A Bayesian Filter for Modeling Traffic at Stop Intersections

Thierry Wyder∗, Georg Schildbach†, Stephanie Lefevre†, Francesco Borrelli†

Abstract— All-way stop intersections are widely used fortraffic management in North America. Therefore, modeling andcontrol of vehicle behavior at stop intersections is fundamentalfor driver assistance systems and autonomous driving. Thispaper presents a method to predict the maneuvers performed byvehicles at arbitrary all-way stop intersections, using noisy sen-sor data. This is required for an autonomous vehicle to decidewhen to enter the intersection, or for a driver assistance systemto decide when to issue a collision warning to the driver. Theproblem is divided into two components. The first componentestimates the maneuver intention of the drivers by means of anaıve Bayesian filter. The second component predicts the orderin which the vehicles will enter the intersection by means of akinematic feedback model. Both algorithms are evaluated usingreal world data collected with laser sensors mounted on a vehi-cle. The Bayesian filter is successfully applied to intersections ofdifferent sizes and geometries. We show that the filter identifiesmaneuvers earlier than a deterministic reference model.

I. INTRODUCTION

A. Motivation

All-way stop intersections are widely used in NorthAmerica, especially in residential neighborhoods and ruralareas as they provide efficient, economical, and safe trafficmanagement for low traffic volumes. Despite that, there areapproximately 700,000 reported vehicle crashes at stop signsin the US annually [1]. Advanced Driver Assistance Systems(ADAS) have the potential to improve vehicle safety at stopintersections by issuing warnings or by direct interventions.Ultimately, this may lead to autonomous vehicles that nav-igate all-way stop intersections successfully without humancontrol.

We present two models of vehicle behavior at all-waystop intersections. One model can be used to predict theright-of-way at the intersection, and the other one to pre-dict the trajectory taken by a specific vehicle. ADAS andautonomous vehicles need this information to recognize andavoid conflicts with other vehicles, and to navigate throughan intersection in a natural and safe manner.

For example, Figure 1 illustrates two situations wherethree cars arrive at an intersection in the order indicated bytheir number. Traffic laws demand that each vehicle has tostop before entering the intersection, and that the vehicles

∗Thierry Wyder is with the Department of Mechanical and ProcessEngineering, Swiss Federal Institute of Technology Zurich, Switzerland(email: thwyder@ethz.ch)†Georg Schildbach, Stephanie Lefevre, and Francesco

Borrelli are with the Department of Mechanical Engi-neering, University of California Berkeley, U.S.A. (email:schildbach|slefevre|fborrelli@berkeley.edu)

This work was partially supported by the National Science Foundationunder grant No. 1239323 and by the Hyundai Center of Excellence at UCBerkeley.

enter the intersection in the order of their arrival at the stopline. In the left scenario, this means that car 3 can enterthe intersection even though cars 1 and 2 have the right-of-way, because their maneuvers do not cross with the intendedtrajectory of car 3. In the right scenario, car 3 has to waitfor cars 1 and 2 to leave the intersection before entering,since their maneuvers cross the intended trajectory of car 3.In order to make the correct decision in these two scenarios,car 3 has to be able to determine the right-of-way and toanticipate the intended maneuvers of the other vehicles.

Humans have the ability to quickly assess the right-of-wayand to estimate the intent of other drivers, but the same task ischallenging for computers. The available measurements areoften limited and noisy, human experience can be difficultto transfer into an algorithm, and some clues such as turnsignals and driver gaze are hard to detect with sensors.

1

32

3

1

2

Fig. 1: Traffic scenarios at all-way stop intersections withoutconflicts (left) and with conflicts (right).

B. Related workA recent survey on vehicle motion prediction classifies the

existing approaches into three groups: physics-based models,maneuver-based models, and interaction-aware models [2].

Physics-based models describe a vehicle’s movementbased on the fundamental laws of motion. Due to theirrelative simplicity, they have received a lot of attention. Theyare commonly based on a bicycle model, a constant turnrate model, or a velocity change rate model. However, thesemodels ignore important factors such as traffic rules, roadgeometry, and driver inputs. Therefore they are limited toshort-term predictions (up to 1 second).

Maneuver-based models try to identify the type of ma-neuver that the driver is performing or about to perform,leading to more reliable motion predictions. Maneuvers areusually identified using prototype trajectories, discriminativeclassifiers, or dynamic Bayesian networks. Trajectory modelscan be learned from training data or can be constructed basedon road geometry.

Interaction-aware models consider the entire vehicle en-vironment, including interactions between multiple vehiclesand the reactions of other drivers. They try to overcome theassumption of the previous two model types that all vehiclesmove independently from each other. However, due to theircomplexity, only few models of this type can be foundin literature. They are usually based on dynamic Bayesiannetworks.

In this paper, our focus lies on maneuver recognition.For that task, the most popular approaches are prototypetrajectories [3], discriminative classifiers such as RelevanceVector Machines [4] or MultiLayer Perceptrons [5], andprobabilistic models. For example, hidden Markov models(HMMs) are used in [6] for estimating a driver’s inten-tion to comply with the traffic rules at an intersection. Aprobabilistic finite state machine is used in [7] to modelthe temporal sequence of different maneuvers, and wassuccessful in distinguishing turn maneuvers from similarmaneuvers on real world data. In [8], a dynamic Bayesiannetwork (DBN) is used to predict vehicle maneuvers atroad intersections, including topographical information froma map. The predictions take the entry lane, the position andthe orientation of the vehicle as well as the road geometryinto account.

C. Contribution

This paper presents a novel approach for traffic estimationat all-way stop intersections. The approach works withdifferent intersections of variable geometries. In particular,we propose a regression-based model for right-of-way as-sessments and design a Bayesian filter for predicting themaneuvers of vehicles. Both algorithms are evaluated withreal-world data that is collected with a laser sensor systemmounted on a test vehicle.

The remainder of the paper is structured as follows. Theexperimental setup is described in Section II, as it is anintegral part of the algorithm development. Then Sections IIIand IV present our algorithms for maneuver prediction andright-of-way estimation respectively, with their experimentalresults shown in Section V. Finally, Section VI concludesthis paper and outlines future work.

II. DATA COLLECTION

All of our algorithms were implemented and tested offlinein a simulation environment with real measurement data. Themeasurements were collected with 6 IBEO laser scannersinstalled on our vehicle, denoted next as ego vehicle. Thesesensors provide a 360° coverage around the vehicle of objectsup to 200 meters away.

The data was collected with the ego vehicle parked nextto 5 different all-way stop intersections, with the layoutsand geometries shown in Figure 2. The laser sensors detectand track the vehicles approaching and going trough theintersection. The position of the vehicles are provided in acoordinate system centered on the ego vehicle. The headingangle of the tracked vehicles can be calculated through thedifferences of their successive positions. Although further

measurement variables are provided by the sensors, theywere found to be too noisy or not useful for our problem.

1

54

32

0 20m

Fig. 2: Schematics of Intersection 1,2,3,4 and 5 used for datacollection.

All the vehicles in the collected dataset complied with thetraffic rules; that is, they never run the stop line or stop inthe middle of the intersection. However, rolling stops, whena vehicle slows down but does not stop completely, are veryfrequent and amount to 65% of all stops at intersections [9].They were therefore taken into account.

III. MANEUVER PREDICTIONThe maneuver intention of drivers is estimated using a

naıve Bayesian filter at discrete time steps k = 0, 1, . . . .The proposed algorithm can be applied to intersections ofarbitrary shape. For simplicity however, it will be illustratedon the simple cross-shaped Intersection 1 in Figure 2. Inthis section, we first describe the method used to generatereference paths before we introduce the filter algorithm.

A. Reference Path Generation & Segment Selection

We assume that a road map is available that specifiesthe geometry of the intersection. First, a reference path isgenerated for every possible maneuver m at the intersection(i.e., a pair of entrance lane and exit lane), based on thegeometric description of the intersection. For the specificexample of Intersection 1 in Figure 2, a vehicle can performthree maneuvers m: left (L), straight (S), or right (R):

m ∈ {L,S,R} . (1)

To generate the reference paths, we use sets of clothoidcurves because they produce smooth paths with smoothchanges in curvature [10] suitable to represent vehicle trajec-tories. The clothoid fitting algorithm used in this work waspublished in [11].

After generating the paths, we divide them into equidistantsegments, as illustrated in Figure 3 on the left hand side. Then-th segment of the reference path pertaining to maneuverm is called sm,n and the curvilinear coordinate along thisreference path is denoted lm,n. The progression of a vehicle iinto the intersection at time k is measured by the curvilineardistance along the vehicle’s trajectory and denoted lik, asshown in Figure 3 on the right hand side. At each timestep k, for each vehicle i, and for each reference path m,a segment n can be found for which the distance |lm,n− lik|

is minimized. This segment will be referred to as the matchedsegment and denoted nmk (i), or in short form as n.

sS,3 sS,8

lR,7lik

nSk = sS,7

nLk = sL,7

nRk = sR,7

Fig. 3: Segmentation of the reference paths and selection ofthe matched segments.

B. Naıve Bayesian Filter

1) Variables: The naıve Bayesian filter estimates thecurrent maneuver of vehicle i at time k, denoted as mk (wewill omit the reference to a specific vehicle i from here on).The estimate is based on all available measurements of thevehicle up to time k. As measurements, we use the positiondifference ∆dmk and the orientation difference ∆φmk betweenthe vehicle and its matched segment nmk (i) on each referencepath m:

∆dmk =

sgnm(xn, yn, xk, yk

)·√

(xn − xk)2 + (yn − yk)2 , (2)

∆φmk = φn− φk . (3)

Here (xk, yk) and φk refer to the vehicle’s position andorientation at time k, respectively, and (xn, yn) and φn referto the position and orientation of the center of the matchedsegment nmk (i). The sign function sgnm defines whether thevehicle is located to the left or the right of the reference path.

In summary, the measurements for a vehicle at time k aredefined as

zk = [∆dLk ,∆φLk ,∆d

Sk ,∆φ

Sk ,∆d

Rk ,∆φ

Rk ] . (4)

For convenience, all measurements up to time k shall bedenoted z1:k.

2) Filtering equation: The naıve Bayesian filter is rep-resented graphically in Figure 4. The recursive filteringequation is:

P(mk

∣∣z1:k

)∝ P

(zk∣∣mk

)·∑

mk−1

P(mk

∣∣mk−1

)P(mk−1

∣∣z1:k−1

), (5)

where ‘∝’ indicates a directly proportional relationship (toyield a normalized probability).

3) Transition probabilities: The transition probability be-tween two consecutive states is defined by:

P(mk

∣∣mk−1

)=

{a if mk = mk−1

1−a2 if mk 6= mk−1

, (6)

∆dSk ∆φSk

mkmk−1 mk+1

∆dRk ∆φRk∆φLk∆dLk

Fig. 4: Naıve Bayesian filter used for maneuver estimation

with a being the probability of staying at the same stateacross one time step. The value of a = 0.45 is set to optimizethe performance of the maneuver predictor for the metricsdefined in Section V-A.2 and the collected training data.

4) Measurement model: Because of the naıvety assump-tion in our Bayesian filter, the measurement probability canbe written as:

P(zk∣∣mk

)= P(∆dLk

∣∣mk

)· P(∆φLk

∣∣mk

)· P(∆dSk

∣∣mk

)· P(∆φSk

∣∣mk

)· P(∆dRk

∣∣mk

)· P(∆φRk

∣∣mk

). (7)

For every segment on a reference path m, we assume thatthe position difference ∆dmk and the orientation difference∆φmk are distributed normally if m = mk and uniformly ifm 6= mk, which is equivalent to writing:

∆dmk ∼ N (0, σ2∆d,n) if m = mk ,

∆dmk = 1 if m 6= mk ,∆φmk ∼ N (0, σ2

∆φ,n) if m = mk ,

∆φmk = 1 if m 6= mk .

The two standard deviations of every segment on everyreference path were fitted from 14,007 data points on 223trajectories collected at Intersection 1 in Figure 2. Bothdepend on the distance lm,n, in order to account that theuncertainty varies with the location along a reference path.Linear fits along the distance lm,n were calculated for bothsets of standard deviations and were then averaged over alltwelve reference paths of the intersection to generalize toarbitrary geometries.

IV. RIGHT-OF-WAY ESTIMATION

The right-of-way at a stop intersection can be estimatedfrom the predicted arrival times of vehicles. Therefore ourapproach to estimate the right-of-way relies on estimatingthe arrival time at the stop line.

We model a vehicle approaching an intersection with thefollowing system of ordinary differential equations:

δk = vk , (8a)vk = αδ · δk + αv · vk , (8b)

where δk is the vehicle’s distance to the stop line, vk is thevehicle’s speed, and αδ and αv are gains learned from databy minimizing the difference between the predicted arrival

times and the true arrival times on a set of training data. Thetraining set consists of twelve intersection approach trajec-tories, and the resulting parameters are αδ = −1.5741 [1/s2]and αv = −1.7820 [1/s].

This model is used with new sensor measurements at everytime step k to predict the future motion of the vehicle andto calculate the time needed by the vehicle to reach the stopline (where δk = 0). This way we can estimate the right-of-way and predict the order in which the vehicles will enterthe intersection.

By applying this right-of-way prediction followed by themaneuver estimation approach from Section III, an intelligentsystem (ADAS or autonomous vehicle) has all the necessaryinformation to make a decision about how to handle theintersection.

V. EXPERIMENTAL RESULTS

This section presents the simulation results of both themaneuver prediction and the right-of-way estimation.

A. Maneuver PredictionIn order to evaluate the maneuver prediction algorithm

and assess the usefulness of the stochastic reasoning, weimplemented a deterministic predictor as a reference modelto compare to.

1) Reference Model: The reference model is a constantturn rate and constant acceleration model:

xT = x0 +

∫ T

0

(v0 + a0τ) cos(φ0 + ω0τ) dτ (9a)

yT = y0 +

∫ T

0

(v0 + a0τ) sin(φ0 + ω0τ) dτ (9b)

with the initial conditions (x0, y0, v0, a0, φ0, ω0) given bythe measurements. In this context, the variable a denotesthe acceleration and ω the angular velocity. We use thismodel to predict the future positions of a vehicle after theintegration time T . The maneuver is estimated by combiningthe position predictions with geometric threshold lines in theintersection. For each entrance road of the intersection, twolines are drawn to split the intersection area into three regionscorresponding to the three possible maneuvers, as illustratedin Figure 5. These lines are defined by the entry point to theintersection and the midpoints between the dots marking theexits of the reference paths.

Threshold LineReference Path

yPo

sitio

n[m

]

x Position [m]5 10 15 20 25 30

−5

0

5

10

15

Fig. 5: Construction of threshold lines.

At each time step, the maneuver of the vehicle can beestimated based on the region the predicted location of thevehicle is in. The forward integration time T in (9) wassubject to an optimization using the training data set. Avalue of T = 0.6 seconds was found to be optimal underthe performance metrics defined in Section V-A.2.

2) Evaluation metrics: To evaluate the maneuver predic-tion, two performance metrics were introduced. The first isthe correct classification rate (CCR) given by

CCR =100

j

j∑k=1

Ck (10)

with

Ck =

{1, if mpred

k = mactk

0, otherwise,(11)

with j the number of measurements, mpredk the maneuver

predicted by the filter and mactk the maneuver actually per-

formed. This metric returns the overall performance of theestimator but weighs all errors equally. In reality however,misclassifications occurring early in the intersection are lesslikely to cause problems than misclassifications occurringlater on. Therefore we also wish to evaluate how early thealgorithm is able to correctly classify maneuvers. To this endwe define the distance until correct classification (DCC),which measures the curvilinear distance between the pointwhere the vehicle entered the intersection and the pointwhere all subsequent maneuver classifications were correct.

3) Results: The data set used for validation stems fromthe five intersections seen in Figure 2 and consists of 544 tra-jectories with a total of 38,224 data points. The data used forthe training was of course removed from the testing dataset,so that all the testing is performed on previously unseentrajectories. The classification results for all these trajectoriescombined are given in Table I. Compared to the constantrate model classifier, the naıve Bayesian filter increases thecorrect classification rate by almost 2%. More importantly,it reduces the tracking distance needed to achieve a correctclassification by 30% for the 90% quantile, the 99% quantile,and the mean.

ConstantRate Model

NaıveBayesian

Filter

Relative Im-provement

CCR 89.90 [%] 91.69 [%] +1.99 [%]

90% Quantile ofDCC

3.20 [m] 2.24 [m] -30.00 [%]

99% Quantile ofDCC

14.02 [m] 9.87 [m] -29.60 [%]

Mean of DCC 1.07 [m] 0.72 [m] -32.54 [%]

StandardDeviation of

DCC2.42 [m] 1.97 [m] -18.74 [%]

TABLE I: Combined results from all validation trajectories.

We will now graphically present the results of two outof the five validation intersections, in order to illustrate the

Reference PathIncorrectCorrect

x Position [m]

yPo

sitio

n[m

]

x Position [m]

yPo

sitio

n[m

]

5 10 15 20 25 305 10 15 20 25 30−5

0

5

10

15

−5

0

5

10

15

Fig. 6: Intersection 1: Maneuver prediction using the naıveBayesian filter.

Threshold LineIncorrectCorrect

x Position [m]

yPo

sitio

n[m

]

x Position [m]

yPo

sitio

n[m

]

10 20 3010 20 30−5

0

5

10

15

−5

0

5

10

15

Fig. 7: Intersection 1: Maneuver prediction using the constantrate model.

ability of the algorithm to generalize to different intersectiongeometries.

The results obtained for Intersection 1 are depicted inFigure 6 for the naıve Bayesian filter and Figure 7 for theconstant rate model. The results obtained for Intersection 3are depicted in Figure 8 for the naıve Bayesian filter andFigure 9 for the constant rate model. The data points areplotted in blue when the maneuver at that point is classifiedcorrectly, independent of what the maneuver was. They arered when the estimator returned the wrong maneuver. Wenotice that the naıve Bayesian filter performs better than theconstant rate model in both cases, even though the filter wastrained using data collected at Intersection 1 and had neverseen Intersection 3.

B. Right-of-Way Estimation

The proposed approach for estimating the right-of-waywas verified with a total of 1,996 data points from a set

Reference PathIncorrectCorrect

x Position [m]

yPo

sitio

n[m

]

x Position [m]

yPo

sitio

n[m

]

10 20 30 4010 20 30 40

−5

0

5

10

15

−5

0

5

10

15

Fig. 8: Intersection 3: Maneuver prediction using the naıveBayesian filter.

Threshold LineIncorrectCorrect

x Position [m]

yPo

sitio

n[m

]

x Position [m]

yPo

sitio

n[m

]

10 20 30 4010 20 30 40

−5

0

5

10

15

−5

0

5

10

15

Fig. 9: Intersection 3: Maneuver prediction using the constantrate model.

of twelve trajectories of cars approaching stop intersections.As seen in Table II, an absolute difference between thepredicted and the actual arrival time, averaged over all datapoints, of 0.70 seconds was achieved. The standard deviationof this error is 0.66 seconds. Figure 10 shows an exampleof the prediction of the time to intersection for a trajectoryof roughly 9 meters. The upper plot displays the measuredvelocity profile of the vehicle approaching the stop sign, setat the zero mark of the x-axis. The blue line in the lower plotshows the time it took the vehicle from that data point untilit crossed the stop sign. The red line is the time predictedby the model. There is an initial prediction error of 0.69seconds at the start of the prediction, when the vehicle is 9.23meters away from the stop line. Averaged over the wholetrajectory, the model has a prediction error of 0.23 secondsand it becomes more accurate, the closer the vehicle gets tothe stop sign.

The limitations of the model become apparent whenconsidering abrupt driver behavior as shown in Figure 11.Initially, the predicted time to arrival is 2.68 seconds lowerthan the actual time. This is caused by the driver’s abruptbreaking maneuver well before the stop line, which theprediction model can not foresee. Later, the prediction timeis overestimated by more than a second because at that point,the velocity of the vehicle used as an initial condition for themodel is very low. In this example, the average absolute erroris 1.33 seconds.

Average Absolute Error 0.7022 [s]

Standard Deviation 0.6618 [s]

TABLE II: Arrival time prediction accuracy.

PredictedMeasured

Tim

eto

Stop

,t[s

]

x Position [m]

Velocity

Vel

ocity

,v[m

/s]

−10 −9 −8 −7 −6 −5 −4 −3 −2 −1 00

1

2

3

2

3

4

5

6

Fig. 10: Time prediction performance I.

PredictedMeasured

Tim

eto

Stop

,t[s

]

x Position [m]

Velocity

Vel

ocity

,v[m

/s]

−8 −7 −6 −5 −4 −3 −2 −1 00

1

2

3

4

5

0

1

2

3

4

5

Fig. 11: Time prediction performance II.

VI. CONCLUSIONS

In this paper, a method for maneuver prediction at stopintersections was developed that is applicable to arbitraryintersections of different geometries and sizes. It makesuse of the geometric description of an intersection andmeasurements from laser sensors. This algorithm is basedon a naıve Bayesian filter and was tuned with real worlddata. It performs well on five verification intersections. Acomparison with a deterministic predictor as a reference

model showed that the Bayesian filter reduces the trackingdistance necessary to classify trajectories correctly by 30%.

Further on, a right-of-way estimator for stop intersectionswas implemented. It predicts the arrival time of vehicles atthe stop sign. A kinematic feedback model was tuned withreal world data of vehicles approaching stop intersections.The model predicted the arrival time with an average absoluteerror of 0.7 seconds.

The combination of the right-of-way prediction and ma-neuver estimation approaches can be used by a drivingassistance system to help drivers in intersection areas, orby an autonomous vehicle to handle arbitrary all-way stopintersections according to the legal transportation code.

Further work is needed to improve the right-of-way esti-mator so it can handle stop-and-go situations like the onepresented in Figure 11. Before implementing the methodspresented on a test vehicle, a strong emphasis will need tobe put on real-time preprocessing of the sensor data.

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