Vehicle System Dynamics: International Journal of Vehicle Mechanics ...

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Nonlinear Design Approach to Four-Wheel-Steering Systems Using Neural Networks MASAO NAGAI , ETSUHIRO UEDA & ANTONIO MORAN Published online: 27 Jul 2007.

To cite this article: MASAO NAGAI , ETSUHIRO UEDA & ANTONIO MORAN (1995) Nonlinear Design Approach to Four-WheelSteering Systems Using Neural Networks, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 24:4-5, 329-342, DOI: 10.1080/00423119508969095 To link to this article: http://dx.doi.org/10.1080/00423119508969095

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Vehicle System Dynamics, 24 (1995), pp. 329-342

0042-3 1 14/95/2404-329$6.00 0 Swets & Zeitlinger

Nonlinear Design Approach to Four-Wheel-Steering Systems Using Neural Networks

Downloaded by [Antonio Moran] at 16:03 12 January 2015

MASAO NAGAI,' ETSUHIRO UEDA and ANTONIO MORAN

SUMMARY Four-wheel-steering (4WS) systems have been studied and developed with remarkable success from the viewpoint of vehicle dynamics. Most of the control methods require a linearized bicycle model of the actual vehicle system which is however strongly influenced by tire nonlinearity. This paper proposes a new method to design the 4WS system taking into account the nonlinear characteristics of tires and suspensions. For this purpose integration of artificial neural network and linear control theory is introduced for the identification and control of a nonlinear vehicle model structured using a software for multi-body dynamic analysis (ADAMS). This model takes into account the nonlinear characteristics of actual vehicles with tires modeled by "magic formula". The results of computer simulations show that the proposed nonlinear approach is efficient in improving the handling and stability of vehicles.

1. INTRODUCTION

In order to improve the handling and stability of vehicles, researches on fourwheel-steering systems (4WS) have been carried out actively [l], and some systems have been applied to actual vehicles successfully. The basic control concept of 4WS is to control the side slip angle (side slip velocity) and yaw rate. And a simplified linear bicycle model of the actual vehicle concerning these two motions has been used in designing the steering systems. However, in order to enhance the active safety of vehicles under certain critical driving situations, such as, emergent accident avoidance, sudden decrease in road surface friction, running on slippery roads, it is important to design the control system taking into account the deterioration of vehicle performance resulting from the nonlinearity of tires. Certain control methods to overcome the tire nonlinearity have been studied, e.g., by controlling the suspension stiffness or the driving and . braking forces of the tires. Apart from these indirect control methods for improving vehicle handling and stability, this paper proposes a new approach using neural network theory

' All correspondence should be addressed to Prof. M. Nagai, Department of Mechanical Systems Engineering Tokyo University of Agriculture and Technology, Nakamachi 2-24-16, Koganeishi, Tokyo 184, Japan. Tel: + 81-423-88-7090 Fax: + 81-423-85-7204 E-mail: [email protected]


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to find out how the deterioration of vehicle performance caused by the tire nonlinearity can be compensated. In recent years, neural networks have been successfully applied to identification and control methods for complex dynamical systems using its nonlinear learning ability. However, difficulty still exists in designing the neural network control systems because the neural network itself is a black box and most applications of it are not based on the analytical models derived from the actual systems [2-51. This paper shows a new approach to design the four-wheel-steering system using neural network theory to overcome the tire nonlinearity. The new 4WS control system is configured by an integration of a linear feedback control law and neural network. The linear feedback control law can be theoretically derived from a so-called linear bicycle model which shows a good approximation of the actual vehicle dynamics in rather small lateral and longitudinal accelerations. On the other hand, the neural network can cope with the vehicle nonlinearity and must be trained by a certain learning method. This paper shows the methods for identifying the nonlinearity and designing the integrated control system, and computer simulation is carried out to verify the effectiveness of the proposed new design method.

2. IDENTIFICATION O F VEHICLE DYNAMICS 2.1. Full Vehicle Model A 2000cc class compact passenger car is modeled by using a software for multibody dynamic analysis ADAMS for this study. The freedoms of the vehicle motions are totally sixteen, including six for the rigid car body, two for the front and rear steering systems, and eight for the four tires' rotation and vertical motions. The front and rear suspensions are described by a double wishbone type, and the tires are modeled by the experimental expression "magic formula" [6] which takes into account the tire factors such as vertical, braking, driving and lateral forces. The full vehicle model is shown in Fig. 1.

Fig. I . A full vehicle model.


NONLINEAR DESIGN APPROACH TO 4WS BY N.N.

Lateral Acceleration [rn/s/sl Fig. 2. Linearity during steady state circular turn.

Fig. 2 shows the relationship between the front steer angle and the lateral acceleration during steady state turn when the rear steer angle is fixed. It indicates that the extent of nonlinearity increases when the lateral acceleration becomes large. Fig. 2 shows the results when the vehicle runs at 80 km/h on a dry paved road surface. The nonlinearity appears when the lateral acceleration is larger than approximately 0.4G, and this nonlinear relationship is mainly resultant from the tire nonlinearity.

2.2. Identification of the Vehicle Dynamics In the range of small lateral acceleration where the linear approximation can be applied, the motions of the vehicle can be expressed by the linear bicycle model. The equations of motions regarding yaw rate and side slip angle can be written as follows:

where, m designates the total vehicle mass, I the yaw moment of inertia, P the side slip angle at the center of gravity of the vehicle, y the yaw rate, and 6, the front and rear steer angles respectively, If and 1, the distances between the center of gravity and the front and rear axles respectively, Kf and K, the concerning powers of the front and rear tires respectively, V the running velocity. Table I shows the baseline parameters of the vehicle. In order to integrate

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MASAO NAGAI ET AL.

Table 1. Vehicle parameters.

m If

1, V

1300 kg 1.0m 1.60m 80km/h

Kf Kr At

3000 kgrn2 44400 N/rad 43600N/rad 0.04 sec


this linear bicycle model with neural network, Eqs. (1) and (2) are transformed to the discrete type of equation as Eqs. (3) and (4).

The coefficients of these discrete equations are calculated using a sampling period. Fig. 3 shows the integrated identification model consisting of neural network and the linear bicycle model to identify the full vehicle model. The input variables of the identification model are the lateral acceleration u (k),

Linear Bicycle Model

f1(x)

Input Layer

f2(x)

f3(x)

1

f4(x)

Hidden Hidden Output Layer1 Layer2 Layer

1 f,(x) = f4(x) = X , Fig. 3. Integrated identification model.

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2

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Fig. 4. Steering input data for training neural network.

the side slip angle p (k), the yaw rate y (k), the front steer angle &(k) and the rear steer angle 6,(k). The output variables are a(k + I), P(k + 1) and y(k + l), respectively. As shown in Fig. 3, the neural network comprises four layers; with the input layer having five neurons, the first hidden layer twelve neurons, the second hidden layer ten neurons, and the output layer three neurons. The output functions of the input and output layers are modeled by a linear function and those of the hidden layers are modeled by a sigmoid function. Fig. 4 shows the time history of the input data to the front steer angle for training the neural network. The input data includes step inputs with different amplitudes 3.4 deg and 5.5 deg, and two different sinusoidal inputs with amplitudes 3.4 deg and 5.5 deg, and frequencies 1.O Hz and 1.5 Hz. On the other hand the input data to the rear steer angle are random wave. The training is done under the condition of the running velocity 80 km/h and the road friction

)

ADAMS

A: Forward learning B: Recurrent learning

.

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Fig. 5. Block diagram for learning identification model.

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334

MASAO NAGAI ET AL. 6.0

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m


Time [sec]

Time [sec] Fig. 6. Responses of identified model and full vehicle model.

coefficient 0.7. The total number of training data is 3500 which are sampled within 140 seconds with the sampling period 0.04 sec. Fig. 5 shows the block diagram of the learning process of the identification model of the full vehicle model. The network is firstly trained by the error back propagation method in a forward manner (denoted by A in Fig. 5), and then trained by the error back propagation method in a recurrent one (B in Fig. 5). Comparing with the former type working fast but inaccurately, the latter learning method gives a high accuracy under external noises [5].

2.3. Identification Results Fig. 6 shows the time history of the yaw rate and the side slip angle when a sinusoidal test input to the front steer angle, which is different from the training input, is added to the identified model. The amplitude of the test input is 5.5 deg, and the frequency of it is 1.25 Hz. The dotted lines represent the responses of the full vehicle model, the chained ones the responses of the linear bicycle model, and the solid ones the responses of the integrated identified model including neural network. It is clearly found from the results that the responses of the linear bicycle model show larger amplitudes than those of the full vehicle model, while those of the integrated identified model shows good agreements with those of the full vehicle model.


335

3. DESIGN O F 4WS CONTROL SYSTEM


This section derives a new 4WS control law for the rear wheel steer using the identified model obtained in the last section. The control system developed in this study is based on an integration of the linear 4WS control law derived from the linear bicycle model and the neural network. The objective of the 4WS control in this study is to bring the side slip angle of the vehicle body to zero. Therefore, the linear control law can be theoretically derived from the linear bicycle model expressed previously. The linear control law using the yaw rate feedback compensation is expressed by Eq. (5).

The integrated control system proposed in this study is based on this linear control law which is integrated by the neural network as shown in Fig. 7. The neural network is composed of three layers, with the input layer having three neurons, the hidden layer ten neurons and the output layer one neuron. The

fl(~)

Input Layer

1 Fig. 7. Integrated control system

fz(x)

f3W

Hidden Layer1

Output Layer

Neuro Control 1

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MASAO NAGAI ET AL.


Fig. 8. Block diagram of integrated 4WS control system.

input variables of the controller are the lateral acceleration cr(k), the yaw rate y(k) and the front steer angle (k) which are easily measured in the actual vehicle. The output of the controller is the rear wheel steer angle 6, (k). In order to minimize the side slip angle, a criterion function expressed by Eq. (6) is introduced. In this function the side slip angle and the rear wheel steer angle are evaluated.

where, pl, p2 are the weighting factors. In the following calculation the values p, = 1.0 and p2 = 0.5 are used. The coefficients and the threshold values of each neuron of the network are obtained through.the training process of the controller so that the criterion function can be minimized. The training method of the controller is the same as used in the training of the identification model.

4. RESULTS AND DISCUSSIONS In this section the integrated controller obtained in the last section is applied to the nonlinear full vehicle model created by ADAMS in order to examine its effectiveness by computer simulation. The block diagram of the control system is shown in Fig. 8. In the following discussion, the performances of the twowheel-steering system (2WS), the linear 4WS control and the integrated 4WS control are respectively calculated and compared with each other. The calculations are carried out under the condition of the running velocity 80km/h and the road friction coefficient 0.7. 4.1. Step Response Fig. 9 shows the transient responses when a step wise input to the front steer angle which reaches from 0 deg to 4.8 deg within 0.2 seconds is added to the



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MASAO NAGAI

ET AL.

Lateral Acceleration [m/s/sl Fig. 10. Linearity during steady state circular turn.

full vehicle model. The dotted lines denote the case of 2WS, the chained lines the linear 4WS control, and the solid lines the integrated 4WS control. In the case of 2WS, both the overshoot and undershoot appear clearly in the response of the side slip angle, and the large overshoot appears in the response of the yaw rate. On the other hand, both the 4WS control systems improve the transient responses significantly. Concerning the offset value of the side slip angle, there exists a large difference between the linear 4WS control and the integrated 4WS control. Although the linear 4WS control can make the side slip angle be zero only within the range of small front steer angle, it leaves an offset value in the side slip angle in this transient response. Nevertheless, the integrated 4WS control whch is expected to be a nonlinear controller can compensate the offset significantly. The responses in Fig. 9 converge to steady state values after a certain time, which mean a steady-state turn situation. Fig. 10 shows the relationship between the lateral acceleration and the front wheel steer angle in a steady state turn under the same condition as shown in Fig. 2. While the nonlinearity in the case of 2WS appears at larger lateral acceleration than approximately 0.4G, the linear 4WS control makes it possible that the linearity between the front steer angle and the lateral acceleration is sustained up to a rather high lateral acceleration value. The integrated 4WS system using the neural network improves this trend to the further extent. 4.2. Responses to Sinusoidal Input The effectiveness of the control system is also verified by computer simulation when the front wheel is steered by sinusoidal waves. Two cases with different amplitudes 3.4 deg and 5.5 deg are studied. Fig. 11 shows the case when the amplitude is 3.4 deg and the frequency 1.25 Hz. According to the result, both the linear control and the integrated control improve the fluctuations in the response of the side slip angle, and the phase delay in the yaw rate response is

NONLINEAR DESIGN APPROACH TO 4WS BY N.N

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also improved, compared with those of 2WS. clear difference between the two control methods does not appear in this case, because a rather small amplitude is used so that the nonlinearity is small enough to be neglected. Fig. 12 compares the frequency responses of the side slip angle to the front steer angle in two cases of input amplitude. As the wave is not a typical

MASAO NAGAI ET AL.

-

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sinusoidal curve caused by the nonlinear effect (cf. Fig. ll), the root mean square values are depicted in the figures. Fig. 12 shows that very little difference exists between the linear and integrated controls when the input amplitude is small, but that the difference becomes distinct when the input amplitude increases. It may be concluded that in the range of large input amplitude the integrated control system with neural network can work effectively and

Table 2. 4WS control effect on the root mean square values. (a) Side slip angle (r.rn.s. deg)

8~ 3.4 deg 5.5 deg

1.0 Hz 1.5 Hz 1.0 Hz 1.5 Hz

2WS

Linear 4WS

0.460 0.473 0.866 0.720

0.0578 0.0798 0.230 0.232

-

0.483 0.874 1.03 1.49

(b) Rear wheel angle (r.m.s. deg) 3.4 deg 5.5 deg

1.0 Hz 1.5 Hz 1.0 Hz 1.5 Hz

Integrated 4WS



0.0

0.5

1.0

i.5

2.0

2.5

3.0

Frequency [Hz] Fig. 13. Frequency responses of yaw rate.

compensate the nonlinearity of tires. Table 2 lists the root mean square values of the side slip angle and rear steering angle in the above two cases. The results shown in Fig. 12 and Table 2 can explain the following items: 1) The larger the input amplitude is, the more effective the integrated approach is, regarding the minimization of the side slip angle when compared with the linear approach, and the former one tends to decrease the rear steer angle. 2) When the input amplitude is small, the integrated nonlinear approach makes the side slip angle value relatively large compared with the linear control. But as the absolute values are very small, i.e.less than 0.1 deg, it can be said that the proposed approach is actually effective. Fig. 13 shows the frequency response of the yaw rate which is important to evaluate the cornering performance. While the gain of the yaw rate decreases significantly in the case of 2WS at high frequency, the linear 4WS control attenuates this decrease. This trend exists almost in the same way regardless of the existence of the neural network, because this effect is generally obtained from the minimization of the side slip angle by the four-wheel-steer control system.

5. CONCLUSIONS Although 4WS is effective to improve the handling and stability of vehicles, it shows a certain control limit owing to the tire nonlinearity. In order to cope with this problem, this paper proposed and verified the effectiveness of a new 4WS control approach which integrates the advantages of the linear 4WS control and the neural network which is capable of nonlinear learning. The advantage of this control approach lies in the fact that it is designed on the basis of the


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MASAO NAGAI ET AL.

knowledge about the vehicle dynamics (represented by the bicycle model), and introduces the neural network to acquire the knowledge about the nonlinearity. The proposed system is applied to the full vehicle model created by the multibody dynamic analysis software. The following conclusions are obtained from the simulation results. 1) While the linear bicycle model cannot fully express the dynamics of the vehicle when the steer angle becomes so large that the tire nonlinearity begins to have obvious influence on vehicle performance, the addition of the neural network makes it possible to exactly express the dynamics in the whole working range of the tires. 2) The nonlinear approach with the addition of neural network turns to be more effective for 4WS control when the amplitude of the steer angle is so large that the nonlinearity shows strong influence. 3) The combination of the linear description of the vehicle model and the nonlinear neural network simplifies and accelerates the learning process by reducing the number of neurons in each layer of the network. The influences of variable running velocity and road surface condition are not discussed in this paper, and it is interesting and necessary to investigate the adaptability of neural networks in future. REFERENCES 1. Y. Furukawa, et al., A Review of Four-Wheel Steering Studies from the Viewpoint of Vehicle Dynamics and Control, Vehicle System Dynamics, 18- 1, (1989), 15 1. 2. T.Shiotsuka, A. Nagamatsu, K. Yoshida, Adaptive Control of 4WS System by Using Neural Network, AVEC'92, Yokohama, 9, (1992), 252. 3. M. Kato, et al., Estimation of Vehicle Side Slip Angle with Artifiial Neural Networks, Proceedings of SAEJ, No. 931, (1993), 49. 4. A. Moran, M.Nagai, Optimal Preview Control of Rear Suspension Using Neural Networks, Vehicle system Dynamics, 22-5, (1993), 161. 5. A. Moran, Identification and Control of Nonlinear Vehicle Dynamic Using Neural Networks, Doctoral Thesis, Tokyo University of Agriculture and Technology, 2 (1994). 6. H. P. Pacejka, et al, A New Tire Model with an Application in Vehicle Dynamics Studies, SAE Paper No. 890087, (1989).