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This paper presents the methods and results for the trajectory design and optimization for the low earth orbit (LEO) satellites in formation to observe the geostationary orbit (GEO) satellites’ beams. The background of the trajectory design mission is the 9
^{th} China Trajectory Optimization Competition (CTOC9). The formation is designed according to the observation demands. The flying sequence is determined by a reference satellite using a proposed improved ephemeris matching method (IEMM). The formation is changed, maintained and transferred following the reference satellite employing a multi-impulse control method (MICM). Then the total observation value is computed by propagating the orbits of the satellites according to the sequence and transfer strategies. Based on the above methods, we have obtained a fourth prize in the CTOC9. The proposed methods are not only fit for this competition, but can also be used to fulfill the trajectory design missions for similar multi-object explorations.

Geostationary orbit (GEO) plays an important role in communication, navigation, and other areas [

The problem is a multi-object optimization problem, which appears in many practical applications. For example, the multi-asteroid exploration, the multi-planet exploration and the multi-debris clearing mission [

This paper is organized as follows. Section 2 gives a brief description of the problem. Section 3 gives the general description of the methods. Application of the methods to the problem is presented in Section 4. Results and conclusions are finally presented in Sections 5 and 6.

In the problem, the GEO constellation consists of 18 satellites, each of which is equipped with a transmitter which directs its beam to the ground station. The diagram of the projection of the GEO satellites’ beams is shown in

An LEO formation consisting of 3 monitoring satellites is inserted into circular orbits of altitude of 400 km and inclination of 42.8˚ at initial time. The task of the three monitoring satellites is to monitor all the beams of the GEO satellites. The “monitoring” operation of a GEO satellite beam is executed only when the three monitoring satellites are in the coverage of the beam. Monitoring proﬁt of a GEO satellite, which is deﬁned by the geometry between the three monitoring satellites and the GEO satellite, is accumulated in the process of monitoring execution. Assuming the three monitoring satellites are in the coverage of a GEO satellite’s beam at a specified time, ^{7} km^{2}), the whole monitoring task is completed. Please refer to Ref. [

There are two optimization objectives of diﬀerent priorities in the problem. The primary objective is to minimize the time duration of the whole monitoring

task. The secondary objective, in case two or more teams achieve the same primary objective, is to minimize the total fuel consumption of the three monitoring satellites.

Each monitoring satellite is subject to the J_{2} perturbation. And the motion equation in the J2000 earth centered inertial coordinate is as follows,

{ x ˙ = v x y ˙ = v y z ˙ = v z v ˙ x = − μ x r 3 ( 1 + 3 2 J 2 ( R e r ) 2 ( 1 − 5 z 2 r 2 ) ) v ˙ y = − μ y r 3 ( 1 + 3 2 J 2 ( R e r ) 2 ( 1 − 5 z 2 r 2 ) ) v ˙ z = − μ z r 3 ( 1 + 3 2 J 2 ( R e r ) 2 ( 3 − 5 z 2 r 2 ) ) (1)

In which, μ is the gravitational constant of the earth. x, y, and z refer to the position of the satellites. v_{x}, v_{y}, v_{z} refer to the velocity. r = x 2 + y 2 + z 2 refers to the range from the earth center to the satellite. R_{e} is the radius of the earth, and J_{2} is the perturbation constant of the earth’s oblateness.

The initial masses of all satellites are 500 kg, and the allowed fuel consumption should not be more than 200 kg. The maneuvers of the monitoring satellites are instantaneous changes of the satellites velocities. After each maneuver, the fuel consumption Δm should be evaluated using the Tsiolkovsky equation.

Δ m = m ( 1 − e − Δ V g e I s p ) (2)

where m is the mass before the maneuver, ΔV is the magnitude of the maneuver, g_{e} is the acceleration of gravity at sea level of Earth, I_{sp} is the specific impulse of the propulsion.

The GEO satellites are assumed to follow non-perturbed Keplerian orbits. The longitude and radius of each GEO satellite remain constant, and the latitude remains zero during the whole mission. For more information of the problem description, please refer to Ref. [

In order to solve the problem effectively, we decomposed the methods used by us into following steps.

Step 1: The formation is designed according to the observation demands by analyzing the observation features.

Step 2: The flying sequence is firstly determined by a specified reference satellite using a proposed improved ephemeris matching method (IEMM). Then, the formation is maintained and transferred following the reference satellite by a multi-impulse control method (MICM). The total monitoring profit is computed by propagating the satellites’ orbits subject to J_{2} perturbation according to the sequence and transfer strategies.

Step 3: The result data is checked and submitted to the competition organizer.

The basic procedure of the methods employed in this paper for solving the problem A of CTOC9 is described in

In order to design stable formations and reduce the maintaining cost, we analyzed the stability of the formations by propagating the differential equations of the elements difference between the satellites. In the presence of the J_{2} perturbation, some orbital elements change, including the drifts in perigee and mean anomaly, and the nodal regression. Employing orbit-averaged quantities [

{ δ a ˙ = δ e ˙ = δ i ˙ = 0 δ Ω ˙ = − 3 μ J 2 R e 2 2 ( 1 − e L 2 ) 2 a L 3.5 [ − sin i L δ i + cos i L ( 4 e L δ e 1 − e L 2 − 7 δ a 2 a L ) ] δ ω ˙ = 3 μ J 2 R e 2 4 ( 1 − e L 2 ) 2 a L 3.5 [ − 5 sin 2 i L δ i + ( 5 cos 2 i L − 1 ) ( 4 e L δ e 1 − e L 2 − 7 δ a 2 a L ) ] δ M ˙ = − 3 2 μ a L 3 δ a a L + 3 μ J 2 R e 2 4 ( 1 − e L 2 ) 1.5 a L 3.5 × [ − 3 sin 2 i L δ i + ( 3 cos 2 i L − 1 ) ( 3 e L δ e 1 − e L 2 − 7 δ a 2 a L ) ] (3)

where μ is the gravity constant of Earth. a, e, i, Ω, ω, Μ are the orbital elements of the satellites, and δa, δe, δi, δΩ, δω, δΜ are the elements’ differences between

the leader and the follower satellites. The elements with subscript L refer to the leader satellites.

Equation (3) shows that the J_{2} perturbation does not change the differences of a, e, i but changes the differences of Ω, ω, Μ between the leader and the follower satellites. If the initial values of the elements’ differences with subscript “0” at time t_{0} are given, the values at any time t can be computed as follows,

{ δ a = δ a 0 , δ e = δ e 0 , δ i = δ i 0 δ Ω = δ Ω 0 + δ Ω ˙ ( t − t 0 ) δ ω = δ ω 0 + δ ω ˙ ( t − t 0 ) δ M = δ M 0 + δ M ˙ ( t − t 0 ) (4)

Equations (3) and (4) obviously show that, when there are differences in the elements a, e or i, the formation is unstable, and changes quickly. As a result, in this paper, the three satellites’ initial elements a, e and i are chosen to be nearly the same.

The initial relations of the three satellites’ elements will be determined by the mission’s requirements, which will be shown in Section 4.1.

For the flying sequence searching, the improved ephemeris matching method (IEMM) is proposed.

From the characteristics of the Lambert problem [_{l}) to the position of the target beam b (propagated to the target time) is, the less energy is needed for the satellite to transfer from b_{l} to flyby b. Thereby we can measure the relative size of the transfer energy consumed by examining the closeness between the target beam’s ephemeris and the satellite’ ephemeris propagated from the leaving beam. In this paper, we define the reference satellite as the leader satellite. When propagating the reference satellite’s orbit, the satellite’s velocity when leaving the beam is not fixed. We add a small Δv along the velocity in range [−25 m/s, 25 m/s] at the departure epoch with a step size of 5m/s and propagate the satellite’s trajectory. We pick up the one closest to the matching target beam (if the monitoring profit for this beam is not fulfill the requirement), record the Δv value, the arrival time t_{f}, and the position R and velocity V of the closest point from the central line of the beam to the reference satellite’s propagated orbit. This process is different from the ephemeris matching method (EMM) in Ref. [

For a segment of the transfer trajectories, the mathematical model of IEMM can be expressed as

Δ v = f ( λ r ‖ R ( t ) − R b ( t ) ‖ , λ v ‖ V ( t ) − V b ( t ) ‖ ) (5)

where f (^{.}) is a monotonic increasing function; R and V are respectively the position and velocity of the satellite’s propagated orbit at arrival time t; R_{b} and V_{b} are respectively the position and velocity of a point on the central line of a beam b. The point is chosen to be with the same height to the satellite’s propagated orbit at the arrival time. λ_{r} and λ_{v} are the weights of the positions’ difference and velocities’ difference. For this paper, the satellites need to flyby (not rendezvous) the target beam, so the velocities between them need not to be equal. Only the positions need to match. Therefore, λ_{v} is set to be 0, and λ_{r} could be 1.

For a transfer segment, when we have set a Δv_{i} (which is determined by step 5m/s in the small range [−25, 25] m/s from lowest to highest), the target beam b_{i} whose ephemeris matches the reference satellite could be found along the propagating satellite’s orbit till t_{i}. Then we identify the target beam and record the corresponding parameters that meet the condition of top 5 values of the net accumulated monitoring profits when propagating the satellites orbits in the segment. Here net profit means when the accumulated monitoring profit of a GEO satellite’s beam has up to the threshold, it will not be added to the total accumulated value computed along this segment. When computing the monitoring profit, the formation is formed theoretically by the geometry relationship between the leader and the follower satellites, because in this sequence searching process, the formation maintaining is not considered.

The procedure of the IEMM algorithm is shown in

Step 1: Propagate the reference satellite’s orbits from the last beams of the recorded sequences with the initial velocity offset Δv_{i} from −25 m/s to 25 m/s with a step size of 5 m/s. There will be 11 orbits to be propagated.

Step 2: Find the 11 target beams whose central line is closest to the satellite’s orbits corresponding to Δv_{i} at the end time t_{i}.

Step 3: Propagate the reference satellite’s orbit and use the theoretical formation (by the geometry relationship between the leader and follower satellites) to compute the net accumulated monitoring profits.

Step 4: For a sequence, if the total monitoring profit meets the condition of all the beams’ monitoring profits up to the threshold, record the new sequence as one of the candidate sequences. If any one of the satellite’s fuel has all been consumed but not all the observation values of the beams have been up to the threshold, the sequence is abandoned. Otherwise, abandon the 6 sequences with the least monitoring profits, record the top 5 sequences for further searching, and turn to Step 1.

Step 5: If all recorded sequences have fulfilled the threshold, stop searching, and pick out the sequence with shortest mission time for further processing.

For the formation maintaining and orbit transfer, a multi-impulse control method (MICM) is employed. The Δvs of all maneuvers are optimized by the differential evolution (DE) algorithm and the J_{2} perturbed Lambert problem solving algorithm. The algorithm procedure is shown in

There are two layers of the loop. The inner layer is the solving of the J_{2} perturbed Lambert problem. The locally optimization shooting method is used to solve the boundary value problem. If the three satellites’ final positions R_{f} and velocities V_{f} equal the values predetermined by the theoretical relationship of the formation, the problem is solved. The predetermined values are given by the sequence searching and formation designing algorithms. When the position and velocity of the reference satellite are provided, based on the relationship between the three satellites in formation, the required positions and velocities of the follower satellites are determined, which are the objective values of the shooting method.

The outer layer of the loop is using the DE algorithm to optimize the velocity increments (Δvs) to be minimal. The optimizing problem is a time-fixed optimal transfer problem. DE is a heuristic searching algorithm introduced by Storn and Price [

J = min { Δ v i , k } (7)

where Δv_{i,k} refers to the sum of the velocity increments of the i^{th} transfer of satellite k. The variables to be determined by the optimization are as follows,

{ Φ 1 , k = [ Δ t 1 , k , Ω 1 , k , ω 1 , k , M 1 , k , Δ t ′ 1 , k , Δ R 1 , x , Δ R 1 , y , Δ R 1 , z ] Φ c , k = [ Δ t c , k , Δ t ′ c , k , Δ R c , x , Δ R c , y , Δ R c , z ] Φ i , k = [ Δ t i , k , Δ ω i , k ] , i ≠ 1 ≠ c (8)

in which Φ_{1,k} indicates the variables relative to the three-impulse transfer for the first maneuver of satellite k; Δt_{1,k} is the waiting time when the satellite is flying along the initial orbit; Ω_{1,k}, ω_{1,k}, and M_{1,k} are respectively the right ascension node, the argument of perigee, and the mean anomaly of the initial orbit; Δt’_{1,k} is the transfer time from the leaving point to a middle point on an initial J_{2} perturbed Lambert orbit; ΔR_{1,x}, ΔR_{1,y}, and ΔR_{1,z} are the position offsets from the middle points. Then two Lambert orbits are computed again to fulfill the transfers from the leaving point to the middle point and from the middle point to the end point.

Φ_{c,k} indicates the variables relative to the three-impulse transfer for formation changing (Why the formation need to be changed? Please refer to section 4.1.) of satellite k; Δt_{c,k} is the waiting time, at the end of which the satellite start to transfer; Δt’_{c,k} is the transfer time from the leaving point to a middle point on an initial J_{2} perturbed Lambert orbit; ΔR_{c,x}, ΔR_{c,y}, and ΔR_{c,z} are the position offsets from the middle points. Then two Lambert orbits are computed again to fulfill the transfers from the leaving point to the middle point and from the middle point to the end point.

Φ_{i}_{,k} indicates the variables relative to the two-impulse transfers for the rest maneuvers of satellite k; Δω_{i,k} is the difference of the argument of perigee between the leader satellite and the follower satellite. When propagating the orbits, only small difference of Δω_{i,k} will be generated by the J_{2} perturbation. In order to save fuel, we do not require the argument of perigee for the 3 satellites exactly equal each other. But when solving the J_{2} perturbed Lambert problem, the offset of the target mean anomaly should be set to have an opposite value to Δω_{i,k}. This will ensure a relatively stable shape to fulfill the observation. The bounds of the variables are shown in

Incorporating all the maneuvers of the 3 monitoring satellites, the observation values W_{j} are computed by propagating the J_{2} perturbed orbits with a step time t_{m} of 1 second using Equation (9) [

W j = ∑ m = 1 M S j ( t m ) g j ( t m ) (9)

where j refers to the label of the GEO satellite’s beam, S_{j} is the area of the triangle formed by 3 monitoring satellites, and g_{k} is sine of the angle from the GEO satellite’s beam to the triangle plane. The results are organized to fit the submission format in order to pass the data checking process. M refers to the total duration (in second) when the triangle is in the beam j.

In order to monitor the 11^{th} GEO satellite whose beam is locating at the highest latitude (which is shown in ^{th} beam could also be monitored, but it will consume too much fuel. So we choose the first way.

Δt_{1,k}/s | Ω_{1,k}/deg | ω_{1,k}/deg | M_{1,k}/deg | Δt’_{1,k}/s | ΔR_{1,x}/km | ΔR_{1,y}/km | ΔR_{1,z}/km | |
---|---|---|---|---|---|---|---|---|

up | ΔT_{1}/2 | 10 | 90 | 90 | 90 | 30 | 30 | 30 |

low | 0 | −10 | −90 | −90 | −90 | −30 | −30 | −30 |

Δt_{c}_{,k}/s | Δω_{c}_{,k}/deg | Δt’_{c,k}/s | ΔR_{c,x}/km | ΔR_{c,y}/km | ΔR_{c,z}/km | |||

up | 180 | 4 | ΔT_{c} + 500 | 50 | 50 | 50 | ||

low | −60 | −4 | ΔT_{c} − 500 | −50 | −50 | −50 | ||

Δt_{i}_{,k}/s | Δω_{i}_{,k}/deg | |||||||

up | 45 | 2 | ||||||

low | −45 | −2 |

Note: ΔT_{1} is the time range from initial to the first beam in the sequence. ΔT_{c} is the time range from the leaving to the arriving point of the formation changing transfer.

When the monitoring profit of the 11^{th} GEO satellite’s beam is near 1 × 10^{7} km^{2}, the formation is changed from the first one to the second one which constitutes a triangle perpendicular to the orbit plane for the observing of the remaining GEO satellites’ beams locating at the low latitudes. The first and the second formations are shown in

When designing the formations, not only the monitoring requirements, but also the stability should be considered to reduce the formation maintaining cost. Finally, the relations of the 3 satellites are determined and shown in

The leader satellites’ elements a_{L}, e_{L}, i_{L}, Ω_{L}, ω_{L}, Μ_{L} are determined by the flying sequence and transferring strategies. The follower satellites are transferred to fit the relations in

For the flying sequence determination, the proposed IEMM described in Section 3.3 is employed. For every searched sequence, the labels of the beams, the epochs when the leader satellite arrive the beams and the positions & velocities of the orbits are recorded. Then the top 5 beams with high monitoring profits are chosen and added to the sequence to form new sequences. For a special sequence, if the total monitoring profit meets the condition of all the beams’ monitoring profits up to the threshold, record the new sequence as one of the candidate sequences. If any one of the satellite’s fuel has all been consumed but not all the monitoring profits of the beams have been up to the thresholds, the sequence is abandoned. Otherwise, abandon the last 6 values, and record the top 5 sequences for further searching. If all recorded sequences have fulfilled the threshold condition, stop searching, and pick out the sequence with shortest mission time for further processing.

The sequence searching result is shown in ^{5} s (27.8 hours). Most of the distances between the

Formation | Satellite | a | e | i | Ω | ω | Μ |
---|---|---|---|---|---|---|---|

Formation 1 | Leader | a_{L} | e_{L} | i_{L} | Ω_{L} | ω_{L} | Μ_{L} |

Follower 1 | a_{L} | e_{L} | i_{L} | Ω_{L} | ω_{L} | Μ_{L} − 2.237 | |

Follower 2 | a_{L} | e_{L} − 0.02 | i_{L} | Ω_{L} + 3 | ω_{L} | Μ_{L} | |

Formation 2 | Leader | a_{L} | e_{L} | i_{L} | Ω_{L} | ω_{L} | Μ_{L} |

Follower 1 | a_{L} | e_{L} | i_{L} | Ω_{L} | ω_{L} | Μ_{L}_{ }+ 2.237 | |

Follower 2 | a_{L} | e_{L} | i_{L} | Ω_{L}+3 | ω_{L} | Μ_{L} − 1 |

reference satellite’s orbits and the central line of the corresponding target beams are no more than 100 km, which means that most of the target beams in addition to the passed beams along the transfer orbit could all be observed by the formation. The consumed fuel by the reference satellite is lower than 160 kg, which means 40 kg left could be used to guarantee the fulfillment of the formation transfer and maintenance along the overall task.

For the formation transfer and maintenance, the MICM described in section 3.4 is employed. The time points, Δvs of all maneuvers are optimized by the DE algorithm and J_{2} perturbed Lambert problem solving algorithm. Therefore, the satellites can fly following the reference satellite and the formation can be maintained. The orbit of the reference satellite is shown in

The changes of the reference satellite’s elements are shown in _{2} perturbation.

The consumed fuel for all the maneuvers are shown in

Incorporating all the maneuvers of the 3 satellites, the monitoring profits (observation values) shown in ^{7} km, which fulfills the requirement of the overall monitoring task.

We have obtained the fourth prize of CTOC9. The total flying time is 26.9611 days, and the total consumed fuel is 549.3799 kg, as shown in

The initial parameters, the number of maneuvers, and the final consumed fuels of the 3 satellites are shown in

The final observation values are shown in

This paper presented the methods and results for the problem A of CTOC9, the trajectory design and optimization problem for the low earth orbit (LEO) satellites in formation to observe the geostationary orbit (GEO) satellites’ beams. Two methods have been proposed to respectively solve the sequence searching and formation transfer & maintenance problems. The first method is the IEMM and the second one is MICM. Compared with other groups with better results, the methods in this paper may be more powerful in solving similar or more complex multi-objective trajectory design problems, because most of the methods used by other groups employed the regressive orbit to solve the special problem A of CTOC9. If the beams pointing to the surface of the earth to be

Rank | Team Name | 1st Object/day | 2nd Object/kg |
---|---|---|---|

1 | Nanjing University of Aeronautics and Astronautics (NUAA) | 20.42817 | 528.15235 |

2 | Research Center of Flight Dynamics and Control (RCFDC), NUDT | 21.40815 | 450.64002 |

3 | Tsinghua University (TU) | 24.97238 | 588.02681 |

4 | Xi’an Satellite Control Center (XSCC) | 26.96105 | 549.37992 |

5 | Northwestern Polytechnical University (NPU) | 27.22356 | 566.65516 |

6 | Laboratory of Space System Dynamics and Simulation Technology (LSSDST), NUDT | 29.06861 | 545.47487 |

7 | China Academy of Launch Vehicle Technology(CALVT) | 41.53081 | 423.36019 |

8 | China Academy of Space Technology (CAST) | 48.96269 | 526.07781 |

Leader | Follower 1 | Follower 2 | |
---|---|---|---|

a (km) | 6778.0676 | 6778.1642 | 6778.1172 |

e | 1.1058 × 10^{−5} | 4.0258 × 10^{−}^{6} | 5.5090 × 10^{−}^{6} |

i (deg) | 42.8000 | 42.8000 | 42.8000 |

Ω (deg) | 224.6348 | 224.5748 | 227.6658 |

ω (deg) | 67.5210 | 274.2653 | 18.3462 |

Μ (deg) | 202.4822 | 358.0967 | 237.9293 |

Number of maneuvers | 116 | 116 | 116 |

Consumed fuel | 175.8506 | 177.3275 | 196.2017 |

GEO beam | Monitoring profit/1 × 10^{7} km^{2} | GEO beam | Monitoring profit/1 × 10^{7} km^{2} |
---|---|---|---|

1 | 1.3684 | 10 | 1.0117 |

2 | 1.2856 | 11 | 1.1730 |

3 | 1.0462 | 12 | 1.2073 |

4 | 1.2803 | 13 | 1.4633 |

5 | 1.1974 | 14 | 1.0422 |

6 | 1.0284 | 15 | 1.0859 |

7 | 1.2502 | 16 | 1.2151 |

8 | 1.2091 | 17 | 1.2087 |

9 | 1.0815 | 18 | 1.2898 |

monitored are not fixed, the regressive orbit may not be an effective candidate tool to solve the problem. But we have attempted to propose general methods which need not any special characteristics of the monitor or the target. The proposed methods may have a wider application area to solve more multi-objective trajectory design and optimization problems.

We are very grateful to the 9^{th} China trajectory optimization competition organizers for the interesting problem, challenge, and fruitful discussions. This work was supported by the Youth Fund Project of the Space Systems Department (No. 2017SY26B0001), the Funds of Natural Science of China (No. 61673390, and No. 61503414 ), and the Fund of The State Key Laboratory of Astronautic Dynamics (No. 2016ADL-DW0202).

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

Lu, Y., Sun, Y., Hou, X.Y. and Meng, Y.H. (2018) Trajectory Design and Optimization for LEO Satellites in Formation to Observe GEO Satellites’ Beams. International Journal of Astronomy and Astrophysics, 8, 368-385. https://doi.org/10.4236/ijaa.2018.84026