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Dynamics Modeling and Parameter Identification for Autonomous Vehicle
Navigation
Kristijan Maˇ
cek∗, Konrad Thoma∗, Richard Glatzel∗, Roland Siegwart∗
∗Swiss Federal Institute of Technology Zurich, Switzerland
email: {kristijan.macek@mavt, thomak@student, glatzelr@student, roland.siegwart@mavt}.ethz.ch
Abstract— This paper focuses on development of a dynamic
model for an Ackermann-like vehicle based on a static tire-road
friction model and laws of technical mechanics. The model
takes as input the steering angle of the wheels in front and
the rotational velocities of the drive wheels in the back of the
vehicle. It delivers a 3-DOF output in terms of CoG vehicle
velocity, body slip angle and the yaw rate of the vehicle in
the x-y plane, as well as estimates on the forces acting on the
system. It is suitable for modeling dynamic vehicle regimes in
e.g. overtaking maneuvers/obstacle avoidance and lane-keeping,
enabling active steering control by stabilizing the dynamics
of the vehicle. The physical model description is based on
previous works combined with a suitable friction model that is
tractable in practice. Experimental verification of the obtained
model is given for the Smart testing vehicle platform, where a
separate analysis is done for directly measured as opposed to
estimated/optimized parameters of the model.
I. INTRODUCTION
In order to develop stabilizing control laws in dynamic
vehicle regimes, a vehicle model that takes dynamics into
account is developed in this paper. The model is suitable
for lane-keeping control and obstacle avoidance maneuvers
when lateral dynamic effects based on wheel slip come into
place, i.e at increased vehicle speeds, steering angles and
decreased road friction. In future work, this model will be
used for the lane-keeping control and obstacle avoidance of
the Smart test vehicle for autonomous vehicle navigation in
our research [1], aiming at improving vehicle traffic safety.
A vehicle can be analyzed as consisting of five individual
subsystems: one vehicle body and four wheels. All the five
bodies can in general move freely with respect to each other
in six degrees of freedom. Without further simplification the
dynamics of a vehicle model would therefore consist of thirty
differential equations. However, for control purposes this is
neither necessary nor efficient. The existing models cope
with this problem by simplifying the model architecture as
much as possible while still satisfying the requirements of
the model. The simplest solution found in literature is the
one-track bicycle model well described by Mitschke in [2].
This model combines the front and rear wheels respectively
and treats the vehicle as a bicycle. It describes the vehicle
motion in three degrees of freedom (x-y position and yaw
rate). More accurate models can be found in [3] and [4]
which are four wheel models describing also the pitch and
roll movements of the vehicle (five DOF).
For the lane-keeping control, our model is required to
generate an accurate prediction of the yaw rate and the
longitudinal/lateral acceleration of the vehicle, therefore the
calculation of the pitch and roll rate is not necessary and will
not be considered here. In consequence, a four wheel three
degrees of freedom model is derived, which describes the
motion of the car in the x- and y- direction on a horizontal
plane and the rotation about the z-axis normal to it.
In Sec. II, the architecture of the model is presented, with
each segment of the model analyzed in detail. The vehicle
model proposed follows closely the derivation of [5] which is
general for simulation of behavior of passenger cars. In Sec.
III, the procedure to measure and estimate the parameters
of the model is given. The relevant unknown parameters
of the model are identified by an optimization routine.
The model/parameter validation is based and validated on
real vehicle data, where a good fit and realistic estimation
of the dynamic vehicle behavior is observed, followed by
conclusions in Sec. IV. The contribution of the paper can be
found mainly in combining an existing systematic vehicle
dynamic model derivation with a choice of a friction model
that suits well the vehicle behavior tested on real data and
is still simple enough for control purposes. Furthermore, a
systematic methodology for model parameter measurement
and estimation is presented.
II. VEH ICL E MODEL
A. Model Architecture
As mentioned earlier, the dynamic model developed here
consists of five connected subsystems: one vehicle body
and four wheels, which are rigidly coupled to the vehicle
body. The model is symmetrical about the vehicle body’s
longitudinal axis. The rear wheels, which are driven by a
torque, cannot be steered and are therefore aligned with the
vehicle body’s longitudinal axis. The front wheels can be
steered according to the Ackermann steering model and the
vehicle motion is constrained to horizontal movements only.
As shown in Fig. 1 different coordinate systems for all
subsystems are defined with respect to an inertial (In) coor-
dinate system, which is defined as a fixed system in which
the x- and y- axis describe a horizontal plane and the z-
axis points upwards. The center of gravity (CoG) coordinate
system is associated with the vehicle body and has its origin
in the CoG of the total system (i.e. vehicle body plus the five
wheels). Each wheel has its own (W) coordinate system. The
wheels always touch the level horizontal plane described by
the xIn - and yIn-axis in one single point (Wheel Ground
Fig. 1. Coordinate Systems: In -
Inertial, CoG - Center of Gravity, W
- Wheel.
Fig. 2. Geometry and defined ve-
locities of the vehicle.
Contact Points). Relative to the CoG-system the wheels are
allowed to rotate freely about their rotational axis yW. The
steering angles δWF L and δWFR of the front wheels and the
driving torque TDrive that acts on both rear wheels are the
physical inputs to the vehicle.
B. Vehicle kinematics
1) Vehicle Body Kinematics: The velocity of the CoG in
the CoG-system is defined by:
−−−→
cvCoG =vC oG ·cos(β)
vCoG ·sin(β)(1)
Here vCoG is the absolute value of the velocity and βis the
angle between the x-axis of the CoG-system and the velocity
vector as shown in Fig. 2. βis known as the body side slip
angle. Transformation of cvCoG to the In-system yields:
−−−→
ivCoG =cos ψ−sin ψ
sin ψcos ψ·−−−→
cvCoG
=vCoG ·cos(β+ψ)
vCoG ·sin(β+ψ)(2)
where the yaw angle ψis the angle between the In-system
and the CoG-system. The acceleration iaCoG of the CoG is
derived in the In-system ([5](p.261)):
−−−→
iaCoG =vC oG(˙
β+˙
ψ)−sin(β+ψ)
cos(β+ψ)+(3)
+ ˙vCoG cos(β+ψ)
sin(β+ψ)
Transforming the iaCoG into the CoG-system is done by
rotating the vector about ψaround the zIn -axis:
−−−→
caCoG =vC oG(˙
β+˙
ψ)−sin β
cos β+ ˙vCoG cos β
sin β
(4)
with additional notation caCoG,x ≡axand caC oG,y ≡ay.
2) Wheel Kinematics: The wheel ground contact point
velocity is defined as the velocity of the ground contact point
of each wheel not taking into account the rotational speed
of the wheel [5] (p.226), representing a stationary point with
respect to the vehicle body ([6](pp.68)):
−−→
cvwn=−−−→
cvCoG +
0
0
˙
ψ
×
xn
yn
0
(5)
The wheel velocities then depend on vCoG , the yaw rate ˙
ψ,
and the geometry of the vehicle, where xnand ynare the
coordinates of the four wheel ground contact points, n=
1,...,4, given in the CoG-system. The geometry of the car
is shown in Fig. 2. The lengths and widths lF, lR, bR, bR
describe the exact position of each wheel ground contact
point with respect to CoG. The four wheel velocities are
given by:
−−−−−→
cvwF L,R =vCoG cos β∓˙
ψbF
2
vCoG sin β+˙
ψlF(6)
−−−−−→
cvwRL,R =vCoG cos β∓˙
ψbR
2
vCoG sin β−˙
ψlR(7)
The tire side slip angle αof each wheel is defined similar
to the vehicle body side slip angle β.αis the angle between
the wheel axis xwnand the wheel velocity vwnas shown in
Fig. 3. The four α’s are calculated by:
Fig. 3. Side slip angle (given for the front left
tire).
Fig. 4. Wheel slip.
αF L,R =δWFL,R −arctan vCoG sin β+˙
ψlF
vCoG cos β∓˙
ψbF
2
(8)
αRL,R =−arctan vCoG sin β−˙
ψlR
vCoG cos β∓˙
ψbR
2
(9)
The wheel equivalent velocity of each wheel, vRn, is given
by the rotational speed of the wheel ωnand the effective
wheel radius, reff [5](pp.249-250):
vRn=ωn·reff (10)
where its direction is the positive x-direction of the wheel
coordinate system, describing motion of the wheel if it rolled
perfectly on the road.
3) Wheel Slip Calculation: The three expressions for vw,
αand vRare now combined in the slip equations. The slip is
a normalized description of the tire movement relative to the
road and is depicted well by Burckhardt in [7]. This relative
movement is solely responsible for the tire friction forces and
is therefore important for the friction forces calculation. The
longitudinal slip sLis defined in the direction of vwand
describes the relative movement in forward direction. The
side slip sSis defined perpendicular to vwand describes
the relative movement in side direction as shown in Fig.
4. The two slip equations are given in Tab. 11 ([5](p.237)).
Burckhardt differentiates between the driving vehicle and the
braking vehicle, thus the slip stays always between -1 and
1.
Braking Driving
vRcos α≤vWvRcos α > vW
Longitudinal slip sL=vRcos α−vW
vWsL=vRcos α−vW
vRcos α
Side slip sS=vRsin α
vWsS= tan α
(11)
C. Vehicle dynamics
The dynamics of the system can be derived using the prin-
ciples of linear and angular momentum in two-dimensions
for the vehicle body and the four wheels ([6](pp.68)).
1) Vehicle Body Dynamics: According to the principle of
linear momentum, the acceleration of the car is determined
by the sum of all forces cFiacting upon it:
k
X
i=1
−→
cFi=mCoG ·−−−→
caCoG (12)
The rotational motion of the wheels is neglected here, thus
the only rotation allowed by the model architecture is the yaw
motion around the z-axis, with Tibeing the torques induced
by the forces acting on the car (angular momentum):
l
X
i=1
Ti=Jz·¨
ψ(13)
2) Wheel Dynamics: The torque produced about the y-
axis of each individual wheel is the engine torque TDrive ,
which is for the front wheels equal to zero and for the rear
wheels counter-balanced by the longitudinal friction forces
FW Lnon each wheel:
Tdriven=JWn˙ωn+ref f ·FWLn(14)
D. Forces description
The most important forces for the dynamics of the vehicle
are the friction forces which are generated at the tire/road
contact area ([8](p.17)). All the relevant forces for the model
acting horizontally upon the system are shown in Fig. 5.
Fig. 5. Forces: FL(longitudinal) and FS(lateral) friction forces, Faer
the aerodynamic force.
1) Aerodynamic Force: The aerodynamic resistance due
to the vehicle’s motion through the air is the major braking
force in the model and is aligned with the xCoG direction of
the car:
−−−→
cFaer =−sgn(vC oG)·caerAL
ρAir
2·v2
CoG ·−→
cex(15)
with ρAir being air density, caer aerodynamic constant and
ALthe effective aerodynamic surface of the vehicle.
2) Normal Forces: The normal forces are calculated at
each wheel ground contact point in the positive zwdirection.
Due to the asymmetry of the vehicle body along the vehicle
body’s lateral axis the normal forces differ already with no
motion of the car and become larger at the rear wheels while
accelerating and larger at the outside wheels while in a turn.
The approach of [5] (pp.306-308) takes into account the
shifting of the wheel load while accelerating or driving in a
turn. Assuming no suspension and roll or pitch motion, the
Fig. 6. Normal Forces Calculation.
dependencies of the normal forces on the longitudinal and
lateral accelerations, axand ay, can be calculated separately.
By balancing different loads on axles due to longitudinal
acceleration and furthermore axle load shift due to driving
in a turn (see Fig. 6), the torque balance at the front left
contact point yields:
FZF R =1
2mCoG ·lR
lg−hCoG
lax+(16)
+mCoG ·lR
lg−hCoG
lax·hCoG ·ay
bF·g
where similar expressions can be derived for other normal
forces as well.
3) Friction Forces: The behavior of friction at the
tire/road contact area is a highly non-linear phenomenon
which is complex to describe. Several approaches to the
friction characteristics have been developed. Among these
are dynamic friction models of which the LuGre model is
among the most promising ([9], [10] and [11]). In contrast,
static friction models empirically approximate the tire char-
acteristics and are well studied. Their output are two forces
for each wheel which act upon the wheel ground contact
point in the direction of vW,FL- the longitudinal friction
force and perpendicular to it, FS- the lateral friction force.
4) Burckhardt Friction Model: The static friction model
that is used in our system was proposed by Burckhardt in [7].
The model aims at obtaining a realistic relationship between
the slip of the tires and the friction coefficients in longitudinal
and lateral direction µLand µS. Two auxiliary parameters
sRes and µRes are introduced:
sRes =qs2
L+s2
S(17)
µRes =c1·1−e−c2·sRes −c3·sRes (18)
with calculation of the slips sLand sSas explained in Sec.
II-B.3.
Fig. 7. Friction coefficient charac-
teristics.
Fig. 8. µLdependency on slip sL
and different αside slip angles.
By choosing values for the coefficients c1,c2and c3
according to different conditions for dry asphalt, wet asphalt
and snow, the Fig. 7 shows the resulting friction coefficient
as given by Eq. 18. As one can see in Fig. 8, Eq. 18 also
provides the dependency of the friction coefficients on the
tire side slip angles α.
Now the longitudinal and lateral friction coefficient µL
and µSare calculated by Eq. 19 and the longitudinal and
lateral friction forces FLand FSare given in Eq. 20 and
Eq. 21.
µL=µRes
sL
sRes
, µS=µRes
sS
sRes
(19)
with the longitudinal and lateral friction force being:
FL=µL·FZ=µRes ·sL
sRes
·FZ(20)
FS=µS·FZ=µRes ·sS
sRes
·FZ(21)
E. Steering
This section deals with the relationship between the angle
on the steering column δSand the two wheel steering angles
δWF L and δWF R on the front wheels.
1) Steering Column Model: For the derivation of the
relationship between δS, the steering column angle and δW,
the steering angle of the hypothetical wheel in the front
center, the model used is a static proportional relationship,
where iTis the transmission coefficient of [12](pp.224).
δW=δS
iT
(22)
2) Ackermann Steering: Since all y-axes of the wheels
should intersect in the instantaneous center of motion (ICM),
the left and right front wheel angles, δWF L and δWF R are a
function of δW:
δWF L,R = arctan l
l
tan δW∓bF
2!(23)
F. Summary of the Model
The complete vehicle model combines individual segments
elaborated so far. The steering and torque as inputs are
propagated through the model, generating forces that produce
following outputs: the CoG-velocity vCoG, the vehicle body
side slip angle βand the yaw rate ˙
ψas shown in the
equations of motion:
˙vCoG =cos β
mCoG hΣFX−caer AL
ρ
2·v2
CoG i(24)
+sin β
mCoG
[ΣFY]
˙
β=cos β
mCoG ·vC oG
[ΣFY](25)
−sin β
mCoG ·vC oG hΣFX−caerAL
ρ
2·v2
CoG i−˙
ψ
JZ¨
ψ= [FY F L +FY FR ]·lF−[FY RL +FY RR]·lR
+ [FXF R −FX F L]·bF+ [FX RR −FXRL]·bR
(26)
where the sum of forces along each coordinate are:
ΣFX=FXF L +FX F R +FXRL +FX RR (27)
ΣFY=FY F L +FY FR +FY RL +FY RR (28)
III. PARAMETER ESTIMATION AND EXPERIMENTAL
MODEL VALIDATION
A. Acquisition of Measurement Data
From the CAN-Bus (Controller Area Network) that is
included serially in the Smart vehicle, the following data
can be obtained:
•δS- steering wheel angle
•wF L,wF R - rotational wheel speed front
left/right
•wRL,wRR - rotational wheel speed rear left/right
The IMU300CC-100 is an external inertial measurement unit
supplementary integrated in the test vehicle with following
measurements taken:
•˙
ψ- yaw rate (range 100◦/s, bias <±2.0◦/s,
scale factor accuracy <1%)
•ax,ay- acceleration in the x- and y- direction
(range ±2g, bias <±30mg, scale factor
accuracy <1%).
Both sensor systems take measurements at a sample rate of
Ts=11ms.
B. Parameter Identification
1) Assured Parameters: In this category of parameters, all
natural constants and tabulated values are considered, as well
as the car geometry which is given by the drivers manual.
The coefficients of the friction model given in Sec. II-D.4
are taken for a dry asphalt road, i.e. c1= 1.2801,c2= 23.99
and c3= 0.52.
2) Measured Parameters:
a) Effective Wheel Radius: reff is a function of the
two static wheel radii r0and rstat, where r0is the radius of
the unloaded wheel and rstat is the compressed wheel radius
as the vehicle stands on its four wheels ([5](pp.250)):
reff =r0·sin(arccos(rstat
r0))
arccos(rstat
r0)(29)
which for Smart vehicle yields reff = 0.273m.
b) Total Mass of the Smart and x-Position of the CoG:
The vehicle was weighted on an industrial scale with an
accuracy of 10kg. To find out the x-position of the CoG
described by the two lengths lRand lF, the law of lever was
used:
lF=l·mrear
mrear +mf ront
, lR=l·mfront
mrear +mf ront
(30)
where the results also depend on the number of passengers.
For the case of no load: m= 760kg,mf ront = 330kg,
mrear = 430kg,lF= 1.025m,lR= 0.787.
3) Estimated Parameters:
a) Transmission Coefficient - iT rans:The transmission
coefficient is important for the car behavior because it
determines the magnitude of the wheel steering angles. The
initial estimate was iT rans= 15, which is in the range given
for other vehicles by [12](pp.224) and was later optimized.
b) Inertial Momenta of the Vehicle Body and the
Wheels: To estimate the value of the inertial moment the
formula ([12] (p.406)) was used:
Jz= 0.1269 ·LT·l(31)
where LTis the total length of the vehicle and lthe length
between the front and rear axis. This formula gives a result
for Jzof about 500kgm2.
For the estimation of the inertial momenta of the front
wheels, the assumption is that the wheel is a solid disk of
radius r=20cm and a mass of m=5kg:
JWF=1
2m·r2(32)
which yields the inertial momentum of a single front wheel
as JWF=0.1kgm2.
4) Optimization of the Parameters: To optimize the es-
timated parameters the difference between a set of exper-
imentally measured and modeled outputs is compared. An
error function is defined as the sum of all normalized,
weighted and squared differences between the five measured
and modeled variables:
err =1
n
1·1
x˙
ψq2
0.5·1
xaxq2
0.5·1
xayq2
0.5·1
xωF Lq2
0.5·1
xωF Rq2
·
n
X
i=1
(∆i(˙
ψ))2
(∆i(ax))2
(∆i(ay))2
(∆i(ωF L))2
(∆i(ωF R))2
(33)
where the difference for each individual measurement is:
∆i(◦) = (◦)measured i−(◦)modeled i(34)
The error of the yaw rate ˙
ψis weighted twice as high as the
errors of the four other variables because of its importance
and the quality of the measurements. The data is normalized
with the difference between the maximum and minimum
value of each measured parameter over the whole test drive
(x˙
ψq,xaxq,xayq,xωF Lq,xωF R q). The MatlabT M function
fminsearch is applied in order to search for the minimum
of the error function. The optimized parameter values were
derived as:
iT rans = 28.5576 , Jz= 1490.3kgm2, JWF= 0.1071kgm2
which is in good correspondence with the Smart vehicle
used, since smaller vehicles types have typically a higher
transmission coefficient.
C. Experimental Model Validation
After identifying the unknown parameters of the model
and optimizing it with measured data, the model is validated
by comparing it to a new test drive of the Smart vehicle. It is
expected that the vehicle model follows the measured data
without any further optimization. The inputs to the model
are shown in Fig. 9, 10 and 11, as front steering wheel
angle δW(with the estimated steering angles on left and
right front wheel δW L,δW R ) and the rear rotational wheel
velocities ωRL,ωRR , respectively. Peak differences in rear
wheel velocities indicate that the driving regime is dynamic.
Note that the wheel dynamics is induced with the drive
torques on rear axis as explained in Sec. II-C.2, however, the
torque quantities cannot be measured directly. They can only
be estimated in a forward manner by an additional model of
vehicle engine and transmission or in a backward manner
according to Eq. 14, therefore the rear wheel velocities are
considered as direct inputs here and represent the driving
quantities.
In Fig. 14, 15 and 16 the yaw rate ˙
ψand accelerations
axand ayare given as a comparison between the mea-
sured and modeled dynamic variables, since they can be
directly measured by the IMU unit. For the given precision
of the IMU, the model errors are: |∆max|˙
ψ= 6.7◦/s,
Fig. 9. Measured input steering an-
gle δWand estimated left and right
front wheel angles δF L and δF R .
Fig. 10. Measured input wheel
velocity rear left ωRL.
Fig. 11. Measured input wheel
velocity rear right ωRR.
Fig. 12. Estimated wheel forces
front left, longitudinal FLF L and
lateral FSF L .
Fig. 13. Estimated wheel forces rear
right, longitudinal FLRR and lateral
FSRR .
Fig. 14. Estimated and measured
yaw rate
˙
ψ.
Fig. 15. Estimated and measured
acceleration in the x-direction ax.
Fig. 16. Estimated and measured
acceleration in the y-direction ay.
σ˙
ψ= 2.3◦/s,|∆max|ax= 0.71m/s2,σax= 0.33m/s2
and |∆max|ay= 1.42m/s2,σay= 0.74m/s2. As can be
seen, the measured and modeled data correspond well, with
maximal model deviations |∆max|due to peaks of highly
dynamic regime. The estimated forces on the front left wheel
of Fig. 12 show that the lateral force FSFL is considerable
and changes direction according to the steering angle. The
longitudinal force FLF L at this wheel is small negative or
close to zero due to the small wheel inertia and net rolling
resistance, opposing the longitudinal motion. The estimated
longitudinal force of the rear right wheel FLRR of Fig.
13 is in contrast to FLF L much bigger since it represents
one of the two driving forces exerted on the vehicle by
the engine and transmission, changing from the acceleration
phases (positive) to the braking phases (negative). The overall
magnitudes of the forces also correspond well to those found
in other literature ([13], [2]) taking into account the scale of
the car class. The developed dynamic model will be used for
feed-forward trajectory simulation within a control scheme
which relies also on the on-line feedback action to account
for model inaccuracies.
IV. CONCLUSION
In this work a 3DOF dynamic vehicle model was devel-
oped describing the motion of vehicle in horizontal plane
by considering lateral and longitudinal body dynamics and
friction forces exerted by the ground surface acting on the
vehicle. A detailed analysis included identification of all the
relevant parameters of the model, either directly measured or
estimated. An experimental verification based on comparison
of the model output and measured data of a test vehicle
proved a good correlation and validity of the model. Future
work will include applying the model for lane-keeping and
obstacle avoidance control of the vehicle and automatic
switching of friction model coefficients based on different
ground surfaces.
V. ACK NOWLEDGM EN T S
The authors would like to thank their colleague Sascha
Kolski for his help in acquiring experimental vehicle data.
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