A novel method of internal ballistics identification and performance prediction for SRMs based on genetic algorithm

Improving the identification accuracy of internal ballistic parameters in the solid rocket motor(SRM) is of great significance in guaranteeing that missiles fulfill their intended operational missions. In practice, the internal ballistic performance is according to the inverse calculation burning area obtained by the measured pressure data of the SRM and the measured burning rate, which still has ascending space for optimization in the prediction accuracy. Accordingly, a genetic algorithm-based method for the identification of internal ballistic parameters and performance prediction for SRMs was proposed. Based on the measured, data of limited test runs, the initial identification of the burning rate coefficient, pressure exponent and propellant density was carried out by GA (Genetic Algorithm). The model was updated on the basis of the inverse calculation burning area obtained by identification results. Then the secondary identification was carried out to modify the key parameters. The Φ50mm laboratory-scale test SRM was analyzed as an example. The internal ballistic performance in the SRM was predicted. The calculation results show that the prediction results obtained by the method are in high agreement with the measured pressure data, which verifies the effectiveness of the method in improving the prediction accuracy of the internal ballistic performance.


Introduction
The basic task of internal ballistic in solid rocket motors is to calculate the variation law of combustion chamber pressure with time and space under various operating conditions of the motors.The zero-dimensional internal ballistic calculation method assumes that the parameters in the combustion chamber are uniform, which is simpler and more intuitive than the one-dimensional and multi-dimensional internal ballistic calculation methods.Therefore, the zero-dimensional internal ballistic calculation method is a more widely used solution method in engineering [1].The internal ballistic characteristics of the SRM directly determine the performance of the entire machine, which affects the flight characteristics of the rockets.Further improving the accuracy of internal ballistic prediction is of great significance to guarantee the working reliability of the aircraft.
In practice, parameters such as combustion velocity characteristics are usually identified based on measured data from single or multiple SRMs and the geometric recession law of the combustion area to obtain more accurate internal ballistic prediction results.At present, scholars have carried out research on the problem of accurate prediction of internal ballistic performance in the SRM from the perspectives of theoretical algorithm modification and parameter identification method improvement.Liu Yang et al proposed a modified combustion velocity coefficient processing method based on the theoretical analysis of the processing error of the measured data of the SRM, which can improve the accuracy of internal ballistic prediction in the SRM with a high pressure-to-strength ratio [2].Li Chunyan et al adapted the existing internal ballistic calculation method from the theoretical point of view for the characteristics of short-time working SRMs, and the obtained predicted results have higher accuracy compared with the measured data [3].
In terms of identification parameter selection, Cavallini found that the nozzle throat ablation rate has a large effect on thrust and combustion chamber pressure by studying the factors affecting the actual flight performance of Vega [4][5].Bianchi took a numerical simulation approach to study the effect of nozzle throat ablation rate and combustion velocity variations on the combustion chamber pressure and found that the change of these two parameters had a significant effect on the pressuretime curve [6].Li Xiaobin et al based on the internal ballistic mathematical model of the SRM working process and measured combustion chamber pressure time-varying data, parameters such as combustion speed and nozzle throat ablation rate can be calculated by the parameter identification method [7].In the improvement of parameter identification methods, traditional identification methods such as least squares and maximum likelihood methods often have problems such as initial point sensitivity.Therefore, methods such as genetic algorithm and particle swarm optimization, which have initial value robustness and global searchability, have been gradually and deeply applied in the field of internal ballistic parameter identification.Fan Chao et al used the genetic algorithm for the SRM parameter identification, obtained the global optimal identification values of the propellant burning rate model and throat diameter change model, and the obtained internal ballistic performance prediction results were in good agreement with the test measurements [8].Pu Xiaohang et al realized the prediction self-correction of the internal ballistic of the SRM by obtaining the prediction parameters through a genetic algorithm based on partially measured pressure data [9].Zhang Xiaoping et al established a formulation characterization method for high-energy solid propellants and used a back propagation neural network optimized by the genetic algorithm (GA-BP) to predict the highpressure combustion performance of the propellant and its changing law [10].
The traditional SRM measured data processing often through the combustion time of the average pressure and propellant average combustion velocity calculation to obtain the combustion velocity coefficient and other parameters.However, the average pressure and the average combustion velocity are not consistent with the combustion velocity modeling correspondence [2].As a result, there is some error in the calculations.Most of the current research work uses methods such as genetic algorithms and neural networks to identify critical internal ballistic parameters.There is no correction and secondary identification of the propellant burning area and internal ballistic parameters based on the measured data, and the accuracy of the identification of the internal ballistic parameters and the performance prediction can be further improved.Therefore, it is necessary to carry out the secondary identification of the key internal ballistic parameters to obtain more accurate identification results, and then improve the accuracy of the internal ballistic prediction of the SRM.
In this paper, a novel method of internal ballistics identification and performance prediction for SRMs based on genetic algorithm was proposed.This method combines the combustion surface backcalculation with genetic algorithm-based parameter identification for the first time.The secondary correction of internal ballistic parameters through the combustion surface back-calculation can improve the accuracy of the parameters, which significantly improves the accuracy of internal ballistic prediction.Based on the measured pressure data of the SRM, the preliminary identification of the burning rate coefficient, pressure exponent, and propellant density is carried out by the genetic algorithm.Based on the identification results, the combustion area data are corrected.On this basis, the secondary parameter identification is carried out to finalize the key internal ballistic parameters.According to the results of the second parameter identification and the modified combustion area date, the internal ballistic performance of the SRM can be obtained by forward prediction.The 50mm laboratory-scale test SRM was analyzed as an example.The accuracy and validity of the internal ballistic performance prediction within the method are verified, and the relevant results can provide technical references for the optimized design of solid rocket motors.

Internal ballistic prediction model
In this paper, the zero-dimensional internal ballistic model [11][12] is used as the basis for the positive prediction of SRM performance, which has the following assumptions: (1) the gas is an ideal gas and the gas flow is a zero-dimensional constant flow; (2) the ignition process of propellant is completed instantaneously.Considering the effect of the ignition powder on the internal ballistic calculation, combining the law of mass conservation in the combustion chamber, the law of energy conservation, the equation of volume conservation in the combustion chamber, the equation of combustion velocity, and the equation of gas state, the derived differential equations for internal ballistics is shown in Eq. (1).
where c is the gas density in the combustion chamber; Vc is the combustion chamber free volume; p m p m is the propellant gas generation rate; i m i m is the ignition powder gas generation rate; t m t m is the nozzle gas flow rate; pc is the combustion chamber pressure; k is the gas specific heat ratio; is the heat loss coefficient; Rc is the gas constant; p T is the propellant adiabatic combustion temperature; fi is the gunpowder of ignition powder; Tc is the gas temperature in combustion chamber; Ab is the burning area of propellant; rp is the burning rate of propellant; Ai is the burning area of ignition powder; ri is the burning rate of ignition powder; a is the burning rate coefficient; n is the burning rate pressure exponent.(2) where p is the propellant density; i is the ignition powder density; ei is the burn-off thickness of ignition powder; ei0 is the diameter of ignition powder; is the function relating to specific heat ratio. (5)

Genetic algorithm-based method for performance prediction
Considering the problem of the SRM test cost, it is of great significance to carry out the study of accurate identification of key parameters of internal ballistic based on the data of one or several test runs to reduce the cost and improve the design efficiency of solid rocket motor design.Based on the identification method of internal ballistic parameters in engineering, this paper proposes a genetic algorithm-based method for the identification and prediction of internal ballistic parameters, which can improve the accuracy of internal ballistic prediction.

Methods for identifying internal ballistic parameters in engineering
The engineering identification of the internal ballistic parameters is roughly based on the measured combustion chamber pressure curve, post-test throat area, and other data to calculate the combustion velocity parameter and back-calculate the burning area.The average burning rate is often calculated by Eq. ( 6).Calculate the burning rate coefficient according to Eq. ( 7).Burning area back-calculation was performed by burning rate coefficients as well as Eq. ( 8) to obtain burning area data.Finally, an internal ballistic forward prediction is performed based on these parameters to obtain the performance prediction curve.
where W is the thickness of the propellant; t is the burning time; At is the area of the nozzle throat; C * is the characteristic speed.

Process of genetic algorithm-based performance prediction method
The above-average burning rate and average pressure do not have actual correspondence.Therefore, there is some error in the calculation.There is still much space for optimization in the accuracy of internal ballistic performance prediction.In this paper, an internal ballistic parameter identification and accurate prediction method is proposed, which is an improvement of the engineering method.
Based on the measured pressure data of a single SRM, selected portions of the motor's stabilized operating period were calculated.Based on the established identification correction method to obtain all the key parameters required for internal ballistic prediction, such as burn rate model parameters and burn area data, internal ballistic performance prediction is carried out.This method can improve the accuracy of internal ballistic prediction, and the specific process is shown in Fig. 1.Starting from the measured pressure data of the SRM, the first parameter identification of the burning rate coefficient, pressure exponent, and propellant density is performed based on the genetic algorithm.The burning area-grain thickness curve was obtained by parameter identification results and the burning area-pressure equation.The computational model for parameter identification is then updated.A second parameter identification is carried out using the resulting burning area data and genetic algorithm.The initial identification parameters are corrected according to the results of the secondary identification.A positive prediction of the ballistic performance in the SRM is carried out based on the obtained parameters and the burning area data.

Genetic algorithm-based identification of internal ballistic parameters
The identification of internal ballistic parameters is mainly based on the inverse back-calculation of the measured pressure data during the SRM operating time.This work is accomplished based on the genetic algorithm to identify three parameters, namely, burning rate coefficient, pressure exponent, and propellant density, in the following steps: (1) Based on the measured data information and design experience, the time interval t is selected as the minimum time microelement for calculating the degree of fit of the two curves; (2) A range of initial values for the given parameters, where the burning rate coefficient and the pressure exponent are also required to satisfy the burning rate model relationship; (3) N sets of data(a, n, p) are randomly given as the initial population of the model for a given range of initial values, and each set of data is an individual of the initial population; (4) A forward prediction mathematical model is established based on zero-dimensional internal ballistic theory, and an internal ballistic performance prediction can be performed based on any set of data.
(5) Since the data from the stabilized operating section of the SRM is more representative, a total of M nodes are selected in the stabilized operating section at intervals of t.Establish the fitness function of the internal ballistic prediction model, which is a row vector consisting of the difference between the predicted and measured pressures at M nodes.The norm of this row vector is used as a judgment of how well the data set is adapted; (6) Elimination of poorly adapted initial individuals and retention of highly adapted initial individuals.Data are exchanged and mutated among the retained individuals to form offspring of the initial population, and the new population is again selected by a fitness function.(7) Repeat the previous step until the results converge.By substituting the data with the highest fit into the internal ballistic prediction model, the pressure curve with the highest fit to the measured data is obtained.
The above are the steps of initial parameter identification, and based on the results of the initial parameter identification, the burning area back-calculation is carried out.Updating the mathematical model of internal ballistic prediction according to the back-calculated burning area data and then correcting the parameter identification.The range of initial values of the parameters for the second identification can be appropriately narrowed to speed up convergence and improve efficiency.

50mm laboratory-scale test SRM
In order to verify the accuracy and validity of the method of internal ballistic parameter identification and performance prediction in SRMs based on genetic algorithm, 50mm laboratory-scale test SRM is selected as the validation case of the study.The specific parameters of this type of SRM are shown in Table 1, Fig. 2 shows its Schematic, and Fig. 3 shows its physical drawing.

First parameter identification
The parameter identification process is realized based on MATLAB software, and the optimal set of data * a , * n , * p is identified by inverse back-calculation of the genetic algorithm.Based on the zero-dimensional internal ballistic prediction model to obtain the prediction pressure curve, the fit between the prediction pressure curve and the measured pressure curve is used as the fitness function, and the minimum value of the fitness function is solved.The steps for establishing the fitness function for the initial parameter identification are as follows: (1) The minimum time interval t = 0.001s was taken for curve fitness calculation; (2) Within the stabilized work section, 2000 points were selected as curve fitness calculation time nodes at time intervals of t; (3) Calculate the difference vector between the predicted and measured pressure values at all nodes, and use Euclid's norm of this row vector as the fitness function.
During the first parameter identification process, the initial value ranges of the three parameters to be identified are set to be 14.5mm/(s•MPa n ) a 15.5mm/(s•MPa n ),0.21 n 0.25,1650kg/m 3 p 1750kg/m 3 .Based on the genetic algorithm for parameter identification, the results converge after several iterations of calculation, and the solution process is shown in Fig. 5.The results of the initial parameter identification are burning rate coefficient a=15.36mm/(s•MPan ), pressure exponent n = 0.231, and propellant density p = 1679 kg/m 3 .The internal ballistic prediction is performed based on the parameter identification results, and the preliminary predicted pressure-time curve can be obtained, as shown in Fig. 6.

Second parameter identification
Based on the results of the initial parameter identification, the burning area back-calculation is performed according to the burning area-pressure equation, and the burning area data of the existing mathematical model is updated.In order to accelerate convergence, the range of initial values of parameters for the second identification is appropriately narrowed compared to the first identification.
The initial value ranges of the three parameters to be recognized are set as follows: 15.0mm/(s•MPa n ) a 15.5mm/(s•MPa n ),0.22 n 0.24,1650kg/m 3 p 1700kg/m 3 .As shown in Fig. 7, the results converge after several iterations of calculations, and the second parameter identification results are as follows: burning rate coefficient a=15.26mm/(s•MPan ), pressure exponent n = 0.235, and propellant density p = 1696 kg/m 3 .Based on the back-calculated burning area data and the results of the second parameter identification, the internal ballistic prediction is performed again, and the modified pressure-time curve shown in Fig. 8 can be obtained.As can be seen from the figure, the accuracy of the second performance prediction is higher than that of the first, and the prediction results are highly consistent with the measured data.

Conclusion
This article proposes a genetic algorithm-based method for internal ballistic parameter identification and prediction in SRMs.Computational analysis using 50mm laboratory-scale test solid rocket motor as a study case.The research conclusions of this article are as follows: (1) Based on the quadratic parameter identification correction and burning area back-calculation, the prediction results which are in high agreement with the experimentally measured pressure data are obtained.The effectiveness of the method for improving the prediction accuracy of internal ballistic performance is verified.
(2) In addition to the case parameters in this paper, several prediction parameters with strong dispersion characteristics under the environmental profile of SRMs can also be identified and corrected.Under the condition of having the relevant parameters of the motor case, propellant, grain, nozzle and the pressure data of the motor test run, the related method is also applicable to the SRM with complex propellant configurations and burning area recession patterns.The application of the method is highly generalized.
(3) Based on this internal ballistic parameter identification and prediction method, the measured data from a limited number of SRM test runs can be fully utilized.It is capable of carrying out the accurate prediction of internal ballistic performance and grasping the dispersion of internal ballistic performance under complex environmental profiles.This method can provide a technical means to improve the efficiency of SRM design.

Figure 1 .
Figure 1.Parameter identification and prediction method process.

Figure 3 .
Figure 3. Physical drawing of 50mm laboratory-scale test SRM.The grain of the 50mm laboratory-scale test SRM is made by modified double-base propellant.Obtaining combustion chamber pressure-time data from the SRM ground static ignition tests to verify the accuracy of parameter identification and performance prediction.Fixing the SRM on the thrust test stand during the pressure test.The pressure sensors are mounted on the head and tail of the combustion chamber.The specific mounting test locations are shown in Figure 4.

Figure 5 .
Figure 5. Calculation progress of first parameters identification.

Figure 6 .
Figure 6.Comparison between first prediction and pressure date measured.

Figure 7 .
Figure 7. Calculation progress of second parameters identification.

Figure 8 .
Figure 8.Comparison between second prediction and pressure date measured.