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HUNGARIAN JOURNAL
OF INDUSTRIAL CHEMISTRY
VESZPRÉM
Vol. 39(1) pp. 153-156 (2011)
TYRE PRESSURE MONITORING WITH WAVELET-TRANSFORM
P. TÓTH , K. ENISZ, D. FODOR
University of Pannonia, Faculty of Engineering, Institute of Mechanical Engineering
10 Egyetem street, Veszprém 8200, HUNGARY
e-mail: petertoth.19810809@gmail.com
The Tire Pressure Monitoring System (TPMS) is considered to be a widespread system in automotive industry. It has two
basic aspects of view – direct and indirect. The first one uses pressure sensors while the second one uses the wheel speed
signals of the ESP system on board to make an estimation of the pressure levels of the tyres.
The aim of this study is to find an indirect method which can be the competitor in digital signal processing of the
most common and currently used Fourier-transform, and can be a match for the classic solution in complexity, necessary
runtime and efficiency.
Another part of this research is to find an appropriate model to generate wheel speed signals that contain the additional
information to detect the pressure change, moreover to develop a function library for the research to support the
measurements and to analyze the results.
Keywords: tyre, pressure, DSP, indirect, wavelet, fourier, complexity
Introduction
Nowadays the Tyre Pressure Monitoring Systems (TPMS)
are almost a standard equipment of mid-size passenger
cars and this solution will be a mandatory feature for
new cars in Europe starting 2012. The reason is simple.
Due to some difference between the ideal and current
pressure the traction and the stability can decrease
dramatically and it can easily lead to an accident.
Methods
There are two methods to measure the pressure in the
tyres.
The simplest one is to place intelligent pressure
sensors in the tyres. Their radio signals are acquired and
the pressure levels can either be displayed in the cockpit
or be used by the driving stability system on board. This
solution, called direct monitoring, is simple and reliable
as long as the batteries of the sensors and the sensors
themselves last.
If the signals of the wheel speed sensors (Fig. 1) of
an ESP system contained some information about the
actual pressure levels, this attribute could be used. The
second method, called indirect monitoring, is based on
this additional information in the speed signals. Moreover
using sensors already available in the vehicle instead of
new ones would reduce the costs.
Figure 1 : A wheels speed sensor of a stability system
Once the tyre is on the wheel and it’s blown to a
pressure level its circumference – apart from extreme
driving manoeuvres and situations – can be considered
to be constant. The only variant in this equation would
be the tyre pressure. If it changes, the circumference
changes as well. Since the wheel speed sensors measure
the rotational speed (ω) of the wheel and that value is
multiplied by a constant radius (R) parameter (v = ω·R)
the real circumferential speeds of wheels with different
pneumatic pressures and equal rotational speeds may be
diverse. The wheel with the smallest circumference runs
the most, the one with the highest runs the least. This
behaviour can be observed and after a few calculations
quite a good tyre pressure feedback can be given. The
weakness of this solution is that it’s less reliable as the
vehicle speed increases.
Still in the indirect method the wheel speed signals
can be observed from another point of view. The wheel
can be considered as a spring-mass system. The spring
is the flexible tyre the mass is the wheel base and some
amount of the suspension. It starts oscillating while
running on the road and as all such systems this one has
154
an eigenfrequency which appears in the wheel speed
signal too (Fig. 2).
Figure 2: The wheel speed signal used in the
experiments
This frequency depends on the amount of mass and
the spring coefficient. The mass is usually constant so
the only variant in this system is the tyre, mostly its
pressure. The less pressure the tyre has the lower frequency
appears and vice versa.
Current Solution
To gather the eigenfrequency of the mentioned system
some kind of frequency analysis must be performed.
The effective and obvious solution for this has been the
Fourier-transform so far. It decomposes the signal to its
constituent frequencies and returns the amplitudes of
them. Its equivalent in digital world is the Fast Fourier-
transform (FFT). Its result can easily be transformed to
an exact form to find the most dominant components.
Figure 3: Partial result of a FFT
Once the transform has been performed the only
work to do is to find the tyre’s eigenfrequency in it which
is usually in the range of 40–60 Hz. The example in
Fig. 3 has been calculated from 32 samples so as in the
case of a FFT the result will have 32 values. The FFT is
symmetric so only the half of the resulting array contains
real information. Therefore there are only 16 values
available in the image (0...15). The sampling frequency
(fs) is 200 Hz so the highest detected frequency component
at this sampling frequency is fs / 2 = 100 Hz. The range
of 0–15 in the result means a frequency range from 0 to
100 Hz. The peak at 7 in the image means
f = (100 Hz/16)·7 = 43.75 Hz. (1)
However there are only 16 values for 100 Hz which
means there is 6.25 Hz for each segment. Therefore the
actual eigenfrequency has to be calculated from all values
instead of simply searching the maximal one. Of course
for higher accuracy more samples are necessary.
Development
In this study the mentioned Fourier-transform has been
replaced by the Wavelet-transform. More specifically by
the Haar-transform which is one of the simplest variants
of the Wavelet-transforms. It still contains the frequency
information, but has time information in the result as
well. To perform the Haar-transform the Haar-functions
have to be calculated first. There always are as many
Haar-functions as many samples. The exact form to
describe all these functions is as follows:
( )
N
th
1
0 =
(2)
( )
( ) ( )
( )
otherwise
qtq
qtq
N
th pp
pp
p
p
k 2/2/5.0
2/5.02/1
|
|
|
0
2
2
1 2/
2/
<≤−
−<≤−
⎪
⎩
⎪
⎨
⎧
−=
(3)
where k = 1...N; p = [log2k]; q = k – 2
p + 1, the time is
transformed to a [0 ; 1]interval, so 0 ≤ t ≤ 1.
A possible output for the Haar-transfom can be seen
in Fig. 4.
If performed on a noiseless signal, the domains in it
are easy to see (Fig. 5). All of them belong to different
frequency components but the different values within
them contain the time information.
Figure 4: Partial result of the Haar-transform
Figure 5: Domains in the Haar-transform
155
Table 1: Domains in the Haar-transform
Interval f Tperiod
0 DC component ∞
1 6.25 Hz 160 ms
2-3 12.5 Hz 80 ms
4-7 25 Hz 40 ms
8-15 50 Hz 20 ms
16-31 100 Hz 10 ms
The time information isn’t important at this point
due to the narrow time interval of the 32 samples.
fs = 200 Hz → Ts = 5 ms → T32samples = 160 ms (4)
Therefore only a mean value is calculated for each
domain and the time information is cancelled. This way
the disturbing noise effects can be damped a bit with a
minimal loss of (unused) information.
Figure 6: Mean values for each domain
After the curve of the mean values is available (Fig. 6)
the different frequency values can be paired with their
corresponding curves. The actual eigenfrequency can be
found based on a frequency-curve table and some
interpolation.
The mean value of the domains described in Table 1
can be seen in Fig. 7 along the frequency range between
40 and 60 Hz. The correlation of the mean values and
the different frequency values is obvious, but the highest
frequency domains have to be used in the calculations to
get a definite result.
Figure 7: Mean values of domains for different
frequencies
Benefits
The benefits of using the Wavelet-transform can be
measured in miliseconds. There is a comparison of the
necessary running times in the following charts. The
running times of a basic Discrete Fourier-transform (DFT)
and a Discrete Haar-transform (DHT) were compared.
Figure 8: Runtime with 32 samples;
red: DHT, blue: DFT
The experiment was performed on an average PC so
the transforms had to be run 400 times to get quite an
accurate measurement. This way the necessary time
with 32 samples for DFT was ~10.5 ms, for DHT just
above 4 ms. The new solution is 2,5 times faster already
but let’s see what would happen if the amount of samples
was increased to 64:
156
Figure 9: Runtime with 64 samples;
red: DHT, blue: DFT
The Haar-transform got about 4 times faster. Let’s
see the same situation with 128 samples:
Figure 10: Runtime with 128 samples;
red: DHT, blue: DFT
For this amount of samples the DHT is at least
7 times faster than the DFT.
The complexity of the DFT is considered to be
O(n·log n). The complexity of the DHT can be reduced
to O(n) with a minimal optimization:
Its generator functions – mostly at higher frequencies –
contain intervals full of zeroes. These parts can be
ignored due to the pointlessness of multiplications with
zero. Otherwise the runtime would be similar to the
runtime of the DFT.
This comparison may seem useless but the stability
controller ECUs in vehicles are usually equipped with
low-end CPUs. Therefore every reduced µs matters.
Figure 11: Runtime with 32, 64, 128 and 256 samples
ACKNOWLEDGEMENT
We acknowledge the financial support of this work by
the Hungarian State and the European Union under the
TAMOP-4.2.1/B-09/1/KONV-2010-0003 project.
REFERENCES
1. F. GUSTAFSSON: Virtual sensors of tire pressure and
road friction, Society of Automotive Engineers.
Warrendale, PN: 2001-01-0796
2. C. JAIDEVA, A. GOSWAMI, K. CHAN: Fundamentals
of Wavelets, John Wiley and Sons
3. N. PERSSON: Event based sampling with application
to vibraton analysis in pneumatic tires. In: Proceedings
of IEEE Int Conf on Acoustics, Speech, and Sig
Proc 2001 in Salt Lake City, USA 6, 3885–3888
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
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/OtherNamespaces [
<<
/AsReaderSpreads false
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(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
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/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
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/ConvertColors /ConvertToCMYK
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/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
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>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice