Measurement of Wheel Radius in an Automated Guided Vehicle - MDPI
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applied
sciences
Article
Measurement of Wheel Radius in an Automated
Guided Vehicle
Miroslaw Smieszek 1 , Magdalena Dobrzanska 1 and Pawel Dobrzanski 2, *
1 Department of Quantitative Methods, Faculty of Management, Rzeszow University of Technology, al.
Powstancow Warszawy 10, 35-959 Rzeszow, Poland; msmieszk@prz.edu.pl (M.S.); md@prz.edu.pl (M.D.)
2 Department of Computer Engineering in Management, Faculty of Management, Rzeszow University of
Technology, al. Powstancow Warszawy 10, 35-959 Rzeszow, Poland
* Correspondence: pd@prz.edu.pl
Received: 25 June 2020; Accepted: 5 August 2020; Published: 8 August 2020
Abstract: In the case of automated guided vehicles using odometry, a very important issue is to
know the actual rolling radius of the wheel used to calculate the position of the vehicle. This radius
is not constant. Its changes depend on the elastic deformation of the band layer and wheel slip.
The theoretical determination of the value of the radius and the nature of the change is very difficult.
For this reason, it was decided to determine the rolling radii by the experimental method. For this
purpose, an appropriate test stand was built and the proposed research method was checked.
Within the tests, there was also obtained a number of interesting results characterizing the material
used to build bands and the ranges of changes rolling radii for the tested material were specified.
Keywords: automated guided vehicle; wheel radius; wheel slip; odometry errors; measurement
methods
1. Introduction
The use of an automated guided vehicle (AGV) is becoming more and more common. In many
cases, the routes implemented by these vehicles are subject to constant changes and are adapted
to the currently assigned task. The basic navigation system for most of these vehicles is odometry.
The navigation process using odometry assumes the constancy of parameters such as rolling radii and
wheelbases. Slips are also not included in this process. In real conditions, when working with different
loads and on different surfaces, these parameters change their values and slides occur. Not taking
these changes into account in the navigation algorithm leads to errors in determining the position.
An extensive review regarding odometry errors is included in papers [1,2]. There are several types
of errors that affect the accuracy of the positioning process. These include systematic and unsystematic
errors. The systematic errors that arise in determining the current position during the movement of the
vehicle are aggregated [1,3]. Depending on the type of surface, the share of the systematic and the
unsystematic errors in the error of position determination may be different. In addition, odometry
errors can be caused by the same movement equations [4]. These equations describe any trajectory
through a series of short sections. The accuracy of this approximation depends on the sampling
frequency and vehicle speed.
Many methods are used to correct errors. The first works related to error correction concern
systematic errors. Correction of these errors and calibration of the system takes place after traveling
the preset route section and measuring the error of the final position [1,2,5–8].
Modern measurement techniques enable online measurements. A variety of techniques may
be used for measurements, a brief review of which is contained in [9–11]. In most solutions using
Appl. Sci. 2020, 10, 5490; doi:10.3390/app10165490 www.mdpi.com/journal/applsciAppl. Sci. 2020, 10, 5490 2 of 13
additional measurement systems, extensive filtration methods are applied. These can be online
methods such as at work [12–14] or offline as at work [15].
All the aforementioned works assume that the wheel and ground are rigid, cooperation between
the wheel and the road takes place at one point and there is no slip. In recent years, there have been
works including changes in rolling radius and occurrence of wheel slip [16–21].
Such changes in the rolling radius have been taken into account in the navigation process of the
vehicle presented in Sec. 4.1 and intended for the carriage of loads. The shape of the route and the
mass of the cargo carried during the operation of the vehicle is subject to change [22]. It depends on
the currently assigned task. In the case of a change in the mass of the transported load or its position
on the vehicle, the loads of individual wheels change. Along with these changes, the rolling radii of the
vehicle wheels are subject to change. Failure to take this fact into account in the calculation algorithm
will lead to errors. In order to reduce these errors, it is necessary to enter into the calculation algorithm
appropriate values of rolling radii taking into account the current wheel load. This operation will be
possible only after examining and determining the dependence of the rolling radius on the wheel load.
As part of the article, it was decided to describe the measurement methodology as well as the course of
experimental research aimed at determining the dependence of the rolling radius on the wheel load in
the vehicle under consideration.
The paper is organized as follows. Section 2 presents a research object and its characteristics and
information about the wheel and its properties. Section 3 presents an overview of selected measurement
methods. As part of the review, errors resulting from failure to maintain ideal measuring conditions were
indicated. Taking into account the measurement capabilities and existing environmental conditions,
the most adequate measurement method for the further stage of experimental research was indicated.
Section 4 presents the results obtained from experimental research. The wheel radii were
determined in these tests. These tests were carried out for various loads. As part of the section,
a statistical analysis of the measurement results was also carried out and characteristic values related to
the actual wheel operating conditions were determined. Section 5 contains a discussion of the obtained
results and their in-depth analysis. Finally, Section 6 presents the conclusions.
2. Determination of the Vehicle Position and the Role of the Rolling Radius
2.1. Movement of a Wheeled Vehicle
The basic method of determining the position in motion of an autonomous vehicle is odometry.
Its rules are described in detail in the works [1–4,12]. There are many ways to drive and steer the vehicle.
Most autonomous transport vehicles use two modes of driving and steering. In the first one,
independently powered rear wheels are used for driving and steering. In the second case, the front
wheel is driven and steered. In both cases, the data necessary to control the vehicle, such as the
rotational speed of the wheels or the steering angle of the front wheel, come from encoders installed at
the respective wheels. Figure 1 shows the vehicle in the X0 O0 Y0 reference system and the individual
quantities characterizing the vehicle movement are marked.Most autonomous transport vehicles use two modes of driving and steering. In the first one,
independently powered rear wheels are used for driving and steering. In the second case, the front
wheel is driven and steered. In both cases, the data necessary to control the vehicle, such as the
rotational speed of the wheels or the steering angle of the front wheel, come from encoders installed
at theSci.
Appl. respective wheels. Figure 1 shows the vehicle in the X0O0Y0 reference system and the individual
2020, 10, 5490 3 of 13
quantities characterizing the vehicle movement are marked.
Figure 1. Vehicle movement in the reference system X0 O0 Y0 .
The position of a selected point O in the base reference system X0 O0 Y0 (Figure 1) is defined by the
state vector X j . The next position j + 1 of the vehicle point O is determined by the state vector X j+1 .
State vector X j+1 in the j + 1 iteration is expressed by:
X j+1 = X j + V j+1 dt (1)
where:
vO,j+1 cos θ j + ω j+1 dt
x j+1 x j
sin θ j + ω j+1 dt
X j+1 = y j+1 X j = y j V j+1 = v . (2)
O, j+1
θ j+1 θj
ω j+1
Velocities vO, j+1 and ω j+1 can be determined from the following relations, for a vehicle using
measured data from the rear wheels:
vO,j+1 = vR,j+1 + vL,j+1 /2, (3)
ω j+1 = vR, j+1 − vL,j+1 /b, (4)
where:
vR, j+1 —speed of the right wheel in j + 1 step;
vL, j+1 —speed of the left wheel in j + 1 step;
ω j+1 —angular velocity of the vehicle relative to the reference system X0 O0 Y0 ;
b—wheelbase of the driven wheels.
For a vehicle using measurement data from the front wheel:
vO,j+1 = vK,j+1 cos α j+1 , (5)
ω j+1 = vK,j+1 sin α j+1 /l. (6)
where:
vK,j+1 —speed of the front wheel in j + 1 step;
αK,j+1 —turning angle of the front wheel in j + 1 step.
To determine the speed of individual wheels: vR,j+1 , vL,j+1 vK,j+1 the data from the encoders and
the assumed values of the radius r of the vehicle wheel are used.
In the above consideration, it was assumed that the wheels are rigid and they roll without slipping,
the contact between the wheel and the floor is a point contact and the radii r of the rear wheels are
the same.K, —turning angle of the front wheel in j + 1 step.
To determine the speed of individual wheels: R, , L, K, the data from the encoders
and the assumed values of the radius r of the vehicle wheel are used.
In the above consideration, it was assumed that the wheels are rigid and they roll without
slipping,
Appl. the10,contact
Sci. 2020, 5490 between the wheel and the floor is a point contact and the radii r of the4 rear
of 13
wheels are the same.
2.2. Wheel and
2.2. Wheel and its
Its Properties
Properties
The
The wheel
wheel radius
radius used
used to
to determine
determine thethe position
position from
from odometry
odometry calculations in real
calculations in real movement
movement
conditions is not a constant value. Its size is influenced by many factors. The vehicle
conditions is not a constant value. Its size is influenced by many factors. The vehicle wheel wheel shown
shown in
in Figure 2a has a flexible belt. Thanks to this, the wheel has the ability to damp vibrations
Figure 2a has a flexible belt. Thanks to this, the wheel has the ability to damp vibrations in selected in
selected
operatingoperating conditions.
conditions. The physical
The physical modelmodel of theofwheel
the wheel is shown
is shown in in Figure2b.
Figure 2b.The
Thewheel
wheel has
has
longitudinal
longitudinal and circumferential elasticity, which was modeled by a spring. Additionally, the model
and circumferential elasticity, which was modeled by a spring. Additionally, the model
features
features damping
damping elements
elements designed
designed toto dissipate
dissipate vibrations.
vibrations.
(a)
Appl. Sci. 2020, 10, x FOR PEER REVIEW (b) 4 of 13
Figure 2. Vehicle
Figure wheel
2. Vehicle (a) (a)
wheel view; (b) (b)
view; model, g—deflection,
model, rS —static
g—deflection, radius.
rS—static radius.
In the case of a stationary wheel, the vertical force loading the wheel reduces the radius of
In the case of a stationary wheel, the vertical force loading the wheel reduces the radius of the
the wheel by the value of deflection g (Figure 2b). In the case of a rolling wheel, circumferential
wheel by the value of deflection g (Figure 2b). In the case of a rolling wheel, circumferential
deformations and slips occur in the wheel’s contact with the ground. The rolling radius of the wheel
deformations and slips occur in the wheel's contact with the ground. The rolling radius of the wheel
changes. These changes are additionally dependent on the type of horizontal forces and moments
changes. These changes are additionally dependent on the type of horizontal forces and moments
acting on the wheels. There may be two cases here. In one of them, the wheel is driven; the other is
acting on the wheels. There may be two cases here. In one of them, the wheel is driven; the other is
braked. The occurring phenomena have been comprehensively described in the works [23,24]. In real
braked. The occurring phenomena have been comprehensively described in the works [23,24]. In real
operating conditions of the wheel there is a slip and it is better than to use the concept of the rolling
operating conditions of the wheel there is a slip and it is better than to use the concept of the rolling
radius rt .
radius rt.
In the range of small slips up to 2–3%, the relationship between pressure forces and slip is linear
In the range of small slips up to 2–3%, the relationship between pressure forces and slip is linear
in both cases of operation. This is shown schematically in Figure 3a.
in both cases of operation. This is shown schematically in Figure 3a.
(a) (b)
Figure
Figure 3. Changes
3. Changes in wheel
in wheel radius
radius (a)(a)
as as a function
a function of static
of static load;
load; (b)(b) during
during motion,
motion, under
under thethe driving
driving
force/braking
force/braking force.
force.
Changing
Changingthethe
slipslip
value affects
value the value
affects of the of
the value rolling radius. radius.
the rolling The course
Theofcourse
changes
of in the rolling
changes in the
radius is shown
rolling radius schematically in Figure 3b
is shown schematically [25]. 3b [25].
in Figure
3. An Overview of Selected Measurement methods
3.1. Introduction
Rolling radius defined as the agreed size of the radius of such a rigid wheel, which at section L
performs the same number of rotations nk. This describes Relationship (7):
L=2πrtnk. (7)
Thus, the rolling radius of the wheel rt is determined by the formula:
L
= . (8)
2 nkAppl. Sci. 2020, 10, 5490 5 of 13
3. An Overview of Selected Measurement methods
3.1. Introduction
Rolling radius defined as the agreed size of the radius of such a rigid wheel, which at section L
performs the same number of rotations nk . This describes Relationship (7):
L = 2πrt nk . (7)
Thus, the rolling radius of the wheel rt is determined by the formula:
L
rt = . (8)
2πnk
Wheel slip of the driven wheel s is:
vs ωr − ωrt rt
s= = = 1− . (9)
vx ωr r
Wheel slip of the braked wheel s is:
vs ωrt − ωr r
s= = = 1− . (10)
vx ωrt rt
where:
rt —rolling radius;
r—free radius;
vs —slip velocity;
vx —wheel center speed.
For the driven wheel in the slip range s from 0 to 1, the rolling radius rt changes from r to 0.
With the case of the rolling radius rt = 0 we deal when the vehicle stands v = 0 and the driven wheels
rotate at an angular speed ω > 0. For a nondriven wheel in the slip range s from 0 to 1, the rolling
radius rt varies from r to ∞. With the case of a rolling radius, rt = ∞ is when the vehicle moves at a
speed v > 0 and the braked wheel is blocked ω = 0. In the range of small slip to s = 0.02, the relationship
between vertical wheel load and circumferential force is linear [22].
There are many advanced theoretical methods to determine the rolling radius as a function of
the type of work and load. The accuracy of these methods depends on the knowledge of specific
parameters. Sometimes, and in many cases, this requires additional experimental testing. For this
reason, it was decided to determine the rolling radii directly on the basis of experimental research.
According to Relationship (8), to determine the rolling radius, it is sufficient to measure the distance
traveled by the road L and read the recorded data on the quantities made by the wheel revolutions nk .
In this measurement method, it is very important to maintain the straight-line nature of the movement.
In real traffic conditions, there will always be smaller or larger deviations from the theoretical route.
These deviations are affected by the control system and incorrect initial position and values of the
rolling radii entered into the navigation algorithm.
3.2. Methodology of Measurements in Curved Motion
With a large deviation of the actual track from the assumed theoretical course, the vehicle performs
arc motion (Figures 4 and 5). Determination of rolling radii from the dependence (8) for such a course
is subject to a significant error. The distance traveled along a curve differs from the length of the section
connecting the beginning and the end of the curve.3.2. Methodology
With a large of deviation
Measurements in Curved
of the actual Motion
track from the assumed theoretical course, the vehicle
performs arc motion (Figures 4 and 5). Determination
With a large deviation of the actual track from the of rolling
assumed radii from the dependence
theoretical (8) for
course, the vehicle
such a course
performs is subject
arc motion to a significant
(Figures 4 and 5). error. The distance
Determination traveled
of rolling along
radii froma the
curve differs from
dependence (8) the
for
length of the section connecting the beginning and the end of the curve.
such a course is subject to a significant error. The distance traveled along a curve differs from the
Appl. Sci. 2020, 10, 5490 6 of 13
length of the section connecting the beginning and the end of the curve.
Figure 4. An example of a curvilinear route.
Figure4.
Figure 4. An
An example
example of
of aa curvilinear
curvilinear route.
route.
In many cases, the recorded measurement data was characterized by high interference. A
fragment
In of such
Inmany
many cases,a course
cases,thethe is shown
recorded in Figure 5.
measurement
recorded Cyclic
data
measurement datadeviations
was andbymeasurement
characterized high by
was characterized highdisturbances
interference. A fragment
interference. are
A
visible
of such on
a the
course course.
is shown This
in type
Figure of
5. data
Cyclichas been
deviations subjected
and to appropriate
measurement filtration
disturbances
fragment of such a course is shown in Figure 5. Cyclic deviations and measurement disturbances are are processes
visible on
described
the
visible oninThis
course. [15].
the type of data
course. This has been
type of subjected
data has to appropriate
been subjectedfiltration processesfiltration
to appropriate describedprocesses
in [15].
described in [15].
Figure 5. View of oscillations and interference
interference on
on aa selected
selected fragment.
fragment.
Figure 5. View of oscillations and interference on a selected fragment.
There are are many
many ways
ways to to determine
determine the
the path
path traveled
traveled in in curvilinear
curvilinear motion.
motion. In the the conducted
conducted
preliminary
preliminary
There are tests,
tests, two
many methods
twoways
methods were used,
were used,
to determine shown
the shown in Figure
in Figure
path traveled 6.
6. The
in curvilinearfirst method determines
methodIndetermines
motion. the conducted the
the
distance
distance ssc between the start and end position of a selected point of the
preliminary tests, two methods were used, shown in Figure 6. The first method determines the
c vehicle. In the
the second
second method,
characteristic
characteristic
distance sc between markers theonstart
markers onthethe
reference
and surface
reference can
of a be
surface
end position used
can be to
selected measure
used
point distance
to the
of measure sIn
w , the
e.g.,second
vehicle.distance door openings.
sw, e.g., door
method,
In both
openings. methods,
characteristic In both the distance
methods,
markers on the d from the
thereference reference
distance dsurface
from the surface is
canreference recorded
be used surface and the counter
is recorded
to measure distance reads
andswthe the data
counter
, e.g., door
from
reads the
the
openings.
Appl. encoders
data
Sci. 2020,In from
10,both of individual
the encoders
x FORmethods,
PEER REVIEW wheels.
of individual wheels.
the distance d from the reference surface is recorded and the counter 6 of 13
reads the data from the encoders of individual wheels.
Figure 6. The
The curve
curve linear
linear run
run and measurement data, R—radius of curvature, d, dnn distance between
vehicle and reference surface, sww, sCC measured path, 00 middle of the arc, half arc angle ϕ
measured path, φ..
Knowledge of
Knowledge value of the radius of curvature R and half arc angle φ
of the value ϕ makes it possible to
determine the lengths of arcs traveled by the wheels and their rolling radii.
rolling radii.
3.3. Methodology
3.3. Methodology of
of Measurements
Measurements in
in Rectilinear
Rectilinear Motion
Motion
In order
In order to
to meet
meet the
the requirements
requirements for
for ensuring
ensuring the
the straight-line movement of
straight-line movement of the
the vehicle,
vehicle, it
it is
is
necessary to properly control and drive the vehicle. This can be obtained, for example,
necessary to properly control and drive the vehicle. This can be obtained, for example, by movingby moving
along aa given
along given reference
referenceplane.
plane.AsAsshown
shownininFigure
Figure7,7,
asaspart
partof of
this movement,
this thethe
movement, fixed distance
fixed of the
distance of
selected vehicle point and the assumed reference plane is maintained.
the selected vehicle point and the assumed reference plane is maintained.3.3. Methodology
3.3. Methodology of
of Measurements
Measurements in
in Rectilinear
Rectilinear Motion
Motion
In order
In order toto meet
meet the
the requirements
requirements for
for ensuring
ensuring the
the straight-line
straight-line movement
movement of of the
the vehicle,
vehicle, itit is
is
necessary to properly control and drive the vehicle. This can be obtained, for example,
necessary to properly control and drive the vehicle. This can be obtained, for example, by moving by moving
along aa given
along given reference
reference plane.
plane. As
As shown
shown inin Figure
Figure 7,
7, as
as part
part of
of this
this movement,
movement, the
the fixed
fixed distance
distance of of
Appl.
the Sci. 2020, 10,
selected 5490 point and the assumed reference plane is maintained.
vehicle 7 of 13
the selected vehicle point and the assumed reference plane is maintained.
Figure 7.
Figure 7. Data
Data recorded
Data recorded during
recorded during the
the movement
movement of
of the
the vehicle.
vehicle.
Due to
Due
Due to the
to the imperfections
the imperfections of
imperfections of the
of the reference
the reference surface
reference surface and
surface and the
and the measuring
the measuring system,
measuring system, there
system, there may
there may be
may be three
be three
three
types
types of
types of errors.
of errors. The first
errors. The first of of these
of these
these isis related
is related
related to to the
to the nonlinearity
the nonlinearity
nonlinearity of of the
of the reference
the reference surface.
reference surface.
surface. The The second
The second
second
error
error is
is related
related to
to the
the structure
structure of
of the
the reference
reference surface.
surface. The third error
error is
error is related to the structure of the reference surface. The third error is related to the accuracy of
is related
related to
to the
the accuracy
accuracy of
of
the
the measuring
measuring device.
device. All
All three errors affect the measurement data
the measuring device. All three errors affect the measurement data which are used in the control which are used in
in the
the control
control
system. This
system.
system. This contributes
This contributesto
contributes tothe
to theformation
the formationofof
formation ofvehicle
vehicleoscillations
vehicle oscillations
oscillations ininin relation
relation
relation to
toto
thethe assumed
assumed
the assumed direction
direction
direction of
of movement.
movement. The The
coursecourse
of such of such oscillations
oscillations is shown isinshown
Figure in
8a.
of movement. The course of such oscillations is shown in Figure 8a. Some of the errors in Figure
Some of8a.
theSome
errors of
in the errors
measurement in
measurement
measurement data
data can be eliminated can
data can be be eliminated
byeliminated
applying by by applying
real-time
applying real-time
filtering.
real-time filtering.
Excessive Excessive
filtering.deviation
Excessive fromdeviation
the set
deviation from
path
from the
thecanset
set
path
be
path can be
partially
can be partially
partially
eliminated eliminated
by the use
eliminated by the
by the use of
of real-time
of real-time
use real-time filtration.
filtration. An example
An example
filtration. An example of using
of using
of using an online
an online
an online
filterfilter
for
filter
for measurement
for measurement
measurement data data
data entered
entered
entered into
intointo the
thethe control
control system
system
control system is presented
presented
is presented
is in inin [12].
[12].
[12]. The The
The benefits
benefits
benefits of
ofof using
using
using such
such
such a
a filter
afilter are
filter are
are shown
shown
shown in
inin Figure
Figure
Figure 8b.
8b.8b.
(a)
(a) (b)
(b)
Figure 8.
Figure
Figure 8. The
8. The route
The route of
route of the
of the vehicle
the vehicle with
vehicle with the
with the enlargement
the enlargement of
enlargement of the
the selected
selected fragment
fragment (a)
(a) before
before filtration;
filtration;
(b)
(b) after
(b) after filtration
after filtration along.
filtration along.
along.
The application of online filtration has reduced deviations from the set driving path and thus
increased the accuracy of the rolling radius determination process. However, it is a big problem to find
the appropriate reference surface in real measurement conditions. Some of the surfaces used in the
research had discontinuities. The effect of these disturbances is visible peaks in Figures 4, 5 and 8.
3.4. Assessment of the Influence of Deviation from the Ideal Trajectory on the Value of the Rolling Radius
In the case of a vehicle with a propulsion system presented in Section 4.1 small errors in the values
of the rolling radii and control system errors cause a slight oscillation A from the set driving path.
For the angle of deviation from the given path ψ = 0, the vehicle moves along the assumed direction
(Figure 9a). For an incorrectly defined initial position of the vehicle in which the deviation angle ψ , 0
the vehicle route deviates from the assumed direction of travel. The distance that can be covered is
limited by the width of the corridor and the value of the angle ψ (Figure 9b).
Errors larger in the values of rolling radii cause a strong deviation C from the assumed trajectory.
The actual vehicle movement is carried out in a curve (Figure 9c). In all cases, oscillations coming from
the control system are visible on the course of the route.For the angle of deviation from the given path ψ = 0, the vehicle moves along the assumed direction
(Figure 9a). For an incorrectly defined initial position of the vehicle in which the deviation angle ψ ≠
(Figure 9a). For an incorrectly defined initial position of the vehicle in which the deviation angle ψ ≠
0 the vehicle route deviates from the assumed direction of travel. The distance that can be covered is
0 the vehicle route deviates from the assumed direction of travel. The distance that can be covered is
limited by the width of the corridor and the value of the angle ψ (Figure 9b).
limited by the width of the corridor and the value of the angle ψ (Figure 9b).
Errors larger in the values of rolling radii cause a strong deviation C from the assumed trajectory.
Errors larger in the values of rolling radii cause a strong deviation C from the assumed trajectory.
The
Appl. Sci.actual vehicle
2020, 10, 5490 movement is carried out in a curve (Figure 9c). In all cases, oscillations coming8 of 13
The actual vehicle movement is carried out in a curve (Figure 9c). In all cases, oscillations coming
from the control system are visible on the course of the route.
from the control system are visible on the course of the route.
(a) (b) (c)
(a) (b) (c)
Figure 9. Vehicle movementon on themeasuring
measuring section: (a) low values ofofrolling error ψ = 0; (b)(b)
low
Figure Vehicle
9. 9.
Figure Vehiclemovement
movement onthe the measuring section: (a)low
section: (a) lowvalues
valuesof rollingerror
rolling ψψ
error = (b)
= 0; 0; lowlow
values
values of of rolling
rolling error
error ψ ψ ≠ 0;
0; (c)
(c) high
high values
values of
of rolling
rolling errors,
errors, L—direct
L—directdistance
distance between
betweenthe start
the and
start and
values of rolling error ψ ≠ 0; (c) high values of rolling errors, L—direct distance between the start and
,
endpoints
endpoints of the route,
of the A—amplitude
A—amplitude of oscillation, C—deviation,
C—deviation, λ—wavelength,
λ—wavelength, ψ—angleψ ψ—angle
endpoints of route,
the route, A—amplitudeof oscillation,
of oscillation, C—deviation, λ—wavelength, deviation.
—angle
deviation.
deviation.
In order to determine the effect of oscillation and deviation from the target track on measurement
accuracy,In order to determine the effect of oscillation and deviation from the target track on
In aorder
seriestoof computer
determine calculations
the effect of were done. and
oscillation deviation from the target track on
measurement accuracy,
The vehicleaccuracy,
moves from a series of
the of computer
starting calculations
point were done.
to the endpoint. The direct distance L between these
measurement a series computer calculations were done.
The vehicle moves from the starting point to the endpoint. The direct distance L between these
points The
is 40vehicle
m. Vehicle
movestrajectory is not ideal
from the starting point and
to thedeviates
endpoint.from Thethe linedistance
direct connecting the start
L between and
these
points is 40 m. Vehicle trajectory is not ideal and deviates from the line connecting the start and
points is Therefore,
endpoints. 40 m. Vehiclethe actual Lr path
trajectory is not ideal and
traveled deviates
by the from
vehicle the linethan
is greater connecting L. The
the start
the distance andfirst
endpoints. Therefore, the actual Lr path traveled by the vehicle is greater than the distance L. The first
endpoints.
series Therefore,
of calculations the actualthe
concerned Lr path traveled
course by theshown
of motion vehiclein is Figure
greater 9a,
thanwhile
the distance L. The
the second first of
series
series of calculations concerned the course of motion shown in Figure 9a, while the second series of
series of calculations
calculations concerned concerned
the course the course shown
of motion of motion shown9c.
in Figure in Figure
In both9a, while
cases, thethe secondtraveled
distance series of by
calculations concerned the course of motion shown in Figure 9c. In both cases, the distance traveled
thecalculations
vehicle underconcerned the course
the assumed ofconditions,
traffic motion shown in Figure
marked
by the vehicle under the assumed traffic conditions, marked
9c. determined.
Lr , was In both cases,Using
the distance traveled
Dependence
Lr, was determined. Using Dependence
(11),
by the
the(11), vehicle
relative under
measurement the assumed
error ∆ traffic
was conditions,
determined. marked L r, was determined. Using Dependence
the relative measurement error Δ was determined.
(11), the relative measurement error Δ was determined.
Lr −−L
∆Δ == − (11)(11)
Δ= L (11)
The
The results
results ofthe
ofof the calculationsare
are shown in
in Figure 10.
The results thecalculations
calculations areshown
shown in Figure 10.
Figure 10.
0,006
0,004 0,006
0,004
0,003 0,004
0,003 0,004
0,002
0,002 0,002
0,001 0,002
0,001
0 0
00 0.02 0.04 0.06 0.08 0.1 00 0.5 1 1.5 2
0 0.02 0.04 A, m0.06 0.08 0.1 0 0.5 C,1m 1.5 2
A, m C, m
(a) (b)
(a) (b)
Figure 10. Relative error in the function: (a) of the amplitude A of oscillation at the λ = 5m and L =
Figure
Figure 10.10.Relative
Relativeerror
error in
in the
the function:
function:(a)(a)ofofthe
40m; (b) of the deviation C at the A = 0 and L= 40m.
amplitude
the A of
amplitude A oscillation at the
of oscillation at λthe λ =and
= 5m 5 mL and
=
40m; (b) of the deviation C at the A = 0 and L= 40m.
L = 40 m; (b) of the deviation C at the A = 0 and L = 40 m.
In both cases, it can be observed that the effect of the deviation C and oscillation A on the relative
error is quite large. To ensure the accuracy the deviation had to be minimized. Hence, it was decided to
use the modification of this method involving manual towing of the vehicle along a designated straight
line. Several dozen attempts at various vehicle loads have been made with this method. The load
values included future vehicle performance parameters.
The measurements were carried out in the corridors of a rigid surface. One of these is showing
in Figure 11a. During the tests, the length of the distance L covered by the vehicle was measured.
The value of the distance was measured by a laser rangefinder and ranged from 34 m to 42 m. The results
of these tests and their analysis are presented in the next chapter. Basic research has been preceded by
many trial tests.intended for transporting goods. The vehicle has three wheels. The front wheel is a drive and steering
wheel. This wheel is connected via gears with two DC electric motors. One of them built inside the
wheel is used to drive the vehicle. The second engine is located on the vehicle frame and, through a
worm gear, regulates the steering angle around the vertical axis of rotation. The rear wheels are
support wheels.
Appl. Sci. 2020, These wheels are connected with encoders (Figures 11b and 2a). Signals 9from
10, 5490 of 13
encoders are used to control the movement of vehicles.
(a) (b)
Figure
Figure 11. The
The tested
tested vehicle
vehicle (a)
(a) view;
view; (b)
(b) scheme.
scheme.
4. Research
All vehicleOutcomes
wheelsandhaveTheir
an Analysis
outer diameter of 200 mm and a width of 50 mm. There are
polyurethane belts around the wheel circumferences. Compared to rubber, polyurethane is
4.1. Research Object
characterized by greater stiffness and therefore less elasticity. Wheels with such belting can be used
The of
for loads research
8500 N. object shown in Figure 11 was built at the Rzeszow University of Technology. It is
intended for transporting
In order to ensure the goods. The vehicle
possibility has three wheels.
of autonomous The front
operation, wheel isisaequipped
the vehicle drive and with
steering
an
wheel. This
onboard wheel isAll
computer. connected via gears
data necessary to with twothe
control DCmovement
electric motors.
of the One
vehicleof them built inside
was supplied the
to the
wheel is used
onboard to drive
computer the the
using vehicle. The second
National engineNI
Instruments is USB-6343
located onmeasuring
the vehicle acquisition
frame and, card.
throughThea
worm gear,
scheme regulates
of the the steering
measuring systemangle around
is shown in the vertical
Figure 11b.axis of rotation.
Encoders wereThe usedreartowheels
measureare angular
support
wheels. and
velocity These wheelstraveled
distance are connected
data ofwith encoders
individual (Figures 11b and 2a). Signals from encoders are
wheels.
usedIn to the
control
casetheof movement
rear wheels, of these
vehicles.
were MHK40 encoders from Autonics Company. Encoders
All vehicle
integrated withwheels
electrichave an outer
motors werediameter
used to of 200 mm and
measure thea width
angular of 50 mm. There
position, speedare and
polyurethane
traveled
belts around
distance of thethe wheel
front circumferences.
wheel. The vehicle Compared to rubber,
uses additional laser polyurethane is characterized
sensors to determine by greater
the position of the
stiffnessrelative
vehicle and therefore
to theless elasticity.
adopted Wheelssurface.
reference with such belting canlasers
DT50-P2113 be used werefor loads
used of to 8500 N. the
measure
In order
distance to ensure the
perpendicular possibility
to the of autonomous
longitudinal axis of theoperation, the vehicle
vehicle, e.g., from the is corridor
equippedwall.withThe
an onboard
DT500-
computer.
A611 All data
laser was usednecessary
to measuretothe control the in
distance movement of the vehicle
the longitudinal was supplied to the onboard
direction.
computer using on
Depending thethe
National Instruments
batteries used, the NI USB-6343
vehicle measuring
in working orderacquisition
had different card. The scheme
weights. For lowof
the measuring
capacity system
batteries, is shown
the weight wasin 100
Figure
kg. 11b.
WithEncoders were used
larger batteries to measure
enabling severalangular
hours ofvelocity and
operation,
distance
the weight traveled
was 160 data
kg. of individual
The maximum wheels.
speed of movement was 1m/s. The expected maximum load is
2500 In
N. the case of rear wheels, these were MHK40 encoders from Autonics Company.
Encoders integrated with electric motors were used to measure the angular position, speed and
traveled distance of the front wheel. The vehicle uses additional laser sensors to determine the position
of the vehicle relative to the adopted reference surface. DT50-P2113 lasers were used to measure
the distance perpendicular to the longitudinal axis of the vehicle, e.g., from the corridor wall. The
DT500-A611 laser was used to measure the distance in the longitudinal direction.
Depending on the batteries used, the vehicle in working order had different weights. For low capacity
batteries, the weight was 100 kg. With larger batteries enabling several hours of operation, the weight was
160 kg. The maximum speed of movement was 1 m/s. The expected maximum load is 2500 N.
In the vehicle from Figure 11, odometry was the primary navigation system. While the vehicle
was moving, fixed rolling radius values were used in the calculation algorithm. These values were
introduced into the calculation system at the beginning of the study. Additional measuring devices
installed, such as laser rangefinders, allowed the current position of the vehicle to be determined in
relation to the surroundings. Using the appropriate calculation systems [26] based on the measurements
obtained, it is possible to correct errors [10,11].
4.2. Research Results and Their Analysis
As part of the basic research, 56 tests in four measuring series were carried out. In the individual
series were the following loads: 790 N, 1093 N, 1376 N and 1800 N. On the basis of the obtained test
results, the influence of the load on the rolling radii of the wheels was determined using the Statistica
package. The average values and standard deviations in a given series were obtained in the individual4.2. Research Results and Their Analysis
As part of the basic research, 56 tests in four measuring series were carried out. In the individual
series were the following loads: 790 N, 1093 N, 1376 N and 1800 N. On the basis of the obtained test
results,
Appl. Sci. the
2020,influence
10, 5490 of the load on the rolling radii of the wheels was determined using the Statistica
10 of 13
package. The average values and standard deviations in a given series were obtained in the
individual measurement series and were presented in Table 1. Coefficients of variation in individual
measurement series
measurement series and
were were presented
presented in Table
in Table 1. Coefficients
2. Apart from theofmeasurements
variation in individual measurement
for the left wheel and
series were presented in Table 2. Apart from the measurements for the left wheel
the 1800 N load, the standard deviation values characterizing the dispersion of measurementsand the 1800 N load,
are
the standard
similar. deviation values characterizing the dispersion of measurements are similar.
Table 1.
Table Average values
1. Average values and
and standard
standard deviations
deviations in
in the
the individual
individual measurement
measurement series.
series.
Left Wheel
Left Wheel RightWheel
Right Wheel
Force
ForceFNF,NN
,N
r,r,mm σ
σx× 10 −6, ,m
10-6 m r,r,m
m 10-6−6
σσx×10 ,m,m
790
790 0.10046
0.10046 25.632
25.632 0.10030
0.10030 23.935
23.935
1093
1093 0.10049
0.10049 25.103
25.103 0.10034
0.10034 24.587
24.587
1376
1376 0.10054
0.10054 20.679
20.679 0.10041
0.10041 18.644
18.644
1800 0.10058 37.081 0.10046 20.101
1800 0.10058 37.081 0.10046 20.101
Table 2.
Table Coefficients of
2. Coefficients of variation
variation in
in individual
individual measurement
measurement series.
series.
Force Force
FN (N)FN (N) 790790 1093
1093 1376 1376 1800 1800
Left wheel0.000255
Left wheel 0.000250 0.000206
0.000255 0.000250 0.000206 0.000369
0.000369
RightRight
wheelwheel0.000239 0.000245 0.000186
0.000239 0.000245 0.000186 0.000200 0.000200
Figure
Figure 12a,b
12a,b shows
shows thethe determined
determined regression
regression curves
curves together
togetherwith
withaa 95%
95% confidence
confidence interval.
interval.
For
For the
the left
left wheel,
wheel, the
thecorrelation
correlation coefficient
coefficient is 0.8617,
0.8617, and
and for
for the
the right
right wheel,
wheel, is
is0.9351
0.9351TheThecorrelation
correlation
coefficient
coefficient for the left
left wheel
wheel assumes
assumes smaller
smaller values
values due
due to
to the
the greater
greater dispersion
dispersion of
of the
themeasurement
measurement
results. The measure of this spread
results. The measure of this spread is is the standard deviation (Table 1).
(a) (b)
Figure 12. Linear
Figure 12. Linear regression
regressionwith
witha 95%
a 95% confidence
confidence interval
interval estimated
estimated forleft
for (a) (a)wheel;
left wheel; (b)wheel.
(b) right right
wheel.
Figure 13 shows a comparison of the obtained rolling radius courses as a function of load for
bothFigure
wheels.13Theseshows a comparison
courses of the slope.
have a similar obtained rolling
Their radius courses
nonparallelism as a function
is basically of load
determined for
by the
both wheels. These courses have a similar slope. Their nonparallelism is basically determined
measurement for the left wheel loaded with 1800 N. This measurement is burdened with the largest by the
measurement
error
Appl. and forx the
the 10,
Sci. 2020, largest left wheel
FORdispersion
PEER loaded with 1800 points
of the measuring
REVIEW N. Thisshown
measurement
in Figureis12a.
burdened with the largest
10 of 13
error and the largest dispersion of the measuring points shown in Figure 12a.
Figure 13.Comparison
Figure13. Comparisonofofthe
theobtained
obtainedrolling
rollingradius
radiuscourses.
courses.
Using
Usingthethedata
datafrom
fromTable
Table1,1,slip
slipcoefficients
coefficientsfor
forboth
bothwheels
wheelswere
weredetermined.
determined.Figure
Figure1414shows
shows
the relationship between the load of a given wheel and slip. The obtained relationships
the relationship between the load of a given wheel and slip. The obtained relationships for for the
the slip,
slip,s
s ==0.1%
0.1%totoss==0.3%
0.3%are
arelinear.
linear.Figure13.
Figure 13.Comparison
Comparisonofofthe
theobtained
obtainedrolling
rollingradius
radiuscourses.
courses.
Usingthe
Using thedata
datafrom
fromTable
Table1,1,slip
slipcoefficients
coefficientsfor
forboth
bothwheels
wheelswere
weredetermined.
determined.Figure
Figure1414shows
shows
therelationship
the relationshipbetween
betweenthe theload
loadofofaagiven
givenwheel
wheelandandslip.
slip.The
Theobtained
obtainedrelationships
relationshipsfor
forthe
theslip,
slip,s s
Appl. Sci. 2020, 10, 5490 11 of 13
==0.1%
0.1%totos s==0.3%
0.3%arearelinear.
linear.
(a)
(a) (b)
(b)
Figure14.
Figure
Figure 14.Load
14. Loaddependence
Load dependenceas
dependence asasaa aslip
slipfunction
slip functionfor
function for(a)
for (a)left
(a) leftwheel;
left wheel;(b)
wheel; (b)right
(b) rightwheel.
right wheel.
wheel.
Aspart
As partofofthe
thestudy
studyalso
alsodetermined
determinedthe
thedeflection
deflectioncharacteristics
characteristicsof
characteristics ofofa aflexible
flexiblebend
bendasasa afunction
function
of load. The graph presenting such characteristics is shown in
of load. The graph presenting such characteristics is shown in Figure Figure 15.
Figure 15.
15.
Figure
Figure15. Deflectioncharacteristics
15.Deflection characteristicsasasa afunction
functionofofload.
load.
The
Thegraph
The graphpresented
graph presented
presented ininin
Figure
Figure151515
Figure isisaisahysteresis loop,
ahysteresis
hysteresis which
loop,
loop, is visible
which
which after
isisvisible
visible thethe
after
after enlargement
theenlargementof the
enlargement ofof
selected fragment.
theselected
the selectedfragment.The dependence
fragment.The
Thedependence of the
dependenceofofthe deflection on
thedeflectionthe
deflectionon pressure
onthe force
thepressure does
pressureforce not
forcedoes differ
doesnot much
notdiffer from
differmuch
much
the
fromlinear
from characteristic.
thelinear
the linearcharacteristic.
characteristic.
5.5.Discussion
Discussion
5. Discussion
The
Theresults
resultsobtained
obtainedfromfromexperimental
experimentalstudies studiesarearecharacterized
characterizedby bysmall
smalldispersion
dispersionand andthus
thus
The results obtained from experimental studies are characterized by small dispersion and thus
high accuracy.
highaccuracy. A comparison
accuracy.AAcomparison of coefficients
comparisonofofcoefficients of variation
coefficientsofofvariation with
variationwith the
withthe values
thevalues obtained
valuesobtained from
obtainedfrom the
fromthe calculation
thecalculation
calculation
high
ofofrelative
relativeerrors
errors(Figure
(Figure10)
10)indicates
indicatesthat appropriate
appropriateprecision
thatappropriate precisionand anddiligence
diligencearearemaintained
maintainedduring
during
of relative errors (Figure 10) indicates that precision and diligence are maintained during
the tests.
thetests. During
tests.During
Duringthethe experimental
theexperimental
experimentaltests, tests, it was
tests,ititwas tried
wastried to make
triedtotomake
makethe the oscillation
theoscillation and
oscillationand deviation
anddeviation values
deviationvalues
values
the
smaller
smallerto A = 0.025 mmand CC==0.3 m. The calculated values of the coefficient ofofvariation Table
Table222for
smaller totoAA==0.025
0.025m andC
and 0.3m.
= 0.3 m.TheThecalculated
calculatedvaluesvaluesof ofthe
thecoefficient
coefficientof variationTable
variation for
for
both
bothwheels,
wheels,except for
exceptfor the
forthe load
theload with
loadwith the
withthe force
theforce
forceF FFNN== 1800
1800N, N, are
N,are comparable.
arecomparable.
comparable.For For aa force
For force load of FNN =
load of
both wheels, except N = 1800 a force load of FFN =
=1800
1800N, N,thethecoefficient
coefficientofofvariation
variationfor forthe
theleftleftwheel
wheel is84% 84%higher.
higher.The Thevalues
values ofthe thecoefficient
coefficient
1800 N, the coefficient of variation for the left wheel isis84% higher. The values ofofthe coefficient ofof
ofvariation
variationare areinfluenced
influenced byby thethe impossible
impossible to completely eliminate deviationsandand oscillations as
variation are influenced by the impossible totocompletely
completely eliminate
eliminate deviations
deviations and oscillations
oscillations asaswell
well
well as,
as,for for example,
forexample, unexpected
example,unexpected
unexpectedaccidental accidental
accidentalslips slips in
slipsininthe the contact
thecontact
contactofoftheof the wheel
thewheel with
wheelwithwiththethe surface
thesurface and
surfaceandand
as,
measurement
measurementerrorserrors in data
errorsinindata recording.
datarecording.
recording.The The calculations
Thecalculations of the
calculationsofofthe slip
theslip coefficients
slipcoefficients s as a function
coefficientss sasasaafunction of load
functionofofload
load
measurement
carried out show their linear relationship (Figure 14). Equation (10) used for calculations takes into
account the elastic deformation of the wheel and microslips in the contact between the wheel and the
surface. According to the information contained in a series of studies and in Figure 3, for low values of
slips s < 2%, the relationship between the force acting on the wheel and the slip is linear.
The diagram in Figure 15 clearly indicates the elastic properties of the belt material. This graph
has the shape of a hysteresis loop. The field contained inside the hysteresis loop indicates the ability to
damp vibrations.
The test results and their courses presented in Figures 14 and 15 are compatible with the examples
shown in many works [23–25]. This proves the correctness of the adopted methodology and diligence
in the course of research.
The results obtained and the relationships developed on their basis are very important for the
automated guided vehicle. Especially in the vehicle shown in Figure 11. In the vehicle shown inAppl. Sci. 2020, 10, 5490 12 of 13
Figure 11, the rear wheels do not carry any driving or braking forces and are used to measure the
distance traveled.
During the tests, the force loading the wheel did not exceed 21% of the maximum wheel load.
By increasing the wheel load by 1010 N from 790 N to 1800 N, the relative increase in the rolling radius
was 0.12%. Because with the slip below 2%, all changes are linear, a further increase in the load by
the force of 1010 N and 2020 N will result in an increase of the rolling radius by 0.24% and 0.36%.
The increments of the rolling radius are related to the value obtained under the load of 790 N. In the
last case, the total force on the wheel will be 3820 N. This is less than half the allowable force.
Assuming that only one wheel is subject to load changes, the other wheel is loaded with a constant
force F = 790 N, using the dependencies (1)–(4), it is possible to determine the deviation from the set route
and the radius of the curve along which the vehicle will move. The results are presented in Table 3.
Table 3. Radii of curvature and deviation from the track.
Additional Loading Force FN Deviation from the Track to the
Radius of the Arc R (m)
Over the Initial Value FN = 790 (N) Road L = 40 m (m)
1010 503 1.59
2020 251 3.20
3030 168 4.84
The analysis of the data in Table 3 shows that the differences in the load on the wheels cause
significant changes in the trajectory carried out by the vehicle. Correction of the above errors can be
made through additional navigation systems or by taking into account corrections made to the control
system, coming from the load measurement system of a given wheel.
6. Conclusions
The conducted experimental tests showed the usefulness of the proposed measurement method.
The main purpose of the research was to determine the relationship between the wheel’s rolling radius
and its load. The main tests were preceded by a description of elastic wheel properties and presentation
of possible measurement methods. After considering the preferred accuracy and availability of
measuring equipment, the most favorable measurement methodology was selected. It is important to
know the actual values of the wheel radii during vehicle operation. This is particularly true when
odometry is the basic navigation system. This problem is even more important when the mass of
loads carried by the vehicle changes within the permissible ranges for a given vehicle. The developed
measurement methodology may also be useful for the initial determination of the value of wheel rolling
radii. This is especially important for newly built vehicles. The wheel manufacturers do not provide
the value of the rolling radii and, in addition, these wheels differ in dimensions. These differences
probably fall within the tolerance of performance. However, from the point of view of odometry,
they are important.
Author Contributions: Conceptualization, M.S., M.D. and P.D.; Methodology, M.S., M.D. and P.D.; Software,
M.S., M.D. and P.D.; Formal Analysis, M.S., M.D. and P.D.; Investigation, M.S., M.D. and P.D.; Resources, M.S.,
M.D. and P.D.; Data Curation, M.S., M.D. and P.D.; Writing—Original Draft Preparation, M.S., M.D. and P.D.;
Writing—Review & Editing, M.S., M.D. and P.D.; Visualization, M.S., M.D. and P.D. All authors have read and
agreed to the published version of the manuscript.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
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