# Robust Backstepping Control Applied to UAVs for Pest Recognition in Maize Crops
## 3 Quadrotor Model  
The quadrotor model (see Figure 1) is obtained by representing a solid body moving in 3D space and being subjected to one force and three moments [31]. The generalized coordinates of the four rotor helicopter \(q=(\gamma,\lambda)\), where \(\gamma=(x,y,z)\in\mathbb{R}^{3}\) represents the relative position of the center of mass of the quadrotor with respect to an inertial reference, and \(\lambda=(\phi,\theta,\psi)\in\mathbb{R}^{3}\) are the Euler angles representing the orientation of the quadrotor, known as roll-pitch-yaw, respectively. The inertial moments in the flying robot are defined by \(I_{x}\), \(I_{y}\), \(I_{z}\) ordered with respect to \(x\), \(y\) and \(z\) axes, while \(m\) denotes its mass, \(l\) is the arm length, and \(\tau_{\phi}\), \(\tau_{\theta}\) and \(\tau_{\phi}\) are the input signals to be applied to the motors.  
The following equations describe the movement of the quadcopter affected by collective throttle \(u_{s}\) and torques \(\tau_{\phi}\), \(\tau_{\theta}\) and \(\tau_{\psi}\)[32]  
\[\begin{array}{l}\dot{x}=-\frac{1}{m}\sin\theta u_{s}\\ \dot{y}=\frac{u_{s}}{m}\cos\theta\sin\phi\\ \dot{z}=\frac{\cos\theta\cos\phi}{m}u_{s}-g\\ \dot{\phi}=\phi\dot{y}\frac{y_{t}-z}{l_{s}}+\frac{l}{l_{s}}\tau_{\phi}\\ \ddot{\theta}=\phi\dot{y}\frac{l_{s}-l_{s}}{l_{s}}+\frac{l}{l_{s}}\tau_{\phi} \\ \dot{\psi}=\phi\dot{\phi}\frac{y_{t}-l_{s}}{l_{s}}+\frac{l}{l_{s}}\tau_{\psi}. \end{array} \tag{11}\]  
The following assumptions, proposed in [11], are considered:  
1. The UAV is considered as a rigid and symmetrical body.
2. The center of gravity (CoG) of the quadrotor is in its origin.
3. The blades are rigid bodies with fixed angle pitch.
4. In the nominal model of the quadrotor, the aerodynamics effects have not been considered.
5. The motor dynamic could be modeled as a first-order transfer function, and it is sufficient to reproduce the dynamics between the propeller's speed set-point and its true speed. As the time constant of this transfer function is small, we can consider that the rotor dynamic is approximately equal to one [33]. This assumption is used to suppose additional dynamics (or uncertain parameters) that represent matched vanishing perturbations in the actuators.
6. The attitude angles, pitch, roll and yaw are restricted in the interval \([-\frac{\pi}{4},\frac{\pi}{4}]\).  
**Remark 1**.: _Additionally to Assumption 5, the effects of the propellers are neglected [34]. However, a more accurate trust model of the vehicle is given in [34], which can be expressed as_  
\[T_{i} = c_{1}\omega_{i}^{2}\bigg{(}c_{2}\bigg{(}1+\frac{3}{2}\mu_{i}^{2} (t)\bigg{)}-\lambda_{i}(t)\bigg{)}\] \[= c_{1}\omega_{i}^{2}c_{2}+\underbrace{c_{1}\omega_{i}^{2}\bigg{[} c_{2}\frac{3}{2}\mu_{i}^{2}(t)-\lambda_{i}(t)\bigg{]}}_{\delta},\]  
_where \(T_{i}\) is the trust of motor \(i^{\text{th}}\), \(\omega_{i}\) is the angular velocity of the motor \(i^{\text{th}}\), \(\mu_{i}(t)\) and \(\lambda_{i}(t)\) are the advance and inflow ratios respectively (nonlinear functions), \(c_{1}\) and \(c_{2}\) are positive constants. So, \(\delta\) could be the second term of \(T_{i}\), among other unmodelled dynamics._  
Figure 1: Quadcopter diagram.  
With the nominal model given by (11), the robust backstepping controller is synthesized in the next section.