REFINEMENT OF THE PARAMETERS OF A MATHEMATICAL MODEL OF

Errors in the calculation of the parameters of quadcopter control models at design stage significantly change the desired aerodynamic properties of the drone and make it difficult to control its flight along the intended path. Therefore, to calculate the adequate operation modes of the blades, it becomes necessary to refine some parameters of the mathematical model of the drone as accurately as possible. This paper shows the possibility of using control parameters (rotational speed of the blades) and information received from navigation devices of the drone to refine the values of the parameters of the mathematical model of the drone. For this purpose, a mathematical model of a quadcopter is built, and the problem of refining the parameters of its dynamic model is investigated based on the information received from navigation devices and the control parameters in the initial period of its flight. From the results obtained from several consecutive measurements, a system of equations expressing a mathematical model is solved. The mean value of the corresponding solutions of the system of three-dimensional linear equations obtained at different time intervals is the refined value of the parameters.


INTRODUCTION
The low cost of unmanned aerial vehicles (drones) has given impetus to their widespread use for military purposes. The possibility of equipping them with appropriate hardware makes it possible for them to perform various tasks. Quadcopters should be mentioned among such currently widely used vehicles. As a rule, quadcopters are equipped with navigation devices (gyroscope, accelerometer, etc.), which allow obtaining information related to the drone's location and aerial orientation and control its flight.
Dynamic mathematical models describe the control and movement of the quadcopter. In this study, the drone is considered as a solid, and its mathematical model expresses the relationship between the thrust generated by the motor and acceleration, air drag, torque, rotational speed and velocity of the drone relative to the earth. There is a wide range of studies dealing with the dynamic modelling of drones. For instance, [1] is devoted to the building of a mathematical model of a drone to solve the problem of eliminating its deviation from the intended flight path. In [2], a mathematical model was built to control the stability and flight path of a quadcopter. In the paper, feedback data related to the orientation of the drone is given in the form of Euler angles. In the mathematical model built in [3], the aerodynamic forces acting on the drone are not considered, and the case of its flight at low speeds was studied. In [4], a mathematical model that defines the spatial position of the quadcopter by Euler angles was developed and used to create a flight simulation program. In [5], a mathematical model was developed, the quadcopter's spatial position being expressed by Euler angles, and a controller stabilising the altitude and orientation of the drone was developed based on this model. Several currently designed drone models provide for the use of MPU-9250 type devices [6]. Since this device does not measure Euler angles, it is impossible to directly apply the results of the above studies to solving problems of quadcopter control when using it. Thus, the accelerometer and gyroscope of MPU-9250 allow calculating the loads that occur during movement, as well as the rate of change of Krylov rotation angles (roll, pitch and yaw) [7, P.9], which determine the spatial position of the drone, and current orientation angles by integrating these speeds.
A mathematical model of the quadcopter control problem is built in this paper based on feedback data received from an MPU-9250 type device. The rotational speeds of the quadcopter blades are used as control parameters. It is assumed that:  the drone has 4 symmetrically fixed identical motors that rotate its blades;  for clarity, if we indicate the drone motors as shown in Fig. 1, blades 1 and 3 rotate clockwise, and blades 2 and 4 counterclockwise;  due to the small angular rotational velocities, the gyroscopic forces created by the movement of the drone can be neglected;  gravity and aerodynamic drag affect the centre of mass of the drone, so these forces do not create a torque;  the mass of the drone is distributed only along its arms, in other words, the drone's rotary inertia matrix has a diagonal shape.
The dynamic model of a quadcopter is expressed by numerous parameters. These parameters are usually calculated using empirical formulas at the drone design stage. Stable drone control requires these parameters to be known as accurately as possible.

Fig. 1. Quadcopter schematic
Therefore, this paper explores the possibility of using the control parameters (rotational speed of the blades) and the data obtained from navigation devices of the drone to refine the values of the parameters of the mathematical model of the drone.

MATHEMATICAL MODEL OF THE DRONE
To determine the position of the quadcopter relative to the ground and build a dynamic model, a fixed-in-the-earth right-handed normal The mathematical model of drone control provides for setting such a relationship between the flight path   for the execution of the given flight path   To specify the physical (inertial) characteristics of the quadcopter, we introduce a rectangular right-handed Oxyz coordinate system fixed to it [7, P.23]. The quadcopter can be described schematically as shown in Fig. 1. Suppose that the origin of coordinates is located in the centre of the drone, the Ox and Oy axes are directed along its "arms", and the Oz axis is directed upward perpendicular to the Oxy plane. As mentioned above, when designing a drone control system, MPU-9250 type devices are used to provide feedback [6]. These devices allow expressing the spatial position (orientation) of the drone by the yaw angle The definition of yaw, pitch and roll angles is given in [7, P.9]. These angles virtually indicate the spatial position of the Oxyz coordinate system fixed to the drone relative to the normal (2) can be used to find in the Oxyz coordinate system the components of the vector given in the fixed-in-the-earth coordinate system. For simplicity, for the angle  under consideration,  S and  C are written instead of  sin and  cos from here next.
To build the drone control model, we write the equations of its movement relative to the Oxyz coordinate system [9, P.128]: The position of the drone relative to the g g g g z y x O coordinate system can be identified with the position of the Oxyz coordinate system relative to . Taking this into account, the angular velocity w and the angular acceleration ε of the drone relative to can be calculated as follows: Let us give the procedure of calculation of the vectors F and M on the right-hand side of the system of equations (3)-(4). Following the above conditions for the direction of rotation of the drone's blades, the propelling force created by its motors is always oriented in the direction of the Oz axis of the fixed Oxyz coordinate system: where k is a coefficient determined experimentally, j  is the rotational speed of the blades of the j -th motor. Here and thereafter, the asterisk indicates the transposition operation.
The aerodynamic drag is directed opposite to the movement of the drone: where a C is the drag coefficient, which is proportional to the area of the projected plane perpendicular to the direction of the drone's movement and depends on its aerodynamic shape. In the general case, the relationship between the coefficient a C and the aerodynamic shape is very complex and depending on the direction of movement for the drone under consideration, its value can be determined experimentally [10]. Usually, for simplicity, this coefficient is assumed to be identical in all directions. Therefore, the force a F can be broken into the following components: Regardless of the spatial position, gravity acting on the drone is always directed vertically to the earth. Using the transformation matrix (1), we can obtain the following formula for expressing gravity relative to the Oxyz coordinate system: As mentioned above, it is assumed that gravity and aerodynamic drag affect the centre of mass of the drone, so these forces do not create a torque. Since the drone's motors are fixed to it, the torque generated by the rotation of the blades can be expressed in the Oxyz coordinate system as follows: where l is the length of the arms of the drone, and b is a coefficient determined experimentally.
Also, according to the condition that the drone rotary inertia matrix has a diagonal shape, it can be written as the following diagonal matrix: According to equality (6) If we write the values in equations (3)-(4) in terms of components, we can obtain the following system of differential equations for the dynamic model of the drone: Based on equations (4), we can write equations expressing the components of the drone's velocity vector relative to the earth: To solve system (1), (6)-(16) by the functions initial values for a certain moment 0 t t  must be given: System (1), (6)-(20), being a Cauchy problem written for a system of ordinary differential equations, expresses the dynamic model of a drone.

STATEMENT OF THE PROBLEM OF PARAMETER REFINEMENT (IDENTIFICATION)
Depending on the tasks performed, devices connected to the quadcopter (photo and video camera, radio repeater, packages of various shapes, etc.) can to some extent change their aerodynamic and technical characteristics. In this regard, the values of some parameters of the mathematical model of a quadcopter can differ from the calculated indicators.
The dynamic model of a quadcopter is expressed by several parameters, including the total mass of the drone, the components of the moment of inertia, the coefficient of proportionality between the rotational speed of the blades and the lifting force that they create, and other quantities. These values, in turn, depend on the size of the drone, the mass of its parts, the distribution of these parts relative to the centre of gravity and other factors. They can be determined experimentally or calculated analytically in the framework of hypotheses put forward in building the mathematical model. However, the errors in finding these parameters significantly change the aerodynamic properties of the drone, which makes it difficult to control its flight along the intended path.
The change in aerodynamic properties is reflected in the fact that the rotation of the blades in design modes is not enough for the drone to move along the intended path, and these modes need to be altered. In essence, it is necessary to refine some parameters of the mathematical model of the drone as accurately as possible to calculate adequate modes of operation of the blades. Mathematically, this is considered an inverse problem. Thus, the possibility of using the control parameters (rotational speed of the blades) and the data from navigation devices of the drone to refine the values of the parameters of the mathematical model of the drone is investigated in the following paragraphs.
An analysis of the dynamic model of a quadcopter written in the form of system (1), (6)-(20) above shows that the quantities in the model are divided into three groups.
The first group includes control parameters