trajectory design for the jaxa moon nano-lander omotenashi

SSC17-III-07Trajectory Design for the JAXA Moon Nano-Lander OMOTENASHI

Javier Hernando-Ayuso, Yusuke OzawaThe University of Tokyo

7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan; [email protected]

Shota TakahashiThe Graduate University for Advanced Studies

3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan; [email protected]

Stefano Campagnola∗

Jet Propulsion Laboratory4800 Oak Grove Drive, La Canada Flintridge, CA 91011, USA

[email protected]

Toshinori IkenagaTsukuba Space Center, Japan Aerospace Exploration Agency

2-1-1 Sengen, Tsukuba-shi, Ibaraki 305-8505, [email protected]

Tomohiro Yamaguchi, Tatsuaki HashimotoInstitute of Space and Astronautical Science, Japan Aerospace Exploration Agency

3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, [email protected]

Chit Hong Yamispace Inc.

3-1-6 Azabudai, Minato-ku, 106-0041 Tokyo, [email protected]

Bruno V. SarliCatholic University of America

620 Michigan Ave NE, Washington, DC 20064 USA; +1 301 286 [email protected]

ABSTRACT

OMOTENASHI (Outstanding MOon exploration TEchnologies demonstrated by Nano Semi-Hard Impactor) is aJAXA 6U cubesat that aims to perform a semi-hard landing at the Moon surface after being deployed into a lunarfly-by orbit by the American Space Launch System, Exploration Mission-1. In this paper, we present the analysisand design of OMOTENASHI trajectory, divided in an Earth-Moon transfer using a cold gas thruster and a landingphase using a solid rocket motor. Strong constrains exist between the two phases, making the mission design a verychallenging task. The flight path angle at Moon arrival must be shallow in order to minimize the effect of delay ofthe deceleration maneuver. This, together with the execution error of the cold gas maneuver, demands a correctionmaneuver to compensate for these errors. Requirements on the ground station tracking are also deduced from thisanalysis, and it was found that the use of DDOR is an enabling technology for a safe lunar landing. Under the currentsubsystems design, we found that the most critical factors in the landing success rate are the maneuver orientation,thrust duration and total delta-v errors. Results suggest accuracy requirements to the landing devices, solid rocketmotor and attitude accuracy, as well as to the transfer phase trajectory design.

∗The work of Stefano Campagnola was carried out as an Interna-tional Top Young Fellow in ISAS/JAXA, Japan

Hernando-Ayuso 1 31st Annual AIA/USUConference on Small Satellites

INTRODUCTIONSmall satellites are being considered for missions ofincreasing complexity and interest. They offer a re-duced cost and development time, which allows to re-spond to technological and science demands in a shortertimescale. Their use in Low Earth Orbit has already beenproven, and there is an growing interest on applying theconcept to interplanetary missions. This was already thecase of PROCYON, the first interplanetary small satel-lite, developed and launched by The University of Tokyoand JAXA in 2014 as a secondary payload of Hayabusa2mission.1

One type of mission that can greatly benefit from the ad-vantages of small satellites is Moon exploration. The useof cubesats detaching from Moon-orbiting spacecraft hasbeen proposed in the past .2 However, if a piggyback op-portunity in a mission that features a lunar flyby is avail-able, the mission scenario can be considerably simplified.

This opportunity will arise in the first launch of Amer-ican Space Launch System (SLS), called ExplorationMission-1 (EM-1). After launch in 2019, thirteen6U cubesats will be injected into a lunar flyby orbit.3

JAXA will seize this opportunity with OMOTENASHI(Outstanding MOon exploration TEchnologies demon-strated by NAnoSemi-Hard Impactor). OMOTENASHImission also seeks to study the radiation environment be-yond Low Earth Orbit in order to support human spaceexploration.4

However, OMOTENASHI is a challenging mission. Oneof the main challenges comes from trajectory, whichmust be robust to execution and navigation errors. As wepresent in this paper, a robust trajectory must have a smallflight path angle (FPA) at Moon arrival. In particular,we found that it must satisfy −7 deg ≤ FPA ≤ 0 deg inorder to be error-robust. To this end, the design of thedifferent arcs of the trajectory cannot be performed inde-pendently, as they are strongly coupled.

After detaching from SLS, OMOTENASHI must per-form two deterministic maneuvers that will make thiscubesat the first one to perform a semi-hard landing onthe Moon. A first maneuver, DV1, will inject OMOTE-NASHI into a Moon-impacting orbit. After perform-ing midcourse trajectory correction maneuvers (TCM) asneeded, a solid rocket motor will be ignited shortly be-fore the expected Lunar surface collision at a speed of ap-proximately 2.5 km/s. After the deceleration maneuver(DV2), OMOTENASHI will experience a free-fall froma low height (close to 100 m) and arrive at the Moon sur-face with a speed of around 20 m/s.4, 5 In order to reducethe mass budget, OMOTENASHI is composed of an or-biting module, a retromotor module and a surface probe.The orbiting module must be ejected at rocket motor ig-

nition to achieve the required deceleration. Finally, thesurface probe will separate from the retromotor moduleat burnout to reduce the load on the energy absorptionmechanisms.

Figure 1 shows the current state of the design of thespacecraft for different parts of the mission. On thetop, Fig. 1a shows the orbiting configuration of OMOTE-NASHI, featuring solar arrays in the +Y face. The solidrocket motor, including its sealing lid, is also visible. Be-fore DV2, OMOTENASHI will deploy its airbag as canbe seen in Fig. 1b. The orbiting module is ejected afterthe solid rocket motor ignition, being the configurationduring the deceleration maneuver as shown in Fig. 1c. Adetailed view of the internal parts of OMOTENASHI ispresented in Fig. 2. Figure 2a shows the Reaction Con-trol System (RCS), attitude control module, communica-tion devices, rocket motor and surface probe. Lookingfrom a different angle, Fig. 2b shows the battery module,the laser diode (LD) used to ignite the motor, and the de-vices in charge of inflation of the airbag: N2 gas tank andshape memory allow (SMA) opener.

In this paper we perform a detailed analysis of a semi-hard lunar lander like OMOTENASHI trajectory, includ-ing the Earth-Moon transfer (DV1, TCM) and the land-ing phase (DV2). We propose a design methodology forDV1 by analyzing the set of feasible solutions that ar-rive at the Moon with a small FPA. Results of sensitivityanalysis under OD and maneuver execution errors sug-gest that a TCM must be considered. Finally, we designthe landing phase imposing zero vertical velocity and aspecified height over the Moon surface at burnout. Weidentify critical errors in the system, which can be seenas requirements for the spacecraft to achieve a safe land-ing.TRAJECTORY OVERVIEWIn order to design a trajectory that leads to a safe semi-hard landing on the surface of the moon, the trajectory isdivided in two arcs: the transfer and the landing phase.

During the transfer phase, OMOTENASHI must modifyits orbit from the Moon fly-by injection orbit to a Moonintersection orbit.6 Additionally, health check-ups andorbit determination are key aspects of this phase. OD is acritical resource during the first hours of operation, as the13 delivered cubesats have the same need of accuratelyassessing the orbit they are flying, and the time before theMoon fly-by/arrival is limited. During the trajectory de-sign process, this was identified as one of the key aspectsthat the small satellite community should address in thenear future.

The landing phase starts minutes before arriving at theMoon surface, and the main event is the deceleration ofthe spacecraft by the use of a solid rocket motor.7 An-


(a) OMOTENASHI in its orbiting configuration

(b) OMOTENASHI with deployed airbag

(c) OMOTENASHI after orbiting module detachment

Figure 1: OMOTENASHI configuration at differentparts of the mission

other important aspect is that in order to successfully de-liver the required deceleration ∆V , the orbiter modulewill be detached from the rest of the spacecraft to reduce

the mass to be decelerated.

In the absence navigation and maneuver execution er-rors, any Moon-intersecting trajectory would lead to asuccessful landing, provided that the surface probe canabsorb the residual kinetic energy after braking. How-ever, the uncertainty on the actual trajectory introducesvery strong constraints between the two phases.

From the point of view of a safe landing, a trajectory witha very shallow FPA is preferred to minimize the effect oftiming errors on the vertical displacement of the space-craft. A high position error on the vertical direction maylead to a premature landing during the solid rocket motorburn, jeopardizing the mission. On the other hand, a veryshallow FPA might cause missing the Moon in the pres-ence of errors. To reduce the fly-by probability, a TCMmay be introduced if necessary. A TCM must be care-fully planned in order not to hinder the orbit accuracyduring the landing, as it reduces the time to perform ODbefore the landing phase.

The current analysis and design were conducted withthe initial conditions provided by Marshall Space FlightCenter8 and shown in Table 1. The position and veloc-ity components are expressed in a Moon-centered refer-ence frame whose axes are parallel to the J2000 Eclipticframe. We considered the Sun, Earth and Moon Grav-ity as point masses and an impulsive DV1 maneuver. Inthe future solar radiation pressure, spherical harmonicsand finite burns will be included, but the results will notqualitatively change.

Table 1: Initial conditions expressed in the Moon-centered J2000 Ecliptic frame

Component ValueEpoch 2018 Oct 07 15:39:16x [km] 341 095.06

y [km] −43 570.46

z [km] −18 326.52

vx [km/s] −3.59

vy [km/s] −2.71

vz [km/s] 0.98

Figure 3 shows the transfer trajectory in an Earth-centered frame that rotates with the Moon. Figure 4shows the landing phase trajectory, including the decel-eration and final free-fall.ORBIT DETERMINATIONNavigation accuracy plays a critical role in the successof a lunar lander like OMOTENASHI. In the first place,the strong FPA constraints at Moon arrival demand aprecise knowledge of the state vector of the satellite at


Rocket Motor

Attitude Control Module

X-band TransmitterUHF-band TransmitterRadiation Monitor

Y

Z X

Airbag (folded)Surface Probe CaseCrushable Material

RCSX-band

Antenna

(a) OMOTENASHI detailed view (+Y face)

LD Module

SMA opener

N2 Gas Tank

Battery Module

Y

XZ

(b) OMOTENASHI detailed view (-Y face)

Figure 2: OMOTENASHI subsystems detailed view

TCM

DV1

Moon landing phaseDeployment

Figure 3: Transfer phase

DV1. Moreover, vertical position errors during landingmay lead to an early impact with the Moon while thesolid motor rocket maneuver is being performed. Timingerrors during landing could also jeopardize the mission,even if the approach trajectory is characterized by a smallFPA.

Moreover, we found that the actual trajectory has a stronginfluence in the position accuracy at Moon arrival. Weobserved differences up to one order of magnitude inthe vertical position error at landing when following dif-ferent orbits for the same tracking strategy. However, asmall-satellite operator has in most cases limited controlover the orbit he is being deployed into. Thus, great caremust be taken when designing the OD strategy to be em-ployed.

We will considered the observables for orbit determina-tion reported in Table 2.

During the transfer phase, the orbit knowledge is criticalwhen designing the trajectory. We are especially con-cerned about the OD errors at DV1 and TCM epoch.We study two different cases that differ in the amountof resources employed, and label them as A and B. Forboth cases we consider 3 hours tracking from Uchinoura

Figure 4: Landing phase

Space Center (USC), in which we perform communica-tions, two-way Doppler measurements and 30 minutes oftwo-way range measurements. Additionally, OMOTE-NASHI team has requested the support of of Goldstone(GDS), part of NASA Deep Space Station (DSN). In caseA, we consider only 30 minutes of two-way Dopplermeasurements, while for case B we increase the dura-tion to 3 hours and also include 30 minutes of two-wayranging. For both cases, we plan on requesting the use ofMadrid (MAD) antenna for DV1 uplink. Figure 5 showsthe observations planning for DV1. The expected posi-tion and velocity errors are reproduced in Table 3.

We also performed a similar analysis to up to TCMepoch. We maintain the same strategy used for A andB cases for an additional day of tracking. Figure 6 showsthe ground stations coverage up to TCM epoch. Onceagain, the uplink of the TCM command is planned withthe support of MAD. The expected navigation errors areshown in Table 4.

After DV1/TCM, a tentative OD analysis was performedusing only JAXA resources to decrease the mission costand complexity. To this end, a campaign of observations


Table 2: Orbit Determination observables

type duration interval 1–σ noise biasX-band 2-way Doppler 3 h 60 s 0.5 mm/s no bias

X-band 2-way range 30 min 60 s 10 m no biasDDOR (GDS-CAN, CAN-MAD) 30 min 600 s 1 ns no bias

Table 3: Orbit Determination 3–σ errors at DV1epoch

Case Error T N H

Aposition [km] 1.5 2.3 14.4

velocity [cm/s] 0.3 1.5 21.1

Bposition [km] 0.1 0.2 2.3

velocity [cm/s] 0.1 0.4 2.6

Blue : Spacecraft elevation, Red : Sun elevation

DV1

Disposal

MAD30 min

USC343 hours

GDS30 min (case 1) or 3 hours (case 2)

DSN Goldstone

JAXA USC34

DSN Madrid

Data-cut-off

Figure 5: Ground station visibility and observationsplanning for DV1

DV2

USC343 hours (all cases)

GDS30 min (case 1) or 3 hours (case 2)

TCMDV1

Figure 6: Ground station visibility and observationsplanning for TCM

from USC was planned and simulated. We performed3 sets measurements spanning 3 hours using two-wayDoppler and including 30 minutes of two-way-ranging.In the simulation, a priori covariance was not consid-ered. The visibility from the USC ground station and

Table 4: Orbit determination 3–σ errors at TCMepoch

Case Error T N H

Aposition [km] 3.3 0.7 23.5

velocity [cm/s] 15.4 4.4 28.3

Bposition [km] 0.8 0.2 1.3

velocity [cm/s] 1.5 0.3 1.5

the communication windows are shown in the upper partof Fig. 7. During the last communication window no ODis planned, as a cut-out time for final computation anduplink of DV2 is introduced. The simulated covariancematrix at DV2 epoch provides a vertical error close to400 m, unacceptable for a safe landing with a free-fallinitial height of around 100 m.

After studying several configurations, it was decided toinclude Delta Differential One-way Ranging (delta-DORor DDOR) measurements using DSN stations in Gold-stone (GDS), Canberra (CAN) and Madrid (MAD). Thismethod requires two stations to be visible at the sametime, and can be accomplished with the combinationsGDS-CAN and CAN-MAD as can be seen in Fig. 7. Themeasurements are characterized by the lower row of Ta-ble 2. Results show a vertical 3–σ error of the order of50 m, which would not jeopardize the landing maneuveras it will be shown later. The position covariance matrixat the solid rocket motor ignition, expressed in the J2000Ecliptic reference system and in km2, is:

CDV2 =

1.3× 10−3 −9.7× 10−4 1.7× 10−3

−9.7× 10−4 7.2× 10−4 −1.3× 10−3

1.7× 10−3 −1.3× 10−3 2.4× 10−3

This makes DDOR a critical element of the mission nec-essary for its success.TRANSFER PHASEIn this section we introduce OMOTENASHI transferphase design strategy. First, we describe the calcula-tion of DV1 maneuvers that lead to a shallow FPA atMoon arrival. Next, we analyze the trajectory sensitiv-ity to navigation and execution errors. Finally, and basedon the previous point, we present the design of the TCM


DV1 DV2TCM DCODDOR x 4

(GDS-CAN, CAN-MAD)

with TCM

without TCM

Not included in OD

DV2 command upload

Figure 7: Ground station visibility and observationsplanning after DV1/TCM. CAN is omitted because itsvisibility is similar to USC

to compensate for errors at DV1 epoch.

DV1

We can characterize a maneuver ∆v = (vx, vy, vz)> by

its magnitude and two orientation angles. To this end, weintroduce the azimuth φ and polar angle θ defined in theJ2000 Ecliptic reference frame as

φ = atan2 (vy, vz) (1)

θ = cos−1 vz‖v‖

(2)

Then, ∆v takes the form

∆v =

v sin θ cosφ

v sin θ sinφ

v cos θ

(3)

We split the design of DV1 into two parts. First, we per-form a coarse grid search for all orientations and differentmagnitudes. Once we fix the magnitude as a compro-mise between fuel consumption and size of the feasibleregion, we use an iterative process to obtain a fine gridof feasible maneuver orientations. In this analysis we areconsidering impulsive DV1 maneuvers, but in the follow-ing months and after the cold gas system design is fixed,finite thrust will be incorporated into our analysis.

Figure 8 shows the result of the grid search for all thrustdirections (12 deg resolution) and magnitudes from 10 to20 m/s. We can determine the nominal DV1 magnitudeby evaluating the number of solutions that have shallowFPA and arrive at the near side of the Moon. While formagnitudes of 10 and 12.5 m/s some of the trajectoriesarrive at the Moon, their FPA is too deep to lead to a

safe landing. Magnitudes close to 15 m/s are promisingwith the current initial condition, as wide regions withshallow FPA are available. Figure 9 shows the effect ofvarying the maneuver orientation for a fixed magnitudeof 15 m/s, where FPA smaller than −20 deg were trun-cated and shown in blue color.

-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

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45

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90

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lat

itu

de

[deg

]

10 m/s

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15 m/s

17.5 m/s

20 m/s

Figure 8: Landing location of DV1 coarse grid search,with 12 deg resolution and different magnitudes

We set 15 m/s as the nominal magnitude and study thenominal orientation by iterative grid refinement. In eachiteration, the grid is refined filtering out unfeasible ori-entations that lead to a too steep FPA, flybys or the farside of the Moon. Figure 10a shows the grid solutionspace of the thrust directions with the magnitude fixed at15 m/s. The non-colored region corresponds to trajecto-ries which will not intersect the lunar surface, performinga flyby. The solutions with shallow FPA exist near theboundary of the colored region. Figure 10b shows theresult after the first grid refinement. This iterative pro-cess consists not only in a refinement of the border so-lutions, but also in discarding unfeasible solutions suchas with steep FPA or on the far-side of the Moon. Fig-ure 10c is the result of 7 refinement iterations (Resolutionis 12/27 ' 0.094deg).

Figures 11 and 12 are DV1 direction solutions and theircorresponding arrival points at the average Moon sur-face, respectively. Red thrust direction solutions (Fig.11) and red arrival points (Fig. 12) represent the trajecto-ries that might be hindered at arrival by the lunar surface,i.e. craters or mountains. Blue solutions correspond tothe trajectories on which the probe can safely reach theMoon surface. Four candidate nominal DV1 solutionsare marked with green symbols in these graphs. Table 5contains the parameters for these four candidates. Notethat we shifted case 3 (FPA = −5.5 deg instead of −5)to avoid landing uphill at the outer wall of crater Zeno P(latitude 43.4 N, longitude 66.1 E).

DV1 Error AnalysisWe studied the errors at DV1 epoch and how they af-fect the Moon arrival using linear theory. A Monte-Carlo


-180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180

Selenograhic longitude [deg]

-90

-75

-60

-45

-30

-15

0

15

30

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lat

itu

de

[deg

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-20

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0

Fli

ght

Pat

h A

ngle

[deg

]

Figure 9: Landing location and flight path angle ofDV1 direction coarse grid search, for 15 m/s. FPAdeeper than−20 deg are truncated and shown in blue

Table 5: Candidate DV1 maneuvers

case ∆v [m/s] φ [deg] θ [deg] FPA [deg]1 15 63.66 45.38 −3

2 15 63.47 45.56 −4

3 15 63.84 45.99 −5.5

4 15 63.38 46.13 −6

simulation performed using linear propagation providesgreat advantages from a computational point of view, asa high number of samples would take an unacceptablylong time if a numerical propagation was used. Thus, nu-merical propagation will only be used for validation, andwe will employ linearization around the nominal orbit toevaluate the errors at Moon arrival.

Considering a reference orbit y (t), the state transitionmatrix (STM) between a initial epoch ti and a final epochtf can be defined as9

Φ (tf , ti) =∂y (ti)

∂y (tf )(4)

By use of the STM, deviations from the reference orbitδy can be calculated as

δy (tf ) = Φ (tf , ti) δy (5)

When tf is the epoch in which the nominal solution ar-rives at the Moon surface, the linear method providesposition vectors not contained on the Moon surface. Tocorrect for this time delay, the samples must be forwardor backward propagated to the Moon surface. Depend-ing on the dispersion of the samples and the degree ofaccuracy needed, several methods are possible. The sim-plest option is to assume a free fall over a flat Moon andsolve the parabolic flight equations. The model can beexpanded to consider the Moon curvature for example.6

If we neglect forces other than the gravitational pull of

(a) 12deg resolution

(b) 6deg resolution

(c) 0.09375deg resolution

Figure 10: Iterative grid search

the Moon considered as a point mass, we can also modelthis correction as a Keplerian orbit and solve for the an-gular position that makes the orbital radius equal to theMoon radius. These two approaches were evaluated toprovide accurate enough results for practical situationsand were validated using high-accuracy numerical prop-agations,6 which yield the most accurate solution to theadjustment of the final points.

The sensitivity to navigation and execution errors at DV1epoch can be evaluated in a reasonable time using the lin-ear approach presented above. We apply independentlyisotropic position, isotropic velocity, DV1 magnitude and


10 20 30 40 50 60 70 80 90 100 110

[deg]

43

43.5

44

44.5

45

45.5

46

46.5

47

[d

eg]

43

21

Figure 11: Final iteration of the direction grid search(Zooming of Fig. 10c. Note that axis are not to scale).Nominal DV1 candidates are marked in green

Figure 12: Moon arrival points with FPA isolines.Nominal DV1 candidates arrival points are markedin green

DV1 direction errors of different magnitudes and ana-lyze the success rate of the transfer phase. We consideras failure those cases in which OMOTENASHI missesthe Moon surface and flybys the Moon, or if the FPAis steeper than −7 deg. Figure 13 shows the result of thesensitivity analysis. The expected errors marked with redellipses, where we considered the 3–σ errors for DV1magnitude and direction as 1% and 1 deg respectively.Orbit determination errors are not critical neither on ODcases A or B as can be seen on Figs. 13a and 13b. How-ever, Figs. 13c and 13d reveal that DV1 execution errormay jeopardize the transfer to the Moon. Figure 13 is apowerful tool for the trajectory design team, as it allowsone to draw as an important conclusion that a TrajectoryCorrection Maneuver must be considered.

Trajectory Correction ManeuverAt TCM epoch, we apply an impulse ∆vTCM to com-pensate for errors in DV1. Our strategy will be based onre-targeting the same landing location as in the nominal

orbit.

To this end, we first assume a Gaussian dispersion at DV1which includes OD and execution error. Next, we samplethe uncertainty and for every point we map it to TCMepoch using linear theory. For each sample we calculatea correction maneuver, and apply OD and execution errorat TCM epoch. The results are finally propagated usingthe STM until Moon arrival. This is sketched in Fig. 14.

To calculate the required TCM, we apply the Fixed Timeof Arrival method9 (FTA). Variations from the nominalestate at Moon arrival δya can be linearly propagatedfrom the deviation at TCM. This is the superimpositionof the variations right before the maneuver (δya), plusthe variation caused by the maneuver itself (δyTCM)

δya =Φ (ta, tc) (δyc + δyTCM )

=

[Φrr Φrv

Φvr Φvv

](δrc

δvc + ∆vTCM

)(6)

where we set δrTCM = 0 because a maneuver is an in-stantaneous change only of velocity. By setting the finalposition variation to zero (δra = 0), Eq. (6) is satisfiedby

∆vTCM = −δvc −Φ−1rv Φrrδrc, (7)

δva = Φrrδrc + ΦvvΦ−1rv Φrrδrc. (8)

For error case A, employing the TCM calculation al-gorithm above, we obtain a mean magnitude of about0.149 m/s and a standard deviation of 0.07 m/s. The mag-nitude of this maneuver is similar to the OD velocity un-certainty. Thus, no effect can be observed after applyingthe TCM as the maneuver is covered by statistical noise.On the other hand, case B has slightly smaller errors atDV1 epoch, which reduces the TCM mean magnitude to0.138 m/s, with a similar standard deviation. In this case,the TCM is proven to effective as a way to improve thetransfer phase, but it requires the use of Goldstone an-tenna during 3 hours. This is shown in Fig. 15 for thefour DV1 candidates.LANDING PHASE

DV2 designThe solid rocket motor is at an early phase of design andits thrust profile is not available yet. Consequently, weassume constant thrust through the total duration of theburn. If the velocity increment ∆v, the specific impulseIsp, the burn duration T , and the initial spacecraft mass


0 20 40 60 80 100

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(d) DV1 orientation error

Figure 13: Error sensitivity at DV1 epoch. Expected errors are marked by the red ellipses

Figure 14: Trajectory Correction Maneuver flow

m0 are known, the thrust magnitude F can be determinedas

F =m0

(1− exp

(−∆v

Isp

))T

Isp. (9)

Moreover, we assume that the thrust is applied along aninertially fixed direction. To give this orientation, weintroduce the Local-Vertical Local-Horizontal referenceframe (LVLH) 〈ur,us,ut〉 of the OMOTENASHI Se-lenocentric orbit at the burn start epoch. Its axes are givenby the unit vectors aligned with the the radial directionur, the out-of-plane direction ut and the transversal di-

rection us:

ur =r

‖r‖

∣∣∣∣t0

, (10)

ut =v × r‖v × r‖

∣∣∣∣t0

, (11)

us = ur × ut. (12)

With this convention, the local vertical plane is definedby the unit vectors ur and us, and the local horizontalplane is defined by the unit vectors us and ut. The thrustdirection is determined by the angle with respect to thelocal horizontal plane α and the angle with respect to thelocal vertical plane β:

F = F(

cosβ sinα ur + cosβ cosα us − sinβ ut

)(13)

After the solid rocket motor parameters have been fixed,the design of the breaking maneuver has three degreesof freedom: the orientation angles α and β, and the mo-tor ignition time t0. In the first place the out-of-vertical-plane angle β is set to 0, otherwise the deceleration effec-tiveness would be reduced and the residual final-velocity


would increase. To fix the two remaining variables, weimpose two constraints at the motor burn-out: zero verti-cal velocity and a target height hf over the Moon surface.With these constraints, the design problem is mathemat-ically closed and the maneuver parameters can be cal-culated. The DV2 design parameters are summarized inTable 6.

Table 6: DV2 parameters

Design parameter Nominal valuet0 fixed by free-fall conditionα fixed by free-fall conditionβ 0 degIsp 260 s∆v 2.5 km/sT 20 s

The final landing velocity depends on the residual hor-izontal velocity, the target height and the local topogra-phy. This velocity is limited by requirements of the im-pact absorption mechanisms. Current design suggests alimit of 30 m/s for the velocity component perpendicularto the ground, and 100 m/s for the component parallel tothe ground.10

DV2 execution errorErrors in the deceleration maneuver execution may causeOMOTENASHI to deliver the payload to the lunar sur-face at an unacceptable velocity. Unless otherwise speci-fied, all the errors are assumed to follow Gaussian distri-butions.

In the first place, the solid rocket motor ignition couldhappen with a delay with respect to the design ignitiontime. This includes both onboard clock errors and ig-nition mechanism delay. The former are estimated tosmaller than 0.01 s, while the latter are estimated to beabout 0.1 s. These errors, together with OD error, de-mand a shallow FPA when approaching the Moon to min-imize the vertical position error.

Next, the solid rocket engine could show performanceoutside its design point. In particular we consider varia-tions in the specific impulse and thrust duration, causedby a different combustion rate and non-uniformities inthe solid fuel. Additionally, initial fuel mass errors orleft-over fuel will lead to a different total ∆v.

Another important factor is to consider the errors on thethrust direction. OMOTENASHI will be spin-stabilizedduring the deceleration maneuver, being the accuracy ofthe spin axis of about 1 deg. The spin axis can also nu-tate due to perturbations, caused by the initial spin state

and the separation of the orbital module right after igni-tion and perturbations during the motor burn. This ef-fect will average over a nutation period and is modeledby a penalty on the total ∆v. This penalty consists ofa negative half-normal Gaussian distribution whose 3–σstandard deviation is equal to a 2% of the total ∆v.

The execution errors and its values are summarized inTable 7.

Table 7: Maneuver execution errors

Design parameter 3–σ errort0 ±0.11 sα ±1 degβ ±1 degIsp ±5 s

∆v±25 m/s−50 m/s (nutation)

T ±2 s

DV2 sensitivity analysisIn order to assess the robustness of the landing phase,we performed an extensive campaign of Monte-Carlo(MC) simulations with nMC = 103 points, samplingfrom the N-dimensional Gaussian distribution that in-cludes the N sources of error, and propagating the orbituntil OMOTENASHI lands on the real Moon surface. Totest different scenarios, the initial free-fall height was se-lected as hf = {80, 130, 180, 230} m.

For the MC samples that arrive at the Moon surface af-ter the motor burnout, we projected the landing veloc-ity into the local ground-tangent and normal directionsusing the local Moon topography. Figure 16 shows theimpact normal velocity cumulative distribution function(cdf), and reveals a landing success rate between 40%and 65% for all the considered heights and cases. Thesuccess rate could be greatly improved if a higher impactvelocity was admissible, and the free-fall height was aug-mented. The advantages of a trajectory with a shallowFPA is also clear in Fig. 16, as the landing success rate ishigher than for deeper FPA trajectories. A trade-off in-volving the transfer and the landing phase will determinethe FPA of the final trajectory. Finally, Figure 17 showsthe impact ground-tangent velocity cdf for the differentinitial heights. The ground-tangent velocity decreases asthe free-fall initial height increases, which suggests thatthe deceleration maneuver is less efficient if the target fi-nal height is low. If only the ground-tangent velocity wasimportant, one should aim for a higher initial height thanthe values we considered.

For a non-negligible portion of the MC samples and for


all the considered heights and FPAs, OMOTENASHImakes contact with the lunar surface during the solidrocket motor burn. This makes the cdf curves not reachthe 100% of the cases, as these cases are considered asfailures.

To assess the importance of each source of error, the anal-ysis performed above with the full error model was re-peated for case 2 (FPA = −4 deg), considering onlyone error source is acting per simulation. For each MCrun, the standard deviations of the ground-tangent andnormal impact velocities were calculated. Table 8 showsthe 3–σ values of the normal impact velocity. It reflectsthe strong influence of the out-of-horizontal-plane angleα and the burn duration T on the landing dispersion. Ta-ble 9 shows the 3–σ values of the ground-tangent impactvelocity: the dispersion is governed by the DV2 magni-tude and the out-of-vertical-plane angle β.

From these results, one may infer that in order to increasethe success rate of the landing phase, one or several of thefollowing strategies could be considered:

1. Increase the structural limit of the landing devices.This would allow to raise the initial free-fall height,and would decrease the number of premature land-ings and augment the probability of the landing ve-locity to be in the feasible range.

2. Improve the attitude accuracy during the solidrocket engine burn. In this way, the efficiency ofthe deceleration maneuver would be increased.

3. Improve the solid rocket engine performance. Thiswould also lead to a more efficient deceleration ma-neuver

All of these options are under study by the OMOTE-NASHI team, carefully trading-off the increased cost andcomplexity for every subsystem with the accepted mis-sion risk.

If these critical factors were improved, the landing suc-cess rate would highly increase. To illustrate it, we chosea scenario in which for case 2 (FPA = −4 deg) thecritical errors (α, β, T , and ∆v – both magnitude andnutation effect) were cut by half. The rest of the errorswere unchanged with respect of the previous simulations.Results are shown in Figs. 18 and 19, and an overall im-provement can be observed. The success rate is above80%, and if the landing velocity structural limit is raisedto 40 m s−1 it increases well over 95%.CONCLUSIONSIn this paper the current state of the trajectory design ofOMOTENASHI mission was presented. The design of

Table 8: Normal landing velocity 3–σ variation [m/s]

hf 80 m 130 m 180 m 230 mall errors 29.67 26.77 26.05 24.69

OD only 5.86 4.77 3.98 3.12

t0 only 3.26 2.50 2.14 1.16

α only 28.15 25.31 22.03 22.74

β only 1.64 1.03 0.64 0.60

Isp only 0.85 0.64 0.57 0.30

∆v only 3.04 2.67 2.42 1.03

T only 20.18 19.80 18.49 15.63

Table 9: Ground-tangent landing velocity 3–σ varia-tion [m/s]

hf 80 m 130 m 180 m 230 mall errors 38.33 38.34 37.26 37.33

OD only 0.20 0.17 0.14 0.12

t0 only 0.09 0.06 0.05 0.02

α only 1.53 1.83 2.05 2.37

β only 18.18 17.30 19.15 18.41

Isp only 0.02 0.02 0.01 0.01

∆v only 38.11 38.39 39.04 38.81

T only 0.52 0.50 0.46 0.37

the transfer phase, including a cold gas maneuver to tar-get the Moon and a trajectory correction maneuver, wasintroduced. After the transfer phase, a deceleration ma-neuver using a solid rocket motor, reaching a final zerovertical-velocity and a specified height over the Moonsurface, will be followed by a ballistic free-fall.

All error sources were identified and characterized. De-viations from the nominal trajectory were studied andthe most critical contributions were determined, which inturn allows to propose requirements to the design of therelated subsystems in order to increase the success rateof the transfer and landing phases of OMOTENASHI.

The analysis of the landing phase shows the need of atrajectory correction maneuver to compensate for exe-cution errors of the deterministic maneuver. Orbit De-termination requirements can also be drawn from thisstudy, and we determined that OMOTENASHI must re-quest support from international partners in order to usetheir ground stations for a safe landing.

Simulation results of the landing phase identify severalcritical error sources that should be further studied in or-der to increase the landing success rate: structural limit


of the landing devices and accuracy of attitude and solidrocket engine. This is currently being considered byOMOTENASHI team. Additionally, we found that em-ploying DDOR tracking is paramount to reduce the ver-tical position error at Moon landing, which could jeopar-dize the mission.

The effect of the flight path angle at Moon arrival wasalso studied. It was found that for this kind of mission itis necessary to design a shallow flight path angle trajec-tory, shallower than−7 deg. This imposes constraints onthe design of the transfer phase, which becomes stronglycoupled with the landing phase.

In future work, and before OMOTENASHI is launched,there are some additional tasks that OMOTENASHI tra-jectory team must address in their work. This includesfurther refinement of the dynamical model (i.e. includ-ing spherical harmonics, finite thrust burns for the coldgas thruster and solar radiation pressure). In addition,the robustness of the entire trajectory should be studied,since in the present work they are analyzed separately.AcknowledgementsThe Ministry of Education, Culture, Sports, Science andTechnology (MEXT) of the Japanese government sup-ported Javier Hernando-Ayuso with one of its scholar-ships for graduate school students.

The authors are thankful to Junji Kikuchi for providingCAD models of OMOTENASHI flight model.References

1. Funase, R., Koizumi, H., Nakasuka, S.,Kawakatsu, Y., Fukushima, Y., Tomiki, A.,Kobayashi, Y., Nakatsuka, J., Mita, M.,Kobayashi, D. et al.: “50kg-class deep spaceexploration technology demonstration micro-spacecraft PROCYON”, Small Satellite Confer-ence, Utah, USA, 2014.

2. Song, Y.-J., Lee, D., Jin, H. and Kim, B.-Y.: “Po-tential trajectory design for a lunar CubeSat im-pactor deployed from a HEPO using only a smallseparation delta-V”, Advances in Space Research,59(2) (2017), pp. 619–630.

3. Schorr, A. A. and Creech, S. D.: “Space LaunchSystem Spacecraft and Payload Elements: Mak-ing Progress Toward First Launch (AIAA 2016-5418)”, AIAA SPACE 2016, 2016.

4. Hashimoto, T., Yamada, T., Kikuchi, J., Ot-suki, M. and Ikenaga, T.: “Nano Moon Lander:OMOTENASHI”, 31st International Symposiumon Space Technology and Science, 2017-f-053,Matsuyama, Japan, 2017.

5. Campagnola, S., Ozaki, N., Hernando-Ayuso, J.,Oshima, K., Yamaguchi, T., Oguri, K., Ozawa, Y.,Ikenaga, T., Kakihara, K., Takahashi, S., Funase,R. and Hashimoto, Y. K. T.: “Mission Analysisfor EQUULEUS and OMOTENASHI”, 31st In-ternational Symposium on Space Technology andScience, 2017-f-044, Matsuyama, Japan, 2017.

6. Ozawa, Y., Takahashi, S., Hernando-Ayuso, J.,Campagnola, S., Ikenaga, T., Yamaguchi, T. andSarli, B.: “OMOTENASHI Trajectory Analy-sis and Design: Earth-Moon Transfer Phase”,31st International Symposium on Space Technol-ogy and Science, 2017-f-054, Matsuyama, Japan,2017.

7. Hernando-Ayuso, J., Campagnola, S., Ikenaga,T., Yamaguchi, T., Ozawa, Y., Sarli, B. V., Taka-hashi, S. and Yam, C. H.: “OMOTENASHI Tra-jectory Analysis and Design: Landing Phase”,26th International symposium on Space FlightDynamics, held together the 31st InternationalSymposium on Space Technology and Science,2017-d-050, Matsuyama, Japan, 2017.

8. Stough, R.: “REVISED - Delivery of InterimOctober 7th 2018 Launch Post ICPS DisposalState Vectors for Secondary Payload Assess-ment”, Technical report, George C. MarshallSpace Flight Center, NASA, 2016.

9. Battin, R.: An introduction to the mathematicsand methods of astrodynamics, Aiaa, 1999.

10. Yamada, T., Tanno, H. and Hashimoto, T.:“Development of Crushable Shock AbsorptionStructure for OMOTENASHI Semi-hard ImpactProbe”, 31st International Symposium on SpaceTechnology and Science, ISTS-2017-f-055, Mat-suyama, Japan, 2017.


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trajectory design for the jaxa moon nano-lander omotenashi

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