Stiffness Modeling and Analysis of Multiple Configuration Units for Parabolic Deployable Antenna

Abstract

Space-deployable antennas have development requirements of an ultra-large aperture, high stiffness, and multi-frequency multiplexing. To address the challenge of stiffness characterization in the multi-closed-loop complex systems of deployable mechanisms, this paper proposes a parametric stiffness modeling method and a static stiffness model is established, ranging from components and limbs to the overall mechanism. The motion/force mapping model of the deployable mechanism is obtained using screw theory, and the stiffness mapping from joint space to workspace is achieved via the Jacobian matrix. A comprehensive stiffness model of the deployable mechanism incorporating joint effects is established based on the principle of virtual work and the superposition principle of deformations, and its validity is verified through finite element simulation. Building on this, stiffness characteristics based on structural configuration are investigated, and structural forms with excellent stiffness performance are selected through comprehensive evaluation. Six configurations of the deployable mechanism are derived topologically from this structure, and the optimal configuration is selected based on stiffness performance. The parametric stiffness modeling method proposed in this study can effectively characterize the contribution of each component to the overall system stiffness. It lays a theoretical foundation for establishing a quantitative relationship between stiffness performance and configuration, enabling performance-based configuration optimization and dimensional optimization.

1. Introduction

Space-deployable antennas have seen increasing application in mobile communications, deep-space exploration, and electronic surveillance. Modular deployable antennas, designed on a modular concept, can achieve various apertures by scaling the size and number of identical modules. Adjusting module stiffness further enables multi-frequency shared operation [1,2,3]. These antennas offer a high folding ratio, favorable stiffness, and light weight, making them promising candidates for future ultra-large-aperture, multi-function antenna systems.

The deployable mechanism serves as the supporting skeleton of the deployable antenna, undertaking the key functions of deployment, positioning, and on-orbit load-bearing. Its deployment process is a special transition from mechanism motion to structural locking. Current research on deployable antennas primarily focuses on novel configuration, deployment strategies, deployment dynamics, and surface accuracy analysis [4,5,6,7,8]. Zhang [9] proposed a novel deployable mesh reflector antenna configuration based on a cable-dome tensegrity structure, designed a new W-type deployable truss, and developed a 2 m aperture prototype to validate the feasibility of the scheme. Sun [10] proposed a novel double-ring deployable truss antenna scheme, significantly improving the stowage ratio in the height direction. Inspired by origami, Russo [11] designed a single-degree-of-freedom self-deploying reflectarray antenna, achieving a stowage efficiency of 8:1 through prototype testing. Kim [12] proposed a deployable truss based on scissor units that can be stowed into a flat configuration, effectively enhancing longitudinal packaging efficiency. Khoroshylov [13] proposed a feedforward-feedback robust control strategy based on two in-orbit identifiable parameters. This strategy addresses the challenges of high-precision attitude control and vibration suppression. Its effectiveness was verified through numerical simulations. Angeletti [14] proposed an optimal design method based on a smart material actuator network to address the micro-vibration control problem, determined the optimal placement positions and velocity feedback gains of the actuators, and verified its effectiveness in suppressing structural vibrations via attitude maneuver simulations.

In the area of configuration synthesis and optimization, Zhu [15] applied the screw theory-based constraint synthesis method to perform mechanism configuration synthesis for triangular pyramid folding-deploying units, optimizing the selection based on indicators such as high stowage ratio and good environmental adaptability. Duan [16] performed a configuration synthesis of deployment mechanisms based on the modified Grübler–Kutzbach formula, selecting configurations considering load-bearing capacity, lightweight design, and structural complexity. Zhao [17] proposed a modular deployable mechanism configuration synthesis method based on screw theory and screened configurations using the stowage ratio as the criterion. Tian [18] employed graph theory for configuration synthesis of deployable mechanisms for mesh surface antennas and used the fuzzy comprehensive evaluation method to optimize the deployable units. Evaluation indicators are essential criteria for assessing the performance of deployable mechanisms. However, existing research in configuration optimization often relies on empirical criteria or single geometric indicators, lacking a systematic evaluation framework with clear physical significance, particularly in selecting configurations based on stiffness, deployment accuracy, and dynamic performance. Therefore, establishing performance indicators with good engineering applicability and clear physical meaning is of significant theoretical importance for the configuration optimization and design of deployable mechanisms.

The structural stiffness of a deployable mechanism after deployment and locking directly influences its on-orbit load-bearing capacity and positional stability. In complex space environments and external disturbances, structural stiffness is not only fundamental to ensuring high-precision pointing and long-term stable operation of the antenna but also plays a critical role in achieving the desired modal behavior. Establishing a parametric stiffness model aims to define the quantitative relationship between the mechanism’s configuration parameters and its overall stiffness performance. This is a crucial prerequisite for achieving performance-based configuration selection and dimensional optimization. Bouzgarrou [19] and Piras [20] employed Finite Element Analysis (FEA) to conduct static stiffness analysis on 3T1R (three translational and one rotational)parallel mechanisms and planar parallel mechanisms, respectively. Balancing computational accuracy and efficiency, Gorgulu [21] utilized the virtual joint method for stiffness modeling of a parallel haptic device. Deblaise [22] and Wu [23] established stiffness models for a Delta mechanism and a 5-DOF (five degrees of freedom)redundant parallel mechanism, respectively, using the Matrix Structural Analysis (MSA) method. In summary, existing stiffness analysis methods primarily include FEA, MSA, the virtual joint method, and the virtual spring method. However, FEA does not readily reveal the influence of design parameters on stiffness, making parametric modeling challenging. MSA is often inadequate for capturing the characteristics of non-standard components [24,25,26,27,28,29,30,31,32].

Addressing the aforementioned issues, this study proposes a parametric stiffness modeling method based on the virtual joint method. An overall stiffness model for the deployable mechanism is established that accounts for joint effects, effectively characterizing the contribution of individual components to the overall system stiffness. The hexagonal prism space-deployable antenna support structure studied in this paper is composed of identical hexagonal prism modules, characterized by high modularity, strong scalability and good interchangeability. While ensuring good stiffness, it achieves a high surface profile accuracy and a larger aperture.

The layout of this paper is structured as follows: Section 2 presents the motion analysis of the deployable mechanism and derives its motion/force mapping model using screw theory. Section 3 proposes an analytical stiffness modeling method, establishing static stiffness models from the component and limb levels up to the complete deployable mechanism, with the model’s validity verified through FEA simulations. Section 4 investigates stiffness characteristics based on structural topology. Structural forms exhibiting excellent stiffness-to-mass ratios are screened through comprehensive evaluation. Six deployable mechanism configurations are then derived topologically, and the optimal configuration is selected using stiffness performance as the key metric.

2. Composition of the Modular Deployable Antenna and the Kinematics of the Rib Unit

2.1. Composition of the Antenna Structure

The modular design principle forms the basis of the modular deployable antenna. By varying the size, shape, combination, and arrangement of the modules, the antenna aperture can be efficiently scaled. This structural configuration is well-suited to meet the multi-band operational requirements of reflector antennas [2]. As shown in Figure 1, the antenna is constructed from 19 modules of identical design, arranged in a three-layer configuration. The first layer contains a single module, surrounded by six modules forming the second layer, and twelve modules constituting the third layer. To fulfil the multi-frequency multiplexing capability of the deployable antenna, the stiffness of each layer is designed to be distinct. When fully deployed, the antenna assembly forms a parabolic profile.

Each module is further composed of structures including a front cable net, a metal reflector mesh, vertical tension ties, a posterior cable net, a supporting truss, and crossing cables. The front cable net is primarily used to fix and tension the metal reflector mesh; the metal reflector mesh serves as the working surface of the deployable antenna for signal transmission and reception; the vertical tension ties connect the front and rear cable nets, precisely adjusting the mesh surface shape; the rear cable net acts as the base layer of the mesh, providing a connection carrier for the vertical ties while also tensioning the front cable net and the metal reflector mesh; the supporting truss of each module consists of six rib units radially arranged around the center. The supporting truss, which serves as the skeleton of the modules, facilitates the deployment and precise positioning of flexible components such as the metal reflector mesh. Tension cables are integrated along the periphery of the support structure to enhance the overall rigidity after deployment.

2.2. Description of the Mechanism Unit

The support truss of the module consists of six rib units arranged radially about the center [2]. As shown in Figure 2, the rib unit serves as the basic deployable element and primarily consists of a slider, a spring, and a set of link members. The topological structure of the mechanism is shown in Figure 3, which features a kinematic chain RR and a chain ((R-PRR)RR-R)R connecting the fixed base (link 1) to the moving platform (link 5). The linear motion of the slider along the central rod drives the folding and deploying movements of the entire system.

To describe the folding and deployment motion of the rib unit, as shown in Figure 4, a fixed coordinate system O-xyz and a moving coordinate system D-uvw, with axes parallel to those of O-xyz, are established at point D. Local coordinate systems A₀–x₀y₀z₀, A₁–x₁y₁z₁, A₃–x₃y₃z₃, A₄–x₄y₄z₄, A₅–x₅y₅z₅, A₆–x₆y₆z₆, A₇–x₇y₇z₇ and A₈–x₈y₈z₈ are established at points A₀, A₁, A₃, A_4, A₅, A_6, A₇ and A₈, respectively. In the deployed state, the x-axis of the local coordinate system A₀–x₀y₀z₀ aligns with the direction of link 3, while the x-axis of A₁–x₁y₁z₁ aligns with the direction of link 7. Similarly, the x-axes of the local coordinate systems A₃–x₃y₃z₃, A₅–x₅y₅z₅, and A₈–x₈y₈z₈ align with the directions of link 8, link 6, and link 4.

The centers of the planar revolute joints within the rib unit are located at points A₀, A₁, A₂, …, A₈. These are planar revolute joints with a single degree of freedom, and their rotation axes are all oriented along the z-axis of the coordinate system.

In Figure 4, α₈ denotes the angle between A₇A₈ and the x-axis in the deployed state, α₅ denotes the angle between A₅A₆ and the x-axis in the deployed state, α₁ denotes the angle between member A₁A₃ and the x-axis in the deployed state, and α₀ denotes the angle between A₀A₂ and the x-axis in the deployed state.

In the deployed state, the rotation matrices R₀, R₁, R₅, and R₈ for the coordinate systems A₀–x₀y₀z₀, A₁–x₁y₁z₁, A₃–x₃y₃z₃, A₅–x₅y₅z₅, and A₈–x₈y₈z₈ relative to the base coordinate system O-xyz can be obtained by successive rotations about the z-axis by angles α₀, α₁, α₅, and α₈, respectively. Their expressions are given as follows:

$$R_0=\left[\begin{array}{ccc}\cos {\alpha }_0& -\sin {\alpha }_0& 0\\ \sin {\alpha }_0& \cos {\alpha }_0& 0\\ 0& 0& 1\end{array}\right],{\mathbf{R}}_1=\left[\begin{array}{ccc}\cos {\alpha }_1& -\sin {\alpha }_1& 0\\ \sin {\alpha }_1& \cos {\alpha }_1& 0\\ 0& 0& 1\end{array}\right],R_5=\left[\begin{array}{ccc}\cos {\alpha }_5& -\sin {\alpha }_5& 0\\ \sin {\alpha }_5& \cos {\alpha }_5& 0\\ 0& 0& 1\end{array}\right],R_8=\left[\begin{array}{ccc}\cos {\alpha }_8& -\sin {\alpha }_8& 0\\ \sin {\alpha }_8& \cos {\alpha }_8& 0\\ 0& 0& 1\end{array}\right]$$

(1)

2.3. Motion/Force Mapping Model

As shown in Figure 3, the motion of the deployable unit mechanism originates from link 1 and is transmitted to the terminal reference point D through closed-loop chain I and chain II. The motion of point D can be expressed as a linear combination of the motions of the single-degree-of-freedom joints within the two chains. Therefore, the instantaneous motion twist of the mechanism at point D can be expressed as

$$\left\{\begin{array}{c}{S}_{\mathrm{t}}={\displaystyle \sum _{j_a=1}^2{\rho }_{a,j_a,1}{\widehat{S}}_{ta,j_a,1}}\\ S_{\mathbf{t}}={\displaystyle \sum _{j_a=1}^2{\rho }_{a,j_a,2}{\widehat{S}}_{ta,j_a,2}}\end{array}\right.$$

(2)

where $S_{\mathrm{t}}$ represents the instantaneous screw motion of point D, ${\widehat{S}}_{ta,j_{a,}1}$ and ${\mathrm{\rho }}_{ta,j_{a,}1}$ are the $j_a$ allowable unit screw and amplitude in the first chain, ${\widehat{S}}_{ta,j_{a,}2}$ and ${\mathrm{\rho }}_{ta,j_{a,}2}$ are the $j_a$ allowable unit screw and amplitude in the second chain.

$${\widehat{S}}_{ta,1,1}=\left(\begin{array}{c}{s}_{\mathrm{z}}\\ r_{\mathrm{D}{\mathrm{A}}_8}\times s_{\mathrm{z}}\end{array}\right),{\widehat{S}}_{ta,2,1}=\left(\begin{array}{c}{s}_{\mathrm{z}}\\ r_{\mathrm{D}{\mathrm{A}}_7}\times s_{\mathrm{z}}\end{array}\right),{\widehat{S}}_{ta,1,2}=\left(\begin{array}{c}{s}_{\mathrm{z}}\\ r_{\mathrm{D}{\mathrm{A}}_6}\times s_{\mathrm{z}}\end{array}\right),{\widehat{S}}_{ta,2,2}=\left(\begin{array}{c}{s}_{\mathrm{z}}\\ r_{\mathrm{D}{\mathrm{A}}_5}\times s_{\mathrm{z}}\end{array}\right)$$

(3)

where $r_{\mathrm{D}{\mathrm{A}}_8}$ is the vector from point D to point A₈ in the coordinate system D-uvw,$r_{\mathrm{D}{\mathrm{A}}_6}$ is the vector from point D to point A₆ in the coordinate system D-uvw, $r_{\mathrm{D}{\mathrm{A}}_5}$ is the vector from point D to point A₅ in the coordinate system D-uvw, $s_{\mathrm{z}}=\left(\begin{array}{ccc}0& 0& 1\end{array}\right)$.

Based on screw theory, the force screw acting on an object can be expressed in Plücker coordinates as ${\mathbf{S}}_{\mathrm{w}}={\left[\begin{array}{cc}{s}_w^{\mathrm{T}}& {(r_w\times s_w)}^{\mathrm{T}}\end{array}\right]}^{\mathrm{T}}$, where $s_w$ represents the unit direction vector of the pure force screw, $r_w$ represents the position vector. Taking the rib unit structure as a free-body for force analysis, a static equilibrium equation is established at point D:

$${\mathbf{S}}_{\mathrm{W},\mathrm{D}}=f_{wa,1}{\widehat{S}}_{wa,1}+{\displaystyle \sum _{kc=1}^4{f}_{wc,kc,1}{\widehat{S}}_{wc,kc,1}}+{\displaystyle \sum _{kc=1}^4{f}_{wc,kc,2}{\widehat{S}}_{wc,kc,2}}$$

(4)

where $S_{\mathrm{W},\mathrm{D}}$ denotes the external wrench acting on point D of the structure, $f_{wa,1}$ and ${\widehat{S}}_{wa,1}$ represent the unit actuation wrench and its amplitude, $f_{wc,k_c,i}$ and ${\widehat{S}}_{wc,k_{c,}i}$ represent the unit constraint wrench and its amplitude for the i-th chain.

Taking the generalized inner product of both sides of Equation (2) with respect to $S_{wa,g_{k,}k}$ or $S_{wc,k_c,i}$, we get

$$\left\{\begin{array}{c}{S}_{wa,g_k,k}^{\mathrm{T}}{\widehat{S}}_t={\rho }_{a,g_{k,}k}{\widehat{S}}_{wa,g_k,k}^{\mathrm{T}}{\widehat{S}}_{ta,g_{k,}k}\\ S_{wc,k_c,i}^{\mathrm{T}}{\widehat{S}}_t=0\end{array}\right.$$

(5)

where k = 1, i = 1, 2, k_c = 1, 2, 3, 4, $S_{wa,g_{k,}k}$, ${\widehat{S}}_{wa,g_{k,}k}$ denote the actuation wrench and the unit actuation wrench of the k-th chain, $S_{wc,k_c,i}$, ${\widehat{S}}_{wc,k_{c,}i}$ denote the constraint wrench and the unit constraint wrench of the k-th chain.

By locking the actuated joint, the reciprocal wrench obtained through the reciprocal product operation from the motion twists of the other kinematic pairs in the chain constitutes the constraint and actuation wrenches. Subsequently, the resultant wrench in the coordinate system D-uvw is expressed as

$$\begin{array}{c}{S}_{wa,1,1}=\left(\begin{array}{c}0\\ s_y\end{array}\right),S_{wc,1,1}=\left(\begin{array}{c}{s}_x\\ 0\end{array}\right),S_{wc,2,1}=\left(\begin{array}{c}{s}_y\\ 0\end{array}\right),S_{wc,3,1}=\left(\begin{array}{c}0\\ s_{\mathrm{z}}\end{array}\right),S_{wc,4,1}=\left(\begin{array}{c}{r}_{\mathrm{D}{\mathrm{A}}_6}\times s_{{\mathrm{A}}_5{\mathrm{A}}_6}\\ s_{{\mathrm{A}}_5{\mathrm{A}}_6}\end{array}\right)\\ S_{wc,1,2}=\left(\begin{array}{c}{s}_x\\ 0\end{array}\right),S_{wc,2,2}=\left(\begin{array}{c}{s}_y\\ 0\end{array}\right),S_{wc,3,2}=\left(\begin{array}{c}0\\ s_z\end{array}\right),S_{wc,4,2}=\left(\begin{array}{c}0\\ s_{{\mathrm{A}}_8{\mathrm{A}}_7}\end{array}\right)\end{array}$$

(6)

Rewriting the inner product of Equation (5) in matrix form, we get

$$J{S}_{\mathrm{t}}=J_{\rho }\rho$$

(7)

$$J=\left[\begin{array}{c}{J}_a\\ J_c\end{array}\right],J_a=\left[\begin{array}{cc}{0}^{\mathrm{T}}& {s_y}^{\mathrm{T}}\end{array}\right],\rho =\left[\begin{array}{c}{\stackrel{\cdot }{\mathrm{t}}}_1\\ \mathbf{0}\end{array}\right]=\left[\begin{array}{c}{\mathrm{\rho }}_{a,2}\\ \mathbf{0}\end{array}\right]$$

(8)

$$J_c=\left[\begin{array}{cc}{s_x}^{\mathrm{T}}& 0^{\mathrm{T}}\\ {s_y}^{\mathrm{T}}& 0^{\mathrm{T}}\\ {(r_{\mathrm{D}{\mathrm{A}}_6}\times s_{{\mathrm{A}}_5{\mathrm{A}}_6})}^{\mathrm{T}}& {s_{{\mathrm{A}}_5{\mathrm{A}}_6}}^{\mathrm{T}}\\ 0^{\mathrm{T}}& {s_{{\mathrm{A}}_8{\mathrm{A}}_7}}^{\mathrm{T}}\\ 0^{\mathrm{T}}& {s_{\mathrm{z}}}^{\mathrm{T}}\end{array}\right],J_{\rho }=\left[\begin{array}{c}1\\ \mathbf{0}\end{array}\right]$$

(9)

From the above analysis, it can be seen that the rib unit exhibits an over-constrained characteristic. Closed-loop Chain I provides one actuation wrench and four constraint wrenches to the terminal reference point D, while Chain II provides four constraint wrenches to the same point. Among these, the constraint couples along the x-axes and y-axes, as well as the constraint force along the z-axis, are identified as redundant constraints.

3. Static Stiffness Modeling and Validation

The structural stiffness of a deployable mechanism after deployment and locking directly determines its on-orbit load-bearing capacity and positional accuracy during service. In the face of complex space environments and external disturbances, stiffness not only serves as the foundation for ensuring high-precision pointing and long-term stable operation of the antenna but also plays a critical role in achieving the desired modal behavior. Therefore, this paper aims to systematically investigate the stiffness performance of deployable mechanisms.

3.1. Modeling Approach for Overall Stiffness

The deformation resistance of the rib unit under external loads is characterized by its static stiffness. The resistance of the rib unit to deformation under external loads is characterized by static stiffness. Given that the deployable antenna operates in a microgravity environment and primarily bears static loads from the metallic cable net, all components remain within the elastic deformation range. The members in the mechanism are slender structures with considerable flexibility, while the joint stiffness is relatively high. Based on this, the modeling process in this paper does not consider the effects of joint flexibility and material nonlinearity.

When establishing the stiffness model of the deployable rib unit mechanism, the flexibilities of the moving platform, the two supporting chains, and the fixed base must be taken into account. By integrating the stiffness of the components with the force/deformation transmission relationships, the overall stiffness of the mechanism can be determined. The stiffness modeling procedure is summarized as follows:

(1)

Based on the structural properties of the components, analytical or finite element analysis methods are employed to determine the compliance matrix of each component in its local coordinate system.

(2)

To obtain the compliance of a supporting chain, the compliance matrices of its constituent components are transformed into the moving coordinate system D-uvw and then linearly superimposed.

(3)

Utilizing the principle of virtual work, Hooke’s law, the motion/force mapping model, and the deformation compatibility conditions, the stiffness matrices of the two chains in the moving coordinate system are derived.

(4)

Considering the compliance of the fixed base and the external links, their compliances are linearly superimposed with those of the branch chains to establish the overall stiffness model of the mechanism at the terminal reference point in the deployed state.

3.2. Stiffness Model of the Chains

3.2.1. Stiffness Matrix of Closed-Loop Chain I

As shown in Figure 3, the deformation of a closed loop can be transmitted from the fixed base to the output end through its two branches. Therefore, the stiffness modeling approach for the closed loop proceeds as follows. First, the compliance of each branch path is calculated based on the compliance matrices of its constituent components using the principle of deformation superposition. Subsequently, by applying Hooke’s law and the deformation compatibility conditions, the stiffness model of the closed loop in its joint space is established.

(1) Stiffness Model of Closed Loop 1

As shown in Figure 5, the equivalent elastic model of Closed Loop 1 indicates that forces or deformations can be transmitted to point A₃ via either path F-A₁A₂A₃ or path B-A₀A₂A₃. According to the principle of deformation superposition, the compliance of path F-A₁A₂A₃ and path B-A₀A₂A₃ are solved respectively as follows:

$$C_{\mathrm{A}1}=C_{1,1},C_{\mathrm{A}0}=C_{2,1}$$

(10)

where C_1,1 denotes the compliance matrix of the short diagonal web member transmitted from point A₃ to point A₂ in the coordinate system D-uvw, C_2,1 represents the compliance matrix of the support rod at point A₂ in the same coordinate system.

If points A₃ and A₂ are rigidly connected, the deformation at point A₂ is the sum of the deformation of the short diagonal web member at A₃ and the rigid-body displacement of A₂A₃. Thus

$$C_{1,1}=T_{1,1}{C}_{1,2}{T}_{1,1}^{\mathrm{T}},T_{1,1}=\left[\begin{array}{cc}{E}_3& 0\\ [{\mathit{l}}_{{\mathrm{A}}_2{\mathrm{A}}_3}\times ]& E_3\end{array}\right]$$

(11)

where C_1,2 denotes the compliance matrix of the link 7 at point A₃, ${\mathit{l}}_{{\mathrm{A}}_2{\mathrm{A}}_3}$ represents the vector from point A₂ to point A₃ in the D-uvw coordinate system, $\left[{\mathit{l}}_{{\mathrm{A}}_2{\mathrm{A}}_3}\times \right]$ is the skew-symmetric matrix of this vector. In this expression,

$${\mathbf{C}}_{1,2}={\mathbf{T}}_{1,2}{\overline{\mathbf{C}}}_{1,2}{\mathbf{T}}_{1,2}^{\mathrm{T}},{\mathbf{C}}_{2,1}={\mathbf{T}}_{2,1}{\overline{\mathbf{C}}}_{2,1}{\mathbf{T}}_{2,1}^{\mathrm{T}}$$

(12)

$${\mathbf{T}}_{1,2}=\left[\begin{array}{cc}{\mathbf{R}}_{1,2}& 0\\ [{\mathit{l}}_{\mathrm{D}{\mathrm{A}}_1}\times ]{\mathbf{R}}_{1,2}& {\mathbf{R}}_{1,2}\end{array}\right],{\mathbf{R}}_{1,2}=\left[\begin{array}{ccc}\cos {\alpha }_1& \sin {\alpha }_1& 0\\ -\sin {\alpha }_1& \cos {\alpha }_1& 0\\ 0& 0& 1\end{array}\right],{\mathbf{T}}_{2,1}=\left[\begin{array}{cc}{\mathbf{R}}_{2,1}& 0\\ [{\mathit{l}}_{\mathrm{D}{\mathrm{A}}_0}\times ]{\mathbf{R}}_{2,1}& {\mathbf{R}}_{2,1}\end{array}\right],{\mathbf{R}}_{2,1}=\left[\begin{array}{ccc}\cos {\alpha }_0& -\sin {\alpha }_0& 0\\ \sin {\alpha }_0& \cos {\alpha }_0& 0\\ 0& 0& 1\end{array}\right]$$

(13)

where ${\overline{C}}_{1,2}$ denotes the compliance matrix of the link 7 at point A₃ in the coordinate system A₁–x₁y₁z₁, ${\overline{C}}_{2,1}$ denotes the compliance matrix of the link 3 at point A₂ in the coordinate system A₀–x₀y₀z₀, ${\mathit{l}}_{\mathrm{D}{\mathrm{A}}_1}$ is the position vector from point D to point A₁ in the D-uvw coordinate system, ${\mathit{l}}_{\mathrm{D}{\mathrm{A}}_0}$ is the position vector from point D to point A₀ in the D-uvw coordinate system.

According to Hooke’s law and the deformation compatibility conditions, we can obtain

$$C_{{\mathrm{A}}_1}{S}_{w,{\mathrm{A}}_1}=S_{\mathrm{t},{\mathrm{A}}_1},C_{{\mathrm{A}}_0}{S}_{w,{\mathrm{A}}_0}=S_{\mathrm{t},{\mathrm{A}}_0}$$

(14)

Establish the virtual work equation of closed loop 1 at point A₂:

$$S_{\mathrm{t},\mathrm{CL},1,1}^{\mathrm{T}}{S}_{w,\mathrm{CL},1,1}=S_{\mathrm{t},{\mathrm{A}}_1}^{\mathrm{T}}{S}_{w,{\mathrm{A}}_1}+S_{\mathrm{t},{\mathrm{A}}_0}^{\mathrm{T}}{S}_{w,{\mathrm{A}}_0}$$

(15)

Therefore, in the coordinate system D-uvw, the stiffness matrix of closed loop 1 is

$$K_{1,1}={\displaystyle \sum _{i=0}^1{C_{{\mathrm{A}}_i}}^{-1}}$$

(16)

(2) Stiffness model of closed loop 2

The schematic of closed loop 2 is shown in Figure 6. Following the approach used for deriving the stiffness matrix of closed loop 1, the compliance of path F-A₁A₂A₃A₄A₆ and path F-A₅A₆ can be written as

$$C_{\mathrm{A}4}=C_{3,1}+C_{4,1},C_{\mathrm{A}5}=C_{5,1}$$

(17)

where C_3,1 denotes the compliance matrix of the link 8 at point A₄ in the D-uvw coordinate system, C_4,1 represents the compliance matrix of closed loop 1 at point A₄ in the D-uvw coordinate system, C_5,1 is the compliance matrix of link 6 at point A₆ in the D-uvw coordinate system.

$$C_{3,1}=T_{3,1}{\overline{C}}_{3,1}{T}_{3,1}^{\mathrm{T}},C_{4,1}=T_{4,1}{K}_{1,1}^{-1}{T}_{4,1}^{\mathrm{T}},C_{5,1}=T_{5,1}{\overline{C}}_{5,1}{T}_{5,1}^{\mathrm{T}},R_{3,1}=R_{1,2}$$

(18)

$$T_{3,1}=\left[\begin{array}{cc}{R}_{3,1}& 0\\ [{\mathit{l}}_{\mathrm{D}{\mathrm{A}}_3}\times ]R_{3,1}& R_{3,1}\end{array}\right],T_{4,1}=\left[\begin{array}{cc}{E}_3& 0\\ [{\mathit{l}}_{A_4{A}_3}\times ]& E_3\end{array}\right],T_{5,1}=\left[\begin{array}{cc}{R}_{5,1}& 0\\ [{\mathit{l}}_{D{A}_5}\times ]R_{5,1}& R_{5,1}\end{array}\right],R_{5,1}=\left[\begin{array}{ccc}\cos {\alpha }_5& -\sin {\alpha }_5& 0\\ \sin {\alpha }_5& \cos {\alpha }_5& 0\\ 0& 0& 1\end{array}\right]$$

(19)

where ${\overline{C}}_{3,1}$ denotes the compliance matrix of link 8 at point A₄ in the A₃–x₃y₃z₃ coordinate system, ${\overline{C}}_{5,1}$ denotes the compliance matrix of link 6 at point A₆ in the A₅–x₅y₅z₅ coordinate system, ${\mathit{l}}_{\mathrm{D}{\mathrm{A}}_3}$ is the position vector from point D to point A₃ in the D-uvw coordinate system, ${\mathit{l}}_{\mathrm{D}{\mathrm{A}}_5}$ is the position vector from point D to point A₅ in the D-uvw coordinate system, and ${\mathit{l}}_{{\mathrm{A}}_4{\mathrm{A}}_3}$ is the vector from point A₄ to point A₃ in the D-uvw coordinate system.

In the D-uvw coordinate system, the stiffness matrix of Closed Loop 2 is given by

$$K_{2,1}={\displaystyle \sum _{i=4}^5{T}_i^{-\mathrm{T}}{C}_{{\mathrm{A}}_i}^{-1}{T}_i^{-1}}, T_1=\left[\begin{array}{cc}{E}_3& 0\\ [{\mathit{l}}_{\mathrm{D}{\mathrm{A}}_6}\times ]& E_3\end{array}\right]$$

(20)

where ${\mathit{l}}_{\mathrm{D}{\mathrm{A}}_6}$ is the vector from point A₆ to point A₄ in the D-uvw coordinate system.

3.2.2. Stiffness Matrix of Chain II

The schematic of Closed Loop 3 is shown in Figure 7. The stiffness matrix of Chain II at point D is expressed as

$$K_{\mathrm{CL},2}=T_2^{-\mathrm{T}}{C}_{\mathrm{CL},2}^{-1}{T}_2^{-1},C_{\mathrm{CL},2}=T_{2,2}{C}_{2,2}{T}_{2,2}^{\mathrm{T}}$$

(21)

$$T_2=\left[\begin{array}{cc}{E}_{3\times 3}& 0\\ [{\mathit{l}}_{\mathrm{D}{\mathrm{A}}_7}\times ]& E_{3\times 3}\end{array}\right],T_{2,2}=\left[\begin{array}{cc}{R}_{2,2}& 0\\ [{\mathit{l}}_{\mathrm{D}{\mathrm{A}}_8}\times ]R_{2,2}& R_{2,2}\end{array}\right],R_{2,2}=\left[\begin{array}{ccc}\cos {\alpha }_8& -\sin {\alpha }_8& 0\\ \sin {\alpha }_8& \cos {\alpha }_8& 0\\ 0& 0& 1\end{array}\right]$$

(22)

where ${\mathit{l}}_{\mathrm{D}{\mathrm{A}}_7}$ is the vector from point D to point A₇, ${\mathit{l}}_{\mathrm{D}{\mathrm{A}}_8}$ is the position vector from point D to point A₈, both defined in the D-uvw coordinate system. C_CL,2 denotes the compliance matrix of the link 4 at point A₇ in the D-uvw coordinate system, while C_2,2 represents the compliance matrix of the link 4 at point A₇ in the A₈–x₈y₈z₈ coordinate system.

3.3. Overall Stiffness Model of the Mechanism

Based on Equation (5), the deformation mapping model of the rib unit is established as follows

$$J_{\mathrm{CL},i}{S}_{\mathrm{t}}=S_{\mathrm{t},\mathrm{CL},i},i=1,2$$

(23)

where $S_{\mathrm{t},\mathrm{CL},i}$ denote the deformations of the two branch chains in the D-uvw coordinate system.

$$J_{\mathrm{CL},1}=\left[\begin{array}{cc}{s}_x^{\mathrm{T}}& 0\\ s_y^{\mathrm{T}}& 0\\ {(r_{\mathrm{D}{\mathrm{A}}_6}\times s_{{\mathrm{A}}_5{\mathrm{A}}_6})}^{\mathrm{T}}& s_{{\mathrm{A}}_5{\mathrm{A}}_6}^{\mathrm{T}}\\ 0& s_y^{\mathrm{T}}\\ 0& s_{\mathrm{z}}^{\mathrm{T}}\end{array}\right],J_{\mathrm{CL},2}=\left[\begin{array}{cc}{s}_x^{\mathrm{T}}& 0\\ s_y^{\mathrm{T}}& 0\\ 0& s_{{\mathrm{A}}_8{\mathrm{A}}_7}^{\mathrm{T}}\\ 0& s_{\mathrm{z}}^{\mathrm{T}}\end{array}\right]$$

(24)

Rewrite Equation (4) in matrix form:

$$S_{\mathrm{W},\mathrm{D}}=J_{\mathrm{CL},i}^{\mathrm{T}}{S}_{\mathrm{W},\mathrm{CL},i},i=1,2$$

(25)

where $S_{\mathrm{W},\mathrm{CL},1}=S_{wa,1}+{\displaystyle \sum _{k_c=1}^4{S}_{wc,k_c,1}}$, $S_{\mathrm{W},\mathrm{CL},2}={\displaystyle \sum _{k_c=1}^4{S}_{wc,k_c,2}}$.

The virtual work equation can be expressed as

$$S_{\mathrm{t}}^{\mathrm{T}}{S}_{\mathrm{W},\mathrm{D}}=S_{\mathrm{t},\mathrm{CL},1}^{\mathrm{T}}{S}_{\mathrm{W},\mathrm{CL},1}+S_{\mathrm{t},\mathrm{CL},2}^{\mathrm{T}}{S}_{\mathrm{W},\mathrm{CL},2}$$

(26)

Based on the above force mapping model and in accordance with Hooke’s law, the stiffness model of the single DOF deployable unit is formulated. Substituting Equations (23) and (25) into Equation (26), we obtain

$${\rm K}={\displaystyle \sum _{i=1}^2{J}_{\mathrm{CL},i}{K}_{\mathrm{CL},i}{J}_{\mathrm{CL},i}^{\mathrm{T}}}$$

(27)

where K_CL,i represents the stiffness matrix of closed-loop chain I or chain II in the D-uvw coordinate system.

3.4. Compliance Matrix of the Deployable Unit Components

As shown in Figure 4, the rotational axes of the revolute joints in this deployable mechanism are all aligned with the z-axis direction. Since the hinges only possess a rotational degree of freedom about the z-axis, they transmit only force, and not torque, in the direction of rotation, meaning the corresponding rotational stiffness is zero. To reflect the influence of this hinge degree of freedom on the overall stiffness characteristics of the mechanism in the stiffness model, during the modeling process, this paper releases the hinge degree of freedom by setting the rotational stiffness of the links about the z-axis to zero, and correspondingly setting the compliance component corresponding to the rotational degree of freedom about the z-axis to infinity in the compliance matrix. Specifically, at points A₂, A₃, A₄, A₆, and A₇, the rotational stiffness of the components about the z-axis direction is zero, causing the compliance matrix to exhibit singularity in the direction of this degree of freedom.

The general expression for the compliance matrix of a component can be written as

$$C={\left[\begin{array}{cccc}{\Delta }_{\alpha u1,\tau u1}& {\Delta }_{\alpha u1,\tau v1}& \dots & {\Delta }_{\alpha u1,fw1}\\ {\Delta }_{\alpha v1,\tau u1}& {\Delta }_{\alpha v1,\tau v1}& \dots & {\Delta }_{\alpha v1,fw1}\\ ⋮& ⋮& \dots & ⋮\\ {\Delta }_{pw1,\tau u1}& {\Delta }_{pw1,\tau v1}& \dots & {\Delta }_{pw1,fw1}\end{array}\right]}_{6\times 6}$$

(28)

where the ith (i = 1,2,⋯,6) column represents linear/angular deformations resulted from unit force/moment.

The compliance matrix of a link can be obtained either through analytical analysis or extracted directly from finite element software. Taking the link 3 as an example and employing the analytical approach, the nonzero entries of the link’s compliance matrix are given as follows:

$$\begin{array}{c}{\mathrm{c}}_{11}={\displaystyle \frac{\mathrm{L}}{\mathrm{G}{\mathrm{I}}_{\mathrm{P}}}},{\mathrm{c}}_{22}={\displaystyle \frac{\mathrm{L}}{\mathrm{E}{\mathrm{I}}_{\mathrm{y}}}},{\mathrm{c}}_{33}={\displaystyle \frac{\mathrm{L}}{\mathrm{E}{\mathrm{I}}_{\mathrm{z}}}},{\mathrm{c}}_{44}={\displaystyle \frac{\mathrm{L}}{\mathrm{E}\mathrm{A}}},{\mathrm{c}}_{55}={\displaystyle \frac{{\mathrm{L}}^3}{3\mathrm{E}{\mathrm{I}}_{\mathrm{y}}}}\\ {\mathrm{c}}_{66}={\displaystyle \frac{{\mathrm{L}}^3}{3\mathrm{E}{\mathrm{I}}_{\mathrm{z}}}},{\mathrm{c}}_{35}={\mathrm{c}}_{53}={\displaystyle \frac{{\mathrm{L}}^2}{2\mathrm{E}{\mathrm{I}}_{\mathrm{z}}}}, {\mathrm{c}}_{26}={\mathrm{c}}_{62}=-{\displaystyle \frac{{\mathrm{L}}^2}{2\mathrm{E}{\mathrm{I}}_{\mathrm{y}}}}\end{array}$$

(29)

where L is the length of the rod member, A is the corresponding cross-sectional area EI_z and EI_z is the bending section modulus, GI_p is the torsional section modulus.

3.5. Model Verification

A stiffness simulation study was conducted using ANSYS 2021, Canonsburg PA, USA to validate the stiffness model established previously. The effectiveness of the theoretical modeling approach was confirmed by comparing the deformation results from the theoretical model with those obtained under similar conditions in the simulation. The structural parameters for the components within the rib unit are set as follows: the material for all links is aluminum alloy 2A12, its elastic modulus E = 70 GPa, density 2840 kg/m³, and Poisson’s ratio are 0.31. The cross-section of the central link has an outer diameter of 12 mm with a wall thickness of 1 mm, while the other links have an outer diameter of 10 mm with a wall thickness of 1 mm. The scale parameters are shown in

Table 1. Dimensions of each member of the rib unit (mm).

${\mathit{l}}_{\mathbf{O}\mathbf{E}}$	${\mathit{l}}_{\mathbf{E}{\mathbf{A}}_{\mathbf{8}}}$	${\mathit{l}}_{\mathbf{O}{\mathbf{A}}_{\mathbf{5}}}$	${\mathit{l}}_{{\mathbf{A}}_{\mathbf{8}}{\mathbf{A}}_{\mathbf{7}}}$	${\mathit{l}}_{{\mathbf{A}}_{\mathbf{7}}\mathbf{D}}$	${\mathit{l}}_{{\mathbf{A}}_{\mathbf{5}}{\mathbf{A}}_{\mathbf{6}}}$	${\mathit{l}}_{{\mathbf{A}}_{\mathbf{1}}{\mathbf{A}}_{\mathbf{3}}}$	${\mathit{l}}_{{\mathbf{A}}_{\mathbf{0}}{\mathbf{A}}_{\mathbf{2}}}$	${\mathit{l}}_{{\mathbf{A}}_{\mathbf{3}}{\mathbf{A}}_{\mathbf{4}}}$	${\mathit{l}}_{\mathbf{D}{\mathbf{A}}_{\mathbf{6}}}$	${\mathit{l}}_{\mathbf{B}{\mathbf{A}}_{\mathbf{0}}}$	${\mathit{l}}_{\mathbf{F}{\mathbf{A}}_{\mathbf{1}}}$
150	20	50	545	20	535	105	83	450	122	20	20

. Based on these parameters, the theoretical values for the diagonal elements of the compliance matrix for each link, obtained via the analytical method, are shown in

Table 2. The theoretical values of the diagonal elements of the compliance matrix of each member.

	Angular Compliance ((rad/(N·m)) × 10−6)			Linear Compliance ((μm/N) × 10−3)
	${\mathrm{c}}_{11}$	${\mathrm{c}}_{22}$	${\mathrm{c}}_{33}$	${\mathrm{c}}_{44}$	${\mathrm{c}}_{55}$	${\mathrm{c}}_{66}$
${\overline{C}}_{1,2}$	6.78	5.176	$\infty$	0.053	19.0	19.0
${\overline{C}}_{2,1}$	2.45	1.873	$\infty$	0.019	0.902	0.902
${\overline{C}}_{5,1}$	34.5	26.34	$\infty$	0.270	2516	2516
${\overline{C}}_{3,1}$	29.06	22.18	$\infty$	0.227	1497	1497
$C_{2,2}$	35.2	26.86	$\infty$	0.27	2660	2660

.

Considering the influence of the revolute joints and the case where this influence is neglected, external loads of F = 10 N, 20 N, 30 N, 40 N, and 50 N were sequentially applied at point D along the y-direction, as illustrated in Figure 4. The linear deformations of reference point D in the coordinate system O-xyz were obtained from both the theoretical model and the FEA model. The resulting deformation curves from the theoretical and finite element models are presented in Figure 8 The deformation and stiffness at point D under a 50 N external load are shown in

Table 3. Deformation and stiffness at point D under an external load of 50 N.

	Theoretical Value (mm)	Simulation Value (mm)	Error	Stiffness Value (N/mm)
Values	0.435	0.512	17.7%	115

.

The results indicate that the deformation trends obtained from the simulation and the theoretical model are in good agreement, with an overall error of approximately 17% between the theoretical and FEA results, thereby validating the effectiveness of the theoretical stiffness model. The main reason for this deviation is that the compliance of the root short beams and external links was not considered in the theoretical stiffness model, where these components were treated as rigid bodies. In contrast, all components were modeled using beam elements in the finite element analysis, resulting in slightly larger deformations in the simulation compared to the theoretical calculations.

4. Stiffness Analysis of Multiple Configurations

4.1. Stiffness Analysis Based on Structural Configuration

As the fundamental unit of deployable antenna support trusses, the single-degree-of-freedom deployable rib unit undergoes a unique mutual transformation between mechanism and structure during its folding and deployment process. Aiming to achieve high stiffness in the deployed state, this study takes a planar deployable mechanism as the research object and constructs a quadrilateral closed frame as the geometric constraint boundary. Based on the presence or absence of support diagonal members (red lines), secondary support diagonal members (blue lines), and their topological layout, three structural configurations are established: a basic structure without diagonal members, a single-diagonal support structure, and a composite-diagonal support structure, as shown in Figure 9, Figure 10 and Figure 11.

Static analysis was performed on 19 structural types. Concentrated loads of 50 N were applied to nodes on opposite end faces along the y-axis and x-axis of the global coordinate system, while the end faces of the short sides of the quadrilateral frame structure were subjected to fully fixed constraints, as shown in Figure 9. Static stiffness characteristic simulation experiments were conducted. Figure 12 shows the deformation results of each structure in different directions. Common structural types exhibiting high stiffness-to-mass ratios were identified from a structural perspective, and these structures were subsequently used to construct new topological configurations.

Based on the comparative deformation results of the structures, it is evident that Structures 2 and 3 within the single-diagonal support configuration exhibit superior stiffness characteristics, while Structures 11, 12, 14, 15, 16, 18, and 19 within the composite-diagonal support configuration achieve higher structural stiffness. This indicates that structural stiffness performance is enhanced when the primary support diagonals are arranged diagonally (connecting the diagonal nodes of the quadrilateral closed frame) or when they follow a topological layout extending from the vertices to the long side of the quadrilateral closed frame. Further comparison of the deformation results for composite-diagonal support structures reveals that arranging the secondary support diagonals in a scissor-like geometric configuration (forming an X-bracing pattern) with the primary diagonals can effectively improve the structure’s resistance to deformation.

4.2. Stiffness Analysis of Mechanism Configuration

Based on the results of the aforementioned structural stiffness study, Structures 2, 3, 11, 12, 14, and 16 were selected as the fundamental frameworks. Six distinct deployable unit mechanisms were constructed by introducing kinematic joints. All mechanisms satisfy the condition for a planar mechanism with a single degree of freedom, expressed by the formula (3N − 2P = 1), where P is the number of joints and N is the total number of links. Based on the parametric stiffness modeling method, the stiffness models for the six configurations are shown in Appendix A. Furthermore, a finite element simulation model was established for each mechanism configuration. A fixed constraint boundary condition was applied to link 1, and a concentrated load of 50 N was applied at the end along the longitudinal axis of the global coordinate system. The von Mises stress contour plots for the six deployable mechanism configurations are presented in Figure 13.

Based on the study of the simulation data, the stress distribution characteristics of the deployable unit mechanism configurations exhibit a consistent pattern: the arrangement of primary support diagonal members in the configuration influences the distribution of the von Mises equivalent stress. The key areas of focus include the primary support diagonal members, the long-side link members of the quadrilateral closed frame rigidly connected to them, and the revolute joints between the secondary and primary support diagonal members. In particular, the long-side link members of the closed frame experience local bending stresses, while the primary support diagonal members are subjected to axial tensile/compressive stresses.

For the six developed deployable unit mechanism configurations, the aforementioned theoretical stiffness modeling method was applied to compute their stiffness. External loads of F = 10 N, 20 N, 30 N, 40 N, and 50 N were sequentially applied at point D along the y-direction. The deformation results of each configuration along the y-direction are shown in Figure 14. The deformation and stiffness at point D for multiple configurations under a 50 N external load are shown in

Table 4. Deformation and stiffness at point D for multiple configurations under a 50 N external load.

	Theoretical Value (mm)	Simulation Value (mm)	Error	Stiffness Value (N/mm)
configuration 2	0.453	0.461	1.8%	110
configuration 3	0.438	0.515	17.5%	114
configuration 11	0.443	0.468	5.6%	112.8
configuration 12	0.435	0.512	17.7%	115
configuration 14	0.44	0.463	5.2%	113.6
configuration 16	0.442	0.482	4%	113

.

As a crucial load-bearing component, the main purpose of the deployable antenna support truss is to ensure the stable operation of the antenna in its deployed state. The deformation in the y-direction is primarily analyzed because the stiffness characteristics in the vertical direction are dominant. It is demonstrated that the stiffness calculation results of the configurations are accurate by looking at the deformation results of the structural types that correspond to these six configurations. A comparison of the deformation results from the finite element simulations and the theoretical stiffness models for the six configurations shows the same overall trend. The maximum error is 18%, with an average error of about 10%. The main reason for this deviation is that the flexibility of the root short beams and external links was not considered in the theoretical stiffness model, where these components were treated as rigid bodies, resulting in larger deformations in the simulation compared to the theoretical calculations. This further verifies the effectiveness of the established theoretical stiffness model of the deployment unit mechanism.

Based on the analysis of the deformation characteristics shown in Figure 14, a comparative study of the six deployable unit mechanism configurations indicates that Configuration 11 exhibits the best structural stiffness performance, followed by Configurations 2 and 14. The findings demonstrate that enhanced stiffness characteristics can be achieved when the moving platform and the fixed base are connected via three supporting branches, particularly when the primary diagonal supporting rods are attached to the diagonal nodes of the quadrilateral closed frame. Furthermore, the comparison among Configurations 11, 2, and 14 reveals that, for an identical topological layout of the primary diagonal supports, the structural stiffness is positively correlated with the number of link members. Increasing the quantity of secondary diagonal supporting rods strengthens the load-bearing capacity of the mechanism and improves the overall configuration’s resistance to deformation.

5. Conclusions

This study proposes an analytical stiffness modeling method. Through screw theory, kinematic analysis of a modular space-deployable mechanism is conducted, and a static stiffness model of the deployable mechanism is established. Based on an investigation of the stiffness characteristics of the structure, six deployable mechanism configurations are derived topologically.

Starting from the kinematic description and position analysis of the deployable mechanism, a motion–force mapping model for the deployed unit mechanism is formulated using screw theory, revealing the intrinsic relationship between force transmission laws and kinematic characteristics in the deployed state. For deployable mechanisms with multi-closed-loop complex configurations, a limb stiffness modeling method based on closed-loop deformation compatibility conditions is introduced. By combining the principle of virtual work and the deformation superposition principle, a parametric stiffness model of the deployable mechanism is constructed. The correctness and effectiveness of the full mechanism stiffness model are verified through finite element simulations. The model accurately reflects the contribution of each component to the system stiffness, providing a basis for extracting stiffness performance indicators.

Under the boundary constraints of a quadrilateral closed-loop truss structure, the stiffness characteristics of the structure are investigated, leading to the selection of a typical structure with an excellent stiffness-to-mass ratio. On this basis, six deployable mechanism configurations are derived topologically. A comparison between finite element simulation results and the analytical stiffness model confirms the accuracy of the stiffness evaluation for each configuration, demonstrating the practical utility of this method in performance-driven configuration selection and design of deployable mechanisms.