Mathematical model of an unmanned aerial vehicle with four propulsors (quadcopter) is indispensable in quadcopter movement simulation and later modelling of the control algorithm. Mathematical model is, at the same time, the first step in comprehending the mathematical principles and physical laws which are applied to the quadcopter system. The objective is to define the mathematical model which will describe the quadcopter behavior with satisfactory accuracy and which can be, with certain modifications, applicable for the similar configurations of multirotor aerial vehicles. At the beginning of mathematical model derivation, coordinate systems are defined and explained. By using those coordinate systems, relations between parameters defined in the earth coordinate system and in the body coordinate system are defined. Further, the quadcopter kinematic is described which enables setting those relations. Also, quadcopter dynamics is used to introduce forces and torques to the model through usage of Newton-Euler method. Final derived equation is Newton's second law in the matrix notation. For the sake of model simplification, hybrid coordinate system is defined, and quadcopter dynamic equations derived with the respect to it. Those equations are implemented in the simulation. Results of behavior of quadcopter mathematical model are graphically shown for four cases. For each of the cases the propellers revolutions per minute (RPM) are set in a way that results in the occurrence of the controllable variables which causes one of four basic quadcopter movements in space.

Altitude and attitude controlled quadcopter model is used for the behavior and control algorithm testing, before the implementation on the experimental setup. The main objective is the control system design which will achieve good task performance in the combination with the quadcopter dynamic model. Also, for the control model, it is important to be relatively easy to modify for the use of other control algorithms and to be easy to implement on the experimental setup. At the beginning of this article, the control system design process is described. Quadcopter dynamics equations are simplified by applying several assumptions and in that form are used for the controller synthesis. Quadcopter control system is split into inner and outer control loop because the quadcopter is underactuated system which means that the direct control of all of the degrees of freedom is not possible. In the second part, the PI-D control algorithm is described which is applied on the simplified quadcopter dynamic model. The inner loop controls roll, pitch and yaw angles together with the quadcopter altitude. Its outputs are four control variables. Outer loop controls quadcopter X and Y position. Its outputs are roll and pitch desired angles. Regulated quadcopter model behavior is shown for the three types of task. First, the achieving of position in space is simulated. Second, the reference trajectory tracking is shown. Last task shown is the reference trajectory tracking with added periodical disturbances. Simulations show bounded positions error of the regulated quadcopter system using PI-D controller for the different types of tasks performed under different conditions.