ACTIVE FOUR WHEEL STEERING SYSTEM FOR ZERO SIDESLIP ANGLE AND LATERAL ACCELERATION CONTROL

ACTIVE FOUR-WHEEL STEERING SYSTEM FOR ZERO SIDESLIP ANGLE AND LATERAL ACCELERATION CONTROL Toshihiro Hiraoka∗ Osamu Nishihara∗ Hiromitsu Kumamoto∗ ∗ Graduate School of Informatics, Kyoto

ACTIVE FOUR-WHEEL STEERING SYSTEM FOR ZERO SIDESLIP ANGLE AND LATERAL

ACCELERATION CONTROL Toshihiro Hiraoka∗ Osamu Nishihara∗

Hiromitsu Kumamoto∗

∗ Graduate School of Informatics, Kyoto University,

Kyoto, JAPAN

Abstract: This paper proposes an active four-wheel steering system It has three

additional points to an active front steering system proposed by the authors: 1)

active rear steering to realize zero sideslip angle, 2) variable steering ratio to

prevent abrupt involution during slowdown, and 3) model following sliding mode

controller that is robust against the system uncertainty Computer simulations

demonstrate good maneuverability of the proposed system

Keywords: Active four wheel steering, Model following sliding mode control,

Variable steering ratio

1 INTRODUCTION

Recently, active steering systems (Ackermann,

1997; Shimada et al., 1997) have been studied and

developed for improvement of active safety

Authors (Hiraoka et al., 2001; Hiraoka et al., 2002)

also proposed an active front steering (AFS) law

based on the dynamics of lateral acceleration at a

center of percussion with respect to rear wheels

By using the control law, lateral acceleration at

the center of percussion can be proportional to

steering wheel angle without influence of vehicle

sideslip angle and yaw rate The AFS improves

maneuverability of the vehicle in the difficult

driving situation such as packed snow road

However, the AFS law has problems: 1)

deterio-ration of response from driver’s steering input to

sideslip angle and yaw rate, 2) abrupt involution

at slowdown, and 3) no consideration of system

uncertainty such as cornering power perturbation

First, active four-wheel steering law is proposed

in this paper by addition of active rear steering

(ARS) to AFS in order to realize zero sideslip

angle, so that the dynamic characteristics of yaw rate and lateral acceleration at a center of gravity become the first order lag system Second, variable steering ratio is introduced to the relationship between driver’s steering input and front wheel angle And last, model following controller is in-troduced for the active four-wheel steering system

to become robust against system uncertainty

2 ACTIVE FRONT STEERING

2.1 2DOF vehicle model

This paper assumes that a vehicle with constant velocity moves with only two degrees of freedom, right and left transition, and yaw rotation Figure

1 shows 2DOF vehicle model used in this paper

sideslip angle, γ is the yaw rate, and u f /u rare the front/rear steering angle A linearized equation of motion of four-wheel steering vehicle becomes

β G

P

v

r

θ

l f

l r

L p

u f

Target

course

Vehicle

u r

p

ε

x

y

x0

y0

Fig 1 2DOF vehicle model

˙

A =

⎡

⎢

⎢

⎣

− 2(K f + K r)

mv2

− 2(K f l f − K r l r)

I z − 2(K f l2f + K r l2r)

I z v

⎤

⎥

⎥

⎦

B =

⎡

⎢

⎣

2K f

mv

2K r

mv 2K f l f

I z − 2K r l r

I z

⎤

⎥

⎦ , C =

⎡

⎢

⎣

1

mv

− l w

I z

⎤

⎥

inertia around the axis of the center of gravity G,

v is the vehicle velocity, K f /K rare the cornering

powers of front/rear tires, l f /l rare the distances

between G to front/rear wheel axles, w is the

lateral disturbance, and l wis the distance between

G to the disturbance load center

2.2 A center of percussion with respect to rear

wheel

In this paper, a center of percussion with respect

to rear wheels is considered as a datum point P

for control: no acceleration is generated at P even

if an impact force acts on the rear wheels

I z /(ml r ) Lateral acceleration a p at P is

a p=− 2K f l

l r m (β +

l f

v − u f) +l r − l w

2.3 Active front steering for lateral acceleration

control at a center of percussion

2.3.1 Active front steering law An active front

steering law based on eq.(3) is proposed by

au-thors (Hiraoka et al., 2001; Hiraoka et al., 2002):

u f ≡ u m + u a= 1

k s δ + (β + l f

component, δ is the driver’s steering wheel angle input, and k sis the steering ratio

2.3.2 Transfer function Substitution of eq.(4)

into eq.(1) yields a transfer function G ap (s) from

δ to lateral acceleration at P.

k s · 2K f l

fre-quency response Therefore, a driver can control lateral acceleration at P without taking influence

of vehicle sideslip angle and yaw rate Driving simulator experiments demonstrated the improve-ment of path following capability on the packed

snow road (Hiraoka et al., 2001).

from δ to β and γ, in the both case of AFS vehicle

and conventional 2WS vehicle Figure 2 shows the

step responses of β and γ of the two vehicles.

Velocity is 90[km/h], step input of steering wheel angle is 1[rad], and other parameters are shown

in Table 2 (see Section 4) These figures illustrate

that the responses from δ to β and γ of AFS are

more vibratory than that of conventional 2WS

2.3.3 Steering wheel angle for constant radius

(radius: r[m]):

δ 0,AF S = k s l r m

2K f l

v2

while conventional 2WS vehicle needs the angle

δ 0,2W S:

δ 0,2W S = k s0 l

r



2l2

K f l f − K r l r

 (7)

Time [s]

-0.03 -0.02 -0.01 0 0.01

From: Steering wheel angle [rad]

Conventional 2WS AFS

0 0.1 0.2 0.3 0.4 0.5

Velocity: 90 [km/h]

Steering wheel input: 1[rad]

Steering ratio: 16 (constant)

Fig 2 Step responses of sideslip angle (upper) and yaw rate (lower)

Table 1 Transfer functions (2WS vs AFS)

G β(s) = G β(0) 1 +T β s

1 +a0s + a1s2, G γ(s) = G γ(0) 1 +T γ s

1 +a0s + a1s2

G β(0) 1

k s · K f(2K r l r l − l f mv2)

2K f K r l2− (K f l f − K r l r)mv2

1

k s · K f(2K r l r l − l f mv2)

K r l r mv2

G γ(0) 1

k s · 2K f K r lv

2K f K r l2− (K f l f − K r l r)mv2

1

k s ·2K f l

l r mv

a0

(K f+K r)I z v + (K f l2+K r l2)mv

2K f K r l2− (K f l f − K r l r)mv2

I z+ml2

l r mv

4K f K r l2− 2(K f l f − K r l r)mv

I z

2K r l r

2K r l r l − l f mv2

2K r l

l f mv

2K r l

to the square of velocity v when the steering

makes vehicle stable even if the original steer

characteristic is oversteer (K f l f − K r l r > 0).

However, an abrupt involution will happen when

the vehicle slows down with the constant steering

wheel angle

3 ACTIVE FOUR-WHEEL STEERING

3.1 Addition of active rear steering for zero

sideslip angle

3.1.1 Active four-wheel steering law An active

four-wheel steering (A-4WS) law is defined as the

summation of AFS shown in eq.(4) and ARS

Here, the ARS consists of linear combination of

steering input δ and vehicle state x.

k s

1

⎡

⎣ 1 l v f

k b k g

⎤

Similarly to the study of ARS (Harada, 1995), the

gains k h , k g are obtained to satisfy zero sideslip

angle (G β (s) = 0).

k h=− K f

K r , k g= mv2− 2K r l r

Therefore, the active four-wheel steering law

with-out the feedback of sideslip angle is defined as the

following equation again

k s

⎡

− K f

K r

⎤

⎥

⎦ , E =

⎡

⎢

⎣

v

0 mv2− 2K r l r

2K r v

⎤

⎥

⎦ (12) Substitution of eq.(11) into eq.(1) yields

A =

⎡

⎢

⎣

− 2(K f + K r)

− 2(K f l f − K r l r)

I z

⎤

⎥

⎦ , (14)

B =

⎡

2K f l

k s I z

⎤

⎥

3.1.2 Transfer function Transfer functions from driver’s steering input to sideslip angle, yaw rate, lateral acceleration at G and P become as follows:

k s · 2K f l

l r mv , T0=

I z

As shown in eq.(16) and eq.(17), G γ (s) and G ag (s)

become the first order lag It represents that A-4WS has better response than conventional 2WS and AFS

3.2 Variable steering ratio

Active four-wheel steering vehicle with the control law (11) requires the same steering wheel angle

δ 0,A4W S as eq.(6) for the constant radius turn

δ 0,A4W S = δ 0,AF S = k s l r m

2K f l

v2

From eq.(7) and eq.(19), δ 0,A4W Sis coincides with

δ 0,2W S when k s is defined as follows

k s = k s0



2K f l2

l r mv2 − K f l f − K r l r

K r l r

 (20)

δ 0,A4W S = k s0 l

when k sis set as

k s = k s0 2K f l2

3.3 Model following sliding mode controller

dy-namics has multi degrees of freedom and

non-linearity Especially, a perturbation of cornering

power and a lateral disturbance such as

cross-wind affect on vehicle lateral motion Define the

Equation of motion for vehicle with the cornering

power perturbation becomes

˙x = Ax + Bu + Cw

3.3.2 Addition of model following controller

This paper proposes the active four-wheel steering

law by addition of model following sliding mode

controller ¯u = [¯ u f u¯r] to the steering law (11)

to guarantee the robustness against the system

uncertainty Figure 3 shows a block diagram of

the proposed system

Substitution of eq.(25) into eq.(24) yields

˙

x = A  x + B  δ + B0u + f (x, t)¯ (26)

Let the reference vehicle model be

˙

er-ror equation is obtained from eq.(26) and eq.(28)

+

x f

u

r

u

Vehicle

-+

m

Reference Vehicle Model

Sliding Mode Controller

Feedback Controller

+ + + +

δ

Feedforward Controller

Model following controller

Active four-wheel steering controller derived from linear vehicle model

f u

r

u

Disturbance

Fig 3 Block diagram of proposed system

˙e = A m e + (A m − A  )x +(B m − B  )δ − B0¯u − f (x, t) (29)

defines the transfer functions G β (s), G γ (s) shown

in eq.(16) as the reference model Then, eq.(29) becomes

This paper designs the model following controller

¯

u to match the actual vehicle state vector x =

[β m γ m]

func-tion σ is defined as follows.

σ =

σ1

σ2

= Se =

1 p

1 q

e1

e2

(31)

Time differentiation of σ appears as

˙σ = S ˙e = S{A m e − B0(¯u + d(x, t))}. (32)

Substitution of ˙σ = 0 to eq.(32) gives the

equiva-lent control input ¯u eq

¯

and nonlinear control input ¯u nl

¯

u = ¯ u eq,0+ ¯u nl

ρ1sgn(σ1

ρ2sgn(σ2

(34)

where ρ1, ρ2 are positive constant values.

the time differentiation of V becomes

˙

Let the parameters p = I z /(l r mv), q = −I z /(l f mv)

in the switching function, eq.(35) becomes

V = −σ12K f0 l

l r mv (ρ1sgn(σ1) + d1

−σ22K r0 l

l f mv (ρ2sgn(σ2) + d2). (36)

Therefore, the convergence of the system to the

switching plane is guaranteed by the Lyapunov

stable theorem because ˙V < 0 when ρ1> |d1| and

ρ2> |d2|.

Finally, we have the control law:

u f

u r

k s

⎡

− K f0

K r0

⎤

⎦ δ +

⎡

⎢

⎣

v

2K r0 − l r

v

⎤

⎥

⎦ x

+

⎡

⎢

⎣

−1 − l r mv

2K f0 l

2K r0 l

⎤

⎥

⎦ e +

ρ1sgn(σ1

A sign function sgn(σ i) of eq.(37) is approximated

by the following equation to prevent systems

chat-tering by smoothing control input

sgn(σ i) |σ σ i

i | + µ i , µ i > 0 (i = 1, 2) (38)

4 SIMULATION This paper performed two simulations using

Car-Sim Ver.5.12 by Mechanical Car-Simulation

Corpo-ration: 1) constant radius turn test with

decel-eration and 2) double lane change test CarSim

is a software package for simulating real vehicle

dynamics by using 19 degrees of freedom vehicle

model Table 2 shows the simulation parameters

4.1 Constant radius turn test with deceleration

In order to verify the effectiveness of VSR

(Vari-able Steering Ratio, eq.(22)), simulations were

performed with five vehicles:

(1) Conventional 2WS

(2) AFS without VSR

(3) A-4WS without VSR, SMC(Sliding Mode

Controller)

(4) A-4WS with VSR, without SMC

(5) A-4WS with VSR, SMC

The vehicle starts with an initial velocity 60[km/h],

and decelerates for 10[s] by -4[km/h] per second

Table 2 Simulation parameters

1707 2741.9 1.014 1.676 68909 51406

ρ1 ,ρ2 µ1 ,µ2 τ τ l k s0

0 20 40 60 80

x[m]

Conventional2WS

A-4WS with VSR,SMC

A-4WS w/o VSR, SMC

A-4WS with VSR w/o SMC

AFS w/o VSR

Initial velocity: 60[km/h]

Final velocity: 20[km/h]

Running time: 10[s]

Steering angle: 0.43[rad]

=Target radius: 100[m]

Fig 4 Constant radius turn test with deceleration Vehicle’s steering angle is 0.43[rad] The angle is

necessary for the constant radius turn r=100[m] that is calculated by substitution of r = 100 into

eq.(21)

Figure 4 demonstrates that VSR prevents the abrupt involution, and also shows that A-4WS with SMC can run along the constant radius path

r=100[m] by the model following controller 4.2 Double lane change test

Double lane change tests were performed to verify the performance of obstacle avoidance and driv-ability Here, three vehicles ran along the path (see Figure 5) under two conditions

Vehicles:

(1) A-4WS with SMC (2) A-4WS without SMC (3) Conventional 2WS Conditions:

(1) Dry asphalt road (µ=0.9), v = 90[km/h] (2) Packed snow road (µ=0.2), v = 60[km/h]

In the simulations, a look-ahead driver model was

used The reference point is L = τ v[m] ahead from

the control datum point P The model outputs a

steering wheel angle δ based on the course error ε

at the reference point

δ(s) = he −τ l s

where h is the proportional gain, and τ lis the dead time

Figure 6 shows that A-4WS with SMC obtains desired responses of sideslip angle and yaw rate

y[m]

x[m]

P 2

(130,3.5)

P 3

(155,3.5)

P 4 (180,0)

Fig 5 Path of double lane change test

0 100 200 300

-2

0

2

4

x [m]

A-4WS with SMC A-4WS w/o SMC Conventional 2WS

-2 0 2 4

x [m]

A-4WS with SMC A-4WS w/o SMC Conventional 2WS

-0.02

0

0.02

Time [s]

A-4WS with SMC A-4WS w/o SMC Conventional 2WS

-0.1 -0.05 0 0.05 0.1

Time [s]

A-4WS with SMC A-4WS w/o SMC Conventional 2WS

-0.5

0

0.5

Time [s]

A-4WS with SMC A-4WS w/o SMC Conventional 2WS

-0.4 -0.2 0 0.2 0.4

Time [s]

A-4WS with SMC A-4WS w/o SMC Conventional 2WS

-10

-5

0

5

10

Time [s]

ag

2 ]

A-4WS with SMC A-4WS w/o SMC Conventional 2WS

(1) µ = 0.9 (Dry asphalt road, v=90[km/h])

-10 -5 0 5 10

Time [s]

ag

2 ]

A-4WS with SMC A-4WS w/o SMC Conventional 2WS

(2) µ = 0.2 (Packed snow road, v=60[km/h])

Fig 6 Double lane change test

even on the packed snow road by the model

follow-ing slidfollow-ing mode controller that works effectively

to compensate a state error Therefore, it has

an adequate path following capability Moreover,

the lateral acceleration response of A-4WS with

SMC becomes faster than others It implies the

improvement of obstacle avoidance capability

Next, A-4WS w/o SMC is compared with

Conven-tional 2WS On the dry asphalt road, a sideslip

angle of A-4WS w/o SMC, that should be zero

in the linear region because of eq.(16), is

equiva-lent to that of Conventional 2WS, but yaw rate

response and path following capability are

bet-ter than Conventional 2WS Furthermore, on the

packed snow road where tire characteristic shows

strong nonlinearity, A-4WS w/o SMC gets worse

in the control results than Conventional 2WS

5 CONCLUSION This paper proposed the active four-wheel

steer-ing system for zero sideslip angle and lateral

accel-eration control at a center of percussion by using

model following sliding mode controller

Theo-retical analysis and computer simulations showed

that the proposed system had a good

maneuver-ability and the robustness against system

uncer-tainty such as cornering power perturbation and lateral disturbance

REFERENCES

pre-vents car skidding IEEE Control Systems

17(3), 23–31.

Harada, H (1995) Control strategy of active rear wheel steering in consideration of system

delay and dead times Transaction of JSAE (in Japanese)26(1), 74–78.

Hiraoka, T., H Kumamoto and O Nishihara (2002) Side slip angle estimation and active front steering system based on lateral accel-eration data at centers of percussion with

respect to front/rear wheels Proceedings of

2002 JSAE Annual Congress (in Japanese).

Hiraoka, T., H Kumamoto, O Nishihara and

K Tenmoku (2001) Cooperative steering sys-tem based on vehicle sideslip angle estimation from side acceleration data at percussion

cen-ters Proceedings of IEEE International Vehi-cle Electronics Conference 2001 pp 79–84.

Shimada, Y., S Nohtomi, S Horiuchi and

N Yuhara (1997) An adaptive LQ con-trol system design for front and rear wheel

steering vehicle Transaction of JSAE (in Japanese)28(4), 111–116.

in the control results than Conventional 2WS

5 CONCLUSION This paper proposed the active four- wheel

steer-ing system for zero sideslip angle and lateral

accel-eration control. .. and lateral disturbance

REFERENCES

pre-vents car skidding IEEE Control Systems

17(3), 23–31.

Harada, H (1995) Control strategy of active rear wheel steering. .. consideration of system

delay and dead times Transaction of JSAE (in Japanese)26(1), 74–78.

Hiraoka, T., H Kumamoto and O Nishihara (2002) Side slip angle estimation and active