Quadrotor Unmanned Aerial Vehicle-Mounted Launch Device Precision Analysis for Countering Intruding Drones

Abstract

In light of the current issue of commercial and modified drones frequently intruding into sensitive locations such as airports, this paper proposes the use of quadrotor drones equipped with a launch device for interception. For an “X”-shaped quadrotor drone equipped with a two-degree-of-freedom gimbal launch device, dynamic, control, and ballistic models are constructed to analyze the impact of single-shot firing on the drone’s attitude and the factors affecting accuracy in rapid-fire scenarios. Simulation results indicate that downward firing has the highest shooting accuracy and is suitable for counter-drone missions. For single-shot firing, lateral downward firing compared to frontal firing can effectively reduce attitude and position changes, which is beneficial for engaging stationary targets. In the case of rapid fire, control of the firing interval is crucial for accuracy; a larger firing interval can significantly enhance shooting precision. When firing small payloads in rapid succession, vertical downward firing has the highest accuracy, while lateral firing results in a larger distribution radius of the impact points. To improve shooting accuracy, frontal firing is recommended. Future research will further explore the dynamic response of drones under different firing conditions and develop more advanced control strategies to enhance their practical performance and reliability.

1. Introduction

Multi-rotor unmanned aerial vehicles (UAVs) have been widely applied in many fields due to their technological maturity, cost-effectiveness, and ease of control [1,2,3,4,5,6,7]. However, commercial and modified drones frequently intrude into sensitive areas such as airports and government facilities, posing a serious threat to security. Traditional anti-drone jammers mainly target drones that follow established communication protocols, and their effectiveness is limited against drones with modified communications [8,9]. In such cases, physical interception becomes an effective countermeasure, such as driving away or destroying intruding drones by launching capture nets or interceptor projectiles. If the goal is to capture drones to obtain detailed information, using a net gun to fire a capture net is a more appropriate choice. Capture nets, due to their large size and slow speed, are more suitable for single-shot interception. Conversely, if the objective is to quickly destroy drones, using a shotgun is recommended. As a type of firearm, shotguns have a wide shooting range and are powerful at close range. Modified versions can achieve rapid fire and have a high interception efficiency. This paper discusses equipping a quadrotor drone platform with a variety of launch systems, such as net guns or shotguns, to perform effective anti-drone missions [10,11].

As a launch platform, quadrotor UAVs exist in two modes: single-shot firing and burst firing. When performing single-shot firing, the primary consideration is the stability of the UAV. Numerous scholars at home and abroad have conducted extensive research on a launch device on UAVs to strike targets. A research team from Imperial College London conducted experiments on deploying wireless sensor networks using launch platforms on miniature UAVs, analyzing the deployment accuracy of the launch system for firing 30 g sensors within 4 m [12]. Zhu Cong from North University of China [13] studied the firing characteristics of miniature UAVs, using a balanced launch device to analyze and study the firing accuracy and stability, concluding that a single launch effect is better than three consecutive launches, and increasing the firing interval can improve the launch effect. Ye Bangsong and Fang Tong from Shanghai University of Engineering Science [14,15] proposed a structure and method for an aerodynamic launch device on a miniature UAV, studying the kinematic state of the UAV platform during bullet firing. Chen Jun from Nanjing University of Science and Technology studied the design and fabrication of a net capture system [16], Lu Sihang designed and conducted a mechanical analysis of firearms specifically for small rotary-wing UAVs [17], and Wang Lintao and others researched the launch of special ammunition by UAVs [18]. In these papers, researchers analyzed the impact of the launch device on the drone’s status in both single-shot and burst-firing modes. However, this research was limited to the situation where the launch device was oriented in the direction of the drone’s forward flight and did not address the impact of firing in other attitudes on the drone’s position and posture. To study the effects of the launch device’s different attitudes in the airframe coordinate system on the drone’s position and posture, we proposed a strike model with the launch device located under the dual-axis gimbal of the drone. We analyzed the impact of the launch device on the drone’s position and posture changes at various launch angles, as well as the impact on strike precision during rapid consecutive firing. We hope that this research will help determine the launch angle that has the least impact on the drone’s position and posture after launch, thereby providing theoretical support for the fixation of the launch device.

Due to the advantages of multi-rotor UAVs over fixed-wing UAVs, such as hover capability, easy operation, and lower cost, this paper chooses multi-rotor UAVs as the platform for carrying a launch device for development research. Multi-rotor UAVs are underactuated systems, with six outputs (covering position and attitude) but only four independent inputs (total thrust and three-axis torque). Therefore, multi-rotor UAVs can only track four desired commands. This paper mainly studies the ability of a quadrotor UAV carrying a launch device to maintain stable flight after fixed-point firing, with particular attention paid to the UAV’s position and heading angle control inputs. Although there are various advanced UAV position control algorithms, such as Model Predictive Control (MPC) [19], Active Disturbance Rejection Control (ADRC) [20], adaptive control [21], and sliding mode control [22], these algorithms can respond quickly to environmental disturbances, but the research process is relatively complex. In contrast, this paper uses a relatively simple PID controller for impact resistance research on quadrotor UAVs, because open-source flight control systems tend to use PID controllers with strong reliability and stability and mature technology for UAV control. There are numerous platforms available for UAV dynamic simulation, including the XTDrone platform developed by the National University of Defense Technology, the Rflysim platform developed by the Beijing University of Aeronautics and Astronautics, and MathWorks’ MATLAB 2023b, among others. These simulation platforms can highly realistically simulate real-world flight environments. The key to simulation lies in building UAV models. This paper utilizes MATLAB’s Simulink module to develop a dynamic model of a UAV launch device and integrates it with the UAV model on the Rflysim platform, thereby verifying the changes in UAV posture and projectile trajectory under various firing conditions.

Section 2 establishes a six-degree-of-freedom model for a quadrotor UAV equipped with a launch device, utilizing a PID controller for position and attitude control to derive the ballistic model of the launch device. Section 3 analyzes the UAV’s launch device under single-shot firing with heavy load conditions, examining the relationship between the UAV’s pose changes and the attitude angles of the two-axis gimbaled platform that connects the launch device and the UAV through theoretical analysis and simulation. Section 4 investigates the relationship between the UAV’s pose changes and the attitude angles of the two-axis gimbaled platform, as well as the accuracy of strikes at a fixed distance, under conditions of rapid firing with light load. Section 5 summarizes the impact of the two-axis gimbaled platform’s attitude angles and firing intervals on the UAV’s pose changes and strike accuracy.

2. Establishment of a Quadrotor UAV Model with a Launch Device

In the process of establishing the mathematical model of a quadrotor UAV with a launch device, we employed the Newton–Euler method for dynamic analysis, thereby obtaining the dynamic and control equations of the UAV [1,2]. At the same time, we have also introduced the attitude angle information of the launch device relative to the airframe coordinate system, further studying the impact of firing at different attitudes on the position and posture of the UAV.

2.1. Six-DOF Dynamics Equations of Multi-Rotor UAVs

Multi-rotor UAVs can be equipped with a corresponding launch device as needed, such as crossbows [23], net guns [16], firearms [17], etc. To accurately describe the dynamic characteristics of the UAV, two coordinate systems are introduced in the text: the world coordinate system and the body coordinate system. The body coordinate system is rigidly connected to the UAV body and changes with the UAV’s attitude. The body coordinate system can be transformed into the world coordinate system through Euler angles. The world coordinate system used in the text follows the “North–East–Down” direction, denoted as $x_e,y_e,z_e$, while the body coordinate system follows the “Forward–Right–Down” direction, denoted as $x_b,y_b,z_b$. Both coordinate systems are right-handed coordinate systems.

The positional relationship between the launch device and the UAV is shown in Figure 1, with two scenarios: below [24] and above the UAV. The difference between these two positions is that the recoil after firing affects the attitude of the UAV, primarily the pitch angle, which in turn affects the position of the UAV. The variable η is used to distinguish between these two situations: when the launch device is located below the UAV, η = 1; when the launch device is located above the UAV, η = −1. This paper focuses on the scenario where the launch device is located below the UAV but still takes this factor into account during the model establishment process.

In the design field of quadrotor UAVs, the frame structures are mainly divided into two configurations: “X”-type and “+”-type. For the “+”-type frame, the motor mounting positions are aligned with the $x_b$ and $y_b$ axes of the vehicle coordinate system. In contrast, the motor layout of the “X”-type frame results in the diagonal lines connecting the motors forming a 45° angle with the vehicle coordinate axes $x_b$ and $y_b$.

Considering that the “X”-type frame has become the industry standard due to its superior symmetry and wide range of applications, this paper will adopt the “X”-type frame as the basis for the analysis. The vehicle structure and coordinate system setup of the UAV are shown in Figure 2, which provides a detailed display of the geometric configuration and the corresponding relationships of the coordinate system, offering an accurate reference framework for dynamic analysis and the formulation of control strategies [3,4].

To facilitate the UAV’s targeting and striking of objectives in various directions, as shown in Figure 2, a two-axis gimbaled mount is installed on the UAV body. The launch device is connected to the body through this mount, enabling rotation about the $y_b$ and $z_b$-axes of the vehicle coordinate system. The projection of the launch device axis in plane $x_b{y}_b$ and $x_b$-axis is denoted α, and the angle with the $z_b$-axis is denoted as β. Due to the constraints of the UAV body and the mounting feet, the range of α is from −30° to 30° (positive direction is counterclockwise), and the range of β is from 0° to 120° (positive direction is counterclockwise). When the launch angle β = 0°, the launch direction aligns with the $z_b$-axis of the body, suitable for targeting objects below the UAV; when α = 0°, targets within the $x_b{z}_b$ plane can be struck; when α = 0° and β = 90°, targets directly in front of the UAV can be aimed; when β > 90°, targets above the UAV can be targeted.

After firing, the total mass of the UAV, including the gimbaled mount and the launch device, is m, and the acceleration due to gravity is denoted as g. In the world coordinate system, the position coordinates of the UAV’s center of mass are $p=[\begin{array}{ccc}{p}_x& p_y& p_z{]}^{\mathrm{T}}\end{array}$, and the distance from the motors to the body’s center of mass is d. The coordinates of the desired position are $p_{\mathrm{d}}=[\begin{array}{ccc}{x}_{\mathrm{d}}& y_{\mathrm{d}}& z_{\mathrm{d}}{]}^{\mathrm{T}}\end{array}$, the UAV’s center of mass velocity is $v=[\begin{array}{ccc}{v}_x& v_y& v_z{]}^{\mathrm{T}}\end{array}$, and the UAV’s attitude $\Theta =[\begin{array}{ccc}\varphi & \theta & \psi {]}^{\mathrm{T}}\end{array}$ is represented by Euler angles, ϕ (roll), θ (pitch), and ψ (yaw). In the body coordinate system, the angular velocities around the body coordinate axes are $\omega ={[\begin{array}{ccc}{\omega }_x& {\omega }_y& {\omega }_z\end{array}]}^{\mathrm{T}}$, the lift force magnitude is F, the thrust provided by propellers 1 to 4 is $F_1,F_2,F_3,F_4$, respectively, and the moments about the $x_b,y_b,z_b$ axes are $M_x,M_y,M_z$, with corresponding moments of inertia $I_x,I_y,I_z$. The recoil force of the launch device is $F_s$, with components in the three body coordinate axis directions being $F_{sx},F_{sy},F_{sz}$, and the moments formed by the recoil force on the body are $M_{sx},M_{sy},M_{sz}$. ${[\begin{array}{ccc}{l}_x& l_y& l_z\end{array}]}^{\mathrm{T}}$ denotes the coordinates of the firing point relative to the UAV’s center of mass in the body coordinate system.

The forces and moments exerted by the launch device on the UAV within the body coordinate system are described as follows [15,18]:

$$\left[\begin{array}{c}{F}_{sx}\\ F_{sy}\\ F_{sz}\end{array}\right]=\left[\begin{array}{c}-F_s\cos \alpha \sin \beta \\ F_s\sin \alpha \sin \beta \\ -F_s\cos \beta \end{array}\right],$$

(1)

$$\begin{array}{c}\left[\begin{array}{c}{M}_{sx}\\ M_{sy}\\ M_{sz}\end{array}\right]=\left[\begin{array}{l}{l}_x\\ l_y\\ l_z\end{array}\right]\times \left[\begin{array}{c}{F}_{sx}\\ \eta F_{sy}\\ F_{sz}\end{array}\right]\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} =\left[\begin{array}{l}{l}_y{F}_{sz}-l_z\eta F_{sy}\\ l_z{F}_x-l_x{F}_{sz}\\ l_x\eta F_{sy}-l_y{F}_{sx}\end{array}\right].\end{array}$$

(2)

The kinematic and dynamic models of the UAV are shown as follows:

$$\dot{p}=v,$$

(3)

$$\left[\begin{array}{c}m{\dot{v}}_x\\ m{\dot{v}}_y\\ m{\dot{v}}_z\end{array}\right]=\left[\begin{array}{c}0\\ 0\\ mg\end{array}\right]-R_b^e\left[\begin{array}{c}{F}_{sx}\\ F_{sy}\\ F+F_{sz}\end{array}\right].$$

(4)

In the equations, m is the mass of the UAV, g is the acceleration due to gravity, F is the resultant force of the thrust from the four propellers, with the downward direction defined as the positive direction, and $R_b^e$ is the rotation matrix from the body coordinate system to the world coordinate system, expressed as follows:

$$R_b^e=R_z^{\mathrm{T}}(\psi )R_y^{\mathrm{T}}(\theta )R_x^{\mathrm{T}}(\varphi ).$$

(5)

In which, $R_x,R_y,R_z$ are defined as follows:

$$\left\{\begin{array}{l}{R}_x(\varphi )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \varphi & \sin \varphi \\ 0& -\sin \varphi & \cos \varphi \end{array}\right]\\ R_y(\theta )=\left[\begin{array}{ccc}\cos \theta & 0& -\sin \theta \\ 0& 1& 0\\ \sin \theta & 0& \cos \theta \end{array}\right]\\ R_z(\psi )=\left[\begin{array}{ccc}\cos \psi & \sin \psi & 0\\ -\sin \psi & \cos \psi & 0\\ 0& 0& 1\end{array}\right]\end{array}\right.$$

(6)

The relationship between the UAV’s attitude rate of change and the body angular velocity is shown as follows:

$$\left[\begin{array}{c}\dot{\varphi }\\ \dot{\theta }\\ \dot{\psi }\end{array}\right]=\left[\begin{array}{ccc}1& \tan \theta \sin \varphi & \tan \theta \cos \varphi \\ 0& \cos \varphi & -\sin \varphi \\ 0& \sin \varphi /\cos \theta & \cos \varphi /\sin \theta \end{array}\right]\left[\begin{array}{c}{\omega }_x\\ {\omega }_y\\ {\omega }_z\end{array}\right],$$

(7)

$$\left[\begin{array}{c}{I}_x{\dot{\omega }}_x\\ I_y{\dot{\omega }}_y\\ I_z{\dot{\omega }}_z\end{array}\right]=\left[\begin{array}{c}{M}_x\\ M_y\\ M_z\end{array}\right]-\left[\begin{array}{c}{\omega }_x{\omega }_y\\ {\omega }_y{\omega }_z\\ {\omega }_x{\omega }_z\end{array}\right]\times \left[\begin{array}{c}{I}_x\\ I_y\\ I_z\end{array}\right]+\left[\begin{array}{c}{M}_{sx}\\ M_{sy}\\ M_{sz}\end{array}\right].$$

(8)

The lift force and torque experienced by the UAV have a specific relationship with the thrust provided by each propeller. The rotor radius of the drone is d. The distance between the propeller’s center and the drone’s center of mass on the $z_e$-axis is $l$, where $l=c_M/c_T$, $c_M$ and $c_T$ are the torque coefficient and thrust coefficient, respectively, both of which can be measured experimentally. The specific expressions for the forces and torques on the UAV are as shown as follows [1].

$$\left\{\begin{array}{l}F=F_1+F_2+F_3+F_4\\ M_x={\displaystyle \frac{\sqrt{2}}{2}}d(F_1-F_2-F_3+F_4)\\ M_y={\displaystyle \frac{\sqrt{2}}{2}}d(F_1+F_2-F_3-F_4)\\ M_z=l(F_1-F_2+F_3-F_4)\end{array}\right.$$

(9)

in which $F_i$ represents the lift force of the i-th propeller, where i = 1, …, 4.

2.2. UAV Low-Level Control

The fully autonomous control closed-loop diagram of the multi-rotor UAV is shown in Figure 3, which is divided into inner-loop control and outer-loop control. The inner-loop controller is the attitude controller, and the outer-loop controller is the position controller. Both the attitude and position control of the UAV adopt a PID control strategy to achieve point control. The PID parameters are obtained using the MATLAB “PID Autotuning for UAV Quadcopter” module [25], and these parameters do not consider the disturbance of the recoil force from the launch device. Detailed PID parameters can be found in

Table 1. UAV parameter specification table.

Category	Parameter	Value
Intrinsic physical parameters	The total mass of the UAV with the launch device, m (kg)	7.5
	Gravity acceleration g (m/s2)	9.8
	Moments of inertia of the UAV with launch device ($I_x,I_y,I_z$) (kg·m2)	(0.1392, 0.1392, 0.28)
	Distance from the center of mass to the recoil force application point ($l_x,l_y,l_z$) (m)	(0, 0, 0.1)
Initial state	Initial position ($p_x,p_y,p_z$) (m)	(0, 0, 0)
	Initial velocity ($v_x,v_y,v_z$) (m/s)	(0, 0, 0)
	Initial Euler angles (ϕ, θ, ψ) (°)	(0, 0, 0)
	Initial angular velocities (${\omega }_x,{\omega }_y,{\omega }_z$) (rad/s)	(0, 0, 0)
Initial input parameter magnitude	Lift force F (N)	0
	Torque ($M_x,M_y,M_z$) (N·m)	(0, 0, 0)
Parameter output limit range	Output lift parameter $U_1$ range (min, max)	(0, 250)
	Output torque parameter $U_2~U_4$ range (min, max)	(−50, 50)
Parameter binding	Position proportional coefficients $(k_{p_x\mathrm{p}},k_{p_y\mathrm{p}},k_{p_z\mathrm{p}})$	(40, 40, 40)
	Position differential coefficients $(k_{p_x\mathrm{d}},k_{p_y\mathrm{d}},k_{p_z\mathrm{d}})$	(0, 0, 0)
	Position integral coefficients $(k_{p_x\mathrm{i}},k_{p_y\mathrm{i}},k_{p_z\mathrm{i}})$	(25, 25, 25)
	Attitude proportional coefficients $(k_{\varphi \mathrm{p}},k_{\theta \mathrm{p}},k_{\psi \mathrm{p}})$	(5, 5, 5)
	Attitude differential coefficients $(k_{\varphi \mathrm{d}},k_{\theta \mathrm{d}},k_{\psi \mathrm{d}})$	(0, 0, 0)
	Attitude integral coefficients $(k_{\varphi \mathrm{i}},k_{\theta \mathrm{i}},k_{\psi \mathrm{i}})$	(1, 1, 1)
	Desired position $(x_{\mathrm{d}},y_{\mathrm{d}},z_{\mathrm{d}})$ (m)	(0, 0, 5)

. Since the quadrotor UAV is an underactuated system [17,18,19,20], it can only control four independent variables. To facilitate aiming, the four controlled variables include three desired position coordinates $p_{\mathrm{d}}$ and one desired yaw angle ${\psi }_{\mathrm{d}}$. The actual position coordinates of the UAV are $p$, and the error between the desired position and the actual position is represented as $p_e=[\begin{array}{ccc}{x}_e& y_e& z_e{]}^{\mathrm{T}}\end{array}$; the actual attitude Euler angles of the UAV are $\Theta$, and the error between the desired attitude angles and the actual attitude angles is represented as ${\Theta }_e=[\begin{array}{ccc}{\varphi }_e& {\theta }_e& {\psi }_e{]}^{\mathrm{T}}\end{array}$.

Firstly, through the position control loop, using the desired position coordinates $p_{\mathrm{d}}$ in conjunction with the PID control algorithm, the control inputs $U_1,U_x,U_y$ can be calculated. Subsequently, the desired Euler angles ${\varphi }_{\mathrm{d}},{\theta }_{\mathrm{d}}$ are calculated using the desired horizontal position coordinates $x_{\mathrm{d}},y_{\mathrm{d}}$. The specific calculation method for the entire process is shown as follows:

$$\left\{\begin{array}{c}{x}_e=x_{\mathrm{d}}-p_x\\ \begin{array}{l}{y}_e=y_{\mathrm{d}}-p_y\\ z_e=z_{\mathrm{d}}-p_z\end{array}\end{array}\right.$$

(10)

$$\left\{\begin{array}{l}{U}_1=k_{p_z\mathrm{p}}{z}_e+k_{p_z\mathrm{i}}{\displaystyle {\int }_0^t{z}_e\mathrm{d}t+k_{p_z\mathrm{d}}{\dot{z}}_e}\\ U_x=k_{p_x\mathrm{p}}{x}_e+k_{p_x\mathrm{i}}{\displaystyle {\int }_0^t{x}_e\mathrm{d}t+k_{p_x\mathrm{d}}{\dot{x}}_e}\\ U_y=k_{p_y\mathrm{p}}{y}_e+k_{p_y\mathrm{i}}{\displaystyle {\int }_0^t{y}_e\mathrm{d}t+k_{p_y\mathrm{d}}{\dot{y}}_e}\end{array}\right.$$

(11)

$$\left[\begin{array}{c}{\varphi }_{\mathrm{d}}\\ {\theta }_{\mathrm{d}}\end{array}\right]=-{\displaystyle \frac{1}{(F+F_{sz})}}\left[\begin{array}{cc}\sin \psi & -\cos \psi \\ \cos \psi & \sin \psi \end{array}\right]\left[\begin{array}{c}{U}_x-F_{sx}\\ U_y-F_{sy}\end{array}\right].$$

(12)

PID related parameter descriptions are provided in Table 1.

Through the attitude control loop, based on the desired Euler angles ${\varphi }_{\mathrm{d}}$ and ${\theta }_{\mathrm{d}}$ calculated by the position control loop, and combining with the given desired yaw angle ${\psi }_{\mathrm{d}}$, the errors between the desired attitude angles and the actual attitude angles of the UAV can be determined as follows

$$\left\{\begin{array}{c}{\varphi }_e={\varphi }_{\mathrm{d}}-\varphi \\ {\theta }_e={\theta }_{\mathrm{d}}-\theta \\ {\psi }_e={\psi }_{\mathrm{d}}-\psi \end{array}\right.$$

(13)

Using these error values, along with the PID control algorithm, the control inputs $U_2,U_3,U_4$ can be further calculated.

$$\left\{\begin{array}{l}{U}_1=F\hfill \\ U_2=M_x=k_{\varphi \mathrm{p}}{\varphi }_e+k_{\varphi \mathrm{i}}{\displaystyle {\int }_0^t{\varphi }_e\mathrm{d}t+k_{\varphi \mathrm{d}}{\dot{\varphi }}_e}\hfill \\ U_3=M_y=k_{\theta \mathrm{p}}{\theta }_e+k_{\theta \mathrm{i}}{\displaystyle {\int }_0^t{\theta }_e\mathrm{d}t+k_{\theta \mathrm{d}}{\dot{\theta }}_e}\hfill \\ U_4=M_z=k_{\psi \mathrm{p}}{\psi }_e+k_{\psi \mathrm{i}}{\displaystyle {\int }_0^t{\psi }_e\mathrm{d}t+k_{\psi \mathrm{d}}{\dot{\psi }}_e}\hfill \end{array}\right.$$

(14)

The output values $U_1,U_2,U_3,U_4$ can be used as the basis for controlling the lift force F and the moments $M_x,M_y,M_z$, along with the recoil force $F_s$ as input parameters substituted into dynamic Equations (3)–(9). Through the calculations of these equations, the real-time position and attitude of the UAV, $p$ and $\Theta$, can be obtained. Utilizing these equations allows for the determination of the real-time position and attitude of the quadrotor UAV when affected by the recoil force.

2.3. Ballistic Equations of the Launch Device

If it can be considered that the direction of the firing point is the same as the direction of the recoil force, then the position of the UAV’s firing point ${[\begin{array}{ccc}{p}_{lx}& p_{ly}& p_{lz}\end{array}]}^{\mathrm{T}}$ can be calculated.

$$\left[\begin{array}{c}{p}_{lx}\\ p_{ly}\\ p_{lz}\end{array}\right]=R_b^e\left[\begin{array}{c}{l}_x\\ l_y\\ l_z\end{array}\right]+\left[\begin{array}{c}{p}_x\\ p_y\\ p_z\end{array}\right].$$

(15)

In the global coordinate system, the initial direction of the projectile -(for ease of description, this paper collectively refers to the objects launched by the launch device as projectiles, including arrows, crossbow bolts, bullets, shotgun shells, capture nets, etc.) -when fired can be represented using ${[\begin{array}{ccc}{K}_x& K_y& K_z\end{array}]}^{\mathrm{T}}$.

$$\left[\begin{array}{c}{K}_x\\ K_y\\ K_z\end{array}\right]=R_b^e\left[\begin{array}{c}-\cos \alpha \sin \beta \\ \sin \alpha \sin \beta \\ -\cos \beta \end{array}\right].$$

(16)

To facilitate the analysis of the projectile’s point of impact accuracy, we have simplified the analysis by neglecting the effect of gravity on the projectile, which is reasonable for projectiles with high initial velocities when conducting short-range accuracy analysis [26,27]. Therefore, we assume the trajectory of the projectile from the launch device to be a straight line, and the point of impact at $x_e$ is given as follows:

$$\left\{\begin{array}{c}{y}_e={\displaystyle \frac{K_y(x_e-p_{l,x})}{K_x}}+p_{l,y}\\ z_e={\displaystyle \frac{K_z(x_e-p_{l,x})}{K_x}}+p_{l,z}\end{array}\right.$$

(17)

When the x-coordinate of the desired point is equal to $x_e$, we can determine that the point of impact lies on the plane where $x_e=x$.

2.4. Model Assumptions

For the sake of computational simplicity, the following assumptions are made:

• The weight of the UAV remains constant after firing;
• During firing, the entire UAV, gimbaled mount, and payload are considered as a single rigid body;
• The motion of the UAV is not affected by other disturbances;
• The effects of the Earth’s rotation and revolution are neglected;
• The safe range of UAV attitude angle changes is limited to between 0° and 55°;
• Within the safe range of UAV attitude angle changes, the effect of air resistance is ignored;
• Due to the short shooting distance, the projectile’s trajectory is assumed to be a straight line, and the projectile’s initial velocity direction is consistent with the aiming and firing direction, unaffected by the UAV body and gimbaled mount.

2.5. Input Parameter Settings

The parameter settings for the UAV are detailed in Table 1. The table lists the following parameters of the UAV: inherent physical parameters, initial state parameters, initial input parameters, output limit range parameters, and set parameters.

The total calculation time for the entire motion process of the UAV is set to 20 s. The desired height of the UAV is set to 5 m. At 5 s into the flight, the launch device begins firing. The variation in the recoil force Fs experienced by the launch device over time is detailed in

Table 2. Recoil force F_s variation data of the launch device during firing.

Single-shot firing	Time (s)	0.000	0.001	0.002	0.003	0.004	0.005	0.006
	Recoil force (N)	0	2500	3200	2500	1500	1000	500
	Time (s)	0.007	0.008	0.009	0.010	0.011	0.012	0.013
	Recoil force (N)	400	300	250	200	200	180	150
	Time (s)	0.014	0.015	0.016	0.017	0.018	0.0019	0.0020
	Recoil force (N)	140	130	120	110	100	50	0
Burst firing	Time (s)	0.000	0.001	0.002	0.003	0.004	0.005	0.006
	Recoil force (N)	0	800	1600	800	400	100	50
	Time (s)	0.007	0.008	0.009	0.010	0.011	0.012	0.013
	Recoil force (N)	40	30	25	20	20	18	15
	Time (s)	0.014	0.015	0.016	0.017	0.018	0.0019	0.0020
	Recoil force (N)	14	13	12	11	10	5	0

and shown in Figure 4 and Figure 5: when the UAV fires a large payload in a single shot, these data simulate the scenario of firing a rocket projectile or a capture net, with a total impulse of 13.53 N·s (see Figure 4); when the UAV fires a small payload in a burst, the total impulse per shot is 3.983 N·s, and the total impulse for 30 shots in a burst is 119.49 N·s (see Figure 5).

3. Analysis of UAV Single-Shot Firing with Heavy Payload

This section begins by analyzing, through simulation, the impact of the two-axis gimbaled platform connecting the UAV and the launch device on the UAV’s attitude and torque changes during heavy-load single-shot firing at various orientations. Subsequently, the analysis examines the effects of firing on the UAV’s position and lift force.

3.1. Patterns of UAV Attitude and Torque Changes During Single-Shot Firing

In the MATLAB Simulink environment, a nonlinear model of the quadrotor UAV and a recoil model of the launch device are constructed [28]. The recoil force of the launch device is used as a disturbance input to the UAV’s six-degree-of-freedom motion module. The UAV uses a PID controller to perform closed-loop control on its position and attitude based on the given desired positions and heading angles. Calculations reveal that when the $z_b$-axis gimbaled mount remains stationary (i.e., α = 0°) and fires forward, as shown in Figure 6 and Figure 7, the roll angle ϕ and yaw angle ψ do not change, and the UAV’s control torques $M_x$ and $M_z$ are also 0. However, when the $z_b$-axis gimbaled mount rotates for lateral firing (i.e., α ≠ 0°, β ≠ 0°), the roll angle ϕ and yaw angle ψ will need to be adjusted. In particular, when the $z_b$-axis gimbaled mount fires in the opposite direction (α is 30° and −30°, respectively), the adjustment directions of the roll angle ϕ and yaw angle ψ are also opposite.

As can be seen from Figure 6 and Figure 7, and

Table 3. Range of attitude angle variation in UAV launch device under single-shot conditions at different shooting angles.

Parameter	Value
Launch device gimbal angles [α, β] (°)	[0, 0]	[0, 45]	[0, 90]	[0, 120]	[30, 45]	[−30, 45]
Roll ϕ (min, max) (°)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−37.37, 29.52)	(−31.05, 37.7)
Pitch θ (min, max) (°)	(0, 0)	(−50.55, 37.65)	(−63.9, 43.61)	(−56.66, 41.48)	(−48.09, 37.94)	(−44.35, 35.96)
Yaw ψ (min, max) (°)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−5.28, 9.22)	(−10.17, 4.32)
Mx (min, max) (N·m)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−50, 15.5)	(−15.48, 50)
My (min, max) (N·m)	(0, 0)	(−50, 20.34)	(−50, 32.58)	(−50, 33.1)	(−50, 20.4)	(−50, 50)
Mz (min, max) (N·m)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−1.97, 2.53)	(−1.6, 2.09)

, when the $z_b$-axis gimbaled mount rotates for lateral firing (α ≠ 0°, β ≠ 0°), the range of motion of the roll angle ϕ is greater than that of the yaw angle ψ (for example, when α = 30° and β = 45°, the variation range of ψ can reach 37.37°, while the variation range of ψ does not exceed 9.22°). Correspondingly, the UAV’s control torque $M_x$ is also greater than $M_z$.

As shown in Figure 8 and Table 3, except for firing vertically towards the ground (gimbal firing angle α = 0°, β = 0°), when firing at any other angle, the UAV’s pitch angle θ will change, as will the UAV’s control torque $M_y$. Under the influence of the recoil force, the pitch angle θ is initially negative, with its maximum and minimum values having a small difference in magnitude. When firing in the horizontal direction (α = 0°, β = 90°), the pitch angle’s range of change is the greatest (θ_min = −63.089°, θ_max = 43.614°). This is due to the maximum torque exerted on the pitch angle by the recoil force. At this point, the attitude angle exceeds 55°, and the UAV may crash due to excessive attitude. It is recommended that the UAV should fire downwards (β < 90°).

3.2. UAV Position and Lift Force Change Patterns

From Figure 9, it can be observed that when the gimbaled mount rotates around the $y_b$-axis and fires in the positive direction of the $x_e$-axis, except for the case of firing straight down (i.e., β = 0°), in other cases, the UAV will initially move in the negative direction of the $x_e$-axis under the influence of the recoil force and then gradually adjust to a balanced position. Particularly when the gimbaled mount on the $y_b$-axis is rotated to a larger angle (β > 45°) for firing, the UAV will first move in the negative direction of the $x_e$-axis and then adjust to the positive direction of the $x_e$-axis. According to the data in

Table 4. Range of position changes in UAV launch device under different angle shooting.

Launch Device Gimbal Angles [α, β] (°)	[0, 0]	[0, 45]	[0, 90]	[0, 120]	[30, 45]	[−30, 45]
$p_x$ (min, max) (m)	(−0.241, 0.009)	(−0.241, 0.009)	(−0.283, 0.09)	(−0.211, 0.129)	(−0.206, 0.009)	(−0.222, 0.009)
$p_y$ (min, max) (m)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−0.015, 0.105)	(−0.105, 0.013)
$p_z$ (min, max) + 5 (m)	(−0.353, 0.013)	(−0.189, 0.123)	(0, 0.364)	(0, 0.428)	(−0.182, 0.22)	(−0.187, 0.216)
F (min, max) (N)	(29.367, 76.452)	(41.997, 83.141)	(72.917, 101.359)	(73.48, 106.787)	(41.975, 87.744)	(42.026, 87.304)

, as the rotation angle of the gimbaled mount on the $y_b$-axis increases, the distance the UAV moves in the positive direction of the $x_e$-axis also increases. For example, when the rotation angle of the gimbaled mount is α = 0°, β = 120°, the UAV moves almost entirely in the positive direction of the $x_e$-axis by 0.129 m.

From Figure 10, it can be seen that when the gimbaled mount on the $z_b$-axis rotates for firing (⍺ ≠ 0°), lateral displacement occurs under the influence of the recoil force, with the displacement direction being opposite to the firing direction.

From Figure 11 and Figure 12, it can be observed that when the gimbaled mount rotates around the $y_b$-axis and fires in the direction of the ground (β ≤ 45°), the lift force F required by the UAV decreases, causing the UAV to rise and move in the direction of the sky (negative direction of the $z_e$-axis), while when β > 45°, the lift force required by the UAV increases. According to the data in Table 4, when α = 0°, β = 0°, F_max = 76.452 N; when α = 0°, β = 120°, F_max = 106.787 N. In this case, the UAV’s height above the ground decreases, moving in the direction of the ground (positive direction of the $z_e$-axis).

It is known from Table 4 that when the rotation angle β of the gimbaled mount on the $y_b$-axis remains constant, the lift force required also increases with the rotation angle α of the gimbaled mount on the $z_b$-axis. For example, when α = 0°, β = 45°, F_max = 83.141 N, and when α = 30°, β = 45°, F_max = 87.744 N. Therefore, when determining the lift force parameters of the UAV, it should be set according to the maximum rotation angles (α_max, β_max).

Unlike the movement of a ground-based launch device after firing, the UAV, in most cases after firing, experiences greater changes in the vertical direction than in the horizontal direction. For example, when firing straight down, the UAV’s upward movement can reach 0.353 m, while when the firing angle is α = 0°, β = 120°, the UAV descends by 0.428 m compared to a maximum horizontal change of only 0.211 m.

This difference is due to the change in the UAV’s attitude angles caused by the recoil force during firing, leading to a change in the direction of the lift force. In the vertical direction, the recoil force disrupts the original equilibrium, while in the horizontal direction, the component of the lift force is opposite to the direction of the recoil force, which can offset part of the recoil force, resulting in less horizontal displacement. Therefore, it is necessary to ensure that there are no obstacles around after firing, especially to pay attention to safety in the vertical direction.

It can also be observed that when the launch device fires downward (β = 45°), compared to lateral firing (α = ±30°), the change in attitude angles is smaller. Therefore, lateral downward firing during single-shot firing helps to reduce the change in the UAV’s attitude angles. Using gimbaled angles of α = 30° and β = 45° can reduce the movement amplitude of the UAV’s attitude and position, thereby reducing the risk of the UAV crashing or colliding.

4. Analysis of UAV Burst Firing with Light Payload

In this section, we first analyze the impact of the two-axis gimbaled platform connecting the UAV and the launch device on the UAV’s attitude and torque changes during low-load rapid firing through simulation. Subsequently, we discuss the effects of firing on the UAV’s position and lift force. Finally, we investigate the accuracy of continuous firing at a fixed distance.

4.1. Patterns of UAV Attitude and Torque Changes During Burst Firing

The UAV fires 30 rounds of ammunition at the 5th second and ceases firing at the 8th second. Figure 13 illustrates the relationship between the UAV’s attitude and time during the 30 round burst, where the horizontal axis represents time and the vertical axis represents the UAV’s three attitude angles. Figure 14 shows the UAV’s attitude during the 5th to 8th second burst firing, with the horizontal axis indicating the round number (1–30) and the vertical axis indicating the UAV’s attitude angle at that time.

Table 5. Range of UAV attitude changes (°) during continuous firing at different angles of the launch device.

Gimbal Angles [α, β] (°)		[0, 0]	[0, 30]	[0, 45]	[0, 90]	[0, 120]	[30, 45]
Roll ϕ (min, max)	t = 0–20 s	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−62.01, 22.28)
	t = 5–8 s *	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−67.25, 37.15)
Pitch θ (min, max)	t = 0–20 s	(0, 0)	(−42.07, 19.9)	(−52.34, 33.54)	(−70.08, 33.02)	(−67.8, 42.71)	(−67.25, 37.15)
	t = 5–8 s *	(0, 0)	(−41.98, 5.78)	(−50.06, 3.42)	(−65.15, 0)	(−61.49, 7.83)	(−61.64, 23.28)
Yaw Ψ (min, max)	t = 0–20 s	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−12.44, 23.57)
	t = 5–8 s *	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−11.86, 22.89)

* Firing.

lists the maximum and minimum values of the UAV’s attitude angles from 0 to 20 s and the maximum and minimum values of the UAV’s attitude angles during the 5–8 s ammunition firing. In the table, a negative sign indicates the opposite direction of a positive sign.

From Figure 13, it can be seen that when the $z_b$-axis gimbaled angle is α = 0° and firing is carried out in the forward direction, there will be no change in the roll angle ϕ and yaw angle ψ. Adjustments to the roll angle ϕ and yaw angle ψ only occur when the $z_b$-axis gimbaled mount is rotated for lateral firing (α ≠ 0° and β ≠ 0°).

For the roll angle ϕ, no change occurs during forward firing. However, during lateral firing, influenced by the recoil force, the roll angle exhibits significant fluctuations near ϕ = 0° during the launch process (5–8 s) but eventually stabilizes.

For the pitch angle θ, among the three attitude angles, it has the largest range of variation. During the firing process (5–8 s), regardless of forward or lateral firing, the pitch angle fluctuates around θ = −25°. After the drone completes a burst of fire (8–20 s), the pitch angle fluctuates around θ = 0° and then gradually adjusts back to the initial attitude (θ = 0°). Throughout the entire process (0–20 s), according to Table 5, when α = 0°, β = 30°, (θ_min, θ_max) = (−42.07°, 19.9°); when α = 0°, β = 90°, (θ_min, θ_max) = (−70.08°, 33.02°). It can be observed that the closer the $y_b$-axis gimbal angle β is to 90°, the greater the fluctuation amplitude of the drone. When the firing angles are α = 0°, β = 90° and α = 0°, β = 120°, the absolute value of the pitch angle θ exceeds 55°, surpassing the UAV’s limited attitude angle, making the UAV prone to crashing. When β = 45°, for forward firing (α = 0°), (θ_min, θ_max) = (−52.34°, 33.54°), while for lateral firing (α = 30°), (θ_min, θ_max) = (−67.25°, 37.15°). It is evident that the pitch angle θ fluctuates more during lateral firing than during forward firing, with the absolute value exceeding 55°, making the UAV prone to crashing.

For the yaw angle ψ, no change occurs during forward firing, and during lateral firing, the yaw angle oscillates around the initial attitude (ψ = 0°).

From Table 5 and

Table 6. Range of UAV position changes during continuous firing at different angles.

Gimbal Angles [α, β] (°)		[0, 0]	[0, 30]	[0, 45]	[0, 90]	[0, 120]	[30, 45]
$p_x$ (min, max) (m) 1	t = 0–20 s	(0, 0)	(−0.085, 0.007)	(−0.088, 0.103)	(−0.165, 0.048)	(−0.353, 0.05)	(−0.302, 0.505)
	t = 5–8 s *	(0, 0)	(−0.085, 0.006)	(−0.087, 0.011)	(−0.159, 0.047)	(−0.245, 0.055)	(−0.302, 0.505)
$p_y$ (min, max) (m) 2	t = 0–20 s	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−0.302, 0.034)
	t = 5–8 s *	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(0, 0)	(−0.301, 0.032)
$p_z+5$ (min, max) (m) 3	t = 0–20 s	(−1.025, 0.037)	(−0.891, 0.067)	(−0.777, 0.103)	(−0.138, 0.112)	(0, 0.515)	(−0.551, 0.171)
	t = 5–8 s *	(−1.019, 0.001)	(−0.884, 0.001)	(−0.773, 0.001)	(−0.133, 0.001)	(0.001, 0.513)	(−0.548, 0.001)

* Firing. ¹ UAV $x_e$ direction coordinate change range. ² UAV $y_e$ direction coordinate change range. ³ UAV $z_e$ direction coordinate change range (adjusted relative to the zero point).

, it can also be observed that the maximum absolute values of the three attitude angles and three position changes in the UAV during the burst firing (5–8 s) are essentially consistent with the maximum absolute values throughout the entire process (0–20 s). That is, when the UAV’s attitude angles and positions are at their maximum absolute values, the UAV is firing or has just finished firing, and due to inertia, the position and attitude angles will continue to increase slightly.

4.2. Patterns of UAV Position and Lift Force Changes

Figure 15 illustrates the relationship between the UAV’s position and time over 0–20 s, where the horizontal axis represents time and the vertical axis represents the UAV’s displacement in three directions. Figure 16 shows the position of the UAV during the firing of 30 rounds between the 5th and 8th seconds, with the horizontal axis representing the round number (1–30) and the vertical axis representing the UAV’s position at the time of firing. Table 6 lists the maximum and minimum values of the UAV’s position changes over 0–20 s, as well as the maximum and minimum values during the 5–8 s of projectile firing, with negative signs indicating the opposite direction.

From Figure 15, it can be observed that when the $z_b$-axis gimbaled angle α is 0° and firing is carried out in the forward direction, the UAV’s center of mass only experiences displacement in the vertical $z_e$ and horizontal $x_e$ directions. When the $z_b$-axis gimbaled mount is rotated for lateral firing (α ≠ 0°, β ≠ 0°), it causes displacement in the $x_b,y_b,z_b$ directions.

For the horizontal $x_e$ direction, during the firing process (5–8 s), under the influence of the recoil force, the UAV’s center of mass fluctuates around $p_x$ ≈ −0.1 m from the initial position ($p_x$ = 0 m). After the firing is completed at the 8th second, the UAV’s position in the $x_e$ direction fluctuates around $p_x$ ≈ 0 m and tends to stabilize. Specifically, between 5 and 8 s, according to Table 6, when α = 0°, β = 30°, there are $p_x$ (min, max) = (−0.085, 0.006) m; when α = 0°, β = 120°, there are $p_x$ (min, max) = (−0.245, 0.055) m, which is 3.3 times the former. It can be seen that the larger the $y_b$-axis gimbaled firing angle β, the greater the fluctuation amplitude of the UAV in the $x_e$ direction, mainly in the negative direction of the $x_e$-axis, and the farther the fluctuation center value is from $p_x$ = 0 m. When β = 45°, for forward firing (α = 0°), $p_x$ (min, max) = (−0.0873, 0.011) m; for lateral firing (α = 30°), $p_x$ (min, max) = (−0.302, 0.505) m, which is 8.2 times the former. It can be seen that the UAV’s position fluctuation amplitude in the $x_e$ direction is greater during lateral firing than during forward firing.

For the horizontal $y_e$ direction, forward firing does not produce changes, but during lateral firing, the position in the $y_e$ direction continuously increases. When α = 30°, β = 45°, $p_y$ (min, max) = (0, 1.501) m; at this time, the absolute value of the coordinate change in the $y_e$ direction exceeds 1 m, and the UAV is prone to crash by touching other obstacles.

For the vertical $z_e$ direction, during the firing process (5–8 s), under the influence of the recoil force, the UAV will rise or fall from the initial position ($p_z$ = −5 m) and tend to stabilize. After the firing is completed at the 8th second, the UAV’s center of mass in the $z_e$ direction will return to $p_z$ = −5 m and tend to stabilize. Specifically, according to Table 6, when α = 0°, β = 0°, there are $p_z+5$ (min, max) = (−1.019, 0.001) m; when α = 0°, β = 90°, there are $p_z+5$ (min, max) = (−0.133, 0.001) m. It can be seen that when the $y_b$-axis gimbaled firing angle β is within the range of (0°, 90°), the UAV’s center of mass will rise under the influence of the recoil force, and the smaller β is, the greater the rise amplitude in the height direction. When α = 0°, β = 0°, the absolute value of the coordinate change in the $y_e$ direction exceeds 1 m, and the UAV is prone to crash by touching other obstacles.

Similarly, it can be known that when the $y_b$-axis gimbaled angle β is within the range of (90°, 120°), between 5 and 8 s, the UAV’s center of mass will fall under the influence of the recoil force, and the greater the value of β is, the greater the fall amplitude in the height direction. When β = 45°, for forward firing (α = 0°), $p_z+5$ (min, max) = (−0.773, 0.001) m; for lateral firing (α = 30°), $p_z+5$ (min, max) = (−0.548, 0.001) m. It can be seen that the UAV’s rise amplitude in the height direction is slightly smaller during lateral firing than during forward firing.

Under the given parameters, whether forward or lateral firing, when firing 15 rounds, the UAV tends to stabilize, with almost no fluctuations, achieving dynamic balance.

4.3. Strike Accuracy at a Fixed Distance

For the target located at $x_e$ = 5 m, when the projectile firing interval is 0.1 s, at different shooting angle conditions, the positional differences between the 30th shot and the 1st shot in the $y_e$ and $z_e$ directions are shown in Figure 17. It can be observed from the figure that lateral shooting produces errors in both the $y_e$ and $z_e$ directions; vertical downward shooting has the highest precision. When shooting at angles from 0° to 90° and 0° to 120°, the dispersion is within 10 m, while dispersion from other angles exceeds 100 m.

When the shooting interval is extended from 0.1 s to 1 s and 2 s, Figure 18 and

Table 7. Maximum position distribution (m) of 30 rounds relative to the first found during burst firing at intervals of 0.1 s, 1 s, and 2 s at different angles.

Firing Interval (s)	Gimbal Angles [α, β] (°)
	[0, 0] 1	[0, 30] 1	[0, 45] 1	[0, 90] 1	[0, 120] 1	[30, 45] 1	[30, 45] 2
0.1	0	298.8	−560.5	9.460	5.982	936.0	−353.1
1.0	0	9.921	3.657	−1.743	−3.530	4.519	−1.450
2.0	0	1.606	1.650	−1.417	1.459	4.069	−1.276

¹ The maximum deviation from the first bullet in the $z_e$ direction. ² The maximum deviation from the first bullet in the $y_e$ direction.

illustrate the positional difference between the 30th shot and the 1st shot in the $z_e$ direction (shooting errors exceeding 50 m are considered meaningless and thus not displayed in Figure 18). The charts show that when the shooting interval is extended to 1 s, the maximum dispersion in the $z_e$ direction for shots taken in various attitudes is less than 10 m; when the shooting interval is extended to 2 s, the maximum dispersion in the $z_e$ direction for shots taken in various attitudes is less than 5 m. Under different shooting intervals, the dispersion effect is optimal when the shooting angles are between 0° and 90° and between 0° and 120°.

5. Conclusions

This study, focusing on the “X”-type quadrotor UAV equipped with a two-degree-of-freedom gimbaled launch device, established dynamic models, control models, and ballistic models to analyze the impact of firing actions on the UAV’s flight status and the probability of hitting the target. The main findings are as follows:

• Whether it is single-shot or continuous firing, the recoil torque after firing causes significant changes in the UAV’s pitch angle, which fluctuates during shooting. When firing horizontally, the firing torque is the greatest and may exceed the allowed angle of the UAV, leading to a crash. Therefore, this angle is suitable for smaller launch loads. For larger loads, it is recommended to fire downwards to reduce the amplitude of attitude changes. At the same time, changes in the UAV’s attitude angles during firing will cause changes in the direction of lift force, leading to greater vertical position changes than horizontal ones. Attention should be paid to the altitude direction after firing to avoid collisions with obstacles.
• When firing a large payload in a single shot, the analysis of six different launch angles revealed that the precision is highest when firing vertically downward (gimbal angles α = 0°, β = 0°), which can be used for counter-UAV purposes. Firing a capture net at a target UAV from above can effectively hit the target. When firing laterally downward (gimbal angles α = 30°, β = 45°), the torque is minimized, and the UAV’s attitude and position changes after firing can be effectively reduced.
• When the UAV fires a small payload in a burst, vertical downward firing also has the highest precision. In open areas, if the UAV’s maneuvering speed is fast, or the target is slow, a downward attack can be adopted. Unlike single-shot firing, the error in lateral firing during burst firing is larger, and it is not recommended. If the UAV needs to improve the precision of burst firing without a gimbaled mount, for targets ahead, the firing device can be installed at angles α = 0°, β = 90°; for targets in a narrow working area below, the firing device can be installed at angles α = 0°, β = 45°. Finally, controlling the burst interval is a key factor in controlling precision. When the interval between bursts is too short, precision is poor; if it is too long, firepower is weak. The optimal firing angles corresponding to different burst intervals are different and require specific calculations.

Future research can further explore the dynamic response of UAVs under different firing conditions, as well as more advanced control strategies, to improve the performance and reliability of UAVs in practical applications.