evaluation of a sliding mode observer for vehicle sideslip angle
TRANSCRIPT
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Control Engineering Practice 1
Evaluation of a sliding mode obs
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Keywords: Vehicle dynamics; Nonlinear models; State observers; Performance evaluation; Qualitative analysis
security actuators, for validating vehicle simulators and for The aim of an observer or virtual sensor is to estimate a
particular unmeasurable variable: sideslip angle. Theliterature describes several observers for sideslip angle.For example, Kiencke and Nielsen (2000) presents linear
ARTICLE IN PRESS
E-mail addresses: [email protected] (J. Stephant),
[email protected] (A. Charara), [email protected] (D. Meizel).
URL: http://www.ensil.unilim.fr/jstephan.1Since September 2005, Joanny Stephant was assistant professor atand nonlinear observers using a bicycle model. Venhovensand Naab (1999) uses a Kalman lter for a linear vehiclemodel. Stephant, Charara, and Meizel (2003) and Stephant
0967-0661/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.conengprac.2006.04.002
ENSIL (16 rue Atlantis Parc dEster Technopole BP 6804; 87068
LIMOGES cedex, France) and XLIM/DMI/MOD laboratory, UMR
6172.advanced vehicle control systems.Braking and control systems must be able to stabilize the
car during cornering. When subject to transversal forces,such as when cornering, or in the presence of a camberangle, tire torsional exibility produces an aligning torquewhich modies the original wheel direction. The difference
particular unmeasurable variable from available measure-ments and a system model. This is an algorithm whichdescribes the movement of the unmeasurable variable bymeans of statistical conclusions from the measured inputsand outputs of the system. This algorithm is applicableonly if the system is observable.This paper presents a sliding mode observer which uses a
simple nonlinear vehicle model along with two measure-ments (yaw rate and vehicle speed) in order to estimate one
Corresponding author. Tel.: +33 (0)5 55 42 37 05;fax: +33 (0)5 55 42 37 10.1. Introduction
A vehicle is a highly complex system bringing together alarge number of mechanical, electronic and electromecha-nical elements. To describe all the movements of thevehicle, numerous measurements and a precise mathema-tical model are required.In vehicle development, knowledge of wheel-ground
contact forces is important. This information is useful for
between a wheels longitudinal axis and wheel speed ischaracterized by an angle known as tire sideslip angledi. The angle between the vehicles longitudinal axis andthe direction of vehicle speed is known as sideslip angled. This is a signicant signal in determining the stabilityof the vehicle (Mammar & Koenig, 2002), and it is the maintransversal force variable. Measuring sideslip angle wouldrepresent a disproportionate cost in the case of an ordinarycar, and it must therefore be observed or estimated.J. Stephanta,,1, A.aHeudiasyc Laboratory UMR CNRS/UTC 6599, UTC, Centre de
bXLIM/DMI/MODUMR 6172 - ENSIL - 16 rue Atlantis Pa
Received 30 September 2
Available on
Abstract
This paper presents a sliding mode observer of vehicle sideslip an
the tire/road interface. The vehicle is rst modelled, and the model i
a validated simulator and real experimental data acquired by th
method. The observer requires a yaw rate sensor and data about
properties of the nonlinear observability matrix condition numbe
error, vehicle speed and tire cornering stiffness are presented.
r 2006 Elsevier Ltd. All rights reserved.5 (2007) 803812
erver for vehicle sideslip angle
araraa, D. Meizelb
cherches de Royallieu BP20529, 60205 Compie`gne cedex, France
Ester Technopole, BP 6804, 87068 LIMOGES cedex, France
; accepted 4 April 2006
5 June 2006
, which is the principal variable relating to the transversal forces at
bsequently simplied. This study validates the observer using both
eudiasyc laboratory car, and also shows the limitations of this
icle speed are required in order to estimate sideslip angle. Some
e discussed, and relations between this variable and observation
www.elsevier.com/locate/conengprac
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(2004) present a comparison of several linear and nonlinear But cornering stiffness is modied as a result of verticalforces on the wheel. Variations in cornering stiffness duringa double lane change maneuver at different speeds areshown in Fig. 2.Lateral tire/road forces Fiy are highly nonlinear.
Various wheel-ground contact force models are to befound in the literature, including a comparison of threedifferent models in Stephant, Charara, and Meizel (2002).
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Notations
C1;2Fd Front, rear wheel cornering stiffness Nrad1F1;2l Longitudinal force in the front, rear wheel
frame (N)F1;2t Transversal force in the front, rear wheel frame
(N)F1;2y Transversal force in the vehicle frame (N)Izz Second moment of mass about the z axis (kgm
2)
L1;2 Center of gravity to front, rear axle distance(m)
mv Vehicle total mass (kg)VG Speed of center of gravity ms1y Lateral acceleration at center of gravity ms2b Front wheel steering angle (rad)d Vehicle sideslip angle (rad)d1;2 Front, rear wheel sideslip angle (rad)_c Yaw rate rad s1
J. Stephant et al. / Control Engineering Practice 15 (2007) 803812804observers, and also show (Stephant, 2004) that the choiceof statespace model impacts the observer method. Basedon the same (4) model, an extended Kalman lter, anextended Luenberger observer and a sliding mode observergive similar results; state trajectories along a vehicles pathare similar. In this paper the sliding mode observer is usedto illustrate the ndings. Performances obtained throughsimulation and experimentation using several classical purelateral dynamics tests are presented. The nonlinearobservability problem and its relation to vehicle speedand cornering stiffness is discussed. Finally, the domain ofvalidity for this observer is specied using the conditionnumber of an observability matrix as a criterion.
2. Vehicle model
Lateral vehicle dynamics has been studied since the1950s. Segel (1956) presented a vehicle model with threedegrees of freedom in order to describe lateral movementsincluding roll and yaw. If rolling movements are neglected,a simple model known as the bicycle model is obtained.This model is currently used for studies of lateral vehicledynamics (yaw rate _c and sideslip). A nonlinearrepresentation of the bicycle model is shown in Fig. 1.Notation is explained in Appendix.A certain number of simplications are used in this
study. Cornering stiffnesses CiFd are taken to be constant.
L2
L1
Ft2
Ft 1
Fl2
Fl1 1
2VG
y0
x0
yx
Fig. 1. Schematic representation of the bicycle model.In this paper, transversal forces are taken to be linear. Thisassumption is reasonable when vehicle lateral acceleration y is less than 0:4g Lechner (2001). Consequently,transversal forces can be written as
Fit CiFd:di; i 1; 2. (1)When projected into the vehicle coordinate system,transversal forces shown in Fig. 1 are written
F1y F1t cosb;F2y F2t ;
8
-
Rear and front tire sideslip angles are calculated usingkinematic relations with vehicle speed VG and yaw rate _c:
d1 b d L1_c
VG;
2 2_c
8>>>>> (3)
bz hNLbx;where L is the observer gain matrix in R32. To coverchattering effects (Perruquetti & Barbot, 2002), thefunction signeq used in this paper is
signeqx arctanl x 2=p.
Coefcient l is a design parameter to adjust the slope of thearctan function as shown in Fig. 4. The greater the lcoefcient, the greater the slope of the signeq function, andthe greater the chattering.For all tests presented in this paper, l 1, and errors in
the different statespace variables are below 0:03 ms1 asregards speed and 3:3 s1 as regards yaw rate.
4. Observability
Using the nonlinear state space formulation, theobservability denition is local and uses the Lie derivative(Nijmeijer & Van der Schaft, 1990). It is a function of
ARTICLE IN PRESS
0.4
0.6atan(1* x)
J. Stephant et al. / Control Engineerd d LVG
::The vehicle model can be expressed in terms of a nonlinearstatespace formulation using Newtonian theory, based onFig. 1, as follows:
_x fNLx; u, (4)with x VG d _cT, u b F1l F2l T and
_x1 1
mvu2 cosx2 u1 u3 cosx2
1mv
C2Fd x2 L2x3
x1
sinx2;
1mv
C1Fd u1 x2 L1x3
x1
sinx2 u1;
_x2 1mvx1
u2 sinu1 x2 u3 sinx2
1mvx1
C1Fd u1 x2 L1x3
x1
cosu1 x2
1mvx1
C2Fd x2 L2x3
x1
cosx2 x3
_x3 1
IzzL1u2 sinu1 L2C2Fd x2 L2
x3
x1
1Izz
L1C1Fd u1 x2 L1x3
x1
cosu1:
8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:
(5)
The state comprises the speed of the center of gravity VG,the sideslip angle d and the yaw rate _c. Inputs are thefront wheel steering angle b and the longitudinal forcesapplied to the front F 1l and rear F2l wheels.
3. Sliding mode observer
From Perruquetti and Barbot, 2002 it is clear that thiskind of observer is useful when working with reducedobservation error dynamics and when seeking a nite timeconvergence for all observable states, as well as robustnesswhen confronted with parameter variations (with respect toconditions). Fig. 3 presents the sliding mode observermethod applied to a nonlinear vehicle model (4).In this paper two measurements are used to estimate the
vehicle sideslip angle: the yaw rate and the speed of theFig. 3. Sliding mode observer method.center of gravity. The rst measurement is available fromthe ESP control unit, and the second can be calculatedfrom the ABS sensors. The observation equation can bewritten
z hNLx h1NLx h2NLxT
x1 x3T VG _cT. 6
The sliding mode observer equations are
b_x fNLbx; u L signeqz bz;((7)
-10 -8 -6 -4 -2 0 2 4 6 8 10
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
x
atan(10* x)*2/pi
Fig. 4. Sign functions for sliding mode observer.0.8
1
sign ( x) and signeq ( x) function
sign()
ing Practice 15 (2007) 803812 805estimated state trajectory and inputs applied to the model.For the system described by Eq. (7) and sensor set (6) the
-
observability function is
ox^; u
h1NLx^Lfh1NLx^; uL2f h1NLx^; u
h2NLx^Lfh2NLx^; uL2f h2NLx^; u
0BBBBBBBBB@
1CCCCCCCCCA, (8)
with
LfhjNLx^
dhjNLx^dx
fNLx^; u;8>>>>>: (9)The observability function is therefore expressed as
o
h1NL
f1NLP3i1
qf1NLqxi
:f iNL
h2NLP3i1
qh2NLqxi
:f iNL
P3i1
qqxi
P3i1
qh2NLqxi
:f iNL
:f iNL
0BBBBBBBBBBBBBBBBB@
1CCCCCCCCCCCCCCCCCA
. (10)
If the o function is invertible at the current state and input,the system is observable. This function is invertible if itsJacobian matrix O has a full rank. This Observabilitymatrix is dened as
O ddx
ox^; u. (11)
It is also possible to give an observability indicator with theinverse of the Jacobian matrix condition number. Thisgives a measure of the sensitivity of the solution to theobservation problem. The condition number of a matrix Ois dened as the ratio of the largest singular value of O to
0 1 2 3 4 5 6 7 8 9
-0.8-0.6-0.4-0.2
00.20.40.6
time (s)
()
sideslip angle 90km/h
(a)
CallasSMOFig. 5. Sideslip angle by sliding mode observer. (a) ISO drecherche sur les transports et leur securite). The Callasmodel takes into account vertical dynamics (suspension,tires), kinematics, elasto-kinematics, tire adhesion andaerodynamics. This vehicle simulator was developed bySERA-CD (http://www.sera-cd.com).The performance of the observer is evaluated on an ISO
double lane change and a slalom. This kind of test isrepresentative of the transient lateral behavior of a vehicle.The double lane change is performed at three differentspeeds: 40, 90 and 105 km/h. The difference between thethree tests is the level of lateral acceleration. At 105 km/hthe level is so high that the simulators virtual driver losescontrol of the car. The slalom is performed at 50, 80 and90 km/h. The virtual driver in the Callas simulatorincreases the frequency applied to the steering wheel from0.01 to 0.2Hz. All tests are performed on a dry track, withthe friction coefcient between ground and tire assumed tobe at its maximum possible value of one.
5.1. Observer results
Fig. 5 presents results of the calculation of vehiclesideslip angle for a double lane change (Fig. 5(a)) and aslalom (Fig. 5(b)) by the sliding mode observer (7).Stephant (2004) presents some similar results. An extendedKalman lter, an extended Luenberger observer and thissliding mode observer are compared. These three kinds ofobserver applied to the same model 4 give similar results.
0 10 20 30 40 50-0.25
-0.2-0.15
-0.1-0.05
00.05
0.10.15
0.20.25
time (s)
()
sideslip angle 80km/h
b)
CallasSMOouble lane change at 90 km/h; (b) slalom at 80 km/h.
-
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(b
late
neerFig. 6 summarizes the performance of the observer inestimating different variables; Fig. 6(a) for the three doublelane changes and Fig. 6(b) for the three slaloms.The three rows of each histogram present the normalized
error attributable to the observer. The normalized error ofa variable z is dened by
z jz^SMO zCallasj100
maxjzCallasj, (13)
400
10
20
30
sideslip angle
0
5
10
15
yaw rate
05
101520
lateralacceleration
00.020.040.060.08
CG speed
0
2
4
6
01
2
3
01
2
3
00.010.020.030.04
0
20
40
60
0
5
10
15
0
10
20
0
1
2
3
x 10-4
max
%|er
ror|
mean %
|error
| va
r %
|error
|
(a)
10590 40 10590 40 10590 40 10590
40 10590 40 10590 40 10590 40 10590
40 10590 40 10590 40 10590 40 10590
Fig. 6. Normalized error by the observer. Errors in sideslip angle, yaw rate,
105 km/h; (b) slalom at 50, 80 and 90 km/h.
J. Stephant et al. / Control Engiwhere z^SMO is the variable calculated by the observer, zCallasis the variable calculated by the Callas simulator andmaxjzCallasj is the absolute maximum value of the variablegiven by the simulator during the test maneuver.The rst row shows the maximum normalized error
during the course of the maneuver. The second row showsthe mean, and the third row the variance. The leftmostcolumn presents the results for sideslip angle observation,the second column the yaw rate estimation, the third thelateral acceleration, and the rightmost column the vehiclespeed.The black, gray and white blocks in Fig. 6(a) represent
the double lane change maneuver at 40, 90 and 105 km/h,respectively, and similarly in Fig. 6(a) regarding the slalommaneuver at 50, 80 and 90 km/h.The lateral acceleration is calculated by
by C1Fdmv
b C
2FdL
2 C1FdL1mvcVG
!b_c C1Fd C2Fdmv
bd,(14)
where variables marked :^ are estimated by the observer.As shown in the last column of Figs. 6(a) and (b), thevehicle speed is calculated precisely by the sliding modeobserver.
5.1.1. Double lane change maneuver
All the variables are correctly estimated for themaneuver at 40 km/h. On average the sideslip angle erroris around 2%, the yaw rate error 1% and the lateralacceleration error 0.5%. As regards the different lateral
500
10
20
30
sideslip angle
max
%|er
ror|
0
5
10yaw rate
01234
lateralacceleration
0
0.02
0.04
0.06CG speed
0
5
10
mean %
|error
|
0
2
4
6
0
0.5
1
1.5
0
0.01
0.02
0.03
0
20
40
60
var
%|er
ror|
0
5
10
0
0.5
1
02468
x 10-5
)
9080 50 9080 50 9080 50 9080
50 9080 50 9080 50 9080 50 9080
50 9080 50 9080 50 9080 50 9080
ral acceleration and vehicle speed. (a) ISO double lane change at 40, 90 and
ing Practice 15 (2007) 803812 807variables, the higher the speed, the greater the maximumnormalized error, and the greater the error variance. Thesame applies to yaw rate and to lateral acceleration. Thelevel of error is around 6.5% for the sideslip angleestimation at 90 and 105 km/h. It should be noted thatfor this kind of path, lateral forces applied are directlylinked to the speed.
5.1.2. Slalom maneuver
Same remarks can be made regarding the slalom tests.The higher the speed, the greater the normalized error in theestimation of yaw rate and lateral acceleration. In the case ofsideslip angle the error behavior may be explained by phaseshift between the simulators and the observers variables.
5.1.3. Remark
Using several nonlinear observers based on the (4) modegives estimated statespace variable trajectories which aresimilar. Results presented in the next section are availablefor observers based on this model.
5.2. Observability results
The rank of the observability matrix (11) is 3 along thethree different paths. The model is observable.
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uble
eerAs discussed in Section 4, the condition number of an
0 2 4 6 8 10 12 14 16 182.6
2.612.622.63
x 10-3 inverse of condition number (-)
0 1 2 3 4 5 6 7 8 9
7.2
7.4
7.6x 10-3
0 0.5 1 1.5 2 2.5 3 3.5 4
9.5
10
10.5x 10-3
time (s)(a)
105km/h
90km/h
40km/h
Fig. 7. Inverse of condition number of observability matrix. (a) ISO do
J. Stephant et al. / Control Engin808observability matrix is used to obtain an indicator ofobservability quality. Fig. 7 presents the inverse of thecondition number of matrix O for the six differentmaneuvers.In the following two sections the relation between the
condition number of the observability matrix and vehiclespeed and tire cornering stiffness is discussed.
5.2.1. Speed and observability matrix
Two conclusions may be drawn about the relation of theobservability matrix to vehicle speed. First, Fig. 9(a)illustrates that the greater the speed, the smaller thecondition number. At 40 km/h the condition number isaround 380, at 90 km/h it is 135, and at 105 km/h itis 100.The curve of the inverse of the condition number is
related to the curve of vehicle speed, as shown in Fig. 8, inparticular for the double lane change maneuver at 40 km/h.The correlation coefcient is close to 0.9 for this test path,as shown in Fig. 9(b). At higher speeds the correlationbetween the two variables is not signicant (below 0.6). Theinuence of speed variations is also visible in the slalommaneuver at 50 km/h, when abrupt speed variations appearsuch as at time t 12 s, which is highlighted by a lightgray box in Figs. 7(b) and 8(b).
5.2.2. Real tire cornering stiffness and observability matrix
The second conclusion can be drawn by comparing theshape of the tire cornering stiffness curve, shown in Fig. 10,with the curve for the condition number of the observa-
3.5053.51
3.5153.52
x 10-3 inverse of condition number (-)
50km/h
6.256.3
6.356.4
6.45x 10-3
80km/h
0 10 20 30 40 507.17.27.37.4
x 10-3
time (s)
90km/h
(b)
lane change at 40, 90 and 105 km/h; (b) slalom at 50, 80 and 90 km/h.
ing Practice 15 (2007) 803812bility matrix, shown in Fig. 7. Variations in both variablesare similar.As shown in Fig. 11, the correlation coefcient between
the real tire cornering stiffness and the inverse of theobservability matrix is above 0.9, except in the case of thedouble lane change maneuver at 40 km/h. This can beexplained as follows: in the model, tire cornering stiffness isassumed to be constant throughout all tests, as shown inFig. 10. The greater the variation in actual corneringstiffness, the greater the condition number of the observa-bility matrix, and the less accurate the model.
5.2.3. Conclusion
The analysis of the evolution of the observability matrixcondition number has shown that this variable is inuencedby the speed of the vehicle and, to an event greater extent,by real variations in tire cornering stiffness.
6. Experimental results
In order to study experimentally the performance of thevehicle sideslip angle sliding mode observer, data werecollected using the Heudiasyc Laboratory vehicle (to bepresented in the following section). The test was a slalomperformed at high speed (80 km/h). While the car is beingcontrolled by a driver, the steering angle amplitude andfrequency increase. With this kind of path, the lateralacceleration applied depends on the steering input. Therst aim of this test was to determine the level of lateral
-
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neer0 2 4 6 8 10 12 14 16 1811.1211.1411.1611.18
speed of center of gravity (m/s)
24.9925
40km/h
J. Stephant et al. / Control Engipressure at which the results yielded by the sliding modeobserver become too high. The second aim was to conrmconclusions regarding the properties of the observabilitymatrix presented in Section 5.2.
6.1. Experimental vehicle
STRADA is the Heudiasyc Laboratorys test vehicle: aCitroen Xantia station wagon equipped with number ofsensors, shown in Fig. 12. The following were used in thetests described in this paper:
0 1 2 3 4 5 6 7 8 924.9724.98
0 0.5 1 1.5 2 2.5 3 3.5 4
29.229.329.4
time (s)
105km/h
(a)
90km/h
Fig. 8. Vehicle speed along the different paths. (a) ISO double lane c
10 15 20 25 30 35100
150
200
250
300
350
400speed and condition number
average speed on the test path (m/s)
con
ditio
n nu
mbe
r of o
bser
vabi
lity m
atrix
(-)
Slalom 80km/h
(a)
ISO Double LaneChange 40km/h
Slalom 50km/h
Slalom 90km/h
ISO Double LaneChange 90km/h
ISO Double LaneChange 105km/h
Fig. 9. Speed and condition number of observability matrix: (a) condition n
coefcient between the condition number of the observability matrix and spee2222
Thoffro
222
24
25
(b)
han
-
Corre
latio
n co
effic
ient
(-)
(b)
umb
d..22
.2313.882
13.884
50km/h
13.886speed of center of gravity (m/s)
ing Practice 15 (2007) 803812 809lateral accelerometer ym,odometry: rotational speeds of the four wheels (ABSsensors) VGm,yaw rate gyrometer _cm,steering angle bm,correvit Sensor dm.
e speed of the center of gravity is calculated as the meanthe longitudinal speeds of the two rear wheels calculatedm their rotational speed (ABS sensor) and their radius.
2.2.21
80km/h
0 10 20 30 40 50
.98
25
.02
time (s)
90km/h
ge at 40, 90 and 105km/h; (b) slalom at 50, 80 and 90 km/h.
0.2
0
0.2
0.4
0.6
0.8
1
1.2
ISO Double Lane Change 40km/h
Correlation coefficient between speed and inverseof condition number
ISO Double Lane Change 90km/h
ISO Double Lane Change 105km/h
Slalom 50km/hSlalom 80km/h
Slalom 90km/h
Maneuver
er of the observability matrix as a function of speed; (b) correlation
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ARTICLE IN PRESS
1
1
eer1.52
1.54
1.56
x 105 Front cornering stiffness (N/rad)
40km/h Callas
x 105
Hypothesis for model
J. Stephant et al. / Control Engin810The Correvit S-400 is a noncontact optical sensormounted at the rear of STRADA on the sprung mass ofthe car. The S-400 sensor provides highly accuratemeasurement of distance, speed and acceleration, sideslipangle, drift angle and yaw angle. The S-400 sensor usesproven optical correlation technology to ensure the mostaccurate possible signal representation. This technologyincorporates a high intensity light source that illuminatesthe test surface, which is optically detected by the sensorvia a two-phase optical grating system.
0.81
1.21.41.6
0 0.5 1 1.5 2 2.5 3 3.5 45
10
15x 104
time (s)(a) (b
Hypothesis for model
Hypothesis for model 105km/h Callas
90km/h Callas
Fig. 10. Front cornering stiffness. Simulator calculation and hypothesis used
105 km/h; (b) slalom at 50, 80 and 90 km/h.
0
0.2
0.4
0.6
0.8
1
ISO Double Lane Change 40km/h
Correlation coefficient between tyre cornering stiffnessand inverse of condition number
ISO Double Lane
ISO Double Lane
Slalom50km/h
Slalom80km/h
Slalom90km/h
Maneuver
Corre
latio
n co
effic
ient
(-)
Change 105km/h
Change 90km/h
Fig. 11. Correlation coefcient between the condition number of the
observability matrix and tire cornering stiffness.Hypothesis for model
.45
1.5
.55
x 105 Front cornering stiffness (N/rad)
1.11.21.31.41.51.6 x 10
5
1.6x 105
80km/h Callas
Hypothesis for model 50km/h Callas
ing Practice 15 (2007) 8038126.2. Observer results
Fig. 13 presents the estimation of sideslip angle usingthe sliding mode observer (7) and the vehicles lateralacceleration calculated using (14). It was shown that thelinear approximation for tire/road transversal forces isvalid when the lateral acceleration does not exceed 0:4g,and that a linear vehicle bicycle model using linear tireforce is representative when the lateral acceleration doesnot exceed 0:3g (Lechner, 2001). In this model, cornering isperformed at constant speed. The speed over the slalomtest was approximately constant at 80 km/h, as shown inthe upper part of Fig. 14. Given these conditions, thenonlinear model (4) shows the same characteristics as thelinear bicycle model.From Fig. 13(a) it can be seen that the error in sideslip
angle estimation attributable to the sliding mode observer
0 10 20 30 40 500.8
11.21.4
time (s))
90km/h Callas Hypothesis for model
for constructing the observer: (a) ISO double lane change at 40, 90 and
Fig. 12. STRADA: the Heudiasyc Laboratorys experimental vehicle.
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ARTICLE IN PRESSneer-10-8-6-4-202468
1012
lateral acceleration at correvit position (m/s/s)
m/s
/sJ. Stephant et al. / Control Engiis less than 0:5 during the rst 6.8 s, and then 1
between 6.8 and 12.5 s. Subsequently the error increases.On the three last peaks the error is 1:7, 2:8 and 5.This error level can be linked to the lateral acceleration.Between 0 and 6.8 s the lateral acceleration remainsless than 0:57g at peak values. Between 6.8 and 12.5 slateral acceleration is between 0:6g and 0:8g. After 12.5 slateral acceleration exceeds 0:8g at peak values, whichmeans that the observer can no longer estimate the sideslip
0 2 4 6 8 10 12 14-12
0 2 4 6 8 10 12 14-10
-5
0
5
time (s)
()
Sideslip angle at correvit position ()
(a)
CorrevitSMO
Fig. 13. Measured lateral acceleration and sideslip angle (sideslip angle estima
lane change at 50 km/h.
0 2 4 6 8 10 12 1421
21.2
21.4
21.6
21.8
(m/s)
speed of center of gravity
0 2 4 6 8 10 12 147
7.58
8.59
9.510
10.5
x 10-3
time (s)
inverse of condition number (-)
(a)
Fig. 14. Inverse of the observability matrix condition number and sideslip angle
and condition number; (b) normalized observation error for sideslip angle.-10-8-6-4-202468
1012
lateral acceleration at correvit position (m/s/s)
(m/s/
s)
ing Practice 15 (2007) 803812 811angle correctly. The same conclusions may be drawnin relation to the sideslip angle estimation error for theVDA double lane change maneuver at 50 km/h shown inFig. 13(b).If a vehicle sideslip angle error of less than 0:5 is
acceptable, then it is possible to use the observer presentedin this paper when lateral acceleration does not exceed 0:6g.However, when the lateral acceleration exceeds 0:6g, thisobserver is not sufciently accurate.
()
64 66 68 70 72-12
64 66 68 70 72-6-4-202468
time (s)
Sideslip angle at correvit position ()
(b)
CorrevitSMO
ted by the sliding mode observer): (a) slalom at 80 km/h; (b) VDA double
0 5 10 15
5
10
15
20
25
30
35
40
45
50
time (s)
(%)
normalized observation error on sideslip angle || (%)
SMO
(b)
observation error for experimental slalom maneuver at 80 km/h: (a) speed
-
nonlinear bicycle vehicle model. The rst conclusion is thatwith a lateral acceleration not exceeding 0:6g, observerresults are quite good. The model used is valid for lateralacceleration less than 0:4g because of the assumption of
Nijmeijer, H., & Van der Schaft, A. J. (1990). Nonlinear dynamical control
ARTICLE IN PRESS
14.414.5
)
speed of center of gravity(m/s)
J. Stephant et al. / Control Engineering Practice 15 (2007) 8038128126.3. Observability result
The rank of the observability matrix is 3 throughout theslalom and the double lane change tests. Using thiscriterion, the model is observable. Figs. 14(a) and 15present the inverse of observability condition number forthe experimental slalom and double lane change maneu-vers. As it has been shown in relation to the validation bysimulation, the condition number is directly linked to the
64 66 68 70 7213.9
1414.114.2(m
64 66 68 70 724.5
55.5
66.5
x 10-3
time (s)
inverse of condition number (-)
Fig. 15. Speed and inverse of the observability matrix condition number
for experimental double lane change maneuver at 50 km/h.real cornering stiffness. In actual slalom tests the resultingcornering stiffness (by axle) decreases with each cornering.The decreasing peaks of the condition number curvecorrespond to the peaks of the vehicle sideslip angle curve.The greater the sideslip angle estimation error, the higherthe condition number, which can be seen if one comparesFig. 14(a) and (b). As discussed in Section 5.2, the inverseof the condition number is related to the vehicle speed. Thisresult is conrmed by the experimental double lane changemaneuver and can be illustrated clearly by comparing thetwo graphics of Fig. 15. The singularity in the vehicle speedat 70.5 s is also found in the inverse of observability matrixcondition number.
7. Conclusion
This paper has presented in detail the properties of avehicle sideslip angle sliding mode observer applied to asystems. Berlin: Springer.
Perruquetti, W., & Barbot, J.-P. (2002). Sliding mode control in
engineering. New York: Marcel Dekker, Inc.
Segel, M. (1956). Theoretical prediction and experimental substantiation
of the response of the automobile to steering control. In Proceedings of
the automobile division of the institute of mechanical engineers (Vol. 7,
pp. 310330).
Stephant, J., & Charara, A. (2005). Observability matrix and parameter
identication: Application to vehicle tire cornering stiffness. Proceed-
ings of the ECC-CDC2005, Sevilla, Spain.
Stephant, J., Charara, A., & Meizel, D. (2002). Force model comparison
on the wheel-ground contact for vehicle dynamics. Proceedings of the
IEEE intelligent vehicle symposium, Versailles, France.
Stephant, J., Charara, A., & Meizel, D. (2003). Vehicle sideslip angle
observers. Proceedings of the European control conference (ECC2003),
Cambridge, UK.
Stephant, J. (2004). Contribution a` letude et a` la validation experimentale
dobservateurs appliques a` la dynamique du vehicule. Ph.D. thesis,
Universite de Technologie de Compie`gne, December.
Venhovens, P., & Naab, K. (1999). Vehicle dynamics estimation using
Kalman lters. Vehicle System Dynamics, 32, 171184.linear lateral tire forces.The second conclusion is that the condition number of
the observability matrix provides an indicator regardingthe quality of the estimation. This methodology has beenapplied, in this paper, with a sliding mode observer. Butsimilar results may be obtained using any nonlinearobserver based on the same vehicle model.Since the condition number is directly related to the
variations in speed and cornering stiffness, and given thatthe speed is known, it would appear possible to identify thereal cornering stiffness from the calculation of thiscondition number, as shown in (Stephant & Charara,2005).Since the condition number is directly related to thevariations in speed and cornering stiffness, and given thatthe speed is known, it would appear possible to identify thereal cornering stiffness from the calculation of thiscondition number, as shown in (Stephant & Charara,2005).
References
Kiencke, U., & Nielsen, L. (2000). Automotive control system. Berlin:
Springer.
Lechner, D. (2001). Analyse du comportement dynamique des vehicules
routiers legers: developpement dune methodologie appliquee a` la securite
primaire. Ph.D. thesis, Ecole centrale de Lyon, Octobre.
Mammar, S., & Koenig, D. (2002). Vehicle handling improvement by
active steering. Vehicle System Dynamics, 38, 211242.14.3/s
Evaluation of a sliding mode observer for vehicle sideslip angleIntroductionVehicle modelSliding mode observerObservabilitySimulation resultsObserver resultsDouble lane change maneuverSlalom maneuverRemark
Observability resultsSpeed and observability matrixReal tire cornering stiffness and observability matrixConclusion
Experimental resultsExperimental vehicleObserver resultsObservability result
ConclusionReferences