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A benchmark study on the modelbased estimation
of the gokart sideslip angle
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ICMSQUARE 2021
Journal of Physics: Conference Series 2090 (2021) 012156
IOP Publishing
doi:10.1088/17426596/2090/1/012156
1
A benchmark study on the modelbased estimation
of the gokart sideslip angle
M D’Inverno1, V M Arricale1, A Zanardi2, E Frazzoli2,
A Sakhnevych1, F Timpone1
1Dept. of Industrial Engineering, University of Naples ”Federico II”, Naples, Italy
2Institute for Dynamic Systems and Control, ETH Zurich, Zurich, Switzerland
Email: vincenzomaria.arricale@unina.it
Abstract. Nowadays, the active safety systems that control the dynamics of passenger cars
usually rely on realtime monitoring of vehicle sideslip angle (VSA). The VSA can’t be measured
directly on the production vehicles since it requires the employment of highend and expensive
instrumentation. To realiably overcome the VSA estimation problem, diﬀerent modelbased
techniques can be adopted. The aim of this work is to compare the performance of diﬀerent
modelbased state estimators, evaluating both the estimation accuracy and the computational
cost, required by each algorithm. To this purpose Extended Kalman Filters, Unscented Kalman
Filters and Particle Filters have been implemented for the vehicle system under analysis. The
physical representation of the process is represented by a singletrack vehicle model adopting
a simpliﬁed Pacejka tyre model. The results numerical results are then compared to the
experimental data acquired within a speciﬁcally designed testing campaign, able to explore
the entire vehicle dynamic range. To this aim an electric gokart has been employed as a
vehicle, equipped with steering wheel encoder, wheels angular speed encoder and IMU, while
an Smotion has been adopted for the measurement of the experimental VSA quantity.
Keywords: Kalman ﬁlters, Particle ﬁlters, vehicle state estimation, vehicle dynamics
modeling, sideslip angle
1. Introduction
The typical active safety systems that control the dynamics of passenger cars rely on realtime
monitoring of vehicle sideslip angle (VSA), but the VSA is not measured directly because it
requires the use of highend instruments, which usually cannot be equipped in the passenger
cars due to the signiﬁcant related costs and bulky instrumentation [1]. However, this is not the
only application ﬁeld, indeed an accurate knowledge of the VSA may improve the ADAS system
or may be used to improve the trajectory in autonomous vehicle [2]. In the motorsport ﬁeld,
the application of the VSA is used to enhance the overall vehicle performance during the race.
In all these application ﬁelds the VSA estimation is increasingly diﬀused, and it is evaluated
employing diﬀerent measurements available onboard, such as wheel velocities, linear and angular
accelerations [3] [4].
The technical literature is plenty of articles about VSA estimation. There are diﬀerent
approaches to estimate the VSA starting from the observerbased methods [5], [6], [7] up to
neural network databased techniques [8], [9]. The observerbased methods are characterized
by the type of state estimator adopted, e.g. Extended Kalman ﬁlters [10] [11], in which the
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Jacobian matrix computation is required and the nonlinear problem is linearized through a
Taylor expansion. In [12] and [13], Unscented Kalman ﬁlters are adopted. The application of
such ﬁlters is widely used due to an easier implementation because they do not require the
computation (analytical or numerical) of the Jacobian matrix. In this work, another type of
state estimator is considered in addition to the ones mentioned above, consisting in the Particle
ﬁlter. The main applications of this latter involve the tracking problems, as reported in [14].
All the ﬁlters are based on the mathematical modeling of the process to estimate, a
mathematical representation of the vehicle is required in this application. Diﬀerent assumption
have to be stated in order to model the vehicle dynamics. Two kinds of vehicle models can
be found in the literature, which are denoted respectively as kinematic and dynamic [15]. The
kinematic model is concerned with the vehicle motion with no reference to forces; thus, it does
not need complex parameters such as those regarding tyres, which often are the cause of the
nonlinearity of the vehicle model. However, the main issue of VSA estimation using a kinematic
vehicle model lays in the fact that it does not work when the vehicle yaw rate is relatively small
or zero, this leads to the system unobservability, as reported in [16]. The dynamic model, on the
other hand, provides a more detailed description of the vehicle dynamics, as it is based on the
equilibrium equations. It can have diﬀerent levels of detail/complexity and hypotheses used, each
of them aﬀects the estimation accuracy. Several authors introduce simplifying hypotheses, such
as a singletrack vehicle model as in [17] and [18]. Additional assumptions may be adopted, as
the availability of the vehicle longitudinal speed or the hypothesis of small steering angles. Quite
often the equilibrium equations are coupled with a tyre model but it is not strictly necessary
as proposed in [19], there are several approaches, the most used are: linear models, Pacejka
models, rational tyre model [20] and Dugoﬀ model [21]. The use of a dynamic model can lead to
a good VSA estimation, however, the accuracy of the results strongly depends on the tyre model
parameters. Unmodeled eﬀects, such as road conditions and tyre wear, can dramatically worsen
the reliability of the estimation, meanwhile other secondary eﬀects as the type of suspension
adopted [22] and the thermodynamics of the tyre inner chamber [23] can be neglected easier in
certain hypothesis. Several authors attempt to deal with this issue employing algorithms which
provide an online update of tyre parameters as in [24] and [25]. A singletrack vehicle model and
a simpliﬁed Pacejka tyre model are adopted in this work, the main reason is that the aim of this
paper is to compare the performance of diﬀerent types of state estimators using the same plant
model. The benchmark is not only based on the estimate accuracy but also on the runtime
capability of each proposed algorithm.
An electric gokart is employed to acquire the dataset. It is equipped with sensors in order
to acquire the necessary input signals to feed the process model of the ﬁlters. In addition, an
Smotion is used to acquire the true VSA and to validate the estimate one [26].
The paper is organized as follows: an overview of the implemented state estimators is reported
in section 2, the mathematical formulation of the vehicle and tyre models is described in section
3. In Section 4, the implementation of the ﬁlters and the experimental data acquisition are
shown. In section 5, the results obtained are shown comparing them towards the experimental
VSA. Finally, the conclusions are reported in section 6.
2. State Estimators
State estimators implemented for VSA estimation are presented in this section, they are based
on the ﬁltering technique and, in particular, their discretetime form is considered. The plant
model is assumed the same for each ﬁlter. It is based on the physical equations presented in the
next section, considered in a discrete time form applying the Euler method. Basically, it is a set
of nonlinear equations that can be summarized by the following equation:
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ˆx−
k=f(ˆx+
k−1, uk−1) + wk−1
zk=h(ˆx−
k, uk) + vk
(1)
Concerning the ﬁrst equation, the term on the left side is the apriori state estimate, on the
right side there are the functions f(·) that represent the nonlinear system of equations and the
process noise term, respectively. ˆx+
k−1is the previous time aposteriori state estimate. The
second equation is a set of nonlinear and linear equations that gives in output the estimate
measurements relying on the apriori state estimate.uk−1and ukare additional inputs at the
previous and current time step, respectively. The terms wk−1and vkrepresent the noise terms; in
this work they are considered additive,white,Gaussian,zeromean and uncorrelated. The noise
covariance matrices are respectively Qand Rand are computed adopting a ”training” method,
that runs the ﬁlter and tries to minimize the prediction error through a surrogate optimization
algorithm [27].
2.1. Kalman ﬁlters
Numerous nonlinear Kalman ﬁlters can be found in literature, however all them descend from
the linear kalman ﬁlter (KF) theory [28]. The KF algorithm can be divided into two steps:
•Time update, it projects the last computed state estimate ahead in time. It is also called
prediction step.
•Measurement update, it adjusts the projected estimate by an actual measurement at that
time. It is also called correction step.
However the KF is adopted only with linear process, in this application nonlinear Kalman ﬁlters
are required. This brieﬂy introduction is necessary in order to point out the basis of the next
KFs, in fact the steps are common to every KFs but each one is characterized by a speciﬁc
strategy to deal with the nonlinearity of the process.
2.1.1. Extended Kalman Filters The EKFs attempt to approximate the nonlinear system
around the state estimate. Three types of EKFs are implemented in this work and they are
brieﬂy presented.
The FirstOrder Extendend Kalman Filter (FOEKF) is based on the linearization of the
nonlinear system around the state estimate using the ﬁrstorder Taylor expansion [28] [29]. It
approximates measurement equations by expanding it in a Taylor series around ˆx−
k, the reason
is because it is the best estimate of xkbefore the measurement at time kis taken into account.
When the aposteriori state estimate is obtained, the best estimate of xkbecomes ˆx+
k.
The basic idea of the Iterated Extended Kalman Filter (IEKF) is to reduce the linearization
error by reformulating the Taylor series expansion around ˆx+
k[28]. This process can be repeated
as many times as desired: although, for most problems the majority of the possible improvements
is obtained by relinearizing only one time. The main diﬀerence between SOEKF and IEKF
is the iteration cycle that reﬁnes the measurement update equations at time k, so the more the
measurement equations are nonlinear the more eﬀective the reﬁnement is.
The Second Order Extended Kalman Filter (SOEKF) perform a second order Taylor
expansion of the process equations, f(·) and h(·) [28].
Summing up, the EKFs rely on the computation of the Jacobian matrices (and also Hessian
matrices in SOEKF), their computation may be quite diﬃcult and and computationally
expensive especially for high nonlinear systems. These matrices can be evaluated analytically
or numerically, in the ﬁrst case the computational burden required is smaller than in the second
case.In this application the ﬁrst method is adopted.
The main EKFs ﬂaw is the linearization of a nonlinear process, in fact the linearized
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transformation is a good estimation method only when error propagation can be well
approximated by a linear model, this could force the use of very small sampling times.
2.1.2. Unscented Kalman Filters Unscented Kalman Filters (UKFs) aims to overcome the
main EKFs’ ﬂaw, it provides a simpler and more immediate way to propagate mean and
covariance of random variables through a nonlinear transformation [30] [31]. The unscented
transformation is used to calculate the statistics of a random variable which undergoes a
nonlinear transformation. The UKFs propagate the mean and covariance of the sigma points
using system nonlinear equations and the apriori state estimate is the weighted mean of them.
As well, predicted measurements for each propagated sigma point can be computed the using
the measurement equations and the predicted measurements vector is the weighted mean of
them. The implemented UKFs diﬀer in number of sigma points and/or weights formulation. An
example of sigma points are reported in (2), they refer to the General Unscented Kalman Filter.
The parameters αand kcontrol the spread of the sigma points around the mean state value.
x(0) = ¯x
x(i)= ¯x+ ˜x(i)i= 1,...,2n
˜x(i)= (pα2(n+k)Pxx)T
ii= 1, . . . , n
˜x(n+i)=−(pα2(n+k)Pxx)T
ii= 1, . . . , n
(2)
The simpliest UKF is here called Simply Unscented Kalman Filter (SUKF), it uses 2nsigma
points and also they have the same weight. Referring to the (2), in this case αis set equal to 1
and kis null.
Based on [32] and [33], it can be shown the same order of mean and covariance estimation
accuracy can be obtained by choosing (2n+ 1) sigma points instead of (2n) as before. This
type of UKF is called General Unscented Kalman Filter (GUKF). This means that the 2n
sigma points are symmetrically distributed around the mean value. The GUKF’s algorithm
and the SUKF’s algorithm are quite the same, the main diﬀerence is the number of sigma
points computed at each time step and their weight factors as shown in [28].
A new set of sigma points and weight factors can be introduced. It can be shown that if
the state vector, x, has nelements then the minimum number of sigma points, that gives the
order of estimation accuracy of the previous section, is equal to (n+ 1) [33]. These sigma points
are called simplex sigma points and for this reason the algorthm is called Simplex Unscented
Kalman Filter (SIMPUKF). However, the number of sigma points can be reduced to (n+ 1)
by choosing one of the weights to be zero. This type of ﬁlter aims to reduce the computational
eﬀort reducing the number of sigma points without losing in estimation accuracy.
The Spherical Unscented Transformation aims to rearrange the sigma points of the simplex
algorithm in order to obtain better numerical stability, as reported in [31] and [33]. The ﬁlter
based on this transformation is called Spherical Unscented Kalman Filter (SPHEUKF).
The UKFs don’t require the computation of Jacobian o Hessian, unlike the EKFs, this is
a great advantage that make them easy to implement for all the nonlinear system. The main
obstacle lays in the fact that the state covariance matrices must be semideﬁnite positive in
order to have real matrices after applying the Cholesky decomposition [28]. This goal can be
achieved tuning the noise covariance matrices.
2.2. Particle Filters
Particle Filter (PF) aims to estimate the state of a nonlinear process investigating the properties
of sets of particles rather than the properties of individual particles as in UKFs. It is a completely
nonlinear state estimator, unlike the UKFs and the EKFs presented before which are based on
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the approximation of the nonlinear system. It is a technique for implementing a recursive
Bayesian ﬁlter by Monte Carlo (MC) simulations [34]. The key idea is to represent the required
posterior density function by a set of random samples with associated weights and to compute
estimates based on these samples and weights. As the number of samples becomes very large, the
MC characterization becomes an equivalent representation to the usual functional description of
the posterior pdf [14]. Therefore, the basic idea is to randomly generate a given number Nstate
vectors based on the initial pdf p(x0), that is known. These state vectors are called particles
and are denoted as x+
0,i, with i= 1, . . . , N. At each time step, each particle is propagated
to the next time step using the system of equations f(·) in (1). Each propagated particle is
characterized by a weight that reﬂects the likelihood that the particle represents the true state
of the system. This weight factor is based on the of the measurement equation and of the pdf
of the measurement noise, the relative likelihood qithat the estimate measurement zkis equal
to a speciﬁc measurement yk, given the premise that xkis equal to the particle ˆx−
k,i, can be
computed as follows:
qi=P[(zk=yK)(xk= ˆx−
k,i)] = P[vk=yK−h( ˆx−
k,i)]
∼1
(2π)m/2R1/2exp−[yk−h(ˆx−
k,i)]TR−1
2[yk−h(ˆx−
k,i)](3)
Aresampling strategy is applied, in other words a brand new set of particles is computed:
the basic idea is to eliminate particles that have small relative likelihood and to concentrate on
particles with large one. A lot of resampling strategy can be found in literature, an exhaustive
collection is done in [35], only three strategies are implemented in this application: Multinomial
Resampling, Stratiﬁed Resampling, Systematic Resampling. Finally, for each time step the
aposteriori state estimate is computed as a weighted sum of the resampled particles.
The main implementation issue consists of the sample impoverishment, this can be overcome
simply increasing the number of particles Ns, but this can quickly lead to an unreasonable
demand of computational cost , and often simply delays the inevitable sample impoverishment.
3. Mathematical Model
Several vehicle and tyre models are present in literature [15]. In this case, considering the main
goal of this work, a singletrack vehicle model and a simpliﬁed Pacejka tyre model has been
chosen [36].
3.1. Vehicle model
The vehicle model adopted in this work is a singletrack vehicle model, which does not require the
knowledge of such not easily measurable vehicle parameters. Taking into account the assumption
behind this model as reported in [15], in this application the hypothesis of small steering angle
is not considered, because the steering angle values reached during the ontrack tests are not
so small to be neglected. The vehicle involved is an electric gokart, in which the braking and
tractive forces aﬀect only the rear axle.
The kinematic equations of the vehicle model are:
ax= ˙u−v r
ay= ˙v+u r (4)
With athe linear acceleration, uthe longitudinal velocity of the CoG along the xaxis, vthe
lateral velocity along the yaxis, and rthe angular velocity around the zaxis.
The vehicle sideslip angle is given by
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Figure 1. Singletrack vehicle model basic scheme
β= tan−1v
u(5)
Considering the equations (4), the equilibrium equations are:
m( ˙u−v r) = Fx1cos(δ1)−Fy1sin(δ1) + Fx2
m( ˙v+u r) = Fy1cos(δ1) + Fx1sin(δ1) + Fy2
Jz˙r=Fy1a1cos(δ1) + Fx1a1sin(δ1)−Fy2a2
(6)
The equations above are deﬁned in the ISO bodyﬁxed reference system, the forces Fxiand Fyi
1
are given by the tyre model adopted and are deﬁned in the ISO tyre reference framework.
3.2. Tyre model
As mentioned, a simpliﬁed version of the Pacejka tyre model, based on the macroparameters
instead of microparameters, has been adopted [36]:
F0=Dsin {Carctan [B xs−E(B xs−arctan (B xs))]}(7)
The formulation is the same for both planar forces, the independent variable xsidentify the slip
ratio or the tyre slip angle. The tunable parameters are not all considered as ﬁxed parameters,
indeed a vertical load dependence is adopted for the peak value D:
D=D(Fz) = µFz= (a1Fz+a2)Fzwith a1<0 (8)
in this case the terms a1and a2are ﬁxed value parameters. The parameters B,Cand Ddeﬁne
the slope in the linear region [36].
The eﬀect of the combined slip case has been taken into account by the Gfunction (or Hill
function), again considering the shift null:
G= cos {Ccarctan[Bcxc−Ec(Bcxc−arctan(Bcxc))]}(9)
1where the index irefers to the axle, 1 for the front axle and 2 for the rear one.
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Note the diﬀerence between the independent variable in (9) and in (7), if xsis the tyre slip angle
so xcis the slip ratio and vice versa. All the macroparameters are computed starting from
the output obtained using a T.R.I.C.K. methodology [37], which provides a virtual telemetry
requiring the acquired data set as input. The obtained forces and slips are used as true values
in order to calibrate the MF model presented above.
Finally the formulation of the MF force in combined slip case is given by:
Fc=F0G(10)
These tangential forces are assumed to be dependent only on: tyre vertical load Fz, tyre slip
angle αand slip ratio k. The dependence on the camber angle γand on the spin slip φas well
is neglected in this work.
The vertical load may be evaluated with diﬀerent levels of accuracy. In this work the
aerodynamics eﬀect is neglected, and because of the chosen vehicle dynamics model (Single
track) only longitudinal load transfer and the static load distribution aﬀect the vertical forces
acting on the single wheel at the front and rear axle.
The mathematical formulation of the slip ratio and the tyre slip angle, for each axle, is given
by:
α1= arctan (v+r a1) cos (δ1)−usin (δ1)
ucos (δ1)+(v+r a1) sin (δ1)
k1=ω1Rr1−ucos (δ1)+(v+r a1) sin (δ1)−
ucos (δ1)+(v+r a1) sin (δ1)
α2= arctan (v−r a2)
u
k2=−ω2Rr2−u
u
(11)
4. Implementation of the ﬁlters and experimental data acquisition
The mathematical formulation of the physical model described in the previous section is
implemented as process of each ﬁlters. In this application, the state vector of each ﬁlter is
constituted by:
xk= [uk, vk, rk]T(12)
longitudinal and lateral vehicle velocity [m/s] and yaw rate [rad/s] respectively. The ﬁlter
requires the knowledge of some measurements acquired at each time step, in order to correct
the apriori state estimate provided by the process. In this application, the elements of the
measurement vector are:
yk=ωEN C ODER
1,1k, ωENCODER
1,2k, rIMU
k, aIM U
yk, aIM U
xkT
(13)
respectively rotational velocity of the frontleft and frontright wheels [rad/s], yaw rate [rad/s]
and longitudinal and lateral vehicle linear acceleration [m/s2]. The elements of this vector are
compared with the corresponding estimate ones, the diﬀerence is proportional to the amount of
correction of the apriori state estimate.
The system equations adopted in each ﬁlter descend from the equilibrium equations (6), they
are computed starting from the (14) and applying the Euler method.
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˙u=v r +1
mFx1cos(δ1)−Fy1sin(δ1) + Fx2
˙v=−u r +1
mFy1cos(δ1) + Fx1sin(δ1) + Fy2
˙r=1
JzFy1a1cos(δ1) + Fx1a1sin(δ1)−Fy2a2
(14)
the expressions on the right side are strongly nonlinear, especially because of the tyre model
chosen. As mentioned above a linearization is needed to implement the FOEKF and the IEKF
or in other words the Jacobian matrix has to be computed, it is given by the partial derivatives
of f(·) with respect to the state vector x:
∂f
∂x ˆx+
=∂f (ˆx+, u, 0)
∂x =
a1a2a3
a4a5a6
a7a8a9
(15)
where the expressions of each term are reported in the Appendix B. The Hessian computation
is required for the implementation of the SOEKF, it is not reported here for sake of simplicity.
The same can be done with the measurement equations h(·) given by the expressions below:
ω11 =cos (δ11)
Rr1
ˆu−+sin (δ11)
Rr1
ˆv−+a1sin (δ11)−t1/2 cos (δ11)
Rr1
ˆr−
ω12 =cos (δ12)
Rr2
ˆu−+sin (δ12)
Rr2
ˆv−+a1sin (δ12)−t1/2 cos (δ12)
Rr2
ˆr−
r= ˆr−
ay=1
mFy1cos(δ1) + Fx1sin(δ1) + Fy2
ax=1
mFx1cos(δ1)−Fy1sin(δ1) + Fx2
(16)
On the right side of the equations above there are the estimate measurements, which are
compared to the acquired ones computed by (13).
The measurements mentioned in (13) are acquired by sensors, in addition some of them
feed the system equations as seen in (1) and are called additional inputs uk−1or uk. These
measurements require sensors which are commonly used for the basic data acquisition, sometimes
they equip common vehicles and are used to prevent critical events. In this application an electric
gokart is used in order to acquire the experimental data, several indoor ontrack tests have been
conducted with the aim to explore the vehicle behaviour in diﬀerent operating conditions. The
vehicle is rear drive and also the brakes aﬀect only the rear axle. The sensors adopted consist
of: an encoder per each front wheel to acquire the rotational velocity whereas the rotational
velocities of the rear ones is measured through the electric motors, an IMU which acquires the
inplane accelerations and the yaw rate, an Smotion provides the true values of the longitudinal
and lateral velocities (thus the side slip angle) and ﬁnally an encoder measures the steering
wheel angle. Starting from this latter, the steering angle of the front wheels is computed using
a nonlinear function identiﬁed using experimental data. All acquired data are subjected to
postprocessing, consisting in the cleaning (deleting the incorrect and/or duplicated samples),
synchronisation, ﬁltering and resampling (20 Hz in this application). The signals acquired by
the IMU and Smotion are transported to the CG using rotational matrices that are based on
the geometrical distances of the sensors from the CG.
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5. Results and validation
In this section the obtained results are presented, comparing each estimate VSA to the
experimental one acquired by the SMotion. All the ﬁlters are ”trained” using the same algorithm
that provides the process and measurements covariance matrices.
As can be seen in the ﬁgures below, the VSA estimation depends on the type of ﬁlter
implemented. They refer to one of the led tests, the other ones are taken into account in
term of mean RMSE of the estimate VSA in the table (1).
Table 1. RMSE mean values for each test.
[deg] TEST 1 TEST 2 TEST 3 TEST 4 MEAN
FOEKF 2.52 1.18 2.23 2.02 2.03
IEKF 2.46 1.26 2.09 2.26 2.02
SOEKF 2.48 1.20 2.36 2.22 2.06
SUKF 2.10 1.32 2.38 1.59 1.85
GUKF 2.40 1.06 2.72 2.87 2.26
SIMPUKF 2.60 1.39 2.95 1.86 2.20
SPHEUKF 2.17 1.43 2.82 2.70 2.28
MULTIresPF 4.62 1.32 3.53 3.80 3.32
SYSTresPF 3.26 1.38 3.24 3.65 2.88
STRAresPF 4.28 1.44 3.88 3.99 3.40
Another ﬁlter’s index of performance taken into account in this work is the runtime. The
mean runtime required by each algorithm is reported in the table (2). This may be useful in
the realtime application, for this reason the time taken by each implemented ﬁlter to estimate
1sof real time acquired data is considered.
Considering each test, the implemented EKFs provide quite similar VSA estimations. The
ﬁgure (2) refers to the test #3, in which the IEKF outperforms the others at the beginning, this
is due to the reﬁning provided by the iteration cycles. The time taken by the EKFs is diﬀerent,
as can be seen in table 2, the FOEKF needs less than the half of the time taken by the others.
The implemented UKFs provide slightly diﬀerent VSA estimations. The best performance in
term of RMSE are achieved by the SUKF and the GUKF, that uses more sigma points than
the others. Note that the SUKF outperforms the GUKF on three out of the four tests. The
time taken by the EKFs is diﬀerent and the same trend is repeated for each test. The GUKF
require more time than the others, as expected, because it adopted 7 sigma points and also their
weight factors are diﬀerent, while in the SUKF they are the same and the number of sigma
points involved is 6. The SIMPUKF and the SPHEUKF require a very similar execution time,
they diﬀer only in the weight factor deﬁnition and both use 5 sigma points.
The implemented PFs always provide noisy VSA estimations and also higher value of RMSE.
In general, the SYSresPF outperforms the others, but they are not so far one each other as
shown in ﬁgure (4). The execution time is very high if compared to the other ﬁlters ones. In
addition, note that if the random number generator is not ﬁxed, the results depends on the
random numbers generated as well, that could be very diﬀerent at each run of the PF.
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Table 2. Time taken by each implemented ﬁlter to estimate 1 sof real time acquired data.
[ms/s] TEST 1 TEST 2 TEST 3 TEST 4 MEAN
FOEKF 8.20 8.86 8.64 7.49 8.30
IEKF 22.91 24.35 25.19 22.37 23.71
SOEKF 18.26 20.63 19.31 17.84 19.01
SUKF 27.78 27.36 27.79 25.952 27.22
GUKF 39.08 37.20 35.98 34.08 36.59
SIMPUKF 25.73 25.13 23.28 22.24 24.10
SPHEUKF 25.08 25.42 23.52 22.45 24.12
MULTIresPF 311.0 310.9 306.0 305.8 308.5
SYSTresPF 309.4 303.2 306.1 304.0 305.7
STRAresPF 308.2 302.9 308.1 306.8 306.5
Figure 2. VSA estimated by EKFs and acquired by Smotion
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Figure 3. VSA estimated by UKFs and acquired by Smotion
The estimated VSA, but also all the other estimated quantities as well, are characterized by a
noisy trend. This is due to the fact that the samples are very close one each other but the same
sample have also diﬀerent weight factors at each time step: this means that at consecutive time
step the estimated value is not so close to the previous as in the previous ﬁlters.
At that point it becomes important to highlight that all the results presented are strongly
dependent on the Q and R matrices. As said, they are computed using a training method
based on an optimization method, the surrogate optimization for global minimization of time
consuming objective functions has been adopted in thesis work, but others can be found in
literature [27]. However, the stopping condition adopted is the maximum number of function
evaluations, that has been set on 50, a relatively low value. In other words, the optimizator
runs 50 times the ﬁlter, changing the Q and R values, storing only the ones that minimize the
residual prediction error computed till now. In addition, this is repeated 2 times: the initial
values of Q and R obtained in the second optimization cycle are the best values provided by
the previous optimization cycle. Best VSA estimation may be achieved if higher limit is set
for the maximum number of function evaluations, but of course this requires much more time,
especially considering the PFs. The reason behind the low value chosen is that the the aim of
this work is to compare the diﬀerent type of ﬁlters, adopting the same boundary conditions, not
looking for the best state estimate that each ﬁlter may provide.
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Figure 4. VSA estimated by PFs and acquired by Smotion
6. Conclusions
A vehicle sideslip angle estimation based on dynamic model is proposed in this work. The
purpose of this paper is to compare the performance of diﬀerent modelbased state estimators
(Extended Kalman Filters, Unscented Kalman Filters and Particle Filters) in terms of estima
tion accuracy and the computational cost for a chosen vehicle. The mathematical model of the
vehicle and the features of each implemented ﬁlter have been mathematically presented.
Concerning the obtained results, the EKFs and the UKFs show a better state estimation
using the vehicle model presented and the Pacejka macroparameters computed. Considering the
mean value of the RMSE computed per each tests, the SUKF exhibits the lowest value whereas
the other UKFs exhibit a value which is about the 20% higher; the EKFs show the same mean
value if compared one each other but this is about the 10% higher than the one reached by the
SUKF; Finally, the SYSTresPF shows the lowest RMSE mean value if compared with the other
implemented PFs, but if compared to the others the PFs exhibit the higher values, in particular
the STRAresPF shows the highest one. Concerning the computational burden required by
each state estimator which can be considered as proportional to the time taken by the ﬁlter
to estimate 1 second of real time, the FOEKF is characterized by the lowest amount of time
required, thanks to its simple algorithm. The other EKFs present a value which is about the
150% higher. Considering the UKFs, the SIMPUKF and the SPHEUKF require lower runtime
than the SUKF and the GUKF, this latter is characterized by the highest one if considering
only the Kalmanbased ﬁlters, it is about the 340% higher than the lowest one. Considering the
PFs, the time required is one order of magnitude higher than the Kalmanbased one, they take
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about onethird of second to estimate 1 second of real time. However, the overall state estimate
is not always accurate, in same cases the estimate VSA is quite diﬀerent than the actual one,
this may be mitigated adopting a complete Magic Formula, with microparameters instead of
macroparameters. Moreover, it may be interesting to consider a tricycle vehicle model instead
of a bicycle one in order to take into account the lateral load transfer eﬀect.
In order to improve the VSA estimation the tyre model may be a key factor, in this work
the macroparameters are ﬁxed, but considering their update during ﬁlter execution, due to
thermodynamics and wear eﬀect, could lead to better results.
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Appendix A
The algorithms of the implemented ﬁlters are here reported.
Algorithm 1 FirstOrder Extended Kalman Filter algorithm
1: ˆx+
0=E[x0]Initial state
2: P+
0=E[(x0−ˆx+
0)(x0−ˆx+
0)T]Initial state Covariance
3: procedure FOEKF(T,{u}T
k=1)
4: for k= 1 →Tdo
5: ˆx−
k=f(ˆx+
k−1, uk−1,0) apriori state estimate
6: P−
k=Fk−1P+
k−1FT
k−1+Q apriori state estimate covariance
7: zk=h(ˆx−
k, uk,0) apriori measurement estimate
8: Kk=P−
kHT
k(HkP−
kHT
k+R)−1Kalman gain
9: ˆx+
k= ˆx−
k+Kk(yk−zk)aposteriori state estimate
10: P+
k= (I−KkHk)P−
kaposteriori state estimate covariance
11: end for
12: end procedure
Algorithm 2 Iterated Extended Kalman Filter algorithm
1: ˆx+
0=E[x0]
2: P+
0=E[(x0−ˆx+
0)(x0−ˆx+
0)T]
3: procedure IEKF(T,{u}T
k=1, N)
4: for k= 1 →Tdo
5: ˆx−
k=f(ˆx+
k−1, uk−1,0)
6: P−
k=Fk−1P+
k−1FT
k−1+Q
7: ˆx+
k,1= ˆx−
k
8: for i= 1 →Ndo
9: zk,i =h(ˆx+
k,i, uk,0) −Hk,i(ˆx−
k−ˆx+
k,i)
10: Kk,i =P−
kHT
k,i(Hk,iP−
kHT
k,i +R)−1
11: ˆx+
k,i+1 = ˆx−
k+Kk,i(yk−zk,i)
12: P+
k,i+1 = (I−KkHk)P−
k
13: end for
14: end for
15: end procedure
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Algorithm 3 Second Order Extended Kalman Filter algorithm
1: ˆx+
0=E[x0]
2: P+
0=E[(x0−ˆx+
0)(x0−ˆx+
0)T]
3: procedure SOEKF(T,{u}T
k=1)
4: for k= 1 →Tdo
5: ˆx−
k=f(ˆx+
k−1, uk−1,0) + 1
2
n
P
i=1
φiT r∂2fi
∂x2ˆx+
k−1
P+
k−1
6: P−
k=Fk−1P+
k−1FT
k−1+Q
7: zk=h(ˆx−
k, uk,0) + 1
2
m
P
i=1
φiTr ∂2hi
∂x2ˆx−
k
P−
k
8: Kk=P−
kHT
k(HkP−
kHT
k+R)−1
9: ˆx+
k= ˆx−
k+Kk(yk−zk)
10: P+
k= (I−KkHk)P−
k
11: end for
12: end procedure
Algorithm 4 Simply Unscented Kalman Filter algorithm
1: ˆx+
0=E[x0]
2: P+
0=E[(x0−ˆx+
0)(x0−ˆx+
0)T]
3: procedure SUKF(T,{u}T
k=1)
4: for k= 1 →Tdo
5: ˆx(i)
k−1= ˆx+
k−1+qnP +
k−1T
i
i= 1, . . . , n
6: ˆx(n+i)
k−1= ˆx+
k−1−qnP +
k−1T
i
i= 1, . . . , n
7: ˆx(i)
k=f(ˆx(i)
k−1, uk−1,0)
8: ˆx−
k=1
2n
2n
P
i=1
ˆx(i)
k
9: P−
k=1
2n
2n
P
i=1
(ˆx(i)
k−ˆx(−)
k)(ˆx(i)
k−ˆx(−)
k)T+Q
10: z(i)
k=h(ˆxi
k, uk,0)
11: zk=1
2n
2n
P
i=1
z(i)
k
12: Py
k=1
2n
2n
P
i=1
(z(i)
k−zk)(z(i)
k−zk)T+R
13: Pxy
k=1
2n
2n
P
i=1
(ˆx(i)
k−ˆx(−)
k)(z(i)
k−zk)T
14: Kk=Pxy
k(Py
k)−1
15: ˆx+
k= ˆx−
k+Kk(yk−zk)
16: P+
k=P−
k−KkP−
kKT
k
17: end for
18: end procedure
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The GUKF’s algorithm is similat to the SUKF’s one, but it involves (2n+ 1) sigma points
as shown in (2) instead of 2n. The formulation of sigma points and weight factors are here
reported:
x(0) = ¯x
x(i)= ¯x+ ˜x(i)i= 1,...,2n
˜x(i)= (pα2(n+k)Pxx)T
ii= 1, . . . , n
˜x(n+i)=−(pα2(n+k)Pxx)T
ii= 1, . . . , n
W(0)
m=α2(n+k)−n
α2(n+k)W(i)
m=1
2α2(n+k)i= 1,...,2n
W(0)
c= (2 −α2+β)−n
α2(n+k)W(i)
c=1
2α2(n+k)i= 1,...,2n
The SPHEUKF and the SIMPUKF adopt a diﬀerent formulation of the sigma points as
reported below:
Algorithm 5 Spherical Unscented Transformation
1: W(0) ∈[0,1) Initial choice
2: W(i)=1−W(0)
n+1 i= 1, . . . , n + 1 Weights initialization
3: σ(1)
0= 0 σ(1)
1=−1
√2W(1) σ(1)
2=1
√2W(1) Sigma vector initialization
4: for j= 2 →ndo
5: σ(j)
i=
"σ(j−1)
0
0#i= 0
σ(j−1)
i
−1
√j(j+1)W(1)
i= 1, . . . , j
"0(j−1)
j
√j(j+1)W(1) #i=j+ 1
Sigma vector building
6: end for
7: x(i)= ¯x+σ(n)
i√Pxx i= 0, . . . , n + 1 Sigma points
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Algorithm 6 Simplex Unscented Transformation
1: W(0) ∈[0,1) Initial choice
2: W(i)=(2−n(1 −W(0))i= 1,2
2i−2W(1) i= 3, . . . , n + 1 Weights initialization
3: σ(1)
0= 0 σ(1)
1=−1
√2W(1) σ(1)
2=1
√2W(1) Sigma vector initialization
4: for j= 2 →ndo
5: σ(j)
i=
"σ(j−1)
0
0#i= 0
"σ(j−1)
i
−1
√2W(j+1) #i= 1, . . . , j
"0(j−1)
j
√2W(j+1) #i=j+ 1
Sigma vector building
6: end for
7: x(i)= ¯x+σ(n)
i√Pxx i= 0, . . . , n + 1 Sigma points
Appendix B
The terms in equations (15) are:
a1= 1 + ∆t
m∂Fx1
∂u +∂Fx2
∂u −∂Fy1
∂u δ1
a2= ∆tˆr++1
m∂Fx1
∂v +∂Fx2
∂v −∂Fy1
∂v δ1
a3= ∆tˆv++1
m∂Fx1
∂r +∂Fx2
∂r −∂Fy1
∂r δ1
a4= ∆t−ˆr++1
m∂Fy1
∂u +∂Fy2
∂u +∂Fx1
∂u δ1
a5= 1 + ∆t
m∂Fy1
∂v +∂Fy2
∂v +∂Fx1
∂v δ1
a6= ∆t−ˆu++1
m∂Fy1
∂r +∂Fy2
∂r +∂Fx1
∂r δ1
a7=∆t
Jz∂Fy1a1
∂u −∂Fy2a2
∂u +∂Fx1a1
∂u δ1
a8=∆t
Jz∂Fy1a1
∂v −∂Fy2a2
∂v +∂Fx1a1
∂v δ1
a9= 1 + ∆t
Jz∂Fy1a1
∂r −∂Fy2a2
∂r +∂Fx1a1
∂r δ1
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