Mission planning for on-orbit servicing through multiple servicing satellites: A new approach

Accepted Manuscript Mission planning for on-orbit servicing through multiple servicing satellites: A new approach K. Daneshjou, A.A. Mohammadi-Dehabadi, M. Bakhtiari PII: DOI: Reference:

S0273-1177(17)30382-4 http://dx.doi.org/10.1016/j.asr.2017.05.037 JASR 13245

To appear in:

Advances in Space Research

Received Date: Revised Date: Accepted Date:

5 October 2016 17 May 2017 23 May 2017

Please cite this article as: Daneshjou, K., Mohammadi-Dehabadi, A.A., Bakhtiari, M., Mission planning for on-orbit servicing through multiple servicing satellites: A new approach, Advances in Space Research (2017), doi: http:// dx.doi.org/10.1016/j.asr.2017.05.037

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Mission planning for on-orbit servicing through multiple servicing satellites: A new approach K. Daneshjou, A.A. Mohammadi-Dehabadi, M. Bakhtiari

Department of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran 16844

Corresponding author e-mail: [email protected]

Abstract In this paper, a novel approach is proposed for the mission planning of on-orbit servicing such as visual inspection, active debris removal and refueling through multiple servicing satellites (SSs). The scheduling has been done with the aim of minimization of fuel consumption and mission duration. So a multi-objective optimization problem is dealt with here which is solved by employing particle swarm optimization algorithm. Also, Taguchi technique is employed for robust design of effective parameters of optimization problem. The day that the SSs have to leave parking orbit, transfer duration between parking orbit and final orbit, transfer duration between one target to another, and time spent for the SS on each target are the decision parameters which are obtained from the optimization problem. The raised idea is that in addition to the aforementioned decision parameters, eccentricity and inclination related to the initial orbit and also phase difference between the SSs on the initial orbit are identified by means of optimization problem, so that the designer has not much role on determining them. Furthermore, it is considered that the SS and the target rendezvous at the servicing point and the SS does not perform any phasing maneuver to reach the target. It should be noted that Lambert theorem is used for determination of the transfer orbit. The results show that the proposed approach reduces the fuel consumption and the mission duration significantly in comparison with the conventional approaches. Keyword: on-orbit servicing, mission planning, Lambert targeting problem, multiple servicing satellites, bi-objective optimization, Taguchi method. 1. Introduction Nowadays on-orbit servicing due to its particular importance has attracted the attention of many researchers. The on-orbit servicing not only increases the life time and the systems performance 1

but also able to reduce the mission costs to a great extent due to the high costs of launching vehicles and satellites. Visual inspection, refueling and on-orbit debris removal are the common on-orbit services which can be done with single or multiple satellites. Generally, the phases of missions related to above-mentioned three categories are the same, i.e. at the first, the servicing satellite (SS) rendezvouses with the desired target and then performs the defined operation on it. As stated above, one of the purposes for on-orbit servicing is space debris removal (SDR). Space debris includes out of service satellites and space bodies. Due to the large number of debris in space, they are a great threat to other satellites, so SDR is considered as an important operation. There are several methods which can be used in order to remove the space debris in an efficient way. Some of these techniques require direct contact with targets while others are contactless. The contact methods include tether ((Zhang et al. , 2015), (Aslanov and Yudintsev, 2015), (Zhang et al. , 2016), (Shan et al. , 2017)), electrodynamic tether ((Iess et al. , 2002a), (Iess et al. , 2002b), (Zhong and Zhu, 2013), (Kawamoto et al. , 2006)), robotic docking ((Smith et al. , 2004), (Bosse et al. , 2004)) and electric sails (Johnson, 2010). As well as, ion beam ((Bombardelli and Pelaez, 2011), (Merino et al. , 2013), (Kitamura, 2010)) and laser ((Phipps et al. , 2012),(Liedahl et al. , 2013),(Wang et al. , 2016),(Hou et al. , 2016),(Sang and Bennett, 2014),(Kirchner et al. , 2013),(Bennett et al. , 2013)) rely on contactless methods. Some of the space missions relate to SDR consists of two main stages which stated above, i.e. rendezvousing the SS with debris or a certain servicing point and then removing the debris. Many studies with respect to the SDR have been done in the past few years. Zuiani and Vasile (Zuiani and Vasile, 2012) proposed a novel approach for preliminary design of low-thrust and many revolution transfers. They reported that their approach reduces the number of the control parameters and increases the computational speed. Madakat et al. (Madakat et al. , 2013) studied the SDR problem to minimize the cost and the duration of the mission. They modeled the problem as a bi-objective time dependent traveling salesman problem. They also used branch and bound approach for debris removal. In their research, they considered one servicing satellite and the Lambert theorem was used for determining the transfer orbit. Cerf (Cerf, 2013) investigated the optimization of both debris selection and trajectory in the SDR missions. He considered the specific impulsive maneuver transfer in order to change the problem to a finite dimension. Meanwhile, he linearized the problem around an initial reference solution by employing the branch and bound algorithm. The proposed method restricted to a single mission and it was suitable only for a small number of the 2

debris. Hence in another work (Cerf, 2015), he taken into account a general transfer strategy (high-thrust to a low-thrust vehicle) and used simulated annealing algorithm for finding the optimal mission scheduling. Yu et al. (Yu et al. , 2014) studied the mission planning for debris removal on LEO orbit through hybrid optimal control theory with focus on some parameters such as communication time window, terminal state, and time distribution constraints. They (Yu et al. , 2015) also considered the

perturbation effect. In another study (Yu, Chen, 2015), they used

hybrid optimal control theory for optimization of SDR scheduling. They assumed that the debris is in a coplanar GEO orbit and the servicing spacecraft initially parked on the debris orbit. In addition, Jing et al. (Jing et al. , 2014) studied bi-objective (cost and time) mission planning for the purpose of debris removal. They have considered both single and multiple servicing spacecrafts to performing the mission and have compared them with each other based on different factors. They supposed that the SSs are initially on the GEO and several debris are on different orbits with different inclinations. Olympio and Frouvelle (Olympio and Frouvelle, 2014) investigated the schedule of mission for space debris removal on the sun-synchronous orbits by consideration of the

2

perturbation and low-thrust propulsion systems. They employed the

techniques of global optimization to find the optimal space debris sequence and mission scenario. Removing multiple debris through chemical and electric propulsion systems have been investigated by Braun et al. (Braun et al. , 2013). For increasing the mission efficiency, they have chosen the targets from a predefined priority list. They indicated that performing the mission with electric propulsion system requires lower fuel mass and more travel time as compared to the chemical propulsion system. However, the mission duration depends on distribution of the targets. Visual inspection is another on-orbit servicing. Some purposes for inspection are planning and on-orbit servicing (because the cost of servicing and maintenance on the orbit is smaller than the other methods), checking the physical health of deployed components such as antenna and solar arrays. In recent years, many researchers have investigated the inspection missions. Horri et al. (Horri et al. , 2012) and Geller (Geller, 2007) studied relative attitude dynamics and control for a satellite inspection mission. Zhai et al. (Zhai et al. , 2013) developed a new type of estimator for the satellite formation for tracking and inspection of on-orbit targets. Zhou et al. (Zhou et al. , 2015c) studied the optimization of mission planning for the visual inspection of multiple satellites that initially placed in a geosynchronous high eccentric orbit. In the mission strategy, 3

they considered that the inspection can be carried out for multiple satellites with different orbital planes. Another important application of on-orbit servicing is refueling. Refueling increases the lifetime of satellites which perform a certain mission. Moreover, some difficulties such as missions around the orbits with high drag or high altitude that restricts the missions can be overcome. Zhou et al. (Zhou et al. , 2015a) studied the optimal mission planning for refueling of multiple geosynchronous satellites with one servicing satellite and one fuel station. They obtained the optimal mission planning with the aim of minimization of the fuel cost by using genetic algorithm and random search to solve the high-level and low-level optimization problem, respectively. They (Zhou et al. , 2015b) additionally designed a new strategy for the refueling mission of the multiple geosynchronous satellites with multiple servicing satellites and fuel stations. The optimization was performed to minimize the fuel cost by employing hybrid particle swarm optimization in high-level and exhaustive search in low-level of the optimization problem. In both of the studies they considered that the servicing satellites can carry just a limited amount of fuel and the servicing satellites and the fuel stations are parked at the GEO orbit. Zhang et al. (Zhang et al. , 2013) optimized the refueling mission of a near-circular LEO multiple spacecrafts by the means of one servicing satellite. In their research, they considered the

2

perturbation and

constraints of rendezvous time. For the purpose of optimization, they employed the hybrid encoding genetic algorithm. Shen and Tsiotras (Shen and Tsiotras, 2005) investigated the problem of peer2peer (P2P) refueling of multiple satellites in a circular constellation. The goal of their research was to equalize the fuel among the satellites in the constellation after one refueling period while minimizing the total fuel cost during the orbital transfers. Also, there are literatures in the area of the cooperative rendezvous maneuver. Prussing and Conway (Prussing and Conway, 1989) studied the optimal terminal maneuver for a cooperative impulsive rendezvous of two space vehicles Coverstone-Carrol and Prussing (Coverstone-Carroll and Prussing, 1993, 1994) investigated the optimal cooperative rendezvous between coplanar circular orbits and stated that the cooperative rendezvous consumes less fuel than the active-passive rendezvous. Their proposed model considered only one-to-one rendezvous. Mifakhraie and Conway (Mirfakhraie and Conway, 1994) developed the method of determining minimum fuel trajectories for the case of fixed-time cooperative rendezvous. They used the primer vector theory in their method. Crispin and Ricour (Crispin and Ricour, 2007) studied the rendezvous problem for two 4

active spacecraft which are orbiting around the sun. They assumed that the spacecrafts have lowthrust propulsion system that provide continues and constant thrust. Dutta and Tsiotras (Dutta and Tsiotras, 2010) studied the general problem of cooperative P2P refueling strategy for satellites where are in a circular orbit. For solving the problem they employed network flow theory. They stated that the proposed cooperative egalitarian P2P strategy is the best amongst all known P2P refueling alternatives until that time. Dutta (Dutta and Tsiotras, 2009) studied the problem of cooperative rendezvous between two satellites. They assumed that the rendezvous completed in a fixed time and the rendezvous orbits are circular. They studied two different types of cooperative rendezvous: one of them is Hohmann transfers and another is Hohmann transfer and a phasing maneuver. They employed their strategies to solve the problem of P2P refueling satellites residing in two different circular orbits. Feng et al. (Feng et al. , 2014) studied the far-distance cooperative rendezvous problem for two spacecrafts. They stated that for two rendezvous spacecrafts which have very similar masses, the cooperative rendezvous can reduce the fuel and the time consumption compared with the active-passive rendezvous. Also, for the case that the flight time is restricted during active-passive rendezvous and it is equal to the cooperative rendezvous, the fuel consumption would increase much more. In another study, Feng et al. (Feng et al. , 2015) investigated the far-distance rapid cooperative rendezvous of two spacecraft with different masses. They stated that when the rendezvous duration is limited to within a certain interval, cooperative rendezvous decreases the fuel consumed significantly. In the case which the duration of rendezvous is unrestricted, the rendezvous time decreases as the thrust increases, but the total fuel consumption varies slightly. Also, Feng et al. (Feng et al. , 2016) studied the fardistance cooperative rendezvous between two spacecraft in a non-Keplerian orbit. Du et al. (Du et al. , 2015) proposed a cooperative maneuver for the scheduling of multi-spacecraft refueling in a circular orbit. In order to optimize the problem, they employed multi-island genetic algorithm and sequential quadratic programming. In their problems, they reveal that the cooperative strategy can save around 27.31% in fuel respect to the non-cooperative strategy. Leigh and Black (Leigh and Black, 2015) proposed a differential correction algorithm in order to deliver an impulsive maneuver to a satellite to place it in an orbit with certain radii, centered around multiple cooperative satellites, duration a limited time. Zavoli and Colasurdo (Zavoli and Colasurdo, 2014) studied the problem of rendezvous between two cooperating spacecraft with time-constrained and finite-thrust. Zhao et al. (Zhao et al. , 2017) investigated the optimization of 5

a LEO cooperative multi-spacecraft refueling mission with considering the

perturbation and

target’s surplus propellant constraint. Moreover, they considered the time and the target spacecraft’s surplus propellant capability constraint. In this study, a novel idea is applied for optimal mission planning of different on-orbit servicing operations which are performed through multiple servicing satellites. In space missions, the eccentricity, the inclination and the location of the SSs on the initial orbit considerably affect the fuel consumption and the mission duration. In the other hand, in the previous studies, the values of the aforementioned parameters were identified by the designer while they may not be the proper values to reach minimum fuel and duration for the mission. In this study to fill the gap in the previous research works, it is considered that the above parameters are obtained through the optimization process and the designer has no role in determining them and he only identifies the governing constraints on the mission. Since the mission planning is performed based on minimization of both the fuel consumption and the mission time, a multi-objective particle swarm optimization (MOPSO) algorithm is applied for this purpose. Also, the Taguchi technique is used for the robust design of control parameters of the MOPSO algorithm. Furthermore the transfer orbits of the SSs is found based on the Lambert targeting method.

2. Mission scenario In order to investigate the proposed approach, a mission planning problem is considered in order to minimize its fuel cost and time. In the proposed mission, the on-orbit servicing (which can be visual inspection, refueling or active debris removal) on

targets is carried out by using

servicing satellites (SSs). Generally, the aforementioned services have the same phases, as described below: The SS leaves the initial orbit and goes on the transfer orbit, then leaves the transfer orbit and rendezvouses with the target at the servicing point and performs the defined task (inspection, refueling, or debris removal) on the target, after that goes toward the next target. Figure 1 shows a schematic view of the whole mission for SSs and multi targets. After the SS revolution on the parking orbit in a certain time (

), the first velocity impulses (

) is

exerted to the SS. Consequently, the SS departures the parking orbit and flies on the elliptical transfer orbit for duration of

(the transfer orbit is obtained through Lambert’s theorem). 6

After that, the second velocity impulses (

) is applied to the SS to leave the transfer orbit

and it rendezvous with the target at the servicing point on the final orbit. After performing the duties on the target, the SS spends a certain time on it with the target until it leaves the target). Then

is the time that the SS rendezvouses makes the SS to leave the target and fly

toward another target. This process continues till end of the mission. The following constraints should be satisfied during the mission: •

Targets and SSs are in two different elliptic non-coplanar orbits.

•

Each SS services at least one target.

•

Each target is serviced only by one SS.

•

At the end of the mission, the SSs are placed with a certain phase shift from each other.

•

Operating time (i.e. elapsed time from the beginning until the end of the mission) is the same for all the SSs.

According to the above assumptions, the mission is a simultaneous collaborative mission.

Figure 1: Schematic view of mission for SSs and multi targets

7

3.

Problem solution

The solution of the problem consists of two main steps: Step 1: Obtaining all the missions in which number of

SSs are able to service

targets.

In this step the whole states are taken into consideration using a new proposed method. For explanation of the method, it is assumed that the number of SSs and targets are 2 and 3, respectively, i.e.

and

(It should be noted that in the following sections the results

are obtained for these values). At first, according to Table 1, for each SS, a number is assigned to each target. Table 1: Assigned number to targets for each SS Target 1

Target 2

Target 3

SS A

1

2

3

SS B

4

5

6

Then, the missions in which each SS is able to service the targets alone are calculated (Table 2 and Table 3). Table 2: The missions in which SS A is able to service the targets alone Possible missions

Servicing order

Mission 1

1

2

3

Mission 2

3

1

2

Mission 3

2

3

1

Table 3: The missions in which SS B is able to service the targets alone Possible missions

Servicing order

Mission 1

4 or target 1

5 or target 2

6 or target 3

Mission 2

6 or target 3

4 or target 1

5 or target 2

Mission 3

5 or target 2

6 or target 3

4 or target 1

8

In the next step, different states for which two SSs are able to service three targets are calculated. Table 4: Different states in which two SSs are able to service three targets Number of targets that each SS is able to service to targets in every state

State number

SS A

SS B

1

3

0

2

2

1

3

1

2

4

0

3

Now, according to Table 2, Table 3 and Table 4, the missions related to each state are calculated such as the provided examples in the following. For example, if one wants to calculate all the possible missions for state 2 in Table 4 (i.e. obtaining all of the possible missions in which SS A services 2 targets and SS B services 1 target), the two first columns from Table 2 which are related to the SS A are combined with the third column of Table 3 refer to SS B (because the number of targets that SS A and SS B is able to service them are 2 and 1, respectively). Table 5: Missions relate to state 2 Possible missions

Targets which SS A

Targets which SS B

service to them

service to them

Mission 1

1

2

6 or target 3

Mission 2

3

1

5 or target 2

Mission 3

2

3

4 or target 1

As another example, in state 3 from Table 4, the first column of Table 2 which corresponds to the SS A are combined with the second and third columns of Table 3 which are related to the SS B (because the number of targets that SS A and SS B is able to service them are 1 and 2, respectively). For states 1 and 4, the missions are the same as those from the SS A and SS B according to Table 2 and Table 3, respectively.

9

Table 6: Missions related to state 3 Possible missions

Targets which SS

Targets which SS B service to

A service to them

them

Mission 4

1

5 or target 2

6 or target 3

Mission 5

3

4 or target 1

5 or target 2

Mission 6

2

6 or target 3

4 or target 1

Step 2: Optimization of the problem in order to select the best mission in terms of minimum fuel and time. Having extracted all of the possible missions for servicing number of

targets by the means of

SSs, these missions are used in the optimization problem in order to select the best mission from them for on-orbit servicing. The optimization method is proposed in Section 5. 4.

Lambert targeting

The transfer orbit associated for each SS is found using Lambert theorem. As seen from Figure 2, the SS and the target are first located at

and

, respectively (The superscripts “s” and “t”

denote the SS and the target, respectively). After passing of time is applied at point

, the first velocity impulse

to enter the SS into the transfer orbit. During the transfer time

SS flies from the parking orbit to the final orbit and arrives to the point during the time of reach to point

, the target goes from point

to

, the

. On the other hand,

. Therefore, the SS and the target

, simultaneously. Finally the second velocity impulse

is applied at point

to make it enter into the final orbit and rendezvous with the target. It should be noted that the time

and the transform time

the positions

and

are obtained from the optimization problem. So, according to

and the time of transformation

, one is able to use the Lambert theory to

obtain the transfer orbit (Chobotov, 2002, Curtis, 2013, Vallado, 2001) and subsequently the required velocity changes. The mathematical formulation is proposed in the following subsections.

10

Figure 2: Schematic of Lambert targeting 4.1

State vector of an object after time

The

dynamics

characterization of , where

inclination,

body on space

is the semi major axis,

is the argument of perigee,

can be

specified

is the eccentricity,

with

vector

is the orbital

is the right ascension of ascending node and

is the

mean anomaly. For a body which revolves on a specific Keplerian orbit, just the mean anomaly changes. To get the state vector of an object after the time state vectors

and

, one must first obtain the initial

from the vector . Next the following relationships are used in order to

calculate the values of , , ,

which are known as Lagrange coefficients:

(1)

In the above equations, 398600

.

is the gravitational parameter and its value for the earth is

is the reciprocal of the semi major axis, and is calculated by the means of

below equation:

11

(2)

is the universal anomaly which is obtained from solving the universal Kepler’s equation (3) by employing iteration procedure. velocity vector

and

are the magnitude of the position vector

and the

, respectively. (3)

and

are Stumpff function that are defined as follow for an ellipse:

(4)

Finally by employing equation (5), the state vector ( , ) after a time of from the initial state vector (

,

can be calculated

) (Curtis, 2013).

(5) 4.2

Lambert targeting formulation

Velocity of the satellite at the beginning of placement on the transfer orbit is velocity when leaving the transfer orbit is points

and

. By calculating the velocities

and the satellite and

at the

on the initial orbit, respectively, the total velocity increment for each

transformation is found as below:

(6)

and

are obtained using the following equations: (7)

and

are Lagrange coefficients and they are the same as equation (1), but just with a

difference that the universal anomaly, , is calculated from equation (8). (8)

12

is the angle between

and

which is obtained for a prograde trajectory by the following

equations (Curtis, 2013):

(9)

It should be noted that the Lambert theory (or Lambert targeting) is not only used for transmission of the SS from the parking orbit to the final orbit but also used for SS transmission from one target to another one. The requirement velocity impulses for whole of the mission is provided by Equation (10) where represents the number of the transfer orbits.

(10)

5. Optimization procedure 5.1

Particle swarm optimization algorithm

Particle swarm optimization (PSO) algorithm is a meta heuristic technique adapted from social behavior of the bird or the fish groups which was introduced by Eberhart and Kennedy (Eberhart and Kennedy, 1995), in 1995. The PSO algorithm has several advantages, for instance it requires only a set of elementary arithmetic operators that makes implementation of this algorithm simple and economical (Bakhtiari et al. , 2017, Fakoor et al. , 2016). 5.2

Multi-objective particle swarm optimization (MOPSO)

Multi-objective particle swarm optimization was introduced by Coello (Coello et al. , 2004, Coello and Lechuga, 2002). In fact, MOPSO is an extension of the PSO algorithm that is used for solving the multi-objective problems. In the MOPSO algorithm a new concept called repository has been added to the PSO. The repository is also known as hall of fame. The repository is an archive of the non-dominated particles that have been found and its members represent the Pareto front. Choosing the best global answer and the best personal recollection for each particle are important steps in the MOPSO algorithm. When the particles want to move, they choose a member of the repository as the leader. It is noted that the leader should be a member of the repository and a non-dominated answer. Therefore, in the PSO algorithm there is only one 13

objective and one particle incorporates the best answer while there are several non-dominated particles in the result set in the MOPSO. For this reason, “globalbest” concept is used in the PSO, whereas the “repository” concept is applied in the MOPSO. The equations for updating the position and the velocity of each particle are as below (Coello, Pulido, 2004):

(11)

Where

is the inertia weight coefficient,

and

are the cognitive and social factors,

respectively. They are random numbers in the range of gives the best recollection of each particle, it is chosen randomly and

denotes the number of particles, represents a member of repository and

is the number of iterations.

The best personal recollection vector is determined based on the following rules: 1) If the new position

dominates the best recollection

, then the new

position is the best recollection position. (12)

2) If the best recollection position dominates the new position, then the best recollection position does not change. (13)

3) If the new position and the best recollection does not dominate each other, one of them is considered randomly as the best position. The steps of the algorithm are as follows: 1) Determination of the required parameters for implementation of the algorithm such as maximum number of iteration, number of particles, number of repository member, values of ,

and

(Section 5.3).

2) Generation the initial population. 3) Determination of the best recollection for each particle.

14

4) Separation of the non-dominated members of population and storing them in the repository. 5) Each particle chooses a leader among the repository members and performs its movement, i.e. the velocity and the position of the particle is updated. 6) The best recollection of each particle is updated. 7) New non-dominated members are added to the repository. 8) Dominated members of the repository are removed. 9) If the stop conditions aren’t satisfied, the algorithm is repeated starting from step 5. Here, the objectives are the fuel consumption and the mission duration. Number of the decision variables varies based on the number of the SSs and the targets. Generally, for each rendezvous the decision variables are as follows: 

The revolution time of each SS on the initial orbit.



Duration of the orbital transfers, either the travel time between the initial or the final orbit or between one target and another target (



).

Total time of performing the assigned task and revolution of the SS on the target (

In the current study, , and

).

parameters associated to the initial orbit have been added to the

decision variables. The constraints governing the mission include the following: 

;



.

  

is the period of the final orbit.

. . .

For the proposed example, i.e. 2 SSs and 3 targets, the decision parameters include: (14)

and the objective functions are as follows (15)

15

5.3

Taguchi method

Performance of the optimization problems such as MOPSO depends on the control parameters significantly, therefore in order to achieve the best response from these algorithms, it is necessary to tune the control parameters based on the problem conditions. In the MOPSO, the main parameters which should be tuned on the best level include: inertia factor ( ), cognitive factor ( ), social factor ( ), number of particles (NumPop), maximum number of iterations (MaxIter), number of repository members (NumRep) and number of grids (NumGrid). Design of experiment (DOE) through Taguchi method is employed for designing the robust control parameters (parameter tuning) in such a way that the variation of the objective functions minimizes or the quality of the responses improves (Myer and Montgomery, 2002). The purpose of the experiment design using Taguchi method is to obtain the maximum volume of the information by performing the minimum number of experiments. Taguchi method uses two important concepts in DOE: 1) Orthogonal array (OA): OA is employed for production of a matrix in different experiments based on different levels that should be changed. OA helps the designer to study different controllable factors on the quality properties in a fast way; 2) Signal to noise ratio (S/N): this concept demonstrates the sensitivity of the quality properties to the controllable parameters of the process. The greater S/N implies that the parameter is more effective. Therefore, through Taguchi method one can determine the best level for each parameter based on the minimum variation of the objective function. More details for Taguchi method and orthogonal array are presented in the reference (Taguchi et al. , 2005). In order to tune the parameters with Taguchi method, a characteristic must be considered that represents rather different qualities of the experimental results. There is only one objective function in the single objective optimization problem. Hence, this objective function is selected as the quality characteristic of the response. While in the multi-objective optimization problem (here deals with a bi-objective optimization problem) there are several objective functions in addition to a Pareto frontier. Therefore, another quality characteristic should be used in order to show the effect of both objective functions on the characteristics. In the multi-objective optimization problems, there are several criteria for assessment, the following ones are applied here (Gao et al. , 2014, Rabiee et al. , 2012):

16

1) Number of Pareto Solution (NPS): this criterion counts the number of points on Pareto frontier or number of non-dominated solutions of the algorithm. 2) Mean Ideal Distance (MID): it denotes the mean distance between Pareto frontier and ideal point (0, 0) which is obtained from the following equation:

(16)

In the above equation

is the number of non-dominated points.

and

represent the

first and second objective functions. Superscripts ‘Pareto’ and ‘total’ denote the particles on Pareto frontier and all of the particles, respectively. 3) Spacing Metric (SM): the distribution uniformity of points on Pareto frontier is measured by this metric according to equation (17).

(17)

Where

is Euclidean distance between two consecutive points on Pareto frontier and

is average of

.

4) Hyper area Difference (HD): This criterion is used for quality assessment of the difference between Pareto solution that is obtained from optimization algorithm and true Pareto solution (the solution which dominates all the possible solutions). It should be noted that the true solution usually does not exist but one can use this criterion to compare qualitatively how much an approximate solution differs to another approximate solution in terms of the true solution. For assessment of this metric, at first, maximum ( minimum (

) and

) points (they are named bad and good points, respectively) are determined

for each objective function in such a way that all of the Pareto solutions are enclosed in a rectangular area. Then the Pareto solutions convert to dimensionless quantities via equation (18).

(18)

17

According to Figure 3, the shaded area represents the value of hyper difference metric for the Pareto frontier. More details about HD is proposed on (While et al. , 2012, Wu and Azarm, 2001).

Figure 3: Dimensionless Pareto solution, Shaded area represents the value of HD In this study, the results are conducted using Minitab 17 software. According to Table 7, three levels are considered for each parameter. Since the number of parameter is 7 and three levels are considered for each parameter, orthogonal array has 27 rows. The results of optimization problem associated to each row of Table 7 are listed in Table 8. Figure 4 illustrates the values of S/N for each parameter.

Table 7: Considered levels for control parameters Control parameters

level 1

level 2

level 3

cognitive factor (C1)

1

1.5

2

social factor (C2)

1

1.5

2

inertia weight (w)

1

1.5

2

number of particle (NumPop)

100

150

200

number of iteration (MaxIter)

50

100

150

number of repository (NumRep)

50

75

100

number of grid size (NumGrid)

5

10

15

18

Table 8: Orthogonal arrays and values of different assessment criteria

Run

Control parameters

Assessment criteria

C1

C2

W

NumPop

MaxIter

NumRep

NumGrid

NPS

MID

SM

HD

1

1

1

1

1

1

1

1

13

0.8797

0.9468

0.3789

2

1

1

1

1

2

2

2

10

0.8553

0.8448

0.3491

3

1

1

1

1

3

3

3

26

1.1403

0.8892

0.4216

4

1

2

2

2

1

1

1

10

0.8244

0.8293

0.3101

5

1

2

2

2

2

2

2

14

0.8363

0.7291

0.3095

6

1

2

2

2

3

3

3

15

0.8251

0.709

0.3084

7

1

3

3

3

1

1

1

16

0.678

0.8234

0.2568

8

1

3

3

3

2

2

2

26

0.6188

0.7025

0.2028

9

1

3

3

3

3

3

3

25

0.681

0.7272

0.2528

10

2

1

2

3

1

2

3

18

0.6992

0.6923

0.2676

11

2

1

2

3

2

3

1

30

0.8401

0.8528

0.3204

12

2

1

2

3

3

1

2

25

0.7799

0.7986

0.2834

13

2

2

3

1

1

2

3

24

0.5777

0.7967

0.1871

14

2

2

3

1

2

3

1

11

0.8511

0.9105

0.3648

15

2

2

3

1

3

1

2

29

0.9963

0.57

0.3942

16

2

3

1

2

1

2

3

28

0.5688

0.9293

0.2499

17

2

3

1

2

2

3

1

20

0.8178

0.8786

0.331

18

2

3

1

2

3

1

2

23

0.5978

1.0466

0.2914

19

3

1

3

2

1

3

2

10

0.7931

0.5907

0.3184

20

3

1

3

2

2

1

3

7

0.8328

0.6142

0.3115

21

3

1

3

2

3

2

1

24

0.6321

0.7976

0.2542

22

3

2

1

3

1

3

2

21

0.7693

0.7152

0.2739

23

3

2

1

3

2

1

3

29

0.6514

0.6161

0.2474

24

3

2

1

3

3

2

1

30

0.8327

0.8138

0.3306

25

3

3

2

1

1

3

2

28

0.7823

0.8112

0.2951

26

3

3

2

1

2

1

3

9

0.9518

0.5992

0.3798

27

3

3

2

1

3

2

1

23

0.6686

0.841

0.2945

19

Figure 4: Values of S/N for different control parameters 5.4

Definition of criterion for result assessment

A reasonable criterion is necessary to compare the results with each other. It is clear that any solution which leads to less fuel consumption and shorter time is more desirable for the mission. This viewpoint is used for definition of the criteria to assess the results. The following example is given to clarify this issue. Figure 5 shows the Pareto fronts which have been obtained from optimization of the problem for two different cases. To select a time ( which is performed at a constant velocity, point

or

) for the mission

is selected because it has shorter time as

compared to the point . On the other hand, if one decides to carry out the mission at the constant time, the point

is selected because it requires less velocity, hence the Pareto front 1 is better for

the mission. Pareto front 1 is closer to the time and the velocity axis, in other words, the Pareto front 1 is said to dominate the Pareto front 2. As a result, generally in the case of two objective min-min optimization problems, if the Pareto front 1 is able to dominate the Pareto front 2, it is better and more desired solution for the problem.

20

Figure 5: Schematic of two different Pareto frontier 6.

Numerical simulations

In the numerical simulations, it is assumed that the SSs and the targets are in a non-coplanar elliptic orbits. The orbital elements associated with the final orbit and the state of the targets are presented in Table 9 which are the same for all the problems. Furthermore, Table 10 gives several cases for the orbital elements and the state of SSs on the initial orbit and it should be noted that the perigee is considered to be constant for all the cases under study. All .

proposed examples (3

targets and 2 spacecrafts) Table 9: Orbital elements relate to the final orbit Orbital elements

0.35

4000

0

True anomaly of targets

35

0

21

45

125

275

Table 10: Orbital elements of the initial orbits for different cases Orbital elements Case 1

0.1

20

40

180

140

Case 2

0.2

20

40

180

140

Case 3

0.2

20

40

80

40

Case 4

0.2

15

40

180

140

500

In order to show the effects of ,

and

45

30

parameters on the fuel and the time, four kinds of

problems are solved through optimization algorithm which include: a) Problem 1: , and

are added to some of the decision parameters and are characterized by

optimization. Table 11 summarizes the results related to the expressed approach in the study from which , and

are added to the decision parameters and their values are identified by the optimization

problem and aiming to yield the minimum fuel consumption and the mission time, so the designer has no role in determining them. The results of this problem are compared to those obtained from three others in order to examine the performance of the proposed idea. Meanwhile, Figure 6 illustrates the mission related to particle 3 on Table 11.

Table 11: Results of problem 1 Decision variables Particle number 1

0.17989

4.224285

269.111

21678.94

46175.57

12272.4

12137.31

13784.92

2

0.16678

1.649979

188.4902

14783.28

33565.74

12157.4

12889

13689.53

3

0.18243

3.423087

250.9735

11725.45

47435.26

13055.36

11522

13175.11

4

0.19161

3.896878

245.5336

11556.56

46468.06

12700.3

11972.23

13805.39

5

0.16761

6.796878

211.5336

10470.85

45412.86

13202.24

11879.99

13303.45

6

0.14463

10.72485

239.6796

7667.471

43358.39

13057.61

11805.14

13181.71

22

Continues of Table 11 Decision variables

Objective functions

Particle number*

Total

Duration

velocity

of

impulses

mission

Best mission

SS A

SS B

1

7392.751

4804.288

13657.07

7015.013

8.276238

48359.1

2

3→1

2

7916.551

5792.153

14762.07

9420.685

10.33468

44277.91

3

1→2

3

5979.737

4044.339

12965.61

5758.847

11.66112

36519.4

2

3→1

4

6398.22

4097.725

12889.94

5760.583

12.92729

36415.66

2

3→1

5

4892.4

5603.544

12707.37

5320.858

13.27578

33886.36

2

3→1

6

4221.12

5708.025

12800.88

4492.327

15.67062

29438.52

1→2

3

* “Particle number” denotes the number in which assigned to the particles where on the Pareto frontier.

For the application problem 1 presented (results of Table 1), Table 12 is presented to give more details on the DV repartition (DV per maneuver for each spacecraft). These successive DV are the useful results to design the vehicles. Table 12: DV per maneuver for each spacecraft in problem 1 Best mission

SS A Departure 1st target and rendezvouses with 2nd target ( ) -

Achieving final position ( )

Particle number

SS A

SS B

1

2

3→1

Departures initial orbit and rendezvouses with 1st target on final orbit ( ) 3.309040

2

3

1→2

4.123951

-

-

3

2

3→1

4.669844

-

-

4

2

3→1

5.975003

-

-

5

2

3→1

5.867106

-

-

6

1→2

3

4.466536

3.436616

-

Continues of Table 12 23

-

SS B Particle number

1 2 3 4 5 6

Departures initial orbit and rendezvouses with 1st target on final orbit ( ) 2.758341 2.656652 3.488896 3.587813 4.104110 4.585926

Departures 1st target and rendezvouses with 2nd target ( )

Achieving final position ( )

(

0.974018 1.413808 1.144581 1.180708 1.215564 -

1.234837 2.140264 2.357801 2.183764 2.089003 3.181540

3.309040 4.123951 4.669844 5.975003 5.867106 7.903152

)

a) Initial position of the SSs and the targets

24

(

)

4.967197 6.210725 6.991278 6.952286 7.408678 7.767467

(

)

8.276238 10.33468 11.66112 12.92729 13.27578 15.67062

b) Trajectory of SS A

c) Trajectory of SS B from initial orbit to target 3

25

d) Trajectory of SS B from target 3 till end of mission Figure 6: Different mission phases for particle 3 according to Table 11 b) Problem 2:

and are constants, while

changes.

Figure 7 depicts the state in which the values of effect of variable

and are constants and

changes. Indeed the

has been investigated. Cases 1 and 2 from Table 10 are used to obtain the

results. According to figure, results for

dominate the

, hence the case with

are conducted to better response in comparison with

. Also, the proposed

approach in this study (problem 1) dominates both

and

. Therefore, it

outperforms them. It should be noted that in general state one can’t conclude that by decreasing the responses are improved. While it depends on the position of the targets on the final orbit and may vary from one problem to another.

26

Figure 7: Results of problem 2:

c) Problem 3:

and are constants, while

and are constant values, but

changes

changes.

Cases 1 and 2 from Table 10 are used to study the effect of eccentricity ( ). By consideration of two different values for

(

and

) and constant values for and

carried and the results are illustrated in Figure 8. In this problem, in other word the first one resulted in a better response.

27

, the mission is

dominates the

,

Figure 8: Results of problem 3: d) Problem 4:

and

and are constant values, but

changes

are constants and changes.

The effect of inclination is shown in Figure 9. The results are obtained from the cases 2 and 4 of Table 10 in which only

changes and

and

remain constant. The fuel consumption and the

mission duration increases when the inclination changes from

Figure 9: Results of problem 4:

and 28

to

.

are constants and changes

7.

Conclusion

On-orbit servicing has large applications and is used for important purposes. The minimization of both the fuel consumption and the mission duration are the most important issues which are considered in these missions. Thus, a new approach is proposed in this study for minimization of both the fuel and the time. Unlike the conventional methods in which the designer identifies the eccentricity, inclination and phase difference of the SSs, the parameters are specified through the optimization problem in the proposed approach. The results show that the eccentricity, the inclination and the phase difference of the SSs on the initial orbit have major effects on the fuel and the time of missions. In the proposed examples, increasing the eccentricity, phase difference between the SSs, and decreasing the inclination lead to a decrease in the fuel consumption and duration of the mission. The results reveal that the new proposed method reduces the fuel and the time significantly, so it can be introduced as an appropriate method for planning of the space missions. Moreover, the fuel consumption is smaller in the missions with longer travel time, because the SS has more opportunities to revolve on the targets and can select an appropriate transfer orbit which needs lower impulsive velocity. Furthermore, the obtained results demonstrate that for a certain mission, if the devoted fuel to the mission increases, changes in the eccentricity, the inclination and the location of the SSs on the parking orbit have less impacts on the mission duration.

29

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33

Highlights  Identifying initial orbit and phase difference between SSs on it by using MOPSO.  Proposing a method for obtaining missions that

satellites service to

targets.

 Employing multi-objective particle swarm optimization for solving the problems.  Employing Taguchi method for robust design of effective parameters in MOPSO.

34