# Equilibrium stresses and rigidity for infinite tensegrities and frameworks

S. C. Power  Dept. Math. Stats.
Lancaster University
Lancaster LA1 4YF
U.K<EMAIL_ADDRESS>
###### Abstract.

Asymptotic equilibrium stresses are defined for countably infinite
tensegrities and generalisations of the Roth-Whiteley characterisation of
first-order rigidity are obtained. Generalisations of prestress stability and
second order rigidity are given for countably infinite bar-joint frameworks
and are shown to give sufficient conditions for continuous rigidity relative
to certain bounded differentiable motions. The proofs are based on a new short
proof for finite frameworks that prestress stability ensures continuous
rigidity.

MSC2020 Mathematics Subject Classification. 52C25

A finite tensegrity in ${\mathbb{R}}^{d}$ can be conceived of as a structure
consisting of inextensible cables, incompressible struts and rigid bars, all
of which are connected at their endpoints in the manner of a linearly embedded
graph $G$. It is denoted as $G(p)$, where $p=\\{p_{1},\dots,p_{n}\\}$ is the
embedding (or placement) of the vertices of $G$ and where it is understood
that certain edges of $G$ correspond to cables, struts and bars, the so-called
members of $G(p)$. Roth and Whiteley [16] obtained a useful characterisation
of their first-order rigidity, also known as infinitesimal rigidity (IR). This
is given in terms of the existence of a proper equilibrium stress together
with an evident necessary condition, namely that the structure with all
members replaced by bars should be first-order rigid. See Theorem 1.1. We
consider here countably infinite tensegrities and obtain generalisations of
the Roth-Whiteley theorem with respect to first-order rigidity relative to
asymptotically decaying motions, such as first-order $c_{0}$-rigidity, for
velocity fields that tend to zero at infinity, and first-order
$\ell^{2}$-rigidity, for finite energy velocity fields. In fact we consider
first-order _${\mathcal{X}}$ -rigidity_, the first-order rigidity with regard
to velocity fields of ${\mathcal{X}}\otimes{\mathbb{R}}^{d}$, where
${\mathcal{X}}$ is one of the classical Banach sequence spaces $\ell^{q},1\leq
q<\infty$, or $c_{0}$.

We also consider equilibrium stresses from the point of view of generalising
the notion of prestress stability to countably infinite tensegrities and bar-
joint frameworks. Recall that the continuous rigidity (CR) of a finite bar-
joint framework requires that there are no nontrivial continuous motions $t\to
p(t)$, for $t\in[0,1]$, which preserve the length constraints on members. It
is well-known that continuous rigidity is ensured by prestress stability (PS)
and also by second order rigidity (2OR). The main references here are Connelly
[3], [4] and Connelly and Whiteley [7]. See also the more recent articles of
Connelly and Gortler [5] and Holmes-Cerfon, Gortler and Theran [9]. We give
new short proofs of these implications which use only Connelly’s stress energy
function. The arguments also adapt readily to the countably infinite setting
and our definitions of boundedly prestress stable (BPS) and boundedly smoothly
rigid (BSR) and we obtain analogous implications.

In Section 1 we define tensegrities and their infinitesimal (first-order)
flexes and give a proof of the Roth-Whiteley theorem. In Section 2 we consider
countably infinite tensegrity frameworks noting first that the existence of a
proper equilibrium stress need not certify first-order rigidity. On the other
hand we show that the existence of a certain _proper asymptotic equilibrium
stress_ (Definition 2.4) does imply first-order ${\mathcal{X}}$-rigidity. In
Section 3, which is essentially independent of the previous sections, we
consider prestress stability for infinite bar-joint frameworks.

The deeper direction of the Roth-Whiteley characterisation is that first-order
rigidity ensures the presence of a proper equilibrium stress. To generalise
this, to asymptotic equlibrium stresses and first-order
${\mathcal{X}}$-rigidity, we consider closed convex cones $C$ in Banach
sequence spaces and make use of Hahn-Banach separation functionals and a
relative second dual cone equality lemma, namely Lemma 2.10.

Much of the basic theory of first-order rigidity for finite bar-joint
frameworks extends in some manner to countably infinite bar-joint frameworks.
See Kitson and Power [11], [12] for example. On the other hand countably
infinite tensegrities have received little attention in this regard. E. B.
Ashton [1] has considered “continuous tensegrities” and certain extensions of
the Roth-Whiteley theorem but the main applications involve tensegrities with
a continuum of struts, in a continuous path, with ends connected by curved
cables. As we have remarked, a literal generalisation of the Roth-Whiteley
characterisation to countably infinite tendesgrities does not hold for general
first order rigidity. The adjustments we make are natural from the perspective
of functional analysis and in particular we restrict attention to countable
tensegrities $G(p)$ which have _uniform structure_ in the sense that there is
an upper bound both to the lengths of members and to the degrees of the
vertices of $G$. This ensures that the rigidity matrix determines a bounded
linear transformation from ${\mathcal{X}}\otimes{\mathbb{R}}^{d}$ to
${\mathcal{X}}\otimes{\mathbb{R}}$ and that its transpose matrix is a bounded
linear transformation in the reverse direction. For other considerations of
the rigidity matrix as a bounded linear transformation see also Owen and Power
[14] and Kastis, Kitson and Power [10].

The proofs below are self-contained apart from standard separation results for
closed cones in Banach spaces. These follow from Theorem 2.8, while for the
Hilbert space ${\mathcal{X}}=\ell^{2}$ we also have the direct argument of
Lemma 1.2. The recent book of Connelly and Guest [6] gives a useful reference
to the rigidity theory of finite bar-joint frameworks and tensegrities.

##  1\. Finite and infinite cable-strut tensegrities

A _tensegrity_ $G(p)$ is a bar-joint framework $(G,p)$ in ${\mathbb{R}}^{d}$
for which certain bars $p_{i}p_{j}$, for $ij$ in a subset $E_{c}$ (resp.
$E_{s}$), have been replaced by _cables_ (resp. _struts_). This arises from a
partition $E=E_{b}\cup E_{c}\cup E_{s}$ of the edges of the underlying simple
graph $G=(V,E)$. The infinitesimal constraint equations, associated with a
finite tensegrity $G(p)$ and the indexing $V=\\{v_{1},\dots,v_{n}\\}$, are the
conditions for a velocity field
$u=\\{u_{1},\dots,u_{n}\\}\subset{\mathbb{R}}^{d}$ and placement
$p=(p_{1},\dots,p_{n})$ which are given by

$\displaystyle(p_{i}-p_{j})\cdot(u_{i}-u_{j})$ $\displaystyle=0,\quad ij\in
E_{b},$ $\displaystyle(p_{i}-p_{j})\cdot(u_{i}-u_{j})$ $\displaystyle\leq
0,\quad ij\in E_{c},$ $\displaystyle(p_{i}-p_{j})\cdot(u_{i}-u_{j})$
$\displaystyle\geq 0,\quad ij\in E_{s}.$

When these conditions hold $u$ is said to be an _infinitesimal flex_ of
$G(p)$. If all infinitesimal flexes $G(p)$ are trivial, that is, are rigid
motion flexes in the usual sense for bar-joint frameworks, then $G(p)$ is said
to be _first-order rigid_. These definitions also apply to countably infinite
tensegrities $G(p)$.

It is convenient to write $\overline{G(p)}$ for the associated bar-joint
framework $(G,p)$ and we note that if ${G(p)}$ is first-order rigid then so
too is $\overline{G(p)}$ since any infinitesimal flex of the bar-joint
framework is also an infinitesimal flex for ${G(p)}$. An _equilibrium stress_
$\omega:E\to{\mathbb{R}}$ for the bar-joint framework $\overline{G(p)}$ is a
scalar field (or _stress field_) $\omega=(\omega_{e})$ such that for each
joint $p_{j}$ we have the equilibrium condition

$\Sigma_{i:ij\in E}\leavevmode\nobreak\ \leavevmode\nobreak\
\omega_{ij}(p_{i}-p_{j})=0.$

An _equilibrium stress_ $\omega:E\to{\mathbb{R}}$ for the tensegrity $G(p)$ is
an equilibrium stress $\omega$ for $\overline{G(p)}$ with the additional
property that $\omega_{ij}$ is nonpositive (resp. nonnegative) if $ij$ indexes
a cable (resp. strut). Also an equilibrium stress $\omega$ of $G(p)$ is said
to be _proper_ if $\omega_{ij}$ is nonzero for all cables and struts.

Since a bar constraint for the pair $u_{i},u_{j}$ is equivalent to a cable
constraint together with a strut constraint, for the same pair $ij$, it is
convenient to consider _cable-strut tensegrities_ $G(p)$ determined by $p$ and
subsets $E_{c},E_{s}$ which are not necessarily disjoint subsets of
$\\{1,\dots,n\\}^{2}$ (or of ${\mathbb{N}}^{2}$ if $G$ is countable). The
constraint system for this cable-strut tensegrity $G(p)$ is then the set of
inequalities

$\displaystyle(p_{i}-p_{j})\cdot(u_{i}-u_{j})$ $\displaystyle\leq 0,\quad
ij\in E_{c},$ $\displaystyle(p_{i}-p_{j})\cdot(u_{i}-u_{j})$
$\displaystyle\geq 0,\quad ij\in E_{s}.$

Note that we retain the fact that $G$ is the associated simple graph, so that
a label $ij$ in both $E_{c}$ and $E_{s}$ corresponds to one edge of $G$. Also,
$\overline{G(p)}$ is the associated bar-joint framework, as before.
Henceforth, and without loss of generality, we consider only cable-strut
tensegrities $G(p)$.

The inequality constraint system for $G(p)$ can be expressed in terms of a
matrix condition for a certain _tensegrity rigidity matrix_ $R(G(p))$. If $G$
is finite then the matrix has $m=|E_{c}|+|E_{s}|$ rows and $n$ edges, where
the row for a member $e$ is equal to

$\hbox{}\vbox{\kern 0.86108pt\hbox{$\kern 0.0pt\kern
2.5pt\kern-5.0pt\left[\kern
0.0pt\kern-2.5pt\kern-5.55557pt\vbox{\kern-0.86108pt\vbox{\vbox{
\halign{\kern\arraycolsep\hfil\@arstrut$\kbcolstyle#$\hfil\kern\arraycolsep&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep&&
\kern\arraycolsep\hfil$\@kbrowstyle#$\ifkbalignright\relax\else\hfil\fi\kern\arraycolsep\cr
5.0pt\hfil\@arstrut$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
v_{i}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
v_{j}$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\\\e$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
sgn(e)(p_{i}-p_{j})$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle
0$\hfil\kern 5.0pt&5.0pt\hfil$\scriptstyle sgn(e)(p_{j}-p_{i})$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle 0$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle\cdots$\hfil\kern
5.0pt&5.0pt\hfil$\scriptstyle$\hfil\kern 5.0pt\crcr}}}}\right]$}},$

where $sgn(e)$ is $-1$ (resp. $+1$) if $e$ is a cable (resp. strut). It
follows that a velocity field $u$ is an infinitesimal flex of $G(p)$ if and
only if the stress field $R(G(p))u$ lies in the cone ${\mathbb{R}}^{m}_{+}$.
Here and below we view $u$ as a column matrix and $R(G(p))u$ as the usual
matrix product. Note also that $R(G(p))$ is obtained from the usual rigidity
matrix $R(G,p)$ by repeating rows corresponding to cable-strut pairs and then
by multiplying the rows corresponding to cables by -1. (The matrix $R(G,p)$
has $|E|$ rows of the form above with all signs positive.) It follows that
$R(G(p))u=0$ if and only if $R(G,p)u=0$. Also, $\omega$ is an equilibrium
stress of $G(p)$ if and only if $|\omega|R(G(p))u=0$.

Let $G$ be a countably infinite simple graph with infinite cable-strut
tensegrity $G(p)$ with $p=(p_{1},p_{2},\dots)$ a vertex placement in
${\mathbb{R}}^{d}$. The infinite rigidity matrix $R(G(p))$ can be viewed as a
linear transformation from the space of velocity fields $u$ in the direct
product vector space $({\mathbb{R}}^{d})^{\infty}$ to the vector space
${\mathbb{R}}^{E}=\prod_{e\in E}{\mathbb{R}}$. We shall assume that each
vertex of $G$ has finite degree. In this case the map $\omega\to\omega
R(G(p))$ is well-defined as a linear transformation.

The Roth-Whiteley theorem for finite cable-strut tensegrities takes the
following form.

######  Theorem 1.1.

Let $G(p)$ be a finite cable-strut tensegrity. Then $G(p)$ is first-order
rigid if and only if $\overline{G(p)}$ is first-order rigid and there exists a
proper equilibrium stress.

The sufficiency direction of the proof is straightforward. For a stress field
$\omega$ with nonnegative values for struts and nonpositive values for cables,
let $\mu_{e}=|\omega_{e}|$. Then $\omega$ satisfies the equilibrium equations
if and only if $\mu R(G(p))=0$. Now let $\overline{G(p)}$ be first-order
rigid, let $\omega$ be a proper equilibrium stress for $G(p)$ and let $u$ be
an infinitesimal flex of $G(p)$, so that $\sigma=R(G(p))u$ lies in the cone
${\mathbb{R}}^{m}_{+}$. Then

$\mu\cdot\sigma=\mu\dot{(}R(G(p)u)=(\mu R(G(p))\cdot u=0.$

Since $\mu_{e}$ is strictly positive for every member $e$ it follows that
$\sigma=0$. Thus $u$ is an infinitesimal flex of $\overline{G(p)}$. By our
assumptions, $u$ is a trivial rigid motion flex and it follows that $G(p)$ is
first-order rigid.

The necessity direction depends on some basic properties of convex cones in
${\mathbb{R}}^{m}$ which are brought into play for the convex cone generated
by the rows of the rigidity matrix $R(G(p))$.

A cone $C$ in ${\mathbb{R}}^{m}$ is a nonempty set closed under multiplication
by nonnegative real numbers. The dual cone of $C$ is the cone $C^{*}$ in the
dual space $({\mathbb{R}}^{m})^{\prime}$ consisting of the set of linear
functionals $x^{*}$ such that $x^{*}(x)\geq 0$ for all $x$ in $C$. A standard
fact is that the second dual cone $C^{**}$ coincides with $C$ (that is, with
the canonical image of $C$ in the second dual space). This follows readily
from the following separation lemma.

######  Lemma 1.2.

Separation lemma. Let $C$ be a closed convex cone in ${\mathbb{R}}^{m}$, or in
a general Hilbert space, and $w\notin C$. Then there is a continuous linear
functional $f$ with $f(w)<0\leq f(x)$ for all $x\in C$.

###### Proof.

Since $C$ is closed, by elementary Hilbert space geometry there is an element
$x_{*}$ of $C$ which is closest to $w$. Since $C$ contains
${\mathbb{R}}_{+}x_{*}$ and $C$ is convex the triangle $wx_{*}O$ is
necessarily a right-angled triangle with $\langle x_{*}-w,x_{*}\rangle=0$. Let
$y=x_{*}-w$ with linear functional $f_{y}(\cdot)=\langle\cdot,y\rangle$. In
particular $f(w)=\langle w,x_{*}-w\rangle=\langle w,x_{*}\rangle-\|w\|^{2}<0.$

Let $H=\\{y\\}^{\perp}$ be the associated hyperplane with closed half spaces
$H_{+},H_{-}$ (where the functional takes nonnegative and nonpositive values).
We show that $f_{y}(x)\geq 0$ for $x\in C$, ie. $C\subseteq H_{+}$. Suppose
not and $x_{1}\in C$ with $\langle x_{1},x_{*}-w\rangle<0$. Then for
$0<\alpha<1$

$\|w-(\alpha
x_{1}+(1-\alpha)x_{*})\|^{2}=\|(w-x_{*})-\alpha(x_{1}-x_{*})\|^{2}=$
$\|w-x_{*}\|^{2}-2\alpha\langle
w-x_{*},x_{1}-x_{*}\rangle+\alpha^{2}\|x_{1}-x_{*}\|^{2}.$

Also $\langle x_{*}-w,x_{*}\rangle=0$ and so for small $\alpha$ this is less
than $\|w-x_{*}\|^{2}$, a contradiction since $C$ is convex. ∎

######  Lemma 1.3.

Dual cone lemma. Let $X=\\{x_{1},\dots,x_{m}\\}$ be a set of vectors in
${\mathbb{R}}^{m}$ with convex cone
$C(X)=\\{\sum_{i}\lambda_{i}x_{i}:\lambda_{i}\geq 0\\}$. Then the following
statements are equivalent.

(i) $C(X)^{*}$ is a subspace.

(ii) $C(X)^{*}=X^{\perp}$.

(iii) $\sum_{i}\mu_{i}x_{i}=0$ for some choice of strictly positive
coefficients $\mu_{1},\dots,\mu_{m}$.

###### Proof.

The equivalence of (i) and (ii) is immediate. If (i) holds then the cone
$C(X)=C(X)^{**}$ is also a subspace. Thus for each index $j$

$-x_{j}=\sum_{i}\lambda_{i,j}x_{i}\quad\mbox{with}\quad\lambda_{i,j}\geq 0,$

and so

$0=\sum_{i}\mu_{i,j}x_{i}\quad\mbox{with}\quad\mu_{i,j}\geq
0,\leavevmode\nobreak\ \leavevmode\nobreak\ \mu_{j,j}>0.$

Adding these $m$ equalities gives the desired coefficients for (iii).

If (iii) holds then for $x^{*}\in C(X)^{*}$ we have
$0=x_{*}(\sum_{j}\mu_{j}x_{j})=\sum_{j}\mu_{j}x_{*}(x_{j}).$ Since
$x_{*}(x_{j})\geq 0$ for each $j$ it follows from the strict positivity of the
coefficients that $x_{*}(x_{j})=0$ for each $j$. Thus $X^{\perp}\subseteq
C(X)^{*}$. Since the reverse inclusion is elementary, (ii) holds. ∎

######  Lemma 1.4.

Dichotomy lemma. Let $A$ be an $m$ by $n$ real matrix. Then precisely one of
the following is true.

(i) There exists $u$ in ${\mathbb{R}}^{m}$ with $Au\neq 0$ and
$Au\in{\mathbb{R}}^{m}_{+}$.

(ii) There exists a row vector $\mu$ with strictly positive entries such that
$\mu A=0$.

###### Proof.

Let $X=\\{x_{1},\dots,x_{m}\\}$ be the set of row vectors of $A$. Then

$\\{u:Au=0\\}=X^{\perp},\quad\quad\\{u:Au\in{\mathbb{R}}^{m}_{+}\\}=C(X)^{*}.$

Note that (i) is false if and only if $C(X)^{*}=X^{\perp}$. By the previous
lemma this occurs if and only if $\sum\mu_{i}x_{i}=0$ for some choice of
strictly positive coefficients and so (i) is false if and only if (ii) is
true. ∎

To complete the proof of Theorem 1.1 let $G(p)$ be first-order rigid. If $u$
is an infinitesimal flex, with $R(G(p))u\in{\mathbb{R}}^{m}_{+}$ then $u$ must
be a rigid motion flex and so $R(G(p))u=0$. This means condition (i) in Lemma
1.4 does not hold for $A=R(G(p))$. Therefore (ii) holds and this strictly
positive vector $\mu$ determines a proper equilibrium stress $\omega$. Also
$\overline{G(p)}$ is first-order rigid since this property holds for $G(p)$.

##  2\. First-order rigidity for infinite tensegrities

That the Roth-Whiteley equivalence fails to hold for countable tensegrities is
evident from the following two examples.

######  Example 2.1.

Let $G(p)$ be the infinite periodic tensegrity framework in ${\mathbb{R}}^{2}$
whose placement is the periodic tiling by equilateral triangles and where all
the members are cables. Then $\overline{G(p)}$ is first-order rigid and there
exist proper equilibrium stresses. Indeed, any stress field which is constant
on any straight line through a cable and takes negative values is such a
stress. However, $G(p)$ is not first-order rigid since, for example, the
contractive affine map $(x,y)\to(x/2,y/2)$ restricts to a velocity field which
is an infinitesimal flex of $G(p)$. On the other hand we shall see that $G(p)$
is first-order $c_{0}$-rigid.

######  Example 2.2.

A countable cable-strut tensegrity $G(p)$ in ${\mathbb{R}}^{2}$ is indicated
in Figure 1. The dashed lines represent cables between the vertex placements
(the joints of $G(p)$) while the solid lines represent cable-strut pairs
(equivalent to bars). The nondiagonal members all have length 1. While the
companion bar-joint framework $\overline{G(p)}$ is first-order rigid, the
tensegrity $G(p)$ has nontrivial infinitesimal flexes in which countably many
of the upper joints have the same leftward translational velocity while all
other joints are fixed. This leads to the observation that the set of
infinitesimal flexes of $G(p)$ is a convex cone which is not finitely
generated. The figure also indicates an equilibrium stress field $\omega$ for
$G(p)$, where a “0” label on a cable-strut pair indicates that the total
stress $\omega_{ij,c}+\omega_{ij,s}$ is equal to 0 for these 2 members.
Choosing nonzero values for these pairs leads to $\omega$ being a proper
equilibrium stress for $G(p)$. A consequence of this example is that there is
no literal counterpart to the dichotomy lemma for an infinite matrix $A$, even
when $A$ has the sparse structure of a rigidity matrix.

Figure 1. A countable cable-strut tensegrity with a proper equilibrium stress.

Let ${\mathcal{X}}$ be one of the Banach spaces $c_{0}$ or $\ell^{q}$, for
$1\leq q<\infty$. Then the tensor product space
${\mathcal{X}}\otimes{\mathbb{R}}^{d}$ may be viewed as a space of velocity
fields for any countable cable-strut tensegrity $G(p)$. More precisely, this
is the vector space of velocity fields $u=(u_{1},u_{2},\dots)$, with
$u_{k}\in{\mathbb{R}}^{d}$, which is a Banach space for a tensor product norm
$\|\cdot\|=\|\cdot\|_{\mathcal{X}}\otimes\|\cdot\|^{\prime}$, where
$\|\cdot\|^{\prime}$ is a norm on ${\mathbb{R}}^{d}$. These norms are
equivalent norms and in this sense independent of $\|\cdot\|^{\prime}$. Let us
take $\|\cdot\|^{\prime}=\|\cdot\|_{q}$ when ${\mathcal{X}}=\ell^{q}$ and
$\|\cdot\|^{\prime}=\|\cdot\|_{\infty}$ when ${\mathcal{X}}=c_{0}$. Then the
normed vector-valued sequence space ${\mathcal{X}}\otimes{\mathbb{R}}^{d}$ is
naturally isometrically isomorphic to ${\mathcal{X}}$.

######  Definition 2.3.

Let $G(p)$ be a countable tensegrity in ${\mathbb{R}}^{d}$ and let
${\mathcal{X}}$ be a classical Banach sequence space. Then $G(p)$ is first-
order ${\mathcal{X}}$-rigid (or ${\mathcal{X}}$-infinitesimally rigid) if
every infinitesimal flex of $G(p)$ in ${\mathcal{X}}\otimes{\mathbb{R}}^{d}$
is trivial.

In particular while the “strip tensegrity” in Example 2.2 is infinitesimally
flexible it is nevertheless $c_{0}$-infinitesimally rigid. To see this note
that the subframework determined by the lower two rows of joints is an
infinitesimally rigid tensegrity. Let $u$ be a general infinitesimal flex of
$G(p)$. By adding a trivial (rigid motion) infinitesimal flex we may assume
that the the velocity $u(p)$ at every joint $p$ of this subframework is zero.
If $u(p)$ is nonzero for some joint $p$ in the upper row of cables then it is
necessarily of the form $(-a,0)$ with $a>0$. It follows that the velocity
$u(p^{\prime})$ for any joint in the top row to the right of $p$ has the form
$(-a^{\prime},0)$ with $a^{\prime}\geq a$, and so $u$ is not in
$c_{0}\otimes{\mathbb{R}}^{d}$.

We now define some proper asymptotic equilibrium stresses for a tensegrity
$G(p)$ which are associated with the Banach sequence space ${\mathcal{X}}$. We
write ${\mathcal{X}}^{\prime}$ for the dual sequence space of ${\mathcal{X}}$.
Recall that the rigidity matrix $R(G(p))$ accommodates the negative signs (for
cables) of an equilibrium stress in the sense that a stress field $\omega$ is
an equilibrium stress if an only if $|\omega|R(G(p))=0$.

######  Definition 2.4.

Let $G(p)$ be a countable cable-strut tensegrity and let $\omega^{(n)}$, for
$n=1,2,\dots,$ be a sequence of finitely nonzero stress fields for $G(p)$ such
that

(i) $\omega^{(n)}_{e}\leq 0$ if $e$ is a cable and $\omega^{(n)}_{e}\geq 0$ if
$e$ is a strut,

(ii) for every member $e$ there is $\delta_{e}>0$ such that
$|\omega^{(n)}_{e}|\geq\delta_{e}$ for sufficiently large $n$.

Then

(a) $\omega^{(n)}$ is a _proper ${\mathcal{X}}^{\prime}$-asymptotic
equilibrium stress_ for $G(p)$ if $|\omega^{(n)}|R(G(p))$ tends to 0 in
${\mathcal{X}}^{\prime}\otimes{\mathbb{R}}^{d}$ as $n\to\infty$,

(b) $\omega^{(n)}$ is a _proper
$({\mathcal{X}}^{\prime},{\mathcal{X}})$-asymptotic equilibrium stress_ for
$G(p)$ if $|\omega^{(n)}|R(G(p))$ tends to 0 in the weak topology of
${\mathcal{X}}^{\prime}\otimes{\mathbb{R}}^{d}$.

Note that there is no requirement that the sequence $\omega^{(n)}$ is subject
to any kind of convergence. The convergence condition in (b) ensures that
$|\omega^{(n)}|R(G(p))u$ tends to 0 when the velocity field $R(G(p))u$ is in
${\mathcal{X}}\otimes{\mathbb{R}}^{d}$, and so for our contexts below (a) is a
stronger requirement than (b).

We consider tensegrities $G(p)$ with _uniform structure_ by which we mean that
there is an upper bound to the degrees of the vertices of $G$ and an upper
bound to the lengths of the members of $G(p)$. In this case $R(G(p))$ is a
bounded linear transformation from ${\mathcal{X}}\otimes{\mathbb{R}}^{d}$ to
${\mathcal{X}}\otimes{\mathbb{R}}$ (since the bar lengths are uniformly
bounded) and the transpose matrix $R(G(p))^{T}$ is a bounded linear
transformation from ${\mathcal{X}}\otimes{\mathbb{R}}$ to
${\mathcal{X}}\otimes{\mathbb{R}}^{d}$ (since $G(p)$ has uniform structure).

In the next lemma we note that a proper equilibrium stress in
${\mathcal{X}}^{\prime}\otimes{\mathbb{R}}$, for ${\mathcal{X}}\neq\ell^{1}$,
provides a proper ${\mathcal{X}}$-asymptotic equilibrium stress.

######  Lemma 2.5.

Let $G(p)$ be a cable-strut tensegrity with uniform structure and let
${\mathcal{X}}$ be $c_{0}$ or $\ell^{q}$, for $1<q<\infty,$ with dual space
${\mathcal{X}}^{\prime}$. Let $\omega=(\omega_{1},\omega_{2},\dots)$ be a
proper equilibrium stress field for $G(p)$ with
$\omega\in{\mathcal{X}}^{\prime}\otimes{\mathbb{R}}$. For each $n$ let
$\omega^{(n)}$ be the stress field $(\omega_{1},\dots,\omega_{n},0,0,\dots)$.
Then the sequence $(\omega^{(n)})$ is a proper
${\mathcal{X}}^{\prime}$-asymptotic equilibrium stress for $G(p)$.

###### Proof.

In view of the hypotheses for ${\mathcal{X}}$ the sequence $\omega^{(n)}$
tends to $\omega$ in ${\mathcal{X}}\otimes{\mathbb{R}}^{d}$ as $n\to\infty$.
Since $G(p)$ has uniform structure,
$|\omega^{(n)}|R(G(p))\to|\omega|R(G(p))=0.$ ∎

Any proper equilibrium stress for the strip framework of Example 2.2 must have
the same negative value $\omega_{e}$ for all the cable members $e$ and so
there is no proper equilibrium stress in
${\mathcal{X}}^{\prime}\otimes{\mathbb{R}}$ when ${\mathcal{X}}$ is $c_{0}$ or
$\ell^{q},1<q<\infty$. However, there do exist proper
$({\mathcal{X}}^{\prime},{\mathcal{X}})$-asymptotic equilibrium stresses which
are obtained by truncating, in the manner of the previous lemma, the
$\ell^{\infty}\otimes{\mathbb{R}}$ equilibrium stress given in Example 2.2.

The next lemma generalises the sufficiency direction of the Roth-Whiteley
theorem.

######  Lemma 2.6.

Let $G(p)$ be a countable cable-strut tensegrity with uniform structure, let
${\mathcal{X}}$ be one of the Banach sequence spaces $c_{0},\ell^{q},$ for
$1\leq q<\infty$, and suppose that $\overline{G(p)}$ is first-order
${\mathcal{X}}$-rigid and there exists a proper
$({\mathcal{X}}^{\prime},{\mathcal{X}})$-asymptotic equilibrium stress. Then
$G(p)$ is first-order ${\mathcal{X}}$-rigid.

###### Proof.

Let $\overline{G(p)}$ be first-order ${\mathcal{X}}$-rigid and let
$\omega^{(n)}$ be a proper $({\mathcal{X}}^{\prime},{\mathcal{X}})$-asymptotic
equilibrium stress for $G(p)$. For each $n$ let $\mu^{(n)}$ be the nonnegative
stress field with $\mu^{(n)}_{e}=|\omega^{(n)}_{e}|$. Let
$u\in{\mathcal{X}}\otimes{\mathbb{R}}^{d}$ be an infinitesimal flex so that
the stress field $\sigma=R(G(p))u$ is nonnegative, with $\sigma_{e}\geq 0$ for
all $e$. We have

$\mu^{(n)}R(G(p))u=\mu^{(n)}\cdot\sigma=\sum_{e}\mu^{(n)}_{e}\sigma_{e},$

where the sum is finite for each $n$ and the terms $\mu^{(n)}_{e}\sigma_{e}$
are nonnegative. On the other hand since $\mu^{(n)}R(G(p))$ tends to zero,
these finite sums also tend to zero, as $n$ tends to infinity, and so
$\mu^{n}_{e}\geq\delta_{e}$ for some $\delta_{e}$ and all $n\geq n_{e}$. It
follows that $\sigma_{e}=0$ for each $e$. Thus $u$ is an infinitesimal flex
for $\overline{G(p)}$ and so $u$ is a rigid motion infinitesimal flex, as
required. ∎

###  2.1. Closed convex cones and first-order ${\mathcal{X}}$-rigidity.

We now obtain the following generalisation of the Roth-Whiteley theorem. Note
that the sufficiency direction follows from Lemma 2.6.

######  Theorem 2.7.

Let $G(p)$ be a countable cable-strut tensegrity with uniform structure and
let ${\mathcal{X}}$ be one of the Banach sequence spaces $c_{0},\ell^{q},$ for
$1\leq q<\infty$. Then $G(p)$ is first-order ${\mathcal{X}}$-rigid if and only
if $\overline{G(p)}$ is first-order ${\mathcal{X}}$-rigid and there exists a
proper ${\mathcal{X}}^{\prime}$-asymptotic equilibrium stress.

A consequence of the Hahn-Banach theorem for locally convex topological vector
spaces is the following general separation theorem. For a proof see Theorem
3.9 in Conway [8].

######  Theorem 2.8.

Let ${\mathcal{X}}$ be a real locally convex topological vector space and let
$A$ and $B$ be two disjoint closed convex subsets of ${\mathcal{X}}$. If $B$
is compact, then $A$ and $B$ are strictly separated.

Strict separation means that there exists a real value $\alpha$ and a
continuous linear functional $f$ such that $f(b)<\alpha<f(a)$ for all $b\in
B,a\in A$. Applying this to a singleton set $B=\\{w\\}$ and a closed convex
cone $A$, with $w\notin A$, we have $\alpha<0=f(0)$. Also, since $A$ is a cone
we cannot have $f(a)<0$ for any $a\in A$ for otherwise $f(\beta a)<\alpha$ for
some $\beta>0$. Thus we have the following separation lemma for closed convex
cones.

######  Lemma 2.9.

. Let $C$ be a closed convex cone in a real locally convex topological vector
space ${\mathcal{X}}$ and let $w\in{\mathcal{X}}\backslash C$. Then there is a
continuous linear functional $f$ on ${\mathcal{X}}$ with $f(w)<0\leq f(x)$ for
all $x\in C$.

When ${\mathcal{X}}$ is a Hilbert space a direct elementary proof of this is
given by the proof of Lemma 1.2.

Henceforth we assume that ${\mathcal{X}}$ is one of the Banach sequence spaces
$\ell^{q},1\leq q<\infty$, or $c_{0}$, and we write $\hat{x}$ for the image of
$x\in{\mathcal{X}}$ under the canonical injection from ${\mathcal{X}}$ to its
second dual space ${\mathcal{X}}^{\prime\prime}$.

As a substitute for the double dual cone identity $\hat{C}=C^{**}$ in finite
dimensions we have the following lemma.

######  Lemma 2.10.

Let $C$ be a closed convex cone in a Banach space ${\mathcal{Y}}$ and let
$\hat{{\mathcal{Y}}}$ be the natural embedding of ${\mathcal{Y}}$ in the
second dual ${\mathcal{Y}}^{\prime\prime}$. Then
$\hat{C}=C^{**}\cap\hat{{\mathcal{Y}}}$.

###### Proof.

The inclusion of $\hat{C}$ in the intersection is clear. Let
$w\in{\mathcal{X}}\backslash C$. By Lemma 2.9 there exists $f$ in $C^{*}$ with
$f(w)<0$. In particular $\hat{w}\notin C^{**}$. ∎

Example 2.1 shows that the dichotomy lemma does not extend in a literal way to
infinite matrices. However we have the following extension to tensegrities
with uniform structure. We write ${\mathbb{R}}^{E}_{+}$ to denote the
nonnegative cone in the direct product space ${\mathbb{R}}^{E}=\prod_{e\in
E}{\mathbb{R}}$.

######  Lemma 2.11.

Let $G(p)$ be a countable tensegrity in ${\mathbb{R}}^{d}$ with uniform
structure which is first-order ${\mathcal{X}}$-rigid. Then precisely one of
the following is true.

(i) There exists $u$ in ${\mathcal{X}}\otimes{\mathbb{R}}^{d}$ with $R(G(p))u$
a nonzero element of the cone ${\mathbb{R}}_{+}^{E}$.

(ii) There exists a proper ${\mathcal{X}}^{\prime}$-asymptotic equilibrium
stress $\omega^{(n)}$ for $G(p)$.

###### Proof.

If (i) holds then (ii) is false by the sufficiency direction of Theorem 2.7.

Assume then that (i) does not hold. Since $G(p)$ has uniform structure
$R(G(p))u$ lies in ${\mathcal{X}}\otimes{\mathbb{R}}$ for any velocity field
$u$ in ${\mathcal{X}}\otimes{\mathbb{R}}^{d}$. Let $X=\\{x_{1},x_{2},\dots\\}$
be the rows of $R(G(p))$ and let $C(X)$ be the closed convex cone in
${\mathcal{X}}^{\prime}\otimes{\mathbb{R}}^{d}$ generated by $X$. As in the
earlier dichotomy lemma, any velocity field $u$ may be viewed as a continuous
linear functional on the cone $C(X)$. The failure of (i) is the assertion that
$C(X)^{*}=X^{\perp}$ where $X^{\perp}$ denotes the annihilator of $X$ in
$({\mathcal{X}}^{\prime}\otimes{\mathbb{R}}^{d})^{\prime}$. In particular the
dual cone $C(X)^{*}$ is a linear subspace of
$({\mathcal{X}}^{\prime}\otimes{\mathbb{R}}^{d})^{\prime}$ and so by, Lemma
2.10, $C(X)$ is also a subspace. In particular if $y$ is the element
$x_{1}+\frac{1}{2}x_{2}+\frac{1}{4}x_{2}+\dots$ in $C(X)$ then $-y$ lies in
$C(X)$ and so $-y$ is a limit in
${\mathcal{X}}^{\prime}\otimes{\mathbb{R}}^{d}$ of a sequence of finite sums
$\sigma^{(n)}=\sum_{i}\lambda_{i}^{n}x_{i}$, for $n=1,2,\dots$, where the
coefficients are nonnegative. It follows that the sequence

$x_{1}+\frac{1}{2}x_{2}+\dots+\frac{1}{2^{n}}x_{n}+\sigma^{(n)},n=1,2,\dots,$

tends to $0$ in ${\mathcal{X}}^{\prime}\otimes{\mathbb{R}}^{d}$. Also, this
sequence has the form

$\mu^{(n)}R(G(p)),n=1,2,\dots,$

where for each $k$ we have $\mu^{(n)}_{k}\geq\frac{1}{2^{k}}$ for $n\geq k$.
Let $\omega_{k}^{(n)}=\mu_{k}^{(n)}$ (resp. $-\mu_{k}^{(n)}$) if $x_{k}$ is
the row for a strut (resp. cable). Then $\omega^{(n)}$ is the desired proper
${\mathcal{X}}^{\prime}$-asymptotic equilibrium stress. ∎

##  3\. Prestress stability, second order rigidity and continuous rigidity

We first consider finite bar-joint frameworks and give direct proofs that
prestress stability implies continuous rigidity and that second order rigidity
is an intermediate property.

###  3.1. PS, 2OR and CR for finite frameworks

A classical result of Asimow and Roth [2] in the rigidity theory of finite
bar-joint frameworks $(G,p)$ is that $(G,p)$ is continuously rigid (CR) if it
is infinitesimal rigidity (IR). The proof depends on the local structure of a
general variety in ${\mathbb{R}}^{N}$. With this structure determined (one
might think of a neighbourhood of $p$ being a finite union of parts of
manifolds through $p$) the existence of a nontrivial continuous motion implies
the existence of a differentiable motion $p(t)$, with derivative
$p^{\prime}(0)$ providing a non rigid infinitesimal flex. It is also true that
infinitesimal rigidity implies continuous rigidity for finite tensegrities, a
fact due to Connelly - see Roth-Whiteley [16] Theorem 5.7.

For a finite _generic_ framework, being continuously rigid is equivalent to
being first-order rigid [2]. The nongeneric bar-joint frameworks of Figure 2
show that this equivalence fails in general. That the first framework is CR
and not IR is clear, since the 3 lower joints are colinear. It is less evident
that the second framework is continuously rigid. One can give a direct ad hoc
proof but better insights come from the consideration of equilibrium stresses
and prestress stability.

Figure 2. CR but not IR.

A general _stress_ of a bar-joint framework $(G,p)$ is any real field
$\omega:E\to{\mathbb{R}}$ on the edges and an _equilibrium stress_ is a stress
with $\omega R(G,p)=0$. This is equivalent to an equilibrium condition at each
vertex, namely $\sum_{j}\omega_{ij}(p_{i}-p_{j})=0,$ for each vertex $v_{i}$.

In the next definition we adopt the convention that $\omega_{ij}=0$ if $i=j$
or if $ij$ is not an edge. Also we have $\omega_{ij}=\omega_{ji}$ and a
summation over pairs $i,j$ with $i<j$ indicates a summation over all the edges
of $G$. We say that an infinitesimal flex is _trivial_ if it is a rigid motion
infinitesimal flex.

######  Definition 3.1.

A finite bar-joint framework $(G,p)$ is _prestress stable_ (PS) if there is an
equilibrium stress $\omega$ such that for each nontrivial infinitesimal flex
$u$,

$\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}>0.$

From the definitions it follows that first-order rigidity implies prestress
stability. To see that prestress stability implies continuous rigidity we make
use of the stress energy functions introduced by Connelly [4].

For a general stress $\omega$ for $(G,p)$ the _stress energy function_
$E_{\omega}:{\mathbb{R}}^{nd}\to{\mathbb{R}}$ is given by

$E_{\omega}(q)=\sum_{i<j}\omega_{ij}|q_{i}-q_{j}|^{2}.$

In the presence of prestress stability, for an equilibrium stress $\omega$,
the next lemma shows that the second derivative at $t=0$ of $E_{\omega}(p+tu)$
is positive for a nontrivial infinitesimal flex $u$. The usefulness of the
lemma in the proof of Theorem 3.3 comes from the stability fact that if
$u^{\prime}$ is close enough to $u$ then the second derivative at $t=0$ of
$E_{\omega}(p+tu^{\prime})$ is also positive.

######  Lemma 3.2.

If $\omega$ is any stress for $(G,p)$ and $q,u$ are vectors in
${\mathbb{R}}^{dn}$ then

$\frac{d^{2}}{dt^{2}}E_{\omega}(q+tu)|_{t=0}=2\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}.$

###### Proof.

Expanding the terms

$\omega_{ij}|(q_{i}+tu_{i}-(q_{j}+tu_{j})|^{2}=\omega_{ij}\langle
q_{i}+tu_{i}-(q_{j}+tu_{j}),q_{i}+tu_{i}-(q_{j}+tu_{j})\rangle$

we have

$E_{\omega}(q+tu)=\sum_{i<j}\omega_{ij}\big{(}|q_{i}-q_{j}|^{2}+2t\langle
q_{i}-q_{j},u_{i}-u_{j}\rangle+t^{2}|u_{i}-u_{j}|^{2}\big{)}$

and the desired identity follows. ∎

Let us also note that the derivative of $E_{\omega}(p+tu)$ at $t=0$ is given
by

$\frac{1}{2}\frac{d}{dt}E_{\omega}(p+tu)|_{t=0}=\sum_{i<j}\omega_{ij}\langle
p_{i}-p_{j},u_{i}-u_{j}\rangle=\sum_{i<j}\omega_{ij}\langle
p_{i}-p_{j},u_{i}\rangle+\sum_{i<j}\omega_{ij}\langle
p_{j}-p_{i},u_{j}\rangle$ $=\sum_{i<j}\omega_{ij}\langle
p_{i}-p_{j},u_{i}\rangle+\sum_{i<j}\omega_{ij}\langle
p_{j}-p_{i},u_{j}\rangle=\sum_{i<j}\omega_{ij}\langle
p_{i}-p_{j},u_{i}\rangle+\sum_{j<i}\omega_{ji}\langle
p_{i}-p_{j},u_{i}\rangle,$

which is zero when $\omega$ is an equilibrium stress.

######  Theorem 3.3.

Prestress stability implies continuous rigidity for finite bar-joint
frameworks.

###### Proof.

Let $(G,p)$ be prestress stable with equilibrium stress $\omega$ as in
Definition 3.1. Assume that $(G,p)$ is not continuously rigid. Then there is a
continuously differentiable motion $p(t)$ of $(G,p)$ with $p^{\prime}(0)$ a
nontrivial infinitesimal flex $u$. The real-valued function $t\to
E_{\omega}(p+tu)$ has vanishing derivative at $t=0$, since $\omega$ is an
equilibrium stress, and so by Lemma 3.2 it has the form $A(u)t^{2}+C$ where
$C$ is constant and
$A(u)=2E_{\omega}(u)=2\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}>0$. In particular
$E_{\omega}(p+tu)$ is strictly increasing on ${\mathbb{R}}_{+}$.

Consider the finite cone $C(u,\delta)$ of velocity fields in
${\mathbb{R}}^{nd}$ given by

$C(u,\delta)=\\{su^{\prime}:s\in[0,1],|u^{\prime}-u|\leq\delta\\}$

where $\delta>0$ is such that $A(u^{\prime})>0$ when
$|u^{\prime}-u|\leq\delta$. As in the previous paragraph
$E_{\omega}(p+tu^{\prime})$ is strictly increasing on ${\mathbb{R}}_{+}$ from
which it follows that $E_{\omega}(q)>E_{\omega}(p)$ for all $q\neq p$ in
$p+C(u,\delta)$.

For any continuously differentiable path $s\to q(s)$ in ${\mathbb{R}}^{nd}$
with $q(0)=p$ and $q^{\prime}(0)=u$ the point $q(s)$ is equal to to $p+su$ to
first order, and so $q(s)$ lies in $p+C(u,\delta)$ for some sufficiently small
$s>0$. Considering this for the motion $p(s)$ gives the desired contradiction
since in this case the bar lengths are preserved and the function $s\to
E_{\omega}(p(s))$ is constant. ∎

###  3.2. Second order rigidity

In view of the entries of the rigidity matrix $R(G,u)$ we see that

$\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}=\sum_{i<j}\omega_{ij}\langle
u_{i}-u_{j},u_{i}-u_{j}\rangle=\omega R(G,u)u.$

Thus, prestress stability may be reworded as the following property.

_There exists $\omega$ with $\omega R(G,p)=0$ and $\omega R(G,u)u>0$ for all
nontrivial flexes $u$._

This compact reformulation is helpful when making comparisons with another
form of rigidity, namely _second order rigidity_ (2OR). The idea of second
order rigidity is that a nontrivial infinitesimal flex $u$ of $(G,p)$ may
exist but for each such $u$ there is no differentiable continuous motion
$p(t)$ starting in the direction $u$, that is, with $p^{\prime}(0)=u$. One way
to effect this obstruction, in view of the proof of Theorem 3.3, is the
presence an equilibrium stress $\omega$ specifically for $u$ in the sense that
$\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}>0.$ This suggests the following
additional definition which is evidently a weakening of prestress stability.

######  Definition 3.4.

A finite bar-joint framework $(G,p)$ is _weakly prestress stable_ (WPS) if for
each nontrivial infinitesimal flex $u$ there is an equilibrium stress $\omega$
with $\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}>0.$

In fact Connelly and Whiteley [7] have shown that this property is equivalent
to second order rigidity as defined in terms of (the nonexistence of)
nontrivial second order flexes. Such a flex for $(G,p)$ is a pair $(u,a)$
where $u,a$ are vectors in ${\mathbb{R}}^{nd}$, representing a nontrivial
velocity flex $u$ and an acceleration vector $a$. This pair should satisfy the
equations arising from the second derivative of the constraint equations for a
smooth motion $p(t)$, and they have the form

$R(G,p)u=0,\quad R(G,p)a+R(G,u)u=0.$

Since the image space $R(G,p){\mathbb{R}}^{nd}$ is orthogonal to $\ker
R(G,p)^{T}$, the nullspace of the transpose, it follows, from the Fredholm
alternative, that either there is a solution $a$ to the equation
$R(G,p)a=-R(G,u)u$ or there exists $\omega$ in $\ker R(G,p)^{T}$ with
$\omega\cdot(R(G,u)u)\neq 0$. This shows that second order rigidity implies
weakly prestress stable. In view of our comments preceding Definition 3.4 the
reverse implication holds and the properties are equivalent. In particular
this gives an alternative to Connelly’s original proof [3] (see also [6]) that
second order rigidity implies continuous rigidity.

To summarise, for finite bar-joint frameworks we have

$IR\implies PS\implies 2OR(=WPS)\implies CR$

######  Example 3.5.

Let us revisit the square within square framework of Figure 4. The outer
joints are $p_{1}=(-1,1),p_{2}=(1,1),p_{3}=(1,-1),p_{4}=(-1,-1)$, the inner
joints are
$p_{5}=(-1/2,1/2),p_{6}=(1/2,1/2),p_{7}=(1/2,-1/2),p_{8}=(-1/2,-1/2)$, and an
equilibrium stress is indicated in Figure 3.

Figure 3. An equilibrium stress.

The nontrivial “parallel motion” infinitesimal flex $z$ is defined by
$z_{1}=z_{2}=z_{5}=z_{6}=(1,0)$, with $z_{k}=(0,0)$ for other values of $k$,
and the infinitesimal flex $w$ is infinitesimal rotation of the inner square,
with $w_{5}=(1,1),w_{6}=(1,-1),w_{7}=(-1,-1),w_{8}=(-1,1)$ and $w_{k}=(0,0)$
for other values of $k$. Up to a scalar factor, and a trivial infinitesimal
flex, a typical nontrivial infinitesimal flex has the form $\alpha z+\beta w$
with $0\leq\alpha,\beta\leq 1$ not both zero. We have

$E_{\omega}(\alpha z+\beta
w)=\sum_{ij}\omega_{ij}\left[\alpha^{2}|z_{i}-z_{j}|^{2}+\beta^{2}|w_{i}-w_{j}|^{2}-2\alpha\beta\langle(z_{i}-z_{j}),(w_{i}-w_{j})\rangle\right].$

We note that with the exception of $i,j=5,8$ or $6,7$ at least one of the
pairs of vectors $z_{i}-z_{j},w_{i}-w_{j}$ is zero. Moreover, for these
exceptional values the pair of vectors is an orthogonal pair, and so there is
no contribution to the sum from the inner product terms. It remains to note
that despite the presence of negative stresses $\omega_{ij}$ (of value -1) on
the outer edges we have $\sum_{ij}\omega_{ij}\alpha^{2}|z_{i}-z_{j}|^{2}>0$ if
$\alpha>0$. It follows that $E_{\omega}(\alpha z+\beta w)$ is positive and so
the framework is prestress stable and hence continuously rigid.

###  3.3. Prestress stability and stress matrices

Prestress stability can also be defined in terms the _stress matrix_ of an
equilibrium stress, as we now demonstrate.

For any velocity vector $u$ and stress field $\omega$ associated with the
edges $ij$ of a finite bar-joint framework $G$ (and with $\omega_{ij}=0$ for a
nonedge) we have

$\omega
R(G,u)u=\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}=\sum_{i<j}\omega_{ij}\langle
u_{i}-u_{j},u_{i}-u_{j}\rangle,$

which in turn is equal to

$\sum_{i<j}\omega_{ij}(\langle u_{i},u_{i}\rangle-2\langle
u_{i},u_{j}\rangle+\langle u_{j},u_{j}\rangle)=\sum_{j:j\neq
i}\omega_{ij}|u_{i}^{2}|-\sum_{i<j}2\omega_{ij}\langle u_{i},u_{j}\rangle.$

On the other hand a general square symmetric $n\times n$ matrix $\Omega_{ij}$,
with entries $a_{ij}$, has a quadratic form

$x^{T}\Omega x=\sum_{i}x_{i}(\Omega
x)_{i}=\sum_{i,j}a_{ij}x_{i}x_{j}=\sum_{i}a_{ii}x_{i}^{2}+2\sum_{i<j}a_{ij}x_{i}x_{j},$

and $\Omega\otimes I_{d}$ similarly has the quadratic form

$u^{T}(\Omega\otimes I_{d})u=\sum_{i,j}a_{ij}\langle
u_{i},u_{j}\rangle=\sum_{i}a_{ii}|u_{i}|^{2}+2\sum_{i<j}a_{ij}\langle
u_{i},u_{j}\rangle.$

So we see that

$\omega R(G,u)u=u^{T}(\Omega\otimes
I_{d})u,\quad\mbox{where}\quad\Omega_{ij}=-\omega_{ij},\quad\mbox{for}\quad
i\neq j,\quad\mbox{and}\quad\Omega_{ii}=\sum_{j:j\neq i}\omega_{ij}.$

In particular the row sums and the column sums of $\Omega$ are zero. The
matrix $\Omega$ is defined to be the _stress matrix_ for $\omega$ and it
follows that we have the following alternative definition of prestress
stability.

######  Proposition 3.6.

A finite bar-joint framework $(G,p)$ in ${\mathbb{R}}^{d}$ is prestress stable
if and only if there exists an equilibrium stress matrix $\Omega$ for $p$ with
$u^{T}(\Omega\otimes I_{d})u>0$ for each nontrivial infinitesimal flex $u$.

Prestress stability can also be defined for finite tensegrities in the same
way to that of Definition 3.1, but in terms of the stress energy function
associated with a proper equilibrium flex. See Connelly [3], Connelly and
Whiteley [7] and Connelly and Guest [6], for example. Here one can also find
examples of tensegrities, such as various Cauchy polygons, which are
continuously rigid by virtue of having stress matrices which are positive
semidefinite on a complementary subspace to the space of rigid motion
infinitesimal flexes.

###  3.4. Prestress stability for infinite frameworks

We first note an elementary extension of Theorem 3.3 to countable bar-joint
frameworks.

######  Definition 3.7.

A countable framework $(G,p)$ is _sequentially prestress stable_ if there is
an increasing sequence of prestress stable subframeworks $(G_{n},p)$,
determined by a chain $G_{1}\subset G_{2}\subset...$ of finite subgraphs whose
union is $G$.

######  Definition 3.8.

A countable framework $(G,p)$ is _sequentially continuously rigid_ if there is
an increasing sequence of continuously rigid subframeworks $(G_{n},p)$,
determined by a chain $G_{1}\subset G_{2}\subset...$ of finite subgraphs whose
union is $G$.

######  Theorem 3.9.

If $(G,p)$ is sequentially prestress stable then it is sequentially
continuously rigid and hence continuously rigid.

This theorem follows readily from Theorem 3.3.

######  Example 3.10.

Figure 4 indicates an infinite bar-joint framework $(G,p)$. The joint
positions are located dyadically; the outer joints are
$p_{1}=(-1,-1),p_{2}=(-1,1),p_{3}=(1,1),p_{4}=(1,-1)$ and the other joints
have the form $\frac{1}{2^{k}}v_{j},j=1,2,3,4,k=1,2,\dots$. The framework
$(G,p)$ has an infinite dimensional flex space, since the quadruple of joints
of the square subframeworks with bar lengths $(1/2)^{k}$, for $k=1,2,\dots$,
is the support of an infinitesimal rotation flex, say $w^{(k)}$, for
$k=1,2,\dots$. We assume, as before, that $|w^{(k)}|=\sqrt{2}$ for each $k$.
It follows from the continuous rigidity of the double square framework of
Example 2 and a simple induction argument that the framework is sequentially
continuously rigid. Also it can be shown that it is sequentially prestress
stable.

Figure 4. A partially defined equilibrium stress.

We also observe that it is possible to define an equilibrium flex $\omega$ for
$(G,p)$ which is summable in the sense that the sequence of weights
$\omega_{e}$ gives an absolutely convergent series. The equilibrium condition
is satisfied for the vertex $v_{1}$ at $A$. Also the equilibrium condition is
satisfied at $B$ if $a\frac{1}{8}+a\frac{1}{2}=1/4$, that is if
$a=\frac{4}{5}$. It follows that there is a rotationally symmetric summable
equilibrium stress $\omega$ with stress values $(\frac{4}{5})^{k}$ for
$k=2,3,\dots$.

A continuous flex $p(t)$ of the single joint framework $(K_{1},p_{1})$ can be
space filling in view of well-known facts about continuous functions. More
strangely, there exists a countable bar-joint framework in ${\mathbb{R}}^{2}$
which has an essentially unique nontrivial continuous flex which fixes a given
base edge $p_{1}p_{2}$ and for which a distinguished joint $p_{3}$ has a space
filling motion $p_{3}(t),t\in[0,1]$. The underlying reason for this, roughly
speaking, is that it is possible to simulate convergent series of traced
algebraic curves within a single infinite framework. (See Owen and Power [13]
and Power [15].) We also note that there are infinitely differentiable
continuous functions on $[-1,1]$ which are nonzero and have all derivatives
vanishing at $0$.

These cautionary observations motivate the specification of stronger forms of
continuous flexibility for countably infinite frameworks the exploration of
their connection with other forms of prestress stability.

Define the space of bounded displacement vectors (or bounded velocity vectors)
$u=(u_{k})_{k\in{\mathbb{N}}}$ as the normed space
$\ell^{\infty}\otimes{\mathbb{R}}^{d}$ with norm given by the supremum, so
that $\|u\|=\sup_{k}|u_{k}|$, with $|\cdot|$ the usual Euclidean norm on
${\mathbb{R}}^{d}$. In the next definition we consider continuous motions
$p(t)$ which are _bounded motions_ , in the sense that $p(t)-p(0)$ belongs to
$\ell^{\infty}\otimes{\mathbb{R}}^{d}$ for all $t$, which are differentiable
coordinatewise, and which satisfies a strong differentiability condition for
$t=0$. For convenience we refer to these continuous flexes as boundedly smooth
motions.

######  Definition 3.11.

A _boundedly smooth motion_ of the countably infinite bar-joint framework
$(G,p)$ is a continuous motion $p(t),t\in[0,1],$ such that $p_{k}(t)$ is
differentiable for each $k$ and the limit

$\lim_{t\to 0}(p(t)-p(0))/t=(p_{k}^{\prime}(0))_{k\in{\mathbb{N}}}$

exists with respect to convergence in the displacement space
$\ell^{\infty}\otimes{\mathbb{R}}^{d}$, with
$p^{\prime}(0)=(p_{k}^{\prime}(0))_{k\in{\mathbb{N}}}$ a bounded infinitesimal
flex. Moreover, a boundedly smooth motion $p(t)$ is _proper_ if
$p^{\prime}(0)$ is nonzero, and is _nontrivial_ if $p^{\prime}(0)$ is
nontrivial.

######  Definition 3.12.

A countable bar-joint framework in ${\mathbb{R}}^{d}$ is _boundedly smoothly
rigid_ (BSR) if it has no nontrivial boundedly smooth motions.

Let us also say that a countable framework $(G,p)$ is _boundedly
infinitesimally rigid_ (BIR) if there exists no infinitesimal flexes which are
nontrivial and bounded. Plainly BIR implies BSR.

######  Definition 3.13.

A bar-joint framework $(G,p)$ with bounded bar lengths is _boundedly prestress
stable_ (BPS) if there is a _summable_ equilibium stress $\omega$, in the
sense that $\sum_{i<j}|\omega_{ij}|$ is finite, and for all bounded nontrivial
infinitesimal flexes $u$ of $(G,p)$,

$\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}>0.$

We note that when the bar lengths are bounded $(G,p)$ has _finite stress
energy for $\omega$_ in the sense that

$E_{\omega}(p)=\sum_{i<j}\omega_{ij}|p_{i}-p_{j}|^{2}<\infty.$

Once again we refer to the function $q\to E_{\omega}(q)$ as the _stress energy
function_ for $\omega$ and $G$.

######  Example 3.14.

The framework $(G,p)$ of Example 4 is boundedly prestress stable for the
summable equilibrium stress $\omega$ given there. To see this let $u$ be a
nontrivial, bounded infinitesimal flex. We may assume that $u$ has zero
velocities at $p_{3},p_{4}$. In this case $u$ admits an infinite sum
representation

$u=\alpha z+\sum_{k}\beta_{k}w^{(k)},$

where $(\beta_{k})$ is a bounded sequence and where $z$ is the “parallel
motion” infinitesimal flex whose velocities are zero on the lower joints of
the square subframeworks and equal to $(1,0)$ on the upper joints. The
positivity and finiteness of $E_{\omega}(u)$ follow as in Example 3.5 since
the resulting series are absolutely convergent.

######  Example 3.15.

Let $(G,p)$ be the lacunary linear framework in ${\mathbb{R}}^{2}$ with bars
corresponding to the intervals
$\dots,[-7,-3],[-3,-1],[-1,0],[0,1],[1,3],[3,7],\dots.$

Figure 5. A lacunary linear framework.

Evidently $(G,p)$ is not BSR. Let $\omega$ be the summable equilibrium stress
with $\omega_{e}=\frac{1}{2^{k}}$ if the bar for edge $e$ has length $2^{k}$.
Every equilibrium stress is in fact a scalar multiple of $\omega$. Note that
for any uniformly bounded infinitesimal flex $u$ (or indeed any bounded
velocity field) the quantity $E_{\omega}(u)$ is finite. On the other hand
$E_{\omega}(p)$ is not finite and so $(G,p)$ is not BPS.

We now show that BPS implies BSR.

######  Lemma 3.16.

Let $\omega$ be a summable stress for a bar-joint framework $(G,q)$ with
bounded bar lengths, and let $u$ be a bounded velocity vector. Then

$\frac{d^{2}}{dt^{2}}E_{\omega}(q+tu)|_{t=0}=2\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}.$

###### Proof.

We have

$\omega_{ij}|(q_{i}+tu_{i}-(q_{j}+tu_{j})|^{2}=\omega_{ij}\langle
q_{i}+tu_{i}-(q_{j}+tu_{j}),q_{i}+tu_{i}-(q_{j}+tu_{j})\rangle,$

and so

$E_{\omega}(q+tu)=\sum_{i<j}\omega_{ij}|q_{i}-q_{j}|^{2}+\sum_{i<j}\omega_{ij}2t\langle
q_{i}-q_{j},u_{i}-u_{j}\rangle+\sum_{i<j}\omega_{ij}t^{2}|u_{i}-u_{j}|^{2}.$

Each of these series is absolutely convergent and so the desired identity
follows. ∎

From the expansion of $E_{\omega}(q+tu)$ above it follows that when $\omega$
is summable and $(G,p)$ has bounded bar lengths then for a bounded
infinitesimal flex $u$ we have

$\frac{d}{dt}E_{\omega}(p+tu)|_{t=0}=2\sum_{i<j}\omega_{ij}\langle
p_{i}-p_{j},u_{i}-u_{j}\rangle,$

where the series is absolutely convergent. In particular, when $\omega$ is a
summable equilibrium stress for $(G,p)$ then, as with finite frameworks, this
derivative is zero.

######  Theorem 3.17.

Let $G$ be a countably infinite bar-joint framework with bounded bar lengths.
If $(G,p)$ is boundedly prestress stable then it is boundedly smoothly rigid.

###### Proof.

Assume that $(G,p)$ is BPS, with associated summable equilibrium stress
$\omega$, and that $(G,p)$ is not BSR. Then there is a boundedly smooth motion
$p(t)$ of $(G,p)$ with $p^{\prime}(0)$ equal to a nontrivial bounded
infinitesimal flex $u=(u_{k})$. As in the proof of Theorem 3.3 we write $A(u)$
for $2\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}$. Note that the function
$u^{\prime}\to A(u^{\prime})$ is continuous in the sense that for all
$\epsilon>0$ there exists $\delta>0$ such that $|A(u^{\prime})-A(u)|<\delta$
for $\|u^{\prime}-u\|<\delta$. To see this note that

$\begin{split}A(u)-A(u^{\prime})&=\sum_{i<j}\omega_{ij}\big{(}|u_{i}-u_{j}|^{2}-|u_{i}^{\prime}-u_{j}^{\prime}|^{2}\big{)}\\\
&\leq\sum_{i<j}\omega_{ij}\big{(}(|u^{\prime}_{i}-u^{\prime}_{j}|+2\delta)^{2}-|u_{i}^{\prime}-u_{j}^{\prime}|^{2}\big{)}\\\
&=\sum_{i<j}\omega_{ij}\big{(}4\delta|u^{\prime}_{i}-u^{\prime}_{j}|+4\delta^{2}\big{)}\end{split}$

Since $u^{\prime}$ is bounded and $\omega$ is summable this is less than
$\epsilon$ for suitably small $\delta$. The same can be said for
$A(u^{\prime})-A(u)$.

As in the proof of Theorem 3.3 it now follows from this continuity that for
some $\delta>0$ the function $q\to E_{\omega}(q)$ is increasing on each ray of
the cone $p+C(u,\delta)$. However, for the motion $p(t)$ the strong
differentiability condition for $t=0$ shows that $\|p(s)-(p+su)\|<\delta s$,
for some small $s>0$. Thus $p(s)$ lies in the cone $p+C(u,\delta)$ and so
$E_{\omega}(p(0))<E_{\omega}(p(s))$, which is the desired contradiction. ∎

We remark that the condition in Definition 3.13 that $p_{k}(t)$ be
differentiable for $t>0$ for each $k$ is not needed in the previous proof.

###  3.5. Weakly boundedly prestress stable

######  Definition 3.18.

A countable bar-joint framework is _weakly boundedly prestress stable_ (WBPS)
if for each nontrivial bounded infinitesimal flex $u$ there is a summable
equilibrium stress $\omega$ with $\sum_{i<j}\omega_{ij}|u_{i}-u_{j}|^{2}>0.$

In parallel with finite frameworks the proof of Theorem 3.17 shows that WBPS
implies BSR and so we have the following hierarchy.

######  Theorem 3.19.

The following implications hold for countable bar-joint frameworks with
bounded bar lengths.

$IR\implies BIR\implies BPS\implies WBPS\implies BSR$

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