An indirect TPMS algorithm based on tire resonance frequency estimated by AR model.
Proper tire pressure is very important for multiple driving performance of a car, and it is necessary to monitor and warn the abnormal tire pressure online. Indirect Tire Pressure Monitoring System (TPMS) monitors the tire pressure based on the wheel speed signals of Anti-lock Braking System (ABS). In this paper, an indirect TPMS method is proposed to estimate the tire pressure according to its resonance frequency of circumferential vibration. Firstly, the errors of ABS wheel speed sensor system caused by the machining tolerance of the tooth ring are estimated based on the measured wheel speed using Recursive Least Squares (RLS) algorithm and the measuring errors are eliminated from the wheel speed signal. Then, the data segments with drive train torsional vibration are found out and eliminated by the methods of correlation analysis. Using the corrected and selected vibration noise, the resonance frequency of the tire vibration system is identified by Maximum Entropy Spectral Estimation (MESE) based on Auto-regressive (AR) model. Finally, the proposed algorithm is verified by test data, and the results show that the resonance frequency can be estimated and the changing of tire pressure can be indicated consequently.
CITATION: Zhao, J., Su, J., Zhu, B., and Shan, J., "An Indirect TPMS Algorithm Based on Tire Resonance Frequency Estimated by AR Model," SAE Int. J. Passeng. Cars - Mech. Syst. 9(1):2016.
As is well known, the improper tire pressure might result in serious wear of tires and high fuel consumption of cars. More seriously, deflated tires are more likely to be blown out and result in severe accidents . Therefore, the study of Tire Pressure Monitoring System (TPMS), which is designed to monitor and warn the abnormal tire pressure online, is necessary for accident prevention of motor vehicles. In some applications which are named as direct TPMS, tire pressure can be measured directly by pressure sensors mounted inside of the tire or on the inflating valve . More economically, tire pressure can be monitored by some kinds of indirect signals and algorithms, and these applications are named as indirect TPMS . In these applications, additional sensors are not necessary and ABS wheel speed signals are used to identify tire pressures according to specific tire characteristics. For instance, rolling radius of a tire will change along with its pressure, which will further result in changes of vehicle states. Thus, the low tire pressure can be monitored by different kinds of vehicle state observers . This method works effectively except when all of the tires of a car are deflated to the same pressure, which means the sizes of tires are almost equal and the difference between tire radiuses cannot be observed . Another application of indirectly TPMS is based on the characteristic between the tire resonance frequency and its pressure. Normally, the higher the tire pressure is, the larger its resonance frequency is. As this method is based on the characteristic of each tire itself, the pressure of a specific tire can be monitored independently, and thus, deflations of any tires can be monitored.
In the frequency based TPMS methods, the tire is look as a vibration system around its circumference, and the noise in wheel speed signal is the response of the vibration system under the excitation of road irregularity. Assuming the road input to be white noise, the resonance frequency of the tire system can be monitored by spectrum analysis methods, and the tire pressure can be estimated consequently. Although there have been plenty of researches about indirect TPMS, the studies on how to improve the accuracy and operating speed of the algorithm are always necessary.
The first essential issue is to find out "effective" data from ABS wheel speed signals. In order to identify the resonance frequency of a tire, the excitation of the tire vibration system must be "good" white noise. If there exist too many disturbances in the excitations, the resonance frequency cannot be identified accurately. The disturbances might be caused by excitation of road, system error of measuring system or torsional vibration of powertrain. That is why the frequency based TPMS won't work when the car is driving on rough road or the torsional vibration is happening in drive line. In this paper, we tried to find out ways to eliminate the "bad" vibration data for tire resonance frequency identification. The manufacturing error of the tooth ring and the drive line torsional vibration are analyzed, estimated and eliminated from the ABS wheel speed signals.
Another important issue in tire resonance frequency identification is the spectrum analysis methods. In the past decades, various spectrum analysis methods are utilized in these TPMS algorithms, and the Fast Fourier Transform (FFT) is the most widely used one . The wavelet based methods are also used in time-frequency analysis of the characteristic between tire resonance frequency and wheel speed signals . Comparing to these methods, the Maximum Entropy Spectral Estimation (MESE) method based on Auto-Regressive (AR) model has higher resolution than FFT and smaller computational complexity than wavelet, and might show good performance in tire resonance frequency identification . In this paper, the AR based MESE algorithm is designed to achieve better performance of resonance frequency estimation.
In this paper, a frequency-analysis based Indirect TPMS is researched to monitor tire pressure more accurately. This paper is organized as follows:
1. The ABS wheel speed was analyzed, and the manufacturing error of the ABS tooth ring is identified. Based on this, the effect caused by the manufacturing error is eliminated from the measured wheel speed signals.
2. Based on the corrected signals, the correlations of the speeds of driving wheel are analyzed. The data segments which show drive line torsional vibration won't be used in the following spectrum estimation.
3. Then, the AR model is established and the MESE method is used to identify the tire resonance frequency and estimate the tire pressure deflation.
Finally, the proposed algorithm is verified by test data samples, and the statistic of these samples is described.
IDENTIFICATION OF TOOTH RING ERROR OF ABS WHEEL SPEED SENSOR
The wheel speed sensor system is one of the most important components of ABS. It is consisted of a toothed ring and an electromagnetic sensor, which is shown in Figure 1. It measures the time span when each tooth rotationally passes over the electromagnetic sensor, and then the wheel rotational speed can be calculated according to the time spans. For the estimation of tire resonance frequency, the vibration noise in wheel speed should be extracted and analyzed by frequency-domain method such as FFT.
A 2Hz high-pass filter is used to filter the wheel speed signals to acquire the vibration noise, and the noise is further analyzed by FFT. Figure 2 is the waterfall plot of the original wheel vibration noise spectrum maps. Due to large differences of the magnitudes of the spectrum on different frequency segments, the energy concentration on frequency domain cannot be seen clearly.
Figure 3 is the 2D version of waterfall in Figure 2. In Figure 3, each line is a PSD map based on a noise data which is measured at a constant vehicle speed. The vehicle speed is shown on the y-axis, and the spectrum maps are plotted on their corresponding vehicle speed. As the magnitudes of spectrums are too small to be distinguished in comparison with vehicle speed, they are magnified 100 times. It is clear that there exist peaks relative to vehicle speed. However, the tire resonance frequency should be actually insensitive to vehicle speed. It means that these peaks are caused by some other disturbances.
Ideally, the teeth should be equally spaced around the circumference of the ring. However, manufacturing error and deformation error, which should be considered as systemic errors of the wheel speed measuring system, exist on each tooth. These errors are not problems for the usage, such as in ABS or ESP control, because the noises caused by them are much smaller than wheel speed. But as shown in
Figure 3, the peaks that might depict tire resonance frequency cannot be seen at all. That means these errors cannot be ignored as their magnitudes are equivalent to the vibration noise of wheel speed. They must be identified and eliminated before estimating the tire resonance frequency preciously.
For a toothed ring with Nt teeth, Nt time spans will be logged per rotation. As shown in Figure 4, let the number of rotation circles be k and the number of teeth be i, and mark the time span of the ith tooth in the k th circle as [DELTA]t(k).
All of the logged [DELTA]t(k) will compose a data series which can be used to identify the error of the ith tooth.
To identify the error of the toothed ring, the linear error model is established . The measured value of the k th error of the ith tooth is
[DELTA][[theta].sub.i](k) = [[theta].sub.i](k)-[[theta].sub.0] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
[??](k) is the average rotational speed of one circle
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
In this paper, Recursive Least Squares (RLS) algorithm is used to identify the errors. Let the estimated value of the k th error of the ith tooth is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the error between the estimated and measured value should be
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] can be updated recursively by the following steps 
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
Where p is recursive factor with p(1) = 1, [lambda] is the forgetting factor and [lambda] [member of] (0,1).
The flow chart of the estimation algorithm of teeth errors is shown in Figure 5. The manufacturing error of a qualified tooth ring product cannot exceed its designed tolerance requirement. The situation that the measured [DELTA][theta] exceed the tolerance restriction might be caused by sharp acceleration or deceleration of the vehicle and the average wheel speed [??] is not accurate. Thus, the recursive procedure should not be started until the initial value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] falls into its tolerance restriction region.
Once the recursive program begins, the error of each tooth is estimated respectively. The iteration of a tooth is deemed to be convergence if
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
The algorithm is converged if the iterations of all of the teeth are converged. Then, the measured time spans are corrected by equation (10).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
To verify the performance of the algorithm, the corrected wheel speed is filtered by a bandpass filter and only the frequency components within 30~60Hz are reserved. Figure 6 is the waterfall plot of the corrected wheel vibration noise spectrum maps, and the spectrum is also magnified. It is clearer if we look along the y-axis in Figure 6, and an energy concentration peak can be seen in the overlaid map, which is shown in Figure 7. That is, if the logged data is long enough, the resonance frequency can be identified.
DATA SELECTION OF DRIVE TRAIN TORSIONAL VIBRATION
Drive train torsional vibration is a typical disturbance in the tire vibration system. In frequency domain, it can result in peaks whose magnitudes are much larger than the peak of resonance frequency. Figure 8 and Figure 9 show examples of frequency-domain analysis of the left driving wheel. Figure 8 shows a data segment with torsional vibration in drivetrain. Figure 8(a) is the unfiltered wheel speed and the low-pass filtered wheel speed, which can be seen as the wheel speed without noise. Figure 8(b) is the noise derived from wheel speed. The regular vibration components can be seen clearly even in time domain figure 8(b). In spectrum map shown in figure 8(c), a sharp peak can be seen, and it results in that the resonance frequency cannot be identified at all. While an ideal data segment and its spectrum map should look like the curves shown in Figure 9, whose resonance frequency is clearer.
As the torsional vibration is applied on the whole drive train, the noises of the two driving wheels will show strong correlation if the torsional vibration exists. Figure 10 and Figure 11 show the wheel speed noises of both two driving wheels corresponding to the same data segments in Figure 8 and Figure 9 respectively. It is seen that correlation of the noises in Figure 10 is much stronger, which indicates that the torsional vibration is occurring in drive train.
In order to describe the correlation of the two driving wheels, the correlation factor [k.sub.cor] is calculated
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
Where [w.sub.i] and [??] are the noise and its mean of the driving wheel, and the left and right driving wheels are denoted by the subscripts l and r respectively. [N.sub.w] is the number of sampling points in a data segment.
For the data segment shown in Figure 8 / Figure 10, the correlation factor of the two driving wheels is 0.6850. While the correlation factor of the data segment shown in Figure 9 / Figure 11 is much smaller, which is 0.1228. The correlation factor [k.sub.cor] of two driving wheels can be used to identify the torsional vibration. In this paper, a data segment can be used in identification of tire resonance frequency only when
[k.sub.cor] < 0.2 (12)
SPECTRUM ESTIMATION AND RESONANCE FREQUENCY IDENTIFICATION
In practice, the Autoregressive (AR) Model, which is also known as the Whole Extremity Model, can be used to describe random processes . If the input of a linear system is a white noise sequence w(n) , its output can be written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
That is, x(n) can be described by the previous p sampling points. p is named as the order of the AR model, and [a.sub.k] are coefficients of the AR model.
After z transformation, the AR Model can be written as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
Based on AR model, the power spectral density (PSD) can be estimated by MESE. If the PSD [P.sub.ww] (z) is equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the output power spectrum of the AR model can be written as follows.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
The parameters [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [a.sub.k] can be calculated by the Yule-Walker equation.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The autocorrelation coefficient [[PHI].sub.XX] (k) is
[[PHI].sub.XX](k)=E[x(n) x (n-k)] (17)
As the computing task is heavy in normal MESE, some kinds of fast algorithms have been developed to solve this problem. In this paper, the Levinson-Durbin Recursive Algorithm is utilized. There are three steps in this algorithm.
Confirm p of AR Model. The order p has great influence on the performance of the spectrum estimation. The method can be found in  and not described in this paper.
Use the first-order Yule-Walker equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
to calculate [a.sub.11] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)
Calculate [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by the recursion formula (21), (22) and (23).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (22)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)
Thus, the power spectrum in equation (15) can be calculated by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and p.
The proposed algorithm was verified by road test data. As the "bad" data might be caused by a lot of reasons, in order to verify the proposed algorithm, the test was carried out under stable conditions. (1) The test was carried out on good paving road, and there are no rough roads or ice roads in the trip. (2) The vehicle was driving stable, and no maneuvers such as panic braking, rapid accelerating or sharp steering were performed. Thus, the errors caused by transient state of tire can be prevented. (3) As the ABS speed sensor cannot provide precious and detailed speed signal when the speed was low, we only used the data when speed was higher than 40kph (4) The temperature is another factor that affects the tire pressure. Before tests the car was driven for about 30min to warm up the tires. (5) The tire pressure was tuned at warm state. The normal pressure was set as 2.3bar, and the deflated pressure was set as 1.8 bar.
The tests for normal pressure and deflated pressure were lasted for 600 seconds respectively. The "effective" test data were divided into segments for spectrum estimation. Each segment contains 4096 sampling points.
An example of the estimation results is shown in Figure 12. The two spectrum maps in Figure 12 are corresponding to the test data of normal and deflated tire pressure respectively. The peaks of the spectrum maps indicate the resonance frequencies of the tire in different pressures.
The statistic of the overall estimation results are shown in Table 1. It is seen that the mean of the resonance frequency of the deflated tire is lower than that of the normal tire. It is also seen that the standard deviations are not small. To ensure the accuracy, sufficient data must be use and the estimated resonance frequency should be the average of plenty of effective data segments. Thus, the abnormal tire pressure can be monitored when its identified tire pressure is lower than a preset threshold.
In this paper, a method of indirect TPMS algorithm based on tire resonance frequency estimation has been introduced. The manufacturing error of the tooth ring is first identified. Base on this, the measured wheel speed signals are corrected. The waterfall plots showed that the errors are removed effectively. Then, the data segments with drive train torsional vibration are eliminated based on correlation factors between noises of two driving wheels. Finally, the spectrum estimation is implemented by MESE based on AR model, and the resonance frequency is extracted according to the spectrum maps. The algorithm is verified by road test data, and the results show that the tire deflation can be identified by the decrease of resonance frequency.
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Jian Zhao, Jing Su, Bing Zhu, and Jingwei Shan
Dr. ZHAO Jian, born in 1978, is currently a professor at State Key Laboratory of Automotive Simulation and Control, Jilin University, China. He received his PhD degree in vehicle engineering from Jilin University, China, in 2007. His research interests include vehicle dynamics analysis and control.
Su Jing, born in 1992, is currently a master candidate at College of Automotive Engineering and State Key Laboratory of Automotive Simulation and Control, Jilin University, China.
ZHU Bing, corresponding author, born in 1982, is currently an associate professor at State Key Laboratory of Automotive Simulation and Control, Jilin University, China. He received the Ph.D. degree in vehicle engineering from Jilin University, China, in 2010. His research interests include vehicle dynamics analysis, vehicle active safety control and optimization as well as bionic engineering.
SHAN Jingwei, born in 1990, is currently a master candidate at College of Automotive Engineering and State Key Laboratory of Automotive Simulation and Control, Jilin University, China.
This work is partially supported by National Natural Science Foundation of China (51105169, 51205156, 51475206, and 51575225).
ABS - Antilock Braking System
AR Model - Auto-Regressive Model
FFT - Fast Fourier Transformation
MESE - Maximum Entropy Spectral Estimation
PSD - Power Spectral Density
RLS - Recursive Least Square
TPMS - Tire pressure monitoring system
Table 1. Statistic of resonance frequency estimation. Normal Deflated Test Time 600s 600s Points for Spectrum Estimation 4096 4096 Effective Segments 16 15 Mean 47.26Hz 45.47Hz Std 1.62Hz 1.16Hz
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|Author:||Zhao, Jian; Su, Jing; Zhu, Bing; Shan, Jingwei|
|Publication:||SAE International Journal of Passenger Cars - Mechanical Systems|
|Date:||Apr 1, 2016|
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