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## 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:
- 2008
- 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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- 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.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
- Rights Management:
- Copyright Vikas, Vishesh. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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- LD1780 2008 ( lcc )
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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 |

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PAGE 1 1 PAGE 2 2 PAGE 4 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 PAGE 5 Table page 3-1SixsolutionsforCase1 ................................ 20 4-1ListofoperationstoobtainSylvesterMatrixforcase2 .............. 23 4-2TwentysolutionsforCase2 ............................. 25 5-1ListofoperationstoobtainSylvesterMatrixforCase3 .............. 34 5-2Twenty-foursolutionsforCase3 .......................... 35 5 PAGE 6 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 PAGE 7 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 PAGE 8 8 PAGE 9 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 PAGE 10 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 PAGE 11 1-1 1-2 )isthebiologicalmotivationofthemechanisminthefollowingresearch.Inthefollowingresearch,aplanarpre-stressedtensegritystructureisexamined.Thestructureconsistsofthreestrutsconnectedbytwoties. Figure1-1. Biologicalmodeloftheknee Figure1-2. Tensegritybasedmodelofcross-sectionofknee 11 PAGE 12 2-1 .Thetopplatform(indicatedbypoints4,5,and6)isconnectedtothebaseplatform(indicatedbypoints1,2,and3)bytwospringelementswhosefreelengthsareL01andL02andbyavariablelengthconnectorwhoselengthisreferredtoasL3.Althoughthisdoesnotmatchtheexactdenitionoftensegrity,thedeviceisprestressedinthesamemannerasatensegritymechanism.Theexactproblemstatementisasfollows: Given: Find:Allstaticequilibriumcongurations Figure2-1. Tensegritymechanism 12 PAGE 13 2-1 .Otherparameterswereinvestigated,butnoneyieldedalesscomplicatedsolutionthanispresentedhere.Theco-ordinatesofpoints1to6are WherethesuperscriptsB;Tdenotetheco-ordinatesystemsxedonthereferenceframesofthebottomplatformandtopplatformsrespectively.Itisalsoknownthat 13 PAGE 14 $1=264S01S01L375;$2=264S02S02L375;$3=264S03S03L375(2{7)and Forcesinthethreelegsaref1;f2;f3whichmaybeexpressedas (2{11) (2{12) Forcesinthespringsandconnectorarewrencheswithzeropitch(i.e.,pureforcesalongthedirectionoftherespectivelines).Forstaticequilibrium,thesumofthethreewrenchesmustbeequaltozero. 14 PAGE 15 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 PAGE 16 16 PAGE 17 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 PAGE 19 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 PAGE 20 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 PAGE 21 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 PAGE 22 Itisdesiredtoformasystemofequations 22 PAGE 23 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 PAGE 24 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 PAGE 25 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 PAGE 26 Bifurcationdiagramforsolutionofx1andvaryingparameterL01 26 PAGE 27 EightrealsolutionsforCase2 27 PAGE 28 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 PAGE 29 Itisdesiredtoformasystemofequations 29 PAGE 30 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 PAGE 31 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 PAGE 32 (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 PAGE 33 (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 PAGE 34 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 PAGE 35 Twenty-foursolutionsforCase3 35 PAGE 36 36 PAGE 37 23 ].Onerelatestothedisplacementofarigidbody.Theotherrelatestoforceswhichactonarigidbody. 1. Anygivendisplacementofarigidbodycanbeeectedbyarotationaboutanaxiscombinedwithatranslationparalleltothataxis. 2. Aforce,andacoupleinaplaneperpendiculartotheforce,constituteanadequaterepresentationofanysystemofforcesappliedtoarigidbody.PluckercoordinateswereintroducedbyJuliusPluckerinthe19thcenturyasawaytoassignsixhomogenouscoordinatestoeachlineinprojective3-space.InScrewTheory,theyareusedtorepresentthecoordinatesofscrews,twistsandwrenches. 37 PAGE 38 (rr1)S=0 (A{2) PluckercoordinatesofthelinearefS;S0Lgandtheysatisfythefollowingconstraints 38 PAGE 39 $=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 PAGE 40 ofdegreesnandmandrootsi;i=1;2;:::n,i;i=1;2;:::mrespectively.Theresultant([ 24 ])isdenedby 40 PAGE 41 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 PAGE 42 42 PAGE 43 [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 PAGE 44 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 PAGE 45 VisheshVikaswasbornonthe31stofMay,1983inNewDelhi,India.HeattendedhishighschoolatDelhiPublicSchoolRKPuram,Delhi.HerecievedhisBachelorinTechnologyinMechanicalEngineeringfromIndianInstituteofTechnology,GuwahatiinMayof2005.Afterthat,heworkedatMAIA,INRIALorraine(LORIA),France.In2007,hejoinedtheCenterforIntelligentMachinesandRobotics(CIMAR)attheUniversityofFlorida,completinghisMastersofSciencedegreeinMechanicalEngineeringinAugustof2008.UponcompletionofhisMS,VisheshwillpursuePhDinDepartmentofAerospaceandMechanicalEngineeringatUniversityofFlorida. 45 |