## 4 Robust Saturated Backstepping Tracking Controller Design
 \tag{34}\]  
Here, the desired heading angle \(\psi_{d}\) is determined by some high-level guidance algorithm that will not be discussed as it is beyond the scope of this paper. Manipulating (34) yields  
\[ \begin{cases}u_{1}=m\sqrt{F_{1}^{2}+F_{2}^{2}+(F_{3}+g)^{2}}\\ \phi_{d}=\arcsin(\frac{m}{u_{1}}[F_{1}\sin(\psi_{d})-F_{2}\cos(\psi_{d})])\\ \theta_{d}=\arctan(\frac{1}{F_{3}+g}[F_{1}\cos(\psi_{d})+F_{2}\sin(\psi_{d})]) \end{cases}  \tag{35}\]  
Let \(\mathbf{e}_{4}\) denote attitude tracking error as  
\[\mathbf{e}_{4}=\mathbf{X}_{4}-\mathbf{\bar{X}}_{4} \tag{36}\]  
with \(\mathbf{\bar{X}}_{4}=[\phi_{d},\theta_{d},\psi_{d}]\) being a virtual control law. Then, the derivative of \(\mathbf{e}_{4}\) can be obtained as  
\[\dot{\mathbf{e}}_{4}=\dot{\mathbf{X}}_{4}-\dot{\mathbf{X}}_{4}=\mathbf{X}_{5}-\dot{\mathbf{X}}_{4} \tag{37}\]A virtual control law for (37) is constructed as  
\[\bar{\mathbf{X}}_{5}=-k_{4}(\hat{\mathbf{X}}_{4}-\bar{\mathbf{X}}_{4})+\hat{\mathbf{X}}_{4} \tag{38}\]  
where \(k_{4}\in\mathbb{R}^{+}\) is a controller gain. Thus, the angular rate tracking error is defined as  
\[\mathbf{e}_{5}=\mathbf{X}_{5}-\bar{\mathbf{X}}_{5} \tag{39}\]  
The following LPF is applied to compute the approximation of \(\hat{\mathbf{X}}_{5}\),  
\[\tau_{5}\hat{\mathbf{X}}_{5f}+\mathbf{X}_{5f}=\bar{\mathbf{X}}_{5} \tag{40}\]  
where \(\tau_{5}\in\mathbb{R}^{+}\). Then,  
\[\hat{\mathbf{X}}_{5}=\hat{\mathbf{X}}_{5f}+\epsilon_{5} \tag{41}\]  
with the approximation error \(\epsilon_{5}\) satisfying \(\parallel\)\(e_{5}\)\(\parallel\)\(\leq\)\(E_{5}\in\mathbb{R}^{+}\). Taking derivative of (39), we have  
\[\dot{\mathbf{e}}_{5}=\dot{\mathbf{X}}_{5}-\dot{\mathbf{X}}_{5}=\mathbf{X}_{6}+f_{5}(\mathbf{X}_{5} )+B_{5}\mathbf{U}_{2}-\dot{\mathbf{X}}_{5} \tag{42}\]  
To address the control input saturation corresponding to \(\mathbf{U}_{2}\), let us consider the following Lyapunov candidate function,  
\[V_{3}=\frac{1}{2}(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T}(\mathbf{e}_{5}-\mathbf{\xi}_{2})+\frac{ 1}{2}\xi_{2}^{T}\xi_{2} \tag{43}\]  
where, \(\mathbf{\xi}_{2}\) is an auxiliary variable relating to the input saturation and its dynamics will be designed later. Differentiating Equation (37) yields  
\[\dot{V}_{3}=(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T}(\dot{\mathbf{e}}_{5}-\dot{\mathbf{\xi}}_{2}) +\xi_{2}^{T}\dot{\mathbf{\xi}}_{2} \tag{44}\]  
Substituting Equation (36) into (38) gives  
\[\dot{V}_{3}=(\mathbf{e}_{5}-\mathbf{\xi}_{2})^{T}(\mathbf{X}_{6}+f_{5}(\mathbf{X}_{5})+B_{5} \mathbf{U}_{2}-\dot{\mathbf{X}}_{5}-\dot{\mathbf{\xi}}_{2})+\xi_{2}^{T}\dot{\mathbf{\xi}}_{2} \tag{45}\]  
To achieve the \(\dot{V}_{3}\) negative definite, the control vector \(\mathbf{U}_{2}\) and the dynamics of \(\mathbf{\xi}_{2}\) are designed as  
\[ \begin{cases}\mathbf{U}_{2}=\mathbf{B}_{5}^{-1}[-f_{5}(\hat{\mathbf{X}}_{5})-k_{5}(\hat\mathbf{{X}}_{5}-\bar{\mathbf{X}}_{5}-\mathbf{\xi}_{2})-\hat{\mathbf{X}}_{6}+\hat{\mathbf{X}}_{5f}+\hat {\mathbf{\xi}}_{2}]\\ \dot{\mathbf{\xi}}_{2}=-k_{6}\mathbf{\xi}_{2}+\Delta\mathbf{U}_{2}\end{cases}  \tag{46}\]  
where \(k_{5},k_{6}\in\mathbb{R}^{+}\) and \(\Delta\mathbf{U}_{2}=\text{sat}(\mathbf{U}_{2})-\mathbf{U}_{2}\). Assume that \(\Delta\mathbf{U}_{2}\) satisfies \(\parallel\Delta\mathbf{U}_{2}\parallel\)\(\leq\)\(v_{2}\) with \(v_{2}\in\mathbb{R}^{+}\).