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155Journal of Asian Architecture and Building Engineering/January 2017/162
A Genetic Algorithm Based Form-nding of Tensegrity Structures
with Multiple Self-stress States
Seunghye Lee1, Jaehong Lee2 and Joowon Kang*3
1 Assistant Professor, Department of Architectural Engineering, Sejong University, Korea
2 Professor, Department of Architectural Engineering, Sejong University, Korea
3 Professor, School of Architecture, Yeungnam University, Korea
Abstract
A form-nding method of tensegrity systems is a process of nding an equilibrium conguration and a key
step in the design of tensegrity. Over the past few years, several studies have been made on the form-nding
methods of tensegrity systems, however, these methods are limited in the tensegrity systems with multiple
self-stress states. In this study, a numerical method is presented for form-finding of tensegrity structures
with multiple states of self-stress by using a force density method combined with a genetic algorithm. The
proposed method can design the desired tensegrity shape through a genetic algorithm with appropriate
constraints. The design variable can be uniquely dened in the case of multiple states of self-stress using only
the constraint of the member types. An eigenvalue decomposition of the force density matrix and a singular
value decomposition of the equilibrium matrix are performed repeatedly in order to determine a feasible
solution for nodal coordinates and force densities. A genetic algorithm is then adopted to uniquely dene a
single integral feasible set of force densities. Several numerical examples are presented to prove efciency in
searching for self-equilibrium congurations of tensegrity structures. In all cases, the single integral feasible
self-stress states can be obtained.
Keywords: tensegrity structure; force density method; form-nding; genetic algorithm; self-stress state
1. Introduction
Tensegrities are spatial, reticulated and lightweight
structures that consist of a set of discontinuous
compressive components inside a set of continuous
tensile components (Ali et al., 2010). The design of
tensegrities is divided into three distinct steps: form-
nding, structural stability and load analysis (Schenk,
2005). A key step in the design of tensegrity structures
is the determination of their equilibrium conguration,
known as form-nding. Recently Tibert and Pellegrino
(2011) announced a review of available form-nding
methods, offering a review and classication of seven
form-nding methods for tensegrity structures.
As pioneering work in form-finding, the force
density method was first proposed by Schek (1974)
for cable structures. The concept of the force density
method is based on the force-length ratio (or force
densities). Estrada et al. (2006) presented a multi-
parameter form-finding procedure for tensegrity
structures using the force-density method. Masic
et al. (2005) extended the force-density method by
explicitly incorporating shape constraints for general
and symmetric tensegrity structures. Zhang and Ohsaki
(2006) presented the adaptive force density method for
the form-nding problem of tensegrity structures. Most
recently, Tran and Lee (2010) proposed a numerical
method for tensegrity structures based only on the
given topology and member types. They also presented
an approach for a form-finding method of tensegrity
structures with multiple states of self-stress (Tran and
Lee, 2011).
Most form-nding methods assume a given topology
and try to nd equilibrium congurations using some
given constraints. A different approach is to find the
topology with a genetic algorithm. A genetic algorithm
provides up-to-the-minute search techniques by
adapting mechanisms found in genetics (Goldberg,
1989). Recently, several studies have researched the
form-finding methods of tensegrity structures using
genetic algorithms for searching self-equilibrium
topology. Paul et al. (2005) used genetic algorithms
to develop from an initial arbitrary topology into a
stable one. Xu and Luo (2010) presented a form-
finding method of irregular tensegrities based on the
genetic algorithm. Yamamoto et al. (2011) proposed a
genetic algorithm based form-nding method to obtain
*Contact Author: Joowon Kang, Professor,
Yeungnam University, 280 Daehak-Ro, Gyeongsan,
Gyeongbuk 38541, Korea
Tel: +82-53-810-2429 Fax: +82-53-810-2036
E-mail: kangj@ynu.ac.kr
( Received April 4, 2016 ; accepted November 18, 2016 )
DOI http://doi.org/10.3130/jaabe.16.155
156 JAABE vol.16 no.1 January 2017 Seunghye Lee
tensegrity structures with fewer design variables.
Koohestani (2012) provided an efcient form-nding
method using a genetic algorithm that is used as an
optimization technique. However, most of these studies
used symmetric constraints in searching suitable forms
of the tensegrities.
In this paper, the work by Tran and Lee (2011)
is extended to obtain unique feasible sets of force
densities by using the force density method combined
with a genetic algorithm. The proposed method can
determine multiple shapes of a tensegrity through a
genetic algorithm with appropriate constraints. That
is, the desired shape of the tensegrity can be obtained
by adjusting suitable constraints. The procedure only
requires the topology and the types of members (i.e.,
either compression or tension). First, the eigenvalue
decomposition (EVD) of the force density matrix is
implemented to determine the nodal coordinates that
compose the equilibrium matrix. Second, the multiple
states of force densities are obtained through the
singular value decomposition (SVD) of the equilibrium
matrix. A genetic algorithm is then performed to
uniquely dene a single integral feasible set of force
densities. Finally, this process is iteratively performed
to determine the range of feasible sets of the nodal
coordinates and the force densities until the required
rank deciencies of the force density and equilibrium
matrices for the case of multiple states of self-stress
are met. Several numerical examples of tensegrity
structures with multiple states of self-stress are
presented to demonstrate the efciency and accuracy
of the proposed method.
2. Formulation of Self-equilibrium Equations
2.1 Basic Assumptions
In this study, the basic assumptions regarding
tensegrity structures are stated as follows:
· The topology of the structure in terms of nodal
connectivity is known.
· Members are connected by pin-joints.
· No external load is applied and the self-weight
of the structures is neglected during the form-
nding procedure.
· There are no dissipative forces acting on the
systems.
· Neither local nor global buckling are considered.
2.2 Self-equilibrium Equations
A topology of d-dimensional (d = 2 or 3) tensegrity
structure with b members and n free nodes can be
expressed by a connectivity matrix C (∈Rb×n) as
discussed in (Tran and Lee, 2010). Since a tensegrity
system does not required any support (xed node), in
this study, only the free node is considered. According
to Fig.1., if member k connects nodes i and j (i < j), the
ith and jth elements of the kth row of the C matrix are
then set to 1 and -1, respectively, as follows:
When the external load and self-weight are ignored,
the equilibrium equations in each direction of a general
tensegrity structure given by (Schek, 1974) can be
stated as
Refer to the Cartesian coordinate system (O-xyz),
the nodal coordinate vector x, y and z are used.
The notation Q (∈Rb×b) is diagonal square matrix,
calculated by
where q (∈Rb×n) suggested by Schek (1974) is the force
density vector. Each component of this vector is the
member force fk to length of element lk (k = 1, 2, 3…,
b) ratio (qk = fk/lk). The equilibrium equations (Eq. (2))
can be rewritten as follows:
where D (∈Rn×n) is the force density matrix (Tiber and
Pellegrino, 2011, Estrada et al., 2006), or stress matrix
(Connelly, 1982).
Eq. (2) can be reorganized as
where A (∈Rdn×b) is known as the equilibrium matrix,
dened by
Fig.1. Illustration of the Schematic Diagram for Two-dimensional
Tensegrity Systems
k
x
y
O
i (xi, yi)
j (xj, yj)
lk
Ck,p
(1)
C
T
QCx
(2a)
CTQC
(2b)
CTQCz
(2c)
Qdiag (3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
C
k,p
(1)
CTQCx (2a)
CTQC (2b)
CTQCz (2c)
Qdiag (3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Ck,p
(1)
CTQCx (2a)
CTQC (2b)
CTQCz (2c)
Qdiag
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Ck,p
(1)
CTQCx (2a)
CTQC (2b)
CTQCz (2c)
Qdiag (3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Ck,p
(1)
CTQCx (2a)
CTQC (2b)
CTQCz (2c)
Qdiag (3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Ck,p
(1)
CTQCx (2a)
CTQC (2b)
CTQCz (2c)
Qdiag (3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
157JAABE vol.16 no.1 January 2017 Seunghye Lee
Eq. (4) shows the relationship between force density
matrix D and nodal coordinates, and Eq. (5) illustrates
the relationship between the equilibrium matrix A
and force densities. Generally, both mechanical and
geometrical parameters are obtained as a result of the
form-nding process.
2.3 Rank Deciency Conditions
In this form-finding procedure for tensegrity
structures, two rank deficiency conditions (rank
deciency conditions of force density and equilibrium
matrices) are required. For a d-dimensional tensegrity
structure with n free nodes, a dimension of rank
deciency of D (null space) can be expressed as
where rD = rank (D). The rank deficiency of D (nD)
has at least one state of self-stress, since the sum of
the elements of row or column of force density matrix
is always equal to zero, and nD has at least d useful
particular solutions (Tran and Lee, 2010). Therefore,
the rank deciency condition is dened as
The second rank deficiency condition is related to a
dimension of null space of the equilibrium matrix A as
follows:
where rA = rank (A). The dimension of null space of
the equilibrium matrix A is identical to "s" known
as the number of independent states of self-stress.
This study is limited to a consideration of tensegrity
structures with multiple states of self-stress, which
ensures the existence of at least two states of self-stress
and can be stated as
3. Algorithm
3.1 Force Density Method
In the proposed method, the dimension size, the
connectivity of nodes, and the type of each member
are only required for a form-nding procedure. Based
on the type of each member, the initial force density
coefficients of cables and struts are automatically
assigned as +1 and -1, respectively.
Firstly, the connectivity matrix C is composed by
Eq. (1) for the given of the connectivity of nodes.
The force density matrix is then calculated from the
initial force density vector and the nodal coordinates
are adopted from the eigenvalue decomposition of the
force density matrix D.
The square force density matrix D can be factorized
as follows by using the eigenvalue decomposition
(Meyer, 2000).
where Φ (∈Rn×n) is the orthogonal matrix whose ith
column is the eigenvector basis
φ
i (∈Rn)is of D. Λ
(∈Rn×n) is the diagonal matrix whose diagonal elements
are the corresponding eigenvalues, i.e., Λii =
λ
i. The
eigenvector
φ
i of Φ corresponds to eigenvalue
λ
i of Λ.
The eigenvalues are in increasing order as
It is clear that the number of zero eigenvalues of D
is equal to the dimension of its null space. The first
d+1 eigenvectors of Φ, corresponding to the first
d+1 smallest eigenvalues, respectively, are chosen as
nodal coordinates [x, y, z] for d-dimensional tensegrity
structure. Subsequently, these nodal coordinates are
substituted into Eq. (5) to select the candidates for a set
of force densities by the singular value decomposition
of the matrix A.
where U and W matrices are orthogonal matrices. V is
a diagonal matrix with non-negative singular values of
A. The matrices W from Eq. (13) can be expressed as
(Pellegrino, 1993).
As a result, the general solution q
¯ of Eq. (5) that lies
in the null space of A is formulated as
where the coefcients ci are arbitrary values and qi (∈Rb,
i = 1,2, …, s) are the particular solutions of Eq. (5).
A genetic algorithm is then used to obtain the
coefficients ci related to a set of force densities that
satisfy Eq. (4). Finally, the process is iteratively
calculated to search for a feasible set of the nodal
coordinates and force densities until Eq. (8) is satised.
Additionally, the vector of unbalanced forces
ε
f (∈Rdn)
dened as follows can be used to evaluate the accuracy
of the results:
The Euclidean norm of
ε
f is used to dene the design
error ∈as
3.2 Genetic Algorithm
As pointed out in the previous section, a genetic
algorithm is used to obtain a set of the coefficients
ci and force densities q
¯ in Eq. (15). These problems
are formulated as constrained optimization problems
that draw feasible solutions from form-finding. The
key point in developing a form-finding method for a
tensegrity structure using genetic algorithms is how to
dene tness functions and constraints. In this study,
the fitness function combined with coefficients ci is
constructed to minimize it using a genetic algorithm;
the magnitude of ci is assumed to be a unit. Also, the
Ck,p
(1)
CTQCx (2a)
CTQC (2b)
CTQCz (2c)
Qdiag (3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Ck,p
(1)
CTQCx (2a)
CTQC (2b)
CTQCz (2c)
Qdiag (3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Ck,p
(1)
CTQCx (2a)
CTQC (2b)
CTQCz (2c)
Qdiag (3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Ck,p
(1)
CTQCx (2a)
CTQC (2b)
CTQCz (2c)
Qdiag (3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
158 JAABE vol.16 no.1 January 2017 Seunghye Lee
fitness function is provided with penalty functions
to significantly improve accuracy. The process of
determining a unique feasible set q
¯ is formulated as a
constrained optimization problem as
Towards a more effective algorithm, three constraints
are added into the objective function shown as follows:
• Constraint 1
• Constraint 2
• Constraint 3
in which Γ denotes the total set of the force density,
Γcable is the set of the force density for cable members,
and Γstrut is the set of the force density for strut
members. In Eq. (18a), qc and qs are force densities
that are allowed to take a value from 0 to + 1 for cables
and from -1 to 0 for struts, respectively. In Eq. (18b),
subscript i and j denote element numbers in the equal k
group, ∈0 is then used to dene the tolerance. Eq. (18a)
indicates the unilateral condition for tensegrity, which
is necessary in order to have a unique value for force
density. On the other hand, Eq. (18b) and Eq. (18c) are
optional constraints for drawing the desired shape of
the tensegrity from the optimization problem. Eq. (18b)
denotes grouping constraints indicating that the force
density values of the members are identical if they are
in the same group k. Eq. (18c) is optional constraints
for two distinct members that are in a linear relation to
each other. In the previous paper (Tran and Lee, 2011),
the members are required to be grouped based on the
symmetry of the structure in order to find the single
integral feasible self-stress state.
These can be time-consuming and it is often
difficult to perform suitable groupings. However, in
this study, the force density can be uniquely defined
for the tensegrity with multiple states of self-stress
even for grouping constraints. If any regular form of
a tensegrity structure is necessary, the other optional
constraints can be applied to the fitness function. In
other words, the specic solutions can be derived in a
versatile manner using the appropriate constraints, and
user-dened shapes can be achieved as well.
3.3 Solution Procedure
An outline of the form-nding process is shown in
Fig.2. The feasible sets of the nodal coordinates and
the single integral force densities of the tensegrity
structures can be simultaneously defined by the
proposed form-nding method through the following
procedure.
• Initialization:
(1) Specify the dimension size, the connectivity
of nodes, and the type of each member (cable
or strut). An initial force density vector set is
needed to determine the rst generation. Assign
the initial force density coefficients of cables
and struts as +1 and −1, respectively.
(2) Dene the connectivity matrix C by Eq. (1).
• Iterations:
(3) By the number of s, perform EVD (Eq. (11))
and SVD (Eq. (13)) to dene nodal coordinates
[x,y,z] and force densities q, respectively.
(4) Perform the genetic algorithm (Eq. (17)) to
dene the coefcients ci through Eq. (15). And
then evaluate accuracy of the results q
¯ through
Eq. (16).
(5) If the design error
ε
f is greater than stopping
criteria, the results are substituted into matrix D
of Eq. (4) and go to Step 4.
• Termination:
(6) The process is terminated until Eq. (16) has
been checked. The nal coordinates and force
density vector are the solutions.
Fig.2. Outline of the Proposed Form-nding Procedure
Input
[x y z] 0
T
D
D
Determine
0
T
Aq
A UVW
( 1, 2, , )
i
qi s
Find
1
s
ii
i
c
qq
GA
Find
Design error < tolerance
no
yes
output
0
,,dCq
[x y z]
Determine
q
i
c
[x y z]
q
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
(17)
(18a)
(18b)
(18c)
159JAABE vol.16 no.1 January 2017 Seunghye Lee
4. Numerical Examples
In this section, numerical examples of tensegrity
structures with multiple states of self-stress are
presented to demonstrate the efciency of the proposed
method. Based on the algorithm developed, both the
connectivity matrix and the force density vector are
simultaneously defined with the nodal connectivity
and the type of each member. Note that all of the force
densities given in tables were normalized with respect
to the force density coefcient of Element 1.
Each run of the genetic algorithm is conducted for
200 maximum generations, using a population size of
200. The convergence of all results was veried using
a stopping criteria
ε
f = 0.01, and the tolerance ∈0 of the
constraints is set to 0.01.
4.1 2D Hexagonal Tensegrity Structure with Eight
Cables
The initial topology of the 2D hexagonal tensegrity
structure with two state of self-stress (s=2) is shown in
Fig.3. The tensegrity comprises three struts and eight
cables, and the structure is composed of six nodes
and 11 members; i.e. n = 6 and b = 11. The only basic
information is the connectivity matrix C and the type
of each member.
Firstly, the initial force density set is applied to
determine the rst generation data, namely, the force
density matrix for the equilibrium equation (Eq.
(4)). The initial force density coefficients of cables
and struts are assigned as +1 and −1, respectively.
Secondly, from the EVD of the force density matrix
D, the nodal coordinates are adopted, which in turn
leads to a new force density set from the SVD of the
equilibrium matrix A. The form-finding process is
terminated until Eq. (16) has been checked.
The member type (Constraint 1) and grouping
constraints (Constraint 2) are applied to the minimum
optimization problems so that the force density output
coincides with those of the previous study (Tran and
Lee, 2011). In the previous study, a linear relation
between member group 2 and 3 is imposed to obtain a
single integral feasible pre-stressed mode. Despite the
fact that the linear relation is not needed in the present
study, the relation (q3 = 1.5q2, 2 and 3 indicate groups)
is imposed to draw the same results for a comparison.
Since the 2D hexagonal tensegrity structure has two
states of self-stress (s=2), two force density sets of the
elements could be obtained as follows:
Finally, the general force density solution set q
¯ of
Eq. (5) is obtained as shown in Table 1. Table 2. shows
that the general force density solution sets are in good
agreement with those of previous studies. All of the
force densities given in Table 2. were normalized with
respect to the force density coefcient of Element 1.
Table 2. Comparison of the Force Densities Obtained by the
Proposed Method with the Previous Study
Group Element Tran and Lee (2011) Present
11-4 1.0 1.0
2 5-6 2.0 2.0
37-8 3.0 3.0
4 9-10 -2.5 -2.5
511 -0.5 -0.5
Table 1. The Force Density Sets and Coefcients for Two State
of Self-stress (s=2). 2D Hexagonal Tensegrity Structure
Elem. No. q1q2c1c2q
¯
1 -0.316 -0.301 0.392 -0.316 0.153
2 -0.316 -0.301 0.153
3 -0.316 -0.301 0.153
4-0.316 -0.301 0.153
50.440 -0.145 0.306
60.440 -0.145 0.306
70.123 -0.446 0.459
80.123 -0.446 0.459
9 -0.281 0.296 -0.382
10 -0.281 0.296 -0.382
11 0.158 0.150 -0.076
Table 3. Comparison of the Force Densities Obtained by the
Proposed Method with the Previous Study. 3D Three-strut
Octahedral Cell
Elem.
No.
Present Tran and Lee
(2011)
Constraint
1
Constraint
1,2
Constraint
1, 2, 3
1 1.0 1.0 1.0 1.0
2 1.0 1.0 1.0 1.0
3 1.0 1.0 1.0 1.0
41.0 1.0 1.0 1.0
50.7 1.4 0.5 0.5
60.8 1.4 0.5 0.5
7 0.7 1.4 0.5 0.5
8 0.8 1.4 0.5 0.5
9 0.7 1.4 0.5 0.5
10 0.8 1.4 0.5 0.5
11 0.7 1.4 0.5 0.5
12 0.8 1.4 0.5 0.5
13 -1.7 -2.4 -1.5 -1.5
14 -1.8 -2.4 -1.5 -1.5
15 -1.5 -2.9 -1.0 -1.0
6 5
1
23
4
(8)
(7)
(1)
(2) (4)
(3)
(11)
(9) (10)
(5) (6)
Fig.3. The Initial Topology of the 2D Hexagonal Tensegrity
Structure with Eight Cables. The Struts and Cables are Depicted
by Thick and Thin Lines, Respectively
(9)
3
1
6
5
4
2
(11)
(10) (12)
(5)
(1)
(2)
(3)
(4)
(7)
(8)
(6)
(15)
(14)
(13)
Fig.4. The Initial Topology of the 3D Three-strut Octahedral Cell
160 JAABE vol.16 no.1 January 2017 Seunghye Lee
4.2 3D Three-strut Octahedral Cell
The 3D three-strut octahedral cell is a tensegrity
with three struts and 12 cables as shown in Fig.4. The
input parameters for this example are n = 6 and b = 15
and d = 3. After investigating the rank deciency of the
force density matrix, the tensegrity is formed to have
three self-stress states (s = 3).
In the previous study (Tran and Lee, 2011), in order
to nd the single integral feasible self-stress state, the
symmetry properties of the structure are required.
The method requires appropriate grouping in
order to obtain a single feasible pre-stressed mode.
Accordingly, a trial and error scheme should be
employed to find an appropriate grouping. In this
study, three cases of this example are performed. First,
only the constraint for the member type (Eq. (18a))
is applied (Constraint [1]). As a second case, the
grouping constraint (Eq. (18b)) is added to the member
type constraint (Constraint [1, 2]). The members are
grouped according to the geometry and symmetry. The
nal case (Constraint [1, 2, 3]) is performed using all
three constraints (member type (Eq. (18a)), grouping
(Eq. (18b)), and force density ratio among the members
(Eq. (18c)).
In the rst and the second cases, the obtained force
densities differ from those of the previous study as in
Table 3. To obtain identical results, the specic force
density ratio constraint of element 1 and 5 (q
¯1 = 2q
¯5)
is additionally imposed to the fitness function. As a
result, the same force density values are obtained,
as presented in Table 3. Fig.5. shows the case of
Constraint [1], Constraint [1, 2], and Constraint [1, 2,
3], respectively. As shown in Fig.5., the different shape
of the tensegrity is achieved by applying different
constraints. Even for a tensegrity with a multiple self-
stress status, form-nding can be achieved by simply
imposing a member type constraint. If the obtained
shape of the tensegrity is not satisfactory, an alternative
shape can be obtained by adjusting grouping or ratio
constraints only. The desired shape of the tensegrity
can be obtained by imposing appropriate constraints.
4.3 3D Six-strut Tensegrity
A 3D six-strut tensegrity has six struts and 24 cables,
and the initial topology has 12 nodes and 30 elements
(Fig.6.). After investigating rank deficiency, the
structure obtained two states of self-stress (s = 2). This
example is also performed using two constraint cases.
Firstly, only the constraint for the member type
(Eq. (18a)) is applied. As the second scenario, two
constraints, member type (Eq. (18a)) and grouping (Eq.
(18b), are provided. The second case used in this study
is the same as that used in the condition of the previous
study (Tran and Lee, 2011). Fig.7.(a) and 7.(b) show a
comparison between the nal topologies obtained by
two cases. Each case achieved a single integral feasible
self-stress state. Table 4. shows the obtained force
density coefcients. According to Table 4., the results
of the second scenario agreed well with those of the
previous study.
For the purpose of designing an arbitrary tensegrity
shape, a new constraint condition is provided for
fitness function. A linear force density ratio between
Element 28 and 30 is additionally imposed q28 = q30. In
this example, since using all three constraints is a strict
restriction for obtaining a single feasible self-stress
state, the grouping constraint is not provided. Fig.7.(c)
shows a newly-designed 3D six-strut tensegrity shape.
This new tensegrity shape differs from other shapes in
Fig.7.; the results of force densities differ signicantly
Fig.5. The Obtained Geometry of the 3D Three-strut Octahedral
Cell by Using (a) Constraint [1], (b) Constraint [1, 2], and
(c) Constraint [1, 2, 3]
(a)
(b)
(c)
Fig.6. The Initial Topology of the 3D Six-strut Tensegrity
(9)
1
(11)
(10)
(12)
(5)
(1)
(2)
(3)
(4)
(7)
(8)
(6)
(15)
(14)
(13)
2
3
4
5
6
7
8
9
10
11
12
(16)
(17)
(18)(19)
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
(30)
161JAABE vol.16 no.1 January 2017 Seunghye Lee
from that of other cases as shown in Table 4. (Constraint
[1, 3]). This indicates that the specic solutions can be
versatile in derivation using the appropriate constraints.
Also, it shows that any number of different shapes
of the single integral feasible self-stress state can be
created.
5. Conclusions
In this study, a numerical method using a force
density method combined with a genetic algorithm has
been proposed as a form-nding process for tensegrity
structures with multiple states of self-stress. The
proposed method comprises equilibrium equations
using a force density method.
Both the eigenvalue decomposition of the force
density matrix and the singular value decomposition of
the equilibrium matrix are iteratively executed to nd
the range of feasible sets of the nodal coordinates and
the force densities. Then a genetic algorithm is used to
nd a unique feasible set of force densities. The method
could be adapted simply to determine the uniquely
defined force density. The feature of the proposed
method is that a grouping or symmetric constraint is
not required to find the single integral feasible self-
stress state. The desired tensegrity shapes can be
designed through a genetic algorithm with appropriate
constraints. Three examples of tensegrity structures
with multiple states of self-stress are performed. A
very good performance of the proposed method has
been shown in the numerical examples; they clearly
show that the specific solutions can be derived with
versatility using the appropriate constraints.
Acknowledgement
This research was supported by a grant (NRF-
2015R1C1A2A01055897) fr om NR F (National
Research Foundation of Korea) funded by MEST
(Ministry of Education and Science Technology) of
Korean government.
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(a) (b)
(c) (d)
Fig.7. The Obtained Geometry of the 3D Three-strut Octahedral
Cell; (a) Constraint [1], (b) Constraint [1, 2], (c) Constraint
[1, 2, 3] (d) Model of the 3D Three-strut Octahedral Cell with
Constraint [1, 2, 3] Condition
Table 4. Comparison of the Force Densities Obtained by the
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Elem.
No.
Present Tran and
Lee (2011)
Constraint
[1]
Constraint
[1, 2]
Constraint
[1, 3]
1 1.0 1.0 1.0 1.0
2 0.6 1.0 6.3 1.0
3 1.0 1.0 1.0 1.0
40.6 1.0 6.3 1.0
5 1.0 1.0 1.0 1.0
6 0.6 1.0 6.3 1.0
71.0 1.0 1.0 1.0
80.6 1.0 6.3 1.0
91.1 1.6 7.5 1.6
10 1.1 1.6 7.5 1.6
11 1.1 1.6 7.5 1.6
12 1.1 1.6 7.5 1.6
13 1.4 1.6 4.3 1.6
14 1.4 1.6 4.3 1.6
15 1.4 1.6 4.3 1.6
16 1.4 1.6 4.3 1.6
17 1.1 1.6 7.5 1.6
18 1.1 1.6 7.5 1.6
19 1.1 1.6 7.5 1.6
20 1.1 1.6 7.5 1.6
21 1.4 1.6 4.3 1.6
22 1.4 1.6 4.3 1.6
23 1.4 1.6 4.3 1.6
24 1.4 1.6 4.3 1.6
25 -1.3 -1.6 -5.9 -1.6
26 -1.3 -1.6 -5.9 -1.6
27 -1.3 -1.6 -5.9 -1.6
28 -1.3 -1.6 -5.9 -1.6
29 -1.6 -2.2 -10.4 -2.2
30 -1.9 -2.2 -5.9 -2.2
162 JAABE vol.16 no.1 January 2017 Seunghye Lee
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