where x = [[x,y,z].sup.T] [member of] [R.sup.3] is the position vector, [m.sub.c] [member of] R is the Chaser mass (known and varying with time), [[omega].sub.0] = [square root of [mu]/[r.sub.HF][member of] R] is the angular velocity of the LVLH frame at a distance [r.sub.HP] from center of Earth, [mu] is the gravitational parameter of Earth (known and constant), and F = [[[F.sub.x], [F.sub.y], [F.sub.z]].sup.T] [member of] [R.sup.3] is the total force vector, which is the sum of the forces due to the thrusters and the forces due to the action of the external environment disturbances affecting the system; that is,
The forces obtained from the thrusters and the external disturbances are transformed from body frame to LVLH frame.
where [F.sup.*] [member of] [R.sup.3] represents the vector of force in body frame ([F.sub.B] frame, see Figure 3), F [member of] [R.sup.3] represents the force vector in LVLH frame (see (2)), and [R.sup.b.sub.o]([phi], [theta], [psi])[member of] [R.sup.(3,3)] is the transformation matrix between these two reference frames.
Note that [q.sub.rel] and [[omega].sub.rel] are relative quaternion and angular rates written in LVLH frame.
where [[omega].sup.b.sub.EO] is the angular speed of the LVLH frame relative to ECI and [[omega].sup.b.sub.OB] is the angular speed of the spacecraft with respect to LVLH frame.
where, under the assumption of circular orbit, angular velocity of the LVLH frame is
where [[omega].sub.0] is the angular velocity of the LVLH frame, as defined in Section 3.1.