An experiment evaluating how the tiny mass eccentricities in spinstabilized projectiles affect the position of impact points

Chuan-lin Chen,Hui Xu,Chen-lei Huang,Zhong-xin Li,Zhi-lin Wu

School of Mechanical Engineering,Nanjing University of Science and Technology,Nanjing,Jiangsu,210094,China

Keywords:Tiny mass eccentricity Small-caliber projectile Bullet Artificial intelligence algorithm Global sensitivity analyses Precision trials ANFIS Sobol’s algorithm

ABSTRACT This study investigates and quantifies some possible sources affecting the position of impact points of small caliber spin-stabilized projectiles(such as 12.7 mm bullets).A comparative experiment utilizing the control variable method was designed to figure out the influence of tiny eccentric centroids on the projectiles.The study critically analyzes data obtained from characteristic parameter measurements and precision trials.It also combines Sobol’s algorithm with an artificial intelligence algorithm-Adaptive Neuro-Fuzzy Inference Systems(ANFIS)-in order to conduct global sensitivity analysis and determine which parameters were most influential.The results indicate that the impact points of projectiles with an entry angle of 0° deflected to the left to that of projectiles with an entry angle of 90°.The difference of the mean coordinates of impact points was about 12.61 cm at a target range of 200 m.Variance analysis indicated that the entry angle-i.e.the initial position of mass eccentricity-had a notable influence.After global sensitivity analysis,the significance of the effect of mass eccentricity was confirmed again and the most influential factors were determined to be the axial moment and transverse moment of inertia(Izz Iyy),the mass of a projectile(m),the distance between nose and center of mass along the symmetry axis for a projectile(Lm),and the eccentric distance of the centroid(Lr).The results imply that the control scheme by means of modifying mass center(moving mass or mass eccentricity)is promising for designing small-caliber spin-stabilized projectiles.

1.Introduction

Spin-stabilized projectiles have been widely used in rifles,grenade launchers,and artillery equipment since World War I.Today,modern military tactics increasingly rely on accuracy to improve first-round hit probability.These tactics have inspired efforts to improve the firing accuracy of spin projectiles,including research on muzzle flow fields[1],studies on the changing geometry of projectiles[2],and work on optimizing the internal mass distribution of projectiles[3].Most researchers have assumed that projectiles are symmetrical and lack mass eccentricity.However,tiny eccentricities are unavoidable characteristics of massproduced 12.7 mm projectiles.The effect of mass eccentricity was first published by F.W.Mann,who plugged rifle bullets to unbalance them deliberately and produce a desired trajectory deflection that he called“X-Error”[4].InModern Exterior Ballistics,Robert L.McCoy called this effect“l(fā)ateral throwoff”and derived equations to calculate this effect by taking a 20 mm steel cone-cylinder projectile with a drilled hole in the bottom as an example[5].However,these studies focused on projectiles that were intentionally unbalanced to make the effect of mass eccentricity perceptible,and only a few studies published the experimental data on 12.7 mm projectiles with tiny mass eccentricity caused by normal manufacture.These experimental data on 12.7 mm projectiles would be the first materials to study the effect of tiny mass eccentricity on trajectory,and also could be referred when one designed small caliber smart bullets that utilized mass eccentricity.

Mass eccentricity is an accepted control scheme which is designed to solve attitude control problems.D.Childs[6]designed a single moving mass to dampen attitude oscillation and counteract torque.To simplify satellites’internal structure,S.Chesi[7]presented an automatic satellite balancing system which featured three balancing masses set along the three orthogonal directions.F.Janssens[8,9]proposed a spring-mass system to avoid the phenomenon of nutation,which often occurs in the upper stage firings with spinning solid rocket engines.C.Li[10-12]proposed that a projectile with a single,internal moving mass that can move on the internal rail and with reaction jets,and a combination bank-to-turn control mode was proposed.The experimental and numerical analysis indicated control authority increases proportionally as the mass ratio increases and as the static margin decreases in magnitude.Although spin-stabilized projectiles are without propulsion and show weak maneuverability,many researchers have changed the position of centroids in an attempt to correct typical trajectory errors and increase projectiles’overall stability.With a partially restrained internal member(PRIM)deviating from the body axis,A.Hodapp’s study[13]provided a passive means for eliminating PRIM-induced instabilities.Hodapp also suggested that this method is ineffective for large-diameter,thin PRIM shapes.C.Murphy[14]studied the pitching and yawing frequency properties of spinning projectiles with two different configurations,in which the internal moving mass set moves linearly along the symmetry axis of projectile and in a circular motion around the symmetry axis.J.Rogers[15]proposed a 7-degree-of-freedom flight dynamic model to investigate the potential of an internal translating mass control mechanism.Rogers’s study found that control authority increases proportionally as a projectile’s spin rate increases and as the distance of the translating mass offset from the projectile shell’s center of mass increases.To improve firing accuracy,Rogers[16]applied this method to study the variable stability of a 155 mm projectile and let this projectile have high stability after launch to resist launch perturbations but decreased stability to obtain greater control authority in the remainder of flight.The results showed that this method efficiently reduces the dispersion of impact points at a range of 5000 m.

The above literature mainly focused on aircrafts and largecaliber munitions(e.g.155 mm)using eccentric centers of mass to change the trajectory or attitude of projectiles and shows the effectiveness of this method.However,while these studies can be informative,there is no evidence that these conclusions are also suitable for small-or medium-caliber projectiles(e.g.12.7 mm).If an experiment can determine and quantify the effect of tiny eccentric centroids on the position of the impact points of 12.7 mm projectiles,engineers will have more confidence in deciding whether to apply the control scheme-mass eccentricity-when designing a small-caliber small projectile.Furthermore,much of the research on the impact point dispersion characteristics of small-or medium-caliber munitions uses statistical methods[17,18].Also,efforts made to quantify the delivery error in smallcaliber weapon systems by investigating the error budget and its components have been continually divided into smaller components as researchers’understanding grew.A.D.Groves divided the error budget of flat-fire weapon systems into Ballistic Coefficient,Within-Lot Muzzle Velocity Variation,Cant Error,Range and Cross Wind Gustiness Aiming Error[19].Building upon Groves’s work,J.M.Weaver introduced the error budget of accessory equipment(e.g.Laser Range Finder and Stadia),and the performance of different projectiles in cross-wind as error components,respectively[20].Next,R.Von.Wahlde applied Jonathan’s work to assess the effectiveness of fire controls that were added to several sniper weapon systems[21];his efforts have been widely used to design target acquisition devices and aim compensation methods,since 1999[22-26].Even though these prior studies on error budget make researchers’evaluation of the accuracy of weapon systems more reliable,they are not suitable for an engineer who seeks to improve projectile design.The reason is that they regard the“Ammunition Dispersion”as random errors(unexplained errors)that are caused by variations between rounds such as mass,shape,and propellant[27].Moreover,owing to an overall lack of raw data on tiny mass eccentricity in 12.7 mm projectiles,there are seldom studies that indicate the particular characteristic parameters(including tiny eccentric centroids)of a 12.7 mm projectile,which are caused by variations in the manufacturing process,that are the most critical and influential in determining impact points.

Therefore,two questions arise.First,do the tiny eccentric centroids of 12.7 mm projectiles,which were caused in the normal manufacturing process,influence the position of impact pointsand if so,can this effect be quantified?Second,if the characteristic parameters(including tiny eccentric centroids)of 12.7 mm projectiles are recorded,can the factors that critically affect the position of impact points be determined?

This paper explores answers to both of these questions.Hence,based on studies of lateral throwoff[5],a comparative experiment to determine the effect of tiny mass eccentricity in 12.7 mm spinstabilized projectiles on the position of the impact points was designed.Also,extending the prior research on the error budget of weapon systems,we regarded the characteristic parameters of 12.7 mm projectiles as”Variable Bias Errors”[27].In order to study the influence of these characteristic parameters,and to enable engineers to adjust outputs toward expected values by fine-tuning influential parameters,we conducted a parameter global sensitivity analysis for projectiles’shooting processes,which are widely regarded as a complex highly nonlinear model with large uncertainties in the parameters[2,3,17,28].

The paper is divided as follows.Section 2 presents the test plan and setup for the measurement of characteristic parameters and the precision trials,and introduces the protocol followed during the experiments.Section 3 provides the method used in this study:a combination of Sobol’s algorithm with an artificial intelligence--Adaptive Neuro-Fuzzy Inference Systems(ANFIS)-to conduct the global sensitivity analysis,which used the data obtained in the trials mentioned in Section 2.Section 4 focuses on the results obtained in the trials and global sensitivity analysis.Question one was studied using an analysis of variance(ANOVA)to investigate the position of every impact point for two specific entry angle groups Question two was answered with global sensitivity indices by analyzing global sensitivity parameters.Section 5 states our conclusions.

2.Experiments on eccentric projectiles

An experimental analysis was designed to determine whether the eccentric centroid of a 12.7 mm projectile influences the position of impact point,and whether this influence can be quantified.

We measured the characteristic parameters associated with each projectile.These included:the mass of a projectile(m),the distance between its nose and center of mass along its symmetry axis(Lm),the eccentric distance of the centroid to the symmetry axis in any plane perpendicular to that axis(Lr),and the eccentric angle(α),the axial moment of inertia about the symmetry axis(Izz),and the transverse moment of inertia(Iyy).The eccentric centroid is determined byLrwithαassociated with our test angle of 0°at the testing coordinate system of the testing platform for the eccentric centroid.However,theLrwas obtained in the measurements and indexed by the sample number,theαfunctioned as a control variable in the shooting experiment by letting the entry angle(β)was 0°or 90°.The characteristic parameters are shown in Fig.1,below.

As shown in Fig.2,the main experimental procedures were given with the purpose of each component of the experiment.The detail process will be introduced in this section.

Fig.1.Characteristic parameters of projectiles.

2.1.Measurement of characteristic parameters

We measured 50 pieces of 12.7 mm projectiles.To ensure that the samples represented the characteristics of the projectile population at large,samples were selected from 5 batches,as shown in Fig.3.

The testing platforms were provided by the Testing Technology Center of Characteristic Parameters at Nanjing University of Science and Technology.To measurem,LmandLrwithα,we used the Double Facade Mass Centroid Testing Equipment(ZLZXPX-10).The test specifics were C3（≤0.0020%）form,less than±0.1 mm forLm,and less than±0.002 mm forLr.We also used Torsional Pendulum Inertia Testing Equipment(CSJGL-10)to measureIzz andIyy,and our test specifics were less than±1% and±0.8% forIzz andIyy,respectively.

As shown in Fig.4,the PC is equipped with a controller which controls the testing platforms.On ZLZXPX-10,station A measuredmwithLm,while station B measuredLrwithα.On the other hand,on CSJGL-10 stations C and D,measuredIzz andIyy,respectively.

A picture of the experimental setup is given in Fig.5.The measurements were done at 21.1°C and 61.3% humidity.The characteristic parameters of each projectile were tested 5 times.The average value of testing data after removing abnormalities was taken as the final result for each parameter.Sample numbers and the testing 0°position were written on each projectile after its eccentric centroid had been measured(Fig.6).This method enabled collation of the testing data(Lrandα)with the sample number and the testing 0°position,to determine the position of each eccentric centroid.Data are recorded in tables below.Table 1 presents a summary of test results.

2.2.Shooting experiment

As mentioned earlier,analyzing the effect of mass eccentricity on the impact point requires measuring both the influence ofLrand the effect ofα.The influence ofLrcould be analyzed conveniently by matching impact points with sample numbers.However,to analyze the effect of the eccentric angleα,the entry angleβ,which determined the direction of eccentric centroid of each projectile in the chamber,was controlled when a projectile matched with the chamber.To avoid abnormal launch conditions,we divided the 50 projectiles into five groups with similar distributions ofLr,where the entry angleβof groups 1-3 was 90°and that of groups 4-5 was 0°,as presented in Table 2.

2.2.1.Assembling projectiles

To assemble the munitions,cartridges were carefully selected to ensure consistency.The mass of the propellant in all projectiles was 16.0±0.01 g.To controlβeasily,the eccentric angleαand the testing 0°position of each projectile were marked on the bottom of each tested cartridge.As shown in Fig.7,the heavy line represents the direction of the eccentric angle,and the fine and dotted lines represent the direction of the testing coordinate system(0°and 90°),respectively.

2.2.2.Eccentric centroid determination andfiring protocol

As shown in Fig.8,the munitions were fired with a precision(Mann)barrel.For the group whoseβwas 90°,the black heavy line on the bottom of cartridge was consistent with a vertical white line on the barrel-i.e.,the eccentric centroid was along the vertical direction and kept a certain distance ofLrassociated with the symmetry axis.Similarly,for the group whoseβwas 0°,the black heavy line was consistent with a horizontal white line-i.e.,the eccentric centroid was along the horizontal direction(see Fig.9).

To eliminate the influence of barrel temperature on firing accuracy,warmer rounds were fired before we fired the test rounds from a cold barrel.The firing cycle period was about 25-30s per firing,and the mean velocity at 25 m along trajectory(V25)was recorded by the sky screen targets.Impact points were measured as coordinates along thexandyaxes of the Cartesian plane on the target.The barrel was washed and cooled with water of a constant temperature after each group had been tested.The length of the indoor ballistic range was 200 m.The temperature in the shooting range was 23.6°C,and the relative humidity was 64.5%.The length of the barrel used in this experiment was 1003 mm with a twist of one turn at 389.1 mm.The setup of the shooting experiment is shown in Fig.10.

3.Sensitivity analysis

Sensitivity analysis(SA)is able to identify the parameters whose variations are expected to be influential to the model output[29].This method enables engineers to adjust outputs toward expected values by fine tuning influential parameters on purpose.This method helps us answer our second question-which of the tested parameters(m,Lm,Lr,α,Izz,Iyy,andV25)had the greatest influence on the positions of impact points.The tested parameters of projectiles and the position of the impact points were modeled using ANFIS.Then,we conducted a global sensitivity analysis using Sobol’s method-which quantifies output uncertainty due to changes in the input parameters(which are taken singly or in combination with others)over their entire domain of variation[30]-to quantify the influence of the tested parameters.

3.2.1.Artificial intelligence algorithm:Adaptive Neuro-Fuzzy Inference Systems(ANFIS)

ANFIS combines the predictive ability of fuzzy inference systems and the learning ability of neural networks,and thus provides a new and effective method of modeling and predicting complex nonlinear systems[31].AFNIS is used widely across different research domains,including geology[32],material sciences[33],thermodynamics[34]and energy[35].

We applied AFNIS to model the relationship between the tested parameters and the position of impact points-i.e.,the input parameters were the characteristic parameters,V25andβ,of the test projectiles,and the corresponding output parameters were thexandycoordinates of the impact points.We ran AFNIS with MATLAB 2016B Toolbox.For a detailed description on the theory behind these techniques,see Jyh.Shing[36,37].

To avoid large truncation errors due to differences in the magnitudes of the parameters,and to meet the requirements of Sobol’s method(which requires a unit input factor space),the parameters for training and checking the ANFIS were presented as dimensionless quantities-i.e.,the maximum was 1,and the minimum was 0.Regarding characteristic parameters,the original domains of variation are shown in Table 1;however,forV25andβ,the original domains of variation were 869.87-914.08 m/s and 0°-90°,respectively.

Fig.2.Experimental procedures regarding mass asymmetric projectiles.

To obtain optimal results for an ANFIS study,H.Y.Wang[38]recommends that the number of items in a study’s input data should be at least five times more than the number of input parameters.With this in mind,we had 44 items in our training data(excluding an outlier-no.5),with five Check Data(one per group)to ensure the accuracy of ANFIS.The Check Data is summarized in Table 3.Finally,the maximum error lies between the impact points predicted by ANFIS and the experimental results is 0.364 cm,as shown in Fig.11.Since the calculation error was within an acceptable range,the trained ANFIS was applied throughout the remainder of this study.

3.2.2.Sobol’s method

Sobol’s method uses the decomposition of variance to calculate Sobol’s sensitivity indices,which are an essential effective indicators in SA[39].The method is explained below,following the procedure outlined in some of Sobol’s work[40,41].To determine the output’s sensitivity to the variation of an input parameter,an input factor space(Ωk=(x｜0≤xi≤1;i=1，…，k))is introduced.The main feature is the decomposition of the function(f(x1，…，xk))into summands of increasing dimensionality,namely:

The total number of summands is 2kterms,and all summands in Eq.(1)are orthogonal,i.e.,

Fig.3.Samples of 12.7 mm projectiles selected for measurement.

The summands in Eq.(1)can be expressed as integrals off(x),then,

and so on.In Eqs.(4)and(5)x～idenotes the all factors butxi,so as tox～(ij).Assuming thatf(x)is square integrable,then all thef1，2，…，k(x1…xk)in Eq.(1)are also square integrable.Squaring Eq.(1)and integrating over in input factor space(Ωk)we get:

and the variances are expressed as Eqs.(7)and(8):

Fig.4.Schematics of the testing platforms.

Fig.5.The experimental setup.

Fig.6.The projectiles,labeled with sample numbers after measurement.

Table 1Test data regarding the characteristics of projectiles.

then the total variance off(x)due to Eqs.(6)-(8)can be computed as

The sensitivity indices are defined as the ratios of Eqs.(8)and(9):

Dividing both sides of Eq.(6)byD,we obtain:

Here,Siis called the first-order sensitivity index for factorxi,which measures the main effects ofxion the output-i.e.,the partial contribution ofxito the variance of(f(x1，…，xk)).Similarly,Sijis called the second-order sensitivity index fori≠j,which measures the effects of interactions betweenxiandxj,and so on.

Therefore,the total sensitivity index,which measures the main effects of a given parameter and all the interactions(of any order)involving that parameter,can be expressed as:

Table 2Summary of the testing groups.

Fig.7.The eccentric angleαand the testing coordinate system marked on the bottom.

Fig.8.Mann barrel.Entry angle of 90°.

Fig.9.Entry angle of 0°.

Fig.10.Setup for shooting experiment.

Table 3Check Data from five groups,dimensionless.

Fig.11.ANFIS and experimental results for positions of impact points.

In practice,Sobol’s method is relatively easy to be implemented by using Monte Carlo-based integration[39].A.Saltelli[42]discussed a quasi-Monte Carlo method for simultaneous computation ofSiandStoti,by which Eqs.((3),(7),(12)and(13)are expressed as shown below by Eqs.(14)-(17):

where A and B are two independent sampling matrices.In matrix AiB,in which all columns are from A except thei-th column(which is from B),Nis the sampling size for quasi-Monte Carlo discretization.In this work,Sobol’s quasi-random(Sobol’s QR)sequences were applied to sample the input factors,since in this method the quasi-random points know the position of previously sampled points and fill gaps between them.Sobol’s QR is characterized by low discrepancy properties and outperforms crude Monte Carlo sampling in the estimation of multi-dimensional integrals[43,44].

Based on the trained ANFIS,we sampled the input parameters(characteristic parameters,V25andβ)using Sobol’s QR to create sampling matrices A and B.Then,the sampling matrices A,B,andwere substituted into ANFIS to get correspondingxandycoordinates of impact points:f(x)in Eqs.(14)-(17).Finally,substituting the results-D,Diandin Eqs.(15)-(17)-into Eq.(12)，(13),producedSiand

4.Results and discussion

Based on the results obtained in the measurement and trials,we conducted an analysis of variance(ANOVA)to investigate our first research question.To answer our second research question,the global sensitivity indices of parameters were given by global sensitivity analyses.

4.2.1.Analysis of variance(ANOVA)

As shown in Fig.12,an ANOVA was carried out for which the Factor was the entry angleβand the Dependent Factors were thexandycoordinates of the impact points to determine the significance of eccentricity of 12.7 mm projectiles.Before conducting this,the data from the impact points were checked by calculating Homogeneity of Variance to confirm whetherβwas suitably selected as the Factor of experiments in section 2.

The calculation of the homogeneity of variance on the experimental data of the impact points was carried out with Levene’s test[45].A singlep-value was computed for the set of populations and was tested and compared against 0.05 in order to confirm or reject the null hypothesis of equality of variances.Four algorithms for calculating Levene’s statistics-ones based on mean,on median,on median and with adjusted degrees of freedom,and on trimmed mean(i.e.mean calculated after removing 5%of the maximum and minimum variable values)-were applied.As presented in Table 4,the significance is larger than 0.05 for each of these algorithms.Therefore,the conclusion of the statistic test is that there is not sufficient evidence to reject the hypothesis that the variances associated with the position of collected impact points are equal along thexaxis and theyaxis,within groups.

Table 4Homogeneity of variance.

Table 5ANOVA on position of impact points with different entry angles.

The findings of our ANOVA are summarized in Table 5.As mentioned earlier,the variances within groups were equal;if the variances between groups were significantly different,this would imply that the independent factor had significant effect on dependent variables.In Table 5,the significance of theycoordinates of the impact points was 0.704 and thus indicates that there was insufficient evidence to confirm that the entry angleβ had an effect on theycoordinates of the impact points.However,the significance of this same entry angle onxcoordinates was 0.000039,implying thatβ(i.e.the initial position of mass eccentricity)had an effect on thexcoordinates of impact points.

Similar results are shown in Fig.13.In this picture,the impact points ofβ=0°are distributed to the left of the impact points of β=90°.Table 6 shows how the mean coordinates of two group impact points were not different on theyaxis but were different on thexaxis by-32.900 cm and-20.286 cm,respectively-a disparity of 12.61 cm.

This experiment provided preliminary verification that the tiny mass eccentricity of 12.7 mm projectiles notably affected the position of impact points.For engineers who seek to improve small caliber projectile design,tiny eccentricities caused in normal manufacture are not negligible.Readers should keep in mind that the data in this work derives from merely one instance,and the results will be different if different entry angles or a slightly longer or shorter barrel are used.In future experiments explaining the cause of this difference,the initial yaw rate,the initial yaw angle,the aimpoint of barrel,and the main aeroballistics quantities should be also recorded in order to confirm whether the lateral throwoff and aerodynamic jump are still suitable to estimate deflection of the 12.7 mm projectiles caused by tiny mass eccentricities.

Fig.13.Distribution of all impact points of different entry angles on target plane,200 m.

4.2.2.Global sensitivity analyses

Before answering the second question,we conducted a sample size independence study to evaluate the possibility of discretization errors of the global sensitivity indices.We did this because the sampling matrices were created by Sobol’s QR.In theory,as the number of samples increase,the possibility of discretization errors reduces and eventually disappears-i.e.,the global sensitivity indices converge to true values.The results of the sample size independence study are shown in Table 7.There,are the total sensitivity indices regarding impact points’ycoordinates andxcoordinates,respectively.From our sample size independence study,we can see that the maximum relative error at the size of 5.0×104was 0.49%,while that at size of 1.0×105was 0.06%-i.e.,when the number of samples was larger than 1.0×105,the total sensitivity indices of the factors considered in this work changed slightly.This implies that the results are close to a stable state.Therefore,the sample size was selected as 5.0×105for the remainder of the study.

Global sensitivity analyses using Sobol’s method conducted at the sample size of 5.0×105in an attempt to answer our second research question.In Fig.14,andare the first-order sensitivity indices regarding impact points’ycoordinates andxcoordinates,respectively.SinceSirepresents the influence of a single factor(theifactor)on output,whilemeasures the main effects of theiparameter and all the interactions(of any order)involving that parameter,ifwhich implies that the model is a nonlinear system.It can be seen that all of the first-order sensitivity indices were lower than the total sensitivity indices,indicating that the interactions among the above parameters had a greater impact on the position of impact points than any single factor alone.

In terms of the first-order sensitivity indices regarding the impact points’xandydirections,theof each factor was largerthan that ofexcept in the cases ofLmandα,whosewere near zero.However,the top three forwere the first-order sensitivity indices ofV25,Iyy,andm.This finding implies that individually changingV25,Iyy,and/ormcould influence the position of impact points’ydirection more significantly than they could influence the position of impact points’xdirection.

Table 6Description of the coordinates of the impact points.

Table 7The total sensitivity indices of factors at different sample sizes.

Fig.14.First-order sensitivity indices and total sensitivity indices of the seven parameters.

5.Conclusions

This study introduced a novel experimental approach to evaluating the effects of tiny eccentric centroids in 12.7 mm projectiles on the positions of those projectiles’impact points.To determine the relationship between all factors and the projectiles’impact points,we applied ANFIS to model the shooting process of the weapon system by using the data obtained in the experiment.We then simulated large numbers of real firings and utilized Sobol’s method to determine the significance of all factors considered in the experiment.

The experiment results indicated that the impact points of two entry angles distributed separately,and the mean disparity between the two groups’impact points’xcoordinates was 12.61 cm.The results of ANOVA showed that the entry angle was the critical factor affecting impact point distribution.This answers our first question:the tiny eccentric centroids caused by normal manufacturing processes have a notable effect on the distribution of impact points.

Regarding our second question,Sobol’s method revealed that theofαwas larger than that ofwhich confirmed the above finding(that the entry angle was the critical factor in determining the distribution of impact points).Meanwhile,infers that changing parameters individually has little influence on the distribution of impact points.Therefore,based on the distribution ofchanging a combination ofIyy,Izz,and/ormare effective ways to increase control authority.However,keeping any combination ofIzzm,andLrandIyy,m,andLmconsistent can minimize the dispersion of impact points’ycoordinates andxcoordinates,respectively.

The results show that even tiny eccentric caused in manufacture,can affect the position of impact points notabl y,let alone intentionally unbalancing 12.7 mm projectile by actuators will control the impact points,which implies that the control method by modifying mass center is promising for small-caliber spin-stabilized projectiles.In future experiments,the characteristics of trajectory such as the initial yaw rate,the initial yaw angle,the aimpoint of barrel,and the main aeroballistics quantities will also be investigated in order to facilitate better understanding of the interaction between tiny eccentric centroids and impact points.Based on these data,Monte-Carlo simulations will be conducted and compared with the ANFIS in order to investigate the error budget of more expansive parameters of 12.7 mm projectiles.

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.

Acknowledgement

The authors are grateful for the reviewers’instructive suggestions and careful proofreading.This work was supported by the Fundamental Research Funds for the Central Universities,China(grant no.30918012203)and the Foundation of National Laboratory,China(grant no.JCKYS2019209C001).