Computers and Structures 88 (2010) 237246Contents lists available at ScienceDirect

Computers and Structures

journal homepage: www.elsevier .com/locate /compstrucAdvanced form-finding of tensegrity structures

Hoang Chi Tran, Jaehong Lee *

Department of Architectural Engineering, Sejong University, 98 Kunja Dong, Kwangjin Ku, Seoul 143-747, Republic of Koreaa r t i c l e i n f o

Article history:Received 22 May 2009Accepted 28 October 2009Available online 30 November 2009

Keywords:Tensegrity structureSingle value decompositionForm-findingEigenvalue problem0045-7949/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.compstruc.2009.10.006

* Corresponding author. Tel.: +82 2 3408 3287; faxE-mail address: jhlee@sejong.ac.kr (J. Lee).a b s t r a c t

A numerical method is presented for form-finding of tensegrity structures. The topology and the types ofmembers are the only information that requires in this form-finding process. The eigenvalue decompo-sition of the force density matrix and the single value decomposition of the equilibrium matrix are per-formed iteratively to find the feasible sets of nodal coordinates and force densities which satisfy theminimum required rank deficiencies of the force density and equilibrium matrices, respectively. Basedon numerical examples it is found that the proposed method is very efficient and robust in searchingself-equilibrium configurations of tensegrity structures.

2009 Elsevier Ltd. All rights reserved.1. Introduction

The tensegrity structures first proposed by Fuller [1] have beendeveloped in recent years due to their innovative forms, light-weight and deployability. They belong to a class of free-standingpre-stressed pin-jointed cablestrut system where contacts are al-lowed among the struts [2]. Their classification is presented asClass 1 (where bars do not touch) and Class 2 (where bars do con-nect to each other at a pivot) [3]. As a pioneering work of form-finding, so-called force density method was proposed by Schek[4] for form-finding of tensile structures. Motro et al. [5] presentedthe dynamic relaxation which has been reliably applied to tensilestructures [6] and many other non-linear problems. Vassart andMotro [7] employed the force density method in symbolic formfor searching new configurations. Recently, Masic et al. [8], Zhangand Ohsaki [9] and Estrada et al. [10] developed new numericalmethods using a force density formulation. Micheletti et al. [11]used a marching procedure for finding stable placements of a giventensegrity, and Zhang et al. [12] employed a refined dynamic relax-ation procedure for form-finding of nonregular tensegrity systems.Most recently, Rieffel et al. [13] introduced an evolutionary algo-rithm for producing large irregular tensegrity structures. Tibertand Pellegrino [14] presented a review paper for the existing meth-ods for form-finding problem of tensegrity structures. The most re-cent review for this problem can be found in Juan and Tur [15].

In most available form-finding methods, the symmetry of struc-ture and/or some of member lengths must be assumed known inadvance. Moreover, element force density coefficients are solvedin symbolic form. For example, (i) symmetric properties arell rights reserved.

: +82 2 3408 3331.employed in a group theory so as to simplify form-finding problemas presented in [8]; (ii) a number of member lengths are specifiedat the start in a dynamic relaxation procedure and non-linear pro-gramming, as presented in [5,6,14]; however, these informationmay not always be available or easy to define at the beginning;and (iii) force density coefficients are considered as symbolic vari-ables which cannot be applied for structures with large number ofmembers [14].

In this paper, a numerical method is presented for form-findingof tensegrity structures. Both Classes 1 and 2 of tensegrity struc-tures are considered. Tensegrities satisfying either stability (i.e.,the tangent stiffness matrix is positive definite) or super stability(i.e., the geometrical stiffness matrix is positive definite) can be ob-tained by present form-finding procedure in a few of remarkableiterations, which is more efficient and versatile than other avail-able methods only dealing with super stable tensigrities (e.g.[9,10]) that are more restrictive in the real mechanical structures.The topology and the types of members, i.e. either compressionor tension, are the only information that requires in this form-find-ing process. In other words, any initial nodal coordinate, symmetricproperties, and member lengths are not necessary for the presentform-finding. The force density matrix is derived from an incidencematrix and an initial set of force densities assigned from proto-types, while the equilibriummatrix is defined by the incidence ma-trix and nodal coordinates. The eigenvalue decomposition of theforce density matrix and the single value decomposition of theequilibrium matrix are performed iteratively to find the feasiblesets of nodal coordinates and force densities which satisfy the min-imum required rank deficiencies of the force density and equilib-rium matrices, respectively. The evaluation of the eigenvalues oftangent stiffness is also included for checking the stability of thetensegrity structures.

Table 1The incidence matrix of the 2-D two-strut tensegrity structure.

C

Member/node 1 2 3 4

(1) 1 1 0 0(2) 0 1 1 0(3) 0 0 1 1(4) 1 0 0 1(5) 1 0 1 0(6) 0 1 0 1

238 H.C. Tran, J. Lee / Computers and Structures 88 (2010) 2372462. Formulation of self-equilibrium equations

2.1. Fundamental assumptions

In this study, the following assumptions are made in tensegritystructures:

The topology of the structure in terms of nodal connectivity isknown.

Members are connected by pin joints. No external load is applied and the self-weight of the structure is

neglected during the form-finding procedure. There are no dissipative forces acting on the system.

2.2. Force density method for the self-equilibrium and free-standingtensegrity structures

For a d-dimensional tensegrity structure with bmembers, n freenodes and nf fixed nodes (supports), its topology can be expressedby a connectivity matrix Cs2 Rbnnf as discussed in [4,16,9].Suppose member k connects nodes i and ji < j), then the ith andjth elements of the kth row of Cs are set to 1 and 1, respectively,as follows:

Csk;p 1 for p i;1 for p j;0 otherwise:

8>: 1

If the free nodes are numbered first, then to the fixed nodes, Cs canbe divided into two parts as

Cs C Cf ; 2

where C2 Rbn and Cf 2 Rbnf describe the connectivities of themembers to the free and fixed nodes, respectively. Let x; y; z2 Rnand xf ; yf ; zf 2 Rnf denote the nodal coordinate vectors of the freeand fixed nodes, respectively, in x, y and z directions. For a simpletwo-dimensional two-strut tensegrity structure as shown in Fig. 1,which consists of six members (b 6, four cables and two struts)and four nodes, the connectivity matrix C2 R64 is given in Table1.

The equilibrium equations in each direction of a general pin-jointed structure given by [4] can be stated as

CTQCx CTQCfxf px; 3aCTQCy CTQCfyf py; 3bCTQCz CTQCf zf pz; 3c

where px;py and pz2 Rn are the vectors of external loads appliedat the free nodes in x; y and z directions, respectively. The symbol,T , denotes the transpose of a matrix or vector. And Q 2 Rbb isdiagonal square matrix, calculated by1

4

3

2

(4)

(1) (2)

(3)

(6)

(5)

Fig. 1. A two-dimensional two-strut tensegrity structure. The thick and thin linesrepresent the struts and cables, respectively.Q diagq; 4

where q2 Rb suggested in [4] is the force density vector, definedby

q fq1; q2; . . . ; qbgT 5

in which each component of this vector is the force fi to length li ra-tio qi fi=li known as force density or self-stressed coefficient in [7].Without external loading, Eq. (3) can be rewritten neglecting theself-weight of the structure as

Dx Dfxf ; 6aDy Dfyf ; 6bDz Df zf ; 6c

where matrices D2 Rnn and Df 2 Rnnf are, respectively, given by

D CTQC; 7aDf CTQCf 7b

or by

D CTdiagqC; 8aDf CTdiagqCf : 8b

It is noted that when external load and self-weight are ignored, atensegrity system does not require any fixed node, and the tenseg-rity geometry is defined by the relative position of the nodes, andthe system can be considered as free-standing, forming a rigid-bodyfree in space [7,16]. In this context, Eq. (7b) and (8b) vanish, and Eq.(6) becomes:

Dx 0; 9aDy 0; 9bDz 0; 9c

where D known as force density matrix [14,10] or stress matrix[1719] can be written directly by [7,18,19] without using Eq. (7)or (8) as follows:

Di;j

qk if nodes i and j are connected by member k;Pk2X

qk for i j;

0 otherwise

8>>>:

10

in which X denotes the set of members connected to node i. Forexample, for the two-dimensional two-strut tensegrity structurein Fig. 1, D can be written explicitly from Eq. (10) as

D

q1 q4 q5 q1 q5 q4q1 q1 q2 q6 q2 q6q5 q2 q2 q3 q5 q3q4 q6 q3 q3 q4 q6

26664

37775:

11

From Eq. (10), it is obvious that D is always square, symmetric andsingular with a nullity of at least 1 since the sum of all components

H.C. Tran, J. Lee / Computers and Structures 88 (2010) 237246 239in any row or column is zero for any tensegrity structure. Differentfrom the matrix D of the cable net, which is always positive definite[4], D of the tensegrity structure is semi-definite due to the exis-tence of struts as compression members, with qk < 0. Consequently,it cannot be invertible. For simplicity, Eq. (9) can be reorganized as

Dx y z 0 0 0: 12

However, by substituting Eq. (8) into Eq. (9) the equilibrium equa-tions of the tensegrity structure can be expressed as

CTdiagqCx 0; 13aCTdiagqCy 0; 13bCTdiagqCz 0: 13c

Eq. (13) can be reorganized as

Aq 0; 14

where A2 Rdnb is known as the equilibrium matrix in [16], de-fined by

A CTdiagCxCTdiagCyCTdiagCz

0B@

1CA: 15

Eq. (12) presents the relation between force densities and nodalcoordinates, while Eq. (14) shows the relation between projectedlengths in x; y and z directions, respectively and force densities. BothEqs. (12) and (14) are linear homogeneous systems of self-equilib-rium equations with respect to nodal coordinates and force densi-ties, respectively.3. Requirement on rank deficiency conditions of force densityand equilibrium matrices

Let q be the vector of force density and C be the incidence ma-trix of a d-dimensional tensegrity structure in self-equilibrium. It iswell known that the set of all solutions to the homogeneous sys-tem of Eq. (12) is the null space of D. The dimension of this nullspace or rank deficiency of D is defined as

nD n rD; 16

where rD rankD. It is obvious that vector I1 f1;1; . . . ;1gT

2 Rn1, is a solution of Eq. (12) since the sum of the elements ofa row or a column of D is always equal to zero. The most importantrank deficiency condition related to semi-definite matrix D ofEq. (12) is defined by

nD P d 1: 17

This condition forces Eq. (12) to yield at least d useful particularsolutions [20] which exclude the above vector I1 due to degener-ating geometry of tensegrity structure [14,9]. These d particularsolutions form a vector space basis for generating a d-dimen-sional tensegrity structure. Therefore, the minimum rank defi-ciency or nullity of D must be d 1 for configuration of anytensegrity structure embedding into Rd, which is equivalent tothe maximum rank condition of D proposed by [1719,16] asfollows:

maxrD n d 1: 18

Similarly, the set of all solutions to the homogeneous system of Eq.(14) lies in the null space of A. Let nA denote dimension of null spaceof the equilibrium matrix A which is computed by

nA b rA; 19

where rA rankA. The second rank deficiency condition which en-sures the existence of at least one state of self-stress can be stated ass nA P 1; 20

where s is known as the number of independent states of self-stress,while the number of infinitesimal mechanisms is computed bym dn rA, as presented in [21,22]. It is clear that Eq. (20) allowsEq. (14) to create at least one useful particular solution [20].

In short, based on these two rank deficiency conditions, Eqs.(17) and (20), the proposed form-finding procedure searches forself-equilibrium configurations that permit the existence of at leastone state of self-stress in the structure. It should be noted thatthese two are necessary but not sufficient conditions which haveto be satisfied for any d-dimensional tensegrity structure to be ina self-equilibrium state [17,14,16]. The sufficient conditions fortensegrity or self-stressed pin-jointed structures can be found in[24,23].4. Form-finding process

The proposed form-finding procedure only needs to know thetopology of structure in terms of the incidence matrix C, and typeof each member, i.e. either cable or strut which is under tension orcompression, respectively. Based on element type, the initial forcedensity coefficients of cables (tension) are automatically assignedas +1 while those of struts (compression) as 1, respectively, asfollows:

q0 1 1 1|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}cables

1 1 1|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}struts

8 nD.

Case 1. The first nD orthonormal eigenvectors of U are directlytaken as potential nodal coordinatesx y z 2 U /1 /2 /nD : 25

The algorithm then iteratively modifies the force density vectorq as small as possible to make the first nD eigenvalues of D be-come null as

ki 0 i 1;2; . . . ;nD: 26

All the projected lengths L2 RbnD of U along nD directions forb members are computed by

L CU C/1 C/2 C/nD 27

to remove one vector /i among nD eigenvector bases of U ifC/i 0 28

or which /i causesdetjLdLdT j 0; 29

where Ld2 Rb which indicates the vector of lengths of b mem-bers from any combination of d eigenvectors among nD aboveeigenvector bases in d-dimensional space (assuming d 3) isgiven by

Ld ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC/i

2 C/j2 C/k

2q

: 30

Eq. (28) shows /i is linearly dependent with the vector I1 whileEq. (29) proves at least one of b members in d-dimensionalstructure has a zero length. If there is no /i which satisfiesEqs. (28) and (29), the first three eigenvectors of U are chosenas nodal coordinates x y z for 3-dimensional tensegritystructure.

Accordingly, D will finally have the required rank deficiency nD

without any negative eigenvalue. It implies D is positive semi-definite, and any tensegrity structure falling into this case issuper stable regardless of material properties and level ofself-stress coefficients [1719,25].

Case 2. Where p > nD, the rank deficiency may be forced to belarger than requirement or enough but D may not be positivesemi-definite during iteration. Additionally, the proposedform-finding procedure will evaluate the tangent stiffnessmatrix of the tensegrity structure which is given in[24,23,26,9] as follows:KT KE KG; 31

where

KE Adiageal

AT ; 32a

KG I3 D 32b

in which KE is the linear stiffness matrix, KG is the geometricalstiffness matrix induced by pre-stressed or self-stressed state;e;a and l 2 Rb denote the vectors of Youngs moduli, cross-sec-tional areas and pre-stressed lengths of b members of thetensegrity structure, respectively; I32 R33 and are the unitmatrix and tensor product, respectively. If the tangent stiffnessmatrix is positive definite, then the structure is stable when itsrigid-body motions are constrained; i.e., the quadratic form ofKT is positive with respect to any non-trivial motion d as

dTKTd > 0 33oreigKT k1 P k2 P P kdnr > 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}positive stiffness

k1 k2 kr 0|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}rigid-body motions

;

34

where r dd 1=2 is the number of independent rigid-bodymotions. Using this criterion, stability of any pre-stressed ortensegrity structure can be controlled by checking eigenvaluesof tangent stiffness matrix of the structure [24,23]. In short,the best scenario of configuration in 3-dimensional space isformed by three best candidate eigenvectors selected from thefirst fourth eigenvector bases which corresponding to the firstfourth smallest eigenvalues, respectively. These eigenvalueswill be gradually modified to be zero by the proposed iterativeform-finding algorithm. In other words, the proposed form-finding procedure has repeatedly approximated equilibriumconfiguration such thatDx y z 0 0 0: 354.2. Single value decomposition of the equilibrium matrix

The equilibrium matrix A is computed by substituting the set ofapproximated nodal coordinates [x y z] from Eq. (35) into Eq.(15). In order to solve linear homogeneous system (Eq. (14)) thesingle value decomposition [20] is carried out on the equilibriummatrix A:

A UVWT ; 36

where U2 Rdndn u1 u2 udn and W2 Rbb w1 w2 wb are the orthogonal matrices. V2 Rdnb is a diagonal ma-trix with non-negative single values of A in decreasing order as

r1 P r2 P P rb P 0: 37

As indicated in Eq. (22b), the iterative form-finding algorithm issuccessful in case of nA 1. Accordingly, there are also two casesfor s during the iterative form-finding procedure:

Case 1. s 0, there exists no null space of A. In other words,the right single value rb of A in V is not equal to zero. Itdenotes that Eq. (14) has no nonzero force density vector qas a solution. In this case, if the right single vector basis wbin W corresponding to smallest singular value rb in V is usedas the approximated q, the sign of q may not match with thatof q0. Thus, all columns of W employed to compute a vector qthat best matches q0 are scanned by form-finding procedure.The procedure stops sign-finding until the sign of all compo-nents of wjj b; b 1; . . . ;1 is identical to that of q0, i.e.signwj signq0. That vector wj is directly taken as theapproximated q. In doing so, the form-finding procedure definesthe approximated q that matches in signs with q0, such thatAq 0: 38Case 2. s 1, it is known [27] that the bases of vector spaces offorce densities and mechanisms of any tensegrity structure arecalculated from the null spaces of the equilibrium matrix. Inthis case, the matrices U and W from Eq. (36) can be expressed,respectively, asU u1 u2 urA jm1 mdnrA 39aW w1 w2 wb1jq1 39b

where the vectors m2 Rdn denote the m dn rA infinitesi-mal mechanisms; and the vector q12 Rb matching in signs

56 (5)

H.C. Tran, J. Lee / Computers and Structures 88 (2010) 237246 241with q0 is indeed the single state of self-stress which satisfiesthe homogeneous Eq. (14).(4)(6)In summary, the eigenvalue decomposition of force density ma-

1

2 3

4

(1)

(2)

(3)

(7)

(8) (9)

Fig. 2. A two-dimensional hexagonal tensegrity structure.

Table 2The force density coefficients of the 2-D hexagonal tensegrity structure.

Force densitycoefficients

Tibert and Pellegrino[14]

Estrada et al.[10]

Present

q1q6 1.0 1.0 1.0q7q9 0.5 0.5 -0.5

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.0-0.2 0.2-0.4 0.4

Fig. 3. The obtained geometry of the two-dimensional hexagonal tensegritystructure.trix D and the single value decomposition of the equilibrium ma-trix A are performed iteratively to find the feasible sets of nodalcoordinates [x y z] and force density vector q which satisfythe minimum required rank deficiencies of the force density andequilibrium matrices as presented in Eq. (22), respectively.

Let the coordinates of node i be denoted as pi xi yi zi 2 R3.It should be noted that since the tensegrity structure without fixednodes is a free body in the space, the force density vector q doesnot change under affine transformation [28,29,8] which transformspi to pi as follows:

pi piT t; 40

where T2 R33 and t 2 R3 represent affine transformation whichis a composition of rotations, translations, dilations, and shears. Itpreserves collinearity and ratios of distances; i.e., all points lyingon a line are transformed to points on a line, and ratios of the dis-tances between any pairs of the points on the line are preserved[30]. Hence, numerous geometries in self-equilibrium are derivedby Eq. (40) with the same force density vector q. For simplicity, inthis paper, it is assumed that t = 0 and T is considered as

T 1 0 00 1 00 0 1

0B@

1CA: 41

Since the tensegrity structure should satisfy the self-equilibriumconditions, the vector of unbalanced forces ef 2 Rdn defined as fol-lows can be used for evaluating the accuracy of the results:ef Aq: 42

The Euclidean norm of ef is used to define the design error as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffief ef T

q: 43

Two set of parameters which are nodal coordinates and force den-sity vector can be simultaneously defined by proposed form-findingprocedure through following algorithm.1

4

3

2

(4)

(1) (2)

(3)

(6)

(5)(7)

(8)

5

Fig. 4. A two-dimensional four-strut tensegrity structure.

-0.2

0.0

0.2

0.4Algorithm

Step 1: Define C by Eq. (1) for the given topology of tensegritystructure.

Step 2: Specify the type of each member to generate initial forcedensity vector q0 by Eq. (21). Set i 0.

Step 3: Calculate Di using Eq. (8). Step 4: Carry out Eq. (23) to define x y zi through Eq. (35). Step 5: Determine Ai by Eq. (15). Step 6: Perform Eq. (36) to define qi1 through Eq. (38). Step 7: Define Di1 with qi1 by Eq. (8). If Eq. (22) is satisfied, the

solutions exist. Otherwise, set i i 1 and return to Step 4. Step 8: If D is positive semi-definite, go to Step 10. Otherwise,

specify material property, force density coefficient for eachmember based on force density vector found.

Step 9: If Eq. (34) is fulfilled, go to Step 10. Otherwise, set i i 1and return to Step 4.

Step 10: The process is terminated until Eq. (43) has beenchecked. The final coordinates and force density vector are thesolutions. Otherwise, set i i 1 and return to Step 4.0.0

-0.4

-0.2-0.4 0.2 0.4

Fig. 5. The obtained geometry of the two-dimensional four-strut tensegritystructure.5. Numerical examples

Numerical examples are presented for several tensegrity struc-tures using Matlab Version 7.4 (R2007a) [31]. Based on algorithm

Table 3Eigenvalues of the tangent stiffness matrix of the 2-D four-strut tensegrity structure.

Axial stiffness k1 k2 k3 k4 k5 k6 k7

eiai 10 11.2930 11.2930 14.1421 20.0000 34.1421 43.1334 43.1334

242 H.C. Tran, J. Lee / Computers and Structures 88 (2010) 237246developed, both nodal coordinates and force density vector aresimultaneously defined with limited information of nodal connec-tivity and the type of the each member. A hexagon, expandableoctahedron and truncated icosahedron belong to Class 1 tensegritystructure, while the remainings are of Class 2.

5.1. Two-dimensional tensegrity structures

5.1.1. HexagonThe initial topology of the hexagonal tensegrity structure com-

prising three struts and six cables (Fig. 2) which was studied by Ti-bert and Pellegrino [14] and Estrada et al. [10] is herein used forverification purpose. Initial nodal coordinates, member lengthsand force density coefficients are unknown in advance. The onlyTable 4The force density coefficients of the expandable octahedron tensegrity structure.

Force densitycoefficients

Tibert and Pellegrino[14]

Estrada et al.[10]

Present

q1q24 1.0 1.0 1.0q25q30 1.5 1.5 1.5

56

78

910

42

1112

1 3

(13)

(23) (22)

(20)

(19)

(7)

(5)

(1)

(9)

(2)

(28)

(17)

(21)

(14)

(3) (4)

(11)

(6)

(12)

(8)

(24)

(18)

(10)

(15)(16)(30)

(29)

(25)

(27)

(26)

Fig. 6. An expandable octahedron tensegrity structure.

0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.40.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.5

7

5

11

1

2

9

8

3

4

6

12

10

a b

Fig. 7. The obtained geometry of the expandable octahedroninformation is the incidence matrix C and the type of each memberwhich is employed to automatically assign the initial force densityvector by proposed form-finding procedure as

q0 fq1q6 1; q7q9 1gT: 44

The obtained force density vector normalized with respect to theforce density coefficient of the cable 1, as presented in Table 2,agrees well with those of Tibert and Pellegrino [14] and Estradaet al. [10]. The associated stable configuration of the structure isplotted in Fig. 3. The form-finding procedure converges in onlyone iteration with the design error 8:028 1016. The structureobtained has only one self-stress state s 1 and one infinitesimalmechanism m 1 when its three rigid-body motions are con-strained indicating it is statically and kinematically indeterminate[22]. The force density matrix D is positive semi-definite, and thestructure is certainly super stable regardless of materials and0.4 0.2 0 0.20.4

0.5

0

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

7

5

11

1

9

2

3

8

4

6

12

10

tensegrity structure. (a) Top view, (b) perspective view.

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Number of iterations toward selfequilibrium

Des

ign

erro

r ()

Fig. 8. The convergence of the proposed iterative algorithm for the expandableoctahedron.

H.C. Tran, J. Lee / Computers and Structures 88 (2010) 237246 243prestress levels [1719,25]. Consequently, the proposed form-find-ing procedure with limited information about the incidence matrixand element prototype is indeed capable of finding a self-equilib-rium stable tensegrity structure by imposing the two necessaryrank deficiency conditions.

5.1.2. Four-strut tensegrityAs a next example, consider a two-dimensional four-strut

tensegrity structure comprising four struts and four cables(Fig. 4). Similarly, the only information is the incidence matrix Cand the type of each member which is employed to automaticallyassign the initial force density vector by proposed form-findingprocedure as

q0 fq1q4 1; q5q8 1gT : 45

The obtained force density vector normalized with respect to theforce density coefficient of the cable 1 is as follows:

q fq1q4 1; q5q8 2gT : 46

The associated configuration of the structure whose nodalcoordinates are obtained and presented in Fig. 5. The form-findingTable 5Main specification in form-finding procedure of the expandable octahedron tenseg-rity structure at every iteration.

Iteration q1q24 q25q30 ls=lc

1 0.9073 1.2658 1.6180 6.321 101

2 0.8856 1.3095 1.6325 1.181 1013 0.8811 1.3179 1.6330 2.330 1024 0.8802 1.3196 1.6330 4.700 1035 0.8800 1.3199 1.6330 9.000 1046 0.8800 1.3200 1.6330 2.000 1047 0.8800 1.3200 1.6330 3.722 1058 0.8800 1.3200 1.6330 7.443 1069 0.8800 1.3200 1.6330 1.489 10610 0.8800 1.3200 1.6330 2.977 10711 0.8800 1.3200 1.6330 5.955 10812 0.8800 1.3200 1.6330 1.191 10813 0.8800 1.3200 1.6330 2.382 109

Table 6The force density coefficients of the truncated icosahedron tensegrity structure.

Force density coefficients Estrada et al. [10] Present

q1q60 1.0000 1.0000q61q90 0.6775 0.6775q91q120 0.3285 0.3285

0.2 0.15 0.1 0.05 0 0.05 0.1 0.15 0.2

0.2

0.15

0.1

0.05

0

0.05

0.1

0.15

0.2

5

4

21

27

1

26

3

222

25

2830

7

16

6

29

23

24

20

17

11

468

12

50

10

51

947

191815

45

41

52

13

5549

14

48

44

42

31

53

36

54

40

32

4335

57

37

58

39

33

56

34

38

59

60

0

0

0

0

a b

Fig. 9. The obtained geometry of the truncated icosahedronprocedure converges in only one iteration with the design error 2:220 1015. The structure achieved has only one self-stressstate s 1 and no infinitesimal mechanism m 0 after con-straining its three rigid-body motions indicating it is statically inde-terminate and kinematically determinate [22].

Contrary to all the other approaches, the proposed form-findingdoes not require the force density matrix positive semi-definite.Particularly in this problem, the force density matrix D is negativesemi-definite (possessing one negative eigenvalue) implying thatthe structure is not super stable. Accordingly, tangent stiffness ofthe structures has been investigated using Eq. (34). For simplicity,all members are assumed to have the same axial stiffness eiai 10.The force density coefficients of the members are particularly setas Eq. (46). The nonzero eigenvalues of KT after neglecting firstthree zero ones corresponding to three rigid-body motions arelisted in Table 3 which shows the smallest is 11.2930; i.e., thestructure is definitely stable.5.2. Three-dimensional tensegrity structures

5.2.1. Expandable octahedronThe tensegrity structure consisting of 6 struts and 24 cables

(Fig. 6) was investigated first by Tibert and Pellegrino [14] and then0.20

0.2 0.2 0.1 00.1 0.2

0.2

.15

0.1

.05

0

.05

0.1

.15

0.2

29

30

28

26

27

51

52

50

8

46

7

4925

55

53

5

1

21

54

9

4724

6

48

4

10

57

2

22

32

23

56

3

45

31

33

58

44

12

6016

13

11

41

2059

35

34

43

17

42

1415

19

3818

39

37

40

36

tensegrity structure. (a) Top view, (b) perspective view.

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Number of iterations toward selfequilibrium

Des

ign

erro

r ()

Fig. 10. The convergence of the proposed iterative algorithm for the truncatedicosahedron.

244 H.C. Tran, J. Lee / Computers and Structures 88 (2010) 237246by Estrada et al. [10] using symbolic analysis and numerical meth-od, respectively. The calculated force density vector after normaliz-ing with respect to the force density coefficient of the cable 1, aspresented in Table 4, agrees well with those of Tibert and Pellegri-no [14] and Estrada et al. [10]. The associated stable configurationof the structure is presented in Fig. 7. All of struts and cables havethe same length, respectively. The length ratio between struts andcables is ls=lc 1:633 coinciding with the one ls=lc 1:63299 sta-ted by Estrada et al. [10]. The form-finding procedure converges in13 iterations. The design error convergence is described in Fig. 8with 2:382 109.

In order to illustrate the proposed form-finding procedure, mainspecification at every step of iteration is presented in Table 5. Ascan be seen from this table, for the practical purpose it seemsthe proposed algorithm sufficiently converges at iteration 6, butfor the demonstration of the efficiency all iterations are displayed.The structure obtained has only one self-stress state s 1 andone infinitesimal mechanism m 1 except for its six rigid-bodymotions. Accordingly, it belongs to statically and kinematicallyindeterminate structure [22]. The force density matrices D is posi-tive semi-definite which leads structure to super stable regardlessof materials and prestress levels [1719,25].5.2.2. Truncated icosahedronMore complicated example is the truncated icosahedron

tensegrity structure with 30 struts and 90 cables which was ana-lyzed by Estrada et al. [10]. The calculated force density vectorafter normalizing with respect to the force density coefficient ofthe cable 1, as presented in Table 6, agrees well with that of Estradaet al. [10]. The associated stable configuration of the structure isplotted in Fig. 9. The design error shown in Fig. 10 which ex-poses the convergence of the proposed form-finding in 39 itera-0.6 0.4 0.2 0 0.2 0.40.5

0.4

0.3

0.2

0.1

0

0.1

0.2

0.3

0.4

0.511

44

56

22

33

0

0

0

0

0

0

a b

Fig. 12. The obtained geometry of the octahedral cell ten

1

23

4

5

6

(1) (2)(4)

(5)(6) (8)

(7)

(9) (10)

(11)(12)

(13)

(3)

Fig. 11. An octahedral cell tensegrity structure.tions (a little bit more effective in convergence in comparisonwith 45 iterations presented by [10]) is 7:572 109. The obtainedstructure possesses one state of self-stress s 1 and 55 infinites-imal mechanisms m 55 excluding its six rigid-body motions.The force density matrix D is positive semi-definite which leadsstructure to super stable regardless of materials and prestress lev-els [1719,25]. It is clear that the introduction of single prestressstiffens all infinitesimal mechanisms to make the structure stable.

5.2.3. Octahedral cellThe system shown in Fig. 11 has five struts and eight cables.

Four of five struts form a quadrangular polygon. The calculatedforce density vector after normalizing with respect to the forcedensity coefficient of the cable 1 is:

q fq1q8 1; q9q12 1; q13 2gT: 47

The associated configuration of the structure whose nodal coordi-nates are obtained and displayed in Fig. 12. The form-finding proce-dure converges in only one iteration with the design error 3:279 1015. The structure obtained has only one state ofself-stress s 1 and no infinitesimal mechanism m 0 afterconstraining its six rigid-body motions indicating it is staticallyindeterminate and kinematically determinate [22].

Similar to the force density matrix D of the four-strut tensegrity,that of the octahedral cell is also negative semi-definite (possess-ing one negative eigenvalue) implying that the structure is notsuper stable. Accordingly, tangent stiffness of the structures hasbeen investigated using Eq. (34). For simplicity, all members areassumed to have the same axial stiffness eiai 2. The force densitycoefficient of each member is directly set as Eq. (47). The nonzeroeigenvalues of KT after neglecting first six zero ones correspondingto six rigid-body motions are listed in Table 7 which shows thesmallest is 0.1002; i.e., the structure is mechanically stable.

5.2.4. Tensegrity domeA three-dimensional tensegrity dome displayed in Fig. 13 has

six struts and 12 cables. The calculated force density vector afternormalizing with respect to the force density coefficient of thecable 1 is:

q fq1q6 1:0000; q7q12 1:4574;q13q15 0:5000; q16q18 0:5426g

T: 48

Fig. 14 shows the obtained self-equilibrium configuration. Theform-finding procedure converges in only one iteration with the de-sign error 1:935 1015. The structure obtained has only one0.6 0.40.2 0

0.2 0.4

0.5

0

0.5

.6

.4

.2

0

.2

.4

.6

44

11

5

6

33

22

segrity structure. (a) Top view, (b) perspective view.

0.4 0.2 0 0.2 0.4 0.6

0.4

0.2

0

0.2

0.4

0.6

11

45

89

67

33

220.4

0.20

0.20.4

0.6

0.40.2

00.2

0.40.6

0.4

0.2

0

0.2

0.4

11

5

46

7

228

9

33

a b

Fig. 14. The obtained geometry of the three-dimensional six-strut tensegrity dome structure (a) Top view, (b) Perspective view.

Table 7Eigenvalues of the tangent stiffness matrix of the octahedral cell tensegrity structure.

Axial stiffness k1 k2 k3 k4 k5 k6 k7 k8 k9 k10 k11 k12

eiai 2 0.1002 1.9605 2.4870 2.8353 3.6434 3.9516 4.1584 4.1802 9.2446 9.4750 9.5693 12.3043

(3)

(2)

(9)(5)

(1)

(8)(6)

(4)

(7)

(10)

(11)

(12) (13)

(14)(15)

(17)

(18)

(16)

3

1

2

4

5

6

7

8

9

Fig. 13. A three-dimensional six-strut tensegrity dome structure.

H.C. Tran, J. Lee / Computers and Structures 88 (2010) 237246 245state of self-stress s 1 and four infinitesimal mechanismsm 4 after constraining its six rigid-body motions indicating itis statically and kinematically indeterminate [22]. The force densitymatrix D is positive semi-definite which leads structure to superstable regardless of materials and prestress levels [1719,25].6. Concluding remarks

The advanced form-finding procedure for both Classes 1 and 2tensegrity structures has been proposed. The force density matrixis derived from the incidence matrix and initial set of force densi-ties formed by the vector of type of member forces. The elementsof this vector consist of unitary entries +1 and 1 for membersin tension and compression, respectively. The equilibrium matrixis defined by the incidence matrix and nodal coordinates. Theeigenvalue decomposition of the force density matrix and the sin-gle value decomposition of the equilibrium matrix are performediteratively to find the range of feasible sets of nodal coordinatesand force densities which satisfy the required rank deficiencies ofthe force density and equilibrium matrices, respectively.Any assumption about initial nodal coordinates, symmetricproperties, member lengths and the positive semi-definite condi-tion of the force density matrix is not necessary in the proposedform-finding procedure. A rigorous definition is given for the re-quired rank deficiencies of the force density and equilibriummatri-ces that lead to a stable non-degenerate d-dimensional tensegritystructure. The eigenvalues of the tangent stiffness are also evalu-ated for checking the stability of the tensegrity structures. In thenumerical examples, a very good convergence of the proposedmethod has been shown for two-dimensional and three-dimen-sional tensegrity structures. The proposed algorithm is stronglycapable of searching novel configurations with limited informationof topology and the members type. As a natural extension of thisresearch, form-finding with more complicated constraints awaitsfurther attention.Acknowledgments

This research was supported by a grant (code#06-R&D-B03)from Cutting-edge Urban Development Program funded by the

246 H.C. Tran, J. Lee / Computers and Structures 88 (2010) 237246Ministry of Land, Transport and Maritime Affairs of Korean govern-ment. The authors also would like to thank the anonymous review-ers for their suggestions in improving the standard of themanuscript.

References

[1] Fuller RB. Synergetics explorations in the geometry of thinking. London,UK: Macmillan; 1975.

[2] Wang BB. Free-standing tension structures. From tensegrity systems to cablestrut systems. London and New York: Taylor and Francis; 2004.

[3] Pinaud JP, Solari S, Skelton RE. Deployment of a class 2 tensegrity boom. ProcSPIE Smart Struct Mater 2004;5390:15562.

[4] Schek HJ. The force density method for form finding and computation ofgeneral networks. Comput Methods Appl Mech Eng 1974;3:11534.

[5] Motro R, Najari S, Jouanna P. Static and dynamic analysis of tensegrity systems.In: Proceedings of the international symposium on shell and spatial structures,computational aspects. Springer; 1986. p. 2709.

[6] Barnes MR. Form finding and analysis of tension structures by dynamicrelaxation. Int J Space Struct 1999;14(2):89104.

[7] Vassart N, Motro R. Multiparametered form finding method: application totensegrity systems. Int J Space Struct 1999;14(2):14754.

[8] Masic M, Skelton R, Gill P. Algebraic tensegrity form-finding. Int J Solids Struct2005;42(1617):483358.

[9] Zhang JY, Ohsaki M. Adaptive force density method for form-finding problemof tensegrity structures. Int J Solids Struct 2006;43(18-19):565873.

[10] Estrada G, Bungartz H, Mohrdieck C. Numerical form-finding of tensegritystructures. Int J Solids Struct 2006;43(22-23):685568.

[11] Micheletti A, Williams WO. A marching procedure for form-finding fortensegrity structures. J Mech Mater Struct 2007;2(5):10126.

[12] Zhang L, Maurin B, Motro R. Form-finding of nonregular tensegrity systems. JStruct Eng ASCE 2006;132(9):143540.

[13] Rieffel J, Valero-Cuevas F, Lipson H. Automated discovery and optimization oflarge irregular tensegrity structures. Comput Struct 2009;87(56):36879.[14] Tibert AG, Pellegrino S. Review of form-finding methods for tensegritystructures. Int J Space Struct 2003;18(4):20923.

[15] Juan SH, Tur JMM. Tensegrity frameworks: static analysis review. Mech MachTheory 2008;43(7):85981.

[16] Motro R. Tensegrity: Structural Systems for the Future. London: Kogan Page;2003.

[17] Connelly R. Rigidity and energy. Invent Math 1982;66(1):1133.[18] Connelly R, Terrell M. Globally rigid symmetric tensegrities. Topol Struct

1995;21:5978.[19] Connelly R. Tensegrity structures why are they stable. In: Thorpe MF, Duxbury

PM, editors. Rigidity Theory and Applications. Kluwer Academic Publishers;1999. p. 4754.

[20] Meyer CD. Matrix analysis and applied linear algebra. SIAM; 2000.[21] Calladine CR. Buckminster Fullers tensegrity structures and Clerk Maxwells

rules for the construction of stiff frames. Int J Solids Struct 1978;14(2):16172.[22] Pellegrino S, Calladine CR. Matrix analysis of statically and kinematically

indeterminate frameworks. Int J Solids Struct 1986;22(4):40928.[23] Ohsaki M, Zhang JY. Stability conditions of prestressed pin-jointed structures.

Int J Nonlinear Mech 2006;41(10):110917.[24] Murakami H. Static and dynamic analyses of tensegrity structures. Int J Solids

Struct 2001;38(20):361529.[25] Connelly R. Generic global rigidity. Disc Comput Geom 2005;33(4):54963.[26] Guest S. The stiffness of prestressed frameworks: a unifying approach. Int J

Solids Struct 2006;43(34):84254.[27] Pellegrino S. Structural computations with the singular value decomposition of

the equilibrium matrix. Int J Solids Struct 1993;30(21):302535.[28] Tarnai T. Duality between plane trusses and grillages. Int J Solids Struct

1989;25(12):1395409.[29] Connelly R, Whiteley W. Second-order rigidity and prestress stability for

tensegrity frameworks. SIAM J Disc Math 1996;9(3):45391.[30] Weisstein EW. Affine transformation; MathWorld, .[31] Yang WY, Cao W, Chung TS. Applied numerical methods using Matlabs. United

States of America: Wiley; 2005.

http://www.mathworld.wolfram.com/AffineTransformation.htmlhttp://www.mathworld.wolfram.com/AffineTransformation.html

Advanced form-finding of tensegrity structuresIntroductionFormulation of self-equilibrium equationsFundamental assumptionsForce density method for the self-equilibrium and free-standing tensegrity structures

Requirement on rank deficiency conditions of force density and equilibrium matricesForm-finding processEigenvalue decomposition of force density matrixSingle value decomposition of the equilibrium matrix

Numerical examplesTwo-dimensional tensegrity structuresHexagonFour-strut tensegrity

Three-dimensional tensegrity structuresExpandable octahedronTruncated icosahedronOctahedral cellTensegrity dome

Concluding remarksAcknowledgmentsReferences