Interactive Freeform Design of Tensegrity
Tomohiro Tachi
The University of Tokyo

Abstract. In this paper, we propose a novel interactive method for flexibly designingtensegrity structures under valid force equilibriums. Unlike previous form-finding
techniques that aim to obtain a unique solution by fixing some parameters such as
the force densities and lengths of elements, our method provides a design systemthat allows a user to continuously interact with the form within a multidimensional
solution space. First, a valid initial form is generated by converting a given polygon mesh surface into a strut-and-cable network that follows the original surface,
and the form is then perturbed to attain an equilibrium state through a two-step
optimization of both node coordinates and force densities. Then, the form can be
freely transformed by using a standard 2D input device while the system updates theform on the fly based on the orthogonal projection of a deformation mode into thesolution space. The system provides a flexible platform for designing a tensegrity
form for use as a static architectural structure or a kinetic deployable system.

1Introduction

1.1

BackgroundA tensegrity structure is a tension-stabilized, self-standing spatial structure composed of struts (compression members) and cables (tension members), where every
strut is isolated or connected to the other struts only via cables. Different formsof tensegrity structures have been explored through experiments by sculptors andarchitects such as Kenneth Snelson and Buckminster Fuller. In addition to being aform of expression, the tensegrity has been applied to a lightweight structural system or a deployable system in the design of architecture. Since a tensegrity structure
is in a self-equilibrium state because of stressed cables, its geometry is constrained
intrinsically. Therefore, when tensegrity structures are to be used in architecturaland spatial designs, a computational form-finding approach is useful.
One of the bases for solving pin-jointed wire-frame structures in equilibriumis the force density method introduced by [Linkwitz and Schek 1971], which successfully solves the form of a prestressed cable network by solving only one linearsystem. However, typically, the direct application of the force density method totensegrity structures yields poor results; the configuration degenerates to a point,line, or plane. This is because the free-standing self-equilibrium state is produced
only by a nontrivial singular set of force densities.

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Figure 1: Tensegritized version of a simplified Stanford Bunny derived by using our algorithm (cross-eyed stereo).
(a)

(b)

(c)

(d)

Figure 2: The process. (a) Given polygonal mesh. (b) Generated topology. (c) Valid initialconfiguration in equilibrium. (d) A freeform variation of (c).

Different numerical form-finding methods for tensegrity structures have been
investigated by many researchers; for a more systematic review in this field, see[Tibert and Pellegrino 2003]. One of the conventional approaches for solving thesingularity of tensegrity is to find a set of force density values that ensure a valid spatial configuration, for example, the approach in the adaptive force density method
proposed by [Zhang and Ohsaki 2006]. Another approach is that used in the dynamic relaxation method for solving a nonlinear problem to converge to one of thesolutions through an iterative physical or pseudo-physical simulation [Motro 1984;Miki and Kawaguchi 2010]. However, existing approaches use some predefined parameters or an objective function, e.g., constant lengths, force densities, or a pseudoenergy definition, in order to converge to some “unique” solution. However, fromthe design point of view, such predefined parameters are not directly relevant to thedesign, but more restrict the design freedom.

Freeform Tensegrity

1.2

Overview

In this paper, we propose a novel computational method that first facilitates freeform
designing of tensegrity structures through the construction of a single-layered
tensegrity structure that follows the given surface and freeform manipulation basedon user interaction (an example is shown in figure 1). Instead of obtaining a“unique” solution, we construct a responsive environment that allows users to interactively explore the continuous solution space of valid tensegrity structures. By
separating the mandatory equilibrium constraints from the optional geometric constraints, the designers have more freedom in selecting different geometric constraints, such as equal-tension conditions. The procedure for the construction ofa tensegrity structure in our method is as follows (Figure 2).
1. Topology Generation Step: The topology and the initial configuration are constructed from a given polygonal mesh. See Section 2 for details.
2. Form-finding Step: The initial configuration is modified to have a valid equilibrium state by performing two-step iterative optimization. See Section 3 fordetails.
3. Interactive Design Step: A user can interactively modify the form by directly
dragging the structure displayed on the screen. The system ensures that thestructure satisfies the equilibrium conditions by orthogonally projecting thedeformation at each frame. See Section 4 for details.

2Topology Generation

A topology of a strut-and-cable network can be constructed as the initial configuration for form-finding in multiple ways. Here, we describe a method that convertsa general given polygonal mesh into a strut-and-cable network that forms a single
layer of a surface. The process is as follows (Figure 3 top).
1. Isolate each edge and independently rotate it by θ (−90◦ < θ < 90◦ ) aboutan axis passing through its midpoint parallel to the surface normal. θ is aparameter selected by the user. These edges are defined as struts.
2. Connect the endpoints with cables to form each m-gonal closed loop corresponding to an m-valency vertex.
3. Connect the endpoints with cables to form each n-gonal closed loop corresponding to an n-gonal facet.
The above construction process ensures that each node is shared by one strut andfour cables. This converts a regular tetrahedron into a well-known six-strut icosahedral tensegrity structure (Figure 3 bottom).

3Form-Finding

The initial configuration of the tensegrity structure is then modified to satisfy thegeometry and force constraints of a tensegrity structure. The configuration of theT. Tachi

θfacevertex

Figure 3: Top: Process of topology generation from a polygon mesh. Bottom: Conversionof regular tetrahedron into a regular icosahedral tensegrity structure in the case of θ = 45◦ .

fe = fi,jxi

xje = i,j = ||xj−xi||Figure 4: Notation of the parameters.
structure is represented by the coordinate xi = {xi , yi , zi }T of each node i and theforce density we = wi, j = fi, j /i, j = fi, j / xi − x j  of each segment e incident tovertices i, j, where fe = fi, j is the axial force applied to the segment (Figure 4).

T T
Overall, the configuration is represented by a 3V vector X = xTand E
1 , · · · , xVvector W = (w1 , · · · , wE )T , where V is the number of vertices (nodes) and E is thenumber of edges (segments). In our constructed configuration, these numbers arerelated as follows: V = 2S and E = 5S, where S is the number of struts, i.e., thenumber of edges on the original polygon mesh. The equilibrium at each node isthen represented asfi =(x j − xi )fi, j x j − xi  = ∑ wi, j (x j − xi ) = 0.
jincident to i
jincident to i∑(1)

In the force density method, W is a fixed constant, and the system is linear. In
our method, W becomes a variable, and the system is nonlinear. Since the sign ofwi, j describes whether the element is a tension or compression member, we assert

Freeform Tensegrity

inequalities

wi, j< −α>αfor a strut
for a wire

(2)Here, the positive value α can be set arbitrarily in the sense that an arbitrary scalingof wi, j is consistent with equation (1). We set wi, j proportional to a scale factor; thedetailed reason is for this setting is discussed in Section 4.2. Our problem definitionis as follows: solve fi = 0 for every node i with respect to xi and wi, j while satisfying
inequality (2).
The solution is not unique, and we find a solution close to theoriginal configuration by the following two steps to make U (X, W) =22∑i fi  + ∑e max (0, α − |we |) = 0 :1. minimize U with respect to W for a fixed geometry X.
2. minimize U with respect to W and X using an iterative method wherein theinitial configuration is given by step 1.
In the first step, the number of variables is E = 5S and 3V = 6S target equilibrium
equations exist. Therefore, the minimization problem converges to the least-squares
solution of an over-constrained system. In the second step, the number of variables
is E + 3V > 3V , and U minimizes to 0; thus, equation (1) is satisfied at every node.
The conjugate gradient method is used for solving these optimization problems. We
have observed that this two-step approach avoids the degenerate trivial configurations, e.g., the one where all points are located on a point.

4

Interactive Freeform Editing

4.1

Projection Method

Our method enables a free exploration of the continuous solution space, instead ofjust obtaining a single solution. In order to find a valid deformation within the solution space, we use a projection-based method using the Moore-Penrose generalized
inverse matrix [Penrose 1955] of a Jacobian matrix, which has been applied forshape analysis of transformable structures [Hangai and Kawaguchi 1991].
 X
The configuration of the structure is represented by a 3V + E vector. SinceWeach node satisfies equation (1), the variables are constrained by a 3V -vector equa
T Ttion F (X, W) = fT
= 0. This forms an underdetermined system where a0 , · · · , fV
multidimensional continuous solution space exists.
A first-order deformation in the solution space can be represented by using theJacobian matrix of the constraints:
ΔXΔX∂F∂FC= ∂X=0(3)∂WΔWΔW

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The elements of the Jacobian matrix are calculated as follows:⎧⎪⎨− ∑kincident to i wi,k if i = j∂fi= wi, j
if j is incident to i
∂x j ⎪⎩
0else∂fi=∂w j,kxk − xi
0if i = j (or x j − xi if i = k)else(4)(5)

One of the solutions of Equation (3) is calculated using the Moore-Penrose generalized inverse C+ of the constraint matrix C as follows: ΔX0ΔX+= I−C C(6)
ΔW
ΔW0

ΔX0
is an arbitrary vector. Since the multiplication of (I − C+ C) in equaHere,ΔW0tion (6) represents
the orthogonal projection to the constrained subspace, the soluΔX
ΔX0tionis the nearest valid defromation to a given.
ΔW
ΔW0
We develop an interactive system based on this projection procedure. In thesystem, the user can freely select and drag the points on the rendered tensegritystructures through a 2D input device to apply a deformation, e.g., a point translation
or a weighed point-set translation or rotation. Before
 the deformation is applied
ΔX0ΔX
to the current state, the deformation vector
is substituted with
to0ΔW
get the constrained valid next state. Since the abovementioned Euler integration
accumulates error, the residual is eliminated using Newton-Raphson’s method foreach step. The generalized inverse solution for each step is calculated using theconjugate gradient method.
4.2

Weight of Parameters

Since the orthogonal projection modifies
the deformation to the least norm solution,
ΔX
it is essential that the configuration
have a consistent dimension. Therefore,ΔW
we represent the parameters W in the length unit (i.e., we use [m] instead of [N/m]).
In order to attain scale invariance, we set the initial values of the parameters proportional to some scale factor such as the average length of the initial form. The
self-equilibrium force can be uniformly scaled independent of the form; however,this results in a change in the response to a given input deformation. When thevalues of force density are scaled down, the proportion of force densities becomeseasier to be changed, so the equilibrium state can adapt to a given deformation; as arelative result, the form follows the original user’s input deformation more strongly.
This scaling factor can also be controlled during freeform editing.

Freeform Tensegrity

original model

3 b
3-cubesbl b
blob

snub-dodecahedronb
bunny

number of nodes

56

204

300

600

number of struts

28

102

150

300

number of cables

112

408

600

1200

initial form-ﬁnding

0.5 [s]2.6 [s]3.3 [s]490 [s]interaction frame rate

24 [fps]

7.5 [fps]

4.8 [fps]

0.85 [fps]Figure 5: Amount of time required for the calculation.

5Design System

5.1

Interface

A software system is implemented using C++ with OpenGL for graphics and BLAS
(with Intel MKL) for linear algebra. The user can first import a mesh model in theOBJ format, let the system convert it to a valid tensegrity form, and then edit theform interactively by directly selecting and dragging the form. Figure 8 shows ascreenshot of the system. The system runs in either the edit mode or the simulationmode, and the user toggles to switch between the modes. In the edit mode, interactive freeform editing described in Section 4 is performed, and the dragging of theform causes changes in the shape and force of the structure. In the simulation mode,the system calculates the deformation under the length constraints that fix the struts’lengths and prevent the cables from extending. We can check whether the tensegrity
structure is stable by examining it in the simulation mode.
5.2

Computation Time

Figure 5 shows the amount of time required for performing calculations for fourmodels with different complexities on a laptop PC with an Intel Core i5-2502M
CPU (2.50 GHz dual core). For models with up to 150 struts, form-finding is sufficiently fast and the form modification is performed at an interactive frame rate.
The slower convergence of the Stanford Bunny model with 300 struts in the initialform-finding step is due to an oscillation that occurred when solving the inequalityconditions. However, the interactivity in the form modification process had an acceptable value of 0.85 fps, in which we can use the system repeatedly to obtain thedesired configuration before actually constructing a model.

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Figure 6: Image of a tensegrity structure that fits the ground.

6

Optional Geometric Constraints for Designs

In our method, we can add optional geometric constraints or penalty forces arbitrarily, provided that adequate degrees of freedom remain in the solution space. This isan advantage of our method over methods based on the convergence to a unique solution or a pseudo-physical simulation, in which cases adding constraints or penalty
forces would destroy the equilibrium. Now, we consider several important geometric constraints and demonstrate their use for design and fabrication of a tensegrity
structure.
Fixing to Reference: The system enables the fixing of selected points to reference
points or to the ground plane. Figure 6 shows an example of a design in which
several points are fixed on the ground so that the whole structure is placed on theground, as intended. Here, the whole structure is merely placed on the ground;
therefore, the joints at the ground bear the weight and the external load only (theprestresses are balanced within the structure).
Fixed Length: The lengths of elements can be fixed by adding a constraint i, j = const for each rigidized edge i, j, whose derivative is given by∂i, j /∂xi = (xi − x j ) /i, j . An interesting usage of the fixed length constraint is thedesign of a transformable tensegrity structure. After designing one configuration,
we select some cables, define them as the actuators, and keep the lengths of theother struts and cables fixed. A transformation that can be obtained in the designsystem can be realized in the constructed structure by controlling the lengths of theactuators (Figure 7).
Equal Tension: By introducing the equal-tension constraint represented by
i, j wi, j −  j,k w j,k = 0, we can simplify the fabrication process. In the construction
process of a tensegrity structure, we install prestresses by tensioning each cable.

Freeform Tensegrity

Figure 7: Transformable tensegrity. Two configurations can be transformed among themselves by controlling the lengths of actuator cables shown in green.

Figure 8: Screenshot of the system. A form with equal tension constraint applied to eachcable loop (left) and its force diagram (reciprocal figure) (right).

However, adjustment of the tension becomes difficult for a complex model because
the change in the tension of one of the cables affects the tensions of the others.
Here, if we know in advance that adjacent cables are to have the same tension,then we can replace them with a continuous cable passing through a hole at theend of the strut; therefore, the joint structure is also simplified. This is because anequal-tension cable does not apply a sliding force to the joint.
An effective way of setting equal tension constraints is to group each closed loopthat is originally derived from the same vertex or facet in the topology generation
step (Section 2) and assign the same tension to each group. Although this constraint
is a trade-off between the freedom of design and the simplicity of construction,
we could still obtain an interesting asymmetric form by using the equal-tension
constraint (Figure 8).
Intersection Avoidance of Struts: In the actual construction, the intersection ofstruts must be avoided. In our method, we can simply use a penalty force calculated from the distance between intersecting struts and use them as the deformation
mode. This is in contrast to a form-finding based on pseudo-physical simulation, in
which introducing a penalty force would affect the equilibrium.

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7Conclusion

We proposed a novel interactive method for freely designing tensegrity structures.
First, we discussed the process of constructing a valid initial tensegrity structurefrom an arbitrary polygon mesh based on topology generation and form-finding
through a two-step optimization. Then, we presented a method for interactive
freeform editing based on solving the equilibrium using projection to the solution
space. Our method succeeds in manipulating a tensegrity structure at an interactive
frame rate for a model with 750 segments by using a standard laptop PC. We further
realized optional geometric constraints, including the fixed length constraint, whichenables the design of transformable tensegrity structures, and the equal-tension constraint, which simplifies the actual fabrication process. The proposed method ensured only the equilibrium and left the evaluation of stability to the user by providing the simulation mode in the system. In future, we intend to investigate thetheoretical approach for ensuring the stability and stiffness of the structure.

Acknowledgments
This research was supported by JST, PESTO.

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and Spatial Structures 51, 4, 291–303.
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P ENROSE , R. 1955. A generalized inverse for matrices. Mathematical Proceedingsof the Cambridge Philosophical Society 51, 406–413.
T IBERT, A. G., AND P ELLEGRINO , S. 2003. Review of form-finding methods fortensegrity structures. International Journal of Space Structures 18, 4, 209–223.
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Author’s address:
Tomohiro Tachi (tachi@idea.c.u-tokyo.ac.jp): The University of Tokyo, 3-8-1
Komaba, Meguro-Ku, Tokyo 153-8902, Japan.