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Vehicle System Dynamics: International
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Vehicle longitudinal velocity estimation
during the braking process using
unknown input Kalman filter
Bijan Moaveni
a
, Mahdi Khosravi Roqaye Abad
b
& Sayyad Nasiri
c
a
School of Railway Engineering, Iran University of Science and
Technology, P.O. Box 16846-13114, Tehran, Iran
b
School of Railway Engineering, Iran University of Science and
Technology, Tehran, Iran
c
Research and Applied Division of Automotive (RADA), Sharif
University of Technology, Tehran, Iran
Published online: 25 Jul 2015.
To cite this article: Bijan Moaveni, Mahdi Khosravi Roqaye Abad & Sayyad Nasiri (2015): Vehicle
longitudinal velocity estimation during the braking process using unknown input Kalman filter,
Vehicle System Dynamics: International Journal of Vehicle Mechanics and Mobility
To link to this article: http://dx.doi.org/10.1080/00423114.2015.1038279
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Vehicle System Dynamics, 2015
http://dx.doi.org/10.1080/00423114.2015.1038279
Vehicle longitudinal velocity estimation during the braking
process using unknown input Kalman filter
Bijan Moavenia∗
, Mahdi Khosravi Roqaye Abadb
and Sayyad Nasiric
aSchool of Railway Engineering, Iran University of Science and Technology, P.O. Box 16846-13114,
Tehran, Iran; bSchool of Railway Engineering, Iran University of Science and Technology,
Tehran, Iran; cResearch and Applied Division of Automotive (RADA), Sharif University of Technology,
Tehran, Iran
(Received 29 May 2013; accepted 1 April 2015)
In this paper, vehicle longitudinal velocity during the braking process is estimated by measuring the
wheels speed. Here, a new algorithm based on the unknown input Kalman filter is developed to esti-
mate the vehicle longitudinal velocity with a minimum mean square error and without using the value
of braking torque in the estimation procedure. The stability and convergence of the filter are analysed
and proved. Effectiveness of the method is shown by designing a real experiment and comparing the
estimation result with actual longitudinal velocity computing from a three-axis accelerometer output.
Keywords: vehicle longitudinal velocity estimation; Kalman filter; unknown input; real-time
systems
1. Introduction
The necessity of reducing the risk of human mistakes and the loss of human life while driving
a vehicle in different conditions such as a wet, snowy and icy road thereby decreasing crashes
is the most important reason for developing electronic stability control systems in cars.[1,2]
One of the main problems in vehicle stability is wheel slip/slide control under traction or
braking conditions. Especially when the friction coefficient between the road and the tyre is
too low and the wheels lock, this issue can endanger the safety of passengers.[1] The antilock
brake system (ABS) can be a good solution to solve this braking process problem.[3–8] The
main task of this system is preventing the wheels from locking. When the wheels lock, the
ability to control the vehicle is reduced and the longitudinal friction force is decreased. So, the
stopping distance increases to the point where it becomes dangerous and directional stabil-
ity decreases. When the wheels are locked, the vehicle velocity is quite different from wheel
speed. The longitudinal slip (λ) is defined as Equation (1) and describes the normalised differ-
ence between the vehicle longitudinal speed vx and the peripheral speed of the wheel ωRω [4]
λ =
vx − ωRω
vx
, (1)
where ω is the wheel angular speed and Rω is the rolling radius of the tyre. It is well known
that the friction force between the road and the tyre is a nonlinear function of the wheel slip.
*Corresponding author. Email: b_moaveni@iust.ac.ir
c 2015 Taylor & Francis

2 B. Moaveni et al.
Figure 1. Friction coefficient vs. wheel slip.
Figure 1 illustrates the tyre–road friction coefficient vs. the wheel slip. It shows that the maxi-
mum friction occurs when the wheel slip is in the range between 0.08 and 0.3. In an ABS, the
controller has to keep the wheel in this range to access the maximum friction force.[4] Thus,
the control algorithm needs to know the angular speed of the tyre and the vehicle longitudinal
velocity.
We can use speedometers, accelerometers, tachometers and gyroscopes to compute wheel
velocity, vehicle acceleration and so on.[2] Speedometers are implemented for conventional
vehicles in two forms: a contact form (like wheel speed sensors) [9] and a contactless form
(like reflection sensors).[10–12] Contact methods used in most vehicles are reliable and eco-
nomically efficient, but may not compute speed accurately through the slipping. Non-contact
methods, such as the spatial filtering method [11] or the optical correlation method [12] are
accurate but very expensive. They are used less in practice due to being heavy and complex
and requiring frequent maintenance.[13]
Although measuring the vehicle longitudinal velocity is very important, there are no reli-
able and economically efficient sensors to measure the longitudinal velocity of the vehicle
accurately during slipping. Due to this, experts and researchers are looking for a suitable
methodology to accurately estimate vehicle velocity.[2]
In scientific communities, different estimators, such as Kalman filter,[1,2,14] the sliding
mode observer,[15,16] the nonlinear adaptive filter [4] and nonlinear observers [17–21] are
investigated to estimate the longitudinal velocity of the vehicle. One of the most popu-
lar estimators is the Kalman filter. The Kalman filter is a well-known state estimator due
to its optimal performance, convenient form for real-time processing, ease of implemen-
tation, convenient measure of estimation accuracy and its wide spread application in data
fusion problems.[14,22] Numerous studies have used this filter in accurate estimation of
the longitudinal velocity of vehicles [23–25] and estimation of other effective variables in
the dynamic model of vehicles.[5,26–32] Ray [1] implemented an extended Kalman filter
(EKF) for estimating state variables, and the lateral and longitudinal forces of the wheel
using a model with nine degrees of freedom. Alvarez [2] considered the longitudinal and
lateral velocity estimation of a car by using the Kalman filter with a seven-degree of free-
dom model in his Master’s thesis. Moreover, Guo et al. [14] used and implemented the EKF

Vehicle System Dynamics 3
on field-programmable gate array in 2013 to estimate the longitudinal velocity of a vehicle.
The necessity of having an accelerometer to measure instantly vehicle acceleration is the
remarkable point of all these studies.
To estimate the states of a dynamic system using an observer, it is usually assumed that all
inputs are known and measurable. But, in practical cases some inputs are not known or not
measurable. In these cases, the standard Kalman filter will fail.[33]
Many researchers have tried to solve the problem of inefficiency of the Kalman filter in the
presence of unknown inputs by introducing the unknown input observers (UIO).[33–39] For
example, in [33], Koenig and Mammar considered controlling the lateral velocity of a vehicle.
They assumed that the vehicle is unstable, nonlinear and containing noisy sensors. In the
work, the steering angle and the trajectory to be followed have been considered as two inputs.
Since detecting changes in the road curvature is difficult, this input has been considered as an
unknown input. In [34], Imsland et al. studied observers for nonlinear systems and illustrated
that the error dynamics for a nonlinear observer with unknown input has the same structure
as the error dynamics of a nonlinear observer without unknown inputs. In the paper, the bank
of the road was considered as an unknown input for estimating the lateral velocity of the car.
Unfortunately, a problem that most of the articles deal with is using an input called braking
torque in the longitudinal velocity estimation procedure. This input is not easily measurable
and its measurement requires additional sensors.[1,2,14] Clearly, by using additional sen-
sors, costs will be raised and we should devote more attention to maintenance and repair
issues. Therefore, we propose a new approach based on UIO to estimate longitudinal veloc-
ity accurately and without using any additional sensors to measure brake torque or other new
variables. Of course, in [2], the authors claimed to present another approach to estimate vehi-
cle velocity by employing the nonlinear deterministic tyre force model and defining new input
variables. But unfortunately, the measured variables are not specified clearly and to measure
the other state variables we require additional new sensors.
In this paper, a new algorithm based on the unknown input Kalman filter (UIKF) and
data fusion concept is proposed for estimating the longitudinal velocity of a vehicle. This
approach requires the wheels speed to estimate the vehicle longitudinal velocity. Then a real-
time estimation of the longitudinal velocity is done by designing a real experiment. Also, the
convergence of the filtering approach to the true value is proved. Since the main goal of this
research is to implement the approach practically, the estimation algorithm is implemented
in a C+ + environment. The most important feature of this algorithm, in contrast to similar
methods, is that it does not use an accelerometer in the velocity estimation procedure and
also the braking torque is considered as an unknown input.
The paper is organised as follows. First, the dynamic model of vehicle is explained in
Section 2 and then, a Kalman filter approach with an unknown input is presented in Section 3.
In this section, the stability of the UIKF is verified and a simplified method is proposed
to implement the UIKF easily. Since the simplified method cannot be used for multiple-
input and multiple-output systems, a data fusion approach is applied to achieve the best
outcome. An experimental test is designed for evaluating the above-mentioned approach in
Section 4. Detailed evaluation of the estimator performance and the estimation results using
an experimental test are provided in Section 5. Conclusions are given in Section 6.
2. Vehicle dynamics modelling
In this paper, we attempt to implement an UIKF in order to estimate the vehicle longitudinal
velocity. Since, the Kalman filter requires the dynamic model of vehicle to estimate the
longitudinal velocity, it is vital to have an accurate but not necessarily complex model of

4 B. Moaveni et al.
(a) (b)
Figure 2. Five degree-of-freedom bicycle model. (a) Top view and (b) side view.
Figure 3. The free diagram of the car in braking.
the vehicle motion. Implementation of the Kalman filter will be restricted in dealing with a
complex model due to the computational load. The vehicle dynamic equations are introduced
in Equation (2) based on the results of Ray,[1] Alvarez [2] and Kiencke and Nielsen.[40]
This model utilises only the measurements of the front and rear wheels, it does not include
roll motion and vertical suspension dynamics. Also, vehicle motion is considered on a flat
road. Figure 2 illustrates a free diagram of the vehicle motion model and signs conventions
corresponding to the five degree-of-freedom vehicle and Figure 3 shows a free diagram of
the car in the braking process. A nomenclature of the system state variables and parameters
are shown in Table 1.
˙vx =
1
m
[−Fxf cos δf − Fyf sin δf − Fxr] + rvy, (2a)
˙vy =
1
m
[Fyf cos δf − Fxf sin δf + Fyr] + rvx, (2b)
˙r =
1
Iz
[Lf(Fyf cos δf − Fxf sin δf) − LrFyr], (2c)
˙ωf =
1
Iω
[RωFxf − KbTb], (2d)

Table 1. A nomenclature of the system state variables and parameters.
Symbol Description Unit Symbol Description Unit
vx Longitudinal vehicle velocity m/s Tb Applied braking torque N m
vy Lateral vehicle velocity m/s Fxf Front longitudinal tyre force N
r Yaw rate rad/s Fyf Front lateral tyre force N
ωf Front wheel angular velocity rad/s Fxr Rear longitudinal tyre force N
ωr Rear wheel angular velocity rad/s Fyr Rear lateral tyre force N
δf Front steering wheel angle rad mb Sprung mass kg
m Total mass of the car kg Lr Distance from COG to the
rear axle
m
mf Mass of front wheels kg Lf Distance from COG to the
front axle
m
mr Mass of rear wheels kg Iz Moment of inertia around
vertical axis
kg m2
Rω Wheel radius m Ip Moment of inertia about
longitudinal axis
kg m2
g The acceleration of gravity m/s2 Kb Fixed proportion of braking
force applied to the front
wheels
–
Iω Moment of inertia of the wheel kg m2 H Height of COG m
Cd Drag coefﬁcient – Ha Height of drag force m
W Car weight N Af Frontal area m2
ρ Air density kg/m3 Dx The deceleration m/s2
˙ωr =
1
Iω
[RωFxr − (1 − Kb)Tb]. (2e)
Equation (3) shows the state variables
x(t) = [vx; vy; r ; ωf; ωr]T
. (3)
Speciﬁcally in Equation (2), the input vector is as follows:
u(t)
d(t)
=
δf
Tb
, (4)
where u(t) is the known input and d(t) is the unknown input. The output equation (or
measurement equation) is as follows:
z1(t)
z2(t)
=
ωf
ωr
+
n1(t)
n2(t)
. (5)
Without loss of generality, we assume that n1(t) and n2(t) are zero-mean white Gaus-
sian measurement noises with variances R1 and R2, respectively. All outputs (wheels angular
velocity) are measured by using a sensor set that is implemented on the rear and front wheels
of the vehicle.
The tyre model has been derived from [2,17,40] to calculate the tyre forces. The rear and
front longitudinal and lateral tyre forces can be accurately calculated by using the following
equations:
Fxr
Fyr
=
μr
λr
Fzr
λxr
λyr
, (6)
Fxf
Fyf
=
μf
λf
Fzf
λxf
λyf
. (7)

6 B. Moaveni et al.
Equations (8) and (9) show the rear and front normal forces at the tyre–road interface
during braking
Fzr =
m ˙vxH
2(Lf + Lr)
+
WLr
2(Lf + Lr)
−
(1/2)ρCdAfv2
xHa
2(Lf + Lr)
, (8)
Fzf = −
m˙vxH
2(Lf + Lr)
+
WLf
2(Lf + Lr)
+
1/2(ρCdAfvx
2
Ha)
2(Lf + Lr)
. (9)
Since to have a tyre model, it is necessary to know the coefﬁcient of friction μ, the resultant
rear and front friction obtained from the Burckhardt equation as follows:
μr = (c1(1 − e−c2λr
) − c3λr)ψ, (10)
μf = (c1(1 − e−c2λf
) − c3λf)ψ, (11)
where ψ is a constant scalar. Equations (12)–(15) show the rear and front longitudinal and
lateral wheel slips and the resultant rear and front wheel slips, respectively
λxr
λyr
=
1
vr
Rωωr cos αr − vr
Rωωr sin αr
, (12)
λxf
λyf
=
1
vf
Rωωf cos αf − vf
Rωωf sin αf
, (13)
λr = λ2
xr + λ2
yr, (14)
λf = λ2
xf + λ2
yf. (15)
The magnitudes of the rear and front axle velocities are as follows:
vr = (vy − Lrr)2
+ v2
x, (16)
vf = (vy + Lfr)2
+ v2
x. (17)
The rear and front wheel slip angles αr and αf are as follows:
αr = −tan−1 νy − Lrr
νx
, (18)
αf = δf − tan−1 νy + Lfr
νx
. (19)
Consequently, the real model of the vehicle is as follows:
˙x(t) = f (x(t), u(t), d(t)), (20)
z(t) = h(x(t)) + n(t), (21)

Figure 4. The block diagram of car’s dynamic model during braking.
where
f (x(t), u(t), d(t)) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1
m
(−Fxf cos δf − Fyf sin δf − Fxr) + rvy
1
m
(Fyf cos δf − Fxf sin δf + Fyr) + rvx
1
Iz
(Lf(Fyf cos δf − Fxf sin δf) − LrFyr)
1
Iω
(RωFxf − KbTb)
1
Iω
(RωFxr − (1 − Kb)Tb)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, (22)
h(x(t)) =
0 0 0 1 0
0 0 0 0 1
x(t). (23)
Figure 4 shows the block diagram of the vehicle dynamic during the braking process based
on Equations (2)–(19).
3. Unknown input Kalman ﬁlter
As explained in the previous section, two state variables (ωf and ωr) of the vehicle model are
completely measurable. But, the important point is that, an input called the braking torque
(Tb) is not measurable. When there is unknown input in the system, the performance of the
Kalman ﬁlter to estimate the longitudinal velocity of the vehicle can be very poor. Therefore,
we have to use UIKF.
In [35], an UIKF approach with unknown input is presented for the following linear
discrete-time stochastic system with unknown inputs:
xk+1 = Axk + Buuk + Bddk + ωk, (24a)
zk = Cxk + nk, (24b)
where xk ∈ Rn
is the state vector, zk ∈ Rm
is the measurement vector, dk ∈ Rp
is the unknown
input and uk ∈ Rq
is the known input. The process noise ωk and the measurement noise nk

8 B. Moaveni et al.
are the zero-mean white noise whose covariance matrices are as Equation (25)
E{ωkωT
l } = Qδkl = Qδ(k − l),
E{nknT
l } = Rδkl = Rδ(k − l),
E{ωknT
l } = 0,
(25)
where ‘δ’ denotes the Kronecker delta function and ‘T’ denotes the transpose operator. This
filter is structured as follows:
• Measurement update:
Fk = CPk|k−1CT
+ R, (26a)
kk = Pk|k−1CT
Fk
−1
, (26b)
Lk = Kk + (I − KkC)Bd × {BT
d CT
Fk
−1
CBd}−1
BT
d CT
Fk
−1
, (26c)
ˆxk|k = ˆxk|k−1 + Lk(zk − Cˆxk|k−1), (26d)
Pk|k = (I − LkC)Pk|k−1(I − LkC)T
+ LkRLT
k . (26e)
• Time update:
ˆxk|k−1 = Aˆxk−1|k−1 + Buuk−1, (26f)
Pk|k−1 = APk−1|k−1AT
+ Q. (26g)
The main feature of this filter is that its performance is not influenced even with the
presence of dk with non-Gaussian values or unknown statistics.[35,36] The condition for
employing this filter is that the number of outputs of the system should be more than the num-
ber of unknown inputs. In other words, it is necessary that the condition rank(C) ≥ rank(Bd)
is satisfied.
It is worth noting that, if nk is not a zero-mean noise, the origins of the state variables and
the outputs can be shifted so that the mean of the noise is zero. Also, if measurement noise is
not white, the noise shaping filter should be designed so that it matches the frequency spec-
trum of the coloured noise. The shaping filter can be incorporated into the vehicle dynamic
model and Kalman filter can be designed for augmented model.[41]
3.1. The filter analysis
Since we are going to employ the filter in order to estimate the longitudinal velocity of the
vehicle in practice, stability of this filter and convergence of the estimated value to the true
value is critical. Therefore, in this section, we will analyse the stability of the filter. For this
purpose, the dynamics of the estimation error will be analysed. The estimation error is defined
as Equation (27)
˜xk−1|k−1 = xk−1 − ˆxk−1|k−1. (27)
Subsequently:
˜xk|k = xk − ˆxk|k. (28)

By substituting Equations (24-a) and (26-d) into Equation (28), the following results will
be obtained.
˜xk|k = Axk−1 + Buuk−1 + Bddk−1 + ωk−1 − ˆxk|k−1 − Lk(zk − Cˆxk|k−1), (29a)
˜xk|k = Axk−1 + Buuk−1 + Bddk−1 + ωk−1 − Aˆxk−1|k−1 − Buuk−1
− Lk(Cxk + nk − Cˆxk|k−1), (29b)
˜xk|k = A˜xk−1|k−1 + Bddk−1 + ωk−1 − Lk(C(Axk−1 + Bddk−1 + ωk−1) + nk − CAˆxk−1|k−1),
(29c)
˜xk|k = A˜xk−1|k−1 − LkCA˜xk−1|k−1 + Bddk−1 + ωk−1 − Lk(C(Bddk−1 + ωk−1) + nk),
(29d)
˜xk|k = (I − LkC)A˜xk−1|k−1 + Bddk−1 + ωk−1 − Lk(C(Bddk−1 + ωk−1) + nk), (29e)
E [˜xk|k] = (I − LkC)A E [˜xk−1|k−1] + E [Bddk−1 + ωk−1]
− LkC E [Bddk−1 + ωk−1] − LkE [nk], (29f)
E [˜xk|k] = (I − LkC)AE [˜xk−1|k−1] + E [Bddk−1] + E [ωk−1] − LkCE [Bddk−1]
− LkCE [ωk−1] − LkE [nk]. (29g)
By considering nk and ωkas the zero-mean white Gaussian noises, Equation (30) can be
established
E [˜xk|k] = (I − LkC)AE [˜xk−1|k−1] + (I − LkC)BdE [dk−1]. (30)
According to Equation (30), the convergence of the estimation error to zero value depends
on the two following conditions:
(1) The convergence condition: the coefficient of E[dk−1] is zero. In other words, (I −
LkC)Bd = 0.
(2) The stability condition: all eigenvalues of (I − LkC)A are placed into a unit circle.
3.2. Analysing the convergence condition
Using Equation (26-c) and multiplying it by the ‘CBd’ from the right side, Equation (31) will
be established that shows the first condition is satisfied.
Lk = Kk + (I − KkC)Bd × {BdH Fk
−1
CBd}−1
B C Fk
−1 ×CBd
⇒ , (31a)
LkCBd = KkCBd + (I − KkC)Bd × {BdC Fk
−1
CBd}−1
BdC Fk
−1
CBd →, (31b)
LkCBd = KkCBd + (Bd − KkCBd) → LkCBd = Bd →, (31c)
(I − LkC)Bd = 0. (31d)
3.3. Simplified filtering method
For single input and single-output (SISO) systems, CBd is scalar. In this case, Equation (31-d)
can be rewritten as Equation (32)
(I − LC)Bd = 0, (32a)

10 B. Moaveni et al.
LCBd = Bd, (32b)
L =
1
CBd
× Bd. (32c)
Therefore, the matrix L is unchanged during the estimation process and it can be computed
offline. In this case, equations of the unknown Kalman filter (26) can be briefed as follows:
ˆxk|k = Aˆxk−1|k−1 + L(zk − C(Aˆxk−1|k−1 + Buuk−1)). (33)
The light computational load is the key benefit of the filter (33). This property makes it
a practical filter. But it is only applicable to SISO systems, while the car dynamic model
has two outputs. To solve this problem, we use two separated filters (Equation (33)), for two
outputs ωf and ωr. It is worth mentioning that the results of using two filters in two separate
estimations of the longitudinal velocities of the car are not necessarily the same. Therefore,
we need a method to fuse the estimation results to give us a better estimation.
3.4. Sensitivity analysis
In previous subsections, we present UIKF for estimating the vehicle longitudinal velocity.
The goal of this methodology is computing the precise slip ratio using Equation (1). But, it is
clear that the estimated velocity has error in relation to the true value. Consequently, in this
section, we analyse the sensitivity of slip ratio with respect to longitudinal velocity. Based
on the definition of sensitivity,[42] the sensitivity of slip ratio with respect to longitudinal
velocity can be computed using Equation (34)
Sλ
vx
=
∂λ
∂vx
×
vx
λ
=
Rω
v2
x
×
vx
λ
=
1 − λ
λ
. (34)
This equation shows that in low values of slip ratio, the sensitivity is large and vice versa.
In other word, the accuracy of velocity estimation in low slip values is more important than
high slip values.
3.5. Multi-sensor data fusion
Combining the multi-sensor information deals with the problem of how to combine data from
different sensors in a sensor network. The combined information is developed to achieve
high-precision measurements or improvement of the estimation process.[43]
Generally speaking, the multi-sensor data fusion approach consists of methods such as
federated Kalman filter,[44] Bar-Shalom/Campo,[45] Millman method,[46] and so on. In this
research, we utilise the Millman method. In general, the Millman method provides a solution
to the problem of combining data from more than two sensors
ˆxk+1|k+1,Mill =
2
i=1
Jk+1,i ˆxk+1|k+1,i, (35)
where i = 1, 2 and Jk+1,i is a weighting matrix with dimension j × j based on the state vector.
The matrix Jk+1,i can be found by solving Equations (36) and (37)
Jk+1,1Pk+1|k+1,1 − Jk+1,2Pk+1|k+1,2 = 0, (36)
2
i=1
Jk+1,i = Ij = diag
j
(1, . . . , 1), (37)

where Ij is the unit matrix with dimensions j × j. Solving these two equations, result in
Jk+1,1 = Pk+1|k+1,2(Pk+1|k+1,1 + Pk+1|k+1,2)−1
, (38a)
Jk+1,2 = Pk+1|k+1,1(Pk+1|k+1,1 + Pk+1|k+1,2)−1
. (38b)
In this case, the state estimation error covariance matrix regardless of the covariance
between the sensors is calculated using the following equation:
Pk+1|k+1,Mill =
2
i=1
Jk+1,iPk+1|k+1,iJT
k+1,i. (39)
Since, by using Equation (33), the matrix Pk|k is not calculated, we can ﬁnd it by using
Equations (26-e) and (26-g) independently of the vehicle velocity estimation procedure.
These two equations can be rewritten as
Pk|k,i = (I − LC)(APk−1|k−1,iAT
+ Q)(I − LC)T
+ LRLT
, (40a)
Pk|k,i = (I − LC)APk−1|k−1,iAT
(I − LC)T
+ (I − LC)Q(I − LC)T
+ LRLT
. (40b)
Subsequently, in steady-state form, we have a discrete-time algebraic Lyapunov
equation as
Pss,i = (I − LC)APss,iAT
(I − LC)T
+ (I − LC)Q(I − LC)T
+ LRLT
. (40c)
Obviously, we can compute Pss,i ofﬂine and independent of estimation procedure.
Therefore, Equations (38) and (35) can be rewritten as Equations (41) and (42), respectively.
Jss,1 = Pss,2(Pss,1 + Pss,2)−1
, (41a)
Jss,2 = Pss,1(Pss,1 + Pss,2)−1
, (41b)
ˆxk+1|k+1,Mill =
2
i=1
Jss,i ˆxk+1|k+1,i. (42)
By using Equations (41) and (42) in data fusion procedure, this method did not add notable
extra computational load to estimation process, while it reduces the estimation error.
4. Experimental test
In this section, an experimental test is designed to evaluate the vehicle velocity estimation
approach during the braking process. The experimental test is started by moving the vehicle
and after speeding up enough (about 80 km/h), the severe braking is applied to the vehicle.
The whole time of the experimental test is done in about 36 s, but the time duration of the
braking process is less than 4 s.
As shown in Figure 5, the estimation process is separated into three main parts consisting of
preparing the system model, calculating the wheel speed, implementing UIKF and calculating
the slip value.
According to Equation (5), ωf and ωr are considered as system outputs, they should be cal-
culated by measuring the wheel speed sensors. Figure 6 shows the overview of how sensors
and data acquisition system are installed on the test vehicle. We have two sets of sensors,

Figure 5. A block diagram of the estimation process.
wheel speed sensors and a three-axis accelerometer. The output of wheel speed sensor is an
analogue pulse wave. By increasing the speed of the wheel the pulse frequency will also
increase. We have recorded the pulses by a data acquisition card with 10 KHz sampling fre-
quency according to the intended maximum initial longitudinal speed of the car in braking
duration (100 km/h) and Nyquist criterion.
A three-axis accelerometer sensor is employed to record the car longitudinal deceleration.
By integrating the deceleration, we can compute the actual velocity of the vehicle. We will
compare the estimation results with the actual velocity to qualify the estimation procedure.
It is worth noting that the accelerometer is mounted close to the centre of gravity (COG) of
the test vehicle. Also, by comparing the output signal of the accelerometer in z-axis with the
gravity acceleration, we compensated the effect of the vehicle pitch motion in measuring the
longitudinal deceleration during the braking process.
After filtering noises out from the pulses, ωf and ωr are calculated using Equation (43).
In this equation, fWss is the pulse frequency received from the wheel speed sensor within a
certain sample period and Nth is the number of teeth on the wheel speed sensor. We employed
an algorithm based on the measurement of a time interval between successive pulses for
calculating fWss [47]
ωwheel =
2πfWss
Nth
. (43)
Figure 7 shows the calculated wheel speed during braking. Using the wheel speed, the
information which is needed to estimate the vehicle longitudinal velocity is provided. To
estimate the longitudinal velocity exactly we need vehicle parameters. Table 2 gives the spec-
ification of the vehicle used for the experiments.[48,49] Also, the initial values required to
apply the filter are ˆx1|0 = [Rωωf, 0, 0, ωf, ωr]T
and P1|0 = 10I5.

Figure 6. Overview of the test vehicle, installed sensors and data acquisition system.
Figure 7. The calculated wheel speed using noiseless pulses.
In Section 3, we introduce linear UIKF for estimating the longitudinal velocity, while
as mentioned in Equations (2)–(23), the car’s dynamic model is completely nonlinear. For
systems described by linear time-invariant (LTI) models, like (24), the state estimation can
be successfully accomplished by using UIKF. But, LTI models only provide a good represen-
tation of nonlinear systems around an operating point and consequently the generated state

Table 2. Speciﬁcations of the test vehicle.
Symbol Value Unit Symbol Value Unit
m 850 kg mb 765 kg
mf 45 kg Lr 1.425 m
mr 40 kg Lf 0.92 m
Rω 0.27 m Iz 1627 kg m2
g 9.81 m/s2 Ip 2035 kg m2
Iω 4.15 kg m2 Kb 0.7 –
ψ 0.85 – Cd 0.32 –
H 0.57 m Ha 1.16 m
W 8330 N Af 1.087 m2
ρ 1.027 kg/m3 Nth 48 –
estimation is not valid far away from this point. Therefore, we employ a decoupled multi-
ple linear model representation of the vehicle in estimation procedure.[50] By linearising the
nonlinear model around several operating points, we achieved three decoupled linear mod-
els as follows which can model the nonlinear vehicle’s dynamic. The matrices A, Bu and Bd
of the discrete-time linear state space model (Equation (24)), for the vehicle under test with
10 KHz sampling frequency, are as
A =
⎡
⎢
⎢
⎢
⎢
⎣
0.99989951 + ε11 0.0 0.0 0.00002110 0.00000632
−0.00000014 0.99999996 −0.00025001 0.00000003 0.0
−0.00000006 −0.0 0.99999998 0.00000001 −0.0
0.00432343 0.00000033 0.00000030 0.99881970 0
0.00129530 −0.00000002 0.00000003 0 0.99964638
⎤
⎥
⎥
⎥
⎥
⎦
,
(44a)
Bu =
⎡
⎢
⎢
⎢
⎢
⎣
−0.0009
0.3335
0.1603
0.0164
0
⎤
⎥
⎥
⎥
⎥
⎦
× 10−4
, (44b)
Bd =
⎡
⎢
⎢
⎢
⎢
⎣
0
0
0
−0.1687
−0.0723
⎤
⎥
⎥
⎥
⎥
⎦
× 10−5
, (44c)
where
ε11 =
⎧
⎪⎨
⎪⎩
4.5 × 10−5
0.25 < λ,
−5 × 10−4
0.08 < λ ≤ 0.25,
0 λ = 0.
(45)
Due to that, the slip ratio is not a known value during the estimation procedure, the wheel
deceleration is used to choose the true value for ε11 as follows:
ε11 =
⎧
⎪⎨
⎪⎩
4.5 × 10−5
|Rω ˙ω| < 50(m/s2
)
,
−5 × 10−4
10(m/s2
)
< |Rω ˙ω| ≤ 50(m/s2
)
,
0 |Rω ˙ω| ≤ 10(m/s2
)
.
(46)

5. Estimation results and analysis
Since, we are going to estimate the vehicle longitudinal velocity during severe braking; the
estimation process is started as braking. Now, ωf and ωr can be considered as outputs and the
estimation process is performed by applying linear models in four scenarios:
Scenario-1: UIKF in two-output mode
In this scenario, ωf and ωr are considered as outputs and subsequently, matrix C is as
follows:
C =
0 0 0 1 0
0 0 0 0 1
. (47)
The vehicle longitudinal velocity is estimated by using Equation (26) and the result is
shown in Figure 8. Also, in Figure 8, the actual vehicle longitudinal velocity computed from
the three-axis accelerometer outputs is shown.
As mentioned in Section 3, we have to ensure the stability of the UIKF approach and
the convergence of the estimated value to the true value. According to Equation (30), for
the convergence of this approach, it is necessary that the coefficient of E[dk] is zero and
all eigenvalues of (I − LkC)A are placed into a unit circle. It is proved by Equation (31)
that the coefficient of E[dk] is zero but to verify the other condition in this scenario, the
eigenvalues of matrix (I − LkC)A are shown in Figure 9. This figure shows the eigenvalues
of (I − LkC)A used in the experimental test throughout the estimation process. As it can be
seen from Figure 9, all the eigenvalues of matrix (I − LkC)A are placed into a unit circle
which represents satisfying the second condition for convergence. Figure 10 illustrates the
convergence of the elements of the matrix L in the scenario. As can be seen, the matrix L is
changed during the estimation process but eventually tend to a constant value.
Scenario-2: UIKF in single-output mode using ωf
In this scenario, ωf is considered as output and subsequently, matrix C is as follows:
C = 0 0 0 1 0 . (48)
Following the above discussion, in this case, matrix L is constant and it can be computed
using Equation (49)
L =
1
CBd
× Bd = 0 0 0 1 0.4286
T
. (49)
Figure 8. Comparing the estimated velocity with actual vehicle velocity in Scenario-1.

Figure 9. Eigenvalues of matrix ((I − LkC)A); Scenario-1.
Figure 10. The convergence of the elements of matrix L; Scenario-1.
Figure 11. The actual value and estimated values of the vehicle longitudinal velocity in Scenarios 2, 3 and 4.
The eigenvalues of (I − LC)A in nominal case (ε11 = 0) are {0.9997, 0.9999, 0.9998,
0.9996, 0} and clearly, all eigenvalues are stable. The estimated velocity and the actual
velocity of the vehicle during the test are shown in Figure 11.
Scenario-3: UIKF in single-output mode using ωr

Figure 12. The estimated longitudinal slip and the actual longitudinal slip.
In this scenario, ωr is considered as output and subsequently, matrix C is as follows:
C = 0 0 0 0 1 . (50)
Similar to Scenario 2, the matrix L is as Equation (51) and all eigenvalues of (I − LC)A in
nominal case (ε11 = 0) are stable
L =
1
CBd
× Bd = 0 0 0 2.3333 1
T
. (51)
Figure 11 shows the estimated longitudinal velocity in comparison with actual velocity.
Scenario-4: Combining the results of Scenario-2 and Scenario-3
As explained in both Scenarios 2 and 3, in both cases, the vehicle velocity is estimated by
using simplified UIKF and just one output accurately. But, unfortunately, these two methods
have no similar estimation results. In this scenario, we are going to combine the results of
Scenario-2 and Scenario-3 using the Millman multi-sensor data fusion method to achieve
a better result. The Millman data fusion approach is recommended for uncorrelated local
estimates.[51] Due to that, we present a correlation analysis of velocity estimation errors
in Equation (52). By comparing the diagonal and off-diagonal elements of the correlation
matrix, as it can be seen the correlation is not strong. In Equation (52), ˜vx−ωf
and ˜vx−ωr
are
estimation errors of vehicle longitudinal velocity in two Scenarios 2 and 3, respectively
cov(˜vx−ωf
, ˜vx−ωr
) =
0.1437 −0.0279
−0.0279 0.1227
. (52)
The results of the estimated velocity by data fusion approach are shown in Figure 11. Also,
the estimated slip and the actual slip are shown in Figure 12.
It is clear that the estimation of the vehicle longitudinal velocity is much closer to the actual
longitudinal speed extracted from the acceleration sensor, although the change in the friction
coefficient is unknown to the estimator. Of course, all tests were done on flat, dry asphalt
with friction coefficient around 0.7 as might be seen in Figure 6. Meanwhile, in previous
research,[1,2,14] similar results were achieved in terms of accuracy of the estimation by
measuring braking torque and using the accelerometer in the estimation process. It should
be noted that, in this paper, a three-axis accelerometer is used to measure the longitudinal
acceleration and its data are only used for comparing the estimated velocity with its actual
value.

Table 3. The MSEs of the estimation and the computational time in each iteration step.
Programming environment
Scenario number
MSE of estimation for
longitudinal velocity Matlab C + +
1 0.1501 2.8 ms 21.65 μs
2 4.2799 0.53 ms 5.61 μs
3 2.2028 0.53 ms 5.61 μs
4 0.1714 0.72 ms 7.27 μs
Table 3 shows the mean square errors (MSEs) of the estimation for vx in all scenarios. This
result shows that in spite of the extremely noisy output and neglecting Tb, the estimation is
suitable.
Since, the computational speed is very critical in the experimental test, the estimation
algorithm has been implemented in both Matlab and C + + environment in order to com-
pare their computational speed. The estimation results are the same in both Matlab and C + +
environment but the computational speed is quite different. Table 3 also contains the compu-
tational time of simulation results in both Matlab and C + + environments for all scenarios.
The results show that the computational speed in the C + + environment is much faster than
the Matlab environment and implementing algorithm in the C + + environment can be suit-
able in the experimental test. Also, comparing the obtained results shows the advantages of
simplified filters.
6. Conclusion
In this paper, an algorithm is proposed to estimate the vehicle longitudinal velocity based
on the UIKF, in the presence of an unknown input called braking torque. In the estimation
process, a bicycle model of the vehicle and the corresponding analytic tyre force equations
are used. The experimental test is performed with an extra three-axis accelerometer sensor. It
is used to measure the actual vehicle longitudinal velocity and its data are used for comparing
the estimated velocity and validating the results of the estimation process. The experimental
results illustrate the acceptable performance of the estimation, despite extremely noisy output
and neglecting the braking torque.
In this study, a multiple linear model was considered for vehicle dynamic because,
the dynamic model of vehicle is nonlinear. Therefore, we employed a standard UIKF for
vehicle velocity estimation. It is recommended that further research be undertaken to employ
unknown input extended Kalman filter (UIEKF) in vehicle velocity estimation. It would be
interesting to compare the results of experiences for both UIKF and UIEKF.
Acknowledgment
The authors would like to thank Mrs Arezoo Shariki Moqadam for her efforts to implement the estimation algorithms
in C++.
Disclosure statement
No potential conflict of interest was reported by the authors.
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