## 2 System Description and Mathematical Modeling
### System Mathematical Modeling  
In formulating a precise physical model for the Parrot Mambo UAV, it is imperative to meticulously account for various aerodynamic influences, including gravity, propeller thrust, air resistance, aerodynamic torque, and the torque generated by gyroscopic motion. In the pursuit of enhancing the model's fidelity to real-world dynamics, it becomes paramount to incorporate random uncertainties applied to the sensors, as discussed in [12]. This imperative consideration contributes to a more accurate representation of the quadcopter's behavior in diverse and unpredictable scenarios. To ensure a comprehensive and nuanced representation, certain assumptions underpin model construction:  
* The quadcopter is treated as a rigid body with a symmetrical structure, facilitating a simplified yet robust characterization. * The alignment of the center of gravity with the quadcopter's geometrical center is assumed. * Euler angles governing the orientation of the UAV (\(\varphi\), \(\theta\), \(\psi\)) are constrained within specific bounds: \(-\pi<\varphi<\pi\), \(-\pi<\theta<\pi\), and \(-\pi<\psi<\pi\). This constraint ensures realistic and physically meaningful variations in orientation. * Both the position and velocity of the UAV are deemed measurable, providing essential parameters for accurate depiction of its motion. * The thrust and drag forces are considered to be proportionally related to the squared velocity of the propellers, aligning with the principles of aerodynamics and motor dynamics. By adhering to these assumptions and considerations, the ensuing physical model not only captures the fundamental dynamics of the Parrot Mambo but also integrates the essential elements necessary for precise analyses, as in [11]. Before constructing a mathematical model for the quadcopter, it is crucial to establish two reference notations: \(R_{g}\bigg{(}O\), \(\overset{\rightarrow}{I}\), \(\overset{\rightarrow}{J}\), \(\overset{\rightarrow}{K}\bigg{)}\) and \(R_{g}\)(\(\alpha\), \(\overset{\rightarrow}{i}\), \(\overset{\rightarrow}{j}\), \(\overset{\rightarrow}{k}\)), where \(R_{g}\) is tied to the ground, providing a fixed \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \(m\) & 0.0630 kg \\ \(l\) & 0.0624 m \\ \(C_{d}(x,y,z)\) & \(10^{-1}\) diag [\(6.2\) 4.5 6.9] N \(\cdot\) s/m \\ \(C_{a(x,y,z)}\) & \(10^{-1}\) diag [\(1\) 1.1.1] N \(\cdot\) s/m \\ \(K_{b}\) & 0.0107 N \(\cdot\) s\({}^{2}\) \\ \(K_{d}\) & 0.7826400 \(\times\)\(10^{-3}\) N \(\cdot\) m \(\cdot\) s\({}^{2}\) \\ \(I_{x}\) & 0.0582857 \(\times\)\(10^{-3}\) kg \(\cdot\) m\({}^{2}\) \\ \(I_{y}\) & 0.0716914 \(\times\)\(10^{-3}\) kg \(\cdot\) m\({}^{2}\) \\ \(I_{z}\) & 0.1 \(\times\)\(10^{-3}\) kg \(\cdot\) m\({}^{2}\) \\ \(I_{r}\) & 0 1021 \(\times\)\(10^{-6}\) kg \(\cdot\) m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parrot Mambo model physical parameters. frame, while \(R_{b}\) is connected to the body of the UAV. The center '\(o\)' of \(R_{b}\) aligns with the quadcopter's mass center, and \((x_{Rb},\,y_{Rb},\,z_{Rb})\) represents its position relative to the Earth. Euler angles, or \((\varphi,\,\theta,\,\psi)\), describe how the quadcopter is tilted or turned. To create a realistic quadcopter model, we must consider all aerodynamic effects, especially those influenced by the mechanical movements of the propellers. These factors play a crucial role in shaping the quadcopter's motion and behavior. Understanding spatial references, positions, orientation dynamics, and aerodynamic effects is essential for building an accurate mathematical model that reflects the real-world behavior of the quadcopter. The modeling approach is based on [PERSON]'s laws and [PERSON]'s theorem. The physical model of the quadcopter can be described by the following equations:  
\[ \begin{cases}\dot{\eta}=f(\eta,u)+w_{k}\\ \zeta=h(\eta)+v_{k}\end{cases}  \tag{1}\]  
\[