Some Practical Vehicle Speed Estimation Methods by a ... - IEEE Xplore

2011 14th International IEEE Conference on Intelligent Transportation Systems Washington, DC, USA. October 5-7, 2011

Some Practical Vehicle Speed Estimation Methods by a Single Traffic Magnetic Sensor Haijian Li, Honghui Dong, Limin Jia, Dongwei Xu and Yong Qin 

Abstract² This paper proposes three practical vehicle speed estimation methods by a single multifunction magnetic sensor. Compared with traditional methods, this algorithm is simple and convenient to be realized. The multifunction magnetic sensor is described and introduced in this work. Next, a vehicle detection algorithm with a linear time complexity is put forward. Through setting two windows and scanning the vehicle waveform, we obtain the points of vehicle approaching and vehicle leaving, which are the bases for vehicle count, headway time, time occupancy, stopping time, detection of vehicle stopping and presence. The detailed detection methods of vehicle stopping and presence are described. We next present some speed estimation methods in detail. According to three speed estimation methods of Vehicle Length based (VLB), Time Difference based (TDB) and Mean Value based (MVB) to get different reference speeds when a vehicle passes over the sensor, we then analyze the applicability of the three methods. At last, we test the speed estimation methods by adoption of field data. Through the comparison with the real speed of 45 vehicles, it shows that the mean absolute errors (MAE) of VLB, TDB and MVB methods are respectively 4.12km/h, 5.90km/h and 4.05km/h and the mean speed errors of the three methods are all less than 1km/h. These errors are suitable for traffic engineering. Key words² ITS; Traffic Magnetic Sensor; Vehicle Detection; Speed Estimate Method

I. INTRODUCTION

I

NCREASING traffic congestion has become a critical problem for many cities. The Intelligent Transportation System (ITS) is the hot research area to solve the traffic problem. Traffic surveillance provides traffic flow information which is got by kinds of traffic sensors for ITS. And ITS takes advantage of traffic flow information to support traffic

Manuscript received May 15, 2011. This work is supported by the National 863 program (2006AA11Z231) and Beijing Science Foundation Plan Project under Grant (D07020601400707), and is also supported by the State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University (RCS2010ZT004). Haijian Li is a PhD student with School of Traffic and Transportation, Beijing Jiaotong University, Beijing 10044, CHINA. E-mail: [email protected], Ph:+ 8601051686441. Honghui Dong is an assistant professor with State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, CHINA. E-mail: [email protected]. Limin Jia is a professor with State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, CHINA. E-mail: [email protected].

978-1-4577-2197-7/11/$26.00 ©2011 IEEE

management and reduce traffic congestion. Many traffic surveillance technologies are introduced and have been studied, like inductive loop [1-5], microwave radar [6], video image processing method [7-11], magnetic sensor [12-17]. Besides, Global Positioning System (GPS), Mobile Phones are also used to get the traffic flow information. To a great extent, traffic flow information is gained from the traffic sensors installed on the road. For traffic data collection, the cost of sensors, installing and maintaining are great, and many researchers have devoted themselves to study the characteristics of different sensors and select suitable and cheap sensors to reduce the cost of ITS [15,20]. Moreover, the accuracy of different sensors is paid more attention [13,14,16,17]. Lots of researchers have developed some algorithms and methods for different sensors to optimize the computation, reduce the algorithm complexity, enhance the computation speed, and so on. According to the comparison of many kinds of traffic sensors, the magnetic sensors have their unique superiority which are not be influenced by the bad climatic, has a low cost and data size, and is convenient to realize wireless transmission [20,22]. What is more, they are potentially better than loops due to installation and maintenance costs, and they are potentially better than videos because they are more flexible and have less data size. Thus, they are becoming one of the most popular kinds of sensors used in ITS. In the past years, many data fusion methods have been put forward to obtain the important traffic flow information, such as speed, flow, occupy, vehicle existence, vehicle classification, and so on. Those traffic data are very important to traffic control and traffic management. The experiment shows that the traffic flow, occupy and speed, all can be got. The vehicle speed is one of the most important traffic parameters. Many researchers have done a lot of studies based on travel time and got the travel time to estimate some kinds of delays [18,19]. But the basis of their study is to get the vehicle speed, no matter instantaneous speed or travel speed. So how to get the vehicle speed exactly is an important research topic. More and more speed estimation methods based on magnetic sensor can be found in papers and those methods have solved some problems [13,15,22]. However, for speed estimation, those traditional methods may have several major drawbacks.

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One is that they always need a group of sensors (two at the least). Vehicle speed is obtained with two magnetic sensors separately placed at a distance D. When a vehicle passes over, the signal change is detected with one sensor. After time delay Td, the signal change is detected with the other sensor. The speed v is then computed as v = D/Td. Secondly, there are always some places (such as bridge, curve, exit and entrance, little area) that cannot be layout more sensors at a specified distance, so those methods will not suitable. Moreover, those methods need more sensors which will increase the cost. In this paper, the magnetic sensor is introduced to obtain the traffic information. A kind of multifunctional traffic magnetic sensor has been designed in our study, and it can get most of the traffic information with the traffic flow, speed, occupy, vehicle classification, stopping time, vehicle existence and headway time. This is an advantage to the conventional loop method. The paper proposes a new speed estimation method by a single multifunctional traffic magnetic sensor. Compared with traditional methods, the method is simple and has a linear time complexity, so it is convenience to be realized on the computer. In practice, this method can obtain a good accuracy. The layout of this paper is as follows. In Section 2, the multifunctional magnetic sensor is introduced. The algorithm for vehicle detection and the methods of speed estimation by a single magnetic sensor are described in Section 3. Section 4 gives the experiment analysis and comparison. At the end, we draw the conclusion in Section 5.

use in the multifunctional traffic magnetic sensor. The multifunctional traffic magnetic sensor designed in this study is a double-chip sensor and there are two AMR chips in the board along the traffic flow direction (Figure 2). Those AMR chips can detect vertically the magnetic field strength simultaneously. The AMR sensor has a Wheatstone bridge device. It is made out of a nickel-iron thin-film deposited on a silicon wafer and patterned as a resistive strip element [15]. In the magnetic field, a change in the bridge resistive elements will causes a corresponding change in voltage across the Wheatstone bridge [15]. Earths magentic flux lines

Fig. 1. TKHGLVWXUEDQFHRI(DUWK¶VPDgnetic flux lines by a vehicle

II. THE MULTIFUNCTIONAL MAGNETIC SENSOR Vehicle data collection is based the multifunctional traffic magnetic sensor designed in our study [21]. Almost all vehicles have significant amounts of iron metals, and the magnetic field disturbance created by a vehicle can provide the traffic information detected by a magnetic sensor, which makes it a good candidate for vehicle detection. The sensor is based on magneto-resistive technology, in which the circuit resistance is changed with the changing magnetic field. Magneto-Resistive Effect makes it easy to design some vehicle detection devices. Magneto-Resistive Effect refers to certain metals or semiconductors resistance changing with the magnetic field changing. Same with the Hall Effect, Magneto-Resistive Effect is due to the Lorentz Force acting on the carrier in a magnetic field. Figure 1 gives a representation of the disturbance of the magnetic flux lines when the earth's magnetic field meets a vehicle. A sensitive magnetic sensor is needed to measure this earth magnetic change. In this paper, one type of the Magneto-Resistive sensors, Anisotropic Magneto-Resistive (AMR) chip is adopted for this work. And it is very suitable for

Fig. 2. The multifunctional magnetic sensor

The magnetic sensor's output voltage is the mV level voltage, and is enlarged to V level voltage by amplifier. After the filter and A/D converter, the sensor obtains the corresponding digital signal. The digital signal changes with the magnetic field when a vehicle passing. Through observing the waveform data of the digital signal, we can obtain the change of the voltage and the magnetic field changing. According to the sampling frequency and chip model of the magnetic sensor, some effective algorithms for speed estimation can be done. The traffic magnetic sensor in this study is a multifunctional sensor with two chips. Two same AMR chips are installed on the board with the direction of the traffic flow at a specific distance (D). The magnetic directions detected by two chips are the vertical directions, and they are respectively set to Z1 and Z2 axis. The two chips can collect the data of vehicle at the same time. The multifunctional traffic magnetic sensor in this study can

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vehicle detection. 2500

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achieve those functions of vehicle speed, vehicle classification, vehicle count, time occupancy, vehicle stopping, vehicle existence and headway time. This paper mainly studies the vehicle speed estimation method based on multifunctional traffic magnetic sensor and simply discusses the method of vehicle count, discriminate of vehicle stopping and existence, stopping time calculation. Field application of traffic magnetic sensor has been done and it is easy to collect field vehicle waveform data for our study. Next section is devoted to vehicle detection and speed estimation method.

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III. VEHICLE DETECTION AND SPEED ESTIMATION METHOD A. Framework The framework of vehicle detection and speed estimation is shown as figure 3. After processing from field data, the vehicle detection and stopping detection are done. Through judging the vehicle passing or stopping, we can do vehicle count and obtain the vehicle waveform data. Next, the final speed will be got from the estimation methods of Vehicle Length based (VLB), Time Difference based (TDB) and Mean Value based (MVB). Field Data Data Process Vehicle Detection Stopping Detection

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In order to improve the time resolution, the waveform data should be processed by linear interpolation. After interpolation, the shape of the vehicle waveform is not changed, and the horizontal axis stretches only. C. Vehicle Detection The dual threshold approach is adopted to detect the vehicle. Only when the magnetic amplitude is enough high and holds on for enough long duration, the vehicle is identified. As shown in figure 5, InWinHeight and InWinWidth mean the start amplitude threshold and the start time threshold. At the same time, only when the magnetic amplitude is low enough and holds on a long duration, it is recognized the vehicle is out of the detector. In figure 5, the OutwinHeight and OutWinWidth are used respectively to describe the end amplitude threshold and the end time threshold. The A area is the vehicle approaching area, and the B area is the vehicle leaving area. Those four model parameters InWinHeigh, InWinWidth, OutWinHeigh, OutWindWidth should be calibrated by field data. According to adjust the value of those model parameters, the most suitable value of those parameters could be got through the maximum rate of accuracy of vehicle count. Because those four model parameters are relative value, once they are calibrated, they can be used in other field data. 2500

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Fig. 3. The framework of the speed estimation method

B. Date Process The magnetic signal output from the AMR sensors cannot be used directly. Figure 4 shows the output data of magnetic signal with the noise, which will influence vehicle detection. It is necessary to design a low-pass filter to get rid of the noise. There, an FIR filter is used for this low-pass design. After the low-pass filter, the data will be more smoothly and is good for

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Fig.5. The discrimination windows of vehicle approaching and leaving

The process of vehicle detection is given in the following. For no vehicle passing, the magnetic amplitude is essentially the same; for vehicle passing, the magnetic amplitude will be a

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big change. If we can find a point where the absolution difference between magnetic amplitude and EnvironValue is greater than InWinHeigh in the waveform from left to right (window A, in figure 5), and in a range of InWinWidth, all of those points satisfy the condition above, then the point is set to be the vehicle approaching point, written as VehStart. By a similar manner, the vehicle leaving point can be found using window B, and the point is written as VehEnd (window B, in figure 5). Select the point VehStart as the point of vehicle approaching and VehEnd as the point of vehicle leaving, the estimate methods of some traffic parameters are described as follows: For vehicle count: A pair of points (VehStart and VehEnd) shows that a vehicle has passed the magnetic sensor. The waveform between VehStart and VehEnd stands for the vehicle data, then traffic flow increased by 1. For vehicle stopping: When a vehicle is above a sensor, there will be two states: stopping and passing. The vehicle state can be got according to the shape of the vehicle waveform. If it is the stopping state, we can set the instantaneous speed of the vehicle to be 0; and if passing state, we will get the instantaneous speed of the vehicle to be a positive number. When the vehicle state is stopping, the magnetic field around the vehicle has a non-tendency change, but it will rise or fall near to some value which is always far from EnvironValue. Figure 6 shows the vehicle waveform of stopping state. A special window (window C) can be set to distinguish stopping or not. The width and height of the window C are set to be StopWinWidth and StopWinHeight, respectively. If there are some successive points in the waveform inside window C, the stopping state exists. To scan the vehicle waveform from left to right using the window C, we can find the start-stopping point (StartStop) and end-stopping point (EndStop). The time of stopping can be obtained by the sampling frequency. For StopWinWidth and StopWinHeight, they can be calibrated by field data, and they are also relative value. Once calibrated, they can be used in other field data similarly. 2500

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For stopping time: The point StartStop is the start point of vehicle stopping and the point EndStop is the end point. So the stopping time can be given as: EndStop  StartStop (1) Tstopping fs Where: fs the sampling frequency; StartStop the start sampling point of vehicle stopping; EndStop the end sampling point of vehicle stopping. The discrimination of vehicle existence is very important for traffic security management and military affairs management. Vehicle existence is not different from vehicle stopping. For vehicle existence, when the waveform data are starting to be collected, the vehicle has been there, so we cannot judge when the vehicle arrives. For vehicle stopping, when the waveform data are starting to be collected, the vehicle has not been there. We know exactly when the vehicle arrives, and we can find the VehStart point and the VehEnd point. Using the algorithm of vehicle stopping, we can do the discrimination of vehicle existence. The vehicle goes near the sensor, slows down and stops. If the number of VehStart is set to be 1, the number of StartStop must to be greater than 1. For vehicle existence, if the number of VehStart is set to be 1, the number of StartStop must to be 1, too. So the algorithm can judge the cases of vehicle existence and vehicle stopping. D. Speed Estimation Methods (1) Vehicle Length based Speed Estimation If the length of the vehicle is known, then we may obtain the speed according to the time that the vehicle passes by the sensor. For the vehicle waveforms of Z1 and Z2 are almost the same, using one of the two axes data is ok (in this paper, select Z1). The formula is: ( LVeh  Lextra ) u1000 u 3.6 u fs (km/h) (2) vA VehEnd  VehStart Where: LVeh the length of vehicle; the extra length of vehicle; Lextra fs the sampling frequency; Vehstart the start sampling point of vehicle detection; VehEnd the end sampling point of vehicle detection; LVeh can be calculated by vehicle classification, and LVeh and fs are two known parameters. VehEnd-Vehstart can be got by vehicle detection algorithm. The parameter Lextra needs to be calibrated based on field data. Because the magnetic amplitude at VehStart is not equal to the magnetic amplitude when the vehicle head is just passing over the sensor and the magnetic amplitude at VehEnd is also not equal to the magnetic amplitude when the vehicle tail is just

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passing over the sensor, the extra length Lextra is existent. Figure 7 shows the existent of Lextra. When the magnetic amplitude is equal to the magnetic amplitude at VehStart, the distance between vehicle and sensor is Lextra1. And when the magnetic amplitude is equal to the magnetic amplitude at VehEnd, the distance between vehicle and sensor is Lextra2. Then Lextra = Lextra1 + Lextra2. For different thresholds, the Lextra will be different. 2500

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Fig. 7. Diagram of extra vehicle length

According to the character of field magnetic, we can know that the waveform of the same vehicle is invariant with different speeds, but the time axis is compressed or stretched. And for the same speed, the waveforms of the different vehicles are different. So the vehicles with a same type must be a same Lextra , and the vehicles with a different type may be a different Lextra .Then based on the field data of different type vehicles, the Lextra of different type vehicles can be got by the real speed. In practice, the Lextra of different type vehicle is close, so we can use an average Lextra of different type vehicle as a common Lextra. (2) Time Difference based Speed Estimation For the magnetic sensor with two chips, when a vehicle is passing the two chips(Z1 and Z2), two similar waveforms will be got. According to the distance between the two chips and the time difference of the two similar waveforms, the speed of the vehicle can be obtained. Because the distance between the two chips is no large enough, the time difference will be very small. In order to reduce the round error, we can do some process for the data of the waveforms. The effective method to deal with the data is interpolation method, so the formula is given as follows:

vB

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L u fs u 3.6 u k (km/h) ds

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Where: L the distance of two chips; fs the sampling frequency; k the times of interpolation; ds the sampling difference of the two similar waveforms.

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After interpolation, the sampling difference (ds) of the two similar waveforms is shown as figure 8(the times of interpolation is 10) and can be got as following methods. Method 1: Moving the abscissa of the two waveforms, we can write down the abscissa difference and calculate the correlation coefficient of the two data vector. We will select the abscissa difference as the sampling difference (ds) when the value of the correlation coefficient is the maximum one. Method 2: Take some special points in the two waveforms, and calculate corresponding abscissa difference of those points, we can get some values of abscissa difference and take the average of those values as the sampling difference (ds).

ds

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Fig. 8. The ds of two similar waveforms with same vehicle (Z1 axis and Z2 axis)

The quotient that the sampling difference is divided by the times of interpolation and the sampling frequency is the time difference that the vehicle passes the two chips one by one. (3) Mean Value based Speed Estimation Mean always possesses some good characters, and it can eliminate some errors. So if the speeds of vA (VLB) and vB (TDB) are got, the Mean Value based speed vC can be given as follows:

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IV. EXPERIMENTS A. Data Collection and Process 7KH H[SHULPHQW VHOHFWV  YHKLFOHV¶ ZDYHIRUP GDWD collected at 11 o'clock on May 12, 2009, in Beijing as this experiment data. We get traffic situation using a live video, and REWDLQWKHYHKLFOHV¶VWDWHDQGVSHHGZKHQWKH\SDVVWKrough the sensor, as shown in figure 9. Starting time is supposed as 0, so the relative time when a vehicle passes over the sensor can be obtained. The length of vehicles is supposed as 4.5m in average. The FIR low-pass filter is designed to process the field data.

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Calibration Detection

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Speedvideo Comparison Speedsensor

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Fig. 9. The data collection method of video and sensor

B. Detection of Vehicle Approaching and Leaving By setting suitable size of the approaching window (InWinHeigh and InWinWidth) and the leaving window (OutWinHeight and OutWinWinth), the waveform data are cut out from the data of all vehicles. The parameters of vehicle detection window A and window B are set as follows: InWinHeigh = 70, InWinWidth = 30, OutWinHeight = 25, OutWinWinth = 120. And the parameters of stopping identification window are set as follows: StopWinHeigh = 10, StopWinWidth = MinStopT × fs. MinStopT is set as 2s (the minimum stopping time). Using these window parameters, we JRW  YHKLFOHV¶ ZDYHIRUPV H[DFWO\ DQG VRPH YHKLFOH waveforms of different state are shown in Figure 10. These waveforms can reflect the state when the vehicles pass over the sensor.Speed estimation and Analysis

Vehicle Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

By the vehicle waveform data and the speed estimate methods, three reference speeds based on different estimation methods are calculated. According to calibration using field data, the common Lextra=1.8m. Table 1 shows the result of different speed estimation methods. The MAE of VLB, TDB and MVB is respectively 4.12km/h, 5.90km/h and 4.05km/h. The comparison chart between the real speed and the estimation speeds of different methods is shown in figure 11. Moreover, the accuracy rate of vehicle count is close to 100% and the number of vehicle stopping is 3 times which are exactly found out. 2200

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Fig. 10. Four vehicle waveforms of different states (passing or stopping)

TABLE I REAL SPEED AND ESTIMATION SPEED OF VLB, TDB AND MVB (speed unit: km/h) Real vA(VLB) vB(TDB) vC(MVB) vA-MAE vB-MAE Speed 33 31 20 25 1.6 12.6 47 34 33 34 12.9 13.6 19 13 22 18 6.2 3.0 0(Stopping) 0(Stopping) 0(Stopping) 0(Stopping) 0.0 0.0 8 8 11 10 0.1 3.2 13 30 29 29 16.8 15.4 14 12 14 13 1.9 0.4 34 33 25 29 1.1 9.1 31 29 22 26 2.2 9.1 25 30 18 24 5.0 6.8 30 44 40 42 14.0 9.9 27 37 17 27 10.2 10.1 11 10 18 14 0.9 7.3 29 21 25 23 7.8 3.9 0(Stopping) 0(Stopping) 0(Stopping) 0(Stopping) 0.0 0.0 12 14 8 11 2.3 4.0 19 17 22 20 1.7 3.4 17 13 18 16 3.7 1.5 27 40 22 31 13.2 4.6 30 25 22 24 5.0 7.8 34 27 25 26 7.1 9.1 19 16 20 18 2.7 1.2 26 26 29 27 0.1 2.7 22 16 25 20 6.1 2.9 21 17 20 18 3.8 0.9 31 32 22 27 0.8 9.1

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vC-MAE 7.1 13.2 1.6 0.0 1.7 16.1 0.8 5.1 5.6 0.9 12.0 0.0 3.2 5.9 0.0 0.9 0.8 1.1 4.3 6.4 8.1 0.8 1.4 1.6 2.3 4.1

27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Average

31 14 0(Stopping) 13 14 16 20 27 24 33 33 23 11 17 23 28 47 9 5 21.40

33 10 0(Stopping) 19 18 16 15 21 29 28 34 16 10 19 21 27 42 12 3 21.07

25 22 0(Stopping) 17 15 29 17 25 33 29 25 20 11 22 22 20 29 29 7 20.49

29 16 0(Stopping) 18 17 22 16 23 31 28 29 18 11 21 22 23 35 20 5 20.78

1.8 3.9 0.0 5.6 3.9 0.4 4.7 5.8 4.8 4.6 1.4 6.7 0.9 1.6 1.7 0.8 4.9 3.2 1.5 4.12

6.3 8.3 0.0 3.2 1.2 12.9 3.1 1.8 9.1 4.1 7.6 2.8 0.2 4.7 0.5 7.8 18.3 19.7 2.1 5.90

2.3 2.2 0.0 4.4 2.5 6.6 3.9 3.8 7.0 4.3 3.1 4.7 0.3 3.2 1.1 4.3 11.6 11.4 0.3 4.05

Fig. 11. Comparison line chart of Real Speed and Estimation speeds

There are some vA-MAEs more than 10km/h such as vehicle 2, 6, 11, 12, 19. Some factors will exist to influence the accuracy rate of VLB method. The length of each vehicle is supposed as 4.5m for short, but it will not be always true. If the length of each vehicle could be obtained exactly and easily through vehicle classification or some other methods, the accuracy rate of VLB will greatly increase. For TDB method, there are also some vB-MAEs more than 10km/h such as vehicle 1, 2, 6, 12, 32, 43, 44. Some factors such as the times of interpolation, interpolation algorithm, noise of waveform data, the distance of two chips will be related to the accuracy rate of 7'%:KDW¶VPRUHWKHPDJQHWLFVHQVRURIRXUH[SHULPHQWLV installed near the traffic light, which increases the difficulty to completely remove the influence of each vehicle for magnetic field. If the field data is collected from highway or the place far from the traffic light, the result will be better. Besides, the MVB method is often better than other methods (the MAE 4.05 is less than both of 4.12 and 5.90), because it can always reduce

the maximum error of other two methods. What should be pointed out is that only the estimation speeds of other two methods are good enough, the MVB method is meaningful and effective, because sometimes one of VLB and TDB will be better than MVB. In table 1, from the view of average speed of 45 vehicles, the average speed of vA is 21.07 km/h and it is better than 20.78 km/h of vC as the real average speed is 21.04 km/h. V. CONCLUSIONS In this paper, we propose a feasible technology to get the vehicle speed and other traffic information such as vehicle count, vehicle stopping using a single magnetic sensor. The speed estimation methods referred in this paper are simple and are convenient to be realized. After data process, the vehicle detection algorithm is described in detail and it is the basis for speed estimation. We adopt three different methods to obtain the speed of each vehicle using a single magnetic sensor. In

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practice, the vehicle detection algorithm and the speed estimation methods are proved that they are practical and effective. The result shows that the average absolute error of different methods is less than 6 km/h, and the numerical error of average speed of all the vehicles is less than 1 km/h. Moreover, the accuracy rate of vehicle count is close to 100% and the number of vehicle stopping is 3 times which are exactly found out. If the length of each vehicle is exactly obtained, the result of VLB method must be better. Fortunately, the length of a vehicle can be exactly calculated through some effective vehicle classification algorithm. To study how to obtain the vehicle length and how to develop effective vehicle classification algorithm will be the next research topic for the magnetic sensor. Moreover, the way of deploying two sensors continuously for a specific distance could get more accurate vehicle speed. Because the time difference will be extend and it will reduce the numerator error remarkably. But it will bring new error for the problem of time synchronization with two sensors. Besides, the cost of sensors will be twice more. In practice, the three speed estimation methods are not always usable at the same, but at least one of those methods is almost usable. If the length of each vehicle can be got easily in practice, we can select VLB method. And if the time difference can be obtained conveniently, the TDB method could be used. In like manner, if both of the methods above are workable, the MVB method may give a better result.

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