LANE-TRACKING CONTROL PERFORMANCE OF FOUR-WHEEL-STEERING AUTOMOBILE

Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain

www.elsevier.com/locate/ifac

LANE-TRACKING CONTROL PERFORMANCE OF FOUR-WHEEL-STEERING AUTOMOBILE Pongsathorn Raksincharoensak*, Hiroshi Mouri**, and Masao Nagai* *Tokyo University of Agriculture and Technology **Vehicle Research Laboratory, Nissan Motor Co.,Ltd. Abstract: This paper investigates the use of four-wheel-steering system for the control of vehicle, in the context of automatic lane-tracking. In this paper, the lane-tracking control system is designed on the basis of the four-wheel-steering automobile whose desired steering response is realized by the application of model matching control. Using simplified linear 2 degree-of-freedom bicycle model, simulation study and theoretical analysis are conducted to evaluate the lane-tracking control performance of active four-wheel-steering automobiles which have different desired steering response. A comparative study clearly reveals their advantages in lane-tracking performance. Copyright © 2002 IFAC Keywords: Automobiles, Four-wheel-steering, Autonomous vehicles, Vehicle dynamics, Model-following control, Optimal control, Linear quadratic regulators, Kalman filters 1. INTRODUCTION

For further enhancing active safety, four-wheelsteering (abbreviated as 4WS) system has drawn the attention of many researchers in the field of vehicle dynamics and control for a long time since it provides two independent control inputs for lateral and yaw motion. Therefore, until now, various types of 4WS systems have been proposed (Furukawa and Abe, 1997). In this paper, to further enhance the lanetracking performance, the lane-tracking controller is designed on the basis of automobile equipped with active four-wheel-steering system. The active fourwheel-steering system is designed to realize the desired steering response with the application of model matching technique which was proposed by Nagai (1989). Then, the lane-tracking controller is designed separately by linear quadratic control theory. Two desired steering responses are proposed, and their effects on lane-tracking control performance are theoretically analyzed and compared to each other.

Recently, researches about Intelligent Transport systems (ITS) are being carried out actively in various fields. The idea of Automated Highway System is motivated to play a key role in increasing traffic volume and relieving driver’s workload. Since the mid 1950s, control algorithms based on not only classical control theories but also modern control theories have been proposed on the design of automated vehicles (Hedrick, 1994). This paper focuses on the control of vehicle to trace the desired trajectory in the lateral direction which is applicable to systems such as automatic lane-keeping system. At the beginning of our research, to make vehicle follow a desired lane, a lane-tracking control system based on conventional front-wheel steering vehicle was designed to regulate lateral deviation at the centre of gravity of vehicle with the application of optimal control theory. As a result, the lateral deviation is well regulated, while yaw dynamics tends to have undesirable damping behaviour (Mouri, et al., 1997, and Marumo, et al., 2000). From the viewpoint of vehicle dynamics and control, however, it is impossible to achieve desirable control performance in both lateral and yaw motion with using the only front-wheel steering angle as control input.

2. VEHICLE MODEL This paper mainly focuses on the normal lane-keeping behaviour of vehicle which the body side slip angle is so small that the velocity is approximately constant.

385

3.1 Model matching controller

The effects of roll, heave, and pitch motions as well as the dynamic characteristics of tyre and steering system are also neglected. The side slip angle of right and left tyres are in the region that the linear tyre model is valid. Based on these assumptions, this paper employs an equivalent bicycle model as shown in Fig. 1 to describe the vehicle dynamics. The earth-fixed coordinate system is used to derive the equations of motions described as follows: (1) myc = 2 F f + 2Fr Iφ = 2l f F f − 2l r Fr

Model matching controller consists of feedforward and feedback controllers which makes the actual vehicle follow the desired steering response. However, since the modelling error in vehicle dynamics is not considered here, the feedback controller design will not be mentioned here. The steering response of the desired model can be determined in many ways. This paper discusses the following two types of steering response, those are:

(2)

lf  y cr   F f = C f  δ f − φ + φ r − (3) V V   lr y cr   Fr = C r  δ r + φ + φ r −  (4) V V   where m denotes the vehicle mass, I the yaw moment of inertia, yc the lateral displacement at centre of gravity, φ the yaw angle (Subscript r denotes the variable relative to reference line.) , V the vehicle velocity, lf and lr the distances from the front and rear axles to the center of gravity, respectively, Ff and F r the front and rear cornering forces, respectively, C f and Cr the front and rear cornering stiffnesses, respectively, δ f and δr the front and rear steering angles, respectively. In the case of curved trajectory with constant radius of curvature ρ , the relative variables with respect to the desired course are expressed as follows: (5) φr = φ − ρV y cr = y c − ρV 2 .

(1) zero-side-slip-angle response which will be referred as “4WS-A”, and (2) steering response which has no phase-lag in lateral acceleration which will be referred as “4WS-B”. First, for 4WS-A vehicle, the desired steering response can be expressed as follows:

β ( s) =0 ; δ (s )

(7)

kr φ( s) = . (8) δ (s ) 1 + τ r s where, β indicates the body side-slip angle, δ the steering wheel angle input. k r and τr are the steadystate gain and the time constant of yaw rate response respectively which are determined from conventional front-wheel steering vehicle as follows: 1 V⋅ 1 kr = 2 ⋅ m(C f l f − C r lr )V l ist (9) 1− 2 2l C f C r

(6)

τ r = kr I 2l f C f

3. CONTROL SYSTEM DESIGN The objective of the control system is to regulate the lateral deviation at the centre of gravity and yaw deviation of the vehicle to be zero. The control system , as shown in Fig. 2, consists of the model matching controller which makes the vehicle follow the desired vehicle response by making use of the active front and rear steering angle and the path tracking controller designed by linear quadratic control theory.

(10) ls

Y lf

lr Gr

Ir φ

Gf

V

ysr

ycr

O

X

Fig. 1. Equivalent bicycle model

Desired lane  φ  φ     yc  +    yc  desired

_

Model Matching 4WS

LQ Lane-Tracking Controller

G

Feedforward Controller

δf

+ +

+ +

Vehicle motion

Desired model

Fig.2. 4WS lane-tracking control system description

386

δr

Feedback βd  Controller φ   d

_

+

Vehicle Model β    φ 

 φ  φ     yc     yc 

Conventional 2WS

2

4WS-B

1.5

-15

1

4WS-A

0.5

Desired Response for 4WS

-20 -25

-2

10-1

10

-1

0

1

10

10

2

3

10

10

Frequency [Hz]

-30 10

0 -2 10 0

Phase [deg]

Gain φ / 䵾δ [dB]

-10

.

Gain δf /δ [rad/rad]

-5

100

101

-20

4WS-A

-30 -40 -2 10

Fig.3. Approximated first-order lag yaw rate response

4WS-B

-10

10

-1

0

10

1

10

2

3

10

10

Frequency [Hz]

(a) Front steering angle where, l indicates the vehicle wheelbase, ist the steering gear ratio. In this case, according to arbitrary steering angle input, the active front and rear steering law can be obtained as follows:

Gain δr /δ [rad/rad]

1

δ f (s ) 2ll f C f k I  I  1  = r  +  ml rV + −   δ (s ) 2lC f  τ r  V τ r  1 + τ r s  ; (11)  I  2llr C r I + − +  ml f V − V τr  τ r 

 1   .  1 + τ r s 

4WS-B

-1

10

10

0

10

1

10

2

3

10

4WS-B

-50 -100

4WS-A

-150

-1

10

10

0

10

1

10

2

3

10

Frequency [Hz]

(b) Rear steering angle Fig. 4. Frequency characteristics of front and rear steering angle by model matching control 3.2 Lane-tracking controller design Practically, for the design procedure, the lane-tracking problem contains yaw rate, yaw angle, lateral velocity and displacement as degrees-of-freedom. In the case of 4WS-A vehicle, the vehicle body side slip angle is always zero, so the heading angle becomes proportional to the lateral velocity; whereas, in the case of 4WS-B vehicle, the vehicle body side slip angle is proportional to the yaw rate. Using 4WS vehicle, to express the vehicle plant by state-space equation, one degree of freedom can be reduced. Here, to make the condition for comparison uniform, yaw rate, yaw angle, and lateral displacement are treated as state variables. Therefore, the vehicle plant can be expressed in the form of state-space equation as follows:

(15)

Therefore, in this case, the steering law for active front and rear steering angle can be obtained as follows: I  + ml rV δ (s ) τ r  2ll f C f I  1  +  + 2lC f τ r −   τ r  1 + τ r s  ;  V (16) δ r (s ) k  I = r − + ml f V δ ( s) 2lC r  τ r =

0.2 -2 10

-200 -2 10

β ( s) kτ = r r (13) δ (s ) 1 + τ r s ; kr φ( s) = . (14) δ (s ) 1 + τ r s Note that the yaw rate response is set identical to 4WSA vehicle. According to this desired steering response, the lateral acceleration response becomes constant gain without phase-lag as follows:

δ f (s )

4WS-A

0.4

Frequency [Hz]

(12) Next, for 4WS-B vehicle, the desired steering response can be expressed as follows:

 β (s ) φ(s )  yc (s )  = k rV . = V  +  δ ( s)  δ ( s ) δ ( s) 

0.6

0

Phase [deg]

δ r (s ) k = r δ ( s) 2lC r

0.8

kr 2lC f

4WS-A vehicle:  φ  − 1 / τ r 0    0  φr  =  1  y cr   0 V   

 2ll C I  1  +  − r r + 2lC rτ r +   τ r  1 + τ r s  . V  (17) 387

0  φ  k r / τ r    0  φ r  +  0 δ 0  y cr   0 

(18)

4WS-B vehicle:  φ     φr  =  y cr   

− 1 / τ r  1   Vτ r

control algorithm, with the application of Kalman filter, for navigating curves by detecting only the vehicle’s lateral deviation, without requiring feedforward of road curvature.

0  φ  k r / τ r    0  φr  +  0 δ 0  y cr   0 

0 0 V

(19)

The change in road curvature is approximated as a first-order system disturbed by gaussian white noise, which can be expressed as: d ρ = − λρ + w (24) dt where w indicates the process noise of the control plant, λ indicates the dynamic characteristics of curvature variation. The lateral deviation at the sensor near the front bumper, employed for estimating vehicle states together with curvature, can be expressed as follows:

The lane-tracking controller, which determines the value of the steering wheel angle input, is designed by linear quadratic control theory. All state variables of vehicle are fed back to determine the steering wheel angle as the following expression.

δ = − K φφ − K φ φr − K yc ycr

(20)

In the above equation, the state feedback gains, K φ , K φ , K yc , are determined to minimize the performance index which regards only lateral deviation from the desired lane as follows: ∞

(

)

J = ∫ qy + rδ dt 2 cr

2

2

y sr = y cr + l s φ r −

(21)

where, q, r indicates the weighting factors of lateral deviation and steering wheel angle respectively. With the conventional 2WS vehicle, the control plant for lane-tracking problem can be expressed as the following fourth-order system:  φ  a 11 a12 a13 0  φ  b11         0 0 0  φ r   0   φr  =  1 + δ  y  a 31 a 32 a 33 0  y  b31  2WS (22) cr cr        0 1 0  y cr   0   y cr   0 where, all elements in the matrices are a11 = − a13 = −

2 r

2(l C f + l C r ) IV 2(l f C f − lr C r )

, a12 =

2(l f C f − lr C r )

, a 31 = −

4WS-A vehicle:  φ   − 1 / τ r     φr  =  1  y cr   0     ρ   0  φ   − 1 / τ r     φr  =  1  y cr   Vτ r     ρ   0

,

I 2(l f C f − lr C r )

δ 2WS = − K φφ − K φ φ − K y c y c − K y c y c

0 0 V 0

0 0   φ   kr / τ r  0         0 − V   φr   0  0 δ +  w + 0  (26) 0 0   ycr   0        0 − λ   ρ   0  1 

4WS-B vehicle:

, mV 2(C f + C r ) a32 = , a33 = − , m mV 2C f l f 2C f b11 = , b31 = . I m The control input in this case is obtained by conducting the feedback of all four state variables as follows: IV 2(C f + C r )

(25)

where, ls indicates the distance from CG to front bumper, and v indicates observation noise. Determining the covariances of process noise and observation noise, the Kalman filter can be designed. To construct the lane-tracking controller with curvature estimation, the curvature will be included as a state variable so that the state-space equation of each type of vehicle can be rewritten as follows:

0

2 f

ls ρ+v 2

0 0 V 0

0  0 0   φ   kr / τ r        0 − V   φr   0  0 δ +  w +   0  (27)  0  0 0  ycr       0 − λ   ρ   0  1 

2WS vehicle:  φ   a11 a12    0  φr   1  ycr  =  a31 a32    0  y cr   0  ρ   0 0   

a13 0 a33 1 0

0 0   φ  b11   0    0 0 − V   φ r   0    2 0 − V   y cr  + b31 δ 2WS + 0 w.       0 0   ycr   0   0    1    0 − λ  ρ   0   

(28) The steering wheel angle can be calculated as follows: δ = − K φ φ − K φ φ r − K yc y cr − K ρ ρ 4WS: (29)

(23)

where, all state feedback gains are determined to minimize the performance index J as indicated in Eq.(21)

2WS: δ 2WS = − Kφφ − Kφ φ r − K yc y cr − K yc ycr − K ρ ρ . (30) where, feedback gains are determined in the same way as described in Section 3.2.

3.3 Road curvature estimation using Kalman filter 4. CLOSED-LOOP SYSTEM ANALYSIS

In the case of curved roadway, road curvature is treated as a disturbance which reflects in steady-state error. To deal with this problem, this section proposes a

This section will discuss the difference in lane-tracking responses due to the different types of desired steering

388

response.

5. SIMULATION

By substituting Eq. (20) into Eq. (18), the closed loop system of 4WS-A vehicle can be obtained as follows:

5.1 Lane-tracking simulation on straight roadway

 1 kr  φ   − τ − τ Kφ r    r 1  φr  =   y   0  cr   

The lateral desired course, step input with magnitude of 0.2 meter, is applied to the vehicle running at constant speed of 100 km/h, at 1 second after simulation started. The simulation result is shown in Fig. 5.

kr Kφ τr 0

−

V

 kr K yc   φ  τr   0   φr . 0   ycr    

(31)

Fig. 5 compares the step desired lane-tracking responses in the case of 4WS-A, 4WS-B, and 2WS vehicle. Clearly, both 4WS-A and 4WS-B vehicles provide the better damping behaviour of both lateral and yaw motions comparing with 2WS vehicle. Especially, in the case of 2WS vehicle, the yaw angle has dominantly oscillatory response. Moreover, the

Substituting the relation between the lateral velocity and yaw angle in this case: ycr = Vφr

(32)

and rearranging Eq.(29), the tracking response of lateral displacement can be expressed as the following third-order transfer function. krVK yc yc = * 3 2 yc τ r s + (1 + k r Kφ ) s + k r Kφ s + krVK yc

Table 1 Parameters used for simulation

(33) Here, yc and y indicates the lateral displacement of vehicle and the desired trajectory respectively. * c

By substituting Eq. (20) into Eq. (19), the closed loop system of 4WS-B vehicle can be obtained as follows: kr Kφ τr 0

−

V

 kr K yc   φ  τr   0   φr    y cr  . 0   

(34)

Substituting the relation between the lateral velocity and yaw motion in this case:

Unit

Value

Distance from C.G. to front axle Distance from C.G. to rear axle Vehicle mass Yaw moment of inertia Equivalent front cornering stiffness Equivalent rear cornering stiffness

lf lr m I Cf Cr

m m kg kgm2 N/rad N/rad

1.358 1.472 1980 3758 50000 90000

and rearranging Eq.(32), the tracking response of 4WSB vehicle can be expressed as follows: k rVK yc = 2 * y c s + k r K φ s + k rVK yc yc

0.2 0.15 0.1 0.05

(36)

2

* 12

* 14

* 31

-0.5

4

0

1

2 Time [s]

3

4

0

1

2 Time [s]

3

4

2

6 4 2

1 0

-1

0 0

1

2 Time [s]

3

-2

4

10 1

10

0

2WS

∞

∫ y cr2 dt

* 32

4WS-B

0

-1

4WS-A

10

(37) * 12

3

0

Fig. 5. Step desired lane-tracking responses

− a (s − a s − a ) − a (a s − a ) yc = * * * * * yc* ( s 2 − a11* s − a )(s 2 − a 33 s − a 34 ) − (a 31 s + a32 )( a13 s + a14* ) * 11

1

0.5

8

On the other hand, the conventional 2WS vehicle achieves tracking response as the following fourthorder transfer function: * 11

0

1

Time [s]

-2

2

4WS-B 4WS-A 2WS

0

With the 4WS-B vehicle, lane-tracking response becomes second-order system. This implies that 4WSB vehicle has potential in providing better responsiveness in lane-tracking comparing with 4WSA vehicle.

* 34 * 12

1.5

(35)

y cr = V τ rφ + V φ r

4WS-B:

Notation

0.25

Lateral d isp.[m]

−

Front steer angle[deg]

 1 kr  φ   − τ − τ K φ r    r 1 φ r  =   y cr   Vτ r    

Model parameter

Rear steer angle[deg]

4WS-A:

Yaw angle [deg]

−

* 13

where, a = a11 − b11Kφ , a = a12 − b11Kφ , a = a13 − b11K yc , *

*

-2

*

a14 = −b11 K yc , a31 = a31 − b31 Kφ , a32 = a32 − b31K φ ,

10 -4 10

* * a 33 = a 33 − b31 K yc , a 34 = − b31 K yc .

10-3

10-2

∞

10-1

∫δ 0

2

10 0

101

dt

Fig. 6. Trade-off curves in lane-tracking control

389

102

4WS-B vehicle also provides better responsiveness and damping behaviour in lane-tracking response when comparing with the 4WS-A vehicle.

6. CONCLUSIONS A theoretical analysis of four-wheel-steering (4WS) system used for automatic lane-tracking control system was conducted. The comparative study on the lanetracking control performance between two types of steering responses of 4WS system was clarified. The major conclusions to be drawn from simulation study are summarized as follows:

The lane-tracking performances of three types of vehicles are evaluated by using “Trade-off curve”. The trade-off curves, shown in Fig. 6, plot the square of lateral error along the vertical axis and plot the square of steering wheel angle along the horizontal axis. Simulation result shows that under the same steering angle control input, the 4WS-B vehicle shows the best performance in lane-tracking with smallest deviation. On the other hand, as can be noticed from the graph, lane-tracking control performance by 4WS-A vehicle is not significantly improved from 2WS vehicle. Moreover, to make the better understanding in the improved tracking response of yaw direction, Fig. 7 plots the square of the lateral deviation along the horizontal axis and the square of the yaw deviation along the vertical axis. Clearly from the graph, 4WSB vehicle shows the best performance in lane-tracking with smallest lateral and yaw deviations.

(1) By applying active 4WS system to lane-tracking task, the lateral and yaw dynamics during lanetracking manoeuvre can be effectively controlled to have desirable characteristics, comparing with 2WS vehicle. (2) It is clarified that active 4WS system which makes the phase-lag of the lateral acceleration be zero (4WS-B) provides the superior lane-tracking performance in lateral direction to the one which makes vehicle body side slip angle be zero (4WSA), however, 4WS-B has a drawback that the heading error of vehicle cannot be regulated during tracking the curved roadway.

5.2 Lane-tracking simulation on curved roadway REFERENCES The simulation under the condition, that vehicle runs straightly at constant speed of 100 km/h for 1 second and enters to the curved trajectory with constant radius of 400 m, is conducted.

Furukawa, Y. and M. Abe (1997). Advanced Chassis Control Systems for Vehicle Handling and Safety. Vehicle System Dynamics, Vol.28, pp.59-86. Hedrick, J.K., M. Tomizuka and P. Varaiya (1994). Control Issues in Automated Highway Systems. IEEE Control Systems, Vol. 12, pp.21-32. Marumo, Y., H. Mouri, Y. Wang, T. Kamada, and M. Nagai (2000). Study on Automatic Path Tracking Using Virtual Point Regulator, JSAE Review, Vol.21, pp.523-528. Mouri, H. and H. Furusho (1997). Automatic Path Tracking Control Using Linear Quadratic Control Theory. IEEE Conference on Intelligent Transportation Systems. Nagai, M. (1989). Active Four-Wheel-Steering System by Model Following Control. Proceedings of 11th IAVSD Symposium, pp.428-439.

Fig. 8 shows the curved lane-tracking responses of three types of vehicle. Since the curvature estimation is conducted, the lateral deviation is regulated to be zero in steady-state. 4WS-B vehicle provides the most satisfactory response of both yaw and lateral directions in transient state whereas the 4WS-A and 2WS still have oscillatory response. 4WS-B also provides the smallest lateral deviation in tracking the curved lane with satisfactory response. However, 4WS-B vehicle has its drawback about the heading error due to its own desired model. On the other hand, 4WS-A vehicle, regulating the side slip angle during cornering, is superior in tracking performance of yaw direction. 0

Yaw rate [deg/s]

2WS -1

10

∞

∫φ

2 r

dt

4WS-A

Head ing error [deg]

10

10

8 6 4 2 0

0

1

4WS-B

-3

10

-1

10

0

∞

∫y

2 cr

10

dt

1

10

0

Fig. 7. Lane-tracking performance of each vehicle

3 2 Time [s]

1 0.5 0 -0.5 -1 0

1

3 2 Time [s]

4

1 0.5 0

-0.5 0

4 Lateral deviation.[m]

-2

10

Side slip angle [deg]

0

1.5

5

1

x 10 -3

2 3 Time [s]

0 4WS-B 4WS-A 2WS

-5 -10 -15 0

1

2 Time [s]

Fig. 8. Step curved lane-tracking responses

390

4

3

4