^{1}

^{1}

^{1}

On the conditions that the spacecraft engine is in finite thrust mode and the maneuver time is given, it takes a long time to compute the minimum duration transfer trajectories of space-to-ground vehicles, which is mainly because the initial values of the adjoint variables involved in the optimization model have no definite physical meanings and the model is sensitive to them. In order to develop space-to-ground transfer trajectory programmes in real time in an uncertain environment for the decision makers, we propose a fast method for computing the minimum duration transfer trajectories of space-to-ground vehicles with the given position of the landing point and the arbitrary maneuver point. First, the optimization model based on the hybrid method is established to compute the minimum duration transfer trajectory. Then, the region composed of maneuverable points is gridded and the initial values of the adjoint variables and the values of partial state variables of the minimum duration transfer trajectories at all gridded points are computed and saved to a database. Finally, the predicted values of the initial values of the adjoint variables and the values of partial state variables at any maneuver point within the region composed of maneuverable points are computed by using a binary cubic interpolation method. Finally, the minimum duration transfer trajectory is obtained by the hybrid method which takes the neighborhood of the predicted values as the search ranges of the initial values of the adjoint variables and the values of partial state variables. Simulation results demonstrate that the proposed method, which requires only 2.93% of the computational time of the hybrid method, can improve substantially the computational time of the minimum duration transfer trajectory of a space-to-ground vehicle under the guarantee of ensuring accuracy. The methodology of converting the time domain into the space domain is well applied in this paper.

The transfer trajectories of spacecraft, such as return satellites, manned spacecraft, space shuttles, space-to-ground kinetic weapons, and other kinds of space-to-ground vehicles, from their orbit to the landing point is uniformly referred to as space-to-ground transfer trajectories. A typical space-to-ground transfer trajectory consists of a transition trajectory segment and a reentry trajectory segment. Depending on the resistance and lift of the spacecraft in the reentry segment, the re-entry trajectories are classified into ballistic, semiballistic, and gliding [

In the existing literature, direct methods [

We study a typical space-to-ground transfer trajectory which only takes into consideration of the energy but not the heat rate, normal load, and dynamic pressure. The typical space-to-ground transfer trajectory adopts the zero angle of attack and ballistic reentry mode (Figure

A sketch diagram of a space-to-ground transfer trajectory.

In this section, we provide the differential equations of motion in the earth-fixed coordinate system and then establish the optimization model for the minimum duration transfer trajectory.

The differential equations of motion are [

The equations for the dimensionless parameters are

The minimum duration transfer trajectory optimization problem based on the hybrid method is a high-dimensional nonlinear optimization problem, which is hard to obtain global optimal solutions. GA has strong global search ability and simple implementation steps, which is suitable for solving the minimum duration transfer trajectory optimization problem. The main idea of the hybrid method for computing the minimum duration transfer trajectory can be summarized as follows. The optimization problem of the transition trajectory segment is converted into a two-point boundary value problem according to Pontryagin’s minimum principle and the reentry trajectory segment needs to satisfy constraints. Then, the initial values of the adjoint variables and the values of partial state variables are adjusted by means of the genetic algorithm (GA) [

The minimum duration optimization index is written as

According to Pontryagin’s minimum principle, the Hamiltonian function is given by

From (

From the sufficient conditions of optimality and formula (

The minimum duration transfer trajectory consists of the transition trajectory segment and the reentry trajectory segment. The dimensionless geocentric distance

The transition trajectory satisfies initial boundary conditions:

The transition trajectory satisfies the terminal boundary constraints:

Therefore, the cross-section conditions are given by

The reentry trajectory satisfies the constraints

Because the index includes time, we have

In order to achieve the minimum index under the given constraints, eleven parameters are taken as optimization variables including the initial values of seven adjoint variables (

The fitness function is shown in formula (

The chromosomes are selected using roulette wheel selection. Some genes on two different chromosomes reciprocally cross according to the crossover probability and others mutate according to the mutation probability. The execution of selection, crossover, and mutation leads to the next generation population.

Steps

The minimum duration transfer trajectory at a given maneuver point can be computed through these steps.

Given the landing point position, the minimum duration transfer trajectory at any one maneuver point can be obtained by the hybrid method, but the computational time is still too long for real-time applications. On the basis of the characteristics of the problem and the hybrid method, the region composed of maneuverable points is gridded. Then, the initial values of the adjoint variables and the values of partial state variables of the minimum duration transfer trajectory at all gridded points are computed and saved to a database. The predicted values of the initial values of the adjoint variables and the values of partial state variable at any one maneuver point in the region composed of maneuverable points are computed by using a binary cubic interpolation method [

After analysing a large number of computational results, we find that the initial values of the adjoint variables and the values of partial state variables of the minimum duration transfer trajectories have characteristics of regular change. Therefore, we come up with the idea of gridding the region composed of maneuverable points and the sizes of discrete grids are uniform. The region composed of maneuverable points is gridded according to the rules described below. The initial values of the adjoint variables and the values of partial state variables of the minimum duration transfer trajectory of at all gridded points are computed and saved to a database.

Figures

Track of sub-satellite points after the spacecraft runs one day.

Track of sub-satellite points after the spacecraft runs five days.

Track of sub-satellite points after the spacecraft runs twenty days.

After the region (shown as Figure

Sketch diagram of the maneuverable region of the spacecraft.

The predicted values of the initial values of the adjoint variables and the values of partial state variables at any maneuver point in the region composed of maneuverable points are computed using a binary cubic interpolation method. A neighborhood

Given the time at any maneuver point, the initial values of the adjoint variables and the values of partial state variables at the ten points which are the nearest to the point

The neighborhoods

The parameters of the vehicle are as follows: the orbital radius is 6678.14km, the flattening is 0, the value of the orbital inclination is 50°, the right ascension of ascending node is 0°, the argument of perigee is 105°, the true anomaly is 0°, the mass of the spacecraft is 500kg, the engine thrust is 100N, the gas jet velocity is 3000m/s, the shape of the vehicle is an axisymmetric cone, and the characteristic area of vehicle is 0.02m^{2}.

The position of the landing point is (-158°, 38°) (the first value is the longitude and the second is the latitude). The region S composed of maneuverable points (shown as Figure

Position of one hundred maneuver points.

No. | | |
---|---|---|

1 | 13.1 | 40.1 |

2 | 13.2 | 40.1 |

3 | 13.3 | 40.1 |

4 | 13.4 | 40.1 |

5 | 13.6 | 40.1 |

6 | 13.7 | 40.1 |

7 | 13.7 | 40.1 |

8 | 13.8 | 40.1 |

9 | 13.9 | 40.1 |

10 | 14.1 | 40.2 |

11 | 14.2 | 40.2 |

12 | 14.3 | 40.2 |

13 | 14.4 | 40.2 |

14 | 14.6 | 40.2 |

15 | 14.7 | 40.2 |

16 | 14.8 | 40.2 |

17 | 14.9 | 40.2 |

18 | 15.1 | 40.2 |

19 | 15.2 | 40.2 |

20 | 15.3 | 40.2 |

21 | 15.4 | 40.3 |

22 | 15.6 | 40.3 |

23 | 15.7 | 40.3 |

24 | 15.8 | 40.3 |

25 | 15.9 | 40.3 |

26 | 16.1 | 40.3 |

27 | 16.2 | 39.6 |

28 | 16.3 | 39.6 |

29 | 16.4 | 39.6 |

30 | 16.6 | 39.6 |

31 | 16.7 | 39.6 |

32 | 16.8 | 39.6 |

33 | 16.9 | 39.6 |

34 | 17.1 | 39.3 |

35 | 17.2 | 39.3 |

36 | 17.3 | 39.3 |

37 | 17.4 | 39.3 |

38 | 17.6 | 39.3 |

39 | 17.7 | 39.3 |

40 | 17.8 | 39.3 |

41 | 17.9 | 39.3 |

42 | 18.1 | 39.1 |

43 | 18.2 | 39.1 |

44 | 18.3 | 39.1 |

45 | 18.4 | 39.1 |

46 | 18.6 | 39.1 |

47 | 18.7 | 39.1 |

48 | 18.8 | 39.1 |

49 | 18.9 | 39.1 |

50 | 14.1 | 40.1 |

51 | 14.2 | 40.1 |

52 | 14.3 | 40.1 |

53 | 14.4 | 40.1 |

54 | 14.6 | 40.1 |

55 | 14.7 | 40.1 |

56 | 14.8 | 40.1 |

57 | 14.9 | 40.1 |

58 | 15.1 | 40.1 |

59 | 15.2 | 40.1 |

60 | 15.3 | 40.1 |

61 | 15.4 | 39.9 |

62 | 15.6 | 39.9 |

63 | 15.7 | 39.9 |

64 | 15.8 | 39.9 |

65 | 15.9 | 39.9 |

66 | 16.1 | 39.9 |

67 | 16.2 | 39.9 |

68 | 16.3 | 39.9 |

69 | 16.4 | 39.9 |

70 | 16.6 | 39.9 |

71 | 16.7 | 39.9 |

72 | 16.8 | 39.9 |

73 | 16.9 | 39.9 |

74 | 17.1 | 39.4 |

75 | 17.2 | 39.4 |

76 | 17.3 | 39.4 |

77 | 17.4 | 39.4 |

78 | 17.6 | 39.4 |

79 | 17.7 | 39.4 |

80 | 17.8 | 39.4 |

81 | 15.4 | 38.9 |

82 | 15.6 | 38.9 |

83 | 15.7 | 38.9 |

84 | 15.8 | 38.9 |

85 | 15.9 | 38.9 |

86 | 16.1 | 38.9 |

87 | 16.2 | 38.9 |

88 | 16.3 | 38.9 |

89 | 16.4 | 38.9 |

90 | 16.6 | 38.9 |

91 | 16.7 | 38.9 |

92 | 16.8 | 38.9 |

93 | 16.9 | 38.9 |

94 | 17.1 | 39.2 |

95 | 17.2 | 39.2 |

96 | 17.3 | 39.2 |

97 | 17.4 | 39.2 |

98 | 17.6 | 39.2 |

79 | 17.7 | 39.2 |

100 | 17.8 | 39.2 |

In Table

The area S where the spacecraft can maneuver.

The idea of the proposed method is as follows. First, the minimum duration transfer trajectories at all grid points are obtained by using the traditional hybrid method. Second, the predicted values of minimum duration transfer trajectories at one hundred maneuver points, excluding the grid points, are computed by using the proposed method. Finally, by taking the neighborhood ranges as the search range, the minimum duration transfer trajectories at one hundred maneuver points are obtained by using the traditional hybrid method. The results obtained by the proposed method are used for comparison with the traditional hybrid method.

The

Initial values of the adjoint variables (partial) and the values of partial state variables at grid points.

| | | | | | | | | | |
---|---|---|---|---|---|---|---|---|---|---|

13 | 40.000 | 11.308 | 3.217 | -19.323 | 14.287 | 0.002 | 1.214 | 8.383 | -7.216 | 4505.6 |

| ||||||||||

13.5 | 40.159 | 11.209 | 3.289 | -19.306 | 14.293 | 0.154 | 1.361 | 8.352 | -7.142 | 4486.5 |

39.732 | 11.176 | 3.295 | -19.342 | 14.248 | -0.283 | 1.792 | 8.439 | -7.357 | 4509.3 | |

| ||||||||||

14 | 40.280 | 11.159 | 3.348 | -19.289 | 14.204 | 0.259 | 1.472 | 8.330 | -7.092 | 4466.1 |

39.904 | 11.111 | 3.357 | -19.308 | 14.165 | -0.123 | 1.848 | 8.407 | -7.282 | 4486.2 | |

39.137 | 11.042 | 3.374 | -19.383 | 14.086 | -0.887 | 2.618 | 8.559 | -7.667 | 4524.4 | |

| ||||||||||

14.5 | 40.603 | 11.103 | 3.412 | -19.167 | 14.136 | 0.571 | 1.139 | 8.268 | -6.943 | 4438.9 |

40.216 | 11.057 | 3.419 | -19.216 | 14.095 | 0.181 | 1.527 | 8.347 | -7.137 | 4459.7 | |

39.757 | 11.001 | 3.432 | -19.274 | 14.043 | -0.283 | 1.990 | 8.439 | -7.366 | 4483.0 | |

39.059 | 10.931 | 3.447 | -19.351 | 13.966 | -0.991 | 2.688 | 8.580 | -7.718 | 4518.3 | |

| ||||||||||

15 | 40.643 | 10.906 | 3.482 | -19.418 | 14.044 | 0.602 | 1.089 | 8.261 | -6.933 | 4425.6 |

40.311 | 10.875 | 3.488 | -19.456 | 14.004 | 0.269 | 1.424 | 8.328 | -7.101 | 4443.3 | |

39.878 | 10.833 | 3.451 | -19.502 | 13.955 | -0.174 | 1.861 | 8.415 | -7.319 | 4466.2 | |

39.125 | 10.752 | 3.469 | -19.587 | 13.870 | -0.928 | 2.615 | 8.568 | -7.697 | 4505.3 | |

38.245 | 10.659 | 3.491 | -19.685 | 13.774 | -1.818 | 3.494 | 8.746 | -8.119 | 4550.3 | |

| ||||||||||

15.5 | 40.727 | 10.518 | 3.540 | -19.775 | 13.948 | 0.678 | 0.996 | 8.246 | -6.902 | 4410.2 |

40.413 | 10.471 | 3.579 | -19.824 | 13.914 | 0.362 | 1.313 | 8.309 | -7.061 | 4427.3 | |

39.758 | 10.401 | 3.604 | -19.896 | 13.847 | -0.301 | 1.969 | 8.442 | -7.391 | 4460.4 | |

39.218 | 10.342 | 3.619 | -19.961 | 13.780 | -0.845 | 2.511 | 8.553 | -7.664 | 4488.8 | |

38.365 | 10.248 | 3.642 | -20.060 | 13.686 | -1.699 | 3.364 | 8.724 | -8.089 | 4531.4 | |

| ||||||||||

16 | 40.830 | 10.414 | 3.651 | -19.836 | 13.859 | 0.769 | 0.880 | 8.227 | -6.862 | 4393.6 |

40.487 | 10.377 | 3.659 | -19.871 | 13.823 | 0.423 | 1.225 | 8.297 | -7.035 | 4411.5 | |

39.873 | 10.313 | 3.674 | -19.941 | 13.756 | -0.188 | 1.843 | 8.422 | -7.345 | 4442.5 | |

39.258 | 10.246 | 3.689 | -20.012 | 13.685 | -0.796 | 2.461 | 8.546 | -7.654 | 4474.4 | |

38.607 | 10.174 | 3.706 | -20.091 | 13.613 | -1.452 | 3.117 | 8.677 | -7.983 | 4506.7 | |

| ||||||||||

16.5 | 40.578 | 10.287 | 3.730 | -19.917 | 13.734 | 0.504 | 1.120 | 8.280 | -7.000 | 4394.8 |

40.192 | 10.247 | 3.739 | -19.964 | 13.690 | 0.115 | 1.508 | 8.358 | -7.196 | 4414.6 | |

39.558 | 10.181 | 3.754 | -20.037 | 13.618 | -0.517 | 2.146 | 8.486 | -7.515 | 4446.3 | |

38.879 | 10.109 | 3.770 | -20.117 | 13.543 | -1.199 | 2.832 | 8.623 | -7.856 | 4480.8 | |

| ||||||||||

17 | 40.260 | 10.157 | 3.804 | -20.013 | 13.596 | 0.175 | 1.431 | 8.346 | -7.171 | 4379.4 |

39.821 | 10.108 | 3.810 | -20.060 | 13.553 | -0.269 | 1.873 | 8.434 | -7.390 | 4402.5 | |

39.372 | 10.059 | 3.822 | -20.114 | 13.508 | -0.721 | 2.324 | 8.525 | -7.615 | 4425.7 | |

38.877 | 10.007 | 3.832 | -20.171 | 13.454 | -1.220 | 2.824 | 8.624 | -7.864 | 4451.6 | |

| ||||||||||

17.5 | 39.827 | 10.010 | 3.881 | -20.119 | 13.459 | -0.278 | 1.860 | 8.435 | -7.399 | 4329.4 |

39.546 | 9.985 | 3.888 | -20.150 | 13.427 | -0.565 | 2.143 | 8.492 | -7.540 | 4343.6 | |

38.931 | 9.913 | 3.903 | -20.223 | 13.364 | -1.012 | 2.763 | 8.616 | -7.849 | 4374.5 | |

| ||||||||||

18 | 39.588 | 9.882 | 3.957 | -20.196 | 13.333 | -0.513 | 2.090 | 8.485 | -7.528 | 4310.4 |

38.896 | 9.814 | 3.973 | -20.272 | 13.262 | -1.210 | 2.785 | 8.624 | -7.876 | 4345.7 | |

| ||||||||||

18.5 | 39.274 | 9.753 | 4.038 | -20.459 | 13.201 | -0.838 | 2.396 | 8.551 | -7.697 | 4315.3 |

| ||||||||||

19 | 39.072 | 9.726 | 4.096 | -20.613 | 13.076 | -1.051 | 2.589 | 8.594 | -7.808 | 4315.7 |

In Table

Landing speed and final mass of vehicles in one hundred experiments.

No. | | |
---|---|---|

1 | 7.641 | 404.2 |

2 | 7.610 | 404.6 |

3 | 7.554 | 404.9 |

4 | 7.502 | 405.1 |

5 | 7.445 | 405.3 |

6 | 7.410 | 405.4 |

7 | 7.399 | 405.6 |

8 | 7.373 | 405.8 |

9 | 7.300 | 405.9 |

10 | 7.276 | 406.1 |

11 | 7.223 | 406.2 |

12 | 7.189 | 406.4 |

13 | 7.067 | 406.6 |

14 | 7.001 | 406.7 |

15 | 6.948 | 406.7 |

16 | 6.902 | 406.9 |

17 | 6.852 | 407.1 |

18 | 6.821 | 407.2 |

19 | 6.794 | 407.3 |

20 | 6.756 | 407.5 |

21 | 6.713 | 407.7 |

22 | 6.682 | 407.7 |

23 | 6.653 | 407.8 |

24 | 6.629 | 408.1 |

25 | 6.206 | 408.2 |

26 | 6.197 | 408.3 |

27 | 7.182 | 406.5 |

28 | 7.154 | 406.7 |

29 | 7.001 | 406.9 |

30 | 6.922 | 407.0 |

31 | 6.879 | 407.0 |

32 | 6.798 | 407.1 |

33 | 6.740 | 406.9 |

34 | 7.165 | 406.6 |

35 | 7.133 | 406.8 |

36 | 7.095 | 406.9 |

37 | 7.010 | 407.4 |

38 | 6.921 | 407.7 |

39 | 6.833 | 408.1 |

40 | 6.795 | 408.2 |

41 | 6.721 | 408.5 |

42 | 7.109 | 407.1 |

43 | 7.056 | 407.3 |

44 | 7.001 | 407.4 |

45 | 6.950 | 407.6 |

46 | 6.923 | 407.6 |

47 | 6.887 | 407.7 |

48 | 6.832 | 407.9 |

49 | 6.821 | 408.0 |

50 | 7.229 | 406.4 |

51 | 7.155 | 406.5 |

52 | 7.102 | 406.7 |

53 | 7.079 | 406.8 |

54 | 7.001 | 407.0 |

55 | 6.938 | 407.3 |

56 | 6.899 | 407.5 |

57 | 6.723 | 407.8 |

58 | 6.686 | 407.9 |

59 | 6.569 | 408.1 |

60 | 6.430 | 408.3 |

61 | 6.442 | 408.1 |

62 | 6.402 | 408.3 |

63 | 6.368 | 408.4 |

64 | 6.321 | 408.5 |

65 | 6.300 | 408.5 |

66 | 6.279 | 408.6 |

67 | 6.225 | 408.8 |

68 | 6.198 | 408.9 |

69 | 6.158 | 409.1 |

70 | 6.113 | 409.2 |

71 | 6.083 | 409.2 |

72 | 6.053 | 409.3 |

73 | 6.002 | 409.5 |

74 | 7.102 | 406.5 |

75 | 7.086 | 406.6 |

76 | 7.026 | 406.8 |

77 | 6.956 | 407.1 |

78 | 6.823 | 407.4 |

79 | 6.791 | 407.5 |

80 | 6.722 | 407.7 |

81 | 6.865 | 407.0 |

82 | 6.823 | 407.2 |

83 | 6.784 | 407.4 |

84 | 6.736 | 407.5 |

85 | 6.695 | 407.6 |

86 | 6.652 | 407.8 |

87 | 6.601 | 407.9 |

88 | 6.539 | 408.2 |

89 | 6.493 | 408.3 |

90 | 6.457 | 408.5 |

91 | 6.401 | 408.7 |

92 | 6.380 | 408.7 |

93 | 6.333 | 408.9 |

94 | 6.555 | 408.3 |

95 | 6.512 | 408.4 |

96 | 6.478 | 408.6 |

97 | 6.402 | 408.8 |

98 | 6.373 | 408.9 |

79 | 6.345 | 409.0 |

100 | 6.291 | 409.2 |

In Table

Spatial distribution of

Spatial distribution of

Spatial distribution of

Spatial distribution of

Spatial distribution of

Spatial distribution of

Spatial distribution of

Spatial distribution of

Spatial distribution of

Transfer trajectory of No.5 experiment.

Transfer trajectory of No.45 experiment.

Transfer trajectory of No.85 experiment.

Figures

Fitting error of

Fitting error of

Fitting error of

Fitting error of

Fitting error of

Fitting error of

Fitting error of

Fitting error of

Fitting error of

Landing point error.

Fitting error of final mass.

Fitting error of total time.

Figure

Comparison of the computational time of two methods.

The simulation results demonstrate that the proposed method is feasible. Although the proposed method still needs some computational time to obtain the accurate transfer trajectories, yet the proposed method improves substantially the computational time. Fast computational of the space-to-ground transfer trajectories lays the foundation for the decision makers to develop space-to-ground transfer trajectory programmes in real time in an uncertain environment.

Fast computational method of the space-to-ground transfer trajectories can lay the foundation for decision makers to develop space-to-ground transfer trajectory programmes in real time in an uncertain environment. In order to overcome the computational cost encountered with the hybrid method, we solve the transfer trajectory by using interpolation scheme and hybrid method. The main idea of this paper is to convert the problem from the time domain into the spatial domain. The simulation results demonstrate that the proposed method has high computational efficiency.

Although the proposed method has high computational efficiency, it still has some limitations that some time is still needed to obtain the accurate transfer trajectory. To further accelerate the computational time of the transfer trajectories, investigating more interpolation methods as well as optimization algorithms in future research would be better. The methods that combine interpolation schemes and hybrid methods can be applied to other problems related to fixed transfer trajectories and finite thrust maneuver mode. Furthermore, the methodology of converting the time domain into the space domain can also be applied to other research areas.

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

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