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Hill’s Problem with Solar Radiation Pressure: Periodic Orbits and their Stabilization Marco Giancotti 1 , Yuichi Tsuda 2 , and Stefano Campagnola 2 1 Sapienza University of Rome ([email protected]) 2 Institute of Space and Astronautical Science Abstract During a rendezvous with an asteroid, the spacecraft’s dynamics are dominated by the effects of solar radiation pressure (SRP), besides other perturbations like an irregular gravity field. This work is a theoretical study of the various periodic orbits that exist around the asteroid. The dynamics are modeled through the circular Hill problem with the addition of an SRP term. The whole family is obtained for each orbit using numerical continuation, and the stability features of each family are analyzed. It is possible to show how several families are connected together at special bifurcation points and that, besides unstable orbits, stable ones also exist. A simple stabilization mechanism is then developed as a linear quadratic regulator to correct any additional perturbations and errors in orbital determination. This work is part of an effort to study feasible orbital strategies for the Hayabusa 2 mission, and the parameters of the asteroid are based on the mission’s target, 1999JU3. 1 Introduction Hayabusa 2, like its predecessor Hayabusa, will re- main in a hovering stance close to its target rather than orbit around it. Upon reaching the asteroid 1999 JU3, the spacecraft will stop at a fixed point called “gate position”, 20 km from the asteroid on the line connecting it with the earth (Kawaguchi et al, 2008). During its phases, it will hover temporarily closer to the asteroid, and eventually touch down on its surface. As a possible extension of the mission, it is useful to study what kind of orbital motion is possible in this environment. This kind of problem has been studied recently, but for different regimes and for a limited number of orbits (Scheeres, 2011; Katherine and Vil- lac, 2010). This work approaches the problem from a theoretical point of view, by simplifying the dynamics with the circular Hill’s problem with an added term for solar radiation pressure. Through a grid search we find many periodic solutions, then we use numer- ical continuation to obtain the orbit families of some of them. Finally, we show a way to stabilize them in the presence of additional perturbations. 2 Hill’s Problem with Radia- tion Pressure If we assume that the asteroid has a regular grav- itational potential and travels around the sun in a circular orbit, the dynamics of a spacecraft close to it can be modeled with the equations of motion of Hill’s problem. We use a rotating reference frame that is centered on the asteroid and where the sun is fixed in the -x 1

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Hill’s Problem with Solar Radiation Pressure: Periodic Orbits andtheir Stabilization

Marco Giancotti1, Yuichi Tsuda2, and Stefano Campagnola2

1Sapienza University of Rome ([email protected])2Institute of Space and Astronautical Science

Abstract

During a rendezvous with an asteroid, the spacecraft’s dynamics are dominated by the effects of solarradiation pressure (SRP), besides other perturbations like an irregular gravity field. This work isa theoretical study of the various periodic orbits that exist around the asteroid. The dynamics aremodeled through the circular Hill problem with the addition of an SRP term. The whole family isobtained for each orbit using numerical continuation, and the stability features of each family areanalyzed. It is possible to show how several families are connected together at special bifurcationpoints and that, besides unstable orbits, stable ones also exist. A simple stabilization mechanismis then developed as a linear quadratic regulator to correct any additional perturbations and errorsin orbital determination. This work is part of an effort to study feasible orbital strategies for theHayabusa 2 mission, and the parameters of the asteroid are based on the mission’s target, 1999JU3.

1 Introduction

Hayabusa 2, like its predecessor Hayabusa, will re-main in a hovering stance close to its target ratherthan orbit around it. Upon reaching the asteroid1999 JU3, the spacecraft will stop at a fixed pointcalled “gate position”, 20 km from the asteroid on theline connecting it with the earth (Kawaguchi et al,2008). During its phases, it will hover temporarilycloser to the asteroid, and eventually touch down onits surface.

As a possible extension of the mission, it is useful tostudy what kind of orbital motion is possible in thisenvironment. This kind of problem has been studiedrecently, but for different regimes and for a limitednumber of orbits (Scheeres, 2011; Katherine and Vil-lac, 2010). This work approaches the problem from atheoretical point of view, by simplifying the dynamics

with the circular Hill’s problem with an added termfor solar radiation pressure. Through a grid searchwe find many periodic solutions, then we use numer-ical continuation to obtain the orbit families of someof them. Finally, we show a way to stabilize them inthe presence of additional perturbations.

2 Hill’s Problem with Radia-tion Pressure

If we assume that the asteroid has a regular grav-itational potential and travels around the sun in acircular orbit, the dynamics of a spacecraft close toit can be modeled with the equations of motion ofHill’s problem.

We use a rotating reference frame that is centeredon the asteroid and where the sun is fixed in the −x

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Figure 1: Examples of periodic orbits resulting from the grid search. Each is shown from different angles.

direction (Szebehely, 1967),

x− 2y = − x

r3 + 3x+ β (1a)

y + 2x = − y

r3 (1b)

z = − z

r3 − z . (1c)

Here the units of time and distance have been set to1/ω and

(µ/ω2)1/3 respectively, where µ is the grav-

itational parameter of the asteroid and ω is its con-stant angular velocity around the sun. The term β inthe first of Eqs. (1) accounts for the effect of solar ra-diation (Scheeres, 2012). It assumes that the exertedforce is always pointing away from the sun and doesnot depend on the attitude of the spacecraft. Theconstant depends on the parameters of the spacecraftand on the mass of the asteroid. Due to uncertainties,this should be in the range 27 < β < 55.

Eqs. (1) admit an integral of motion, the Jacobiconstant, defined as

C = 12(x2 + y2 + z2)− 1

r− 3

2x2 + 1

2z2 − βx .

3 Linear-Quadratic RegulatorGiven a system described by the equations

x = f (x(t)) (2)

in the presence of small deviations δx from a nominalorbit, it is possible to linearize the motion as

δx = Jδx +Bu . (3)

Here J = ∂f/∂x is the Jacobian matrix of f . Ourpurpose is to correct the state deviations with active

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Figure 2: Five steps of families a, c and dd, respectively from the top.

control, which is included as a term Bu in (3). Theoptimal control problem we have to solve is to find thecontrol strategy that minimizes the cost functional

J =∫ t1

t0

(δxTQδx + uTRu

)dt+ δxT

1 Sδx1 , (4)

with δx1 = δx(t1) and Q, R and S adequate weightmatrices.

The linear state feedback law in the theory of LQRstates that the optimal control u∗ depends linearly onthe state (δx in this case) with the following relation(Athans and Falb, 2006):

u∗(t) = −R−1BTPδx∗ ,

and that the matrix P is a solution of the Riccatidifferential equation

P = −PJ − JTP −Q+ PBR−1BTP , (5)

subject to the terminal condition P (t1) = S.

4 Grid Search for Periodic Or-bits

The symmetries of Hill’s problem (1) provide sim-ple sufficient conditions for trajectories to be periodic(Miele and Mancuso, 2001). In this work we make useof a symmetry with respect to the x–z plane. Any

trajectory that crosses that plane twice perpendic-ularly must be periodic due to this symmetry. Weuse this condition to perform a wide grid search forperiodic orbits. From a grid of initial conditions onthe x–z plane with perpendicular initial velocity, wepropagate systematically and look for new orthogonalcrossings with the same plane.

To reduce the computational burden, we restrictthe initial grid to a half-circle on the x–z plane, cen-tered on the asteroid and with a radius of 20 km.Thus, the grid extends only on the angle inside thehalf-circle and on the magnitude of the initial veloc-ity. We then seek new crossings with the plane thatsatisfy the conditions x = 0 and z = 0 simultane-ously.

The search is repeated for β = (27, 28, 29, 30, 31,32, 33, 34, 35, 55) to maximize the number of solu-tions. This results into 37 unique periodic orbits.

Figure 1 shows four representative examples ofthese orbits. The periods, numbers of revolutionsand general shapes vary from orbit to orbit. Amongthese we select 9 families of interest for further studyin the following sections.

5 Periodic Orbit FamiliesFor the selected orbits, we apply an efficient numer-ical continuation technique, called pseudo-arclengthcontinuation, to obtain new solutions for different

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rio

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T

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Figure 3: Bifurcation plots showing the period withrespect to C of some of the families. The familyshown as a dashed line has its period multiplied by afactor 3.

values of C. This leads to the construction of wholefamilies of trajectories, one for each initial orbit, witha continuum of parameters like period, maximum-approach distance and energy. We selected the solarradiation pressure parameter β = 33 for all our con-tinuation efforts. Three examples are displayed inFigure 2.

Having the whole families available is useful whenstudying the stability of the orbits. In general, withina single family exist ranges of C for which the orbitsare stable or unstable. Because stable orbits are morerobust in the presence of small errors in orbit deter-mination, being able to choose the stable regimes ofa family may be an advantage for the orbit design.

By tracing the orbits along their families we canreveal the connections between different types of mo-tion. For instance, Figure 3 shows the periods of fivefamilies with respect to C. Families e and ee haveexactly the same periods, and so do also families dwith dd and a with aa. The line for a/aa, showndashed, represents the triple of their actual periods.It is evident from the figure that e/ee and d/dd all bi-furcate from a/aa, that is, there are some values of Cfor which the families coincide. We can confirm thisby looking at Figure 2, where families c and dd bothend with circular shapes, characteristic of a. When,in these cases, the period of the bifurcated orbits is

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pericente

r dis

tance

a

cc

c

dd

bd

D

T

D

T

ebb

Figure 4: Bifurcation diagram showing the distanceof closest approach to the asteroid of all of the se-lected orbit families with respect to C. The thickline is family a.

triplicated, the bifurcations are called period-tripling(T in the figure).

The families are also shown in Figure 4 for whatconcerns the minimum approach distance along oneperiod, rmin. Again, all of the families bifurcate froma with either period-doubling (D) or period-tripling(T) bifurcations.

6 Stabilization with the LQRThe final part of this study regards the task of sta-bilizing the periodic orbits when they are disturbedby perturbations other than solar radiation pressure.Here we test the LQR method with two types of per-turbation: random instantaneous errors and gravityirregularities.

6.1 Random Instantaneous ErrorsIn the first simulation, we introduce errors in the po-sition and velocity of the spacecraft at regular inter-vals. These errors are applied almost instantaneously,and their directions and magnitudes are chosen atrandom using normal distributions. While this typedisturbance is not something that occurs in reality,it simulates the result of a non-continuous orbit de-termination scheme. Because of the restrictions on

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positio

n d

ispla

cem

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]

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Figure 5: Random Instantaneous errors controlledby LQR. Thrust and displacement histories over oneperiod.

the attitude of the spacecraft relative to the aster-oid, for long intervals of time it may not possible tomeasure its position and velocity in a satisfying way.When such measurements do occur, the discoveredstate may be substantially off from the nominal value.These simulations test the response of the LQR algo-rithm to suddenly-discovered errors of this kind.

After solving Eq. (5) over one full period of eachorbit, all the orbits tested show a behavior similar tothat of family c, given as an example in Figure 5.The error in position (δx, δy, δz) is of the order of

1000 km, and that in velocity is close to 0.01 m/s.All the examples given have been produced with thesame weights of the cost function. For this choiceof the weights, in all cases it takes about 1.5 days tobring the trajectory back to its nominal path after anerror has been introduced.

For all orbits the necessary thrust u is of the orderof 0.1 mN, while the total ∆v during a whole period isabout 0.3 m/s. This similarity in the results is prob-ably explained by the size of the errors, the controlof which is of a larger scale with respect to the otherforces acting naturally in the system.

6.2 Gravity IrregularitiesFor orbits that approach the asteroid very closely,the effect of the gravitational irregularities may be-come important. The shape model of Hayabusa 2’starget, 1999 JU3, is not yet known in detail, but itsproperties have been somewhat constrained throughlight-curve observations (Muller et al, 2010). Herewe approximate its shape as a triaxial ellipsoid withup to the 4th degree and order polynomial terms,and propagate the periodic orbits with the LQR al-gorithm.

The assumed dimensions of the asteroid are

[899.8 m, 869.9 m, 808.6 m] ,

whereas the period of rotation is 7.63 h and the axisof rotation has ecliptic longitude and latitude, respec-tively, λecl = 73.1◦ and βecl = −62.3◦.

Figure 6 shows the thrust and displacement histo-ries of an orbit of family c in this situation. Type corbits (top-right orbit in Figure 1) have three closeapproaches to the asteroid during one period. Forthis reason, most of the gravitational perturbationsare felt during these three short time intervals, as vis-ible in Figure 6. The thrust shows peaks of the orderof 0.3 mN, while the displacement from the nominalorbit never exceeds 80 m. The closest approach to theasteroid, for this orbit, is 1117 m, or approximately680 m of altitude. Clearly, lower orbits would expe-rience stronger perturbations, and vice versa. Thenecessary control, however, remains for most casesaround or below the order of 1 mN, easily attainable

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even with ion engines. We should note that the ratiobetween thrust effort and state displacement dependson the weights Q, R and S in Eq. (4): lower thrustsare possible if we accept higher position and velocityerrors, and the converse is obviously also true.

7 ConclusionsIn this work we identify a number of periodic orbitsexisting in a system similar to that of Hayabusa 2 andits target. We describe some properties of selectedorbit families, discussing their relations and parame-ters. Finally, we show how a simple linear-quadraticregulator is sufficient to stabilize these orbits whenone of two different perturbations are present. Weshow that these orbits can in general be stabilizedwithin short time frames and with low thrust.

ReferencesAthans M, Falb P (2006) Optimal Control: An Intro-

duction to the Theory and Its Applications. DoverPublications

Katherine YYL, Villac BF (2010) Periodic orbitsfamilies in the hill’s three-body problem with solarradiation pressure. In: Advances in the Astronau-tical Sciences Series 136, vol 1

Kawaguchi J, Fujiwara A, Uesugi T (2008) Hayabusa- Its technology and science accomplishment sum-mary and Hayabusa-2. Acta Astronautica 62(10-11)

Miele A, Mancuso S (2001) Optimal Trajectoriesfor Earth-Moon-Earth Flight. Acta Astronautica49(2)

Muller T, et al (2010) Thermo-physical propertiesof 162173 (1999 JU3), a potential flyby and ren-dezvous target for interplanetary missions. Astron-omy and Astrophysics

Scheeres DJ (2011) Orbital Mechanics Sbout SmallBodies. Acta Astronautica 72

Scheeres DJ (2012) Orbital Motion in Strongly Per-turbed Environments. Springer

Szebehely V (1967) Theory of Orbits. The RestrictedProblem of Three Bodies. Academic Press

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Figure 6: Spherical harmonic perturbations con-trolled by LQR. Thrust and displacement historiesover one period.

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