Models and strategies for J2-perturbed orbital pursuit–evasion games

First, authors discuss the maneuver method under J₂ perturbation. Since the long-term variations for semimajor axis (a), inclination (i), and eccentricity (e) are zero in the presence of J₂ perturbation, the inclination is assumed not time-varying. The variations for ω and Ω are in proportion to a^-7/2(1-e²)^-2. Assuming that there is a satellite on a circular orbit with limited ΔV, when using J₂ perturbation to change Ω under a fixed inclination, the transfer orbit used can be treated as a circular orbit. This also implies that ω is not worth analyzing for this variable is only valuable under a noncircular orbit. Six orbital elements can be simplified to “three orbital elements”, that is, the semi-major axis (a), the RAAN (Ω), and the inclination (i). The maneuvering method through a “gliding orbit” is shown in Fig. 2. The entire transfer process contains three steps: (1) Two impulsive dogleg maneuvers, taking the satellite from initial status to a “gliding orbit”, which could be simplified to a circular orbit, as shown in the previous section. (2) Waiting on the gliding orbit to let J2 perturbation make progress. (3) Two impulsive dogleg maneuvers take the satellite from gliding orbit to the target satellite. When it comes to changing Ω, the spacecraft can use a ΔV in 2 ways: change its Ω directly and instantaneously through thrust, or change its altitude and/or inclination and wait for J₂ perturbation to change its Ω over a long period.

Fig. 2. Diagram of orbit transfers using J₂. (A) Change of Ω over time. (B) Change of i and a over time.

Then, authors discuss circular and inclination-fixed LTOPEGs, including the maneuverability of both satellites described by their reachable domains (RDs), the game model, and the winning conditions. The maneuverability under J₂ perturbation can be divided into 2 parts based on the amount of time consumed. The first part uses J₂ perturbation to perform a long-duration gliding maneuver, and the second part uses the spacecraft’s thrusters for an “immediate” impulsive maneuver. The RD, measured by Δr and ΔΩ, represented in polar coordinates has properties similar to that of an ellipse, while the nonlinear properties caused by Kepler dynamics can be observed as the ability to change the orbital radius. The more ΔV used, the more obvious the asymmetry is, as the “+Δr side” of the ellipse tends to be larger. The “fixed-inclination OPEG” can help us unveil some dynamic properties under J₂ perturbation, in which 6 orbital elements can be simplified into 2: Ω and a, which can be represented visually as “conveyor belt”. Under the influence of perturbation, Ω is constantly varying, just like the conveyor belt moving at different speeds. The closer the satellite is to Earth, the faster the conveyor belt’s speed. A rendezvous process under this model is shown in Fig. 5 (not to scale). Moreover, this paper takes the evader’s mission execution ability as the most critical indicator of the game. The winning conditions are defined as “the evader has used up all of its ΔVs” or “the evader’s payload is damaged.”

Fig. 5. Maneuver with target orbit as the reference.

Finally, authors discuss strategies for the J₂-perturbated OPEG. Strategies for the evader are divided into three cases by Φ which is the ratio of ΔV used by both sides. Meanwhile, the “catchable domain” (CD) is defined, which represents the viability to ensure capture after “instant” dogleg maneuvers. When the evader’s orbit is within the CD, it will lose the game. (1) When Φ < 2.2, it is unwise for the evader to conduct LTOPEGs, as the evader has greater advantages in STOPEGs. (2) When Φ = ∞, where the pursuer does not need to consider the evader’s STOPEG capability at all, and the RD of the pursuer is equal to the CD. The strategy of the pursuer is the same as the direct approach strategy in Winning conditions of the game. (3) When 2.2 < Φ < ∞, the pursuer’s CD should change after the evader has made a maneuver. Due to the assumption of complete information, the change of the CD should be considered before the evader carries out the maneuver. Specifically, there are three plans to escape regarding (2) and (3), namely Plan A, B, and C. Plan A tries to use dogleg maneuvers to reach an orbit higher than the threatened area to avoid the CD. The evader will return to the mission orbit after its RAAN has made a full revolution around Earth. Plan B is similar to Plan A but tries to reach an orbit lower than the threatened area. Plan C tries to use an impulse to reach an orbit higher than the threatened area and then uses another impulse to return to the mission orbit after the CD of the pursuer passes by. Besides, Plan C will use maneuvers to overcome the influence of J₂ perturbation during this period and therefore, the evader need not glide around Earth for a full revolution. Among three plans, Plan C costs more ∆V than the other two. Apart from RDs and CDs, a problem arises with “arrival time matching”. It can be envisioned that if the pursuer deliberately matches the arrival time, the evader will also attempt to avoid it, then the pursuer will match again, and so on, resulting in a game of matching the arrival time. Φ_B = ∆V_E/∆V_P is obtained considering a slight ∆a_E for both sides to determine which side requires more ΔV to rematch/avoid an existing “resonant arrival time”. As for the pursuer’s response, four choices are summarized as follows. (1) Do nothing. (2) Maneuver to cover the mission orbit again for a shorter time. (3) Maneuver to occupy the mission orbit. (4) Maneuver to change its orbit to a position between the initial approaching orbit and mission orbit.

Fig. 6. CD, threatened area, and escape strategy in LTOPEG when Φ = ∞.