Method for Determining the Conditions for Ensuring Stable Rearward
Movement of a Semi-trailer Combination with Non-turning Semi-trailer
Wheels
MARYNA KOLISNYK1, OLEKSANDR PISKACHOV1, IRYNA PISKACHOVA2
1Department of Computer Systems, Networks and Cybersecurity,
National Aerospace University “KhAI”,
Kharkiv,
UKRAINE
2Department of Automation and Computer-Integrated Technologies,
State Biotechnological University,
Kharkiv,
UKRAINE
Abstract: - The efficiency of organising freight transport and using motor transport in a Smart City depends on
a set of its properties, which in the process of operation determine its suitability for use in given operating
conditions. Vehicles have different overall dimensions and designs, which determine their maneuverability and
directional stability, their ability to make a curve-line movement in the city, as well as to move safely in
reverse. Nowadays, tractor-trailer combinations with non-turning semi-trailer wheels are widely used for
freight transport. Such a scheme of vehicle construction does not ensure its directional stability when reversing,
which can lead to the folding of the tractor and semi-trailer and loss of mobility of the combination. In the
absence of additional special systems, the driver controls the rearward movement of the road train using the
steering wheel and rear-view mirrors, and the accuracy of delivery to the object depends on the level of the
driver's training. The research of driver's reaction time spent on situation assessment, decision making, and
realisation has been carried out. It has been revealed that with manual control, there are difficulties connected
with training of drivers to estimate the parameters of movement and control of the road train, in particular, with
the process of parking and its accurate delivery to the object. With the help of the developed mathematical
model of the rearward movement of the road train and the proposed new turn control law, the research of
maneuvering of the road train during its delivery to the object has been carried out and the results confirming
the possibilities of building an automated system allowing to reduce the influence of the human factor on the
parking process, as well as to reduce the time required to perform the maneuver have been obtained. The
conducted analysis of system stability with the help of Routh–Hurwitz criterion has determined the conditions
of providing stability of the system moving in reverse. The method of controlling the backward movement of
the road train and the design of the device realising it has been developed. The results of simulation modeling
allowed us to find the necessary values for the control law for the movement of road trains of different lengths,
as well as the preferred variants of the initial positions of the road train for the realisation of its precise delivery
to the object.
Key-Words: - Smart City, Maneuverability, Road Train, Instability, Semi-road train, Laplace operator, Routh–
Hurwitz criterion, Method.
Received: April 11, 2024. Revised: August 13, 2024. Accepted: September 7, 2024. Published: October 24, 2024.
1. Introduction
1.1 Motivation and Goal of the Paper
The creation of industrial systems of the Internet of
Things, Smart City systems and Smart Intersection
systems implies the solution of many problems.
These include network problems, logistics
problems, optimisation problems, traffic control
problems, and others.
When solving the problems of logistics and
debugging the transport system operation, it is
necessary to solve a number of problems, in
particular, the organisation of large-size trucks and
road trains movement within the city.
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In the conditions of the Smart City system,
there is a subsystem of smart transport. It includes
a number of subsystems that provide logistic tasks,
signal transmission and traffic safety. The vehicle,
which is the subject of research in this paper, has
different maneuverability and directional stability
when moving forward and backward, which
complicates logistic tasks in the Smart City system
and can lead to a violation of traffic safety. In
particular, there is a problem in the delivery of the
road train for loading: it is necessary to accurately
deliver the side of the road train to the platform for
loading or loading the cargo into the mine. For this
purpose, a method must be devised to ensure safe
maneuvering on the smallest possible footprint.
Road trains with semi-trailers are mostly used
for the transport of long goods. The basic
regularities of motion of such road trains with
steerable and unsteerable wheels of semi-trailers on
different types of turns are now practically
important, [1], [2], [3], [4], [5], [6]. However, the
majority of scientific works are related to the study
of their maneuverability and stability when moving
forward. Up to now, the question of estimating the
maneuvering properties of a road train moving in
reverse remains insufficiently developed, [7], [8],
[9], [10], [11], [12], [13], [14]. In order to deploy a
road train on narrow roads, it has to maneuver
using both forward and reverse movements to
accurately deliver to the object. It is difficult to
steer the combination when reversing, as even a
slight turn of the steering wheel causes the
combination to fold. When reversing, the position
of the semi-trailer must be monitored and if the
direction of travel of the semi-trailer changes
slightly, it must be immediately leveled in the axis
of travel. To steer the semi-trailer, turn the steering
wheel in the direction of the change of direction. It
can also be leveled by moving it forward.
However, this maneuvering of the vehicle does not
allow the semi-trailer to turn toward the object
accurately and quickly. The task is especially
difficult when the base length of the semi-trailer
exceeds the base length of the tractor.
The main reason for the development of
unmanned means of cargo transport was the need
to exclude humans from driving the vehicle, due to
the large influence of the human factor on the
driving process. Currently, the majority of road
trains are manually driven. Manual driving poses
difficulties in training drivers to drive the truck
and, in particular, in the process of parking. To
make the driver's job easier, special markings on
the maneuvering area can be used to facilitate a
precise approach to the unloading point. There are
tasks of positioning the road train so that the
wheels on the left or right side of the vehicle are on
the same line. This line is drawn on the road.
This line is called the collision line. The
difficulty of accurately reversing of a road train to
an object indicates the need to automate the process
of driving a road train, which would reduce the
influence of human factor on the process of
parking, as well as reduce the time required to
perform the maneuver, [5]. At the moment there
are no systems of automatic control of a road train,
which could provide the process of perpendicular
parking of a semi-trailer in a given sector.
Therefore, it is urgent to create such a system that
provides autonomous performance of this
operation. Such a system is intended for use on
tractors with semi-trailers, as well as on other types
of lorry transport having a semi-trailer in its
composition. The movement of the semi-trailer
behind the tractor has some peculiarities, which
must be taken into account when controlling the
movement. The main feature is that the wheel axle
of the semi-trailer moves at a smaller radius than
that of the tractor. This difference depends on three
main parameters: the length of the tractor; the
length of the semi-trailer; and the angle between
the axles of the tractor and the semi-trailer. In
automated and non-automated traffic control
systems, the driver is involved. Based on the
information received by the senses, the driver acts
on the tractor controls to perform a certain
maneuver. The driver receives most of the
information visually. The driver determines the
position of the road train relative to the collision
line. The time taken by the driver to assess the
situation, the position of the road train relative to
the object of approach, and to make and implement
a decision is called the driver's reaction time, we
suggest calculating the time value by the formula,
[1]: ,
where: – latent time, i.e. the time elapsing from
the moment when the driver fixes a certain
situation to the beginning of the implementation of
the decision; – motor reaction time (turning the
steering wheel); - reaction time, which depends
on the driver's experience, fatigue, psychological
state, situation, speed and varies widely.
The tasks performed by the driver at this time
can be divided into logical and reflexive tasks.
Logical solutions arise in the process of thinking.
In reflexive actions, the role of conditioned reflex -
automaticity of behavior developed on the basis of
experience - is great.
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The driver's reaction in non-automated traffic
control systems is complex. He has to evaluate the
situation, choose a certain solution from several
possible solutions, and perform a motor action.
Therefore, the driver's reaction time can vary
widely. Numerous statistical studies have shown
that with probability p = 0.9 this time is between
0.1 and 1.5 seconds. Significant reaction time leads
to untimely or erroneous actions, and thus to an
incorrect approach to the object. It is particularly
difficult to control the movement of a road train if
the semi-trailer is long-wheelbase and its width
significantly exceeds the width of the tractor. Some
vehicle control systems depend on supporting
infrastructure (e.g. the use of traffic management
systems, sensors embedded in the road, Smart
Traffic Lights). Through the use of various
measurement tools, video cameras, satellite
navigation systems, and radar, advanced
technologies make it possible to simulate human
presence at the level of decision-making about
vehicle orientation and speed. There are two main
directions for creating such systems: complex
automation of a car and automation of individual
modes of vehicle movement (parking, traffic jams,
driving on highways, passing through ‘smart’
traffic lights).
Modern cars contain electronic driver
assistance systems with varying degrees of
automation, such as stability control, collision
avoidance, cruise control, parking distance control,
and others. Electronic systems provide part of the
vehicle control functions, such as automatic control
of speed, acceleration, turning, and parking
maneuvering. When solving logistic tasks, it is
necessary to take into account the movement
parameters of large vehicles. They can make
maneuvers moving backward, for example, in a
warehouse for loading, or they can move at an
intersection and create an obstacle on the road for
other types of transport. It is especially important to
take into account the maneuvers of such transport
vehicles in the case of the automatic type of control
- with the help of auto-pilot because the wrong
trajectory of both the vehicle and its semi-trailer
can lead to catastrophic consequences, [1].
The results obtained in this work can be used in
the development of road trains, including
unmanned ones, which have non-turning wheels of
semi-trailers.
1.2 Work-related Analysis
Vehicle maneuverability is one of its main
operational properties, which are described in a
number of works.
The paper [1] presents the developed
mathematical models of road train movement in a
given direction in reverse and the results of their
study.
In [2], decision making about drivers' behavior
when receiving speed recommendations related to
energy consumption and safety-free speed is
modeled through vehicle-infrastructure
communication.
In [3], the results of modeling the rearward
movement of a trailed road train are presented and
technical solutions for automatic folding prevention
are proposed, which allows for automated parallel
parking in combination with a path-planning
algorithm.
In [4], the reverse motion of an articulated
vehicle, namely a tractor-trailer with a single trailer
on the axle, is analyzed and a fully autonomous
driving system is developed that allows reverse
parking in the presence of static obstacles.
In [5], it is shown how a vehicle with passive
trailers can be easily driven using the proposed
driver assistance system and traffic control scheme.
Since the keypad is an optional device for the
driver assistance system, the proposed scheme can
be implemented using conventional trucks without
many hardware modifications. A manual push-pull
control strategy is established. A kinematic scheme
of a vehicle with trailers for push-pull control is
proposed.
In [6] it is shown that the complexity of
controlling a road train is due to pronounced
nonlinearities, as well as instability of the control
object when reversing, often leading to the
phenomenon known as a folding knife.
In [7] based on kinematic and dynamic models
three control approaches for dynamic stabilization
in a road train configuration are proposed, as well
as a methodology for tuning the control gains using
three possible actuators.
Paper [8] presents the results of developing a
path tracking and cascade controller for controlling
an unmanned tractor-trailer vehicle during
reversing.
In [9], a deep learning method based on an
auto-vehicle model is proposed to determine the
parameters of front and rear stiffness of an auto-
vehicle when cornering.
In [10], taking into account drifting methods
applicable to front-wheel drive (FWD) drivetrain
configurations, the cornering balance was
calculated using a car model with front-wheel drive
and rear wheels ‘locked’ at zero angular velocity
with a handbrake applied using a controller.
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In [11], a V2I architecture is proposed that can
operate in both autonomous and manually
controlled vehicles and coordinate their movement
based on V2I communication.
In [12], the reverse motion of a vehicle and
trailer combination is investigated. A single-vehicle
path model is used with a quasi-static tyre model to
develop a simple feedback linear controller that can
provide a stable reversing motion along a straight
path.
In [13], technical solutions are proposed to
control the reversing motion of a trailed road train.
For this purpose, two feedback controllers are
developed that support the driver with auto-
automatic steering inputs in different situations.
In [14], the problem of path tracking control of
a mobile robot with two trailers is solved, for
which a path tracking control algorithm (using LOS
(Line of sight) method and PID (Proportion,
differential and integral) control algorithm) is
proposed.
The analysis of the literature has shown that no
one deals with the solution of such a problem by
the method proposed in this paper, namely,
bringing an unstable system to a stable controlled
motion along the guideway.
The goal of our research is to improve the
efficiency and safety of traffic control in the system
Smart City by reducing the maneuvering time and
providing accurate accident-free rearward delivery
of the road train to the object or to the object.
2 Theoretical Justification of Road
Train Maneuverability
Let us consider the initial position of the semi-
trailer train with the wheels of the semi-trailer not
rotated relative to the frame before the beginning of
its movement back to the object. Let us replace the
real wheels of the road train links with conditional
average reduced wheels located in the longitudinal
planes of the links, which allows us to consider
kinematic links in the form of a ‘bicycle’ scheme,
[1], [6], [8].
Let the steering wheels of the tractor are rotated
relative to its frame by angle φ0, the frame of the
tractor is rotated by angle β0, and the frame of the
semi-trailer is rotated relative to the longitudinal
line of the collision by angle α0. The wheels of the
semi-trailer are displaced relative to the
longitudinal line of the collision by the value Z0
(Figure 1), [1]. Let us make the following
assumptions: the motion of the road train is flat,
parallel to a fixed support surface, with constant
speed; there is no lateral drift of elastic tyres; the
influence of the suspension on the trajectory is not
taken into account. Let us use the known system of
equations describing the motion of the road train.
The velocity of the road train links is given by the
vector of the tractor's driving wheels velocity
.
Fig. 1: Kinematic diagram characterizing the
position of the vehicle before reversing (X axis
coincides with the collision line1), [1]
Determine the values of projections of velocity
vectors V1...V6 (Figure 1), [1]:
Let's write down the equations of linkages of
the road train:
where: l – tractor base length;
L – semi-trailer base length;
- angular velocity of folding of the tractor
frame with the semi-trailer frame;
- angular velocity of the semi-trailer axle to the
longitudinal collision line;
– rate of change of displacement of semi-trailer
wheels relative to the longitudinal collision line;
– wheel speed of the semi-trailer to the object.
Taking into account the velocity values, the
equations of motion of the links can be rewritten in
the following form [1], [2]:
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(1)
In order to provide control over the reversing
movement of the road train to the object in reverse,
the system of equations (1) is supplemented with a
feedback equation, assuming that the control over
φ(t) should depend on the values β(t), α(t), Z(t)
measured both in the initial position of the road
train relative to the object before the start of
movement and during its movement. Taking into
account the assumptions made, let us choose a
linear law of controlling the rotation of the steering
wheels of the tractor, different from the one used in
[1].
(2)
where - gain by β(t); - gain by α(t); - gain
by Z(t).
Then the complete system of equations of
motion of the road train taking into account the
control will take the following form:
(3)
Thus, we have obtained a mathematical model
of the movement of a road train with non-turning
wheels relative to the frame of a semi-trailer in
reverse to the object. Let us determine the
necessary and sufficient conditions for finding the
coefficients K1...K3. For this purpose we linearise
the system of equations (3), taking into account that
at small values of angles cosφ=1, sinφ=1, cosβ=1,
sinβ=β, cosα=1, sinα=α. Let us study the system of
equations describing the motion of the road train
along the Y axis only, as well as its angular
displacements.
Then the system of equations (3) can be
represented in the following form:
(4)
To simplify the calculation of automatic
control systems, the operator method of description
is used. The equations of dynamics are written in
the form of images of functions obtained by means
of the direct Laplace transform (operator form of
equation writing).
Such transformations allow us to pass from
differential equations to algebraic equations.
Let us represent the system (4) by means of
Laplace transforms with zero conditions, [1]:
The image σ(s) of the original σ(t) is a function
of the complex variable s given by the integral:
Using the substitution rule, let us exclude the
variables , β, α from the system (5). To do this,
we first substitute expression (2) into the first
equation of system (5), and then substitute
expressions for β, α into the second and third
equations of system (5), respectively. Let us
exclude from (5) the variables , β, α.
Such transformations allow us to pass from
differential equations to algebraic equations.
In this case, the differentiation operation
(original)
is replaced by an image ,
integration operation
– depiction
etc. Let us represent the system (4) by means of
Laplace transformations with zero conditions:
(5)
Using the rule of substitution, let us exclude
the variables , β, α from the system (5). To do
this, we first substitute the expression (2) into the
first equation of the system (5), and then substitute
the expressions for β, α into the second and third
equations of the system (5), respectively.
Excluding from (5) the variables , β, α.
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(6)
Since a control system with a linear control law
of the front wheels of the tractor is chosen to
control the movement of the tractor, we use the
Routh–Hurwitz criterion to assess the conditions of
its stability, which refers to algebraic stability
criteria that impose restrictions on the coefficients
of the characteristic equation. The characteristic
equation of the considered closed system is the
denominator in equation (6) and has the form:
It can be visualized in a different way:
(8)
where:
Since the characteristic equation (7) is
determined by (8), it is necessary and
sufficient for the stability of the linear control
system that the three Routh–Hurwitz determinants
are positive. For the considered characteristic
equation of the third order, the determinant will
have the following form:
All diagonal minors of the Routh–Hurwitz
determinant must be greater than zero, i.e.:
hence,
Hence, in this case not only the positivity of all
coefficients of the characteristic equation is
required, but also the observance of the condition:
Whence it follows that
From the stability conditions should be
greater than zero, i.e.:
Hence,
The determinant is found as follows:
Hence
Thus, in order to ensure a stable movement of a
semi-trailer combination with wheels not rotated
relative to the frame of the semi-trailer in reverse to
the object at the selected law of controlling the
rotation of the front steerable wheels of the
tractor (2), the following conditions must be met:
(9)
Fulfillment of the obtained conditions of
stability of operation on the movement will provide
the road train with stable movement along the
longitudinal collision line from different initial
positions of its location relative to the collision
line. In this case, the time of arrival to the stable
motion will be the less, the more successfully the
proportionality coefficients K1...K3 will be selected
and depends, as it can be seen from (9), on the
geometrical dimensions of the tractor and semi-
trailer. For different tractors and semi-trailers, the
coefficients will be different.
Let us again turn to the system of equations (3).
There are five unknowns in the system (3) of five
equations. However, it is not possible to solve the
system of equations by direct integration. To find
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solutions, we can use a method of numerical
integration of the system of differential equations
with high accuracy, for example, the Runge-Kutta
method. The theoretical studies carried out earlier
can serve for the development of algorithms for
solving the set problems. In addition, they have
shown that the road train can move to the object in
different ways.
a)
b)
c)
d)
e)
f)
Fig. 2: Calculated cases of road train positioning
relative to the collision line: β0=0, α0=0, Z0=0;
b) β0=0, α0=0, 0<Z0≤2 m; c) 0<β0≤π/3, α0=0, Z0=0;
d) - π/3≤β0≤-π/6, π/6≤α0≤π/3, -0,8 m≤Z0≤0,8 m; e)
β0=0, π/6≤α0≤π/3, 0≤Z0≤1,5 m; f) 0≤β0≤π/3, -
π/3≤α0≤-π/6, 0<Z0≤1,5 m
At one ratio of coefficients, the road train is able
to come to a stable movement along the collision
line for a short time, at another - the road train will
make oscillatory movements relative to the
collision line for a long time before it comes to a
stable movement along the collision line. In this
paper, we consider the scheme of a specific road
train, for which the proportionality coefficients are
chosen based on the geometric dimensions of the
tractor with the base from the front wheels to the
support point of the semi-trailer (l = 5.2 m) on the
tractor and semi-trailer with the base taking into
account the conditions (9). The study of the
character of maneuvering for the case of its
approach to the object from one possible initial
position relative to the collision line does not allow
us to fully evaluate the work of the control system
with the adopted control law (2). Therefore, it is
necessary to analyse the rearward movement of the
road train from other possible initial positions
relative to the collision line. By investigating the
system (3) under different initial conditions, taking
into account the calculated limitation on the
steering angle of the driven wheels of the road train
and the assumptions made, the most unfavorable
variants of the position of the road train have been
determined, at which the control system cannot
bring the road train to a stable movement along the
collision line in a relatively short time. Six variants
of possible initial positions of the road train relative
to the collision line are shown in Figure 2, [1].
The nature of change of parameters
for all considered variants of the
initial placement of the road train is shown in
Figure 3, Figure 4 and Figure 5.
In the first variant (Figure 2(a)) the wheels are
on the longitudinal line of the collision, i.e.:
. At this initial position, the road
train will move steadily along the collision line.
In the second variant (Figure 2(b)) the road
train is stretched in a line, having
and be at a distance of up to 2 m
from it. Calculations have shown that at initial
values of displacements the road
train will always move steadily along the line of
approach to the object. As can be seen from the
figures, the road train comes to a stable motion
along the collision line in 40 seconds at a given
driving speed V = 0.3 m/s. In this case, the tractor
performs such a maneuver, at which the
displacement of the semi-trailer wheels relative to
the collision line gradually decreases.
The third variant (Figure 2(c)) assumes the
placement of the semi-trailer of the road train
relative to the collision line similar to the second
variant, i.e. and , except that
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the tractor is rotated relative to the semi-trailer by
the folding angle, which varies within the most
possible limits during operation of the
combination: . Under these
conditions, it takes approximately 40 seconds for
the road train to reach a stable movement relative
to the collision line, maneuvering with a slight
increase in the area required. As the initial offset
increases, the steady-state time increases to 45 to
50 seconds.
In the fourth variant (Figure 2(d)), the position
of the road train is set within the following limits:
, ,
. If , the
road train will come to a stable movement in 40 -
50 seconds, while the time of coming to a stable
movement increases to 70 80 seconds. Increasing
the maneuvering time to a stable movement along
the guide leads to an increase in the distance from
the initial position to the object required for an
accurate approach. Therefore, it is desirable to set
the road train in the initial position at angles not
exceeding 45 degrees.
The fifth variant (Figure 2(e)) is a special case
of the variant and is characterized by the same
regularities. Sixth option (Figure 2(f)) at
; ;
is the most unfavorable for the control
system. For this position of the combination
vehicle relative to the collision line, the
maneuvering time will be long and the combination
vehicle may not have time to come to a stable
movement along the guide rail, having approached
the object inaccurately.
Fig. 3: Dependence of the folding angle β = f(t)
when moving to the approach object from the
initial positions (Figure 2)
Fig. 4: Dependences of angles of rotation of the
semi-trailer frame relative to the longitudinal
collision line α=f(t) when moving to the object of
approach from the initial positions (Figure 2)
Fig. 5: Dependence of displacements of semi-trailer
wheels relative to the collision line z = f(t) when
moving to the object of approach from the initial
positions (Figure 2)
3 Discussion of Research Results of
Modeling Processes of Road Train
Movements
Investigations of the processes of modelling the
motions of the road train, analysis of dependencies
(Figure 3, Figure 4 and Figure 5) were carried out
β=f(t), α=f(t), Z=f(t), obtained under different
initial conditions, taking into account the
restrictions on the rotation of the steerable wheels
of the tractor, which reduce the functional
capabilities of the control system to bring the road
train to a stable reversible movement along the line
of approach to the object, it is necessary to
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determine the maneuvering properties of the road
train in the design, taking into account the design
factors of its systems.
For the example under consideration, it is
reasonable to exclude setting the semi-trailer to
angles within the range when setting the road train
in the initial position π/6≤α≤-π/6. The first option is
the most favorable for the operation of the control
system. The use of the road train control system
allows you to significantly reduce the time and area
of maneuvering with a high accuracy of approach
to the object in reverse. Auxiliary automated
control systems or an autopilot can be used to
control the reversing motion of the road train.
Based on the conducted research and taking into
account [1] a new design of the electrical scheme
for the construction of the motion control system
has been developed, shown in Figure 6.
Let us consider an example of a control system
for a road train with non-rotating wheels of a semi-
trailer traveling in reverse (Figure 6). The device
contains the control object 1, sensor 2 of the angle
of folding of the tractor with semi-trailer β, sensor
3 of the angle of inclination of the semi-trailer axis
to the longitudinal collision line α, sensor 4 of the
displacement of the left wheels of the semi-trailer
relative to the longitudinal collision line Z,
amplifiers 5 -7, adders 8 and 9, zero-indicator 10,
steering drive 11, the tracking system 12 of wheel
rotation and sensor 13 of the angle of rotation of
the steered front wheels of the tractor. The steering
actuator 11, the tracking system 12, and the sensor
13 are located on the tractor. Sensors 2, 3, 4 fulfil
the function of feedback, i.e. having measured the
values of β, α, Z, they transmit them in the form of
signals to the adder 8. The inputs of sensors 2...4
are connected to the control object, and the input of
sensor 13 is connected to the output of tracking
system 12 for wheel rotation. The outputs of
sensors 2 - 4 are connected respectively to the
inputs of amplifiers 5...7, and the output 13 is
connected to the second input of the adder 9.
Amplifiers 5 - 7 with gain coefficients K1...K3,
their outputs are connected respectively to the first,
second and third inputs of the adder 8. The output
of the adder 8 is connected to the first input of the
adder 9. The output of the adder 9 is connected to
the input of the zero indicator 10. The output of the
zero indicator 10 is coupled to the steering actuator
11, and the steering actuator 11 is coupled to an
input of the wheel tracking system 12. The output
of the wheel tracking system 12 is coupled to the
control object 1.
When reversing the road train, the device
works as follows. Before starting the approach of
the road train to the object in reverse, it is set near
the longitudinal line of the collision, with the axis
of the semi-trailer inclined to it at an angle α0, the
left wheels of the semi-trailer are displaced relative
to it by the value Z0, and the tractor and semi-trailer
have a folding angle β0. The wheels of the tractor
are rotated relative to the frame by the value φ0. At
the moment when the transport vehicle starts
reversing to the object, the device is switched on.
In this case, sensors 2 - 4 measure the values of α, β
and Z and output them in the form of electrical
signals to amplifiers 5 - 7 respectively. Amplifiers
5 - 7 amplify the signals α, β and Z to the values
K1β, K2α and K3Z and feed them to the first, second
and third adders 8 respectively. In the adder 8, the
values K2α and K3Z are algebraically subtracted
from the value K1β. The total signal is given
by the adder 9. The sensor 13 measures the actual
angle of rotation of the steerable front wheels of the
tractor relative to the frame and outputs an
electrical signal to the second input of the adder 9.
From the output of the adder 9 the resulting signal
is fed to the input of the zero-indicator 10, causing
a deviation of its arrow from zero value by an
amount equal to the difference between the
specified angle of rotation of the steered front
wheels of the tractor and the actual real .
By means of the steering drive the driver of the
road train influences the tractor wheel steering
system so that the arrow of the zero indicators is
constantly at zero. With the help of the tracking
system of wheel turning the tractor makes a set
maneuver along the longitudinal line of the
collision, thus, acting on the control object, it will
carry out the approach of the road train to the
object so that the left wheels of the tractor and
semi-trailer were on the longitudinal line of the
collision. This will reduce the time for
maneuvering and convergence.
The considered system of controlling the
movement of the road train in reverse along the
longitudinal line of the collision with the proposed
parameter gauges allows the mobile system, in
compliance with the stability conditions (9), to
reach the collision line for the minimum
maneuvering time on the minimum area. In
addition, the maneuvering time and, accordingly,
the maneuvering area can be smaller if the design
restrictions on the rotation of the front steerable
wheels of the tractor are reduced.
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Fig. 6: Schematic diagram of the system of controlling the rearward movement of a road train
The use of such train control systems will
significantly reduce the maneuvering time and
maneuvering area with high accuracy of
approaching the object, especially when the length
of the semi-trailer is long.
Another option for equipping (equipping) a
tractor with a semi-trailer with non-turning semi-
trailer wheels can be the construction of a traffic
control system with autopilot, which allows
maneuvering in a limited area without the driver's
participation.
4 Features of Unmanned Cars
Maneuverability
The process of creation and development of
ground-based drones shows that their creation and
development follows the following main directions
[9], [10], [11], [12], [13], [14]: introduction and
expansion of functionality of driver assistance
systems; creation of methods and systems of drone
traffic control, which are both at the stage of
development and testing of prototypes and in
operation.
Many global car manufacturers, especially in
the USA, Germany, Japan, Italy, China, Great
Britain, France, and Korea (General Motors, Ford,
Mercedes Benz, Volkswagen, Audi, BMW, Volvo,
Caddilac) are working on the development of
unmanned cars.
Realisation of the advantages of unmanned
vehicles is impossible without the efficient
operation of motion control systems, which may be
limited by the speed of measuring, computing, and
actuating devices. An integrated approach to the
creation of an unmanned car is currently realised
only by some companies, for example, Google. It
should be noted that single-car parking systems
have existed for a long time. Toyota, Volkswagen,
Valeo, and Ford have achieved the greatest success
in creating such systems. Currently, various
automatic parking systems are being developed and
implemented, which provide parallel or
perpendicular parking of a car in automatic mode.
Toyota, BMW, Ford, Mercedes-Benz, Nissan,
Opel, and Volkswagen have parking autopilot.
The process of further improvement of the
adaptive cruise control system is ongoing, which in
the future will allow to realize an automatic mode
of car driving in traffic jams. Audi, Ford are
conducting research in this direction.
Developments of BMW, and Cadillac on
automation of movement of cars and road trains on
motorways are based on existing active safety
systems. A modern car contains electronic driver
assistance systems with various degrees of
automation of the vehicle control process, such as
directional stability, warning, cruise control,
parking distance control, and others. Electronic
systems provide part of the vehicle control
functions, such as automatic speed control, turning,
and parking maneuvering functions.
To ensure safe maneuvering, e.g. when
parking, in addition to the autopilot, distance
sensors (e.g. ultrasonic or laser sensors, which have
already been developed and are commercially
available) should be installed at points on the road
train that define its overall lane when driving in a
curve, including in reverse. Similar sensors should
also be installed on the front part of the tractor and
the rear part of the semi-trailer.
The analysis of control systems for unmanned
vehicles has revealed a large number of problems
arising before the developers in the process of their
creation and in determining the requirements to the
motion control system, which is caused by the
following objective factors: design features of
vehicles; sufficiently high error of measuring the
motion parameters; impossibility of most systems
to take into account the external conditions,
continuously changing in the process of motion;
functional limitations of control systems.
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Perpendicular parking of a semi-trailer is one of the
most complicated maneuvers of a road train and is
divided into several stages.
It is obvious that the quality of control system
operation directly determines traffic safety, and the
developers at the stage of engineering design are
obliged to determine the operational capabilities of
vehicles, at which the probability of an emergency
situation is reduced to a minimum. Therefore, the
task of preliminary prediction and evaluation of
curvilinear motion characteristics of vehicles
(including unmanned vehicles) at the design stages
is relevant and important.
5 Conclusions
1. The conducted research allowed us to achieve
the set research goal and show the ways to
achieve it. The obtained modeling results can
be used in the development of Smart City
systems, Smart Traffic Lights, in solving
logistic problems of industrial systems of the
Internet of Things.
2. The mathematical model of reversing
movement of a truck with a semi-trailer with
non-rotating wheels of the semi-trailer has been
developed, which allowed to provide and
describe the accurate accident-free delivery of
cargo to the object by the road train.
3. On the basis of the analysis of the character of
movement the main parameters influencing the
stable movement of the road train in reverse
have been determined, and the law of control of
the tractor wheels providing its stable
movement by means of introduction of
feedbacks has been justified. As well as the
construction and application of the design of
the electrical scheme of the vehicle control
system.
4. Analysing the stability of the system by means
of the Routh–Hurwitz criterion has allowed us
to obtain the necessary and sufficient
conditions for ensuring the stability of the
moving system.
5. Modelling of the road train movement with the
help of the simulation model allowed to
determine the nature and parameters of its
movement relative to the collision line from
different initial positions relative to the object,
as well as to choose the preferred options of the
initial location of the vehicle before starting the
movement.
6. On the basis of the research results the new
method of controlling the directional
movement of the road train and the device
realising it are developed.
7. The obtained graphical dependences based on
the results of modelling from different initial
positions of the road train relative to the
collision line allowed to determine the
necessary ratios of the parameters of the law of
controlling the rotation of the tractor wheels
taking into account the design restrictions on
the rotation of the wheels.
8. The results of simulation have shown that
when designing the schemes of road trains it is
necessary to take into account the overall
dimensions of the tractor and semi-trailer to
ensure the stability of movement, the required
maneuverability on a limited area of
maneuvering in a short time.
9. According to the results of the work done, a
method of determining the conditions for
ensuring the stable movement of a semi-trailer
train in reverse with non-turning wheels of the
semi-trailer has been developed.
The practical significance of the developed
method - the results obtained in this work can be
used in the stage of development of road trains. It
could be used also for unmanned vehicles, with
non-turning wheels of semi-trailers.
The future research is planned to continue
work on improvement of mathematical apparatus
for specification of procedure of definition of
values of coefficients of the control law by means
of methods of optimisation, and also to work out
possibility of full automation of process of control
of backward movement of a road train of the
considered type.
The practical significance of the developed
method - the results obtained in this work can be
used in the stage of development of road trains. It
could be used also for unmanned vehicles, with
non-turning wheels of semi-trailers.
Future research is planned to continue work on
the improvement of mathematical apparatus for
specification of procedure of the definition of
values of coefficients of the control law by means
of methods of optimisation, and also to work out
the possibility of full automation of the process of
control of backward movement of a road train of
the considered type.
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Volume 19, 2024
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The authors equally contributed in the present
research, at all stages from the formulation of the
problem to the final findings and solution.
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Scientific Article or Scientific Article Itself
Self-financing.
Conflict of Interest
No conflicts of interest in the research.
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