## 4 Almost Global Reduced-Attitude Tracking Control for Fixed-Wing UAVs  
### Error Functions  
The goal is to design a state-feedback control law \(u\in\mathbb{R}^{3}\) to make the reduced attitude \(\eta\in\mathbb{S}^{2}\) asymptotically track a smooth, time-varying reference \(\eta_{d}\in\mathbb{S}^{2}\) and at the same time drive \(\omega^{\parallel}\) to \(\omega^{\parallel}_{d}\), where \(\omega^{\parallel}_{d}\in\mathrm{N}_{\eta}\mathbb{S}^{2}\) denotes the desired value of \(\omega^{\parallel}\), yet to be specified. Furthermore, let the desired reduced-attitude vector \(\eta_{d}\) satisfy the reference model  
\[\dot{\eta}_{d}=\eta_{d}\times\omega^{\perp}_{d}, \tag{11}\]  
where \(\omega^{\perp}_{d}\in\mathrm{T}_{\eta_{d}}\mathbb{S}^{2}\).  
**Assumption 5**.: _The angular velocity references \(\omega^{\perp}_{d}\), \(\omega^{\parallel}_{d}\) and their derivatives \(\dot{\omega}^{\perp}_{d}\triangleq\frac{d}{d}\omega^{\perp}_{d}\), \(\dot{\omega}^{\parallel}_{d}\triangleq\frac{d}{d}\omega^{\parallel}_{d}\) can be bounded a priori by_  
\[\begin{split}\|\omega^{\parallel}_{d}\|&\leq c_{ \omega^{\parallel}_{d}}\quad\|\dot{\omega}^{\parallel}_{d}\|\leq b_{\omega^{ \parallel}_{d}}\|\omega\|+c_{\dot{\omega}^{\perp}_{d}}\\ \|\omega^{\perp}_{d}\|&\leq c_{\omega^{\perp}_{d}} \quad\|\dot{\omega}^{\perp}_{d}\|\leq c_{\dot{\omega}^{\perp}_{d}},\end{split} \tag{12}\]  
_where \(c_{\omega^{\parallel}_{d}},c_{\dot{\omega}^{\perp}_{d}},c_{\omega^{\perp}_{d}},c_{\dot{\omega}^{\perp}_{d}},b_{\omega^{\perp}_{d}}\in\mathbb{R}_{+}\) are appropriate constant parameters._  
Let a smooth configuration error function \(\Psi\colon\mathbb{S}^{2}\times\mathbb{S}^{2}\to\mathbb{R}\) be defined by  
\[\Psi(\eta,\eta_{d})=1-\eta^{\top}_{d}\eta=1-\cos\
u, \tag{13}\]  
where \(\
u\) is the angle between \(\eta\) and \(\eta_{d}\). The function \(\Psi\) measures the \"distance\" between two points \(\eta\) and \(\eta_{d}\) on \(\mathbb{S}^{2}\), and is clearly positive definite with respect to \(\eta=\eta_{d}\). There are two critical points: A minimum when \(\eta=\eta_{d}\), and a maximum when \(\eta=-\eta_{d}\). In subsequent Lyapunov analysis, \(\Psi\) is used as a pseudo-potential energy term in Lyapunov functions.  
We proceed by defining the following error vectors:  
\[e_{\eta} \triangleq\eta\times\eta_{d}\in\mathrm{T}_{\eta}\mathbb{S}^{2} \tag{14}\] \[e_{\omega} \triangleq\omega-\omega_{d}\in\mathbb{R}^{3}, \tag{15}\]  
where \(\omega_{d}=\Pi^{\perp}_{\eta}\omega^{\perp}_{d}+\omega^{\parallel}_{d}\). The error vector \(e_{\eta}\) can be viewed as a gradient vector field on \(\mathbb{S}^{2}\) induced by the potential function \(\Psi\)[53]. As \(\|e_{\eta}\|=|\sin\
u|\), \(e_{\eta}\) vanishes at the critical points of \(\Psi\). The error terms \(e_{\eta}\) and \(e_{\omega}\) are also compatible in the sense that \(\dot{\Psi}=e^{\top}_{\omega}e_{\eta}\), which will cancel with the proportional feedback term defined later when calculating the derivative of a Lyapunov function. The error vector \(e_{\eta}\) is geodesic in the sense that its direction defines an axis of rotation which connects \(\eta\) and \(\eta_{d}\) with the shortest possible curve on \(\mathbb{S}^{2}\).  
Differentiating \(e_{\eta}\) gives  
\[\dot{e}_{\eta} =-S(\eta_{d})S(\eta)\omega^{\perp}+S(\eta)S(\eta_{d})\omega^{ \perp}_{d} \tag{16}\] \[=-S(\omega^{\perp}_{d})e_{\eta}-S(\eta_{d})S(\eta)e_{\omega}, \tag{17}\]  
where we have used (15), the fact that \(\eta\times\omega^{\parallel}=0\) and the identity \(S(S(a)b)=S(a)S(b)-S(b)S(a)\) for any \(a,b\in\mathbb{R}\) (which can be derived using the Jacobi identity of vector cross products).  
From (10), the derivative of \(e_{\omega}\) satisfies  
\[J\dot{e}_{\omega}=S(J\omega)\omega+V_{a}D\omega+V_{a}^{2}B[u-u_{\mathrm{ trim}}]+\Delta(v_{r},t)-J\dot{\omega}_{d}. \tag{18}\]