Low Friction Emulation of Lateral VehicleDynamics Using Four-Wheel Steer-by-Wire

Holly E. B. Russell1,2 and J. Christian Gerdes1

Abstract— Low friction surfaces such as ice are challengingfor drivers to navigate safely because the limited tire forcecapability drastically alters the vehicle dynamics comparedto dry roads. Experiments on real or emulated low frictionsurfaces are important for testing control systems, under-standing driver-vehicle interactions, and training drivers. Asa way of enabling these experiments with the added flexibilityof a tunable friction coefficient, this paper presents a force-based approach to emulating the lateral dynamics of a vehicleon a low friction surface using four-wheel steer-by-wire. Thesteer angle commands are computed with a combination offeedforward and state feedback, and the controller explicitlyhandles rear wheel actuator limitations. Experimental resultsfrom an asphalt surface confirm this controller successfullytracks the lateral dynamics of the low friction reference model.

I. INTRODUCTION

In most driving situations, vehicles operate far from thephysical limits of the tires and respond predictably andlinearly to driver commands. Low friction surfaces such asicy roads create a challenging environment for drivers tonavigate, because they change the handling limits of thevehicle and its response to driver commands in an unfamiliarand nonlinear manner. Drivers in this type of situationstruggle to judge the grip of the tires, and therefore havedifficulty responding appropriately to the road conditions, asdiscussed by Forster [1].

Envelope control schemes, such as those developed byBeal and Gerdes [2] and Bobier and Gerdes [3], and Elec-tronic Stability Control systems [4] have been developedto help drivers stabilize vehicles in situations where theyreach the handling limits. Since the vehicle dynamics dependon the surface friction, validation of these systems requireslow friction surfaces. In addition, when developing driverassistance and safety systems, it is important to understandthe interaction between the driver and the vehicle. It istherefore useful to evaluate how drivers respond to lowfriction conditions.

The simplest way to perform these types of experimentsis to drive on a wet or icy surface, which truly has lowfriction. This is the approach taken by Falcone et. al. [5] intheir design of active front steering control for autonomousvehicle path tracking, as well as Gao et. al. [6] in theirwork on obstacle avoidance for semi-autonomous vehicles.Another possibility is to make the vehicle handle as if it wereon a low friction surface, while actually driving on a highfriction surface such as dry asphalt. This is the idea behind

1Department of Mechanical Engineering, Stanford University, Stanford,CA 94305, USA

2Corresponding author: heb@stanford.edu

the commercially available SKIDCAR driver training system,which supports most of the vehicle’s weight on a frame withfree-castering wheels to reduce the normal forces on the tires.This causes the tires to quickly reach their handling limitson a high friction surface and allows drivers to experiencethese limits without being on a true low friction surface.

Another way to enable these experiments is to use four-wheel steer-by-wire to emulate the dynamics of a vehicleon a low friction surface. Front steer-by-wire has been usedpreviously to change vehicle handling characteristics by Yihand Gerdes [7], who combined full state feedback and thedriver’s steering command to change the effective frontcornering stiffness of the vehicle, and by Yamaguchi andMurakami [8], who extended this approach to include vehicleparameter estimation. Adding rear steer-by-wire expands thepossibilities for vehicle handling modification beyond frontcornering stiffness modification. In particular, the vehiclecan emulate low friction dynamics for a range of frictioncoefficients, rather than being limited to the conditions ofone specific road surface. Although the vehicle is emulat-ing nonlinear dynamics, it is operating on a high frictionsurface and is far from its handling limits. This means thatexperiments can be carried out while maintaining the abilityto brake if needed for safety. This is in contrast to a truelow friction surface, where the vehicle may actually be at itslimits and unable to safely stop.

This paper demonstrates a method for using four-wheelsteer-by-wire to emulate the lateral dynamics of a vehicleon a low friction surface. This approach can be useful fortesting control systems, studying human-vehicle interactionnear the limits of handling, and training drivers to respondappropriately to changing conditions. The low friction em-ulation controller is designed to track the lateral dynamicsof a low friction reference model by exploiting knowledgeof the nonlinear tire forces. Simulation and experimentalresults confirm the controller successfully reproduces thelateral dynamics of the reference model, and emulates beingnear the limits of handling.

II. CONTROLLER OVERVIEW

The goal of the low friction emulation controller is tomake a vehicle on a high friction surface have the samelateral dynamics as if it were on a low friction surface.This is accomplished by developing a reference model ofthe low friction lateral dynamics of a front-wheel steeringvehicle, then designing a controller to track these desireddynamics. This section provides a brief introduction to thecontrol scheme, which is depicted in the block diagram

reference

model

plant (vehicle)

Vx controller δdriver

Fyf

r

Fyr

β

δr

δf

r

β

~ ~

~ ~

Fig. 1. Block diagram for the low friction emulation control scheme.

in Fig. 1. In the first block, the reference model lateraldynamics are simulated using a planar bicycle model. Theinputs to this model are the vehicle speed Vx and the driver’sdesired front steer angle δdriver, which is computed fromthe steering wheel angle divided by the steering ratio. Themodel generates the reference sideslip angle β, yaw rate r,and front and rear lateral tire forces Fyf and Fyr, whichare later used by the controller. A key requirement of thereference model is capturing the nonlinear behavior of thetires, which reflects the limited force capability on the lowfriction surface. The particular tire model chosen for thiswork meets this requirement with only two parameters, thecornering stiffness and one friction coefficient; it would bea minor change to instead use a more complex model, forexample with two friction coefficients to allow a clear peakin the tire force curve.

Using the reference model dynamics, the controller com-putes the necessary front and rear steer angle commandsto apply to the vehicle to achieve the same dynamics onthe actual, high friction surface. Since tire forces determinethe evolution of vehicle dynamics, the problem of trackingthe reference model dynamics can be framed as generatingappropriate tire forces on the experimental vehicle. Thecontroller uses the reference model tire forces to generatefeedforward steer angle commands that result in approxi-mately the same tire forces on the experimental vehicle. Toensure tracking of the reference dynamics in the presenceof modeling uncertainty and disturbances, the total steerangle commands δf and δr include full state feedback terms.The steer angle commands are applied to the experimentalvehicle, and the process repeats at each execution step of thecontroller.

Although it would be possible to approach this problemusing only feedback, this force-based feedforward-feedbackapproach is advantageous in the way it affects unmodeleddynamics including vehicle body roll. Like the lateral dy-namics, roll dynamics are also determined by tire forces. Ifthe tire forces of the experimental vehicle and the referencemodel are the same, in addition to having the same lateraldynamics, they will have very similar roll dynamics.

III. VEHICLE DYNAMICS AND TIRE FORCE MODELS

The lateral dynamics of both the low friction referencemodel and the high friction experimental vehicle are modeledwith the two-state planar bicycle model depicted in Fig. 2.The linearized equations of motion for sideslip β and yaw

b a

αr

δr

αf

δf

Vy

Vx

V β

r Fyr Fyf

Front of car

Fig. 2. The planar bicycle model with four-wheel steering.

rate r are functions of the front and rear lateral tire forcesFyf and Fyr, where m is the mass, Iz is the yaw moment ofinertia, a and b are the distances from the center of gravityto the front and rear axles, respectively, and Vx is the vehiclespeed, assumed constant.

β =Fyf + FyrmVx

− r r =aFyf − bFyr

Iz(1)

For the low friction reference model, lateral tire forceis modeled using the nonlinear Fiala brush tire model,assuming a single coefficient of friction and a parabolic forcedistribution, as formulated by Pacejka [9]. In this model, thelateral tire forces Fyf and Fyr are functions of the front andrear tire slip angles αf and αr,

Fy =

−Cα tan α+

C2α

3µFz| tan α| tan α

− C3α

27µ2F 2ztan3 α, |α| < αsl

−µFzsgn α, |α| ≥ αsl

(2)

αf = arctan

(β +

ar

Vx

)− δdriver (3)

αr = arctan

(β − br

Vx

)(4)

αsl = arctan

(3µFzCα

)(5)

Fzf =mgb

a+ bFzr =

mga

a+ b(6)

where Cα is the cornering stiffness, µ is the tire-road frictioncoefficient, g is the gravitational constant, αsl is the tire slipangle corresponding to full sliding, and Fzf and Fzr are thefront and rear tire static normal loads.

For the high friction experimental vehicle, both the tire slipangles and the lateral tire forces are modeled as linear. Themaximum lateral tire forces will be the friction limit for thereference model, or Fy,max = µFz . Since the experimentalvehicle is on a dry asphalt surface with a friction coefficientof about 0.9, the lateral forces will be much less than themaximum capability of the vehicle, so the linear tire modelis appropriate.

αf = β +ar

Vx− δf αr = β − br

Vx− δr (7)

Fyf = −Cαfαf Fyr = −Cαrαr (8)

Tire Slip Angle

La

tera

l T

ire

Fo

rce

Ma

gn

itu

de

Fy

α α

µ = 0.3

(reference)

µ = 0.9

(actual)

Fig. 3. Force matching. The reference model tire force Fy is computedfrom the model tire slip angle α using the nonlinear tire model in (2). Thisforce is used to calculate the desired tire slip angle α of the high frictionexperimental vehicle.

The steer angles δf and δr come from the controller block,and the driver’s desired steer angle δdriver impacts theexperimental vehicle dynamics only through the referencemodel dynamics.

IV. CONTROLLER DETAILS

A. Nominal Controller

Given the lateral dynamics and tire forces of the referencemodel, the controller uses feedforward steer angle commandsto match the reference model tire forces, and feedback steerangle commands to track the reference sideslip and yaw rate.The force-matching step is depicted in Fig. 3, which showsthe lateral force versus slip angle for the experimental vehicle(µ = 0.9) and the reference model (µ = 0.3). The referencemodel and the experimental vehicle have the same lateralforce Fy , but this force occurs at different slip angles αand α. For this example, Fy is near the maximum possiblelateral force for the reference model, but well within thelinear regime of the tire force curve for the experimentalvehicle. Therefore, the linear slip angle and tire force modelsin (7) and (8) are used to compute the necessary feedforwardsteer angles δf,FFW and δr,FFW to achieve the desired tireforces.

δf,FFW = −αf + β +ar

Vx=FyfCαf

+ β +ar

Vx(9a)

δr,FFW = −αr + β − br

Vx=FyrCαr

+ β − br

Vx(9b)

To compensate for errors from disturbances, parameteruncertainty, and unmodeled effects, state feedback is usedto ensure tracking of the lateral dynamic states β and r.This results in additional steer angle terms δf,FB and δr,FB .

δf,FB = Kβf (β − β) +Krf (r − r) (10a)

δr,FB = Kβr(β − β) +Krr(r − r) (10b)

The feedback gains Kβf , Kβr, Krf and Krr were initiallycomputed from pole placement with natural frequency 8Hz,

damping ratio 1, and nominal vehicle speed Vx = 8m/s,then tuned experimentally to ensure good performance inthe combined feedback system including the steer-by-wiresystem. The gains used for this paper are Kβf = 0.1312,Kβr = 0.16949, Krf = 0.01193, and Krr = −0.01361.

The total steer angle commands are given by

δf =FyfCαf

+ β +ar

Vx+Kβf (β − β) +Krf (r − r) (11a)

δr =FyrCαr

+ β − br

Vx+Kβr(β − β) +Krr(r − r) (11b)

where the total steer angles are a summation of the feedfor-ward and feedback steer angles.

B. Actuator Limitations

The low friction emulation controller developed aboveworks well for tracking the lateral dynamics of the referencemodel in simulation. However, the steer angles of a realvehicle are limited by actuator constraints, so in some casesthe steer angle commands developed in (11) will be saturated,and therefore it will be impossible to generate the desiredlateral tire forces.

The steer angle limits typically become problematic whenthe reference model sideslip β becomes large. This occurswhen the rear tire force of the reference model is saturatedat its maximum value Fyr,max = µFzr. The maximumachievable sideslip angle is related to the maximum rearsteer angle, so the rear wheels reach their steering limitbefore the front wheels. When the rear steer angles saturate,they lose the ability to generate additional lateral tire force,leaving only the front steering capable of changing the lateraldynamics of the vehicle.

When rear wheel saturation occurs, it is important toensure that the vehicle continues to behave in a way thatis intuitive to the driver, and preferably feels as much like alow friction response as possible. Since there is now only oneactuator, it is only possible to directly control one state of thevehicle, and a new control objective must be chosen. In thiswork, when the rear wheels saturate, the controller objectiveis to track the reference model yaw rate. This choice allowsthe vehicle to continue turning the same direction with thesame yaw rate as it did before the rear angle saturated. Sincethe achievable sideslip angle is fundamentally limited by themaximum wheel angle limits, it would be impossible to trackunbounded sideslip in any case.

This new objective fits nicely into the control structureused for the nominal controller, with feedforward and feed-back components to the front steer angle command. Theactual rear lateral tire force Fyr is approximated with thelinear tire force model (8), which results in the followingexpression when the rear wheels are at their actuation limits.

Fyr = −Cαrαr = −Cαr(β − br

Vx− (±δr,max)

)(12)

In the nominal controller, the feedforward steer angles arecalculated from the reference model tire forces. In the rearwheel saturation case, the feedforward front steer angle is

still calculated from a desired tire force, but rather thandirectly coming from the reference model, this desired forceis found by equating the yaw dynamics of the referencemodel ˙r and the experimental vehicle r from (1), includingthe actual rear lateral force found in (12).

Fyf,des = Fyf −b

aFyr +

b

aFyr (13)

δf,FFW =Fyf,desCαf

+ β +ar

Vx(14)

The kinematic relationship in (14) is formulated in termsof β instead of β, which allows the controller to meet itsobjective of tracking yaw rate while leaving sideslip to evolveuncontrolled. State feedback is still needed to compensateunmodeled effects, and now consists of a single gain on yawrate error.

δf,FB = Krf (r − r) (15)

The total front steer angle command is the sum of thefeedforward and feedback terms derived above, and the rearsteer angle is the maximum value allowed by the actuatorlimitation.

δf =Fyf,desCαf

+ β +ar

Vx+Krf (r − r) (16a)

δr = ±δr,max (16b)

Fig. 4 demonstrates the performance of the controller insimulation. The driver commands a 10◦ slalom for the first8 seconds, then holds the steering wheel at 10◦. Duringthe first 14 seconds, the experimental vehicle tracks boththe sideslip and the yaw rate of the reference model. Afterthis, the rear wheels reach their actuation limit, and thecontroller continues to track yaw rate but allows sideslip todeviate. The reference model tire forces are saturated severaltimes during the maneuver, but the sideslip only grows largeafter the forces have been saturated for several seconds.The simulation results give promise that in experiment thecontroller will track the reference sideslip and yaw ratewhenever the rear wheels are within their actuation limits,and will otherwise successfully track the reference yaw rate.

V. EXPERIMENTAL RESULTS

The low friction emulation controller is demonstratedexperimentally using the X1 research vehicle, pictured inFig. 5. X1 is an electric vehicle designed and built bystudents at Stanford University. It is equipped with a four-wheel independent steer-by-wire system for lateral control,and drive-by-wire and four-wheel independent brake-by-wire for longitudinal control. Vehicle sideslip, yaw rate, andlongitudinal speed are measured using a NovAtel SPAN-SEsystem that uses tight integration of an IMU and GNSS. Thephysical parameters used to model the vehicle are listed inTable I.

The vehicle’s on-board computer, a dSPACE MicroAuto-Box II, is used for both vehicle control and simulation ofthe reference model in real-time. The reference model usesthe same physical parameters as the experimental vehicle,

0 5 10 15−20

−10

0

10

20

(a)

Ste

er A

ngle

s (d

eg)

Time History Plots (time in seconds)

Driver

Front

Rear

Rear Limit

0 5 10 15

−20

−10

0

10

20

(b)

Sid

eslip

Ang

le (

deg)

Model

Actual

0 5 10 15−30

−15

0

15

30

(c)

Yaw

Rat

e (d

eg/s

)

Model

Actual

0 5 10 15−4

−2

0

2

4

(d)

Ref

. Lat

. Tire

For

ces

(kN

)

Front

Rear

Limits

Fig. 4. Simulated ramp steer maneuver with reference model friction µ =0.3 and longitudinal speed Vx = 10m/s: (a) steer angles, (b) sideslip angle,(c) yaw rate, and (d) reference model lateral tire forces.

TABLE IVEHICLE PARAMETERS

Parameter Symbol Value UnitsMass m 1973 kgYaw moment of inertia Iz 2000 kg ·m2

Distance from CG to front axle a 1.55 mDistance from CG to rear axle b 1.21 mFront cornering stiffness Cαf 140 kN/radRear cornering stiffness Cαr 170 kN/radMaximum front steer angle δf,max 18 degMaximum rear steer angle δr,max 14 deg

Fig. 5. X1 research testbed, equipped with four-wheel steer-by-wire.

except the cornering stiffnesses Cαf and Cαr are reduced toreflect the characteristic of the reference low friction surface.The friction coefficient µ can be changed on-the-fly to allowemulation of different surfaces, although all results presentedhere use µ = 0.3 for consistency.

The experiments for this paper took place at ThunderhillRaceway Park in Willows, CA. The testing surface is anasphalt skidpad with a friction coefficient µ of about 0.9 andan average grade of about 2.5◦.

Fig. 6 is an experimental slalom maneuver that demon-strates the dynamic capabilities of the controller at moderateexcitation of the reference model. The reference lateral tireforces approach their limits, but do not saturate. The rearsteer angle stays below the actuator limit throughout, whichmeans that the nominal controller (11) is always active. Theexperimental vehicle successfully tracks both the referencesideslip and yaw rate during the entire maneuver, even in thepresence of variable longitudinal speed.

The maneuver shown in Fig. 7 is more dramatic becausethe driver holds the steering wheel at large angles for longerperiods of time. In the early part of the maneuver, thedynamics look similar to the previous maneuver. At aboutt = 16s, the driver steers hard to the right and holds thesteering wheel for several seconds. This causes both thefront and rear reference lateral tire forces to saturate, andas a result the sideslip starts to grow rapidly, indicating thata spin is imminent. The driver then countersteers at aboutt = 20s to recover. Although the dynamics of the maneuverare dramatic, the experimental vehicle tracks the referencevehicle sideslip and yaw rate well throughout, excepting theperiods when the rear wheels reach their actuation limits,resulting in uncontrolled sideslip.

These experiments validate the effectiveness of the force-based feedforward-feedback control structure in trackinga low friction reference model. They show that the lowfriction emulation controller can track the reference modelsideslip and yaw rate nearly to the reference model’s forcelimits. Yaw rate is tracked when the reference model is atits limits, and the sideslip could be tracked further if the

5 10 15−20

−10

0

10

20

(a)

Ste

er A

ngle

s (d

eg)

Time History Plots (time in seconds)

Driver

Front

Rear

Rear Limit

5 10 15

−10

0

10

(b)

Sid

eslip

Ang

le (

deg)

Model

Actual

5 10 15−30

−15

0

15

30

(c)

Yaw

Rat

e (d

eg/s

)

Model

Actual

5 10 15−4

−2

0

2

4

(d)

Ref

. Lat

. Tire

For

ces

(kN

)

Front

Rear

Limits

5 10 150

5

10

(e)

Spee

d (m

/s)

Fig. 6. Experimental maneuver on asphalt: (a) steer angles, (b) sideslipangle, (c) yaw rate, (d) reference model lateral tire forces and (e) longitu-dinal speed. The reference model has µ = 0.3. Since the steer angles staywithin the actuator limits, yaw rate and sideslip are both tracked throughoutthe maneuver.

test vehicle had additional rear steering capability. Usingsimple dynamics and tire models, this controller convincinglyreproduces the lateral dynamics of a vehicle on a low frictionsurface.

0 10 20 30 40−20

−10

0

10

20

(a)

Ste

er A

ngle

s (d

eg)

Time History Plots (time in seconds)

Driver

Front

Rear

Rear Limit

0 10 20 30 40

−10

0

10

(b)

Sid

eslip

Ang

le (

deg)

Model

Actual

0 10 20 30 40−30

−15

0

15

30

(c)

Yaw

Rat

e (d

eg/s

)

Model

Actual

0 10 20 30 40−4

−2

0

2

4

(d)

Ref

. Lat

. Tire

For

ces

(kN

)

Front

Rear

Limits

0 10 20 30 400

5

10

(e)

Spee

d (m

/s)

Fig. 7. Experimental maneuver on asphalt: (a) steer angles, (b) sideslip an-gle, (c) yaw rate, (d) reference model lateral tire forces and (e) longitudinalspeed. The reference model has µ = 0.3.

VI. CONCLUSION

Using four-wheel steer-by-wire to emulate the lateral dy-namics of a vehicle on a low friction surface is feasible witha force-based control approach. Furthermore, this approachshould accurately reproduce unmodeled roll dynamics, sincethe tire forces are the same as those in the reference model.

The control structure described in this paper is readilyextensible to a more complex scheme that includes longitu-dinal modeling and control. Since the controller is formulateddirectly using a nonlinear lateral tire model, this modelcan be modified to include the reduction in lateral forcecapability that occurs during braking and acceleration. Themodel can be further expanded to include a fully coupledlateral-longitudinal tire model and longitudinal dynamics.As the next step in the development of this controller, thedrive-by-wire and brake-by-wire capabilities of X1 will beused to add longitudinal control to make the low frictionemulation even more realistic. Finally, the controller willenable experiments to evaluate driver response to low frictionconditions.

ACKNOWLEDGMENT

The authors would like to thank the NISSAN MO-TOR Co., Ltd. and project team members Susumu Fujita,Yoshitaka Deguchi, and Hikaru Nishira for sponsoring thisresearch. We would also like to thank the members ofthe Dynamic Design Laboratory at Stanford for help withexperimental testing.

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