## 2 Dynamics Modeling  
In this paper, to address the design issue clearly, we focus on 3-DoF quadrotor dynamics. Hence, the simplified 3-DoF dynamics model is adopted rather than the 6-DoF quadrotor systems. It is worth noticing that the considered 3-DoF model is a classical example to illustrate the control difficulty for such cascade nonlinear systems subject to both the matched and mismatched disturbances. To address this control problem, the 3-DoF dynamic model of a quadrotor is considered in this study to provide a deeper understanding of the proposed algorithm. The dynamic configuration for a 3-DoF quadrotor system is shown in Figure 1. The geometry parameters are summarized in Table 1.  
The quadrotor system contains two brushless direct currents (BLDC) motors to provide the thrusts to drive the quadrotor system. In practical scenarios, each BLDC motor is \begin{table}
\begin{tabular}{c c c} \hline
**Quantity** & **Description** & **Unit** \\ \hline \(m\) & Mass & kg \\ \(J\) & Moment of inertia & kg-m2 \\ \(g\) & Gravitational acceleration & m/s2 \\ \(L\) & Distance between motors and & \\  & center of mass & m \\ \hline \end{tabular}
\end{table} 
Table 1: Explanation of the physical quantities of the quadrotor planar motion system. Figure 1: The dynamic configuration of the 3-DoF quadrotor system. controlled by the electric speed controller (ESC). The ESC controls the rotation speed of the BLDC motor to the desired speed according to given pulse width modulation (PWM) duty cycles. In other words, there exists a simple mapping relationship between desirable rotor speeds and the given PWM duty cycles. Refer to [15; 30; 31], and the thrust \(F_{i}\) generated by \(i\)-th rotor with the speed \(\Omega_{i}\), \(i=1,2\), is modeled as  
\[F_{i}=C_{T}\Omega_{i}^{2} \tag{1}\]  
where \(C_{T}\) is the thrust coefficient. On the basis of (1), the total force \(F\) and torque \(M\) can be expressed as  
\[ \begin{bmatrix}F\\ M\end{bmatrix} =\mathbf{\Gamma} \begin{bmatrix}\Omega_{2}^{2}\\ \Omega_{2}^{2}\end{bmatrix}  \tag{2}\]  
where the force distribution matrix \(\mathbf{\Gamma}\) is defined as  
\[\mathbf{\Gamma}= \begin{bmatrix}C_{T}&C_{T}\\ LC_{T}&-LC_{T}\end{bmatrix}  \tag{3}\]  
The inverse mapping of (2) is given by  
\[ \begin{bmatrix}\Omega_{2}^{2}\\ \Omega_{2}^{2}\end{bmatrix} =\mathbf{\Gamma}^{-1} \begin{bmatrix}F\\ M\end{bmatrix}  \tag{4}\]  
where  
\[\mathbf{\Gamma}^{-1}= \begin{bmatrix}\frac{1}{2C_{T}}&\frac{1}{2C_{T}L}\\ \frac{1}{2C_{T}}&-\frac{1}{2C_{T}L}\end{bmatrix}  \tag{5}\]  
The system kinetic energy \(\mathcal{T}\) includes the translation kinematic energy \(\frac{1}{2}m(\dot{x}^{2}+\dot{z}^{2})\) and the rotational kinematic energy \(\frac{1}{2}f\dot{\theta}^{2}\); the potential energy \(\mathcal{V}\) with respect to the datum \(z=0\) is \(mgz\). Thus, the Lagrangian \(\mathcal{L}=\mathcal{T}-\mathcal{V}\) can be represented by  
\[\mathcal{L} =\mathcal{T}-\mathcal{V}\] \[=\left[\frac{1}{2}m(\dot{x}^{2}+\dot{z}^{2})+\frac{1}{2}f\dot{ \theta}^{2}\right]-mgz \tag{6}\]  
The Lagrange equation is  
\[\frac{d}{dt}\left(\frac{\partial\mathcal{L}}{\partial\dot{\eta}_{j}}\right)- \frac{\partial\mathcal{L}}{\partial\eta_{j}}=N_{j},\quad j=1,2,3 \tag{7}\]  
where the generalized coordinate \(\eta_{j}\) represents \(x\), \(z\), and \(\theta\); the generalized force \(N_{j}\) is \(F\mathrm{s}_{\theta}+d_{x}\), \(F\mathrm{c}_{\theta}+d_{z}\), and \(M+d_{\theta}\), respectively. The notations \(\mathrm{c}_{(\cdot)}\) and \(\mathrm{s}_{(\cdot)}\) represent \(\cos(\cdot)\) and \(\sin(\cdot)\). The following time-varying and state-dependent disturbance models of the external disturbances \(d_{x}\), \(d_{z}\), and \(d_{\theta}\) are considered:  
\[ \begin{split} d_{x}(t,\dot{x})&=f_{x}(t)-C_{dx}\dot{ x}\\ d_{z}(t,\dot{z})&=f_{y}(t)-C_{dz}\dot{z}\\ d_{\theta}(t,\dot{\theta})&=f_{\theta}(t)-C_{d \theta}\dot{\theta}\end{split}