RFT-based Analysis of Mobility Performanceof Hopping Rover on Soft Soil for Planetary Exploration

Kosuke Sakamoto1, Masatsugu Otsuki2, Takashi Kubota2, Yoshiki Morino1

1Waseda University, 3-4-1 Okubo Sinjuku Tokyo, Japan, E-mail: k_sakamoto@ac.jaxa.jp, morino@waseda.jp2ISAS,JAXA, 3-3-1 Yoshinodai Chuo Sagamihara Kanagawa, Japan, E-mail: otsuki.masatsugu@jaxa.jp,

kubota@jaxa.jp

ABSTRACTFuture planetary exploration requires rovers to ex-hibit advanced surface mobility. Various remoteobservations reveal that a number of scientificallysignificant sites are located on rough terrain such ascliff, steep slopes, or moon hole. Instead of improv-ing wheel’s performance, this paper studies the per-formance of a contact-based hopping mechanism onsoft soil terrain. In order to design the concept of ajumping mechanism for challenging planetary envi-ronments, the movement of rovers on loose regolithis studied based on Resistive Force Theory (RFT).Particularly, this paper analyzes the effect of stepperangles against the ground plane through RFT-basedsimulation and experiments. Vertical jumping testswere performed on silica sand. The trend of sim-ulation and experiment is consistent, while smallnumerical errors can be observed. The results showthat RFT is effective to get trend of mobility perfor-mance on soft soil.

1 INTRODUCTIONA lot of planetary exploration missions have carriedout by a lot of countries so far. In the late 1960s andthe early 1970s, the Soviet Union’s Lunokhod 1 and2 [1] had reached the moon for the first time in his-tory, and the U.S.’s Lunar Roving Vehicle (LRV) [2]of the Apollo missions made contribution to the suc-cess of the human explore missions. As more recentexample, China’s Jade Rabbit [3] had also reachedthe moon in 2013. As other examples of rovers,NASA’s Sojouner [4] of Mars Pathfinder, Mars Ex-ploration Rovers (Spirit and Opportunity) [5] andMars Science Laboratory (Curiosity) [6] have ex-plored the red martian surface. These rover’s loco-motion mechanism is wheeled platforms due to theirsimplicity, energy efficiency, and reliability.However, various remote observations discoveredthat a number of scientific significant terrains, suchas cliffs, steep slopes, and moon hole [7], are thoughthard to traversed by conventional wheeled rovers.For example, Figure. 1 shows MER spirit wheel got

Figure 1: MER Spirit wheels got stuck in softsoil (Image by NASA/JPL-Caltech)

stuck in soft soil. Due to this adversity, it is difficultfor Spirit to continue mobile exploration. In addi-tion, to explore small solar system bodies that havelow or ultra-low gravity is interesting for many coun-tries recently. However, these bodies are hard to ex-plore for conventional wheeled rover because of theirlow-gravity level. There are two approaches to solvethese challenges: improving wheel’s performance,and developing different locomotive systems. Asan example of the former approach, NASA/JPL-Caltech developed two wheeled tether rover, calledAxel [8], that was developed to provide access andget samples of extreme terrain. Axel’s wheels arelarge diameter and have grousers. Rovers whichhave large diameter grouser wheels can get over ob-stacles as large as wheel’s diameter without suspen-sion system like a rocker-bogie.On the other hand, non-wheeled rovers are also de-veloped in a lot of countries or research institutions.The PrOP-F Phobos probe [9] was a hopping roverthat was used for to jump on low or ultra-low gravitybodies. DLR’s MASCOT lander [10] was also ableto hop using reaction wheels. The Hedgehog [11]that was developed by NASA/JPL-Caltech can per-form pivoting, slipping and hopping by using threereaction wheels and spikes attached its cubical body.

The Comet Hopper (CHopper) [12] was developedfor NASA Discovery Program mission, which couldhop multiple times on the surface during descents.JAXA’s MINERVA [13] that was deployed fromHayabusa had two wheels to induce hopping forceand determine hopping direction. The Highland Ter-rain Hopper (HOPTER)[14] had a originality loco-motive system to explore small solar system bodiesthat are lower gravity level than the Earth. Researchand development of locomotive mechanism is one ofthe most important points for planetary explorationrovers that have to survive the harsh environmentdiffered from Earth, such as strong cosmic radia-tion, thermal cycling and thermal extremes due to athin or lack of an atmosphere and gravity levels. Atthe point of moving efficiency, hopping is more use-ful than wheeled platform in lower gravity levels.The contribution of this paper is analysis of hop-ping performance on soft soil. Some parts of plan-etary terrain are soft soil, such as regolith of theMoon. The experiment that is supposed lower ormicrogravity level planets is difficult to test underthe Earth environment due to difference of grav-ity levels. As such, microgravity or lower gravitytesting becomes confined to few key experimentsunder well-defined conditions. Therefore, modelsconstruction and simulations of mobility platformsin different environments of kind of planets help usto develop better devices for planetary exploration.

2 DYNAMICS MODELThere are few conventional mobility platform mod-els on soft soil which field of study is called ter-ramechanics. The classical terramechanics was de-veloped by M. G. Bekker [15], later extended by J.Y. Wong [16] or A. R. Reece [17].Recently, the new terramechanical method is pro-posed by Li et al [18]. The proposed approachstands for Resistive Force Theory (RFT)[19]. Theproposed method is also based on empirical modelsfrom experimental results, where they measure theResistive Force (RF) of small plate that is movingfreely in granular media. And, there is only singleparameter, called “scaling factor (SF)” that we needto measure on each terrain type. The new modelis more effective to simulate the dynamic motion ofhopping rover (hopper), because their model dosenot rely on conventional terramechanical models.According to [15], or [17], Only vertical pressure ofhorizontal plate is determined by the conventionalmodels. In addition, the conventional method is ap-plied to only wheel model. According to [18], thereaction or resistive force (RF) exerted leg elements,

Figure 2: Moving of micro-plate in granularmedia (original figure in [18])

called stepper, is given by:

FRFT = ζ

∫S

α (β, γ) z dAS (1)

where α(β, γ) denotes the resistive stress per unitdepth, dAS denotes the area of stepper, and z de-notes the sinkage depth from the soil surface. αdenotes the function of β, attack angle of stepper, γ,intrusion angle of stepper, and ζ denotes the SF thatindicates soil feature. In this paper, as a stepper, flatplate was employed.Figure. 3 shows the schematic of hopping rover. Pa-rameters are defined as follows: z0 denotes an initialsinkage depth, d denotes an initial sinkage of spring,φ denotes a slope angle of stepper, Mb denotes amass of body, m j denotes a mass of stepper, and ks ,cs denote spring and damping constants. The areaof stepper, As , is given by:

As = h0w (2)

where h0 denotes height of stepper, w denotes widthof stepper.The schematic of vertical hopping experiment isshown in Figure. 4. In this paper, hopping direc-tion is only vertical. Hence, only vertical heighth is measured and simulated. On the other hand,motions and rolling of horizontal direction are notconsidered. As a result, the equations of motion isexpressed by:

Mb ( zb (t) + g) + cs(zb (t) − z j (t)

)+ks

(zb (t) − z j (t)

)= 0 (3)

m j

(z j (t) + g

)− cs

(zb (t) − z j (t)

)−ks

(zb (t) − z j (t)

)= FRFT (4)

Figure 3: Schematic of hopper

Figure 4: Schematic of vertical hopping exper-iment

where each zb and z j denote coordinate axises alongthe vertical direction of hopper body and stepper,and the RF, FRFT, is rewritten as:

FRFT = ζ

∫S

α(β, γ)(−z j (t)

)sin φdAS (5)

where α(β, γ) is given by following equation.

α(β, γ) =1∑

m=−1

1∑n=0

[Am,n cos 2π

(mβπ+

nγ2π

)+Bm,n sin 2π

(mβπ+

nγ2π

)](6)

where Am,n and Bm,n are parameters appeared in[18], and in vertical hopping case, β and γ is indi-

Table 1: Parameters for simulation and exper-iment

Terms Symbol Value UnitGravitational acceleration g 9.81 m/s2

Mass of body Mb 0.074 kgMass of stepper m j 0.016 kgSpring coefficient ks 1025 N/mDamping coefficient cs 0.87 Ns/mInitial sinkage of spring d 0.037 mInitial position z0 0.001 mWidth of stepper w 0.08 mHeight of stepper h0 0.06 mThickness of stepper b 0.004 mSF ζ 2.0 -

cated as following equations.

β =π

2− φ (7)

γ =π

2(8)

Note that Eq. (5) is only completed in z j ≤ 0, be-cause FRFT is the resistive force that is generated onlywhen objects move in granular media. Therefore, ifz j ≥ 0, Eq. (5) is rewritten as:

FRFT = 0 (9)

In addition, after the spring length becomes naturallength, the spring is fixed. Therefore, only the grav-ity acts to hopper body and stepper. As a result, theequations of motion when the hopper and stepperare in air, are expressed by:

Mb ( zb (t) + g) = 0 (10)

m j

(z j (t) + g

)= 0 (11)

3 NUMERICAL SIMULATIONThe numerical simulation result of the time historiesof hopping height is shown in Figure. 5. All the pa-rameters that were used in this numerical simulationand experiment are indicated in Table. 1. To calcu-late Eq.5 and Eq. (4), the Runge-Kutta methods wasused. Figure. 5 shows that the slope angle of thehighest point is φ = 90 [deg]. And, as the φ is re-duced, the height is reduced proportionately. Also,the rate of decrease of maximum hopping height isnot constant related to φ. The reason is consideredas follows: FRFT is proportional to sin φ. Hence, ifFRFT is not large enough, stepper sink into the sanduntil forces are balanced. As a result, it is consid-ered that the initial potential energy is dissipated in

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

time [s]

0

0.1

0.2

0.3

0.4

0.5

0.6

Ho

pp

ing

hei

gh

t [m

]

φ = 15

φ = 30

φ = 45

φ = 60

φ = 75

φ = 90

Figure 5: Time history of hopping height of thehopper

0 2 4 6 8

time [s] ×10-3

-25

-20

-15

-10

-5

0

5

Hoppin

g h

eight

[m]

×10-3

φ = 25

φ = 30

φ = 45

φ = 60

φ = 75

φ = 90

Figure 6: Time history of the sinkage of stepper

proportion to sinkage depth.In terms of the sinkage of stepper, as the φ is in-creased, the sinkage depth is decreased, as shown inFigure. 6. If the sinkage is increased, the dissipa-tion of energy is also increased. If the dissipationof energy is proportion to ks , hmax is also propor-tion to ks . However, Figure. 7 shows that hmax isnot proportion to ks . The result indicates that thedissipation of energy has some sort of non-linearparameters.

4 HOPPING EXPERIMENTSIn this section, Vertical hopping experiments andthese experimental results are shown to verify thenumerical simulations results of section 3. Afterdescribing those, the simulated results and experi-mental results are compared and considered.

4.1 Vertical hopping experimentsThe hopper is shown in Figure. 9. The developedhopper is attached two springs, a motor, and a step-

30 40 50 60 70 80 90

φ [deg]

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Max

imum

Hoppin

g h

eight

[m]

ks=800

ks=1000

ks=1200

ks=1400

ks=600

Figure 7: The comparison of maximum hoppingheight with the value of ks

Figure 8: Stepper

per. Springs are taken up by the motor, then re-leased. Using springs forces and stepper, the hopperhops vertically. To measure the hopping height, mo-tion capture cameras, as shown in Figure. 10, andoptical motion capture software were used. Thesoftware is “Motive: Tracker” of OptiTrack andthese cameras are “OptiTrack S250:E”. The hop-per body is a commercial item that is a “ParrotMINIDRONES-Jumping sumo”. As shown in Fig-ure. 8, the steppers were modified by using 3D-printer with PLA resin, and prepared seven forms;φ = 90, 75, 60, 45, 30, 25, 20. The testbed soil is sil-ica sand whose SF is two. Each parameters that areused in this experiments are shown in Table. 1.The results are shown in Table. 2. The values of hop-ping height are indicated in mean ± s.d. These ex-periments were tested ten times each φ. The resultsindicate that, in φ = 90, 75, 60, 45, 30, 25[deg], thehopping height is increased with increase in φ. How-ever, at the φ = 20[deg], the hopping height is higherthan the hopping height at the φ = 25, 30[deg]. The

Figure 9: Experimental setup of hopper

Figure 10: Motion capture cameras

comparison of results of vertical hopping experi-ments and numerical simulations is shown in Fig-ure. 11. The simulated results are shown as blueline, and the experimental results are shown as redsquare with error bar. In 25 ≤ φ ≤ 90[deg], thesame trend of experimental values and simulatedvalues are confirmed from these results. This resultshows that RFT is useful to research the trend ofmotion on soft soil.On the other hand, the error between experimentalvalues and simulated values are also confirmed fromsame results. As the trend of error, it is confirmedthat the smaller the values of φ become, the largerthe values of error.

4.2 DiscussionsThe two reasons of errors and trend of error areconsidered. Firstly, as shown in Figure. 12, if theφ is small enough, i.e., smaller than specific value,a part of stepper sinks in the sand. Therefore, thispart of stepper is not on surface of sand, the valuesof FRFT are different between experiment and sim-ulation. Especially, a part of stepper that is under

Table 2: Vertical hopping height with varyingstepper angles

Stepper angle φ [deg] Hopping height h [cm]90 63.5 ± 1.0975 62.2 ± 1.0260 59.7 ± 0.27445 53.1 ± 0.79030 39.6 ± 0.40025 30.4 ± 0.64220 46.6 ± 0.927

10 20 30 40 50 60 70 80 90

φ [deg]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Ho

pp

ing

hei

gh

t [m

]

Simulation

Experiment

Figure 11: Comparison of vertical hoppingheight between experiment and sim-ulation

the sand makes experimental values of FRFT biggerthan simulated value, because FRFT is proportionalto sinkage depth. Secondly, as shown in Figure. 13,if the slope angle becomes smaller than an angle ofrepose of sand, a part of stepper is above the surfaceof sand. This part of stepper makes the experimen-tal value of FRFT smaller than the simulated value.Figure. 14 shows that conditions of stepper are onsand each angles. The red circle points indicatenon-modeled parts. In the using model, these partswere not modeled. Hence, the errors between ex-periments and simulations were caused.In these two reasons, the first one has a great effecton these errors, because almost all experimental val-ues are larger than the simulated values. Especially,at the φ = 20[deg], large error is observed. Themost of the errors are caused by the part of stepperthat is in sand. As a result, the experimental FRFT ismuch bigger than the simulated one.Figure. 15 and Figure. 16 are the comparisons of

Figure 12: Sand on stepper

Figure 13: Separation of sand and stepper

maximum hopping height when the initial sinkageof spring d, and scaling factor ζ are changed. In-creasing the initial sinkage of spring, the differenceof trend between the values of simulations and ex-periments are increased, as shown in Figure. 15. Thedifference of trend is also confirmed from Figure. 16by increasing ζ . These results indicate that the errorwas caused by effects of surface.

5 CONCLUSIONThe mobility performance of hopping rover is an-alyzed by using RFT that is a new method of ter-ramechanics. The numerical simulation and verticalhopping experiments are conducted. As the simu-lation, the maximum hopping height is increasedwhile the stepper angle φ is increased. The sametrend was also observed by vertical hopping exper-iments. However, the errors between the simulatedvalues and the experimental values were also con-firmed. These errors seem to be caused by the im-proper placement of the stepper on loose sand. Inthis experiments, it is considered that main part ofthe errors were the former reason’s effect, so almost

(a) Horizontally (b) A little oblique

(c) An oblique (d) A large oblique

Figure 14: Images of conditions of stepper eachangles

20 30 40 50 60 70 80 90

φ [deg]

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Max

imum

Hoppin

g h

eight

[m]

Experiment

d = 0.001

d = 0.002

d = 0.003

d = 0.004

d = 0.005

Figure 15: The comparison of maximum hop-ping height with the value of d

all the experimental values were bigger than the sim-ulated values. Although the errors were confirmed,RFT is useful to study the trend of motion on softsoil.

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