Deployment Dynamic Modeling and Driving Schemes for a Ring-Truss Deployable Antenna

Abstract

Mesh reflector antennas are widely used in space tasks owing to their light weight, high surface accuracy, and large folding ratio. They are stowed during launch and then fully deployed in orbit to form a mesh reflector that transmits signals. Smooth deployment is essential for duty services; therefore, accurate and efficient dynamic modeling and analysis of the deployment process are essential. One major challenge is depicting time-varying resistance of the cable network and capturing the cable-truss coupling behavior during the deployment process. This paper proposes a general dynamic analysis methodology for cable-truss coupling. Considering the topological diversity and geometric nonlinearity, the cable network’s equilibrium equation is derived, and an explicit expression of the time-varying tension of the boundary cables, which provides the main resistance in truss deployment, is obtained. The deployment dynamic model is established, which considers the coupling effect between the soft cables and deployable truss. The effects of the antenna’s driving modes and parameters on the dynamic deployment performance were investigated. A scaled prototype was manufactured, and the deployment experiment was conducted to verify the accuracy of the proposed modeling method. The proposed methodology is suitable for general cable antennas with arbitrary topologies and parameters, providing theoretical guidance for the dynamic performance evaluation of antenna driving schemes.

1 Introduction

Spaceborne antennas have been widely used in various fields, such as space communications, military reconnaissance, and deep space exploration, as the essential equipment for transmitting and receiving signals [1,2,3,4,5]. In these applications, large spaceborne antennas are required to be stowed during launch and then fully deployed after entering the desired orbit. Deployable antennas have received more attention and development, along with the acceleration of human exploration of outer space [6,7,8,9,10,11]. Cable antennas are currently the focus of mainstream research owing to their high surface accuracy, structural simplicity, and stowage efficiency [12,13,14,15,16]. Take the ring-truss mesh reflector antenna as an example. It consists of five parts: the ring truss, front cable net, rear cable net, tension ties, and metal mesh, as shown in Figure 1 [17,18,19]. The truss of the antenna contains several parallelogram units composed of the transverse bar, longitudinal bar, thick oblique bar, thin oblique bar, five-dimensional hinge, three-dimensional hinge, and pulley. Figure 1 describes the driving mode of the ring-truss mesh reflector antenna, which is deployed using motors to contract the driving ropes in the oblique bars or by the torsion springs in the connecting hinges of the parallelogram units [20,21,22,23]. The three-dimensional hinge is equipped with synchronization gears so that each unit can be deployed synchronously. The challenges of ensuring the smooth and successful deployment of the antenna focus on accurately capturing the dynamic behavior of the antenna and revealing the influence of driving parameters on the dynamic performance. The key to solving this problem is the dynamic modeling and analysis of the cable-truss coupling effect during deployment.

Figure 1

Ring-truss mesh reflector antenna

The boundary nodes of the cable network are connected to the truss, which stretches the cable network to unfold by connecting nodes [24,25,26,27,28]. In this process, each cable constantly switches between the slacked and tensioned states. The cables interact with each other, resulting in time-varying cable tensions and geometric configurations. The resistance applied to the truss by the boundary cables also varies during deployment [29,30,31,32,33]. The influence of the cable network on the deployment of the truss should be fully captured in the cable-truss coupling dynamic analysis. Nevertheless, the strong geometric nonlinearity and superposition of the small deformation of cables on their significant motion lead to the modeling and solution of problems [34,35,36,37]. In addition, the cable network topology, truss configuration, and antenna parameters vary with applications, challenging the generality [38,39,40,41,42,43]. Therefore, a general and stable modeling and analysis method for cable truss mechanisms applicable to various cable antennas is urgently required.

The cable network contains a large number of cables whose tension varies with time during deployment. Many scholars have studied the deployment process of the cable network [44,45,46,47,48]. Greco et al. [49] modeled cable segments using catenary elements. Nie et al. [50] used a spring-damper element to establish the cable network model. Ambrosio et al. [51] studied the optimal deployment of a radar antenna using the finite element method. He et al. [20] proposed a general deployment analysis method for soft cable networks, ranging from slacked to tensioned. However, these studies used numerical solution algorithms based on time discretization, and the time-varying tensions of the boundary cables during deployment were not derived.

Researchers have also studied the coupling effects between cable networks and trusses [52,53,54,55,56]. Zhang et al. [57] established a mapping relationship among the antenna structure, deployment trajectory, and control system. Lu et al. [58] modeled the cable wheel system of an antenna by considering the long-distance force transmission. Jiang et al. [59] established a dynamic model of a ring-truss antenna using an absolute nodal coordinate formulation. Nie et al. [60] established a truss deployment model using the Lagrange equation. Although the above studies realized the integrated calculation of the truss and cable network by numerical algorithm, an analytical method capturing the cable-truss coupling effects and applied to various cable-truss antennas has not been well studied.

Many scholars have extensively researched antenna deployable mechanism designs, deployment kinematics, and dynamic analyses. Meng et al. [61] proposed a deployable truss configuration with good dynamic performance. Wu et al. [62] derived the governing equations for the deployment motion and boundary conditions. Wang et al. [63] designed a deployment experiment to investigate the antenna deployment process. Liu et al. [64] developed a novel equivalent dynamic model for vibration analysis. However, the existing research focuses on the dynamic analysis of the deployment process. There are no driving modes and parameter design approaches aimed at dynamic performance.

In summary, there are still no general and effective analytical models for the deployment analysis of cable-truss coupling mechanisms, leading to problems in dynamic performance optimization and driving design. In this study, we propose a parameterized cable-truss integrated deployment analysis method by introducing the graph theory to depict the topology of the cable network and truss. A scaled prototype was manufactured, and a deployment experiment was conducted to verify the accuracy of the proposed dynamic analysis method. The driving parameters are designed based on the required driving force and energies. The influence of the driving mode and parameters on the dynamic behavior of the antenna were thoroughly studied and discussed to provide theoretical guidance to improve the dynamic performance of ring-truss deployable antennas.

2 Kinematic Analysis of the Truss

In this section, we take the ring truss as an example to analyze its deployment motion, which provides fundamental variables for the dynamic modeling of cable-truss coupling. As shown in Figure 2, the ring truss is composed of several similar parallelogram units consisting of bars and hinges. The truss expands from a fully stowed state to a deployed state by contracting the oblique bar. The acute angle between the transverse and longitudinal bars is defined as the deployment angle φ that represents the antenna’s deployment phase, which varies from 0 to π/2 rad. In Figure 2, A_j, B_j, C_j, and D_j denote the truss nodes in parallelogram unit j. C_j and D_j coincide with A_j+1 and B_j+1 of parallelogram unit (j+1). R₁, R₂, R₃, and R₄ denote the lengths of the transverse, longitudinal, thick oblique, and thin oblique bars, respectively. The acute angle between the two adjacent units is defined as θ = 2π/n, where n is the number of parallelogram units (n ≥ 4, and n is an even number).

Figure 2

The ring truss and the parallelogram unit

As shown in Figure 2, A₁B₁ is considered as the fixed bar, and the frame (x_j, y_j, z_j) is taken as the local coordinate system of parallelogram unit j. The local coordinate system of the first parallelogram unit is chosen as the global coordinate system. If the position vector of an arbitrary point P in parallelogram unit j under the local coordinate system (x_j, y_j, z_j) is ${\varvec{r}}_{P}^{j}$, its position vector under the local coordinate system (x_i, y_i, z_i) is derived as

$${\varvec{r}}_{P}^{i} = {\varvec{T}}^{ij} {\varvec{r}}_{P}^{j} ,$$

(1)

$$\begin{gathered} {\varvec{T}}^{ij} = \hfill \\ \left[ {\begin{array}{*{20}c} {\cos \left[ {\left( {j - i} \right)\theta } \right]} & 0 & {\sin \left[ {\left( {j - i} \right)\theta } \right]} & {\left\{ {\sum\limits_{k = i + 1}^{j} {\cos \left[ {\left( {k - i - 1} \right)\theta } \right]} } \right\}R_{1} \sin \varphi } \\ 0 & 1 & 0 & {\frac{1}{2}\left[ {1 - \left( { - 1} \right)^{j - 1} } \right]R_{1} \cos \varphi } \\ { - \sin \left[ {\left( {j - i} \right)\theta } \right]} & 0 & {\cos \left[ {\left( {j - i} \right)\theta } \right]} & {\left\{ { - \sum\limits_{k = i + 1}^{j} {\sin \left[ {\left( {k - i - 1} \right)\theta } \right]} } \right\}R_{1} \sin \varphi } \\ 0 & 0 & 0 & 1 \\ \end{array} } \right], \hfill \\ \end{gathered}$$

(2)

where T^ij is the coordinate transformation matrix.

The velocity vector of the truss node is obtained by taking the derivative of the position vector. Assuming that the velocity vector of an arbitrary point in parallelogram unit j under the local coordinate system (x_j, y_j, z_j) is ${\varvec{r}}_{P}^{{j^{\prime}}}$, its velocity vector under the local coordinate system (x_i, y_i, z_i) is denoted by

$${\varvec{r}}_{P}^{{i^{\prime}}} = {\varvec{T}}^{{ij^{\prime}}} {\varvec{r}}_{P}^{j} + {\varvec{T}}^{ij} {\varvec{r}}_{P}^{{j^{\prime}}} ,$$

(3)

The position vectors of nodes A_j, B_j, C_j, and D_j in the local coordinate system are derived as

$$\left\{ \begin{gathered} {\varvec{r}}_{{A_{j} }}^{j} \, = \,\left[ {\begin{array}{*{20}c} 0 & 0 & {{\text{R}}_{2} } & 1 \\ \end{array} } \right]^{{\text{T}}} ,\,{\varvec{r}}_{{C_{j} }}^{1} = {\varvec{r}}_{{B_{j + 1} }}^{1} , \hfill \\ {\varvec{r}}_{{B_{j} }}^{j} \, = \,\left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 1 \\ \end{array} } \right]^{{\text{T}}} ,\,{\varvec{r}}_{{D_{j} }}^{1} = {\varvec{r}}_{{A_{j + 1} }}^{1} , \hfill \\ \end{gathered} \right.$$

(4)

In this way, the position and velocity vectors of the truss nodes obtained in the global coordinate can be obtained. Assuming that the mass of the bars was evenly distributed along the length direction, their centroids were the geometric centers. The position vectors of the centroids are denoted by the average position vectors of the endpoints, as shown in Eq. (5).

$$\left\{ \begin{gathered} {\varvec{r}}_{{A_{j} B_{j} }}^{1} = \frac{{{\varvec{r}}_{{A_{j} }}^{1} + {\varvec{r}}_{{B_{j} }}^{1} }}{2}, \, {\varvec{r}}_{{B_{j} C_{j} }}^{1} = \frac{{{\varvec{r}}_{{B_{j} }}^{1} + {\varvec{r}}_{{C_{j} }}^{1} }}{2}, \hfill \\ \, {\varvec{r}}_{{C_{j} D_{j} }}^{1} = \frac{{{\varvec{r}}_{{C_{j} }}^{1} + {\varvec{r}}_{{D_{j} }}^{1} }}{2}, \, {\varvec{r}}_{{D_{j} A_{j} }}^{1} = \frac{{{\varvec{r}}_{{D_{j} }}^{1} + {\varvec{r}}_{{A_{j} }}^{1} }}{2}, \hfill \\ {\varvec{r}}_{{A_{j} C_{j} }}^{1} = {\varvec{r}}_{{A_{j} }}^{1} + \hfill \\ \frac{{m_{4} \frac{1}{2}R_{4} + m_{3} \left( {\sqrt {R_{1}^{2} + R_{2}^{2} + 2R_{1} R_{2} \cos \varphi } - \frac{1}{2}R_{3} } \right)}}{{\left( {m_{3} + m_{4} } \right)\sqrt {R_{1}^{2} + R_{2}^{2} + 2R_{1} R_{2} \cos \varphi } }}\left( {{\varvec{r}}_{{C_{j} }}^{1} - {\varvec{r}}_{{A_{j} }}^{1} } \right), \hfill \\ {\varvec{r}}_{{B_{j} D_{j} }}^{1} = {\varvec{r}}_{{B_{j} }}^{1} + \hfill \\ \frac{{m_{3} \frac{1}{2}R_{3} + m_{4} \left( {\sqrt {R_{1}^{2} + R_{2}^{2} + 2R_{1} R_{2} \cos \varphi } - \frac{1}{2}R_{4} } \right)}}{{\left( {m_{3} + m_{4} } \right)\sqrt {R_{1}^{2} + R_{2}^{2} + 2R_{1} R_{2} \cos \varphi } }}\left( {{\varvec{r}}_{{D_{j} }}^{1} - {\varvec{r}}_{{B_{j} }}^{1} } \right), \hfill \\ \end{gathered} \right.$$

(5)

where m₃ and m₄ are the mass of the thick and thin oblique bars, respectively.

Similarly, the centroid velocity and rotational angular velocity of the bars are denoted by

$$\left\{ \begin{gathered} {{\varvec{\upnu}}}_{{A_{j} B_{j} }}^{1} = \frac{{{\dot{\varvec{r}}}_{{A_{j} }}^{1} + {\dot{\varvec{r}}}_{{B_{j} }}^{1} }}{2},{{\varvec{\upnu}}}_{{B_{j} C_{j} }}^{1} = \frac{{{\dot{\varvec{r}}}_{{B_{j} }}^{1} + {\dot{\varvec{r}}}_{{C_{j} }}^{1} }}{2},{{\varvec{\upnu}}}_{{C_{j} D_{j} }}^{1} = \frac{{{\dot{\varvec{r}}}_{{C_{j} }}^{1} + {\dot{\varvec{r}}}_{{D_{j} }}^{1} }}{2} \hfill \\ \, {{\varvec{\upnu}}}_{{D_{j} A_{j} }}^{1} = \frac{{{\dot{\varvec{r}}}_{{D_{j} }}^{1} + {\dot{\varvec{r}}}_{{A_{j} }}^{1} }}{2}, \hfill \\ \, \hfill \\ {{\varvec{\upnu}}}_{{A_{j} C_{j} }}^{1} = {\dot{\varvec{r}}}_{{A_{j} }}^{1} + \frac{{m_{4} \frac{1}{2}R_{4} + m_{3} \left( {L - \frac{1}{2}R_{3} } \right)}}{{\left( {m_{3} + m_{4} } \right)L}}\left( {{\dot{\mathbf{r}}}_{{C_{j} }}^{1} - {\dot{\mathbf{r}}}_{{A_{j} }}^{1} } \right), \hfill \\ {{\varvec{\upnu}}}_{{B_{j} D_{j} }}^{1} = {\dot{\varvec{r}}}_{{B_{j} }}^{1} + \frac{{m_{3} \frac{1}{2}R_{3} + m_{4} \left( {L - \frac{1}{2}R_{4} } \right)}}{{\left( {m_{3} + m_{4} } \right)L}}\left( {{\dot{\varvec{r}}}_{{D_{j} }}^{1} - {\dot{\varvec{r}}}_{{B_{j} }}^{1} } \right), \hfill \\ {{\varvec{\upomega}}}_{{A_{j} B_{j} }}^{1} = \frac{{{\dot{\varvec{r}}}_{{A_{j} }}^{1} - {\dot{\varvec{r}}}_{{B_{j} }}^{1} }}{{R_{2} }},{{\varvec{\upomega}}}_{{B_{j} C_{j} }}^{1} = \frac{{{\dot{\mathbf{r}}}_{{C_{j} }}^{1} - {\dot{\varvec{r}}}_{{B_{j} }}^{1} }}{{R_{1} }},{{\varvec{\upomega}}}_{{C_{j} D_{j} }}^{0} = \frac{{{\dot{\varvec{r}}}_{{D_{j} }}^{1} - {\dot{\varvec{r}}}_{{C_{j} }}^{1} }}{{R_{2} }}, \, \hfill \\ {{\varvec{\upomega}}}_{{D_{j} A_{j} }}^{0} = \frac{{{\dot{\varvec{r}}}_{{D_{j} }}^{1} - {\dot{\varvec{r}}}_{{A_{j} }}^{1} }}{{R_{1} }},{{\varvec{\upomega}}}_{{A_{j} C_{j} }}^{1} = \frac{{{\dot{\varvec{r}}}_{{C_{j} }}^{1} - {\dot{\varvec{r}}}_{{A_{j} }}^{1} }}{L},{{\varvec{\upomega}}}_{{B_{j} D_{j} }}^{1} = \frac{{{\dot{\varvec{r}}}_{{D_{j} }}^{1} - {\dot{\varvec{r}}}_{{B_{j} }}^{1} }}{L}, \hfill \\ \end{gathered} \right.$$

(6)

where L is the length of drive rope in the parallelogram unit.

According to the above derivation, we obtained the displacement, velocity, and angular velocity of the bars and hinges in the global coordinate under the arbitrary deployment angle. The deployment process of the truss was accurately depicted.

3 Cable-Truss Coupling Dynamic Analysis Method

In the deployment of the cable antenna, the complex motion of the truss, geometric nonlinearity, time-varying resistance, and cable-truss real-time coupling effect significantly influenced the dynamic performance. This section introduces a cable-truss coupling dynamic analysis method to accurately capture the dynamic behavior of the antenna during deployment. The cable network was discretized into many cable elements according to its topological connection. Parameterized equilibrium equations were derived by depicting the connection of the cables with a topology matrix. In this way, the explicit expression of the time-varying tension of the boundary cables was derived, which can be coupled with the truss model. Then, we derived the expressions of the kinetic and gravitational potential energy of the truss based on the kinematic analysis in Section 2, and established the dynamic model of the cable antenna using the Lagrange equation of the first kind.

3.1 The Cable Network Modeling

During the antenna’s deployment, each cable switches between the slacked and tensioned states under the stretching of the truss. The cables interact with each other during the unfolding process, resulting in time-varying cable tensions and geometric shapes. Additionally, the cables’ small deformations are coupled with their large-scale overall motion, along with the cable’s strong geometric nonlinearity, leading to problems in the derivation of the time-varying tensions of the boundary cables, which are also the resistance load of the truss.

Figure 3 shows the cable network. The cable nodes connected to the truss are defined as boundary nodes, while the other cable nodes are free nodes. The cables connected to the truss are defined as boundary cables, while the other cables are inner cables. The constitutive model of the cable element adopts Hooke’s law with variable stiffness, which depends on the slacked and tensioned states of the cable element. When the distance between the two end nodes of the cable element is larger than the cable’s unstressed length, the cable is in a tensioned state, and the stiffness is calculated using the geometric parameters and elastic modulus. Once the distance between the two endpoints of the cable element was less than or equal to the unstressed length of the cable, the cable was slaked, and the stiffness was set to zero. In this way, all the constitutive equations of the cable element can be derived to form systematic equilibrium equations at arbitrary moments of the unfolding process. In the following derivation, the connection of the cable elements and cable nodes is depicted by the topology matrix so that the systematic equilibrium equations of the cable network can be parameterized for application to various topologies and parameters. Based on cable network modeling, we derived the explicit expression for the time-varying tension of boundary cables.

Figure 3

The cable network of cable antennas

Assuming that the cable network consists of m cable elements and n cable nodes with n_f free nodes and n_b boundary nodes, number these cables and nodes in any order and the corresponding topology matrix are obtained by Eq. (7). Suppose that cable i connects nodes j_a and j_b (j_a ≤ j_b), then set the j_ath and j_bth elements of the ith row of the topology matrix D to 1 and -1, and set other elements in the matrix to zero. Taking the simplified cable network in Figure 4(a) as an example, the black numbers represent the nodal labels, and the numbers inside the red circles represent the cable labels. Figure 4(b) shows the corresponding topology matrix of the cable network in Figure 4(a) according to Eq. (7).

$$\user2{D}\left( {i,j} \right){\text{ = }}\left\{ \begin{gathered} 1,{\kern 1pt} \quad j = j_{{\text{a}}} , \hfill \\ - 1,{\text{ }}\quad j = j_{{\text{b}}} , \hfill \\ 0,\quad {\text{others}}{\text{.}} \hfill \\ \end{gathered} \right.{\text{ }}$$

(7)

Figure 4

A simplified cable network with 10 nodes and 15 cables. a Serial number of cable elements and cable nodes. b The cable network’s topological matrix

The topology matrix is used to describe the cable network and truss in this study. Therefore, the matrix operation can be used to solve the nonlinear equations, and its stability and computational efficiency are excellent.

The mass of the cable network is evenly distributed at the corresponding nodes, and the external forces applied to the free nodes during unfolding are always equal to the gravity of the equivalent node. Thus, the equilibrium equations of the free nodes at an arbitrary moment can be expressed as

$$- {\varvec{D}}_{{\text{f}}}^\text{T} \text{diag}\left\{ {{\varvec{Q}}\left( t \right)\emptyset {\varvec{L}}\left( t \right)} \right\}{\varvec{D}}_{{\text{f}}} {\varvec{S}}_{{\text{f}}} \left( t \right) = {\varvec{G}}_{{\text{f}}} ,$$

(8)

where Q(t) is the cable tension vector at time t; L(t) is the tensile length vector of the a cables at time t; S_f (t) is the coordinate matrix of the free nodes at time t; and G_f is the n_f×3-dimensional equivalent gravity matrix, whose elements are the equivalent gravity at each node.

The cable tension vector Q(t) is

$${\varvec{Q}}\left( t \right) = {\varvec{K}}\left( t \right) \circ \user2{\delta l}\left( t \right),$$

(9)

where $\circ$ is the Hadamard product, which represents the product of the elements of the corresponding positions of the two matrices, and

$${\varvec{K}}\left( t \right)\, = \,\left[ {\begin{array}{*{20}c} {k_{1} \left( t \right)} & {k_{2} (t)} & \cdots & {k_{i} \left( t \right)} & \cdots & {k_{m} \left( t \right)} \\ \end{array} } \right],$$

(10)

$$k_{i} \left( t \right) = \left\{ \begin{gathered} k_{i} , \, l_{i} \left( t \right) > l_{i}, \hfill \\ 0, \, l_{i} \left( t \right) \le l_{i}, \hfill \\ \end{gathered} \right.$$

(11)

$$\user2{\delta l}\left( t \right)\, = \,\left[ {\begin{array}{*{20}c} {\delta l_{1} \left( t \right)} & {\delta l_{2} \left( t \right)} & \cdots & {\delta l_{i} \left( t \right)} & \cdots & {\delta l_{m} \left( t \right)} \\ \end{array} } \right],$$

(12)

$$\delta l_{i} \left( t \right) = \left\{ \begin{gathered} \left( {l_{i} \left( t \right) - l_{i} } \right),\quad l_{i} \left( t \right) > l_{i}, \hfill \\ 0,{\kern 1pt} \quad l_{i} \left( t \right) \le l_{i}, \hfill \\ \end{gathered} \right.$$

(13)

where K(t) is the cable stiffness vector at time t; ${\varvec{\delta l}}(t)$ is the cable elongation vector at time t; k_i is the linearized stiffness of cable i; l_i is the unstressed length of cable i; and l_i(t) is the tensile length of the cable at time t, which is the element in the tensile length vector of the cables shown in Eq. (14).

$${\varvec{L}}\left( t \right) = \sqrt {\left[ {{\varvec{DX}}\left( t \right)} \right]^{2} + \left[ {{\varvec{DY}}\left( t \right)} \right]^{2} + \left[ {{\varvec{DZ}}\left( t \right)} \right]^{2} } .$$

(14)

Up to now, coordinates of the free nodes can be calculated by substituting Eqs. (9) and (14) into Eq. (8), and the coordinates of the boundary nodes are equal to the corresponding coordinates of the truss nodes derived in Section 2.

According to the derivation above, we can obtain the explicit expression for the time-varying tension of the boundary cables using Eq. (15). The real-time coupling effect between the soft cables and truss was reflected by the time-varying boundary cable tension and boundary node coordinates.

$$\begin{gathered} \left[ {\begin{array}{*{20}c} {{\varvec{F}}_{{\text{A}}} } & {;{\varvec{F}}_{{\text{B}}} } \\ \end{array} } \right] = \hfill \\ \left[ {\begin{array}{*{20}c} { - {\varvec{D}}_{{\text{b}}}^\text{T} \text{diag}\left\{ {{\varvec{Q}}\left( t \right)\emptyset {\varvec{L}}\left( t \right)} \right\}{\varvec{D}}_{{\text{b}}} {\varvec{S}}_{{\text{d}}} \left( t \right)} & {{\mathbf{0}}_{{n_{{\text{b}}} \times 1}} } \\ \end{array} } \right], \hfill \\ \end{gathered}$$

(15)

where S_d(t) is the coordinate matrix of the boundary nodes at time t.

The elastic energy of the cable network can be denoted by

$$V_{{\text{c}}} \left( t \right)\,\, = \frac{1}{2}{\varvec{Q}}(t)\user2{\delta l}\left( t \right)^{{\text{T}}} .$$

(16)

3.2 The Truss Modeling

During deployment, the generalized driving force and resistance load of the boundary cables lead to variations in the energy of the truss, including the kinetic energy, gravitational potential energy, and systematic dissipation energy. The kinetic energy of the truss consists of the kinetic energies of the bars and hinges, as shown in Eq. (17). Assuming that the mass of the bars is evenly distributed along the length direction, their mass is concentrated in the centroids. The kinetic energy of the bars is denoted by Eq. (18). The kinetic energy of the hinges is expressed by Eq. (19).

$$T = T_{{\text{b}}} + T_{{\text{h}}} ,$$

(17)

$$\begin{gathered} T_{{\text{b}}} = \frac{1}{2}\sum\limits_{j = 1}^{n} {\left[ {m_{1} \left( {\left| {{\varvec{v}}_{{A_{j} D_{j} }} } \right|^{2} + \left| {{\varvec{v}}_{{B_{j} C_{j} }} } \right|^{2} } \right) + m_{2} \left| {{\varvec{v}}_{{A_{j} B_{j} }} } \right|^{2} } \right.} \hfill \\ + \left( {m_{3} + m_{4} } \right)\left( {\left| {{\varvec{v}}_{{A_{j} C_{j} }} } \right|^{2} + \left| {{\varvec{v}}_{{B_{j} D_{j} }} } \right|^{2} } \right) \hfill \\ + J_{1} \left( {\left| {{{\varvec{\upomega}}}_{{A_{j} D_{j} }} } \right|^{2} + \left| {{{\varvec{\upomega}}}_{{B_{j} C_{j} }} } \right|^{2} } \right) + J_{2} \left| {{{\varvec{\upomega}}}_{{A_{j} B_{j} }} } \right|^{2} \hfill \\ \left. { + J_{3} \left| {{{\varvec{\upomega}}}_{{A_{j} C_{j} }} } \right|^{2} + J_{4} \left| {{{\varvec{\upomega}}}_{{B_{j} D_{j} }} } \right|^{2} } \right], \hfill \\ \end{gathered}$$

(18)

$$T_{{\text{h}}} = \frac{1}{2}\sum\limits_{j = 1}^{n} {\left( {m_{{\text{t}}} \left| {{\varvec{v}}_{{{\text{A}}_{j} }}^{1} } \right|^{2} + m_{{\text{f}}} \left| {{\varvec{v}}_{{{\text{B}}_{j} }}^{1} } \right|^{2} } \right)} ,$$

(19)

where T_b and T_h denote the kinetic energies of the bars and hinges, respectively. m₁ and m₂ are the masses of the transverse and longitudinal bars, respectively. J₁, J₂, J₃, and J₄ denote the rotational inertia of the transverse, longitudinal, odd unit oblique, and even unit oblique bars, respectively, and were calculated by Eq. (20). m_t and m_f are the masses of the three-dimensional and five-dimensional hinges, respectively.

$$\left\{ \begin{gathered} J_{1} = \frac{1}{3}m_{1} R_{1}^{2} , \, J_{3} = \frac{1}{3}m_{3} R_{3}^{2} + \frac{1}{12}m_{4} R_{4}^{2} + m_{4} \left( {L - \frac{{R_{4} }}{2}} \right)^{2} {, } \hfill \\ J_{2} = \frac{1}{3}m_{2} R_{2}^{2} , \, J_{4} = \frac{1}{3}m_{4} R_{4}^{2} + \frac{1}{12}m_{3} R_{3}^{2} + m_{3} \left( {L - \frac{{R_{3} }}{2}} \right)^{2} , \hfill \\ \end{gathered} \right.$$

(20)

Except for the kinematic energy variation, the centroid of the antenna varied during deployment, which leads to gravitational potential energy variation. The fully deployed state of the antenna is defined as the reference configuration for zero gravitational potential energy. Then, gravitational potential energy of the truss, including the potential energies of the bars and hinges, is shown in Eq. (21).

$$G = \left[ {\frac{n}{2}\left( {m_{2} + m_{{\text{t}}} + m_{{\text{f}}} + m_{{3}} + m_{{4}} } \right) + nm_{1} } \right]gR_{1} \cos \varphi ,$$

(21)

where g is the gravitational acceleration, and g is set according to the working conditions.

The antenna deployment process is relatively slow, and the damping of the hinges can be considered viscous damping proportional to the deployment angular velocity. According to the Rayleigh dissipation formula, the systematic dissipation energy can be expressed as

$$\Phi = \sum\limits_{{j = 1}}^{n} {4\int_{0}^{v} {\xi \varphi ^{\prime}} d\varphi ^{\prime} =\, } 2n\xi \varphi ^{\prime}\,^{2} ,$$

(22)

where $\xi$ represents the dissipation coefficient.

If the antenna is driven by the torsion springs in the three-dimensional hinge, the angular variation of the torsion springs corresponds to the angular displacement during deployment, and their generalized driving torques are denoted by Eq. (23). If the antenna is deployed by the driving rope in the oblique bars, then the length variation of the driving rope corresponds to the telescopic length variation of the oblique bars. The generalized driving torque is given by Eq. (24).

$$M_{{\text{s}}} { \,=\, }nk_{{\text{s}}} \left( {\varphi_{{\text{d}}} - \varphi } \right) - \kappa \varphi^{\prime},$$

(23)

$$M_{{\text{r}}} = F_{{\text{r}}} \frac{\partial L}{{\partial \varphi }},$$

(24)

where M_s and M_r denote the generalized driving torque of the torsion springs and driving rope; k_s, φ_d, and $\kappa$ are the torsion spring’s stiffness coefficient, initial pre-deflection angle, and damping coefficient, respectively; and F_r is the driving force of the driving rope.

3.3 Cable-Truss Coupling Dynamic Modeling

The cable-truss coupling dynamic model is established by the Lagrange equation of the first kind in this section. The deployment angle φ is chosen as the generalized coordinate. During deployment, the kinetic and potential energies of the truss as well as the dissipation energy are derived, which are always equal to the work done by the driving torque and resistance load applied to the cable-truss connecting nodes, as shown in Eq. (25). The time-varying tension of the boundary cables applied to the connecting nodes on the truss is derived in Section 3.1, while the energy variation of the truss is modeled in Section 3.2. In summary, the cable-truss coupling dynamic model is established by substituting Eqs. (15), (17), and (21)–(24) into Eq. (25), and the real-time coupling of the interaction forces and positional constraints between the cable network and truss in the deployment are fully captured.

$$\frac{d}{\text{d}t}\frac{\partial T}{{\partial \varphi^{\prime}}} - \frac{\partial T}{{\partial \varphi }} + \frac{\partial G}{{\partial \varphi }} + \frac{\partial \Phi }{{\partial \varphi^{\prime}}} = M + \sum\limits_{j = 1}^{n} {\left( {{\varvec{F}}_{{A_{j} }} \frac{{\partial {\varvec{r}}_{{A_{j} }} }}{\partial \varphi } + {\varvec{F}}_{{B_{j} }} \frac{{\partial {\varvec{r}}_{{B_{j} }} }}{\partial \varphi }} \right),}$$

(25)

where T denotes the kinetic energy of the truss; G denotes the gravitational potential energy of the truss; Φ denotes the systematic dissipation energy; M denotes the generalized driving torque; and F_Aj and F_Bj denote the tension vector of the 1×4-dimensional boundary cables applied to the nodes A_j and B_j of the jth parallelogram unit.

4 Comparison and Experimental Verification

This section verifies the proposed dynamic modeling and analysis method based on literature and experiments. The calculated results were compared with the previous literature and experimental data.

4.1 Comparison with Other Literature

A similar simulation was conducted for comparison with previous literature to verify the modelling method used in this study. Figure 5 shows the antenna configuration and cable network serial numbers used in the simulation. We calculated the required driving force under the desired deployment motion in Ref. [60]. The results are shown in Figures 6, 7 and 8. These results were compared with those in Ref. [60]. The antenna parameters are listed in Table 1. The unstressed cable lengths are listed in Table 2.

Figure 5

The cable network serial numbers of a mesh reflector antenna with twelve parallelogram units

Figure 6

Predesigned deployment motion [50]

Table 1 Parameters of the antenna with twelve parallelogram units

Table 2 The cables’ unstressed length (mm)

Assume the antenna is deployed in a predesigned deployment motion, whose deployment angle and angular velocity curves are shown in Figure 6. The deployment time was set to 10000 s, mainly for comparison with the calculation results in Ref. [50] to verify the accuracy of the modeling method. The deployment angular velocity gradually increased and then decreased to zero before locking, which could effectively avoid impact and vibration to achieve the desired smooth deployment.

We calculated the time-varying tension of the cable network under the predesigned deployment motion in Figure 6. Then, the cable tension-time curve was transformed into the cable tension-deployment angle curve according to the predesigned motion for comparison with the results in Ref. [50].

Figure 7 shows the obtained cable tension. Figure 8 shows the deviation between the cable tensions obtained in this study and those reported in Ref. [50]. According to Figure 8, the maximum deviation was 1.1 N, which was less than 1.4% of the results. This demonstrates that the calculated cable tensions agree well with the results reported in Ref. [50], which verifies the accuracy of the calculation.

Figure 7

Cable tension-deployment angle curve

Figure 8

Deviation of the cable tensions

The comparison of the required driving force for the predesigned deployment motion is shown in Figure 9. The maximum deviation is 24.2 N, which is less than 6.6% of the results in Ref. [60]. This indicates that the required driving force agree well with the previous literature, which verifies the accuracy of the proposed cable-truss coupling dynamic analysis method. As the calculation in Ref. [60] is based on the quasi-static analysis of the unfolding process of the cable network, and cable truss is decoupled in the modeling, it does not fully capture the coupling effects between the cable network and truss, leading to cable tension deviations.

Figure 9

Comparison between the required driving force under the predesigned deployment motion

In this study, numerical nonlinear and integral algorithms were used to solve the dynamic model. The numerical convergence and efficiency of the solution process were ensured by variable step size technology.

4.2 Experimental Verification

We made a scaled prototype of a ring-truss deployable antenna and designed an experimental system to verify the established deployment dynamic model. As shown in Figure 10, the prototype comprises six parallelogram-deployable units. The aperture is 0.75 m, height is 0.41 m, and the cable network contains 43 cables. A vertical rod was fixed on the frame, and a servomotor drove the roller to rotate and contract the driving rope along the diagonal line of the parallelogram. The driving force was transmitted by the diagonal rope to achieve the deployment of each unit. In the experiment, a constant driving force of 12 N was provided, the inertial measurement unit measured the acceleration of the truss node, and the measured data were transmitted wirelessly to the computer signal acquisition software for data processing.

Figure 10

The prototype machine of the ring truss deployable antenna and test system

The hinges and rods were made of resin and aluminum alloy tubes, respectively. Table 3 lists the truss parameters, while Figure 11 shows the cable network topology. Table 4 lists the original length of each cable.

Table 3 Parameters of the antenna with twelve parallelogram units

Figure 11

The cable network serial numbers of the prototype

Table 4 The cables’ unstressed length (mm)

Figure 12 shows the three states in the deployment process of the scaled prototype. The video of the entire deployment process and curves of the deployment angle, node displacement, and acceleration during the experiment are provided in the Supplementary Material.

Figure 12

The deployment process of the scaled prototype

The corresponding deployment angle was calculated using the proposed approach for comparison with the experimental results. The acceleration of the measuring point in Figure 11 was obtained, and the corresponding deployment angle was converted using Eq. (26), according to the geometric relationship shown in Figure 11. Gravity was considered in the simulation to maintain consistency with the experiment without the gravity unloading device. As shown in Figure 13, oscillations occurred in the experimental curve owing to uncontrollable friction and motor vibration. However, the overall trend agreed well with the calculation results, verifying the correctness and accuracy of the proposed cable-truss coupled dynamic analysis method. The deployment angle varied slowly at the beginning and middle stages, and increased rapidly at the end of the stage. This is dependent on the antenna configuration. The acceleration at the end of deployment can be clearly observed in the video provided.

$$\left\{ \begin{gathered} \Delta h = \left| {\iint {a\text{d}t}} \right|, \hfill \\ \varphi = \arccos \left( {\frac{{R_{1} \cos \varphi_{0} - \Delta h}}{{R_{1} }}} \right), \hfill \\ \end{gathered} \right.$$

(26)

where a represents the acceleration of the measuring point; φ₀ is the initial value of the deployment angle; and R₁ is the length of the transverse bar.

Figure 13

The deployment angle curves by the experiment and simulation

5 Driving Scheme Design for the Ring Truss Deployable Antenna

Driving schemes significantly affect the deployment dynamic behavior. The real-time coupling effect between the soft cables and rigid truss makes the driving mode and parameter design more complex than the truss mechanism. This section introduces a driving scheme design method based on the proposed cable-truss coupled dynamic analysis model, discusses the characteristics of the rope and spring drives, and displays a series of numerical examples for the driving-scheme design.

5.1 Driving Schemes for the Ring Truss Deployable Antenna

Deployment can be regarded as a process in which the driving force or driving energy competes with the resistance load or energy. Figure 14 shows the variations in the driving and resistance forces and energies during deployment. Because the cable network is the main resistance load and most of the work done by the driving force/torque is converted into the elastic energy of the cable network, the elastic potential energy of the cable network is shown by the blue dotted curve in Figure 14. It is obtained by the approach in Ref. [50] and provides the basis for the driving scheme design.

Figure 14

Schematic diagram for the driving schemes design

The slope of the elastic energy curve at each deployment angle/driving rope length represents the resistance torque/force generated by the cable network at that state. If the driving force/torque is designed according to the maximum resistance force/torque (the maximum slope of the elastic energy curve), successful deployment can be achieved, but the deployment speed may be difficult to control. Nevertheless, if the driving force/torque is determined by the required energy, there is a risk of unsuccessful deployment, although the conservation-of-energy principle is satisfied. Therefore, the driving schemes based on the above design principles should be evaluated by the dynamic analysis in Section 3 and optimized based on the analysis results to ensure a smooth and successful deployment. The detailed driving scheme design is as follows. The antenna is driven by torsion springs in the initial deployment stage, and then deployed by driving rope. The design method proposed in this study can be used to design the driving torsion spring and rope to complete the actual engineering drive.

5.1.1 Driving Parameters Design for the Rope Drive

(1) The driving parameter design based on the required driving force. Constant-force driving is a simple and commonly used driving mode. If the constant driving force is provided by the driving rope, then the value of the constant force should not be less than the maximum resisting force, depicted by the blue curve in Figure 14(a), to ensure deployment success. The constant force provided by the driving rope can be calculated by Eq. (27), corresponding to the driving curve ① in Figure 14(a). A correction coefficient is added to the equation to offset the impact of system friction and damping on the deployment. The corresponding coefficients also need to be added to solve for the driving force in other driving schemes.

$$F_{{{\text{rc}}}} = \alpha_{1} \times \max \left( {\frac{\partial E}{{\partial L}}} \right),$$

(27)

where F_rc is the constant force provided by the driving rope; α₁ is the correction coefficient; E is the cable network’s elastic potential energy; and L is the driving rope length.

(2) The driving parameter design based on the required energy. If the required energy in the deployment is taken as the basis for the driving-force calculation, the work done by the driving rope should be equal to the elastic energy of the cable network, kinetic energy of the truss, and the rest of the energy consumption. The driving force in the rope can be obtained by considering the derivative of the systematic total energy with respect to the length variation of the driving rope, as expressed in Eq. (28). It should be noted that the driving rope can only be tensioned and not compressed, and cannot provide a negative driving force. Thus, the calculated driving force corresponds to the driving curve ② in Figure 14(a).

$$F_{{{\text{rv}}}} = \alpha_{2} \times \max \left( {\frac{{\partial \left( {E + T} \right)}}{\partial L},0} \right),$$

(28)

where F_rv is the driving force provided by the driving rope, and α₂ is the correction coefficient.

5.1.2 Driving Parameters Design for the Torsion Spring Drive

(1) The driving parameter design based on the required driving force. Driving torsion springs are commonly arranged in five-dimensional hinges. In the folded state, the compression of the torsion spring was the largest, and then gradually decreased with deployment, accompanied by the decrease in the driving torque. To ensure a successful deployment, the driving torque at the arbitrary deployment angle should not be less than the maximum resisting torque at the corresponding deployment angle. In Figure 14(b), the maximum resisting torque occurs at the deployment angle φ₁ and the resisting torque equals 0 at the deployment angle φ₂, so the stiffness of the torsion spring can be determined by Eq. (29), which corresponds to the driving curve ③ in Figure 14 (b).

$$k_{1} = \alpha_{3} \times \frac{{\max \left( {\frac{\partial E}{{\partial \varphi }}} \right)}}{{\varphi_{2} - \varphi_{1} }},$$

(29)

where k₁ is the torsion spring stiffness; α₃ is the correction coefficient; φ₁ is the deployment angle corresponding to the maximum slope of the cable network’s elastic potential energy; and φ₂ is the deployment angle when the cable network’s potential energy reaches the maximum value.

(2) The driving parameter design based on the required energy. The elastic potential energy stored in the torsion springs is converted into the kinetic energy of the truss and elastic energy of the cable network during deployment. As shown by the blue curve in Figure 14(b), the cable network’s elastic potential energy reaches the maximum value at the deployment angle φ₂, so the torsion springs are required to expand the antenna before their deployment angle reaches φ₂. After that, the cable network’s elastic potential energy begins to decrease, and the released elastic potential energy can provide the antenna’s kinetic energy, which continues to drive the rest of the deployment process, as shown by the driving curve ④ in Figure 14(b), and the torsion spring stiffness is determined by Eq. (30).

$$k_{2} = \alpha_{4} \times \frac{{2\left( {E_{1} + T_{1} } \right)}}{{\varphi_{2}^{2} }},$$

(30)

where k₂ is the torsion spring’s stiffness; α₄ is the correction coefficient; E₁ is the cable network’s maximum elastic potential energy; and T₁ is the truss’s kinetic energy when the deployment angle is equal to φ₂.

(3) The driving and braking torsion spring design based on the required energy. The braking spring stores energy as the antenna unfolds, in contrast to the driving spring, and its installation method is similar to that of the driving spring. As shown in Figure 14(b), the cable network’s elastic energy increased before the deployment angle φ₁ and then decreased between the deployment angles φ₂ and φ₃. Correspondingly, the resisting torque is positive before the deployment angle φ₁ and then becomes negative between the deployment angle φ₂ and φ₃. To avoid the impact before locking, we attempt to deploy the antenna and reduce the deployment angular velocity at the final stage with two types of torsion springs: one type provides driving torque and the other provides braking torque. The stiffnesses of the driving and braking torsion springs were designed based on variations in the resistance load. The driving torsion spring has the same stiffness as the driving curve ④ (calculated by Eq. (30)), whereas the stiffness of the braking spring was determined by Eq. (31). The braking springs were used to absorb the released elastic energy and prevent all the released elastic energy from being transformed into the truss’s kinetic energy, so as to effectively slow down the deployment, as shown by the driving curve ⑤ in Figure 14 (b).

$$k_{3} = \alpha_{5} \times \frac{{2\left[ {\left( {E_{2} - E_{1} } \right) + \left( {T_{2} - T_{1} } \right)} \right]}}{{\left( {\varphi_{3} - \varphi_{2} } \right)^{2} }},$$

(31)

where k₃ is the braking spring’s stiffness; α₅ is the correction coefficient; φ₃ is the deployment angle at the deployed state; E₂ is the cable network’s elastic energy at the deployed state; and T₂ is the truss’s kinetic energy at the deployed state.

5.2 Numerical Examples for Various Driving Schemes

This section provides several numerical examples to analyze the dynamic performance under different driving schemes. The time-varying tension of the boundary cable and the dynamic behaviors of the antenna under various driving parameters were analyzed to discuss their influence on the dynamic performance. The dissipative energy was used to simulate the loss caused by friction. The stiffness of the truss was neglected to improve the computational efficiency.

5.2.1 Deployment Dynamic Analysis Under the Rope Drive

(1) Dynamic simulation of the rope drive based on the required driving force (driving curve ① in Figure 14(a)).

In this section, we analyze the dynamic behavior of the antenna when the constant force determined by Eq. (27) is provided by the driving rope. Figure 15 shows the driving force.

Figure 15

Constant force in the driving rope

Figure 16 shows the tension of the boundary cables, which are the main resistance load during deployment. Figure 17 shows the curves of the deployment angle and angular velocity under the constant driving force in Figure 15. It can be found that the deployment angular velocity increased throughout the entire process and increased sharply at the end of deployment, which was accompanied by the decreased tension of the boundary cables. The acceleration at the end of deployment may have a large impact before locking. Therefore, the simple constant-force drive is unsuitable for the desired smooth deployment pulled by the rope.

Figure 16

Boundary cables’ time-varying tensions

Figure 17

Deployment motion under the constant driving force

(2) Dynamic simulation of the rope drive based on the required energy (driving curve ② in Figure 14(a)).

In this section, we applied the driving force shown in Figure 18, which was determined by Eq. (28), to the same antenna to analyze the deployment motion under the rope drive.

Figure 18

Driving force determined by the driving curve ② in Figure 14(a) and Eq. (28)

Figure 19 shows the calculated time-varying tensions of the boundary cables, which are also the time-varying resistance load in the deployment. Figure 20 shows the calculated deployment angle and angular velocity under the driving force in Figure 18. It can be found that the antenna’s motion is smooth in the early and middle stages, which is highly consistent with the predesigned deployment motion shown in Figure 6. The angular velocity increased sharply at 8574 s, which was caused by the release of the elastic potential energy of the cable network and the feature that the rope drive could not provide a braking force. Therefore, the rope drive may lead to a significant impact before locking.

Figure 19

Boundary cables’ time-varying tensions

Figure 20

Deployment motion under the driving force in Figure 18

5.2.2 Deployment Dynamic Analysis Under the Torsion Spring Drive

(1) Dynamic simulation of torsion spring drive based on the required driving force (driving curve ③ in Figure 14(b)).

The torsion-spring driving parameters are calculated according to Eq. (29). The calculated driving torque is shown in Figure 21.

Figure 21

The driving torque-deployment angle curve for the torsion springs

The boundary cable tension, deployment angle, and angular velocity under the torsion-spring driving torque in Figure 21 were calculated by the proposed dynamic approach. The results are shown in Figures 22 and 23. The driving torque of the torsion springs decreased gradually with the increase in the deployment angle owing to the characteristics of the torsion spring drive. However, the required maximum driving torque occurred in the late stage. Therefore, the driving torque provided by the torsion spring must be designed to be far more than that required in the early and middle stages to meet the requirement at the late stage, leading to a quick deployment, as shown in Figure 23.

Figure 22

Boundary cables’ time-varying tensions

Figure 23

Deployment motion under the torsion spring drive

(2) Dynamic simulation of torsion spring drive based on the required energy (driving curve ④ in Figure 14(b)).

The driving torque-deployment angle curve for the torsion springs is calculated by Eq. (30), as shown in Figure 24.

Figure 24

The designed driving torque-deployment angle curve for the torsion springs

We conducted the deployment dynamic analysis for the mesh reflector antenna driven by the torsion spring with the stiffness shown in Figure 24. The calculated boundary cable tension, deployment angle curve, and deployment angular velocity curves are shown in Figures 25 and 26.

Figure 25

Boundary cables’ time-varying tensions

Figure 26

Deployment motion under the torsion spring drive

According to the torsion spring characteristics, the driving torque gradually decreased with an increase in the deployment angle. However, the main resistance of the antenna during deployment, provided by the boundary cables, occurred at the final stage. This means that the residual torque of the torsion spring should be sufficiently large to cover the maximum resistance load of the cable network at the end of the deployment, which leads to a waste of driving energy in the early and middle stages. As shown in Figure 26, the driving torque of the torsion spring was sufficiently large to deploy the truss before time t₁, and the resistance of the boundary cables was very small, as shown in Figure 25. Therefore, the deployment angle and angular velocity increased rapidly before time t₁. Subsequently, the resistance of the boundary cables increased, and the driving torque was less than the required torque, which led to a decrease in the deployment velocity, as shown in Figure 26(b). The boundary cable resistance increased within times t₁ and t₂, which correspond to the slow increase in the deployment angle in Figure 26(a) and the rapid decrease in the deployment angular velocity in Figure 26(b). The torque of the torsion spring was set to zero at the deployment angle where the cable network’s elastic potential energy reached the maximum value to save energy and slow down deployment. However, after time t₂, the resistance of the boundary cables began to decrease, and the deployment velocity increased correspondingly, as shown in Figure 26(b). Most importantly, the antenna deployment velocity in the later stages is relatively high, which may lead to a large impact before locking.

(3) Dynamic simulation under the driving and braking torsion spring drive based on the required energy (driving curve ⑤ in Figure 14(b)).

Based on the stiffness values calculated using Eq. (31), the driving torque-deployment curve of the braking spring was obtained, as shown in Figure 27. The curve before the deployment angle φ₂ represents the driving torque of the driving spring, while the curve between the deployment angle φ₂ and φ₃ represents the braking torque of the braking spring.

Figure 27

Driving torque provided by the driving and braking torsion springs

The boundary cable tension, deployment angle, and angular velocity under the torsion spring drive with driving torque in Figure 27 were calculated by the proposed dynamic method. The results are shown in Figures 28 and 29. As the braking spring began to act at time t₂, the variations in the deployment angle and deployment angular velocity before time t₂ in Figure 29 were the same as those in the curves in Figure 26. Because the braking torque was initially small, the deployment angular velocity of the antenna continued to increase within time t₂ and t₃. Then, the braking springs’ torque balances the released elastic energy of the cable network at time t₃, and the antenna deployment angular velocity started to decrease from time t₃ until the fully deployed moment, and finally reached zero. Obviously, the braking torsion springs can significantly slow down the deployment at the final stage, which avoids the impact before locking.

Figure 28

Boundary cables’ time-varying tensions

Figure 29

Deployment motion under the driving and braking torsion springs

6 Conclusions

• (1) This study proposed a general cable-truss coupling dynamic analysis methodology for cable antennas, which provides an effective tool for analyzing and improving the deployment dynamic behaviors of cable antennas with varying topologies and parameters.

• (2) We derived the explicit expression of the time-varying resistance provided by the boundary cables, which helped achieve real-time coupling between the soft cables and truss. The topological diversity, geometric nonlinearity, and continuous switching between the slacked and tensioned states of the cable network were fully captured during modeling.

• (3) A scaled prototype was fabricated, and a deployment experiment was conducted to verify the accuracy of the proposed cable-truss coupled dynamic analysis method.

• (4) The effects of the driving modes and parameters on the dynamic performance were well studied, and the driving scheme design, including the rope drive and torsion spring drive for a cable–mesh antenna, were displayed.

• (5) The uncontrollable acceleration at the end of deployment caused by the cable network’s features was effectively overcome by adding the braking torsion springs, which avoided the commonly occurring impact of cable antennas caused by the cable network’s elastic energy release before locking.