AVEC ‘06

Steering control using model predictive control and multiple

internal models

Steven D. Keen

David J. Cole

Cambridge University Engineering Department

Trumpington Street

Cambridge, UK CB2 1PZ

Phone: (+44) 1223 765201

Fax: (+44) 1223 332662

E-mail: djc13@cam.ac.uk

Progress in understanding the driver/vehicle interface requires an improved model of driver skill in the

steering control task. A possible explanation for driver steering skill involves driver usage of multiple

internal models of the vehicle dynamics, where each internal model represents the actual vehicle dynamics

within a limited range of the performance envelope of the vehicle. This paper presents the derivation of a

nonlinear driver steering controller using multiple linearized models of the vehicle dynamics and Model

Predictive Control (MPC) theory. The resulting Multiple Linear Internal Model (MLIM) driver steering

controller was implemented in a simulation environment with a nonlinear vehicle model, and tested

performing a closed-loop steering task. The resulting path-following abilities of the MLIM controller are

reviewed and compared to a MPC driver model using a single internal model.

Driver modeling, multiple models, model predictive control

1. INTRODUCTION

The driver/vehicle interface continues to be a subject of

intense interest within the automotive field. A complete

understanding of the interaction between driver and vehicle

will almost certainly lead to significant improvements in

vehicle safety and comfort, by informing vehicle design

engineers with an understanding of how drivers behave in

various driving situations.

In light of this continuing need, exploration of driver

steering behavior continues to be an area of interest to

researchers. One topic of interest concerns mathematical

modeling of driver steering behavior, using modern control

methodologies and relevant knowledge from the areas of

neuroscience and physiology. Accurate and

comprehensive models of driver steering behavior could

play a role in improving vehicle performance and safety

requirements, help improve driver comfort and control while

driving, and inform efforts to develop autonomous vehicles.

The goal of the research presented here is to develop a

mathematical steering model capable of reproducing

measured driver behavior, with the model development

based on both modern control methodologies and relevant

themes from neuroscience.

2. MULTIPLE INTERNAL MODEL CONCEPTS FOR

HUMAN MOTOR CONTROL

Research in the field of neuroscience has been

undertaken in recent decades with the goal of understanding

how humans plan and carry out physical motion tasks

through the use of the central nervous system (CNS) and the

neuromuscular system (NMS). Research in this area of

neuroscience has increasingly pointed to the conclusion that

the human CNS learns and stores multiple sets of ‘internal

models’ for use when interacting with the external world [1].

The internal model paradigm postulates that the CNS

generates and stores models that represent the dynamics of

the physical systems of interest to the CNS, allowing the

CNS to then recall and use these models to predict the

behavior of the dynamic system in question. Therefore, by

possessing the ability to estimate future system behavior by

recalling and using a set of internal models, the CNS makes

use of a predictive or ‘feed-forward’ form of control when

performing a motor control task. Research has shown that

the human CNS learns and stores internal models of its own

sensory organs, and uses these organs to interpret sensory

signals received by the CNS [2]. Research has also

observed human learning and usage of internal models for

dealing with motor control tasks [3],[4].

Constructs for representing the CNS learning, storage

and usage of multiple internal models for sensorimotor

control have been proposed in the literature [1],[5],[6],[7].

Haruno et al in [7] propose a multiple model construct for

how the CNS could use multiple pairs of forward and

inverse models to learn motor tasks and adapt to novel

environments and dynamic conditions.

The usage of a model of the vehicle dynamics in the

derivation of a driver steering controller, as presented in [8],

[9] and [10], has implications in relation to internal model

concepts concerning human motor control. Applying CNS

internal model concepts to the problem of driver steering

control, it is proposed that driver skill level be defined as the

extent to which the driver possesses an understanding of the

Proceedings of AVEC ‘06

The 8th International Symposium on Advanced Vehicle Control, August 20-24, 2006, Taipei, Taiwan

AVEC060095

AVEC ‘06

nonlinear vehicle dynamics. It is proposed that the level of

driver understanding or knowledge of the vehicle dynamics

is directly related to how comprehensively the set of internal

models possessed by the driver covers the entire vehicle

dynamic performance range. A skilled driver who

possesses a comprehensive understanding of the

nonlinearities inherent in the vehicle dynamics will possess

an extensive set of internal models covering the entire range

of the vehicle dynamics. An unskilled driver, on the other

hand, will have only a very limited set of internal models,

and perhaps possess only a simple linear representation of

the vehicle dynamics for making steering control decisions.

The driver steering controller model presented uses

multiple linearized models of the vehicle and tire dynamics

to estimate the future vehicle trajectory, and then determines

an appropriate steering command intended to minimize the

path-following error of the closed-loop driver/vehicle system.

A schematic of the driver model derived on the basis of

multiple model principles is provided below in Fig 1. The

steering control model is shown in a closed-loop with the

driver physiology and the vehicle dynamics. The steering

controller observes both the vehicle state values relevant to

the steering control task (xt) and the preceding road path (Tk).

The controller uses the state feedback to select a steering

command from one of a set of steering control laws, and

commands the steering angle to the driver NMS (δD). The driver NMS acts as a low-pass filter on the steering

command passed on to the vehicle (δSW).

Fig 1. Block Diagram of Nonlinear Multiple Internal

Model Driver Steering Controller

This paper first reviews the derivation of a linear driver

steering controller based on Model Predictive Control

(MPC), and then proceeds to describe the derivation of a

nonlinear steering control law based on multiple linearized

vehicle and NMS models using Model Predictive Control.

The implementation of the resulting driver steering model in

a simulation environment is described, and simulation

results of the multiple linear internal model steering

controller performance in comparison to the single linear

model controllers available in the literature are provided.

3. VEHICLE AND TIRE MODELS

For this paper, a 2-wheel bicycle model idealization of a

4-wheeled vehicle was used for the derivation of the driver

steering controller, and for use in subsequent closed-loop

diver/vehicle simulations. The model is implemented in

conjunction with a standardized nonlinear tire model

formulation as presented in [11]. The vehicle and tire model

coefficients were generated through parameterization of

real-world test data obtained from a test vehicle using

standard methods for model fitting [12]. Details of the

vehicle and tire models are provided in the Appendix.

4. LINEAR DRIVER STEERING CONTROLLER

USING MODEL PREDICTIVE CONTROL (MPC)

Optimal control methods have been applied to the driver

steering control problem. A driver steering controller using

a general LQR formulation was derived by Sharp et al in [8],

while Model Predictive Control was employed by MacAdam

[13],[14], and by Ungoren and Peng in [9]. More recently,

Model Predictive Control (MPC) has been applied to the

driver steering control problem by Cole et al in [10],

resulting in the derivation of a linear driver steering

controller. In [10], the solution of a cost function results in

a MPC steering control law based on state feedback and

future path preview for a discrete domain controller:

[ ]

⋅Ψ⋅−=

k

k

wwDkT

xKKδ

(1)

The steering control law is dependent on two inputs: the

present vehicle states (xk), and the future road path

previewed by the driver from the vehicle position up to a

designated prediction horizon (Tk). Variables and

parameters for the controller are provided in Table 1 below.

A visual schematic of the driver road preview function is

provided in Fig 2. Starting from the present vehicle position,

the driver model observes the lateral path position yk+i and

path angle ψk+i relative to the vehicle frame of reference up

to the prediction horizon Np. The resulting path lateral

position and angle values are stored sequentially in the

preview vector Tk and act as an input to the driver steering

control law defined in (1).

Table 1 Variables and Parameters for MPC Driver Steering

Controller

xk Vehicle system states at time ‘k’ (refer to Appendix);

δD Driver CNS steering wheel command;

yk+i Relative lateral path position of road centerline at

preview point ‘k+i’;

ψκ+ι Relative path angle of road centerline at preview point

‘k+i’;

Np Number of preview points used by the driver to

observe the preceding road path;

Nu Number of preview points used by the driver to

formulate the controller gains;

Ψ Free response matrix of dynamic system from present

states xk, projected up to the prediction horizon;

Υ Forced response matrix of the dynamic system from

present steer angle δD(k-1), projected up to the prediction horizon;

Tk Preview vector containing sequential vehicle lateral

path error yk+i and heading error ψκ+ι values, from

present position up to the prediction horizon Np;

Kw

Controller preview gain vector derived using MPC

formulation.

Ts Driver steering controller sample rate

AVEC ‘06

Fig 2. Driver Road Path Preview up to Prediction

Horizon

5. NONLINEAR STEERING CONTROLLER

ARCHITECTURE USING MPC AND MULTIPLE

INTERNAL VEHICLE MODELS

A nonlinear driver steering controller is derived using

multiple models of the nonlinear vehicle dynamics and

Model Predictive Control, following the methodologies

provided in [15]. The 4-state bicycle model used in [10] is

combined with a 2-state model of the driver’s NMS. The

NMS, representing the dynamics of driver’s limbs and upper

torso when performing the steering task, is implemented as

an under-damped 2nd order system:

=

−−=

=

⋅+⋅=

0

01

2

2

NMS

2

NMSNMSNMSNMS

NMS

SW

SW

NMS

DNMSNMSNMSNMS

gfx

gxfx

ωωωζδ

δ

δ&

&

Details of the NMS model are given in the Appendix.

Combining the vehicle and NMS models results in a 6-state

system:

( ) ( )

( ) ( ) ( ) ( )

=

=

=

⋅+⋅=

14VehVeh4x1

2x4NMS

0

,0

0,

x

NMS

RFF

RF

Veh

NMS

t

Dtt

gG

fg

fF

x

xx

GxFx

ααααα

δ& (2)

The system matrix F is nonlinear, due to the varying

coefficients of fVeh, which are functions of the front and rear

tire slip angles αF and αR. Converting the system in (2)

into the discrete domain results in:

( ) DkkRkFkk xx δαα ⋅Γ+⋅Φ=+ ,1 (3)

The combined system states of the model are:

[ ]TkkkykSWkSWkk yvx ψψδδ &&=

The methodology used to generate the control law given

in (1) is combined with the nonlinear vehicle/NMS model

defined in (3) to generate a driver steering controller capable

of controlling the nonlinear vehicle. It is postulated that

the driver learns and stores an understanding of the

nonlinear vehicle dynamics by linearizing the dynamics

around incremental set points in the nonlinear regime.

MPC is used to formulate a control law valid in a small

range about the designated set point. The skill level

exhibited by an experienced driver is represented by

linearizing the nonlinear system dynamics about multiple

operating points within the nonlinear regime.

It was decided to linearize the nonlinear vehicle

dynamics about set values of front and rear wheel lateral slip

angle, (αF(m), αR

(n)). The slip angle linearization points are

defined as two vectors, with evenly spaced linearization

points:

[ ][ ]

0 :Where 11

321

321

==

≡

≡

RF

N

R

n

RRRR

NOM

R

N

F

m

FFFF

NOM

F

l

l

αα

αααααα

αααααα

LL

LL (4)

Fig 3 provides a graphical description of the chosen

linearization points, showing the linearization set points

covering the full range of the tire lateral coefficient curves

up to tire saturation.

Fig 3. Front and Rear Tire Cornering Coefficient Curves

and Linearization Points at Defined Values of αF, αR

For each set point combination, a nominal tire force

coefficient and tire force coefficient slope are calculated

from the tire force model. Therefore, for a given set point

combination (m,n), the nominal tire force coefficients are

defined as:

( ) ( ) ( ) ( ) n

RRm

FFRyR

n

yRFyF

m

yF CCCCαααα

αα==

≡≡ 00

The nominal tire force coefficient slopes at a given set point

(m,n) are defined as:

( )

( )( )

( )

∂∂

≡∂

∂

∂∂

≡∂

∂== n

RRm

FFRyR

R

n

R

yR

FyF

F

m

F

yFC

CC

Cαααα

ααα

ααα

To compute a set of steering controllers covering all

possible combinations of front and rear tire linearization set

points αF(m), αR

(n), the nonlinear system dynamics described

in (3) must be linearized about each set point. For each

(m,n) combination of set points, front and rear slip angle

perturbations are defined as:

n

RRk

n

Rk

m

FFk

m

Fk αααααα −=∆−=∆

The vehicle dynamics are linearized through a Taylor

Series expansion about each combination of set points,

retaining the first two terms of the expansion. Performing

the Taylor Series expansion and collecting terms results in

the following linearized system dynamics. Variables and

parameters are described in Table 2.

( )( ) ( )

Dkk

nmnm

NOMkk xxx δ⋅Γ+⋅Φ+= ++,,

11 (5)

AVEC ‘06

Table 2 Nominal Trajectory and System Matrices for Linearized

Vehicle/NMS Dynamics

( )( )nm

NOMkx,

1+ Nominal state trajectory for set points (αF

(m), αR(n))

Φ(m,n) Vehicle and NMS state matrix linearized about set

points (αF(m), αR

(n))

Γ Vehicle and NMS input matrix

Ξ(m,n) Nominal free response matrix of linearized dynamic

system from present nominal state rate-of-change,

projected up to the prediction horizon;

Nl Number of linearization points for each tire cornering

coefficient curve

Each linearized model obtained from (5) is then used to

compute a steering control law using MPC. Expanding on

the MPC driver steering controller solution provided in [1]

through the inclusion of the non-zero nominal state

trajectory and the previous calculation cycle steering

command to the formulation, the following cost function is

used to generate a MPC steering control law:

( ) ( )

2.0000005010

0===

=

∆⋅⋅∆+⋅⋅= ∑∑+

=++

+

=++

ψψ

δδεε

q.qRq

qQ

RQJ

yi

y

i

Nk

ki

ikDiikD

Nk

ki

iki

T

ikk

PP

Where the matrices Qi and Ri are weightings on the output

and control effort, respectively. The weighting ‘qy’

determines the weighting placed by the cost function on

minimizing the vehicle lateral path error, while ‘qψ’ is the

weighting placed on minimizing the yaw angle error

between the vehicle and road path. The variable εk+i contains the vehicle path and angle error [ ]Tikikik y +++ = ψε ,

defined for every preview point up to the prediction horizon.

The variable ∆δD(k+i) is the steering angle command change

to be optimized though solution of the cost function.

Solution of the cost function over the preview distance

results in the following control law:

( ) [ ]( )

( )

⋅+=−

+

−

k

kD

k

nm

NOMk

kD

nm

Dk

T

x

x

kkkk)1(

,

1

4321)1(

,

δδδ

(6)

( ) ( ) ( ) ( )

( ) ( ) ( )nm

w

nmnm

w

nmnm

w

nmnm

w

KkKk

KkKk,

4

,,

3

,,

2

,,

1

=Υ⋅−=

Ψ⋅−=Ξ⋅=

Terms in the steering controller are analogous to those

given in Table 1, with the addition of the terms described in

Table 2, where the superscript ‘(m,n)

’ indicates the controller

resulting from linearization of the system dynamics about set

points (αF(m), αR

(n)).

The derivation provided above results in a set of linear

steering control laws, each valid within a small region of

(αF, αR) about their designated linearization set points (αF(m),

αR(n)). The final steering angle command for a given

calculation cycle ‘k’ is determined by the driver controller

employing a switching function to select the best steering

command from the set of available steering controllers. The

switching function selects the steering controller with the

least amount of error between the controller set points (αF(m),

αR(n)) and the present vehicle slip angles (αF, αR):

( )nm

DkDk

,δδ = (7)

Where:

( ) ( ){ } ( ) ( ){ }lj

R

n

Rl

i

F

m

F NjNi KK 1 ,min ,1 ,min =∆=∆=∆=∆ αααα (8)

The resulting controller is designated as the ‘Multiple

Linear Internal Model (MLIM) driver steering controller’,

due to the usage of multiple linearized models of the vehicle

dynamics in computing the control law. The resulting

driver model is nonlinear due to the usage of multiple

controllers and the switching/selection function.

6. SIMULATION

The MLIM driver steering controller defined above in

(6), (7) and (8) has been implemented in the

Matlab/Simulink environment in conjunction with a vehicle

bicycle model and nonlinear tire model analogous to that

used for deriving the steering controller. A schematic of

the simulation is provided in Fig 4.

Fig 4. Schematic of MLIM Driver Steering Controller

Simulation Setup

A series of closed-loop driver/vehicle simulations was

performed, using MLIM controllers generated with various

numbers of linearization points Nl, shown in Table 3.

Table 3 List of Driver Steering Controllers used in Closed-Loop Simulation

Controller # of internal

models

Notes

Single linear internal

model (SLIM)

1 Analogous to

controller in [1]

MLIM4x4 16 Nl = 4

MLIM6x6 36 Nl = 6

MLIM8x8 48 Nl = 8

MLIM10x10 100 Nl = 10

These various driver models were simulated controlling the

nonlinear vehicle model along the centerline of a designated

road path, shown in Fig 5. The initial vehicle speed vx was

set at 16.8m/s to ensure that the vehicle operated in the

nonlinear range of the dynamics. The goal of the

simulation work was to compare the resulting path-holding

performance of the MLIM steering controller with that

exhibited by the single linear internal model (SLIM) driver

steering controller.

7. SIMULATION RESULTS AND DISCUSSION

The driver models listed in Table 3 were tested for their

path-following ability while negotiating a defined road path.

The vehicle was initially positioned along the centerline of

the designated path with zero lateral path error (Fig 5). The

path-following ability of each driver model was then tested

by allowing the driver model to control the vehicle along the

test path for a time of 13 s. Fig 6 shows the resulting

AVEC ‘06

steering angle commands of each driver model. The

resulting vehicle lateral acceleration and path following

error are presented in Fig 7 and Fig 8, respectively.

Fig 5. Road Path Centerline and Vehicle

Simulated Trajectory, for SLIM and

MLIM10x10 Driver Steering Controllers

Fig 6. Steering angle command of various driver

models during path following maneuver

Fig 7. Vehicle lateral acceleration levels during path

following maneuver

The driver model ‘MLIM10x10’, employing 100 internal

models of the vehicle dynamics, employs the largest steering

angle of all the steering controllers, and also generates the

highest levels of lateral acceleration (ay) while controlling

the vehicle. Review of Fig 8 also reveals that driver model

‘MLIM10x10’ maintained the minimum path-following

error from all driver models during the high-g portion of the

steering maneuver (tsim < 7 s). However, in the

post-maneuver portion of the test (tsim > 7 s), driver model

‘MLIM10x10’ gave a slightly larger path error than

‘MLIM8x8’. Overall, each MLIM driver steering

controller tested attained an improved level of

path-following ability over that displayed by the standard

SLIM driver steering controller.

Fig 8. Vehicle lateral path following error levels

during path following maneuver

8. CONCLUSIONS

Using CNS internal model concepts from neuroscience,

a driver steering controller has been derived. This steering

controller uses multiple linearized models of the nonlinear

vehicle dynamics to generate a steering controller for each

linearized model. A switching and selection function is

used to select the most relevant steering controller available

from this set of controllers, based on the driver’s estimate of

the present vehicle and tire lateral slip angles. The

resulting MLIM driver steering controller structure is

flexible in allowing the combination of any number of

linearized internal models for use in generating a set of

feedback steering control laws. The MLIM controller has

been implemented and tested in a closed-loop simulation,

where improved path following capability over a linear

driver model has been demonstrated. The positive results

of this work indicate that there is continued room for

substantial improvement in understanding and modeling

driver skill based on the internal model concepts reviewed in

the work. Future research in this area will concentrate on

improvements to the future path preview abilities of the

driver model, and will involve the inclusion of constraints on

the road path width and their effect on the steering controller

gains. REFERENCES [1] Kawato, M. (1999). "Internal models for motor control and

trajectory planning." Current Opinion in Neurobiology:

718-727.

[2] Merfeld, D. M., Zupan, L. and Peterka, R. J. (1999).

"Humans use internal models to estimate gravity and linear

acceleration." Nature 398: 615-618.

[3] Imamizu, H., Kuroda, T., Miyauchi, S., Yoshioka, T. and

Kawato, M. (2003). "Modular organization of internal

models of tools in the human cerebellum." Proceedings of

the National Academy of Sciences of the United States of

AVEC ‘06 America 100(9): 5461-5466.

[4] Imamizu, H., Miyauchi, S., Tamada, T., Sasaki, Y., Takino,

R., Pütz, B., Yoshioka, T. and Kawato, M. (2000). "Human

cerebellar activity reflecting an acquired internal model of a

new tool." Nature 403: 192 - 195.

[5] Wolpert, D. M. and Ghahramani, Z. (2000).

"Computational Principles of Movement Neuroscience."

Nature Neuroscience Supplement 3: 1212-1217.

[6] Wolpert, D. M. and Kawato, M. (1998). "Multiple paired

forward and inverse models for motor control." Neural

Networks 11(Special Issue): 1317-1329.

[7] Haruno, M., Wolpert, D. M. and Kawato, M. (2001).

"MOSAIC model for sensorimotor learning and control."

Neural Computation 13: 2201 - 2220.

[8] Sharp, R. S. and Valtetsiotis, V. (2001). "Optimal Preview

Car Steering Control." Vehicle System Dynamics

35(Supplement): 101 - 117.

[9] Ungoren, A. and Peng, H. (2005). "An adaptive lateral

preview driver model." Vehicle System Dynamics 43(4):

245-259.

[10] Cole, D. J., Pick, A. J. and Odhams, A. M. C. (2006).

"Predictive and Linear Quadratic Methods for Modelling

Driver Steering Control." Vehicle System Dynamics 44:

259-284.

[11] Pacejka, H. B. (2002). Tyre and Vehicle Dynamics,

Butterworth-Heinemann.

[12] Schindler, E. (2004). Validierung von Fahrzeugmodellen,

Fachhochschule Esslingen - Hochschule für Technik.

[13] MacAdam, C. (1980). "An Optimal Preview Control for

Linear Systems." Journal of Dynamic Systems, ASME

102(3).

[14] MacAdam, C. (1981). "Application of an Optimal Preview

Control for Simulation of Closed-Loop Automobile

Driving." IEEE Transactions on Systems, Man and

Cybernetics 11.

[15] Maciejowski, J. M. (2002). Predictive Control: With

Constraints. Harlow, Pearson Education Limited.

[16] Mitschke, M. and Wallentowitz, H. (2004). Dynamik der

Kraftfahrzeuge. Berlin Heidelberg, Springer-Verlag.

APPENDIX

Tire Force Model

Lateral tire forces are modeled using a standard magic

tire formula as presented in [11]. For situations where the

tire possesses no longitudinal slip, the tire model is defined

as a nonlinear relationship between the tire slip angle αi and

the tire cornering coefficient Cyi(αι). The lateral tire force

generated is then a function of the cornering coefficient and

the normal load on the tire, Fzi:

( )iyiziyi CFF α⋅=

Linearized lateral tire slip angles are defined as the ratio

between longitudinal and lateral velocities at the tire contact

patch [16]:

x

y

R

G

SW

x

y

Fv

bv

iv

av ψα

δψα

&& ⋅−=−

⋅+=

Vehicle Bicycle Model

As derived in [1], the vehicle bicycle model is defined as

follows. ( ) ( ) SWFVehRFVehVeh gxfx δααα ⋅+⋅= ,&

( )

( )

⋅

⋅⋅⋅

⋅

=

=

=

0

0

2

2

0010

001

00

00

2221

1211

GI

CaGm

F

gv

aa

aa

fy

v

zz

Fyf

Fyf

Veh

x

Veh

y

Veh

α

α

ψ

ψ&

( ) ( )( ) ( ) ( )( )

( ) ( )( ) ( ) ( )( )zzx

RyrFyf

zzx

RyrFyf

x

x

RyrFyf

x

RyrFyf

Iv

FbFaa

Iv

bFaFa

vmv

bFaFa

mv

FFax

αααα

αααα

22

2221

1211

2222

2222

−−=

−−=

−−

−=+

−=

The vehicle states modeled are:

Table 4 Vehicle Model States

vyk Lateral velocity m/s

kψ& Yaw rate rad/s

yk Lateral path displacement m

ψk Yaw angle rad

The bicycle model is nonlinear, due to the nonlinear

characteristics of the tire force curve terms. The model is

valid for steady-state forward velocity (vx = Const.)

conditions with zero longitudinal wheel slip (λF,R = 0).

Table 5 Vehicle Model Parameters

ms Vehicle mass 1754 kg

mf Vehicle mass, front axle 1046 kg

mr Vehicle mass, rear axle 708 kg

Izz Moment of inertia, z-axis 2282 kg m2

a Longitudinal distance, c.g. to

front axle

1.11 m

b Longitudinal distance, c.g. to

rear axle

1.64 m

iG Steering wheel gain 15.8

t Vehicle track width 1.5 m

CyF(α F = 0) Linearized front cornering

coefficient

10.23

CyR(αR = 0) Linearized rear cornering

coefficient

23.03

A vehicle model diagram is provided in Fig 9 below.

Fig 9. Vehicle Model Diagram

Neuromuscular System Model

Table 6 Neuromuscular Model States and Parameters

SWδ& Steering angle rate-of-change rad/s

δSW Steering angle rad

ζNMS NMS damping 0.707

ωNMS NMS natural frequency 18.85 rad/sec