## 2 Model Description and Transformation
### Dynamics Model
,\forall\mathbf{x}=[x_{1},x_{2},x_{3}]^{\mathrm{T}}\in \mathcal{R}^{3}.\]  
Consider the additive parameter uncertainties of the dynamics system as (2); the model parameters are assumed to be \(m=m_{0}+\Delta_{m},J_{x}=J_{x0}+\Delta_{J_{x}},J_{y}=J_{y0}+\Delta_{J_{y}},J_{ z}=J_{z0}+\Delta_{J_{z}}\), \(k_{1}=k_{10}+\Delta_{k_{1}},k_{2}=k_{20}+\Delta_{k_{2}},k_{3}=k_{30}+\Delta_{k _{3}},k_{4}=k_{40}+\Delta_{k_{4}},k_{5}=k_{50}+\Delta_{k_{5}},k_{6}=k_{60}+ \Delta_{k_{6}}\), \(m_{0}\) denotes the nominal total mass. \(\mathbf{J}_{0}=\text{diag}\big{\{}J_{x0},J_{y0},J_{z0}\big{\}}\) denotes the nominal inertia matrix. \(k_{10},k_{20},\ldots,k_{60}\) denote the nominal value of the aerodynamic friction coefficients, respectively, and \(\Delta_{*}\) denotes the corresponding parameter uncertainties. Combining (1) and (2), one can further obtain the position and attitude subsystems in the form of  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}_{11}=\mathbf{x}_{12},\\ \dot{\mathbf{x}}_{12}=-g\mathbf{q}-\mathbf{P}_{1}\mathbf{x}_{12}+\frac{1}{m_{0}}(\mathbf{u}+ \mathbf{d}_{1}),\end{array} \right. \tag{3}\]  
\[\left\{ \begin{array}{l}\dot{\mathbf{x}}_{21}=\mathbf{x}_{22},\\ \dot{\mathbf{x}}_{22}=-\mathbf{P}_{2}\mathbf{x}_{22}+J_{0}^{-1}(\mathbf{\tau}+\mathbf{d}_{2}) \end{array} \right. \tag{4}\]  
Figure 1: Mechanical structure of a quadrotor UAV. with \(\mathbf{x}_{11}=\mathbf{\zeta},\mathbf{x}_{12}=\mathbf{\dot{\zeta}}\), \(\mathbf{x}_{21}=\mathbf{\Theta},\mathbf{x}_{22}=\mathbf{\dot{\Theta}}\), and \(\mathbf{q}=[0,0,1]^{\mathrm{T}}\). \(\mathbf{u}=\mathbf{R}_{B}^{\mathrm{E}}\mathbf{U}_{f}\) and \(\mathbf{\tau}=\mathbf{\Phi}_{B}^{\mathrm{E}}\mathbf{U}_{t}\) represent the control inputs under the frame \(\xi_{E}\). \(P_{1}\) and \(P_{2}\) are defined as  
\[P_{1} =\tfrac{1}{m_{0}}\mathrm{diag}\{k_{10},k_{20},k_{30}\},\] \[P_{2} =-\dot{\mathbf{\[PERSON]}}_{B}^{\mathrm{E}}(\mathbf{\[PERSON]}_{B}^{\mathrm{E}})^{-1 }+J_{0}^{-1}\mathbf{F}_{1}+\mathbf{\[PERSON]}_{B}^{\mathrm{E}}J_{0}^{-1}\big{(}(\mathbf{\[PERSON]}_{ B}^{\mathrm{E}})^{-1}\dot{\mathbf{\Theta}}\big{)}_{\times}J_{0}(\mathbf{\[PERSON]}_{B}^{ \mathrm{E}})^{-1}\]  
with \(\mathbf{F}_{1}=\mathrm{diag}\{k_{40},k_{50},k_{60}\}\). \(\mathbf{d}_{1}=\mathbf{u}_{d}-\mathbf{f}_{p}\) and \(\mathbf{d}_{2}=\mathbf{\tau}_{d}-\mathbf{f}_{a}\) represent the lumped uncertainties, where \(\mathbf{u}_{d}=\mathbf{R}_{B}^{\mathrm{E}}\mathbf{D}_{f}\), \(\mathbf{\tau}_{d}=\mathbf{\Phi}_{B}^{\mathrm{E}}\mathbf{D}_{t}\); the parameter uncertainties \(\mathbf{f}_{p}\) and \(\mathbf{f}_{a}\) are defined as  
\[\mathbf{f}_{p} =\Delta_{m}\mathbf{\tilde{\Theta}}q+\Delta_{m}\mathbf{\tilde{\zeta}}+ \Delta_{P_{1}}\mathbf{\dot{\zeta}},\] \[\mathbf{f}_{a} =\mathbf{\Delta}_{f}\mathbf{\tilde{\Theta}}+\Delta_{P_{3}}\mathbf{\tilde{ \Theta}}+\Delta_{P_{1}}\mathbf{\tilde{\Theta}}\]  
with  
\[\mathbf{\Delta}_{P_{1}} =\mathrm{diag}\big{\{}\Delta_{k_{1}},\Delta_{k_{2}},\Delta_{k_{3 }}\big{\}},\] \[\mathbf{\Delta}_{F_{1}} =\mathrm{diag}\big{\{}\Delta_{k_{4}},\Delta_{k_{5}},\Delta_{k_{6 }}\big{\}},\] \[\mathbf{\Delta}_{P_{3}} =-\mathbf{\Phi}_{B}^{\mathrm{E}}\mathbf{\Delta}_{f}(\mathbf{\Phi}_{B}^{ \mathrm{E}})^{-1}+\mathbf{\Phi}_{B}^{\mathrm{E}}\Big{(}\big{(}\mathbf{\Delta}_{f}( \mathbf{\Phi}_{B}^{\mathrm{E}})^{-1}\dot{\mathbf{\Theta}}\big{)}_{\times}+\mathbf{F}_{2} \Big{)}(\mathbf{\Phi}_{B}^{\mathrm{E}})^{-1},\] \[\mathbf{F}_{2} =[0,J_{r}\bar{\omega},0;-J_{r}\bar{\omega},0,0;0,0,0].\]  
Since \(\mathbf{\Delta}_{P_{3}}\) is a Coriolis-like matrix, we have the following result according to the properties of the Coriolis matrix [39]. **Property 1**. : \(\big{\|}\mathbf{\Delta}_{P_{3}}\big{\|}\leq\alpha_{1}\big{\|}\dot{\mathbf{\Theta}} \big{\|}\) _with \(\alpha_{1}\) being a positive constant._