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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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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:

xk¼Axk1þBukþWwk1

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^

xk1þBuk

P

k¼APk1ATþWQWTð2Þ

where ^

x

kindicates the a-priori estimated state at time

step k,Pk1the state covariance at time step (k1),

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

kþKkzkH^

x

k

Pk¼IKkHðÞP

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

k1;uk1;wk1

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

^

xk1;uk1;0ðÞ

Wki;j½

¼ofi½

owj½

^

xk1;uk1;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^

xk1;uk1;0ðÞ

P

k¼AkPk1AT

kþWkQk1WT

k

ð6Þ

and the a-posteriori equations are:

Kk¼P

kHT

kHkP

kHT

kþVkRkVT

k

1

^

xk¼^

x

kþKkzkh^

x

k;0

Pk¼IKkHk

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

k1¼^

xk1

XiðÞ

k1¼^

xk1þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

NþwðÞPk1

p

ifor i ¼1;2;...;N

XiðÞ

k1¼^

xk1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

NþwðÞPk1

p

iNfor 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

NþwðÞPk1

p

istands for the i-th column of the

argument, which is calculated through the Cholesky

decomposition. The UKF parameter wis deﬁned as

w¼r2NþjðÞNwhere 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

k1;uk1;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

wþNþ1r2þc

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

Kk¼PxkzkP1

zk

^

xk¼^

x

kþKkzk^

z

k

Pk¼P

kKkPzkKT

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

þvy10

vx01

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

W¼vy10

vz01

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

2Fy21bFy22 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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Meccanica

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 2kðÞkk\1

1k1

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

2lmaxh

2lmB1ayþ1

4qv2

xCz1Sa

Fz12 ¼mgb

2lmaxh

2lþmB1ayþ1

4qv2

xCz1Sa

Fz21 ¼mga

2lþmaxh

2lmB2ayþ1

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

hd1þd2d1

ðÞa

l

B2¼1

tw2

a

ld2þKr2

Kr1þKr2

hd1þd2d1

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 ¼darctan vyþra

vxrtw1=2

a12 ¼darctan vyþra

vxþrtw1=2

a21 ¼arctan vyrb

vxrtw2=2

a22 ¼arctan vyrb

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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Meccanica

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 u¼d½and 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

k1¼^

xk1

X1ðÞ

k1¼^

xk1þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

NþwðÞPk1

p

1

X2ðÞ

k1¼^

xk1þﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

NþwðÞPk1

p

2

X3ðÞ

k1¼^

xk1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

NþwðÞPk1

p

3

X4ðÞ

k1¼^

xk1ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

NþwðÞPk1

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

mvxkrk

Dt

rkþ1¼rkþFy11kþFy12k

cos dk

ðÞaþFy11kFy12k

sin dk

ðÞ

tw1

2Fy21kþFy22k

b

hi

Dt

Jz

ð26Þ

123

Meccanica

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

123

Meccanica

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

123

Meccanica

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 drtw1

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

^

bkbk

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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Meccanica

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

123

Meccanica

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

123

Meccanica

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:

vykþ1¼1Dt

mvxk

C1þC2

ðÞ

vykþDt

mvxk

C1aC2bðÞDtvxk

rkþDtC1

mdk

rkþ1¼Dt

Jzvxk

C1aC2bðÞ

vykþ1Dt

Jzvxk

C1a2þC2b2

rkþDtC1a

Jz

dk

8

>

>

>

<

>

>

>

:

zk¼

rk¼rk

ayk¼C1þC2

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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