The paper focuses on the problems of mathematical model development that allow simulating the motion of airborne objects of aircraft, missile and helicopter types. The simulation results of various spatial maneuvers are given

Рассмотрены вопросы построения математической модели, позволяющие моделировать движение воздушных объектов самолетного, ракетного и вертолетного типов. Приведены результаты моделирования в случае выполнения различных пространственных маневров

Although many research papers [1-7] are devoted to air raid simulation, the implementation of a task-oriented mathematical model based on modern programming languages is still required. The statement of the air raid simulation problem may vary depending on different applications. As a result, both mathematical model and its software implementation need certain corrections. Due to these factors, the problem of software implementation of algorithms for generating trajectory parameters is relevant.

The operator using a computer-aided design (CAD) system for radar station (RS) is supposed to be able to generate an air strike scenario for simulation experiment purposes. A strike combines groups of air attack weapons (AAW), each group comprising a certain number of particular AAW deployed as a formation. AAW shall move in accordance with the formation, following a certain operator-defined route and performing various spatial maneuvers such as S-turn, corkscrew maneuver, terrain following, reaching a flight level, reaching particular velocity, etc. The route consists of an array of waypoints expressed in terms of latitude, longitude, and altitude above seal level to be reached by AAW. Hence, the problem related to the generation of trajectory parameters belongs to the type of boundary value problems that certainly impose some limitations on the problem-solving approach. As a rule, the simulation of a flight of airborne objectives involves numerical integration of a system of equations that can describe motion in a certain coordinate system [1-7]. As for the Cauchy problem, such an approach is very efficient and allows to obtain trustworthy results, but direct numerical integration cannot be applied to solve a boundary value problem. That is why, researchers have to rely on implicit integration schemes, which lead to solving a system of linear algebraic equations or to the shooting method. Due to its peculiarity associated with highly accurate plotting, the vector approach discussed herein meets boundary conditions and involves much less computations than the above-mentioned methods. Taking into account the actual problem statement, this method looks more attractive. It should be noted that the vector approach to computation of trajectory parameters has already been developed by the Central Aerodynamic Institute (TsAGI) research teams [

Below we will discuss an approach for generating the flight trajectory of a single AAW following waypoints and performing various spatial maneuvers.

The mathematical model is based on so- called vector method of trajectory parameters generation. Let us assume that there is an inertial

coordinate system with the orthonormal basis (i, j, k), as well as a wind coordinate system with the orthonormal basis (ν, λ, μ), where unit vector v is directed away from the centre of mass along the vector of velocity of the simulated AAW; unit vector λ is directed perpendicular to V, and its direction coincides with the direction of the lift force action; unit vector μ completes the triple to form a right-handed one. Relative positions of the described coordinate systems are shown in Fig. 1 (the origins of coordinates are aligned for clarity).

Vector v describes the current direction of AAW motion. By changing the vector, we can also control the direction of AAW motion, making it pass through waypoints and perform spatial maneuvers.

Let us assume that in addition to the current direction of motion determined by unit vector νn, the required direction of motion has been selected and described by unit vector νn. If the route generated by the operator of the radar CAD does not include spatial maneuvers, vector Vn is oriented toward the next waypoint en route (Fig. 2, a). If there are some spatial maneuvers, vector νn changes its orientation in the corresponding segment of the trajectory within the time period according to the predetermined maneuver. For example, when making S-turn, vector νn alternately turns in different directions relative to the direction to the next waypoint (Fig. 2, b).

Thus, the method of motion direction vector control is based on the control law generation algorithm for vector v based on vector νn.

To ensure that AAW passes through a certain waypoint, the current and required direction vectors shall be co-directional. For this purpose the current direction vector may be turned toward the required direction vector until both vectors align. This transition is available by generating the derivative of the current direction vector. Physically, the current motion direction vector is turned by generating the required normal acceleration spatially oriented in such a way that AAW together with vector V turns in the required direction. In real flight conditions, such a turn is performed by deflecting the corresponding airfoils, for example, through elevator deflection for maneuvering in the vertical plane or through aileron deflection for maneuvering in the horizontal plane. Taking into account the objective of flight simulation for various AAW, a detailed simulation, including simulation of controls deflection, is irrelevant in terms of either computational complexity or requirements to the characteristics of trajectories to be computed.

Let us assume that between current direction vector ν and required direction vector νn there is a mismatch that can be described by the value of angle φ between the vectors. Angular velocity ων of the current direction vector is proportional to the mismatch value with proportionality coefficient kφ, which is called the mismatch reduction coefficient. The relation of three parameters mentioned above can be represented as the following equation:

ων = kφφ.

Therefore, it is necessary to obtain the value of the current direction vector’s derivative, which will ensure that the current direction vector makes a turn toward the required direction vector; moreover, the mismatch reduction coefficient will help control the rate of turn, providing the required value of normal acceleration.

If we analyse motions within a wind coordinate system, the said turn of the current direction vector toward the required direction vector will result in generating a derivative for the required direction vector. To determine the derivative of the required direction vector, the vector may be represented as a linear combination of the required and current direction vectors:

where Vn - time derivative of vector.

This is the condition required to ensure that a mismatch between the current and required directions of motion will be compensated in the plane of the required and current direction vectors.

To determine coefficients a and b, two conditions can be used:

Both conditions give the following values a and b:

Taking this into account, the expression for the required rate of change of vector v will be represented as follows:

It is worth mentioning that this expression contains an exception at φ = 0, but it can be expanded as follows:

The research paper [

where g - free fall acceleration;

V - target velocity modulus;

ny - normal acceleration;

j - vertical unit vector of Earth-based coordinate system;

λ - unit vector directed along the lift force.

The following expression for the required value of normal acceleration and the direction of its action can be derived from the condition of equality of the current rate of change of vector V and the rate of change of vector Vn [

Now, using the difference expression of the derivative, we can write the calculation pattern to determine the current direction vector at the next integration step:

Vk+1 =Vk + V∆t,

where ∆t - integration step.

To control the current flight speed, a certain flight path acceleration may be selected if needed, depending on the condition whether the flight speed needs to be increased or decreased. The finite difference scheme for determination of the flight speed at the next time step is as follows:

Vk+1 = Vk + g(nx - (ν, j))∆t.

The approach described above allows to get control over the current direction vector, which enables passing of an AAW through all waypoints, as well as execution of assigned combat maneuvers. The described scenario may be used for motion simulation if some or all source data is not available.

For more realistic AAW flight simulation, aerodynamic peculiarities of each AAW shall be taken into account. In particular, the approach described above leaves out a maximum normal acceleration limitation.

In case the value of required normal acceleration is exceeded, the mismatch reduction coefficient may be corrected; therefore, the current direction vector will be turned at such rate that the normal acceleration value lies within the permissible range. The expression (1) may be represented as follows:

If (A, A) > n2у mах, where ny max - maximum value of normal acceleration, the following quadratic equation may be solved in order to determine the mismatch reduction coefficient:

The solution to the equation (3) is such a value of the mismatch reduction coefficient that ensures the AAW turn with the selected maximum value of normal acceleration:

If data on aerodynamic coefficients is available, they may be taken into account as part of the vector approach. For this purpose we can write the following expressions in a general form to determine normal and flight path acceleration:

where P(H, M, n) - drag depending on flight altitude H, Mach number M, power plant throttle ratio n;

α - incidence angle;

m - AAW weight;

q - ram air pressure;

S - AAW reference area;

Cy (α, M) - lift force coefficient depending on incidence angle and Mach number;

Cx (Cy, M) - drag coefficient depending on Mach number and lift force coefficient.

If normal acceleration value ny is obtained and flight path acceleration value nx (not greater than the maximum permissible limit) is predetermined, we can determine the incidence angle values and the power plant throttle ratio, i. e., the drag value, by solving a system of equations. It is worth mentioning that the normal acceleration value is calculated in accordance with the expressions (2) and (3), but the flight path acceleration is equal to zero by default, and its value changes only if a user-defined maneuver at a certain velocity is performed while the flight path acceleration is ramping up to the maximum permissible limit.

It should be noted that a detailed simulation of the AAW motion with regard to a velocity modulus variation based on the above-mentioned algorithm requires available information on dependencies of aerodynamic coefficients on flight conditions. In particular, this requires information on the dependence of the lift force coefficient on the incidence angle and Mach number Cy(α, M), dependence of the drag coefficient on the lift force coefficient and Mach number Cx (Cy, M), as well as on power plant altitude-velocity performance P (H, M, n).

Finally, the finite difference scheme is represented by the following formulae:

where rk - AAW center-of-gravity position radius vector at k-integration step.

Besides, the described approach can be used to compute trajectory parameters in order to simulate terrain following for a maneuvering fighter or cruise missile.

Based on the mathematical model, the software component has been designed in order to provide an API for trajectory computation. The software component has been designed within the object-oriented programming paradigm, the main component of which is the class with a set of methods. The class constructor allows to convey boundary conditions (waypoint coordinates, maneuvers) and aircraft performance. The class goes with a computational method, which allows to initiate the trajectory computation with conditions predetermined by means of the class constructor. After the computational method is executed, the user may apply methods that return computation results.

The basis of the architecture concept implies further integration of the component into a common integration platform, plus an option to work with groups of AAW. In addition to computational classes and interfaces, a massive database is implemented to save AAW performance data, as well as simulation results and information on groups. The key tool for software development is the Qt library [

Fig. 4 shows an example of AAW group generation. First of all, the user shall select elements to be included in a group (Fig. 4, a), then set up the formation (Fig. 4, b).

Fig. 5 shows an example of displayed computed trajectories for the group comprising three AAW, which are deployed as a wedge formation in accordance with Fig. 4 (b) without maneuvering. These AAW are supposed to fly through waypoints selected in the window shown in Fig. 6.

Fig. 7 shows more detailed computation results, which include images of time-dependant geocentric coordinates (three upper graphs), time- dependant geodetic coordinates (three middle graphs), pitch and roll angles, and velocity-time relation (three lower graphs). It is worth mentioning that the flight speed remains unchanged during flight as maneuvers at a certain speed have not been selected.

The paper describes the basic principles for generating trajectory parameters based on control over the AAW motion direction vector. This method, on the one hand, allows to calculate trajectories for AAW with a very limited set of known source data; on the other hand, it allows to perform highly accurate computations if detailed information on aerodynamic characteristics is available.

The statement of the problem described herein requires that the boundary conditions are met. This takes much more effort to apply the approaches based on the integration of a system of equations in comparison to the vector approach. It is the boundary-like behaviour of the problem that makes the vector approach one of the best possible methods for generating trajectory parameters.

The authors declare that there are no conflicts of interest present.