The purpose of the study was to consider the motion features of kinetic lateral control interceptors and obtain dynamic equations which describe their motion in a flat rectangular coordinate system by resolving the acting forces with respect to the interceptor velocity vector. We took into consideration the nonuniformity of the interceptor motion and obtained formulas for evaluating indicators of its maneuvering potential. As an example, we assessed the capabilities of the THAAD interceptor missile (USA) in regards to the choice of missed interception

Рассмотрены особенности движения кинетических перехватчиков, использующих для управления полетом двигатели поперечного управления. Получены динамические уравнения, описывающие движение кинетического перехватчика в плоской прямоугольной системе координат путем разложения действующих сил относительно вектора скорости перехватчика. Получены формулы для оценки показателей маневренных возможностей перехватчика с учетом неравномерности его движения. В качестве примера выполнена оценка возможностей кинетического перехватчика противоракеты комплекса противоракетной обороны THAAD (США) по выбору промаха

At the present time, the literature on exterior ballistics and missile guidance methods offers fairly detailed descriptions (see, for example, [1-4]) of the equations of missile motion in the atmosphere, employing aerodynamic methods of flight control (through application of aerodynamic vanes and ram-air lift force control). Less detailed elaboration is provided on the flight motion under jet reaction control, i. e., using gas vanes, located in the missile engine nozzle, and nozzle vectoring. At the same time, the known literature does not offer in any explicit form the equations of motion for the method of missile combat stages’ (kinetic interceptors’) flight control implying application of a lateral flight control motor, which has become a mainstream feature in a number of foreign anti-missile defence systems. Besides, it is customary in the literature to assess the manoeuvring potential of missiles (anti-aircraft, guided) on the premise that a missile is moving with a uniform acceleration, which, in case of a kinetic interceptor, whose motion at the final stage of the flight is taking place under continuous expenditure of propellant, resulting in the intensive change of available accelerations, may lead to significant errors in estimation of chances to eliminate the miss distance.

In this paper we determined the basic relationships for mathematical description of kinetic interceptor’s motion in the vertical plane using a lateral flight control motor, as well as the relationships for assessing manoeuvring potential of the interceptor with consideration of its motion non-uniformity. Such approach makes it possible to study the motion and assess interceptor’s manoeuvring potential at the initial design stages. Use of more complex mathematical models of motion may complicate the analysis of specific features of motion control and the simulation results.

The principle of using lateral control motors for controlling missile flight consists in creation of thrust and, due to it, displacement of missile body in a direction perpendicular to its longitudinal axis, without changing the direction of the axis itself. Arranged in the missile centre of mass, along the perimeter of the missile body, are several motor nozzles whose axes are oriented perpendicularly to the longitudinal axis. The motor combustion products get into some or other nozzle, depending on the desired direction of missile body displacement. Thrust vector Рдпу of the motor is directed oppositely to the currently open nozzle or to the resultant force of several nozzles opened simultaneously.

Lateral control motors are applied, as a rule, in the interception stages (kinetic interceptors) of anti-missiles, e. g., in the interception stage of THAAD anti-missile (Fig. 1), interceptor LEAP of SM-3 anti-missile series, interceptor EKV of the GBI anti-missile (USA).

The lateral control motors may run on liquid (THAAD, EKV) or solid (LEAP) propellants. Normally, a solid-fuel motor operates continuously for tens of seconds, starting from the moment of interceptor separation from the launch vehicle and until hitting the target. In liquid-fuel motors, a possibility is provided for their periodic activation through fuel feed control.

Kinetic interceptor flight control by means of a lateral control motor is characterised by small response delay and a possibility for interceptor moving to a comparatively long distance within a short time. This is conditioned primarily by small mass of the interceptor and, at the same time, relatively high thrust of its motor, as well as by the fact that motor thrust is directed virtually along the normal to the velocity vector.

Lateral control does not change the direction of interceptor’s longitudinal axis. The longitudinal axis direction in exoatmospheric flight is corrected by means of special attitude-control jet reaction motors, accommodated at a certain distance from the interceptor centre of mass, normally in its tail section, so as to provide a thrust arm. The attitude-control motors usually operate in the pulse mode, providing for interceptor axis turning in the direction to target and creating conditions for target observation by the airborne direction finder.

With application of the lateral control motors, the interceptor flight path towards target can be corrected, and the miss distance eliminated (nulled) in the immediate proximity to the target. Considering high relative closing-in speeds, limited capabilities of interceptor’s small-size direction finder to measure target parameters, when target data can be confidently determined immediately before the hit, as well as taking into account a possibility of target manoeuvre, the lateral control is an efficient method for eliminating (nulling) miss distance when firing fast-moving targets.

To describe the motion of an interceptor steered with the use of a lateral control motor, we shall apply the known approach of resolving the acting forces with respect to the velocity vector [1, 2]. The resolving will be done in the starting coordinate system, i. e., in a flat rectangular coordinate system Oxy, whose origin is aligned with the starting point of missile motion (launch point), axis Ox is directed horizontally towards target, and axis Oy - vertically upwards. The interceptor flight path is characterised by radius- vector rп = (x, y)T, dynamically changing in the course of movement, where x, y - current coordinates.

With the known initial state, the interceptor flight path is fully defined by its velocity vector V, which is directed tangentially to the trajectory and characterised by the absolute value V and tilt angle Θ. Vector V is reckoned relative to the positive direction of axis Ox. Parameters V, Θ, which depend on the forces acting on the missile, determine the trajectory character. In a controlled flight, the tilt angle of velocity vector Θ is selected by the interceptor guidance method.

Missile trajectory proper is defined in the starting coordinate system by two differential equations

х = V cos Θ, у = V sin Θ, (1)

integrated in a unified system together with the differential equations describing variations of magnitudes V and Θ depending on their influencing factors.

The current value of interceptor’s velocity vector depends on the initial state (at the moment of separation from the launch vehicle) and speed variations caused by the forces acting in flight. The forces acting on the interceptor, with their resolving along the velocity vector directing and orthogonal to it, are shown in Fig. 2. Here, the following designations of variables are used:

Рдпу - thrust vector of the lateral control motor, directed orthogonally to the interceptor longitudinal axis;

P - thrust vector of the gases discharged through the tail nozzle (or thrust of a pre-acceleration module, if such is available in the interceptor), directed along the longitudinal axis;

X - ram air resistance, i. e., directed opposite to the velocity vector;

Y - ram-air lift force, i. e., the vector directed orthogonally to the velocity vector;

Gr - Earth’s gravity vector at a distance r from its centre, directed towards the Earth’s centre of mass;

α - angle of interceptor longitudinal axis deflection from the velocity vector;

r - distance from the Earth’s centre to the interceptor:

R3 - Earth’s radius.

The Earth’s gravity depends on the current mass ofthe interceptor m and gravity acceleration &.:

where g0 – gravity acceleration near the Earth’s surface.

The Earth’s gravity vector can be resolved into two components: tangential component Grт, acting along the velocity vector, and normal component Grн, orthogonal to it,

Gr = Grт + Grн,

each one of which can be determined through respective component of the gravity acceleration:

Grт = mgrт Grн = mgrн.

Resolving of gravity acceleration into the tangential and normal components is given in [

Aerodynamic forces X, Y depend both on the missile parameters and those of the Earth’s atmosphere. The values of forces are determined from the following formulas [1-3]:

Х = cxqS, Y = cyqS,

where сх, су - non-dimensional aerodynamic coefficients;

- ram air pressure;

ρ - air density in a given trajectory point;

S - interceptor’s midsection (largest cross-section) area.

Let us consider the process of interceptor motion (see Fig. 2).

Having separated from the launch vehicle, the interceptor is moving in the rarefied air layers, and then in the airless space. At the moment of separation, the interceptor motion is defined by the velocity vector, which is characterised by given values of the absolute magnitude of V and tilt angle Θ.

In flight, the lateral control motor creates a lateral (steering) acceleration equal to the ratio of motor thrust to interceptor mass:

Thrust power Рдпу, of the lateral control motor, which determines normal accelerations of the interceptor, is a steering force.

The values of normal accelerations required for moving along the selected trajectory are determined by the interceptor guidance method. In accordance with the guidance method, the direction and thrust power Рдпу are regulated by opening some or other nozzles of the motor, i. e., thrust power Рдпу is also a control parameter of flight, which distinguishes controlled flight with the use of a lateral control motor from other flight control methods. For example, under the aerodynamic flight control method the steering force is ram-air lift force, and the control parameter is the incidence angle.

For liquid-propellant motor, it is also possible to regulate motor thrust through dosed supply of firel into the combustion chamber. A portion of the excess gases can be discharged from the combustion chamber of the motor through the tail nozzle, creating thrust P, directed along the longitudinal axis of the interceptor.

In a general case, the longitudinal axis of the interceptor is oriented at an angle a relative to the velocity vector ensuring the necessaty conditions for target observation by the airborne direction finder. Angle α is an independent magnitude, it is created by the aerodynamic controls of the launch vehicle at the moment of atmospheric flight termination and can be corrected in subsequent (exoatmospheric) flight by means of special attitude-control motors of the interceptor. At the final leg of flight towards target this angle is close or equal to zero.

A system of dynamic equations of interceptor motion, based on the described resolving of forces, appears as

where WT - tangential acceleration, directed tangentially to the trajectory;

WH - normal acceleration, orthogonal to the trajectory’s tangential line.

Expanding dependencies for the vectors of forces, we obtain equations

An analysis of the components of dynamic equations of motion (3), (4) shows as follows.

In exoatmospheric flight, variation of the interceptor speed imparted to it at the moment of separation is determined basically by the action of the tangential component of gravity acceleration grт and by acceleration Р/m under the effect of jet thrust of gases discharged through the tail nozzle (or pre-acceleration module thrust).

Considering air thinness, the trajectory shape is determined basically by the joint action of acceleration PДПУ/m, created by the lateral control motor, and normal component of gravity acceleration grн.

Let the trajectory shape be assigned by the interceptor guidance method, i. e., normal accelerations characterising trajectory curvature are known

W н.тр =V Θ.

Then it can be possible to determine from (4) the interceptor accelerations created by the lateral control motor as required to ensure interceptor motion along the trajectoty of an assigned shape:

In this case the motor thrust used in formula (3) is equal to

Рдпу = m W потр. дпу

Analysing formula (5), it can be seen that, when controlling interceptor flight by means of lateral control motor, controlling parameter Рдпу is a function of angle Θ and its derivative , assigned by the guidance method. Formula (5) determines requirements to the lateral control motor proceeding from a necessity to ensure motion along the trajectory of a selected shape and to compensate for trajectory distortions conditioned by the action of gravity force, air resistance, and thrust of the gases discharged through the tail nozzle.

Equations (3), (5), supplemented by kinematic equations (1) and relationship equations of the variables used, allow to calculate the trajectory of interceptor controllable flight.

For the case of interceptor passive flight (without pre-acceleration module operation, P = 0) in the airless space (no ram air pressure, q = 0), with zero angle of longitudinal axis tilt a, the equations of motion take the following view:

i. e., interceptor speed variation is fully determined by the action of gravity force, and the trajectory shape - by joint action of the lateral control motor and gravity.

For this case, required accelerations of the lateral control motor will amount to

Wпотр.дпу = Wн.тр + g rн (6)

i. e., the motor must create accelerations allowing to ensure both motion along a selected trajectory and compensation of gravity.

From formula (6), dependencies for some characteristic trajectories follow:

Interceptor’s manoeuvring potential characterises its capability to timely move to a specified distance in a plane orthogonal to the velocity vector (i. e., to eliminate the guidance error of a specified magnitude and compensate for a sudden manoeuvre of the target). Hereinafter, such displacement will be referred to as miss elimination h.

It is customary to evaluate the manoeuvring potential of anti-aircraft missiles by the magnitude of addition to the characteristic speed, which, as applies to a kinetic interceptor, with consideration of the above variables, is defined by integral

where τдпу - total running duration of the motor.

Indicator Vx is a technical one in the sense that it does not depend on a particular flight trajectory and the current moment of time. It determines the limit speed capabilities of the interceptor, and proceeding from it one can make a choice in favour of some or other interceptor; however, it does not allow to estimate interceptor capabilities for miss elimination when flying along a particular trajectory and in a certain moment of time.

For assessment of the manoeuvring potential of a kinetic interceptor for the current moment of flight along a particular trajectory we shall use the margin of available accelerations ∆Wдпу and the time required for elimination of miss of a specified magnitude Th.

Indicators ∆Wдпу and Th are determined as follows.

Over the entire period of interceptor controlled flight its available normal accelerations must exceed the required ones, i. e., the following inequality shall be satisfied:

Wдпу > Wпотр.дпу.

The excess of available accelerations over required ones

ΔWдпу = Wдпу - Wпотр.дпу

determines the margin of interceptor normal accelerations that can be used in miss elimination. At that, magnitude Wдпу is calculated in a general case by formula (2), and Wпотр.дпу - by formula (5), or, for the case of passive flight considered above, formula (6).

For a case of uniform fuel consumption, occurring in practice at the final leg of interceptor flight, magnitude Wдпу can be determined as follows.

The current mass m of the interceptor is

m = mo - mт.дпуt, (7)

where m0 - initial mass of the interceptor;

m т.дпу - per-second mass fuel consumption of the motor.

Under uniform fuel depletion, the magnitude of per-second mass consumption

where mт.дпу - initial fuel mass.

Motor thrust can be estimated by the formula

where u - gas discharge speed.

As a result, the available interceptor accelerations can be calculated by the formula

The time of miss elimination can be determined, in an elementary case, from the following considerations. It is known that the distance that a body moving with constant acceleration W can cover over time t is equal to . Hence, the time taken by a body to cover distance S under constant acceleration W can be determined by the formula

By virtue of dependence (8), under uniform motor fuel consumption during interceptor flight, the function of accelerations Wдпу (t) is growing hyperbolically, reaching the maximum by the moment of total fuel depletion. Considering that fuel load amounts to tens of percent of the interceptor mass proper, the change of function Wдпу (t) is characterised by high intensity, which does not allow to use formula (9) to estimate the time required for elimination of large misses.

Let us obtain a formula for calculation of a distance that can be covered by a non-uniformly moving body over time period Δt starting from moment t, given the known function of accelerations W (t).

Speed that the body gains over time period Δt starting from moment t is equal to

The distance that the body will cover over this period of time is equal to

Combining the two formulas, we obtain an equivalent expression for calculation of a distance that the body will cover over time Δt starting from moment t:

In case of interceptor, not all of the available normal accelerations are used for compensation of miss h, but only that part of them which exceeds the accelerations required for moving along the trajectory assigned by the guidance method. Considering this, the distance covered by interceptor in the plane orthogonal to the velocity vector over specified time t3 starting from moment t is equal to:

Equation (10) is a formula for the magnitude of miss eliminated over time t3 starting from moment t.

It is not feasible to explicitly express from formula (10) a dependence for calculation of miss elimination time Th. In this respect, for estimation of time Th by formula (10), one should obtain for a certain moment of time t a set of miss estimates h0 for different values of t3 and select as an estimate of time Th the value under which miss h0 is the closest to specified miss value h.

For approximate estimation of miss elimination time Th for a moment of time t, proceeding from formula (9), the following expression can be used:

Initial data. As an example, we shall consider kinetic interceptor of the THAAD anti-missile. The basic anti-missile specifications are known [

After separation from the launch vehicle, interceptor is moving towards the target along virtually straight trajectory. Let us consider five trajectories with tilt angles to the horizon of 40°, 50°, 60°, 70°, 80°, overlapping the kill zone. The starting point of interceptor trajectories is at an altitude of 40 km (altitude of interceptor separation from the launch vehicle). Interceptor starting speeds are equal to the speeds of carrier missile at an altitude of 40 km and depend on the trajectoty tilt. Preliminary simulation of missile flight through the dense atmosphere made it possible to determine interceptor starting speeds for each of the trajectories as lying in a range between 1870 m/s (tilt angle of 40° - “slow” trajectory because of long passing through the dense atmosphere) and 2660 m/s (tilt angle of 80° - “fast” trajectory due to a fast climb).

The fuel mass of THAAD interceptor is unknown. Based on the analysis of interceptor equipment composition, as seen in its cross-section mockup (see Fig. 1), an assumption was made that the fuel mass could make from one-third to half of the interceptor mass, i. e., the value of fuel mass is within mтдпу = 30-45 кг. kg. In accordance with this, we shall assess interceptor capabilities taking two extreme values of the above range of mass mтдпу.

Results of calculations and assessment. For calculations, a system of equations (3), (5), (1), describing interceptor controllable motion, was used. The thrust was taken equal to zero P = 0 (interceptor’s passive flight), interceptor longitudinal axis orientation angle α = 0.

Fig. 3 shows estimated trajectories of a kinetic interceptor moving at different angles and their respective normal components of gravity acceleration grн.

An analysis of the obtained dependencies demonstrates that the normal component of gravity acceleration for the considered range of THAAD interceptor application altitudes variation depends primarily not on the altitude but rather on the trajectory tilt angle. The greater the trajectory tilt angle, the smaller the gr н value (however, at the same time there is a growdh of the magnitude of gravity acceleration tangential component gr т as an addition to the resultant acceleration gr). With trajectory tilt angle of 80°, the magnitude of grн amounts to ~1.6.. .1.5 m/s2, whereas with the angle of 40° it is within ~7.3.. .7.0 m/s2. It follows from this that the requirements to lateral control motor during interceptor flight along trajectories close to vertical ones are substantially lower than in a horizontal flight. In a vertical flight, gravity exerts a braking action, slowing down the interceptor speed (due to the effect of gravity component gr т), however, it does not affect the capabilities of lateral control motor to eliminate miss in the horizontal plane.

Shown in Fig. 4, 5 is the variation in the interceptor flight dynamic characteristics at the initial fuel mass of 30 kg and 45 kg, respectively. The calculations were performed by formulas (7), (8), (10).

With firel mass of 30 kg, the available interceptor accelerations are growing, along with fuel depletion, from ~10.4 to 15.5 mis2. These accelerations are sufficient for compensation of the gravity normal component when moving along any of the considered trajectories, but for elimination of miss occurring due to target’s manoeuvre, they may be not. The calculations show that over a time of lateral control motor operation equal to 1 s, a miss of -~...7 m can be eliminated, depending on the trajectory tilt angle and interceptor current mass, and over 2 s - -~...27 m.

With fuel mass of 45 kg, the interceptor has much higher manoeuvring characteristics, with the available normal accelerations, depending on fuel depletion, lying within a range of -16...31 mis2. Over a time period of 1 s, a miss of 4... 14.5 m can be eliminated, and 2 s - ~17.58 m.

A comparative assessment of the results of miss calculation by exact formula (10) and approximate formula (11) shows that the latter yields substantial errors: with miss elimination time exceeding 20 s, the error is over 10 %. Such considerable miss elimination time demand can be characteristic of long-range interceptors, such as EKV and LEAP. For this case it is necessary to make calculations by formula (10). Given a miss elimination time of 10 s, the error is about 1 %. If miss elimination time amounts to the units of seconds, the error is insignificant, and for practical calculations formula (11) can be used.

The dynamic equations obtained in the paper, describing kinetic interceptor motion on the exoatmospheric flight segment, are formally similar to the known equations for atmospheric flight, but they have different constituents and other physical content. The main specific feature of the equations of motion is that their constituents include the lateral control motor thrust as a flight control parameter and, at the same time, a controlling force. Obtaining a dependence of velocity vector parameters on said force made it possible to determine lateral control motor accelerations required to support interceptor motion along a trajectory of a selected shape with account of the necessity to compensate for the action of gravity force, air resistance, as well as the thrust of gases discharged through the tail nozzle.

In addition to that, in this paper we obtained mathematical expressions for assessing manoeuvring potential ofa kinetic interceptor with consideration of its motion non-uniformity, which is characteristic of interceptor, since its available accelerations change non-linearly in the process of flight as a result of fuel depletion.

An aggregate of the obtained expressions makes it possible at the initial design stages to perform analytical simulation of kinetic interceptor motion in the vertical plane, assess its manoeuvring potential and capabilities for miss elimination, as well as formulate requirements to the interceptor lateral control motor, namely, its thrust, run duration, and fuel load.

The solved practical task on assessing lateral control motor characteristics and capabilities of THAAD anti-missile kinetic interceptor confirms functional capacity of the proposed mathematical apparatus.

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