# A Comparative Study of Control Methods for X3D Quadrotor Feedback Trajectory Control
## 2 Mathematical Modeling of X3d Quadrotor  
Kinematics and dynamics are the two parts of the X3D model system and are described using the Newton-Euler theorem rules as: (1) A quadrotor has a symmetrical and rigid frame, (2) the quadrotor's center of gravity is the same as the body's fixed frame origin, (3) the propellers have a rigid design, (4) the square of the propeller's speed determines thrust and drag. The X3D quadrotor parameter list is mentioned in Table 2. The complexity of the re-evaluated X3D model, as illustrated in Figure 1, has been significantly decreased. Equation (1) gives the onboard controller's input vector.  
\[U=\left[U_{\phi}\,,U_{\theta}\,,U_{\psi}\,,U_{thrust}\right] \tag{1}\]  
The angular velocity (\(\mathbf{S}_{\omega}\)) and collective thrust (\(\mathbf{S}_{c}\)) outputs of the onboard controller are delivered from the kinematics of the reference body X3D, which results in the system's final output in terms of translation position (\(\boldsymbol{P}\)) and orientation (\(\boldsymbol{\Theta}\)). The Newton-Euler formulation provides a comprehensive mathematical account of quadrotor dynamics [14]. Position, Euler angle, linear velocity, and angular velocity are among the 12-degree-of-freedom output states described by Equation (2).  
\[12-DOF=\left[x,y,z,\phi,\theta,\psi,\dot{x},\dot{y},\dot{z},\dot{\phi},\dot{ \theta},\dot{\psi}\right] \tag{2}\]  
The right-handed system is the inertial (Earth) frame of reference \(E=\left[O_{c},X_{c}\,,Y_{c},Z_{c}\right]\) and denotes the origin, which is the center of the earth. Its purpose is to determine the quadrotor's location. Right-handedness is reflected in the bodily frame of reference as \(B=\left[O_{B},X_{B}\,,Y_{B},Z_{B}\right]\) and denotes the origin, which is located at the quadrotor's center of gravity. It is used to figure out the quadrotor's orientation with the earth frame.  
In the body frame, the torque, the force \(F_{B}\), angular velocity \(\omega_{B}\), and linear velocity \(v_{B}\) are all computed. The body and the earth reference frames are shown in Figure 2. The coordinates of the quadrotor (body frame) can be aligned with the earth frame in the  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameters** & **Symbol** & **Value** \\ \hline Quadrotor Mass & m & 0.54 kg \\ \hline Gravity Acceleration & g & 9.807 m/s\({}^{2}\) \\ \hline Arm length of Quadrotor & L & 0.225 m \\ \hline Inertia Moment & \(I_{xx}\) & 0.022 kg.m\({}^{2}\) \\  & \(I_{yy}\) & 0.022 kg.m\({}^{2}\) \\  & \(I_{zz}\) & 0.0018 kg.m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: X3D Quadrotor parameter list.  
Figure 1: X3D Quadrotor re-evaluation model.  
following sequence: \(Z_{e}\) on the earth, the frame is aligned to the yaw angle on the body frame \(Z_{B}\) (positive \(\psi\)), \(Y_{e}\) is aligned to the pitch angle on \(Y_{B}\) (positive \(\theta\)), and \(X_{e}\) is aligned to the roll angle on \(X_{B}\) (positive \(\varphi\)). Rotation matrices (\(x\phi D,y\phi D,z\psi D\)) about the three axes (roll, pitch, and yaw) are described in Equation (3).  
\[\begin{array}{l}x\phi D\ \ =\begin{bmatrix}1\ 0\ 0\ 0\ cos\ cos\ \phi\ \ \ sin\ sin\ \phi\ \ \ 0\ -\ sin\ sin\ \phi\ \ \ cos\ \phi\ \ \ \ \ \ \ \ \ \ \ \ \ \  
where \(RotorBF=1 BF+2 BF+3 BF+4 BF\).  
The upward forces created by the X3D rotors are \(1 BF\)_to_\(4 BF\), and \(F_{g}\) is the force impacting the body. The angular velocity of rotors affects the \(RotorBF\).  
The nonlinear Simulink model of the x3D quadrotor is presented in Figure 3. The nonlinear dynamic equations of the x3D quadrotor are linearized using a first-order Taylor approximation to implement the linear controller. The X3D near the hover position's aggregate linearized equations may be expressed as Equation (14):  
\[\begin{array}{c}\dot{x}=V_{x}\,\ \dot{y}=V_{y}\,\ \dot{z}=V_{z}\\ \dot{\phi}=p,\ \dot{\theta}=q\,\ \dot{\psi}=r\\ \dot{V}_{x}=-\phi\dot{g}\ \dot{V}_{y}=\theta g\ \dot{V}_{z}=\frac{F_{z}+1_{f} \cdot V_{Z}}{m}\end{array} \tag{12}\]