Synchronous Spray Effect Based on Dual Plant-Protection UAV Collaboration in Corn Fields

Abstract

It has become common to apply multiple drones to conduct plant-protection in large-scale farms, where dual-UAV synchronisation is representative. However, current studies are mainly dedicated to the spray quality of a single UAV, and it remains unclear whether synchronous operation affects spray effectiveness. This paper focuses on the spray efficacy and coupling effects of dual-UAV collaboration. Five-factor orthogonal four-level tests were conducted using the developed UAV collaboration system, and the results were compared with those of asynchronous and ideal linear superposition. It is indicated that (1) spray uniformity was impacted by the relative height between the UAVs and the flight speed of the UAVs (all the p-values < 0.02), whilst the deposition amount was affected by the relative horizontal spacing between the UAVs and the height of the left UAV relative to the forward flight direction (all the p-values < 0.04); (2) the proportion of high-quality spray in the coupling areas had a negative relation with the relative horizontal distance of the two UAVs, and the threshold of the effective coupling distance was 5 m; and (3) synchronous coupling should be avoided. If it is not, the left-side UAV (referring to the forward direction of flight) should be at a higher altitude (5 m or 6.5 m), be 0.5 m higher than the right and fly with a low or medium flight speed (3.5 m/s–4.5 m/s). The research can give a reference to the real spray operation by multiple UAVs.

1. Introduction

It is of high efficiency to utilise multiple plant-protection Unmanned Aerial Vehicles (UAVs) to prevent large-scale pests and diseases (P&Ds) [1], and dual-UAV synchronous spray is a typical mode. During collaboration, the spray effect is determined by the two-UAV fog field that is affected by UAV operation parameters [2] (Figure 1). Therefore, it is essential to analyse the spray efficacy combined with parameter variations and derive an optimal strategy for dual-UAV collaborations.

As shown in

Table 1. The studies related to UAV spray effect in the last 5 years in agriculture.

Perspectives	Main Approaches	Study Points	Benefits	Limitations	Citations
The relation between operation parameters and droplet distribution	Real tests based on water-sensitive papers or fluorescence	Parameter optimisation for a quad-rotor UAV in sugarcane fields	Optimal parameters for different heights of canopies	Single UAV without a unified parameter	[3]
		Optimisation of the chemical amount used for P20-type UAV	Prevention effect of different chemicals compared with manual spray	Single UAV and not related to UAV operation parameters	[4]
		Spray deposition of all-day operation by P20 and the prevention effect on cotton aphid	Enhancing the prevention advantages during night	Single UAV	[5]
		parameter optimisation for coverage rate performance based on fluorescence	Comparing fluorescence and water-sensitive papers	Single UAV	[6]
The consistency from wind Fields to droplet distribution	CFD models with droplets and winds	Impact of wing tip vortex on drifts in different flight speeds of an S-40 UAV	Drifts compared with real tests	Single UAV and Tail vortex not validated	[7]
		Impact of wind fields on drifts based on an SLK-5 UAV	CFD accuracy improved	Single UAV and drift in wake not validated	[8,9]
	Real tests	Direct measurement by anemometers to verify the consistency between wind and droplets	Analysing the effect of operation parameters on deposition	Single UAV and open-field tests	[10,11]

, single-UAV spray effect is a heated topic in recent 5-year studies. Two perspectives are mainly considered. In terms of the relation between operation parameters and droplet distribution, real tests were conducted based on different objects, such as sugarcane [3], rice [4], cotton [5] or no-crop conditions [6], and statistics was used to optimise operation parameters. For the consistency from wind fields to droplet distribution, Computational Fluid Dynamics (CFD) modelling in UAV hovering is the key approach [7,8,9,10,11], and enhancing model accuracy based on verifications is the purpose, not related to operation optimisation. Irrespective of which aspect is conducted, the multi-UAV spray effect is rarely discussed.

However, there have been relatively few retrievable studies on multi-UAV operations (summarised in

Table 2. The studies related to multi-UAV collaboration in agriculture.

Perspectives	Questions	Algorithms	Study Points	Benefits	Limitations	Citations
The optimisation of collaborative paths	Multiple flights based on a single UAV	TSP	Optimal path planning and transfer for adjacent multi-fields	Simulation combined with actual field shapes	Based on the nominal spray swath and lack of verifications	[12]
		SA	Optimal path based on the minimum power consumption in hilly areas	Simulation combined with actual orchard terrains	Based on the nominal spray swath and lack of verifications	[13]
		The optimal modelling by mileage, energy and load	Energy-optimal path to minimise total range, payload and safe operation	Field tests	Regular rectangular area verified and simplified verification conditions	[14]
	Multi-UAV operation simultaneously	PSO	Minimising the total number of operation sorties, total return supply time, total time consumption, supply time interval and total boundary return points	Algorithm simulation and comparison based on the prescription map and actual plot features	Based on the nominal spray swath and lack of verifications	[15]
		Optimal probability, GT, artificial potential field, etc.	Path coverage and formation control strategy for complex polygonal regions with obstacles	Complex regions and formation control involved	Based on the nominal spray swath and lack of verifications	[16]
Collaborative spray strategies	Multi-UAV operation simultaneously	Autonomous control for multi-agent non-uniform spraying by the Heat Equation Driven Area Coverage	A multi-UAV non-uniform spraying control system combined with CFD fog spectrum and Dubins motion model	Novel thinking based on CFD results	Lack of field tests	[17]

) in the past 5 years, primarily focusing on the optimisation of collaborative path and spray strategies. For the optimisation of collaborative paths, the previous research focuses on planning a full-coverage operation path that can adapt to various shapes of field parcels. Typical machine learning algorithms were optimised and applied, such as the Travelling Salesman Problem (TSP) [12], Simulated Annealing (SA) [13], optimal modelling [14], Particle Swarm Optimisation (PSO) [15] and Graph Theory (GT) [16]. Meanwhile, the objective was to find the shortest path or minimise energy consumption. In terms of collaborative spray strategies, Ivic et al. [17] used CFD to obtain simulation results of droplet distributions of nozzles and then developed an autonomous control system for multi-UAV non-uniform spray. Nonetheless, their limitations are as follows: (1) emphasis on simulations but lack of verifications and (2) without considering the actual droplet distributions [18]. Thus, it is a novel research direction to combine spray effectiveness with multi-UAV parameter optimisation.

This paper focuses on the spray quality and effects of dual-UAV collaboration, which have received very little attention. The main contributions are as follows: (1) clearly demonstrating the significant factors and the synchronisation effect and (2) based on the inadequacy of synchronised spray, providing a suitable operating strategy. The findings can serve as a reference for multi-UAV collaboration.

2. Materials and Methods

2.1. Hardware

A system for two quad-rotor UAVs (manufactured by Zhengfeng Co., Ltd., Shenzhen, China) was developed (Figure 2a), integrating a Real-Time Kinematic (RTK) ground base and two laptops. The UAVs were equipped with Beidou RTK (localisation errors of 0.8 cm ± 1 ppm in the horizontal plane and 1.5 cm ± 1 ppm in height), and the acceleration of each UAV was fixed as 2 m/s², which enabled repeated flight paths and heights and minimised collaborative error. Moreover, TCP Client (IP: 192.168.1.1) was used to establish the data link between the UAVs. Based on the control interface on the laptop, key operational parameters (flight height, flight speed and waypoint) could be set (Figure 2b).

2.2. Experiments and Data Processing

2.2.1. Experiment Design

The spray experiment using dual-UAV collaboration was conducted at the Zhuozhou Experimental Station, China Agricultural University, Hebei Province, China. The corn was in the Pustulation Stage, with row spacing of approximately 0.50 m and plant spacing of approximately 0.40 m. Meanwhile, the UAV payload was 5 kg, and the spray swath was about 3 m. There were two nozzles (VP 11002, produced by Licheng, Ningbo, Zhejiang Province, China) under each UAV. The spray pressure was about 0.48 MPa and the volume of flow was about 3.5 L/min (by DP-521, produced by Anshi Agri Tech, Anqing, Anhui Province, China).

As shown in Figure 3, 26 rows of corn were used in the experiment, and water-sensitive papers were set at a height of 0.70 m (Bottom Canopy), 1.40 m (Middle Canopy) and 2.10 m (Top Canopy), respectively. At each layer was set two pieces of paper. Considering the UAV spray swath, the reference UAV (UAV 1) was kept to fly over Row 6, whilst the flight path of the other UAV (UAV 2) was adjusted. The length of the flight preparation area was 50 m, and that of the spray area was about 30 m. The natural wind was a southeast wind, less than 0.40 m/s, and the relative humidity was about 75%.

Furthermore, an orthogonal experiment, L₁₆(4⁵), was designed. Five factors with four levels were considered, which were as follows:

• the collaborative speed (v) under 3.5 m/s, 4 m/s, 4.5 m/s and 5 m/s;
• the flight height of the reference UAV (h) under 5 m, 5.5 m, 6.0 m and 6.5 m;
• the relative height of the two UAVs (Δh) under −0.5 m, 0 m, 0.5 m and 1 m;
• the relative distance between front and back (Δx) under −0.5 m, 0 m, 0.5 m and 1 m;
• the relative distance between the UAVs in horizon (Δy) under 4.0 m, 5.0 m, 6.0 m and 7.0 m.

Meanwhile, because Δx = 0 during asynchronous and single-UAV spraying, the 4 groups with Δx = 0 were extracted and then compared with the asynchronous and single-UAV spray tests under the same conditions below, where a single-UAV spray was prepared for ideal linear superposition:

• v under 3.5 m/s, h under 5.5 m, Δh under 0 m and Δy under 5 m;
• v under 4 m/s, h under 5 m, Δh under 1 m and Δy under 6 m;
• v under 4.5 m/s, h under 6.5 m, Δh under 0.5 m and Δy under 4 m;
• v under 5 m/s, h under 5.5 m, Δh under −0.5 m and Δy under 7 m.

After the experiment, water-sensitive papers were scanned as grey images, and iDAS [19] was utilised to obtain droplet coverage (%) and deposition density (number/cm²).

2.2.2. Responses

According to the China National Standard (GB/T 24677.2-2009), in terms of the crop in the $i^{th}$ ($1\le i\le 26$) row, the sample height in canopies (k) was 0.70, 1.40 or 2.10 ($k\in \left\{0.70, 1.40, 2.10\right\}$). There were two pieces of water-sensitive papers in each layer. If the deposition densities were $N_{d_{i}1}^k$ and $N_{d_{i}2}^k$, and the coverage rates were $C_{i 1}^k$ and $C_{i 2}^k$, the responses (dependent variables) were as follows:

• the layered droplet deposition number in average ($N_{d_i}^k$):

$$N_{d_i}^k={\displaystyle \frac{1}{2}}{\displaystyle \sum _{r=1}^2{N}_{d_{i}r}^k}$$

(1)

• the droplet deposition number of the entire plant in average (${\overline{N}}_{d_i}$):

$${\overline{N}}_{d_i}={\displaystyle \frac{1}{3}}{\displaystyle \sum _{k\in \left\{0.70,1.40,2.10\right\}}{N}_{d_i}^k}$$

(2)

• the layered coverage rate in average ($C_i^k$):

$$C_i^k={\displaystyle \frac{1}{2}}{\displaystyle \sum _{r=1}^2{C}_{i r}^k}$$

(3)

• the droplet deposition number of the entire plant in average (${\overline{C}}_i$):

$${\overline{C}}_i={\displaystyle \frac{1}{3}}{\displaystyle \sum _{k\in \left\{0.70,1.40,2.10\right\}}{C}_i^k}$$

(4)

• and the coefficient of variation in the entire plant (CV):

$$C{V}_i={\displaystyle \frac{\sqrt{\frac{1}{3}{\displaystyle {\sum }_{k\in \left\{0.70,1.40,2.10\right\}}{\left(C_i^k-{\overline{C}}_i\right)}^2}}}{{\overline{C}}_i}}$$

(5)

2.2.3. Definition of Wind Coupling Area

The wind coupling area (Ω) was defined as: the area enclosed by the crop numbers corresponding to the trajectory centreline of each drone. In other words, there were four kinds of relative distance between the UAVs in horizon (Δy), so the Ω should be as follows: ${\Omega |}_{\Delta y=4}=\left\{6,7,8,\dots ,12,13\right\}$, ${\Omega |}_{\Delta y=5}=\left\{6,7,8,\dots ,14,15\right\}$, ${\Omega |}_{\Delta y=6}=\left\{6,7,8,\dots ,17,18\right\}$ and ${\Omega |}_{\Delta y=7}=\left\{6,7,8,\dots ,19,20\right\}$.

2.3. Main Effect Analysis and Trend Modelling of Synchronous Spray

2.3.1. Data Distribution Trend and Main Effect Analysis

(1)

Diagnostic and Statistical Methods for Data Distribution

Due to the typical skewed, heavy-tailed and heteroscedastic characteristics of spray data, histograms and Q–Q plots of whole-plant droplet deposition (N_d) and the penetration uniformity index (CV) were drawn to assess data normality. If the data showed skewness, heavy tails or substantial deviation from the hypothesised quantile lines, the rank-based non-parametric test would be employed to enhance robustness against outliers, long-tailed distributions and unequal variances.

(2)

Main Effect Analysis and Significance Testing

For the determination of the main effects of the factors and responses in the whole-plant scale, the Kruskal–Wallis H Test (K-W Test) was applied to assess the between-group differences in N_d and CV with respect to the five factors. Any factor within the significance level (p < 0.05) was further tested by Dunn’s Test, and the Bonferroni correction was used to adjust the p-value to control the Family-wise Error Rate (FER). This allowed the identification of the preferred level under each single-objective criterion (e.g., maximising N_d or minimising CV).

To further analyse the impact of each factor on the responses, the median and interquartile range (IQR) were used as the primary descriptive statistics to present the central tendency and distribution characteristics of each group. In terms of the significant factor levels, visualisations such as box plots and heatmaps were generated to illustrate the differences among levels. Box plots were to display the median, IQR and potential outliers for each factor level to demonstrate the distribution of the responses. Meanwhile, heatmaps based on the p-value matrix adjusted by the Bonferroni correction were drawn, with different colour intensities to indicate the significance for each pairwise comparison of factor levels.

2.3.2. Trend Modelling and Operation Parameter Optimisation of Synchronous Spray

To demonstrate the continuous trend of the responses across the factor distributions, Response Surface Methodology (RSM) was used to develop Multi-Factor Quadratic Trend Modelling, which provided a unified framework for the continuous description of response profiles, optimal ranges and boundary sensitivity.

If the variables found to be significant for CV were $v_{cv}^{\ast }$, $h_{cv}^{\ast }$, $\Delta h_{cv}^{\ast }$, $\Delta y_{cv}^{\ast }$ and $\Delta x_{cv}^{\ast }$, the model $CV=f_{cv}\left(v_{cv}^{\ast },h_{cv}^{\ast },\Delta h_{cv}^{\ast },\Delta y_{cv}^{\ast },\Delta x_{cv}^{\ast }\right)$ was established. Since the discrete CV may exhibit heteroscedasticity, HC3 Robust Standard Errors (HC3 RSEs) were applied to mitigate the impact of heterogeneous errors on statistical inference.

Meanwhile, if the variables found to be significant for N_d were $v_{N_d}^{\ast }$, $h_{N_d}^{\ast }$, $\Delta h_{N_d}^{\ast }$, $\Delta y_{N_d}^{\ast }$ and $\Delta x_{N_d}^{\ast }$, the model $N_d=\log \left(f_{N_d}\left(v_{N_d}^{\ast },h_{N_d}^{\ast },\Delta h_{N_d}^{\ast },\Delta y_{N_d}^{\ast },\Delta x_{N_d}^{\ast }\right)+1\right)$ was established by considering the potential for a Skewed Distribution. Then, Duan Smearing was utilised to modify inverse transformation deviations.

2.4. Joint Evaluation Indicators: Quadrant Division and High-Quality Spray Rate

To address the “dual index trade-off” and compare cross-factor levels and operation modes under different conditions, a joint evaluation method integrating N_d and CV was employed.

The medians of ${\overline{N}}_d$ and $CV$ of all the samples were used as threshold values (${\overline{N}}_{d_T}$, $C{V}_T$) for orthogonal segmentation. Hence, each sample (${\overline{N}}_{d_i}$, $C{V}_i$) was mapped into the four quadrants below:

• Quadrant A: high deposition with high uniformity (${\overline{N}}_{d_t}\ge N_{d_T} \& C{V}_i\le C{V}_T$), which means high-quality spray;

• Quadrant B: low deposition with high uniformity (${\overline{N}}_{d_t}<N_{d_T} \& C{V}_i\le C{V}_T$), which means droplet uniformity was qualified, but deposition was not sufficient;

• Quadrant C: high deposition with low uniformity (${\overline{N}}_{d_t}\ge N_{d_T} \& C{V}_i>C{V}_T$), which means spray deposition was qualified, but uniformity was not sufficient;

• Quadrant D: low deposition with low uniformity (${\overline{N}}_{d_t}<N_{d_T} \& C{V}_i>C{V}_T$), which means a high-risk operation due to non-ideal deposition and uniformity.

The number of each crop falling into Quadrants A, B, C and D under different horizontal distances (Δy) was counted, and bar charts were obtained for each Δy level to identify dominant intervals of operation parameters further.

2.5. Analysis of Synchronous Effect

2.5.1. Effect Difference Between Synchronous and Asynchronous Spray

(1)

Analysis of the Entire Spray Area

The effect difference between synchronous and asynchronous spray on CV and N_d was analysed from two dimensions: layered and whole-plant. For both perspectives, the Wilcoxon Signed-rank Test was first applied, followed by the Pratt Test for zero-difference samples. Then, the Holm–Bonferroni correction was used to adjust the p-values. Finally, the Benjamin–Hochberg FDR correction was employed to control the false-positive rate, thereby allowing significance differences to be assessed. The p-Holm (the p-value by the Holm–Bonferroni correction) < 0.05 indicated statistical significance.

(2)

Analysis of the Coupling Spray Area

In terms of the four kinds of coupling areas, ${\Omega |}_{\Delta y=4}=\left\{6,7,8,\dots ,12,13\right\}$, ${\Omega |}_{\Delta y=5}=\left\{6,7,8,\dots ,14,15\right\}$, ${\Omega |}_{\Delta y=6}=\left\{6,7,8,\dots ,17,18\right\}$ and ${\Omega |}_{\Delta y=7}=\left\{6,7,8,\dots ,19,20\right\}$, the same process in (1) was used, where (p-Holm) < 0.05 illustrated significance.

Moreover, heat maps and line charts were drawn for further demonstration.

2.5.2. Effect Differences Between Ideal Linear Superposition Spray and Synchronous and Asynchronous Spray

(1)

Deposition Comparison

If the deposition of the ith whole plant under a single-UAV spray was ${\overline{N}}_{d_{i single}}$, the ideal linear superposition should be made as ${\overline{N}}_{d_{i linear}}=2\times {\overline{N}}_{d_{i single}}$. Then, the gain of collaboration G_M was calculated by the following:

$$G_M={\displaystyle \frac{{\overline{N}}_{d_i}^M-{\overline{N}}_{d_{i linear}}}{{\overline{N}}_{d_{i linear}}}}$$

(6)

where M is a marker that should be the synchronous mode or the asynchronous mode. If G_M > 0, G_M ≈ 0 or G_M < 0, the synchronous spray has a gain, is close to or becomes weakened compared to the ideal linear mode.

(2)

Uniformity Comparison

If the spray uniformity of the ith whole plant under a single-UAV spray was $C{V}_{i single}$, the improvement rate $I_{CV}^M$ was calculated by the following:

$$I_{CV}^M={\displaystyle \frac{C{V}_{i single}-C{V}_i^M}{C{V}_{i single}}}$$

(7)

where M is a marker that should be the synchronous mode or the asynchronous mode. If $I_{CV}^M$ > 0, $I_{CV}^M$ ≈ 0 or $I_{CV}^M$ < 0, the uniformity improved, remained close to or deteriorated compared to the ideal linear spray.

Additionally, the Wilcoxon Signed-rank Test was used to assess the significance of the difference, with p < 0.05 indicating statistical significance.

3. Results and Discussions

3.1. Significance and Modelling of Variables for Synchronous Spray

3.1.1. Main Effect and Significance of Synchronous Operation Parameters

Figure 4 shows the histogram and Q–Q plot of the two responses (N_d and CV) of synchronous spray. It is evident that the distribution of these two responses was abnormal, and parameter estimation under normality and homogeneity-of-variance assumptions was not suitable as the primary inference method. The overall distribution was biased, with samples concentrated at lower values and a long tail toward higher values. Meanwhile, the data showed a heavy-tailed characteristic and might be affected by extreme values due to a systematic deviation in the mid-to-high quantiles from the theoretical quantile line in the Q–Q plots.

Furthermore,

Table 3. K-W Test results of factors on CV and N_d in the synchronous spray.

Factors	Responses (Independent Variables)
	CV			Nd
	KW_H (df = 3)	p-Value K-W	Significance	KW_H (df = 3)	p-Value K-W	Significance
Speed v	13.1227	0.00438	**	7.4475	0.0589	ns
Reference UAV Height h	4.6041	0.203	ns	12.6688	0.00541	**
Relative Distance between Front and Back Δx	7.4161	0.0598	ns	4.0697	0.254	ns
Relative Distance in Horizontal Δy	3.1668	0.366	ns	9.8417	0.02	*
Relative Height between the UAVs Δh	12.0595	0.00718	**	7.1765	0.0665	ns

ns: p ≥ 0.05; *: p < 0.05; ** p < 0.01.

lists the K-W Test results for the two responses, which demonstrates that different factors had significant effects on the response variables (CV and N_d). In terms of spray uniformity, the flight speed (v) and the relative height between the UAVs (Δh) had significant impact on CV (p-value = 0.004379 and 0.007182, respectively), while the droplet deposition of whole plants (N_d) was mainly influenced by the flight height of the reference UAV (h) and the relative distance in horizon (Δy) (p = 0.005410 and 0.019961, respectively).

Moreover, it can be further known that the factors presented a characteristic of “index decoupling”, which means that adjusting one response would not change the result of the other, since the variables used to vary CV and N_d were different. The deposition level (N_d) could be adjusted efficiently by changing h and Δy, whereas CV could be varied by choosing v and Δh. It has the potential to enable the production of different spray qualities during synchronous spray.

Based on the main-effect results above, the factors that reached significance were used for the Dunn Test, Bonferroni correction, joint evaluation and analysis of pattern/coupling area differences.

Figure 5 presents the results of the Bonferroni correction between CV and v & Δh. In terms of the relation between CV and v, the dominant effect of speed on uniformity was mainly within the transition from low to medium speed, as the significance was from the pairs of (3.5 m/s, 4.0 m/s) (p-value = 0.0247) and from (3.5 m/s, 4.5 m/s) (p-value = 0.0049). Meanwhile, according to the forward direction of the flight, the left UAV should be 0.5 m higher than the right (with the lowest CV and p-value < 0.05). Hence, a speed of 4.5 m/s and a height difference of 0.5 m should be optimal when minimising CV is the target.

Similarly, it can be known from Figure 6 that the influence of height of the reference UAV (h) on deposition was more pronounced over a larger height span, with the pair (5.0 m, 6.5 m) reaching a highly significant level (p-value = 0.0024). Moreover, to ensure a higher deposition quantity, the relative distance in horizontal (Δy) should be within the range from 4 m to 5 m, since significant differences primarily arose between Δy = 4 m and Δy = 6 m (p-value = 0.0347), and the median of N_d was higher when Δy was between 4 m (18.867) and 5 m (19.417), respectively.

Overall, Figure 5 and Figure 6 not only demonstrate the main effects of each parameter for synchronous spray but also show that “index decoupling” is further supported by the levels determined and these graphs.

3.1.2. Trend Modelling of Synchronous Spray

Figure 7 illustrates the relation between the responses and the significant variables. It can be seen that the two relations belonged to a convex optimisation problem, which means that an optimal solution could be found by performing a stationary analysis and a grid search with respect to this nonlinear characteristic.

The models were finally determined as follows:

$$CV=0.4142-0.0717v-0.0334\Delta h+0.1042v^2+0.0891\Delta h^2+0.0403v\cdot \Delta h$$

(8)

with the adjusted p-value = 0.0006, and

$$\ln \left(N_d+1\right)=2.4521+0.2528h-0.1440\Delta y+0.2533h\cdot \Delta y$$

(9)

with the adjusted p-value = 0.0144.

Table 4. The regression results of the responses with significant variables.

Models	Variables	p-Value > |t|	Coefficient
$CV=f_{CV}\left(v,\Delta h,v^2,\Delta h^2,v\cdot \Delta h\right)$	β1	1.32027 × 10−24 ***	—
	v	0.004357689 **	−0.071719234
	Δh	0.182649286	—
	v2	0.013366593 *	0.104236158
	Δh2	0.034364246 *	0.089058887
	v·Δh	0.230381292	—
$N_d=f_{N_d}\left(h,\Delta y,h^2,\Delta y^2,h\cdot \Delta y\right)$	h	0.000865575 ***	0.2528
	Δy	0.056713162	−0.1440
	h2	0.977033137	—
	Δy2	0.209835348	—
	h·Δy	0.012747849 *	0.2533

*: p < 0.05; ** p < 0.01; ***: p < 0.001.

gives the specific indices of the models. Combined with Figure 7, the following is clear:

(1)

The adopted modelling method improved the ability to capture global evolutionary laws, as HC3 RSE ensured the reliability of characterising the average response trend, and the logarithmic transformation effectively weakened the effect of extreme values.

(2)

The optimal operation parameters could be further determined. In terms of CV, the optimal condition should be v ≈ 4.486 m/s and Δh ≈ 0.342 m (predicted theoretical minimum CV ≈ 0.4007), which means that the UAVs under moderate speed with moderate relative height are beneficial for the overall uniformity. For improving N_d, the optimal condition occurred at the boundary condition (h = 6.5 m and Δy = 4.0 m), with the predicted N_d corrected ≈ 44.32/cm². These ranges are close to the main effects.

3.2. Joint Evaluation Based on Quadrant Division of Spray Quality

Figure 8 shows the spray-quality distribution for each crop in the N_d–CV joint quadrant at various levels of Δy.

From Figure 8, the following can be attested:

(1)

The increase in the relative horizontal distance (Δy) reduced the proportion of high-quality spray in the coupling areas. When Δy was 4 m and 5 m, high-quality spray (high N_d with low CV) accounted for about 40% and 43%, respectively, whilst it decreased to 25% and 18% when Δy was 6 m and 7 m. In particular, when Δy was high (6 m and 7 m), the spray in the coupling areas showed deterioration in uniformity or insufficient sedimentation.

(2)

The effective coupling distance for synchronous collaboration showed threshold characteristics at about Δy = 5 m. When Δy was 4 m and 5 m, high-quality spray was concentrated at the centre of the coupling areas (Crops 7 to 9 and Crops 11 to 12, respectively). When Δy increased to 6 m, spray quality declined markedly, particularly for the poor-quality spray (Quadrant D) of Crops 12 to 16. When Δy increased to 7 m, the contribution of the synergistic stacking of the two UAVs to deposition was further weakened, and the coupling area showed insufficient sedimentation.

3.3. Synchronous Effect Results

3.3.1. Results of the Effect Difference Between Synchronous and Asynchronous Spray

(1)

Effect Difference in the Entire Spray Area

Figure 9 presents the N_d distribution heatmaps and CV line charts, and

Table 5. The differences in deposition and uniformity for synchronous vs. asynchronous spray in the whole spray area.

Relative Horizontal Distance Δy	Synchronous vs. Asynchronous Nd								Synchronous vs. Asynchronous CV
	$\mathbf{The} \mathbf{Whole} \mathbf{Crop} {\overline{\mathit{N}}}_{\mathit{d}}$		Top-Layer Nd		Middle-Layer Nd		Bottom-Layer Nd
	Magnitude Size	${\mathit{p}}_{\mathit{h}\mathit{o}\mathit{i}\mathit{m}}$	Magnitude Size	${\mathit{p}}_{\mathit{h}\mathit{o}\mathit{i}\mathit{m}}$	Magnitude Size	${\mathit{p}}_{\mathit{h}\mathit{o}\mathit{i}\mathit{m}}$	Magnitude Size	${\mathit{p}}_{\mathit{h}\mathit{o}\mathit{i}\mathit{m}}$	Magnitude Size	${\mathit{p}}_{\mathit{h}\mathit{o}\mathit{i}\mathit{m}}$
4 m	Syn < Asy	0.368	Syn < Asy	0.174	Syn < Asy	0.583	Syn > Asy	0.583	Syn < Asy	0.003 (**)
5 m	Syn < Asy	0.011 (*)	Syn < Asy	0.024 (*)	Syn < Asy	0.011 (*)	Syn < Asy	0.009 (**)	Syn > Asy	0.867
6 m	Syn < Asy	0 (*)	Syn < Asy	0.006 (*)	Syn < Asy	0 (*)	Syn < Asy	0 (*)	Syn < Asy	0.671
7 m	Syn < Asy	0 (*)	Syn < Asy	0 (*)	Syn < Asy	0 (*)	Syn < Asy	0 (*)	Syn > Asy	0.15

*: p < 0.05; ** p < 0.01.

reports the significance of the difference between synchronous and asynchronous collaboration.

Overall, in terms of deposition volume, asynchronous spray outperformed synchronous spray across all the relative horizontal distances (Δy), with this advantage being more pronounced at medium-to-large spacings. Moreover, the difference between the two modes in CV (uniformity) was spacing-dependent: synchronous performance was superior only under close-range (strong-coupling) conditions. Specifically:

(1)

Deposition across the whole spray area shows that the asynchronous mode produced higher N_d than the synchronous mode at all the four Δy levels. The increase in N_d was evident at Δy = 4 m, whereas the improvements were statistically significant (p-value by Holm-corrected < 0.05) at Δy = 5–7 m. Meanwhile, the asynchronous spray could form more continuous medium-to-high-value bands across the top, middle and bottom layers, which might result in a stable increase in deposition intensity across the entire plant zone.

(2)

In terms of layered-specific deposition, the advantage of the asynchronous spray shifted from being “localised to certain layers” to “consistent enhancement across all the layers” with the increase in Δy. The layered results showed a clear progression: at Δy = 4 m, differences in N_d between the two modes were not significant in any layer. At Δy = 5, 6 and 7 m, N_d in all the three layers was significantly higher for the asynchronous mode (Holm-corrected p < 0.05), demonstrating that the gain was no longer confined to individual layers but extends consistently throughout the crops.

(3)

From the perspective of CV: only under a strong-coupling condition (Δy = 4 m) was the synchronous mode significantly better than the asynchronous mode. The difference in CV exhibited a distinct “spacing dependence”: at Δy = 4 m, CV was significantly lower in the synchronous mode (p < 0.01), indicating that synchronisation was more effective at maintaining coverage uniformity at close spacing. At Δy = 5–7 m, the increase in deposition by the asynchronous spray was not accompanied by systematic deterioration in uniformity (all the p-values > 0.05). Moreover, the asynchronous curve fluctuated at Δy = 4 m in the line charts, whereas the fluctuation of the two modes became similar at Δy ≥ 5 m.

In summary, at the whole-crop scale, compared with synchronous spray, asynchronous collaboration could deliver significantly greater deposition (Δy ≥ 5 m) and essentially unchanged uniformity (Δy ≥ 5 m). Synchronous spray with greater uniformity was observed only at a short horizontal distance (Δy = 4 m).

(2)

Effect Difference in the Coupling Spray Area

Table 6. The differences in deposition and uniformity for synchronous vs. asynchronous spray in the coupling spray area.

Relative Horizontal Distance Δy	Synchronous vs. Asynchronous Nd								Synchronous vs. Asynchronous CV
	$\mathbf{The} \mathbf{Whole} \mathbf{Crop} {\overline{\mathit{N}}}_{\mathit{d}}$		Top-Layer Nd		Middle-Layer Nd		Bottom-Layer Nd
	Magnitude Size	${\mathit{p}}_{\mathit{h}\mathit{o}\mathit{i}\mathit{m}}$	Magnitude Size	${\mathit{p}}_{\mathit{h}\mathit{o}\mathit{i}\mathit{m}}$	Magnitude Size	${\mathit{p}}_{\mathit{h}\mathit{o}\mathit{i}\mathit{m}}$	Magnitude Size	${\mathit{p}}_{\mathit{h}\mathit{o}\mathit{i}\mathit{m}}$	Magnitude Size	${\mathit{p}}_{\mathit{h}\mathit{o}\mathit{i}\mathit{m}}$
4 m	Syn < Asy	0.195	Syn < Asy	0.195	Syn < Asy	0.195	Syn < Asy	0.25	Syn < Asy	0.219
5 m	Syn < Asy	0.029 (*)	Syn < Asy	0.041 (*)	Syn < Asy	0.029 (*)	Syn < Asy	0.129	Syn < Asy	0.625
6 m	Syn < Asy	0.032 (*)	Syn > Asy	0.488	Syn < Asy	0.040 (*)	Syn < Asy	0.031 (*)	Syn < Asy	0.488
7 m	Syn < Asy	0.0003 (***)	Syn < Asy	0.0003 (***)	Syn < Asy	0.001 (***)	Syn < Asy	0.0003 (***)	Syn > Asy	0.489

*: p < 0.05; ***: p < 0.001.

shows the differences in the coupling areas.

Combined with Table 6 and Figure 9, the following is justified:

(1)

In terms of the whole-crop deposition, greater flight spacing (Δy ≥ 5 m) more effectively demonstrates the advantages of asynchronous operation in the coupling spray area, as the deposition amount of the asynchronous spray was larger than the synchronous, and all the p-values (Δy ≥ 5 m) were < 0.05.

(2)

For the layered deposition, the deposition by asynchronous spray gradually shifted from being “predominantly in the top and middle layers” (p-value = 0.041 and 0.029, respectively) to “predominantly in the middle and bottom layers” (p-value = 0.040 and 0.031, respectively), and eventually extended to “all the layers” (p-value < 0.01 in all the three layers). This provides more detail than the perspective of the whole spray area.

(3)

The primary contribution of the asynchronous spray in the coupling zone was to enhance deposition intensity and spatial coverage, since CV did not show significant differences across the four Δy conditions (p-value > 0.05).

From the conclusions above, it can be clarified that asynchronous spraying not only increased the deposition volume across the whole crop but also did not easily lead to a significant decrease in penetration uniformity. This is because, when the horizontal distance between the two synchronous UAVs is very small, the overall wind field is coupled stably, thereby facilitating the formation of a relatively uniform deposition pattern. As horizontal spacing increases, the wind-field coupling between the two drones weakens or disappears. However, during asynchronous operations, the second drone utilises time misalignment to re-cover the canopy within a relatively independent airflow channel, which can be understood as two temporally separated supplementary sprays on the plants. Therefore, multiple UAVs should maintain a relatively long horizontal distance if used in corn fields. This can significantly increase deposition across the entire crop and each layer without sacrificing uniformity and has greater potential to ensure operational efficiency.

3.3.2. Effect Differences Between Ideal Linear Superposition and Synchronous and Asynchronous Spray

Table 7. The effect differences between the ideal linear superposition spray and synchronous and asynchronous spray.

Spray Mode				Effect Differences in Nd							Effect Differences in CV
Syn (A)	Asy (B)	Lin (C)	Δy	${\overline{\mathit{N}}}_{\mathit{d},\mathit{L}\mathit{i}\mathit{n}}$	${\overline{\mathit{N}}}_{\mathit{d},\mathit{S}\mathit{y}\mathit{n}}$	${\overline{\mathit{N}}}_{\mathit{d},\mathit{A}\mathit{s}\mathit{y}}$	GA	GB	${\mathit{p}}_{{\mathit{N}}_{\mathit{d}} \left|\mathit{S}\mathit{y}\mathit{n} \mathbf{vs} \mathit{L}\mathit{i}\mathit{n}\right.}$	${\mathit{p}}_{{\mathit{n}}_{\mathit{d}}\left|\mathit{A}\mathit{s}\mathit{y}\right. \mathbf{vs} \mathit{L}\mathit{i}\mathit{n}}$	${\overline{\mathit{C}\mathit{V}}}_{\mathit{L}\mathit{i}\mathit{n}}$	${\overline{\mathit{C}\mathit{V}}}_{\mathit{S}\mathit{y}\mathit{n}}$	${\overline{\mathit{C}\mathit{V}}}_{\mathit{A}\mathit{s}\mathit{y}}$	${\mathit{I}}_{\mathit{C}\mathit{V}}^{\mathit{S}\mathit{y}\mathit{n}}$	${\mathit{I}}_{\mathit{C}\mathit{V}}^{\mathit{A}\mathit{s}\mathit{y}}$	${\mathit{p}}_{\mathit{C}\mathit{V}\left|\mathit{S}\mathit{y}\mathit{n}-\mathit{L}\mathit{i}\mathit{n}\right.}$	${\mathit{p}}_{\mathit{C}\mathit{V}\left|\mathit{A}\mathit{s}\mathit{y}-\mathit{L}\mathit{i}\mathit{n}\right.}$
A	B	C	4	108.003	29.699	47.685	−0.725	−0.558	0.087	0.144	0.416	0.294	0.522	0.292	−0.256	0.279	0.381
A	B	C	5	98.511	29.805	56.658	−0.697	−0.425	0.066	0.540	0.497	0.725	0.627	−0.459	−0.262	0.322	0.741
A	B	C	6	102.886	13.593	60.303	−0.868	−0.414	0.0005	0.106	0.685	0.534	0.499	0.219	0.270	0.120	0.093
A	B	C	7	80.450	20.403	54.790	−0.746	−0.319	0.172	0.838	0.353	0.484	0.372	−0.373	−0.055	0.182	0.937

Syn, Asy and Lin mean synchronous spray, asynchronous spray and ideal linear spray, respectively.

lists the differences in effects between the ideal linear superposition and the synchronous and asynchronous sprays. The following can be illustrated:

(1)

Asynchronous coordination could yield gains over single-UAV spray at most spacings, though it fell short of achieving the ideal doubling effect. However, synchronous spraying tended to reduce performance relative to a single UAV. The synergistic gain coefficients (G_A and G_B) for both synchronous and asynchronous modes across the four horizontal spacings (Δy = 4–7 m) were negative, indicating that overall crop deposition under dual-UAV spray was generally lower than the linear superposition. That is, neither synchronous nor asynchronous spray could achieve ideal doubling. Moreover, the synchronous mode showed a more substantial weakening effect (GA = −0.697 to −0.868), particularly at Δy = 6 m (p < 0.05). In contrast, the asynchronous mode achieved 44.2–68.1% of the ideal doubling target. When Δy was 5–7 m, it still provided a deposition gain relative to single-UAV operation (+15.0–36.2%), though it did not reach the ideal linear superposition.

(2)

Regarding penetration uniformity, no significant differences were observed between linear superposition and the two coordination modes. However, to improve uniformity, at larger horizontal spacings, synchronous dual-UAV spray might increase local deposition at the expense of penetration uniformity, whereas the asynchronous mode showed greater potential for balancing deposition volume and uniformity. The improvement rate (I_CV) relative to single-UAV operation varied with Δy. When Δy was 4 m, CV decreased slightly for the synchronous mode but increased for the asynchronous mode. At Δy = 5 m, both modes exhibited higher CV than the single unit. If Δy was 6 m, both synchronous and asynchronous modes presented reduced CV, with a larger improvement under asynchronous operation. When Δy was 7 m, the synchronous mode showed markedly higher CV than the single UAV, while the asynchronous mode remained close to the single-UAV level. Only at Δy = 7 m did the synchronous mode display a notable tendency towards uniformity deterioration (p ≈ 0.08).

From the above findings, it can be concluded that asynchronous coordination more closely approached the target state of linear superposition (the weakening effect was smaller) and could achieve deposition gains relative to single-UAV operation without significantly sacrificing uniformity (Δy ≥ 5 m). Conversely, synchronous coordination in most combinations exhibited stronger synergistic weakening and a trend towards uniformity degradation at specific spacings. Therefore, from the objective of increasing deposition while maintaining uniformity, asynchronous coordination may have greater potential at medium and larger horizontal spacings.

3.4. Spray Strategy of Multi-UAV Collaboration

Given the significant factors in synchronous spray, the results of joint evaluation of synchronous spray qualities and the comparison of coupling effects, the following can be summarised:

(1)

One of the key factors in multi-UAV spraying is to control the horizontal spacing between drones, and synchronous coupling should be avoided as much as possible. It is recommended that the horizontal spacing between drones be at least the spray swath (5 m in the study). Moreover, asynchronous collaborative work can be utilised to improve the amount and uniformity of droplet deposition.

(2)

If synchronous collaboration is unavoidable, the minimum spacing (4 m in the study) should be used as much as possible while maintaining flight safety. In addition, based on the forward direction of flight, the left-side UAV should fly at a higher altitude (5 m or 6.5 m in the study) and be slightly higher than the right (0.5 m in the study). Meanwhile, combined with low or medium flight speed (3.5 m/s–4.5 m/s in the study), high-quality spray (high N_d with low CV) can be achieved.

3.5. Discussions

This paper focuses on the spray efficacy and coupling effect during dual-UAV synchronous spraying. It is found that the relative height between the UAVs (Δh) and the flight speed of the UAVs (v) can be used to adjust spray uniformity (CV), whilst the relative horizontal spacing between the UAVs (Δy) and the height of the left UAV relative to the forward flight direction (h) can be utilised to vary spray deposition amount (N_d). More importantly, synchronous flight should be avoided during spraying, and asynchronous collaboration is recommended to improve deposition uniformity and amount.

According to the literature, previous studies have primarily converged on the wind-field and spray effects of a single UAV or on the scheduling and planning of multiple drones, without integrating the spray effects of multi-UAV collaboration and the optimisation of operation parameters [20,21,22,23]. In other words, they have not indicated whether it is reasonable to plan multi-UAV paths based on the ideal spray width, nor have they indicated whether synchronous operations will improve the spray effectiveness. This paper clearly addresses the questions through orthogonal tests and detailed data analyses. Multiple statistical methods and charts have been used to illustrate these issues. The proposed spray strategy for multi-UAVs can serve as a reference for practice.

Moreover, water-sensitive papers combined with image processing are commonly used in spray field tests, but it is a challenge for image analysis systems to accurately measure coverage densities if the coverage rate is larger than about 17% [24]. In this study, iDAS software was used to process the data, as its performance is better than DepositScan^® if analysing high-coverage water-sensitive images (with an error of at most 12% vs. at least 39%), while the performance is close when analysing medium and low coverage images [19]. Even though the coverage of some water-sensitive papers was more than 17% in this study, the error was acceptable.

Furthermore, Figure 8 and Figure 9 show an L-shaped bias in the high-quality spray, which is concentrated on the right side of each UAV, from high to low. This means UAV spray may have an “edge effect”, where the spray efficacy on the right side may be higher than that on the left. It is consistent with the conclusions previously drawn by the research team in the corn-field experiment [18]. Due to limitations in the cost and feasibility of experimental quantities, full tests were not conducted; however, this does not affect the validity of the conclusions. Thus, comparisons among different types of UAVs can further demonstrate them.

4. Conclusions

This study focuses on the spray characteristics of both the entire spray area and the coupling area in the synchronous collaboration of dual UAVs. Orthogonal and comparative tests were conducted, and significant factors, coupling effects and spray strategies for multi-UAVs were identified. The conclusions are as follows:

(1)

The relative height between the UAVs and the flight speed of the UAVs were significant for spray uniformity (all the p-values < 0.02), whilst the relative horizontal spacing between the UAVs and the height of the left UAV relative to the forward flight direction were significant for spray deposition amount (all the p-values < 0.04). These mean that using two factors could independently adjust one response without changing the outcome of the other.

(2)

The increase in the relative horizontal distance of the two UAVs (Δy) reduced the proportion of high-quality spray in the coupling areas, and the effective coupling distance for synchronous collaboration showed threshold characteristics at about Δy = 5 m. Meanwhile, asynchronous spray outperformed synchronous spray across all the Δy levels.

(3)

Combined with the joint evaluation of spray qualities and coupling effects, synchronous coupling should be avoided as much as possible. If not, the left-side UAV (according to the forward direction of flight) should fly at a higher altitude (5 m or 6.5 m in the study), be slightly higher than the right (0.5 m higher in the study) and work with a low or medium flight speed (3.5 m/s–4.5 m/s in the study). This can achieve high-quality spray (high deposition amount and uniformity) during synchronous collaboration.