# Quadcopter Adaptive Trajectory Tracking Control:
## 2 Preliminaries and Mathematical Model  
The following notations and symbols will be used throughout this paper. The notations \(x\) or \(X\) are the vector and bold letter \(\mathbf{X}\) is the matrix. Next, \(\mathbb{R}^{n}\) stands for set of real \(n\)-dimensional vector, \(\mathbb{R}^{n\times m}\) denotes set of real \(n\)-row and \(m\)-column matrix, and \(\mathbf{I}_{n}\) is the \(n\times n\) identity matrix. \(c\phi=\cos(\phi)\), \(s\phi=\sin(\phi)\), \(c\theta=\cos(\theta)\), \(s\theta=\sin(\theta)\), \(c\psi=\cos(\psi)\), and \(s\psi=\sin(\psi)\) are denoted. \(\mathrm{vec}\left(\mathbf{X}\right)\) denotes a vectorization of a matrix \(\mathbf{X}\), e.g., if \(\mathbf{X}=\left[X_{1},X_{2},X_{3}\right]\) then \(\mathrm{vec}\left(\mathbf{X}\right)=\left[X_{1}^{T},X_{2}^{T},X_{3}^{T} \right]^{T}\). \(\mathrm{vec}\left(\mathbf{\dot{X}}\right)\) represents the first time-derivative of each its elements. Lastly, \(\left|\bullet\right|\) and \(\otimes\) are used to represent the absolute and Kronecker product operators, respectively. In this section, the model, the parameters, and the variables of Unmanned Arial Vehicles, specifically Plus-Configuration are introduced in Figure 1, Tables 1 and 2, respectively. In order to meet the requirements based on path tracking, a control law is expectedly designed to automatically change four different rotor speeds \(\Omega_{i}\). The intermediate control inputs [12] will be shown as follows:  
\[ \begin{split} U_{1}&=b\left(\Omega_{1}^{2}+\Omega_ {2}^{2}+\Omega_{3}^{2}+\Omega_{4}^{2}\right)\\ U_{2}&=b\left(-\Omega_{2}^{2}+\Omega_{4}^{2}\right) \\ U_{3}&=b\left(\Omega_{1}^{2}-\Omega_{3}^{2}\right) \\ U_{4}&=d\left(-\Omega_{1}^{2}+\Omega_{2}^{2}- \Omega_{3}^{2}+\Omega_{4}^{2}\right)\end{split}  \tag{1}\]  
where \(U_{1}\) impacts on vertical movement, \(U_{2}\) and \(U_{3}\) are able to drive this system move forward, back, turn left and right in the horizontal plane, and \(U_{4}\) makes this system rotate at a standing point. Figure 1: Plus-Configuration of a quadcopter. First, to achieve the mathematical model of the considered quadcopter, we use the Euler-Lagrange approach to describe the system position \(\zeta=\left[x,y,z\right]^{T}\) in the Earth-fixed frame. The Euler-Lagrange equations are applied into the system following a translational \(E_{trans}\) and a potential energy \(E_{pot}\)[12] as follows:  
\[\frac{d}{dt}\left[\frac{\partial}{\partial\zeta}L_{trans}\left( \zeta,\zeta\right)\right]-\frac{\partial}{\partial\zeta}L_{trans}=\mathbf{R}f_ {B} \tag{2}\] \[L_{trans}\left(\dot{\zeta},\zeta\right)=E_{trans}+E_{pot}\] (3) \[E_{trans}=\frac{m}{2}\dot{\zeta}^{T}\dot{\zeta}\] (4) \[E_{pot}=-mgz\] (5) \[\mathbf{R}= \begin{bmatrix}c\psi c\varrho&c\psi\theta s\phi-s \psi c\varrho&c\psi s\theta c\phi+s\psi s\phi\\ s\psi c\theta&s\psi s\theta s\phi+c\psi c\varrho&s\psi s\theta c\phi-c\psi s \phi\\ s\theta&c\theta s\phi&c\theta c\phi\end{bmatrix} \] (6) \[f_{B}=\left[0,0,U_{1}\right]^{T} \tag{7}\]  
where matrix \(\mathbf{R}\) in (6) is the transformation matrix from body to Earth-fixed frame and control input \(f_{B}\) (7) contains lift force \(U_{1}\) in \(z-axis\) of body frame in (1). Substituting (3), (4), (5), (6), and (7) into (2) to obtain the second ordinary differential equations of dynamics in (8):  
\[\frac{d}{dt}\left(\dot{\zeta}\right)=\frac{U_{1}}{m} \begin{bmatrix}\left(c \psi s\theta c\phi+s\psi s\phi\right)\\ \left(s\psi s\theta c\phi-c\psi s\phi\right)\\ \left(c\theta c\phi\right)\end{bmatrix} + \begin{bmatrix}0\\ 0\\ -g\end{bmatrix}  \tag{8}\]  
Considering the dynamic rotation model based on the energy of rotation through Euler-Lagrange method with hovering assumption. The second ordinary differential equations of Roll, Pitch, and Yaw angles will be described with several unknown parameters in Table 1:  
\[ \begin{array}{l}\ddot{\phi}=a_{1}\dot{\theta}\dot{\psi}-\mu_{1}\dot{\phi}+b_ {1}U_{2}\\ \ddot{\theta}=a_{2}\dot{\phi}\dot{\psi}-\mu_{2}\dot{\theta}+b_{2}U_{3}\\ \ddot{\psi}=a_{3}\dot{\phi}\dot{\theta}-\mu_{3}\dot{\psi}+b_{3}U_{4}\end{array}  \tag{9}\]