Equilibrium and stability of tensegrity structures: A convex analysis approach

In this paper, tensegrity structures are modeled by introducing 
suitable energy convex functions. These allow to enforce both 
ideal and non-ideal constraints, gathering compatibility, 
equilibrium, and stability problems, as well as their duality 
relationships, in the same functional framework. Arguments of 
convex analysis allow to recover consistently a number of basic 
results, as well as to formulate new interpretations and analysis 
criterions.


Introduction.
A tensegrity structure is a space truss made up of struts and guys. Truss-type structures were developed in XIX century together with metallurgic industry. In the last part of the XX century, due to advances in technology and imagination of artists (see the works by K. Snelson), architects and engineers [1,2,3,4], a special truss variety, the tensegrity, was conceived and implemented. In tensegrities, struts are usually made by bars, that is structural members carrying both tension and compression (namely, bilateral members), and guys are realized by cables, that is unilateral members able to carry only tension. An essential contribution to the rigorous analysis of tensegrities has been provided by mathematicians [5,6,7,8].
A tensegrity structure can be modeled as a discrete system of points in space (nodes) whose positions are restrained through frictionless constraints described by linear equalities and inequalities. When a structural member is modeled as inextensible, the corresponding constraint is said to be ideal and tensegrity models accounting only for ideal (opposite, non-ideal) members, are denoted as ideal (opposite, non-ideal). Otherwise, when both ideal and non-ideal constraints occur, a mixed-type tensegrity will be here referred to. Although many energy-based frameworks are available for non-ideal structures (wherein possible ideal members are treated by introducing a fictitious extensibility) [6], a variational framework explicitly accounting for ideal behavior has been proposed only very recently in [9]. There, following [10] and within the mathematical framework of the convex analysis [11,12], restrictions corresponding to ideal members are regarded as internal constraints for the variational formulation of kinematic and static structural problems. These internal constraints are defined by means of suitable free-energy contributions, depending on nodal configuration and fulfilling convexity requirements.

FRANCO MACERI, MICHELE MARINO AND GIUSEPPE VAIRO
In this paper, tensegrity structures modeled considering both ideal and non-ideal (herein assumed as at most linearly viscoelastic) constraints are addressed by means of a consistent and non-conventional variational approach. The internal constraints describing ideal members are enforced by introducing suitable dissipative pseudopotential contributions, depending on nodal velocities.
Several classical results are consistently recovered by energy arguments, in the context of both kinematic and static problems of ideal, mixed-type and non-ideal tensegrities. The novel perspective herein focused allows to gather engineering and mathematical approaches. Accordingly, basic results are coupled with nonconventional energy-based physical interpretations, even in the case of ideal or mixed-type structures, wherein classical regular energy arguments cannot be applied. Following such a formulation, an operative stability criterion is introduced, enabling the stability analysis of mixed-type tensegrities.
2. Notation. Let E be the three-dimensional Euclidean space, V := {v = P − Q, ∀ P, Q ∈ E} the vector space associated with E, endowed with the usual inner product a · b ∈ R between a, b ∈ V, and let · be the Euclidean norm on V. Let (O, ξ 1 , ξ 2 , ξ 3 ) be a time-independent Cartesian frame in E, {ξ 1 , ξ 2 , ξ 3 } being an orthonormal basis for V.
Let t ∈ [0, +∞) be the time variable and assume that all time-dependent functions are as regular as needed in time.
Let p j (t) ∈ E, with j ∈ {1, . . . , n p } and N n p > 1, be the point occupied by the node j at time t, and p j (t) = p j (t) − O its position vector in V at time t. Let N be a set of n p nodes, such that p j = p k ∀ j, k ∈ {1, . . . , n p } with j = k. Denote by p(t) := {p 1 (t), . . . , p np (t)} ∈ V np a configuration of N at time t, with p o := p(0). Moreover, letṗ j (t) ∈ V be the velocity of p j at time t (that is,ṗ j := dp j /dt), p(t) := {ṗ 1 (t), . . . ,ṗ np (t)} ∈ V np , andṗ o :=ṗ(0).
Let now consider n e bilateral scalar constraints for nodal positions as well as n b bilateral (namely, corresponding to bar-type structural elements) and n u unilateral (that is, cable-type members) constraints restricting the relative position of couples of nodes in N , and let define the following sets: I b := {(i, j, k) | ∃ bar k between nodes i and j, i < j} (2) Whenever necessary, apex b (respectively, u or e) will denote in the following quantities associated to bilateral bar-type constraints (respectively, unilateral cable-type constraints or bilateral external supports).
Definition 2.1. The tensegrity T r is the set of nodes collected in N and of constraints identified by E, I b and I u .
3. Tensegrity modeling and problems statement. For a given tensegrity structure T r , several models could be conceived depending on the description of kinematic and static features of the involved members, as well as depending on the modeling of their joints. Structural members restrict nodal positions or velocities by means of reaction forces. These restrictions (namely, constraints) can be external or internal, depending on whether they are applied to single nodes or to couples of nodes. Kinematics and statics of constraints herein addressed are described in what follows.
3.1. Constraints modeling. For each internal constraint among node i and j, let define the length as b and the unit vector identifying the constraint axis as while e o m ∈ V (with e o m = 1) identifies the axis of the m th external constraint.
3.1.1. Kinematic modeling. The following three types of constraints are herein considered where b o k and c o h are the reference lengths of the k th bilateral and h th unilateral constraint, respectively, and where δ b , and χ h (t) ∈ R + ∪ {0} depend on constraints constitutive response, which will be defined in Section 3.2.1.
3.1.2. Static modeling. Denoting as r q s the reactive force exerted by node s on constraint q (opposite to the reaction of q on s), the constraints' static behavior is assumed to be described by: where ν m (t), λ b k (t), λ u h (t) ∈ R have the physical meaning of reaction force values.
3.2. Tensegrity structure modeling. The structure T r is assumed to be loaded only by external forces at nodes and any inertial effect is disregarded. Among possible models for T r , let T be the one wherein bars (respectively, cables) are assumed to be massless and to enforce frictionless pin-jointed internal bilateral (respectively, unilateral) constraints. Internal constraints are assumed to behave as elastic, viscoelastic or ideal, while external constraints are assumed to be ideal. The ideal behavior is defined by: Definition 3.1. A bilateral (respectively, unilateral) constraint is said to be ideal if its reaction force acting on nodes makes a zero virtual work (respectively, non negative) for any admissible (that is, satisfying the constraint's kinematic restriction) virtual displacement of the constrained nodes.
For T , letÎ b ⊆ I b andÎ u ⊆ I u (of cardinalityn b andn u , respectively) be the sets identifying the non-ideal constraints, so that the setsI b = I b \Î b andI u = I u \Î u collect all constraints modeled as ideal (withn b = n b −n b andn u = n u −n u , respectively).
3.2.1. Constitutive modeling. For (i, j, k) ∈Î b and (i, j, h) ∈Î u , reactive force values λ b k and λ u h are split in two contributes: dissipative (ζ b k and ζ u h ) and nondissipative (f b k and f u h ). Non-ideal constraints are herein modeled as linearly viscoelastic. Therefore, let κ b k (respectively, η b k ) and κ u h (η u h ) be the stiffness constants (respectively, damping coefficients) of the corresponding structural member. Assuming κ b k , κ u h ∈ R + and η b k , η u h ∈ R + ∪ {0}, the constraints constitutive behavior is assumed to be described by: where · γ h (t) and H(x−x o ) is the Heaviside function centered in x o . It is worth pointing out that both dissipative and non-dissipative contributions of cable response are assumed to be unilateral.
Function χ h (t), introduced in Eq. (10) and governing the kinematics of the h th unilateral constraint, is defined as: Accordingly, χ h has the mechanical meaning of reactive force value (respectively, non-dissipative part) of the h th internal unilateral ideal constraint (respectively, non-ideal). By employing Eqs. (15) and (17), The elastic behavior is recovered by considering in Eqs. (16) Since such a constitutive model, the reference lengths b o k and c o h acquire the physical meaning of the unstressed lengths of the k th bar and of the h th cable, respectively. Furthermore, from Definition 3.1 it follows that internal ideal constraints correspond to inextensible structural members, ensuring that the distance between the constrained nodes is always equal to (for bilateral) or not greater than (for unilateral) the unstressed length.
3.2.2. Linearized model. As customary in structural theories and useful in some applications, a linearized model can be obtained by considering statics at t = 0 and a first-order approximation in time around t = 0 of the kinematical restrictions.
and V nr := V np \ V r the set of non-rigid-body velocities.
Let the following problems be introduced: Definition 3.2. The velocityv is said to be kinematically admissible for T when v is a solution of Problem 1.

Definition 3.3 (Rigidity).
T is said to be rigid when Problem 2 has no solution.

3.3.2.
Statics. Denote by f j ∈ V the resultant of the active forces on node j, and by f = {f 1 , . . . , f np } ∈ V np the set of nodal forces on N . Moreover, let the set of the self-equilibrated nodal forces be defined as: Denote also by r j ∈ V the nodal resultant of a set (generally not unique if it exists) of reactive forces exerted on constraints by node j according to Eqs. (11), (12) and (13), and let r = {r 1 , . . . , r np } ∈ V np be a set of nodal reactions.
The equilibrium problem for model T is: Problem 3. Given f ∈ V np acting on T in a given state (p(t),ṗ(t)), find r such that r = f .

Problem 3 can be formulated in a variational form through the application of the Principle of Virtual Powers:
Definition 3.4. T in (p(t) ,ṗ(t)) is said to be in equilibrium under nodal forceŝ f (t) when Problem 3 has solution with f =f (t).
Definition 3.5. T is said to be in steady-state equilibrium under time-independent nodal forcesf when Problem 3 has a solution at t = 0 with f =f andṗ o = ∅ np .
Definition 3.6. A setr of reactive forces is said to be self-equilibrated whenr is solution of Problem 3 with null nodal forces, that is f = ∅ np .
Definition 3.7 (Static-rigidity). T is said to be static-rigid when it is in steadystate equilibrium for any set of nodal forces such that f ∈ F o .
Definition 3.8 (Pre-stressability). T is said to be pre-stressable when T is in steady-state equilibrium under null nodal forces (f = ∅ np ) and Remarks. Since previous definitions, the following remarks hold: Remark 1. Rigidity, static-rigidity and pre-stressability properties do not depend on parameters κ b k , κ u h , η b k and η u h . In fact, static-rigidity and pre-stressability problems (both static), do not involve material behavior. Addressing rigidity (kinematic problem), quantities η b k and η u h do not affect the statement of Problem 2. Moreover, due to the constitutive response in Eqs. (14) and (15), request (22) ensures that stiffness constants do not intervene in Problem 2. As a consequence, it results: Remark 2. Rigidity, static-rigidity and pre-stressability are properties of T r , independent on the choice of the model T (e.g., ideal, linearly elastic, linearly viscoelastic).
Remark 3. It should be pointed out that rigidity is a kinematic property, whereas static-rigidity is a static property. They are proved to be equivalent, as shown in [9].
3.4. Stability problem and tangent response. Non-ideal (n b =n u = 0) or mixed-type (when both ideal and non-ideal internal constraints occur) tensegrities are addressed in this section.
Let the time-dependent positive-defined scalar quantity L(t) be defined as: and let introduce:  In many engineering problems, structural equilibrium is combined with the condition that all cables have to be tensioned (namely, there does not exist a slack cable). This is equivalent to require that at a given time t In this case, as an engineering task, it is frequently asked to identify the effects of small perturbations superimposed upon the equilibrium configuration of the structure. Two problems can be formulated: . For a tensegrity model T in the state (p(t),ṗ(t)) and satisfying the condition (26), find the nodal force variation δf ∈ V np which ensures the equilibrium when perturbations δp, δṗ ∈ V np admissible with ideal constraints are assigned.
Problem 6 (Inverse tangent response). For a tensegrity model T in a stable steady-state equilibrium and satisfying the condition (26) in the reference configuration (that is, for t = 0), find the variation δp ∈ V np admissible with ideal constraints and which ensures the equilibrium for a given perturbation δf ∈ V np of applied nodal forces.
Problems 5 and 6 will be solved by adopting a constructive approach, described in the following. In particular, it will be shown that under condition (26) unilateral constraints behave as bilateral in a neighborhood of the structure configuration, and thereby each of previous problems has an unique solution.
4. An energy-based approach. Statics and kinematics of tensegrity structures will be now recovered within a variational approach based on convex analysis arguments enabling to model both ideal and non-ideal constraints in a unique framework. The definition of free-energy and dissipative potential functions in the case of non-ideal members is coupled with a non-standard variational description of ideal constraints. This non-standard description is physically motivated by the fact that a motion along a non-admissible direction would imply the dissipation of an infinite amount of energy. 4.1. Convex framework. As customary in convex analysis [12], defineR = R ∪ {+∞}, where the regular addition is completed by the rules: a + (+∞) = +∞ (∀ a ∈ R) and +∞ + (+∞) = +∞, while multiplication by positive numbers is completed by a × (+∞) = +∞ (∀ a ∈ R + ). Let I o (x) be the indicator function of the zero of R (that is I o (x) = 0 if x = 0 and I o (x) = +∞ elsewhere) and I − (x) be the indicator function of the non positive numbers. If g is a convex function defined on a convex part X of a real vector space, denote by ∇g(x) and ∂g(x) a subgradient and the subdifferential set of g at point x ∈ X, respectively. Omitting the proof (see [9]), the following result is recalled:  The internal forces r j are split in dissipative r d j and non dissipative r nd j components, defined through the following constitutive laws: where Ψ(p) is the free-energy (gathering all the time-independent properties of the system) and Φ is the pseudo-potential of dissipation of T . As defined by Jean Jacques Moreau [12], pseudo-potential of dissipation function Φ is a positive convex function ofṗ, with value 0 forṗ = 0. In this paper, function Φ(p,ṗ) is chosen as dependent also on p in order to take into account the influence of the actual state on such a dissipation. Denoting by Ψ ε the strain-energy function associated with the actual configuration, by Φε (respectively,Φ) the dissipative pseudo-potential associated with non-ideal (respectively, ideal) constraints, the constitutive laws are defined by: In turn, by employing the previously-introduced apex rule, Ψ ε is herein defined as: where ideal constraints do not contribute to Ψ. Moreover, Φε is chosen as: andΦ as:Φ (p,ṗ) := 4.3. Ideal constraints. For (i, j, k) ∈I b , the kinematic constraint in Eq. (9) combined with Eq. (14) reads also as: For a given configuration p, the set of velocities v ∈ V np satisfying (p j − p i ) · (v j − v i ) = 0 is a vector space. Therefore, the function is convex in V np and endowed with generalized derivative: where Thereby, the definition set D of ∂Φ b k (p, v) is: and, from Eqs. (27), the generalized derivative ofΦ b k corresponds to r k s for s = i, j.
Similarly, for (i, j, h) ∈I u , the kinematic constraint in Eq. (10) combined with Eq. (15) reads as: For a given configuration p(t) and introducing the function Thereby, the definition set of ∂Φ u h is: and, from Eqs. (27), the generalized derivative ofΦ u h corresponds to r h s for s = i, j.

Remark 4.
If ∂Φ u h /∂v j > 0 then (p j −p i )·(v j −v i ) = 0. Therefore, an unilateral ideal member with a non-trivial reaction force kinematically behaves as a bilateral ideal one in the neighborhood of p(t).
As regards external constraints identified by (j, m) ∈ E, the function is convex in V np and its generalized derivative is with Therefore, the definition set of ∂Φ e m is D(∂Φ e m (p, v)) : and, from Eqs. (27), the generalized derivative ofΦ e m corresponds to r e j .

4.4.
Non-ideal constraints. The free-energy and dissipative pseudo-potential contributions for a bilateral non-ideal constraint (i, j, k) ∈Î b are respectively: The constraint reaction is obtained by differentiation of Eq. (47) as: Denoting by ∇ j and∇ j the gradient operators with respect to the components of p j andṗ j , respectively, simple algebra allows to prove that where I is the second-order identity tensor and symbol ⊗ indicates the dyadic product. Accordingly, at time t, the variation of r k j with respect to p j andṗ j leads to where δp j = [β k ⊗ β k ]δp j (analogously, δṗ j ) is the component of δp j parallel to β k and δp ⊥ j = [I − β k ⊗ β k ]δp j the one orthogonal to β k . For a bilateral non-ideal member, the tangent stiffness matrix K b k and the tangent damping matrix D b k can defined as: From Eq. (51), two contributions at the tangent elastic stiffness can be identified: the material stiffness K M k := κ b k [β k ⊗β k ] referred to the reference configuration, and the geometric stiffness K G k : accounting for the reorientation of stressed members.
Alternatively, tangent stiffness matrix can be arranged as from which well-established results for the purely elastic case [13] can be simply recovered.
If the linearization of the equilibrium problem is performed, disregarding orthogonal contributes to β k , then K b k = K M k . The free-energy and dissipative pseudo-potential contributions for the unilateral non-ideal constraint (i, j, h) ∈Î u are respectively Therefore, by differentiation of Eqs. (54) and (55), the corresponding constraint reaction results in and tangent stiffness and damping matrices are: where the scalar quantity α results from: Remark 5. When u h (p) > c o h , unilateral non-ideal member behaves as a bilateral member in the neighborhood of p(t).

4.6.
Ideal model: Rigidity, static-rigidity and pre-stressability. As a consequence of Remark 1, rigidity, static-rigidity and pre-stressability can be addressed by assuming an ideal behavior for each structural member. In this case, from Eqs.
(28), the model for T r withn b =n u = 0,n b = n b andn u = n u , namely the ideal model, is characterized by: withΦ(p,ṗ) defined as in Eq. (31) whenI b = I b andI u = I u .
It is worth remarking that, starting from p o , the motions admissible with all the kinematic restrictions of the ideal model for T r are described by the closed convex cone D (∂Φ id (p o , v)).
In order to embed the rigidity concept in a variational framework, let the following problem be considered: Problem 7 is a minimization problem over the set V o nr , which is non convex. This is a pitfall for finding a solution. In order to skip this drawback, let E + be the set collecting fictitious linearly independent external ideal constraints which prevent rigid-body motions of the structure and let Φ + e (p,ṗ) the corresponding contribution to the pseudo-potential of dissipation. An explicit definition of E + (and then of Φ + e ) cannot be provided because it depends on the external constraints in E. Nevertheless, if the tensegrity structure is a free-body in the space (i.e., n e = 0 and E = ∅, as in the case of tensegrity moduli) with n p ≥ 3, a possible choice for Φ + e is: being I, J, H ∈ N three not-aligned nodes, ξ N = n/ n and n = (p J − p I ) ∧ (p J − p H ). By considering a new form of the pseudo-potential function as: it is possible to introduce: From the convexity of V np and ofΦ(p o , v) over V np , it immediately follows:  On the basis of Proposition 1 and Lemma 4.2, as well as accounting for Remarks 8 and 9, it is immediate to prove: T r is rigid ⇐⇒ Problem 8 has a unique solution.
Since Remark 6 and Eq. (67), the following remark holds: If Φ + e = 0, Remark 10 reads as: Let the set R := {∇Φ u 1 , . . . , ∇Φ u nu } ∈ (R + ∪ {0}) nu be introduced. Observing that ∇Φ u h is proportional to the reaction force of the h th cable and that the condition ∅ np ∈ ∂Φ id (p o , ∅ np ) means that there exists a reactive set r in a steady-state selfequilibrium, then Remark 11.
Duality between kinematic and static arguments in tensegrities is a well-established result [5]. It has been recently recovered through an energy-based approach developed in the context of the convex analysis [9]. For the sake of completeness and omitting proofs (see [5,9]), the following results are recalled: T r is rigid ⇐⇒ T r is static-rigid Theorem 4.5. Denoting byT the bar-truss model obtained from T r by assuming as bilateral all the unilateral constraints, then: T r is rigid ⇐⇒T is rigid and T r is pre-stressable The rigidity of the space trussT can be classically determined by means of simple linear algebra, while the pre-stressability problem of T r requires to find admissible solutions of a system of 3n p equilibrium equations. The latter problem can be faced by means of a quadratic optimization approach [9]. Therefore, Theorem 4.5 allows to move towards the development of effective algorithms enabling the solution of the rigidity problem.
Owing to the definition of non-ideal constraints, function Ψ(p) is twice differentiable in a neighborhood of p. Moreover, assume that condition (26) is satisfied, which is equivalent to prescribe pre-stressability if t = 0 andṗ o = ∅ np . Since Remarks 4 and 5, both ideal and non-ideal unilateral members behave as bilateral in the neighborhood of p. Thereby, since the kinematic restrictions of unilateral constraints reduce to linear equalities, the spaceV(p) of non-rigid body motions admissible with ideal constraints, that is can be obtained by solving the homogeneous linear system where [[v]] ∈ R 3np andC(p) is the compatibility matrix at p, obtained considering only ideal constraints (i.e., identified by the sets E, E + ,I b andI u ) expressed by a classical bilateral format [15].
The special caseV(p) = ∅ np leads to a number of trivial results and it will be herein not addressed.
where K is the tangent stiffness matrix and D is the tangent damping matrix Matrices K and D are symmetric and can be assembled starting from the tangent stiffness matrices of each bar and cable, by embedding matrices in Eqs. (51) and (57) within a global reference system for the whole structure [13].
When δṗ = ∅ np , that is when only a perturbation with respect to nodal positions is considered, the nodal force variation results in Proof. Within a first-order approximation with respect to nodal positions and letting: it holds: Accordingly, stability condition can be stated as: and the positive definiteness of K(p o , ∅ np ) ensures stability.
In the specialized literature, such a stability notion is often referred to as prestress stability [15]. This is related to the decomposition of K(p o , ∅ np ) in its material and geometric parts (K M and K G ). For conventional materials, K M is at least positive semidefinite (strictly positive only for suitable arrangement of structural members). On the contrary, geometric stiffness matrix K G can have negative eigenvalues depending on pre-stress. Thereby, K(p o , ∅ np ) can be positive definite or not depending on the ratios between material coefficients and pre-stress.
There exists a special class of tensegrities which is stable whatever the material choice and pre-stress level. This occur when K(p o , ∅ np ) is positive definite for any value of pre-stress, that is when K M and K G are both positive definite.
To verify stability when K(p o , ∅ np ) is not positive definite (or, equivalently, when Ψ(p) is non strictly convex in p o ), necessary conditions are required [16]. To this aim, consider the following items. It is worth pointing out that P projects vectors in R 3np on ker[C(p)] (that is, on the space of non-rigid body motions, admissible with ideal constraints), and J on and Eq. (84) is the same as IntroducingK(p,ṗ) := J(p) + P(p)K(p,ṗ)P(p), referred to as the augmented stiffness of T , the following necessary and sufficient stability condition arises as an operative result: T is stable ⇐⇒K(p o , ∅ np ) is positive definite.
Proof. First of all, note that P, J and K are symmetric, and that PP = P, JJ = J.
If T is stable then, for any δp = ∅ np , two cases may occur:  Accordingly, Eq. (89) provides an algebraic solution of Problem 6.

5.
Conclusions. In this paper, a convex analysis approach for the analysis of both ideal and non-ideal or mixed-type tensegrities is proposed. A complete framework has been developed able to describe statics, kinematics and stability of such a kind of structures. The general framework promises an evolution towards the development of analysis and design algorithms based on linear algebra, quadratic programming and linear matrix inequality methods.