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sensors
Article
Investigation on the Model-Based Control Performance in
Vehicle Safety Critical Scenarios with Varying Tyre Limits
Aleksandr Sakhnevych 1,* , Vincenzo Maria Arricale 1, Mattia Bruschetta 2, Andrea Censi 3, Enrico Mion 3,
Enrico Picotti 2and Emilio Frazzoli 3
Citation: Sakhnevych, A.;
Arricale, V.M.; Bruschetta M.,
Censi, A.; Mion, E.; Picotti, E.;
Frazzoli E. Investigation on the
Model-Based Control Performance in
Vehicle Safety Critical Scenarios with
Varying Tyre Limits. Sensors 2021,21,
5372. https://doi.org/10.3390/
s21165372
Academic Editor: Juan A. Cabrera
Received: 21 June 2021
Accepted: 2 August 2021
Published: 9 August 2021
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conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1Department of Industrial Engineering, University of Napoli Federico II, 80125 Naples, Italy;
vincenzomaria.arricale@unina.it
2Department of Information Engineering, University of Padova, Via Gradenigo, 6/B, 35131 Padova, Italy;
mattia.bruschetta@dei.unipd.it (M.B.); picottie@dei.unipd.it (E.P.)
3Institute for Dynamical Systems and Control, ETH Zurich, 8092 Zurich, Switzerland;
acensi@idsc.mavt.ethz.ch (A.C.); enmion@ethz.ch (E.M.); efrazzoli@ethz.ch (E.F.)
*Correspondence: ale.sak@unina.it
Abstract:
In recent years the increasing needs of reducing the costs of car development expressed by
the automotive market have determined a rapid development of virtual driver prototyping tools that
aims at reproducing vehicle behaviors. Nevertheless, these advanced tools are still not designed to
exploit the entire vehicle dynamics potential, preferring to assure the minimum requirements in the
worst possible operating conditions instead. Furthermore, their calibration is typically performed
in a pre-defined strict range of operating conditions, established by specific regulations or OEM
routines. For this reason, their performance can considerably decrease in particularly crucial safety-
critical situations, where the environmental conditions (rain, snow, ice), the road singularities (oil
stains, puddles, holes), and the tyre thermal and ageing phenomena can deeply affect the adherence
potential. The objective of the work is to investigate the possibility of the physical model-based
control to take into account the variations in terms of the dynamic behavior of the systems and of
the boundary conditions. Different scenarios with specific tyre thermal and wear conditions have
been tested on diverse road surfaces validating the designed model predictive control algorithm
in a hardware-in-the-loop real-time environment and demonstrating the augmented reliability of
an advanced virtual driver aware of available information concerning the tyre dynamic limits. The
multidisciplinary proposal will provide a paradigm shift in the development of strategies and a solid
breakthrough towards enhanced development of the driving automatization systems, unleashing the
potential of physical modeling to the next level of vehicle control, able to exploit and to take into
account the multi-physical tyre variations.
Keywords:
model-based control; vehicle dynamic potential; tyre thermodynamics; tyre wear;
weather influence; vehicle safety; double lane change; safety optimization
1. Introduction
The information concerning the vehicle’s non-linear physical limits depending on the
thermal and wear states of tyres, the pavement characteristics, and the boundary conditions
(wet or icy ground, under-inflated or worn tyre, etc.) represents a fundamental additional
value for the optimal behavior of safety- and performance-oriented control logics [1–3].
Virtual driver prototyping is becoming an increasingly exploited tool, allowing the
car manufacturer to perform the majority of the testing campaign already in the design
phase of the vehicle. Specific prototyping choices can be reproduced and evaluated in any
condition within the virtual environment, also at the limit of performance, minimizing the
time-to-market and connected costs [4,5].
In this field, closed-loop control strategies have been widely studied in past years to
address the problem of path following for autonomous driving cars. Examples can be found
Sensors 2021,21, 5372. https://doi.org/10.3390/s21165372 https://www.mdpi.com/journal/sensors
Sensors 2021,21, 5372 2 of 21
in [
6
], where a nested PID steering control has been designed for the lane-keeping task, and
more recently in [
7
], where a pure pursuit controller has been specifically developed for
path tracking. The most recent VD implementations rely on a vehicle controller based on a
non-linear model predictive control (NMPC) technique, which is a model-based control
strategy able to compute the optimal sequence of control inputs over a prediction horizon,
by minimizing a tailored cost function [
8
,
9
]. The control technique is applied in a receding
horizon mode and is capable of handling constraints and the intrinsic non-linearities of the
vehicle model [10].
The main advantages of the NMPC approach are the capability of the controller of
handling all significant features of the process dynamics directly: in this way, the constraints
on variables involved in the task (track limits, actuator constraints) can be easily integrated
into the optimal control problem, hence guaranteeing the maximal exploitation of vehicle
capabilities. Moreover, it is a predictive technique that allows optimizing the vehicle
behavior over a future horizon in time, and therein system states and controls. In this
way, the controller is allowed to retrieve information about future vehicle behavior and
about possible dangerous situations, aiming at anticipating actions and providing suitable
controls for challenging vehicle handling.
The objective of the work consists in the integration of the information concerning
the tyre dynamic limits within the definition of a virtual driver (VD), implemented as a
vehicle controller aiming at testing the vehicle behavior at limit of handling condition,
and demonstrating the advantages in terms of both enhanced active safety and optimized
performance. An interesting VD definition that addresses the problem of real-time obstacle
avoidance on low-friction road surfaces has been proposed in [
11
], where the code gener-
ation tool ACADO [
12
] has been used to define and solve the NMPC problem. Another
similar implementation of such a controller for an autonomous ground vehicle has been
proposed in [
13
], where the controller has been also validated in co-simulation with a
hard real-time dSPACE DS1005 Autobox system. The vehicle model employed in the both
implementations has consisted of a four-wheel vehicle, where tyres have been described
by means of a linear tyre model and Fiala tyre model for longitudinal and lateral dynamics,
respectively [
14
,
15
]. The inputs are the steering angle and the front/rear braking ratios,
while the bounds are defined through specifically defined spatial constraints. A different
virtual driver definition, especially designed for high performance vehicles, has been de-
veloped in [
16
]: here, the vehicle model integrates longitudinal load transfer and Pacejka’s
lateral tyres forces model. The controller implementation has been tested in a real-time
co-simulation with a commercial software VI-CarRealTime (VI-CRT) within a double lane
change (DLC) maneuver, where the abilities of the controller have been demonstrated with
high speed operating conditions and a challenging track geometry. The NMPC strategy has
also been applied in racing environment as the autonomous vehicle controller for handling
1:43 scale, RC electric vehicles [
17
] and autonomous racecar [
18
], with the specific purpose
of achieving aggressive maneuvering and lap time minimization.
In this work, the authors aim to investigate the possibility to employ the model-based
strategies to control the non-linear time-dependent system, i.e., the full vehicle model with
temperature and wear sensitive tyres operating in completely different environmental
conditions. To perform the study, the standardized DLC maneuver, currently employed for
the validation of virtual driver and advanced driving assistance systems
(ADAS) [16,19,20]
,
is implemented in Matlab/Simulink virtual environment. The vehicle and tyre models
have been characterized and validated for a reference GT vehicle, identifying the requisite
complex tyre–road coupled phenomena concerning the temperature, wear, and road pave-
ment dependencies [
21
,
22
]. Four different roads, i.e., dry, wet, snowy, and icy, two diverse
tyre mileages, i.e., new and worn, and three thermal tyre conditions are combined and
analyzed in the study to understand which could be the advantages of the employment
of the model-based controllers, aware of the tyre instantaneous characteristics, bound-
ary operating and weather conditions, and overall vehicle dynamic potential [
23
]. The
model-based control logics, able to make use of the additional information concerning
Sensors 2021,21, 5372 3 of 21
the dynamic limits of the system, is tested in co-simulation with a 14 degrees-of-freedom
vehicle plant model, where the tyres are described by means of a Pacejka’s magic formula
(MF) model. The vehicle controller is based on a robust and computationally effective
non-linear model-predictive-control (NMPC), implemented in the open-source NMPC
software MATMPC [
24
], able to take into account the additional instantaneous information
concerning the varying adherence potential and the vehicle non-linearities. The infor-
mation concerning the vehicle non-linear physical limits depending on the thermal and
wear states of tyres, the pavement characteristics and the boundary conditions (wet or icy
ground, under-inflated or worn tyre, etc.) represents a fundamental additional value for
the optimal behavior of safety- and performance-oriented control logics [
25
–
27
], as it allows
to maximize the potential to avoid obstacles and to reduce the severity of collisions [28].
The authors aim to lay the foundation of the future advanced driving systems, sen-
sitive to environmental conditions and adaptive to continuously varying characteristics
of the underlying non-linear system. Being currently mainly based on mere empirical
calibration, the physical model-based estimation can represent a crucial factor towards the
improvement of the pedestrians’ and passengers’ active safety, enabling the management
of the activation threshold ranges on the basis of the instantaneous operating and the
environmental boundary conditions [
29
,
30
]. This can be already employed in the current
ADAS to communicate to the driver the necessity to co-act in specific situations, but it also
constitutes a fundamental root for the future driving automatization [31,32].
The paper is organized as follows: Section 2introduces the problem description
concerning complex phenomena linked with the tyre–road interaction and their influence
on the overall vehicle dynamics; Section 3describes the advanced methodologies developed
to characterize, model, and reproduce the dynamic behavior of the real system in the
virtual environment, and introduces the adopted model-based control, evidencing the
peculiarities of the designed cost function; in Section 4the outputs of the conducted
simulations employing different road surfaces, in adverse boundary conditions and with
diverse states of the tyres are discussed, addressing particular attention towards the control
strategies. Finally, in Section 5, a discussion on the next developments and the conclusions
are presented.
2. Problem Description
A proper understanding of the tyre dynamic behavior and of its multiple intrinsic
dependencies is a crucial topic for tyre manufacturers, to improve tyre performance and
durability, for users, to set the optimal working conditions, and for researchers, to develop
computationally efficient mathematical models able to represent the experimental behavior
with a high degree of accuracy. Friction phenomenon, arising at the tyre–road interface,
originates from three physical contributions: the adhesive term relative to molecular
Van der Waals links arising between the two counter surfaces in mutual contact, the
hysteretic term linked to the deformation losses within the elastomeric material, and the
wear term [
33
,
34
]. All of them are deeply interconnected and dependent on the specific tyre
working conditions, in terms of sliding velocity, temperature, and pressure distributions,
arising at the tyre contact patch as a result of different excitation spatial frequency spectra,
representative of diverse types of road pavement [
35
]. Furthermore, tyres may deeply
modify their dynamic behavior over time due to ageing effects, influencing the dynamic
potential of the overall vehicle [22].
The enrichment of the vehicle state with the information concerning the tyre in-
stantaneous and potential friction will allow, taking into account the tyre multi-physical
variations (Figures 1and 2), represents a key point in the development of control logics,
able to adapt to sudden variation in boundary conditions in order to guarantee the vehicles
higher stability in critical scenarios . Indeed, in the Figure 2it is possible to observe how
the adherence ellipse changes in three tyre thermal ranges.
Sensors 2021,21, 5372 4 of 21
(a) (b)
Figure 1.
Tyre behavior variations. (
a
) Compound temperature influence on the characteristic
interaction shape. (b) Wear effect on available grip.
(a) (b) (c)
Figure 2.
MF-based standard and evo tyre models compared with the experimental points in three thermal ranges: under-
heating condition (
a
), optimal temperature (
b
), over-heating condition (
c
), (camber angle =
−
2
deg
|vertical load = 3000
N
).
The aim of the proposed adaptive control will be to avoid collisions and to minimize
risks in any environmental condition, validating all the scenarios on interest in a highly ac-
curate simulation environment. The information concerning the vehicle non-linear physical
limits depending on the thermal and wear states of tyres, the pavement characteristics and
the boundary conditions (wet or icy ground, under-inflated or worn tyre, etc.) represents a
fundamental additional value for the optimal behavior of safety- and performance-oriented
control logics. The non-linear model predictive control approach is employed to integrate
the tyre varying dynamic parameters within the definition of physical constraints of the ve-
hicle, guaranteeing the stability of the system and allowing to achieve the optimal solution
for the defined vehicle instantaneous dynamic limits.
3. Physical Model, Physical Model-Based Control, and Virtual Scenario
To parametrize the vehicle and the tyres’ model, the authors have collected data with
a chosen GT vehicle in a specific test session on track. Due to a non-disclosure agreement
with the industrial research partner, the vehicle and the track will not be specified.
The track session has consisted of handling tests in the widest possible range of tyre
operating conditions in terms of temperature, pressure, and wear level. Following the vehicle
model parametrization and the tyre parameters’ estimation procedures described in [
36
,
37
],
the vehicle non-linear system has been completely characterized in all the conditions of
interest, being able to faithfully reproduce the experimental data in the virtual environment.
3.1. Vehicle Parametrization
The 14 degrees of freedom (DoF) vehicle model, based on the mathematical represen-
tation described in [
38
], has been modeled in a
MATLAB/Simulink
environment
as follows:
Sensors 2021,21, 5372 5 of 21
•
6 DoF to reproduce longitudinal, lateral, vertical, pitch, roll, and yaw motion of the
vehicle body;
•
4 DoF concerning the wheel rotation and 4 DoF for the wheel normal displacement,
with the hypothesis that the degrees of freedom to the relative motion between
the wheel and the vehicle body can be neglected along the longitudinal and lateral
directions, allowing only the independent rotational and vertical displacements.
Furthermore, the parameterized vehicle is rear-wheel drive with front steering and internal
combustion engine. The tyre model is described by Pacejka’s magic formula model, whose
parameters have been characterized for different conditions of temperature, pressure, and
wear. Per each road surface under study (dry, wet, snowy, and icy), the tyre-road friction
coefficient has been supposed constant and is applied as an additional scaling factor of
the
λµx
and
λµy
parameters [
39
], linearly combining the tyre characteristics identified on a
reference road with the ones potentially achievable on diverse pavement surfaces.
The vehicle dynamic behavior in the reference tyre conditions has been validated in a
slow-ramp-steer maneuver, whose parameters are summarised in the Table 1and outputs are
illustrated in the Figure 3, feeding the model with the steering input presented in the
Figure 4a):
Table 1. Slow-ramp-steer inputs.
Description Value Unit
start time 13.26 s
end time 20.3 s
initial velocity 27.9 m/s
initial gear 3 -
ramp duration 7.04 s
initial steer 0 deg
slope steer −22.29 deg/s
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
ay [ m/s2 ]
-160
-140
-120
-100
-80
-60
-40
-20
0
- l/R [ ° ]
Outdoor
Simulated
(a)
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
ay [ m/s2 ]
0
0.2
0.4
0.6
0.8
1
1.2
1.4
[ ° ]
Outdoor
Simulated
(b)
Figure 3.
Comparison between outdoor acquisitions and simulation output. (
a
) Steering angle vs.
lateral acceleration diagram. (b) Sideslip angle vs lateral acceleration diagram.
For the validation purpose, lateral acceleration
ay
, steering angle
δ
, side slip angle
β
have been compared for the same inputs. Figure 3shows the comparisons between
experimental data and model outputs shown on the classic
ay−δ
and
ay−β
diagrams. An
aspect that is worth pointing out is the difference between the black dashed and continuous
lines: the first one is obtained using the starting parameters provided by the research
partner, the second one is obtained employing the calibration procedure described in [
37
].
In particular, the starting under-steering characteristics (dashed lines) have been revised
better identifying the parameters linked to the anti-roll bars stiffness and the steering maps,
leading to a less under-steering behavior within the handling diagram, in agreement with
the experimental data.
Sensors 2021,21, 5372 6 of 21
0 2 4 6 8 10 12 14 16 18 20
Time (sec)
-160
-140
-120
-100
-80
-60
-40
-20
0
20
[ ° ]
Outdoor
Simulated
(a)
-50 0 50 100 150 200 250 300
Longitude
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
20
Latitude
(b)
Figure 4.
Example of lateral maneuver’s input reproduction. (
a
) Experimental and simulation
steering angle comparison. (b) Slow-ramp-steer trajectory in virtual environment.
The enhanced parametrization has led to a higher slope in the linear section
(Figure 3a
),
but also higher lateral grip and side-slip angle values, related to the rear axle behavior
(Figure 3b). Once the vehicle and the tyres’ subsystems have been properly characterized
in the specific range of temperature, pressure, and wear, the validity range of the MF tyre
model has been extended adopting the MF-Evo one, described in [
40
]. In particular, the
tyre model calibration process can be summarized in three fundamental steps: the first one
is related to the pre-processing of the experimental data (which allows to discern useful
information contained in the acquired data and to eliminate the non-physical outliers); the
second one concerns the identification of the standard MF micro-coefficients in a specific
range of temperature, pressure, and wear; the third step aims at the calibration of the
additional multi-physical analytical formulations, taking into account of the entire dataset
and, thus, extending the tyre model towards thermal and degradation phenomena.
The calibration results are visible in terms of adherence ellipse in the Figure 1, where
the experimental data have been compared towards the MF and MF-evo outputs within
different temperature working ranges of the tyre. Finally, the parameters of the MF-evo
model have been further modified to extend the applicability of the tyre model on different
road surfaces, modifying the identified friction factors towards the pavement characteristics,
as reported in the Table 2. The resulting interaction characteristics for different tyres, in
diverse thermodynamic conditions and in contact with different road surfaces have been
summarized in Figures 5and 6.
-15 -10 -5 0 5 10 15
slip angle [deg]
-1
-0.5
0
0.5
1
y [-]
Dry road
Icy road
Snowy road
Wet road
(a)
-1 -0.5 0 0.5 1
x [-]
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
y [-]
(b)
Figure 5.
New tyre in optimal thermal condition in contact with different road surfaces. (
a
) Lateral
interaction characteristics. (b) Adherence ellipse.
In steady-state conditions, the global force exerted by the tyres is in a dynamic equi-
librium with the centrifugal force, as a function of the longitudinal velocity of the vehicle
Sensors 2021,21, 5372 7 of 21
v
and the instantaneous cornering radius
R
, relating the lateral acceleration
ay
and the
longitudinal velocity vof the vehicle’s center of mass (CM) by the equation:
ay=v2
R; (1)
To demonstrate the potential influence of the road surface characteristics on the
overall vehicle behavior, a set of simulations has been conducted with different tyre
parameters described in Figure 5in a steady-state lateral slow-ramp-steer (SRS) maneu-
ver. The maximum achievable value of the forward velocity
v
for a given curvature and
ay−δ
characteristics are reported for dry, wet, snowy, and icy pavement conditions in the
Figure 7a,b, respectively.
-15 -10 -5 0 5 10 15
slip angle [deg]
-1
-0.5
0
0.5
1
y [-]
Tyrenew - Toverheated
Tyrenew - Tcold
Tyrenew - Toptimal
Tyreworn - Toverheated
Tyreworn - Tcold
Tyreworn - Toptimal
(a)
-1 -0.5 0 0.5 1
x [-]
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
y [-]
(b)
Figure 6.
New and worn tyres in diverse thermal conditions in contact with the dry road. (
a
) Lateral
interaction characteristics. (b) Adherence ellipse.
(a) (b)
Figure 7.
SRS maneuver on different road surfaces. (
a
) Vehicle understeer characteristics. (
b
) Maxi-
mum velocity achieved.
Table 2. Summary of the velocity maximum values assumed for each road scenario.
Friction Coefficient µLateral Acceleration ayLongitudinal Velocity v
[−][m/s2] [m/s]
0.35 0.35 5.92
0.55 0.70 8.37
0.80 1.50 12.2
1.00 2.52 15.9
3.2. Internal Vehicle Model
A four-wheel vehicle model based on the description in [
16
] has been used as the in-
ternal model for the NMPC controller. Specific characterization of load transfers, gear shift
Sensors 2021,21, 5372 8 of 21
predictions, longitudinal force saturation, and an ellipsoidal tyre friction constraint have
been also introduced in the model definition to improve the overall prediction capabilities
of the controller. Finally, the model dynamics have been reformulated in spatial coordinates
with respect to the curvilinear abscissa
s
along the track. In this way, track constraints can
be defined with respect to space and the time can be considered as a minimization variable,
as already highlighted in previous works [16,41,42].
The continuous-time dynamics model is described as
˙
ξ=φ(ξ(t),u(t);p(t)), (2)
where the state is represented by
ξ(t)∈Rnx
,
u(t)∈Rnu
is the input, whereas the time-
varying parameter vector is p(t)∈Rnp.
¨
x=˙
y˙
ψ+1
m ∑
i,j
Fxi,j−Fd
x!,¨
y=−˙
x˙
ψ+1
m ∑
i,j
Fyi,j!,
¨
ψ=1
Iz"a ∑
j
Fyf,j!−b ∑
j
Fyr,j!+c ∑
i
Fxi,r−∑
i
Fxi,l!#,
(3)
where longitudinal and lateral positions are
x
,
y
, while
ψ
is the yaw angle.
m
and
Iz
are
the mass and the inertia around the vertical axis of the vehicle, respectively.
a
,
b
,
c
are the
vehicle dimensional parameters, front wheels to CM longitudinal distance, rear wheels to
CM longitudinal distance, and wheels to CM lateral distance, respectively.
F{x,y}{i,j}
are
the lateral and longitudinal forces on the wheels and
Fd
x
is the longitudinal drag force in
the vehicle’s reference frame. Subscripts
i∈ { f
,
r}
refer to front or rear wheels,
j∈ {l
,
r}
left or right wheels. Figure 8illustrates the physical quantities involved and the reference
systems chosen.
δf
is the steering angle of the front wheels, assumed the same for the both
front tyres, and
βf,j
is the side slip angle of the
f,j
-th tyre. The projection of cornering and
longitudinal forces in the vehicle frame, the position and the dynamics of the vehicle’s CM
in the inertial frame
X
,
Y
, and the vehicle side slip angle
β
are described in [
16
], whereas
the longitudinal drag force and the down-force are modeled as [43] pp. 97–98.
Figure 8. Internal vehicle model for control.
Differently from [
16
], the longitudinal tire forces in each wheel reference frame are
computed as
Fli,j=fengi,j−fbrki,j, (4)
Sensors 2021,21, 5372 9 of 21
where the engine and braking forces are
fengi,j=satτengi
rw,µFzi,j,fbrki,j=satτbrki
rw,µFzi,j, (5)
where
µ
is the tyre friction coefficient,
rw
is the wheel radius and the saturation function is
defined in (8). Then, the engine and braking torques at the wheels are:
τengi=γt(τmax
eng,i−τmin
eng,i) + τmin
eng,i,τbrki=γbτmax
brk,i, (6)
where
γt,b
are the normalized throttle and braking efforts,
τmax
brk,i
is the maximum torque
given by the braking system to front/rear wheels,
τmax
eng,i
and
τmin
eng,i
are the maximum and
minimum torque values expressed by the engine at front/rear wheels at a given gear and
are changed as a time-varying parameter to the actual model gearshift. To compute the
torques in the prediction horizon, an iterative strategy predicting the engine rpm, and,
hence, gearshift, based on the predicted velocity is used [
16
]. Specifically, the engine rpm
quantity is computed as
rpmpred =vpred
x
rw
diffratio
gearratio
60
2π, (7)
where
diffratio
and
gearratio
are the input/output torque ratios at the differential and at
the gearbox (in a specific gear), respectively. The dependence of
τmax,min
eng,i
w.r.t. the engine
rotational velocity has been neglected. Finally, the saturation function is defined as:
sat(fa,fb) = fb
1+exp(−5(fa
fb−1
2)) . (8)
The normal forces
Fzi,j
are modeled considering the load transfer in steady-state
condition as described in [
44
]. The algebraic loop in the model has been avoided by
considering
Fsat0
x
(total longitudinal force expressed in the vehicle frame saturated at
nominal
Fz
) and
Fstatic
y
(the sum of the lateral forces computed at nominal
Fz
on each wheel)
as the forces used for the load transfer dynamics.
Finally, the lateral forces model is based on the simplified MF model described in [
39
]
pp. 187–188, expressed by means of the macro-parameters B,C,D,E.
3.3. NMPC Algorithm
The goal of the NMPC controller for the virtual driver is to compute a reliable sequence
of steering, throttle, brake commands in a prediction horizon, given a tailored cost function.
The NMPC algorithm is based on
MATMPC
[
10
,
45
], an open source software built in
MATLAB
for real-time NMPC solution.
In
MATMPC
, a non-linear programming problem (NLP) is formulated at sampling instant
i
by applying direct multiple shooting [
46
] to an optimal control problem (OCP) over the
prediction horizon
S= [s0
,
sf]
, which is divided into
N
shooting intervals
[s0
,
s1
,
. . .
,
sN]
,
as follows
min
ξ·|i,u·|i
N−1
∑
k=0
1
2khk(ξk|i,uk|i)k2
W+1
2khN(ξN|i)k2
WN(9)
s.t. 0 =ξ0|i−ˆ
ξ0, (10)
0=ξk+1|i−φk(ξk|i,uk|i;pk|i),k∈[0, N−1], (11)
rk|i≤rk(ξk|i,uk|i)≤rk|i,k∈[0, N−1], (12)
rN|i≤rN(ξN|i)≤rN|i(13)
where
ξ·|i= (ξ>
0|i
,
ξ>
1|i
,
. . .
,
ξ>
N|i)>
, and
u·|i= (u>
0|i
,
u>
1|i
,
. . .
,
u>
N−1|i)>
, while
ˆ
ξ0
represents
the measurement of the current state. System states
ξk|i∈Rnξ
are defined at the discrete
Sensors 2021,21, 5372 10 of 21
arc-length point
sk
for
k=
0,
. . .
,
N
and the control inputs
uk|i∈Rnu
for
k=
0,
. . .
,
N−
1
are piece-wise constant. Their definitions are given in
(14)
and
(15)
. Here,
(12)
is defined
by
r(ξk|i
,
uk|i):Rnξ×Rnu→Rnr
and
r(ξN|i):Rnξ→Rnl
with lower and upper bound
rk|i
,
rk|i
. Equation
(11)
refers to the continuity constraint where
φk(ξk|i
,
uk|i
;
pk|i)
is a numerical
integration operator that solves
(16)
with initial condition
ξ(
0
) = ξ0|i
and returns the
solution at
sk+1
. The time has been included as a state variable with the following ODE
˙
t=1
˙
s
to fulfil the minimization of the travel time over the prediction horizon. The full state
vector is then given by:
ξ= [ ˙
x,˙
y,˙
ψ,eψ,ey,δf,γt,γb,t]T, (14)
where
eψ
,
ey
are orientation and lateral error of the vehicle with respect to the center-line of
the path, respectively. The input computed by the algorithm is then:
u= [ ˙
δf,˙
γt,˙
γb,eslip,eerr ,egg]T, (15)
where
˙
δf
,
˙
γt
,
˙
γb
are the derivatives of the actual input to the vehicle and
e
are slack vari-
ables. This formulation allows a smooth action of the controller and avoids too aggressive,
unrealistic behaviors.
The dynamics equation of the model used in the NMPC algorithm can be compactly
written as
ξ0=φ(ξ(s),u(s);p(s)), (16)
where p(s) = hζ(s),τMAX,min
eng,i(s)iT.
The real-time iteration scheme (RTI) [
47
] is employed to reduce the time required
to solve the
(9)
problem. Moreover, a non-uniform grid strategy [
48
] has been used for
lowering the computational burden and let the controller predict a sufficiently long horizon
(chosen 400 m in advance for the specific vehicle).
The cost function for the NMPC is defined as:
hk(ξk,uk) = [β,γt·γb,ζ·γt,t,˙
δf,˙
γt,˙
γb,eslip,eerr ,egg]>,
hN(ξN) = [β,γt·γb,ζ·γt,t,ey−eref
y,˙
ey,eψ−eref
ψ+
+β,˙
eψ]>.
(17)
The penalty on the vehicle side slip
β
is used to limit the sliding behavior of the
vehicle; simultaneous throttling and braking are penalized by the cost
γt·γb
. The
ζ·γt
cost
is included to make the controller accelerate smoothly during the final phase of the track
corner exit. The objective variable time
t
is added to minimize the time on the prediction
horizon. Smooth control actions are ensured by the objective terms on the inputs. The
three slack variables are also adopted to define the soft constraints [49], which increase the
robustness of the overall procedure. Finally, the terms related to errors
eyand eψ
, used only
as terminal objective variables, are introduced to integrate information about the trajectory
over the prediction horizon.
The constraints are defined as
rk= [δf,γt,γb,˙
δf,˙
γt,˙
γb,eslip,eerr ,egg,β+eslip ,
ey+eerr,(µx
¨
xext
g)2+ (µy
¨
yext
g)2+egg]>,
rN= [δf,γt,γb]>,
(18)
where the constraints on
δf
,
γt, and γb
are intrinsic bounds of the actual vehicle controls,
while those on
˙
δf
,
˙
γt, and ˙
γb
are added in order to improve the smoothness of the computed
inputs and can be used to easily tune the aggressivity of the NMPC driving commands.
Additionally, the slack variables have been constrained in order to help the optimization
procedure restricting the search space of the inputs. The slack variables are used for defining
the soft constraints: the first one is introduced on the side-slip of the vehicle and helps
Sensors 2021,21, 5372 11 of 21
the controller to regain control of the vehicle in case of high skidding; the second one
is used to correct the trajectory when the vehicle is out of track; the third one instead
is designed to make the controller respect the required gg diagram, which represent the
maximum combined longitudinal-lateral acceleration that can be induced by the combined
longitudinal-lateral behavior of the specific tyre spec [
50
].
µx
and
µy
are the longitudinal
and lateral friction coefficient of the tyres, respectively, whereas the considered accelerations
on the vehicle are
¨
xext =∑i,jFxi,j−Fx
d
m,
¨
yext =∑i,jFyi,j
m.
(19)
At the
i
-th sampling instant, considering that the QP solution is
∆ui∗
, the control input
is updated by
ui∗=ui−1∗+∆ui∗, (20)
The first sample of
ui∗
is applied to the vehicle, the prediction horizon is shifted
forward and the optimization procedure is repeated with updated state measurement.
3.4. Co-Simulation Environment
The co-simulation platform, represented in the Figure 9, is composed of the
following subsystems:
•
Plant model: a 14 DoF vehicle model reproducing the overall vehicle
dynamics behavior;
•
Road pavement: a boundary condition module concerning the asphalt condition
and computing the tire-road friction coefficient to reproduce dry, wet, snowy, and
icy contact;
•
Tyre: MF-Evo tyre model reproducing the tyre dynamic behavior in different thermal
and wear conditions;
•
Path reference: the track geometrical representation defined by the specific maneuver
and employed to compute the cost function.
Figure 9. Co-simulaton platform.
The maneuver chosen for the current study is the emergency double-lane-change
maneuver, generally performed on the highway to overtake another vehicle [
51
]. The test
is commonly adopted because it correlates the ability of controlling the vehicle at the limits
Sensors 2021,21, 5372 12 of 21
of handling with an enhanced safety for the vehicle occupants in scenarios concerning
the presence of obstacles on the path [
52
]. Given the criteria for ideal lane-change path,
prescribing a minimal length path with a smooth and continuous curvature at a given
vehicle forward velocity, the trajectory of the DLC maneuver is computed without violating
the track boundaries and assuring that all the tyres remain always in contact with the road
surface (possible lift motions are avoided with constraints modelled within the maximum
load transfers, as described in [9]).
The co-simulation is conducted in
MATLAB/Simulink
environment, coupling the plant
model with NMPC controller and performing the dynamic simulation of the plant model
at
fsim
= 1000 Hz, while the control action is updated by NMPC at
fctrl
= 100 Hz. The
simulations have been computed on a
Windows 10
machine with Intel(R) Core(TM) i7-
7700HQ @ 2.80GHz CPU.
4. Analysis and Results
The knowledge of the instantaneous and potential grip directly on board and in real-
time potentially allows the vehicle control logic to maximize the probability of avoiding
obstacles and reducing the severity of collisions. To investigate the possible outcomes of a
model-based control within a vehicle safety-linked scenario, the authors have performed
within the DLC maneuver a complete design of experiment comprehending:
•
Case A: the adoption of two different sets of NMPC weights (best and global) in the
definition of the cost function.
The best NMPC set of weights addresses the maximum achievable performance of
the underlying vehicle plant model, specifically calibrated for a new tyre working in
the optimal thermal range in contact with the dry road, whereas the global NMPC
set of weights represents the trade-off solution to guarantee ability of the vehicle to
complete the DLC maneuver in the worst proposed dynamic scenario, i.e., a worn
cold tyre in contact the icy road surface. In this case, the parameters of the plant and
the controller models are the same for each simulation;
•
Case B: the analysis of the vehicle dynamic response in case of different tyre thermal
and ageing conditions on the same road and in case of the tyre with a specific thermal
and wear state on different pavements. In this case, the parameters of the plant and
the controller models are the same for each simulation;
•
Case C: the possibility to employ the non-linear model predictive controller calibrated
with the average set of weights in conditions where the parameters of the controller
model can be updated in real-time on the basis of the actual state of the plant model or
can be constant and with an estimation on the friction value affected by a percentage
error respect the real value.
This particular scenario has been conducted to highlight the importance of the correct
estimation of the parameters of the controller model, potentially aware of the actual
knowledge of tyre-road friction. The simulation outputs with average tyre parameters
within the controller model have been compared towards the ones obtained with the
instantaneous parameters of the co-simulated vehicle plant to put in evidence the
importance of the correct information concerning the tyre friction and stiffness for the
vehicle dynamics control.
The simulation outputs have been compared in terms of the vehicle trajectory, the
forward velocity, the vehicle side slip angle, and yaw angle.
4.1. Case A
In this section, the impact of two possible sets of weights, defined within the NMPC
cost function, is investigated. Both the plant and controller models share the same model
parameters of a new tyre in the optimal thermal window in contact with the dry road.
The best set of NMPC weights represents the most suitable solution to perform the
DLC maneuver with both the plant model and the controller model in the maximum
performance conditions of the tyre, corresponding the the maximum dynamic limits of
Sensors 2021,21, 5372 13 of 21
the vehicle. The global set of NMPC weights stands for the conservative trade-off solution,
calibrated to guarantee the accomplishment of the maneuver in all the possible tyre-linked
and boundary conditions, in which the plant and controller models share the same physical
parameters (i.e., the performance of the vehicle controller is limited by the worst possible
dynamic scenario of a cold and worn tyre in contact with the icy road).
In the Figure 10a the trajectories of the vehicle with the best (red) and global (black)
sets of NMPC weights are compared. It is easy to observe that the optimized set of
weights allows the vehicle performing at a larger trajectory and achieving significantly
higher velocities both in the first part of the curves and at the end of the DLC maneuver
(
Figure 10b
). It is worth highlighting that the best set also demonstrates higher side slip and
yaw angles (Figure 11a,b), because it is specifically optimized to perform in the scenario of
a new optimal tyre in contact with the dry road, therefore allowing the vehicle to reach the
actual friction limits. Furthermore, the best set allows the vehicle to approach to the DLC
manuever and to end the scenario 6.62 and 8.34 s before, respectively (Figure 11c).
(a) (b)
Figure 10.
(
a
) Vehicle trajectory performed in the DLC maneuvers in a different road surface (dry in
black, wet in red, snow in blue, and icy in light blue), but with the same tyre (new tyre in optimal
range temperature) for a NMPC tuned to better perform the maneuver in all road surface, tyre, and
temperature condition. (b) Vehicle velocity.
(a) (b) (c)
Figure 11. (a)βangle. (b) Yaw angle. (c) Time.
4.2. Case B
In this section, only the global set of NMPC weights has been employed to compare
the dynamic response of the vehicle in two scenarios: (1) different road characteristics
(dry, wet, snowy, and icy) with the new tyre within the optimal thermal range, and (2)
different tyre thermal and ageing conditions in contact with the dry road. The plant and
the controller models share the same physical parameters for each iteration.
•Scenario B1
In the Figure 12a it is possible to observe how the vehicle maneuver characterized
by the highest friction coefficient (dry pavement) performs the DLC with a largest
Sensors 2021,21, 5372 14 of 21
trajectory and the highest velocity Figure 12b in minimum amount of time Figure 13c
and Table 3.
(a) (b)
Figure 12. (a) Vehicle trajectory. (b) Vehicle velocity.
(a) (b) (c)
Figure 13. (a) Side slip angle. (b) Yaw angle. (c) Time.
Table 3. Summary of time’s maneuver for each scenario.
Road Surface Time [s]
Dry 24.8
Wet 26.2
Snowy 40.8
Icy 51.7
Since the global NMPC set is limited by the most critical dynamic condition (worn
cold tyre in contact with the icy road), the Figure13a shows higher values in terms of
side slip angle for snowy and icy road surfaces, foreseeing the possibility to perform
the maneuver in more aggressive way for dry and wet road conditions.
Such a conservative behavior can be motivated by the fact that the global set of weights
is a result of a trade-off between completely different dynamic scenarios in the respect
of vehicle maneuverability and safety.
•Scenario B2
The comparison between a same road condition (dry) performing with different tyre
condition (new or worn, in the optimal temperature range, cold or overheated) are
shown in the following figure. Regarding the analysis of trajectories, shown in the
Figure 14a it is possible to observe how they are too similar each other due to the same
road pavement, however in the new tyre condition a little largest trajectory has been
carried out to achieve an highest velocity Figure 14.
Sensors 2021,21, 5372 15 of 21
(a) (b)
Figure 14.
(
a
) Vehicle trajectory performed in the DLC maneuvers in a dry road, with different tyre
condition (New tyre (continuous lines) and worn tyre (dashed lines) in optimal (black), cold (blue),
and overheated (red) temperature range. (b) Vehicle velocity.
(a) (b) (c)
Figure 15. (a) Side slip angle. (b) Yaw angle. (c) Time.
Table 4. Summary of time’s maneuver for each scenario.
Tyre Condition Time s
New −Topt 23.0
New −Tcold 23.7
New −Toverheated 23.44
Worn −Topt 24.24
Worn −Tcold 25.8
Worn −Toverheated 24.8
The analysis side slip angle show a dependence of
β
angle with the tyre stiffness,
indeed the highest value of
β
has been performed to highest cornering stiffness Figure 15a.
Finally, in the Table 4are shown the performing time for each condition.
4.3. Case C
•Scenario C1
In this paragraph the aim of the authors is to argue the following query:
If the plant and the controller do not share the same model parameters, i.e., the parameters
of the controller model are not updated by a specific co-simulated estimator of the vehicle
parameters and state, and of the tyres’ and the road conditions are not known a priori, how a
controller model with an average "parameters’" configuration could perform with different
plant model employment scenarios within the DLC maneuver?
With this purpose, the controller model has been fed with the parameters of
friction and stiffness corresponding the mean value of the all possible tyre-road
conditions explored.
It is worth highlighting that, as expected, it is not possible to perform the DLC
maneuver with the icy road with the above configuration. Indeed, as appears clear in
Sensors 2021,21, 5372 16 of 21
the Figure 16, the rear axle achieves the maximum slip ratio, not allowing to complete
the simulation in safety.
Figure 16. Slip ratio achieved for the four tyres.
For this reason, in the following figures, only dry, wet, and snowy road conditions
are reported. In the Figure 17a,b it is possible to observe how the difference between
the three pavement surfaces are less pronounced towards the results discussed in
Scenario B. Moreover, the vehicle in contact with the wet road achieves a maximum
velocity, even higher than with the dry surface, completing the maneuver in less
time Figure 18c).
(a) (b)
Figure 17.
(
a
) Vehicle trajectory performed in the DLC maneuvers in a dry, wet, and snow road, with
new tyre in optimal range temperature. (b) Vehicle velocity
(a) (b) (c)
Figure 18. (a) Side slip angle. (b) Yaw angle. (c) Time.
The reason for such behavior can be conducted to the conservative control action,
particularly visible in dry boundary condition, since the absolute difference in terms
of the friction limit is particularly high between the plant and the controller mod-
els in this scenario. Indeed, in the Figure 18a the the side slip angle is similar for
three conditions explored. Furthermore, even the maneuver in snow conditions is
achieved in a comparable time period, since the friction limit of the average con-
Sensors 2021,21, 5372 17 of 21
troller model is similar to the one of the plant model working in snowy boundary
conditions Figure 18b.
•Scenario C2
Remarking that an accurate online friction coefficient estimation becomes absolutely
necessary to allow exploiting the vehicle dynamics in maximum performance condi-
tions within a combined DLC maneuver, in this paragraph the aim of the authors is
to argue another possible query: In a real scenario, where an onboard tyre-road friction
estimator able to estimate (among others) the grip parameter with a certain degree of accuracy
and to update the control model parameters in run-time, is available, how a controller model
with a percentage error concerning the vehicle instantaneous conditions could perform within
the same maneuver?
With this purpose, the parameter concerning the tyres’ friction of the controller model
has been considered with an intrinsic error with a supposed standard deviation of
±15% respect to the actual grip value of the vehicle plant model.
It is worth noting that in a scenario where the grip factor is overestimated, the con-
troller with the global configuration of the cost function computes more aggressive control
actions leading to out-of cones trajectories and undesirable sliding effect. To avoid this
issue, a robust global configuration has been introduced in order to let the controller being
effective in managing the vehicle behavior in overestimated grip-linked scenarios. The
above new configuration leads to more conservative actions and, consequently, to a consid-
erable loss of performance in terms of velocity. In particular, the loss in performance in
terms of the average speed (in percentage) in the four cases analyzed has been objectively
quantified in Table 5. In Figures 19 and 20, the performance obtained by the two configura-
tions in terms of trajectories, speed, side-slips, and yaw angles are also compared. Notably,
the side-slip in Figure 5reaches peaks of 5 degrees, confirming that the configurations
obtained controls the vehicle at the limit of handling.
(a) (b)
Figure 19.
(
a
) Vehicle trajectory performed in the DLC maneuvers in conservative vs global configu-
ration. (b) Vehicle velocities
(a) (b)
Figure 20. (a) Side slip angles. (b) Yaw angles.
Sensors 2021,21, 5372 18 of 21
Table 5.
Summary of the difference in velocity mean values (%) and lateral error assumed for each
road scenario.
Road Surface Friction Estimation Longitudinal Velocity v
[−] [−] [%]
Icy correct −
Icy overestimation −16.02
Snowy correct −
Snowy overestimation −8.36
Wet correct −
Wet overestimation −8.30
Dry correct −
Dry overestimation −21.00
5. Conclusions
The global autonomous mobility industry, growing at a rapid pace, is an intrinsically
multidisciplinary field that aims at designing advanced onboard control logics by inte-
grating principles from different disciplines including mechanical, control, and computer
science engineering, legal, social, and economic fields. The multidisciplinary proposal
will provide a paradigm shift in the development of strategies and a solid breakthrough
towards enhanced development of the driving automatization systems.
In the proposed work, the authors have investigated how accurate information re-
garding the state of the real system of the parameters concerning the controller model
could affect the behavior of the real system, represented in the form of the high-fidelity
validated plant model. The influence of the tyre thermal dynamics, the impact of the
possible ageing effects and the contact with different road pavements have been examined.
Wrong parameters in the definition of the internal model of the NMPC might compromise
the control performance, especially when the vehicle is supposed to drive at the limit of
handling conditions. Specifically, a controller characterized by an overestimation of the
grip conditions is forced to compute too aggressive control actions that might bring the
vehicle in unstable and unsafe conditions, that are very difficult to handle for the controller
itself. On the contrary, an underestimation of the grip might reduce the performance of
the controller, which is forced to compute too conservative control actions. Moreover, the
parameters of the cost function play an important role in defining the level of performance
that the controller is required to achieve. A high weight on the travel time forces the vehicle
to drive fast along the path, hence requiring effective proper internal model parameters to
describe the vehicle behavior at the limit of handling. Instead, a more conservative tuning
(i.e., high weights on side-slip, lateral error, orientation error) can be effective with also less
precise coefficients, as the vehicle is not supposed to travel at the limit of handling. Due to
these statements, in future work, the effects of including a real-time estimate of the tyre
and the environment states, along with an adapting strategy for the weights of the cost
function will be included in the whole analysis.
The investigation constitutes a part of the broader panorama of studies on novel model-
based ADAS systems, which could become adaptive to the system state and boundary
conditions, by means of a real-time physical model-based estimator of adherence, sensitive
to environmental conditions and to the instantaneous state of tyres. In this way, the physical
limits of the real system could become totally exploited by the control logic, minimizing
the safety-related risks, and unleashing the potential of physical modeling to the next level
of vehicle control.
Sensors 2021,21, 5372 19 of 21
Author Contributions:
Conceptualization, A.S.; methodology, A.S., M.B. and E.M.; software, V.M.A.,
E.M. and E.P.; validation, A.S. and E.M.; formal analysis, A.S., E.M. and A.C.; investigation, V.M.A.,
E.M. and E.P.; resources, A.S., M.B., A.C. and E.F.; data curation, V.M.A. and E.M.; writing, A.S., V.M.A.
and E.M.; visualization, M.B. and A.C.; supervision, A.S., M.B. and E.F.; project administration, A.S.
M.B. and E.F.; funding acquisition, A.S. and E.F. All authors have read and agreed to the published
version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: Data available on request due to industrial confidential agreements.
Conflicts of Interest: The authors declare no conflicts of interest.
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