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 J₂ 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 Figure 3.

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 J₂ perturbation, some orbital elements change, including the drifts in perigee and mean anomaly, and the nodal regression. Employing orbit-averaged quantities [13] [14] , differences in the orbital elements between the leader and the follower satellites are given as below:

{ 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 J₂ 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 t₀ 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 [15] , it is not hard to understand that, the closer distance from the satellite’s position (propagated to the target time without any control from the leaving beam b_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. [16] , in which the trajectory is propagated without Δv along the velocity, so we call it improved ephemeris matching method (IEMM).

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 Figure 4, and is described as follows:

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 J₂ perturbed Lambert problem solving algorithm. The algorithm procedure is shown in Figure 5.

There are two layers of the loop. The inner layer is the solving of the J₂ 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 [17] . Its remarkable performance as a global optimization algorithm on continuous numerical minimization problems has been extensively explored [18] . As with other evolutionary algorithms, DE solves optimization problems by evolving a population of candidate solutions using alteration and selection operators. For the problem in this paper, the objective function is as follows:

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 J₂ 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 J₂ 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 J₂ 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 J₂ 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 Table 1.

Incorporating all the maneuvers of the 3 monitoring satellites, the observation values W_j are computed by propagating the J₂ perturbed orbits with a step time t_m of 1 second using Equation (9) [6] :

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 Figure 6), we raise the apogees of the three orbits and make the formation be a triangle parallel with the orbit plane to reach the beam. Of course if we raise the inclinations of the orbits, the 11^th beam could also be monitored, but it will consume too much fuel. So we choose the first way.

Note: ΔT₁ 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 × 10⁷ km², 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 Figure 7.

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 Table 2.

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 Table 2. The values of differences between the elements of the leader and the followers are determined by optimization to find the highest mean monitoring profit for passes of the GEO satellite’s beams.

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 Figure 8, which shows that most of the durations between two target beams of the ephemeris matching processes are no longer than 1 × 10⁵ s (27.8 hours). Most of the distances between the

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 J₂ 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 Figure 9, in which the red pluses refer to the maneuvers.

The changes of the reference satellite’s elements are shown in Figure 10, in which the semi-major and the orbital eccentricity are firstly raised by the initial transfer, and then oscillate in the following transfers. The orbital inclination nearly unchanged to save fuel. And the right ascension node changes from about 225 deg to 100 deg due to the J₂ perturbation.

The consumed fuel for all the maneuvers are shown in Figure 11. For every transfer, the velocity increment and the consumed fuel are minimized by the DE algorithm. There are 3 transfers whose consumed fuels are much more than others. The first one is for the transfer from the initial circular orbit to the elliptical orbit, the second one is for the formation changing, and the third one is for the final flying by to monitor the last beam in order to make the total time to be the least while consuming nearly all the fuel of each satellite.

Incorporating all the maneuvers of the 3 satellites, the monitoring profits (observation values) shown in Figure 12 are computed using Equation (9) by propagating the orbits. Figure 12 also shows that all the monitoring profits of the GEO satellites’ beams are larger than the threshold of 1 × 10⁷ km, which fulfills the requirement of the overall monitoring task.