Modeling for Libration Point Relative Motion

In the same manner as the Two-Body problem is applied to describe Earth orbit spacecraft motion, the Restricted Three Body Problem (RTBP) is the most common and simplest model used for the relative motion near the Sun-Earth/Moon (SEM) libration point. It accounts for the gravitational forces from the major primary body, the Sun, and second body, the Earth and Moon combination. This system, while simple, has no simple analytical solution to use to analyse the dynamic motion of spacecraft.

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Figure 1.12: Lagrange Libration Points of the Sun-Earth System

To obtain a closed-form analytical solution of relative motion to the RTBP, there are three critical assumptions currently used to simplify the model. First, the mass of the third body (spacecraft) is assumed to be infinitesimally small relative to those of the other two bodies. Second, the Sun and the Earth-Moon combination are each considered to be point masses for the purposes of modeling the gravitational forces. In addition, the motion of one primary to the other is an elliptical orbit around the barycentre of the system. With these assumptions, the formulation of the RTBP can be investigated effectively. However, the elliptical motion is still complex. If the eccentricity of an elliptical orbit is near zero, the RTBP will reduce to the Circular Restricted Three Body Problem (CRTBP), whose solution can be expressed simply as three second-order scalar differential equations. Setting the first and second derivatives to zero in these equations leads to five solutions, called the Lagrange libration points (five points in the space) as shown in Fig. 1.12. All of these points are in the plane as the Sun-Earth motion. Three of them are collinear with the line that connects the Sun and the Earth, and the other two are located at the vertices of two different equilateral triangles whose other vertices are the centre of the Sun and the Earth-Moon. Substituting these points into the differential equations, the stability property of each Lagrange point can be determined: the former three collinear points are unstable, while the other two points are stable (Luquette, 2006).

Although there is no general analytical solution to the CRTBP, several re- searchers have attempted to develop approximate analytical solutions. Using a linearization method, it has been shown that there are periodic or quasi-periodic

1.2. Libration Point Formation Flying Control Overview

trajectories in the vicinity of the collinear libration points (Richardson, 1980). The most general motion is the set of Lissajous trajectories which belong to three- dimensional quasi-periodic solutions.

In the 1980s, Richardson (1980) developed the analytical approximation for periodic motion near the collinear libration points in the SEM system. In his pa- per, the CRTBP differential equations were developed with respect to a collinear libration point. Using Taylor series expansion of the nonlinear terms, a hill-like linearized motion equation with its relevant analytical solution was obtained by truncating the higher order terms. Selecting suitable initial conditions, the solu- tion gives rise to HALO orbit motion, which is a periodic orbit. Furthermore, in order to obtain a more accurate approximation for a HALO orbit over long periods of time, a third-order analytical solution was also presented in his paper by using the classical Lindstede-Poincare method. With this solution, an amplitude con- straint relationship was derived between the out-of-plane and in-plane motions, and using this approach spacecraft can be initialized at any position on a HALO orbit. For the same problem, Howell & Barden (1999) employed manifold theory of dynamical systems to obtain periodic or quasi-periodic solutions to the CRTBP. In their paper, the motion around collinear libration points was considered in the context of centre manifolds.

Considering the formation flying of two spacecraft for the CRTBP, several au- thors have developed relative motion models in the vicinity of the stable libration points based on these quasi-periodic or periodic libration orbits, which has also been done for spacecraft formation flying motion in Earth orbit using a similar approach.

Following the approach of Richardson, Segerman & Zedd (2003) used a mod- ified Lindstedt-Poincare method to develop a third order solution of the relative motion in a HALO-type reference orbit. Using Richardson’s analytical solution as a reference orbit, Roberts (2005) obtained an expression for the gravity gradient and derived linearized relative motion equations. This motion model was then compared to the Satellite Tool Kit numerical orbit propagator and Segerman’s higher order model. The author concluded that this linear gravity gradient model was sufficiently accurate for controller design rather than using other higher order models. Some researchers have selected unstable orbits as the reference orbit for dynamic modeling. Without considering disturbances, Collange & Leitner (2004)

designed Lissajous trajectories of relative motion using analogous methodologies. The dynamics of the relative motion in unstable orbits is also studied in Scheeres & Vinh (2000).

In order to apply linear control methods, Luquette & Sanner (2004) developed the linearized dynamics of relative motion for the CRTBP and further extended them to the RTBP. The dynamics equations are formulated in both the inertial frame and the rotating frame. The relative motion equations were used directly to produce an adaptive Lyapunov control law. The controller could then compen- sate for modeling errors of linearization and other disturbances and improve the performance of the relative motion.

Generating the dynamic equations for formation flying for the CRTBP will provide a good model to use for the design of control and navigation systems. However, models which only include the nonlinear gravitational forces in the model is inadequate for high precision interferometry missions. To improve the model accuracy, the perturbations from the Sun the Earth and the Moon should be included in the model. Therefore, the RTBP is transformed into the more complex n-body ephemeris problem, where the time invariance properties of the CRTBP are lost and precise periodic orbits will not exist in the vicinity of libration points. Furthermore, any formation flying control algorithms must be tested and validated using the n-body ephemeris model.

After reviewing the natural dynamics observed on the envelope of HALO orbits in the CRTBP, Barden & Howell (1999) investigated the relative motion of more complex dynamical models, including the perturbations from the Sun and the Moon. Further, Marchand & Howell (2003) extended these results into an n- body ephemeris model by considering other gravitational perturbations as well as solar radiation pressure. Later a summary of collinear libration point formation flying was presented in Howell & Marchand (2005), in which the natural and

non-natural spacecraft formations near the SEM L1 and L2 libation points were

discussed. Hamilton et al. (2002) developed a high fidelity dynamics model called Generator, which included the effects of eccentricity, an independent moon, the other planets and solar radiation pressure.

Reviewing the above papers and models, all of them are only concerned with the translational motion while assuming the spacecraft as a point mass. For the modeling of rotational motion, the effect of translational motion is neglected,

1.2. Libration Point Formation Flying Control Overview

hence the translational motion and rotational motion are usually modeled sepa- rately as uncoupled systems. However, such an assumption is not possible for precision formation missions and the coupling effect caused by the displacement and misalignment of actuators needs to be modeled. In Pan & Kapila (2001) a coupled translation and rotation dynamic model was developed using a vector formalism approach, which is used for the Earth orbit formation flying but can also be applied to the relative motion in the vicinity of libration points. The modeling of relative translational and rotational dynamics was also considered in Gaulocher (2005) which was based on an interest in high precision measurement for the in- terferometry mission (Pegase). They linearized the coupled dynamics model with respect to the nominal configuration and used a linear fractional transformation to model varying parameters and their tolerances. However, this paper assumed that the formation state was very close to a nominal state and didn’t concentrate on the influences of gravitational forces and other disturbances. Focusing on the coupling effect between translational and rotational control, Luquette (2006) modeled a coupled dynamics successfully by using a coupled matrix of thrusters with the misalignment and misplacement.