Specification of vehicle model
An automotive performance has improved from the demand of ride comfort and driving stability. Many research have proposed various control system design methods for active and semi-active suspension systems. To improve vehicle response by steering, a roll angle control , a distribution control of the suspension stiffness of front and rear , and pitch angle control  have been proposed. Furthermore, to improve vehicle response due to disturbance on the road, a tire vertical load control [4-6] caused by road disturbance was proposed. In recent year, it is reported that roll and lateral vehicle motions are affected by the road input. These motions are affected by not only road displacement but also the change of tire side force [8-11]. As shown in Figs. 1 and 2, the tire side force is caused by a toe change and a scuff due to the roll motion and tire side force. However, there are few research about a suspension control method which takes into consideration with a suspension characteristic that is change of tire side force caused by suspension stroke and tire side force.
In this research, we designed a suspension control system which reflects a suspension characteristic, the change of tire side force caused by suspension stroke and tire side force. A new semi-active suspension control method is proposed to reduce the vehicle vibration and vehicle lateral motion due to the road input.
To design such a system, the time delay of the road input from the front wheel to the rear wheel needed to be modeled. In the vehicle control, there are several control systems which take into consideration with this delay [12-16]. However, the purpose of these research is to find a way to reduce the vehicle vertical vibration. Oraby evaluated motion of the vehicle lateral direction . However, the dynamics of the lateral direction do not be considered in the control system design.
In order to design the controller, a vehicle model including the tire side force change caused by road disturbance is constructed. Moreover, the road input from the front wheel to the rear wheel time delay is modelled with Pade approximation. These suspension characteristics and the time delay are modelled with liner model. The disturbance accommodating
2.1. Modeling of the vehicle
Figure 2 shows a full vehicle model which is equipped with semi-active suspension between each wheel and the vehicle body. The weight of the vehicle body is supported by the spring. We assume that a vehicle model is a generic sedan car as shown in Table 1. The equations of motion which are, lateral, bounce, roll, pitch, yaw, and each unsprung motion are as follows:
As shown in Fig.2, tire side force is caused by the toe change and the scuff. The toe change and the scuff are caused by the suspension stroke and the tire side force. The tire side force of each tire,
where, a value in the parenthesis in Eq.(3) means the tire slip angle of each tire.
2.2. Vehicle response caused by antiphase road disturbance
The state equation of the vehicle model which reflects change the tire side force is defined as Eq. (6). This model is an LTI model when the vehicle velocity is fixed.
We confirmed that the vehicle responsed to an antiphase road disturbance. In this research, the phases of road input of left-right are zero and π. Zero means a coordinate phase, and π means the antiphase. In the antiphase road, the road displacement at the vehicle velocity,
The transfer function of the vehicle responce caused by the antiphase road disuturbance at 33.3 m/s (120 km/h) is shwon in Fig. 3. Here, Figure 3 (a) is the date of previous study. Figure 3 (b) is the data of our study. In these figures, specification of the vehicle model of each study is different. However, from the result showing the same tendency for vehicle responses, we found that the our model is appropriate.
2.3. Design of damper coefficient
In general, the performance of the controller is affected by the damping coefficients of the suspension. In this section, we design the damping coefficient for the evaluation values which are the vertical acceleration at the coordinate phase road and the lateral acceleration at the antiphase road. In the next section, we design the control system which uses the damping coefficients designed in this section. The road condition is shown in Figs. 4 and 5. The road displacement is assumed such that the Power Spectral Density (PSD) characteristic of the road surface is C class defined by ISO . The lateral accelerations are shown in Fig. 8 and the lateral acceleration of vehicle body are derived from a geometric relation as follows.
The sensitivity curves of the lateral and the vertical acceleration are defined by ISO . We designed the damping coefficient for the suspension to minimize the Root Mean Square (RMS) which considers the sensitivity curves. A frequency response of the filter  which modeled sensitivity curve with the transfer function is shown in Fig. 7. We calculated the RMS values using the time history of the vertical and lateral accelerations which pass this filter. The simulation results when both front and rear damping coefficients are changed are shown in Fig. 8, which is a contour diagram of the RMS values. There are minimum point in Fig. 8 (a), (b) and (c). The evaluation value is the sum of vertical and lateral acceleration of RMS shown in Fig. 8. (c). From the simulation results, the damping coefficients of the suspension are 2000 and 1300 N/m/s.
3. Controller design
We used the controller designed based on the linear
3.1. Approximation of time delay of road disturbances
When the front and rear tire passes the same path, road disturbance affects the rear tire has the delay to the front tire. As shown in Fig. 10, the road input from the front wheel to the rear wheel is the delayed. The delay is modeled with a third-ordered Pade approximation. As shown in Fig. 10 (b), we consider the effect of vehicle velocity in the control system design by using a linear model. To check whether the Pade approximation is correct, we carried out the numerical simulations. The response of the vehicle model which includes the Pade approximation when the vehicle runs on the antiphase road disturbance is shown in Fig. 11. The vehicle velocity is 16.7 m/s (60km/h). In the lower frequency of the resonance frequency of the tire, we found that the result of the simulation model which uses the time delay and Pade approximation are almost the same.
3.2. Disturbance-accommodating control
We found that feedforward control of disturbance information in the finite frequency range and feedback control improve performance . The power spectral density of the actual velocity of disturbances had flat characteristics in a low frequency, and decreased according to frequency at a region of high frequency. We assumed that it regarded as the colored noise formed by shaping filter which has a transfer function with low-pass characteristics. This filter of the each wheel is based on the road condition which defined by ISO . The filter is as follows:
The feedforward control of disturbances resulted in worse accuracy outside the assumed frequency . Furthermore, because each resonance frequency of the vehicles and tire differs, the control system design considering each resonance frequency is needed. Therefore, the control system was designed by using the
We integrated each state variable of the road disturbance model and frequency weights for controlled values. The frequency weights are as follows:
where, is a minimum of
In this reserch, we assumed that the vehicle speed is constant. The measured outputs,
A bandpass filter,
3.4. Comparison of the response of the frequency weight
To check the effect of the suspension model which includes the tire side force caused by the road disturbance, we compared two control systems, one which considers the tire side force and the control system, and one which does not consider one. The design of two general control methods changed the controlled value of the lateral acceleration of the vehicle body. Frequency weights,
The RMS value changed by the frequency weight,
4.1. Simulation condition
To clarify the effect of the suspension model which includes the tire side force, we compare the two control systems. The two control systems are designed in Cheater 3. As for the control system which inflects the tire side force (
In the simulation, we combine the control system and the vehicle model which includes the tire side force shown in Chapter 2. The vehicle velocity is 16.7 m/s (60km/h). Moreover, we assume that the range of the damping coefficient of the semi-active suspension is 100-10000 N/m/s in Fig. 15. The semi-active damper has the first order delay. The cut off frequency is 10 Hz. We used MATLAB (The Math Work Inc.) to calculate the Runge-Kutta method for the differential equations. There were two road conditions. One is the coordinate phase road in Fig. 5 (a), and the other is the antiphase road in Fig. 5 (b). To demonstrate the control effect, we show the responses which does not control he semi-active damper (Non control).
4.2. Simulation results
In check whether the proposed method reduced vertical and lateral motion, we did the numerical simulations. The time history of the vertical acceleration caused by the coordinate phase and the lateral acceleration caused by the antiphase road is shown in Fig. 14. The Lissajous figure for the suspension velocity and the damping force of the front suspension are shown in Fig. 15. The PSD of the vertical acceleration and the lateral acceleration is shown in Fig. 16.
In the coordinate phase road, there are few difference in the time history and the Lissajous figure. As for the PSD, two control systems can reduce the vibration better than the non control one near the resonance frequencies of the vertical direction of the vehicle body and vertical direction of the human head. On the other hand, in the antiphase road, there are differences in the time history and the Lissajous figure. The PSD of lateral acceleration, two control systems can reduce the vibration near the frequency range with a high level of lateral acceleration sensitivity. Additionally, the control system which includes suspension model of the tire side force change caused by road disturbance can reduce the vibration better than the control system which does not include the tire side force change in the same frequency range. The simulation results confirmed that using the proposed control system reduces the vertical and the lateral motion of the vehicle caused by the disturbance in the road.
This research proposed disturbance accommodating