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Acta Astronautica

Acta Astronautica 96 (2014) 2341

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journal homepage: www.elsevier.com/locate/actaastro

Representations of higher-dimensional Poincar maps withapplications to spacecraft trajectory design$

Amanda F. Haapala n, Kathleen C. HowellSchool of Aeronautics and Astronautics, Purdue University, 701 W. Stadium Ave., West Lafayette, IN 47907-2045, USA

a r t i c l e i n f o

Article history:Received 25 May 2013Received in revised form11 November 2013Accepted 16 November 2013Available online 25 November 2013

Keywords:Circular restricted three-body problemPoincar mapsMulti-body dynamicsInvariant manifoldHeteroclinic

65/$ - see front matter & 2013 IAA. Publishex.doi.org/10.1016/j.actaastro.2013.11.019

paper was presented during the 63rd IAC iesponding author. Tel.: 1 765 494 5117; faail addresses: ahaapala@purdue.edu (A.F. Hapurdue.edu (K.C. Howell).

a b s t r a c t

The Circular Restricted Three-Body Problem (CR3BP) offers a useful framework forpreliminary trajectory design in multi-body regimes. While the CR3BP offers a simplifiedmodel with useful symmetries, trajectory design in this dynamical regime is oftennontrivial. It is essential to gain insight into the available solution space to generatetrajectories that meet a variety of constraints. The Poincar map is a powerful tool that, incombination with a constraint on the energy level, allows a reduction in dimension suchthat, for the planar problem, the system is reduced to two-dimensions and the phasespace is fully represented by the projection onto a plane. In the spatial problem, however,Poincar maps must represent at least four states and are therefore challenging tovisualize. In this investigation, a method to represent the information contained in higher-dimensional Poincar maps using a two-dimensional image for visualization is exploredand is applied for trajectory design. Four-dimensional map representations are demon-strated to compute transfers, including heteroclinic and homoclinic connections, betweenperiodic and quasi-periodic orbits in the spatial problem. Alternative maps, such as theperiapse Poincar map, require that at least five Cartesian states be represented. Maprepresentations are generated to visualize the full state associated with crossings of theperiapse map, and are employed to locate families of periodic orbits about the Moon, aswell as transfers to these orbits via transit trajectories.

& 2013 IAA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

The Circular Restricted Three-Body Problem (CR3BP)serves as a useful framework for preliminary trajectorydesign in a multi-body force environment. Libration pointorbits are well known solutions that have been employedin numerous missions. Missions to libration point orbits inthe SunEarth system include observatories in the vicinityof L1 (ISEE-3 [33], SOHO [35], ACE [34], WIND [34], Genesis[34]), and near L2 (WMAP [36] and the Herschel andPlanck Space Observatories [13]). ARTEMIS was the firstlibration point mission in the EarthMoon system; two

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n Naples.x: 1 765 494 0307.apala),

spacecraft were maintained in large quasi-periodic orbitsabout the L1 and L2 points before entering long-term lunarorbits [16]. While the existence of periodic and quasi-periodic solutions near the libration points in the CR3BP isgenerally well understood, the design of trajectories thatincorporate these orbits is nontrivial. To generate trajec-tories that meet a variety of constraints, tools that offerinsight into the available solution space are essential. ThePoincar map is a powerful tool that is useful to evaluatethe solution space and to compute trajectories withspecified characteristics. In combination with a constrainton the energy level, Poincar maps allow a reduction indimension such that, for the planar problem, the system isreduced to two dimensions and the state space is fullyrepresented via a planar projection. In the spatial problem,Poincar maps must represent at least four state variables.In this investigation, methods to represent the available

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A.F. Haapala, K.C. Howell / Acta Astronautica 96 (2014) 234124

information in higher-dimensional Poincar maps areexplored and are applied for trajectory design.

Previous researchers have employed a variety of tech-niques to facilitate trajectory design in the spatial problem.To examine behavior in the vicinity of the collinearlibration points, Jorba and Masdemont [28], as well asGmez et al. [19], obtain higher-order normal form expan-sions of the Hamiltonian in the vicinity of the equilibriumpoints to decouple the stable and unstable motion. Remov-ing unstable behavior via a reduction to the center mani-fold yields a system with two degrees of freedom; thus,maps associated with periodic and quasi-periodic orbits inthe spatial problem for a particular energy level are two-dimensional. Gmez et al. [18] compute the stable/unstable manifold asymptotic to the center manifoldassociated with the collinear points for a particular energylevel and employ a Poincar map to reduce the dimensionof the problem. By selecting two additional parameters,the four-dimensional maps are reduced to two-dimen-sions. Thus, the state space is fully represented by project-ing the map onto a plane, and the problem of locatingtrajectories with prescribed characteristics is more tract-able. Paskowitz and Scheeres [38] employ periapse mapsto classify trajectory behavior in the spatial problem; theseauthors represent higher-dimensional maps by plottingvectors such that the basepoint represents the three-dimensional position state and the magnitude and direc-tion of the vector indicate the velocity at periapsis.

The problem of representing multivariate data setsgraphically is treated extensively in the field of datavisualization. Strategies to represent complicated andinterconnected information support the exploration ofhigher-dimensional data sets by exploiting the humanability to perceive and recognize patterns. While algo-rithms may be employed for data analysis, the develop-ment of such algorithms often requires a priori knowledgeconcerning the solutions of interest. Exploiting the cap-abilities of human perception allows the potential to revealnew or unexpected solutions. From a trajectory designperspective, a visual representation of the data allows thedesigner to both develop intuition about the solutionspace and to recall trends and conclusions. Visual repre-sentations for Poincar maps additionally prove useful tofacilitate an interactive trajectory design approach, basedon user interaction with visual tools. The field of datavisualization offers many strategies to aid in visuallyrepresenting multivariate data sets. Parallel coordinates[27], mosaic plots [23], scatterplot matrices, and trellisdisplays [4] are examples of modifications to the conven-tional two-axis plot that enable visualization of multi-variate data. An alternative approach offered from datavisualization is the use of glyphs, i.e., graphical objectswhose physical characteristics are determined by a dataset. A variety of glyph definitions have been previouslydemonstrated, and are discussed by Ward [46].

Glyphs prove useful for the development of a visualrepresentation for higher-dimensional Poincar maps.These map representations provide a wealth of insightinto the solution space, and facilitate the location ofa variety of new solutions. In this investigation, the useof glyphs is demonstrated to visualize four or more state

variables associated with the crossings of a Poincar map,and the resulting map representations are demonstratedfor several mission design scenarios, including the searchfor maneuver-free transfers between libration point orbitsin the spatial problem in both the EarthMoon and SunEarth systems.

Poincar maps are also useful to locate solutions suchas transit and non-transit trajectories [8,18,30] by propa-gating a batch of initial conditions and examining subse-quent returns to the map. The periapse Poincar map hasbeen employed by several researchers to investigate thedynamics in the vicinity of the smaller primary in boththe planar and spatial problems [9,10,21,22,26,38,45]. Inthe planar problem, the projection of the periapse mapinto configuration space provides a complete and intuitiverepresentation of the state space. Such maps offer insightinto the dynamics facilitating temporary-capture andescape in the vicinity of the smaller primary. To fullyrepresent the periapse map in terms of Cartesian coordi-nates in the spatial problem, at least five states must berepresented. Sample periapse map representations aregenerated to display the full six-dimensional Cartesianstate space, and these maps are employed to examineregions of transit, as well as regions of long-term capture.Periodic orbits about the Moon are located by visualinspection of the structures associated with nearby cap-ture orbits on the map, and are used to initialize continua-tion schemes to compute families of periodic orbits. Byobserving map crossings associated with transit trajec-tories in the vicinity of the periodic