actuators

Article
Development of a Universal Adaptive Control Algorithm for an
Unknown MIMO System Using Recursive Least Squares and
Parameter Self-Tuning
Hanbyeol La and Kwangseok Oh *

School of ICT, Robotics and Mechanical Engineering, Hankyong National University,

Anseong-si 17579, Republic of Korea; byeol0515@hknu.ac.kr
* Correspondence: oks@hknu.ac.kr; Tel.: +82-31-670-5117

Abstract: This study proposes a universal adaptive control algorithm for an unknown multi-input
multi-output (MIMO) system using recursive least squares (RLS) and parameter self-tuning. The
issue of adjusting the control and system parameters in response to changes in the platform was
discussed. The development of a control algorithm that can consistently achieve reliable and robust
control performance in various systems is important. This study aimed to develop a control algorithm
that can track the reference value for any unknown MIMO system. For the controller design, an
nth-order differential error dynamic model was designed, and an RLS with a scale factor was used to
estimate the coefficients of the error dynamics. In the current scenario, the numbers of control inputs
and error states in the error dynamics were assumed to be equal. It was designed such that the control
input is derived based on the Lyapunov stability concept using the estimated coefficients. The scale
factor in the RLS and injection term in the control input based on the sliding-mode approach were
computed using a self-tuning methodology. The performance of the proposed universal adaptive
control algorithm was evaluated using an actual DC motor and CarMaker (version 8.1.1) software
tests under various scenarios.

Keywords: recursive least squares; universal adaptive control; MIMO system; self-tuning; scale
factor; sliding-mode approach
Citation: La, H.; Oh, K. Development
of a Universal Adaptive Control
Algorithm for an Unknown MIMO
System Using Recursive Least Squares 1. Introduction
and Parameter Self-Tuning. Actuators With the advancement in technology in various platforms, such as autonomous driving
2024, 13, 167. https://doi.org/ mobility, there is an increasing demand for the control of multiple actuators. In recent
10.3390/act13050167 years, these systems have provided assistance to users; however, in future, automatic
Academic Editor: Anyang Lu control is expected to be replaced by automated functions because of their autonomy while
performing different activities. However, to achieve this, accurate knowledge of the system
Received: 15 March 2024 parameters and sophisticated sensors are required because the system parameters affect
Revised: 25 April 2024
the control performance. These sensors are expensive and require control to track target
Accepted: 27 April 2024
values without sensors. In addition, modification of the controller with respect to the
Published: 1 May 2024
variation in the number of actuators or platforms is extremely inefficient in terms of time.
Therefore, regardless of the unknown system, the user requires a control technology that
can be applied to various systems and can provide stable control.
Copyright: © 2024 by the authors.
Liu et al. proposed a robust controller design method that combines particle swarm
Licensee MDPI, Basel, Switzerland. optimization and computational methods to control multi-input multi-output (MIMO)
This article is an open access article systems [1]. Homaeinezhad et al. proposed a methodology that merges backstepping and
distributed under the terms and sliding-mode controls to manage nonlinear loads and parameter changes in MIMO sys-
conditions of the Creative Commons tems [2]. Pongfai et al. proposed a parameter design methodology based on a proportional-
Attribution (CC BY) license (https:// integral-derivative (PID)-based adaptive cluster learning process for the optimal control
creativecommons.org/licenses/by/ of MIMO systems [3]. Homaeinezhad et al. proposed a Lyapunov-stability-based algo-
4.0/). rithm as a nonpredictive methodology for designing nonlinear MIMO control systems [4].

Actuators 2024, 13, 167. https://doi.org/10.3390/act13050167 https://www.mdpi.com/journal/actuators

Actuators 2024, 13, 167 2 of 20

Recent studies have proposed adaptive fuzzy controllers based on sliding-mode control
methods to control electric mobility, hexacopter drones, and electric wheelchairs [5–7].
Huang et al. proposed a robust control system that incorporates design procedures such
as back-steps to alleviate the controllability conditions of nonlinear MIMO systems with
actuator defects [8]. Thanh et al. proposed a super-twist sliding technique for the finite-time
stability of uncertain nonlinear systems using MIMO [9]. Zeghlache et al. demonstrated
robust control against wind gusts and external disturbances using the Lyapunov method
to stabilize a MIMO system and ensure that the desired signal is accurately controlled [10].
As the aforementioned studies defined a control system limited to motors or vehicles
and evaluated the proposed algorithm, it was difficult to apply it to various unknown
systems [2,5,7]. This study aims to develop a universal controller for application to var-
ious unknown systems. Usman et al. estimated the parameters in real time using an
observer with upper- and lower-limit ranges based on the parameters of an actual DC
motor [11]. Laid and Boubekeur proposed a model-independent control approach using
a hyperlocal model identification method for position control of uncertain and unmod-
eled DC motors [12]. Mynar et al. proposed an extended Kalman filter-based parameter
estimation as a sensorless control methodology for efficient and cost-effective motor con-
trol [13]. Jiang et al. developed a model-independent adaptive control algorithm that uses
front-and-rear-wheel drive control distribution strategies for autonomous vehicles [14].
Wang et al. developed a model-independent four-wheel steering control algorithm for a
four-wheel independent steering vehicle to ensure the steering stability based on the input
and output data of the vehicle without using a mathematical model [15]. In addition, Fliess
and Join proposed a control methodology to minimize the parameter adjustment process
without using a mathematical model with a model-independent intelligent PID control
algorithm [16]. Liu et al. used sliding-mode control-based uncertainty. Moreover, as a
strategy to secure robust control performance against external disturbances, a position-
control algorithm that does not consider the constraints of the mathematical model has
been proposed [17]. Moreno Gonzalez et al. proposed the lateral control of a vehicle over
a wide speed range [18]. Farhan et al. performed a sensorless current prediction control
for a motor by estimating the position and speed of the rotor using an extended Kalman
filter [19]. Wang and Wang developed an independent data-driven model without any
learning process [20]. Numerous studies have confirmed that researchers are developing
control algorithms to ensure steering stability by utilizing the extended Kalman filter for
model-independent adaptive controls. To use the Kalman filter or enhanced Kalman filter,
other system information such as dynamic parameters may be required. To overcome this
problem, this study aims to develop an adaptive control that does not use system informa-
tion. Zhang et al. used camera-based lane detection and side-error calculations to propose
a path-tracking algorithm for intelligent electric vehicles. This algorithm integrated a linear
quadratic regulator based on error dynamics and sliding-mode control [21]. Research is
also underway to devise path tracking for mobile robots that incorporates the robustness
of sliding-mode control [22,23] and the uncertainties and disturbances inherent in mathe-
matical models and parameters and reflects the physical properties of the robot based on
the model predictive controller [24,25]. Controllers based on mathematical models that are
used for path-tracking autonomous vehicles and robots may have a negative impact on
control performance owing to uncertainties in the parameters, mathematical models, and
disturbances in various driving situations. Research is currently underway to improve the
path-tracking control performance by compensating for the uncertainties and disturbances
in the parameters and mathematical models. This is achieved using neural networks and
deep reinforcement [26–28]. Moudoud et al. used an adaptive sliding-mode controller
with a fuzzy logic system to track the trajectory of a mobile robot with uncertainties and
disturbances [29]. Aware et al. proposed a two-timescale technique based on sliding-mode
control to perform lateral control, in which the system obtains a stable yaw rate and sideslip
angle [30]. In previous studies, sliding-mode control was used to develop control algo-
rithms for systems with model uncertainties and disturbances. From the perspective of
Actuators 2024, 13, x FOR PEER REVIEW 3 of 20

Actuators 2024, 13, 167 3 of 20

to develop control algorithms for systems with model uncertainties and disturbances.
From the perspective of performance advancement, research is being conducted to im-
prove control performance using fuzzy-logic-based adaptive control, neural networks,
performance advancement, research is being conducted to improve control performance
and deep reinforcement. In this study, we propose a recursive least square (RLS)-based
using fuzzy-logic-based adaptive control, neural networks, and deep reinforcement. In this
mathematical model-free control algorithm as a methodology for performing universal-
study, we propose a recursive least square (RLS)-based mathematical model-free control
purpose control of unknown systems. The proposed algorithm designs error dynamics
algorithm as a methodology for performing universal-purpose control of unknown sys-
based on RLS and estimates the coefficients of error dynamics; therefore, it does not re-
tems. The proposed algorithm designs error dynamics based on RLS and estimates the
quire system information. The control algorithm was designed based on the Lyapunov-
coefficients of error dynamics; therefore, it does not require system information. The control
stability-based sliding-mode scheme, and the magnitude of the injection term required to
algorithm was designed based on the Lyapunov-stability-based sliding-mode scheme, and
derive
the the control
magnitude input
of the was self-tuned.
injection The main
term required contributions
to derive of this
the control study
input wasare summa-
self-tuned.
rizedmain
The as follows:
contributions of this study are summarized as follows:
(1) The
(1) The proposed
proposed algorithm
algorithm estimates
estimates the RLS-based error
the RLS-based error dynamics
dynamics coefficients
coefficients and
and
does
does not require information regarding the system. Therefore, it can be used as aa
not require information regarding the system. Therefore, it can be used as
universal-purpose
universal-purpose controller
controller in in various
various unknown
unknown systems.
systems.
(2) In this study, a virtual test drive simulator,
(2) In this study, a virtual test drive simulator, CarMaker,
CarMaker,andand
an actual DC motor
an actual plat-
DC motor
form were used to evaluate the reasonable performance of the proposed
platform were used to evaluate the reasonable performance of the proposed universal universal
controller.
controller. In
Inthe
thecase
caseofofthethe
CarMaker-based
CarMaker-based evaluation, thisthis
evaluation, study attempted
study attemptedto ver-
to
ify the performance of the proposed algorithm in various systems
verify the performance of the proposed algorithm in various systems using front- using front-wheel
steering vehicles
wheel steering and front-and-rear-wheel
vehicles and front-and-rear-wheel steering vehicles.
steering vehicles.
The remainder of this paper is organized as follows: Section 22 introduces
The remainder of this paper is organized as follows: Section introduces thethe concept
concept
of an adaptive control algorithm using RLS and parameter self-tuning. Section
of an adaptive control algorithm using RLS and parameter self-tuning. Section 3 discusses 3 discusses
the performance
performance of of the
theproposed
proposedalgorithm
algorithmusing
usingMATLAB/Simulink
MATLAB/Simulink (version
(version2019a), an
2019a),
actual
an DCDC
actual motor,
motor,andand
thethe
CarMaker
CarMaker (version 8.1.1)
(version software.
8.1.1) Finally,
software. Sections
Finally, 4 and
Sections 5 pre-
4 and 5
sent conclusions
present conclusionsandand
future
futurework,
work,respectively.
respectively.

2. Adaptive Control Algorithm Using RLS and Parameter Parameter Self-Tuning

Self-Tuning
Figure
Figure 1 depicts
depicts a schematic of the proposed
proposed adaptive
adaptive controlcontrol algorithm.
algorithm. The control
error
error is
is calculated
calculatedusing
usingthe
thedifference
differencebetween
betweenthe thetracking
tracking reference
reference value and
value thethe
and state of
state
the plant.
of the TheThe
plant. calculated error
calculated is used
error in the
is used error
in the dynamics
error dynamics for the
for RLS-based
the RLS-basedcoefficient
coeffi-
estimation. At this
cient estimation. Atstage, the error
this stage, the dynamics assumeassume
error dynamics that thethat
numbers of control
the numbers inputs
of control
and errors are equal. RLS estimates the coefficients
inputs and errors are equal. RLS estimates the coefficients (C ij , D ) required to derive the adaptive
i (𝐶𝑖𝑗 , 𝐷𝑖 ) required to derive the
control
adaptive input andinput
control uses and
the residuals to self-tune
uses the residuals the parameters
to self-tune for the injection
the parameters term of
for the injection
the sliding-mode approach based on the Lyapunov direct method.
term of the sliding-mode approach based on the Lyapunov direct method. The adaptive The adaptive control
algorithm, which iswhich
control algorithm, designed based onbased
is designed the sliding-mode approachapproach
on the sliding-mode and self-tuning injection,
and self-tuning
determines the control input.
injection, determines the control input.

Figure 1.
Figure 1. Block
Block diagram
diagram of
of the
the adaptive
adaptive control
control algorithm.
algorithm.

2.1. MIMO System Error Dynamics

This study explores systems that can be represented by the following error dynamics.
𝑒̇1 = 𝐶11 𝑒1 + 𝐶12 𝑒2 + 𝐶13 𝑒3 + ⋯ + 𝐶1𝑛 𝑒𝑛 + 𝑢1 + 𝐷1
(1)
𝑒̇2 = 𝐶21 𝑒1 + 𝐶22 𝑒2 + 𝐶23 𝑒3 + ⋯ + 𝐶2𝑛 𝑒𝑛 + 𝑢2 + 𝐷2
Actuators 2024, 13, 167 4 of 20

2.1. MIMO System Error Dynamics

This study explores systems that can be represented by the following error dynamics.
.
e1 = C11 e1 + C12 e2 + C13 e3 + · · · + C1n en + u1 + D1
.
e2 = C21 e1 + C22 e2 + C23 e3 + · · · + C2n en + u2 + D2
.
e3 = C31 e1 + C32 e2 + C33 e3 + · · · + C3n en + u3 + D3 (1)
..
.
.
en = Cn1 e1 + Cn2 e2 + Cn3 e3 + · · · + Cnn en + u p + Dn

where e = n denotes the control error and u = p denotes the input, and we assume that the
following conditions are satisfied:
(A1) All the control errors have a complex influence on each other.
(A2) At this stage, n = p and the number of control errors and control inputs are the same.
According to assumption (2), the control input of Equation (1) can be expressed as
Equation (2). The terms for each of the n errors can be expressed in the above equation,
along with the weighting coefficient of the error. The error dynamics for a MIMO nth-order
system can then be considered as follows:
.
n o
n
ei = ∑ j=1 Cij e j + ui + Di (i = 1, 2, 3, · · · , n) (2)

Because this study considers an unknown MIMO system, the coefficients C and D in
Equation (2) are unknown matrices. The C and D values were estimated in real time using
the recursive least squares method. The next section describes the coefficient estimation.

2.2. RLS-Based Coefficient Estimation

An RLS method with multiple forgetting factors was used to estimate the coefficients.
Depending on the number of multiple inputs and outputs, the relationship function for
each equation can be designed as shown in Equation (3). Equations (4)–(7) represent the
RLS equations used for the coefficient estimation. In the virtual relationship function, y, ϕ,
and θ represent the output, regressor, and estimated value, respectively.

yi = ϕiT θi (3)
.  T  T
yi = ei − ui , ϕi = e j 1 , θi = Cij Di (4)
In this study, the cost function V was defined using Equation (5) to design a recursive
least squares algorithm with multiple forgetting factors between zero and one. The recursive
least squares method with multiple forgetting factors treated in this study was designed
with reference to previous research [31].

1 k
2
V θ̂, k = ∑ λk−i y(i ) − ϕθ̂ (k)


(5)
2 i =1

The estimated value that minimizes the sum of the squares of the residuals, which is
the cost function, was calculated. The calculation process is shown in Equation (6).


θ̂i,k = θ̂i,k−1 + Li,k yi,k − ϕi,k θ̂i,k−1 (6)

It should be noted that the optimal gain L was used for the estimation and the covari-
ance matrix P was used for calculating the optimal gain L; during each sampling phase,
Actuators 2024, 13, 167 5 of 20

these were repeatedly updated using a forgetting factor and covariance update factor, as
shown in Equation (7).
Li,k = Pi,k−1 ϕi,k (λ + ϕi,k Pi,k−1 )
(7)
Pi,k = S( I − Li,k ϕi,k ) Pi,k−1 / λ
A covariance update factor with a value of one or more is applied to the covariance
matrix in Equation (7) to intentionally increase the variance, further giving more weight
to the measurements, and is expressed as Equation (8). The scale factor is calculated and
.
applied using the change in error from elow,th , the lower limit of the change in error, to
.
eup,th , the upper limit. When the error change is less than the lower limit, the scale factor is
applied as the smin constant value, which is the lower limit of the scale factor. When the
change in error is greater than the lower limit, the opposite situation applies.
 . .
 s smin

− s
,
. .
.
e < elow,th
. .
max
S = e. up,th −e. low,th e − elow,th + smin , elow,th ≤ e < eup,th
min
(8)
 . .
smax , eup,th ≤ e


2.3. Derivation of Control Input Based on the Lyapunov Direct Method

In this study, the Lyapunov direct method was used to ensure the stability of the
control algorithm. The defined sub cost function was expressed as Js to minimize each
control error. Each sub cost function (Js ) for the error equation of the MIMO system on the
left side was organized on the right side.
n
J = Js1 + Js2 + Js3 + · · · + Jsn = ∑i=1 Jsi (9)

Equation (10) presents the Lyapunov candidate function and two conditions for the
error to converge to zero within a finite time. Weights were applied to the Lyapunov
candidate function to adjust the weights for each error.
wi 2
Jsi = e
2 i
.
Condition 1 : J si ≤ −αJsi 1/2 , α > 0 (10)
Condition 2 : lim Jsi = ∞
|ei |→∞

For asymptotic stability, the rate of change in the Lyapunov candidate function with
respect to time must always be negative. Therefore, the Lyapunov derivative is derived
with respect to time. The rate of change of error defined in Equation (2) is substituted as in
Equation (11).
. .
n
n
o 
J si = wi ei ei = wi ei ∑ j=1 Cij e j + u i + Di (11)

The control input to converge the error to zero and ensure control stability is designed
as shown in Equation (12) using the injection term denoted.
 
n
ui = −∑ j=1 Ĉij e j − D̂i + v
(12)
v = −ρi sign(ei )

The rate of change in the cost function can be obtained from Equation (13) using
Equations (11) and (12). Here, the residual between the actual system value and the
estimated value is given by Equation (14). Therefore, Equation (13) is rearranged into
Equation (15) using the residual.
. n
n
o n
n
o 
J si = wi ei ∑ j=1 Cij e j − ∑ j=1 Ĉij e j + Di − D̂i − ρi sign(ei ) (13)
Actuators 2024, 13, 167 6 of 20

n o n o
n n
Ri = ∑ C e
j=1 ij j
− ∑ Ĉ e
j=1 ij j
+ Di − D̂i (14)
.
J si = wi ei ( Ri − ρi sign(ei )) (15)
By defining the boundary value Lbi using Equations (16) and (17), Equation (15) can be
expressed as Equation (18). Based on this, Equation (13) can be organized into Equation (19)
with a finite-time convergence condition of one:

| Ri | + ηi = Lbi (16)

| Ri | ≤ Lbi (17)
.
J si = wi ei ( Ri − ρi sign(ei )) ≤ wi (|ei | Lbi − |ei |ρi ) = −wi |ei |(ρi − Lbi ) (18)
r
. 1 wi
J si ≤ −αJ 2 = − α |e | (19)
2 i i
Because the right-hand sides of Equations (18) and (19) can be assumed to be the same,
the magnitude of the injection term ρi is determined accordingly.
α α
ρi = Lbi + √ i = | Ri | + ηi + √ i (20)
2wi 2wi

Equation (12) can be expressed as Equation (21) using the magnitude of the injection
term derived from Equation (20). The control input ui generates unrealistic chattering,
depending on the sign of the error. Therefore, to obtain a continuous control signal, the
discontinuous function was replaced with a sigmoid function, as shown in Equation (22).
The adaptive control input is summarized in Equation (23).
  
n αi
ui = −∑ j=1 Ĉij e j − D̂i − | i|
R + η i + √ f ( e, m ) (21)
2wi
me
= sigmoid f unction = f (e, m) (22)
1 + m|e|
n   
αi me
ui = − ∑ Ĉij e j − D̂i − | R i | + ηi + √ (23)
j =1 2wi 1 + m|e|

The next section describes the performance evaluation results of the control algorithm
in various scenarios based on actual DC motors and CarMaker (software).

3. Performance Evaluation
To evaluate the proposed control algorithm for unknown systems, evaluations were
performed on DC motors, a front-wheel steering vehicle, and a front-and-rear-wheel steer-
ing vehicle. The control algorithm was implemented in a MATLAB/Simulink environment.

3.1. Performance Evaluation of DC Motor-Based Adaptive Speed Control

Figure 2 depicts a schematic of the model used for the performance evaluation of the
proposed control algorithm. An actual DC motor test platform (QUBE-Servo 2) was used to
evaluate the performance of the proposed control algorithm. The DC motor was equipped
with an optical rotation encoder to measure the angular displacement. The DC motor and
.
laptop were linked using a USB cable; the angular velocity (θ) was estimated using the
angular position (θ), which was measured by the DC motor using a Kalman filter.
Actuators
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Actual
Figure2.2.Actual
Figure DCmotor
DC motorplatform
platform(QUBE-servo
(QUBE-servo2).2).

The Theevaluation
evaluationscenario
scenariowas
wasregarding
regardingreference
referenceangular
angularvelocity
velocitytracking,
tracking,andandthe
the
error dynamics are defined as Equation (24), with the control input represented
error dynamics are defined as Equation (24), with the control input represented as voltage as voltage
𝑉.V.
𝑒𝜃̇ eisθ. the
is the error
error between
between the the currently
currently estimated
estimated angular
angular velocity
velocity andreference
and the the reference
an-
angular
gular velocity.
velocity. TheThe system
system parameters
parameters of the
of the DCDC motor
motor areare listed
listed in in Table1.1.The
Table Thedesign
design
parametersand
parameters andthe
theadaptive
adaptivevoltage
voltagecontrol
controlalgorithm
algorithmare
arelisted
listedininTable
Table2.2.
.
𝑒̇ 𝜃̇ =eθ𝐶= C11 e . + D1 + V (24)
(24)
.
11 𝑒𝜃̇ + θ𝐷1 + 𝑉

Table 1. DC motor system parameters.

Table 1. DC motor system parameters.
Parameter Unit Value
Parameter Unit Value
Resistance Ω 8.4
Resistance
Torque constant Ω Nm/A 8.4
0.042
Torque constant Nm/A 0.042
Motor Motor back-EMF
back-EMF constantconstant V/(rad/s) V/(rad/s) 0.042
0.042
Rotor inductance
Rotor inductance mH mH 1.16
1.16
−6 −6
Inertia Inertia Kgm2 Kgm2 4.6××
4.6 1010

Table
Table2.2.Design
Designparameters
parametersvelocity
velocitytracking
trackingcontrol.
control.
Parameter
Parameter Unit Unit Value
Value
Decay rate of the Lyapunov function (𝛼)
Decay rate of the Lyapunov function (α) -
- 0.0001
0.0001
Reachability factor
Reachability factor (η) (𝜂) - - 0.0001
0.0001
D̂̂11 ̂
1 ) , 𝐷1 ) (0.1, −−2.73)


Initialvalue
Initial value
of of estimated
estimated states
states (Ĉ11 ,(𝐶 - - 0.1, 2.73

Initial
Initialvalue
valueof of
covariance
covariance P12
(P11 ,(𝑃 )
11 , 𝑃12 )
- - 0.00008, 0.00009
(0.00008, 0.00009)


Forgetting factor (λ11 , λ12 ) - 0.99995, 0.99993
(0.99995, 0.99993)
Forgetting factor (𝜆11 ,. 𝜆12 )
-
Lower scale factor threshold ( elow,th - 0.01
Lower scale factor threshold . (𝑒̇𝑙𝑜𝑤,𝑡ℎ ) - 0.01
Upper scale factor threshold ( eup,th - 0.08
Upper scale factor threshold (𝑒̇𝑢𝑝,𝑡ℎ ) - 0.08
Normalized threshold (smax ) - 1.001
Normalized threshold (𝑠𝑚𝑎𝑥 ) - 1.001

Theseparameters
These parameterswere
weredetermined
determinedusing
usingaatrial-and-error
trial-and-errormethod.
method.In Incontrol
controlparam-
parame-
ter determination,
eter determination, determining
determiningparameters
parametersusing
usingthethetrial-and-error
trial-and-error technique
technique is relatively
is rela-
time time
tively consuming due to
consuming duemultiple failedfailed
to multiple attempts. Ref. [32]
attempts. Ref.proposed an adaptive
[32] proposed neuro-
an adaptive
fuzzy inference methodology to determine parameters that improve
neuro-fuzzy inference methodology to determine parameters that improve the stability the stability of of
the
the system without careful tuning to achieve optimal performance. In the future, we aimto
system without careful tuning to achieve optimal performance. In the future, we aim
toadvance
advancethe
theparameter
parameter determination
determination method
method andand plan
plan to toprove
provethetherobustness
robustnessofofthe the
control algorithm by comparing and analyzing various determined
control algorithm by comparing and analyzing various determined parameter sets for parameter sets for
evaluationscenarios,
evaluation scenarios,
asas
inin [33].
[33]. AsAs depicted
depicted in Figure
in Figure 2, the
2, the inertia
inertia of the
of the rotating
rotating diskdisk
is
is relatively small and friction is low; therefore, micro-tuning of the control parameters
relatively small and friction is low; therefore, micro-tuning of the control parameters is
is considered necessary for velocity tracking. Figure 3 depicts the angular velocity and
considered necessary for velocity tracking. Figure 3 depicts the angular velocity and con-
control error results of the DC motor.
trol error results of the DC motor.
 Actuators
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Actuators 2024, 167 820of 20

(a)
(a)
(a) (b)
(b)
(b)
Figure
Figure
Figure 3.3.3.
Figure Velocity
Velocity
Velocity tracking
tracking
tracking
3. Velocity result:
result:
result:
tracking (a)
(a)
(a)
result: Velocity;
Velocity;
Velocity;
(a) (b)
(b)
(b)
Velocity; Error.
Error.
Error.
(b) Error.

FromFrom
From From
thetheangular
the angular
the angular
angular velocity
velocity
velocity
velocity tracking
tracking
tracking
tracking results
results
results depicted
results depicted
depicted
depicted inFigure
inin Figure
in
Figure 3,3,3,ititit
Figure isisis
3, it evident
is evident
evident
evident that
that
that thethe
that
the
the
errors
errors
errors
errors were
were
were were approximately
approximately
approximately
approximately 318318
318
318 deg/s
deg/sdeg/s
deg/s inthe
inin the
the initial
in initial
the
initial transition
initial transition
transition
transition section.
section.section.
section. After
After
After 222s,s,s,the
After the
2the error
s, error
the
errorerror
decreased
decreased to to
a a value
value closer
closer to to zero
zero but
but but chattering
chattering
chattering occurred
occurred
occurred (RMS
decreased to a value closer to zero but chattering occurred (RMS value of 5.34 deg/s). To To
decreased to a value closer to zero (RMS (RMS value
value value of of 5.34
of
5.34 5.34 deg/s).
deg/s). deg/s).
ToTo
ensure
ensure
ensure
ensure initial
initial
initial convergence
initial convergence
convergence
convergence performance,
performance,
performance,
performance, aaahigh-grade
high-grade
a high-grade
high-grade sigmoid
sigmoid
sigmoid
sigmoid function
function
function
function orororinitial
initial
or initial
initial value
value value
value of of
ofof
RLS
RLS RLS
RLScan can
canbecan be applied;
be applied;
beapplied; however,
however,
applied;however,
however,this this
this
thismaymay
may
maylead lead
lead to
tounreasonable
unreasonable
leadtotounreasonable
unreasonableresults results
resultsas as
as chattering
chattering
aschattering
chatteringthat that
that
thatoc- oc-
occurs
oc-
curs
curs
curs in
ininin
the
thethe
the actual
actual
actual
actual platform
platform
platform
platform also also
also
also increases.
increases.
increases.
increases. FigureFigure
Figure
Figure 444depicts
depicts
4 depicts
depictsthethe thescale
scale
the scale
factor
scale factor
factor
factor results results
results
results calcu- as
calculated
calcu-
calcu-
lated
latedthe
lated as as
as thetheabsolute
absolute
the absolute
of the
absolute of
ofof theerror
error
the
the errorusing
using
error using
using thethethreshold
the threshold
the threshold
values
threshold values(red
(red line)line)
values(red
values(red line)listed
listed
line) listed
in Table
listed inTable
inin Table
2. 2.2.2.
Table

(a)
(a)
(a) (b)
(b)
(b)
Figure
Figure
Figure 4.4.4.
Figure Path
Path
Path tracking
tracking
tracking
4. Path result:
result:
result:
tracking (a)
(a)
(a)
result: Absolute
Absolute
Absolute
(a) value
value
value
Absolute of
ofof
value error
error
error differential;
differential;
differential;
of error (b)
(b)
(b)
differential; Scale
Scale
Scale
(b) factor.
factor.
factor.
Scale factor.

TheThe
The Theerror
error
errorerror graph
graph
graphgraph indicates
indicates
indicates
indicates thechattering
the
the chattering
the chattering
chattering phenomenon.
phenomenon.
phenomenon.
phenomenon. InInFigure
In Figure
In
Figure 4,4,4,this
Figure this isisisillustrated
4, this
this illustrated
is illustrated
illustrated
in
ininmore more
in more
more detail
detaildetail
detail by by indicating
by indicating
byindicating
indicating the the rate
the
raterate
therate of change
of change
ofofchange
change of
ofofthethe
the chattering
of chattering
the chattering
chattering error.
error.error.
error. ThisThis
This This is because
is because
isisbecause
because thethe
the
the
scale
scale
scale scale factor
factor
factor
factor applied
appliedapplied
applied to
totothethe
the covariance
to covariance
the covariance
covariance update
updateupdate
update of
ofofRLSRLS
of RLS
RLS uses
uses
uses uses it. The
it. The
it.it.The
The scale
scale
scalescalefactor
factor
factor factor also
alsoalso
also resulted
resulted
resulted
resulted
inchattering
inin chattering
in chattering
chattering between
between between
between theupper
the
the upper
the
upper upperand
and
and andlower
lower
lowerlowerlimits.
limits.
limits.
limits. This
This
ThisThisresults
results results
results inmore
inin more
in more
more sensitive
sensitive
sensitivesensitive estimates
estimates
estimates
estimates
compared
comparedcompared
compared tothat
toto that
to
that thatofRLS,
ofof RLS,
of
RLS, RLS,which
which
which which applies
applies
applies
applies ananexisting
an existing
an existing
existing covariance
covariance
covariance
covariance ofone.
ofof one.
of
one.one. Figure
Figure
Figure Figure 5a 5adepicts
5a depicts
5a depicts
depicts
thethe the
theerror error dynamics
error
errordynamics dynamics
dynamicscoefficientscoefficients
coefficients
coefficientsthat thatthat
thatare are estimated
are estimated
areestimated
estimatedbased based
based
basedon on
onRLS RLS
on RLS
RLSusing using
using forgetting
usingforgetting forgetting
forgettingfactors. factors.
factors.
factors.
Although
AlthoughAlthough
Althoughthis thisthisis
thisisisan an
is an extremely
extremely
anextremely
extremelysmall small
small
smallvalue,value,
value,
value,the the coefficients
coefficients
thecoefficients apparently
apparently
coefficientsapparently
apparentlychange. change.
change.
change.Constant Constant
ConstantC11 ,
Constant
𝐶
𝐶𝐶1111 , which
which
, ,which
11 is is multiplied
multiplied
whichisismultiplied
multipliedby by by
the
bythe the error,
error,
theerror, has
error,has has
a a significant
significant
hasaasignificant impact
impact
significantimpactimpacton on on
the
onthe the control
control
thecontrol
controlinputinputinput depend-on
depending
inputdepend- depend-
ingingon
ing the
on error;
onthe
the therefore,
theerror;
error;
error; therefore,
therefore,
therefore, it was estimated
itititwas
wasestimated
was estimatedas aas
estimated value
asaaavalue
as smaller
value
value than
smaller
smaller
smaller D1 .𝐷𝐷1𝐷.1. .
than
than
than 1

(a)
(a)
(a) (b)
(b)
(b)
Figure
Figure 5.5.5.
Figure
Figure Velocity
Velocity
Velocity tracking
5. Velocity
tracking
tracking result:
tracking
result:
result: (a)
result:
(a)
(a) Estimated
(a)
Estimated
Estimated coefficient
Estimated
coefficient
coefficient𝐶𝐶
coefficient𝐶11
11 ;C ;(b)
(b)(b)
Estimated
11 ;Estimated
;(b) coefficient
Estimated
Estimated 𝐷𝐷𝐷. 1. .D1 .
coefficient
coefficient
coefficient 11
11
 Actuators2024,
Actuators 2024,13,
13,x xFOR
FOR PEER REVIEW 9 9ofof92020
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The
TheThecoefficient
coefficient estimation
coefficient estimation
estimation residual
residual
residual isisdepicted
depicted
is ininFigure
depicted Figure
in Figure6a,6a,
6a, and
and andthethe
the magnitude
magnitude
magnitude ofofthe
thethe
of
injection
injection term
injection
termtermthat was
thatthat
waswascalculated
calculated
calculated using
using
using the
the theabsolute
absolute
absolute value
valuevalueof the
of the
of the residual
residual
residual is depicted inin in
is depicted
is depicted
Figure
Figure
Figure 6b.6b.
6b. The
The Thecontrol
control inputs
control inputs
inputs using
using self-tuned
using self-tuned
self-tuned injection
injection
injection terms
terms
terms and
and andestimation
estimation
estimation coefficients,
coefficients,
coefficients,
and
and andcurrents,
currents,
currents, areare
are depicted ininFigure
depicted
depicted Figure
in 6c6cand
Figures and
6c and Figure 6d,respectively.
6d, 6d,
Figure respectively.
respectively.

(a)
(a) (b)
(b)

(c)
(c) (d)
(d)
Figure
Figure 6.6.Velocity
Figure Velocity tracking
tracking
6. Velocity result:
result:
tracking (a)Residual;
(a)
result: Residual;
(a) (b)Magnitude
(b)
Residual; Magnitude
(b) ofofinjection
Magnitude injection term;
term;
of injection (c)Voltage;
(c)
term; Voltage; (d)Current.
(d)
(c) Voltage; Current.
(d) Current.

InInthe
the
In thecase
case ofofof
case aaDC
aDC
DC motor system,
system,asaspreviously
motorsystem,
motor previouslymentioned,
previously mentioned,the
mentioned, theinertia
inertiafor
inertia for
for control
control
control and
and friction
andfriction is significantly
frictionisissignificantly
significantlysmall.small. Therefore,
small.Therefore,
Therefore,they they react
theyreact sensitively
reactsensitively to
sensitivelytotosmallsmall control
smallcontrol inputs
controlinputs
inputs and
and
and exhibit
exhibit
exhibit chattering.
chattering.
chattering. To
ToTo control
control
control thethe
the sensitive
sensitive
sensitive system
system
system ininin amore
more
a amore stable
stable
stable manner,
manner,
manner, arela-
rela-
a arelatively
tively
small
tively small
small covariance
covariance
covariance and and andestimated
estimated
estimated initial
initialinitial
values values wereapplied.
were applied.
values were applied. Theforgetting
The forgetting
The forgetting
factor was factor
set in
factor
was set
close in close
proximity proximity
to one to
to one to
remember remember
plenty plenty
of of
previousprevious
data, data,
and
was set in close proximity to one to remember plenty of previous data, and the scale factor theand the
scale scale
factor factor
updates
updates
updates thecovariance
the covariance
the covariance
so that soso that
the
that theresulting
resulting
the resulting
system system does
does not
system does notbecome
become
not become insensitive.
insensitive.
insensitive.

3.2. 3.2.
3.2. Performance
Performance
Performance Evaluation
Evaluation
Evaluation of CarMaker-Based
ofofCarMaker-Based
CarMaker-Based Adaptive
Adaptive
Adaptive Path
Path
Path Tracking
Tracking
Tracking Control
Control
Control
The
The performance
performance of
thethe proposed algorithm was evaluated usinganan IPG/CarMaker-
The performance ofofthe proposed
proposed algorithm
algorithm was
was evaluated
evaluated using
using an IPG/CarMaker-
IPG/CarMaker-
based simulation.
basedsimulation.
simulation.The The trajectory
Thetrajectory
trajectoryofofthe of the target
thetarget S-curved
targetS-curved
S-curvedroadroad is depicted
roadisisdepicted in Figure
depictedininFigure 7. Table
Figure7.7.Table
Table 3
based 33
lists
liststhe the vehicle
thevehicle parameters
vehicleparameters
parametersused used
usedfor for
forthe the simulation-based
thesimulation-based performance
simulation-basedperformance
performanceevaluation.evaluation.
evaluation.TheTheve-The
ve-
lists
velocity
locity was was
set set
to 30tokmph
30 kmphand andCarMaker
the the CarMaker driver
driver model
model was was
usedused
for forpedal
the the pedal control.
control.
locity was set to 30 kmph and the CarMaker driver model was used for the pedal control.

(a)
(a) (b)
(b)
Figure7.7.Evaluation
Figure
Figure Evaluation environment:
7. Evaluation
environment: (a)Vehicle
environment:
(a) Vehicle
(a) used
Vehicle
used ininIPG/CarMaker;
used IPG/CarMaker;
in (b)Vehicle
IPG/CarMaker;
(b) Vehicle
(b) trajectory.
Vehicle trajectory.
trajectory.
Actuators 2024, 13, 167 10
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20

Table 3. Vehicle parameters for proposed controller.

Table 3. Vehicle parameters for proposed controller.
Parameter Unit Value
Parameter Unit Value
Mass (m) Mass (𝑚) kg kg 2108
2108
Wheelbase (l) m 2.97
Wheelbase (𝑙)
Distance between CG * and front/rear axle (l f , lr ) m m 1.47,2.97
1.5
Distance between CG * and
Z-axis rotational inertia (Iz )front/rear axle (𝑙 , 𝑙 )
𝑓 𝑟 kg·m 2 m 1.47,
3170.6 1.5
Z-axisstiffness
Estimated cornering rotational inertia (𝐼𝑧 )
of front/rear kg ∙ m2 3170.6
N/rad 118, 270, 117, 990
Estimated cornering (C f , Cr ) of front/rear wheels (𝐶 , 𝐶 )
wheels stiffness N/rad 118,270, 117,990
𝑓 𝑟
**CG
CGisisthethe
center of gravity.
center of gravity.

3.2.1.
3.2.1. Front-Wheel
Front-WheelSteering
SteeringVehicle-Based
Vehicle-BasedAdaptive
AdaptivePath
PathTracking
TrackingControl
Control
As
As depicted in Figure 8, the control input derived for path tracking was
depicted in Figure 8, the control input derived for path tracking was applied
applied
equally
equally to the front wheels of the CarMaker’s vehicle model to evaluate the versatilityof
to the front wheels of the CarMaker’s vehicle model to evaluate the versatility of
the
theproposed
proposedalgorithm
algorithmforforfront-wheel
front-wheelsteering
steeringvehicles.
vehicles.

Figure8.8.Concept
Figure Conceptdiagram
diagramof
offront-wheel
front-wheelsteering
steeringvehicle
vehicleconfiguration.
configuration.

Theintegrated
The integratederror
error(e(𝑒
int ) wasdesigned
) was
𝑖𝑛𝑡 designed using
using the
the lateral
lateral preview
preview error
error andandyawyaw an-
angle
error, as shown
gle error, in Equation
as shown in Equation (25). The
(25). error
The dynamics
error dynamicsareare
defined
definedininEquation
Equation(26).
(26).The
The
design
designvariables
variablesofofthe
theadaptive
adaptivesteering
steeringcontrol
controlalgorithm
algorithmarearelisted
listedininTable
Table4.4.

eint= =
𝑒𝑖𝑛𝑡 y+
𝑒𝑦e+ 𝑒𝜓eψ (25)
(25)
.
e = C eint + D1 + δ f (26)
(26)
𝑒̇𝑖𝑛𝑡int= 𝐶1111
𝑒𝑖𝑛𝑡 + 𝐷1 + 𝛿𝑓

Table4.4.Design
Table Designparameters
parametersofoffront-wheel
front-wheelsteering
steeringvehicle
vehicleadaptive
adaptivesteering
steeringcontrol.
control.

Parameter
Parameter Unit Unit Value
Value
Decay rate of the Lyapunov
Decay rate of the Lyapunov functionfunction
(α) (𝛼) - - 1 1
Reachability
Reachability factor
factor (η) (𝜂) - - 0.010.01
Initial value
Initial valueof estimated
of estimatedstates
states D̂̂1 ) , 𝐷
(Ĉ11 , (𝐶 ̂ - 0, 0,0 0
11 1 ) -
Initial value of covariance (P11 , P12 ) - 0.001, 0.001
Initial value of covariance (𝑃11 , 𝑃12 ) - 0.001, 0.001
Forgetting factor (λ11 , λ12 ) - 0.9999, 0.9999
Lower Forgetting factor (𝜆11
scale factor threshold ( e,low,th
. 𝜆 )
12


- - 0.9999,
0.03 0.9999
Lower scale factor threshold . (𝑒̇𝑙𝑜𝑤,𝑡ℎ ) - 0.10.03

Upper scale factor threshold ( eup,th -
Normalized threshold (smax ) (𝑒̇𝑢𝑝,𝑡ℎ )
Upper scale factor threshold - - 0.1
1.008
Normalized threshold (𝑠𝑚𝑎𝑥 ) - 1.008

Compared to the previous DC motor platform, the vehicle had a large system inertia;
Compared to the previous DC motor platform, the vehicle had a large system inertia;
therefore, the design parameters were set to relatively large values. The designed path is
therefore, the design parameters were set to relatively large values. The designed path is
an S-curve that includes straight and curved roads, as depicted in Figure 9.
an S-curve that includes straight and curved roads, as depicted in Figure 9.
Actuators 2024, 13, x FOR PEER REVIEW 11 of 20

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(a) (b)
Figure 9. Path tracking result: (a) Trajectory; (b) Longitudinal velocity.
(a) (b)
In Figure 10,(a)the vehicle oscillation is visible (b) for approximately 5 s in the section
Figure 9. Path
Figure tracking
9. Path trackingresult: (a) Trajectory;
result: (a) Trajectory; (b) (b)
Longitudinal
Longitudinal velocity.
velocity.
where
Figure 9.a Path
straight
trackingroad and
result: (a) aTrajectory;
curved (b) road change; velocity.
Longitudinal however, the magnitude is small. At
approximately
In Figure
In Figure 33
10, s,
10,the
the the robot
vehicle
vehicle appears to react
oscillation
oscillation is to
is visible theforchanging
visible for road surface
approximately
approximately 5 s 5insinthe
inreal
the time
section
section
and In Figure
gradually 10, the stability
finds vehicle oscillation
as it adapts.is road
visible
Figure for11approximately
depicts the 5 s in
lateral the sectionand
preview yaw
where a straight road and a curved road change; however, the magnitude is small. At At
where a straight road and a curved change; however, the magnitude is small.
where
angle a straight
errors.
approximately From road the
33 and thea curved
s,lateral preview
robot road errors,
appears change; ithowever,
to react is evident
to the the magnitude
that
changingsymmetry
roadis small.
appears
surface Atinfor
realthetime
approximately 33 s, the robot appears to react to the changing road surface in real time
approximately
curved paths. 33
The s, the robot
scale-factor appears to
calculated react
usingto the
the changing
absolute road
valuessurface
of thein real
error time
differential
and and gradually finds stability as it adapts. Figure 11 depicts the lateral preview and yaw
and gradually
gradually findsfindsstability
stability as as it adapts.
it adapts. FigureFigure 11 depicts
11 depicts the lateral
the lateral previewpreview
and yawand yaw
is depicted
angleangle in Figure
errors.
errors. From From 12.
the theThe scale
lateral
lateral factor
preview
preview is calculated
errors,
errors, in
it isitevident real
is evident time
thatthat with the
symmetry
symmetry absolute value
appears for the
angle errors. From the lateral preview errors, it is evident that symmetry appearsappears
for the for the
of the
curved error
curvedpaths.within
paths.
The Thethe threshold
scale-factor
scale-factor range
calculated of
calculated 0.03
using and
usingthe 0.1
the (red lines
absolute
absolute
curved paths. The scale-factor calculated using the absolute values of the error differential
in Figure
values
values of of
the 12).
the
error error differential
differential
is is depicted
is depicted
depicted in in
inFigure Figure
Figure12.12. The12.
The The
scalescalescale
factor
factor factor is calculated
is calculated
is calculated in realintimein real
realwith time
timethewith with the absolute
the absolute
absolute value value value
of of the
of the error error
errorwithin within
withinthe the the threshold
threshold
threshold range
range range of
of 0.03
of 0.03 0.03
and 0.1 and
and(red 0.1
0.1lines (red
(redinlineslines in Figure
in Figure
Figure 12). 12).12).

(a) (b)
(a) (b)
(a) (b)

(c)
(c) (d) (d)
Figure
Figure 10. Path
Figure Path tracking
10. tracking
Path result:
result:
tracking (a) (a)
Yaw
result:Yaw
(a) rate;
rate; (b) Lateral
(b) Lateral
Yaw rate; velocity;
velocity;
(b) (c) Longitudinal
(c) Longitudinal
Lateral velocity; acceleration;
(c)acceleration;
Longitudinal (d)
(d) acceleration;
acceleration. (c)
Lateral acceleration.
Lateral (d)
(d) Lateral acceleration.
Figure 10. Path tracking result: (a) Yaw rate; (b) Lateral velocity; (c) Longitudinal acceleration; (d)
Lateral acceleration.

(a) (b)
(a) (b)
Figure 11. Cont.
(a) (b)
Actuators
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(c)
(c)
(c) (d)
(d)
(d)
Figure
Figure
Figure 11.
Figure
11. Path
11.
11. Path
Path tracking
Path
tracking
tracking result:
tracking
result:
result: (a)
result: Lateral
(a) Lateral
(a) Lateral preview
(a) Lateral
preview
preview error;
preview
error;
error; (b)
error; Yaw
(b) Yaw
(b) Yaw angle
(b) Yaw
angle
angle error;
angle
error;
error; (c)
error; Integrated
(c) Integrated
(c) Integrated error;
(c) Integrated error;
error;
error;
(d)
(d) Integrated
(d)
Integrated error
Integrated differential.
error
error differential.
differential.
(d) Integrated error differential.

(a)
(a)
(a) (b)
(b)
(b)
Figure
Figure
Figure 12.
12. Path
12. Path
Path tracking
tracking
tracking result:
result:
result: (a) Absolute
(a) Absolute
(a) Absolute value
value
value of error
of error
of error differential;
differential;
differential; (b) Scale
(b) Scale
(b) Scale factor.
factor.
factor.
Figure 12. Path tracking result: (a) Absolute value of error differential; (b) Scale factor.

Based
Based
BasedBased on
on the
on the
on threshold
thethe
threshold
threshold
threshold set
set listed
set set
listed in
listed
listed in Table
in Table
Table 4,
in Table
4, the
4, the
the scale
scale
4, the factor
factor
scale
scale factor of the
of the
factor
of the value
value
of the
value smaller
smaller
value smaller
smaller than
than
thanthan
the
the lower
lower
the limit
limit
lower of
of
limit 0.03
0.03
of is
is
0.03 1
1 and
and
is 1 larger
larger
and than
than
larger the
thanthe upper
upper
the upper limit
limit of
of
limit 0.1
0.1
of is
is
0.1
the lower limit of 0.03 is 1 and larger than the upper limit of 0.1 is 1.008. Figure 13a depicts1.008.
1.008.
is Figure
Figure
1.008. 13a
13a
Figure depicts
depicts
13a depicts
the
the coefficients
thethe
coefficients
coefficients
coefficients estimated
estimated
estimated
estimated using
using
usingusingRLS.
RLS.
RLS. The
TheThe
RLS.
The RLS
RLS
RLS estimation
estimation
RLS estimation
estimation coefficient
coefficient
coefficient
coefficient is
is updated
is updated
updated in
is updated curved
in curved
in curved
in curved
driving
driving
driving
driving and
and
and exhibits
and exhibits
exhibits
exhibits convergence
convergence
convergence
convergence results
results
results
results in
in straight
inin straight
straight
straight driving.
driving.
driving. Figure
driving.Figure
Figure 13b
Figure13b
13b depicts
13bdepicts the
depictsthe
depicts resid-
theresid-
the residual
resid-
ual
ualat
ual at
at this
atthis
this time.
thistime. This
This
time.This
time. indicates
indicates
Thisindicates
indicatesthat that residuals
that residuals occur
residuals occur during
occur during curved
during curved driving.
curveddriving.
driving.TheThe magnitudes
Themagnitudes
The magnitudes of
magnitudes
of the
of the
of the injection
theinjection
injection terms
injectionterms
termsare
terms are depicted
aredepicted
are depicted in in Figure
in Figure 13c,d
Figure 13c,d
13c,d as as control
as control inputs.
inputs.
control inputs.

(a)
(a)
(a) (b)
(b)
(b)

(c)
(c)
(c) (d)
(d)
(d)
Figure
Figure
Figure 13.
13. Path
13.Path
Path tracking
tracking
tracking result:
result:
result: (a) Estimated
(a)Estimated
(a) Estimated coefficient;
coefficient;
coefficient; (b) Residual;
(b)Residual;
(b) Residual; (c) Injection;
(c)Injection;
(c) Injection; (d) Control
(d)Control
(d) Control input.
input.
input.
Figure 13. Path tracking result: (a) Estimated coefficient; (b) Residual; (c) Injection; (d) Control input.
Actuators 2024, 13, x FOR PEER REVIEW 13 of 20
Actuators 2024, 13, 167 13 of 20

As mentioned in Equation (22), in the above performance evaluation, the sigmoid

functionAswasmentioned
applied toin the
Equation
injection(22), in the
term for above performance
chattering evaluation,
attenuation. the sigmoid
In this study, the
functionphenomenon
chattering was applied to ofthe injection
control inputterm
wasfor chattering
analyzed attenuation.
using In thisvalues
two gradient study,of the chat-
the
tering function
sigmoid phenomenonusedoffor
control input was
the injection analyzed
term, using
and the two gradient
evaluation values
results are of
as the sigmoid
follows.
function
Figure used the
14 shows for derived
the injection term,and
trajectory and longitudinal
the evaluation resultsresults
velocity are as of
follows.
tracking Figure
per- 14
shows the
formance derived trajectory
by applying and longitudinal
the two gradient values (0.3velocity
and 1)results of tracking
of the sigmoid performance
function for the by
applying
injection the two gradient values (0.3 and 1) of the sigmoid function for the injection term.
term.

(a) (b)
Figure 14. 14.
Figure PathPath
tracking result:
tracking (a) Trajectory;
result: (b) (b)
(a) Trajectory; Longitudinal velocity.
Longitudinal velocity.

In Figure
In Figure15,15,
there is no
there significant
is no difference
significant difference in the longitudinal
in the longitudinalvelocity, butbut
velocity, as the
as the
slope
slope of the
of the sigmoid
sigmoid function
function increases,
increases, it itisisobserved
observedthatthatthe
theoscillation
oscillationofofyaw
yaw rate,
rate, lat-
lateral
eralvelocity,
velocity,andandlateral
lateralacceleration
acceleration also
alsoincreases
increases generally.
generally. InInparticular, onon
particular, thethe
straight
straightroad
after
road afterthe
thecurved
curvedroad,
road,itit can
can be seen that
be seen thatrelatively
relativelylarge
largeoscillations
oscillations occur
occur in in
bothboth
thethe
negative
negative andand positive
positive directions,
directions, with
with thetheerror error centered
centered atIn
at 0. 0. Figure
In Figure
16, 16,
thethe magnitude
magnitude
of the
of the integrated
integrated error
error waswas reduced,
reduced, butbutit isitdifficult
is difficult to analyze
to analyze thethe control
control stability
stability onlyonly
using control error due to the oscillation of the
using control error due to the oscillation of the steering angle. steering angle.

(a) (b)

(c) (d)
Figure 15. Path
Figure tracking
15. Path result:
tracking (a) Yaw
result: (a) rate;
Yaw (b) Lateral
rate; velocity;
(b) Lateral (c) Longitudinal
velocity; acceleration;
(c) Longitudinal (d)
acceleration;
Lateral acceleration.
(d) Lateral acceleration.
 Actuators
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14 of 20

(a) (a) (b) (b)

(c) (c) (d) (d)

Figure
Figure
Figure 16. Path
16. 16.
PathPath tracking
tracking result:
result:
tracking (a) Lateral
Lateral
(a) Lateral
result: (a) preview
preview error;
error;
preview (b) Yaw
(b) (b)
error; Yaw Yaw angle
angle error;
error;
angle (c) Integrated
Integrated
(c) Integrated
error; (c) error;
error;
error;
(d)Control
(d) (d) Control
Control input.
input.
input.

Through
Through
Through analysis
analysis
analysis of the
of the
of the
aboveabove
above provided
provided
provided results,
results, ititwas
it was
results, was confirmed
confirmed
confirmedthat that an increase
an an
that increase
increase in
in in
the
thethe gradient
gradient
gradient of the
of the
of the sigmoid
sigmoid
sigmoid function
function
function can
cancan cause
cause an increase
increase
an increase
cause an of the
of the
of chattering phenomenon.
chattering phenomenon.
BasedBased
Based onthe
on on
the the results
results
results analysis,
analysis,
analysis, itit was
it was was found
found
found that
that
that the
thethe using
using
using the
thethe sigmoid
sigmoid
sigmoid function
function
function can
cancan play
a aa
play
play
role
rolerole for
forfor chattering
chattering
chattering attenuation.
attenuation.
attenuation.

3.2.2.
3.2.2.
3.2.2. Front-and-Rear-Wheel
Front-and-Rear-Wheel
Front-and-Rear-Wheel Steering
Steering
Steering Vehicle-Based
Vehicle-Based
Vehicle-Based Adaptive
Adaptive
Adaptive Path
PathPath Tracking
Tracking
Tracking Control
Control
Control
As
As As depicted
depicted
depicted in Figure
in Figure
in Figure 17,
17, 17, the
the the front-and-rear-wheel
front-and-rear-wheel
front-and-rear-wheel control
control
control inputs
inputs
inputs derivedforfor
derived
derived for path
path
path
tracking
tracking
tracking were
were
were applied
applied
applied equally
equally
equally to the
to the
to the left
leftleftand
andand right wheels,
right
right respectively.
wheels,
wheels, respectively.
respectively. Because
Because
Because the vehicle
the the vehi-
vehi-
cleis
cledriven
is driven at
at aaat
is driven relatively
relatively low
a relatively speed,
lowlow thethe
speed,
speed, the front-and-rear-wheel
front-and-rear-wheel
front-and-rear-wheel steering inputs
steering
steering areare
inputs
inputs are derived in
derived
derived
opposite
in phases.
opposite
in opposite phases. phases.

Figure
Figure
Figure 17. Concept
17. 17. Concept
Concept diagram
diagram offront-and-rear-wheel
front-and-rear-wheel
of front-and-rear-wheel
diagram of steering
steering vehicle
vehicle
steering configuration.
configuration.
vehicle configuration.

In InIn this
this this section,
section,
section, thethe the error
error
error dynamics
dynamics
dynamics of the
of
of the the front
front
front and
andand rear
rear
rear wheels
wheels
wheels are
areare designed
designed
designed using
using
using
thethe
the lateral
lateral
lateral preview
preview
preview errorerror
errorandand
and
yawyaw
yaw angle
angle error,
error,
error, respectively,
respectively,
respectively, as
as expressed
expressed
as expressed in
in Equation
Equation
in Equation (27).
(27).
TheThe
The control
control
control input 𝑢1ufor
input
input 𝑒1 ewas
𝑢1 1for
for 1𝑒1was
was definedasasthe
defined
defined the
as thefront-wheel
front-wheel
front-wheel steering
steering
steering angle
angleangle 𝛿𝑓 and
and
𝛿δ𝑓f and thethe
the control
con-
con-
input
trol u
input for
𝑢
trol input 𝑢2 2 for e was
for 𝑒
2 2𝑒2 was defined
was as
defined the
as rear-wheel
the rear-wheel steering angle
steering
2 defined as the rear-wheel steering angle 𝛿𝑟 , and angle
δr , and
𝛿 , the
and adaptive
the steering
adaptive
𝑟 the adaptive steer- steer-
ingcontrol
ing
control input
controlinput based
input
based on
based onEquation
on Equation
Equation (23)
(23)was
(23) derived
waswas asasshown
derived
derived ininEquation
as shown
shown in Equation
Equation (28).
(28).(28).
.
e=
𝑒̇𝑦𝐶
𝑒̇𝑦 = y = 𝐶𝑦11 y+
C+𝑒11𝑦𝐶e+ C+
𝐶𝜓12 ψ+
𝑒12𝜓𝛿e𝑓+ 𝛿𝑓δ f
11 𝑒 12 𝑒
.
eψ = C21 ey + C22 eψ + δr (27)(27)
(27)
𝑒̇𝜓𝐶=
𝑒̇𝜓 = 𝐶𝑦21+𝑒𝑦𝐶+
21 𝑒 𝐶𝜓22+
22 𝑒 𝑒𝜓𝛿+ 𝑟 𝛿𝑟
Actuators 2024, 13, x FOR PEER REVIEW 15 of 20

Actuators 2024, 13, 167 15 of 20

2
𝛼 𝑚𝑒1
𝛿𝑓 = − ∑ 𝐶̂1𝑗 𝑒𝑗 − ((|𝑅1 | + 𝜂 + ) )
√2𝑤 1
1 + 𝑚|𝑒1 | 
𝑗=12  
α me1
δ f = − ∑ Ĉ1j e j − | R1 | + η + √ (28)
2 2w 1 1 + m | e1 |
j =1 𝛼 𝑚𝑒2
𝛿𝑟 = − ∑2 𝐶̂2𝑗 𝑒𝑗 − ((|𝑅
 2 | + 𝜂 + ) ) (28)
√2𝑤α 1 + me2 |
𝑚|𝑒
δr = −𝑗=1∑ Ĉ2j e j − | R2 | + η + √2w 1 + m|e2 |
2

j =1 2
As listed in Table 5, the same values were applied to the decay rate of the Lyapunov
functionAs
andlisted in Table factor
reachability 5, the same values
to derive thewere applied to the decay
front-and-rear-wheel rate angles.
steering of the Lyapunov
function and reachability factor to derive the front-and-rear-wheel steering angles.
Table 5. Design parameters for front-wheel steering vehicle adaptive steering control.
Table 5. Design parameters for front-wheel steering vehicle adaptive steering control.
Parameter Unit Value
Decay rate of theParameter
Lyapunov function (𝛼) Unit
- Value
1.2
Reachability
Decay rate factor function
of the Lyapunov (𝜂) (α) -- 0.01
1.2
Initial value Reachability
of estimated factor (η) (𝐶̂11 , 𝐷
states ̂1 ) -- (0, 0.01
0)

Initial value of estimated states
Initial value of estimated states (𝐶11 ( Ĉ , D̂̂12) )
̂21 , 𝐷 -- (0,0, 0) 0

Initial value of estimated states (Ĉ21 , D̂2 ) - 0, 0
Initial value of covariance (𝑃11 , 𝑃12 ) (0.001, 0.001)

Initial value of covariance (P11 , P12 ) 0.001, 0.001

Initial value
Initial valueofofcovariance
covariance (P 21, ,P𝑃
(𝑃21 22
22 )) (0.00001,
0.00001, 0.00001)
0.00001

Forgetting factor (𝜆
Forgettingfactor ( 11
λ 11 , λ𝜆12)
12 ) -- (0.9999,
0.9999, 0.9999)
0.9999

Forgettingfactor
factor (𝜆 21 ,, λ
(λ21 𝜆22 ) -- 0.99999, 0.99999)
(0.99999, 0.99999
Forgetting . )
22

Lower scale factor threshold ( elow,th - 0.03
Lower scale factor threshold (𝑒̇. 𝑙𝑜𝑤,𝑡ℎ ) - 0.03
Upper scale factor threshold ( eup,th - 0.1
Upper scale factor threshold (𝑒̇𝑢𝑝,𝑡ℎ ) - 0.1
Normalized threshold (smax ) - 1.008
Normalized threshold (𝑠𝑚𝑎𝑥 ) - 1.008

TheThe parameters
parameters were
were determined
determined using
using a trial-and-error
a trial-and-error method.
method. Compared
Compared to the
to the
previously
previously mentioned
mentioned DCDC motor
motor platform,the
platform, thevehicle
vehiclehas
hasaalarge
large system
system inertia; therefore,
there-
thethe
fore, forgetting factor
forgetting andand
factor initial value
initial were
value set to
were setrelatively largelarge
to relatively values. Figure
values. 18 depicts
Figure 18
the results
depicts of driving
the results alongalong
of driving the target path path
the target without lane departure.
without lane departure.

(a) (b)
Figure 18. Path
Figure tracking
18. Path result:
tracking (a) Trajectory;
result: (b) (b)
(a) Trajectory; Longitudinal velocity.
Longitudinal velocity.

In Figure 19, 19,

In Figure thethe
results of the
results of thefront-and-rear-wheel
front-and-rear-wheel steering
steeringcases indicate
cases thatthat
indicate thethe
lateral velocity
lateral is lower
velocity than
is lower thatthat
than of the front-wheel
of the front-wheel steering
steeringcase. Figure
case. 20 depicts
Figure 20 depicts thethe
lateral preview
lateral previewandand yaw angle
yaw errors
angle of the
errors of thetarget path.
target However,
path. However, active steering
active steeringoccurs
occurs
because
because of the
of the rear-wheel
rear-wheel steering
steering intervention,
intervention, andandthethe signofofthe
sign thelateral
lateralpreview
previewerrorerror is
reversed compared to the front-wheel
is reversed compared to the front-wheel steering case. steering case.
Figure
Figure 21 depicts
21 depicts thethe scale-factor
scale-factor results
results calculated
calculated overover a threshold
a threshold range
range (red(red line)
line)
set set using
using thethe absolute
absolute value
value of the
of the error
error difference.
difference. ThisThis assists
assists in updating
in updating thethe covariance
covariance
so that
so that estimates
estimates cancan respond
respond andand converge
converge quickly
quickly in curved
in curved driving
driving sections.
sections. Figure
Figure 22a22a
depicts
depicts thethe coefficients
coefficients estimated
estimated using
using RLSRLS in real
in real time.
time. TheThe estimated
estimated coefficients
coefficients for for
thethe
rear wheels were smaller than those for the front wheels. Figure
rear wheels were smaller than those for the front wheels. Figure 22b depicts the residual 22b depicts the residual
estimated
estimated at this
at this time.time.
TheThe magnitude
magnitude of the
of the injection
injection term term calculated
calculated using
using thethe residual
residual
is depicted in Figure 22c, and the control input result calculated using this is depicted in
Figure 22d. This shows that the rear-wheel steering angle is in the opposite phase and is
small compared to the front-wheel steering angle.
Actuators
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20

is depicted in Figure 22c, and the control input result calculated using this is depicted in
is
is depicted
depicted in
in Figure
Figure 22c,
22c, and
and the
the control
control input
input result
result calculated
calculated using
using this
this is
is depicted
depicted in
in
Figure 22d. This shows that the rear-wheel steering angle is in the opposite phase and is
Actuators 2024, 13, 167 Figure
Figure 22d.
22d. This
This shows
shows that
that the
the rear-wheel
rear-wheel steering
steering angle
angle is
is in
in the
the opposite
opposite phase
phase and
and16is
isof 20
small compared to the front-wheel steering angle.
small
small compared
compared to to the
the front-wheel
front-wheel steering
steering angle.
angle.

(a) (b)
(a)
(a) (b)
(b)

(c) (d)
(c)
(c) (d)
(d)
Figure 19. Path tracking result: (a) Yaw rate; (b) Lateral velocity; (c) Longitudinal acceleration; (d)
Figure
Figure 19.
19. Path
Figure 19. tracking
PathPath
tracking result:
result:
tracking (a)
(a) Yaw
result: (a) rate;
Yaw Yaw (b)
rate; (b) Lateral
Lateral
rate; velocity;
velocity;
(b) Lateral (c)
(c) Longitudinal
Longitudinal
velocity; acceleration;
acceleration;
(c) Longitudinal (d)
(d)
acceleration;
Lateral acceleration.
Lateral
Lateral acceleration.
acceleration.
(d) Lateral acceleration.

(a) (b)
(a)
(a) (b)
(b)
Figure 20. Path tracking result: (a) Lateral preview error; (b) Yaw angle error.
Figure
Figure 20.
20. Path
Figure Path
20. tracking
tracking
Path result:
result:
tracking (a)
(a) Lateral
result: Lateral preview
preview
(a) Lateral error;
error;
preview (b)
(b) Yaw
error; Yaw
(b) angle
angle
Yaw error.
error.
angle error.

(a) (b)
(a)
(a) (b)
(b)
Figure 21. Path
Figure tracking
21. Path result:
tracking (a) Absolute
result: value
(a) Absolute of error
value differential;
of error (b) Scale
differential; factor.
(b) Scale factor.
Figure
Figure 21.
21. Path
Path tracking
tracking result:
result: (a)
(a) Absolute
Absolute value
value of
of error
error differential;
differential; (b)
(b) Scale
Scale factor.
factor.
Actuators 2024,
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2024, x FOR
167 PEER REVIEW 17 of 20
17 of 20

(a) (b)

(c) (d)
Figure 22. 22.
Figure Path tracking
Path result:
tracking (a) Estimated
result: coefficient;
(a) Estimated (b) (b)
coefficient; Residual; (c) Injection;
Residual; (d) (d)
(c) Injection; Control input.
Control input.

Proof . First, the cost function designed is as follows:

Proof. First, the cost function designed is as follows:
𝑤𝑖
𝐽𝑠𝑖 = 𝑒𝑖2
2w
Jsi = i ei2
𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 1 ∶ 𝐽si̇ ≤ .−𝛼𝐽𝑠𝑖21/2 , 𝛼 > 0
Condition 1 : J si ≤ −αJsi 1/2 , α > 0
𝐶𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛 2 ∶ 𝑙𝑖𝑚 𝐽𝑠𝑖 = ∞
Condition |𝑒𝑖 2 : lim Jsi = ∞
|→∞
|ei |→∞
For stable control, the derivative of the cost function with respect to time must be
negative.
ForBystable
differentiation
control, the of derivative
the cost function of the costwithfunction
respect to time
with and substitution
respect to time mustof be
thenegative.
control input term with the
By differentiation designed
of the controlwith
cost function input, the following
respect to time and equations can of
substitution bethe
derived.
control input term with the designed control input, the following equations can be derived.
. .
n 𝑛 o 
n
𝐽𝑠𝑖̇ J si==𝑤w𝑖 𝑒i𝑖e𝑒̇i𝑖ei==𝑤w
𝑖 𝑒i𝑖ei({∑ ∑ j =1
𝑗=1
𝐶C
𝑖𝑗 𝑒
ij e
𝑗 }
j ++𝑢u𝑖 +
i +𝐷D𝑖 )i

 
n
ui = − ∑ 𝑛 Ĉ e − D̂ − ρi sign(ei )
j=1̂ ij j ̂ i
𝑢 = (− ∑ 𝐶𝑖𝑗 𝑒𝑗 − 𝐷𝑖 − 𝜌𝑖 𝑠𝑖𝑔𝑛(𝑒𝑖 ) )
. n 𝑖 o𝑗=1 n o 
n n
J si = wi ei ∑ j=1 Cij e j − ∑ j=1 Ĉij e j + Di − D̂i − ρi sign(ei )
𝑛 𝑛
𝐽𝑠𝑖̇ = 𝑤𝑖 𝑒= n 𝑒 } − {∑
𝐶̂ 𝑒 } + 𝐷 − 𝐷 ̂ − 𝜌(𝑖e𝑠𝑖𝑔𝑛(𝑒

wi ei ∑𝐶j=
𝑖 ({∑ 𝑖𝑗 𝑗Cij − Ĉij e j +
1
𝑖𝑗 D𝑗i − D̂i𝑖 + ρi 𝑖sign i ) 𝑖 ))
𝑗=1 𝑗=1

Here, the difference between the estimated and the measured value is converted into
𝑛
a residual, and the=magnitude
𝑤𝑖 𝑒𝑖 (∑ of(𝐶 the ̂
defined injection̂ term is substituted.
𝑖𝑗 − 𝐶𝑖𝑗 ) 𝑒𝑗 + 𝐷𝑖 − 𝐷𝑖 + 𝜌𝑖 𝑠𝑖𝑔𝑛(𝑒𝑖 ))
𝑗=1
   
. αi
Here, the differenceJbetween
si = w e
i i the
R i − | i|
estimated
R + and
η i + √
the measured
sign ( value
e i ) is converted into
2wi
a residual, and the magnitude of the defined injection term is substituted.
From the derived cost function, control stability 𝛼𝑖 is analyzed according to the signs of
𝐽𝑠𝑖̇ =
the residuals and errors (five 𝑖 (𝑅𝑖 − (|𝑅𝑖 | + 𝜂𝑖 +
𝑤𝑖 𝑒cases). ) 𝑠𝑖𝑔𝑛(𝑒𝑖 ))
√2𝑤𝑖
From the derived cost A1, Ri ≥ 0control
function, > 0, Ri <
and ei stability ρi (designed
is analyzed )
according to the signs of
.
the residuals and errors (five Jcases).
= wi ei ( Ri − ρi ) < 0, ∵ sign(e) = 1
si
A1, 𝑅𝑖 ≥ 0 𝑎𝑛𝑑 𝑒𝑖 > 0, 𝑅𝑖 < 𝜌𝑖 (𝑑𝑒𝑠𝑖𝑔𝑛𝑒𝑑)
Actuators 2024, 13, 167 18 of 20

A2, Ri ≤ 0 and ei > 0, Ri < ρi (designed)

.
J si = wi ei ( Ri − ρi ) < 0, ∵ sign(e) = 1
A3, Ri ≥ 0 and ei < 0, Ri < ρi (designed)
.
J si = wi ei ( Ri + ρi ) < 0, ∵ sign(e) = −1
A4, Ri ≤ 0 and ei < 0, Ri < ρi (designed)
.
J si = wi ei ( Ri + ρi ) < 0, ∵ sign(e) = −1
A5, ei = 0
.
J si = 0
It has been proven that the cost function is less than or equal to zero in all possible
residual and error signs. □

4. Discussion
The performance of the proposed universal-purpose adaptive control algorithm for
the MIMO system was evaluated using a DC motor, front-wheel steering vehicle, and front-
and-rear-wheel steering vehicle. These scenarios were set to simulate an environment that
can improve work efficiency and engineer convenience through universal applications on
various platforms with the advancement in technology. No information from the system is
required for target-value tracking. For RLS-based coefficient estimation, only a few control
parameters, such as initial values and forgetting factors, are required. The evaluation results
confirm that a reasonable target-value tracking performance of the adaptive control input
is finally derived through the RLS-based real-time estimated coefficients. The DC motor
system has a significantly small inertia and friction to control; therefore, it reacts sensitively
to the control input and chattering occurs. As such, the RLS-based coefficient estimation
is sensitive to the initial parameter settings. This was performed using a trial-and-error
method and a micro-tuning methodology. Furthermore, an oscillation phenomenon was
confirmed in the straight driving section during the last curved drive. The oscillation
worsened in the rear-wheel steering scenarios. Oscillation, which is revealed as a vehicle
condition quantity, has a relatively small value; however, it can affect target-value tracking
when driving at high speeds. The goal is to solve this problem through a more stable
control in the future, and an evaluation of the actual mobility.

5. Conclusions
In this study, we proposed a universal-purpose adaptive control algorithm using
a sliding-mode approach and parameter self-tuning. We defined several error dynam-
ics based on the control errors. The control error of each system can be integrated and
controlled by applying weights. To design a sliding-mode approach without system in-
formation, we estimated the coefficients for the RLS-based error dynamics. The control
error and estimation coefficients were then used to derive the magnitude of the injection
term and adaptive control input. The purpose of this study was to easily apply the same
controller to various platforms and achieve target-value tracking performance with the
advancement in technology. Adaptive control input derivation does not require any in-
formation from the system. This is achieved by estimating the coefficients of the error
dynamics using RLS in real time. In addition, by applying a self-tuning injection term
according to the magnitude of the control error, the control error can be quickly reduced
to zero while complying with the finite-time convergence condition. The evaluation was
conducted using actual DC motor platforms, CarMaker-based front-wheel steering vehicles,
and front-and-rear-wheel steering vehicles. The evaluation results demonstrated that the
proposed control algorithm tracks the target value reasonably without any information
from the system. This is expected to improve the engineers’ efficiency and convenience
Actuators 2024, 13, 167 19 of 20

by universally applying the same controller to various systems. However, chattering was
observed in the control input and system states. Chattering affected the residuals used
in control input calculations, and thus led to chattering of the control inputs. Our goal
was to develop algorithms that minimize chattering by deriving stable control inputs. In
addition, our goal was to advance the control algorithm by applying an integral term to
rapidly converge to relatively large errors during the initial stage. Moreover, we intend to
expand the application of the universal controller to environmental configurations, such as
high-speed driving and evaluation using actual mobility platforms.

Author Contributions: Conceptualization, K.O.; methodology, H.L. and K.O.; software, H.L.; valida-
tion, H.L.; formal analysis, H.L. and K.O.; investigation, H.L.; resources, H.L.; data curation, H.L.;
writing—original draft preparation, H.L.; writing—review and editing, H.L. and K.O.; visualization,
H.L.; supervision, K.O.; project administration, K.O.; funding acquisition, K.O. All authors have read
and agreed to the published version of the manuscript.
Funding: This research was funded by the National Research Foundation of Korea
(NRF-2022R1F1A1075167) and government funding (Ministry of Science and ICT) in 2022.
Data Availability Statement: Data sharing is not applicable to this article.
Conflicts of Interest: The authors declare no conflict of interest.

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