## Abstract

Tensegrity systems are prestressed frameworks composed of bars and cables. A particular elastic tensegrity system is examined. This system can be bistable in two fundamentally different ways, one depending on its geometric dimensions, and the other one depending on the initial deformation, or prestrain, of the elastic elements. A reduced-order semi-analytical model is derived, and its predictions are verified with a full-order numerical model. In particular, the critical geometry and prestrain at which the system switches from one regime to another are determined. This case study provides a benchmark and new insights on this class of structures.

## 1. Introduction

A tensegrity structure is a pin-connected framework composed of bars and cables; with bars typically in compression, and cables necessarily in tension, such that the system is in a self-stressed state before the application of external loads.

Tensegrity systems were considered for the first time in 1948 by Snelson, when he was a student of Fuller. In the same period, Emmerich was also independently investigating structures of this kind. Starting in the 1960s, Snelson built several outdoor tensegrity sculptures, with bars never connected to each other. His term, ‘floating-compression’, emphasizes this feature. Fuller instead used the words ‘tensile integrity’ to highlight the fact that cables constitute a connected set. More details about the origin of tensegrities can be found in the study of Gómez Jáuregui [1].

Since the late 1970s, tensegrity systems have been rigorously studied by mathematicians in rigidity theory [2,3], and by structural engineers [4–8], with steadily growing interest from the scientific community up to the present days. Owing to their versatility and to their peculiar properties, applications for tensegrity structures have been sought in many areas, especially in civil engineering, aerospace engineering and robotics. Some representative references in these fields are the following: Rastorfer [9], Motro [10], Yuan *et al.* [11], Skelton & de Oliveira [12], Moored & Bart-Smith [13], Zolesi *et al.* [14], Paul *et al.* [15], Shibata & Hirai [16]. Tensegrities are particularly suitable for variable geometry applications, such as deployable structures or adaptive systems, elements of which can be used as sensor or actuators. In addition, tensegrity systems possess a highly nonlinear mechanical behaviour [17–19], which can be exploited for designing new materials [20].

We focus on an aspect which is seldom examined in the literature, namely, the property of a tensegrity structure of being bistable or multistable. In the literature on tensegrities, the first occurrence of a bistable system appeared in Calladine [4], as the classic two-bar system exhibiting snap-through instability. In cases like this, it occurs that by exchanging cables with bars, the energy passes from having a single well to having a double well. A more significative example has been given by Defossez [21], who reported the case of an elastic tensegrity structure with multiple stable equilibrium configurations. Ranganathan *et al.* [22] showed that the classic tensegrity prisms can pass from one configuration to another with opposite orientation, if large elastic deformations of its members are allowed. In the same paper, other cases were also presented. More recently, Xu & Luo [23,24] found multiple equilibrium configurations for various tensegrity systems. In Zhang [25] and Zhang *et al.* [26], another tensegrity system which can have more than one equilibrium configuration has been presented.

In the present study, we analyse a particular structure (figure 1) which can pass from one to two stable equilibrium configurations, either by a change in the overall geometry of the system, or by a change of prestrain, i.e. the initial deformation associated with the self-stress in the system. In the next section, we review the basic definitions and notions about tensegrity systems that we need in order to perform our computations. In the third section, we present the actual analysis of the bistable structure.

## 2. Preliminaries

We first introduce the basic notions of selfstress and mechanism for frameworks, which are independent of constitutive assumptions. Then we briefly review the notions of stiffness and stability which are usually employed for tensegrity systems. Many of the statements we make will be justified in the last subsection, where we present the detailed elastic formulation employed in the successive analysis.

### (a) Frameworks, selfstresses and mechanisms

A *framework* is defined as a set of *N* points, called *nodes*, in the three-dimensional Euclidean space, together with a set of *E* *edges* connecting pairs of nodes. We will say that is the edge connecting nodes . Let **p**_{I} be the position vector of node *I* with respect to a certain reference frame. The *configuration* of a framework is given by the 3*N*-dimensional vector **p** which groups together all the nodal position vectors. Similarly, we can associate with each node a load vector and a displacement vector, so that **f** and **u** are the 3*N*-dimensional vectors containing all nodal loads and nodal displacements, respectively. The edge *IJ* is associated with the axial force it carries, *t*_{IJ}, and with its per cent elongation, *e*_{IJ}, so that **t** and **e**, respectively, are the corresponding *E*-dimensional vectors for the whole framework.

In the linear theory of bar frameworks [5], the *equilibrium operator*, **A**, a function of **p** only, provides the linear relation between axial forces and external loads,
2.1while its transpose, the *kinematic compatibility operator*, **A**^{T}, links ‘small’ displacements to ‘small’ elongations,
For the purposes of this paper, we consider only the case where the equilibrium operator does not have full rank, meaning that both its nullspace and the nullspace of **A**^{T} are non-empty: there exist *selfstresses*, **t**_{s}, which are balanced by null loads, and *mechanisms*, **u**_{m}, nodal displacements which do not change the length of the edges
We say that a mechanism is *non-trivial* if it does not correspond to a rigid-body motion of the framework.

### (b) Stiffness and stability

In the linearized theory of elastic frameworks [27], the *tangent stiffness operator*, **K**_{T}, provides the linear relation between displacement increments and load increments,
This operator is equal to the Hessian of the potential elastic energy with respect to the parameters chosen to identify a configuration. Given an equilibrium configuration satisfying (2.1), a general stability condition would require **K**_{T} to be positive definite. The tangent stiffness operator can be decomposed as
where **K**_{M} is the *material stiffness operator*, which is always positive semidefinite and depends on the elastic stiffness of the edges, and **K**_{G} is the *geometric stiffness operator*, which depends on the axial forces in the edges. The material stiffness comes into play only when edges change in length; the geometric stiffness comes into play only when edges change in direction.

For tensegrity systems, there are two typical notions of stability. Consider a framework whose equilibrium matrix does not have full rank, and it is subjected to no external forces. We have that **K**_{M}**u**=**0** if and only if **u** is a mechanism, i.e. when edges do not change in length. We say that this framework is *prestress-stable* if it admits a selfstress for which
If the opposite inequality holds for some mechanism, we say that the system is *prestress-unstable*.

Prestress-stability does not imply stability: even if the prestress-stability condition is satisfied, it might happen that there are some **u**, which are not mechanisms, for which **K**_{T}**u**⋅**u**<0, necessarily having **K**_{G}**u**⋅**u**<0, resulting in an unstable system. We say that a tensegrity system is *superstable* if it is prestress-stable and
so that **K**_{T} must be positive definite. In other words, for a superstable system, the geometric stiffness is positive along non-trivial mechanisms, and non-negative otherwise.

We see that superstability is a stronger condition than prestress-stability. However, when a framework is prestress-stable, it is always possible to build a corresponding physical structure by making the edges very stiff with respect to the selfstress, or, vice-versa, by applying a very low selfstress with respect to the elastic stiffness of the edges. A prestress-stable system, which is not superstable, can be stable for one choice of members' stiffness and rest-length and be unstable for another choice. For this reason, it is usually stated that *a superstable framework is stable independently of the selfstress level and material properties* [28,29].

### (c) Elastic formulations

Each edge is modelled as a linear spring. We associate to each pair of vertices the scalar *k*_{IJ}, with *k*_{IJ}>0 if , and *k*_{IJ}=0 if . The axial force of an edge is then given by , with *λ*_{IJ} being the length of the edge, *λ*_{IJ}=∥**p**_{J}−**p**_{I}∥, and being the rest length of the corresponding spring. In this way, the elastic energy of the framework is written as

We first consider the case of the configuration **p** being a function of *n* parameters, or Lagrangian coordinates, grouped in the vector **x**≡(*x*_{1},…,*x*_{n}). Note that we use lowercase subscripts for these parameters, not to be confused with the uppercase subscripts employed for nodes and edges. Assuming that no loads are applied to the framework, equilibrium configurations satisfy ∇_{x}*U*=**0**, where ∇_{x} denotes the gradient operator with respect to **x**. This is a set of *n* equations of the form
2.2where (⋅)_{,i} denotes the partial derivative with respect to *x*_{i}. The last expression shows the form of the equilibrium operator, when compared with the compact equation **A****t**=**0**. We see that **t** must be a selfstress for the framework. The equilibrium equations are trivially satisfied when **t**=**0**, that is, when the framework is unstressed. This corresponds to having for all the edges.

The tangent stiffness operator, **K**_{T}=∇^{2}_{x}*U*, can be obtained from (2.2), we have in components
2.3where the two terms in the summation contribute, respectively, to **K**_{M} and **K**_{G}, and (⋅)_{,ij} denotes the second partial derivative with respect to *x*_{i} and *x*_{j}.

We see that **K**_{G}=**0** when the framework is unstressed. To see that **K**_{M} is positive semidefinite, it is enough to consider the framework unstressed, so that *U*=0. Since *U* cannot be negative, energy variations can only be positive. Since the first variation is null, ∇_{x}*U*=**0**, the second variation must be non-negative, which means that **K**_{M} is positive semidefinite. From the first term in (2.3), we have that **K**_{M} depends on the material properties of the structure through the spring constants, and it is directly related to changes in length of members, being null only for mechanisms. From the second term in (2.3), we have that **K**_{G} depends directly on the axial forces, a fact which is true even for different choices of the constitutive behaviour. Moreover, **K**_{G} is related to changes in direction of members, a fact which we justify below.

As it often happens in numerical models, the configuration **p** can just be a function of the Cartesian coordinates of the nodes. If this is the case, the stiffness operators can be expressed as follows. Let **W**_{IJ} be the linear operator
2.4represented by a 3-by-3 matrix. Here, we have introduced: **n**_{IJ}, the unit vector parallel to edge *IJ*, **n**_{IJ}=(**p**_{J}−**p**_{I})/*λ*_{IJ}; **1**_{3}, the identity in a three-dimensional vector space; the scalar *ω*_{IJ}, which is the so-called *stress* of edge *IJ*, given by *ω*_{IJ}=*t*_{IJ}/*λ*_{IJ}. The symbol ⊗ represents the dyadic product, defined by the relation
with **a**, **b** and **c** being arbitrary three-dimensional vectors, and the dot indicating the inner product. The tangent stiffness operator is then represented by a 3*N*-by-3*N* matrix, partitioned into 3-by-3 blocks, with the block in position *IJ* given by
2.5The two terms in (2.4), *k*_{IJ}**n**_{IJ}⊗**n**_{IJ} and *ω*_{IJ}(**1**_{3}−**n**_{IJ}⊗**n**_{IJ}), contribute, respectively, to **K**_{M} and **K**_{G}. We remark that, when **K**_{T} is applied to **u**, each **W**_{IJ} is applied to the displacement vector of either node *I* or *J*. Therefore, since **n**_{IJ}⊗**n**_{IJ} and (**1**_{3}−**n**_{IJ}⊗**n**_{IJ}) project vectors, respectively, along the direction of the edge *IJ* and on the plane orthogonal to it, we have that **K**_{M}**u** is non-null when edges change in length, while **K**_{G}**u** is non-null when edges change in direction.

We further observe that (2.5) can be viewed as the expression of a discrete weighted Laplacian for the underlying graph of the framework [30], where weights are not scalar but given by the operators **W**_{IJ}'s.

We now discuss some alternative forms of (2.5). First, (2.4) can be rewritten as
2.6This way, **K**_{T} takes the form
2.7with and obtained according to (2.5) from the contribution of the first and second term of (2.6), respectively. Note that can be seen as the material stiffness operator of a fictitious elastic framework whose spring constants have the form . The operator is related to the so-called *stress matrix*, ** Ω**, an

*N*-by-

*N*matrix, defined component-wise as 2.8This is actually a standard weighted Laplacian of the underlying graph, with the stresses as scalar weights.

For stiff frameworks, i.e. for every edge *IJ*, we have that
Sometimes in the literature the tangent stiffness operator is computed as
which corresponds to replacing the **W**_{IJ}'s with
This is probably done to simplify formulas; however, the approximation made is seldom stated explicitly. It is easy to see that, for a mechanism **u**_{m}, , so that the prestress-stability condition can be given in term of in the same way as before. The definition for superstability usually found in the literature is given in terms of the stress matrix, by requiring ** Ω** to be positive semidefinite, with the dimension of its nullspace equal to four, plus an additional condition [28]. Note that this corresponds to having a positive semidefinite , with the dimension of its nullspace equal to 12. This condition guarantees the framework to be stable only if is positive semidefinite, which is true only if for every edge

*IJ*[27]. In general, it is possible to have springs with negative rest lengths, e.g. prestressed springs [31]. In such a case, this definition of superstability does not imply overall stability. The definition of superstability given in §2

*b*is consistent even if prestressed springs are employed. We conclude this section by observing that, no matter which definition is adopted, if we allow for negative rest-lengths, then a prestress-stable system, which is not superstable can always become unstable for sufficiently small values of the rest-lengths, i.e. for sufficiently large selfstress. The structure we analyse in §3 is an example where this happens even for positive rest-lengths.

## 3. The bistable system

The tensegrity system object of this paper is depicted in figure 2. Figure 2*a* shows a *high-symmetry* equilibrium configuration for this system. High-symmetry configurations are prestress-stable only for some choices of the geometric parameters. In such case, there are two possibilities: for low levels of selfstress the equilibrium configuration is stable, and it is unique; for high levels of selfstress the configuration is unstable, and there are two more equilibrium configurations which are stable. These are *low-symmetry* configurations, one of which is shown in figure 2*b*. When the high-symmetry configuration is prestress-unstable, the system still possesses two additional stable, low-symmetry, equilibrium configurations.

These constitute the two bistable regimes that are available to this structure. In the following, we first analyse high-symmetry configurations to determine whether they are prestress-stable or not. Then, we construct a reduced-order model for low-symmetry configurations, to look at the energy landscapes in each of the above cases. Finally, we confirm the result of the reduced-order model with a full-order finite-element numerical model.

### (a) High-symmetry configurations

We remark that in our analysis we consider the structure as a conventional bar-framework, we only use the terms *bar* and *cable* as labels for the edges of the framework. We check *a posteriori* that the stress in cables is always non-negative in the configurations considered.

Parallel views of a high-symmetry configuration are shown in figure 3. The system is composed of five bars and 16 cables. One bar of length 2*H* is placed on a central vertical axis. Two bars of length 2*L* are placed on the two opposite edges of a horizontal rectangle, while two cables of length 2*l*, with *l*<*L*, form the other two edges of that rectangle. The plane of the rectangle intersects the vertical bar at a distance *h*<*H* from the midpoint of the bar. Four cables connect the closer end-node of the vertical bar to the nodes at the vertices of the rectangle. There is another rectangle, identical to the previous one but rotated by an angle of *π*/2 about the central vertical axis, placed at the same distance *h* from the mid-point of the vertical bar, but on the opposite side. Four cables connect the other end-node of the vertical bar to the nodes at the vertices of the latter rectangle. Finally, four more cables connect the end-nodes of the horizontal bars which are closer to each other.

Given the ratio *h*/*H*, the ratio *l*/*L* is determined by the self-equilibrium conditions when horizontal cables have null stress; we have
For each choice of the geometric parameters satisfying this condition, we can construct the equilibrium operator, **A**, and verify that it is rank-deficient, thus admitting self-stresses and mechanisms. We find that there is only one independent self-stress state **t**_{s}. With this, we can compute the geometric stiffness operator **K**_{G} and test it for prestress-stability. In this way, we constructed the prestress-stable region for high-symmetry configurations shown in figure 4. Later, we will examine two cases, represented by points *A* and *B* on this plot.

### (b) Low-symmetry configurations, reduced-order model

We define low-symmetry configurations as those which remain unchanged under the following symmetry operations (figure 5): a rotation of *π* about the axis *z*; an improper rotation of *π*/2 about the axis *z*, that is, a rotation of *π*/2 about the axis *z* plus a reflection with respect to the plane *x*–*y*. To remove rigid-body motions, we require the vertical bar to remain fixed and parallel to the axis *z*, while the other bars remain parallel to the coordinate planes *x*–*z*, *y*–*z* (figure 5).

If bars are rigid, the number of parameters necessary for identifying each configuration is four. These can be chosen as the three coordinates, *x*_{C},*y*_{C},*z*_{C}, of the centre *C* of a horizontal bar, say *AB*, plus the angle *φ* between this bar and the horizontal plane, as shown in figure 5. With this choice, the expressions of nodal coordinates are reported in table 1.

Note that high-symmetry configurations are obtained for the following values of the parameters: 3.1

In our model, we consider only eight cables to be linearly elastic, while the other members are inextensible cables and rigid bars. The elastic cables are those labelled as *a* and *b* in figure 5, plus those symmetrically placed. This assumption reduces the number of independent parameters to two, due to the additional constraints given by the inextensibility of cables *d* and *e*. The lengths *l*_{d} and *l*_{e} of these cables, are given by
3.2and
3.3By taking *φ* and *x*_{C} as independent parameters, we can solve (3.2) and (3.3) for *y*_{C} and *z*_{C},
with and expressed as
and
Differentiating (3.2) and (3.3), we have
and
By substituting the high-symmetry values (3.1) into these expressions and considering that *dl*_{d}=*dl*_{e}=0, we obtain
and
Then *dy*_{C}=*dz*_{C}=0 holds in a high-symmetry configuration, which means that small displacements from this configuration occur only with variations in *x*_{C} and *φ*.

The lengths of the elastic cables are expressed by
and
We will denote by *λ*_{0} the common value of these lengths at high-symmetry configurations, *λ*_{0}=*λ*_{a}(0,0)=*λ*_{b}(0,0).

By differentiating the above expressions at high-symmetry configurations, we can look for the displacements causing null elongations, which correspond to the mechanism. Considering the first expression, we have
and by setting *dλ*_{a}=0 and *dy*_{C}=*dz*_{C}=0, we find
3.4Since similar computations for *λ*_{b} give the same result, high-symmetry configurations possess a mechanism and its direction is given by (3.4).

At this point, we can compute the potential elastic energy for given rest-lengths and elastic constants of cables *a* and *b*:

### (c) Examples

In the following, we assume same spring constants, *k*_{a}=*k*_{b}=*k*, and same natural lengths, , for the elastic cables. We give examples whose geometric parameters correspond to points *A* and *B* in the two different regions shown in figure 4

— case A: a prestress-unstable system with

*L*=1,*H*=1/2 and*h*/*H*=0.65. The elastic constant*k*is equal to 10, while the ratio is equal to 0.1.— case B: a prestress-stable system with

*L*=1/2,*H*=1/2, and*h*/*H*=0.15. The elastic constant is the same as in case A,*k*=10.We further consider two situations: (case B1) and (case B2).

Figures 6–8 show the energy contour plots for cases A, B1 and B2, respectively. In case A, the system is bistable, with the high-symmetry configuration being unstable. The two stable configurations are placed along the direction of the mechanism, represented by a dashed line. In case B1, there is one stable configuration, which is the high-symmetry configuration. In case B2, the system is bistable. Again the high-symmetry configuration is unstable. The two stable configurations are placed far away from the direction of the mechanism.

### (d) Numerical models, critical prestrain

To assess the validity of these results, obtained with a reduced model with only two degrees of freedom, we performed some numerical computations on a full finite-element model. Rigid bars and inextensible cables have been assigned very large spring constants.

First, we computed the 30-by-30 tangent stiffness matrix in the high-symmetry configuration. This matrix was then reduced to a 2-by-2 matrix by considering only the two displacement vectors which were compatible with our assumptions on symmetry and inextensibility. The solid lines shown in figures 6–8 correspond to the directions of the eigenvectors of this reduced tangent stiffness matrix.

In case A (figure 6), the high-symmetry configuration is unstable. The direction of the mechanism, represented by the dashed line, is close to the solid line corresponding to the negative eigenvalue. This reflects the prestress-instability condition: the curvature of the energy is negative along the dashed line. In case B, the dashed line is not close anymore to the solid lines. The high-symmetry configuration is prestress-stable both in case B1 (figure 7) and in case B2 (figure 8), with the energy having positive curvature along the dashed line. However, the eigenvalues of the reduced matrix are both positive only in case B1. This shows that there is a critical value of the prestrain, , for which one of the eigenvalues becomes negative, with the high-symmetry prestress-stable configuration becoming unstable. Figure 9 shows the contour plot of the critical prestrain obtained numerically.

To verify that the reduced model effectively captures the behaviour of a corresponding physical system, we applied to the finite-element model a dynamic relaxation procedure with kinetic damping. Dynamic relaxation is a form-finding procedure to obtain the equilibrium configuration of tensile-structures and tensegrity structures [32,33]. The equilibrium configurations, which have been found by employing this procedure are in agreement with the reduced model. These configurations are marked with a star in figures 6 and 8. Moreover, we verified that the stress in all cables was positive in these configurations.

## 4. Concluding remarks

This case study showed that it is possible to change the mechanical behaviour of a tensegrity system in two different ways. A certain stable system can become unstable, while displaying additional stable configurations, either by changing its geometry, or by increasing its prestress. In the latter situation, the prestress-stable configuration is actually unstable. Usually, this can happen when a prestress-stable system is not superstable and we allow the rest-lengths to be negative, as for the case of prestressed springs. In the present system, this occurs while rest-lengths remain positive. One open question would be whether additional stable configurations always exist or not, and it can be the subject of further studies.

In relation to this, the definition of superstability given in this paper differs from the one found in the literature, in that it is consistent no matter what is the sign of the rest-lengths: according to our definition, superstability always implies stability.

Regarding the reduced model, the choice of the elastic and inextensible cables is crucial in the analysis. With a different choice, we might not be able to catch the bistable behaviour. In addition, in a bistable case, like case A or case B2, the energy values at the two minima do not necessarily need to be the same, since different values can be obtained by choosing *k*_{a}≠*k*_{b} for the spring constants.

Lastly, every definition of stability given here or in the literature is *local*, i.e. it applies to a particular configuration. Paralleling the definition of global rigidity [34], we can envisage a notion of *global stability* for elastic systems possessing a unique stable configuration. Globally, stable structures are certainly useful in applications. A characterization of global stability would then allow one to distinguish between globally stable systems and multistable ones, such as those already found in the literature.

## Acknowledgements

The author thanks Simon Guest for providing useful comments on a draft version of this paper. The author also thanks Walter Whiteley for bringing up the remark on the choice of the elastic and inextensible cables.

- Received January 27, 2013.
- Accepted March 5, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.