanding of the nature of these pre-stressed mechanisms.

The work can also be easily extended to study the human niusculoskeletal system and

microlevel study of cellular hardware using the cellular tensegrity theory.









K(INEMATIC ANALYSIS OF A PLANAR TENSEGRITY MECHANISM WITH
PRE-STRESSED SPRINGS


















By
VISHESH VIK(AS


A THESIS PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2008










APPENDIX B
SYLVESTER MATRIX

In mathematics, a Sylvester matrix (named after English Mathematician James

Joseph Sylvester) is a matrix associated to two polynomials that gives some information

about those polynomials. If two polynomials have a common factor, then the determinant

of the associated Sylvester Matrix is equal to zero.

Given polynomials

p(x) = axnx+ anx- ...ax+a (B-1)


q(x) = bmxm + bm-ixm-l... blx + bo (B-2)


of degrees n and m and roots asi, i = 1, 2, .. n, pi, i = 1, 2, .. m respectively. The

resultant([24]) is defined by



i= 1 j= 1

This is also given by the determinant of the corresponding Sylvester matrix. It can be

observed that for the resultant to be zero, the determinant of the Sylvester matrix should

vanish.

To construct the Sylvester matrix for the system p(x) = 0, q(x) = 0, equations of

the form Zkp(x) = 0 and Zkg(x) = 0 may be added to the system. The enlarged system

will have exactly the same solutions as the original system of two equations. Consider the










[16] Tibert A. Deplo;,rlal.1, T 00 ,li;l // Structures for Space Applications. PhD thesis, Royal
Institute of Technology, 2003. PhD Thesis.

[17] K~enner H. Geodesic Iabth and How to Use It. University of California Press, Berkeley
and Los Angeles, CA, 1976.

[18] Stern I.P. Development of design equations for self-deploi- ll-lM n-strut tensegrity
systems. Alaster's thesis, University of Florida, Gainesville, FL, 1999.

[19] K~night B.F. D. ~1 pl.;;ald.- Antenna Kinemartic~s using T i,.Uti~ Structure Design. PhD
thesis, University of Florida, Gainesville, FL, 2000.

[20] Aldrich J. Control So;.//;, .: for a C'lasms of Light and Agile Robotic T --i,;./ /l~
Structures. PhD thesis, University of California, 2004. PhD Thesis.

[21] Roberts J., Lipson H., Paul C. and F. Cuevas. Gait production in a tensegrity based
robot. In Proceedings of the 2005 International C'onference on Advanced Robotics,
2005.

[22] Crane C. and Duffy J. Kinemartic Aiel;,ims of Robot Iaftnipulators. Cambridge
University Press, March 1998.

[23] Ball R.S. A Treatise on the Theory of Screws. Cambridge University Press, 1998.

[24] Weisstein E. Resultant. From MathWorld-A Wolfram Web Resource.
http: //mathworld.wolfram. com/Resultant .html.

[25] Rao A.V. D;,n. : of Particles and Rigid Bodies: A S;,;l-/. mal.. Approach. Cam-
bridge University Press, 2nd edition, 2006.

[26] Vikas V., B~i-,t J., Crane C. and R. Roberts. K~inematic analysis of a planar
tensegrity mechanism with pre-stressed springs. Advances in Robot Kinemartic~s, 2008.










Values of xl that correspond to each of the six solutions of x2 can be determined from

observing
X1~~- -112

x o 1 0 P P P3 .(3-12)

X] Qi Q2 3 0

The corresponding value for xl is the 3rd value of the solution vector on the left side of

Equation 3-12. Unique corresponding values for yl and y2 are calculated for each value of

xl and x2 from Equation 2-23


yi = 2 tan- (xi) i = 1, 2 (3-13)


3.2 Numerical Example

The following values were selected for a numerical example

L12 = 10 m

p~z -3 m, p3y =

L45 = 5 m

p6z = -1.1990 m, p6y = -2.2790 m

L3 = 7.56 m

ki = 1.5 N/lm, Lol = 0 m

k2 = 3.7 N/lm, LO2 = 0 m

Coefficients Ai, Bi(i = 1,...9) are evaluated numerically and a sixth degree poly-

nomial in, x2 l~,;, 1S,,,,, obtained, by exadn qain311 and dividing; it, by (1 +2 x ) Si

solutions of yl, y2 arT liSted in Table 3-1. The four real solutions are shown in Figure 3-1.

The two complex solutions are shown to satisfy Equations 3-4 and 3-5.









Abstract of Thesis Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Master of Science

K(INEMATIC ANALYSIS OF A PLANAR TENSEGRITY MECHANISM WITH
PRE-STRESSED SPRINGS

By

Vishesh Vikas

August 2008

C'I .Ir~: Carl Crane
Major: Mechanical Engineering

This thesis presents the equilibrium analysis of a planar tensegrity mechanism. The

device consists of a base and top platform that are connected in parallel by one connector

leg (whose length can he controlled via a prismatic joint) and two spring elements whose

linear spring constants and free lengths are known. The thesis presents three cases: 1) the

spring free lengths are both zero, 2) one of the spring free lengths is zero and the other is

nonzero, and 3) hoth free lengths are nonzero. The purpose of the thesis is to show the

enormous increase in complexity that results front nonzero free lengths. It is shown that

six equilibrium configurations exist for Case 1, twenty equilibrium configurations exist for

Case 2, and no more than sixty two configurations exist for Case 3.


















































02 3 4 5 6 7 8 9 10

L,1


Figure 4-1. Bifurcation diagram for solution of .ri and varying parameter Lol


are eight (as numerically calculated), at some value of Lol, the number of solutions

increase to 9 or 10 also. Thus, it is possible to move from one value of .ri to another by

slowly changing the free-length.


Lol1YS Xl


__ir___C__r___~_
_r__r__r___C_

i.-.----------- _______~_________-~~--------------------
~III:I:::::II3I=~~=---;r_:_;:II::rr-
~-~~~~--~~~-~i~------------------~~____:

~~C--_-_--I'Lr:Ir=r:~~=~l~_______1I==


-----~-~---r_~_________

I/ //
_Jk:l~. --

~~r~~r~ /I


__r___rC______________
____C____________1___~__




_rC~_rC
_r___rC__
~~-I-_____~_____~_~~


_~____r--~rr---
-~~~~'-~i-I:~~---L~--c-~_~____
-----r~-~ -~-~c-`--~~------









Substitutingf Equations 2-7 to 2-12 into Equation 2-15 gives


ki(1 A )BL1 + k2~( 2 XBL2 3 Bp4

ki(1 Az)(BP2 BP5) + k2~( 2 X)B 3 BP6)


(2-16)

(2-17)


where At = Lol/dl, X2 = LO2 d2. As the problem is planar, vector Equation 2-16 is

equivalent to two scalar equations and vector Equation 2-17 is equivalent to one scalar

equation. Therefore, there are three unknowns (dl, d2, f3) and three equations. One

scalar equation in dr, d2 can be obtained from Equation 2-16 by performing a cross

product operation with BP4 to both sides of the equation to yield


ki(1 Ai)BP4 X BL1 + k2~( 2- XBP4 X BL1 = 0 .


(2-18)


It is desired to solve for y,, y2 from Equations 2-13, 2-14, 2-17 and 2-18. Substituting

Equations 2-1, 2-3 and 2-6 into these equations yields.


ki(1 Az) (c1S2L45 -- S1C2L45 + 1L12)

k2~ 2 X) (C1~as +26 C_ p3y) S1 C296m S~I', -- p3z)

ki(1 A )L12 (S1L3 + 2L45)

k2~ 2 Xa (32 81L3 + 2P6z C_;*, ) -- p3y C1L3 C2P6z S.}' )

d~ + ((2p3y 28296z 2c_; )Si + (2p3, 2c2962 + 2s_; )c1)L3
(pa 2 2 2 ) + (2p62P3y 2p36yf33m 2 + (2p62p3z + 2p6yP3y)C2

d2 L2- L2 + ((L1 2c2L5)C -, 2S1S2L45)L3 + 2c2L45L12 Ls


= (2-19)



= (2-20)



= (2-21)


0 (2-22)


The equations can now be analyzed for three different cases -

1. BOTH FREE LENGTHS ARE ZERO. (i.e., X1 = X2 = 0)

2. ONE FREE LENGTHS IS ZERO, ANOTHER NON-ZERO. (i.e., At = 0, X2 / 0)

3. BOTH FREE LENGTHS ARE NON-ZERO. (i.e., At / 0, X2 / 0)

Equations 2-19 to 2-22 are nonlinear functions of sin yl, cos yl, sin y2 and cOS y2. The

concept of 'tan-half angles', ([22]) converts these four equations into four nonlinear






















Table 5-2. Twenty-four solutions for Case :3


1
-7. 1517:3
+0.27141
+5.10000
t2.751:37
t12.882:3
t5.10000
-8.34619
t10.40:30
+9.741:32
-8.84:36:3
-9.91765
+20.12:31
-20.5096
+20.0955
-20.790:3
+11.2958
+4.80076
+4.89600
-7. 26801
+17.6764
+15.7880
-19.3426
+15.3028
-9.5678:3


+14.4690
+8.44826

t5.59:366
t10.32:32

t0.16126
t1.31517
+1.8:3447
+1.:3626:3
+2.90077
-1.62:330
-1.25716
-1.2:32634
-0.50454
+5.5:3768
+17.7501
-17. 7508
-17.5157
-12.25:30
+14.0808
-11.5960
+14.:338:3
+16.6180


.1 1
+0.06791
+0.24086
+0.37697
t0.46257
t0.48104
t0.49871
t1.08109
t1.17125
+1.24786
+1.24970



+1.91068
t2.01111
t2.1:3470
t2.462:32
-0.64549
-0.65507
-0.92267
-1.02457
-1.34644
-1.39506
-1.42815
-1.46480


+0.47666
-0.41749
-1.16995
-0.50:356
+0.81960
-0.09772
-0.49:30:3
-0.79791
-0.02641
-0.2:30:39
-0.24201
-:31.1760
+0.38270
+0.85549
+2.97004
-0.17160
+0.47906
+0.47724
+0.48824
-2.77:350
+2.15516

-0.58529
+0.25896










CHAPTER 2
PROBLEM STATEMENT AND APPROACH

The mechanism analyzed here is shown in Figure 2-1. The top platform (indicated by

points 4, 5, and 6) is connected to the base platform (indicated by points 1, 2, and 3) by

two spring elements whose free lengths are Lol and LO2 and by a variable length connector

whose length is referred to as L3. Although this does not match the exact definition of

tensegrity, the device is prestressed in the same manner as a tensegrity mechanism. The

exact problem statement is as follows:

Given:

L12 distance between points 1 and 2

p~z, P3y COordinates of point 3 in coordinate system 1

L45 distance between points 4 and 5

p6z, P6y COordinates of point 6 in coordinate system 2

L3 distance between points 1 and 4

kl, Lol spring constant and free length of spring 1

k2, LO2 Spring constant and free length of spring 2

Find:

All static equilibrium configurations








Lox L3 ,








Figure 2-1. Tensegrity mechanism










Table 4-2. Twenty solutions for Case 2


2.5 11l:1.11 0.411872i
2.5 ilI :1.1. 0.411872i
1.833882 0.139340i
1.833882 + 0.139340i
1.674852 + 0.000000i
1.492124 + 0.000000i
0.944817 + 0.000000i


-0.307013 1.456473i
-0.307013 + 1.456473i
-1.464914 0.949033i
-1.464914 + 0.949033i
+72.13693 + 0.000000i
-4.068392 + 0.000000i
+0.519148 + 0.000000i
+0.272806 0.408042i
+0.272806 + 0.408042i
+0.459257 + 0.000000i
+0.835170 + 0.288061i
+0.835170 0.288061i
-0.828087 + 0.306801i
-0.828087 0.306801i
-0.455891 + 0.000000i
+2.406830 + 0.000000i
-0.425974 + 0.000000i
-0.276958 0.470503i
-0.276958 + 0.470503i
+2.885304 + 0.000000i


+26.02618 + 1.476993i.
+26.02618 1.476993i.
-20.35270 1.239320i.
-20.35270 + 1.239320i.
-19.el s : ;+ 0.000000i
+19.64973 + 0.000000i
-7.534905 + 0.000000i
-3.7-111. 2 1.228946i.
-3.7-111. 2 + 1.228946i.
+5.926762 + 0.000000i
-0.516827 + 0.424 190i.:
-0.516827 0.424 190i.:
+0.450188 + 0.304645i.
+0.450188 0.304645i.
-5.417888 + 0.000000i
-16.46027 + 0.000000i
+6.719935 + 0.000000i
+1.780312 0.214965i.
+1.780312 + 0.21 1'll ..
+18.04674 + 0.000000i


-0.928707
-0.928707
-0.766478
-0.172483
-0.172483
+0.169610
+0.169610


- 0.056278i
+ 0.056278i
+ 0.000000i
- 0.153043i
+ 0.153043i
- 0.150949i
+ 0.150949i


+0.711628 + 0.000000i
+0.843106 + 0.000000i
+0.869530 + 0.000000i
+1.002755 0.055558i
+1.002755 + 0.055558i
+1.091412 + 0.000000i


*Four solutions correspond to when points 2, 5 are coincident (i.e., dl = 0). Value of
xl for dr = 0 may be determined from system of equations Equations 4-5, 4-6, 4-5

D4 Ds D6 X 0
E4 Es E6 2 (-
F4 F5 F6i 1 0 O 4
Sdl=


1)


forml(X )


| Sdi=o |


(4-22)

0 (4-23)


S((L2 -2 L +L2 + 2L3L12)X2 + L +L2


Ls, 2L3L12)


Remaining 24th degree polynomial can be divided by [(L2 -2 L L2 + 2L3L12)~
L~ + L 2 Ls, 2L3L12 2 TOSultingr in a 20th degree polynomial.-- TU1

*Remaining 20 solutions satisfy Equations 4-5, 4-6, 4-7. For this particular numerical
example there are 8 real and 12 complex solution sets for xl, x2, 1-

A bifurcation diagram between the solution xl and varying parameter Lol is shown in


Figure 4-1. It is interesting to observe that the four solutions for Lol


LO2 = 0 case


bifurcate as the free length Lol is varied. At Lol


2.3 m, the total number of solutions










ACKENOWLED GMENTS

I would like to express profound gratitude to my advisor, Prof. Carl Crane, for his

invaluable support, encouragement, supervision and useful so__~-r;-- me.. throughout this

research work. His moral support and continuous guidance enabled me to complete my

work successfully.

I would like to thank Prof. .John Schueller and Prof. Warren Dixon for serving on

my committee. I am also grateful to Prof. .Jay Gopalakrishnan and Dr. .Jahan B ,v ,f

for listening to my queries and answering my questions regarding the research work. I

would like to acknowledge the support of the Department of Energy under grant number

DE-FGO4-86NE37967.

I am especially indebted to my parents, Dr. Om Vikas and Mrs. Pramod K~umari

Sharma, for their love and support ever since my childhood. I also wish to thank my

brother, Pranay, for his constant support and encouragement. I thank my fellow students

at the Center for Intelligent Machines and Robotics. From them, I learned a great deal

and found great friendships. I would also like to thank Nicole, Piyush, Rakesh, Rashi and

Sreenivas for their friendships.










polynomial equations. It defines


xi = tan i
2


(2-23)


for i = 1, 2


thus ,


1 -X2
ce = -
C"1+ xy


2xe
Si = -
1+ X2'


(2-24)


for i = 1, 2 .


Each of the four cases will be analyzed in the chapters that follow.









where SBFL, XBFL are TOSpectively the square matrix(Sylvester matrix) and vector of

unknown coefficients. To obtain minimum dimension of SBFL, Equations 5-2, 5-3 are

divided by dld2, 5-4 is divided by d 5-5 is divided by d They are rewritten as


(F~x + F2 2 F3 2ai 4" 52 2g F6 li (7X 8sx 2 F9) = 0 (5-11)


(Gix + G2 2 G3 2i +(Gqx + GgZ2 G6 dli +(G7x + GgZ2 + G9) = 0 (5-12)

(Hl, x + H2 C2 H 13) + (H/ + H5 + H6 i = ~1 0 (5-13)

(Ix +2, I2 2 3 74X 5 2X 6 Idi = 0 (5-14)


where dli = 1/dl, dali ld2. SBFL, XBFL can be obtained by performing the operations

listed in Table 5.1 The determinant of the Sylvester Matrix is zero for common roots of

Equations 5-11, 5-12, 5-13, 5-14 and thus


fBFL(X1) =| SBFL |= 0 (5-15)


This results in a 104 degree polynomial in xl. The solutions for xl can be obtained by

solving this polynomial equation. To calculate corresponding values of xa2 1l 2, let SBFL

be the same as SBFL Without its first column and last row, SBFL be first column of SBFL

without its last element and XBFL be same as XBFL Without its first element(which is 1).


SBL(2x2) SBFL (51x1) SBFL(51x51) XBL1x)(1x1) (-6
x(lxl) x(1xst) XF(11

Thus, the first 51 equations of Equation 5-10 may be written as


SBFLXBFL = -SBFL (5-17)


Each solution for xl is substituted into SBFL, SBFL and XBFL iS SOlVed as


BFL BF-1 BFL(5-18)


dl, d2, 2a are elements of vector XBFL









CHAPTER 1
INTRODUCTION

The human musculoskeletal system is often described as combinations of levers and

pulleys. However, at a number of places (particularly the spine) this lever-pulley-fulcrum

model of the musculoskeletal system calculates such extreme amount of forces that will

tear muscles off the bones and shear hones into pieces. This, however, does not happen

in real life and can he explained by the concept of 'tensegrity'. Tensegrity (abbreviation

of 'tensional integrity [1], [2]), is synergy between tension and compression. Tensegrity

structures consist of elements that can resist compression (e.g., struts, hones) and elements

that can resist tension (e.g., ties, muscles). The entire configuration stands by itself

and maintains its form (equilibrium) solely because of the internal arrangement of the

struts and ties ([3],[4]). No pair of struts touch and the end of each strut is connected to

three non-coplanar ties ([5]). 1\ore formal definition of tensegrity is given by Roth and

Whiteley([6]), introducing a third element, the bar, which can withstand both compression

and tension. Tensegrity structures can he broadly classified into two categories, prestressed

and geodesic, where continuous transmission of tensional forces is necessary for shape

stability or single entity of these structures([7]). 'Prestressed tensegrity structures', hold

their joints in position as the result of a pre-existing tensile stress within the structural

network. 'Geodesic tensegrity structures', triangulate their structure members and

orient them along geodesics (minimal paths) to geometrically constrain movement. Our

bodies provide a familiar example of a prestressed tensegrity structure: our hones act

like struts to resist the pull of tensile muscles, tendons and ligaments, and the shape

stability (stiffness) of our bodies varies depending on the tone (pre-stress) in our muscles.

Examples of geodesic tensegrity structures include Fullers geodesic domes, carbon-based

buckminsterfullerenes (buckyballs), and tetrahedral space frames, which are of great

interest in astronautics because they maintain their stability in the absence of gravity

and, hence, without continuous compression. Idea of combining several basic tensegrity









Table 4-1. List of operations to obtain Sylvester Matrix for case 2
Equations # Equations # Unknowns Added Unknowns
{ (4-5), (46) } 3 9(x,2
(4-7) (x x2, 1 d


{(4-5),(4-6)}-d1
{(4-5),(4-6)}-d
(4-7)-dl
{ (4-5),(4-6) }*Z2
(4-7) -x2
{(4-5),(4-6)}-dlZ2
{(4-5),(4-6)}-d 2
(4-7)-dl


S1


4.1 The matrix SoFL and vector XoFL are written as


D3 0 0 Ds

E3 0 0 Es

0 0 FI Fs

D6 0 D1 0

E6 0 E1 0

0 D3D04 0

0 E3 E4 0

F6 F3 0 0

0 0 0 D6

0 0 0 E6

0 0 F2 F6

0 0 D2 0

0 0 E2 0

0 0 Ds 0

0 0 Es 0

00 0 0


D4 D1D2 0

E4 E1E2 0

F4 0 0 F3

0 D4DsD 3

0 E4 EsE3

0 0 0 D6

0 0 0 E6

0 F4 F5 0

DsD02D3 0

Es E2 E3 0

F 0 0 0

0 Ds D6 0

0 Es E6 0

0 00 0

0 00 0

0 Fs F6 0
(4- 5)


0 0 0

0 0 0

0 0 F2

0 0 D2

0 0 E2

0 0 Ds

0 0 Es

0 0 0

D1 0 0

El 0 0

0 FI F3

D4 D1D3

E4 E1 E3

0 D4 D6

0 E4 E6

F4 00


0 0

0 0

0 0

0 0

0 0

D1 D2

El E2

FI F2

0 0

0 0

0 0

0 0

0 0

D2D03

E2 E3

F2 F3


SonL









The representation of a screw is


SSOL + hS A6

Five quantities are required to specify a screw, of these 4 are required to specify a line.

The fifth one in the pitch of the screw, b.

A twist is represented as 8 $1. A twist requires six algebraic quantities for its

complete specification, of these five are required for complete specification of a screw. The

sixth quantity, the amplitude of twist (0) expresses the angle of rotation. The distance of

translation is the product of amplitude of twist and pitch of the screw. If pitch is zero,

the twist reduces to pure rotation around the screw $. If pitch is infinite, then finite twist

is not possible except the amplitude he zero, in which case the twist reduces to pure

translation parallel to the screw $.

A wrench is represented as f $1. A wrench requires six algebraic quantities for its

complete specification, of these five are required for complete specification of a screw. The

sixth quantity, the intensity of wrench ( f) expresses the magnitude of force. The moment

of couple is the product of intensity of wrench and pitch of the screw. If pitch is zero, the

wrench reduces to pure force along the screw $. If pitch is infinite the wrench reduces to

couple in a plane perpendicular to the screw $.










LIST OF FIGURES

Figure page

1-1 Biological model of the knee ......... .. 11

1-2 Tensegrity based model of cross-section of knee .. .. .. 11

2-1 Tensegrity mechanism .. ... ... .. 12

:3-1 Four real solutions for Case 1 .. ... ... 20

4-1 Bifurcation diagram for solution of .ri and varying parameter Lol .. .. .. 26

4-2 Eight real solutions for Case 2 .. ... .. 27

5-1 Twenty-four real solutions for Case :3 (cases 1 to 12) ... .. .. .. :32

5-2 Twenty-four real solutions for Case :3 (cases 1:3 to 24) .. .. 3:3










APPENDIX A
SHORT INTRODUCTION TO THEORY OF SCREWS

Screw theory was developed by Sir Robert Stawell Ball in 1876, for application in

kinematics and statics of mechanisms (rigid body mechanics). It is a way to express

displacements, velocities, forces and torques in three dimensional space, combining both

rotational and translational parts.

The Theory of Screws is founded upon two celebrated theorems [23]. One relates to

the displacement of a rigid body. The other relates to forces which act on a rigid body.

1. REDUCTION OF THE DISPLACEMENT OF A RIGID BODY TO ITS SIMPLEST FORM.
Fundamental theorem discovered by ChI I-1. -; states
Any given displacement of a rigid body can he effected hv a rotation about
an axis combined with a translation parallel to that axis.

2. REDUCTION OF OF A SYSTEM OF FORCES APPLIED TO A RIGID BODY TO ITS
SIMPLEST FORM. Fundamental theorem discovered by Poinsot states
A force, and a couple in a plane perpendicular to the force, constitute an
adequate representation of any system of forces applied to a rigid body.

Picker coordinates were introduced by Julius Plicker in the 19th century as a way to

assign six homogenous coordinates to each line in projective 3-space. In Screw Theory,

they are used to represent the coordinates of screws, twists and wrenches.









Table 3-1. Six solutions for Case 1


71 (radians)
1 +0.66861
2 -1116
3 -0.51037
-1.7079i
4 -0.51037
+1.7079i
5 +0.81733
6 a


y2 (radians)
-0.63297
-0.20769
-0.05764
-1.0903i
-0.05764
+1.0903i
+1.88221
+2.54351


i


Figure 3-1. Four real solutions for Case 1

















j/


(a) Case 1










(d) Case 4


(b) Case 2


(c) Case 3 fl = f~ = ()










(f) Case fi ft = f2 = )


(e) Case 5


(g) Case 7 (h) Case 8 (i) Case 9










(j) Case 1() (k) Case 11 (1) Case 12

Figure 5-1. Twenty-four real solutions for Case 3 (cases 1 to 12)

































S2008 Vishesh Vikas











TABLE OF CONTENTS

page

LIST OF TABLES ......... ... . 5

LIST OF FIGURES ......... .. . 6

ACK(NOWLEDGMENTS .......... . .. .. 7

ABSTRACT ............ .......... .. 8

CHAPTER

1 INTRODUCTION ......... ... .. 9

2 PROBLEM STATEMENT AND APPROACH .... .. 12

3 BOTH FREE LENGTHS ARE ZERO . ..... 17

3.1 Equilibrium Analysis ......... . 17
3.2 Numerical Example ......... .. 19

4 ONE FREE LENGTH IS ZERO ............ ...... 21

4.1 Equilibrium Analysis ......... . 21
4.2 Numerical Example ......... .. 24

5 BOTH FREE LENGTHS ARE NON-ZERO ..... .... 28

5.1 Equilibrium Analysis ......... . 28
5.2 Numerical Example ......... .. 31

6 CONCLUSION ......... . . 36

APPENDIX

A SHORT INTRODUCTION TO THEORY OF SCREWS ... .. .. 37

B SYLVESTER MATRIX ......_._. ... .. 40

REFERENCES ......._._.. ........_._.. 44

BIOGRAPHICAL SK(ETCH ....._._. .. .. 45









D1 = k2 93y + 6y)X: + 2(k l(L45 + L12) + k~2 96z + 32) 1 k~2 96y + 3y)

D2 = -2(kilL45 + k~2P62)m q~'.i' F + 2(k lL45 + k~2p62

D3 = k~2 93y p6y)(X: 1) 2(k2 9p3z p62) + k l(L12 L45 )1

D4 = -2Lolkl(L45 + L12 X1

Ds = 2k lLoiL45(-1

D6 = 2k lLoi(L45 L12)X 141

El=k2 (3y 96z + L3) P32P6y)X: + 2L3(k L12 + k293m 1l
k2 (3y(L3 p6z P32P6y)
E2 = 2(1 + x )(kilL12L45 + k~2P3yf6y + .I' l's

E3 k2 (3y(L3 p62) + 32P6y)X: + 2L3(klL12 + k~2p32 1
k2 (3y(L3 + 6z) P32P6y)
E4 = -2kilL12LoiL3 1

Es = -2kilL12LoiL45( )

E6 = -2kilL12LoiL3 1 (4-12)

F, = 1 + x~

F2 =

F3 :

F4 = 2(L45 L12)L3(1 x2 )~ L L 2L45L12 +2 L ( 2 2

Fs = -8L3 1L45

F6 = (2L12 2L45)L3(1 x ) (L~ 2L45L12 + Lsg + L~~( 2 ~ 43


It is desired to form a system of equations


SoFLXoFL = 0 (4-14)


where SoFL, XoFL are respectively the square matrix(Sylvester matrix) and vector of

'unknown coefficients. They can be obtained by performing the operations listed in Table























Vakratunda 1!!., !. I.,!..,\-.,
Koti soorya samaprabhaa
Nirvigfhnam kurume deva
Sarva karyeshu sarvadaa.





(a) Case 13


(b) Case 14


(c) Case 15


(d) Case 16


(e) C'ase 17


(f) C'ase 18


(g) Case 19


(h) Case 20


(i) Case 21











(1) Case 24


(j) Case 22


(k) Case 23


Figure 5-2. Twenty-four real solutions for Case 3 (cases 13 to 24)












Table 5-1. List of operations to obtain Sylvester Matrix for Case 3


Equations
{ (5-11), (5-12) }
(5-13)
(5-14)


{(5-11),(5-12)}-du
{(5-11),(5-12)}-di
{(5-11),(5-12)}- di d~i
(5-13)-d~i
(5-14)-dii
{(5-11),(5-12)}-d
(5-13)-dli
{(5-11),(5-12)}-d
(5-14)-dli


(5-13)-d i
(5-14)-di
(5-13)-di d~i
(5-14)-di dai
{(5-11),(5-12)}-x2
(5-13)xZ2
(5-14)-x2
{(5-11),(5-12)}-d~i 2
{(5-11),(5-12)}-dlid2i 2
(5-13)-d~i 2
(5-14)-diix2
{(5-1),(-1)}.1a
(5-13)dlil2
{(5-11),(5-12)} .1 2iX

(5-14)-dlizx2


(5-13)-dlid2i 2
(5-14)-dlid2i 2


# Equations
4




8

12


15

18

26






52


# Unknowns
15




18

24


27

30

39






52


Added Unknowns

(x x2,)
(x x2 ldi
(x x2 2di
(x x2 ld i
(x x2, i dia



(x x2 i 2i~


(x x2 ld i

(x x2, i dia

(x~ x2 did~i


lii

d2i1

ai
dli 2i

di 2


2ii









structures to form a more complex structure has been analyzed([8], [9]) and different

methods to do so have also been studied([10]).

There has been a rapid development in static and dynamic analysis of tensegrity

structures in last few decades ([11]). This is due to its benefits over traditional approaches

in several fields such as architecture ([12]), civil engineering, art, geometry and even hi-

ology. Benefits of tensegrity structures are examined by Skelton et al.([13]). Tensegrity

structures display energy efficiency as its elements store energy in form of compression

or tension; as a result of the energy stored in the structure, the overall energy required

to activate these structures will be small([l14). Since compressive members in tensegrity

structures are dl;id..ini large displacements are allowed and it is possible to create deploy-

able structures that can he stored in small volumes. Deploi- .1.1.' antennas and masts are

notable space applications ([15],[16]). K~enner established the relation between the rotation

of the top and bottom ties. Tobie ([3]), presented procedures for the generation of tensile

structures by physical and graphical means. Yin ([5]) obtained K~enner's ([17]) results

using energy considerations and found the equilibrium position for unloaded tensegrity

prisms. Stern ([18]) developed generic design equations to find the lengths of the struts

and elastic ties needed to create a desired geometry for a symmetric case. Knight ([19])

addressed the problem of stability of tensegrity structures for the design of deploi- .1.1.*

antennae. On macro level, tensegrity structures are used to model human musculoskele-

tal system, deploi-,1.1-- antennae, architecture structures, etc.; on cellular level, Donald

Ingher ([7]) proposes Cellular Tensegrity Theory in which the whole cell is modelled as

a prestressed tensegrity structure, although geodesic structures are also found in the cell

at smaller size scales. Stephen Levin ([14]) proposed a truss-tensegrity model of the spine

mechanics. Benefits of tensegrity structures make them interesting for designing mobile

robots. Aldrich ([20]) has built and controlled robots based on tensegrity structures and

Paul et al. ([21]) have built triangular prism hased mobile tensegrity robot.s









system of equations


p(x) = 0

Xp(X) = 0


x(m-l)p(x) = 0, (B-4)

q (x) = 0


x("- )q(x) = 0

The system may be written as a matrix equation

a, a _i .. O 0 x (n+m-1) 0



0 0 -- ai ao x(m-l) 0
(B-5)
bm bm-1 (m- 2) 0



0 0 --- --- bi bo 1 0



The Sylvester matrix(S,,,) associated with polynomials p(x) and q(x) is a square matrix of

dimension (n + m) x (n + m). The determinant of the Sylvester matrix will vanish when

p(x) and q(x) have a common root. The converse is also true. In order for there to be a

common root for Eqns (B-1) and (B-2) it is necessary that


det(S,,,) = 0 (B-6)









GI1 = k2LO2[( 93fy-9y62 L9m1+ 1-x)33

G2' = -2k2LO2[s )()TG 93f+ P3yj~y)
Gr3 = k2LO2[( 93yfm 3fy 2LH3s93m 1 + (1-x)L3P3y]

G4 = -2kilL12LoiL3 1

Gg = -2(x: + 1)kiL45L12Lol

G6 = -2kilL12LoiL3 1

G7 = [2k x L12 + (lJ2p~ 1 \ 131y:1)k2] L~3 3y+ (] 2\3m f_ ly rrk2

GS = %2(x + 1)kilL45L12 + 2(xr + 1)(p3riV62 -t3yf6y)k2
G9s = [2k~~l~l x L12\ + (2p 1 + (31k L3 Z9i3zfay 36)k (5-7)

H1 = (1 + x )

H2 = 0

H3 =( ~

H4 (LI3 +L12 +L45 2(x + 1) 4(L12 + L45)L3

Hs = -8L3X L45

H6 = (L3 + L12- L45 2(x~ + 1) + 4(L12 J-L45)L3 (5-8)

II = (1 + x )


I2 = 01+ ~


Id = [(L3 93r 96m~ 2 (p3y -t 6y 2] + 3y + 62) 1 + 3z +62] L3

Is 4(p)3yfljm P3zP6y L396y)T + 96 jPy 6rZ 1)L3

IB = [(L3 93r 96m2f C13y 96y 2] 2 9y 96y Ir3m -962]L3 (.5-9

It is desired to form a systern of equations


(5-10)


SBFLXBFL = 0










Equations :34, :35 can he rewritten as


Prx + P2x 1 ~3

Qlxr + Q2x 1 Q3


(36)

(37)


where


P, = (A4 t+ 2 2~X A3 P2 4AX 5g~ 2 A6 P3 (7X 8 As2 9 ,(8

Q1 = (B1Xr B272 3) 2a 4 5BX 2gX B6) 3 (7X 872, 9 (


We form the Sylvester s 1\atrix(Appendix B) by multiplying Equations :36, :37 with ri

and write


3, 0 xr 0

P2 3 x:r 0
(:310)
Q3 0 :ri 0

&2 Q3 1 0

is zero for common roots of Equations :36, :37 and


Pr P2



&1 2

0 Qi

The determinant of Sylvester matrix

thus


SP, 2~ P3

fZFL 2 P P 0 (:311)
01 &2 Q3 0

0 Qi 02 Q3

Expansion of Equation :311 ( fZFL 72)) yields an eighth degree polynomial in x:1 2*

was found that this eighth degree polynomial could be divided symbolically by the
term (1 +\ ",cr,, ),, without, an rmandr eslting in a,,,l sixth degree, poyoma n ,2 The

coefficients of this polynomial have been obtained symbolically, but are not listed here due

to their length.










XoFL = L1[1, x ~ di, dx~ x 2,d, d1, 2 2 .1 2,1 2 2 d1, X2d1, d ]T(4-16)

The determinant of the Sylvester matrix is zero for common roots of Equations 4-5, 4-6,

4-7 and thus

foFL(X1) =| SonL |= 0 (4-17)

This results in a 32 degree polynomial in xl. The solutions for xl can be obtained by

solving this polynomial equation. To calculate corresponding values of x2, dl, let SonL

be the same as SoFL without its first column and last row, SonL be first column of SonL

without its last element and XoFL be same as XoFL without its first row. Thus,



SOFL(16x16) Son(1sxl) Son1xs OFL(16x1)(1x1
x(lxl) x(1xis) XF(si

Thus, the first 15 equations of Equation 4-14 may be written as


SoFLXoFL = -SoFL (4-19)


Each solution for xl is substituted into SoFL, SoFL and XoFL is solved as


XoFL = -SoFt- SoFt (4-20)


Values of dl and x2 Which correspond to a solution of xl are the 6th and 9th elements of

vector XoFt.

4.2 Numerical Example

Values of the given parameters were selected to be the same as in the previous

example (Sec 3.2) with the exception of the free length of spring 1 which is now set to

Lol = 2.3 m. Equations 4-11, 4-12, 4-13 are evaluated and substituted into Equation

4-15. Equation 4-17 is obtained and solved to obtain values of xl. Corresponding values

of x2, d1 arT Obtained from Equation 4-20. Solutions of Equation 4-17 were as follows

*Eight of the solutions of xl were either fi. Thus, Equation 4-17 may be divided
through by (1 + x 4 T"ulting, in a2th degree polynomial.









CHAPTER 5
BOTH FREE LENGTHS ARE NON-ZERO

5.1 Equilibrium Analysis

Both non-zero free length implies


X1= Lol/di, X2 = LO2 d2 .


(5-1)


It is important to observe that Equations 2-19, 2-20 are coupled to both Equations 2-22,

2-21.

After applying the 'tan-half angle concept'(Equation 2-24), the following equations


are obtained.


(Fix+ 2 2 F3 kd1 (42 r52 2 F6 d2 + ~ 82 2 F9 dld2 = 0


(5-2)


(Gix~ + 4G2 2 G3 d1 + G 2 GgZ2 G6 d2 (G7x2 GgZ2 + G9 did2

(H x 2 \2 + H) (H4x~ H5Z2 H6) = 0


0 (5-3)

(5-4)

(5-5)


(11xc + I2 C2 31 4 526)


where


FI = LOak2 96py 93iy)(1
F2=2LOak2 (16r(x- 1)


F3 = LO2k2 9r3y 96y)(1


- x) 2(pfimi 13r) 1

-2p)6yT1)

- x ) 2(pY3z -6 Y~C11


F4 = -2(L45 + L12)Lo1Jkixt

Fs=2klLoiL45(X-1

F6 = 2(L45 L12)Lo1Jk xi

F- (p6y 93~ 2(x-1 2 [(p3z + 62)k~2 + (L45 + L12)kil] X1


'/' ,A1


Fs = 2(k2962 + kilL45 (


F9 (p3y P6y)k~2(X 1) + 2 [(p3z P62)k~2 + (L12 L45)kil] Xi


(5-6)










Defining some basic terms used in Screw Theory


PITCH OF SCREW. Rectilinear distance through which the nut is translated parallel
to axis of the screw, while the nut is rotated through the angular unit of circular
measure. Pitch is thus a linear magnitude

SCREW. A straight line with which a definite linear magnitude termed the pitch
is associated. In rigid body dynamics, velocities of a rigid body and the forces and
torques acting upon it can he represented hv the concept of a screw.

TwlsT. A screw representing the velocity of a body. A body is said to receive a
twist when it is rotated uniformly about the screw, while it is translated parallel to
the screw, through a distance equal to the product of the pitch and circular measure
of angle of rotation.

WRENCH. A screw representing forces and torques on a body. It denotes a force
and couple in a plane perpendicular to the force. One way to conceptualize this is to
consider someone who is fastening two wooden boards together with a metal screw.
The person turns the screw (applies a torque), which then experiences a net force
along its axis of rotation.

A straight line can he defined by two points. Assuming two points(1, 2) with point

vectors rl, r2 the equation of an arbitrary point lying on the line made by points 1, 2 can

he written as

(r2 rl)
S = (A-1)
|r2 rll
(r rl) x S = 0 (A-2)

Sr xS =r x S (A-3)


Picker coordinates of the line are {S: SOL} and they satisfy the following constraints


|S| = 1, S SOL = 0 (A-4)


It should be noted that dimensions of S and SOL are different. Also, only four algebraic

quantities are required to define a line due to the two constraints


|S| = 1, S SOL = 0 (A-5)

























li\


Figure 4-2. Eight real solutions for Case 2










LIST OF TABLES

Table page

3-1 Six solutions for Case 1 ......... . 20

4-1 List of operations to obtain Sylvester Matrix for case 2 .. .. .. 23

4-2 Twenty solutions for Case 2 ......... .. 25

5-1 List of operations to obtain Sylvester Matrix for Case 3 ... .. .. 34

5-2 Twenty-four solutions for Case 3 ......... .. 35











Most of the papers in the field of tensegrity assume zero free length of springs. In

real life systems, esp. biological systems, such assumption does not hold true. Following

research shows that this assumption is not trivial, complexities to find solutions increase

tremendously and the number of static equilibrium configurations also increase. Cross-

section modeling of the human knee joint (Figure 1-1, 1-2) is the biological motivation of

the mechanism in the following research. In the following research, a planar pre-stressed

tensegrity structure is examined. The structure consists of three struts connected by two

ties.



,flilsil111 1 mil.I








..I. II .r na .

Il l



Figure 1-1. Biological model of the knee



















Lipansesst tendon modeled
an inertensible ~ee


Figure 1-2. Tensegrity based model of cross-section of knee









CHAPTER 4
ONE FREE LENGTH IS ZERO

4.1 Equilibrium Analysis

The free length of one spring is zero (Lol = 0) implies


At = Lol/di, X2 = 0 .


(4-1)


It is important to observe that Equations 2-19, 2-20 are coupled only to Equation 2-22

(i.e., the terms containing d2 VanIlSh) and these equations may be written as


((kilL45 + k~2p62) 2C1 S1C2) + X~' C1C2 S1S82

+kiL1281 + k2 93mil P3yC1 1d + kilLoi(L45 81C2 -- C1S2)


0 (4-2)



0 (4-3)


0.(4-4)


CL1281


(L3(ki1L12 + k~2p3z k~2L3P3yC1 81 + k~2 932P6y p3yp62)C2

+(kilL12L45 + k~2 93yf6y + 32P6m 2 1S) kilL12LoiL381 kilL12LoiS2L45
d12 + (2L3L12 2L3C2L45)C1 L 2 +2c2L45L12 2L3S1S2L45 Lsg L


Applying the 'tan-half angle concept' (Equation 2

following equations are obtained


Ald + A2 =

Bld + B2 =

Cld + C2


-24), to Equations 4-5, 4-6 and 4-7, the


(4-5)

(4-6)


where


Dia + D2 2 D3



=F x~ + F2x 2 F3


D4x~ + D5Z2 + D6

SE4x~ + E5Z2 + E6

-F4X 5 2g~ F6


(4-9)

(4-10)










The concept of Sylvester Matrix can be extended to more than 2 equations. More

importantly, it can be extended to more than one variable. Given a set of a polynomial

equations in m variables (pl(xl, xm), p,(xl, xm), it is possible to construct

Sylvester Matrix of p dimension by multiplying equations by combinations of xl, -, xm e.g.,

xl1 2 91 1, sm). There is no definite algorithm to construct the Sylvester Matrix, the

process solely depends on the nature of the polynomial equations. The Sylvester Matrix

with the minimum dimension yields a non-zero determinant in the embedded variable.

Let St (xi) is the minimum dimension Sylvester Matrix for the given set of a polynomial

equations, then, for any Sylvester Matrix Sy (xi) with a greater dimension than that of

St (xi), determinant of the Sylvester Matrix vanishes and does not give any information

about the embedded variable. So, the Sylvester Matrix is unique by its dimension. More

precisely, the determinant of the Sylvester Matrix is unique. To construct this minimum

dimension Sylvester Matrix, it may be required to perform tricks on the set of given

equations and in the process, change the nature of the polynomial. As the theory for

Sylvester Matrix for multivariable, multi-polynomial-equation system is not developed,

there may be introduction of 'extraneous solutions', that do not satisfy the set of given

equations .









where cs, aS are abbreviations for cos 3, sin 3 (i

calculated as

BPj = BP4 TB TP,


1, 2) respectively. Points 5, 6 may be


.) = 5, 6 .


(2-5)


Calculatingf coordinates of points 5, 6 gives

L3C1 + L45C2

BPs = L3S1 L45S2B

0 B

The Pliicker coordinates of the three lines are


$1 = I Sol i 2 =
Sort


L3C1 + 62C2 P6yS2

L3S1 6m2S + 6yC2






$3 SO3
SO3L


p6





SO2

SO2L


(2-6)


(2-7)


and


BPs- Bp2
Soi =
| Ps BP2
SO2 = IBg-B3

BP4
SO3 =
IBP4|

Forces in the three legs are fl, f2,


BLi
1
BL1
d2
BP4
L3


BP2X BP5
So1L = Bp2 x Sol =

BP3 BP6
SO2L BP3 x SO2 =

SO3L = 0 .


(2-8)

(2-9)

(2-10)


f3 Which may be expressed as


(2-11)

(2-12)

(2-13)

(2-14)


Forces in the springs and connector are wrenches with zero pitch (i.e., pure forces along

the direction of the respective lines). For static equilibrium, the sum of the three wrenches

must be equal to zero.


flix + f2 2 + 3 3 = 0 .


(2-15)


fi = kl(dl Loi)

f2 = k2 2~ LO2)

d~ = | |"Ps -- Bp2 2

d~ = | |"P6 BP3 12









A solution approach of satisfying force and moment conditions for equilibrium is

considered. It is apparent that since the length L3 is given, the device has two degrees

of freedom. Thus, there are two descriptive parameters that must be selected in order to

define the system. For this an~ lli--- the descriptive parameters are chosen as the angles

71 and y2 Which are respectively the angle between X axis of coordinate system attached

to the base and line defined by points 1 and 4 and angle between X axis of coordinate

system attached to the base and line defined by points 4 and 5 as shown in Figure 2-1.

Other parameters were investigated, but none yielded a less complicated solution than is

presented here.

The co-ordinates of points 1 to 6 are

0 L12 p3,

fPI = 0 fP2 = 0 fP3 = p3w (2-1)

0 0 0

0 L45 p6;,

TP4 = 0 TPs = 0 O 6P = p6y (2-2)

0 0 0

Where the superscripts B, T denote the co-ordinate systems fixed on the reference frames

of the bottom platform and top platforms respectively. It is also known that



BP4 = 3 1 (2-3)



c2 -s2 0
R 8 C (2-4)

0 01









CHAPTER 3
BOTH FREE LENGTHS ARE ZERO

3.1 Equilibrium Analysis

Zero free length for both the springs (Lol = LO2 = 0) implies


At = a2 = 0 (3-1)


It is important to observe that Equations 2-19 and 2-20 are decoupled from Equations 2-

22, 2-21 (i.e., terms containing dr, d2 VanIlSh). Substituting Equation 3-1 into Equations

2-19 and 2-20 gives


k_; (cic2 + 182) + (klL45 + k2p62) 82C1 S1C2) + (k2p3z + klL12 81 k2P3yC1 = 0 (3-2)

(kilL3L12 + k~2L3p32 81 + (kilL12L45 + k~2P3yP6y + k~2P32P62 82 k~2L3C1P3y
= 0. (3-3)
+(k293mf6y k293yf6,)C2

Applying 'tan-half angle concept',(Equation 2-24) to Equations 3-2 and 3-3 yields


(Alx~ + A2 2 + A3)X: + (Aqx~ + AgZ2 + A6 X1 + (A7x~ + AgZ2 + A9) = 0 (3-4)

(B1 x + B2 2 3 4, 5 2BX 6 Bg,+B)1 7 8L-x 2 9),+ g = 0 (3-5)


where Ai, Bi(i = 1,2,..., 9) are defined in terms of the given parameters as
Al= k2 3y + 6y) 1= k2 933y6z 32P6y + L3P3y)

A2=-2(k2962 + kilL45) 2=2(kilL12L45 + k~2 93yP6y + 32P62)

A3= k2 33y 6y) 3= k2 932P6y + L I' p3yp62)

A4=2((L45 + L12)kil + (p6z + 32)k~2) 4=2k lL3L12 + 2k~2L3p3z

A = 4 1. F. B = 0

A6=2((L12 L45)kil + (p3z 62)k~2) 6 4

A7=-(p6yp3y)k~2 B7 83

A8=2(klL45 + k2p62) s8 2

A9 9P6y 93y) k2 9 1 -1




Full Text

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page LISTOFTABLES ..................................... 5 LISTOFFIGURES .................................... 6 ACKNOWLEDGMENTS ................................. 7 ABSTRACT ........................................ 8 CHAPTER 1INTRODUCTION .................................. 9 2PROBLEMSTATEMENTANDAPPROACH ................... 12 3BOTHFREELENGTHSAREZERO ....................... 17 3.1EquilibriumAnalysis .............................. 17 3.2NumericalExample ............................... 19 4ONEFREELENGTHISZERO .......................... 21 4.1EquilibriumAnalysis .............................. 21 4.2NumericalExample ............................... 24 5BOTHFREELENGTHSARENON-ZERO .................... 28 5.1EquilibriumAnalysis .............................. 28 5.2NumericalExample ............................... 31 6CONCLUSION .................................... 36 APPENDIX ASHORTINTRODUCTIONTOTHEORYOFSCREWS ............. 37 BSYLVESTERMATRIX ............................... 40 REFERENCES ....................................... 44 BIOGRAPHICALSKETCH ................................ 45 4

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Table page 3-1SixsolutionsforCase1 ................................ 20 4-1ListofoperationstoobtainSylvesterMatrixforcase2 .............. 23 4-2TwentysolutionsforCase2 ............................. 25 5-1ListofoperationstoobtainSylvesterMatrixforCase3 .............. 34 5-2Twenty-foursolutionsforCase3 .......................... 35 5

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Figure page 1-1Biologicalmodeloftheknee ............................. 11 1-2Tensegritybasedmodelofcross-sectionofknee .................. 11 2-1Tensegritymechanism ................................ 12 3-1FourrealsolutionsforCase1 ............................ 20 4-1Bifurcationdiagramforsolutionofx1andvaryingparameterL01 26 4-2EightrealsolutionsforCase2 ............................ 27 5-1Twenty-fourrealsolutionsforCase3(cases1to12) ................ 32 5-2Twenty-fourrealsolutionsforCase3(cases13to24) ............... 33 6

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Iwouldliketoexpressprofoundgratitudetomyadvisor,Prof.CarlCrane,forhisinvaluablesupport,encouragement,supervisionandusefulsuggestionsthroughoutthisresearchwork.Hismoralsupportandcontinuousguidanceenabledmetocompletemyworksuccessfully.IwouldliketothankProf.JohnSchuellerandProf.WarrenDixonforservingonmycommittee.IamalsogratefultoProf.JayGopalakrishnanandDr.JahanBayatforlisteningtomyqueriesandansweringmyquestionsregardingtheresearchwork.IwouldliketoacknowledgethesupportoftheDepartmentofEnergyundergrantnumberDE-FG04-86NE37967.Iamespeciallyindebtedtomyparents,Dr.OmVikasandMrs.PramodKumariSharma,fortheirloveandsupporteversincemychildhood.Ialsowishtothankmybrother,Pranav,forhisconstantsupportandencouragement.IthankmyfellowstudentsattheCenterforIntelligentMachinesandRobotics.Fromthem,Ilearnedagreatdealandfoundgreatfriendships.IwouldalsoliketothankNicole,Piyush,Rakesh,RashiandSreenivasfortheirfriendships. 7

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1 ],[ 2 ]),issynergybetweentensionandcompression.Tensegritystructuresconsistofelementsthatcanresistcompression(e.g.,struts,bones)andelementsthatcanresisttension(e.g.,ties,muscles).Theentirecongurationstandsbyitselfandmaintainsitsform(equilibrium)solelybecauseoftheinternalarrangementofthestrutsandties([ 3 ],[ 4 ]).Nopairofstrutstouchandtheendofeachstrutisconnectedtothreenon-coplanarties([ 5 ]).MoreformaldenitionoftensegrityisgivenbyRothandWhiteley([ 6 ]),introducingathirdelement,thebar,whichcanwithstandbothcompressionandtension.Tensegritystructurescanbebroadlyclassiedintotwocategories,prestressedandgeodesic,wherecontinuoustransmissionoftensionalforcesisnecessaryforshapestabilityorsingleentityofthesestructures([ 7 ]).`Prestressedtensegritystructures',holdtheirjointsinpositionastheresultofapre-existingtensilestresswithinthestructuralnetwork.`Geodesictensegritystructures',triangulatetheirstructuremembersandorientthemalonggeodesics(minimalpaths)togeometricallyconstrainmovement.Ourbodiesprovideafamiliarexampleofaprestressedtensegritystructure:ourbonesactlikestrutstoresistthepulloftensilemuscles,tendonsandligaments,andtheshapestability(stiness)ofourbodiesvariesdependingonthetone(pre-stress)inourmuscles.ExamplesofgeodesictensegritystructuresincludeFullersgeodesicdomes,carbon-basedbuckminsterfullerenes(buckyballs),andtetrahedralspaceframes,whichareofgreatinterestinastronauticsbecausetheymaintaintheirstabilityintheabsenceofgravityand,hence,withoutcontinuouscompression.Ideaofcombiningseveralbasictensegrity 9

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8 ],[ 9 ])anddierentmethodstodosohavealsobeenstudied([ 10 ]).Therehasbeenarapiddevelopmentinstaticanddynamicanalysisoftensegritystructuresinlastfewdecades([ 11 ]).Thisisduetoitsbenetsovertraditionalapproachesinseveraleldssuchasarchitecture([ 12 ]),civilengineering,art,geometryandevenbi-ology.BenetsoftensegritystructuresareexaminedbySkeltonetal.([ 13 ]).Tensegritystructuresdisplayenergyeciencyasitselementsstoreenergyinformofcompressionortension;asaresultoftheenergystoredinthestructure,theoverallenergyrequiredtoactivatethesestructureswillbesmall([ 14 ]).Sincecompressivemembersintensegritystructuresaredisjoint,largedisplacementsareallowedanditispossibletocreatedeploy-ablestructuresthatcanbestoredinsmallvolumes.Deployableantennasandmastsarenotablespaceapplications([ 15 ],[ 16 ]).Kennerestablishedtherelationbetweentherotationofthetopandbottomties.Tobie([ 3 ]),presentedproceduresforthegenerationoftensilestructuresbyphysicalandgraphicalmeans.Yin([ 5 ])obtainedKenner's([ 17 ])resultsusingenergyconsiderationsandfoundtheequilibriumpositionforunloadedtensegrityprisms.Stern([ 18 ])developedgenericdesignequationstondthelengthsofthestrutsandelastictiesneededtocreateadesiredgeometryforasymmetriccase.Knight([ 19 ])addressedtheproblemofstabilityoftensegritystructuresforthedesignofdeployableantennae.Onmacrolevel,tensegritystructuresareusedtomodelhumanmusculoskele-talsystem,deployableantennae,architecturestructures,etc.;oncellularlevel,DonaldIngber([ 7 ])proposesCellularTensegrityTheoryinwhichthewholecellismodelledasaprestressedtensegritystructure,althoughgeodesicstructuresarealsofoundinthecellatsmallersizescales.StephenLevin([ 14 ])proposedatruss-tensegritymodelofthespinemechanics.Benetsoftensegritystructuresmaketheminterestingfordesigningmobilerobots.Aldrich([ 20 ])hasbuiltandcontrolledrobotsbasedontensegritystructuresandPauletal.([ 21 ])havebuilttriangularprismbasedmobiletensegrityrobot.s 10

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1-1 1-2 )isthebiologicalmotivationofthemechanisminthefollowingresearch.Inthefollowingresearch,aplanarpre-stressedtensegritystructureisexamined.Thestructureconsistsofthreestrutsconnectedbytwoties. Figure1-1. Biologicalmodeloftheknee Figure1-2. Tensegritybasedmodelofcross-sectionofknee 11

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2-1 .Thetopplatform(indicatedbypoints4,5,and6)isconnectedtothebaseplatform(indicatedbypoints1,2,and3)bytwospringelementswhosefreelengthsareL01andL02andbyavariablelengthconnectorwhoselengthisreferredtoasL3.Althoughthisdoesnotmatchtheexactdenitionoftensegrity,thedeviceisprestressedinthesamemannerasatensegritymechanism.Theexactproblemstatementisasfollows: Given: Find:Allstaticequilibriumcongurations Figure2-1. Tensegritymechanism 12

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2-1 .Otherparameterswereinvestigated,butnoneyieldedalesscomplicatedsolutionthanispresentedhere.Theco-ordinatesofpoints1to6are WherethesuperscriptsB;Tdenotetheco-ordinatesystemsxedonthereferenceframesofthebottomplatformandtopplatformsrespectively.Itisalsoknownthat 13

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$1=264S01S01L375;$2=264S02S02L375;$3=264S03S03L375(2{7)and Forcesinthethreelegsaref1;f2;f3whichmaybeexpressedas (2{11) (2{12) Forcesinthespringsandconnectorarewrencheswithzeropitch(i.e.,pureforcesalongthedirectionoftherespectivelines).Forstaticequilibrium,thesumofthethreewrenchesmustbeequaltozero. 14

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2{7 to 2{12 intoEquation 2{15 gives (2{16) (2{17) where1=L01=d1;2=L02=d2.Astheproblemisplanar,vectorEquation 2{16 isequivalenttotwoscalarequationsandvectorEquation 2{17 isequivalenttoonescalarequation.Therefore,therearethreeunknowns(d1;d2;f3)andthreeequations.Onescalarequationind1;d2canbeobtainedfromEquation 2{16 byperformingacrossproductoperationwithBP4tobothsidesoftheequationtoyield 2{13 2{14 2{17 and 2{18 .SubstitutingEquations 2{1 2{3 and 2{6 intotheseequationsyields. (2{19) (2{20) (2{21) Theequationscannowbeanalyzedforthreedierentcases1. 2. 3. 2{19 to 2{22 arenonlinearfunctionsofsin1;cos1;sin2andcos2.Theconceptof`tan-halfangles',([ 22 ])convertsthesefourequationsintofournonlinear 15

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16

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2{19 and 2{20 aredecoupledfromEquations 2{22 2{21 (i.e.,termscontainingd1;d2vanish).SubstitutingEquation 3{1 intoEquations 2{19 and 2{20 gives (3{2) (k1L3L12+k2L3p3x)s1+(k1L12L45+k2p3yp6y+k2p3xp6x)s2k2L3c1p3y+(k2p3xp6yk2p3yp6x)c2=0: Applying`tan-halfangleconcept',(Equation 2{24 )toEquations 3{2 and 3{3 yields (A1x22+A2x2+A3)x21+(A4x22+A5x2+A6)x1+(A7x22+A8x2+A9)=0 (3{4) (B1x22+B2x2+B3)x21+(B4x22+B5x2+B6)x1+(B7x22+B8x2+B9)=0 (3{5) whereAi;Bi(i=1;2;:::;9)aredenedintermsofthegivenparametersas

PAGE 18

3{4 3{5 canberewrittenas (3{6) (3{7) where (3{8) WeformtheSylvester'sMatrix(Appendix B )bymultiplyingEquations 3{6 3{7 withx1andwrite 3{6 3{7 andthus 3{11 (fZFL(x2))yieldsaneighthdegreepolynomialinx2.Itwasfoundthatthiseighthdegreepolynomialcouldbedividedsymbolicallybytheterm(1+x22)withoutanyremainderresultinginasixthdegreepolynomialinx2.Thecoecientsofthispolynomialhavebeenobtainedsymbolically,butarenotlistedhereduetotheirlength. 18

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3{12 .Uniquecorrespondingvaluesfor1and2arecalculatedforeachvalueofx1andx2fromEquation 2{23 p3x=3m;p3y=7m L45=5m p6x=1:1990m;p6y=2:2790m L3=7:56m k1=1:5N=m;L01=0m k2=3:7N=m;L02=0mCoecientsAi;Bi(i=1;:::9)areevaluatednumericallyandasixthdegreepoly-nomialinx2isobtainedbyexpandingEquation 3{11 anddividingitby(1+x22).Sixsolutionsof1;2arelistedinTable 3-1 .ThefourrealsolutionsareshowninFigure 3-1 .ThetwocomplexsolutionsareshowntosatisfyEquations 3{4 and 3{5 19

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SixsolutionsforCase1 1+0:668610:6329721:161660:2076930:510370:057641:7079i1:0903i40:510370:05764+1:7079i+1:0903i5+0:81733+1:8822161:57259+2:54351 Figure3-1. FourrealsolutionsforCase1 20

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2{19 2{20 arecoupledonlytoEquation 2{22 (i.e.,thetermscontainingd2vanish)andtheseequationsmaybewrittenas ((k1L45+k2p6x)(s2c1s1c2)+k2p6y(c1c2+s1s2)+k1L12s1+k2(p3xs1p3yc1))d1+k1L01(L45(s1c2c1s2)L12s1)=0 (4{2) (L3(k1L12+k2p3xk2L3p3yc1)s1+k2(p3xp6yp3yp6x)c2+(k1L12L45+k2(p3yp6y+p3xp6x))s2)d1k1L12L01L3s1k1L12L01s2L45=0 (4{3) Applyingthe`tan-halfangleconcept'(Equation 2{24 ),toEquations 4{5 4{6 and 4{7 ,thefollowingequationsareobtained (4{5) (4{6) (4{7) where 21

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Itisdesiredtoformasystemofequations 22

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ListofoperationstoobtainSylvesterMatrixforcase2 Equations#Equations#UnknownsAddedUnknowns 4{5 ),( 4{6 )g39(x22;x2;1)( 4{7 )(x22;x2;1)d1(x22;x2;1)d21f( 4{5 ),( 4{6 )gd159f( 4{5 ),( 4{6 )gd21812(x22;x2;1)d31( 4{7 )d1 4{5 ),( 4{6 )gx21616x32( 4{7 )x2x32d1f( 4{5 ),( 4{6 )gd1x2x32d21f( 4{5 ),( 4{6 )gd21x2x32d31( 4{7 )d1 ThematrixSOFLandvectorXOFLarewrittenas 23

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4{5 4{6 4{7 andthus 4{14 maybewrittenas 3.2 )withtheexceptionofthefreelengthofspring1whichisnowsettoL01=2:3m.Equations 4{11 4{12 4{13 areevaluatedandsubstitutedintoEquation 4{15 .Equation 4{17 isobtainedandsolvedtoobtainvaluesofx1.Correspondingvaluesofx2;d1areobtainedfromEquation 4{20 .SolutionsofEquation 4{17 wereasfollows 4{17 maybedividedthroughby(1+x21)4resultingina24thdegreepolynomial. 24

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TwentysolutionsforCase2 4{5 4{6 4{5 {z }Sd1=024x22x2135=2400035(4{21) 4{5 4{6 4{7 .Forthisparticularnumericalexamplethereare8realand12complexsolutionsetsforx1;x2;d1.Abifurcationdiagrambetweenthesolutionx1andvaryingparameterL01isshowninFigure 4-1 .ItisinterestingtoobservethatthefoursolutionsforL01=L02=0casebifurcateasthefreelengthL01isvaried.AtL01=2:3m,thetotalnumberofsolutions 25

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Bifurcationdiagramforsolutionofx1andvaryingparameterL01 26

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EightrealsolutionsforCase2 27

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2{19 2{20 arecoupledtobothEquations 2{22 2{21 .Afterapplyingthe`tan-halfangleconcept'(Equation 2{24 ),thefollowingequationsareobtained. (F1x22+F2x2+F3)d1+(F4x22+F5x2+F6)d2+(F7x22+F8x2+F9)d1d2=0(5{2) (G1x22+G2x2+G3)d1+(G4x22+G5x2+G6)d2+(G7x22+G8x2+G9)d1d2=0(5{3) (H1x22+H2x2+H3)d21+(H4x22+H5x2+H6)=0(5{4) (I1x22+I2x2+I3)d22+(I4x22+I5x2+I6)=0(5{5)where 28

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Itisdesiredtoformasystemofequations 29

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5{2 5{3 aredividedbyd1d2, 5{4 isdividedbyd21, 5{5 isdividedbyd22.Theyarerewrittenas (F1x22+F2x2+F3)d2i+(F4x22+F5x2+F6)d1i+(F7x22+F8x2+F9)=0(5{11) (G1x22+G2x2+G3)d2i+(G4x22+G5x2+G6)d1i+(G7x22+G8x2+G9)=0(5{12) (H1x22+H2x2+H3)+(H4x22+H5x2+H6)d21i=0(5{13) (I1x22+I2x2+I3)+(I4x22+I5x2+I6)d22i=0(5{14)whered1i=1=d1;d2i=1=d2.SBFL;XBFLcanbeobtainedbyperformingtheoperationslistedinTable 5.1 ThedeterminantoftheSylvesterMatrixiszeroforcommonrootsofEquations 5{11 5{12 5{13 5{14 andthus 5{10 maybewrittenas 30

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3.2 )withtheexceptionofthefreelengthsofspring1and2whicharenowsettoL01=5:1m;L02=6:6309m.Equations 5{6 5{7 5{8 5{9 arecalculatednumerically.Next,theSylvesterMatrix(SBFL)wasobtainedbyoperationsexplainedinSection( 5.1 ).Finally,Equation 5{15 issolvedforx1.Observationsforsolutionsofx1were 5{2 5{3 5{4 5{5 5{2 5{3 5{4 5{5 .Fortwoofthesesolutions(Case3andCase6)-f1;f2arezero. 31

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(b)Case2 (c)Case3-f1=f2=0 (d)Case4 (e)Case5 (f)Case6-f1=f2=0 (g)Case7 (h)Case8 (i)Case9 (j)Case10 (k)Case11 (l)Case12Figure5-1. Twenty-fourrealsolutionsforCase3(cases1to12) 32

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(b)Case14 (c)Case15 (d)Case16 (e)Case17 (f)Case18 (g)Case19 (h)Case20 (i)Case21 (j)Case22 (k)Case23 (l)Case24Figure5-2. Twenty-fourrealsolutionsforCase3(cases13to24) 33

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ListofoperationstoobtainSylvesterMatrixforCase3 Equations#Equations#UnknownsAddedUnknowns 5{11 ),( 5{12 )g415(x22;x2;1)( 5{13 )(x22;x2;1)d1i( 5{14 )(x22;x2;1)d2i(x22;x2;1)d21i(x22;x2;1)d22if( 5{11 ),( 5{12 )gd1i818(x22;x2;1)d1id2if( 5{11 ),( 5{12 )gd2if( 5{11 ),( 5{12 )gd1id2i1224(x22;x2;1)d21id2i( 5{13 )d2i(x22;x2;1)d1id22i( 5{14 )d1if( 5{11 ),( 5{12 )gd21i1527(x22;x2;1)d31i( 5{13 )d1if( 5{11 ),( 5{12 )gd22i1830(x22;x2;1)d32i( 5{14 )d1if( 5{11 ),( 5{12 )gd21id2i2639(x22;x2;1)d31id2if( 5{11 ),( 5{12 )gd1id22i(x22;x2;1)d1id32i( 5{13 )d22i(x22;x2;1)d21id22i( 5{14 )d21i( 5{13 )d1id2i( 5{14 )d1id2i 5{11 ),( 5{12 )gx25252x32( 5{13 )x2x32d1i( 5{14 )x2x32d2if( 5{11 ),( 5{12 )gd2ix2x32d21if( 5{11 ),( 5{12 )gd1id2ix2x32d22i( 5{13 )d2ix2x32d1id2i( 5{14 )d1ix2x32d21id2if( 5{11 ),( 5{12 )gd21ix2x32d1id22i( 5{13 )d1ix2x32d31if( 5{11 ),( 5{12 )gd22ix2x32d32i( 5{14 )d1ix2x32d31id2if( 5{11 ),( 5{12 )gd21id2ix2x32d1id32if( 5{11 ),( 5{12 )gd1id22ix2x32d21id22i( 5{13 )d22ix2( 5{14 )d21ix2( 5{13 )d1id2ix2( 5{14 )d1id2ix2

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Twenty-foursolutionsforCase3 35

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36

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23 ].Onerelatestothedisplacementofarigidbody.Theotherrelatestoforceswhichactonarigidbody. 1. Anygivendisplacementofarigidbodycanbeeectedbyarotationaboutanaxiscombinedwithatranslationparalleltothataxis. 2. Aforce,andacoupleinaplaneperpendiculartotheforce,constituteanadequaterepresentationofanysystemofforcesappliedtoarigidbody.PluckercoordinateswereintroducedbyJuliusPluckerinthe19thcenturyasawaytoassignsixhomogenouscoordinatestoeachlineinprojective3-space.InScrewTheory,theyareusedtorepresentthecoordinatesofscrews,twistsandwrenches. 37

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(rr1)S=0 (A{2) PluckercoordinatesofthelinearefS;S0Lgandtheysatisfythefollowingconstraints 38

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$=264SS0L+hS375:(A{6)Fivequantitiesarerequiredtospecifyascrew,ofthese4arerequiredtospecifyaline.Thefthoneinthepitchofthescrew,h.Atwistisrepresentedas$1.Atwistrequiressixalgebraicquantitiesforitscompletespecication,ofthesevearerequiredforcompletespecicationofascrew.Thesixthquantity,theamplitudeoftwist()expressestheangleofrotation.Thedistanceoftranslationistheproductofamplitudeoftwistandpitchofthescrew.Ifpitchiszero,thetwistreducestopurerotationaroundthescrew$.Ifpitchisinnite,thennitetwistisnotpossibleexcepttheamplitudebezero,inwhichcasethetwistreducestopuretranslationparalleltothescrew$.Awrenchisrepresentedasf$1.Awrenchrequiressixalgebraicquantitiesforitscompletespecication,ofthesevearerequiredforcompletespecicationofascrew.Thesixthquantity,theintensityofwrench(f)expressesthemagnitudeofforce.Themomentofcoupleistheproductofintensityofwrenchandpitchofthescrew.Ifpitchiszero,thewrenchreducestopureforcealongthescrew$.Ifpitchisinnitethewrenchreducestocoupleinaplaneperpendiculartothescrew$. 39

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ofdegreesnandmandrootsi;i=1;2;:::n,i;i=1;2;:::mrespectively.Theresultant([ 24 ])isdenedby 40

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Thesystemmaybewrittenasamatrixequation {z }Sp;q2666666666666664x(n+m1)...x(m1)x(m2)...13777777777777775=26666666666666640...00...03777777777777775:(B{5)TheSylvestermatrix(Sp;q)associatedwithpolynomialsp(x)andq(x)isasquarematrixofdimension(n+m)(n+m).ThedeterminantoftheSylvestermatrixwillvanishwhenp(x)andq(x)haveacommonroot.Theconverseisalsotrue.InorderfortheretobeacommonrootforEqns( B{1 )and( B{2 ),itisnecessarythat det(Sp;q)=0:(B{6) 41

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[1] FullerR.Synergetics:TheGeometryofThinking.MacMillanPublishingCo.,Inc.,NewYork,1975. [2] EdmondsonA.AFullerExplanation:TheSynergeticGeometryofR.BuckminsterFuller.Birkhauser,Boston,1987. [3] TobieR.S.Areportonaninquiryintotheexistence,formationandrepresentationoftensilestructures.Masterofindustrialdesignthesis,PrattInstitute,NewYork,1976. [4] PughA.AnIntroductiontoTensegrity.UniversityofCaliforniaPress,1976. [5] DuyJ.,YinJ.andCraneC.Ananalysisforthedesignofself-deployabletensegrityandreinforcedtensegrityprismswithelasticties.InternationalJournalofRoboticsandAutomation,SpecialIssueonComplianceandCompliantMechanisms,17,2002. [6] RothB.andWhiteleyW.Tensegrityframeworks.InAmericanMathematicalSociety,editor,TransactionsoftheAmericanMathematicalSociety,page419446,1981. [7] IngberD.http://www.childrenshospital.org/research/ingber/tensegrity.html.HavardMedicalSchool. [8] HanaorA.Aspectsofdesignofdoublelayertensegritydomes.JournalofSpaceStructures,7(2):101{113,1992. [9] HanaorA.Geometricallyrigiddouble-layertensegritygrids.JournalofSpaceStructures,9(4):227{238,1994. [10] MotroR.Tensegritysystems:thestateoftheart.JournalofSpaceStructures,7(2):75{83,1992. [11] Juan,S.H.andMiratsTur,J.M.Tensegrityframeworks:Staticanalysisreview.MechanismandMachineTheory,2007.inpress. [12] FuF.Structuralbehavioranddesignmethodsoftensegritydomes.JournalofConstructionalSteelResearch,61(1):25{35,2005. [13] HeltonJ.,AdhikariR.,PinaudJ.,Skelton,R.andChanW.Anintroductiontothemechanicsoftensegritystructures.InIEEE,editor,Proceedingsofthe40thIEEEconferenceonDecisionandcontrol,page42544258,2001. [14] S.Levin.Thetensegrity-trussasamodelforspinemechanics:Biotensegrity.JournalofMechanicsinMedicineandBiology,2(3&4):375{388. [15] FuruyaH.Conceptofdeployabletensegritystructuresinspaceapplications.JournalofSpaceStructures,7(2):143{151,1992. 43

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TibertA.DeployableTensegrityStructuresforSpaceApplications.PhDthesis,RoyalInstituteofTechnology,2003.PhDThesis. [17] KennerH.GeodesicMathandHowtoUseIt.UniversityofCaliforniaPress,BerkeleyandLosAngeles,CA,1976. [18] SternI.P.Developmentofdesignequationsforself-deployablen-struttensegritysystems.Master'sthesis,UniversityofFlorida,Gainesville,FL,1999. [19] KnightB.F.DeployableAntennaKinematicsusingTensegrityStructureDesign.PhDthesis,UniversityofFlorida,Gainesville,FL,2000. [20] AldrichJ.ControlSynthesisforaClassofLightandAgileRoboticTensegrityStructures.PhDthesis,UniversityofCalifornia,2004.PhDThesis. [21] RobertsJ.,LipsonH.,PaulC.andF.Cuevas.Gaitproductioninatensegritybasedrobot.InProceedingsofthe2005InternationalConferenceonAdvancedRobotics,2005. [22] CraneC.andDuyJ.KinematicAnalysisofRobotManipulators.CambridgeUniversityPress,March1998. [23] BallR.S.ATreatiseontheTheoryofScrews.CambridgeUniversityPress,1998. [24] WeissteinE.Resultant.FromMathWorld{AWolframWebResource.http://mathworld.wolfram.com/Resultant.html. [25] RaoA.V.DynamicsofParticlesandRigidBodies:ASystematicApproach.Cam-bridgeUniversityPress,2ndedition,2006. [26] VikasV.,BayatJ.,CraneC.andR.Roberts.Kinematicanalysisofaplanartensegritymechanismwithpre-stressedsprings.AdvancesinRobotKinematics,2008. 44

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VisheshVikaswasbornonthe31stofMay,1983inNewDelhi,India.HeattendedhishighschoolatDelhiPublicSchoolRKPuram,Delhi.HerecievedhisBachelorinTechnologyinMechanicalEngineeringfromIndianInstituteofTechnology,GuwahatiinMayof2005.Afterthat,heworkedatMAIA,INRIALorraine(LORIA),France.In2007,hejoinedtheCenterforIntelligentMachinesandRobotics(CIMAR)attheUniversityofFlorida,completinghisMastersofSciencedegreeinMechanicalEngineeringinAugustof2008.UponcompletionofhisMS,VisheshwillpursuePhDinDepartmentofAerospaceandMechanicalEngineeringatUniversityofFlorida. 45