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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: [email protected]
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
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