Benest (1976), Dermott & Murray (1981) and Lidov & Vashkov’yak (1994a,b). Most of this early work was completed in the context of the restricted elliptic three-body problem.

For an asteroid following a quasi-satellite orbit, the key parameter to study is the difference between the mean longitudes of the asteroid and its host planet or relative mean longitude. The mean longitude of an object is given by $\lambda$ = $M$ + $\Omega$ + $\omega$, $M$ is the mean anomaly, $\Omega$ is the longitude of ascending node and $\omega$ is the argument of perihelion. When the relative mean longitude librates around 0$^{\circ}$, the object is in the quasi-satellite dynamical state, if it librates around 60$^{\circ}$, the objects is called an $L_4$ Trojan, when it librates around -60$^{\circ}$ (or 300$^{\circ}$), it is an $L_5$ Trojan, if the libration amplitude is larger than 180$^{\circ}$, it is said that the object follows a horseshoe orbit and when the relative mean longitude circulates (does not oscillate around a certain value or oscillates freely) we say that the object is no longer in a 1:1 mean motion resonance with the planet, i.e. it becomes a passing object.

In Fig. \[qs\] we plot the motion of 2002 VE$_{68}$ 