two-dimensional sensor system for

178 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 15, NO. 1, FEBRUARY 2014

Two-Dimensional Sensor System forAutomotive Crash Prediction

Saber Taghvaeeyan and Rajesh Rajamani

Abstract—This paper focuses on the use of magnetoresistive andsonar sensors for imminent collision detection in cars. The mag-netoresistive sensors are used to measure the magnetic field fromanother vehicle in close proximity, to estimate relative position,velocity, and orientation of the vehicle from the measurements.First, an analytical formulation is developed for the planar vari-ation of the magnetic field from a car as a function of 2-D positionand orientation. While this relationship can be used to estimateposition and orientation, a challenge is posed by the fact that theparameters in the analytical function vary with the type and modelof the encountered car. Since the type of vehicle encountered isnot known a priori, the parameters in the magnetic field functionare unknown. The use of both sonar and magnetoresistive sensorsand an adaptive estimator is shown to address this problem.While the sonar sensors do not work at very small intervehicledistance and have low refresh rates, their use during a short initialtime duration leads to a reliable estimator. Experimental resultsare presented for both a laboratory wheeled car door and for afull-scale passenger sedan. The results show that planar positionand orientation can be accurately estimated for a range of relativemotions at different oblique angles.

Index Terms—Crash detection, crash sensors, extended Kalmanfilter (EKF), magnetic sensors, sensor fusion, vehicle positionsensors.

I. INTRODUCTION

THE work in this paper is motivated by the need to developan inexpensive sensor system for an automobile that can

predict an imminent collision with another vehicle, just beforethe collision occurs. The prediction needs to occur at least100 ms before the collision, so that there is adequate timeto initiate active passenger protection measures to protect theoccupants of the vehicle during the crash. Examples of simpleoccupant protection measures that can be initiated based on theprediction include pretightening of seat belts and gentler infla-tion of air bags. In addition, active crash space enhancementsystems such as active bumpers [1], [2] and rapid active seatback control [3] can be utilized.

It should be noted that active occupant protection measuresinvolve considerable cost, discomfort, and even a small riskto the occupants. For example, deployment of air bags is

Manuscript received January 13, 2013; revised May 23, 2013 and July 10,2013; accepted July 22, 2013. Date of publication September 23, 2013; date ofcurrent version January 31, 2014. This work was supported in part by fundingfrom the Intelligent Transportation Systems Institute, University of Minnesota.The Associate Editor for this paper was U. Ozguner.

The authors are with the Department of Mechanical Engineering, Universityof Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TITS.2013.2279951

an expensive action resulting in considerable cost. Likewise,rapid seatback motion control during driving can be significantannoyance and a danger to the driver, if triggered unnecessarily.Therefore, these measures can be initiated only if the collisionprediction system is highly reliable. A false prediction of colli-sion has highly unacceptable costs.

Traditionally, radar and laser systems have been used oncars for adaptive cruise control and collision avoidance [4]–[9]. These sensors typically work at intervehicle spacing greaterthan 1 m. They do not work at very small intervehicle spacingand further have a very narrow field of view at small distances[6]. Collision prediction based on sensing at large distancesis unreliable. For example, even if the relative longitudinalvelocity between two vehicles in the same lane is very high,one of the two vehicles could make a lane change resulting inno collision. An imminent collision can be reliably predictedenough to inflate air bags only when the distance betweenvehicles is very small and when it is clear that the collisioncannot be avoided under any circumstances. Radar and lasersensors are not useful for such small distance measurements.A radar or a laser sensor can cost well over $1000. Hence, itis also inconceivable that a number of radar and laser sensorsbe distributed all around the car in order to predict all thepossible types of collisions that can occur. It should be notedthat camera-based image processing systems suffer from someof the same narrow field of view problems for small distancesbetween vehicles.

Therefore, this paper focuses on the development of a sensorsystem that can measure relative vehicle position, velocity, andorientation at very small intervehicle distances. The main ideaof the new proposed sensing system is to use the inherentmagnetic field of a vehicle for position estimation. A vehicleis made of many metallic parts (for example, chassis, engine,body, etc.), which have a residual magnetic field and/or getmagnetized in the Earth’s magnetic field. These magnetic fieldscreate a net magnetic field for the whole vehicle, which can beanalytically modeled as a function of vehicle-specific parame-ters and position around the vehicle. By measuring the magneticfield using anisotropic magnetoresistive (AMR) sensors, theposition of the vehicle can be estimated if the vehicle-specificparameters are known. However, the specific parameters varyfrom one type of vehicle to another type of vehicle, bothdue to varying amounts of ferromagnetic material and varyinglevels of magnetization. Hence, these parameters need to beestimated along with the position for every encountered vehicle.This challenge is addressed by the use of both sonar andmagnetoresistive sensors and an adaptive estimator. While thesonar sensors do not work at very small intervehicle distances

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TAGHVAEEYAN AND RAJAMANI: TWO-DIMENSIONAL SENSOR SYSTEM FOR AUTOMOTIVE CRASH PREDICTION 179

and have low refresh rates, their use during a short initialtime duration leads to a reliable estimator. Magnetoresistivesensors offer the advantages of being able to work at verysmall distances, down to 0 m, having a very high refreshrate, and being highly inexpensive and compact. To the bestof the authors’ knowledge, this is the first journal publicationthat utilizes magnetoresistive sensors for 2-D vehicle positionestimation.

An AMR sensor has a silicon chip with a thick coatingof piezoresistive nickel–iron. Every car has a magnetic field,which is a function of the vehicle position around the vehicle.The presence of an automobile in close range to an AMRsensor causes a change in the magnetic field, which changes theresistance of the nickel–iron layer. By measuring the resistance,obtaining the real-time magnetic field, and developing modelsthat relate magnetic field to intervehicle position, estimationsystems for intervehicle distance estimation can be developed.The HMC2003 three-axis magnetic sensor boards from Hon-eywell are utilized for the system developed in this paper.Each sensor board contains core HMC100x AMR sensingchips, which cost about $10. Application note AN218 fromHoneywell describes the use of the AMR chips for vehicledetection and traffic counting applications (neither of whichinvolves vehicle position estimation) [10].

A custom-designed sonar system is also used, which consistsof one transmitter and two receivers, and measures not only thedistance to the objects but also the orientation of the object. Thissystem is described in detail in later sections.

This paper is organized as follows. In Section II, a briefreview of authors’ previous work on 1-D position estimation us-ing AMR sensors is provided. In Section III, the problem of 2-Dvehicle position estimation and the challenges in using mag-netic field for that application are discussed. This discussion isfollowed by the derivation of the 2-D magnetic field equationsfor a car presented in Section IV. The custom-designed sonarsystem that has been developed is discussed in Section V. Theextended Kalman filter (EKF) used for 2-D position estimationadopting measurements from the magnetic sensors and thesonar sensor is discussed in Section VI. Results of differentexperiments are presented in Section VII. Finally, this paperis concluded in Section VIII. Thus, this paper develops themagnetic field equations for a car, develops an EKF for relativeposition and orientation estimation in close proximity, based onmeasurements from magnetic sensors and a custom-designedsonar system, and verifies the performance of the developedestimator through different experiments.

II. ONE-DIMENSIONAL POSITION ESTIMATION

Every car has a magnetic field, which is a function of theposition around the vehicle. This was demonstrated earlierpurely for 1-D vehicle position estimation in [11]. This paperconsiders 2-D position estimation, building on a previous con-ference publication by the same authors [12]. The parametersin the magnetic field versus position function vary with thetype and model of encountered vehicle. Since the type ofvehicle encountered is not known a priori, the parameters ofthe magnetic field function are unknown. Therefore, adaptive

Fig. 1. General scenario of the experiment.

Fig. 2. Developed PCB for capturing AMR and sonar sensors data.

estimation algorithms are used for estimation of 2-D positionand orientation from magnetic field measurements.

For 1-D case, a number of tests with different vehicles werefirst performed in order to experimentally investigate the mag-netic field generated by an encountered vehicle as a functionof 1-D distance. Fig. 1 shows a general schematic of the tests.An AMR sensor and a sonar sensor were packaged on a printedcircuit board (PCB) together with a microprocessor (see Fig. 2)that reads the sensors signals and transmits their values to acomputer.

The outputs of the AMR and sonar sensors were sampledat the rate of 2 kHz using a dsPIC microcontroller with a12-bit analog-to-digital converter (ADC). Fig. 3 shows therelationship between the magnetic field (in the X-direction)and the actual distance obtained from a sonar sensor for atypical test using a Volkswagen Passat vehicle. Magnetic fieldis plotted in arbitrary voltage units, the same as what was readfrom the ADC of the microcontroller. It can be seen that there isobviously a nonlinear relation between the measured magneticfield and distance.

Based on the preceding experimental data and an analyticalexpression developed in [11], it was observed that below athreshold distance, i.e., xth, the following relation holds be-tween magnetic field and distance:

B =p

x+ q (1)

where B is the magnetic field, x is the distance of the vehiclefrom the sensors, and p and q are the vehicle-dependent param-eters. This equation was fit to experimental data from variousvehicles. Fig. 3 shows the fitting results from an experimentwith a Volkswagen Passat vehicle. The vehicle was moved froman initial distance toward the sensors. In Fig. 3, data set 1 isthe set of data points obtained after a certain time when thevehicle gets closer than xth to the sensors. This data set wasused for curve fitting. Data set 2 is the set of data points from


Fig. 3. Results of the experiment with a Volkswagen Passat vehicle and thefitted curve.

TABLE IRESULTS FROM CURVE FITTING

the same experiment where the vehicle is farther than xth fromthe sensors and plotted for comparison.

The equation was also verified against data from the sametype of experiment with a Chevrolet Impala, a Hyundai Elantra,and a Honda Accord vehicle. Table I summarizes the results ofthe experiments showing the coefficient of determination (R2)of the fitted line and estimated xth for various vehicles.

The next step would be to adopt the proposed equation forproximity sensing. However, the values of the parameters p andq vary with the vehicle, being constant for a specific vehicle butchanging from one model to another. Since the type of vehicleencountered is not known a priori, these parameters have to beadaptively updated in real time. This has been discussed in [11].

III. TWO-DIMENSIONAL POSITION ESTIMATION

For 1-D motion, in which the vehicle is moving directlytoward or away from the sensors, we found that, below a thresh-old distance xth, the relation in (1) can be assumed betweenmagnetic field and distance. That relationship could be usedtogether with an adaptive estimator for real-time estimation ofposition just prior to a frontal or rear-end collision. However,an impact due to collision can occur at any location aroundthe car body. In fact, side impact and oblique collisions atrural intersections are a significant source of fatalities [13].It is therefore necessary to be able to estimate not only therelative position but also the orientation of the colliding vehicleanywhere in the 2-D plane.

To further investigate the 2-D motion problem, first considerthe simplified case in which the vehicle is moving toward the

Fig. 4. Vehicle moving toward sensors at a constant angle.

Fig. 5. Two-dimensional position estimation and the parameters to beestimated.

sensors at a constant angle θ, as shown in Fig. 4. In this case,if the angle θ is known, the magnetic field along the directionof motion of the vehicle can be expressed in terms of the AMRsensor’s measurements as

B = BxAMR cos θ +ByAMR sin θ =p

r+ q (2)

where r is the distance measured along the direction of motion.This equation can potentially be used with an adaptive algo-rithm to estimate r.

However, if θ is not constant or if the colliding vehicle ismoving toward the sensors at an offset (meaning that its centerline does not pass through the center of the AMR sensor), thepreceding approach cannot be adopted.

Hence, to fully identify and classify a crash in 2-D motion,we need to estimate xA, yA, v, θ, and ω, as shown in Fig. 5,where xA and yA are the position of point A with respect tothe coordinate frame attached to the approaching car, v is thelongitudinal velocity of the approaching car along its x-axis, θis the orientation of the approaching car relative to the host car(in other words, it is the angle between the x-axis of the coor-dinate frame attached to the approaching car and the X-axisof the coordinate frame at point A), and ω is the rotationalvelocity of the approaching car.

It is worth mentioning that another way of expressing theposition of the objects would be to express the position ofpoint O with respect to the coordinate frame attached to pointA. However, using this coordinate frame, the measurementequations, which are derived later on, will be more complicated.

Use of magnetic field for estimation of vehicle position in the2-D motion is much more complicated than in the 1-D motionnot only because of the additional degree of freedom for thevehicle but also because of the complex pattern of the vehicle’smagnetic field in 2-D space. The magnetic field lines are notparallel to each other, and they curve out to the sides. This is


Fig. 6. Magnetic field lines of a magnetic bar.

Fig. 7. Door of a Ford vehicle used for experiments.

Fig. 8. Magnetic field of the door while moving toward AMR sensors at 45◦.

the same with any type of magnetic objects. Fig. 6 shows themagnetic field lines of a rectangular magnet.

This phenomenon can be also observed in the experimentswith the door of a Ford passenger sedan. The door is mountedon a wheeled platform, as shown in Fig. 7, and can be maneu-vered and moved easily toward or away from the sensors indifferent directions.

Fig. 8 shows a case where the door is coming toward a setof four AMR sensors at a 45◦ angle. The sensor readings atfour different locations of the door during its motion are shown.As shown in the figure, initially, the direction of the magneticfield (shown with a green dashed line) is very different fromthe normal direction of the door. Hence, it is not possible toobtain the orientation of the door, i.e., θ, by only determiningthe direction of the magnetic field. It can be also seen that as thedoor gets closer, the magnetic field magnitude increases, and its

Fig. 9. Analysis of a magnetic field around a magnetic block in twodimension.

Fig. 10. Analysis of a magnetic field of line of dipoles.

direction changes to become in line with the normal directionof the door.

To further investigate a vehicle’s magnetic field, a mathe-matical expression for the field in a 2-D plane created by arectangular magnetic body is derived in the next section.

IV. DERIVATION OF A MATHEMATICAL EXPRESSION FOR

MAGNETIC FIELD IN TWO DIMENSIONS

Modeling a vehicle as a rectangular block of magneticdipoles, as shown in Fig. 9, we want to obtain a mathematicalexpression for the magnetic field at an arbitrary point A. Whilesuch a magnetic field approximation for a vehicle body might atfirst appear crude, the model obtained from such an assumption,together with parameter estimation, is likely to be significantlymore useful than a purely empirical function obtained withoutany magnetic field models. The goal is to later use the derivedmagnetic field equations and estimate vehicle position by mea-suring Bx and By using an AMR sensor at point A. In thefollowing derivations, the height of the block da is assumedto be small with respect to its width and length.

As a first step, the magnetic field of a line of magnetic dipoles(see Fig. 10) is obtained.

According to [14], the following relations can be writtendown:

dBr =μ0dm0

2πr3cosα (3)

dBα =μ0dm0

4πr3sinα. (4)

In order to obtain Bx and By , we need to integrate dBx anddBy, which are given by the following equations:

dBx = dBr cosα− dBα sinα (5)dBy = dBr sinα+ dBα cosα. (6)


Expressing r and α in terms of x and y, we have

r =(x2A + (yA − y)2

) 12 (7)

α = atan

(yA − y

xA

)(8)

sinα =yA − y

(x2A + (yA − y)2)

12

(9)

cosα =xA

(x2A + (yA − y)2)

12

. (10)

Hence

dBx =μ0dm0

4πr3(2 cos2 α− sin2 α)

=μ0dm0

4π2x2

A − (yA − y)2

(x2A + (yA − y)2)

52

(11)

dBy =μ0dm0

4πr33 cosα sinα

=μ0dm0

4π3xA(yA − y)

(x2A + (yA − y)2)

52

. (12)

The magnetic field generated by the line of magnetic dipolesis then obtained by integrating the preceding terms as

Bx−l(xA, yA)=μ0m0dadx

4π

b∫−b

2x2A − (yA − y)2

(x2A + (yA − y)2)

52

dy

=μ0m0dadx

4π

(2x2

A(yA + b) + (yA + b)3

x2A (x2

A + (yA + b)2)32

−2x2A(yA − b) + (yA − b)3

x2A (x2

A + (yA − b)2)32

)

By−l(xA, yA)=μ0m0dadx

4π

b∫−b

3xA(yA − y)

(x2A + (yA − y)2)

52

dy

=m0dadx

4π

(− xA

x2A (x2

A + (yA + b)2)32

+xA

x2A (x2

A + (yA − b)2)32

).

In order to obtain the total magnetic field created bythe whole bar, Bx−l and By−l can be integrated in theX-direction as

Bx =

xA+L∫xA

Bx−l(x, yA)

=μ0m0da

4π

(− yA + b

(xA + L) ((xA + L)2 + (yA + b)2)12

Fig. 11. Analysis of magnetic field of line of dipoles.

+yA − b

(xA + L) ((xA + L)2 + (yA − b)2)12

+yA + b

xA (x2A + (yA + b)2)

12

− yA − b

(xA (x2A + (yA − b)2)

12

)

By =

xA+L∫xA

By−l(x, yA)

=μ0m0da

4π

(1

((xA + L)2 + (yA + b)2)12

− 1

((xA + L)2 + (yA − b)2)12

− 1

(x2A + (yA + b)2)

12

+1

(x2A + (yA − b)2)

12

). (13)

The equations for the magnetic field derived in (13) are com-plex and not suitable for real-time estimation of both equationparameters and position. However, it is possible to simplify (13)for the colored region shown in Fig. 11 for which we haveyA � b and xA ≤ xth.

Assuming xA + L � xA, (13) can be simplified to

Bx =μ0m0da

4π

(yA + b

xA (x2A + (yA + b)2)

12

− yA − b

xA (x2A + (yA − b)2)

12

)

By =μ0m0da

4π

(− 1

(x2A + (yA + b)2)

12

+1

(x2A + (yA − b)2)

12

). (14)


Next, if we assume that yA is small and close to zero, (14)can be further simplified to the following equations:

Bx =μ0m0da

2πb

xA (x2A + b2)

12

+ f(p, b, xA)y2 + · · ·

=μ0m0da

2πb

xA (x2A + b2)

12

(15)

By =μ0m0da

2πb

(x2A + b2)

32

y + f(p, b, xA)y3 + · · ·

=μ0m0da

2πby

(x2A + b2)

32

. (16)

It should be noted that (15) is the same as the equationobtained earlier for the 1-D case, replacing da with 2a. Forsmall values of xA, the preceding equations can be simplifiedfurther as

Bx =μ0m0da

2π1

xA

(1 +

(xA

b

)2) 12

∼= μ0m0da

2πxA=

p

xA(17)

By =μ0m0da

2πby

(x2A + b2)

32

=pby

(x2A + b2)

32

. (18)

If there exists any static magnetic field at point A, like theEarth’s magnetic field, a constant needs to be added to thepreceding equations to obtain the total magnetic field to bemeasured by the sensors. Thus

Bx =p

xA+ qx (19)

By =pby

(x2A + b2)

32

+ qy. (20)

It is also possible to subtract the static magnetic field at thelocation of the AMR sensors from the measurements to avoidadding a constant to readings from the AMR sensors. However,for readings of Bx, it was observed that, even when subtractingthe static magnetic field from the measurements, using (19)results in a better fit compared with using (17). Therefore, thefollowing equations are being used for position estimation inthe next sections of this paper:

Bx =p

xA+ q (21)

By =pby

(x2A + b2)

32

. (22)

In order to use (21) and (22) for position estimation bymeasuring the magnetic field, we need to make sure that theAMR sensors are in the colored region shown in Fig. 11 forwhich we have simplified the magnetic field equations. For thepoints inside this region, the ratio |By/Bx| is small. Hence, wecan install multiple AMR sensors around the host vehicle andcheck the ratio |Byi/Bxi| for each AMR sensor and pick thesensors with the lowest ratio as the most appropriate sensor toutilize for position estimation. This concept is shown in Fig. 12.

Fig. 12. AMR sensors arrangements for 2-D position estimation.

Each AMR sensor measures the magnetic field along its X-and Y -axes, i.e., BxAMRi and ByAMRi. Hence, the magneticfield of the approaching vehicle at the location of AMR sensori expressed in the approaching vehicle coordinate frame can beobtained as{

Bxi = BxAMRi cos(θ) +ByAMRi sin(θ)Byi = −BxAMRi sin(θ) +ByAMRi cos(θ).

(23)

As can be seen from (23), to calculate the ratio |Byi/Bxi| inorder to select the most appropriate AMR sensors for measure-ment update, we need at least an initial estimate of θ. Therefore,a custom-designed sonar system has been developed, whichmeasures both the distance to the object as well as its orien-tation. The sonar sensor detects the objects at longer distancescompared with AMR sensors. Hence, when the AMR sensorsrespond to the presence of a car, the estimated θ from thesonar system can be used to select the most appropriate AMRsensors for measurement updates. The sonar sensor is also usedto initialize the magnetic field parameters, and this is shown tospeed up the convergence of the magnetic field parameters. Thesonar system is described in the following section.

V. SONAR MEASUREMENT SYSTEM

The developed sonar measurement system includes onetransmitter, i.e., T , at point A and two receivers, i.e., R1 and R2,at distances d1 and d2 from A arranged in the order shown inFig. 13. This configuration of the transmitter and the receiversmakes it possible to measure the orientation of the target and itsdistance from the sensors.

Measuring the travel time of sound for receivers 1 and 2,the distance that the echo pulse has traveled can be calculated.Using the fact that the incident and reflected angles of soundare the same (similar to light when it reflects from a mirror), itcan be concluded that the measured distances equal l1 and l2,as shown in Fig. 13. In other words, l1 and l2 equal the distancefrom the image of transmitter T at point B to the receivers R1

and R2, respectively.Knowing l1 and l2 and the distance between the transmitter

and the receivers, i.e., d1 and d2, the angle θ2 can be calculatedusing the cosine rule as

ds = d1 + d2 (24)


Fig. 13. Sonar system with one transmitter and two receivers.

l21 = d2s + l22 − 2dsl2 cos(90 − θ2) (25)

θ2 = asin

(d2s + l22 − l21

2dsl2

). (26)

Knowing θ2, the length ls, i.e., the distance between thetransmitter and its image (AB), can be calculated using thecosine rule as

l2s = d22 + l22 − 2d2l2 cos(90 − θ2). (27)

Then xs (AC), i.e., the sonar estimate of xA, can be calcu-lated from ls since

xs =ls2.

Applying the cosine rule one more time, θs, i.e., the sonarestimate of θ, can be also calculated as

l22 = d22 + l2s − 2d2ls cos(90 + θs) (28)

θs = asin

(l22 − d22 − l2s

2d2ls

). (29)

In practice, the measured signals from the sonar sensors arel1m and l2m, where

l1m = l1 + n1 n1 ∼ N(0, σs)

l2m = l2 + n2 n2 ∼ N(0, σs).

If we want to use xs and θs in the estimator, we should alsocalculate the covariance of noise in the measurements. Sincexs and θs have nonlinear relations with n1 and n2, we need tocalculate the derivatives of xs and θs with respect to n1 and n2,which would be

∂xs

∂n1= − l1m

2ls

d2ds

(30)

∂xs

∂n12=

l2m2ls

d1ds

(31)

∂θs∂n1

=cos(y2)

cos(θs)

l1ml2mdsl2s

(32)

∂θs∂n2

=cos(y1)

cos(θs)

l1ml2mdsl2s

. (33)

Similar to any sonar sensor used for 1-D position measure-ments, the developed sonar system will not also work at closeproximities (typically below 0.5 m). In addition, the sonarsystem will not work if the orientation of the object, i.e., θ,increases so that θ1 and θ2 are greater than the beam angles ofthe transducers. It should be mentioned that the threshold for θis not constant and slightly changes with the distance from theobject. Hence, there cannot be a constant value for θthreshold.However, an easier way of detecting outliers is at the beginningof the calculation, by looking at the values obtained for θ1 andθ2. If they have imaginary values, then an outlier is detected.This method for detecting outliers is described here later.

It is also worth mentioning here that the sonar system aloneis not able to determine yA. In other words, the object can movearbitrary along its Y -axis and the sonar readings will remain thesame. Finally, the sonar system has a much lower refresh rate(20 Hz) compared with AMR sensors (1 Hz). Hence, the use ofthe AMR sensors is critical.

Addressing Crosstalk and Multipath Problems of the SonarSystem: Problems with crosstalk and multipath returns are ad-dressed in three different layers of the proposed system: designlevel, sonar sensor level, and estimator level.

1) Design level: The sonar transmitter is set to send out asignal every 50 ms, and each receiver hears back the firstsignal that returns to it. After the first signal has been de-tected, the receivers ignore any returning signals until thenext signal is transmitted.The voltage level of the signalsent to the transmitter and the time duration of 50 ms aredesign parameters. The voltage level is set such that thesonar system is not able to detect objects more than ∼6 maway. Since the speed of sound is about 340 m/s, all thepossible returns are within dt = 2 × 6/340 = 35 ms ofthe transmitter. Hence, transmitting signals every 50 msensures that there will not be any returns from a previoustransmit to interfere with the current measurement.

2) Sonar sensor level: The designed sonar system includesone transmitter and two receivers. Using the two re-ceivers, it is possible to measure the distance to the objectas well as its orientation. On the other hand, havingtwo receivers provides another level of robustness. As anexample, assume that there is an object at a distance ofxA = 2 m in front of the sonar system at an angle ofθ = 20◦. The raw readings from the two receivers willbe l1 = 3.95 m and l2 = 4.05 m, from which xs and θscan be calculated as described by (24)–(29). Now, let usassume that due to a multipath or a crosstalk error, l1 ismeasured as l′1 = 3 m. This will cause the argument ofthe “asin” function in (26) to be larger than 1, causingan imaginary value for θ2. Hence, in general, by lookingat the values obtained for θ1 and θ2, we can determine ifthe readings are valid or not. This adds another level ofrobustness against multipath and crosstalk errors.


Fig. 14. Crash prediction system sensor arrangement.

3) Estimator level: Finally, in the estimator, the“Mahalanobis distance” is used to reject false readings ofthe sonar sensing system. The Mahalanobis distance isdefined as

dmah−k =((

xs−k − x−k

) (P−x−k

)−1 (xs−k − x−

k

))0.5

(34)

where xs−k is the measured distance from the sonarsystem, x−

k is the a priori estimated distance, and P−x−k

is the a priori covariance of the estimated distance. Fora given P−

x−k, the farther the measurement is from thepredicted estimate, the larger is the Mahalanobis distance.For a certain distance between the measurement and thepredicted estimate, the smaller the uncertainty in the pre-dicted estimate is, the larger is the Mahalanobis distance.In other words, as we are more certain about the currentestimate, we reject a measurement with a shorter distanceto the predicted estimate compared with the case whenwe are less certain about the current estimate and weallow a larger distance between the measurement and theestimated distance. Fig. 21 in Section VII shows how theMahalanobis distance is used to reject sonar outliers inthe experiment with the vehicle door.

VI. EKF FOR TWO-DIMENSIONAL POSITION AND

ORIENTATION ESTIMATION

As mentioned earlier, the current 2-D crash prediction systemis based on four AMR sensors and a sonar system consisting ofthree sonar transducers, as shown in Fig. 14.

The sonar sensing system works at larger distances comparedwith AMR sensors; however, it does not work at short distances.To account for the different working ranges of the sensors,a state machine shown in Fig. 15 is utilized. In state 0, theestimator will use the sonar sensor to update position since theAMR sensors are not yet affected by the approaching vehicle.As soon as the AMR sensors respond to the approachingvehicle, updates would be done using both sonar and AMRsensors (state 1). When the vehicle enters a distance wherethe sonar readings are not valid any more due to very smalldistances or other reasons, updates would be done using onlythe AMR sensors (state 2).

For transitions between the states in the finite-state machine,xth would be the best variable to utilize for switching from state 0

Fig. 15. Two-dimensional positioning state diagram.

to state 1, but there is no prior knowledge about xth when anew vehicle is approaching. Therefore, the covariance of theAMR data at predetermined time intervals can be used instead.Starting from state 0, whenever the covariance is higher than athreshold, the estimator switches to state 1 in order to switchfrom sonar to sonar–AMR updates. To obtain more meaningfulinitial values for the states p and q, a least squares (LS) fittingcan be performed at the switching time. The estimated valuesand their covariance are used as initial values for p and q andtheir covariance. While in state 1, xth can be calculated in realtime and used for determining if the vehicle is moving out of theview of the AMR sensors or the sonar sensor and if the systemshould switch back to state 0 or switch to state 2.

The new sonar system also measures the orientation of theobject, i.e., θ; and if θ increases beyond a threshold, it willnot be able to measure the orientation due to the limitationsin the beam width of the sonar transducers. This also results ina transition from state 1 to state 2.

The EKF [15], [16] is used for estimation of magnetic fieldequation parameters, as well as position, orientation, and veloc-ity of the approaching vehicle. There are some tight time con-straints with the real-time 2-D positioning system. Sensors dataare captured through two microchip dsPIC microcontrollers andtransferred to MATLAB running on a PC at 500 Hz via serialport. Since the system is working in real time, there would be atime period of 2 ms for transferring data from microcontrollersto MATLAB (taking about 0.5 ms), analyzing data, and visu-alization. Having an EKF running in MATLAB (described inlater sections), it means that there is about 1.5 ms to performtime and measurement update steps of the EKF. Therefore,the measurement equations should be simplified as much aspossible. For instance, it is possible to use equations obtainedearlier for the magnetic field in two dimension without anysimplifying assumptions. However, calculating the Jacobian re-quired for EKF measurement update will become complicatedand computationally intense. It is also worth mentioning thatthe ordinary unscented Kalman filter [17], [18] fails in thisproblem mainly because of the discontinuity at x = 0.

With the aforementioned explanations, we will derive theequations for the EKF. It should be noted that the equationspresented here are the general equations used in state 1, whereboth sonar and AMR sensor measurements are available. Incase of states 0 or 2, the Kalman filter measurement equations


will be a reduced version of the measurement equations derivedhere. The state vector to be estimated is

X = [x y v a θ ω α p q]T (35)

where x, y, and θ express the position and orientation of the ob-ject; v and ω are the longitudinal and rotational velocities of theobject, respectively; a and α are the longitudinal and rotationalaccelerations, respectively; and p and q are the magnetic fieldequation constants.

A. Time Update Equations

In order to obtain the time update equations of the EKF,we need to derive the dynamic equations of the system. Con-sidering Fig. 5, it is assumed that the object moves with alongitudinal velocity v along its x-axis and a rotational velocityω. Using the transport theorem, the dynamic equations can bewritten down as

rA = (rA)rel + ω × rA ⇒ xA = −v + ωyA & yA = −ωxA.(36)

Discretizing the preceding equations, dropping the A sub-script, and including longitudinal and rotational accelerations,the following equations for the dynamics of the system areobtained:

Xk = f(Xk−1, wk−1), wk ∼ (0, Qk)

xk =xk−1 − vk−1dt+ ωk−1yk−1dt

yk = yk−1 − ωk−1xk−1dt vk = vk−1 + ak−1dt

ak = ak−1 + w1k−1 θk = θk−1 + ωk−1dt

ωk =ωk−1 + αk−1dt αk = αk−1 + w2k−1

pk = pk−1 + w3k−1 qk = qk−1 + w4

k−1. (37)

The time update equations will be as in (38), shown at thebottom of the page.

Next, the measurement update equations of the EKF arederived.

B. Measurement Update Equations

Here, the measurement update equations for state 1 arederived. States 0 and 2 have a reduced version of the measure-ment update equations derived here. In state 1, the availablemeasurements are the distance xs and the orientation θs fromthe sonar system and eight measurements from four AMRsensors BxAMRi, ByAMRi (i = 1, 2, 3, 4), which are measuredwith respect to the coordinate frame of each AMR sensor. Ateach measurement update, first, the measured magnetic fieldsby each AMR sensor expressed in the coordinate frame of theapproaching vehicle, i.e., Bxi and Byi, are calculated as

Bxi =BxAMRi cos θ−k +ByAMRi sin θ

−k (39)

Byi = −BxAMRi sin θ−k +ByAMRi cos θ

−k , i = 1, 2, 3, 4.

(40)

Second, the ratio between the magnetic fields in Y - andX-directions, i.e., BRATi = |Byi/Bxi|, is calculated for eachsensor. Then, the two sensors that have lower values of BRAT

are selected (named m and n), and the corresponding valuesof Bxi and Byi of those two sensors are assigned to the newvariables Bxm, Bxn, Bym, and Byn to be used in measure-ment update. This procedure is performed in order to use thesimplified magnetic field equations (21) and (22). The resultwill be the measurement update equations in (41) and (42),shown at the bottom of the next page, where xi = x+ di cos θ,i = m,n; dm and dn are the distances of AMR sensors m and

X−k = fk−1

(X+

k−1, 0)

P−k =Fk−1P

+k−1F

Tk−1 +Gk−1Qk−1G

Tk−1

Fk−1 =∂fk−1

∂X|X+

k−1

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 ω+k−1dt −dt 0 0 y+k−1dt 0 0 0

−ω+k−1dt 1 0 0 0 −x+

k−1dt 0 0 00 0 1 dt 0 0 0 0 00 0 0 1 0 0 0 0 00 0 0 0 1 dt 0 0 00 0 0 0 0 1 dt 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Gk−1 =

⎡⎢⎣0 0 0 1 0 0 0 0 00 0 0 0 0 0 1 0 00 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 1

⎤⎥⎦T

. (38)


n, respectively, from point A; and values in the Rk matrixrepresent the covariance of noise in measurements and aredetermined through experiments. ymeas is an estimate of y andis obtained from equations of magnetic field in two dimen-sion as

Bx =μ0m0da

2πb

xA (x2A + b2)

12

By =μ0m0da

2πby

(x2A + b2)

32

⇒

BRAT =Bx

By=

yxA

x2A + b2

. (43)

For each one of the selected sensors, i.e., m and n, we havethe following relations:

BRATi =yi(x+ di sin θ)

(x+ di sin θ)2 + b2, i = m,n (44)

where yi = y + di cos θ, i = m,n.Now, using x−

k and θ−k after each time update and thepreceding equations, we can get an estimate of ym and yn fromthe following linear equations:⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

BRATmTm − ym

(x−k + dm sin θ−k

)= 0

BRATnTn − yn

(x−k + dn sin θ

−k

)= 0

ym − yn = (dm − dn) cos θ−k

Tm − Tn =(x−k + dm sin θ−k

)2

−(x−k + dn sin θ

−k

)2

(45)

where

Ti =(x−k + di sin θ

−k

)2

+ b2, i = m,n. (46)

Solving for ym, yn, Tm, and Tn, we can obtain an estimationof y from the following equation:

ymeas = ym − dm cos θ−k . (47)

It should be noted that, while calculating the Jacobian for theEKF measurement update, the effect of uncertainties in x−

k andθ−k on ymeas is ignored, and ymeas is assumed to have a zeromean Gaussian noise to make the Jacobian simpler.

VII. EXPERIMENTAL RESULTS

The developed estimator was tested with the Ford vehicledoor shown in Fig. 6 and a Mazda Protégé 1999 sedan. Testingwith the door has the advantage that more complicated scenar-ios can be implemented since it is easier to move it around.

A. Results From the Tests With a Ford Door

The estimator was first verified with tests using the Fordvehicle door. The door is mounted on a wheeled platform, asshown in Fig. 7, and can be maneuvered and moved easilytoward or away from the sensors in different angles.

Fig. 16 shows snapshots of the estimated position of the doorat different time intervals during an experiment. The color ofthe line used to represent door position goes from black togray from the beginning toward the end of the experiment. Thered rectangles show the position of the AMR sensors (sonartransducers are not shown). The door has been moved towardthe sensors from point A to point B and then rotated to θ ∼=−35◦ at point B, moved closer to the sensors to point C, movedback to point B and rotated to θ ∼= 0◦, and, finally, moved closerto the sensors to point D.

Z =h(X,n)

Z = [xs θs 0 0 ymeas ]T

h(X,n) =

[x+ nx θ + nθ Bxm −

(p

xm+ q

)+ (nxAMRm cos θ + nyAMRm sin θ) . . .

. . . Bxn −(

p

xn+ q

)+ (nxAMRn cos θ + nyAMRn sin θ) y + ny

]TKk =P−

k HTk

(HkP

−k HT

k +Rk

)−1

X+k = X−

k +Kk

[Zk − hk

(X−

k , 0)]

p+k =(I −KkHk)P−k (41)

Hk =∂h

∂X

∣∣∣∣X−

k

=

⎡⎢⎢⎢⎢⎣1 0 0 0 0 0 0 0 00 0 0 0 1 0 0 0 0

− pxm

0 0 0 Bym + pdm cos θxm

0 0 − 1xm

−1

− pxn

0 0 0 Byn + pdn cos θxn

0 0 − 1xn

−10 1 0 0 0 0 0 0 0

⎤⎥⎥⎥⎥⎦∣∣∣∣∣∣∣∣∣∣X−

k

Rk =diag([σxs σθs σB σB σy−meas] (42)


Fig. 16. Snapshots of the estimated position of the door at different timeintervals.

Fig. 17. Estimated “x” over time.

Fig. 18. Estimated “y” over time.

Figs. 17–20 show some of the estimated states over time. Itis shown in Figs. 17 and 19 that, when the door rotates to morethan a threshold angle, at t ∼= 11 s, the sonar measurementsfor distance and orientation are not valid anymore, and theestimator switches from state 1 to state 2 and only updateswith the AMR sensors. When the door rotates back toward θ =0, the sonar measurements become valid again, at t ∼= 17 s,

Fig. 19. Estimated orientation over time.

Fig. 20. Estimated “p” over time.

and the estimator switches to state 1. In addition, when laterin the experiment, at t ∼= 22 s, the door gets very close to thesensors, the estimator switches to state 2. Also, from Fig. 18,we can see that, at the beginning of the experiment, since theAMR sensors are not available yet, updates are performed onlywith sonar sensors (State 0), and hence, an estimate of y isnot available. However, as soon as the AMR sensors becomeavailable (t ∼= 5.5 s), estimation of y becomes possible as well.

Fig. 21 shows how the Mahalanobis distance is used to rejectsonar outliers in this experiment with the vehicle door. In thisexample, as dmah−k goes over the predefined threshold, thesonar measurement is ignored.

B. Results From the Tests With a Mazda Protégé Vehicle

Tests were conducted with a full-scale passenger sedan(Mazda Protégé) to evaluate the performance of the developedsystem at various orientation angles. In tests with the Mazdacar, it was moved toward the sensors at a fixed angle. Datawere obtained at various fixed angles to verify performance fordifferent orientations. Figs. 22–26 show the result from one ofthe tests, where the car moved at an orientation of about θ =−20◦ toward the sensors and then back away from the sensors.In these figures, the pink triangles indicate the time interval thata vehicle is detected and active positioning is performed. The


Fig. 21. Mahalanobis distance used to reject sonar outliers.

Fig. 22. Estimated “x” over time.

Fig. 23. Estimated longitudinal velocity over time.

blue circles indicate the time interval that the AMR sensors areresponding and the estimator is in state 1 or 2 depending on theavailability of sonar measurements. In addition, in this test, xth

for the sonar sensor has been set to 1 m, and hence, the estimatorswitches from state 1 to state 2 when x < xsonar−th = 1 m,However, the sonar measurements are valid up to about 0.3 mfrom the transducers. With this higher threshold, we can eval-

Fig. 24. Estimated orientation over time.

Fig. 25. Estimated “p” over time.

Fig. 26. Error in “x” estimation over time.

uate the estimator performance by comparing sonar systemand estimator values. The error between the AMR estimation(state 2) and the sonar measurement is shown in Fig. 26.

It should be noted that the tests here were carried out usinga single vehicle. Additional vehicles could be parked close tothe test vehicle if multipath and robustness issues need to beevaluated.


VIII. CONCLUSION

This paper has focused on the development of a noveland unique automotive sensor system for the measurement ofrelative position and orientation of another vehicle in closeproximity. The sensor system is based on the use of AMRsensors, which measure magnetic field. While AMR sensorshave previously been used to measure traffic flow rate and todetect vehicles in parking spots, they are used here to measurethe relative 2-D position of the vehicle.

Analytical formulations were developed to predict the rela-tionships between position and magnetic field for 2-D relativemotion. The use of these relationships to estimate vehicleposition is complicated by the fact that the parameters in therelationships vary with the type of vehicle under consideration.Since the type of vehicle encountered is not known a priori, theparameters for the magnetic field–position relationship have tobe adaptively estimated in real time.

A system based on the use of multiple AMR sensors anda custom-designed sonar system together with an EKF wasdeveloped to estimate vehicle parameters, position, and orien-tation. The use of the combined sensors results in a reliablesystem that performs well without the knowledge of vehicle-specific magnetic field parameters. Test results with a wheeledlaboratory test rig consisting of a door and tests with a full-scale passenger sedan were presented. The experimental resultsin this paper confirm that the developed sensor system is viableand that it is feasible to adaptively estimate vehicle positionand orientation even without knowledge of vehicle-dependentparameters.

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Saber Taghvaeeyan received the B.S. degree inelectrical engineering from the University of Tehran,Tehran, Iran, in 2009. He is working toward thePh.D. degree in mechanical engineering at the Uni-versity of Minnesota, Minneapolis, MN, USA.

His research interests include sensors, control sys-tems, and vehicle dynamics.

Mr. Taghvaeeyan was the recipient of the 2013–2014 University of Minnesota Doctoral DissertationFellowship and the 2013 ITS Minnesota Best StudentPaper Award.

Rajesh Rajamani received the B.Tech. degreefrom the Indian Institute of Technology at Madras,Chennai, India, in 1989 and the M.S. and Ph.D.degrees from the University of California, Berkeley,CA, USA, in 1991 and 1993, respectively.

He is a Professor of mechanical engineering withthe University of Minnesota, Minneapolis, MN,USA. He has coauthored over 95 journal papersand is the author of Vehicle Dynamics and Control(Springer Verlag). He is a coinventor on eight patentapplications. His research interests include sensors

and control systems for automotive and biomedical applications.Dr. Rajamani is a Fellow of the American Society of Mechanical Engineers

(ASME). He has served as the Chair of the IEEE Contol Systems SocietyTechnical Committee on Automotive Control. He has served on the editorialboards of the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

and the IEEE/ASME TRANSACTIONS ON MECHATRONICS. He has been arecipient of the CAREER award from the National Science Foundation, the2001 Outstanding Paper award from IEEE TRANSACTIONS ON CONTROL

SYSTEMS TECHNOLOGY, the Ralph Teetor Award from the Society of Au-tomotive Engineers, and the 2007 O. Hugo Schuck Award from the AmericanAutomatic Control Council.