Sinking Velocity Impact-Analysis for the Carrier-Based Aircraft Using the Response Surface Method-Based Improved Kriging Algorithm

,e deck landing sinking velocity of carrier-based aircraft is affected by carrier attitude, sea condition, aircraft performance, etc. Its impact analysis is a complex nonlinear problem, and there even is some contradictory phenomenon that when the approach velocity increases, the sinking velocity decreases under certain circumstances. Aiming at exploring the impact of the various related deck landing parameters on sinking velocity for carrier-based aircraft in the actual environment, response surface methodbased improved Kriging algorithm (IK-RSM) is proposed based on genetic algorithm and Kriging model. Based on the deck landing measured data of the F/A-18A aircraft in the actual operating environment, the impact degree of the 15 deck landing parameters on the sinking velocity is explored, respectively, by using the partial correlation analysis of multivariate statistical theory and the IK-RSM. It can be found that the 4 parameters are strongly correlated with the sinking velocity; that is, the aircraft glide angle and deck pitch angle are highly correlated with the sinking velocity; next, the approach velocity and the engaging velocity are moderately correlated with the sinking velocity. ,e 4 parameters above could be used to establish the impact analysis model of the sinking velocity. ,e genetic algorithm is applied to the correction coefficients optimization of the IK-RSM’s kernel functions, and the IK-RSM of the F/A-18A aircraft sinking velocity is formed. Compared with the Kriging model and the empirical formula, the sinking velocity prediction accuracy indexes of IK-RSM are the best; for example, the determination coefficient is 0.981, the mean relative error is 1.813%, and the maximum relative error is 6.771%. Furthermore, based on the sinking velocity IK-RSM and the sensitivity analysis method proposed, we have explained the reason for the contradictory phenomenon that when the approach velocity increases, the sinking velocity decreases at some samples. It could provide certain technical support for the flight attitude control related to the sinking velocity during the actual flight of carrier-based aircraft.


Introduction
Sinking velocity is defined as the component of the aircraft velocity in vertical direction in the deck landing process of the carrier-based aircraft, which is an important design parameter for the landing gear [1]. e sinking velocity is the indication of the impact degree in the aircraft landing, and the range of which will seriously influence the weight of the landing gear and airframe structure. If the design value of sinking velocity is too small, this may lead to the aircraft structure being too weak and unable to reach the reliability requirement.
Reversely, if the design value of sinking velocity is too large, this may lead to the aircraft structure being too heavy and further affects the aircraft flight performance. e deck landing sinking velocity of carrier-based aircraft is affected by many factors, such as aircraft carrier attitude, sea condition, and aircraft performance. So, the uncertainty will be too great to assess the value of sinking velocity under the influencing factors above. erefore, it is urgent to carry out the impact-analysis research of the sinking velocity to reasonably design the relevant parameters of landing gear and guarantee the safety of carrier-based aircraft furthermore.
ere have been quite a large body of literatures on carrierbased aircraft, and various theories and techniques have been developed. References [2,3] studied the statistical features of the sinking velocity for carrier-based aircraft, which include the distribution characteristics and the empirical formula for calculating the mean and standard deviation. Micklos [4] investigated running state of carrier-based aircraft under considering different operational conditions and further provided the measured data. e statistical data show that there are many factors influencing the sinking velocity including 15 deck landing parameters, and the relationship between the sinking velocity and the corresponding influencing factors is highly nonlinear. In certain cases, there also is a contradictory phenomenon that the approach velocity increases and the sinking velocity decreases. Geng et al. [5] applied flight dynamics model of carrier aircraft to analyze the effect of the response time of engine, wave-off requirements, elevator efficiency, and deflection rate on the sinking velocity. Xia et al. [6] proposed an improved linearization method to correctly emulate aircraft groundspeed variations. Wang et al. [7] established the landing dynamics model and used finite element method to study the aircraft deck landing. Zhu et al. [8] built an actual model of arresting hook to investigate the influence on collision process under the deck friction. Wang et al. [9] employed realistic mechanisms and strategies to establish a model for carrier landing operations and studied the sinking velocity with pilot behavior. Zhang et al. [10] discussed the carrier-based aircraft landing laws that landed on the carrier by using the dynamics model of carrier-based aircraft landing gears that landed on moved deck. Wang et al. [11] and Yang et al. [12] evaluated the safety carrier-based aircraft ski-jump takeoff by the integrated dynamic simulation models of multibody system and nonlinear model. Yue et al. [13] studied the flow field of exhaust jets and its impact on the flight deck to provide some references for suitability of carrierbased aircrafts. Yin et al. [14] used simulated model to verify the deck movement and ship wake in the landing process of carrier-based aircraft. Wang et al. [15] developed an adaptive disturbance rejection algorithm to discuss the carrier-based aircraft dynamics and the linearized longitudinal model under turbulence conditions. Although the above efforts investigated the carrier-based aircraft from different perspectives, there are still some shortcomings on the sinking velocity study: (1) it has only provided the measured data of different kinds of carrierbased aircrafts considering different working conditions without including the relevant influence factors on the sinking velocity; (2) it has only finished the deterministic analysis of carrier-based aircraft by dynamic analysis theories without considering the randomness of influential factors, especially the sinking velocity; (3) most of works investigate the sinking velocity of carrier-based aircraft on the basis of theoretical methods without verifying the feasibility and applicability in engineering compounded with the measured data. Additionally, multiple disciplines need to be considered in the landing process of carrier-based aircraft, which include flight dynamics, aircraft control, and structural design. Hence, it is necessary to study the influence of the related parameters by numerical method, for instance, flight attitude, aircraft type, landing time (day/night), carrier motion, and sea conditions, on the sinking velocity. e sinking velocity of the carrier aircraft will be accurately determined. Due to the large number of parameters involved in the analysis and the high dimension, the traditional Kriging model is probable to fall into the local optimum in the process of solving the correction coefficient, which leads to some error between the prediction result and the measured value [16,17]. erefore, it is necessary to find a suitable method to establish a mathematical model between various factors and the sinking velocity to study the influence degree of each factor. erefore, the goal of this paper is oriented to explore an analytical technique to study the sinking velocity of carrier-based aircraft, response surface methodbased improved Kriging algorithm (IK-RSM), which integrates genetic algorithm and Kriging model, for the impact analysis and sensitivity analysis of carrier-based aircraft sinking velocity.
Based on the deck landing statistical data of the carrierbased aircraft F/A-18A, by using multivariate partial correlation analysis method, the key correlated parameters of the sinking velocity are identified, including the approach velocity, the engaging velocity, the glide angle, and the deck pitch angle. en, the Kriging interpolation model and genetic algorithm are used to establish the IK-RSM model between the sinking velocity and the key correlated parameters mentioned above. e analysis of the sensitivity of each parameter to sinking velocity is carried out. An impact-analysis method for the carrier-based aircraft's sinking velocity based on measured data is developed and will provide technical support for the carrierbased aircraft design and research.

Partial Correlation Analysis Method for Determining the Key Correlated Parameters
Correlation analysis is firstly applied to study the correlation relationship among natural phenomena and explore the correlative direction and degree among the correlated stochastic phenomena [18,19]. Subsequently, the relationship of multiple parameters is performed by correlation analysis. ere are various technologies for correlation analysis, which mainly include chart analysis, covariance analysis, correlation coefficient analysis, regression analysis, and information entropy analysis. e correlation coefficient analysis is selected to finish the correlation analyses of the sinking velocity of carrier-based aircraft because this technique can judge and determine the correlation degrees between research object and influential factors. For multiple parameters involved in the process of influential parameter analyses for the sinking velocity of carrier-based aircraft, partial correlation analysis method is utilized. e basic principle of partial correlation analysis method is to individually investigate the correlation degree of two factors without considering the effects of other influential factors. e correlation coefficient is the statistic index for the correlative degree between two variables, whose range is [−1, 1]. Hereinto, the correlation coefficients are positive or negative values. e positive value reveals that the outputs produce a positive variety with input variable, and vice versa. e correlation coefficient r � 1 illustrates that the two variables have completely linear correlation, r � −1 reveals that the two variables have perfect negative correlation, and r � 0 expresses that the two variables do not have certainly linear dependence. |r| ≥ 0.8 indicates that the variables are highly correlated, 0.5 ≤ |r| < 0.8 means the variables are moderately correlated, 0.3 ≤ |r| < 0.5 indicates that the variables are lowly correlated, and |r| < 0.3 denotes the variables are weakly correlated [20]. e measure indexes of partial correlation analysis are, respectively, first-order partial correlation coefficient, second-order partial correlation coefficient, and higher-order partial correlation coefficient, and the computational theories of these coefficients are listed as follows.
(1) First-order partial correlation coefficient reflects the correlative degree between two variables without considering the influence of another variable, which is expressed as where r ij is the correlation coefficient between x i and x j ; r ih is the correlation coefficient between x i and x h ; r jh is the correlation coefficient between x j and x h . (2) Second-order partial correlation coefficient indicates the correlative degree between two variables without considering the effects of two other variables, x h and x m , which is expressed as where r ij−hq is the second-order partial correlation coefficient indicating the correlative degree between two variables without considering the effects of two other variables, x h and x q . (3) Higher-order partial correlation coefficient is applied to study the correlative degree between two variables considering the existence of multiple variables. Assuming that there are k variables, namely, x 1 , x 2 , . . ., x k , the g(g ≤ k − 2)-order partial correlation coefficient between x i and x j can be expressed as follows: where r ij−l 1 l 2 ···l g−1 , r il g −l 1 l 2 ···l g−1 , and r jl g −l 1 l 2 ···l g−1 are the g − 1 order partial correlation coefficients.
e correlation between two variables can be determined by the hypothesis test; that is, and here m is the number of samples; m − k − 2 is the degree of freedom; r is the g-order partial correlation coefficient. If the associated probability value p is less than the significant level, the null hypothesis is rejected that the correlation between the two variables is significant; otherwise, the null hypothesis is accepted due to no significant correlation between the two variables.

Parameter Impact Analysis Method Based on Improved Kriging Response Surface Model
With the help of the good interpolation characteristics of the IK-RSM, the influential factors of the sinking velocity and their laws will be discussed.
where F(β, x) and z(x) are quadratic polynomial regression model and random function, respectively. F(β, x) is denoted by vector expression; that is, f T · β is the polynomial re- T is the regression basis function for x and is expressed as the polynomial of x. β is the coefficient vector of the regression basis function, and p is the number of regression basis functions, which is related to the number of independent variables and the form of the regression basis function selected. z(x) is the error item formed by a stochastic process to correct the model. erefore, the Kriging model y(x) is the sum of the deterministic regression f T · β and the approximation deviation z(x).

Error
Correction. e stochastic process z(x) included in the mathematical model of IK-RSM (see equation (5)) (e.g., IK-RSF) is normal distribution; the mean and variance are 0 and σ 2 , respectively. However, the value of covariance is not equal to 0; the mean, the variance, and covariance of z(x) are written as follows: where R(θ, x i , x j ) is the correlation model for the arbitrary samples x i and x j with parameter θ; {x i } i�1, 2, . . ., m and {x j } j�1, 2, . . ., m are the vectors of the ith and jth random input variable; m is the number of training samples; σ 2 indicates Advances in Materials Science and Engineering 3 the process variance. e form of R(θ, x i , x j ) is denoted in equation (7).
n is the number of variables, where x i,k and x j,k are the kth components of ith and jth random input variable, respectively; R(θ k , x i,k − x j,k ) is kernel function. Kernel function has many forms like Gaussian, exponential, linear, cubic, and so forth. Gaussian function is selected in this paper due to high computational accuracy in Kriging algorithm. us, equation (7) is restructured by where θ is the correction coefficient for the kernel function. en, the kernel function matrices R could be gained by training samples. R is displayed by and here R(θ, x i , x j ) can be required from equation (8).

Polynomial Regression Basis Function.
ere are three forms for the regression polynomial models with the orders 0, 1, and 2 for the IK-RSM. e details are as follows.
3.1.4. Error Assessment. In summary, at arbitrary site x, the local deviation of the prediction results given by the IK-RSM is represents the correlation degree between arbitrary site x and the known sample point. According to the least squares estimation principle, the variance estimate value σ 2 of the Kriging model can be expressed as

Correction Coefficient Optimization of the Kernel Function Based on Genetic Algorithm.
To accurately calculate the correction coefficient θ i , the genetic algorithm is applied to find the optimum values of correction coefficients. For the m samples from n-dimensional variable, the Kriging model based on genetic algorithm is formulated as follows: e minimum fitness function is taken as the optimization objective, and the difficulty is to ensure that the error between the predicted value and the measured value is as small as possible in the process of parameter optimization, so as to ensure that the prediction accuracy of the established model meets the engineering requirements. It is possible for the objective function φ(θ) of the correction coefficient θ to be local optimum [21,22]. Considering that the genetic algorithm is good at global optimization, based on the sample of sinking velocity, the correction coefficient θ of kernel function of Kriging model is optimized by using the genetic algorithm. e specific process is as follows in Figure 1.
As known from Figure 1, the specific solution process of Kriging correction coefficient based on genetic algorithm is as follows: according to the obtained data, select the sample data of relevant variables and normalize the data; define the optimization variable θ in the solution process, code the variables, and initialize the population; calculate the fitness value according to the fitness 4 Advances in Materials Science and Engineering function; select the excellent individual in the population; generate the next generation of individuals to form the population through crossing operation; then carry out mutation operation, and calculate the objective function value; judge whether the objective function value meets the termination criterion; if not, perform the fitness value calculation again; if the criterion is met, the optimization parameter value is output.

Parameter Sensitivity
Analysis. e gradient of the IK-RSM's interpolation results of the n-dimensional variable is Specifically, it can be expressed as where c � R − 1 (Y − fβ); J f (x) and J r (x) are the partial derivative matrices of regression basis function vector f and kernel function vector r to the variable x, respectively. en, Meanwhile, J r (x) can be expressed as where erefore, the estimated value of the total differential for response y(x) is approximated as (zy(x)/zx i )dx i is the sensitivity of the response with the increasement dx i for the ith variable. Approximately,

Correlated Parameter Determination of the Sinking
Velocity. In [4], 252 valid deck landing parameters samples of F/A-18A carrier-based aircraft are provided. Combined with the above partial correlation analysis, the results of the partial correlation coefficient calculation of the ship state parameters of the F/A-18A carrier aircraft (excluding the sinking velocity, a total of 15) and the sinking velocity of the ship are listed in Table 1.
From the partial correlation analysis results, it is found that the aircraft glide angle and deck pitch angle are highly correlated with the sinking velocity; next, the approach velocity and the engaging velocity are moderately correlated with the sinking velocity. e aircraft pitch rate, carrier velocity, landing weight, deck roll angle, aircraft roll angle, aircraft roll rate, aircraft pitch rate, aircraft pitch angle, aircraft yaw angle, ramp to touchdown distance, and offcenter engaging distance are not correlated with the sinking velocity.
erefore, the key correlated parameters of the sinking velocity V S of the F/A-18A carrier-based aircraft could be determined, including the aircraft glide angle C G , the deck pitch angle C P , the approach velocity V A , and the engaging velocity V E . e IK-RSM of these four parameters with the sinking velocity is further established, and the sensitivity of each correlated parameter will be calculated.  Tables 2 and 3, including 5 variables: the sinking velocity V S , the aircraft glide angle C G , the deck pitch angle C P , the approach velocity V A , and the engaging velocity V E . e 126 samples from Table 2 are used to construct the IK-RSM of the sinking velocity, and the other 126 samples in Table 3 are used to verify the validation of the above model. e mean and standard deviation of each deck landing parameters are shown in Table 4.

IK-RSM of the Sinking Velocity.
Considering the nonlinear relationship between the sinking velocity V S and the other parameters including aircraft glide angle C G , deck pitch angle C P , approach velocity V A , and engaging velocity V E , the quadratic regression basis function is determined for the IK-RSM of the sinking velocity. According to the sample in Table 2, in the sinking velocity IK-RSM established, the sample number is m � 126, the variable number is n � 4, and the number of the regression basis functions is p � 1/2(n + 1)(n + 2) � 15. According to the sample, combined with equations (8), (9), and (11)-(13), the optimization model of correction coefficient θ for the kernel function of IK-RSM is as follows: e optimization process of objective function is as follows.
By using the Lagrange multiplier method, the objective function converges to 0.065692 finally, and the convergence process is shown in Figure 2.
Furthermore, the genetic algorithm is employed, and the evolution process of the objective function is shown in Figure 3. e convergence of the objective function is achieved through 100 generations iterated, the convergence result is 0.060538, and the correction coefficient matrix θ � 363.683 0.008 493.786 327.062] T . erefore, R and β are as follows: R �       where the matrix R is practically the "identity matrix": 1 in the main diagonal and about 0 as other coefficients.

Error Assessment and Comparison with Other Models.
126 samples in Table 3 are used to verify the IK-RSM of sinking velocity. Meanwhile, the sinking velocity empirical formula provided by [20] and the ordinary Kriging model are compared with the IK-RSM. e measured data of aircraft glide angle C G , deck pitch angle C P , approach velocity V A , and engaging velocity V E in Table 3 are replaced in the above models, respectively. e measured data of sinking velocity and the prediction results of various models are shown in Figure 4. e error comparisons are shown in Figure 5 and Table 5. e compared results in Table 5 show that, compared with the empirical formula provided by [20] and the Kriging model, the sinking velocity prediction accuracy indexes of IK-RSM are the best, including the coefficient of determination (the coefficient of determination is more close to 1, which means higher precision), the mean relative error, and the maximum relative error. erefore, by using the IK-RSM, we can get the sinking velocity prediction model with the various parameters including aircraft glide angle C G , deck pitch angle C P , approach velocity V A , and engaging velocity V E .
It is too complex to deduce the sinking velocity according to the related theory of the aircraft conceptual design and flight mechanics, which is unable to accomplish analysis effectively. Based on the measured data, through the partial correlation analysis and the IK-RSM model of the sinking velocity in this paper, the internal relationship between the sinking velocity and the other deck landing parameters can be primarily obtained; that is, the sinking velocity is the vertical component of the engaging velocity, and it is also related to the glide angle and the deck pitch angle of the aircraft during the deck landing process and also including the random error compensation of Kriging model. Compared with the empirical formula, the predicted results from the IK-RSM model are more consistent with the measured value. e sensitivity analysis of the above parameters can be further carried out.

Sensitivity Analysis
Results. According to equation (18), the gradient of sinking velocity can be obtained at all correlated parameters. If the correlated parameters have a certain increment, the corresponding increment of sinking velocity can be obtained according to the above gradient, that is, the sensitivity of each variable. Furthermore, according to equation (22), we can get the increment of sinking velocity when all correlated variables vary at the same time. Hence, Table 6 shows the influence degree of sinking velocity due to 1% increment at the mean of each correlated parameter. e samples in Tables 1 and 2 have been sorted from small to large according to the sinking velocity, and all 252 samples are determined. According to the sensitivity analysis of the parameters related to the sinking velocity of all 252 samples, the increment of the sinking velocity can be calculated at 1% of the variation of each variable from each sample. e calculation results are shown in Figure 6. e average influence degree of the correlated parameters can be calculated from the 252 samples, as shown in Figure 7. It can be seen that the sensitivity of the engaging velocity V E is the greatest, that of the aircraft glide angle C G is the second, that   of the deck pitch angle C P is the third, and that of the approach velocity V A is the least. e above analysis means that the sinking velocity is most sensitive to the engaging velocity V E . From the sensitivity curve of Figure 6, it can be seen that there is a big sudden change in the influence degree on the sinking velocity at the No. 225 and No. 242 samples, and the increment of the sinking velocity reaches 0.2527 m/s and 0.31 m/s, respectively. e above two sample points and their adjacent samples are listed in Table 7. e parameters of the two samples and their adjacent samples also have a big sudden change, including the approach velocity, the engaging velocity, the aircraft glide angle, and the deck pitch angle. erefore, the influence degree of each parameter on sinking velocity has a sudden change, which leads to a big change in sensitivity analysis results at the two sample points.

No. of the sinking velocity samples
Measured data Empirical formula provided by [20] Kriging model IK-RSM No. of the sinking velocity samples Figure 5: Error comparison of the various models for the sinking velocity. s to 75 m/s, and, according to the general proportional relationship between the approach velocity and the sinking velocity, the sinking velocity should increase. However, the measured value of sinking velocity decreases from 3.8 m/s to 3.6 m/s. Here, it can be seen that, in the sensitivity analysis results of various variables correlated with sinking velocity, the increment of sinking velocity is −0.0957 m/s when the engaging velocity V A increases 1%, which indicates that the sinking velocity has a decreasing trend under these circumstances. Similarly, for samples No. 187 and No. 168, the approach velocity increased from 74 m/s to 76 m/s; however, the measured value of sinking velocity decreases from 4.2 m/ s to 4 m/s. It also can be seen that, in the sensitivity analysis results of various variables correlated with sinking velocity, the increment of sinking velocity is −0.0441 m/s when the engaging velocity V A increases 1%. erefore, through the sinking velocity IK-RSM and the sensitivity analysis method proposed, we have explained the contradictory phenomenon that when the approach velocity increases, the sinking velocity decreases at some samples. It could provide certain technical support for the flight attitude control related to the sinking velocity during the actual flight of carrier-based aircraft.

Conclusion
e deck landing sinking velocity of carrier-based aircraft is affected by aircraft carrier attitude, sea condition, and aircraft performance. Based on the deck landing measured data of the F/A-18A aircraft, the influence degree of the deck landing parameters on the sinking velocity under the actual operating environment has been explored, by using the partial correlation analysis of multivariate statistical theory and the IK-RSM of the sinking velocity, and the following conclusions are formed: (1) It can be found that the 4 parameters are strongly correlated with the sinking velocity: the aircraft glide angle, the deck pitch angle, the engaging velocity, and the approach velocity, and the mentioned parameters could be used to establish the impact analysis model of the sinking velocity. (2) e genetic algorithm is applied to the correction coefficients optimization of the kernel functions, and the IK-RSM of sinking velocity is formed and can be used to predict the sinking velocity of the carrierbased aircraft. Compared with the empirical formula and the Kriging model, the sinking velocity prediction accuracy indexes of IK-RSM are the best; for example, the coefficient of determination is 0.981 (the coefficient of determination is more close to 1, which means higher precision), the mean relative error is 1.813%, and the maximum relative error is 6.771%.   (3) rough the sensitivity analysis of the IK-RSM of the F/A-18A aircraft sinking velocity, it can be seen that the sensitivity of the engaging velocity V E is the greatest, that of the aircraft glide angle C G is the second, that of the deck pitch angle C P is the third, and that of the approach velocity V A is the least. Furthermore, through the sensitivity analysis method proposed and the gradient of sinking velocity at all correlated parameters, we have explained the contradictory phenomenon that when the approach velocity increases, the sinking velocity decreases at some samples. It could provide certain technical support for the flight attitude control related to the sinking velocity during the actual flight of carrier-based aircraft.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.