∎

11institutetext: Alexandre M. Ribeiro 22institutetext: Department of
Computational Mechanics, School of Mechanical Engineering, Mendeleyev Street
200, 13083-860 Campinas, Brazil
22email<EMAIL_ADDRESS>33institutetext: Alexandra Moutinho
44institutetext: Institute of Mechanical Engineering - IDMEC, LAETA, Instituto
Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon,
Portugal
44email<EMAIL_ADDRESS>55institutetext: André R.
Fioravanti 66institutetext: Department of Computational Mechanics, School of
Mechanical Engineering, Mendeleyev Street 200, 13083-860 Campinas, Brazil
66email<EMAIL_ADDRESS>77institutetext: Ely C. de Paiva
88institutetext: Department of Integrated Systems, School of Mechanical
Engineering, Mendeleyev Street 200, 13083-860 Campinas, Brazil
88email<EMAIL_ADDRESS>
# Estimation of Tire-Road Friction for Road Vehicles: a Time Delay Neural
Network Approach

Alexandre M. Ribeiro  Alexandra Moutinho  André R. Fioravanti  Ely C. de Paiva

(Received: date / Accepted: date)

###### Abstract

The performance of vehicle active safety systems is dependent on the friction
force arising from the contact of tires and the road surface. Therefore, an
adequate knowledge of the tire-road friction coefficient is of great
importance to achieve a good performance of different vehicle control systems.
This paper deals with the tire-road friction coefficient estimation problem
through the knowledge of lateral tire force. A time delay neural network
(TDNN) is adopted for the proposed estimation design. The TDNN aims at
detecting road friction coefficient under lateral force excitations avoiding
the use of standard mathematical tire models, which may provide a more
efficient method with robust results. Moreover, the approach is able to
estimate the road friction at each wheel independently, instead of using
lumped axle models simplifications. Simulations based on a realistic vehicle
model are carried out on different road surfaces and driving maneuvers to
verify the effectiveness of the proposed estimation method. The results are
compared with a classical approach, a model-based method modeled as a
nonlinear regression.

###### Keywords:

Road friction estimation Artificial neural networks Recursive least squares
Vehicle safety Road vehicles

††journal: Journal of the Brazilian Society of Mechanical Sciences and
Engineering

##  1 Introduction

One of the primary challenges of vehicle control is that the source of force
generation is strongly limited by the available friction between the tire
tread elements and the road. In order to better understand vehicle handling
due to force generation mechanisms, several research activities related to
vehicle dynamics and control fields are oriented towards estimation of all
components of the tire-ground contact.

The knowledge of specific tire-road contact operation points, such as the
saturation point where the tire generates the maximum force available from
friction, could lead to a new range of applications in vehicle control.
Moreover, current commercial vehicle safety systems, such as Anti-lock Brake
Systems (ABS), Traction Control Systems (TCS) and Electronic Stability Control
(ESC) could have a significant improvement in performance by the knowledge of
the full vehicle states and operating conditions that are still limited by the
lack of information singh2014Estimation . Therefore, to reach this full
potential the recognition of the tire’s limit handling is indispensable. With
this is mind, we emphasize the importance of estimating the vehicle road
conditions, specifically the Tire Road Friction Coefficient (TRFC).

Friction estimation often relies on a model-based estimator using a well-
defined and interpretable mathematical model for the purpose of capturing the
inherent friction effects under the tire dynamics (forces and moments). The
most common model-based approaches use the steering system model Liang2019 ;
hsu2006method , quarter-car model AlZughaibi2018 , four-wheel vehicle dynamic
model ahn2011robust , powertrain and wheel dynamics model Castro2010 .

For the estimation problem, in pasterkamp1997  the correlation between the
self-aligning moment of the steering wheels and road friction was firstly
examined. Lately, hsu2009estimation ; hsuJournal ; ahn2011robust ;
ahnJournal2013  explored the use of a nonlinear Recursive Least Squares method
employed as a mean for identification of tire-road friction through an
observed data composed by the self-aligning moment and also expanded for a
lateral dynamic force analysis.

The wheel dynamic model can also be utilized with a tire model to estimate the
friction. In Muller2003 ; Hsiao2011 ; Rajamani2012  the wheel rolling motion
is used to detect the longitudinal force and longitudinal friction adopting
the powertrain configuration and wheel drive engine. The estimator is built
primarily exploring the force-slip ratio plane and its relationship with the
road friction coefficient.

Another model-based approach discussed in literature is a slip-slope
algorithm. This method is based on the assumption that the low slip-plane zone
(linear region of the force-slip plane, characterizing normal driving
conditions) can be used to estimate the tire-road friction. Distinct studies
Li2007 ; Qi2015 ; Xia2016  have shown this methodology.

Despite the majority of model-based methods, a number of algorithms have been
studied based on different concepts to estimate the surface condition. In
Casselgren2007 ; Tuononen2008 , an optical sensor is used as a tire sensor
that can measure the road ahead and the tire carcass deflections which may be
exploited in the estimation of friction potential. Cameras are also used to
identify different surfaces. The detection is based on the light polarization
changing when reflected from road surface Jokela2009 . Also, Jonsson2011
proposed a method that merges weather data and road images taken by a camera
on the vehicle. More recently, based on the hypothesis that the friction
coefficient affects the natural frequency of the vehicle systems, such as in-
wheel motor drive system or steering system, the road-friction is estimated
through frequency analysis Chen2015 ; Chen2017 .

In the field of Machine Learning and Artificial Intelligence the work
presented in Pasterkamp1998  primarily designed a feedforward neural network
optimizing its topology by means of Genetic Algorithms to determine the actual
tire forces from the measured signals. The estimated values were shown to have
potential of exploring the nuances about TRFC.

In Matuko2008 , friction force estimation is presented in a stricted
accelerating/decelerating maneuver. A two-layer feedforward neural network
with linear output is used with the vehicle longitudinal dynamics. The
proposed estimator is validated through a noiseless signal data.

In Zareian2015  and Zhang2017  a feedforward neural network is synthesized
using the vehicle response to both longitudinal and lateral excitations.
Therefore, the estimation is investigated under acceleration or brake maneuver
while cornering.

Among the model-based methods that implement TRFC estimation using neural
networks, some prior works also estimate friction through road conditions. It
is the case of vision-based works used to estimate the road conditions ahead
of the vehicle, where the task becomes a classification problem. In
Panahandeh2017 , an image dataset was used to train three machine learning
models including logistic regression, support vector machine, and neural
networks to predict the friction class. In a similar form Roychowdhury2018
proposes a convolutional neural network model to learn region-specific
features for road surface condition classification and how to infer the
friction coefficient. Other works also implemented the classification by
analyzing the road texture Yang2018 ; Du2019 .

In this study, the presented estimation process focuses on the dynamic
characteristics of a rear-motorized-wheels electric vehicle to achieve the
tire road friction estimation and contributes in the following aspect: the
estimator is developed by means of a time delay neural network (TDNN) as a way
to identify the TRFC based exclusively on the lateral force information and
the estimates are compared with a nonlinear least squares (NLS) estimator
based on a moving data window.

Although TDNN have been used for diverse purpose such as speech recognition
Wu2019 , movement behavior Sarah19 , joint angle estimation Triwiyanto18  and
nonlinear behavior prediction of a time series Ashwini18 , no other work has
studied the TDNN as a reliable procedure to estimate or measure the effect of
TRFC on a vehicle dynamic. Our main motivation for using TDNN is that the
time-delay arrangement structure enables the network to effectively capture
the temporal vehicle response due to a change in road friction.

This paper is organized as follows: Section 2 presents a vehicle theory
development with mathematical models for the tire force models which are used
in the estimation method. Section 3 details the least squares regression
method. In Section 4, the TDNN estimation algorithm proposed is described.
Simulation results are shown and analyzed in Section 5. Finally, this paper is
concluded in Section 6.

##  2 Tire-Ground Contact Model

When sufficient excitations exist in the lateral direction, vehicle lateral
dynamics can be used as the basis of the TRFC estimation. The most common tire
friction models used in the literature are those of algebraic tire slip angle
and force relationships. Although many approaches to the tire-road friction
modeling can be found, for this work we selected three analytical models.
These models were chosen for their clear and simple formulation. They have
fewer tunning parameters and have a good representation of the tire forces
nonlinearities.

As mentioned, the force generated between the tire and the road is related to
the slip angle and it is of fundamental importance for the knowledge of how
the lateral forces arise during a curve. The slip angle $\alpha$ is the angle
between the orientation of the tire and the orientation of the velocity vector
of the wheel, as depicted in Fig 1.

Figure 1: Schematic diagram of the planar vehicle and tire slip angle
representation

The formal definition of slip angle can be derived via kinematic analysis of a
planar four-wheel vehicle and is usually defined as:

$\begin{split}\alpha_{fl,fr}=&\arctan\Big{(}\frac{v+ar}{u\pm
cr}\Big{)}-\delta_{fl,fr},\\\ \alpha_{rl,rr}=&\arctan\Big{(}\frac{v-br}{u\pm
cr}\Big{)},\end{split}$ (1)

where $u$ and $v$ are vehicle longitudinal and lateral velocity components,
$r$ is the vehicle yaw rate, $a$ and $b$ are the distances from the vehicle
center of gravity to front and rear axles, respectively, and $c$ is half of
the wheelbase distance. $\delta_{i}$ stand for tire steering angles and the
subscripts $fl$, $fr$, $rl$ and $rr$ denote quantities corresponding
respectively to the front left, front right, rear left and rear right wheels.
In the vehicle model sign convention, the steering angle is negative for left
turns and positive for right turns.

###  2.1 Mathematical Formulation

Tire models express the relationship between tire forces and moments with slip
ratio and slip angle.

Different mathematical tire models have been developed in the literature. The
most widely used model is the semi-empirical tire model introduced by Pacejka
pacejka2006tyre , called Pacejka tire model or Magic formula. In a simplified
form, the formulation of this tire model for lateral force is as follows:

$F_{y}=D\sin[C\arctan\\{B\alpha-E(B\alpha-\arctan B\alpha)\\}]+S_{v_{y}},\\\ $
(2)

where D, C, B, E are the Magic formula semi-empirical parameters based on tire
measurement data, $S_{v_{y}}$ is the vertical offset of the characteristic
curve and $\alpha$ is the slip angle.

A second model, known as Dugoff’s tire model, was developed in 1969 by Dugoff
et al. dugoff1969tire . In its simplest form, the lateral force is expressed
as:

$\displaystyle F_{y}=$ $\displaystyle-C_{\alpha}\,\tan\alpha\,f(\lambda),$ (3)
with $\displaystyle f(\lambda)$
$\displaystyle=\begin{cases}(2-\lambda)~{}\lambda\text{,}~{}&\text{if}~{}\lambda<1\\\
1&\text{otherwise}\end{cases}$ $\displaystyle\lambda$ $\displaystyle=\frac{\mu
F_{z}}{2C_{\alpha}\lvert\tan\alpha\rvert},$

where $F_{z}$ is the normal tire load, $\mu$ is the friction coefficient and
$C_{\alpha}$ the cornering stiffness. Conceptually, cornering stiffness is a
property of the tire that changes slowly with time due to tire wear, inflation
pressure, and temperature fluctuations hsu2009estimation .

Finally, another widespread model is the Brush model pacejka2006tyre , which
defines the lateral force as follows:

$\displaystyle F_{y}=\begin{cases}-3\mu
F_{z}\theta_{y}\sigma_{y}\big{\\{}1-\lvert\theta_{y}\sigma_{y}\rvert+\frac{1}{3}(\theta_{y}\sigma_{y})^{2}\big{\\}}\text{,}&\lvert\alpha\rvert<\alpha_{sl}\\\
-\mu F_{z}\text{sign}(\alpha)&\text{o/w}\end{cases}$ (4) where
$\displaystyle\theta_{y}=\frac{C_{\alpha}}{3\mu F_{z}},$
$\displaystyle\sigma_{y}=\tan\alpha,$
$\displaystyle\alpha_{sl}=\arctan\left(\nicefrac{{1}}{{\theta_{y}}}\right).$

Although this article only introduces the most popular and widely used
approaches in tire-road friction estimation, there are many valuable studies
that have tried to develop new friction models. This subject is addressed in
broader texts and books about ground vehicle dynamics such as pacejka2006tyre
; Wong2008theory

The lateral force characteristic curve for each of the presented models is
shown in Fig. 2 for several friction coefficients. Initially, the lateral
forces increase linearly with the slip angle until it reaches saturation,
which represents the tire force limits.

(a)

(b)

(c)

Figure 2: Tire lateral force characteristic curve for each model, varying
normal force $F_{z}$ and three levels of friction $\mu$

These models show similar behaviors when the slip angle is small. However,
they may deviate from each other when high values of vertical force $F_{z}$
and friction $\mu$ are available. These characteristics suggest that, in the
estimation processes, these models could lead to errors due to model
discrepancy.

##  3 Tire-road Friction Identification Through Parametric Regression

As seen in the previous section, the lateral force can be characterized by
three fundamental parameters: tire slip angle $\alpha$, normal force $F_{z}$
and tire-road friction $\mu$.

When a sufficiently large lateral excitation is detected during a vehicle
maneuver, the of road friction estimation can be achieved using the measured
signals and the analytical models (2)-(4). This methodology can be seen as a
problem of fitting experimental data to a nonlinear analytical function, as
addressed in hsu2009estimation ; hsuJournal ; ahn2011robust ; ahnJournal2013 .
The method allows the formulation of the problem as one of unconstrained
nonlinear least-squares (NLS) optimization.

In other words, we desire to investigate how well we can identify our lateral
tire parameters using lateral force information. This requires a good
measurement of the lateral tire forces, as well as the knowledge of individual
tires. However, if unavailable, the use of an estimate of the axle forces
(lumped forces) may hold a lumped friction estimate.

The nonlinear curve-fitting in a least-squares problem consists of finding
decision variables $x$ that solve the problem:

$\displaystyle~{}~{}~{}x^{*}$ $\displaystyle=\arg\min_{x}\left\lVert
F(x)-\overline{F}\right\rVert_{2}^{2},$ (5)
$\displaystyle=\arg\min_{x}\sum_{k}^{N}\Big{(}F(x)-\overline{F}_{k=1}\Big{)}^{2},$
(6)

where $x^{*}$ is the optimum value that minimizes the objective function, with
$F(x)$ the parametric function and $\overline{F}$ representing the measured
data.

Assuming that the tire analytical models are a good representation of the
lateral tire force behavior, they can be used as a parametric function of the
NLS method with $N$ sets of observed data (in this case, groups of $F_{y}$ and
$\alpha$).

Despite the promising results of this approach hsuJournal ; ahnJournal2013 ,
the NLS method has some drawbacks. It requires a long computation time and
sometimes this process fails to converge to the true optimal values. The
estimator based on NLS generally shows stable estimation results, but does not
always guarantee stability and it is difficult to quantify the stability and
convergence ahn2011robust . Furthermore, a critical drawback of the NLS is
that it is computationally heavy. In a low-speed microprocessor, it may not
sustain the same level of performance.

As an alternative to this methodology, we propose solving the problem using
neural networks. The theme is approached in a similar form, with the same
window of $N$ observable data applied in a time-delayed neural network.

##  4 Estimation of Tire-road Friction Coefficient Using Neural Networks

This section proposes a time delay neural network to detect the TRFC. Two main
benefits are expected from this method: firstly, a TDNN can establish network
connections and the relationship between input and output instead of storing
an entire complex tire model in the controller, which can significantly reduce
the computations, guarantee the real time performance and avoid model errors
due to model discrepancy; secondly, because the TDNN is trained by measured
data, it is able to create a mapping from input parameters to the friction
coefficient and accurately capture the temporal structure hidden in the data
Zhang2017 .

As the analytical models (2)-(4) show, the lateral force is dependent of
$\alpha$, $F_{z}$ and $F_{y}$ and these are therefore the parameters selected
to feed the neural network. Fig. 3 shows the overall structure used for the
TRFC estimation.

Figure 3: Block diagram of the proposed hierarchical estimator

Although the existence of tire force sensors, the forces are still hard to
measure and the sensors are very expensive. As solution, a Kalman filter
variant is used for tire force estimation. Here, we use the approach presented
in Cordeiro2016 ; Cordeiro2017  to estimate $F_{y}$ and $F_{z}$ using ordinary
vehicle sensors, such as GPS, inertial measurement unit (IMU) and encoders.

The observers are developed based on nonlinear vehicle dynamic models with the
Extended Kalman Filter (EKF) algorithm. Three estimators are used in a cascade
structure to decouple the forces dependency. For vertical forces a 3D vehicle
model is used to create a 12-states nonlinear system. For longitudinal and
lateral forces the planar vehicle model is used to create two 7-states
nonlinear systems. Compromising precision and simplicity, the Dugoff tire
model is chosen as the analytical model.

All measures needed for the estimation process are listed in Table 1.

Table 1: Signals and description of measures Signal | Description
---|---
$a_{x}$, $a_{y}$, $a_{z}$ |  Vehicle longitudinal, lateral and vertical accelerations
u, v, w |  Vehicle longitudinal, lateral and vertical linear velocities
p, q, r |  Vehicle roll, pitch and yaw angular rates
$h_{z_{ij}}$ | Suspension deflection
$\delta_{ij}$ | Tire steering angle

The aforementioned works about the filter and observer showed that the overall
EKF forces estimator is robust against TRFC variations. It turns out that the
update stage uses inertial measurements (acceleration) and is capable to
correct the forces errors induced by prediction stage. The intuition behind
this is that Newton’s second law equations prediction is efficiently corrected
by a proper acceleration measurement. Therefore, a constant $\mu=0.8$ is
applied for the EKF forces estimation.

With this approach the forces are detected individually, which holds the
potential of detecting the TRFC independently for each tire. The wheel slip
angle $\alpha$ is calculated straightforward using (1).

Also, a supplementary consideration should be taken to ensure the algorithm
outputs reasonable estimates. Due to sensors noise and the inherent
perturbation on lateral forces, and specially on slip angle calculation, the
TDNN inputs should be low-pass filtered to prevent the high frequency
disturbances from being propagated to the estimate. A unit gain 5 Hz low-pass
filter is applied to the estimated forces and slip angle, as shown in Fig. 3.

Before advancing into the learning process of the proposed neural network, it
is important to make one addition to the model. When considering the
correlation between friction and each tire measure, the correlation
coefficient between the normalized lateral force $\nicefrac{{F_{y}}}{{F_{z}}}$
is significantly higher if compared with each force separately, as listed in
Table 2. According to (1), the slip angle $\alpha$ is determined by vehicle
velocities and normal force $F_{z}$ is mainly affected by the roll over
effect. These measures are only affected indirectly by friction, thereby, a
low correlation is expected.

Table 2: Correlation coefficients between the specified variables | $\alpha$ | $F_{y}$ | $F_{z}$ | $\nicefrac{{F_{y}}}{{F_{z}}}$
---|---|---|---|---
$\mu$ | 1.681e-7 | 0.2955 | 1.186e-8 | 0.4076

Given this fact, the normalized forces $\nicefrac{{F_{y}}}{{F_{z}}}$should be
selected as one input to feed the neural network instead of $F_{y}$ and
$F_{z}$ separately. The basis for this choice also lies on the friction circle
concept in which the maximum value of the resultant force is determined along
a circle (directly influenced by friction), and this value can be decomposed
into the limits of the normalized forces Gim1991 .

It is important to note that a positive correlation was observed for positive
lateral forces (data obtained from a right hand maneuver). A negative
correlation with similar magnitude is expected for negatives forces. The
signal, thus, is a consequence of the reference frame.

The TDNN architecture is depicted in Fig. 4. Two inputs were selected: a
normalized lateral force $F_{y}/F_{z}$, obtained from the kalman estimator,
and the calculated slip angle.

Figure 4: Time-delayed neural network architecture with a single hidden layer

The configuration of the proposed TDNN for TRFC is as follows (see Fig. 4): 2
inputs with 50 samplings delay (observation window of size N = 50) and one
single hidden layer with 50 neurons. The neurons differentiable transfer
function is nonlinear, properly selected as a Tan-Sigmoid transfer function.

In the neural network data collecting stage, about 200,000 original data are
obtained from simulation with a 100 Hz sampling rate. The range of variation
of the network input parameters to the tire model is bounded as described in
Table 3. The friction coefficient is set with different levels and the vehicle
response (data of $\alpha$, $F_{y}$ and $F_{z}$) is obtained.

Table 3: Data training parameters and space dimension Input parameter |  Variation
---|---
Friction coefficient $\mu$ |  0.3 to 1.2 at intervals of 0.1
Slip angle $\alpha$ |  [-0.12 0.12] rad
Lateral Force $F_{y}$ |  [-2.8 2.8] kN
Normal force $F_{z}$ |  [2 4.4] kN

The observation window size was chosen based on the nature of our system. The
rise time response due to a change in friction is shown in Fig. 5. The slip
angle response (and consequently lateral tire forces, see Fig. 2) shows an
average rise time of 0.2 s. In an attempt to capture the vehicle behavior
during these transitions, we choose the window size to be approximately double
the rising time. Therefore, it is defined N = 50 that corresponds to an
observation window of 0.5 seconds.

(a)  Figure 5: Rise time due to a change in friction. At instant 60 s friction
changes from 0.8 to 0.6 and at instant 80 s from 0.6 to 0.9. Rise time is 0.21
s and 0.19 s for left and right plot, respectively.

As in the classical Neural Network, the Time Delay Neural Network also has a
training phase. Training was achieved using Matlab Neural Network Toolbox. The
Levenberg-Marquardt algorithm is used with 1000 epochs of training iterations
with 70% of the collected data randomly taken as the training set, 15% used
for validation set and 15% as the test set.

Training phase and further evaluations were carried out on a desktop
architecture which features a four-core 4.00 GHz Intel Core i7-6700K
Processor, NVIDIA GeForce GTX 950, 32GB of RAM and Ubuntu 16 LTS OS. The
system took 7002 seconds to train the aforementioned dataset.

Fig. 6 shows the error residuals of the trained neural network. About 88% of
the total data are distributed around zero.

Figure 6: Error histogram of the TDNN dataset

The correlation coefficient (R-value), which is a linear regression between
the TDNN predicted values and the targets, and the mean squared error (mse)
are shown in Table 4. With R values around 0.94 and 0.95, suggests that the
models has a considerable potential of matching the network response to the
friction change patterns. Notice also that both R and mse values are close for
Training, Validation and Test dataset, which indicates that the network is not
overfitting the data.

Table 4: Correlation coefficient R-value and mse error. Values of R closer to 1 indicate better agreement between targets and predicted values | R | mse
---|---|---
Training | 0.94355 | 1.0680e-3
Validation | 0.94123 | 1.2597e-3
Test | 0.94059 | 1.3500e-3
Total | 0.94276 | 1.1391e-3

##  5 Results

The simulation results presented in this section are obtained using
Matlab/Simulink. A representative and realistic full-vehicle multibody
dynamics model (including a steering system, powertrain system, suspension
system and the Pacejka tire model for tire ground interactions), was used
consisting of the following motions:

  * •

Longitudinal, lateral and vertical body motion;

  * •

Wheels rotation;

  * •

Unsprung masses motion;

  * •

Pitch, roll and yaw body rotation.

The physical parameters of the car used in this study are listed in Table 5.
All values are extracted from rafaelcordeirothesis  where a complete vehicle
modeling and data validation are given. The simulator dynamics is formulated
as a 32-states model. The vehicle is configured to simulate a independent rear
wheel drive vehicle, providing references of a vehicle state and measured
signals. Gaussian noises are added (according to the commercial MTi Xsens
sensor specifications (MTi-G-700)) in the simulated measurements to
realistically reproduce a real application.

Table 5: List of vehicle main physical parameters rafaelcordeirothesis  Parameter name |  Value
---|---
Vehicle mass |  1100 kg
Yaw inertia moment |  1350 $\nicefrac{{kg}}{{$m^{2}$}}$
Roll inertia moment |  337.5 $\nicefrac{{kg}}{{$m^{2}$}}$
Pitch inertia moment |  1350 $\nicefrac{{kg}}{{$m^{2}$}}$
Distance from CG to front wheels |  1.5 m
Distance from CG to rear wheels |  1.9 m
Wheelbase |  1.8 m
Wheel rotational inertia |  1 $\nicefrac{{kg}}{{$m^{2}$}}$
Wheel radius |  0.25 m
Height of CG |  0.5 m

The simulation results of three representative maneuvers are presented here.
Table 6 gives the details and purpose of each maneuver. Fig. 7 provides the
physical representation of each proposed scenario where the color designates
the friction coefficient. The change in friction is time-dependent and occurs
at successive time intervals.

During simulation, the system is set on an equilibrium point with constant
longitudinal velocity. It is achieved by a PID cruise controller Osman09 . In
fact, with constant longitudinal speed, the longitudinal tire forces are
reactive with low magnitude and would not have significant impact on lateral
forces due coupling effect Hindiyeh09 .

Each case was performed on a different theoretical surface, where a
theoretical $\mu$ = 1.0 surface roughly corresponds to driving on a dry
pavement, $\mu$ = 0.8 on a wet pavement and $\mu$ = 0.6 corresponds to driving
on gravel Wang2004 .

Table 6: Details and investigative purpose of each simulated maneuver Simulated Maneuver | Test Surface | Purpose
---|---|---
Ramp steer with constant friction coefficient (Fig. 7a) |  Constant dry pavement ($\mu$ = 0.8) |  Investigate the early friction sensing ability of the estimator. During the maneuver, lateral forces are gradually increasing, eventually reaching saturation.
Constant steer - case 1: step varying $\mu$ for all tires (Fig. 7b) |  Five levels of TRFC with unequal steep sizes ($\mu$ = 1.0 to 0.6) |  Determine the estimator response to a quick change in road surface. Friction coefficient is varied at five new levels.
Constant steer - case 2: different step varying $\mu$ for left/right tires (Fig. 7c) |  Each wheel is subjected simultaneously to two different frictions ($\mu_{FL}$ = 0.9, 0.8, $\mu_{FR}$ = 0.8, 0.7) |  Validate the TRFC detection under small friction variations. Also, each wheel is exposed to different surfaces at the same time (distinct $\mu$ for FL and FR tires).

(a)

(b)

(c)

Figure 7: Schematic road layout with time-dependent changing of friction
coefficient

It is valid to remark that, even though our training and test sets included
both left and right turns, due to the symmetry of the system, right turns have
a mirror response of left turns that is also included in the network as the
result of a data augmentation process.

###  5.1 Ramp Steer Maneuver

Fig. 8 displays the data resulting from the simulation of a left-hand ramp
steer maneuver. The steering angle $\delta$ goes linearly from 0 to -18
degrees at the roadwheels reference.

(a)

(b)

(c)  (d)

Figure 8: Front right lateral force and slip angle for a ramp steer maneuver

To ensure that there is enough data to be meaningful for the NLS fit and the
TDNN approach, first the process is initialized by placing a slip angle
threshold $\alpha_{thres}$. The slip angle data must exceed $\alpha_{thres}$
before the estimation begins, otherwise the fitting optimization may not
guarantee a reliable solution.

The TDNN estimator will be here compared with the NLS approach. To show the
dependency of the NLS fit with the mathematical model, the regression is
performed choosing the Dugoff and Brush models as a parametric function of the
nonlinear regression. The window is selected with size $N=50$ and will be used
in all cases showed from here on.

Using the estimation algorithm with $\alpha_{thres}$ = 1 degree, the NLS and
TDNN algorithm waits until the front tire slip angle exceeds $\alpha_{thres}$
at t = 12 s before fitting the force-slip data (see Fig. 9). The estimated
value $\hat{\mu}$ is the optimum solution $x^{*}$ of the optimization problem
(6) and before instant 12 s the estimation simply holds $\hat{\mu}$ at initial
value $\mu_{0}=0.5$.

(a)

Figure 9: Friction estimates from ($F_{y}$,$\alpha$) data for different
approaches

The slight increase in the friction estimate as the maneuver progresses is
expected. Initially, during linear tire regime operation, lateral forces
measurements have yet to reach their peak value and both methods underestimate
the friction coefficient. As more lateral force measurements become available,
the peak force limit is reached and the friction estimate reaches a final
estimate. Therefore, adequately large slip angles are required for stable and
accurate estimation in both TDNN and NLS methods and the slip angle data
threshold is indispensable.

The model error also becomes apparent on the NLS fit, where Brush and Dugoff
models show different convergence values due to model discrepancy. This
divergence arises due to the difference between Pacejka model, used to
generate the data, with the models used on the estimators (as discussed in
section 2).

###  5.2 Constant Steer: Case 1

In this setup, the tire-road friction coefficient is set at five levels,
varying randomly from 1.0 to 0.6. The transitions occur during successive
equal time intervals of 10 seconds. The vehicle is set on an equilibrium point
in a constant left turn maneuver. The steering angle of the front left and
right tires is set to -18.36 and -15.82 degrees. These values follow the
steering Ackerman Geometry. As a consequence, the inner-turn wheel reaches
higher tire side slip and lateral force if compared to the outer-turn wheel,
as shown in Fig. 10. Also, vertical force is higher on the right tire due to
the load transfer. This response appears as a result of the roll over effect
measured as a change in vehicle center of mass location relative to the
wheels.

(a)

(b)  (a) front left (fl)

(b)

(c)

(b) front right (fr)

Figure 10: Slip angle, vertical and lateral forces of the tires: Constant
steer case 1

The TRFC estimation results are shown in Fig. 11. One can note the front left
estimative is more accurate and less oscillating than the front right
estimative. A necessary condition for good estimation results, as shown
previously, is a large lateral excitation (high slip angle).

(a)

(b)

(c)

Figure 11: Front left and front right friction estimates in a time varying
friction scenario

This scenario highlights the dependency of the NLS to the parametric function.
The NLS final estimates of each interval show a constant error bias while the
TDNN produces a solid and concise estimative. Naturally, there are
discrepancies between the dynamic behavior of the real tire system and the
derived mathematical model (see Fig. 2). Therefore, a constant error bias
should be expected on the NLS model-based approach.

Table 7 displays the root mean square (RMS) error of the estimates of the
front left and right tires. Although very similar, a high overall estimation
accuracy is achieved for both techniques and it shows great promise for a real
implementation.

Table 7: RMS error of the estimated friction - case 1 Methodology | $\mu_{fl}$ | $\mu_{fr}$
---|---|---
TDNN | 0.0346 | 0.0350
$\text{NLS}_{\text{Dugoff}}$ | 0.0421 | 0.0357
$\text{NLS}_{\text{Brush}}$ | 0.0546 | 0.0395

Moreover, the rate of convergence is slightly higher on the TDNN approach.
Fig. 12 highlights this response by zooming in Fig. 11a on three intervals of
friction transitions. The TDNN estimates converges to a more accurate values
with faster responses than the NLS estimates. This behavior lies in the fact
that the relationship between input and output was correctly mapped on the
database and therefore can be observed on the following results.

(a)

Figure 12: Friction estimates rate of convergence comparison

###  5.3 Constant Steer: Case 2

While case 1 showed an equal change for both wheels, here, the tire-road
friction coefficient is set at two different levels for each tire. The
transition occur during an equal time interval of 10 seconds. Left wheel
friction undergoes a transition from 0.9 to 0.8 while right TRFC goes from 0.8
to 0.7.

The vehicle is set on an equilibrium point in a constant left turn maneuver.
The knowledge of forces and friction of individual tires is desirable and
would offer stability control systems with most needed information. Thus, this
change maneuver is conducted to verify that the estimator can identify the
friction for each tire individually and certify that the estimation is indeed
independent for each wheel.

Fig. 13 displays the simulated slip angle, lateral and vertical forces from
the proposed right-hand steer maneuver. At instant 10 s, the road surface
adhesion coefficient decreases to a different value for each tire. Again,
vertical forces are maintained constant due to the maneuver nature, with the
changing in friction mostly affecting lateral forces and slip angle.

(a)

(b)

(a) front left (fl) (b)

(c)  (b) front right (fr)

Figure 13: Slip angle, vertical and lateral forces of the tires: Constant
steer case 2

Fig. 14 shows the estimation results. With the TDNN approach, the individual
wheel friction is confidently estimated with high accuracy. Since slip angles
are larger than for the case 1 scenario, it satisfies the required large
lateral forces excitation and gives very accurate estimates. On the NLS
estimation results, however, a constant error are still apparent and can be
seen during the transitions. This characteristic should be considered when the
estimated result is used for control purposes.

(a)

(b)

(c)

Figure 14: Front left and front right friction estimates in a time varying
friction scenario with different friction for each wheel

The TDNN shows a slightly better estimation quality, as seen in the RMS error
listed in Table 8. Note that the regression based method still exhibits the
hindsight bias caused by the inevitable model differences.

Table 8: RMS error of the estimated friction - case 2 Methodology | $\mu_{fl}$ | $\mu_{fr}$
---|---|---
TDNN | 0.0638 | 0.0477
$\text{NLS}_{\text{Dugoff}}$ | 0.0652 | 0.0493
$\text{NLS}_{\text{Brush}}$ | 0.0666 | 0.0486

A significant difference in processing time is found when comparing
performance and computational efficiency. Fig. 15 shows the normalized
histogram of the computation time for both techniques. These values, computed
at each time instant, were obtained from 14000 data signals.

It is clear that the computation time of TDNN algorithm is lower than NLS. In
summary, TDNN averages 0.594 milliseconds to process the signal (corresponding
to one iteration) compared to 3.379 milliseconds of the NLS. This has the
potential to allow a higher sample frequency of the complete system.

(a)

(b)

Figure 15: Normalized histogram of computational time for NLS and TDNN methods

##  6 Conclusions

In this study, we presented a hierarchical TRFC estimation method based on a
time delay neural network and compared it with a classical nonlinear
regression approach, using the same data observation window. The overall
estimation algorithm was evaluated on varying road surfaces with three
different scenarios. A high-fidelity and realistic full-vehicle multibody
dynamics model for Matlab/Simulink platform was used in the simulations.

Although road-friction was accurately identified using both algorithms, there
is a primary shortcoming in the presented lateral-force based friction
estimation: it requires sufficient levels of lateral excitation for the
correct friction identification. An earlier knowledge of the TRFC is
desirable, however, both approaches showed a similar behavior: a necessary
waiting time for the tire slip angle to fulfill the observation window
satisfying the specified excitation threshold.

Nonetheless, as an algebraic methods, the NLS method relies more heavily on an
accurate model which may be a source for estimation errors. Indeed, the NLS
needs a prespecified parametric function for precise tunning parameters and,
as a consequence, stationary estimation errors may be expected if the given
function is not adequate.

On the other hand, the TDNN method is independent of any mathematical tire
model, however, requires a sufficient and representative database. In this
study, the TDNN was also able to provide estimates with lower RMS errors
compared with the classical regression approach. It also demands less
computation time at each time instant and may be the best alternative for a
real time implementation in embedded systems.

Since the proposed method is only analyzed theoretically and validated via
simulation, an actual benchmark or field test is needed in the subsequent work
to verify the proposed approach. Future works may also include the design of a
time delay neural network containing not only lateral information but also
longitudinal forces, slip ratio and self-moment align.

###### Acknowledgements.

This work was supported by FCT Portugal, through IDMEC under projects LAETA
(UID/EMS/ 50022/2019). The mobility of A. Ribeiro has been possible with the
Erasmus Mundus SMART2 support (Project Reference: 552042-EM-1-2014-1-FR-ERA
MUNDUS-EMA2) coordinated by CENTRALESUPELEC. The authors also acknowledge the
support of FAPESP through Regular project AutoVERDE N. 2018/04905-1, Ph.D.
FAPESP 2018/05712-2 and CNPq grant 305600/2017-6.

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