### Related Work  
The existing work in the literature that shares the most similarities with this paper can be found in [58; 59; 60; 61], where nonlinear attitude controllers for fixed-wing UAVs are developed using quaternions, and that also use a model of the rotational dynamics. In [58, 59], the translational and rotational subsystems are decoupled by estimating the higher-order derivatives of the angle of attack and sideslip angle. This enables controllers for the two subsystems to be designed separately. In [60], a nonlinear PID controller for fixed-wing UAV (full) attitude control is presented. The control law is based on unit quaternions and compensates for aerodynamic coupling effects using integral action. This approach is extended in [61] to apply also to rudderless (i.e., underactuated in attitude) fixed-wing UAVs by using a projection of the quaternion error to a yaw-free subspace. In [62], a gain-scheduled attitude controller based on Euler angles is given. An algorithm for automatic tuning is provided, and the control system is verified experimentally in a wind tunnel.  
Reduced-attitude control has been extensively applied for thrust-vector control of _multirotor_ UAVs, e.g., [48]. _Fixed-wing_ UAVs on the other hand, are subject to additional aerodynamic forces and moments that make control of such vehicles fundamentally different. Besides, the reduced-attitude representation used in this paper (gravity direction represented in body-fixed frame) is different than the thrust-direction of multirotors (body-fixed axis represented in the inertial frame). The representation used here is similar to that used to stabilize the inverted equilibrium manifold of the 3-D pendulum in [31, 63].  
The idea of separately controlling reduced attitude and another variable that is decoupled from the reduced-attitude vector is not new. In [64], the reduced attitude is steered along a geodesic path, while the full attitude is stabilized. In [65, 66], the attitude control of a quadrotor is decoupled into thrust-vector control on \(\mathbb{S}^{2}\), and control of the angle of rotation about the thrust vector. A similar approach is taken in [67] with a control allocation strategy that allows to prioritize reduced-attitude correction over yaw errors. Different rotational error metrics for quadrotor control, defined in terms of both full and reduced attitude are compared in [68]. In [69], a vector-projection algorithm is used for trajectory tracking for an agile fixed-wing UAV (where aerodynamics are dominated by the propeller). The roll angle is decoupled from the reference attitude such that thrust and lift forces can be pointed such that position tracking is achieved. Compared to these works, we simultaneously control reduced attitude and an _angular velocity_ around the reduced-attitude vector.  
While the present work employs Lyapunov-based methods to develop lightweight control laws with stability guarantees, other approaches using optimal control algorithms have also been proposed, using deep reinforcement learning [70] and nonlinear model predictive control [71].  
Preliminary results of the work presented in this paper have previously been reported in [72], and some initial work towards extending this by applying tools from hybrid control can be found in [73].