TELKOMNIKA Telecommunication Computing Electronics and Control Vol.
No.
April 2026, pp.
ISSN: 1693-6930.
DOI: 10.
12928/TELKOMNIKA.
Performance assessment of an adaptive model predictive control with torque braking for lane changes Zulkarnain1.
Irwin Bizzy1.
Armin Sofijan2.
Mohd Hatta Mohammed Ariff3 Department of Mechanical Engineering.
Faculty of Engineering.
Universitas Sriwijaya.
Palembang.
Indonesia Department of Electrical Engineering.
Faculty of Engineering.
Universitas Sriwijaya.
Palembang.
Indonesia Advanced Vehicle System.
Malaysia Japan International Institute of Technology.
Universiti Teknologi Malaysia.
Kuala Lumpur.
Malaysia
Article Info
ABSTRACT
Article history:
The growing demand for autonomous vehicles requires robust control systems that can maintain safety during complex maneuvers like lane However, a significant research gap exists in developing controllers that effectively manage the combined challenges of steering and braking across diverse and unpredictable driving conditions, such as varying speeds and low-friction road surfaces.
This research addresses this gap by proposing and evaluating an adaptive model predictive control (MPC) system integrated with a torque braking distribution strategy.
The key advantage of our adaptive method is its ability to continuously update its internal model in real-time, allowing it to anticipate and respond to changing road friction and vehicle dynamics more effectively than a static controller.
In simulations of a lane change maneuver across speeds of 10-25 m/s and road friction levels 3 .
to 1 .
ry asphal.
, the proposed system demonstrated a substantial performance improvement.
The proposed framework demonstrated a 52.
8% average reduction in lateral tracking error and enhanced stability by reducing the yaw rate by up to 41.
8% on low-friction surfaces, compared to a non-adaptive MPC baseline.
These results quantitatively confirm that our frameworkAos synergistic coordination of steering and braking significantly enhances the safety, precision, and reliability of autonomous lane change maneuvers.
Received May 3, 2025 Revised Dec 30, 2025 Accepted Jan 30, 2026 Keywords:
Adaptive model predictive Autonomous vehicle Braking torque Lane change Steering This is an open access article under the CC BY-SA license.
Corresponding Author:
Irwin Bizzy Department of Mechanical Engineering.
Faculty of Engineering.
Universitas Sriwijaya Jalan Raya Palembang Prabumulih Km 32.
Ogan Ilir.
Palembang.
Indonesia Email: irwin@ft.
INTRODUCTION
The study of autonomous vehicles has developed into a quickly expanding topic of study because to major industry and academic effort.
Particularly in the last ten years, interest in this topic has significantly increased .
Autonomous vehicles have the ability to lessen the role that driver error and irresponsibility play in collisions between vehicles.
A self-learning method for autonomous vehicles includes environment recognition, real-time localisation, course planning, and motion tracking.
The path tracking control, which concentrates on the vehicle control in the lateral and longitudinal directions in order to follow a specified path or trajectory, is one of the key components of any autonomous vehicle .
Numerous renowned automakers have been performing research and advancing technology to produce smart autonomous vehicles .
Kang et al.
, autonomous vehicle systems must be the requirements that they can know where they are, understand the goals they must achieve, and protect themselves from experiencing mishaps while Journal homepage: http://journal.
id/index.
php/TELKOMNIKA TELKOMNIKA Telecommun Comput El Control Given the aforementioned requirements, it follows that the entire autonomous system should be composed of several modules and tasks to carry out, as shown in Figure 1 .
, .
Figure 1.
Schematic diagram of autonomous vehicle building components .
One of the control systems that is still being developed is path following.
This control systemAos goal is to reduce the vehicleAos lateral and heading errors relative to the predetermined course.
The steering correction input is sent to the vehicle by the control system in order to accomplish this goal.
The vehicle has three different types of actuators: the steering, throttle, and braking systems.
The steering controlAos primary goal is to create a control strategy that can accurately estimate the path to the intended target.
This control system also keeps the vehicleAos directional stability while following the path .
The inclusion of model uncertainty and outside disturbances makes path following control for autonomous vehicles more Variations in the vehicle/environment parameters and the vehicle state may cause model The simulation stage is where control method is often implemented, with the controlled plant being modelled to produce simulated responses.
The majority of these papers deal with non-linear or linear control techniques with linearized vehicle models.
When using vehicle linear models, some vehicle-related effects are disregarded.
Consequently, path tracking control may not always give outcomes that are satisfactory .
, .
Theoretically, an autonomous vehicle ought to be able to recover from significant disturbances without assistance from a person.
In a geometric/kinematic form of controller called the geometric controller is used for those purposes.
The configuration and deployment of geometric/kinematic controllers is But according to earlier studies, this kind of controllers may require adequate tuning to operate effectively, and a tuned controller might only operate with a specific kind of trajectory and speed range .
, .
Additionally, the majority of geometric/kinematic controllers, including the Stanley controller, are incapable of handling sharp corners.
Therefore, a suggestion for improvement is made to incorporate a dynamic vehicle that will modify the controllerAos settings in accordance with road circumstances .
, .
Several references contain a common way to develop the control law.
Researchers .
, .
detailed the application of the linear quadratic regulator (LQR) controller in route tracking control with dynamic feedback.
However, the controller struggled because there was no path feedback.
A feed forward term of steering angle to zero steady state error was added to the original technique.
Since the look-ahead distance is still absent and the controller can only be reactive, the outcome was still worse.
The ideal preview Performance assessment of an adaptive model predictive control with torque braking for A (Zulkarnai.
A ISSN: 1693-6930 method suggested by .
, .
in at least three articles can be used to solve this issue.
Adaptive controllers are developed for use in systems that require strong robustness against specific disturbance and uncertainty The development and evaluation of adaptive trajectory tracking control have been documented in reference .
A reference signal with linear velocity and angle is used by this controller.
The reference signal is complemented with an adaptive parameter update law to prevent drift parameters.
Although this concept is being implemented on a mobile robot that rides a unicycle, it can be adjusted to work on a vehicle that has single point control.
Adaptive rule was added to the back-stepping kinematic/dynamic theory that employed .
, .
It aims to assist a mobile robot with unidentified features.
In this study, a dynamic controller along with a torque controller were proposed.
There are several notable adaptive controllers available, each of which can address a distinct component of the operating environment for a vehicle.
An adaptive controller investigation on slippery road conditions was done .
Another adaptive controller used in the vehicle for unidentified skidding is also covered in .
To stabilize yaw on the front and rear wheels, research is conducted.
An adaptive law that adjusts cornering stiffness by tracking the dynamic wheel sideslip angle eliminates the effect of wheel-road sliding on slick roads.
Position tracking control system for an autonomous vehicle is the main goal of this study.
METHOD
A non-holonomic system can be categorized as an autonomous vehicle.
The term Auholonomic systemAy describes the relationship between the total number of four degrees of freedom (DOF), for the dynamic system and the controllable DOF.
The controllable DOF of a non-holonomic system was less than the total DOF.
According to .
, .
, a typical vehicle is regarded as non-holonomic because it has a total of four DOF, including motion in the two Cartesian coordinates of direction and heading, but only two DOF that can be controlled: the longitudinal .
orward and backwar.
and lateral directions .
ounded steering The majority of studies on path tracking controllers use kinematic vehicle modelling as shown in Figure 2.
Due of its simplicity and crucial relevance to explain the motion of the vehicle, this model is commonly utilized.
The position and orientation of the vehicle with regard to its local axis and the global axes, are described using this model.
For this, a straightforward kinematic model is frequently employed, as shown in Figure 3.
Local .
and global (X-Y) axes are used to define the vehicle, yc represents the vehicleAos longitudinal velocity, is its heading in relation to the local coordinates, and yue is its orientation in relation to the global coordinates .
ycaycuyc yue
c ] = [ ycycnycu yue
ycycu = yc ycaycuyc yuE
ycyc = yc ycycnycu yuE
Oe ycycnycu yue ycycu
][ ]
ycaycuyc yue ycyc The dynamic vehicle model takes into account external and internal forces, energy or momentum within the system to describe the motion of the vehicle in terms of its position, velocity, and acceleration as shown Figure 3.
This model takes into account the mass of the vehicle body as well as tire forces .
Figure 2.
Full vehicle in kinematic model Figure 3.
7DOF Autonomous vehicle model TELKOMNIKA Telecommun Comput El Control.
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April 2026: 696-706 TELKOMNIKA Telecommun Comput El Control Oc ycAycycnyc [Oeyaycycyc Oe yaycycyco ]ycoyc yaycuyceyco ycycnycu yu yaycyceyco ycaycuyc yu ]yco yaycuyceyc ycycnycu yu yaycyceyc ycaycuyc yu yce = yayc,yaya yueO yaycuyceyc ycaycuyc yu Oe yaycuyceyco ycaycuyc yu ( yaycyceyco ycycnycu yu Oe yaycyceyc ycycnycu yu Oe yaycuycyco yaycuycyc .
yaycuycyc yaycuycyco yaycuyceyco ycaycuyc yu yaycyceyco ycycnycu yu yaycuyceyc ycaycuyc yu yaycyceyc ycycnycu yu = ycoyca ycaycu yaycycyc yaycycyco Oe yaycuyceyco ycycnycu yu Oe yaycuyceyc ycycnycu yu yaycyceyco ycaycuyc yu yaycyceyc ycaycuyc yu = ycoyca ycayc Researchers have introduced and employed a variety of tire models.
However.
Pacejka tire model has also been widely employed in addition to linier estimation of tire response.
This non-linear, semiempirical tire model can replicate the forces produced by the tires.
Function P may represent yaycu, yayc, or ycAyc in .
, where yaA, ya, ya, ycIEa, and ycIyc stand for the functionAos stiffness factor, shape factor, peak value, curvature factor, horizontal shift, and vertical shift, respectively.
Using the Calspan tire formula, researchers can delve into a wealth of additional insights and findings related to tire performance and behavior .
, .
This model uses a polynomial formulation of a saturation function, yce(), to estimate the normalized lateral and longitudinal forces, yaycu and yayc , respectively.
The tyre vertical force, yayc , the coefficient of friction for the road surface, the lateral stiffness coefficient, yayc, the modified longitudinal stiffness coefficient, yaAoyca, the constant for the tyre camber angle, ycU, and the tire camber angle, must all be known in order to use this model.
The normalized lateral and longitudinal forces can be calculated using .
ayc , yu, yu.
= ya ycycnycu.
a ycaycycaycycaycu.
aAyu.
) ycIyc yuNyayc yuNyayc yayc ycycaycu yu Ooyayc 2 ycycaycu2 yu yaycaAyc 2 ycUyu yu yaycaA yc Ooyayc 2 ycycaycu2 yu yaycaAyc 2 Controller development The location monitoring system of an autonomous vehicle will be given a new control structure.
This calls for the use of two basic subsystem control strategies: steering control to track lateral motion and speed control to track longitudinal motion.
Without the use of a path planning module, the vehicle is directed to follow a pre-defined trajectory by the steering control system.
Position tracking control will focus on lateral and longitudinal motion control, with little or no attention paid to vertical vehicle motions.
As a result, this thesis wonAot discuss suspension control.
Figure 4 outlines the core elements for autonomous vehicle control, focusing on steering and braking torque.
Reference inputs from higher-level controllers include desired lateral position and yaw angle.
The state estimator uses sensor data to estimate lateral position and yaw angle.
The adaptive model predictive control (MPC) system employs adaptation laws to adjust control parameters and dynamic models in response to changing conditions.
Computed braking torque and steering angle are then sent to the vehicleAos control A feedback loop compares actual state with reference inputs, enabling real-time adjustments.
Figure 4.
The application of an adaptive (MPC) system in an autonomous car A multi-step prediction control law is adopted by the adaptive MPC controller, which corrects faults with feedback while repeatedly optimizing.
The adaptive MPC controller optimizes stability and performance while having positive effects on controls, high robustness, and better balancing of control objectives and handling restrictions of the system.
These benefits are crucial factors to take into account while creating automatic trajectory tracking controllers for use with autonomous vehicles.
Trajectory tracking is used to Performance assessment of an adaptive model predictive control with torque braking for A (Zulkarnai.
A ISSN: 1693-6930 drive the car toward the middle of the road by controlling the steering wheel .
State-space modelling is used in the adaptive MPC controllerAos vehicle plant model.
This state-space model can be a closed-loop property-analysis and has various advantages for handling multivariable systems.
In .
illustrates how a time-invariant linear equation (LTI) can be used to characterize the system to be controlled in this scenario as a discrete-time state-space model.
= yaycc ycu.
yaAycc yc.
yc( yc.
= yaycc ycu( yc.
= ycuE ya.
Oe ycuE ) yaA.
Oe ycE ) yc( yc.
= ycI ya( ycu( yc.
Oe ycuE ) .
The state-space model on the compact form is represented by .
Where yc.
, ycu.
, and yc.
are, respectively, the control input, state variable, and prediction output of step yco.
The yaw, yaw rate, and global lateral position arrange the vehicle state in .
The wheel steering angle and yaw moment serve as the control input.
The .
is expressing the approximate dynamics of a system using a first-order Taylor series expansion around the nominal state and control input.
ItAos a linearization that allows for local analysis of the systemAos behavior.
yuiyce yuiyce ycuN = yce.
cu0 , yc0 ) [| | ] .
cu Oe ycu0 ) [| | ] .
c Oe yc0 )} ayuiycu ycu0,yc0 ayuiyc ycu0 ,yc0 ya.
The steering and yaw moment inputs to the vehicle model will be calculated using the adaptive MPC that has been composed and constructed.
The goal of this control input is to steer the vehicle along the predetermined course.
The amount of lateral position and heading error is what determines how accurately the vehicle follows the defined trajectory.
The adaptive MPC controller makes use of a cost function to do The cost function is a quadratic program (QP) with minimized optimal control input and the desired error functions .
, .
The adaptive MPC must satisfy the objective and constraints of .
Its design is challenging due to the vehicleAos non-holonomic nature.
The tracking accuracy is governed by the cost functionAos first term, while its second and third terms regulate control effort aggressiveness.
The controller must then compute the optimal steering and yaw moment that jointly minimize this cost function while adhering to all defined A recursive least squares (RLS) estimator updates cornering stiffness parameters online, directly linking the Adaptive MPC to the vehicleAos dynamic state.
The core adaptation mechanism is realized through an RLS algorithm, which recursively updates the estimation of critical parameters the tire-road friction coefficient .
uN) at each timestep .
via the update .
= y.
Oe IycN .
Oe .
= P.
Oe.
( IycN .
Oe.
C .
= C .
Oe .
= .
/)[P.
Oe .
Oe K.
IycN .
Oe .
] Key parameters are now explicitly provided, including a prediction horizon of 20 steps, control horizon of 5, and empirically tuned weighting matrices .
cE, ycI and ycI) optimized for path-tracking, with a 50 ms sampling time.
The road friction coefficient .
uN) is intrinsically embedded in the calculation of the nonlinear tire forces, which form the foundation of .
Specifically, in .
, the self-aligning moment .
cAy.
is a direct function of the lateral tire forces .
ya ya.
, yc.
, iyc.
) = ycy Ocycn=1 Anyc.
co yc.
Oe yc.
co yc.
An2ycE yca Ocya ycn=1(Aniyc.
co ycn Oe .
AnycI Anyc.
co ycn Oe 1AnycI ) yuycoycnycu O yu O yuycoycaycu ycAyc,ycoycnycu O ycAyc O ycAyc,ycoycaycu yc = .
cyc yue yueN ycU] yc = .
u ycAyc ] TELKOMNIKA Telecommun Comput El Control.
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April 2026: 696-706 TELKOMNIKA Telecommun Comput El Control Longitudinal tyre forces Braking torque generates longitudinal tire forces, enabling vehicle control.
The required torque is derived from the yaw moment needed to maintain stability, a common stability control method.
These forces originate at the tire-road contact patch and must obey the friction circle law to prevent traction loss.
Standard formulas can calculate lateral and longitudinal forces, and algorithms can optimize torque distribution per tire for trajectory tracking.
The braking module stabilizes the vehicle during sharp turns while preventing excessive lateral acceleration.
According to the algorithm described in .
, the braking torque utilized to determine the longitudinal tire forces is estimated.
The braking torques of a tire were calculated using .
ycNycayceyco = ycNycayceyc = ycNycaycyco = ycNycaycyc = .
cAyc .
cIyc yco ycoyce 2 yco 2 ycycnycu( ycycaycuOe1 ( yc Oeyuyce )Ooycoyce yc cAyc .
cIyc yco 2 yco 2 ycycnycu( ycycaycuOe1 ( yc Oeyuyce )Ooycoyce yc cAyc .
cIyc yco 2 yco 2 ycycnycu( ycycaycuOe1 ( yc Oeyuyce )Ooycoyce yc cAyc .
cIyc yco 2 yco 2 ycycnycu( ycycaycuOe1 ( yc Oeyuyce )Ooycoyce yc ycAyc > 0, yuyce Oe yuyc O 0 ycAyc < 0, yuyce Oe yuyc O 0 ycAyc > 0, yuyce Oe yuyc > 0 ycAyc < 0, yuyce Oe yuyc > 0 Trajectory model This specific coordinate point denotes a precise geographical location, either in terms of latitude and longitude or vice versa, to accurately delineate the path of the road.
As the vehicle undergoes the maneuver, its real-time position on the global axis is compared with this coordinate, enabling the computation of lateral and heading errors.
To evaluate the controllers, this study employs four distinct road trajectory models, as depicted in Figure 5.
These trajectory is designated based on the specific characteristics and contours of the roads, with some conforming to the lane change trajectories outlined in the research papers by .
, .
The selection of this specific trajectory is highly intentional.
It is not a random path but a condensed representation of a common yet complex driving task that, when performed at the limits of vehicle dynamics, becomes a primary safety-critical scenario.
Testing this allows engineers to ensure that a vehicle can handle both everyday driving and emergency situations effectively.
Figure 5.
Lane change trajectory for controller testing RESULT AND DISCUSSIONS A high-fidelity vehicle dynamics model was implemented in the simulation to accurately capture the nonlinear behavior of the autonomous vehicle.
The core of this model relies on the critical physical parameters detailed in Table 1, which define key properties such as mass distribution, inertia, and tire The comparative performance of the proposed adaptive MPC integrated with torque braking distribution versus a baseline solely adaptive MPC is presented in Figure 6.
The evaluation uses a double-lanechange maneuver at a speed of 15 m/s on a surface with a friction coefficient .
uN) of 0.
Figures 6.
illustrate the trajectory tracking and lateral error, respectively.
The results confirm that the integrated controller delivers superior tracking accuracy and stability.
A key demonstration of this advantage occurs at yc = 5.
seconds, where the torque braking distribution proactively corrects the vehicleAos path, as seen in the trajectory (Figure 6.
) and the corresponding dip in lateral error (Figure 6.
This corrective action is a direct application of direct yaw moment control, a well-established stabilization method for critical maneuvers Performance assessment of an adaptive model predictive control with torque braking for A (Zulkarnai.
A ISSN: 1693-6930 by .
The improved performance stems from the coordinated reduction of lateral error through synergistic steering and braking, a principle supported by recent studies on integrated chassis control .
The braking torque responses for the front-left (TbFL) and front-right (TbFR) wheels are shown in Figures 6.
The front-left wheel applies higher corrective torque, peaking at approximately 6 Nm, while the front-right wheel shows moderated peaks between 3-4.
5 Nm.
This differential torque generation creates the necessary yaw moment for stabilization.
The magnitude and timing of this intervention align with findings from hardwarein-the-loop (HIL) tests on direct yaw control (DYC) systems, which report similar torque differentials for effective lateral error correction during lane changes .
Furthermore, the 52.
8% average reduction in lateral error achieved by our method represents a significant improvement over the performance benchmarks reported for non-integrated adaptive MPC controllers in similar scenarios .
While simulation results are promising, the real-world efficacy of such torque-vectoring strategies has been validated in experimental autonomous vehicles, confirming their critical role in maintaining stability under low-friction conditions .
Table 1 The vehicle parameters were used in simulation Symbol yco ycoyce ycoyc yayc yayce yayc yayui ycIyui ycoyc yci Parameter Vehicle mass Front axle to CoG Rear axle to CoG Mass inertia Front tire cornering stiffness Front tire cornering stiffness Wheel moment of inertia Wheel radius Vehicle track width Gravitational acceleration Value Unit kg/m2 N/rad N/rad m/s2 .
Figure 6.
Simulation results: .
path tracking at a speed of 15 m/s in the double lane change (BLC) maneuver trajectory with friction surface 0.
6, .
lateral error using adaptive MPC integrated with torque braking distribution .
ontroller A) compared to MPC .
ontroller B), .
rear left braking torque, .
rear right braking torque, .
front left braking torque, and .
front right braking torque Yaw and lateral error with varying speed and surface friction In order to evaluate the control performance under various road conditions and vehicle speeds, a root mean square error (RMSE) analysis was conducted for both yaw angle and lateral position.
The results, as illustrated in Figure 7, compare the RMSE values at different velocities for three distinct road friction TELKOMNIKA Telecommun Comput El Control.
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April 2026: 696-706 TELKOMNIKA Telecommun Comput El Control coefficients .
uN =1, 0.
6, and 0.
This comparison provides a clear understanding of how tireAeroad interaction affects the accuracy and stability of vehicle motion control at increasing velocities.
The Figure 7.
shows RMSE of yaw angle .
and the Figure 7.
shows RMSE of lateral position for three road friction coefficients .
uN = 1, 0.
6, 0.
For low-to-moderate speeds .
Ae20 m/.
RMSE values remain small for all yuN, with the best performance at yuN = 1 and progressively worse errors as yuN Above 22.
5 m/s the errors rise sharplyAiyaw RMSE increases markedly and lateral-position RMSE exhibits a dramatic jumpAiespecially for yuN = 0.
6 and yuN = 0.
3, indicating loss of tracking accuracy at high speed and low friction.
This trend suggests tyre force saturation and stronger nonlinear tyre dynamics .
nd likely model mismatc.
at high speeds and low yuN, which the current controller cannot fully compensate.
Practically, the results imply the control design requires either gain scheduling or adaptive/robust control, improved tire/vehicle modeling .
r friction estimatio.
, and possibly higher control bandwidth or actuator authority to maintain stability and tracking at high speeds and low-friction conditions.
RMSE Yaw .
Miu = 1 Miu = 0.
Miu = 0.
10 m/s 5 m/s 15 m/s 5 m/s 20 m/s 5 m/s 25 m/s Velocity .
Figure 7.
RMSE of lateral error and yaw rate across a lane change scenario at various velocities: .
friction yuN = 0.
6 and .
friction yuN = 0.
CONCLUSION
This study proposed and validated a novel integrated control framework, combining adaptive MPC with a proactive torque braking distribution strategy, to enhance autonomous vehicle safety during lane change maneuvers.
The primary scientific contribution is the synergistic, real-time coordination of steering and braking within a unified predictive optimization loop.
Quantitative simulation results across diverse speeds .
-25 m/.
and friction levels .
uN = 0.
3 to 1.
confirm the frameworkAos efficacy.
The adaptive MPC demonstrated a 52.
8% average reduction in lateral tracking error and improved stability by reducing yaw rate by up to 41.
8% on low-friction surfaces, compared to a non-adaptive baseline.
These improvements are attributed to the controllerAos ability to update critical parameters in real-time and proactively generate stabilizing yaw moments.
However, performance limitations were observed at very high speeds (>22.
5 m/.
on low-friction roads.
The primary limitation of this work is its reliance on simulation-based validation.
Future work must therefore address uncertainties from model simplifications and unmodeled hardware effects through experimental validation using HIL systems and prototype vehicle testing.
Performance assessment of an adaptive model predictive control with torque braking for A (Zulkarnai.
A ISSN: 1693-6930
FUNDING INFORMATION
Authors state no funding involved.
AUTHOR CONTRIBUTIONS STATEMENT
This journal uses the Contributor Roles Taxonomy (CRediT) to recognize individual author contributions, reduce authorship disputes, and facilitate collaboration.
Name of Author Zulkarnain Irwin Bizzy Armin Sofijan Mohd Hatta Mohammed Ariff C : Conceptualization M : Methodology So : Software Va : Validation Fo : Formal analysis ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue I : Investigation R : Resources D : Data Curation O : Writing - Original Draft E : Writing - Review & Editing ue ue ue ue ue ue ue ue ue ue ue ue ue ue ue Vi : Visualization Su : Supervision P : Project administration Fu : Funding acquisition CONFLICT OF INTEREST STATEMENT Authors state no conflict of interest.
DATA AVAILABILITY
Data availability is not applicable to this paper as no new data were created or analyzed in this REFERENCES