# Closed-Form Solutions for the Form-Finding of Regular Tensegrity Structures by Group Elements

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Force Density Method and Self-Equilibrium Condition

#### 2.1. Assumptions

- (a)
- Members are connected using pin joints;
- (b)
- No external loads are included, and the gravity is ignored;
- (c)
- Member failure is not considered: i.e., struts do not yield and cables do not bear compressive forces;
- (d)
- The structure is self-equilibrated and free-standing without any support;
- (e)
- The topology (connectivity between the nodes and members) of the tensegrity structure is known, and the geometrical configuration of the structure can be described in terms of nodal coordinates only.

#### 2.2. Force Density Method

_{k}and l

_{k}are the force and length of member k, respectively.

**E**as the force density matrix, which can be written directly from the force densities q

_{k}[31,50] as

**E**can be written as

**x**,

**y**, and

**z**are the coordinates of the structural system. In other words, the equilibrium condition of tensegrity structures can be expressed in terms of force densities.

#### 2.3. Non-Degeneracy Condition

**E**is always square and symmetric, and positive-definite if all members are in tension. So, for cable-nets, which only possess positive values of force densities,

**E**is invertible, and nodal coordinates can be uniquely determined by solving the linear Equation (3) [18]. However, there are struts in tensegrity structures, which means negative elements of

**E**. So, matrix

**E**is invariably singular with rank deficiency. Define the rank deficiency h of

**E**as

**E**should satisfy the following condition

## 3. Stable Conditions

**d**is any nodal displacement vector, and

**K**is the tangent stiffness matrix. It should be notable that the rigid-body motions are not included in

**d**. The tangent stiffness matrix

**K**of a tensegrity structure can be divided into two parts

**K**

^{E}is the linear stiffness matrix, and

**K**

^{G}is the geometrical stiffness matrix. Substitute Equation (7) into Equation (6)

- (i)
- the geometry matrix is full rank;
- (ii)
- the matrix
**E**has (d+1) nonzero eigenvalues; - (iii)
- the matrix
**E**is positive semi-definite;

**E**is not positive semi-definite, geometrical stiffness matrix

**K**

^{G}will have negative eigenvalues and become dominant over the linear stiffness matrix

**K**

^{E}at sufficiently large force densities in the elements. So the tangent stiffness matrix

**K**fails to be positive definite, and the structure becomes unstable. Condition (ii) guarantees the self-equilibrium condition. Conditions (ii) and (iii) can be examined by the total number of zero-eigenvalues and the sign of the minimum eigenvalue respectively after eigenvalue decomposition of the force density matrix. According to Zhang [38], condition (i) will be satisfied in most cases if the structure is divisible.

## 4. Form-Finding Method

#### 4.1. Assumptions

_{1}, q

_{2}, q

_{3}, ...; otherwise, we use letters, i.e., q

_{s}, q

_{b}, ... Moreover, the number of variables of force densities can be reduced by making use of symmetry properties.

#### 4.2. Equilibrium Analysis

**E**. With the force densities assumed, the force density matrix

**E**can be obtained according to Equation (2). Then the eigenvalues λ

_{1}, λ

_{2}, λ

_{3}, ... λ

_{n}of matrix

**E**can be determined by solving the equation [38]

**I**is the unit matrix. The expanded form of Equation (9) is shown as

_{0}(·), A

_{1}(·), ..., A

_{n-1}(·) denote the polynomial functions in terms of the force densities and Equation (10) can be obtained by eigenvalue decomposition.

- Step 1: Calculate the rank deficiency h of the force density matrix
**E**(Equation (4)). - Step 2: Notice the number of expressions that are equal to zero and record it as t
_{1}. Usually, there is only one expression of eigenvalue equal to zero after the decomposition. We assume this eigenvalue as λ_{1}. - Step 3: Find all the expressions of eigenvalues that are definitely greater than zero and denote the number of eigenvalues as t
_{2}. - Step 4: Select a few of the rest expressions of eigenvalues equal to zero. Record the number of expressions that are assumed to zero as t
_{3}, and t_{3}+ t_{1}≥ h should be satisfied. - Step 5: Get the relationship between force densities from the equations in Step 4.

#### 4.3. Super-Stability Examination

_{3}+ t

_{1}) zero eigenvalues, t

_{2}positive eigenvalues and (m-t

_{1}-t

_{2}-t

_{3}) uncertain eigenvalues. If all of the uncertain eigenvalues are not less than zero, the structure is super stable. N1, N2, N3, ... represent nodes while M1, M2, ... denote members. A planar tensegrity structure, three types of prismatic tensegrity structures (triangular prism, quadrangular prism, pentagonal prism) and a star-shaped tensegrity structure will be analyzed in the next section.

## 5. Examples

#### 5.1. Planar Tensegrity

_{1}, q

_{2}, and q

_{s}. q

_{1}represents the force density of cable elements M1, M3, and M5; q

_{2}denotes the force density of cable elements M2, M4, and M6; q

_{s}indicates the force density of struts. Then the force density matrix

**E**can be given as

_{s}is a negative value. Assume that λ

_{2}is greater than zero. Therefore, λ

_{3}and λ

_{5}are always greater than zero. Then assume λ

_{4}and λ

_{6}equal to zero, so relationship among q

_{1}, q

_{2}, and q

_{s}is obtained as

_{1}is positive, q

_{2}must be greater than -q

_{s}. q

_{1}is known by Equation (13), and therefore λ

_{2}is greater than 0. In conclusion, λ

_{2}, λ

_{3}, and λ

_{5}in Equation (12) are greater than zero while the other three are equal to zero, which satisfies the super-stability condition of tensegrity structures.

#### 5.2. Three-dimensional prismatic tensegrity structures

#### 5.2.1. Triangular Prism

_{1}, q

_{2}, q

_{b}, and q

_{s}. q

_{1}represents the force density of cable element M1, M2, M3; q

_{b}denotes the force density of M4, M5, M6; q

_{2}denotes the force density of M7, M8, M9, and q

_{s}denotes the force density of struts. Then the force density matrix

**E**can be given as

**E**in Equation (14) are shown as

_{1}, q

_{2}, q

_{b}are all positive values, and q

_{s}is negative because of the compression in struts. Set λ

_{2}equal to zero to satisfy the non-degeneracy condition. Then, we have

_{3}and λ

_{5}are always greater than zero. Set λ

_{4}and λ

_{6}also equal to zero. Thus, the relationship among q

_{1}, q

_{2}, q

_{b}, and q

_{s}is obtained as

_{3}and λ

_{5}in Equation (15) are greater than zero; the other eigenvalues in Equation (15) are all equal to zero so that the structure also satisfies the super-stability condition for tensegrity structures. The results are found the same with analytical solutions derived by Pellegrino and Tibert [27]. The relationship between variables q

_{1}, q

_{2}, and q

_{s}are plotted in the three-dimensional vector graph, shown in Figure 4.

#### 5.2.2. Quadrangular Prism Tensegrity Structure

_{1}, q

_{2}, q

_{b}, and q

_{s}. q

_{1}represents the force density of elements M1, M2, M3, M4; q

_{2}represents the force density of M5, M6, M7, M8; q

_{b}represents the force density of M9, M10, M11, M12; and q

_{s}represents that of struts. Then the force density matrix

**E**can be given as

**E**in Equation (19) are

_{1}, q

_{2}and q

_{b}, the force densities of cable members, are all positive values, and q

_{s}, the force density of struts, is negative. Presume that λ

_{2}equals to zero. Then

_{3}, λ

_{5}and λ

_{7}are always greater than zero. Thus, the sign of λ

_{1}, λ

_{2}, λ

_{3}, λ

_{5}, and λ

_{7}are determined. Set λ

_{6}and λ

_{8}or λ

_{4}equal to zero.

_{6}and λ

_{8}equal to zero, the relationship among q

_{1}, q

_{2}, q

_{b}, and q

_{s}is obtained as

_{4}

_{1}, λ

_{2}, λ

_{6}, and λ

_{8}, are zero while the other eigenvalues are greater than zero. So the super-stability condition for tensegrity structures is satisfied.

_{4}equal to zero, the relationship between variables by substituting Equation (21) is shown as

_{6}and λ

_{8}

_{1}, q

_{2}, and q

_{s}are plotted in the three-dimensional vector graph, shown in Figure 6.

#### 5.2.3. Quadrangular Prism Tensegrity Structure

_{1}, q

_{2}, q

_{b}, and q

_{s}; q

_{1}represents the force density of elements M1, M2, M3, M4, M5; q

_{2}represents the force density of M11, M12, M13, M14, M15; q

_{b}represents the force density of M6, M7, M8, M9, M10; and q

_{s}represents the force density of struts. Then the force density matrix

**E**can be given as

**E**in Equation (25) are shown as

_{1}, q

_{2}, q

_{b}are all positive values, λ

_{3}, λ

_{5}, λ

_{7}, and λ

_{9}are always greater than zero. However, the sign of λ

_{4}, λ

_{6}, λ

_{8}and λ

_{10}cannot be determined yet.

_{4}and λ

_{8}equal to zero, the relationship among q

_{1}, q

_{2}, q

_{s}is obtained as

_{6}and λ

_{10}equal to zero, the relationship among q

_{1}, q

_{2}, q

_{s}is obtained as

_{4}and λ

_{8}are shown as

_{1}, λ

_{2}, λ

_{6}, and λ

_{10}, are zero while the other eigenvalues are greater than zero. So the super-stability condition for tensegrity structures is satisfied. The relationship between variables q

_{1}, q

_{2}and q

_{s}are plotted in the three-dimensional vector graph, shown in Figure 8.

#### 5.3. Star-Shaped Tensegrity Structure

_{1}, q

_{2}, q

_{b}, and q

_{s.}q

_{1}represents the force density of elements M1, M2, M3; q

_{2}represents the force density of elements M4, M5, M6; q

_{b}represents the force density of elements M7, M8, M9, M10; and q

_{s}represents that of struts. Then the force density matrix

**E**can be given as

**E**in Equation (33) are shown as

_{2}equal to zero. We obtain

_{1}, q

_{2}, q

_{b}are all positive value, λ

_{3}, λ

_{4}, λ

_{5}, and λ

_{7}are always greater than zero. Set both λ

_{6}and λ

_{8}equal to zero, the relationship among q

_{1}, q

_{2}, q

_{s}is obtained as

_{1}, q

_{2}, and q

_{s}are plotted in the three-dimensional vector graph, shown in Figure 10.

## 6. Conclusions

**E**are zero (two-dimensional tensegrities and three-dimensional tensegrities, respectively). A closed-form solution of the force densities is induced by setting the necessary number of eigenvalues to zero. Then, the super-stability conditions for tensegrity structures are examined.

**E**, a closed-form solution can be conjectured according to the non-degeneracy condition. The closed-form solution can help in understanding the design of regular tensegrity structures in many fields, both in robotics and architecture, as well as deployable structures. Though the proposed method is efficient enough to solve the examples mentioned above, more advanced methods should be developed for form-finding of more complex tensegrities, which will be studied in the future.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Veenendaal, D.; Block, P. An overview and comparison of structural form finding methods for general networks. Int. J. Solids Struct.
**2012**, 49, 3741–3753. [Google Scholar] [CrossRef] [Green Version] - Ali, N.B.H.; Rhode-Barbarigos, L.; Albi, A.A.P. Design optimization and dynamic analysis of a tensegrity-based footbridge. Eng. Struct.
**2010**, 32, 3650–3659. [Google Scholar] - Fu, F. Non-linear static analysis and design of Tensegrity domes. Steel Compos. Struct.
**2006**, 6, 417–433. [Google Scholar] [CrossRef] [Green Version] - Li, P.C.; Wu, M.E. Stabilities of cable-stiffened cylindrical single-layer latticed shells. Steel Compos. Struct.
**2017**, 24, 591–602. [Google Scholar] - Cai, J.G.; Zhang, Q.; Feng, J.; Lee, S.H. Structural evaluation of a foldable cable-strut structure for kinematic roofs. Steel Compos. Struct.
**2018**, 29, 669–680. [Google Scholar] - Tibert, A.G.; Pellegrino, S. Deployable tensegrity reflectors for small satellites. J. Spacecr. Rocket.
**2002**, 39, 701–709. [Google Scholar] [CrossRef] [Green Version] - Li, T.; Jiang, J.; Deng, H. Form-finding methods for deployable mesh reflector antennas. Chin. J. Aeronaut.
**2013**, 26, 1276–1282. [Google Scholar] [CrossRef] [Green Version] - Ingber, D.E. Tensegrity I: Cell structure and hierarchical systems biology. J. Cell Sci.
**2003**, 116, 1157–1173. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Stamenović, D. Effects of cytoskeletal prestress on cell rheological behavior. Acta Biomater.
**2005**, 1, 255–262. [Google Scholar] [CrossRef] [PubMed] - Cai, J.G.; Yang, R.G.; Wang, X.Y.; Feng, J. Effect of initial imperfections of struts on the mechanical behavior of tensegrity structures. Compos. Struct.
**2019**, 207, 871–876. [Google Scholar] [CrossRef] - Connelly, R.; Back, A. Mathematics and Tensegrity: Group and representation theory make it possible to form a complete catalogue of “strut-cable” constructions with prescribed symmetries. Am. Sci.
**1998**, 86, 142–151. [Google Scholar] [CrossRef] - Bliss, T.; Werly, J.; Iwasaki, T. Experimental validation of robust resonance entrainment for cpg-controlled tensegrity structures. IEEE Trans. Control Syst. Technol.
**2013**, 21, 666–678. [Google Scholar] [CrossRef] - Li, Y.; Feng, X.Q.; Cao, Y.P. A Monte Carlo form-finding method for large scale regular and irregular tensegrity structures. Int. J. Solids Struct.
**2010**, 47, 1888–1898. [Google Scholar] [CrossRef] [Green Version] - Cai, J.; Feng, J. Form-finding of tensegrity structures using an optimization method. Eng. Struct.
**2015**, 104, 126–132. [Google Scholar] [CrossRef] - Xu, X.; Wang, Y.; Luo, Y. Finding member connectivities and nodal positions of tensegrity structures based on force density method and mixed integer nonlinear programming. Eng. Struct.
**2018**, 166, 240–250. [Google Scholar] [CrossRef] - Schek, H.J. The force density method for form finding and computation of general networks. Comput. Methods Appl. Mech. Eng.
**1974**, 3, 115–134. [Google Scholar] [CrossRef] - Aboul-Nasr, G.; Mourad, S.A. An extended force density method for form finding of constrained cable nets. Case Stud. Struct. Eng.
**2015**, 3, 19–32. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.Y.; Ohsaki, M. Adaptive force density method for form-finding problem of tensegrity structures. Int. J. Solids Struct.
**2006**, 43, 5658–5673. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.Y.; Ohsaki, M.; Kanno, Y. A direct approach to design of geometry and forces of tensegrity systems. Int. J. Solids Struct.
**2006**, 43, 2260–2278. [Google Scholar] [CrossRef] [Green Version] - Sánchez, J.; Serna, M.Á.; Morer, P.A. Multi-step force–density method and surface-fitting approach for the preliminary shape design of tensile structures. Eng. Struct.
**2007**, 29, 1966–1976. [Google Scholar] [CrossRef] - Tran, H.C.; Lee, J. Advanced form-finding of tensegrity structures. Comput. Struct.
**2010**, 88, 237–246. [Google Scholar] [CrossRef] - Feng, X.; Guo, S. A novel method of determining the sole configuration of tensegrity structures. Mech. Res. Commun.
**2015**, 69, 66–78. [Google Scholar] [CrossRef] - Motro, R.; Najari, S.; Jouanna, P. Shell and Spatial Structures: Computational Aspects; Lecture Notes in Engineering 26; Springer: Berlin, Germany, 1987. [Google Scholar]
- Zhang, L.; Maurin, B.; Motro, R. Form-Finding of Nonregular Tensegrity Systems. J. Struct. Eng.
**2006**, 132, 1435–1440. [Google Scholar] [CrossRef] - Pagitz, M.; Tur, J.M.M. Finite element based form-finding algorithm for tensegrity structures. Int. J. Solids Struct.
**2009**, 46, 3235–3240. [Google Scholar] [CrossRef] - Ohsaki, M.; Zhang, J.Y. Nonlinear programming approach to form-finding and folding analysis of tensegrity structures using fictitious material properties. Int. J. Solids Struct.
**2015**, 69–70, 1–10. [Google Scholar] [CrossRef] - Pellegrino, S.; Tibert, A.G. Review of Form-Finding Methods for Tensegrity Structures. Int. J. Space Struct.
**2003**, 18, 209–223. [Google Scholar] - Juan, S.H.; Tur, J.M.M. Tensegrity frameworks: Static analysis review. Mech. Mach. Theory
**2008**, 43, 859–881. [Google Scholar] [CrossRef] [Green Version] - Motro, R. Tensegrity: Structural Systems for the Future; Elsevier: San Diego, CA, USA, 2003. [Google Scholar]
- Koohestani, K.; Guest, S.D. A new approach to the analytical and numerical form-finding of tensegrity structures. Int. J. Solids Struct.
**2013**, 50, 2995–3007. [Google Scholar] [CrossRef] [Green Version] - Connelly, R.; Terrell, M. Globally rigid Symmetric Tensegrities. Struct. Topol.
**1995**, 21, 59–78. [Google Scholar] - Crane, C.D.; Duffy, J.; Correa, J.C. Static Analysis of Tensegrity Structures. J. Mech. Des.
**2005**, 127, 257–268. [Google Scholar] [CrossRef] [Green Version] - Murakami, H.; Nishimura, Y. Static and dynamic characterization of regular truncated icosahedral and dodecahedral tensegrity modules. Int. J. Solids Struct.
**2001**, 38, 9359–9381. [Google Scholar] [CrossRef] - Raj, R.P.; Guest, S.D. Using Symmetry for Tensegrity Form-Finding. J. Int. Assoc. Shell Spat. Struct.
**2006**, 47, 245–252. [Google Scholar] - Zhang, J.Y.; Guest, S.D.; Ohsaki, M. Symmetric prismatic tensegrity structures: Part I. Configuration and stability. Int. J. Solids Struct.
**2009**, 46, 1–14. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.Y.; Guest, S.D.; Ohsaki, M. Symmetric prismatic tensegrity structures. Part II: Symmetry-adapted formulations. Int. J. Solids Struct.
**2009**, 46, 15–30. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.Y.; Ohsaki, M. Self-equilibrium and stability of regular truncated tetrahedral tensegrity structures. J. Mech. Phys. Solids
**2012**, 60, 1757–1770. [Google Scholar] [CrossRef] - Zhang, L.Y.; Li, Y.; Cao, Y.P. Self-equilibrium and super-stability of truncated regular polyhedral tensegrity structures: A unified analytical solution. Proc. R. Soc. A
**2012**, 468, 3323–3347. [Google Scholar] [CrossRef] [Green Version] - Zhang, L.Y.; Li, Y.; Cao, Y.P. A unified solution for self-equilibrium and super-stability of rhombic truncated regular polyhedral tensegrities. Int. J. Solids Struct.
**2013**, 50, 234–245. [Google Scholar] [CrossRef] [Green Version] - Zhang, L.Y.; Zhu, S.X.; Li, S.X.; Xu, G.K. Analytical form-finding of tensegrities using determinant of force-density matrix. Compos. Struct.
**2018**, 189, 87–98. [Google Scholar] [CrossRef] - Estrada, G.G.; Bungartz, H.J.; Mohrdieck, C. Numerical form-finding of 2D tensegrity structures. Int. J. Solids Struct.
**2006**, 43, 6855–6868. [Google Scholar] [CrossRef] [Green Version] - Masic, M.; Skelton, R.E.; Gill, P.E. Algebraic tensegrity form-finding. Int. J. Solids Struct.
**2005**, 42, 4833–4858. [Google Scholar] [CrossRef] - Burkhardt, R.; Solutions, T. The Application of Nonlinear Programming to the Design and Validation of Tensegrity Structures with Special Attention to Skew Prisms. J. -Int. Assoc. Shell Spat. Struct.
**2006**, 150, 3. [Google Scholar] - Connelly, R. Rigidity and energy. Invent. Math.
**1982**, 66, 11–33. [Google Scholar] [CrossRef] - Micheletti, A.; Williams, W.O. A marching procedure for form-finding for tensegrity structures. J. Mech. Mater. Struct.
**2007**, 2, 857–882. [Google Scholar] [CrossRef] [Green Version] - Xu, X.; Luo, Y. Form-finding of nonregular tensegrities using a genetic algorithm. Mech. Res. Commun.
**2010**, 37, 85–91. [Google Scholar] [CrossRef] - Yamamoto, M.; Gan, B.S.; Fujita, K.A. Genetic Algorithm Based Form-Finding for Tensegrity Structure. Procedia Eng.
**2011**, 14, 2949–2956. [Google Scholar] [CrossRef] [Green Version] - Koohestani, K. Form-finding of tensegrity structures via genetic algorithm. Int. J. Solids Struct.
**2012**, 49, 739–747. [Google Scholar] [CrossRef] [Green Version] - Seunghye, L.; Buntara, S.G.; Jaehong, L. 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] - Maxwell, J.C. On the calculation of the equilibrium and stiffness of frames. Philos. Mag.
**1864**, 27, 294–299. [Google Scholar] [CrossRef] - Calladine, C.R. Buckminster Fullews “tensegrity” structures and Clerk Maxwell’s rules for the construction of stiff frames. Int. J. Solids Struct.
**1978**, 14, 161–172. [Google Scholar] [CrossRef] - Deng, H.; Kwan, A. Unified classification of stability of pin-jointed bar assemblies. Int. J. Solids Struct.
**2005**, 42, 4393–4413. [Google Scholar] [CrossRef] - Guest, S. The stiffness of prestressed frameworks: A unifying approach. Int. J. Solids Struct.
**2006**, 43, 842–854. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.Y.; Ohsaki, M. Stability conditions for tensegrity structures. Int. J. Solids Struct.
**2007**, 44, 3875–3886. [Google Scholar] [CrossRef] [Green Version] - Chen, Y.; Feng, J.; Sun, Q.Z. Lower-order symmetric mechanism modes and bifurcation behavior of deployable bar structures with cyclic symmetry. Int. J. Solids Struct.
**2018**, 139–140, 1–14. [Google Scholar] [CrossRef] - Chen, Y.; Sareh, P.; Feng, J.; Sun, Q.Z. A computational method for automated detection of engineering structures with cyclic symmetries. Comput. Struct.
**2017**, 191, 153–164. [Google Scholar] [CrossRef] - Zhang, J.Y.; Ohsaki, M. Tensegrity Structures: Form, Stability, and Symmetry; Mathematics for Industry 6; Springer: Berlin, Germany, 2016. [Google Scholar]
- Estrada, G.G. Analytical and Numerical Investigations of Form-finding Methods for Tensegrity Structures. Ph.D. Thesis, University of Stuttgart, Stuttgart, Germany, 2007. [Google Scholar]

**Figure 2.**Two-dimensional rotary symmetry planar tensegrity structure with six cables (black lines) and three struts (green lines).

**Figure 3.**Triangular prism rotary symmetry tensegrity structure with nine cables (black lines) and three struts (green lines).

**Figure 4.**Vector graph of the solutions for the relationship between variables q

_{1}, q

_{2}, and q

_{s}of the triangular prism rotary symmetry tensegrity structure.

**Figure 5.**Quadrangular prism rotary symmetry tensegrity structure with 12 cables (black lines) and 4 struts (green lines).

**Figure 6.**Vector graph of the solutions for the relationship between variables q

_{1}, q

_{2}, and q

_{b}of the quadrangular prism rotary symmetry tensegrity structure.

**Figure 7.**Pentagonal prism rotary symmetry tensegrity structure with 15 cables (black lines) and 5 struts (green lines).

**Figure 8.**Vector graph of the solutions for the relationship between variables q

_{1}, q

_{2}, and q

_{b}of the pentagonal prism rotary symmetry tensegrity structure.

**Figure 9.**Star-shaped rotary symmetry tensegrity structure with nine cables (black lines) and three struts (green lines).

**Figure 10.**Vector graph of the solutions for the relationship between variables q

_{1}, q

_{2}, and q

_{b}of the star-shaped rotary symmetry tensegrity structure.

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## Share and Cite

**MDPI and ACS Style**

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.
https://doi.org/10.3390/sym12030374

**AMA Style**

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(3):374.
https://doi.org/10.3390/sym12030374

**Chicago/Turabian Style**

Zhang, Qian, Xinyu Wang, Jianguo Cai, Jingyao Zhang, and Jian Feng.
2020. "Closed-Form Solutions for the Form-Finding of Regular Tensegrity Structures by Group Elements" *Symmetry* 12, no. 3: 374.
https://doi.org/10.3390/sym12030374