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Model Predictive Control (MPC) and Proportional
Integral Derivative Control (PID) for Autonomous Lane
Keeping Maneuvers: A Comparative Study of Their
Efficacy and Stability
*Ahsan Kabir Nuhel1, Muhammad Al Amin1, Dipta Paul1, Diva Bhatia2, Rubel Paul3
and Mir Mohibullah Sazid1
1American International University-Bangladesh
1{nuhel7050, aminmuhammadal2, diptapal4, sazidarnob}@gmail.com
2 Vellore Institute of Technology, Vellore, India
2divabhatia13@gmail.com
3 Sonargaon University
3rubelpaul695@gmail.com
Abstract. The escalating frequency of fatal crashes has led to an enhanced
focus on road safety, resulting in the creation of diverse driver assistance
systems. Several instances of these systems encompass active braking, lane
departure warning, cruise control, lane maintaining, and numerous additional
examples. However, the primary objective of this research is to examine the
effectiveness and reliability of a model predictive control (MPC) and a
proportional integral derivative (PID) control in executing lane keeping
maneuvers within an autonomous vehicle. In this paper, a custom controller for
autonomous lane-changing maneuvers is developed by utilizing the Model
Predictive Control (MPC) and Proportional-Integral-Derivative (PID)
controllers. Different trajectory models are employed to assess the overall
effectiveness of the designed model, showcasing its superiority over existing
models.
Keywords: Autonomous car, trajectory models, MPC, PID, Model
Prediction.
1 Introduction
Autonomous vehicle is one of key technological advancement to keep the road safe
and decrease the number of fatal accidents [1]. Apart from than, most of the modern
cars come with Advance Driving Assistance System (ADAS) to help the driver in the
road. The main functions for these kinds of systems are autonomous breaking, lane
keeping assist, object detection, lane departure warning and so on. In this project, a
custom controller have designed to keep the car in the lane [2]. At first, Kinematic
Bicycle model is discussed which is used to design the MPC controller.
2
While designing the MPC controller, as inputs, this method considers the current
condition of the vehicle, including its position, velocity, and heading, as well as the
positions of surrounding vehicles [3]. System dynamic is also taken into consideration
to fulfill the control objective. Different trajectory models are generated to observe
the models and checking the outputs with the pre-existing models [4]. The results are
analyzed with reference trajectory, straight and curve trajectory for different
attributes.
The paper is divided into five sections. In Section 2, the paper describes earlier
fundamental research concepts. Section 3 describes the proposed methodology and
experimental setup. Section 4 analyzes the results, and Section 5 discusses potential
future applications.
2Literature Review
In the research of Trajectory Tracking Control using MPC Controller for of
Quadcopter [5], G. Ganga and co-author Meher Madhu Dharmana constructed a
quadcopter by utilising the dynamic equation in 2017. They proceeded to devise a
linear Model Predictive controller and a Proportional-Integral-Derivative controller to
address the issue of trajectory tracking for the quadcopter. The findings of this study
demonstrate the superiority of the Linear Model Predictive controller over the
Proportional-Integral-Derivative controller in the context of design objectives.
In a paper Hengyang Wang, Biao Liu, Xianyao Ping, and Quan worked on
Autonomous Vehicles Path Tracking Control Based on an Improved Model Predictive
Controller. They proposed an enhanced MPC controller that incorporates fuzzy
adaptive weight control. The objective of their study is to address the challenge of
lane tracking in autonomous vehicles. The fuzzy adaptive control algorithm is
employed to implement this controller, which primarily involves the cost function by
dynamically increasing the weight in the classical model predictive control (MPC)
approach.
In 2020 Shuping Chen, Huiyan Chen, and Dan Negrut worked on the study of Path
Tracking for Autonomous Vehicles with the Implementation of Model Predictive
Control Incorporating Three Vehicle Dynamics Models with Varying Fidelities. They
proposed the practical application of path tracking for autonomous vehicles. In the
study conducted by [7], an MPC controller was developed utilising three distinct
models: an 8-DOF model, the bicycle model, and a 14-DOF model. The reference
paths employed in the experiment consisted of a straight line and a circular trajectory.
The researchers also conducted a comparative analysis of the performances exhibited
by various models.
Ak Nuhel, MM Sazid and MNM Bhuiyan designed a level 5 autonomous car using
machine learning, deep learning and CNN [8]. Apart from that, MPC controller is
used to design the path of the car using different trajectory analysis. An overview of
vehicle safety was also discussed.
In the paper, Eugenio Alcalá, Vicenç Puig, and Joseba Quevedo worked on "LPV-
3
MPC Control for Autonomous Vehicles" [9] in 2019. They addressed a trajectory
tracking issue pertaining to autonomous vehicles. The approach employs a cascade
control strategy, wherein an external loop is utilised for position control through the
implementation of a Learning Parameter Varying (LPV) Model Predictive Control
(MPC) controller. The distinction between the LPV-MPC controller and the
Nonlinear (NL) MPC controller is demonstrated, and the superior performance of the
LPV-MPC controller is discussed.
3 Methodology
3.1 Vehicle Model
According to Limebeer and Massaro [10], the vehicle model serves as a fundamental
basis for investigating the features of vehicle control. Consequently, an accurate
vehicle model is crucial for developing a reliable and precise model of vehicle
control. Both the kinematic bicycle model and the dynamic bicycle model are
analyzed in detail in this research, for the purpose of implementing autonomous
driving capabilities in our vehicle. The utilization of models is imperative for
conducting an in-depth analysis of vehicles, particularly for the purpose of modeling
the controller.
Kinematic Bicycle model.
The motion of a bicycle can be conceptualized within the theoretical framework of
the Kinematic Bicycle Model.
Fig. 1. Kinematic Bicycle model
The studies by Rajamani and Zhang [2] show that the kinematic bicycle model is
commonly used in the study of vehicle features [11]. Figure 1 demonstrates the
kinematic bicycle model. There are four wheels total, two in the front and two in the
back, however in this model they function as one. The forward lumped wheel sits
precisely in the geometric center of the front axle, whereas the rear lumped wheel is
centered on the back axle. The kinematic model can be quantitatively expressed by
equations (1) through (5) if it is assumed that the front wheel is the only part
responsible for steering.
4
vsin
(
ψ+β
)
,
(
2
)
v
lr
(
β
)
,
(
3
)
The inertial frame's center of mass is represented by the coordinates (X,Y), where X
and Y are variables. The direction of the self-driving car's movement is indicated by
ψ
and its velocity is marked by the variable v. Distances from the center of mass to
the front and rear wheels are represented by the
lf
and
lr
variables, respectively. The
present model incorporates acceleration
(
a
)
and the stecring angle
(
δ
)
as control
inputs. In order to simplify the analysis and facilitate practical application, it is
common practice to assume that the rear wheels of a vehicle have zero steering angle
δr=0
, with the focus being solely on the steering of the front wheel. The present
kinematic model exhibits a relatively elementary structure in comparison to
alternative models that incorporate additional physical factors such as aerodynamic
drag, gravitational force, and frictional resistance. The kinematic model can be
identified with only two parameters, namely
(
lf
and
lr
), rendering it applicable for
both longitudinal and lateral control purposes.
3.2 Controller Design
The controller utilizes Model Predictive Control (MPC) as its fundamental approach.
This method takes into account the present states of the vehicle, including its position,
velocity, and heading, as well as the positions of neighboring vehicles, as inputs [12].
The output of the MPC is a set of acceleration and steering angle values.
Fig.
2. Stabilization via Model-Predictive Controlling High-Speed Autonomous Ground Vehicle
vcos
(
ψ+
β
)
,
(
1
)
˙
x=
˙
y=
˙
ψ=
(
5
)
˙
v
=a
(
4
)
β=
arctan
(
lf
l
+
r
lr
+
tan
(
δ
f
)
)
)
5
System Dynamics.
The study utilizes a nonlinear kinematic bicycle model to depict the characteristics of
vehicle dynamics [3]. The kinematics are rewritten here for completeness from
equation 6 to 10:
˙
x
˙
y
lr
lf+lr
tan
(
δ
)
)
(
10
)
Cartesian coordinates
(
x , y
)
represent the location of the vehicle's center of mass,
whereas the inertial heading
ψ
is the direction in which the vehicle is moving. The
variable
v
stands for the speed of the car, while the letter
a
stands for the acceleration
felt by the vehicle's center of mass in the direction of the speed. Distances from the
vehicle's center to the front and rear axles are denoted by the
lf
and
lr
variables. Both
the front wheel's steering angle
(δ)
and the vehicle's acceleration
(a)
are used as
inputs for control. The following mathematical representation captures the essence of
the discrete-time dynamical model derived by the Euler discretization method:
z
(
t+1
)
=f
(
z
(
t
)
,u
(
t
)
)
(
11
)
where
z=
[
x y ψ v
]
⊤
and
u=
[
a δ
]
⊤
(12) for time
t
.
Control Objective.
The principal goal of the control system is to effectively integrate into the designated
lane, while concurrently avoiding any potential collisions with other vehicles. We
have a predilection for changing lanes at a prior intersection. Enhanced driving
comfort is typically associated with a preference for smooth accelerations and
steering. The presentation of the objective function formulation is as follows:
J=∑
l=t
t+T
❑λ¿
(
x
(
l∣t
)
; xend
)
D
(
l∣t
) (
12
)
+∑
l=t
t+T
❑λv
‖
v
(
l∣t
)
−vref
‖
2
(
13
)
+∑
l=t
t+T−1
❑λδ∥δ
(
l∣t
)
∥2
(
14
)
+∑
l=t
t+T−1
❑λa∥a
(
l∣t
)
∥2
(
15
)
+∑
l=t+1
t+T−1
❑λΔδ ∥δ
(
l∣t
)
−δ
(
l−1∣t
)
∥2
(
16
)
=
vcos
(
ψ
+
β
)(
6
)
=
vsin
(
ψ
+
β
)(
7
)
v
lr
˙
ψ=
˙
v=
sin
(
β
)(
8
)
a
(
9
)
β
=
tan−1
(
6
+∑
l=t+1
t+T−1
❑λΔa ∥a
(
l∣t
)
−a
(
l−1∣t
)
∥2(17)
The temporal value of
l
is determined by the data collected at time
t
, and is denoted
by the symbol
(
l∨t
)
. The latitude coordinate of the road's terminus is represented by
the symbol
xend
. The ego vehicle's distance norm to the target lane at time
l
is
denoted by
D
(
l∨t
)
. The symbol
vref
refers to the reference velocity. The
regularization of each penalty is accomplished by employing
λdiv
,
λv, λδ, λa, λ Δδ
,
and
λΔa
,, correspondingly. The timely lane change is incentivized through the
utilization of a dynamic weight, denoted as
λdiv
,, which is expressed as a convex
function. Specifically,
λdiv
is represented as
‖
1
xend−x
‖
. The aforementioned
expression denoted by (12) serves to penalize the deviation of the car's center from
the vertical midpoint of the designated lane. The expressions denoted by (14) and (15)
serve to impose a penalty on the exertion of control in relation to the steering angle
and acceleration, respectively. To improve ride quality, the steering rate and
disturbance is impacted by the equations labelled as (16) and (17), respectively.
Transitional Model Predictive based Controlling for Lane Changing.
To avoid rear-end crashes during overtaking maneuvers, autonomous trajectory path
tracking using Model Predictive Control is implemented [14]. In Figure 4 we see a
model of the control architecture used by the autonomous vehicle. for making a lane
change, the major focus for designing trajectories is to reduce the amount of yaw
acceleration the vehicle experiences. The constraints pertaining to the dynamics of
vehicles and the boundaries of the roadside are articulated as limitations within a set
of convex optimization problems. The acquisition of reference positions and
velocities is achieved through the utilization of a convex optimization algorithm.
Model Predictive Control (MPC) controllers, like the one seen in Figure 4, make use
of mathematical models of autonomous cars to anticipate how the system would
evolve in the future. This methodology is employed to enhance future vehicle
performance and minimize the discrepancy between the planned trajectory route and
the actual route.
7
Gc=Kp+Ki
s+Kds
(18)
Kp, K i
and
Kd
values were obtained using the built-in tuning tool in MATLAB
SIMULINK and was imported in coding script., as presented in Table 2. In order to
evaluate the efficacy and purpose of control techniques during vehicular maneuvers,
distinct control parameters are established for each test velocity to enhance the
system's response.
Fig.
3.
Architecture for the management of systems in autonomous vehicles.
Fig.
4.
The Lateral Control configuration of the proposed model
PID Controller
The study presents a PID controller that is specifically designed to ensure that the
vehicle adheres to a desired yaw rate [13],
˙
ψ
, while simultaneously minimizing the
sideslip,
β
. Equation (18) pertains to the PID control law. This consists of
K
p,
K
i,
and
K
d
, the proportional, integral, and derivative gains, respectively. These three
parts work together to form the
Gcblock of the controller transfer function.
8
Table 1.
PID Tuning Parameters
Control
Parameters
Speed
Slow
30 km /h
High
80 km /h
Kd
3 0.03
Kp
7 0.81
Ki
5 8.10
3.3
Development of trajectory generator and tracking controller
The development of a trajectory tracking controller is a crucial area of research in the
field of control systems. This controller aims to enable precise tracking of desired
trajectories by a dynamic system. By utilizing advanced control algorithms and sensor
feedback, the trajectory tracking controller enhances the system's ability to follow
predefined
paths
accurately,
facilitating
applications
in
various
domains
such
as
robotics,
autonomous
vehicles,
and
aerospace.
Prediction
Model
Development
employs a nonlinear dynamic system that takes into account the output:
˙
ξ
(
t
)
=f
(
ξ
(
t
)
,
μ
(
t
)
)
(
19
)
η
(
t
)
=h
(
ξ
(
t
)
,
μ
(
t
)
)
(
20
)
The following expression includes the state transition function f(.,.), a state variable
ξ(t)
with n dimensions, a control variable
μ(t)
with m dimensions, and an output
variable
η(t)
with p dimensions.
Convert the continuous systems of Equations (17) and (18) into a linear time-varying
system.
ξ
(
k
+1
)
=Ak
,
t
ξ
(
k
)
+Bk
,t
μ
(
t
)
+dk
,t
(21)
η
(
t
)
=C
k
,t
ξ
(
k
)
+
Dk
,
t
μ
(
t
)
+ek
,t
(
22
)
Taking into account the following presumptions:
9
´
Ak ,t
´
Bk ,t
(
24
)
´
Ck ,t
´
Dk , t
´
d
(
k∨t
) (
28
)
The symbol
0m ×n
represents a zero matrix with dimensions
m × n
, while
Im
denotes
an identity matrix with dimensions m.
When designing the trajectory tracking controller, it is imperative to take into account
the constraints imposed by the vehicle dynamics.
3.4 The constraint of sideslip angle at the mass center.
Significant changes in driving stability can occur when the mass center's sideslip
angle β is outside the linear range of the lateral force, necessitating the imposition of
constraints. The mass center slip angle's constraint range is commonly represented by
the arctangent function.
−arctan
(
0.02 μg
)
≤ β ≤ arctan
(
0.02 μg
)(
29
)
The constraint of lateral acceleration
The amount of grip provided by a car's tires on the road has a direct impact on the
vehicle's performance. The presence of different coefficients of adhesion on the road
results in the generation of distinct longitudinal and lateral forces that are applied to
the Tire by the ground. This research establishes a connection between forward
velocity, sideways velocity, and road adhesiveness, highlighting the presence of
inequality in this relationship. The inequality of the given expression is as follows.
√
ax
2+ay
2≤ μg
(
30
)
can be presented, where the acceleration along the longitudinal axis is denoted by
ax
and the acceleration perpendicular to it is denoted by
ay
. The longitudinal velocity of
the vehicle showed almost no variation during a relatively short time period. It is
reasonable to hypothesize that the vehicle maintains a constant longitudinal speed. As
a result, Equation (30) can be simplified in the subsequent manner.
|
ay
|
≤ μg
(31)
=
[
Ck
,t
Dk
,
t
]
(
25
)
=
Dk
,t
(
26
)
´
ξ
(
k∨t
)
=
[
μ
(
ξ
k
(
−
k∨
1∨
t
)
t
)
]
(
27
)
d
(
k
0m
∨t
)
]
=
[
=
[
B
I
k
m
,t
]
=
[
0
A
m
k
×
,t
n
B
I
k
m
,t
]
(
23
)
10
Parameters Values
Mass (m) 1500 g
Mass moment of inertia (Iz) 3000 g
Front wheel cornering stiffness (Caf) 19000 N/m
Back wheel cornering stiffness (Car) 33000 N/m
The gap between the front wheel and
the mass (Lf)
2 c.m
The gap between the back wheel and
the mass (Lr)
3 c.m
Sampling time (Ts) 0.02 sec
Lane width 7 meters
Number of lanes 5
Reference Trajectory frequency 0.01 Hz
The inability to accurately calculate can be ascribed to the constraint conditions being
either overly inclusive or overly restrictive, depending on the specific road adhesion
circumstances. A relaxation coefficient is included in to provide the adaptive modifi-
cation of constraint conditions in response to the solution scenario of each control
iteration, making the restriction on lateral acceleration a flexible constraint. This in -
equality
holds
for
every
value
of
the
lateral
acceleration
|
a
y
|
≤
μg:
a
y
,
min−ε
≤
a
y
≤
a
y
,max
+ε
. (32)
The
maximum
and
minimum
lateral
accelerations,
a
y
,
maxand
a
y
,
min,
are
shown
below, where
ε
stands for the relaxation factor.
3.5
Experimental Setup
The experiment was set up in simulation using the command prompt of Windows
version 11 to execute the main file and a support file for backing up library functions.
Python version 3.7, along with NumPy and Matplotlib libraries, was utilized for the
process, while separate animation scripts were incorporated for visualizations. The
experiment
involved
constants
related
to
an
autonomous
vehicle
and
model
parameters,
which
are
detailed
in
Table
2,
allowing
researchers
and
engineers
to
analyze their impact on the vehicle's behavior and the model's performance.
Table
2.
Dimensions of the car model and other environmental attributes
4
Results Analysis
The author of the study devised three distinct trajectory models to assess the efficacy
and
performance
of
the
overall
system
under
investigation (Figure 5).
These
systems
were
meticulously
engineered
and
implemented
to
evaluate
different
aspects and functionalities of the overarching system. The accompanying visual aids,
11
(a) Car following reference trajectory in a straight line
(b) Car following reference trajectory in a curvy line
(c)
Car following reference trajectory in a hybrid (Mixed trajectory of sinusoidal and
parabolic) line
Fig.
5.
Car’s attributes in following difference reference trajectories
presented
below,
depict specific instances where a car adeptly tracks and follows its
designated
reference trajectory.
In these illustrations,
the reference trajectory is
represented by
the blue color,
while the actual position of the vehicle is denoted
by red color.
These
visual
representations
vividly
showcase
the
successful
execution
of
the
implemented
control
algorithms
and
highlight
the
capability
of
the
system
to
accurately adhere to the desired trajectory while maintaining the desired position).
12
Fig.
6.
Different attributes of car at following curvy line trajectory
The front wheel demonstrated smooth movement and effectively adjusted its position
to
accommodate
both
positive
and
negative
slopes
as
required
by
the
system,
effectively avoiding overshooting. When analyzing the hybrid parabola scenario, a
minor delay was observed during the initial response, and the front wheel exhibited
aggressive behavior. This behavior can be attributed to the fact that, initially, the
vehicle
was
oriented
in
the
x-direction
with
a
yaw
angle
of
0
radians,
while
the
reference yaw angle was set to 0.5 radians at time 0 seconds. Consequently, there was
a noticeable deviation in tracking the predetermined setpoint. As the vehicle gradually
aligned
itself
with
the
desired
setpoint,
its
velocity
decreased,
leading
to
a
more
consistent speed. Additionally, it was noted that the steering wheel angle reached its
maximum limit of
pi/6
radians during this specific instance (showcases in Figure 6).
Fig.
7.
Model attributes at frequency 0.01 Hz
From
the
analysis
presented
in
Figure
7,
it
becomes
evident
that
the
controller's
performance deteriorates when operating at higher frequencies. This degradation can
be attributed to the fact that the car's longitudinal velocity remains unchanged, leaving
insufficient time for the system to stabilize, despite the amplitude of the input signal
not
being
significantly
marginal.
To
overcome
this
limitation,
the
longitudinal
13
velocity was subsequently adjusted to a value of 20 m/s. Particularly, the experiments
show that the vehicle can successfully follow a reference input's optimally trajectory.
Moving
on
to
Figure
8,
we
observe
the
output
pertaining
to
variations
in
higher
weight
matrices
within
the
cost
function
for
both
state
and
final
horizon
period
outputs. The outcomes reveal a proportional relationship between the weight values
and
the
minimization
of
specific
errors
associated
with
either
the
yaw
reference
angles or the vehicle's position. Consequently, the error in the yaw angle significantly
diminishes in comparison to the larger positional error, thereby validating the efficacy
of the approach.
Fig.
8.
Outputs concerning changing in the weight matrices by breaking the identity law (by
prioritizing Yaw angle error)
According to the findings presented in Figure 9, the incorporation of the Proportional-
Integral-Derivative
(PID)
controller
resulted
in
observable
oscillation
in
both
the
steering wheel and yaw reference angle. This phenomenon arises due to the fact that
the PID controller solely considers errors within a single sampling time and lacks the
ability
to
take
into
account
the
entire
time
horizon,
unlike
the
Model
Predictive
Control
(MPC)
controller.
Therefore,
the
task
of
fully
mitigating
overshooting
becomes a challenging attempt for the PID controller. On the other hand, due to the
MPC
controller's
ability
to
make
more
informed
decisions
by
utilising
future
predictions
derived
from
the
system
model,
the
oscillations
and
error
reduction
achieved are significantly smoother in comparison to those obtained with the PID
controller.
14
Fig.
10.
Results pertaining exponential route
Fig.
9.
Output concerning PID controller.
Furthermore, to assess the overall effectiveness of the proposed model, three distinct
trajectories were formulated. Among these trajectories, only the exponential and cubic
polynomials
were
considered,
and
their
corresponding
outcomes
are
depicted
in
figures 10 and 11. The results exhibit a satisfactory tracking performance characterized
by
consistent
and
accurate
adherence
to
the
prescribed
reference
setpoints,
while
maintaining a desirable level of smoothness. It is worth noting that the vehicle's initial
course angle is set to zero, whereas the initial course angle of the reference trajectory
exceeds zero. Consequently, the vehicle demonstrates the ability to adjust its direction
during this phase.
This
modification enables
accurate monitoring of
the
reference
trajectory, thereby improving both the resilience and precision. As the vehicle's speed
increases and the adhesion coefficient of the road surface reduces, the aforementioned
simulation findings show that noticeable deviations from the desired trajectory are
noticed
when
using
double-shifting.
Moreover,
such
deviations
may
give
rise
to
hazardous situations where the vehicle loses control over its direction.
15
Fig. 11. Results pertaining cubic polynomial route
5 Conclusion
The paper uses a MPC controller to track the path of any cars. It can be used in ADAS
or Autonomous cars for keeping the vehicle in lane. For the modern safety
requirements, it important have ADAS in every modern vehicle [14]. This MPC
controller can meet one of the key features in ADAS system. The paper analysis the
different trajectory of the car using the custom made MPC controller and shows how
it is better than pre-existing controller such as PID controller. The vehicle movement
and speed also taken into consideration while doing the maneuvers so that the cars
can keep their lane. Using the MPC controller, Computer vision and Deep learning,
more advance ADAS or Autonomous Vehicle can be designed for which will make
our roads safer.
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