A new method for evaluating the firing precision of multiple launch rocket system based on Bayesian theory

Yunfei Mio , Guoping Wng , Wei Tin

a College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu, China

b Institute of Launch Dynamics, Nanjing University of Science and Technology, Nanjing, Jiangsu, China

Keywords:Multiple launch rocket system Bayesian theory Simulation credibility Mixed prior distribution Firing precision

ABSTRACT How to effectively evaluate the firing precision of weapon equipment at low cost is one of the core contents of improving the test level of weapon system.A new method to evaluate the firing precision of the MLRS considering the credibility of simulation system based on Bayesian theory is proposed in this paper.First of all,a comprehensive index system for the credibility of the simulation system of the firing precision of the MLRS is constructed combined with the group analytic hierarchy process.A modified method for determining the comprehensive weight of the index is established to improve the rationality of the index weight coefficients.The Bayesian posterior estimation formula of firing precision considering prior information is derived in the form of mixed prior distribution, and the rationality of prior information used in estimation model is discussed quantitatively.With the simulation tests,the different evaluation methods are compared to validate the effectiveness of the proposed method.Finally, the experimental results show that the effectiveness of estimation method for firing precision is improved by more than 25%.

1.Introduction

At present, most of the estimation methods of firing precision test are based on the classical statistical theory.Only when the sample size is large enough, can the test results have high confidence.As a destructive test, the firing precision evaluation test of weapon system will consume a lot of manpower and material resources, and the cost is very high.Therefore, for the expensive weapon system,the classical statistical theory[1,2]cannot meet the requirements of low-cost firing precision test evaluation.In fact,for modern complex weapon systems, the contradiction between test cost and expenditure has become increasingly prominent.In this case, Bayes statistical inference method has been widely used [3].

In addition to establishing environmental simulation tests and improving the accuracy of measurement equipment,the advanced Bayes analysis method has been used to improve the precision test method.In 1984,the United States adopted the Bayesian method in the precision evaluation of the "Panxing II" missile [4].Since the 1970s,many domestic scholars have carried out various theoretical and applied research in the field of aircraft test analysis and identification by using Bayesian method.Zhang[5],Tang[6],Zhang[7],Zhang [8], Jin [9] and others have applied Bayesian method to the aerospace field,and have achieved rich research results.The multisource information Bayes fusion technology, Bayes sequential posterior weighted test method, Bayes sequential truncation method and other aircraft precision test and analysis methods are established,which are successfully applied to missile precision and reliability test design.Fu et al.put forward the overall inference method of missile hit accuracy, and realized the evaluation of missile firing accuracy when there is only one miss distance test data[10].Wang et al.discussed the acquisition and reasonable use of prior information in firing precision evaluation[11,12].Song et al.proposed the Bayes parameter self-assessment method for missile firing precision[13].By using the pre-test information as each stage for transmission, Wang et al.proposed an evaluation method combined with multi-stage Bayes information fusion for HWIL simulation [14].Ning et al.established a guidance accuracy evaluation method of field small sample test based on equivalent sample and real simulation fusion [15,16].However, in the field of conventional weapons, such as self-propelled artillery and multiple launch rocket system (MLRS), the test method based on the classical statistical theory is still used at present.The test level and evaluation effectiveness is insufficient.What’s more, the current methods for considering simulation information are mostly based on single impact points data, without considering the simulation credibility of the overall system,resulting in inaccurate estimation of equivalent samples during evaluation.

In Bayes statistical inference, the accuracy of estimated results usually depends on credibility of prior information [17].With the extensive use of simulation technology and its increasing credibility, simulation information is gradually becoming an important source of information in addition to the range flight test.The credibility evaluation methods of simulation models can generally be divided into subjective analysis methods and objective analysis methods.In terms of subjective analysis methods,Ahn et al.applied group decision technology and Delphi method to evaluate the credibility of simulation model [18].Jebeile et al.discussed the importance of empirical consistency in model validation [19], and also pointed out the limitations of subjective analysis methods.For objective analysis methods, Hauduc et al.and Crochemore et al.summarized various methods for consistency between simulation results and test data [20-22].Kwag et al.verified the complex simulation model composed of multiple sub-models by using Bayesian network method [23].Li et al.used Bayesian method to verify the model by introducing prior information [24].When the simulation credibility evaluation problem began to develop into a comprehensive evaluation problem of multiple indicators, the single evaluation method of simulation model is unable to meet the evaluation requirements.Comprehensive evaluation methods of simulation model credibility have also been gradually proposed,such as analytic hierarchy process [25], fuzzy comprehensive evaluation [26], similarity theory, Bayesian network [27], etc.Comprehensive evaluation methods can give full play to the advantages of qualitative analysis and quantitative analysis to obtain simulation credibility evaluation of complex systems.

In view of the above problems, a comprehensive evaluation index system model for the credibility of simulation system is proposed in this paper; an improved method for determining the comprehensive weight of the index is developed;with the form of mixed prior distribution,the Bayesian posterior estimation formula of firing precision considering prior information is deduced, and the rationality of prior information in precision evaluation is quantitatively discussed.Finally, the effectiveness of the proposed method is verified by the simulation test and firing precision test.

2.Credibility evaluation of simulation system

2.1.Credibility evaluation model of firing precision simulation system of the MLRS

According to the previous research, the MLRS firing precision simulation system mainly includes three main modules: vibration characteristics simulation module, launch dynamics simulation module, and flight dynamics simulation module.Therefore, based on the composition of the simulation system,this paper adopts the group analytic hierarchy process (GAHP) method to establish the credibility evaluation index model of the MLRS firing precision simulation system,as shown in Fig.1.The credibility of simulation system is divided into three main indicators, namely, the simulation credibility of vibration characteristics, the simulation credibility of launch dynamics, and the simulation credibility of flight dynamics.Then the main factors that affect the above indicators are determined according to the analysis results of the influencing factors.At the same time,combine with the existing test methods,the main indicators are decomposed into sub-indicators.The simulation credibility of vibration characteristics mainly includes the simulation credibility of natural frequency and vibration mode;the simulation credibility of launch dynamics mainly includes the simulation credibility of dynamics responses of launch device and rocket; the simulation credibility of flight dynamics mainly includes the simulation credibility of rocket flight speed and impact points of rocket.

Fig.1.The credibility evaluation index model of the MLRS firing precision simulation system.

2.2.The comprehensive weight of evaluation index

In GAHP, the determination of index weight is one of the most important content.The multiple judgment matrices of the indicator system are usually determined from different analysis methods or expert qualitative analysis.Then the more reasonable comprehensive weights of indicators are obtained through comprehensive analysis.

For multiple judgment matrices, the index weight is composed of three aspects: 1) the consistency level of the judgment matrix,which reflects the consistency relationship between the judgment matrices;2)the deviation of the individual of the judgment matrix from the group, which reflects the difference between the individual and the group of the judgment matrix; 3) the difference of judgment matrix caused by different methods or experts.The weight coefficients are determined from the above three aspects,respectively and then the comprehensive weight of the indicators is obtained by the convex combination method.For the consistency level of the judgment matrix and the deviation of individuals relative to the group, the weight coefficients can be calculated by the method in Ref.[25].However, there is no relevant research on the differences brought by different methods on the judgment matrix.Therefore, combined with Bayesian method, this paper proposes an improved method to determine the comprehensive weight of indicators from multiple judgment matrices.The specific implementation steps are as follows:

Assuming that there are indicators xi(i =1,2,…,n);the sources of judgment matrix are recorded as E = {e1,e2,…,em};evaluating the relative importance of ek(k=1,2,…,m) to indicators and constructing corresponding judgment matrix Ak=(a(k)ij)n×n;using eigenvalue method to check the consistency level.Under the condition of passing the inspection, the following analysis is carried out:

2.3.Numerical example

In order to verify the effectiveness of the above method,take the credibility evaluation of simulation system in Ref.[25] as an example for comparative analysis,and the judgment matrices given are as follows:

The eigenvalue method is used to check the consistency of the judgment matrix, and all CR values are less than 0.1, which meets the consistency requirements.Based on the above judgment matrices, the index weights determined by a single judgment matrix, by the method in Ref.[25] and by the improved weight method are compared,as shown in Table 1.The comparison resultsdenote that the index weight vector obtained by the proposed method is in a reasonable space in all dimensions, the CR value is the smallest, and the consistency is higher, which verifies the effectiveness of the method.

Table 1 Comparison of different index weight methods.

2.4.The comprehensive credibility of simulation system

After obtaining the comprehensive weight coefficients of the indicators of system, it is also necessary to analyze the consistency of the simulation results and test results of each indicator.The credibility of the simulation results of each module is obtained through static and dynamic data analysis, which is marked as Pi.Then the credibility of the simulation system can be computed by Eq.(11).

3.Estimation method for firing precision considering prior distribution

3.1.Mixed prior distribution of firing precision of the MLRS

Assume that the longitudinal and lateral impact points are independent of each other and obey normal distribution.First, take the longitudinal impact point as an example for discussion.Assuming that X ～N(μ,E2/(2ρ2)), the sample and the population are independently and identically distributed.With the parameters(μ,E2), the joint probability density of the subsample isE2).According to the principle of conjugate distribution, π(μ,E2)can be regarded as a normal-inverse gamma distribution,which is recorded as

Assuming that the priori sample size is n0,and there is X =(x1,x2,…,xn).Take the distribution parameter of inverse gamma as

Then the prior distribution is

where λ1=1-λ0and π0(μ,E2) is a normal-inverse gamma distribution, and its parameters are obtained from prior information;π1(μ,E2)is Jeffrey's non-information prior distribution,that is π1(μ,E2) = (1/E2)3/2.When λ0= 0, π(μ,E2) = π1(μ,E2); when λ0= 1,π(μ,E2) =π0(μ,E2).

3.2.Posterior distribution density

The above formula is recorded as

where

ω0and ω1are weighting coefficient, which represents the weight of prior information and sample information in the posterior estimation.

3.3.Posterior distribution and estimation of parameters

Based on the posterior distribution density of random variables(μ,E2), the posterior estimates can be obtained according to the posterior edge distribution of μ and E2.The posterior edge distribution of (μ,E2) under the conjugate prior condition

That is

Then the posterior expectation estimation of μ and E2are respectively

Eq.(27) is the Bayesian posterior estimation of longitudinal density, which is marked as ^Ex.Similarly, the Bayesian posterior estimation of lateral density is^Ez.According to the definition,when there is no systematic error, the Bayesian posterior estimation of firing precision is

The deduction of estimation method for firing precision considering prior distribution is shown in Fig.2.

4.Discussion on the rationality of mixed prior information

In the posterior estimation of firing precision, the weight of prior information depends on Eq.(21),which includes two aspects of simulation system credibility λ0and edge density ratio J.Fig.3 shows the change of weight coefficient with prior information credibility and edge density ratio.When the credibility is fixed,the weight coefficient decreases with the increase of the J value;When J value is fixed,the weight coefficient increases with the increase of credibility.

Fig.2.The deduction of estimation method for firing precision considering prior distribution.

Fig.3.The influence of credibility and edge density ratio on weighting coefficient.

According to Eq.(21), the edge density ratio includes prior sample size n0, field test sample size n1, prior sample information μ0, σ0and field test sample information μ1, σ1.The weighting coefficients of prior information are discussed as follows:

1) When n1is fixed and n1= 18, the change of edge density ratio and weighting coefficient with priori sample size is shown in Fig.4.With the increase of the priori sample size, J gradually decreases and ω0increases.When n0→∞, J and ω0converge to a certain constant,which reduces the influence of priori sample size on the posterior estimation of parameters,so as to avoid large priori sample size flooding the field sample information.When μ0=μ1,σ0= σ1, J = 1, there is ω0=λ0at n0= n1, that is, the weighting coefficient is equal to the credibility of the simulation information.At this time, the posterior estimation of firing precision is

Fig.4.The change of J and ω0 with n0 of priori samples information.

Fig.5.The change of J and ω0 with μ0 of priori samples information.

Fig.6.The change of J and ω0 with σ0 of priori samples information.

To sum up,in the case of considering the limit condition,when the prior sample information is completely inconsistent with the field sample informationthe prior weighting coefficient is taken as zero, and the mixed prior distribution become the non-information prior distribution,which effectively reduces the influence of prior information on the parameter posterior estimation.When the prior sample data and field test data are completely consistent, the weight coefficient is equal to the credibility of the prior information, which can effectively evaluate the prior information and test information.Therefore,the mixed prior distribution method can effectively use prior information and sample information for parameter estimation.

5.Simulation and test verification

5.1.Simulation verification

Assume that the overall distribution is N(0,0,E2x,E2z), the prior distribution is N(0,0,E2x0,E2z0), the simulation credibility is λ0, and define E0= (1 -λ0)E+ E.Extract test samples from N(0,0,E2x,E2z)and extract simulation samples from N(0,0,E2x0,E2z0).The relative error ECEPand mean square error MSECEPare used to verify the effectiveness of the method.The calculation equation is expressed as

where ︿CEP is the estimated value of CEP,M is the number of Monte Carlo tests.

Fig.7.Comparison results of different methods when λ0 = 0.2.

Fig.8.Comparison results of different methods when λ0 = 0.95.

In the simulation experiment,with σx=10,σz= 25,N0= 18,N1= 18, the real value Ex= 6.74, Ez=16.86 and CEP = 20.2.The relative errors of different estimation methods under different simulation credibility are compared.Fig.7 shows a case where the simulation credibility is low (λ0= 0.2), thus the consistency between the simulation data and the test data is poor, and the evaluation results should be more inclined to the test data.However,the traditional Bayesian method does not consider the simulation credibility,but only integrates the test data and the simulation data,resulting in the evaluation results deviating from reality.The relative error of the proposed method is closer to the real value,which can effectively prevent erroneous prior information from flooding the field test information.

As shown in Fig.8,when the simulation credibility is high(λ0=0.95), it can be seen that the proposed method is close to the traditional Bayesian method at this time.When the simulation results are completely consistent with the test results,this method is the same as the traditional Bayesian method, but in the actual process,there are always errors between the simulation results and the test results.Therefore, it is necessary to fully consider the credibility of simulation data.

Further, the MSE of each method was simulated, and the number of Monte Carlo tests was 10,000.The simulation results are listed in Table 2 and shows that when the simulation credibility is low, the error of the estimated value of the simulation sample is largest.The error of estimated value by using Bayesian method gradually decreases as the simulation credibility increases.When the simulation credibility is low(λ0≤0.6),the estimated results of the proposed method are less affected by the simulation data and are close to the test results.When the simulation credibility is high,the proposed method has the smallest error and higher accuracy.

5.2.Test verification

A typical test scheme for the firing precision of the MLRS is carried out to verify the proposed method.The specific steps are as follows:

(1) Conduct firing tests,and measure launch and flight test data of rocket by using photoelectric systems, radar, and highspeed cameras.

(2) Input the test conditions, including system structural parameters, meteorological parameters, etc., into the simulation system to obtain launch and flight simulation data of rocket, including natural frequency B1, vibration mode B2,swing angular velocity of pitching body B3 and rocket B4,impact points B5 and flight velocity B6.

(3) Obtain the simulation credibility of each sub-index B1-B6 by data analysis.

(4) Calculate the credibility of simulation system through the credibility evaluation model.When the credibility does not meet the requirements, check whether there are any abnormalities in the test.If no,check and modify the simulation model until the simulation credibility meets the requirements;

(5) Evaluate the firing precision of the MLRS through Eqs.(27)and (28).

Table 2 The MSE of different method.

(6) Conduct three groups firing tests to obtain three evaluation results,and verify the effectiveness of the proposed method by compared with classical method.

The test scene is shown in Fig.9.The distribution of impact points of three groups are given in Fig.10.Finally, combined with test data,the credibility of simulation system is computed through the credibility evaluation model,proposed in section 1,and there is P = 0.92.

Then the firing precision evaluation results by using classical method and Bayesian method are listed in Table 3.The classical method is the mean value of evaluation results of all groups,that is

Fig.9.The firing test scene.

Fig.10.The impact points of three groups.

Table 3 Comparison of evaluation results of firing precision of the MLRS.

The smaller the standard deviation, the higher the estimation accuracy of the estimate value, and the more effective the estimation method.The test results show that the effectiveness of the firing precision estimation with the proposed method is improved by more than 25%.

6.Conclusions

Based on Bayes theory, a new method to evaluate the firing precision of the MLRS considering the credibility of simulation system is proposed in this paper.With the idea of group analytic hierarchy process,a comprehensive index system of the credibility of the simulation system is constructed, and a comprehensive evaluation method of the simulation credibility is established.A modified method for determining the comprehensive weight of the index is developed, which improves the rationality of the index weight.Combined with the credibility of the simulation system, a Bayesian posterior estimation formula of firing precision considering prior information is derived in the form of mixed prior distribution, and the rationality of prior information in this model is discussed quantitatively.With the simulation test, the different evaluation methods are compared to verify the effectiveness of the proposed method.Finally, the experimental results of a typical firing precision test show that this method improves the effectiveness of firing precision estimation by more than 25%.

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.

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant Nos.11972193 and 92266201).