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This paper presents a novel approach for the preliminary design of Low-Thrust, many-revolution transfers. The main feature of the novel approach is a considerable reduction in the control parameters and a consequent gain in computational speed. Each spiral is built by using a predefined pattern for thrust direction and switching structure. The pattern is then optimised to minimise propellant consumption and transfer time. The variation of the orbital elements due to the thrust is computed analytically from a first-order solution of the perturbed Keplerian motion. The proposed approach allows for a realistic estimation of

One of the most critical issues related to the exploitation of space around the Earth is the threat posed by space debris. Since the beginning of the space era in the late 1950s, an increasing number of man-made, inert objects has been orbiting the Earth. Recent statistics revealed around 15000 trackable objects, for a total of some 6000 tons of material. Some of these objects are simply spent upper stages of launch vehicles, some others are satellites which are no longer active due to failures or to having reached their end of life. Others, however, are the results of past collisions. It is easy to imagine that even a single collision between two objects is likely to generate tens of smaller objects as a result. The outcome of a collision in an already crowded environment could generate a cascade of collisions generating an exponentially increasing volume of space debris. In fact, the debris produced by a collision is itself likely to collide with other objects, thereby producing other debris which will generate further collisions, and so on. This chain reaction, known as the

Recent guidelines issued by international spacer regulatory institutions such as the United Nations Committee for the Peaceful Uses of Outer Space (COPUOS) [

There have been various proposals on how to remove inert objects from space. They can be generally classified in two major groups: contactless and with direct physical contact. In the latter category, one can find methods based on some form of docking with or capturing the object. Once the removing spacecraft and the piece of debris are attached, the latter is dragged into a re-entry trajectory or to a graveyard orbit. Technical problems related to the attitude state of motion of the piece of debris and the fragility of appendices and cover material (including paint) make this removal solution complicated. A potentially interesting solution is represented by Project ROGER [

A recent idea simultaneously proposed by Bombardelli and Peláez [

Assuming a scenario in which a single IBS needs to de-orbit multiple pieces of debris, one would need to solve an interesting mission design problem: the optimisation of the de-orbit sequence and trajectories for multiple target objects in minimum time and with minimum propellant. In the hypothetical mission scenario which is analysed in this work, it is assumed that a number of pieces of debris have been shortlisted as priority targets due to the threat they pose to satellites operating in LEO. For example, Liou and Johnson [

The paper is organised as follows: Section

As shown by Bombardelli and Peláez [

Ion Beam Shepherd spacecraft.

Since it is necessary to keep the Shepherd spacecraft at a constant distance from the debris, the thrust ^{-7 }km/s^{2}. If one considers instead the spacecraft alone, the acceleration achievable would be slightly higher, 5·10^{-7 }km/s^{2}. Given this order of magnitude, the thrust acceleration can be considered as a perturbative force compared to the Earth’s gravitational force, and therefore, the analytical approach to the propagation of the low-thrust motion described in [

The objective of this study is that of optimising the performance and cost of a debris de-orbiting mission performed by a single spacecraft. As mentioned in the introduction, it is assumed that there are five pieces of debris of different masses and lying in circular orbits with different radii and orientations. It is assumed that the IBS spacecraft departs from a low-Earth parking orbit, rendezvous with the first object, transfers it to an elliptical re-entry orbit, rendezvous with the second object, transfers it to a second elliptical re-entry orbit, and so forth until all five pieces of debris are removed. One important issue is defining in which order the pieces of debris need to be de-orbited. In the following, all possible sequences are generated a

Each fetch and de-orbit operation is split in two phases:

A

A

Given the magnitude of the available thrust acceleration, both phases require a spiral orbit transfer. If a direct transcription approach is used to optimise each spiral the number of parameters that needs to be defined is very high leading to high computational times. The latter fact would make the solution of a multiobjective optimisation of all possible de-orbiting sequences computationally intractable. Thus, in this paper, a simplified, highly efficient, trajectory model is proposed for each one of the two phases.

The objective of the de-orbiting phase is that of lowering an initial circular orbit such that its perigee is equal or below 300 km, which basically translates into a perigee lowering manoeuvre. Therefore, it is appropriate to assume that in general, as soon as the initial circular orbit becomes slightly eccentric, one keeps thrusting around the apogee in order to lower the perigee. The thrust level will also be kept at its maximum in order to minimize gravity losses. Moreover, since the de-orbit condition is independent of the final orbit’s orientation, one can reasonably assume that the perigee lowering will be performed in-plane. In this sense, the only Keplerian parameters which need to be altered are the semimajor axis and eccentricity. By analysing the structure of Gauss’ variational equations:

Thrusting arc around apogee with thrust directed along transverse direction.

In order to obtain a fast propagation of the thrusting arcs, the analytical propagation of perturbed motion with finite perturbative elements in time (FPET) derived in [

Motion propagation with FPET is based on a first-order analytical solution of perturbed Keplerian motion. In this formulation, the state is expressed in nonsingular equinoctial elements:

As explained above, the only nonzero component of the acceleration will be

Since the thrust magnitude and direction are fixed, the only free control parameter is the semiamplitude

In order to evaluate the time and Δ

Compute the set of initial Equinoctial parameters

Initialise the number of orbits, the total

Set

Initialise the mass of the IBS spacecraft:

While

Interpolate the amplitude of the thrusting arc in the current orbit, that is,

Compute the acceleration

Compute the time of flight

Compute the equinoctial parameters after the thrusting arc

Compute the current perigee radius

Compute the thrusting time

Update the total time of flight:

Update the IBS spacecraft mass:

Set

Back track the value of the longitude

At this point, one gets the

Bounds and number of samples for the de-orbit parameters.

Lower bound | 0 | 0 | |

Upper bound | |||

Samples | 8 | 50 | 50 |

Given the limited number of decision variables, for each piece of debris, one has 20000 de-orbit instances to propagate. Since each instance requires typically 1·10^{−2} s of CPU time, with a code implemented in MATLAB and running on a 3.16 GHz, 4 GB desktop PC running Windows 7, the whole computation can be completed in roughly five minutes. The set of de-orbit

(a)

(a) final semi-major axis and (b) eccentricity after de-orbit with respect to

Now, it is desirable that the surrogate model returns the

3D Plot of surrogate models for

According to the scenario presented in Section

First, given the limited acceleration provided by low-thrust propulsion systems, one should consider that the orbit transfer will require a high number of multiple revolutions around the Earth, typical in the range of hundreds to few thousands. In this sense, it is possible to argue that achieving the proper phasing to transfer from the initial to final orbit would not be a major issue. Even a small variation of

Thus, in order to account for

With these assumptions, the main issue in designing the multirevolution transfer will be that of achieving the required change in the apogee and perigee radii in order to match those of the final orbit, and to achieve the required rotation of the orbit plane.

The control of eccentricity and semimajor axis, required to match the target perigee and apogee altitudes, can be obtained by inserting two thrust arcs per revolution, one around the apogee and one around the perigee. This methodology is analogous to what was done in the previous section for the perigee lowering. In the same way, the radial component of the thrust acceleration is set to zero. The transverse component this time can have either positive or negative sign (

Since a plane change is required, the out-of-plane component of the thrust acceleration is nonzero. Thanks to this, the control parameters can be reduced to the semiamplitude of the apogee and perigee thrusting arcs,

To define the actual values of

Thrusting arcs around apogee and perigee.

Given a set of control parameters

Compute the set of initial Equinoctial parameters

Compute the set of target Equinoctial parameters

Initialise the total Δ

Set

Initialise the mass of the IBS spacecraft:

While

Compute the interpolated values for

Compute

Compute the current acceleration acting on the spacecraft:

Compute the time of flight

Compute the Equinoctial parameters after the thrusting perigee arc

Compute the thrusting time at perigee

Update

Update the IBS spacecraft mass:

Set

Compute the current acceleration on the spacecraft:

Compute the time of flight

Compute the equinoctial parameters after the thrusting apogee arc

Compute the thrusting time at apogee

Update

Update the IBS spacecraft mass:

Set

Back-track the point at which

Compute the mismatch between the actual final conditions and the target orbit:

Summarizing, the TPBVP has been reduced to an optimisation problem in the form:

This problem can be solved with a gradient-based optimisation algorithm like MATLAB’s

In the following, an example of transfer from an elliptical orbit with 300 km perigee altitude and eccentricity 0.031 (corresponding to the final orbit of a de-orbiting strategy) to a circular orbit of 1100 km altitude (corresponding to the orbit of the next debris in an hypothetical removal sequence). Parameters of the two orbits are reported in Table

Parameters of departure and arrival orbits.

Departure | 6892.24 | 0.031 | 0 |

Arrival | 7478.16 | 0 | 10 |

First, it is considered the case of a coplanar transfer, that is,

(a) Variation of semi-major axis, (b) eccentricity, (c) perigee, and apogee radiuses for multi-revolution orbital transfer (coplanar case).

Control parameters for multi-revolution orbital transfer (coplanar case): (a) thrust arc length; (b) azimuth and elevation.

The same problem, but this time with the 10° plane change specified in Table

(a) variation of semi-major axis, (b) eccentricity, (c) plane change, (d) perigee, and apogee radiuses for multirevolution orbital transfer (10° plane change).

Control parameters for multirevolution orbital transfer (10° plane change): (a) thrust arc length; (b) azimuth and elevation.

The aim is now that of optimising the timing and sequence of a removal mission by means of a single IBS spacecraft. It is assumed that the spacecraft departs from an LEO with a 250 km semi-major axis altitude and coplanar with respect to the first piece of debris in the sequence. The five target objects have the orbital parameters and mass reported in Table

Mass, initial orbit parameters, and minimum de-orbit time of the debris.

Debris nr. | mass (kg) | |||||
---|---|---|---|---|---|---|

1 | 500 | 6828.16 | 0 | 1 | 65 | 2.67 |

2 | 120 | 7128.16 | 0 | 2 | 150 | 3.36 |

3 | 300 | 6978.16 | 0 | −2 | 200 | 3.68 |

4 | 400 | 7478.16 | 0 | −1 | 90 | 11.12 |

5 | 800 | 7178.16 | 0 | 0 | 45 | 12.25 |

Table

Relative inclination change

Debris number | 2 | 3 | 4 | 5 |
---|---|---|---|---|

1 | 2.16 | 1.47 | 1.95 | 1 |

2 | — | 3.63 | 2.65 | 2 |

3 | — | — | 2.52 | 2 |

4 | — | — | — | 1 |

The de-orbit sequence is defined by the order according to which the five pieces of debris are removed, the time needed to rendezvous with _{Tot}. The latter is computed simply as

The domain

Optimisation boundaries.

Parameter | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Lower Bound | 5 | 5 | 5 | 5 | 5 | |||||

Upper Bound | 100 | 50 | 100 | 50 | 100 | 50 | 100 | 50 | 100 | 50 |

Each biobjective optimisation problem is solved with MACS, a hybrid-memetic stochastic optimisation algorithm [_{Tot}. Figure

Pareto fronts for sequences starting with debris nr. 1.

Pareto fronts for sequences starting with debris nr. 2.

Pareto fronts for sequences starting with debris nr. 3.

Pareto fronts for sequences starting with debris nr. 4.

Pareto fronts for sequences starting with debris nr. 5.

From a visual inspection of the fronts, it is possible to see that sequences starting from debris number 1 seem to present the best

Global Pareto front.

One can see that the global Pareto front is composed by individual solutions belonging exclusively to sequence 13452, which is, therefore, globally dominant. In order to rank the degree of optimality of each sequence, it is proposed to use an approach inspired by the performance metrics for optimisation algorithms proposed in [

Conv is given by averaging the distance of each point of

Ranking of the de-orbit sequences.

Rank | Conv( | Rank | Conv( | Rank | Conv( | |||
---|---|---|---|---|---|---|---|---|

1 | 13452 | 0 | 41 | 42513 | 21.18 | 81 | 52143 | 31.43 |

2 | 13542 | 5.14 | 42 | 15234 | 21.26 | 82 | 32145 | 31.62 |

3 | 13524 | 6.61 | 43 | 32451 | 21.46 | 83 | 54123 | 31.72 |

4 | 12453 | 6.78 | 44 | 52134 | 21.46 | 84 | 54132 | 31.83 |

5 | 12543 | 7.25 | 45 | 34521 | 21.52 | 85 | 42135 | 32.43 |

6 | 31542 | 9.41 | 46 | 35142 | 21.79 | 86 | 52314 | 32.70 |

7 | 31452 | 9.85 | 47 | 35214 | 21.99 | 87 | 42531 | 33.05 |

8 | 34512 | 11.59 | 48 | 34251 | 22.02 | 88 | 21435 | 33.96 |

9 | 24513 | 12.15 | 49 | 52431 | 22.04 | 89 | 54231 | 34.05 |

10 | 15243 | 12.16 | 50 | 45132 | 22.13 | 90 | 23145 | 34.31 |

11 | 12534 | 12.33 | 51 | 54312 | 23.39 | 91 | 23514 | 34.56 |

12 | 31254 | 12.37 | 52 | 21543 | 23.60 | 92 | 53421 | 34.67 |

13 | 15432 | 13.24 | 53 | 24315 | 23.62 | 93 | 25341 | 34.71 |

14 | 35124 | 13.87 | 54 | 41352 | 23.81 | 94 | 14325 | 34.91 |

15 | 13254 | 14.22 | 55 | 43152 | 23.90 | 95 | 41253 | 35.20 |

16 | 31524 | 14.36 | 56 | 12435 | 24.40 | 96 | 32514 | 35.42 |

17 | 15342 | 14.48 | 57 | 34125 | 24.53 | 97 | 14235 | 35.65 |

18 | 13425 | 16.30 | 58 | 15324 | 24.89 | 98 | 32541 | 36.42 |

19 | 24531 | 16.53 | 59 | 53142 | 24.90 | 99 | 51234 | 36.81 |

20 | 14523 | 16.65 | 60 | 23154 | 25.61 | 100 | 42153 | 36.91 |

21 | 14352 | 16.69 | 61 | 53124 | 25.67 | 101 | 51423 | 38.10 |

22 | 34152 | 17.16 | 62 | 51243 | 25.80 | 102 | 54321 | 38.25 |

23 | 25134 | 17.17 | 63 | 43512 | 25.83 | 103 | 45231 | 40.16 |

24 | 12354 | 17.47 | 64 | 31425 | 25.95 | 104 | 51432 | 40.98 |

25 | 14253 | 17.63 | 65 | 12345 | 25.96 | 105 | 41523 | 41.91 |

26 | 31245 | 17.81 | 66 | 21453 | 26.01 | 106 | 45321 | 44.72 |

27 | 15423 | 17.85 | 67 | 52413 | 26.09 | 107 | 32415 | 45.05 |

28 | 51342 | 17.88 | 68 | 51324 | 26.56 | 108 | 42351 | 45.38 |

29 | 14532 | 18.05 | 69 | 35241 | 26.68 | 109 | 43521 | 45.43 |

30 | 25413 | 18.07 | 70 | 25143 | 26.77 | 110 | 53214 | 45.72 |

31 | 54213 | 18.17 | 71 | 24153 | 26.93 | 111 | 43251 | 45.87 |

32 | 35412 | 18.48 | 72 | 34215 | 27.52 | 112 | 23541 | 46.11 |

33 | 21345 | 19.32 | 73 | 21534 | 28.00 | 113 | 52341 | 46.89 |

34 | 25431 | 19.43 | 74 | 32154 | 29.65 | 114 | 41325 | 47.14 |

35 | 13245 | 19.55 | 75 | 43125 | 30.17 | 115 | 41532 | 47.50 |

36 | 35421 | 19.56 | 76 | 23451 | 30.40 | 116 | 53241 | 48.31 |

37 | 25314 | 19.97 | 77 | 24351 | 30.97 | 117 | 23415 | 48.84 |

38 | 45213 | 19.98 | 78 | 45123 | 31.07 | 118 | 42315 | 48.85 |

39 | 45312 | 20.07 | 79 | 24135 | 31.18 | 119 | 43215 | 52.91 |

40 | 21354 | 20.07 | 80 | 53412 | 31.22 | 120 | 41235 | 65.42 |

As one would expect, sequence 13452 has the lowest Conv since it coincides with part of the global Pareto front. Sequences 13524, 13542 and 12543 have also a low Conv index and thus they are quite close to the globally optimal solution, as shown in Figure

Pareto fronts corresponding to the four best sequences according to Conv

Table

Best

12345 | 2.30 | 108.17 | 24513 | 2.17 | 106.64 | 42315 | 2.53 | 116.99 |

12354 | 2.26 | 107.81 | 24531 | 2.26 | 105.94 | 42351 | 2.63 | 114.24 |

12435 | 2.27 | 106.25 | 25134 | 2.24 | 102.96 | 42513 | 2.28 | 104.48 |

12453 | 2.13 | 100.63 | 25143 | 2.41 | 109.53 | 42531 | 2.36 | 109.50 |

12534 | 2.18 | 105.81 | 25314 | 2.30 | 107.66 | 43125 | 2.42 | 109.26 |

12543 | 2.13 | 103.26 | 25341 | 2.49 | 107.22 | 43152 | 2.34 | 107.36 |

13245 | 2.22 | 102.72 | 25413 | 2.26 | 104.36 | 43215 | 2.67 | 116.73 |

13254 | 2.11 | 103.03 | 25431 | 2.27 | 107.97 | 43251 | 2.56 | 113.93 |

13425 | 2.15 | 103.12 | 31245 | 2.20 | 100.73 | 43512 | 2.43 | 108.12 |

13452 | 31254 | 2.10 | 103.46 | 43521 | 2.53 | 111.63 | ||

13524 | 2.07 | 101.03 | 31425 | 2.30 | 106.79 | 45123 | 2.46 | 106.61 |

13542 | 2.02 | 100.08 | 31452 | 2.12 | 97.810 | 45132 | 2.33 | 102.12 |

14235 | 2.45 | 115.10 | 31524 | 2.15 | 104.32 | 45213 | 2.25 | 102.99 |

14253 | 2.21 | 104.45 | 31542 | 2.12 | 100.52 | 45231 | 2.42 | 110.85 |

14325 | 2.42 | 112.87 | 32145 | 2.44 | 111.30 | 45312 | 2.27 | 101.57 |

14352 | 2.21 | 105.30 | 32154 | 2.35 | 107.73 | 45321 | 2.55 | 111.39 |

14523 | 2.25 | 107.07 | 32415 | 2.51 | 115.31 | 51234 | 2.49 | 110.19 |

14532 | 2.27 | 105.43 | 32451 | 2.33 | 107.85 | 51243 | 2.38 | 107.21 |

15234 | 2.29 | 107.13 | 32514 | 2.38 | 107.70 | 51324 | 2.36 | 106.79 |

15243 | 2.14 | 102.56 | 32541 | 2.40 | 109.14 | 51342 | 2.25 | 103.38 |

15324 | 2.27 | 106.71 | 34125 | 2.42 | 109.68 | 51423 | 2.53 | 113.27 |

15342 | 2.17 | 102.45 | 34152 | 2.33 | 104.78 | 51432 | 2.55 | 112.77 |

15423 | 2.26 | 109.45 | 34215 | 2.36 | 112.19 | 52134 | 2.29 | 106.29 |

15432 | 2.24 | 106.63 | 34251 | 2.30 | 107.17 | 52143 | 2.44 | 108.36 |

21345 | 2.31 | 103.98 | 34512 | 2.18 | 101.86 | 52314 | 2.46 | 116.72 |

21354 | 2.24 | 103.07 | 34521 | 2.24 | 104.80 | 52341 | 2.61 | 112.97 |

21435 | 2.58 | 115.15 | 35124 | 2.27 | 103.81 | 52413 | 2.30 | 106.23 |

21453 | 2.38 | 106.26 | 35142 | 2.30 | 105.62 | 52431 | 2.37 | 108.24 |

21534 | 2.40 | 113.76 | 35214 | 2.32 | 109.19 | 53124 | 2.29 | 103.32 |

21543 | 2.32 | 110.97 | 35241 | 2.37 | 111.19 | 53142 | 2.36 | 108.06 |

23145 | 2.47 | 113.45 | 35412 | 2.28 | 101.50 | 53214 | 2.60 | 114.17 |

23154 | 2.36 | 107.94 | 35421 | 2.29 | 108.91 | 53241 | 2.62 | 116.59 |

23415 | 2.63 | 114.64 | 41235 | 53412 | 2.45 | 106.79 | ||

23451 | 2.48 | 111.27 | 41253 | 2.47 | 107.83 | 53421 | 2.46 | 112.91 |

23514 | 2.55 | 114.91 | 41325 | 2.55 | 113.15 | 54123 | 2.49 | 112.69 |

23541 | 2.54 | 111.22 | 41352 | 2.37 | 108.25 | 54132 | 2.38 | 105.83 |

24135 | 2.42 | 107.59 | 41523 | 2.54 | 111.57 | 54213 | 2.24 | 104.67 |

24153 | 2.43 | 108.50 | 41532 | 2.57 | 112.14 | 54231 | 2.42 | 115.17 |

24315 | 2.38 | 110.73 | 42135 | 2.44 | 115.59 | 54312 | 2.22 | 107.06 |

24351 | 2.42 | 108.28 | 42153 | 2.47 | 108.75 | 54321 | 2.50 | 116.42 |

Table

Debris removal sequence and timing for minimum

Phase | Final Keplerian elements | Duration (days) | mass (kg) | ||||

Departure | 6628.16 | 0.010 | 1 | 65 | — | — | 1000 |

Nr. 1 reached | 6828.16 | 0 | 1 | 65 | 5 | 0.115 | 996.11 |

Nr. 1 de-orbited | 6752.69 | 0.011 | 1 | 65 | 22.06 | 0.043 | 993.21 |

Nr. 3 reached | 6978.16 | 0 | −2 | 200 | 88.10 | 0.239 | 985.17 |

Nr. 3 de-orbited | 6826.44 | 0.022 | −2 | 200 | 25.96 | 0.084 | 980.63 |

Nr. 4 reached | 7478.16 | 0 | −1 | 90 | 66.71 | 0.476 | 964.88 |

Nr. 4 de-orbited | 7055.54 | 0.053 | −1 | 90 | 34.33 | 0.221 | 951.69 |

Nr. 5 reached | 7178.16 | 0 | 0 | 45 | 55.89 | 0.241 | 943.91 |

Nr. 5 de-orbited | 6912.18 | 0.034 | 0 | 45 | 30.77 | 0.144 | 931.48 |

Nr. 2 reached | 7128.16 | 0 | 2 | 150 | 56.98 | 0.297 | 922.12 |

Nr. 2 de-orbited | 6901.39 | 0.032 | 2 | 150 | 33.99 | 0.124 | 917.24 |

By analysing in more detail the Δ

Table

Debris removal sequence and timing for minimum TOF_{Tot}.

Phase | Final Keplerian elements | Duration (days) | Mass (kg) | ||||

Departure | 6628.16 | 0.010 | 1 | 65 | — | — | 1000 |

Nr. 1 reached | 6828.16 | 0 | 1 | 65 | 5 | 0.115 | 996.11 |

Nr. 1 de-orbited | 6685.24 | 0.001 | 1 | 65 | 4.04 | 0.081 | 990.61 |

Nr. 3 reached | 6978.16 | 0 | −2 | 200 | 8.59 | 0.312 | 980.14 |

Nr. 3 de-orbited | 6701.87 | 0.004 | −2 | 200 | 6.29 | 0.154 | 971.87 |

Nr. 4 reached | 7478.16 | 0 | −1 | 90 | 14.79 | 0.664 | 950.17 |

Nr. 4 de-orbited | 6789.06 | 0.016 | −1 | 90 | 17.13 | 0.362 | 928.76 |

Nr. 5 reached | 7178.16 | 0 | 0 | 45 | 7.99 | 0.281 | 919.92 |

Nr. 5 de-orbited | 6715.72 | 0.006 | 0 | 45 | 15.9 | 0.252 | 898.39 |

Nr. 2 reached | 7128.16 | 0 | 2 | 150 | 9.87 | 0.466 | 884.28 |

Nr. 2 de-orbited | 6725.90 | 0.007 | 2 | 150 | 6.75 | 0.221 | 875.87 |

A final note is devoted to the assumption mentioned in Section

One can see that, once the transfer type is defined, the left side of (

If the transfer type is not quasicircular but involves spirals with nonnegligible eccentricity, then an arbitrary delay cannot be introduced without altering the position of the lines of nodes. However, it is still possible to introduce an arbitrary number of coasting arcs of duration equal to the orbital period of the osculating orbit, that is, one full revolution. The phase variation obtained by one such revolution is

Note that, given the orbits involved in the transfer,

This will bring the phase difference to a quantity which is lower than the maximum phase variation per revolution achievable, leaving a residual phase difference:

A last coasting revolution is inserted to delete the residual when the semi-major axis which gives the proper angular velocity

The total delay introduced in the worst case is, therefore, given by the sum of the periods of the coasting revolutions:

By applying the above strategies to the minimum

This work presented a novel computational approach for the preliminary design of multispiral trajectories. The approach was applied to the design of an orbit debris removal mission by means of an IBS spacecraft. The models proposed here for the computation of low-thrust many-revolution transfers, allowed for a considerable reduction in control parameters and at the same time for a fast propagation of low-thrust motions thanks to the analytical propagation with FPET. Thanks to the reduced computational cost for the evaluation of a single fetch and deorbit operation, a multiobjective optimisation problem could be solved in which thousands of different debris removal sequences were examined to find the optimal ones with respect to

Future work will deal with comparing the proposed approach with similar methods like orbit averaging. In addition, although the proposed method has been applied to the special case of a debris removal mission, it is suitable to be extended and applied to more general trajectory design problems which involve many-revolution transfers from elliptical to circular or from elliptical to elliptical orbits. Current developments are incorporating gravity perturbations in the analytical solution to allow the computation of more accurate solutions.

In the following, the first-order solution for perturbed Keplerian motion is reported (see also [

18 Radial-transverse reference frame.

For