Evaluation of a sliding mode observer for vehicle sideslip angle

ARTICLE IN PRESS

Control Engineering Practice 15 (2007) 803–812 www.elsevier.com/locate/conengprac

Evaluation of a sliding mode observer for vehicle sideslip angle J. Ste´phanta,,1, A. Chararaa, D. Meizelb a

Heudiasyc Laboratory UMR CNRS/UTC 6599, UTC, Centre de Recherches de Royallieu BP20529, 60205 Compie`gne cedex, France b XLIM/DMI/MODUMR 6172 - ENSIL - 16 rue Atlantis Parc d’Ester Technopole, BP 6804, 87068 LIMOGES cedex, France Received 30 September 2005; accepted 4 April 2006 Available online 5 June 2006

Abstract This paper presents a sliding mode observer of vehicle sideslip angle, which is the principal variable relating to the transversal forces at the tire/road interface. The vehicle is first modelled, and the model is subsequently simplified. This study validates the observer using both a validated simulator and real experimental data acquired by the Heudiasyc laboratory car, and also shows the limitations of this method. The observer requires a yaw rate sensor and data about vehicle speed are required in order to estimate sideslip angle. Some properties of the nonlinear observability matrix condition number are discussed, and relations between this variable and observation error, vehicle speed and tire cornering stiffness are presented. r 2006 Elsevier Ltd. All rights reserved. Keywords: Vehicle dynamics; Nonlinear models; State observers; Performance evaluation; Qualitative analysis

1. Introduction A vehicle is a highly complex system bringing together a large number of mechanical, electronic and electromechanical elements. To describe all the movements of the vehicle, numerous measurements and a precise mathematical model are required. In vehicle development, knowledge of wheel-ground contact forces is important. This information is useful for security actuators, for validating vehicle simulators and for 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 camber angle, tire torsional flexibility produces an aligning torque which modifies the original wheel direction. The difference Corresponding author. Tel.: +33 (0)5 55 42 37 05; fax: +33 (0)5 55 42 37 10. E-mail addresses: [email protected] (J. Ste´phant), [email protected] (A. Charara), [email protected] (D. Meizel). URL: http://www.ensil.unilim.fr/jstephan. 1 Since September 2005, Joanny Ste´phant was assistant professor at ENSIL (16 rue Atlantis Parc d’Ester Technopole BP 6804; 87068 LIMOGES cedex, France) and XLIM/DMI/MOD laboratory, UMR 6172.

0967-0661/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2006.04.002

between a wheel’s longitudinal axis and wheel speed is characterized by an angle known as ‘‘tire sideslip angle’’ ðdi Þ. The angle between the vehicle’s longitudinal axis and the direction of vehicle speed is known as ‘‘sideslip angle’’ ðdÞ. This is a significant signal in determining the stability of the vehicle (Mammar & Koenig, 2002), and it is the main transversal force variable. Measuring sideslip angle would represent a disproportionate cost in the case of an ordinary car, and it must therefore be observed or estimated. The aim of an observer or virtual sensor is to estimate a particular unmeasurable variable from available measurements and a system model. This is an algorithm which describes the movement of the unmeasurable variable by means of statistical conclusions from the measured inputs and outputs of the system. This algorithm is applicable only if the system is observable. This paper presents a sliding mode observer which uses a simple nonlinear vehicle model along with two measurements (yaw rate and vehicle speed) in order to estimate one particular unmeasurable variable: sideslip angle. The literature describes several observers for sideslip angle. For example, Kiencke and Nielsen (2000) presents linear and nonlinear observers using a bicycle model. Venhovens and Naab (1999) uses a Kalman filter for a linear vehicle model. Ste´phant, Charara, and Meizel (2003) and Ste´phant

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Notations

L1;2

C 1;2 Fd F 1;2 l

mv VG y€ b d d1;2 c_

F 1;2 t F 1;2 y I zz

Front, rear wheel cornering stiffness ðN rad1 Þ Longitudinal force in the front, rear wheel frame (N) Transversal force in the front, rear wheel frame (N) Transversal force in the vehicle frame (N) Second moment of mass about the z axis (kg m2)

(2004) present a comparison of several linear and nonlinear observers, and also show (Ste´phant, 2004) that the choice of state–space model impacts the observer method. Based on the same (4) model, an extended Kalman filter, an extended Luenberger observer and a sliding mode observer give similar results; state trajectories along a vehicle’s path are similar. In this paper the sliding mode observer is used to illustrate the findings. Performances obtained through simulation and experimentation using several classical pure lateral dynamics tests are presented. The nonlinear observability problem and its relation to vehicle speed and cornering stiffness is discussed. Finally, the domain of validity for this observer is specified using the condition number of an observability matrix as a criterion. 2. Vehicle model Lateral vehicle dynamics has been studied since the 1950s. Segel (1956) presented a vehicle model with three degrees of freedom in order to describe lateral movements including 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 vehicle _ and sideslip). A nonlinear dynamics (yaw rate ðcÞ representation of the bicycle model is shown in Fig. 1. Notation is explained in Appendix. A certain number of simplifications are used in this study. Cornering stiffnesses ðC iF d Þ are taken to be constant.

Center of gravity to front, rear axle distance (m) Vehicle total mass (kg) Speed of center of gravity ðm s1 Þ Lateral acceleration at center of gravity ðm s2 Þ Front wheel steering angle (rad) Vehicle sideslip angle (rad) Front, rear wheel sideslip angle (rad) Yaw rate ðrad s1 Þ

But cornering stiffness is modified as a result of vertical forces on the wheel. Variations in cornering stiffness during a double lane change maneuver at different speeds are shown in Fig. 2. Lateral tire/road forces ðF iy Þ are highly nonlinear. Various wheel-ground contact force models are to be found in the literature, including a comparison of three different models in Ste´phant, Charara, and Meizel (2002). In this paper, transversal forces are taken to be linear. This assumption is reasonable when vehicle lateral acceleration € is less than 0:4g Lechner (2001). Consequently, ðyÞ transversal forces can be written as F it ¼ C iF d :di ;

i ¼ 1; 2.

(1)

When projected into the vehicle coordinate system, transversal forces shown in Fig. 1 are written 8 < F 1y ¼ F 1t cosðbÞ; (2) : F 2y ¼ F 2t ; where ðbÞ is the front wheel steering angle.

x 104 Callas : real front tire cornering stiffness (N/rad) 16 14

y

δ1

y0 ψ

x

β

(N/rad)

12

F1l

10 8

x0 δ

Ft1

VG δ

2

2

4

ψ L1

2 Fl

Ft

40km/h 90km/h 105km/h

6

L2

Fig. 1. Schematic representation of the bicycle model.

2

0

50

100

150

200

posx (m) Fig. 2. Real front tire cornering stiffness from the Callas simulator. Double lane change maneuver at 40, 90 and 105 km/h.

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Rear and front tire sideslip angles are calculated using _ kinematic relations with vehicle speed ðV G Þ and yaw rate c: 8 _ > 1 1 c > > > 2 2 c > ¼ d þ L : d : VG

sign (εx) and signeq (εx) function 1 0.8

0.4 0.2

-0.2

x_ ¼ f NL ðx; uÞ,

-0.4

_ T , u ¼ ðb F 1 F 2 ÞT and with x ¼ ðV G d cÞ l l 8 1 > > u2 cosðx2  u1 Þ þ u3 cosðx2 Þ x_ 1 ¼ > > mv > >   > > > 1 2 > 2 x3 > þ C x þ L sinðx2 Þ; > 2 > mv F d x1 > > >   > > 1 1 > 1 x3 > > C u  x  L þ sinðx2  u1 Þ; 1 2 > > mv F d x1 > > > > > 1 > > _ > < x2 ¼ mv x1 u2 sinðu1  x2 Þ  u3 sinðx2 Þ   1 1 > 1 x3 > C u 1  x2  L cosðu1  x2 Þ þ > > mv x1 F d x1 > > >   > > 1 x3 > > C 2F d x2 þ L2 þ cosðx2 Þ  x3 > > > m x x1 v 1 > >    > > > 1 > 1 2 2 2 x3 > _ ¼ L u sinðu Þ  L C x þ L x > 3 2 1 2 Fd > I zz x1 > > >   > > 1 x3 > > > þ L1 C 1F d u1  x2  L1 cosðu1 Þ: : I zz x1

sign() atan(1*εx) atan(10*εx)*2/pi

0.6

The vehicle model can be expressed in terms of a nonlinear state–space formulation using Newtonian theory, based on Fig. 1, as follows: (4)

805

0

-0.6 -0.8 -1 -10

-8

-6

-4

-2

0 x

2

4

6

8

10

Fig. 4. Sign functions for sliding mode observer.

(5)

The state comprises the speed of the center of gravity ðV G Þ, _ Inputs are the the sideslip angle ðdÞ and the yaw rate ðcÞ. front wheel steering angle ðbÞ and the longitudinal forces applied to the front ðF 1l Þ and rear ðF 2l Þ wheels. 3. Sliding mode observer From Perruquetti and Barbot, 2002 it is clear that this kind of observer is useful when working with reduced observation error dynamics and when seeking a finite time convergence for all observable states, as well as robustness when confronted with parameter variations (with respect to conditions). Fig. 3 presents the sliding mode observer method 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 the

center of gravity. The first measurement is available from the ESP control unit, and the second can be calculated from the ABS sensors. The observation equation can be written z ¼ hNL ðxÞ ¼ ðh1NL ðxÞ h2NL ðxÞÞT _ T. ¼ ðx1 x3 ÞT ¼ ðV G cÞ The sliding mode observer equations are ( b_ ¼ f NL ðb x x; uÞ þ L signeq ðz  bzÞ; bz ¼ hNL ðb xÞ;

ð6Þ

(7)

where L is the observer gain matrix in R32 . To cover chattering effects (Perruquetti & Barbot, 2002), the function signeq used in this paper is signeq ðx Þ ¼ arctanðl  x Þ  2=p. Coefficient l is a design parameter to adjust the slope of the arctan function as shown in Fig. 4. The greater the l coefficient, the greater the slope of the signeq function, and the greater the chattering. For all tests presented in this paper, l ¼ 1, and errors in the different state–space variables are below 0:03 ðm s1 Þ as regards speed and 3:3 ð s1 Þ as regards yaw rate.

4. Observability

Fig. 3. Sliding mode observer method.

Using the nonlinear state space formulation, the observability definition is local and uses the Lie derivative (Nijmeijer & Van der Schaft, 1990). It is a function of estimated state trajectory and inputs applied to the model. For the system described by Eq. (7) and sensor set (6) the

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observability function is 1 0 ^ h1NL ðxÞ C B ^ uÞ C B ðLf h1NL Þðx; C B C B ðL2 h1 Þðx; f NL ^ uÞ C B ^ uÞ ¼ B oðx; C, ^ C B h2NL ðxÞ C B C B ðLf h2 Þðx; NL ^ uÞ A @ 2 2 ^ uÞ ðLf hNL Þðx;

the smallest. The condition number and its inverse have no units. condðOÞ ¼

5. Simulation results

(9)

The observability function is therefore expressed as 1 0 h1NL C B C B f 1NL C B  1  C B 3 P qf NL i C B :f NL C B C B qx i i¼1 C B C B 2 hNL o¼B C. C B 2 C B 3 P qhNL i C B :f NL C B C B i¼1 qxi C B 3  2 C BP 3 q P qhNL i i A @ :f NL :f NL i¼1 qxi i¼1 qxi

(10)

If the o function is invertible at the current state and input, the system is observable. This function is invertible if its Jacobian matrix O has a full rank. This ‘‘Observability matrix’’ is defined as d ^ uÞ. oðx; dx

(11)

It is also possible to give an observability indicator with the inverse of the Jacobian matrix condition number. This gives a measure of the sensitivity of the solution to the observation problem. The condition number of a matrix O is defined as the ratio of the largest singular value of O to

The Callas simulator provides simulations which can be used to study the performance of the sliding mode observer of vehicle sideslip angle. This software is a realistic simulator validated by car manufacturers and research institutions including INRETS (‘‘Institut national de recherche sur les transports et leur se´curite´’’). The Callas model takes into account vertical dynamics (suspension, tires), kinematics, elasto-kinematics, tire adhesion and aerodynamics. This vehicle simulator was developed by SERA-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 is representative of the transient lateral behavior of a vehicle. The double lane change is performed at three different speeds: 40, 90 and 105 km/h. The difference between the three tests is the level of lateral acceleration. At 105 km/h the level is so high that the simulator’s virtual driver loses control of the car. The slalom is performed at 50, 80 and 90 km/h. The virtual driver in the Callas simulator increases the frequency applied to the steering wheel from 0.01 to 0.2 Hz. All tests are performed on a dry track, with the friction coefficient between ground and tire assumed to be at its maximum possible value of one. 5.1. Observer results Fig. 5 presents results of the calculation of vehicle sideslip angle for a double lane change (Fig. 5(a)) and a slalom (Fig. 5(b)) by the sliding mode observer (7). Ste´phant (2004) presents some similar results. An extended Kalman filter, an extended Luenberger observer and this sliding mode observer are compared. These three kinds of observer applied to the same model 4 give similar results.

sideslip angle 80km/h

Callas

0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8

SMO

δ (°)

δ (°)

sideslip angle 90km/h

0 (a)

(12)

(8)

with 8 > ^ dhjNL ðxÞ j > > ^ ^ uÞ; L f NL ðx; h ð xÞ ¼ < f NL dx > ^ dLif hjNL ðxÞ > Liþ1 hj ðxÞ > ^ uÞ: f NL ðx; : f NL ^ ¼ dx

O¼

lmax ðOÞ . lmin ðOÞ

1

2

3

4 5 6 time (s)

7

8

0.25 0.2 0.15 0.1 0.05 0 -0.05 -0.1 -0.15 -0.2 -0.25

Callas SMO

0

9 (b)

10

20

30

40

50

time (s)

Fig. 5. Sideslip angle by sliding mode observer. (a) ISO double lane change at 90 km/h; (b) slalom at 80 km/h.

ARTICLE IN PRESS J. Ste´phant et al. / Control Engineering Practice 15 (2007) 803–812

20

10

10

5

0

0 40 90 105

6

mean %|error|

20 15 10 5 0 40 90 105

0.08 0.06 0.04 0.02 0 40 90 105 0.04

3

2

2

0.02

2

1

1

0.01

0

0

0

0

40 90 105

40 90 105

20 10

0.03

40 90 105

0 50 80 90

50 80 90

0.04 0.02 0 50 80 90

50 80 90

6

10

1.5

0.03

1

0.02

2

0.5

0.01

0

0

4 5 0

40 90 105

CG speed 0.06

4 3 2 1 0

5

0

lateral acceleration

yaw rate 10

30

40 90 105

3

4

sideslip angle

CG speed max %|error|

15

30

lateral acceleration

yaw rate

mean %|error|

max %|error|

sideslip angle

807

50 80 90

50 80 90

0 50 80 90

50 80 90 x 10-5

x 10-4

5

0

0

(a)

2

10

20 40 90 105

3

20

15

40

var %|error|

var %|error|

60

10

1 0

0 40 90 105

40 90 105

60

10

(b)

1

6

40 20

5

0.5

0

0

4 2

0 40 90 105

8

50 80 90

50 80 90

0 50 80 90

50 80 90

Fig. 6. Normalized error by the observer. Errors in sideslip angle, yaw rate, lateral acceleration and vehicle speed. (a) ISO double lane change at 40, 90 and 105 km/h; (b) slalom at 50, 80 and 90 km/h.

Fig. 6 summarizes the performance of the observer in estimating different variables; Fig. 6(a) for the three double lane 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 of a variable z is defined by z ¼ j^zSMO  zCallas j

100 , maxðjzCallas jÞ

(13)

where z^SMO is the variable calculated by the observer, zCallas is the variable calculated by the Callas simulator and maxðjzCallas jÞ is the absolute maximum value of the variable given by the simulator during the test maneuver. The first row shows the maximum normalized error during the course of the maneuver. The second row shows the mean, and the third row the variance. The leftmost column presents the results for sideslip angle observation, the second column the yaw rate estimation, the third the lateral acceleration, and the rightmost column the vehicle speed. 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 slalom maneuver at 50, 80 and 90 km/h. The lateral acceleration is calculated by b y€ ¼



 C 1F d bþ mv

!   C 2F d L2  C 1F d L1 b_ C 1 þ C 2F d b d, c þ  Fd cG mv mv V (14)

where variables marked ð^:Þ are estimated by the observer.

As shown in the last column of Figs. 6(a) and (b), the vehicle speed is calculated precisely by the sliding mode observer. 5.1.1. Double lane change maneuver All the variables are correctly estimated for the maneuver at 40 km/h. On average the sideslip angle error is around 2%, the yaw rate error 1% and the lateral acceleration error 0.5%. As regards the different lateral variables, the higher the speed, the greater the maximum normalized error, and the greater the error variance. The same applies to yaw rate and to lateral acceleration. The level of error is around 6.5% for the sideslip angle estimation at 90 and 105 km/h. It should be noted that for this kind of path, lateral forces applied are directly linked 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 the estimation of yaw rate and lateral acceleration. In the case of sideslip angle the error behavior may be explained by phase shift between the simulator’s and the observer’s variables. 5.1.3. Remark Using several nonlinear observers based on the (4) mode gives estimated state–space variable trajectories which are similar. Results presented in the next section are available for observers based on this model. 5.2. Observability results The rank of the observability matrix (11) is 3 along the three different paths. The model is observable.

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inverse of condition number (-)

x 10-3 2.63

3.52 3.515 3.51 3.505

40km/h

2.62 2.61 2.6 0

2

4

6

8

10

12

14

16

x 10-3 6.45 6.4 6.35 6.3 6.25

7.4 90km/h

7.2 1

2

3

4

5

6

7

50km/h

18

7.6

0

8

x 10-3

80km/h

9 x 10-3

x 10-3 10.5

7.4

10

7.3 105km/h

7.2

9.5

90km/h

7.1 0 (a)

inverse of condition number (-)

x 10-3

0.5

1

1.5

2 2.5 time (s)

3

3.5

4

0

10

20

(b)

30 time (s)

40

50

Fig. 7. Inverse of condition number of observability matrix. (a) ISO double lane change at 40, 90 and 105 km/h; (b) slalom at 50, 80 and 90 km/h.

As discussed in Section 4, the condition number of an observability matrix is used to obtain an indicator of observability quality. Fig. 7 presents the inverse of the condition number of matrix O for the six different maneuvers. In the following two sections the relation between the condition number of the observability matrix and vehicle speed and tire cornering stiffness is discussed. 5.2.1. Speed and observability matrix Two conclusions may be drawn about the relation of the observability matrix to vehicle speed. First, Fig. 9(a) illustrates that the greater the speed, the smaller the condition number. At 40 km/h the condition number is around 380, at 90 km/h it is 135, and at 105 km/h it is 100. The curve of the inverse of the condition number is related to the curve of vehicle speed, as shown in Fig. 8, in particular for the double lane change maneuver at 40 km/h. The correlation coefficient is close to 0.9 for this test path, as shown in Fig. 9(b). At higher speeds the correlation between the two variables is not significant (below 0.6). The influence of speed variations is also visible in the slalom maneuver at 50 km/h, when abrupt speed variations appear such as at time t ¼ 12 ðsÞ, which is highlighted by a light gray 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 the shape of the tire cornering stiffness curve, shown in Fig. 10,

with the curve for the condition number of the observability matrix, shown in Fig. 7. Variations in both variables are similar. As shown in Fig. 11, the correlation coefficient between the real tire cornering stiffness and the inverse of the observability matrix is above 0.9, except in the case of the double lane change maneuver at 40 km/h. This can be explained as follows: in the model, tire cornering stiffness is assumed to be constant throughout all tests, as shown in Fig. 10. The greater the variation in actual cornering stiffness, the greater the condition number of the observability matrix, and the less accurate the model. 5.2.3. Conclusion The analysis of the evolution of the observability matrix condition number has shown that this variable is influenced by 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 the vehicle sideslip angle sliding mode observer, data were collected using the Heudiasyc Laboratory vehicle (to be presented in the following section). The test was a slalom performed at high speed (80 km/h). While the car is being controlled by a driver, the steering angle amplitude and frequency increase. With this kind of path, the lateral acceleration applied depends on the steering input. The first aim of this test was to determine the level of lateral

ARTICLE IN PRESS J. Ste´phant et al. / Control Engineering Practice 15 (2007) 803–812

speed of center of gravity (m/s) 11.18 11.16 11.14 11.12

809

speed of center of gravity (m/s) 13.888 13.886

40km/h

50km/h

13.884 13.882 0

2

4

6

8

10

12

14

16

18

25

22.23

24.99

22.22

24.98

22.21

90km/h

0

1

2

3

4

5

6

80km/h

22.2

24.97 7

8

9 25.02

29.4

25

29.3 105km/h

29.2

24.98 90km/h

0

0.5

1

1.5

(a)

2 2.5 time (s)

3

3.5

4

0 (b)

10

20

30 time (s)

40

50

Fig. 8. Vehicle speed along the different paths. (a) ISO double lane change at 40, 90 and 105 km/h; (b) slalom at 50, 80 and 90 km/h.

Correlation coefficient between speed and inverse of condition number

speed and condition number

1

350

300 Slalom 50km/h

250

200 Slalom 80km/h

ISO Double Lane Change 40km/h

0.8 0.6

ISO Double Lane Change 105km/h

0.4 ISO Double Lane Change 90km/h

0.2 Slalom 50km/h Slalom 80km/h

Slalom 90km/h

150

100 10 (a)

1.2 ISO Double Lane Change 40km/h

Correlation coefficient (-)

condition number of observability matrix (-)

400

0 ISO Double Lane Change 90km/h

ISO Double Lane Change 105km/h

15 20 25 30 average speed on the test path (m/s)

Slalom 90km/h

-0.2

35 (b)

Maneuver

Fig. 9. Speed and condition number of observability matrix: (a) condition number of the observability matrix as a function of speed; (b) correlation coefficient between the condition number of the observability matrix and speed.

pressure at which the results yielded by the sliding mode observer become too high. The second aim was to confirm conclusions regarding the properties of the observability matrix presented in Section 5.2.

 

lateral accelerometer ðy€ m Þ, odometry: rotational speeds of the four wheels (ABS sensors) ðV Gm Þ, _ Þ, yaw rate gyrometer ðc m steering angle ðbm Þ, correvit Sensor ðdm Þ.

6.1. Experimental vehicle

  

STRADA is the Heudiasyc Laboratory’s test vehicle: a Citroe¨n Xantia station wagon equipped with number of sensors, shown in Fig. 12. The following were used in the tests described in this paper:

The speed of the center of gravity is calculated as the mean of the longitudinal speeds of the two rear wheels calculated from their rotational speed (ABS sensor) and their radius.

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x 105

Front cornering stiffness (N/rad)

x 105

Front cornering stiffness (N/rad) 1.55

1.56

1.5

1.54

40km/h Callas

Hypothesis for model

x 105 1.6 1.4 1.2 1 0.8

1.6 1.5 1.4 1.3 1.2 1.1

90km/h Callas

Hypothesis for model

x 104

1.6 1.4 1.2 1 0.8

15 10 Hypothesis for model

105km/h Callas

5 0

0.5

1

1.5

(a)

2 2.5 time (s)

3

3.5

4

50km/h Callas

Hypothesis for model

1.45

1.52

x 105

Hypothesis for model

80km/h Callas

x 105

90km/h Callas

0

10

(b)

20

Hypothesis for model

30 time (s)

40

50

Fig. 10. Front cornering stiffness. Simulator calculation and hypothesis used for constructing the observer: (a) ISO double lane change at 40, 90 and 105 km/h; (b) slalom at 50, 80 and 90 km/h.

Correlation coefficient between tyre cornering stiffness and inverse of condition number

Correlation coefficient (-)

1

ISO Double Lane Change 90km/h

ISO Double Lane Change 105km/h

0.8

Slalom 90km/h

Slalom 50km/h Slalom 80km/h

0.6 ISO Double Lane Change 40km/h

0.4 Fig. 12. STRADA: the Heudiasyc Laboratory’s experimental vehicle.

0.2

6.2. Observer results 0 Maneuver Fig. 11. Correlation coefficient between the condition number of the observability matrix and tire cornering stiffness.

The Correvit S-400 is a noncontact optical sensor mounted at the rear of STRADA on the sprung mass of the car. The S-400 sensor provides highly accurate measurement of distance, speed and acceleration, sideslip angle, drift angle and yaw angle. The S-400 sensor uses proven optical correlation technology to ensure the most accurate possible signal representation. This technology incorporates a high intensity light source that illuminates the test surface, which is optically detected by the sensor via a two-phase optical grating system.

Fig. 13 presents the estimation of sideslip angle using the sliding mode observer (7) and the vehicle’s lateral acceleration calculated using (14). It was shown that the linear approximation for tire/road transversal forces is valid when the lateral acceleration does not exceed 0:4g, and that a linear vehicle bicycle model using linear tire force is representative when the lateral acceleration does not exceed 0:3g (Lechner, 2001). In this model, cornering is performed at constant speed. The speed over the slalom test was approximately constant at 80 km/h, as shown in the upper part of Fig. 14. Given these conditions, the nonlinear model (4) shows the same characteristics as the linear bicycle model. From Fig. 13(a) it can be seen that the error in sideslip angle estimation attributable to the sliding mode observer

ARTICLE IN PRESS J. Ste´phant et al. / Control Engineering Practice 15 (2007) 803–812 lateral acceleration at correvit position (m/s/s)

(m/s/s)

m/s/s

lateral acceleration at correvit position (m/s/s) 12 10 8 6 4 2 0 -2 -4 -6 -8 -10 -12 0

2

4

6

8

10

12

12 10 8 6 4 2 0 -2 -4 -6 -8 -10 -12 64

14

66

Sideslip angle at correvit position (°) 5

811

68

70

72

Sideslip angle at correvit position (°)

Correvit

8

Correvit

SMO

6

SMO

4 (°)

(°)

0

2 0

-5

-2 -4 -6

-10 0

2

4

6

8

(a)

10

12

14

64

66

68

(b)

time (s)

70

72

time (s)

Fig. 13. Measured lateral acceleration and sideslip angle (sideslip angle estimated by the sliding mode observer): (a) slalom at 80 km/h; (b) VDA double lane change at 50 km/h.

speed of center of gravity

normalized observation error on sideslip angle |εδ| (%)

21.8 50 (m/s)

21.6

SMO

45 21.4 40 21.2 35 21 2

x 10-3

4

6

8

10

12

30

14 (%)

0

inverse of condition number (-)

20

10.5 10 9.5 9 8.5 8 7.5 7

15 10 5 0

(a)

25

2

4

6

8 time (s)

10

12

14

0 (b)

5

10

15

time (s)

Fig. 14. Inverse of the observability matrix condition number and sideslip angle observation error for experimental slalom maneuver at 80 km/h: (a) speed and condition number; (b) normalized observation error for sideslip angle.

is less than 0:5 during the first 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 remains less than 0:57g at peak values. Between 6.8 and 12.5 s lateral acceleration is between 0:6g and 0:8g. After 12.5 s lateral acceleration exceeds 0:8g at peak values, which means that the observer can no longer estimate the sideslip

angle correctly. The same conclusions may be drawn in relation to the sideslip angle estimation error for the VDA double lane change maneuver at 50 km/h shown in Fig. 13(b). If a vehicle sideslip angle error of less than 0:5 is acceptable, then it is possible to use the observer presented in this paper when lateral acceleration does not exceed 0:6g. However, when the lateral acceleration exceeds 0:6g, this observer is not sufficiently accurate.

ARTICLE IN PRESS J. Ste´phant et al. / Control Engineering Practice 15 (2007) 803–812

812

(m/s)

speed of center of gravity(m/s) 14.5 14.4 14.3 14.2 14.1 14 13.9 64

66

68

70

72

inverse of condition number (-)

x 10-3 6.5 6 5.5 5 4.5 64

66

68

70

72

time (s) Fig. 15. Speed and inverse of the observability matrix condition number for experimental double lane change maneuver at 50 km/h.

nonlinear bicycle vehicle model. The first conclusion is that with a lateral acceleration not exceeding 0:6g, observer results are quite good. The model used is valid for lateral acceleration less than 0:4g because of the assumption of linear lateral tire forces. The second conclusion is that the condition number of the observability matrix provides an indicator regarding the quality of the estimation. This methodology has been applied, in this paper, with a sliding mode observer. But similar results may be obtained using any nonlinear observer based on the same vehicle model. Since the condition number is directly related to the variations in speed and cornering stiffness, and given that the speed is known, it would appear possible to identify the real cornering stiffness from the calculation of this condition number, as shown in (Ste´phant & Charara, 2005).Since the condition number is directly related to the variations in speed and cornering stiffness, and given that the speed is known, it would appear possible to identify the real cornering stiffness from the calculation of this condition number, as shown in (Ste´phant & Charara, 2005).

6.3. Observability result References The rank of the observability matrix is 3 throughout the slalom and the double lane change tests. Using this criterion, the model is observable. Figs. 14(a) and 15 present the inverse of observability condition number for the experimental slalom and double lane change maneuvers. As it has been shown in relation to the validation by simulation, the condition number is directly linked to the real cornering stiffness. In actual slalom tests the resulting cornering stiffness (by axle) decreases with each cornering. The decreasing peaks of the condition number curve correspond to the peaks of the vehicle sideslip angle curve. The greater the sideslip angle estimation error, the higher the condition number, which can be seen if one compares Fig. 14(a) and (b). As discussed in Section 5.2, the inverse of the condition number is related to the vehicle speed. This result is confirmed by the experimental double lane change maneuver and can be illustrated clearly by comparing the two graphics of Fig. 15. The singularity in the vehicle speed at 70.5 s is also found in the inverse of observability matrix condition number. 7. Conclusion This paper has presented in detail the properties of a vehicle sideslip angle sliding mode observer applied to a

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