## 2 Mathematical Modeling of UAV
### Problem Statement
#### 2.2.1 Wind Disturbance Model of the Multi-rotor UAV  
The aerodynamic performance of the multi-rotor UAV with varying axle distance is usually disturbed by lateral wind along the direction of hovering and flying, and the aircraft stability may even be affected in severe cases. UAVs in agriculture, power line inspection, and other special operations usually work in windy environments (typical wind force of 4-7). In that case, the actuator needs to provide greater forces to produce a greater angular movement to maintain aircraft stability. [PERSON] and [PERSON] performed a wind disturbance study of a multi-rotor UAV in gentle wind [29]. Studies showed that the UAV undergoes significant horizontal drift due to lateral and horizontal winds. The lift provided by the actuator may be insufficient due to changes in air pressure in the propeller's upper and lower surfaces, which further result in the change ub attitude angle. Here, the influence of gusts in different directions on the lift of each rotor was considered [29] and is given by the following equation:  
\[f_{i}=f_{w,i}+f_{m,i}\quad(i=1,2,3,4)\, \tag{5}\]  
where \(f_{wi}\) is the extra thrust generated by the transverse air flow. To estimate the influence of the extra thrust on the dynamic model of the aircraft, we considered the induction wind velocity of each propeller [30]:  
\[V_{i}=\sqrt{\frac{f_{i}}{2\rho S_{i}}}\, \tag{6}\]  
where \(\rho\) is the air density and \(S_{i}\) is the cross-sectional area of the rotating propeller. Further, the thrust generated by each actuator is given by  
\[f_{i}=2\rho S_{i}V_{i}^{tol}V_{i}^{m}\, \tag{7}\]where \(V_{i}^{m}\) is the induction wind velocity when the actuator is in normal operation and \(V_{i}^{tol}\) is the total induction wind velocity of the actuator disturbed by gusts. Considering the influence of \(V_{i}^{w}\) gust disturbance, \(V_{i}^{tol}\) is given by  
\[V_{i}^{tol}=\sqrt{\left(V_{i}^{w}\cos\alpha+V_{i}^{m}\right)^{2}+\left(V_{i}^{ w}\sin\alpha\right)^{2}}\, \tag{8}\]  
where the angle \(\alpha\) is the angle included between the direction of the transverse wind and the axial direction of the rotor. Based on the ratio between the velocity of the disturbing wind force and that of the wind generated by the propeller (the velocity of wind force 7 is 15 m/s and that of the wind generated by the propeller is above 200 m/s), the parameter \(r_{i}=v_{i}^{w}/v_{i}^{m}\), \(r_{i}\in(0,0.1)\) was defined. From Equations (6) and (7), the following was obtained:  
\[f_{i}=2\rho S_{i}(V_{i}^{m})^{2}\sqrt{1+r_{i}^{2}+2r_{i}\cos\alpha}\, \tag{9}\]  
From this, we have  
\[\begin{array}{l}f_{w,i}=f_{i}-f_{m,i}\\ =2\rho S_{i}\big{(}V_{i}^{m}\big{)}^{2}\sqrt{1+r_{i}^{2}+2r_{i}\cos\alpha}-2 \rho S_{i}\big{(}V_{i}^{m}\big{)}^{2}\ \.\\ \leq 2\rho S_{i}\big{(}V_{i}^{m}\big{)}^{2}\cdot r_{i}\end{array} \tag{10}\]  
Thus, the term \(d_{i}^{w}=2\rho S_{i}\big{(}V_{i}^{m}\big{)}^{2}\cdot r_{i}\) defines the wind force acting on each propeller. Since the torsion generated by rotor rotation was just the opposite to the aerodynamic resistance generated by rotor rotation, the aerodynamic resistance was defined as \(d_{i}^{\rm T}=\rho S_{i}\big{(}V_{i}^{w}\big{)}^{2}\big{/}2\) Correspondingly, \(d_{w}=\big{[}d_{1}^{tol}\ d_{2}^{\Phi}\ d_{3}^{\Phi}\ d_{4}^{\Phi}\big{]}^{T}\) was the disturbance vector caused by gusts, which is defined as  
\[d_{w}=\left[\begin{array}{c}d_{1}^{tol}\\ d_{2}^{\Phi}\\ d_{3}^{\Phi}\\ d_{4}^{\Phi}\end{array}\right]=\left[\begin{array}{c}\big{(}d_{1}^{w}+d_{2} ^{w}+d_{3}^{w}+d_{4}^{w}\big{)}/m\\ \big{(}d_{1}^{w}+d_{3}^{w}-d_{1}^{w}-d_{4}^{w}\big{)}/J_{XX}\\ \big{(}d_{1}^{w}+d_{2}^{w}-d_{3}^{w}-d_{4}^{w}\big{)}/J_{YY}\\ \big{(}d_{1}^{\rm T}+d_{2}^{\rm T}+d_{3}^{\rm T}+d_{4}^{\rm T}\big{)}/J_{ZZ} \end{array}\right]\, \tag{11}\]