An optimization model of parameter matching for aircraft catapult launch

Zeyang ZHOU, Jun HUANG

School of Aeronautic Science and Engineering, Beihang University, Beijing 100083, China

KEYWORDS

Abstract To efficiently and fully utilize aircraft carrier resources, an optimization model is presented to deal with parameter matching between aircraft and carrier in the process of aircraft catapult launch. Based on carrier aircraft longitudinal dynamic equations and theorem of kinetic energy in catapult launch course, the work characteristics of different forces are learned and a theory model of parameter matching is deduced.In view of the uncertainty of the model parameters of the theory model and the low matching accuracy of the approximate model,an optimization model of parameter matching is introduced in line with the structure of theory model and the approximate model and is generated by the proposed immune genetic algorithm. Compared with the original genetic algorithm and immune algorithm, the proposed algorithm has better calculation accuracy and convergence. The calculation results show that the optimization model occupies certain application value of engineering estimation from the comparison with the relevant literature data and has higher precision than the approximate models.The validity of the proposed approach is verified with numerical case study on a carrier based aircraft.

1. Introduction

With the development and innovation of the carrier catapult,the existing steam catapults are gradually replaced by electromagnetic launch technology.Aircraft-to-carrier adaptability is one of the core contents of the overall design of ship-borne aircraft1and is related to the utilization of ship’s resources and flight safety.The research of parameter matching between aircraft and carrier is of great importance in engineering application.

The problem of parameter matching between aircraft and carrier is a major concern in catapult launch because it is of importance for carrier aircraft design and launch engineering application.1,2The takeoff mass is a basic parameter of the carrier aircraft and is related to the fuel and weapon mass that could be loaded.2The catapult force is the output of the aircraft carrier catapult and determines the great kinetic energy required for the aircraft to take off. The take-off speed is an important performance during the take-off of the aircraft3and directly affects the climb and track safety after leaving the carrier. The matching relationship between these three parameters has always been the research focus of the carrier aircraft and its theoretical connection is urgently needed to be explored. To deal with the problem, a set of parameter matching values is obtained to manifest the adaptation characteristics between catapult energy, nose gear fast-extension force and elevator preset angel.3The influences of the single and combined parameters including aircraft and circumstance were evaluated and analyzed.4–6The catapult launch environment is a high lift, dynamic environment,7which suggests the importance of studying flight safety.8,9A catapult launch criteria is put forward to determine the critical domain of the parameters,10including aircraft engine thrust, aircraft mass,catapult force et al. The indexes including the maximal subsidence,the rate of climb at leaving time and the minimum altitude from sea level,11are solved to indicate the launching security of the aircraft,12based on the multi-agent system simulation model.13Therefore, it could be noted that the Theory Model (TM) of parameter matching between aircraft and carrier captures less attention especially on the three basic parameters, aircraft mass, catapult force and takeoff velocity. The optimal solution of the parameter matching model could be determined by Genetic Algorithm(GA)or Immune Algorithm(IA), where the convergence of the optimal solution could be improved by the idea of immune algorithm.

Based on the above issue analysis, genetic algorithm and immune algorithm with parallel computing and global solution search behavior14are studied to handle the matching model and parameters optimization.15,16The research status of genetic algorithm is discussed in the aspects of genetic operators,parameters identification and convergence.17A surrogate model based on Radial Basis Function (RBF) neural network and combined with genetic algorithm is presented18and an Immune Genetic Algorithm (IGA) based on the theory of immunity in biology is proposed.19A full description of the algorithm (IGA) is presented before investigating its application in the context of software test data generation using some benchmark programs.20The genetic algorithm is suitable for searching global optimal solutions, regardless of whether the search area is continuous or not and whether the target parameters are independent of each other.15–17In addition, with the rise of electromagnetic catapults, a multi-segmented longstroke dual-stator pulsed linear induction motor is put forward for electromagnetic catapult.21–23The ElectroMagnetic Aircraft Launch System (EMALS) seems to be able to overcome the limits of steam catapults, and a number of configurations have been proposed.24,25Based on the commonness of the two type of catapults, the theoretical model of parameter matching between aircraft and carrier is of great importance in engineering application. Although the Approximate Model(AM) and the Optimization Model (OM) are derived from the theoretical model, the solution of the model parameters still requires an algorithm with parallel computing power.Comprehensive consideration of the independence of model parameters, the global search ability of GA and the convergence characteristics of IA, an immune genetic algorithm is designed to solve the model parameters of the matching models.

The existing research results cover some parameters match problem including nose gear fast-extension, elevator control and catapult energy,but the field still is quite vibrant and alive when considering the theoretical relationship and matching model containing the important parameters of aircraft and carrier.To begin with,we derive a theoretical matching model containing aircraft mass,catapult force,takeoff velocity,work ratio and model coefficient,which provides theoretical support for the approximate models. Secondly, a more flexible optimization model is designed to overcome the shortcomings of the approximate models.On this basis,this manuscript considers the parallelization,global selection and convergence ability and establishes a reasonable aircraft-carrier parameter matching model solving algorithm.

Regarding to the organization of this article,the theoretical model is derived in Section 2. The optimization algorithm is presented in Section 3. Simulation and model results are compared and discussed in Section 4 and finally, the conclusions are presented in Section 5.

2. Equations and models

Harboring the idea that the theoretical model of the problem could be established from initial equations based on the characteristics of the catapult launch. The launch process of carrier-based aircraft could be described in schematic Fig.1,whereWis the aircraft weight,Tcis the catapult force,Vtis the takeoff speed,t0is the start time of the catapult process,ttis the takeoff time,Vcis the carrier velocity,w0is constant wind,Tis the aircraft engine thrust,Lis the aerodynamic lift,Dis the aerodynamic drag, α is angle of attack, θ is the pitch angle,γ is the track angle,φtis the aircraft engine thrust angle.

At the beginning point of the catapult stroke,the aircraft to be launched is hold by the release bar.Then the catapult force is increased rapidly until the release bar is broken.26The aircraft is pulled by the piston of the catapult to accelerate in the catapult stroke on carrier deck. After the catapult bridle is released, the aircraft begins to slide freely due to inertia and engine thrust force until it reaches the takeoff velocity or leave the aircraft carrier.The aircraft then begins the climbing flight,which indicates the end of the entire launch process.

The longitudinal dynamics equations of aircraft catapult launch in the ship coordinate systemOxzcould be described as:

Fig.1 Problem of aircraft catapult launch parameters matching.

wherewis the deck wind.During the free taxiing stage,the aircraft takes off when it rushes out of the carrier deck orNm=0. In the aircraft catapult take-off process, the theorem of kinetic energy in catapult launch could be expressed as the following form:

where ΔEkis the kinetic energy change of the aircraft,Ecis the work done byTc,Esis a defined sum work,Etis the work done by aircraft engine thrust,Xtis the displacement of forceT,Edis the work done by aerodynamic drag,Xdis the displacement of forceD,Eris the work done by ram drag,Xris the displacement of forceDr,Efis the work done by the friction between landing gears and carrier deck,Xnis the displacement of forcefnandXmis the displacement of forcefm. The dynamic equations for aircraft catapult simulations and the aircraft data are shown in Ref.10.

Generally, the catapult force participate in parameters matching in the form of average catapult force or catapult energy1,4not the change of catapult force becauseTcis the function of catapult stroke rather than a variable or a constant. HereTcis divided into two parts as the following form:

whereT0is the base catapult force still as the function of catapult stroke,Fcis the change of catapult force,Xkis the length of catapult stroke,E0is the work done by base catapult force.According to the work characteristic in catapult launch,EcorE0is much larger than the other four work, even much larger thanEs.The work characteristic in the launch process could be defined as:

where ε is a work ratio representing the work characteristic.

2.1. Theory model

The motion of a carrier aircraft is usually represented by a multi-degree-of-freedom kinematics equation, which makes it difficult to directly display the matching relationships of certain parameters. In order to obtain a more concise model, an operation of natural logarithm of both sides of Eq. (4) is performed to obtain the Theoretical Model(TM)as the following form:

wheremis the aircraft mass,C0is a model parameter. TM gives the basic solution to the parameter matching problem.The method for solving the model parameters and the changes are presented,which lays the foundation for the following text.Because of its theoretical derivation, TM is generally applicable to steam and electromagnetic catapult and the paired carrier aircrafts in full takeoff mass range.Taking a difference for each variable of Eq. (11), the following equations could be used for analysis:

This difference equation predicts the link between the percentages of catapult parameters when analyzing the effects of parameter changes. The change of takeoff velocity, aircraft mass, catapult force andE0could be directly reflected in the Eq. (14), other parameters such as deck wind, carrier motion and disturbance affecting the ε value.

2.2. Approximate model

Considering that ε is a positive decimal close to 0,the Approximate Model (AM) of the matching problem could be expressed as follows:

where ΔC0≈0 is the simplifying condition for the derivation of the approximate model,mnis near to standard takeoff mass,mmis the maximum takeoff mass. Compared with TM, AM appears simple and contains only one model parameterC0.AM requires ΔC0≈0 or ε ≈0 where the work done by catapult force (Ec) is much greater than the sum work done by the other forces (Es) from preload to take-off. When solvingC0by optimization method, AM needs at least one (m,Fc,Vt) sample point. The disadvantage of this model is that theC0value is not unique and it causes errors in the calculation results.

2.3. Optimization model

However,it is obviously cumbersome to calculate the parameter matching by the theoretical model. The direct use of the approximate model to calculate the matching problem is not accurate enough due to the uncertainty of theC0value. In order to achieve high efficiency and precision application of TM and eliminate the drawbacks of AM, an Optimization Model (OM) with three undetermined coefficients could be developed as:

whereA,BandCare undetermined parameters. OM could overcome the uncertainty of AM’sC0value and the complexity of TM. ΔC0≈0 or ε ≈0 are also required for OM and there are at least three (m,Fc,Vt) sample points when solving the model by optimization method. OM is an engineering estimation method for catapult parameters matching. Its prototype comes from the theoretical model with clear physical meaning.

3. Immune genetic algorithm

To solve the OM by the proposed algorithm,the optimization goal could be defined as:

whereFvis the fitting goodness of the velocityVt,Fvmis the maximum ofFv,A*,B* andC* are the optimal solution ofA,BandCseparately whenFvreaches the maximum value.The flowchart of the genetic algorithm with step immune operator is shown in Fig.2.In order to search for the optimal solution as soon as possible, the algorithm is terminated whenn＞Nor the fitness reaches an expected value wherenis the current evolution number,Nis the total evolution number.

3.1. Algorithm formula

The initialization ofA,BandCvalue is proceeded as the following form:

whereVi,miandFiare small amounts of catapult launch simulation results,Viis takeoff speed,miis aircraft mass,Fiis catapult force change,Lmis the number ofmi. The initial values of parameterA,BandCcould be determined as:

whereNpis the size of the initial population,ris a true random operator to generate a random value in the given range.

Fig.2 Flowchart of proposed genetic algorithm with step immune operator.

The value of takeoff speed and the fitness of each individual could be determined as:

The total fitness (FS) of the population and the selection probability (Ps) could be calculated as:

3.2. Step immune operator

The step immune operator could be described as the following steps:

whenn=1, performing the genetic algorithm without the step immune operator;

whenn=2,3,...,N.

Step 1.Sorting the current population according to the fitness of each individual in ascending order; the population is recorded asPn.

Step 2.Selecting the individual with maximum fitness of each generation in the previousn0generations; then0value is favorable for accelerating the increase in the average fitness of the next generation population where the initial value ofNais equal ton; the current step population is recorded asP’.

Step 3.Replacing the firstn0individuals ofPwithP’; the population is recorded asP’’.

Step 4.Upsetting the populationP′′with a random order for the crossover operator in the next evolution. The optimal solutionA*,B* andC* values could be obtained to verify the validity of the parameter matching model in full takeoff mass range when the algorithm is terminated. The proposed method could also be used to acquire the model AMc that optimizes onlyC.

4. Results and discussion

Fig.3 supports that as the aircraft mass increases,the value of the aircraft takeoff speed decreases but the flight path sink value gets the adverse change because a larger takeoff mass will increase the burden of the catapult, and only less kinetic energy could be obtained under the same work conditions,which in turn will lead to insufficient aircraft climbing ability.The aircraft model is A6‘Intruder’carrier based aircraft with a standard takeoff mass 24.282 t (1 t=1000 kg), a maximum takeoff mass 26.581 t and a steam-catapult launch mode.Simulation results are used to calculate and compare the results of various models including AM, AMc and OM.

Table 1 shows the difference between the two simulation results which could be used to calculate the approximate model. The work done by the catapult force is much larger than the other four work,even much larger than the sum workEsbecause catapult force is so huge that ε value is approximately equal to 0. Concerning that the value of ε is a positive decimal less than 0.1, the approximate model could be obtained based on the simplifying condition ΔC0=-0.0020.However, both the twoFvvalues of the approximate model are not large enough to maintain the accuracy of the matching results and therefore the approximate model have to be optimized. Comparing the steam catapult force in Ref. 10 and the electromagnetic catapult force in Ref.23,the theory model is suitable for the electromagnetic catapult launch when the two conditions ε ≈0 and ΔC0≈0 are satisfied.

4.1. Rationality of the optimization model

Fig.4 supports that the OM results agree well with the given data and the absolute value of the relative error is less than 0.6379% where the global solution (A*,B*,C*) of OM model generated by the proposed algorithm is (0.4980, 0.4976,4.4556) and the given data comes from the Table 1 in Ref. 3.This proves that the OM model is applicable and accurate for this parameter matching problem, while OM is also easier to be determined through limited experimental data or sample points because OM comes from TM and is relatively flexible.

4.1.1. Compared with AM

Fig.5 shows that the results of the approximate model are parallel with the simulation results and the relative error values of the approximate model are a bit large.The approximate model bring a large relative error around-7%where all the absolute error values of takeoff velocity are between-5 m/s and-4 m/s because of the instability of theC0value and the large range of aircraft takeoff mass and catapult force. It could be noticed that the Approximate Model (AM) is not accurate enough to estimate the takeoff velocity for the carrier-based aircraft in the catapult launch process. According to the drawbacks of the approximate model, it is necessary to solve and verify the optimization model for the problem.

Table 2 shows that theA,B,Cvalues of the optimization model generated by the presented algorithm are obviously different from that of the approximate model. Particularly,Cvalue increases to about 3.51 and there is a slight change in the values ofAandB. Moreover, the maximum fitness values of the optimization models significantly increase compared with the approximate model.

Fig.6 reveals that theVtcurves of OM fit more closely with the curve of simulation than AMVtcurve while theVtrelative error values of OM are far less than those of AM because OM has more accurate and reliable model parameters after the iterative solution by the proposed algorithm. All theVtrelative error values of OM are in the range [-0.326%, 0.3017%],showing that the OM’s interpretation of the takeoff speed is accurate enough.

Fig.3 Aircraft catapult launch simulation curves with various m, Fc=100%, w=8 Kn (1 Kn=0.514 m/s).

Table 1 Simulation results for solving the approximate model, m=24.282 t, w=8 Kn.

Fig.4 Data of Ref. 3 to verify optimization model.

Fig.5 Results of approximate models AM with C0=3.4395.

Table 2 Global solution of A*, B* and C* generated by proposed algorithm.

Fig.6 Comparison between optimization models and approximate model.

Fig.7 supports that the OM method has advantages in improving the curves of velocity and flight track compared with AM. Due to the large error of AM, the track curve is not so safe and the velocity curve of the first two stages of catapult process are also significantly lower than other ones. The velocity-time curve determined by OM is obviously better than the other three. The larger take-off speed is accompanied by a stronger climbing ability, which makes the flight path more secure as shown by thez-time curve.

4.1.2. Compared with AMc

It is necessary to discuss theCvalue optimization of AM because the other two model parameters of AM and OM are very close.A model(AMc)that only optimizesCvalue is generated by the proposed algorithm whereC(AMc)=3.5114.TheVtdifference(Dv)between OM and AMc could be defined as follows:

Fig.8 manifests that the difference inVtbetween OM and AMc is small but most of the OM results are slightly larger than AMc in the full mass range (Appendix A Table A1).There are 49 points whereSv=1 are positive in the total 65 sample points and the minimumDvvalue is just -0.0113 m/s in those points whereSv=-1. Larger takeoff velocity also ensures the safety of aircraft flight track because most pilots expect a catapult launch without flight sinkage.10This shows that the OM model is superior to the AMc model in terms of pilot life support and carrier-based aircraft launch safety.

Fig.9 supports that the OM method still has advantages in improving the curves of velocity and flight track compared with AMc.AMc matching results could catch up with the original performance curves, but it is obviously not perfect compared with OM. Compared with the other three curves, OM match demonstrates unparalleled speed and track advantages,which makes the aircraft faster and safer to leave the carrier.

4.2. Comparison among different algorithms

Fig.7 OM match method to solve catapult force increase, Fc0=100%, w=8 Kn.

Fig.8 Vt difference between OM and AMc.

Fig.9 OM match method to solve the mass increase, m0=24.282 t, w=8 Kn.

Fig.10 Influence of step immune operator on individuals.

Fig.10 supports that step immune operator has a strong normative role in selecting the next generation individuals. To clearly show the individual’s evolutionary process, 10 initial values ofCare randomly generated as a small-scale populations to perform 20 evolutions. From Fig.10(a) to Fig.10(b), 4 of the random 10 individuals are selected in the range[3.514, 3.516] and the rest are relocated. 7 individuals are selected over 3.512 with generations increasing from generation=2 to generation=4. There are only 2 individuals left below 3.510 when the generations reach 20.This demonstrates the powerful immune-selective effect of step immune operators on individuals so that individuals with high fitness are more likely to evade conventional selection operations.

4.2.1. GA results

Fig.11 manifests that the promotion of total fitness and the randomness and diversity of solution of the original GA is worth learning and improving. It takes the original GA more than 200 steps to make the total fitness exceed 59.99595 and the maximum fitness over 0.9999905 because GA is a kind of evolutionary algorithm which draws on some phenomena in evolutionary biology, including inheritance, mutation, natural selection and hybridization.These operations make GA better search for the optimal solution,but cannot guarantee the convergence of the search process.

4.2.2. IA results

Fig.12 manifests that the total fitness of IA exceeds 59.99595 and the maximum fitness exceeds 0.9999925 after about 100 steps while the maximum fitness is better in convergence because of the immune operator to retain the best individual of the previous generation. However, the convergence characteristics of IA’s total fitness still need improvement. The optimal individual source algebra and quantity of the original IA need to be reasonably processed to ensure the convergence of the algorithm and the diversity of the optimal solution.Compared with the original GA, the original IA has a higher total fitness level, and the maximum fitness converges better.The advantages of GA and IA could be the development ideas of IGA operators, including immunization, replacement and reset operators in the immune link.

4.2.3. The presented algorithm

Fig.13 manifests that both the total fitness and the maximum fitness have been greatly improved with the increase of generations. Within 0–100 generations, the total fitness increases rapidly and fluctuates a little because the number of outstanding individuals has not yet been accumulated enough. Afterwards, the rate of increase in total fitness slows but becomes more stable. The maximum fitnessFscquickly exceeds 0.5 and then increases slowly with small fluctuations. This shows that the proposed algorithm is effective for improving the total fitness and it could improve and maintain the maximum fitness very well.

Fig.14 supports that the matching connection between the three launch parameters (m,FcandVt) presents a ladder type linear relation and the takeoff speed values of the optimization model are consistent with the simulation results.In the stage of small catapult force, there is a little gap ofVtvalues between simulation results and the optimization results because the absolute value of ΔC0increases leading to minor fluctuations inCvalues when the catapult force decreases but this adverse impact has been well improved in the stage of a more powerful catapult force.This shows that OM is accurate and reliable to estimate the takeoff velocity and deal with the parameter matching problem.

Fig.11 Total and maximum fitness of original GA, where Fsc=(Fs-59.99)×10000, Fmc=max(Fv)×105–99999.

Fig.12 Total and maximum fitness of original IA, where Fsc=(Fs-59.99)×10000, Fmc=max(Fv)×105–99999.

Fig.13 Total and maximum fitness of presented algorithm, where Fsc=(Fs-59.99)×10000, Fmc=max(Fv)×105–99999.

Fig.14 Testing optimization model in full mass range.

Fig.15 presents the change or loss of catapult force caused by steam leakage or the staleness and instability of the catapult, which is collectively referred to as steam leakage. Compared with the stability and linear controllability of the electromagnetic catapult, the steam catapult exhibits low efficiency and instability.Considering the influence of steam leakage (l), the catapult force could be expressed as:

Fig.15 Catapult force with steam leakage.

Fig.16 Impact of steam leakage, l=0:0.01:0.3, Fc=85:5:110(unit :%), w=8 Kn.

Fig.17 OM match method to solve steam leak, Fc=100%, w=8 Kn.

Fig.16 shows that a 30%steam leakage could cause a huge loss of takeoff velocity, and theVtvalues calculated by OM also have a large error. Only 29.5669% of the 186 sample points with steam leakage are allowed when the absolute relative error ofVtis less than 3%.This shows that the steady output of the steam catapult force is of critical importance for the aircraft launch safety and the applicability of the OM model. Thanks to the Eq. (14) of TM to adjust aircraft mass and the solution of OM to match the results,the takeoff velocity distribution is restored to the initial state in the face of the catapult force leak.

Fig.17 supports that steam leak would result in reduced catapult power, which in turn reduces aircraft velocity and affects aircraft flight track.At this time,the flight track sinkage is greatly increased and there is a danger of falling into the sea.According to the matching result of TM and OM,the velocity curve of the aircraft is recovered and the flight track curve is improved because the theory model could determine the links between the catapult parameters and the optimization model could give the accurate matching results to guarantee the save launch. This shows that the theory and optimization model could deal with the parameters matching problem well.

5. Conclusions

(1) The theory model of the parameter matching problem is presented, which shows the general connection of aircraft mass,percentage coefficient and leak of catapult force,takeoff velocity,work ratio,the defined sum work and the model constant.It gives the model prototype for the establishment of approximate models and optimization model and provides the commonality of parameter matching between aircraft and carrier both in steam catapult launch and in electromagnetic catapult launch.

(2) Drawing on the advantages of GA and IA,IGA exhibits better convergence characteristics and global search capabilities. Generated by the proposed IGA, the optimization model is accurate and reliable to finish the engineering estimation of the matching problem.

(3) The paper shows that a method of combining theoretical derivation and immune genetic algorithm could be still exploited for solving parameter matching problem in catapult launch.

Acknowledgements

This work was supported by the Excellence Foundation of BUAA for PhD. and the National Natural Science Foundation of China (No. 91641123).

Table A1 Simulation results in the full takeoff mass range.

Appendix A

Table A1 shows that increasing the takeoff velocity could be achieved by increasing the catapult force or reducing the takeoff mass, where the effect of increasing the catapult force is more pronounced. A light takeoff mass (24.055 t) is selected to guard against possible risks or unfavorable launch conditions,27,28such as sudden change of deck wind, adverse ship disturbance movement29–31or catapult lack of power due to steam leaks.