A new preliminary trajectory design method for asteroid rendezvous mission using multiobjective optimization techniques is proposed. This method can overcome the disadvantages of the widely employed Pork-Chop method. The multiobjective integrated launch window and multi-impulse transfer trajectory design model is formulated, which employes minimum-fuel cost and minimum-time transfer as two objective functions. The multiobjective particle swarm optimization (MOPSO) is employed to locate the Pareto solution. The optimization results of two different asteroid mission designs show that the proposed approach can effectively and efficiently demonstrate the relations among the mission characteristic parameters such as launch time, transfer time, propellant cost, and number of maneuvers, which will provide very useful reference for practical asteroid mission design. Compared with the PCP method, the proposed approach is demonstrated to be able to provide much more easily used results, obtain better propellant-optimal solutions, and have much better efficiency. The MOPSO shows a very competitive performance with respect to the NSGA-II and the SPEA-II; besides a proposed boundary constraint optimization strategy is testified to be able to improve its performance.
The optimization of interplanetary trajectories to an asteroid continues to arouse a great deal of interest [
This type of method is very intuitionistic and easily executed. However, only the two-impulse trajectory is investigated in this method, and as demonstrated by Lawden’s theory [
In this paper, a new preliminary trajectory design method for asteroid rendezvous mission using multiobjective optimization techniques is proposed. This method can overcome the disadvantages of the Pork-Chop method. The multiobjective integrated launch window and multi-impulse transfer trajectory design model is formulated, which employs minimum-fuel cost and minimum-time transfer as two objective functions. In this model, the Earth departure date, hyperbolic velocity, and the interplanetary transfer impulses are all chosen optimization design variables. The multiobjective particle swarm algorithm is employed to locate the Pareto solution, by which the relationships characteristics among the overall mission parameters can be effectively revealed.
The particle swarm optimization (PSO) algorithm firstly introduced by Kennedy and Eberhart [
The implementation of the PSO algorithm adopts a population of particles, whose behavior is affected by either the best local (i.e., within a certain neighborhood) or the best global individual. The relative simplicity of PSO and the fact that it is a population-based technique have made it a natural candidate to be extended for multiobjective optimization. In a survey paper in 2006 on multiobjective particle swarm optimization (MOPSO) [
In summary, the main contribution of this paper is twofold. (1) A novel asteroid rendezvous mission design method using the multiobjective techniques is proposed. Compared with the current widely employed Pork-Chop method, the proposed approach is demonstrated to be able to provide much more easily used results, obtain better propellant-optimal solutions, and have much better efficiency. (2) As far as we know, it is the first time to apply the MOPSO to spacecraft trajectory optimization. The MOPSO proves to be quite effective in finding the Pareto-optimal solutions to asteroid rendezvous multiobjective optimization problems. The MOPSO shows a very competitive performance with respect to two highly competitive multiobjective evolutionary algorithms: the nondominated sorting genetic algorithm-II (NSGA-II) [
The interplanetary transfer trajectory is always divided into three different segments, that is, planet departure segment, heliocentric transfer segment, and capture segment by using the concept of influencing sphere. As the influencing sphere of the planet is much smaller than that of the sun, the flight path and flight time of the departure and capture segments are much small compared with that of the heliocentric transfer. Therefore, for a preliminary trajectory design of an asteroid exploration mission, the design emphasis is firstly focused on the heliocentric transfer, while the planet departure is assumed as instantaneous process with an impulsive maneuver and the capture segment is omitted.
This paper studies the asteroid rendezvous problem departure from the Earth.
Let
The heliocentric transfer trajectory is modeled by a two-body dynamic model with the following governing equations:
The thrust acceleration
In order to avoid dealing with the equality constraints described as (
The last two impulses are determined by solving the Lambert problem constrained by (
The total velocity characteristic is chosen as the first objective function
The heliocentric transfer time is chosen as the second objective function
The constraint on the time of impulse is considered. The general constraint on
As the transfer time is one of the objective functions,
A general multiobjective optimization problem is to find the design variables that optimize a vector objective function over the feasible design space. The objective functions are the quantities that the designer wishes to minimize, maximize, or attain a certain value. The problem formulation in standard form for a minimization is given here, which is similar for the other cases:
For the multiobjective asteroid rendezvous design problem, the two objective functions are described by (
The classical optimization method for a multiobjective optimization problem is the weighting method. In recent years, the multiobjective evolutionary algorithms (MOEA) have been greatly investigated in the domain of multiobjective optimization. There are many variants of MOEA reported in the literature; a recent survey on MOEA and their application in aeronautical and aerospace engineering has been made in [
In the study, except for the MOPSO, we also test two other mostly popular MOEA. The first is the NSGA-II algorithm which is proposed by Deb et al. [
The second is the SPEA-II proposed by Zitzler et al. [
The MOPSO applied in this study is the algorithm proposed by Pulido and Coello [
The MOPSO is based on the use of Pareto ranking and a subdivision of decision variable space into several subswarms which is done using clustering techniques. The complete execution process of this algorithm can be divided into three stages: initialization, flight, and generation of results [
At the first stage, every swarm is initialized. Each swarm creates and initializes its own particles and generates the leaders set among the particle swarm set by using Pareto ranking. In the second stage, it firstly performs the execution of the flight of every swarm; next, it applies a clustering algorithm to group the guide particles. This is performed until reaching a total of
Details of this algorithm can be found in [
(1) Initialize its particles (2) Initialize the set of global leaders: (3) Select a leader (4) Perform the flight (5) Update the value If it is a leader then add to the (6) Store leaders in (7) Assign each leader group to a swarm
The MOPSO algorithm requires the following parameters: (1)
The multiobjective asteroid mission design problem is a highly constrained problem whose constraints are described in (
The total constraint is calculated by making use of a nondifferentiable penalty function. For the general constrained problem in (
In the present work
For most optimization problems, each design variable has its own upper and low values. Thus, a strategy to maintain the particles within the search space in case they go beyond their boundaries is necessary for the MOPSO algorithm.
In [
However, our simulation experiments show that this strategy is not very effective in solving our problems. Therefore, another simple strategy is proposed. When a decision variable goes beyond its boundaries, the decision variable takes a random value from its feasible design space. The probability-based disposal to boundary constraint could enrich the diversity of swarm flight.
The MOPSO with these two different boundary constraint optimization strategies is, respectively, called as MOPSO-I and MOPSO-II. Their performance is compared in Section
In order to allow a quantitative assessment of the performance of the MSOPO, we adopted the following two metrics.
The first one is the epsilon indicator [
The second one is the hypervolume indicator [
The statistical test chosen for result evaluation is the Mann-Whitney test [
In order to testify the effectiveness of the proposed method, two different asteroid mission designs are illustrated. Table
Orbit elements of asteroid.
Index | Asteroid 1 | Asteroid 2 |
---|---|---|
Name | 1999YR14 | 2340 |
|
1.65365126892224 | 0.84421076388332195 |
|
0.40069261757759106 | 0.44975834146342486 |
|
3.7221930161441943 | 5.8547882390182853 |
|
3.1338963493744654 | 211.50460158030430 |
|
9.4143875285008676 | 39.994195753797953 |
|
114.73402134869427 | 240.44827444641544 |
Epoch (MJD2000) | 3255.0000 | 3255.000000 |
Design space of variables.
Variables | Space | Units |
---|---|---|
|
|
MJD2000 |
|
|
km/s |
|
|
n/a |
|
|
n/a |
|
|
n/a |
|
||
|
||
|
|
km/s |
|
|
days |
From our experiments, this multiobjective asteroid rendezvous design problem is very difficult to be solved, and the true Pareto fronts of this multiobjective problem are not known. In order to obtain the Pareto fronts as possible close to the true ones, the MOPSO is executed with a much larger number of function evaluations. The parameters of the MOPSO are
Considering the stochastic characteristic of the MOPSO, 10 independent runs for each test case are completed. All the Pareto solutions of the 10 independent runs are compared, and the repeated and non-Pareto solutions are deleted, and the revised Pareto solutions are selected as the final solutions. In the following examples, the figured Pareto fronts are all obtained in the same method. The MOPSO with different boundary constraint optimization methods, that is, the MOPSO-I and MOPSO-II, is both tested. The performance comparisons between the MOPSO-I and MOPSO-II will be analyzed in the next section.
Figure
Pareto fronts of Asteroid 1.
Produced by MOPSO-I
Produced by MOPSO-II
Pareto fronts of Asteroid 2.
Produced by MOPSO-I
Produced by MOPSO-II
In order to explain this phenomenon, the propellant-optimal solutions are obtained by using the approach employed in [
Relations between transfer time and total characteristic velocity (propellant-optimal solutions, Asteroid 1).
The influence of different number of impulses on the total
As seen from Figures
It is not easily to determine which one is much closer to the true Pareto fronts from Figure
Pareto fronts produced by MOPSO with different boundary constraint optimization.
Asteroid 1, two-impulse
Asteroid 2, three-impulse
Therefore, the quantitative metrics are calculated and the statistical results are provided in Table
MOPSO performance with different boundary constraint optimization.
Test problem | Algorithm | Epsilon | Hybervolume | ||||
---|---|---|---|---|---|---|---|
Mean | std |
|
Mean | std |
| ||
Asteroid 1, |
MOPSO-I | 0.011987 | 0.008378 | 0.1620 | 0.972618 | 0.006829 | 0.0257 |
MOPSO-II | 0.006833 | 0.004066 | 0.977958 | 0.001301 | |||
|
|||||||
Asteroid 2, |
MOPSO-I | 0.025630 | 0.0278051 | 0.1620 | 0.87509 | 0.308357 | 0.6232 |
MOPSO-II | 0.0399274 | 0.026254 | 0.968537 | 0.016548 |
Through the comparisons provided in Table
Our simulation experiments show that
MOPSO parameters configurations.
Index | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | Case 7 | Case 8 | Case 9 |
---|---|---|---|---|---|---|---|---|---|
|
400 | 400 | 400 | 200 | 200 | 200 | 100 | 100 | 100 |
|
400 | 200 | 100 | 400 | 200 | 100 | 400 | 200 | 100 |
Ten independent runs for the MOPSO-II with each group of parameters in solving the three-impulse, Asteroid 1 problem, are executed.
Figure
Pareto fronts produced by MOPSO-II with different number of function evaluations.
Cases 1, 5, and 9
Cases 2, 5, and 8
Cases 7, 8, and 9
Figure
MOPSO-II performance with different parameters (Cases 2 and 4).
Test problem | Algorithm | Epsilon | Hybervolume | ||||
---|---|---|---|---|---|---|---|
Mean | std |
|
Mean | std |
| ||
Asteroid 1, |
Case 2 | 0.010692 | 0.006797 | 0.6776 | 0.984063 | 0.002738 | 0.1405 |
Case 4 | 0.012918 | 0.012848 | 0.985643 | 0.002736 |
MOPSO-II performance with different parameters (Cases 6 and 8).
Test problem | Algorithm | Epsilon | Hybervolume | ||||
---|---|---|---|---|---|---|---|
Mean | std |
|
Mean | std |
| ||
Asteroid 1, |
Case 6 | 0.006045 | 0.002645 | 0.4727 | 0.99544 | 0.000816 | 0.0757 |
Case 8 | 0.007806 | 0.004042 | 0.994456 | 0.001074 |
MOPSO-II performance with different parameters (Cases 3, 5, and 7).
Test problem | Algorithm | Epsilon | Hybervolume | ||
---|---|---|---|---|---|
Mean | std | Mean | std | ||
Asteroid 1, |
Case 3 | 0.00583413 | 0.0036399 | 0.994062 | 0.00135846 |
Case 5 | 0.00438474 | 0.00129004 | 0.994755 | 0.00100777 | |
Case 7 | 0.00564449 | 0.00274183 | 0.994917 | 0.00119178 |
Case 3 | Case 5 | Case 7 | |
---|---|---|---|
Case 3 |
|
|
|
Case 5 | 0.3447 |
|
|
Case 7 | 0.2123 | 0.7337 |
Bold indicates epsilon; the other is hybervolume.
Pareto fronts produced by MOPSO-II with same number of function evaluations.
Cases 2, 4
Cases 6, 8
Cases 3, 5, and 7
The comparisons between Case 2 and Case 4 show that larger size of swarm can obtain better average epsilon indicator with a
Although the result obtained from this experiment seems to be inconclusive on which parameter is the best, we can argue that the MOPSO’s performance is not sensitive to the main PSO parameters, under the condition that the total number of function evaluation retains a large value. We have obtained competitive results in most cases without paying special attentions on PSO parameters.
To demonstrate the performance of the employed MOPSO, two other popular algorithms are tested and compared: the NSGA-II and the SPEA-II.
In the following examples, the total number of function evaluations was set to 20000 for all the algorithms compared. The NSGA-II and the SPEA-II were run using a population size of 200, a maximum number of generations of 100. The MOPSO-I and MOPSO-II used 200 particles, a maximum number of generations of 20, a maximum number of generations per swarm of 5, and a total of 5 swarms. The source code of the NSGA-II, and SPEA-II provided in the EMOO repository is also employed in this study.
The two test problems are the same to those employed in Section
Performance comparisons of MOPSO, NSGA-II, and SPEA-II.
Test problem | Algorithm | Epsilon | Hybervolume | ||
---|---|---|---|---|---|
Mean | std | Mean | std | ||
Asteroid 1, |
MOPSO-I | 0.035041 | 0.038848 | 0.876849 | 0.026781 |
NSGA-II | 0.113438 | 0.094049 | 0.821205 | 0.080834 | |
SPEA-II | 0.100881 | 0.032094 | 0.811347 | 0.035008 | |
|
|||||
Asteroid 2, |
MOPSO-II | 0.159179 | 0.081796 | 0.875023 | 0.078886 |
NSGA-II | 0.0594614 | 0.058546 | 0.838198 | 0.301149 | |
SPEA-II | 0.12172 | 0.055917 | 0.902724 | 0.072393 |
MOPSO-I | NSGA-II | SPEA-II | |
---|---|---|---|
MOPSO-I |
|
|
|
NSGA-II | 0.0073 |
|
|
SPEA-II | 0.0017 | 0.1620 |
Bold indicates epsilon; the other is hybervolume.
MOPSO-II | NSGA-II | SPEA-II | |
---|---|---|---|
MOPSO-II |
|
|
|
NSGA-II | 0.3447 |
|
|
SPEA-II | 0.4274 | 0.5205 |
Bold indicates epsilon; the other is hybervolume.
Pareto fronts produced by MOPSO, NSGA-II, and SPEA-II.
Asteroid 1, two-impulse
Asteroid 2, three-impulse
Evidently seen from Figure
Also evidently seen from Figure
In conclusion, the MOPSO showed a better performance with respect to the NSGA-II and the SPEA-II for the two test problems. For other test problems, the MOPSO is not always better, but its performance is very competitive.
The widely used Pork-Chop (PCP) method is also tested here for comparison. The
Relations between total characteristic velocity, transfer time, and earth departure time.
Asteroid 1, two-impulse
Asteroid 2, two-impulse
We analyze the comparisons between the proposed multiobjective optimization approach and the PCP method in the following three aspects.
In the PCP method, the contours of the
In contrast to the PCP method, the proposed multiobjective optimization approach can provide friendly the designer of this information. The relations between total characteristic velocity, transfer time, and earth departure time of those obtained Pareto-optimal solutions for two test cases are provided in Figure
The minimum-propellant solution searched by the PCP method for the first asteroid mission is calculated with a
Contours of time of flight corresponding to the optimal transfers (Asteroid 1).
The PCP is in essence of an exhaustive searching method. In our test, the search space for departure time and arrive time is
The paper formulates the asteroid rendezvous preliminary trajectory design as a multiobjective optimization problem and employs the multiobjective particle swarm optimization (MOPSO) algorithm to locate the Pareto-optimal solution set. Compared with the widely employed Pork-Chop method, the proposed approach is demonstrated to be able to provide much more easily used results, obtain better propellant-optimal solutions, and have much better efficiency. The results show that the proposed approach can effectively and efficiently demonstrate the relations among the mission characteristic parameters such as launch time, transfer time, propellant cost, and number of maneuvers, which will provide useful reference for practical asteroid mission design. The MOPSO proves to be quite effective in finding the Pareto-optimal solutions and its performance can be improved by a proposed boundary constraint optimization strategy. The MOPSO is found to be very competitive with respect to two highly competitive multiobjective evolutionary algorithms: the NSGA-II and the SPEA-II.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (no. 11222215), the 973 Project (no. 2013CB733100), and the Hunan Provincial Natural Science Foundation of China (no. 13JJ1001).