Vector method in generating trajectory parameters in the air raid simulation task

Introduction

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.

Objective

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 [8], but it has been used only to solve the problem related to aerial combat simulation and implemented based on outdated programming languages.

Problem-oriented mathematical model

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).

Fig. 1. Relative positions of wind and inertial coordinate systems

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 ν_n, the required direction of motion has been selected and described by unit vector ν_n. If the route gene­rated by the operator of the radar CAD does not include spatial maneuvers, vector V_n is oriented to­ward the next waypoint en route (Fig. 2, a). If there are some spatial maneuvers, vector ν_n 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 ν_n alternately turns in dif­ferent directions relative to the direction to the next waypoint (Fig. 2, b).

Fig. 2. Examples of generation of the required direction vector: а - route without maneuvers (1-3 - numbers of waypoints); b - route with S-turn

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

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 ν_n there is a mismatch that can be described by the value of angle φ between the vectors. An­gular velocity ω_ν of the current direction vector is proportional to the mismatch value with pro­portionality coefficient k_φ, which is called the mismatch reduction coefficient. The relation of three parameters mentioned above can be rep­resented as the following equation:

ω_ν = k_φφ.

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 V_n - 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:

1. the derivative of the current direction vector is orthogonal to the vector itself, i. e. ;
2. the modulus of the vector V_n is propor­tional to the mismatch angle between the current and required directions of motion, i. e. = kφ.

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 [8] proves that the equation for the current rate of change of vector V is as follows

where g - free fall acceleration;

V - target velocity modulus;

n_y - 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 V_n [8]:

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:

V^k+1 =V^k + 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:

V^k+1 = _V^k + g(n_x - (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) > n²_у mах, where n_{y 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;

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

C_x (C_y, M) - drag coefficient depending on Mach number and lift force coefficient.

If normal acceleration value n_y is obtained and flight path acceleration value n_x (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 C_y(α, M), dependence of the drag coefficient on the lift force coefficient and Mach number C_x (C_y, 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 r^k - 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.

Peculiarities of software implementation

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 [9]. Figures 3-6 illustrate debugging graphic inter­face elements (the integration platform of the CAD system for radar features its own graphic interface with a computational model to be integrated only). The main interface window displays a terrain map where the user can plot waypoints and maneuvers. The top toolbar of the interface accommodates tools for database administration, performance settings for new AAW, tools for editing AAW saved to database, AAW grouping, computation initiation and result review (see Fig. 3).

Fig. 3. Debugging graphic interface toolbar

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. 4. AAW group settings: group elements (a), formation (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. 5. Example of algorithm execution

Fig. 6. Waypoints

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.

Fig. 7. Detailed results of trajectory computation

Conclusion

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.