## 4 Almost Global Reduced-Attitude Tracking Control for Fixed-Wing UAVs
### Coordinated-Turn Equation  
The coordinated-turn equation provides an approximation of the relationship between heading rate and the roll angle during banked-turn maneuvers, and is given by [1],  
\[\dot{\psi}=\frac{g}{V_{a}}\tan\phi, \tag{20}\]  
where \(g\) is the acceleration of gravity, \(V_{a}>0\), and the roll angle \(\phi\) has to satisfy \(|\phi|\
eq\pi/2\). With \(\omega=[p\ \ q\ r]^{\top}\), the heading rate can be also be written as a function of \(\omega\) and the Euler angles roll and pitch as  
\[\dot{\psi}=q\frac{\sin\phi}{\cos\theta}+r\frac{\cos\phi}{\cos\theta},\quad| \theta|\
eq\frac{\pi}{2}. \tag{21}\]  
From (3) and (21) we can relate \(\omega^{\parallel}\) to \(\dot{\psi}\) as follows:  
\[\omega^{\parallel}=(\eta^{\top}\omega)\eta=(-p\sin\theta+\dot{\psi}\cos^{2} \theta)\eta. \tag{22}\]Although the heading rate \(\dot{\psi}\), given by (21), is not well defined for \(|\theta|=\pi/2\), \(\omega^{\parallel}\) is globally defined. Furthermore, if \(|\theta|\
eq\pi/2\), the body-fixed roll rate \(p\) satisfies \(p=\dot{\psi}-\dot{\psi}\sin\theta\). Therefore, for \(|\theta|<\pi/2\), Equations (21) and (22) can be combined to obtain  
\[\omega^{\parallel}=(\dot{\psi}-\dot{\phi}\sin\theta)\eta. \tag{23}\]  
Motivated by (23) and the coordinated-turn Equation (20), we propose the following design for \(\omega^{\parallel}_{d}\) that satisfies Assumption 5:  
\[\omega^{\parallel}_{d}=\bigg{(}\frac{g}{V_{a}}\tan\phi_{d}-\dot{\phi}_{d}\sin \theta_{d}\bigg{)}\eta, \tag{24}\]  
where \(\theta_{d}\) and \(\phi_{d}\) are consistent with \(\eta_{d}\) (in the sense that (3) is satisfied). Clearly, since (23) is only valid for \(|\theta|\
eq\pi/2\), and (24) contains \(\tan\phi_{d}\), we restrict the desired reduced-attitude as follows:  
**Assumption 6**. : _The desired reduced-attitude \(\eta_{d}\) is such that \(|\theta_{d}|\leq c_{\theta_{d}}<\pi/2\) and \(|\phi_{d}|\leq c_{\phi_{d}}<\pi/2\), for some \(c_{\theta_{d}},c_{\phi_{d}}\in\mathbb{R}_{+}\), and \(\eta_{d},\phi_{d},\theta_{d}\) satisfy Equation (3)._  
**Remark 6**. : _We stress that the mentioned singularities at \(\phi=\pm\pi/2\) and \(\theta=\pm\pi/2\) are only present for the reference angles. The allowed reference orientations cover most typical flight conditions, except for certain aerobatic maneuvers. The controller design, however, is globally defined, which enables recovery from large reduced-attitude errors, e.g., resulting from large wind gusts._  
Alternative design choices for \(\omega^{\parallel}_{d}\):  
It is possible to consider some variations of the preceding design of \(\omega^{\parallel}_{d}\). We now present a few of these options, but leave it as an exercise to the reader to fully explore these possibilities. * An alternative to (24) is to define \(\omega^{\parallel}_{d}\) in terms of \(\eta_{d}\) and then project to \(\mathrm{N}_{\eta}\mathbb{S}^{2}\): \[\omega^{\parallel}_{d}=\Pi^{\parallel}_{\eta}\bigg{[}\bigg{(}\frac{g}{V_{a}} \tan(\phi_{d})-\dot{\phi}_{d}\sin\theta_{d}\bigg{)}\eta_{d}\bigg{]}=\bigg{(} \frac{g}{V_{a}}\tan(\phi_{d})-\dot{\phi}_{d}\sin\theta_{d}\bigg{)}(\eta^{ \top}_{d}\eta)\eta.\] The extra term \(\eta^{\top}_{d}\eta=\cos(\
u)\) puts less emphasis on turn coordination when errors in reduced attitude are large. * Equation (24) only satisfies the coordinated-turn Equation (20) asymptotically, as \(\eta\to\eta_{d}\) (and \(\phi\to\phi_{d}\)). One might consider to instead use the actual value of \(\phi\) instead of \(\phi_{d}\), but in the case, we cannot guarantee a priori that \(\omega^{\parallel}_{d}\) and its derivative are bounded. This means that the subsequent stability analysis needs to be adjusted. A pragmatic solution could be to use a saturation function in combination with (24). To summarize, the expression for the total desired angular velocity \(\omega_{d}\) in (15) is  
\[\omega_{d}=\Pi^{\perp}_{\eta}\omega^{\perp}_{d}+\omega^{\parallel}_{d},\]  
where \(\omega^{\parallel}_{d}\in\mathrm{N}_{\eta}\mathbb{S}^{2}\) is given by (24), and \(\omega^{\perp}_{d}\in\mathrm{T}_{\eta_{d}}\mathbb{S}^{2}\).