Optimization of Elliptical Double-Beta Spray Gun Model Under the Control of Fan Air Pressure

Abstract

The air spray gun model for painting robots is a mathematical model that describes the performance and behavior of the air spray gun in a spray-painting system and simulates the spraying process. Currently, film uniformity and spraying efficiency are key factors in evaluating the spray performance of this model. To further enhance the accuracy and controllability of spray gun modeling, this study used the elliptical double-beta spray pattern model to investigate the key parameters influencing its performance. Fan air pressure was selected as the optimization variable. A fixed-point spraying experimental platform was established where spraying experiments were conducted under six different pressures, and coating thickness data were collected. The optimal fitting function was obtained using data processing software. Experimental verification showed that the amplitude error was within 3 mm and the film thickness error was within 4 µm. The results indicate that fan air pressure can accurately predict film thickness, significantly improving paint utilization, with a high engineering application value. This provides new theoretical support for precise control in the spraying process and optimization of automated spraying systems.

1. Introduction

With the advent of Industry 4.0, automation and artificial intelligence technologies have gradually permeated various industries [1]. The furniture manufacturing industry, as a typical example of this transformation, has achieved full automation in many areas, greatly enhancing production efficiency [2]. However, despite automation being implemented in many manufacturing processes, the furniture spray-painting industry still largely relies on traditional manual spraying methods. Prolonged exposure to such working conditions not only poses a serious threat to workers’ health [3] but also results in over 47% paint waste, with low efficiency and significant resource consumption [4]. In contrast, automated spray systems offer significant advantages in improving production efficiency, reducing costs, and minimizing personnel exposure to hazardous environments, especially with their immense potential in enhancing film uniformity [5]. Therefore, optimizing existing automated spray equipment and expanding the coverage of automated spraying systems is crucial for promoting the sustainable development of the furniture manufacturing industry.

The air spray gun is one of the key devices in spraying technology, and its performance is primarily evaluated based on the uniformity of the coating film and the spray volume [6], which are the core indicators of spray gun performance. During the spraying process, accurately controlling the distribution of film thickness and the spraying area has become a focal point of current research [7]. Traditional spray gun models can only describe the spatial distribution of the coating film thickness and cannot accurately predict the actual film thickness. Moreover, during spraying, these models operate with a fixed spray width and cannot dynamically adjust the spray pattern according to the shape of the target surface. This not only increases the complexity of trajectory planning in automated spray systems but also leads to a significant reduction in paint utilization. Therefore, optimizing traditional spray gun models to accurately predict film thickness and automatically adjust the spraying area according to different-shaped workpieces is crucial for improving spraying efficiency and accuracy [8].

To accurately capture the distribution characteristics of paint film thickness within an elliptical spray area, this study employed the elliptical double-β spray model to represent the spraying behavior of the air spray gun. Although the model demonstrates strong mathematical expressiveness and can describe the variation trend in film thickness with the spatial position, it still presents limitations in practical applications, such as the inability to directly predict the actual film thickness or to adaptively adjust to the size of the target surface. Therefore, optimization is required to improve its applicability and accuracy. Through an analysis of the key factors affecting model precision, it has been found that variations in fan air pressure during spray gun operation may induce slight mechanical vibrations, causing fluctuations in the spray height. Although these fluctuations are negligible in practical spraying, even minor deviations can introduce uncertainty and nonlinearity from a modeling perspective, thereby reducing the accuracy and stability of the fitted model. To ensure high spray quality, the spray gun is typically kept perpendicular to the target surface and moved at a uniform speed, making the spray angle and velocity essentially constant and, therefore, not suitable as optimization variables. Among the adjustable parameters, fan air pressure can be precisely controlled via an electric–pneumatic pressure proportional valve [9] and dynamically adjusted within an automated system, offering ideal control conditions. Other parameters can be held constant through the trajectory optimization of the robotic manipulator holding the spray gun. Therefore, this study selected fan air pressure as the optimization variable for the elliptical double-β model. By adjusting this parameter, the model’s prediction accuracy for film thickness distribution and control precision of the spray width could be enhanced, which not only improves the coating quality but also reduces material waste, aligning with the modern spray industry’s demands for high efficiency, low energy consumption, and advanced automation.

Based on the experimental principles, this study builds upon the traditional three-air automatic spray gun system, incorporating the precise control of spray pattern pressure through an electric–pneumatic pressure proportional valve. A timed spray gun control system based on the STM32 control chip [10] was designed and implemented, along with the establishment of a test environment suitable for spot-based spraying experiments. To obtain a more accurate spraying model, spraying experiments were conducted on pine boards under six different pressure levels, and the experimental data were fitted using a genetic algorithm to optimize the key parameters in the elliptical double-β spray gun model [11]. An error analysis was performed to derive the parameter variation functions under different pressures, which were then incorporated into the elliptical double-β model, resulting in a new model with fan air pressure as an independent variable. Finally, by comparing the actual spraying results under different pressure conditions with the predicted values from the model, the accuracy of the proposed elliptical double-β spray gun pressure model was validated, and its potential for application in practical engineering was discussed.

2. Construction of Spot Spraying Experimental Platform

To meet the requirements of the spot spraying experimental environment, a simple spot spraying platform needed to be constructed based on the working conditions of the spray gun before conducting the spraying experiments. This platform could securely fix both the spray gun and the workpiece to be sprayed.

2.1. Equipment Selection

For the overall platform framework, lightweight, high-strength, and corrosion-resistant aluminum square tubes and profiles were selected for assembly [12]. To maintain a constant distance between the spray gun and the target surface, a pulley mechanism was installed on the frame, allowing for both horizontal and vertical adjustments of the spray gun. This setup ensured that the distance remained unchanged even when the target workpiece was replaced. Additionally, it guaranteed that the central axis of the spray gun remained perpendicular to the surface of the workpiece during repeated fixed-point spraying experiments. As this study did not involve curved surface spraying, such an alignment ensured consistency in the experimental setup.

As the experiment was conducted under varying spray pressure conditions, a traditional two-air automatic spray gun, which only has an atomization pressure port (CAP) and a control air pressure port (CYL), could not adjust the spray pattern size by controlling the pressure. Therefore, a three-air automatic spray gun with an additional fan air pressure port (FAN) was selected for this experiment.

To ensure accurate control of the pressure and allow for real-time adjustments based on the spraying area of the workpiece, a standard manual pressure regulator could not be used. After comparing the parameters of various automatic spray guns, the SMC ITV2030-312L electric–pneumatic pressure proportional valve was ultimately selected to control the pressure at the FAN port.

In the control of the three-air automatic spray gun, based on the number of pneumatic solenoid valves and the performance of the microcontroller, the STM32F405RGT6 microcontroller was selected as the control chip. A pneumatic control board capable of supporting the required components was designed, and the necessary ports were soldered onto the control board [13]. The detailed design of the pneumatic control board is shown in Figure 1a,b.

2.2. Configuration and Installation

Based on the control program developed in the FreeRTOS operating system in Reference [14], the spraying process shown in Figure 2 was designed. The spot spraying experiment was triggered by pressing a button to complete a 0.5 s spraying task. Once the control board was powered with 24 V, the system initialized and entered the command reception state. The desired spray pressure was set and transmitted to the electric–pneumatic pressure proportional valve via the voltage signal from the microcontroller. When the control board button was triggered, the operating system called the spraying task, activated the pneumatic solenoid valves connected to the spray gun’s CAP, CYL, and FAN interfaces, opened the air inlet connected to the spray gun, and entered the spraying state. After the timer had run for 0.5 s, the spray gun was turned off, completing a spot spraying experiment. The workpiece was then replaced for the repeated experiment under the same pressure conditions.

After the control system configuration was completed, based on the working principle of the three-air automatic spray gun (as shown in Figure 3), the control board was connected to the solenoid valves and the electric–pneumatic pressure proportional valve. The solenoid valve’s air inlet was connected to the pressure regulator, and the solenoid valve that connected to the FAN port had its air inlet linked to the electric–pneumatic pressure proportional valve. The solenoid valve’s exhaust ports were connected to the three air inlets of the automatic spray gun. Finally, the compressor and paint tanks were connected to the experimental platform.

3. Point Spraying

To obtain the thickness distribution of the coating and the dimensional parameters of the elliptical spray area, a planar fixed-point spraying experiment was designed. A pinewood board with dimensions of 400 mm × 200 mm × 1.5 mm was selected as the spraying substrate. To improve the adhesion of the water-based paint and to reduce the influence of substrate porosity on coating absorption, a transparent anti-rust sealing primer was uniformly applied to the surface. This primer not only served as a barrier layer that stabilized the paint–substrate interaction and minimized the variability in film thickness caused by wood absorption differences but also helped to homogenize the paint adhesion rate under different spray pressures. Once the primer had completely dried, the surface was polished using 400 CW sandpaper to further enhance the flatness and uniformity of the coating surface.

During the spraying process, the distance between the three-air automatic spray gun and the wooden board was fixed at 220 mm. The paint tank pressure was set to 0.35 MPa, the flow control valve opening was set to 1, and both the atomizing pressure and the switch pressure were maintained at 0.35 MPa. Considering that fan air pressure typically varies between 0.10 MPa and 0.40 MPa in water-based paint applications for wooden furniture, six pressure levels—0.10 MPa, 0.12 MPa, 0.14 MPa, 0.16 MPa, 0.18 MPa, and 0.20 MPa—were selected for the experiments. To achieve precise control of the fan air pressure, an electric–pneumatic pressure proportional valve was introduced into the system. The paint used was a dark-green water-based coating mixed with 5% distilled water, which facilitated the observation of the film thickness distribution within the sprayed area. The spraying duration for each trial was set to 500 ms. The results of the spraying experiments are shown in Figure 4.

As shown in Figure 4, the paint mist particles adhered to the workpiece surface, forming an elliptical region. As the fan air pressure increased, the elliptical area expanded. After the water-based paint was completely dry, sampling points were taken along the x-axis at a 5 mm interval and along the y-axis at a 10 mm interval to measure the film thickness D(x, y) at each point, as shown in Figure 5.

Referring to the measurement principle of the KEYENCE VHX-6000 ultra-depth 3D microscope system described in Reference [15], the paint film samples were measured using the KEYENCE VHX-5000 ultra-depth 3D microscope system, which operates based on the same principle. The measurement accuracy of this system can reach 0.01 μm, fully meeting the 1 μm standard for spraying inspections [16]. For each measurement point, the film thickness was measured at three adjacent locations within the microscope’s field of view. The average value of these measurements was taken as the result for that point. The specific procedure is shown in Figure 6.

To better express the distribution of film thickness, the film thickness data were visualized using the Seaborn library (Jupyter Notebook, python 3) [17], generating a heatmap. The color intensity of the blocks in the heatmap reflected the values of the film thickness, as shown in Figure 7, where the x- and y-axes represent the horizontal and vertical positions on the substrate surface in millimeters (mm), respectively, with the origin (0, 0) at the geometric center of the spray area.

4. Elliptical Double-β Spray Gun Model Optimization

4.1. Elliptical Double-β Spray Gun Model

An automatic air spray gun compresses the air delivered by the air pump, atomizing the coating into fine particles that adhere to the wood surface, ultimately forming a continuous coating film. Due to the regulating air action of the outer ring of the air cap in the automatic air spray gun, the spray pattern is elliptical in shape. Within this elliptical region, the paint film thickness gradually decreases from the center toward the edges. This distribution characteristic is well-described by the β distribution, a continuous probability distribution defined on the interval [0, 1], which is suitable for modeling asymmetric bounded phenomena. The elliptical double-β spray model integrates β distributions in both the x- and y-directions and, therefore, it can more accurately characterize the spatial distribution of paint film thickness during spraying, as shown in Figure 8. The model equation is as follows [18]:

$$F\left(x,y\right)=F_{\max }{\left(1-{\displaystyle \frac{x^2}{a^2}}\right)}^{{\beta }_1-1}{\left[1-{\displaystyle \frac{y^2}{b^2\left(1-x^2/a^2\right)}}\right]}^{{\beta }_2-1}$$

(1)

$$-a\le x\le a,-b\sqrt{1-x^2/a^2}\le y\le b\sqrt{1-x^2/a^2}$$

(2)

In the equation, $F(x,y)$ represents the thickness distribution of the coating on the surface, $F_{max}$ is the maximum film thickness, $a$ and $b$ correspond with the semi-major and semi-minor axes of the elliptical spraying region, and ${\beta }_1$ and ${\beta }_2$ correspond with the thickness distribution exponents in the x- and y-directions.

4.2. Parameter Estimation

Due to the splashing of paint mist, the boundaries of the spray pattern can become blurred, making it difficult to accurately determine the major and minor axes of the elliptical region. To obtain precise data, the lsqcurvefit() function in MATLAB (version R2024a) can be used to perform least squares fitting. The lsqcurvefit() function takes a model function, initial parameter values, and data points as inputs, then attempts to find a new set of parameter values that minimize the sum of squared errors between the model function and the data points [19]. Although the maximum film thickness can be measured, it is also treated as an unknown variable for fitting due to potential human measurement errors [20]. The fitness function for the genetic algorithm is set as follows:

$$f_1=-{\displaystyle \sum _{i=1}^n{\left[D_I-D\left(x_i,y_i\right)\right]}^2}$$

(3)

In the equation, $n$ represents the total number of experimental data points and $D_I$ refers to the measured data for $({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}})$.

Taking the section F (x, y) = (−20, 10) at 0.10 MPa as an example, the fitted curves are shown in Figure 9.

To further validate the fitting performance presented in Figure 9, during the estimation of the fitting parameters of the elliptical double-β spray model using the nonlinear least squares method, the Jacobian matrix of the residuals for the 95% confidence interval was calculated using the Nlparci function in MATLAB. This function accounts for both the parameter estimation and covariance matrix in determining the confidence interval, thereby reflecting the accuracy of the model fitting process. The resulting 95% confidence intervals of the parameters are listed in

Table 1. The 95% confidence intervals of the fitted parameters at 0.10 MPa under F (x, y) = (−20, 10).

	Fmax/μm	a/mm	b/mm	${\mathsf{\beta }}_1$	${\mathsf{\beta }}_2$
x = −20 mm	[178.27, 181.44]	/	[44.06, 48.51]	/	[1.1924, 1.2072]
y = 10 mm	[224.83, 227.72]	[89.21, 92.73]	/	[1.8742, 1.9005]	/

.

We analyzed the parameter confidence interval ranges in Table 1. The fitting was independently performed along the x-axis (y = 10 mm) and y-axis (x = −20 mm) and, therefore, only direction-relevant parameters were presented. Parameters such as a and β₁ were omitted in the y-direction fitting, and b and β₂ in the x-direction. An analysis of the parameter confidence interval ranges indicated that the maximum deviation fell within ±2.23 μm, which was below the ±5 μm tolerance specified in Reference [16], demonstrating the high accuracy and reliability of the proposed model.

For each fan air pressure condition, the data in the x- and y-directions were separately fitted. To further improve the fitting accuracy, the x- and y-direction data were combined into a complete two-dimensional thickness distribution dataset, and the elliptical double-β spray model was applied to fit the entire dataset. The coefficient of determination (R²) was calculated based on the statistical information provided during the MATLAB fitting process to evaluate the overall fitting performance of the two-dimensional model under each pressure condition.

The coefficient of determination (R²) [21] is a standard used to measure the relationship between a fitted curve and the actual data points. It reflects the degree to which the model explains the data. The value of R² ranges from 0 to 1, with values closer to 1 indicating a better fit of the model. The R² value can be calculated using the following formula:

$$R^2=1-{\displaystyle \frac{{\displaystyle {\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}}{{\displaystyle {\sum }_{i=1}^n{\left(y_i-\overline{y}\right)}^2}}}$$

(4)

In the formula, $y_i$ is the actual observed value, ${\widehat{y}}_i$ is the fitted predicted value, and $\overline{\mathrm{y}}$ is the mean of the actual data.

By comparing the model fitting data with the confidence intervals, outliers that significantly deviated from the predicted results were eliminated. The parameters within the fitting interval were then statistically averaged to obtain more accurate model parameters. The final results are summarized in

Table 2. Fitted parameter values and R² results for each fan air pressure from 0.10 to 0.20 MPa.

P/MPa	Fmax/μm	a/mm	b/mm	${\mathsf{\beta }}_1$	${\mathsf{\beta }}_2$	R2
0.10	309.18	93.41	46.15	1.8887	1.2018	0.98831
0.12	273.36	101.13	47.99	1.9586	1.1955	0.99274
0.14	256.17	112.73	51.46	2.0119	1.1803	0.98944
0.16	233.95	120.23	53.93	2.1263	1.1747	0.97920
0.18	217.71	129.19	55.77	2.3347	1.1675	0.98601
0.20	206.09	136.72	58.25	2.5610	1.1598	0.98395

, where the calculated coefficients of determination (R²) are also listed.

4.3. Model Optimization

After obtaining the model parameters corresponding with each pressure level, a functional model could be established based on the relationship between the experimental data [22]. To achieve an accurate functional model, it was necessary to analyze the functional relationships between the parameters and determine which type of function could best describe the relationship between the parameters.

Taking the film thickness parameters as an example, it was observed that the values of the film thickness parameters generally showed a decreasing trend. Common functions that describe such decreasing trends include linear, exponential, and power functions. To analyze this, a regression analysis was performed in MATLAB using the Polyfit function and the Nlinfit function [23].

The Polyfit function in MATLAB is commonly used for polynomial fitting and is particularly suitable for fitting models in the form of polynomials. The Polyfit function uses the least squares method to fit the data [24], finding a polynomial model that minimizes the error (usually the sum of squared errors) between the actual data points and the fitted curve. It transforms the data (x and y) into the parameters (i.e., coefficients) of the polynomial model, which can be used to describe the data trend.

On the other hand, the Nlinfit function is used for nonlinear fitting and allows the fitting of more complex models, and is typically used in situations where a nonlinear function form needs to be specified. Unlike Polyfit, Nlinfit is not suitable for fitting polynomials but is designed to fit user-defined nonlinear models. The Nlinfit function uses the least squares method or other optimization algorithms (such as the trust region algorithm) to fit the data and minimize the error between the fitted model and the actual data [25]. The function adjusts the model parameters (such as β) to fit the data, where β represents the parameters of the fitted model. Users can provide custom nonlinear model functions not limited to simple polynomials.

In the experiment, taking the maximum film thickness as an example, pressure was set as the independent variable (x) and maximum film thickness as the dependent variable (y). The Polyfit and Nlinfit functions in MATLAB were used to fit different types of functions, and the goodness of fit for different models was compared to select the most suitable model. The specific fitting curves are shown in Figure 10.

After model fitting has been completed, evaluating the goodness of fit is a key step in selecting the optimal model. In addition to the coefficient of determination (R²) discussed earlier, the fitting error is also a commonly used and effective evaluation metric that can be employed to further assess the model’s fitting accuracy [26].

The fitting error was evaluated using the root mean square error (RMSE), which is derived as the square root of the mean squared error (MSE) [27]. The RMSE provides a direct measure of the average deviation between fitted and observed values in the same units as original data. A lower RMSE indicates better fitting performance.

The mean squared error (MSE) was calculated as follows:

$$MSE={\displaystyle \frac{1}{n}}{\displaystyle {\sum }_{i=1}^n{\left(y_i-{\widehat{y}}_i\right)}^2}$$

(5)

The root mean square error (RMSE) was calculated as follows:

$$RMSE=\sqrt{MSE}$$

(6)

In MATLAB, both Polyfit and Nlinfit provided fitting statistics for the model. These statistics were incorporated into the analysis function to obtain specific values, as shown in

Table 3. Analysis function values.

	Linear	Exponential	Power	Polynomial
R2	0.96995	0.98539	0.99664	0.99533
RMSE	6.0514	4.2201	2.024	2.3869

.

From the analysis of the data in the table, it could be seen that the power function’s values were closer to the true function compared with other types of functions. Therefore, the power function was selected as the most suitable model for the film thickness parameter function. The resulting function was as follows:

$$F{\left(P\right)}_{\max }=80.4986P^{-0.5827}$$

(7)

For the remaining parameters, a regression analysis was similarly conducted, and the most suitable functions were determined based on the coefficient of determination and fitting errors. Under the influence of fan air pressure, the variations in the major and minor axes of the spray width were ascertained as follows:

$$a\left(P\right)=-516.66P^2+595.30P+38.50$$

(8)

$$b\left(P\right)=-149.60P^2+168.21P+30.57$$

(9)

To simplify the expression of the model, we defined ${\beta }_1\left(P\right)={\beta }_1-1$ and ${\beta }_2\left(P\right)={\beta }_2-1$, where ${\beta }_1$ and ${\beta }_2$ are the original shape parameters of the elliptical double-β spray model along the x- and y-axes, respectively. This adjustment simplified the exponent expressions in the model equation without affecting its physical meaning or accuracy.

$${\beta }_1\left(P\right)={\beta }_1-1=62.6071P^2-12.2047P+1.4959$$

(10)

$${\beta }_2\left(P\right)={\beta }_2-1=1.1161P^2-0.7628P+0.2679$$

(11)

By substituting the obtained functions into the original elliptical double-β spray gun model and eliminating unnecessary decimal places, the film thickness model related to the pressure variable P could be derived as follows:

$$F\left(x,y,P\right)=F{\left(P\right)}_{\max }\ast {\left(1-{\displaystyle \frac{x^2}{a{\left(P\right)}^2}}\right)}^{{\beta }_1\left(P\right)}\ast {\left[1-{\displaystyle \frac{y^2}{b{\left(P\right)}^2\left(1-x^2/a{\left(P\right)}^2\right)}}\right]}^{{\beta }_2\left(P\right)}$$

(12)

$$a\left(P\right)=-516.7P^2+595.3P+38.5$$

(13)

$$b\left(P\right)=-149.6P^2+168.2P+30.6$$

(14)

$$-a\left(P\right)\le x\le a\left(P\right),-b\left(P\right)\sqrt{1-x^2/a{\left(P\right)}^2}\le y\le b\left(P\right)\sqrt{1-x^2/a{\left(P\right)}^2}$$

(15)

In the equation,

$$F{\left(P\right)}_{\max }=80.5P^{-0.6};$$

$${\beta }_1\left(P\right)=62.6P^2-12.2P+1.5;$$

$${\beta }_2\left(P\right)=1.1P^2-0.8P+0.3.$$

5. Spray Gun Model Validation

The fan air pressure was adjusted to 0.30 MPa. As the predicted spray width closely matched the dimensions of the original wooden board, a new pine board with dimensions of 450 mm in length, 250 mm in width, and 1.5 mm in thickness was used. All other experimental conditions remained the same as the first experiment. Spot-based spraying was performed at a height of 220 mm, and the resulting spray pattern is shown in Figure 11.

The wooden board was then cut in the same manner as the first experiment. The film thickness data were measured using the Keyence VHX-5000 super-depth 3D microscope system (Keyence, Osaka, Japan). The obtained data were then converted into a heatmap, as shown in Figure 12.

Similarly, the film thickness data for each row and column were fitted using a genetic algorithm (GA) in MATLAB. After averaging the values, the accurate values for each parameter were obtained. As an example, fitted curves for the data at F(x, y) = (10, −10) are shown in Figure 13.

A similar uncertainty analysis was conducted, as shown in Figure 13. The 95% confidence intervals of the fitted parameters are summarized in

Table 4. Confidence intervals of the fitted parameters for F (x, y) = (10, −10) at 0.30 MPa.

	Fmax/μm	a/mm	b/mm	${\mathsf{\beta }}_1$	${\mathsf{\beta }}_2$
x = 10 mm	[145.66, 150.73]	/	[60.72, 67.88]	/	[1.1286, 1.1507]
y = −10 mm	[128.29, 134.06]	[166.29, 175.73]	/	[4.4398, 4.4532]	/

, using the same method applied in Figure 9.

The obtained parameter values were then filled into the table and, based on the optimized model, the theoretical values for the film thickness, major axis, minor axis, and β at 0.30 MPa were also derived, as shown in

Table 5. Comparison of predicted and fitted values.

	Fmax/μm	a/mm	b/mm	${\mathsf{\beta }}_1$	${\mathsf{\beta }}_2$
Predicted	162.36	170.59	62.56	4.4691	1.1395
Fitted	158.39	167.73	65.16	4.4498	1.1447

.

From the data comparison in Table 5, it can be seen that the maximum film thickness error was 3.97 μm, which was below the standard ±5 μm threshold [16], while the errors for the major axis a and minor axis b were both within 3 mm, meeting the ±5 mm acceptance criterion for custom furniture [28]. Lastly, the differences for ${\mathsf{\beta }}_1$ and ${\mathsf{\beta }}_2$ were only 0.0193 and 0.0052, showing minimal deviation from the predicted values and indicating that the elliptical double-β pressure spray gun model aligned well with the actual spraying conditions.

6. Conclusions

Based on an elliptical double-β spray model, this study designed and constructed an experimental platform. By analyzing the influencing factors of the model, fan air pressure was selected to optimize the original model, resulting in the development of an elliptical double-β pressure spray gun model, which was subsequently validated through experiments. The proposed model effectively addressed the limitations of traditional spraying processes, such as the inability to automatically adjust the spray pattern and accurately predict the paint film thickness. By dynamically adjusting the fan air pressure in real time, the system could automatically match the spray parameters according to the surface dimensions of the workpiece and the required coating thickness, thereby significantly improving film uniformity and reducing material waste. This demonstrates high engineering applicability.

The main objective of this study was to adjust the spray pattern width and improve the uniformity of the paint film, rather than to investigate the influence of each model parameter during the spraying process. Therefore, fan air pressure was selected as the optimal control variable to construct the basic spray gun model. To ensure the adaptability of the model in practical applications, it is necessary to integrate the trajectory control and posture adjustment systems of the spraying robot arm so that the spray gun can maintain a stable spraying distance and perpendicular angle with the target surface under different heights. When applied to curved surface spraying scenarios, due to the complexity of the surface geometry and the uncertainty of the paint mist particle flow, further research is required in combination with trajectory optimization algorithms of the robotic arm. This will ensure that the spray gun trajectory remains aligned with the surface normally, thereby achieving more precise atomization control and improving coating uniformity. As the model was designed for small-area furniture spraying, the spray width is limited. When applied to large surface coating tasks, the model needs to be integrated with robotic path planning, such as grid-type trajectories, to achieve adaptive coverage of the entire area. By predicting the film thickness through the model and dynamically adjusting the spray width in real time, the system can ensure uniform overlapping and spacing, thereby maintaining a consistent coating thickness across large surfaces.