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Form-finding of tensegrity structures based on
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MATMA-2023
Journal of Physics: Conference Series 2691 (2024) 012026
IOP Publishing
doi:10.1088/1742-6596/2691/1/012026
1
Form-finding of tensegrity structures based on genetic
algorithm
Mingxiang Zheng 1, Zhengyin Du 1, Hui Li 1, Min Lin 1, and Zhifei Ji 1, 2,3
1 College of Marine Equipment and Mechanical Engineering, Jimei University, Xiamen,
Fujian Province, 361021, China
2 Fujian Province Key Laboratory of Ocean Renewable Energy Equipment, Xiamen,
Fujian, 361021, China
3Corresponding author’s e-mail: jizhifei@jmu.edu.cn
Abstract: We develop a novel form-finding method that utilizes force density and a genetic
algorithm. Firstly, the equilibrium equation is derived by using the force density method. Next,
the force density matrix is decomposed through QR decomposition. Subsequently, an
optimization objective function is introduced, which incorporates information about the force
density values. The optimal solution for the objective function is obtained through the use of a
genetic algorithm. We determine a suitable set of force density values that satisfy the
requirements of the equilibrium matrix rank deficiency, force density rank deficiency, and
precision, thereby establishing the equilibrium configuration of the structure. The simulation
results verify the reliability of the proposed method. With its advantages of straightforward
calculations, rapid convergence, and high precision, this method proves to be well-suited for the
form-finding of both regular and irregular tensegrity structures.
1. Introduction
The concept of “tensegrity” was originally presented by Fuller in the 1960s. Tensegrity structures
generally consist of discrete bars and continuous cables, forming a rigid-flexible coupling structure [1].
Due to their lightweight nature, strong impact resistance, excellent mechanical properties, and
collapsibility, these structures have found applications in fields such as architecture [2], robotics [3],
and aerospace [4]. Finding the geometric configuration of tensegrity structures, known as form-finding,
is crucial in their structural design as they exhibit a self-equilibrium without the presence of external
forces. Form-finding involves determining the initial prestress and structure shape. Existing form-
finding methods generally consider topological connections as known conditions, with node coordinates
and prestress as variables. These methods can be categorized as topological form-finding and prestress
form-finding [5]. Commonly adopted form-finding strategies include the dynamic relaxation method
[6], force density method [7], energy minimum method [8], and intelligent optimization algorithms
derived from these approaches.
The force density method, widely applied for its ease of understanding and simplicity of operation,
represents a typical form-finding approach. The force density method involves the utilization of the
force density coefficient, which expresses the relationship between force and length. The force density
matrix is computed by applying the association matrix. It is derived by utilizing an initial set of force
density values. The equilibrium matrix is determined by taking into account both the association matrix
and the coordinates of nodes within the system. Iterative solutions are used to determine the singular
MATMA-2023
Journal of Physics: Conference Series 2691 (2024) 012026
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doi:10.1088/1742-6596/2691/1/012026
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values of the force density matrix. Ultimately, it is possible to obtain the node coordinates and set of
force densities that meet the minimum rank-deficiency requirement [5]. Both regular and irregular
tensegrity structures can be analyzed. However, traditional force density methods encounter challenges
when dealing with complex tensegrity structures. To address this issue, intelligent optimization
algorithms derived from the force density method have been developed, such as the combination of the
force density method based on GA [9], NN [10], and AFSA [11].
In summary, extensive advancements have been made in the study of form-finding methods for
tensegrity structures, and the initial form of a tensegrity structure can be quickly obtained. However, for
complex tensegrity structures, a combination of intelligent algorithms is required for form-finding. This
article introduces an intelligent form-finding method for tensegrity structures. Firstly, the equilibrium
equations are established. Then, the force density matrix is subjected to QR decomposition, and an
optimization objective function incorporating the force density values is introduced. To achieve the
optimal solution for the objective function, we utilize the genetic algorithm in our approach, thereby a
suitable set of force densities is determined, which satisfies the rank conditions. As a result, we can
ascertain the balanced configuration of the structure. The efficacy of the form-finding method is
showcased through illustrative examples.
The structure of this article is organized as follows. Section 2 introduces the self-equilibrium
configuration and rank deficiency condition of tensegrity structures. Section 3 provides a detailed
explanation of the GA-based form-finding process for tensegrity using QR decomposition. Section 4
shows several examples to demonstrate the effectiveness of this form-finding method.
2. Force-density form-finding method and rank deficiency condition
2.1. Assumptions
The geometric topology and connection relationships of the structure are known.
All nodal connections of the individual elements are assumed to be hinged.
External loads are not considered in the analysis.
The weight of the structure is neglected.
The deformation caused by structural bending is neglected.
The consideration of dissipative forces’ impact on the structural system is omitted.
2.2. Self-equilibrium equation of tensegrity structure
The detailed content of force density can be found in [7]. For any tensegrity structure with b components
and n nodes, the connectivity matrix 𝑪
can be determined based on the topological connection
relationships. It is assumed that Component k is connected between the nodes i and j, and the kth row of
the connectivity matrix C will have elements 1 and -1 corresponding to the ith and jth positions,
respectively. If the nodes i and j are not connected by Component k, the corresponding element in C will
be set to zero. Therefore, the connectivity matrix can be presented as follows:
𝑪
,
1 for p=i
1 for p=j
0 otherwise
(1)
We suppose that the force density of the tensegrity structure is 𝒒𝑞
,𝑞
,…,𝑞
,…,𝑞
. The force
density of the member k can be obtained by the following equation.
𝑞
(2)
where 𝑓
is the internal force of the member k, and 𝑙
is the length of the member k.
Therefore, the force density of the whole structure 𝑸∈𝑹
can be defined as follows:
𝑸𝑑𝑖𝑎𝑔𝒒 (3)
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Journal of Physics: Conference Series 2691 (2024) 012026
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Combining Equations (1) to (3), the equilibrium equations of tensegrity structure in each direction are
defined as follows:
𝑪
𝑸𝑪𝒙 0 (4)
𝑪
𝑸𝑪𝒚0 (5)
𝑪
𝑸𝑪𝒛0 (6)
The force density matrix 𝑬∈𝑹
are given in the following equations.
𝑬𝑪
𝑸𝑪 (7)
After rearranging the equations, the equations define the equilibrium of a tensegrity structure as
follows:
𝑬𝒙 𝒚 𝒛
0 0 0
(8)
Equations (4) to (6) can be rearranged to form the force density equations for the tensegrity structure.
𝑨𝒒0 (9)
𝑨𝑪
diag𝑪𝒙
𝑪
diag𝑪𝒚
𝑪
diag𝑪𝒛 (10)
The tensegrity structure is a self-balancing structure system, so the unbalanced force 𝜖 is adopted as
an index to evaluate the accuracy of the calculation results, which is defined as follows:
𝜖 |𝑨𝒒| (11)
In this paper, we control design error within 𝜖 10
.
2.3. Two rank deficiency conditions
The solution space, which characterizes the equilibrium equation of a tensegrity structure, can be defined
as the null space of matrix E. The rank deficiency of E defines the dimension of the null space.
𝑛
𝑬
𝑛𝑟
𝑬
(12)
𝑟
𝑬
𝑟𝑎𝑛𝑘𝑬 (13)
In Equation (8), the row or column of matrix E exhibits a summation of elements equal to zero, so
Equation (8) has at least one eigenvalue of zero, and the corresponding vector 𝑰
= {1,1,…,1
is a
solution of Equation (8), which obviously cannot be used as a coordinate vector of a feasible node.
Therefore, the first rank deficiency condition must be satisfied for semi-definite matrix E of D-
dimensional tensegrity structure.
𝑛
𝑬
𝑑1 (14)
The null space of Matrix A captures the entire solution set of Equation (9). We define the dimension
of matrix A as 𝑛
𝑨
, representing the equilibrium matrix, which can be obtained by using the following
equation.
𝑛
𝑨
𝑏𝑟
𝑨
(15)
where 𝑟
𝑨
𝑟𝑎𝑛𝑘𝑨. The condition for the existence of at least one self-equilibrating mode is
expressed as the second rank-deficiency condition.
𝑠𝑛
𝑨
1 (16)
Based on the above two rank deficiency conditions, the proposed form finder search allows for a
self-equilibrium configuration in which at least one self-stress state exists in a tensegrity structure.
MATMA-2023
Journal of Physics: Conference Series 2691 (2024) 012026
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3. Form-finding of tensegrity structure based on genetic algorithm
3.1. QR decomposition
In the preceding section, the equilibrium equations for the tensegrity structure have been derived, and
by solving these equations, the null space of the solutions can be obtained. However, Matrix E needs to
meet the requirement of having a rank that should be equal to or greater than d+1. In this paper, QR
decomposition is used to compute the rank of the force density matrix to satisfy the first rank deficiency
condition. QR decomposition decomposes a matrix into the product of two matrices, where one is an
orthogonal matrix and the other is an upper triangular matrix, for example, E=QR. The column vectors
of Q form an orthonormal basis and satisfy the property of 𝑸𝑸
𝑰, where 𝑰 is the identity matrix. R
is an upper triangular matrix with all zero elements below the diagonal and positive diagonal elements.
It is an effective method for solving linear systems of equations.
𝑹⎣
⎢
⎢
⎢
⎡
𝑟
𝑟
𝑟
…𝑟
𝑟
𝑟
…𝑟
𝑟
…𝑟
⋱⋮
𝑟
⎦
⎥
⎥
⎥
⎤
(17)
For matrices with arbitrary entries, it is generally not possible to form such a decomposition. This is
because before performing QR decomposition, the matrix needs to undergo Schmidt orthogonalization,
which requires ensuring the linear independence of the column vectors. However, this issue of linear
dependence can be bypassed by using Householder transformations. Householder transformation is an
orthogonal transformation capable of converting an arbitrary n-dimensional vector, denoted as Vector
a, into any desired n-dimensional vector b. In other words, for any two vectors a and b (𝒂,𝒃∈
𝑹
and ||𝒂||
||𝒃||
), it is possible to find a Householder transformation P such that b = Pa.
We set 𝝎𝒂𝒃/||𝒂𝒃||
, and the reflection matrix 𝑷
can be defined as:
𝑷
𝑰2𝝎𝝎
(18)
where I is the unit matrix.
𝑷
…𝑷
𝑬𝑹 (19)
We set 𝑷𝑷
…𝑷
, we can obtain:
𝑷𝑩𝑹 (20)
By left-multiplying both sides of Equation (20) by 𝑷
, we can obtain:
𝑬𝑷
𝑹𝑸𝑹 (21)
where 𝑷
is equivalent to an orthogonal matrix Q, and R is the upper triangular matrix.
3.2. Objective function
The rank deficiency of E for tensegrity structures is equal to or greater than d+1. When the tensegrity
structure is two-dimensional or three-dimensional, the rank should be n-3 or n-4, respectively. By
forcing the elements in the n-3, n-2, and n-1, nth rows in the n-2 and n-1, and nth rows of the upper
triangular matrix R to be zero, the rank-deficiency condition for 2D or 3D tensegrity structures can be
satisfied. Therefore, according to the mean square error loss function, the objective function is as follows:
𝑀𝑖𝑛 𝛱
𝜶𝜷 (22)
𝑆.𝑡. 0𝒒
1（𝒒
∈𝜞
） (23)
1𝒒
0（𝒒
∈𝜞
） (24)
MATMA-2023
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5
𝜳𝑞
,𝑞
,⋯,𝑞
𝒉 (25)
where w represents the number of elements in the upper triangular matrix R that need to be set to zero;
α is defined as 𝜶 ∑
∑
𝒓
, which represents the mean square deviation between 𝑟
and 0; β
is defined as 𝜷∑
1/|𝒒
|, which ensures that each force density value is not close to or not equal to
zero, while not affecting the minimization of α; 𝜞 represents the set of force density; 𝜞
represents
the collection of force densities associated with the cable members; 𝜞
is the set of force density of
bar members; 𝒉 is the number of members in a group with the same power density value. Equation (25)
is an optional constraint, and most of the tensegrity structures in engineering applications are symmetric
structures, that is, there are one or more groups of members with equal force density values.
GA is a kind of optimization algorithm that simulates the natural evolution process. It has advantages
in global search. In this paper, GA is utilized to encode the force density in real numbers, and a series
of operations such as selection, crossover, and mutation are carried out to optimize the solution of the
objective function step by step.
3.3. Form-finding process
The entire geometric programming algorithm proceeds as follows:
(1) Initialization: We set the connectivity matrix, specify the design error tolerance, and define the
parameters for the genetic algorithm.
(2) Optimization algorithm design: A set of force density values that meet the constraints (23) and
(24) is randomly generated, the force density is encoded as a real number, and the force density value is
obtained by selection and crossover and mutation operations, meanwhile Matrix E was formed
according to Equation (7). QR decomposition of Matrix E is performed to iteratively get a collection of
force densities and minimize the objective function.
(3) Equation system construction: We utilize the connectivity matrix and force densities to construct
the equilibrium equation system and the force density equation system.
(4) Evaluation of three conditions: We check if Matrix E satisfies Equation (14), verify if Matrix A
satisfies Equation (16), and evaluate if the error meets the design requirements. If all of these conditions
are met, we will output the force density values and nodal coordinates. Otherwise, we proceed to Step
(2).
4. Example
In this section, four numerical cases of regular 3-bar 6-cable, irregular 3-bar 6-cable, regular 6-bar 24-
cable, and irregular 6-bar 24-cable tensegrity structures are given to confirm the feasibility and
effectiveness of the method. The results of the examples show that the method is suitable for both 2D
and 3D tensegrity structures, the design error is up to the requirements, and the accuracy is high. The
main parameters of GA are set as follows.
Population size: 100
Crossover probability: 0.8
Probability of mutation: 0.1
Iterations: 100
4.1. A 3-bar 6-cable tensegrity structure
The planar 3-bar 6-cable tensegrity structure consists of 9 members and 6 nodes, with each node
connected to 2 cables and 1 bar. Both regular and irregular 3-bar 6-cable tensegrity structures can be
determined based on Equation (25). The force density is divided into three groups: the first group
consists of rod force density, the second group contains four equal cable force densities, and the third
group contains two equal cable force densities, resulting in a regular 3-bar 6-cable tensegrity structure.
As shown in Figure 1, it is evident that the objective function rapidly converges from an initial value of
0.0681 to approximate zero, and the objective function value converges to 1.9425e-8 after performing
100 iterations. With an error of 3.0575e-10, the force density values for the bars are -0.4169, among
MATMA-2023
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doi:10.1088/1742-6596/2691/1/012026
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which one group of cables is 0.8338 and the other group of cables is 0.4169. After normalization, the
force density of the rod and the two groups of cables are -1, 2, and 1 respectively, which is consistent
with the research results in [2] and verifies the validity of our study. Through the coordinate position of
each node, we can get the relationship between the length of the rod and the cable, and the designer can
design and make the whole tensioning structure according to the length relationship. The structural
diagram (Figure 2) can be plotted based on the connectivity matrix and node coordinates, where bold
lines indicate bar components (1), and thin lines represent cable components (2). From Figure (2), it can
be seen that the structure has a symmetric distribution of bars and cables. The resulting structure is a
regular 3-bar 6-cable tensegrity structure.
Figure1. Iterative process for regular 3-bar
6-cable tensegrity structure for
m
-finding.
Figure 2. Regular 3-bar 6-cable tensegrity
structure.
Figure 3. Iterative process for irregular 3-
bar 6-cable tensegrity structure form-
finding.
Figure 4. Irregular 3-bar 6-cable tensegrity
structure.
We can obtain an irregular 3-bar 6-cable tensegrity structure without using Equation (25). As shown
in Figure 3, the objective function converges quickly, and after 100 iterations, the objective function value
converges to 5.1685e-10 with an error of 3.7896e-6. We found that the accuracy of the irregular structure
decreases compared to the regular 3-bar 6-cable tensegrity structure due to the different force density
values for the bars and cables. However, it still satisfies the rank deficiency condition and can also form
a stable structure. The force density values for the bars are 𝑞
= [-0.1581, -0.3058, -0.3248], and for the
cables, they are 𝑞
= [0.5095, 0.8742, 0.7332, 0.3827, 0.1961, 0.4122]. By substituting these force
density values into Equation (8), we can calculate the node coordinates of the tensegrity structure. The
structure diagram (Figure 4) is created based on the connectivity matrix and node coordinates, with bold
lines representing the bar components (1) and thin lines representing the cable components (2). From
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Figure 4, we can observe that the lengths of the bars and cables are not equal, forming an irregular
tensegrity structure.
4.2. A 6-bar 24-cable tensegrity structure
The spatial 6-bar 24-cable tensegrity structure is comprised of 30 units and 12 nodes, with each node
connected to 4 cables and 1 bar. This structure is a regular configuration and can be divided into 2 groups
of units: one group consists of cable units with equal force densities, and the other group consists of bar
units with equal force densities. Real number coding is applied to encode the force densities. As shown
in Figure 5, it is evident that the objective function rapidly converges. After 15 iterations, the objective
function value reaches 6.94e-09. After 100 iterations, the objective function value decreases to 1.31e-
31 with an error of 1.01e-15. The force density value for the bars is -0.5661, and the force density value
for the cables is 0.3774. After normalization, the force density of the rods and the cables are -1 and 1.5,
respectively, and the consistency between the normalized force densities and the results reported in [2]
and [9] further supports the effectiveness of this algorithm. By substituting these force density values
into Equation (8), the node coordinates of the tensegrity structure can be calculated. Through the
coordinate position of each node, we can get the length of the rod and the cable: 0.8659 and 0.5449. We
can get the length of the rod as 1.5891 times the length of the cable, and the designer can design and
make the entire tension structure according to the length relationship. The structural diagrams (Figure
6) can be plotted based on the connectivity matrix and node coordinates, where bold lines indicate bar
components (1) and thin lines represent cable components (2).
Figure 5. Iterative process for regular 6-bar
24-cable tensegrity structure for
m
-finding.
Figure 6. Regular 6-bar 24-cable tensegrity
structure.
Figure 7. Iterative process for irregular 6-
bar 24-cable tensegrity structure form-
finding.
Figure 8. Irregular 6-bar 24-cable tensegrity
structure.
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When Equation (25) is removed, the resulting 6-bar 24-cable tensegrity structure becomes an
irregular shape. If we use the given genetic algorithm with 100 iterations, it may not achieve the desired
tensegrity configuration due to the insufficiently small objective function. Therefore, we increase the
iteration number to 200 (Figure 7). The results derived from running the entire algorithm vary each time,
but all of them produce force densities that meet the rank-deficiency conditions and successfully achieve
the desired error. This means that there are many possible shapes for irregular 6-bar 24-cable tensegrity
structures, and we randomly choose one for demonstration. As shown in Figure 7, the convergence speed
of the objective function is fast, and the resulting objective function is 3.9285e-09 with an error of
8.4033e-6. The force density for bars is 𝑞
= [-0.6243, -0.6194, -0.6744, -0.5318, -0.7408, -0.6771],
and the force density for cables is 𝑞
= [0.2535, 0.5536, 0.6140, 0.6223, 0.4070, 0.7004, 0.5398, 0.7585,
0.4552, 0.7698, 0.5557, 0.5907, 0.3704, 0.6210, 0.2303, 0.3106, 0.4608, 0.6407, 0.2365, 0.3852, 0.4104,
0.2227, 0.5726, 0.6059]. Similarly, we can find the coordinates of each node by Equation (8) based on
the obtained rod and cable force density value. Based on these node coordinates, we can plot the structure
diagram. The irregular 6-bar 24-cable tensegrity structure is depicted in Figure 8, where bold lines
indicate bar components (1) and thin lines represent cable components (2). In Figure 8, we can see that
the lengths of each rod and cable are not consistent, so this is an irregular tensegrity structure.
5. Conclusion
Tensegrity structures have the advantages of being lightweight and shock-resistant. They thus have
found extensive application in the domains of biomimetics, aerospace technology, and material science.
Recently, biomimetic legged robots and space exploration robots based on tensegrity structures have
been proposed. Allowing for the rapid generation of stable configurations for tensegrity structures, the
form-finding method requires only knowledge of the connectivity of rods and cables. We establish an
optimization model for tensegrity form-finding based on the force density method. The model is solved
by using a genetic algorithm to obtain the force density values, as well as the node coordinates. We
validate our findings by using four examples of planar and spatial tensegrity structures. The results
demonstrate that the force density-based form-finding method using QR decomposition and GA
effectively solves the form-finding problem for tensegrity structures. It applies to both spatial and planar
tensegrity structures, including regular and irregular configurations. Therefore, designers can utilize our
research method to obtain node coordinates and corresponding force density values, to facilitate
tensegrity structure assembly and internal force analysis.
In engineering applications, regular tensegrity structures are the main structural forms of tensegrity
structures. Our research can find regular tensegrity structures with high precision. During the form-
finding process of an irregular tensegrity structure, we observed that the accuracy of form-finding
decreased. By modifying some parameters of GA, the accuracy of the algorithm was improved, and the
accuracy requirements were met. Future work will focus on the study of more complex tensegrity
structures, especially irregular tensegrity structures. This would involve speeding up the convergence of
genetic algorithms or utilizing large-scale optimization methods.
Acknowledgments
The authors gratefully acknowledge the support of the Natural Science Foundation of Fujian Province,
China (Grant No. 2023J01792), the Scientific Research Foundation of Jimei University (Grant No.
ZQ2017005), the Young and Middle-aged Teacher Education Scientific Program of Fujian Province
(Grant No. JAT200250), and the Xiamen Science and Technology Subsidy Project (Grant No.
2023CXY0316).
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