Linear dynamics of tensegrity structures - Virginia...

Engineering Structures 24 (2002) 671–685www.elsevier.com/locate/engstruct

Linear dynamics of tensegrity structures

Cornel Sultana,*, Martin Corlessb, Robert E. Skeltonc

a Harvard University, Boston, MA 02115, USAb Purdue University, West Lafayette, IN 47907-1282, USAc University of California, La Jolla, CA 92093-0411, USA

Received 11 December 2000; received in revised form 10 August 2001; accepted 24 September 2001

Abstract

The linearized equations of motion for tensegrity structures around arbitrary equilibrium configurations are derived. For certaintensegrity structures which yield particular equilibrium configurations of practical interest, the linearized models of their dynamicsaround these configurations are presented. Evidence which indicates that these equilibria are stable is given and some stiffness anddynamic properties of these structures are investigated. 2002 Elsevier Science Ltd. All rights reserved.

Keywords:Tensegrity structures; Dynamics; Vibration; Damping

1. Introduction

Since their introduction by Snelson, Fuller andEmmerich around 1948, tensegrity structures have beena matter of surprise and fascination. Their ability tomaintain an equilibrium shape under no external force (aproperty of tensegrity structures called prestressability),their flexibility achieved through an ingenious designwhich combines rigid elements and elastic tendons, andtheir very low weight, aroused the interest of artists,scientists, and engineers.

Several documents show that patents for tensegritystructures were applied for, almost simultaneously, byR.B. Fuller in the US and by D.G. Emmerich in France.What happened afterwards was a source of controversybetween Snelson, Fuller and Emmerich regarding thepriority of the invention (interested readers are referredto articles published in the International Journal of SpaceStructures 11(1–2), 1996). It should be emphasized thatthe structural system described by the three authors isthe same: a simple structure comprising three struts andnine tendons. The tensegrity concept was later gen-eralized by other researchers [1–7].

Tensegrity structures are lattices that form spatial net-works depending on the arrangement of the vertices:

* Corresponding author. Tel.:+1-617-381-0013.E-mail address:[email protected] (C. Sultan).

0141-0296/02/$ - see front matter 2002 Elsevier Science Ltd. All rights reserved.PII: S0141-0296 (01)00130-4

tower-like structures, layered networks, or crystallinetype networks according to the number of directions theydevelop. They consist of a set ofsoftmembers (e.g. elas-tic tendons), and a set ofhard members (e.g. bars). Aperspective view of a tensegrity structure composed of6 bars and 24 tendons is given in Fig. 1. It is apparentthat tensegrity structures are very flexible, being capableof large displacement; their shape can be easily changedby modifying the tendons lengths.

In tensegrity structures research, more attention hasbeen paid so far to the statics of these structures than totheir dynamics. Due to substantial contributions by vari-ous researchers [3,8,2,9–15,7,16,17] tensegrity staticsresearch has reached a certain stage of maturity, whereasthese structures dynamics is still an emerging field. It isour strong conviction that research in tensegrity struc-tures should focus considerably more on their dynamicsand control. This is so because tensegrity structures areexcellent candidates for controllable structures, with theactuators and sensors easily embedded in the structures(for example the tendons can carry both actuating andsensing functions). Future applications of tensegritystructures are mostly in the field of controllable struc-tures capable of large motions, which would exploitthese structures flexibility and versatility: deployablestructures, robotic arms, space telescopes, motion simu-lators, etc. These applications development is not poss-ible without thorough investigation and understanding oftensegrity structures dynamics.

672 C. Sultan et al. / Engineering Structures 24 (2002) 671–685

Fig. 1. A tensegrity structure.

Very few articles on tensegrity structures dynamicshave been published so far. Motro [11] presents someresults of a linear dynamics experimental analysis of atensegrity structure composed of three bars and nine ten-dons. The experiment determines the dynamic character-istics of the structure by harmonic excitation of a nodeand measurement of the responses of the other nodes.In another article, Furuya [18] analyzes the vibrationalcharacteristics of some tensegrity structures using finiteelement programs and concludes that the modal fre-quencies increase as the pretension increases. Murakami[19–21] uses the Lagrangian and Eulerian approach toderive the equations of motion of a large class of struc-tures and applies it to some tensegrity structures fornumerical simulations and modal analysis. RecentlyOppenheim and Williams [22] tackled the dynamics oftensegrity structures analytically. In their article thedynamics of a tensegrity structure composed of threebars and six tendons is examined showing that, if onlylinear kinetic damping in the tendons is assumed, thegeometric flexibility of the structure leads to a muchslower rate of decay of vibrations than might beexpected. In order to improve the structure’s dampingproperties, linear kinetic damping of the angular motionbetween structural members in contact is assumed, lead-ing to an exponential rate of decay. In another article byOppenheim and Williams [23] these remarks arereinforced leading to the conclusion that friction in therotational joints of the structure is a more importantsource of damping than the damping in the tendons.

Sultan [16] derives mathematical models of the non-

linear dynamics for a large class of tensegrity structures.The resulting equations are second order ordinary differ-ential ones, illustrating an important advantage of thesestructures: their behavior can be described accuratelyenough by ordinary differential equations, which aremuch easier to deal with than partial differential equa-tions, usually used to analyze classical (truss) flexiblestructures.

In the area of active control the published literature issparse. Djouadi [24] presents a tensegrity antenna forwhich a quadratic optimal controller to minimize thenodal displacements is designed. Sultan [25] simul-taneously designs the control system and a tensegritystructure, showing that the resulting system has betterperformance than if the sequential approach (structuredesign followed by controller design) is applied. In aseries of articles, Sultan, Corless and Skelton proposeand investigate various applications of tensegrity struc-tures in motion simulators (see Sultan [6,26]),deployable structures [27], sensors [28], or space tele-scopes [29].

In this article we present linearized models of thedynamics of a class of tensegrity structures around refer-ence solutions of the nonlinear equations of motion.Some particular tensegrity structures are then studied;the reference solutions are equilibrium ones, previouslyinvestigated by Sultan [7]. Explicit expressions of thelinearized models matrices are given. Using these mod-els, conclusions about some important properties of thestructures investigated (stability of the correspondingequilibrium configurations, stiffness, etc.) are drawn.

2. Linearized dynamics of tensegrity structures

In the following we present some results on nonlinearequations of motion of a class of tensegrity structuresderived by Sultan [16] then we present the correspondinglinearized equations of motion.

Consider a tensegrity structure composed of E elasticand masslesstendons and R rigid bodies. The followingmathematical modeling assumptions are made: all con-straints on the system are holonomic, scleronomicandbilateral, the external constraint forces are workless, theforces exerted on the structure by external force fields(e.g. gravitational) are neglected, the joints of the systemare affected at most by linear kinetic friction(linear kin-etic friction means that the friction forces and torques areproportional to the relative linear and angular velocitiesbetween the members in contact), the tendons are at mostaffected by linear kinetic damping(linear kinetic damp-ing means that the damping force introduced by a tendonis proportional to the time derivative of its elongation).Under these assumptions the equations of motion for thissystem can be derived as shown by Sultan [16] yielding:

M(q)q � c(q,q) � A(q)T(q) � C(q)q � H(q)F (1)

673C. Sultan et al. / Engineering Structures 24 (2002) 671–685

� 0

where

� q � [q1…qN]T is the vector of independent gen-eralized coordinates used to describe the structure’sconfiguration.

� M(q) is the mass matrix.� c(q,q) is a vector of quadratic functions in q, whose

components can be expressed as:

ci � �Nj � 1

�Nn � 1

�∂Mij

∂qn

�12

∂Mjn

∂qi�qjqn, i � 1,…,N. (2)

�A(q)T(q) is the vector of elastic generalized forces

where A[n,j] �∂lj∂qn

, n � 1,…, N, j � 1,…,E, Tj and lj

being the tension and length, respectively, of tendon j.� C(q)q is a vector of generalized damping forces where

C(q) is the damping matrix (we note that this formof generalized damping forces is valid only for thelinear kinetic friction and damping assumptions).

� H(q)F is a vector of generalized forces due to externalforces and torques where F is the vector of externalforces and torques and H(q) is the disturbance matrix.The forces and torques included in F are not con-servative or damping ones (e.g. they are externalforces acting on the rigid bodies).

We consider that the dynamics of the structure is con-trolled through tendons. This task can be accomplishedby motors attached, for example, to the rigid bodies.These motors work as follows. For example, the motorpulls a tendon and rolls it over a small wheel in such away that its active length is shortened: the part of thetendon which is rolled over the motor wheel no longercarries force. Hence this control procedure works as ifthe rest-length of the tendon would be shortened. Simi-larly, when the motor reverses its direction of rotation,a portion of the inactive tendon becomes active, carryingforce; hence the rest-length of the tendon increases. Wecall this procedure of tendon control, rest-length control.

Let qe denote an equilibrium solution of Eq. (1), thatis qe � qe � 0. The linearized equations of motionwhich describe the approximate dynamics of the struc-ture in the neighborhood of qe are derived using Taylorseries expansion. This yields

Meq � Ceq � Keq � Beu � Hef � 0 (3)

where

q � q�qe, u � l0�l0e, f � F�Fe

and Me, Ce, Ke, Be, and He are the mass, damping, stiff-ness, control, and disturbance matrices, respectively,evaluated at qe.

The control variables are denoted by u, whereas l0c

represents the vector of rest-lengths at equilibrium. Thedisturbances are denoted by f, Fe being the vector ofexternal forces and torques at equilibrium.

3. Two stage SVD tensegrity structures

In the following we present the linearized equationsof motion for a certain class of tensegrity structures,coined two stage SVD tensegrity structures. These struc-tures have great opportunities for industrial applicationsas shown by Sultan [25,27,28]. Their static propertieshave been thoroughly investigated by Sultan [7,16].

3.1. Description

A two stage SVD tensegrity structure is composed ofsix bars, labeled AijBij, i � 1,2,3, j � 1,2, a rigid top(B12B22B32), a rigid base (A11A21A31), and 18 tendons (seeFigs. 2 and 3). A stage is composed of bars with thesame second index (e.g. A11B11, A21B21, and A31B31

belong to the first stage). The acronym ‘SVD’ comesfrom the notation we introduce for the tendons: Bi1Aj2

will be called saddle tendons, Aj1Bi1 and Aj2Bi2 verticaltendons, and Aj1Ai2 and Bj1Bi2 diagonal tendons respect-ively.

The assumptions made for the mathematical modelingof two stage SVD tensegrity structures are: the tendonsare massless, linear elastic, and not-damped (not affec-ted by damping), the base and the top are rigid bodies,the bars are rigid, axially symmetric, and for each barthe rotational degree of freedom around the longitudinalaxis of symmetry is neglected. This is a reasonableassumption if the points of attachment of the bars to therigid bodies and to the tendons belong to the axes ofsymmetry of the bars, so that no longitudinal torque isgenerated, or, if we assume that the thickness of any baris negligible. Linear kinetic friction torques (proportionalto the relative angular velocity) act at the six jointsbetween the base and the rigid top with the bars, beinggiven by

Fig. 2. Two stage SVD tensegrity structure.


Fig. 3. Symmetrical prestressable configuration.

M→

fj � dj(w→

a�w→

b), j � 1,…,6, (4)

where M→

fj, is the friction torque exerted by body ‘b’ onbody ‘a’ due to the relative angular motion between bod-ies ‘a’ and ‘b’ (in our case ‘a’ is the base or the rigidtop and ‘b’ is one of the bars). The scalar dj�0 is the

coefficient of friction at joint j while w→∗ is the angularvelocity of rigid body ‘*’ . We neglect the forces exertedon the structure by external force fields (e.g.gravitational). A force and a torque act on the rigid top.We remark that these assumptions are particular casesof the modeling assumptions made for the derivation of

tensegrity structures nonlinear and linearized equationsof motion. Thus these equations for two stage SVD ten-segrity structures have the form given by Eqs. (1) and(3) respectively.

The inertial system of reference, b1, b2, b3, is a dextralset of unit vectors, whose center coincides with the geo-metric center of triangle A11A21A31. The axis b3 is orthog-onal to A11A21A31 pointing upward while b1 is parallel toA11A31, pointing towards A31. The top’s dextral referenceframe, t1, t2, t3, is located at the geometric center of tri-angle B12B22B32, labeled Ot, while t3 is orthogonal toB12B22B32 pointing upward and t1 is parallel to B12B32

pointing towards B32. For simplicity it is assumed thatthe top reference frame is central principal for the toprigid body.

The 18 independent generalized coordinates used todescribe the motion of this holonomic system, are: y, f,q, the Euler angles for a 3 1 2 sequence to characterizethe inertial orientation of the top reference frame, X, Y,Z, the inertial Cartesian coordinates of the origin of thetop reference frame (Ot), dij and aij, the declination andthe azimuth of the longitudinal axis of symmetry of barAijBij, measured with respect to the inertial frame anddefined as follows: dij is the angle made by vectorAijBij

J→with b3 and aij is the angle made by the projection

of this vector onto plane (b1, b2) with b1 (see Fig. 2).The vector of generalized coordinates is

q � [d11a11d21a21d31a31d12a12d22a22d32a32 y f q X Y Z]T (5)

3.2. Reference solutions: prestressable configurations

In the following we assume that the bars are identical(their length, mass, and transversal moment of inertiabeing respectively labeled as l, m, and J) and that thebase and top triangles are equal equilateral triangles ofside length b.

The reference solution, qe, is a prestressable con-figuration. A prestressable configuration is an equilib-rium one for which all tendons are in tension and whichoccurs under no external forces or torques (henceFe � 0). The prestressability problem consists in findinga prestressable configuration (see Sultan [7]). The parti-cular prestressable configurations we consider here arecoined symmetrical prestressable configurations forwhich the tensions in the saddle, vertical, and diagonaltendons are respectively equal. The geometry of such aconfiguration is characterized as follows: all bars havethe same declination, d, the vertical projections of pointsAi2, Bi1, i � 1,…,3, onto the base make a regular hexa-gon, and planes A11A21A31 and B12B22B32 are parallel (seeFig. 3). In addition all saddle tendons have the sametension, TS, all vertical tendons have the same tension,TV, all diagonal tendons have the same tension, TD.

We solved the prestressability problem for this classof configurations (see Sultan [7]) and proved that this


set can be parameterized by two quantities: the azimuthof bar A11B11, which we call a, and the common decli-nation of the bars, d, yielding

qe � [d a d a �4p3d a �

2p3d a �

2p3d a d a (6)

�4p3

5p3

0 0 0 0 2l cos(d)�h]T

where

h � � cos(d)

2 sin(d) cos �a �p6��

b

�3� p � �b2

3�3p2� ifa �

p3

l cos(d)2

ifa �p3

(7)

with p � lsin(d)cos(a �p6

). These solutions exist if and

only if a and d satisfy the following conditions:

p6

� a �p2

, 0 � d �p2

, l sin(d)� cos(a �p6

)� (8)

�b

2�3and sin(a �

p6

) �3l sin(d)

2b.

This set can be represented in the (a, d, h) space by anequilibrium surface (see Sultan [7] for details).

We also proved that the tensions are completelydetermined up to a multiplicative positive scalar, P,called the pretension coefficient (see Sultan [7]):

[TSTVTD]T � P[T0ST0V

T0D]T (9)

Detailed expressions of the basis tensions, T0S, T0V

, T0D,

are given in Appendix A.

3.3. Linearized equations of motion

The linearized equations of motion around these pre-stressable configurations are given by Eq. (3):


where q � q�qe, u � l0�l0e, f � F (since Fe � 0) andMe, Ce, Ke, Be, He are the mass, damping, stiffness, con-trol, and disturbance matrices, respectively, at qe.

For output feedback control, measurement equationsare also necessary. We consider that the measurements,z, are the tendons lengths. Through the same lineariz-ation procedure, the measurement equations can be writ-ten as

z � Geq, with Ge[i,j] � (∂li∂qj

)(qe). (11)

Here z � z�ze, ze is the vector of tendons lengths at qe,and Ge is the measurement matrix at qe.

For simplicity we consider that all the damping coef-ficients have the same value, d�0. We also consider thatall saddle tendons are identical, of rest-length S0 andstiffness kS, that all vertical tendons are identical, of rest-length V0 and stiffness kV, that all diagonal tendons areidentical, of rest-length D0 and stiffness kD. Here wedefine the stiffness of tendon i, called ki, as the productbetween its cross section, Ai, and modulus of elasticity,Ei (hence ki � AiEi). In the following Mt denotes themass of the rigid top and J1, J2, J3 its principal momentsof inertia with respect to the top reference frame.

3.3.1. Mass, damping, and disturbance matricesThe particular mass, damping, and disturbance matr-

ices (Me, Ce, and He, respectively), are obtained by eval-uating the general mass, damping, and disturbance matr-ices (M(q), C(q), H(q) respectively; see Sultan [16] fordetails on these matrices) at qe, where qe has beenexpressed in terms of a and d in Eq. (6). This yields:

Me � Md � �0 0 0

0 0 Ma

0 MTa 0 (12)

where

Md � diag([J, Js2(d), J, Js2(d), J, Js2(d), J, Js2(d), (13)

J, Js2(d), J, Js2(d),J3, J1, J2, M,M,M])

with

J � J �mb2

4, J1 � J1 �

mb2

2, J2 � J2 �

mb2

2, J3 (14)

� J3 � mb2, M � Mt � 3m.

In these formulas we used the following notation:c(∗) � cos(∗), s(∗) � sin(∗). Matrix Ma is

Ma �ml2

a �

e

�3e c(d)s(a �

p6

) �c(d)c(a �p6

) s(d)

g 0 0 s(d)c(a �p6

) s(d)s(a �p6

) 0

a 2e

�30 �c(d)c(a) �c(d)s(a) s(d)

g 0 0 s(d)s(a) �s(d)c(a) 0

a �e

�3�e c(d)c(a �

p3

) c(d)s(a �p3

) s(d)

g 0 0 �s(d)s(a �p3

) s(d)c(a �p3

) 0

(15)

where a �b

�3c(d)c(a �

p3

), e �b2s(d), g � �

b

�3s(δ)s(α �

π3

).


The damping matrix, Ce, is

Ce � Cd � �0 0 0

0 0 Ca

0 CTa 0 (16)

where

Cd � (17)

�d diag([1,1,1,1,1,1,1,1,1,1,1,1,3,3,3,0,0,0])

and

Ca � �d�0 �s(a) c(a) 0 0 0

�1 0 0 0 0 0

0 c(a�p6

) s(a�p6

) 0 0 0

�1 0 0 0 0 0

0 �c(a �p6

) �s(a �p6

) 0 0 0

�1 0 0 0 0 0

. (18)

The disturbance matrix, He, is

He � 0

Ha�, Ha �

0 0 �1 0 0 0

�1 0 0 0 0 0

0 �1 0 0 0 0

0 0 0 �12

��3

20

0 0 0�3

2�

12

0

0 0 0 0 0 �1

(19)

for the following distribution of external forces and tor-ques: F � [M1 M2 M3 F1 F2 F3]T where M∗ /F∗, is thecomponent along the t∗ axis of the external torque/forceacting on the rigid top.

3.3.2. Stiffness matrixFor the derivation of the stiffness matrix, Ke, we con-

sider the potential energy of the structure which, for lin-ear elastic tendons, is

U � �18

i � 1

ki

2l0i

(li�l0i)2 �

kS

2S0�6

i � 1

(Si�S0)2 (20)

�kV

2V0�6

i � 1

(Vi�V0)2 �kD

2D0�6

i � 1

(Di�D0)2.

Since there are no external forces acting on the structurewhen it is in equilibrium at qe (Fe � 0), Ke is just theHessian of the potential energy evaluated at qe:

Ke[j,n] � (∂2U

∂qj∂qn

)(qe) � (kS

S0�6

i � 1

(∂Si

∂qj

∂Si

∂qn

� (Si

�S0)∂2Si

∂qj∂qn

) �kV

V0�6

i � 1

(∂Vi

∂qj

∂Vi

∂qn

� (Vi�V0)∂2Vi

∂qj∂qn

) (21)

�kD

D0�6

i � 1

(∂Di

∂qj

∂Di

∂qn

� (Di�D0)∂2Di

∂qj∂qn

))(qe).

Using the expressions for the rest-lengths, S0, V0, D0,corresponding to a symmetrical prestressable configur-ation with equal saddle, vertical, and diagonal tendonstensions respectively, derived by Sultan [7],

S0 �kSS

T0SP � kS

, V0 �kVV

T0VP � kV

, (22)

D0 �kDD

T0DP � kD

,

we obtain

Ke[j,n] � kVKV[j,n] � kSKS[j,n] � kDKD[j,n] (23)

� PKP[j,n]

where

KS[j,n] �1S�

6

i � 1

(∂Si

∂qj

∂Si

∂qn

)(qe),

KV[j,n] �1V�

6

i � 1

(∂Vi

∂qj

∂Vi

∂qn

)(qe), (24)

KD[j,n] �1D�6

i � 1

(∂Di

∂qj

∂Di

∂qn

)(qe).

Here P is the pretension coefficient, Si, Vi, Di, i � 1,…,6,are the lengths of the saddle, vertical, and diagonal ten-dons respectively expressed in terms of the generalizedcoordinates, q, and S, V, D, the values these functionstake for q � qe (S, V, D are given in Appendix B). Matr-ices KS, KV, KD can be written as:

KS �1SASAT

S, KV �1V

AVATV, KD �

1D

ADATD (25)

where, for example,

AS[i,j] � (∂Sj

∂qi

)(qe), j � 1,…,6, i � 1,…,18. (26)

It is important to remark that, since KS, KV, and KD areexpressed as the product of a matrix and its transposeand a positive number, they are all positive semidefinitematrices and that the stiffness matrix, Ke, is linear inpretension and the tendons stiffnesses. Matrices AS, AV,AD are too complicated to be given here (see Sultan[16]).


The elements of matrix KP can be written as:

KP[j,n] � T0S(KS[j,n] � �6

i � 1

(∂2Si

∂qj∂qn

)(qe))

� T0V(KV[j,n] � �6

i � 1

(∂2Vi

∂qj∂qn

)(qe)) � T0D(KD[j,n] (27)

� �6

i � 1

(∂2Di

∂qj∂qn

)(qe)).

This yields:

KP � KPd� �0 KT

P3KT

P1

KP30 KT

P2

KP1KP2

0 . (28)

See Sultan [16] for detailed expressions of matricesKPd

, KPi, i � 1,2,3.

3.3.3. Control and measurement matricesThe control matrix can be derived from the potential

energy as

Be[i,j] � (∂2U

∂qi∂l0j

)(qe) (29)

yielding

Be � [BSBVBD] (30)

where

BS � �(kS � PT0S

)2

kSSAS, BV � (31)

�(kV � PT0V

)2

kVVAV, BD � �

(kD � PT0D)2

kDDAD.

We remark that Be is quadratic in pretension.If we assume that the measurement vector is

z � [S1S2…S6V1V2…V6D1D2…D6]T (32)

where Si, Vi, and Di denote the length of the ith saddle,vertical, and diagonal tendon, respectively, then themeasurement matrix is

Ge[i,j] �∂li∂qj

(qe)⇒Ge � [ASAVAD]T. (33)

Hence Ge depends only on the geometry of the structure.

3.4. Stability and stiffness properties

In the following we analyze some stability and stiff-ness properties of two stage SVD tensegrity structuresyielding symmetrical prestressable configurations with

equal tensions in all saddle, vertical, and diagonal ten-dons respectively.

The general mass matrix, M(q), was derived alongwith the total kinetic energy of the system, Ttot, whichis a quadratic positive definite form in q: Ttot �12qTM(q)q (see Sultan [16] for details on Ttot and M(q)).

Hence Me, a particular value of M(q), Me � M(qe), ispositive definite, Me�0.

We shall prove directly that Ce is semipositive defi-nite, Ce�0, through successive application of Schurcomplements:

Ce�0⇔��dI6 0 0

0 �dI6 Ca

0 CTa Cb

�0⇔Cb �1d

CTaCa�0 (34)

where

Cb � �3dI3 0

0 0�. (35)

Using Ca expression from Eq. (18), Eq. (34) becomes

Cb �1d

CTaCa�0⇔3��w�2 vTw

vTw 3��v�2��0 (36)

where

v � [c(a) s(a�p6

) s(a �p6

)], w � [s (a)�c(a�p6

) (37)

�c(a �p6

)].

Since 3��w�2 � 0, Eq. (36) is equivalent to

�v�2�w�2�(vTw)2�0 (38)

which is the well known Schwartz inequality.The stiffness matrix is given by

Ke � PKP � kSKS � kVKV � kDKD. (39)

As mentioned before, matrices KS, KV, and KD arepositive semidefinite. Matrix kS KS+kV KV+kD KD is posi-tive definite on all space except for the kernel of [AS AV

AD]T. Numerous numerical experiments indicated that KP

is positive definite at all symmetrical prestressable con-figurations with equal saddle, vertical, and diagonal ten-dons tensions respectively. This has been ascertained asfollows: for various values of l, b, a, and d, we computedthe eigenvalues of KP. In all cases these eigenvalueswere strictly greater than 0. For example in Fig. 4 wegive the variation of the minimum eigenvalue of KP witha and d for l � 0.4 m and b � 0.27 m over all the afore-mentioned prestressable configurations. We ascertainthat this eigenvalue is strictly greater than zero in the


Fig. 4. Minimum eigenvalue of KP for two stage SVD.

interior of the equilibrium surface. The only points atwhich this eigenvalue is zero are on one boundary of

the equilibrium surface, namely for d �p2

, but these are

not feasible prestressable configurations (see Eq. (8):

0 � d �p2

).

Thus Ke is positive definite: Ke�0.These results yield several important consequences:

� Ke � 0⇒det(Ke) � 0. Since Ke is the Jacobian ofA(q)T(q) � H(q)F evaluated at a solution, qe, ofA(q)T(q) � H(q)F � 0, by the implicit function the-orem, it follows that in the neighborhood of eachelement of the class of symmetrical prestressable con-figurations with equal tensions in the saddle, vertical,and diagonal tendons respectively, the solution of thestatic problem, A(q)T(q) � H(q)F � 0, can beexpressed as a single valued function of the externalforces and torques: q � q(F). Hence the staticresponse is well determined (no static bifurcationphenomena are possible).

� Me � 0, Ce�0, and Ke�0 imply linearized stability ofeach symmetrical prestressable configuration withequal saddle, vertical, and diagonal tendons tensionsrespectively. Actually our numerical experiences indi-cated that these solutions are asymptotically stable.This fact has been ascertained by applying the follow-ing result: a system described by Eq. (3) and forwhich Me � 0, Ke � 0, Ce�0 is asymptotically stableif and only if rank([2Me�Ke; Ce]) � rank(Me) forevery natural frequency, , defined by det(2Me�Ke) � 0 and �0. We checked the rank of[2Me�Ke;Ce] for various values of the structure’sparameters and over the whole class of prestressableconfigurations of interest; this rank was always equalto the rank of Me, N � 18. An analytical proof of this

fact seems very difficult because of the complicatedstructure of Ke. This shows that only with kinetic fric-tion at the joints these prestressable configurations areasymptotically stable. We remark that this result isin agreement with the conclusions of Oppenheim andWilliams [22,23] obtained for a simpler tensegritystructure and using a different mathematical model,that linear kinetic friction at the joints results inasymptotically stable equilibria, leading toexponential rate of decay of perturbed motions andbeing more efficient than linear kinetic damping inthe tendons.

� KP�0 shows that the stiffness matrix, Ke, becomesmore positive definite if P increases (that is if P1�P2

then Ke1� Ke2

where P1 and P2 are two values of thepretension coefficient and Ke∗ are the correspondingstiffness matrices). Hence increasing the pretensionresults in a stiffer structure with higher natural fre-quencies.

3.5. Linear versus nonlinear dynamics

We next present numerical simulations of a two stageSVD tensegrity structure’s linear and nonlinear dynam-ics. This will help us evaluate the differences betweenthe nonlinear and linear models and the influence of pre-tension upon these differences.

The structure analyzed here is characterized by thefact that all bars are identical, of length l � 0.4 m, massm � 0.8 kg, and transversal moment of inertia J � 1.2kg m2. The base and top triangles are equal, equilateraltriangles of side length b � 0.27 m. The coefficients offriction at all joints are equal, di � �4, i � 1,…,6. Themass of the rigid top is Mt � 10 kg and its moments ofinertia J1 � 30 kg m2, J2 � 40 kg m2, J3 � 50 kg m2.The tendons are assumed to have the same stiffness,ki � 500 N, i � 1,%,18.

We consider that the structure is in equilibrium, yield-ing a symmetrical prestressable configuration for whichthe tensions in all saddle, all vertical, and all diagonaltendons are respectively equal. This configuration ischaracterized by:

a11 � a22 �p3

, a21 � a32 �5p3

,

a31 � a12 � p, y �5p3

, f � q � X � Y � 0, (40)

Z � 0.3 m,dij �p3

for i � 1,2,3, j � 1,2,3.

For a given pretension coefficient, P, the rest-lengths ofthe tendons which assure this prestressable configurationcan be determined using Eq. (22).

Figs. 5 and 6 represent the responses of the linear andnonlinear models of the structure to a perturbation in the


Fig. 5. Initial conditions response for P � 500: nonlinear (- - -) andlinear models (———).

Fig. 6. Initial conditions response for P � 2500: nonlinear (- - -) andlinear models (———).

initial conditions. The perturbation consists of a verticalvelocity of the rigid top equal to 2 m s�1: Z(0) �2 m s�1. These graphs represent time histories of threegeneralized coordinates (d11, a11, and Z). Fig. 5 corre-sponds to the case of a pretension coefficient of P �500, while Fig. 6 corresponds to P � 2500. The timehistories of the other generalized coordinates and gen-eralized velocities are qualitatively similar. We have alsochecked that the tendons are always in tension and thatthe bars do not collide with each other throughout themotion.

These simulations illustrate the differences betweenthe nonlinear and linear models. We ascertain that thepretension coefficient increases the frequencies of thelinear response (the higher the pretension coefficient thehigher these frequencies are). We also note that at higherpretension the differences between the linear and nonlin-

ear models are less significant. Hence the linear approxi-mation is better at high pretension. We also note that thesame differences are more significant at high generalizedvelocities. This is so because of the quadratic terms ingeneralized velocities which appear in the nonlinearmodel, c � c(q,q), but do not appear in the linear model.We also note that these responses are typical of anasymptotically stable equilibrium: the motion dies out intime and settles down to the unperturbed equilibriumstate.

The same conclusions are reached if we consider dif-ferent initial conditions perturbations and different refer-ence solutions in the class of symmetrical prestressableconfigurations with equal tensions in all saddle, all verti-cal, and all diagonal tendons respectively.

3.6. Dynamic characteristics

In the following we investigate the dynamic character-istics of a two stage SVD tensegrity structure yieldingsymmetrical prestressable configurations with equalsaddle, vertical, and diagonal tendons tensions respect-ively.

Consider a two stage SVD tensegrity structure havingthe following geometrical, inertial, and elastic character-istics: l � 0.4 m, b � 0.27 m, m � 0.8 kg, J � 1.2 kgm2, Mt � 10 kg, J1 � 30 kg m2, J2 � 40 kg m2, J3 �50 kg m2, ki � 500 N, i � 1,…,18. The pretension isP � 500. The natural frequencies of a system whose lin-earized dynamics is described by Eq. (10) are definedby the square roots of the eigenvalues of the matrixM�1

e Ke. The variations of the minimum and maximumnatural frequencies of this structure over the whole classof symmetrical prestressable configurations of interestare given in Figs. 7 and 8. We ascertain that the dynamicrange (defined as the difference between the maximumnatural frequency and the minimum one) increases for

Fig. 7. Minimum natural frequency for two stage SVD.


Fig. 8. Maximum natural frequency for two stage SVD.

small declination (d) because of the dramatic increase ofthe maximum natural frequency.

It is also interesting to evaluate the influence of thedamping and pretension coefficients on the dynamiccharacteristics of the structure by plotting the minimumand maximum modal dampings and the minimum andmaximum modal frequencies of the structure over thedamping and pretension coefficients. Recall that for afirst order linear system whose dynamics is described by

x � Apx � Bpu � Hpf (41)

the modal dampings and the modal frequencies aredefined by the eigenvalues of the system matrix, Ap: theabsolute value of the real part of an eigenvalue is themodal damping whereas the absolute value of theimaginary part is the modal frequency. We can easilycast our vector second order system given by Eq. (10)in first order form by introducing the vector of state vari-ables x � [qT qT]T. Hence

Ap � 0 I

�M�1e Ke �M�1

e Ce�, (42)

Bp � 0

�M�1e Be

�, Hp � 0

�M�1e He

�.

Consider a two stage SVD tensegrity structure whosegeometrical, inertial, and elastic characteristics are thesame as before and which yields a symmetrical prestress-able configuration for which the tensions in the saddle,vertical, and diagonal tendons are respectively equal and

characterized by a �p3

and d �p3

. The variations of the

minimum and maximum modal dampings with the pre-tension and damping coefficients for this configurationare given in Figs. 9 and 10 and those of the minimumand maximum modal frequencies in Figs. 11 and 12respectively. As expected, the minimum and maximum

Fig. 9. Minimum modal damping for two stage SVD.

Fig. 10. Maximum modal damping for two stage SVD.

Fig. 11. Minimum modal frequency for two stage SVD.


Fig. 12. Maximum modal frequency for two stage SVD.

modal dampings increase with the damping. It is inter-esting to remark that, for a given damping coefficient,at small pretension the maximum modal dampingincreases as the pretension decreases. We also remarkthat the range of modal damping (defined as the differ-ence between the maximum modal damping and theminimum one) increases as the damping in the structureincreases. Also the modal dynamic range (defined as thedifference between the maximum modal frequency andthe minimum one) increases with the pretension coef-ficient.

It is worth mentioning that for some combination ofdamping and pretension the minimum modal frequencyis zero (Fig. 11). This means that some modes of thelinear system are pure exponentially decaying ones (ofthe type exp(�zt), z being the modal damping and t thetime). Also, since the maximum modal damping isgreater than zero for any combination of damping andpretension in the range under investigation (Fig. 12),there are always exist oscillatory modes whose envelopeis exponentially decaying (of the type exp(�zt) cos(wt)), w being the modal frequency).

4. Two stage SVDT tensegrity structures

In the following we analyze another class of tensegritystructures, coined two stage SVDT. A two stage SVDTtensegrity structure is obtained from a SVD one if wereplace the top rigid body with tendons connecting thetop nodal points, B12, B22, and B32. These new tendons—B12B22, B22B32, B32B12—will be called top tendons, hencethe acronym ‘SVDT’ used for these structures.

We make the same modeling assumptions as for twostage SVD tensegrity structures and we define the iner-tial reference frame in the same way. The independentgeneralized coordinates used to describe this system’sconfiguration are the Cartesian coordinates, xi2, yi2, zi2,

of the mass center of bar Ai2Bi2, i � 1,2,3, with respectto the inertial reference frame, the azimuth aij and thedeclination dij, which characterize the orientation of barAijBij, i � 1,2,3, j � 1,2, with respect to the inertialframe (these angles are defined in the same way as thedeclination and azimuth used for two stage SVD tensegr-ity structures). The vector of generalized coordinates is:

q � [d11a11d21a21d31a31x12y12z12d12 (43)

a12x22y22z22d22a22x32y32z32d32a32]T.

4.1. Reference solutions

In the following, as in the SVD case, we consider thatall bars have the same length, l, mass, m, and transversalmoment of inertia, J. The reference solutions are sym-metrical prestressable configurations for which the ten-sions in the saddle, vertical, diagonal, and top tendonsare respectively equal. The geometry of these configur-ations is identical to that of the symmetrical prestressableconfigurations of the two stage SVD tensegrity structurespreviously investigated. It has been proved by Sultan [7]that the solution of the prestressability problem for theseconfigurations is given by:

a11 � a22 � a, a21 � a32 � a �4p3

, a31

� a12 � a �2p3

, x12 �l4

sin(d) cos(a)

��3

4l sin(d) sin(a)�

b2, y12 �

b

2�3

��3

4l sin(d) cos(a) �

l4

sin(d) sin(a), z12

�32l cos(d)�h, x22 �

b2�

l2

sin(d) cos(a), y22 (44)

�b

2�3�

l2

sin(d) sin(a), z22 �32

l cos(d)�h, x32

�l4

sin(d) cos(a)��3

4l sin(d) sin(a), y32

�l4

sin(d) sin(a) ��3

4l sin(d) cos(a)�

b

�3, z32

�32l cos(d)�h, dij � d, i � 1,2,3, j � 1,2,

where h is given by the same formula as in the SVDcase (Eq. (7)) and b represents the length of the baseand top triangles side (these triangles are equilateral atthe prestressable configurations considered here). This


set is identical with the set of symmetrical prestressableconfigurations with equal tensions in the saddle, vertical,and diagonal tendons, respectively, of two stage SVDtensegrity structures (see Sultan [7] for the proof).

We have also proved that the tensions at these con-figurations are completely determined up to a multipli-cative, positive, scalar (see Sultan [7]):

[TSTVTDTT]T � P[T0ST0V

T0DT0T

]T (45)

where P is the pretension coefficient, TS, TV, TD, TT arethe tensions in the saddle, vertical, diagonal, and top ten-dons respectively, and T0S

, T0V, T0D

, T0Tare given in

Appendix C.

4.2. Linearized equations of motion

The linearized equations of motion around the afore-mentioned reference solutions are given by Eq. (3):


For simplicity, we shall not consider the case of externaldisturbances in the following (hence F � Fe � f � 0)and we shall not be interested in the disturbance matrix,He. The controls are assumed to be the rest-lengths ofthe tendons. As in the SVD case we also consider thatall the damping coefficients have the same value, d�0,all saddle tendons are identical, of rest-length S0 andstiffness kS, all vertical tendons are identical, of rest-length V0 and stiffness kV, all diagonal tendons are ident-ical, of rest-length D0 and stiffness kD, and all top ten-dons are identical, of rest-length T0 and stiffness kT.

4.2.1. Linearized equations of motion matricesThe mass matrix is diagonal:

Me � diag([J �ml2

4, (J �

ml2

4)s2(d), J �

ml2

4, (47)

(J �ml2

4)s2(d), J �

ml2

4, (J �

ml2

4)s2(d),

m,m,m,J,Js2(d),m,m,m,J,Js2(d),m,m,m,J,Js2(d)]).

The damping matrix is also diagonal:

Ce � (48)

�ddiag([1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0])

The stiffness matrix can be derived from the potentialenergy as in the SVD case yielding

Ke � PKP � kSKS � kVKV � kDKD � kTKT. (49)

where matrices KP, KS, KV, KD, KT are given by similarformulas as in the SVD case (see Eqs. (24) and (27)).

Analogously, the control matrix is

Be � [BSBVBDBT] (50)

Fig. 13. Minimum natural frequency for two stage SVDT.

where BS, BV, BD, BT are given by similar formulas asin the SVD case (see Eq. (31)), hence Be is quadraticin pretension.

It is important to remark that these results, regardingthe structure of the stiffness and control matrices at pre-stressable configurations, can be easily generalized toother tensegrity structures characterized by the modelingassumptions stated in this article. Namely the stiffnessmatrix is linear in the level of pretension and the tendonsstiffnesses whereas the control matrix is quadratic inpretension.

4.3. Stability properties and dynamic characteristics

It is easy to observe that at these prestressable con-figurations the mass matrix is positive definite whereasthe damping one is semipositive definite: Me � 0, Ce�

0 (since 0 � d �p2

). It has also been numerically ascer-

Fig. 14. Maximum natural frequency for two stage SVDT.


tained (similarly with the SVD case) that KP�0. HenceKe�0, implying, as in the SVD case, stability of theseconfigurations. Actually all our numerical experimentsindicated that these are asymptotically stable equilib-rium configurations.

Figs. 13 and 14 show the variations of the minimumand maximum natural frequencies with a and d over thewhole class of symmetrical prestressable configurationswith equal tensions in the saddle, vertical, diagonal, andtop tendons respectively for a two stage SVDT tensegritystructure with the following parameters: l � 0.4 m,b � 0.27 m, m � 0.8 kg, J � 1.2 kg m2, P � 500,kS � kV � kD � kT � 500 N. As in the SVD case, weascertain how the dynamic range dramatically increasesas d decreases.

The influence of pretension and damping on the samestructure at a symmetrical prestressable configurationwith equal saddle, vertical, diagonal, and top tendons

tensions, respectively, characterized by a �p3

and d �

p3

is also evaluated. The variations of the minimum and

maximum modal dampings with the pretension anddamping coefficients are plotted in Figs. 15 and 16. Weobserve that the range of the modal damping increaseswith the damping in the structure. The minimum andmaximum modal frequencies variations with the preten-sion and damping coefficients are given in Figs. 17 and18. As in the SVD case, the modal dynamic rangeincreases with the level of pretension. We also remarkthat the maximum modal frequency is greater than zero,leading to the existence of oscillatory modes whoseenvelope is exponentially decaying, and that, for somecombination of damping and pretension, the minimummodal frequency is zero, leading to the existence of pureexponentially decaying modes (see Figs. 17 and 18).

Fig. 15. Minimum modal damping for two stage SVDT.

Fig. 16. Maximum modal damping for two stage SVDT.

Fig. 17. Minimum modal frequency for two stage SVDT.

Fig. 18. Maximum modal frequency for two stage SVDT.


5. Conclusions

Linear models which describe the approximatedynamics of tensegrity structures in the neighborhood ofequilibrium reference solutions have been derived. Fortwo classes of tensegrity structures these equations havebeen presented. The reference solutions around whichthese linear models are built are prestressable configur-ations of practical interest. The stiffness matrix at theseconfigurations is linear in the level of pretension and thetendons stiffnesses whereas the control matrix is quad-ratic in pretension. The mass and stiffness matrices arepositive definite and the damping matrix is positive sem-idefinite. Hence these reference solutions are stable.Numerical experiences indicated that they are actuallyasymptotically stable, showing that only linear kineticfriction at the rigid to rigid joints is sufficient to assureasymptotical stability of these configurations. The modaldynamic range generally increases with the pretensionwhereas the modal damping range increases with thedamping.

Appendix A

Two stage SVD tensegrity structures basis tensions are:

[T0ST0V

T0D] �

[TrSTr

VTrD]

�6�[TrSTr

VTrD]�

(51)

where

TrV � �

VD

1

�3 cos(a �p6)��l cos(d)

h�1� sin�a�

p6�� cos(a)�Tr

D ifa �p3

VTrD

D �3l2b

sin(d)�1� ifa �p3

,

(52)

TrS � �

SD�l cos(d)

h�1�Tr

D if a �p3

TrD if a �

p3

,

(53)

TrD � 1. (54)

Appendix B

Saddle, vertical, and diagonal tendons lengths at sym-metrical prestressable configurations:

S (55)

� �h2 �b2

3� l2 sin2(d)�

2

�3lb sin(d) cos(a�

p6

),

V � �b2 � l2�2lb sin(d) sin(a �p6

), (56)

D � �h2 �b2

3� l2�

2

�3lb sin(d) sin(a)�2lh cos(d). (57)

Appendix C

Two stage SVDT tensegrity structures basis tensions are:

[T0ST0V

T0DT0T

] �[Tr

STrVTr

DTrT]

�6�[TrSTr

VTrD�0.5Tr

T]�(58)

where TrV, Tr

S, and TrD are given by Eqs. (52)–(54)

whereas TrT is

TrT � (59)

�TrD

6D

3l2 sin(d) cos(d) � 6bh cos �a�p3�

�6lh sin(d)�2�3bl cos(d) sin(a)

�3h cos �a�p6�

if a �p3

TrD

6D2b2�9lb sin(d) � 9l2 sin2(d)

bif a�

p3

.

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