representations of invariant manifolds for applications in ... · have investigated approximation...

Representations of InvariantManifolds for Applications in

Three-Body Systems

K.C. Howell,1 M. Beckman,2 C. Patterson,3 and D. Folta4

Abstract

The lunar L1 and L2 libration points have been proposed as gateways granting inexpen-sive access to interplanetary space. To date, individual transfers between three-body systemshave been determined. The methodology to solve the problem for arbitrary three-body sys-tems and entire families of orbits is currently being studied. This paper presents an initialapproach to solve the general problem for single and multiple impulse transfers. Two dif-ferent methods of representing and storing the invariant manifold data are presented. Someparticular solutions are presented for two types of transfer problems, though the emphasis ison developing the methodology for solving the general problem.

Introduction

With the increasing interest in missions involving Sun-Earth and Earth-Moonlibration points, it is necessary to further develop numerical, and possibly semi-analytical, tools to assist in trajectory design in multi-body regimes. Halo orbits arewell-known examples of periodic orbits in such regions of space. Thus far, Earth-to-halo transfers, as well as halo-to-Earth arcs, have been computed using anumber of different numerical procedures including exploitation of the invariantmanifolds associated with a particular periodic halo orbit (or quasi-periodic Lissa-jous trajectory). More recently, transfers between different three-body systems area new focus for potential mission scenarios. Developing transfers between Earthand a halo orbit or between different three-body systems involves the numericalintegration of different sets of initial conditions near a desired halo orbit manifolduntil a trajectory is identified that is most suitable for the application of interest. Of

The Journal of the Astronautical Sciences, Vol. 54, No. 1, January–March 2006, pp. 000–000

1

1Hsu Lo Professor, School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana,47907; Fellow AAS.2Flight Dynamics Engineer, Bldg. 11, Room E 121, NASA Goddard Space Flight Center, Greenbelt, Mary-land, 20771.3Graduate Student, School of Aeronautics and Astronautics, Purdue University, West Lafayette, Indiana,47907.4Senior Flight Dynamics Engineer, Bldg 11, Room S116, NASA Goddard Space Flight Center, Greenbelt,Maryland, 20771.

course, the stable/unstable invariant manifolds that correspond to a single periodichalo orbit are infinite in number, but do reside on the surface of a tube, as seen inFig. 1. Nevertheless, continuous computation of individual manifolds using numeri-cal integration is not efficient or even practical for some applications. In the analy-sis of trajectories to/from a halo orbit, for example, the size of the most usefulperiodic orbit may be unknown and its amplitude may serve as a free parameter.The design space includes not just a single tube corresponding to the invariantmanifolds of one halo orbit; rather, it becomes a volume consisting of many tubes.For the related problem of system-to-system transfers, the goal may be the inter-section of two manifold tubes–one from each system. A maneuver at an intersec-tion point will shift the vehicle from one tube to the other. In the planar CircularRestricted Three-Body Problem (CR3BP), the flow in this region of space can alsobe visualized by noting that the tube is a separatrix, i.e., it bounds different regionsof the flow [1]. Thus, transit trajectories might be sought that pass inside the tubesand shift from one tube to the next. In this case, the spatial intersection of thetubes is still the key information to complete the computations. Computing theseintersections is facilitated by efficient but accurate approximations of the tubes. So,the objective of this work is twofold: (i) representations/approximations of the in-variant manifolds associated with periodic libration point orbits that are efficientand accurate in an automated process; (ii) demonstration that the approximationcan be used to generate solutions to representative problems.

It is generally impossible to determine an analytical expression for stable and un-stable manifolds, so accurate numerical computations are required. This methodusually proceeds by using one set of initial conditions and numerically integratinga single trajectory. Increasingly, the required trajectory length is long and accuracysuffers if the step size is sufficiently small to accommodate potentially sharpchanges along the path. Hobson offers some consideration of this issue and amethod for approximating the manifold that includes an estimate of the errors [2].Hobson’s method is an improved approach to globalize the manifold in terms of

2 Howell, Beckman, Patterson, and Folta

FIG. 1. Stable Manifold Tube Approaching the Earth Associated with a Halo Orbit of Out-of-PlaneAmplitude Az � 270,000 km.

step size, and it still employs the standard first-order approximation to initiate theprocess. A single trajectory may not, however, reflect the actual complex dynami-cal behavior of the system. In fact, the problems of interest here include a completeset of manifolds associated with a periodic orbit as part of the potential solutionspace. This set is comprised of the stable or unstable manifolds that are locatedalong the surface of a tube corresponding to the periodic halo orbits in the CR3BP(Fig. 1). Either the numerical computation of some minimum number of orbits isrequired to represent the tubes or individual trajectories must be computed or re-computed as needed. If such data must be available for analysis, it may require alarge amount of storage and retrieval capability. The specific requirements dependupon the application, of course. Relatively recently, a number of mathematicianshave investigated approximation techniques. Guder et al. [3] introduce cell map-ping for the prediction of long term behavior and global analysis of nonlinear dy-namical systems. Dellnitz and Hohmann [4] use these cell mapping techniquescombined with a subdivision procedure to approximate unstable manifolds andglobal attractors. However, they are not interested in determining the full global be-havior; rather, a numerical method that allows the approximation of the global at-tractor up to a specified accuracy is detailed. A box in is specified in which thedynamical behavior is to be analyzed. The box is subdivided and boxes discardedthat do not contain part of the relative global behavior. A continuation method isfurther developed by Dellnitz and Hohmann to approximate the types of invariantmanifolds that are of interest here. This method is also called an “outer approxi-mation” of the unstable manifold. It is very useful to obtain a global view, how-ever, it does not offer a parameterization of the manifold via arc length andsmall details along the path may cause difficulty. Another technique by Krauskopfand Osinga [5, 6] is designed to specifically produce a list of points on the mani-fold by adding new points at prescribed distances from the last point and essentially“growing” the manifold. It is difficult to search this type of solution space, however,for the state information that is desired for the applications presented in this paper.

Approximating the manifold tubes for application to mission design is the firststep in this current investigation. The process must be straightforward and offer ac-curate representations of the position and velocity states. It might also be used torepresent the volume that includes multiple manifold surfaces to aid in determininga specific halo orbit for a given scenario. This process is applicable to error analy-sis, recovery, and design for contingencies. Recent studies suggest that these typesof analyses would greatly benefit from another parameter such as halo orbit size; ifthe manifold is represented in an easily accessible form, it can be accomplished ina automated manner [7, 8].

Application of these approximations to the system-to-system problem of recentinterest is also used to demonstrate the approach. As noted by Farquhar et al., [9]the libration points may become a primary hub for future human activities in theEarth’s neighborhood. The Sun-Earth L2 point is expected to become the locationof a number of large astronomical observatories. These L2 telescopes may requirehuman servicing and repair as the missions become more ambitious and, thus, morecomplex. Closer to Earth, the Earth-Moon L1 libration point is suggested as thestaging node for the missions to the Sun-Earth L2 point as well as the Moon, Mars,and the rest of the solar system. Folta et al. [10] produce various trajectories tosupport the human servicing role and offer other examples as well. But, improvedand automated schemes for computation of such designs will be necessary for

IRn

Representations of Invariant Manifolds for Applications in Three-Body Systems 3

efficiency and larger studies. Lo and Ross [11] also support the Earth-Moon L1

point as a “portal” to move beyond the Earth’s neighborhood. The manifold tubesare introduced by Lo and Ross as the basis for the design strategy to produce thetrajectories to move between these systems. Koon et al. [12] describe the conceptin more detail for the Earth-Moon to Sun-Earth problem. It has also been examinedin the Jovian system [13]. In any case, intersections may ultimately be sought be-tween many tubes from many different halo orbits in each system; the complexityforces a new look at the computations. Of course, this problem creates a very com-plicated solution space and some representation of this set of solutions is sought tofacilitate an automated design procedure. This study examines both splines and acell structure to represent the manifolds.

Manifold Approximation: Splines

To consider the intersection of a single trajectory (or a group of trajectories) witha surface that defines an invariant manifold corresponding to some halo orbit, themanifold must be generated and stored in an accessible format. One straightforwardmethod investigated is the use of splines. The existing algorithms within theMATLAB™ Spline Toolbox™ are incorporated.

The data set that corresponds to a manifold associated with one periodic haloorbit in some arbitrary system is first generated numerically using a variable steppropagator and the differential equations that model the CR3BP. Multiple individ-ual trajectories along one tube are generated to produce a good approximation tothe entire manifold surface. For this analysis, N points along the halo orbit are iden-tified and then globalized to generate N manifolds identified with one tubular sur-face corresponding to a specified periodic halo orbit. The initial state vectors arepropagated and then the states are identified farther along each trajectory, each atthe same time, resulting in vector elements that together represent a mani-fold surface. This data set is placed in a grid for later computations. The grid iscomprised of N columns and M rows (M number of integration points, each at thesame time). Spline knots are placed along each trajectory at the location of eachisolated state vector, measured in time from the initial state at the halo orbit. Thepolynomial spline function is generated from the knot sequence, andthe spline polynomial coefficients,

(1)

A bivariate manifold spline is then generated for a manifold surface correspondingto a single halo orbit. The bivariate spline function yields manifold position data fora given trajectory, �, or equivalently the phase angle or time of the originating haloorbit (a total of N trajectories), and the time from the initiation of the manifold, �(a total of � points)

(2)

This process enables calculation of position components given two parameters:number of the trajectory (or trajectory tag number) and time from the periodic orbit.The result appears in Fig. 2.

�x

y

z� � cs��, �� N

u�1 �Mv�1

pu�� pv��

j � 1�npj �x� � �k

i�1 �x � � j�k�icji,

cji

� j,pj��

N � 6


The spline function can be expanded to include a third variable— the size of theoriginating halo orbit, . The resulting trivariate spline now represents a volume inconfiguration space and may be used to search for solutions where the size of thehalo orbit is not fixed. For position

(3)

The velocity data can then be stored in a separate spline function

(4)

State information can be now obtained from the spline approximations. In thiswork, the transfers are all computed using the alternate representation (cells).

Manifold Approximations: Cells

Also investigated is a representation of the manifolds in terms of a relativelystraightforward cell structure such that each cell includes a polynomial model ofone or more manifolds. Thus, the potential set of solutions may incorporate mani-folds from various halo orbits. The cell structure then includes volume elementsthat can be searched for intersections with other single trajectories or intersectionswith other cells.

In the Sun-Earth system, consider a stable manifold tube that is computed nu-merically. With this initial surface, the manifold tubes that correspond to several ad-ditional, nearby halo orbits fill a volume in space as the individual manifoldsurfaces wrap around each other. To model any particular tube, exploit the fact thatover small regions the stable manifold of a halo orbit in the circular restricted three-body problem is nearly flat and can be represented in position and velocity by loworder functions. A volume of space through which several manifold surfaces passand that is sufficiently small to be approximated in this manner is denoted here asa cell. (See Fig. 3.)

�x

y

z� � csv��, �, � � �Z

w�1 �Nu�1

�Mv�1

pw��pu�� pv��

�x

y

z� � cs��, �, � � �Z

w�1 �Nu�1

�Mv�1

pw��pu�� pv��


FIG. 2. Position Splines.

Within a cell, position data obtained from a single stable manifold tube are ap-proximated using the fit function from Mathematica© to produce an analytical func-tion of a surface. This process is repeated with other nearby manifolds until surfaceapproximation functions for several manifolds passing through the cell are avail-able. With this same position data, velocity data are also used to generate a singlefit of each velocity component within the cell as a function of position. Thus, if Nstable halo manifolds pass through a single cell, N fits to approximate the manifoldsurfaces are produced, but only three fits are generated to approximate velocity, onefor each component.

Fitting and Cell Shapes

The velocity components are fit as third-order polynomial functions of positionin the standard rotating coordinate frame associated with the circular restrictedthree-body problem, that is

(5)

Thus, the approximation for the velocity vector is For thesurface (position) fits, which often globally exhibit significant curvature, the posi-tion data are transformed to a spherical coordinate set and the surface is defined bya fit of radial data as a function of angular coordinates as

(6)

where approximates radial distance and , � are angles that orient the radialdirection in 3D. The origin for the spherical set of coordinates is selected as theaverage center of curvature computed from every data point within the cell. Thistransformation is accomplished by defining unit vectors in the average tangential,normal, and binormal directions from the mean center of the data and placing thecell origin along the normal direction at a distance equal to the average radius ofcurvature. With a small data set, this choice encourages a cell shape that is indica-tive of the local shape of the manifold itself and a smooth surface fit with little vari-ation in r. In some areas of the tube, however, manifold trajectories tend to lie in

r

� a13 sin�� sin�� a14 sin2�� a9 sin2�� a10 sin�� a11 cos�� sin�� a12 cos�� sin�� a5 cos2�� a6 sin�� a7 cos�� sin�� a8 sin�� cos��

r�, �� a0 � a1 cos�� a2 cos2�� a3 cos�� a4 cos�� cos��

V�x, y, z� � �vx, vy, vz�.

� a14xyz � a15y2z � a16z2 � a17xz2 � a18yz2 � a19z 3

� a7y2 � a8xy 2 � a9y 3 � a10z � a11xz � a12x2z � a13yz

vx,y,z�x, y, z� � a0 � a1x � a2x2 � a3x 3 � a4y � a5 xy � a6x2y


FIG. 3. Cells are Volumes in Space Defined to Contain Small Regions of Manifold Trajectories.

the same plane as the path representing the centers of curvature for the data; for acell to contain several such trajectories, this choice will result in a large variation inr. In this case, the origin is shifted to a location along the binormal direction. Withthe data points transformed to this frame, the cell boundaries are defined as the min-imum and maximum values of r, , and �. The fits are reasonably accurate every-where except very near the Earth where the curvature of the actual manifold is high.Along the manifolds, a high curvature region requires the use of smaller cellswithin which the manifolds can be approximated as nearly flat. Also, the volume ofspace contained in a cell should not extend too far beyond the volume defined bythe manifold. Consequently, cell sizes and shapes are tailored to the local shapeof the manifold data. Then, position and velocity functions for the manifold datawithin the cells can be determined with standard fitting algorithms. This processhas been automated for examples of the type examined here.

Consider a halo orbit near the Sun-Earth L1 libration point with out-of-planeamplitude and its associated stable manifold tube as it appears inFig. 4. The data on the manifold tube are compared with fits generated from tubesassociated with halos ranging in amplitude from to

The root-mean-square (RMS) of residual errors is calculated foreach of the four fit functions in each cell. Residual RMS values for the surface fitsrepresenting this manifold are typically less than 10 km, though for some cells nearthe Earth, the values are as high as 7000 km. In Fig. 4(a), the surface fits appearwith their RMS values. The largest RMS values extend beyond the range of the dis-play but occur only in cells very near the location of the Earth, while more moder-ate errors are evident in all other cells. In Fig. 4(b) to 4(d), the RMS values for the

Az � 880,000 km.Az � 800,000 km

Az � 840,000 km


FIG. 4. SEL1 Manifold Position and Velocity Fits: Three Halos Used in Fit Range in Size fromAz � 800,000 km to Az � 880,000 km.

residuals of each of the three velocity fits in each cell are indicated. In most cells,these values are less than Again, large errors in the velocity fits occuronly in cells nearest to the Earth. The RMS values in these cells extend beyond therange of the display from to For the applications demonstratedin this study, accurate fits are required in regions of intersection between manifoldsfor transfer trajectory design, specifically intersections between manifolds associ-ated with both the Earth-Moon and the Sun-Earth systems. These intersections donot typically occur near the Earth itself, thus the high accuracy of the approxima-tions more distant from the Earth has proven sufficient for the purposes of thiswork. When it is necessary to represent manifolds near the Earth, a collection ofcells can be generated to a greater accuracy by fitting over yet smaller and smoothersets of data points in that region.

Cell Creation

Approximation of an entire volume of manifolds involves slicing that volumeinto many such cells. The creation of the cells begins with a single tube of trajec-tories, each propagated backwards to its first periapsis point. The cells cover theentire area of the tube and are kept relatively small so that the analytical approxi-mations remain accurate. Also, they are designed to overlap at their edges so thatthe fits within one cell mesh smoothly with the fits of a neighboring cell. Conse-quently, the fits within the cells collectively represent a more continuous approxi-mation of the manifold.

To create the cells, a manifold tube is sliced into two types of sections, as seenin Fig. 5. Section A is formed by truncating the tube close to the periodic orbit. Thatis, the manifold is computed for a specified length of time from the initial point nearthe halo orbit. In the first step, this time corresponds to a distance close to oneperiod of the halo orbit. This length of the manifold is denoted as a “band.” Using

1100 ms.12 ms

0.10 ms.


FIG. 5. Dividing a Manifold for Cell Creation.

the libration point as an origin or a point farther along the stable manifold associ-ated with the libration point itself, the orbit is geometrically sliced into 64 sectors(typically) and the data within each sector are used to define a new cell and frame(Fig. 6). This process is then repeated a predetermined number of times to createtwo or three additional bands. Section B is comprised of the rest of the tube. Themethod of cell creation is then modified in this second section to more closely fol-low the evolution of the manifold trajectories. A limited number of neighboring tra-jectories on the manifold are collected and denoted here as a “ribbon.” It is thenpropagated backward from the last section to periapsis and sliced evenly in distancealong its length. Each slice of data taken from the ribbon is used to define a newcell. The process is repeated using the last trajectory of the previous ribbon as thefirst trajectory of a new ribbon until the entire tube is represented. Note that beforethis step is complete, some cells may require resizing to maintain accuracy. A max-imum volume is selected and any cell larger than the maximum is sliced in halfalong its largest side. When placing data in cells it is also important that a minimumamount of data from a minimum number of trajectories always be present so thatsurface fits can be accurate. To accomplish this, manifold tubes have actually beenpropagated twice; the second time containing three times the number of trajectoriesthan the first, and with many more points. The manifolds with the sparse data are usedto create most of the cells, but when the minimum point and trajectory requirementsare not met within a cell, the more densely populated arc is incorporated.

Adding Manifold Data

After initial cell creation, the cells contain the data corresponding to merely onemanifold tube. Data from additional manifold tubes can be added with certaincaveats. The cell shapes at this point are tailored to the original tube shape and size,so the trajectories of any other manifold tube will not be fully contained within theirvolumes. Therefore, to contain as much data as possible from a nearby tube, thecells may be required to increase in size. However, the fitting process is premisedon the idea that manifold data can be approximated only over small regions, so cellgrowth is limited to encompass only the new tube data. A single cell originally con-tains all manifold data from one tube within a bounded volume. The data of a sec-ond tube is transformed to the original cell’s spherical coordinate frame, and theangular bounds of the cell are increased a small amount. Any part of the new tube’sdata that fits within the new angular bounds of the cell is considered as data for asecond surface within the cell. The bounds of the cell are then reset to reflect theincrease in the volume of the stored data. In Fig. 7 is an example of multiple SEL1


FIG. 6. Dividing a Band Into Sectors.

manifold tubes that have been placed in the same set of cells. They range in sizefrom to Originally, only three tubes areplaced in the cells in Fig. 7, however, vacant areas appear. In these areas, the ve-locity fits will be less accurate. This problem can be solved by adding more mani-fold tubes without allowing cell growth, but maintaining the minimum number ofpoints and trajectories. (Fig. 7 includes five manifolds through the cells.) If, in asingle cell, the minima are not met, then the data for that surface do not get addedto the cell. So some cells may contain more manifold tubes from a wider range ofhalos than other cells. It is also possible to approximate the wider range of halos bystacking cells, essentially implementing the cell creating process for the manifoldsof one set of halo orbits, then creating entirely new cells for a larger or smaller setand repeating.

Example: Using Cells to Find Halo Transfers in the CR3BP

With approximations for Sun-Earth L1 stable manifolds within the cells, theprocess of computing transfers to halo orbits is simplified. For any spacecraft tra-jectory that intersects a cell at a point the velocity is labeled where bold de-notes a vector. Assuming a manifold passes through the intersection point, theapproximation for manifold velocity at that point is determined and is labelled

where the velocity fits are used to estimate each component of velocity andcombined to yield the vector approximation. Thus, an initial guess for a maneuverthat would be necessary to complete a transfer is simply It isassumed that the continuous distribution of periodic orbits in a family yields man-ifolds that exist on a continuum of surfaces passing through the cells, and thatvelocity on those surfaces varies continuously throughout the volume. Analyticalapproximations for only a few of the surfaces of position are planned to be avail-able, but a single velocity approximation is continuous over the entire cell volume.So, even if lies between or outside the known manifold surfaces within the cells,it is still identified with some member of the manifold continuum, thus, the veloc-ity fit can be applied and used as an initial guess for the velocity on a manifold tra-jectory passing through that point. Moreover, it is not necessary that a particularhalo orbit be specified in advance. Indeed, if the best intersection point within the

rint

Vcell � v�rint� � vint.

v�rint�,

vint,rint,

Az � 976,000 km.Az � 800,000 km


FIG. 7. Five Tubes Through a Set of Cells.

cell is to be selected from among all possible points in the volume, and that bestpoint is not known a priori, then any halo orbit in or near the range of those usedto initially generate the manifolds and create the cells may serve as the ideal targethalo orbit for the transfer. So, rather than selecting a specific halo orbit and at-tempting to locate intersections with its stable manifold, an easier approach may bethe determination of the lowest-cost intersection in a volume of stable manifolds as-sociated with a set of periodic orbits. The location of the ideal intersection pointcould then be used to determine a target orbit for the transfer.

The Use of Surface Fits in Determining a Destination Halo Orbit. An intersec-tion point inside a cell has spherical coordinates This point may be in-side, outside, or somewhere between the surfaces defined within the cell thatcorrespond to known manifold tubes as is apparent in Fig. 8. To discover its loca-tion relative to the existing surface fits, the surface fit functions, are eval-uated where i ranges from 1 to n, the number of surfaces within the cell. Eachsurface, and thus each value of is associated with a specific halo. Thesehalo orbits are characterized by their initial conditions. For example,

is the initial state of the i-th halo orbit at the x-z plane crossing.Each such initial state is fit as a quadratic function of directly associatingthe halo orbits to the sizes of the manifold tubes at within the cell. This pro-cedure yields functions that can be solved for a set of initial conditions of a desti-nation halo orbit as a function of the location r within the cell, that is

�, ��ri�, ��,

�x0i , 0, z0

i , 0, y0i, 0�

ri�, ��,

ri�, ��,

�r, , ��.rint


FIG. 8. Locating the Intersection Point Within a Cell.

(7)

The coefficients can be determined by solving the three matrix equations

(8a,b)

(8c)

If the number of surfaces this is accomplished with a least squares fit. Ifthen a less accurate linear fit must suffice. However, an effort has been made

to include a minimum of three manifold surfaces in every cell.Approaching the Destination Halo Orbit. Approximations from the fit functions

are generally most successful when applied near or within the bounds of the regionwhere the data are available. This region is defined by the surfaces within the cell.Thus, the surfaces nearest and farthest from the cell origin define the region. Thedistance between these two surfaces at any point in the cell defines a width,

and any trajectory intersection point that is more than a distance w fromthe surfaces is ignored as a potential transfer point for the given cell (but, it wouldre-emerge in another cell). These low order velocity fits within the cells are gener-ally sufficient to produce a trajectory arc from a point within the cell to the vicin-ity of the libration point orbit, but it is likely that it will not directly insert into aperiodic orbit. The sensitivity of the problem demands that some amount of target-ing be used to complete the transfer. However, with the approximation, an addi-tional automated step results in successful transfers to halo orbits in the circularrestricted problem.

Assume that an initial trajectory originates from somewhere in or beyond thecurrent three-body system. There is no restriction on the source of the incoming leg.Assuming that this path will intersect one or more cells, they are identified and allintersecting points within the cell are tagged. For each intersection point, the ve-locity discontinuity is calculated, that is, seeking the location

that minimizes The calculated is then applied. The point of closestapproach to the halo is then numerically determined. Since is an approxima-tion, the transfer arc will not approach the periodic orbit asymptotically. A haloorbit insertion (HOI) point is selected along the halo orbit and targeted from This results in an additional velocity adjustment, at the intersection pointin the cell. Finally, a maneuver, is included at the HOI point. This completesthe transfer with a total cost This cost is afunction of the HOI location. For convenience, the MATLAB™ OptimizationToolbox™ function fminbnd can be utilized to determine cost as a function of theHOI location and identify a minimum.

Numerical values for a test case appear in Table 1 and Fig. 9. Three halos atand are used to generate

manifold cell volumes, and a fourth stable manifold to a halo orbit of amplitudeone that is not incorporated in the fit process, is used to test theAz � 945,000 km,

Az � 880,000 kmAz � 840,000 km,Az � 800,000 km,

C � Vcell � Vtargeter � VHOI. VHOI,

Vtargeter,rint.

Vcell

Vcell Vcell.rint

Vcell � v�rint� � vint

rint

w�, ��,�, ��

n � 3,n � 3,

�1

..

1

r1

..

r3

r12

..

r32��ay0

ay1

ay2

� � �y01

..

y0n�

�1

..

1

r1

..

r3

r12

..

r32��az0

az1

az2

� � �z01

..

z0n��1

..

1

r1

..

rn

r12

..

rn2��ax0

ax1

ax2

� � �x01

..

x0n�,

y0 � ay0 � ay1r � ay2r2

z0 � az0 � az1r � az2r 2

x0 � ax0 � ax1r � ax2r 2


transfer procedure. A point along this actual stable manifold is perturbed with a value equal to at a point within a cell. At this precise intersection point, theapproximations within the cells are employed to estimate the manifold velocity atthis point, approximate the appropriate and determine the specific halo thatwould minimize the cost of a transfer. The actual manifold should be the best solu-tion and the results of the approximations appear in the table. A good estimate ofthe destination halo is achieved and the transfer is accomplished with a cost verysimilar to the perturbation.

Alternative to Direct Halo Orbit Insertion. In some examples, the HOI selectionphase of the process yields results that reflect the sensitivity of the problem and thefact that the space is rich with solutions. For some cases, the optimization proce-dure successfully and accurately targets the HOI points, but the magnitude of theresulting is high despite a reasonably close and inexpensive initial guess fora halo orbit approach arc. The resulting transfer arc may also bear little resemblanceto the initial guess for the approach arc. This type of result is not surprising with alinear targeting algorithm in a nonlinear and sensitive regime. The process can thenbenefit from an alternate insertion scheme. An algorithm is implemented that isbased on a multi-point targeting method employed by Howell and Pernicka14 aswell as Howell and Gordon.15 The procedure was previously exploited for station-keeping via impulsive maneuvers to maintain a vehicle in the vicinity of a haloorbit; thus, for this application, it requires that the approaching trajectory pass rea-sonably close to the halo orbit at some point. At such a point near the halo, a

Vtargeter

20 ms

V

20 ms V


FIG. 9. Transfer Determined Using Cells.

TABLE 1. Targeting a Halo Orbit

Destination Total atHalo Orbit Intersection,

Exact Manifold 945,481 20.00 —Intersection

Approximation Result 945,176 19.84 1.21at Same Intersection Point

VHOI, msmsAz, km V

single maneuver, alters the spacecraft path to track the halo orbit although itmay not, in fact, insert directly into the reference orbit. Observe that this iscomputed based on the predicted deviations of the current path relative to the ref-erence at three future times. This replaces the in the original formula-tion. Note again that the spacecraft is not delivered precisely into the destinationorbit with this maneuver, however, the resulting path remains close to the halo orbit.The use of this as an estimate for an insertion maneuver is sufficient to bringthe solution to a higher-order model.

System-to-System Transfers

An even more challenging problem is the system-to-system transfer. In particu-lar, for demonstration here, the transfer from orbits near EML1 or EML2 to orbitsnear SEL1 or SEL2 is considered. As noted previously, similar types of transfersare the focus of a number of researchers.9–12,16,17 In this work, an efficient method ofcomputation is sought using approximations to the manifolds. The ultimate objec-tive is to use the approximation to produce the transfers in full, 3-D ephemeris mod-els that may include other perturbations as well. This problem is consideredconceptually in terms of splines; detailed computations are accomplished with thecell structures.

Using Splines to Determine the Intersections of Earth-Moon and Sun-Earth Tubes

Using the manifold position spline functions, intersections can be computed be-tween invariant manifolds from different three-body systems. Assuming that thegoal is to shift directly from one surface to another, these intersections result inone-impulse transfers between the three-body systems. Such would be the case ifit was desired to depart an EML2 orbit and reach a SEL1 halo orbit, for example.Figure 10 includes an example of a large SEL1 stable manifold tube and variousEML2 unstable manifold tubes at different phases of the Moon. From this figure,it is apparent that the tubes intersect. Using a Moon phase of 210°, an intersectionbetween the manifolds is assured at greater than one point. The MATLAB™ Opti-mization Toolbox™ is employed to determine the intersections. The MATLAB™function fmincon produces the minimum of a constrained nonlinear multivariatefunction. The solution yields a position on each manifold, which differ by only 43 km

V*

VHOI V*

V* V*,


FIG. 10. Intersections Between Tubes.

for this example. Repeated uses of fmincon for numerous values of � for one man-ifold will result in the entire intersection ring. An actual ring of intersection is suc-cessfully computed and is apparent in Fig. 11. Once the intersection is determinedin configuration space, the velocity differences are also available, and the cost canbe computed. Figures 10 and 11 illustrate this concept quite well in three dimen-sions. But, the problem can be extrapolated to manifold volumes as well.

Using Cells to Approximate a Transfer and Shift into the Ephemeris Model

The same Earth-Moon to Sun-Earth system-to-system transfer is also computedwith complete numerical results using the cell structure. In this case, the problemis solved in two steps. First, an approximate solution is obtained in the circular re-stricted problem using the cells. Then, with the software package Generator devel-oped at Purdue University, this approximation produces an end-to-end trajectoryfrom an Earth-Moon libration point orbit to a Sun-Earth libration point orbit in anephemeris four-body model simultaneously including the Sun, Earth, and Moon. Itis noted that other researchers are also seeking transfers of this type in the fullephemeris model with alternate approaches.10–12,16,17

In the circular restricted Earth-Moon system, unstable manifolds for an L1 haloorbit of amplitude are propagated. These manifolds form a tube,of course. Any of the trajectories can be transformed to Sun-Earth coordinatesand intersections are determined with cells that correspond to stable manifoldsassociated with the SEL2 halo orbits. A set of cells is generated for SEL2 stablemanifolds representing the halo orbits ranging from to

as seen in Fig. 12. These cells contain position and velocity fitsfor the data from only three tubes. Initially, the Moon is 45° out-of-phase with theSun-Earth system. It is also noted that the Earth-Moon frame is rotated aboutthe Sun-Earth x-axis to more accurately represent the inclination of the Moon’sorbit. In Fig. 12, all EML1 manifold trajectories are overlaid with the SEL2 cellboundaries. The estimated total cost for the “best” transfer is as deter-mined in the circular restricted problem and noted in Table 2. The transfer appearsin Fig. 13 in both the circular restricted Earth-Moon view (Fig. 13(a)) and the cir-cular restricted Sun-Earth view (Fig. 13(b)). In the Earth-Moon view, it is observedthat the manifold that departs from the vicinity of L1 encircles L2 as it departs thesystem. This characteristic is indicative of an L2 transit trajectory.

24.52 ms,

�5�

Az � 220,000 kmAz � 170,000 km

Az � 39,000 km


FIG. 11. Ring of Intersection.


FIG. 12. EML1 Paths Intersect SEL2 Cells.

TABLE 2. Solution for Transfer to Halo Orbit

Destination Total atHalo Orbit Intersection,

Approximation Result 123,265 km 21.146 3.372at Best Intersection Point

VHOI, msmsAz, km V

FIG. 13. Transfer from Earth-Moon L1 Halo Orbit to a Sun-Earth L2 Halo Orbit.

Recall that the is calculated in the Sun-Earth frame with no regard for the dy-namical effects of the Moon. This solution is then transferred into the four-bodymodel with a complete ephemeris formulation. The Generator software incorpo-rates an RK 8/9 integrator; a gravity model including the ephemeris states for thecelestial bodies, but no solar radiation pressure force, is incorporated. The first arcalong the transfer, as viewed in the Earth-Moon frame, appears the same as that inFig. 13(a). The Sun-Earth view is plotted in Fig. 14 and some differences relativeto the solution in the circular problem (Fig. 13(b)) now appear. The cost at the in-tersection point in the ephemeris model is there is no longer any libra-tion orbit insertion maneuver, i.e.,

Finally, patch points are automatically introduced into Generator (as marked inFig. 15) and all legs are simultaneously corrected to yield a zero cost solution. Ofcourse, the final libration point orbits are not periodic but are quasi-periodic Lis-sajous trajectories. The final size of the halo orbits in the full model in the Earth-Moon system corresponds to in the Sun-Earth system, theout-of-plane amplitude is In the Sun-Earth view, the final zerocost solution that is determined in the ephemeris model appears in Fig. 16. Notethat it is very similar to the plot in Fig. 14, obtained before the cost reductionprocess. It is interesting to compare the Earth-Moon departure path from the ap-proximation in the circular problem to the final ephemeris result corresponding tothe no cost result. This comparison appears in Fig. 17. The general characteristics

Az � 163,000 km.Az � 39,600 km;

VHOI � 0.15.27 ms;

V


FIG. 14. Transfer in the Sun-Earth Frame: Ephemeris Model.

are maintained. As this example is reviewed, observe that every step of the processyields a libration point orbit of different size. This flexibility appears to aid thesearch for a solution.

Additional Examples

A number of sample transfers have been computed that originate at quasi-periodicLissajous orbits near libration points in the Earth-Moon system and arrive at librationpoint orbits (LPO’s) in the vicinity of the Sun-Earth L2 point. Earth-Moon manifolds


FIG. 15. Patch Points in the Cost Reduction Algorithm.

FIG. 16. Final Solution from an Earth-Moon L1 Lissajous Trajectory to a Sun-Earth L2 LissajousTrajectory in the Full Model (Zero Cost).

(typically in the four-body model) are computed and transformed to the Sun-Earthrotating frame. Initial estimates for the transfer arcs, computed in the Sun-Earth circu-lar restricted three-body system, employ the cell approximations of the manifolds.Thus, the initial transfer is computed by patching together a manifold in an ephe-meris four-body model with a transfer in a CR3BP model. Initial approximations tothe transfers also employ a two-burn scheme, i.e., Themulti-point targeting routine is employed in a number of cases, as indicated. All re-sults are ultimately transitioned to the full four-body ephemeris model and, whereindicated, the cost is reduced to a minimal value. Since precisely periodic librationpoint orbits do not exist in the ephemeris model, all orbits indicated in theephemeris model are actually quasi-periodic Lissajous trajectories computed usingperiodic halo orbits from the circular restricted model as first approximations. Theamplitudes listed for these quasi-periodic “halo-like” orbits are averages valuesover four revolutions; though the “halo-like” Lissajous trajectories generally pos-sess only small variations in amplitude over time. The cost reduction procedure em-ploys a differential corrections scheme that necessarily alters the path of thetrajectory in the search for a lower cost solution. For large reductions in cost, vari-ations in amplitudes of quasi-periodic Lissajous trajectories can become substan-tial. In these cases, the amplitudes are noted only as “more variable.”

Departure from a Northern EML2 Libration Point Orbit. Table 3 includes trans-fers from northern EML2 libration point orbits that are designed to reach LPO’snear SEL2. Originating in EML2 orbits ranging in size from to

the initial estimate for the total cost is observed to range from 70to All of these solutions are reduced to free transfers in the full ephemerismodel after a cost reduction process. Case 3c appears in Fig. 18 with both a viewof the Earth-Moon and the Sun-Earth rotating frames.

Departure from a Southern EML2 Libration Point Orbit. For comparison,two similar transfers are computed from southern orbits near EML2, and the resultsappear in Table 4. Although the initial approximation is somewhat higher in cost,

89 ms.Az � 40,000 km,

Az � 10,000 km

Vcell � Vtargeter � VHOI.


FIG. 17. Comparison of the Circular Approximation to the Departure Trajectory from the Earth-MoonSystem with the Final Ephemeris Solution.

free transfers between the Earth-Moon and Sun-Earth libration point orbits aredetermined.

Variation in the Start Time. The transfer identified as Case 3 in Table 3 is initi-ated with a start date of 01/08/2010. This marks the initial date along the Lissajousorbit in the Earth-Moon system, i.e., several revolutions before departure along theunstable manifold. This starting time can be varied to improve the cost and so threeother transfers are computed for spacecraft either lagging or leading the vehicle as-sociated with that in Case 3. The resulting transfers appear in Table 5. Smallchanges in the initial start time of the EML2 orbit will produce considerablechanges in the geometry of manifold intersections in the Sun-Earth frame, leadingto significantly larger or smaller costs for transfers. The procedure for reducing thecost of the transfer trajectories demonstrates considerable robustness in its abilityto minimize very expensive transfers such as those in Cases 7 and 8, although thisis accomplished at the expense of a large alteration in the characteristics of the ini-tial and final LPO’s. For example, the original design of the transfer in Case 7


FIG. 18. Case 3c, Free Transfer From an Orbit Near EML2 to an Orbit Near SEL2.

TABLE 3. Comparing Transfers from Northern EML2 LPO’s of Various Amplitudes

Earth-Moon LPO Final Sun-Earth Total atCase and Model Used LPO Intersection,

1a. Eph. & CR3BP 40,000 127,000 69.90 2.351b. Full Ephemeris 120,000 70.00 2.381c. with Cost Reduction More variable 114,000 0 0




VHOI, msmsAz, kmAz, km V

departes an Earth-Moon LPO of but when the cost was reducedto generate a free transfer, the shape of the originating orbit was altered so thatthe Az amplitude of the Earth-Moon departure Lissajous trajectory ranges from38,000 km to 47,000 km.

Transfers from EML1. Table 6 contains data on two transfers from the unstablemanifold of an LPO near EML1 with Any unstable manifold tubedeparts in two directions. In Case 10, the manifold tube is propagated toward theEarth. This directs the spacecraft to the interior of the Earth-Moon system (andaway from the Moon) and, thus, does not immediately take advantage of theL1-Moon-L2 corridor. Without this apparent advantage, a large maneuver is thennecessary to exit the Earth-Moon system along Sun-Earth manifolds. This transferappears in Fig. 19.

In Case 11, the manifolds are propagated towards the Moon, rather than towardthe Earth. The transfer representing Case 11, as computed in the ephemeris modelappears in Fig. 20. Note the full revolution of the Moon before transiting EML2 forescape in the Earth-Moon system.

Before full cost reduction is attempted, adjustments to the initial start date canalter the geometry and modify the cell intersections creating lower-cost options.Some results are displayed in Table 7. Various solutions at both lower and highercosts are computed and all are transitioned to the full ephemeris model. In eachcase, the best option involves a departure from an EML1 orbit, a revolution around

Az � 15,000 km.

Az � 15,000 km,


TABLE 4. Comparing Transfers from Southern EML2 LPO’s

Earth-Moon LPO Final Sun-Earth Total atCase and Model Used LPO Intersection,



VHOI, msmsAz, kmAz, km V

TABLE 5. Varying Start Date of Transfer in Case 3

Variation in Start Final Sun-Earth Total atCase and Model Used Time, days LPO Intersection,

7a. Eph. & CR3BP �6 127,000 161.36 1.337b. Full Ephemeris 127,000 164.95 5.007c. with Cost Reduction 111,000 0 0

8a. Eph. & CR3BP �3 128,000 187.36 12.368b. Full Ephemeris 129,000 172.29 14.998c. with Cost Reduction 92,000– 0 0

148,000

9a. Eph. & CR3BP �3 125,000 40.11 1.609b. Full Ephemeris 118,000 34.82 2.479c. with Cost Reduction 130,000 0 0

VHOI, msmsAz, km V

the Moon, and then transiting past EML2 to escape the Earth-Moon system beforethe intersection with the Sun-Earth manifold.

Reducing the cost in Case 16 (and others like it) with the differential correctionsprocess presents challenges. First, the orbit of the spacecraft around the Moon isparticularly sensitive to disturbances and, second, the spacecraft must be con-strained not to impact the Moon itself. The constraint can be incorporated, but thislimits the available solution space. A solution exists whereby the lunar orbit is


TABLE 6. Transfers From an EML1 LPO

Departure Final Sun-Earth Total atCase and Model Used Direction LPO Intersection,

10a. Eph. & CR3BP Toward Earth 147,000 433.68 8.0710b. Full Ephemeris 148,000 439.04 8.4910c. with Cost Reduction 100,000 320.00 8.50

11a. Eph. & CR3BP Toward Moon 124,000 131.26 1.7611b. Full Ephemeris 124,000 126.99 3.00

VHOI, msmsAz, km V

FIG. 19. Case 10b, Departing an Orbit Near EML1 Toward Earth Before Boosting to anOrbit Near SEL2.

FIG. 20. Case 11, Intersecting the Sun-Earth Manifold After Lunar Flyby.

bypassed entirely to achieve a free transfer. In Fig. 21, the original ephemeris solu-tion in Case 16b is plotted in the Earth-Moon frame, along with the new arc thatbypasses the lunar revolution. The lunar revolution is replaced with two new ma-neuvers (as indicated in Fig. 21), but these can be removed in a later step. In the fig-ure, and accomplish the bypass where marks the intersection with theSun-Earth manifold (in a cell). The reduced cost solution undergoes significant cor-rection to remove all maneuvers. The final result appears in Fig. 22. The Sun-Earthlibration point orbit does increase in size (as indicated in Table 7).

v3 v2 v1


TABLE 7. Varying Transfer Start Date in Case 11

Variation in Final Sun-Earth Total atCase and Model Used Start Time, days LPO Intersection,

12a. Eph. & CR3BP �6 130,000 529.37 15.0112b. Full Ephemeris 134,000 529.31 16.00

13a. Eph. & CR3BP �9 104,000 1634.39 147.1213b. Full Ephemeris 102,000 1634.84 140.83

14a. Eph. & CR3BP �11.75 172,000 131.89 0.3414b. Full Ephemeris 186,000 140.00 2.00

15a. Eph. & CR3BP �12.75 137,000 107.65 0.1515b. Full Ephemeris 148,000 104.89 2.00

16a. Eph. & CR3BP �13.75 125,000 101.48 2.4816b. Full Ephemeris 129,000 88.50 4.9916c. with Cost Reduction 155,000 0 0

VHOI, msmsAz, km V

FIG. 21. Bypassing a Lunar Revolution to Facilitate Cost Reduction in Case 16.

Summary

The objective of this work is the development of approximations for stable andunstable manifolds and the use of these representations in mission design applica-tions. Besides numerical integration, a number of different approaches have beenexplored by other researchers to approximate stable and unstable manifolds. Theapplications here necessitate a formulation that offers a parameterization that canbe exploited for targeting and design. Both splines and cell structures are investi-gated and prove successful in different types of computations.

The system-to-system transfers are a challenging application of the approxima-tion tool. The immediate goal for extension of this work is generalization for thesystem-to-system application. Further developments are planned that include awider range of applications as well.

Acknowledgments

Portions of this work were supported by NASA and the Goddard Mission Services EvolutionCenter under contract numbers NCC5-358 and NAG5-11839.

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FIG. 22. The Zero Cost Transfer in Case 16.

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