Set Oriented Approximation of Invariant Manifolds: Review of Concepts for Astrodynamical Problems

Abstract

During the last decade set oriented methods have been developed for the approximation and analysis of complicated dynamical behavior. These techniques do not only allow the computation of invariant sets such as attractors or invariant manifolds. Also statistical quantities of the dynamics such as invariant measures, transition probabilities, or (finite-time) Lyapunov exponents, can be efficiently approximated. All these techniques have natural applications in the numerical treatment of problems in astrodynamics. In this contribution we will give an overview of the set oriented numerical methods and how they are successfully used for the solution of astrodynamical tasks. For the demonstration of our results we consider the (planar) circular restricted three body problem. In particular, we approximate invariant manifolds of periodic orbits about the L 1 and L 2 equilibrium points and show an extension to the application of a continuous control force. Moreover, we demonstrate that expansion rates (finite-time Lyapunov exponents), which so far have mainly been applied in fluid dynamics, can provide useful information on the qualitative behavior of trajectories in the context of astrodynamics. The set oriented numerical methods and their application to astrodynamical problems discussed in this contribution serve as further important steps towards understanding the pathways of comets or asteroids and the design of energy-efficient trajectories for spacecraft.

Combined low-thrust propulsion and invariant manifold trajectories to capture NEOs in the Sun–Earth circular restricted three-body problem

“…Recently, some works present a method to derive optimal low-thrust orbits in n-body dynamical frameworks using invariant manifolds as first guess solutions for low-thrust trajectories (Anderson and Lo 2009), later on optimised with sophisticated algorithms (Whiffen and Sims 2002). Moreover, low-thrust propulsion has been used within the restricted three-body problem to design interplanetary transfers (Dellnitz et al 2006(Dellnitz et al , 2007Mingotti and Gurfil 2010;Mingotti et al 2011a), transfers to the Moon (Mingotti et al 2007(Mingotti et al , 2009aMingotti 2010;Mingotti and Topputo 2011;Pierson and Kluever 1994;Kluever and Pierson 1995;Herman and Conway 1998;Yang 2007) and transfers to LPOs (both in the Sun-Earth and Earth-Moon systems) (Starchville 1997;Topputo 2007;Mingotti et al 2007;Ozimek and Howell 2010;Martin et al 2010;Sukhanov and Eismont 2002;Senent et al 2005) The latter take advantage of the three-body dynamics, where lowthrust orbits have to target a point belonging to the stable manifold associated to the final orbit.…”

Earth–Mars transfers with ballistic escape and low-thrust capture

“…(11) is not given like in [22,23], but rather in this approach it represents an unknown that is found by solving an optimal control problem (t i , t f are the initial, final times, respectively). T is determined in such a way that a certain state is targeted and a certain objective function is minimized at the same time.…”

Eight-shaped Lissajous orbits in the Earth-Moon system

“…In [2], a stability technique based on local Lyapunov exponents is applied for maneuver design and navigation in the three-body problem. It is shown in [9] that finite-time Lyapunov exponents can provide useful information on the qualitative behavior of trajectories in the context of astrodynamics. Local Lyapunov exponents are used to determine the behavior of nearby trajectories in finite time.…”