Yaw Rate Estimation for Vehicle Control Applications

N. Sivashankar Scientific Research Labs, Ford IVlotcr Company, 20000 Rotunda Drive, MD 2036 Dearborn, Ml 48121

A. G. Ulsoy Mechanical Engineering and Applied Mechanics Department, The University of Michigan, Ann Arbor, Ml 48109

Yaw Rate Estimation for Vehicle Control Applications* This paper describes a method for vehicle yaw rate estimation using two accelerometers and a steer angle sensor. This yaw rate estimate can be used as an inexpensive alternative to commercial yaw rate sensors in vehicle control applications. The proposed method combines two complementary approaches to yaw rate estimation using accelerometers. This new method is superior to either method used by itself. This paper presents the new approach, supporting analyses, simulation results and experimental validation. The simulation results are based upon both linear and nonlinear vehicle dynamics models and include important effects such as sensor drift and noise, disturbances acting on the vehicle, and model uncertainties. The experimental validation is based on test data from a specially instrumented vehicle driven on a test track. These results indicate that the proposed yaw rate estimation scheme performs well for a wide range of operating conditions and is not difficult to implement.

1

Introduction In recent years, there has been a tremendous increase in interest in advanced safety features in an automobile. This has led to the development of advanced vehicle chassis control systems such as anti-lock brakes (ABS), traction control, four wheel steer, electronic stability program, etc. Yaw rate is an important element of the vehicle dynamics that influences the driver's (and passenger's) perception of vehicle handling and safety features. Hence, a considerable level of effort is being directed towards developing reliable and accurate methods to monitor and control the yaw rate of an automobile. This paper describes a reliable and inexpensive new approach for estimating the yaw rate of a vehicle using accelerometers. Consider the schematic of a vehicle shown in Fig. 1. The vehicle heading angle, i/-'(f)> is defined as the angle between the inertial X-axis (of the inertial X-Y-Z coordinate frame) and the body fixed A;-axis (of the vehicle-fixed x-y-z coordinate frame). The time rate of change of i/'(0. with respect to the vehicle-fixed coordinate system x-y-z, is the yaw rate, r{t). It is necessary to measure the vehicle yaw rate in various vehicle control applications, such as brake-steer (Pilutti et al, 1998), lateral dynamics control and safety warning systems such as collision warning and roadway departure warning systems. In order to measure the yaw rate, sensors are commercially available for use in vehicle control research. Currently, these sensors cost several hundred dollars each even in mass production quantities. This cost is extremely prohibitive for use in mass produced automobiles. Although it is expected that the cost of such sensors will be dramatically reduced in the future, it is of interest to consider methods to estimate the yaw rate accurately for near term use in vehicles. Besides being used to generate an approximate yaw rate measurement for vehicle control applications in the near term, these estimators can be used in the future as redundant (i.e., "backup") sensors for diagnostic purposes in a vehicle with a yaw rate sensor. We have used two accelerometers and a steering angle sensor along with the vehicle forward speed measurement to estimate the yaw rate. The yaw rate is primarily influenced by the steering input of the driver. So, a steering angle sensor would be available as part of an active safety or a vehicle control system * This work was done while the second author was a Visiting Academic with the Scientific Research Labs, Ford Motor Company, Dearborn, MI 48121. Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript

received by the DSCD September 18, 1996. Associate Technical Editor: S. D. Fassois.

where the yaw rate is monitored or controlled. Hence, it is not unreasonable to assume that a steering angle sensor would be available on such a vehicle. The vehicle forward speed measurement does not, have to be extremely accurate. Our yaw rate estimator is sufficiently robust to utilize the speed information fed to the speedometer or (preferably) that used in the ABS module. In principle, the yaw rate can be estimated using just two accelerometers. This method is purely based on vehicle kinematics and is affected significantly by problems such as sensor drift and noise. The yaw rate can also be estimated using a single accelerometer and a steering angle sensor. This method uses a dynamic Kalman filter as an estimator and is affected by vehicle disturbances such as wind gusts, superelevation, etc. Since this estimator uses a linear model of the vehicle dynamics, the estimate is poor during extreme maneuvers of the vehicle. Our approach exploits the advantages of these two complementary methods. It has shown good results in linear and nonlinear simulations as well as actual vehicle tests. We think that an inexpensive approach to (approximately) measure the yaw rate will be important for future commercialization of certain advanced vehicle control systems such as intelligent chassis control systems, collision warning and avoidance systems. The problem of estimating the yaw rate from vehicle acceleration measurements has already been considered (Hitachi, 1993; Soltis, 1993;Soltisetal., 1994; Zarembaetal., 1994). However, the approaches to date for yaw rate measurement using accelerometers have been based purely upon vehicle kinematics. Consequently, these methods are inherently sensitive to measurement noise. We are not aware of any methods that use both kinematic and model based dynamic approaches to estimate yaw rate. Hence, this new approach is the main contribution of this paper. We describe the proposed new approach in Section 2. This is followed by a discussion of linear and non-linear simulation results and of experimental results in Section 3. Finally, we summarize our work and present some concluding remarks in Section 4. 2

Proposed Approach

2.1 Overview. The proposed new approach to yaw rate measurement combines two complementary approaches. Approach 1: A kinematic estimate of the yaw rate can be obtained using two lateral accelerometers that are placed at the left and right sides of the vehicle (see Fig. 1). If aî and 0^2

Journal of Dynamic Systems, Measurement, and Control Copyright © 1998 by ASME Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 11/17/2014 Terms of Use: http://asme.org/terms

JUNE 1998, Vol. 120 / 267

vehicle speed measurement

steering angle sensor

y

BODY FIXED

lat. accelerometer # 1 accelerometer # 2

ze

kinematic and the dynamic estimate. In the limit, when the yaw rate is really small, the filter output is primarily a function of the dynamic estimate. On the other hand, when the yaw rate is very large, the filter output depends primarily on the kinematic estimate. Our combined approach to yaw rate estimation is described in Section 2.4. Since our combined estimator uses the two degree-of-freedom linear model for yaw rate estimation, it needs information on the forward speed. The speed information can be used either in a continuously time-varying filter to estimate the yaw rate or in a gain scheduled time-invariant filter. We have chosen a gain scheduled time-invariant filter to estimate the yaw rate. A complete description of this gain scheduled filter follows in Section 2.5. In this paper, we have modeled the accelerometer dynamics using a first order high-pass filter and used an additive measurement noise Wy to corrupt the measured acceleration. This model is represented as

(2)

= KXtty - X,) + Wy

INERTUL

where the actual lateral acceleration is denoted by Uy and the sensor output by flyj. The first-order high pass filter has a gain of Ks and a time constant of (1 / a j . The sensor state is denoted represent the lateral accelerations measured by the two acceler- as Xs. The high pass nature of the sensor model reflects the fact ometers, Oy the acceleration at CG and Ly the distance between that inexpensive accelerometers have DC drift, and therefore the two accelerometers along the width of the vehicle then the the DC information is filtered as part of the sensor signal amplification to eliminate this problem (DoebUn, 1990; Jurgen, kinematic estimate of the yaw rate is 1994). However, the DC information is lost in this process. Thus, the sensed (as opposed to the actual) accelerations have ayi — ay2\ ro = sign (Oy) (1) a subscript "s" and these include the effects of sensor noise and dynamics. where sign (•) represents the sign (+ve or - v e ) of the variable. 2.2 Kinematic Estimate. As we described in the introThis estimate is relatively insensitive to disturbances acting on the vehicle. However, this estimate is good only if (in addition duction to this section, the kinematic estimate of the yaw rate to other conditions) the signal-to-noise ratio in the difference is obtained from two lateral accelerometers that are placed on signal (Oyi - Oyz) is large. During vehicle maneuvers when the the vehicle as shown schematically in Fig. 1. One can use other yaw rate is small, the noise completely dominates the difference configurations to estimate the yaw rate using two acceleromesignal and the estimate is poor. As we discuss in detail in ters. A discussion of various schemes for kinematically estimatSection 2.2, another drawback of using the relationship in (1) ing the yaw rate and a discussion on the selection of the approis its inability to predict the sign of the yaw rate ( + ve or - v e ) priate sensor configuration is presented in (Sivashankar et al., under transient conditions. Hence, this approach may not pro- 1995). Consider the accelerometer configuration in Fig. 1. These vide satisfactory results. accelerometers measure the following quantities along the body Approach 2: A dynamic estimate of the yaw rate can be fixed y-axis (assuming sufficiently small roll and pitch rates): derived using a simple linear model of the vehicle and a lateral flj,i = ay + Lxif + Lyir^ = i) + ur + L^\f + Ly^r^ (3) acceleration measurement at the CG. The lateral acceleration at the CG can be used as a measurement to a linear model based flj,2 = tty + L^^f ~ l^yir^ = ii + ur + L^2'' ~ l^yi^^ (4) Kalman filter to estimate the yaw rate. The advantage of this approach is that the effect of sensor noise can be minimized where Uy is the lateral acceleration at the center of mass, r is during the filter design process. Also, the dynamic estimate the yaw rate, Lyi, Ly2 are the distances of the two accelerometers predicts the sign of the yaw rate well during transients. The primary limitation of this approach is that the linear estimator model of the vehicle breaks down for larger values of yaw rate a,i a„i due to dominance of the non-linear effects such as those due acCciciuiiicici It 1 to forward speed and tire nonlinearities. Hence, this approach is inappropriate for extreme maneuvers of the vehicle when the yaw rate is large. We describe the mathematical details of this 6 u method in Section 2.3. Fig. 1

Measurements on the vehicle

1

\

Our Combined Approach: It is clear from the above discussion that the measurements from the two approaches complement each other. While the dynamic estimate (Approach 2) is good for low values of yaw rate, the kinematic estimate (Approach 1) is good for higher values of yaw rate. We have exploited this complementary property in our filter design (see Fig. 2). The filter uses the kinematic yaw rate estimate fa, the vehicle lateral acceleration a,, the steering angle and the vehicle speed information as inputs. Depending on the measurement noise characteristics, it chooses a judicious combination of the

„ / K I -0„2

•"•V

^.

Oj,

'",

Filter

aj/l+aj/2

2

' •.—1

1 estimated yaw rate a»2

1

ji-1

"ja

n z, Fig. 2 Configuration of the estimator, S: steering angle, u: forward speed, s,: sign of a.

268 / Vol. 120, JUNE 1998

Downloaded From: http://dynamicsystems.asmedigitalcollection.asme.org/ on 11/17/2014 Terms of Use: http://asme.org/terms

Transactions of the ASME

from the center of mass along the body fixed y-axis, Lî, L^2 are the distances of the two accelerometers from the center of mass along the body fixed x-axis. Notice that L, = Ly, + Ly2Using the sensed lateral accelerations, the lateral acceleration at the center of mass of the vehicle can be computed as: "ysl

+ a'ys2

i) + ur + (Lî + L^2)r.

6: front wheel steer angle;

Uo'. vehicle forward speed.

A linear two degree-of-freedom vehicle dynamics model is used as the basis for the Kalman filter design. The two degree-offreedom bicycle model (see Ulsoy, 1992; Wong, 1993 for details of the derivation of this model) of the vehicle is represented

(5) X = Ax + B6 +

We will assume throughout this paper that the accelerometers are placed close to the center of mass along the body fixed xaxis, i.e., L^ = Lî + L^2 ^ 0. If this condition is not satisfied then Qy cannot be accurately estimated (measured) as in Eq. (5). The kinematic estimate of the yaw rate can be computed from (3) and (4) as in Eq. (1): ro

sign (fly,)

flyil

-l-ysl I

where x := [v r x^]' is the vector of state variables defined above, 6 is the steer angle input, and w := [w\ W2 Wj]' is the process noise. Note that the velocity terms are only used as part of the state vector in the standard two degree-of-freedom bicycle model while the position terms are ignored. The body roll and pitch are assumed to be small and hence their behaviors are also ignored. The matrices A and B are described as

(6) U^/

2.

3.

The accelerometers are aligned along the length of the vehicle. If this condition is not satisfied then the quantity (flysi — a^.,2) is corrupted by other vehicle variables and Eq. (6) is a poor estimate of vehicle yaw rate. The pitch and roll rates of the vehicle are small compared to the yaw rate. If not, a^si and 0^52 are corrupted by other vehicle variables and Eqs. ( 3 ) - ( 4 ) are poor approximations. the signal-to-noise ratio in the difference signal (a^ji aj,,2) is large.

The conditions (1) and (2) are easily satisfied for most of the maneuvers of the vehicle by appropriate placement of the accelerometers. However, condition (3) is not satisfied for vehicle maneuvers which result in lower values of yaw rate. This is because the noise gets amplified while taking the difference between two almost similar values. Consider the kinematic estimate of the yaw rate in (6). Although the sign of the yaw rate is the same as the sign of the lateral accelerations from the two sensors in steady state (Hitachi, 1993) the signs may be different under transient conditions. Hence, this approach to yaw rate estimation may be inaccurate under transient conditions. It is clear from the above discussion that the kinematic estimate, ro: is robust in the presence of external disturbances such as wind gusts and superelevation. However, as detailed above, the estimate degrades considerably in the presence of sensor noise. 2.3 Yaw Rate Estimate With One Acceleration Measurement. As an alternative to the kinematic estimate in (6), one can generate an estimate of the yaw rate by utilizing a dynamic model of the vehicle and applying state estimation (e.g., Kalman filtering) techniques. It utilizes a Kalman filter with the following three states: v.

the lateral velocity (m/s); jc.,:

r:

the yaw rate (rad/s);

sensor state

and a single measurement as the input to the Kalman filter: o„.

the measured lateral acceleration at the center of mass.

The steer angle input to the vehicle (at the front wheel) and vehicle forward speed (preferably obtained from the ABS module in the vehicle) are also assumed to be measured and used in the estimator:

'

âr

^âf

^âr

muo

This relationship is independent of the speed of the vehicle, the location of the yaw axis of rotation of the vehicle and disturbances such as wind gusts acting on the vehicle. From the above equation, it seems as though the yaw rate can be obtained by a simple operation on the accelerometer outputs. However, the above formula gives a good estimate of the yaw rate if: 1.

(7)

A =

tflUQ

mU(,

aCqf - bC^,

a'^Cgf + b^Cg^

OsiCgf + C„,)

a,{aC^f - bC„,.)

0

B =

where C„/ and Ca,. are the effective cornering stiffness for the front and rear tires, respectively, m is the mass of the vehicle, /j is the moment of inertia of the wheel, a and b are the distances of the front and rear axles from the mass center along the body fixed ;c-axis. This model incorporates the accelerometer sensor model as described in (2), where a^ and K^ represent the eigenvalue and gain of the accelerometer high pass dynamics, respectively. The measured output is the lateral acceleration at the center of mass (a,,,) and it is represented in terms of the states as y = Uy,, = c x+ d6 +

(8)

where Ks(Caf

+ Car) KîaCaf

muo

—

mua

bC^r)

-K,

d =

"•sâj

and Wy is the measurement noise. The resulting Kalman filter equations are X = Ax + B6 + k{y - y)

(9)

(10) y = c'x + d5. where x is the vector of estimates of the state variables in x defined above, y = Uys is the measurement, y is an estimate of the measurement and k is the Kalman filter gain. The matrices A, B, c, and d are as derived above for the two degree-offreedom bicycle model with the accelerometer sensor dynamics. The process and measurement noise terms in ( 7 ) - ( 8 ) are assumed to be independent, zero-mean, and normally distributed. The selection of these noise terms is instrumental in determining the specific value of the Kalman filter gains in (9), k = [k, k2 ^3]'. Essentially, the smaller the variance of the measurement noise (relative to the process noise), the larger the gains associated with that measurement signal.

Journal of Dynamic Systems, iVIeasurement, and Control


JUNE 1998, Vol. 120 / 269

Although the dynamic estimator is effective, it is sensitive to unknown lateral and yaw disturbances acting on the vehicle. If one attempts to also model these disturbances, by augmenting the above state equations with additional disturbance states, the model becomes unobservable. This loss of observability, when the basic model is augmented with states, appears to be the dual of the problem discussed by Porter and Power (Porter et al., 1970). This observabiUty issue is discussed further in Sivashankar et al. (1995). Note that a single accelerometer, placed to measure a^, is required to generate the estimate f. However, the approach presented in the next section requires two accelerometers (to measure Oyi and 0,2) such that both Oy and fo can be calculated. 2.4 Combined Estimator. In this subsection, we describe another estimator which combines the complementary features of the two approaches described previously in Subsections 2.2 and 2.3. The combined estimator is schematically illustrated in Fig. 2. It utilizes a Kalman filter with the following two measurements: Oy/. the measured lateral acceleration at the center of mass; Po'. the kinematic yaw rate estimate and the following four states: v. e/.

the lateral velocity;

r:

the yaw rate;

yaw disturbance;

x,,:

sensor state;

(11)

A linear two degree-of-freedom vehicle dynamics model is again used and the Kalman filter is described by state Eq. (9), where x is the vector of state variables defined above in (11), 6 is the steer angle input and the coefficients of the matrices are aCg, - bC„ muo

0

0

0

-a.

muo

aCqf — bCg, A = 0

0 Us (Cgf + Cgf)

as(aCar ~ bCg,.)

muo

muo

two complementary approaches described in Sections 2.2 and 2.3. Essentially, this combined Kalman filter provides an optimal trade-off between the kinematic estimate fo that is corrupted by noise and the smoother dynamic estimate that is sensitive to vehicle disturbances and modeling errors. There are several important aspects of the Kalman filter design presented in this subsection. Compared to the Kalman filter in Section 2.3, we have included one additional input (fo) and one additional state (e,). These are important differences, and require some explanations. Most important is the addition of the noise corrupted kinematic estimate of the yaw rate as a measurement input to the Kalman filter. This not only improves the estimate of the yaw rate, but changes the observability conditions and enables the inclusion of a yaw disturbance state (e,) in the model. This overcomes the loss of observability issue alluded to in Section 2.3. Note that the two required measurements, i.e., ttys and fo, can be obtained from the two lateral accelerometer measurements 0^,1 and Uyî as shown previously in (5) and (6). 2.5 Gain-Scheduled Estimator. The combined estimator developed in the previous subsection incorporates a linear two degree-of-freedom model of the vehicle. Recall that the linear two-degree-of-freedom model is obtained by linearizing the non-linear equations at the operating forward speed. Hence, a single linear model of the vehicle is insufficient to describe the motion of the vehicle for a wide range of speed variations, for example, speeds ranging from MQ = 15 m/s (a!33 mi/hr) to uo = 35 m/s («!78 mi/hr). The combined estimator, designed at a nominal forward speed, does not have good performance over the whole range of speed variations. It was also found that the performance of the combined estimator, described in the previous section, degraded when the vehicle experienced large yaw rates. This is because the vehicle dynamics departs considerably from the linear model under large yaw rate conditions. Hence, a single Kalman filter was found to be insufficient to operate accurately for a wide range of conditions. In order to address these issues, we have developed combined estimators for a few nominal values within the range of operating conditions and switch between these estimators depending on the conditions. The gain-scheduled estimator, which has a switchable gain, ensures good performance over all vehicle operating conditions. We describe this estimator and the design process in this section. We divided the domain of operation of the vehicle into four distinct regions. 1. low forward speed and low yaw rate 2. low forward speed and high yaw rate

(12)

B =

3. high forward speed and low yaw rate 4. high forward speed and high yaw rate We designed a model based combined Kalman estimator for each of these four operating conditions. We used the forward Notice that the yaw disturbance (e^) is modeled as a step disturspeed («o) and the relative magnitude of the accelerometer difbance (with units of rad/s^) acting on the yaw velocity state. ference signal (|(flj,.,i -flyj2)/(«}..!i+ ays2)\) as gain-scheduling This represents vehicle disturbance such as wind gusts which variables. If the accelerometer difference signal is found noisy, induce yaw motion on the vehicle. The measured outputs, Uy^ then one could use the steering angle as a gain-scheduling variand ro, are represented in terms of the states as in (10) where able that captures extreme maneuvers of the vehicle. the coefficients are The design procedure is as follows. For a fixed low forward KsiCgf Ks{aCgf " OCg,) speed (Mo = 20 m/s), we get a linear time-invariant model of cf + Cgr) 0 the vehicle (Eqs. (12) and (13)). Using this linear model, we mua designed two Kalman filter gains, one for the low yaw rate 1 0 J 0 condition and the other for the high yaw rate condition. Since 0 the process model is fixed, the filter gains are strictly a function ' "sâ. of the process and measurement noise characteristics. We d = (13) picked the noise characteristics to reflect the following facts. m L

0

Consequently, the combined estimator utilizes both of the

1.

At low yaw rate values, the signal-to-noise ratio of the accelerometer difference signal is small. The kinematic

270 / Vol. 120, JUNE 1998


Transactions of the ASME

2.

yaw rate estimate (ro) is poor. The filter should primarily depend on the vehicle model and the measurement of the acceleration at the CG (i.e., Uy, = (ayî + ay.,2)/2) to estimate the yaw rate. Hence, the process noise covariance and the noise covariance of the Oy^ measurement are chosen smaller than the noise covariance of the PQ measurement in the filter design process. At high yaw rate values, the signal-to-noise ratio of the fo measurement is large and the vehicle may be operating outside the linear region of operation. Hence, the noise covariance of the PQ measurement is chosen smaller than the process noise covariance and the noise covariance of the Uy,, measurement in the filter design process.

This method of picking the noise characteristics was repeated for a model of the vehicle linearized at a higher forward speed (MO = 30 m/s). Thus, the structure of the gain-scheduled combined estimator is as follows. X = Aiuo)x + B6 + kiiy - y);

y = c'(uo)x + dS

(14)

where B and d are as in Eqs. (12) and (13), A{uo) and c{uo) are similar to A and c in Eqs. (12) and (13). However, these two matrices can take one of two values depending on the gain scheduled parameter Uo. The matrix k, takes one of four values and is defined as follows:

\aysl-ay,2\ \ciyal+ay32\

Pol < £2

h

^

|aii3i-l-a!/32t

ko

«o| > £2 The constants £i and £2 are fixed thresholds for the two gain scheduling variables. This is a realization of a two variable gain scheduled Kalman filter. As will be shown using simulation results in Section 3, this estimator operates well over a wide range of operating conditions. Results and Discussion In this section, we present both simulation and experimental results for yaw rate estimation utilizing the approach described in Section 2. First a comprehensive set of results are presented using a linear simulation model and a nonlinear simulation model. Finally, we describe the estimation results obtained using experimental data.

Table 1 Standard set of parameters used in the simulations

Parameter Description

Parameter Value and Units

Total Mass, m

2000 kg

Moment of inertia, It

2712 kg m^

Forward velocity, u^

20 m/s

Wheelbase,; = a + i

2.69 m

Distance from mass center to front axle, 0

1.04 m

Distance from mass center to rear axle, b

1.65 m

Front tire cornering stiffness, CQ}

-137509.92 N/rad

Rear tire cornering stiffness, CaT

-117800.16 N/rad

Distance between acceierometers, Ly

1.5 m

Acceleration due to gravity, g

9.8 m/s*

Accelerometer model gain, Kg

1.0

Accelerometer model eigenvalue, a.

l/{27r)

Standard deviation of measurement noise, o-y = [CTJ,I 0^2]

[10^2 10-^]

Standard deviation of process noise, a = [ffi 02 0-3 a\ 0-5]'

[10-3 lQ-2 10-3 10-2 io-«]

the kinematic estimate {r_0_hat) in the presence of disturbances and measurement noise. This conclusion is also borne out in a more extensive set of simulations reported in (Sivashankar et al., 1995), but omitted here due to space limitations. The linear two degree-of-freedom vehicle model is suitable for use as the basis of the linear Kalman filter design. However, it is desirable to evaluate this linear Kalman filter in simulations using a more complex vehicle dynamics model. This will allow us to assess the significance of modeling errors on the Kalman filter performance. Here we utilize a seven degree-of-freedom model which also includes a nonlinear (Pacejka) tire model (Bakker et al, 1989; Venhovens, 1992). The model states are: longitudinal velocity («), lateral velocity (v), yaw velocity ( r ) , and the four tire rotational speeds (fi,, ; = 1, 2, 3, 4). The tire forces are nonlinear functions of the sideslip angle ( a ) at each tire, and the longitudinal slip (K) at each tire. Combined braking and steering maneuvers can be handled. The model is described in Venhovens (1995).

3

3.1 Linear and Nonlinear Simulation Studies. A linear two degree-of-freedom vehicle model, as described in Section 2.4 above, is used here not only as the basis for the Kalman filter, but also to simulate the actual vehicle dynamics. The standard set of parameter values used in the simulations is given in Table 1 and any changes from these values are explicitly noted with each set of results. The simulations are for a sinusoidal steer angle input at the front wheels of magnitude 1 degree (i.e., approximately 15-20 degrees at the steering wheel) with a frequency of 0.25 Hz, i.e., 6(t) = 60 sin (-Kt/2) and simulations are carried out for a period of 2.5 seconds. A constant yaw rate disturbance is also simulated in the plant model. The measurement and process noise terms are generated using MatLab's "rand" command as normally distributed zero-mean values. The simulation results are shown in Fig. 3. In this figure, " r " represents the true yaw rate of the vehicle, "r_0_hat" is the kinematic estimate and "rjiat" is the combined estimate. It is clear that the new yaw rate estimate {rjiat) is better than

•

-t—

i 1/ .

/'

\

/!

- m l i '"1 M 1 ^ 'J

A if Jf1> ' i 1 ' 1 Kt; r f f /1

(-

1 ri h

rJM

- - - r_0_l«

1

R1'K

j \

•\l>

r

""•••

.......4 .>;

1

jU L

\ 1

'

1

lii

• 1 ' , 0

.3

.S

.9

1.2

1.S

I.S

2.1

2.4

2.7

TIME IN SECONDS

Fig. 3

Performance of the combined estimator

Journal of Dynamic Systems, Measurement, and Control


JUNE 1998, Vol. 120 / 271

; 32 30 «

29

2

26

Q lU Uj

24

&«

22 20

ia 16 3

»

1

^

: : ;

; i

:

1

:

3

; Jp>^\ \

1.5

yf

.5

\

2

:

r

• W

I

a z

0

L..&^.i • ' \ J

a:

5 ^, Lli

(A

.2

T^^Si^ SAWrf^7*v^ \

Fig. 4

1.5

.5

2 2.S TIME IN SECONDS

"yj2

.^ys\

+ a.ytil

1.5

2

2J

3

3.5

4

4.5

Fig. 5 Performance of gain-sclieduled estimator

Forward speed and steer angle variations

^ys2

2a„, and then choose

1

^ÂSE^ '

TIME IN SECONDS

Using the theory outlined in Section 2.5, we designed four linear estimators and gain scheduled them on two variables— forward speed and relative magnitude of the accelerometer difference signal. To test this gain-scheduled estimator, we simulated the nonlinear vehicle model over a wide range of forward speeds and steering angles. The forward speed of the vehicle increased from 17 m/s to 34 m/s while the steering angle was a sinusoidal-type function with the peak value changing from 3 degrees (at lower speeds) to 1 degree (at higher speeds). The forward speed and steering angle input to the vehicle are plotted in Fig. 4. For the sake of convenience, we approximate. "ysl

i

jlyv;

iiw-\

0

3.5

i*^ ' ^^"""ftiJi

•