Active Formation Flying Along an Elliptic Orbit

THE relative motion of a follower satellite with respect to the leader in a given circular orbit is described by autonomous nonlinear differential equations. The linearized equations around the null solution are known as Hill-Clohessy-Wiltshire (HCW) equations [1-3]. The HCW equations were used by many authors to study rendezvous problems (see [4] and references therein). The Tschauner-Hempel (TH) equations replace the HCW equations, when the orbit of the leader is eccentric [2,5]. Rendezvous problems along an eccentric orbit were studied in [4,5]. The HCW equations possess periodic solutions, which are useful as temporary orbits before mission and for proximity operations such as inspection and repair. The leader-follower formation and reconfiguration problems based on the periodic solutions were studied by many authors [6-10]. The TH equations also have periodic solutions. They are characterized by Inalhan et al. [11], and the initialization procedure to periodic motion is given. Periodic solutions also follow from the transition matrix of the TH system given by Yamanaka and Ankersen [12]. The effects of eccentricity on the shape and size of relative orbits are studied by Sengupta and Vadali [13]. Periodic solutions of the TH equations are used for formation flying [14,15] because no control efforts are needed to maintain them. However, their period is fixed and is equal to that of the leader orbit, which would be inconvenient for a quick inspection of the leader. The shape of periodic solutions is irregular compared with that of the HCW equations, which would be undesirable for some missions.