With increasing and more stringent requirements for advanced vehicle integration, including vehicle dynamics and control, traditional control and optimization strategies may not qualify for many applications. This is because, among other factors, they do not consider the nonlinear characteristics of practical systems. Moreover, the vehicle wheel model has some inadequacies regarding the sideslip angle, road adhesion coefficient, vertical load, and velocity. In this paper, an adaptive neural wheel network is introduced, and the interaction between the lateral and vertical dynamics of the vehicle is analyzed. By means of nonlinear analyses such as the use of a bifurcation diagram and the Lyapunov exponent, the vehicle is shown to exhibit complicated motions with increasing forward speed. Furthermore, electric power steering (EPS) and active suspension system (ASS), which are based on intelligent control, are used to reduce the nonlinear effect, and a negotiation algorithm is designed to manage the interdependences and conflicts among handling stability, driving smoothness, and safety. Further, a rapid control prototype was built using the hardwareintheloop simulation platform dSPACE and used to conduct a real vehicle test. The results of the test were consistent with those of the simulation, thereby validating the proposed control.
When a car is travelling at a high speed, a slight variation of the steering could result in a crash. Moreover, steering instability is primarily caused by the lateral force acting on the steering wheels, which is particularly affected by the sideslip angle [
Currently, vehicle dynamics is mostly studied by means of nonlinear dynamics. For example, Wu and Sheng [
Shi et al. [
Liu et al. [
Ono et al. [
The steering and suspension are two important subsystems of the chassis of a vehicle, and both directly affect the overall vehicle performance, including handling stability, driving smoothness, and safety. The two systems are coupled and interact with each other. The suspension causes significant variations in the vertical force acting on the wheel, whereas the steering affects the lateral force, which in turn affects the overall lateral dynamics [
Studies have been conducted on the integrated control of electric power steering (EPS) and active suspension system (ASS) for complex nonlinear timevarying vehicle dynamics. The lateral and vertical dynamics of the vehicle were evaluated using different indexes, control strategies, and wheel dynamics. According to the linear superposition principle, if the lateral and vertical dynamics are separately controlled, the determined comprehensive properties would not be accurate. Over the past few years, diverse intelligent control strategies for integrated control have been proposed. March and Shim [
Chen et al. [
In the present study, a fuzzy control method was used to reduce the nonlinear effect, and a coordination mechanism was introduced to manage the interdependences and conflicts among handling stability, driving smoothness, and safety. Moreover, computer simulations were used to validate the bifurcation and chaotic motions implied by the integrated nonlinear differential equations that describe the vehicle dynamics. Finally, to conduct a real vehicle test, a rapid control prototype was built using the hardwareintheloop simulation platform dSPACE. The results of the test were consistent with those of the simulation, thereby validating the proposed control.
To develop the vehicle dynamics model based on the steering working condition, the effect of the ground tangential force on the cornering properties of the wheel and the aerodynamics were ignored, as shown in Figure
Vehicle dynamics model.
During the steering process of the vehicle, the effect of the roll angle and yaw velocity on changes in the sideslip angle cannot be ignored. The equations of motion of the vehicle were therefore derived by taking the effect of the roll angle into consideration and are as follows:
lateral movement of the vehicle:
Taking into consideration the effect of the stabilizer bar on the inclination angle of the vehicle body, the resultant force of the suspension is given by the following:
When the pitch angle
When a real vehicle is traveling on a nonflat road, the effect of the road on the wheel varies, which makes it necessary to obtain a dynamic model that is adaptable to environmental changes. The adaptive neural network (ANN) is more appropriate and is as follows:
ANN includes the input layer, hidden layer, and output layer. The input layer contains five variables, namely, the sideslip angle of the wheel
The input of the neuron in the hidden layer of the network is given by
The relationship between the input and output of the hidden layer is given by the following sigmoid function, which is an expression of the output of the neuron in the hidden layer:
The output layer consists of one neuron, the input function of which is as follows:
The relationship between the input and output of the output layer is given by the following sigmoid function:
Let us assume that the inclination angles of the left and rightside wheels are equal and that the inclination angles of the front and rear wheels can be expressed as
For our experiments, we used a 165/65R13 Radial Tyre (Hankooktire). The equipments used for the experiments included a Vehicle Tyre Road Rotating Test Stand (Figure
Analytical tire contact system.
Based on the characteristics of the Magic formula model, a comparative analysis of the fitting performances of the adaptive and Magic formula models was conducted using experimental tire data for different loads and with the assumption of steady working conditions, namely, a wheel speed of 20 km/h and inflation pressure of 250 kPa (Figure
Model fitting.
Figure
Interaction between the suspension and steering systems.
It can be seen from (
An analytical solution of the nonlinear dynamic system of a coupled neural network is extremely cumbersome, which makes its actual application difficult. It is thus necessary to obtain an approximate solution by a numerical method, that is, to convert the differential differential equation to a differential equation. In this study, Matlab/Simulink was used to develop a simulation model, the calculation flow of which is shown in Figure
Vehicle parameters.
Parameter  Value 

Kerb weight/kg  900 
Maximum total weight/kg  1330 
Unsprung mass (front/rear)/kg  35/33 
Front axle weight (empty/full)/kg  540/640 
Wheel base/mm  2335 
Front axlecentred distance/mm  955 
Rear axlecentred distance/mm  1380 
Front/rear wheel tread/mm  1360/1355 
Vehicle body 
3400/1575/1670 
Wheel model/mm  165/65R13 
Calculation flowchart.
The maximum Lyapunov exponent method and the phase plane method are the major methods used to examine the nonlinear system in this paper. In mathematics, the Lyapunov exponent characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, the rate at which two trajectories separated by
The rate of separation may be different for different orientations of the initial separation vector. Thus, there is a spectrum of Lyapunov exponents, the number of which is equal to the dimensionality of the phase space. The largest exponent is commonly referred to as the maximal Lyapunov exponent (MLE) because it determines the predictability for a dynamic system. A positive MLE is usually considered as an indication that the system is chaotic (provided some other conditions are met, e.g., the compactness of the phase space). It should be noted that an arbitrary initial separation vector would typically have a component in the direction of the MLE, and the exponential growth rate would obliterate the effect of the other exponents over time.
The maximal Lyapunov exponent is defined as follows:
When computing the maximal Lyapunov exponent, the time series should be reconstructed to determine the closest point
The separation interval is defined as follows:
It is assume that
For every point
Assuming that the closest point of
A phase plane is a visual display of certain characteristics of certain types of differential equations. However, a coordinate plane is a plane whose axes represent two state variables such as
The yaw angle rate
In the bifurcation diagram (Figure
Bifurcation diagram.
Figure
Lyapunov exponent diagram.
The input signals and system state are specified by
Description of control mode.
Identification  Rule  Working condition 

1 

Assist 
2 

Damping 
3 

Aligning 
Conditions of EPS hybrid control system.
Identification  Logical expression 













The schedule of the fuzzy rules is based on the working condition. In the assist condition, a seamless velocity assist mode is employed using a velocity range of 0–200 km/h and intervals of 20 km/h. For each velocity, the assist provided by the motor can be divided into three segments, which are as follows. When the steering wheel input torque is in the range of 01 Nm, there is a boosting dead zone and no assist is provided by the motor. When the input torque is in the range of 1–6 Nm, there is an assist zone. When the torque is beyond this range, there will be a 6 Nm assist saturated zone. The four parameters used for designing linear assist are the steering wheel input torque
Characteristics of a linear assist.
Based on the requirements and objectives, the aligning condition is composed of two parts. One is the aligning control, the major function of which is the provision of necessary assist for easy steering of the wheel back to the central position. In this process, the aligning control functions as a PI adjuster for adjusting the target steering wheel position
The other part is damping control, the major function of which is to bring the vehicle back to the central position and avoid the shimmy events under the damping condition. During this process, a certain control voltage is generated based on the angular velocity of the aligning process, which produces a certain damping torque in the motor. The quicker the steering wheel spins, the higher the control voltage generated and the larger the damping torque is. Conversely, the slower the steering wheel spins, the smaller the damping torque is. The steering wheel alignment speed can therefore be adjusted by adjusting the damping coefficient:
The alignment control and the damping control can be integrated in one PID returning control algorithm. Moreover, by adjusting the coefficient, alignments that produce different effects can be obtained as expressed by the following:
In the damping condition, the operation status of the system is determined by the value of the deviation
The input variable
Fuzzy rule table.

NB  NM  NS  ZO  PS  PM  PB 

NB  PB  PB  PM  PM  PS  ZO  ZO 
NM  PB  PB  PS  ZO  NS  ZO  NS 
NS  PM  PM  PM  PS  ZO  NS  NS 
ZO  PM  PM  PS  ZO  NS  NM  NM 
PS  PS  PS  ZO  NS  NS  NM  NM 
PM  PS  PS  ZO  NS  NM  NM  NB 
PB  ZO  ZO  NM  NM  NM  NB  NB 
Membership function of variable
In the external state
To rank the changing amplitudes of
Steering
Medium and low speed
High speed
The fuzzy logic system can be described as follows:
In this formula,
s.t. If “stability”
If “comfort”
If “safety”
The fuzzification and defuzzification processes of the fuzzy control of this study were designed using the fuzzy logic toolbox in MATLAB/Simulink (see Figure
Fuzzy controller interface in Matlab/Simulink.
Figure
Schematic diagram.
The logical process of the negotiation algorithm is as follows (see Figure
input: road noise
output: the controlled current
State flow of the controlling strategy.
Categorize the operation conditions into steering control, suspension control, and coordination control based on the external input
With the whole vehicle model, after obtaining signals
After receiving orders for the subtargets, the EPS receives signals for the steering wheel torque
The controller coprocesses all the subsystems collected during the sequences of vague relational weights as follows:
input: the proposals for
output: the proposals for
Compute
If
Compute the concession
Output the proposal.
Using the above task integration, the return results for the entire operation condition can be obtained to output the controlled current
The tests were performed using the control system shown in Figure
Rapid control system.
Entire control loop.
Main parts of the test and control systems.
Speed sensors installed on the vehicle
Test car
Gyroscope
Data acquisition box
To validate the control, a step input road and various Sshaped operation conditions were used to perform joint simulations for different speeds of a passive and active vehicle, respectively. The test vehicle was equipped with the original passive suspension system and then with the active suspension system with chaos control.
A simulation analysis of the angle step steering input operation conditions can be used to illustrate the improved status of the ASS achieved by stabilizing the handling of the vehicle.
The road surface was simulated using a vehicle speed of
A simulation analysis of the angle step steering input operation conditions on a pulse road can be used to illustrate the improved status achieved by stabilizing the handling and ride comfort of the vehicle. The operation condition used for the simulation consisted of a velocity of 120 km/h (33 m/s) and front wheel steering angle of 6°. For a dry asphalt road, when the vehicle arrived at the 130 m point after 4 s, the front and rear axle wheels were excited by the pulse of the road with an amplitude of 5 cm.
Figures
Vehicle results for angle step steering input.
Slip angle of vehicle body
Yaw velocity of vehicle body
Phase portrait without control
Phase portrait with intelligent control
Meanwhile, it can be seen from the phase portraits in Figures
Under the Sshaped operation condition, the slip angle and yaw velocity when the intelligent controller is applied are significantly better than the others as shown in Figures
Vehicle results for Sshaped input.
Slip angle of vehicle body
Yaw velocity of vehicle body
Phase portrait without control
Phase portrait with intelligent control
In Table
Lyapunov exponents for road test.
Measurement points  Lyapunov exponents  

Slip angle  Step input  PID control  1.89 
Intelligent control 


Snake input  PID control  2.29  
Intelligent control 




Yaw velocity  Step input  PID control  3.18 
Intelligent control 


Snake input  PID control  5.16  
Intelligent control 

It can be seen from Figures
Results for steppulse road.
In this paper, we considered the coupling mechanism of advanced vehicle integration, as well as adaptive neural network of the vehicle wheels. By nonlinear analysis such as the use of bifurcation diagram and the Lyapunov exponent, it was shown that the lateral dynamics of the vehicle was characterized by complicated motions with increasing forward speed. Electric power steering and active suspension system based on intelligent control were used to reduce the nonlinearity, and a negotiation algorithm was designed to manage the interdependences and conflicts among handling stability, driving smoothness, and safety. The results of rapid control prototyping confirmed the feasibility of the proposed control method. By comparing the time response diagrams, phase portraits, and Lyapunov exponents for different operation conditions, we observed that the slip angle and yaw velocity of the lateral dynamics entered a stable domain and the chaotic motions were successfully suppressed. It was also shown that the safety was significantly improved. Under the angle step steering input operation conditions on the pulse road, the proposed intelligent control significantly improved the ride comfort and handling stability. The uneven pulse excitation of the road surface had almost no effect on the manipulation of the vehicle. Future work will focus on optimizing and improving the robustness of the control and the information constraints and faults of the integrated vehicle dynamics.
Qualities of the vehicle and the suspension, respectively
Velocity
Slip angle and slip angular speed, respectively
Height of the center of gravity of the vehicle under roll condition
Roll angle, roll angular speed, and roll angular acceleration of the vehicle, respectively
Lateral force on four wheels
Rotary inertia of the vehicle
Yaw rate and yaw acceleration, respectively
Distances from the front and rear wheels to the center of the body, respectively
Vertical displacement, vertical velocity, and vertical acceleration of the vehicle body, respectively
Composite force exerted by the suspension on the vehicle body
Pitching rotary inertia and roll rotary inertia of the vehicle, respectively
1/2 of the track
Unsuspended mass
Stiffness of the wheels
Displacement of the road
Vertical displacement, vertical velocity, and vertical acceleration of the unsuspended mass
Stiffness of the stabilizer bar angle of the front and rear suspensions, respectively
Stiffness of the suspension
Damping coefficient of the suspension
Vertical displacement, vertical velocity, and vertical acceleration of the suspended mass, respectively
Stiffness of the stabilizer rods of the front and rear suspensions, respectively
Input of the jth neuron of the hidden layer
Output of the input layer
Weight of the connection between the input and hidden layers
Output of the
Weight of the connection between the hidden and output layers
Output of the neuron of the output layer
Inclination angles of the leftside and rightside wheels of the front axle, respectively
Inclination angles of the leftside and rightside wheels of the rear axle, respectively
Lyapunov exponent
Returning control voltage of the motor
Angle of the steering wheel
Deviation between the target and actual steering angles
Proportionality coefficient
Integral coefficient
Returning control voltage of the motor
Integral coefficient
Aligning control voltage of the motor
Deviation between the target and actual damping currents
Armature current for the target damping torque
Maximum deviation between the target and actual damping currents
PWM duty ratio corresponding to
PWM duty ratio corresponding to
The authors declare that there is no conflict of interests regarding the publication of this paper.
This project was supported by the National Natural Science Foundation of China (Grant No. 50875112, 51305167), the Ph.D. Programs Foundation of the Ministry of Education of China (Grant no. 20093227110013, 20103227120011), the Jiangsu Municipal Natural Science Foundation (Grant no. BK2010337), and the Natural Science Foundation of Higher Education of Jiangsu (Grant no. 09KJA580001).