MATHEMATICAL MODELLING OF THE UNMANNED AERIAL VEHICLE DYNAMICS Belarusian National

The article gives a classification of the main components of unmanned aerial vehicle (UAV) systems, gives the areas in which the application of UAVs is actual in practice today. Further, the UAV is considered in more detail from the point of view of its flight dynamics analysis, the equation necessary for creating a mathematical model, as well as the model of an ordinary dynamic system as a non-stationary nonlinear controlled object, is given. Next, a description of the developed software for modeling and a description of program algorithm are given. Finally, a conclusion describes the necessary directions for further scientific researches.


Introduction
Unmanned aerial vehicle -a fairly new direc tion for research and are of interest not only for practical civil application like in Urban Search and Rescue Operations [1], but also for military operations with a high degree of success [2].
According to the classification of Global Hawk RYAN aeronautical center USA, the sche matic structure of unmanned aerial vehicle is (Fig. 1): There are unmanned aerial vehicles for vari ous purposes, a variety of aerodynamic schemes and with a variety of tactical and technical char acteristics.

Theory analysis
In general, the UAV can be used in a wide va riety of human activities, such as: -archeology (search under a layer of sand); -architecture (autonomously survey the ter rain and create 2D-and 3D-maps and terrain models); -aerial photography (UAVs allow you to cre ate digital maps with virtually any resolution, ranging from a few centimeters to a point); -safety monitoring (for example, during con struction works); -urban infrastructure (search for unautho rized dumps, detection of illegal buildings, quali ty control of the road surface, taking air samples, measuring radio emission levels); -Forestry (fighting poachers, identifying fires, smoke, monitoring obstacles, monitoring ani mals); -meteorology (search and / or study of hurri canes and other natural phenomena), etc.An unmanned aerial vehicle can, for example, provide search for accidents (accidents) of techni cal equipment and missing groups of people.The search is conducted according to a pre-set flight task or by an operative route of the flight.Usual ly, the UAV is equipped with guidance systems, airborne radar systems, sensors and video camer as.Advanced structure of the UAV systems [3] is shown in Figure 2.
The success of the UAV is primarily due to the rapid development of microprocessor-based computers, control systems, navigation, informa tion transfer and even artificial intelligence ele ments.Achievements in these areas make it possi ble to fly in automatic (semi-automatic) mode from take-off to landing, to solve a very wide range of tasks [4].
Consider the UAV in more detail from the point of view of the analysis of its flight dynamics.
Unmanned aerial vehicle, like other aircraft, uses its power point and aerodynamic forces for its flight in the atmosphere.
In general, the UAV movement is represented by two aspects: the forward movement of its cen ter of mass and the rotational motion around its center of mass.
To simulate an unmanned aerial vehicle, the necessary requirements, according to V. S. Moi seev, is: «... except for their adequacy and suffi cient for practical application of accuracy, we will consider the simplicity and intelligibility of mod els to the specialists in managing UAVs... ... The fulfillment of this requirement is con ditioned by the necessity of their active participa tion in the development on the basis of these models of effective control laws for UAVs.
In addition, the simplicity of the applied UAV motion models implies, as practice has shown, the relatively low laboriousness of mathematical methods and algorithms used in the formation of such laws...» [5, pp. 35-37].
The model chosen should correspond to the tasks in the simulation and adequately reflect the relationship between them.
The functioning of a complex system occurs when input random signals (influences, functions, processes) are affected, as well as various random disturbances (interference).
The mathematical model of a stochastic sys tem is described on the basis of information inter action between its constituent parts.Such a model can be represented physically by various mathe matical models, depending on the completeness and degree of detail of the processes and the final research task.In engineering practice and scientific research to describe systems from the point of view of the completeness of the description, the object is de scribed in terms of the state space.The state of an object is understood to mean the totality of the values of xi, which completely determine its posi tion at a given instant of time.
The most common model of dynamic objects are differential equations.It is most convenient to consider objects that are described by ordinary differential equations.The order of the system of differential equations describing the model of the object is not directly determined by the number of inputs and outputs, but depends on the operators that convert the input signals to the output [6].
Considering the above, we will describe the equation of an unmanned aerial vehicle, which is necessary for creating a mathematical model: The state of systems for which initial states and input effects are known can be described by the following equation: where x(t) -a set of values of the state vector that describes the region of possible states of the sys tem; f[¼] -transfer function of the system state; ξ(…) -function describing the pattern of the input effects, according to the type of solving problem.In the modeling of dynamic systems it is im portant to consider the system structure and inter action of the system with the environment.ξ(…) parameters must be taken into account [7].
The totality of the effects of the environment to the object can be divided into two groups, ac cording to the type of the influence of the envi ronment to the state variables (phase coordinates) of the object.
The first group includes influences that addi tively change the state variables at the application point.
This means that signals proportional to these influences are summed with corresponding state variables.
The second group of environmental influenc es changes the state variables of the object indi rectly, usually not additively.
Those effects change the operator of the object (system).This means the transformation law of input effects into output variables of the object [6].
An example of a model of an ordinary dynamic system as a nonstationary nonlinear control-led object with perturbations ξ (...) can be as fol lows: ( ) where u(t) -the control vector or input variables; y(t) -vector of output variables of systems.
To obtain a correct stochastic model of the state of a system with continuous time, it is nec essary to use stochastic differential equations.Consequently, after appropriate transformations, the stochastic differential equation for the consid eration model will have the following form [7]: Further, in order to simulate an unmanned ae rial vehicle, it is necessary to determine the coor dinate system.As a basis, take the trajectory coor dinate system, the projection of which axes on the UAV looks as follows (Figure 3).
As mentioned in the beginning, the UAV mo tion can be simplified as a motion of its center of mass, which will serve as a reference point in the trajectory reference system (Ot) [9].
Denote the axes of the trajectory system Xt, Yt and Zt, respectively; the roll angle from the axis Xt -Kx, the angle of yaw or slip (the angle of the path) -Ry, the pitch or attack angle (inclination of the trajectory) -Tz, then the dynamic equation of motion in general form will look as follows [10] Figure 4 shows a comparison of the trajectory and terrestrial coordinate systems, as well as the

Software development
The resulting equations of flight dynamics were used as the basis for the creation of a mathemati cal model of the UAV flight of.The mathematical model was implemented on an object-oriented high-level language -C # 5.0 [11,12].
To build a graphical user interface, the pro gram used WPF technology [13].
The program is designed to measure the dis tance of flight of an unmanned aerial vehicle with a limited fuel, on ideal conditions flight with rec tilinear motion in steady state (after all the adjust ments associated with flying to a given course) and to compare the distance of flight of the UAV with wind influence.
The algorithm of the program includes the following basic steps: 1) input of initial data, including: -current speed (km / h); -direction of flight (in degrees); -minimum and maximum values of wind power (m/s); -flight time (Tmax) (s), etc.
2) calculation of a number of parameters (the main ones are listed): -power point speed (according to the condi tion of the simulation, they correspond to the set speed); -wind power at each iteration (a randomly generated sequence of values between the limits (WPmin and WPmax), specified by user entering data).
The wind model obtained at this stage is shown in Figure 5; -the distance in meters (until the end of the UAV fuel or the completion of the preset simula tion time); 3) the output of the results in a tabular (in the form of a series of columns of parameters grouped by pairs (for comparison) describing the UAV flight in ideal and with influence of wind condi tions) and, in part, graphically.
The main program window is shown in Figure 6.
An example of the output results from the program is shown in Figure 7.

Conclusion
In this paper, the UAV was considered as de scription of its dynamics during the flight in ideal conditions and with wind influence to it.
Further research will continue in the field of modeling the automatic control system in rela tion to the mathematical model of an unmanned aerial vehicle described in this paper, and the de scribed model is improved to construct a more realistic wind model with consideration of longi tudinal overload (occurs when the power point thrusting and flight shifting) and lateral overload (occurs when flying with slip) and other influ ences, and power point, navigation equipment, etc.

Fig. 3 .
Fig.3.Projection of the axes of the trajectory coordinate system on the UAV[8]

Fig. 4 .Fig. 5 .Fig. 6 .
Fig. 4. Comparison of the trajectory and terrestrial coordi nate systems with the projections of the forces acting on the UAV