The Influence of Machining Deformation on the Pointing Accuracy of Pod-Type Space Self-Deployable Structures

Abstract

As key driving and supporting components of spacecraft, pod-type space self-deployable structures have terminal pointing accuracy that directly affects overall spacecraft performance. To clarify the influence of the structure’s machining deformation on its pointing accuracy, this study focuses on two key processes, namely laser welding and hot forming. Based on the bionic symmetric structural characteristics of pod-type structures, a laser welding finite element model with a surface Gaussian heat source and a hot forming constitutive model coupled with creep aging were established. An orthogonal experimental design was adopted: for laser welding, three parameters, namely laser power, spot diameter, and welding speed, each with three levels, were selected, and an L₉(3³) orthogonal table was constructed to conduct nine groups of simulations; for hot forming, two parameters, namely processing temperature and holding time, each with three levels, were chosen, and nine groups of simulations were designed based on the first two columns of the L₉(3⁴) orthogonal table. The combined method of residual analysis and analysis of variance was used to quantitatively identify the influence of each process parameter on pointing accuracy. The results show that in laser welding, welding speed has the most significant impact on deformation, followed by laser power, and spot diameter has the least; in hot forming, processing temperature and holding time have similar effects on deformation. Physical machining verification was performed, and the actually measured deformations are 0.164 mm and 0.034 mm, which are close to the simulation results of 0.176 mm and 0.047 mm, meeting the index requirement that the terminal pointing deformation of a single pod structure is less than 0.2 mm. The results can provide a theoretical basis and engineering reference for the actual machining of such structures.

1. Introduction

With the gradual increase in requirements for deep space exploration, space communication, and other missions, the spatial configuration of spacecraft is becoming larger and more complex [1]. The increasing complexity of exploration missions has directly promoted the upgrading of technologies related to spacecraft structures, such as large antennas and satellite power supply [2]. For example, the significant extension of detection distance is likely to lead to a sharp increase in communication signal attenuation, requiring larger-diameter antennas to compensate for the loss. The expansion of spacecraft size will lead to an increase in energy demand, requiring larger-area satellite deployment wings to obtain solar energy.

As key payload support structures for such spacecraft structures, large antenna trusses and satellite deployment drive rods [3] are not only increasing in size with the expansion of spacecraft structures, but their pointing and positioning accuracy requirements are also gradually increasing. For example, the antenna of NASA’s NISAR satellite [4] is nearly 12 m long, adopting a hinged structure, realizing multi-stage deployment through shape memory actuators, and the final positioning accuracy reaches ±2 mm; the Tianma Radio Telescope [5] is nearly 65 m, achieving a pointing accuracy better than 3″ through high-precision design and manufacturing of welding orbits, multi-sensor pointing calibration and other technologies; Zhao et al. [6] addressed difficulties such as high assembly and integration accuracy and many connection links existing in multiple drive structures of the antenna, and through the spatial multi-axis precision assembly and adjustment method, the spatial axial position accuracy of the drive mechanism was better than 0.01°, and finally the pointing accuracy after three-axis assembly and commissioning of the antenna mechanism was better than 0.042°. High pointing and positioning accuracy can effectively ensure the normal operation of such spacecraft structures.

To adapt to the launch environment, the driving and supporting structures of large-size key payloads are often designed based on the folding-deployment principle [7]. They are tightly stored in a limited space before launch, and deployed by driving after the spacecraft enters orbit to realize functions, which can effectively improve the space utilization rate in the launch cabin and save launch costs. Space deployment mechanisms can be divided into spring-driven, motor-driven, pneumatic-driven, and self-driven according to the driving method [8]. Among them, the self-driven type has received extensive attention because it does not require external power and can deploy autonomously. Pod-type self-driven mechanisms have been widely studied [9]. Through a special bionic pod structure design, self-driven power is generated by the elastic deformation formed when the material is bent, and they are generally made of high-performance titanium alloy metals. The mechanism has high transverse stiffness and excellent stored elastic performance, which is very conducive to realizing the deployment and support functions of spacecraft spatial structures.

The pod structure is a typical thin-walled, special-shaped symmetrical structure. For the processing technology of such structures, the hot forming process is often used. The material is heated to a certain temperature and die-cast into shape using a mold. For example, Wang et al. [10] studied the influence of mold design and different process parameter combinations on the generation of forming defects in TC4 titanium alloy thin-walled double-sided curved flanging structure, and finally obtained ideal defect-free parts; Tao et al. [11] studied the precision control technology of titanium alloy thin-walled hyperbolic structures during the hot forming process of skins, and finally obtained the relationship between the instability characteristics and defect characteristics of thin-walled hyperbolic skins during the forming process; Guo et al. [12] completed the analysis of hot deep drawing forming of titanium alloy box parts using finite element simulation. Pod-type self-driven structures are often formed by superimposing and combining multiple pod structures, similar to a sandwich, and are often manufactured using welding processes. For example, Shao et al. [13] explored the influence of laser welding deformation on curved sandwich structures, and the final structural deformation was within 0.6 mm; Zhao et al. [14] studied the influence of laser welding deformation on frame-truss rocket tank wall panel structures, established a welding finite element model and optimized the welding sequence and direction; Feng et al. [15] studied the influence of welding processes on the weld formation and bearing capacity of thin-walled annular titanium alloys. However, the above studies did not directly quantify the sensitivity of different processing parameters to the processing deformation of thin-walled special-shaped and sandwich structures, nor did they analyze the influence degree of processing processes on the pointing accuracy of pod-type mechanisms. Therefore, it is necessary to analyze the processing flow and parameters of such structures, determine the influence degree of different processing parameters on the deployment pointing accuracy, clarify the priority of processing sensitive parameters, and fill the gap in processing parameter guidance for practical manufacturing applications.

Based on this, the core research objectives of this paper are clearly defined as follows: firstly, to establish an accurate simulation model of machining processes for the bionic symmetric characteristics of pod-type structures, providing a reliable platform for process parameter analysis; secondly, to quantitatively identify the sensitivity of key machining process parameters to pointing accuracy via orthogonal experiments and statistical analysis, clarifying the priority of parameter regulation; finally, to optimize the process parameter combination that meets terminal pointing accuracy requirements and verify its reliability through physical machining, offering theoretical support and engineering reference for the high-precision manufacturing of such structures.

Therefore, Section 2 of this paper introduces the structural characteristics and corresponding processing methods of pod-type structures, determines the laser welding and hot forming simulation models, and designs the orthogonal experiment scheme. Section 3 analyzes the simulation results and obtains the sensitivity of different processing parameters to the deployment pointing accuracy using a combination of residual and variance analysis methods. Section 4 summarizes the conclusions of the entire paper.

2. Simulation Methods and Experiment Design

2.1. Structure Introduction and Processing Methods

The pod-type space self-deployable structure used in this paper, which is formed by arraying single pod structures, is shown in Figure 1. The pod structure can be regarded as a combination of multiple micro-grooves, curved surfaces, and cover plates. The micro-grooves are composed of multiple tiny filaments, which can increase the transverse stiffness of the structure; the curved surface can deform under force to generate elastic force; and the cover plate can disperse the force on the curved surface and ensure the overall stiffness of the structure. At the same time, to meet the lightweight requirements of the spacecraft, the entire structure is made of TC4 titanium alloy metal material commonly used in aerospace [16]. The material parameters are based on the standard GB/T 2965-2023 [17], as shown in

Table 1. Parameters of the TC4 material used.

Elastic Modulus (GPa)	Density (g/cm3)	Poisson’s Ratio	Thermal Conductivity (W/(m·K))	Specific Heat Capacity (J/(kg·K))	Linear Expansion Coefficient (/°C)	Yield Strength (MPa)
88	4.49	0.34	7.8	610	9.5 × 10−6	860

. The length and width of the plates used in the structure are both 220 mm, the thickness of a single plate is 0.2 mm, and the radius R of a single circle on the curved surface is 22.8 mm.

The pod-type space self-deployable structure is interlocked layer by layer, and can be regarded as being formed by superimposing and combining different repetitive structural features, with mutually symmetrical structures. Therefore, analyzing the influence of the processing technology of a single pod structure can be extended to the overall pod-type space self-deployable structure. The manufacturing process of a single pod structure is divided into a combination of cutting process, welding process, and hot forming process, as shown in Figure 1. First, the cutting process cuts the TC4 titanium alloy plate into a reasonable shape and cuts out the multi-micro-groove filament structural features; then, the welding process is used to weld two cut plates together; next, the hot forming process is used to form the curved surface structure; finally, the welding process is used to weld a single pod structure. Repeating this process, multiple pod structures are superimposed to form the required pod-type space self-deployable structure. As pointed out in Yang et al.’s research [8], the processing quality of multi-micro-groove filaments has a great influence on the transverse stiffness of the structure, but the influence on the pointing accuracy is negligible, and the subsequent hot forming process also needs to perform die stamping and pressing on its structure. Therefore, this paper does not consider the cutting process, but mainly analyzes the influence of the process parameters of the welding process and hot forming process on the deployment pointing and positioning accuracy.

2.2. Simulation Model Construction

2.2.1. Welding Process

Laser welding technology is adopted for the welding process because of its high welding efficiency and good welding quality, which are more suitable for the welding of titanium alloy materials [18]. The overall thickness of the plate is 0.2 mm. Due to the thinness of the welded plate, the surface Gaussian heat source model is selected as the welding heat source model. The simulation analysis software Ansys2024 is used to calculate the laser energy distribution. The formula of the model is as follows:

$$Q(X,Y)=(2{\displaystyle \frac{P}{\pi R_0^2}})\exp (-{\displaystyle \frac{2X^2}{R_0^2}})\exp (-{\displaystyle \frac{2Y^2}{R_0^2}})$$

(1)

where $Q(X,Y)$ is the instantaneous heat flux density from the center of the laser spot, P is the laser output power, $X,Y$ is the coordinate of any point in the plane, and $R_0$ is the characteristic radius of the laser spot. The coefficient 2 is the energy normalization coefficient of the surface Gaussian heat source, ensuring that the total energy of the laser heat source is consistent with the actual output power P.

In the software, two titanium alloy thin plates are stacked together, and a laser beam irradiates the parts to be welded from above, as shown in Figure 2a. Processing parameters, including the welding heat source, welding trajectory, and welding speed, are set, and four welds are planned. Welds 1 and 2 are located in the middle of the structure, while welds 3 and 4 are symmetrically distributed on the left and right. This layout ensures that the two workpieces are tightly welded together, and the left and right ends can undergo subsequent hot forming processing. To ensure minimal deformation, a welding strategy of welding the middle welds first and then the end welds is adopted, as shown in Figure 2b. Meanwhile, after welding one weld, the next weld is performed only after the completed weld cools completely. Finally, the laser welding finite element model of titanium alloy thin plates is obtained.

2.2.2. Hot Forming Process

The curved surface is a hyperboloid structure, which is manufactured by the hot forming process. Usually, there is a certain constitutive relationship between the flow stress of metal during hot deformation and the deformation temperature, deformation rate, and deformation degree, which can be described by the Arrhenius equation. The equation is expressed as follows under low stress conditions:

$$\ln \sigma =-{\displaystyle \frac{1}{n}}\ln A+{\displaystyle \frac{1}{n}}\ln \epsilon +{\displaystyle \frac{1}{n}}{\displaystyle \frac{Q}{RT}}$$

(2)

where $\sigma$ is the flow stress corresponding to a specified strain time; n is the stress index; A is the material constant; $\epsilon$ is the strain rate; Q is the activation energy of thermal deformation; R is the gas constant; and T is the absolute temperature.

Combined with the equation, the high-temperature forming constitutive model of the structure can be established, and the simulation software is used for analysis. The thickness of the formed TC4 curved surface thin plate is 0.2 mm, which is much smaller than the dimensions in the length and width directions. To reflect the deformation and springback of the plate along the length direction and show the stress and strain distribution in the thickness direction of the plate, a two-dimensional simulation is adopted in this study. Moreover, due to the symmetrical structure of the part, only 1/4 of the structure is simulated. The cross-section of the plate along the length direction is taken as the simulation object, and only the sheet metal is set as the deformable body. The other parts are assumed to be rigid bodies because their strength is very high relative to the blank. At the same time, during the hot forming process of the titanium alloy plate, the entire blank and mold are carried out in a heating furnace, so the temperature can be considered very uniform and constant, and the established analysis model is shown in Figure 3. First, the explicit analysis method is used to calculate the shape, stress, and strain distribution of the sheet metal after forming, and then the calculation results are used as the initial state for creep aging analysis. The contact algorithm adopts the general contact algorithm of the penalty function method. For the setting of friction conditions, the Coulomb friction condition is selected, the friction coefficient is 0.1, and the normal behavior is set as hard contact, that is, separation after contact is allowed. Finally, the finite element model of hot forming of the curved surface workpiece is obtained.

2.3. Orthogonal Experiment Design

2.3.1. Scheme Design

To clarify the influence of the two processing processes on the structural pointing accuracy, this paper mainly analyzes the laser welding process parameters, including laser power, spot diameter, and welding speed, and the hot forming process parameters, including processing temperature and holding time, and determines the value range of each processing parameter as follows:

In the laser welding process, Jing et al. [19] used a laser with a minimum spot diameter of 0.4 mm for 0.8 mm titanium alloy plates to obtain good weld mechanical properties. Liang et al. [20] used a laser with a welding power of 3.2 kW and a welding speed of 1.67 mm/s for 5 mm titanium alloy plates to ensure that the plates are fully welded. Therefore, for the thin-walled structure in this paper, the processing reference values can be set as a laser power of 0.2 kW, a spot diameter of 3 mm, and a welding speed of 5 mm/s. These machining parameter benchmark values are reference experimental values. These processing parameter reference values can ensure that the plates are fully welded during processing, with good weld mechanical properties and controlled deformation. In the hot forming process, according to the research of Li et al. [21], to reduce the springback of metal during hot stretch bending, the forming temperature range of TC4 titanium alloy should be 500 °C to 750 °C. The research of Du et al. [22] pointed out that the residual stress of TC4 titanium alloy tends toward a certain limit value after 2000 s. Therefore, the forming temperature of 600 °C and the holding time of 3600 s can be selected as the processing reference values. These machining parameter benchmark values are reference experimental values. These reference values can ensure that the metal springback is reduced and the structure tends toward the stress relaxation limit.

To reduce the simulation workload while ensuring parameter coverage, the orthogonal experiment method is used to construct the processing parameter combination matrix of the two processing processes separately. For each processing parameter, two levels above and below are selected according to the reference value. Among them, the L₉(3³) orthogonal table is selected for laser welding to construct nine groups of simulation experiments, and the parameter combinations of each group are shown in the left part of

Table 2. Orthogonal experiment parameter combinations of laser welding and hot forming processes.

Experiment No.	Laser Welding			Hot Forming
	Laser Power P (kW)	Spot Diameter d (mm)	Welding Speed v (mm/s)	Forming Temperature T (°C)	Holding Time t (s)
1	0.18	2.5	4	550	2400
2	0.18	3.0	5	550	3600
3	0.18	3.5	6	550	4800
4	0.20	2.5	5	600	2400
5	0.20	3.0	6	600	3600
6	0.20	3.5	4	600	4800
7	0.22	2.5	6	650	2400
8	0.22	3.0	4	650	3600
9	0.22	3.5	5	650	4800

. For the hot forming process, the first two columns of the L₉(3⁴) orthogonal table are selected to construct nine groups of simulation experiments, and the parameter combinations of each group are shown in the right part of Table 2.

2.3.2. Evaluation Indicators

Taking the processing pointing accuracy of the structural terminal as the core goal, the deformation after processing of the two processes is taken as the evaluation indicators, respectively, which are specifically defined as follows:

• Laser Welding

The deformation displacement values $\mathrm{\Delta }{x}_{a1},\mathrm{\Delta }{y}_{a1},\mathrm{\Delta }{x}_{a2},\mathrm{\Delta }{y}_{a2}$ in the x and y directions of the left and right end faces 1 and 2 of the welded thin plate. Since the middle part of the thin plate will be greatly reduced in the mold during the subsequent hot forming process, the deformation of the left and right end faces will be superimposed and affect the subsequent processing procedures. Therefore, the deformation of the two end faces needs to be controlled.

At the same time, the residual stress value $\max ({\sigma }_{a1},{\sigma }_{a2})$ of the processed cross-section 50 mm away from the upper and lower end faces is used as the standard to measure the processing quality of the welding process.

2.

Hot Forming

A terminal feature point is selected at the curved forming end of the bottom end face, and the difference between its relative distance to the other forming end and the standard distance of the model $\mathrm{\Delta }{x}_b$ and $\mathrm{\Delta }{z}_b$ is used as the measurement standard, which can directly measure the deformation degree of the curved surface processing.

At the same time, the maximum value $\max ({\sigma }_b)$ of the residual stress after overall processing is used as the standard to measure the processing quality of the hot forming process.

3.

Overall Pointing Accuracy

The pointing accuracy of the pod-type space self-deployable structure in this paper is defined as the centroid position deviation $\mathrm{\Delta }x$ and $\mathrm{\Delta }y$ between the projection of the terminal face of the structure onto the starting face, as shown in Figure 4c. Due to the symmetry of the structure, for the laser welding process, the deformation amount of end face 1 or 2 can be taken as the maximum of the two. At the same time, due to the different machining positions of the two processes, the maximum laser welding deformation amount and the hot forming deformation amount are simply added together, which can be used as the reference value to measure the process simulation deformation amount, so as to obtain the pointing deformation accuracy $\mathrm{\Delta }{\delta }_x$ and $\mathrm{\Delta }{\delta }_y$ of a single pod structure. At the same time, according to the research of Zuo et al. [23], the eccentricity error of the designed spatially deployable diffractive telescope after deployment is less than 0.3 mm, and the antenna surface accuracy designed by Huang et al. [24] is also less than 0.5 mm. Therefore, the accuracy control range of the above similar structures can be referred to, and the index requirement is specified such that the pointing deformation accuracy ≤ 0.2 mm, that is, 200 μm. To ensure the transverse stiffness, the pod-type space self-deployable structure adopts an electromagnetic-like orthogonal design, and the coordinate system of adjacent pod structures needs to be flipped by 90°. Therefore, the pointing accuracy calculation formulas of a single pod structure and the overall structure can be obtained as follows:

$$\left\{\begin{array}{c}\mathrm{\Delta }{\delta }_x=\max (\mathrm{\Delta }{x}_{a1},\mathrm{\Delta }{x}_{a2})+\mathrm{\Delta }{x}_b\\ \mathrm{\Delta }{\delta }_y=\max (\mathrm{\Delta }{y}_{a1},\mathrm{\Delta }{y}_{a2})\\ \mathrm{\Delta }x=\mathrm{\Delta }\delta x(1)+\mathrm{\Delta }\delta y(2)+\mathrm{\Delta }\delta x(3)+\cdots +\mathrm{\Delta }\delta y(n)\\ \mathrm{\Delta }y=\mathrm{\Delta }\delta y(1)+\mathrm{\Delta }\delta x(2)+\mathrm{\Delta }\delta y(3)+\cdots +\mathrm{\Delta }\delta x(n)\end{array}\right.$$

(3)

where $\mathrm{\Delta }{x}_{a1},\mathrm{\Delta }{y}_{a1},\mathrm{\Delta }{x}_{a2},\mathrm{\Delta }{y}_{a2}$ is the laser welding deformation value, $\mathrm{\Delta }{x}_b$ is the hot forming deformation value, and n is the number of single cells.

3. Results and Analysis

3.1. Simulation Results

The simulation calculation and analysis were completed according to the above orthogonal experiment scheme. The thermal strain changes during the processing of laser welding condition 5 are shown in Figure 5, and the other conditions are not listed as examples. It can be seen from the figure that the thermal strain diffuses symmetrically from the weld core area to the two ends of the structure, showing a gradient decreasing trend along the direction away from the weld, and the thermal strain distribution characteristics at the symmetric positions of the structure are consistent, reflecting the constraint effect of structural symmetry on the transmission of thermal deformation. The simulation results of the creep stress distribution of the material under hot forming condition 5 are shown in Figure 6, and the other conditions are not listed as examples. It can be seen from the figure that at the initial stage of simulation, the residual stress is mainly concentrated in the middle area of the curved surface; with the extension of holding time, the stress gradually transfers to the edge and is continuously released, the distribution state tends to be uniform rather than concentrated, and the stress value also gradually decreases; after removing the mold constraint, the blank contour maintains the designed shape, and the residual stress is mainly retained in the transition part with large curvature, without obvious stress concentration. All results are summarized in

Table 3. Summary of simulation results of the two processing processes.

Laser Welding	Condition	$\mathbf{\Delta }{\mathbf{x}}_{\mathit{a}\mathbf{1}}$/μm	$\mathbf{\Delta }{\mathbf{y}}_{\mathit{a}\mathbf{1}}$/μm	$\mathbf{\Delta }{\mathbf{x}}_{\mathit{a}\mathbf{2}}$/μm	$\mathbf{\Delta }{\mathbf{y}}_{\mathit{a}\mathbf{2}}$/μm	$\mathbf{max}\mathbf{(}{\mathbf{\sigma }}_{\mathit{a}\mathbf{1}}\mathbf{,}{\mathbf{\sigma }}_{\mathit{a}\mathbf{2}}\mathbf{)}$/Mpa
	1	−124.3	95.214	134.51	43.211	113.37
	2	−89.401	71.554	95.692	27.501	85.621
	3	−65.251	46.925	137.29	42.671	57.878
	4	−108.44	84.524	116.76	32.176	95.57
	5	−83.334	64.982	89.853	26.224	74.979
	6	−114.91	89.032	123.86	34.883	100.3
	7	−99.155	76.608	106.94	31.1	88.472
	8	−135.08	103.79	145.54	40.407	116.44
	9	−102.43	80.64	109.35	27.598	90.74
Hot Forming	Condition	$\mathrm{\Delta }{\mathrm{x}}_b$/μm		$\mathrm{\Delta }{\mathrm{y}}_b$/μm		$\max ({\sigma }_b)$/Mpa
	1	78.2		99.1		25.76
	2	52.4		64.6		24.13
	3	44.5		53.8		21.52
	4	65.4		80.6		24.04
	5	46.9		58.0		23.47
	6	41.9		51.9		22.38
	7	42.9		53.2		21.91
	8	40.0		49.2		20.78
	9	39.1		47.8		18.78

below, and it can be seen that the deformation results all meet the index requirements, and the residual stress results are all less than the material yield limit.

3.2. Sensitivity Analysis of Process Parameters

To quantitatively evaluate the significance of the influence of each processing parameter on the pointing deformation, based on the simulation results in Table 3, this study uses the range analysis method and the analysis of variance (ANOVA) method to process the orthogonal experiment results.

The range analysis method can initially and qualitatively analyze the sensitivity of each processing parameter. The average value of the results under the same level of each factor can be calculated, and the difference between the maximum and minimum values of the results under different levels of the same parameter can be calculated. By comparing the differences obtained from different parameters, the sensitivity ranking of different processing parameters can be obtained. The larger the difference, the greater the influence of the level change of the factor on the output result, that is, the higher the sensitivity.

At the same time, to further quantitatively analyze the parameter sensitivity, the analysis of variance method can be used to judge whether these influences are statistically significant. The core of the analysis of variance method is to decompose the total variation of the data into the variation caused by the change of factor levels and the variation caused by random errors, and evaluate the significance of the factors by comparing the two. For the deformation of the left and right sides of laser welding in the x and y directions, the maximum value of the two can be taken for analysis. The main calculation formulas are as follows:

The total sum of squares statistic $S{S}_T$ is used to reflect the total fluctuation of the data:

$$S{S}_T={\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}$$

(4)

where n is the total number of experiments, $y_i$ is the observed value of the i-th experiment, and $\overline{y}$ is the total average value of all observed values.

The factor $S{S}_j$ sum of squares is used to reflect the fluctuation caused by different levels of a certain factor j:

$$S{S}_j={\displaystyle \frac{{\displaystyle {\sum }_{k=1}^P(T_{jk}^2)}}{m}}-{\displaystyle \frac{{({\displaystyle {\sum }_{i=1}^n{y}_i})}^2}{n}}$$

(5)

where P is the number of levels of each factor, $T_{\mathrm{jk}}$ is the sum of all test results of the j-th factor at the k-th level, m is the number of repeated tests at the same level, n is the total number of experiments, and y_i is the observed value of the i-th experiment.

The error $S{S}_e$ sum of squares reflects the fluctuation caused by random errors:

$$S{S}_e=S{S}_T-{\displaystyle \sum S{S}_j}$$

(6)

The F-test value is used to judge the significance of the factor effect:

$$F_j={\displaystyle \frac{M{S}_j}{M{S}_e}}={\displaystyle \frac{S{S}_j/f_j}{S{S}_e/f_e}}$$

(7)

where $F_j$ is the F-test value of the j-th factor, $M{S}_j$ is the mean square of the j-th factor, which is obtained by dividing the sum of squares of the factor $S{S}_j$ by its degrees of freedom, $M{S}_e$ is the mean square of errors, which is obtained by dividing the sum of squares of errors $S{S}_e$ by its degrees of freedom $f_e$ $f_j$ is the degrees of freedom of the j-th factor, which is usually $f_j=p-1$, $f_e$ is the degrees of freedom of errors, and the calculation formula is $f_e=f_T-{\displaystyle \sum f_j}$, where f_T = n – 1 is the total degrees of freedom.

Compare the calculated value $F_j$ with the critical value of the F-distribution under a specific significance level. If $F_j>F_{\alpha }(f_j,f_e)$, the null hypothesis is rejected, and it is considered that the factor has a statistically significant influence on the output response.

The calculation results of the above range analysis method and analysis of variance method are summarized in

Table 4. Summary of range analysis results.

Type	Indicator	Parameters	Level	Laser Power	Spot Diameter	Welding Speed
Laser Welding	$\max (\mathrm{\Delta }{\mathrm{x}}_{a1},\mathrm{\Delta }{\mathrm{x}}_{a2})$/μm	Average Value	1	122.5	119.4	134.64
			2	110.16	110.36	107.27
			3	120.61	123.5	111.36
		Range R		12.34	13.14	27.37
		Optimal Level		2	2	2
	$\max (\mathrm{\Delta }{\mathrm{y}}_{a1},\mathrm{\Delta }{\mathrm{y}}_{a2})$/μm	Average Value	1	71.23	85.45	96.01
			2	79.51	80.11	78.91
			3	87.01	72.2	62.84
		Range R		15.78	13.25	33.17
		Optimal Level		1	3	3
	$\max ({\sigma }_{a1},{\sigma }_{a2})$/MPa	Average Value	1	85.62	99.14	110.04
			2	90.28	92.35	90.64
			3	98.55	82.97	73.78
		Range R		12.93	16.16	36.26
		Optimal Level		1	3	3
Type	Indicator	Parameter	Level	Forming Temperature		Holding Time
Hot Forming	$\mathrm{\Delta }{\mathrm{x}}_b$/μm	Average Value	1	58.37		62.17
			2	51.4		46.43
			3	40.67		41.83
		Range R		17.7		20.33
		Optimal Level		3		3
	$\max ({\sigma }_b)$/MPa	Average Value	1	23.8		23.9
			2	23.3		22.79
			3	20.49		20.89
		Range R		3.31		3.01
		Optimal Level		3		3

and

Table 5. Summary of analysis of variance results.

Type	Indicator	Process Parameter	$\mathit{S}{\mathit{S}}_{\mathit{j}}$	$\mathit{M}{\mathit{S}}_{\mathit{j}}$	F	p
Laser Welding	$\max (\mathrm{\Delta }{\mathrm{x}}_{a1},\mathrm{\Delta }{\mathrm{x}}_{a2})$/μm	Laser Power	265.081	132.54	0.245	0.803
		Spot Diameter	271.15	135.575	0.251	0.8
		Welding Speed	1307.595	653.798	1.209	0.453
	$\max (\mathrm{\Delta }{\mathrm{y}}_{a1},\mathrm{\Delta }{\mathrm{y}}_{a2})$/μm	Laser Power	373.897	186.949	69.508	0.014
		Spot Diameter	266.632	133.316	49.567	0.020
		Welding Speed	1651.277	825.639	306.976	0.003
	$\max ({\sigma }_{a1},{\sigma }_{a2})$/MPa	Laser Power	257.194	128.597	18.226	0.052
		Spot Diameter	395.281	197.641	28.011	0.034
		Welding Speed	1975.407	987.704	139.983	0.007
Hot Forming	$\mathrm{\Delta }{\mathrm{x}}_b$/μm	Forming Temperature	477.029	238.514	3.764	0.12
		Holding Time	682.142	341.071	5.382	0.073
	$\max ({\sigma }_b)$/MPa	Forming Temperature	19.112	9.556	22.563	0.007
		Holding Time	13.902	6.951	16.412	0.012

, respectively. Table 4 intuitively reflects the influence degree of each process parameter on different evaluation indicators through the range R value, and Table 5 quantitatively judges the statistical significance of the influence of each parameter through the F-value and p-value. Together, they provide a quantitative basis for the parameter sensitivity ranking.

To more clearly present this sensitivity difference and significance characteristics, the core data of Table 4 and Table 5 are visualized to obtain Figure 7. Figure 7a,b show the visualization results of the range analysis of laser welding and hot forming processes, respectively, which can intuitively show the relative size of the range of each parameter; Figure 7c,d show the visualization results of the F-value of the analysis of variance, which can clearly compare the statistical significance of the influence of each parameter.

Combined with the quantitative data in Table 4 and Table 5 and the visualization results in Figure 7, the sensitivity rules of each process parameter can be systematically sorted out:

In the laser welding process, it can be intuitively seen from the range data in Table 4 and the range analysis results of laser welding in Figure 7a that the range corresponding to welding speed is the largest among all parameters, followed by laser power, and the range of spot diameter is the smallest. This indicates that the level change of welding speed has the most significant impact on the fluctuation of machining deformation and residual stress. This is consistent with the conclusion of “strain rate-related parameters dominate thermal deformation evolution” found by Chen et al. [25] in their study on superalloy machining. Essentially, welding speed directly determines the material heat input efficiency and cooling rate. When the welding speed increases, the heat input per unit length of the weld decreases, the thermal expansion and contraction effect weakens, and the residual stress decreases synchronously; while laser power regulates deformation by changing the heat input intensity, and spot diameter only affects the range of the heat-affected zone. The regulatory effect of both on deformation is weaker than that of welding speed. From the visualization results of the analysis of variance in Figure 7c, the F-value of welding speed is much higher than that of laser power and spot diameter. Combined with the p-value in Table 5, it can be seen that its influence on the deformation amount $\max (\mathrm{\Delta }{\mathrm{y}}_{a1},\mathrm{\Delta }{\mathrm{y}}_{a2})$ and residual stress $\max ({\sigma }_{a1},{\sigma }_{a2})$ exhibits p < 0.05, indicating statistical significance, while laser power only has a significant influence on some indicators, and the significance degree of spot diameter is relatively low.

The weak significance of some indicators is mainly due to two reasons: first, the two ends of the structure are provided with fixed boundary constraints to simulate the fixtures at both ends of the actual machining. This constraint inhibits the deformation in the x direction, making the response of $\max (\mathrm{\Delta }{\mathrm{x}}_{a1},\mathrm{\Delta }{\mathrm{x}}_{a2})$ to the change of process parameters limited; second, the welding sequence of left first and then right leads to the welding of the right end after cooling, resulting in a smaller deformation base, which further weakens the significance of the parameter influence. Microscopically, the cooling rate dominated by welding speed directly affects the β-phase transformation process of TC4 titanium alloy. When the cooling rate increases, the β-phase transformation is more sufficient, the grain size is more uniform, and the structural deformation is easier to control; whereas changes in laser power and spot diameter have a relatively indirect regulatory effect on the phase transformation, only affecting the local tissue state by changing the temperature field distribution of the heat-affected zone. Therefore, their sensitivity and statistical significance are weaker than those of welding speed. Thus, the subsequent process optimization can focus on the indicator results of the deformation amount $\max (\mathrm{\Delta }{\mathrm{y}}_{a1},\mathrm{\Delta }{\mathrm{y}}_{a2})$ and residual stress $\max ({\sigma }_{a1},{\sigma }_{a2})$. The optimal levels corresponding to both are consistent, that is, a laser power of 180 W, a spot diameter of 3.5 mm, and a welding speed of 6 mm/s.

In the hot forming process, the range data in Table 4 shows that the ranges of forming temperature and holding time are similar, with comparable sensitivity, and holding time is slightly more sensitive to deformation. This phenomenon is consistent with the conclusion of “temperature and holding time synergistically affect tissue uniformity” found by Li et al. [26] in their research on high-temperature forming of TC4 alloy. An increase in forming temperature improves the dislocation activity and grain boundary sliding rate of the material, reducing the flow stress; the extension of holding time is conducive to stress relaxation and tissue uniformity. The combined effect of the two makes the material deformation in a “stable stage”, so the significant difference in deformation amount is not obvious. From the results of analysis of variance, the influence of both on the residual stress $\max ({\sigma }_b)$ shows p < 0.05, indicating statistical significance, while the influence on the deformation amount $\mathrm{\Delta }{\mathrm{x}}_b$ does not reach the significance level. Overall, to achieve the minimum deformation control, the optimal reference parameters for the hot forming process are a forming temperature of 650 °C and a holding time of 4800 s.

It should be pointed out that this study has certain limitations: First, the simulation does not consider the possible multiphase structure of TC4 titanium alloy and the influence of its evolution on parameter sensitivity, and the analysis is conducted solely based on the single-phase material assumption. However, research by Zhou et al. [27] shows that the evolution of a multiphase structure will change the mechanical response and thermal conductivity of the material, which may affect the parameter sensitivity ranking. Second, the process parameter range does not cover extreme temperatures and holding times, such as conditions higher than 650 °C or longer than 4800 s, and the sensitivity rules under the boundary values of parameters have not been explored. Third, the coupled influence of parameters on micro-damage has not been analyzed in combination with the viscoplastic damage model [25], which should be further addressed in subsequent studies.

In summary, for the laser welding process, in practical engineering applications, the welding speed parameter can be optimized first, the welding power parameter can be adjusted synergistically, and the spot diameter can be considered finally. The optimal levels are a laser power of 180 W, a spot diameter of 3.5 mm, and a welding speed of 6 mm/s. For the hot forming process, the two processing parameters of forming temperature and holding time have similar influence degrees, and the influence of holding time on deformation amount is slightly larger. A forming temperature of 650 °C and a holding time of 4800 s can be used as reference values for physical machining parameters.

3.3. Machining Experiment Verification

To verify the accuracy and authenticity of the above results, the obtained reference values of processing parameters are applied to actual physical machining, and pod-type space self-deployable structure test pieces are obtained for physical test verification. The laser welding experiment uses a laser welding head of model HW60-P(V3), whose maximum laser output power can reach 6.0 kW, ensuring high-precision welding of thin-walled titanium alloy plates; the hot forming experiment adopts a vertical vacuum annealing furnace with an effective working size of 700 mm × 900 mm, a maximum working temperature of 1300 °C, and a vacuum degree of up to 5.0 × 10⁻³ Pa. The temperature uniformity is controlled within ±5 °C, which meets the temperature stability and environmental requirements of TC4 titanium alloy hot forming; precision measurements are performed using a TrackScan-P42 photogrammetry instrument in Beijing, which has a maximum measurement accuracy of 0.002 mm and a maximum scanning area of 500 mm × 600 mm, and can accurately capture the centroid position deviation of the structural terminal, ensuring the reliability of the measurement data.

Finally, a test piece of the pod-type space self-deployable structure is obtained for physical test verification, as shown in Figure 8. The photogrammetry instrument is used to measure the plane centroid position deviation of the deployment terminal. To avoid errors, three measurements are performed in total, and the actual measurement results are shown in

Table 6. Comparison of physical machining and simulation analysis results.

Measurement No.	X-Direction Component (mm)	Y-Direction Component (mm)
1	0.167	0.030
2	0.172	0.035
3	0.154	0.038
Mean Value	0.164	0.034
Simulation Result	0.176	0.047

. According to the above simulation analysis results, $\mathrm{\Delta }X=0.176\mathrm{mm}$ and $\mathrm{\Delta }Y=0.047\mathrm{mm}$, the average values of the three measurements, 0.164 mm and 0.034 mm, are taken as the comparison and measurement results. The results all meet the index requirement of ≤0.2 mm. It can be seen that the simulation results are close to the actual measurement results, which well conform to the actual processing position deviation degree. Due to the idealization of the simulation analysis and the error of the measurement technology, there are certain deviations between the results, but it can still prove the accuracy and authenticity of the simulation analysis results.

4. Conclusions

Aiming at the problem of pointing accuracy control caused by machining deformation of pod-type space self-deployable structures, this study clarifies the influence rules of key process parameters and optimizes the machining scheme through simulation modeling, orthogonal experiments, and experimental verification. The core conclusions are as follows:

(1)

The sensitivity ranking of process parameters to pointing accuracy is quantified. In laser welding, welding speed is the most significant factor affecting structural deformation, followed by laser power, and spot diameter has the weakest influence; in hot forming, processing temperature and holding time have similar effects on deformation, and the sensitivity of holding time is slightly dominant.

(2)

The optimal parameter combination for machining pod-type structures is proposed. Laser welding adopts a laser power of 180 W, a spot diameter of 3.5 mm, and a welding speed of 6 mm/s; hot forming adopts a temperature of 650 °C and a holding time of 4800 s. Through physical machining verification, the measured terminal pointing deformation of a single pod structure is 0.164 mm and 0.034 mm, both meeting the engineering index of ≤0.2 mm and showing high consistency with the simulation results.

(3)

The engineering practical value of the research is clarified. This parameter combination can be directly applied to the machining and production of key components such as large spacecraft antennas and satellite solar wings, effectively controlling the machining deformation and residual stress of thin-walled special-shaped structures, reducing the trial-and-error cost of high-precision manufacturing, and providing reliable technical support for the stable application of high-pointing accuracy structures in missions such as deep space exploration and space communication.