## 2 Quadrotor Dynamics Model and Problem Formulation  
In this section, we briefly present the quadrotor dynamics model, as it was clearly introduced and verified in many previous studies [34; 43; 50; 51]. Figure 1 illustrates the quadrotor configuration. The four rotors of the quadrotor generate four thrust forces \(F_{i}(i=1,...,4)\) that have relationship with the control inputs as  
\[ \begin{cases}u_{1}=F_{1}+F_{2}+F_{3}+F_{4}\\ u_{2}=F_{2}-F_{4}\\ u_{3}=F_{3}-F_{1}\\ u_{4}=-F_{1}+F_{2}-F_{3}+F_{4}\end{cases}  \tag{1}\]  
The cascaded dynamics model of the quadrotor is described as  
\[ \begin{cases}m\ddot{\mathbf{P}}=-\mathbf{\Lambda_{1}}\dot{\mathbf{P}}-\mathbf{G}+\mathbf{B _{1}}u_{1}+\mathbf{d_{1}(t)}\\ J\ddot{\mathbf{\Theta}}=-\mathbf{\Lambda_{2}}\dot{\mathbf{\Theta}}+\mathbf{B_{2}}\mathbf{U_{2}}+\mathbf{ d_{2}}(t)\end{cases}  \tag{2}\]where \(\mathbf{P}=[x,y,z]^{T}\) and \(\mathbf{\Theta}=[\phi,\theta,\psi]^{T}\), respectively, denote the position and attitude of quadrotor in the inertial frame \(E\). \(m\in\mathbb{R}^{+}\) the mass. \(\mathbf{G}=[0,0,mg]^{T}\) with \(g\) being the gravitational acceleration. \(\mathbf{J}=diag(J_{1},J_{2},J_{3})\in\mathbb{R}^{3\times 3}\) the nominal inertia momentum matrix. \(\mathbf{\Lambda_{1}}=diag(\lambda_{x},\lambda_{y},\lambda_{z})\) and \(\mathbf{\Lambda_{2}}=diag(\lambda_{\phi},\lambda_{\theta},\lambda_{\psi})\) represent the damping matrices, with \(\lambda_{i}(i=x,y,...,\psi)\) being nominal drag coefficients. \(\mathbf{B_{1}}=[c(\phi)s(\theta)c(\psi)+s(\phi)s(\psi),s(\phi)s(\theta)c(\psi)c( \theta)]^{T}\) the input matrix of position subsystem, where \(s(x),c(x)\) is an abbreviation for \(\sin(x),\cos(x)\), respectively. \(\mathbf{B_{2}}=diag(l,l,c_{d})\) is a symmetric positive definite constant matrix, where \(l,c_{d}\) are the quadrotor arm length and nominal force-to-moment coefficients, respectively. \(u_{1}\in\mathbb{R}\) and \(\mathbf{U_{2}}=[u_{2},u_{3},u_{4}]^{T}\) the control inputs. \(\mathbf{d_{1}}(t)\) and \(\mathbf{d_{2}}(t)\) are the lumped disturbances, including parameterized uncertainties and external disturbances, in position and orientation dynamics, respectively. To take the control input saturation into account, the control input vector \([u_{1},u_{2},u_{3},u_{4}]^{T}\) is restricted by \(\text{sat}(\mathbf{u})\) with  
\[\text{sat}(\mathbf{u})= \begin{bmatrix}\text{sat}(u_{1})\\ \text{sat}(u_{2})\\ \text{sat}(u_{3})\\ \text{sat}(u_{4})\end{bmatrix}  \tag{3}\]  
where  
\[sat(u_{i})= \begin{cases}u_{i}^{\text{max}},&u_{i}>u_{i}^{\text{max}}\\ u_{i},&-u_{i}^{\text{max}}\leq u_{i}\leq u_{i}^{\text{max}},\quad(i=1,...,4)\\ -u_{i}^{\text{max}},&u_{i}<-u_{i}^{\text{max}}\end{cases}  \tag{4}\]  
with \(u_{i}^{\text{max}}\) being the maximum control authority in the \(i\)-th component that can be provided by the actuators. To address the observer and controller design, the quadrotor dynamics model (2) is transformed into the augmented dynamics in translational and rotational loops as  
\[ \begin{cases}\dot{\mathbf{X}}_{1}=\mathbf{X}_{2}\\ \dot{\mathbf{X}}_{2}=-\mathbf{\Lambda_{1}}\mathbf{X}_{2}/m+(\mathbf{B}_{1}u_{1}-\mathbf{G})/m+\mathbf{ X}_{3}\\ \dot{\mathbf{X}}_{3}=\mathbf{h}_{p}\\ \dot{\mathbf{X}}_{4}=\mathbf{X}_{5}\\ \dot{\mathbf{X}}_{5}=-\mathbf{J}^{-1}\mathbf{\Lambda_{2}}\mathbf{X}_{5}+\mathbf{J}^{-1}\mathbf{B}_{2} \mathbf{U}_{2}+\mathbf{X}_{6}\\ \dot{\mathbf{X}}_{6}=\mathbf{h}_{\theta}\end{cases}  \tag{5}\]  
where the system state variables are denoted as \(\mathbf{X}_{1}=\mathbf{P}\), \(\mathbf{X}_{2}=\dot{\mathbf{P}}\), \(\mathbf{X}_{4}=\mathbf{\Theta}\), and \(\mathbf{X}_{5}=\dot{\mathbf{\Theta}}\). We extend the lumped disturbances as additional state variables of the system with \(\mathbf{X}_{3}=\mathbf{d}_{1}(t)/m\) and \(\mathbf{X}_{6}=\mathbf{J}^{-1}\mathbf{d}_{2}(t)\).