PSfrag replacements

DAMPING OF VEHICLE ROLL DYNAMICS BY GAIN SCHEDULED ACTIVE STEERING J¨ urgen Ackermann, Dirk Odenthal DLR, German Aerospace Center, Institute of Robotics and System Dynamics, Oberpfaffenhofen, D-82230 Wessling, Germany kR Fax: +49-8153-28 1847 and e-mail: [email protected], [email protected] 1

Keywords: Rollover avoidance, active steering, gain scheduling, parameter space approach, robust control. δc

Abstract Active steering is applied to robustly reduce the rollover risk of vehicles with an elevated center of gravity. An actuator sets an auxiliary steering angle which is mechanically added to the steering angle commanded by the driver. The control law presented, is based on feedback of the roll rate and the roll acceleration. The controller gains are scheduled with the speed and the vehicle’s CG height. The controller gains are found by the parameter space approach and constrained optimization in frequency domain. Robust reduction of transient rollover risk is shown by evaluation of the sensitivity function at various operating points. Simulation of a double lane change maneuver illustrates the benefits in time domain.

1

Introduction

Accident analysis [1, 2] of vehicles with an elevated center of gravity (CG) have shown that the ratio of the track width and the height of CG is the most important parameter affecting vehicle rollover risk. The track width is a fixed parameter while the height of CG is varying subject to different loads. Recent vehicle dynamics control systems using individual wheel braking (e.g. Electronic Stability Program, ESP [3]) or active steering [4] for stabilization of the yaw movement are established primarily for passenger cars with a low CG. These concepts can in general help to avoid critical situations and thus indirectly help to reduce rollover risk. Based on active steering, a new approach was presented in [5], particularly considering the uncertain height of CG. The aim is to robustly decrease the rollover risk in the transient roll overshoot of the vehicle’s body during lane change or obstacle avoidance maneuvers. The assumed controller structure shown in Fig. 1 consists of proportional feedback of both the roll rate φ˙ and the roll ac¨ An actuator sets a small auxiliary angle δc celeration φ.

φ˙

δf

δs

vehicle

actuator

v, h δa

δφ˙ −

kp (v, h) + kd (v, h) s

Fig. 1: Controller structure. which is mechanically added to the driver’s steering angle δs . The actuator set point is δa = −δφ˙ according to the control law. In [5] the controller gains were constant and set according to considerations in parameter space and time domain. The resulting controller was shown to robustly reduce the rollover risk in transient maneuvers. However, only at high velocity the reduction was significant. In order to achieve significant rollover risk reduction over the entire operating domain, in this paper the controller is improved by scheduling the gains with both velocity and height of the CG. The synthesis of the controller gain scheduling law is realized by means of the parameter space approach and constrained parameter optimization. The performance of the closed loop system is illustrated by stroboscopic visualization of a simulated double lane change maneuver. Moreover, sensitivity functions are analyzed to demonstrate the robust roll dynamics improvement.

2

Vehicle model

Fig. 2 illustrates a simple model of the vehicle which will be used for the considerations in this paper. This multibody system consists of two rigid bodies. Body 1 with mass m1 is composed of the front and rear axles, the four wheels and the frame. Body 2 is the sprung mass (m2 ). The position of the vehicle’s roll axis depends on the suspension kinematics. The model assumes a fixed roll axis parallel to the road plane in the longitudinal direction of the vehicle at a height hR above the street. Hence, body 2

Rt Rt trix. The triple { 0 vx dt, 0 vy dt, ψ} describes the vehicle location and orientation. It does not affect the vehicle dynamics. Thus, since vx is assumed constant, for state space representation of the lateral, yaw and roll dynamics ˙ T. a reasonable choice of the states is x1 = φ, x2 = [vy r φ] Introducing submatrices with appropriate dimensions eq. (1) can be reduced to the following descriptor state space model         0 x1 0 T x˙ 1 1 0 δ , (2) + = S f −L −D − G x2 0 M x˙ 2

z1

z2

φ CG2

m2 ay,2 m2 g

roll axis

h · cos φ

y2

φ

y1 hR

where

CG1 m1 g

Fz,R

road

Fz,L

M

=

T

D

Fig. 2: Vehicle rollover model. is linked to body 1 with a one degree of freedom joint. The roll movement of the roll body is damped and sprung by suspensions and stabilizers with an effective roll damping coefficient dφ˙ and roll stiffness cφ . The CG of body 1, i.e. CG1 , is assumed to be in the road plane below CG2 , since its contribution to the roll movement is considered to be negligible. For body 1 the same assumptions as for the single-track model [6] are used in order to represent the main features of vehicle steering dynamics in the horizontal plane. Linear tire characteristics were assumed. All model parameters are compiled in Tab. 1. The multibody system describes the vehicle’s longitudinal, lateral, yaw and roll dynamics. A similar description of a vehicle model can e.g. be found in [7]. Applying Jourdain’s principle of virtual power, the nonlinear equations of motion are obtained according to a proper choice of minimal velocities and minimal coordinates respectively, e.g. z˙

=

z

=

 T vx vy r φ˙ hR Rt t 0 vx dt 0 vy dt ψ

φ

iT

,

where vx and vy are the velocity components of body 1 in longitudinal and lateral direction, ψ is the yaw angle, r = ψ˙ is the yaw rate of body 1, φ is the roll angle and φ˙ is the roll rate of body 2 relative to body 1. Linearization for straight driving at constant speed vx = v results in the following linear time-invariant 2nd order vector differential equation ¯ z¨ + (D ¯ + G) ¯ z˙ + Lz ¯ = Sδ ¯ f , M

(1)

which from a mechanical point of view is a gyroscopic dis¯ is the mass matrix (symmetric, sipative system where M ¯ ¯ positive definite), D the damping matrix (symmetric), G ¯ the gyroscopic matrix (skew-symmetric), L the conserva¯ the operating mative location matrix (symmetric) and S

D12

=

"

m 0 −h m2



0 Jz 0

(cf + cr ) µ/v  D12 0

#

−h m2 0 , J2,x + h2 m2 D12
cf lf 2 + cr lr 2 µ/v −h m2 v/2

= (cf lf − cr lr ) µ/v + m v/2 , "

G = L = S

=

T

=

0 −m v/2 0

m v/2 0 −h m2 v/2

0



0 −h m2 v/2 , dφ˙

#

h m2 v/2 , 0

 T 0 0 c φ − m2 g h ,  T c f µ c f lf µ 0 ,   0 0 1 .

The matrices of the output equation     x1 y = C1 C2 + F δf , x2

(3)

i.e. C 1 , C 2 and F , will be defined later on in this paper. For the simulations numerical parameter values of the vehicle, shown in Tab. 1, are taken from [8]. cf = 582 kN/rad cr = 783 kN/rad cφ = 457 kN m/rad dφ˙ = 100 kN/rad g = 9.81 m/s2 hR = 0.68 m h = 1.15 m J2,x = 24201 kg m2 Jz = 34917 kg m2 lf = 1.95 m lr = 1.54 m m = 14300 kg m2 = 12487 kg µ=1 T = 1.86 m

front cornering stiffness rear cornering stiffness roll stiffness of passive suspension roll damping of passive suspension acceleration due to gravity height of roll axis over ground nominal height of CG2 over roll axis roll moment of inertia, sprung mass overall yaw moment of inertia distance front axle to CG1 distance rear axle to CG1 overall vehicle mass sprung mass road adhesion coefficient track width

Table 1: Numerical vehicle data. For derivation of the transfer function from δf to an arbitrary single output y Cramer’s formula is applied. Laplace transformation of eq. (2) yields the regular ma-

trix equation      s −T 0 x1 0  L M · s + D + G 0  x2  =  S  δ f . −C 1 −C 2 1 y F

3

Design of a gain scheduled roll dynamics controller

(4)

The control design objective is to find a control law which decreases the overshoot of the rollover coefficient during transient maneuvers. The challenge is the robustness of The transfer function from δf to y can be computed as replacements PSfrag the control law with respect to a wide speed range v ∈ [20 km/h, 100 km/h] and an interval of the CG height, i.e. s −T 0 h ∈ [0.77 m, 1.53 m]. Fig. 3 shows the respective operating M · s + D + G S det L domain Q. The efficiency of feedback of various vehicle T −C 1 −C 2 F . (5) Gyδf (s) = s h −T V4 V3 det L M ·s+D+G 1.53 m

Substitution of the numerical values for the fixed vehicle parameters (e.g. from Tab. 1) into eq. (5) results in a parametric transfer function Gyδf (s, q) which depends on v and h, i.e. the vector of uncertain parameters is q T = [v h].

Q

0.77 m

V1 20 km/h

V2 100 km/h v

Fig. 3: Operating domain. Rollover coefficient From the equilibrium of vertical forces, i.e. gravitation forces of body 1 and 2, m1 g and m2 g respectively, and tire vertical loads Fz,L and Fz,R (front and rear), and balance of moments w.r.t. CG1 (see Fig. 2) a rollover coefficient is defined as Fz,R − Fz,L Fz,R + Fz,L   2 m2 ay,2 = (hR + h cos φ) + h sin φ , mT g

R=

(6)

with ay,2 derived from the dynamical model, i.e. ay,2 = v˙ y + vr − hφ¨ .

(7)

When Fz,R = 0 (Fz,L = 0) the right (left) wheels lift off the road and the rollover coefficient takes on the value R = −1 (R = 1). For straight driving on a horizontal road for the tire vertical loads it holds that Fz,R = Fz,L which means that R = 0. Note, that the vehicle model is only valid until one or more wheels lift off the road. Assuming m1  m2 and φ to be small, eq. (6) results in R≈

2(hR + h) ay,2 2h + φ, T g T

(8)

which equals the rollover coefficient definition given in [8]. Further, assuming the second term of eq. (8) to be negligible against the first one yields R ≈ 2(hR + h)/T · ay,2 /g, which matches the rollover coefficient definition given in [2]. This approximation reflects the accident analysis results in [1], where it was stated that the track width ratio 2(hR + h)/T is the most important vehicle parameter affecting rollover risk.

dynamics variables to front wheel steering, aiming at roll damping increase, has been investigated preliminary to this paper. This analysis suggests to apply proportional feedback of both the roll rate and the roll acceleration. For robust control design the parameter space approach [9] and constrained optimization are applied. The operating domain is represented by a grid of 81 points (representatives) respectively, i.e. nine velocities times nine heights of CG2 . All controller design and analysis steps were performed with these 81 representatives. However, for the sake of clarity in this paper only the four vertices of the operating domain Vi , i = 1 . . . 4 (marked as circles with different linestyle in Fig. 3) are considered. The results for the 81 representatives are corresponding. For the four vertices we first want to find controller gains which improve the roll dynamics performance in terms of roll damping. At the same time the system shall not become slower. For this purpose the parameter space approach is applied. The proceeding is to first specify the demands on the closed loop system as eigenvalue specifications. However, it is not possible to formulate these restrictions in a continuous way, since this would require to compute the symbolical dependency of the damping of the roll dynamics eigenvalues on both, the uncertain height of CG2 and the velocity. Using the parameter space approach, the roots of the closed loop system for the respective vertex Vi must be located in a desired socalled Γi -stable region in the eigenvalue plane. Mapping the boundaries ∂Γi of this region to the controller pa(i) rameter plane yields controller regions KΓ which meet the respective eigenvalue specifications for this vertex in a nonconservative way. This procedure will be made clearer in the next section. A gain scheduled control law with the speed v and the height h is then designed by applying Γi -stability and

further specifications in frequency domain to get significant improvement for all vertices or representatives respectively of the plant.

jω ∂Γ(q) Γ(q) σ0 (q)

Controller structure

σ

The control law of Fig. 1 is described by feedback of the roll rate through an ideal PD-controller Gc (s, k) = kp + kd s

(9)

with k = [kp kd ]T . The differentiating part in the controller (9) means proportional feedback of the roll acceleration. It may be obtained by numerically differentiating the roll rate signal. Alternatively, it can be computed from the signals of two lateral accelerometers being mounted to the bottom and top of the roll body, respectively. The steering actuator is modelled as a third order low pass filter ωa3 (10) (s2 + 2 da ωa + ωa2 )(s + ωa ) √ with damping fixed to da = 1/ 2 and a bandwidth of 5 Hz (ωa = 10π rad/s). The transfer function from δf to φ˙ is computed symbolically from eq. (5) with C 1 = 0, C 2 = [0 0 1] and F = 0. The characteristic polynomial of the closed-loop system results in n o ˙ p(s, k, q) = Numerator 1 + Gφδf (s, q) Gc (s, k) Ga (s) .

Fig. 4: Definition of the Γ - stable region. roots lie within a sector formed by two lines of constant damping Dφ,0 ˙ . The composite of these demands ∂Γ is presented in Fig. 4. The ∂Γ-parameters σ0 and Dφ,0 are ˙ chosen separately for each operating condition (the vertices in Fig. 3) as given in Tab. 2. This guarantees that damping and real part in the closed-loop cannot be worse than the respective open-loop values for the vertex operating conditions. The eigenvalue locations of the yaw and lateral mode are not considered separately.

Ga (s) =

(11)

Eigenvalue specifications Tab. 2 shows damping coefficients Dφ,0 ˙ , natural frequencies ω0 in rad/s and maximum real parts σ0 of the corresponding roll dynamics eigenvalues of the four vertices of the operating domain of the uncontrolled system.

h = 1.53 m

h = 0.77 m

v = 20 km/h Dφ,0 ˙ = 0.44 V3 : ω0 = 2.19 σ0 = −0.96 Dφ,0 ˙ = 0.48 V1 : ω0 = 3.43 σ0 = −1.61

v = 100 km/h Dφ,0 ˙ = 0.52 V4 : ω0 = 2.11 σ0 = −1.10 Dφ,0 ˙ = 0.55 V2 : ω0 = 3.29 σ0 = −1.82

Table 2: Roll dynamics of the conventional vehicle. The mathematical description of the performance specifications can be represented by boundaries ∂Γ in the eigenvalue plane of the characteristic polynomial of the closed loop system as  ∂Γ := s|s = σ(ω) + jω, ω ∈ [ω − ; ω + ] . (12)

An upper limit for the settling time −1/σ0 is ensured if the roots of the characteristic polynomial are located to the left of a shifted imaginary axis s = σ0 + jω. A certain minimum damping of the system is guaranteed if the

Γ-stability boundaries in parameter space In the next controller design step, appropriate gains kp and kd are determined for each vertex Vi . The parameter space approach is used, to map the ∂Γ-boundaries into the kp -kd -plane. The resulting plane cuves divide the (i) kp -kd -plane into a finite number of regions. KΓ denotes the very set of (kp , kd )-values for which the closed loop eigenvalues match the Γi -specifications. The pair (kp , kd ) = (0, 0) lies exactly on the boundary of this set since, by construction, the uncontrolled vehicle characterizes a limit case of the specifications. The intersection T4 (i) KΓ = describes the set of controller gains i=1 KΓ that simultaneously Γ-stabilizes all four vertices of the operating domain. In [5] a roll dynamics controller with fixed gains kp , kd out of the set KΓ was introduced (considering a different smaller speed interval). However, for the extended operating domain, considered in the present paper, further investigations of the closed loop system showed unsatisfactory performance. Therefore, beneficial gain scheduling kp (v, h) and kd (v, h) is derived to achieve (i) better performance. The KΓ regions are shown in Fig. 5. The linestyles correspond to the circles in Fig. 3. For more details about the parameter space approach the reader is referred to [9]. The controller design was performed using the Matlab-based toolbox PARADISE ([10], http://www.op.dlr.de/FF-DR-RR/paradise/paradise.html). (i) Within the Γi -stable regions KΓ we want to find the controller gains which yield optimal roll damping. A further constraint is based on the sensitivity function. Gain scheduled controller design by sensitivity constrained optimization The parameter space approach delivered sets of controllers for each vertex such that additional requirements can now

0.03

grid by solving the constrained optimization problem

0.15 0.02 (3) Γ

kd

k

d

K

0 −0.05

min 1 − Dφ˙ (v ∗ , h∗ , kp , kd )

(4)

0.01

KΓ

0 0

1 kp

2

0

0.04 kp

0.08

subject to performance constraints

0.03

σ(v ∗ , h∗ , kp , kd ) − σ0 (v ∗ , h∗ ) ≤ 0

0.15 0.02 kd

K

0 0

1 k

2

0

0.04 k

p

0.08

p

Fig. 5: Γ-stability boundaries in the plane of controller gains kp , kd . be considered when picking the gains from this set. Therefore, an optimization approach is performed at PSfrag aiming replacements good closed loop performance. The sensitivity function is given by the transfer function between the roll rate reference value φ˙ ref = 0 and the ˙ i.e. control error e = φ˙ ref − φ, S(s, q, k) = =

(13)

1 ˙

1 + Gc (s, k) Ga (s) Gφδf (s, q)

∞ 0

−3

x 10

0.2

5

0 20

0.77 1.15

60

100 1.53

0 20

h (m)

0.77 1.15

60

100 1.53

v (km/h)

h (m)

Fig. 6: Gain scheduling law for kp and kd .

φ˙ ref (s) .

The sensitivity function equals the ratio of the closed loop transfer function to the transfer function of the conventional system from steering input or any other (disturbance) input to the roll rate, i.e. S(s, q, k) = ˙ ˙ ˙ Gφδs (s, q, k)/Gφδf (s, q), where Gφδs (s, q, k) denotes the closed loop transfer function from steering wheel angle ˙ The transfer function of the openδs to roll rate φ. loop system, i.e. the complementary sensitivity function ˙ P (s, q, k) = Gc (s, k) Ga (s) Gφδs (s, q, k), is a stable and rational function of relative degree three. Therefore, the sensitivity function must satisfy Bode’s theorem [11] in the following form Z

where Dφ˙ (Dφ,0 ˙ )/σ (σ0 ) denotes the damping of the roll eigenvalues/maximum real part of all eigenvalues of the controlled (conventional) system. The asterisk denotes the respective grid point. Fig. 6 shows the optimization results for ω + = 3 Hz and a grid of 9 × 9 representatives. It turns out that the influ-

v (km/h)

e(s)

(17)

kd

0 −0.05

(16)

∗

max ln |S(j ω, v , h , kp , kd )| ≤ 0

ω∈(0,ω + ]

(2) Γ

0.01

ln |S(jω, q, k)|dw = 0 .

ence of h is of minor importance. Thus, if a measurement or estimation of height h is not available, the gain scheduling with h can be omitted. Then, h is set to its nominal value (see Tab. 1). A robustness analysis in parameter space shows that the performance specifications are still satisfied. Therefore, the intersection of the Γi -stable regions for each of the two velocities which correspond to the vertices is formed as KΓI KΓII

(14)

This means that steering inputs and disturbance impacts are reduced at some frequencies and increased at others. The set of reasonable driver steering excitations lies in the range between 0 Hz and about 2.5 Hz. Thus, to ensure transient steering reduction and disturbance attenuation of the closed loop system the zero crossing of ln |S(jω, q, k)| must be at frequencies above 2.5 Hz. To obtain a suitable gain scheduling law, v and h are gridded. Then, the controller parameters kp and kd are computed for fixed velocities v ∗ and heights h∗ along this

(1)

(3)

(2) KΓ

(4) KΓ

= K Γ ∩ KΓ =

∩

(18) .

(19)

Fig. 7 shows, that the pairs of controller gains (kp , kd ) taken from Fig. 6 and marked as crosses in Fig. 7 for velocities v and heights h corresponding to the four vertices lie within KΓI and KΓII respectively. Thus, even if the 0.03 0.15 0.02

0.1 kd

k

d

K(1) Γ

0.05

∗

kp

0.1

(15)

kp ∈ kd ∈

I

KΓ

0.05

kd

0.1 0.05

0 −0.05

KIIΓ

0.01 0

0

1 kp

2

0

0.04 kp

0.08

Fig. 7: Selection of Γ-stabilizing controller parameters for different velocities. contoller gains are not scheduled with h the closed-loop system is still Γ-stable and the performance is improved considerably.

4

Closed loop system analysis

Simulation results

The closed-loop system analysis in time and frequency domain shows the improvement with regard to reducing the maximum overshoot of the rollover coefficient. Roll dynamics analysis In Fig. 8 damping coefficients and maximum real parts of the roll dynamics eigenvalues are compared to those of the conventional vehicle by means of plotting the damping coefficients and the maximum real parts respectively versus

−1

σ0 σ, 0.77 60 v (m/s)

1.15 100 1.53 h (m)

−2 0.77

20 60 v (m/s)

1.15 100 1.53 h (m)

Fig. 8: Characteristic roll dynamic values. velocity and height of the CG2 . The black surface plot belongs to the controlled system while the gray one belongs to the conventional system. From this it can be seen that the controller is not only robustly Γ-stable with respect to the representatives considered but also to the entire operating domain. Significant improvement is achieved w.r.t. both, damping of the roll mode and settling time of the roll mode. Sensitivity analysis Fig. 9 shows that the zero-crossing of ln |S(jω, q)| takes place at about 3 − 4 Hz for all vertices. Since the set of 0.1 0

ln | S(j ω) |

lacements

Dφ,0 ˙

,

Dφ˙

0.65 0.6 0.55 0.5 0.45 20

−0.1 −0.2 −0.3 −0.4 0

2

4

6

8

10

Frequency (Hz)

Fig. 9: Magnitude of the sensitivity function. suitable driver steering excitations is assumed to lie in between 0 Hz and about 2.5 Hz, this means that transient steering reduction can be assured for all representatives of the plant. Furthermore, transient steering reduction is particularly distinctive for the input range close to the natural frequency ω0 of the roll dynamics of the conventional system (marked as crosses in Fig. 9 with values for ω0 taken from Tab. 2).

Simulations were performed using the nonlinear kinematic vehicle model mentioned in section 2. Fig. 10 shows exemplary simulation results of a double lane change maneuver for v = 60 km/h and an unfavourable large height of CG2 , i.e. h = 1.53 m. The steering angle commanded by the driver is given by π δs (t) = δˆs sin(ωt) h(t) + h(t − ) ω π π
− h(t − 2 ) − h(t − 3 ) , ω ω

where h(t) denotes the step function defined by  0 for t ≤ Td h(t − Td ) = 1 for t > Td .

(20)

(21)

The steering wheel angle amplitude δˆs of excitation of the conventional and the controlled vehicle respectively is chosen such that the wheels of the conventional vehicle just lift off the road while the weels of the controlled vehicle do not. The excitation frequency ω is below the natural frequency of the conventional vehicle at the considered operating point. In Fig. 10 the simulation results are shown in truescale stroboscopic visualizations, time and location plots. The top stroboscopic pictures, taken at a time interval of ∆t = 0.6 s, show the course of the conventional/controlled vehicle during the double lane change maneuver. The response of the conventional vehicle is only drawn until one wheel lifts off the road. For the sake of clarity the vehicle’s location is plotted on the lower right subplot. The lifting off of the conventional vehicle from the road in the stroboscopic pictures of the roll movement is overdrawn with a small additional angle. The arrows plotted below the tires depict the tire vertical loads Fz,L and Fz,R . The stroboscopic pictures on the left correspond to the pictures at the top. However, they are only plotted until the conventional vehicle’s wheels lift off the road. From the time plots it can be seen, that at the cost of additional steering effort (compare gray line and dashed-dotted/dotted line in the upper left subplot) at the same rollover risk (upper right subplot), the maximum lateral displacement ymax of the controlled vehicle is bigger than that of the conventional vehicle. This means that at the same rollover risk the driver of the controlled vehicle can apply a bigger amplitude to drive around obstacles.

5

Conclusions

Active steering with PD-feedback of the roll rate was investigated. This control was shown to significantly reduce the rollover risk of a vehicle with a high center of gravity (CG) in transient steering maneuvers, e.g. double lane change. The roll dynamics is robustly improved despite

0m

∆t = 0.60 sec

112.0 m

conventional car controlled car

ymax = 2.47 m

steering effort (controlled car) 0.5

1 0 −1 −2 −3

0 R

δf (deg) , δs (deg)

con nal ventioal convention nal tio conventional conventio nal nvl entio conven cona

−0.5

0

2

4 time (s)

−1 0

6

6

y (m)

ay / g

2 0

−0.2

ymax = 2.69 m

4 time (s)

3

0.2 contro d lledcontrolledcontrolled controlled controlle lled contro

2

1 0

0

2

4 time (s)

6

−1 0

50 x (m)

100

Fig. 10: Double lane change maneuver at v = 60 km/h and h = 1.53 m. the height of the CG is uncertain due to varying payloads. Controller gain scheduling with speed achieves good performance as a result of a combined approach in parameter space and by constrained optimization. In [12] this concept is expanded by an emergency rollover avoidance system based on simultaneous active steering and braking.

References [1] R. W. Allen, H. T. Szostak, D. H. Klyde, T. J. Rosenthal, and K. J. Owens, “Vehicle dynamic stability and rollover,” tech. rep., Systems Technology, Inc., Hawthorne, CA, 1992. U.S.-D.O.T., NHTSA. [2] D. N. Wormley, “Analysis of automotive roll-over dynamics.” Course at Carl Cranz Gesellschaft, Oberpfaffenhofen, Germany, 1992. [3] A. v. Zanten, R. Erhardt, and G. Pfaff, “FDR - die Fahrdynamikregelung von Bosch,” Automobiltechnische Zeitschrift, vol. 96, pp. 674–689, 1994. [4] J. Ackermann, “Robust control prevents car skidding,” IEEE Control Systems Magazine, pp. 23–31, June 1997. Bode Prize Lecture 1996. [5] J. Ackermann and D. Odenthal, “Robust steering control for active rollover avoidance of vehicles with elevated center of gravity,” in Proc. Interna-

tional Conference on Advances in Vehicle Control and Safety, (Amiens, France), July 1998. [6] P. Riekert and T. Schunck, “Zur Fahrmechanik des gummibereiften Kraftfahrzeugs,” Ingenieur Archiv, vol. 11, pp. 210–224, 1940. [7] L. Segel, “Theoretical prediction and experimental substantiation of the response of the automobile to steering control,” in IMechE, pp. 310–330, 1956-1957. [8] R. C. Lin, D. Cebon, and D. J. Cole, “Optimal roll control of a single-unit lorry,” in Proc. IMechE, vol. 210, Part D, pp. 45–55, 1996. [9] J. Ackermann, A. Bartlett, D. Kaesbauer, W. Sienel, and R. Steinhauser, Robust control: Systems with uncertain physical parameters. London: Springer, 1993. [10] W. Sienel, J. Ackermann, and T. B¨ unte, “Design and analysis of robust control systems in PARADISE,” in Proc. IFAC Symposium on Robust Control Design, (Budapest, Hungary), 1997. [11] J. C. Doyle, B. A. Francis, and A. R. Tannenbaum, Feedback control theory. Englewood Cliffs, NJ: Macmillan Publishing Company, 1992. [12] D. Odenthal, T. B¨ unte, and J. Ackermann, “Nonlinear steering and braking control for vehicle rollover avoidance,” in European Control Conference, (Karlsruhe, Germany), 1999.