(2020) 7:38
Fujiwara et al. Robomech J
https://doi.org/10.1186/s40648-020-00183-0

Open Access

RESEARCH ARTICLE

Estimation of resistance force at steady‑statesinkage for cylindrical wheel‑typed lunar/
planetary exploration rovers with functionof push–pull locomotion
Daisuke Fujiwara1* , Naoki Tsujikawa2, Tetsuya Oshima2 and Kojiro Iizuka2

Abstract
Planetary exploration rovers have required a high traveling performance to overcome obstacles such as loose soil androcks. Push-pull locomotion rovers is a unique scheme, like an inchworm, and it has high traveling performance onloose soil. Push-pull locomotion uses the resistance force by keeping a locked-wheel related to the ground, whereas
the conventional rotational traveling uses the shear force from loose soil. The locked-wheel is a key factor for travelingin the push-pull scheme. Understanding the sinking behavior and its resistance force is useful information for estimating the rover’s performance. Previous studies have reported the soil motion under the locked-wheel, the traction, andthe traveling behavior of the rover. These studies were, however, limited to the investigation of the resistance force
and amount of sinkage for the particular condition depending on the rover. Additionally, the locked-wheel sinks into
the soil until it obtains the required force for supporting the other wheels’ motion. How the amount of sinkage andresistance forces are generated at different wheel sizes and mass of an individual wheel has remained unclear, and its
estimation method hasn’t existed. This study, therefore, addresses the relationship between the sinkage and its resistance force, and we analyze and consider this relationship via the towing experiment and theoretical consideration.
The results revealed that the sinkage reached a steady-state value and depended on the contact area and mass ofeach wheel, and the maximum resistance force also depends on this sinkage. Additionally, the estimation model did
not capture the same trend as the experimental results when the wheel width changed, whereas, the model captured
a relatively the same trend as the experimental result when the wheel mass and diameter changed.
Keywords: Push-pull locomotion, Resistance force, Loose soil, Terramechanics
Introduction
NASA, ESA, and JAXA have investigated past water
activity or a clue to life on Martain or Lunar surfaces[1–5]. For investigating such planetary surfaces, surface
mobility is one of the key factors for achieving a reliable mission, and many rovers have been developed in
the world. The rover that is equipped with the functionof a push-pull locomotion is a unique scheme, like an*Correspondence: nb16107@shibaura‑it.ac.jp
1
Department of Functional Control Systems, Shibaura Instituteof Technology, 307, Fukasaku, Minuma‑ku, Saitama 337‑8570, Japan
Full list of author information is available at the end of the article

inchworm locomotion, and has a good traveling performance in the cylindrical wheel-typed rovers [6, 8, 10].
This scheme is a scheme that one side (front or rear)
wheels are locked against ground, and they support apush or pull locomotion of the other side wheels using
resistance force from the ground.
The autonomous traveling on a planet requires information between the wheels and soil. Understanding theinteraction between a wheel and soil helps to design thetraveling performance and model of a rover. The interaction between the locked-wheel and soil is the key factor
for the push-pull locomotion rovers.

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Fujiwara et al. Robomech J(2020) 7:38

For this reason, many researchers have investigatedthis interaction for each push-pull locomotion roveror wheel walking robot. Sybel et al. [7] investigated theresistance force of each locked-wheel for a wheel walking robot around the 1960s. In 1998, Andrade et al. tried
to estimate the traveling performance of this locomotion
by constructing the locked-wheel model against soil for
Marsokhod rover [8]. Wong [9] investigated soil flow
beneath a locked-wheel during towing. Creager et al. [10]also observed the soil flow under a wheel and measured
the traction and sinkage of Scarab rover during traveling.
According to Sybel et al. [7], the locked-wheel sinks into
the soil with an increasing sinkage, and this sinkage reachto close value to the steady-state value. This phenomenonis also observed in our previous study using the testbed
rover [11].
Slipping of a locked (supported) wheel is a negative
effect on this locomotion, whereas the resistance force
against soil increases with an increasing sinkage. A large
resistance force helps to support the repositioning of theother wheels’ motion. Analyzing especially the steadysinkage of the locked-wheel can lead to understanding
the maximum resistance force. This analysis can contribute to constructing the accurate model for estimating
the behavior of the locked-wheel and push-pull locomotion rovers. Although many studies have investigatedthe locked-wheel phenomenon and the traveling performance of push-pull locomotion, the amount of sinkage
and its resistance force of each locked-wheel at different
wheel seize and mass is yet to be determined. Additionally, the estimation model has not been developed.
This paper firstly aims to investigate sinkage and its
resistance force of each locked-wheel on loose soil during towing through an experiment. Furthermore, thispaper considers theoretically this phenomenon based on
the terramechanics and soil mechanics and tries to estimate the steady-state sinkage and resistance force of eachlocked-wheel in the next.

Push‑pull locomotion traveling mechanism
Figure 1 shows the schematic view of push-pull locomtoion testbed rover that is owned by our laboratory, andFig. 2 also shows actual images of the traveling experiment on the surface with slope using this testbed.
The front wheels of the rover are firstly fixed to theground at the initial position (Fig. 2a). Next, the rover
can shrink its wheel-base using the resistance force of thefront wheels (Fig. 2a–b). In the next, the rear wheels of
the rover are fixed to the ground. Then, the rover extends
its wheel-base using the resistance of the rear wheels(Fig. 2b–c). The rover can move to forward direction byrepeating this scheme.

Page 2 of 13

Fig. 1 Schematic view of the testbed rover with function of
push-pull locomotion [11]When the required resistance force that supports theother wheels’ motion is small, the necessary accumulated
soil behind the wheels is small, whereas the locked-wheelneeds large accumulated soil that supports the otherwheel’s motion when the required force is large.

The behavior of the locked‑wheel
As mentioned earlier, when the push-pull locomotion
rover moves, the rover uses the resistance force generated by keeping the wheel related to the ground. Whenthe resistance force generated by sinkage that depends ona vertical load of the wheel is less, the wheel sinks into
the soil until the required resistance force rises.
This section describes the force component of thetotal resistance force and sinking mechanism of thelocked-wheel.
Resistance force

According to Wong’s study [9], when the locked-wheel
acts on the soil, a wedge-shaped soil (area A) is formedin front of the wheel, and it pushes the soil mass area B,
behaving like a bulldozing blade. Based on Wong’s study[9], this study assumes a virtual plane between soilwedge area A and B and assumes that the soil-wedge area
A moves together with the wheel and pushes soil-wedge
area B. As shown in Fig. 3, this paper consider that theresistance force consists of five forces as follows:

Ftotal = F1 + F2 + F3 + F4 + F5

(1)

where, F1 is the passive earth force of area B, which is calculated as the force of a virtual plate. F2 is the shear force
under the soil wedge of area A. F3 is the side friction force
in area A. F4 is the friction force on the side surface of thewheel, and F5 is the friction force between the wheel surface and soil. In this study, the wheel for the experimenthas a smooth side surface; therefore, the friction force F4
is expected relatively small. Hence, force F4 is negligible.

Fujiwara et al. Robomech J(2020) 7:38

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Fig. 2 Actual image of the traveling experiment on the soil. The rover firstly shrinks its wheel-base, and the front wheel is fixed to the ground andsupports the rear wheels’ motion (a–b). Next, the rover extends its wheel-base, and the rear wheel is also fixed to the ground and supports the front
wheels’ motion (b–c)

Fig. 3 Locked-wheel on the soil surface based on Wong’s
observation [9]. The soil wedge (area A) beneath the wheel isassumed to move together with the wheel. The surface between soil
wedge areas A and B is assumed as a virtual surface

Fig. 4 Relationship between the plate and soilp(z) =
z0 indicates the sinkage of the wheel, and it increases
depending on an increasing towing distance.
Wheel sinkage based on terramechanics theory

Terramechanics studies defined the interaction
between a wheel and soil. In 1969, Bekker defined
the pressure-sinkage relationship of a plate as shownin Fig. 4. According to Bekker [12], the normal stressp(z) acts on a plate when a plate sinks into loose soil asshown in Eq. (2).

kc
+ kφ z nb(2)where kc , kφ , n, are the pressure-sinkage parameters that
depend on soil types. b indicates the width of a rectangular contact area of a plate.
Additionally, Bekker defined the rigid wheel-soil
interaction model on loose soil based on the pressuresinkage model as shown in Fig. 5. The phenomenon ofa wheel differs from a plate because sinkage varies in its
position. The sinkage under the wheel is large, whereas
the sinkage that is close to the surface of the contact
area is small. The equilibrium equations for the vertical

Fujiwara et al. Robomech J(2020) 7:38

Page 4 of 13

Wh = ww



kc
+ kφ
ww

  z0 
0


√
zn D
√
dz
2 (z0 − z)Then, using z0 − z = t 2, then dz = −2tdt , Eq. (7) is√√  z0kc
Wh = w w+ kφ(z0 − t 2 )n dt
D
ww
0

(7)

(8)

Expanding (z0 − t 2 )n, and using the first two terms(z0 n − nz0 n−1 t 2 ), Eq. (8) is as follows:

√
z0 D
ww wkwc + kφ
(9)Wh =z0 n (3 − n)3
Rearranging Eq. (9) it becomes,[2/(2n+1)]
3W
h
�√ 
�
z0 = 
ww (3 − n) wkwc + kφD

Fig. 5 Wheel-soil interaction on loose soil [12](10)Equation 10 indicates the sinkage of a wheel depends on
its vertical load [9].
direction of a towed rigid wheel can be writtern as follows Eq. (3): θsWh = w w
σ r cos θdθ(3)0where σ is the nromal pressure. ww and r are the wheel
width and radius. Wh is the wheel weight. θs is the contact
angle. The normal pressure σ acting on the wheel rim isassumed that it is equal to the normal pressure p beneaththe plate at the same depth z. Hence, σ r cos θ dθ equals
to p dx. Using the pressure-sinkage relationship Eq. (2),the equilibrium equation for the vertical direction is asfollows: z0p(x)dxWh = − w w0 z0 
(4)kc
= − ww+ kφ z n dx
ww0
where z0 is the sinkage. When the wheel sinkage is small,from the geometry shown in Fig. 5, the position x is asfollows:x2 = [D − (z0 − z)](z0 − z)
= D(z0 − z)(5)where D is the wheel diameter.

2xdx = −Ddz

Then, substituting Eq. (6) into Eq. (4).

(6)

Estimation method for the steady‑state sinkageof the locked‑wheel
This study applies the wheel-soil interaction model
defined by Bekker to estimate the steady-state sinkageof the locked-wheel, especially using Eq. (10). When thewheel is towed, the wheel moves dynamically. However,previous studies indicated that the resistance force acting on a plate from soil has weak velocity dependency
when the velocity is small [13, 14]. Although the wheelsoil interaction model is constructed based on the pressure-sinkage relationship, when the velocity is small, thephenomenon can be assumed close to a static. Thus, thispaper postulates the wheel-soil model can be applied to
the locked-wheel.
Estimation method for the maximum resistanceforce of the locked‑wheel
The method was based on an observation by Wong [9].
As mentioned eariler, Wong’s observation assumed thatsoil wedge area A behaved like a plate and acted on soil
mass area B. This paper, therefore, assumes that the soil
wedge area A move together with wheel and pushes soil
wedge area B, and the slipsurface beneath the soil wedge
is a constant shape. This section describes each force acting on the locked-wheel as shown in Fig. 3.
Classical analytical model for calculating force F1
against area B

There are several analytical models for predicting force ofplate tools against soil mass have been widely studied sinceFujiwara et al. Robomech J(2020) 7:38

Page 5 of 13

the 1960s [15–20]. Analytical models have been developedfor estimating or calculating earthmoving, excavation, andcutting force of a bucket. These models were summarizedby [21], and the models have been verified by several typesof research [22–24]. Furthermore, Yeomans et al. and Scottet al. [14, 25] proposed the leg model for planetary rovers
and estimated its resistance force using these analyticalmodels.
The analytical model for applying to the locked-wheel
model requires the ability to consider the accumulated
soil behind the wheel. The analytical models, however,
have different abilities depending on the model. According to Blouin et al. [21], Osman, Mckyes, Grisso, Swick andPerumpral, and Zeng [15, 17–20] model can consider theaccumulated soil. Osman model, however, contains indeterminate parameter referring to [22], and Mckeys andSwick & perumpral model represents equivalent results
according to [23].
Furthermore, our previous plate bulldozing tests confirmed that Mckyes model indicates a close value to theexperiments. Form this result, we chose Mckyes model for
calculating force F1 in this paper.
Mckyes model

In 1985, Mckyes et al. [17] proposed the earthmoving
model that was first introduced by Reece [26]. Mckyesmodel can consider the effects of the soil-tool adhesionCa, soil cohesion c, soil weight Wb, blade width ww, andsurcharge q. The model assumes that the soil slip on thetool surface and within the soil itself. Then, the frictional
component of shear strength on the two slip lines havebeen combined with perpendicular forces to form resultant
forces FT and R, and cohesional resistance forces Ca L andcL1 as shown in Fig. 6 [17]. The total force FT acting on theplate is derived using Terzaghi’s soil bearing capacity factor
N as follows:

FT = (γ gz02 Nr + cz0 Nc + qz0 Nq + Ca z0 Nca )ww (11)(cot β + cot ρ)Nr =2[cos(β + δ) + sin(β + δ) cot(ρ + φ)](12)(1 + cot ρ cot(ρ + φ))cos(β + δ) + sin(β + δ) cot(ρ + φ)

(13)

Nc =

Nq = 2Nγ
Nca =1 − cot β cot(ρ + φ)[cos(β + δ) + sin(β + δ) cot(ρ + φ)]

(14)
(15)

where β is the rake angle. g is the earth gravity. ρ is theshear plane failure angle. φ is the internal friction angle.

γ is the soil density. δ is the external friction angle. q is
the soil surcharge. Herein, this paper assumes that thesoil surcharge q is constant along the wheel width, and ρ
is generally found based on the assumption that the soil
failure will occur at the angle ρ which give the weakest
resistance force. This can be estimated by determining
the value of ρ at which Nr is minimized.
FTh = FT sin(β + δ)

(16)

where FTh represents the horizontal force of the total
force FT acting to the plate. Herein, force FTh indicates
F1 of area B (Fig. 3). This force can be generated by acting soil mass of area A, like a plate. The parameters used
for the analytical model as shown in list of notations(Table 1).
Shear and friction force F2, F3, F5 of area A

In area A, shear strength F2 beneath the soil wedge isderived from Coulomb’s failure criterion, and side friction force F3 is calculated based on earth pressure atrest coefficient as follows:

F2 = cww ls + (Ws + Wh ) tan φ(17)where Ws is the soil weight of the soil wedge and Wh is
the wheel weight. For the side friction force calculation,
vertical normal stress at depth z0 is:σv = γ gz0 + σb(18)

where σb is the wheel weight per unit area. Then, the horizontal normal stress is:

σh = K0 σv(19)where K0 = 1 − sin φ [27] is the coefficient of earth pressure at rest. Side friction force F3 is derived as follows:

F3 = 2(c + σh tan φ)Aarea(20)

where Aarea is an area of the side surface of the soil wedge
in area A.
The frictional force F5 is determined by the shear
stress τ , which is calculated based on Mohr-Coulomb
model, in the tangential direction at an arbitrary pointon the wheel surface as follows [12]:τ = cw + σ tan δ(21)

where σ is the normal pressure on the wheel surface. cw
is the soil-wheel adhesion. δ is the tool-soil friction angle.
Therefore, the sum of the horizontal component of theshear force τ cos(θ) along the wheel surface is the force F5
between the wheel surface and soil as follows:

Fujiwara et al. Robomech J(2020) 7:38

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Table 1 List of notations
Description (unit)

SymbolEach area of soil wedge in area A (m2 )Aarea

Cohesion of the soil (N/m2 )cSoil-tool adhesion (N/m2 )Ca

Soil-wheel adhesion (N/m2 )cw

Diameter of the wheel (m)D

Total resistance force of the locked-wheel (N)FtotalForce against soil mass area B (N)F1

Shear force beneath the soil wedge area A (N)F2Side friction force of the soil wedge area A (N)F3Side friction force of the side surface of the wheel (N)F4Friction force on the wheel surface (N)F5

Total force acting on the plate (N)FTHorizontal component of the total force (N)FThEarth gravity (m/s2 )gCoefficient of earth pressure at rest (−)

K0

Soil modulus of deformation depend on c (N/m(n+1))

kc

Internal friction angle modulus (N/m(n+2))

kφ

Tool length (m)lSlip surface length (m)L1Base line length of the soil wedge in area A (m)lsSinkage ratio (−)

nTerzahgi’s soil bearing capacity factors (−)

Nγ , Nc , Nq , Nca

Normal stress (N/m2 ))pSurcharge on the soil surface (N/m2 )qSurcharge loading (N)Q

Resultant force acting on slip surface (N)R

Wheel radius (m)rWidth of the plate or wheel (m)b, wwWeight of the plate (N)Wp

Weight of the soil wedge (N)Ws, WbWeight of the wheel (N)Wh

Position toward the horizontal direction of the wheel(m)xSinkage (m)z0 , zRake angle (◦ )β

Soil density (kg/m3 )γExternal friction angle (◦ )

δContact angle (◦ )

θ

Static contact angle (◦ )θs

Shear plane failure angle (◦ )

ρNormal stress acting on wheel surface (N/m2 )σHorizontal normal stress (N/m2 )σhWheel weight per unit area (N/m2 )σbF5 = ww r θoτ cos(θ)dθ

0

(22)

where ww is the wheel width. r is the wheel radius, and θoindicates the wheel contact angle at the soil surface. The
calculation is performed using the soil and wheel parameters Tables 2, 3

ExperimentFig. 7 depicts the schematic view of the experimental setup for the towing test. The soil bin area is width,
length, and high of 0.3, 1.2, and 0.18–0.2 m, respectively,
and Silica Sand No. 5 fills that area. The size of the wheel
is set as a basis of the rover testbed (Fig. 1) [11]. For the
towing test at different wheel sizes, the wheel size is 0.17,
0.20, 0.28, and 0.35 m in diameter, 0.04, 0.06, 0.08, and 0.1
m in width (Fig. 8). The wheel mass is from 1.0 to 2.5 kg .
Tables 2, 3 summarizes the experimental conditions, thesoil parameters, and variables for calculating the resistance force. In this system, the wheel can move to a vertical direction freely; that is, the towing motion does notaffect the vertical motion. The wheel sinkage depends onthe vertical load of the wheel. The wheel unit connects
to the towing unit via the towing rope. For investigating
the effect of the towing velocity on the force, the towing
speed is 0.00341, 0.017, 0.0341, and 0.051 m/s. The rotational speed of the towing motor shaft is 100, 500, 1000,
and 1500 rpm by PID control, and the towing speed of
the wheel unit becomes the abovementioned velocities.
The Motion Capture System measures the displacement,
and the force sensor also measures the resistance forceof the wheel. Tables 4, 5 summarize the specification ofeach system. The experimental trials are 5–10 times ineach condition. The detailed procedure is as follows:• A leveling plate with spikes stirs up the soil at first.
Then, the leveling plate smooths the soil surface
along the sidewall of the soil box without compaction.
• The wheel slowly and carefully is set on the soil surface.
• The rope tows the wheel unit at each constant speed.

Vertical normal stress (N/m2 )σv

Experimental result

Shear stress on the wheel surface (N/m)τ

Raw data

Internal friction angle (◦ )φFigure 9 provides the actual image of the towing experiment. Figure 10 also shows the raw data of the typicalexperiment. Each graph indicates the sinkage and resistance force. As shown in Fig. 9, 10, the wheel sinks into
the soil with an increasing sinkage, and the resistanceFujiwara et al. Robomech J(2020) 7:38

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R

(a)

(b)

Fig. 6 The wedge theory of passive soil failure. a Soil wedge. b Forces on soil wedge

Table 2 Experimental conditions for the locked-wheel
towing test
Description (unit)

Value

Slope angle (◦ )

0

Soil

Silica Sand No. 5

Velocity dependency

Towing speed (m/s) 0.00341, 0.017,
0.0341, 0.051

Figure 11 shows the relationship between the resistanceforce and sinkage at different towing velocity. The difference between each value is small; therefore, the velocity dependency of the resistance force is weak under the
towing velocities adopted in this paper.

0.00341

Rotational speed of
motor shaft (rpm)100, 500, 1000, 1500 100

Wheel diameter
D (m)

0.17

0.17

Wheel width ww (m) 0.04

0.2, 0.28, 0.35

0.06, 0.08, 0.1 0.04

1.0, 1.5, 2.0, 2.5

Wheel mass (kg)force also rises. Then, the sinkage reaches a steady-statevalue.

2.5

2.5

Sinkage at the steady‑state condition

Figures 12, 13, 14 present the steady-state sinkage at different wheel mass, width, and diameter. The continuouslines indicate the theoretical results calculated by thesinkage model Eq. (10), and each plotted points indicatesthe experimental results of the sinkage at each steadystate condition.
The amount of sinkage increases with an increasing wheel mass, whereas it decreases with an increasing

Table 3 Soil parameters and values for the calculation of the wheel force
Modulus

Value

Unit

Name of parameters

Referencesc762(N/m2 )

Soil cohesion

[31]

Ca

762

(N/m2 )

Soil-tool adhesion

[31]

cw

0

(N/m2 )

Soil-wheel adhesion

–

g

9.81

(m/s2)

Earth gravity

–

kc

1000

(N/m(n+1))

Soil modulus of deformation depend on c

Decided by experimentkφ

500000(N/m(n+2))

Internal friction angle modulus

Decided by experiment

n

1.1

(–)

Sinkage ratio

Decided by experimentq

Measured(N/m2 )

Surcharge on the soil surface

Measured by experiment

δ

15

(◦ )

External friction angle

Decided by plate towing experiment

β

90

(◦ )

Rake angle

–

γ

1430

(kg/m3)

Soil density

Measured by experiment

φ

22.3

(◦)

Internal friction angle[31]

Fujiwara et al. Robomech J(2020) 7:38

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Table 4 Specification of motion capture system
Description (unit)

Value

Model type (–)

OptiTrack Prime13

Frame rate (FPS)

100

Resolution (pixel)

1280 × 1024

Accuracy (m)Operation range (m)View angle (◦ )≤ 0.001
1–12

Horizontal FOV:56,
Vertical FOV:46

Table 5 Specification of force sensor
Description (unit)

Value

Model type (–)

Leptrino
PFS080YS102U6S

Rating capacity Fz (N)±1000

Rating capacity Fx , Fy (N)Rating capacity Mx , My , Mz (Nm)Sampling frequency (Hz)Resolution (–)

±500
±30
100

±1/4000

Fig. 7 Single wheel tester

wheel width and diameter. This is because when thewheel width and diameter increases, the ground contact
area becomes large. Consequently, the amount of sinkage
can be reduced. The calculated results capture relatively
the same trend as the experimental results.
Resistance force at the steady‑state condition

Figures 15, 16, 17 also provide the resistance force at
different wheel mass, diameter, and width. The continuous line indicates the theoretical value calculated by theresistance force model Eq. (1) and the data points indicate the resistance force when the sinkage becomes a
steady-state value.
The resistance forces of the experimental results
increase with an increasing wheel mass, whereas when

the wheel diameter and width increase, the trend of theresistance force indicates a relatively decreasing trend as
shown in the graphs Figs. 16, 17. The decreasing trend
can be considered to be caused by decreasing the steadystate sinkage with an increasing wheel width and diameter. Although the theoretical value captures the trendof the resistance force depending on the wheel mass anddiameter, the trend of the resistance force depending on
the wheel width is relatively different. Furthermore, thechange of the resistance forces depending on the wheel
width and diameter is relatively small.

Discussion
Steady‑state sinkage and resistance forceof the locked‑wheel

According to Sybel et al. [7], the sinkage increased withan increasing distance where a wheel was towed, andthe sinkage and force finally reached to a steady-value.
The same increasing trend was confirmed as shown in
Fig. 10. Additionally, experimental results in this paper
confirmed that the steady-state sinkage depended on
the contact area and wheel mass (Figs. 12, 13, 14), andthe maximum resistance force depended on this sinkage (Fig. 15, 16, 17). Although the sinkage becomes large,
the wheel mass especially contributes to improving theresistance force. When the wheel width and diameterincreases, the contact area becomes large. However, theamount of sinkage decreases. Consequently, the changeof the resistance force at steady-state sinkage dependingon the wheel width and diameter can be considered small.
To decide the required resistance force and the allowable
steady-sinkage, this information can help to estimate thetraveling performance of push-pull locomotion.
Estimation model of the steady‑state sinkage
and resistance force

Previous studies mainly confirmed the sinking behavior and resistance force experimentally [7, 9, 10, 28].
Andrade et al. [8] developed the estimation model of theresistance force especially for the locked-wheel of the
Mrasokhod rover. However, theoretical consideration
between the steady-state sinkage and maximum resistance force at different wheel size has remained unclear.
From experimental results, the velocity dependencyof the resistance force was weak under the velocity
adopted in this paper. Previous studies confirmed thatthe dependence of the plate force on the velocity at thelow-speed range 1 or 10–50 mm/s [13, 14] was weak.
Additionally, the dependence of the towing speed at
20–180 cm/s for the plate force was confirmed small
effect [29]. The experimental results indicated thesame trend as the previous studies and suggested thatthe velocity dependency on the resistance force of theFujiwara et al. Robomech J(2020) 7:38

Page 9 of 13

Fig. 8 Each wheel for the test. a Diameter 0.17 m, width 0.04 m, b Diameter 0.2 m, width 0.04 m, c Diameter 0.28 m, width 0.04 m. d Diameter 0.35m, width 0.04 m. e Diameter 0.17 m, width 0.06 m. f Diameter 0.17 m, width 0.08 m. g Diameter 0.17 m, width 0.1 m

Fig. 9 Actual image of the wheel towing experiment. Diameter 0.17 m; width 0.04 m; mass 2.5 kg

locked-wheel was weak. From this, the pressure-sinkage relationship of the locked-wheel can be considered as a static phenomenon under the towing velocity
in this paper. This result suggested that the wheel-soil
interaction model defined by Bekker can be applied toestimate sinkage of the locked-wheel.

The theoretical calculation for the steady-state sinkage based on terramchanics theory represented thesame trend as the experimental results of the steadystate sinkage at different wheel mass, diameter, andwidth (Figs. 12, 13, 14, 15, 16, 17). This result suggested

Fujiwara et al. Robomech J(2020) 7:38

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Force (N)Sinkage (m)Rawdata

0.00341 (m/s)
0.017 (m/s)
0.0341 (m/s)
0.051 (m/s)

Sinkage (m)Time (s)Fig. 11 Relationship between the wheel sinkage and resistance force
at the different towing velocityaRawdataSinkage (m)Force (N)Experiment
TheoryTime (s)bFig. 10 Raw data of the sinkage and resistance force. Diameter 0.17m; width 0.04 m; initial sinkage 0.01 m; wheel mass 2.5 kg; towing
speed 0.00341 m/s. a Sinkage, b Resisatnce force

that the model can be used for estimating the steadystate sinkage of the locked-wheel.
Although the trend of resistance force at different
wheel width calculated by Eq. (1) relatively differedfrom the experimental results (Fig. 16), the theoreticalresult at different wheel diameter and mass indicatedrelatively the same trend of the experimental results(Figs. 15, 17).
The model indicated Eq. (1) includes some assumption. For example, the accumulated soil toward thetowing direction was assumed uniform. However,
our previous observation confirmed that the shape ofan embankment behind the wheel along the wheel’s
width direction is nonuniform, like a fan-shaped.

Wheel mass (kg)Fig. 12 Mean ±SD steady-state sinkage at different wheel mass.
Diameter 0.17 m; width 0.04 m

Furthermore, Higa et al. [30] indicated that the stress
distribution beneath the wheel along the wheel’s width
direction was also nonuniform. That is, the phenomenon and geometry in the soil wedge along the wheel
width direction can be present nonuniform.
Furthermore, the theoretical model assumed a virtual plane between the soil wedge area A and B andassumed a constant slip surface beneath the soil wedge.
Further improvement in the locked-wheel model
for steady-state values should consider the nonuniform phenomenon along the wheel width and validate
assumption of the soil wedge, and slip surface. These
considerations may allow an even closer value to theexperimental values.

Fujiwara et al. Robomech J(2020) 7:38

Page 11 of 13

Experiment
TheoryForce (N)Sinkage (m)Experiment
TheoryWheel mass (kg)

Wheel width (m)Fig. 13 Mean ±SD steady-state sinkage at different wheel width.
Diameter 0.17 m; mass 2.5 kg

Fig. 15 Mean ±SD resistance force of steady-state sinkage at
different wheel mass. Diameter 0.17 m; width 0.04 m

Experiment
TheoryWheel diameter (m)Fig. 14 Mean ±SD steady-state sinkage at different wheel diameter.
Width 0.04 m; mass 2.5 kg

Conclusion
Locked-wheel behavior is a key factor for designing a
push-pull locomotion rover. Understanding steadystate sinkage and its resistance force lead to understanding the maximum force that the locked wheel
can generate. For this reason, this paper investigated
the relationship between steady-state sinkage and theresistance force of the locked-wheel at different wheel
mass, diameter, and wide. Additionally, we estimatedthe steady-state sinkage using the wheel-soil interaction model based on terramechanics at first, then tried
to estimate the resistance force against its sinkage
using the locked-wheel model based on soil mechanics. The experimental results clarified the steady-statesinkage and resistance force at different wheel sizes.

Wheel width (m)Fig. 16 Mean ±SD resistance force of steady-state sinkage at
different wheel width. Diameter 0.17 m; mass 2.5 kg

Experiment
TheoryForce (N)Sinkage (m)Force (N)Experiment
TheoryWheel diameter (m)Fig. 17 Mean ±SD resistance force of steady-state sinkage at
different wheel diameter. Width 0.04 m; mass 2.5 kg

Fujiwara et al. Robomech J(2020) 7:38

Furthermore, the estimation of the sinkage and resistance force gave theoretical consideration. The main
conclusions are as follows:• The steady-state sinkage depends on the contact
area and mass of the locked wheel. As the widthand diameter increases, the steady-state sinkage
becomes small, whereas when the width and diameter become small, the sinkage also becomes large.
As the mass increases, the stady-state sinkage also
increases.
• The maximum resistance force especially depends
on the steady-state sinkage. Although the sinkage
becomes large, increasing the wheel mass contributes to an increasing resistance force. Additionally, as an increasing wheel width and diameter, theresistance force becomes small because of decreasing the steady-state sinkage.
• The wheel-soil interaction model based on terramechanics captured relatively the same trend as
the steady-state sinkage of the experimental results.
Although the further consideration of the accuracy ofthe model and validating assumptions included in themodel are necessary, the resistance force model using
steady-state sinkage indicated relatively the same
trend as the experimental results at different wheel
diameter and mass.
This knowledge can contribute to the design for thetraveling performance of push-pull locomotion. Furthermore, when the accurate model of the locked-wheel will
be constructed through further consideration, the model
can be used for unmanned control of the rovers. Further
works comprise validating the accuracy of each model,
considering the gravity effect, analyzing the behaviorof the locked-wheel on any soil types such as regolith
simulant of the planet, and constructing the push-pull
locomotion model that uses the locked wheel model for
autonomous control.
Acknowledgements
This work was supported by JKA and its promotion funds from KEIRIN RACE.
Authors’ contributions
DF developed the methodology of the resistance force model and performedexperiments and analyzed data. DF also drafted the manuscript. NT developed
the methodology of estimating the steady-state sinkage. TO developed thetowing experimental system. KI edited and revised manuscripts. All authors
read and approved the final manuscript.
Funding
The development of the experimental system, rover system, and analysis of
this study were funded by JKA.
Availability of data and materials
The datasets used and analyzed in this study are available from the corresponding author upon request.

Page 12 of 13

Competing interests
The authors declare that they have no competing interests.
Author details
Department of Functional Control Systems, Shibaura Institute of Technology,
307, Fukasaku, Minuma‑ku, Saitama 337‑8570, Japan. 2 Department of Machinery and Control Systems, Shibaura Institute of Technology, 307, Fukasaku,
Minuma‑ku, 337‑8570 Saitama, Japan.

1

Received: 21 April 2020 Accepted: 3 October 2020

References
1. NASA (2020) NASA MARS 2020 Mission. https​://mars.nasa.gov/mars2​020/.
Accessed 15 Apr 2020
2. Orosei R, Lauro SE, Pettinelli E, Cicchetti A, Coradini M, Cosciotti B, Di
Paolo F, Flamini E, Mattei E, Pajola M, Soldovieri F, Cartacci M, Cassenti F,
Frigeri A, Giuppi S, Martufi R, Masdea A, Mitri G, Nenna C, Noschese R,
Restano M, Seu R (2018) Radar evidence of subglacial liquid water on
mars. Science 361(6401):490–493. https​://doi.org/10.1126/scien​ce.aar72​
68
3. ESA (2020) ESA ExoMars mission 2022. http://www.esa.int/Scien​ce_Explo​
ratio​n/Human​_and_Robot​ic_Explo​ratio​n/Explo​ratio​n/ExoMa​rs. Accessed
15 Apr 2020
4. JAXA (2020) JAXA Hayabusa2. http://www.hayab​usa2.jaxa.jp/. Accessed
15 Apr 2020
5. Tsuda Y, Yoshikawa M, Saiki T, Nakazawa S, ichiro Watanabe S (2019)
Hayabusa2-sample return and kinetic impact mission to near-earth
asteroid ryugu. Acta Astronautica 156:387–393. https​://doi.org/10.1016/j.
actaa​stro.2018.01.030
6. Azkarate M, Zwick M, Hidalgo Carrió J, Nelen R, Wiese T, Poulakis P,
Joudrier L, Visentin G (2015) First experimental investigations on wheelwalking for improving triple-bogie rover locomotion performances. In:
Conference: ASTRA 2015-13th ESA Workshop on Advanced Space Technologies for Robotics and Automation, At Noordwijk, The Netherlands
7. Sybel HV, Grosse-Scharmann F (1961) Triebkraftsteigerung bei geländefahrzeugen durch das schub-scritt-verfahren (increasing of driving forcesin terrain through push-walk process). In: Proc First International Conference on Terrain-Vehicle Systems, Edizioni Minerva Techinca, pp 895–912
8. Andrade G, Amar FB, Bidaud P, Chatila R (1998) Modeling robot-soil interaction for planetary rover motion control. In: Proceedings. 1998 IEEE/RSJ
International Conference on Intelligent Robots and Systems. Innovationsin Theory, Practice and Applications (Cat. No.98CH36190), vol 1, pp 576 –
581. https​://doi.org/10.1109/IROS.1998.72468​09. Wong J (1978) Theory of ground vehicles. Wiley, New Jersey
10. Creager C, Johnson K, Plant M, Moreland S, Skonieczny K (2015) Push-pull
locomotion for vehicle extrication. J Terramech 57:71–80. https​://doi.
org/10.1016/j.jterr​a.2014.12.001
11. Fujiwara D, Iizuka K, Asami D, Kawamura T, Suzuki S (2019) Study on traveling performance for variable wheel-base robot using subsidence effect.
Int J Mech Eng Robotics Res. https​://doi.org/10.18178​/ijmer​r.8.2.233-238
12. Bekker G (1969) Introduction to terrain-vehicle systems. University ofMichigan Press, Ann Arbor
13. Albert R, Pfeifer MA, Barabási AL, Schiffer P (1999) Slow drag in a granularmedium. Phys Rev Lett 82:205–208. https​://doi.org/10.1103/PhysR​evLet​
t.82.205
14. Yeomans B, Saaj CM, Winnendael MV (2013) Walking planetary roversexperimental analysis and modelling of leg thrust in loose granular soils. J
Terramech 50(2):107–120. https​://doi.org/10.1016/j.jterr​a.2013.01.006
15. Osman MS (1965) The mechanics of soil cutting blades: osman,m.s., jour. agric engr res, 9, 4, 1965. J Terramech 2(1):98. https​://doi.
org/10.1016/0022-4898(65)90115​-1
16. Gill WR, Vanden Berg GE (1967) Soil dynamics in tillage and traction.
Washington, D.C. : Agricultural Research Service, U.S. Dept. of Agriculture,includes index
17. McKyes E (1985) Soil cutting and tillage. Developments in agricultural
engineering. Elsevier Science, Amsterdam

Fujiwara et al. Robomech J(2020) 7:38

18. Grisso R, Perumpral J (1985) Review of models for predicting performance of narrow tillage tool. Trans ASA. https​://doi.org/10.13031​/2013.32388​
19. Swick W, Perumpral J (1988) A model for predicting soil-tool interaction. J
Terramech 25(1):43–56. https​://doi.org/10.1016/0022-4898(88)90061​-4
20. Zeng X, Burnoski L, Agui J, Wilkinson A (2007) Calculation of excavation
force for isru on lunar surface. https​://doi.org/10.2514/6.2007-1474
21. Blouin S, Hemami A, Lipsett M (2001) Review of resistive force models
for earthmoving processes. J Aerospace Eng. https​://doi.org/10.1061/(ASCE)0893-1321(2001)14:3(102)22. Wilkinson A, DeGennaro A (2007) Digging and pushing lunar regolith:classical soil mechanics and the forces needed for excavation andtraction. J Terramecha 44(2):133–152. https​://doi.org/10.1016/j.jterr​
a.2006.09.001
23. King R, Susante PV, Gefreh M (2011) Analytical models and laboratory
measurements of the soil-tool interaction force to push a narrow tool
through jsc-1a lunar simulant and ottawa sand at different cutting
depths. J Terramecha 48(1):85–95. https​://doi.org/10.1016/j.jterr​
a.2010.07.003
24. Xi B, Jiang M, Cui L, Liu J, Lei H (2019) Experimental verification on
analytical models of lunar excavation. J Terramech 83:1–13. https​://doi.
org/10.1016/j.jterr​a.2019.01.002
25. Scott G, Saaj C (2012) The development of a soil trafficability model for
legged vehicles on granular soils. J Terramech 49:133–146. https​://doi.
org/10.1016/j.jterr​a.2011.12.002

Page 13 of 13

26. Reece AR (1964) Paper 2: the fundamental equation of earth-moving
mechanics. Proceedings of the Institution of Mechanical Engineers,
Conference Proceedings 179(6):16–22. https​://doi.org/10.1243/
PIME_CONF_1964_179_134_02
27. Jaky J (1944) The coefficient of earth pressure at rest. in hungarian (anyugalmi nyomas tenyezoje). J Soc Hungarian Arch Eng 78(22):355–358
28. Scott J Moreland DWCCVA Krzysztof Skonieczny (2011) Soil motion
analysis system for examining wheel-soil shearing. In: 17th International
Conference of the International Society for Terrain Vehicle Systems 2011
(ISTVS 2011), Blacksburg, Virginia
29. Wieghardt K (1975) Experiments in granular flow. Ann Rev Fluid Mech
7(1):89–114. https​://doi.org/10.1146/annur​ev.fl.07.01017​5.00051​3
30. Higa S, Nagaoka K, Nagatani K, Yoshida K (2015) Measurement and modeling for two-dimensional normal stress distribution of wheel on loose
soil. J Terramech 62:63–73. https​://doi.org/10.1016/j.jterr​a.2015.04.001
31. Noriaki M (2013) Study on development locomotion performancebased on terramechanics for wheeled rover on soft ground. PhD thesis,
SOKENDAI

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