Design of a robust controller for rollover prevention with active ...

Journal of Mechanical Science and Technology 26 (1) (2012) 213~222 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-011-0915-9

Design of a robust controller for rollover prevention with active suspension and differential braking† Seongjin Yim* Advanced Institutes of Convergence Technology, Seoul National University, 864-1, Iui-dong, Yeongtong-gu, Suwon-si, Gyeonggi-do 443-270, Korea (Manuscript Received May 18, 2011; Revised August 22, 2011; Accepted August 28, 20011) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract This paper presents a method to design a robust controller for rollover prevention. Several types of controllers for rollover prevention have been proposed in such a way to minimize the lateral acceleration and the roll angle. The rollover prevention capability of these controllers can be enhanced if the controlled vehicle system is robust to the variation of the height of the center of gravity (C.G.) and the speed of the vehicle. With this idea, a robust controller is designed with linear quadratic static output feedback and parameter sensitivity reduction scheme. Differential braking and an active suspension system are adopted as actuators that generate yaw and roll moments, respectively. The proposed method is shown to be effective in preventing rollover through simulations on the nonlinear multi-body dynamic simulation software, CarSim®. Keywords: Rollover prevention control; Robust control; Parameter sensitivity reduction; Linear quadratic static output feedback; Active suspension; Differential braking ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction In the last decade, a widespread supply of SUVs (Sports Utility Vehicles) with high center of gravity (C.G.) has increased the frequency of rollover accidents. Most rollover accidents were fatal. Though rollovers occurred in about 3% of all crashes in 2002, 33% of all fatalities were caused by rollovers [1]. Although this portion of fatalities has been reduced from 36.3% to 33.7% in the last eight years, rollovers still account for a large portion of all deaths caused by passenger vehicle crashes [2]. Hence, vehicle rollover should be prevented for passenger safety. The major factors that influence rollover are lateral acceleration ay, the height of the center of C.G. hs and lateral force Fy. Untripped rollover occurs due to large lateral acceleration by excessive steering at high speed. On a low friction road or at low speeds, rollovers cannot occur since the lateral acceleration is small. On this account, it is necessary to reduce the lateral acceleration and lateral force to prevent rollover. Following this idea, several control schemes were proposed. The most common scheme for rollover prevention is to reduce the reference yaw rate through differential braking or active steering in order to make a vehicle have under-steer characteristics †

This paper was recommended for publication in revised form by Associate Editor Kyongsu Yi Corresponding author. Tel.: +82 31 888 9023, Fax.: +82 31 888 9040 E-mail address: [email protected] © KSME & Springer 2012 *

[3-6]. However, these approaches did not take the variation of the height of C.G. and the vehicle speed into account since the height of C.G. has been regarded as invariant. The rollover prevention capability of these control schemes can be enhanced by designing a controller that considers the variation of these factors. Gaspar et al. used 2DOF bicycle and 1DOF roll model with longitudinal velocity as a time-varying parameter in the design of a gain-scheduled controller for rollover prevention, while unmodelled dynamics was regarded as an unstructured uncertainty [7]. To enhance rollover prevention capability, it is necessary to design a controller which is robust to the variation of the height of C.G. and vehicle speed. In this paper, these parameters are treated as structured variables, which is normally more difficult to handle than an unstructured one. In order to design a controller which is robust against these uncertain parameters, a parameter sensitivity reduction scheme is adopted. The key idea of this parameter sensitivity reduction scheme is to assign the weights of the linear quadratic (LQ) objective function in order to reduce the trajectory sensitivity of a state. The trajectory sensitivity is defined as a derivative of the state with respect to a particular parameter at its nominal value [8-10]. It is known that the trajectory sensitivity is satisfactory up to ±30% variation of a nominal value [10]. For given nominal statespace models, it is easy to obtain the trajectory sensitivity model. The reduction of the trajectory sensitivity makes the system robust to parameter variations. It was proven that the

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S. Yim / Journal of Mechanical Science and Technology 26 (1) (2012) 213~222

follows [18]:

(a) 2-DOF bicycle model

Lateral motion: ma y − ms hsφ&& = Fyf + Fyr

(1)

Yaw motion: I zγ& = l f Fyf − lr Fyr + M B

(2)

Roll motion: t t I xφ&& − ms hs a y = − ⋅ f1 + ⋅ f 2 − Cφφ& − Kφφ + ms ghsφ . 2 2

(3)

In these equations, MB, f1, and f2 are the control yaw moment by differential braking and the active suspension forces, respectively. In Eqs. (1) and (3), the lateral acceleration ay is defined as follows: a y = v& y + γ vx . (b) 1-DOF roll model Fig. 1. 3-DOF vehicle model.

minimum trajectory sensitivity is equivalent to the minimum eigenvalue sensitivity [11]. The parameter sensitivity reduction scheme has been widely used to design a robust controller [12-16]. In the research of Tuel, the trajectory sensitivity was incorporated into LQ objective function and then minimized using the linear quadratic regulator (LQR) [9]. LQR requires all the states for feedback. To avoid introducing an observer or an estimator for LQR, it is desirable to design a controller based on available outputs only. Hence, the linear quadratic static output feedback (LQ SOF) method is adopted to design the robust controller [15]. The parameter sensitivity reduction scheme can be easily integrated into LQ SOF by modifying the structure of the weights on states in the LQ cost function. This paper is organized as follows: In Section 2, the yaw and roll motion of a vehicle is modeled with 3DOF vehicle model, and a design method for the rollover prevention controller is presented. A design method for robust rollover prevention controller is also proposed utilizing LQ SOF with the parameter sensitivity reduction scheme. In Section 3, a linear analysis and simulations are performed on a linear vehicle model and nonlinear model based on a commercial multibody dynamics software, Carsim® [17]. The conclusions are given in Section 4.

2. Design of a rollover prevention controller 2.1 Vehicle model The vehicle model used in this paper is a 3-DOF model shown in Fig. 1. This model consists of 2-DOF bicycle and 1DOF roll model to describe the yaw and the lateral motion, and the roll motion, respectively. The equations of motion for this vehicle model are given as

(4)

In Eqs. (1) and (2), the lateral tire forces Fyf and Fyr are assumed to be linearly proportional to the tire slip angle for small α, as shown in Eq. (5). The tire slip angle α is defined as the difference between the direction of wheel velocity and the steering angle, as given in Eq. (6), which can be obtained through the approximation tan −1 (θ ) ≈ θ . Fyf = −C f α f , Fyr = −Crα r

αf =

vy + l f γ vx

− δ f , αr =

(5)

v y − lr γ vx

(6)

The cornering stiffness Cf and Cr are valid within the linear region where α is small. If α goes over the saturated region of Fy, the constant cornering stiffness assumption is not valid any more. Moreover, Fy varies according to the variation of μ. To capture this feature of the lateral tire forces, Cf and Cr should be treated as an uncertain parameter in controller design procedure. The reference yaw rate γd generated by the driver’s steering input δf is modeled with a first-order system as follows:

γd

⎛ K ⎞ = ⎜ γ ⎟δ f ⎝τ s +1⎠ =

⎛ δf ⎞ ⎜ ⎟ C f ⋅ Cr ⋅ L2 + m ⋅ vx2 ⋅ ( lr ⋅ Cr − l f ⋅ C f ) ⎝ τ s + 1 ⎠ C f ⋅ Cr ⋅ L ⋅ v x

(7)

where τ is the time constant, and Kγ is the steady-state yaw rate gain determined by the speed of vehicle [19]. The state-space representation of Eq. (7) is as follows: 1

K

γ&d = − γ d + γ δ f . τ τ

(8)

The error of yaw rate eγ is defined as the difference between the actual yaw rate γ and the reference one γd.

215


eγ = γ − γ d

(9)

The state x, the control input u and the disturbance w are defined as follows: x ⎡⎣ v y u [M B

γ φ& φ γ d ⎤⎦ f1

T

f2 ]

T

.

(10)

w δf

From these definitions and equation of motions, the statespace equation of the vehicle model is obtained as follows: x& = Ax + B1w + B 2u = E −1A e x + E−1B e1w + E −1B e 2u ⎡ m ⎢ 0 ⎢ E ⎢ − ms h s ⎢ ⎢ 0 ⎢ 0 ⎣

− ms h s

0 Iz 0 0 0

0 Ix 0 0

0⎤ 0 ⎥⎥ 0⎥ , ⎥ 0⎥ 1 ⎥⎦ 0 0 ms gh s − Kφ 0

0 0 0 1 0

a12 0 ⎡ a11 ⎢a a22 0 ⎢ 21 ⎢ 0 ms h s vx −Cϕ Ae ⎢ 0 1 ⎢0 ⎢ 0 0 0 ⎢0 ⎣ ⎡ Cf ⎤ 0 0⎤ ⎡0 ⎢l C ⎥ ⎢1 f f 0 0 ⎥⎥ ⎢ ⎥ ⎢ ⎢ 0 ⎥ ⎢ t t ⎥ B e1 ⎢ ⎥ , Be 2 ⎢0 − 2 2⎥ ⎢ 0 ⎥ ⎢0 0 0⎥ ⎢ K ⎥ ⎢ ⎥ ⎢ r ⎥ 0 0 ⎦⎥ ⎣⎢0 ⎣ τ ⎦ a11 = −

where a21 = −

C f + Cr vx

, a12 = −

l f C f − lr C r vx

In Eq. (12), qi is the weight of each objective. Through tuning the value of qi, it is possible to emphasize each objective. The weights qi of the LQ cost function are set by the relation qi = 1/ηi2 from Bryson’s rule, where ηi represents the maximum allowable value of each term [20].

l f C f − lr C r

, a22 = −

vx

0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥, 0 ⎥ 1⎥ − ⎥ τ⎦

⎡ a y ⎤ ⎡ a11 a12 + vx ⎢ φ& ⎥ ⎢ 0 0 y=⎢ ⎥=⎢ ⎢γ ⎥ ⎢ 0 1 ⎢ ⎥ ⎢ 0 ⎣γ d ⎦ ⎣ 0

a14 0 0 0

a15 ⎤ 0 ⎥⎥ x ≡ Cx 0⎥ ⎥ −1⎦

(13)

Assuming that the static output feedback controller u=Ky is used, the LQ SOF problem is to find K that minimizes the LQ cost function J. The available outputs for SOF are the roll rate, the yaw rate, the lateral acceleration, and the reference yaw rate, given as Eq. (13). In Eq. (13), a1i is the elements of the first row of the matrix A from Eq. (11). These signals are easy to measure by sensors. The LQ cost function, Eq. (12), can be converted into the following form: ∞

J = ∫ ( C2 x + D2u ) ( C2 x + D2u ) dt T

0

(11)

=∫

∞

0

(14)

{x Qx + u N x + x Nu + u Ru}dt T

T

T

T

T

where Q CT2 C2 , N CT2 D2 , R DT2 D2

− mvx ,

l 2f C f + lr2Cr

.

vx

As shown in Eq. (11), the longitudinal velocity vx is a timevarying parameter. In order to deal with the time-varying parameter, it is necessary to design a controller for time-varying system or gain-scheduled controller. However, its procedure is complex. For this reason, a robust controller should be designed through treating the time-varying parameter vx as an uncertainty. 2.2 LQ SOF controller design for rollover prevention The LQ cost function for rollover prevention is defined as follows:

⎡ q1 a11 ⎢ ⎢ 0 ⎢ ⎢ 0 C2 ⎢ 0 ⎢ ⎢ 0 ⎢ ⎢ 0 ⎢ 0 ⎣ ⎡ q1 b2,11 ⎢ ⎢ 0 ⎢ 0 ⎢ D2 ⎢ 0 ⎢ ⎢ q5 ⎢ ⎢ 0 ⎢ 0 ⎣

K

(12)

q1 ( a12 + vx )

q1 a14

q2

0

0

0

q3

0

0 0 0 0

0 0 0 0

q4 0 0 0

q1 b2,12 0 0 0 0 q6 0

q1 a13

q1 a15 ⎤ ⎥ − q2 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥, 0 ⎥ ⎥ 0 ⎥ 0 ⎥⎦

q1 b2,13 ⎤ ⎥ 0 ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥ q6 ⎥⎦

.

In Eq. (14), b2,1i is the element of the first row of the matrix B2 from Eq. (11). With this notation, LQ SOF is formulated as the following optimization problem [21]: min

2 2 2 &2 ∞ ⎛ q1eγ + q2 a y + q3φ + q4φ ⎞ J =∫ ⎜ ⎟ dt . 2 2 2 0 ⎜ ⎟ ⎝ + q5 M B + q6 f1 + q6 f 2 ⎠

a13 1 0 0

1 J = trace ( P ) 2

( A + B 2KC )

T

s.t.

P + P ( A + B 2KC )

+ Q + CT K T NT + NKC + CT K T RKC = 0

.

(15)

216


For an arbitrary K, the LQ cost function J can be readily computed by solving the Lyapunov equation, given as the constraint in Eq. (15). There have been several methods to compute the optimal K [21, 22]. However, this problem has not been proven to have a global optimum. In this situation, a heuristic search such as a genetic algorithm or an evolutionary strategy is a good alternative to the classical gradient based search. For this reason, the evolutionary strategy, CMA-ES, is used to find the optimal K [23].

Table 1. Parameters and values of SmallSUV model in CarSim. M

p = [ p1

(17)

pr ]

p2 L

p 0 = ⎡⎣ p10

pr0 ⎤⎦

p20 L

(18)

where p and p0 are the vectors of system parameters and its nominal values, respectively. r is the number of uncertain parameters. Let us assume that this system is controllable or stabilizable by full-state feedback. This is a necessary condition for the existence of the solution of LQR. The first-order trajectory sensitivity with respect to the parameter pi is defined as follows: σ i (t , p 0 ) ≡

∂x(t ) ∂pi

.

(19)

pi = pi0

984.6

Cf

Ix

442

Cr

27000

Iz

1302

Cφ

1000

lf

0.88

Kφ

60000

lr

1.32

vx

60

⎛ ∂w ⎞ ⎛ ∂B 2 ⎞ 0 ⎛ ∂u ⎞ + B1 ( p 0 ) ⎜ ⎟+⎜ ⎟ u + B2 (p ) ⎜ ⎟ p p ∂ ∂ ⎝ i⎠ ⎝ i ⎠ ⎝ ∂pi ⎠ = Aσ i + A pi x + B1, pi w + B1w pi + B 2, pi u + B 2u pi

In Eq. (19), the subscript pi = p will be omitted hereafter for simplicity. With the above definition of the trajectory sensitivity, the actual state x can be represented as follows: r

x ( t , p ) = x ( t , p 0 ) + ∑ σ i ( t , p 0 ) Δpi .

σi

(20)

In Eq. (20), Δpi is the variation of the parameter pi. According to Eq. (20), the trajectory sensitivity indicates how much the parameter variation Δp has the effect on the state [10]. Hence, it is necessary to reduce the trajectory sensitivity for the purpose of designing a robust controller. It is known that the trajectory sensitivity is satisfactory for up to ±30% variation of a nominal value [10]. For given nominal state-space models, it is easy to obtain the trajectory sensitivity model. Contrary to the quadratic stabilization or polytopic uncertainty [24], the trajectory sensitivity model does not require information about the range of the parameter variations. By differentiating the state equation in Eq. (16) with respect to the parameter pi at its nominal value pi0, the equation of trajectory sensitivity is easily obtained as follows: ⎛ ∂A ⎞ ⎛ ∂B1 ⎞ σ& i = A ( p 0 ) σ i + ⎜ ⎟x + ⎜ ⎟w ∂ p ⎝ i⎠ ⎝ ∂pi ⎠

∂x ∂w ∂u , w pi , u pi ∂pi ∂pi ∂pi

A A ( p 0 ) , B1 B1 ( p 0 ) , B 2 B 2 ( p 0 ) . A pi

∂A ∂B ∂B , B1, pi 1 , B 2, pi 2 ∂pi ∂pi ∂pi

Since the disturbance w is independent of the parameter pi in Eq. (21), let w p = 0 . Using the relation (22) obtained from u=Ky, the closed-loop trajectory sensitivities are obtained as Eq. (23). i

∂u ∂y ∂x =K = KC = KCσ i ∂pi ∂pi ∂pi

(22)

(

)

σ& i = ( A + B 2KC ) σ i + A pi + B 2, pi KC x r

(

Q ⇒ Q + ∑ ρi A pi + B 2, pi KC i =1

i =1

(21)

where

u pi = 0 i

0.51

ms

Let us consider the following linear time-invariant system: (16)

hs

27000

2.3 Robust LQ SOF controller design for rollover prevention

x& ( t , p ) = A ( p ) x ( t ) + B1 ( p ) w ( t ) + B 2 ( p ) u ( t )

1146.6

) (A T

pi

+ B 2, pi KC

(23)

)

(24)

In Eq. (23), the state x acts as a disturbance to the trajectory sensitivity σ. To reduce the trajectory sensitivity, it is necessary to reduce the last term of Eq. (23). To do this, the last term of Eq. (23) is incorporated into the weighting matrix Q in the LQ cost function, Eq. (14), as shown in Eq. (24) [15]. ρi is the tuning parameter that determines which uncertain parameter is emphasized. The tuning parameter ρi is set to the squared inverse of the largest singular value of the matrix A p . Since only the weighting matrix Q in the LQ cost function is modified, the optimum K can be easily computed by the same procedure of the previous subsection, CMA-ES. i

3. Simulation The linear controllers are designed based on the linear vehicle model, shown in Eq. (11). The nominal values of parameters in the linear vehicle model are referred from SmallSUV model in CarSim®, as given in Table 1. Since the rollover prevention occurs at high speeds over 60 km/h, the longitudinal velocity was set to 60 km/h.

217


Table 2. Weights in LQ cost function.

η1

η2

η3

η4

η5

η6

7 m/s2

5 deg

10 deg/s

0.08 rad/s

5000 N⋅m

2000 N

To avoid rollover in cornering situation, a vehicle should follow the reference yaw rate γd with a small lateral acceleration ay. To follow the reference yaw rate, the yaw rate error eγ should be reduced. To prevent rollover, the lateral acceleration ay, roll angle φ and roll rate φ& should be reduced. For these purposes, the weights q1, q2, q3 and q4 in Eq. (12) should be set to higher values. If the weights on the roll angle and the roll rate are highly emphasized for rollover prevention, the yaw rate error increases due to the lateral load transfer caused by active suspension, and this can cause a loss of the maneuverability or lateral stability [25]. To cope with this problem, an electronic stability control (ESC) was adopted in the previous work [25]. On the other hand, if the weight on the yaw rate error is highly emphasized for maneuverability or lateral stability, the roll angle cannot decrease [25]. Therefore, the LQ objective function given in this paper should simultaneously consider these effects. For this reason, the roll angle, the roll rate and the yaw rate error are highly emphasized in the LQ objective function.

From the relationship in Eq. (4), the reduction of the lateral acceleration requires that the yaw rate and the longitudinal velocity should be reduced. This increases the yaw rate error because a small yaw rate means that a vehicle cannot follow the reference one. In other words, the reduction of the yaw rate indicates that the yaw rate tracking performance is deteriorated. From this fact, it is expected that the rollover prevention can be achieved by reducing the lateral acceleration at the expense of the increase of the yaw rate error. The values of ηi for the weights in the LQ cost function are given in Table 2. The uncertain parameters to be considered for rollover prevention are the cornering stiffness Cf and Cr, the height of C.G hs, and the vehicle speed vx. For rollover prevention, the height of C.G hs and the vehicle speed vx should be treated as uncertain since these parameters have the critical effects on the rollover, as pointed out in Section 1. The cornering stiffness Cf and Cr are treated as uncertain since the variation of Cf and Cr has a large effect on the lateral behavior of a vehicle, as pointed out in Section 2. 3.1 Linear system analysis for the designed controllers Fig. 2 shows the Bode plots that are drawn based on the designed controllers. In these plots, the input is the steering angle,

Steering -> Roll Angle

5

Steering -> Roll Rate

20 15

0

10 5 Magnitude (dB)

Magnitude (dB)

-5 -10 -15 -20

-10 -15

Open-Loop

Open-Loop

-20

LQR

-25

0 -5

LQ SOF

LQR LQ SOF

-25

Robust LQ SOF

-30 -2 10

10

Robust LQ SOF

-1

10

0

10

-30 -2 10

1

Frequency (Hz)

-1

0

10 Frequency (Hz)

10

1

10

2

(b) Bode plot from δf to φ&

(a) Bode plot from δf to φ Steering -> Lateral Acceleration

40

10

25.4

Steering -> Yaw Rate Error

25.2

35

25 24.8 Magnitude (dB)

Magnitude (dB)

30

25

20

24.6 24.4 24.2 24

Open-Loop

15

LQR LQ SOF

23.8

Open-Loop

23.6

LQ SOF

LQR Robust LQ SOF

Robust LQ SOF

10 -2 10

10

-1

0

1

10 10 Frequency (Hz)

(c) Bode plot from δf to ay Fig. 2. Bode plots from the steering input to each output.

10

2

10

3

23.4 -1 10

0

10 Frequency (Hz)

(d) Bode plot from δf to eγ

10

1

218


Eq. (25) shows the geometric relation between the tire forces and the positive MB. In Eq. (25), the matrix H is called the effectiveness matrix [26]. t ⎤ ⎡ Fx1 ⎤ ⎡ t ⎢ − 2 cos δ f + l f sin δ f − 2 ⎥ ⎢ F ⎥ = M B ⎣14444 x3 ⎦ 4244444 3⎦ ⎣{ H z (a) Positive MB

(b) Negative MB

The objective function of WLS is defined as follows:

Fig. 3. Coordinate system corresponding to tire forces.

JWLS =

and the outputs are the roll angle, the roll rate, the lateral acceleration and the yaw rate error. In this figure, the legends LQR, LQ SOF, and Robust LQ SOF stand for the controller designed by LQR, LQ SOF and LQ SOF with a parameter sensitivity reduction scheme, respectively. Since the frequency of the fishhook maneuver with the maximum steering angle of 221 degrees is near 0.5 Hz, frequency responses below 0.5 Hz should be checked [25]. As shown in Fig. 2(a) and (b), LQ SOF and Robust LQ SOF show better performance in controlling the roll angle and roll rate, compared to LQR. On the other hand, these controllers show nearly the same performance in reducing the lateral acceleration and the yaw rate error, as shown in Fig. 2(c) and (d). From these results, it can be concluded that the static output feedback controllers, LQ SOF and Robust LQ SOF, concentrate their efforts to reduce the roll angle and the roll rate. As shown in Fig. 2(c) and (d), the designed controllers give reduced lateral acceleration and increased yaw rate error against the open-loop system. This results from the fact that large weights are set to q2 and q3 in Eq. (12), as shown in Table 2. As shown in Fig. 2, Robust LQ SOF has the minimum roll angle and roll rate, and the maximum yaw rate error against the open-loop system, LQR, and LQ SOF. Generally, a rollover prevention control system makes a vehicle exhibit under-steer characteristics because of the reduction of the lateral acceleration caused by that of the yaw rate. From this result, it is concluded that Robust LQ SOF concentrates to control the roll motion at the expense of the increased yaw rate error or the under-steer characteristic of the controlled vehicle, compared to LQR and LQ SOF. 3.2 Yaw moment distribution with weighted least square To show the effectiveness of the proposed controller in rollover prevention, the simulations were performed on a nonlinear multi-body dynamic simulation software, CarSim®. To simulate the designed controllers on CarSim, it is necessary to distribute the yaw moment MB to four wheels. For this purpose, the weighted least square (WLS) was adopted [26]. Fig. 3 shows the coordinate system corresponding to tire forces to generate the control yaw moment MB. As shown in Fig. 3, the braking force should be applied to left or right wheels according to the sign of MB. For instance, if the sign of MB is positive, the braking forces of the left wheels, Fx1 and Fx3, should be applied.

(25)

Fx21 Fx23 + = zT Wz Fz21 Fz23

(26)

⎛ 1 1 ⎞ , 2 ⎟. 2 ⎝ Fz1 Fz 3 ⎠

where W = diag ⎜

In Eq. (26), the inverse of the vertical tire forces means that the larger the vertical tire forces the larger the longitudinal tire force, and vice-versa. By the virtue of this inverse, it is not necessary to limit the tire forces by a friction circle. In Eq. (26), the vertical tire forces should be estimated because these values cannot be measured or the sensor is too expensive. The vertical tire forces are estimated with the longitudinal and lateral acceleration, as given in Ref. [27]. The problem is the quadratic programming with an equality constraint [26]. Using the Lagrange multiplier technique on this problem, the optimum solution can be easily obtained as follows: z opt = W −1HT ( HW −1HT ) M B . −1

(27)

If the sign of MB is negative, the braking forces of the right wheels, Fx2 and Fx4, can be obtained by the identical method that is described above. The relationship between the braking force Fx and brake pressure PB on a wheel is assumed as follows: PB =

rw Fx . KB

(28)

3.3 Rollover prevention control for CarSim vehicle model With the designed controllers, the simulation was performed on the nonlinear vehicle model, SmallSUV in CarSim®. In the simulation, the steering input is the fixed fishhook maneuver with the maximum angle of 221 deg [28]. The vehicle SmallSUV rolls under the fixed fishhook maneuver with the maximum angle of 221 deg at 63 km/h. The initial speed of the vehicle is set to 80 km/h, and the tire-road friction coefficient is set to 1.1. There are no speed controls to maintain a constant speed. The actuators of the brake and the active suspension are modeled as a first-order system with the time constant of 0.12 and 0.08, respectively. To prevent the locking of a brake, the ABS provided in CarSim is used. The operating range of the slip ratio in the ABS is between 0.05 and 0.15. Figs. 4, 5 and 6 show the responses of SmallSUV model and

219

S. Yim / Journal of Mechanical Science and Technology 26 (1) (2012) 213~222 10

No Control LQR LQ SOF Robust LQ SOF

10 5

Roll Angle [deg]

Lateral Accleration [g]

15

0 -5 -10 -15 0

2

4

6

8

5 0 -5 -10

10


0

2

4

time [sec]

(a) Lateral acceleration (m/s2) Longitudinal Velocity [km/h]

Yaw Rate Error [deg/s]

0 No Control LQR LQ SOF Robust LQ SOF

-20

0

2

4

8

10

(b) Roll angle (deg)

20

-40

6 time [sec]

6

8

10

80

LQR LQ SOF Robust LQ SOF

60 40 20 0

0

2

4

time [sec]

6

8

10

time [sec]

(c) Yaw rate error (deg/s)

(d) Longitudinal velocity

1

x 10

4

10 Brake Pressure [MPa]

Yaw Moment Input [N-m]

Fig. 4. Responses of SmallSUV model for each controller.

LQR LQ SOF Robust LQ SOF

0.5 0 -0.5 -1

0

2

4

6

8

6 4 2 0

10

FL FR RL RR

8

0

2

4

(a) Yaw moment input for each controller 10

FL FR RL RR

8 6 4 2 0

2

4

6

10

8

6 4 2 0

10

FL FR RL RR

8

0

2

4

time [sec]

6

8

time [sec]

(c) Applied brake pressure of LQ SOF Active Suspension Input [N]

8

(b) Applied brake pressure of LQR

Brake Pressure [MPa]

Brake Pressure [MPa]

10

0

6 time [sec]

time [sec]

(d) Applied brake pressure of Robust LQ SOF

1000 500 0 -500 LQR LQ SOF Robust LQ SOF

-1000 -1500

0

2

4

6 time [sec]

(e) Active suspension input Fig. 5. Control Input in CarSim simulation for each controller in Fig. 4.

8

10

10

220


10


750

C.G. Height [mm]

0 -10

Y [m]

-20 -30 -40

700 650 600 550

-50

No Control LQR

-60 -70 -10

500

LQ SOF Robust LQ SOF

0

10

75

80

85

90

95

100

Vehicle Speed [km/h] 20

30

40

50

60

X [m]

Fig. 7. Rollover thresholds for each controller.

(a) Vehicle trajectories for each controller

0 Real Ref.

-20

Y [m]

Y [m]

0

-40

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-60

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Fig. 8. Rollover speeds for each controller.

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(b) Reference and real trajectories for each controller Fig. 6. Vehicle trajectories of each controller based on CarSim vehicle model.

control inputs under the fixed fishhook maneuver with 221 deg at 80 km/h, respectively. As shown in Fig. 4(c), the proposed controller shows poor yaw rate tracking performance due to the largest braking input. Accordingly, the lateral acceleration and roll angle are decreased, as shown in Figs. 4(a) and (b). From these results, we can conclude that the proposed controller makes the vehicle have an under-steer characteristic, just like the implication of the Bode plots in Fig. 3. As shown in Fig. 6(a), the yaw moment input of the proposed controller is larger than that of the LQR and LQ SOF controller. The active suspension inputs are reversed. Figs. 5(a)-(e) show the control yaw moment input, the applied brake pressures and the active suspension inputs for each controller. In these figures, the legends FL, FR, RL, RR represent the front left, front right, rear left, rear right wheel, respectively. As shown in Fig. 5(a)(d), the control yaw moment and the corresponding braking input of the proposed controller are larger than those of LQR and LQ SOF because of its larger yaw rate error. The active

suspension inputs are reversed. Fig. 6 shows the trajectories of the vehicles with each controller. As shown in Fig. 6(a), the vehicle with the proposed controller has the smallest braking distance and cornering radius than that of LQR and LQ SOF due to the larger braking input generated from larger yaw rate error. This fact can be also confirmed in Fig. 6(b). With the fixed nominal values of the system parameters, a rollover threshold is defined as the pair of the minimum C.G. height and vehicle speed when the rollover occurred. Fig. 7 shows the rollover thresholds for each controller. For a particular controller, the rollover cannot occur if the C.G. height and vehicle speed are below the thresholds given in Fig. 7. As shown in Fig. 7, the controlled vehicle shows better performance in rollover prevention than the uncontrolled one. For example, in Fig. 7, the vehicle height of rollover occurring at the same speed is increased to the maximal extent of 23%, 33%, and 36% compared to the uncontrolled case for LQR, LQ SOF, and Robust LQ SOF, respectively. From these results, it can be concluded that the proposed controller, Robust LQ SOF, considering the parameter uncertainties, is superior to LQR and LQ SOF. With fixed nominal values of vehicle height, mass and inertia, a rollover speed is defined as the minimum vehicle speed at the time of the rollover. Fig. 8 shows rollover speeds for each controller. As shown in Fig. 8, the rollover speed is increased to the extent of 38%, 46%, and 52% compared to the uncontrolled case for LQR, LQ SOF, and Robust LQ SOF,


respectively. From this result, it can be concluded that the proposed controller gives the best performance in rollover prevention.

4. Conclusions In this paper, a rollover prevention controller was proposed for vehicle systems with a large C.G. height such as SUVs and vans. The differential braking and the active suspension were adopted as an actuator. The rollover prevention controller was designed with LQ SOF based on the nominal values of parameters to reduce the lateral acceleration and the roll angle. From the results of simulation, it was shown that LQ SOF controller makes the vehicle avoid the rollover by enhancing the under-steer characteristic. This controller showed good performance in rollover prevention in that the vehicle height of rollover occurring at the same speed was increased to the maximal extent of 33% compared to the uncontrolled case through the simulation on the nonlinear CarSim® vehicle model. Since the key parameters to affect the vehicle rollover are the C.G. and the speed of vehicles, the rollover prevention capability can be enhanced by the controller that is designed to be robust against the variation of these parameters. To design the robust controller, the parameter sensitivity reduction was adopted. The design problem of the robust controller was formulated as LQ SOF with the parameter sensitivity reduction scheme, and solved by the heuristic search, CMA-ES. The controllers were designed based on the linear vehicle model, and were applied to the nonlinear CarSim® vehicle model. Through the simulation in CarSim®, the proposed controller was shown to improve the rollover prevention capability in that the rollover threshold and the rollover speed were increased to the maximal extent of 10% and 4% compared to LQR and LQ SOF, respectively. From these results, it can be concluded that the controller robust to the height of C.G. and vehicle speed enhances the rollover prevention capability.

Acknowledgment This work is supported by Advanced Institutes of Convergence Technology (2011-P3-08), Seoul National University, Korea.

Nomenclature-----------------------------------------------------------------------ay Cf, Cr Cφ eγ Fyf, Fyr Fx, Fz f1, f2 H g h, hs

: Lateral acceleration (m/s2) : Cornering stiffness of front/rear tire (N/rad) : Roll damping coefficient (N⋅m⋅s/rad) : Yaw rate error (rad/s) : Lateral tire forces of front/rear wheel (N) : Longitudinal and lateral tire forces (N) : Active suspension forces (N) : Effectiveness matrix : Gravitational acceleration constant (9.81 m/s2) : Height of C.G. from ground and roll center (m)

Ix, Iz J JWLS K KB Kγ Kφ LQ LQR SOF lf, lr MB m, ms PB qi rw SOF t vx, vy W αf, αr

δf γ, γd μ ηi φ, φ& ρi σi τ

221

: Roll/yaw moment of inertia about roll/yaw axis (kg⋅m2) : Linear quadratic cost function : Objective function of WLS : LQ static output feedback gain : Pressure-force constant (N/MPa) : Steady state gain of reference yaw rate : Roll stiffness (N⋅m/rad) : Linear quadratic : Linear quadratic regulator : Static output feedback : Distance from C.G. to front/rear axle (m) : Control yaw moment (N⋅m) : Vehicle total and sprung mass (kg) : Brake pressure (MPa) : Weights on the terms in J : Radius of a wheel (m) : Static output feedback : Track width (m) : Longitudinal and lateral velocity of a vehicle (m/s) : Weighting matrix of WLS : Tire slip angle of front/rear wheel (rad) : Steering angle of a front wheels (rad) : Yaw rate and reference yaw rate (rad/s) : Tire-road friction coefficient : Tuning factors of the weights in J : Roll angle (rad) and roll rate (rad/s) : Weights on the sensitivity terms in J : Trajectory sensitivity of a parameter pi : Time constant of the first-order system for γd

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Seongjin Yim received a B.S. degree in mechanical engineering from Yonsei University, Korea, in 1995, and M.S. and Ph.D. degrees in mechanical engineering from the Korea Advanced Institute of Science and Technology (KAIST) in 1997 and 2007, respectively. Since 2008, he has been a postdoctoral researcher in the BK21 School for Creative Engineering Design of Next Generation Mechanical and Aerospace System at Seoul National University, Korea. He currently is a research professor in Advanced Institutes of Convergence Technology, Seoul National University. His research interests are robust control, vehicle rollover prevention, and integrated chassis control systems of hybrid and electrical vehicles.