Adaptive fuze-warhead coordination method based on BP artificial neural network

Peng Hou,Yang Pei,Yu-xue Ge

School of Aeronautics, Northwestern Polytechnical University, Xi'an, 710072, Shaanxi, China

Keywords: Aircraft vulnerability Fuze-warhead coordination BP artificial neural network Damage probability Initiation delay

ABSTRACT The appropriate fuze-warhead coordination method is important to improve the damage efficiency of air defense missiles against aircraft targets.In this paper,an adaptive fuze-warhead coordination method based on the Back Propagation Artificial Neural Network (BP-ANN) is proposed,which uses the parameters of missile-target intersection to adaptively calculate the initiation delay.The damage probabilities at different radial locations along the same shot line of a given intersection situation are calculated,so as to determine the optimal detonation position.On this basis,the BP-ANN model is used to describe the complex and highly nonlinear relationship between different intersection parameters and the corresponding optimal detonating point position.In the actual terminal engagement process,the fuze initiation delay is quickly determined by the constructed BP-ANN model combined with the missiletarget intersection parameters.The method is validated in the case of the single-shot damage probability evaluation.Comparing with other fuze-warhead coordination methods,the proposed method can produce higher single-shot damage probability under various intersection conditions,while the fuzewarhead coordination effect is less influenced by the location of the aim point.

1.Introduction

Anti-aircraft missiles usually cannot hit the aircraft directly,but pass near the target with a certain miss distance [1].Fragment warhead,being the main force of air defense warheads,uses highspeed fragmentation groups to penetrate aircraft components,and can still cause damage to targets at a certain miss distance[2,3].To improve the damage efficiency,the warhead fragments should fall on the critical components of the aircraft as much as possible [4].Therefore,it is of great significance to study the coordination between fragment warhead and proximity fuze in the final stage of missile-target engagement.

The fuze-warhead coordination is to use the information of environment,target and guidance to detonate the warhead during the engagement between the missile and the target,to cause the best damage effect to the target [5].The relevant research mainly focuses on the mathematical model of fuze-warhead coordination,the optimization of the fuze-warhead coordination method,the evaluation of fuze-warhead coordination efficiency,and so on.

In the mathematical model of fuze-warhead coordination,Zhang [6]established the fuze-warhead coordination model of anti-radiation missile attacking radar target.Based on the engagement parameters of the missile and the target,an adaptive initiation strategy is proposed to calculate the initiation delay time of proximity fuze.Li[7]established the fuze-warhead coordination model of Explosive Formed Projectile (EFP),which calculates the time of ammunition detonation under the constraints of ammunition attitude angle.Yang [8]generalized the model from stationary ground targets to moving ground targets.For aircraft targets,Zhao[9]constructed a fuze-warhead coordination model of fragment warhead attacking early warning aircraft,and analyzed the influence of typical parameters on the fuze-warhead coordination effect.

In the optimization of the fuze-warhead coordination method,Nyongesa[10]uses a genetic algorithm to calculate the delay time of anti-air missile proximity fuze,which greatly improves the efficiency of delay-time calculation.Huang [11]adaptively adjusted the beam angle of the fuze antenna to coincide with the fuze actuation zone and the fragment dispersing center.It improves the fuze-warhead coordination efficiency with different targets.Chopper[12]used the Guidance Integrated Fuzing(GIF)technique to improve the accuracy of evaluating the initiation delay time.On this basis,Wang [13]adopted the hit-function as the control strategy to improve the hit probability and damage efficiency of the missile.Zhang14established the control model of the optimal initiation delay under arbitrary encounter conditions.The model adopts the Second order Debased Converted Measurement(SDCM)tracking algorithm based on radar measurement,which greatly improves the accuracy of initiation delay estimation.The GIF-based research [12-14]aim at delay-time estimation and time-to-go prediction,which can help the missile to approach the target.However,the ideal detonation point can be different from the closest meeting point between the missile and the target.Therefore,Chakravarti [15]proposed end-game algorithm for ballistic and maneuvering targets,which can make fragments reach the vulnerable center of the target and ensure the maximum damage to the target.Zhao [16]proposed the fuze-warhead cooperation method of fortification storming/heat missiles attacking air target based on the GIF.The method accurately determined the optimal initiation delay,and controlled the front-back fragment clusters of the two-stage warhead to hit the critical components of the target.

In the evaluation of fuze-warhead coordination efficiency,Ren[17]proposed to use semi-physical simulation.Practical experiments are not always easy to carry out,simulation is a new method[18].Wang [19]takes single-shot damage probability as the index to evaluate the combat effectiveness of the missile through simulation,so as to guide the design of fuze-warhead coordination.

Most of the above researchs [5-19]focus on how to make the missile warhead accurately hit the given position of the target,and usually simplify the target model.However,the aircraft structure is complex,and the differences between aircrafts are relatively large.To study the fuze-warhead coordination of such targets,it is necessary to establish a refined model.On this basis,accurately identifying the vulnerable zone of the aircraft and determine the optimal initiation delay is still the main research direction[20].

Fuze initiation delay is the time between the detection of the target and the initiation of the warhead.The general models are the fixed [21]and the adaptive [22]delay-time models.Most of the previous researches take the center of the target as the reference point when calculating the delay time,ignoring the air drag to fragments.For aircraft targets with many critical components,this simplification may produce errors when calculating the fuze initiation delay.

In this paper,the high-precision vulnerability model of the aircraft is established.Taking the typical intersection parameters as the input and the optimal detonation position of the missile as the output.A BP-ANN model is constructed.BP-ANN is widely used in mapping various nonlinear relationships because of its powerful computational advantages such as self-learning,adaptive and generalized fault-tolerant abilities [23].The objective of BP-ANN model is to establish the mathematical correlation between userdefined inputs and outputs [24].Once the BP-ANN model is constructed,the proposed method can quickly obtain the optimal initiation delay time according to the engagement parameters.Compared with other fuze-warhead coordination methods,the method proposed in this paper can ensure a better fuze warhead coordination effect on the basis of fast calculation of initiation delay.

The main structure of this paper is as follows.Section 2 introduces the evaluation index of fuze-warhead coordination effect.Section 3 proposes an adaptive fuze-warhead coordination method based on BP-ANN.Section 4 constructs a series of simulations to analyze the fuze-warhead coordination efficiency of the proposed method and compare it with other common methods.The main conclusions and future prospects are shown in Section 5.

2.Evaluation index of fuze-warhead coordination effect

When anti-aircraft missile strikes aircraft target,the single-shot damage probability is the main basis for evaluating the fuzewarhead coordination effect.The single-shot damage probability of anti-aircraft missile can be expressed as

wherepk（x，y，z）is the damage probability of aircraft after the warhead detonates at point（x，y，z）in the relative velocity coordinate system;fg（y，z）is the guidance error distribution of missile;ff（x/y，z）is the radial distance distribution of fuze initiation point;pf（y，z）is the fuze actuation probability under a given miss error.ff（x/y，z）,pf（y，z）is usually expressed in the form offf（x/ρ，θ）andpf（ρ，θ）based on the miss distance ρ,miss azimuth θ,and ρ=θ=arctan（y/z）.Coordinate damage probabilitypk（x，y，z）is related to the warhead damage characteristic,the target vulnerability characteristic,and the engagement parameters.

The warhead damage characteristic can be expressed by the dispersion model of fragments,which can give the initial velocity and dispersion direction of fragments at the detonation-time.1The dispersion field of the preformed fragment warhead missile is concentrated in the static dispersion zone.The static dispersion zone of the fragment can be represented by the dispersion cone,which is formed by the front dispersion angle φfand the back dispersion angle φb.The dynamic dispersion zone is obtained by superimposing the missile velocity at the detonation-time on the static dispersion zone,as shown in Fig.1.The dynamic dispersion angle is calculated as

Fig.1.Dispersion model of fragments.

where φ and φ′are the static and dynamic dispersion angle;v0is the static initial velocity of fragments;vmis the velocity of the missile.This equation can be used to calculate the dynamic front dispersion angleand the back dispersion angleof the fragments.

The dynamic initial velocity of fragment vfcorresponding to each dispersion angle in the dispersion zone at the detonation-time is expressed as

The target vulnerability characteristics can be represented by the target vulnerability model,which mainly includes the surface element model,the damage criterion of critical component,and the damage tree[2].

The surface element model of aircraft can describe the position,the size,the material and other information of the main system components,which is generally composed of configuration model,structure model and system model.

The damage criterion model of critical components is used to determine the damage probability under a given damage level.The damage criteria can be categorized as combustion criterion for fuel critical components,explosion criterion for ammunition critical components and penetration criterion for mechanical damage components [2].

The damage tree model uses the tree graph to establish the logical relationship between the damage of critical components and the damage of the whole aircraft target [25].residual mass and the residual velocity of the fragment decrease gradually.

Residual velocity and residual mass of the fragment after penetrating aircraft component can be calculated by the THOR penetration equation [27].

After the missile is detonated,the engagement parameters are used to determine the intersection of fragments and aircraft target.As the fragments fly towards the target,the fragment velocity gradually decreases because of drag,while the aircraft velocity remains constant.When the velocity direction of the fragment is not collinear with that of the aircraft,the velocity direction of the fragment relative to the aircraft will change continuously.If the fragment meets the target,the hitting position and speed are determined.Due to the effect of drag,the fragment dispersion velocity gradually decreases from vf0to vfh,so the trajectory of the fragments relative to the aircraft is a curve[1],as shown in Fig.2,

Fig.2.The dispersion process of a fragment.

If the fragment intersects with the target,the shot line scanning method [26]can be used to track the movement trajectory of the fragment inside the aircraft.The intersection of the shot line and the internal components of the target can be determined.The movement of typical fragment inside the aircraft is shown in Fig.3.As the number of fragment penetrating component increases,the where vsis velocity of the fragment before perforation;vris residual velocity of the fragment after perforation;tis the thickness of the surface element;Afis the collision area of the fragment;θ is the angle between the fragment penetration direction and the normal direction of the surface element;c1-c10is the coefficient related to the target plate material;mfis the mass before the fragment penetrates the surface element;mris the residual mass after the fragment penetrates the surface element.

Fig.3.Schematic diagram of fragment movement inside the aircraft.

The damage probabilitypkiof critical components is calculated according to the motion parameters of each fragment and the damage criterion of critical component.

wherepkiis the damage probability of critical componentiunder the attack of the warhead;pki/jis the damage probability of the critical componentiunder the attack of fragmentj;niis the number of fragments hitting critical componenti.Based on the damage probability of each critical component and the damage tree,the damage probability of the whole aircraft under a given strike can be obtained.

Guidance error distributionfg（y，z）of air defense missile is generally expressed by the two-dimensional normal distribution.

wheremyandmzare the mathematical expectation of distribution center;σyand σzare the standard deviation of the detonation point from the distribution center.Whenmy=mz=0 and σy=σz=σ,the guidance error distribution can be expressed by circular normal distribution.

The relationship between the standard deviation σ and missile hit accuracy index CEP can be expressed as CEP=1.1774σ.

The distributionff（x/ρ，θ）of the fuze initiation point and the fuze actuation probabilitypf（ρ，θ）under a given miss error need to be determined by the fuze-warhead coordination model.

The guidance error distributionfg（ρ，θ）,the fuze initiation point distributionff（x/ρ，θ）and the fuze actuation probabilitypf（ρ，θ）are used to determine the position of missile detonation point.The position of the missile detonation point is generally given in the relative velocity coordinate systemOXrYrZr[28],as shown in Fig.4.The position of the detonation point relative to the aircraft target can be determined by the miss distance ρ,the miss azimuth θ and the radial deviationxr.The miss distance ρ and the miss azimuth θ are determined by the guidance error distributionfg（ρ，θ）.The radial deviationxris determined by the fuze-warhead coordination model.Under the given miss error distribution,the location distribution of detonation point can be calculated by the fuze-warhead coordination model.Different detonation point positions will result in different single-shot damage probabilities.Therefore,the singleshot damage probability is selected as the index to evaluate the fuze-warhead coordination effect.

Fig.4.The missile detonation point position.

3.Adaptive fuze-warhead coordination method based on BPANN

Fuze-warhead coordination is that the proximity fuze selects the most favorable initiation point on the terminal engagement trajectory according to the intersection parameters,so as to maximize the single-shot damage probability.In this section,an adaptive fuze-warhead coordination method based on BP-ANN is proposed.Then several common fuze-warhead coordination methods are introduced for comparison.Finally,a calculation model of aircraft damage probability considering fuze detection simulation is established.

3.1. Adaptive fuze-warhead coordination method based on BP-ANN(method 1)

This paper proposes a fuze-warhead coordination method(method 1) that uses a BP-ANN model to predict the optimal detonation position of the missile,and determines the initiation delay.The BP-ANN model is constructed with the intersection parameters as the input and the corresponding optimal radial distance obtained from simulation as the output.In the actual engagement process,according to the intersection parameters of the missile and the aircraft target,the constructed BP-ANN model is used to determine the optimal radial distance of the detonation point adaptively,and then the optimal initiation delay is determined.To sum up,the method 1 is mainly composed of two parts,which are the construction of BP-ANN model and the adaptive determination of initiation delay based on actual intersection parameters using the constructed BP-ANN model.

3.1.1.Construction of the BP-ANN model

The BP-ANN model is built by the following steps.(i)According to the given missile-target intersection parameters,sample points are generated according to certain sampling rules.(ii) For each sample point,the damage probability of the aircraft when the missile detonates at different radial distances is calculated by the shot line scanning method.29(iii)The optimal radial distance of the missile at each sample point is calculated according to the damage probability values at different radial distances.(iv) The BP-ANN model is constructed with the intersection parameters as the inputs and the optimal radial distance as the output.

This section takes miss errors(miss distance ρ and miss azimuth θ)under given strike direction as typical intersection parameters,to construct the BP-ANN model.

The guidance error of missile striking aircraft is determined by Eq.(8).The probability that the location of the detonation points in the miss distance plane falling into the 3σ circle is 0.9889.Therefore,during sampling,the miss distance ρ is uniformly sampled in the interval [0，3σ]according to the interval Δρ,and the miss azimuth θ is uniformly sampled in the interval[0，2π）with the interval Δθ.Therefore,the sample space is composed ofNs=（3σ/Δσ）·（2π/Δθ）+1 sample points on the miss distance plane.

For these sample points,the simulation interval[xrwmin，xrwmax]of each detonation point positionxrwalong theXr-axis is set respectively,and uniform sampling is conducted according to the interval Δxrw.Thenn=（xrwmax-xrwmin）/Δxrwradial distance sample points are obtained.Finally,the position coordinates of the detonation point are generated in the form of（xrw，ρ，θ）,and the damage probabilitypkxwat the given coordinates is calculated through simulation,as show in Fig.5.

Fig.5.Calculation of damage probability at each radial distance of a given sample point.

After obtaining the damage probability ofnradial distance points under the given miss error（ρ，θ）,the radial distanceof the optimal detonation point under a given miss error（ρ，θ）can be calculated.

wherexrwiis theith radial distance value of the detonation point under a given miss error（ρ，θ）;pkiis the damage probability of the aircraft after the missile detonating at the corresponding position.

The optimal radial distance ofNsgroups of sample points is obtained by Eq.(9).The mapping relationship between miss error(sample input) and optimal radial distance (sample output) is complex and highly nonlinear,which is related to the missile power field,the target vulnerability and the intersection parameters.

Therefore,this paper constructs a BP-ANN model to express this mapping relationship.BP-ANN has good nonlinear expressing ability and can realize any complex nonlinear mapping.It is a multilayer feedforward network trained by error back propagation algorithm [30,31].BP-ANN model consists of the input layer,the hidden layer and the output layer.A typical 3-layer BP-ANN structure is shown in Fig.6.To train the network,the Bayesian regularization algorithm is used to repeatedly adjust the weights ωij,ωjand thresholds of neurons in input layer and hidden layer,hidden layer and output layer.When the accuracy of the network output meets the given requirements,the weight and the threshold of the current network will be saved.Then the current network will be tested.If the error meets the requirements,the BP-ANN model is completed.

Fig.6.BP-ANN structure diagram.

The intersection parameters such as ρ and θ are inputs,and the corresponding optimal radial distanceis the output.Regularization algorithm is used to prevent over-fitting phenomenon.‘tansig’ is the neuron transfer function in the hidden layer,and‘purelin’ is the neuron function of the output layer.The Mean Square Error,MSE,and correlation coefficient,R,are used to evaluate the accuracy and performance of the network.The MSE is defined as

whereyiis the output value of the BP-ANN corresponding to theith sample input;is the sample output value corresponding to theith sample input;nis the sample capacity.Rreflects the fitting degree of the BP-ANN.TheRis defined as

whereCov（y，）is the covariance between the BP-ANN prediction output and the sample output;Var（y）andVar（）are the variances of BP-ANN prediction output and sample output respectively.

In case of constructing BP-ANN model with more typical intersection parameters as input,the construction process is basically the same as that with miss distance and miss azimuth as input.However,due to the increase of input parameters,the number of sample points used to construct BP-ANN model should also be increased accordingly.

3.1.2.Adaptive determination of initiation delay based on intersection parameters

After the BP-ANN model is constructed,the fuze detection simulation model can be used to adaptively determine the initiation delay based on the intersection parameters.In the terminal intersection section,the proximity laser fuze emits laser beams around.If the target is irradiated by the laser beam,the signal begins to accumulate.When the signal accumulates to a certain extent (After a certain signal accumulation time τ1),the fuze is activated.After the fuze is activated,the warhead will be detonated after a certain time delay τ2.Two parameters are mainly considered from fuze activation to warhead detonation,that is,fuze activation time τ1and fuze initiation delay τ2.The fuze-warhead coordination model mainly determines the initiation delay τ2.

The proximity laser fuze can achieve omnidirectional detection by arranging multiple sets of lasers in the circumferential direction.The optical fuze has a narrow field of view,so its beam inclination Ωfis usually taken as a constant value.Therefore,the fuze detection field can be represented by a fuze detection cone,[32]which is formed by the detection angle ηmax,the field of view angle λ,and the maximum action distanceRmaxof the fuze,as shown in Fig.7.WhereLis the distance between the fuze center and the warhead center.

Fig.7.Simulation principle of fuze detection field: (a) Schematic diagram of the fuze detection field;(b) Simulation diagram of the fuze detection cone.

The area corresponding to the field of view angle in the detection field is the actual detectable area of the fuze.In the fuze detection coordinate system,each coordinate axis is parallel to the missile body coordinate system,and the origin takes the fuze center.The fuze detection model uses discrete scan lines instead of the fuze detection beam to simulate the fuze detection process.The scan lines start from the center of the fuze,and the angle between the scan line andXf-axis is η (scan line inclination).The angle between the projection of the scan line onYfZfthe projection plane andYf-axis is ω(scan line azimuth).Each scan line is constrained in the detection beam.That is,the corresponding η is in [ηmax-λ/2，ηmax+λ/2],ω is in [0，2π）.

The principle of fuze detection simulation is shown in Fig.8.First,2π/Δω scan line sampling planes passing through theXf-axis and perpendicular to theYfZfplane are generated with the azimuth sampling interval Δω.Then λ/Δη scan lines are generated at intervals Δη in each sampling plane.Finally,Nsl=（2π/Δω）·（λ/Δη）+1 scan lines are generated.The intersection of these scan lines and all surface elements of the aircraft is calculated respectively.If the intersection of any scan line and any surface element is inside the surface element,and the distance between the intersection and the center of the fuze is less than the maximum action distanceRmaxof the fuze,the fuze is triggered.After the signal accumulation time τ1,if there is still any surface element detected by any scan line,it is considered that the fuze detects the aircraft and the fuze is activated[33].

Fig.8.Fuze detection simulation schematic.

For the terminal engagement with a given miss error,the radial distance simulation interval [xrmin，xrmax]of the fuze center in the relative velocity coordinate system is set according to the aircraft size and the maximum action distanceRmaxof the fuze.Starting fromxrminwith Δxras the sampling step,it is gradually judged whether the aircraft is detected according to the fuze detection simulation model.If the aircraft is not detected even if the radial distance of the fuze center reachesxrmax,it is considered that the detection fails and the missile misses the target.If in theith sampling step,the fuze is activated after the fuze detects the aircraft.At this moment,the radial distance of the fuze center in the relative velocity coordinate system isxri=xrmin+i·Δxr+vr·τ1.

Under the given intersection conditions,the optimal radial distanceof the warhead center is calculated by using the constructed BP-ANN model.On this basis,the optimal initiation position of the fuze center can be determined asComparing the relationship between the optimal detonation positionand the fuze activation point positionxri.If≤xri,then the initiation delay τ2=0,that is,the warhead will detonate immediately after the fuze is activated.If>xri,the initiation delay τ2is set according to the optimal radial distanceof the fuze center.

Considering that the missile with proximity fuze cannot penetrate the aircraft target,it is necessary to perform collision detection when setting the initiation delay.As shown in Fig.9,when the fuze is activated,the radial distance of the fuze center in the relative velocity coordinate system isxri.At this time,a ray is generated from the front section of the missile along the relative motion trajectory and intersects with the aircraft surface element model.If there is an intersection point,it indicates that a collision may occur.The distance between the front section of the missile and the collision point is calculated asRc.Then the radial distancexcof the collision point in the relative velocity coordinate system is

Fig.9.Schematic diagram of collision detection.

wheredis the distance between the fuze center and the front section of the missile;θcis the angle between the axial direction of the missile and the velocity vrof the missile relative to the aircraft.To prevent the missile from penetrating the target,the position of the fuze center when the warhead is detonated cannot exceed=xc-d·cos θc=xri+Rc.According to the fuze-warhead coordination simulation model based on BP-ANN,the optimal initiation position of the fuze center isthe fuze initiation delay τ2should be corrected,and the corrected initiation delayis

Method 1 is based on the results of aircraft damage assessment.Using the damage results of different radial distances under a given miss error,the radial distance of the optimal detonation point is calculated.According to the mapping relationship between the intersection parameters and the radial distance of the optimal detonation point,a BP-ANN model is constructed.Finally,the BPANN model is used to predict the optimal initiation delay under the given intersection conditions.Because method 1 is based on the damage result of the missile to the aircraft,it can ensure that each attack has a relatively large damage probability value.In addition,after the BP-ANN model is constructed,the intersection parameters can be used to quickly determine the initiation delay without additional simulation calculation.

3.2. Common simulation methods of fuze-warhead coordination

3.2.1.Fuze-warhead coordination method based on fuze actuation zone(method 2)

The fuze actuation zone is generally given in the relative velocity coordinate system.Given the miss distance ρ and miss azimuth θ,it is generally considered that the probability density function of the fuze actuation point along theXr-axis obeys one-dimensional normal distribution.

wheremxand σxare the dispersion mathematical expectation and standard deviation of the fuze actuation point along theXr-axis respectively.The change of fuze activation probability with the distance from the target obeys the normal integral distribution[19].

Method 2 simulates the fuze initiation delay by adjusting the mathematical expectationmxof the fuze actuation point position.The fuze-warhead coordination process is completed by maximizing the single-shot damage probability calculated by the fuze activation probability and the coordinate damage probability.However,the establishment of the fuze actuation zone model usually requires the fuze ground flying test and simulation test.In addition,the actual fuze-warhead coordination process needs to consider various engagement situations,which is costly and timeconsuming.

3.2.2.Fuze-warhead coordination model based onfixed delay-time(method 3)

The fuze initiation delay τ2based on the fuze-warhead coordination method with fixed delay time is a constant value[29].After the fuze detects the target,this fixed delay time is used to determine the position of the missile detonation.Method 3 can improve the single-shot damage probability by setting an appropriate fixed delay time.However,the optimal fixed delay time is generally different with the change of intersection parameters.

3.2.3.Adaptive fuze-warhead coordination model based on aim point(method 4)

After the fuze detects the aircraft,the adaptive fuze-warhead coordination method based on aim point adaptively sets the delay time τ2according to the intersection parameters,so that the fragment central dispersion zone passes through the aim point[22].When the aim point is selected as the aircraft vulnerability center,a better damage effect can be obtained.However,the fuzewarhead coordination effect of method 4 will be affected by the location of the aim point.

3.3. Calculation method of single-shot damage probability based on Monte Carlo simulation

The fuze-warhead coordination method takes single-shot damage probability as the evaluation index.The coordinate damage probability is determined by the method given in Section 2.The guidance error is determined by the missile CEP.The radial distance of the detonation point and fuze actuation probability are determined by the fuze-warhead coordination model.It is difficult to accurately express the single-shot damage probability directly from Eq.(1) under the given intersection conditions and the damage level.Therefore,this paper uses Monte Carlo simulation to calculate the damage probability.

The calculation process of aircraft damage probability under given intersection conditions and given damage level is shown in Fig.10.It mainly includes 6 steps.

Fig.10.Aircraft damage probability calculation flowchart.

Fig.11.3D model of the hypothetical aircraft.

(i) Set simulation parameters.The aim point position and the Monte Carlo simulation timesNmcare set respectively.

(ii) Set miss error.The CEP of the missile is used to determine the standard deviation σ of the guidance error distribution.Two groups of random numbers that follow the normal distributionN（0，σ2）are respectively generated as the miss error（yrwi，zrwi）of theith sample point.On this basis,the miss distance ρiand miss azimuth θiare calculated.

(iii) Fuze detection simulation.Methods 1,3 and 4 all use the fuze detection simulation model to judge whether the aircraft is detected,and method 2 judges whether the fuze is activated according to fuze activation probability.For these four methods,if the detection fails,the damage probabilitypkiof the sample point is 0.If the target is detected,the initiation delay τ2is further calculated to determine the detonation point position.

(iv) Determine the position of the detonation point.Method 1 uses the constructed BP-ANN model combined with the intersection parameters of theith sample point to calculate the initiation delay τ2.The radial distanceof the warhead center is further calculated.Method 2 generates a random number subject to a normal distributionN（mx，）as the radial distance of the fuze center at theith sample point.From this,the radial distanceof the warhead center is calculated.Method 3 calculates the radial distanceof the warhead center at theith sample point through the fixed delay-time τ2.Method 4 determines the radial distanceof the warhead center by adaptive initiation delay τ2.According to the miss error（ρi，θi）and the radial distancethe coordinatesof the detonation point corresponding to theith sample point in the relative velocity coordinate system can be obtained.

(v) Coordinate damage probability calculation.The damage probabilitypkiat theith sample point is calculated according to the locationof the detonation point and the coordinate damage probability analysis method given in Section 2.If the simulation times do not reach the upper limitNmc,repeat steps (ii)～(v).Otherwise,if the simulation times reach the upper limit,the average damage probability can be determined.

(vi) Average damage probability calculation.The single-shot damage probabilitypkcan be calculated by averaging the damage probability ofNmcsample points.

4.Case study

This section takes a hypothetical aircraft like MQ-1 UAV as the example.The 3D model is shown in Fig.11.The anti-aircraft missile has a prefabricated fragmentation warhead and laser fuze.Taking the single-shot damage probability result under C-level as the evaluation index,the effectiveness of the fuze-warhead coordination method is analyzed.

The BP-ANN model is first constructed.On this basis,the effects of the adaptive fuze-warhead coordination method based on BPANN and the other three fuze-warhead coordination methods are compared.Finally,the influence of the aim point position on the two adaptive fuze-warhead coordination methods is analyzed.

4.1. BP-ANN model construction

In this section,the BP-ANN model is constructed with miss azimuth and miss distance as inputs.To generate the sample points,four attack directions from the lower hemisphere of the attack ball are selected as typical attack directions.The pitch angles of the four directions are all 45°,and the azimuth angles are 45°,135°,225°and 315°,respectively.The aim point is the center of the body,as shown in Fig.12.The main parameters of the missile and the UAV are shown in Table 1.

Table 1 Intersection parameters and simulation parameters.

Fig.12.Attack direction.

After determining the terminal intersection parameters,the miss distance is sampled at the interval of 1 m on[0,15 m],and the miss azimuth is sampled at the interval of π/9 on[0,2π).Finally,271 sample points are generated.For each sample point,the radial distance sampling point of the detonation point is set respectively according to the radial distance sampling interval,and the damage probability of each detonation point is calculated.The results of attack direction 1 are shown in Fig.13.

Fig.13.Damage probability cloud map of sample points: (a) Damage probability at different radial distances for a given sample point;(b)Damage probability of different radial distances for each sample point.

The radial distance of the optimal detonation point corresponding to each sample point is calculated by Eq.(9).The azimuth represented by the miss azimuth of 2π is the same as that represented by the miss azimuth of 0.However,increasing the data points of miss azimuth 2π is beneficial to the construction of BPANN model.Therefore,the sample data is expanded,and finally 287 groups of sample data points are obtained.Typical sample data are shown in Table 2.The spatial position distribution of sample points in the relative velocity coordinate systemXrYrZris shown in Fig.14.CoordinateXrrepresents the optimal radial distance coordinate of sample points,whileYrandZrrepresent the position of sample points in the miss distance plane.The optimal radial distance coordinates of sample points gradually decrease with the increase of miss distance.The distribution of sample points presents a cone.

Table 2 Sample data value (attack direction 2).

Fig.14.Spatial distribution of sample points (attack direction 2).

Taking the miss distance and miss azimuth of attack direction 3 as the input,and the optimal radial distance as the output,the BPANN model is constructed.80% of the sample data is used as training data,5% as validation data,and the remaining 15% as testing data.BP-ANN model has a single hidden layer structure.The training function is the ‘trainbr’ function based on Bayesian regularization algorithm.The transfer function of the hidden layer is a hyperbolic tangent S-type function‘tansig’,and the output layer is a linear transfer function‘purelin’.MSE and correlation coefficient R are the network performance evaluation index.The maximum training epoch is 1000.To achieve the expected accuracy,the number of hidden layer neurons is first set to 2 during network training,and then the number of neurons is continuously adjusted.Finally,the number of hidden layer neurons is adjusted to 16 by testing multiple groups of networks,which is more beneficial to the research problem.The fitting effect index of attack direction 3 is shown in Fig.15.

Fig.15.BP-ANN fitting effect index (attack direction 3): (a) MSE;(b) Training: R=0.99892;(c) Test: R=0.99926;(d) All: R=0.99898.

It can be seen from Fig.15(a) that after 584 iterations,theMSEvalue of the training sample of attack direction 3 reached the minimum value of 0.0069615,and theMSEvalue of the testing sample is basically equal to that of the training sample.This shows that the deviation between the predicted value of the BP-ANN model and the experimental value is small,and the prediction effect is great.As can be seen from Fig.15(b)to 15(d),the correlation coefficientsRof training samples,test samples and all samples are 0.99892,0.99926 and 0.99898 respectively,which are close to 1.It indicates that the model has a high degree of fitting and can be used for data prediction.According to the same method and network parameter setting,BP-ANN models of attack direction 1,2 and 4 can be constructed respectively.The fitting effect indexesMSEof the models are 0.0082833,0.008602 and 0.0057314 respectively.The correlation coefficientsRof all sample are 0.99855,0.98757 and 0.99888 respectively.This shows that the BP-ANN model constructed with four attack directions has good accuracy and network performance,which can be used in the subsequent adaptive fuzewarhead coordination process.

4.2. Influence of multiple input parameters on the fuze-warhead coordination effect of method 1

In this section,the BP-ANN model is constructed with miss error and attack direction parameters as input and the optimal radial distance as output.The parameters of the UAV and the missile are the same as those in Table 1 except for the pitch angle and yaw angle of the missile,which are determined by the attack direction.

The CEP of the missile is 5 m,the miss distance is sampled at a sampling interval of 1 m on [0,10 m],and the miss azimuth is sampled at a sampling interval of π/6 on[0,2π).The missile attack direction is given in the form of intersection angleQand intersection azimuthA.WhereQis the complementary angle between the missile velocity vmand the aircraft velocity vt,andAis the angle between the projection of the missile velocity vmin theYtvZtvplane and theYtvaxis,as show in Fig.16.The setting of attack directions is shown in Table 3.2904 sample points are obtained by sampling according to the above method.Similarly,the data of miss azimuth 2π is augmented with the data of miss azimuth 0,and finally 3144 sample points are obtained.

Table 3 Attack direction setting.

Fig.16.Definition of missile velocity direction.

The method in Section 3.1 is used to obtain the radial distance of the optimal damage detonation point for each sample point.The sample data is formed with the intersection angle,intersection azimuth,miss distance,and miss azimuth as inputs and the optimal radial distance of the corresponding situation as output.BP-ANN model is used to fit the sample data.90% of the sample data are randomly selected as training data,5% as validation data,and 5% as testing data.Through multiple trials,the network structure is set as a BP-ANN model with double hidden layers.The training function of the network is also set as the ‘trainbr’ function based on Bayesian regularization.The transfer function of each hidden layer selects the hyperbolic tangent S-type function ‘tansig’,and the transfer function of the output layer selects the linear function‘purelin’.MSE and correlation coefficient R are chosen as the network performance evaluation indexes.The maximum training epoch is 1000.The neurons number of each hidden layer is determined by trial-and-error process.The number of neurons in the first hidden layer is set to 50,and the number of neurons in the second hidden layer is set to 25.The final fitting effect index is shown in Fig.17.

Fig.17.Fitting effect index: (a) MSE;(b) All: R=0.98886.

Compared with the BP-ANN fitting effect index of Fig.15 for two input parameters.The fitting effect of BP-ANN model with four input parameters is basically the same.TheMSEof the training sample is 0.014219,and theRof all sample is 0.98886.However,as the number of input parameters increases,more sample data are needed to build the BP-ANN model.The structure of the network becomes more complex,and subsequently more tests are required to construct a suitable BP-ANN model.

For the BP-ANN model constructed with four input parameters,four attack directions are randomly generated to test the fuzewarhead coordination effect.The attack directions are given in the form of two-tuple (Q,A),respectively 1-(40°,190°),2-(120°,200°),3-(60°,140°),4-(30°,130°).The cloud maps of damage probability in these four directions are obtained through simulation,as shown in Fig.18.Under these four attack directions,the sample points where the target is detected have a high damage probability.This indicates that method 1 can still have a great effect of fuze-warhead coordination after adding the attack direction into the input parameters,which expands the applicability of the method.

Fig.18.Cloud map of damage probability under different strike directions: (a) 1-(40,190);(b) 2-(120,200);(c) 3-(60,140);(d) 4-(30,130).

4.3. Comparison of 4 fuze-warhead coordination methods

In order to verify the effectiveness of method 1,the BP-ANN model established in Section 4.1 and the damage probability calculation method in Section 3.3 are used as the basis.And the Monte Carlo simulation test is used to analyze the probability of the missile damaging the aircraft along the four strike directions shown in Fig.12.The aim point is selected as the center of the body,and the simulation times are 500.Finally,the cloud map of damage probability in 4 directions when CEP is 5 is shown in Fig.19.

Fig.19.Cloud map of damage probability when CEP=5: (a) Attack direction 1;(b) Attack direction 2;(c) Attack direction 3;(d) Attack direction 4.

The damage probability results of the missile striking the aircraft in these four directions are shown in Table 4.It can be seen that the damage probability of each attack direction has little difference when the CEP value is given.However,the damage probability decreases with the increase of CEP when the direction is given.As can be seen from Fig.19,for the sample points where the aircraft is detected,the damage probability is relatively high (in red).This is because BP-ANN model can give the optimal radial distance of the detonation point.For the sample points where the aircraft is not detected,it is considered that the missile misses the aircraft and the damage probability is 0 (in yellow).

Table 4 Damage probability results of method 1.

When CEP=3 and CEP=8,similar results are shown in Fig.20.The change in the damage probability is mainly caused by the number of sample points where the aircraft is not detected.For the sample points where the aircraft is detected,method 1 can control the missile to detonate at the optimal explosion point,so that the damage probability is relatively high.This indicates that method 1 proposed in this paper has a good fuze-warhead coordination effect.

Fig.20.Cloud map of damage probability in typical direction when CEP=3,8.(a) CEP=3,direction 2;(b) CEP=8,direction 4.

Aiming at the terminal intersection process of missile and aircraft,Monte Carlo simulation tests are respectively created by using the other three fuze-warhead coordination methods.The single-shot damage probability under the attack directions shown in Fig.12 is calculated.These damage probability results are compared with the damage probability obtained by method 1.According to Section 3.2,the simulation parameters of these methods are set respectively.For method 2,the standard deviation σxof the detonation point along theXr-axis is set to 2 m,and the mathematical expectationmxis set to -8,-6,-4,-2 m,respectively.For method 3,the fixed delay-time τ2is set to 5,8,10 and 12 ms respectively.Method 4 has no additional parameter settings.Other intersection parameters and simulation parameters of these three methods are shown in Table 1.The final damage probability results are shown in Table 5.

Table 5 Damage probability of different fuze-warhead coordination methods(method 2-4).

According to the damage probability data in Tables 4 and 5,the damage probability curves under different fuze-warhead coordination methods are obtained,as shown in Fig.21.

Fig.21.Damage probability curve under different fuze-warhead coordination method: (a)Attack Direction 1;(b)Attack Direction 2;(c)Attack Direction 3;(d) Attack Direction 4.

It can be seen in Fig.21 that under a given attack direction,the maximum damage probability corresponding to each method gradually decreases with the increase of missile CEP.The maximum damage probability of method 2 and method 3 under a given CEP in a given attack direction is very close,and the damage probability of method 1 and method 4 is very close.Method 1 constructs BP-ANN model based on target damage results,and then uses it to adaptively determine the initiation delay.Therefore,each sample point can detonate at the optimal detonation position,which ensures that each sample point has a high damage probability.

As can be seen in Fig.21,under a given CEP,when the mathematical expectationmxof the detonation point position dispersion is the same,the damage probability of method 2 under different attack directions is different.However,the optimal damage probability corresponding to the fuze-warhead coordination process is not much different,and they are all obtained whenmx=-2 m.With a given CEP,the optimal fixed delay-time corresponding to the maximum damage probability of method 3 will change with the change of attack direction.In addition,under the same attack direction,the optimal fixed delay-time corresponding to different CEP may also be different.When the aim point is the center of the body,method 4 has a higher damage probability in each attack direction under different CEP.

The maximum damage probability obtained by methods 1 and 2 is the overall optimal,which cannot make each sample point have a high damage probability.Method 1 and method 4 can make each sample point that detects the target have a high damage probability.Therefore,under the same conditions,the damage probability obtained by methods 1 and 4 is higher than that of methods 2 and 3.As shown in Fig.22.

Fig.22.Cloud map of damage probability under different methods（CEP5-attack direction 3）: (a) Method 1;(b) Method 2;(c) Method 3;(d) Method 4.

The fuze-warhead coordination effect of methods 1 and 4 is relatively better,and a simulation example is set to further study the details of these two methods.The intersection direction is the attack direction 1,the miss distance is 5 m,and the miss azimuth is 270°.Finally,the location where fragments hit the aircraft is shown in Fig.23.The delay time obtained by the two methods is different,resulting in different positions of fragments hitting the aircraft,but they can both cause the aircraft to be damaged.This is because the radial distance that can cause aircraft damage under a given miss error is an interval,as shown in Fig.13(a).Although method 1 and method 4 have different delay time,the positions of the detonation points are both within the damage interval.

Fig.23.The hit position of the adaptive delay method: (a) Method 1;(b) Method 4.

Method 1 uses Eqs.(9)and(12)to determine the delay time,and the detonation point is located in the center of the damage interval.Even if there is a certain deviation in the position of the detonation point during the actual engagement,as long as the position of the detonation point does not deviate from the damage interval,the aircraft can still be damaged.

To analyze the complexity of the constructed BP-ANN model to predict the initiation delay,1000 sets of miss distance and miss azimuth are randomly generated for the attack direction 3 shown in Fig.12.The time taken to calculate the fuze initiation delay is used as the basis for assessing the complexity of the algorithm.The constructed BP-ANN model calculates the initiation delay for these 1000 sample points and the computational time is calculated.After 100 independent runs,the average of the computation time is taken as the computation time of these 1000 sample points.The average computation time for these 1000 sample points is 35.381 ms,and the average computation time for 1 sample point is 3.538e-2 ms.As a comparison,the average computation time for these 1000 sample points using method 4 is calculated to be 2.19e-2 ms,and the average time of 1 sample point is 2.19e-5 ms.

In terms of the calculation efficiency of the algorithm,both method 4 and method 1 take far less than 1 ms for a single calculation,and the calculation efficiency of method 4 is much higher than that of method 1.However,method 4 simplifies the relative dispersion direction of fragments in the central dispersion zone when adaptively calculating the initiation delay.When the aircraft velocity has a great influence on the relative dispersion direction of fragments,the calculation accuracy of method 4 will decrease.The damage probability calculation of method 1 considers the general scenario of missile-target intersection,and the BP-ANN model is constructed using the offline damage probability results.The fuze initiation delay can be quickly determined according to the intersection parameters,which supports online decision-making under given engagement conditions.

4.4. Influence analysis of aim point position

According to the analysis in Section 4.3,when the aim point is set as the center of the body,method 1 and method 4 have a great fuze-coordination effect,and it can be seen from Tables 4 and 5 that the damage probability of method 1 and method 4 under the given four attack directions is relatively close.To study the influence of aim point position on method 1 and method 4,Taking the attack direction 1 in Fig.12 as an example,the missile CEP is 5 m.The nose and the tail on the body are selected as the aim points,and the position diagram of the aim point is shown in Fig.24.

Fig.24.Sketch of aim point position.

Taking the nose and the tail as the aim points,the damage probability cloud map with CEP=5 is calculated by method 1 and method 4.The results are shown in Figs.25 and 26.

As can be seen in Fig.25,for the sample points where the target is detected,method 1 can give a suitable detonation point position.Whether the nose or the tail is selected as the aim point,the damage probability of the sample point where the target is detected is relatively large.When the aim point is the nose,the damage probability of the aircraft calculated by method 1 is 0.651184,and when the aim point is the tail,it is 0.783114.

Fig.25.Damage probability cloud map at a given aim point with method 1: (a) Aim point: Nose;(b) Aim point: Tail.

As can be seen in Fig.26,when the aim point is the nose,the damage probability of some sample points where the target is detected is low.As a result,the aircraft damage probability obtained by method 4 at this aim point is low,which is 0.480471.This indicates that when the aim point is the nose,method 4 cannot ensure that each sample point where the aircraft is detected has a great fuze-warhead coordination effect.When the aim point is the tail,it can be seen that the damage probability corresponding to the sample point where the aircraft is detected is relatively high.This indicates that the fuze-warhead coordination effect of method 4 is good at this aim point,and the corresponding damage probability is 0.779413.The reason for these results is that the nose is where the mission system is located,which is low vulnerability,and the probability of aircraft damage caused by mission system damage is relatively low,and the tail is where the engine system is located,which is highly vulnerability,and the probability of aircraft damage caused by engine system damage is relatively high.

Fig.26.Damage probability cloud map at a given aim point with method 4: (a) Aim point: Nose;(b) Aim point: Tail.

Based on the above analysis and the damage probability results when the aim point is the center of the aircraft in Section 4.3,it can be seen that method 4 has certain requirements for the position of the aim point.When the vulnerable center or the critical component with high vulnerability is selected as the aim point,the fuzewarhead coordination effect of method 4 is better.However,method 1 is less affected by the position of the aim point,because it is based on the damage results of different miss errors.On the premise that the fuze can detect the target,method 1 can give the appropriate detonation point position,so that the damage probability of the aircraft is high.

5.Conclusions

(1) For method 1,when constructing BP-ANN model,if the miss error is the input parameter,the correlation coefficient of each attack direction is 0.99.If the miss error and attack direction are the input parameters,it is necessary to increase the number of hidden layers and neurons in the hidden layer to make the correlation coefficient of the model reach the same value.

(2) When the aim point is the body center,the single-shot damage probability calculated by methods 2 and 3 is statistically optimal,which does not guarantee that the damage probability of each sample point is relatively high.Method 1 and method 4 use the intersection parameters to adaptively determine the detonating position of sample points,which has high fuze-warhead coordination efficiency.

(3) Compared with method 4,the proposed method 1 can ensure that the fragments fall in the critical area of the aircraft and is less affected by the position of the aim point.In addition,method 1 uses the constructed BP-ANN model to realize the fast calculation of fuze initiation delay,which can guide the fuze-warhead coordination process in real time according to the intersection parameters.

Since the accuracy of the fuze-warhead coordination process simulation highly depends on the model,developing the BP-ANN model is an interesting topic.Future work can be carried out on adding inputs to the model and optimization of sample data.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.