# A probabilistic model for estimating driver behaviors and vehicle trajectories in traffic environments

**ABSTRACT** This paper presents a filter that is able to simultaneously estimate the behaviors of traffic participants and anticipate their future trajectories. This is achieved by recognizing the type of situation derived from the local situational context, which subsumes all information relevant for the drivers decision making. By explicitly taking into account the interactions between vehicles, it achieves a comprehensive situational understanding, inevitable for autonomous vehicles and driver assistance systems. This provides the necessary information for safe behavior decision making or motion planning. The filter is modeled as a Dynamic Bayesian Network. The factored state space, modeling the causal dependencies, allows to describe the models in a compact fashion and reduces the computational complexity of the inference process. The filter is evaluated in the context of a highway scenario, showing a good performance even with very noisy measurements. The presented framework is intended to be used in traffic environments but can be easily transferred to other robotic domains.

**2**Bookmarks

**·**

**116**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**With the objective to improve road safety, the automotive industry is moving toward more “intelligent” vehicles. One of the major challenges is to detect dangerous situations and react accordingly in order to avoid or mitigate accidents. This requires predicting the likely evolution of the current traffic situation, and assessing how dangerous that future situation might be. This paper is a survey of existing methods for motion prediction and risk assessment for intelligent vehicles. The proposed classification is based on the semantics used to define motion and risk. We point out the tradeoff between model completeness and real-time constraints, and the fact that the choice of a risk assessment method is influenced by the selected motion model.Robomech Journal. 07/2014; 1(1). - SourceAvailable from: Stephanie Lefevre[Show abstract] [Hide abstract]

**ABSTRACT:**This work tackles the risk estimation problem from a new perspective: a framework is proposed for reasoning about traffic situations and collision risk at a semantic level, while classic approaches typically reason at a trajectory level. Risk is assessed by estimating the intentions of drivers and detecting conflicts between them, rather than by predicting the future trajectories of the vehicles and detecting collisions between them. More specifically, dangerous situations are identified by comparing what drivers intend to do with what they are expected to do according to the traffic rules. The reasoning about intentions and expectations is performed in a probabilistic manner, in order to take into account sensor uncertainties and interpretation ambiguities. This framework can in theory be applied to any type of traffic situation; here we present its application to the specific case of road intersections. The proposed motion model takes into account the mutual influences between the maneuvers performed by vehicles at an intersection. It also incorporates information about the influence of the geometry and topology of the intersection on the behavior of a vehicle, and therefore can be applied to arbitrary intersection layouts. The approach was validated with field trials using passenger vehicles equipped with Vehicle-to-Vehicle wireless communication modems, and in simulation. The results demonstrate that the algorithm is able to detect dangerous situations early and complies with real-time constraints.Inria research report. 10/2013; RR-8379. - SourceAvailable from: Sebastian Brechtel
##### Conference Paper: Learning Context Sensitive Behavior Models from Observations for Predicting Traffic Situations

[Show abstract] [Hide abstract]

**ABSTRACT:**Abstract — Estimating and predicting traffic situations over time is an essential capability for sophisticated driver assistance systems or autonomous driving. When longer prediction hori- zons are needed, e.g., in decision making or motion planning, the uncertainty induced by incomplete environment perception and stochastic situation development over time cannot be neglected without sacrificing robustness and safety. Especially describing the unknown behavior of other traffic participants poses a complex problem. Building consistent probabilistic models of their manifold and changing interactions with the environment, the road network and other traffic participants by hand is error- prone. Further, the results could hardly cover the complete diversity of human behaviors. This paper presents an approach for learning continuous, non-linear, context dependent process models for the behavior of traffic participants from unlabeled observations. The resulting models are naturally embedded into a Dynamic Bayesian Network (DBN) that enables the prediction and estimation of traffic situations based on noisy and incomplete measurements. Using a hybrid state repre- sentation it combines discrete and continuous quantities in a mathematically sound way. Experiments show a significant improvement in estimation and prediction accuracy by the learned context dependent models over standard models, which only consider vehicle dynamics.International IEEE Conference on Intelligent Transportation Systems (ITSC); 01/2013

Page 1

A Probabilistic Model for Estimating Driver Behaviors

and Vehicle Trajectories in Traffic Environments

Tobias Gindele, Sebastian Brechtel and Rüdiger Dillmann

Institute for Anthropomatics

Karlsruhe Institute of Technology

D-76128 Karlsruhe, Germany

Email: {gindele | brechtel | dillmann}@kit.edu

Abstract—This paper presents a filter that is able to si-

multaneously estimate the behaviors of traffic participants and

anticipate their future trajectories. This is achieved by recog-

nizing the type of situation derived from the local situational

context, which subsumes all information relevant for the drivers

decision making. By explicitly taking into account the interac-

tions between vehicles, it achieves a comprehensive situational

understanding, inevitable for autonomous vehicles and driver

assistance systems. This provides the necessary information for

safe behavior decision making or motion planning. The filter is

modeled as a Dynamic Bayesian Network. The factored state

space, modeling the causal dependencies, allows to describe the

models in a compact fashion and reduces the computational

complexity of the inference process. The filter is evaluated in

the context of a highway scenario, showing a good performance

even with very noisy measurements. The presented framework

is intended to be used in traffic environments but can be easily

transferred to other robotic domains.

I. INTRODUCTION

Autonomous vehicles need a comprehensive representa-

tion of their environment in order to make optimal behavior

decisions or to conduct behaviors in a safe manner. The

process of acquiring the necessary information is hindered

by the fact that most of the environmental states are not

directly observable and therefore need to be inferred. E.g.

there exists no sensor which can perceive the plans of traffic

participants. Today’s sensory information is often limited

to noisy measurements of pose, velocity and some basic

geometric features.

While human drivers naturally put themselves into the

position of other traffic participants to reason about their

behaviors, machine tracking algorithms are usually limited to

physical models and simplest heuristics. This is not sufficient

for robust long term predictions — and consequently —

anticipatory driving. The incorporation of semantics and

context information is one key aspect when considering

forward-looking driver assistance and autonomous vehicles.

The use of probabilistic methods helps to connect the

symbolic and metric behavior representation, at the same

time providing a decent way to cope with uncertainty and

inaccuracy of a semantic formulation. The combination of

both degrees of abstraction allows on the one hand to

estimate the state of objects more exactly, because they are

enriched with a complex understanding of situations and their

context. On the other hand the system does not only obtain

a symbolic situation classification, but also predicts it in the

future, which is the basis for probabilistic decision making.

II. RELATED WORK

The state estimation problem of traffic participants is often

done with a standard filtering approach. Usually the pose

of a vehicle is the only state of interest, which allows the

use of classical tracking methods like Kalman Filters or

Extended Kalman Filters. In the area of multi-target tracking,

objects are often assumed to move independently to employ

a process model that only models the systems dynamics.

While this assumption induces some beneficial mathematical

properties — since the objects can be tracked individually —

it is definitely not valid for traffic scenarios, where transitions

of vehicle states are highly coupled. To solve this problem

it is necessary to model these interactions and different

behaviors.

Dagli et al [1] [2] realized this problem and proposed a

motivation based approach. Other than relying solely on the

system dynamics, they conclude that it is necessary to infer

the motivations and goals of driver in order to predict his be-

havior. Another approach is chosen by Zhang et al [3] which

estimates the behaviors of vehicles in intersection scenarios.

They choose an hierarchical approach that first estimates the

likelihoods of possible paths through the intersection on the

abstraction level of lanes to identify possible conflicts. In

the second stage the behaviors are estimated with a Dynamic

Bayesian Network (DBN). Forbes et al [4] already suggested

the use of a DBN to estimate the states of traffic participants

but their focus lied more on the following decision process of

the autonomous vehicle, rather than the state estimation prob-

lem. The subproblem of situation recognition is addressed

by Meyer et al [5] who use a relational hidden Markov

model to recognize different classes of traffic situations based

on a semantic representation. While this approach has the

ability to predict situations on a high abstraction level, it

can not estimate the quantiative properties like the pose

of a vehicle. The problem of behavior estimation is often

addressed by using Hidden Markov Models. A special variant

are interacting multiple models filters (IMM) [6], which

explain the observed behavior of an object as an interaction

of several models corresponding to different behavior modes.

This shows some similarities to our approach since we also

use multiple models to describe different behaviors.

!"#"$#%&'$()&*+),&-.),/$(000

1))2,/$3.)4*+*)5*$.)$()&*//-6*)&$7+,)89.+&,&-.)$:;8&*<8

=,>*-+,$(8/,)>?$@.+&26,/?$:*9&*<A*+$#BC!!?$!"#"

D1EF%

BGHC#CE!EECGIJHCBK#"KL!IF""$M!"#"$(000#I!J

Page 2

III. BAYESIAN MODEL

A. Overview

Our approach to the estimation problem of the desired

properties is to formulate it as a time discrete filtering

problem and to apply an appropriate probabilistic inference

method to reason about the hidden states.

The main idea is a chain of reasoning steps that result

in the prediction of the next state of the global situation.

Beginning with the extraction of implicit context information

from the vehicle states – like which vehicle drives in front

of another – one is able to classify the type of situation

each vehicle resides in. Each situation class implies certain

behaviors that are likely to be conducted by a driver in this

situation. Based on the behavior and context information

the metric realization in form of a desired trajectory can

be inferred. This ultimately closes the loop by enabling the

prediction of the next vehicle states.

As can be seen, we use different levels of abstraction to

achieve the goal, thereby combining symbolic and subsym-

bolic representations. The following sections describe the

necessary models and representations in detail.

B. vehicle model

For representing a vehicle state we apply a kinematic

one-track model which virtually replaces the front and the

rear wheels by one in the middle. Neglecting any slipping

effects, the car rotates around its instantaneous center of

rotation (ICR). Therefore the car’s state depends directly on

the orientation of the wheels and the length of the vehicle

as depicted in Figure 1.

λ

ICR

δ

δ

l

Fig. 1.One-track model of a non-holonomic car

The set of vehicle states X ⊆ R5comprises the position

�x1,x2

the steering angle δ. A vehicle state is therefore defined as

x =

given by l and assumed constant.

The basic controls that a driver can directly influence are

the acceleration a and steering rate ω of the vehicle. So a

��of the vehicle together with the gear angle ψ, the

�x1,x2,ψ,v,δ��∈ X. The length of the vehicle is

longitudinal velocity v in direction of the gear angle and

control u is defined as: u =�a,ω��∈ U. The total system

C. Trajectories

A trajectory is a time ordered set of states of a dynamical

system. In our case a trajectory defines a vehicle state over

a time interval. Given a function of the vehicle position

equations are provided by

˙ x1

˙ x2

˙ψ

˙ v

˙δ

=

v cosψ

v sinψ

vtanδ

a

ω

l

(1)

y(t) : [ts,te] → R2

together with its first and second derivative and the systems

equations (1) all dimensions of the vehicle state x ∈ X are

defined.

The trajectories we need to represent are realizations

of actual behaviors and therefore have to fulfill the non-

holonomic constraints. To get a compact representation we

further state that all trajectories are entirely defined by their

start and end states and the corresponding time interval.

y =�xs,xe,ts,te

This constrains the complexity of behaviors, since the space

of possible trajectories is restricted.

The intermediate values are obtained by interpolation. We

chose a Bézier interpolation scheme. Higher order Bézier

curves have proven to be useful in the context of trajectory

representation [7]. Besides their compact representation they

produce jerk continuous trajectories which obey the non-

holonomic constraints of the vehicle. Other interpolation

mechanisms like spline interpolation that generate suffi-

ciently smooth curves could be used as well.

A Bézier curve of degree n with control points

{P0,...,Pn} is given by

��∈ {X × R}2= T

P[ts,te](t) =

n

�

i=0

Bn

i(t)Pi

(2)

where the Bernstein polynomials Bn

iare defined as

Bn

i(t) =

�n

i

�

(t − ts

te− ts)i(te− t

te− ts)n−i.

The derivative itself is again a Bézier curve with modified

control points Diof degree n − 1:

n−1

�

with

Di=

te− ts(Pi+1− Pi)

To fully define the cubic Bézier trajectory function yB(t)

given y the two inner control points need to be defined. They

can be derived from the velocity and gear angle constraints

˙P[ts,te](t) =

i=0

Bn−1

i

(t)Di

(3)

n

(4)

#I!I

Page 3

of the start and end state together with the first derivative of

the Bézier curve (n = 3):

�vscosψs

�vecosψe

D. Behaviors

Behaviors are symbolic representations of context depen-

dent motion primitives that a vehicle is able to conduct. We

assume that a finite set of behaviors is sufficient to describe

all possible maneuvers of a real driver. We chose the classes

based on semantic equivalents that humans use to describe

the maneuvers of a vehicle. This is not an optimal choice

but has the advantage that the estimations can be interpreted

by humans.

For the evaluation of the approach we modeled several

basic behaviors for highway driving situations. Naturally

the set of behaviors can be extended to other domains like

urban traffic where e.g. behaviors for intersection handling

are needed. It is important that the situation description

comprises all features necessary to discriminate behavior

classes.

The set of behaviors B consists of {free_ride, following,

acceleration_phase, sheer_out, overtake, sheer_in} which

are sufficient to describe the most common maneuvers in

highway scenarios. Examples of the different behavior pat-

terns together with corresponding trajectories are shown in

Figure 2.

P1= P0+

vssinψs

�te− ts

�te− ts

3

P2= P3−

vesinψe

3

A

B

C

D

E

F

Fig. 2.

acceleration_phase, D sheer_out, E overtake, F sheer_in

Behaviors with trajectories. A free_ride, B following, C

a) free_ride: This behavior is applied if there are no

other vehicles or obstacles ahead. The main aspects a driver

has to consider are speed limits, sight conditions and the

curvature of the road.

b) following: A vehicle is in following mode if another

vehicle drives ahead on the same lane. Unlike the free_ride

behavior the driver has to accommodate to the velocity of

the up front vehicle to keep the necessary safety distance.

c) acceleration_phase: The complete overhauling pro-

cess is separated in four phases. The first phase is the

acceleration phase where the driver accelerates to become

fast enough to drive past the up front vehicle.

d) sheer_out: In the sheer out phase the car is often

still accelerating. The car moves smoothly in a S-shaped

trajectory to the opposite lane. When it reaches the center

of the opposite lane the overtake phase begins.

e) overtake: In the overtake phase the overhauling car

drives on the opposite lane until it is far enough ahead of

the other vehicle to sheer back in.

f) sheer_in: By sheering in the car changes back from

the opposite lane to the original lane. As soon as the car

reaches the original lane it changes back to free_ride or

following.

All the different behaviors induce constraints on the cor-

responding trajectories. A requirement for a trajectory of

the sheer_out behavior is e.g. that the end point lies on the

left opposite lane and that the velocity is significantly faster

than that of the overhauled vehicle. These constraints are

expressed by a conditional probability distribution function

(cdf) over the trajectory space depending on the behavior

and the situational context.

E. Situation context

The situational context describes the properties of the

vehicle surroundings. It consists of relations between objects,

which are in our case mainly distances. The relations are

extracted by analyzing the implicit features of the situation,

thereby making them explicit. These relations are the direct

basis for the drivers decisions. E.g. if the distance between

two car falls below the safety distance, the the driver of the

following car will decelerate.

dlat

centerline

dlon

Fig. 3.

the baseline for the distance calculations

Distance relations between two cars. The dashed line represents

Since the situational context is only needed to infer the

behaviors, only those relations need to be considered that are

relevant for the drivers decision making. They are evaluated

for every car in the scene. Figure 3 illustrates the distance

relations between two cars. For the modeled set of behaviors

B the following distance relations are relevant:

longitudinal distance: The longitudinal distance dlon,f

describes the distance of a vehicle to the next vehicle ahead.

The distance is measured between the projected positions

of the vehicles on the centerline of the lane. If there is no

vehicle ahead in the observable area then the distance is

set to infinity. The analog distance relation dlon,bmeasures

the distance to the next vehicle behind. Since the centerline

follows the curvature of the road it is a better measure than

the Euclidean distance.

#I!G

Page 4

lateral distance: The lateral distance dlatmeasures the

displacement between the position of a car and the centerline.

Again it’s calculated by projection of the position onto the

centerline.

relative velocity: Regarding a vehicle and the next

vehicle ahead, the relative velocity ddv,f describes the dif-

ference between their velocities. Analogous ddv,bdescribes

the relative velocity to a vehicle behind.

The basic set of context relations is defined as C =

{dlat,dlon,f,dlon,b,ddv,f,ddv,b}. Since for some driving ma-

neuvers, like lane changing or overtaking, the objects on

adjacent lanes are also relevant, similar relations are included

in the overall set of context features. If the set of behaviors

is extended for other applications, C needs to be extended

to contain all relevant context features.

F. Statespace

The state space of the filter consists of several random

variables that describe the different aspects of the situation.

We assume a finite number of n vehicles in the scene. The

random variables are defined as follows:

X: Vector containing the vehicle states.

X =�X1 ... Xn

C: Vector containing the features of local situational

context, primarily consisting of relevant distances

between the traffic participants and other objects,

e.g. the distance to next vehicle ahead.

C =�C1 ... Cn

S:Vector containing the recognized situations. S is

the set of all situations

S =�S1 ... Sn

B:Vector of behaviors. B is the set of all behaviors

B =�B1 ... Bn

T:Vector of the trajectories, representing the realiza-

tion of the behaviors.

T =�T1 ... Tn

Z:Measurement vector providing information about

the pose. The actual form depends on the sensor

configuration. We assume the sensor signals can

be linearly mapped onto a subspace of the vehicle

pose to ease the formulation of the measurement

model. A reasonable assumption is the observabil-

ity of the position (x1,x2), orientation ψ and speed

v of the vehicles.

Z =�Z1 ... Zn

All variables marked with a minus like X−correspond to

the random variables at time index t − 1. Since we assume

that the first order Markov assumption is met, we only need

to consider random variables with time index t and t − 1.

��,

Xi∈ X

��,

Ci∈ C

��,

��,

Si∈ S

Bi∈ B

��,

Ti∈ T

��,

Zi=�x y ψ v�

G. Decomposition of the Joint Distribution

The filter is modeled as a first order Markov process.

The process satisfies the Markov property since the state

space subsumes all relevant information needed to infer the

next system state. This property can be achieved for all

higher order Markov processes [8]. To exploit the conditional

independencies between random variables it is beneficial to

model the process as a Dynamic Bayesian Network (DBN)

[8].

Assuming several conditional independencies the joint

distribution can decomposed as follows:

P(X,X−,C,S,B,B−,T,T−,Z) =

P(X−)P(B−)P(T−)P(X|T−)P(C|X)

P(S|C)P(B|B−,S)P(T|T−,B,C,X)P(Z|X)

The cdfs are explained in section III-H.

Figure 4 shows the resulting DBN. Continuous nodes are

depicted by circles and discrete nodes by rectangles. Solid

lines represent direct dependencies within a time frame while

dashed lines state dependencies between time steps. The

behaviors e.g. depend on the current situations and the last

conducted behaviors.

(5)

X

B

Z

C

S

T

Fig. 4.

while solid lines represent causal dependencies within a time frame

Bayesian Network. Dashed Lines represent temporal dependencies

H. Filtering Models

The factors of the decomposition of the joint distribution

(5) are referred to as the models of the filter. They describe

the dependencies between the random variables in form of

cdfs. Detailed descriptions and interpretations of the models,

as well as the definitions of the corresponding cdfs, are given

in the following sections.

1) Dynamics Model - P(X|T−): The dynamics model

describes the change of the vehicles states after a time

step ∆t, given their last planned trajectory. This model is

often referred to as the motion model. Since we neglect

the explicit handling of collisions between vehicles in the

dynamics model, we are able to regard the individual motions

as conditionally independent. This simplifies the cdf to

P(X|T−) =

n

�

i=1

P(Xi|T−

i)

#I!H

Page 5

Since the planned trajectory already accounts for the non-

holonomic motion constraints, the new vehicle state can

be derived from the planned trajectory and current time t

using (2). Allowing some small deviations from the intended

motion, represented by the error term ex, the cdf can be

stated as follows:

P(Xi|T−

i= y) = gt(y) + ex

where gt(y) : T → X calculates the new vehicle state

given the curve parameters y. We assume exto be normal

distributed with ex∼N(0,Σx)

2) Context Model - P(C|X): The context model de-

scribes how the local situational context is derived from the

vehicle states. The context describes all relevant information

needed by a driver to choose an appropriate behavior. This

result in a local view for each vehicle, allowing us to

decompose the model as follows:

P(C|X) =

n

�

i=1

P(Ci|X)

The distance relations are evaluated with the helper function

dmap(x) as described in III-E. Therefore a virtual map of

the road network is employed, which allows to match the

vehicles to corresponding lanes, as well as finding the next

vehicles ahead or behind. It also provides the centerlines

needed for the projection. The cdf for a specific x is defined

as a Dirac distribution

P(C|X = x) = δdmap(x).

Since the distances can be calculated without uncertainty,

given the exact states of vehicles and assuming a perfect

map, no error term is needed.

This model couples all vehicle states together. By extract-

ing all relevant information from the global object state and

mapping it to the local context states, one is able to formulate

the remaining models for every object individually. This

simplifies the formulation of the other models drastically and

enables fast convergence of machine learning methods and

better generalization, since the models need not to be learned

for different numbers of objects.

3) Situation Model - P(S|C): If the local context of a

vehicle is known, it is possible to distinct several classes

of situations. The matching of contexts to the finite set

of possible situations S is done by the situation model.

This matching is in fact a discretization that transfers the

knowledge about the situation to a symbolic level. All

contexts that are grouped in a situation class should share

the characteristic, that they imply similar distributions over

behaviors. This maximizes the information a situation holds

about the drivers behavior decision.

Since the type of situation is recognized individually

for each vehicle on the basis of its local context, we can

decompose the situation model to:

P(S|C) =

n

�

i=1

P(Si|Ci)

To define the likelihood function for a situation class, we

use a weighted multivariate normal pdf, so all likelihood

functions together can be interpreted as a finite normal mix-

ture. Figure 5 illustrates the relationship between a context

and situation likelihoods. In this example the context only

consists of the distance to a vehicle driving ahead and two

situation classes are distinguished, namely that the vehicle is

close and that the vehicle is far away.

dlat

P(S|C)

vehicle far

vehicle close

Fig. 5. Exemplary likelihood functions of a simplified situation model with

two classes.

4) Policy Model - P(B|S,B−): On the basis of the last

conducted behaviors and considering the type of situation

that a vehicle currently resides, the policy model describes

the behavior decision of a driver in a probabilistic way. For

each situation class it defines a distribution over the behaviors

likely to be chosen, which can be seen as set of probabilistic

rules covering every combination of behavior and situation.

To simplify the model, we decompose the cdf:

�

Regarding the behavior decisions as independent is in general

a valid assumption, because we can presume the drivers do

not communicate to arrange their behaviors. But expecting

that the current behavior decision is not influenced by the

last behaviors of the other traffic participants is a strong

assumption which is not always true. This is only done for

complexity reasons and analyzing the impact of this decision

will be part of future research.

Currently the set of rules is defined by a human expert.

An even better way to obtain the model – in contrast to

engineering – is to use machine learning methods or to

employ a planner. This would ensure that the model fits the

real policies of drivers.

5) Behavior Realization Model - P(T|B,X,C,T−):

The behavior realization model makes the actual step from

abstract symbolic behaviors to concrete trajectories, thereby

taking into account the overall situation. The cdf of trajecto-

ries is also biased by the planned trajectories of the last time

step.

In order to reduce the model complexity the cdf is decom-

posed like the policy model, stating

n

�

#I!B

P(B|S,B−) =

n

i=1

P(Bi|Si,B−

i)

P(T|B,X,C,T−) =

i=1

P(Ti|Bi,Xi,Ci,T−

i).

Page 6

Neglecting the planned trajectories of the other traffic par-

ticipants is a critical choice, since the model isn’t able to

enforce non-overlapping trajectories directly. This is accom-

plished only indirectly by means of the policy model which

favors behavior decisions that do not result in overlapping

trajectories.

The most likely trajectories are the ones that optimize

several criteria set by the driver. The most important are

the length of a path to reach a goal, how economic and

comfortable the trajectory is and of course the safety prop-

erties. These facts need to be considered in the cdf, e.g if a

vehicle drives at high speeds the trajectory on an overtaking

maneuver has a flatter shape than one driven at low speeds.

Since we use a Bézier curve representation for the tra-

jectories based on control points as described in III-C, the

cdf defines a conditional distribution over the control point

parameter space. To define the cdf we use deterministic

functions kbfor each behavior type b together with additive

white noise eb∼ N(0,Σb). This results in

P(T|B= b,X= x,C= c,T−= y) = kb(x,c,y) + eb.

The function kbcalculates the control points parameters for a

specific behavior b. Like the context model this function also

makes use of the virtual road map to generate control points

according to the centerlines of the lanes. The main difficulty

is to find the appropriate end control point, since the values of

the start control point are constrained by the vehicle state in

order to produce smooth change-overs. E.g. for the sheer_in

behavior, the function picks a position and orientation of the

end control point based on the centerline of the opposite lane

together with an appropriate desired velocity.

These relationships can become rather complex, so for

more complex domains, like inner city environments, it

is a good idea to utilize a motion planner that generates

distributions over likely trajectories. Using machine learning

methods is another way to generate the model

6) Observation Model - P(Z|X): The observation model

describes the coherence between the overall system state and

the likelihood of an observation. We assume that we receive

independent observations zifor each vehicle individually, i.e.

we don’t have to cope with the data association problem [9]

and are able to state the cdf as

P(Z|X) =

n

�

i=1

P(Zi|Xi).

We assume a linear mapping H between vehicle state and

expected observation to stay sensor agnostic. The sensor

noise ez is assumed to be normally distributed with ez ∼

N(0,Σz). The cdf of the vehicle observation model can then

be stated as

P(Z|Xi= xi) = Hxi+ ez.

IV. FILTER EQUATIONS

Given the joint distribution of all variables (5), the recur-

sive update formulas can be derived using Bayes’rule. The

distribution of the posterior state is estimated by marginal-

izing over the prior state and weighting with the likelihood

of the measurement z. Thereby the update can be separated

in two steps – classically called Prediction and Correction.

The resulting update equation is

P(X,C,B,T,U|z) ∝

P(z|X)

�

B−,T−

P(X,C,S,B,T|B−,T−)P(B−,T−)

(6)

where the integral corresponds to the prediction and the other

part to the correction.

The integral of the prediction step is not solvable in

its general form. The existence of an analytical solution

depends strongly on the distribution types and the cdfs.

Classical representatives of this kind are e.g. Kalman filters

and histogram filters [10].

Since we are dealing with a mixed state space covering

discrete behaviors and continuous vehicle states in addition

to non-linear cdfs, we use a particle filter framework for

inference. Particle filters approximate the true posterior and

have been used successfully in many robotics applications

[10]. The inference is achieved with likelihood weighting

[8] where samples are drawn for all unobserved random vari-

ables according to their parent states. The overall particles are

weighted with the evidences according to the corresponding

cdfs. Since the number of samples is finite, the integral of

the prediction is replaced with a finite sum over all prior

particles.

V. EXPERIMENTS

To evaluate the presented filter we implemented a slightly

simplified version of the model, using only a subset of

the named behaviors. The behaviors used are free_ride,

following, sheer_out and overtake. These behaviors are suf-

ficient to show the principal properties of the filter. The

traffic scenario we used for the evaluation was a two lane

road scenario with two vehicles, as depicted in Figure 6.

In the simulated runs, the vehicle coming from behind is

approaching the slower vehicle on the right and changes the

lane, so it can pass the other vehicle. For the evaluation we

Fig. 6.

overtaking the other

Scenario of the experiment. Two lane with two cars, where on is

assumed the measurability of the position, gear angle and

velocity of the vehicles. The measurements come at a fre-

quency of 10Hz. To have a valid ground truth, we simulated

the scenario several times and added white Gaussian noise

to the measurements. The accuracy of the filter is measured

by estimating the average root mean square error (RMSE) of

the position of the overtaking vehicle. As position estimation

#I%"

Page 7

we used the expectation of the particle distribution. The

expectation shows a slightly more accurate and more stable

performance than the use of the best particle. The particle

inference employed 500 particles.

A. Filtering Results

The results of the average estimation error of the vehicles

position for different settings of the noise parameters is

shown in Tab.I. The filter yields a small average estimation

TABLE I

AVG. RMS ERROR OF THE MEAN POSITION FOR DIFFERENT NOISE

PARAMETERS

σx1,x2

0.050

0.100

0.150

0.200

0.250

σψ

0.025

0.050

0.075

0.100

0.125

σv

0.250

0.500

0.750

1.000

1.250

avg. RMS position error

0.06

0.09

0.13

0.18

0.21

error of 6cm of the vehicles position for low measurement

noise. As can be seen, the estimation error increases slower

than the applied noise, showing the smoothing effect of the

process model. This means that the filter is able to estimate

the vehicle states even under heavy noise due to the accuracy

of his models. Tab.II shows the average likelihood of the true

TABLE II

AVG. LIKELIHOOD OF THE TRUE CONDUCTED BEHAVIOR FOR

DIFFERENT NOISE PARAMETERS

σx1,x2

0.050

0.100

0.150

0.200

0.250

σψ

0.025

0.050

0.075

0.100

0.125

σv

0.250

0.500

0.750

1.000

1.250

avg. behavior likelihood

0.9168

0.9226

0.9059

0.8934

0.8748

conducted behavior, measuring the quality of the behavior

estimation. As expected the reliability of the estimation

decreases slowly with increasing noise. In all considered sets

of noise parameters a maximum likelihood estimator would

classify the behavior correctly in almost every case.

VI. CONCLUSION

In this paper we presented a filter that allows to estimate

simultaneously the current pose and behavior as well as

the anticipated trajectory of traffic participants. These es-

timations are needed by an autonomous vehicle for decision

making and motion planning. By first recognizing the type

of situation based on the local situational context, the filter

is able to infer the likely behaviors.

The filter is modeled as a Dynamic Bayesian Network,

thereby exploiting the conditional independencies, which

allows to describe the models in a compact way and to

calculate the inference more efficiently. The probabilistic

approach makes the estimation robust under noisy measure-

ments. A distinctive feature is that the expected trajectories

are estimated via filtering without enrolling the DBN into

the future. This saves a lot of computational effort since it

scales linearly with the number of enrolled time slices.

The evaluation showed a good performance of the filter

in the domain of highway scenarios. For an application of

this approach in an urban environment the models have to be

more complex to deal with the variety of possible situations.

An interesting direction to explore, is the estimation of

plans of traffic participants. We expect that this will improve

the accuracy of the estimation of the other states. On the

computational side, the use of Rao-Blackwellized methods

[11] could reduce the complexity of the inference.

ACKNOWLEDGMENT

The authors gratefully acknowledge the contribution of the

German collaborative research center ”SFB/TR 28 – Cogni-

tive Cars” granted by Deutsche Forschungsgemeinschaft.

REFERENCES

[1] I. Dagli, M. Brost, and G. Breuel, “Action recognition and prediction

for driver assistance systems using dynamic belief networks,” Lecture

Notes in Computer Science, pp. 179–194, 2003.

[2] I. Dagli and D. Reichardt, “Motivation-based approach to behavior

prediction,” in Proceedings of the IEEE Intelligent Vehicles Sympo-

sium, 2002.

[3] J. Zhang and B. Rössler, “Situation Analysis and Adaptive Risk

Assessment for Intersection Safety Systems in Advanced Assisted

Driving,” in Proceedings of the 21th Fachgespräch Autonome Mobile

Systeme, 2009, pp. 249–258.

[4] J. Forbes, T. Huang, K. Kanazawa, and S. Russell, “The batmobile:

Towards a bayesian automated taxi,” in International Joint Conference

on Artificial Intelligence, vol. 14, 1995, pp. 1878–1885.

[5] D. Meyer-Delius, C. Plagemann, G. Von Wichert, W. Feiten, G. Law-

itzky, and W. Burgard, “A Probabilistic Relational Model for Charac-

terizing Situations in Dynamic Multi-Agent Systems,” in Proceedings

of the 31st Annual Conference for Data Analysis, Machine Learning

and Applications, 2007.

[6] E. Mazor, A. Averbuch, Y. Bar-Shalom, and J. Dayan, “Interacting

multiple model methods in target tracking: a survey,” IEEE Transac-

tions on Aerospace and Electronic Systems, vol. 34, no. 1, pp. 103–

123, 1998.

[7] J. Choi and G. Elkaim, “Bézier Curve for Trajectory Guidance,” in

World Congress on Engineering and Computer Science, San Francisco,

CA, Oct. 22-24 2008.

[8] K. Murphy, “Dynamic Bayesian Networks: Representation, Inference

and Learning,” Ph.D. dissertation, University of California, 2002.

[9] I. Cox, “A review of statistical data association techniques for motion

correspondence,” International Journal of Computer Vision, vol. 10,

no. 1, pp. 53–66, 1993.

[10] S. Thrun, W. Burgard, and D. Fox, Probabilistic Robotics (Intelligent

Robotics and Autonomous Agents).

sachusetts, USA, 2005.

[11] A. Doucet, N. de Freitas, K. Murphy, and S. Russell, “Rao-

Blackwellised particle filtering for dynamic Bayesian networks,” Pro-

ceedings of the Sixteenth Conference on Uncertainty in Artificial

Intelligence, pp. 176–183, 2000.

MIT press, Cambridge, Mas-

#I%#

#### View other sources

#### Hide other sources

- Available from Rüdiger Dillmann · May 27, 2014
- Available from uka.de