Modelling and Simulation in Engineering
Volume 2021 (2021), Article ID 6644339, 15 pages
https://doi.org/10.1155/2021/6644339

Research Article

Deployment of On-Orbit Service Vehicles Using a Fuzzy Adaptive Particle Swarm Optimization Algorithm

Yaxiong Li, Xinglong Sun, Xinxue Liu, Jian Wu, and Qingguo Liu

Xi’an High-Tech Institute, Xi’an 710025, China

Correspondence should be addressed to Xinglong Sun; 1229594814@qq.com

Received 25 October 2020; Revised 9 April 2021; Accepted 5 June 2021; Published 21 June 2021

Academic Editor: Feng Xiao

Copyright © 2021 Yaxiong Li et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

On the basis that satellites given fixed count and orbit elements can be served in bounded time when an on-orbit serving mission order is set at any uncertain time in a given time interval, the deployment of on-orbit service vehicle (OSV) serving satellites becomes a complex multiple nested optimization problem, and the essence of deployment is to determine the count and orbit elements of OSVs. In consideration of the characteristics of this deployment problem, we propose a fuzzy adaptive particle swarm optimization (FAPSO) algorithm to solve this problem. First, on the basis of double pulse rendezvous hypothesis, a transfer optimization model of a single OSV serving multiple satellites is established based on genetic algorithm (GA), and this is used to compute the indexes of the subsequent two optimization models. Second, an assignment optimization model of OSVs is established based on the discrete particle swarm optimization (DPSO) algorithm, laying the foundation of the next optimization model. Finally, the FAPSO algorithm, which improves the performance of PSO algorithm by adjusting the inertia weight, is proposed to solve the deployment problem of multiple OSVs. The simulation results demonstrate that all optimization models in this study are feasible, and the FAPSO algorithm, which has a better convergence result than that obtained using the other optimization algorithms, can effectively solve the deployment problem of OSVs.

1. Introduction

The on-orbit service of a spacecraft is a type of space operation that humans, robots, or both extend lives and improve their abilities in performing tasks of various spacecraft [1–3]. The main tasks of on-orbit service vehicles (OSVs) include on-orbit assembly, on-orbit maintenance, and logistic support [4]. As humans explore deeper in space, more and more diverse satellites are deployed in space. The main reasons why satellites are abandoned are that some components are broken and the fuel to maintain their orbits is not enough. The on-orbit service of a spacecraft is an effective method to reduce the waste of satellite resource and avoid producing space junk. Using OSVs to extend the lives of satellites deployed and improve the ability to perform tasks would be an important research direction.

The basic work of deployment of OSVs includes transfer orbit optimization and assignment optimization. In the existing literature, direct methods [5, 6], indirect methods [7], and hybrid methods [8] are used to study the optimization of transfer trajectories of different types of spacecraft. OSV transfer orbit is one kind of transfer orbits. Although the direct methods suffer from the lack of high precision, its computation is efficient. Considering the deployment and assignment optimization need to calculate a lot of transfer orbits under different states, we use direct method to compute transfer orbits. The literature related to assignment optimization and the deployments of OSVs are as follows. A novel approach for the mission planning of on-orbit servicing such as visual inspection is proposed by Daneshjou et al. [9]. The on-orbit servicing task allocation is solved by a hybrid discrete particle swarm optimization algorithm which is proposed by Zhang and Zhang [10]. Pradeepmon et al. [11] solved quadratic assignment problems by using discrete particle swarm optimization (DPSO) algorithm.

Deployment of OSVs serving satellites, which is a complex multiple nested optimization problem, is almost not systematically studied in the existing literatures. Deployment of OSVs is the application of deployment problem. The optimization methods of other related deployment problems mainly include some intelligent algorithms. Tesarova & Vokalova [12] used genetic algorithm (GA) to deploy radio stations. Singh et al. [13] used particle swarm optimization (PSO) algorithm to deploy directional sensors. Ahmadian et al. [14] solved the large dimension electric distribution network problem by using a hybrid algorithm combining PSO and TS (Tabu search) algorithm (PSO+TS). Zhang et al. [15] obtained optimal deployment of flow monitors based on a hybrid algorithm combining GA and SA (simulated annealing) algorithm (GA+SA). The convergences of these intelligent algorithms are not good to compute complex multiple nested optimization problem. Thus, we propose a fuzzy adaptive particle swarm optimization (FAPSO) algorithm to improve the convergence of the results by balancing global and local search capability of PSO algorithm. The study background of the present study is that multiple satellites given fixed count and orbit elements need to be maintained by multiple OSVs in bounded time when an on-orbit serving mission order is set at any uncertain time in a given time interval (on-orbit mission). The main content of the present study consists of four parts (Sections 2–5). The framework of the present study is shown in Figure 1.

Figure 1: Framework of the present study.

Part A (Section 2): on the basis of the double pulse rendezvous hypothesis and taking the maximum weight of remaining fuel as the optimization index (optimization index A), a transfer optimization model of single OSV serving multiple satellites is established based on the GA, which is used to calculate the indexes of the subsequent optimization models.

Part B (Section 3): at the time of an on-orbit serving mission order set and taking the maximum value of both the count of satellites that can be served and the minimum value of a single OSV’s optimization index A among all OSVs as the optimization index (optimization index B), an assignment optimization model of multiple OSVs serving multiple satellites is established based on DPSO algorithm.

Part C (Section 4): first of all, taking the time in a given time interval as the optimization variable and taking the minimum value of optimization index B as the optimization index (optimization index C), the optimization model to determine whether OSVs given initial count and orbit elements can achieve on-orbit mission or not is established based on the pattern search method and this is used to calculate the optimization index of the deployment model. Then, given the count of OSVs and taking the maximum value of optimization index C as the optimization index (optimization index D), the deployment optimization model of multiple OSVs serving multiple satellites is established based on the fuzzy adaptive particle swarm optimization (FAPSO) algorithm.

Part D (Section 5): the feasibility of all optimization models in the present study is verified through comparison with other intelligent algorithms, and the FAPSO algorithm is more efficient in solving the deployment problem of OSVs.

2. Transfer Orbit Optimization Model

A single OSV can serve multiple satellites. The procedure of the double pulse rendezvous method is that a single OSV manoeuvres because of the 1st pulse thrust at some time which is called manoeuvre time, flies without thrust for some time, and manoeuvres because of the 2nd pulse thrust to get the same velocity as the first satellite at some time which is called rendezvous time when both the OSV and the first satellite reach the same position. The OSV begins to serve the first satellite after the double pulse rendezvous procedure. After serving the first satellite, the OSV serves other satellites in the same way as it served the first satellite.

The procedure of the OSV serving the 1st satellite by double pulse rendezvous method is shown in Figure 2. The orbit of the OSV is original orbit, and the orbit of a satellite is target orbit. The position of the OSV at the manoeuvre time is point , and the position of the OSV at the rendezvous time is point . When the OSV manoeuvres from point on the original orbit to point on target orbit in seconds to serve the satellite, and are its position vector and velocity vector, respectively, at point , and and are its position vector and velocity vector, respectively, at point . The determination of the 1st velocity increment (Formula (1)) of the OSV serving the 1st satellite at point on the original orbit is called the Lambert problem, and the method to solve the Lambert problem is studied by Schumacher et al. [16]. The 2nd velocity increment of the OSV serving the 1st satellite at point on the target orbit is written as Formula (2). After serving the 1st satellite for a certain time, the OSV begins to serve the 2nd satellite in the same procedure as it served the 1st satellite.

Figure 2: The procedure of the double pulse rendezvous method.

In Formula (1) and Formula (2), is the velocity of the OSV at point on the original orbit, is the velocity of the OSV at point on the transfer orbit after pushing by the 1st pulse thrust, is the velocity of the OSV at point on the transfer orbit, and is the velocity of the OSV at point on the target orbit after pushing by the 2nd pulse thrust.

Given the sequence of serving satellites and taking the maximum weight of the remaining fuel as the optimization index, the transfer orbit optimization model of a single OSV serving multiple satellites is established by the GA. The manoeuvre and rendezvous times are searched globally using the GA. Optimization index A can be obtained by calculating the velocity increments of the procedures of an OSV serving multiple satellites.

2.1. Mathematical Model

From the viewpoint of energy, problems correlated to multiple OSVs serving multiple satellites are investigated in the present study. The weight of a single OSV includes the weight of the fuel it carries and the weight of its other parts.

In Formula (3), is the weight of a single OSV, is the weight of the fuel carried by the OSV, and is the weight of the other parts.

Given the sequence of serving satellites, the mathematical model can be written as

In Formula (4), is the optimization index A, is the weight of the remaining fuel after the on-orbit mission is achieved, are the manoeuvre times, are the rendezvous times, is the maximum time interval for finishing the on-orbit mission, is the time interval for the OSV serving one satellite, is the count of the transfer orbit segments (the th segment corresponding to the th satellite served by the OSV), is the weight of the consumed fuel in the th segment, and is the weight of fuel used to keep the orbit. can be written as Formula (5) and can be written as Formula (6).

In Formulas (5) and (6), is the count of satellites served, is the count of satellites that single OSV can serve, is the weight of the remaining fuel after a single OSV serves satellites, is the specific impulse of the OSV, is the acceleration of the earth, is the weight of the remaining fuel after the OSV serves the th satellite, is the 1st velocity increment of the OSV serving the th satellite at the manoeuvre time, and is the 2nd velocity increment of the OSV serving the th satellite at the rendezvous time.

2.2. Optimization Model

The GA is an intelligence algorithm which is widely used in many fields [17–19]. Taking the maximum value of the remaining fuel weight (max ) as the optimization index, the transfer optimization model of a single OSV serving multiple satellites is established based on the GA. The implementation steps are as follows.

Step 1. In order to meet the constraint conditions in Formula (4), the variables are written as

In Formula (7), is the count of the transfer orbit segments. Furthermore, are taken as genes encoded to a chromosome (Figure 3). The initial populations are randomly generated. The population size , the crossover probability , the mutation probability , and the maximum number of iterations are set.

Step 2. The object function of Formula (4) is taken as the fitness function of the GA. Each chromosome has a fitness function value. The fitness function values of all chromosomes in the current generation are calculated.

Step 3. The chromosomes are selected by means of roulette wheel selection. Some genes on two different chromosomes reciprocally cross according to the crossover probability. Furthermore, some genes mutate according to the mutation probability.

Step 4. Steps 2 and 3 are repeatedly executed until the GA converged () or the maximum number of iterations is reached.

Figure 3: Coding of the chromosome.

Given the sequence of serving satellites, the maximum could be obtained through the GA.

3. Assignment Optimization Model

3.1. Mathematical Model

Assumption 1. There are OSVs that are remarked as , there are satellites that are remarked as , a single OSV serves satellites at most, and a single satellite can be served once at most. The assignment matrix is written as

In Formula (8), , . The mathematical model is written as

In Formula (9), is the optimization index B. Furthermore, is the count of satellites that a single OSV can serve, is the weight of a single OSV, is the weight of the remaining fuel after the th OSV serves the satellites, and is the weight of the fuel used to keep the orbit.

Optimization index B includes two parts.

The first part (optimization index B₁) is written as the formula , which refers to the count of satellites served. This part evaluates the use ratio of OSVs.

The second part (optimization index B₂) is written as the formula , which refers to the minimum weight () of a single OSV’s remaining fuel among all OSVs divided by the weight of the OSV (). The purpose of the present study is to use the minimum count OSVs to serve the satellites given the count and their orbits based on the on-orbit mission. The formula refers to the weight of the remaining fuel after a single OSV serves satellites, and this remaining fuel can be used to keep the orbit before the OSV execute its on-orbit mission, which evaluates the ability of a single OSV keeping its orbit. If the value of is larger, the overall ability of all OSVs to keep orbits would be better. The maximum value of the formula is , which the value is less than 1. In optimization index B, optimization B₁ is the main optimization index. The value of increasing or decreasing one satellite (the definition of optimization index B₁) is larger than the maximum value of optimization index B₂.

3.2. Optimization Model

The assignment optimization model of multiple OSVs serving multiple satellites is established based on the DPSO [20, 21], and the main components of the DPSO algorithm are as follows.

3.2.1. Coding of a Particle

The particles are encoded in decimal system. The length of each particle is . The coding of a particle is presented as Figure 4. In Figure 4, are OSVs, and refers to the th satellite served by the th OSV. The value of is obtained from the set and 0 means that no satellite is served.

Figure 4: Coding of particle.

3.2.2. Calculation of the Fitness Function

The object function of Formula (9) is taken as the fitness function shown as

In Formula (10), is the th particle.

3.2.3. Particle’s Position and Velocity Update Formulas

The position of particle swarm optimization (PSO) is determined through the particle velocity, individual extreme, and global extreme, and the particle’s position and velocity update formulas of the PSO algorithm are written as [22, 23]

In Formula (11), is the individual extreme in the th iteration, is the global extreme in the th iteration, is the fitness function value of the th particle in the th iteration, is the fitness function value of the th particle in the th iteration, is the particle velocity value of the th particle in the th iteration, is the particle velocity value of the th particle in the th iteration, is the inertia weight that reflects the extent that the particle maintains the present velocity, is the cognitive coefficient that reflects the weight of the particle getting close to the individual extreme, is the social coefficient which reflects the weight the particle getting close to the global extreme, is a random number between 0 and 1, and is a random number between 0 and 1.

On the basis of the PSO, the particle position and velocity update formulas for the DPSO algorithm is written as follows:

In Formula (12), is the fitness function value of the th particle in the th iteration, and is the fitness function value of the th particle in the th iteration. This formula includes three formulas which are written as

is the value of the th particle in the th iteration after the calculation of . The formula means that a real number between 0 and 1 is randomly generated. When the condition is met, the formula is executed which means that two integers and are randomly generated in interval and the values on the th particle at positions and are interchanged. When the condition is met, the formula is executed which means that the coding values on the th particle are unchanged.

is the value of the th particle in the th iteration after the calculation of . is the individual extreme in the th iteration. When the condition is met, the formula is executed which means that two integers and are randomly generated in interval , and the values on at positions and as well as the values on at positions and are interchanged. When the condition is met, the formula is executed which means that the coding values on are unchanged.

is the global extreme in the th iteration. When the condition is met, the formula is executed which means that two integers and are randomly generated in interval , and the values on at positions and as well as the values on at positions and are interchanged. When the condition is met, the formula is executed which means that the coding values on are unchanged.

The implementation steps of the DPSO algorithm are as follows.

Step 1. Set parameters: the population size of the particle swarm, the count of OSVs, the maximum count of satellites served by a single OSV, the maximum number of iterations , the inertia weight , the cognitive coefficient , the social coefficient , and .

Step 2. The particle swarm population is initialized, the values on the particles are randomly generated, the fitness function values of all particles are calculated, and the individual extreme and global extreme are recorded.

Step 3. , the position of the particles is updated through Formula (12), the fitness function values of all particles are calculated, and the individual extreme and global extreme are updated.

Step 4. Step 3 is executed repeatedly until () or the maximum number of iterations is reached.

4. Deployment Model

4.1. Problem Description

The deployment problem of OSVs includes two parts: one is to determine the count of OSVs, while the other is to determine the orbit elements of all OSVs.

A single OSV serves satellites at most. If satellites need to be served, the initial count of satellites can be calculated:

In Formula (16), the function means rounding up to the nearest integer. After the count of OSVs is determined by Formula (16), the orbit element orbit radius (), orbit eccentricity ratio (), orbit inclination angle (), right ascension of the ascending node (), argument of perigee (), and true anomaly () of OSVs are generated according to the orbit elements of given satellites.

The optimization model to determine whether OSVs’ given count and orbit elements can achieve the on-orbit mission or not is established and its optimization index is called optimization index C, which is the foundation of the deployment optimization model. If the given OSVs could not achieve the on-orbit mission, the orbit elements of all OSVs are adjusted to determine whether the on-orbit mission can be achieved and obtain the best optimization index. If the on-orbit mission could still not be achieved, the count of OSVs is increased and the orbit elements of all OSVs are adjusted again to obtain the best optimization index.

4.2. Optimization Index

Given the count and orbit elements of OSVs and taking the time interval as the searching region for optimization variable , the optimization model to determine whether OSVs’ given count and orbit elements can achieve on-orbit mission or not is established based on the pattern search method and its optimization index is called optimization index C (). In the entire searching region, optimization index B of the assignment optimization model at any time point can be calculated and minimum optimization index B is called optimization index C. The optimization index of the deployment optimization model is called optimization index D ().

In Formula (17), is optimization index B.

The implementation steps of the pattern search [24] are as follows.

Step 1. . The function value at base point in the time interval is calculated. The iteration step size and the convergence precision are set.

Step 2. The temporary point is determined through

Formula (18) means that the temporary point is determined in point , point , and point which has a minimum value of . The temporary point is defined as base point . th is the base point pointing to base point .

Step 3. If the condition is met, the th vector is twice longer than the th vector. . The temporary point is written as

The temporary point is defined as the base point which is taken as the starting point of the th vector. If the condition is met, the th vector is half of the th vector and its starting point becomes the base point .

Step 4. Steps 2 and 3 are repeatedly executed. If the value of remains unchanged, the iteration step size will be half until it reaches a certain degree of accuracy. The algorithm ends.

4.3. Optimization Model

Given the count of OSVs, the optimized orbit elements are obtained by adjusting orbit elements of all OSVs. The deployment problem of OSVs is a high-dimension nonlinear problem which nests the optimization problems in Sections 2.2, 3.2, and 4.2.

If the inertia weight is bigger, the global search capability of the PSO algorithm will be stronger. If the inertia weight is smaller, the local search capability of the PSO algorithm will be stronger. Considering that the searching space of deployment problem of OSVs is large and the optimization variables are sensitive to the initial values, a larger inertial weight will be chosen to enhance the global search capability. Considering that the fuel taken by the OSV is expensive, enhancing the local search capability to obtain the better local results will also be very important. Hence, the fuzzy control system is introduced to adjust the inertia weight , which can balance the global search capability and local search capability.

The essence of the FAPSO algorithm is updating the inertia weight through the fuzzy controller. The updated inertial weight is determined through the fuzzy controller in three steps.

Step 1. The input variables of the fuzzy controller are translated into fuzzy variables. The two input variables are the inertial weight and the normalized fitness function values of particles in the present iteration.

In Formula (20), is the fitness function value of one particle in the present iteration, is the minimum fitness function value in the present iteration, and is the maximum fitness function value in the present iteration.

The fuzzy variables of these two input variables are calculated through the membership functions [25, 26].

Formulas (21), (22), and (23) are the membership functions that correspond to fuzzy language “small,” “medium,” and “large,” respectively. and are the constants when two input variables or are within the range of fuzzy language “small.” and are the constants when two input variables or are within the range of fuzzy language “medium.” and are the constants when two input variables or are within the range of fuzzy language “large.”

Step 2. The values of two input variables and and their fuzzy variables are used to calculate the fuzzy control variable according to the fuzzy reasoning algorithm and the fuzzy control rule base.

The fuzzy reasoning algorithm can be expressed through

In Formula (24), is the input variable , is the input variable , is the set where is in, is the set where is in, is the implication relationship between and , is the membership function value of , is the membership function value of , and “” is the operational symbol for choosing the smaller value between and .

The fuzzy control rule base is described in Table 1.

Step 3. The fuzzy control variable is translated into the precise value after the defuzzification procedure. The precise value of a particle is the centre point’s -coordinate value of the area of the curve in Step 2 and the -coordinate.

Table 1: The fuzzy control rule base.

Number

1
Small
Small
Medium

2
Small
Medium
Small

3
Small
Large
Small

4
Medium
Small
Large

5
Medium
Medium
Medium

6
Medium
Large
Small

7
Large
Small
Large

8
Large
Medium
Medium

9
Large
Large
Small

In Formula (25), is , is the area of the curve in Step 2 and the -coordinate, and is the -coordinate value. The updated inertial weight is written as

The implementation steps of the FAPSO algorithm are as follows.

Step 1. Set parameters: the population size of the particle swarm, the count of OSVs, the maximum number of iterations , the inertia weight , the cognitive coefficient , the social coefficient , and . The coding of the th particle is , the expression refers to the th orbit element of the th OSV in the th particle, and corresponds to the six orbit elements , , , , , and , respectively.

Step 2. The particle swarm population is initialized, the values on the particles are randomly generated, the fitness function values of all particles are calculated using Formula (17), and the individual extreme and the global extreme are recorded.

Step 3. The inertial weight and the normalized fitness function values in the th iteration are calculated and the updated inertial weight is determined through the fuzzy controller in three steps. . The updated inertial weight is placed into Formula (11) to calculate the fitness function value of the particle in the th iteration. The fitness function values of all particles are calculated, and the individual extreme and global extreme are recorded.

Step 4. Step 3 is repeatedly executed until () or the maximum number of iterations is reached.

Step 5. If the optimization result cannot achieve the on-orbit mission, one OSV is added to the present OSVs and Steps 1–4 are executed. If the optimization result can achieve the on-orbit mission, the cycle calculation is ended.

5. Simulations and Analyses of Results

5.1. Simulations

The initial orbit elements of all satellites are presented in Table 2. The search range orbit elements (, , , , , and ) of the OSVs determined according to the orbit elements of all satellites are [6700 km,11000 km], [0, 1), [0°, 180°], [0°, 360°], [0°, 360°], and [0°, 360°], respectively. The initial orbit elements of OSVs are presented in Table 3. The weight of a single OSV is 1000 kg. A single OSV serves three satellites at most and carries 600 kg of fuel. The specific impulse is 3000 seconds. The duration of time to serve a satellite is more than an hour. The on-orbit mission should be achieved within 72 hours.

Table 2: Initial orbit elements of satellites.

Satellite

S1
9306
0.238550
28.321
224.138
307.927
135.862

S2
7059
0.000225
98.084
76.815
185.081
283.332

S3
9237
0.000165
52.006
171.386
350.005
188.701

S4
7828
0.001219
101.618
87.562
293.441
282.681

S5
7707
0.054088
99.031
56.562
347.938
308.882

S6
7554
0.002882
90.045
245.844
85.215
113.968

S7
8428
0.201645
82.982
262.555
29.901
44.341

S8
9501
0.213200
28.226
34.607
152.421
325.392

S9
9401
0.016556
64.822
302.192
337.387
56.304

S10
9939
0.321639
56.917
231.629
218.498
128.371

S11
10687
0.355917
57.022
347.224
317.457
122.776

S12
6874
0.000555
97.420
104.012
316.284
204.781

S13
7125
0.003924
42.084
51.815
255.081
30.200

S14
8838
0.000249
98.178
69.604
155.872
150.009

S15
9123
0.013793
52.674
109.398
23.270
200.020

S16
7685
0.004693
37.477
52.707
158.017
60.000

S17
9215
0.489300
55.371
128.871
305.501
80.000

S18
7453
0.004218
23.618
77.562
293.441
50.101

S19
9675
0.054089
99.031
56.562
187.938
160.032

S20
7534
0.003002
91.145
239.735
84.325
113.000

S21
6903
0.000549
96.390
110.033
325.324
203.070

Table 3: Initial orbit elements of OSVs.

OSV

O1
7751
0.000120
32.223
56.232
76.223
255.520

O2
8400
0.000225
78.002
102.332
253.010
122.243

O3
8526
0.004120
76.924
154.257
73.020
243.000

O4
9201
0.370056
55.223
180.922
223.001
95.620

O5
7002
0.004021
87.235
102.556
330.765
88.598

O6
8002
0.529030
145.667
156.007
200.321
270.321

O7
9923
0.052301
120.023
254.124
122.232
55.220

O8
9001
0.006322
78.336
155.336
95.663
45.692

In the simulation of the transfer orbit optimization model, the parameters of GA are taken from literature written by Tesarova & Vokalova [12]. The population size is 20, the weight of the fuel used to keep the orbit is 50 kg, the maximum number of iterations is 50, the crossover probability is 0.85, and the mutation probability is 0.1.

In the simulation of assignment optimization model, the parameters of DPSO are taken from literature written by Pradeepmon et al. [11]. The population size is 10, the maximum number of iterations is 50, the inertia weight is 0.75, the cognitive coefficient is 2, and the social coefficient is 2.

In the simulation of deployment model, the parameters of PSO are taken from literature written by Singh et al. [13]. The population size is 10, the maximum number of iterations is 50, the inertia weight is 0.75, the cognitive coefficient is 2, and the social coefficient is 2. The particle velocities corresponding to the six orbit elements , , , , , and are 50000, 0.00001, 10, 10, 10, and 10, respectively. In the optimization model to determine whether OSVs’ given count and orbit elements can achieve on-orbit mission or not, the iteration step size of the pattern search is () seconds and the time interval is () seconds. The parameters of the fuzzy controller are shown in Table 4.

Table 4: Parameters of the fuzzy controller.

Parameter
Fuzzy language

Small
0
0.06

Medium
0.05
0.4

Large
0.3
1

Small
0.2
0.6

Medium
0.4
0.9

Large
0.6
1.1

Small
-0.12
-0.02

Medium
-0.04
0.04

Large
0.00
0.05

We design the comparison experiments between the proposed FAPSO algorithm and GA, PSO, PSO+TS, and GA+SA algorithms. The parameters of these algorithms are successively taken from the existing literatures mentioned in the present paper [12–15]. 30 independent runs are performed for each algorithm.

5.2. Analyses of Results

The results of the 10th experiment presented in Tables 5–7 and Figures 5–13 are obtained using the FAPSO algorithm. The obtained orbit elements of OSVs are presented in Table 5, and the parking orbits of OSVs are presented in Figure 5. The deployment results of OSVs are presented in Table 6 including the value of optimization D (), the time to obtain , and the assignment results of OSVs. On the basis of results presented in Table 6, OSVs’ manoeuvre time, rendezvous time, and the weight of remaining fuel are presented in Table 7. The transfer orbits of OSVs are presented in Figures 6–13. The convergence curves (the average values of 30 independent experiments) of the FAPSO algorithm, GA, PSO algorithm, PSO+TS algorithm, and GA+SA algorithm are presented in Figure 14.

Table 5: Orbit elements of OSVs after simulations.

OSV

O1
7509
0.000221
28.100
55.221
125.122
305.600

O2
8144
0.000198
82.998
115.419
320.385
65.768

O3
7792
0.000221
100.325
112.366
102.981
200.421

O4
8879
0.250022
55.348
200.000
250.887
120.133

O5
6998
0.005411
96.951
156.052
349.870
149.551

O6
7331
0.452239
38.002
200.333
250.554
35.010

O7
10049
0.240000
53.239
300.256
320.457
225.000

O8
9514
0.151000
97.000
265.107
145.932
102.000

Table 6: Results of deployment.

Time (h)
OSV
Satellite

9336.450
O1
S8, S1, S18
21.106

O2
S7, S20, S6

O3
S2, S4

O4
S10, S17, S15

O5
S12, S21, S5

O6
S13, S16

O7
S11, S9, S3

O8
S19, S14

Table 7: OSVs’ manoeuvre time, rendezvous time, and the weight of remaining fuel.

OSV

O1
51692.558
67975.303
104345.873
109193.762
193120.182
198267.663
199.102

O2
175202.944
187625.628
219612.145
225239.147
250263.743
255990.129
111.257

O3
17001.794
148722.117
218123.229
239686.355
—
—
105.759

O4
15887.24
26912.829
31971.051
37423.464
47426.589
59549.761
126.350

O5
3605.5
39687.429
51749.981
63848.707
—
—
150.445

O6
33806.931
60783.397
97927.029
107616.136
179462.553
216913.012
232.828

O7
22523.519
44028.819
105397.328
126423.851
151691.717
186117.106
107.141

O8
31522.046
72061.837
75662.262
130570.746
—
—
164.904

Figure 5: Parking orbits of all OSVs.

Figure 6: Transfer orbit of O1 serving S8, S1, and S18.

Figure 7: Transfer orbit of O2 serving S7, S20, and S6.

Figure 8: Transfer orbit of O3 serving S10, S17, and S15.

Figure 9: Transfer orbit of O4 serving S12, S21, and S5.

Figure 10: Transfer orbit of O5 serving S11, S9, and S3.

Figure 11: Transfer orbits of O6 serving S13 and S16.

Figure 12: Transfer orbit of O7 serving S19 and S14.

Figure 13: Transfer orbit of O8 serving S2 and S4.

Figure 14: The convergence curves of the FAPSO algorithm and other algorithms.

In Table 6, the time to obtain optimization index D () is after 9336.450 hours. O1, O2, O4, O6, and O7 each serve 3 satellites. O3, O5, and O8 each serve 2 satellites. O1 serves S8, S1, and S18 in sequence. O2 serves S7, S20, and S6 in sequence. O3 serves S2 and S4 in sequence. O4 serves S10, S17, and S15 in sequence. O5 serves S12, S21, and S5 in sequence. O6 serves S13 and S16 in sequence. O7 serves S11, S9, and S3 in sequence. O8 serves S19 and S14 in sequence. In Table 7, and are the manoeuvre time and rendezvous time to serve the 1st satellite, respectively, and are the manoeuvre time and rendezvous time to serve the 2nd satellite, respectively, and are the manoeuvre time and rendezvous time to serve the 3rd satellite, respectively, and is the weight of remaining fuel after the th OSV serving the satellites. In Figure 14, the convergence results of the FAPSO algorithm, PSO algorithm, PSO+TS algorithm, GA, and GA+SA algorithm are 21.106, 21.083, 21.084, 21.080, and 21.094, respectively. After the convergence results are entered into the objection function of Formula (9), the weight of the remaining fuel obtained through the FAPSO algorithm is 23 kg, 22 kg, 26 kg, and 12 kg more than that obtained by the PSO algorithm, PSO+TS algorithm, GA, and GA+SA algorithm, respectively.

6. Conclusion and Further Research

The deployment of multiple OSVs is a complex multiple nested optimization problem, which is systematically solved by a series of intelligent algorithms in the present study. The deployment model includes the transfer orbit optimization model and the assignment optimization model. The transfer orbit optimization is a continuous optimization problem, which is solved by the intelligence algorithm GA. The assignment optimization is a discrete optimization problem, which is solved by the intelligence algorithm the DPSO algorithm. The deployment is a continuous optimization problem, which is solved by the FAPSO algorithm. The FAPSO algorithm, which has a better convergence result than that obtained using the other similar optimization algorithms, can effectively solve the deployment problem of OSVs.

In further studies, some aspects should be considered such as the finite thrust engine of the OSV, the influence on the parking orbits because of the disturbing force, and optimization algorithms getting the simulation results faster.

Notations

: | Uncertain time
: | Time interval
: | Manoeuvre times
: | Rendezvous times
: | Maximum time interval for finishing the on-orbit mission
: | Time interval for the OSV serving one satellite
: | Position vector at point
: | Position vector at point
: | Velocity of the original orbital at point
: | Velocity vector at point
: | Velocity vector, respectively, at point
: | Velocity of the target orbit at point
: | The 1st velocity increment of the OSV serving the th satellite at the manoeuvre time
: | The 2nd velocity increment of the OSV serving the th satellite at the rendezvous time
: | Weight of a single OSV
: | Weight of the fuel carried by the OSV
: | Weight of the other parts
: | Weight of the remaining fuel after the on-orbit mission is achieved
: | Weight of the consumed fuel
: | Weight of fuel used to keep the orbit
: | Weight of the remaining fuel after a single OSV serves satellites
: | Weight of the remaining fuel after the th OSV serves the satellites
: | Count of the transfer orbit segments
: | Count of satellites served
: | Count of satellites that single OSV can serve
: | th particle
: | Fitness function value of the th particle in the th iteration
: | Fitness function value of the th particle in the th iteration
: | Individual extreme in the th iteration
: | Global extreme in the th iteration
: | Optimization index B
: | Fitness function value of one particle in the present iteration
: | Minimum fitness function value in the present iteration
: | Maximum fitness function value in the present iteration
: | Value of the th particle in the th iteration after the calculation of
: | Value of the th particle in the th iteration after the calculation of
: | Inertia weight
: | Input variable
: | Input variable
: | Set where is in
: | Set where is in
: | Membership function value of
: | Membership function value of
: | Operational symbol for choosing the smaller value between and .

Data Availability

The simulation data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

Acknowledgments

This article is supported by the special research fund of Xi’an High-Tech Institute and cosupported by the National Natural Science Foundation of China (61603398).

References

X. Q. Chen, J. P. Yuan, W. Yao, and Y. Zhao, On-Orbit Service for Spacecraft, China Astronautic Publishing House, Beijing, China, 2009.

F. Sellmaier, T. Boge, J. Spurmann, S. Gully, T. Rupp, and F. Huber, “On-orbit servicing missions: challenges and solutions for spacecraft operations,” in In Proceedings of the AIAA space operations 2010 conference, space operations: exploration, scientific utilization, and technology development, pp. 213–233, 2010.

D. Walt, On-Orbit Servicing of Space Systems, Krieger Publishing Company, Florida, USA, 1993.

Y. Li, J. B. Yao, C. J. Xin, and X. C. Su, “The approaching process analysis of on-orbit service vehicle to circular co-planarity orbit target,” Aerospace Control, vol. 33, no. 4, pp. 56–61, 2015.

J. A. Andrew, B. S. David, and J. H. Terry, “Early mission design of transfers to halo orbits via particle swarm optimization,” Journal of the Astronautical Sciences, vol. 63, no. 2, pp. 81–102, 2016.

M. C. F. Bazzocchi and M. R. Emami, “Stochastic optimization of asteroid three-dimensional trajectory transfer,” Acta Astronautica, vol. 152, pp. 705–718, 2018.

M. J. Grant and R. D. Braun, “Rapid indirect trajectory optimization for conceptual design of hypersonic missions,” Journal of Spacecraft and Rockets, vol. 52, no. 1, pp. 177–182, 2015.

B. Andrea and C. Christian, “A hybrid, self-adjusting search algorithm for optimal space trajectory design,” Advances in Space Research, vol. 50, no. 4, pp. 471–488, 2012.

K. Daneshjou, A. A. Mohammadi-dehabadi, and M. Bakhtiari, “Mission planning for on-orbit servicing through multiple servicing satellites: a new approach,” Advances in Space Research, vol. 60, no. 6, pp. 1148–1162, 2017.

Y. Zhang and Q. Zhang, “On-orbit servicing task allocation for multi-spacecrafts using HDPSO,” Applied Mechanics and Materials, vol. 538, pp. 150–153, 2014.

T. G. Pradeepmon, R. Sridharan, and V. V. Panicker, “Development of modified discrete particle swarm optimization algorithm for quadratic assignment problems,” International Journal of Industrial Engineering Computations, vol. 9, no. 4, pp. 491–508, 2018.

B. Tesarova and A. Vokalova, “Using a genetic algorithm in a process of optimizing the deployment of radio stations,” Future Data and Security Engineering, vol. 106, no. 46, pp. 110–118, 2017.

P. Singh, S. Mini, and K. Sabale, “Particle swarm optimization for the deployment of directional sensors,” Swarm, Evolutionary, and Memetic Computing, vol. 98, no. 73, pp. 167–175, 2016.

A. Ahmadian, A. Elkamel, and A. Mazouz, “An improved hybrid particle swarm optimization and tabu search algorithm for expansion planning of large dimension electric distribution network,” Energies, vol. 12, no. 16, p. 3052, 2019.

J. Zhang, X. H. Zhang, and H. X. Wu, “Genetic simulated annealing algorithm for optimal deployment of flow monitors,” in In Proceedings of the third international conference on natural computation, pp. 398–405, 2007.

P. W. J. Schumacher, C. Sabol, C. C. Higginson, and K. T. Alfriend, “Uncertain Lambert problem,” Journal of Guidance, Control, and Dynamics, vol. 38, no. 9, pp. 1573–1584, 2015.

V. A. Kostenko and A. V. Frolov, “Self-learning genetic algorithm,” Journal of Computer and Systems Sciences International, vol. 54, no. 4, pp. 525–539, 2015.

B. D. Liu, “Genetic algorithms,” Theory and Practice of Uncertain Programming, vol. 239, pp. 9–17, 2009.

E. Mehmed, “A new approach for the genetic algorithm,” Journal of Statistical Computation and Simulation, vol. 79, no. 3, pp. 275–297, 2009.

R. Karthi, S. Arumugam, and K. R. Kumar, “Discrete particle swarm optimization algorithm for data clustering,” Nature Inspired Cooperative Strategies for Optimization, vol. 236, 2009.

A. Saman and S. Rifat, “Discrete particle swarm optimization method for the large-scale discrete time–cost trade-off problem,” Expert Systems with Applications, vol. 51, pp. 177–185, 2016.

A. Ismail and A. P. Engelbrecht, “Self-adaptive particle swarm optimization,” Simulated Evolution and Learning, vol. 7673, pp. 228–237, 2012.

S. Kiranyaz, T. Ince, and M. Gabbouj, “Particle swarm optimization,” Multidimensional Particle Swarm Optimization for Machine Learning and Pattern Recognition, vol. 15, pp. 45–82, 2014.

R. K. Sahu, S. Panda, and S. Padhan, “A novel hybrid gravitational search and pattern search algorithm for load frequency control of nonlinear power system,” Applied Soft Computing, vol. 29, pp. 310–327, 2015.

E. Kayacan, A. Sarabakha, S. Coupland, R. John, and M. A. Khanesar, “Type-2 fuzzy elliptic membership functions for modeling uncertainty,” Engineering Applications of Artificial Intelligence, vol. 70, pp. 170–183, 2018.

T. Mitsuishi and K. Saigusa, “Defuzzification of periodic membership function on circular coordinates,” International Journal of Mathematical, Computational, Natural and Physical Engineering, vol. 8, no. 10, pp. 1246–1249, 2014.