Interception probability simulation and analysis of salvo of two electromagnetic coil launched anti-torpedo torpedoes

Bo-qi Wu ,Gung-sen Wng ,Xio-cun Gun ,*,Bio Wu

a National Key Laboratory of Science and Technology on Vessel Integrated Power System,Naval University of Engineering,Wuhan,430033,China

b Jiangsu Automation Research Institute,Lianyungang,222006,China

Keywords:Interception probability Electromagnetic coil launch Anti-torpedo torpedo Adaptive PSO

ABSTRACT Electromagnetic coil launch is an important branch of electromagnetic launch(EML)technology,which is suitable for launching anti-torpedo torpedo(ATT).This paper focuses on the EML parameters and the interception probability of the EML two ATTs salvo.Based on the launching model of a multi-stage coil launcher,the trajectory model of the ATT and the attacking torpedo,a calculation method for the EML two ATTs salvo parameters is proposed,with the conditions of capture and interception given reasonably.An adaptive particle swarm optimization(APSO)algorithm is proposed to calculate the optimal launching parameters,by designing the adaptive inertia weight and time-varying study factors.According to the analysis of the simulation with Monte Carlo method,EML will improve the interception probability effectively,and the interception probability is affected by the launching range.The results demonstrate good performance of the proposed APSO in calculating EML parameters for the two ATTs salvo in certain combat situation.Implications of these results are particular regarding the command and decision in the anti-torpedo combat.

1.Introduction

Electromagnetic launch(EML)technology is a new launching technology that converts electric energy into kinetic energy of the projectile,which is characterized by its controllability,high energy utilization rate,and the large initial velocity of the projectile[1].According to the preset control strategy,the projectile is accelerated by the pulsed power supplies charging,to achieve an ultrahigh speed at the muzzle[2,3].

The anti-torpedo torpedo(ATT)is a kind of hard kill method of torpedo defense for the surface warships,and it is regarded as the last barrier to intercept the attacking torpedo.The intercept modes can be classified into brief intercept,advanced point direct intercept,head-on intercept et al.With any mode,the search coverage is limited with one single ATT.Two ATTs salvo can effectively broaden the searching area and reduce the probability of attacking torpedo escaping.The salvo can be divided into sector salvo and parallel course salvo[4].With the gradual improvement of the torpedo weapon performance,the sailing parallelism can satisfy the tactical requirements,and parallel course salvo is widely used[5].The key problem of parallel course salvo is to calculate the launching parameters of the ATTs.The conventional parameters calculation refers to the geometric method.In Ref.[6],the mathematical model of wake flow homing torpedo salvo was established,but the proposed calculation method is not suit for the acoustic homing ATT.The calculation method of launching parameters is simplified in Ref.[7]by simplifying the turning process of the secondary angle of salvo.Transferring the random error of the target position and movement to the heading,an analytical model is established,which can be used to calculate the capture probability of parallel course salvo of torpedoes[8].However,there is no study of launching parameters calculation for the two ATTs EML salvo to date,and the impact of the EML on the interception probability is not quantitatively analyzed.At the same time,the spacing distance of the two ATTs is mainly decided based on experience according to the effective homing distance conventionally,and the calculated launching parameters may be not the optimal in various situations.

We aim to propose a suitable calculation method and evaluate the interception probability of EML two ATTs salvo against an attacking torpedo.A simulation with the Monte Carlo method is carried out in the settings of a typical anti-torpedo combat situation.The multi-stage electromagnetic coil launching model,the trajectory models of ATT and attacking torpedo are established respectively.The guidance mode of the attacking torpedo is assumed to be wire and acoustic homing when establishing the trajectory model.A calculation method for EML two ATTs salvo parameters is proposed based on the geometric method for conventional launch.To optimize the launching parameters of the salvo of two ATTs,including the launching range,parallel spacing distance,and the searching heading,an adaptive PSO algorithm is proposed by designing the inertia coefficient and study factors based on the standard PSO,to obtain the optimal interception probability.Through the numerical simulation,the influence of the EML method and the launching range on interception probability is analyzed,and compared with the geometric method,the improvement of the proposed APSO algorithm is verified.The arrangement of this study is shown in Fig.1.

2.Mathematical model

Referring to the development characteristics of self-defense weapons and equipment,and focusing on the requirements of the caliber and weight for launching capability[9],multi-stage electromagnetic coil launching can be adopted to replace the conventional launching method.

2.1.Multi-stage EML model

Multi-stage electromagnetic induction coil launcher is mainly composed of power supply,drive coil,circuit switch,gun barrel,and other parts.It is essentially a coaxial induction motor.With multiple independent capacitor group as power supply,power supply at each stage respectively supply the coil at the same stage.When the power supply at each stage discharges in turn,the current goes through the drive coil in turn,with the superposition of strong magnetic field generated by drive coil in the instantaneous change of magnetic field effect on the armature[10].With the induced eddy current of the armature in the transient magnetic field,the armature is pushed to the muzzle direction by Lorentz force.The armature is accelerated to the muzzle under the action of stepwise thrust by independent switches that control the circuits to be switched on in turn[11].The principle of which can be described by the circuit equation,thrust equation,and motion equation[12,13].

The Equivalent circuit of multi-stage coil launcher is shown as Fig.2.The circuit equation of thekth stage coil can be given as

Fig.1.The arrangement of this study.

Fig.2.Equivalent circuit of a multi-stage EM coil launcher.

where the meaning of each variable is shown in Table 1.

Table 1Parameters of the equivalent circuit.

Thrust equation can be expressed as

whereFk(t)is the force exerted on the armature at timet,andWkis the energy storage of thekth stage coil under ideal conditions,which is only related to the mutual inductance gradient between thekth stage coil and the armature dWk/dz.

During the launch process,ATT motion equation can be expressed as

whereaktandvktrepresent the acceleration and velocity of ATT respectively at timet.

2.2.ATT salvo trajectory model

The warship can detect the attacking torpedoes with the sonar system,obtaining the distance and bearing of the target,of which the accuracy is determined by the error level of the sonar system.After filtering the detected data,the speed and heading of the attacking torpedo can be calculated,and the launching parameters can be obtained.ATT executes its navigation actions after entering the water according to parameter command.

The calculation method of the launching parameters is related to the interception mode,which includes the brief interception,direct interception,and head-on interception as mentioned[14].Direct interception,its underwater trajectory can be divided into searching segment and tracing segment.

The searching trajectory can be described as

where,(XAtt,YAtt)represents the position of the ATT at timet;CAtrepresents the searching heading,which is calculated by the fire control device;VAtrepresents the speed of ATT;ζ is the advanced angle of the ATT[15,16].

The tracing trajectory can be described as

where,CAtis decided by the guidance law of the ATT after capturing the target.

For a single ATT,an attacking torpedo is considered to be captured successfully when the target enters the homing sector of the ATT.The following conditions should be met

whereDT-At_irepresents the distance between the attacking torpedo and ATT;QT_i=CAt_i-BT-At_irepresents the bearing of the attacking torpedo detected by the ATT.

If the distance between the attacking torpedo and ATT is less than the explosion damage radius of ATT,i.e.DT-At_i＜Rdes_i,ATT is considered to have completed the mission of intercepting the attacking torpedo.

2.3.Attacking torpedo trajectory model

When building the trajectory model,its motion in the vertical direction is also ignored.The trajectory of the attacking torpedo can be described as

where,(XT(t),YT(t))represents the coordinates of the attacking torpedo at timet;VTandCTrepresent the speed and heading of the attacking torpedo respectively.

Suppose that the attacking torpedo is a wire-guided type torpedo,which will switch to acoustic homing mode to capture and trace the warship after satisfying certain conditions.Supposing that the attacking torpedo traces the position of the warship with wireguided guidance,therefore the speed and the course of the warship will affect the searching course and heading velocity of the attacking torpedo.

3.Optimization for ATT salvo parameters based on APSO

3.1.ATT salvo parameters

It can be seen from(1-3)that,for the certain ATT,the optimal control timing sequence is specified,and its muzzle velocityvmis mainly determined by the charging voltageU0,which can be described as

Fig.3.The ATTs separate in the air trajectory,and turn into parallel searching after diving.

Under certain environmental conditions,the ATT flight trajectory is only decided byvm.The ATT horizontal launching range can be changed by adjusting the launching pitch angle

According to the analysis,the launching range is controlled jointly by the charging voltage and launching pitch angle.In the application,the charging voltage is maximized to increase the initial speed of the ATT by default,and to reduce the flight time in the air.And the horizontal launching range of the ATT is mainly determined by θ.At the same time,to capture the target,ATT should receive the searching headingCAtcommand before launching from the fire control device.And the launching azimuth angles can be obtained based onCAtand the overlap coefficient of the ATTs homing sectors.

The launching parameters of the EML two ATTs salvo include the launching azimuth angle,pitch angle,initial voltage,and the underwater searching course of the trajectory.When launching ATT,the air trajectory can be used to realize the spread,so that the two ATTs are parallel when searching for targets in the water,and ATT"s homing capability can be utilized as adequately as possible,as shown in Fig.3.Thus,the parallel spacing distance should be suitable to guarantee the searching area and avoid mutual guidance.

Define the overlap coefficient of the homing sector as

As Fig.3 shows,ρ∈(0,1).ddenotes the transverse projection overlapping distance between the homing sectors of the ATTs;DAtdenotes the parallel spacing distance between the two ATTs,which can be expressed

whereLldenotes the launching range in the flight,which belongs to[Llmin,Llmax];δ denotes the spread angle from the searching course.

In the conventional geometric calculation,DAtis set as a fixed value according to the experience.The searching width is enough and targets will not be omitted between the homing sectors of the two ATTs,on the premise of no mutual guide between each other.

Thus,the opening angle of the two ATTs can be calculated

where,Rguideis the effective homing distance,and ?guideis the effective angle.

From(4-6),it can be seen that with the larger horizontal air range segment of ATTs,δ is smaller,of which the variation trend is consistent with that described in Fig.3 and the spacing distance can be adjusted by ρ,the launching azimuth angles of the two ATTs are obtained

whereCAtis the main search course of the two ATTs after diving in the water.

As the anti-torpedo combat process shows,the control of the EML ATTs salvo refers to the diving positions,the searching course and the interval spacing distance of the two ATTs.According to the given parameter calculation,the control is realized by adjusting the EML ranges,the searching headings and the overlap coefficient.Thus,the EML parameter set is(Ll,ρ,CAt).To simplify the research of the parameters,the influence of theLlcan be considered firstly by quantitative comparison and analysis.

3.2.Adaptive PSO

To obtain the optimal launching parameters of the salvo of the EML two ATTs salvo,a set of launching parameters is defined as a particle,and the PSO algorithm is adopted to achieve the best interception probability.According to the former research,the capture probability of a single EM launched ATT can be described as an integral analytic model which obeys normal distribution[17].As the analytic model is a unimodal function,standard PSO is suitable to search the optimal launching parameters.For the EML ATTs salvo,the interception probability is not unimodal,due to the overlap area of the two ATTs during parallel searching trajectory.Thus,the launching parameters optimization of the salvo of ATTs is more complicated than the single ATT.However,each parameter has specific range according to the combat situation,and the PSO is applicable with proper improvement.

The core thought of PSO algorithm is the bionic evolution.The positions of the particles are controlled in the searching area by updating the speeds to obtain the best fitness[18].As PSO is an iterative algorithm based on swarm,it might be confined to the local optimal solution prematurely and failed to find the global optimal value when solving multi-peak optimization problems[19].

In this study,particleiis defined as(CAt,ρ),and the dimension of the particled=2.The update law of the velocity and position of thejth dimension at generation t can be expressed as

Constant coefficients were used in the standard PSO.According to the existing studies,ω=0.7,c1=c2=1.4 can achieve an acceptable effect in most problems,but the constant coefficients cannot achieve an optimal effect in various application scenarios[20,21],and it is usually associated with premature convergence.To avoid the particles falling into the local optimal position prematurely,the adaptive design of inertia coefficient ω and the timevarying coefficient design of study factorsc1andc2are carried out.

Taking the ratio of the optimal fitness of the swarm and the individual average as a reference,the adaptive inertia coefficient is adopted,and the study factors that vary with the generations are designed as

whereNrepresents the population scale andTis the generations of iteration.As shown in(10),during the initial iterationvaries a lot,and ω is large to promote the convergence of the swarm.In the later iterationgets close to theand ω is smaller.The particles has a strong local search ability to find the optimal solution.

c1andc2vary in the range of[c1min,c1max]and[c2min,c2max]respectively.c1decreases fromc1maxtoc1min,andc2increases fromc2mintoc2maxwhen updating.In this way,particles can have a strong global search ability in the later stage of the iteration,so as to avoid falling into the local solution.

Fig.4.Flowchart of APSO algorithm.

Fig.5.Simulation process of the EML ATTs salvo against the attacking torpedo.

3.3.Framework of APSO

The launching parameters are optimized by the proposed APSO,as(9)and(10)shown.The particle is defined asx=(ρ,CAt),and the interception probability is taken as the fitness.The process of APSO implementation are as follows:

Step 1.Initialization:swarm scaleN=20;Maximum generationsT=50;Study factors limitsc1min=c2min=1,c1max=c2max=3;ATT parallel searching headingCAt∈(Cw,QT);The overlap coefficient ρ∈(0,1);vCmin=-5,vCmax=5,vρmin=-0.05,vρmax=0.05;The initial position of the particle(CAt0,ρ0)is generated randomly in the range.

Step 2.For the initialized particle swarm,the fitness valueof particleiis calculated.The optimal position of particleican be findpbesti=The optimal position of the swarm is expressed bygbest,s.t.f(gbest)=max(f(pbesti)).

Step 3.Update the particles according to(9);The swarm of

second generation is obtained and the fitnessof each particle is calculated.

Step 4.Compare the fitness valueof the second generation with the value ofpbesti.If f(xi2)＞f(pbesti),let

Step 5.Compare the fitness value of the second generation with

Step 6.Repeat step 3 to 5 until the finish condition is met,which is generally set as generation reachingTor f(gbest)reaching the expectation;

7.Outputgbest.

According to the process above,its flow chart can be described as shown in Fig.4.

4.Numerical experiments and analysis

4.1.Simulation method and settings

The simulation was carried out by the means of Monte Carlo method to calculate the interception probability of EML ATTs salvo.With conventional launching method,the two ATTs expands a certain distance after diving into the water and turns to the searching heading for parallel searching.The simulation was carried out according to the anti-torpedo operation process,as shown in Fig.5.The proposed APSO method and geometric conventional method was utilized in the launching parameters calculation section respectively.

The simulation was conducted for 2000 times in each situation,and the interception probability was calculated according to the times of capture and interception.Table 2 shows the parameters setting of the combat scenario and simulation.

4.2.Influence of EML range

To analyze the influence of the launch method and EML launching rangeLl,the error level of the platform detection(Derror,Berror)is set as(5%,5°),and the other parameters are as shown in Table 2.The interception probabilities with different alarming distances and EML launching ranges are shown as Fig.6.

Table 2Parameters setting of the simulation.

With the launching range of 1200 m,the interception probability with the two launching methods is shown in Fig.6(a).When the initial alarming distanceDTvaries from 3 km to 8 km,there existsPEML＞PCL.Furtherly,when the alarming distanceDTis 3 km,the interception probability is relatively high,withPEMLreaching 0.93 andPCLreaching 0.91.WhenDTgets 8 km,the interception probability is lower than the other situations,PEMLandPCLdecreases to 0.87 and 0.83 respectively.The interception probability decreases overall with an increasingDT,and the difference betweenPEMLandPCLincreases simultaneously.Adopting the EML technology,part of the underwater trajectory under the traditional launching method is replaced by the flight section,and ATT approach to the incoming torpedo rapidly.Thus,the position dispersion of the incoming torpedo is reduced,and the deviation of the predicted intercept point,caused by the calculation error ofCT,is also reduced.There existsPEML＞PCL.

Fig.6.Interception probability comparison.

Fig.7.(ρ,CAt)and interception probability obtained from the APSO.

Fig.8.Interception probability with parameters obtained from conventional and APSO methods.

The launching rangeLlcan be controlled from 600 m to 1200 m with interval of 100 m by adjusting the voltage of the power supplies and the launching pitch angle,and the simulation results ofPEMLare shown in Fig.6(b).When the alarming distanceDTremains constant,PEMLincreases with a largerLl.However,the improvement amplitude decreases at the same time.PEMLincrease more obviously with a largerDT.WhenDTis 3 km,PEMLgets higher than the other situations,reaching 0.93.Meanwhile,PEMLgets 0.90 whenDTis 4 km-7 km.WhereasPEMLgets 0.85-0.88 whenDTis 8 km.

4.3.Improvement effect of APSO

When analyzing ρ andCAt,the detection error level(Derror,Berror)of the platform is divided into six levels as(5%:5%:30%,5°:5°:30°).The alarming distanceDT=8000 m,and the EML range is set as the maximumLl=Llmax.With each error level,the proposed APSO algorithm is used to optimize the launching parameters,and the results are shown in Fig.7.

As Fig.7 shows,the optimized(ρ,CAt)and interception probabilityPAPSOare different under different error levels.Fig.7(a)shows that the overlap coefficient ρ decreases with the increasing error.When the error level is I,the optimization result of ρ is the largest,which is 0.670.When the error level is VI,ρ gets 0.257,which is the smallest.As Fig.7(b)illustrates,there is no obvious variation trend of the searching headingCAtwith the changing error level,and the optimized results are of(59.85°,62.26°)for each error range.PAPSOdecreases with the increase of the error as shown in Fig.7(c).With error level I,PAPSOis 0.883,and the value is 0.757 with level VI.The result reveals that the APSO algorithm can optimize(ρ,CAt)parameters by taking interception probability as fitness at each level.ρ andCAtis updated under the proposed APSO algorithm respectively,and the higher interception probability can be obtained after each update.At the same time,the algorithm is capable of searching for optimization at the later stage of iteration.The curve of(20%,20°)shows that an update occurs at the 47th generation.

With the same combat settings,the conventional geometric method is used to calculate the EML parameters.And the interception probability was obtained through simulation.A results comparison with the APSO method is shown in Fig.8.

Through the comparison at each error level,the interception probability obtained from the APSO algorithm is higher than that from the conventional geometric method.With a larger error,the interception probability improves more significantly.The interception probability is improved slightly at level I,which is 0.044%;the interception probability is improved by nearly 1%at level II and III;at level IV and V,the improvement is about 1.5%;the interception probability increases more than 1.8% at level VI.The comparison illustrated that the EML parameters optimized from the proposed APSO algorithm can effectively improve the interception probability.

5.Discussion

On account of the wire-guided method of the attacking torpedo,the headingCTis affected by the movement of the warship,especially in the case that the bearing of the torpedo verges on being perpendicular to the warship"s.Significantly,theCTchanges less when the ATTs approach the torpedo quickly when EML method is adopted,and the probability of the torpedo escaping from the searching of the ATT decreases,thus there exitsPEML＞PCL.And the probability of the torpedo"s escaping increases whenCTchanges more with a largerD.which can be suppressed by the EML method.On the other hand,the absolute error,representing the deviation between the detected position and the actual position,increases with a largerDT.Large error of the original data will affect the processed results of the fire control device,which could lead the launching azimuth angle and the heading of ATT deviates from the ideal searching heading,and the ATTs fail to capture the target as a result.

According to the previous analysis,Llis capable of decreasing the probability of the wire-guided torpedo to escape from the ATTs’searching,and the EML ATTs will approach the torpedo more quickly with a largerLl,accompanied byCTchanging less.PEMLincreases with a largerLlas a result.PEMLdeviates within 0.05 whenDTvaries from 3 km to 7k m with a differentLl,however,with an obvious decrease whenDTis 8 km using a differentLl.It is significant for the combat command in which the sonar detected a target with a distance of 8 km,the interception probability can be improved distinctly when the ATTs are launched until the target sails to the distance of 7 km to the warship.

As Fig.7 shows,the optimization result of ρ from APSO decreases with the increasing error.As the location dispersion of the target is larger when the detection error is larger,to achieve coverage of target location as possible,the spacing distance between the ATTs in the searching trajectory should expand,making full use of the homing sectors of the two ATTs,and ρ is less as a result.On the other hand,Fig.7(b)shows that at different error levels,the optimization results ofCAtbelong to a certain range of(59.85°,62.26°),which indicates that the detection error level has little influence onCAtunder a certain combat situation set.And theCAtcalculated by the conventional calculation is in the range,that is to say the APSO optimization results is close to the conventional calculation onCAt,with a deviation yet.On one hand,the adopted Monte Carlo method in the simulation is subject to the simulation times.Meanwhile,the conventional calculation method takes the geometric center of homing sector as reference.However,in different combat situation,take the fixed geometric center into calculation formula cannot guarantee the optimal interception probability.Thus,the results of the conventional calculation can not be regarded as the theoretical optimalCAt.

As the comparison shown in Fig.8,the APSO algorithm optimized interception probability,decreases with the increasing error.By analyzing the specific optimization results,the interception probability is mainly affected by ρ.For error II and III,the deviation between optimization result and experienced value is less than 0.05;the deviation is about 0.15 and 0.18 at error level IV and V respectively;however,the deviation of ρ is about 0.17 at level I.For the high interception probability before optimization,the improvement is not obvious.In practical application,the launching parameters can be optimized according to the error level of the sensor.

6.Conclusion

To study the interception probability of EML two ATTs salvo,the EML model,the ship motion model,the trajectory model of the attacking torpedo and ATT is established respectively.Based on the conventional launching method,a method of parameters calculation is proposed for the EML two ATTs salvo.And a kind of APSO algorithm is presented to improve the interception probability by optimizing the EML parameters.The Monte Carlo method is utilized in the numerical simulation and analysis of anti-torpedo combat.The main finding s are as followings:

a)EML technology can improve the interception probability of two ATTs salvo.The improvement is more obvious when target alarming distance is larger,and the interception probability achieves higher with a larger launching range.

b)The proposed APSO algorithm can effectively optimize the EML salvo parameters,and the obtained interception probability is higher than the conventional geometric calculation results at different error levels.The improvement is more obvious when the error is larger.

c)A higher interception probability can be obtained by adjusting the overlap coefficient at different error levels.Therefore,the overlap coefficient should be taken into consideration in the EML two ATTs salvo.

Declaration of competing interest

There are no conflicts of interest with other reports or authors.

Acknowledgements

The authors would like to acknowledge National Natural Science Foundation of China(Grant No.51777212).