Enumeration–screening method for the design of simple polygonal tensegrities

Abstract

Tensegrities, consisting of axially pre-compressed bars and pre-stretched strings, hold broad applications in the design of, for instance, architectures, soft robotics and metamaterials. In this paper, we propose an enumeration–screening method to design planar tensegrities of simple polygonal shapes. In such a polygonal tensegrity, the strings are joined pair-wise to form a simple polygon (a planar shape consisting of straight, non-intersecting line segments) and only one bar is added at each node. The total number of simple polygonal tensegrities designed by this scheme increases exponentially with the number of bars. Moreover, we demonstrate that each of these designed topologies can produce a self-equilibrated and stable tensegrity configuration. This work helps understand the topological features of simple polygonal tensegrities, which can be used as elementary cells to design some novel two- and three-dimensional tensegrity structures.

1. Introduction

As a class of pre-stressed structures, tensegrity comprises a discontinuous set of axially compressed bars and a continuous set of axially stretched strings [1]. Owing to their unique architectural and mechanical features, tensegrity structures have found a wide variety of technologically important applications, ranging from light-weight architectures, advanced mechanical devices, functional structural metamaterials, to molecular/cellular biomechanical models [2–6].

With the increasing demand for new types of tensegrities, their topological design arouses the interest of many researchers. Through a systematic study of the topological characteristics of prismatic tensegrities, Zhang & Ohsaki [3] derived a new type of structures, called star-shaped tensegrities [7,8]. By combining the ground structure method and mixed integer programming, Ehara & Kanno [9] proposed a novel optimization-based method for topological design of tensegrities. Thereby they obtained some new polyhedral tensegrities and their assemblies. Furthermore, Kanno [10–12] extended this method to the topology optimization under self-weight and external loads/constrains. Recently, Xu et al. [13,14] proposed an improved method based on Kanno's studies [10–12] and constructed tensegrity topologies with more than one bar at a single node. Using the force density method combined with a genetic algorithm, Lee & Lee [15,16] developed an automatic search method for tensegrity topological design. As a freeform design method, Tachi [17] proposed a polygonal mesh-based scheme by converting the surface meshes of a topology into a bar-and-string network. In addition, large-scale tensegrities can be assembled from various shapes of elementary cells [18–20].

Recently, planar or two-dimensional (2D) tensegrities have been proposed as popular prototypes in various scenarios and widely applied in the design of active and passive mechanical systems. Much effort has been directed toward understanding their static and kinematic properties under different boundary and loading conditions [21–23]. By embedding sensors and actuators into their components (either bars or strings), 2D tensegrities can be designed as intelligent robots with variable stiffness for vibration control, motion bionics and energy harvesting [24–26]. They have been also exploited to design novel compliant systems, for instance multi-stable devices, self-similar structures and mechanical metamaterials [27–29]. In addition, some biomechanical models have been proposed on the basis of 2D tensegrities to investigate the dynamic behaviour of proteins, cells, tissues and organs [30–32]. To date, however, only three main types of 2D tensegrities have been reported: quadrilateral, hexagonal and octagonal [33–35]. This limitation of structural types hinders their further applications, since some required functions may not be achieved by the existing classical 2D structures.

In the present paper, therefore, we propose an enumeration–screening method to construct 2D tensegrities with simple polygonal topologies (the simplest but commonly used type of planar geometries), which consist of straight and non-intersecting edges that are joined pair-wise to form a closed path. These simple polygonal tensegrities have the following topological features: the strings are connected end-to-end at the nodes to form a simple polygon, and each node has only one bar to exactly fulfil Pugh's definition of tensegrity [36]. This paper is organized as follows. We first describe the topological design method of simple polygonal tensegrities in §2. Then we give some examples of tensegrities designed by this method, and discuss their structural topologies and form-finding configurations in §3. Finally, the main conclusions are summarized in §4.

2. Topological design

The structural topology of a tensegrity defines the connectivity of its nodes, strings and bars, but it does not contain any size informations (e.g. the lengths of bars and strings). The tensegrity topology is generally described either by a connectivity matrix or a geometric graph, and the latter will be used in this paper. For clear illustration, we will draw the topological graphs of simple polygonal tensegrities following regular polygons. The topologies of all simple polygonal tensegrities can be constructed by our enumeration–screening procedure described in the sequel.

(a). Enumeration

First, we generate all possible topological graphs by using an enumeration scheme. The topological graph of a simple polygonal tensegrity can be drawn as follows. Its nodes and strings are located at the vertices and edges of a regular polygon. Then the bars are added following two basic rules: (i) each bar connects two un-neighbouring nodes to avoid the overlapping of strings and (ii) neither of two bars has shared nodes to comply with the strict definition of tensegrity [36]. The polygon should have an even number of edges, and thus a simple polygonal tensegrity with n bars possesses 2n nodes and 2n strings [19]. The number n is the only parameter we need for the topological design.

Let the nodes be numbered from 1 to 2n in the counter–clockwise direction and the bars from 1 to n according to their order of addition. The starting and ending nodes of the ith bar is denoted as s_i and e_i (i = 1, 2, … , n), respectively. s_i is specified to be the minimum number of the remaining, unused nodes, while e_i can be a number of the remaining unused nodes from s_i + 2 to n + 1 (i = 1) or 2n (i = 2, 3, … , n). This scheme is formulated as

$$s_1=1,e_1=\{3,4,\dots ,n+1\},$$ 2.1

and

$$\begin{array}{r}\begin{array}{l}{s}_i=min(\{s_{i-1}+1,\dots ,2n\}\mathrm{\setminus }{\{e_j\}}_{1\le j<i})\\ e_i=\{s_i+2,\dots ,2n\}\mathrm{\setminus }{\{e_j\}}_{1\le j<i}\end{array}(i=2,3,\dots ,n)\end{array}\},$$ 2.2

where the symbol ‘∖’ denotes the elimination of the numbers at its right side from the number set at its left side. Here the flip operation of the topological graph is considered to shrink the value range of the ending node e₁ of the first bar. As shown in figure 1, the starting and ending nodes of the five bars in each decagonal graph are marked to illustrate the order of the bar linking operation.

Figure 1.

Three decagonal topological graphs. The graph in (a) can be overlapped by that in (b) via rotation, or by that in (c) via flip. The orders of adding bars are indexed by the subscripts of nodal marks s_i and e_i. The numbers on dashed circles are nodal indices defined by equations (2.3) and (2.4) for uniqueness screening. Here and in the sequel, the thick and thin solid lines of topological graphs denote bars and strings, respectively. (Online version in colour.)

Using equations (2.1) and (2.2), one can readily enumerate all topological graphs that may generate simple polygonal tensegrities. The total numbers of the obtained graphs consisting of two to nine bars are calculated, respectively, as tabulated in table 1.

Table 1.

Total number of topological graphs obtained in §2a–c with different numbers of bars.

	total number of topological graphs
number of bars	all casesa	topologically unique casesb	mechanically feasible casesc	ratio between the numbers of feasible and unique graphs (%)
2	1	1	1	100
3	3	2	2	100
4	19	7	6	85.7
5	168	29	25	86.2
6	1849	196	165	84.2
7	24 108	1788	1542	86.2
8	362 671	21 994	19 343	87.9
9	6 181 035	326 115	292 482	89.7

^aAll cases refer to the graphs enumerated by using equations (2.1) and (2.2) in §2a.

^bTopologically unique cases refer to the graphs selected from all graphs via the uniqueness screening in §2b.

^cMechanically feasible cases refer to the topologically unique graphs satisfying the mechanical feasibility condition in §2c.

(b). Uniqueness screening

Though each graph obtained in §2a has a unique group of node numbers defined by equations (2.1) and (2.2), some of these graphs actually correspond to an identical topology. For example, three decagonal graphs shown in figure 1 have different nodal pairs denoted by s_i and e_i, indicating the different orders of their adding bars. However, it can be seen that the graph in figure 1a can be overlapped by that in figure 1b via rotation about the normal axis off the plane by 2π/5, or by that in figure 1c via flip about the vertical vector in the plane. Therefore, the three graphs in figure 1 in fact have the same topology and are called isomorphic in this work. There exist several routines to judge the graph isomorphism, which can be described by a group of abstract mathematical expressions [37]. Here, we propose a simple scheme for graph isomorphism judgement, which consists of two steps of operations: grouping and screening.

We formulate a set of nodal indices to identify the duplication of graphs and then choose one of them as the delegate. The indices of the paired nodes connected by one bar are defined by the subtraction of their numbers, that is

$$\begin{array}{r}\begin{array}{l}\mathrm{\Gamma }(s_i)=e_i-s_i-{\delta }_{i}n\\ \mathrm{\Gamma }(e_i)=-\mathrm{\Gamma }(s_i)\end{array}(i=1,\dots ,n)\end{array}\},$$ 2.3

where the coefficient δ_i is

$${\delta }_i=\{\begin{array}{ll}0& \text{if}\hspace{0.17em}{e}_i-s_i<n,\\ 1& \text{if}\hspace{0.17em}{e}_i-s_i=n,\\ 2& \text{if}\hspace{0.17em}{e}_i-s_i>n.\end{array}$$ 2.4

Note that the condition e_i > s_i is preset by equations (2.1) and (2.2). The first and third cases in equation (2.4) produce − n < $\mathrm{\Gamma }$(e_i) < 0 < $\mathrm{\Gamma }$(s_i) < n and − n < $\mathrm{\Gamma }$(s_i) < 0 < $\mathrm{\Gamma }$(e_i) < n, respectively, which ensure the span of non-diametral bars in a small semicircle and along the counter–clockwise direction from its positive nodal index to the negative one. In the second case, one has $\mathrm{\Gamma }$(s_i) = $\mathrm{\Gamma }$(e_i) = 0 that makes the span of diametral bars non-directional. These operations avoid possible confusions caused by the rotation congruent transformation (e.g. figure 1a,b). The second case also renders all nodal indices in the opposite signs when the graph takes a flip transformation (e.g. figure 1a,c). Therefore, for two isomorphic topological graphs, they will take the same nodal indices in the same order along the counter–clockwise direction, or, alternatively, their nodal indices will be opposite and in the reverse order (e.g. figure 1). This will be used as the basic criterion of uniqueness screening operation to screen the graphs.

The above criterion exactly screens the topologically unique graphs, but its direct application is quite time-consuming and troublesome. To improve the calculation efficiency, we introduce a grouping index $\pi$ to divide these graphs into several groups, and then apply the criterion to each group. Here, a sum number of 2n exponential functions is suggested as an example

$$\mathrm{\Pi }=\sum _{I=1}^{2n}{a}^{\mathrm{\Gamma }(I)-\mathrm{\Gamma }(I-1)}$$ 2.5

with the setting $\mathrm{\Gamma }$(0) = $\mathrm{\Gamma }$(2n), where the base a > 1 is an integer (e.g. a = 2), and the exponent is calculated by the subtraction of two neighbouring nodal indices defined in equations (2.3) and (2.4). The topological graphs with the same index in equation (2.5) are considered as a group. In each group, the first graph is picked out as the delegate, and the rest are then compared with it by using the basic criterion of uniqueness screening. This process will stop when all graphs have been proved isomorphic; otherwise, the operation repeats for a new group formed by the non-isomorphic graphs with respect to the delegated one in their original group. Finally, we gather the delegated graphs selected by the above operation as the topologically unique graphs.

The above grouping–checking scheme shows a high efficiency for the uniqueness screening of our constructed topological graphs. Following this step, the duplicates are discarded and each polygonal topology is finally described by a single graph. The topologically unique graphs are greatly less than those obtained in §2a, as listed in table 1.

(c). Evaluation of mechanical feasibility

A stable tensegrity should be in a complementary equilibrium state of pre-stretched strings and pre-compressed bars. In what follows, this mechanical principle will be used to check the feasibility of the obtained topologies such that the self-equilibrium of the tensegrity configurations can be guaranteed. In a simple polygonal tensegrity, the end-to-end strings form a closed polygon, restricting the possible relative separation of the nodes and the pre-tensions of the strings take non-zero components in any radial direction. To suppress the approaching motions of nodes, the bars should also be pre-compressed such that they have continuous projections in all directions. A regular polygon has a certain number of symmetric axes through the midpoints of the paired edges, by which we can define a condition for screening the mechanical feasibility of simple polygonal topologies as follows. If the projections of bars fill the internal segments of all these symmetric axes in its graph, the topology is considered to be mechanically feasible; otherwise, it is unfeasible and will be ruled out. Note that this is a necessary condition to find a self-equilibrated tensegrity and, in other words, a topology violating this condition does not correspond to a tensegrity in self-equilibrium. Taking the decagonal topology in figure 2 as an example, the projections of its five bars take an unfilled region in the internal segment of vertical axis (figure 2a) and thus this topology is mechanically unfeasible. In this case, one can divide the structure into two parts, top and bottom, without any connecting bars between them. When the two parts approach each other along the vertical direction, the connecting strings will be slack (figure 2b), indicating that this tensegrity is impossible to hold a self-equilibrated state. Additionally, the above condition could not be sufficient for the self-equilibrium of a simple polygonal tensegrity, because only the polygonal symmetric axes have been considered in checking the projections of bars. To be sufficient, it needs further examination of the bar projections in all directions in the plane.

Figure 2.

A mechanically unfeasible decagonal topology. (a) The projections of its five bars take an unfilled region along the vertical axis. (b) The connecting strings become slack when the top and bottom parts approach each other. (Online version in colour.)

The condition of mechanical feasibility screening is formulated as follows. For a topological graph consisting of 2n strings, there exist n projection axes that can be used to examine the projections of its n bars. These axes are numbered from J = 1 to n, each of which passes the midpoints of two paired strings. Firstly, we analyse the perpendicular projections of the nodes in the graphs. The Jth axis starts from the midpoint of the string linking nodes J and J + 1, and ends at the midpoint of the string linking nodes J + n and J + n + 1. On this axis, the projections ξ in the range of 1 ≤ ξ ≤ J are held by the paired nodes J + ξ and J + 1 − ξ, and the other projections ξ in the range of J + 1 ≤ ξ ≤ n are held by the paired nodes J + ξ and J + 1 + 2n − ξ. Note that the latter case will vanish when J = n. The nodes from 1 to 2n take n projections, dividing the axis into n − 1 parts. The above projection relations of the nodes are illustrated in the diagram in figure 3.

Figure 3.

Diagram of nodal projections on the Jth projection axis. The numbers in the circles indicate the projection points. (Online version in colour.)

Next, the projections of bars, defined by the segments linking the perpendicular projections of their starting and ending nodes, can be determined on each projection axis. According to the projection relation of nodes described above, the projection of the ith bar on the Jth axis is the segment between the projections of its starting node s_i and ending node e_i,

$$\xi (I)=\{\begin{array}{ll}J+1-I& \text{if}\hspace{0.17em}1\le I\le J\\ I-J& \text{if}\hspace{0.17em}J<I\le J+n\\ J+1+2n-I& \text{if}\hspace{0.17em}J+n<I\le 2n\end{array}(I=s_i,e_i;1\le J\le n).$$ 2.6

For example, two decagonal topological graphs and their projection axes are shown in figure 4, in which the projections of all bars are illustrated by segments.

Figure 4.

Decagonal topological graphs that are (a) mechanically feasible and (b) unfeasible. All projection axes in (a) are filled by the projections of bars, while in (b), there exists an unfilled region on the fifth axis. (Online version in colour.)

Finally, the mathematical union of the projection segments of bars on each projection axis is calculated and compared with the entire range of the axis. When all projection axes are filled by the corresponding projections of bars, the topological graph is regarded as a feasible result that can produce a self-equilibrated tensegrity; otherwise, it will be ruled out. Thus, the decagonal topology in figure 4a is mechanically feasible, while that in figure 4b should be excluded because it has an unfilled region on the projection axis numbered by J = 5. It should be noticed that in figure 4b, there exist two isolated parts (nodes 1, 2, 9 and 10 form a part, and nodes 3–8 form another) without any connecting bars between them. The relative approaching motions of these isolated parts cannot be restricted (figure 2), and the corresponding structure is impossible to satisfy the self-equilibrium conditions of tensegrity.

After operating this step, one can determine the mechanically feasible graphs that describe the topologies of all simple polygonal tensegrities. The total numbers of these topologies consisting of two to nine bars are listed in table 1. We also calculate the ratio between the number of mechanically feasible graphs and the number of topologically unique graphs. The ratio keeps 100% for two or three bars, but varies in the range of 84–90% when the number of bars is between 4 and 9.

In the above design process, we employ integer comparisons, selections and additions as major operations to reduce the mathematical complexity and improve the computational efficiency. Additionally, it is worth mentioning that although the structural topologies in this section are built on the basis of regular polygon-like graphs, our design method also works when these topologies are expressed by irregular graphs. A tensegrity topology is defined only by the connectivity of its nodes, strings and bars. When the lengths of the components are varied in a graph, the corresponding topology remains unchanged. Thus, before employing our proposed operations for a topology taking an irregular polygon, one merely needs to transform it into a regular morphology by adjusting the geometric sizes of edges.

3. Simple polygonal tensegrities

(a). Structural topologies

Now we can enumerate the topologies of all simple polygonal tensegrities using the topological design method proposed in §2. Let m denote the total number of the structural topologies consisting of n bars. The points determined by the paired numbers (m, n) are plotted in figure 5, and can be well fitted by an exponential function

$$m=\mathrm{int}\{\exp (0.176n^2-0.125n-0.499)\},$$ 3.1

where the symbol $\mathrm{int}\{\}$ denotes integer operation. For comparison, the factorial function (n − 1)!, a typical function describing rapid growth, is also depicted in figure 5. It can be clearly seen that the total number of structural topologies increases rapidly with the number of bars, even faster than the rate of the factorial growth. Equation (3.1) shows that the topologies consisting of 10 bars may be close to the magnitude order of 10 million. Thus, the present method can supply a huge number of candidates for simple polygonal tensegrities.

Figure 5.

Total number of simple polygonal topologies (m) versus the number of bars (n) described by solid circles. Their relation can be fitted by an exponential function. The stars refer to factorial integers (n − 1)!. (Online version in colour.)

All structural topologies of the simple polygonal tensegrities consisting of two to five bars are illustrated in Figures 6a–9a, respectively. With the increase in the number of bars, the topological graphs become more complicated. All graphs with two, three and four bars can hold symmetric transformations of rotation or flip (figure 6a–8a), while some graphs with five bars cannot (figure 9a(21)–9a(25)). We have obtained more types of topologies can be obtained in comparison with some previous schemes (e.g. [33–35]).

Figure 7.

Two simple hexagonal tensegrities: (a) structural topologies and (b) form-finding configurations. (Online version in colour.)

Figure 6.

A simple quadrilateral tensegrity: (a) structural topology and (b) form-finding configuration. (Online version in colour.)

Figure 9.

Twenty-five simple decagonal tensegrities: (a) structural topologies and (b) form-finding configurations. (Online version in colour.)

Figure 8.

Six simple octagonal tensegrities: (a) structural topologies and (b) form-finding configurations. (Online version in colour.)

(b). Structural configurations

We proceed to determine the self-equilibrated configurations of simple polygonal tensegrities by using the stiffness-matrix based form-finding method [38]. For the form-finding analysis, the input parameters include the structural topology, the natural lengths and the axial stiffnesses of components (i.e. bars and strings). The topology defines the connectivity relation of nodes, bars and strings, as well as their total numbers. The assigned lengths of components should make strings under tension and bars under compression in finally solved configuration, which can be ensured by shortening the strings or lengthening the bars. The specified lengths and stiffnesses of the components can also adjust the final morphology of tensegrity. The following assignment scheme is taken as an example. Consider a specific morphology with the nodes distributed uniformly on a circle of unit radius. The natural lengths of the bars and strings are then specified as 105 and 95% of their lengths in this configuration, respectively. The axial stiffnesses of bars and strings are set to be 10 and one, respectively.

After the above input data have been specified, the form-finding analysis is implemented to determine the tensegrity configuration in self-equilibrium. Taking the topologies in figures 6a–9a as examples, their corresponding configurations are shown in figure 6b–9b, respectively. Most of these tensegrities finally take morphologies like regular polygons, while some others do not (e.g. figure 8b(5), 9b(9) and 9b(22)–9b(25)). These tensegrity configurations are all proved to be stable, because the positive definiteness of their structural stiffness matrices can be confirmed by the form-finding method [38]. Additionally, it is verified that all the strings are under pre-tension while all the bars are under pre-compression in each of the above configurations. This indicates that these configurations are globally rigid and thus also stable [39,40]. It is noticed that the tensegrities in figures 6b–9b may have intersections of bars, which can be avoided by using narrow sheets as bars or fabricating a local arc at the centre of bars [23].

Therefore, each simple polygonal topology studied in this section finally produces a self-equilibrated and stable tensegrity. Though the mechanical feasibility condition in §2c is defined on the basis of some polygonal symmetric axes rather than the consideration of all directions, it can also ensure the existence of a self-equilibrated state for each tensegrity in figures 6–9. For other simple polygonal tensegrities, the sufficiency of this condition needs further verification.

4. Concluding remarks

In this paper, we have developed an enumeration–screening method to design simple polygonal tensegrities. The total number of these tensegrities increases exponentially with the number of bars that is the only parameter we need for the topological design. Furthermore, some novel 2D and 3D tensegrities can be constructed by using these planar tensegrities as elementary cells following the structural assembly schemes in, for example, [41–43]. This work provides a basic route to construct many types of tensegrity structures that may be applied in mechanical metamaterials, intelligent robots and biomechanics.

Finally, it is noticed that due to the employment of integer comparisons, selections and additions as major operations, the present enumeration–screening design method shows a high computational efficiency in determining a structural topology. It may take long time to enumerate all topologies consisting of a large number of bars. However, one usually does not need to determine all topologies but only a limited number of them, and thus this method is still efficient. The computational cost of this procedure mainly depends on the required number of topologies rather than the number of bars.

Data accessibility

Source code of our method for designing polygonal tensegrities with, for instance, seven bars is available in the electronic supplementary material.

Authors' contributions

L.Y.Z., G.K.X. and X.Q.F. designed the study. L.Y.Z., Y.L. and G.K.X. implemented the method and constructed the tensegrities. All authors analysed the results and wrote the manuscript.

Competing interests

We declare we have no competing interests.

Funding

This work was supported by the National Natural Science Foundation of China (grant nos 11872106, 11672227 and 11502125), the Fundamental Research Funds for the Central Universities of China (grant no. FRF-TP-18-006A3) and the State Key Laboratory of Structural Analysis for Industrial Equipment (grant no. GZ18101).