RU2164654C2 - Method for homing of flight vehicles on ground targets - Google Patents

Abstract

FIELD: homing of flight vehicles with on-board radars using synthesizing of antenna aperture or Doppler beam narrowing of antenna directivity pattern on ground targets. SUBSTANCE: the method consists in measurement of the values of range from the homed flight vehicle to the ground target, rate of closure of the homed flight vehicle and ground target, relative bearing of the ground target and angular velocity of the sighting line in the horizontal plane, as well as lateral acceleration of the homed flight vehicle in the horizontal plane, formation of the control signal in the horizontal plane in the form of the sum of the difference of the required lead angle weighed by a variable gain and the relative bearing of the ground target and the angular velocity of the sighting line weighed by a variable gain, and the value of acceleration of the homed flight vehicle. The values of variable gains are calculated with due account made for the current values of range from the homed flight vehicle to the ground target and the rate of their closure, as well as for the gains determining the accuracy of homing by the relative bearing of the ground target and angular velocity of the sighting line, as well as determining the efficiency of the process of homing, and the values of the required lead angle are formed by special way so that the conditions of use would be taken into account, and stabilization of the required resolution in the horizontal plane would be provided. EFFECT: enhanced accuracy and efficiency of homing, enhanced stabilization of the required linear resolution in the horizontal plane at homing of the flight vehicle on ground targets with the aid of on-board radars using synthesizing antenna aperture or Doppler beam narrowing. 15 dwg

Description

 The invention relates to homing systems, in particular to homing systems of aircraft (LA) on ground objects using radar tools installed on board the aircraft using aperture synthesis (SA) of the antenna or Doppler beam beam (DOL) antenna radiation pattern.

 There are several traditional methods of targeting aircraft on ground objects: the method of vane guidance [1, p. 181]; a method of successive lead-time [1, p. 182].

 In the general case, the guidance method refers to the law of the formation of the required phase trajectory of the induced control object, guidance on which will hit the target or object. The phase coordinates of the required motion are found by converting the phase coordinates of the relative and absolute motion of the target (ground object) and the control object (induced aircraft). The rule for generating control signals, also referred to as mismatch parameters, which determines the mismatch between the real phase coordinates of the induced aircraft and their required values, is called the trajectory control algorithm. It should be noted that in aircraft homing systems for ground targets control signals (mismatch parameters) are usually formed in horizontal and vertical planes, since it is in these planes (course and pitch) that their steering organs are located.

The essence of the method of vane guidance is that with the direction to the ground object, the air velocity vector of the induced aircraft is combined. If the mismatch parameters for the course and pitch planes are formed in a spatially stabilized measuring coordinate system, then the trajectory control algorithm with the method of vane guidance is determined by the following relationships:
Δ _{T FG} = φ + αsinγ, Δ = φ _{The PV} + αcosγ, (1)
where Δ _FG, Δ _PV - process control signals when a feathered guidance in the horizontal and vertical planes respectively;
φ _G , φ _B - on-board bearings of a ground object in horizontal and vertical planes, measured by the goniometer of an on-board radar station with a space-stabilized antenna;
α, γ are the angles of attack and roll, respectively.

The need to take into account the second terms in (1) is due to the fact that the instantaneous direction of the aircraft’s movement coincides with the direction of its velocity vector, which differs from the position of the aircraft’s construction axis by the value of the angle of attack. In the presence of a roll, the goniometer with a stabilized antenna still measures the projections of the side bearing in the horizontal φ _G and vertical φ _B planes, while the spatial position of the angle of attack changes. The latter causes the appearance of projections αsinγ and αcosγ of the angle of attack on the horizontal and vertical planes. Due to its simplicity, the method of weathervane guidance is widely used when pointing aircraft at stationary and inactive land objects, but its main drawback is the rather strong influence of the side wind on the accuracy of guidance. If necessary, parry the wind usually use the method of sequential lead.

 When using the sequential lead method, sometimes called the chase method with an additional lead angle, the control signals in the horizontal and vertical planes along with the terms proportional to the projections of the side bearing of the ground object also contain projections of the additional lead angle, which is chosen proportional to the angular velocity of the line of sight. The values of the proportionality coefficients are selected, as a rule, so that the guidance path is close to straightforward.

 It should be noted that the specifics of the methods of guidance of aircraft with airborne radar systems using SA or DOL on ground objects is the use of purely curved guidance trajectories in the horizontal plane. As a rule, these trajectories should provide the required linear resolution in the horizontal plane in the initial section, and high accuracy of pointing at the ground, generally moving object, in the final section. With such guidance trajectories, the energy expenditures for aircraft control sharply increase, and therefore the issue of improving the efficiency of guidance procedures also becomes relevant (economics here refers to the property of a guidance system that characterizes its ability to spend as little energy as possible on control signals).

 However, to meet the conflicting requirements of high accuracy and cost-effectiveness of guidance while stabilizing the required linear resolution in the horizontal plane, in conditions of real restrictions on the available transverse overloads of the aircraft, using any traditional guidance method is practically impossible. It should be emphasized that the alternate empirical use of different guidance methods at different parts of the trajectory does not allow obtaining acceptable results due to the difficulty in determining the moment of transition from one method to another, the occurrence of significant transient processes and the difficulty of providing guidance economy.

где Δ_{ПУ Г} - сигнал управления в горизонтальной плоскости;

соответствующие постоянные коэффициенты усиления;
φ_Г - значение бортового пеленга наземного объекта в горизонтальной плоскости;
ω_Г - значение угловой скорости линии визирования в горизонтальной плоскости.Of the known methods of pointing aircraft at ground objects, the closest (prototype) is the method of successive lead-ins [1, p. 182], since with this method of pointing the control signal in the horizontal plane contains terms proportional to the values of the direction-finding bearing and the angular velocity of the line of sight, as well as in the present invention. The trajectory control algorithm in the horizontal plane when using the sequential lead method can be represented as follows:

where Δ _{PU G} is the control signal in the horizontal plane;

corresponding constant gain factors;
φ _G - the value of the side bearing of a ground object in the horizontal plane;
ω _G - the value of the angular velocity of the line of sight in the horizontal plane.

выбираются таким образом, чтобы траектория наведения была близка к прямолинейной.Odds

are selected so that the guidance path is close to straight.

However, the use of this method when aiming an aircraft at ground-based objects using radars operating in SA or DOL modes is difficult. This is due to the following main disadvantages of the prototype:
the impossibility of stabilizing the required linear resolution of radar in the horizontal plane (since when using the method of successive anticipations the guidance trajectory is close to straightforward, it is almost impossible to ensure the use of SA or DOL, for which the fundamental condition is the non-zero value of the side bearing of the ground object, which, in turn, , determines the curvature of the guidance paths);
the lack of consideration of the requirements for the efficiency of the guidance process, especially when using curved guidance paths (since using constant amplification factors it is impossible to take into account the energy costs for control signals that change in the general case during the guidance process);
the lack of consideration of the conditions of use (since when changing operating conditions, for example, when changing the distance from an aircraft to a ground object, approach speed, etc., constant gain factors need to be tuned in the general case, that is, they are not adaptive to the conditions of use).

 Thus, the objective of the present invention is to develop a method of pointing aircraft to ground objects, while simultaneously providing high accuracy of guidance, stabilization of the required linear resolution in the horizontal plane when using SA or DOL in radar and high efficiency control process.

 The problem is achieved in that in the method of sequential lead-time, which consists in the fact that the values of the side bearing of the ground object and the angular velocity of the line of sight in the horizontal plane are measured, the control signal in the horizontal plane is formed by summing the values of the side bearing of the ground object weighted by the corresponding gain factors and the angular velocity of the line of sight (2) additionally measure the distance from the induced aircraft to the ground object and the speed of their approach, and that induced same aircraft speed and the lateral acceleration in the horizontal plane; calculate the required lead angle, providing the required linear resolution in the horizontal plane, an additional term is introduced into the control signal in the horizontal plane, taking into account the flow rate of current overloads (accelerations); variable amplification factors are formed that take into account the operating conditions (the distance from the induced aircraft to the ground object and the speed of their approach), the weight of the guidance errors along the side bearing and the angular velocity of the line of sight, as well as the requirements for guidance efficiency. The latter makes it possible in the guidance process to automatically redistribute priorities between accuracy, guidance efficiency and stabilization of the required linear resolution.

где Δ_Г - сигнал управления в горизонтальной плоскости;

коэффициенты, определяющие точность наведения по бортовому пеленгу и угловой скорости линии визирования в горизонтальной плоскости:

- коэффициент, определяющий экономичность наведения в горизонтальной плоскости;
φ_Г - значение бортового пеленга в горизонтальной плоскости;
ω_Г - значение угловой скорости линии визирования в горизонтальной плоскости;
Д - значение дальности от наводимого ЛА до наземного объекта;
V_СБ - значение скорости сближения наводимого ЛА с наземным объектом;
V - значение скорости наводимого ЛА;
j_Г - значение поперечного ускорения наводимого ЛА в горизонтальной плоскости;
λ - длина волны БРЛС;
ΔF - полоса пропускания доплеровского фильтра:
φ_ГТ - требуемый угол упреждения, при котором обеспечивается требуемое линейное разрешение Δl_T в горизонтальной плоскости.The proposed guidance method is carried out by the following trajectory control algorithm in the horizontal plane:

where Δ _G is the control signal in the horizontal plane;

coefficients that determine the accuracy of guidance on the side bearing and the angular velocity of the line of sight in the horizontal plane:

- coefficient determining the efficiency of guidance in the horizontal plane;
φ _G - the value of the side bearing in the horizontal plane;
ω _G - the value of the angular velocity of the line of sight in the horizontal plane;
D is the value of the distance from the induced aircraft to the ground object;
V _SB - the value of the convergence rate of the induced aircraft with a ground object;
V is the value of the speed of the induced aircraft;
j _G - the value of the lateral acceleration of the induced aircraft in the horizontal plane;
λ is the radar wavelength;
ΔF - bandwidth of the Doppler filter:
φ _GT is the required lead angle at which the required linear resolution Δl _T in the horizontal plane is provided.

 In FIG. 1, the dashed line shows the guidance path in the sequential lead method, and the solid line shows the guidance path for the proposed guidance method.

In FIG. 2 shows a simplified block diagram of a possible variant of a system for guiding an aircraft to ground objects that implements the proposed guidance method, where;
1 - radar antenna system:
2 - radar receiver / transmitter,
3 - radar goniometer;
4 - computer control signal;
5 - control system:
6 - aircraft;
7 - automatic range and speed approach radar:
8 - calculator of the required lead angle;
9 - accelerometer.

 FIG. 3 illustrates the geometric relationships between the coordinates of the absolute and relative motion of the induced aircraft and a ground object in the horizontal plane.

 FIG. 4-15 illustrate the performance of the invention.

 Consider how one of the possible options for the guidance system of aircraft on ground objects when using the proposed method of guidance (Fig. 2).

 The radar antenna system 1 performs spatial selection of the signal reflected from the ground object, which then goes to the input of the radar receiver 2, in which the signal reflected from the ground object is extracted from the noise due to narrow-band Doppler filtering when using SA or DOL, from the output receiver 2, the signal is fed to the input of the automatic range selector and approach speed 7, in which the distance measurements from the induced aircraft to the ground object and their approach speed, as well as at the input q goniometer 3, wherein the bead is formed measurements bearing ground object and the angular velocity of the line of sight on the horizontal plane. Using the accelerometer 9, the transverse acceleration of the aircraft is measured in the horizontal plane. The generated measurements of the distance and approach speed are sent to the calculator of the required lead angle 8, in which, according to relation (4), the required lead angle is formed, which ensures the required linear resolution in the horizontal plane. The value of the required lead angle and all generated measurements are sent to the control signal computer 4, in which the control signal is generated in the horizontal plane according to relation (3). In the control system 5, the generated control signal is converted in the horizontal plane to the corresponding control actions that are supplied to the control organs of the aircraft 6 itself.

 Consider how the formation of the control signal in the horizontal plane when using the proposed method of guidance in dynamics.

(3) сигнала управления. Влияние этого слагаемого становится особенно значительным на малых расстояниях до наземного объекта. Следовательно, в процессе полета, по мере приближения ЛА к наземному объекту, происходит плавное перераспределение влияния ошибок наведения по бортовому пеленгу наземного объекта φ_Г-φ_ГТ, стабилизирующих разрешающую способность в горизонтальной плоскости на начальном участке траектории наведения, к ошибкам по угловой скорости линии визирования ω_Г, минимизирующим текущий промах на конечном участке траектории.At large distances, when D is large and ω _Г ≈ 0, the control is determined by the first term (3). In such a situation, the flight of the aircraft will be carried out at an angle φ _Г ≈ φ _ГТ to the ground object, providing stabilization of the required linear resolution in the horizontal plane. In this case, as D decreases and ω _G increases, the influence of the second term begins to increase

(3) control signal. The influence of this term becomes especially significant at small distances to a ground object. Therefore, during the flight, as the aircraft approaches the ground object, there is a smooth redistribution of the influence of guidance errors along the side bearing of the ground object φ _Г -φ _ГТ , which stabilize the resolution in the horizontal plane at the initial portion of the guidance path to errors in the angular velocity of the line of sight ω _G , minimizing the current miss on the final section of the trajectory.

При этом выбором k_j - коэффициента, определяющего экономичность наведения, добиваются выполнения условия j_Г ≅ j_{Г ДОП}, при котором текущие управляющие перегрузки не превышают предельно допустимых значений. Манипулируя значениями коэффициентов q_φ и q_ω/ можно варьировать моментом времени перехода от преимущественного наведения по бортовому пеленгу к наведению по угловой скорости линии визирования.The weight of errors along the side bearing and the angular velocity of the line of sight depends on the ratio of the coefficients

At the same time, by choosing k _j , the coefficient that determines the efficiency of guidance, the condition j _G ≅ j _{G DOP} is satisfied, under which the current control overloads do not exceed the maximum permissible values. By manipulating the values of the coefficients q _φ and q _{ω /,} it is possible to vary the time moment of the transition from predominant guidance along the side bearing to guidance along the angular velocity of the line of sight.

The geometric relationships between the coordinates of the absolute and relative motion of the aircraft and the ground object in the horizontal plane using the proposed guidance method are shown in FIG. 3, on it:
XOZ - terrestrial non-rotating coordinate system;
About _LA - the current location of the aircraft;
About _BUT - the current position of the ground object:
V _BUT is a vector characterizing the speed and direction of movement of a ground object;
V _T and V are the required and actual values of the velocity vectors of the induced aircraft;
D is the range from the aircraft to the ground object;
φ _ГT is the required value of the side bearing (lead angle) at which the required resolution Δl _T in the horizontal plane is provided:
φ _G - the actual value of the side bearing of the ground object in the horizontal plane;
j _G is the lateral acceleration of the aircraft in the horizontal plane;
ε _G is the angle of sight of the ground object in the horizontal plane.

где φ_Г - бортовой пеленг наземного объекта в горизонтальной плоскости,
ω_Г - угловая скорость линии визирования в горизонтальной плоскости,
Д - дальность от ЛА до наземного объекта,
V_СБ - скорость сближения ЛА с наземным объектом,
j_Г - поперечное ускорение ЛА в горизонтальной плоскости,

- соответственно шумы возмущений по бортовому пеленгу наземного объекта и угловой скорости линии визирования. При получении (5), (6) было учтено, что

и тот факт, что поперечное ускорение наземного объекта равно нулю.By analyzing the geometric relationships of the absolute and relative phase coordinates, we can obtain the relations:

where φ _G - side bearing of a ground object in the horizontal plane,
ω _G - the angular velocity of the line of sight in the horizontal plane,
D is the range from the aircraft to the ground object,
V _SB - the speed of approach of the aircraft with a ground object,
j _G - transverse acceleration of the aircraft in the horizontal plane,

- respectively, disturbance noises along the side bearing of the ground object and the angular velocity of the line of sight. Upon receipt of (5), (6), it was taken into account that

and the fact that the lateral acceleration of a ground object is zero.

где Д - дальность от ЛА до наземного объекта,
ω_Г - угловая скорость линии визирования в горизонтальной плоскости,
V_СБ - скорость сближения ЛА с наземным объектом;
текущее значение линейного разрешения в горизонтальной плоскости определялось соотношением

где Д - дальность от ЛА до наземного объекта,
V - скорость наводимого ЛА,
λ - длина волны БРЛС,
ΔF - полоса пропускания доплеровского фильтра,
φ_Г - бортовой пеленг наземного объекта в горизонтальной плоскости,
ω_Г - угловая скорость линии визирования в горизонтальной плоскости;
шумы возмущений в (5), (6) отсутствуют и управление передается безынерционно:
скорость наземного объекта равна нулю.To assess the effectiveness of the proposed guidance method, a simulation was conducted of the aircraft guidance system on a ground object. In the modeling process, the following assumptions were made:
as one of the indicators of the quality of the functioning of the trajectory control algorithm, the current miss in the horizontal plane was adopted, which was determined by the ratio

where D is the distance from the aircraft to the ground object,
ω _G - the angular velocity of the line of sight in the horizontal plane,
V _SB - the speed of approach of the aircraft with a ground object;
the current value of the linear resolution in the horizontal plane was determined by the ratio

where D is the distance from the aircraft to the ground object,
V is the speed of the induced aircraft,
λ is the radar wavelength,
ΔF is the bandwidth of the Doppler filter,
φ _G - side bearing of a ground object in the horizontal plane,
ω _G is the angular velocity of the line of sight in the horizontal plane;
disturbance noises in (5), (6) are absent and control is transmitted inertialessly:
the speed of the ground object is zero.

The purpose of the simulation was to study the capabilities of the proposed guidance method to provide the required linear resolution in the horizontal plane and the required accuracy of guidance. The study was carried out in several stages. At the first stage, the influence of the values of the coefficients q _φ , q _ω and k _j , taking into account the requirements for accuracy and economy, on the functioning of the trajectory control algorithm was studied. At the second stage, the influence of the required linear resolution Δl _{T was} studied.
The studies were carried out by time modeling of the following dependences: D is the range from the aircraft to the ground object, φ _G is the side bearing of the ground object in the horizontal plane, ω _G is the angular velocity of the line of sight in the horizontal plane, Δ _G is the parameter of the mismatch in the horizontal plane, Δl - the current linear resolution, h - the current miss and x, z - the coordinates of the aircraft in the Earth's fixed coordinate system XOZ. Moreover, φ _Г and ω _{Г were} calculated by (5), (6) using the first-order Euler method, the control signal by (3), (4), the current linear resolution by (8), and the current miss by ( 7), and the current location of the aircraft was determined by reckoning coordinates.

In the simulation, it was assumed (unless specifically stated) that the initial value of the distance from the aircraft to the ground object is D ₀ = 15,000 m, the initial position of the aircraft in the XOZ plane was determined by the coordinates x ₀ = 15,000 m, Z ₀ = 0 m, the position of the ground object was determined by the coordinates x _HO = 0 m, Z _HO = 0 m, the radar wavelength λ = 0.008 m, the Doppler filter passband ΔF = 40 Hz, the required linear resolution in the horizontal plane Δl _T = 10 m.

 The simulation results are shown in FIG. 4-15: in FIG. 4, 7, 10, 13 show the dependences of the current linear resolution Δl on the guidance time, in FIG. 5, 8, 11, 14 - the dependence of the current miss h on the guidance time, and in FIG. 6, 9, 12, 15 - guidance trajectories in the XOZ coordinate system.

штрихпунктирной - q_ω= q_ω3(q_ω1<q_ω2<q_ω3). На фиг. 10-12 приведены результаты исследований влияния коэффициента k_j, определяющего экономичность наведения на функционирование алгоритма траекторного управления. На этих фигурах сплошная линия соответствует k_j=k_j1, пунктирная - k_j= k_j2, штрихпунктирная - k_j = k_j3 (k_j1 < k_j2 < k_j3).In FIG. Figures 4-6 show the results of studies of the influence of the coefficient q _φ , which determines the accuracy of guidance on the side bearing on the functioning of the trajectory control algorithm. In these figures, the solid line corresponds to q _φ = q _φ1, dashed - q _φ = q _φ2, dot-dash _{- q φ = q φ3 (q} φ1 <q φ2 <q φ3). In FIG. Figures 7 - 9 show the results of studies of the influence of the coefficient q _ω , which determines the accuracy of pointing along the angular velocity of the line of sight on the main indicators of the functioning of the trajectory control algorithm. In these figures, the solid line corresponds to q _ω = q _ω1 , the dashed -

dash-dotted - q _ω = q _ω3 (q _ω1 <q _ω2 <q _ω3 ). In FIG. 10-12 are the results of studies of the influence of the coefficient k _j , which determines the efficiency of guidance on the functioning of the trajectory control algorithm. In these figures, the solid line corresponds to k _j = k _j1, dashed - k _j = k _j2, dot-dash _{- k j = k j3 (k} j1 <k j2 <k j3).

Analysis of the results of the first stage of research allows us to draw the following conclusions:
an increase in the coefficient q _φ leads to a faster development of the initial errors on the side bearing, while the value of the control signal at the initial stage increases, as a result of which the missile trajectory deviates more from the straight line, while the required linear resolution is more stable:
an increase in the coefficient q _ω practically does not affect the initial guidance portion, however, in the final portion it leads to a faster tendency of the current value of the angular velocity of the line of sight to the required zero value (at ω _Г = 0, the current miss is zero (7)), thereby more quickly the current miss is selected, while on the whole, the guidance path q _{ω is} affected a little:
the value of the coefficient k _j , since it is included both in the weight of the guidance error along the side bearing and in the weight of the guidance error in the angular velocity of the line of sight, affects the entire length of the guidance section of the aircraft. Moreover, a decrease in k _j at the initial stage leads to a faster development of the initial errors in the required on-board bearing, and at the final stage, to a faster development of errors in the angular velocity of the line of sight. All these circumstances lead to the fact that the required value of the linear resolution is better stabilized, and in the initial section the guidance path deviates more from the straight line, and in the final one, the aircraft is faster displayed on the ground object.

The results of studies of the effect of the required linear resolution Δl _T on the functioning of the trajectory control algorithm are shown in FIG. 13-15. In these figures, the solid line corresponds to Δl _T = 7 m, the dashed Δl _T = 10 m, and the dot-dash line Δl _T = 13 m. Analysis of the results of the second stage of studies allows us to draw the following conclusion: decreasing the required resolution Δl _T significantly increases the deviation of the aircraft trajectory from the straight line - so, at Δl _T = 13 m, the maximum lateral deviation from the straight path is z _max ≈ 1750 m, at Δl _T = 10 m-z _max ≈ 2400 m, and at Δl _T = 7 m-z _max ≈ 4250 m.
In General, according to the simulation results of the proposed guidance method, we can draw the following conclusions.

 The simulation results confirmed the efficiency of the proposed method of targeting aircraft on ground objects.

 The proposed method allows for the stabilization of the required linear resolution in the horizontal plane at the initial guidance site, and the current missed aircraft minimization at the final site.

 Varying the values of the coefficients that take into account the requirements for accuracy and economy makes it possible to fairly easily and smoothly redistribute priorities in guidance between ensuring the required linear resolution in azimuth in the horizontal plane and allowable misses of the aircraft and allows achieving an acceptable quality of functioning of the proposed guidance method as a whole.

 The use of the invention in comparison with the prototype due to the introduction of an additional adaptive lead angle and gain factors that adaptively take into account the application conditions, as well as the requirements for accuracy and cost-effectiveness of the guidance process will allow, as the simulation showed, to simultaneously provide high accuracy and cost-effectiveness of guidance while stabilizing the required linear resolution with azimuth in the horizontal plane when using in radar systems CA or DOL.

 In addition, the use of the claimed method of pointing aircraft to ground objects does not impose any additional restrictions on the element base and does not impose any significant requirements on the speed and memory size of the computers.

Literature
1. Merkulov V.I., Lepin V.N. Aircraft radio control systems, part 1, part 2. - M .: Radio and communications, 1997.

Claims (1)

A method of guiding aircraft to ground objects, which consists in measuring the side bearing of a ground object and the angular velocity of the line of sight in the horizontal plane, generating a control signal in the horizontal plane as the sum of the weighted values of the side bearing of the ground object and the angular velocity line of sight in the horizontal plane, characterized in that they measure the range from the induced aircraft (LA) to ground of the object and the speed of convergence, and the values of the velocity induced by the aircraft and its acceleration in the horizontal plane, generating a control signal in a horizontal plane by the ratio

where Δ _G is the control signal in the horizontal plane;
q _φГ , q _ωГ - coefficients that determine the accuracy of guidance along the side bearing and the angular velocity of the line of sight in the horizontal plane;
k _jГ - coefficient determining the efficiency of pointing in the horizontal plane;
D is the value of the distance from the induced aircraft to the ground object;
V _SB - the value of the convergence rate of the induced aircraft with a ground object;
φ _G - the value of the side bearing in the horizontal plane;
ω _G - the value of the angular velocity of the line of sight in the horizontal plane;
j _G - the value of the lateral acceleration of the induced aircraft in the horizontal plane;
φ _GT - the required lead angle, providing the required linear resolution in the horizontal plane, which is calculated by the ratio

where λ is the wavelength of the airborne radar station;
ΔF is the bandwidth of the Doppler filter;
V is the value of the speed of the induced aircraft;
Δl _T is the required linear resolution in the horizontal plane.

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