- Email: [email protected]
Model Based Tire Pressure Diagnosis
Copyright ~ IFAC Fault Detection, Supervision and Safety for Technical Processes, Kingston Upon Hul~ UK, 1997
MODEL BASED TIRE PRESSURE DIAGNOSIS U . Kiencke, R. Eger, H. Mayer
Institute for Industrial Information Systems University of K arlsruhe, Germany Hertzstr. 16, 76187 Karlsruhe email: [email protected] lax: 0049-721-755788
Abstract: This paper investigates the posibility of using the spectral infomation of wheel rotation to develop a tire pressure monitor. Different models are presented to explain pressure dependent resonant frequencies. Measurements are carried out to verify the models experimentally. The different system inputs of varying road surfaces and driving conditions, which influent the estimated spectra, do not allow for simple classification methods. In order to overcome the classification problem neural networks are used for tire pressure classification. Copyright © 1998 IFAC Keywords: Tire Pressure Monitoring, Wheel Speed Analysis, Tire Belt Dynamics, Neural Network Classification
1. INTRODUCTION
pressure on board. Based on the rotational speed of the wheels, we try to mark pressure dependent resonant frequencies which then can be used for classification. As the wheel speed is used in most modem automobiles by the Anti-Lock-BrakingSystem, no additional sensors would be necessary. This promises a relatively cheap solution for the onboard diagnosis of incorrectly inflated tires.
The safety of passengers in automobiles vitally depends on the correct inflation of the tires. A loss in tire pressure results in a greater stress that may eventually lead to the destruction of the tire due to thermal effects. Furthermore the vehicle's side stability is reduced. Extreme driving conditions, such as sharp curves can therefore be very dangerous. In modern tires a loss of pressure is usually a very slow process. It can take up to several weeks until the pressure loss results in a 'flat tire'. This explains why the driver of a car is in most cases insensible to this faulty condition. In recent years research has been done to develop technical systems that detect a certain loss in tire pressure (usually 0.6· 105 Pa loss) and inform the driver of this condition (Folger, 1989). The most obvious solution is to measure the tire pressure directly using pressure sensors, i.e. Michelin MTM system (Herbert, 1990). That these systems have to be located inside the rotating wheel and the required use of sophisticated sensors results in a very complex and expensive system. In this article, we present a new approach in detecting the loss of tire
2. SPECTRAL ANALYSIS OF WHEEL ROTATION The use of wheel speeds for tire pressure classification has been proposed by Williams (1992) and Mayer (1995a). As the radius of the wheel decreases with the loss of pressure, the wheel must turn more frequently to cover the same distance than a correctly inflated wheel. By comparing the mean rotational speed of all four wheels, it is possible to detect the defect one. The analysis of the dynamics puts the focus on the system characteristic of each individual wheel. The necessity of comparative diagnosis becomes obsolete.
795
n [1/sJ
p
':~I 3.5 0
0.5
1
t[SJ
x[n]
0.5
1
t[SJ
1.5
2: akx[n -
r.[O]
1.5
[
k=l
+ 2: bku[n -
a2 r.[2] . =.
... r.[-(p-2)]
.
r.[~-I]
".
.
: ap
: r.[M-p)
:
r.[p]
(4) (5)
The results of an AR spectral estimation using a model order of 20 is shown in figure 2. Both front and rear wheels show two characteristic frequencies, one below 20 H z and the other around 40 H z. The rear wheel shows an additional resonance above 80 H z. Comparing the two spectra at different pressures one recognizes a shift of the resonant frequencies. If this shift can be proven to be pressure dependent, a tire fault diagnosis based on the dynamics of the wheel rotation is possible. IHI I %(dBJfront :
·10
,.;~.
;
!
"'-""!
:
·········~':\::,l,···;;;.!···~\,,·····)··············~·· · ·..···· 1l1=2,2·10 55Pa !-
! -- Pz=1,S'10
Pa
:: ::::::::::::::]::::::::::::::+:::::::... ::::~:~:'~~::::::J.: . .: .=:::::~::~~.:::~::::
4OL---~----~----~----~--~----~
o
20
40
so
80
100 f (Hz) 120.
40~--~----~----~----~--~----~
o
q
k]
... r.[_(p_l)]][a1] [r'[I]]
r.[l]
The idea to have a closer look on the oscillations of the wheel speeds results from measuring the rotational speed together with the vertical acceleration of the axle. Appel and Hohnheiser (1995) as well as Mayer (1995b) performed a vibration signal analysis to develop a tire pressure monitor. As this method requires a separate acceleration sensor to be installed, the system costs are quite high. If the wheel speed shows similar vibration characteristics, it could be used for tire pressure monitoring as welL The test vehicle was driven over a sequence of equally spaced pulses to generate a strong excitation of the system. The results are shown in figure 1. Besides the expected vertical acceleration the rotational speed showed strong oscillations too. In order to find characteristic frequencies that might even be pressure dependent, a spectral analysis was performed. Instead of using the classical periodogram spectral estimator, linear parametric models were used. This has the great advantage that the parameters of the model contain all the spectral information. These parameters can therefore be used for a classification. The input sequence u[k] that represents the road excitation was assumed to be white noise with a variance of (12. For the general ARMAmodel (eqn. 1) the spectrum can be calculated by evaluating (2) along the unit circle. x[n] = -
(3)
To improve the estimation of the unknown parameters ak the Yule-Walker equation was expanded (4) using more samples of the autocorrelation sequence Tz[k]. This over determined set of equations can then be solved using a least squares estimator (5).
Fig. 1. Rotational speed n and vertical acceleration a of one wheel, generated by pulses on the road.
p
+ urn]
k]
k=I
~E:!EI o
= - 2: akx[n -
20
40
60
80
100 f (Hz) 120
Fig. 2. Spectrum of wheel rotation at different pressures (front and rear wheel)
k] (1)
k=O
(2)
3. MODELS FOR WHEEL SPEED RESONANCE The problem is to estimate the unknown parameters ak and bk. To avoid the nonlinear equations for estimating the parameters bk the simpler ARmodels were used.
In this chapter different models are derived to explain the occurrence of resonant frequencies in the wheel speed spectrum.
796
3.1 Drive based oscillations
The wheel speed shows oscillations in the same range. To explain the wheel speed oscillations, the vertical vibration model is extended to the one shown in figure 4b. Two effects are now taken into
The first oscillations to be discussed are the frequencies above 80 H z . These frequencies appear on the rear wheels only. As the vehicle has rear wheel drive, these oscillations might be generated by the oscillating engine torque that is transmitted trough the power train to the rear wheels. To verify this assumption, the engine speed as well as the speeds of rear wheels and differential was sampled (see figure 3) . Before the clutch delivers the engine torque to the drive shaft (t < 0.6 sec) the oscillations appear only in the engine speed. If the torque is delivered though the clutch to the drive shaft, the oscillations appear on the rear axle as well. The oscilations can be seen very well at the differential gear, as they are only slightly damped by the drive shaft. On the wheels, this oscillation appears as well, but its amplitude is reduced due to the lowpass characteristics of the rear axle, which connects the differential gear with the wheels. The frequency depends only on the engine speed and the transmission gear. It is not pressure dependent as figure 2 suggests and cannot be used for a tire fault diagnosis.
a)
Fig. 4. Quarter-car model and extended model to simulate wheel speed oscillations around 15
Hz. account. First the dynamic tire radius. The change of the radius can be expressed by Tdyn=ZR-h
(7)
While the speed of ' the heavy vehicle remains unchanged, the radius change results in different wheel speeds. The oscillations have therefore the same dynamics as the dynamic radius. The second effect is the dynamic wheel load. It can be calculated by
I
5.0 clutch
4.8
(
(8)
4.6
4.4
The resulting tractive force on the ground depends on the friction coefficient IJ of the road and the vertical load.
4.2
Fx
3.2
3L-----~----~----____~======~ 0.4
0.6
0.8
1.2
FN,stat = (mF
t Is)
3.2 Suspension of the vehicle Resonant frequencies around 15 H Z occur in the transfer function of the vehicle's suspension system. The dynamic behavior of a quater-car model, shown in figure 4a, can be described by the following equations:
mRzR = CF(ZF-ZR)
+ dF(zF -
+ mR) . 9
(9)
The dynamic behavior of (8) and (9) are described by (6), assuming a constant IJ. As the tractive force results in a torque MA at the wheel, the wheel speed must show the same dynamics as the suspension system. Figure 5 shows the simulation results of the extended model. The pressure loss was modeled by changing the tire stiffness CR. The frequency shift in wheel speed is pressure dependent and can be used to detect a faulty tire.
Fig. 3. Oscillations of engine speed transmitted to the rear wheels through the power train.
mFZF = -CF(ZF - ZR) - dF(zF - ZR)
= IJ (FN,stat + FN,dyn)
3.3 Tire belt dynamics To explain the resonance around 40 H z, a more complex model is necessary. The simple assumption to represent the tire by a single spring is not sufficient. In reality, the tire is a highly complex system with distributed parameters. Many models have been derived to describe the dynamics of the tire belt, see (B6hm and Kollatz, 1988;
(6)
ZR)- CR(ZR - h)
In this model the tire is represented by the pressure dependent stiffness CR. The resonance drifts to lower frequencies due to a loss of pressure.
797
figure 7, these parameters can be expressed with the pressure p as
J!i!. [dB) IHoI 0 -2 -4
CR
~ 2
-6
C
P
T
tan (00)
~ 2 pL cos (O~) (12) ooh (1-
4R)
-8 -10 -12 -14 -16 -18
0
5
10
15
20
25
30
35 40 f [Hz)
Fig. 5. Spectrum of simulated wheel speed at two different tire pressures. Schulze 1988) . The following model is based on the tire model of Schulze (1988) but is extended to the interesting influence on the wheel speed. Figure 6. shows the tire belt with the displacement of one differential mass. Taking the forces onto the mass into account, the following partial differential equations with respect to the time t and the angle
Fig. 7. Tire model and geometry to determine the parameters CR and CT . The resulting torque on the wheel can be expressed by integrating the resulting tangential forces around the belt using (11) .
pV = ~ (v"
+ v) - {b (u' + v) - dR (v + Ov') - CRV - p[20 W + 0 2 (v" + 2u' - v - R) + n(v' -
pU = -
1& (u" + v') CT U -
+ n(R + v + u') -
(10)
u) - a f
~]
+ v) + 0 2 (u" + 2v' af
at
=j
r (
u (
d
(13)
To simulate the system, equation (10) and (11) were transformed into a set of ordinary differential equations using 12 equally spaced masses around the belt to represent the original system. The magnitude response of the simulated system is shown in figure 8. The frequency range matches with the results of the measured wheel spectra. As the resonance generated by tire belt vibrations are pressure dependent, they can be used for a pressure classification.
e:,]
u)
(11)
with
of (
=p
o u)
- dT (u + Ou')
P [20 (u'
J 211"
M (t)
of (
IGM~2ltnl (dB]
a
o ·1
::::::~r::=~. ~.:1>: ;I=:::r:::::::t: ::~:: ~ :~~~~: : : : : : : :~::::r::::::::r::::::I: : :::T:~:. ..C==:~:~:::=:=t=·=· .
w
~
~
~
~
...... ....:,
~
ro
~
00
~
f [Hz)
Fig. 8. Simulated magnitude response of torque generated by belt-dynamics.
4. PRESSURE CLASSIFICATION Fig. 6. Model of tire belt with the displacement of a differential belt mass.
The results of chapter 3 promise an easy implementation of a system for tire pressure classification. The frequency shift of resonance could be detected by applying a simple threshold_ Unfortunately the situation in the real system is not as easy as assumed. Figure 9 shows the pole locations of the estimated AR-models. The estimation was
The radial and the tangential force on the differential mass were modeled by differential springs with the stiffness of CR and c,. . Huang and Su (1992) described these parameters as pressure dependent. With the geometric parameters shown in
798
carried out using a moving window of 200 samples for the estimator. The two semicircles show all the resulting poles from two different driving tests.
NeU/ll1 Network
0.8 0.6
Fig. 10. Structure of pressure classification using neural networks. To generate an appropriate network for the classi· fication of tire pressure it is necessary to have a set of input vectors for the training period. This set should contain many different driving situations with different tire pressures to cover the influence of different road surfaces and driving conditions.
N
]"0.4 0.2
• p,=2,2 '10SPa S o Pz=l ,6 '1 0 Pa
0-1~=-~0.~8===:;-o::';.6~-o~.~4--;-o~.2;;---;;0:--~0~.2~0~.4~""'~~K>--!
·
a)
Re{z}
0.8
z-plane '"
,-.,.
=200Hz
The input vectors to the neural network represent the estimated AR coefficients. A model order of 10 was used. A moving window containing 200 samples that is equivalent to 1 second was applied to the wheel speed data. The resulting AR coefficients of different tests were put together to form the training data set.
/'-37Hz
0.6
..··
"_.... -_ .... -0.2
0 Relz}
0.2
'
_.-••. -'.10 Hz 0.4
0.6
A neural network with 10 input and 10 hidden neurons was trained using the back propagation algorithm. Input vectors belonging to nominal tire pressure were assigned the output value of '1'. A loss in tire pressure of 0.6 . 105 Fa was assigned the network output of '0'. To verify the capability of the trained network to classify the tire pressure correctly, a test data set was applied to the neural network. The applied input vectors were not part of the training data set. The resulting network output is presented in figure 11. The threshold to classify for tire pressure loss
Fig. 9. Pole location of estimated AR spectra for different wheel exitations (different road pavements) . While fig. 9b allows classification using a threshold method, the results shown in fig. 9a require a more complex treatment. There are several reasons why the pole locations of the estimated AR-models vary like this. The first reason results from the uncertainty of the spectral estimation process itself. The time variant road surfaces are of strong influence as well. The assumption in chapter 2 to describe the system input coming from the excitation of the road as white noise is not met. The input sequence is rather given as white noise that passes a time variant filter which generates the different road excitations. For the spectral estimator this filter becomes part of the estimated system. Its changing characteristic is responsible for the varying pole locations.
1 front axle left
o
0.8 0.6
1;:::======::::;---.,.---I
v
la
0.4 5 0.2 0 p,=2,2'10 • P2=1,6'10 Pa 00
50
100
150
200
250
';
o p,=2,2'10 5Pa • P2=1,6'10sPa
Neural networks can be used in applications where the transformation between input and output values is not known analytically. During a training period the network is adapted to transform the input values to the required output. Generally a nonlinear transformation between input space and output space is generated by the network. The classification problem can be solved quite well using neural networks. The structure of tire pressure classification using neural networks is shown in figure 10.
00
50
100
...........
300
350
400
•
I( I [s]
•
. ~ • ....~.......Ill"."...--.
0
150
...
,. • •
•• •••
5. NEURAL NETWORK CLASSIFICATION
•
•
200
•250•
400 I[S]
Fig. 11. Output of trained neural network using a test data set of untrained parameters. is indicated with a dashed line. It can be seen that the neural network is capable of classifying most input vectors correctly. Some output values are very close to the threshold or even in the wrong class. This would lead to false alarms of the system. To improve the classification the network output was filtered through a lowpass
799
filter . A moving average architecture was chosen for easy implementation on a microcontroller. A filter length of 30 seconds gives the results shown in figure 12. A classification is now possible without any false alarms. Due to the MA-filter the dynamic is reduced. This puts no significant disadvantage to the classification system as a loss in tire pressure usually occurs very slowly. It can also be seen, that a simultaneous pressure loss at two wheels can be detected as well. ,~fr~on~t~~~e~le~ft~__________~____________-,
References Appel, U., F. Hohnheiser (1995), On-line Vehicle Tire Pressure Monitoring by Means 0/ Vibration Signal Analysis, 5th EAEC International Congress, Strasbourgh B6hm, F., M. Kollatz, (1988), Some Theoretical Models for Computation 0/ Tire Nonuni/ormities, VDI-Berichte, Reihe 12, Nr. 124, VDIVerlag Folger, J., H. Wallentowitz (1989), Electronic Tyre Pressure Control, International EAEC Conference on New Developments in Power Train and Chassis Engineering, Strasbourg Herbert, J. (1990), Michelin tire monitor (M. T.M), SAE Technical Paper No. 905241, 23rd FISITA Congress, Turin Huang, S.C., C.K. SU, (1992), In-plane Dynamics 0/ Tires on the Road Based on an Experimentally Verified Rolling Ring Model, Vehicle System Dynamics, International Journal of Vehicle Mechanics and Mobility, Vol. 21, p. 247-267 Mayer, H. (1995a), Comparative Diagnosis 0/ Tyre Pressure, 3rd IEEE Conference on Control Applications, p. 627-632, Glasgow Mayer, H. (1995b), Classification 0/ Tyre Pressure using Vertical Axis Vibration, Signal Processing in Industrial Applications, p. 99113, Deutsche Hochschulschriften, HanselHohenhausen, Frankfurt Schulze, D.H., (1988), Zum ..Schwingungsverhalten des Curtelrei/ens beim Uberrollen kurzwelliger Bodenunebenheiten, VDI-Berichte, Reihe 12, Nr. 98, VDI-Verlag Williams, R. (1992), DWS ein neues DruckluftWarnsystem fii.r Automobilrei/en, ATZ Automobiltechnische Zeitschift No. 94, p. 336- 340
0.8 O.6b-_ _ _~=;_-------~t_-___j
o p,=2,2'10SPa 5 " Pz=l,6'10 Pa
\ •
'50
200
250
300
350
400 t [5]
"".,-V'"
0.8 ...
0.6~_ _~=-=.,-___........._--::;;.-----=-_--1 0.4 0.2 00
p,=2,2'10SPa 5 "Pz=l,6'10 Pa
Cl
50
'00
'50
200
250
300
350
400
1(5]
Fig. 12. MA-filtered output for classification
6. CONCLUSION The vibration analysis of wheel speeds provides a new method for tire pressure monitoring. Sophisticated tire models were applied to show that the resonant frequencies around 15 Hz, coming from the suspension system of the vehicle and the frequency range of 40 H z, generated by the tire belt vibration are pressure dependent. Measurements were carried out to verify the results experimentally. Due to the variety of road surfaces that results in different excitations a simple pressure classification using frequency thresholds was not feasible. The AR coefficients that contain all the spectral information were then used to train neural networks. These networks were then capable of classifying the tires even on different road surfaces. With this method incorrectly inflated tires can be detected. It is furthermore possible to detect more than one faulty tire at the same time as no comparative evaluation is necessary.
800
MODEL BASED TIRE PRESSURE DIAGNOSIS U . Kiencke, R. Eger, H. Mayer
Institute for Industrial Information Systems University of K arlsruhe, Germany Hertzstr. 16, 76187 Karlsruhe email: [email protected] lax: 0049-721-755788
Abstract: This paper investigates the posibility of using the spectral infomation of wheel rotation to develop a tire pressure monitor. Different models are presented to explain pressure dependent resonant frequencies. Measurements are carried out to verify the models experimentally. The different system inputs of varying road surfaces and driving conditions, which influent the estimated spectra, do not allow for simple classification methods. In order to overcome the classification problem neural networks are used for tire pressure classification. Copyright © 1998 IFAC Keywords: Tire Pressure Monitoring, Wheel Speed Analysis, Tire Belt Dynamics, Neural Network Classification
1. INTRODUCTION
pressure on board. Based on the rotational speed of the wheels, we try to mark pressure dependent resonant frequencies which then can be used for classification. As the wheel speed is used in most modem automobiles by the Anti-Lock-BrakingSystem, no additional sensors would be necessary. This promises a relatively cheap solution for the onboard diagnosis of incorrectly inflated tires.
The safety of passengers in automobiles vitally depends on the correct inflation of the tires. A loss in tire pressure results in a greater stress that may eventually lead to the destruction of the tire due to thermal effects. Furthermore the vehicle's side stability is reduced. Extreme driving conditions, such as sharp curves can therefore be very dangerous. In modern tires a loss of pressure is usually a very slow process. It can take up to several weeks until the pressure loss results in a 'flat tire'. This explains why the driver of a car is in most cases insensible to this faulty condition. In recent years research has been done to develop technical systems that detect a certain loss in tire pressure (usually 0.6· 105 Pa loss) and inform the driver of this condition (Folger, 1989). The most obvious solution is to measure the tire pressure directly using pressure sensors, i.e. Michelin MTM system (Herbert, 1990). That these systems have to be located inside the rotating wheel and the required use of sophisticated sensors results in a very complex and expensive system. In this article, we present a new approach in detecting the loss of tire
2. SPECTRAL ANALYSIS OF WHEEL ROTATION The use of wheel speeds for tire pressure classification has been proposed by Williams (1992) and Mayer (1995a). As the radius of the wheel decreases with the loss of pressure, the wheel must turn more frequently to cover the same distance than a correctly inflated wheel. By comparing the mean rotational speed of all four wheels, it is possible to detect the defect one. The analysis of the dynamics puts the focus on the system characteristic of each individual wheel. The necessity of comparative diagnosis becomes obsolete.
795
n [1/sJ
p
':~I 3.5 0
0.5
1
t[SJ
x[n]
0.5
1
t[SJ
1.5
2: akx[n -
r.[O]
1.5
[
k=l
+ 2: bku[n -
a2 r.[2] . =.
... r.[-(p-2)]
.
r.[~-I]
".
.
: ap
: r.[M-p)
:
r.[p]
(4) (5)
The results of an AR spectral estimation using a model order of 20 is shown in figure 2. Both front and rear wheels show two characteristic frequencies, one below 20 H z and the other around 40 H z. The rear wheel shows an additional resonance above 80 H z. Comparing the two spectra at different pressures one recognizes a shift of the resonant frequencies. If this shift can be proven to be pressure dependent, a tire fault diagnosis based on the dynamics of the wheel rotation is possible. IHI I %(dBJfront :
·10
,.;~.
;
!
"'-""!
:
·········~':\::,l,···;;;.!···~\,,·····)··············~·· · ·..···· 1l1=2,2·10 55Pa !-
! -- Pz=1,S'10
Pa
:: ::::::::::::::]::::::::::::::+:::::::... ::::~:~:'~~::::::J.: . .: .=:::::~::~~.:::~::::
4OL---~----~----~----~--~----~
o
20
40
so
80
100 f (Hz) 120.
40~--~----~----~----~--~----~
o
q
k]
... r.[_(p_l)]][a1] [r'[I]]
r.[l]
The idea to have a closer look on the oscillations of the wheel speeds results from measuring the rotational speed together with the vertical acceleration of the axle. Appel and Hohnheiser (1995) as well as Mayer (1995b) performed a vibration signal analysis to develop a tire pressure monitor. As this method requires a separate acceleration sensor to be installed, the system costs are quite high. If the wheel speed shows similar vibration characteristics, it could be used for tire pressure monitoring as welL The test vehicle was driven over a sequence of equally spaced pulses to generate a strong excitation of the system. The results are shown in figure 1. Besides the expected vertical acceleration the rotational speed showed strong oscillations too. In order to find characteristic frequencies that might even be pressure dependent, a spectral analysis was performed. Instead of using the classical periodogram spectral estimator, linear parametric models were used. This has the great advantage that the parameters of the model contain all the spectral information. These parameters can therefore be used for a classification. The input sequence u[k] that represents the road excitation was assumed to be white noise with a variance of (12. For the general ARMAmodel (eqn. 1) the spectrum can be calculated by evaluating (2) along the unit circle. x[n] = -
(3)
To improve the estimation of the unknown parameters ak the Yule-Walker equation was expanded (4) using more samples of the autocorrelation sequence Tz[k]. This over determined set of equations can then be solved using a least squares estimator (5).
Fig. 1. Rotational speed n and vertical acceleration a of one wheel, generated by pulses on the road.
p
+ urn]
k]
k=I
~E:!EI o
= - 2: akx[n -
20
40
60
80
100 f (Hz) 120
Fig. 2. Spectrum of wheel rotation at different pressures (front and rear wheel)
k] (1)
k=O
(2)
3. MODELS FOR WHEEL SPEED RESONANCE The problem is to estimate the unknown parameters ak and bk. To avoid the nonlinear equations for estimating the parameters bk the simpler ARmodels were used.
In this chapter different models are derived to explain the occurrence of resonant frequencies in the wheel speed spectrum.
796
3.1 Drive based oscillations
The wheel speed shows oscillations in the same range. To explain the wheel speed oscillations, the vertical vibration model is extended to the one shown in figure 4b. Two effects are now taken into
The first oscillations to be discussed are the frequencies above 80 H z . These frequencies appear on the rear wheels only. As the vehicle has rear wheel drive, these oscillations might be generated by the oscillating engine torque that is transmitted trough the power train to the rear wheels. To verify this assumption, the engine speed as well as the speeds of rear wheels and differential was sampled (see figure 3) . Before the clutch delivers the engine torque to the drive shaft (t < 0.6 sec) the oscillations appear only in the engine speed. If the torque is delivered though the clutch to the drive shaft, the oscillations appear on the rear axle as well. The oscilations can be seen very well at the differential gear, as they are only slightly damped by the drive shaft. On the wheels, this oscillation appears as well, but its amplitude is reduced due to the lowpass characteristics of the rear axle, which connects the differential gear with the wheels. The frequency depends only on the engine speed and the transmission gear. It is not pressure dependent as figure 2 suggests and cannot be used for a tire fault diagnosis.
a)
Fig. 4. Quarter-car model and extended model to simulate wheel speed oscillations around 15
Hz. account. First the dynamic tire radius. The change of the radius can be expressed by Tdyn=ZR-h
(7)
While the speed of ' the heavy vehicle remains unchanged, the radius change results in different wheel speeds. The oscillations have therefore the same dynamics as the dynamic radius. The second effect is the dynamic wheel load. It can be calculated by
I
5.0 clutch
4.8
(
(8)
4.6
4.4
The resulting tractive force on the ground depends on the friction coefficient IJ of the road and the vertical load.
4.2
Fx
3.2
3L-----~----~----____~======~ 0.4
0.6
0.8
1.2
FN,stat = (mF
t Is)
3.2 Suspension of the vehicle Resonant frequencies around 15 H Z occur in the transfer function of the vehicle's suspension system. The dynamic behavior of a quater-car model, shown in figure 4a, can be described by the following equations:
mRzR = CF(ZF-ZR)
+ dF(zF -
+ mR) . 9
(9)
The dynamic behavior of (8) and (9) are described by (6), assuming a constant IJ. As the tractive force results in a torque MA at the wheel, the wheel speed must show the same dynamics as the suspension system. Figure 5 shows the simulation results of the extended model. The pressure loss was modeled by changing the tire stiffness CR. The frequency shift in wheel speed is pressure dependent and can be used to detect a faulty tire.
Fig. 3. Oscillations of engine speed transmitted to the rear wheels through the power train.
mFZF = -CF(ZF - ZR) - dF(zF - ZR)
= IJ (FN,stat + FN,dyn)
3.3 Tire belt dynamics To explain the resonance around 40 H z, a more complex model is necessary. The simple assumption to represent the tire by a single spring is not sufficient. In reality, the tire is a highly complex system with distributed parameters. Many models have been derived to describe the dynamics of the tire belt, see (B6hm and Kollatz, 1988;
(6)
ZR)- CR(ZR - h)
In this model the tire is represented by the pressure dependent stiffness CR. The resonance drifts to lower frequencies due to a loss of pressure.
797
figure 7, these parameters can be expressed with the pressure p as
J!i!. [dB) IHoI 0 -2 -4
CR
~ 2
-6
C
P
T
tan (00)
~ 2 pL cos (O~) (12) ooh (1-
4R)
-8 -10 -12 -14 -16 -18
0
5
10
15
20
25
30
35 40 f [Hz)
Fig. 5. Spectrum of simulated wheel speed at two different tire pressures. Schulze 1988) . The following model is based on the tire model of Schulze (1988) but is extended to the interesting influence on the wheel speed. Figure 6. shows the tire belt with the displacement of one differential mass. Taking the forces onto the mass into account, the following partial differential equations with respect to the time t and the angle
Fig. 7. Tire model and geometry to determine the parameters CR and CT . The resulting torque on the wheel can be expressed by integrating the resulting tangential forces around the belt using (11) .
pV = ~ (v"
+ v) - {b (u' + v) - dR (v + Ov') - CRV - p[20 W + 0 2 (v" + 2u' - v - R) + n(v' -
pU = -
1& (u" + v') CT U -
+ n(R + v + u') -
(10)
u) - a f
~]
+ v) + 0 2 (u" + 2v' af
at
=j
r (
u (
d
(13)
To simulate the system, equation (10) and (11) were transformed into a set of ordinary differential equations using 12 equally spaced masses around the belt to represent the original system. The magnitude response of the simulated system is shown in figure 8. The frequency range matches with the results of the measured wheel spectra. As the resonance generated by tire belt vibrations are pressure dependent, they can be used for a pressure classification.
e:,]
u)
(11)
with
of (
=p
o u)
- dT (u + Ou')
P [20 (u'
J 211"
M (t)
of (
IGM~2ltnl (dB]
a
o ·1
::::::~r::=~. ~.:1>: ;I=:::r:::::::t: ::~:: ~ :~~~~: : : : : : : :~::::r::::::::r::::::I: : :::T:~:. ..C==:~:~:::=:=t=·=· .
w
~
~
~
~
...... ....:,
~
ro
~
00
~
f [Hz)
Fig. 8. Simulated magnitude response of torque generated by belt-dynamics.
4. PRESSURE CLASSIFICATION Fig. 6. Model of tire belt with the displacement of a differential belt mass.
The results of chapter 3 promise an easy implementation of a system for tire pressure classification. The frequency shift of resonance could be detected by applying a simple threshold_ Unfortunately the situation in the real system is not as easy as assumed. Figure 9 shows the pole locations of the estimated AR-models. The estimation was
The radial and the tangential force on the differential mass were modeled by differential springs with the stiffness of CR and c,. . Huang and Su (1992) described these parameters as pressure dependent. With the geometric parameters shown in
798
carried out using a moving window of 200 samples for the estimator. The two semicircles show all the resulting poles from two different driving tests.
NeU/ll1 Network
0.8 0.6
Fig. 10. Structure of pressure classification using neural networks. To generate an appropriate network for the classi· fication of tire pressure it is necessary to have a set of input vectors for the training period. This set should contain many different driving situations with different tire pressures to cover the influence of different road surfaces and driving conditions.
N
]"0.4 0.2
• p,=2,2 '10SPa S o Pz=l ,6 '1 0 Pa
0-1~=-~0.~8===:;-o::';.6~-o~.~4--;-o~.2;;---;;0:--~0~.2~0~.4~""'~~K>--!
·
a)
Re{z}
0.8
z-plane '"
,-.,.
=200Hz
The input vectors to the neural network represent the estimated AR coefficients. A model order of 10 was used. A moving window containing 200 samples that is equivalent to 1 second was applied to the wheel speed data. The resulting AR coefficients of different tests were put together to form the training data set.
/'-37Hz
0.6
..··
"_.... -_ .... -0.2
0 Relz}
0.2
'
_.-••. -'.10 Hz 0.4
0.6
A neural network with 10 input and 10 hidden neurons was trained using the back propagation algorithm. Input vectors belonging to nominal tire pressure were assigned the output value of '1'. A loss in tire pressure of 0.6 . 105 Fa was assigned the network output of '0'. To verify the capability of the trained network to classify the tire pressure correctly, a test data set was applied to the neural network. The applied input vectors were not part of the training data set. The resulting network output is presented in figure 11. The threshold to classify for tire pressure loss
Fig. 9. Pole location of estimated AR spectra for different wheel exitations (different road pavements) . While fig. 9b allows classification using a threshold method, the results shown in fig. 9a require a more complex treatment. There are several reasons why the pole locations of the estimated AR-models vary like this. The first reason results from the uncertainty of the spectral estimation process itself. The time variant road surfaces are of strong influence as well. The assumption in chapter 2 to describe the system input coming from the excitation of the road as white noise is not met. The input sequence is rather given as white noise that passes a time variant filter which generates the different road excitations. For the spectral estimator this filter becomes part of the estimated system. Its changing characteristic is responsible for the varying pole locations.
1 front axle left
o
0.8 0.6
1;:::======::::;---.,.---I
v
la
0.4 5 0.2 0 p,=2,2'10 • P2=1,6'10 Pa 00
50
100
150
200
250
';
o p,=2,2'10 5Pa • P2=1,6'10sPa
Neural networks can be used in applications where the transformation between input and output values is not known analytically. During a training period the network is adapted to transform the input values to the required output. Generally a nonlinear transformation between input space and output space is generated by the network. The classification problem can be solved quite well using neural networks. The structure of tire pressure classification using neural networks is shown in figure 10.
00
50
100
...........
300
350
400
•
I( I [s]
•
. ~ • ....~.......Ill"."...--.
0
150
...
,. • •
•• •••
5. NEURAL NETWORK CLASSIFICATION
•
•
200
•250•
400 I[S]
Fig. 11. Output of trained neural network using a test data set of untrained parameters. is indicated with a dashed line. It can be seen that the neural network is capable of classifying most input vectors correctly. Some output values are very close to the threshold or even in the wrong class. This would lead to false alarms of the system. To improve the classification the network output was filtered through a lowpass
799
filter . A moving average architecture was chosen for easy implementation on a microcontroller. A filter length of 30 seconds gives the results shown in figure 12. A classification is now possible without any false alarms. Due to the MA-filter the dynamic is reduced. This puts no significant disadvantage to the classification system as a loss in tire pressure usually occurs very slowly. It can also be seen, that a simultaneous pressure loss at two wheels can be detected as well. ,~fr~on~t~~~e~le~ft~__________~____________-,
References Appel, U., F. Hohnheiser (1995), On-line Vehicle Tire Pressure Monitoring by Means 0/ Vibration Signal Analysis, 5th EAEC International Congress, Strasbourgh B6hm, F., M. Kollatz, (1988), Some Theoretical Models for Computation 0/ Tire Nonuni/ormities, VDI-Berichte, Reihe 12, Nr. 124, VDIVerlag Folger, J., H. Wallentowitz (1989), Electronic Tyre Pressure Control, International EAEC Conference on New Developments in Power Train and Chassis Engineering, Strasbourg Herbert, J. (1990), Michelin tire monitor (M. T.M), SAE Technical Paper No. 905241, 23rd FISITA Congress, Turin Huang, S.C., C.K. SU, (1992), In-plane Dynamics 0/ Tires on the Road Based on an Experimentally Verified Rolling Ring Model, Vehicle System Dynamics, International Journal of Vehicle Mechanics and Mobility, Vol. 21, p. 247-267 Mayer, H. (1995a), Comparative Diagnosis 0/ Tyre Pressure, 3rd IEEE Conference on Control Applications, p. 627-632, Glasgow Mayer, H. (1995b), Classification 0/ Tyre Pressure using Vertical Axis Vibration, Signal Processing in Industrial Applications, p. 99113, Deutsche Hochschulschriften, HanselHohenhausen, Frankfurt Schulze, D.H., (1988), Zum ..Schwingungsverhalten des Curtelrei/ens beim Uberrollen kurzwelliger Bodenunebenheiten, VDI-Berichte, Reihe 12, Nr. 98, VDI-Verlag Williams, R. (1992), DWS ein neues DruckluftWarnsystem fii.r Automobilrei/en, ATZ Automobiltechnische Zeitschift No. 94, p. 336- 340
0.8 O.6b-_ _ _~=;_-------~t_-___j
o p,=2,2'10SPa 5 " Pz=l,6'10 Pa
\ •
'50
200
250
300
350
400 t [5]
"".,-V'"
0.8 ...
0.6~_ _~=-=.,-___........._--::;;.-----=-_--1 0.4 0.2 00
p,=2,2'10SPa 5 "Pz=l,6'10 Pa
Cl
50
'00
'50
200
250
300
350
400
1(5]
Fig. 12. MA-filtered output for classification
6. CONCLUSION The vibration analysis of wheel speeds provides a new method for tire pressure monitoring. Sophisticated tire models were applied to show that the resonant frequencies around 15 Hz, coming from the suspension system of the vehicle and the frequency range of 40 H z, generated by the tire belt vibration are pressure dependent. Measurements were carried out to verify the results experimentally. Due to the variety of road surfaces that results in different excitations a simple pressure classification using frequency thresholds was not feasible. The AR coefficients that contain all the spectral information were then used to train neural networks. These networks were then capable of classifying the tires even on different road surfaces. With this method incorrectly inflated tires can be detected. It is furthermore possible to detect more than one faulty tire at the same time as no comparative evaluation is necessary.
800