3.1 Decomposition of modular antennas

Because the assembly errors occur during the assembly process, the assembly positions of modules are actually inaccurate. In general, the assembly error of the module is equal to the error of two matching faces, as shown in Fig. 4(a).

Figure 4. Matching faces between two modules.

In the actual assembly process, the matching faces are used to determine the relative position of two modules. As shown in Fig. 4, the relative position between module 1 and module 2 is determined by the pair of matching faces 1. The relative position between module 1 and module 3 is determined by the pair of matching faces 2. Since the positions of module 2 and module 3 are determined, the pair of matching faces 3 does not affect the relative position between two modules. In summary, the matching faces can be divided into active matching faces and passive matching faces. The active matching faces determine the relative position between two modules, and the passive matching faces only represent the connection relationship.

According to the type of matching faces, assembly errors can be divided into active errors and passive errors. Active errors and passive errors appear at the active matching face and passive matching face, respectively. During the assembly process, the distribution of the active assembly is determined by the assembly sequence. Meanwhile, the distribution of the active/passive errors also be determined. All modules are divided into the assembly zone based on the active errors, as shown in Fig. 5(a). There is an active error in the gap where the red line goes through, and the passive errors appear in the other gaps. The 12 nodes of a single module are numbered as shown in Fig. 5(b).

Figure 5. Assembly errors classification.

Because the positions of modules are inaccurate, their positions need to be adjusted. In order to get good adjustment effect, the active connections are installed at the active errors. The soft connection is installed in the passive error position to ensure the mobility of the antenna. The structure of soft connection is shown in Fig. 6.

Figure 6. Structure of the connection.

The active connection consists of a linear actuator, two pedestals and two sphere joints. The linear moto, inchworm actuator and ultrasonic moto also can be used as linear actuator [Reference Dong, Li, Wang and Ning24, Reference Li, Deng, Zhang, Hu and Liu25]. The soft connection consists of a sleeve, a slider, two springs and two pedestals. The slider is drove when the one of pedestal is compressed or stretched. Two springs are mounted on the upper and lower sides of the slider.

3.2 Exponential product expression of assembly error

Ideal node coordinates of the modular antenna can be obtained by applying the design method in Section 2. The geometric centre of the module is

(3) \begin{align}{P_{Mi}} = \frac{1}{4}\left( {P_i^1 + P_i^4} \right) + \frac{1}{4}\left( {P_i^7 + P_i^{10}} \right) + \left[ {\begin{array}{*{20}{c}}0\quad 0\quad { - h/2}\end{array}} \right]\end{align}

where $P_i^1$ , $P_i^4$ , $P_i^7$ and $P_i^{10}$ represent the node 1, 4, 7, 10 of the module $i$ , $h$ is the height of the module.

Because the geometry of mounting faces is triangle, the geometric centre of the triangle is calculated as

(4) \begin{align} P_{Ti}^k = \frac{1}{2}\left( {\left.P_i^n + P_i^{n + 1}\right) + \left[ {\begin{array}{*{20}{c}}0\quad 0\quad { - {\rm{h}}/2}\end{array}} \right]} \right)\end{align}

where $P_{Ti}^k$ is the geometric centre of the mounting face of the module i. Because the module includes six mounting faces, the value of $k$ ranges from 1 to 6. The n represents the node numbers of the module, which ranges from 1 to 12.

When modules i and $j$ are connected through the active mounting faces, the geometric centres of the mounting faces are $P_{Ti}^k$ and $P_{Tj}^k$ . Then, the centre point between the active mounting faces of modules $i{\rm{\;}}$ and $j$ is defined as

(5) \begin{align}V_{ij}^k = \frac{1}{2}\left( {P_{Ti}^k + P_{Tj}^k} \right)\end{align}

Each module is regarded as a rigid body, and the relationship between the module nodes and the geometric centre point is fixed. The assembly error transfer model is shown in Fig. 7. The $V_{ij}^k$ represents the virtual assembly point between the modules $i{\rm{\;}}$ and $j$ . Point ${P_{Mi}}$ is the origin of the global coordinate system.

Figure 7. Assembly error transfer model.

During the assembly mission, the assembly error is a variable with six degrees of freedom. Thus, exponential product theory is adopted to describe the assembly errors [Reference Chen, Wang and Lin26, Reference Fu, Fu and Shen27].

A virtual coordinate system is established at the virtual assembly point between two modules, and the directions of the virtual coordinate system are defined. The X-axis and Y-axis are equal to $\frac{{P_i^n - P_i^{n + 1}}}{{\left| {P_i^n - P_i^{n + 1}} \right|}}$ and $\frac{{P_{Ti}^k - P_{Tj}^k}}{{\left| {P_{Ti}^k - P_{Tj}^k} \right|}}$ , their expressions are ${\rm{\;}}{X_{ei}}$ and ${Y_{ei}}$ , respectively. The Z-axis is equal to $\frac{{{X_{e1}} \times {Y_{e1}}}}{{\left| {{X_{e1}} \times {Y_{e1}}} \right|}}$ .

When two modules are assembled, there will be displacement and rotation errors along the X-Y-Z axes. According to the screw theory, the principal part $\boldsymbol{{s}}$ of the screw is $\left( {\begin{array}{*{20}{c}}{{X_{e1}}}\quad {{Y_{e1}}}\quad {{Z_{e1}}}\end{array}} \right)$ . The physical meaning the rotation or displacement direction of the assembly error.

The subpart ${s_0}$ of the screw is

(6) \begin{align}{{\bf{s}}_{\boldsymbol{{0}}}} = \boldsymbol{{s}} \times \boldsymbol{{q}}\end{align}

where the $\boldsymbol{{q}}$ is the point coordinate position vector of the virtual assembly point $\left( {\begin{array}{*{20}{c}}{{x_{V_{ij}^k}}}\quad {{y_{V_{ij}^k}}}\quad {{z_{V_{ij}^k}}}\end{array}} \right)$ .

The screw axis of the assembly module in the global coordinate system is obtained.

The 4 × 4 matrix $\hat{\boldsymbol\xi} $ is defined as

(7) \begin{align}\hat{\boldsymbol\xi} = \left[ {\begin{array}{*{20}{c}}{\hat{\boldsymbol{{s}}}}\quad {{\boldsymbol{{s}}_{\boldsymbol{{0}}}}}\\[4pt] 0\quad 0\end{array}} \right]\end{align}

where the $\hat s$ is an antisymmetric matrix spanned by $\boldsymbol{{s}}$ .

Defining the rotation angle about the axis of virtual coordinates as d, the exponential product of rotation about this axis is

(8) \begin{align}{e^{\hat \xi \theta }} = \left[ {\begin{array}{c@{\quad}c}{{e^{\hat s\theta }}} & {\theta \boldsymbol{{I}} + \left( {1 - \cos \theta } \right)\hat{\boldsymbol{{s}}} + \left( {\theta - \sin \theta } \right){{\hat{\boldsymbol{{s}}}}^2}}\\[4pt] 0 & 1\end{array}} \right]\end{align}

Defining the displacement distance along the axis of virtual coordinates as d, the exponential product of the movement along the axis of translation is

(9) \begin{align}{e^{\hat \xi d}} = \left[ {\begin{array}{*{20}{c}}{{\boldsymbol{{I}}_{3X3}}}\quad {\boldsymbol{{s}}d}\\[4pt] 0\quad 1\end{array}} \right]\end{align}

When the assembly module rotates or moves along the virtual coordinate system, its position is

(10) \begin{align}{g_{ab}}\left( \theta \right) = {e^{\hat \xi \theta }}{g_{ab}}\left( 0 \right)\end{align}

where ${g_{ab}}\left( 0 \right)$ represents the ideal position of the assembly module, ${g_{ab}}\left( \theta \right)$ represents the position of the module with considering the assembly error.

The assembly error chain includes multiple modules, the geometric centre of the last module is

(11) \begin{align}{g_{ab}}\left( \theta \right) = {e^{{{\hat \xi }_1}{\theta _1}}}{e^{{{\hat \xi }_2}{\theta _2}}} \cdots {e^{{{\hat \xi }_n}{\theta _n}}}{g_{ab}}\left( 0 \right)\end{align}

The practical position of single module is obtained by Equation (11).

The above method can only be applied to analyse the rotation and displacement assembly error along the three directions of the virtual coordinate system. However, the actual assembly error is random variable in the actual on-orbit assembly mission. Thus, the concept of assembly error ball is proposed to describe the deformation surface.

Figure 8 shows the proposed assembly error ball, which is centred on the virtual assembly point. A random point on the surface of the ball is selected as the actual assembly point of the module. Moreover, the direction vector between the random point and the virtual assembly point is regarded as the direction of the displacement error of the module. Another point is selected randomly, the direction vector between the point and the virtual point is regarded as the rotation axis of the rotation error. The rotation is selected within a specified range.

Figure 8. Error ball model.

The formulation of the error ball used here is as follows

(12) \begin{align}{\left( {x - {x_{V_{ij}^k}}} \right)^2} + {\left( {y - {y_{_{ij}^k}}} \right)^2} + {\left( {z - {z_{V_{ij}^k}}} \right)^2} = R_0^2\end{align}

where ${R_0}$ is the maximum displacement error in the assembly process, which can be adjusted based on actual assembly level. A point ${\boldsymbol{{q}}^{\textbf{1}}} = \left( {\begin{array}{*{20}{c}}{{x_{r1}}}\quad {{y_{r1}}}\quad {{z_{r1}}}\end{array}} \right)$ on the surface of the error ball is randomly selected. The direction vector is given by

(13) \begin{align}{\boldsymbol{{s}}^{\bf 1}} = \frac{{\left( {\begin{array}{*{20}{c}}{{x_{r1}} - {x_{V_{ij}^k}},}\quad {{y_{r1}} - {y_{V_{ij}^k}},}\quad {{z_{r1}} - {z_{V_{ij}^k}}}\end{array}} \right)}}{{\left| {\left( {\begin{array}{*{20}{c}}{{x_{r1}} - {x_{V_{ij}^k}},}\quad {{y_{r1}} - {y_{V_{ij}^k}},}\quad {{z_{r1}} - {z_{V_{ij}^k}}}\end{array}} \right)} \right|}}\end{align}

The screw is used to describe the rotation error. The principal part is ${\boldsymbol{{s}}^{\bf 1}}$ , and the subpart ${{\boldsymbol{{s}}^{\bf 1}}} \times {{\boldsymbol{{q}}^{\bf 1}}}$ .

The rotation angle θ is a random value. The exponential product of rotation is obtained by Equation (8).

Similar, the point ${{\boldsymbol{{q}}^{\bf 2}}} = \left( {\begin{array}{*{20}{c}}{{x_{r2}}}\quad {{y_{r2}}}\quad {{z_{r2}}}\end{array}} \right)$ is randomly selected. The direction vector is

(14) \begin{align}{{\boldsymbol{{s}}^{\bf 2}}} = \frac{{\left( {\begin{array}{*{20}{c}}{{x_{r2}} - {x_{V_{ij}^k}},}\quad {{y_{r2}} - {y_{V_{ij}^k}},}\quad {{z_{r2}} - {z_{V_{ij}^k}}}\end{array}} \right)}}{{\left| {\left( {\begin{array}{*{20}{c}}{{x_{r2}} - {x_{V_{ij}^k}},}\quad {{y_{r2}} - {y_{V_{ij}^k}},}\quad {{z_{r2}} - {z_{V_{ij}^k}}}\end{array}} \right)} \right|}}\end{align}

The principal part is ${{\boldsymbol{{s}}^{\bf 2}}}$ , and the subsidiary part ${{\boldsymbol{{s}}^{\bf 2}}} \times {{\boldsymbol{{q}}^{\bf 2}}}$ . Based on Equation (9), the exponential product formula of the displacement vector is obtained. Its displacement is equal to ${\rm{\lambda }}\left| {\left( {\begin{array}{*{20}{c}}{{x_{r2}} - {x_{V_{ij}^k}},} \quad {{y_{r2}} - {y_{V_{ij}^k}},}\quad {{z_{r2}} - {z_{V_{ij}^k}}}\end{array}} \right)} \right|$ , where λ is a random value between 0 and 1.

Finally, the ideal assembly model is applied to generate the exponential product of the rotation error and movement error by Equation (11) simultaneously.

(15) \begin{align}{g_{ab}}\left( {\theta ,d} \right) = {e^{{{\hat \xi }_1}{\theta _1}}}{e^{{{\hat \xi }_2}{d_2}}} \cdots {e^{{{\hat \xi }_n}{\theta _n}}}{g_{ab}}\left( 0 \right)\end{align}

When the assembly error appears on the modular antennas, the top and bottom surfaces generate the different deformations. Meanwhile, the thickness of module will affect the deformation. Because the performance of modular antenna is mainly affected by the deformation of the top surface. the deformation of the top surface is shown in this paper. The calculation process of the error model is shown in Fig. 9.

Figure 9. Flowchart of the modular antenna error analysis.

3.3 Shape adjustment of modular antenna

In order to ensure the node of each module places in the ideal paraboloid, active connections are installed between parts of the modules to adjust their positions as shown in Fig. 10.

Figure 10. Actuators installation of modular antenna.

From Fig. 11, it can be seen that two mounting faces and three actuators can be regarded as a parallel mechanism. The mounting faces 1 and 2 are taken as the static platform and the moving platform of the parallel mechanism, respectively.

Figure 11. Definition of the assembly face.

The nodes that consider the assembly errors are regarded as deformation surfaces. In order to minimise the error between the deformed and ideal surface, the least square formula is applied as

(16) \begin{align}F\left( {R,T} \right) = {\rm{min}}\mathop \sum \limits_{i = 1}^N \|{Q_i} - {\left( {\boldsymbol{{R}}{P_i} + \boldsymbol{{T}}} \right)^2}\|\end{align}

where ${P_i}$ is the deformed data point set, ${Q_i}$ is the ideal data point set. $\boldsymbol{{R}}$ is the 3 × 3 rotation matrix, $\boldsymbol{{T}}$ is the 3 × 1 translation vector. The singular value decomposition is applied to solve Equation (16).

Barycentric coordinates of two data sets are

(17) \begin{align}\bar P = \frac{1}{N}\mathop \sum \limits_{i = 1}^N {P_i}{\rm{\;\;\;\;\;\;\;\;\;}}\bar Q = \frac{1}{N}\mathop \sum \limits_{i = 1}^N {Q_i}\end{align}

Through barycentric coordinates of two data sets, all remaining points are transformed as

(18) \begin{align}P_i^{kv} = P_i^k - \bar P{\rm{\;\;\;\;\;}}Q_i^{kv} = Q_i^k - \bar Q\end{align}

where the points $P_i^k$ and $Q_i^k$ belong to the corresponding two data sets.

A matrix is constructed

(19) \begin{align}\boldsymbol{{H}} = \mathop \sum \limits_{i = 1}^n P_i^{kv}\, {}^{\rm{*}}\,{\left( {Q_i^{kv}} \right)^T}\end{align}

Singular value decomposition (SVD) is performed on the matrix ${\rm{H}}$ .

(20) \begin{align}\boldsymbol{{H}} = \boldsymbol{{U}}\boldsymbol\Sigma {\boldsymbol{{V}}^{\bf T}}\end{align}

The rotation matrix R and translation matrix T are obtained.

(21) \begin{align}\boldsymbol{{R}} = \boldsymbol{{V}}{\boldsymbol{{U}}^{\boldsymbol{{T}}}},\, \boldsymbol{{T}} = \bar Q - \boldsymbol{{R}}^{\rm{*}}\bar P\end{align}

The matrix ${T_m}$ is formed with $T$ and ${\rm{R}}$ .

(22) \begin{align}{\boldsymbol{{T}}_{\boldsymbol{{m}}}} = \left[ {\begin{array}{*{20}{c}}\boldsymbol{{R}}\quad \boldsymbol{{T}}\\[4pt] 0\quad 1\end{array}} \right]\end{align}

Three new points $P_{ja}^N$ of the transformed moving platform can be obtained by multiplying the old three points $P_{ja}^O$ with ${T_m}$ . The length between two points of the new moving platform and the static platform is

(23) \begin{align}l_1^N = \left| {P_{ja}^N - P_{ja}^O} \right|\end{align}

The length change of the actuator is

(24) \begin{align}\Delta l = l_1^O - l_1^N\end{align}

The adjustment process is shown in Fig. 12.

Figure 12. Diagram of shape adjustment process.

4.1 Node generation

The parameters of the modular antenna are shown in Table 1. Projecting the assembly gap of the paraboloid onto the plane, the designed gap on the plane is obtained. The graph of ideal parabola and the gap distance projected to the plane are shown in Fig. 13(a) and (b), respectively.

Table 1. Model parameter

Figure 13. Ideal paraboloid and gap distance on the plane.

According to the gap distance on the plane, each point’s coordinates of the hexagon are calculated, as shown in Fig. 14(a), and the details of the gap are shown Fig. 14(b). The Fig. 14(a) shows the projection of the modular antenna on the projection plane. Every module is the same size on the projection plane. Based on the theory in Section 2, it can be known that the gap distances in the projection plane are different. In order to obtain the equal gap distances between two modules in the paraboloid, the gap distances in the projection plane are projected to the paraboloid.

Figure 14. Plane projection of the modules and gap.

The partitioned hexagons are projected to the ideal paraboloid, as shown in Fig. 15(a). When the partitioned hexagons are projected to the ideal paraboloid, the shape of part hexagon will change. Because the module consists of two hexagons and some sway rods, their shape change can be achieved by adjusting the machine size. Based on the coordinates of the upper hexagon, the coordinates of the other components are obtained, as shown in Fig. 15(b).

Figure 15. Antenna structure.

4.2 Assembly error analysis and verification

This paper takes the assembly errors between modules 1–6 and the module 0 as the objects to study the influence of rotation assembly error in three directions on the antenna, as shown in Table 2.

Table 2. Direction and value of assembly errors

When the assembly error of six modules is 0.001rad along the X-axis, Y-axis and Z-axis, respectively, the deformation is shown in Fig. 16.

Figure 16. Surface deformation (the first module layers).

Figure 16 shows that the maximum deformation of the antenna is 0.035m, where an assembly error (0.001rad) appears in the first layer. And the assembly error of Y-axis has the least influence on the antenna reflector. From the Fig. 16, it can be obtained that the assembly errors will spread to outside modules, and the value of the error is tripled. In order to decrease assembly error of the antenna, the assembly accuracy of internal modules should be ensured.

When the assembly error of the second module layers is 0.001rad along the X-axis, Y-axis and Z-axis, respectively, the deformation is shown in Fig. 17. From the Fig. 17, the assembly errors do not influence the inside modules. Compared with the results in Fig. 16, the maximum is smaller. We can know that the inner errors seriously affect the assembly error. Meanwhile, we can observe that the soft connection can effectively block error transmission.

Figure 17. Surface deformation (the second modules layers).

When the assembly error of the third module layers is 0.001rad along the X-axis, Y-axis and Z-axis, respectively, the deformation of modular antenna is shown in Fig. 18. The results show that the assembly errors of the X-axis and Y-axis are bigger than that of the Z-axis. However, the maximum deformation is still smaller than that of the Fig. 17.

Figure 18. Surface deformation (the third modules layers).

The maximum deformation of the surface caused by them is shown in Fig. 19 and Table 3. When the equal rotation errors appear at different assembly layers. It can be seen from the Table 3 that the first modules layers assembly error caused the maximum deformation. Therefore, the modules of the first modules layers should be assembled accurately in practical engineering application.

Figure 19. Deformation of the different module layers.

Table 3. Maximum deformation of different module layers

In order to verify the accuracy of the theorical model, the ideal antenna model is established in the dynamics simulation software, as shown in Fig. 20(a). The rotation pair is established at the ideal virtual installation point.

Figure 20. 3D model of modular antenna.

The rotation error with the same value is introduced into the software simulation model and the theoretical model, respectively. The random two points are measured. The 3D coordinates of the simulation and theoretical model are shown in Table 4. The results show that the theoretical model can describe antenna assembly errors accurately.

Table 4. The 3D coordinate of the measured point

4.3 Analysis of adjustment algorithm

Firstly, the assembly position errors are 0.001m and the assembly rotation error is 0.001rad, respectively. According to the assembly error ball model, the deformed surface with the displacement and rotation assembly error is generated. Because the inner assembly error has the greatest influence on the deformation of the modular antenna, the actuators are installed on the inner module, as shown in Fig. 21(a).

Figure 21. The deformation of the modular antenna.

Based on the error ball concept, the original deformation is generated, as shown in Fig. 21(b). The RMS error of the modular antenna is 0.97mm. In order to decrease the RMS error, the adjustment method is applied to decrease this deformation. The adjusted deformation surface is shown in Fig. 21(c), its RMS error is 0.64mm. Comparing the Fig. 21(b) with Fig. 21(c), the area of larger deformation is significantly decreased.