Design and accuracy analysis of new cable network topology configuration of umbrella-shaped deployable antenna

To reduce the back-pillow effect of the cable-net antenna and improve the accuracy and error uniformity of the umbrella antenna surface, a new cable-net topology of the umbrella antenna is proposed. Firstly, the cable network segment is divided to obtain a new topological configuration of the umbrella antenna. Secondly, starting from the cable net accuracy calculation method, the Gauss-Legendre integral method is used to calculate the accuracy of the umbrella sky clue network surface method. Finally, the new cab-net configuration is compared and analyzed by example, and it is of great significance to improve the accuracy of the antenna surface and improve the uniformity of error distribution.


Introduction
Due to the limitation of launch vehicle loading space, the deployable antenna [1] is the most effective way to achieve a large aperture, and mesh antenna has become the main development trend of satellite antenna due to the high stowage ratio.With the development of satellite payload technology, the aperture requirements of antennas are getting larger and larger, and the accuracy requirements of antennas are getting higher and higher, among which umbrella antenna is the main structure form of high-precision mesh antenna [2] due to its advantages of high accuracy, reliable deployment, and high stowage ratio.Figure 1 shows Chang'e-4 relay satellite structure [3] .The emerging load technologies represented by data and relay satellites and high-throughput communication satellites have put forward higher requirements on the profile accuracy and error distribution uniformity of umbrella antennas.For high-precision umbrella antennas, it is necessary to study the effect of the anti-pillow effect [4] on surface accuracy.At present, to reduce the surface error caused by the anti-pillow effect, there are usually two types of methods.One is to change the antenna structure form of the cable network to increase the tension array; the other is to use the reflective surface without material to increase the reflective surface The cable network configuration, cable-network tension, cable-network equilibrium state, and other factors are the main factors to determine the final surface accuracy of the umbrella antenna.Therefore, the research on the high-precision mesh cable network configuration is important to realize the highprecision surface of the antenna and the uniformity of error distribution, which plays an important supporting role to solve the key technology in major projects.

Umbrella Antenna segmentation
The cable network system [5] of the antenna is the support structure of the flexible metal reflector network, and the configuration in the unfolded state will affect the structural stability and electromagnetic performance of the cable network antenna.Because the metal reflector network is directly attached to the reflector cable network, its equilibrium configuration will directly determine the design accuracy of the reflector surface, so the design of the reflector cable network configuration is an important element of the cable network system design [6] .
Agrawal et al. [7] proposed a pre-design method for the length of the cable segments similar to the triangular cable network method and derived equations for the relationship between the length and accuracy of the cable segments of the quadrilateral cable network as well as the hexagonal cable network.The expression provides a basis for the calculation of the parameters of the initial geometry finding for different face piece configurations of cable network antennas.
Referring to a design method of quadrilateral cable segment parameters based on double primary Bessel surfaces proposed by Jin et al. [8] , the relationship equation between the length of quadrilateral mesh cable segments and the accuracy of the profile is obtained.

New cable network topology design for umbrella antenna
According to the theory of the anti-pillow effect, when the circumferential elastic modulus of the metal mesh is close to zero, it can be regarded as a tension line instead of the metal mesh surface, as shown in Figure 2, which plays a certain role in suppressing the anti-pillow effect [9] .This new configuration is based on the circumferential rope network configuration with the addition of several radial ropes perpendicular to the circumferential rope network, which is equivalent to improving the anti-pillow effect of the strip circumferential rope network from the geometric configuration level of the rope network.The node position is shown in Figure 3(a), and the projection of the front grid surface in the vertical aperture direction is shown in Figure 3(b).Since the cable network of the umbrella antenna is a rotating paraboloid, only a single sector is usually chosen for the study.The node position of the single-sector cable network is shown in Figure 4.

Accuracy calculation of new cable network configuration of umbrella antenna
The cable network unit fitting error is a large surface error source of the mesh antenna, that is, the principle error or design error.The cable section between the nodes of the cable network is straight in the case of no gravity, the reflective surface of the metal mesh is divided into a series of cable network units by the cable network system, and the deviation between the cable network units and the ideal reflective surface is the principle error.
In the calculation of the principle error, the definition domain needs to be discretized.According to the distribution of nodes on the reflecting surface, the reflecting surface is discretized into a finite triangular region, and the Delaunay triangular dissection algorithm is chosen to discretize the reflecting surface.
For a single triangular region of the reflective surface, the root-mean-square error values are: Therefore, the projected area is used to weigh the accuracy of the profile, and the results obtained are more in line with the actual working conditions.The calculation equation of the overall rope network principle error based on the weighting of the grid projection area is: where RMSi


is the RMS of the ith grid, and i S is the projected area of the ith grid on the projection plane.
The key to this method is to find the RMS (root mean square) error of each face piece.Since the Gaussian integration method is the most accurate interpolation type of numerical integration, this paper uses the Gauss-Legendre integral method to interpolate and fit the plane to approximate the RMS error of the antenna main grid plane.
The Gauss-Legendre integral method on the interval is: The integral function in Equation ( 3) is transformed by coordinates to convert the triangular integral region to the Gaussian Lejeune integral region, and the following equation [10] is derived: , , x y x y where J is the determinant of the Jacobi matrix.
The shape function of the interval transformation is: The mapping from the triangle transformation to the Gauss-Legendre integral region is: , Numerical integration through Gauss-Legendre integral method gives Equation ( 8): where M is the number of Gaussian interpolation points in  direction, same as in  direction;

Error analysis of new cable network configuration of umbrella antenna
The following analysis of the calculation example is combined with the above calculation method of type surface error.
In this paper, the nodal coordinates of the main reflective surface are determined using the method of cutting the rotating paraboloid in the vertical direction for modeling the main cable network.The design parameters of the reflective surface are 0 r m s 1470 , 4200 , 400 , 0.3 , 18 f mmD mmr mm mmn For the traditional cable network structure of the umbrella antenna, the geometric configuration of the integration interval in the projection plane is shown in Figure 5(a).And the relationship between the RMS error of each facet and the facet serial number is obtained as shown in Figure 5(b).The number of summarized points is 211, and the accuracy of the cable mesh shape surface is 0.2162, which meets the design index.For the new cable network structure of the umbrella antenna, the geometric configuration of the integration interval in the projection plane is shown in Figure 6(a).And the plot of the RMS error of each facet against the facet serial number is obtained as shown in Figure 6(b).The number of summarized points is 221, and the accuracy of the cable mesh shape surface is 0.2339, which meets the design index.From analyzing the data in Figure 6, we obtained that the error distribution of the new cable network configuration is at 0.11~0.25 mm, the error in the conventional cable network configuration is distributed over 0.16~0.28mm, and the new configuration has a smaller range where the error lies.Further analysis shows from the figure that the new configuration grid cell errors are mainly concentrated on the 0.23~0.25 mm, quadrilateral grid cells located in the central region on the sector structure, while the rest of the grid cell errors are less than 0.23 and account for a smaller percentage of the triangular grid cells located immediately adjacent to the antenna rib region in the sector structure.
The main reason is that the principle error is the same because the size of the quadrilateral grid cells in the new configuration is the same, and the error is the same in most areas of the sector because the quadrilateral grid occupies a larger proportion in the whole sector.The areas immediately adjacent to the ribs are divided according to the new configuration to form triangular mesh cells, and the number and area are much smaller than the quadrilateral mesh, so their errors are smaller and account for a smaller proportion of the whole sector.
The error distribution of the grid cells in the new configuration is analyzed.The more concentrated errors in the figure are located in the 0.235~0.255mm, and the percentage of the total sector area is 85.2%.On the other hand, the error of 76.3% of the area of the sector of the traditional cable mesh structure is located in the range of 0.194~0.24mm.The comparative analysis shows that the new topology has an obvious advantage in the uniformity of the mesh error.

Conclusion
Comparing and analyzing the two cable network configurations, the new cable network configuration proposed in this paper shows certain advantages in profile accuracy and error uniformity, especially in the profile error uniformity, which is of great significance to achieving high precision profile and error distribution uniformity of the antenna.

Figure 2 .Figure 3 .
Figure 2. Suppression of circumferential zero moduli of elasticity.Meanwhile, to improve the error uniformity of the umbrella antenna network, this paper proposes a new umbrella antenna network configuration as shown in Figure3below, which is called the radial reinforced cable configuration to improve the error uniformity of the umbrella antenna as well as the anti-pillow effect.

Figure 4 .
Figure 4.A new topological configuration of single-sector cable network.

Figure 5 .
Figure 5. Traditional cable network structure of umbrella antenna.

Figure 6 .
Figure 6.New cable network structure of umbrella antenna.
bending stiffness to reduce the anti-pillow effect.This paper adopts the first method to change the structure form of the cable network, design a new umbrella antenna cable network configuration, and analyze its accuracy.