## 2 Model Description and Transformation
### Model Transformation  
Define the reference trajectories of the position and attitude subsystems as \(\mathbf{\zeta}_{d}=[x_{d},y_{d},z_{d}]^{\mathrm{T}}\) and \(\mathbf{\Theta}_{d}=[\phi_{d},\theta_{d},\psi_{d}]^{\mathrm{T}}\). Due to the under-actuated characteristics of a quadrotor UAV, the desired roll and pitch can be generated by  
\[\mathbf{\phi}_{d}=\arcsin\biggl{(}\frac{u_{q1}\sin\psi_{d}-u_{q2}\cos\psi_{d}}{ \big{\|}u_{q}\big{\|}}\biggr{)},\theta_{d}=\arctan\Bigl{(}\frac{u_{q1}\cos \psi_{d}+u_{q2}\sin\psi_{d}}{u_{q3}}\Bigr{)},\]  
where \(\mathbf{u}_{q}=\frac{\mathbf{u}}{m_{0}}\) with \(u_{q}=[u_{q1},u_{q2},u_{q3}]^{\mathrm{T}}\). Then, the tracking errors can be represented as \(\mathbf{\zeta}_{e}=[x_{e},y_{e},z_{e}]^{\mathrm{T}}=\mathbf{\zeta}-\mathbf{\zeta}_{d}\) and \(\mathbf{\Theta}_{e}=[\phi_{e},\theta_{e},\psi_{e}]^{\mathrm{T}}=\mathbf{\Theta}-\mathbf{ \Theta}_{d}\). Combining with (3) and (4), one can further obtain the error subsystems:  
\[\left\{ \begin{array}{l}\dot{\mathbf{e}}_{11}=\mathbf{e}_{12},\\ \dot{\mathbf{e}}_{12}=-g\mathbf{q}-\mathbf{P}_{1}\mathbf{e}_{12}+\tfrac{1}{m_{0}}(\mathbf{u}+\mathbf{d }_{1})-\mathbf{\tilde{\zeta}}_{d}-\mathbf{P}_{1}\mathbf{\dot{\zeta}}_{d},\end{array} \right. \tag{5}\]  
\[\left\{ \begin{array}{l}\dot{\mathbf{e}}_{21}=\mathbf{e}_{22},\\ \dot{\mathbf{e}}_{22}=-\mathbf{P}_{2}\mathbf{e}_{22}+J_{0}^{-1}(\mathbf{\tau}+\mathbf{d}_{2})-\mathbf{ \tilde{\Theta}}_{d}-\mathbf{P}_{2}\mathbf{\dot{\Theta}}_{d},\end{array} \right. \tag{6}\]  
where \(\mathbf{e}_{11}=\mathbf{\zeta}_{e},\mathbf{e}_{12}=\mathbf{\tilde{\zeta}}_{e},\mathbf{e}_{21}=\mathbf{ \Theta}_{e}\), and \(\mathbf{e}_{22}=\mathbf{\dot{\Theta}}_{e}\). For subsequent optimal controller design, divide the control inputs \(\mathbf{u}\) and \(\mathbf{\tau}\) into two parts (i.e., \(u=u_{0}+u_{1},~{}\mathbf{\tau}=\mathbf{\tau}_{0}+\mathbf{\tau}_{t}\)). By setting two feedforward controllers \(\mathbf{u}_{l}=m_{0}(g\mathbf{q}+\mathbf{\tilde{\zeta}}_{d}+P_{1}\mathbf{\dot{\zeta}}_{d})\) and \(\mathbf{\tau}_{l}=J_{0}(\mathbf{\tilde{\Theta}}_{d}+P_{2}\mathbf{\dot{\Theta}}_{d})\), (5) and (6) can be formulated as  
\[\dot{\mathbf{e}}_{i}=\mathbf{f}(\mathbf{e}_{i})+\mathbf{g}_{i}(\mathbf{\chi}_{i}+\mathbf{d}_{i}). \tag{7}\]As shown in (7), \(i=1,2\) represent the error subsystems of position and attitude, respectively. In addition, \(\mathbf{e}_{i}=[\mathbf{e}_{i1},\mathbf{e}_{i2}]^{\top}\), \(\chi_{1}=\mathbf{u}_{0}\), \(\chi_{2}=\tau_{0}\):  
\[\mathbf{f}(\mathbf{e}_{i})= \begin{bmatrix}\mathbf{e}_{i2}\\ -\mathbf{P}_{i}\mathbf{e}_{i2}\end{bmatrix} ,\ \mathbf{g}_{1}= \begin{bmatrix}\mathbf{0}_{3 \times 3}\\ \frac{1}{m_{0}}\mathbf{I}_{3}\end{bmatrix} ,\ \mathbf{g}_{2}= \begin{bmatrix}\mathbf{0}_{3 \times 3}\\ J_{0}^{-1}.\end{bmatrix} \]  
On the basis of the above transformation of the dynamical model, the UAV trajectory tracking control problem is transformed into a stabilization problem. To the end, we first designed FTNNOBs to estimate the uncertainties in fixed time and give online compensation for the approximate optimal control policies. Then, the approximate optimal control policies for the nominal systems \(\dot{\mathbf{e}}_{i}=f(\mathbf{e}_{i})+\mathbf{g}_{i}\chi_{i}\) are obtained using the ADP technique. The block diagram of the control systems is shown in Figure 2.  
**Lemma 1** ([40]). : _Consider a nonlinear system in the form:_  
\[\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}),\ \mathbf{f}(\mathbf{0})=\mathbf{0}, \tag{8}\]  
_where \(\mathbf{f}\ :\ \mathbf{U_{0}}\to\mathcal{R}^{n}\) is continuous in an open neighborhood \(\mathbf{U_{0}}\) of the origin. Consider the system (8); if a Lyapunov function \(V(\mathbf{x})\) satisfies \(\dot{V}(\mathbf{x})\leq-\alpha V^{p}(\mathbf{x})-\beta V^{q}(\mathbf{x})+\vartheta\), where \(\alpha\), \(\beta\) and \(\vartheta\) are positive constants, and \(0<p<1\), \(q>1\), then the trajectory of the system is practical fixed-time stable within settling time \(t_{f}\), which satisfies \(t_{f}\leq\frac{1}{\alpha\omega(1-p)}+\frac{1}{\beta\omega(q-1)}\) with \(0<\omega<1\).