NONLINEAR STEERING AND BRAKING CONTROL

FOR VEHICLE ROLLOVER AVOIDANCE

Dirk Odenthal, Tilman Bunte, Jurgen Ackermann

DLR, German Aerospace Center, Institute of Robotics and System Dynamics,Oberpfaffenhofen, D-82230 Wessling, Germany

Fax: +49-8153-28 1847 and e-mail: Dirk.Odenthal@dlr.de, Tilman.Buente@dlr.de, Juergen.Ackermann@dlr.de

Keywords: Vehicle dynamics control, rollover avoid-ance, active steering, robust control, absolute stability.

Abstract

Steering and braking control is applied to avoid rollover ofroad vehicles. The control concept presented is composedof three feedback loops: Continuous operation steeringcontrol, emergency steering control and emergency brak-ing control. In continuous operation the roll rate and theroll acceleration are fed back by velocity scheduled gains tothe front wheel steering angle. Thereby, the vehicles rolldamping is robustly improved for a wide range of speedand height of the center of gravity. The latter may changefor example with a truck from ride to ride. A rollover co-efficient is defined that basically depends on the lateral ac-celeration at the center of gravity of the vehicles sprungmass. For critical values of this variable the emergencysteering and braking system is activated. The rollover co-efficient is also used for nonlinear feedback to the frontwheel steering angle. The control concept is evaluated bylinear sensitivity analysis and by simulations. Addition-ally, absolute stability of the steering control concept isverified using Popovs criterion.

1 Introduction

There are typical driving situations which can directly orindirectly induce vehicle rollover. Examples are excessivespeed when entering a curve, severe lane change or obsta-cle avoidance maneuvers (in particular when the center ofgravity (CG) is high) or disturbance impact like sidewind.One may distinguish two different categories of situationsfrom which rollover can arise: In the first case rollover iscaused directly, this is called rollover on a plane surface.In the other case (tripped rollover) after the vehicle hasalready entered a skidding state, rollover may occur if thewheels hit an obstacle.

Vehicles with an elevated CG are especially threatenedby rollover. Also, rollover accidents very often result frommisinterpretation of the vehicle dynamics by the driver,

in particular when the CG height varies severely accord-ing to different payloads. From common sense it is clearthat the ratio of the track width and the CG height isthe most important parameter affecting vehicle rolloverrisk. The track width is a fixed parameter whereas theCG height is either (nearly) fixed (e.g. passenger cars) oruncertain subject to varying loadings (e.g. trucks). In [1]an online estimation method was presented which allowsto determine the height of the CG. Hence, we assume theCG height to be known and constant during operation.

Present vehicle dynamics control systems using individ-ual wheel braking (e.g. Electronic Stability Program, ESP[2]) or active steering (e.g. Robust Steering Control, [3])have been primarily established for passenger cars with alow CG. These concepts can in general help to avoid skid-ding and thus help to avoid tripped rollover. However,until now, the primary task of individual wheel brakingand active steering has been the stabilization of the yawmotion.

In [4] a new approach was presented focussing onrollover avoidance by active steering. There, an actuatorsets a small auxiliary front wheel steering angle in addi-tion to the steering angle commanded by the driver. Theaim was to robustly decrease the rollover risk due to tran-sient roll overshoot of the vehicles body when performinglane change or obstacle avoidance maneuvers. The controllaw consists of proportional feedback of both the roll rateand the roll acceleration. The gains were fixed accord-ing to robustness and performance considerations in pa-rameter space and time domain. The resulting controllerwas shown to robustly reduce the maximum roll angleovershoot after steering input steps for large variations ofthe CG height in particular at high velocity. Moreover,the roll damping was robustly improved. In [5] this con-troller was modified by gain scheduling against velocityand CG height to achieve comparably good results also atlow speeds and different heights of CG. With this linearcontrol concept, however, the vehicle may still roll over incase of too large steering wheel inputs.

In this paper a control concept is presented where thelinear steering control is extended by nonlinear emergencysteering and braking control. Section 2 describes a lin-ear vehicle model which is used for the subsequent linear

and nonlinear steering control synthesis and analysis. Insection 3 the three control loops which form the rolloveravoidance control concept are explained. The performanceof the resulting system is investigated by means of a non-linear simulation in section 4. There the controlled vehicleis compared with the conventional vehicle when enteringa curve at risky speed.

2 Vehicle model

The main features of vehicle steering dynamics in a hor-izontal plane can be described by the single track model[6]. To take into account the influence of the height of theCG, this model is extended by the vehicles roll dynamics.For straight driving at constant speed the following lineardifferential equations represent the vehicles lateral, yawand roll dynamics:

m v m2 h =((cr lr cf lf )

vm v

)r

(cf + cr) + cf f(1)

Jz r = (cf lf 2 + cr lr2)v

r

+ (cr lr cf lf ) + cf lf f(2)

(J2,x + m2 h

2)

+ d + (c m2 g h) = m2 h ay,1(3)

where ay,1 is the lateral acceleration of the unsprung mass

ay,1 = v ( + r) . (4)

The system states are the side slip angle of the unsprungmass, the yaw rate r, the roll angle and the roll rate .The system input is the front wheel steering angle f .Numerical values of the parameters of the model, shownin Tab. 1, are taken from [7]. In the sequel we assume dryroad ( = 1) and the deviation of the height h from itsnominal value to be known (e.g. estimated according to[1] at the start of each ride).

cf = 582 kN/rad front cornering stiffness

cr = 783 kN/rad rear cornering stiffness

c = 457 kNm/rad roll stiffness of passive suspension

d = 100 kN/rad roll damping of passive suspension

g = 9.81 m/s2 acceleration due to gravity

hR = 0.68 m height of roll axis over ground

h = 1.15 m nominal height of CG2 over roll axis

J2,x = 24201 kg m2 roll moment of inertia, sprung mass

Jz = 34917 kgm2 overall yaw moment of inertia

lf = 1.95 m distance front axle to CG1lr = 1.54 m distance rear axle to CG1m = 14300 kg overall vehicle mass

m2 = 12487 kg sprung mass

= 1 road adhesion coefficient

T = 1.86 m track width

Table 1: Numerical vehicle data.

z2

y2

z1

y1

roll axis

CG2m2 ay,2

m2 g

road

hR

h cos

T

Fz,R Fz,Lm1 g

CG1

Figure 1: Vehicle rollover model.

A more detailed description of the model can be foundin [5].

Rollover coefficient

Fig. 1 illustrates some further physical assumptions forthe derivation of a rollover coefficient. The tire verticalloads are denoted Fz,L and Fz,R. From the equilibriumof vertical forces and balance of roll moments a rollovercoefficient R is defined as

R =Fz,R Fz,LFz,R + Fz,L

=2 m2m T

((hR + h cos)

ay,2g

+ h sin

).

(5)

If Fz,R = 0 (Fz,L = 0), then the right (left) wheels liftoff and the rollover coefficient takes on the value R = 1(R = 1). For straight driving on a horizontal road andsymmetric load R equals zero because Fz,R = Fz,L. Note,that the vehicle model is only valid if |R| 1, which meansthat all wheels have road contact.

Assuming m1 m2, h sin (hR +h cos)ay,2/g andthe roll angle to be small, eq. (5) is approximated by

R 2(hR + h)T

ay,2g

, (6)

which matches the definition of a rollover coefficient in [8].According to this definition the ratio of track width T andthe height of CG2 hR + h is the most important vehicleparameter affecting rollover risk. This corresponds wellwith the results of an accident analysis [9]. The lateralacceleration ay,2 at CG2 is related with the state variablesof the model by

ay,2 = v( + r) h . (7)

3 Rollover avoidance control

The assumed controller structure, shown in Fig. 2, consistsof three feedback loops: Continuous operation steeringcontrol, emergency steering control and emergency brak-ing control. In addition to the steering angle s com-manded by the driver, an auxiliary steering angle c is setby an actuator, i.e. f = s +c. The actuator is modelledas a third order dynamical filter

Ga(s) =3a

(s2 + 2 da a s + 2a)(s + a)(8)

with a = 2pi 5 Hz and da = 1/

2. The actuator setpoint a is formed by the sum of the continuous opera-tion steering control signal and the emergency steeringcontrol signal R. The latter and the braking force fxare zero as long as the vehicle remains in a rollover stablemargin. This means that emergency steering and brakingcontrol are only activated for |R| > R. The value of thethreshold R is chosen with regard to safety considerationsand subject to the quality of the rollover coefficient signalR. The latter mainly depends on the quality of the lateralacceleration signals and the reliability of the CG heightestimation. In this paper the threshold is set to R = 0.9.In the sequel, the steering control concept shown in Fig. 2

PSfrag replacements

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

emergency braking control

actuator

fs

a

R

fx

c

kR

1

fx,d

h, v

R

R

R

continuous operationsteering control

emergency steering control

vehicleZusatzlenkwinkel-

Aktuator

Steer-by-wire-

Aktuator

Emergency braking pressure control

fahrdynamischeGren

1

Bremsaktuator

Figure 2: Controller structure.

is described and investigated. First, only the continuousoperation steering control concept is studied. Then this

concept is extended by adding emergency steering con-trol and finally additional emergency braking control isapplied.

3.1 Continuous operation steering control

The task of the controller design presented in [5] was toreduce the rollover risk for a wide speed range v [v, v+]and a known (or even uncertain) height h [h, h+]. Thecorresponding operating domain is shown in Fig. 3. This

PSfrag replacements

Q

h

0.77 m

1.53 m

v

V1 V2

V3 V4

20 km/h 100 km/h

Figure 3: Operating domain.

aim was met by improving the roll damping through gainscheduled feedback of the roll rate and the roll accelera-tion, i.e. by the control law a = with

= kp(v, h) + kd(v, h) . (9)

The scheduling law is described in detail in [5]. There, asensitivity analysis shows the robust performance of theclosed loop system.

With the feedback of and to the front wheel steer-ing angle the roll damping of the vehicle was improvedconsiderably. In fact, the steering transfer function hasbeen shaped such that the roll mode is excited less in thefrequency range of the roll resonance frequency. Thus,the risk of causing a rollover by steering excitation hasbeen reduced. However, even the controlled vehicle canroll over if the steering input is large enough.

3.2 Rollover emergency steering control

The nonlinear control introduced in this section can be in-terpreted as an intelligent steering angle limitation suchthat rollover on a plane road can be completely avoided.The key idea is that rollover avoidance is given priorityover lanekeeping because a tipped vehicle is no longersteerable. To drive the narrowest curve which is physi-cally possible, maximum lateral acceleration must be ap-plied. The lateral acceleration is limited, however, by theboundary where rollover occurs. This boundary is reachedif the vehicle is steered such that the inner wheels arejust about to lift off the road, corresponding to |R| = 1.The optimal strategy to keep the narrowest curve pos-sible while avoiding rollover would be to keep |R| = 1.With some safety margin, this idea is implemented ina nonlinear steering control law. Therefore, if the mag-nitude of R exceeds R, then the overstepping difference

R = kR sign(R) (|R| R) is fed back to the front wheelsteering angle f such that the curvature of the course isslightly reduced and rollover is avoided, i.e. the emergencysteering control feedback is described by the relation

R =

{kR sign(R) (|R| R) |R| > R

0 |R| R .(10)

This strategy works very well as will be shown in section4. In order to implement the prescribed effect, a deadzone element is introduced into the emergency steeringfeedback loop. The black line in Fig. 4 shows the charac-teristics of the dead zone with an absolute value thresholdof R and a slope of kR. This corresponds to the dead zone

PSfrag replacements

R

R

R

R kR

1

1

1

Figure 4: Dead zone element and Popov sector.

element in the emergency steering feedback loop in Fig. 2.However, this nonlinear element in the loop induces therisk of limit cycles. Therefore, a stability analysis is per-formed using Popovs sufficient criterion on absolute sta-bility [10]. This criterion is briefly illustrated: We considera controlled system with one nonlinear function f in theloop (the rest of which is linear and has the stable transferfunction G0(s)). The characteristics of f lies within a sec-tor [0, k], which is limited by the abscissa f1(u) = 0 andby the line f2(u) = k u (k corresponds to kR in Fig. 4).Popov proved that the system is absolutely stable, if thelocus

z = Re G0(j) + j Im G0(j), 0 (11)

(called the Popov plot) lies in the complex z-plane com-pletely on the right hand side of a straight line (calledPopov line)

Im {z} = (

Re {z}+ 1k

)(12)

with arbitrary slope .To verify absolute stability for the nonlinear steering

control, Fig. 4 shows a Popov sector with slope kR (plottedgray) which covers the characteristics of the dead zoneelement. The depicted Popov plots in Fig. 5 belong to thevertices of the operating domain. The different linestylescorrespond to those used in Fig. 3. For this analysis, G0(s)in eq. (11) is determined as the open loop transfer functionfrom R to R in Fig. 2. The vehicle and actuator dynamics

10 5 0 5 10100

80

60

40

20

0

201/kR

*

Re { z }

Im { z

}

Figure 5: Popov line and Popov plots for the vertices ofthe operating domain.

are represented by eqns. (1)-(3) and eq. (8) respectivelywith the parameter values in Tab. 1. Note, emergencybraking is not used for this analysis. Fig. 5 shows a Popovline which touches twice the Popov plot corresponding tothe most critical vertex V4. This choice for the Popov lineyields the largest intersection with the real axis 1/kR =2.54 and therefore the maximum allowable Popov sectorwith kR = 0.39. For the investigations in the sequel, aslope of kR = 0.35 in the dead zone element was chosen.Hence, the system is absolutely stable at all investigatedoperating points. This is true because all Popov plots lieto the right hand side of the Popov line. Thus, it is ensuredthat no limit cycles occur in the nonlinear steering controlloop.

3.3 Rollover emergency braking control

Applying braking control requires the application of a non-linear dynamic model of the vehicle with longitudinal ve-locity v as an additional state variable and the brakingforce fx as an additional input (see Fig. 2). The presenta-tion of this model is omitted here for the sake of brevity.Note, however, that linearization for straight driving atconstant speed yields eqns. (1) - (3). fx is assumed to acton CG1 in the vehicles longitudinal direction. The timedelay effect of the brakes is modelled by a first order lagwith a time constant of 0.1 s. The intention of emergencybraking is to make the deviation from the desired coursebeing induced by emergency steering control as small aspossible. This task is realized by decelerating the vehicleas soon as the rollover coefficient becomes critical. Thebraking action is described by the following relation:

fx =

{0 |R| R

m ax,max |R| > R(13)

Note, that fx,d in Fig. 2 describes the brake pedal force setby the driver. Alternatively, in a refined realization a dy-namic characteristics is applied to distinguish between de-creasing (R sign(R) > 0) and increasing (R sign(R) < 0)rollover stability. Assuming decreasing rollover stability,braking shall be implemented as fast as possible while

in increasing rollover stability the breaking force shall bewithdrawn. Such a dynamical relation is e.g. given by

fx =

0 |R| Rm ax,max |R| > R R sign(R) > 0

|R|RRR

m ax,max |R| > R R sign(R) < 0,

(14)

where R denotes the (dynamic) maximum absoluterollover coefficient which is stored in a memory while inincreasing rollover stability state.

For ax,max a value of 0.4 g is set. Emergency brak-ing and steering control have been integrated such thatrollover is avoided for a wide input range while even keep-ing the deviation from the desired course small.

Future development will be made on a refined strategyfor taking the right dose of braking impact. Not nec-essarily maximum deceleration has to be applied for theachievement of minimum tracking error. Then the goalwill be to coordinate the steering and braking action ina precise manner such that the lateral displacement fromthe course as intended by the driver gets minimal.

4 Simulation results

The simulations were performed using the nonlinear dy-namic vehicle model mentioned in section 3.3, assumingdry road and an unfavourably large height, h = 1.53 m.Fig. 6 shows the responses of the conventional (dashedline) and the controlled vehicle (solid line) when a ramp-like input signal is applied to the steering wheel angles. Both braking control approaches are investigated.The black solid line corresponds to braking action due toeq. (13), the gray line is according to eq. (14). This ma-neuver is similar to driving through a highway exit with in-creasing curvature (clotoidal transition). After about 3.5 sthe conventional vehicle rolls over. The dashed line endswith the vehicle rollover, but for the sake of comparabilitythe simulation is continued until the end of the maneuver(dotted linestyle). Note that the simulation model is nolonger valid if |R| > 1. The difference of both vehiclesuntil 3.5 s indicates the effect of the continuous operationsteering control.

Emergency steering and braking control is switched onafter about 3.3 s when the rollover coefficient R impliesthat the vehicle is close to rollover, i.e. |R| > 0.9. Dueto the fast and precise steering intervention the rolloveris avoided. However, only little track error occurs in thevehicles position plot (x, y) in Fig. 6 because the vehicleis simultaneously decelerated by the emergency brakingsystem.

Comparably advantageous results were obtained whenother maneuvers, e.g. lane change maneuvers, and varia-tions of v and h were investigated in further simulations.

0 2 4 6 80

1

2

3

time (s)

f (de

g)

0 2 4 6 8

45

50

55

60

time (s)

v (k

m/h)

0 2 4 6 80

0.5

1

time (s)

R

0 30 60 900

30

60

x (m)

y (m

)

0 2 4 6 80

5

10

15

(de

g)

time (s)0 2 4 6 8

0

5

10

15

time (s)

r (d

eg/s)

0 2 4 6 8

2

0

2

4

time (s)

d/d

t (de

g/s)

0 2 4 6 80

0.10.20.30.4

time (s)a

y / g

Figure 6: Simulation results for a driver steering inputramp.

5 Conclusions

A vehicle dynamics control concept composed of steeringand braking control was presented which significantly re-duces the rollover hazard caused by steering inputs. Gainscheduled continuous steering control as described in [5]forms an inner control loop which achieves improved rolldynamics. For rollover emergency case, an outer nonlin-ear steering control loop avoids rollover at the cost of somecourse deviation. This lane tracking error is, however, re-duced by simultaneous deceleration which also supportsrollover counteraction. An exemplary simulation of a ma-neuver with a rampwise steering input illustrates the func-tionality of the control concept. The Popov criterion wasused to prove robust absolute stability of the nonlinearsteering control.

References

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