## 2 Preliminaries
### Reduced-Attitude Representation  
Let \(R\in\text{SO}(3)\) be the rotation matrix transforming vectors from \(\{b\}\) to \(\{n\}\), and let \(e_{3}=[0\;\;0\;\;1]^{\top}\) represent the inertial \(z\)-direction (direction of gravitational acceleration). We employ the following reduced-attitude representation:  
\[\eta=R^{\top}e_{3}\in\mathbb{S}^{2}, \tag{2}\]  
which is interpreted as the inertial \(z\)-axis, expressed in \(\{b\}\). By expanding (2) using the roll-pitch-yaw Euler-angle parametrization of \(R\)[1], the reduced-attitude vector \(\eta\) can be expressed in terms of the roll angle \(\phi\in[-\pi,\pi]\) and pitch angle \(\theta\in(-\pi/2,\pi/2)\) as follows:  
\[\eta=\begin{bmatrix}-\sin(\theta)\\ \cos(\theta)\sin(\phi)\\ \cos(\theta)\cos(\phi)\end{bmatrix}. \tag{3}\]  
Observe that this particular choice of attitude representation is invariant to changes in the heading/yaw angle \(\psi\). The reduced attitude representation is illustrated in Figure 2, where a section of the sphere corresponding to \(\theta=0\) is shown. Figure 3 shows another section where the aircraft is shown from the side with a possible nonzero roll angle. As shown, the vector \(\eta\) is expressed in the body-fixed frame and points towards the ground.  
The reduced-attitude representation (2) is the same as the one considered for the 3-D pendulum in [31], but different compared to the one used for thrust-vector control for multirotor UAVs, which is the thrust direction in the inertial-frame [48].  
Let \(\omega\in\mathbb{R}^{3}\) be the angular velocity of the body-fixed frame relative to the inertial frame, expressed in the body-fixed frame. The reduced-attitude vector \(\eta\) satisfies  
\[\dot{\eta}=\eta\times\omega, \tag{4}\]  
which can be derived from (2) and the relation \(\dot{R}=RS(\omega)\)[74].  
Using (1), we can perform an orthogonal decomposition of the angular velocity \(\omega\) with respect to \(\eta\) such that \(\omega=\omega^{\perp}+\omega^{\parallel}\), where  
\[\omega^{\perp}\triangleq\Pi^{\perp}_{\eta}\omega\in\mathrm{T}_{\eta}\mathrm{ S}^{2}\qquad\omega^{\parallel}\triangleq\Pi^{\parallel}_{\eta}\omega\in \mathrm{N}_{\eta}\mathrm{S}^{2}. \tag{5}\]  
Applying this decomposition of \(\omega\) in combination with (4) gives  
\[\dot{\eta}=\eta\times(\omega^{\perp}+\omega^{\parallel})=\eta\times\omega^{ \perp}. \tag{6}\]  
The parallel component \(\omega^{\parallel}\) is the angular velocity about the inertial \(z\)-axis (expressed in the body-fixed frame) and does not influence \(\dot{\eta}\).  
Figure 2: Reduced-attitude representation illustrated with a section of the two-sphere corresponding to \(\theta=0\).  
**Remark 1**.: _Note that since the two-sphere \(\mathbb{S}^{2}\) is a two-dimensional manifold, in principle two degrees of freedom (DOFs) are sufficient to control reduced attitude. However, since \(\eta\) is fixed in the inertial frame, the two required DOFs (control directions) vary with the orientation of the vehicle and are thus not fixed in \(\{b\}\). Therefore, we need three actuators to make the reduced attitude fully controllable throughout the configuration space. In this paper, we consider only UAVs with fully actuated rotational dynamics, and at each time instant, we use the remaining DOF to control \(\omega^{\parallel}\)._