J. Cent. South Univ. (2016) 23: 1142-1151

DOI: 10.1007/s11771-016-0364-9

Design of an optimal active stabilizer mechanism for enhancing vehicle rolling resistance

Yaghoub Pourasad¹, Mehdi Mahmoodi-k², Majid Oveisi³

1. Faculty of Electrical Engineering, Urmia university of Technology, Urmia, Iran;

2. Young Researches and Elite Club, Tabriz Branch, Islamic Azad University, Tabriz, Iran;

3. Department of Marine Engineering, Chabahar Maritime University, Chabahar, Iran

Central South University Press and Springer-Verlag Berlin Heidelberg 2016

Abstract: Improving rollover and stability of the vehicles is the indispensable part of automotive research to prevent vehicle rollover and crashes. The main objective of this work is to develop active control mechanism based on fuzzy logic controller (FLC) and linear quadratic regulator (LQR) for improving vehicle path following, roll and handling performances simultaneously. 3-DOF vehicle model including yaw rate, lateral velocity (lateral dynamic) and roll angle (roll dynamic) were developed. The controller produces optimal moment to increase stability and roll margin of vehicle by receiving the steering angle as an input and vehicle variables as a feedback signal. The effectiveness of proposed controller and vehicle model were evaluated during fishhook and single lane-change maneuvers. Simulation results demonstrate that in both cases (FLC and LQR controllers) by reducing roll angle, lateral acceleration and side slip angles remain under 0.6g and 4° during maneuver, which ensures vehicle stability and handling properties. Finally, the sensitivity and robustness analysis of developed controller for varying longitudinal speeds were investigated.

Key words: vehicle dynamics; rollover; handling; FLC; LQR

1 Introduction

Vehicle ride comfort, handling and rollover resistance are the main performance criteria for the vehicles industry. When vehicle is in motion, the vibration from the road surface negatively impacts the ride comfort, handling stability and speed. Better performance in rollover resistance tests could be achieved by one or more of the four ways: 1) Increasing vehicle under steerability properties by reducing distance between center of gravity to rear axle or increasing rear suspension system stiffness in contrast to front axle suspension system. This configuration may make the vehicle spin in limit maneuvers and possibly reduce riding comfort; 2) Lowering the roll center height of the vehicle by optimizing suspension geometry which increases side slip before rollover occurs; 3) Increasing the suspension stiffness and/or damping will reduce the body roll of the vehicle, thereby increasing vertical load transfer on the tires and decreasing the lateral force between the tires and the road; 4) An additional mechanism and active control system to increase roll stiffness, which can be achieved by applying anti-roll bar system or active/semi active suspension system.

There are two perspectives for handling ratings. One perspective is how safe the vehicle is to drive. The other is how well the vehicle gives an enthusiast driver a pleasurable sense of control. Suspension system and its components (stiffness and damping rates) have significant role in vehicle motion and handling properties. Coil springs which usually are used in passenger vehicle suspension system are less stiff with better ride comfort compared to leaf springs. Therefore, the roll stiffness of the vehicle with coil springs is usually less than that in vehicles with leaf springs. In order to improve the roll characteristic of coil springs, an antiroll bar mechanism must be used (Fig. 1) [1].

Various mechanisms such as optimal geometry passive [2], semi active [3], active [4] suspensions and anti-roll bar [5] systems have been studied to compromise the trade-offs between ride and handling. Due to active suspension systems cost, energy usage and packaging difficulties, they are used on a limited basis for luxury and special vehicles. Whilst, anti-roll bar recently has become a practical solution to improve vehicle handling, roll-stability and ride comfort.

Also, active control systems such as direct yaw moment control (YDC), active front steering (AFS) and vehicle dynamic control (VDC) are developed in previous researches to improve vehicle stability and roll resistance [6-10]. PAPELIS et al [7] utilized electronic stability control (ESC) to assist driver in maintaining vehicle control and prevention of vehicle loss of control.

Their results showed that using ESC reduces 25% loss of control in severe maneuvers.

Fig. 1 Anti-roll bar system in passenger car suspension [1]

Previous researches studied the mechanism and systems for prevention of rollover while ensuring not to satisfy handling and ride properties. In Ref. [8], a novel FLC was proposed to improve vehicle safety by regulating dampers in semi active suspension system. The controller stabilized the sprung mass which improved vehicle handling and reduced the possibility of overturning. SINGH et al [9] presented an active roll control system using an active suspension based on FLC for enhancement of vehicle roll dynamics.

MASHADI et al [10] developed a DYC system based on LQR for vehicle path following considering the driver performance. They used 2-DOF vehicle model for developing of LQR controller. Their results showed that LQR controller could be properly applied for vehicle path following and improving handling characteristic. They tried to maintain the lateral acceleration and velocity at 0.8 m/s² and 40 m/s, respectively.

In previous studies, various control strategies were utilized to control vehicle path control, improving ride, handling and rolling resistance systems [11-16]. WU et al [11] developed a robust chassis controller to improve vehicle handling performance and lane keep ability by utilizing H_∞ controller. MASHADI et al [10, 12] proposed an optimal LQR and PID controllers for vehicle path following based on intelligent optimization method. They utilized genetic algorithm procedure to adapt a set of optimized controller parameters suitable for various driving styles, road conditions and the initial errors of vehicle position and orientation.

In Ref. [17], the integrated control system based on LQR and sliding mode theories was exploited for yaw moment and wheel slip controller design, respectively. Also the validation of proposed controller was done by Hardware In-the-Loop Simulation (HILS) in various maneuvers. SONG and CHE [18] proposed a braking yaw motion controller and a steering yaw motion controller to improve the yaw rate response and body slip angle response of the vehicle. They used 15-DOF vehicle model, simplified steering system model and driver model to evaluate the proposed controllers. Since steering control system had four wheel steering, so it showed better performance than the yaw control system. However, it should be noted that yaw moment is most appropriate even when the vehicle passes a critical maneuver in the non-linear regime [19-20]. A unified chassis control is proposed in Ref. [21] to prevent vehicle rollover, and to improve vehicle maneuverability and its lateral stability by integrating electronic stability control (ESC) and AFS. The rollover prevention was achieved through speed control, and the vehicle stability was improved via yaw rate control. CAO et al [22] proposed the integrated controller of the yaw and rollover stability controls using nonlinear prediction model. The linearization tire model was built up as the rollover predictive model and its accuracy obtained by real vehicle tests.

These researches were focused on improving handling, ride or rollover resistance separately by active control systems. In this work, a tradeoff between handling properties and rollover resistance compromised through anti roll bar moment system which was controlled by two strategies based on FLC and LQR.

YIM and YI [23] presented a combination of an active roll control system with chassis control for controlling yaw and roll dynamics of hybrid electric vehicles. They verified the effectiveness integrated controller by simulations on the vehicle simulation software CarSim. SHIM and VELUSAMY [24] used active geometry suspension system to maintain stability of economy vehicles over emergency maneuvers. They evaluated the effects of passive MacPherson and multi-link suspension systems in front and rear axles respectively, to achieve critical hardpoints and geometry and improve roll stability of the vehicle.

HER et al [25] described an integrated control strategy of the differential braking, the semiactive suspension damper and the active roll moment to modify the agility and the stability of a vehicle. Yaw and moment in upper and brake distribution torque in lower controllers were developed based on sliding mode theory.

The purpose of a vehicle active control system is to balance the tradeoff between roll resistance and handling performance. This is achieved by controlling the momentum and motion between the vehicle body and wheels in order to follow driver desired path with maximum stability. Control strategies (LQR and FLC) provide the best tradeoff between roll quality and handling performance by considering 3-DOF roll and 2-DOF lateral dynamics, respectively. The proposed control structures are typically based on combination of the steer angle, roll angle, longitudinal speed, yaw and roll moments, lateral acceleration and velocity, which controls the yaw rate and the roll motion simultaneously.

2 Vehicle modelling

In order to investigate the roll over resistance and handling behavior of vehicle simultaneously, integrated lateral and roll dynamics should be considered. So, two vehicle models were utilized in this work. At first, a 3-DOF roll model for controller designing is developed whose degrees includes: lateral velocity, yaw and roll. In addition to realize the simulating of the vehicle model, a comprehensive vehicle model (an 8-DOF model) in Ref.[5] is utilized to simulation.

2.1 An 8 DOF model

To simulate the control of a vehicle during roll tests, as demonstrated in Fig. 2, a non-linear 8-DOF is used. The DOFs associated with this model are the longitudinal velocity u, the lateral velocity v, the yaw rate r, the roll Φ, and four wheel rotational speeds, ω_fl, ω_fr, ω_rl and ω_rr.

                            (1)

                     (2)

                   (3)

                   (4)

                            (5)

                           (6)

     (7)

       (8)

                   (9)

Fig. 2 8-DOF vehicle model including roll, yaw and longitudinal model

In this work, a typical front wheel steering passenger car is utilized whose steering angles are

                     (10)

2.2 Tire model

Tire is the connection element of vehicle to road, and the whole forces acting on vehicle are applied through tire, so modeling of nonlinear tire properties is critical in design process. The mathematical model proposed by PACEJKA [26] is used to the analysis of lateral force due to side slip of the tire. It is commonly called PACEJKA’s theory and is related to the tire cornering characteristics which overally takes into account the interactions between longitudinal and side forces as:

                (11)

Vertical loads acting on tires can be deduced as [27]:

     (12)

     (13)

 (14)

 (15)

2.3 Vehicle roll model

In order to evaluate the roll behavior of vehicle during sever maneuvers, the main parameters which effects the vehicle roll stability should be considered as: variation of center of gravity, vehicle track, tire and suspension properties. As shown in Fig. 4, a 3-DOF vehicle model which combines 2-DOD bicycle model (r and v) with 1-DOF roll model (f) is developed.

By imposing Newton’s second law of motion along the y-axis and the torque balance around the z-axis and x-axis, the equation of motion for lateral, yaw and roll dynamic can be expressed respectively as:

                (16)

                 (17)

   (18)

where a_y, r and f are lateral acceleration, yaw rate and roll angle, respectively. M_f and m_s are sprung and roll mass, respectively, and h_s is the distance between central of gravity to roll center of sprung mass. Also, F_yf and F_yr are lateral forces of front and rear tires, respectively. K_f and C_f are roll stiffness and roll damping coefficients of suspension system which are calculated through damping (C_s), springs (K_s) and anti-roll bar (K_arb) stiffness of rear and front axels:

          (19)

                        (20)

where t is vehicle track.

Fig. 3 3-DOF vehicle model (yaw and roll):

2.4 Load transfer

If the vehicle body rolls, the left and right wheels at both front and rear axles will increase in load at one side and decrease at the other side. This is called the load transfer due to roll. Defining the load transfer for the axle load △F; the roll moment around the roll center at the wheels in the plane perpendicular to the vehicle longitudinal direction has to be in equilibrium, as shown in Fig. 4, the following equations are derived:

        (21)

These equations give the load transfer between the left and right wheels due to a constant lateral acceleration. The equations show that a higher vehicle center of gravity distance from the roll axis, h_s, results in a larger load transfer at the front and rear wheels. Furthermore, a load transfer at the front and rear wheels is basically proportional to the front and rear roll stiffness ratios to the total roll stiffness, respectively.

Fig. 4 Vehicle load transfer in roll mode

The original rollover resistance ratings are determined solely from the vehicle’s static stability factor (SSF, F_ss). The SSF of a vehicle is a calculation of rollover resistance, based on its most important geometric properties. As shown in Fig. 4, vehicle’s SSF is calculated taking the moments about the center of contact for the outside tires yields:

            (22)

                            (23)

                                 (24)

The lower the SSF number is, the more likely the vehicle is to roll over in a tripped single-vehicle crash. A higher SSF value equates to a more stable and less top-heavy vehicle. SSF values across all vehicle types typically range from around 1.00 to 1.50.

3 Controller design

In this work, two types of controller are developed, one is based on fuzzy logic controller for nonlinear vehicle model and the other is based on linear quadratic regulation for linear vehicle roll model. In both cases, front steering angle and momentum deduced by controller for prevention of increasing roll angle are defined as system inputs. State variables include lateral velocity, yaw rate, roll angle and roll rate:

                           (25)

By considering 3 DoF vehicle model in previous section, state space for designing controller is described as

 (26)

Since the main objective is to reduce roll angle or increase rollover threshold (stability), so controller should regulate the lateral load transfer (LLT, T_ll) in its optimal condition. LLT criteria as a function of lateral acceleration, roll and roll rate is obtained as

      (27)

When one of the tires losses its contact with road and the applied force on tire eliminates, rollover occurs. In this condition, LLT becomes 1 and rollover happens. In other words, when the forces at the both sides are equal, LLT becomes zero and vehicle is in the most stable condition.

3.1 Optimal control

According to optimal control theory, cost function based on rolling angle should be minimized formulated as

   (28)

where X_d is desired state variables. Due to enhancement the desired roll angle, linearized state space solves in steady state condition. Since and should be zero in steady state condition, so according to Eq. (18) for vehicle roll dynamic, desired roll angle could be calculated as

                          (29)

Also, Q is a positive semi-definite state weighting matrix, R is the positive semi-definite control weighting matrix which is calculated by trial and error. Initial value of these weighting factors are considered as:

         (30)

                                  (31)

Minimum performance index is obtained by solving riccati equation:

                (32)

where P is:

                                (33)

By analytical solving, controller input is calculated as:

                     (34)

Afterwards, linear feedback coefficients are derived for all state variables, therefore, the optimum moment value, which could be produced by anti-roll bar system, is regulated for prevention of vehicle instability.

3.2 Fuzzy controller development

Owing to its simple practical nature, fuzzy type controllers shown in Fig. 5 are developed for roll resistance of the vehicle model. This type of controllers has the advantages of using expert’s attitude and experimental results. The anti-roll moment control law for the desired state space vector X_d=[y_d v_d Φ_d r_d]^T is achieved by this control system.

In this work, control strategy is developed by FLC, applying Mamdani implication (Fig. 5). The main objective of this controller is to maintain vehicle stability and prevention of vehicle roll over during maneuvers.

There are four steps of a fuzzy rule-based system, fuzzification, rule base design, approximate reasoning and defuzzification [28], which are illustrated as

Fuzzification: It is the process of changing a real scalar value into a fuzzy value. In this work, based on state variables and control parameters, five conditions are considered for fuzzification which demonstrates the status of control strategy parameters: very low, low, normal, high and very high.

Fuzzy rule base: The rules are composed of a set of if–then rules, from which an inference mechanism is formed. In other words, rules describe the relation between state variable and control parameters.

Approximate reasoning: The operators within the logic rules form the inference engine. Generally, logic rules use AND or OR as connecting operators between state variables to specify sufficient control parameter.

Defuzzification: In a fuzzy control rule-base system, after obtaining a final fuzzy set, it is required to defuzzify set to get a numerical output as the control signal. The most common defuzzifier is the center of area, or the mean of maxima, which is utilized in this work. The centroid defuzzification technique can be expressed as

                             (35)

where x^* is the defuzzified output, μ_i(x) is the aggregated membership function and x is the output variable. As shown in Fig. (6), a fuzzy logic controller is developed to design a control strategy for PHEV, which has two state variables as inputs (roll rate and lateral acceleration) and one control parameter as an output (anti-roll moment). The antiroll-bar controller is modeled in the FLC and their membership functions of the inputs and outputs are shown in Fig. 6.

Fig. 5 Fuzzy logic controller

The designed FLC has 2 inputs, 3 MFs for each input and 9 fuzzy rules for each input. Requested anti roll moment is normalized so that the number 0.5 is allocated to optimum moment. Therefore, the center of MFs arelocated in 0.5. Also, as shown in Fig. 3, rules are set to respect the rules overlap. The surface of them are shown in Fig. 7.

Rules based on vehicle roll model are regulated whenever lateral acceleration and roll-rate of vehicle are increased, controller commands anti-roll bar system to produce more moment to improve vehicle handling properties and prevention of rollover.

4 Simulation results

To evaluate the vehicle roll and handling dynamic behavior and the effectiveness of anti-roll bar over various maneuvers (Fishhook and single lane change), numerical evaluations are conducted using MATLAB/ SIMULINK environment. Nonlinear 8-DOF vehicle model performances are investigated under FLC and LQR controllers. This simulation code is validated and used extensively [27]. Steering angle as an input signal to vehicle closed loop model in fishhook and lane change maneuvers are illustrated in Figs. 8 and 9, respectively.

Fig. 6 Membership functions of two modes in fuzzy logic controller:

Fig. 7 Two-mode rules surface

4.1 Fishhook maneuver

At the first stage, vehicle path following (path, lateral deviation), roll (load transfer, roll rate, roll and lateral acceleration), handling performance (side slip angle, lateral velocity and acceleration) and control effort(roll moment) at a constant speed of 120 km/h over Fishhook maneuver are presented in four cases (Figs. 10-17 respectively): FLC, LQR, without control at 90 km/h and 120 km/h.

Fig. 8 Steering angle in fishhook maneuver

Fig. 9 Lane change maneuver (T: Vehicle width)

Vehicle path following and deviation of desired path demonstrate that controllers can properly control the vehicle dynamic variables to track driver intended path with minimize deviation. Also, as shown in Figs.10 and 11, increasing longitudinal velocity during fishhook maneuver results in more deviation of vehicle from track.

Fig. 10 Fishhook maneuver path

Fig. 11 Lateral deviation in fishhook maneuver

Fig. 12 Roll angle in fishhook

Fig. 13 Roll rate in fishhook

Fig. 14 Load transfer in fishhook

Fig. 15 Controller effort (anti-roll moment) in fishhook

Simulation results of vehicle roll variables including roll angle, roll rate, load transfer and controller moment over fishhook maneuver are depicted in Figs. 12-15 respectively. FLC has the minimum amount and oscillations of control effort and roll responses. Also, it is obvious that in cases without control, vehicle tends to rollover and with increasing the vehicle longitudinal speed from 90 to 120 km/h, turning over occurs at t= 3.8 s.

Furthermore, from the handling properties including lateral acceleration of Fig. 16 and side slip angle of Fig. 17, it can be concluded that both the controllers achieve the optimal handling conditions. In controlled cases, lateral acceleration and side slip angles remain under 0.6g and 4 degree during maneuver, which ensures vehicle stability and handling properties.

Fig. 16 Lateral acceleration in fishhook

Fig. 17 Side slip angle in fishhook

4.2 Lane change

In the next simulation, vehicle travels at the constant longitudinal speed of 90 km/h on single lane change maneuver. Vehicle properties and variables on FLC, LQR and no control one are illustrated in Fig. 18. Figures clearly show that the vehicle without controller has an undesirable oscillatory response with a growing side-slip and roll angle. The responses are not converged to steady-state values at a certain time and remain severely oscillatory, so that the probability of rollover and instability is caused. However, in the presence of controllers, vehicle tracks lane change maneuver with minimum error in path. Also, controllers by applying the momentum improve the vehicle stability and roll properties during lane change maneuver. Moreover, the results demonstrate that vehicle steady state response shows the least oscillation in FLC case.

Fig. 18 Lane change maneuver at speed of 90 km/h

5 Conclusions

1) FLC and LQR theories based on vehicle 3-DOF roll dynamic are developed to produce optimal momentum. Controllers are trained to follow desired sideslip, roll angle, and yaw rates during various maneuvers such as fishhook and lane change.

2) An 8-DOF vehicle model in MATLAB/Simulink environment is utilized to simulate the controller performance and vehicle behavior. Simulation results clarify control strategies hold the lateral acceleration and sideslip angles less than limited critical overturning and instability for various speeds and maneuvers.

3) Moreover, the FLC roll and handling controller show slightly better results compared to the LQR one, with smaller control effort, roll and side slip angles in severe maneuvers.

4) Finally, concerning that in quick emergency maneuvers, vehicle has a greater likelihood to roll over; the sensitivity analysis of FLC to longitudinal speed is conducted.

5) Results show that proposed controller could be more effective in high speed, which prevents the vehicle overturning in quick maneuvers.

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(Edited by FANG Jing-hua)

Received date: 2015-04-28; Accepted date: 2015-10-12

Corresponding author: Mehdi Mahmoodi-k, PhD Candidate; Tel: +98-9335622512, E-mail: m_mahmoodi_k@iust.ac.ir