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Particle swarm optimization (PSO) is a population-based stochastic optimization technique in a smooth search space. However, in a category of trajectory optimization problem with arbitrary final time and multiple control variables, the smoothness of variables cannot be satisfied since the linear interpolation is widely used. In the paper, a novel Legendre cooperative PSO (LCPSO) is proposed by introducing Legendre orthogonal polynomials instead of the linear interpolation. An additional control variable is introduced to transcribe the original optimal problem with arbitrary final time to the fixed one. Then, a practical fast one-dimensional interval search algorithm is designed to optimize the additional control variable. Furthermore, to improve the convergence and prevent explosion of the LCPSO, a theorem on how to determine the boundaries of the coefficient of polynomials is given and proven. Finally, in the numeral simulations, compared with the ordinary PSO and other typical intelligent optimization algorithms GA and DE, the proposed LCPSO has traits of lower dimension, faster speed of convergence, and higher accuracy, while providing smoother control variables.

Swarm intelligence is a collective dynamic behavior of distributed, self-organized systems, natural or artificial, employed in work on artificial intelligence. It introduces many simple agents with very general rules to achieve an “intelligent” global optimal behavior. Swarm intelligence-based techniques can be used in a number of applications on optimization. The US military is investigating the swarming techniques to control unmanned vehicles. The European Space Agency is thinking about an orbital swarm for self-assembly and interferometry. NASA is investigating the use of swarm technology for planetary mapping.

In particular, trajectory optimization problem is one of the most important tasks in the preliminary design of the next generation of high speed vehicles, such as NASA’s X-43 unmanned hypersonic vehicle (HV), and has a great effect on the choice of conceptual design [

Trajectory optimization with multiconstraints has been linked with some stochastic search algorithms. The typical approaches include genetic algorithms (GA) [

Under the guidance of thought of the hybrid algorithm [

In LCPSO, the following improvements make it promising in solving complex problems. Firstly, the search space is divided into certain subspaces and different swarms are arranged to optimize the different parts of the search space. In this way, the optimized scale for each swarm can be reduced directly. The algorithm is suitable for the problems with larger scale and higher dimension.

Secondly, for the optimal control problem by PSO, the discretization is necessary before solving the parameter optimization problem. Herein, the Legendre orthogonal polynomial approximation can achieve higher smoothness of control variables with lower dimensions. Additionally, a theorem on how to find the precise range of optimal parameters is given and proven, as well as how to find the boundaries of the coefficient of polynomials. Therefore, the proposed LCPSO is expected to realize the optimization with higher accuracy and efficiency.

Thirdly, in order to solve the optimal control problems with arbitrary final time rather than the fixed one, the proposed Legendre orthogonal polynomial approximation method introduces an additional control variable to transcribe the original optimal problem to the one with fixed final time. Then, a traditional one-dimensional search method based on the interval analysis is proposed to optimize the additional control variable. This way, the specific optimal problem with single boundary can be solved.

Finally, we use two typical trajectory optimization problems to illustrate efficiency of the proposed LCPSO algorithm. One is the ascent trajectory optimization of X-43 hypersonic vehicle, and the other one is the classic optimal orbit transfer problem. The simulation results demonstrate the advantages of the LCPSO in terms of solution accuracy and convergence rate by comparing with some traditional intelligent optimization algorithms.

The organization of the remainder of this paper is as follows. Section

HV is one of the main workhorses for most of the nations of the world for scientific studies and military and commercial applications [

The 3-DOF dynamics described in (

The control variables

This way, the dynamic parameter optimization problem is transformed into a static one before the PSO algorithm performed.

The evaluation of the performance index starts when the scramjet is launched. The vehicle releases from the boost phase and then enters into the later ascent and cruise phase. Here, we define the switching time as

In this problem, the optimal time of ascent stage is free but satisfying some constraints. Thus, an additional control variable is introduced to transcribe the proposed problem to a fixed final time problem. The switching step is available and then the ascent trajectory can be divided into two phases. In the first phase, only the fuel throttle opening needs to be optimized to guarantee that it is a minimized constant value at the beginning of the second phase. In order to simplify the equations and problem, we give a reasonable assumption that the switching step satisfies

The design space

Here, the left two parts in the vector

To seek the solutions of both the continuous input

Considering that the performance function here is selected to minimize the fuel expendable ratio (FER) through the control variables

Furthermore, the ascent trajectory terminate conditions can be represented by the following inequalities:

The terminal conditions are added to the performance function to ensure that all the constraints are satisfied. Then, the performance function of LCPSO is defined as follows:

Considering that the proposed LCPSO provides two groups of particles based on Legendre orthogonal polynomial approximation, the assumption of the fixed time interval

The proposed LCPSO is designed as a double iteration scheme: the inner iteration of dual cooperative PSO (CPSO) [

The pseudocode of CPSO algorithm is reported in Pseudocode

The mathematical formulation of the one-dimensional search method based on interval analysis is designed as

The one-dimensional search algorithm based on the interval analysis is reported in Pseudocode

The midpoint and the initial width of the switching step interval are defined as

The method starts with the initial switch step interval, which is split into multiple subdivisions. Then, those subdivisions are either sent to the solution list which are considered later, or removed from further test list by certain cut-off condition. The above process is repeated by choosing a new switch step interval until no switch step could be considered or a global optimal point is found.

The Legendre orthogonal polynomials can be generated using Gram-Schmidt orthonormalization [

Then, using Legendre polynomial approximation, the original optimal control problem can be transformed into a parameter optimization problem: to find the optimal coefficients

Obviously, the appropriate ranges of the coefficients

Assume that

According to the assumption, we have the following equations:

According to the orthogonal relationship of the Legendre polynomials, we have

Substituting (

Assume that

Since the range of

According to (

Moreover, according to (

The closed form for the orthogonal polynomials is given in (

Substituting (

Consider that the control value

Firstly, we use the model of the X-43 vehicle [

In Table

Results after 1 time numerical simulation.

Iteration time | Switching step ratio | Corresponding step | Optimal cost |
---|---|---|---|

1 | 0.450 | 90.0 | 320.849 |

1 | 0.550 | 110.0 | 314.069 |

1 | 0.650 | 130.0 | 291.451 |

1 | 0.750 | 150.0 | 433.650 |

1 | 0.850 | 170.0 | 491.392 |

2 | 0.610 | 122.0 | 293.675 |

2 | 0.630 | 126.0 | 254.625 |

2 | 0.650 | 130.0 | 275.729 |

2 | 0.670 | 134.0 | 294.822 |

2 | 0.690 | 138.0 | 348.983 |

3 | 0.622 | 124.4 | 285.098 |

3 | 0.626 | 125.2 | 259.907 |

3 | 0.630 | 126.0 | 254.625 |

3 | 0.634 | 126.8 | 262.411 |

3 | 0.638 | 127.6 | 278.003 |

The optimal flight path angle profile is shown in Figure

The profile of opening of fuel valve.

The profile of the angle of attack.

The profile of the angle of flight path.

In Table

Coefficient range selection results.

| 0.5 times range | 2 times range | 5 times range | |
---|---|---|---|---|

Ratio of feasible results | | 59.1 | 63.5 | 24.5 |

Least consumption of fuel | | 55.9% | 53.6% | 56.4% |

Mean consumption of fuel | | 61.1% | 64.5% | 70.5% |

Result distribution with standard and half times range.

Result distribution with standard, double times, and five times range.

The orbit transfer problem is another kind of hotspot issues on trajectory optimization research [

The Earth-Mars transfer orbit can be divided into three stages: geocentric escape section, heliocentric transition section, and areosynchronous capture section. Each segment has different constraints. In the geocentric escape section, the initial state constraints are

The LCPSO algorithm parameters are set as follows: the population size

Convergence processes of the fitness functions.

The final resulting candidate solutions by LCPSO, DE, and GA are shown in Figures

Final times for the geocentric, heliocentric, and areocentric segments.

Segment | Final time (days) | ||
---|---|---|---|

| GA | DE | |

1 (geocentric) | | 34.6 | 34.5 |

2 (heliocentric) | | 182.4 | 174.7 |

3 (areocentric) | | 20.3 | 19.6 |

Whole flight time | | 237.3 | 228.8 |

State variables in geocentric segment.

State variables in heliocentric segment.

State variables in areocentric segment.

In this paper, a novel interval analysis based Legendre cooperative PSO algorithm is proposed and applied to solve the trajectory optimization problems. The Legendre orthogonal polynomial approximation is synthesized with the dual cooperative particle swarms to transfer the arbitrary final time optimal problem into a two-point boundary value problem with fixed terminal one. Then, a fast one-dimensional interval search method is provided in each selected interval to reduce the search space of the particles in the iterations. Furthermore, a theorem that determines the range of the parameters of the Legendre polynomial is investigated, and the problem can be solved to get closer to the global optimal solution.

Lastly, in numeral simulation, the results demonstrate that the LCPSO algorithm can solve the unsmooth trajectory optimization problem of X-43 effectively and obtain a smooth control variable. And the appropriate range of parameter values will significantly reduce the complexity of the optimization search. Moreover, in the solution to the orbit transfer problem, the comparisons with the existing GA and DE algorithms represent the notion that the proposed LCPSO method has better performance in the speed of convergence, final accuracy, and constraints satisfaction.

The flaw of the proposed LCPSO algorithms is that the parameters of the Legendre polynomial are sensitive in adjustment. Nevertheless, in the trajectory optimization problem of HV, the variety of parameters is a relatively gentle process because of the dynamic characters of HV, and the parameters are adjusted less frequently in iterations. Normally, the parameters will remain the same after they are adjusted once before. The optimization between parameter sensitivity and optimal solution will be studied to make the method more practical in engineering in our future work.

Cooperative particle swarm optimization

Differential evolution

Degree of freedom

Fuel expendable ratio

Hypersonic vehicle

Genetic algorithm

Legendre cooperative particle swarm optimization

Particle swarm optimization.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported in part by the National Natural Science Foundation of China (Grants nos. 61573161 and 61473124).