Trajectory Design for the JAXA Moon Nano-Lander OMOTENASHI

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

Javier Hernando-Ayuso, Yusuke Ozawa The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan; +81-42-336-24309 javier.hernando@ac.jaxa.jp

Shota Takahashi The Graduate University for Advanced Studies 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan; +81-42-336-23042 [email protected]

Stefano Campagnola∗ Jet Propulsion Laboratory 4800 Oak Grove Drive, La Canada˜ Flintridge, CA 91011, USA Stefano.Campagnola@jpl.nasa.gov

Toshinori Ikenaga Tsukuba Space Center, Japan Aerospace Exploration Agency 2-1-1 Sengen, Tsukuba-shi, Ibaraki 305-8505, Japan [email protected]

Tomohiro Yamaguchi, Tatsuaki Hashimoto Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency 3-1-1 Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan [email protected]

Chit Hong Yam ispace Inc. 3-1-6 Azabudai, Minato-ku, 106-0041 Tokyo, Japan [email protected]

Bruno V. Sarli Catholic University of America 620 Michigan Ave NE, Washington, DC 20064 USA; +1 301 286 0353 [email protected]

ABSTRACT

OMOTENASHI (Outstanding MOon exploration TEchnologies demonstrated by Nano Semi-Hard Impactor) is a JAXA 6U cubesat that aims to perform a semi-hard landing at the Moon surface after being deployed into a lunar ﬂy-by orbit by the American Space Launch System, Exploration Mission-1. In this paper, we present the analysis and design of OMOTENASHI trajectory, divided in an Earth-Moon transfer using a cold gas thruster and a landing phase using a solid rocket motor. Strong constrains exist between the two phases, making the mission design a very challenging task. The ﬂight path angle at Moon arrival must be shallow in order to minimize the effect of delay of the deceleration maneuver. This, together with the execution error of the cold gas maneuver, demands a correction maneuver to compensate for these errors. Requirements on the ground station tracking are also deduced from this analysis, and it was found that the use of DDOR is an enabling technology for a safe lunar landing. Under the current subsystems 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 rocket motor 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/USU Conference on Small Satellites INTRODUCTION nition to achieve the required deceleration. Finally, the Small satellites are being considered for missions of surface probe will separate from the retromotor module increasing complexity and interest. They offer a re- at burnout to reduce the load on the energy absorption duced cost and development time, which allows to re- mechanisms. spond to technological and science demands in a shorter timescale. Their use in Low Earth Orbit has already been Figure 1 shows the current state of the design of the proven, and there is an growing interest on applying the spacecraft for different parts of the mission. On the concept to interplanetary missions. This was already the top, Fig. 1a shows the orbiting configuration of OMOTE- case of PROCYON, the first interplanetary small satel- NASHI, featuring solar arrays in the +Y face. The solid lite, developed and launched by The University of Tokyo rocket motor, including its sealing lid, is also visible. Be- and JAXA in 2014 as a secondary payload of Hayabusa2 fore DV2, OMOTENASHI will deploy its airbag as can mission. 1 be seen in Fig. 1b. The orbiting module is ejected after the solid rocket motor ignition, being the configuration One type of mission that can greatly benefit from the ad- during the deceleration maneuver as shown in Fig. 1c. A vantages of small satellites is Moon exploration. The use detailed view of the internal parts of OMOTENASHI is of cubesats detaching from Moon-orbiting spacecraft has presented in Fig. 2. Figure 2a shows the Reaction Con- been proposed in the past . 2 However, if a piggyback op- trol System (RCS), attitude control module, communica- portunity in a mission that features a lunar flyby is avail- tion devices, rocket motor and surface probe. Looking able, the mission scenario can be considerably simplified. from a different angle, Fig. 2b shows the battery module, the laser diode (LD) used to ignite the motor, and the de- This opportunity will arise in the first launch of Amer- vices in charge of inflation of the airbag: N2 gas tank and ican Space Launch System (SLS), called Exploration shape memory allow (SMA) opener. Mission-1 (EM-1). After launch in 2019, thirteen 6U cubesats will be injected into a lunar flyby orbit. 3 In this paper we perform a detailed analysis of a semi- JAXA will seize this opportunity with OMOTENASHI hard lunar lander like OMOTENASHI trajectory, includ- (Outstanding MOon exploration TEchnologies demon- ing the Earth-Moon transfer (DV1, TCM) and the land- strated by NAnoSemi-Hard Impactor). OMOTENASHI ing phase (DV2). We propose a design methodology for mission also seeks to study the radiation environment be- DV1 by analyzing the set of feasible solutions that ar- yond Low Earth Orbit in order to support human space rive at the Moon with a small FPA. Results of sensitivity exploration. 4 analysis under OD and maneuver execution errors suggest that a TCM must be considered. Finally, we design However, OMOTENASHI is a challenging mission. One the landing phase imposing zero vertical velocity and a of the main challenges comes from trajectory, which specified height over the Moon surface at burnout. We must be robust to execution and navigation errors. As we identify critical errors in the system, which can be seen present in this paper, a robust trajectory must have a small as requirements for the spacecraft to achieve a safe land- flight path angle (FPA) at Moon arrival. In particular, ing. we found that it must satisfy −7 deg ≤ FPA ≤ 0 deg in TRAJECTORY OVERVIEW order to be error-robust. To this end, the design of the In order to design a trajectory that leads to a safe semi- different arcs of the trajectory cannot be performed inde- hard landing on the surface of the moon, the trajectory is pendently, as they are strongly coupled. divided in two arcs: the transfer and the landing phase.

After detaching from SLS, OMOTENASHI must per- During the transfer phase, OMOTENASHI must modify form two deterministic maneuvers that will make this its orbit from the Moon fly-by injection orbit to a Moon cubesat the first one to perform a semi-hard landing on intersection orbit. 6 Additionally, health check-ups and the Moon. A first maneuver, DV1, will inject OMOTE- orbit determination are key aspects of this phase. OD is a NASHI into a Moon-impacting orbit. After perform- critical resource during the first hours of operation, as the ing midcourse trajectory correction maneuvers (TCM) as 13 delivered cubesats have the same need of accurately needed, a solid rocket motor will be ignited shortly be- assessing the orbit they are flying, and the time before the fore the expected Lunar surface collision at a speed of ap- Moon fly-by/arrival is limited. During the trajectory de- proximately 2.5 km/s. After the deceleration maneuver sign process, this was identified as one of the key aspects (DV2), OMOTENASHI will experience a free-fall from that the small satellite community should address in the a low height (close to 100 m) and arrive at the Moon sur- near future. face with a speed of around 20 m/s. 4, 5 In order to reduce the mass budget, OMOTENASHI is composed of an or- The landing phase starts minutes before arriving at the biting module, a retromotor module and a surface probe. Moon surface, and the main event is the deceleration of The orbiting module must be ejected at rocket motor ig- the spacecraft by the use of a solid rocket motor. 7 An-

Hernando-Ayuso 2 31st Annual AIA/USU Conference on Small Satellites the mass to be decelerated.

In the absence navigation and maneuver execution errors, any Moon-intersecting trajectory would lead to a successful landing, provided that the surface probe can absorb the residual kinetic energy after braking. How- ever, the uncertainty on the actual trajectory introduces very strong constraints between the two phases.

From the point of view of a safe landing, a trajectory with a very shallow FPA is preferred to minimize the effect of timing errors on the vertical displacement of the spacecraft. A high position error on the vertical direction may (a) OMOTENASHI in its orbiting conﬁguration lead to a premature landing during the solid rocket motor burn, jeopardizing the mission. On the other hand, a very shallow FPA might cause missing the Moon in the pres- ence of errors. To reduce the ﬂy-by probability, a TCM may be introduced if necessary. A TCM must be carefully planned in order not to hinder the orbit accuracy during the landing, as it reduces the time to perform OD before the landing phase.

The current analysis and design were conducted with the initial conditions provided by Marshall Space Flight Center 8 and shown in Table 1. The position and velocity components are expressed in a Moon-centered reference frame whose axes are parallel to the J2000 Ecliptic frame. We considered the Sun, Earth and Moon Grav- ity as point masses and an impulsive DV1 maneuver. In the future solar radiation pressure, spherical harmonics (b) OMOTENASHI with deployed airbag and ﬁnite burns will be included, but the results will not qualitatively change.

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

Component Value Epoch 2018 Oct 07 15:39:16 x [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- (c) OMOTENASHI after orbiting module detachment centered frame that rotates with the Moon. Figure 4 shows the landing phase trajectory, including the decel- Figure 1: OMOTENASHI configuration at different eration and final free-fall. parts of the mission ORBIT DETERMINATION Navigation accuracy plays a critical role in the success other important aspect is that in order to successfully de- of a lunar lander like OMOTENASHI. In the first place, liver the required deceleration ∆V , the orbiter module the strong FPA constraints at Moon arrival demand a will be detached from the rest of the spacecraft to reduce precise knowledge of the state vector of the satellite at

Hernando-Ayuso 3 31st Annual AIA/USU Conference on Small Satellites Airbag (folded) Y Attitude Control Module Surface Probe Case Z X Crushable Material LD Module

RCS SMA opener X-band N2 Gas Tank Antenna Y

Z X X-band Transmitter UHF-band Transmitter Radiation Monitor Battery Module

Rocket Motor

(a) OMOTENASHI detailed view (+Y face) (b) OMOTENASHI detailed view (-Y face)

Figure 2: OMOTENASHI subsystems detailed view

Deployment Moon landing phase

DV1

TCM

Figure 3: Transfer phase Figure 4: Landing phase DV1. Moreover, vertical position errors during landing may lead to an early impact with the Moon while the solid motor rocket maneuver is being performed. Timing Space Center (USC), in which we perform communica- errors during landing could also jeopardize the mission, tions, two-way Doppler measurements and 30 minutes of even if the approach trajectory is characterized by a small two-way range measurements. Additionally, OMOTE- FPA. NASHI team has requested the support of of Goldstone (GDS), part of NASA Deep Space Station (DSN). In case Moreover, we found that the actual trajectory has a strong A, we consider only 30 minutes of two-way Doppler inﬂuence in the position accuracy at Moon arrival. We measurements, while for case B we increase the dura- observed differences up to one order of magnitude in tion to 3 hours and also include 30 minutes of two-way the vertical position error at landing when following dif- ranging. For both cases, we plan on requesting the use of ferent orbits for the same tracking strategy. However, a Madrid (MAD) antenna for DV1 uplink. Figure 5 shows small-satellite operator has in most cases limited control the observations planning for DV1. The expected posi- over the orbit he is being deployed into. Thus, great care tion and velocity errors are reproduced in Table 3. must be taken when designing the OD strategy to be employed. We also performed a similar analysis to up to TCM epoch. We maintain the same strategy used for A and We will considered the observables for orbit determina- B cases for an additional day of tracking. Figure 6 shows tion reported in Table 2. the ground stations coverage up to TCM epoch. Once again, the uplink of the TCM command is planned with During the transfer phase, the orbit knowledge is critical the support of MAD. The expected navigation errors are when designing the trajectory. We are especially con- shown in Table 4. cerned about the OD errors at DV1 and TCM epoch. We study two different cases that differ in the amount After DV1/TCM, a tentative OD analysis was performed of resources employed, and label them as A and B. For using only JAXA resources to decrease the mission cost both cases we consider 3 hours tracking from Uchinoura and complexity. To this end, a campaign of observations

Hernando-Ayuso 4 31st Annual AIA/USU Conference on Small Satellites Table 2: Orbit Determination observables

type duration interval 1–σ noise bias X-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 bias DDOR (GDS-CAN, CAN-MAD) 30 min 600 s 1 ns no bias

Table 3: Orbit Determination 3–σ errors at DV1 Table 4: Orbit determination 3–σ errors at TCM epoch epoch

Case Error T N H Case Error T N H position [km] 1.5 2.3 14.4 position [km] 3.3 0.7 23.5 A A velocity [cm/s] 0.3 1.5 21.1 velocity [cm/s] 15.4 4.4 28.3 position [km] 0.1 0.2 2.3 position [km] 0.8 0.2 1.3 B B velocity [cm/s] 0.1 0.4 2.6 velocity [cm/s] 1.5 0.3 1.5

Blue : Spacecraft elevation, Red : Sun elevation

DSN Goldstone Data-cut-off the communication windows are shown in the upper part of Fig. 7. During the last communication window no OD GDS JAXA USC34 30 min (case 1) or is planned, as a cut-out time for ﬁnal computation and 3 hours (case 2) uplink of DV2 is introduced. The simulated covariance

Disposal matrix at DV2 epoch provides a vertical error close to USC34 DSN Madrid 3 hours 400 m, unacceptable for a safe landing with a free-fall initial height of around 100 m.

MAD DV1 30 min After studying several conﬁgurations, it was decided to include Delta Differential One-way Ranging (delta-DOR Figure 5: Ground station visibility and observations or DDOR) measurements using DSN stations in Gold- planning for DV1 stone (GDS), Canberra (CAN) and Madrid (MAD). This method requires two stations to be visible at the same GDS 30 min (case 1) or time, and can be accomplished with the combinations 3 hours (case 2) GDS-CAN and CAN-MAD as can be seen in Fig. 7. The measurements are characterized by the lower row of Ta- ble 2. Results show a vertical 3–σ error of the order of USC34 50 m, which would not jeopardize the landing maneuver 3 hours (all cases) as it will be shown later. The position covariance matrix at the solid rocket motor ignition, expressed in the J2000 2 DV1 Ecliptic reference system and in km , is: TCM DV2   1.3 × 10−3 −9.7 × 10−4 1.7 × 10−3  −4 −4 −3 CDV2 = −9.7 × 10 7.2 × 10 −1.3 × 10  1.7 × 10−3 −1.3 × 10−3 2.4 × 10−3

This makes DDOR a critical element of the mission nec- Figure 6: Ground station visibility and observations essary for its success. planning for TCM TRANSFER PHASE In this section we introduce OMOTENASHI transfer from USC was planned and simulated. We performed phase design strategy. First, we describe the calcula- 3 sets measurements spanning 3 hours using two-way tion of DV1 maneuvers that lead to a shallow FPA at Doppler and including 30 minutes of two-way-ranging. Moon arrival. Next, we analyze the trajectory sensitiv- In the simulation, a priori covariance was not consid- ity to navigation and execution errors. Finally, and based ered. The visibility from the USC ground station and on the previous point, we present the design of the TCM

Hernando-Ayuso 5 31st Annual AIA/USU Conference on Small Satellites DV1 TCM DDOR x 4 DCO DV2 safe landing. Magnitudes close to 15 m/s are promising (GDS-CAN, CAN-MAD) with the current initial condition, as wide regions with shallow FPA are available. Figure 9 shows the effect of varying the maneuver orientation for a ﬁxed magnitude Not included in OD DV2 command upload of 15 m/s, where FPA smaller than −20 deg were truncated and shown in blue color.

with TCM 90 without TCM 75 60 45 30 15 0 -15 Figure 7: Ground station visibility and observations -30 10 m/s planning after DV1/TCM. CAN is omitted because its -45 12.5 m/s Selenograhic latitude [deg] -60 15 m/s visibility is similar to USC 17.5 m/s -75 20 m/s -90 -180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 Selenograhic longitude [deg] to compensate for errors at DV1 epoch. Figure 8: Landing location of DV1 coarse grid search, DV1 with 12 deg resolution and different magnitudes > We can characterize a maneuver ∆v = (vx, vy, vz) by We set 15 m/s as the nominal magnitude and study the its magnitude and two orientation angles. To this end, we nominal orientation by iterative grid refinement. In each introduce the azimuth φ and polar angle θ defined in the iteration, the grid is refined filtering out unfeasible ori- J2000 Ecliptic reference frame as entations that lead to a too steep FPA, flybys or the far side of the Moon. Figure 10a shows the grid solution φ = atan2 (v , v ) (1) y z space of the thrust directions with the magnitude fixed at 15 m/s. The non-colored region corresponds to trajectories which will not intersect the lunar surface, performing v θ = cos−1 z (2) a flyby. The solutions with shallow FPA exist near the kvk boundary of the colored region. Figure 10b shows the Then, ∆v takes the form result after the first grid refinement. This iterative process consists not only in a refinement of the border so-   v sin θ cos φ lutions, but also in discarding unfeasible solutions such as with steep FPA or on the far-side of the Moon. Fig- ∆v = v sin θ sin φ (3)   ure 10c is the result of 7 refinement iterations (Resolution v cos θ is 12/27 ' 0.094deg).

Figures 11 and 12 are DV1 direction solutions and their We split the design of DV1 into two parts. First, we per- corresponding arrival points at the average Moon sur- form a coarse grid search for all orientations and different face, respectively. Red thrust direction solutions (Fig. magnitudes. Once we fix the magnitude as a compro- 11) and red arrival points (Fig. 12) represent the trajecto- mise between fuel consumption and size of the feasible ries that might be hindered at arrival by the lunar surface, region, we use an iterative process to obtain a fine grid i.e. craters or mountains. Blue solutions correspond to of feasible maneuver orientations. In this analysis we are the trajectories on which the probe can safely reach the considering impulsive DV1 maneuvers, but in the follow- Moon surface. Four candidate nominal DV1 solutions ing months and after the cold gas system design is fixed, are marked with green symbols in these graphs. Table 5 finite thrust will be incorporated into our analysis. contains the parameters for these four candidates. Note that we shifted case 3 (FPA = −5.5 deg instead of −5) Figure 8 shows the result of the grid search for all thrust to avoid landing uphill at the outer wall of crater Zeno P directions (12 deg resolution) and magnitudes from 10 to (latitude 43.4 N, longitude 66.1 E). 20 m/s. We can determine the nominal DV1 magnitude by evaluating the number of solutions that have shallow FPA and arrive at the near side of the Moon. While for DV1 Error Analysis magnitudes of 10 and 12.5 m/s some of the trajectories We studied the errors at DV1 epoch and how they af- arrive at the Moon, their FPA is too deep to lead to a fect the Moon arrival using linear theory. A Monte-Carlo

Hernando-Ayuso 6 31st Annual AIA/USU Conference on Small Satellites 90 0 75 60 45 -5 30 15 0 -10 -15 -30

-45 -15 Flight Path Angle [deg]

Selenograhic latitude [deg] -60 -75 -90 -20 -180 -150 -120 -90 -60 -30 0 30 60 90 120 150 180 Selenograhic longitude [deg]

Figure 9: Landing location and ﬂight path angle of (a) 12 deg resolution DV1 direction coarse grid search, for 15 m/s. FPA deeper 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 (b) 6 deg resolution simulation performed using linear propagation provides great advantages from a computational point of view, as a high number of samples would take an unacceptably long time if a numerical propagation was used. Thus, numerical propagation will only be used for validation, and we will employ linearization around the nominal orbit to evaluate the errors at Moon arrival.

Considering a reference orbit y (t), the state transition matrix (STM) between a initial epoch ti and a ﬁnal epoch 9 tf can be deﬁned as (c) 0.09375 deg resolution ∂y (ti) Φ (tf , ti) = (4) ∂y (tf ) Figure 10: Iterative grid search By use of the STM, deviations from the reference orbit δy can be calculated as

δy (tf ) = Φ (tf , ti) δy (5) the Moon considered as a point mass, we can also model this correction as a Keplerian orbit and solve for the an- When tf is the epoch in which the nominal solution ar- gular position that makes the orbital radius equal to the rives at the Moon surface, the linear method provides Moon radius. These two approaches were evaluated to position vectors not contained on the Moon surface. To provide accurate enough results for practical situations correct for this time delay, the samples must be forward and were validated using high-accuracy numerical prop- or backward propagated to the Moon surface. Depend- agations, 6 which yield the most accurate solution to the ing on the dispersion of the samples and the degree of adjustment of the final points. accuracy needed, several methods are possible. The sim- plest option is to assume a free fall over a flat Moon and The sensitivity to navigation and execution errors at DV1 solve the parabolic flight equations. The model can be epoch can be evaluated in a reasonable time using the lin- expanded to consider the Moon curvature for example. 6 ear approach presented above. We apply independently If we neglect forces other than the gravitational pull of isotropic position, isotropic velocity, DV1 magnitude and

Hernando-Ayuso 7 31st Annual AIA/USU Conference on Small Satellites 47 orbit.

46.5 To this end, we ﬁrst assume a Gaussian dispersion at DV1 4 46 3 which includes OD and execution error. Next, we sample 45.5 2 1 the uncertainty and for every point we map it to TCM 45 epoch using linear theory. For each sample we calculate [deg] 44.5 a correction maneuver, and apply OD and execution error at TCM epoch. The results are ﬁnally propagated using 44 the STM until Moon arrival. This is sketched in Fig. 14. 43.5

43 10 20 30 40 50 60 70 80 90 100 110 To calculate the required TCM, we apply the Fixed Time [deg] of Arrival method 9 (FTA). Variations from the nominal estate at Moon arrival δy can be linearly propagated Figure 11: Final iteration of the direction grid search a from the deviation at TCM. This is the superimposition (Zooming of Fig. 10c. Note that axis are not to scale). of the variations right before the maneuver (δy ), plus Nominal DV1 candidates are marked in green a the variation caused by the maneuver itself (δyTCM)

δya =Φ (ta, tc)(δyc + δyTCM ) " # ! Φ Φ δr (6) = rr rv c Φvr Φvv δvc + ∆vTCM

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

−1 ∆vTCM = −δvc − Φrv Φrrδrc, (7)

Figure 12: Moon arrival points with FPA isolines. −1 Nominal DV1 candidates arrival points are marked δva = Φrrδrc + ΦvvΦrv Φrrδrc. (8) in green For error case A, employing the TCM calculation al- gorithm above, we obtain a mean magnitude of about DV1 direction errors of different magnitudes and ana- 0.149 m/s and a standard deviation of 0.07 m/s. The mag- lyze the success rate of the transfer phase. We consider nitude of this maneuver is similar to the OD velocity un- as failure those cases in which OMOTENASHI misses certainty. Thus, no effect can be observed after applying the Moon surface and ﬂybys the Moon, or if the FPA the TCM as the maneuver is covered by statistical noise. is steeper than −7 deg. Figure 13 shows the result of the On the other hand, case B has slightly smaller errors at sensitivity analysis. The expected errors marked with red DV1 epoch, which reduces the TCM mean magnitude to ellipses, where we considered the 3–σ errors for DV1 0.138 m/s, with a similar standard deviation. In this case, magnitude and direction as 1% and 1 deg respectively. the TCM is proven to effective as a way to improve the Orbit determination errors are not critical neither on OD transfer phase, but it requires the use of Goldstone an- cases A or B as can be seen on Figs. 13a and 13b. How- tenna during 3 hours. This is shown in Fig. 15 for the ever, Figs. 13c and 13d reveal that DV1 execution error four DV1 candidates. may jeopardize the transfer to the Moon. Figure 13 is a powerful tool for the trajectory design team, as it allows LANDING PHASE one to draw as an important conclusion that a Trajectory Correction Maneuver must be considered. DV2 design The solid rocket motor is at an early phase of design and Trajectory Correction Maneuver its thrust proﬁle is not available yet. Consequently, we

At TCM epoch, we apply an impulse ∆vTCM to com- assume constant thrust through the total duration of the pensate for errors in DV1. Our strategy will be based on burn. If the velocity increment ∆v, the speciﬁc impulse re-targeting the same landing location as in the nominal Isp, the burn duration T , and the initial spacecraft mass

Hernando-Ayuso 8 31st Annual AIA/USU Conference on Small Satellites 1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6 Nominal Case 1 Nominal Case 1 Nominal Case 2 Nominal Case 2 Transfer success probability 0.5 Nominal Case 3 Transfer success probability 0.5 Nominal Case 3 Nominal Case 4 Nominal Case 4 0.4 0.4 0 20 40 60 80 100 1 2 3 4 5 6 7 8 9 10 3 position error [km] 3 velocity error [cm/s] (a) Orbit determination position error (b) Orbit determination velocity error

1 1

0.9 0.9

0.8 0.8

0.7 0.7

0.6 0.6 Nominal Case 1 Nominal Case 1 Nominal Case 2 Nominal Case 2 Transfer success probability 0.5 Transfer success probability 0.5 Nominal Case 3 Nominal Case 3 Nominal Case 4 Nominal Case 4 0.4 0.4 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 3 DV1-size error [%] 3 DV1 orientation error [deg] (c) DV1 magnitude error (d) DV1 orientation error

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

rection us:

r ur = , (10) krk t0

v × r ut = , (11) kv × rk t0

us = ur × ut. (12)

With this convention, the local vertical plane is defined by the unit vectors ur and us, and the local horizontal Figure 14: Trajectory Correction Maneuver flow plane is defined by the unit vectors us and ut. The thrust direction is determined by the angle with respect to the local horizontal plane α and the angle with respect to the local vertical plane β: m0 are known, the thrust magnitude F can be determined as F = F cos β sin α ur + cos β cos α us − sin β ut ∆v m0 1 − exp − I (13) F = sp I . (9) T sp Moreover, we assume that the thrust is applied along an After the solid rocket motor parameters have been fixed, inertially fixed direction. To give this orientation, we the design of the breaking maneuver has three degrees introduce the Local-Vertical Local-Horizontal reference of freedom: the orientation angles α and β, and the mo- frame (LVLH) hur, us, uti of the OMOTENASHI Se- tor ignition time t0. In the first place the out-of-vertical- lenocentric orbit at the burn start epoch. Its axes are given plane angle β is set to 0, otherwise the deceleration effec- by the unit vectors aligned with the the radial direction tiveness would be reduced and the residual final-velocity ur, the out-of-plane direction ut and the transversal di-

Hernando-Ayuso 9 31st Annual AIA/USU Conference on Small Satellites would increase. To ﬁx the two remaining variables, we and the separation of the orbital module right after igni- impose two constraints at the motor burn-out: zero verti- tion and perturbations during the motor burn. This ef- cal velocity and a target height hf over the Moon surface. fect will average over a nutation period and is modeled With these constraints, the design problem is mathemat- by a penalty on the total ∆v. This penalty consists of ically closed and the maneuver parameters can be cal- a negative half-normal Gaussian distribution whose 3–σ culated. The DV2 design parameters are summarized in standard deviation is equal to a 2% of the total ∆v. Table 6. The execution errors and its values are summarized in Table 6: DV2 parameters Table 7.

Design parameter Nominal value Table 7: Maneuver execution errors

t0 fixed by free-fall condition Design parameter 3–σ error α fixed by free-fall condition t ±0.11 s β 0 deg 0 α ±1 deg Isp 260 s β ±1 deg ∆v 2.5 km/s I ±5 s T 20 s sp ±25 m/s ∆v −50 m/s (nutation) The final landing velocity depends on the residual hor- T ±2 s izontal velocity, the target height and the local topography. This velocity is limited by requirements of the impact absorption mechanisms. Current design suggests a DV2 sensitivity analysis limit of 30 m/s for the velocity component perpendicular In order to assess the robustness of the landing phase, to the ground, and 100 m/s for the component parallel to 10 we performed an extensive campaign of Monte-Carlo the ground. 3 (MC) simulations with nMC = 10 points, sampling from the N-dimensional Gaussian distribution that in- DV2 execution error cludes the N sources of error, and propagating the orbit Errors in the deceleration maneuver execution may cause until OMOTENASHI lands on the real Moon surface. To OMOTENASHI to deliver the payload to the lunar sur- test different scenarios, the initial free-fall height was se- face at an unacceptable velocity. Unless otherwise speci- lected as hf = {80, 130, 180, 230} m. fied, all the errors are assumed to follow Gaussian distri- butions. For the MC samples that arrive at the Moon surface after the motor burnout, we projected the landing veloc- In the first place, the solid rocket motor ignition could ity into the local ground-tangent and normal directions happen with a delay with respect to the design ignition using the local Moon topography. Figure 16 shows the time. This includes both onboard clock errors and ig- impact normal velocity cumulative distribution function nition mechanism delay. The former are estimated to (cdf), and reveals a landing success rate between 40% smaller than 0.01 s, while the latter are estimated to be and 65% for all the considered heights and cases. The about 0.1 s. These errors, together with OD error, de- success rate could be greatly improved if a higher impact mand a shallow FPA when approaching the Moon to min- velocity was admissible, and the free-fall height was aug- imize the vertical position error. mented. The advantages of a trajectory with a shallow FPA is also clear in Fig. 16, as the landing success rate is Next, the solid rocket engine could show performance higher than for deeper FPA trajectories. A trade-off in- outside its design point. In particular we consider varia- volving the transfer and the landing phase will determine tions in the specific impulse and thrust duration, caused the FPA of the final trajectory. Finally, Figure 17 shows by a different combustion rate and non-uniformities in the impact ground-tangent velocity cdf for the different the solid fuel. Additionally, initial fuel mass errors or initial heights. The ground-tangent velocity decreases as left-over fuel will lead to a different total ∆v. the free-fall initial height increases, which suggests that the deceleration maneuver is less efficient if the target fi- Another important factor is to consider the errors on the nal height is low. If only the ground-tangent velocity was thrust direction. OMOTENASHI will be spin-stabilized important, one should aim for a higher initial height than during the deceleration maneuver, being the accuracy of the values we considered. the spin axis of about 1 deg. The spin axis can also nu- tate due to perturbations, caused by the initial spin state For a non-negligible portion of the MC samples and for

Hernando-Ayuso 10 31st Annual AIA/USU Conference on Small Satellites all the considered heights and FPAs, OMOTENASHI Table 8: Normal landing velocity 3–σ variation [m/s] makes contact with the lunar surface during the solid rocket motor burn. This makes the cdf curves not reach hf 80 m 130 m 180 m 230 m the 100% of the cases, as these cases are considered as all errors 29.67 26.77 26.05 24.69 failures. OD only 5.86 4.77 3.98 3.12 t only 3.26 2.50 2.14 1.16 To assess the importance of each source of error, the anal- 0 ysis performed above with the full error model was re- α only 28.15 25.31 22.03 22.74 peated for case 2 (FPA = −4 deg), considering only β only 1.64 1.03 0.64 0.60 one error source is acting per simulation. For each MC Isp only 0.85 0.64 0.57 0.30 run, the standard deviations of the ground-tangent and ∆v only 3.04 2.67 2.42 1.03 normal impact velocities were calculated. Table 8 shows the 3–σ values of the normal impact velocity. It reﬂects T only 20.18 19.80 18.49 15.63 the strong inﬂuence of the out-of-horizontal-plane angle α and the burn duration T on the landing dispersion. Ta- ble 9 shows the 3–σ values of the ground-tangent impact Table 9: Ground-tangent landing velocity 3–σ varia- velocity: the dispersion is governed by the DV2 magni- tion [m/s] tude and the out-of-vertical-plane angle β. hf 80 m 130 m 180 m 230 m From these results, one may infer that in order to increase all errors 38.33 38.34 37.26 37.33 the success rate of the landing phase, one or several of the OD only 0.20 0.17 0.14 0.12 following strategies could be considered: t0 only 0.09 0.06 0.05 0.02 α only 1.53 1.83 2.05 2.37 1. Increase the structural limit of the landing devices. β only 18.18 17.30 19.15 18.41 This would allow to raise the initial free-fall height, I only 0.02 0.02 0.01 0.01 and would decrease the number of premature land- sp ings and augment the probability of the landing ve- ∆v only 38.11 38.39 39.04 38.81 locity to be in the feasible range. T only 0.52 0.50 0.46 0.37

2. Improve the attitude accuracy during the solid rocket engine burn. In this way, the efficiency of the deceleration maneuver would be increased. the transfer phase, including a cold gas maneuver to target the Moon and a trajectory correction maneuver, was 3. Improve the solid rocket engine performance. This introduced. After the transfer phase, a deceleration ma- would also lead to a more efficient deceleration maneuver using a solid rocket motor, reaching a final zero neuver vertical-velocity and a specified height over the Moon surface, will be followed by a ballistic free-fall. All of these options are under study by the OMOTE- All error sources were identified and characterized. De- NASHI team, carefully trading-off the increased cost and viations from the nominal trajectory were studied and complexity for every subsystem with the accepted mis- the most critical contributions were determined, which in sion risk. turn allows to propose requirements to the design of the related subsystems in order to increase the success rate If these critical factors were improved, the landing suc- of the transfer and landing phases of OMOTENASHI. cess rate would highly increase. To illustrate it, we chose a scenario in which for case 2 (FPA = −4 deg) the The analysis of the landing phase shows the need of a critical errors (α, β, T , and ∆v – both magnitude and trajectory correction maneuver to compensate for exe- nutation effect) were cut by half. The rest of the errors cution errors of the deterministic maneuver. Orbit De- were unchanged with respect of the previous simulations. termination requirements can also be drawn from this Results are shown in Figs. 18 and 19, and an overall im- study, and we determined that OMOTENASHI must re- provement can be observed. The success rate is above quest support from international partners in order to use 80%, and if the landing velocity structural limit is raised their ground stations for a safe landing. to 40 m s−1 it increases well over 95%. CONCLUSIONS Simulation results of the landing phase identify several In this paper the current state of the trajectory design of critical error sources that should be further studied in or- OMOTENASHI mission was presented. The design of der to increase the landing success rate: structural limit

Hernando-Ayuso 11 31st Annual AIA/USU Conference on Small Satellites of the landing devices and accuracy of attitude and solid 5. Campagnola, S., Ozaki, N., Hernando-Ayuso, J., rocket engine. This is currently being considered by Oshima, K., Yamaguchi, T., Oguri, K., Ozawa, Y., OMOTENASHI team. Additionally, we found that em- Ikenaga, T., Kakihara, K., Takahashi, S., Funase, ploying DDOR tracking is paramount to reduce the ver- R. and Hashimoto, Y. K. T.: “Mission Analysis tical position error at Moon landing, which could jeopar- for EQUULEUS and OMOTENASHI”, 31st In- dize the mission. ternational Symposium on Space Technology and Science, 2017-f-044, Matsuyama, Japan, 2017. The effect of the flight path angle at Moon arrival was also studied. It was found that for this kind of mission it 6. Ozawa, Y., Takahashi, S., Hernando-Ayuso, J., is necessary to design a shallow flight path angle trajec- Campagnola, S., Ikenaga, T., Yamaguchi, T. and tory, shallower than −7 deg. This imposes constraints on Sarli, B.: “OMOTENASHI Trajectory Analy- the design of the transfer phase, which becomes strongly sis and Design: Earth-Moon Transfer Phase”, coupled with the landing phase. 31st International Symposium on Space Technol- ogy and Science, 2017-f-054, Matsuyama, Japan, In future work, and before OMOTENASHI is launched, 2017. there are some additional tasks that OMOTENASHI tra- 7. Hernando-Ayuso, J., Campagnola, S., Ikenaga, jectory team must address in their work. This includes T., Yamaguchi, T., Ozawa, Y., Sarli, B. V., Taka- further refinement of the dynamical model (i.e. includ- hashi, S. and Yam, C. H.: “OMOTENASHI Tra- ing spherical harmonics, finite thrust burns for the cold jectory Analysis and Design: Landing Phase”, gas thruster and solar radiation pressure). In addition, 26th International symposium on Space Flight the robustness of the entire trajectory should be studied, Dynamics, held together the 31st International since in the present work they are analyzed separately. Symposium on Space Technology and Science, Acknowledgements 2017-d-050, Matsuyama, Japan, 2017. The Ministry of Education, Culture, Sports, Science and 8. Stough, R.: “REVISED - Delivery of Interim Technology (MEXT) of the Japanese government sup- October 7th 2018 Launch Post ICPS Disposal ported Javier Hernando-Ayuso with one of its scholar- State Vectors for Secondary Payload Assess- ships for graduate school students. ment”, Technical report, George C. Marshall Space Flight Center, NASA, 2016. The authors are thankful to Junji Kikuchi for providing CAD models of OMOTENASHI flight model. 9. Battin, R.: An introduction to the mathematics References and methods of astrodynamics, Aiaa, 1999. 1. Funase, R., Koizumi, H., Nakasuka, S., Kawakatsu, Y., Fukushima, Y., Tomiki, A., 10. Yamada, T., Tanno, H. and Hashimoto, T.: Kobayashi, Y., Nakatsuka, J., Mita, M., “Development of Crushable Shock Absorption Kobayashi, D. et al.: “50kg-class deep space Structure for OMOTENASHI Semi-hard Impact exploration technology demonstration micro- Probe”, 31st International Symposium on Space spacecraft PROCYON”, Small Satellite Confer- Technology and Science, ISTS-2017-f-055, Mat- ence, Utah, USA, 2014. suyama, Japan, 2017.

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

3. Schorr, A. A. and Creech, S. D.: “Space Launch System 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 Symposium on Space Technology and Science, 2017-f-053, Matsuyama, Japan, 2017.

Hernando-Ayuso 12 31st Annual AIA/USU Conference on Small Satellites 55 0 1 Nominal 0.9 TCM (Colormap) -2 50 No TCM (Gray) 0.8 Topography (Red) -4 0.7 45 0.6 -6 0.5

40 FPA cdf 0.4 -8 0.3 35 -10 0.2 Flight Path Angle [deg] TCM 0.1 No TCM Selenographic latitude [deg] 30 -12 0 55 60 65 70 75 80 85 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Selenographic longitude [deg] Flight Path Angle [deg] (a) Nominal DV1 candidate No.1

55 0 1 Nominal 0.9 TCM (Colormap) -2 50 No TCM (Gray) 0.8 Topography (Red) -4 0.7 45 0.6 -6 0.5

40 FPA cdf 0.4 -8 0.3 35 0.2

-10 Flight Path Angle [deg] TCM 0.1 No TCM Selenographic latitude [deg] 30 -12 0 55 60 65 70 75 80 85 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Selenographic longitude [deg] Flight Path Angle [deg] (b) Nominal DV1 candidate No.2

55 0 1 Nominal TCM (Colormap) 0.9 -2 50 No TCM (Gray) 0.8 Topography (Red) -4 0.7 45 0.6 -6 0.5

40 FPA cdf 0.4 -8 0.3 35 0.2 -10 Flight Path Angle [deg] TCM 0.1

Selenographic latitude [deg] No TCM 30 -12 0 55 60 65 70 75 80 85 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Selenographic longitude [deg] Flight Path Angle [deg] (c) Nominal DV1 candidate No.3

55 0 1 Nominal TCM (Colormap) 0.9 -2 50 No TCM (Gray) 0.8 Topography (Red) -4 0.7 45 0.6 -6 0.5

40 FPA cdf 0.4 -8 0.3 35 0.2 -10 Flight Path Angle [deg] TCM 0.1

Selenographic latitude [deg] No TCM 30 -12 0 55 60 65 70 75 80 85 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 Selenographic longitude [deg] Flight Path Angle [deg] (d) Nominal DV1 candidate No.4

Figure 15: Effect of TCM for knowledge error case B

Hernando-Ayuso 13 31st Annual AIA/USU Conference on Small Satellites 1 1 80 m 80 m 0.9 130 m 0.9 130 m 180 m 180 m 0.8 230 m 0.8 230 m

0.7 0.7 cdf cdf

0.6 0.6

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0.3 0.3 perpendicular,impact perpendicular,impact V V 0.2 0.2

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0 0 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 V [m/s] V [m/s] perpendicular,impact perpendicular,impact (a) Case 1 (FPA = −3 deg) (b) Case 2 (FPA = −4 deg) 1 1 80 m 80 m 0.9 130 m 0.9 130 m 180 m 180 m 0.8 230 m 0.8 230 m

0.7 0.7 cdf cdf

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0 0 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 V [m/s] V [m s -1] perpendicular,impact perpendicular,impact (c) Case 3 (FPA = −5.5 deg) (d) Case 4 (FPA = −6 deg)

Figure 16: Normal impact velocity cdf considering the full error model

Hernando-Ayuso 14 31st Annual AIA/USU Conference on Small Satellites 1 1 80 m 80 m 0.9 130 m 0.9 130 m 180 m 180 m 0.8 230 m 0.8 230 m

cdf 0.7 cdf 0.7

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0.3 0.3 ground-tangent,impact ground-tangent,impact V V 0.2 0.2

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0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 V [m/s] V [m/s] ground-tangent,impact ground-tangent,impact (a) Case 1 (FPA = −3 deg) (b) Case 2 (FPA = −4 deg) 1 1 80 m 80 m 0.9 130 m 0.9 130 m 180 m 180 m 0.8 230 m 0.8 230 m

cdf 0.7 cdf 0.7

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0 0 0 10 20 30 40 50 60 70 80 90 100 0 10 20 30 40 50 60 70 80 90 100 V [m/s] V [m/s] ground-tangent,impact ground-tangent,impact (c) Case 3 (FPA = −5.5 deg) (d) Case 4 (FPA = −6 deg)

Figure 17: Ground-tangent impact velocity cdf considering the full error model.

Hernando-Ayuso 15 31st Annual AIA/USU Conference on Small Satellites 1 80 m 0.9 130 m 180 m 0.8 230 m

0.7 cdf

0.6

0.5

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0.3 perpendicular,impact V 0.2

0.1

0 0 5 10 15 20 25 30 35 40 45 50 V [m/s] perpendicular,impact

Figure 18: Normal impact velocity cdf with improvement on critical errors (FPA = −4 deg)

1 80 m 0.9 130 m 180 m 0.8 230 m

cdf 0.7

0.6

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0 0 10 20 30 40 50 60 70 80 90 100 V [m/s] ground-tangent,impact

Figure 19: Ground-tangent impact velocity cdf with improvement on critical errors (FPA = −4 deg)

Hernando-Ayuso 16 31st Annual AIA/USU Conference on Small Satellites