# Passivity-Based Control for a Micro Air Vehicle Using Unit Quaternions
## 1 Introduction  
Many efforts have been made to model and control quad-rotor Unmanned Aerial Vehicles (UAVs). In recent years, a wide number of strategies have been developed to solve the stabilization problem for this type of system using different methodologies in terms of control techniques and modeling approaches, such as [1; 2; 3; 4]. Several strategies based on Euler angles provide an intuitive and easy way to visualize the quad-rotor kinematic and dynamical properties, but some obstacles are found with this representation, for instance, certain orientations can not be defined due to mathematical singularities, also a problem known as \"gimbal lock\" exists, which consists on an infinite number of possible representations for a single rotation. A potential solution for this problem is the use of unit quaternions to represent the rigid body rotations. In the last years, several researchers have explored unit quaternions as an alternative to avoid these kind of issues. For instance, a quaternion-based nonlinear \(P^{2}\) controller, for solving the attitude problem of a quad-rotor is proposed in [5], the proposed control scheme performs very well with a very small overshoot and a very good reference tracking, however only numerical simulations are presented to prove the efficiency of the suggested scheme. In [6], a quaternion-based feedback is developed for the attitude stabilization of a quad-rotor. The control design takes into account a priori input bound and is based on nested saturations. The authors forced the closed-loop trajectories to enter in a fixed neighborhood around the origin in a finite time and remain thereafter. A quaternion-basednonlinear robust output feedback tracking controller is developed in [7] to address the attitude and altitude tracking problem of a quad-rotor, here, approximation components based on neural network are introduced to estimate the model uncertainties and robust feedback components are designed to compensate for external disturbances. Also, [8] investigates the attitude control design using backstepping technique for a quad-rotor represented by unit quaternions, external disturbance torques were taken into consideration, the attitude and the angular velocity of the quad-rotor were governed to be uniformly ultimately bounded by the proposed controller. Meanwhile, experimental results and implementation of a quaternion based attitude robust controller for quad-rotors are presented in [9]. Here, a state feedback controller which compensates nonlinearities and uncertainties is proposed, attenuating their effects in the closed-loop control system. A robust controller based on quaternion for the attitude control problem of quad-rotors with a nominal controller and a robust compensator is proposed in [10]. A quaternion based control scheme for the attitude control problem of a quad-rotor has been presented and experimentally evaluated in [11]. Recent works of quad-rotor's control based on quaternions can be found in [12; 13; 14; 15; 16]. Few works based on PBC (Passivity-Based Control) strategies applied to UAVs can be found in the literature. For example in [17], the authors presented a nonlinear control technique based on passivity to solve the path tracking problem for the quad-rotor, but only one control loop was considered in their work. For this work [PERSON], et al. showed that the PBC formulation leads to a set of partial differential equations constraints due to the underactuation degree of the system. In [18] the authors proposed a controller based on an extension of Interconnection and Damping Assignment-Passivity Based Control (IDA-PBC) framework, for shaping the whole physical characteristics of a quad-rotor and for obtaining a desired interactive behavior between the vehicle and the environment. In [19] a passivity-based approach is designed for this vehicle. The authors reduced the problem to the longitudinal plane to avoid solving complicated PDEs (Partial Differential Equations).