almaz-anteyJournal of «Almaz – Antey» Air and Space Defence Corporation2542-0542"Almaz - Antey" Air and Space Defence Corporation, Joint Stock Company10.38013/2542-0542-2018-4-91-99Research ArticleThe method of exact solution of the problem of short-range guidance with two turnsFeliksonA. E.noemail@neicon.ru“Almaz - Antey” Air and Space Defense Corporation, Joint Stock Company30122018049199

The task of determining the guidance parameters by the “two-turn maneuver” method is related to numerical solution of an equation defining the trajectory of the fighter-interceptor. We propose an easy-to-implement in software approach which is based on the exact solution of the problem at each step of the iterative process constructed by one of the parameters

The problems of providing automated control of fighter aviation, solving navigation tasks and tasks of fighter interceptors’ (FI) guidance con­trol have been dealt with in a number of papers .

To solve the problem of selecting parame­ters of FI guidance trajectory with a single turn, different methods have been proposed , in­cluding those based on the linear programming method.

At the same time, the FI guidance trajecto­ries with a single turn in some cases do not allow to perform target interception from the best en­gagement position, and at times yield no solution at all .

Trajectories with two turns provide more possibilities. However, as can be seen from papers , to calculate FI flight trajectory parame­ters, various presumptions and simplifications are applied, including a premise about the smallness of the angles of turn and discreteness of turning radii change.

To reduce expenditure of computing resources in the automated calculation of FI flight trajectory parameters, in the most elementary case, two levels of the turning radius are considered.

First, a solution for the recommended turning radius is sought. If it is not found, a solution is sought for the minimum turning radius. If no solu­tion is available again, it is assumed that guidance by a given method is impossible.

Such a procedure of FI flight trajectory pa­rameters calculation entails an increase of task solving time; therewith, the more turning radius gradations are used, the higher the expenditure of computing resources.

Moreover, in certain conditions such pre­sumptions lead to ill-founded selection of the minimum turning radius, which is achieved by in­creasing aircraft roll angle with increased g-load, which is extremely undesirable.

This paper introduces a method that makes it possible to simplify problem solution and is free from the said drawbacks.

The problem under consideration is that of FI short-range guidance to target. The short- range guidance is understood as the final stage of ground-controlled guidance, when FI is brought into a point located at a specified distance from target, ensuring that the specified FI heading rela­tive to target heading is maintained.

For FI guidance, it is necessary to determine its flight trajectory parameters. The trajectory of FI motion under the “two-turn manoeuvre” method consists of the first turn, straight-line flight seg­ment, and second turn.

The initial data for solving the problem of determining FI flight trajectory parameters are the initial positions, headings, and velocities of target and FI, position of the final point of FI motion tra­jectory relative to target, FI final velocity, aspect of target engagement (FI heading relative to target in the final point).

In the above statement, the problem of de­termining FI motion trajectory parameters belongs to the subfield of mechanics called kinematics. Kinematics describes motion of bodies, without investigating its causes.

Normally, motion is considered in some or other coordinate system. Paper  provides basic

information on the coordinate systems used in calculations associated with motion trajectories of aerial vehicles.

The tasks of carrying out navigation calcu­lations and guidance in computer-aided systems are more convenient to solve in a target-related coordinate system .

The origin of coordinates in this system coincides with the target position. Axis П of the rectangular coordinate system is directed along the target flight heading. The positive direction of axis П coincides with the direction of target mo­tion. Axis Б is directed perpendicular to the target heading. The positive direction of axis Б - to the right of the direction of target motion.

We shall consider only positive values of Б, as the case of Б < 0 is reduced to the case of Б > 0 by respective change of the coordinates and heading.

The initial data on the target and FI are given, as a rule, in the local rectangular coordinate sys­tem of one of the controlling command posts (CP). We shall further assume that axis X of the CP coordinate system is directed to the north and axis Y - to the east.

Taking into account the turning angles of axes and the offsets of coordinate origins, recalculation of data from the local rectangular coordinate system of the CP (X, Y) into coordinate system of the target (Б, П) is performed by means of expressions

where Q - FI heading relative to axis X of the CP coordinate system;

X, Y - FI coordinates in the CP coordinate system;

Хц, Yц - target coordinates in the CP coordi­nate system;

Б,П - FI coordinates in the target coordinate system;

Qц - target heading relative to axis X of the CP coordinate system;

If the calculated value γ < 0, then it is as­sumed that γ * = γ + 2π.

If the calculated value Б < 0, then it is as­sumed that Б* = -Б, γ * = 2π - γ .

A reverse calculation is done by the for­mulas

Let us consider FI flight trajectory in the target coordinate system when implementing the “two-turn manoeuvre” method.

We shall use the following denotations:

Vц - target velocity;

Vn - programmed velocity of FI flight;

α1 - first turn angle;

α2 - second turn angle;

Vk - FI velocity in the final point of guidance trajectory;

Б, П - FI initial coordinates in the tar­get coordinate system (Б - lateral component, П - longitudinal component);

С - length of straight-line segment of FI tra­jectory;

Sk - distance from FI to target in the final point of guidance trajectory;

p - FI aspect relative to target (FI heading relative to target heading) in the final point of guidance;

m1 - 1st turn direction marker;

m2 - 2nd turn direction marker;

Y1 - FI heading on the straight-line segment of flight trajectory;

γ0 - FI heading in the initial point of guidance trajectory;

L - target location at the moment of FI en­tering the final point of guidance trajectory.

Fig. 1 shows FI flight trajectory in the target coordinate system.

Fig. 1. FI flight trajectory:

Ц - target at the moment of FI being in the initial guidance point; Цк - target at the moment of FI being in the final guidance point; И - FI in the initial guidance point; Ик - FI in the final guidance point

https://cdn.elpub.ru/assets/journals/almaz-antey/2018/4/XNFVhuR40LnRl7xKTWD6dU8oEVLnqW1gcbURFk3r.png

Each of the turn direction markers m1, m2 takes the value of+1 if FI makes a clockwise turn (right turn) and -1 if it makes a counter-clockwise turn (left turn).

The FI velocity is assumed constant from the initial point till the second turn start, and equal to the programmed velocity of fighter-interceptor flight Vn.

On the second turn segment, FI velocity changes uniformly from programmed flight ve­locity Vn to specified velocity in the final point Vk.

The FI heading is reckoned counter-clock­wise from 0 to 2ф.

To determine FI flight trajectory parameters, let us compose balance equations according to flight coordinates and time. For that purpose we shall find projections of the components of FI and target flight trajectories on the coordinate axes Б, П.

The FI initial coordinates are equal to Б, П.

FI displacements along the coordinate axes during FI movement are as follows:

Target displacements along the coordinate axes over the time of FI movement are as follows:

Target offset along the coordinate axes rela­tive to FI position in the final trajectory point:

Considering the above components of FI and target flight trajectories, we can write down a system of balance equations as per coordinates and time which ensures FI acquiring the speci­fied position with the specified heading relative to target:

After substitution of the found components in the system of equations (1), we have

The system of equations (2) includes FI turning angles α1 and α2. To determine modulus of the first turn angle, we set

Four cases, as shown in Fig. 2, are possible:

if m1 = +1, γ1 <γ0 (Fig. 2, а), then α1 = 2π-Δγ1;

if m1 = +1, γ1 >γ0 (Fig. 2, b), then α1 = 2π-Δγ1;

if m1 = -1, γ1 < γ0 (Fig. 2, c), then α1 = Δγ1;

if m1 = -1, γ1 > γ0 (Fig. 2, d), then α1 = Δγ1

Fig. 2. Angle of the first turn:a – right turn from greater heading value to smaller one; b – right turn from smaller heading value to greater one; c – left turn from greater heading value to smaller one; d – left turn from smaller heading value to greater one

https://cdn.elpub.ru/assets/journals/almaz-antey/2018/4/HU4lyXNwmDr9lA9ATLPC91JnXHrbSLUGzvo1L4yX.png

Analysis of the expressions for the first turn angle shows that the sign before increment of heading ∆γ1 is opposite to the sign of turn direction marker m1. Term 2≠ appears in case ofm1 = +1. Therefore,

we obtain an expression for the second turn angle:

For successful detection and lock-on of tar­get by the on-board radar, with subsequent launch of missiles, FI heading p in the final point must conform to certain limitations.

For tail-chase engagement

2π - Δρ ≤ p < 0 или 0 ≤ p ≤ Δρ.

π-Δρ < p <π + Δρ.

It is usually assumed that

Δv = π / 6.

With the known FI initial position (Б, Π), velocities of FI Vn, Vk and target Vц , FI initial heading γ0, radii R1,R2 and directions m1, m2 of turns, FI final headingp, final distance Sk between FI and target, two unknowns remain in system (2): heading Y1 on the straight-line trajectory segment and length С of this segment.

As follows from expressions (5), (8), turning angles α1, α2 depend on the unknown pa­rameter γ1 .

Substituting in system (2) expressions (5), (8) for α1, α2 and excluding parameter С from it, we have

Hence, we obtain an equation for determining FI heading γ1 on the straight-line segment of flight trajectory:

The length of straight-line segment С is de­termined by substituting the value of heading Y1 in one of the expressions (9).

For the problem solution to exist, the length of straight-line trajectory segment С must be a non-negative magnitude.

Selection of the final solution from those possible is done based on the parameter of the best position for target engagement:

This expression defines target position Цк at the moment of FI entering the final point. This is the path covered by target over the time of FI flight from the initial to the final trajectory point.

The turning radii we assume to be equal:

R1 = R2 = R.

In the method being considered, to solve equation (10) it is proposed to use as the ini­tial approximation the FI trajectory with turning radii R = 0. In this case equation (10) for the initial approximation has the view

Then the unknown magnitude of Y1 remains under the sign of trigonometric functions only, and the equation is reduced to a quadratic one.

To obtain solution to the problem with a real turning radius, it is gradually increased from zero to a specified value, with successive determination of FI flight parameters at each step of the turning radius change.

Considering that the process of transition from the initial solution to the final one occurs as a result of gradual deliberate change of an addi­tionally selected variable, the latter can be called a master parameter.

In this paper, the FI turning radius is selected as a master parameter.

A tentative value of the FI heading at any given step is determined using a derivative from the FI heading with regard to the turning radius:

where ∅R - step as per the turning radius accepted for calculations.

For computations by formula (12), it is necessary to find an expression of derivative  from function γ1 (R) , which is implicitly specified by means of equation (10). For that purpose, we perform derivation of equation (10) by parameter R:

Since expression (12) considers only the linear part of FI heading Y1 change, the found value of Y1 will not be a solution to equation (10). However, equation (10) can be solved relative to the turning radius R.

Considering denotations accepted in equa­tion (10), we have

Substituting these values in equation (10), we find

For the value of heading γ1( k+1) found with the help of expression (12), there is a possibility to determine the exact value of turning radius R from expression (14).

The pair (R, γ,) satisfies equation (10). Suc­cessive transition from step to step ensures ap­proximation to the sought solution, corresponding to the specified turning radius R3.

As a result of the described process, we find equation solution

F(γ1) = R3 - R (γ1) = 0,                                         (15)

where R3 - specified turning radius value;

R (y1) is determined by formula (14).

Equation (15) is equivalent to equation (10) and provides solution to the task if it exists. If there is no solution to the task for R3, then the iteration process enables to find possible solution options at turning radii R different from R3.

If we use the Newton’s method instead of the constant step of 0R, then, as demonstrated by numerous examples, it takes 2-7 iterations to determine the sought solution.

In this case, instead of formula (12), to de­termine the value of heading Y1 at a given step of the iteration process, it is necessary to apply the expression

Here the derivative from function R (y1) is expressed through a derivative determined by for­mula (13), from function γ, (R inverse to it.

The turning radius changes at each step of the iteration process, so that the pair (R, γ1) satis­fies expression (10).

To demonstrate specific features and bene­fits of the method proposed in the paper, let us consider a numerical example.

Let the FI be in point Б = 80 km, П = 200 km, its heading being γ0 = 250° and ve­locity Vn = 0.35 km/s.

In the final point of its flight trajectory the FI is to be at a distance of Sk = 50 km from target, with heading p = 150° (target head-on engagement) and velocity Vk = 0.25km/s.

The velocity of target to which the FI is guided is Vц =0.30km/s.

Guidance is performed by the “two-turn manoeuvre” method. The first turn is to the right ( m1 = +1) ,and the second turn - to the left (m2 = -1). The recommended turning radius is equal to Rp = 25 km, the minimum one - Rmin =10 km.

To compare the processes of computation and the obtained results, let us make calculations by two methods.

New method (master parameter method) : the turning radius is a variable value and is accept­ed as a master parameter in the iteration process. Equation F(y1) = R3 - R(y1) = 0 is solved.

Old method (constant radius): the turning ra­dius retains a constant value, equal to the specified

one, at all iteration process steps. Equation f (γ1 ) = e cos γ1 + (γ1 + d2 )sin γ1 + s = 0 is solved.

Fig. 3 shows turning radius dependence on the FI heading on the straight-line segment of flight trajectory, built for the given numeri­cal example with the help of formula (14). With R3 = Rp = 25 km, there is no solution, and with R3 = Rmin = 10 km, a solution exists (see Fig. 3). Given a priori possession of those data, let us assess performance of both of the considered cal­culation methods.

Fig. 3. Dependence of the turning radius on FI heading on the straight-line segment of flight trajectory

https://cdn.elpub.ru/assets/journals/almaz-antey/2018/4/n2Po7zuLJm53k5WfBzEi2xYopu6Ik8BRRflvu08B.png

We summarise the calculation results in tables, which contain calculations of heading Y1 on the straight-line segment of flight trajectory, turning radius R, straight-line segment length С, turning angles α1, α2, location L, functions f (Y1) and F (γ1).

Table 1 gives the data for the newly pro­posed master parameter method with constant step ΔR = 2,5км при R3 = Rp = 25 км.

Table 1

Iteration process with constant step ΔR = 2,5, km at R3 = Rp = 25 km

Iteration numberHeading Y1, degTurning radius R, kmf (Y1), kmF ( Y1), kmС, kmα1, degα2, degL, km
0144.170025.0093.96105.835.8380.53
1141.812.39022.6190.36108.198.1981.67
2139.234.77020.2386.46110.7710.7782.90
3136.377.13017.8782.18113.6313.6384.25
4133.159.47015.5377.46116.8516.8585.74
5129.4911.79013.2172.24120.5120.5187.39
6125.2114.07010.9366.37124.7924.7989.25
7120.0816.3108.6959.68129.9229.9291.38
8113.6618.4906.5151.86136.3436.3493.89
9105.1020.5504.4542.34144.9044.9096.95
1092.3222.3902.6129.79157.6857.68100.90
1167.4923.6001.409.12182.5182.51106.22

Table 2 gives the data obtained at each step of the iteration process with variable step for R3 = Rp = 25 km.

Table 2

Iteration process with variable step at R3 = Rp = 25 km

MethodIteration numberγ1, degR, kmf (γ1), kmF (γ1), kmС, kmα1, degα2, degL, km
New (master parameter method)  0144.170025.0093.96105.835.8380.53
1120.6116.1008.9060.36129.3929.3991.17
298.3021.6803.3235.44151.7051.7099.14
374.4423.4701.5314.60175.5675.56105.09
1105.2625.0019.550----
276.8625.008.120----
327.9925.007.080----

It follows from Tables 1, 2 that the new method, in spite of the absence of solution at R3 = = Rp = 25 km, makes it possible to accept, as a solu­tion to the task, the results that conform the most to the specified initial data, namely, γ1 = 74.44°, R = 23.47 km, C = 14.60 km, α1 = 175.56°, α2 = 75.56°, L = 105.09 km.

The FI flight trajectory corresponding to these data is shown in Fig. 1.

The old method does not yield a solution, as the values of heading y1, obtained in the process of iterations, do not satisfy equation (10).

Table 3 gives the data obtained at each step of the iteration process when seeking a solution for R3 = Rmin = 10 km.

Table 3

Iteration process with variable step at R3 = Rmin = 10 km

MethodIteration numberγ1, degR, kmf (γ1), kmF (γ1), kmС, kmα1, degα2, degL, km
New (master parameter method)  0144.170010.0093.96105.835.8380.53
1134.758.3501.6579.79115.2515.2585.00
2132.489.9200.0876.49117.5217.5286.04
3132.3610.000076.32117.6417.6486.10
1132.9810.001.290----
2132.3610.000.00076.32117.6417.6486.10

Table 3 shows that for R3 = Rmin = 10 km, both methods yield the same result.

This example shows that the new approach makes it possible not just to do without superfluous computations, but to find a more acceptable solu­tion from the viewpoint of g-loads affecting the aircraft.

The volume of the program that imple­ments the proposed method is 100 operators of the C language. Therewith, the program part run with­in a cycle makes 25 operators. The total volume of an executable file, with account of the initial con­ditions, data preparation, and computation mana­gement, is 30 Kbyte, which is by 15 % less than the volume of the previously applied program .

Under the proposed method, the average task completion time for a set of tasks is by 30 % less than under the previously applied one, with the errors of FI bringing to the final point of flight trajectory not exceeding 10-2 km.

The results of the undertaken study are as follows:

The author expresses his gratitude to the reviewers for their comments that facilitated im­provement of the substantiation and presentation quality of the material of this paper.

References