OR2‐2微小重力天体着陸時の探査機の推進剤のスロッシングのメカニカルモデルの提案と検証
Mechanical Model of Spacecraft Propellant Sloshing
during Decent Operation on Microgravity Celestial Body
高橋正樹 1, 岡本朔弥 1, 古市侑太郎 2, 八幡祐成 2, 姫野武洋 2, 大槻真嗣 3, 馬場満久 3, 今田高峰 3
Masaki TAKAHASHI1, Sakumi OKAMOTO1, Yutaro FURUICHI2, Yusei YAHATA2, Takehiro
HIMENO2, Masatsugu OTSUKI3, Mitsuhisa BABA3 and Takane IMADA3
慶應義塾大学，Keio University,
東京大学，The University of Tokyo,
3 宇宙航空研究開発機構，Japan Aerospace Exploration Agency,
12

1. Background
JAXA plans to explore a Martian satellite, which is a microgravity object in the Martian atmosphere, basedon its experience in microgravity object exploration, as represented by the Hayabusa project1). This plan isreferred to as Mars Moons Exploration (MMX; Martian Moons eXploration)2). One of the challenges in landingis that the spacecraft has a large amount of liquid propellant and oxidizer, and there are concerns regarding
the effects of sloshing 3). Additionally, landing on a slope needs to be considered, and the effects of sloshingmust be investigated. Computational Fluid Dynamics (CFD) fluid analysis is required to reproduce sloshingaccurately; however, it is computationally time‐consuming. Therefore, if there are multiple possible landing
scenarios, it is necessary to propose a reduced model that can accurately reproduce sloshing with a smaller
computational load than that of CFD.
Because three landing scenarios are assumed in MMX, the proposed model should be able to accommodatechanges in the sloshing phenomenon owing to the different amounts of propellant and oxidizer remaining.
This study proposes a mechanical model that can reproduce the sloshing phenomenon of a propellant
onboard a spacecraft while landing on a microgravity object. The values of the model parameters weredetermined according to the operational scenario using a genetic algorithm. Numerical calculations wereperformed and compared with the CFD analysis to validate the model with the obtained parameters.
2.

Degenerate model of propellant

2.1. The results of CFD analysis
The CFD results for each evaluation item of the axisymmetric sloshing of the propellant for the input
acceleration during landing are shown in Figures 1 and 2. Focusing on the shape transition of a propellant
1 of 8

with a nominal liquid level as an example, the following observations were made.
①

At approximately 16–17 s, a Z‐positive reaction force is generated owing to the downward movementof the liquid surface.

②

At approximately 17–19 s, a Z‐negative reaction force was generated owing to the upward movement ofthe liquid surface.

③

At approximately 20 s, the load peak occurred owing to the collision of the upper end of the liquidcolumn.

④

At approximately 20–24 s, the upward movement of the liquid through the liquid column and the Z‐
positive load due to the collision continued.

⑤

After approximately 23 s, part of the colliding liquid returns and hits the liquid surface.

⑥

After 24 s, the load decreased while the wave propagated across the interface.

Figure 1. Animation of CFD results in nominal scenario.

①Fz [N]

10③④⑥0⑤②-1005

1015

20

2530

202530time [s](a) Tank reaction force.

zg [m]

0.5

0-0.5

05

1015

time [s](b) Center of gravity
Figure 2. CFD results during 1st decent operation on microgravity celestial body.
2 of 8Fz [N]100-10051015202530202530time [s](a) Tank reaction force.

zg [m]

0.50-0.5051015time [s](b) Center of gravity
Figure 3. CFD results during 2nd decent operation (nominal scenario) on microgravity celestial body.

Fz [N]

100-10051015202530202530time [s](a) Tank reaction force.

zg [m]

0.50-0.5051015time [s](b) Center of gravity
Figure 4. CFD results during 3rd decent operation on microgravity celestial body.

3 of 8(a) Front side(b) Top side
Figure 5. Proposal model(a) Front side(b) Top side

Figure 6. Axisymmetric sloshing of the proposed model

2.2.

Evaluation item to tune parameters of degenerate model

We propose a propellant degeneracy model that can accurately analyze propellant sloshing by comparing
and examining CFD results. The following three evaluation items were compared with the CFD results:①

Reaction force exerted on the tank by propellant sloshing②

The point of reaction force exerted by the propellant sloshing on the tank should be specifically adjusted.

③The center of gravity of the propellant

Figures 2 to 4 show the CFD results for each evaluation item of axisymmetric propellant sloshing for theinput acceleration during the first, second, and third landings, respectively.
3.

Degenerate model

3.1.

Definition of model

The proposed model was constructed using the Simscape Multibody in MATLAB. The proposed model andits principles of operation are illustrated in Figures 5 and 6, respectively. Axisymmetric sloshing is represented
by the rotation around the axis of the link joints, and pendulums with different masses and link lengths areattached to represent the oscillations of axisymmetric sloshing, especially at points that need to be adjusted.
3.2.

Parameters of degenerate model

As shown in Figure 7 and Table 1, there are many parameter variables in the proposed model, such as the4 of 8

tankjointP1
k1x, k1z,c1x, c1z

m1fuelOsmallpendulum

mfuel

Z

r1

Y

jointPkx, kz,cx, cz

R1R
mcenterrlargependulum

XFigure 7. Parameters of the proposed model

mass of the pendulum, link length, and damping coefficient of the joints, and it is difficult to determine theoptimal parameters through trial and error.
Therefore, we introduce a genetic algorithm that can derive an optimal or sub‐optimal solution in a relatively
short time.
3.3.

Genetic Algorithm

In this study, the parameters were determined using a genetic algorithm. The degree of adaptation isexpressed by the following equation:
𝑓𝑓𝑓

(1)
1

𝑓𝑓,

∑𝑓𝑓
1

∑

where 𝑓𝑥 ,𝑥 ,

(2)

,

(3)

,

is the fitness value of the reaction force of the propellant sloshing and 𝑓

for the center of gravity of the spacecraft with the propellant. 𝑓
sloshing derived from the simulation, and 𝑓
the CFD. 𝑥 ,and 𝑥 ,is the fitness value

is the reaction force of the propellant

is the reaction force of the propellant sloshing derived fromare the centers of gravity of the spacecraft with the propellant derived from thesimulation and CFD, respectively.
where N is the total number of samples, N1 is the number of samples at the point where the transverse
sloshing should be aligned, N2 is the number of samples at the point where the axisymmetric sloshing shouldbe aligned, n1 is the sample start point at the point where the transverse sloshing should be especially aligned,
and n2 is the sample starting point at the point where the axisymmetric sloshing should be especially aligned.
n2 denotes the sample starting point of the axisymmetric sloshing, which is particularly aligned. The
simulation time was 40 s, and the data sampling interval was 0.01 s. The number of generations was set to 100,the crossover rate to 0.8, and the mutation rate to 0.03. The crossover method was a one‐point crossover withrandom crossover points, and the crossover individuals were determined using a tournament strategy.
4.

Simulation Results
In this section, the proposed degenerate model and CFD results are compared. Table 2 shows the analyticalconditions, and Table 3 shows the values of the model parameters obtained by the genetic algorithm. Figures
8–10 show the results of the numerical analysis using the model parameters listed in Table 3. It can be seen
5 of 8

that the proposed model is more consistent with the CFD results, with smaller errors in the center of gravity
position, tank reaction force, and especially in all the points that need to be aligned. In addition, theconventional model is characterized by the fact that adaptability 3, which is the point to be especially adjusted,is small in all cases, making it difficult to adjust adaptability 3. However, the proposed model has much larger
adaptivity, which enables highly consistent matching. As we were able to verify the effectiveness of theproposed model, we constructed a coupled analytical model incorporating the degenerate model proposed in
section 3‐2 in section 3‐3.
Table 1. Model parameters
Symbolmcenter

Parameter

UnitkgFixed massmfuel

Large pendulum slosh masskg

m1fuelSmall pendulum slosh mass

kg

R

Large pendulum cross link lengthm

R1

Small pendulum cross link length

mrLarge pendulum tip link length

mr1

Small pendulum tip link lengthmkx

Large pendulum lateral sloshing link spring coefficient

Nm/radkz

Large pendulum axisymmetric sloshing link spring coefficient

Nm/radk1x

Small pendulum lateral sloshing link spring coefficient

Nm/radkz

Small pendulum axisymmetric sloshing link spring coefficient

Nm/radk1x

Large pendulum lateral sloshing link damping coefficient

Nms/radk1z

Large pendulum axisymmetric sloshing link damping coefficient

Nms/radcx

Small pendulum lateral sloshing link damping coefficient

Nms/radcz

Small pendulum axisymmetric sloshing link damping coefficient

Nms/radTable 2. Weight of propellant and oxidizer in operational phase.
Liquid type

1st Landing

Nominal

3rd Landing

propellant

488 kg404.5 kg

321 kgoxidizer

214.5 kg

180.5 kg

146.5 kg

Table 3. Weight of propellant and oxidizer in operational phase.
Parameters

Unit

Range

1st Landing

Nominal

3rd Landing

mcenterkg

100 ‐ 250

191

138

101

5457

mfuel

kg

20 – 60

45rm

0.20 – 0.30

0.28

0.25

0.25

r1

m

0.020 – 0.30

0.045

0.056

0.023

kx

Nm/deg0.000050 – 0.0010

0.000830.000670.00022kz

Nm/deg0.000050 – 0.0010

0.000160.000094

0.00006

k1x

Nm/deg0.000050 – 0.0010

0.000370.000180.00095kz1

Nm/deg0.000050 – 0.0010

0.000410.000130.00028cx

Nms/deg

0.0030 – 0.050

0.012

0.00400.0181

cz

Nms/deg

0.0030 – 0.050

0.050

0.00470.0317c1x

Nms/deg

0.000010 – 0.0010

0.000900.00060

0.00069

c1z

Nms/deg

0.000010 – 0.0010

0.000420.000540.00081
6 of 8(a) Tank reaction force.

(b) Center of gravity
Figure 8. Comparison results of the proposed model during 1st decent operation on microgravity celestial body.

(a) Tank reaction force.

(b) Center of gravity
Figure 9. Comparison results of the proposed model during 2nd decent operation on microgravity celestial body.

7 of 8(a) Tank reaction force.

(b) Center of gravity
Figure 10. Comparison results of the proposed model during 3rd decent operation on microgravity celestial body.

4.

Conclusions

In this paper, we propose a mechanical model that can reproduce the sloshing phenomenon of apropellant onboard a spacecraft during landing on a microgravity object. The values of the model parameters
were obtained using a genetic algorithm based on an operational scenario. A numerical analysis wasperformed using the model with the determined parameters, and the validity of the model was verified bycomparison with the results of CFD analysis.
References
1)

Y. Kawakatsu, et al.: Preliminary design of Martian Moons eXploration (MMX). Acta Astronautica, 202 (2023) 715,
DOI: https://doi.org/10.1016/j.actaastro.2022.09.009.

2)

JAXA, Martian Moons eXploration (MMX), https://www.isas.jaxa.jp/missions/spacecraft/developing/mmx.html (asof 2023/9/6).

3)

Romero Calvo, A., García Salcedo, A. J., Garrone, F., Rivoalen, I., Cano Gómez, G., Castro Hernández E., and Maggi,
F. StELIUM: A student experiment to investigate the sloshing of magnetic liquids in microgravity. Acta Astronautica,173 (2020)., 344 355, https://doi.org/10.1016/j.actaastro.2020.04.013.

© 2023 by the authors. Submitted for possible open access publication under theterms and conditions of the Creative Commons Attribution (CC BY) license(http://creativecommons.org/li censes/by/4.0/).

8 of 8