Analytical equations for the connectivity matrices and node positions of minimal and extended tensegrity plates

Abstract

Tensegrity structures are three-dimensional networks of truss members loaded in tension or compression. The location of the end points of the truss members, denoted as the nodes, and the associated node-member connectivity matrices are the fundamental descriptors in the modeling and design of tensegrity structures. This paper presents systematic analytical formulas for such node locations and connectivity matrices for tensegrity plates of two different topologies. The formulas apply to plates of any thickness, diameter, and complexity. As application examples, dynamic simulations demonstrating a strategy for morphing the planar plates toward domes are studied. The presented formulas allow for efficient computations and can be employed in the numerical analysis and design of shape-controllable antennas and mirrors, architectural constructions, and other applications based on tensegrity plate and dome-like structures.

Keywords

3 bar prism connectivity matrices node positions tensegrity tensegrity dome tensegrity plate

Introduction

A tensegrity structure is a three-dimensional (3D) network of truss members loaded in tension (denoted as strings or cables) or compression (denoted as bars or struts). The truss members are assumed to be connected through frictionless ball joints, so as to not impose torques among adjacent members. Many studies have shown that by systematically arranging these truss members, one can design structures that provide optimal properties such as minimal mass to support a set of applied forces¹ or absorb and dissipate mechanical energy.^2,3 Due to these characteristics, tensegrity structures are being widely studied in civil engineering,^4-7 robotics,⁸ and aerospace engineering.^9,10 In this work, plates and domes formed by the repetition of a tensegrity unit known as the 3 bar prism are studied. Tensegrity structures formed by this prism have been the subject of several studies including form-finding¹¹ and static and dynamic analyses.^12–15 Domes formed by 3 bar prism can be obtained from planar tensegrity plates by shortening the strings on either side of the plate. For example, Chan et al.¹⁶ experimentally demonstrated the ability of a flat tensegrity plate formed by seven prisms to morph toward a dome via string length control. Tensegrity domes¹⁷ have been employed to create satellite reflectors¹⁸ and large civil structures.^7,19 In the analysis and design of tensegrity structures, the position of the end points of its truss members, denoted as the nodes, and the associated node-member connectivity matrices are the fundamental descriptors, particularly for topology optimization. Despite the many studies on tensegrity plates in the literature, systematic sets of analytical expressions for the position of their nodes and the node-member connectivity matrices are still not available. Specifying the connectivity matrices is equivalent to characterizing a particular “weave” when making basket, cloth, or net, except that the compressive members involved in the tensegrity “material” allow 3D “fabrics” to be designed. Characterizing the connectivity matrices still does not specify the geometrical shape. To fully define the geometry of a tensegrity structure, the connectivity matrices along with the node locations must be provided.

This work derives analytical formulations for the node positions and connectivity matrices of tensegrity plates with two different topologies. In the first topology, the plates are formed by minimal 3 bar prisms, where “minimal” indicates that the prisms have the minimum of number of strings required to generate a pre-stressable stable structure. The second topology considers plates formed by new 3 bar prisms that include additional members to potentially increase the stiffness in directions that have large compliance in conventional minimal prisms. To form plates, the prisms are arranged in concentric rings, where the number of concentric rings forming the plate is termed as the complexity. The derived analytical expressions for the node positions and connectivity matrices are applicable to plates of any complexity. The Supplemental Material of this article includes MATLAB^® scripts that output the derived node positions and connectivity matrices for tensegrity plates given their size parameters and complexity.

Few of the plethora of tensegrities explored in the literature have a common connectivity rule,²⁰ which makes the task of designing tensegrity structures for specific applications a very challenging task. Among those few tensegrity topologies that posses a common connectivity rule, tensegrity plates are some of the most used in engineering applications as previously mentioned. Analytical formulations for the node positions and connectivity matrices of tensegrity plates would allow for efficient computations and facilitate future research efforts in their analysis and design. Although one can use computer-aided design (CAD) software to create complex topologies through operations such as patterned repetition and mirror, and thus can generate node positions and connectivity matrices for tensegrities, analytical expressions provide the following advantages:

The analytical formulas allow the user to change the shape and topology of the plates by simply changing the value of a handful of scalar parameters such as plate thickness, diameter, and complexity.

The analytical formulas can be readily written as scripts for the importation of the plate geometry and topology into CAD software; just as in the Supplemental Material of this article that includes MATLAB scripts that output the derived node positions and connectivity matrices of the plates.

The analytical expressions readily allow for the distinction of bars and strings and subsets of bars and strings by systematically arranging those subsets of members along the connectivity matrices. This is not the case in most CAD software where the user cannot systematically adjust the ordering of the connectivity matrices, or this requires significant efforts from the user. The same argument applies to the ordering of the nodes.

The aforementioned ordering of the connectivity and node position matrices readily allows the user to alter or eliminate specific subsets of bars or strings, assign properties to specific subsets of members, select a specific subset of strings as control elements, and change the positions of specific subsets of nodes in a systematic fashion.

Nagase et al.²¹ provided connectivity matrices for tensegrity structures constructed from repetitions of minimal 3 bar prisms. Connectivity matrices that include overlapping nodes were derived first, which were then transformed to connectivity matrices without overlapping nodes in a subsequent step.²¹ However, analytical expressions for the node positions of tensegrity plates formed by the minimal prisms were not provided in Nagase et al.²¹ Such expressions of the node position vectors are derived in this article for the first time for tensegrity plates of any complexity. Furthermore, as previously stated, this article presents the connectivity matrices of a new type of tensegrity plates generated by adding further members to tensegrity plates comprised minimal 3 bar prisms. The additional members provide higher stiffness in multiple directions and thus allow for more structurally robust tensegrity plates. Also, connectivity matrices without overlapping nodes are derived here directly and systematically for both minimal and extended tensegrity plates. As application examples, dynamic simulations demonstrating a strategy for morphing the planar tensegrity plates into domes are presented. Detailed dynamic models are used to simulate the morphing process.

Modeling of tensegrity structures

The mathematical model that describes the dynamics of tensegrity structures is summarized here. The complete derivation of the model is provided in Goyal and Skelton.²² The purpose of this section is to indicate the way that the node positions and connectivity matrices derived in the subsequent section are applied in the modeling of tensegrity structures.

Model preliminaries

A typical portion of a tensegrity structure is illustrated in Figure 1. The bars and strings are connected at nodes. The number of nodes, bars, and strings in the structure are denoted as $N$ , $B$ , and $S$ , respectively. The position vector of the ith node is denoted as $n_{i} \in R^{3}$ . The vector along the length of the jth bar is denoted as $b_{j} \in R^{3}$ and the vector along the length of the kth string is denoted as $s_{k} \in R^{3}$ .

Figure 1.

Portion of a tensegrity structure showing the position vector of the ith node ( $n_{i}$ ), the vector along the jth bar ( $b_{j}$ ), and the vector along the kth string ( $s_{k}$ ).

The matrices $N \in R^{3 \times N}$ , $B \in R^{3 \times B}$ , and $S \in R^{3 \times S}$ are constructed by, respectively, placing the node position vectors, bar vectors, and string vectors in each of their columns as $N = [\begin{matrix} \dots & n_{i} & \dots \end{matrix}]$ , $B = [\begin{matrix} \dots & b_{j} & \dots \end{matrix}]$ , and $S = [\begin{matrix} \dots & s_{k} & \dots \end{matrix}]$ . The bar and string connectivity matrices are denoted as $C_{B} \in R^{B \times N}$ and $C_{S} \in R^{S \times N}$ , respectively, and their components are defined as follows

[C_{B}]_{ji} = {\begin{matrix} - 1; & if n_{i} is the start point of b_{j} \\ 1; & if n_{i} is the end point of b_{j} \\ 0; & if n_{i} is not connected to b_{j} \end{matrix}

(1)

[C_{S}]_{ki} = {\begin{matrix} - 1; & if n_{i} is the start point of s_{k} \\ 1; & if n_{i} is the end point of s_{k} \\ 0; & if n_{i} is not connected to s_{k} \end{matrix}

(2)

The bar and string matrices B and S are related to the node matrix N through these connectivity matrices as follows

B = {NC}_{B}^{T}, S = {NC}_{S}^{T}

(3)

Dynamics of tensegrity structures

The resultant vector of externally applied forces at the ith node is denoted as $w_{i} \in R^{3}$ . The matrix $W \in R^{3 \times N}$ is constructed by placing the external force vectors in its columns as $W = [\begin{matrix} \dots & w_{i} & \dots \end{matrix}]$ . The strings are assumed to be axially deformable truss members with negligible mass. The magnitude of the tensile force of the kth string is denoted as $f_{k}$ , $k = 1, \dots, S$ . To simplify the model formulation, the tensile force per unit length $γ_{k}$ (often referred to as force density) is introduced, as it is usually done in the analysis of tensegrity structures.^21,23 It is defined as the tensile force of the string $f_{k}$ divided by its current length $∥ s_{k} ∥$ (i.e. $γ_{k} = f_{k} / ∥ s_{k} ∥$ ). It is assumed that $f_{k}$ is decomposed as $f_{k} = f_{e_{k}} + f_{v_{k}}$ , where $f_{e_{k}}$ and $f_{v_{k}}$ are the contributions from string stiffness and damping, respectively. These contributions are given as

\begin{matrix} f_{e_{k}} = {\begin{matrix} y_{k} (∥ s_{k} ∥ - μ_{k} s_{0_{k}}); & μ_{k} s_{0_{k}} \leq ∥ s_{k} ∥ \\ 0; & μ_{k} s_{0_{k}} > ∥ s_{k} ∥ \end{matrix} \\ f_{v_{k}} = {\begin{matrix} c_{k} \frac{s_{k} \cdot {\overset{\cdot}{s}}_{k}}{∥ s_{k} ∥}; & μ_{k} s_{0_{k}} \leq ∥ s_{k} ∥ \\ 0; & μ_{k} s_{0_{k}} > ∥ s_{k} ∥ \end{matrix} \end{matrix}

(4)

where the scalars $s_{0_{k}}$ , $y_{k}$ , and $c_{k}$ are, respectively, the initial length, extensional stiffness, and extensional damping of the kth string. The rest length ratio $μ_{k}$ , initially with a value of 1, can be manipulated via spools or material actuation to change the current rest length ( $μ_{k} s_{0_{k}}$ ) of a string and thereby change its tensile force. The extensional stiffness and damping are functions of the elastic modulus $E_{k}$ , damping coefficient $ζ_{k}$ , cross-sectional area $A_{k}$ , and initial length $s_{0_{k}}$ ²³ written as $y_{k} = E_{k} A_{k} / s_{0_{k}} = π E_{k} r_{k}^{2} / s_{0_{k}}$ and $c_{k} = ζ_{k} A_{k} / s_{0_{k}} = π ζ_{k} r_{k}^{2} / s_{0_{k}}$ , where $r_{k}$ is the radius of the circular cross-sectional area of the kth string.

The diagonal matrix containing the tensile force per unit length of each string in its diagonal components is denoted $\hat{γ} \in R^{S \times S}$ . Point masses are placed at the nodes that connect to only strings and are not connected to bars, here called string nodes for brevity (see, for example, the node at the start of $s_{k}$ in Figure 1). The number of string nodes in a tensegrity structure is denoted as $P$ . The mass of the lth string node is denoted as $m_{p_{l}}$ . The diagonal matrix that contains the mass of each string node in its diagonal components is denoted as ${\hat{m}}_{p} \in R^{P \times P}$ . The present model assumes that the bars are rigid and have non-zero mass. The compressive force per unit length, the constant length, and the mass of the jth bar are denoted $λ_{j}$ , $l_{j} = ∥ b_{j} ∥$ , and $m_{b_{j}}$ , respectively. The diagonal matrices containing the compressive force per unit length, the constant length, and the mass of each bar in their diagonal components are, respectively, denoted as $\hat{λ}$ , $\hat{l}$ , and ${\hat{m}}_{b} \in R^{B \times B}$ .

The matrix containing the position vectors of the nodes at the end points of the bars is denoted as $N_{B} \in R^{3 \times (N - P)}$ and the matrix containing the position vectors of the string nodes is denoted as $N_{S} \in R^{3 \times P}$ . The connectivity matrices $C_{nb} \in R^{(N - P) \times N}$ and $C_{ns} \in R^{P \times N}$ are used to obtain $N_{B}$ and $N_{S}$ from the node matrix N as follows

N_{B} = {NC}_{nb}^{T}, N_{S} = {NC}_{ns}^{T}

(5)

The bar matrix can then be obtained as follows

B = N_{B} C_{b}^{T} = {NC}_{nb}^{T} C_{b}^{T}

(6)

where $C_{b} \in R^{B \times (N - P)}$ describes the connectivity between the nodes in $N_{b}$ and the bar members. Considering equations (3) and (6), the bar connectivity matrix $C_{B}$ defined in equation (1) can be formulated in terms of $C_{nb}$ and $C_{b}$ as follows

B = {NC}_{B}^{T} = {NC}_{nb}^{T} C_{b}^{T} \to C_{B}^{T} = C_{nb}^{T} C_{b}^{T} \to C_{B} = C_{b} C_{nb}

(7)

The string matrix S is obtained from $N_{B}$ and $N_{S}$ as follows

S = [\begin{matrix} N_{B} & N_{S} \end{matrix}] [\begin{matrix} C_{sb}^{T} \\ C_{ss}^{T} \end{matrix}] = N [\begin{matrix} C_{nb}^{T} & C_{ns}^{T} \end{matrix}] [\begin{matrix} C_{sb}^{T} \\ C_{ss}^{T} \end{matrix}]

(8)

where $C_{sb} \in R^{S \times (N - P)}$ and $C_{ss} \in R^{S \times P}$ are matrices that, respectively, identify the bar-to-string and string-to-string connections. Considering equations (3) and (8), it follows that the string connectivity matrix $C_{S}$ defined in equation (2) can be formulated in terms of $C_{nb}$ , $C_{ns}$ , $C_{sb}$ , and $C_{ss}$

C_{S}^{T} = [\begin{matrix} C_{nb}^{T} & C_{ns}^{T} \end{matrix}] [\begin{matrix} C_{sb}^{T} \\ C_{ss}^{T} \end{matrix}] \to C_{S} = [\begin{matrix} C_{sb} & C_{ss} \end{matrix}] [\begin{matrix} C_{nb} \\ C_{ns} \end{matrix}]

(9)

Having introduced the aforementioned matrices, we now proceed to summarize the equations that describe the dynamics of tensegrity structures. In the model considered here, the nodes may be subject to linear constraints $NP = D$ , where $P \in R^{N \times C}$ and $D \in R^{3 \times C}$ are arbitrary constant matrices that define the linear constraints and $C$ denotes the number of linear constraints. The dynamics of tensegrity structures are then compactly described as follows²²

\overset{\cdot\cdot}{N} M + NK = W + Ω P^{T}

(10)

where $Ω \in R^{3 \times C}$ is the matrix of Lagrange multipliers arising from the linear constraints. The matrices $M \in R^{N \times N}$ and $K \in R^{N \times N}$ are given as $M = [\begin{matrix} C_{nb}^{T} ((1 / 12) C_{b}^{T} {\hat{m}}_{b} C_{b} + C_{r}^{T} {\hat{m}}_{b} C_{r}) & C_{ns}^{T} {\hat{m}}_{p} \end{matrix}]$ and $K = [\begin{matrix} C_{S}^{T} \hat{γ} C_{sb} - C_{nb}^{T} C_{b}^{T} \hat{λ} C_{b} & C_{S}^{T} \hat{γ} C_{ss} \end{matrix}]$ , where $C_{r} = (1 / 2) | C_{b} |$ is the matrix that identifies the center of mass of the bars as a function of the node positions (|·| denotes the operator that returns a matrix whose components are the absolute values of the components of the input matrix). Details on the calculation of the force in the bars $\hat{λ}$ and Lagrange multipliers $Ω$ are provided in Goyal and Skelton.²²

The node matrix N and the bar and string connectivity matrices $C_{B}$ and $C_{S}$ defined in equations (1) and (2) are required to determine the configuration of all the members in the structure. It is indicated in equations (7) and (9) that $C_{B}$ and $C_{S}$ can be formulated in terms of the matrices $C_{nb}$ , $C_{ns}$ , $C_{b}$ , $C_{sb}$ , and $C_{ss}$ . In the subsequent section, analytical expressions for the node matrix N along with $C_{nb}$ , $C_{ns}$ , $C_{b}$ , $C_{sb}$ , and $C_{ss}$ (and hence $C_{B}$ and $C_{S}$ ) are derived for tensegrity plates of two different topologies.

Node positions and connectivity matrices of tensegrity plates

The node positions and connectivity matrices required to fully define the configuration of tensegrity plates are presented here. Such data are provided for the minimal tensegrity plate and for the new extended tensegrity plate.

Minimal tensegrity plate

The tensegrity plate described in this section is formed by the repetition of a tensegrity unit known as the minimal regular 3 bar prism, which is illustrated in Figure 2. This unit has been used to form tensegrity plates,^21,24,25 towers,^26,27 beams,²⁸ and masts.¹¹ The unit is called minimal because it has the minimum number of strings (nine) required to form a stable pre-stressable prism with 3 bars. It is called regular because its top and bottom faces are regular polygons, which in this work are equilateral triangles. As shown in Figure 2(a), the top and bottom triangles lie in parallel planes and are contained within circumscribing circles of radii $r_{t}$ and $r_{b}$ , respectively. In Skelton and de Oliveira,¹ it was proven that the twist angle $α$ formed by the projection of the top and bottom parallel triangles must be uniquely $π / 6$ radians ( $30 °$ ) to keep the prism self-balanced and pre-stressable. Thus, a value of $α$ of $30 °$ is adopted in the derivation of the node positions of all the prisms forming the tensegrity plates in this work to ensure that they are self-balanced in pre-stressable. The distance separating the top and bottom faces shown in Figure 2(b) is denoted as the thicknessh.

Figure 2.

Geometry of the minimal regular 3 bar prism: (a) radii of top ( $r_{t}$ ) and bottom ( $r_{b}$ ) faces and twist angle $α$ , (b) thickness h, and (c) local coordinate system and node labels.

The local coordinate system illustrated in Figure 2(c) is established for the prism and its origin is located at the center of the bottom triangular face. The position vector of the origin is denoted as $c \in R^{3}$ and the position vectors of the top and bottom nodes are denoted as $n_{t_{i}} \in R^{3}$ and $n_{b_{i}} \in R^{3}$ , $i = 1, 2, 3$ , respectively. The angle between the vector $n_{b_{1}} - c$ and the x-axis is denoted as $β$ . The node matrix $N_{0} \in R^{3 \times 6}$ of a 3-bar prism is given as

N_{0} = [N_{t_{0}} N_{b_{0}}]

(11)

where $N_{t_{0}} \in R^{3 \times 3}$ and $N_{b_{0}} \in R^{3 \times 3}$ are the top and bottom node matrices given as follows

\begin{matrix} N_{t_{0}} = [n_{t_{1}} n_{t_{2}} n_{t_{3}}] \\ = r_{t} [\begin{matrix} - \frac{1}{2} \sin (β) - \frac{3^{\frac{1}{2}}}{2} \cos (β) & \sin (β) & - \frac{1}{2} \sin (β) + \frac{3^{\frac{1}{2}}}{2} \cos (β) \\ - \frac{3^{\frac{1}{2}}}{2} \sin (β) + \frac{1}{2} \cos (β) & - \cos (β) & \frac{3^{\frac{1}{2}}}{2} \sin (β) + \frac{1}{2} \cos (β) \\ \frac{h}{r_{t}} & \frac{h}{r_{t}} & \frac{h}{r_{t}} \end{matrix}] \end{matrix}

(12)

\begin{matrix} N_{b_{0}} = [n_{b_{1}} n_{b_{2}} n_{b_{3}}] \\ = r_{b} [\begin{matrix} \cos (β) & - \frac{3^{\frac{1}{2}}}{2} \sin (β) - \frac{1}{2} \cos (β) & \frac{3^{\frac{1}{2}}}{2} \sin (β) - \frac{1}{2} \cos (β) \\ \sin (β) & - \frac{1}{2} \sin (β) + \frac{3^{\frac{1}{2}}}{2} \cos (β) & - \frac{1}{2} \sin (β) - \frac{3^{\frac{1}{2}}}{2} \cos (β) \\ 0 & 0 & 0 \end{matrix}] \end{matrix}

(13)

A flat tensegrity plate can be formed by assembling 3 bar prisms as shown in Figure 3(a). All the prisms comprising the plate have the same thickness h and the same top and bottom radii $r_{t} = r_{b} = r$ . For every pair of adjacent prisms, two of the nodes of one prism lie on the top and bottom strings of the other. The distance $d_{c}$ between the centers of two adjacent prisms shown in Figure 3(b) is given as follows

d_{c} = 3 {(2 - 3^{\frac{1}{2}})}^{\frac{1}{2}} r

(14)

The prisms are arranged in concentric rings, where the number of rings is denoted as the complexityq of the plate. For example, the complexity of the plate in Figure 3 is 4. A plate of complexity q contains $1 + 3 q (q - 1)$ prism units. The diameter of the circle circumscribing the plate is denoted as d. The relation between the plate diameter d and the prism radius r is given as follows¹

d = 2 r {(9 (2 - 3^{\frac{1}{2}}) (q - {1)}^{2} + 3 (q - 1) + 1)}^{\frac{1}{2}}

(15)

Figure 3.

Geometry of a tensegrity plate: (a) top and side views showing the global coordinate system, plate diameter d, and thickness h and (b) prism radius r and distance between adjacent units $d_{c}$ .

The geometry of a tensegrity plate is defined by its diameter d, thickness h, and complexity q. To establish systematic analytical formulations for the node positions and connectivity matrices, the prisms are divided into four parts as shown in Figure 4. Part 0 has only one prism, the central unit, and each of the other three parts (Parts 1, 2, and 3) have $q (q - 1)$ prisms. Parts 1, 2, and 3 are placed in a three-fold symmetric arrangement (i.e. $120 °$ rotation symmetry). Each prism unit is uniquely labeled as $(j, i, k)$ , where j is the part index (1, 2, or 3), $i = 0, \dots, q - 2$ is the row index, and $k = 0, \dots, q - 1$ is the column index. The central unit is labeled as $(0, 0, 0)$ . The labels of the units in Part 1 are shown in Figure 5. Parts 2 and 3 have the same prism arrangement of Part 1. The global coordinate system shown in Figure 3(a) is established with its origin located at the center of the bottom triangle of unit $(0, 0, 0)$ . The z-axis of the global coordinate system is orthogonal to the plate faces.

Figure 4.

Groups of prism units forming a tensegrity plate.

Figure 5.

Labels of prism units in Part 1 (refer to Figure 4).

The top and bottom node matrices of a unit $(j, i, k)$ are, respectively, denoted as $N_{t} (j, i, k) \in R^{3 \times 3}$ and $N_{b} (j, i, k) \in R^{3 \times 3}$

\begin{matrix} N_{t} (j, i, k) = [\begin{matrix} n_{t_{1}} (j, i, k) & n_{t_{2}} (j, i, k) & n_{t_{3}} (j, i, k) \end{matrix}] \\ N_{b} (j, i, k) = [\begin{matrix} n_{b_{1}} (j, i, k) & n_{b_{2}} (j, i, k) & n_{b_{3}} (j, i, k) \end{matrix}] \end{matrix}

(16)

The position vectors of the top and bottom nodes of each part are placed in the matrices $N_{t} (j), N_{b} (j) \in R^{3 \times 3 q (q - 1)}$ as follows

\begin{matrix} N_{t} (j) = [\begin{matrix} N_{t} (j, 0, 0) & \dots & N_{t} (j, 0, q - 1) & \dots & N_{t} (j, q - 2, q - 1) \end{matrix}] \\ N_{b} (j) = [\begin{matrix} N_{b} (j, 0, 0) & \dots & N_{b} (j, 0, q - 1) & \dots & N_{b} (j, q - 2, q - 1) \end{matrix}] \end{matrix}

(17)

and $N_{t} (0) = N_{t} (0, 0, 0)$ and $N_{b} (0) = N_{b} (0, 0, 0)$ . As indicated in Figure 6, for every pair of adjacent prisms, two of the nodes of one prism are located on the top and bottom strings of the other. The edges of the top and bottom triangles of every prism in the plate are thus each subdivided into two strings. Therefore, each prism in the plate has six top strings, six bottom strings, three vertical strings, and three bars as labeled in Figure 7. The bar matrix of unit $(j, i, k)$ is denoted as $B (j, i, k) \in R^{3 \times 3}$

B (j, i, k) = [\begin{matrix} b_{1} (j, i, k) & b_{2} (j, i, k) & b_{3} (j, i, k) \end{matrix}]

(18)

The matrix $B (j) \in R^{3 \times 3 q (q - 1)}$ is constructed by collecting the bar matrices of the units in Part j as follows

B (j) = [\begin{matrix} B (j, 0, 0) & \dots & B (j, 0, q - 1) & \dots & B (j, q - 2, q - 1) \end{matrix}]

(19)

and $B (0) = B (0, 0, 0)$ .

Figure 6.

Units adjacent to prism $(1, i, k)$ .

Figure 7.

Bar and string members of a prism in a tensegrity plate.

The top, bottom, and vertical string matrices of unit ( $j, i, k$ ) are denoted as $S_{t} (j, i, k) \in R^{3 \times 6}$ , $S_{b} (j, i, k) \in R^{3 \times 6}$ , and $S_{v} (j, i, k) \in R^{3 \times 3}$ , respectively

\begin{matrix} S_{t} (j, i, k) = [\begin{matrix} s_{t_{1}} (j, i, k) & \dots & s_{t_{6}} (j, i, k) \end{matrix}] \\ S_{b} (j, i, k) = [\begin{matrix} s_{b_{1}} (j, i, k) & \dots & s_{b_{6}} (j, i, k) \end{matrix}] \\ S_{v} (j, i, k) = [\begin{matrix} s_{v_{1}} (j, i, k) & s_{v_{2}} (j, i, k) & s_{v_{3}} (j, i, k) \end{matrix}] \end{matrix}

(20)

The matrices $S_{t} (j) \in R^{3 \times 6 q (q - 1)}$ , $S_{b} (j) \in R^{3 \times 6 q (q - 1)}$ , and $S_{v} (j) \in R^{3 \times 3 q (q - 1)}$ are formed by, respectively, collecting the top, bottom, and vertical string matrices of the units in Part j as follows

\begin{matrix} S_{t} (j) = [\begin{matrix} S_{t} (j, 0, 0) & \dots & S_{t} (j, 0, q - 1) & \dots & S_{t} (j, q - 2, q - 1) \end{matrix}] \\ S_{b} (j) = [\begin{matrix} S_{b} (j, 0, 0) & \dots & S_{b} (j, 0, q - 1) & \dots & S_{b} (j, q - 2, q - 1) \end{matrix}] \\ S_{v} (j) = [\begin{matrix} S_{v} (j, 0, 0) & \dots & S_{v} (j, 0, q - 1) & \dots & S_{v} (j, q - 2, q - 1) \end{matrix}] \end{matrix}

(21)

and $S_{t} (0) = S_{t} (0, 0, 0)$ , $S_{b} (0) = S_{b} (0, 0, 0)$ , and $S_{v} (0) = S_{v} (0, 0, 0)$ . To allow each prism in the plate to have the same number of members, boundary nodes are placed at the $6 (2 q - 1)$ boundary strings as shown in Figure 8 so that each of them is divided into two strings, just as the top and bottom string segments at the interior of plate. Therefore, $2 q - 1$ top boundary nodes $n_{p_{t}} (j, l)$ and $2 q - 1$ bottom boundary nodes $n_{p_{b}} (j, l)$ are added to Parts 1, 2, and 3 (where $j = 1, 2, 3$ and $l = 0, \dots, 2 q - 2$ ). The boundary nodes connect only to strings and hence are the only string nodes in the plate. The position vectors of the top and bottom boundary nodes of Part j are, respectively, placed in the matrices $N_{p_{t}} (j) \in R^{3 \times (2 q - 1)}$ and $N_{p_{b}} (j) \in R^{3 \times (2 q - 1)}$

\begin{matrix} N_{p_{t}} (j) = [\begin{matrix} n_{p_{t}} (j, 0) & \dots & n_{p_{t}} (j, 2 q - 2) \end{matrix}] \\ N_{p_{b}} (j) = [\begin{matrix} n_{p_{b}} (j, 0) & \dots & n_{p_{b}} (j, 2 q - 2) \end{matrix}] \end{matrix}

(22)

Figure 8.

Top and bottom nodes at the boundary of a minimal plate.

The number of prism units in the plate is denoted $U$ and is given as follows

U = 1 + 3 q (q - 1)

(23)

In total, a minimal tensegrity plate has $N = 6 U + 6 (2 q - 1)$ nodes, $B = 3 U$ bars, $S = 15 U$ strings, and $P = 6 (2 q - 1)$ string nodes. The node matrixN of the tensegrity plate is obtained by collecting the matrices $N_{t} (j)$ , $N_{b} (j)$ , $N_{p_{t}} (j)$ , and $N_{p_{b}} (j)$ from equations (17) and (22)

N = [\begin{matrix} N_{t} & N_{b} & N_{p} \end{matrix}]

(24)

where $N_{t} \in R^{3 \times 3 U}$ , $N_{b} \in R^{3 \times 3 U}$ , and $N_{p} \in R^{3 \times 6 (2 q - 1)}$ are, respectively, the matrices of top nodes, bottom nodes, and boundary nodes

\begin{matrix} N_{t} = [\begin{matrix} N_{t} (0) & N_{t} (1) & N_{t} (2) & N_{t} (3) \end{matrix}] \\ N_{b} = [\begin{matrix} N_{b} (0) & N_{b} (1) & N_{b} (2) & N_{b} (3) \end{matrix}] \\ N_{p} = [\begin{matrix} N_{p_{t}} (1) & N_{p_{t}} (2) & N_{p_{t}} (3) & N_{p_{b}} (1) & N_{p_{b}} (2) & N_{p_{b}} (3) \end{matrix}] \end{matrix}

(25)

Similarly, the bar matrixB of the tensegrity plate is given as

B = [\begin{matrix} B (0) & B (1) & B (2) & B (3) \end{matrix}]

(26)

where the matrices $B (j)$ are defined in equation (19). The string matrixS is given as

S = [\begin{matrix} S_{t} & S_{b} & S_{v} \end{matrix}]

(27)

where $S_{t} \in R^{3 \times 6 U}$ , $S_{b} \in R^{3 \times 6 U}$ , and $S_{v} \in R^{3 \times 3 U}$ are the matrices of top, bottom, and vertical strings, respectively

\begin{matrix} S_{t} = [\begin{matrix} S_{t} (0) & S_{t} (1) & S_{t} (2) & S_{t} (3) \end{matrix}] \\ S_{b} = [\begin{matrix} S_{b} (0) & S_{b} (1) & S_{b} (2) & S_{b} (3) \end{matrix}] \\ S_{v} = [\begin{matrix} S_{v} (0) & S_{v} (1) & S_{v} (2) & S_{v} (3) \end{matrix}] \end{matrix}

(28)

and the matrices $S_{t} (j)$ , $S_{b} (j)$ , and $S_{v} (j)$ are defined in equation (21). After defining the labels and ordering of the nodes, bars, and strings in their associated matrices, the next step is to provide the formulas for the node positions and the connectivity matrices that relate the nodes to the members.

Node positions

The node positions of tensegrity plates are calculated through the four steps outlined in this section.

Step 1. Calculate the node positions of each prism in its local coordinate system. The radius r of each prism is calculated from the complexity q and plate diameter d using equation (15). The node matrices of each prism in its local coordinate system ( $N_{t_{0}}$ and $N_{b_{0}}$ ) are determined by substituting the unit radius r, thickness h, and angle $β$ into equations (12) and (13). To obtain the plate configuration illustrated in Figures 3 and 4, the angle $β$ is selected as $7 π / 12$ radians ( $105 °$ ).

Step 2. Determine the location of the origin of the local coordinate systems for prisms in Parts 0 and 1 in the global coordinate system. The origin of the local coordinate system of the single unit of Part 0, $c (0, 0, 0)$ , coincides with the origin of the global coordinate system

c (0, 0, 0) = {[\begin{matrix} 0 & 0 & 0 \end{matrix}]}^{T}

(29)

The origin of the local coordinate system of prisms in Part 1, $c (1, i, k)$ , are determined using the vectors $d_{c_{1}}, d_{c_{2}} \in R^{3}$ shown in Figure 9

\begin{matrix} c (1, i, k) = c (0, 0, 0) + (i + 1) d_{c 1} + k d_{c 2} \\ i = 0, \dots, q - 2, k = 0, \dots, q - 1 \end{matrix}

(30)

where $d_{c_{1}}$ and $d_{c_{2}}$ are given as

d_{c_{1}} = d_{c} {[\begin{matrix} - \frac{1}{2} & - \frac{3^{\frac{1}{2}}}{2} & 0 \end{matrix}]}^{T}, d_{c_{2}} = d_{c} {[\begin{matrix} 1 & 0 & 0 \end{matrix}]}^{T}

(31)

and the distance between adjacent units, $d_{c}$ , is calculated using equation (14).

Figure 9.

Vectors $d_{c_{1}}$ an $d_{c_{2}}$ connecting the centers of adjacent units.

Step 3. Determine the node positions for prisms in Parts 0 and 1. The node positions of prisms in Parts 0 and 1 are obtained as follows

\begin{matrix} N_{t} (0, 0, 0) = N_{t_{0}}, N_{t} (1, i, k) = N_{t_{0}} + c (1, i, k) \otimes [\begin{matrix} 1 & 1 & 1 \end{matrix}] \\ N_{b} (0, 0, 0) = N_{b_{0}}, N_{b} (1, i, k) = N_{b_{0}} + c (1, i, k) \otimes [\begin{matrix} 1 & 1 & 1 \end{matrix}] \\ i = 0, \dots, q - 2, k = 0, \dots, q - 1 \end{matrix}

(32)

where ⊗ denotes the Kronecker product, $c (1, i, k)$ is calculated via equation (30), and $N_{t_{0}}$ and $N_{b_{0}}$ are given in equations (12) and (13). The locations of the boundary nodes of Part 1, $n_{p_{t}} (1, l)$ and $n_{p_{b}} (1, l)$ , are calculated as follows

\begin{matrix} n_{p_{t}} (1, l) = {\begin{matrix} n_{t_{3}} (0, 0, 0) + q d_{c_{1}} + (l + 1) d_{c_{2}}; & l = 0, \dots, q - 1 \\ n_{t_{3}} (0, 0, 0) + (2 q - l - 1) d_{c_{1}} + q d_{c_{2}}; & l = q, \dots, 2 q - 3 \\ n_{t_{2}} (0, 0, 0) + (q - 1) d_{c_{1}} - d_{c_{2}}; & l = 2 q - 2 \end{matrix} \\ n_{p_{b}} (1, l) = {\begin{matrix} n_{b_{1}} (0, 0, 0) + q d_{c_{1}} + (l + 1) d_{c_{2}}; & l = 0, \dots, q - 1 \\ n_{b_{1}} (0, 0, 0) + (2 q - l - 1) d_{c_{1}} + q d_{c_{2}}; & l = q, \dots, 2 q - 3 \\ n_{b_{3}} (0, 0, 0) + (q - 1) d_{c_{1}} - d_{c_{2}}; & l = 2 q - 2 \end{matrix} \end{matrix}

(33)

Step 4. Determine the node positions for units in Parts 2 and 3. The node locations for prisms in Parts 2 and 3 are calculated by applying rotation transformations to the node locations of Part 1 with respect to the z-axis of the global coordinate system. First, define $Q (ϕ) \in R^{3 \times 3}$ as the matrix that represents a rotation of $ϕ$ with respect to the z-axis

Q (ϕ) = [\begin{matrix} \cos (ϕ) & - \sin (ϕ) & 0 \\ \sin (ϕ) & \cos (ϕ) & 0 \\ 0 & 0 & 1 \end{matrix}]

(34)

It is observed in Figure 4 that the positions of the nodes in Parts 2 and 3 can be determined by rotating those of Part 1 by $2 π / 3$ radians ( $120 °$ ) and $4 π / 3$ radians ( $240 °$ ), respectively, about the z-axis. Therefore, the node matrices $N_{t} (2, i, k)$ , $N_{t} (3, i, k)$ , $N_{b} (2, i, k)$ , and $N_{b} (3, i, k)$ and the boundary nodes $n_{p_{t}} (2, l)$ , $n_{p_{t}} (3, l)$ , $n_{p_{b}} (2, l)$ , and $n_{p_{b}} (3, l)$ are calculated as follows

\begin{matrix} N_{t} (2, i, k) = Q (\frac{2 π}{3}) N_{t} (1, i, k) & N_{b} (2, i, k) = Q (\frac{2 π}{3}) N_{b} (1, i, k) \\ N_{t} (3, i, k) = Q (\frac{4 π}{3}) N_{t} (1, i, k) & N_{b} (3, i, k) = Q (\frac{4 π}{3}) N_{b} (1, i, k) \\ i = 0, \dots, q - 2, k = 0, \dots, q - 1 \end{matrix}

(35)

\begin{matrix} n_{p_{t}} (2, l) = Q (\frac{2 π}{3}) n_{p_{t}} (1, l) & n_{p_{b}} (2, l) = Q (\frac{2 π}{3}) n_{p_{b}} (1, l) \\ n_{p_{t}} (3, l) = Q (\frac{4 π}{3}) n_{p_{t}} (1, l) & n_{p_{b}} (3, l) = Q (\frac{4 π}{3}) n_{p_{b}} (1, l) \\ l = 0, \dots, 2 q - 2 \end{matrix}

(36)

where $N_{t} (1, i, k)$ and $N_{b} (1, i, k)$ are calculated using equation (32) and $n_{p_{t}} (1, l)$ and $n_{p_{b}} (1, l)$ are calculated using equation (33). All the node positions derived in this section are substituted into equations (24) and (25) to establish the node matrix N of the tensegrity plate.

Connectivity matrices

The subdivision of the node matrix N in equation (24) allows us to find the matrices of nodes that connect to bars, $N_{B}$ , and string nodes, $N_{S}$ , through the following expressions

\begin{matrix} N_{B} = [\begin{matrix} N_{t} & N_{b} \end{matrix}] = N [\begin{matrix} I_{2 B} \\ O_{P \times 2 B} \end{matrix}], \\ N_{S} = N_{p} = N [\begin{matrix} O_{2 B \times P} \\ I_{P} \end{matrix}] \end{matrix}

(37)

where $I_{m}$ is the $R^{m \times m}$ identity matrix and $O_{n \times m}$ is the $R^{n \times m}$ zero matrix. The number of bars $B$ and string nodes $P$ in the plate are provided in the above equation (24). Using equations (5) and (37), the matrices $C_{nb}$ and $C_{ns}$ are written as

C_{nb} = [\begin{matrix} I_{2 B} & O_{2 B \times P} \end{matrix}], C_{ns} = [\begin{matrix} O_{P \times 2 B} & I_{P} \end{matrix}]

(38)

As shown in the node and member labels of Figures 2(c) and 7, the lth bar of a prism connects the lth bottom node and the lth top node in the same unit, $l = 1, 2, 3$ . Therefore, the relation between the bar matrix $B (j, i, k)$ , the top node matrix $N_{t} (j, i, k)$ , and the bottom node matrix $N_{b} (j, i, k)$ of a prism in the plate is

B (j, i, k) = - N_{t} (j, i, k) + N_{b} (j, i, k)

(39)

Equations (39), (37), and (26) allow us to write the relation between $N_{B}$ and the bar matrix B as follows

B = - N_{t} + N_{b} = N_{B} [\begin{matrix} - I_{B} \\ I_{B} \end{matrix}]

(40)

Using equations (6) and (40), the matrix $C_{b}$ is written as

C_{b} = [\begin{matrix} - I_{B} & I_{B} \end{matrix}]

(41)

The bar connectivity matrix $C_{B}$ defined in equation (1) is determined from $C_{nb}$ and $C_{b}$ using equation (7)

C_{B} = C_{b} C_{nb} = [\begin{matrix} - I_{B} & I_{B} & O_{B \times P} \end{matrix}]

(42)

The construction of the string connectivity matrix $C_{S}$ is more challenging than that of $C_{B}$ . This is because the top and bottom strings of a unit $(j, i, k)$ may be connected to nodes within the same unit, nodes of adjacent units, or boundary nodes as shown in Figures 5 and 6. The identification of such nodes depends on the region where the unit is located within the Part. Referring to Figure 5, there are nine distinct regions in Parts $j = 1, 2, 3$ : corner units $(j, 0, 0)$ , $(j, 0, q - 1)$ , $(j, q - 2, 0)$ , and $(j, q - 2, q - 1)$ ; boundary units $(j, 0, k)$ , $(j, i, 0)$ , $(j, q - 2, k)$ , and $(j, i, q - 1)$ ; and interior units $(j, i, k)$ , where $i = 1, \dots, q - 3$ and $k = 1, \dots, q - 2$ . The top string matrix for the single unit of Part 0 is determined as follows

\begin{matrix} S_{t} (0, 0, 0) = N_{t} (0, 0, 0) Θ_{00} + N_{t} (1, 0, 1) Θ_{13} \\ + N_{t} (2, 0, 1) Θ_{23} + N_{t} (3, 0, 1) Θ_{33} \end{matrix}

(43)

and the top string matrices for the corner units, boundary units, and interior units of Parts $j = 1, 2, 3$ are determined as

\begin{matrix} S_{t} (j, 0, 0) = N_{t} (j, 0, 0) Θ_{00} + N_{t} (j, 1, 1) Θ_{13} \\ + N_{t} (0, 0, 0) Θ_{2 j} + N_{t} (j + 2, 0, 2) Θ_{33} \\ S_{t} (j, 0, q - 1) = N_{t} (j, 0, q - 1) Θ_{00} + n_{p_{t}} (j, 2 q - 3) Θ_{01} \\ + N_{t} (j + 1, q - 2, 0) Θ_{23} + N_{t} (j, 0, q - 2) Θ_{32} \\ S_{t} (j, q - 2, 0) = N_{t} (j, q - 2, 0) Θ_{00} + n_{p_{t}} (j, 0) Θ_{01} \\ + N_{t} (j, q - 3, 0) Θ_{21} + n_{p_{t}} (j, 2 q - 2) Θ_{03} \\ S_{t} (j, q - 2, q - 1) = N_{t} (j, q - 2, q - 1) Θ_{00} + n_{p_{t}} (j, q - 1) Θ_{01} \\ + N_{t} (j, q - 3, q - 1) Θ_{21} + N_{t} (j, q - 2, q - 2) Θ_{32} \end{matrix}

(44)

\begin{matrix} S_{t} (j, 0, k) = N_{t} (j, 0, k) Θ_{00} + N_{t} (j, 0, k + 1) Θ_{13} \\ + N_{t} (j + 1, k - 1, 0) Θ_{23} + N_{t} (j, 0, k - 1) Θ_{32} \\ S_{t} (j, i, 0) = N_{t} (j, i, 0) Θ_{00} + N_{t} (j, i + 1, 1) Θ_{13} \\ + N_{t} (j, i - 1, 0) Θ_{21} + N_{t} (j + 2, 0, i + 2) Θ_{33} \\ S_{t} (j, q - 2, k) = N_{t} (j, q - 2, k) Θ_{00} + n_{p_{t}} (j, k) Θ_{01} \\ + N_{t} (j, q - 3, k) Θ_{21} + N_{t} (j, q - 2, k - 1) Θ_{32} \\ S_{t} (j, i, q - 1) = N_{t} (j, i, q - 1) Θ_{00} + n_{p_{t}} (j, 2 q - 3 - i) Θ_{01} \\ + N_{t} (j, i - 1, q - 1) Θ_{21} + N_{t} (j, i, q - 2) Θ_{32} \\ i = 1, \dots, q - 3, k = 1, \dots, q - 2 \end{matrix}

(45)

\begin{matrix} \begin{matrix} S_{t} (j, i, k) = N_{t} (j, i, k) Θ_{00} + N_{t} (j, i + 1, k + 1) Θ_{13} \\ + N_{t} (j, i - 1, k) Θ_{21} + N_{t} (j, i, k - 1) Θ_{32} \end{matrix} \\ i = 1, \dots, q - 3, k = 1, \dots, q - 2 \end{matrix}

(46)

where the matrices $Θ_{mn}$ are defined as

\begin{matrix} Θ_{00} = [\begin{matrix} - e_{1} & e_{2} & - e_{2} & e_{3} & - e_{3} & e_{1} \end{matrix}] \\ Θ_{01} = [\begin{matrix} 1 & - 1 & 0 & 0 & 0 & 0 \end{matrix}] & Θ_{11} = [\begin{matrix} e_{1} & - e_{1} & e_{0} & e_{0} & e_{0} & e_{0} \end{matrix}] \\ Θ_{02} = [\begin{matrix} 0 & 0 & 1 & - 1 & 0 & 0 \end{matrix}] & Θ_{12} = [\begin{matrix} e_{2} & - e_{2} & e_{0} & e_{0} & e_{0} & e_{0} \end{matrix}] \\ Θ_{03} = [\begin{matrix} 0 & 0 & 0 & 0 & 1 & - 1 \end{matrix}] & Θ_{13} = [\begin{matrix} e_{3} & - e_{3} & e_{0} & e_{0} & e_{0} & e_{0} \end{matrix}] \\ Θ_{21} = [\begin{matrix} e_{0} & e_{0} & e_{1} & - e_{1} & e_{0} & e_{0} \end{matrix}] & Θ_{31} = [\begin{matrix} e_{0} & e_{0} & e_{0} & e_{0} & e_{1} & - e_{1} \end{matrix}] \\ Θ_{22} = [\begin{matrix} e_{0} & e_{0} & e_{2} & - e_{2} & e_{0} & e_{0} \end{matrix}] & Θ_{32} = [\begin{matrix} e_{0} & e_{0} & e_{0} & e_{0} & e_{2} & - e_{2} \end{matrix}] \\ Θ_{23} = [\begin{matrix} e_{0} & e_{0} & e_{3} & - e_{3} & e_{0} & e_{0} \end{matrix}] & Θ_{33} = [\begin{matrix} e_{0} & e_{0} & e_{0} & e_{0} & e_{3} & - e_{3} \end{matrix}] \end{matrix}

(47)

and the vectors $e_{0}, e_{1}, e_{2}, e_{3} \in R^{3}$ have the following definitions

\begin{matrix} e_{0}^{T} = [\begin{matrix} 0 & 0 & 0 \end{matrix}], e_{1}^{T} = [\begin{matrix} 1 & 0 & 0 \end{matrix}], \\ e_{2}^{T} = [\begin{matrix} 0 & 1 & 0 \end{matrix}], e_{3}^{T} = [\begin{matrix} 0 & 0 & 1 \end{matrix}] \end{matrix}

(48)

The bottom string matrices for all the units are obtained by making the following substitutions in equations (43)–(46)

\begin{matrix} S_{t} (j, i, k) \to S_{b} (j, i, k), N_{t} (j, i, k) \to N_{b} (j, i, k), \\ n_{p_{t}} (j, l) \to n_{p_{b}} (j, l) \end{matrix}

(49)

\begin{matrix} Θ_{00} \to Θ_{00}, Θ_{01} \to Θ_{02}, Θ_{02} \to Θ_{03}, Θ_{03} \to Θ_{01} \\ Θ_{11} \to Θ_{22}, Θ_{12} \to Θ_{23}, Θ_{13} \to Θ_{21} \\ Θ_{21} \to Θ_{32}, Θ_{22} \to Θ_{33}, Θ_{23} \to Θ_{31} \\ Θ_{31} \to Θ_{12}, Θ_{32} \to Θ_{13}, Θ_{33} \to Θ_{11} \end{matrix}

(50)

In the above equations (49) and (50), the right arrow symbol “→” is used to denote substitution. For example, $A \to Z$ indicates that matrix A is substituted or replaced by matrix Z. This notation is employed in the remainder of this article.

Unlike the top and bottom strings, the vertical strings are connected only to the top and bottom nodes of the same unit (see Figure 2). Therefore, the vertical string matrix is written as follows

S_{v} (j, i, k) = - N_{t} (j, i, k) + N_{b} (j, i, k) [\begin{matrix} e_{2} & e_{3} & e_{1} \end{matrix}]

(51)

After determining the string-node connectivity for all the prisms of the plate, the next step is to use such information to formulate the string connectivity matrix $C_{S}$ . First, $C_{S}$ is partitioned as follows

C_{S}^{T} = [C_{S_{t}}^{T} C_{S_{b}}^{T} C_{S_{v}}^{T}]

(52)

where $C_{S_{t}} \in R^{6 U \times N}$ , $C_{S_{b}} \in R^{6 U \times N}$ , and $C_{S_{v}} \in R^{3 U \times N}$ are the top, bottom, and vertical string connectivity matrices, respectively

C_{S_{t}}^{T} = [\begin{matrix} Ψ_{t}^{t} \\ Ψ_{b}^{t} \\ Ψ_{p_{t}}^{t} \\ Ψ_{p_{b}}^{t} \end{matrix}], C_{S_{b}}^{T} = [\begin{matrix} Ψ_{t}^{b} \\ Ψ_{b}^{b} \\ Ψ_{p_{t}}^{b} \\ Ψ_{p_{b}}^{b} \end{matrix}], C_{S_{v}}^{T} = [\begin{matrix} Ψ_{t}^{v} \\ Ψ_{b}^{v} \\ Ψ_{p_{t}}^{v} \\ Ψ_{p_{b}}^{v} \end{matrix}]

(53)

Similarly, the matrices $C_{sb}$ and $C_{ss}$ that, respectively, identify the bar-to-string and string-to-string connections are divided as

C_{sb}^{T} = [\begin{matrix} Ψ_{t}^{t} & Ψ_{t}^{b} & Ψ_{t}^{v} \\ Ψ_{b}^{t} & Ψ_{b}^{b} & Ψ_{b}^{v} \end{matrix}], C_{ss}^{T} = [\begin{matrix} Ψ_{p_{t}}^{t} & Ψ_{p_{t}}^{b} & Ψ_{p_{t}}^{v} \\ Ψ_{p_{b}}^{t} & Ψ_{p_{b}}^{b} & Ψ_{p_{b}}^{v} \end{matrix}]

(54)

The matrices $Ψ_{t}^{t} \in R^{3 U \times 6 U}$ , $Ψ_{b}^{t} \in R^{3 U \times 6 U}$ , $Ψ_{p_{t}}^{t} \in R^{3 (2 q - 1) \times 6 U}$ , and $Ψ_{p_{b}}^{t} \in R^{3 (2 q - 1) \times 6 U}$ indicate the connectivity of the top nodes, bottom nodes, top boundary nodes, and bottom boundary nodes, respectively, with the top strings. As indicated in equations (43)–(46), the top strings are not connected to the bottom nodes or the bottom boundary nodes. Therefore

Ψ_{b}^{t} = O_{3 U \times 6 U}, Ψ_{p_{b}}^{t} = O_{3 (2 q - 1) \times 6 U}

(55)

Also, using the information of equations (43)–(46), the matrices $Ψ_{t}^{t}$ and $Ψ_{p_{t}}^{t}$ are written as

Ψ_{t}^{t} = [\begin{matrix} Φ_{00}^{t} & Φ_{10}^{t} & Φ_{20}^{t} & Φ_{30}^{t} \\ Φ_{01}^{t} & Φ_{11}^{t} & Φ_{21}^{t} & Φ_{31}^{t} \\ Φ_{02}^{t} & Φ_{12}^{t} & Φ_{22}^{t} & Φ_{32}^{t} \\ Φ_{03}^{t} & Φ_{13}^{t} & Φ_{23}^{t} & Φ_{33}^{t} \end{matrix}]

(56)

Ψ_{p_{t}}^{t} = [\begin{matrix} O_{(2 q - 1) \times 6} & Φ_{14}^{t} & O_{(2 q - 1) \times 6 q (q - 1)} & O_{(2 q - 1) \times 6 q (q - 1)} \\ O_{(2 q - 1) \times 6} & O_{(2 q - 1) \times 6 q (q - 1)} & Φ_{24}^{t} & O_{(2 q - 1) \times 6 q (q - 1)} \\ O_{(2 q - 1) \times 6} & O_{(2 q - 1) \times 6 q (q - 1)} & O_{(2 q - 1) \times 6 q (q - 1)} & Φ_{34}^{t} \end{matrix}]

(57)

where the matrices $Φ_{mn}^{t}$ , $m = 0, \dots, 3$ , $n = 0, \dots, 4$ , are expressed as follows

\begin{matrix} Φ_{00}^{t} = Θ_{00} \in R^{3 \times 6}, Φ_{01}^{t} = [\begin{matrix} O_{3 \times 6} \\ Θ_{13} \\ O_{(3 q (q - 1) - 6) \times 6} \end{matrix}] \in R^{3 q (q - 1) \times 6} \\ Φ_{02}^{t} = [\begin{matrix} O_{3 \times 6} \\ Θ_{23} \\ O_{(3 q (q - 1) - 6) \times 6} \end{matrix}] \in R^{3 q (q - 1) \times 6} \\ Φ_{03}^{t} = [\begin{matrix} O_{3 \times 6} \\ Θ_{33} \\ O_{(3 q (q - 1) - 6) \times 6} \end{matrix}] \in R^{3 q (q - 1) \times 6} \end{matrix}

(58)

Φ_{10}^{t} = [\begin{matrix} Θ_{21} & O_{3 \times (6 q (q - 1) - 6)} \end{matrix}] \in R^{3 \times 6 q (q - 1)}

(59)

\begin{matrix} Φ_{11}^{t} = [\begin{matrix} Λ_{0}^{t} & Λ_{2}^{t} & \dots & O_{3 q \times 6 q} & O_{3 q \times 6 q} \\ Λ_{1}^{t} & Λ_{0}^{t} & \dots & O_{3 q \times 6 q} & O_{3 q \times 6 q} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ O_{3 q \times 6 q} & O_{3 q \times 6 q} & \dots & Λ_{0}^{t} & Λ_{2}^{t} \\ O_{3 q \times 6 q} & O_{3 q \times 6 q} & \dots & Λ_{1}^{t} & Λ_{0}^{t} \end{matrix}] \\ \in R^{3 q (q - 1) \times 6 q (q - 1)} \end{matrix}

(60)

Φ_{12}^{t} = [\begin{matrix} Λ_{3}^{t} & O_{3 q (q - 1) \times 6 q (q - 2)} \end{matrix}] \in R^{3 q (q - 1) \times 6 q (q - 1)}

(61)

\begin{matrix} Φ_{13}^{t} = [\begin{matrix} O_{6 \times 6 q} & \dots & O_{6 \times 6 q} & O_{6 \times 6 q} \\ Λ_{4}^{t} & \dots & O_{3 \times 6 q} & O_{3 \times 6 q} \\ ⋮ & ⋱ & ⋮ & ⋮ \\ O_{3 \times 6 q} & \dots & Λ_{4}^{t} & O_{3 \times 6 q} \\ O_{3 q (q - 2) \times 6 q} & \dots & O_{3 q (q - 2) \times 6 q} & O_{3 q (q - 2) \times 6 q} \end{matrix}] \\ \in R^{3 q (q - 1) \times 6 q (q - 1)} \end{matrix}

(62)

Φ_{14}^{t} = [\begin{matrix} \begin{matrix} O_{q \times 6 q} & \dots & O_{q \times 6 q} \\ O_{1 \times 6 q} & \dots & Λ_{5}^{t} \\ ⋮ & ⋱ & ⋮ \\ Λ_{5}^{t} & \dots & O_{1 \times 6 q} \\ O_{1 \times 6 q} & \dots & O_{1 \times 6 q} \end{matrix} & Λ_{6}^{t} \end{matrix}] \in R^{(2 q - 1) \times 6 q (q - 1)}

(63)

\begin{matrix} Φ_{20}^{t} = [\begin{matrix} Θ_{22} & O_{3 \times (6 q (q - 1) - 6)} \end{matrix}] \in R^{3 \times 6 q (q - 1)} \\ Φ_{21}^{t} = Φ_{13}^{t}, Φ_{22}^{t} = Φ_{11}^{t}, Φ_{23}^{t} = Φ_{12}^{t}, Φ_{24}^{t} = Φ_{14}^{t} \end{matrix}

(64)

\begin{matrix} Φ_{30}^{t} = [\begin{matrix} Θ_{23} & O_{3 \times (6 q (q - 1) - 6)} \end{matrix}] \in R^{3 \times 6 q (q - 1)} \\ Φ_{31}^{t} = Φ_{12}^{t}, Φ_{32}^{t} = Φ_{13}^{t}, Φ_{33}^{t} = Φ_{11}^{t}, Φ_{34}^{t} = Φ_{14}^{t} \end{matrix}

(65)

The matrices $Λ_{m}$ , $m = 1, 2, 3, 4, 5, 6$ , employed in equation (63) are given as

Λ_{0}^{t} = [\begin{matrix} Θ_{00} & Θ_{32} & \dots & O_{3 \times 6} & O_{3 \times 6} \\ O_{3 \times 6} & Θ_{00} & \dots & O_{3 \times 6} & O_{3 \times 6} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ O_{3 \times 6} & O_{3 \times 6} & \dots & Θ_{00} & Θ_{32} \\ O_{3 \times 6} & O_{3 \times 6} & \dots & O_{3 \times 6} & Θ_{00} \end{matrix}] \in R^{3 q \times 6 q}

(66)

Λ_{1}^{t} = [\begin{matrix} O_{3 \times 6} & O_{3 \times 6} & \dots & O_{3 \times 6} & O_{3 \times 6} \\ Θ_{13} & O_{3 \times 6} & \dots & O_{3 \times 6} & O_{3 \times 6} \\ ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ O_{3 \times 6} & O_{3 \times 6} & \dots & O_{3 \times 6} & O_{3 \times 6} \\ O_{3 \times 6} & O_{3 \times 6} & \dots & Θ_{13} & O_{3 \times 6} \end{matrix}] \in R^{3 q \times 6 q}

(67)

Λ_{2}^{t} = [\begin{matrix} Θ_{21} & \dots & O_{3 \times 6} \\ ⋮ & ⋱ & ⋮ \\ O_{3 \times 6} & \dots & Θ_{21} \end{matrix}] \in R^{3 q \times 6 q}

(68)

\begin{matrix} Λ_{3}^{t} = & [\begin{matrix} O_{3 \times 6} & Θ_{23} & \dots & O_{3 \times 6} \\ O_{3 (q - 1) \times 6} & O_{3 (q - 1) \times 6} & \dots & O_{3 (q - 1) \times 6} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ O_{3 \times 6} & O_{3 \times 6} & \dots & Θ_{23} \\ O_{3 (q - 1) \times 6} & O_{3 (q - 1) \times 6} & \dots & O_{3 (q - 1) \times 6} \end{matrix}] \\ \in R^{3 q (q - 1) \times 6 q} \end{matrix}

(69)

Λ_{4}^{t} = [\begin{matrix} Θ_{33} & O_{3 \times 6 (q - 1)} \end{matrix}] \in R^{3 \times 6 q}

(70)

Λ_{5}^{t} = [\begin{matrix} O_{1 \times 6 (q - 1)} & Θ_{01} \end{matrix}] \in R^{1 \times 6 q}

(71)

\begin{matrix} Λ_{6}^{t} = [\begin{matrix} Θ_{01} & O_{1 \times 6} & \dots & O_{1 \times 6} \\ O_{1 \times 6} & Θ_{01} & \dots & O_{1 \times 6} \\ ⋮ & ⋮ & ⋱ & ⋮ \\ O_{1 \times 6} & O_{1 \times 6} & \dots & Θ_{01} \\ O_{(q - 2) \times 6} & O_{(q - 2) \times 6} & \dots & O_{(q - 2) \times 6} \\ Θ_{03} & O_{1 \times 6} & \dots & O_{1 \times 6} \end{matrix}] \\ \in R^{(2 q - 1) \times 6 q} \end{matrix}

(72)

The matrices $Ψ_{t}^{b} \in R^{3 U \times 6 U}$ , $Ψ_{b}^{b} \in R^{3 U \times 6 U}$ , $Ψ_{p_{t}}^{b} \in R^{3 (2 q - 1) \times 6 U}$ , and $Ψ_{p_{b}}^{b} \in R^{3 (2 q - 1) \times 6 U}$ indicate the connectivity of the top nodes, bottom nodes, top boundary nodes, and bottom boundary nodes, respectively, with the bottom strings. These matrices are expressed as follows

Ψ_{t}^{b} = O_{3 U \times 6 U}, Ψ_{p_{t}}^{b} = O_{3 (2 q - 1) \times 6 U}

(73)

Ψ_{b}^{b} = [\begin{matrix} Φ_{00}^{b} & Φ_{10}^{b} & Φ_{20}^{b} & Φ_{30}^{b} \\ Φ_{01}^{b} & Φ_{11}^{b} & Φ_{21}^{b} & Φ_{31}^{b} \\ Φ_{02}^{b} & Φ_{12}^{b} & Φ_{22}^{b} & Φ_{32}^{b} \\ Φ_{03}^{b} & Φ_{13}^{b} & Φ_{23}^{b} & Φ_{33}^{b} \end{matrix}]

(74)

Ψ_{p_{b}}^{b} = [\begin{matrix} O_{(2 q - 1) \times 6} & Φ_{14}^{b} & O_{(2 q - 1) \times 6 q (q - 1)} & O_{(2 q - 1) \times 6 q (q - 1)} \\ O_{(2 q - 1) \times 6} & O_{(2 q - 1) \times 6 q (q - 1)} & Φ_{24}^{b} & O_{(2 q - 1) \times 6 q (q - 1)} \\ O_{(2 q - 1) \times 6} & O_{(2 q - 1) \times 6 q (q - 1)} & O_{(2 q - 1) \times 6 q (q - 1)} & Φ_{34}^{b} \end{matrix}]

(75)

The matrices $Φ_{mn}^{b}$ , $m = 0, \dots, 3$ , $n = 0, \dots, 4$ , are determined by making the substitutions indicated in equation (50) to the expressions for $Φ_{mn}^{t}$ from equations (58) to (72).

The matrices $Ψ_{t}^{v} \in R^{3 U \times 3 U}$ , $Ψ_{b}^{v} \in R^{3 U \times 3 U}$ , $Ψ_{p_{t}}^{v} \in R^{3 (2 q - 1) \times 3 U}$ , and $Ψ_{p_{b}}^{v} \in R^{3 (2 q - 1) \times 3 U}$ indicate the connectivity of the top nodes, bottom nodes, top boundary nodes, and bottom boundary nodes, respectively, with the vertical strings. Using equation (51), these matrices are given as

\begin{matrix} Ψ_{t}^{v} = - I_{3 U}, Ψ_{p_{t}}^{v} = O_{3 (2 q - 1) \times 3 U} \\ Ψ_{b}^{v} = I_{U} \otimes [\begin{matrix} e_{2} & e_{3} & e_{1} \end{matrix}], Ψ_{p_{b}}^{v} = O_{3 (2 q - 1) \times 3 U} \end{matrix}

(76)

This concludes the determination of the connectivity matrices for minimal tensegrity plates of any complexity.

Extended tensegrity plate

A new type of plate is proposed herein on the basis of the minimal tensegrity plate. The fundamental unit of the new plate is obtained by adding three top inner strings, three bottom inner strings, and three vertical inner strings to a minimal regular 3 bar prism. This prism is denoted as extended regular 3 bar prism and is illustrated in Figure 10. The additional strings increase stiffness in directions that have large compliance in conventional minimal prisms. A directional stiffness comparison between the minimal and the new prism is not provided here but is recommended as future work. The definitions of top radius $r_{t}$ , bottom radius $r_{b}$ , and thickness h provided for the minimal prism are also applicable for the new topology. The labels of the added members in an extended regular 3 bar prism are indicated in Figure 11. The vectors along the length of the top inner strings, bottom inner strings, and vertical inner strings are denoted as ${\bar{s}}_{t_{i}}$ , ${\bar{s}}_{b_{i}}$ , and ${\bar{s}}_{v_{i}}$ , respectively, $i = 1, 2, 3$ .

Figure 10.

Extended regular 3 bar prism: (a) top view and (b) perspective view.

Figure 11.

Nodes and members of an extended 3 bar prism: (a) nodes and (b) added members. The labels of the other members are provided in Figure 7.

The extended tensegrity plate is formed by arranging extended 3 bar prisms in concentric rings as shown in Figure 12. The diameter d, distance between centers of adjacent prisms $d_{c}$ , and complexity q introduced for the minimal plate are also applicable here. To allow each prism in the plate to have the same topology, boundary vertical strings are added as shown in Figure 13. The total number of boundary vertical strings is denoted as $V = 3 (2 q - 1)$ .

Figure 12.

Top view of a tensegrity plate formed by extended regular prisms.

Figure 13.

Boundary nodes and boundary vertical strings of the extended 3-bar prism plate.

The division of the plate into four parts as illustrated in Figure 4 is also employed here to define the labels of nodes and members for the 3 bar prism extended plate. The nodes and bar members of the 3 bar prism extended plate are identical to those of the minimal plate. Therefore, the node matrix N and bar matrix B of the extended 3 bar prism plate correspond to those provided in equations (24) and (26), respectively. The process for the determination of the node positions for the extended 3 bar prism plate is also the same as that outlined for the minimal plate. The string matrix S for the extended plate, however, is different than that of the minimal plate because further strings are added in the extended plate. Let ${\bar{S}}_{t} (j, i, k), {\bar{S}}_{b} (j, i, k), {\bar{S}}_{v} (j, i, k) \in R^{3 \times 3}$ be the top, bottom, and vertical inner strings matrices of unit $(j, i, k)$

\begin{matrix} {\bar{S}}_{t} (j, i, k) = [\begin{matrix} {\bar{s}}_{t_{1}} (j, i, k) & {\bar{s}}_{t_{2}} (j, i, k) & {\bar{s}}_{t_{3}} (j, i, k) \end{matrix}] \\ {\bar{S}}_{b} (j, i, k) = [\begin{matrix} {\bar{s}}_{b_{1}} (j, i, k) & {\bar{s}}_{b_{2}} (j, i, k) & {\bar{s}}_{b_{3}} (j, i, k) \end{matrix}] \\ {\bar{S}}_{v} (j, i, k) = [\begin{matrix} {\bar{s}}_{v_{1}} (j, i, k) & {\bar{s}}_{v_{2}} (j, i, k) & {\bar{s}}_{v_{3}} (j, i, k) \end{matrix}] \end{matrix}

(77)

The matrices ${\bar{S}}_{t} (j), {\bar{S}}_{b} (j), {\bar{S}}_{v} (j) \in R^{3 \times 3 q (q - 1)}$ , $j = 1, 2, 3$ , are constructed by collecting the inner string matrices of units in Part j as follows

\begin{matrix} {\bar{S}}_{t} (j) = [\begin{matrix} {\bar{S}}_{t} (j, 0, 0) & \dots & {\bar{S}}_{t} (j, 0, q - 1) & \dots & {\bar{S}}_{t} (j, q - 2, q - 1) \end{matrix}] \\ {\bar{S}}_{b} (j) = [\begin{matrix} {\bar{S}}_{b} (j, 0, 0) & \dots & {\bar{S}}_{b} (j, 0, q - 1) & \dots & {\bar{S}}_{b} (j, q - 2, q - 1) \end{matrix}] \\ {\bar{S}}_{v} (j) = [\begin{matrix} {\bar{S}}_{v} (j, 0, 0) & \dots & {\bar{S}}_{v} (j, 0, q - 1) & \dots & {\bar{S}}_{v} (j, q - 2, q - 1) \end{matrix}] \end{matrix}

(78)

and ${\bar{S}}_{t} (0) = {\bar{S}}_{t} (0, 0, 0)$ , ${\bar{S}}_{b} (0) = {\bar{S}}_{b} (0, 0, 0)$ , and ${\bar{S}}_{v} (0) = {\bar{S}}_{v} (0, 0, 0)$ . Also, let ${\bar{S}}_{t}, {\bar{S}}_{b}, \bar{S_{v}} \in R^{3 \times 3 U}$ be matrices of top, bottom, and vertical inner strings of the plate

\begin{matrix} {\bar{S}}_{t} = [\begin{matrix} {\bar{S}}_{t} (0) & {\bar{S}}_{t} (1) & {\bar{S}}_{t} (2) & {\bar{S}}_{t} (3) \end{matrix}] \\ {\bar{S}}_{b} = [\begin{matrix} {\bar{S}}_{b} (0) & {\bar{S}}_{b} (1) & {\bar{S}}_{b} (2) & {\bar{S}}_{b} (3) \end{matrix}] \\ {\bar{S}}_{v} = [\begin{matrix} {\bar{S}}_{v} (0) & {\bar{S}}_{v} (1) & {\bar{S}}_{v} (2) & {\bar{S}}_{v} (3) \end{matrix}] \end{matrix}

(79)

The vectors along the boundary vertical strings (illustrated in Figure 13) located in Part j, $j = 1, 2, 3$ , are denoted ${\tilde{s}}_{v} (j, l)$ , $l = 0, \dots, 2 q - 2$ . The boundary string vectors of Part j are placed in the matrix ${\tilde{S}}_{v} (j) \in R^{3 \times (2 q - 1)}$ as follows

{\tilde{S}}_{v} (j) = [{\tilde{S}}_{v} (j, 0) \dots {\tilde{S}}_{v} (j, 2 q - 2)]

(80)

The matrix of boundary string vectors of the entire plate, ${\tilde{S}}_{v} \in R^{3 \times V}$ , is constructed as follows

{\tilde{S}}_{v} = [\begin{matrix} {\tilde{S}}_{v} (1) & {\tilde{S}}_{v} (2) & {\tilde{S}}_{v} (3) \end{matrix}]

(81)

After defining the previous matrices, the string matrix $S \in R^{3 \times (24 U + V)}$ of the extended tensegrity plate is constructed as follows

S = [\begin{matrix} S_{t} & {\bar{S}}_{t} & S_{b} & {\bar{S}}_{b} & S_{v} & {\bar{S}}_{v} & {\tilde{S}}_{v} \end{matrix}]

(82)

where $S_{t}$ , $S_{b}$ , and $S_{v}$ have the same definitions as those of the minimal plate provided in equation (28).

Connectivity matrices

As previously mentioned, the node matrix N and the bar matrix B of the extended plate are identical to those of the minimal one; therefore, the bar connectivity matrix $C_{B}$ and the matrices $C_{nb}$ , $C_{ns}$ , and $C_{b}$ for the extended plate correspond to those provided in equations (42), (38), and (41). The string connectivity matrix $C_{S}$ of the extended plate is divided as follows

C_{S}^{T} = [C_{S_{t}}^{T} {\bar{C}}_{S_{t}}^{T} C_{S_{b}}^{T} {\bar{C}}_{S_{b}}^{T} C_{S_{v}}^{T} {\bar{C}}_{S_{v}}^{T} {\tilde{C}}_{S_{v}}^{T}]

(83)

where $C_{S_{t}} \in R^{6 U \times N}$ , $C_{S_{b}} \in R^{6 U \times N}$ , and $C_{S_{v}} \in R^{3 U \times N}$ are the top, bottom, and vertical string connectivity matrices, respectively, which are provided in equation (53). The matrices ${\bar{C}}_{S_{t}} \in R^{3 U \times N}$ , ${\bar{C}}_{S_{b}} \in R^{3 U \times N}$ , ${\bar{C}}_{S_{v}} \in R^{3 U \times N}$ , and ${\tilde{C}}_{S_{v}} \in R^{V \times N}$ are the top inner, bottom inner, vertical inner, and boundary vertical string connectivity matrices, respectively. These matrices are subdivided as

{\bar{C}}_{S_{t}}^{T} = [\begin{matrix} {\bar{Ψ}}_{t}^{t} \\ {\bar{Ψ}}_{b}^{t} \\ {\bar{Ψ}}_{p_{t}}^{t} \\ {\bar{Ψ}}_{p_{b}}^{t} \end{matrix}], {\bar{C}}_{S_{b}}^{T} = [\begin{matrix} {\bar{Ψ}}_{t}^{b} \\ {\bar{Ψ}}_{b}^{b} \\ {\bar{Ψ}}_{p_{t}}^{b} \\ {\bar{Ψ}}_{p_{b}}^{b} \end{matrix}], {\bar{C}}_{S_{v}}^{T} = [\begin{matrix} {\bar{Ψ}}_{t}^{v} \\ {\bar{Ψ}}_{b}^{v} \\ {\bar{Ψ}}_{p_{t}}^{v} \\ {\bar{Ψ}}_{p_{b}}^{v} \end{matrix}], {\tilde{C}}_{S_{v}}^{T} = [\begin{matrix} {\tilde{Ψ}}_{t}^{v} \\ {\tilde{Ψ}}_{b}^{v} \\ {\tilde{Ψ}}_{p_{t}}^{v} \\ {\tilde{Ψ}}_{p_{b}}^{v} \end{matrix}]

(84)

Similarly, the matrices $C_{sb}$ and $C_{ss}$ that respectively identify the bar-to-string and string-to-string connections are divided as

C_{sb}^{T} = [\begin{matrix} Ψ_{t}^{t} & {\bar{Ψ}}_{t}^{t} & Ψ_{t}^{b} & {\bar{Ψ}}_{t}^{b} & Ψ_{t}^{v} & {\bar{Ψ}}_{t}^{v} & {\tilde{Ψ}}_{t}^{v} \\ Ψ_{b}^{t} & {\bar{Ψ}}_{b}^{t} & Ψ_{b}^{b} & {\bar{Ψ}}_{b}^{b} & Ψ_{b}^{v} & {\bar{Ψ}}_{b}^{v} & {\tilde{Ψ}}_{b}^{v} \end{matrix}]

(85)

C_{ss}^{T} = [\begin{matrix} Ψ_{p_{t}}^{t} & {\bar{Ψ}}_{p_{t}}^{t} & Ψ_{p_{t}}^{b} & {\bar{Ψ}}_{p_{t}}^{b} & Ψ_{p_{t}}^{v} & {\bar{Ψ}}_{p_{t}}^{v} & {\tilde{Ψ}}_{p_{t}}^{v} \\ Ψ_{p_{b}}^{t} & {\bar{Ψ}}_{p_{b}}^{t} & Ψ_{p_{b}}^{b} & {\bar{Ψ}}_{p_{b}}^{b} & Ψ_{p_{b}}^{v} & {\bar{Ψ}}_{p_{b}}^{v} & {\tilde{Ψ}}_{p_{b}}^{v} \end{matrix}]

(86)

The matrices ${\bar{Ψ}}_{t}^{t} \in R^{3 U \times 3 U}$ , ${\bar{Ψ}}_{b}^{t} \in R^{3 U \times 3 U}$ , ${\bar{Ψ}}_{p_{t}}^{t} \in R^{3 (2 q - 1) \times 3 U}$ , and ${\bar{Ψ}}_{p_{b}}^{t} \in R^{3 (2 q - 1) \times 3 U}$ indicate the connectivity of the top nodes, bottom nodes, top boundary nodes, and bottom boundary nodes, respectively, with the top inner strings. These matrices are expressed as follows

{\bar{Ψ}}_{b}^{t} = O_{3 U \times 3 U}, {\bar{Ψ}}_{p_{b}}^{t} = O_{3 (2 q - 1) \times 3 U}

(87)

{\bar{Ψ}}_{t}^{t} = [\begin{matrix} {\bar{Φ}}_{00}^{t} & {\bar{Φ}}_{10}^{t} & {\bar{Φ}}_{20}^{t} & {\bar{Φ}}_{30}^{t} \\ {\bar{Φ}}_{01}^{t} & {\bar{Φ}}_{11}^{t} & {\bar{Φ}}_{21}^{t} & {\bar{Φ}}_{31}^{t} \\ {\bar{Φ}}_{02}^{t} & {\bar{Φ}}_{12}^{t} & {\bar{Φ}}_{22}^{t} & {\bar{Φ}}_{32}^{t} \\ {\bar{Φ}}_{03}^{t} & {\bar{Φ}}_{13}^{t} & {\bar{Φ}}_{23}^{t} & {\bar{Φ}}_{33}^{t} \end{matrix}]

(88)

{\bar{Ψ}}_{p_{t}}^{t} = [\begin{matrix} O_{(2 q - 1) \times 3} & {\bar{Φ}}_{14}^{t} & O_{(2 q - 1) \times 3 q (q - 1)} & O_{(2 q - 1) \times 3 q (q - 1)} \\ O_{(2 q - 1) \times 3} & O_{(2 q - 1) \times 3 q (q - 1)} & {\bar{Φ}}_{24}^{t} & O_{(2 q - 1) \times 3 q (q - 1)} \\ O_{(2 q - 1) \times 3} & O_{(2 q - 1) \times 3 q (q - 1)} & O_{(2 q - 1) \times 3 q (q - 1)} & {\bar{Φ}}_{34}^{t} \end{matrix}]

(89)

The matrices ${\bar{Φ}}_{mn}^{t}$ , $m = 0, \dots, 3$ , $n = 0, \dots, 4$ , are determined by making the following substitutions to the expressions for $Φ_{mn}^{t}$ in equations (58)–(72)

\begin{matrix} Θ_{00} \to {\bar{Θ}}_{00}, Θ_{01} \to {\bar{Θ}}_{01}, Θ_{02} \to {\bar{Θ}}_{02}, Θ_{03} \to {\bar{Θ}}_{03} \\ Θ_{11} \to {\bar{Θ}}_{11}, Θ_{12} \to {\bar{Θ}}_{12}, Θ_{13} \to {\bar{Θ}}_{13} \\ Θ_{21} \to {\bar{Θ}}_{21}, Θ_{22} \to {\bar{Θ}}_{22}, Θ_{23} \to {\bar{Θ}}_{23} \\ Θ_{31} \to {\bar{Θ}}_{31}, Θ_{32} \to {\bar{Θ}}_{32}, Θ_{33} \to {\bar{Θ}}_{33} \end{matrix}

(90)

where

\begin{matrix} {\bar{Θ}}_{00} = [\begin{matrix} e_{0} & e_{0} & e_{0} \end{matrix}], \\ {\bar{Θ}}_{01} = [\begin{matrix} 0 & 1 & - 1 \end{matrix}], & {\bar{Θ}}_{02} = [\begin{matrix} - 1 & 0 & 1 \end{matrix}], & {\bar{Θ}}_{03} = [\begin{matrix} 1 & - 1 & 0 \end{matrix}] \\ {\bar{Θ}}_{11} = [\begin{matrix} e_{0} & e_{1} & - e_{1} \end{matrix}], & {\bar{Θ}}_{12} = [\begin{matrix} e_{0} & e_{2} & - e_{2} \end{matrix}], & {\bar{Θ}}_{13} = [\begin{matrix} e_{0} & e_{3} & - e_{3} \end{matrix}] \\ {\bar{Θ}}_{21} = [\begin{matrix} - e_{1} & e_{0} & e_{1} \end{matrix}], & {\bar{Θ}}_{22} = [\begin{matrix} - e_{2} & e_{0} & e_{2} \end{matrix}], & {\bar{Θ}}_{23} = [\begin{matrix} - e_{3} & e_{0} & e_{3} \end{matrix}] \\ {\bar{Θ}}_{31} = [\begin{matrix} e_{1} & - e_{1} & e_{0} \end{matrix}], & {\bar{Θ}}_{32} = [\begin{matrix} e_{2} & - e_{2} & e_{0} \end{matrix}], & {\bar{Θ}}_{33} = [\begin{matrix} e_{3} & - e_{3} & e_{0} \end{matrix}] \end{matrix}

(91)

The matrices ${\bar{Ψ}}_{t}^{b} \in R^{3 U \times 3 U}$ , ${\bar{Ψ}}_{b}^{b} \in R^{3 U \times 3 U}$ , ${\bar{Ψ}}_{p_{t}}^{b} \in R^{3 (2 q - 1) \times 3 U}$ , and ${\bar{Ψ}}_{p_{b}}^{b} \in R^{3 (2 q - 1) \times 3 U}$ indicate the connectivity of the top nodes, bottom nodes, top boundary nodes, and bottom boundary nodes, respectively, with the bottom inner strings. These matrices are expressed as follows

{\bar{Ψ}}_{t}^{b} = O_{3 U \times 3 U}, {\bar{Ψ}}_{p_{t}}^{b} = O_{3 (2 q - 1) \times 3 U}

(92)

{\bar{Ψ}}_{b}^{b} = [\begin{matrix} {\bar{Φ}}_{00}^{b} & {\bar{Φ}}_{10}^{b} & {\bar{Φ}}_{20}^{b} & {\bar{Φ}}_{30}^{b} \\ {\bar{Φ}}_{01}^{b} & {\bar{Φ}}_{11}^{b} & {\bar{Φ}}_{21}^{b} & {\bar{Φ}}_{31}^{b} \\ {\bar{Φ}}_{02}^{b} & {\bar{Φ}}_{12}^{b} & {\bar{Φ}}_{22}^{b} & {\bar{Φ}}_{32}^{b} \\ {\bar{Φ}}_{03}^{b} & {\bar{Φ}}_{13}^{b} & {\bar{Φ}}_{23}^{b} & {\bar{Φ}}_{33}^{b} \end{matrix}]

(93)

{\bar{Ψ}}_{p_{b}}^{b} = [\begin{matrix} O_{(2 q - 1) \times 3} & {\bar{Φ}}_{14}^{b} & O_{(2 q - 1) \times 3 q (q - 1)} & O_{(2 q - 1) \times 3 q (q - 1)} \\ O_{(2 q - 1) \times 3} & O_{(2 q - 1) \times 3 q (q - 1)} & {\bar{Φ}}_{24}^{b} & O_{(2 q - 1) \times 3 q (q - 1)} \\ O_{(2 q - 1) \times 3} & O_{(2 q - 1) \times 3 q (q - 1)} & O_{(2 q - 1) \times 3 q (q - 1)} & {\bar{Φ}}_{34}^{b} \end{matrix}]

(94)

The matrices ${\bar{Φ}}_{mn}^{b}$ , $m = 0, \dots, 3$ , $n = 0, \dots, 4$ , are determined by making the following substitutions to the expressions for $Φ_{mn}^{t}$ in equations (58)–(72)

\begin{matrix} Θ_{00} \to {\bar{Θ}}_{00}, Θ_{01} \to {\bar{Θ}}_{01}, Θ_{02} \to {\bar{Θ}}_{02}, Θ_{03} \to {\bar{Θ}}_{03} \\ Θ_{11} \to {\bar{Θ}}_{12}, Θ_{12} \to {\bar{Θ}}_{13}, Θ_{13} \to {\bar{Θ}}_{11} \\ Θ_{21} \to {\bar{Θ}}_{22}, Θ_{22} \to {\bar{Θ}}_{23}, Θ_{23} \to {\bar{Θ}}_{21} \\ Θ_{31} \to {\bar{Θ}}_{32}, Θ_{32} \to {\bar{Θ}}_{33}, Θ_{33} \to {\bar{Θ}}_{31} \end{matrix}

(95)

where the matrices ${\bar{Θ}}_{mn}$ are provided in equation (91). The matrices ${\bar{Ψ}}_{t}^{v} \in R^{3 U \times 3 U}$ , ${\bar{Ψ}}_{b}^{v} \in R^{3 U \times 3 U}$ , ${\bar{Ψ}}_{p_{t}}^{v} \in R^{3 (2 q - 1) \times 3 U}$ , and ${\bar{Ψ}}_{p_{b}}^{v} \in R^{3 (2 q - 1) \times 3 U}$ indicate the connectivity of the top nodes, bottom nodes, top boundary nodes, and bottom boundary nodes, respectively, with the vertical inner strings. These matrices are expressed as follows

\begin{matrix} {\bar{Ψ}}_{t}^{v} = [\begin{matrix} {\bar{Φ}}_{00}^{v} & {\bar{Φ}}_{10}^{v} & {\bar{Φ}}_{20}^{v} & {\bar{Φ}}_{30}^{v} \\ {\bar{Φ}}_{01}^{v} & {\bar{Φ}}_{11}^{v} & {\bar{Φ}}_{21}^{v} & {\bar{Φ}}_{31}^{v} \\ {\bar{Φ}}_{02}^{v} & {\bar{Φ}}_{12}^{v} & {\bar{Φ}}_{22}^{v} & {\bar{Φ}}_{32}^{v} \\ {\bar{Φ}}_{03}^{v} & {\bar{Φ}}_{13}^{v} & {\bar{Φ}}_{23}^{v} & {\bar{Φ}}_{33}^{v} \end{matrix}], {\bar{Ψ}}_{b}^{v} = [\begin{matrix} {\underline{Φ}}_{00}^{v} & {\underline{Φ}}_{10}^{v} & {\underline{Φ}}_{20}^{v} & {\underline{Φ}}_{30}^{v} \\ {\underline{Φ}}_{01}^{v} & {\underline{Φ}}_{11}^{v} & {\underline{Φ}}_{21}^{v} & {\underline{Φ}}_{31}^{v} \\ {\underline{Φ}}_{02}^{v} & {\underline{Φ}}_{12}^{v} & {\underline{Φ}}_{22}^{v} & {\underline{Φ}}_{32}^{v} \\ {\underline{Φ}}_{03}^{v} & {\underline{Φ}}_{13}^{v} & {\underline{Φ}}_{23}^{v} & {\underline{Φ}}_{33}^{v} \end{matrix}] \end{matrix}

(96)

\begin{matrix} {\bar{Ψ}}_{p_{t}}^{v} = [\begin{matrix} O_{(2 q - 1) \times 3} & {\bar{Φ}}_{14}^{v} & O_{(2 q - 1) \times 3 q (q - 1)} & O_{(2 q - 1) \times 3 q (q - 1)} \\ O_{(2 q - 1) \times 3} & O_{(2 q - 1) \times 3 q (q - 1)} & {\bar{Φ}}_{24}^{v} & O_{(2 q - 1) \times 3 q (q - 1)} \\ O_{(2 q - 1) \times 3} & O_{(2 q - 1) \times 3 q (q - 1)} & O_{(2 q - 1) \times 3 q (q - 1)} & {\bar{Φ}}_{34}^{v} \end{matrix}] \\ {\bar{Ψ}}_{p_{b}}^{v} = [\begin{matrix} O_{(2 q - 1) \times 3} & {\underline{Φ}}_{14}^{v} & O_{(2 q - 1) \times 3 q (q - 1)} & O_{(2 q - 1) \times 3 q (q - 1)} \\ O_{(2 q - 1) \times 3} & O_{(2 q - 1) \times 3 q (q - 1)} & {\underline{Φ}}_{24}^{v} & O_{(2 q - 1) \times 3 q (q - 1)} \\ O_{(2 q - 1) \times 3} & O_{(2 q - 1) \times 3 q (q - 1)} & O_{(2 q - 1) \times 3 q (q - 1)} & {\underline{Φ}}_{34}^{v} \end{matrix}] \end{matrix}

(97)

The matrices ${\bar{Φ}}_{mn}^{v}$ , $m = 0, \dots, 3$ , $n = 0, \dots, 4$ , are determined by making the following substitutions to the expressions for $Φ_{mn}^{t}$ in equations (58)–(72)

\begin{matrix} Θ_{00} \to {\tilde{Θ}}_{00}, Θ_{01} \to {\tilde{Θ}}_{01}, Θ_{02} \to {\tilde{Θ}}_{02}, Θ_{03} \to {\tilde{Θ}}_{03} \\ Θ_{11} \to {\tilde{Θ}}_{11}, Θ_{12} \to {\tilde{Θ}}_{12}, Θ_{13} \to {\tilde{Θ}}_{13} \\ Θ_{21} \to {\tilde{Θ}}_{21}, Θ_{22} \to {\tilde{Θ}}_{22}, Θ_{23} \to {\tilde{Θ}}_{23} \\ Θ_{31} \to {\tilde{Θ}}_{31}, Θ_{32} \to {\tilde{Θ}}_{32}, Θ_{33} \to {\tilde{Θ}}_{33} \end{matrix}

(98)

and the matrices ${\underline{Φ}}_{mn}^{v}$ , $m = 0, \dots, 3$ , $n = 0, \dots, 4$ , are determined by making the following substitutions to the expressions for $Φ_{mn}^{t}$ in equations (58)–(72)

\begin{matrix} Θ_{00} \to {\tilde{Θ}}_{00}, Θ_{01} \to {\tilde{Θ}}_{02}, Θ_{02} \to {\tilde{Θ}}_{03}, Θ_{03} \to {\tilde{Θ}}_{01} \\ Θ_{11} \to {\tilde{Θ}}_{22}, Θ_{12} \to {\tilde{Θ}}_{23}, Θ_{13} \to {\tilde{Θ}}_{21} \\ Θ_{21} \to {\tilde{Θ}}_{32}, Θ_{22} \to {\tilde{Θ}}_{33}, Θ_{23} \to {\tilde{Θ}}_{31} \\ Θ_{31} \to {\tilde{Θ}}_{12}, Θ_{32} \to {\tilde{Θ}}_{13}, Θ_{33} \to {\tilde{Θ}}_{11} \end{matrix}

(99)

where the matrices ${\tilde{Θ}}_{mn}$ are given as

\begin{matrix} {\tilde{Θ}}_{00} = [\begin{matrix} e_{0} & e_{0} & e_{0} \end{matrix}], \\ {\tilde{Θ}}_{01} = [\begin{matrix} 1 & 0 & 0 \end{matrix}], & {\tilde{Θ}}_{02} = [\begin{matrix} 0 & 1 & 0 \end{matrix}], & {\tilde{Θ}}_{03} = [\begin{matrix} 0 & 0 & 1 \end{matrix}] \\ {\tilde{Θ}}_{11} = [\begin{matrix} e_{1} & e_{0} & e_{0} \end{matrix}], & {\tilde{Θ}}_{12} = [\begin{matrix} e_{2} & e_{0} & e_{0} \end{matrix}], & {\tilde{Θ}}_{13} = [\begin{matrix} e_{3} & e_{0} & e_{0} \end{matrix}] \\ {\tilde{Θ}}_{21} = [\begin{matrix} e_{0} & e_{1} & e_{0} \end{matrix}], & {\tilde{Θ}}_{22} = [\begin{matrix} e_{0} & e_{2} & e_{0} \end{matrix}], & {\tilde{Θ}}_{23} = [\begin{matrix} e_{0} & e_{3} & e_{0} \end{matrix}] \\ {\tilde{Θ}}_{31} = [\begin{matrix} e_{0} & e_{0} & e_{1} \end{matrix}], & {\tilde{Θ}}_{32} = [\begin{matrix} e_{0} & e_{0} & e_{2} \end{matrix}], & {\tilde{Θ}}_{33} = [\begin{matrix} e_{0} & e_{0} & e_{3} \end{matrix}] \end{matrix}

(100)

The matrices ${\tilde{Ψ}}_{t}^{v} \in R^{3 U \times V}$ , ${\tilde{Ψ}}_{b}^{v} \in R^{3 U \times V}$ , ${\tilde{Ψ}}_{p_{t}}^{v} \in R^{3 (2 q - 1) \times V}$ , and ${\tilde{Ψ}}_{p_{b}}^{v} \in R^{3 (2 q - 1) \times V}$ indicate the connectivity of the top nodes, bottom nodes, top boundary nodes, and bottom boundary nodes, respectively, with the vertical boundary strings. These matrices are expressed as

\begin{matrix} {\tilde{Ψ}}_{t}^{v} = O_{3 U \times V}, & {\tilde{Ψ}}_{b}^{v} = O_{3 U \times V}, & {\tilde{Ψ}}_{p_{t}}^{v} = I_{V}, & {\tilde{Ψ}}_{p_{b}}^{v} = - I_{V} \end{matrix}

(101)

This concludes the presentation of the node positions and connectivity matrices for the minimal and extended tensegrity plates. It must be noted that rows of the connectivity matrices $C_{B}$ and $C_{S}$ can be removed to eliminate bar or string members. This adds a higher level of flexibility in the topology design of tensegrity plates using the formulation presented here. The expressions for the node position vectors are derived in this work for the first time for tensegrity plates of any complexity. The derived formulas allow inputs such as plate diameter, thickness, and complexity (number of rings of prism units forming the plate). Furthermore, the connectivity matrices of a new type of tensegrity plates (the extended tensegrity plate) generated by adding further members to tensegrity plates comprised minimal 3 bar prisms are derived for the first time. The additional members provide higher stiffness in multiple directions and thus allow for more structurally robust tensegrity plates. Also, connectivity matrices without overlapping nodes are derived here directly and systematically for both minimal and extended tensegrity plates. Sub-matrices that distinguish among subsets of strings in different locations of the plate and between string nodes (corresponding to point masses) and mass-less nodes are derived for the first time. The string connectivity matrix $C_{S}$ for the minimal plate is divided into three main sub-connectivity matrices, which are related to top, bottom, and vertical strings. The string connectivity matrix for extended plate is formed by adding four additional sub-connectivity matrices, which are related to inner top, inner bottom, inner vertical, and boundary vertical strings. The ordering of the connectivity and node position matrices readily allows the user to alter or eliminate specific subsets of bars or strings, select a specific subset of strings as control elements, or change the positions of specific subsets of nodes in a systematic manner.

Implementation example: morphing tensegrity plates into domes

This section provides a demonstration of the implementation of the node positions and connectivity matrices derived in the previous sections. The demonstration entails dynamic simulations of tensegrity plates morphing into domes. In-depth analysis of the dynamics problem, material selection, or structural optimization of the resulting domes are not performed here, but are recommended for future studies. The dynamics model for tensegrity structures used here was summarized earlier in this article. Such dynamic equations are implemented in MATLAB^® using ODE4.

The shape of a tensegrity plate can be altered by changing the rest lengths of its strings. To transform a plate into a dome, the rest lengths of all the bottom strings are shortened simultaneously, while the rest lengths of all the top and vertical strings are kept constant throughout the transformation. The transformation of a tensegrity plate toward a dome obtained from such a strategy is illustrated in Figure 14.

Figure 14.

Transformation of a tensegrity plate into a dome performed by shortening the rest lengths of the bottom strings of the plate: (a) perspective view and (b) side view.

The following expression for the rest length ratios $μ_{i}$ of the bottom strings is employed

μ_{i} (t) = {\begin{matrix} 1 + \frac{t}{t_{e}} (μ_{e} - 1); & t \leq t_{e} \\ μ_{e}; & t > t_{e} \end{matrix}

(102)

where $μ_{e}$ is the final value of the rest length ratios and $t_{e}$ is the time when such a final ratio is achieved, here called shortening time. The initial value of time t is 0 in all the simulations. The expression for $μ_{i} (t)$ of the bottom strings in equation (102) is illustrated in Figure 15. The value of $μ_{i} (t)$ is constant and equal to 1 for all other strings in the plate (top and vertical strings). The initial position of the nodes, contained in the matrix N, is obtained using the procedure presented earlier. The initial velocity vector for all the nodes is assumed as the zero vector. The nodes on the top of the central prism in the plate are fixed in space to prevent rigid body motion (see Figure 14).

Figure 15.

Relationship between rest length ratio $μ_{i}$ versus time t for the bottom strings from equation (102).

The geometric and material parameters used in the dynamic simulations are provided in Table 1. All the strings have radius $r_{s}$ , elastic modulus $E_{s}$ , and damping coefficient $ζ_{s}$ . All the bars have mass density $ρ_{b}$ and radius $r_{b}$ , and all the point masses at the string nodes have mass $m_{σ}$ . Future studies that address specific applications of tensegrity plates/domes should consider material selection for the bars and strings.

Table 1.

Geometric and material parameters assumed in the dynamic simulations.

Parameter	Value	Units
d	10.0	m
h	0.5	m
$r_{s}$	1.0E−3	m
$E_{s}$	1.0	GPa
$ζ_{s}$	1.0E−3	GPa s
$r_{b}$	5.0−3	m
$ρ_{b}$	1000.0	kg/m³
$m_{σ}$	1.0E−2	kg

To observe the influence of the string shortening rate on the dynamics of the plates/domes, we show two simulations of a dome transformed from the same initial plate configuration using the same final rest length ratio $μ_{e} = 0.8$ but two different shortening times: $t_{e} = 1 s$ and $t_{e} = 100 s$ . Figure 16 highlights the node whose position history is compared in Figure 17 for the two cases. It is observed in Figure 17 that using $t_{e} = 1 s$ results in significant vibration of the plate prior to converging to its final configuration, while $t_{e} = 100 s$ shows a steady transformation toward the final configuration. Both cases converge to a configuration of static equilibrium since the damping coefficient of the strings, $ζ_{s}$ , is non-zero, see Table 1. The final configuration in both cases is the same, which can be observed in the position where the highlighted node converges in Figure 17(a) and (b). Analogous behavior is observed in any other node of the plate. Such results are also observed in dynamic simulations of minimal and extended plates with other complexities, although not shown here for the sake of brevity. Therefore, for the examples shown subsequently, the rate at which the strings are shortened does not affect the obtained final shape.

Figure 16.

Schematic of an extended tensegrity plate of complexity $q = 2$ . The position history of the highlighted node is plotted in Figure 17.

Figure 17.

Position history of the node highlighted in Figure 16: (a) shortening time $t_{e}$ of 1 s and (b) shortening time $t_{e}$ of 100 s.

In the next examples, the shape of the domes generated by shortening the bottom strings of the plates through equation (102) is compared with a parabola. Such a parabolic shape may be useful in potential applications such as antennas, mirrors, or civil constructions. Minimal and extended plates of complexities $q = 2, 3, 4$ and rest length ratios of $μ_{e} = 0.8, 0.85, 0.9, 0.95$ are considered. The studied plates are illustrated in Figure 18.

Figure 18.

Schematics of the studied plate topologies and complexities.

The final shape of the bottom face of the domes is compared against a parabola with the following formulation

z = - \frac{κ}{2} (x^{2} + y^{2}) + z_{0}

(103)

where $κ$ is the curvature and $z_{0}$ is the distance of the parabola vertex to the origin. To quantify how close is the shape of the bottom face of the generated domes to a parabola, the root-mean-squared error ( $E_{rms}$ ) of the z-position of the nodes with respect to the parabola is employed

E_{rms} = ‖ A {[\begin{matrix} κ & z_{0} \end{matrix}]}^{T} - b ‖

(104)

where the matrix $A \in R^{3 U \times 2}$ and the vector $b \in R^{3 U \times 1}$ are given as

A = [\begin{matrix} ⋮ & ⋮ \\ - \frac{1}{2} (x_{i}^{2} + y_{i}^{2}) & 1 \\ ⋮ & ⋮ \end{matrix}], b = [\begin{matrix} ⋮ \\ z_{i} \\ ⋮ \end{matrix}]

(105)

and $x_{i}$ , $y_{i}$ , and $z_{i}$ are the coordinates of the ith node. It is remarked that only the coordinates of the bottom nodes are included in A and b. The following equation is employed to determine $κ$ and $z_{0}$ of the parabola that best approximates the final shape of the dome, that is, that minimizes $E_{rms}$

[\begin{matrix} κ \\ z_{0} \end{matrix}] = A^{+} b

(106)

where $A^{+}$ denotes the pseudoinverse of A. Once $κ$ and $z_{0}$ are determined via equation (106), such values are substituted into equation (104) to determine the root-mean-squared error $E_{rms}$ .

Results showing $κ$ versus percentage rest length ratio $μ_{e} \times 100$ and normalized percentage error $(E_{rms} / d) \times 100$ versus final curvature $κ$ are provided in Figure 19. For all the considered topologies (minimal and extended), complexities, and rest length ratios, $(E_{rms} / d) \times 100$ is lower than 2%. Such a result indicates that the strategy employed here to morph a plate into a dome yields final shapes that are very close to parabolas. It is observed in Figure 19(a) that extended plates with any of the considered complexities achieve a higher curvature $κ$ for the same value of $μ_{e}$ than the minimal plates. The plots for normalized percentage error $(E_{rms} / d) \times 100$ versus $κ$ in Figure 19(b) indicate that similar errors are observed between minimal and extended plates for the same achieved curvature. Such an error decreases by increasing the complexity q of the plates.

Figure 19.

Comparison of the final dome shape of minimal and extended tensegrity plates for all the considered complexities: (a) final curvature $κ$ versus percentage rest length ratio $μ_{e} \times 100$ and (b) normalized percentage error versus final curvature $κ$ .

Schematics of curvature $κ$ versus percentage rest length ratio $μ_{e} \times 100$ such as those presented in Figure 19(a) can be used as guidance on the selection of the rest length ratio $μ_{e}$ that allows a plate to achieve a goal curvature value. Given the plate topology (complexity q and minimal or extended plate topology) and its dimensions, one can generate such plots through simulations to find the achievable curvature values of the plate and the associated rest length ratios that allows it to reach such curvatures.

Configurations of minimal and extended tensegrity plates of complexity $q = 3$ under a rest length ratio $μ_{e} = 0.8$ are shown in Figures 20 and 21, respectively. Figures 20(c) and 21(c) show the final configuration of the domes and their corresponding parabolic surface fits. Since the final dome configurations are in static equilibrium, their node positions and string rest lengths determined from the model can be recorded and be used to construct static, stable tensegrity domes.

Figure 20.

Configurations of a minimal tensegrity plate of complexity $q = 3$ under a rest length ratio $μ_{e} = 0.8$ : (a) initial configuration, (b) final configuration, and (c) final configuration and best parabolic surface fit with $κ = 0.16 m^{- 1}$ and $(E_{r m s} / d) \times 100 = 1.30 %$ .

Figure 21.

Configurations of an extended tensegrity plate of complexity $q = 3$ under a rest length ratio $μ_{e} = 0.8$ : (a) initial configuration, (b) final configuration, and (c) final configuration and best parabolic surface fit with $κ = 0.18 m^{- 1}$ and $(E_{r m s} / d) \times 100 = 1.75 %$ .

Conclusion

This article presented systematic analytical equations for the node positions and connectivity matrices of tensegrity plates with two different topologies. Such node positions and connectivity matrices are the fundamental descriptors in the modeling and design of tensegrity structures. We first considered the minimal tensegrity plate formed by the repetition of a tensegrity unit known as the minimal regular 3 bar prism. The unit is called minimal because it has the minimum number of strings (nine) required to form a stable pre-stressable prism with 3 bars. The minimal tensegrity plate has been studied in the literature, but sets of analytical formulas for their node positions and connectivity matrices were not available. Such analytical formulations allow for efficient computations and would facilitate future research efforts in the analysis and design of tensegrity plates. In addition, we provided the connectivity matrices of an extended tensegrity plate. The new plate has additional strings that can potentially increase stiffness in directions that have large compliance in conventional minimal plates. A directional stiffness comparison between the minimal and the new plate is not provided here but is recommended as future work.

To derive the node positions and connectivity matrices, the prisms in the hexagonal plates were divided into four parts, where Part 0 contains only the prism at the center and the surrounding three parts have the same number of prisms and are placed in a three-fold symmetric arrangement. This division simplified the derivation of analytical formulas for the connectivity matrices and node positions. The derived formulas allow inputs such as plate diameter, thickness, and complexity (number of rings of prism units forming the plate). The Supplemental Material of this article includes MATLAB^® scripts that output the derived node positions and connectivity matrices for tensegrity plates given their size parameters and complexity. Rows of the connectivity matrices can be removed to eliminate bar or string members. This adds a higher level of flexibility in the topology design of tensegrity plates using the formulation presented here. Finally, a demonstration of the implementation of the node positions and connectivity matrices in the modeling of tensegrity structures was briefly provided. The demonstration consisted of dynamic simulations of tensegrity plates morphing into domes.

Supplemental Material

extended_prism_plate – Supplemental material for Analytical equations for the connectivity matrices and node positions of minimal and extended tensegrity plates

Supplemental material, extended_prism_plate for Analytical equations for the connectivity matrices and node positions of minimal and extended tensegrity plates by Shuhui Jiang, Robert E Skelton and Edwin A Peraza Hernandez in International Journal of Space Structures

Supplemental Material

minimal_prism_plate – Supplemental material for Analytical equations for the connectivity matrices and node positions of minimal and extended tensegrity plates

Supplemental material, minimal_prism_plate for Analytical equations for the connectivity matrices and node positions of minimal and extended tensegrity plates by Shuhui Jiang, Robert E Skelton and Edwin A Peraza Hernandez in International Journal of Space Structures

Footnotes

Acknowledgements

S.J. would like to acknowledge the financial support from the National Natural Science Foundation of China (NSFC) Grant No. 51578492. Any opinions, findings, and conclusions expressed in this material are those of the authors and do not necessarily reflect those of the National Natural Science Foundation of China. E.A.P.H. acknowledges the start-up faculty support from the Department of Mechanical and Aerospace Engineering at the University of California, Irvine.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Edwin A Peraza Hernandez

Supplemental material

Supplemental material for this article is available online.

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Supplementary Material

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