A Model-Based Control of Self-Driving Car Trajectory for Lanes
Change Maneuver in a Smart City
IGOR ASTROV
Department of Software Science,
Tallinn University of Technology,
Akadeemia tee 15a, 12618, Tallinn,
ESTONIA
Abstract: - High-quality computer control of autonomous vehicles in various environments is a priority for
cyber-physical systems (CPS), Industry 4.0, and the global economy as a whole. The paper discusses the
linearized control model of a Self-Driving Car (SDC) with a weight of 1160 kg. For safe maneuvering with
obstacle avoidance, we employ an optimal control by Linear Quadratic Regulator (LQR) using a
Simulink/MATLAB environment that is capable to demonstrate the satisfiability of LQR control for this
maneuver using a 3D simulation environment under changing urban conditions in a smart city. This controller
is easy for engineering implementation.
Key-Words: - bicycle model, computer simulation, LQR control, mathematical model, self-driving cars, smart
city.
Received: December 28, 2022. Revised: September 2, 2023. Accepted: October 7, 2023. Published: October 27, 2023.
1 Introduction
Unmanned autonomous vehicles (AVs) are
becoming increasingly important in CPS, Industry
4.0, and the global economy as a whole, [1]. AV
needs to be developed and applied using
mathematical models that should replace the
ordinary mental models that have developed in the
head of a human driver of conventional vehicles,
[2].
In Tallinn University of Technology (TalTech),
during a noticeable time one of the priority study
fields has been the research of SDCs, [3], [4], [5].
Modeling and simulation for very different types of
aerial AVs and aquatic (both underwater and surface)
AVs have been a notable research theme for several
years now, [6], [7], [8], [9]. These activities have
recently been included in the development of the
Smart City environment, which expands the
cooperation concept for the twin city Tallinn-
Helsinki, [10], [11].
Urban AVs in Smart Cities operate in a higher
range of speeds and accelerations than commonly
used mobile robots. This fact necessitates the use of
dynamic systems techniques to make the control
safer and smoother.
To achieve autonomous motion in driving a
series of components are necessary. First, an AV
sensor network collects all information about the
AV and environment and makes the necessary
measurements. Secondly, the trajectory planning
component generates the desired travel route using
the actual and desired position of the AV. Finally,
the automatic control system generates control
actions on the drives.
Reliable operation of the automatic control
system is one of the most important tasks for AV
because it guarantees its autonomous motion.
Therefore, it is necessary to use rigorous
mathematical models of AV.
The problems of traffic planning and control are
two different, but closely related tasks for SDCs.
The first task is to calculate the possible trajectory
of the vehicle avoiding surrounding obstacles such
as pedestrians, other vehicles, or nonmoving objects.
The second task is the impact on the executive
mechanisms (accelerator pedal, brake pedal, and
steering wheel) to follow the trajectory obtained by
the motion planner, while the system maintains
stability and, if possible, a smooth ride.
It is assumed that the trajectory set by the traffic
planner is safe within certain limits, and then the
goal of the controller is to follow the given safe
trajectory as best as possible, without considering
obstacles. If the controller fails to follow safe
trajectories, it will the entire system in jeopardy.
This work aims to create a mathematical model
of SDC ISEAUTO and implement a control system
for the resulting model in the Simulink environment
WSEAS TRANSACTIONS on SYSTEMS and CONTROL
DOI: 10.37394/23203.2023.18.36
Igor Astrov
E-ISSN: 2224-2856
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by the guidelines of the allocated grant for these
studies.
The level of complexity of the task of controlling
the guidance of a vehicle can be characterized by
two facets: the type of control (lateral, longitudinal,
or mixed) and the complexity of the used model
(kinematic, linear-dynamic, or non-linear dynamic).
This article deals with complex configurations for
solving a nonlinear dynamic problem. Here, we
model a custom-designed LQR control system that
uses the initial non-linear dynamic model of the
SDC.
The LQR controller is chosen here as the most
suitable for the obtained mathematical model of the
SDC with many inputs and many outputs, as well as
for a simpler obtaining of the output trajectory
without additional transformations. This LQR
approach greatly reduces the computational cost for
real-time operation because it works in a much
faster way. This article shows the neat efficiency of
the output coordinate stabilization task.
2 Nonlinear Model of SDC
The bicycle model was used to simulate the SDC to
receive the desired accuracy and avoid expensive
computational costs.
As stated in Figure 1, the transverse dynamics
and longitudinal dynamics have been considered.
The bicycle model is obtained from a four-
wheeled SDC by gluing the front and rear wheels
into a single front and rear wheel, respectively,
along the longitudinal axis of the SDC without the
inclusion of pitch and roll dynamics.
Fig. 1: Bicycle model
The goal is to be able to control the SDC's
trajectory to reach the desired position as fast and
safely as possible.
The kinematics of SDC can be represented in the
form of such equations, as, [12].
(1)
(2)
(3)
where are the coordinates of the center of mass
in the earth-fixed frame, respectively, is the yaw
angle, are the longitudinal and lateral speeds
in the body frame, respectively, and is the yaw
rate.
The bicycle model of SDC can be described by
the following equations, [13].
(4)
(5)
(6)
where is the mass, is the yaw inertia, and
are the lateral tire forces of the front and rear
wheels, respectively; is the driving force, is the
front steering angle; and are the distances from
the center of mass to the front and rear wheel axes,
respectively.
The driving force in (4) can be written as
(7)
where is an acceleration.
The forces and can be found so, [12]
(8)
(9)
where the and are the constant tire stiffness
parameters.
The coordinates and yaw angle can be found by
integrating
(10)
It can be seen that the vector drawn up of
coordinates and yaw angle can be calculated from
obtained equations (1)-(10).
3 Parameters of the Nonlinear
Model of SDC
It is known that some parameters of the SDK
strongly depend on environmental conditions.
Therefore, even small changes in these parameters
can significantly affect the movement of the SDK.
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As a result, for example, the estimation of the
values of parameters of SDC for the needs of
accident reconstruction is very important.
The main parameters of such SDC as ISEAUTO,
[14], are calculated here.
The following parameters of this vehicle are
known
The moment of inertia was calculated so, [15]:
(11)
Hence, assuming that length and
width from (11), we find
Now it is necessary to evaluate the values of
and .
Assuming a small yaw angle θ in a stable state,
we find
(12)
The yaw rate can be obtained in this way, [16]
(13)
From (8), (9), (12), and (13), we find
(14)
(15)
The next inequalities can be done
(16)
(17)
Assuming that we have a small angle of rotation δ
in the stable state, using that ,and
(14)-(17), we get
(18)
(19)
Assuming that we have a small constant yaw
angle θ and a small steering angle in a stable state,
we get
(20)
Combining (6) and (20), we obtain
(21)
Combining (18)-(19) and (21), we obtain
(22)
Assuming that we have a small steering angle
and is constant in a stable state, we get
(23)
Combining (5) and (27), we obtain
(24)
From (3), (13), (18), (19), (22), and (24), and
assuming that we have a small at stable
state, we get
(25)
The vehicle's path depends on many parameters,
including tire, vehicle, and road characteristics. For
example, the stiffness of a wheel in a turn changes
from 50% to 150% of the calculated value, [16].
Consequently, the maximum ratio of from
(25) becomes:
(26)
where is a maximum of velocity .
Hence, from (22), (26) and assuming that
, we find
4 State-Space Model of SDC
From (4) and (7), we find
(27)
It is possible to consider as small values
then is a negligible value at a stable state, and
from (27), we obtain
(28)
Also, it is possible to consider the as a zero in
(28), i.e.
(29)
where is a fixed constant.
We linearize (1)-(2) by using the Taylor series.
From (1), (2), and (29), we find
(30)
(31)
The lateral tire force for front wheels and
the lateral tire force for the rear wheels can be
calculated so, [12]
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(32)
(33)
Combining (29) and (32)-(33), we obtain
(34)
(35)
From (5), (34), (35), and believing that
at stable state, we obtain
(36)
From (6), (34), (35), and believing that
at stable state, we obtain
(37)
The state space model of SDC has the next form
(38)
(39)
Then, the state vector and input signal are
defined so
X
, U= (40)
The matrix in (38) can be described as
(41)
where the next elements of matrix are obtained
from (36)-(37) so
The matrix in (38) can be described as
(42)
where the next elements of matrix are obtained
from (36)-(37) so
The matrices in (39) are denoted so
(43)
5 Control System
When the mathematical model of the controlled
system is linear and the cost function is quadratic,
we have a continuous-time LQR optimal control
problem that provides the best possible performance
concerning the given cost function.
The problem of LQR control over infinite time
for a linear, stationary, stabilized, and detectable
model is to calculate the optimal matrix of the
feedback coefficients K such that the optimal
feedback control law
U (44)
minimizes such cost function
(45)
which was applied to (38).
The matrices Q in (45) can be chosen by
applying the following rules:
(46)
(47)
where is a maximum acceptable value for the
output signal and is a maximum acceptable
value for the input signal.
From (39), (40), (43), (46), and (47), we find
(48)
The matrixes in (48) are chosen so
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6 Simulation Results
In this case, the SDC trajectory consists of two given
lines to avoid obstacles.
A block diagram of the control system for the
case of velocity from (29) in Simulink is
shown in Figure 2 (Appendix).
The full speed of the vehicle can be found so
Let us evaluate the movement restriction area for
safe vehicle maneuvers with obstacle avoidance.
Equation of motion under the action of constant
velocity can be expressed so
(49)
The full-time can be divided into two intervals
such as where these intervals are
times for achieving of the reference signal and
reference signal respectively.
From (49), we find
(50)
where is a safe distance
The prohibited area for movement can be
evaluated as a rectangle with a length and
a width
The input control signal is determined by the
steering angle of the front wheel.
The output control signals are determined by ,
, and .
Velocity
from (29), the references
for coordinate as the
setting time from (50) and the gain matrix
from (44) as
were applied
during this maneuver. Note that
Simulation results for input and output signals
are shown in Figure 3 and Figure 5 (Appendix),
respectively. The yaw angle for this maneuver is
presented in Figure 4. The motion way of SDC is
shown in Figure 6. A smooth transition to the
desired lines was noticeable without an output signal
spike.
Note that the expected accuracy of the output
coordinate regulation in Figure 5 (Appendix) lies
within 5%, and the resulting accuracy of the output
coordinate control in Figure 5 (Appendix) does not
exceed 0.19%.
A possible representative form of SDC motion is
shown in Figure 7. The Simulation 3D Vehicle
block implements three-dimensional animation of a
tire force four-wheel SDC. This animation block
uses the coordinates and angles of SDC to adjust the
rotation for each wheel and make the motion follow
the terrain. The Simulation 3D Scene Configuration
block implements a virtual 3D simulation
environment for the used SDC model.
Fig. 3: Steering angle of SDC
Fig. 4: Yaw angle of SDC
Fig. 6: SDC trajectory
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Fig. 7: The visual display of SDC motion
7 Conclusions
A simulation model for ISEAUTO SDC was
obtained and tested using Simulink software.
The proposed methodology can be used to
improve the design of the SDC simulation model for
applications that involve various control problems.
After setting up the given scenario, we simulate
the two lanes change maneuver and get a realistic
and expressive 3D video. The simulation results
show an impressive quality of LQR control for the
proposed SDC model, ensuring the smoothness and
safety of the SDC motion trajectories. It can be
expected also that the obtained control system may
be applied easily to other types of SDCs as well.
The benefits of his research lie in the fact that the
soft and reliable trajectory of the SDC can be
realized during the transition between two selected
lines of motion.
The limitations of this study were revealed in the
fact that it is impossible to obtain the desired control
time from the output coordinate and, thus, it is
impossible to change the boundaries of the SDC safe
movement zone.
The suggested improvement of this work can be
the development of a control system for the MDC,
which will allow obtaining the desired values for the
time of regulation of the output coordinate and the
possibility of a given change in the safety zone n
when the MDC is moving.
A future direction may be research related to the
development of a simulation model that will consider
the influence of weather conditions on the movement
of the SDC on the base of the ISEAUTO platform in
TalTech in a smart city environment.
Acknowledgment:
This work was partially supported by H 2020 grant
No. 856602 and ERDF grant No. 2014 - 2020. 4. 01.
20 - 0289.
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Appendix
Fig. 2: Control system for SDC
Fig. 5: Coordinate of SDC
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author completed all stages of this research.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This work was partially supported by H 2020 grant
No. 856602 and ERDF grant No. 2014 - 2020. 4.
01. 20 - 0289.
Conflict of Interest
For the author no conflict of interest regarding the
content of this paper.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.e
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