LQR Control for a Quadrotor using Unit Quaternions: Modeling and Simulation Elias Reyes-Valeria, Rogerio Enriquez-Caldera, Electronics Department INAOE Puebla, Mexico rogerio@inaoep.mx Sergio Camacho-Lara and Jose Guichard Astrophysics Department CRECTEALC/INAOE Puebla, Mexico jguichard@inaoep.mx Abstract— Nowadays, quadrotors have become a popular UAV research platform because full control can be achieved through speed variations in each and every one of its four rotors. Here, a non-linear dynamic model based on quaternions for attitude is presented as well as its corresponding LQR Gain Scheduling Control. All considerations for the quadrotor movements are described through their state variables. Modeling is carried out through the Newton-Euler formalism. Finally, the control system is simulated and the results shown in a novel and direct unit quaternion. Thus, a successful trajectory and attitude control of a quadrotor is achieved. Keywords—Quadrotor, UAV, LQR, Unit Quaternion I. INTRODUCTION Interest is growing around Unmanned Aerial Vehicles (UAV’s) due to their industrial and military uses. Aerial photography, mapping, surveillance, search and rescue are among of many intelligence duties that can be carried out without risking the life of an operator. Many research communities continue to develop a great variety of controllers for UAV. Nevertheless, in most of practical control implementations, position control is done by a remote operator that obtains visual feedback information using an on board camera. Meanwhile, attitude control is automatically performed by a local control coupled to the aircraft. However, it is also common for the Control to use rotation matrices based on Tait-Bryan angles whose main flaw is the existence of certain critical points with repercussion in the lost of degrees of freedom. The alternative to such a problem is to make the attitude control using a unit quaternion. These kind of aircrafts make use of four motors i m to produce the required forces for corresponding maneuver on the UAV as depicted in Fig. 1. Thus, the angular movement around the x-axis, known as roll φ , is produced by the resulting torque of 2 m and 4 m . Similarly, the resulting torque of 1 m and 3 m produce an angular movement around the y- axis, known as pitch θ . The movement around z-axis, known as yaw ψ , is due to the torque around z-axis when incrementing the speed of motors along x-axis while lowering speed of motors along y-axis if the principal uplifting force is kept constant. Such movements can be understood from Fig. 1. This article is organized as follows: Section II briefly describes quaternions and all mathematical expressions used in this paper. Section III, introduces the necessary quaternion Kinematics. Section IV, describes the dynamic model for the quadrator under study taking into account the proper considerations. In Section V, the quadrator model is linearized with the purpose to later use Linear Control. Section VI, develops the corresponding LQR Gain Scheduling Control. Section VII, shows the simulated results for the designed system which was implemented using Simulink for the observer and feedback. Finally, in Section VIII, conclusions are given as well as future recommendations. Figure 1. Quadrotor: Forces and movements