# Application of Stewart Platform in the Low-Frequency Vibration Characteristic Test of Space Truss Deployable Antenna on Satellite

Authors: Hui Wang; Jiang Zhao; Yonggang Xue; Zhirong Huang; Shikun Zheng; Xiaofei Ma
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

The space truss deployable antenna has low natural frequency, multi-closed-loop, and multiredundant structure characteristics, which is a kind of flexible antenna. It is hard to obtain the vibration characteristics under low frequency by finite element analysis or conventional shaking table test. To verify the structure characteristics of the antenna in the folded state below 2 Hz, a low-frequency vibration system is established based on the 6-DOF Stewart platform. Rotation matrix analysis method was adopted to obtain the platform’s velocity and acceleration characteristics. Then, a low-frequency vibration test was carried out on the platform. The results show that the natural frequency of the antenna is 1.51 Hz and the maximum dynamic displacement is 140.6 mm, which provides a certain foundation for the further development of the space truss antenna.


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## Body

## 1. Introduction

The space truss deployable antenna is a large-scale mesh deployable truss structure with high storage ratio and large deployment stiffness. The main structure is composed of multiple tetrahedral frame elements with energy storage springs. These basic tetrahedral retraction and deployment elements share nodes, and the retraction movement is associated to form a multi-closed-loop and multiredundant space structure [1–3]. The antenna is folded and fixed before launch, and the expansion is realized by a spring driving on the orbit to form a paraboloid. The antenna has good strength, stiffness, and high deployment reliability [4, 5]. It can flexibly construct the truss system by changing the number of basic truss elements or adopting different connection methods. Due to the stiffness characteristics of the structure and the freedom of the configurable truss structure, many research institutions have carried out relevant technical research at present [6–9].Due to the limitation of rocket payload space, the large aperture antenna needs to be folded. The antenna must be able to withstand the vibration from the rocket or itself without damage during launch. The general method is to use the finite element analysis method or shaking table test to simulate the real vibration environment during transportation [10].The truss antenna contains a large number of rods, ropes, and hinges, and its dynamic model is complex, which brings great challenges to the finite element solution [11]. At the same time, to reduce the weight of the antenna, the lightweight carbon fiber composite is usually used as the rod material, which will inevitably reduce the strength of the rod and make the flexibility of the antenna more obvious. The fundamental frequency of the antenna in the folded state is usually below 2 Hz. Therefore, the vibration test of the truss antenna puts forward higher requirements for the working frequency and output load of the shaking table.Limited by the current low-frequency shaking table technology, the frequency range of the small shaking table is 2 ~ 10 kHz and that of the large shaking table is 5 ~ 2 kHz. Therefore, it is difficult to realize the low-frequency vibration assessment [12]. The Stewart platform is a spatial 6-DOF parallel mechanism. By controlling the independent movement of six telescopic cylinders, it can simulate various attitude responses of the spacecraft during space flight, especially the low-frequency vibration during launch. It has become an important simulation test device for dynamic reliability research of aerospace equipment [13, 14].In this paper, a low-frequency vibration characteristic test platform suitable for the space truss antenna is built based on Stewart platform, and relevant low-frequency tests are carried out to obtain the fundamental frequency and dynamic displacement in the stowed state, which provide a certain foundation for the further development of the space truss antenna.


## 2. Antenna’s Structure Characteristics

The antenna is composed of reflector, truss, and a metal mesh. The reflector truss is composed of several frame units. Each unit adopts a tetrahedral element, which is composed of web members, folding rods, connection nodes, and torsion springs, as shown in Figure1.Figure 1
Frame units with tetrahedral element.The three folding rods can be folded symmetrically around the middle, and the middle of each folding rod is provided with a hinge, which contains a spring. When the tetrahedron is folded, the spring element of the folding rod stores energy and acts as the driving force of the tetrahedron unit when it is deployed. The deployable state and fold state of the antenna are illustrated in Figure2.Figure 2
Fold state of the antenna.


## 3. Kinematics Analysis of Stewart Platform

### 3.1. 6-DOF Stewart Platform

The 6-DOF Stewart parallel platform is mainly composed of the load platform, base platform, and six driving cylinders. Each driving cylinder is connected to the load platform and base platform, respectively, by a hinge. According to different hinge modes, it can be divided into ball joint S-P-S (sphere joint-prismatic joint-sphere joint) type and universal joint U-P-S (universal joint-prismatic joint-sphere joint) type.The working platform includes an upper platform, lower platform, load mounting plate, upper and lower connecting hinges, servo actuator, auxiliary cylinder, and base, as shown in Figure3. The servo actuator is composed of an electric cylinder and motor. The servo actuator receives the control signal and works on the platform to make the upper platform realize longitudinal, heel, yaw, lifting, transverse, and lateral movement and drive the load to realize the simulated motion.Figure 3
6-DOF Stewart parallel platform.


### 3.2. Rotation Matrix Method

The attitude description of Stewart platform generally adopts the multirigid body rotation coordinate method [15]. In this paper, the Cardan angular coordinate is adopted based on the Cartesian coordinate system, and three coordinate axes are selected in a certain order for three consecutive rotations.The Cardan angular coordinates are shown in Figure4. The coordinate system o−xyz first rotates the angle α around the x-axis counterclockwise to o−xy′z′, then rotates the angle β around the y′ axis to o−x′y′Z, and finally rotates the angle γ around the Z-axis to the position of the coordinate system o−XYZ. Each rotation relationship can be described by a matrix. From the coordinate system o−xyz to the coordinate system o−xy′z′, the point P can be expressed as y,z in the coordinate system o−XYZ and y′,z′ in the coordinate system o−xy′z′. The relationship is expressed by the following equation.
(1)y′=Ycosγ−Zsinγ,(2)z′=Ysinγ+Zcosγ.Figure 4
Cardan angle coordinate system schematic diagram.In matrix form,(3)xyz=1000cosα−sinα0sinβcosαxy′z′.Similarly,(4)xy′z′=cosβ0sinβ010−sinβ0cosβx′y′Z,(5)x′y′Z=cosγ−sinγ0sinγcosγ0001XYZ.Therefore,(6)xYZ=cβcγ−cβsγsβcαsγ+sαsβcγcαcγ−sαsβsγ−sαcβsαsγ−cαsβcγsαcγ+cαdβdγcαcβXYZ,where c represents cos () and s represents sin ().Therefore, the vector between the coordinate systemo−XYZ and the coordinate system o−xyz can be converted by the direction cosine matrix R, which can be expressed as
(7)R=cβcγ−cβsγsβcαsγ+sαsβcγcαcγ−sαsβsγ−sαcβsαsγ−cαsβcγsαcγ+cαdβdγcαcβ.


### 3.3. Inverse Kinematics of Stewart Platform

To accurately control the position and attitude of Stewart platform, the six cylinders’ length needs to be solved according to the predetermined position and attitude of the upper load platform, namely, three linear coordinate parameters (x,y,z) and three rotational coordinate parameters (α,β,γ).Two reference pointsOp and Ob, which are associated with the load platform and base platform, are selected to establish the Cartesian coordinate systems P and B, respectively. The coordinate system B is considered fixed, in which other coordinates can be represented expediently.The vector from the origin pointOb of the coordinate system B to the origin point Op of the coordinate system P can be represented as t, where t=x,y,z. The Cardan angle between P and B is θ, where θ=α,β,γ.It is assumed that the vector fromOp to the connection point between the load platform and each electric cylinder is Pi, the vector from Ob to the connection point between the base platform and each cylinder is bi, and the vector of the connection points at both ends of the six cylinders is Si, and the relation between the vectors above can be illustrated in Figure 5.Figure 5
Coordinate system of Stewart platform.Then, the following formula can be obtained:(8)Si=Rpi+t−bi,where R is the rotating cosine matrix (i=1,2,3,4,5,6).The length of electric cylinder is(9)Li=Si=Rpi+t−biRpi+t−biLi=Si.And the unit vector ofSi is
(10)si=SiSi.The formula above establishes the relationship between the pose of the load platform and the cylinder, that is, the inverse kinematics of Stewart platform.The Jacobian matrix is related to the cylinder extension velocitySi and the velocity vector χ̇, and the attitude of the load platform is a function of x,y,z,α,β,γ, then,
(11)ω=θ̇,v=ṫ,χ̇=vTωTT,qi=Rpi.The sliding speed of each cylinder is(12)Ṡi=siv+ω×qi=siTqi×siTvω=Jχ̇.Then, the Jacobian matrix is(13)J=siTqi×siT.


## 3.1. 6-DOF Stewart Platform

The 6-DOF Stewart parallel platform is mainly composed of the load platform, base platform, and six driving cylinders. Each driving cylinder is connected to the load platform and base platform, respectively, by a hinge. According to different hinge modes, it can be divided into ball joint S-P-S (sphere joint-prismatic joint-sphere joint) type and universal joint U-P-S (universal joint-prismatic joint-sphere joint) type.The working platform includes an upper platform, lower platform, load mounting plate, upper and lower connecting hinges, servo actuator, auxiliary cylinder, and base, as shown in Figure3. The servo actuator is composed of an electric cylinder and motor. The servo actuator receives the control signal and works on the platform to make the upper platform realize longitudinal, heel, yaw, lifting, transverse, and lateral movement and drive the load to realize the simulated motion.Figure 3
6-DOF Stewart parallel platform.


## 3.2. Rotation Matrix Method

The attitude description of Stewart platform generally adopts the multirigid body rotation coordinate method [15]. In this paper, the Cardan angular coordinate is adopted based on the Cartesian coordinate system, and three coordinate axes are selected in a certain order for three consecutive rotations.The Cardan angular coordinates are shown in Figure4. The coordinate system o−xyz first rotates the angle α around the x-axis counterclockwise to o−xy′z′, then rotates the angle β around the y′ axis to o−x′y′Z, and finally rotates the angle γ around the Z-axis to the position of the coordinate system o−XYZ. Each rotation relationship can be described by a matrix. From the coordinate system o−xyz to the coordinate system o−xy′z′, the point P can be expressed as y,z in the coordinate system o−XYZ and y′,z′ in the coordinate system o−xy′z′. The relationship is expressed by the following equation.
(1)y′=Ycosγ−Zsinγ,(2)z′=Ysinγ+Zcosγ.Figure 4
Cardan angle coordinate system schematic diagram.In matrix form,(3)xyz=1000cosα−sinα0sinβcosαxy′z′.Similarly,(4)xy′z′=cosβ0sinβ010−sinβ0cosβx′y′Z,(5)x′y′Z=cosγ−sinγ0sinγcosγ0001XYZ.Therefore,(6)xYZ=cβcγ−cβsγsβcαsγ+sαsβcγcαcγ−sαsβsγ−sαcβsαsγ−cαsβcγsαcγ+cαdβdγcαcβXYZ,where c represents cos () and s represents sin ().Therefore, the vector between the coordinate systemo−XYZ and the coordinate system o−xyz can be converted by the direction cosine matrix R, which can be expressed as
(7)R=cβcγ−cβsγsβcαsγ+sαsβcγcαcγ−sαsβsγ−sαcβsαsγ−cαsβcγsαcγ+cαdβdγcαcβ.


## 3.3. Inverse Kinematics of Stewart Platform

To accurately control the position and attitude of Stewart platform, the six cylinders’ length needs to be solved according to the predetermined position and attitude of the upper load platform, namely, three linear coordinate parameters (x,y,z) and three rotational coordinate parameters (α,β,γ).Two reference pointsOp and Ob, which are associated with the load platform and base platform, are selected to establish the Cartesian coordinate systems P and B, respectively. The coordinate system B is considered fixed, in which other coordinates can be represented expediently.The vector from the origin pointOb of the coordinate system B to the origin point Op of the coordinate system P can be represented as t, where t=x,y,z. The Cardan angle between P and B is θ, where θ=α,β,γ.It is assumed that the vector fromOp to the connection point between the load platform and each electric cylinder is Pi, the vector from Ob to the connection point between the base platform and each cylinder is bi, and the vector of the connection points at both ends of the six cylinders is Si, and the relation between the vectors above can be illustrated in Figure 5.Figure 5
Coordinate system of Stewart platform.Then, the following formula can be obtained:(8)Si=Rpi+t−bi,where R is the rotating cosine matrix (i=1,2,3,4,5,6).The length of electric cylinder is(9)Li=Si=Rpi+t−biRpi+t−biLi=Si.And the unit vector ofSi is
(10)si=SiSi.The formula above establishes the relationship between the pose of the load platform and the cylinder, that is, the inverse kinematics of Stewart platform.The Jacobian matrix is related to the cylinder extension velocitySi and the velocity vector χ̇, and the attitude of the load platform is a function of x,y,z,α,β,γ, then,
(11)ω=θ̇,v=ṫ,χ̇=vTωTT,qi=Rpi.The sliding speed of each cylinder is(12)Ṡi=siv+ω×qi=siTqi×siTvω=Jχ̇.Then, the Jacobian matrix is(13)J=siTqi×siT.


## 4. Low-Frequency Vibration Test

A low-frequency vibration test system based on 6-DOF Stewart platform is established as shown in Figure6. In the system, the main function of the electrical control part is to receive the command data output by the control system and transmit it to the servo driver. The signal is amplified to control the motor movement, which will drive the electric cylinder to realize various postures of the platform.Figure 6
Experimental schematic diagram of Stewart platform.In the test, the truss antenna is installed on the platform, and the vibration load acts on the platform along three mutually orthogonal directions, respectively. The measuring points on the structure are showed in Figure7.Figure 7
Measuring points on the antenna.The test conditions are determined as shown in Table1. Moreover, a constant frequency vibration test was carried out at 1.5 Hz with 0.2 g load according to the actual needs.Table 1
Input conditions for low-frequency vibration test.
DirectionFrequency (Hz)MagnitudeZ0.4 ~ 1.80.04~0.45 g1.8 ~ 60.45 g6 ~ 80.45~0.65 gX0.4 ~ 2.10.1 ~ 0.33 g2.1 ~ 8 Hz0.33 gY0.8 ~ 50.2 gScanning rate0.05 Hz/sFigures8–10 show the motion curves of the six cylinders during the vibration test.Figure 8
Motion curves of the six cylinders during the test in theX-direction.Figure 9
Motion curves of the six cylinders during the test in theY-direction.Figure 10
Motion curves of the six cylinders during the test in theZ-direction.The following figures (Figures11–13) show the response curve in each direction during the vibration, and the tables (Tables 2–4) give the maximum strain and dynamic displacement response and fundamental frequency information of the above main measuring points. The fundamental frequency information is obtained by time conversion, and the dynamic displacement is obtained by high-speed photogrammetry system, and the measurement accuracy is better than 0.1 mm.Figure 11
Response curve of point S5 inX-direction vibration.Figure 12
Response curve of point S2 inY-direction vibration.Figure 13
Response curve of point S1 inZ-direction vibration.Table 2
Test result of each measuring point inX-direction.
Measuring pointX-directionTime (s)Initial frequency (Hz)Fundamental frequency (Hz)Amplitude (με)Initial timeFirst peak timeS1-5042.912.6910.44.02S2-79.829.792.7S3-104.154.674.8S4-895182S5-82.38680.3S6-2764090S7-1348786S8-75137100S9-224191.9S10-15734485S11-278591.9S12-294692.6Dynamic displacementdx: -8.2 ~ 8.75 mm; dy: -4.1 ~ 4.3 mm; dz: -61.8 ~ 64.3 mmTable 3
Test result of each measuring point inY-direction.
Measuring pointY-directionTime (s)Initial frequency (Hz)Fundamental frequency (Hz)Amplitude (με)Initial timeFirst peak timeS1-68.41199.822.20.81.51S2-462022.9S3-1201622.2S4-1102222.9S5-6412320.1S6-17910221.3S7-4393824.0S8-37419524.0S9-9.614823.3S10-1226822.9S11-9.64022.2S12-33.55922.2Dynamic displacementdx: -1.73~12.37 mm; dy: -140.6 ~ 117.1 mm; dz: -6.76~3.67 mmTable 4
Test result of each measuring point inZ-direction.
Measuring pointZ-directionTime (s)Initial frequency (Hz)Fundamental frequency (Hz)Amplitude (με)Initial timeFirst peak timeS1-29.733.521.3187.40.44.02S2-45.742.6110.6S3-45.483.294.4S4-129.4161.788.4S5-48.478.795.3S6-365.1255.793.7S7-167276.994.0S8-298.298.893.6S9-39.333.893.9S10-109.7215.993.8S11-21.823.494.0S12-47.217.393.9Dynamic displacementdx: -8.7 ~ 9.25 mm; dy: -4.6 ~ 4.6 mm; dz: -62.4 ~ 65.1 mmAccording to the test results, the response inX-direction and Z-direction of the truss antenna is coupled, and the fundamental frequency is 4.02 Hz, while the fundamental frequency in Y-direction is the lowest, which is 1.51 Hz as shown in Figure 3 and Table 4. The maximum dynamic displacement of the antenna is 140.6 mm. Figure 14 shows the gap between the antenna and the fairing in a certain application, which is much greater than the dynamic displacement.Figure 14
Installation status of antenna in the fairing.


## 5. Conclusion

The truss deployable antenna has the remarkable characteristics of low fundamental frequency and flexible nonlinearity, which makes it difficult to use the conventional shaking table to investigate the vibration characteristics in the low frequency band. In this paper, a low-frequency test system based on 6-DOF Stewart parallel platform is established, and the structural characteristics of the antenna in the stowed state below 2 Hz are analyzed. It shows that the lowest fundamental frequency is 1.51 Hz inY-direction and the dynamic displacement is 140.6 mm. The results are of great value to the further engineering application of the antenna.



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*Source: 8063786-2022-07-04.xml*