# Bifurcation of Lane Change and Control on Highway for Tractor-Semitrailer under Rainy Weather

Authors: Tao Peng; Zhiwei Guan; Ronghui Zhang; Jinsong Dong; Kening Li; Hongguo Xu
Journal: Journal of Advanced Transportation
 (2017)

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## Abstract

A new method is proposed for analyzing the nonlinear dynamics and stability in lane changes on highways for tractor-semitrailer under rainy weather. Unlike most of the literature associated with a simulated linear dynamic model for tractor-semitrailers steady steering on dry road, a verified 5DOF mechanical model with nonlinear tire based on vehicle test was used in the lane change simulation on low adhesion coefficient road. According to Jacobian matrix eigenvalues of the vehicle model, bifurcations of steady steering and sinusoidal steering on highways under rainy weather were investigated using a numerical method. Furthermore, based on feedback linearization theory, taking the tractor yaw rate and joint angle as control objects, a feedback linearization controller combined with AFS and DYC was established. The numerical simulation results reveal that Hopf bifurcations are identified in steady and sinusoidal steering conditions, which translate into an oscillatory behavior leading to instability. And simulations of urgent step and single-lane change in high velocity show that the designed controller has good effects on eliminating bifurcations and improving lateral stability of tractor-semitrailer, during lane changing on highway under rainy weather. It is a valuable reference for safety design of tractor-semitrailers to improve the traffic safety with driver-vehicle-road closed-loop system.


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## Body

## 1. Introduction

Tractor-semitrailer, as an articulated vehicle system, has complex nonlinear dynamic characteristics. Vehicle dynamics bifurcation mechanisms and driving stability have basically been confirmed based on nonlinear theory and tire mechanics. Early works on this subject have established three degrees of freedom (DOF) dynamic model with simplified nonlinear tire formula, which are essential for a clear understanding of the system’s nonlinear stability [1, 2]. Due to the special articulated structure, jackknife and coordination stability have drawn wide attention. Dunn et al. rigorously developed a 15-state model using Lagrange’s method and studied in detail the effects on jackknife stability when using brakes under different road adhesion coefficients [3–5]. To protect the vehicle from spinning and realize improved cornering performance, van de Molengraft-Luijten et al. employed the Routh-Hurwitz stability criterion and bifurcation theory method to get a system equilibrium approximation near the zero value for the forward speed and front wheel steering angle, which revealed the vehicle’s instability under high speeds and substantial steering [6, 7]. Similarly, Ding et al. explained some instability phenomena of the vehicle system such as jackknifing, sideslip, and spinning by correlating them with the behavior in the neighborhood of unstable fixed points based on analysis of eigenvectors, phase trajectories, and status of lateral tire force saturation [8]. Sadri and Wu investigated Lyapunov concept exponents and applications to analyze the stability for the nonlinear vehicle in plane motion with a third-order polynomial tire model [9]. In addition, the existing literatures focus on bifurcation application analysis for the stability assessment of a complex railway vehicle model [10, 11].Moreover, vehicle/driver and vehicle/driver/road cooperation systems and influence parameters were introduced for tractor-semitrailer nonlinearity investigations [12–15]. To study vehicle Hopf bifurcation and chaos characteristics and variation in pilot model parameters and to discuss the qualitative motion behavior near the critical speed, Rossa et al. proposed a simple 3DOF closed-loop vehicle/driver model [16, 17]. Li et al. presented a nonlinear vehicle-road coupled model and analyzed the dynamic behaviors of a nonlinear system using a numerical integration method [18]. With a closed-loop system of articulated heavy vehicles with driver steering control, Liu et al. employed an integration method to derive an analytical periodic solution of the system in the neighborhood of the critical speed and analyzed supercritical and subcritical Hopf bifurcations [19]. Koglbauer et al. paid close attention to drivers’ workload and effort in different road conditions [20]. Bie et al. proposes a weather factor model built by introducing weather factors to free flow speed, capacity, and critical density and plugged it into traffic control [21]. Tang et al. investigated the road condition and driver characteristics on the lane change driving trajectory and traffic safety, such as lane number, driver’s time delay, and perception ability effect [22–27]. Related researches are helpful to master the interactions of vehicle, driver, and work environment.Furthermore, with the rapid development of control theory, a variety of methods are available in the tractor-semitrailer stability control system and product, such as ABS (Antilock Brake System) and ESC (Electronic Stability Control). By an optimal control approach for driver model, Liu studied the stability of a time delayed dynamical tractor-semitrailer, using a numerical method by computing the eigenvalues near the imaginary axis [13]. Lin et al. proposed optimal linear quadratic control algorithm to improve the yaw stability of a tractor semitrailer using active semitrailer steering [28]. Palkovics and El-Gindy applied RLQR/H∞ approach to ensure the vehicle’s performance in the presence of parametric uncertainties and discussed the influence of different control strategies on the vehicle’s directional and roll stability during severe path-follow lane change manoeuvre [29]. Chang et al. designed a H∞ loop shaping robust controller robust of tractor-semitrailer, which provides a novel mean or method for explore tracking control problem of the tractor-semitrailer [30]. Liang et al. proposed a specified fuzzy sliding mode controller to guarantee the robust control in the presence of system uncertainties [31]. By active wheel braking, Takenaga et al. proposed a novel fuzzy logic based yaw moment controllers to track the reference yaw rate of the tractor and the hitch angle and demonstrated the robust and effects in stabilizing the severe instabilities such as jackknife and trailer oscillation in the chosen simulation scenarios [32–34]. Zong et al. applied a multiobjective stability control algorithm to improve the vehicle stability of a tractor semitrailer by using differential braking [35]. Recently, more and more people pay attention to intelligent vehicle based on Cooperative Vehicle Infrastructure System [36–43]. Besides, vehicle control and system model was combined with transportation simulation [44–52].Many of the conclusions of the classical literature play a positive role in vehicle handing stability. However, there are still some limitations, especially in particular research backgrounds. First, it is clear that the vehicle instability in critical conditions is mainly due to the nonlinearity of tires, but most of traditional vehicle dynamics models are deduced by simplified tire formulas. The convenient calculations still deviate greatly from reality. Second, previous research on vehicles’ characteristics has concentrated on steady steering [53–56]. However, lane changes and overtaking by heavy trucks on highways, as a frequent driving behavior, could gain more attention in the near future. Furthermore, most of the traditional efforts have focused on vehicle stability on common road [57, 58]. In fact, rainy weather and lower adhesion coefficient should be taken into consideration, especially for heavy trucks.Tractor-semitrailer, as a very important and highly effective means of freight transportation, is more dangerous than general passenger cars because of their articulated structure, large size, and difficult handling. Accidents involving tractor-semitrailers tend to cause serious casualties and losses. Therefore, active safety and traffic transportation researchers have paid more attention to tractor-semitrailers’ driving stability in recent years. In particular, lane changing is a common driving behavior on highways, but high speeds and large-scale continuous steering can easily cause stability losses, such as sideslips and nonlinear oscillatory, for tractor-semitrailers. Last but not least, considering the large-scale rainy season in the south of our country, vehicles driving on lower adhesion efficient road and harsh environment are common, which cause increased instability and traffic accidents [59–63]. Therefore, nonlinear analysis and control of tractor-semitrailers’ lane changing on highways under rainy weather have important practical significance in the vehicle dynamics research field and provide a valuable reference for active safety design.This work deals with the nonlinear dynamics and control of tractor-semitrailers changing lanes on highways under rainy weather, which is a highly relevant real-life situation. In order to master the essence of stability and instability mechanism of tractor-semitrailer changing lanes at high speed, Rough-Hurwitz stability criterion is developed to analyze vehicle transient stability, and the vehicle bifurcation characteristics under steady and transient steering are studied based on nonlinear theory. In view of the vehicle’s instability mechanism, feedback linearization control model is deduced and controller combined with AFS (Active Front Steering) and DYC (Direct Yaw Control) was established and presented. It is significant to provide a valuable reference for safety design of tractor-semitrailers, to improve the traffic safety of driver-vehicle-road closed-loop system.This text is structured as follows: after a brief description of the vehicle model and validation in Section2, Section 3 analyzes bifurcations with steady steering and sinusoidal steering to simulate the main process of lane changing. Section 4 deals with an investigation of feedback linearization control for lane changing. Section 5 proposes a conclusion and further work. Conflicts of interest and acknowledgements are dealt with at the end of the article.


## 2. Vehicle System Model

By defining ground coordinate systemXOY, Figure 1 illustrates the plane motion force diagrams of tractor and semitrailer vehicle coordinate systems xtotyt and xsosys. The xt and xs axes, respectively, point forward and lie both in the ground plane and in the plane normal to the ground of tractor and semitrailer. The yt and ys axes point to the left. Positions of ot and os are, respectively, centroid of the vehicles. As motion variables we define the longitudinal velocities vxt and vxs, lateral velocities vyt and vys of tractor and semitrailer centroids, the slip angles βt and βs are respectively introduced by longitudinal and lateral velocities of tractor and semitrailer, the yaw angles φt and φs are respectively included angles between longitudinal axes and xs and xt axes of tractor and semitrailer.Figure 1
Schematic diagram of tractor-semitrailer plane motion.As shown in Figure1, when front wheels are steering, the vehicle is not only moving along the longitudinal direction, but also introduced the lateral motion around the steering center, and the yaw motion caused by dynamic centrifugal force at centroid. Thus, 5DOF model involves steering system motion, tractor, and semitrailer’s longitudinal and lateral motions, and yaw motions could be depicted in Figure 1. During the process, forces acting on the tires provide centripetal force of vehicle motion. In common situation, the centrifugal force is balanced with the centripetal force, and the plane motions of the vehicle will be in a regular way. Otherwise, if the centrifugal force is greater than the centripetal force, the vehicle will sideslip.


### 2.1. Vehicle Dynamics Model

Considering the research vehicle is moving on highway when steering without active braking and the road is flat, but wet and slippery under rainy weather, vertical, roll, and pitch motion can be ignored in this situation. Thus, the established vehicle model could be simplified as 5DOF model. The corresponding equations of the vehicle model are as follows:(1)
Tractor,(1)mtv˙xt+vytφ˙t=Fxtf+Fxtr-Fxth,mtv˙yt+vxtφ˙t=Fytf+Fytr-Fyth,Iztφ¨t=atFytf-btFytr+cFyth.(2)
Semitrailer,(2)msv˙xs+vysφ˙s=Fxs+Fxsh,msv˙ys+vxsφ˙s=Fys+Fysh,Izsφ¨s=-bsFys+asFysh.(3)
Steering system [64],(3)Iwδ¨f+Cwδ˙f=Kwδswi-δf-εFytf.Meanwhile, conditions of saddle balance approximately satisfy(4)vxt=vxs,vys=vxtθ+vyt-c+asφ˙t+asθ˙,Fxth=Fxsh,Fyth=Fysh.Taking a domestic tractor-semitrailer as an example, relevant vehicle parameters and symbols are listed in Table1. Here, the mass and structure values of the vehicle refer to test data, inertia, and stiffness and the other performance data are selected based on database of Trucksim.Table 1
Tractor-semitrailer parameters and symbols.
Symbol
Parameter name
Value
Unit
m
t
Tractor mass
5800
kg
m
s
Semitrailer mass
19080
kg
a
t
Distance between tractor centroid and front axle
2.0
m
b
t
Distance between tractor centroid and rear axle
1.4
m
c
Distance between tractor centroid and saddle point
0.8
m
a
s
Distance between semitrailer centroid and saddle point
2.0
m
b
s
Distance between semitrailer centroid and rear axle
1.4
m
I
z
t
Tractor yaw moment of inertia
9737
kg⋅m2
I
z
s
Semitrailer yaw moment of inertia
138437
kg⋅m2
I
w
Moment of inertia of steering wheel around spin
60
kg⋅m2
C
w
Damping coefficient of steering wheel around spin
10250
—
K
w
Steering system stiffness
2000
Nm/rad
ε
Tractor front tire pneumatic trail
0.01
m
i
Steering radio
28
—
v
x
t
Tractor longitudinal velocity
—
m/s
v
x
s
Semitrailer longitudinal velocity
—
m/s
v
y
t
Tractor lateral velocity
—
m/s
v
y
s
Semitrailer lateral velocity
—
m/s
φ
t
Tractor yaw angle
—
rad/s
φ
s
Semitrailer yaw angle
—
rad/s
θ
Articulated angle
—
rad
δ
sw
Tractor steering wheel angle
—
rad
δ
f
Tractor front wheel steering angle
—
rad
F
x
tf
Tractor front wheel longitudinal force
—
N
F
x
t
r
Tractor rear wheel longitudinal force
—
N
F
y
tf
Tractor front wheel lateral force
—
N
F
y
t
r
Tractor rear wheel lateral force
—
N
F
x
s
Semitrailer wheel longitudinal force
—
N
F
y
s
Semitrailer front wheel lateral force
—
N
F
x
th
Longitudinal force from saddle to tractor
—
N
F
y
th
Lateral force from saddle to tractor
—
N
F
x
sh
Longitudinal force from saddle to semitrailer
—
N
F
y
sh
Lateral force from saddle to semitrailer
—
N


### 2.2. Nonlinear Tire Model

In addition to supporting the vehicle steering dynamics, the tires provide the lateral forces necessary to change the speed and direction of the vehicle. There exist many tire models to describe tire behaviors beyond the linear region. One model commonly used in vehicle dynamics simulations was developed by Pacejka of the Delft University of Technology [64, 65]. The lateral slip angle α generates a lateral force, Fy, at the tire-ground interface, which could be expressed as a magic formula:(5)Fy=Dsin⁡Catan⁡Bα-EBα-atan⁡Bα,where α is the tire slip angle and B, C, D, and E are, respectively, the stiffness factor, shape factor, peak factor, and curvature factor.As we know, sideslip instability of a tractor-semitrailer is caused by exceeding the lateral tire force saturation. Lower lateral tire force saturation arises from lower adhesion coefficient under rainy weather, which easily causes severe nonlinear performance of the vehicle. Normally, on wet and slippery road, the absolute value of peak factorD must be lower and the stiffness and shape factors (B and C) are higher than that on general roads. Relevant parameters valued as all wheels of each axis on wet road (μ=0.3) for the tires are listed in Table 2 [66].Table 2
Values ofB, C, D, and E for a nonlinear tire.
Tire position
Tire number
B
C
D
E
Tractor front wheel
2
3.32
1.2
−12987.6
−1.45
Tractor rear wheel
4
11.35
1.2
−16327.2
−1.45
Semitrailer wheel
8
19.62
1.2
−25792.8
−1.45These slip angles may be obtained using the following handing relations:(6)αtf=arctan⁡vyt+atωtvxt-δf,αtr=arctan⁡vyt-btωtvxt,αs=arctan⁡vys-bsωsvxs,where αtf, αtr, and αs are, respectively, the tractor front and rear wheel and the semitrailer wheel’s slip angle, vyt and vys are, respectively, the tractor and semitrailer’s lateral velocity, vxt and vxs are, respectively, the tractor and semitrailer’s longitudinal velocity, at is the distance between tractor centroid and front axle, bt is distance between tractor centroid and rear axle, bs is the distance between semitrailer centroid and rear axle, and ωt and ωs are, respectively, the tractor and semitrailer’s yaw rate.


### 2.3. Vehicle Dynamics Model Validation

The double-lane change is deemed the general method of overtaking. The whole process can be divided into five steps: straight driving in original lane, approximately sinusoidal steering, straight driving in the other lane, and approximately sinusoidal steering and steer returning in original lane. Thus, the tractor-semitrailer comes under complex and severe dynamics, with changes in lane position, large-scale continuous steering, and high speed.To validate the vehicle dynamics model, applying a domestic tractor-semitrailer, we conducted a physical double-lane change experiment on an even and wet road, just as a tractor-semitrailer overtaking under rainy weather on highway. Tests were carried out at a professional test site in the Key Laboratory of Operation Safety Technology for Transport Vehicles, at the Research Institute of the Highway Ministry of Transport, Beijing, China. Details of the test environment are shown in Figure2.Figure 2
Image of vehicle test environment.The tested vehicle was driven along the double-change channel on an even and wet road, with its velocity kept to about 60 km/h. The integrated test system mainly involved a VBOX III for tractor, a VBOX II and an inertial sensor for semitrailer, a vehicle-handling force and angle meter for vehicle steering, a data collection device, and a power supply device, whose function was to measure the steering wheel angle input and the output vehicle motion parameters, including velocity, acceleration, and angular velocity.Setting the simulation inputs the same as test velocities and steering wheel angles, in addition, the time step is 0.05 s in accord with test sampling frequency 20 Hz. Furthermore, the orientations of simulated output variables keep consistent with test.Under the same conditions, tractor and semitrailer lateral accelerations, yaw rates, and joint angle were obtained using vehicle testing and Matlab simulation and compared, as shown in Figure3.Figure 3
Comparison of simulation and vehicle test.
(a)
Input
(b)
Output of tractor lateral acceleration
(c)
Output of semitrailer lateral acceleration
(d)
Output of tractor yaw rate
(e)
Output of semitrailer yaw rate
(f)
Output of joint angle
(g)
Output of roll angles of testClearly, the vehicle’s lateral and yaw dynamics in Figure3 exhibit good consistency between the simulation and the vehicle test. The validation of the model basis is given, using limited data. Moreover, according to the test results, roll angles of the tractor and semitrailer shown in Figure 3(g) are very small (less than 1 deg), so vehicle roll motions on wet and slippery road could be ignored reasonably. Therefore, it can be conservatively concluded that the established 5DOF nonlinear dynamics model closely reflects the physical properties.


## 2.1. Vehicle Dynamics Model

Considering the research vehicle is moving on highway when steering without active braking and the road is flat, but wet and slippery under rainy weather, vertical, roll, and pitch motion can be ignored in this situation. Thus, the established vehicle model could be simplified as 5DOF model. The corresponding equations of the vehicle model are as follows:(1)
Tractor,(1)mtv˙xt+vytφ˙t=Fxtf+Fxtr-Fxth,mtv˙yt+vxtφ˙t=Fytf+Fytr-Fyth,Iztφ¨t=atFytf-btFytr+cFyth.(2)
Semitrailer,(2)msv˙xs+vysφ˙s=Fxs+Fxsh,msv˙ys+vxsφ˙s=Fys+Fysh,Izsφ¨s=-bsFys+asFysh.(3)
Steering system [64],(3)Iwδ¨f+Cwδ˙f=Kwδswi-δf-εFytf.Meanwhile, conditions of saddle balance approximately satisfy(4)vxt=vxs,vys=vxtθ+vyt-c+asφ˙t+asθ˙,Fxth=Fxsh,Fyth=Fysh.Taking a domestic tractor-semitrailer as an example, relevant vehicle parameters and symbols are listed in Table1. Here, the mass and structure values of the vehicle refer to test data, inertia, and stiffness and the other performance data are selected based on database of Trucksim.Table 1
Tractor-semitrailer parameters and symbols.
Symbol
Parameter name
Value
Unit
m
t
Tractor mass
5800
kg
m
s
Semitrailer mass
19080
kg
a
t
Distance between tractor centroid and front axle
2.0
m
b
t
Distance between tractor centroid and rear axle
1.4
m
c
Distance between tractor centroid and saddle point
0.8
m
a
s
Distance between semitrailer centroid and saddle point
2.0
m
b
s
Distance between semitrailer centroid and rear axle
1.4
m
I
z
t
Tractor yaw moment of inertia
9737
kg⋅m2
I
z
s
Semitrailer yaw moment of inertia
138437
kg⋅m2
I
w
Moment of inertia of steering wheel around spin
60
kg⋅m2
C
w
Damping coefficient of steering wheel around spin
10250
—
K
w
Steering system stiffness
2000
Nm/rad
ε
Tractor front tire pneumatic trail
0.01
m
i
Steering radio
28
—
v
x
t
Tractor longitudinal velocity
—
m/s
v
x
s
Semitrailer longitudinal velocity
—
m/s
v
y
t
Tractor lateral velocity
—
m/s
v
y
s
Semitrailer lateral velocity
—
m/s
φ
t
Tractor yaw angle
—
rad/s
φ
s
Semitrailer yaw angle
—
rad/s
θ
Articulated angle
—
rad
δ
sw
Tractor steering wheel angle
—
rad
δ
f
Tractor front wheel steering angle
—
rad
F
x
tf
Tractor front wheel longitudinal force
—
N
F
x
t
r
Tractor rear wheel longitudinal force
—
N
F
y
tf
Tractor front wheel lateral force
—
N
F
y
t
r
Tractor rear wheel lateral force
—
N
F
x
s
Semitrailer wheel longitudinal force
—
N
F
y
s
Semitrailer front wheel lateral force
—
N
F
x
th
Longitudinal force from saddle to tractor
—
N
F
y
th
Lateral force from saddle to tractor
—
N
F
x
sh
Longitudinal force from saddle to semitrailer
—
N
F
y
sh
Lateral force from saddle to semitrailer
—
N


## 2.2. Nonlinear Tire Model

In addition to supporting the vehicle steering dynamics, the tires provide the lateral forces necessary to change the speed and direction of the vehicle. There exist many tire models to describe tire behaviors beyond the linear region. One model commonly used in vehicle dynamics simulations was developed by Pacejka of the Delft University of Technology [64, 65]. The lateral slip angle α generates a lateral force, Fy, at the tire-ground interface, which could be expressed as a magic formula:(5)Fy=Dsin⁡Catan⁡Bα-EBα-atan⁡Bα,where α is the tire slip angle and B, C, D, and E are, respectively, the stiffness factor, shape factor, peak factor, and curvature factor.As we know, sideslip instability of a tractor-semitrailer is caused by exceeding the lateral tire force saturation. Lower lateral tire force saturation arises from lower adhesion coefficient under rainy weather, which easily causes severe nonlinear performance of the vehicle. Normally, on wet and slippery road, the absolute value of peak factorD must be lower and the stiffness and shape factors (B and C) are higher than that on general roads. Relevant parameters valued as all wheels of each axis on wet road (μ=0.3) for the tires are listed in Table 2 [66].Table 2
Values ofB, C, D, and E for a nonlinear tire.
Tire position
Tire number
B
C
D
E
Tractor front wheel
2
3.32
1.2
−12987.6
−1.45
Tractor rear wheel
4
11.35
1.2
−16327.2
−1.45
Semitrailer wheel
8
19.62
1.2
−25792.8
−1.45These slip angles may be obtained using the following handing relations:(6)αtf=arctan⁡vyt+atωtvxt-δf,αtr=arctan⁡vyt-btωtvxt,αs=arctan⁡vys-bsωsvxs,where αtf, αtr, and αs are, respectively, the tractor front and rear wheel and the semitrailer wheel’s slip angle, vyt and vys are, respectively, the tractor and semitrailer’s lateral velocity, vxt and vxs are, respectively, the tractor and semitrailer’s longitudinal velocity, at is the distance between tractor centroid and front axle, bt is distance between tractor centroid and rear axle, bs is the distance between semitrailer centroid and rear axle, and ωt and ωs are, respectively, the tractor and semitrailer’s yaw rate.


## 2.3. Vehicle Dynamics Model Validation

The double-lane change is deemed the general method of overtaking. The whole process can be divided into five steps: straight driving in original lane, approximately sinusoidal steering, straight driving in the other lane, and approximately sinusoidal steering and steer returning in original lane. Thus, the tractor-semitrailer comes under complex and severe dynamics, with changes in lane position, large-scale continuous steering, and high speed.To validate the vehicle dynamics model, applying a domestic tractor-semitrailer, we conducted a physical double-lane change experiment on an even and wet road, just as a tractor-semitrailer overtaking under rainy weather on highway. Tests were carried out at a professional test site in the Key Laboratory of Operation Safety Technology for Transport Vehicles, at the Research Institute of the Highway Ministry of Transport, Beijing, China. Details of the test environment are shown in Figure2.Figure 2
Image of vehicle test environment.The tested vehicle was driven along the double-change channel on an even and wet road, with its velocity kept to about 60 km/h. The integrated test system mainly involved a VBOX III for tractor, a VBOX II and an inertial sensor for semitrailer, a vehicle-handling force and angle meter for vehicle steering, a data collection device, and a power supply device, whose function was to measure the steering wheel angle input and the output vehicle motion parameters, including velocity, acceleration, and angular velocity.Setting the simulation inputs the same as test velocities and steering wheel angles, in addition, the time step is 0.05 s in accord with test sampling frequency 20 Hz. Furthermore, the orientations of simulated output variables keep consistent with test.Under the same conditions, tractor and semitrailer lateral accelerations, yaw rates, and joint angle were obtained using vehicle testing and Matlab simulation and compared, as shown in Figure3.Figure 3
Comparison of simulation and vehicle test.
(a)
Input
(b)
Output of tractor lateral acceleration
(c)
Output of semitrailer lateral acceleration
(d)
Output of tractor yaw rate
(e)
Output of semitrailer yaw rate
(f)
Output of joint angle
(g)
Output of roll angles of testClearly, the vehicle’s lateral and yaw dynamics in Figure3 exhibit good consistency between the simulation and the vehicle test. The validation of the model basis is given, using limited data. Moreover, according to the test results, roll angles of the tractor and semitrailer shown in Figure 3(g) are very small (less than 1 deg), so vehicle roll motions on wet and slippery road could be ignored reasonably. Therefore, it can be conservatively concluded that the established 5DOF nonlinear dynamics model closely reflects the physical properties.


## 3. Bifurcation of Steering

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations. Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden “qualitative” or topological change in its behavior. The name “bifurcation” was first introduced by Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior [67, 68].In view of the lane change handling steps, steering at a high speed on a highway under rainy weather is the primary extensive condition within the whole process. Considering the specific features of tractor-semitrailer, the vehicle nonlinear dynamics with large-scale steering on wet road are more complex than single vehicle under the normal situation and are analyzed in the following subsections.


### 3.1. Vehicle Model Transformation

In order to efficiently analyze tractor-semitrailer’s bifurcation characteristics in typical work conditions and simplify calculation process, steering wheel is not applied as the input of steering, front wheel instead. Based on the idea, the vehicle system is transformed as(7)X˙=fX,δf,where X=(vxt,vyt,φ˙t,θ˙,θ)T.In the case of the vehicle steering motion, suppose the vectorXe=(vxte,vyte,φ˙te,θ˙e,θe)T is an equilibrium point for the system described in (7). To assess the system behavior after stability loss, (7) is expanded into a multidimensional Taylor series in xi around Xe up to order 3, yielding(8)X˙=AX+FX,δf+OX4,where A is the Jacobian matrix of the nonlinear function f(X,δf), computed at the equilibrium point Xe and F(X,δf) contains nonlinear terms of order 3 at Xe:(9)A=DxfX,δfvxte,vyte,φ˙te,θ˙e,θeT.The fifth-order characteristic equation of Jacobian matrixA is as follows:(10)λ5+c1λ4+c2λ3+c3λ2+c4λ+c5=0.According to Hurwitz stability criterion, if and only if all the roots of Jacobian matrixA have strictly negative real part, the system is stable. The criterion identifies the conditions when the poles of a polynomial cross into the right hand half plane and hence would be considered as unstable in control engineering [67–69].


### 3.2. Bifurcation of Steady Steering

Steady steering not only is a typical working condition, for exhibiting vehicle steady characteristics, but also is the basic of transient. Good steady characteristics predict better transient performance to a certain extent.With initial longitudinal velocity 20 m/s, the calculated five eigenvalues (Re represents the real part and Im represents the imaginary part) of Jacobian matrixA with front wheel steering angle varying in the range [0.01,0.049] rad are listed in Table 3.Table 3
Calculated eigenvalues of Jacobian matrixA under various front wheel steering angles.
δ
f/rad
Eig. 1
Eig. 2
Eig. 3
Eig. 4
Eig. 5
Re
Im
Re
Im
Re
Im
Re
Im
Re
Im
0.01
−1.8135
3.3965
−1.8135
−3.3965
−1.3298
2.1452
−1.3298
−2.1452
−0.0001
0
0.02
−1.7838
3.3517
−1.7838
−3.3517
−1.3154
2.1192
−1.3154
−2.1192
−0.0006
0
0.03
−1.7011
3.2258
−1.7011
−3.2258
−1.2801
2.0400
−1.2801
−2.0400
−0.0014
0
0.04
−1.4962
2.9357
−1.4962
−2.9357
−1.2203
1.8070
−1.2203
−1.8070
−0.0032
0
0.049
−0.8385
2.2125
−0.8385
−2.2125
0
0.1102
0
−0.1102
−2.1906
0As shown in Table3, when the steering angle is less than 0.049 rad, there are five eigenvalues with negative real parts. According to the nonlinear theory, in these cases, the vehicle system is stable and the equilibrium points are called nodal points. Once the steering angle reaches 0.049 rad, two pairs of imaginary eigenvalues with zero real parts appear, which means that a Hopf bifurcation may exist [69].According to [68], there is a pair of pure imaginary conjugate eigenvalues and the other eigenvalues have nonzero real part; thus, the corresponding equilibrium point is nonhyperbolic. In addition, it is clearly seen that the real part of the eigenvalues varies as the steering angle increases, and the transverse condition is satisfied as follows:(11)dRe⁡λdδfδf=0.049=Re⁡dλdδfδf=0.049=0--1.22030.049-0.04=135.59≠0.Therefore, it is proved that the Hopf bifurcation is taking place at the equilibrium point when δf=0.049 rad.The equilibrium point could be calculated with the vehicle model by Matlab. Input parameters areδf=0.049 rad and initial vx=20 m/s; the system will be stable within a limited period of time; then the equilibrium point could be obtained according to the stable variables; that is, Xe,δf=-1.3162,0.0761,0,0.0576,19.86,0.049.Once Hopf bifurcation is taking place, under reasonably generic assumptions about the dynamical system, as steering angle slightly rises, a periodic solution called limit cycle branches from the fixed point [67, 68]. When the steering angle is 0.05 rad, limit circles of tractor’s velocities with time are shown in Figure 4.Figure 4
Limit circle of tractor’s velocities withδf=0.05 rad.
(a)
Tractor longitudinal velocity
(b)
Tractor lateral velocityIt can be seen from Figure4, as time increases, a large-scale periodic vibration of the tractor’s velocities, which represents a periodic solution of vehicle system. So there is limit circle when the steering angle is 0.05 rad, after Hopf bifurcation has taken place, which shows large range conversion of vehicle longitudinal and lateral kinetic energy. Obviously, the energy conversion could not be realized in reality because of frictional loss and motion constraint between tractor and semitrailer. However, the articulated vehicle’s conservation of energy and dynamic trends of stability loss is clearly revealed.For the 4DOF model with longitudinal velocity changes, the state variable of quasiperiodic vibration shows energy conservation. Moreover, the eigenvalues change from imaginary with negative real part to pure imaginary in accordance with the manifolds, predicting the Hopf bifurcation phenomenon. It is verified that a pair of conjugate roots, first crossing on the imaginary axis, translates into oscillatory behavior leading to instability.


### 3.3. Bifurcation of Sinusoidal Steering

Sinusoidal steering is introduced to approximately represent the steering process for lane changing. Unlike steady steering, the front wheel steering angle is represented as(12)δf=Assin⁡2πft,where As is the steering amplitude and f is steering frequency. Compared with steady steering, the vehicle with sinusoidal steering is considered a time-changing system. The equilibrium points and corresponding eigenvalues vary with time and the steering angle.Figure5 shows the changes in the steering angle, lateral velocity, and calculated eigenvalues when the initial longitudinal velocity is 20 m/s, and the front wheel steering amplitude and frequency are, respectively, 0.06 rad and 0.2 Hz. With As=0.06 rad, f=0.2 Hz, the calculated eigenvalues present a double wave shape with sinusoidal steering. According to the tractor lateral velocity, the vehicle system is still stable, although transit values of eigenvalue 1 and eigenvalue 3 are positive or have positive real parts during steering. In addition, when t>6.3 s, all of the eigenvalues are negative as steer returning takes place and eigenvalue 5 is approximately equal to 0. Therefore, it can be concluded that the negative eigenvalue is sufficient, but not necessary, for judging stability. Furthermore, comparing sinusoidal and steady steering, when the front wheel angle is greater than 0.05 rad, the vehicle system with steady steering loses stability, but that with sinusoidal steering remains stable. Specifically, vehicle stability is influenced by the frequency and the transient state.Figure 5
Changes in steering angle, lateral velocity, and calculated eigenvalues withAs=0.06 rad, f=0.2 Hz.
(a)
Steering angleAs=0.06 rad, f=0.2 Hz
(b)
Vehicle lateral velocity change with time
(c)
Real part of eigenvalues change with timeFigure6 shows, as the amplitude is increased from 0.06 rad to 0.07 rad, the changes in the steering angle, lateral velocity, and calculated eigenvalues.Figure 6
Changes in steering angle, lateral velocity, and real parts of calculated eigenvalues withAs=0.07 rad, f=0.2 Hz.
(a)
Steering angleAs=0.06 rad, f=0.2 Hz
(b)
Vehicle lateral velocity change with time
(c)
Real part of eigenvalues change with timeIt can be seen from Figure6 that, as the steering amplitude increases to 0.07 rad, not only transit values of eigenvalue 1 and eigenvalue 3 are positive or have positive real parts during steering, just as As=0.06 rad, but also transit values of eigenvalue 5 are positive eigenvalues after steer returning. Meanwhile, there is a pair of conjugate eigenvalues (eigenvalues 1 and 2), approximately lying on the imaginary axis, which indicates the existence of Hopf bifurcation. Figure 7 shows the unstable limit circle of the vehicle typical parameters with continuous sinusoidal steering when As=0.07 rad, f=0.2 Hz.Figure 7
Limit circle of vehicle’s velocity withAs=0.07 rad, f=0.2 Hz.
(a)
Tractor longitudinal velocity
(b)
Tractor lateral velocityJust as in Figure4 for the case of steady steering, Figure 7 shows an unstable limit circle of vehicle dynamic motions with critical sinusoidal steering. It clearly shows there are obvious differences in fluctuation range of vehicle’s lateral velocity between steady and sinusoidal steering. Besides this, the vehicle’s dynamic bifurcation and oscillatory behavior are identical.


## 3.1. Vehicle Model Transformation

In order to efficiently analyze tractor-semitrailer’s bifurcation characteristics in typical work conditions and simplify calculation process, steering wheel is not applied as the input of steering, front wheel instead. Based on the idea, the vehicle system is transformed as(7)X˙=fX,δf,where X=(vxt,vyt,φ˙t,θ˙,θ)T.In the case of the vehicle steering motion, suppose the vectorXe=(vxte,vyte,φ˙te,θ˙e,θe)T is an equilibrium point for the system described in (7). To assess the system behavior after stability loss, (7) is expanded into a multidimensional Taylor series in xi around Xe up to order 3, yielding(8)X˙=AX+FX,δf+OX4,where A is the Jacobian matrix of the nonlinear function f(X,δf), computed at the equilibrium point Xe and F(X,δf) contains nonlinear terms of order 3 at Xe:(9)A=DxfX,δfvxte,vyte,φ˙te,θ˙e,θeT.The fifth-order characteristic equation of Jacobian matrixA is as follows:(10)λ5+c1λ4+c2λ3+c3λ2+c4λ+c5=0.According to Hurwitz stability criterion, if and only if all the roots of Jacobian matrixA have strictly negative real part, the system is stable. The criterion identifies the conditions when the poles of a polynomial cross into the right hand half plane and hence would be considered as unstable in control engineering [67–69].


## 3.2. Bifurcation of Steady Steering

Steady steering not only is a typical working condition, for exhibiting vehicle steady characteristics, but also is the basic of transient. Good steady characteristics predict better transient performance to a certain extent.With initial longitudinal velocity 20 m/s, the calculated five eigenvalues (Re represents the real part and Im represents the imaginary part) of Jacobian matrixA with front wheel steering angle varying in the range [0.01,0.049] rad are listed in Table 3.Table 3
Calculated eigenvalues of Jacobian matrixA under various front wheel steering angles.
δ
f/rad
Eig. 1
Eig. 2
Eig. 3
Eig. 4
Eig. 5
Re
Im
Re
Im
Re
Im
Re
Im
Re
Im
0.01
−1.8135
3.3965
−1.8135
−3.3965
−1.3298
2.1452
−1.3298
−2.1452
−0.0001
0
0.02
−1.7838
3.3517
−1.7838
−3.3517
−1.3154
2.1192
−1.3154
−2.1192
−0.0006
0
0.03
−1.7011
3.2258
−1.7011
−3.2258
−1.2801
2.0400
−1.2801
−2.0400
−0.0014
0
0.04
−1.4962
2.9357
−1.4962
−2.9357
−1.2203
1.8070
−1.2203
−1.8070
−0.0032
0
0.049
−0.8385
2.2125
−0.8385
−2.2125
0
0.1102
0
−0.1102
−2.1906
0As shown in Table3, when the steering angle is less than 0.049 rad, there are five eigenvalues with negative real parts. According to the nonlinear theory, in these cases, the vehicle system is stable and the equilibrium points are called nodal points. Once the steering angle reaches 0.049 rad, two pairs of imaginary eigenvalues with zero real parts appear, which means that a Hopf bifurcation may exist [69].According to [68], there is a pair of pure imaginary conjugate eigenvalues and the other eigenvalues have nonzero real part; thus, the corresponding equilibrium point is nonhyperbolic. In addition, it is clearly seen that the real part of the eigenvalues varies as the steering angle increases, and the transverse condition is satisfied as follows:(11)dRe⁡λdδfδf=0.049=Re⁡dλdδfδf=0.049=0--1.22030.049-0.04=135.59≠0.Therefore, it is proved that the Hopf bifurcation is taking place at the equilibrium point when δf=0.049 rad.The equilibrium point could be calculated with the vehicle model by Matlab. Input parameters areδf=0.049 rad and initial vx=20 m/s; the system will be stable within a limited period of time; then the equilibrium point could be obtained according to the stable variables; that is, Xe,δf=-1.3162,0.0761,0,0.0576,19.86,0.049.Once Hopf bifurcation is taking place, under reasonably generic assumptions about the dynamical system, as steering angle slightly rises, a periodic solution called limit cycle branches from the fixed point [67, 68]. When the steering angle is 0.05 rad, limit circles of tractor’s velocities with time are shown in Figure 4.Figure 4
Limit circle of tractor’s velocities withδf=0.05 rad.
(a)
Tractor longitudinal velocity
(b)
Tractor lateral velocityIt can be seen from Figure4, as time increases, a large-scale periodic vibration of the tractor’s velocities, which represents a periodic solution of vehicle system. So there is limit circle when the steering angle is 0.05 rad, after Hopf bifurcation has taken place, which shows large range conversion of vehicle longitudinal and lateral kinetic energy. Obviously, the energy conversion could not be realized in reality because of frictional loss and motion constraint between tractor and semitrailer. However, the articulated vehicle’s conservation of energy and dynamic trends of stability loss is clearly revealed.For the 4DOF model with longitudinal velocity changes, the state variable of quasiperiodic vibration shows energy conservation. Moreover, the eigenvalues change from imaginary with negative real part to pure imaginary in accordance with the manifolds, predicting the Hopf bifurcation phenomenon. It is verified that a pair of conjugate roots, first crossing on the imaginary axis, translates into oscillatory behavior leading to instability.


## 3.3. Bifurcation of Sinusoidal Steering

Sinusoidal steering is introduced to approximately represent the steering process for lane changing. Unlike steady steering, the front wheel steering angle is represented as(12)δf=Assin⁡2πft,where As is the steering amplitude and f is steering frequency. Compared with steady steering, the vehicle with sinusoidal steering is considered a time-changing system. The equilibrium points and corresponding eigenvalues vary with time and the steering angle.Figure5 shows the changes in the steering angle, lateral velocity, and calculated eigenvalues when the initial longitudinal velocity is 20 m/s, and the front wheel steering amplitude and frequency are, respectively, 0.06 rad and 0.2 Hz. With As=0.06 rad, f=0.2 Hz, the calculated eigenvalues present a double wave shape with sinusoidal steering. According to the tractor lateral velocity, the vehicle system is still stable, although transit values of eigenvalue 1 and eigenvalue 3 are positive or have positive real parts during steering. In addition, when t>6.3 s, all of the eigenvalues are negative as steer returning takes place and eigenvalue 5 is approximately equal to 0. Therefore, it can be concluded that the negative eigenvalue is sufficient, but not necessary, for judging stability. Furthermore, comparing sinusoidal and steady steering, when the front wheel angle is greater than 0.05 rad, the vehicle system with steady steering loses stability, but that with sinusoidal steering remains stable. Specifically, vehicle stability is influenced by the frequency and the transient state.Figure 5
Changes in steering angle, lateral velocity, and calculated eigenvalues withAs=0.06 rad, f=0.2 Hz.
(a)
Steering angleAs=0.06 rad, f=0.2 Hz
(b)
Vehicle lateral velocity change with time
(c)
Real part of eigenvalues change with timeFigure6 shows, as the amplitude is increased from 0.06 rad to 0.07 rad, the changes in the steering angle, lateral velocity, and calculated eigenvalues.Figure 6
Changes in steering angle, lateral velocity, and real parts of calculated eigenvalues withAs=0.07 rad, f=0.2 Hz.
(a)
Steering angleAs=0.06 rad, f=0.2 Hz
(b)
Vehicle lateral velocity change with time
(c)
Real part of eigenvalues change with timeIt can be seen from Figure6 that, as the steering amplitude increases to 0.07 rad, not only transit values of eigenvalue 1 and eigenvalue 3 are positive or have positive real parts during steering, just as As=0.06 rad, but also transit values of eigenvalue 5 are positive eigenvalues after steer returning. Meanwhile, there is a pair of conjugate eigenvalues (eigenvalues 1 and 2), approximately lying on the imaginary axis, which indicates the existence of Hopf bifurcation. Figure 7 shows the unstable limit circle of the vehicle typical parameters with continuous sinusoidal steering when As=0.07 rad, f=0.2 Hz.Figure 7
Limit circle of vehicle’s velocity withAs=0.07 rad, f=0.2 Hz.
(a)
Tractor longitudinal velocity
(b)
Tractor lateral velocityJust as in Figure4 for the case of steady steering, Figure 7 shows an unstable limit circle of vehicle dynamic motions with critical sinusoidal steering. It clearly shows there are obvious differences in fluctuation range of vehicle’s lateral velocity between steady and sinusoidal steering. Besides this, the vehicle’s dynamic bifurcation and oscillatory behavior are identical.


## 4. Bifurcation Control with Feedback Linearization

It is obvious that, with the Jacobian matrixA induced by vehicle’s nonlinear model, the eigenvalues could reveal the bifurcation characteristics and stability. Thus, basic ideal is proposed to ensure the eigenvalues with negative real parts, by proper control method.Feedback Linearization is an approach to nonlinear control design which has attracted a great deal of research interest in recent years. Algebraically transform a nonlinear system into a (fully or partly) linear one, so that linear control techniques can be applied.This differs from the conventional linearization in that feedback linearization is achieved by exact state transformations and feedback, rather than by linear approximations of the dynamics [63, 69–72]. Considering the nonlinearity of tractor-semitrailer, based on feedback linearization theory, an integrated controller is introduced to realize vehicle system asymptotically stable and tracking performance in this section.


### 4.1. Vehicle Control Model

Supposing the tractor longitudinal velocity constant, the vehicle model introduced in Section2 is deduced as follows:(13)X˙=fX,δsw,where X=x1,x2,x3,x4,x5,x6T=vyt,φ˙t,θ˙,θ,δ˙f,δfT.In order to linearize nonlinear systems by feedback linearization method, feedback parameters, as control outputs, not only are controllable, but also ensure linearization. Therefore, setting tractor yaw rateφ˙t and the difference of joint angle and steering angle θ-δf as control output variables, applying AFS and DYC combined control method, the feedback linearization control model could be described as(14)X˙=fX,δsw+g1u1+g2u2,y1=h1x=φ˙t,y2=h2x=θ-δf,where (15)fX,δsw=a11Fyf+a12Fyr+a13Fys+a14vxtx2a21Fyf+a22Fyr+a23Fysa31Fyf+a32Fyr+a33Fysx3-Cwx5Iw-Kwx6Iw-εFytfIw+KwiIwδswx5,(16)g1=0,0,0,0,-KwIw,0T,g2=g21,g22,g23,0,0,0T,(17)u1=δswc,u2=Mc.Variablesa11, a12, a13, and so on in (15) and g21, g22, and g23 in (16) could be expressed by known variables, as follows:(18)a11=a1+ccm2Iz2+Iz1Iz2+a22m2Iz1S,a12=c-b1cm2Iz2+Iz1Iz2+a22m2Iz1S,a13=Iz2-a2b2m2Iz1S,a14=-a22Iz1+c2Iz2m1m2-m1+m2Iz1Iz2S,a21=a1a22m1m2+a1m1+a1m2+cm2Iz2-a2m2Iz1S,a22=cm2-b1m1-b1m2Iz2-a22b1m1m2S,a23=cm1a2b2m2-Iz2S,a31=a1a2c+a2m1m2+a1m1+a1m2+cm2Iz2-a2m2Iz1S,a32=-a2b1c+a2m1m2+cm2-b1m2-b1m1Iz2-a2m2Iz1S,a33=a2Iz1-cIz2m1+b2m1+m2Iz1+cc+a2m1m2S,g21=a2Iz1-cIz2m2S,g22=-a2+ca2m1m2-m1+m2Iz2S,g23=-a2+c2m1m2-m1+m2Iz1+Iz2S,S=a22Iz1+c2Iz2m1m2+m1+m2Iz1Iz2.According to conditions of feedback linearization theory, relative degree of vehicle system should be confirmed as(19)Lg1h1x=∂x2∂X·g1=0;Lg2h2x=∂x2∂X·g2=g22≠0;Lg1h2x=∂x4-x6∂X·g1=0;Lg2h2x=∂x4-x6∂X·g2=0;Lfh1x=∂x2∂X·fX=a21Fytf+a22Fytr+a23Fys;Lfh2x=∂x4-x6∂X·fX=x3-x5;Lg1Lfh2x=∂x3-x5∂X·g1=kwIw≠0;Lg2Lfh2x=∂x3-x5∂X·g2=g23≠0;Lf2h2x=∂x3-x5∂X·fX=a31+εIwFytf+a32Fytr+a33Fys+CwIwx5+kwIwx6-kwIwδsw.It is concluded that relative degree of vehicle system is 3 less than 6, which satisfies partial feedback linearization.Coordinate transformation is necessary for the model, as(20)Z=z1,z2,z3,z4,z5,z6T=x2,x4-x6,x3-x5,-g22x1g21+x2,-g23x1g21+x3,x6,X=x1,x2,x3,x4,x5,x6T=g21z1-z4g22,z1,z5+g21z1-z4g22,z2+z6,z5-z3+g23z1-z4g22,z6T.Then, partial feedback linearization control model is as follows:(21)ξ˙=Acξ+BcγXu-αX,η˙=q0η,ξ,y=Ccξ,where ξ=z1,z2,z3T; η=z4,z5,z6T.That is,(22)ξ˙=z˙1z˙2z˙3=000001000z1z2z3+Lfh1x+Lg1h1xu1+Lg2h2xu20Lf2h2x+Lg1Lfh2xu1+Lg2Lfh2xu2,η˙=z˙4z˙5z˙6=a21-g22a11g21Fytf+a22-g22a12g21Fytr+a23-g22a13g21Fys-g22a14x2g21a31-g23a11g21Fytf+a32-g23a12g21Fytr+a33-g23a13g21Fys-g23a14x2g21x5,y=100010000z1z2z3.According to (14) and (22), it is satisfied that x2=x3=x4=x5=x6=0 if ξ=z1,z2,z3≡0. Thus, the zero dynamic equation η˙=z˙4,z˙5,z˙6T is asymptotic stable at the origin. So it is proved that the vehicle feedback linearization model could be effectively controlled by ξ˙=z˙1,z˙2,z˙3T.Defining control input variablev, as(23)v=v1v2=Lfh1x+Lg1h1xu1+Lg2h1xu2Lf2h2x+Lg1Lfh2xu1+Lg2Lfh2xu2,then initial system could be transformed as a linear system ξ˙=Acξ+Bcv.


### 4.2. Feedback Linearization Controller Design

A linear 4DOF model of tractor-semitrailer is introduced to obtain a reference for control output. Just as the controlled model (13), the reference 4DOF vehicle model also considers tractor and semitrailer’s lateral motions and yaw motions and steering wheel rotation. The only difference is that it is applied linear tire model, as (24)Fy=kα,where k is tire slip stiffness and α is tire slip angle.Based on first-order Taylor expansion of magic formula (5), k of each tractor-semitrailer’s tire could be calculated by ki=BiCiDi, whose values are shown in Table 2.The reference 4DOF vehicle model is describes as(25)X˙r=ArXr+Brδsw,where the reference state variables Xr=x1r,x2r,x3r,x4r,x5r,x6rT=vytr,φ˙tr,θ˙r,θr,δ˙fr,δfrT and the reference outputs are y1r=x1r and y2r=x4r-x6r. Ar and Br are as follows:(26)Ar=mt+ms-c+asmsasms000cmtIzt0000asmtIzs-Izs0000001000000Iw0000001-1×ktf+ktr+ks-ms+mtvxt-atktf-btktr-c+as+bsksvxtas+bsksvxtks0-ktfat+cktf+c-btktr-cmtvxt+atat+cktf+btbt-cktrvxt000-at+cktfasktf+ktr-bsks-asmtvxt+asatktf-btktr+bsc+as+bsksvxt-bsas+bsksvxt-bsks0-asktf001000-εktfvxt-atεktfvxt00-Cwεktf-Kw000010,Br=0000Kwi0.Figure8 shows a designed feedback tracking controller.Figure 8
Structure of feedback tracking controller.The control input variablev is defined as (27)v1=-λ1e1+y˙1r,v2=-λ2e2-λ3e˙2+y¨2r,where e1 and e2 are tracking errors, e1=y1-y1r and e2=y2-y2r.Equation (27) is satisfied as (28)e˙1+λ1e1=0,e¨2+λ3e˙2+λ2e=0.Therefore, the closed-loop system errorse1 and e2 are exponentially stable by reasonable choice of λ1, λ2, and λ3.


### 4.3. Control Performance Analysis

The specific control effects are mainly considered by the control stability, rapidity, and accuracy. Typical effect parameters contain peak timetp, overshoot σ%, and regulation time ts.Applying the proposed vehicle dynamic model, driving circles of step steering and single-lane changing with 20 m/s are conducted to simulate tractor-semitrailer lane changing on highway under rainy weather.Step steering is used to select control parametersλ1, λ2, and λ3 by contrasting control effects. Choosing 4 sets of data of λ1, λ2, and λ3 control effects of each set and are shown in Figure 9.Figure 9
Control effects with 4 sets of data.
(a)
Tractor yaw rate errore1
(b)
Difference of joint angle and steering angle errore2
(c)
Tractor yaw rate response and local amplification
(d)
DYC torque inputIt can be seen from Figure9 that there are slight difference among 4 sets of data. Then, maximum values of e1, e2, typical effect parameters tp, σ%, and ts and DYC torque input are, respectively, calculated, as listed in Table 4.Table 4
Calculated controlled effect parameters and DYC input torques with 4 sets of data.
No.
λ
1
λ
2
λ
3
e
1
m
a
x/rad/s
e
2
m
a
x/rad
Tractor yaw rate response
DYCTmax/N
t
p/s
σ%
t
s/s
(
1
)
500
3000
6
0.0052
0.0303
1.11
47.77
2.04
809.0810
(
2
)
300
5000
6
0.0071
0.0296
1.08
46.59
2.08
779.4590
(
3
)
200
8000
6
0.0088
0.0290
1.07
46.02
2.16
751.6507
(
4
)
100
10000
6
0.0115
0.0281
1.06
45.26
3.34
1.2294
e
+
003Table4 indicates the influence of control parameters for vehicle system control response and accuracy. It clearly showed some diametrically opposed change trend of control effects, such as e1 and e2 and σ% and ts. In consideration of response time, control input size, and system stability, when λ1=300, λ2=5000, and λ3=6, values of the peak time tp=1.08 s and regulation time ts=2.08 s are all smaller, and the DYC torque input Tmax=779.4590 N is also lower correspondingly. Because of the balance of control effects, Group 2 is determined to be as final control parameters.Based on the selected control parameters, critical scenarios of steady steering (δf=0.05) and single-lane changing (As=0.07 rad, f=0.2 Hz), just as introduced in Section 3, are simulated with the established feedback linearization controller. Typical contract of vehicle dynamic variables is shown in Figures 10 and 11.Figure 10
Typical contract of vehicle dynamic variables in critical steady steering.
(a)
Tractor yaw rate control
(b)
Joint angle controlFigure 11
Typical contract of vehicle dynamic variables in critical single-lane changing.
(a)
Tractor yaw rate control
(b)
Joint angle controlFigures10 and 11 clearly show the control effects in critical steady steering and single-lane changing conditions. It can be seen from Figure 10 that if it is without control, tractor and semitrailer yaw motions increase sharply and joint angle will reach to 0.7 rad (about 40.1 deg) in 3 seconds, which indicates that there is jackknife instability in critical steady steering. But vehicle will reach to an ideal steady state value with steady steering if applying the designed controller. Similarly, in Figure 11, vehicle dynamics is stable and the yaw motions change regularly with control in critical single-lane changing, but if it is without control, it is also obvious that the vehicles yaw sharply as yaw rate and joint angle increase rapidly (tractor yaw rate reaches about 2 rad/s in 5 seconds and joint angle reaches about 150 deg), which indicates serious sideslip and jackknife instability.Looking back at Hopf bifurcation phenomenon and instability in critical steady steering or single-lane changing, it can be seen from Figures10 and 11 that, by feedback linearization controller, combined with AFS and DYC strategy, vehicle’s dynamic characteristics could be effectively controlled and ensure stability. In Figure 12, taking the critical single-lane changing as an example, real parts of calculated eigenvalues with control are all lying in the negative half plane with As=0.07 rad, f=0.2 Hz, which indicates the control effects in eliminating bifurcation and ensure stability.Figure 12
Changes in real parts of calculated eigenvalues with control in critical single-lane changing.


## 4.1. Vehicle Control Model

Supposing the tractor longitudinal velocity constant, the vehicle model introduced in Section2 is deduced as follows:(13)X˙=fX,δsw,where X=x1,x2,x3,x4,x5,x6T=vyt,φ˙t,θ˙,θ,δ˙f,δfT.In order to linearize nonlinear systems by feedback linearization method, feedback parameters, as control outputs, not only are controllable, but also ensure linearization. Therefore, setting tractor yaw rateφ˙t and the difference of joint angle and steering angle θ-δf as control output variables, applying AFS and DYC combined control method, the feedback linearization control model could be described as(14)X˙=fX,δsw+g1u1+g2u2,y1=h1x=φ˙t,y2=h2x=θ-δf,where (15)fX,δsw=a11Fyf+a12Fyr+a13Fys+a14vxtx2a21Fyf+a22Fyr+a23Fysa31Fyf+a32Fyr+a33Fysx3-Cwx5Iw-Kwx6Iw-εFytfIw+KwiIwδswx5,(16)g1=0,0,0,0,-KwIw,0T,g2=g21,g22,g23,0,0,0T,(17)u1=δswc,u2=Mc.Variablesa11, a12, a13, and so on in (15) and g21, g22, and g23 in (16) could be expressed by known variables, as follows:(18)a11=a1+ccm2Iz2+Iz1Iz2+a22m2Iz1S,a12=c-b1cm2Iz2+Iz1Iz2+a22m2Iz1S,a13=Iz2-a2b2m2Iz1S,a14=-a22Iz1+c2Iz2m1m2-m1+m2Iz1Iz2S,a21=a1a22m1m2+a1m1+a1m2+cm2Iz2-a2m2Iz1S,a22=cm2-b1m1-b1m2Iz2-a22b1m1m2S,a23=cm1a2b2m2-Iz2S,a31=a1a2c+a2m1m2+a1m1+a1m2+cm2Iz2-a2m2Iz1S,a32=-a2b1c+a2m1m2+cm2-b1m2-b1m1Iz2-a2m2Iz1S,a33=a2Iz1-cIz2m1+b2m1+m2Iz1+cc+a2m1m2S,g21=a2Iz1-cIz2m2S,g22=-a2+ca2m1m2-m1+m2Iz2S,g23=-a2+c2m1m2-m1+m2Iz1+Iz2S,S=a22Iz1+c2Iz2m1m2+m1+m2Iz1Iz2.According to conditions of feedback linearization theory, relative degree of vehicle system should be confirmed as(19)Lg1h1x=∂x2∂X·g1=0;Lg2h2x=∂x2∂X·g2=g22≠0;Lg1h2x=∂x4-x6∂X·g1=0;Lg2h2x=∂x4-x6∂X·g2=0;Lfh1x=∂x2∂X·fX=a21Fytf+a22Fytr+a23Fys;Lfh2x=∂x4-x6∂X·fX=x3-x5;Lg1Lfh2x=∂x3-x5∂X·g1=kwIw≠0;Lg2Lfh2x=∂x3-x5∂X·g2=g23≠0;Lf2h2x=∂x3-x5∂X·fX=a31+εIwFytf+a32Fytr+a33Fys+CwIwx5+kwIwx6-kwIwδsw.It is concluded that relative degree of vehicle system is 3 less than 6, which satisfies partial feedback linearization.Coordinate transformation is necessary for the model, as(20)Z=z1,z2,z3,z4,z5,z6T=x2,x4-x6,x3-x5,-g22x1g21+x2,-g23x1g21+x3,x6,X=x1,x2,x3,x4,x5,x6T=g21z1-z4g22,z1,z5+g21z1-z4g22,z2+z6,z5-z3+g23z1-z4g22,z6T.Then, partial feedback linearization control model is as follows:(21)ξ˙=Acξ+BcγXu-αX,η˙=q0η,ξ,y=Ccξ,where ξ=z1,z2,z3T; η=z4,z5,z6T.That is,(22)ξ˙=z˙1z˙2z˙3=000001000z1z2z3+Lfh1x+Lg1h1xu1+Lg2h2xu20Lf2h2x+Lg1Lfh2xu1+Lg2Lfh2xu2,η˙=z˙4z˙5z˙6=a21-g22a11g21Fytf+a22-g22a12g21Fytr+a23-g22a13g21Fys-g22a14x2g21a31-g23a11g21Fytf+a32-g23a12g21Fytr+a33-g23a13g21Fys-g23a14x2g21x5,y=100010000z1z2z3.According to (14) and (22), it is satisfied that x2=x3=x4=x5=x6=0 if ξ=z1,z2,z3≡0. Thus, the zero dynamic equation η˙=z˙4,z˙5,z˙6T is asymptotic stable at the origin. So it is proved that the vehicle feedback linearization model could be effectively controlled by ξ˙=z˙1,z˙2,z˙3T.Defining control input variablev, as(23)v=v1v2=Lfh1x+Lg1h1xu1+Lg2h1xu2Lf2h2x+Lg1Lfh2xu1+Lg2Lfh2xu2,then initial system could be transformed as a linear system ξ˙=Acξ+Bcv.


## 4.2. Feedback Linearization Controller Design

A linear 4DOF model of tractor-semitrailer is introduced to obtain a reference for control output. Just as the controlled model (13), the reference 4DOF vehicle model also considers tractor and semitrailer’s lateral motions and yaw motions and steering wheel rotation. The only difference is that it is applied linear tire model, as (24)Fy=kα,where k is tire slip stiffness and α is tire slip angle.Based on first-order Taylor expansion of magic formula (5), k of each tractor-semitrailer’s tire could be calculated by ki=BiCiDi, whose values are shown in Table 2.The reference 4DOF vehicle model is describes as(25)X˙r=ArXr+Brδsw,where the reference state variables Xr=x1r,x2r,x3r,x4r,x5r,x6rT=vytr,φ˙tr,θ˙r,θr,δ˙fr,δfrT and the reference outputs are y1r=x1r and y2r=x4r-x6r. Ar and Br are as follows:(26)Ar=mt+ms-c+asmsasms000cmtIzt0000asmtIzs-Izs0000001000000Iw0000001-1×ktf+ktr+ks-ms+mtvxt-atktf-btktr-c+as+bsksvxtas+bsksvxtks0-ktfat+cktf+c-btktr-cmtvxt+atat+cktf+btbt-cktrvxt000-at+cktfasktf+ktr-bsks-asmtvxt+asatktf-btktr+bsc+as+bsksvxt-bsas+bsksvxt-bsks0-asktf001000-εktfvxt-atεktfvxt00-Cwεktf-Kw000010,Br=0000Kwi0.Figure8 shows a designed feedback tracking controller.Figure 8
Structure of feedback tracking controller.The control input variablev is defined as (27)v1=-λ1e1+y˙1r,v2=-λ2e2-λ3e˙2+y¨2r,where e1 and e2 are tracking errors, e1=y1-y1r and e2=y2-y2r.Equation (27) is satisfied as (28)e˙1+λ1e1=0,e¨2+λ3e˙2+λ2e=0.Therefore, the closed-loop system errorse1 and e2 are exponentially stable by reasonable choice of λ1, λ2, and λ3.


## 4.3. Control Performance Analysis

The specific control effects are mainly considered by the control stability, rapidity, and accuracy. Typical effect parameters contain peak timetp, overshoot σ%, and regulation time ts.Applying the proposed vehicle dynamic model, driving circles of step steering and single-lane changing with 20 m/s are conducted to simulate tractor-semitrailer lane changing on highway under rainy weather.Step steering is used to select control parametersλ1, λ2, and λ3 by contrasting control effects. Choosing 4 sets of data of λ1, λ2, and λ3 control effects of each set and are shown in Figure 9.Figure 9
Control effects with 4 sets of data.
(a)
Tractor yaw rate errore1
(b)
Difference of joint angle and steering angle errore2
(c)
Tractor yaw rate response and local amplification
(d)
DYC torque inputIt can be seen from Figure9 that there are slight difference among 4 sets of data. Then, maximum values of e1, e2, typical effect parameters tp, σ%, and ts and DYC torque input are, respectively, calculated, as listed in Table 4.Table 4
Calculated controlled effect parameters and DYC input torques with 4 sets of data.
No.
λ
1
λ
2
λ
3
e
1
m
a
x/rad/s
e
2
m
a
x/rad
Tractor yaw rate response
DYCTmax/N
t
p/s
σ%
t
s/s
(
1
)
500
3000
6
0.0052
0.0303
1.11
47.77
2.04
809.0810
(
2
)
300
5000
6
0.0071
0.0296
1.08
46.59
2.08
779.4590
(
3
)
200
8000
6
0.0088
0.0290
1.07
46.02
2.16
751.6507
(
4
)
100
10000
6
0.0115
0.0281
1.06
45.26
3.34
1.2294
e
+
003Table4 indicates the influence of control parameters for vehicle system control response and accuracy. It clearly showed some diametrically opposed change trend of control effects, such as e1 and e2 and σ% and ts. In consideration of response time, control input size, and system stability, when λ1=300, λ2=5000, and λ3=6, values of the peak time tp=1.08 s and regulation time ts=2.08 s are all smaller, and the DYC torque input Tmax=779.4590 N is also lower correspondingly. Because of the balance of control effects, Group 2 is determined to be as final control parameters.Based on the selected control parameters, critical scenarios of steady steering (δf=0.05) and single-lane changing (As=0.07 rad, f=0.2 Hz), just as introduced in Section 3, are simulated with the established feedback linearization controller. Typical contract of vehicle dynamic variables is shown in Figures 10 and 11.Figure 10
Typical contract of vehicle dynamic variables in critical steady steering.
(a)
Tractor yaw rate control
(b)
Joint angle controlFigure 11
Typical contract of vehicle dynamic variables in critical single-lane changing.
(a)
Tractor yaw rate control
(b)
Joint angle controlFigures10 and 11 clearly show the control effects in critical steady steering and single-lane changing conditions. It can be seen from Figure 10 that if it is without control, tractor and semitrailer yaw motions increase sharply and joint angle will reach to 0.7 rad (about 40.1 deg) in 3 seconds, which indicates that there is jackknife instability in critical steady steering. But vehicle will reach to an ideal steady state value with steady steering if applying the designed controller. Similarly, in Figure 11, vehicle dynamics is stable and the yaw motions change regularly with control in critical single-lane changing, but if it is without control, it is also obvious that the vehicles yaw sharply as yaw rate and joint angle increase rapidly (tractor yaw rate reaches about 2 rad/s in 5 seconds and joint angle reaches about 150 deg), which indicates serious sideslip and jackknife instability.Looking back at Hopf bifurcation phenomenon and instability in critical steady steering or single-lane changing, it can be seen from Figures10 and 11 that, by feedback linearization controller, combined with AFS and DYC strategy, vehicle’s dynamic characteristics could be effectively controlled and ensure stability. In Figure 12, taking the critical single-lane changing as an example, real parts of calculated eigenvalues with control are all lying in the negative half plane with As=0.07 rad, f=0.2 Hz, which indicates the control effects in eliminating bifurcation and ensure stability.Figure 12
Changes in real parts of calculated eigenvalues with control in critical single-lane changing.


## 5. Discussion and Conclusion

In this paper, considering the high degree of danger and complex maneuvers for tractor-semitrailer on low adhesion coefficient road, the nonlinear lane changing behaviors on highways under rainy weather have been studied with a verified 5DOF mechanical model based on a test experiment, and a feedback linearization controller by DYC and AFS strategy is designed to ensure stability in critical scenarios. Points of discussion and conclusions can be summarized as follows:(1)
Nonlinear theory is used to analyze the relationship between the eigenvalues of the Jacobian matrix and vehicle stability. Considering the transient response, the eigenvalues of Jacobian matrix may be positive or have positive real parts as steering wheel angle approaches peak, but the vehicle system is still stable with the angle returns. Therefore, unlike steady steering, transient stability is satisfied if there are transient positive real parts eigenvalues, so it is definitely improved that eigenvalues lying in the negative half plane is sufficient but not necessary for judging stability. It is also clearly concluded that Hopf bifurcation occurs with critical steady and sinusoidal steering, which translates into an oscillatory behavior leading to instability.(2)
It is realized for application feedback linearization theory in tractor-semitrailer stability control. Design process of an integrated controller is detailed presented with vehicle mechanical model, which contains model building, feedback linearization, and control parameters selection. The consequences of performance analysis introduce new insight into the vehicle nonlinear stability control. The eigenvalues analysis results also provide a substantive understanding on control principle.(3)
Through analyzing the vehicle nonlinear stability in special driving conditions, the dynamic rule and stability mechanism for tractor-semitrailers changing lanes at high speeds under rainy weather are well mastered. The bifurcation analysis and nonlinear control provided could be informative for research into vehicle dynamics and active safety control design.Furthermore, tractor-semitrailers have complex nonlinear behaviors, especially considering driver and environment characteristics. The following could be considered in the future:(1)
The driver is always an early decision-maker for vehicle movement; it is essential to study the lane-changing nonlinear behaviors of tractor-semitrailers associated with driver characteristics.(2)
As the study on adhesion coefficient caused by rainy weather, the influence of the other meteorological effects, such as rain for driver sight distance and high winds for vehicle dynamic, should be taken into account to comprehensively analyze the driving stability of tractor-semitrailers.(3)
Synergetic thought should be introduced creatively to tractor-semitrailers, not only for single tractor and semitrailer internal relationship, but also for cooperative operation management of a dynamic motorcade, under a driver/vehicle/road integrated system.As we know, traffic safety is the interaction results of human, vehicle, and environment. First, based on the essence of stability mechanism and nonlinear dynamic characteristics of tractor-semitrailer, advanced active control devices similar to ABS and ESC, should be developed and equipped, especially on slippery pavement. Second, large vehicle drivers’ physiological and psychological characteristics should be paid more attention and it is necessary to research and apply ADAS (Advanced Driver Assistance Systems), such as fatigue warning, auxiliary brake, and Lane keeping assist system. Last but not least, with the development of Internet technology, ITS (Intelligent Traffic System) should be widely applied to build a harmonious running state.



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*Source: 3506053-2017-10-19.xml*