Rotor Speed Prediction Model of Multi-Rotor Unmanned Aerial Spraying System and Its Matching with the Overall Load

: During continuous spraying operations, the liquid in the pesticide tank gradually decreases, and the ﬂ ight speed changes as the route is altered. To maintain stable ﬂ ight, the rotor speed of a multi-rotor unmanned aerial spraying system (UASS) constantly adjusts. To explore the variation law of rotor speed in a multi-rotor UASS under objective operation a tt ributes, based on indoor and outdoor experimental data, this paper constructs a mathematical model of the relationship between rotor speed and thrust. The model ﬁ tt ing parameter ( R 2 ) is equal to 0.9996. Through the neural network, the rotor speed prediction model is constructed with the real-time ﬂ ight speed and the payload of the pesticide tank as the input. The overall correlation coe ﬃ cient ( R 2 ) of the model training set is 0.728, and the correlation coe ﬃ cients ( R 2 ) of the veri ﬁ cation set and the test set are 0.719 and 0.726, respectively. Finally, the rotor speed is matched with the load of the whole UASS through thrust conversion. It is known that the single-axis load capacity under full-load state only reaches about 50% of its maximum load capacity, and the load increase is more than 75.83% compared with the no-load state. This study provides a theoretical and methodological reference for accurately predicting the performance characterization results of a power system during actual operation and investigating the dynamic feedback mechanism of a UASS during continuous operation.


Introduction
With the continuous development of modern aviation application technology, electric multi-rotor unmanned aerial vehicles (UAVs) have become the primary platform for agricultural aerial plant protection spraying in the Asia-Pacific region, owing to their advantages of ease of operation, high operational efficiency, low operational costs, and reduced exposure risks for operators [1][2][3][4][5].An electric multi-rotor UAV is primarily composed of a flight control system, a remote control system, and a power system, among others.Under the correct operation of the remote control system, the flight control system sends flight parameter commands to the power system; then, eventually, with the proper functioning of the power system, the UAV is able to fly normally [6,7].In this process, the power system serves as the terminal for the implementation of flight commands, and its operational state will directly impact the flight parameters of the UASS [8].Furthermore, due to the typical characteristics of UASS spraying operations, the high-speed rotation of the power system rotors ensures the normal flight of the UASS, and the high-speed rotation of the rotor generates a significant downwash wind field.This downwash wind field not only affects the three-dimensional spatial distribution of target crop leaves [9][10][11] but also, due to the turbulence generated by the wind field, results in different motion characteristics of the spray droplets, which are ultimately deposited on the target crops [12][13][14][15].The rotor speed of the power system is one of the factors affecting the distribution and intensity characteristics of the downwash field of the UASS, and the downwash airflow will ultimately affect the deposition distribution of sprayed droplets [16][17][18].Therefore, it can be seen that the performance of the power system directly affects the effectiveness of the plant protection spray operations conducted by an electric multi-rotor UASS [19,20].When the operational parameters and the type of electric multi-rotor UASS are fixed, the continuous reduction in the payload in the pesticide tank during crop protection operations becomes the primary factor that affects the aerodynamic performance of the power system [21].By matching the real-time payload of the UASS pesticide tank with the rotor speed, it becomes easier to intuitively predict and understand the performance of the UASS power system during its continuous operation.As can be seen from the above, the performance of the UASS power system undergoes real-time changes during plant protection operations, which, in turn, affects its flight condition.Additionally, the rotor wind field, which indirectly represents the power system's performance, also impacts the droplet deposition distribution characteristics.Currently, research on the power system of electric multi-rotor UAVs primarily focuses on the simulation and optimization of aerodynamic performance, interactions between downwash flow and spray droplets, etc. [1,[22][23][24][25].Wang Bin et al. [26] designed an eight-rotor electric UASS by selecting and matching key components such as power system motors, electronic speed regulators, and rotors and determined its technical parameters through test flights.In order to optimize the aerodynamic layout of the entire UAV and improve the energy consumption rate of a multi-rotor electric UAV, Li Jiyu et al. adopted a tandem optimization method and adjusted the rotor layout of the power system [27].Zhang Hao et al. [28] established a wireless simulation parameter measurement system (WSPM-System) to obtain the power parameters during the operation of the UASS in order to improve the accuracy of the simulation.Wang Ling et al. used CFD simulation to study the impact of multiple characteristic parameters on the distribution characteristics of the downwash airflow of a UASS, providing a reference for exploring the downwash airflow field and the droplet deposition rules in complex airflow fields when a UASS sprays [29].Coomes et al. conducted relevant research on the impact of the downwash airflow of a quad-rotor UASS on the spray droplets at different rotor speeds and concluded that the distribution characteristics of the downwash airflow at different positions are different, which, in turn, affects the distribution of spray droplets [19].A hexarotor DJI Matrice 600 equipped with T-Motor "15 × 5" carbon fiber blades was tested numerically using computational fluid dynamics (CFD) and experimentally in a wind tunnel by Carreño Ruiz et al.The effects of parameters such as flight speed, nozzle type, and injection pressure on spray distribution were analyzed at different rotor speeds [30].The abovementioned scholars conducted relevant research on aerodynamic layout optimization and the effects of rotor speed on droplet deposition distribution characteristics of a multi-rotor UASS.However, previous research did not consider the impact of payload variations on the performance of power systems during continuous operations, nor did it explain the underlying mechanisms behind these performance changes.With the increasing use of large-scale UASSs, it is clear that the disparities in the power system performance of multi-rotor UASSs become more pronounced during the initial and final phases of operations as the capacity of the pesticide tanks increases..In turn, it will affect the rotor wind field performance of the multi-rotor UASS, which may eventually lead to different operation effects in the whole spraying process [31].
In order to make it easier to predict and grasp the performance of the power system during continuous operation of a UASS, this study conducted indoor and outdoor tests to obtain power performance indicators under various parameters.By taking into account the structural parameters of the UASS and the flight operation parameters, a neural network was established to predict the rotor speed.This allowed for the power system rotor speed of the UASS to be aligned with the payload so as to grasp the power performance parameters of the UASS in real time, clarifying the variations in the power system during the operation of an electric multi-rotor UASS.This research provides data support for further analysis of rotor wind field characteristics.

Laboratory Test Equipment and Method
Due to the ease of controlling parameters in indoor tests, the study initially focused on the individual rotor power system of the multi-rotor UASS before progressing to outdoor tests.Through theoretical analysis and indoor testing, the equation of the relationship between the rotor speed of the power system and thrust was established.The effective payload of the UASS pesticide tank was then matched with the rotor speed, clarifying the impact of changes in the UASS payload on the performance of the power system.This study conducted the relevant indoor tests using a specialized UAV power testing system, as shown in Figure 1.The system, model LY-30KGF (Lingyi Flight (Tianjin) Technology Co., Ltd.), is capable of measuring the power of an integrated propeller with a maximum thrust force of 30 kg and can accommodate rotors with a maximum diameter of 101.6 cm inches.The parameters of the test bench are shown in Table 1.Additionally, the testing system is equipped with a test platform and data acquisition software.During the testing process, the software can display data such as the rotor speed, thrust, torque, voltage, and current of the tested power system and can also calculate and display realtime parameters such as total (electric) power, motor efficiency, propeller force efficiency, system force efficiency, and power consumption.The software enables the control of accelerator opening and the real-time display and recording of test data.As illustrated in Figure 2, the operational principle of the UAV power system test bench is as follows.This test bench achieves motor speed control through computer-based power system control and data acquisition software.Relevant data are collected through voltage, current, thrust, torque, and temperature sensors.The data are summarized by the central microcontroller and ultimately transmitted back to the computer via a wireless data return module.The primary parameters of the power system for the field test model of the TopXGun F16 are presented in Table 2.This power system primarily consists of a TopXGun T80 motor, a HobbyWing XRotor-Pro-80A-HV electronic speed controller, and a 36,115 polymer carbon fiber propeller.The rated power of the T80 single motor is 1100 W. The HobbyWing speed controller provides a quicker throttle response, making it easier to control the motor speed.The carbon fiber propeller, made of polymer, demonstrates a higher elastic modulus and excellent impact resistance [32,33].As shown in Table 3, using the UAV power system testing platform, the performance parameters of the power system for the TopXGun F16 multi-rotor UASS were tested at an accelerator duty cycle of 0 to 100%, with a 5% interval.

Outdoor Test Equipment and Method
Due to the inability to obtain parameters related to the power system during the entire operation process of the UASS in indoor tests, this study included real-time monitoring and data collection of the power parameters and payload of the UASS under different initial payloads and operational parameters.Furthermore, we analyzed the change patterns of the power parameters and payload during the continuous operation process of the UASS.During the continuous operation process, when the UASS enters and switches routes, in order to achieve the target flight speed, the power system needs to provide additional power to achieve hovering, thus achieving the acceleration and uniform speed of the UASS.Additionally, in order to match different initial payloads and the power system, adjustments need to be made.Dynamic changes caused by the continuous spraying of PPPs require the power system of the UASS to provide real-time feedback.As shown in Table 4, this study primarily considers flight speed, initial payload, and spraying flow rate as variables, with each variable set to three levels.To ensure the reliability of subsequent modeling, this experiment utilized an orthogonal experimental design with three factors at three levels, encompassing a total of 27 experimental groups.As shown in Figure 3, the UASS model used in the outdoor test is the Nanjing TopXGun F16, and its specific parameter indicators are shown in Table 5.The formula for calculating propeller thrust (; unit: N) from references [34,35] is utilized to theoretically analyze the correlation between rotor speed and power performance.This analysis aims to establish a model that describes the relationship between rotor speed and parameters of power performance.
where N (unit: RPM, r/min) is the rotational speed of the propeller,  (unit: m) is the diameter of the propeller,  is the dimensionless tension coefficient, and is the air density of the flight environment.
For CT in Equation ( 1), the solution result is abstractly represented as in Equation ( 2) as follows: where Θp denotes the set of all propeller parameters, such as propeller diameter ( ), pitch ( ), and number of blades ( ).The derivation step of   is shown in Equation (3) as follows [36]: where A is the span-chord ratio, where A = Dp/cp, cp is the mean geometric chord length of the paddle, taking the range of 5~8; ε is the correction factor due to the downwash effect, with εℝ+ taking values in the range of 0.85~0.95;λ is the correction factor, the value of which is in the range of 0.7~0.9;ζ is the reference chord's spreading position, the empirical factor of which is taken as 0.4~0.7 (The reference geometric chord of the airfoil refers to the geometric chord of the blade spreading to a certain definite position during the blade design process); α0 indicates the zero-thrust headway angle (unit: rad) in the range of −π/36~0; and K0 is the slope of the wing thrust coefficient curve, approximately equal to 6.11 per radian or 0.107 per degree.
According to the theoretical analysis formula presented in Section 2.3.1, it can be observed that the thrust of the rotor power system of the UASS is dependent on the airfoil parameters, pitch, diameter, number of blades, and rotor speed.Multi-rotor UASSs mostly use fixed-pitch rotors.For a fixed rotor, when hovering under conditions of a high Reynolds number and low subsonic Mach number, the thrust coefficient (CT) and rotor diameter remain unchanged; the rotor thrust performance is mainly related to rotor speed and air density.The air density is primarily influenced by altitude and temperature.Throughout the test, the altitude and temperature remain essentially constant (The average temperature is 295.90K, with a variation range of 295.75 K-296.45K; the average atmospheric pressure is 101.37 kPa, with a variation range of 101.34 kPa-101.39kPa).Hence, in analyzing the correlation between rotor speed and thrust, Equation (1) from Section 2.3.1 is simplified to Equation (4): where  is the thrust model parameter and N is the rotor speed (in RPM, r/min).Combined with the above analysis, there is a quadratic correlation between the thrust and the rotor speed for the same type of UASS power system.Using Matlab software, the least squares method is used to fit and analyze the test data.In addition, the accuracy of the fitting results and the mathematical model is further analyzed and verified by comparing the differences between the test results from the UAV power system test platform, the estimation results of the mathematical model, and the Matlab fitting results.In order to compare the differences between the fitting and the estimated results of the mathematical model, Equation ( 5) is used to calculate the relative error percentage.

𝐸
* 100, where E denotes the relative error percentage, T1 represents the thrust value output by the fitting and mathematical model, and Texp represents the laboratory test thrust value.

Rotor Speed Model and Match to Overall Load
In the continuous operation process, the changes in the flight speed of the UASS and the payload of the pesticide tank are the primary factors affecting the rotor speed [37].However, modeling the rotor speed based on these factors is a typical nonlinear problem, and there is no specific and comprehensive mathematical expression to integrate these factors to obtain the real-time rotor speed results of the UASS during continuous operation.However, neural networks have a strong ability to handle such integration, so we chose a basic data processing method, namely a BP neural network.The BP network model was established on a 64-bit Windows operating system using PyTorch.A BP (backpropagation) neural network is a multi-layer feedforward neural network trained through the error backpropagation algorithm.It consists of an input layer, hidden layer, and output layer.The nodes in each layer are connected through weighted connections, and the weight values are continuously adjusted through the learning process to minimize the output error.Since the AdamW algorithm adds a weight attenuation term, which can effectively prevent over-fitting problems, an early stopping strategy was adopted.The algorithm development environment and hyperparameter settings were determined through experiments, as shown in Table 6.The learning rate is set to 0.0006, the patience is set to 30, and the delta is set to 0.001.The hidden layer uses the ReLU linear activation function.Since it is a regression problem, the output layer does not require an activation function.The CPU is an Intel(R) Core (TM) i5-12400F@2.50GHz.The random access memory is 16 GB.The graphics card is an Intel (R) Arc (TM) A770 16G.In addition, referring to the model of the relationship between the rotor speed and thrust of the power system in the multi-rotor UASS described in Section 2.3.1, the changes of rotor speed and thrust of the UASS with different models and load capacities are different in the actual operation process.According to the change range of the payload of the pesticide tank under the normal operation of the UASS model corresponding to the power system, reference Equation ( 6) and ( 7) are used to convert the load change of a single power system under full-load and no-load conditions so as to match the overall load and rotor speed.Referring to the model of the relationship between rotor speed and thrust established in Section 2.3.1, the rotor speed and the overall load of the UASS are matched, providing a reference parameter value for subsequent research.
where N is the number of power systems (rotors), Fg is the load of the whole UASS in the full-load state of the pesticide tank, and Fsg is the single-axis load under full-load state of the pesticide tank.
where Ng is the load of the whole UASS in the unloaded state of the pesticide tank, and Nsg is the single-axle load under the unloaded state of the pesticide tank.
where Pg is the percentage change in single-axle load.

Modeling of Rotor Speed and Thrust
The fitting results of the rotor speed and thrust of the power system for the UASS are shown in Figure 4.In addition, the appropriate parameters are selected to calculate the thrust of the power system in the UASS at the corresponding speed based on the theoretical analysis formula of propeller dynamic performance presented in Section 2.3.1.In Figure 4, "" is the laboratory test value, the dashed line is the fitting curve, and "☆" is the theoretical estimation value.Various methods are employed to ascertain the correlation among the performance parameters of the power system.Table 7 displays the relevant parameters of the fitting curve for the test data.Among them, SSE represents the sum of squared errors between the fitting data and the original data, RMSE is the square root of the mean sum of squared errors between the corresponding points in the fitting data and the original data, R-square is the correlation coefficient for the fit, and Adj R-sq represents the adjusted R 2 .In a univariate linear regression model, the commonly used performance evaluation index is R-squared.A value closer to 1 indicates a stronger explanatory power of the variables in the equation for y.This indicates a better fit of the model to the data.According to the fitting parameters listed in Table 7, the R 2 fitting parameter for the rotor speed-thrust curve of the power system of the F16 UASS is greater than 0.999.This indicates that the model curve fits the test data well.It can be observed that there is a strong quadratic correlation between the rotor speed and thrust.Figures 5 and 6 illustrate the error percentage and average error of the thrust fitting and estimation values for the F16 power system at various speeds compared to the test values.For TopXGun F16, its rotor speed is greater than 1500 rpm under normal operating conditions, and the rotor speed in this state is defined as high speed.Figure 5 reveals that at low speeds, the relative error between the thrust obtained by the three methods is higher than at high speeds.With increasing speed, the relative error decreases and stabilizes below 5%.Referring to the research findings of Carreño et al. [30], the primary reason is that low rotor speeds result in correspondingly low Reynolds numbers.This reduction leads to lower rotor thrust measurements in indoor tests, thereby causing discrepancies between theoretical calculations and indoor test values, with a relatively large relative error.In addition, factors such as platform measurement accuracy and idealized theoretical calculations can also lead to the existence of errors.It can be seen from Figure 6 that the average errors of the fitted and calculated values, relative to the test values, are within 5%.Therefore, the fitted curve and mathematical model can effectively express the relationship between the rotor speed and thrust of the power system in a UASS during normal operation.They can also accurately predict the thrust of the power system under actual operating conditions.

Neural Network Model of the Rotor Speed
During the spraying operation, the continuous spraying of the liquid from the pesticide tank and the need to maintain a specific flight speed are the main factors that cause the rotor speed of the UASS to constantly fluctuate.Therefore, in this paper, a fully connected neural network is established with real-time flight speed and pesticide tank payload as inputs (instantaneous data) and four motor speeds as outputs.Hyperparameter optimization is carried out using the grid search method.As shown in Figure 7, the structure of the fully connected neural network is selected as follows: three hidden layers from input to output, and the number of nodes in each hidden layer is 117, 103, and 76, respectively.The data set is randomly divided into a training set, validation set, and test set, following a ratio of 0.75:0.15:0.15.

Match of the Rotor Speed and Overall Load
In this paper, TopXGun F16 is used as the test model.Referring to Table 5 for UASS rotor and payload index parameters and Section 2.3.2Equations ( 6)-( 8), the study calculates and analyzes changes in the single power system of the F16 UASS under fullload and no-load conditions.According to the calculation results, the load of the single power system under UASS full load (Fsg) is 9.28 kg, while the load of the single power system under no load (Nsg) is 5.28 kg.The single-axis load of the UASS increases by more than 75.83% under full load compared to the no-load condition.During the continuous spraying operation, it can be observed that the single-axis load of the UASS undergoes significant changes.Changes in the single-axial load will alter the downwash airflow intensity of the UASS, thereby affecting the movement of the spray droplets and ultimately influencing their deposition characteristics on the target crop.
In order to more intuitively express the changes in the performance of a single power system during full-load and no-load flight operation of a UASS, according to the change in single-axis load from full load to no-load state of the F16 UASS and in combination with the performance test data of the 3.1 power system, the load change curve and speed change range of the single rotor under full-load and no-load flight operation of the UASS are determined.Here, in order to match the overall load capacity of the UASS, the thrust unit of a single power system is converted to kilograms (kg).In addition, when combined with the rotor speed prediction model constructed in Section 3.2.1, the four motor speeds of the UASS can be calculated and output for a specific flight speed and pesticide tank payload.The results are shown in Figure 10.It can be seen from Figure 10 that in order to ensure flight flexibility and facilitate attitude adjustment and safe flight of a UASS, the load capacity of a single power system for the UASS under full load only reaches about 50% of its maximum capacity.Additionally, the power redundancy is strong.During the operation of the UASS, the speed and thrust of the power system decrease nonlinearly with the continuous reduction in liquid pesticide in the tank.In addition, the scatter plot in Figure 10 shows the average speed of the four rotors under specific flight parameters as calculated by the neural network model.It can be observed from the figure that the output value of the neural network is largely consistent with both the test value of the power system test platform and the theoretical calculation.

Conclusions
During the continuous operation of the UASS, the amount of liquid in the pesticide tank gradually decreases, and the flight control system continuously provides feedback on the control demand of the flight attitude to the power system, which results in the power system parameters of the UASS constantly changing.In this paper, combined with the structural parameters and flight operation parameters of the UASS, the thrusting force of the power system at different rotor speeds is measured by using the UAV power system test platform.At the same time, the dynamic parameters and payload of the UASS are monitored and collected in real time under various operating conditions, and the neural network is used to construct a rotor speed prediction model for the power system of the UASS.Combined with the model parameters of the UASS, the rotor speed is matched with the overall load, and the variation range and law of the aerodynamic performance parameters of the power system during the operation of the UASS are clarified.The main conclusions are summarized as follows: (1) Through theoretical and data analysis, the model of the relationship between the rotor speed and thrust of the power system for the TopXGun F16 UASS is T = 2.011e-05N 2 .The fitting parameter (R 2 ) is 0.9996.By comparing the theoretical calculations, test results, and fitting results of thrust at the same rotor speed, it is known that the relative difference between the three is consistently within 5%.This indicates that the fitting curve and mathematical model effectively represent the relationship between the rotor speed and thrust of the power system during normal operation.Furthermore, they can accurately predict the thrust of the power system at different rotor speeds.(2) In this paper, the real-time flight speed and payload of the pesticide tank during the operation of the UASS are used as inputs.The four rotor speeds are used as outputs to establish a fully connected neural network for constructing a rotor speed prediction model.The overall correlation coefficient (R 2 ) of the training set is 0.728.At the same time, the model performs well on both the validation set and the test set, with correlation coefficients (R 2 ) of 0.719 and 0.726, respectively.The model takes the real-time load and flight speed as input parameters and can basically accurately predict the rotor speed under current conditions.(3) For the TopXGun F16 UASS, through the matching of the rotor speed and the overall load, it is observed that the single-axis load in the full load state increases by more than 75.83% compared with the single-axis load in the no-load state.Under full-load conditions, the load capacity of the single power system of the UASS only reaches approximately 50% of its maximum capacity, and it has strong power redundancy.
In addition, the rotor speed values calculated by the neural network output, power system platform test, and matching load theory under different test parameters are compared, and it is found that the three are essentially consistent.
For agricultural multi-rotor UAVs, it is of positive significance to explore the change law of aerodynamic performance of the power system with objective operational attributes during operation so as to clarify the change mechanism of the UAV power system.The stable flight of a UASS is the result of the coordinated work of various rotors.In the future, based on the output rotor speed values of the model, by analyzing the impact of flight speed on each rotor speed, it can be explained how a UASS adjusts each rotor's speed to achieve changes in flight speed, whereas the stability and robustness of the UAV control algorithm can be verified in reverse.At the same time, this research can provide a reference for indoor simulation tests UASS and the exploration of the characterization law of the wind field of UASS during operation.
Copyright: © 2024 by the authors.Submitted for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/license s/by/4.0/).

Experimental
Factors Flight Speed (m/s) Initial Payload (Kg) Total Flow Rate (L/min)

Figure 5 .
Figure 5. Thrust fitting and estimation error under different speeds.

Figure 7 .
Figure 7. Model structure of a fully connected neural network.

Figure 8
Figure 8 shows the change in loss rate of the neural network model.Figure 9 is the training result of the neural network.The convergence speed of the training set is faster, the loss rate reaches the lowest value (0.27) in the 147th round, and the overall correlation coefficient (R 2 ) of the training set is 0.728.At the same time, the model performs well on the validation set and the test set, with correlation coefficients (R 2 ) of 0.719 and 0.726, respectively, indicating that the network achieves good fitting performance.With arbitrary real-time load and flight speed as parameters, the model can accurately predict the rotor speed under the current conditions.

Figure 8 .
Figure 8. Change in loss rate.

Figure 9 .
Figure 9.The neural model of the rotor speed.

Figure 10 .
Figure 10.Thrust at different rotor speeds and its matching with the overall UASS load.

Table 1 .
Related parameters of UAV power system test bench.

Table 2 .
TopXGun F16 power system components and corresponding parameters.

Table 4 .
Experimental factors and levels.

Table 6 .
Algorithm development environment and hyperparameter settings.

Table 7 .
Rotor speed-lift fitting parameters of power system.