## 4 Almost Global Reduced-Attitude Tracking Control for Fixed-Wing UAVs
### Control Objective  
From our definition of \(e_{\omega}\), Equation (15), note that \(e_{\omega}\) can be decomposed into two orthogonal parts:  
\[e_{\omega}=\underbrace{(\omega^{\perp}-\Pi^{\perp}_{\eta}\omega^{\perp}_{d})} _{\in\mathrm{T}_{\eta}\mathbb{S}^{2}}+\underbrace{(\omega^{\parallel}- \omega^{\parallel}_{d})}_{\in\mathrm{N}_{\eta}\mathbb{S}^{2}}). \tag{19}\]This means that as \(e_{\omega}\) converges to zero, \(\omega^{\perp}\to\Pi_{\eta}^{\perp}\omega_{d}^{\perp}\) and \(\omega^{\parallel}\to\omega_{d}^{\parallel}\), in a decoupled manner. If in addition \(e_{\eta}=0\), then \(\Pi_{\eta}^{\perp}\omega_{d}^{\perp}=\omega_{d}^{\perp}\).  
The reduced-attitude error vector \(e_{\eta}\) is zero when \(\eta=\eta_{d}\). However, this is also the case when \(\eta=-\eta_{d}\). Naturally, this choice of configuration error leads to an additional undesired equilibrium point at \(\eta=-\eta_{d}\), but due to the topology of the sphere (it is a compact manifold), this is unavoidable when using continuous feedback [26]. The presence of more than one equilibrium prevents us from designing globally stabilizing feedback laws. A suitable notion of stability in this context is the concept of _almost global_ asymptotic stability.  
**Definition 1**.: _An equilibrium solution of a dynamical system is said to be almost globally asymptotically stable if it is asymptotically stable with an almost global domain of attraction, i.e., the domain of attraction is the entire state space excluding a set of Lebesgue measure zero [56]._  
As we consider continuous feedback on a compact configuration manifold, almost global asymptotic stability is the best possible achievable result [27]. In our setting, if the equilibrium point \((\eta,e_{\omega})=(\eta_{d},0)\) is almost globally asymptotically stable, then almost all trajectories converge to it, except for those with initial velocity (depending on the initial configuration error) that are _exactly_ such that \(\omega^{\perp}(t)-\Pi_{\eta}^{\perp}\omega_{d}(t)^{\perp}=0\) when \(\eta(t)=-\eta_{d}(t)\). This set of initial conditions has a dimension lower than the dimension of the state space, and therefore has measure zero.  
We now presicely state the control objective as follows:  
_Almost Global Reduced-Attitude Tracking_  
Design a state-feedback control law \(u\) such that for almost all \(e_{\eta}(t_{0}),e_{\omega}(t_{0}),\eta(t)\to\eta_{d}(t)\) and \(e_{\omega}(t)\to 0\) as \(t\to\infty\).  
**Remark 5**.: _Other configuration error vectors (with corresponding potential functions) on \(\mathbb{S}^{2}\) could be used in place of (14), without changing the general approach considered in this paper. The advantage of using (14) for proportional feedback is that it is simple, smooth, and globally defined. However, there are some performance issues, since for initial reduced-attitudes arbitrarily close to \(-\eta_{d}\), the control action will be close to zero, and the reduced attitude will stay there for an extended period before converging to the desired reduced attitude. Some alternative error vectors that do not vanish when approaching \(-\eta_{d}\), but are not defined at this point, are given in [45; 51; 63]._  
Before continuing with the controller design, we proceed with a discussion on different design choices for \(\omega_{d}^{\parallel}\).