THREE-DIMENSIONAL STRESS DISTRIBUTION OF
A RIGID WHEEL ON LUNAR REGOLITH SIMULANT*Shoya Higa1 , Kazumasa Sawada2 , Keigo Teruya3 , Kenji Nagaoka4 , Kazuya Yoshida5
1

Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: shoya@astro.mech.tohoku.ac.jp
Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: sawada@astro.mech.tohoku.ac.jp
3
Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: teruya@astro.mech.tohoku.ac.jp
4
Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: nagaoka@astro.mech.tohoku.ac.jp
5
Tohoku University, Aoba 6-6-01, Aramaki, Aoba-ku, Sendai, Japan, E-mail: yoshida@astro.mech.tohoku.ac.jp
2

ABSTRACTFor estimation and control of the wheeled mobility of a lunar/planetary exploration rover, three-dimensional stress
distribution measurements and a theoretical model of arigid wheel on loose soil is quite important. We developed a single wheel testbed for use in comprehensive experiments, to understand the wheel-soil interaction, andmeasured the normal, circumferential, and lateral stress
distributions beneath the wheel on lunar regolith simulant.
These experiments yielded estimates of the shear stressdirection and the soil flow directions, by combining thetwo-directional shear stress distributions. We validated
the measurements by integrating the vertical componentsof the normal and shear stress distributions. The classicalshear stress model was also compared with the measurements. We found no significant diﬀerence between themodel and the measurements performed on lunar regolith
simulant under three diﬀerent slip conditions.

1 INTRODUCTION
Most lunar/planetary exploration rovers employ wheelbased locomotion because of its mechanical simplicity, reliability, controllability, and eﬃciency. However, wheels
easily slip on surfaces covered with fine granular regolith,
as found on the moon or Mars. Wheel slippage hinders
robotic operations such as localization or motion along apath. Furthermore, the risk of a rover getting stuck is a
critical problem in lunar/planetary exploration missions.
In fact, one of the twin Mars Exploration Rovers, named
Spirit, of the National Aeronautics and Space Administration (NASA), has been unable to continue planned mission since its wheel got stuck in the loose Martian regolith,
as it attempted to traverse a hilly region [1]. Future rovers
therefore require mechanisms that will prevent such problems in loose soil.
To avoid a sand trap, it is necessary to understandthe process of wheel slippage itself. Generally, the trap
occurs in situations that favor the digging of wheel into
the ground and slippage. These phenomena interact at

the mechanical interface between the traveling wheel andthe soil. From a perspective of terramechanics (research
field concerned with terrain-vehicle systems), loose soil
received much attention [2, 3]. The forces acting on awheel (e.g., the normal force and the drawbar pull) havebeen modeled using the normal and shear stress distributions at the contact patch. The stress distributions on
the wheel surface have been measured and modeled semiempirically [4, 5, 6, 7, 8]. Classical models in terramechanics consider mainly heavy vehicles with large wheelsand do not therefore apply to small exploration rovers [9].
Thus, new model of the stress distributions relevant torovers is needed to estimate their performance precisely.
Several researchers have also studied the stress distributions beneath a wheel of lightweight vehicles [10,11, 12, 13]. We developed an in-wheel measurement device that can measure two-dimensional stress distributions
along both the circumferential and axial direction [14].
Although the developed device cannot detect the shear
stress distribution, its measurement is essential for understanding the underlying physics of the wheel. We therefore improved the device to detect the shear stress distribution on the wheel surface [15].
On the other hand, the stress distributions of a wheel
depend strongly on the soil characteristics. A precise estimate of the mobility performance in loose soil requires
experiments done on soil that can appropriately simulate
the physical and mechanical characteristics of the terrainof interest. In consideration of future lunar exploration
missions, several researchers therefore conducted experiments to clarify the slip characteristics when traveling ona lunar regolith simulant [16, 17, 18]. However, the associated stress distributions were not discussed. In thisstudy, we therefore measured the three-dimensional stress
distributions beneath a wheel traveling on lunar regolith
simulant.
This paper first describes the experimental setup of
our comprehensive single wheel testbed. We then discuss
the measured stress distributions beneath the wheel traveling on lunar regolith simulant under various slip conditions. We also validate the measurements by comparing

Wheel load cancellation mechanism
Linear encoder

#01
#02

Linear encoder

Wheel surface#03

Micro-controller

#04

Wheel

#05

Conveying motor(removable)

Wheel motor
Micro-controller

Cover

Contact-parts

SandboxFigure 2 : Overview of the single wheel testbed.

Measurement device(a F/T sensor + a contact-part)

Power line

Signal line

XBee

XBee

Figure 1 : Measurement device inside a wheel.
PCthe actual and estimated wheel loads.
3.3 V

2 EXPERIMENTAL SETUP
To measure the three-dimensional stress distribution on arigid wheel surface, we developed a single wheel testbed
containing a fixed sensing system. We first describe thismeasurement device and then the single wheel testbed.

12 V

Micro-controller

Micro-controller

Linear encoder

Linear encoder

Motor driver

Motor driver

Conveying motor
Encoder

Wheel motor
EncoderLongitudinal unit

Vertical unit

3.3 V

18 V

2.1 Measurement Device
The measurement device used in this study, shown in Figure 1, consists of a six-axis force and torque (F/T) sensor and five types of “contact-parts.” The F/T sensor
detects the three-axial force and three-axial torque data
independently. The five types of contact-parts are designed to contact with ten specific measurement points on
the wheel surface. The sensing area of each contact-part
is 10 mmf × 10 mm. Each contact-part exposed to thewheel surface yields a measurement. Contact was madeat locations ±0–10 mm, ±10–20 mm, ±20–30 mm, ±30–40 mm, and ±40–50 mm from the wheel center. We fixed
a contact-part on the F/T sensor, and installed this component inside the wheel (the same wheel that was used in aprevious study [15]). The areas of the wheel that were notexposed to the contact-parts were covered with the samematerial as on the wheel surface.
Sand cloth was also pasted onto the wheel surface andto the contact-part exposed to the wheel surface. The sandcloth roughness is nearly same as the average grain diameter of Toyoura standard sand [19]. The particle diameter distribution of lunar regolith simulant (FJS-1 [20])is wider than Toyoura standard sand and its range of thegrain diameters of is smaller than Toyoura standard sand.
Lunar regolith simulant adhered to the sand cloth that was
pasted to the wheel surface throughout the experiments.
Friction on the wheel surface was therefore regarded to beequal to or greater than lunar regolith simulant. Hence,
we consider that there is no slippage of the wheel surface

Figure 3 : System diagram of the single wheel testbed
throughout traveling experiment. We calibrated the sensor
output using a balance weight and pulley before performing the stress measurements.

2.2

Single Wheel Testbed

To conduct an eﬀective and comprehensive experiment on
the wheel-soil interaction, we developed the single wheel
testbed, shown in Figure 2. The testbed, which includes
the sandbox, is 2.50 m long, 1.05 m wide, and 1.25 m
high. Figure 3 outlines the single wheel testbed system,
consisting of a vertical unit, longitudinal unit, and a sandbox. This section describes the system in detail.

2.2.1

Vertical Unit

The vertical unit consists of two vertical slide shafts andbushes, a wheel that is controlled by a given rotational
speed profile, a wheel load cancellation mechanism, and alinear encoder for doing the wheel sinkage measurement.
The wheel can move freely in the vertical direction via thevertical slide guide.
A micro-controller maintains a constant rotational velocity via a motor driver under PI control at 10 ms intervals, and measures the wheel sinkage by counting thelinear encoder signals at the same intervals. The linearencoder resolution for the wheel sinkage measurement
was 25 µm. Measurements were calibrated by assigning

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Figure 4 : Measurement process of three-dimensional stress distribution on the wheel surface.
the zero level to the soil surface.

for each slip ratio s using the following procedure:

2.2.2 Longitudinal Unit

(1) Set #1 contact-part onto the F/T sensor and fix a measurement device into the wheel.

The longitudinal unit consists of two longitudinal linear
guides, a conveying motor, and a linear encoder for use in
the wheel traveling distance measurement. The conveying motor and timing pulley were coupled, and the timing
pulley tightly connected to the vertical unit via a timing
belt. The mode of experiments (forced or free slip experiment mode using several traction loads) can be selectedby connecting additional rope to the timing belt.
The linear encoder for the wheel traveling distancemeasurement was connected to the vertical unit. The
conveying motor was also PI-controlled using a microcontroller via a motor driver at 10 ms intervals. The
micro-controller also counted the wheel traveling distanceby the same intervals. The resolution of the linear encoder
for the wheel traveling distance was 50 µm.

2.2.3 Sandbox(2) Mix and rake the soil in the sandbox to form a flat
surface.
(3) Measure the normal, circumferential, and lateralstress distributions with each angle of the wheel rotation.
(4) Repeat step (2) and (3) three times, and average threemeasurements under the same conditions.
(5) Replace the contact-part at each measurement point,
and repeat step (2)–(4) up to the point #10.
(6) Obtain the entire three-dimensional stress distribution of the wheel by superimposing the threedimensional stress distributions at all ten points on
the wheel surface.

The testbed was mounted onto a sandbox of length 1.6 m,
width 0.3 m, and depth 0.2 m. Two such sandboxes canbe selected to perform experiments on two types of soil.
One sandbox is filled with Toyoura standard sand, and theother with lunar regolith simulant (FJS-1).
We mounted the testbed onto the sandbox filled withthe lunar regolith simulant, and measured the threedimensional stress distribution on the wheel surface. The
lunar regolith simulant provides a good approximation to
the soil characteristics at the moon surface.

The slip ratio s is a key experimental parameter defined asvx, (0 ≤ s ≤ 1),
s=1−
(1)
rω
where r is the wheel radius, ω the wheel angular velocity,and v x the wheel traveling speed.

2.3 Measurement Method

This section describes the measurement of the threedimensional (normal, circumferential, and lateral) stress
distribution on a rigid wheel surface.

As shown in Figure 4, the normal stress distribution andtwo orthogonal shear stress distributions were measured

3

MEASUREMENT OF
THREE-DIMENSIONAL STRESSDISTRIBUTION ON A RIGID WHEEL

(a) Normal stress (s = 0%)(b) Normal stress (s = 40%)(c) Normal stress (s = 80%)(d) Circumferential stress (s = 0%)(e) Circumferential stress (s = 40%)(f) Circumferential stress (s = 80%)(g) Lateral stress (s = 0%)(h) Lateral stress (s = 40%)(i) Lateral stress (s = 80%)

Figure 5 : Measurement results of three-dimensional stress distribution on rigid wheel surface.

3.1 Experimental Condition
We measured the three-dimensional stress distribution on
lunar regolith simulant for three types of slip ratio.
• The wheel diameter and its width were 250 mm and100 mm.
• The wheel circumferential velocity was controlled toremain at 2 cm/s throughout the experiment.
• Three types of slip ratios (0%, 40%, and 80%) were
input.
• The wheel load was set to 50 N using a wheel loadcancellation mechanism.
• The terrain condition was flat.

3.2 Measurement Results and Discussion
In this study, we measured the normal, circumferential,
and lateral stress distributions at contact patch on lunar

regolith simulant. The three-dimensional stress distributions were measured at each angle of the wheel rotation.
The sensing domain of the contact-parts was exposed to a
specific point on the wheel surface at a time. The experiments were carried out under three types of forced slip
conditions.
Figure 5 shows the measurement results under threeslip conditions. The x-axis in each graph corresponds tothe wheel rotation angle θ, the y-axis to the positions y in
the wheel axial direction, and the z-axis and color bar tothe stress values.
Each stress distribution is generated mainly in thefront portion of the wheel. The normal stress distribution decreases and the circumferential stress distribution
increases with increasing the slip ratio.
The normal stress distribution in the axial direction,
when traveling on the lunar regolith simulant under 0%
slip condition, showed that the stress generated near thewheel center is greater than near the edge because thesoil near the edge is pushed toward the lateral side of the

50

40

40

30

30

30

20

20

20

10

10

10

0
−10

y [mm]

50

40

y [mm]y [mm]

50

0
−10

0
−10

−20

−20

−20

−30

−30

−30

−40

−40

−40

−50
−40 −30 −20 −10−50−40 −30 −20 −10

010θ [◦ ]

2030

40(a) s = 0%010θ [◦ ]

20

30

40

−50
−40 −30 −20 −10(b) s = 40%

010θ [◦ ]

2030

40(c) s = 80%

Figure 6 : Generating directions of the shear stress on the wheel surface.
wheel. However, the distributions in the axial direction
under the other slip conditions are almost flat.
Under the conditions used in this study, the lateralstress distributions were smaller than the other stress
distributions, because the wheel has no steering angle.
Whereas the soil cohesion diﬀers, the distribution for Toyoura standard sand showed similar trend [14, 15].
Figure 6 shows the generating directions of the shear
stresses, which were combined two-axial shear stresses(circumferential and lateral stresses). The horizontal axis
represents the wheel rotation angle θ, and the vertical axis
the measurement position in the axial direction. The arrows indicate the shear stress directions on the wheel surface.
The generating directions of the shear stress are related because the it is induced by the soil movement beneath the traveling wheel. The results suggest that mostof the soil moves against the direction of the wheel rotation. In other words, the soil is propelled forward by
the traveling wheel. Moreover, the shear stress magnitude
is reduced and almost vanishes as the wheel rotation angle approaches the zero position. In other words, the soil
motion switches direction and the soil is mostly removed
from the wheel surface.
On the other hand, the soil is pushed to lateral side of
the wheel under the 0% slip condition. This only occurs
under low slip condition, because almost all of the soil
beneath the wheel is conveyed backward by the wheel rotation under other slip conditions.

4 EVALUATION OF THE
MEASUREMENT RESULTS
To evaluate the measured stress distribution, we integrated
the vertical components of the normal and circumferential stress distributions and compared this result with thewheel load. We also evaluated the classical shear stressmodel by comparing the measured circumferential stress
distribution with the model-based shear stress distribution.

4.1

Calculation of the Normal Force of theWheel

The normal force acting on the wheel can be calculated byintegrating the vertical components of the normal stressσ(θ) and the circumferential stress τ x (θ) at all contact
patch with the lunar regolith simulant. The normal force
Fz is calculated as∫ b/2 ∫ θ f
Fz = r{τ x (θ) sin θ + σ(θ) cos θ}dθdy,
(2)
−b/2

θr

where b is the wheel width and θ f and θr are the entry
and the exit angles into and out of the soil, respectively.
The wheel rotation angle θ is zero below the wheel andincreases in the counterclockwise direction. We measured
the circumferential stress distribution at ten points, andtherefore the vertical components at these points were also
integrated in the axial direction. This calculation neglects
the lateral stress distribution τy (θ).

4.2

Shear Stress Model

Janosi and Hanamoto proposed a shear stress model basedon the direct shear test results [22]. From their model, we
can calculate the shear stress at the contact patch using thenormal stress and several soil parameters as follows:τ(θ) = (c + σ(θ) tan ϕ)[1 − e− jx (θ)/kx ],(3)

Table 1 : Calculation parameters for Lunar regolith simulant (FJS-1)
Parameter
c
ϕ
kx
θf
θr
θf
θr
θf
θr

Value
8
37.2
0.02
20
-6
28
-10
32
-15

Unit
[kPa]
[◦ ]
[m]
[◦ ]
[◦ ]
[◦ ]
[◦ ]
[◦ ]
[◦ ]

Source
[20]
[20]
[21]
Experiment (s = 0%)
Experiment (s = 0%)
Experiment (s = 40%)
Experiment (s = 40%)
Experiment (s = 80%)
Experiment (s = 80%)

Table 2 : Actual wheel load vs. estimated wheel load on lunar regolith simulant.
Slip ratio [%]Actual wheel load [N]

0
40
80

50
50
50

a

Estimated wheel load [N]
(I)a
(II)b
55.3
52.9
50.2
51.9
53.4
54.7

Error [%]
(I)a
(II)b
10.6 5.8
0.5
3.8
6.8
9.4

The values of (I) are calculated from all of the measured data.

b The values of (II) are calculated from the measured normal and shear stress distributions calculated with

Equation 4.

where c is the soil cohesion, ϕ an internal friction angle ofthe soil, k x a soil deformation modulus, and j x (θ) the shear
displacement at the wheel rotation angle θ calculated asj x (θ) = r[θ f − θ − (1 − s)(sin θ f − sin θ)].

(4)

4.3 Estimation Error of the Wheel Load fromthe Stress Distributions
The relative error of the estimation result, Eest , between
the actual wheel load Wact and the estimated wheel loadWest is defined as()
West
Eest =− 1 × 100.
(5)Wact
The equations mentioned above were calculated using
the parameters listed in Table 1.

4.4 Evaluation Results
This study compared the actual wheel load with an estimated wheel load, calculated using the measurement results. The initial rise and fall of the each stress distribution
were taken as marking the entry and exit angles, respectively.
Table 2 shows the relative errors between the estimated weights under each condition. All estimations weregreater than the actual wheel load, but all errors were lessthan 11%. This signifies that the shear stress can be estimated reliably, provided the normal stress distribution
and the soil parameters are known. There is therefore no
significant diﬀerence between the model-based and measured shear stress.
Previously, we performed three-dimensional stressmeasurements using Toyoura standard sand, and confirmed similar trends. We therefore conclude that ourmeasurement method is eﬀective for any type of dry sand.

5 CONCLUSION
This study examined the three-dimensional stress distribution on a rigid wheel traveling on lunar regolith simulant.
We first described the single wheel testbed that was used
for conducting comprehensive experiments. We then performed three-dimensional stress measurements on a wheel
using lunar regolith simulant.

The measurements showed almost same trends as theresults in previous works using Toyoura standard sand.
The evaluation results indicated that there is no significant
diﬀerence between the model-based and measured shear
stresses.
Future work will consider additional experimentswith diﬀerent slip conditions to model the stress distribution.

References[1] R. A. Kerr. Planetary science. mars rover trapped insand, but what can end a mission? Science (NewYork, N.Y.), 324(5930):998, May 2009.
[2] M. G. Bekker. Theory of Land Locomotion. University of Michigan Press, Ann Arbor, MI, 1956.
[3] J. Y. Wong. Theory of Ground Vehicles. John Wiley& Sons, Inc., Hoboken, New Jersey, fourth edition,
2008.
[4] O. Onafeko and A.R. Reece. Soil stresses and deformations beneath rigid wheels. Journal of Terramechanics, 4(1):59–80, January 1967.
[5] J. Y. Wong.
Behaviour of soil beneath rigid
wheels. Journal of Agricultural Engineering Research, 12(4):257 – 269, 1967.
[6] G. Krick. Radial and shear stress distribution beneath rigid wheels and pneumatic tyres on yielding
soils with regard to tyre deformation. Journal of Terramechanics, 6(3):73 – 98, 1969.
[7] A. Oida, A. Satoh, H. Itoh, and K Triratanasirichai.
Three-dimensional stress distributions on a tiresand contact surface. Journal of terramechanics,
28(4):319–330, 1991.
[8] S. Wanjii, T. Hiroma, Y. Ota, and T. Kataoka. Prediction of wheel performance by analysis of normal andtengential stress distributions under the wheel-soil
interface. Journal of Terramechanics, 34(3):165–
186, may 1997.
[9] M. G. Bekker. Introduction to terrain-vehicle systems. University of Michigan Press, Ann Arbor, MI,
1969.
[10] K. Nagatani, A. Ikeda, K. Sato, and K. Yoshida. Accurate estimation of drawbar pull of wheeled mobile robots traversing sandy terrain using built-in
force sensor array wheel. In Proceedings of the2009 IEEE/RSJ International Conference on Intelligent Robots and Systems, pages 2373–2378, St.
Louis, MO, USA, Oct. 2009.

[11] G. Meirion-Griﬃth and M. Spenko. A modified
pressuresinkage model for small, rigid wheels on
deformable terrains. J. Terramechanics, 48(2):149–
155, apr 2011.
[12] C. Senatore and K. Iagnemma. Analysis of stress
distributions under lightweight wheeled vehicles.
Journal of Terramechanics, 51:1–17, February 2014.
[13] T. Shirai and G. Ishigami. Development of in-wheel
sensor system for accurate measurement of wheel
terrain interaction characteristics. Journal of Terramechanics, 62:51–61, 2015.
[14] S. Higa, K. Nagaoka, K. Nagatani, and K. Yoshida.
Measurement and modeling for two-dimensional
normal stress distribution of wheel on loose soil. J.
Terramechanics, 62:63–73, dec 2015.
[15] S. Higa, K. Sawada, K. Nagaoka, K. Nagatani, andK. Yoshida. Three-dimensional stress distribution ona rigid wheel surface for a lightweight vehicle. In
Proceedings of the 13th European Conference of the
ISTVS, pages 383–391, Rome, 2015.
[16] S. Wakabayashi, H. Sato, and S. Nishida. Design andmobility evaluation of tracked lunar vehicle. Journalof Terramechanics, 46(3):105–114, 2009.
[17] T. Kobayashi, Y. Fujiwara, J. Yamakawa, N. Yasufuku, and K. Omine. Mobility performance of a rigid
wheel in low gravity environments. Journal of Terramechanics, 47(4):261–274, 2010.
[18] M. Sutoh, K. Nagaoka, K. Nagatani, and K. Yoshida.
Design of wheels with grousers for planetary rovers
traveling over loose soil. Journal of Terramechanics,
50(5-6):345–353, October 2013.
[19] Toyoura Keiseki Kogyo Co., Ltd. About toyoura sand. http://www.toyourakeiseki.com/english/
about.html, visited on 2014/06/17.
[20] H. Kanamori, S. Udagawa, T. Yoshida, S. Matsumoto, and K. Takagi. Properties of Lunar Soil
Simulant Manufactured in Japan. Space 98 (6th),
April 26-30, pages 462–468, 1998.
[21] M. Sutoh. Traveling Performance Analysis of Lunar/Planetary Robots on Loose Soil. PhD thesis, Tohoku University, March 2013.
[22] Z. Janosi and B. Hanamoto. The analytical determination of drawbar pull as a function of slip for
tracked vehicles in deformable soils. In Proceedings of the 1st International Conference on TerrainVehicle Systems, pages 707–736, Turin, Italy, 1961.