Angular acceleration compensation guidance law for passive homing missiles

Jixin Li; Xiaogeng Liang; Keqiang Xia; Yilin You

doi:10.1515/astro-2022-0039

Article Open Access

Angular acceleration compensation guidance law for passive homing missiles

Jixin Li , Xiaogeng Liang , Keqiang Xia and Yilin You

Published/Copyright: September 5, 2022

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From the journal Open Astronomy Volume 31 Issue 1

Abstract

Proportional navigation guidance law (PNG) is widely used for passive homing missiles. But PNG often suffers from inaccurate tracking of target or even failure when the target is maneuvering quickly or releasing artificial decoys. In order to solve this problem, the angular acceleration compensation guidance law (AACG) is constructed by using line of sight (LOS) angular velocity and LOS angular acceleration. The stability analysis with Lyapunov stability theory shows that AACG system is asymptotically stable on a large scale under certain stability constraint conditions. AACG command is designed to be perpendicular to the LOS and optimized by gravity compensation. AACG is neither dependent on target acceleration nor the distance between the missile and the target. The numerical simulation results indicate that AACG has good guidance performance on air-to-air missiles when the target maneuvers violently or releases artificial decoys in the endgame term.

Keywords: guidance; missile; proportional navigation; angular acceleration

1 Introduction

Many types of tactical missiles adopt passive homing guidance systems, such as infrared guided air-to-air missiles, air-to-ground/anti-tank missiles, surface-to-air missiles, cruise missiles, surface-to-surface missiles; semi-active laser guided air-to-ground/anti-tank missiles; and passive radar guided anti-radiation missiles. These passive homing missiles have the advantages of high guidance accuracy, low technical complexity and low cost, among which infrared imaging guidance has strong anti-interference ability (Fei and Liu 2006, Gao et al. 2015, Li and Wang 2018, Liu and Wang 2019, Luo and Shi 2009, Ye et al. 2007). Being simple in form, proportional navigation guidance (PNG) is widely used in early passive homing missiles, because the guidance accuracy can meet the requirements for intercepting non-maneuvering targets with no interference (Paul 2012). However, miss distance of PNG will increase significantly in several seconds before the missile encounters the target (that is the guidance sensitive area), if the target suddenly makes a high maneuver, or releases artificial decoys to produce large disturbance to the seeker output (Arthur and Gerrit 2009, Huang et al. 2017, Li et al. 2019, Paul 2012, Zhou et al. 2019). Some missiles, such as the early AIM-9L Sidewinder air-to-air missile, were unable to compensate for gravity due to the lack of an inertial navigation system. Sometimes, although the missile has a strapdown inertial navigation system, the gravity compensation design is not sophisticated enough.

Nowadays, with the improvement in the inertial measurement technology and the reduction in device price, passive homing missile widely adopts strapdown inertial navigation system for inertial measurement and navigation calculation, and adopts autopilot to achieve missile attitude stability and overload control (Feng 2019, Luo et al. 2012). Kalman filter is also widely used to estimate target motion information (Grewal and Andrews 2010, Zhou et al. 1991). Conditions are ripe for improving guidance performance through some advanced guidance law.

The general form of PNG is shown in Eq. (1), the general form of target acceleration compensation guidance law (or augmented proportional navigation, APN) is shown in Eq. (2), and the general form of optimal guidance law (OGL) is shown in Eqs. (3)–(5) (Paul 2012).

(1) n c = N V C q ̇ ,

(2) n c = N V C q ̇ + N 2 n T = N y + y ̇ t go t go 2 + 0.5 n T ,

(3) n c = N ′ y + y ̇ t go − n L T 2 ( e − x + x − 1 ) t go 2 + 0.5 n T ,

(4) x = t go T ,

(5) N ′ = 6 x 2 ( e − x + x − 1 ) 2 x 3 − 6 x 2 + 6 x + 3 − 12 x e − x − 3 e − 2 x ,

where n _c is the guidance acceleration command; N is the guidance gain; V _C is the approaching velocity between missile and target; q ̇ is the LOC angle rate; n _T is the target acceleration; y is the miss distance or position relative to the reference line; y ̇ is the miss distance rate; t _go is the time to go; N′ is the effective navigation ratio and T is the guidance system time constant.

Theoretically, for attacking non-maneuvering targets, PNG is optimal, and the optimal gain of PNG is 3, ignoring guidance system time constant. For attacking a target with constant acceleration, APN is optimal, and the optimal gain of APN is 3, ignoring guidance system time constant, too. In fact, OGL is optimal on considering target maneuvering and lag of projectile overload control, while PNG and APN performance will be significantly reduced. The reason why OGL is better than PNG is that, OGL compensates the target’s maneuvering acceleration, while PNG only uses LOS angular velocity information. The reason why OGL is superior to APN is that, the gain of OGL changes with the remaining flight time t _go, and the time constant of missile autopilot is compensated.

OGL has the best performance but requires more guidance information such as target position, speed and acceleration. APN also requires accurate target acceleration information. The passive homing missile guidance system can only measure the angular channel information, but not the velocity and distance information, so it is difficult to obtain the target acceleration and range information with high accuracy. Therefore, OGL and APN are not suitable for passive homing missiles. Other methods must be researched.

Hecht (1991) proposed angular acceleration guidance (AAG) for infrared homing missiles to intercept ballistic missile targets with large acceleration in boost/ascending phase at medium or long range. Its general form is the weighted combination of PNG and the LOS AAG law when the target acceleration is constant. The advantage is that it can accurately intercept the maneuvering target with variable acceleration, and it does not need accurate target acceleration information. But two problems remain: first, the algorithm of its guidance parameter k is very complex, and it is specifically designed for the missiles which intercept the ballistic missiles flying in boost/ascending phase, and that means the target motion information is required; second, the information of distance and missile-target approaching speed is also required. So, the information of acceleration, distance and approaching velocity should be measured or estimated. Degradation of information accuracy will lead to the degradation of guidance accuracy.

Sreeja and Hablani (2016) studied a kind of precise guidance system of air-to-ground precision guidance ammunition without engine propulsion using infrared and millimeter wave dual-mode seeker in order to attack the moving tank target. The guidance law is “PNG + ‘target acceleration + gravity + drag’ compensation,” which is essentially “APN + ‘gravity + drag’ compensation.” The guidance command is perpendicular to the LOS. The simulation results indicate that, comparing with “PNG + target acceleration compensation” and “PNG + ‘target acceleration + drag’ compensation,” PNG + ‘target acceleration + gravity + drag’ compensation” can lead to the minimum miss distance and a greater terminal speed inclination angle. In fact, because the guidance command is an overload command, just like the target maneuver needs to be compensated, gravity compensation is indeed necessary, too. But the drag perpendicular to the LOS is a part of the overload response of the missile’s autopilot, so it is a controllable part of the missile’s autopilot. It does not need to be and should not be compensated in the guidance command.

Kumar and Ghose (2013) and Kumar et al. (2014) studied the sliding mode guidance law (SMG) with impact angle or impact time constraint. To solve the vibration problem of SMG, (Zhang et al. 2012) researched a kind of guidance law adopting disturbance observer technology and backstepping method. Cao and Zhang (2020) studied the method based on feedback/feedforward theory to improve the response performance of air-to-air missile guidance loop. Tu et al. (2020) studied the method of Lie group theory to solve the problem of coupling guidance design for air-to-air missile. However, these guidance laws are not very satisfying or applicable to passive homing missile in complicated guidance scenarios.

Because passive homing missile can only measure angular channel information, but cannot obtain target acceleration and range information with high accuracy, based on the equivalent idea, angular acceleration compensation guidance law (AACG) is derived from the two-dimensional plane interception geometry model to avoid the explicit inclusion of target acceleration in the guidance law. Then, based on the idea of automatic control, the distance term in AACG is changed into a function of the velocity modulus, so as to avoid using distance information. Under certain conditions, the missile-target approaching velocity can be estimated by Kalman filter algorithm based on angular measurement information, or the approximate value can be obtained by navigation calculation with enough precision to meet the requirement of guidance. The designed parameters of guidance law are given. The gravity compensation method of AACG guidance law is derived by using block diagram method. Frequency domain and time domain analyses are carried out. Numerical simulations are carried out, too. System analysis and simulation results show that the proposed AACG guidance law performs well even if there occurs target’s large maneuver or big guidance signal disturbance.

2 AACG with novel structure

2.1 Derivation of two-dimensional plane guidance law

APN law (gravity acceleration is not considered) is shown in Eq. (6):

(6) a MC ⊥ = N V C q ̇ + N 2 a T ⊥ ,

where a _M is the normal acceleration of the missile (perpendicular to the missile speed); a _T is the normal acceleration of the target (perpendicular to the missile speed); a M ⊥ is the component of the missile acceleration perpendicular to the LOS and a T ⊥ is the component of the target acceleration perpendicular to the LOS.

The two-dimensional plane interception geometric model is shown in Figure 1.

Figure 1

Two-dimensional plane interception geometric model.

According to the two-dimensional plane interception geometric model shown in Figure 1, the following formula derivation process can be obtained:

(7) R ̇ = − V M cos ( q − θ M ) + V T cos ( q − θ T ) , R q ̇ = V M sin ( q − θ M ) − V T sin ( q − θ T ) ,

(8) 2 R ̇ q ̇ + R q ̈ = V ̇ M sin ( q − θ M ) − V ̇ T sin ( q − θ T ) − V M θ ̇ M cos ( q − θ M ) + V T θ ̇ T cos ( q − θ T ) ,

(9) a M = V M θ ̇ M ,

(10) a T = V T θ ̇ T ,

(11) 2 R ̇ q ̇ + R q ̈ = V ̇ M sin ( q − θ M ) − V ̇ T sin ( q − θ T ) − a M cos ( q − θ M ) + a T cos ( q − θ T ) ,

(12) a M ⊥ = − V ̇ M sin ( q − θ M ) + a M cos ( q − θ M ) ,

(13) a T ⊥ = − V ̇ T sin ( q − θ T ) + a T cos ( q − θ T ) ,

(14) 2 R ̇ q ̇ + R q ̈ = a T ⊥ − a M ⊥ .

Therefore,

(15) a T ⊥ = 2 R ̇ q ̇ + R q ̈ + a M ⊥ .

Eq. (15) is substituted in Eq. (6), therefore,

(16) a MC ⊥ = N V C q ̇ + N 2 ( 2 R ̇ q ̇ + R q ̈ + a M ⊥ ) .

Because V C = − R ̇ , therefore,

(17) a MC ⊥ = N 2 ( R q ̈ + a M ⊥ ) .

Eq. (17) is the equivalent form of APN and it uses LOS angular acceleration, distance and the actual acceleration of the missile.

APN generally assumes that the target acceleration a T ⊥ is constant. The actual situation is that the target maneuvering acceleration a T ⊥ is the projection of the total target acceleration in the vertical direction of the LOS. No matter what maneuvering mode the target adopts, it usually changes, and when the target maneuvering acceleration a T ⊥ is zero, the proportional guidance itself is optimal. Considering this problem, a coefficient can be introduced to compensate the target acceleration in the actual guidance command calculation of APN, and Eq. (6) becomes:

(18) a MC ⊥ = N V C q ̇ + N 2 k a T ⊥ ,

where, k is positive.

Apply Eq. (15) to Eq. (18) to obtain the LOS AACG

(19) a MC ⊥ = ( 1 − k ) N V C q ̇ + k N 2 ( R q ̈ + a M ⊥ ) .

Eq. (19) is equivalent to the weighted combination of proportional guidance and LOS AAG law. The weighting coefficient of LOS AAG law is k and the weighting coefficient of proportional guidance is 1 − k.

2.2 Missile acceleration low-pass filtering

For the fifth order linearized guidance system, assume that the guidance system time constant is T, autopilot transfer function is 1 T 5 s + 1 3 , LOS angular velocity transfer function is q ̇ out q ̇ = 1 T 5 s + 1 2 and LOS angular acceleration transfer function is q ̈ out q ̈ = 1 T 5 s + 1 2 , where q ̇ out is the actual LOS angular velocity output; q ̇ is the ideal LOS angular velocity; q ̈ out is the actual LOS angular acceleration output and q ̈ is the ideal LOS angular acceleration.

Ignore the delay of a M ⊥ measurement in order to match the phase between a M ⊥ and R q ̈ , replace a M ⊥ with a Mf ⊥ , which is the output of the first-order low-pass filter whose input is a M ⊥ and the time constant is T ₁.

(20) a MC ⊥ = ( 1 − k ) N V C q ̇ + k N 2 ( R q ̈ + a Mf ⊥ ) ,

(21) a Mf ⊥ = 1 T 1 s + 1 a M ⊥ ,

(22) T 1 = 2 T 5 .

2.3 Distance parameter R

Because it is difficult for the passive homing missile to obtain the distance, we try to eliminate the distance parameter R. Based on the idea of automatic control, set the distance parameter R in Eq. (20):

(23) R = k 0 V C .

Therefore,

(24) a MC ⊥ = ( 1 − k ) N V C q ̇ + k N 2 ( k 0 V C q ̈ + a Mf ⊥ ) ,

where k ₀ can be assigned as a positive constant.

2.4 Compensation coefficient k

The compensation coefficient k is greater than 0 and less than 1.

Obviously, when k = 0, AACG is equivalent to PNG.

2.5 Stability analysis

Using Lyapunov stability theory, the stability of AACG can be proved, and the necessary conditions which guarantee the stability can be derived. For simplicity, we choose the first-order linearized AACG guidance system model, which is shown in Figure 2, to make the analysis. The equivalent first-order linearized AACG guidance system model is shown in Figure 3.

Figure 2

First-order linearized AACG guidance system model.

Figure 3

Equivalent first-order linearized AACG guidance system model.

The state space block diagram of the equivalent first-order linearized AACG guidance system is shown in Figure 4.

Figure 4

State space block diagram of first-order linearized AACG guidance system.

The functions of the state space shown in Figure 4 is:

(25) x ̇ 1 = − 1 T 1 1 + k k 0 N 2 t go − k N 2 x 1 + 1 T 1 x 2 + k 0 T 1 t go u , x ̇ 2 = − ( 1 − k ) N t go x 1 + ( 1 − k ) N t go u , y = x 1 .

The homogeneous equation of the system is:

(26) x ̇ 1 = − 1 T 1 1 + k k 0 N 2 t go − k N 2 x 1 + 1 T 1 x 2 , x ̇ 2 = − ( 1 − k ) N t go x 1 .

The origin is the unique equilibrium point of the system.

Set Lyapunov function as:

(27) V ( x ⇀ ) = 1 2 ( x 1 2 + m x 2 2 ) ,

where m > 0.

When x ⇀ ≠ 0 0 , V ( x ⇀ ) > 0 .

When x ⇀ → ∞ , V ( x ⇀ ) → ∞ .

(28) V ̇ ( x ⇀ ) = x 1 x ̇ 1 + m x 2 x ̇ 2 ,

(29) V ̇ ( x ⇀ ) = − 1 T 1 1 + k k 0 N 2 t go − k N 2 x 1 2 + 1 T 1 − m ( 1 − k ) N t go x 1 x 2 ,

when

(30) 1 T 1 − m ( 1 − k ) N t go = 0 .

That is,

(31) m = t go ( 1 − k ) N T 1 .

Because k ≥ 0, k < 1, t _go > 0, N > 0 and T ₁ > 0, therefore m > 0.

Then,

(32) V ̇ ( x ⇀ ) = − 1 T 1 1 + k k 0 N 2 t go − k N 2 x 1 2 ,

when

(33) 1 − k N 2 ≥ 0 .

That is,

(34) k ≤ 2 N .

Then,

(35) 1 + k k 0 N 2 t go − k N 2 ≥ k k 0 N 2 t go > 0 ,

where k ₀ > 0.

Therefore, except the origin point,

(36) V ̇ ( x ⇀ ) ≤ 0 .

So, V ̇ ( x ⇀ ) is semi-negative definite.

When x ⇀ ≠ 0 0 , V ̇ ( x ⇀ ) is not always zero. So, the system is asymptotically stable at the origin, the only equilibrium point of the system.

When x ⇀ → ∞ , V ( x ⇀ ) → ∞ . So, it is asymptotically stable on a large scale.

That is, when 0 ≤ k ≤ 2 N , the AACG guidance system is asymptotically stable on a large scale.

We already know that N should be bigger than 2 to make PNG guidance system to be stable in maneuvering target interception. In AACG guidance, we should also ensure that N > 2 is satisfied. Furthermore, according to V ̇ ( x ⇀ ) = − 1 T 1 1 + k k 0 N 2 t go − k N 2 x 1 2 and 0 ≤ k ≤ 2 N , positive parameter k ₀ should be set to an appropriately large positive value, and k can be set to a positive value which should be smaller than 2 N . For example, the parameters can be set as: N = 3, k = 0.35, k ₀ = 1.5 and T ₁ = 0.15, and the AACG guidance system is asymptotically stable on a large scale.

2.6 Guidance information

The guidance information mainly includes the modulus of missile-target approaching velocity V _C, LOS angular velocity q ̇ and LOS angular acceleration q ̈ , and the missile acceleration a M ⊥ in the normal direction of LOS.

The LOS angular velocity q ̇ and LOS angular acceleration q ̈ can be estimated by Kalman filter algorithm based on angle measurement. Kalman filter is used to estimate the target position, velocity and acceleration. The first-order time correlation model, the so-called Singer model (Zhou et al. 1991), is adopted as the target maneuver model. Passive seeker only has angle and angular velocity measurement information. Although it cannot ensure the absolute accuracy of position, velocity and acceleration, Kalman filter can ensure the estimation accuracy of LOS angular velocity and LOS angular acceleration information. Moreover, under certain conditions, the accuracy of target velocity estimation can meet the requirements. At this time, the estimated value of modulus of missile-target approaching velocity V _C is more accurate. When the accuracy of target velocity estimation cannot be guaranteed, the modulus of the projection of missile velocity vector on the missile target LOS can also be taken as V _C. The missile velocity is calculated by strapdown inertial navigation system.

a M ⊥ can be calculated according to the missile attitude, the LOS vector and the missile overload output from the strapdown inertial navigation system, the angle measurement information output of the seeker, and the local gravity acceleration information.

3 Frequency domain and time domain analyses using second-order linearized guidance system

For convenience and clearness of analysis, referring to the method of (Paul 2012), second-order linearized guidance system model is established for frequency domain and time domain analyses. The second-order linearized AACG guidance system model is shown in Figure 5. It is a guidance system with time constant T ₁ + T ₂, where T ₁ is the time constant of the LOS angular velocity and T ₂ is the time constant of the autopilot.

Figure 5

Second-order linearized AACG guidance system model.

The model shown in Figure 6 is equivalent to the second-order linearized AACG guidance system shown in Figure 5.

Figure 6

Equivalent model of second-order linearized AACG guidance system.

Defining Y as output and Y _t as input, the equivalent system’s forward transfer function is G(s) = 1 and feedback transfer function is H(s), then the open-loop transfer function of the loop is:

(37) G ( s ) H ( s ) = H ( s ) = ( 1 − k ) N t go − k N 2 k 0 k 2 ( 1 − k ) s + 1 T 1 T 2 1 − k N 2 t go s 2 + T 1 + T 2 1 − k N 2 t go s + 1 s .

Bode diagram of open-loop transfer function is shown in Figure 7. Compared with PNG, the phase margin of AACG at T ₁ = 0.15 s, T ₂ = 0.2 s, t _go = 1 s, k = 0.35 and k ₀ = 1.5 operating points increases from 42.7° to 61.9°, the amplitude margin increases from 11.8 dB to positive infinity, and the cut-off frequency increases from 2.51 to 2.89 rad s⁻¹. Frequency domain analysis shows that AACG guidance system has larger cut-off frequency, amplitude margin and phase margin than PNG.

Figure 7

Bode diagram of open loop system, t _go = 1 s.

The closed-loop transfer function with forward transfer function G(s) = 1 and feedback transfer function H(s) is:

(38) Φ ( s ) = G ( s ) 1 + G ( s ) H ( s ) = 1 1 + H ( s ) .

The unit step response y of the closed-loop system is shown in Figure 8. The meaning of the input Y _t is the position of the target relative to the reference line, and the output Y represents the miss distance.

Figure 8

Unit step response of closed-loop system Φ(s).

Obviously, when the homing terminal guidance eliminates the initial guidance error, AACG responds faster than PNG and has less overshoot.

Defining Y _m as output and Y _t as input, the equivalent system’s closed-loop transfer function with H(s) as forward transfer function and G(s) = 1 as feedback transfer function is:

(39) Φ ⁎ ( s ) = H ( s ) 1 + G ( s ) H ( s ) = H ( s ) 1 + H ( s ) .

The unit step response Y _m of the closed-loop system is shown in Figure 9. The input is still Y _t, which means the position of the target relative to the reference line, and the output Y _m represents the position of the missile relative to the reference line.

Figure 9

Unit step response of closed-loop system Φ*(s).

Obviously, from another perspective, it can also be seen that when the homing terminal guidance eliminates the initial guidance error, the AACG response is faster than PNG and the overshoot is smaller.

4 Simulation of fifth order linearized guidance system

In the environment of MATLAB/Simulink, simulation models of fifth order linearized guidance system (Paul 2012) of PNG, APN and AACG are established, respectively (as shown in Figures 10–12). The miss distance is Y. The total time constant T of the guidance system is set to be T = 0.25 s, the modulus of the missile-target approaching speed is set to be V _C = 1,000 m s⁻¹ and the target acceleration a _T is a step signal, and the aim point jump ΔY in the process of terminal guidance is also a step signal. The guidance coefficients of PNG, APN and AACG are all determined as N = 3, and the guidance parameters of AACG are: k = 0.35 and k ₀ = 1.5. The influence of target maneuver and aim point jumping on miss distance of the linearized guidance system are researched, respectively. The Saturation block set the guidance command not to be bigger than 60 g, where g is 9.8 m s⁻².

Figure 10

Linearized PNG guidance system.

Figure 11

Linearized APN guidance system.

Figure 12

Linearized AACG guidance system.

First, it is assumed that the aim point keeps stable and only the influence of target’s step maneuver is considered. The impact of step maneuver (amplitude of 9 g) on miss distance is shown in Figure 13. It can be clearly seen from Figure 13 that AACG is significantly better than PNG and APN. The miss distance peaks of AACG, APN and PNG are 1.45, 5.4 and 7.58 m, respectively. Moreover, the miss distance of AACG attenuates to near zero when t _go is greater than 1 s, and PNG and APN attenuate to near zero when t _go is greater than 2 s, that is, the guidance sensitive area of PNG is about 2 s and the guidance sensitive area of APN is also about 2 s, while the guidance sensitive area of AACG is only 1 s, which is about 50% smaller than PNG and APN.

Figure 13

Target maneuver beginning t _go – miss distance.

Second, assuming that the target does not maneuver, only the influence of aim point jumping is considered. The aim point jump is set to 0.3°. The influence of different aim point jumping t _go (0–3 s) on miss distance is shown in Figure 14. It can be clearly seen from Figure 14 that AACG is significantly better than PNG. The miss distance peaks of PNG and AACG were 2.55 and 0.64 m, respectively. Moreover, the miss distance of AACG attenuates to near zero when t _go is greater than 1 s, and PNG attenuates to near zero when t _go is greater than 2 s, that is, the guidance sensitive area of PNG is about 2 s, while the guidance sensitive area of AACG is only 1 s, which is 50% smaller than PNG. It should be noted that the miss distance here is the miss distance of the missile relative to the new aim point after jumping relative to the aim point.

Figure 14

Aim point jumping t _go – miss distance.

5 Gravity compensation

The block diagram method is used to deduce the gravity compensation of the guidance law. Assuming no gravity, the linearized guidance system is shown in Figure 15. In Figure 15, a _MC is the acceleration command, a _M is the acceleration response of the missile. The linearized guidance system considering no gravity is shown in Figure 16. Here a _MC is acceleration command, a _{MC_OL} is overload command (i.e., acceleration command minus gravity acceleration), a _M is missile acceleration response and a _{MC_OL} is the missile overload response (i.e., acceleration response minus gravity acceleration g). The linearized guidance system equivalent to Figure 16 is shown in Figure 17. The gravity acceleration can be treated as the target maneuver with the same magnitude and the opposite direction to the gravity acceleration. Here a _MC is the acceleration command, a _{MC_OL} is the overload command (i.e., acceleration command minus gravity acceleration g) and a _{M_OL} is the missile overload response (i.e., acceleration response minus gravity acceleration g). All the above acceleration commands, acceleration response, overload command and overload response are perpendicular to the missile-target LOS.

Figure 15

Linearized guidance system assuming no gravity.

Figure 16

Linearized guidance system considering gravity.

Figure 17

An Equivalent system to Figure 16.

Here the overload command of the APN perpendicular to LOS is:

(40) a MC_OL = N V C q ̇ + N 2 ( k a T ⊥ − k ⁎ g ⊥ ) ,

where g ^⊥ is the component of gravity acceleration perpendicular to the missile-target LOS, and a T ⊥ − g ⊥ is the component of equivalent target acceleration perpendicular to the LOS. By substituting a T ⊥ − g ⊥ = 2 R ̇ q ̇ + R q ̈ + a M in Eq. (25), then

(41) a MC_OL = ( 1 − k ) N V C q ̇ + k N 2 ( R q ̈ + a M ) − N 2 ( k ⁎ − k ) g ⊥ .

Due to the short flight time and short flight distance of the missile, it can be considered that the gravity is constant.

By substituting R = k ₀ V _C further and replacing a _M with a _Mf, we can get:

(42) a MC_OL = ( 1 − k ) N V C q ̇ + k N 2 ( k 0 V C q ̈ + a Mf ) − N 2 ( k ⁎ − k ) g ⊥ ,

where a _Mf is the output of the first-order low-pass filter (the time constant is T ₁) whose input is a _M.

(43) a M = a M_OL + g ⊥ ,

where a _M is the component of the missile acceleration response perpendicular to the LOS, and a _{M_OL} is the component of the missile’s overload response perpendicular to the LOS. And a _{M_OL} can be obtained from accelerometer’s measurement and the strapdown navigation information.

Similarly, considering gravity compensation, AACG is also equivalent to PNG when k = 0.

6 Numerical simulation of six degrees of freedom guidance system

Using the numerical simulation model of the six degrees of freedom guidance system of an infrared air-to-air missile, PNG and AACG are, respectively, considered, and typical attack scenarios are simulated to compare the performance of PNG and AACG. The numerical simulation model of the six degrees of freedom guidance system, which is shown in Figure 18, generally includes: seeker, Kalman filter, guidance law, stability control algorithm, steering engine, inertial measurement unit, strapdown navigation, missile aerodynamics and kinematics, target’s motion and target’s motion relative to missile. According to the LOS angle channel information measured by the seeker, combined with the missile motion information calculated by strapdown navigation, the guidance information is extracted through Kalman filtering, and the guidance command is generated according to the guidance law. The autopilot outputs the rudder deflection angle command according to the guidance command, operates the rudder to stabilize the missile attitude, and lets the overload response follow the overload command calculated by the guidance law with better dynamic characteristics, so that the missile can attack the target accurately. In the numerical simulation model of the six degrees of freedom guidance system, the influence of gravity is considered, and the compensation method is shown in Eq. (42).

Figure 18

Numerical simulation model of six degrees of freedom guidance system.

First, it is assumed that the aim point remains stable and only the influence of target’s maneuver is considered. A set of 56 simulation conditions are set in this way: at the zero time of autonomous flight after missile launch, the missile’s altitude is 5,000 m, the flight speed is 0.9 Ma, the yaw angle is 0°, the pitch angle is 0°, the distance is 11,500 m, the target azimuth is 0°, the height angle is 0°, the target’s altitude is 5,000 m, the flight speed is 0.9 Ma, the speed yaw angle is 180°, the speed pitch angle is 0°, and the target keeps level flight and does not maneuver. When the distance decreases to a given value, the target starts circling in the horizontal plane. The distance at the beginning of the maneuver is set to 50, 100, …, 2,750 and 2,800 m respectively, with a total of 56 gears. In the six degrees of freedom guidance system simulation model, the target maneuver is set as circling maneuver in the horizontal plane, the amplitude is 9 g, and the maneuver time constant is 0 s.

In Figure 19, the influence of different t _go when the target starts maneuvering on miss distance is that AACG is significantly better than PNG, and the miss distance peaks of AACG and PNG are 3.14 and 9.65 m respectively. The miss distance of AACG attenuates below 1 m when t _go is greater than 0.6 s, and PNG’s miss distance attenuates below 2 m when t _go is greater than 1.4 s. The missile speed at the time of encountering the target (hereinafter referred to as the missile terminal speed) is shown in Figure 20. Terminal speed of the missile using AACG is 702–763 m s⁻¹, and that of PNG is 683–769 m s⁻¹. When t _go is greater than 1 s, the terminal speed of AACG is about 20 m s⁻¹ higher than that of PNG.

Figure 19

Maneuver beginning t _go – miss distance.

Figure 20

Maneuver beginning t _go – terminal velocity.

Then, assuming that the target does not maneuver, only the influence of seeker’s aim point jumping is considered. A set of 56 simulation conditions are set in this way: at the zero time of autonomous flight after missile launch, the missile altitude is 5,000 m, the flight speed is 0.9 Ma, the yaw angle is 0°, the pitch angle is 0°, the distance is 11,500 m, the target azimuth is 0°, the height angle is 0°, the target altitude is 5,000 m, the flight speed is 0.9 Ma, the speed yaw angle is 180° and the speed pitch angle is 0°, and the target keeps level flight and does not maneuver. When the distance is reduced to a given value, the aim point of the seeker jumps by 0.3° and keeps tracking the new aim point until it encounters the target. The distance at the jumping time of the aim point of the seeker is set to 50, 100, …, 2,750 and 2,800 m respectively, a total of 56 gears.

The influence of aim point jumping (0.3°) on miss distance is shown in Figure 21. For example, the terminal speed of missile and AACG is significantly better than PNG, and the miss distance peaks of AACG and PNG are 2.29 and 5.76 m, respectively. AACG’s miss distance decays below 1 m when t _go is greater than 1.1 s, and PNG’s miss distance decays below 1 m when t _go is greater than 1.7 s. In Figure 22, the terminal speed performance of AACG is similar to the PNG.

Figure 21

Aim point jumping t _go – miss distance.

Figure 22

Aim point jumping t _go – terminal velocity.

7 Conclusion

In this article, LOS AACG for passive homing missile is proposed. AACG’s novel structure makes the parameter k independent on target’s acceleration, so the parameter k which can be set as a positive constant is easy to design, and AACG is rather robust in different intercept missions. AACG only needs LOS angular velocity, LOS angular acceleration information and missile-target approaching velocity information, and does not need to obtain accurate target acceleration and distance information. Even if the missile-target approaching velocity information cannot be estimated, it can also be replaced by the modulus of the component of missile velocity in LOS direction and AACG can also work perfectly. So AACG is especially suitable for the terminal guidance of passive homing tactical missile with strapdown inertial navigation system. Numerical simulation results show that AACG still has good guidance performance, which means smaller miss distance under complex attack conditions. For two types of errors, large maneuver and typical interference of angle measurement of the seeker, the guidance accuracy of AACG is obviously better than PNG. Moreover, the guidance sensitive area of AACG is significantly smaller than that of PNG.

Funding information: This research was supported by China Airborne Missile Academy in some missile projects.
Author contributions: Jixin Li: writing – review and editing, investigation, methodology, and formal analysis; Xiaogeng Liang: writing – original draft, writing – review and editing, and investigation; Keqiang Xia: writing – original draft, investigation, and formal analysis; Yilin You: writing – review and editing and resources.
Conflict of interest: Authors state no conflict of interest.

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Received: 2022-06-17

Revised: 2022-07-29

Accepted: 2022-08-01

Published Online: 2022-09-05

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/astro-2022-0039

Keywords for this article

guidance; missile; proportional navigation; angular acceleration

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