## 3 UAV Rotational Dynamics
### Control-Oriented Model  
To summarize, the UAV rotational dynamics can be written  
\[J\dot{\omega}=S(J\omega)\omega+h(v_{r})+V_{a}D\omega+V_{a}^{2}Bu+M_{p}. \tag{7}\]In horizontal, level flight, the angular velocity \(\omega\) is zero. To ensure equilibrium flight (\"trim conditions\"), define  
\[u_{\text{trim}}=\frac{1}{(V_{a}^{*})^{2}}B^{-1}\Big{[}-h(v_{r}^{*})-M_{p}^{*} \Big{]},\]  
where \(V_{a}^{*}\), \(v_{r}^{*}\), \(M_{p}^{*}\) is the trim airspeed, trim relative velocity and trim propeller moment (corresponding to the trim throttle setting), respectively. If \(u=u_{\text{trim}}\) and \(\omega=0\), then \(\dot{\omega}=0\) during trimmed flight. Now define  
\[\Delta(v_{r},t)=V_{a}^{2}Bu_{\text{trim}}+h(v_{r})+M_{p} \tag{8}\]  
which represents the deviation from trimmed flight. We can now combine (8), the rotational dynamics (7) and the reduced-attitude kinematics (4) to obtain the following model that is the basis for control design:  
\[\dot{\eta} =\eta\times\omega \tag{9}\] \[J\dot{\omega} =S(J\omega)\omega+V_{a}D\omega+V_{a}^{2}B[u-u_{\text{trim}}]+ \Delta(v_{r},t). \tag{10}\]  
The state is represented by \((\eta,\omega)\in\mathbb{S}^{2}\times\mathbb{R}^{3}\), the control input is \(u\in\mathbb{R}^{3}\), and we consider \(v_{r}\) (and thus \(V_{a}\)) as an exogenous bounded input.  
To facilitate control design, we will assume the following:  
**Assumption 1**.: _The airspeed \(V_{a}\) is strictly positive and bounded with bounded derivative \(0<V_{\text{min}}\leq V_{a}\leq V_{\text{max}}\)._  
**Assumption 2**.: _The moment vector \(\Delta(v_{r},t)\) and its derivative \(\dot{\Delta}(v_{r},t)\) are bounded._  
**Assumption 3**.: _The control effectiveness matrix \(B\) is invertible._  
**Assumption 4**.: _The damping matrix \(D\) satisfies \(x^{\top}Dx\leq 0,\forall x\in\mathbb{R}^{3}\)._  
**Remark 2**.: _Assumption 2 is an assumption on the translational dynamics, which is assumed to affect the rotational dynamics through the exogenous signal \(v_{r}\) (may also be considered as \"internal dynamics\"). In practice, since we are dealing with a physical system, \(\Delta(v_{r},t)\) and \(\dot{\Delta}(v_{r},t)\) will always be of bounded magnitude. However, since the control input is bounded, we would want these bounds to be relatively small. In particular, during nominal flight, the angle of attack \(\dot{\alpha}\) is usually small, and the lift coefficient is such that a perturbation in a tends to be restored [1]. However, if the stall angle of attack is reached, the slope of the lift coefficient changes such that the \(\dot{\alpha}\)-dynamics might go unstable, which in turn results in a high aerodynamic moment \(\Delta(v_{r},t)\)._  
**Remark 3**.: _A square matrix \(B\) corresponds to a fixed-wing UAV that has fully actuated rotational dynamics, i.e., three independent actuators. Now \(B\) is invertible if it has full rank. It can be shown that the full rank condition corresponds to primary control coefficients being larger than the coefficients associated with secondary roll-yaw coupling effects. The full rank assumption is therefore reasonable for most common fully actuated control surface configurations._  
**Remark 4**.: _Assumption 4 is a dissipation assumption and is equivalent to requiring that \(\text{sym}(D)\) has nonpositive eigenvalues. In nominal flight conditions, this will be true for most airframes [77] but can be relaxed by using a higher derivative gain (adding damping to the system). See Remark 8._