## 2 Preliminaries  
### UAV Dynamics and Problem Formulation  
A hexacopter UAV has a mechanism for generating the required forces and torques. The nonlinear dynamics of a hexacopter, as presented in Figure 1[10], is expressed as follows:  
\[\dot{\mathbf{P}}=\mathbf{v}_{b}, \tag{1}\] \[m\dot{\mathbf{v}}_{b}=\mathbf{R}_{i}\mathbf{F}-m\mathbf{g}\mathbf{e}_{z}+\mathbf{F}_{d},\] \[\dot{\mathbf{\Theta}}=\mathbf{R}_{r}\mathbf{\omega}_{b},\] (2) \[\mathbf{f}_{b}\dot{\mathbf{\omega}}_{b}=-\mathbf{\omega}_{b}\times\mathbf{f}_{b} \mathbf{\omega}_{b}+\mathbf{M}_{b}+\mathbf{M}_{g}+\mathbf{M}_{d},\]  
where \(\mathbf{P}=[x,\ y,\ z]^{T}\) denotes position with respect to the inertia frame, \(\mathbf{\Theta}=[\dot{\mathbf{\rho}},\ \theta,\ \psi]^{T}\) denotes Euler angles, \(\mathbf{v}_{b}=[u,v,w]^{T}\) and \(\mathbf{\omega}_{b}=[p,\ q,\ r]^{T}\) denote linear and angular velocities, respectively, in the body-fixed frame. \(\mathbf{f}_{b}=diag(\{j_{x},\ y_{y},\ j_{z}\})\) denotes a symmetric positive definite moment of inertia matrix, \(\mathbf{e}_{z}=[0\ \ 0\ \ 1]^{T};\mathbf{M}_{g}\) denotes the resultant torque due to the gyroscopic effect, and \(\mathbf{F}_{d}\) and \(\mathbf{M}_{d}\) denote uncertainties, including parameter change, asymmetric structure, payload variation, rotor fluctuation, and aerodynamic drag, which are denoted as \(\mathbf{F}_{d}=[d_{x}\dot{x},\ d_{y}\dot{y},\ d_{z}\dot{z}]^{T},\mathbf{M}_{g}=\sum \limits_{i=1}^{6}J_{r}(\mathbf{\omega}_{b}\times\mathbf{e}_{z})\overline{\Omega}\), and \(\mathbf{M}_{d}=[d_{\phi}\dot{\mathbf{\phi}},\ d_{d}\dot{\theta},\ d_{\phi}\dot{\mathbf{ \psi}}]^{T}\). \(J_{r}\) denotes the moment of inertia of each rotor and \(\overline{\Omega}=\Omega_{1}-\Omega_{2}+\Omega_{3}-\Omega_{4}\ \Omega_{5}+\Omega_{6}\), and \(\Omega_{i}\) denotes the speed of rotor \(i\). The translation force \(\mathbf{F}_{d}\) contains gravity, main thrust force,and other force components. The airframe orientation in space is provided using a rotation matrix \(\mathbf{R}_{t}\) from the body frame to the navigation frame, and \(\mathbf{\omega}_{b}\) is expressed as follows:  
\[\mathbf{R}_{t}=\left[ \begin{array}{ccc}c\partial c\psi&s\phi s\partial c\psi-c \phi s\psi&c\phi s\partial c\psi+s\phi s\psi\\ c\partial s\psi&s\phi s\partial s\psi+c\phi c\psi&c\phi s\phi-s\phi c\psi\\ -s\theta&s\partial c\theta&c\phi c\theta\end{array} \right], \tag{3}\]  
\[\mathbf{\omega}_{b}=\mathbf{R}_{r}=\left[ \begin{array}{ccc}1&0&-s\theta\\ 0&c\phi&c\theta s\phi\\ 0&-s\phi&c\theta c\phi\end{array} \right]\left[ \begin{array}{c}\dot{\phi}\\ \dot{\theta}\\ \dot{\psi}\end{array} \right], \tag{4}\]  
where \(s(\cdot)\) and \(c(\cdot)\) denote \(\sin(\cdot)\) and \(\cos(\cdot)\), respectively. The lift force \(\mathbf{F}=\left[0,\,0,\,u_{f}\right]^{T}\) and torque \(\mathbf{M}_{b}=\left[u_{\phi},\,u_{\theta},\,u_{\varphi}\right]^{T}\) are generated by rotors with respect to body-fixed frame as follows:  
\[ \begin{array}{c}u_{f}=\sum\limits_{i=1}^{6}F_{i},\\ u_{\phi}=-\frac{1}{2}F_{i}l+\frac{1}{2}F_{2}l+F_{3}l+\frac{1}{2}F_{4}l-\frac {1}{2}F_{5}l-F_{6}l,\\ u_{\theta}=-\frac{\sqrt{3}}{2}F_{1}l-\frac{\sqrt{3}}{2}F_{2}l+\frac{\sqrt{3} }{2}F_{4}l+\frac{\sqrt{3}}{2}F_{5}l,\\ u_{\psi}=C_{d}(-\Omega_{1}^{2}+\Omega_{2}^{2}-\Omega_{3}^{2}+\Omega_{4}^{2}- \Omega_{5}^{2}+\Omega_{6}^{2}),\end{array}  \tag{5}\]  
where \(F_{i}=C_{p}\Omega_{i}^{2}\) denotes the force generated in the \(i\)-th rotor, \(C_{d}>0\) and \(C_{p}>0\) denote coefficients of drag and thrust of the propeller, respectively, and \(l\) denotes the distance from rotors to the center of mass. Given the perturbation of the mass of the body due to the loading of the carrying object, (1) is expressed as follows:  
\[m_{0}\dot{\mathbf{v}}_{b}=\mathbf{R}_{t}\mathbf{F}-m_{0}\mathbf{g}\mathbf{e}_{z}+\mathbf{U}_{b}, \tag{6}\]  
where \(\mathbf{U}_{b}=-\Delta m\dot{\mathbf{v}}_{b}+\mathbf{F}_{d}\) and \(\Delta m\) denotes a mass perturbation.