Friction-Adaptive Integrated Position Control for Vehicles on Curved Paths

Subject Areas : robotics

Hadi Sazgar ¹ , ali keymasi khalaji ²

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2 -

Received: 2024-01-06 Accepted : 2024-08-04 Published : 2024-10-20

Keywords: Integrated Longitudinal and Lateral Control, Kinetic Control, Kinematic Control, Nonlinear Tire, Seven Degrees of Freedom Dynamic Model, Tire-Road Friction Estimation,

Abstract :

In critical manoeuvres where the maximum tire-road friction capacity is used, the vehicle's dynamic behaviour is highly nonlinear, and there are strong couplings between longitudinal and lateral dynamics. If the tire-road friction conditions change suddenly during these manoeuvres, the vehicle control will be very complicated. The innovation of this research is a control algorithm to manage vehicles on a curved path with sudden tire-road friction change. The main advantage of the proposed controller is that it is robust to the change of the friction coefficient and other unmodeled uncertainties and ensures vehicle stability with low computational volume. The evaluation of the proposed adaptive controller has been done using the full vehicle model in CarSim software and by defining three different manoeuvres, moving at a constant speed on a curved road, lane-change, and lane-change with braking. Also, in the obtained results, the noise of the yaw speed signals and longitudinal and lateral accelerations are considered. The estimation of the longitudinal and lateral velocities is also done using these data. The obtained results showed that the proposed integrated control can manage the highly nonlinear dynamics of the vehicle in the existence of a sudden and significant change in the friction coefficient.

References:

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Full-Text:



Int.  J.   Advanced Design and Manufacturing Technology, 2024, Vol. 17, No. 3, pp. 49-64

DOI:  10.30486/ADMT.2024.873778                                   ISSN: 2252-0406 https://admt.isfahan.iau.ir

 

Friction-Adaptive Integrated Position Control for Vehicles on Curved Paths

Hadi Sazgar *

Department of Mechanical Engineering,

Iranian Research Organization for Science and Technology (IROST), Iran

E-mail: [e-mail]

*Corresponding author

Ali Keymasi-Khalaji

Department of Mechanical Engineering, Faculty of Engineering,

Kharazmi University, Iran

E-mail: [e-mail] 

Received: 6 January 2024, Revised: 12 June 2024, Accepted: 4 August 2024

Abstract: In critical manoeuvres where the maximum tire-road friction capacity is used, the vehicle's dynamic behaviour is highly nonlinear, and there are strong couplings between longitudinal and lateral dynamics. If the tire-road friction conditions change suddenly during these manoeuvres, the vehicle control will be very complicated. The innovation of this research is a control algorithm to manage vehicles on a curved path with sudden tire-road friction change. The main advantage of the proposed controller is that it is robust to the change of the friction coefficient and other unmodeled uncertainties and ensures vehicle stability with low computational volume. The evaluation of the proposed adaptive controller has been done using the full vehicle model in CarSim software and by defining three different manoeuvres, moving at a constant speed on a curved road, lane-change, and lane-change with braking. Also, in the obtained results, the noise of the yaw speed signals and longitudinal and lateral accelerations are considered. The estimation of the longitudinal and lateral velocities is also done using these data. The obtained results showed that the proposed integrated control can manage the highly nonlinear dynamics of the vehicle in the existence of a sudden and significant change in the friction coefficient. 

Keywords: Integrated Longitudinal and Lateral Control, Kinetic Control, Kinematic Control, Nonlinear Tire, Seven Degrees of Freedom Dynamic Model, Tire-Road Friction Estimation,

Biographical notes: Hadi Sazgar earned his BSc degree from Shahid Bahonar University of Kerman (SBUK), Kerman, Iran, in 2007, followed by his MSc and PhD in Mechanical Engineering from K. N. Toosi University of Technology (KNTU), Tehran, in 2009 and 2019, respectively. He is an assistant professor in the Mechanical Engineering Department at the Iranian Research Organization for Science and Technology (IROST) in Tehran. His research interests focus on the modeling and control of mechanical systems, vehicle dynamic control, and advanced driver assistance systems (ADAS). Ali Keymasi-Khalaji received his BSc from Iran University of Science and Technology (IUST), Tehran, Iran, in 2007, and his MSc and PhD in Mechanical Engineering from K. N. Toosi University of Technology (KNTU), Tehran, in 2009 and 2014 respectively. He has been an associate professor with the Mechanical Engineering Department at Kharazmi University (KHU) in Tehran since 2015. His research interests include modeling and control of mechanical systems, nonlinear control, and adaptive and robust control with applications to mobile robotic systems and mechatronics.

 

1        Introduction

Autonomous driving started in the 1980s [1-2] and has developed and progressed significantly over the past few decades [3]. Automatic driving can save time and reduce air pollution by reducing traffic. In addition, self-driving vehicles can significantly increase the comfort and safety. Based on accident statistics and the proportion of accidents caused by human error, there is a growing need for self-driving cars. The World Health Organization states that more than 1.35 million people die, and 20 to 50 million people are injured due to annual accidents worldwide. Economic studies also show that the costs caused by accidents are more than 3% of the gross domestic product of the countries [4]. A study by the National Highway Traffic Safety Administration (NHTSA) in 2015 shows that human error was effective in about 94 percent of all car accidents [5].

The self-driving car has different parts, one of the most important of which is the control algorithm. The control section must perform the path tracking with the desired accuracy and also ensure vehicle stability. In a critical maneuver in which the longitudinal and lateral movements are performed simultaneously, the tire is in its saturated range. During the manoeuvre, the tire-road friction conditions also change. Due to uncertainties and very strong couplings between longitudinal and lateral dynamics in several levels of dynamics, kinematics, and tire forces, the problem of path following and maintaining the stability of the vehicle in a critical manoeuvre will be very challenging and complex.

Literature review: So far, various control approaches have been proposed to solve this problem. One of the important manoeuvres in which longitudinal and lateral movements are performed at the same time is the lane change maneuverer. In [6-7], a review of the methods presented in the field of lane change control has been done. Depending on the desired manoeuvre, different controllers can be used. For simple manoeuvres where tire slip is negligible, a kinematic model can be used, and a controller can be designed for that [8-10]. Assuming that the slip of the tires is small, the relationship between force and slip is linear and it will be logical to use the linear dynamic model. In some references, the control algorithm has been studied based on a linear dynamic model [11-18]. One of the very important manoeuvres that play a role in reducing accidents is the collision avoidance manoeuvre. Numerous researchers have studied the problem of longitudinal and lateral integrated control with an emphasis on collision avoidance [19-23]. Model predictive control has been used many times in various areas of vehicle control due to its high capability in managing multi-objective constrained systems with uncertainty [24-26]. In some cases, predictive control has been used along with other methods. In [27], the direct Lyapunov approach has been used for longitudinal velocity control and nonlinear predictive control (NMPC) for lateral control. Despite having many capabilities, predictive control also brings challenges. By growing the model order, nonlinear terms, and constraints, the computational cost increases, and the possibility of getting stuck in the local minimum while doing the optimization problem increases.

In [28], a comprehensive approach that combines planning and control mechanisms to enhance the quality of trajectories produced by intelligent vehicles is introduced. The trajectory planning component is engineered using the principles of the Iterative Linear Quadratic Regulator (ILQR), which incorporates the vehicle's nonlinear dynamics to optimize trajectory planning. In [29], a combined H∞ control approach is designed to enhance both the path-following capabilities and the lateral stability of autonomous in-wheel-motor-driven electric vehicles (AIEVs) is presented.

Motivation and innovation: The prior surveys show that valuable research has been conducted in this field. However, the problem of integrated longitudinal and lateral control of the vehicle in critical manoeuvres still needs more investigation. The primary objective of this work is to develop an adaptive control algorithm capable of providing integrated longitudinal and lateral control for highly nonlinear vehicle dynamics, while being robust to sudden changes in the friction coefficient and other uncertainties. A key goal is also to enable the control system to effectively follow curved paths, in addition to managing linear manoeuvres, by separating the control architecture. These objectives are motivated by the need to enhance vehicle safety and performance in challenging driving conditions where the available tire-road friction can vary unexpectedly. In [30-31], the motion control and integrated longitudinal and lateral control in the critical lane change manoeuvre on the highways have been done. To manage the variations of tire and road friction conditions, an adaptive control algorithm is proposed in [32]. This control algorithm uses the sliding mode approach, and by considering the nonlinear tire dynamics, it updates the tire forces in the control law by changing the friction conditions of the tire and the road. Despite having many advantages such as low computational cost, stability, and high tracking accuracy, this controller also has a fundamental limitation.

This control algorithm is only used for lane change manoeuvres on the highways, where the lateral position variation is small compared to the longitudinal position. This article proposes a new control algorithm for critical manoeuvres to overcome this limitation. This algorithm can be used for the integrated longitudinal and lateral dynamics control of the vehicle on all roads (straight and curved roads). The accuracy of trajectory tracking of this algorithm is very high, and it also guarantees vehicle stability. This controller also can adapt to variations in the tire-road friction conditions. This algorithm is robust to unmodeled uncertainties and parameter changes, and it will be helpful for manoeuvres where the vehicle dynamics are highly nonlinear and the tire capacity is in the saturated range. The details related to the design of the integrated controller, as well as the estimation algorithm of tire forces and friction coefficient, will be presented in sections 3 and 4, respectively.

2        Vehicle dynamic model

In high-speed manoeuvres, it is possible to analyse the real behaviour of the vehicle only with a nonlinear dynamic model with many degrees of freedom. On the other hand, due to unmodeled uncertainties and the unknown value of the parameters, it is practically impossible to design a model-based controller based on complex dynamic models with high degrees of freedom. Considering these considerations, in this research, 7 degrees of freedom model, including three degrees for movements in the yaw plane (,  and ) and four degrees of freedom for four wheels, are used. The overview of this model is shown in “Fig. 1”. This model considers longitudinal and lateral load transfer caused by longitudinal and lateral accelerations. The simulations performed on a complete vehicle model show that the design of the controller based on the 7-degree-of-freedom model ( was practical and appropriate (Section 5).

 

 

Fig. 1        The 7 degrees of freedom model.

 

In “Fig. 1”,  are the inertial coordinate axes, and x-y are the local coordinate axes connected to the centre of mass (CG). Subscripts f and r refer to the front axle and rear axle of the vehicle. Also, the pairs (), (), (), and ()refer to the left front, right front, right rear, and left rear tires, respectively.  and  represent the longitudinal and lateral forces of the tire, respectively. The symbol  also indicates the rotation around the wheel spinning axis. The longitudinal and lateral speeds at the CG are measured in local coordinates and are introduced by variables ,  and , respectively. , , and  stand for the front wheel steering angle, vehicle yaw angle, and vehicle side-slip angle, respectively. The description and values of dynamic model parameters are presented in “Table 1”. The values of this table are extracted from D-Class Sedan vehicle in the CarSim software.

In the following, the details of other parts of the dynamic model are stated.

 

Table 1 Vehicle dynamic model parameters [30]

2.1. Vehicle Motion Equations [33]

 

Description	Unit	Value	Parameter
The total mass of the vehicle		1530
Vehicle yaw moment of inertia		2315
Front axle CG distance		1.11
Rear-axle CG distance		1.67
Track width		1.55
Height of CG		0.52
Aerodynamic drag coefficient	---	0.3
Braking gain		700
Brake actuator time constant	---	0.06
Brake delay	msec	31
Driveline efficiency	---	0.85
Driveline gain	---	4.1
Effective wheel radius		0.325
Wheel's moment of inertia		0.9
Rolling resistance coefficient	---	0.015

 

Where,  and  are the rotational velocity and acceleration of the vehicle, respectively.  also represents the aerodynamic force applied to the front surface of the vehicle defined by “Eq. (4)”, [33]:

 

(1)
(2)
(3)

(4)

 

Where,  is the air density and  is the wind speed.  also represents the effective front surface of the vehicle and is equal to , [33].

2.2. Tire Model

For the behaviour of the tire model to be close to the real behaviour, complex and accurate models should be used [33-35]. Accurate models have many parameters that must be identified. Identifying these parameters will be a major challenge considering the variations in tire and road conditions. Even if these parameters are identified, due to the accumulation of dynamic modelling errors and identification errors, the resulting tire and road friction model may significantly differ from the actual friction behaviour. In addition, the selected model for controlling the base model should be as simple as possible. Fortunately, if robust control approaches are used, it can be ensured that the unmodeled uncertainties of the tire will be well covered. Considering these considerations, based on the friction circle, a version of Pacejka's tire model, which has been used for control applications in this field [36-37], will be used [38]. Of course, for integrated control with friction estimation, this tire model will be used with modifications that will be explained in section 4.

 

(5)

 

In this Equation,  represents the vertical load of each tire and  stands for the longitudinal or lateral tire-road friction coefficient.  also represents the total tire slip, which is a function of the longitudinal slip () and lateral slip () of the tire () [38]. By considering the effect of load transfer caused by longitudinal and lateral accelerations using D'Alembert's principle, the vertical force of tires can be approximated with the following relationship. It must be recognized that the effect of vehicle suspension has been ignored.

 

 

2.2.1. Longitudinal Slip Ratio [38]

 

(6)
(7)
(8)
(9)

(10)

 

In the above Equations,  is the longitudinal velocity of the tire contact point with the road surface and  is the longitudinal velocity equivalent to the rotation of the wheel [34].

2.2.2. Tire Slip Angle [33]

 

2.3. Wheel Dynamics

The wheel motion Equation establishes the relationship between the longitudinal force and the traction and braking torques [33].

 

(11)
(12)

 

Where,  is the angular acceleration of the wheel,  is the traction torque,  is the braking torque applied to the wheel.It is assumed that the torque applied to the left and right wheels of the front axle are equal to each other. This assumption is also valid for the wheels of the rear axle. It must be recognized that the traction torque is applied only to the front wheels. It is also assumed that in the braking mode, the distribution ratio of the rear-to-front braking torques is equal to γ. The parameter γ is determined using the pressure distribution valve of the hydraulic system [33]. Therefore, the torques applied to the wheels can be defined as “Eq. (14) and Eq. (15)”.

(13)

 

It should be noted that  and  represent the sum of traction and braking torques applied to the wheels, respectively.

2.4. Power Train Model

It can be presumed that the torque converter is completely locked in high-speed maneuvers. So, the relationship between the engine torque () and the total traction torque () can be expressed as “Eq. (16)”, [33].

 

(14)
(15)

(10)

 

Where,  is the gear transmission ratio.

Also, the throttle opening percentage is a function of the engine's net torque () and its rotation speed (). This function is available as a lookup table ().

2.5. Brake Dynamics

The relationship between the brake torque and the pressure in the main cylinder () can be approximately expressed by “Eq. (17)”, [39]:

 

(17)

3        Longitudinal and lateral integrated control

The proposed control method is a robust control approach that can also be used for curved roads. This controller includes two kinematic and dynamic parts. Separating the system control into two kinematic and dynamic parts makes it possible to control the system position variables. Therefore, in this way, it becomes possible to follow curved paths. The general structure of the controller is such that first, in the kinematics section, the desired longitudinal and lateral velocity is determined based on the tracking longitudinal and lateral position errors. Then, in the dynamic control, appropriate inputs are calculated to reach the desired velocities required in the kinematic control, considering the vehicle dynamics.

3.1. Kinematic Control

The desired position vector is , which represents the desired longitudinal and lateral positions, respectively. Also, the vehicle position vector is defined as  (“Fig. 2”). The error vector, e, is also expressed as the difference between the desired position vector and the vehicle position vector.

 

Fig. 2        Description of inertial and local coordinate system: (a): positions, (b): velocities (β=0), (c): accelerations, and  (d): velocities (β≠0).

 

 

 

The error dynamics for kinematic control is also expressed as:

 

(18)

 

Where,  and  are positive definite control gain matrixes. According to “Fig. 2” and assuming , the velocity components in the inertial coordinates can be related to the components of the local coordinates as:

 

(19)

 

By combining “Eq. (18) and Eq. (19)”, the desired velocity vector  can be written as:

(20)

3.2. Dynamicic Control

The goal of dynamic control is to make  and  tend to values  and  respectively. Therefore, the error of longitudinal and lateral velocities can be expressed as:

 

The error dynamics of longitudinal and lateral velocities can be defined as: 

(21)
(22)

 

Where , ,  and  are positive definite control gain matrixes.

3.2.1. Longitudinal Control

Equation (23) can be used to calculate the longitudinal control input (throttle opening rate or brake cylinder pressure). By simplifying, we have:

(23)
(24)

(25)

 

On the other hand, by combining the Equation of longitudinal motion and wheel dynamics (“Eq. (1) and Eq. (13)”),  can be defined as:

 

 

By setting the right side of “Eq. (25) and Eq. (26)” equal, the total torque applied to the front and rear wheels can be calculated as:

(26)

(27)

 

The dynamics of the brake system and the power train are different, and each has its inputs. So, in the following steps, extracting the control input for each one is described separately.

 

 

 

3.2.1.1. Braking Mode

In braking mode, the ratio of the braking torque of the rear wheels to the front is assumed to be γ. In addition, the torque applied to the left and right wheels is the same in each of the axles. According to these assumptions and “Eq. (27)”, the braking torque applied to each can be expressed by:

 

(28)

 

By determining the total braking torque, the brake cylinder pressure can be determined using “Eq. (17)”.

3.2.1.2. Traction Mode

As mentioned, torque is not applied to the rear wheels in traction mode. According to “Eq. (16) and Eq. (27)”, the required engine torque can be expressed as:

 

(29)

 

The throttle opening percentage can be determined by determining the engine torque and the engine speed.

3.2.2. Lateral Control

First, we multiply both sides of “Eq. (2)” by  and add the resulting Equation with Equation (3).

 

(30)

 

By combining “Eq. (5), Eq. (11), and Eq. (30)”,  can be defined as “Eq. (31)”.

 

(31)

 

Where . On the other hand, using “Eq. (24)”,  can be written as:

 

(32)

 

By setting the right side of “Eq. (31) and Eq. (32)” equal, the steering angle  can be determined as:

 

(33)

4        Online estimation algorithm of tire forces and tire-road friction coefficients

It can be seen carefully in “Eq. (28), Eq. (29), and (33)” that the tire forces must also be known to calculate the control inputs. Since the complex dynamics of tires depend on environmental changes, tire wear, and unpredictable road conditions, the online estimation of these forces is necessary. In references [40-44], methods of identifying parameters of complex tire models, friction coefficient, and tire forces have been reviewed. Generally, past research in this field can be classified into cause-based and effect-based estimation methods. The concentration of cause-based methods is on studying and diagnosing effective factors in the tire-road interaction, and the friction coefficient is identified using specific analytical theories [43]. Effect-based techniques also use vehicle response to determine the friction coefficient [44].

In this research, according to the concept of friction circle and the use of vehicle kinematic characteristics that can be measured or estimated by sensors, tire forces, and friction coefficients are computed online with a straightforward algebraic algorithm and updated in the control law.

 

Since the proposed method works based on the vehicle response, it is an effect-based estimation technique. The details of the proposed method are presented below.

4.1. Calculation of The Longitudinal and Lateral Tire Forces

For simplicity, the vehicle's motion Equations are first rewritten in the following form:

 

Research paper

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© 2024 by the authors. Licensee Islamic Azad University Isfahan Branch. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution 4.0 International (CC BY 4.0) (https://creativecommons.org/licenses/by/4.0/)

 

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