Vehicle longitudinal motion modeling for nonlinear control
K. El Majdoubc, F. Giria,n, H. Ouadic, L. Dugardb, F.Z. Chaouic
aGREYC Lab, Universite´ de Caen Basse-Normandie, UMR CNRS 6072, Caen, France
bGIPSA Lab, UMR CNRS, INPG, Grenoble, France
cENSET, Universite´ de Rabat-Agdal, Rabat, Marocco
a r t i c l e i n f o
Article history:
Received 26 November 2010 Accepted 23 September 2011 Available online 15 October 2011 Keywords:
Vehicle longitudinal motion Longitudinal slip
Tire Kiencke’s model Speed control
a b s t r a c t
The problem of modeling vehicle longitudinal motion is addressed for front wheel propelled vehicles.
The chassis dynamics are modeled using relevant fundamental laws taking into account aerodynamic effects and road slop variation. The longitudinal slip, resulting from tire deformation, is captured through Kiencke’s model. A highly nonlinear model is thus obtained and based upon in vehicle longitudinal motion simulation. A simpler, but nevertheless accurate, version of that model proves to be useful in vehicle longitudinal control. For security and comfort purpose, the vehicle speed must be tightly regulated, both in acceleration and deceleration modes, despite unpredictable changes in aerodynamics efforts and road slop. To this end, a nonlinear controller is developed using the Lyapunov design technique and formally shown to meet its objectives i.e. perfect chassis and wheel speed regulation.
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1. Introduction
Vehicle longitudinal motion control aims at ensuring passenger safety and comfort. It is an important aspect in dynamic collabora- tive driving i.e. when multiple vehicles should coordinate to share road efficiently while maintaining safety. In this respect, several works have been devoted to what is commonly referred to adaptive cruise control of the main objective of which is maintain- ing a specified headway between vehicles (Ioannou and Chien, 1993;Moon, Moon, &Yi, 2009). Different control techniques have been used in these works including linear and adaptive control (You, Hahn, & Lee, 2009), genetic fuzzy control (Poursamad and Montazeri, 2008), sliding mode control (Liang, Chong, No, & Yi, 2003;Nouveliere and Mammar, 2007), anti-sliding control (Fang et al., 2011), scheduling gain control involving PIDs (Ren, Chen, &
Chen, 2008) and estimating some state variables such as sideslip and tire force (Baffet, Charara, & Lechner, 2009). However, most previous works on longitudinal control were based on simple models neglecting important nonlinear aspects of the vehicle such as rolling resistance, aerodynamics effects and road load. In some studies, the controller performances were not formally analyzed (Ren et al., 2008). InYamakawa, Kojima, and Watanabe (2007), longitudinal vehicle control has been studied focusing on torque management for independent wheel drive. It is worth noticing that in all previous studies on longitudinal vehicle control, the control design has been based on simple models not accounting for the tire–road interaction.
In the present study, the problem of longitudinal vehicle control is revisited, for front wheel propelled vehicles, focusing on speed regulation. The aim is to design a controller that is able to tightly regulate the chassis and wheel velocities, in both acceleration and deceleration driving modes, despite changing and uncertain driving circumstances. This problem has not been dealt with previously. A further originality of the present paper is that the control design relies upon a more complete model that accounts for most vehicle nonlinear dynamics including the tire–
road interaction. That is, the study includes two major contribu- tions. First, a suitable control model is developed for the vehicle longitudinal behavior. In this respect, recall that a convenient model is one that is sufficiently accurate but remains simple enough to be utilizable in control design. To meet the accuracy requirement, the model must account not only for aerodynamic phenomena but also, and especially, for tire–road friction. Model- ing the tire/road contact is a quite complex issue involving multiple aspects relevant to tire characteristics (e.g. structure, pressure) and to environmental factors (e.g. road load, tempera- ture). Several tire models have been proposed in the literature e.g.
Guo’s model (Guo and Ren, 2000), Pacejka’s model (Pacejka and Besselink, 1997), Dugoff’s model (Dugoff and Segel, 1970), Gim’s model (Gim and Nikravesh, 1990), Kiencke’s model (Kiencke and Nielsen, 2005). In the present work, Kiencke’s model is retained because it proves to be a good compromise between accuracy and simplicity. Vehicle modeling is completed with chassis dynamics equations. It is carried out according to the bicycle model principle, applying the fundamental dynamics and aerodynamics laws. The full vehicle model turns out to be a combination of two nonlinear state-space representations describing, respectively, the acceleration and deceleration longitudinal driving modes.
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Control Engineering Practice
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doi:10.1016/j.conengprac.2011.09.005
nCorresponding author.
E-mail address:[email protected] (F. Giri).
Nevertheless, it proves to be utilizable in vehicle control design.
Such model development is in fact a major achievement of the present study. The other contribution is the design of a nonlinear controller that ensures global stabilization and longitudinal speed regulation during acceleration/deceleration driving modes. This is carried out based on the Lyapunov design technique (Khalil, 2002). It is formally proved that the developed controller actually achieves the stability and regulation objectives it was designed to.
Furthermore, it is observed through numerical simulations that the controller is quite robust with respect to uncertainties on environmental characteristics.
The paper is organized as follows: Section 2 is devoted to modeling the acceleration/deceleration vehicle longitudinal beha- vior; the obtained model is used inSection 3to design a controller and to analyze the resulting closed-loop system; the controller performances are illustrated inSection 4 by numerical simula- tions. Conclusion is inSection 5.
2. Modeling of chassis longitudinal motion
Except for aerodynamic forces, all external efforts acting on a vehicle are generated at the wheel–road contact. The under- standing and modeling of the forces and torques developed at wheel–road contact is essential for studying properly the vehicle dynamics. These are discussed in the forthcoming sections. In this respect, recall that the vehicle motion is composed of two types of displacements: translations along the x,y,zaxes and rotations around these same axes (Fig. 1).
2.1. Kiencke’s tire modeling
The tire is a main component of the wheel–road contact as it ensures three important functions (Kiencke and Nielsen, 2005):
(i) bearing the vertical load and absorbing road deformations; (ii) producing longitudinal acceleration efforts and contributing to vehicle braking; (iii) producing the required transversal efforts that help the vehicle turning.
The efforts generated at the wheel–road contact include long- itudinal (acceleration/deceleration) forces, lateral guiding forces and self-alignment torque. The effect of these efforts on the vehicle behavior is determined by the tire–road adhesion. For small load variations, the longitudinal coefficient of friction is characterized by the following ratio:
m¼Ftx
Fv
ð1Þ whereFtxdenotes the longitudinal effort andFvthe vertical load.
The ratiomis called longitudinal adhesion or friction coefficient.
The value of this coefficient depends on the tire slip resulting from the deformation of the tire in contact with the road (Kiencke and Nielsen, 2005). The longitudinal slip is characterized by the coefficientldefined as follows.
In acceleration mode, i.e.VvoVw, one has l¼1Vv
Vw
¼1 Vv
ref fOw
ð2Þ
In deceleration mode, i.e.VvZVw, one has l¼Vw
Vv
1¼ref fOw
Vv
1 ð3Þ
wherereffdenotes the effective wheel radius,Owdesignates the wheel angular velocity,!V
wis the speed of the tire–road contact,
!V
v is the linear velocity of the wheel center (Fig. 2). A similar deformation occurs when the wheel presents a slip angleai.e. the resulting lateral slip produces a lateral force!F
ty.
Modeling the efforts at the wheel–road contact has been given a great deal of interest over the last years. In this respect, several tire models have been developed with quite different properties, e.g.
Guo and Ren (2000),Pacejka and Besselink (1997),Dugoff and Segel (1970),Gim and Nikravesh (1990), andKiencke and Nielsen (2005).
For control design use, the most suitable tire model is one that presents the best accuracy/simplicity compromise. From this view- point, Kiencke’s model turns out to be a quite satisfactory choice (Kiencke and Nielsen, 2005). Indeed, this model is sufficiently accurate as it accounts for the main features such as the vertical loadFv, slip anglea, slip coefficientl. On the other hand, it has already proved to be useful in designing simple estimators for state variables like slip angle and lateral efforts (You et al., 2009). In the present paper, this model will prove to be useful in control design.
2.2. Kiencke’s model
This was developed in Kiencke and Nielsen (2005) using Burckhardt’s extended model to compute the friction coefficient
m. Accordingly, the latter is a function of the combined longitudinal/
lateral slip coefficientkand the forces acting on the tire.Fig. 3gives a schematic representation of Kiencke’s wheel model where:
m¼C1ð1expðC2kÞÞC3kexpðkC4VGÞð1C5F2vÞ ð4Þ
k¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l2þk2y
q
ð5Þ
l¼1Vv
Vw andky¼ ð1lÞtanðaÞ ðaccelerationÞ ð6Þ
Longitudinal motion
Vertical motion
x y
z ψψ
Roll φ Pitch
Yaw
ϕ
Fig. 1.Degrees of freedom of a vehicle.
y
z
Ft
F
F V
V F
Fty
r
M Vrv
y z
Fig. 2.Forces applied on the wheel.
Fv
Ftx
Fty
Fig. 3.Kiencke’s wheel model.
l¼Vw
Vv
1 andky¼tanðaÞ ðdecelerationÞ ð7Þ The other parameters and variables contained in Eqs. (4)–(7) have the following meanings:
C1,C2,C3 Parameters depending on the road state.
C4 Coefficient depending on the maximal driving speed.
C5 Coefficient depending on the wheel allowed maximal load.
k Global (longitudinal/lateral) slip.
ky Lateral slip.
VG Speed of the vehicle center of gravity.
Fv Vertical load.
In Kiencke’s model, the longitudinal and lateral efforts are, respectively, described by (8) and (9):
Ftx¼mFv
kðkcosðaÞCmtkysinðaÞÞ ð8Þ Fty¼mFv
kðCmtkycosðaÞ þksinðaÞÞ ð9Þ where Cmt denotes a weighting coefficient taking values in the interval [0.9 0.95]. In longitudinal motion, the following simplifi- cations are used:
(i) The tire behavior is independent of the vehicle maximal speed. That is, the term inC4is neglected in Eq. (4).
(ii) The tire model depends linearly on the maximal load. Then, the term inC5is neglected in (4).
(iii) The vehicle moves along a straight line. That is, the slip angle
ais null.
Using these remarks, Eqs. (4)–(9) simplify to
m¼C1ð1expðC2lÞÞC3l, k¼l, ky¼0 ð10Þ
k¼landky¼0 ð11Þ
Ftx¼mFv ð12Þ
Fty¼0 ð13Þ
2.3. Rolling resistance and aerodynamic resistance
It is obvious that tire deformation causes mechanical losses.
Radial deformations are caused by the vertical load.Fig. 4shows that the vertical force distribution (along the deformation area) is not uniform. The resulting force moves from the central pointIto a point located at a distancedrr,max (Fig. 4). This is called vertical load maximal trail and is defined as follows:
drr,max¼mrrref f ð14Þ
when the wheel starts moving, a torque is generated provided that drr,maxa0. To bring back the vertical load strength to the central pointI, a compensating torqueMrrmust be applied (by the driving motor) on the wheel. This is called rolling resistance
torque and is given by
Mrr¼Fvdrr,max¼Fvmrrref f ð15Þ
2.4. Aerodynamic resistance
Aerodynamic resistance has naturally an impact on energy consumption onboard. Fluid mechanics laws are resorted to explain air flow around a moving vehicle. Accordingly, vehicle forms are continuously reinvented to improve aerodynamic performances (Powers and Nicastri, 2000). Aerodynamic efforts come in direct interaction with the vehicle, producing various forces (drag, lift and lateral) which in turn generate torques (yaw, roll and pitch). The aerodynamic drag force is significant when the vehicle moves at grand speed. The aerodynamic lift effort increases the rolling resistance (because the surface of tire–road contact grows up) improving tire steerability. On the other hand, the aerodynamic lift force decreases the vehicle adhesion which reduces its stability and security (Milliken and Milliken, 1995). In presence of front-wind, the aerodynamic resistance is represented by two forces: the aerodynamic drag force !F
aex and the aero- dynamic lift force!F
aez. These are defined as follows:
Faex¼12rCxSðVvþVaÞ2 Faez¼12rCzSðVvþVaÞ2 8<
: ð16Þ
with,ris the air density depending on atmosphere pressure and ambient temperature;Cxis the aerodynamic drag coefficient;Czis the aerodynamic lift coefficient; Sis the frontal projection area vehicle;Vvis the vehicle speed;Vais the wind speed (positive in presence of front-wind, negative in presence of rear-wind).
2.5. Modeling of a wheel submitted to driving couple
Fig. 5shows a one-wheel vehicle with massMv. The wheel is driven by a coupleMm. LetJdenotes the inertia resulting from the wheel, the transmission shaft and the driving motor. Invoking the dynamic fundamental principle, one gets the following equation:
Mm¼JdOw
dt þFtref fþMrr ð17Þ
where Eq. (15) has been used to account for the rolling resistance.
Using the relationVw¼reffOw, one gets V_w¼ref f
J ðMmFtref fFvref fmrrÞ ð18Þ Eq. (18) together with (1)–(3) and (10) describe the one-wheel vehicle behavior. This is further illustrated by the schematic representation ofFig. 6.
2.6. Model of two-wheel vehicle with one driving wheel
Longitudinal and transversal behaviors can be assumed to be decoupled when the steering angle is small. Then, one makes use of the vehicle symmetry to perform a projection (of all forces) on
Fv
d reff
w
I
Fig. 4.Rolling resistance.
Body of mass Mv
Ωw
Ft
Fv
Fig. 5.One-wheel vehicle model.
the longitudinal axis reducing thus the four-wheel model into a (two-wheel). Fig. 7 illustrates the forces involved in a bicycle model. The involved notations are described inTable 1. Now, the focus will be made on the bicycle model considering that only one wheel is submitted to a motor torqueMm. For vehicle stability purpose, the front wheel is driving. The distribution of the vehicle load over the tires is computed applying the two dynamic fundamental laws. Doing so, one gets the following equations:
FtfðMvgsinyþFaexÞ ¼MvV_v ð19Þ FvfþFvrðMvgcosyFaezÞ ¼0 ð20Þ FvflfFvrlrþ ðFtfþFtrÞh¼0 ð21Þ
Solving Eqs. (20) and (21) forFvfandFvryields:
Fvf¼Mvg ð1CÞcosyw V_v
g þsiny
!
ð1CÞFaez
MvgwFaex
Mvg
" #
ð22Þ
Fvr¼Mvg Ccosyþw V_v
g þsiny
! þCFaez
MvgþwFaex
Mvg
" #
ð23Þ
wherew¼h=ðlfþlrÞand C¼lf=ðlfþlrÞ. From Eq. (19) it is easily seen that the vehicle speed undergoes the following equation:
V_v¼ 1 Mv
ðFtfðMvgsinyþFaexÞÞ ð24Þ
2.7. State-space representation of vehicle longitudinal behavior The equations obtained so far are now combined together to build-up a state-space representation of the vehicle longitudinal
acceleration/deceleration behavior. The vehicle longitudinal dynamics are characterized by two state variables, i.e. vehicle (chassis) speedVv
and front-wheel speed Vw. As the slip coefficient depends on the current driving mode (acceleration or deceleration), the vehicle is characterized by two state-space representations. Each representation describes the vehicle in the corresponding operation mode.
2.7.1. State-space representation in deceleration mode VwrVv
Combining (3), (10), (11), (13) and (15)–(17), one obtains the following state-space representation:
V_w¼a1Mmþ a2þa3
Vw
Vv
þa4exp aVw
Vv
1
a5þa6
Vw
Vv
þa7ðVvVaÞ2þa8
Vw
Vv
ðVvVaÞ2
þa9exp aVw
Vv
þa10ðVvVaÞ2exp aVw
Vv
ð25aÞ
V_v¼a11þa12ðVvVaÞ2þ a2þa3
Vw
Vv
þa4exp aVw
Vv
1
a13þa14
Vw
Vv
þa15ðVvVaÞ2þa16
Vw
Vv
ðVvVaÞ2
þa17exp aVw
Vv
þa18ðVvVaÞ2exp aVw
Vv
ð25bÞ where the various parameters are defined in Table 2. A more compact representation of (25a,b) is obtained introducing the notations:u¼Mm,x1¼Vw,x2¼Vvand
f1ðx1, x2Þ ¼ a2þa3
x1
x2
þa4exp ax1
x2
1
a5þa6
x1
x2
þa7ðx2VaÞ2þa8
x1
x2
ðx2VaÞ2
þa9exp ax1
x2
þa10ðx2VaÞ2exp ax1
x2
ð26aÞ
f2ðx1, x2Þ ¼a11þa12ðx2VaÞ2þ a2þa3
x1
x2
þa4exp ax1
x2
1
a13þa14
x1
x2
þa15ðx2VaÞ2þa16
x1
x2
ðx2VaÞ2
þa17exp ax1
x2
þa18ðx2VaÞ2exp ax1
x2
ð26bÞ With the above notations, the state-space representation (25a,b) is given the following usual compact form:
x_1¼a1uþf1ðx1,x2Þ x_2¼f2ðx1,x2Þ (
ð27Þ
2.7.2. State-space representation in acceleration mode VvoVw
Using (2), (10), (11), (13) and (15)–(17), one gets the following state-space representation:
V_w¼a01Mmþ a02þa03Vv
Vw
þa04exp a0Vv
Vw
1
"
a05þa06Vv
Vw
þa07ðVvVaÞ2þa08Vv
Vw
ðVvVaÞ2
þa09exp a0Vv
Vw
þa010ðVvVaÞ2exp a0Vv
Vw
ð28aÞ
V_v¼a011þa012ðVvVaÞ2þ a02þa03
Vv
Vw
þa04exp a0Vv
Vw
1
þ a013þa014Vv
Vw
þa015ðVvVaÞ2þa016Vv
Vw
ðVvVaÞ2
þ þa017exp a0Vv
Vw
þa018ðVvVaÞ2exp a0Vv
Vw
ð28bÞ h
l
l Faex
Ftf
Faez
F F
F CoG
g M
Fig. 7.The forces acting on a bicycle type vehicle.
Table 1
Notations of vehicle longitudinal model.
lf Distance betweenCoGand the front wheel base (m) lr Distance betweenCoGand the rear wheel base (m) l Distance between the bases of the two wheels (m)
h Height of the gravity center (m)
Faex Aerodynamic drag force (N)
Faez Aerodynamic carrying force (N)
g Gravity acceleration (m s2)
Mv Vehicle mass (kg)
Ftf,Ftr Front and rear wheel drive force (N) Fvf,Fvr Load on the front and rear wheel (N)
y Road slop (rad)
Fv Vv
Vw
Ft Mm
Fig. 6.Schematic representation of a one-wheel vehicle submitted to a driving couple.
where the various parameters are defined in Table 2. Let us introduce the notations:
f01ðx1, x2Þ ¼ a02þa03x2
x1
þa04exp a0x2
x1
1
a05þa06x2
x1
þa07ðx2VaÞ2
þa08x2
x1
ðx2VaÞ2þa09exp a0x2
x1
þa010ðx2VaÞ2exp a0x2
x1
ð29aÞ
f02ðx1, x2Þ ¼a011þa012ðx2VaÞ2þ a02þa03x2
x1
þa04exp a0x2
x1
1
a013þa014
x2
x1
þa015ðx2VaÞ2þa016
x2
x1
ðx2VaÞ2
þa017exp a0x2
x1
þa018ðx2VaÞ2exp a0x2
x1
ð29bÞ whereu,x1 and x2 are as in (26a,b). With these notations, the state-space representation (28a,b) is given the following more compact form:
x_1¼a01uþf01ðx1,x2Þ x_2¼f02ðx1,x2Þ (
ð30Þ
2.8. Control- and simulation-oriented models for two-wheel vehicle with single driving wheel
2.8.1. Control-oriented models
2.8.1.1. Control-oriented model accounting for tire dynamics.
Combining the mode-dependent state space representations (27) and (30), one gets a single model that represents the vehicle in all operation modes i.e.
x_1¼an1ðx1,x2Þuþg1ðx1,x2Þ x_2¼g2ðx1,x2Þ
(
ð31Þ
with
g1ðx1,x2Þ ¼sðx1,x2Þf1ðx1,x2Þ þ ð1sðx1,x2ÞÞf01ðx1,x2Þ ð32aÞ g2ðx1,x2Þ ¼sðx1,x2Þf2ðx1,x2Þ þ ð1sðx1,x2ÞÞf02ðx1,x2Þ ð32bÞ
an1ðx1,x2Þ ¼sðx1,x2Þa1þ ð1sðx1,x2ÞÞa01 ð32cÞ
sðx1, x2Þ ¼1signðx1x2Þ
2 ð32dÞ
In addition to vehicle aerodynamics, this model does account for the tire–road contact effect. Furthermore, it will prove to be useful in control design (Section 3).
2.8.1.2. Control-oriented model ignoring tire dynamics. To better appreciate the benefit of accounting of tire–road contact, a comparison will be performed in Section 4 between the controller obtained from (31) and (32) and the one obtained from a simplified model neglecting tire–road contact. Specifically, the simplified model is obtained letting l¼0 (tire sliding negligence) which immediately implies thatVv¼Vw (i.e.x1¼x2).
Ignoring also the rolling resistance mrr, one gets invoking the dynamic fundamentals principles for translation and rotation:
V_w¼ref f
J ðMmFtfref fÞ ð33Þ
V_v¼ 1 Mv
ðFtfðMvgsinyþFaexÞÞ ð34Þ whereFaexis defined by (16). It is readily seen from (33) and (34) that the vehicle speed undergoes the following equation:
V_v¼ ref f
Jþr2ef fMv
Mm r2ef fMv
Jþr2ef fMv
gsinyþ 1 2Mv
rSCxðVvVaÞ2
ð35Þ
where the different parameters are defined inTable 5. Introduce the following notations:
x¼ ref f
Jþr2ef fMv
ð36aÞ Table 2
Definition of the model parameters.
Vehicle parameters in deceleration Vehicle parameters in acceleration
a C2 a0 C2
a1 ref f
J a01 ref f
J
a2 1þwkv(C1þC3) a02 1þwkv(C1C3)
a3 kvwC3 a03 kvwC3
a4 kvwC1expðC2Þ a04 kvwC1expðC2Þ
a5
ð1CÞMvgcosyðmrrþkvðC1þC3ÞÞr
2 ef f J
a05
ð1CÞMvgcosyðmrrþkvðC1C3ÞÞr
2 ef f J
a6
ð1CÞMvgcosykvC3 r2ef f
J
a06
ð1CÞMvgcosykvC3 r2ef f
J
a7 1
2ð1CÞrSCzðmrrþkvðC1þC3ÞÞr
2 ef f J
a07 1
2ð1CÞrSCzðmrrþkvðC1C3ÞÞr
2 ef f J
a8
12ð1CÞrSCzkvC3 r2ef f
J
a08 1
2ð1CÞrSCzkvC3 r2ef f
J
a9
ð1CÞMvgcosykvC1expðC2Þr
2 ef f J
a09
ð1CÞMvgcosykvC1expðC2Þr
2 ef f J
a10 12ð1CÞrSCzkvC1expðC2Þr
2 ef f J
a010
12ð1CÞrSCzkvC1expðC2Þr
2 ef f J
a11 gsinðyÞ a011 gsinðyÞ
a12 2M1
vrSCx a012 2M1
vrSCx
a13 ð1CÞgcosykvðC1þC3Þ a013 ð1CÞgcosykvðC1C3Þ a14 ð1CÞgcosykvC3 a014 ð1CÞgcosykvC3
a15 2M1
vð1CÞrSCzkvðC1þC3Þ a015 2M1
vð1CÞrSCzkvðC1C3Þ
a16 1
2Mvð1CÞrSCzkvC3 a016 2M1
vð1CÞrSCzkvC3
a17 ð1CÞgcosykvC1expðC2Þ a017 ð1CÞgcosykvC1expðC2Þ
a18 1
2Mvð1CÞrSCzkvC1expðC2Þ a018 2M1
vð1CÞrSCzkvC1expðC2Þ
