# Aerodynamic Parameter Identification of Projectile Based on Improved Extreme Learning Machine and Ensemble Learning Theory

Authors: Tianyi Wang; Wenjun Yi; Youran Xia
Journal: International Journal of Aerospace Engineering
 (2023)

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## Abstract

The firing accuracy of the projectile has a positive relation with aerodynamic parameters. Due to the complex dynamic characteristics of projectiles, there is an overfitting risk when a single extreme learning machine (ELM) is used to identify the aerodynamic parameters of the projectile, and the identification results oscillate transonic region. To obtain the aerodynamic parameters of the projectile accurately, an aerodynamic parameter identification model based on ensemble learning theory and ELM optimized by improved particle swarm optimization is proposed. The improved particle swarm optimization algorithm (IPSO) with an adaptive update strategy is used to optimize the weight and threshold of ELM. Combined with the ensemble learning theory, the improved ELM neural network is regarded as a weak learner to generate a strong learner. The structural parameters of the strong learner were continuously optimized through training, and an aerodynamic parameter identification model of projectile based on ensemble learning theory is obtained. The simulation results show that the introduction of the IPSO and ensemble learning theory enables the model to exhibit excellent generalization ability. The proposed identification model can accurately describe the variation of aerodynamic parameters with the Mach number.


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## Body

## 1. Introduction

Aerodynamics is the decisive factor affecting the ballistic trajectory and flight stability of the projectile [1]. The projectile’s firing accuracy positively relates to aerodynamic parameters [2]. Currently, numerical computation, wind tunnel test, and shooting test are common technical means to obtain aerodynamic parameters [3]. The result of the numerical calculation method depends on the accuracy of the ballistic model. Still, due to the strong coupling of the projectile flight motion and external disturbance, it is difficult to obtain a completely accurate ballistic model. The wind tunnel experiment simulates the genuine flight attitude of the projectile by changing the attitude and velocity of the model, which is generally used to test and correct the shape parameters and aerodynamic characteristics of the projectile. The parameter identification method is used to process the measured data (provided by shooting test) and indirectly extract the aerodynamic parameters of projectile. Research on the aerodynamic parameter identification method has significant practical engineering application value [4]. The least squares method (LSM), maximum likelihood method (ML), Kalman filtering method (KF), and intelligent algorithm are mature algorithms in the field of parameter identification [5–7].LSM [8] is a classical estimation method in aircraft parameter identification. Dunkel [9] realized the identification of aerodynamic derivative and stability derivative by LSM. However, LSM is susceptible to extreme outliers. To mitigate the effects of outliers, Su and Song [10] used recursive LSM with fading memory to improve the identification performance. Kamali et al. [11] proposed improved LSM and successfully identified Dutch roll movement parameters and segment period parameters of aircraft. Mu et al. [12] combined the model reduction technology with LSM for parameter identification. Due to the complex motion characteristics of the projectile, the traditional identification equation (based on LSM) needs to be better posed. Yang et al. [13] combined LSM with an engineering test to obtain the aerodynamic parameters of projectile.ML has been widely used due to its unbiasedness [14–17]. Carnduff and Cooke [18] applied ML to reconstruct the aerodynamic model of an unmanned aerial vehicle (UAV) with an unconventional fuselage structure. However, ML highly relies on model accuracy, and the variance is significant at high latitudes, so it is often used in combination with other methods. To improve the performance of ML, Kumar and Rao [19] combined the output error method with ML, which can accelerate the speed of convergence. Zou and Li [20] introduced the interior point optimization method into ML to reduce the severe error caused by second-order numerical differentiation.By extending the parameters (to be identified) to system states, the problem of parameter identification can be transformed into the issue of optimal state estimation so that KF can be applied to parameter identification [21–23]. In nonlinear systems, KF cannot accurately estimate the state matrix. It was later developed into the extended Kalman filter (EKF) and unscented Kalman filter (UKF). Zheng et al. [24] proved that UKF is better than EKF in projectile parameter identification because of unscented transformation (U-transformation). To reduce the computational complexity and improve the accuracy, many other improved KF algorithms have been proposed. Menon et al. [25] combined the differential vortex lattice algorithm with EKF to realize flight path reconstruction. Majeed and Kar [26] proposed adaptive UKF to improve identification accuracy. Shen et al. [27] combined EKF with the aerodynamic semiempirical method to identify derivative residuals.Due to the complex environment and unknown external interference, getting a completely accurate aircraft model is complex. To solve the modeling error of the traditional identification method, intelligent algorithms and their variants have been widely used in aerodynamic parameter identification. Du et al. [28] combined particle swarm optimization algorithm (PSO) with real-coded genetic algorithm (GA) to obtain the resistance coefficient of the projectile. Based on the maximum likelihood criterion, Li et al. [29] applied the neural network Newton method to extract the zero-lift drag coefficient of the projectile. To improve accuracy and accelerate convergence, Wang et al. [30] introduced an elite crossover strategy into GA. Guan et al. [31] combined GA with ML to identify the zero-lift resistance coefficient of a high-speed rotating projectile based on the speed data of the projectile. Aiming at the problem that gradient descent optimization algorithm is easy to fall into local optima in traditional aerodynamic parameter identification, Han et al. [32] put forward a double backpropagation (BP) neural network, Pu et al. [33] put forward a method of sample expansion and neural network parameter online fast correction based on support vector machine (SVM), and Li et al. [34] proposed an improved teaching-learning-based optimization (ITLBO) for aerodynamic parameter identification. To avoid the initial value estimation, Yan et al. [35] proposed a derivative method for identifying the aerodynamic parameters of aircraft by the three-layer neural network. Hou et al. [36] applied a differential evolution algorithm to weaken initial value sensitivity. Ji-gang et al. [37] combined the advantage of PSO in the initial value section and the advantages of the Newton iteration method in precise iteration and successfully identified the drag coefficient of the projectile. Mohamad et al. [38] put forward the concept of dynamic parameter estimation (DAPE).Extreme learning machine (ELM) [39] is an algorithm for training single hidden layer feedforward neural networks (SLFNs). The structural parameters (input weights and hidden thresholds) of ELM are generated randomly and require no iterative adjustment. Owing to it, ELM has low computational complexity and good real-time performance and has been widely used in cloud computing, data visualization, and random projection [40–42]. Akusok et al. [41] applied ELM to identify the drag coefficient of the projectile for the first time. Affected by uncertain factors such as the actual combat environment and external meteorological conditions, the ballistic trajectory data is characterized by solid nonlinearity, time-dependent nature, and susceptibility to random noise. Randomly generated structural parameters lead to the oscillation of ELM identification results [42]. In addition, when a single ELM is used to identify the aerodynamic parameters of the projectile, all the given training samples are often used to model the global situation. In other words, it is easy to make insufficient use of the sample data and cause overfitting. In summary, using a single ELM to identify aerodynamic parameters has the following limitations:
(a)
The identification result oscillates (especially in the transonic region) due to randomly generated structural parameters(b)
When a single ELM is used to identify projectile aerodynamic parameters, it is hard to make sufficient use of the local information of the sample data and then causes overfittingTo overcome the above problems and then accurately obtain the aerodynamic parameters of the projectile, a large number of documents are referenced. For problem (a), the classical idea is to apply PSO, GA, and other optimization algorithms to optimize the structural parameters of ELM [43–49]. However, iterative optimization increases the time complexity of the algorithm. Then, the adaptive update strategy is introduced to improve the performance of PSO. For problem (b), Schapire [50] proved that multiple weak learners could generate a strong learner with good generalization performance by ensemble theory. Considering that the projectile parameter identification problem is a regression problem, AdaBoost. RT algorithm [51] is used as the integration framework. Above all, this paper puts forward an aerodynamic parameter identification model of projectile based on improved ELM and ensemble learning theory (we named IPSO-ELM-AdaBoost). The proposed IPSO-ELM-AdaBoost is a comprehensive application of multiple algorithms. In short, the functions of the hybrid algorithm can be generalized as follows:
(a)
ELM (function as a weak learner) establishes the mapping relationship between ballistic data and aerodynamic parameters(b)
Improved PSO (IPSO) provides ELM with optimized structural parameters(c)
AdaBoost. RT is responsible for integrating multiple weak learners into strong learnersThe rest of the paper is arranged as follows: in Section2, the concrete expression of the ballistic equation is given. ELM-AdaBoost. RT aerodynamic parameter identification model based on improved particle swarm optimization (IPSO-ELM-AdaBoost) is described in detail in Section 3. The simulation results under standard meteorological conditions are analyzed in Section 4. Ultimately, conclusions are summarized in Section 5.


## 2. Ballistic Trajectory Model

Before parameter identification, the ballistic trajectory model (6DOF) [52] must first be solved to obtain the ballistic data. 6DOF treats the projectile motion as a rigid body motion. It considers the three degrees of freedom of the projectile centroid motion and the three degrees of freedom of the angular motion. Ignoring the dynamic unbalance of the projectile, the aerodynamic eccentricity, and the Coriolis inertial force caused by the Earth’s rotation, 6DOF can be mathematically described as
(1)dvdt=1m−ρvr2Scx0v−wxV−ρvr2Scx2δr2v−wxV+ρS2cyvr2cosδlrcosδudsinδr−ρS2cyvrεv−wxVsinδr−ρvr2SczwzVcosδlrsinδudsinδr+ρvr2SczwyVsinδlrsinδr−mgsinθVcosψV,(2)dθVdt=1mρvrScxwyV2vcosψV+ρScyvr2cosδlrsinδud+vrζwyV2vcosψVsinδr−ρvr2Scy′δNcosγ12vcosψV+ρvrSczv−wxVsinδlr+wzVcosδlrcosδud2vcosψVsinδr−mgcosθVvcosψV+2ΩEmvvcosψVsinψVcosθVcosΛcosαd+sinθVsinψVsinΛ+cosψVcosΛsinαd,(3)dψVdt=1mρvr2vScxwzV+ρS2vcy1sinδrvr2sinδlr+vrξwzV−ρv2Scy′δNsinγ12−ρvr2v2cz1sinδrwyVcosδlrcosδud−ρvr2vScz1sinδrv−wxVcosδlrsinδud+1vmgsinθVsinψV+2ΩEmsinΛcosθV−cosΛsinθvcosαd,(4)dωεdt=−ρSld2Cmxz′vrωε,(5)dωηdt=1AρSl2vrmz1sinδrvrξ−ρSld2vrmzz′ωη−ρSld2my′′1sinδrωεvrη−ρv2Slmz′δMsinγ22−CAωεωξ+ωξ2tanψa,(6)dωξdt=1AρSl2vrmz1sinδrvrη−ρSld2vrmzz′ωξ−ρSld2my′′1sinδrωεvrξ+ρv2Slmz′δM2+CAωεωη−ωηωξtanψa,(7)dθadt=ωξcosψa,(8)dψadt=−ωη,(9)dγdt=ωε−ωξtanψa,(10)dxdt=vcosψVcosθV,(11)dydt=vcosψVsinθV,(12)dzdt=vsinψV,where v is the speed; vr is the relative velocity; vrε, vrη, and vrξ are the components of relative velocity along ε, η, and ξ axes; x is the distance; y is the altitude; z is the lateral distance; ωε, ωη, and ωξ are the projected components of rotational speed along ε, η, and ξ axes; θa is the elevation angle in coordinate system of projectile axes; S is the characteristic area; m is the mass of projectile; l is the reference projectile length; d is the reference projectile diameter; g is the acceleration of gravity; ρ is the air density; wxV, wyV, and wzV are the projected components of wind velocity along xV, yV, and zV; δr is the relative angle of attack; Λ is the latitude; ΩE is the rotational angular velocity of the earth; αd is the angle of departure; δr is the relative angle of attack; θV is the elevation angle in velocity coordinate system; ψV is the direction cosine in velocity coordinate system; ψa is the direction cosine in coordinate system of projectile axes; δM is the aerodynamic malalignment of additional moment; δN is the aerodynamic malalignment of additional force; δud is the horizontal component of angle of attack; δld is the longitudinal component of angle of attack; cx is the drag coefficient; cy is the lift coefficient; cy′ is the lift coefficient derivative; cz is the Magnus force coefficient; mxz′ is the rolling damping moment coefficient derivative; mz is the static moment coefficient; mzz′ is the oscillating damping moment coefficient derivative; and my″ is the Magnus moment coefficient derivative.


## 3. IPSO-ELM-AdaBoost

The proposed IPSO-ELM-AdaBoost is a comprehensive application of multiple algorithms: ELM (function as a weak learner) is used to extract aerodynamic parameters from ballistic data. PSO variants provide ELM with optimized structural parameters. AdaBoost. RT algorithm (function as an integration framework) is responsible for integrating multiple weak learners into strong learners and outputting the projectile’s final aerodynamic parameter identification results. In this section, ELM and AdaBoost. RT will be briefly introduced as prior knowledge, and the idea of adaptive update strategy in IPSO will be presented in detail.


### 3.1. ELM

ELM [39] is a special SLFN without iterative adjustment of structural parameters. The working process of ELM can be divided into learning and prediction. For ELM with M input layer, L hidden neurons, O output layer, and activation function σW1,X,b (the activation function can be any nonzero function), the structure of ELM is shown in Figure 1.Figure 1
The structure of ELM.


#### 3.1.1. Training Process

Given input vectorXM×1,
(13)XM×1=x1,⋯,xMT.The structural parameters of ELM are randomly generated as follows:(14)WL×M1=w1,11⋯w1,M1⋮⋱⋮wL,11⋯wL,M1,(15)bL×1=b1,⋯,bLT.Then, the output of the hidden layer is as follows:(16)HL×1=σWL×M1·XM×1+bL×1.The output weight matrixWO×L2 is mathematically described as
(17)WO×L2=w1,12⋯w1,L2⋮⋱⋮wO,12⋯wO×L2.The outputY^O×1 can be calculated as
(18)Y^O×1=WO×L2⋅HL×1.GivenN training samples XM×N,YO×N, the output of ELM is as follows:
(19)Y^O×N=WO×L2⋅HL×N.The training goal is mathematically described as(20)YO×N−Y^O×N=0.The output weight matrix can be calculated as(21)WO×L2=YO×N⋅HL×N+,where HL×N+ is the Moore–Penrose generalized inverse of HL×N.In order to improve the generalization ability of ELM and avoid overfitting, based on the ridge regression principle [53], a regularization term [54] is introduced in (21):
(22)WO×L2=YO×N⋅HL×NT⋅HL×N+IC−1⋅HL×NT,where I is the identity matrix and C is the regularization factor.


#### 3.1.2. Prediction Process

When ELM completes the training process, the output matrixWO×L2 can be calculated by equation (22). Given P predicting samples XM×P, the output of ELM is
(23)Y^O×P=WO×L2⋅HL×P.


### 3.2. ISPO

PSO is a metaheuristic algorithm proposed by Kennedy and Eberhart [55]. Compared with other metaheuristic algorithms, such as GA [56] and ant colony algorithm [57], the PSO algorithm has a simple structure, easy implementation, global solid searchability, and fast convergence speed [58]. Based on the global best position gbest and the individual best position pbest, the particles are iteratively updated until convergence. The steps of PSO can be described as follows:
(1)
Set hyperparameterHyperparameters that need to be set include the following: population sizee, particle dimension D, velocity inertia weight ωv, learning factors c1 and c2, maximum iteration kmax, minimum error Δmin, maximum particle velocity Vmax, and maximum position Xmax.
(2)
InitializationInitialize positionXi=xi1,xi2,⋯,xiDT and velocity Vi=vi1,vi2,⋯,viDT, i=1,2,⋯e.
(3)
Iterative updateThe fitness functionFitness⋅ is selected to calculate the fitness value of particles and find out the individual and global optimum of particles. For optimization problems, the update rule is
(24)pibestk=pibestk−1,Fitnesspibestk>Fitnesspibestk−1,Xik,Fitnesspibestk≤Fitnesspibestk−1,(25)gbestk=argminFitnesspibestki=1,2,⋯,e,where pibsetk=pi1bestk,pi2bestk,⋯,piDbestkT specifies the individual optima of the ith particle in the kth iteration and gibsetk=g1bestk,g2bestk,⋯,gDbestkT specifies global optima in the kth iteration.Particle is iteratively updated by(26)vidk+1=ωvvidk+c1r1pidbestk−xidk+c2r2gdbestk−xidk,(27)xidk+1=xidk+vidk+1,where d=1,2,⋯,D. r1 and r2 are random numbers subject to uniform distribution.
(4)
Iteration stopWhen the algorithm converges, the optimization result is output. If not, step (3) is transferred to continue the iteration.The structural parameters of ELM that are optimized by PSO can contain more training sample information than randomly one and can effectively improve the identification accuracy. However, the iterative optimization of PSO increases the time complexity of the algorithm. Theωv in equation (26) can keep the motion inertia of particles and make them tend to expand the search space, which has an important influence on the optimization performance of the algorithm. Dynamic ωv can obtain better optimization results than fixed one [59]. A larger ωv can improve the global search ability of the algorithm, while a smaller one can improve the local search ability of the algorithm. In order to improve the convergence speed of the algorithm, the adaptive update strategy is introduced in IPSO. Formula (28) calculates the average distance dgk from the global optimal particle to other particles in the kth iteration and then maps dgk to the interval [0, 1] by formula (29) to obtain the adaptive factor f. f describes the state of the population. In other words, the larger the f, the farther the particle is from the global optimal particle, and the particle needs a larger ωv to quickly approximate the global optimal solution. The smaller the f, the closer the particle is to the global optimal particle, and a smaller ωv is required to limit the particle to the vicinity of the global optimal solution and improve the search accuracy. The adaptive update strategy (based on f) of ωv can be described as
(28)dgk=1e−1∑i=1egbestk−Xik2,(29)fk=dgk−dminkdmaxk−dmink,(30)ωvk=1−fkωvmin+fkωvmax.In order to verify the effectiveness of the IPSO, the standard test functionsfSp and fSc are selected to conduct the test independently for 100 times and compared with the PSO algorithm. The expressions of test functions fSp and fSc are shown in equations (31) and (32), and the related parameter settings of the improved particle swarm are shown in Table 1. The results after 100 independent tests are given in Table 2. Test results show that the introduction of adaptive updated ωv in IPSO can effectively improve the accuracy and convergence speed of the algorithm. In the case of the same number of particles, for the same test function, the results of IPSO are closer to the actual value, and the average convergence speed is faster.
(31)fSp=∑i=130μi2,μi∈−100,100,(32)fSc=0.5+sinμ12+μ221+0.001μ12+μ222,μi∈−10,10.Table 1
Parameter setting of IPSO.
ParameterValuekmax1000ωmax0.8ωmin0.2c11.59c21.83Vmax0.4Xmax1Table 2
Test results of different test functions.
FunctionAlgorithmSizeMaxMinMeanIterationTheoretical valueSpherePSO500.5150.0340.13060001000.0500.0040.01840001500.0060.0010.0023500IPSO500.1780.0230.09645001006.35×10−81.47×10−115.41×10−930001503.83×10−128.79×10−176.26×10−132000SchafferPSO500.0370.0080.01165001000.0370.0090.01045001500.0091.98×10−170.0084000IPSO500.0130.0050.01050001000.0015.73×10−129.14×10−730001501.54×10−52.87×10−136.27×10−72000The structure of ELM optimized by IPSO (IPSO) is shown in Figure2.Figure 2
The structure of IPSO-ELM.


### 3.3. AdaBoost. RT

AdaBoost. RT [60] is proposed for regression problems. The idea is to filter out examples with a relative estimation error higher than the preset threshold value and then follow the AdaBoost procedure. Its basic idea can be as follows: preset threshold φ, train weak learners (IPSO-ELM) based on training samples, and update the weight of training samples according to the current prediction error of IPSO-ELM. In the next round of training, the training samples with significant prediction error will have larger weights and continue to train weak learners based on the new weight. After M training rounds, M weak learners are obtained, and finally, the output of all weak learners is weighted to get the final prediction result. The implementation process of IPSO-ELM-AdaBoost is described as follows:
(1)
InitializationRandomly selectN training samples from the dataset xi,tii=1N, initialize the weight Dji=1/Ni=1,2,⋯,N of the training samples, and set the threshold φ of the algorithm, the initial prediction error rate ε1, and the iteration round number T.
(2)
Train weak learnersj≤TThejth weak learner hj⋅ is trained by training data, and calculate the error of each training sample (Eij) and weak learner (εj).
(33)Eij=hjxi−ti,(34)εj=∑Dji,i:Eij>φ,where hjxi is the prediction result of the jth weak learner (jth round) in the ith training data and ti is the actual value.
(3)
Update the weight of training samplesThe updated formula is as follows:(35)Dj+1i=DjiBj⋅εj2,Eij≤φ,DjiBj⋅1εj2,Eij>φ,where Bj is the normalization factor.
(4)
Repeat the trainingRepeatT rounds of the step to obtain T weak learners and weighting to obtain the output of the strong learner:
(36)hx=∑j=1Tln1/εt2hjx∑j=1Tln1/εt2.The flow chart of IPSO-ELM-AdaBoost is shown in Figure3.Figure 3
The structure of IPSO-ELM-AdaBoost.


## 3.1. ELM

ELM [39] is a special SLFN without iterative adjustment of structural parameters. The working process of ELM can be divided into learning and prediction. For ELM with M input layer, L hidden neurons, O output layer, and activation function σW1,X,b (the activation function can be any nonzero function), the structure of ELM is shown in Figure 1.Figure 1
The structure of ELM.


### 3.1.1. Training Process

Given input vectorXM×1,
(13)XM×1=x1,⋯,xMT.The structural parameters of ELM are randomly generated as follows:(14)WL×M1=w1,11⋯w1,M1⋮⋱⋮wL,11⋯wL,M1,(15)bL×1=b1,⋯,bLT.Then, the output of the hidden layer is as follows:(16)HL×1=σWL×M1·XM×1+bL×1.The output weight matrixWO×L2 is mathematically described as
(17)WO×L2=w1,12⋯w1,L2⋮⋱⋮wO,12⋯wO×L2.The outputY^O×1 can be calculated as
(18)Y^O×1=WO×L2⋅HL×1.GivenN training samples XM×N,YO×N, the output of ELM is as follows:
(19)Y^O×N=WO×L2⋅HL×N.The training goal is mathematically described as(20)YO×N−Y^O×N=0.The output weight matrix can be calculated as(21)WO×L2=YO×N⋅HL×N+,where HL×N+ is the Moore–Penrose generalized inverse of HL×N.In order to improve the generalization ability of ELM and avoid overfitting, based on the ridge regression principle [53], a regularization term [54] is introduced in (21):
(22)WO×L2=YO×N⋅HL×NT⋅HL×N+IC−1⋅HL×NT,where I is the identity matrix and C is the regularization factor.


### 3.1.2. Prediction Process

When ELM completes the training process, the output matrixWO×L2 can be calculated by equation (22). Given P predicting samples XM×P, the output of ELM is
(23)Y^O×P=WO×L2⋅HL×P.


## 3.1.1. Training Process

Given input vectorXM×1,
(13)XM×1=x1,⋯,xMT.The structural parameters of ELM are randomly generated as follows:(14)WL×M1=w1,11⋯w1,M1⋮⋱⋮wL,11⋯wL,M1,(15)bL×1=b1,⋯,bLT.Then, the output of the hidden layer is as follows:(16)HL×1=σWL×M1·XM×1+bL×1.The output weight matrixWO×L2 is mathematically described as
(17)WO×L2=w1,12⋯w1,L2⋮⋱⋮wO,12⋯wO×L2.The outputY^O×1 can be calculated as
(18)Y^O×1=WO×L2⋅HL×1.GivenN training samples XM×N,YO×N, the output of ELM is as follows:
(19)Y^O×N=WO×L2⋅HL×N.The training goal is mathematically described as(20)YO×N−Y^O×N=0.The output weight matrix can be calculated as(21)WO×L2=YO×N⋅HL×N+,where HL×N+ is the Moore–Penrose generalized inverse of HL×N.In order to improve the generalization ability of ELM and avoid overfitting, based on the ridge regression principle [53], a regularization term [54] is introduced in (21):
(22)WO×L2=YO×N⋅HL×NT⋅HL×N+IC−1⋅HL×NT,where I is the identity matrix and C is the regularization factor.


## 3.1.2. Prediction Process

When ELM completes the training process, the output matrixWO×L2 can be calculated by equation (22). Given P predicting samples XM×P, the output of ELM is
(23)Y^O×P=WO×L2⋅HL×P.


## 3.2. ISPO

PSO is a metaheuristic algorithm proposed by Kennedy and Eberhart [55]. Compared with other metaheuristic algorithms, such as GA [56] and ant colony algorithm [57], the PSO algorithm has a simple structure, easy implementation, global solid searchability, and fast convergence speed [58]. Based on the global best position gbest and the individual best position pbest, the particles are iteratively updated until convergence. The steps of PSO can be described as follows:
(1)
Set hyperparameterHyperparameters that need to be set include the following: population sizee, particle dimension D, velocity inertia weight ωv, learning factors c1 and c2, maximum iteration kmax, minimum error Δmin, maximum particle velocity Vmax, and maximum position Xmax.
(2)
InitializationInitialize positionXi=xi1,xi2,⋯,xiDT and velocity Vi=vi1,vi2,⋯,viDT, i=1,2,⋯e.
(3)
Iterative updateThe fitness functionFitness⋅ is selected to calculate the fitness value of particles and find out the individual and global optimum of particles. For optimization problems, the update rule is
(24)pibestk=pibestk−1,Fitnesspibestk>Fitnesspibestk−1,Xik,Fitnesspibestk≤Fitnesspibestk−1,(25)gbestk=argminFitnesspibestki=1,2,⋯,e,where pibsetk=pi1bestk,pi2bestk,⋯,piDbestkT specifies the individual optima of the ith particle in the kth iteration and gibsetk=g1bestk,g2bestk,⋯,gDbestkT specifies global optima in the kth iteration.Particle is iteratively updated by(26)vidk+1=ωvvidk+c1r1pidbestk−xidk+c2r2gdbestk−xidk,(27)xidk+1=xidk+vidk+1,where d=1,2,⋯,D. r1 and r2 are random numbers subject to uniform distribution.
(4)
Iteration stopWhen the algorithm converges, the optimization result is output. If not, step (3) is transferred to continue the iteration.The structural parameters of ELM that are optimized by PSO can contain more training sample information than randomly one and can effectively improve the identification accuracy. However, the iterative optimization of PSO increases the time complexity of the algorithm. Theωv in equation (26) can keep the motion inertia of particles and make them tend to expand the search space, which has an important influence on the optimization performance of the algorithm. Dynamic ωv can obtain better optimization results than fixed one [59]. A larger ωv can improve the global search ability of the algorithm, while a smaller one can improve the local search ability of the algorithm. In order to improve the convergence speed of the algorithm, the adaptive update strategy is introduced in IPSO. Formula (28) calculates the average distance dgk from the global optimal particle to other particles in the kth iteration and then maps dgk to the interval [0, 1] by formula (29) to obtain the adaptive factor f. f describes the state of the population. In other words, the larger the f, the farther the particle is from the global optimal particle, and the particle needs a larger ωv to quickly approximate the global optimal solution. The smaller the f, the closer the particle is to the global optimal particle, and a smaller ωv is required to limit the particle to the vicinity of the global optimal solution and improve the search accuracy. The adaptive update strategy (based on f) of ωv can be described as
(28)dgk=1e−1∑i=1egbestk−Xik2,(29)fk=dgk−dminkdmaxk−dmink,(30)ωvk=1−fkωvmin+fkωvmax.In order to verify the effectiveness of the IPSO, the standard test functionsfSp and fSc are selected to conduct the test independently for 100 times and compared with the PSO algorithm. The expressions of test functions fSp and fSc are shown in equations (31) and (32), and the related parameter settings of the improved particle swarm are shown in Table 1. The results after 100 independent tests are given in Table 2. Test results show that the introduction of adaptive updated ωv in IPSO can effectively improve the accuracy and convergence speed of the algorithm. In the case of the same number of particles, for the same test function, the results of IPSO are closer to the actual value, and the average convergence speed is faster.
(31)fSp=∑i=130μi2,μi∈−100,100,(32)fSc=0.5+sinμ12+μ221+0.001μ12+μ222,μi∈−10,10.Table 1
Parameter setting of IPSO.
ParameterValuekmax1000ωmax0.8ωmin0.2c11.59c21.83Vmax0.4Xmax1Table 2
Test results of different test functions.
FunctionAlgorithmSizeMaxMinMeanIterationTheoretical valueSpherePSO500.5150.0340.13060001000.0500.0040.01840001500.0060.0010.0023500IPSO500.1780.0230.09645001006.35×10−81.47×10−115.41×10−930001503.83×10−128.79×10−176.26×10−132000SchafferPSO500.0370.0080.01165001000.0370.0090.01045001500.0091.98×10−170.0084000IPSO500.0130.0050.01050001000.0015.73×10−129.14×10−730001501.54×10−52.87×10−136.27×10−72000The structure of ELM optimized by IPSO (IPSO) is shown in Figure2.Figure 2
The structure of IPSO-ELM.


## 3.3. AdaBoost. RT

AdaBoost. RT [60] is proposed for regression problems. The idea is to filter out examples with a relative estimation error higher than the preset threshold value and then follow the AdaBoost procedure. Its basic idea can be as follows: preset threshold φ, train weak learners (IPSO-ELM) based on training samples, and update the weight of training samples according to the current prediction error of IPSO-ELM. In the next round of training, the training samples with significant prediction error will have larger weights and continue to train weak learners based on the new weight. After M training rounds, M weak learners are obtained, and finally, the output of all weak learners is weighted to get the final prediction result. The implementation process of IPSO-ELM-AdaBoost is described as follows:
(1)
InitializationRandomly selectN training samples from the dataset xi,tii=1N, initialize the weight Dji=1/Ni=1,2,⋯,N of the training samples, and set the threshold φ of the algorithm, the initial prediction error rate ε1, and the iteration round number T.
(2)
Train weak learnersj≤TThejth weak learner hj⋅ is trained by training data, and calculate the error of each training sample (Eij) and weak learner (εj).
(33)Eij=hjxi−ti,(34)εj=∑Dji,i:Eij>φ,where hjxi is the prediction result of the jth weak learner (jth round) in the ith training data and ti is the actual value.
(3)
Update the weight of training samplesThe updated formula is as follows:(35)Dj+1i=DjiBj⋅εj2,Eij≤φ,DjiBj⋅1εj2,Eij>φ,where Bj is the normalization factor.
(4)
Repeat the trainingRepeatT rounds of the step to obtain T weak learners and weighting to obtain the output of the strong learner:
(36)hx=∑j=1Tln1/εt2hjx∑j=1Tln1/εt2.The flow chart of IPSO-ELM-AdaBoost is shown in Figure3.Figure 3
The structure of IPSO-ELM-AdaBoost.


## 4. Simulation Verification

In this section, a series of simulation tests are carried out under standard weather conditions to validate the feasibility and excellence of IPSO-ELM-AdaBoost in aerodynamic parameter identification. In simulation test 1, IPSO-ELM-AdaBoost is compared with three other machine learning algorithms (ELM, IPSO-ELM, and ELM-AdaBoost), and the performance of mentioned machine learning algorithms is stated in the quantitative ways. In simulation test 2, IPSO-ELM-AdaBoost, LSM, ML, and UKF are used to reconstruct the trajectory, respectively. The performance of the four algorithms is evaluated from the point of fall and side deflection.


### 4.1. Data Preprocessing

Under standard meteorological conditions, 6DOF in Section2 is solved, and 10000 ballistic data are obtained. Table 3 shows the initial launch parameters of 6DOF, and the variation laws of the X, Y, Z, and V of the projectile with time are shown in Figures 4–7.Table 3
Initial launch parameters of 6DOF.
Initial launch parameterValuev0 (m/s)930θa (°)45ψa (°)95ω0 (rad/s)188.5Figure 4
X‐T curve.Figure 5
Y‐T curve.Figure 6
Z‐T curve.Figure 7
V‐T curve.The ballistic trajectory data contains the flight velocity, position, and attitude of the projectile, and different information has different dimensions. To eliminate the influence of different dimensions on data analysis, the min-max normalization is used to preprocess 10000 original datasets. The formula is as follows:(37)xj∗=xj−xminxmax−xmin,where xj represents the original data, xmin is the minimum value of data, xmax represents the maximum value of data, and xj∗ represents the normalized data.


### 4.2. Simulation Test 1

Figures8–12 are graphs of aerodynamic parameter identification results. In the figure, the abscissa represents the Mach number (Ma), and the ordinate represents aerodynamic parameters to be identified. The identification result of IPSO-ELM-AdaBoost can accurately fit the real aerodynamic parameter curves in the whole trajectory range. For the same parameter to be identified, the introduction of the IPSO and ensemble learning theory enables the model to exhibit excellent generalization ability. Take the identification results of mxz′ (Figure 12) as an example, the identification result curve of ELM oscillates, especially in the transonic region 0.8<Ma<1.2. The identification results of IPSO-ELM and ELM show that the structural parameters optimized by IPSO contain more sample information, which leads to higher curve fitting degree. Comparing the identification results of ELM-AdaBoost and ELM, it is found that the identification result curve considering multiple weak learners can better describe the variation law of aerodynamic parameters with Ma. It is worth mentioning that (by comparing the identification results of IPSO-ELM and ELM-AdaBoost), although IPSO can improve the performance of the model, respectively, ISPO-ELM still has the risk of overfitting because it only takes a single learner into consideration. For different aerodynamic parameters (to be identified), the linearity between aerodynamic parameters and Ma affects the fitting degree. The stronger the linearity between aerodynamic parameters to be identified and Ma, the higher the curve fitting degree of identification results.Figure 8
Identification results ofcx.Figure 9
Identification results ofcy.Figure 10
Identification results ofmz.Figure 11
Identification results ofmzz′.Figure 12
Identification results ofmxz′.Table4 shows the model structure and average identification time (after 100 independent tests) of the four algorithms. From the perspective of model structure, the introduction of IPSO and ensemble learning theory can effectively reduce the number of neurons in the hidden layer and simplify the model. From the point of identification time, although the IPSO algorithm and the introduction of ensemble learning theory simplify the complexity of the model, the iterative optimization process of IPSO and the serial training process of the weak learner by AdaBoost. RT both lead to the increase in identification time.Table 4
Results of four algorithms.
ModelStructureTime (s)ELM12-95-58.7IPSO-ELM12-70-520.3ELM-AdaBoost12-65-531.9IPSO-ELM-AdaBoost12-50-533.5To further assess the performance of the model, three well-known measures are taken [61–64]. The description of these measures is as follows:
(a)
Correlation Coefficient (R). R is a statistical measure of the strength of a linear relationship between two variables. Its values can range −1,1. R=1 shows a perfect positive correlation, or a direct relationship. R=−1 describes a perfect negative, or inverse, correlation, with values in one series rising as those in the other decline and vice versa. Moreover, when R=0, there is no possible agreement between experimental results and numerical ones(b)
Mean Square Error (MSE). As a common model performance evaluation function, MSE can reflect the difference between the observed and the predicted ones. The smaller the order of magnitude of MSE, the higher the identification accuracy(c)
Mean Absolute Percentage Error (MAPE). MAPE is a relative error measure that uses absolute values to avoid positive and negative errors canceling out. For each data series, MAPE value is a positive ratio of error value (difference between predicted output and observed one) to observed valueThe specific expression of the above-mentioned measures can be described as(38)R=∑t=1Ppredictedt−predicted¯observedt−observed¯∑t=1Ppredictedt−predicted¯2∑t=1Pobservedt−observed¯2,(39)MSE=1P∑t=1Pobservedt−predictedt2,(40)MAPE=100P∑t=1Ppredictedt−observedtobservedt,where P represents the number of predicted samples, observedt represents the tth actual observation value, predictedt represents the tth model prediction value, predictedt¯ is the mean of predictedt, and observedt¯ is the mean of observedt.The statistical results are collected in Table5. Under the combined action of ensemble theory and structural parameters (optimized by IPSO), IPSO-ELM-AdaBoost has the best generalization performance among the four machine learning algorithms. The structural parameters of ELM optimized by IPSO can make the input weight and threshold contain more information of input samples, which can effectively improve the identification accuracy. AdaBoost can also improve the identification accuracy by comprehensively considering the learning results of multiple weak learners. The results of IPSO-ELM and ELM-AdaBoost show that ensemble theory improves more on the performance of the algorithm. This is because structural parameters (optimized by IPSO) contain more sample information, but on the other hand, they also increase the risk of overfitting.Table 5(a)
R of four algorithms
cxcymzmzz′mxz′ELM0.610.750.890.160.14IPSO-ELM0.720.770.920.570.62ELM-AdaBoost0.830.810.940.890.86IPSO-ELM-AdaBoost0.900.850.970.920.91(b)
MSE of four algorithms
cxcymzmzz′mxz′ELM7.59×10−39.10×10−76.44×10−142.96×10−33.63×10−3IPSO-ELM6.47×10−41.31×10−73.17×10−145.81×10−51.58×10−6ELM-AdaBoost3.20×10−105.36×10−109.44×10−161.03×10−91.54×10−10IPSO-ELM-AdaBoost7.59×10−114.83×10−112.84×10−168.30×10−101.07×10−10(c)
MAPE (%) of four algorithms
cxcymzmzz′mxz′ELM25.828.49.655.271.2IPSO-ELM29.623.95.547.715.9ELM-AdaBoost10.220.75.810.39.2IPSO-ELM-AdaBoost7.118.44.110.08.3


### 4.3. Simulation Test 2

In order to further verify the excellent performance of the proposed algorithm in projectile aerodynamic parameter identification, IPSO-ELM-AdaBoost is compared with the traditional projectile parameter identification algorithm (LSM, ML, and UKF). Figure13 shows the trajectory reconstruction results of the four algorithms. The trajectory reconstructed by IPSO-ELM-AdaBoost has the highest fitting degree with the actual trajectory. Among the three traditional algorithms, the reconstructed result of UKF is the closest to the actual trajectory. The fitting degree of the four algorithms from low to high is as follows: LSM, ML, UKF, and IPSO-ELM-AdaBoost. In practical engineering, more attention is paid to the landing point and side deflection of the projectile, so the landing point (X) and side deflection (Z) of the four algorithms are also presented in Table 6. Compared with the traditional algorithm, the proposed algorithm has the highest accuracy. The drop point error of IPSO-ELM-AdaBoost is 6 m, and the sideway error is 2 m. In terms of the drop point error, the drop point error of LSM and ML is more than 50 meters, which can no longer meet the need of actual engineering. From the perspective of side deviation error, UKF and IPSO-ELM-AdaBoost have good performance, and their side deviation error is less than 10 m.Figure 13
Results of ballistic reconstructions.Table 6
Results of four algorithms.
X (m)Z (m)Observed290131230LSM29127 (+114)1213 (-17)ML28952 (-61)1247 (+17)UKF28980 (+33)1239 (+9)IPSO-ELM-AdaBoost29019 (+6)1228 (-2)


## 4.1. Data Preprocessing

Under standard meteorological conditions, 6DOF in Section2 is solved, and 10000 ballistic data are obtained. Table 3 shows the initial launch parameters of 6DOF, and the variation laws of the X, Y, Z, and V of the projectile with time are shown in Figures 4–7.Table 3
Initial launch parameters of 6DOF.
Initial launch parameterValuev0 (m/s)930θa (°)45ψa (°)95ω0 (rad/s)188.5Figure 4
X‐T curve.Figure 5
Y‐T curve.Figure 6
Z‐T curve.Figure 7
V‐T curve.The ballistic trajectory data contains the flight velocity, position, and attitude of the projectile, and different information has different dimensions. To eliminate the influence of different dimensions on data analysis, the min-max normalization is used to preprocess 10000 original datasets. The formula is as follows:(37)xj∗=xj−xminxmax−xmin,where xj represents the original data, xmin is the minimum value of data, xmax represents the maximum value of data, and xj∗ represents the normalized data.


## 4.2. Simulation Test 1

Figures8–12 are graphs of aerodynamic parameter identification results. In the figure, the abscissa represents the Mach number (Ma), and the ordinate represents aerodynamic parameters to be identified. The identification result of IPSO-ELM-AdaBoost can accurately fit the real aerodynamic parameter curves in the whole trajectory range. For the same parameter to be identified, the introduction of the IPSO and ensemble learning theory enables the model to exhibit excellent generalization ability. Take the identification results of mxz′ (Figure 12) as an example, the identification result curve of ELM oscillates, especially in the transonic region 0.8<Ma<1.2. The identification results of IPSO-ELM and ELM show that the structural parameters optimized by IPSO contain more sample information, which leads to higher curve fitting degree. Comparing the identification results of ELM-AdaBoost and ELM, it is found that the identification result curve considering multiple weak learners can better describe the variation law of aerodynamic parameters with Ma. It is worth mentioning that (by comparing the identification results of IPSO-ELM and ELM-AdaBoost), although IPSO can improve the performance of the model, respectively, ISPO-ELM still has the risk of overfitting because it only takes a single learner into consideration. For different aerodynamic parameters (to be identified), the linearity between aerodynamic parameters and Ma affects the fitting degree. The stronger the linearity between aerodynamic parameters to be identified and Ma, the higher the curve fitting degree of identification results.Figure 8
Identification results ofcx.Figure 9
Identification results ofcy.Figure 10
Identification results ofmz.Figure 11
Identification results ofmzz′.Figure 12
Identification results ofmxz′.Table4 shows the model structure and average identification time (after 100 independent tests) of the four algorithms. From the perspective of model structure, the introduction of IPSO and ensemble learning theory can effectively reduce the number of neurons in the hidden layer and simplify the model. From the point of identification time, although the IPSO algorithm and the introduction of ensemble learning theory simplify the complexity of the model, the iterative optimization process of IPSO and the serial training process of the weak learner by AdaBoost. RT both lead to the increase in identification time.Table 4
Results of four algorithms.
ModelStructureTime (s)ELM12-95-58.7IPSO-ELM12-70-520.3ELM-AdaBoost12-65-531.9IPSO-ELM-AdaBoost12-50-533.5To further assess the performance of the model, three well-known measures are taken [61–64]. The description of these measures is as follows:
(a)
Correlation Coefficient (R). R is a statistical measure of the strength of a linear relationship between two variables. Its values can range −1,1. R=1 shows a perfect positive correlation, or a direct relationship. R=−1 describes a perfect negative, or inverse, correlation, with values in one series rising as those in the other decline and vice versa. Moreover, when R=0, there is no possible agreement between experimental results and numerical ones(b)
Mean Square Error (MSE). As a common model performance evaluation function, MSE can reflect the difference between the observed and the predicted ones. The smaller the order of magnitude of MSE, the higher the identification accuracy(c)
Mean Absolute Percentage Error (MAPE). MAPE is a relative error measure that uses absolute values to avoid positive and negative errors canceling out. For each data series, MAPE value is a positive ratio of error value (difference between predicted output and observed one) to observed valueThe specific expression of the above-mentioned measures can be described as(38)R=∑t=1Ppredictedt−predicted¯observedt−observed¯∑t=1Ppredictedt−predicted¯2∑t=1Pobservedt−observed¯2,(39)MSE=1P∑t=1Pobservedt−predictedt2,(40)MAPE=100P∑t=1Ppredictedt−observedtobservedt,where P represents the number of predicted samples, observedt represents the tth actual observation value, predictedt represents the tth model prediction value, predictedt¯ is the mean of predictedt, and observedt¯ is the mean of observedt.The statistical results are collected in Table5. Under the combined action of ensemble theory and structural parameters (optimized by IPSO), IPSO-ELM-AdaBoost has the best generalization performance among the four machine learning algorithms. The structural parameters of ELM optimized by IPSO can make the input weight and threshold contain more information of input samples, which can effectively improve the identification accuracy. AdaBoost can also improve the identification accuracy by comprehensively considering the learning results of multiple weak learners. The results of IPSO-ELM and ELM-AdaBoost show that ensemble theory improves more on the performance of the algorithm. This is because structural parameters (optimized by IPSO) contain more sample information, but on the other hand, they also increase the risk of overfitting.Table 5(a)
R of four algorithms
cxcymzmzz′mxz′ELM0.610.750.890.160.14IPSO-ELM0.720.770.920.570.62ELM-AdaBoost0.830.810.940.890.86IPSO-ELM-AdaBoost0.900.850.970.920.91(b)
MSE of four algorithms
cxcymzmzz′mxz′ELM7.59×10−39.10×10−76.44×10−142.96×10−33.63×10−3IPSO-ELM6.47×10−41.31×10−73.17×10−145.81×10−51.58×10−6ELM-AdaBoost3.20×10−105.36×10−109.44×10−161.03×10−91.54×10−10IPSO-ELM-AdaBoost7.59×10−114.83×10−112.84×10−168.30×10−101.07×10−10(c)
MAPE (%) of four algorithms
cxcymzmzz′mxz′ELM25.828.49.655.271.2IPSO-ELM29.623.95.547.715.9ELM-AdaBoost10.220.75.810.39.2IPSO-ELM-AdaBoost7.118.44.110.08.3


## 4.3. Simulation Test 2

In order to further verify the excellent performance of the proposed algorithm in projectile aerodynamic parameter identification, IPSO-ELM-AdaBoost is compared with the traditional projectile parameter identification algorithm (LSM, ML, and UKF). Figure13 shows the trajectory reconstruction results of the four algorithms. The trajectory reconstructed by IPSO-ELM-AdaBoost has the highest fitting degree with the actual trajectory. Among the three traditional algorithms, the reconstructed result of UKF is the closest to the actual trajectory. The fitting degree of the four algorithms from low to high is as follows: LSM, ML, UKF, and IPSO-ELM-AdaBoost. In practical engineering, more attention is paid to the landing point and side deflection of the projectile, so the landing point (X) and side deflection (Z) of the four algorithms are also presented in Table 6. Compared with the traditional algorithm, the proposed algorithm has the highest accuracy. The drop point error of IPSO-ELM-AdaBoost is 6 m, and the sideway error is 2 m. In terms of the drop point error, the drop point error of LSM and ML is more than 50 meters, which can no longer meet the need of actual engineering. From the perspective of side deviation error, UKF and IPSO-ELM-AdaBoost have good performance, and their side deviation error is less than 10 m.Figure 13
Results of ballistic reconstructions.Table 6
Results of four algorithms.
X (m)Z (m)Observed290131230LSM29127 (+114)1213 (-17)ML28952 (-61)1247 (+17)UKF28980 (+33)1239 (+9)IPSO-ELM-AdaBoost29019 (+6)1228 (-2)


## 5. Conclusion

Accurate aerodynamic parameters can effectively improve the firing accuracy of the projectile. Affected by uncertain factors such as actual combat environment and external meteorological conditions, the actual ballistic trajectory data is characterized by strong nonlinearity, time-dependent nature, and susceptibility to random noise. Take a single ELM for aerodynamic parameters, and the identification result curve oscillates in transonic region. To accurately obtain aerodynamic parameters of projectile, a new aerodynamic parameter identification model based on improved ELM and ensemble theory is constructed. Under standard weather conditions, the model is trained by trajectory data, and the mapping relationship between trajectory data and aerodynamic parameters is established. The simulation results show that the generalization ability of the model can be effectively improved by optimizing the structural parameters of ELM with IPSO and integrating several weak learners into one strong learner. Although the identification time is increased because of the iterative optimization of IPSO and serial training of AdaBoost. RT, the identification time of IPSO-ELM-AdaBoost is still the same order as that of ELM. The proposed IPSO-ELM-AdaBoost has excellent performance in the aerodynamic parameter identification of projectile and can be extended to the other aircraft.



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*Source: 3854295-2023-02-27.xml*