Optimal mission planning of active space debris removal based on genetic algorithm

With the increasing space activities, the space debris has seriously deteriorated the safe performance of the on-orbit spacecraft and has attracted much attentions. To reduce the rising influence of the space debris and improve the safe performance of the space mission, the three-stage removal strategy for space debris is proposed in this paper. Firstly, the multiple spacecrafts including one main spacecraft and some following spacecraft for space debris removal mission is developed. Then, the fuel, time and the quantity of the following spacecraft are defined as the constraints. Moreover, using the minimum fuel consumption as the optimal object, the mathematical model of the debris removal problem is established. Finally, the genetic algorithm is applied to solve this problem. Compared with the prior space debris removal strategy, the proposed three-stage space debris removal can effectively reduce the fuel consumption. Numerical simulation verifies the effectiveness of the proposed space debris removal scheme.


Introduction
Due to the human space activities, space debris becomes the major sources of the space pollutions. The space debris contains the scrapped rocket arrow and satellite body, the jet of the rocket, the discards during the execution of the space mission, the fragments generated by the collision between the space objects and so on. Furthermore, because of the launch of the satellite, the amount of space debris has considerable increase. According to the reports from Space Surveillance Net (SSN), more than 18,000 space targets have been catalogued. Moreover, 90% of the space targets are space debris [1] and the large amount of space debris threats the safety of the spacecraft in orbit.
For example, Iridium 33 satellite collided with the Russian scrap satellite Cosmos 2251 on February 10, 2009, which seriously influenced the American Iridium constellation. In 1991, Kessler proposed the Kessler Syndrome: Collision of space targets caused a large amount of space debris. Subsequently, the resulting space debris may collide with other spacecraft to lead to the mission failure and produce more debris. Therefore, the number of space debris will continue to increase. Furthermore, when space debris grows to the alert values, even that the humans cannot implement any space activities, space collisions will occur and the amount of space debris will still grow [2]. Based on the related research by NASA and ESA, the fragments should be actively removed every year since This article adopts two-impulse rendezvous maneuvering model under J2000 coordinate frame. Given the location of the two targets of space and the orbital maneuver time, the transfer orbit between the two targets can be obtained by solving the loopy Lambert problem.
In this paper, the method of solving the loopy Lambert problem proposed by Han Chao is used [11].
Define   t z as shown in Eq. (1): x z S z y z is universal variable z , and  is gravitational constant of earth. Figure 1 shows the relationship between   t z and the variation of the universal variable z . From figure 1, transfer time t and the universal variable z has the relationship of approximate periodicity. When given the transfer time t  , there may be one or more corresponding universal variable z 。Calculate the 1 2 , v v   corresponding to the universal variable, then the impulse required for the orbital transfer can be calculated by Eq. (2) Take the orbit with the smallest speed impulse as the maneuvering transfer orbit, that is, the loopy Lambert transfer orbit with minimum fuel consumption.
However, this method can only calculate the impulse for a given transfer time. When the given transfer time varies, the impulse will change with it. Therefore, there is a certain time to minimize the velocity impulse, and the maneuver time need to be designed. We optimize this problem with particle swarm optimization algorithm (PSO) and obtain the optimal transfer orbit with the smallest impulse, meeting the time constraint. Through priori calculation, the PSO will converge to better results when iterating 6-7 times. In order to speed up the calculation, in this paper, the number of particle groups is set to 20, and the maximum number of iterations is 10. Each Particle swarm optimization calculation takes 0.025s. So far, the required impulse j i v  and time j i t from the i-th debris to the j-th debris can be obtained.  the j-th debris, clearing time for the j-th debris and refueling time respectively. The maneuver from the i-th debris to the j-th debris can be divided into two cases: a. maneuver to the next target directly In this situation, fuel and propulsive satellite are enough. Once maneuver between two targets need twice impulse. (

Mission costs
are fuel consumption of twice impulse respectively. j i dM is the fuel consumption from i-th to j-th debris.
Propulsive satellite is needed for one-time removing mission. The mass change caused by the consumption of propulsive satellite is p The whole mass decrement of mission satellite is shown in Eq. (6): The time for one debris is got by Eq. (7) where j i t and j t are maneuver time from the i-th debris to the j-th debris and clearing time for the j-th debris respectively.
b. maneuver to the space depot for replenishment In this situation, fuel or propulsive satellite is not enough. According to the orbit model, the optimal transfer orbit can be calculated through PSO, that is, the maneuver impulse to space depot and impulse from depot to the next debris. Record the four maneuver impulses as 1 , 2 , 1 , 2 in accordance with the time. When maneuvering to space depot, the mass of mission satellite returns to the original state. P represents space depot, the fuel consumption can be calculated by Eqs (8) where r M is the remaining fuel of mission satellite. The time for one debris is got by Eq. (15) where f t and j t are refueling time and clearing time for the j-th debris respectively. P i t and j P t are transfer time from i-th debris to space depot and from space depot to j-th debris respectively.

Optimization model
The number of space debris to be cleaned is n, and they are numbered 1-n in order. The target of the task is all the n debris are completely removed. The order of removing the depot can be represented by an array of the n numbers. For example, N = [3,4,1,2] means that the 3rd, 4th, 1st, and 2nd debris are cleaned in order. All the possible paths are the full arrangement of the n numbers. The optimal array or the optimal path can be found after calculating the fuel consumption in each arrangement. Set the initial position of the mission satellite the same as the first debris.

Calculation process
When a sequence of N, that is, the order of debris removal is given, from the first debris, the mission spacecraft execute the orbital maneuver in sequence to remove debris. After removing the first debris, the fuel consumption and propulsive spacecraft consumed by orbital maneuvers and debris removal can be obtained according to the maneuver strategy. When the next target is to be removed, a judgment is needed: if the mission satellite carries insufficient propulsive satellites or the remaining fuel is insufficient to transfer to the next target, it will maneuver to the space depot for replenishment. The maneuver impulse and time consumption for the mission satellite transfer from debris to debris or space depot can be determined by the orbital model. Through such process, the fuel and time consumption for the given mission sequence can be obtained.
The calculation process can be summarized as the following steps: Step1: Remove the first debris.
Step2: Judge whether fuel or propulsive satellite is enough.
Step3: If the condition of step 2 is met, remove the next space debris.
Step4: If the condition of step 2 is not met, maneuver to space depot for replenishment and remove the next space debris.
In all the steps, the particle swarm optimization algorithm is needed to find the optimal transfer orbit.

Constrains
In this problem of mission planning, some constrains need to be satisfied.
Let max t be the maximum transfer time for each piece of debris. Therefore, the transfer time need to

Genetic algorithm design
It can be analyzed that the problem to be solved in this paper is a TSP problem in fact. When there are more debris to be removed, the problem of "combination explosion" will occur, which greatly increases the amount of calculation. In the method of solving the TSP problem, the genetic algorithm has great advantages in solving this problem. This paper uses genetic algorithm to solve this problem.
Step1. Code In this paper, positive integer coding is used. The chromosomal gene represents the order of debris removal. For example, there are 7 debris to be removed, then the chromosomes [6,5,7,2,3,1,4] indicate that these targets are removed in order.

Step2. Select
The random sampling method is selected for selection. The fitness function is ( ) , which represents the fuel consumption.
Step3. Crossover The crossover method is selecting a position in chromosomal gene randomly, and crossing over the parental gene after the selected position. If the chromosome after the crossover has a repeated gene, the repeated gene is replaced with other gene. E.g. select a position in chromosomal gene randomly FatherN1 = [3,6,8,7,4

Step4. Mutation
The mutation operation is selecting two points on the chromosome randomly and exchanging the genes represented by the two points.

Simulation and analysis
All the spacecrafts run on near-circular orbit. Select 9 pieces of space debris on SSO at the height of 850 km. Orbital inclination is 98.8212 and RAAN is 70. Space depot runs on the same orbital plane 100km under the debris orbit. The true anomaly of debris is shown in table1. . The maneuver time constraint is less than four orbital periods.
Before optimization, the fuel and time consumption are calculated in the condition that the mission schedule is not planned. Set the debris removal path N = [1,2,3,4,5,6,7,8,9]. The results are as follows, mission satellite need replenishment for 3 times and the fuel and time consumption of the entire task is 624.28kg and 352830s respectively. [1,2,3,4,5,6,7,8,9] 624.28     figure 3, the optimal debris removal path [5, 6, 7,8,9, 2,3,  consumption during the refueling process is 402.14 kg. 280.02kg of fuel is supplied by space depot. Mission satellite needs to be replenished three times. Total task time is 316330s. It can be seen from the experimental results that the optimal fuel consumption will reduce by 35.6%(222.14kg) compared to the initial removal path, and the total mission time reduces by 10.3%.  Table 2 shows the detailed consumption of the whole mission. The plus sign indicates mission satellite maneuver to space depot for replenishment. From the figure, fuel consumption is large when satellite need replenishment. The optimal removing sequence of the mission satellite is in the order of sequentially adding a true anomaly angular position. When the service spacecraft returns to the space depot, the order of removing is no longer in full compliance with this law. Under the constraint of propulsive satellite, mission satellite needs to maneuver to space depot more often, increasing fuel consumption of the mission. Transfer time is close to the critical value as longer maneuvering time bring smaller impulse.

Conclusion
In this paper, the active space debris removal mission planning is studied and the three-stage system is proposed to solve this problem. Firstly, the multiple spacecraft including one main spacecraft and some following spacecraft for space debris removal mission is developed. Then, the fuel, time and the quantity of the following spacecraft are defined as the constraints. Moreover, using the minimum fuel consumption as the optimal object, the mathematical model of the debris removal problem is established. Finally, the genetic algorithm is applied to solve this problem. Compared with the prior space debris removal sequence, the optimal three-stage space debris removal sequence can effectively reduce the fuel consumption. Numerical simulation verifies the effectiveness of the proposed space debris removal scheme. Furthermore, the debris removal mission planning strategy can also be applied to refueling problem. However, this developed strategy ignores the influence of perturbation. In the future, the influence of the perturbation and the safe constraints should be studied.