Surface Adjustment Method Based on Fuzzy Theory for Cable Net Structures under Multi-Uncertainties

Because of manufacturing errors, measuring errors, and unpredictable service environment, the cable net structure to be further adjusted is in an uncertainty state. In this paper, the uncertain factors including elastic deformation, thermal deformation, and measurement uncertainties are considered as fuzzy variables, which are equivalent into fuzzy tensions to simplify calculation. A fuzzy force density method is developed for accuracy analysis of the cable net structures under multi-uncertainties, and an optimization model is developed for surface adjustment. The above method is applied to numerical model adjustment of circular truss cable net structure. The results show that the adjusted surface accuracy is signi ﬁ cantly enhanced and its fuzziness is concentrated compared with the initial surface accuracy, which verify the validity of the proposed method.


Introduction
Cable net structures have been widely applied to space deployable reflector antennas such as the AstroMesh antenna, the TerreStar antenna, the antenna of JAXA Engineering Test Satellite, and the SkyTerra antenna [1][2][3][4][5]. High-accuracy surface is a prerequisite for ensuring the electrical performance of the antennas. However, limited by manufacturing and assembling technology, artificial surface adjustment is an essential and tiresome step to improve the surface accuracy, which has been revealed very sensitive to manufacturing errors and environmental changes [6][7][8].
There have been many researchers that have done a lot to improve the efficiency of adjustment strategies. Hiroaki and Natori [9] proposed a shape control method based on the concept of self-equilibrated stresses to improve the control efficient. Du et al. [10] presented a shape adjustment procedure based on optimization and then converted the procedure into a sequential quadratic programming problem to make it more easily. Niu et al. [11] established an optimal adjustment model that an influence coefficient matrix was treated as one target. The above adjustment methods regarding the current configuration of the cable net structure can be accurately obtained. Actually, because of the limitation of measurement accuracy and changeable environment, there must be some uncertain factors such as nodal positions, cable pretensions, material parameters, and environmental temperature [12,13]. Under the influence of these uncertain factors, how does the surface accuracy of the cable net structure change and how to ensure the surface accuracy in a reliable range becomes particularly important.
Nowadays, the methods for dealing with uncertainty can be divided into three categories including the probability theory [14][15][16], the fuzzy algorithm [17], and the interval method [18]. The probability theory is a helpful tool in modeling situations where the primary source of uncertainty is randomness [19,20]. But sometimes, we argue that uncertainty takes other forms; instead of asking whether something is true, we ask how much of it is true and how much a certain property is exhibited in a particular instance. In our previous work [21], an interval force density was proposed and an optimization adjustment model was established for the surface adjustment of cable net structures. However, it is found that only mean and marginal cases can be obtained by the interval method and probability distribution functions need to be further studied. Probability method needs to study the probability distribution of uncertain variables based on a large number of statistical data, while fuzzy algorithm can study uncertainty model by membership function which can be estimated by experience. In order to improve the efficiency of cable net structure adjustment considering uncertainty in engineering, it is necessary to study how to apply fuzzy algorithm to adjustment. Thus, this paper proposes a fuzzy force density method to deal with the surface adjustment problem of the cable net structures under multi-uncertainties. The paper has a guiding significance for the adjustment of cable net structure considering uncertainty in the case of few samples in engineering.

Brief Summary of the Fuzzy Set Theory
Define a fuzzy subset of U as function A: U ⟶ ½0, 1, that is, a characteristic function from U into interval ½0, 1. The value AðuÞ is called the membership of point u in the fuzzy set U or the degree to which point u belongs to set A.
where λ A is called the λ-cut of the fuzzy set A, λ ∈ ½0, 1.
If the lower bound u l ðλÞ and upper bound u u ðλÞ are given, the fuzzy number A can be obtained by summing all λ-cut sets as Definingx as a fuzzy variable and its fuzzy number as AðλÞ,x can be described bỹ where x c ðλÞ and x r ðλÞ are the midvalue and the amplitude of x, respectively, and where Then, the fuzzy variablex can be described by the interval variables λ and δ, where λ ∈ ½0, 1 and δ ∈ ½−1, 1. When the cut level λ is given, the fuzzy variablex becomes an interval variable. Therefore, the operation of the fuzzy variablẽ xðλ, δÞ can be discretized into the operation of interval variables.

Mathematical Models for Fuzzy Cable Net Structures
Cable net structures inevitably suffer from multiple sources of uncertainty in the process of manufacture, assembly, and on-orbit service. Limited by our ability to get informa-tion, parameters of the structures like nodal positions, cable pretensions, material parameters, and environmental temperatures must be uncertain. Thus, we use some fuzzy variables to describe these uncertainties. In order to reveal the influence of these uncertainties on the surface accuracy of the cable net structures, the mathematic models are firstly established based on the force density method and the fuzzy theory in this section.

Equivalent Fuzzy Cable Tensions for a Cable Net
Structure under Multi-Uncertainties. For the cable net structure whose geometric forms are given, the sources of uncertainty can be divided into three categories: elastic deformation uncertainty, thermal deformation uncertainty, and measurement uncertainty [21], among which the uncertainties which would cause cable tension changes can be equivalent into a total fuzzy tension to simplify calculation. According to Hooke's Law, elastic property of a cable can be expressed by the following equation.
where ε is the cable strain, F is the cable tension, E is the elastic modulus, A is the cross-sectional area, L is the stretched length, and L 0 is the unstretched length.

Uncertainty of Elastic Deformation.
Because of multiuncertainties, the axial tension, the elastic modulus, and the cross-sectional area are fuzzy variables. Thus, the above equation can be rewritten as whereε is the fuzzy strain;Ẽ is the fuzzy elastic modulus;Ã is the fuzzy cross-sectional area;L 0 is the fuzzy unstretched length; ΔF E ðλ, δÞ is the fuzziness of the cable tension caused by uncertainty of elastic deformation; see below.
According to the fuzzy set theory, the above equation can be rewritten as where l c 0 and l r 0 are the midvalue and the amplitude of the unstretched length; E c and E r are the midvalue and the amplitude of the elastic modulus; A c and A r are the midvalue and the amplitude of the cross-sectional area.
3.1.2. Uncertainty of Thermal Deformation. According to the thermoelasticity theory, the thermal strain ε T of a cable is 2 International Journal of Aerospace Engineering directly proportional to the temperature difference ΔT, specifically as follows.
where α is the coefficient of thermal expansion. The tension uncertainty caused by thermal deformation can be equivalent to Considering the uncertainties of the temperature difference, the coefficient of thermal expansion, the elastic modulus, and the cross-sectional area, the uncertainty of the tension caused by the thermal deformation can be obtained as where ΔF ΔT is the fuzziness of the cable tension caused by the uncertainty of the temperature difference;α is the fuzzy coefficient of thermal expansion; ΔT is the fuzzy temperature difference. According to the fuzzy set theory, the above equation can be rewritten as where α c and α r are the midvalue and the amplitude of the coefficient of thermal expansion, respectively; ΔT c and ΔT r are the midvalue and the amplitude of the temperature difference.

Uncertainty of Tension
Measurement. There will be some uncertainties when applying and measuring cable tensions. Defining the fuzzy tension caused by manufacture and measure as ΔF M , the following equation can be obtained.
where ΔF c M and ΔF r M are the midvalue and the amplitude of the fuzzy tension caused by manufacture and measure. (8), (12), and (13), the equivalent fuzzy tension can be obtained asF

Equivalent Fuzzy Tension. Combining Equations
where F~t otal is the equivalent fuzzy tension.

Surface Accuracy Analysis for a Fuzzy Cable Net
Structure. For an arbitrary node j connected by some cables, the force balance equations can be obtained as whereF ij denotes the fuzzy tension of the cable connected to nodes i and j; ðx i ,ỹ i ,z i Þ is the fuzzy coordinates of node i; S j  3 International Journal of Aerospace Engineering is the set of all cables connected to node j;L ij is the fuzzy length of the cable connected to nodes i and j.
According to Section 3.1, the fuzzy tensionF ij can be obtained by Equation (14).
In order to consider the effects of geometric and tension uncertainties simultaneously, a fuzzy force density is introduced as follows.q Referring to the force density method [8], the static equilibrium equations can be obtained as follows.
whereQ is a diagonal matrix containing fuzzy force densities; C s is the incidence matrix of the cable net structure;xðλ, δÞ is a column vector of x-coordinates;ỹðλ, δÞ is a column vector of y -coordinates;zðλ, δÞ is a column vector of z-coordinates. If some nodal coordinates are given, e.g., these nodes are attached to a foundation, C s can be partitioned as where the restrained nodes have been put at the end of the numbering sequence. Equation (18) can be rewritten as wherex u ,ỹ u , andz u are the column vectors of unknown x-, y-, and z-coordinates;x f ,ỹ f , andz f are the column vectors of the given x-, y-, and z-coordinates, respectively. Taking the ideal coordinates fx 0 , y 0 , z 0 g as a reference, the fuzzy root-mean-square error (RMS), which can be used to evaluate the structure accuracy, can be obtained as follows.
where N u is the number of the nodes with unknown coordinates.

Optimization Model for Adjustment of a Cable Net Structure under Multi-Uncertainties
The force density of an adjustable cable connected to nodes i and j can be modified as where a ij is the adjustment amount of the cable. Substituting Equation (22) into Equation (20), the cable net structure after adjustment can be obtained and the surface accuracy can be then calculated by Equation (21). Base on this, we establish the following optimization model, which can be solved by the advance and retreat algorithm [21], for the adjustment of the cable net structure under multi-uncertainties:   denotes an inequality constraint that the a ij cannot be smaller than the minimum adjustment amount a 0 which is dependent on the engineering practice; and g ij 2 denotes an inequality constraint that the force density of the cables must be positive due to the fact that the cable has no compressive rigidity.
The adjustment progress of the cable net structure under multi-uncertainties is as follows.
Step 1. Uniformly discrete the horizontal cut set λ into λ i (i = 1, 2, 3, ⋯). When k = 0 and horizontal cut set is equal to λ i , solve initial node coordinates and initial cable tensions by membership function of fuzzy variables.
Step 5. Record the optimal objective function value and design variable value under level cut set λ i . When λ i = 1, turn to Step 6; else, let i = i + 1; turn to Step 1.
Step 6. The distribution of membership function of the adjusted RMS is obtained by curve fitting.

Numerical Example
Take a cable net structure which has been applied to the hoop truss reflectors as an example to illustrate the proposed method. As shown in Figure 1, the cable net structure is composed of a front cable net, a back cable net, and tension ties, among which the front cable net is usually used to support the wire mesh to reflect electromagnetic wave and tension ties are adjustable cables. The geometrical parameters and the material parameters of the cable net structure are shown in Table 1, and the uncertain parameters are shown in Table 2. In this example, the membership functions of the fuzzy variables obey a normal distribution of which the function can be written as where a and σ are the mean and the standard deviation of the fuzzy variable, respectively.

Equivalent Fuzzy Cable Tensions.
When the λ-level cut value is specified as 0.01, 0.02 ... 0.99, and 1.0, respectively, for the fuzzy variables, the interval value of the equivalent fuzzy cable tensionF ij ðλ, δÞ corresponding to the λ-level cut set can be obtained by Equation (16). Then, the discrete intervals can be connected and fitted to obtain the membership distribution curve of the equivalent fuzzy cable tensioñ F ij . Taking cable 258 which is connected to nodes 49 and 5 as shown Figure 1 as an example, the cores of the cable length and the cable tension are 0.6132 m and 9.4495 N, respectively. The distributions of the membership functions for the cable tensions caused by elastic deformation uncertainty (k = 1), thermal deformation uncertainty (k = 2), and tension measurement uncertainty (k = 3) can be obtained by Equations (13), (18), and (19) and drawn in Figure 2. It can be seen from the figure that the fuzziness of the three equivalent tensionsF k ij ∈ ½−0:01, 0:01N (i = 49, j = 5, k = 1, 2, 3) when the cut level λ > = 0:6, that is to say the membership degrees of the equivalent fuzzy tensions are greater than 0.6 when jF k ij j < = 0:01 (i = 49, j = 5, k = 1, 2, 3). By Equation (23), the distribution of the membership function for the total equivalent fuzzy tension can be obtained and shown as Figure 3, from which it can be seen thatF ij ∈  International Journal of Aerospace Engineering ½9:44, 9:46 when the cut level is greater than 0.6 and the curve obeys the normal distribution with 9.4495 N mean and 0.0131 N standard deviation.

Surface Accuracy Analysis.
Due to the measurement error, the nodal positions are also of uncertainty. It is assumed that the membership function for the nodal coordinates obeys the normal distribution with zero mean and 1 mm standard deviation. According to Equation (23), the membership function for RMS can be obtained as shown in Figure 4. The curve is an addition of multiple Gauss curves but not a single normal distribution curve. It can be seen from the figure that RMS < 60:3079 mm when the degree of the membership is greater than 0.6 and RMS < 26:2857 mm when the degree of the membership is greater than 0.8. The fuzziness of the surface accuracy is dispersed, and further adjustment should be carried out to increase the surface accuracy.

Adjusting the Cable Net Structure.
To calculate the adjustment amount of the tension ties, the advance and retreat algorithm is used to solve the optimization model introduced in Section 4, and the RMS value changes in the interactive process are drawn in Figure 5 with the level cut set specified as 0.01, 0.6, 0.8, and 1.0, respectively, for example. From the iterative curves, it can be seen that the convergence speeds are fast in the first 10 steps and the RMS are rapidly decreased, but after that the convergence speed slows down; repetitive adjustment work is needed. What is more, it can be seen from Figure 5, the closer the λ-level cut value is to 1, the smaller the deviation of the RMS is. When λ = 1:00 with a small initial RMS, the coupling effect of front nodes is significant. It may occur that adjusting the tension tie can make the front nodes directly connected to it closer to the ideal position, but the nodes around tensioning tie are affected to deviate from the ideal position, thus making the adjusted RMS larger. This phenomenon makes the adjustment efficiency of advance and retreat algorithm reduced when RMS is small.
The results after 20 iterations are chosen to be the adjustment amount. After adjusting, the RMSs at different cut levels are drawn in Figure 6. It can be seen that RMS < 3:8975 mm when the degree of the membership is greater than 0.6, and RMS < 2:9868 mm when the degree of the membership is greater than 0.8. Compared with the initial surface accuracy, the fuzziness of the surface accuracy is concentrated and the surface accuracy is increased.

Conclusion
We have developed a surface adjustment method for the cable net structures under multi-uncertainties including elastic deformation uncertainty, thermal deformation uncertainty, and measurement uncertainty.
The main contributions of this paper are presented as follows. (1) The uncertain variables are considered as the fuzzy values, membership functions, and λ-level cut sets are introduced to describe the fuzzy values. (2) The elastic deformation uncertainty, thermal deformation uncertainty, and tension measurement uncertainty are equivalent into a total equivalent fuzzy tension to simplify calculation. (3) The force density method is applied to modeling static equilibrium equations for cable net structures with fuzzy parameters. (4) An optimization model for the adjustment of cable net structures under multi-uncertainties is established.
According to the numerical example, the following conclusions can be summarized. (1) Using the advance and retreat algorithm to solve the optimization model can achieve fast convergence. (2) The adjustment efficiency of advance and retreat algorithm will be reduced when the initial RMS is small. (3) This method can be used to obtain the uncertainty distribution of the surface accuracy of cable net structure without sample only by engineering experience. (4) The fuzziness of the surface accuracy can be concentrated effectively by the proposed method.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.