Cross-combined UKF for vehicle sideslip angle estimation with a modified Dugoff tire model: design and experimental results

Abstract

The knowledge of key vehicle states is crucial to guarantee adequate safety levels for modern passenger cars, for which active safety control systems are lifesavers. In this regard, vehicle sideslip angle is a pivotal state for the characterization of lateral vehicle behavior. However, measuring sideslip angle is expensive and unpractical, which has led to many years of research on techniques to estimate it instead. This paper presents a novel method to estimate vehicle sideslip angle, with an innovative combination of a kinematic-based approach and a dynamic-based approach: part of the output of the kinematic-based approach is fed as input to the dynamic-based approach, and vice-versa. The dynamic-based approach exploits an Unscented Kalman Filter (UKF) with a double-track vehicle model and a modified Dugoff tire model, that is simple yet ensures accuracy similar to the well-known Magic Formula. The proposed method is successfully assessed on a large amount of experimental data obtained on different race tracks, and compared with a traditional approach presented in the literature. Results show that the sideslip angle is estimated with an average error of 0.5 deg, and that the implemented cross-combination allows to further improve the estimation of the vehicle longitudinal velocity compared to current state-of-the-art techniques, with interesting perspectives for future onboard implementation.

Full text

Cross-combined UKF for vehicle sideslip angle estimation
with a modified Dugoff tire model: design and experimental
results
Elvis Villano .Basilio Lenzo .Aleksandr Sakhnevych
Received: 27 December 2020 / Accepted: 6 June 2021
The Author(s) 2021
Abstract The knowledge of key vehicle states is
crucial to guarantee adequate safety levels for modern
passenger cars, for which active safety control systems
are lifesavers. In this regard, vehicle sideslip angle is a
pivotal state for the characterization of lateral vehicle
behavior. However, measuring sideslip angle is
expensive and unpractical, which has led to many
years of research on techniques to estimate it instead.
This paper presents a novel method to estimate vehicle
sideslip angle, with an innovative combination of a
kinematic-based approach and a dynamic-based
approach: part of the output of the kinematic-based
approach is fed as input to the dynamic-based
approach, and vice-versa. The dynamic-based
approach exploits an Unscented Kalman Filter
(UKF) with a double-track vehicle model and a
modified Dugoff tire model, that is simple yet ensures
accuracy similar to the well-known Magic Formula.
The proposed method is successfully assessed on a
large amount of experimental data obtained on
different race tracks, and compared with a traditional
approach presented in the literature. Results show that
the sideslip angle is estimated with an average error of
0.5 deg, and that the implemented cross-combination
allows to further improve the estimation of the vehicle
longitudinal velocity compared to current state-of-the-
art techniques, with interesting perspectives for future
onboard implementation.
Keywords Vehicle dynamics Tire modeling 
Sideslip angle Kalman filter Experiments
Abbreviations
ADynamic matrix
aVehicle front semi-wheelbase
a
x
Longitudinal acceleration of the center of
mass
a
y
Lateral acceleration of the center of mass
a
y,m,s
Standard deviation of the measurement noise
on a
y
BControl matrix
B Lateral load transfer coefficient
bVehicle rear semi-wheelbase
CAxle cornering stiffness
C
a
Tire model parameter
C
z
Downforce aero coefficient
dAxle height of the roll center
F
x
Longitudinal force
F
y
Lateral force
E. Villano B. Lenzo
Department of Engineering and Maths, Sheffield Hallam
University, Sheffield S1 1WB, UK
E. Villano A. Sakhnevych
Department of Industrial Engineering, Universita
`di
Napoli Federico II, 80125 Naples, Italy
B. Lenzo (&)
Department of Industrial Engineering, University of
Padova, 35131 Padua, Italy
e-mail: basilio.lenzo@unipd.it
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Meccanica
https://doi.org/10.1007/s11012-021-01403-6(0123456789().,-volV)(0123456789().,-volV)

fDynamic function
G
a
Tire model parameter
HMeasurement matrix
hMeasurement function
IIdentity matrix
J
z
Vehicle moment of inertia (vertical axis)
KKalman gain
K
r
Axle roll stiffness
lVehicle wheelbase
mVehicle mass
NNumber of states in x
nNumber of time samples
PCovariance matrix of the estimated state
P
zk
Measurement covariance matrix
P
xkzk
Cross-covariance matrix
pTire model function
QProcess noise covariance matrix
RMeasurement noise covariance matrix
rYaw rate
r
m,s
Standard deviation of the measurement noise
on r
r
s
Standard deviation of the process noise on r
S
a
Vehicle frontal area
tTime
t
w
Axle track width
uControl input
VMeasurement noise matrix
vMeasurement noise
v
M
Measured wheel speed
v
x
Longitudinal velocity of the center of mass
v
y
Lateral velocity of the center of mass
v
y,s
Standard deviation of the process noise on v
y
WProcess noise matrix
wProcess noise
w
dyn
Weight of the dynamic filter
w
kin
Weight of the kinematic filter
w
r
Noise on the measurement of r
w
ax
Noise on the measurement of a
x
w
ay
Noise on the measurement of a
y
XSigma-point
xSystem state vector
ZMeasurement sigma-point
zMeasurement vector
aTire slip angle
bVehicle sideslip angle
b
e
Root mean square error on ^
b
cUKF parameter
DtDiscretization (sample) time
dWheel steering angle
jUKF parameter
kTire model parameter
l
max
Friction coefficient
qAir density
rUKF parameter
wUKF parameter
Subscripts
for Bi;Czi;di;Fyij ;Fzij ;Kri;twi ;^
vx;ijþ;^
vM;ij;aij

iAxle index: 1 = left, 2 = right
jSide index: 1 = front, 2 = rear
Superscripts/accents
^ Estimated value
-A-priori value
aAugmented
1 Introduction
In a modern social context requiring increasing
possibilities to move fast and on long distances,
vehicle safety is of vital importance to considerably
reduce the number of fatal accidents. To respond to
these urgent societal challenges, in 2011 the European
Commission adopted an ambitious Road Safety Pro-
gramme aiming to halve the chances of deaths in
Europe in the following decade. The programme set
out a mix of initiatives, both at European and national
level, focusing on a considerable improvement of
active safety (onboard vehicle controls), passive safety
(structural and infrastructural enhancements) and
preventive safety (analysing and detecting road users’
behavior) [1].
In the specific context of active safety, the future of
the mobility on wheels is going towards the develop-
ment of advanced control algorithms for enhancing
vehicle interaction both with the road and with the
vehicle network. A full and accurate knowledge of the
vehicle states is required for onboard control logics to
guarantee a correct and effective performance. Current
vehicle control systems of passenger cars rely on
available measurements such as longitudinal velocity,
lateral/longitudinal accelerations, yaw rate. That is the
case of, e.g., the Electronic Stability Control (ESC),
nowadays installed in all passenger cars.
The availability of additional vehicle states would
allow the development of more advanced active
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vehicle controllers, further enhancing vehicle safety.
That is the case of vehicle sideslip angle, a vehicle
state defined as the angle between the vehicle longi-
tudinal axis and the direction of the vehicle velocity at
the center of mass [2]. The availability of this state
would be dramatically helpful [3–5]. However, the
possibility to measure sideslip angle directly on board
is limited. Optical and GPS-based sensors, usually
employed to this purpose, are expensive and quite
uncomfortable for a large-scale adoption. Alterna-
tively, the sideslip angle can be estimated via real-time
software modeling techniques. Several observers have
been developed to this aim, yet sideslip angle estima-
tion is still an open issue in the automotive field [6,7].
Three modeling categories can be identified in the
literature: kinematic models, dynamic models and
combined models. Most of the proposed methods to
estimate sideslip angle need only signals that are easily
measurable within the integrated set of sensors already
available in a standard passenger car.
The first category is based on kinematic relation-
ships involving yaw rate, lateral and longitudinal
velocities and their derivatives. No vehicle or tire
parameters are involved. However, kinematic-based
estimators become unobservable when the yaw rate
approaches zero, and usually provide noisier estima-
tions. Nevertheless, kinematic models are more suit-
able for transient maneuvers and they work well in the
nonlinear region of the tire [8]. Some examples of
kinematic models are shown in [9], where a simple yet
effective logic is adopted to correct the unobservabil-
ity and prevent possible sideslip angle drifting in
straight roads due to yaw rate and lateral acceleration
sensor offsets (which are unavoidable). Selmanaj et al.
[9] correct the approach presented in [8] with a
heuristically calculated term. An heuristic function is
evaluated through the use of bivariate Gaussian
distributions and a set of three signals (steering angle,
yaw rate and sideslip rate) with their derivatives.
Because of their disadvantages, it is infrequent to
come across estimators purely based on kinematic
models. Instead, either a dynamic model is used, or a
combination of kinematic and dynamic models.
The second category is very frequently adopted in
the literature, as it is based on the equilibrium
equations of the vehicle, often described by means
of a single-track model [8,10–18]. However, exam-
ples with a four-wheel configuration vehicle model
can also be found [19–24]. Often an Extended Kalman
filter [25] is used, but in [22–24] the Unscented version
of the Kalman filter is applied when the model
becomes strongly nonlinear. Dynamic approaches
are very sensitive to the tire model adopted within
the estimator. Some papers choose a linear tire model
[10,11,13,14], while others use Pacejka’s Magic
Formula [12,15,23] and a significant group adopt
even different tire models such as the Dugoff tire
model [19,22,24] and the Rational tire model [21,26].
In [10], an extended adaptive Kalman filter is used,
integrated with an estimation/adaptation algorithm for
the tire parameters. On the other hand, [21] proposes a
dual extended Kalman filter, where two Kalman filters
are used in a recursive way. The first one estimates
vehicle parameters that are fed to the second Kalman
filter which estimates the vehicle state. In [13] another
interesting two-stage structure is investigated. The
first stage is an Extended Kalman filter which provides
information about the vehicle state, and the second one
employs the Extended Kalman filter results to obtain
an estimation of the tire parameters. In [23], a new
approach for the vehicle state estimation based on a
detailed vehicle model and an Unscented Kalman filter
is presented. The mathematical model relies on a
planar two-track model extended by an advanced
vertical tire force calculation method. Doumiati et al.
[22] and [24] also apply the Unscented version of the
Kalman filter, but [22] also employs a set of deflection
sensors installed on the suspension system, which is
not a common sensor set for a passenger vehicle. [24]
estimates the tire-road friction coefficient by intro-
ducing it in the state vector. The authors of [15]
employ the Magic Formula in an innovative way,
coupled with an Extended Kalman filter and with the
addition of new tuning parameters which control the
shape of lateral tire forces. Interestingly, [26] intro-
duces cornering stiffnesses directly in the state vector,
hence obtaining their real-time estimation. [27]
applies the same idea to the parameters of the Rational
tire model.
The third category features a mixed approach,
employing a well-thought combination of kinematic
and dynamic modeling. The study presented in [28]
combines a kinematic approach with a dynamic
formulation to overcome the problem of the unob-
servability when the yaw rate is around zero. In [29]
the kinematic and dynamic models are cleverly
combined to make the most of each formulation, and
a steady-state index is defined to properly weight the
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outputs of the two models. Moreover, a simple
cornering stiffness identification method is proposed.
The authors of [30] apply their model to an electric
vehicle featuring multi-sensing hub units, which are
sensor units able to provide a very accurate measure-
ment in terms of tire lateral forces. The authors also
estimate sideslip and roll angles with a coupled
approach including a Recursive Least Squares (RLS)
method and a Kalman filter. In [31] a mixed approach
is applied, with a mostly kinematic-based model: an
algorithm to estimate tire-road friction is presented,
that is activated only if the lateral velocity derivative
is sufficiently high and the yaw rate is above a
predefined threshold, or when the ESC is on.
This paper proposes a new method to estimate
vehicle sideslip angle, with the following novelties:
•the development of an innovative combination of
kinematic and dynamic modeling, denoted as
cross-combined approach, which introduces a
mutual influence between the two approaches;
•the development of an Unscented Kalman filter
framework based on the cross-combined approach
and a modified Dugoff tire model;
•the validation of the proposed approach on a large
set of experimental data acquired on a rear-wheel-
drive motorsport car equipped with an optical
sensor for the measurement of sideslip angle, along
with an Inertial Measurement Unit (IMU), wheel
speed sensors, and a steering wheel sensor;
•a comparison between the proposed method and a
traditional method for sideslip angle estimation.
The remainder of the paper is organized as follows.
Section 2provides a description of the Kalman filter
and its main variants. Section 3describes the pro-
posed estimator with specific focus on the kinematic
filter, the dynamic filter, and the concept of cross-
combination. Section 4presents results based on
experimental tests in which the proposed approach
is compared to a traditional one. Section 5draws the
main conclusions.
2 The Kalman filter (KF)
The Kalman filter is named after Rudolph E. Kalman,
who first described a new solution to the discrete-data
linear filtering problem in 1960 [25]. Theoretically,
the Kalman filter is an estimator for the linear
quadratic Gaussian problem, i.e. estimating the instan-
taneous state of a linear dynamic system perturbed by
Gaussian white noise, by using measurements linearly
related to the state, also corrupted by Gaussian white
noise. The resulting estimator is statistically optimal
with respect to any quadratic function of the estima-
tion error [32]. The name ‘‘filter’’ derives from the fact
that, practically, it allows to remove the known and
unknown noise components in the measurements and
in the description of the system. Several versions of
the KF exist. Some of the most important versions are
described here, including versions that allow to deal
with nonlinear systems, as is the case of vehicle
dynamics.
2.1 The basic Kalman filter (KF) for linear
systems
The original Kalman filter formulation is designed to
deal with linear systems, estimating the state x2RN
of the observed system. The generic linear process can
be described in discrete-time form by means of
process and measurement equations, respectively:
xk¼Axk1þBukþWwk1
zk¼HxkþVvk
ð1Þ
where xkand ukare respectively the state vector and
the input at the generic time step k,Athe dynamic
matrix, Bthe control matrix, Hthe measurement
matrix, Wthe process noise matrix, Vthe measure-
ment noise matrix. xkis a column vector with N
elements. wkand vkrepresent the process and mea-
surement noise with Qand Rbeing the correspondent
covariance matrices. The matrices A;B;H;W;V
allow to relate state, input, and noises to the
subsequent (propagated) state and to the measure-
ments. The equations of the recursive algorithm are
divided into time update equations and measurement
update equations. The time update equations describe
the evolution of the system a-priori, i.e., only based on
the model of the system:
^
x
k¼A^
xk1þBuk
P
k¼APk1ATþWQWTð2Þ
where ^
x
kindicates the a-priori estimated state at time
step k,Pk1the state covariance at time step (k1),
P
kthe a-priori state covariance at time step k.
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The measurement update equations allow to correct
the a-priori estimation based on the gathered mea-
surement, hence providing the a-posteriori estimation
of the state:
Kk¼P
kHTHP
kHTþVRVT

1
^
xk¼^
x
kþKkzkH^
x
k

Pk¼IKkHðÞP
k
ð3Þ
where ^
xkis the estimated state at time step k,Pkthe
covariance at time step k, and Kkis denoted as Kalman
gain. Note that covariance matrices P,Wand Vare
semi-positive definite.
2.2 Extended Kalman filter (EKF) and Unscented
Kalman filter (UKF)
The main drawback of the basic Kalman filter is its
suitability for the estimation of the state of a process
governed only by a linear set of stochastic difference
equations. Yet, it is well known that real processes are
often far from linear. For nonlinear systems, the so-
called Extended Kalman filter can be adopted
[25,33,34]. Equation (1) can be generalized as:
xk¼fx
k1;uk1;wk1
ðÞ
zk¼hx
k;vk
ðÞ ð4Þ
which entails generic functions fand h. The key idea
of the EKF is to linearize the system, at each time step,
around the estimated state of the system at the previous
time step:
Aki;j½
¼ofi½
oxj½
^
xk1;uk1;0ðÞ
Wki;j½
¼ofi½
owj½
^
xk1;uk1;0ðÞ
Hki;j½
¼ohi½
oxj½
^
x
k;0

Vki;j½
¼ohi½
ovj½
^
x
k;0

ð5Þ
where Aki;j½
,Wki;j½
,Hki;j½
,Vki;j½
represent the generic
element of, respectively, Ak,Wk,Hk,Vk, on row iand
column j, and fi½
,hi½
,xi½
,vi½
,wi½ represent the i-th
element of, respectively, f,h,x,v,w. Essentially
Eq. (5) contains the Jacobian matrices of the partial
derivatives of the process and measurement functions
with respect to the state and the noise. As a result, the
following time update equations can be used for the a-
priori evolution of the EKF (note that in these
expressions the two covariance matrices are also
assumed non-constant, for more generality):
^
x
k¼f^
xk1;uk1;0ðÞ
P
k¼AkPk1AT
kþWkQk1WT
k
ð6Þ
and the a-posteriori equations are:
Kk¼P
kHT
kHkP
kHT
kþVkRkVT
k

1
^
xk¼^
x
kþKkzkh^
x
k;0

Pk¼IKkHk
ðÞP
k
ð7Þ
Despite the EKF is an elegant, efficient and
recursive way to estimate the state of a nonlinear
system, it has important flaws:
•The calculation of Jacobian matrices may be
computationally expensive, especially in situations
where the partial derivatives are to be calculated
online at each time step.
•The linearized transformation provides good
results only when the error propagation can be
relatively well approximated by a linear model.
This problem is deeply discussed in [35,36].
To overcome the drawbacks related to the lin-
earization process, many studies have been carried
out. Attempts include the development of high-order
Kalman filters [37] and more sophisticated versions of
the EKF [38]. A widely appreciated solution is the
Unscented Kalman filter (UKF), which provides a
relatively simple and immediate way to propagate
mean and covariance variables of random signals
through a nonlinear transformation, without the need
for linearization. The UKF is founded on the intuition
that it is easier to approximate a probability distribu-
tion than it is to approximate an arbitrary nonlinear
function or transformation [39]. The state distribution
is represented with a set of deterministically chosen
sample points, denoted as ‘‘sigma-points’’. The sigma-
points are a set of 2Nþ1 potential guesses of the state
of the system, with a given mean and covariance
reflecting the same characteristics of the state to be
estimated. In case of additive process and measure-
ment noise, the 2Nþ1 sigma-points can be obtained
as:
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X0ðÞ
k1¼^
xk1
XiðÞ
k1¼^
xk1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NþwðÞPk1
p

ifor i ¼1;2;...;N
XiðÞ
k1¼^
xk1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NþwðÞPk1
p

iNfor i ¼Nþ1;Nþ2;...;2N
ð8Þ
where in general XiðÞ
krepresents the i-th sigma-point
(i¼0;1;2;...;2N) at time step k, which is a column
vector with Nelements—just as the state vector, for
which a sigma-point is a potential guess. The notation
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NþwðÞPk1
p

istands for the i-th column of the
argument, which is calculated through the Cholesky
decomposition. The UKF parameter wis defined as
w¼r2NþjðÞNwhere rand jare further UKF
parameters, discussed below. At each time step, every
sigma-point is propagated through the nonlinear
dynamic function f:
^
XiðÞ
k¼fX
iðÞ
k1;uk1;0

for i ¼0;1;2;...;2N
ð9Þ
where ^
XiðÞ
kis the i-th a-priori propagated sigma-point,
so it is also a column vector with Nelements. Thus, the
a-priori estimated mean and covariance can be com-
puted as [40–42]:
^
x
k¼X
2N
i¼0
WiðÞ
m
^
XiðÞ
k
P
k¼X
2N
i¼0
WiðÞ
c
^
XiðÞ
k^
x
k
no
^
XiðÞ
k^
x
k
no
T
þQ
ð10Þ
based on the appropriate weights
W0ðÞ
m¼w
wþN
W0ðÞ
c¼w
wþNþ1r2þc

WiðÞ
m¼WiðÞ
c¼1
2wþNðÞ
for i ¼1;2;...;2N
ð11Þ
For the calibration phase of the filter:
•jrepresents the tailedness of the probability
distribution, a default starting point can be j¼0
[43];
•0\r\1;
•c[0 (for a Gaussian distribution the optimal
value is c¼2[41]).
The measurement update equation set is:
^
ZiðÞ
k¼h^
XiðÞ
k;0

for i ¼0;1;2;...;2N
^
z
k¼X
2N
i¼0
WiðÞ
m
^
ZiðÞ
k
Pzk¼X
2N
i¼0
WiðÞ
c
^
ZiðÞ
k^
z
k
no
^
ZiðÞ
k^
z
k
no
TþR
Pxkzk¼X
2N
i¼0
WiðÞ
c
^
XiðÞ
k^
x
k
no
^
Z
k^
z
k

T
Kk¼PxkzkP1
zk
^
xk¼^
x
kþKkzk^
z
k

Pk¼P
kKkPzkKT
k
ð12Þ
where ^
ZiðÞ
kis the i-th a-priori measurement vector
corresponding to the i-th a-priori propagated sigma-
point ^
XiðÞ
k, and Pzkand Pxkzkare the measurement
covariance matrix and the cross-covariance matrix,
respectively. The above version of the UKF is
exploited in this paper. However, for completeness,
it is worth noting that in the general case of non-
additive process and measurement noise, the UKF
entails an augmented state vector xa
kand covariance
matrix Pa
k, defined as:
xa
k¼
xk
wk
vk
2
6
43
7
5
Pa
k¼
Pk00
0Q0
00R
2
6
43
7
5
ð13Þ
The corresponding state update and measurement
update equations are reported in [44].
3 Design of the estimator
The proposed estimator is based on an innovative
combination of a standard kinematic filter and a novel
dynamic filter. The following subsections describe in
detail: (1) the kinematic filter; (2) the dynamic filter;
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(3) the cross-combined method to merge kinematic
and dynamic filters.
3.1 Kinematic filter
The kinematic filter only exploits kinematic quantities
related to vehicle motion, so that no tire model is
needed to estimate the lateral velocity, hence the
sideslip angle. By simply manipulating the expres-
sions of longitudinal and lateral acceleration as
functions of longitudinal velocity, lateral velocity,
and yaw rate, and by choosing the state as
x¼vxvy

T, the ideal (noise-free) process is
described by [7]:
_
vx
_
vy

¼0r
r0

vx
vy

þ10
01

ax
ay
 ð14Þ
Regarding the description of the noise, the insight-
ful yet simple approach described in [28] was chosen
since it allows to include directly (in Q) the measure-
ment noise covariances of yaw rate, lateral and
longitudinal acceleration. As a result, the process is
described by:
_
vx
_
vy

¼0r
r0

vx
vy

þ10
01

ax
ay

þvy10
vx01

wr
wax
way
2
43
5ð15Þ
which uses the measurement of yaw rate directly in the
dynamic matrix, and the measurements of longitudinal
and lateral accelerations as inputs. The actual mea-
surement equation of the filter is straightforward:
z¼Hx ¼10½
vx
vy
 ð16Þ
Because both the process and the measurement
equations are linear, the state can be estimated with the
basic KF (2–3) using the matrices:
A¼0r
r0

B¼10
01

H¼10½
W¼vy10
vz01

ð17Þ
The forward Euler method is applied to perform the
calculation in discrete time. As already assessed in [8],
kinematic approaches are unobservable when the yaw
rate is close to zero. A simple reset logic is applied to
correct lateral velocity estimation, by forcing vyto
zero when the magnitude of ris sufficiently small.
Finally, at each time step, the sideslip angle is
calculated, by definition, as b¼arctan vy=vx

.
3.2 Dynamic filter
A dynamic filter is based on the equilibrium equations
of the vehicle, which need a constitutive law (tire
model) to explicitly express the tire forces as functions
of relevant parameters. The subsequent paragraphs
describe respectively: i) vehicle model and tire model;
ii) the UKF implementation of the filter.
3.2.1 Vehicle model and tire model
A double-track vehicle model is adopted, as shown in
Fig. 1. The lateral equilibrium equation and the yaw
balance equation for this model are:
may¼m_
vyþvxr

¼Fy11 cos dðÞþFy12 cos dðÞ
þFy21 þFy22
Jz_
r¼Fy11 cos dðÞaþFy11 sin dðÞ
tw1
2þFy12 cos dðÞa
Fy12 sin dðÞ
tw1
2Fy21bFy22 b
ð18Þ
Note that the steering angles of the front left and
front right wheels are assumed to be the same (d) and
that because longitudinal interactions typically have
small effects, these are neglected in the lateral
dynamics equations. On the other hand, the proposed
double-track schematization allows to consider effects
such as individual wheel slip angles and lateral load
transfers. These effects help grasping a fairly accurate
vehicle behavior, benefiting the estimator accuracy,
unlike the single-track vehicle model adopted in many
other estimators.
The lateral forces are expressed by a nonlinear tire
model, considering key aspects of tire behavior such as
nonlinearity, saturation, and dependency on the ver-
tical load. In particular, the version of the Dugoff tire
model presented in [45] is chosen, as it presents a very
similar behavior to the well-known—yet more
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complex—Magic Formula. For a single wheel, the tire
lateral force can be expressed as:
Fy¼Caðtan aÞpkðÞGað19Þ
where pkðÞis a nonlinear function defined as:
pkðÞ¼ 2kðÞkk\1
1k1

with k¼lmaxFz
2Catan a
jj
ð20Þ
and Gais a correction term, function of the wheel slip
angle and the tire-road friction coefficient:
Ga¼lmax 1:6ðÞtan aþ1:155 ð21Þ
However, with this formulation of Ga[45], same
values of abut with opposite signs would not result in
the same magnitude of lateral force (note that this
formulation does not account for camber). To correct
that, in this paper Eq. (21) is modified as follows:
Ga¼lmax 1:6ðÞjtan ajþ1:155 ð22Þ
which ensures a symmetrical behavior for positive and
negative values of a:
The selected tire model also requires the vertical
load on each tire:
Fz11 ¼mgb
2lmaxh
2lmB1ayþ1
4qv2
xCz1Sa
Fz12 ¼mgb
2lmaxh
2lþmB1ayþ1
4qv2
xCz1Sa
Fz21 ¼mga
2lþmaxh
2lmB2ayþ1
4qv2
xCz2Sa
Fz22 ¼mga
2lþmaxh
2lþmB2ayþ1
4qv2
xCz2Sa
ð23Þ
Each expression in Eq. (23) includes, in order:
static load contribution; longitudinal load transfer
contribution; lateral load transfer contribution; down-
force contribution. For the lateral load transfer,
according to [46], it is:
B1¼1
tw1
b
ld1þKr1
Kr1þKr2
hd1þd2d1
ðÞa
l

B2¼1
tw2
a
ld2þKr2
Kr1þKr2
hd1þd2d1
ðÞa
l

ð24Þ
where Kr1and Kr2are the roll stiffness values of,
respectively, the front and rear axle, and d1and d2are
the heights of the roll centers of, respectively, the front
and rear axle.
Finally, the congruence equations [2] provide the
relationship between kinematic quantities and slip
angles:
a11 ¼darctan vyþra
vxrtw1=2

a12 ¼darctan vyþra
vxþrtw1=2

a21 ¼arctan vyrb
vxrtw2=2

a22 ¼arctan vyrb
vxþrtw2=2

ð25Þ
From the above equations, it is clear that the vehicle
longitudinal velocity is required. That is estimated
based on measurements including wheel speed sensors
and accelerometers, as discussed in Sect. 4.
Fig. 1 Double track vehicle model (adapted from [2])
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3.2.2 UKF filter implementation
Based on the vehicle model in Eq. (18), the state
vector is chosen as x¼vyr

T,soN¼2. The input
vector is u¼d½and the measurement vector is z¼
ra
y

T(both variables can be easily measured with
standard sensors). By discretizing Eq. (18) with the
forward Euler method, the system dynamics is
expressed as:
together with Eqs. (19–25). Regarding the relation-
ships between zand x,ris straightforward because it
appears directly both in zand x, while aycan be related
to the vehicle state at each time step through:
ayk¼1
mFy21kþFy22kþFy11kþFy12k

cos dk
ðÞ
hi
ð27Þ
together with Eqs. (19–25). The matrices Qand Rare:
Q¼v2
y;s0
0r2
s
 ð28Þ
R¼r2
m;s0
0a2
y;m;s
 ð29Þ
where vy;sis the standard deviation of the process noise
on vy,rsis the standard deviation of the process noise
on r,rm;sis the standard deviation of the measurement
noise on r, and ay;m;sis the standard deviation of the
measurement noise on ay.
Based on Eq. (8), the 2Nþ1¼5 sigma-points are:
X0ðÞ
k1¼^
xk1
X1ðÞ
k1¼^
xk1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NþwðÞPk1
p

1
X2ðÞ
k1¼^
xk1þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NþwðÞPk1
p

2
X3ðÞ
k1¼^
xk1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NþwðÞPk1
p

3
X4ðÞ
k1¼^
xk1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
NþwðÞPk1
p

4
ð30Þ
where the square root of a matrix can be calculated
with the Cholesky factorization. The estimated state
vector at each time step can then be obtained using
Eqs. (9–12), noting that for each sigma-point Eq. (9)
is Eq. (26), while Eq. (27) is used in the first of
Eq. (12). Finally, at each time step, again
b¼arctan vy=vx

.
3.3 Cross-combination
The described kinematic filter and dynamic filter run at
the same time. The final estimate of the sideslip angle
is calculated as a weighted average of the sideslip
angle obtained through the two filters according to the
following procedure (Fig. 2): (1) the measured value
of a
y
is stored in a 0.1 s buffer; (2) a steady-state index
is calculated, mainly depending on the Root Mean
Square (RMS) of the stored samples of ay; (3) the
steady-state index is used to compute a weight for the
kinematic contribution, wkin, and a weight for the
dynamic contribution, wdyn, with wkin þwdyn ¼1. The
rationale is that, as suggested in [29], kinematic and
dynamic models are better suited for, respectively,
transient and steady-state conditions.
For ay
1, the root mean square (RMS) value of
the lateral acceleration is computed over the 0.1 s
buffer (e.g., 10 samples for a frequency of 100 Hz). A
membership function is used to calculate the value of
the steady-state index: if the computed RMS value is
lower than 0.4 m/s
2
, meaning that aydoes not vary
vykþ1¼vykþFy11kþFy12k

cos dk
ðÞþFy21kþFy22k
mvxkrk

Dt
rkþ1¼rkþFy11kþFy12k

cos dk
ðÞaþFy11kFy12k

sin dk
ðÞ
tw1
2Fy21kþFy22k

b
hi
Dt
Jz
ð26Þ
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significantly, the steady-state index is set equal to 1;
between 0.4 m/s
2
and 0.6 m/s
2
the membership func-
tion produces a linearly variable output from 1 to 0; for
values larger than 0.6 m/s
2
the maneuver is assumed to
be in transient conditions, thus the steady-state index
is set to 0. Instead, if ayis within ±1m/s
2
, the steady-
state index is set to 1, to prevent possible fluctuations
of the sideslip angle estimation in nearly straight-line
conditions due to the kinematic filter. wdyn is 1 when
the steady-state index is 1, while it varies linearly from
1 to 0.7 corresponding to values of the steady-state
index from 1 to 0.
This paper also proposes, for the first time, to cross-
combine the variables in common between the output
of one filter and the input of the other. Precisely the
variables in common are rand vx, in that:
•the kinematic filter needs ras input and produces vx
as output;
•the dynamic filter needs vxas input and produces r
as output.
Normally, for the kinematic filter, ris taken directly
from a sensor, and for the dynamic filter, vxis
calculated as a function of the measured wheel speeds.
While both values are affected by sensor noise, the
values for the same quantities obtained as output of
each filter are expected to be more accurate. The
kinematic filter should produce a better estimation of
vxthan the value calculated through wheel speed
sensors, and the dynamic filter should produce a better
estimation of rthan the measured value obtained
through the sensor – note that unmodeled effects, such
as pitch and roll motion, do affect the accuracy of the
yaw rate measurement. So, these values of vxand rcan
be used as inputs of the kinematic and dynamic filter,
respectively. This new idea, denoted as cross-combi-
nation and shown in Fig. 3, has the potential to
improve the accuracy of the estimation of vyand thus
of the sideslip angle.
4 Results
The proposed cross-combined filter was tested on a
large set of data obtained through a performance-
oriented rear-wheel-drive car, mounting front tires
Fig. 2 (top) Schematic of the methodology used to calculate the weights of the kinematic and the dynamic filter; (bottom) Detail of the
membership functions
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30/68 (tread band width in cm/external tire diameter in
cm) on an R18 (radius in inches) rim, and rear tires
31/71 mounted on an R18 rim. The vehicle (Fig. 4)
was equipped with:
•an Inertial Measurement Unit (IMU) OXTS 3000
[48], providing longitudinal acceleration, lateral
acceleration, yaw rate, with the following main
specifications: accelerometer, bias stability 2 lg,
Servo technology, range 10 g; gyroscope, bias
stability 2/h, MEMS technology, range 100/s;
•four wheel speed sensors Bosch HA-M [49] with
the following main specifications: max frequency
4.2 kHz, accuracy repeatability of the falling edge
of tooth \4%;
•a steering angle sensor Bosch LWS [50] with the
following main specifications: range 780,
absolute physical resolution 0.1;
•a Correvit S-Motion Type 2055A sensor [51],
providing vehicle longitudinal velocity and side-
slip angle, with the following main specifications:
range 400 km/h, linear velocity measurement
accuracy \|0.2%| (%FSO—Full Span Output),
angle resolution \0.01.
The main vehicle parameters are reported in
Table 1.
Starting from the wheel speed sensor data, two
options were considered to calculate the longitudinal
vehicle velocity. A simple and straightforward option
was to calculate the average speed of the front wheels,
because for a rear-wheel-drive car the front wheels
undergo lower slip values than the rear wheels. A more
sophisticated solution was actually implemented.
Denoting with vM;ij the measured speed at wheel ij,
estimates of the vehicle longitudinal velocity, ^
vx;ij, can
Fig. 3 Schematic of the proposed filtering approach with cross-combination
Fig. 4 Test vehicle
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be obtained from each wheel based on rigid body
kinematics [47] as follows:
^
vx;11 ¼vM;11 cos dþrtw1
2
^
vx;12 ¼vM;12 cos drtw1
2
^
vx;21 ¼vM;21 þrtw2
2
^
vx;22 ¼vM;22 rtw2
2
ð31Þ
Compared to the calculation of the average of ^
vx;ij,
this allows to depurate: (1) the yaw rate effect due to
the wheels being located with a lateral offset with
respect to the vehicle longitudinal axis; (2) the steering
angle effect, as the measured wheel speed is aligned
with the wheel and not necessarily the vehicle
longitudinal axis. Then, the following logic is imple-
mented to identify, among the four wheels, the one
with the lowest slip, based on the measurement of ax:
•if ax[0:5 m/s
2
, i.e. the vehicle accelerates, wheel
speeds are larger than the longitudinal vehicle
speed so the lowest value is the closest to the actual
vehicle speed: the vehicle longitudinal speed is
estimated as min ^
vx;11;^
vx;12;^
vx;21;^
vx;22

•if ax\0:5 m/s
2
, i.e. the vehicle decelerates,
wheel speeds are smaller than the longitudinal
vehicle speed so the largest value is the closest to
the actual vehicle speed: the vehicle longitudinal
speed is estimated as max ^
vx;11;^
vx;12;^
vx;21;^
vx;22

•for low values of acceleration, i.e. if ax
jj
0:5m/
s
2
, the vehicle longitudinal speed is estimated
through a weighted average of ^
vx;ij., with weights
calculated according to [9].
Thanks to the availability of the sideslip angle
measurement, the root mean square error (RMSE)
method was selected as the performance index for
assessing the quality of the estimation:
be¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
nX
n
k¼1
^
bkbk

2
sð32Þ
where nis the number of time samples. The proposed
filtering technique was also compared to a traditional
technique using a linear dynamic model, for which
equations are reported in the Appendix. In the
following figures, the two techniques are referred to
as, respectively, KF (Kalman filter, linear dynamic)
and UKF-CC (Cross-combined kinematic and UKF
dynamic).
Table 1 Vehicle
parameters Quantity Symbol Value Unit
Mass m1345 kg
Wheelbase l2.713 m
Front semi-wheelbase a1.250 m
Front track width t
w1
1.726 m
Rear track width t
w2
1.710 m
Height of the center of mass h0.380 m
Front roll center height d
1
0.01 m
Rear roll center height d
2
0.015 m
Yaw moment of inertia J
z
1869.4 kg m
2
Front downforce coefficient C
z1
0.35 –
Rear downforce coefficient C
z2
0.75 –
Frontal area Sa2.05 m
2
Front axle relative roll stiffness K
r1
/(K
r1
?K
r2
)0.5287 –
Rear axle relative roll stiffness K
r2
/(K
r1
?K
r2
)0.4713 –
Dugoff tire parameter – front axle Ca160,000 N
Dugoff tire parameter – rear axle Ca2105,000 N
Dugoff friction coefficient lmax 1.4 –
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Figures 5and 6depict the measured and estimated
sideslip angle along four race tracks. Each figure cor-
responds to a single lap, which is representative of the
corresponding track since each filter behaved consis-
tently along all laps. These experimental tests include
a variety of testing conditions: multiple runs were
carried out, with the same vehicle, tires, and
equipment, in European and Asian race tracks, in
different seasons of the year, on dry tarmac. For all of
the tracks, the KF is able to follow the general trend
but with significant discrepancies all round, up to
around 5 deg. Instead, the UKF ?CC provides a
much more reliable and smooth tracking. In terms of
RMSE, the KF is normally above 1 deg, while the
Fig. 5 Measured and estimated sideslip angle for one lap of: (left) Track 1; (right) Track 2
Fig. 6 Measured and estimated sideslip angle for one lap of: (left) Track 3; (right) Track 4
Table 2 Performance comparison of a traditional filter (linear dynamic) and the proposed filter
Filter b
e
(deg)
Track 1 Track 2 Track 3 Track 4 Average
Kalman filter, linear dynamic (KF) 1.08 1.14 0.98 1.27 1.12
Cross-combined kinematic and UKF dynamic (UKF-CC) 0.47 0.64 0.48 0.54 0.53
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UKF ?CC settles on an average value of 0.53 deg,
with an improvement of around 50% with respect to
the KF (Table 2).
Figure 7compares different methods to estimate vx,
against the measured value. While all of the methods
perform fairly well—because wheel speeds are rather
informative measurements anyway—important
remarks can be made. For the method using the
average speed of the front wheels, the rationale was to
pick the wheels with lower slips for a rear-wheel-drive
car. However, that is no longer ideal in braking
scenarios, when the front wheels undergo significant
slips, even more than for the rear wheels. This is
evident in Fig. 7just before 296 s. On the other hand,
the method using all of the wheel speeds and ax
provides a more reliable result, though sometimes
affected by discontinuities due to the rule-based nature
of the method. The vxoutput of the kinematic filter,
instead, is the smoothest signal and is the closest to the
measured profile. This further supports the idea of the
cross-combination, because a better vxis given as
input to the dynamic filter, contributing to the quality
of the estimation of the sideslip angle.
5 Conclusion
This paper presented a novel approach for the
estimation of vehicle sideslip angle. The analyses
presented in this paper lead to the following main
conclusions:
•the kinematic and dynamic models for the estima-
tion of sideslip angle can be cross-combined, by
feeding part of the output of each filter as input to
the other filter;
•the cross-combination allows to further improve
the estimation of the vehicle longitudinal velocity
compared to current state-of-the-art techniques, in
turn benefiting the precision of the sideslip angle
estimation;
•the modified Dugoff tire model is a simple yet
effective constitutive model and produces the same
lateral force—slip angle behavior regardless of the
sign of the slip angle;
•the proposed cross-combined kinematic and UKF
dynamic filter allows to estimate the vehicle
sideslip angle with an average RMSE of around
0.5 deg on experimental data.
Future developments will deal with: (1) tire longi-
tudinal dynamics and combined interactions; (2)
effects of roll and bank angles; (3) effects of tire
temperature; (4) the potential investigation of method-
ologies for coping with variable road friction condi-
tions; (5) further experimental tests with possible real-
time implementation of the filter.
Funding Open access funding provided by Universita
`degli
Studi di Padova within the CRUI-CARE Agreement.
Fig. 7 Comparison of longitudinal vehicle speed estimation
methods, Track 4: measured speed through optical sensor (blue),
average of the front wheel speeds (red), technique inspired to [9]
explained at the beginning of Sect. 4(yellow), kinematic filter
used in the proposed method (purple). Left: entire lap; Right:
detail
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Declarations
Conflict of interest The authors declare that they have no
conflict of interest.
Open Access This article is licensed under a Creative
Commons Attribution 4.0 International License, which
permits use, sharing, adaptation, distribution and reproduction
in any medium or format, as long as you give appropriate credit
to the original author(s) and the source, provide a link to the
Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are
included in the article’s Creative Commons licence, unless
indicated otherwise in a credit line to the material. If material is
not included in the article’s Creative Commons licence and your
intended use is not permitted by statutory regulation or exceeds
the permitted use, you will need to obtain permission directly
from the copyright holder. To view a copy of this licence, visit
http://creativecommons.org/licenses/by/4.0/.
Appendix 1
The traditional KF approach used as a comparison in
this paper is a dynamic model. It uses a single-track
vehicle model and a linear tire model, similar to
[26,29,30]. The state vector, input vector, and
measurement vectors are the same as for the proposed
dynamic model, i.e. x¼vyr

T,u¼d,
z¼ra
y

T. The filter equations in discrete time
are the following:
vykþ1¼1Dt
mvxk
C1þC2
ðÞ

vykþDt
mvxk
C1aC2bðÞDtvxk

rkþDtC1
mdk
rkþ1¼Dt
Jzvxk
C1aC2bðÞ

vykþ1Dt
Jzvxk
C1a2þC2b2


rkþDtC1a
Jz
dk
8
>
>
>
<
>
>
>
:
zk¼
rk¼rk
ayk¼C1þC2
mvxk
C1aþC2b
mvxk
rkþC1
mdk
8
<
:ð33Þ
Because of the adopted tire model, the system is
linear, hence the basic Kalman filter may be used.
The values for C1and C2are respectively
110,000 N/rad and 192,500 N/rad. Finally, the
expression of the matrices Qand Rare the same seen
in Eqs. (28) and (29).
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