# Form-Finding of Spine Inspired Biotensegrity Model

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## Featured Application

## Abstract

## 1. Introduction

## 2. Geometrical Input of Spine

#### 2.1. Vertebrae Anatomical Parameters

#### 2.2. Natural Sagittal Curvature

## 3. Search Strategies for Spine Biotensegrity Models

#### 3.1. Geometry of the Spine Biotensegrity Model

#### 3.1.1. Definition of Basic Cell

_{o}-X

_{o}Y

_{o}Z

_{o}. The position vector of nodal coordinates is denoted as

**X**. The origins of the local Cartesian system O

_{i – 1}-X

_{i - 1}Y

_{i - 1}Z

_{i - 1}and O

_{i}-X

_{i}Y

_{i}Z

_{i}are positioned at the centroid of lower (

**X**

_{6i-5}

**X**

_{6i-4}

**X**

_{6i-3}) and upper triangular surfaces (

**X**

_{6i-2}

**X**

_{6i-1}

**X**

_{6i}) of ith (i = 1, 2, …, n) triangular cell, respectively.

#### 3.1.2. Definition of Basic Cell

_{i}B and L

_{i}T, respectively. The triangular cell that is stacked up on the current ith triangular cell is denoted as the (i + 1)th cell. All the elements are classified into specific groups. The struts in every stage are represented by bold lines. The saddle cables (red color) connecting the interface between two triangular cells are represented by irregular dash-dotted lines. Horizontal cables in regular dashed lines join the nodes only at the base and the top of the structures. There are three categories of diagonal cables; namely diagonal cables 1 (cyan color) connecting nodes L

_{i}B and L

_{i}

_{+ 1}B, diagonal cables 2 (pink color) connecting nodes L

_{i}T and L

_{i}

_{+ 1}T, and diagonal cables 3 (blue color) connecting nodes L

_{i}B and L

_{i}T. On top of that, reinforcing cables in zig-zag lines at the first stage are also included.

#### 3.1.3. Assemblage of Spine Biotensegrity Models

**X**

_{6i-5}and

**X**

_{6i-2}of ith triangular cell (Figure 5) are positioned on global Y axis (x = 0), prior to the twisting process (Figure 7a).

#### 3.2. Equilibrium Equations of Spine Biotensegrity Model

#### 3.2.1. Basic Assumptions

- Self-weight of struts and cables are neglected due to the reason that the stress resulted from self-weight is insignificantly small compared to the level of pre-stress [19].
- There is no external force acting on the models. It is necessary to ensure the self-equilibrium position as shown under the topology of biotensegrity model (Figure 7) is rational. Against any disturbance, the self-equilibrium and stability of the obtained topology should be retained. Analysis to study the capability of the established spine biotensegrity to undergo change in shape is not the scope of the study.

#### 3.2.2. Equilibrium Equations: Formulation

_{o}nodes, n

_{c}supports and n

_{u}= 3(n

_{o}-n

_{c}) unconstrained degree of freedoms. Consider an element k of the spine biotensegrity model connecting node i and j (i < j) in a Cartesian coordinate system O-XYZ as shown in Figure 8. The vectors of nodal coordinates of node i and j are denoted as ${\stackrel{-}{\mathrm{x}}}_{i}$ and ${\stackrel{-}{\mathrm{x}}}_{j}$, respectively.

**λ**of an element k is defined as follows:

**f**

_{i}and

**f**

_{j}, respectively (where

**f**

_{i}= -

**f**

_{j}), which are in equilibrium with n. Considering equilibriums at all nodes, the following equation can be obtained:

**B**is a n

_{u}× m matrix consisting of the vector of directional cosines

**λ**of all elements in the spine biotensegrity model with respect to x, y, z axes,

**F**is a vector of nodal forces with size n

_{u}and

**n**is a vector of axial forces for all elements with size m.

_{u}and

**B**

^{+}is Moore–Penrose generalized inverse of matrix

**B**. Singular value decomposition method is used in this study to compute the Moore–Penrose generalized inverse of the matrix

**B**with rank deficiency (i.e., rank < n

_{u}).

**F**=

**0**. Under such condition, the solution for the vector of axial forces

**n**for the spine biotensegrity model is given by the following equation:

**β**is a vector of arbitrary coefficient of size m.

**I**

_{m}−

**B**

^{+}

**B**] in Equation (6) as matrix

**G**of size m × m and applying eigenvector basis decomposition method, the real symmetric matrix

**G**can be expressed as follows:

**Λ**is a diagonal matrix whose diagonal elements Λi (i = 1, 2,…, m) correspond to eigenvalues of matrix

**G**;

**Φ**is an orthogonal matrix whose ith column

**Φ**

_{i}(i = 1, 2,…, m) corresponds to eigenvector associated with eigenvalue Λi. Since matrix

**G**is an orthogonal projection matrix, the eigenvalues Λ

_{i}returns the value of either 0 or 1. The eigenvectors

**Φ**

_{i}correspond to vectors of independent self-equilibrium stress modes for biotensegrity models considered in this study. It is noted that the number of vectors of independent self-equilibrium stress modes is equal to the number of non-zero eigenvalues of matrix

**G**or rank [

**G**].

**Φ**

_{i}

^{−1}as follows:

**n**can be obtained:

**β*** =

**Φ**

^{T}

**β**.

_{i}* is ith component of vector

**β***, which corresponds to vector of coefficients for the linearly independent self-equilibrium stress modes.

**n**is dependent on the choice of the combinations of coefficients β

_{i}*

**,**which might be more than one. In this study, only one of the possible combinations of coefficients β

_{i}

*****was used to determine the vector of axial forces

**n**.

#### 3.2.3. Equilibrium Equations: Solutions

**β**

_{i}

*****for an assumed configuration of spine biotensegrity model. The other is the modification of the assumed configuration until one consistent self-equilibrated configuration is found. For the determination of the trial set of β

_{i}*, an unconstrained minimization problem as shown in Equation (11) is adopted where the conjugate gradient method is used:

_{i}* (i = 1,2,3) determined by solving Equation (11) is used as the trial values to determine the possible solution for

**n**(see Equation (10)), which satisfies the inequality constraints given in Equations (12) and (13) using an iterative process:

_{c}, E

_{c}and A

_{c}are axial forces, yield stress and cross sectional area for cable elements, respectively; n

_{s}, E

_{s}, I

_{s}, L

_{s}, σ

_{s}and A

_{s}are axial forces, Young modulus, moment inertia of section, length, yield stress and cross sectional area for strut elements, respectively. In this study, A

_{c}= 3.14 mm

^{2}, A

_{s}= 50.30 mm

^{2}, E

_{s}= 200 GPa, and I

_{s}= 201 mm

^{4}.

#### 3.3. Algorithm for Form-Finding of Spine Biotensegrity Model

**n**satisfying self-equilibrium condition, as shown in Equation (10), is established. The eigenvector basis decomposition method is adopted to determine the independent self-equilibrium stress modes.

_{i}* through the minimization problem as shown in Equation (11). It is noted that the combination of coefficients β

_{i}* is searched through an iterative form-finding process to satisfy the Equations (12) and (13). After obtaining the combination of the coefficients β

_{i}*, the substitution of the coefficients β

_{i}* into Equation (10), gives the axial forces n in the self-equilibrium state.

_{i}* and nodal coordinates. As the objective function and the constraint conditions cannot be explicitly expressed by design variables, the procedure of form-finding implies an iteration. Both approaches involve only minor modification of nodal coordinates due to the priority of searching for models with close resemblance to the geometry of the human spine.

## 4. Search Results for Spine Biotensegrity Models

#### 4.1. Analysis Results Using Approach One

_{i}is defined by the twist angles for the i-th stage’s triangular cells. Arrangement of the spine biotensegrity models with α

_{i}$\ge $ 60° is found to be not feasible due to the problem of closely spaced struts. Figure 11 shows the form-finding analysis with the application of the same twist angle 10° to 50° for all the triangular cells in the spine biotensegrity models (i.e., cases 10101010, 20202020, 30303030, 40404040, and 50505050). The case 10101010 denotes the model as α

_{1}=α

_{2}= α

_{3}= α

_{4}= 10°. The cables in compression are slacked and should be removed. The slackened cables of SB1 are indicated by the red line (Figure 11). From the form-finding analysis, it is found that SB1 with α

_{1}= α

_{2}= α

_{3}= α

_{4}= 50° yields the least degree of slackness (9 slackened cables). Larger twist angles tend to reduce the number of slacked cables, but at α

_{1}= α

_{2}= α

_{3}= α

_{4}= 20°, the number of slacked cables is maximum. For 10° $\le $ α

_{i}$\le $ 40°, slackened cables are found to occur throughout the model. For α

_{i}= 50°, more slackened cables are found to occur at the first and third stages. The information about slackened cables at each trial case is an important input in the iterative analysis.

_{1}for the triangular cell at stage 1 is less than 50° lead to a lesser number of slackened cables in comparison with other combinations of α

_{2}, α

_{3}, α

_{4}(Figure 12a). A similar effect of such combinations of twist angles is found in SB2 and SB3. When α

_{1}changes while the other α

_{i}are held at 50°, the numbers of slackened cables ranged from three to nine for SB1, SB2, and SB3 are found (Figure 12b). In the model SB1 and SB3, the numbers of slackened cables are maximum at α

_{1}= 20°. In the case of α

_{1}= 10

^{o}in SB1 and α

_{1}= 40° in SB3, the number of slackened cables is minimum. On the other hand, SB2 is not sensitive to α

_{1}. In SB1, as α

_{2}and α

_{3}are assigned larger values, the number of slackened cables decreases. Figure 13 illustrates the numbers and position of slackened cables for parameter α

_{1}= 10°−40° and α

_{2}= α

_{3}= α

_{4}= 50° in SB1. At α

_{1}= 10°, number of slackened cables decreases to five and is uniformly located throughout the stages.

#### 4.2. Iterative Results Using Approach Two

#### 4.3. Spine Biotensegrity Models

_{i}are −1399.907, −2490.990, −131.972 in SB1, 786.626, −2855.643, 462.793 in SB2 and −1368.241, 1188.504, 74.294 in SB3. The above sets of coefficients beta β

_{i}(i = 1, 2, 3), are used to calculate the vector of axial forces (see Equation (10) that satisfy the inequality constraint represented by Equations (12) and (13). It is emphasized that the solution obtained for the self-equilibrated configuration for SB1, SB2, and SB3 represents one possible configuration with close resemblance to the geometry of the human spine. The self-equilibrated configuration could be further refined through solution of constrained optimization problem using maximum stiffness, minimum weight or other suitable criteria as objective functions.

_{i}*, the combinations of coefficients β

_{1}*, β

_{2}* and β

_{3}* that lead to tension in struts or compression in cables are rejected. Non-acceptance of the trial values of coefficients β

_{i}*(i = 1,2,3) obtained from solution of problem defined by Equation (11) is always due to non-satisfaction of Equation (12) (the existence of slackened cables) within the trial configuration of spine biotensegrity models.

## 5. Conclusions

^{o}. The slacking of cables is very sensitive to twist angles near the thoracic and lumbar region. The sensitivity to twist angle of the lowest cell is dependent on the shape of the cell. This study has successfully established for the first time the possible self-equilibrated configurations of biotensegrity models inspired by the human spine. Future works such as shape change analysis, building and controlling real biotensegrity models and conducting experiments considering external loading and operational challenges as well as formulation of self-collision and external collision of the spine biotensegrity models should be carried out. The outcome of such future works could then lead to the possible application of spine biotensegrity in area of robotic tools and deployable structures.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

A_{c} | Cross sectional area of cable |

A_{s} | Cross sectional area of strut |

B | Matrix of directional cosines |

B^{+} | Moore-Penrose generalized inverse of matrix B |

β* | Vector of arbitrary coefficients for linearly independent self-equilibrium stress modes |

D | Triangle altitude |

E_{s} | Young modulus of strut |

F | Vector of nodal forces |

Φ | Eigenvector associated with the eigenvalue Λ |

G | Matrix for existence of solution for Equation (2) (see Equation (7)) |

I_{s} | Moment inertia of section |

L_{s} | Length of strut |

L_{T} | Length of cable or strut element |

Λ | Eigenvalues of matrix G |

λ | Vector of directional cosines |

n | Vector of axial forces |

n_{c} | Axial forces of cable |

n_{s} | Axial forces of strut |

θ | Half of the vertex angle |

σ_{c} | Yield stress of cable |

σ_{s} | Yield stress of strut |

$\stackrel{-}{\mathrm{x}}$ | Vector of nodal coordinates |

## Appendix A

Nodes | SB1 | SB2 | SB3 | ||||||
---|---|---|---|---|---|---|---|---|---|

x | y | z | x | y | z | x | y | z | |

1 | 9.33 | −1.23 | 0 | 9.33 | -1.23 | 0 | 9.33 | −1.23 | 0 |

2 | −24.01 | −48.3 | 0 | −35.34 | −45.27 | 0 | −49.24 | -41.54 | 0 |

3 | 14.67 | −58.67 | 0 | 26 | −61.7 | 0 | 39.91 | −65.43 | 0 |

4 | −8.9 | 9.49 | 123.78 | −8.9 | 9.49 | 123.78 | −8.9 | 9.49 | 123.78 |

5 | −14.71 | −55.48 | 141.36 | −25.95 | −48.49 | 141.36 | −39.72 | −52.18 | 141.36 |

6 | 23.61 | −35.21 | 141.36 | 34.85 | −44.2 | 141.36 | 48.62 | −45.51 | 141.36 |

7 | −33.22 | -39 | 52.83 | −33.22 | −39 | 52.83 | −33.22 | −39 | 52.83 |

8 | 11.48 | −71.52 | 70.41 | 8.47 | −82.75 | 70.41 | 4.77 | −96.53 | 70.41 |

9 | 21.75 | −33.19 | 70.41 | 24.76 | −21.96 | 70.41 | 28.45 | 1.82 | 70.41 |

10 | −10.04 | 52.87 | 382.82 | −10.04 | 52.87 | 382.82 | −10.04 | 59.87 | 382.82 |

11 | −7.19 | 14.89 | 372.81 | −14.34 | 11.56 | 372.81 | −23.12 | 0.46 | 372.81 |

12 | 17.23 | 30.28 | 372.81 | 24.38 | 29.61 | 372.81 | 33.16 | 33.7 | 372.81 |

13 | 14.15 | 63.71 | 332.85 | 14.15 | 63.71 | 332.85 | 14.15 | 63.71 | 332.85 |

14 | −18.11 | 34.13 | 332.85 | −24.57 | 45.65 | 332.85 | −32.51 | 51.21 | 332.85 |

15 | 3.96 | 18.67 | 332.85 | 10.42 | 16.15 | 332.85 | 18.36 | 10.49 | 332.85 |

16 | −5.08 | 10.05 | 531.04 | −5.08 | 10.05 | 531.04 | −5.08 | 10.05 | 531.04 |

17 | −7.84 | −21.15 | 533.31 | −13.91 | −22.77 | 533.31 | −21.37 | −24.77 | 533.31 |

18 | 12.91 | −15.59 | 533.31 | 18.99 | −13.96 | 533.31 | 26.44 | −11.96 | 533.31 |

19 | −18.29 | 2.21 | 527.57 | −18.29 | −9.79 | 527.57 | −18.29 | −4.79 | 527.57 |

20 | 6.37 | −27.52 | 519.59 | 4.74 | −28.59 | 519.59 | 2.74 | −36.05 | 519.59 |

21 | 11.93 | −1.77 | 519.59 | 13.56 | 4.31 | 519.59 | 15.55 | 4.77 | 519.59 |

22 | −7.58 | 9.77 | 564.29 | −7.58 | 9.77 | 564.29 | −7.58 | 9.77 | 564.29 |

23 | −5.12 | −18.78 | 564.29 | −10.33 | −21.21 | 564.29 | −16.74 | −24.21 | 564.29 |

24 | 12.7 | −10.47 | 564.29 | 17.91 | −8.04 | 564.29 | 24.33 | −5.05 | 564.29 |

## References

- Levin, S.M. Continuous tension, discontinuous compression: A model for biomechanical support of the body. Bull. Struct. Integr.
**1982**, 8, 1. [Google Scholar] - Ingber, D.E.; Madri, J.A.; Jamieson, J.D. Role of basal lamina in neoplastic disorganization of tissue architecture. Proc. Natl. Acad. Sci. USA
**1981**, 78, 3901–3905. [Google Scholar] [CrossRef] [Green Version] - Stamenović, D.; Fredberg, J.J.; Wang, N.; Butler, J.P.; E Ingber, D. A Microstructural Approach to Cytoskeletal Mechanics based on Tensegrity. J. Theor. Boil.
**1996**, 181, 125–136. [Google Scholar] [CrossRef] - Wang, N.; Naruse, K.; Stamenović, D.; Fredberg, J.J.; Mijailovich, S.M.; Tolić-Nørrelykke, I.M.; Polte, T.; Mannix, R.; Ingber, D. Mechanical behavior in living cells consistent with the tensegrity model. Proc. Natl. Acad. Sci. USA
**2001**, 98, 7765–7770. [Google Scholar] [CrossRef] [Green Version] - Swanson, R.L. Biotensegrity: A unifying theory of biological architecture with applications to osteopathic practice, education, and research&x#2014;A review and analysis. J. Am. Osteopat. Assoc.
**2013**, 113, 34–52. [Google Scholar] [CrossRef] [Green Version] - Ingber, D.E.; Wang, N.; Stamenović, D. Tensegrity, cellular biophysics, and the mechanics of living systems. Rep. Prog. Phys.
**2014**, 77, 046603. [Google Scholar] [CrossRef] [Green Version] - Oh, C.L.; Choong, K.K.; Low, C.Y. Biotensegrity Inspired Robot–Future Construction Alternative. Procedia Eng.
**2012**, 41, 1079–1084. [Google Scholar] [CrossRef] [Green Version] - Lian, O.C.; Choong, K.K.; Nishimura, T.; Kim, J.Y.; Hassanshahi, O. Shape change analysis of tensegrity models. Lat. Am. J. Solids Struct.
**2019**, 16. [Google Scholar] [CrossRef] [Green Version] - Panjabi, M.M. The Stabilizing System of the Spine. Part I. Function, Dysfunction, Adaptation, and Enhancement. J. Spinal Disord.
**1992**, 5, 383–389. [Google Scholar] [CrossRef] - Levin, S.M. The tensegrity-truss as a model for spine mechanics: Biotensegrity. J. Mech. Med. Boil.
**2002**, 2, 375–388. [Google Scholar] [CrossRef] - Flemons, Tom. Available online: https://tensegritywiki.com/w/index.php?title=Flemons,_Tom&oldid=1315 (accessed on 21 July 2020).
- Chen, T.-J.; Wu, C.-C.; Su, F.-C. Mechanical models of the cellular cytoskeletal network for the analysis of intracellular mechanical properties and force distributions: A review. Med. Eng. Phys.
**2012**, 34, 1375–1386. [Google Scholar] [CrossRef] - Ingber, D.E. Cellular tensegrity: Defining new rules of biological design that govern the cytoskeleton. J. Cell Sci.
**1993**, 104, 613–627. [Google Scholar] - Fraldi, M.; Palumbo, S.; Carotenuto, A.R.; Cutolo, A.; Deseri, L.; Pugno, N.M. Buckling soft tensegrities: Fickle elasticity and configurational switching in living cells. J. Mech. Phys. Solids
**2019**, 124, 299–324. [Google Scholar] [CrossRef] [Green Version] - Tibert, A.G.; Pellegrino, S. Review of Form-Finding Methods for Tensegrity Structures. Int. J. Space Struct.
**2011**, 26, 241–255. [Google Scholar] [CrossRef] - Zhang, Q.; Wang, X.; Cai, J.; Zhang, J.; Feng, J. Closed-Form Solutions for the Form-Finding of Regular Tensegrity Structures by Group Elements. Symmetry
**2020**, 12, 374. [Google Scholar] [CrossRef] [Green Version] - Chen, Y.; Yan, J.; Feng, J.; Sareh, P. A hybrid symmetry–PSO approach to finding the self-equilibrium configurations of prestressable pin-jointed assemblies. Acta Mech.
**2020**, 231, 1485–1501. [Google Scholar] [CrossRef] - Arcaro, V.; Adeli, H. Form-finding and analysis of hyperelastic tensegrity structures using unconstrained nonlinear programming. Eng. Struct.
**2019**, 191, 439–446. [Google Scholar] [CrossRef] - Ohsaki, M.; Zhang, J. Nonlinear programming approach to form-finding and folding analysis of tensegrity structures using fictitious material properties. Int. J. Solids Struct.
**2015**, 69, 1–10. [Google Scholar] [CrossRef] - Cai, J.; Feng, J. Form-finding of tensegrity structures using an optimization method. Eng. Struct.
**2015**, 104, 126–132. [Google Scholar] [CrossRef] - Maddio, P.D.; Meschini, A.; Sinatra, R.; Cammarata, A. An optimized form-finding method of an asymmetric large deployable reflector. Eng. Struct.
**2019**, 181, 27–34. [Google Scholar] [CrossRef] - Lee, S.; Gan, B.S.; Lee, J. A fully automatic group selection for form-finding process of truncated tetrahedral tensegrity structures via a double-loop genetic algorithm. Compos. Part B Eng.
**2016**, 106, 308–315. [Google Scholar] [CrossRef] - Li, X.; Kong, W.; He, J. A task-space form-finding algorithm for tensegrity robots. IEEE Access
**2020**, 2020, 1. [Google Scholar] [CrossRef] - Busscher, I.; Ploegmakers, J.J.W.; Verkerke, G.J.; Veldhuizen, A.G. Comparative anatomical dimensions of the complete human and porcine spine. Eur. Spine J.
**2010**, 19, 1104–1114. [Google Scholar] [CrossRef] [Green Version] - Mejia, E.A.; Hennrikus, W.L.; Schwend, R.M.; Emans, J.B. A prospective evaluation of idiopathic left thoracic scoliosis with magnetic resonance imaging. J. Pediatr. Orthop.
**1996**, 16, 354–358. [Google Scholar] [CrossRef] - Been, E.; Kalichman, L. Lumbar lordosis. Spine J.
**2014**, 14, 87–97. [Google Scholar] [CrossRef] - Anatomy of the Spine. Available online: https://www.sonsa.org/spine-surgery/spine-anatomy/ (accessed on 21 July 2020).
- Roussouly, P.; Pinheiro-Franco, J.L. Sagittal parameters of the spine: Biomechanical approach. Eur. Spine J.
**2011**, 20, 578–585. [Google Scholar] [CrossRef] [Green Version] - Masharawi, Y.; Salame, K.; Mirovsky, Y.; Peleg, S.; Dar, G.; Steinberg, N.; Hershkovitz, I. Vertebral body shape variation in the thoracic and lumbar spine: Characterization of its asymmetry and wedging. Clin. Anat.
**2007**, 21, 46–54. [Google Scholar] [CrossRef] - Knox, J.B.; Lonner, B.S. Sagittal Balance. In Minimally Invasive Spinal Deformity Surgery; Springer: Berlin/Heidelberg, Germany, 2014; pp. 33–37. [Google Scholar]
- Vrtovec, T.; Pernuš, F.; Likar, B. A review of methods for quantitative evaluation of spinal curvature. Eur. Spine J.
**2009**, 18, 593–607. [Google Scholar] [CrossRef] [Green Version] - Skelton, R.E.; de Oliveira, M.C. Tensegrity Systems; Springer: Berlin/Heidelberg, Germany, 2009. [Google Scholar]
- Hangai, Y.; Kawaguchi, K. General Inverse and Its Application to Shape Finding Analysis; Baifukan: Tokyo, Japan, 1991. [Google Scholar]

**Figure 2.**Typical vertebra parameters [24].

**Figure 5.**Definition for the typical triangular cell and triangular surfaces for spine biotensegrity models.

**Figure 7.**Assemblage of a spine biotensegrity model (

**a**) fitting of triangular surfaces perpendicular to the established curvature line, (

**b**) twisting of cells shown in elevation and plan view, (

**c**) incorporation of struts and cables according to the element connectivity pattern.

**Figure 8.**A three-dimensional pinned jointed spine biotensegrity model element with the nodal forces.

**Figure 10.**Methods used to generate trial configurations of spine biotensegrity model: (

**a**) approach one and (

**b**) approach two.

**Figure 12.**Effect of various combinations of twist angles on number of slackened cables for (

**a**) SB1 and (

**b**) SB1, SB2 and SB3.

**Figure 13.**Effect of various twist angles for triangular cell at stage one on number of slackened cables in SB1.

**Figure 14.**Effect of different modifications of nodal coordinates on the number of slackened cables in SB1.

**Figure 15.**Self-equilibrated configuration for spine biotensegrity models (

**a**) SB1, (

**b**) SB2 and (

**c**) SB3. Note: (+/−7) denotes either positive or negative displacement of 7 mm at the selected node.

Modification of Nodal Coordinates (mm) in Y-Direction for Cases A1-H1 | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Case | Nodes | Case | Nodes | ||||||||||||||||||||||||

6 | 20 | 15 | 12 | 13 | 14 | 5 | 4 | 19 | 6 | 20 | 15 | 12 | 13 | 14 | 5 | 4 | 19 | ||||||||||

A1 | −10 | −5 | −7 | 3 | −3 | −7 | 0 | 0 | 0 | E1 | 0 | −5 | −7 | 4 | −3 | −7 | −10 | 0 | 0 | ||||||||

B1 | 0 | −5 | −7 | 3 | −3 | −7 | −10 | 0 | 0 | F1 | 0 | −5 | −7 | 4 | −3 | −7 | −10 | 0 | 2 | ||||||||

C1 | 0 | −5 | −7 | 3 | −3 | −7 | −15 | 0 | 0 | G1 | 0 | −5 | −7 | 4 | 0 | −7 | −10 | 0 | 2 | ||||||||

D1 | 0 | −5 | −7 | 3 | −3 | −7 | −10 | −5 | 0 | H1 | 0 | −5 | −7 | 4 | 0 | −7 | −10 | 0 | 7 | ||||||||

Axial Force (N) for Case 30505050 and Cases A1-H1 | |||||||||||||||||||||||||||

Cable | 30505050 | A1 | B1 | C1 | D1 | E1 | F1 | G1 | H1 | ||||||||||||||||||

40 | −7.73 | 1.65 | 3.94 | 10.00 | 10.00 | 5.00 | 10.00 | 5.00 | 6.07 | ||||||||||||||||||

42 | 10.00 | −2.74 | 0.11 | 1.58 | −2.24 | 1.93 | −7.74 | 1.85 | 0.79 | ||||||||||||||||||

50 | −8.82 | 18.54 | 13.10 | 17.25 | 18.81 | 13.31 | 13.63 | 10.88 | 3.82 | ||||||||||||||||||

51 | −1.11 | −2.27 | −2.77 | −4.06 | −3.73 | −5.45 | −4.13 | −4.44 | 0.03 | ||||||||||||||||||

57 | −28.56 | −2.52 | −1.69 | −1.44 | −2.10 | 8.42 | 12.57 | 14.38 | 22.26 | ||||||||||||||||||

58 | 12.93 | −8.27 | −8.49 | −12.13 | −11.63 | −5.71 | 0.45 | 6.28 | 30.40 | ||||||||||||||||||

59 | −6.35 | 0.43 | 0.68 | 0.93 | 0.85 | −0.59 | 0.16 | −1.07 | 4.27 | ||||||||||||||||||

65 | −5.27 | −13.28 | −9.38 | −12.76 | −13.66 | −14.13 | −11.70 | −12.91 | 0.03 | ||||||||||||||||||

66 | 10.00 | −5.29 | −5.64 | −8.06 | −7.67 | −3.06 | 1.97 | 5.88 | 24.13 | ||||||||||||||||||

67 | −2.89 | −1.41 | −0.81 | −0.10 | 0.05 | 0.13 | −0.60 | 0.03 | 0.03 | ||||||||||||||||||

68 | −7.86 | 10.00 | 6.65 | 10.91 | 11.04 | 2.91 | 10.00 | 3.09 | 6.57 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chai Lian, O.; Kok Keong, C.; Nishimura, T.; Jae-Yeol, K.
Form-Finding of Spine Inspired Biotensegrity Model. *Appl. Sci.* **2020**, *10*, 6344.
https://doi.org/10.3390/app10186344

**AMA Style**

Chai Lian O, Kok Keong C, Nishimura T, Jae-Yeol K.
Form-Finding of Spine Inspired Biotensegrity Model. *Applied Sciences*. 2020; 10(18):6344.
https://doi.org/10.3390/app10186344

**Chicago/Turabian Style**

Chai Lian, Oh, Choong Kok Keong, Toku Nishimura, and Kim Jae-Yeol.
2020. "Form-Finding of Spine Inspired Biotensegrity Model" *Applied Sciences* 10, no. 18: 6344.
https://doi.org/10.3390/app10186344