## 2 Literature Review
### Mathematical Modeling of Quadcopter Motion  
The orientation of a rigid body in a coordinate system can be characterized based on the three Euler angles, which can also be used to translate a point's coordinates from one reference frame to another and to explain the relationship between two reference frames [32; 33; 34; 35]. The Euler angles, which show how a body rotates around the axes of a coordinate system, are represented by the symbols for the roll, pitch, and yaw angles. Figure 1: The working principles of aerial-vehicle-guided agriculture. The position of a quadcopter body is described by six coordinates that are divided into two reference frames. The x, y, and z coordinates, in the cardinal directions of north, east, and down, represent the initial reference frame, which is the fixed coordinate system known as the inertial or Earth frame. The \(\phi\), \(\theta\), and \(\psi\) angles with respect to the body's center of gravity represent the body frame, a movable coordinate system that serves as the second reference frame. The quadcopter has four inputs and six outputs, making it an underactuated nonlinear system [36]. By giving one rotor (or two neighboring rotors) more thrust and the diametrically opposed rotor less thrust, a quadcopter can change its pitch or roll (Figure 2). The thrust force \(\left(T_{l},T_{b},T_{f},T_{r}\right)\) is the force that propels a drone forward. It is generated by the rotation of the drone's propellers. The weight (mg) of a drone due to gravity is determined by its mass and acceleration due to gravity. Torque \(\left(\tau_{l},\tau_{b},\tau_{f},\tau_{r}\right)\) is a measure of the force that causes a drone motor to rotate. In drones, torque is generated by the motors that power the propellers. The amount of torque produced by a motor is determined by its speed and the size of its propeller. Drones use a variety of mechanisms to balance momentum \(\left(M_{l},M_{b},M_{f},M_{r}\right)\), including propellers, rotors, and wings. By carefully controlling the speed and direction of these components, drones can achieve stable flight and maneuverability. In the present analysis, the quadcopter was modeled based on the following assumptions in order to reduce the complexity of the system. The structure is rigid  
The structure is axis-symmetrical  
The center of gravity and the body-fixed frame origin coincide  
The propellers are rigid  
Trust and drag are proportional to the square of the propeller's speed. The three angles that [PERSON] devised to characterize the orientation of a rigid body in a coordinate system are referred to as the Euler angles. They can be used to translate a point's coordinates from one reference frame to another and to explain the relationship between two reference frames. The Euler angles, which show how a body rotates around the axes of a coordinate system, are represented by the symbols \(\phi\), \(\theta\), and \(\psi\) for the roll, pitch, and yaw angles, respectively [37; 38; 39; 40]. The three fundamental Euler angles can be combined to achieve any orientation for a rigid body. The following  
Figure 2: Schematic diagram of reaction torques on each motor of a quadcopter [22]. The quadcopter has a symmetrical structure with the four arms aligned with the body's x- and y-axes. Therefore, the inertia matrix \(I\) is a diagonal matrix in which \(Ixx=Iyy\),  
\[I= \begin{bmatrix}I_{xx}&0&0\\ 0&I_{yy}&0\\ 0&0&I_{zz}\\ \end{bmatrix}  \tag{9}\]  
Gyroscopic forces \(\Gamma\) are caused by the combined rotation of the four rotors and the quadcopter body,  
\[\Gamma=\sum_{i=1}^{4}J_{\tau}\omega_{B}\Lambda\hat{e_{3}}(-1)^{1+i}\omega_{ri} \tag{10}\]  
where  
\[\left\{ \begin{aligned} \Gamma=J_{\tau}\omega_{B}\omega_{r} \\ \omega_{\tau}=\left(-\omega 1+\omega 2-\omega 3+\omega\right)\\ T=\left[\tau_{\phi}\tau_{\phi}\tau_{\phi}\right]^{T}\\ \end{aligned} \right. \tag{11}\]  
When  
\[\left\{