This article investigates the dependence between the accuracy of target designation issued by a ground radar system to an onboard radar station based on the measured target and missile coordinates and on the relative missile position (relative to the radar and the target). Simulation and calculation methods are proposed for estimating the accuracy of target designation in terms of the observation angle and the range and speed of the target taking into account both the errors in measuring the target and missile coordinates by a ground radar system and the relative position of the missile. The influence of the errors in measuring the target and missile coordinates and the relative position of the missile on the target designation accuracy is investigated. The problem of tracing the optimal trajectory of missile guidance to the target is formulated taking into account the errors of target designation to an onboard radar station and other factors.

General characteristics of the problem. The current problem to be solved is to upgrade the tactical air defence missile system. Probable design concepts and certain aspects of the development of a new air defence system are described in [1][2]. As one of the key reasons for weapon system upgrade, we can single out the application of new types of targets such as medium-range ballistic missiles (targets are missile warheads), as well as a sustainable complication of the target environment where air defence weapons have to operate.

To kill missile warheads (high-speed small targets) by using, among other means, kinetic energy-based interception in operational conditions associated with application of countermeasures (false targets and jamming), air defence weapons shall meet stringent requirements, first and foremost, those related to the ability of an onboard radar to ensure SAM guidance accuracy and target selection. To meet the requirements, with account for a small mid-section of a missile, an onboard radar shall operate in the millimetre or optical waveband. Due to high dynamics of the interception process and the lack of time for target search, the radar shall enable assured (searchless) target lock-on using angular coordinates. This task is difficult to accomplish because of a narrow field of view in the selected waveband. Moreover, as the enemy may apply countermeasures, an onboard radar shall have high selectivity in range and speed (Doppler frequency), while its target designation accuracy shall conform to tough selectivity requirements.

Since weapons systems under development shall meet more stringent requirements, this involves a number of problems to be solved. Such problems were of not great importance for weapon systems operated in an extended long-wave range and exposed to less intense enemy countermeasures. They include:

– determination of the desired accuracy of onboard radar’s coordinate support by a ground radar station in order to enable the assured target lock-on by angle, as well as high-precision target selection by range and speed in conditions associated with active enemy’s countermeasures to prevent target detection;– determination of the presence and estimation of degree of dependence of target designation (TD) accuracy on the relative position of the missile (relative to the radar station and target); – determination of the preferable (optimal) SAM movement trajectory in the presence of dependence of TD accuracy on the relative position of the missile;– reaching a compromise on errors in angle, range and speed target designation during the missile guidance process with account for mutual dependence of the errors, etc.

Problem statement. The task is to analyse the process of SAM guidance onto a target. At its initial flight section, the missile is guided using data of a ground radar station tracking both the missile and the target. To estimate the missile position, data from the onboard inertial navigation system (OINS) can be used as well. As the missile approaches the target, the onboard radar will be actuated at some distance to the target. The radar actuation range depends on its performance figure regarding a particular target and operating conditions. The decisive operating conditions are the presence of electronic jamming and false targets. The onboard radar shall lock on a target for automatic tracking (AT) and track it during the process of subsequent high-precision homing on a target.

To enable target lock-on for AT, target designation data are generated based on target and missile coordinates measured by a ground radar station (with account for OINS data), represented as relative target coordinates (angular coordinates, range and speed) calculated with regard to the current missile position. The target designation accuracy depends on errors in relative target coordinate estimates (their root-mean-square (RMS) deviations from true values).

Below we show that relative target coordinates are determined with errors that depend both on target and missile coordinate measurement errors (with account for OINS data) and on the relative position of the missile.

For development of a missile guidance system intended for assured target lock-on based on angular coordinates and high-precision selection of the target signal received by the onboard radar station with regard to range and speed (Doppler frequency), the degree of dependence of TD accuracy upon the relative position of the missile shall be estimated. In this respect, the task is to develop methods and estimate the accuracy of target designation data issued to an onboard radar, depending on errors in estimating target and missile coordinates by a ground radar and also on the relative position of the missile (relative to the radar station and target).

Eventually, results of such investigations can be used for determining the optimal trajectory to bring the missile to the target lock-on point.

1.1. Physical content of target designation errors

To explain the content of the problem and issues raised above, let us analyse the physical content of errors in target designation issued to an onboard radar regarding target angle and range designation in the vertical plane. The explanatory diagram is shown in Figure 1.

Fig. 1. Arcs of dispersion of SAM coordinate estimates (blue) and target coordinate estimates (red) and relations between individual points of arcs of dispersion (orange). There positions of SAM are shown – a), b), c). Arcs of dispersion are the result of statistical simulation of normally distributed target and missile coordinates: the number of simulation repetitions nм = 1000 at target and missile range and angle estimate RMSE, respectively: σRц = 200 m, σRр = 40 m, σεц = 2о , σεр = 0.8о . (For clarity, coordinate RMSE are raised too high in comparison with real values. Missile coordinate RMSE is less than target coordinate RMSE with account for the use of OINS data.) Dashed lines show the boundaries of angle ranges ±3σεц, ±3σεр for the target and two positions of the missile – а) and c)

Due to errors in measurements of angular coordinates and target and missile range measured by a ground radar, the target and missile positions relative to the RS are described with ellipses of dispersion of coordinate estimates (to be more precise, as will be significant for further detailed studies, with ellipsoid-shaped arcs of dispersion). Accordingly, the target designation for an onboard radar, generated based on target and missile coordinates measured with errors, also contains an error caused by both errors in missile and target coordinated measured by a radar and by transition of the target coordinate origin from the RS station point to the missile location point, and therefore, by the relative position of the missile.

To clearly demonstrate the content of target designation errors, Figure 1 shows two extreme cases with the position of the missile being guided onto a head-on target: case a) – missile is on the RS – target line, and case c) – missile is on the RS – target line orthogonal to the RS – target line. Other possible variants of the missile position while guiding it onto an approaching target are basically located between the specified lines. Point b) is shown as an example to describe such an intermediate variant of the missile position. All three positions of SAM as shown in Figure 1 are equidistant from the target. This arrangement will be applied for further estimation of errors in target designation for the radar.

Target range designation is statistically determined by the scatter of distances between individual points of the arc of dispersion of estimated missile coordinates and the points of the arc of dispersion of estimated target coordinates; target angle designation is determined by the scatter of angular directions between the point of arc of dispersion relative to the average value.

Figure 1 shows distances between individual points of arcs of dispersion as orange segments (they represent possible directions of the axis of the radar field of view once the target is locked for AT). The range TD error is determined by the scatter of their segment lengths; the target angle designation error – by the scatter of their angular directions.

According to the Figure, with the similar accuracy of missile and target coordinate measurements for cases a) and c), the range TD error in case a) will be less than that in case c) due to a smaller scatter of distances between points of arcs of dispersion of missile and target coordinates. On the contrary, due to an angularly extended shape of arcs of dispersion, the target angle designation with the missile being in position c) is more accurate than that in position a) (the width of the beam consisting of orange segments from missile position c) towards the target is considerably smaller than from position a)).

For other variants of missile and target positions, errors in target designation for an onboard radar will also depend on target and missile coordinate measurement errors and on the relative position of the missile, while values of target range and angle designation errors are likely to be within limits that correspond to cases a) and c), as shown in Figure 1.

The given description of the physical content of target designation errors using a 2D diagram shown in Figure 1, is true for 3D representation with the use of 3D arcs (ellipsoids) of dispersion of target and missile coordinates. In 3D representation, TD errors also depend on target and missile coordinate measurement errors and on the relative position of the missile, and the nature of the dependence by each angle and range is similar to 2D representation. In this respect, we will analyse 2D representation below.

1.2. Calculation of target designation grid

Target designation for an onboard radar is the result of conversion of target and missile coordinates from the RS coordinate system to the missile coordinate system. Target designation errors are determined by the shape of ellipses of dispersion of target and missile coordinates and their relative orientation.

In implemented missile guidance systems, multiple coordinate conversion is conducted to generate target designation data. As a rule, a ground RS measures target and missile coordinates in a spherical or biconical coordinate system, the missile is guided in wind fixed coordinates, and an onboard radar operates in its own antenna coordinate system [3]. The peculiarity of target designation generation depends on peculiarities of implemented methods of physical measurement of coordinates aboard the missile.

To study the content of target designation errors alone, let us analyse the most primitive variant of target designation generation involving conversion of target coordinates from the spherical coordinate system of the RS directly to a spherical coordinate system of an onboard radar, considering this method of coordinate generation equivalent to multiple conversion of coordinates.

The diagram showing relations between RS, target and missile coordinates at the time of target lock-on for AT is given in Figure 2.

Fig. 2. Diagram showing relations between RS, target and missile coordinates at the time of target lock-on for AT

The position of all objects is analysed in the xOy rectangular coordinate system with its origin at point О where the RS is located.

The RS measures spherical coordinates of the target and missile – their ranges Rц, Rр reckoned from the RS station point, and elevation angles εц, εр reckoned relative to the horizontal axis Ох. Based on these measurements, target designation’s spherical coordinates are calculated: missile – target range Rр-ц reckoned from the missile location point and missile – target elevation angle εр-ц reckoned relative to the horizon.

Target and missile coordinates are determined with errors, which are defined by RMSE of relevant coordinates against their true values σRц, σRр, σεц, σεр. We should note that missile coordinates determination errors (RMSE σRр, σεр) may be considerably smaller than target coordinates determination errors (RMSE σRц, σεц), because the OINS data is taken into account and the missile location is determined with active response.

Thus, the position of the missile and target is defined by random target range vector Rц and random missile range vector Rр with known values of RMSE of their spherical coordinates. The actual problem of target designation accuracy estimation consists in estimation of RMSE of spherical coordinates of target designation vector Rр-ц which is determined as the difference between the specified vectors

Rр-ц = Rц – Rр. (1)

Spherical coordinates of target designation vector Rр-ц are calculated by formulae

(2)

(3)

where ΔХ = Хц – Хр , ΔY = Yц – Yр – projections of differences between target and missile coordinates onto Ох and Оу axes; Хц = Rцcosεц, Хр = Rрcosεр, Yц = Rцsinεц, Yр = Rрsinεр.

Taking into account that the missile onboard radar operates in a spherical coordinate system with the origin at the SAM location point, we shall determine RMSE of target designation’s spherical coordinates σRр-ц, σεр-ц based on known parameters σRц, σRр, σεц, σεр of vectors Rц and Rр.

For further formulations, we will also introduce the formulae for calculating some intermediate values – the angles shown in Figure 2.

(4)

and the missile range to be calculated with account for these angles

(5)

2.1. Estimation of target designation accuracy using the statistical simulation method

The method involves multiple computer-aided numerical experiments (generally, nм times) to simulate the determination (measurement) of random target and missile coordinates Rц, Rр, εц, εр with the predetermined distribution laws; calculation of target designation coordinates at each experiment by formulae (2) and (3); sampling values of random parameters Rр-ц, εр-ц in the amount equal to nм and sampling-based calculation of RMSE of σRр-ц, σεр-ц using methods of mathematical statistics.

To analyse the influence of the accuracy of target and missile coordinate measurements and the relative position of the missile on the value of RMSE of σRр-ц, σεр-ц, let us perform practical calculations using the statical simulation method.

Assume that estimates of target and missile coordinates acquired by the radar (for the missile, including OINS data) are normally distributed and their mathematical expectations Rц, Rр, εц, εр and RMSE of σRц, σRр, σεц, σεр are known. Below we will analyse the case with the target location, as shown in Figure 1: Xц = 30 km, Yц = 60 km, (Rц ≈ 67.1 km), εц = 63.4о .

Suppose the accuracy of target coordinates estimation is defined by values σRц = 0.5 m, σεц = 0.05о . Assume that a given target can be locked by the radar for AT (with account for radar performance figure) at distance Rр-ц = 14 km.

Let us analyse (according to Figure 1) three positions of the missile at the time of target designation reception for target lock-on for AT at a distance of 14 km:

а) Missile is on the RS – target line (α = 0о ): εр = εц, Δε = 0о , Rр = Rц – Rр-ц ≈ 53.1 km ≈ 0.8Rц, Xр ≈ 23.7 km, Yр ≈ 47.5 km; b) Missile is in the intermediate position (α ≈ 22.2о ): εр ≈ 69о , Δε ≈ 5.58о , Rр ≈ 54.4 km, Xр ≈ 19.5 km, Yр ≈ 50.7 km; c) Missile is on the line orthogonal to the RS – target line (α = 90о ): εр ≈ 75.2о ,Δε ≈11.8о , Rр ≈ 68.5 km, Xр ≈ 17.5 km, Yр ≈ 66.3 km.

We will use two variants of missile coordinates accuracy estimation for each position of the missile:

1) missile coordinates RMSE values are similar to those for the target: σRр = 0.5 m, σεр = 0.05о ; 2) missile position determination errors are negligibly small, σRр = 0 m, σεр = 0о .

To provide high accuracy, we will perform a large number of numerical experiments equal to nм = 10,000.

The results of target designation estimation in accordance with the above statistical simulation method for different variants of initial data are given in Table 1.

Table 1

Results of target designation accuracy estimation for various initial data sets for the missile, using statistical simulation methods at nм = 10,000

According to the table, the target designation accuracy considerably depends on errors in target and missile coordinates estimates as well as on the relative position of the missile.

The calculations have proven the physical content of target designation errors – target range designation RMSE σRр-ц grows as angles Δε, α defining the relative position of the missile increase; target angle designation RMSE σεр-ц decreases at the same time.

Simulation demonstrates that random target designation coordinates Rр-ц and εр-ц, calculated by formulae (2), (3) based on normally distributed missile and target coordinates, have near-normal distribution laws. As an example, Figure 3 shows the histograms of Rр-ц and εр-ц values for the intermediate position of SAM (variant b). The figure also shows normal distribution curves, which have average and RMSE values identical to those in resulted histograms. The figure shows a good agreement between histograms and the normal distribution law frequency curve.

Fig. 3. Histograms of random values Rр-ц, εр-ц for missile position b) and normal frequency curves (dotted curves). Calculation are made for σRр = 0.5 m, σεр = 0.05о , nм = 10,000

2.2. Analytical estimation of target designation accuracy

The statistical simulation method is more accurate than the method of analytical estimation of target designation errors (with a sufficient number of experiments), but the analytical method with an acceptable accuracy allows to reduce the computation time, and more importantly, to obtain functional dependences of target designation parameters on coordinate measurement errors with account for the relative position of the missile.

As the basis of analytical estimation, we will apply the method of random argument linearisation, which is well-known in the probability theory [4]. The method does not require knowledge of argument distribution laws. According to the method, the function of random arguments shall be expanded in a Taylor series in the range of values of arguments’ mathematical expectations – the second term of a Taylor series will define the dispersion of initial function. If there is a large discrepancy between statistical and analytical estimates of function dispersion, using the third term of a series is recommended to enhance the estimation accuracy.

In this case, functions Rр-ц, εр-ц (formulae (2), (3)) of random arguments Rц, Rр, εц, εр with known RMSE values are subject to linearisation.

According to the method, the dispersion of target range designation Rр-ц is generally determined by formula

As a result of differentiation and simplification, we get the following expression in an explicit form

(6)

Similarly: the dispersion of target angle designation εр-ц (formula (3)) is generally represented as follows:

By differentiating we get the following expression in an explicit form:

(7)

Using the resulted formulae (6), (7), we have managed to calculate estimates of the target angle designation accuracy for initial data of calculations performed under item 2.1.

The results are given in Table 2. The table also contains deviations from the results obtained under item 2.1, using the statistical simulation method.

Table 2

Results of target designation accuracy estimation using an analytical method for different sets of initial data on the missile

The completed calculations show a good agreement of statistical and analytical simulation. This allows to consider the linearisation method to be fit for estimation of target designation accuracy.

2.3. Probabilities of target selection and radar lock-on

We will determine the probability of target selection by an onboard radar regarding each coordinate for target designation, based on resulted RMSE estimates, assuming they are normally distributed (acceptability of the assumption is proven by the results of statistical simulation specified above).

Generally, Х is designated as a random coordinate for target designation (angular direction of the target εр-ц, its range Rр-ц or speed Vр-ц). Mathematical expectation μ and RMSE σ of value X are known (for coordinates Rр-ц and εр-ц, mathematical expectations are determined by formulae (2), (3); the RMSE is determined by applying one of the methods described above. For speed Vр-ц, formulae (14), (15) intended to calculate mathematical expectation and RMSE are validated below).

The probability that values Х will be within interval μ ± l (the probability of target selection in the specified range) is calculated by formula [4]:

Р(|Х – μ| < l) = 2Ф(l/σ), (8)

where Ф(х) – Laplace’s function.

We will estimate the possibility of assured target selection and assured target lock-on for AT using the “3σ” criterion:

Р(|Х – μ| < 3σ) > 0.9973.

The “3σ” criterion allows to determine the minimum width of selection zone by X-coordinate, based on target designation: if false targets have parameters beyond zone [μ – 3σ, μ + 3σ], the target is reliably selected.

At the same time, the “3σ” criterion defines the possibility of searchless target lock-on: the target can be reliably locked for AT by Х-coordinate without fine search if the lock-on zone overlays the range [μ – 3σ, μ + 3σ].

Thus, if the optical radar’s field of view is equal to Δθ = 2о , then, based on relationship

3σεр-ц < Δθ/2, (9)

the value of RMSE σεр-ц shall be less than 1/3 deg.

Condition (9) is fulfilled for all the initial data represented in Tables 1 and 2 regarding the missile, but the degree of its fulfilment substantially depends on the relative position of SAM and on values of RMSE σεр-ц. If SAM is on the RS – target line, even with a minor increase in coordinate measurement errors in comparison with those given in Tables 1 and 2, RS will fail to provide the required target designation accuracy for assured target angle lock by an optical radar. Actually, the influence of some factors not mentioned herein, such as, for example, onboard radar’s attitude instrument errors, may lead to failure of assured target lock-on.

Below we will analyse the influence of the relative position of the missile and errors in target and missile coordinate estimates on the target designation accuracy.

For the analysis, we will apply the analytical method and statistical simulation method.

In order to find out how a target designation error is generated and to reveal its dependence on the relative position of the missile, we will start with the predetermined extreme position of the missile: SAM is on the RS – target line, and SAM is on the normal to this line. We will also analyse two cases: 1) the exact missile position is known and 2) the missile position is determined with errors.

3.1. SAM is on the RS – target line

When the exact missile position is known (σRр = 0 m, σεр = 0о ), taking into account that Δε = 0о , α = 0о , we get the expression for the target angle destination error from formula (7):

(10)

Fig. 4. Generation of target angle designation error with the missile located on the RS – target line for case a), when exact missile position is known, and b) when the known missile position is determined with errors. The figure shows arcs of dispersion of missile coordinates estimates (blue) and target coordinates estimates (red) and relations between individual points of arcs of dispersion (orange)

The obtained formula corresponds to the physical content of the target angle designation error, namely (see Fig. 4A): the arc of dispersion of target coordinates within the observation angle limited by values ±σεц is equal to 2σεцRц; the onboard radar of the missile, located on the RS – target line, receives target coordinates with target angle designation error σεр-ц. In this case, the following equation is true

2σεр-цRр-ц = 2σεцRц,

from which formula (10) is derived.

According to formula (10), if the missile is on the RS – target line, the target angle designation error is proportional to the ratio of the ranges from the RS to the target and from the missile to the target. The shorter the distance to the target, at which the onboard radar receives target designation data for lock-on for AT by angle, the less accurate this data. The longer range Rц for target engagement, the less accurate the target destination data. Also, we should take into account that the value of onboard radar error σεц generally increases proportionally to an increase in target range Rц, because the RS receiver input signal/noise ratio is reduced.

If the onboard radar has a narrow field of view Δθ that hampers the assured target lock-on, the radar performance figure shall be increased and a target shall be locked at longer ranges Rр-ц and at shorter ranges Rц.

To estimate the possibility of assured lock-on in accordance with the “3σ” criterion in formula (10), value 3σεц shall be used accordingly.

Figure 5 shows graphs calculated by formula (10), which illustrate the possibility of assured target lock-on by angle based on the “3σ” criterion, depending on range Rц (from target to RS) and range Rр-ц (from target to radar).

According to the figure, if the optical radar’s field of view is equal to Δθ = 2о , then, based on equation (9), the assured lock-on of a target located at distance Rц = 70 km from the radar is enabled if distance Rр-ц between the radar and the target is not less than 10 km (if the radar’s capacity allows), and for Rц = 140 km – at Rр-ц > 20 km.

When the missile position is determined with errors, we get the following expression for the dispersion of target angle designation from (7):

(11)

Figure 4b) explains the physical content of the obtained formula. Actually, the formula defines an average angular deviation from the direction to the target, i.e. the segment with its ends positioned randomly, straying within arcs of dispersion of target and missile coordinates (the segment representations are shown in orange).

We should note that formula (10) is a special case of formula (11).

Figure 6 shows curves depicting the range for assured target lock-on for AT, calculated in accordance with (11), using initial data considered above.

Fig. 6. Target angle designation errors at σεр = 3×0,05о and other data shown in Figure 5

3.2. SAM is on the line orthogonal to the RS – target line

In this case, angle α = 90о , angle Δε and missile range Rр are determined according to equations (4), (5).

If the exact missile position is known (σRр = 0 m, σεр = 0о ), we can obtain the target angle designation error from from formula (7):

(12)

Figure 7a explains the physical content of formula (12): values of target range σRц measured by the radar are scattered in the radar observation sector equal to 2σεр-цRр-ц, i.e. 2σRц = 2σεр-цRр-ц. This allows to derive formula (12).

Fig. 7. Generation of target angle designation error with the missile located on the line orthogonal to the RS – target line for case a), when the known missile position is proven, and b), when the missile position is determined with errors. The colour of arcs of dispersion of missile and target coordinate estimates corresponds to Figure 4

Comparing formulae (10) and (12) to estimate target designation errors with the missile located in different relative positions, we can see that as σεцRц > σRц is always true for radars (angular cross resolution is always lower than range resolution), at the same distance Rр-ц target designation for a missile located on the RS – target line is always less accurate than target designation with a missile located on the orthogonal line.

When the missile position is determined with errors, we get the following expression from formula (7) to determine the dispersion of target designation angle:

The form of the obtained formula is determined by the form of ellipsoid-shaped arcs of dispersion of coordinate estimates and their mutual orientation. Figure 7b explains the physical content of the formula. This content is similar to the content of formula (11) described above, but the missile position is different.

Formula (12) is a special case of formula (13). It is interesting to reveal the behaviour of target angle designation error with different intermediate variants of the missile position (between the position on the RS – target line and on the orthogonal line), when angle α changes in the range of 0о to 90о and further.

According to calculations by formula (7), when the exact missile position is known (σRр = 0 m, σεр = 0о ), with the missile successively changing its position from the location on the RS – target line (α= 0о ) to the location on the orthogonal line (α = 90о ) and further (up to α = 180о , corresponding to the missile position behind the target relative to RS), with the same distance Rр-ц between the missile and the target Rр-ц the target angle designation error changes as per the cosine law, acquiring its minimum value at α = 90о and its maximum at α = 0о and180о (see Fig. 8).

Fig. 8. Dependence of target angle designation RMSE σεр-ц on the relative position of the missile (missile position changes circumferentially along radius Rр-ц = 14 km around the target) at Rц = 70 km, σRц = 0.5 m, σεц = 0.05о , for two variants of missile coordinates estimation accuracy: 1 – σRр = 0 m, σεр = 0о ; 2 – σRр = 0.5 m, σεр = 0.05о

A variation of angle α, when the missile position is determined with errors (σRр ≠ 0 m, σεр ≠ 0о ), with lock-on distance Rр-ц remaining constant, shows that the target angle designation error has its minimum value in the range near α = 90о (to be more precise, a bit smaller than 90о ) and its maximum at α = 0о and 180о . Such dependence of the target angle designation error is determined by the form of arcs of dispersion of target and missile coordinates and their mutual orientation.

The presence of a clearly defined dependence of the target angle designation on the relative position of the missile requires the problem of determination of the best suitable missile trajectory to be formulated and solved. Solving this problem allows to enhance the probability of target lock-on if the radar field of view is limited.

3.3. Target angle designation error band

For better understanding of the dependence of target angle designation error on the missile position, we will perform calculation in a wide range of missile coordinates variations with the target being in a fixed position.

For calculations, we will apply two methods described above – statistical and analytical (by formula (7)).

Figure 9 shows calculation results represented in the form of contour maps with the target designation angle error band in the RS coordinate system xOy for one of the variants of target position.

Fig. 9. Results of target angle designation RMSE calculation (in degrees): а – statistical method, nм = 1000; b – analytical method. Initial data: Хц = 30 km, Yц = 60 km, σRр = 0.5 m, σεр = 0.05о , σRц = 0.5 m, σεц = 0.05о , figure c shows 3D view of error function

In the figure, the error value is designated in colours: violet corresponds to the minimum error, red – to the maximum error, intermediate colours from blue to red correspond to a successive increase in the error value. The single-colour area defines a certain range of target designation errors.

According to the figure, missile positions unfavourable for target lock-on by angle are concentrated in the sector of around ±30о with account for initial data specified above.

The angle error reaches its maximum on the RS – target line and grows as the missile approaches the target. Thus, at a distance ~30 km to the target, the error is σεр-ц ≈ 0.12о , at a distance of ~15 km σεр-ц ≈ 0.21о , at a distance of ~5 km σεр-ц ≈ 0.38о .

Calculations made by statistical and analytical methods, are in a good agreement, allowing to prefer the analytical method as it is more time-saving and representative.

Since this analysis is similar to the analysis of the angular error in target designation described above, we will specify final results at once:

1. SAM is on the RS – target line (Δε = 0о , α = 0о ). If we know the exact missile position (σRр = 0 m, σεр = 0о ), we will get the expression for the target designation value from formula (6):

σRр-ц = σRц.

If the missile position is determined with errors, we will get the following expression from formula (6)

σ2 Rр-ц = σ2 Rр + σ2 Rц.

The given formulae have an easily understandable physical content, as shown in Figure 4.

2. SAM is in the line orthogonal to the RS – target line (α = 90о ).

If we know the exact missile position (σRр = 0 m, σεр = 0о ), we will get the following expression from formula (6):

σRр-ц = Rцσεц.

If the missile position is determined with errors, we will get the following expression from formula (6)

σ2 Rр-ц = σ2 Rр sin(Δε)2 + R2 ц(σ2 Rц + σ2 Rр).

Figure 7 explains the physical content of the formula.

Let us form the band of range designation errors for initial data used for forming the band of target angle designation errors shown above (Fig. 9). Results are shown in Figure 10.

Fig. 10. Results of target range designation RMSE calculation (in meters), c – 3D view of error function

According to the figure, the minimum error in target range designation for selected initial data is ensured when the missile is located in a narrow sector around ±5о relative to the line directed from the target to the RS. The error value rapidly increases outside the sector.

Unlike the target angle designation error, the maximum error in target range designation takes place when the missile is located near the line orthogonal to the RS – target line.

Based on results of the analysis of the target angle and range designation error band, we can make the following conclusions:

Target speed (Doppler frequency) designation as well as target range and angle designation is a relative value. It is the target speed calculated as per target and missile coordinates measurement data relative to the current position of the missile. Figure 11 shows the ratio of missile and target velocity vectors (Vр and Vц) at the time of target lock-on. The difference between the projections of these vectors on the missile – target line is target velocity designation Vр-ц = Vр.пр – Vц.пр.

Fig. 11. Relation between missile velocity vector and target velocity vector at the time of lock-on

The RS measures radial velocities of the missile and target (Vр.РЛС and Vц.РЛС) as per additional Doppler frequencies of received signals, but we cannot determine the velocity projections Vр.пр, Vц.пр. In this respect, we use the trajectory method for calculating target velocity Vр-ц: the relative missile velocity in time interval Δt is estimated by formula (14)

(14)

where Rр-ц(t) and Rр-ц(t – Δt) – relative range values at the relevant time to be calculated by formula (2), based on results of target and missile coordinates measurements.

Values Rр-ц(t) and Rр-ц(t – Δt) are independent random values with dispersions (σRр-ц (t))2 and (σRр-ц (t – Δt))2 , respectively, to be calculated by formula (6).

Based on dispersion properties for independent random values, we get the dispersion of relative target speed estimate [5, p. 333]:

(15)

According to the resulted formula, the dependence of target speed designation error on the missile position is similar to the dependence of target range designation error: for accepted initial data in the example discussed above, the target speed designation error has its minimum on the RS – target line and takes maximum values near the line orthogonal to the RS – target line.

All the calculations and conclusions represented above are obtained for fixed position of the missile and target. Let us analyse the influence of the movement factor on target designation errors. We will study radar’s target designation errors along the trajectory of a moving missile being guided onto a target. As a target, we will analyse the ballistic target falling in the RS area.

The following target trajectory parameters are selected (see Fig. 12): the target is falling along an arching trajectory and at the time of missile launch to engage, it has the coordinates Xц = 60 km, Yц = 120 km and speed of 3000 m/s. As the target keeps moving, its speed decreases with acceleration of – 2g (simulation of atmospheric deceleration); at the time of missile impact, the speed is 2420–2430 m/s (depending on impact point).

Fig. 12. а – ballistic target trajectory (red) and three missile trajectories (blue), b – trajectories at impact point

The missile is launched in the vertical direction. The missile speed is set as a variable – the operation of a booster engine and subsequent motion with deceleration is simulated (deceleration is caused by gravity forces and air resistance). The velocity curve is shown in Figure 13. The trajectory of a missile being guided onto a target is plotted using the proportional navigation method. The form of the missile trajectory is determined by the method proportionality coefficient. Figure 12 shows three missile flight trajectories corresponding to different method coefficients.

Fig. 13. Missile velocity curve

Target interception takes place within a time interval of 28.6–29 s (depending on the guidance trajectory). The average missile velocity during flight to impact point is ~2.1 km/s.

Let us calculate errors in target angle, range and speed designation for each missile trajectory. Taking into account a good agreement of the results of statistical and analytical methods of target designation error estimation, we will apply the analytical method for estimation (using formulae (6), (7), (15)).

First, we will carry out calculations, assuming that target coordinate estimation errors are constant values throughout the entire flight time, irrespective of the target range.

Suppose the target coordinates estimate RMSE are equal to σRц = 0.5 m, σεц = 0.05о . We assume missile coordinate determination errors by an order of magnitude smaller: Rр = 0.05 m, σεр = 0.005о .

Figure 14 shows the results of target designation error calculations, depending on current values of distances between the target and missile.

Fig. 14. TD error curves depending on the distance between the target and missile for three missile flight trajectories shown in Fig. 12, at constant values of coordinate estimate errors σRц = 0.5 m, σεц = 0.05о , σRр = 0.05 m, σεр = 0.005о

Calculations are made through numerical integration of missile guidance equations with an interval of 0.1 s. This may be interpreted as operation of the guidance loop and target and missile data update with frequency of 10 Hz. A decrease in the integration period leads to significant variation in the results, except the target speed designation error – it changes in inverse proportion to the integration period as per dependence (15).

According to the plotted graphs (Fig. 14), for the accepted initial data, the missile movement along the trajectory near the RS – target line (trajectory 3) provides very low values of target range and speed designation errors throughout the entire missile flight. This guarantees high-precision target selection based on these coordinates. If the missile deviates from the RS – target line (trajectories 1, 2), target range and speed designation errors are considerably higher and increase as the missile approaches the target, because the missile trajectory passes near the line orthogonal to the RS – target line.

Target angle designation errors are less dependent on the missile trajectory but they substantially depend on the distance to target; that is why, the radar performance figure shall provide target lock-on for AT by angle in ranges matching the width of the radar field of view for searchless lock-on.

Further, we will take into account that as the target approaches RS, the accuracy of target coordinate determination increases due to a higher signal/noise ratio at the RS receiver input.

We know [6] that the coordinate estimate RMSE is inversely proportional to the square root of the signal/noise ratio at the RS receiver input. In its turn, in intense radar detection and ranging conditions, the signal/noise ratio value with all other values being equal (signal power, its duration, antenna aperture size, receiver noise coefficient, etc.) is inversely proportional to the target range to the fourth power Rц 4 . Therefore, if the target range changes, the target coordinates estimate RMSE changes proportionally to the ratio of squared ranges.

In this respect, to estimate the accuracy of coordinates of the target approaching RS for RMSE values σRц, σεц, we will use multiplier kцR = (Rц.оп/Rц) 2 , where Rц.оп is the reference range, which is assumed to be the range of target lock-on for AT, Rц is the current target range, Rц ≤ Rц.оп.

Figure 15 shows graphs of target designation errors, depending on current distances between the target and missile, obtained with account for the reduced RMSE of target coordinate estimates as the target approaches (proportionally to multiplier kцR). In calculations, we assume that when the target is located at the initial point (at the time of missile launch to engage), the values of target coordinate estimate RMSE are equal to σRц = 2.85 m, σεц = 0.285о . At the target impact point, these RMSE values are reduced to values σRц ≈ 0.5 m, σεц ≈ 0.05о due to a decrease in the target range. Missile coordinate estimate RMSE values are taken as constant values equal to σRр = 0.05 m, σεр = 0.005о .

Fig. 15. Diagrams of TD errors, depending on the distance between the target and missile for three missile trajectories shown in Figure 12, with account for the dependence of coordinate estimate RMSE on target range

The plotted graphs show that the ratio of target designation errors remains unchanged, similarly to the case analysed above, i.e. target coordinate estimation errors do not depend on the target range. In this case, target designation errors at long ranges have increased due to increased errors in target coordinate estimation because of a low signal/noise ratio.

According to the conducted studies, target designation errors largely depend on the form of the trajectory of the missile moving towards the target. This circumstance shall be taken into account when optimising the trajectory of the missile guided onto the target with a minimum miss. The most significant factors that define the selection of the trajectory, along with the target designation errors under consideration, include dynamic and fluctuating guidance errors caused by the peculiarities of the missile guidance loop, as well as the action of the air resistance force when the missile flies through the dense atmosphere at a high speed.

The influence of various factors on the selection of the missile trajectory is mostly in conflict with the influence of target designation errors. Thus, for example, in order to reduce dynamic errors in missile guidance, we need to increase the data refresh rate regarding the target and the missile, but this leads to a greater target speed designation error. Or, in order to reduce the target range designation errors, it is necessary to increase the proportional navigation method coefficient, thus ensuring the fastest delivery of the missile to the RS – target line, but an increase in the coefficient requires the expansion of the guidance loop bandpass, resulting in higher fluctuating guidance errors.

In this respect, the trajectory shall be selected with account of the influence of all the specified factors and their joint effect on the resulting tactical parameters and criteria of missile firing, such as, for example, minimisation of missed interception, target discrimination by an onboard radar amidst false targets, minimisation of the time required for the missile to intercept the target, and at the same time, enabling target hit in the maximum range in order to increase the capacity of fire weapon system.

This paper represents the first research that reveals, and gives an in-depth analysis of, the dependence of the accuracy of target designation issued by a ground radar system, based on the measured target and missile coordinates, on the relative position of the missile (relative to the RS and the target).

The mathematical apparatus (represented as two methods – statistical and analytical) has been developed to determine the dependence and estimate it from the quantitative point of view. The analytical method allowed to obtain new formulae that describe the specified dependence for target range, angle and speed. The obtained formulae have been refined to make them suitable for engineering application for two special (extreme) cases: when SAM is on the RS – target line and when it is located on the line orthogonal to the latter. This allows to use these formulae, for instance, for estimates at the SAM guidance system concept design stage. We have studied the physical content of the obtained formulae.

The dependence of the accuracy of target designation data issued to an onboard radar on the relative position of the missile is proven by high consistence of the results, obtained by applying the statistical and analytical methods in calculations with variable initial data within a wide range, as well as by the presence of comprehensible physical content of the obtained formulae for two extreme cases regarding the missile position.

Based on the near real data, we made calculations and proved high importance of consideration to be given to the accuracy of target designation data issued to an onboard radar, depending on the relative position of the missile in order to enable the assured (searchless) target lock-on by angular coordinates due to a narrow field of view of advanced radars operating in the millimetre or optical waveband.

In order to estimate the nature of the dependence of target designation for an orboard radar, we formed target designation error bands, based on different coordinates with the target being in a fixed position. The study proves that the target range and speed designation error can be minimized if the missile is located on the RS – target line, while the target angle designation error, on the contrary, will reach its maximum. When the missile is located near the direction to target, orthogonal to the RS – target line, we will get the maximum target range and speed designation error and the minimum target angle error. These properties of error bands are defined by the shape of arcs of dispersion of target and missile coordinates.

As SAM approaches the target, target angle designation errors increase (more intensely for the SAM positions near the RS – target line). Therefore, the onboard radar performance figure shall be enhanced in order to enable the assured long-range target lock-on by angular coordinates.

We also analysed the behaviour of target designation errors in the dynamics of missile guidance onto a moving target with account for the specific features of the missile guidance trajectory and the dependence of the accuracy of target and missile coordinates measurements performed by an onboard radar on the distance to them. The study proves that in order to optimize the SDAM flight trajectory, we should consider errors in target designation data issued to an onboard radar in combination with other factors due to the controversial behaviour of their mutual influence.

The developed methods allow to estimate the accuracy of target designation for an missile onboard radar with known (preset) errors in target and missile coordinates measured by a ground radar, and at the same time, to solve the inverse problem – to determine the desired accuracy of target and missile coordinate measurements performed by a ground radar to enable searchless target lock-on by angles and required capabilities regarding target selection by range and speed in an environment exposed to hostile countermeasures.