## 2 Preliminaries  
### Modeling  
A brief discussion on quadrotor modeling is presented in this section. The model is derived using Newton-Euler methodology, which has subsequently been used to derive a model-based flight controller. Referring to Figure 1, two reference frames are used to describe the motion of the aerial vehicle. The frame fixed to body is represented by \(B= \begin{bmatrix}b_{x}&b_{y}&b_{z}\end{bmatrix} \), while the frame attached to the earth is characterized by \(E= \begin{bmatrix}e_{x}&e_{y}&e_{z}\end{bmatrix} \). The earth-fixed frame is also known as inertial reference frame in most of the studies. The rotational motions across \(x\), \(y\) and \(z\)-axes are called roll, pitch and yaw, respectively. These are symbolized by vector \( \begin{bmatrix}\phi&\theta&\psi\end{bmatrix} ^{T}\), and the corresponding body angular rates are represented by \(\omega_{B}= \begin{bmatrix}p&q&r\end{bmatrix} ^{T}\). From Newton-Euler equations, the position \(\zeta_{O}^{E}\) of the center of quadrotor body and velocity \(V_{O}^{E}\) can be expressed as:  
\[\frac{d_{\zeta_{O}^{E}}}{dt}=V_{O}^{E} \tag{1}\]  
\[\frac{dV_{O}^{E}}{dt}=-ge_{z}+R\frac{F_{B}}{m}e_{z} \tag{2}\]  
Figure 1: Quadrotor model with reference frames. \[\frac{dR}{dt}=R\ S(\omega_{B}) \tag{3}\]  
where \(g\) is gravity, \(F_{B}\) is vertical thrust, \(m\) is body mass, \(R\) is rotation matrix and \(S(\omega_{B})\) is a skew symmetric matrix. \[S(\omega_{B})= \begin{bmatrix}0&-r&q\\ r&0&-p\\ -q&p&0\end{bmatrix}  \tag{4}\]  
In the inertial reference frame, the rate of change of angular momentum of the quadrotor body relative to its origin is equal to the total moment that results from all external forces that acts on the quadrotor body. This can be written as:  
\[\frac{d^{E}H_{O}^{E}}{dt}=\tau_{B} \tag{5}\]  
The \(\tau_{B}\) is the net moment of the body produced by the external forces and torques about the origin \(O\), while \(H_{O}^{E}\) is the angular momentum in inertial reference frame \(E\) and is equal to \(I_{O}\omega_{B}^{E}\). The term \(I_{O}\) denotes an inertia tensor with \(O\) as its origin; \(\omega_{B}^{E}\) is body angular rates in the inertial reference frame and is equal to \(pb_{x}+qb_{y}+rb_{z}\). Equation (5) can be written in the body reference frame as follows:  
\[\frac{d^{E}H_{O}^{E}}{dt}=\tau_{B} \tag{6}\]  
where \(\frac{d^{B}H_{O}^{B}}{dt}=I_{x}\dot{p}b_{x}+I_{y}qb_{y}+I_{x}rb_{z}\), \(\omega_{B}\times H_{O}^{B}\) is the Coriolis effect and \(\omega\times \begin{bmatrix}0\\ 0\\ I_{I}\Omega_{r}\end{bmatrix} \) is gyroscopic effect. \[ \begin{bmatrix}I_{x}&0&0\\ 0&I_{y}&0\\ 0&0&I_{z}\end{bmatrix}\begin{bmatrix}\dot{p}\\ q\\ \dot{r}\end{bmatrix} + \begin{bmatrix}0&-r&q\\ r&0&-p\\ -q&p&0\end{bmatrix}\begin{bmatrix}I_{x}&0&0\\ 0&I_{y}&0\\ 0&0&I_{z}\end{bmatrix}\begin{bmatrix}p\\ q\\ r\end{bmatrix} + \begin{bmatrix}qI_{r}\Omega_{r}-0\\ 0-pI_{r}\Omega_{r}\\ 0\end{bmatrix} = \begin{bmatrix}\tau_{B\theta}\\ \tau_{B\theta}\\ \tau_{B\theta}\end{bmatrix}  \tag{7}\]  
As the name suggests, a quadrotor possesses four actuator entities called rotors, which produce two major dynamic terms. One is vertical thrust, or the lift force, and the other is the moment around the axis of rotation of the rotors. Lift force is central to performing any type of quadrotor maneuver, and its various combinations give the following four control terms:  
\[\left. \begin{array}{l}F_{B}=b\left(\Omega_{1}^{2}+\Omega_{2}^{2}+\Omega_{3} ^{2}+\Omega_{4}^{2}\right)\\ \tau_{B\phi}=lb\left(\Omega_{4}^{2}-\Omega_{2}^{2}\right)\\ \tau_{B\theta}=lb\left(\Omega_{3}^{2}-\Omega_{1}^{2}\right)\\ \tau_{B\psi}=d\left(\Omega_{4}^{2}+\Omega_{2}^{2}-\Omega_{3}^{2}-\Omega_{1}^{2 }\right)\end{array} \right\} \tag{8}\]  
where \(b\) and \(d\) represent thrust and drag factors, respectively, while \(\Omega_{i}\) represents propeller speed. The rotation sequence \(R(z,\psi)\times R(x,\phi)\times R(y,\theta)\) is used for rotation matrix R and the transformation from Euler angular rates to body angular rates is \( \begin{bmatrix}p\\ q\\ r\end{bmatrix} =