Cornering Stiffness Adaptation for Improved Side Slip Angle Observation

Copyright © IFAC Advances in Automotive Control Salemo, Italy, 2004

ELSEVIER

IFAC PUBLICATIONS www.elsevier.comllocatelifac

CORNERING STIFFNESS ADAPTATION FOR IMPROVED SIDE SLIP ANGLE OBSERVATION Marcus Hiemer" Uwe Kiencke· Takanori Matsunaga" Kazushi Shirasawa ..

.. Institute of Industrial Information Tf'.chnology, University of KarL5ruhe (TH) .... Automotive Electronic.5 Development Center, Mitsubishi Electric Corporation

Abstract: The vehicle body side slip angle (VBSSA) represents a key variable in vehicle dynamics. It allows to assess vehicle stability in critical driving situations. As it cannot directly be measured with standard sensors it has to be determined using a vehicle model including an observer. This paper describes a method to increase the accuracy of VBSSA-estimation using adaptation of the cornering stiffnesses. The adaptation bases on a simple but efficient nonlinear approximation equation of the lateral wheel force. It is capable to describe the tire behavior up to the stability limit by means of two independent parameters. Copyright © 2004 IFAC Keywords: Adaptive systems, automotive control, nonlinear models, observers, optimization, vehicle dynamics

1. INTRODUCTION

approaches. Based on the nonlinear approximation of the lateral forces, the cornering stiffnesses are adapted according to the current driving situation. This takes into account , that the cornering stiffnesses become smaller, if the wheel load decreases and the tire side slip angle is growing. Therefore, the nonlinear tire characteristics even at high lateral accelerations or slippery ground can be described using linear adaptation. In contrast to existing solutions without adaptation, the application range of the model therefore can be extended significantly. Simulation results presented in this paper show that the observation results for the VBSSA can be improved using the presented method of cornering stiffness adaptation.

The vehicle body side slip angle (VBSSA) is defined as the deviation between vehicle longitudinal axis and vehicle velocity direction. It can be used as an input for future generations of driver assistance systems. Furthermore, knowledge about the VBSSA might be used in accident reconstruction in order to achieve information about the vehicle dynamics before a road traffic accident occurs. The VBSSA cannot directly be measured with standard sensors. It is very small for stable driving situations and is increasing with gTowing lateral accelerations. Therefore, a nonlinear vehicle model including a VBSSA-observer will be introduced in this paper. The vehicle model must be capable to describe the vehicle dynamics up to the stability limit. Thus , especially the nonlinear relationship between the tire side slip angle and the lateral wheel force including wheel load changes is considered in this paper. The lateral force cannot be described with sufficient accuracy using linear

667

lino approx. area! Z

..:.:

z .so u.';

,

k1 : 12000, k2= Q.4S

- -- -~

"""..---------, ...... Fz

= l.lkN

k1

-

-I

>=1 Fig. 1. Nonlinear characteristic line for lateral wheel force 2. LATERAL WHEEL FORCE APPROXIMATION

=12000, k2= O.3e

"""..-----------, .. ..,

. .'... . I

.. ...... -----:-- .. -

,-_

'0 ~ in deg

Fig. 2. Different tire characteristics by varying k characteristic maps of a tire. The parameter for the five curves is the wheel load, starting at the bottom with Fz = 1.1kN and ending on top with Fz = 7.1kN. Changes of kl and k2 in equation 2 allow to model different tire characteristics. In the upper left corner of figure 2 the " best fit" of the approximation (solid) with respect to the reference (dashed) is plotted. These " best-fitsr can be varied to model different tire characteristics: In the upper right hand corner of figure 2, kl was increased and k2 was decreased simultaneously. For large TSSAs, the force transmission is increased compared to the reference, whereas for small 0: it is rather small. This is typical for a tire which has a reduced tendency for understeering at high lateral accelerations, (Reimpell and Sponagel, 1988). The characteristic map of the tire in the lower left corner represents a tire with a strong understeering tendency at high lateral accelerations. This behavior results for decreasing kl and increasing k2 compared to the " best-fit values" . In the lower right hand side, a tire is characterized which in general is not capable to transmit forces as well as the " best-fit-tire" of figure 2. This behavior can be achieved by simultaneously decreasing kl and k 2 • The examples show; how variations of k allow to describe tires with different self-steering behavior.

Usually, the relationship between the tire side slip angles (TSSA) O:ij and the lateral wheel forces FSij is assumed to be linear. The index i j describes the individual wheels of the vehicle: i E {f~ R} , front / rear, j E {L , R} , left/ right. However , this assumption is only valid for lateral accelerations below 4m/ 52 (Mitscbke, 1989). Figure 1 shows, that for growing TSSAs the linear approximation is false . The cornering stiffnesses C;j describe the slope of the straight line in figure 1. Furthermore, this figure shows, that the lateral wheel force not only depends on the TSSA. It is also significantly influenced by the wheel load Fz ij . Hence, the lateral wheel forces are the basis to describe the VBSSA , i.e. the lateral vehicle dynamics in curves. The lateral wheel forces in the wheel coordinate system are described by the following linear relation :

(1) In order to describe the lateral vehicle dynamics accurately for changing wheel loads and large TSSAs , the cornering stiffnesses have to be adapted in every time-step. Therefore, first an approximation equation is presented, which allows cornering stiffness adaptation. 2. 1 Approximation equation elementary functions such as Several f (o:) = ..jQ or rational functions have been analyzed to meet the shape of the Fs (o:)-{;urve of figure 1. Finally, the arctan(x )-function coupled with a factor including varying wheel loads best fit the reference data set:

Fs (o:, Fz , k ) =

(1 - ~:)

2.2 Optimization of kl and k2 for best-fit

In order to achieve " best-fits" of the curves using equation 2, parameter optimization must be carried out. The goal is to determine the parameter vector k such that the quadratic error between approximated and real characteristic map is minimized. As function 2 is nonlinear, standard least squares (LS- ) methods CarlDot be applied. Therefore , a nonlinear quality function must be minimized with nonlinear LS-techniques. First, the

Fz arctan(k 2 . 0:) (2)

Equation 2 depends only on a two element parameter vector k = (k 1 , kz )T. Figure 2 shows four

668

k ---- -------.-

m · aXCh

.. m·g

h

/FZFR Fig. 3. Wheel load shift due to accelerations Fig. 4. Wheel load shift due to cornering

deviation between approximation and reference is defined (3)

Using equation 2 provides Fr(k ) = Frcf.r -

(1 -

F:;r) FZ,r arctan(k2Q r ) (4)

FS.r represents the approximation for data point r (r ~ l.. .n). j<~cf,r describes the r - th value of the characteristic map of the tire to be approximated. Next , a quality function is setup, which sums up the quadratic errors in equation 4. The optimization goal is to minimize the quality function

where FZF is the wheel load on the front axle, m the vehicle mass, I the wheel base , lR the distance from rear axle to the cent er of ma..';s, h the height of the center of mass over ground, 9 earth's acceleration and aXCh the cha.s..<;is longitudinal acceleration. aXCh is the inertial reaction of the cha..o;sis to vehicle accelerations. For the rear axle, again the torque balance can be carried out, as the two axles are a..<;sumed to be independent. With a virtual ma..o;s m· which represents the mass of the front axle, the torque balance around the front left wheel can be set up for the rolling axle a..<; well (see figure 4) F

(5)

m· . aYCh . h FZF ZFR=-?-+ b -

(7)

F

Inserting equation 6 into equation 7 provides for the load on the front left wheel:

The Levenberg-lvlarquardt-Optimization, a numerical method which was applied here, represents a combination of the Gauss-Newton-:Method and the method of gradient descent. The solution space is limited which guarantees the existence of an optimization solution. The quality of the solution depends on the initial values. In order to reduce the space of initial values, i.e. to achieve faster and better convergence, a heuristic method was used to determine the initial values. The Levenberg-Marquardt-algorithm quickly converges and for the reference tire map of figure 2 yields kl = 14386 and k2 = 0.33555. Starting with these values, other characteristic tire maps can be described with optimized values kl ' k2 and equation 2.

. (8)

In equation 8, bF is the track on the front axle, aYCh represents the lateral vehicle chassis acceleration. The torque balances are to be carried out for the remaining wheels as well, (Kiencke and Dai<;s, 1994). The resulting equations are simple approximations of wheel load changes. Although some simplifications are employed, they provide reasonable results which influence the accuracy of lateral wheel force approximation in equation 2. Figure 5 shows the wheel load change at the front left wheel for a sine drive , starting with acceleration, ending with braking. The measurement was carried out with a 1987 Ford Scorpio test vehicle. The straight line in this plot represents the static wheel load at the front right wheel. At the beginning, when the vehicle is accelerating, the load at the front axle is reduced. Then the car enters the sine driving phase. The wheel load changes due to cornering. At the end, after about 14.5 seconds , the vehicle is braking. The wheel load moves to the front axle, thus, the vertical force on the front right wheel is increased. The figure shows that longitudinal and lateral accelerations cause significant changes in the vertical wheel forces.

2. 3 Determination of changing wheelload.s According to the driving situation, the load on each individual wheel changes due to accelerations. Neglecting couplings between rolling and pitching, these motions can be analyzed separately. When the ~'USpension dynamics is disregarded, the torque balance at the rear axis contact point (see fig1lre 3) provides

(6)

669

time in s Fig. 5. Wheel load shift for accelerated sine driving maneuver 2.4 Tire side slip angles

0.-

The TSSAs are determined by means of a single track model, (Mitschke, 1989). Investigations by (Dai..<;s, 1996) have shown that for TSSAdetermination the simplification of the single track model does only little affect the model accuracy. For the front and rear tire side slip angles OF , OR results

Fig. 6. Top view of the double track vehicle The indices ij are defined as mentioned above, Fe is the centripetal force, F Xij , F Yij are the longitudinal and lateral forces in the undercarriage coordinate system, Fw is the wind force and nLi are the wheel casters (see figure 6). The FYij are significantly influenced by varying cornering stiffnesses. The first two equations can be transformed and and 1; can be isolated

(9)

/3

(10)

v~ ~ [(f,1FX'j -FW) =~

8w is the wheel turn angle, 1/J the yaw rate, f3 the VBSSA, and v the vehicle velocity. With equations 8, 9, 10 and the optimized parameter vector k the lateral forces can be approximated.

+

~' .) J<'y;j sin f3]

(14)

3. VEIDCLE MODEL

In order to calculate the tire side slip angle, the VBSSA has to be determined. Therefore, a nonlinear vehicle model is set up. Afterwards, a state space observer is designed in order to achieve the VBSSA. For the vehicle model, only the three degrees of freedom in the road surface plane are regarded. Force balances in x- and y-direction and torque balance around the z-axi..<; yield (see figure 6):

mx = mv cos f3 =

L

FXij - Fc sin f3

-

Eliminating the mutual interdependencies of 1; and and using ;f provides after some algebraic transformations the nonlinear state !;pace model, (Kiencke and Nielsen, 2000)

/3

Fw (11 )

x=f(x , u) =

i.j

my = mil sin f3 = L

.h'ij

+ J
(12)

= c(x , u )

(17)

4. NONLINEAR VBSSA-OBSERVER

R + (FXRR - FxRL )' b 2 2

(16)

!3 (v, /3 ,t/J; hij ,OW )

The forces FLij are the longitudinal wheel forces in the wheel coordinate ~-ystem ,

lz;f = (FYFR + FyFL )' (IF - nLFcosOw ) -(Fy RR + J<'y RL ) . (lR + nLR)

FXFL) ' bF

( t/J y

i,j

+ (FXFR -

V) (!l (V,/3,-if;; FLi j ,O\"") ) ~ = h(v,/3,~; hij,OW)

For the nonlinear ~-ystem 16, 17, a VBSSAobserver has to be designed. Due to the non-

(13)

670

Fz tJ·· .. I ~,.,." .~ :J. ~'<:"'.t.

::Id.:;)CI;)t' :7I

--

c( t )

(t :;:

FSij ( t 1 ) t-------,>r"

" r'''''(;.~

FSij (t2 )

Fig. 7. Overview over cornering stiffness adaptation linearity, the linear observer design is extended, (Zeitz, 1977) . Therefore the system 16, 17 is Taylor-approximated around the actual estimated state vector x(k) and the Taylor-expansion is curtailed after the linear term. The observer design is carried Ollt mth the linearized model equations. The elements of the observer gain matrix L are achieved solving the following design equation for

Fig. 8. Adaptation of cornering stiffness These C;j are time-variant parameters for the vehicle model 16, 17 and for the designed VBSSAobserver. The procedure of adaptation is illustrated in the lateral force map of figure 8.

x=x 6. SIMULATIO:\ RESvLTS The quality of VBSSA-observdtion represents a criterion to point out improvements gained with cornering stiffness adaptation.

The Ai (i=l, 2, 3) are the poles which can be freely placed. In (Kiencke and Daiss, 1996), poles Al = A2 = -200 are suggested. Using these poles, the elements of the observer matrix can be determined

oh -

-AI

OXl

L=

6.1 Klotoide simulation Figure 9 shows the simulation results of a Klotoide driving maneuver: at the beginning, the vehicle is driving straight at a velocity of 50 m / so At t = Os the wheel turn angle is linearly increased by the driver from 0 to 1 deg. The velocity remains unchanged. The vehicle afterwards moves on a stationary circle. Figure 9 (top) illustrates, that the cornering stiffnesses of inner and outer track significantly deviate due to wheel load. Therefore. the assumption of time-invariant cornering stiff~ ne&Ses is false. On bottom of figure 9, a comparison of different simulations for the VBSSA is shown. The observer is not capable of capturing the dynamics of (3 sufficiently, if the cornering stiffnesses are not adapted. If so, even the transient phase of the side slip angle signal can be described very well. Furthermore, the stationary value for 3 i.<; more accurate. The relative error concerning the minimum value of {3 is approximately 2% with the adaptation method and about 18% mthout it. The relative error with respect to the stationarv value is below 0 ..5% mth adaptation and approximately 8% v.ithout it. The lateral acceleration for this driving maneuver is between 5 and 9m! s2. Adapting the cornering stiffnesses therefore provides significantly more accurate results.

oh OX3

oh

oh

OXl

OX3

oh

oh _ A3

OXl

OX3

(19)

For the nonlinear double track system, a nonlinear VB SS A-observer v..as designed and the estimated VBSSA is applied to the cornering stiffness adaptation unit .

5. CORl'i"CRING STIFF:--""CSS ADAPTATlO:\ Figure 7 gives an overview over the cornering stiffness adaptation system. The changing vertical wheel forces as well as the TSSA are Howing into the nonlinear approximation block represented by equation 2. Here, according to the tire characteristic map specified by the parameter vector k the lateral wheel forces on each indi,idual wheel are calculated in every calculation step. Using the tire side slip angle determined by means of existing sensor information and the VBSSA from the last calculation step, the cornering stiffnesses can be adapted

6.2 Sign change of VBSSA The velocity of algebraic sign change of the VBSSA was simulated. According to (Gillespie, 1992), for stationary curve driving at low speeds

(20)

671

2 0 ,""",;;:::-~-~-~-~---_-_---

-

V na d (f3

Cl) ' .

!: ,•

.

.

= 0) = 15.tmj s

/

..

:

. ,.. ;:>

'2

Va d(f3

= 0) = 12 .3m/ 5

' ~~~~-7--7--.7 . --.'~O--= '2-~-~ '.~~

tlDle

, VBSSA with and without ada

~ :~

no a7tation

-

o~

~~:r

lD S

f31,ad~ . \

--- . .

(3ad = 0

~o~-~-7--~-7.-~ 'O--= ' 2-~ " -~ ' .~~'.

time in s

time in s Fig. 9. Simulation results: Klotoide

Fig. 10. Velocity of VBSSA algebraic sign change

the rear axle describes a smaller radius than the front axle. This behavior inverts for higher speeds, i.e. the algebraic sign of the VBSSA changes. For a certain velocity v(f3 = 0), the VBSSA will be zero

stiffnesses were adapted and applied to a nonlinear state space model in every simulation step. At the end, simulation results have shown that adaptation of the cornering stiffnesses improve the quality of VBSSA-observation using a nonlinear state space observer. In a next step, measurements with a reference sensor must be employed to validate the method in real-world environment.

V(f3 = 0) =

g ' IR '

CRL FZRL

+ CRR + FZRR

(21 )

Figure 10 shows a Klotoide simulation starting at a velocity of 20 m / so At t = 45 the vehicle enters a curve with a steering wheel angle 6w = 4.5 deg.

REFERENCES

Then, the vehicle brakes down in the curve. The velocity decreases and simultaneously the VBSSA increases. At a certain velocity, the VBSSA-signal crosses the 0 degree line. For both systems with and without adaptation, v(f3 = 0) was calculated with equation 21 and then compared with the simulated system. Adapting the cornering stiffness (index ad ) provides a calculated velocity V a d = 12.3 m / s, the simulation yields Vad = 12.2. Without adaptation (index nad ), i.e. with fixed , constant cornering stiffness, the calculation by means of equation 21 provides V nad = 14.8 m / s, whereas the simulation shows Vnad = 15.7 m / so Furthermore, the VBSSA-values simulated with the nonlinear system appear to be too large. The minimum value of the side slip angle is above 0.5 degrees. That means the simulation using adapt ation shows significantly better results compared to the calculated reference.

Daiss, A. (1996). Bl'.obachtung fahrdynami scher Zustiinde und Verbesserung einer ABSund Fahrdynamikregelung. VDIFortschrittberichte, Reihe 12, Nr. 283. VDIVerlag. Diisseldorf. Gillespie, T . (1992). FundamentaLs of vehicle dynamics. 1st ed .. Society of automotive engi.neers. Warrendale, PA, USA. Kiencke, U. and A. Daiss (1994). Estimation of tyre friction for enhanced ABS-systems. In: Proceedings of the In ternational Symposium on advanced vehicle control, VoU , p.1S-1 B. SAE Japan. Tokyo. Kiencke, U. and A. Daiss (1996). Estimation of tyre slip during combined cornering and braking observer supported fuzzy estimation. In: Preprints of 13th IFAC World Congress, Volume Q, p. 41-46 (Kiencke et al., Ed.) . IFAC. San Francisco. Kiencke, U. and L. Nielsen (2000). A utomotive Control System.s. Springer-Verlag. Berlin, Heidelberg, New York. Mitschke, M. (1989). Dynamik der Kraftfahrzeuge, Band C. Springer-Verlag. Berlin, Heidelberg, New York. Reimpell , R. and P. Sponagel (1988). Fahrwerktechnik: Reifen und Riider. Vogel Fachbuchgruppe: Fahrwerktechnik. 2nd ed .. Vogel Buchverlag. Wiirzburg. Zeitz, M. (1977). N ichtlineare Beobachter for chemi.sche Reaktoren. VDI Verlag.

7. CONCLUSION An equation was presented which allows to approximate the nonlinear tire behavior by means of two parameters. The pair of parameters can be adapted to different tire characteristics using a nonlinear optimization method. In order to increase accuracy of this approximation, wheel load changes have been modelled. By means of the approximation equation, the time-variant cornering

672