# Deep Neural Networks for Form-Finding of Tensegrity Structures

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## Abstract

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## 1. Introduction

## 2. Form-Finding Process Using the Force Density Method (FDM)

#### 2.1. Connectivity and Force Density Matrices

**D**, is then square and symmetric. In linear algebra, a sufficient condition for a symmetric matrix to be invertible is that the matrix is positive definite. The positive definite means that the scalar ${\mathbf{a}}^{T}\mathbf{D}\mathbf{a}$ is strictly positive for every non-zero column vector $\mathbf{a}$. The nodal coordinate vectors, $\mathbf{x},\mathbf{y}$ and $\mathbf{z}$, of the free node can be then obtained. However, because of the existence of struts (${q}_{k}<0$) in the tensegrity structure, the force density matrix of the tensegrity structure is positive semi-definite. The sum of the elements of a row or a column of the force density matrix,

**D**, is always equal to zero for a free-standing tensegrity without fixed nodes. It is obvious that a vector $\overline{\mathbf{I}}={\{1,1,\cdots ,1\}}^{T}$$(\in {\mathbb{R}}^{n\times 1})$ is a solution of Equation (6).

**D**matrix (Equation (5)) equals four, the rank deficiency of the matrix is then three. Only full rank matrices have an inverse [25]. The force density matrices of tensegrity structures have at least d particular solutions except for the above vector $\overline{\mathbf{I}}$. Hence, the minimum rank deficiency of the

**D**matrix must be ($d+1$). Because the force density matrix is positive semi-definite, the

**D**matrix can be factorized as follows by an eigenvalue decomposition (EVD) [26]:

#### 2.2. Selection of Nodal Coordinates

**D**matrix, minimal length and non-zero length conditions can be used to select nodal coordinates from the candidates. For 3D system, the total squared length of the entire structure can be calculated by

#### 2.3. New Set of Force Densities from Equilibrium Matrix

**A**($\in {\mathbb{R}}^{dn\times b}$) is known as the equilibrium matrix, defined by

**A**is given in Table 2.

**A**is not square, another decomposition method is required. A singular value decomposition (SVD) makes it possible to solve the problem [22]. An SVD is a factorization of a real or complex matrix that generalizes the eigen decomposition of a square or any rectangular matrices.

**A**. The rank deficiency, ${h}_{A}$, of the equilibrium matrix is computed by

**A**in decreasing order as

**W**from Equation (17) can be expressed as

**A**is shown in Figure 3, where the notation $m\phantom{\rule{3.33333pt}{0ex}}=dn-rank\left(A\right)$ denotes inextensional mechanisms including both possible infinitesimal mechanisms and rigid body motions.

## 3. DNN-Based Form-Finding Method

#### 3.1. Deep Neural Network Architectures for Training

#### 3.1.1. Data Preparation

#### 3.1.2. Training Phase

#### 3.2. Optimization Algorithm

#### 3.3. Prediction for Lengths and Designation of Coordinates

## 4. Numerical Examples

#### 4.1. Two-Dimensional Two-Strut Tensegrity

#### 4.1.1. Data Preparation and Training Phase

#### 4.1.2. Optimization Algorithm

#### 4.1.3. Prediction for Lengths and Designation of Coordinates

#### 4.2. Three-Dimensional Truncated Tetrahedron

#### 4.2.1. Data Preparation and Training Phase

#### 4.2.2. Optimization Algorithm

#### 4.2.3. Prediction for Lengths and Designation of Coordinates

#### 4.3. Three-Dimensional-Truncated Icosahedron Tensegrity

#### 4.3.1. Data Preparation and Training Phase

#### 4.3.2. Optimization Algorithm

#### 4.3.3. Prediction for Lengths and Designation of Coordinates

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The total number of articles and papers on the Web of Science homepage. Results using subjects, ‘Tensegrity’ and ‘Tensegrity + Form-finding’.

**Figure 2.**A 2D two-strut tensegrity structure in which cables and struts are denoted by the thin and thick lines.

**Figure 4.**Flowchart of the conventional form-finding process of tensegrity structures using the FDM.

**Figure 5.**The DNN models with input and output layers denoted for the force densities and lengths of the elements, respectively.

**Figure 6.**Flowchart of the DNN-based form-finding process of tensegrity structures to eliminate the calculation of EVD and SVD.

**Figure 8.**The final results of 2D two-strut tensegrity structure using the (

**a**) FDM and (

**b**) DNN-based form-finding model with DE algorithm.

**Figure 11.**Comparison of the validation loss for various mini-batch sizes with log-scale for 3D-truncated tetrahedron tensegrity.

**Figure 12.**Comparison of the force densities of the 3D truncated tetrahedron tensegrity using an FDM, a DNN-based form-finding model with a DE algorithm and a analytical solution [6].

**Figure 13.**Relationship between the iteration and error for the 3D-truncated tetrahedron tensegrity using an FDM and a DNN-based form-finding method with a DE algorithm.

**Figure 14.**Three steps of the coordinate designation for the 3D-truncated tetrahedron tensegrity. (

**a**) Step 1: Nodes 1 to 6. (

**b**) Step 2: Nodes 7 to 9. (

**c**) Step 3: Nodes 10 to 12.

**Figure 15.**Final geometry of five iterations of the 3D truncated tetrahedron tensegrity using an FDM and a DNN-based form-finding model with a DE algorithm. (

**a**) FDM. (

**b**) DNN-based form-finding model.

**Figure 16.**3D-truncated icosahedron tensegrity in which the red thick lines denote strut members, and others are cable members. (

**a**) Geometry. (

**b**) Isometric view.

**Figure 17.**Connectivity of the 3D-truncated tetrahedron tensegrity in which the blue dotted lines denote vertical cables, and others represent edge cables.

**Figure 19.**Comparison of the validation loss for various hidden layer sizes with log-scale. The number of neurons is 512 for the 3D-truncated icosahedron tensegrity.

**Figure 21.**Comparison of the training loss and validation loss for various activation functions with log scale. Three hidden layers and 512 neurons were used for the 3D-truncated icosahedron tensegrity example. (

**a**) Training loss. (

**b**) Validation loss.

**Figure 23.**Rectangular coordinates of the 3D-truncated icosahedron tensegrity in a $2\times 2\times 2$ cube, where $a=(3+\sqrt{5})/6$, $b=(\sqrt{5}-1)/3$, and $c=\sqrt{5}/3$. (

**a**) Front view. (

**b**) Top view.

**Figure 24.**Comparison of the force densities of the 3D truncated icosahedron tensegrity using FDM, DNN-based form-finding model with DE algorithm and analytical solution [6].

**Figure 25.**Relationship between the iteration and error of the 3D truncated icosahedron tensegrity using an FDM and a DNN-based form-finding model with a DE algorithm.

**Figure 26.**Final geometry of eight iterations of the 3D-truncated icosahedron tensegrity module using an FDM and a DNN-based form-finding model with DE algorithm. (

**a**) FDM. (

**b**) DNN-based form-finding model.

Elements | Nodes | |||
---|---|---|---|---|

1 | 2 | 3 | 4 | |

1 | 1 | −1 | 0 | 0 |

2 | 0 | 1 | −1 | 0 |

3 | 0 | 0 | 1 | −1 |

4 | 1 | 0 | 0 | −1 |

5 | 1 | 0 | −1 | 0 |

6 | 0 | 1 | 0 | −1 |

$\mathit{n}\times \mathit{d}$ | Total Number of Elements (b) | |||||
---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | |

1 | ${x}_{1}-{x}_{2}$ | 0 | 0 | ${x}_{1}-{x}_{4}$ | ${x}_{1}-{x}_{3}$ | 0 |

2 | ${y}_{1}-{y}_{2}$ | 0 | 0 | ${y}_{1}-{y}_{4}$ | ${y}_{1}-{y}_{3}$ | 0 |

3 | $-({x}_{1}-{x}_{2})$ | ${x}_{2}-{x}_{3}$ | 0 | 0 | 0 | ${x}_{2}-{x}_{4}$ |

4 | $-({y}_{1}-{y}_{2})$ | ${y}_{2}-{y}_{3}$ | 0 | 0 | 0 | ${y}_{2}-{y}_{4}$ |

5 | 0 | $-({x}_{2}-{x}_{3})$ | ${x}_{3}-{x}_{4}$ | 0 | $-({x}_{1}-{x}_{3})$ | 0 |

6 | 0 | $-({y}_{2}-{y}_{3})$ | ${y}_{3}-{y}_{4}$ | 0 | $-({y}_{1}-{y}_{3})$ | 0 |

7 | 0 | 0 | $-({x}_{3}-{x}_{4})$ | $-({x}_{1}-{x}_{4})$ | 0 | $-({x}_{2}-{x}_{4})$ |

8 | 0 | 0 | $-({y}_{3}-{y}_{4})$ | $-({y}_{1}-{y}_{4})$ | 0 | $-({y}_{2}-{y}_{4})$ |

Parameter | Value |
---|---|

Bounds of variables | $[0,1]$ |

Population size | 20 |

Recombination rate | 0.9 |

Tolerance | ${10}^{-4}$ |

**Table 4.**The final results of the 2D two-strut tensegrity structure using the FDM and DNN-based form-finding model with the DE algorithm.

DE + FDM | DE + DNN | ||
---|---|---|---|

∣ f${}_{mean}$–f${}_{best}$ ∣ | 0.66 | 0.56 | |

Solution [cable, strut] | [1.00, −0.97] | [1.00, −1.00] | |

Force density * [cable, strut] | [1.00, −0.97] | [1.00, −1.00] | [1.00, −1.00] |

**Table 5.**The final results of the 3D-truncated tetrahedron tensegrity using an FDM and a DNN-based form-finding model with a DE algorithm.

DE + FDM | DE + DNN | |
---|---|---|

Iteration | 5 | 5 |

∣ f${}_{mean}$–f${}_{best}$ ∣ | 0.22 | 0.14 |

Solution [${q}_{e}$, ${q}_{v}$, ${q}_{s}$] | [0.96, 0.70, −0.38] | [1.00, 1.00, −0.58] |

Force density * [${q}_{e}$, ${q}_{v}$, ${q}_{s}$] | [1.00, 0.73, −0.39] | [1.00, 1.00, −0.58] |

Element | i | j | Element | i | j | Element | i | j |
---|---|---|---|---|---|---|---|---|

91 | 37 | 58 | 101 | 24 | 26 | 111 | 13 | 40 |

92 | 14 | 16 | 102 | 27 | 49 | 112 | 12 | 34 |

93 | 2 | 20 | 103 | 23 | 50 | 113 | 33 | 36 |

94 | 1 | 15 | 104 | 22 | 44 | 114 | 32 | 59 |

95 | 5 | 10 | 105 | 43 | 46 | 115 | 8 | 35 |

96 | 6 | 29 | 106 | 42 | 57 | 116 | 9 | 11 |

97 | 28 | 55 | 107 | 19 | 21 | 117 | 7 | 54 |

98 | 48 | 51 | 108 | 18 | 45 | 118 | 31 | 53 |

99 | 4 | 30 | 109 | 38 | 41 | 119 | 47 | 56 |

100 | 3 | 25 | 110 | 17 | 39 | 120 | 52 | 60 |

**Table 7.**Cartesian coordinates of the 3D-truncated icosahedron tensegrity using the connectivity shown in Figure 17, where $a=(3+\sqrt{5})/6$, $b=(\sqrt{5}-1)/3$, and $c=\sqrt{5}/3$.

No. | x | y | z | No. | x | y | z |
---|---|---|---|---|---|---|---|

1 | 0 | −b/2 | 1 | 31 | −b | a | 1/3 |

2 | −1/3 | −b | a | 32 | −b/2 | 1 | 0 |

3 | −b/2 | −c | 2/3 | 33 | −b | a | −1/3 |

4 | b/2 | −c | 2/3 | 34 | −c | 2/3 | −b/2 |

5 | 1/3 | −b | a | 35 | −c | 2/3 | b/2 |

6 | 0 | b/2 | 1 | 36 | −1 | 0 | −b/2 |

7 | 1/3 | b | a | 37 | −a | 1/3 | −b |

8 | b/2 | c | 2/3 | 38 | −2/3 | b/2 | −c |

9 | −b/2 | c | 2/3 | 39 | −2/3 | −b/2 | −c |

10 | −1/3 | b | a | 40 | −a | −1/3 | −b |

11 | −2/3 | −b/2 | c | 41 | −b/2 | −c | −2/3 |

12 | −2/3 | b/2 | c | 42 | −1/3 | −b | −a |

13 | −a | 1/3 | b | 43 | 0 | −b/2 | −1 |

14 | −1 | 0 | b/2 | 44 | 1/3 | −b | −a |

15 | −a | −1/3 | b | 45 | b/2 | −c | −2/3 |

16 | −b | −a | 1/3 | 46 | a | −1/3 | −b |

17 | −c | −2/3 | b/2 | 47 | 2/3 | −b/2 | −c |

18 | −c | −2/3 | −b/2 | 48 | 2/3 | b/2 | −c |

19 | −b | −a | −1/3 | 49 | a | 1/3 | −b |

20 | −b/2 | −1 | 0 | 50 | 1 | 0 | −b/2 |

21 | b | −a | 1/3 | 51 | c | 2/3 | b/2 |

22 | b/2 | −1 | 0 | 52 | c | 2/3 | −b/2 |

23 | b | −a | −1/3 | 53 | b | a | −1/3 |

24 | c | −2/3 | −b/2 | 54 | b/2 | 1 | 0 |

25 | c | −2/3 | b/2 | 55 | b | a | 1/3 |

26 | 2/3 | −b/2 | c | 56 | b/2 | c | −2/3 |

27 | a | −1/3 | b | 57 | 1/3 | b | −a |

28 | 1 | 0 | b/2 | 58 | 0 | b/2 | −1 |

29 | a | 1/3 | b | 59 | −1/3 | b | −a |

30 | 2/3 | b/2 | c | 60 | −b/2 | c | −2/3 |

**Table 8.**The final results of the 3D-truncated icosahedron tensegrity using an FDM and a DNN-based form-finding model with a DE algorithm.

DE + FDM | DE + DNN | |
---|---|---|

Iteration | 8 | 8 |

∣ f${}_{mean}$–f${}_{best}$ ∣ | 0.14 | 0.04 |

Solution [${q}_{e}$, ${q}_{v}$, ${q}_{s}$] | [1.00, 1.00, −1.00] | [1.00, 0.80, −0.35] |

Force density * [${q}_{e}$, ${q}_{v}$, ${q}_{s}$] | [1.00, 1.00, −1.00] | [1.00, 0.80, −0.35] |

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**MDPI and ACS Style**

Lee, S.; Lieu, Q.X.; Vo, T.P.; Lee, J.
Deep Neural Networks for Form-Finding of Tensegrity Structures. *Mathematics* **2022**, *10*, 1822.
https://doi.org/10.3390/math10111822

**AMA Style**

Lee S, Lieu QX, Vo TP, Lee J.
Deep Neural Networks for Form-Finding of Tensegrity Structures. *Mathematics*. 2022; 10(11):1822.
https://doi.org/10.3390/math10111822

**Chicago/Turabian Style**

Lee, Seunghye, Qui X. Lieu, Thuc P. Vo, and Jaehong Lee.
2022. "Deep Neural Networks for Form-Finding of Tensegrity Structures" *Mathematics* 10, no. 11: 1822.
https://doi.org/10.3390/math10111822