**1. Introduction**

Synthetic aperture radar (SAR), which can provide high-resolution earth images regardless of weather conditions or time of day has recently been used in various fields of domestic and international observation satellites [1,2]. In the case of SAR antennas, to maximize resolution and power gain in acquiring high-resolution images, a large deployable panel is accommodated. However, to deploy such a large space deployment structure in orbit, it is essential to use specific mechanisms that allow the structure to be appropriately folded and stored inside the launch-vehicle fairing and fully deployed in orbit [1–6].

Furthermore, when a satellite performs a mission, line-of-sight (LOS) pointing stability must be secured. During ground imaging, if the relatively large antenna is shaken by external or internal disturbances, the quality of the captured images will be degraded. Therefore, the stiffness of the rotational spring hinges of deployable panels is often increased in order to obtain high deployment stability. This mechanism shortens the deployment time but leads to a high impact load when fully deployed and latched. This deployment impact load could cause damage to structures, payloads, and solar cells. By contrast, decreased stiffness of the rotational spring hinges reduces the deployment impact load by increasing the deployment time, but cannot guarantee successful full deployment due to harness resistance and mechanical friction of deployment devices at low temperatures [7].

**Citation:** Choi, H.-S.; Kim, D.-Y.; Park, J.-H.; Lim, J.H.; Jang, T.S. Modeling and Validation of a Passive Truss-Link Mechanism for Deployable Structures Considering Friction Compensation with Response Surface Methods. *Appl. Sci.* **2022**, *12*, 451. https://doi.org/ 10.3390/app12010451

Academic Editors: Adel Razek

Received: 13 November 2021 Accepted: 1 January 2022 Published: 4 January 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In response to these issues, a variety of methods have been proposed to allow stable deployment and reduce deployment impact loads, such as a torsion spring-latching hinge [8], tape spring hinge [9], and shape memory alloy (SMA)-based hinge [10]. However, these deployment mechanisms still have limitations including low deployment stiffness of the relatively large deployable structures. To address this, truss-link mechanisms [11] have been applied to many satellite missions to increase the deployment stiffness: ERS-1 [12], RADARSAT-1 [13], RADARSAT-2 [14], RADARSAT-Constellation [15], ALOS-2 [16], SEASAT [17], WorldView-4 [18], and Sentinel-5P [19]. Among these, the RADARSAT mission employed the truss-link mechanism to increase deployment stiffness and attempted to reduce the number of truss-links as the follow-up mission progressed to reduce weight. Comparing the number of truss-links in satellites, RADARSAT-1 (1995) has 30 links, RADARSAT-2 (2007) has 22 links, and RADARSAT-Constellation (2019) has four links. The reduction in the number of links is the result of a trend to build up small satellites for constellation missions. In addition, the applications of the truss-link mechanisms have been diversified. Wang et al. proposed a pyramid deployable truss structure (PDTS) design for deployable SAR antennas through an analytical approach based on weighted graphs and kinematic chain techniques [20]. They also proposed a modular deployable truss structure (MDTS) for large SAR antennas [13] with a scaled model. A difference was found between the deployment test and the simulation, however, due to uneven and indeterminate ground friction as well as joint friction. Furthermore, to eliminate interference during the deployment process of the truss structure, the optimal motor driving torque was determined through the design of experiments (DOE) method. Han et al. proposed a large-scale deployable ring truss that composes a space antenna by complementing a rope-based actuation and cable net system [21]. Based on the absolute node coordinate formulation (ANCF), a method for simulating the rigid body motion of the antenna support arm was proposed, and the friction and motion equations of the flexible cable net were presented using the Lagrange equation.

Most of the aforementioned works essentially assume that an active driving mechanism such as motors with cables is employed for large-scale SAR antenna structures. However, such an active mechanism is hardly employed for small satellites due to deployment costs and complexity. In addition, the friction effects cannot be neglected as they generate a difference between deployment tests and simulations.

The goal of this work is to propose a passive truss-link mechanism for large deployable structures of small satellites. In contrast to the many truss-link mechanisms [11–17], we do not accommodate any active driving mechanisms because they are not simple and not sufficiently reliable. As driving mechanisms, only conventional torsion spring hinges with proper latching mechanisms were employed in this work. To achieve successful deployment, an inverse identification technique for equivalent friction torque (EFT) was proposed based on the results of the response surface method combined with the central composite design technique. Finally, a torque margin analysis was conducted to predict whether the deployment would be successful.

This works consisted of three sections: first, we discussed the configuration design from conceptual design to detailed modeling considering deployment kinematics. Second, deployment dynamics simulations and tests were conducted. To minimize the discrepancy between the test and simulation, the friction-compensation technique was introduced, and results were verified by a torque margin analysis. Finally, we provided concluding remarks.

#### **2. Configuration Design of Truss-Link Mechanism**

#### *2.1. Concept Design*

For full deployment of deployable structures with truss-links, the total degrees of freedom (*DOF*) has to be checked to determine whether the structure is over-constrained. If the *DOF* of the deployable structure becomes negative, the deployment will fail due to over-constraint problems. First, we analyzed the *DOF* by referring to RADARSAT-1 and RADARSAT-2 satellites using truss-links [13,14]. Two-dimensional (2D) designs of

one payload wing of the satellite were considered, Type 1 (RADARSAT-1) and Type 2 (RADARSAT-2), to analyze the *DOF*, as seen in Figure 1. We employed Gruebler's equation, as shown in Equation (1) to evaluate the total *DOF* [22].

$$\text{SysDOF} = \mathfrak{Z}(L-1) - \mathfrak{Z}I\_1 - I\_2 \tag{1}$$

**Figure 1.** Configuration of 2-D truss-link mechanism: (**a**) Type 1 (RADARSAT-1) (**b**) Type 2 (RADARSAT-2).

Here, *SysDOF* represents the total *DOF* of the system, *L* represents the number of bodies, *J*<sup>1</sup> represents the number of joints with 1 *DOF* such as a revolute joint, and *J2* represents the number of joints with 2 *DOF* such as a universal joint. As a result, Type 1 has nine *DOF*, and Type 2 has seven *DOF*. The greater number of *DOF* in Type 1 compared with Type 2 is advantageous for preventing over-constraint but has the disadvantage of making the structure heavier owing to a greater number of links than Type 2. Thus, the deployment simulation model is designed based on Type 2 because it is lightweight due to the small number of truss-links and short total length. The total *DOF* calculation results are summarized in Table 1.

**Table 1.** Evaluation of total DOF.


## *2.2. Configuration Design*

The configuration is simplified to a two-dimensional domain to obtain the truss-link's dimensions in the stowed configuration (Figure 2). By simplifying to a two-dimensional geometry, it is possible to reduce the *DOF* and variables used to obtain dimensions and coordinates of the body compared to traditional three-dimensional (3D) methods. The dimensions and coordinates were defined in the formula in the reference [23], which was derived using geometric construction methods and bar-groups methods. Upon verification of the formula's results, it was found that there were some errors, which were fixed accordingly in this study as presented in Equations (2)–(10).

With the panel folded, the length and angle of each truss-link are derived from input variables in Tables 2 and 3. Input variables including *ϕ*1, *ϕ*4, *ϕ*6, *L*1, *L*4, *L*6, *XA*, *XC*, *XD*, *XF*, and *XG* are defined by the mechanical designer considering the size of the panel according to the design requirements. Through Equations (2)–(10), output variables, including *ϕ*2, *ϕ*3, *ϕ*5, *ϕ*7, *L*2, *L*3, *L*5, *L*7, *XB*, *XE*, and *XH*, were determined.

Joint O corresponds to the origin of absolute coordinates as the rotation center of the hinge installed between the bus and the inner panel; Joint I is the rotation center between the inner panel and outer panel; Joint F is the intersection point of all truss-links. Furthermore, to keep the deployment state, three latching points, B, E, and H, were considered, as shown in Figure 2b.

**Figure 2.** Configuration design of truss-link mechanism in (**a**) stowed configuration (**b**) deployed configuration with latching points: B, E, H.

**Table 2.** Design variables of the truss-link mechanism.


**Table 3.** Values of the input and output design variables of truss-link mechanism.


$$L\_3 = \sqrt[2]{\left(x\_F - x\_\mathbb{C}\right)^2 + \left(z\_F - z\_\mathbb{C}\right)^2} \tag{2}$$

$$(\mathbf{x}\_H, \mathbf{z}\_H) = (\mathbf{x}\_G + L\_6 \cos \varphi\_6, \mathbf{z}\_G - L\_6 \sin \varphi\_6) \tag{3}$$

$$L\_{\mathsf{T}} = \sqrt[2]{\left(\mathbf{x}\_{\mathsf{F}} - \left(\mathbf{x}\_{\mathsf{G}} + L\_{\mathsf{G}} \cos \varphi\_{\mathsf{G}}\right)\right)^{2} + \left(z\_{\mathsf{F}} - \left(z\_{\mathsf{G}} - L\_{\mathsf{G}} \sin \varphi\_{\mathsf{G}}\right)\right)^{2}} \tag{4}$$

$$(\ldots \ldots \ldots) = (\ldots \quad \mathbf{I} \quad \ldots \ldots \quad \mathbf{z} \quad \rightarrow \mathbf{I} \quad \ldots \ldots \tag{5}$$

$$(\mathbf{x}\_B, z\_B) = (\mathbf{x}\_A - L\_1 \cos \varphi\_1, z\_A + L\_1 \sin \varphi\_1) \tag{5}$$

$$L\_2 = \sqrt[2]{(\mathbf{x}\_A - L\_1 \cos \varphi\_1) - \mathbf{x}\_F}^2 + (z\_A + L\_1 \sin \varphi\_1) - z\_F \tag{6}$$

$$\varphi\_2 = \cos^{-1}(\frac{\mathbf{x}\_B - \mathbf{x}\_F}{L\_2}) \tag{7}$$

$$(x\_E, z\_E) = (x\_D - L\_4 \cos \varphi\_4, z\_D + L\_4 \sin \varphi\_4) \tag{8}$$

$$L\_5 = \sqrt[2]{(x\_D - L\_4 \cos \varphi\_4) - x\_F} \\ \left( -z\_D + L\_4 \sin \varphi\_4 \right) - z\_F \tag{9}$$

$$\varphi\_{\mathfrak{I}} = \cos^{-1}(\frac{\mathfrak{x}\_F - \mathfrak{x}\_{\mathbb{C}}}{L\_{\mathfrak{I}}}) \tag{10}$$

$$\varphi\_5 = \cos^{-1}(\frac{\mathcal{X}\_F - \mathcal{X}\_E}{L\_5}) \tag{11}$$

$$q\_{\mathcal{T}} = \cos^{-1}(\frac{\mathcal{X}\_F - \mathcal{X}\_H}{L\_{\mathcal{T}}}) \tag{12}$$

For practical design, a 3D configuration design was conducted by CATIA. The 3D model consists of an inner panel, an outer panel, and 12 truss-links, with dimensions of 1620 mm × 800 mm in the fully deployed configuration (see Figure 3) and a total mass of 11.45 kg as shown in Table 4. The mass of hinges is included in the mass of the panel, and the mass of the bracket is included in the mass of each link. The truss-links connection was finally realized as illustrated in Figure 4a, with several connection angles, as shown in Figure 4b. The connection angles according to the truss-links are summarized in Table 5. The material of the truss-link is aluminum 6061, which has a Young's modulus of 68.9 GPa, a Poisson's ratio of 0.33, and density of 2700 kg/m3 [24]. The cross-section of the truss was assumed as a tube-type with an outer diameter of 10 mm and an inner diameter of 8 mm.

**Figure 3.** Configuration of a deployment analysis model.



**Table 5.** Connection angle of truss-links.


**Figure 4.** Configuration of truss-link mechanism: (**a**) 3-D truss, (**b**) truss-links with connection angles θ.

#### *2.3. Latching Mechanism*

Once the truss-links are deployed, they must keep their positions and angles through proper latching mechanisms, which are the most critical factors in maintaining the deployment configuration. As shown in Figure 5, the latching structure consists of a neodymium magnet, a ball plunger, and a stopping protrusion. The neodymium magnet and the ball plunger are installed on the lower bracket, and the stopping protrusion is installed on the upper bracket. If the truss-links rotate, the stopping protrusion meets the ball plunger, slides over the ball plunger and locks. In addition, neodymium magnets were installed on the wall to prevent the stopping protrusion from bouncing by the reaction force.

**Figure 5.** Configuration of latching mechanism: (**a**) Neodymium magnet, ball plungers (**b**) Latching system.

## **3. Deployment Dynamics**

#### *3.1. Modeling of Kinematic Joints and Latching Mechanism*

To present the deployment motion, dynamic modeling was conducted with the aid of commercial multibody dynamics (MBD) software, RecurDyn. There were three kinds of joints: revolute joints for one-axis revolution, fixed joints to eliminate rigid-body motion, and on-off joints for latching truss-links. In contrast to the two-dimensional type-2 design, in the actual 3D design, when many truss-links were connected at a rotation joint (see Figure 4a) such as joint F, the over-constrained mechanisms cannot be avoided according to Equation (1). To address this problem, we used the bushing force function of RecurDyn instead of the revolute joint [25]. The bushing force can model any joint with 6 DOF by adjusting the stiffness of six springs independently. In the case of a revolute joint with the bushing force, the translation stiffness was set to be sufficiently high that the translation motion would be constrained; one of the rotational stiffnesses was set to zero

for the revolution while the other rotation stiffness was set high. The proposed truss-link mechanism must be latched on each joint, B, E, and H, in a fully deployed state at the end of the deployment process. The on/off joints provided by RecurDyn are used to address these problems, which allows operation when certain conditions are met [26], as shown in Table 6. Using this, we modeled the two links connected to the joint to be fixed without generating additional motion when they are aligned.

**Table 6.** Expression function for latching mechanism.

