Rendezvous Missions toTemporarily-Captured Near Earth Asteroids

S. Brelsford1, M. Chyba1, T.Haberkorn2, and G.Patterson1

1Department of Mathematics, University of Hawaii, Honolulu,Hawaii, 96822

2MAPMO-Fédération Denis Poisson, University of Orléans,45067 Orléans, France

October 2, 2018

Abstract

Missions to rendezvous with or capture an asteroid present signifi-cant interest both from a geophysical and safety point of view. Theyare key to the understanding of our solar system are as well step-ping stones for interplanetary human flight. In this paper, we focuson a rendezvous mission with 2006RH120 , an asteroid classified as aTemporarily Captured Orbiter (TCO). TCOs form a new populationof near Earth objects presenting many advantages toward that goal.Prior to the mission, we consider the spacecraft hibernating on a Haloorbit around the Earth-Moon’s L2 libration point. The objective isto design a transfer for the spacecraft from the parking orbit to ren-dezvous with 2006RH120 while minimizing the fuel consumption. Ourtransfers use indirect methods, based on the Pontryagin MaximumPrinciple, combined with continuation techniques and a direct methodto address the sensitivity of the initialization. We demonstrate thata rendezvous mission with 2006RH120 can be accomplished with lowdelta-v. This exploratory work can be seen as a first step to identifygood candidates for a rendezvous on a given TCO trajectory.

Keywords: Temporarily Captured Objects, Three-body Problem, Opti-mal control, Indirect Numerical Methods

MSC2010: 49M05, 70F07, 93C10

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1 IntroductionThe motivation for our work is to study asteroid capture missions for a spe-cific population of near Earth objects. The targets, Temporarily CapturedOrbiters (TCO), are small asteroids that become temporarily-captured ongeocentric orbits in the Earth-Moon system. They are characterized as sat-isfying the following constraints:

• the planetocentric Keplerian energy Eplanet < 0;

• the planetocentric distance is less than three Earth’s Hill radii (e.g.,3RH,⊕ ∼ 0.03 AU);

• it makes at least one full revolution around the Earth in the Earth-Sunco-rotating frame, while satisfying the first two constraints.

In regard to the design of a round trip mission, the main advantage of theTCOs lies in the fact that those objects have been naturally redirected toorbit the Earth. This contrasts with recently proposed scenarios to design,for instance, a robotic capture mission for a small near-Earth asteroid andredirect it to a stable orbit in the EM-system, to allow for astronaut visitsand exploration (e.g. the Asteroid Redirect Mission (ARM)).

In this paper we focus on 2006RH120 , a few meters diameter near Earthasteroid, officialy classified as a TCO. 2006RH120 was discovered by theCatalina Sky Survey on September 2006. Its orbit from June 1st 2006 toJuly 31st 2007 can be seen on Figure 1, generated using the Jet PropulsionLaboratory’s HORIZONS database which gives ephemerides for solar-systembodies. The period June 2006 to July 2007 was chosen to include the portionof the orbit within the Earth’s Hill sphere with a margin of about one month.We can also observe that 2006RH120 comes as close as 0.72 normalized unitsfrom Earth-Moon barycenter. In [14], the authors investigate a populationstatistic for TCOs. Their work is centered on the integration of the tra-jectories for 10 million test-particles in space, in order to classify which ofthose become temporarily-captured by the Earth’s gravitational field – overeighteen-thousand of which do so. This suggests that 2006RH120 is not theonly TCO and that it is relevant to compute a rendezvous mission to thisspecific asteroid to get insight whether TCOs can be regarded as possibletargets for transfers with small fuel consumption, and thus cost.

In this paper, our goal is to solve the optimal control problem that consistsin performing a rendezvous mission from a Halo orbit around the Earth-Moon libration point L2 to the 2006RH120 asteroid, while minimizing the fuelconsumption. The choice of targets for our rendezvous mission sets us apart

2

−4 −2 0 2 4 6−5

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z (L

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Moon

2006RH120

on

June 1, 2006

Earth

2006RH120

on

July 31, 2007

−6 −4 −2 0 2 4−2

0

2

4−4

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x (LD)y (LD)

z (L

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2006RH120

on

June 1, 20062006RH

120 on

July 31, 2007

Moon

Earth

Figure 1: Orbit of 2006RH120 in the Earth-Moon CR3BP rotating (left) andinertial (right) reference frame.

from the existing literature where transfers are typically designed betweenelliptic orbits in the Earth-Moon or other systems [4, 19] and [20], or to aLibration point [11, 25] and [24, 27]. Rendezvous missions to asteroids inthe inner solar system can be found in [10, 17] but they concern asteroids onelliptic orbits which is not the case for us since TCOs are presenting complexorbits and therefore require a different methodology. Our assumption onthe hibernating location for the spacecraft, a Halo orbit around the Earth-Moon unstable Libration points L2, is motivated in part from the successfulArtemis mission [22, 23] and in part from the constraint on the duration ofthe mission, mostly impacted by the time of detection of the asteroid. Indeed,the Arthemis mission demonstrated low delta-v (∆v) station keeping on Haloorbits around L1 and L2. While this first study focuses on similar Haloorbits to the Artemis mission, the study needs to be extended to analyze avariation of Halo orbits both in shape (to include the eight-shaped Lissajousorbits) and also in z-excursion or energy. As a first approach, in this workwe search the best departure point on the prescribed Halo orbit withoutpractical consideration related to synchronization with the asteroid orbit.The primary objective for the mission is to maximize the final mass, butsince the Earth capture duration of the asteroid is limited it imposes a timeconstraint which need to be addressed eventually. In this paper however, weconsider rendezvous missions possibly over long period of time to determinewhat is the correlation between the required ∆v and the duration of themission. A forthcoming paper will bridge our results to the TCOs detectionaspect.

Many methodologies have been developed over the past decades to designoptimal transfers in various scenarios. Due to the complexity of the TCO

3

orbits and the nature of the mission, techniques based on analytical solutionssuch as in [16] for circular Earth orbits are not suitable and we use a numericalapproach. A survey on numerical methods can be found in [8], and forreasons related to the specifics of our problem we choose to use a deterministicapproach based on tools from geometric optimal control versus an heuristicmethod such as in [2, 9] and [27, 29]. More precisely, we use an indirectmethod based on the maximum principle, as well as a direct method andcontinuation techniques to address the difficulties of initialization for ournumerical scheme. Additionally, we fix the structure of the control norm to bea piecewise constant function with the magnitude of the thrust either zero ormaximum with at most three maximum magnitude arcs. Our techniques areillustrated on the 2006RH120 TCO and four others coming from the databaseof [14]. Validation of our approach can be seen by comparing the work to[10], in which the authors develop a low ∆v asteroid rendezvous missionthat makes use of a Halo orbit around Earth-Moon L2. Their situation is,however, different from ours, in that they have carefully chosen a specificasteroid for rendezvous. With a one-year transfer time, the ∆v value theyrealize is 432 m/s, which is comparable to the ∆v values presented here in aless-ideal scenario.

2 Optimal Control Problem and Numerical Al-gorithm

2.1 Model

In this paper, the circular restricted three-body problem [18] is used to ap-proximate the spacecraft dynamics. This is justified by the fact that a TCOcan be assumed of negligible mass, and that the spacecraft evolving in theTCO’s temporary capture space is therefore attracted mainly by two primarybodies, the Earth and the Moon.

The CR3BP model is well known, let us however go through some nota-tions fo the sequel of the paper. We denote by (x(t), y(t), z(t)) the spatialposition of the spacecraft at time t. In the rotating coordinates system, andunder proper normalization, the primary planet identified here to the Earth,has mass m1 = 1− µ and is located at the point (−µ, 0, 0); while the secondprimary, identified to the Moon, has a mass of m2 = µ and is located at(1 − µ, 0, 0), where µ = 1.2153e − 2. The distances of the spacecraft withrespect to the two primary bodies are given by %1 =

√(x+ µ)2 + y2 + z2,

%2 =√

(x− 1 + µ)2 + y2 + z2 respectively. The potential and kinetic energy,respectively V and K of the system are given by

4

V =x2 + y2

2+

1− µ%1

+µ

%2+µ(1− µ)

2, K =

1

2(x2 + y2 + z2). (1)

We assume a propulsion system for the spacecraft is modeled by adding termsto the equations of motion depending on the thrust magnitude and someparameters related to the spacecraft design. The mass of the spacecraft isdenoted by m. Under those assumptions, we have the following equations ofmotion:

x− 2y =∂V

∂x+Tmax

mu1, y + 2x =

∂V

∂y+Tmax

mu2, z =

∂V

∂z+Tmax

mu3 (2)

where u(.) = (u1(.), u2(.), u3(.)) is the control, and satisfies the constraint‖u‖ =

√u21 + u22 + u23 ≤ 1, and with Tmax/m normalized. A first integral of

the free motion is given by the energy of the system E = K − V . We willlater use this energy value to analyze the choice of the rendezvous point andthe parking orbit for the spacecraft. It is well know that the uncontrolledmotion of the dynamical system has five equilibrium points defined as thecritical points of the potential V . Three of them L1, L2 and L3 are alignedwith the Earth-Moon axis and have been shown to be unstable, while thetwo others are stable and are positioned to form equilateral triangles in theplane of orbit with the two primaries. Since our goal is to maximize the finalmass we must include the differential equation governing the variation of themass along the transfer:

m = −βTmax‖u‖ (3)

where the parameter β, the thruster characteristic of our spacecraft, is givenby β = 1

Ispg0(it is the inverse of the ejection velocity ve), with Isp the specific

impulse of the thruster and g0 the acceleration of gravity at Earth sea level.

2.2 Optimal Control Problem

Let q = (qs, qv)t where qs = (x, y, z)t represents the position variables and

qv = (x, y, z)t the velocity ones. From section 2.1, our dynamical system isan affine control system of the form:

q = F0(q) +Tmax

m

3∑i=1

Fi(q)ui (4)

where the drift is given by F0(q) = (x, y, z, 2y+x− (1−µ)(x+µ)%31

− µ(x−1+µ)%32

,−2x+

y − (1−µ)y%31− µy

%32,− (1−µ)z

%31− µz

%32)t, and the control vector fields are Fi(q) = ~e3+i

with ~ei forming the orthonormal basis of IR6.

5

We consider the rendezvous transfer from an initial point qrdv(t0) ona parking orbit O0 ∈ IR6 to a final position and velocity qrdv(tf ) on the2006RH120 orbit. Note that the initial and final positions and velocities arevariables of the global optimization problem. The criterion to maximize isthe final mass which is equivalent to minimizing the fuel consumption or the∆v =

∫ tft0

Tmax‖u(t)‖m(t)

dt. Since the mass evolves proportionally to the norm ofthe thrust, our criterion is equivalent to the minimization of the L1-norm ofthe control:

minu∈U

∫ tf

t0

‖u(t)‖dt, (5)

where U = {u(.); measurable bounded and ‖u(t)‖ ≤ 1 for almost all t}, t0and tf are respectively the initial and final time. Remark that since we chooseO0 to be a Halo orbit around a libration point, it is uniquely determined bya single point of the orbit using the uncontrolled CR3BP dynamics. Thisfact will play an important role for one of the necessary optimality condi-tions below. The large number of variables in our problem significantly addscomplexity to the search for a solution. In particular, in the case of free finaltime we expect an infinite time horizon with a control structure that mimicsimpulsive maneuvers. To simplify our optimal problem we have two options,either we fix the transfer time or we fix the structure of the control. If wefix the final time, the sensitivity of the shooting method can be addressed byusing the solution of a smoother criterion than the L1-norm, for instance theL2-norm, and linking it to the target criterion by a continuation procedure,see [13]. In this paper, however, we take a different approach and decide tofix the structure of the control. In the sequel we focus on designing transfersassociated to controls with a piecewise constant norm with value in {0, 1}and four switchings. More precisely, we consider control functions that arepiecewise continuous such that there exists times t0 ≤ t1 ≤ t2 ≤ t3 ≤ t4 ≤ tfwith

‖u(t)‖ =

{1 if t ∈ (t0; t1) ∪ (t2; t3) ∪ (t4; tf )0 if ∈ (t1; t2) ∪ (t3; t4).

(6)

Here the final time tf is free. Note that our numerical method will be ableto select the best control strategy even if it has less than three boosts.

2.3 Necessary Conditions for Optimality

The maximum principle provides first order necessary conditions for a tra-jectory to be optimal [26]. Details regarding the application of the maxi-mum principle to orbital transfers can be found in many references, includ-ing [4, 26]. We denote by X(t) = (q(t),m(t)) ∈ IR6+1 the state, where

6

Figure 2: Control function modeling thrust impulses over time.

q = (x, y, z, x, y, z) is the position and velocity of the spacecraft with mits mass. The maximum principle applied to our optimal control prob-lem states that if (q(·),m(·), u(·)) = (X(·), u(·)) is an optimal solution de-fined on [t0, tf ], then there exists an absolutely continuous adjoint state(p0, pX(·)) = (p0, pq(·), pm(·)), defined on [t0, tf ] such that:

• (p0, pX(·)) 6= 0, ∀t ∈ [t0, tf ], and p0 ≤ 0 is a constant.

• Let H, the Hamiltonian, be H(t,X(t), p0, pX(t), u(t)) = p0‖u(t)‖ +〈pX(t), X(t)〉, then

X(t) =∂H

∂pX(t,X(t), p0, pX(t), u(t)), for a.e. t ∈ [t0, tf ], (7)

pX(t) = −∂H∂X

(t,X(t), p0, pX(t), u(t)), for a.e. t ∈ [t0, tf ], (8)

where 〈, 〉 denotes the inner product.

• H(t,X(t), p0, pX(t), u(t)) = max‖ν‖≤1H(t,X(t), p0, pX(t), ν), ∀t s.t. ‖u(t)‖ =1 (maximization condition).

• Ψ(ti) = 0 for i = 1, · · · , 4.

• H(tf , X(tf ), p0, pX(tf ), u(tf )) = 0, if tf is free.

• 〈pq(t0), F0(q(t0))〉 = 0 (initial transversality condition).

• pm(tf ) = 0.

The function Ψ(·) is the so-called switching function corresponding to theproblem with an unrestricted control strategy and we have Ψ(t) = p0 +

Tmax

(‖pv(t)‖m(t)

− pm(t)β).

7

The maximization condition of the Hamiltonian H is used to computethe control on [t0; t1] ∪ [t2; t3] ∪ [t4; tf ] and we have u(t) = pv(t)

‖pv(t)‖ for all t ∈[t0; t1] ∪ [t2; t3] ∪ [t4; tf ]. The initial transversality condition reflects the factthat the initial departing point is free on the Halo orbit O0. Remark thatsince the data for the TCO’s trajectory are given as ephemerides, there are nodynamics equations to describe those orbits in the CR3BP. Thus, we cannotcompute the tangent space to a TCO point and we cannot extract a transver-sality condition for pq(tf ) at the rendezvous point. Since we expect numerouslocal extrema for this optimal control problem, it is however preferable tosolve the problem for fixed rendezvous points on a discretization of the TCOorbit.

2.4 Shooting Method

Our numerical method is based on the necessary conditions given in section2.3. Let Z(t) = (X(t), pX(t)), t ∈ [t0; tf ], and u(q, p) the feedback controlexpressed using the maximization condition. Then, we have Z(t) = Φ(Z(t))where Φ comes from equations (7) and (8). The goal is to find Z(t0), t1, t2,t3, t4 and tf such that the following conditions are fulfilled:

1. Ψ(ti) = 0 for i = 1, · · · 4;

2. X(tf ) is the prescribed rendezvous point;

3. X(t0) ∈ O0 and the initial transversality condition is verified;

4. H(tf ) = 0.

The problem has been transformed into solving a multiple points boundaryvalue problem. More specifically, we must find the solution of a nonlinearequation S(Z(t0), t1, t2, t3, t4, tf ) = 0, where S is usually called the shootingfunction. The evaluation of the shooting function is performed using thehigh order numerical integrator DOP853, see [15]. The search for a zero ofthe shooting function is done with the quasi-Newton solver HYBRD of theFortran minpack package. Since S(.) is nonlinear, the Newton-like methodis very sensitive to the initial guess. This leads us to consider heuristic ini-tialization procedures. We combine two types of techniques, a direct methodand a continuation method. The discretization of the TCO’s orbit requiresthe study of thousands of transfers, we use a direct method for a dozen ofrendezvous and expand to other points on the orbit using a continuationscheme. The motivation is that direct methods are very robust but timeconsuming while continuation method succeeded for our problem in most

8

cases very rapidly. The direct method uses the modeling language Ampl,see [12], and the optimization solver IpOpt, see [28], with a second order ex-plicit Runge-Kutta scheme. Details on advanced continuation methods canbe found in [13] and [5], but we use a discrete continuation which is enoughfor our needs.

Moreover, note that in order to fulfill the initial transversality condition,we first prescribe X(t0) on the parking orbit and find a zero of the shootingfunction satisfying all the other conditions. Afterward, we do a continuationon X(t0) along the departing parking orbit in the direction that increases thefinal mass. Once a framing of the best X(t0) with respect to the final mass isfound, we perform a final single shooting to satisfy the initial transversalitycondition along with the other conditions. This decoupling in the search of anextremal is motivated by the fact that we could very likely find a local maxi-mum on X(t0) rather than a local minimum because of the periodicity of theinitial parking orbit. We avoid this fact by first manually ensuring that theX(t0) we find will be the one for the best final mass and not the worst. How-ever, our continuation procedure on X(t0) does not always succeed, mainlybecause of the high nonlinearity of the shooting function, as the trajectorieswe find can be very long. Even if some of the extremals we find do not satisfythe initial transversality condition with the aimed accuracy (typically a zeroof the shooting function is deemed acceptable if ‖S(Z(t0), t1,2,3,4,f )‖ ≤ 10−8),they are still rather close to satisfy it (of the order of 10−4).

3 Results

3.1 Rendezvous to 2006RH120

The objective of this section is to present an analysis of the evolution of thefuel consumption with respect to the location of the rendezvous point on the2006RH120 orbit. As mentioned before, we restrict the study to transferswith at most three boosts. The primary goal is to obtain insights on thevariation of fuel consumption based on the features of the rendezvous pointsuch as its distance from L2, or its energy. The spacrecraft characteristics areassumed to be an initial mass of 350 kg, a specific impulse Isp of 230 s. anda maximum thrust Tmax of 22 N . The Halo orbit from which the spacecraftis departing is chosen to have a z−excursion of 5000 km around the EMlibration point L2, see Figure 3. The choice for this specific Halo orbit is mo-tivated by the successful Artemis mission [22, 23] that used Halo orbits withsimilar characteristics. The point corresponding to the positive z−excursionis qHaloL2 = (1.119, 0, 0.013, 0., 0.180, 0), and the period of this particular

9

Halo orbit is tHaloL2 = 3.413 in normalized time units or approximatively14.84 days.

0

0.2

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−0.05

0

0.05

−0.010

0.01

x (LD)

y (LD)

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LD

)

Earth

L2

Moon

Halo L2

Az = 5000 km

period = 14.84 days

Figure 3: Halo orbit from which the spacecraft is departing, z−excursion of5000 km around the EM libration point L2.

During the period represented in Figure 1, asteroid 2006RH120 does 17clockwise revolutions around the origin of the CR3BP frame, and 3.6 rev-olutions in Earth inertial reference frame. The evolution of the energy of2006RH120 and its distance to the Earth-Moon libration point L2 as the as-teroid evolves on its orbit are given in Figure 4.

0 50 100 150 200 250 300 350 400−2.5

−2

−1.5

−1

−0.5

0

0.5

E (

no

rm.)

time (days)

0 50 100 150 200 250 300 350 4000

2

4

6

8

10

12

time (days)

‖r−

rL2‖(L

D)

Spacecraft to L2

Earth to L2

Moon to L2

Figure 4: Evolution of the energy (left) and of the Euclidean distance to theL2 Libration point (right) for 2006RH120 . For energy, the horizontal linerepresents the energy of the Halo periodic orbit around L2 which is about-1.58.

To analyze the variations of the fuel consumption with respect to therendezvous point on the orbit, we discretize uniformly the orbit of 2006RH120

10

using 6 hours steps. For each rendezvous point of this discretization, wecompute an extremal transfer (i.e. a solution of the maximum principle)with free final time using the techniques explained in section 2. Figure 5shows the evolution of the final mass, the ∆v and of the duration of thetransfer with respect to the rendezvous point on the 2006RH120 orbit for aspacecraft departing from the Halo L2 orbit and corresponding to the threeboosts control strategy. As explained in section 2.4, some of the departurepoint on the Halo orbit are not fully optimized – this is the case for abouttwo thirds of them. Also note that a departure point different from qHaloL2

implies a drift phase whose duration is not included in the transfer duration.It can be observed that the final mass has many local extrema and thatthe variation of the duration of the mission is not continuous (contrary towhat we would expect). It is most likely due to local minima or to the factthat the value function of the problem can present discontinuities. Figure 6

0 50 100 150 200 250 300 350 400220

230

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mf

(kg

)

t(RH120) (days)

Worst transfer

Best transfer

0 50 100 150 200 250 300 350 400200

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700

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900

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∆v(m

/s)

t(RH120) (days)

Worst transfer

Best transfer

0 50 100 150 200 250 300 350 400100

150

200

250

300

350

400

450

t f (d

ays)

t(RH120) (days)

Worst transfer

Best transfer

Figure 5: Evolution of the final mass (top left), of the ∆v (top right) andtransfer duration (bottom) from the Halo orbit around L2 with respect torendezvous points on the orbit of 2006RH120 .

shows the evolution of the departure point on the hibernating orbit for thespacecraft with respect to the rendezvous point on 2006RH120 (left) as wellas the three most frequent departure points on the initial Halo orbit (right).More precisely, we represents the optimized argument of (y, z)(0) (up to the

11

quadrant: arctan(z(0)/y(0))). Note that the initial position on the Haloorbit directly depends on the optimized initial drift time. Since this initialdrift phase has not always been successfully optimized, this figure has to beinterpreted with caution. However, we can see that the departure positionon the initial periodic orbit seems to always be close to a multiple of π.

0 50 100 150 200 250 300 350 400

−3

−2

−1

0

1

2

3

tRH120 (days)

arc

tan(z

0/y

0)

(rad)

11.02

1.041.06

1.081.1

1.121.14

1.161.18

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

−0.01

0

0.01

x (LD)y (LD)

z (

LD

)

Most frequentdeparture point

MoonThird most frequent

departure point

Second most frequentdeparture point

L2

Figure 6: The left picture represents the evolution of (y, z)(0) argument withrespect to time of rendezvous on 2006RH120 . The right picture representsthe three most frequent departure points on the initial Halo orbit.

Comparing the evolution of the final mass from Figure 5 and the evolutionof 2006RH120 energy from Figure 4, we can see that the best final massesare obtained on the first half of 2006RH120 trajectory, that is for rendezvouspoints which energies are closer to the departing energy. For the best transfer,the energy difference between the rendezvous point on 2006RH120 and thedeparting point on the Halo orbit is about 0.046. The rendezvous pointwith the closest energy to the initial orbit is only slightly before the optimalrendezvous point. Its final mass is 319.3 kg which is only 0.5 kg worst thanthe best final mass. This remark suggests strongly that a small difference inenergy between the rendezvous point on 2006RH120 orbit and the departingpoint for the spacecraft on the Halo orbit is advantageous.

The best transfer is represented on Figures 7, and some relevant datais also presented in Table 1. This transfer has a ∆v of 203.6 m/s. Therendezvous takes place on June 26th 2006 and lasts 415.5 days which wouldrequire to detect and launch the mission about 14 months before June 1st2006. On Figure 8, we display the orbit of 2006RH120 in both the rotatingand inertial frame with the rendezvous point for the best transfer. The besttransfer exhibits 14 revolutions around the origin (in the rotating frame) andhas a significant variation in the z−coordinate with respect to the EM plane.

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Table 1: Best Transfer Data from Halo-L2

Parameter Symbol Value

Transfer Duration tf 415.8 daysFinal Mass mf 319.8 kgDelta-v ∆v 203.6 m/sFinal Position qrdvp (2.25, 3.21, -1.04)Final Velocities qrdvv (2.92, -2.02, 0.46)Max Distance from L2 dmax

L210.63 LD

Table 1: Data for the best transfer from qHaloL2 to asteroid 2006RH120 .

−8 −6 −4 −2 0 2 4−10

−5

0

5−6

−4

−2

0

2

4

x (LD)y (LD)

z (L

D)

Spacecraft before rendezvousAsteroid before rendezvousCraft and asteroid in rendezvous

Departureboost

Moon atdeparture

Secondboost

Rendezvousboost

Earth

−10 −5 0 5 10−10

0

10−6

−5

−4

−3

−2

−1

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x (LD)y (LD)

z (L

D)

Spacecraft before rendezvousAsteroid before rendezvousCraft and asteroid in rendezvous

Moon

Departure boost

Second boost

Earth

Rendezvous boost

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

−0.1

−0.05

0

0.05

−0.01

0

0.01

x (LD)y (LD)

z (

LD

)

Halo Orbit

Beginning of transfer

Moon

Departure Point

Figure 7: Best 3 boosts rendezvous transfer to 2006RH120 from a Halo orbitaround L2: Inertial Frame (left). Rotating Frame (Right). Bottom: zoomon the start of the departure from the Halo orbit (rotating frame).

In particular, the z−coordinate of the rendezvous point is −1.04 normalizedunits, that is about 400 thousand km, and the maximum z−coordinate along

13

−6 −4 −2 0 2 4−2

0

2

4−4

−3

−2

−1

0

1

2

3

x (LD)y (LD)

z (L

D)

Asteroid before rendezvousAsteroid after rendezvous

Earth

Moon

Rendezvous onJune 26, 2006

June 1, 2006

July 31, 2007

−4 −2 0 2 4 6−5

0

5−4

−3

−2

−1

0

1

2

3

x (LD)y (LD)

z (L

D)

Asteroid before rendezvousAsteroid after rendezvous

Rendezvous

MoonEarth

June 1, 2006

July 31, 2007

Figure 8: Orbit of 2006RH120 in rotating (right) and inertial (left) referenceframe with the rendezvous point corresponding to the best transfer.

the trajectory is 5.34 normalized units, that is about 2 million km. Thedeparture point on the parking orbit occurs 4.5 days after qHaloL2 . The controlstrategy consists of a first boost of 19.7 s, a second boost starting after154.7 days and lasting 51.1 min , finally, the last boost starting 261.1 daysafter the second boost and lasts 13.8 s, see Figure 9. Despite a seeminglyshort initial boost, this trajectory does not exhibit the profile of an initialjump on an unstable invariant manifold. However, this trajectory exploitsthe fact that a small initial boost leads to a far location where the gravityfield of the two primaries is small and where the second boost can efficientlyaim at the rendezvous point. In particular, we expect the existence of otherlocal minima with a larger final time, going further away from the initial andfinal positions and providing an even better final mass. Also note that thiskind of strategy could not have been obtained if we had restricted the controlstructure to have two boosts rather than three.

To contrast with the best transfer we represent the worst transfer in Fig-ures 10. This transfer has a ∆v of 962.5 m/s (final mass ismf = 228.4kg) anda transfer time of 37.9 days. The final position is qrdvp = (−0.65, 0.35, 0.92)and the final velocity is qrdvv = (−0.61, 1.21,−0.28). The energy at the worstrendezvous point is about −0.168, and the maximum distance to L2 is 6.21LD.

4 Conclusion and Future WorkThe numerical approach using an indirect method produces a low delta-vtransfer to asteroid 2006RH120 departing from a Halo orbit around EM-L2.Further calculations on four synthetic TCOs obtained from the databased

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0 50 100 150 200 250 300 350 400−1.59

−1.58

−1.57

−1.56

−1.55

−1.54

−1.53

−1.52

E

Trajectory energy

Initial energy

Rendezvous energy

0 50 100 150 200 250 300 350 4000

0.5

1

1.5

||u

||

time (days)

Figure 9: The bottom graph represents the control strategy for the besttransfer to 2006RH120 , and the top graph shows the evolution of the energyfor this transfer. The energy of initial Halo orbit EHaloL2 = -1.581360, theenergy after 1st boost = -1.581335, which means that the first boost raisedthe energy for about 2.46e − 05. The energy after 2nd boost = -1.535022which is an increase of about 0.046313 and the energy at the rendezvouspoint = -1.535804 which is a decrease of about = 7.814e-04. The total energydifference is about 0.0456.

produced in [14] suggest that similar results can be expected on a largesample of TCOs. Indeed, in Table 4 we display data regarding the bestthree boosts transfer for four other TCOs. These transfers have a durationof about one year each and produce delta-v values between 223.9 m/s and344.2 m/s. The best transfer is for TCO1 and occurs for an energy differencebetween terminal configurations of about 0.13, for TCO16 the difference isabout 0.6, 1.5 for TCO19 and 2.3 for TCO11. This reinforce the idea of arelation between the final mass of the transfer and the energy difference ofthe rendezvous point with respect to the one from the Halo orbit. This willbe explored on a larger population of TCOs with the goal to characterizewhich asteroids are better suited for a rendezvous mission with a low delta-v. It can also be noted that the maximum distance of the spacecraft fromL2 during the transfer is similar for all four transfers which indicates thatthe long drift is used to pull away from the two primaries attraction fields tomake the second boost more efficient.

Forthcoming work will also add the Sun as a perturbation into the model,such as in [1, 21], to determine its impact on the transfers. Moreover, forpractical reason it would be very interesting to consider the spacecraft parked

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−3.5−3

−2.5−2

−1.5−1

−0.50

0.51

1.5

−1

−0.5

0

0.5

1

1.5

−5

−4

−3

−2

−1

0

1

x (LD)y (LD)

z (

LD

)

EarthSecond Boost

Moon at departure

Rendezvousboost Departure

boost

−4−3

−2−1

01

23

4

−4

−3

−2

−1

0

1

2

3

4

−5

−4

−3

−2

−1

0

1

x (LD)y (LD)

z (

LD

)

Spacecraft

RH120 before rendezvousRendezvous Boost

Earth

Departure Boost

Second Boost

Moon

1

1.05

1.1

1.15

1.2

1.25

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

−0.01

0

0.01

x (LD)y (LD)

z (

LD

)

Halo Orbit

Beginning of transfer

Moon

Departure point

Figure 10: Worst 3 boosts rendezvous transfer to 2006RH120 from a Haloorbit around L2: Inertial Frame (left). Rotating Frame (Right). Bottom:zoom on the start of the departure from the Halo orbit (rotating frame)

Table 2: Best Transfer Data from Halo-L2 for Selected TCOsParameter Symbol TCO1 TCO11 TCO16 TCO19

Transfer Duration (days) tf 362.0 386.6 362.2 364.9Final Mass (kg) mf 316.9 300.5 311.0 307.1Delta-v (m/s) ∆v 223.9 344.2 266.1 294.6Max Distance from L2 dmax

L212.7 11.5 11.5 12.8

Time to dmaxL2

(days) t(dmaxL2

) 232.9 196.6 197.3 232.3

Table 2: Best transfer data from qHaloL2 to selected TCOs.

on a Halo orbit around the ES L1 libration point.

5 AcknowledgementsWe would like to specially thank Robert Jedicke and Mikael Granvik for theirgenerous access to the database of synthetic TCOs, and in general, their helpand support for this research. Geoff Patterson and Monique Chyba were

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partially supported by the National Science Foundation (NSF) Division ofMathematical Sciences, award #1109937.

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