8th International DAAAM Baltic Conference
"INDUSTRIAL ENGINEERING 19-21 April 2012, Tallinn, Estonia
WHEEL MOTION RESISTANCE AND SOIL THRUST TRACTION
OF MOBILE ROBOT
Petritsenko, A.; Sell, R.
Department of Mechatronics, Tallinn University of Technology, Ehitajate tee 5, 19086
Tallinn, Estonia
Abstract: The research presented in this
paper is focusing on the wheel-terrain
interaction of mobile robots. In particular
four most common interaction cases are
analysed and mathematical relationships
in terms of forces are pointed out. The
purpose of defining the wheel-terrain
intaraction forces for different cases are to
transfer them into the design library for
helping designer in early design process.
The design library is a part of early design
framework
and
the
mathematical
relationships, in addition to other
relations, are defined by novel System
Eningeering description language - SysML.
As a result of this study the library consists
of mathematical models for most common
wheel-terrain interactions of wheeled
robots.
Key words: mobile robot, motion, terrain,
traction, modelling.
1. INTRODUCTION
It is well known, that at beginning of
design process of wheeled robot several
parameters must be chosen and finally their
compliance to the functional and
performance
requirements
verified.
Usually verification of chosen parameters,
specified in requirements, is carried out by
means of simulations of mathematical
models with different complexity level or
field tests.
At the beginning and during the design
process many parameters and mutual
interaction must be evaluated, compared
and finally selected for specific robotic
applications. One of the crucial subsystems of mobile robot is a locomotion
sub-system. The design and selection of
conceptual locomotion system fix many
further robot parameters, including key
performance and terrainability parameters.
In wheeled mobile robot, obviously one of
the key components is wheel itself. When
selecting wheel parameters (material,
coverage, geometry, etc.) it is important to
analyse the required terrain capabilities of
the robot and performance criteria of
passing the obstacles and holes. In
addition, designer has to keep the mind
open and do not relay only the
conventional solution. For example in
mobile robotics, many unconventional
locomotion solutions are applied to
improve the terrainability and flexibility in
small scale locomotion end actuator. New
solutions and wheel-terrain studies can be
found in literature [1], [2] including
patented invention from Tallinn University
of
Technology,
Department
of
Mechatronics [3]. In this paper we are
focusing on the detail analyses of wheelterrain interaction and forces influencing
on the different conditions in off-road
terrain. The research results are
mathematical models formulated into novel
engineering description language – SysML
(System Modeling Language) [4]. This
approach enables to reuse the common
cases on later mobile robot design,
including early stage simulations and
candidate solution evaluation process.
2. WHEEL MOTION RESISTANCE
2.1 Motion resistance due to wheel-terrain
interaction (R R )
When a robot moves on paved surfaces and
roads it consumes energy to overcome the
rolling resistance between the tires and the
ground, as well as gravitational and inertial
forces. Obstacles are dealt as a specific
case and are simulated separately.
Rolling resistance between the tire and the
ground is attributed to tire slip, scrubbing
in contact patch, deflection of the road
surface and energy losses due to tire
adhesion on the road and hysteresis.
Rolling resistance varies with the type and
material of tire tread, velocity of vehicle,
and environmental parameters, such as
temperature and humidity [5].
There exists four general wheel-terrain
interaction cases [6] shown in Fig. 1.
Fig. 1. Possible wheel-terrain interaction
cases
Notations for Fig.1
A) rigid wheel travelling over rigid terrain,
B) deformable wheel travelling over rigid
terrain,
C) rigid wheel travelling over deformable
terrain,
D) deformable wheel travelling over
deformable terrain.
These
four
different
wheel-terrain
interaction cases involve different wheel
resistance forces involved in mathematical
models. Simulation models are generated
according to mathematical model. This
allows designer to select different cases
according to simulation interest.
Case A – rigid wheel travelling over rigid
terrain
This model is simplified approach of
wheel-terrain interaction. The model
assumes that rolling resistance of the tire
neglected and wheel-terrain interacts to
each other at single contact point. This
kind of model suits for metallic wheels or
wheels with solid non-metallic tires.
Case B – deformable wheel travelling over
rigid terrain
In this wheel-terrain interaction case
pneumatic tires are usually considered.
Deformation of the terrain is generally
neglected and it is considered that rolling
resistance is caused primarily by the tire
hysteresis, i.e. due to deflection of carcass
of tire. When tire is rolling the carcass of
tire is deflected in the area of ground
contact patch and centre of normal pressure
shifts in direction of rolling. The shift of
centre of normal pressure causes increase
of rolling resistance.
Complex relationships between design and
operational parameters of tire and its
rolling resistance make it difficult to
develop an analytical method for predicting
of rolling resistance [7]. Determination of
rolling resistance, therefore, relies almost
on experiments. For modelling purposes,
design library is composed consisting of
different tire manufacturer’s data, where
average rolling resistances of pneumatic
tires are given. Additionally, design library
consists of data, where rolling resistance of
tire at different inflation pressure is given
as well. In the absence of tire
manufacturer’s
data,
the
empirical
formulas to calculate the rolling resistance
of tire are used and given in [8].
Case C - rigid wheel travelling over
deformable terrain
In this model it is assumed that wheel
travelling over off-road terrain is subject to
sinkage. Wheel is classified as rigid if
deflection of under static loading is much
lower than deformation of soil/terrain. This
model fits to metallic wheels, solid nonmetallic tires and high-pressure pneumatic
tires operating on weak soils. In case of
pneumatic tires, the rolling resistance due
to tire hysteresis is neglected. The main
cause of the motion resistance is due the
deformation of the soil. Motion resistance
of tire consists then from two major
contributions: the compaction resistance
and the bulldozing resistance.
Compaction resistance
This form of wheel motion resistance can
be analyzed considering the mechanics of a
rigid wheel rolling into soft terrain. The
motion resistance of rigid wheel is
produced by vertical work done in making
a rut of depth z as [7]
Rc =
 3W 


 D
(2n+2)
(2n+1)
( 2n+2 )
2n+1
1
(2n+1)
1
 kc
 ( 2n+1) )
(3-n)
(n+1)b
 +k f 
 b

(1)
where b – width of wheel contact area; k c,
k f and n – pressure-sinkage parameters of
the specific terrains; W – vertical force
exerted by the tire to the terrain; D – wheel
diameter.
The pressure-sinkage parameters for
different soils are found from literature and
included to the design library for modeling
purposes.
Bulldozing resistance
Bulldozing resistance is developed when a
substantial soil mass is displaced by a
wheel. This type of resistance is very
common when a wheel compresses the
surface layers of soil and pushes
compacted soil fore and aft of the tire [7].
Soil bulldozing phenomenon is apparent in
the case of a wide wheel traversing very
loose soils and has been estimated to cause
a significant increase in total motion
resistance for sinkage values greater than
1/6 of the wheel diameter. The bulldozing
resistance on narrow tires is mitigated by
the fact that a portion of the soil bulk is
pushed to the sides of wheel. The
bulldozing resistance can be calculated by
implementing the theory of bearing
capacity of soils subject to various criteria
of failure. The equation to calculate the
bulldozing resistance is given in [9] and
not repeated here.
Case D – deformable wheel travelling over
deformable terrain
In this model it is assumed that tire is
subject to sinkage and deflection of tire is
in same order as deformation of off-road
terrain. If the maximum contact pressure
that terrain can support is greater than
combined inflation pressure of tire and
pressure due to the stiffness of the carcass,
then the tire flattens at contact patch with
the terrain. This model suits best for low to
medium pressure pneumatic tires travelling
over off-road terrains [7], [8].
The compaction resistance of deformable
wheel travelling over deformable terrain is
derived in [8] and is
( n+1)
Rc =
bpgr
( n+1) 
n
1
n

(2)
kc
+k f 
 b

where p gr – average ground pressure which
is usually provided by the tire
manufacturer for a given inflation pressure
and wheel loading. Eq. (2) shows that the
compaction resistance of a flexible wheel
is solely a function of width of contact
patch (minimum width of tire in this case)
and geophysical properties of soil.
Additional resistance that has to be
accounted is rolling resistance due to tire
hysteresis as it was pointed out in case B.
2.2 Motion resistance due to slopes (R S )
Ground slopes add a component to the
motion resistance of wheel which is
proportional to the component of parallel
to slope.
2.3 Aerodynamic motion resistance (R A )
Aerodynamic drag is dependent on
aerodynamic factor and square of speeds.
Aerodynamic drag of wheel can be
neglected due to its low influence to wheel
motion resistance.
2.4 Inertia resistance (R in )
The generation of inertia forces is closely
related to traction forces that are available
at wheel-terrain contact surface. Traction is
limited by mechanical properties of terrain
and loading at wheel/soil interface patch.
Thus inertia properties of wheel and robot
can
be
properly
evaluated
after
determination of maximum tractive effort
available at wheel-terrain contact patch.
3. SOIL THRUST AND TRACTION
Vehicle motion relevant to the terrain is
produced through traction (F K ). Caused by
a physical process of adhesion and
deformation, traction develops at the
interface of a powered wheel with the
ground. The maximum produced traction is
limited by the adhesion between wheel and
ground [7] and the torque-speed
characteristics of vehicle’s prime mover
and drivetrain which basically determine
that maximum torque and power
transmitted to the wheel.
z
Direction
of travel
T
W
F
x
r
G
N
Fk
f
G - weight of the wheel,
W- portion of the robotics weight
applied to the wheel
N- normal reaction of wheel,
FR - sum of resistance forces due
to wheel-terrain interaction,
Fk - traction force,
T - applied torque,
R - radius of the wheel
FR
Fig. 2 Forces acting to the driven wheel
For locomotion in soft soils traction is
limited by mechanical properties of soil
and loading at wheel/soil interface patch.
Maximum force that can be sustained by
the soil before excessive slippage occurs is
known as soil thrust. The study of
mechanics of traction generation provides
equations for configuration of a robot’s
wheel and overall robot geometry. Forces
acting on driven wheel generally are shown
in Fig.2
Case A – rigid wheel travelling over rigid
terrain
This model assumes that wheel-terrain
interacts to each other at single contact
point. Only force acting at wheel-terrain
interaction point is frictional force. No slip
of tire allowed in this model. Maximum
traction developed is proportional to the
wheel load and is limited due to Coulomb’
friction force
(3)
FK =μ s (G+W)
Where µ s - sliding coefficient of friction,
G and W – weight of wheel and load to
wheel respectively.
This relationship is well known from
classical mechanics of rigid bodies and is
the simplest model to evaluate a maximum
traction force.
Case B – deformable wheel travelling over
rigid terrain
When tire is rolling, carcass of tire is
deflected in the area of ground contact
patch. Deformation of rigid surface is
neglected. Due to tire deflection the
distance that tire travels when subject to
driving torque will be less that in free
rolling and longitudinal slip of tire occurs.
The traction force developed is given then
as simplified function of longitudinal slip
of tire as [7].
 μ p (G+W) 
FK =μ p (G+W) 1 (4)
4C
i
i


where µ p - coefficient of adhesion; i – slip
of tire; C i – longitudinal stiffness of tire.
CASE C and D – rigid and flexible wheel
travelling over deformable terrain
For rigid and flexible wheel travelling in
soft soils traction is limited by mechanical
properties of soil and loading at wheel/soil
interface patch. Maximum tractive force F
is limited by the thrust produced by soil,
which in turn is proportional to mechanical
strength of soil. Data on the stress/strain
relationships of disturbed soils, sand, snow,
and saturated clays have verified the
appropriateness
of
Janosi-Hanamoto
relationship to describe shear stress-strain
behavior of unprepared terrain [7],[10] as
 -j 
τ(θ)= ( c+p(θ)tanφ ) 1-e K 




(5)
θ0
3.1 Net traction
Net traction of the driven wheel shown in
Fig. 2 is the force equal to the difference
between the tractive effort F K developed by
the wheels and resisting forces as
n
where c and φ are modulus of cohesion
and angle of internal friction of soil, p(θ) –
normal pressure distribution in soil as
function of contact angle between wheel
and soil; j - the shear displacement and is
function of slip velocity; K - shear
deformation parameter,
Integration of the Eq. (5) over contact
patch area leads to total tractive effort as
FK = ∫ τ ( θ )cosθdθ
Of critical importance to the amount of
forward thrust developed at tire-soil
interface is geometric shape of the tire,
which has to be determined before
evaluation of the integral in Eq. (5).
(6)
0
=
F FK − ∑ Ri
(7)
i =1
Net traction is available force that can be
used to generate acceleration of the wheels
and robotics.
4. MODEL LIBRARY
The framework supporting early design
process is developed during the previous
research [11]. One of the key factors of this
framework is a design library which
consists of unified mathematical models
par WheelTerrainInteraction [Resistance]
Do not have significant
impact on low speed
 3W 


 D
Rc =
(2n+2)
( 2n+2)
2n+1
1
1
k
 ( 2n+1) )
(3-n) (2n+1) (n+1)b (2n+1)  c +k f 
 b

=
Rslope
( Gi + Wi ) sin α
Aerodynamic:Resistance
( n+1)
bp gr
Rc =
Case C
n
( n+1) 
Slope:Resistance
Case D
Total:Resistance
ΣR=FR
Compaction:Resistance
Bulldozing:Resistance
Givn in Ref. 9
Obstacle:Resistance
Snow:Resistance
Internal:Resistance
Dependent on the
Component selection
Separate analysis
1
kc
n
+k f 
 b

Separate analysis
Fig. 3 Simplified SysML parametric diagram of resistances in wheel-terrain interaction
transferred into the SysML graphical
diagram format. The model design library
offers pre-defined models for designer to
start fast and efficient early design process
on mobile robotic domain. In this paper we
have
described
the
wheel-terrain
interaction cases which are the ground
issue of the wheeled robots and have to
have therefore well formulated and
verified. The corresponding unified SysML
parametric diagram is shown in Fig. 3.
where the key factors are highlighted. This
graphical representation can be used as a
input for the simulations or transferred
automatically
into
the
analytical
calculations.
5. CONCLUSIONS
In this paper the major resistance forces
and generation of traction forces in
different basic wheel-terrain interaction
cases are introduced. The mathematical
description of them are generated and
transferred into graphical diagram format
of SysML. It allows performing efficient
and optimal early design stage by offering
the necessary simulation models at the
beginning of the design stage, when many
wheel parameters and wheel-terrain
interaction cases must be evaluated,
compared and finally selected for specific
robotic application.
longitudinal interaction for developing
electric vehicle control systems, Vehicle
System Dynamics, 2010, 49:3, 433-447.
3. Sell, R., Kaeeli, M., Wheel-Leg (Wheg),
Patent no EE05283B1, 2009
4. OMG SysML - System modeling
Language
specification,
OMG
document formal/10-06-02, 2010
5. Gillespie, T., D. Fundamentals of
vehicle
dynamics.
Society
of
Automotive Engineers, Inc, 1992.
6. Iagnemma, K., Dubowsky, S. Mobile
Robots in Rough Terrain. SpringerVerlag Berlin Heideberg, 2004.
7. Wong, J., Y. Theory of Ground Vehicles
4th Ed. John Wiley & Sons, Inc;
Hoboken, New-Jersey, 2008.
8. Saarilahti M. Soil interaction models,
In: Haarlaa R, Salo J, University of
Helsinki, Pub. 31, 2002.
9. Genta, C. Introduction to the Mechanics
of Space Robots. Springer, 2011.
10.
Ray, L.R., Brande, D.B., Lever,
J.H. Estimation of net traction for
differential-steered wheeled robots. J. of
Terramechanics, 2009, 46, 75-87.
11.
Sell R., Coatanea E., Christophe F.
Important aspects of early design in
mechatronic,
6th
International
Conference
of
DAAAM
Baltic
Industrial Engineering, Tallinn, 2008
8. ADDITIONAL DATA ABOUT
AUTHORS
6. ACKNOWLEDGEMENT
This research was supported by funding of
the Estonian Science Foundation grant No.
8652
7. REFERENCES
1. Zheng, L., et. Al. A Novel High
Adaptability Out-door Mobile Robot
with Diameter-variable Wheels, Proc. of
the IEEE Int. Conference on
Information
and
Automation
Shenzhen, China, 2011
2. Chengbin Ma, Min Xu & Hao Wang,
Dynamic emulation of road/tyre
Raivo Sell, Ph.D, senior researcher, Tallinn
University of Technology, Department of
Mechatronics, Ehitajate tee 5, Tallinn,
[email protected]
Andres Petritsenko, Ph.D, researcher,
Tallinn
University
of Technology,
Department of Mechatronics, Ehitajate tee
5, Tallinn, [email protected]