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DYNAMICAL ISSUES ASSOCIATED WITH RELATIVE CONFIGURATIONS OF MULTIPLE SPACECRAFT NEAR THE SUN-EARTH/MOON L1 POINT

B. T. Barden and K. C. Howell School of Aeronautics and Astronautics

Purdue University West Lafayette, Indiana 47907-1282

AAS/AIAA Astrodynamics Specialist Conference

Paper 99-450 AAS

Girdwood, Alaska August 1999

AAS Publications Office, P.O. Box 28130, San Diego, CA 92198

AAS 99-450

DYNAMICAL ISSUES ASSOCIATED WITH RELATIVE CONFIGURATIONS OF MULTIPLE SPACECRAFT NEAR

THE SUN-EARTH/MOON POINT 1L

B. T. Barden* and K. C. Howell

The recent interest in applying Dynamical Systems Theory to spacecraft trajectory design in the three-body problem has resulted in new insights. In addition to increased understanding of the rich dynamics in this region of space, the opportunity for new mission concepts has emerged as well. One proposed option is that of flying multiple spacecraft in some specified relative configuration near the collinear libration points. In this investigation, some fundamental issues associated with this concept are explored further. Beginning with a review of the natural dynamics observed on tori that envelope halo orbits in the circular restricted problem, similar motions are pursued in the more complex dynamical model that includes ephemerides for the positions of the Sun, planets and Moon. These results are then compared to additional non-natural configurations.

INTRODUCTION In recent years, the scientific and public interest in libration point missions has increased. A number of proposals for NASAs Discovery and Midex programs involve mission concepts derived from trajectory options available in the region of space near the Sun-Earth/Moon collinear libration points. Current missions that include this type of baseline motion include SOHO, ACE, and WIND. Other planned missions that are also scheduled to spend a considerable length of time in these regions of space include Triana, MAP, and Genesis, which will be launched in November 2000, December 2000, and January 2001, respectively. Other missions, such as NGST, are also considering such mission options. While trajectory design for many of these missions involves more traditional trial-and-error methods, the mission designers for Genesis and Triana successfully incorporate more modern techniques into the design process.1-3 The mathematical foundation for these techniques is based in Dynamical Systems Theory (DST); DST offers a means to establish an initial guess for a trajectory in a region of space where conic approximations are not adequate. Although originally developed within the structure of the circular restricted problem, this technique is now applied successfully in a more complex dynamical model of the solar system, i.e., one that includes solar radiation pressure, as well as ephemerides for the positions of the planets and moons (hereafter called the real model).2 One of the primary advantages of DST is the immediate insight available from the geometry of the phase space in the vicinity of Lissajous trajectories and halo orbits. This is a critical component in the Genesis trajectory design since a return of the spacecraft from an libration point orbit to the Earth is a required element of the mission.

1L1,2 This insight has also been instrumental in exposing new concepts that had not

* Graduate Student, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN

47907-1282 Professor, School of Aeronautics and Astronautics, Purdue University, West Lafayette, IN 47907-1282

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previously been considered. In Barden and Howell,4 the concept of multiple spacecraft flying in some specified configuration near a collinear libration point is discussed in the context of the fundamental motions that exist in the circular restricted three-body problem. The current investigation is then a preliminary application of those results. Specifically, having used DST to establish the existence of certain types of natural motions that would be amenable to a cluster of spacecraft flying in formation, these motions are now examined in the context of the real model and are compared to other non-natural configurations. Because of the existence of precisely periodic orbits, in addition to the symmetries and time-invariance associated with the idealized circular restricted model, it is very natural to use aspects of DST in the circular problem. These properties vanish, however, when shifting the focus to the real system. So, it becomes necessary to carefully justify not only the use of similar techniques with a more complex model, but to clarify the expectations that may be reasonable in terms of motion similar to that seen in the circular problem. This specifically implies that it is necessary to gain some understanding of the evolution of the fundamental motions in the circular problem under perturbation. Thus, along with a review of those tenets of DST that are germane to this study, consideration is also given to certain topological aspects of the surfaces under perturbation. After justifying the methods and anticipated results, a sample formation of six particles (i.e., particles that potentially represent spacecraft) is simulated where the configuration is selected to take advantage of the natural dynamics in the center manifold, as accomplished in the circular restricted problem. Given such a reference, additional formations are selected (without regard to the dynamics) and simulated. Comparisons are then presented between the various formations with regard to cost in V and in terms of the actual evolution of the configurations between maneuvers. Some combination of these types of formations could provide baseline information for analysis of the general behaviors of more complicated or extensive spacecraft configurations in this regime. DYNAMICAL SYSTEMS THEORY Investigations utilizing DST usually begin with special solutions. These might include equilibrium points, periodic orbits, quasi-periodic motions, and homoclinic as well as heteroclinic motions. Each of these solutions is an example of one of the fundamental models for the phase space, i.e., invariant manifolds. An

-dimensional manifold is analogous to a two-dimensional surface in . The concept of an invariant manifold can be simply described as follows: a collection of orbits that start on a surface and stay on that surface for the duration of their dynamical evolution. This basic definition can be used to characterize a variety of behaviors. In addition to the examples already mentioned, there exist invariant manifolds that asymptotically approach or depart other invariant manifolds. These are called stable and unstable manifolds, respectively.

m nR

In the circular restricted three-body problem, the stable and unstable manifolds associated with periodic halo orbits have been the key to progress in the transfer problem (and the results can be successfully extended to a more complex model2). While the task of developing expressions for these nonlinear surfaces is formidable, it is also unnecessary within the context of their intended role in the design process. The computation of the stable and unstable manifolds associated with a particular halo orbit can actually be accomplished numerically in a straightforward manner.1,2,5-7 In this investigation, a similar technique is used to identify specific types of motion near the halo orbit and in the center manifold. Specifically, a local approximation of any invariant manifold (stable, unstable, or center) is available by exploiting the knowledge that the nonlinear manifolds are tangent to their respective eigenspaces, i.e. the subspaces associated with the monodromy matrix (the variational matrix after one period of the motion) of a periodic orbit.8,9 Initially, the invariant subspaces of the monodromy matrix are used to qualify the types of nearby motion. These subspaces can then be used to compute approximations of the motion on the corresponding nonlinear invariant manifolds.

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As a demonstration, consider the motion in the center manifold near periodic halo orbits. The eigenvalues of the monodromy matrix associated with the halo orbit indicate that the center subspace is four-dimensional (with one-dimensional stable and unstable subspaces comprising the balance of the six-dimensional phase space). Specifically, two of the center eigenvalues are real and equal to one, indicating periodic motion, and two are complex conjugates with magnitude equal to one. Let 6543 ,,, eeee be the four eigenvectors associated with the center subspace, where 3e and 4e are associated with the two real eigenvalues, and 5e and 6e are associated with the two complex eigenvalues. Any six-dimensional vector in the center subspace can be expressed as a linear combination of these eigenvectors, i.e., 6655443 ececececY eeW

c

+++= , (1) where i are complex scalars and the superscript is the usual notation representing the center manifold. The values of the scalars, i , determine the type of nearby motion. Specifically, if and are both zero and and are complex conjugates, i.e.,

c cWc 3c 4c

5c 6c 6*555 ececY eW

c

+= , (2) then the resulting vector provides a means to approximate, relative to the halo orbit, nearby quasi-periodic motion on a torus. The actual state can be expressed as,

c

cc

Wp

W

cHW

y

YdXX += , (3)

where HX represents the six-dimensional state on the periodic halo orbit that defines the beginning and, ideally, the end of a period for computation of the monodromy matrix; the vector

cWpy is defined as a three-

e