Effects of Model Complexity on the Performance of Automated Vehicle Steering Controllers Model Development Validation and Comparison

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Effects of Model Complexity on the Performance of Automated Vehicle Steering Controllers: Model Development, Validation and Comparison a

DIRK E. SMITH & JOHN M. STARKEY

b

a

2508 CEBA Mechanical Engineering, Louisiana State University, Baton Rouge, LA, 70803-6413, USA b

1288 Mechanical Engineering Building, Purdue University, West Lafayette, IN, 47907-1288, USA Version of record first published: 27 Jul 2007.

To cite this article: DIRK E. SMITH & JOHN M. STARKEY (1995): Effects of Model Complexity on the Performance of Automated Vehicle Steering Controllers: Model Development, Validation and Comparison, Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility, 24:2, 163-181 To link to this article: http://dx.doi.org/10.1080/00423119508969086

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Vehicle System Dynamics, 24 (1995), pp. 163-181

0042-3 I 14/95/2402-163%6.00 0Swets & Zeitlinger

Effects of Model Complexity on the Performance of Automated Vehicle Steering Controllers: Model

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Development, Validation and Comparison DIRK E. SMITH* and JOHN M. STARKEY**

SUMMARY Recent research on autonomous highway vehicles has begun to focus on lateral control strategies. The initial work has focused on vehicle control during low-g maneuvers at constant vehicle speed, typical of lane merging and normal highway driving. In this paper, and its companion paper, to follow, the lateral control of vehicles during high-g emergency maneuvers is addressed. Models of the vehicle dynamics are developed, showing the accuracy of the different models under low and high-g conditions. Specifically, body roll, tire and drive-train dynamics, tire force saturation, and tire side force lag are shown to be important effects to include in models for emergency maneuvers. Current controllers, designed for low-g maneuvers only, neglect these effects. The follow on paper demonstrates the performance of lateral controllers during high-g lateral emergency maneuvers using these vehicle models.

I. INTRODUCTION

Future generations of highway vehicles will likely have steering controllers that work in harmony with brake and throttle controllers to drive vehicles automatically. Not only would such a system increase safety by relieving drivers from tedious tasks and reduce driver error, it would also allow closer spacing of vehicles on highways and increased speeds. This would ultimately provide higher traffic flow capacity [1,2]. Most researchers currently developing vehicle steering controls for these future vehicles are focused on low-lateral acceleration conditions [3,4,5,6,7,8,9,10]. Under these conditions, low-order vehicle models and linear tire models are adequate, and controllers developed using these models perform well. It is also well known that vehicle behavior under moderate to hlgh-g maneuvers is not accurately predicted by these low-order models. But * 2508 CEBA Mechanical Engineering, Louisiana State University, Baton Rouge, LA 70803-6413, USA. ** 1288 Mechanical Engineering Building, Purdue University, West Lafayette, IN 47907-1288, USA.


164

D.E. SMITH AND J.M. STARKEY

robust control strategies based on coupled nonlinear vehicle models are not yet available in the literature. The lateral control of vehicles during high-g maneuvers is addressed here in two parts. The first part, covered in this paper, explores the effects of vehicle model order and tire model complexity on vehicle response. It reaffirms that under normal driving conditions (below 0.2 g's lateral acceleration) the traditional two degree-of-freedom "bicycle" model with linear tires is completely adequate. It will then show the effects of including tire rotation degrees of freedom, body roll, tire force saturation, and tire side force lag on vehicle response in emergencies (lateral accelerations greater than 0.5g's, possibly combined with longitudinal acceleration or braking). The nonlinear eightdegree-of-freedom model presented here is shown to be a valid vehicle model for emergency maneuvers. The relationship between model complexity and simulation accuracy has been addressed before. This study extends these ideas to investigate which aspects of the vehicle models, developed here, are the most important for controller development. The control studies are the topic of the follow on paper. 2. VEHICLE MODEL DESCRIPTIONS

For this study, three vehicle models and two tire models are used. The vehicle models include the two degree-of-freedom "bicycle" model (2D) with yaw and side slip, a five degree-of-freedom model (5D) with yaw, side slip, longitudinal acceleration, and front and rear wheel rotations, and an eight degree-offreedom model (8D) that adds body roll and separate degrees of freedom for each of the four tire rotations. The tire models are the linear model and Dughoffs tire model [11,12]. Tire side force lag is also irlcluded in the study. The equations for the linearized 2D model, similar to that presented by Ellis [13], are derived from Figure 1.

Fig. 1 . Bicycle model of vehicle dynamics.

AUTOMATED VEHICLE STEERING CONTROLLERS


Fig. 2. Wheel rotation.

Here F, is side tire force at the front and rear. Some researchers choose to normalize equation (1) to include vehicle,side slip, P = tan-'(v/u), in place of lateral velocity. Forward speed, U, is assumed to be constant in these equations, so no tractive force or drive-train models are needed. Several researchers considering lateral control have held the vehicle speed constant and have not considered longitudinal dynamics and throttle control. During low-g maneuvers this assumption may be valid. However, during high-g emergency maneuvers, the vehicle speed decreases for a constant throttle setting because of the extreme yaw attitude of the vehcle. In such maneuvers the desired vehicle speed may not be attainable even with full throttle. Also, during automated emergency maneuvers it is very likely that braking will be needed [14]. For these reasons a five degree-of-freedom model is presented to include wheel rotations and forward velocity as variables, similar to the one presented by Koepele and Starkey [6]. Figures 1 and 2 are used to derive the equations for the 5D model. The equations from Figure 1 are

1 v =[-mUr m

+ 2Ftfsin Gf + 2Fsfcos Gf + 2F,,]

1 r = - [2aFtfsin Gf 1,

+ 2aFSfcos Gf

-

2bFs,]

To increase the accuracy of the model and to account for large steering angles, equations (3) through (5) have not been linearized. The longitudinal aerodynamic drag is included to increase. the accuracy of the required throttle input at high speeds. Due to low lateral velocities, the lateral force and yaw moment due to aerodynamics are small compared to the tire forces, and were therefore not included.

166


The wheel rotational equation of motion is based on Figure 2,

where i designates a front or rear tire (i = f for front, r for rear), and IWiis the equivalent inertia of the drive train, for one wheel. When accelerating, the wheel inertia and half of the engine's rotational inertia are combined as


I

= I

+q

I

Accelerating

(7)

For the non-driven wheel or when braking the engine inertia is not included.

IWi = Iti

Non-driven or Braking

(8)

The drive torque, T, at one wheel, is a function of engine speed (a,),gear ratio (cj), drive train efficiency (qj), and throttle position (WOT). The simple model used here is

where

The coefficients, c,, c2, and c3, were chosen to represent a parabolic engine torque curve. For this study all the simulations are with rear wheel drive vehicles. For the 5D model, T is applied at the one rear wheel equation and the 2F, accounts for the total drive force of two wheels. For the 8D model, T is applied to each rear wheel equation of motion. WOT can take on any value from 0.0 (no engine torque at its current speed) to 1.0 (maximum torque the engine can put out, at its current speed). The total brake torque, Tbrk, is assumed to be distributed according to a brake biasing constant, kbf, such that Tbrkf = kbfTbrk

(11)

Tbrk, = (1 - kbf)Tbrk

(12)

where Tbrk is a driver input. Tbrk can take on any positive value, up to and including the value at which wheel lock-up occurs. Any experimental validation of these models would, of course, require additional models to relate WOT and Tbrk to measurable quantities.



'Roll Axis

Fig. 3. Eight degree-of-freedom vehicle model

The 5D model includes longitudinal load transfer governed by

These equations account for front-rear load shift due to vehicle accelerations, but neglect contributions from road grade (assumed zero) and aerodynamic lift and drag. The drag and lift terms contribute negligible load shifts, less than 5%, compared with b/e and a/e. The acceleration load shift, however, can be significant for high-g maneuvers, and therefore has been included. The 2D and 5D models are a good representation of vehicle dynamics, on a flat road, for low lateral accelerations. However, they lack a roll degreeof-freedom, lateral load transfer, and roll steer effects. Steering compliance is not a factor since the steering input is the front wheel angle. To account for the effects of high-g lateral maneuvers an eight degree-of-freedom model (8D) is presented. It has an added roll degree of freedom and a total of four wheel rotation equations. Figure 3 is used in the derivation of these

168


equations


(IS)

The x and y tire forces are related to the tractive and side forces by the following equations Fxi= F,i cos 6wi - F,, sin Eiwi (19)

F,, = Ftisin Ziwi

+ FSicos 2iwi

(20)

The total steering angles of the wheels, including roll steer are

In addition to longitudinal load transfer, this model includes a quasi-static lateral load transfer due to both lateral acceleration and roll angle. The front roll stiffness ratio, KRSF,determines the frontlrear distribution of the total lateral load transfer.

Fzr~

~

=

z = r

mg a (U - V r )b gT [j f g e

, mg a ~

[? +

(U - Vr) hcg g e

+

$:(

mse sin &)] (1 - KRSF) -- - mTw

(

(1 - KRSF) -- -

mTw



169

The vertical loads on the tires include the effects of the body roll degree of freedom, +,, but neglect the forelaft lags due to the pitch inertia. Since the controller studies presented here focus on high-g lateral maneuvers, the pitch degree of freedom has been neglected. Though the validation example in this paper shows good comparisons with experimental results without including pitch, the effects of pitch inertia should be investigated throughly if braking and throttle controls are added which cause high forelaft accelerations. For all three vehicles, the vehicle-to-global coordinate transformations are given by x = ~ c o s +- -sin+ (28)

3. TIRE MODELS

The tire forces are determined from tire properties and slip models. Tire models are typically based on the slip angle, a, shown in Figure 1, and longitudinal slip, is. The following equations define the slip angles of the front and rear tires for the 2D and 5D models.

For the 8D model, each wheel has an independent slip angle. am = 6wf - tan-'

(,Y-+&) ' (J+~L)

=

6wr - tan-

=

6w,

+ tan-'

(."'-; .)

a r ~=

&Wr

+ tan-'

($L;;~J

O+R

170

D.E. SMITH A N D J.M. STARKEY

The longitudinal slip depends on whether the vehicle is accelerating or braking.

u t Acceleration is = 1 - Rw Ro is = 1 - - Braking

(36)

ut


Here U, is the speed of the wheel center in the direction of the tire heading. For the 2D and 5D models this speed is defined as Utf = U cos Gf

+ (V + ar) sin Gf

(38)

For the 8D model this speed is independent for each wheel. UtfR = (U - i ~ ~cosr Gwf ) U,n = (U

+ $T,,,r)

cos Gwr

+ (V + ar) sin tiwf + (V + ar) sin Gwf

UtrR = (U - i T w r )cos tiw, - (br - V) sin Gw, UtrL = (U

+ i T w r )cos Gwr

- (br - V) sin Gw,

(40) (41) (42)

(43)

The linear tire model used to calculate lateral tire forces is given by

This model is decoupled from the tractive force, and is assumed to be independent of the normal force on the tire. The tractive force is calculated using the non-linear tire model, discussed below. When studying low-g lateral dynamics only, the linear lateral tire model gives good results. The problem arises when there is slip in both directions, as is expected to occur in automated emergency maneuvers, or when a becomes large and tire forces saturate at the friction limit. Dugoff et al. [ l l ] developed a nonlinear tire model based on the friction ellipse idea, that accounts for these two effects.

f (S) =

S(2 - S) if S < 1 ifS>1

AUTOMATED VEHICLE STEERING CONTROLLERS Linear or Non-linear Lateral T i e Model Model Degree \

T i Side Force BD-p)L-L /Lag

Included

Fig. 4. Notation.


C , tan cr F, = 1 - is f (S)

Dugoff s tire model has been widely studied, and used for nonlinear simulations [12,15].

Another important aspect of tire modeling is tire side force lag. Several researchers have shown the importance of tire lag on vehicle response [16,17,18]. In this study tire side force lag is modeled as a first order time lag on the side force.

Tire lag has typically not been included in the models for development and testing of lateral control strategies. This has not proven to be a problem during field tests because the automated maneuvers have been very low-g maneuvers. The effects of tire, lag at high g's will be shown here. Figure 4 shows the notation that defines the vehicle and tire models used in this study. 4. MODEL VALIDATION

Before conclusions can be drawn about the accuracy of any of these models, they must be compared with known references. The strategy here is to validate the 8D model by comparing it with published experimental and analytical results. Once the validity of the 8D model is demonstrated, then the effects of simplifying to the 5D and 2D models can be shown. Because of the lack of well documented vehicle response measurements that include all of the required data for accurate simulation, there are relatively few published results with which to compare these models. Two notable exceptions are El-Gindy and Ilosvai [15] and Allen et al. [19]. These works include sufficient data to drive the simulations, and give experimental vehicle responses with which to compare the simulations.

172

D.E. SMITH A N D J.M. STARKEY 10

I

El-Gindy Experimental El-Gindy Simulation 8D-NLL

8-

22

2

-


Oo

-

I

0.5

1

1.5

2

2.5

3

Time (sec) Fig. 5. Braking in a turn validation example.

One transient vehicle validation testing the high-g behavior of the 8D model is shown in Figure 5. Here, the 8D model is compared under conditions of combined braking and steering. Accelerations up to 0.4g's lateral and 0.7 g's longitudinal are experienced during the simulation. The figure shows that the 8D model with non-linear tires and tire side force lag matches the experimental results well. The data used for the model are in Table 1 in the appendix. Figure 6 compares the yaw rate frequency response of the 8D model with nonlinear tires, with and without tire lag, to the experimental and simulation results obtained by Allen et al. [19]. Here the steering input is sinusoidal and the amplitude is varied, at each frequency, to obtain a peak lateral acceleration of 0.6 g's. Although the differences in the response with and without tire lag are small, the lagged model is a better match, particularly when comparing phase lag. Based on these comparisons, the 8D model with nonlinear tires and tire lag is believed to be sufficiently valid for the maneuvers in this study.

5. VEHICLE SIMULATION COMPARISONS

The three vehicle models presented here are compared in a variety of driving maneuvers. In addition, the effects of the linear and nonlinear tire models are shown, as are the effects of tire lag. The goal of these studies is to identify conditions under which higher order models are needed to adequately predict the

173

AUTOMATED VEHICLE STEERING CONTROLLERS 10'.

-Ex ............

6

.

l!. % .

-.9e

erimental et al. Sirnulation ID-NL

-7 ............

.- .- .- .-

..\.>,\

j

"-""

">.

0


lo0 lo0

10'

1 02

Frequency (rad1se.c) Fig. 6 . Comparison of yaw rate response for 0.6g lateral acceleration, U = 14.75 m/s.

response of automatically controlled vehicles, and to determine what factors are important in designing lateral controllers for high-g emergency maneuvers. The equations were integrated using the Runge-Kutta method with primary time steps of 0.001 seconds. The integration routine took even smaller time steps, if required, to meet the error tolerance of 1.OE-6. Primary time steps smaller than 0.001 seconds had a negligible effect on accuracy, but increased computational time. Figure 7 shows the effects of model complexity under low-g maneuvers. Here three vehicleltire combinations are given a sawtooth steering input while traveling at 10m/s. The models range from the simple 2D vehicle with linear tires and no tire lag to the complex 8D vehicle with nonlinear tires and tire lag. As expected, at low g's, their responses are nearly identical. Figure 8 compares the three vehicle models, all using the nonlinear tire model, at 0.6 g's. From the figure it is clear that the higher order model is necessary at moderate to high-g maneuvers, even when the nonlinear tire model is used. An automated steering control system based on the 2D or 5D models may produce unexpected results during a moderate evasive maneuver. Figure 9 shows the effect of using a nonlinear versus a linear tire model. Here, the 8D model is used with the linear and nonlinear tire models. From Figure 8 it is evident that a linear tire model is insufficient to predict vehicle response during high-g maneuvers. As a increases the nonlinear tires saturate and are unable to generate the lateral force necessary to complete the lane change


Time (sec)


0.2 n Y)

0

1

2 Time (sec)

3

4

-0.2 0

1

2 Time (sec)

3

Fig. 7. Vehicle responses to low g maneuver, U = 10m/s.

Time (sec)

X (m)

Time (sec)

Time (sec)

Fig. 8. Vehicle responses to high g maneuver using the nonlinear tire model, U = 20 m/s.

4


-101


0

1

2 Time (sec)

Time (sec)

3

175

1

4

Time (sec)

Fig. 9. Effect of linear versus non-linear tire models on vehicle response during high g maneuvers, U = 20 m/s.

maneuver. The linear tires do not model the tire saturation. In fact the linear tires predict vehicle lateral accelerations near 0.9 g's, but the friction coefficient is only 0.85. Because of nonlinear tire effects during high-g maneuvers, a control system designed for emergency maneuvers must account for these effects. A control system based on a linear tire model may produce a favorable response for low-g maneuvers, but during evasive maneuvers the actual vehicle response may be unexpected and undesirable. Figure 10 shows the effect of vehicle model degree on response while accelerating in a turn. Here the steering is held constant at 4' and the engine is at full throttle. The vehicle starts from rest with excessive wheel spin. The sharp change in direction for the speed and slip curves occur during gear shifting. As the speed and lateral acceleration increase, the 8D vehicle begins to spin out as indicated by the side slip (P). As the lateral acceleration increases, the lateral load shift on the rear wheels reduces the total cornering force available there. The 5D model does not predict spin-out in this case because it does not model each rear wheel separately. Though for other cases the 5D and 8D models may agree, this example shows that there are vehicle conditions for which the models do not agree. Therefore, the 5D model is not totally reliable. Braking while steering is a very common evasive maneuver. Figure 11 compares the responses of the 5D and 8D models to just such a maneuver.

176



x (m)

Time (sec)

0 Time (sec)

5

10

Time (sec)

Fig. 10. Constant steering, full throttle model comparison.

The vehicles start at 10 m/s and the throttle is controlled to maintain this speed. The steering is ramped up to 5' as shown in the figure. The vehicle has reached steady state in the turn by 2 seconds, and the throttle is turned off and the brakes are applied. From Figure 11 it is apparent that the vehicles respond quite differently, and the interaction between the lateral and longitudinal tire forces become very pronounced. Rear tire side-slip (a,) and vehicle side slip, 0, indicates that the 5D niodel does not spin out, however, the 8D model does. The inside, right rear tire locks-up shortly after the brakes are applied. This is due to the lateral load transfer, to the outside tires, caused by the lateral acceleration and body roll. The 5D model shows no rear tire lock-up, therefore the rear tires are able to develop the necessary side force and prevent the vehicle from spinning-out. Again, this case indicates that there are operating conditions for which the 5D model is not accurate. Although anti-lock brakes (ABS) and Traction Control can eliminate the spin out of the 8D vehicle, the point of these maneuvers is to show that the 8D-NL-L model is necessary to accurately model vehicle response near the limit capabilities of the vehicle. The last modeling aspect to investigate is tire side force lag. Figure 7 showed that during low-g maneuvers tire lag had a negligible effect on vehicle response. Figure 12, however, shows tire lag to be important during high-g maneuvers.


177

steering (deg)

'BrAe (m-m)


0

1

2 Time (sec)

3

4

T i e (sec)

Time (sec)

Time (sec)

Fig. 1 1 . Braking in a turn, U = I0 m/s.

The response of the lagged vehicle is delayed during the negative steer input. This causes the vehicle to have a different position and heading at the end of the maneuver. These studies have shown the differences in vehicle behavior predicted by low and high-order vehicle models. Though substantial differences in model responses are shown it does not necessarily follow that steering controllers designed using the 2D model with linear tires would not perform adequately with the 8D-NL-L model, and ultimately with an actual vehicle. The next paper in this series will address this question by building controllers based on the 2D-L model, and investigate its effectiveness in controlling the various vehicle models presented here.

6. CONCLUSIONS

Several lateral control systems have been developed and simulation-tested using the linear vehicleltire model [8,9,20]. This paper has shown the vehicle models upon which these controllers are based are inadequate for high-g maneuvers. Lower order vehicle models are insufficient to accurately predict vehicle response at high-g's, because of the lack of lateral load transfer and body roll

178


X (m)


Time (sec)

-

50

2

O

6I! 3

$ -50 0

1

2 Time(sec)

3

4

0

1

2 Time (sec)

3

4

Fig. 12. Effect of tire side force lag during a high g maneuver, U = 10 m/s.

dynamics. Also, in high-g maneuvers the right and left tires can have significantly different loads and slip values. Models must include these effects to be valid for emergency maneuvers. Tire modeling has also been shown to be important as the severity of maneuvers increase. The saturation in tire force at high slip angles is not predicted by the linear tire model, nor is the interaction between longitudinal and lateral slip. Tire side force lag has been shown to have an effect on vehicle response during an evasive maneuver. Not including tire lag in the development of automated vehicle controllers could lead to unexpected and potentially dangerous results.

REFERENCES I . Shladover, S.E., Potential Freeway Capacity Etfects of Advanced Vehicle Control Systems. In Proceedings of the 2nd International Conference on Advanced Technologies in Transportation Engineering, pages 21 3-21 7, Minneapolis, MN, August 18-21, 1991. 2. Frank, A.A., Liu, S.J., Liang, S.C., Longitudinal Control Concepts for Automated Automobiles and Trucks Operating on a Cooperative Highway, 1989. SAE Paper No. 891708. 3. Fenton R.E., Mayhan, R.J.. Automated Highway Studies at The Ohio State University - An Overview. IEEE Transactions on Vehicular Technology, 40(1): 100-1 13, 1991. 4. Shladover, S.E., Desoer, C.A., Hedrick, J.K., Tomizuka, M., Walrand, J., Zhang, W., McMahon, D.H., Peng, H., Sheikholeslam, S., McKeown, N.. Automatic Vehicle Control



179

Developments in the PATH Program. IEEE Transactions on Vehicular Technology, 40(1): 114-130, 1991. 5. Peng, H., Tomizuka, M., Vehicle Lateral Control for Highway Automation. In Proceedings of the 1990 American Control Conference, pages 788-794, San Diego, CA, May 23-25 1990. IEEE. Vol. I. 6. Koepele, B., Starkey J., Closed-Loop Vehicle and Driver Models for High Speed Trajectory Following. In Transportation Systems - 1990, pages 59-68. ASME, 1990. AMDVol. 108. 7. Lee, A.Y., A Preview Steering Autopilot Control Algorithm for Four-Wheel-Steering Passenger Vehicles. In Advanced Automotive Technologies - 1989, pages 83-98. ASME, 1989. DSCVol. 13. 8. Dickmans, E.S., Zapp, A., Autonomous High Speed Road Vehicle Guidance by Computer Vision. Preprints IFAC-Congress 1986, 4: 232-237, 1986. 9. Hatwal, H., Mikulcik, E.C., An Optimal Control Approach to the Path Tracking Problem for an Automobile. Transactions of the Canadian Society for Mechanical Engineering, lO(4): 233-241, 1986. 10. Fenton, R.E., Melocik, G.C., Olson, K.W., On the Steering of Automated Vehicles Theory and Experiment. IEEE Transactions on Automatic Control, AC-21(3): 306-3 15, 1976. 1I. Dugoff, H., Fancher, P.S., Segel, L., An Analysis of Tire Traction Properties and Their Influence on Vehicle Dynamic Performance, SAE Transactions, 79: 341-366, 1970. SAE Paper No. 700377. 12. Wong, J.Y., Theory of Ground Vehicles, pages 32-33. John Wiley and Sons Inc., New York, 197R - - .-.

13. Ellis, J.R., Vehicle Dynamics. Business Books Limited, London, 1969. 14. Rouse, Jr, R.J., Hoberock, L.L., Emergency Control of Vehicle Platoons: Control of FollowingLaw Vehicles. Journal of Dynamic Systems, Measurement, and Control, pages 239-244, September 1976. Transactions of the ASME. 15. El-Gindy, M., Ilosvai, L., Computer Simulation Study on a Vehicle's Directional Response in Some Severe Manoeuvres Part 1: Rapid Lane-Change Manoeuvres. International Journal of Vehicle Design, 4(4): 386-40 I, 1983. 16. Allen, R.W., Rosenthal, T.J., Szostak, H.T., Steady State and Transient Analysis of Ground Vehicle Handling, 1987. SAE Paper No. 870495. 17. Heydinger, G.J., Garrott, W.R., Chrstos, J.P., The Importance of Tire Lag on Simulated Transient Vehicle Response, 1991. SAE Paper No. 910235. 18. Heydinger, G.J., Garrott, W.R., Chrstos, J.P., Guenther, D.A., The Dynamic Effects of Tire Lag on Simulation Yaw Rate Predictions. In Transportation Systems - 1990, pages 77-86. ASME, 1990. AMD-Vol. 108. 19.Allen, R.W., Szostak, H.T., Rosenthal, T.J., Johnston, D.E., Test Methods and Computer Modeling for the Analysis of Ground Vehicle Handling, 1986.. SAE Paper No. 861 115. 20. Fenton, R.E., Selim, I., On the Optimal Design of an Automotive Lateral Controller. IEEE Transactions on Vehicle Technology, 37(2): 108-1 13, 1988.

7. APPENDIX

The following vehicle and tire data were used to validate the 8D model in Figure 5. The vehicle was run at 40 km/hr, with a ramp steer input up to 5' over 0.7 seconds. At 1.4 seconds, the brake torque was ramped up to 3000 N-m over 0.3 seconds. Table 2 contains the vehicle and tire data used to validate the 8D model in Figure 6 . Data that was changed, from Table 1, for the vehicle comparisons of figures 7 through 12 are shown in Table 3.

180


Table 1. Vehicle data used for model validation. Nm rad

Nm

apz Nlrad N/U& slip i'


Er

k,front k, rear c1 c2 c3

Nm a= & rad

N-m

Table 2. Vehicle data used for yaw rate response validation. 66185.8

)I Er

k, front k, rear Cl

c2 c3 Csl KRSF Cd d 'll

'l2 rl3 '14

'-I s

3511.6 29000 52526 0.9 0.015 0.20 -0.20 -6.0 59.16 25.0 1.38 0.552 0.32 0.014 0.85 0.85 0.85 0.85 0.85

& rad

$k N/rad N/unit slip s/m rad/rad rad/rad Nm a Nm

rad N-m

m

Table 3. Vehicle data used for model comparisons. CU

ci

30000 50000

N/rad N/unit slip

kbr

0.5

AUTOMATED VEHICLE STEERING CONTROLLERS Table 4. List of symbols.


U V A, \1, r

Forward Velocity Lateral Velocity Acceleration in the y direction Yaw angle Yaw rate 4 Sprung mass roll angle p Roll rate m Vehicle total mass m, Vehicle sprung mass I, Vehicle moment of inertia (Z axis) I,, Vehicle moment of inertia (roll axis) I, Sprung mass product of inertia I, Rotating tire inertia I, Rotating engine inertia I, Equivalent inertia at one wheel Tractive force of one tire F, Side force of one tire F, F, Vertical load on one tire FSI Lagged side force of one tire rSl Side force lag time constant CSI Side force lag constant 6( Steering input to the front wheels Steered angle of a wheel WOT Percent of Wide Open Throttle Tbrk Total brake torque T Drive torque kbf Front brake proportioning constant pa Density of air Tbrkf Total front brake torque Tbrk, Total rear brake torque

g a b

e

T, e

a is

a R o o,

Cj qj C,

Ci

P KRsF

If., X Y Af Cd e,

Acceleration due to gravity Distance from cg to front axle Distance from cg to rear axle Longitudinal wheelbase Height of cg above ground Lateral wheelbase (track width) Distance from sprung mass cg to roll axis Lateral tire side slip Longitudinal slip Roll axis torsional stiffness Roll axis torsional damping Tire rolling radius Axle rotational speed Engine rotational speed Gear ratio of gear j (j = 1 . . . 5) Drive train efficiency of gear j (j = 1 . . .5) Cornering (lateral) stiffness of one tire Longitudinal stiffness of one tire Nominal friction coefficient between tire and ground Ratios of front roll stiffness to the total roll stiffness Roll steer coefficient Global x velocity Global y velocity Vehicle frontal area Drag coefficient Road adhesion reduction factor

Effects of Model Complexity on the Performance of Automated Vehicle Steering Controllers Model Development Validation and Comparison

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