1. Introduction
The surface terrain of the Moon is largely covered
with a fine-grained loose soil called regolith. On such
loose soil, the wheels of a lunar exploration rover easily
slip and lose traction. Therefore, investigations on
the contact and traction mechanics between wheels
and soil are necessary in order to better understand
the motion behavior of a rover on loose soil.
Wheel-soil interaction mechanics have been well
studied in the field of terramechanics. In this field,
the principle of the wheel-soil interaction mechanics
and the empirical models of the stress distributions
beneath the wheels have been previously investigated
[1]-[3]. Recently, these terramechanics-based models
have been successfully applied to the motion analysis
of planetary rovers [4]. We have developed a
terramechanics-based dynamics model for exploration
rovers by considering the slip and traction forces of a
rigid wheel on loose soil [5][6]. Further, the authors
have also elaborated upon a wheel-and-vehicle model
to address the motion dynamics of the rover [7][8].
In this paper, locomotion mechanics of lunar exploration
rovers are addressed for numerical analyses
of motion behaviors of the rover. This research studies
two models to deal with the motion behaviors.
First, a wheel-soil contact model is developed to deal
with the wheel slip/skid behaviors, and subsequently,
the motion dynamics of the rover are numerically obtained
by using a wheel-and-vehicle model.
The wheel-soil contact model enables calculation of
3-axis wheel forces, namely drawbar pull, side force,
and vertical force. Then, the relationships among the
slip ratio (a measure of the longitudinal slip), slip angle
(a measure of the lateral slip), drawbar pull, and
side force are derived. The validity of the wheel-soil
contact model and the theoretical relationships is con-
firmed through experiments using a single-wheel test
bed. In these experiments, lunar regolith simulant is
utilized, which simulates the soil on the lunar surface.
The motion behavior of the entire rover is numerically
evaluated by using a wheel-and-vehicle model in
which the wheel-soil model is incorporated into an articulated
multibody model for describing the motion
dynamics of the vehiclefs body and chassis. The simulation
results obtained with the wheel-and-vehicle
model are compared with the results of experiments
conducted using a four-wheeled rover test bed on the
lunar regolith simulant. The experiments on horizontal
steering and slope traversing are carried out.
This paper is organized as follows: the following
section, Section 2, describes the models for the wheelsoil
contact mechanics. The single wheel experiments
and numerical simulations are described in Section 3
along with a discussion on the validity of the wheelsoil
models. In Section 4, the model for the wheeland-
vehicle dynamics is introduced. The experiments
using a rover test bed are addressed in Section 5, and
the validity of the proposed model is then discussed.
r x
z
vx
soil surface
x
y
slip angle:
vx
vy
v
Fig. 1: Wheel coordinate system
2. Wheel-Soil Contact Model
The following analysis concerns a rigid wheel rotating
on loose soil. A wheel coordinate system is defined
using a right-hand frame, as shown in Fig. 1; in this
system, the longitudinal direction is denoted by x, the
lateral direction by y, and the vertical direction by z.
The coordinate frame rotates according to the steering
action of the wheel (the yaw rotation around the
z axis) but does not rotate with the driving motion
of the wheel (the pitch rotation around the y axis).
2.1 Slip ratio and slip angle
Slips are generally observed when a rover travels
on loose soil. In addition, during steering or slopetraversing
maneuvers, slips in the lateral direction are
also observed. The slip in the longitudinal direction
is expressed by the slip ratio s, which is defined as a
function of the longitudinal traveling velocity of the
wheel vx and the circumference velocity of the wheel
r (r is the wheel radius and represents the angular
velocity of the wheel)
s = (r ? vx)/r (if |r| > |vx| : driving)
(r ? vx)/vx (if |r| < |vx| : braking).
(1)
The slip ratio assumes a value in the range from ?1
to 1.
On the other hand, the slip in the lateral direction
is expressed by the slip angle , which is defined by
using vx and the lateral traveling velocity vy as follows:
= tan?1(vy/vx). (2)
2.2 Wheel stress distribution
When a wheel rotates on loose soil, normal and
shear stresses are generated under the wheel. These
stresses are quite necessary to calculate wheel forces.
Based on terramechanics models, the stresses under a
rotating wheel can be modeled as shown in Fig. 2-(a).
The normal stress () is determined by the following
equation [5]:
() = ?
?
rn kc
b + k[cos ? cos f ]n (m ? < f )
rn kc
b + kcos{f ? ?r
m?r
(f ? m)} ? cos f n
(r < ? m),
(3)
where, f and r are the entry and exit angles, respectively.
The wheel contact patch on loose soil is
m
r
x()
y ()
(Slip angle)
x
b
y
vx
vy v
x
z
h h
()
Wheel-soil contact area
vx
Normal stress distribution
r vx
x
b
y
x
z
h
Fx (Traction force)
Fy (Side force)
Fz (Vertical force)
x()
y ()
(Slip angle)
vx
vy
v
(a) : Stress model (b) : Force model
Fig. 2: Wheel-soil contact model
defined by the region between the entry and exit angles.
kc, k, and n are the soil-specific parameters.
b is the width of wheel. Further, m is the specific
wheel angle at which the normal stress is maximized:
m = (a0 + a1s)f , (4)
where a0 and a1 are parameters that depend on the
wheel-soil interaction. Their values are generally assumed
as a0 ? 0.4 and 0 ? a1 ? 0.3 [9].
The shear stresses x() and y() are expressed
using identical expressions [10]:
x() = (c + () tan )[1 ? e?jx()/kx], (5)
y() = (c + () tan )[1 ? e?jy()/ky ]. (6)
In these equations, c represents the cohesion stress of
the soil; , the internal friction angle of the soil; and
kx and ky, the shear deformation modules.
Further, jx and jy, which are the soil deformations,
can be formulated as functions of the wheel angle
[6][9]:
jx() = r[f ? ? (1 ? s)(sin f ? sin )], (7)
jy() = r(1 ? s)(f ? ) E tan . (8)
2.3 Drawbar pull : Fx
A general force model for a rigid wheel on loose soil
is presented in Fig. 2-(b). Using the normal stress
() and the shear stress in the x direction x(), the
drawbars pull Fx, which acts in the direction from
the soil toward the wheel, is calculated by integrating
from the entry angle f to the exit angle r [9]:
Fx = rb
f
r {x() cos ? () sin }d. (9)
2.4 Side force : Fy
The side force Fy acts along the lateral direction of
a wheel when the vehicle makes a steering maneuver.
We model the side force as follows [6]:
Fy =
f
r {rb E y() + Rb E (r ? h() cos )}d. (10)
Rb is the reaction resistance generated by the bulldozing
phenomenon on the side face of the wheel. Rb
is given as a function of a wheel sinkage h. Detailed
formulation of the side force can be seen in [8].
Motor for conveyance
F/T Sensor
Linear
potentiometer
Wheel
Lunar Regolith Simulant
1.50 [m]
Diameter : 0.18 [m]
Width : 0.11 [m]
Slide guides
Soil depth
= 0.12 [m]
Steering part
0.45[m]
Motor for driving
Fig. 3: Schematic view of the single wheel test bed
2.5 Vertical force : Fz
The vertical force should be equal to the normal
load of the wheel. The vertical force Fz is obtained
by the same method as described in equation (9) [9]:
Fz = rb
f
r {x() sin + () cos }d. (11)
3. Single Wheel Experiment and
Simulation
Single wheel experiments are carried out to validate
the wheel-soil contact model. The experimental
results are compared to numerical simulation results
obtained from the wheel-soil contact model. In particular,
the characteristics of both the drawbar pull
and the side force were confirmed.
3.1 Single wheel test bed
Fig. 3 shows the schematic view of the single wheel
test bed. The test bed comprises both a conveyance
unit and a wheel-driving unit. The steering angle
(which is equivalent to the slip angle in this test bed)
is set between the conveyance unit and the wheel.
The translational velocity and angular velocity of the
wheel are calculated based on the data obtained by
the encoders that are mounted on the conveyance motor
and wheel-driving motor, respectively. The forces
and torques generated by the wheel locomotion are
measured using a six-axis force/torque sensor located
between the steering part and the wheel. The wheel
sinkage is also measured by using a linear potentiometer.
A wheel with a diameter of 0.18 [m] and a width
of 0.11 [m] is covered with paddles having heights of
0.01 [m]. The load of the wheel is approximately 6.6
[kg].
The vessel of the single-wheel test bed is filled with
12 [cm] (depth) of loose soil, lunar regolith simulant
which is equivalent to FJS-1 [11]. The simulated lunar
soil consists of material components and mechanical
characteristics similar to those of the real lunar soil.
In the following experiments, the wheel is made
to rotate with a controlled constant velocity (0.030
[m/s]) by the driving motor, which is mounted inside
the wheel. The translational velocity of the wheel is
also controlled such that the slip ratio of the wheel is
set from 0.0 to 0.8 in steps of 0.1. The slip ratio is constant
during each run. Further, the value of the slip
Table 1: Simulation parameters and values
parameter value unit
c 0.80 [kPa]
37.2 [deg]
kc 1.37 ~ 103 [N/mn+1]
k 8.14 ~ 105 [N/mn+2]
n 1.00
kx 0.043~ + 0.036 [m]
ky 0.020~ + 0.013 [m]
angle of the wheel is varied from 5? to 30?. Multiple
test runs were conducted for a single set of the abovementioned
conditions; the total number of runs was
more than 100. In addition, during each run, more
than 100 data points were extracted for the analysis.
3.2 Numerical simulation procedure
The simulations using the wheel-soil contact model
were performed under the same conditions as those of
the single-wheel experiments. The parameters used in
the simulations are listed in Table 1. Each parameter
is experimentally determined. The drawbar pull and
side force are calculated by equations (9) and (10),
respectively.
3.3 Results and discussion
Experimental measurements of the drawbar pull
and side force are plotted in Fig. 4-(a) and Fig. 4-
(b), respectively, for each slip angle from 5? to 30?.
As mentioned above, hundreds of data points were obtained
from a single test run. Each plot corresponds
to the average value of these data points. The theoretical
curves calculated by the wheel-soil contact
model are also plotted in these figures.
From Fig. 4-(a), it is seen that the drawbar pull
increases with the slip ratio. The reason for this behavior
is that the soil deformation (shear stress) in
the longitudinal direction of the wheel increases with
the slip ratio. This results in a large soil deformation
which, in turn, generates a large drawbar pull.
In the range of slip ratios from 0 to 0.3, the drawbar
pull becomes smaller with increasing slip angles.
This is because the shearing motion in the longitudinal
direction decreases as the lateral slip (slip angle)
increases.
The differences between the experimental and theoretical
values are relatively small in the range of slip
angles from 5?-20?; however, relatively larger differences
are observed for larger slip angles (? 25?). The
reason for this is considered that the soil beneath the
wheel becomes fluidized and different mechanics may
dominate the phenomena in high slip ratio and high
slip angle conditions. In practice, however, such large
slip angles are rarely experienced under normal steering
maneuvers.
Fig. 4-(b) shows that the side force decreases along
with the slip ratio and increases according to the slip
angle. The larger the slip angle, the larger the lateral
velocity on the wheel, which leads to a larger side
force. In addition, it is observed that the side force
has its maximum value at s = 0.0 because the lateral
25
20
15
10
5
0
-5
Drawbar pull : Fx [N]
0.8 0.6 0.4 0.2 0.0
Slip ratio : s
= 5 [deg]
= 10 [deg]
= 20 [deg]
= 30 [deg]
(a) : Slip ratio - Drawbar pull
40
30
20
10
0
Side force : Fy [N]
0.8 0.6 0.4 0.2 0.0
Slip ratio : s
= 5 [deg]
= 10 [deg]
= 20 [deg]
= 30 [deg]
(b) : Slip ratio - Side force
Fig. 4: Experimental and simulation results
velocity, in proportion to the longitudinal velocity, is
maximized at s = 0.0 for each slip angle. The theoretical
curves agree well with the plotted experimental
results.
These results confirm that the wheel-soil contact
model proposed in this paper is able to represent
the motion behavior of the wheel and the contact/
traction forces with appropriate accuracy.
4. Wheel-and-Vehicle Model
To describe the motion dynamics of the rover, a
wheel-and-vehicle model is developed. In this model,
the rover is modeled as an articulated body system to
calculate the motion dynamics of its body and chassis.
Furthermore, the contact forces on each wheel of the
rover can be obtained by using the wheel-soil contact
model.
4.1 Rover test bed
The vehicle in the simulation was tested in our rover
test bed, as shown in Fig. 5. The rover test bed has
the dimensions 0.68 [m] (length) ~ 0.44 [m] (width)
~ 0.32 [m] (height) and weighs approximately 35 [kg]
in total. Each wheel has the same configuration as
that in the single-wheel experiments.
4.2 Definition of wheel-and-vehicle model
The dynamics model of the rover shown in Fig. 5
is completely equivalent to the rover test bed.
The dynamic motion of the rover for given traveling
mg
fw4 fw3
fw2 fw1
Steering
Driving
xb
yb
Zb
Main Body
(Rover test bed) (Vehicle dynamics model)
Fig. 5: Rover test bed and vehicle dynamics model
(a) : Steering experiment (b) : Slope traversing experiment
x y
x
y
Fig. 6: Experimental overviews
and steering conditions are numerically obtained by
successively solving the following motion equation:
H ?
?v
0
?
0
Nq
??
+ C +G = ?
F0
N0
??
+ JT Fe
Ne , (12)
where H represents the inertia matrix of the rover;
C, the velocity depending term; G, the gravity term;
v0, the translational velocity of the main body; 0,
the angular velocity of the main body; q, the angle
of each joint of the rover; F0 and N0, the external
forces and moments acting at the centroid of the main
body; , the torques acting at each joint of the rover;
J, the Jacobian matrix; Fe = [fw1, . . . , fwm], the external
forces acting at the centroid of each wheel; Ne,
the external moments acting at the centroid of each
wheel.
Note that each external force fwi is calculated by
the wheel-soil contact model, as mentioned in Section
2. The subscript i denotes the number of wheels (in
this case, i=1, . . ., 4).
5. Experiment using Rover Test Bed
The experiments on horizontal steering and slope
traversing were conducted in order to validate the
wheel-and-vehicle model. The dynamics simulations
using the proposed model were also carried out and
the simulation results were compared to the corresponding
experiments.
5.1 Experimental setup
Fig. 6 shows overviews of the experimental setup
with our rover test bed. The test field, located at
Japan Aerospace Exploration Agency (JAXA), consists
of a flat rectangular vessel measuring 1.5 ~ 2.0
[m], evenly filled with 10 [cm] (depth) of the lunar
regolith simulant. The vessel can be inclined up to 30
[deg].
The rover test bed travels with a given angular veTable
2: Experimental conditions
(a) Steering experiment
Case
[deg]
Front Rear
a 15 0
b 30 0
(b) Slope traversing
Experiment
Case
[deg]
Front Rear
A 0 0
B 15 0
C 15 15
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
Y position [m]
1.0 0.8 0.6 0.4 0.2 0.0
X position [m]
Case-a
Case-a
Experiment
Simulation
(a) case-a
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-0.1
Y position [m]
1.0 0.8 0.6 0.4 0.2 0.0
X position [m]
Case-b
Experiment
Simulation
(b) case-b
Fig. 7 : Comparison of the simulated and experimental steering
trajectories
locity and steering angle. Each wheel is controlled
to travel with a constant wheel angular velocity and
steering angle by an on-board computer. In every
experiment, the given angular velocity of each wheel
was maintained at 0.3 [rad/s] (2.86 [rpm]). The average
traveling velocity of the rover was around 0.03
[m/s]. The steering trajectories of the rover are measured
using a 3D optical sensor system fixed on the
ceiling.
5.2 Steering experiment and discussion
The conditions for two typical cases in the steering
experiments are listed in Table 2-(a): in case-a, the
steering angles of the left and right front wheels were
fixed at 15?, whereas, in case-b, they were fixed at
30?. The steering angles of the left and right rear
wheels were 0? in both cases.
The experimental results regarding the steering trajectories
of the rover are shown in Fig. 7-(a) and
Fig. 7-(b). The steering trajectories obtained from
the wheel-and-vehicle model are also plotted in the
same figures.
From Fig. 7, it can be observed that the wheeland-
vehicle model well simulates the experimental results.
Taking into account wheel slippage, the proposed
model calculates better steering trajectories,
which almost agree with the experimental results.
The root mean square (RMS) error of the proposed
model is negligible (less than 0.04 [m]). The accuracy
of the proposed model is better than 0.06 [m] (less
than the wheel width) even in the final state. The
proposed model simulates the experimental steering
trajectories with relatively better accuracy from the
viewpoint of the error percentages.
Throughout the experiments, it was observed that
the slip ratios were in the range from 0.1 to 0.3 and
the slip angles were in the range from ?7? to 15?.
Despite such dynamic wheel behavior, our model is
able to calculate the wheel slippage and the steering
motion of the rover with a remarkably good accuracy.
5.3 Slope traversing experiment xand discussion
In the experiment, the rover is given three different
steering configurations as listed in Table 2-(b): in
case-A, no steering anlges were given to any wheels,
in case-B, the steering angles of the front wheels were
fixed at 15?, whereas the rear wheels were 0?, and in
case-C, every steering angle was fixed at 15?. Further,
the slope angle was given from 5? to 15? in steps of
5?.
The experimental results regarding the slope
traversing trajectories of the rover are shown in Fig. 8
with different slope anlges. The trajectories obtained
from the wheel-and-vehicle model are also plotted in
the same figures.
From Fig. 8, in both case-A and case-C, it can be
seen that the trajectories calculated by the wheeland-
vehicle model agree well with the corresponding
experimental results. The errors between the experiment
and simulation are less than a few percents in
regard to the total travel distance of the rover. The
proposed model can simulate such dynamic behavior
on slopes of loose soil.
It is found that there is relatively large error in
case-B when the rover traversed on slopes of 10 [deg]
and 15 [deg]. This error is considered to be due to
the modeling error of the vehicle dynamics model. In
the simulation, the rover is modeled as an articulated
body system, and then, each part of the rover is assumed
to have completely rigid connections with adjacent
parts. The rover test bed has a flexible suspension
mechanism and this mechanism is also modeled
as a spring-damper model. However, the deformation
of this suspension became much larger than the model
expects since the cornering force of the rover in case-
B was larger than that in the other cases. Then, it is
deduced that dynamics parameters for the suspension
was not appropriate so that the large error of the trajectory
must be generated. Note that this issue can
be improved once the suspension mechanism in the
vehicle dynamics model is refined so as to correspond
with the suspension of the rover.
0
0.2
0.4
0.6
0.8
-0.4
-0.2
0
0.2
0.4
-0.04
0
0.04
X positioin [m] Y positioin [m]
Z positioin [m]
Simulation
Experiment
Case-A (Slope angle = 5 [deg])
0
0.2
0.4
0.6
0.8
-0.4
-0.2
0
0.2
0.4
-0.05
0
0.05
X positioin [m] Y positioin [m]
Z positioin [m]
Simulation
Experiment
Case-A (Slope angle = 10 [deg])
0
0.2
0.4
0.6
0.8
-0.4
-0.2
0
0.2
0.4
-0.1
0
0.1
X positioin [m] Y positioin [m]
Z positioin [m]
Simulation
Experiment
Case-A (Slope angle = 15 [deg])
(Case-A)
0
0.2
0.4
0.6
0.8
-0.4
-0.2
0
0.2
0.4
-0.04
0
0.04
X positioin [m] Y positioin [m]
Z positioin [m]
Simulation
Experiment
Case-B (Slope angle = 5 [deg])
0
0.2
0.4
0.6
0.8
-0.4
-0.2
0
0.2
0.4
-0.05
0
0.05
X positioin [m] Y positioin [m]
Z positioin [m]
Simulation
Experiment
Case-B (Slope angle = 10 [deg])
0
0.2
0.4
0.6
0.8
-0.4
-0.2
0
0.2
0.4
-0.1
0
0.1
X positioin [m] Y positioin [m]
Z positioin [m]
Simulation
Experiment
Case-B (Slope angle = 15 [deg])
(Case-B)
0
0.2
0.4
0.6
0.8
-0.4
-0.2
0
0.2
0.4
-0.04
0
0.04
X positioin [m] Y positioin [m]
Z positioin [m]
Simulation
Experiment
Case-C (Slope angle = 5 [deg])
0
0.2
0.4
0.6
0.8
-0.4
-0.2
0
0.2
0.4
-0.05
0
0.05
X positioin [m] Y positioin [m]
Z positioin [m]
Simulation
Experiment
Case-C (Slope angle = 10 [deg])
0
0.2
0.4
0.6
0.8
-0.4
-0.2
0
0.2
0.4
-0.1
0
0.1
X positioin [m] Y positioin [m]
Z positioin [m]
Simulation
Experiment
Case-C (Slope angle = 15 [deg])
(Case-C)
Fig. 8: Comparison of the simulated and experimental trajectories on slope traversing
6. Conclusion
In this paper, locomotion mechanics of lunar exploration
rovers has been addressed for numerical analyses
of motion behaviors of the rover. To deal with
dynamic motion behavior of the rover, the wheel-anddynamics
model has been developed. The proposed
model consists of two models. The wheel-soil contact
model has been developed to calculate the slip/skid
of wheels rotating on loose soil, and subsequently,
the motion dynamics of the rover have been numerically
obtained by incorporating the wheel-soil contact
model into the vehicle dynamics model.
The wheel-soil contact model has been developed
based on terramechanics and then quantitatively verified
using a single wheel test bed. In addition,
through the comparison between experiment and simulation,
it has been confirmed that the wheel-andvehicle
model demonstrates good accuracy in predicting
motion behaviors.
The models developed in this paper are useful in
performing off-line computation of the vehicle motion
trajectories under slipping/skidding conditions. Further,
the model can also be applied to evaluate the
performance of the vehiclefs climbing/traversing capabilities.
Acknowledgment
The authors would like to express their best thanks
to Dr. K. Matsumoto and Dr. S. Wakabayashi at
Japan Aerospace Exploration Agency (JAXA), for
providing opportunities of the steering experiments
using their facility.
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