Rodney L. Anderson

Role of Invariant Manifolds in Low-Thrust Trajectory Design

This paper demonstrates the significant role that invariant manifolds play in the dynamics of low-thrust trajectories moving through unstable regions in the three-body problem. It shows that an optimization algorithm incorporating no knowledge of invariant manifolds converges on low-thrust trajectories that use the invariant manifolds of unstable resonant orbits to traverse resonances. It is determined that the algorithm could both change the energy through thrusting to a level where the invariant manifolds could more easily be used, as well as use thrusting to move the trajectory along the invariant manifolds. Knowledge of this relationship has the potential to be very useful in developing initial guesses and new control laws for these optimization algorithms. In particular, this approach can speed up the convergence of the optimization process, retain the essential geometric and topological characteristics of the initial design, and provide a more accurate estimate of the A V and fuel usage based on the initial trajectory.

The role of invariant manifolds in lowthrust trajectory design (part III)

This paper is the third in a series to explore the role of invariant manifolds in the design of low thrust trajectories. In previous papers, we analyzed an impulsive thrust resonant gravity assist flyby trajectory to capture into Europa orbit using the invariant manifolds of unstable resonant periodic orbits and libration orbits. The energy savings provided by the gravity assist may be interpreted dynamically as the result of a finite number of intersecting invariant manifolds. In this paper we demonstrate that the same dynamics is at work for low thrust trajectories with resonant flybys and low energy capture. However, in this case, the flybys and capture are effected by continuous families of intersecting invariant manifolds.

A survey of ballistic transfers to low lunar orbit

A simple strategy is identified to generate ballistic transfers between the Earth and Moon, i.e., transfers that perform two maneuvers: a translunar injection maneuver to depart the Earth and a lunar orbit insertion maneuver to insert into orbit at the Moon. This strategy is used to survey the performance of numerous transfers between varying Earth parking orbits and varying low lunar target orbits. The transfers surveyed include short three to six day direct transfers, longer three to four month low-energy transfers, and variants that include Earth phasing orbits and/or lunar flybys.

A Survey of Ballistic Transfers to the Lunar Surface

In this study techniques are developed which allow an analysis of a range of different types of transfer trajectories from the Earth to the lunar surface. Trajectories ranging from those obtained using the invariant manifolds of unstable orbits to those derived from collision orbits are analyzed. These techniques allow the computation of trajectories encompassing low-energy trajectories as well as more direct transfers. The range of possible trajectory options is summarized, and a broad range of trajectories that exist as a result of the Suns influence are computed and analyzed. The results are then classified by type, and trades between different measures of cost are discussed.

Dynamical Systems Analysis of Planetary Flybys and Approach: Planar Europa Orbiter

In this analysis, the relationship between a planar Europa Orbiter trajectory and the invariant manifolds of resonant periodic orbits is studied. An understanding of this trajectory with its large impulsive maneuvers should provide basic tools that can be extended to cases that approximate low thrust with many small maneuvers. This study therefore represents a step in understanding low-thrust trajectories. Unstable resonant orbits are computed along with their invariant manifolds in order to examine the resonance transitions that the planar Europa Orbiter trajectory travels through. The stable manifold of a Lyapunov orbit at the L 2 libration point is used to show why a 5:6 resonance is necessary at this energy for capture around Europa.

A Dynamical Systems Analysis of Resonant Flybys: Ballistic Case

In this analysis, resonant flybys were explored within the context of the circular restricted three-body problem using dynamical systems theory. The first step in this process involved the construction of a flyby trajectory continuously transiting between 3:4 and 5:6 resonances in the Jupiter-Europa circular restricted three-body problem. An examination of this trajectory revealed that it followed the invariant manifolds of resonant orbits during these transitions. It was discovered that these transitions occurred for specific energies where the invariant manifolds of the 3:4 and 5:6 resonant orbits were closely related. The potential of the information obtained from this analysis for use in mission design was demonstrated by developing resonance transition trajectories using resonant orbit homoclinic and heteroclinic connections.

Approaching Moons from Resonance via Invariant Manifolds

In this work, the final approach typical of a trajectory traveling from the last resonance of an endgame scenario in a tour down to a moon is examined within the context of invariant manifolds. Previous analyses have usually solved this problem either by using numerical techniques or by computing a catalog of suitable trajectories. The invariant manifolds of a selected set of libration orbits and unstable resonant orbits are computed here to serve as guides for desirable approach trajectories. The analysis focuses on designing an approach phase that may be tied into the final resonance in the endgame sequence while also targeting desired conditions at the moon.

Navigation Between Geosynchronous and Lunar L1 Orbiters

Linked Autonomous Interplanetary Satellite Orbit Navigation (LiAISON) is a new technique that takes advantage of the asymmetrical gravity field present in a three-body system in order to perform absolute tracking of satellites using only relative satellite-to-satellite observations. Previous studies have demonstrated LiAISONs practical applications for lunar missions, including a satellite in a halo orbit about either the Earth-Moon L1 or L2 point. This paper studies the viability of applying LiAISON measurements between a lunar halo orbiter and a satellite in a geosynchronous orbit. Simulations demonstrate that the absolute positions and velocities of both satellites are observable using only relative measurements with an achieved uncertainty on the order of observation noise.

Preliminary Study of Geosynchronous Orbit Transfers from LEO using Invariant Manifolds

The invariant manifolds of libration point orbits (LPOs) in the Sun-Earth/Moon system are used to construct low-energy transfers from Low Earth Orbits (LEOs) to geosynchronous orbits. A maneuver is performed in LEO to insert onto a stable manifold trajectory of an LPO. The spacecraft travels to the host LPO and then follows an unstable manifold trajectory back to a geosynchronous orbit, where an orbit insertion maneuver is performed. The gravitational effects of the Sun-Earth/Moon three-body system act in such a way that large plane changes between the initial and final orbits at Earth may be realized without the execution of any plane change maneuvers. The maneuver costs of the transfers that employ invariant manifolds are compared to those using traditional techniques. The transfers that employ manifold trajectories can lower the cost of traditional Hohmann transfers by up to 3.15 km/s for transfers involving large differences in initial and final inclinations. The decrease in fuel expenditure is accompanied by an increase in time of flight; transfer durations are slightly over one year.

Transfers to Lunar Libration Orbits

This chapter focuses on the performance of low-energy transfers to lunar libration orbits and other three-body orbits in the Earth–Moon system. This chapter presents surveys of direct transfers as well as low-energy transfers to libration orbits, and provides details about how to construct a desirable transfer, be it a short-duration direct transfer or a longer-duration low-energy transfer. The work presented here uses lunar halo orbits as destinations, but any unstable three-body orbit may certainly be used in place of those example destinations. For illustration, Figs. 3-1 and 3-2 show some example direct and low-energy transfers to lunar halo orbits, respectively. One can see that these transfers are ballistic in nature: they require a standard trans-lunar injection maneuver, a few trajectory correction maneuvers, and a halo orbit insertion maneuver. One may also add Earth phasing orbits and/or lunar flybys to the trajectories, which change their performance characteristics. Many thousands of direct and low-energy trajectories are surveyed in this chapter. Table 3-1 provides a quick guide for several types of transfers that are presented here, comparing their launch energy costs, the breadth of their launch period, that is, the