# Optimization Plug Mode of External Fuel and Weapons for Less Changing Aircraft Center of Gravity

Authors: Zeyang Zhou; Jun Huang; Mingxu Yi; Guo Zhong
Journal: Mathematical Problems in Engineering
 (2017)

---

## Abstract

To achieve the minimum change of aircraft center of gravity, plug mode optimization was carried out. Plug mode, including weight and position of external fuel and weapon load, was selected as design mode. Numerical model for calculating aircraft center of gravity was established and used to generate mode samples. Random search algorithm (RSA) was designed and applied for solving the optimization plug mode, and the precise solution of plug mode was determined by full array method (FAM). Compared with typical plug mode, the optimization mode can reach a minimum change in aircraft center of gravity with fewer samples than the solution by FAM. RSA coupled with FAM is an effective scheme for the optimization of aircraft plug mode.


---

## Body

## 1. Introduction

Demands for more fuel and weapons plug and smaller change of aircraft center of gravity have caused aircraft designers and staff to develop system or model to determine aircraft center of gravity. Han et al. developed an adaptive nonsingular fast terminal sliding mode control for aircraft with center of gravity variations to improve the dynamic and the quality characteristics of the aircraft [1]. Yang et al. established aircraft center of gravity estimation using Gaussian process regression models [2]. An aircraft weight and center of gravity indicator system was developed [3]. Gupta et al. proposed a finite element based structural model of a small flexible aircraft, focusing on obtaining a simple model suitable for control design based on a two-step procedure [4]. Li et al. analyzed the position of center of gravity for an unmanned aerial vehicle based on load task [5].Much work has been done on the influence of aircraft center of gravity on flight performance and aerodynamic characteristics. Li et al. did research on effects of partial fuel pump failure on center of gravity control for high-speed aircraft to gain potential performance benefits. Besides, the influence of partial fuel transfer line leak on active center of gravity control was analyzed [6, 7]. Jungo et al. investigated the development of the CEASIOM aircraft design environment for novel aircraft configurations and analyzed static and dynamic stability of the aircraft [8]. Mardanpour et al. did research on engine placement effect on nonlinear trim and stability of flying wing aircraft [9]. Zhen probed into the CG position of minimum drag to check stability boundary and front control boundary for J-7 aircraft [10]. An analytical expression was developed which showed that wing/tail interference drag was determined by wing downwash at downstream infinity [11, 12]. Wang et al. did optimization design of unmanned aerial vehicle with considering the uncertainty of the center of gravity and the influence of the center of gravity on the navigation time is greatly reduced [13]. Diekmann did trim analysis of nonlinear flight dynamics for a civil aircraft with active high-lift system [14]. Tang et al. put forward a calculation and analysis method for aircraft center of gravity to improve flight performance and safety [15].Developing an efficient design and optimization method for plug systems is attracting more and more attention from researchers. Dancila et al. researched fuel burn prediction algorithm for cruise, constant speed, and level flight segments and considered the continuous fuel burn rate variation with time caused by the gross weight modification due to the fuel burn process itself [16]. Patron et al. put forward low calculation time interpolation method on the altitude optimization algorithm for the FMS CMA-9000 improvement on the A310 and L-1011 aircraft [17]. Ivester et al. worked on compact aircraft galley and lavatory arrangement and articulating lavatory partition for an aircraft [18]. Jia et al. developed airborne weapon CAPP system and met the application of an aviation enterprise [19]. Xia et al. investigated the research of release safety analysis method for external stores of weapon [20–22]. Luo developed 1553B bus error correction technique in the management system of aircraft weapon to increase the adaptability between the aircraft and external plug [23].Concentrating on the plug mode based on various missions for fighters with much heavy fuel load or quite a few weapons, a method based on RSA is designed and used to solve the optimization plug mode for minimum changing aircraft center of gravity.In this paper, the design mode and objective function are presented in Section2; and then the method for solving the optimization problem is performed in Section 3; the optimization results are analyzed in detail in Section 4; finally, the manuscript is concluded in Section 5.


## 2. Design Mode and Objective Function

The plug mode is defined as the corresponding relations between external loads and plug positions where the external loads include fuel and weapons loads such as common size auxiliary fuel tanks, various missiles, and bombs. The schematic of plug and spare load for a fighter on active service in current research is shown in Figure1. European EF-2000 is selected as the fighter support, with empty weight 11.150 ton, 13 plug positions, and two types of external fuel tank. The coordinate of aircraft center of gravity can be calculated as(1)XC=Me·Xe+Mif·Xif+∑1pMefi·Xefi+∑1qMewj·XewjMs,(2)Ms=Me+Mif+∑1pMefi+∑1qMewj,where XC is the coordinate of aircraft center of gravity, Me is the aircraft empty weight, Xe is the coordinate of aircraft center of gravity with empty weight, Mif is the total weight of internal fuel, Xif is the equivalent coordinate of internal fuel, Mef is the external fuel weight, Xef is the coordinate of the external fuel plug, Mew is the weight of external weapon, Mew is the coordinate of external weapon, Ms is the total weight of the current aircraft.Figure 1
Plug and spare load schematic, symmetrical loading.Generally, the position of aircraft center of gravity is defined as a percentage of mean aerodynamic chord length:(3)C=XL-XCLmac×100%,Lmac=XL-XT,where C is the position of aircraft center of gravity, XL is the coordinate of the leading edge of mean aerodynamic chord, Lmac is the length of mean aerodynamic chord, and XT is the trailing edge of mean aerodynamic chord.By using symmetrical loading, there are 7 plug positions to carry weapons or fuel tanks as shown in Figure1. With specific task, external fuel and weapons can be determined. Then plug mode can be designed as(4)M=M1,…,M7,Mi∈Mef1,…,Mefp,Mew1,…,Mewq,X=X1,…,X7.The plug mode, namely, the one-to-one correspondence betweenMi and Xi, i=1,2,…,7, can be solved by determining the array of Mi and the center of gravity can be calculated by the combination of M and X based on formula (1).


## 3. Methods for Solving Optimization Problem

Because of various types of missiles and different missions, the group ofM and X can reach hundreds of thousands or more, from the theoretical perspective regardless of plug position limit, including the fixed position of heavy external fuel tank or some anti-ship missile. Supposing the weight of internal fuel equals zero and all the plug positions mounted with weapons or fuel tanks, the optimization plug mode can be solved by the following methods.


### 3.1. Full Array Method

For giving mission and fixed type of external fuel load and weapon, the number of the group ofM and X is an arrangement number, from the theoretical perspective. And a fixed plug position can be considered as cutting the length of the array of X. Then the matrix of all the group of M and X can be expressed as(5)F=FAMM,N0=Amn,N1=Ann,N2≤N010,where FAM is a function to list all the array of M, F is N1×7 matrix, N0 is the number of plug modes in different missions, N1 is the number of plug modes in giving missions, N2 is the number of plug modes by RSA, n is the number of active plugs, and m is the number of all the types of external fuel tanks and weapons. According to (1)–(3), the center of gravity and minimum change of center of gravity can be calculated as(6)ΔXCj=XCj-XC0,ΔXCn1=minΔXC1,…,ΔXCN1,where XC0 is the aircraft center of gravity with a typical plug mode under the giving mission.


### 3.2. Random Search Algorithm

Taking into account EF-2000 fighter, the value ofm can reach 17; then it is formidable for FAM to couple with the huge number N0. Therefore, RSA is designed to search the optimization plug mode in a small samples. For a giving mission and optional types of external fuel load and weapons, RSA is designed and used to solve the optimization plug mode as shown in Figure 2.Figure 2
The flow chart of the random search algorithm and results test.HereN2 is a random number or a giving number, ε is critical error of center of gravity change, and R is N2×7 random matrix by arrangement of M as described in (7). Similarly, the center of gravity and minimum change of center of gravity can be calculated as(7)R=RSAM,(8)ΔXCj=XCj-XC0,(9)ΔXCn2=minΔXC1,…,ΔXCN2.The RSA operator can proceed as follows.Step 1.
Obtain or updateN2 value.Step 2.
For eachi=1,2,…,N2,R
i
=
t
[
M
1
,
M
2
,
…
,
M
j
], 1≤j≤7; here j is determined by the number and type of plugs and t is a true random operator to generate a set of nonrepeating permutations to simulate the various plug mode.Step 3.
CalculateC and ε for each Ri.Step 4.
Determine the optimal plug mode with the minimum absolute value ofε among R.Concerning that the valve ofN2 is not large enough to find the optimization solution, N2′ is used to update N2 to search one more time until meeting the error test. Even if the above 4 steps still cannot get the answer, resetting ε can be available to continue the algorithm. The updating and resetting operator for N2 can be described as follows:(10)N2′=r·N2,r=5ΔXCn2>ε0ΔXCn2≤ε.The optimization plug mode can be determined based on (9) with the termination of the random search algorithm. In addition, the ultimatum solution of the optimal plug mode should pass the results test such as flight qualities and operational and wing structural constrains as shown in Figure 2.


## 3.1. Full Array Method

For giving mission and fixed type of external fuel load and weapon, the number of the group ofM and X is an arrangement number, from the theoretical perspective. And a fixed plug position can be considered as cutting the length of the array of X. Then the matrix of all the group of M and X can be expressed as(5)F=FAMM,N0=Amn,N1=Ann,N2≤N010,where FAM is a function to list all the array of M, F is N1×7 matrix, N0 is the number of plug modes in different missions, N1 is the number of plug modes in giving missions, N2 is the number of plug modes by RSA, n is the number of active plugs, and m is the number of all the types of external fuel tanks and weapons. According to (1)–(3), the center of gravity and minimum change of center of gravity can be calculated as(6)ΔXCj=XCj-XC0,ΔXCn1=minΔXC1,…,ΔXCN1,where XC0 is the aircraft center of gravity with a typical plug mode under the giving mission.


## 3.2. Random Search Algorithm

Taking into account EF-2000 fighter, the value ofm can reach 17; then it is formidable for FAM to couple with the huge number N0. Therefore, RSA is designed to search the optimization plug mode in a small samples. For a giving mission and optional types of external fuel load and weapons, RSA is designed and used to solve the optimization plug mode as shown in Figure 2.Figure 2
The flow chart of the random search algorithm and results test.HereN2 is a random number or a giving number, ε is critical error of center of gravity change, and R is N2×7 random matrix by arrangement of M as described in (7). Similarly, the center of gravity and minimum change of center of gravity can be calculated as(7)R=RSAM,(8)ΔXCj=XCj-XC0,(9)ΔXCn2=minΔXC1,…,ΔXCN2.The RSA operator can proceed as follows.Step 1.
Obtain or updateN2 value.Step 2.
For eachi=1,2,…,N2,R
i
=
t
[
M
1
,
M
2
,
…
,
M
j
], 1≤j≤7; here j is determined by the number and type of plugs and t is a true random operator to generate a set of nonrepeating permutations to simulate the various plug mode.Step 3.
CalculateC and ε for each Ri.Step 4.
Determine the optimal plug mode with the minimum absolute value ofε among R.Concerning that the valve ofN2 is not large enough to find the optimization solution, N2′ is used to update N2 to search one more time until meeting the error test. Even if the above 4 steps still cannot get the answer, resetting ε can be available to continue the algorithm. The updating and resetting operator for N2 can be described as follows:(10)N2′=r·N2,r=5ΔXCn2>ε0ΔXCn2≤ε.The optimization plug mode can be determined based on (9) with the termination of the random search algorithm. In addition, the ultimatum solution of the optimal plug mode should pass the results test such as flight qualities and operational and wing structural constrains as shown in Figure 2.


## 4. Results and Analysis

### 4.1. The Results of Typical Plug Mode

For comparison of the solutions between RSA and FAM, a typical plug load including 10 missiles and 3 external fuel tanks is given as shown in Table1. There are 10 missiles and 3 fuel tanks for the fighter to possess plenty of missiles and fuel. By using symmetrical loading, the typical plug mode is presented in Table 2.Table 1
A typical plug load.
Weapon
Fuel
AIM-9
AIM-120
Paveway
Tank 1
Tank 2
Weight/ton
0.0910
0.1507
0.9070
0.7750
1.1625
Number
2
4
4
2
1Table 2
The distribution of the typical plug load.
Typical plug mode
X
1
2
3
4
5
6
7
Plug
AIM-9
Paveway
Tank 1
Paveway
AIM-120
AIM-120
Tank 2Regarding (1)–(3), the center of gravity in the typical plug mode is calculated to be 17.7205%, where Ce=19.50%. Then ΔC0=-1.7795%.


### 4.2. The Results and Analysis of FAM

To obtain the precise solution of the typical plug mode, FAM is used to produce all the array ofM matrix. Concerning the fact that there is a unique position (X7) for fuel tank 2 as shown in Figure 1 and Table 2, the value of N1 is 720. With this in mind, the precise plug mode can be determined as presented in Figure 3. The results of center of gravity are obtained as shown in Figure 4 with a range of 13.1343%–27.8909%. The change of C is presented in Figure 5 in the scope [−6.3657, 8.3909]. To observe the range of C, the results of center of gravity is sorted as shown in Figure 6. Besides, there are 164 plug modes with less changing of the value of C than the typical plug mode. Then the minimum change of C can be determined as 0.0594% where C=19.5594% with the precise plug mode. Because F matrix is a full arrangement of M, the precise solution can be considered reliable.Figure 3
Precise plug mode.Figure 4
Center of gravity by FAM.Figure 5
Change ofC by FAM.Figure 6
Sorting results of center of gravity by FAM.


### 4.3. The Results and Analysis of RSA

Figure7 shows that it is formidable to search the robust results of plug mode when the initial size of N2 is small. Thanks to the action of updating and resetting operator of N2, the relative error of the optimal C is significantly reduced when the value of N2 increases to 20 as shown in Table 3. Figure 8 presents the fact that there are more solutions close to the value C=19.5594% by FAM. In addition, the error is significantly improved when N2≥20 from Table 3 and the proposed algorithm is theoretically and practically convergent. Figure 9 supports the fact that a new plug mode can be applied to finish the external load distribution when it is suitable for X6 plug position to carry Paveway missile, including the strength of the plug pylons and the space around the plug point. Unlike the other random search method for producing some single values, the RSA operator concentrates on generating a certain number of nonrepeating combinations to simulate the various and promising plug modes. It can be noted from Figure 10 that the center of gravity ranges from 12.4865% to 22.4288% close to the result by FAM. Figure 11 shows that the value of ΔC is between −7.0135% and 2.9288%. It could be remarkable from Figure 12 that there are 59 plug modes which changes center of gravity less than the typical plug mode by comparing the absolute value of ΔC between RSA and the typical plug mode. The minimum value of center of gravity position can reach 19.5605% with ΔC=0.0605% which is close to the solution by FAM. Furthermore, it just costs RSA 100 initial plug modes to get the optimization solution by searching only one time. Compared with FAM, RSA is really fast and efficient with enough precision.Table 3
Center of gravity generations of the optimization plug mode by RSA.
Generations
1
2
3
4
N
2
4
20
100
500
C
/
%
22.3846
19.9799
19.5605
19.5605
Δ
C
/
%
2.8846
0.4799
0.0605
0.0605
ε
/
%
14.748
2.454
0.309
0.309Figure 7
Center of gravity by RSA,N2=4.Figure 8
Center of gravity by RSA,N2=20.Figure 9
The optimization plug mode,N2=100.Figure 10
Center of gravity by RSA,N2=100.Figure 11
Change ofC by RSA, N2=100.Figure 12
Sort center of gravity by RSA,N2=100.Figure13 supports the fact that there are 13 plugs occupying all the plug positions not only to simulate aircraft high payload but also to test the proposed algorithm. By pouring attention to the weight of Paveway missile heavier than that of fuel tank 1 in spite of the former volume smaller than the later, X2 or X4 plug position is strong enough to carry the other plugs lighter than fuel tank 1 from the perspective of wing structure and this is operable. In addition, the high weight of Paveway plug at X2 position is not good for rolling manipulation, thus affecting the lateral flight qualities. Figure 14 indicates that a precise plug mode with the minimum changing center of gravity is obtained by FAM. Although the longitudinal flight qualities are improved because of the minimum changing aircraft center of gravity, the heavy weight of Paveway missile at X1 position is extremely detrimental to the lateral flight qualities as shown in FAM results. In addition, high load often occupying the wing tip is unfavorable for aircraft wing structure and easily leads to structural fatigue along with the aircraft taking off, climbing, and maneuvering. Thanks to the optimal solution by RSA, the disadvantages above are well improved as shown in RSA results. As previously demonstrated, X2 is strong enough to carry fuel tank 1 lighter than the Paveway missile and the auxiliary tank could be preferentially exhausted and abandoned; thus the flight qualities could be improved significantly. The remaining operation is placing the Paveway missile at X6 position and this is convenient and feasible.Figure 13
The schematic of aircraft, plug, plug position, and typical plug mode.Figure 14
Comprehensive analysis and comparison of the results by FAM and RSA.


## 4.1. The Results of Typical Plug Mode

For comparison of the solutions between RSA and FAM, a typical plug load including 10 missiles and 3 external fuel tanks is given as shown in Table1. There are 10 missiles and 3 fuel tanks for the fighter to possess plenty of missiles and fuel. By using symmetrical loading, the typical plug mode is presented in Table 2.Table 1
A typical plug load.
Weapon
Fuel
AIM-9
AIM-120
Paveway
Tank 1
Tank 2
Weight/ton
0.0910
0.1507
0.9070
0.7750
1.1625
Number
2
4
4
2
1Table 2
The distribution of the typical plug load.
Typical plug mode
X
1
2
3
4
5
6
7
Plug
AIM-9
Paveway
Tank 1
Paveway
AIM-120
AIM-120
Tank 2Regarding (1)–(3), the center of gravity in the typical plug mode is calculated to be 17.7205%, where Ce=19.50%. Then ΔC0=-1.7795%.


## 4.2. The Results and Analysis of FAM

To obtain the precise solution of the typical plug mode, FAM is used to produce all the array ofM matrix. Concerning the fact that there is a unique position (X7) for fuel tank 2 as shown in Figure 1 and Table 2, the value of N1 is 720. With this in mind, the precise plug mode can be determined as presented in Figure 3. The results of center of gravity are obtained as shown in Figure 4 with a range of 13.1343%–27.8909%. The change of C is presented in Figure 5 in the scope [−6.3657, 8.3909]. To observe the range of C, the results of center of gravity is sorted as shown in Figure 6. Besides, there are 164 plug modes with less changing of the value of C than the typical plug mode. Then the minimum change of C can be determined as 0.0594% where C=19.5594% with the precise plug mode. Because F matrix is a full arrangement of M, the precise solution can be considered reliable.Figure 3
Precise plug mode.Figure 4
Center of gravity by FAM.Figure 5
Change ofC by FAM.Figure 6
Sorting results of center of gravity by FAM.


## 4.3. The Results and Analysis of RSA

Figure7 shows that it is formidable to search the robust results of plug mode when the initial size of N2 is small. Thanks to the action of updating and resetting operator of N2, the relative error of the optimal C is significantly reduced when the value of N2 increases to 20 as shown in Table 3. Figure 8 presents the fact that there are more solutions close to the value C=19.5594% by FAM. In addition, the error is significantly improved when N2≥20 from Table 3 and the proposed algorithm is theoretically and practically convergent. Figure 9 supports the fact that a new plug mode can be applied to finish the external load distribution when it is suitable for X6 plug position to carry Paveway missile, including the strength of the plug pylons and the space around the plug point. Unlike the other random search method for producing some single values, the RSA operator concentrates on generating a certain number of nonrepeating combinations to simulate the various and promising plug modes. It can be noted from Figure 10 that the center of gravity ranges from 12.4865% to 22.4288% close to the result by FAM. Figure 11 shows that the value of ΔC is between −7.0135% and 2.9288%. It could be remarkable from Figure 12 that there are 59 plug modes which changes center of gravity less than the typical plug mode by comparing the absolute value of ΔC between RSA and the typical plug mode. The minimum value of center of gravity position can reach 19.5605% with ΔC=0.0605% which is close to the solution by FAM. Furthermore, it just costs RSA 100 initial plug modes to get the optimization solution by searching only one time. Compared with FAM, RSA is really fast and efficient with enough precision.Table 3
Center of gravity generations of the optimization plug mode by RSA.
Generations
1
2
3
4
N
2
4
20
100
500
C
/
%
22.3846
19.9799
19.5605
19.5605
Δ
C
/
%
2.8846
0.4799
0.0605
0.0605
ε
/
%
14.748
2.454
0.309
0.309Figure 7
Center of gravity by RSA,N2=4.Figure 8
Center of gravity by RSA,N2=20.Figure 9
The optimization plug mode,N2=100.Figure 10
Center of gravity by RSA,N2=100.Figure 11
Change ofC by RSA, N2=100.Figure 12
Sort center of gravity by RSA,N2=100.Figure13 supports the fact that there are 13 plugs occupying all the plug positions not only to simulate aircraft high payload but also to test the proposed algorithm. By pouring attention to the weight of Paveway missile heavier than that of fuel tank 1 in spite of the former volume smaller than the later, X2 or X4 plug position is strong enough to carry the other plugs lighter than fuel tank 1 from the perspective of wing structure and this is operable. In addition, the high weight of Paveway plug at X2 position is not good for rolling manipulation, thus affecting the lateral flight qualities. Figure 14 indicates that a precise plug mode with the minimum changing center of gravity is obtained by FAM. Although the longitudinal flight qualities are improved because of the minimum changing aircraft center of gravity, the heavy weight of Paveway missile at X1 position is extremely detrimental to the lateral flight qualities as shown in FAM results. In addition, high load often occupying the wing tip is unfavorable for aircraft wing structure and easily leads to structural fatigue along with the aircraft taking off, climbing, and maneuvering. Thanks to the optimal solution by RSA, the disadvantages above are well improved as shown in RSA results. As previously demonstrated, X2 is strong enough to carry fuel tank 1 lighter than the Paveway missile and the auxiliary tank could be preferentially exhausted and abandoned; thus the flight qualities could be improved significantly. The remaining operation is placing the Paveway missile at X6 position and this is convenient and feasible.Figure 13
The schematic of aircraft, plug, plug position, and typical plug mode.Figure 14
Comprehensive analysis and comparison of the results by FAM and RSA.


## 5. Conclusions

RSA for solving the optimization plug mode was carried out. A typical plug mode was used to be the comparison of reference between RSA and FAM. FAM is precise and comprehensive to calculate the precise solution with computing a large number of the arrangements ofM. RSA is accurate and efficient for generating the optimization plug mode with searching among a small number of samples of the arrangement of M. From this research, it can be found that RSA coupled with FAM is an efficient tool for optimization of aircraft plug mode.



---

*Source: 1409829-2017-06-18.xml*