Bayesian Train Localization with Particle Filter, Loosely Coupled GNSS, IMU, and a Track Map

Abstract

Train localization is safety-critical and therefore the approach requires a continuous availability and a track-selective accuracy. A probabilistic approach is followed up in order to cope with multiple sensors, measurement errors, imprecise information, and hidden variables as the topological position within the track network. The nonlinear estimation of the train localization posterior is addressed with a novel Rao-Blackwellized particle filter (RBPF) approach. There, embedded Kalman filters estimate certain linear state variables while the particle distribution can cope with the nonlinear cases of parallel tracks and switch scenarios. The train localization algorithm is further based on a track map and measurements from a Global Navigation Satellite System (GNSS) receiver and an inertial measurement unit (IMU). The GNSS integration is loosely coupled and the IMU integration is achieved without the common strapdown approach and suitable for low-cost IMUs. The implementation is evaluated with real measurements from a regional train at regular passenger service over 230 km of tracks with 107 split switches and parallel track scenarios of 58.5 km. The approach is analyzed with labeled data by means of ground truth of the traveled switch way. Track selectivity results reach 99.3% over parallel track scenarios and 97.2% of correctly resolved switch ways.

Full text

Research Article
Bayesian Train Localization with Particle Filter,
Loosely Coupled GNSS, IMU, and a Track Map
Oliver Heirich
DLR (German Aerospace Center), Institute of Communications and Navigation, 82234 Oberpfaenhofen, Germany
Correspondence should be addressed to Oliver Heirich; oliver.heirich@dlr.de
Received  October ; Revised  February ; Accepted  March 
Academic Editor: Yassine Ruichek
Copyright ©  Oliver Heirich. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Train localization is safety-critical and therefore the approach requires a continuous availability and a track-selective accuracy.
A probabilistic approach is followed up in order to cope with multiple sensors, measurement errors, imprecise information, and
hidden variables as the topological position within the track network. e nonlinear estimation of the train localization posterior is
addressed with a novel Rao-Blackwellized particle lter (RBPF) approach. ere, embedded Kalman lters estimate certain linear
state variables while the particle distribution can cope with the nonlinear cases of parallel tracks and switch scenarios. e train
localization algorithm is further based on a track map and measurements from a Global Navigation Satellite System (GNSS) receiver
and an inertial measurement unit (IMU). e GNSS integration is loosely coupled and the IMU integration is achieved without
the common strapdown approach and suitable for low-cost IMUs. e implementation is evaluated with real measurements from a
regional train at regular passenger service over  km of tracks with  split switches and parallel track scenarios of . km. e
approach is analyzed with labeled data by means of ground truth of the traveled switch way. Track selectivity results reach .%
over parallel track scenarios and .% of correctly resolved switch ways.
1. Introduction
Train localization inside a railway network is necessary for
a collision-free operation and mainly addressed by central-
ized trac control, signaling, and sensors in the railway
infrastructure. Onboard train localization in combination
with communications enables distributed and train centric
assistant systems such as collision avoidance, coupling, and
autonomous train operation. is localization system con-
cept focuses on exclusive onboard computation and sensors
without any additional railway infrastructure.
Future railway systems such as a train centric collision
avoidance system [, ] require a localization system with
continuous availability and a track-selective accuracy. Track
selectivity is the ability to identify the correct track, especially
in the critical, parallel track scenario aer a ride on a divisive
switch way. e track selectivity is the technical challenge and
also the major requirement of train localization.
e goal of train localization is to determine the position
of the train in the track network by topological coordinates,
which are hidden variables and cannot be measured directly.
Single sensor systems, such as global navigation satellite
systems (GNSS), are very benecial for localization, but
themeasurementaccuracyandlackofavailabilityinparts
of the railway environment do not fulll the safety-critical
requirements.
Research on train localization with onboard sensors
focuses on the following question: how is a train-borne,
safety-critical, and onboard localization system designed
andanalyzedintermsofdataprocessingwithcontinuous
availability and a track-selective accuracy? e approach
should cope with hidden variables as the topological position,
imprecise information of measurements from multiple sensor
sources, outages, statistical noise, and systematic measure-
ment errors.
is paper presents a train localization approach by a
Rao-Blackwellized particle lter (RBPF). Figure  shows the
setup with onboard sensor data of an inertial measurement
unit (IMU) and a GNSS receiver. e RBPF estimates the
linear state variables of the one-dimensional train transition
with Kalman lters within each particle of a particle lter.
Furthermore, a novel empirical evaluation methodology
Hindawi Publishing Corporation
Journal of Sensors
Volume 2016, Article ID 2672640, 15 pages
http://dx.doi.org/10.1155/2016/2672640

Journal of Sensors
Track
map
IMU
GNSS
Sequential
Bayesian
lter
Track ID,
track position,
train frame direction,
train speed
(i) Collision avoidance
(ii) Speed monitoring
(iii) Radio
(iv) Train control
F : Bayesian train localization setup with GNSS, IMU, and a track map.
is dened for track selectivity which is not specic to a
certain train localization approach. is paper is considered
as a follow-up of the theoretic probabilistic approach with
a particle lter [–] using satellite range measurements.
e novel parts are the extension of the particle lter for
train localization with a Rao-Blackwellization as well as an
evaluation framework for track selectivity. e RBPF and
a reference map-match approach are evaluated in terms of
track-selective accuracy with real train data of a regional
train.
e outline of this paper starts with a related work review,
a general information for a map-based train localization in
Section , and the description of the used sensors in Section .
Section  contains the derivation of the Rao-Blackwellization
and the RBPF implementation is given in Section . e
track-selective evaluation (Section ) evaluates the RBPF
approach with data from train runs (Section ). Sections 
and  show the results and the discussions on results.
2. Related Work
ere are multiple approaches of train localization in the liter-
ature with onboard sensors and a map. A selection of studies
are chosen, which focus on track selectivity or specic sensor
studies, which can identify the switch way. e dierent
approaches vary in sensor types or combinations, processing
methods, and evaluation scope and will be presented in these
categories.
Inertial sensors are oen used in combination with an
integrated navigation system for GNSS position aiding, for
example,in[,].eyawturnratecanalsobeusedfor
the switch way identication that has been used in [, ]
and analyzed in [, ]. Approaches with GNSS and IMU
are found in [–] and extensions with eddy current sensor
in [, , ]. e eddy current sensor, in principle a metal
detector for characteristic railway features, can be used for
a switch way detection, as speed or displacement sensor.
Sensors such as cameras [, ] or LIDAR [, ] can directly
identify the dierent switch ways and contribute to the track-
selective result. A study of a tightly coupled localization with
rawGNSSdataandatrackmapwasshownin[,].A
tightly coupled approach considers pseudorange, Doppler,
andphasemeasurementsandhastheadvantagetoprocess
location information even with less than four satellites in
view. However, a tightly coupled approach is typically more
complex to implement than a loosely coupled approach and
a user clock oset needs to be estimated additionally. e
processing methods of each study are dierent and dependent
on the used sensors, ltering method, and algorithmic inte-
gration of the map. Saab proposed a train localization using
a map-matching technique with a correlation of the curva-
ture signature []. For the track selectivity, there are two
classes with map integration and estimation of importance:
a multiple hypothesis lter handles and maintains multiple
estimates on several tracks in the vicinity and is commonly
used [, , , ]. e particle lter uses usually a large
set of particles, which are location hypotheses on a map,
and handles the dierent track hypotheses by particles. e
particlelterwithonboardsensorsandamapwasproposed
in []. Fouque and Bonnifait [] dened a marginalized
particle lter with raw GNSS data for the identication of the
carriageway and along position of a road vehicle. Hensel et
al. [] showed a particle lter approach for railways based on
just an eddy current sensor and a map.
Onlyfewapproachesevaluatedatasetswithstatistics
about track selectivity: Lauer and Stein [] used GNSS
and a velocity sensor and showed a gain in track-selective
accuracy and condence between a simple map-match and a
proposed estimation algorithm. Hensel et al. [] showed no
direct gures on switch way resolution but improved switch
detection (.%) and classication (.%) with an eddy
current sensor of  switches. is study focused on switches
as position input and classications on merging and splitting
switch runs. B¨
ohringer[]evaluatedanintegratednavigation
system (GNSS, IMU) in combination with an eddy current
sensor for switch way identication. Even with a moderate
switch detection rate of %, the results received .% of
track-selective accuracy. ese results are considered as most
suitable for a comparison and are based on real train runs of
 km with  switches.
3. Map-Based Railway Navigation
3.1. Topological Coordinates. e goal of train localization
is to estimate the train position in the track network by
topological coordinates as well as the train speed V. A unique
and discrete track ID () identies the track and the track
length variable is the one-dimensional position on that
track. Each track has an origin and a direction indicates
if a train is oriented with or against the track denition. e
topological coordinates are
topo ={,,}.()
Tracks are connected by switches, crossings, or diamond
switch crossings. A track is dened between connec-
tions with a unique ID; that is, it contains no switch or

Journal of Sensors 
T : Train directions.
Train-track frame
direction
 Tra in vel o city
direction
=sign (V)Train-track frame
motion direction
sign ()
+Forward (+) +
−Forward (+) −
+Backward (−)−
−Backward (−)+
+Stop (0)0
−Stop (0)0
crossing. is denition ensures that a track is always
one-dimensional and limited by the two endings of track
beginning and track end.
3.2. Coordinate Frames. e sensors measure in their specic
sensor frame. For further processing, these measurements
are converted in the train frame according to the mounting
parameters. e map and especially the geometry of the
tracks are expressed in the track frame.Anysamplepointofa
track contains a geographic position (WGS) and the track
attitude angles are dened from a local frame in north, east,
anddown(NED).Onceanglesorcurvaturesareusedinthe
map, there is an ambiguity about the direction at which the
track ending the origin is dened. erefore, it is necessary
to dene a start point and consequently a pointing direction
of the track. A train (train frame) can be placed in two
orientations on a track and can move forward or backwards.
Alternatively to V, the train speed can be expressed between
train and track frame by .eabsolutevelocitiesofVand 
arethesame.esignof indicates either an increasing (+)
or decreasing (−)changeofthepositionof the current track
and depends on the track denitions.
roughout this approach, the map information of a
specic location is converted and processed in the train
frame. It should be noted that there is an alternative way to
convert all variables to the track frame.Tableshowsthe
conversions of the train direction variables: train-to-track
frame direction (), the motion direction of the train (),
and the direction of motion between train and track frame by
the sign of . e motion direction depends on a forward
or backward velocity Vregarding the train frame denition of
front and rear. It is possible to compute any of these directions
from the two others by Table . Direction keeps its value
during standstill. e motion direction remains the same
during a train run, while the sign of and can alternate
aer a change of tracks during a train run.
3.3. Along Track and Cross Track. Asuitableanalysisfor
topologicallocalizationperformanceistheapproachbyalong
and cross track. Along track addresses the continuous D
localization on a track. Cross track focuses on discrete,
dierent tracks and track selectivity is the ability of a correct
cross track localization. Sensors can contribute to along and
cross localization with relative or absolute measurements
of displacement or track features. As shown in Figure ,
Cross at switch Absolute Absolute
along (s)cross (id)
Realtive
along (Δs)
(idleft,id
right)
F : Along and cross track denitions.
measurements can contribute to train localization in four
dierent ways:
(i) relative along (): odometry,
(ii) absolute along (): diverse along-track features,
(iii) cross at switch (): competing switch way track
features,
(iv) absolute cross (): diverse cross/parallel track fea-
tures.
Odometry is the processing of relative along measurements,
such as wheel turns, speed, and train acceleration. Depending
on track features, parallel tracks show oen very similar along
and cross track features. Measurements may contribute to
absolute along or cross in a local vicinity or even globally.
3.4. Railway Track Features. A suitable track feature is an
unchanging property of the track which can be measured
by a sensor. Over dierent locations, there are unique track
features as well as repeated, ambiguous features possible.
Here, the features are the track geometry by geographic
position (latitude ,longitude), track attitude (i.e., heading
),andcurvature.eheading()changes over the run
of the track, which is represented by the heading curvature
𝜓=()/. Additional features can be extended, provided
that there is a reproducible signal over dierent locations and
sensors can measure these features.
3.5. Railway Switch. e switch way identication is a critical
process in railway navigation, especially if the tracks are
parallel aer the train passes a splitting switch. As a special
property of a switch, the two tracks of the competing switch
ways dier in geometric characteristics of curvatures 𝜓,
headings , and geopositions (latitude ,longitude). e
geopositions of le and right switch way are located apart by
the cross track distance CT.eswitchwaytrackpositions
and headings increase slowly from the switch start, while
the curvature is already present from the switch start. ere
can be many more switch way features, as shown in other
approaches based on dierent sensors [].
3.6. Track Map. e railway map works as a coordinate trans-
formation between topological coordinates and track fea-
tures (e.g., geometric coordinates). e map model contains
and connects information on topology and track features.
ese track features are parametrized by the D-position 
andstoredindiscretepoints(trackpoints).Acontinuous
representation of track features is achieved by interpolations

Journal of Sensors
between these points. e track features can be obtained from
the map by the topological pose and converted to train frame:
map (,,)=

,

geo position,

attitude,𝜓

curvature

train .()
According to the train direction ,thesignofthecurvature
𝜓changes, while changes by ∘.isspecialmapcanbe
constructed from train-side sensor data [, ] or extracted
from an existing geodatabase.
An open street map (OSM) [] geodatabase is used as
datasourceforthetrackmap.emainadvantagesofOSM
are the availability and the completeness of data points of
many tracks of the desired railway track network. For an
adequate map, there is some additional preparation by track
separation and topological connection (e.g., at switches) as
well as geometry processing being necessary. An OSM data
contains geodata points (,)and the track geometry of
heading ()andheadingcurvature(𝜓) is derived from these
positions. e data is usually collected from various sources,
such as a GNSS hand-held or extracted from aerial or satellite
photo with dierent and undened accuracy. erefore, an
OSM data based map should not be considered as highly
accurate. An analysis of many train runs showed lateral
deviations up to m between an averaged GNSS trace and
the OSM track.
3.7. Train State Estimation. e estimation state of railway
localization is formulated with the following random vari-
ables:
(i) track ID: (discrete),
(ii) position: (continuous,only within a track),
(iii) train direction on track: (binary),
(iv) train speed: V(continuous),
(v) train acceleration: (continuous),
(vi) correlated sensor properties: biases: (continuous).
e estimation state vector or train state est
𝑘for one discrete
time step is dened by
est
𝑘=

,,

topological pose,V,

train motion

𝑘.()
e train-to-track frame speed can be computed by Table :
𝑘=

V𝑘,if =+,
−V𝑘,if =−. ()
e bias vector 𝑘contains correlated sensor errors. ese
biases change over time due to a random dri, which cannot
be calibrated in advance. Additionally to the state variables,
an auxiliary variable is used to represent the vehicle motion
by ∈{forward,stop,backward}. e goal for a train
localization algorithm is to estimate and resolve est
𝑘and 𝑘.
e strong track-train constraint allows to predict the
train position, attitude, and inertial state from the known
track geometry of the map. ese extended train states are
computed from the map by the actual topological position
estimate and contains the geometry in train frame (and 𝜓
transformation according to ):
ext
𝑘=,,,𝜓train
𝑘.()
3.8. Train Control. e train control consists of cross track
control and the along-track control by the train driver:
=





sw

cross control,
control center ,acc,𝑚

along control,
train driver






.()
e cross control inuences the travel path of the train
by the selected switch way (sw :{left,right}). is is
usually controlled by a train control center or sometimes by
the train driver at shunting yards or industrial tracks. e
train driver controls the general train motion 𝑚and the
acceleration acc by the traction and brake lever. e general
train motion is the travel direction selector as well as a train
stop (e.g., activated parking brakes). e control center has
also inuence on the along-track control via signaling.
3.9. Simple Map-Matching. In contrast to state estimation a
reference approach by simple map-matching is described.
e simple map-matching is a snapshot based, nearest neigh-
bor method, which uses no information about the prior
position. e nearest position on track is computed with the
map from a GNSS position measurement:
{,}

topo position =map-match ,

geo position.()
It should be noted that this approach would be sucient if
the position measurement (e.g., ideal GNSS) is continuously
available with an accuracy always better than half of the
distance of parallel tracks. It will be shown in the results
that this approach has its problems with real GNSS position
measurements.
4. Train-Side Sensor Measurements
is paper focuses only on GNSS and IMU measurements,
but other train-born sensors can extend the proposed
approach. e used sensors are considered as inexpensive and
their combination as complementary in terms of measure-
ment errors.
4.1. Global Navigation Satellite Systems. e approach uses
the standard GNSS receiver output by position-velocity-time
(PVT) and is considered as loosely coupled. at means
that the internal GNSS related computations can be kept
transparent, up-to-date with actual receiver technology and
out of the train localization approach. e used receiver was

Journal of Sensors 
a u-blox  GPS receiver, but also other commercial o-the-
shelf receivers could be used.
A drawback of GNSS is the lack of availability and
degraded accuracy in parts of the railway environment. GNSS
data is not available in tunnels or below station roofs. e
accuracy is further aected by multipath, signal loss, and
poor geometry in urban environments, next to acoustic noise
barriers or in dense forests.
4.1.1. GNSS Speed. e GNSS speed measurement con-
tributes to the train odometry in terms of relative along-track
estimation and is part of the PVT data. A single antenna
mounting is invariant of a horizontal rotation; that is, the
speed and also heading measurements are independent of the
mounted yaw angle between antenna and train. In principle,
the GNSS speed is computed from the vector norm of the
antenna motion in north and east component. erefore,
theGNSSspeedisalwayspositiveandthetrainmotion
() must be considered in the use-case for train speed. e
measurement model contains the train speed , additional
white noise , and a conversion for the train motion:
GNSS,V
𝑘=

−V𝑘+V
𝑘,if =backwards,
V𝑘+V
𝑘,if =forwards.()
4.1.2. GNSS Positions. GNSS positions are a favorable mea-
surement for absolute along and cross resolution of the train
location. e GNSS position model includes D positions of
latitude and longitude and additional white noise pos :
GNSS,pos
𝑘=𝑘+pos
𝑘.()
4.1.3. GNSS Heading. e GNSS heading angle contributes
to the switch resolution which was described as cross at
switch contribution in Section .. ere are multiple ways
of the internal heading computation of the receiver: a simple
dierentiation of two consecutive positions or by computa-
tion of a receiver motion vector from Doppler and/or phase
measurements of each satellite. e actual internal method
is unknown, but state-of-the-art methods use positions and
Doppler within a Kalman lter []. e used receiver
outputs an ECEF (earth centered, earth xed) velocity vector,
and the GNSS heading measurement is the angle to north of
the horizontal part of this vector in east and north direction.
e receiver estimates further an accuracy of the velocity
vector (ECEF) better than . km/h in % of all runs and a
resulting heading accuracy of .∘in % of all runs above
 km/h. e GNSS heading is worse at low speeds, so the
heading measurements are used above  km/h. e GPS data
of the experiment and the results showed a good heading
repeatability with usually less than ∘of dierent runs over
the same positions. In comparison, a heading angle between
two consecutive positions showed much worse results.
e heading model contains the heading angle of the
train(trainframe)whichisderivedfromtheestimateofthe
topological pose and the map. A GNSS heading measurement
with single antenna is the heading of the antenna motion and
requires a conversion to train frame by the train motion 
(see also Table ):
GNSS,𝜓
𝑘=

𝑘++𝜓
𝑘,if V<0(=backwards),
𝑘+𝜓
𝑘,if forwards.()
It should be noted that there are no heading measurements
for very low speeds or stopped trains.
4.2. Inertial Sensor. e IMU measurements provide contin-
uous and interference-free data. Two measurements are of
particular interest: the longitudinal acceleration (𝑥)andthe
yaw turn rate (𝑧). e measurements are aligned with the
train frame by prior calibration.
4.2.1. Longitudinal Acceleration. e longitudinal accelera-
tion (𝑥) measures the train acceleration measurement and
it contributes to the relative along localization (odometry).
e along acceleration measurement model considers train
acceleration ,thegravityportion by the slope angle of
the track, and accelerometer bias 𝑎𝑥:
IMU,𝑎𝑥
𝑘=𝑘+⋅sin 𝑘+𝑎𝑥
𝑘

̃
𝑏𝑘+𝑎𝑥
𝑘.()
Inertial sensors are aected by a bias which is changing
overtime,calleddri.Anybiascausesgrowingerrors,as
the train acceleration is integrated over time to speed and
position. e presented approach does not contain a slope
estimation or slope prole in the map. e slope angles of
railway tracks are relatively small and the bias estimation in
the following handles the dierence. A continuous estimation
of the combined oset 𝑘is processed and contains bias and
gravity portion from slope. Extensions with slope estimation
by integration or a slope prole in the map are possible.
4.2.2. Yaw Turn Rate. Rails are a strong constraint between
track geometry and train trajectory. It is possible to measure
the geometric characteristics with a train-side IMU of a
moving train. A certain curvature 𝜓of a track causes yaw
turn rates depending on the train speed:
𝜓=V.()
A complete model of train kinematics for turn rates and
centripetal accelerations is presented in []. Trains are
exposed to low pitch and roll angles in general and especially
at switches. As a consequence, the horizontal heading turn
rate canbeapproximatedbytheyawrateofthetrain:
≈𝑧.()
e curvature as measurable track feature indicates absolute
along locations and resolves dierent switch ways (cross at

Journal of Sensors
switch). is approach is less sensitive to dri as it does
not rely on integration of the inertial measurements. is
property is advantageous for the use of low-cost MEMS
gyroscopes. e measurement model 𝜔of the yaw rate
measurement 𝑧is dened by
IMU,𝜔𝑧
𝑘=𝜓
𝑘⋅V𝑘+𝑔𝑧
𝑘

ℎ𝜔𝑧(𝑇𝑘,𝐵𝑘)+𝜔𝑧
𝑘.()
At this point, the frame and motion denitions are vital
(Table ). e curvature is translated to train frame by train
direction and the speed contains the motion direction 
in its sign. e bias of the gyroscope (𝑔𝑧)iscalibratedonly
during stop phases of the train:
𝑔𝑧
𝑘≈𝑧,if =stop.()
e turn rates are assumed to be zero and the low earth turn
rate is neglected. e resulting small error is not integrated in
the following (e.g., as in strapdown approaches) and results
in a negligible error at the weighting process by a likelihood.
5. Probabilistic Train Localization
e following denitions can be used for a multihypothesis
lter, a particle lter or Rao-Blackwellized particle lter. e
posterior is already specied for GNSS and IMU sensors.
5.1. Train Localization Posterior. e train localization pos-
terior represents the estimation problem. In previous works
[,,],Bayesianmethodsarepresentedwithadynamic
Bayesian network denition for the train localization prob-
lem and the factorization of the posterior is shown in steps.
e posterior of all train states (0:𝑘)andsensorbiases(0:𝑘)
over time steps are estimated given all measurements (0:𝑘),
train control inputs (0:𝑘), and the map (). e map is actu-
ally known in advance and does not change over time for the
train localization problem. It is included in the conditional
part of the distributions to indicate where information of the
map is needed. e unknown train control is also included
to indicate where a train driver or train control can inuence
the train states. e posterior is factorized in a recursive form
in order to compute the posterior practically with estimation
algorithms:
0:𝑘,0:𝑘 |1:𝑘,0:𝑘
∝GNSS
𝑘|𝑘⋅IMU
𝑘|𝑘,𝑘

measurements
⋅𝑘|𝑘−1,𝑘,

train transition ⋅𝑘|𝑘−1

bias transition
⋅0:𝑘−1,0:𝑘−1 |1:𝑘−1,1:𝑘−1,

recursion .()
e factorized posterior is proportional (∝)totheposterior,
because of a missing normalization factor. is normalization
canbecomputedseparately,asallprobabilitiessumuptoone.
ere are dierent considerable lter approaches, which
are able to estimate the nonlinear train transition and the
dierent hypotheses of the topological pose.
5.2. Along Track. One extension to the previous particle
lter approaches [, ] is the separation of states, where
certain states are estimated dierently. In combination with
particle lters, this process is called Rao-Blackwellization
[]. In particular the linear states can be separated from
the nonlinear states of a particle hypotheses and estimated
byamoreoptimallter,astheKalmanlter.etrainstate
contains linear and nonlinear parts: 𝑘={𝑙
𝑘,𝑛
𝑘}. e train
transition of  is split in a linear and nonlinear part via chain
rule and conditional independencies that are removed:
𝑘|𝑘−1,𝑘,=𝑛
𝑘,𝑙
𝑘|𝑛
𝑘−1,𝑙
𝑘−1,𝑘,
=𝑛
𝑘|𝑙
𝑘,𝑛
𝑘−1,𝑘,

nonlinear train transition on tracks
⋅𝑙
𝑘|𝑙
𝑘−1,𝑘

linear train transition .()
5.2.1. Train Odometry Filter. e linear and one-dimensional
train transition is separated from the nonlinear transition on
themapandestimatedwithaKalmanlter.eestimation
of the linear state variables of acceleration , speed V,and
displacement will be called odometry in the following.
e odometry combines also the along acceleration bias
estimation with updates from acceleration and speed mea-
surements:
𝑙
𝑘|𝑙
𝑘−1,𝑘⋅ax
𝑘|ax
𝑘−1

odometry prediction
⋅GNSS,V
𝑘|𝑙
𝑘⋅IMU,ax
𝑘|𝑙
𝑘,ax
𝑘

odometry update .()
e state transition of the odometry prediction step is dened
here as a D transition DWPA model (discrete white noise
constant acceleration) []. e linear D train transition and
the acceleration bias are estimated by the discrete model:
V
𝑘=
02
20
010
0010
0001



system matrix
V
𝑘−1 .()
is model is propagated by state-of-the-art Kalman lter and
updated with GNSS speed and longitudinal IMU acceleration
measurements.
5.2.2. Track Transition Model. e track transition ensures
that the estimates (hypotheses, particles) exist and stay
exclusively on tracks. e nonlinear train transition of ()

Journal of Sensors 
estimates a topological coordinate from the linear displace-
ment, the previous coordinate, an unknown switch way, and
the map: 𝑛
𝑘|𝑙
𝑘,𝑛
𝑘−1,𝑘,
=topo
𝑘|topo
𝑘−1 ,𝑘,sw
𝑘,. ()
e function of the map for each hypothesis (particle) is
topo
𝑘=map,trans.topo
𝑘−1 ,𝑘. ()
is model considers the discontinuity at a track change and
in case of a splitting switch scenario the next track is sampled
from a discrete uniform distribution (le or right). According
to the motion state of standstill (=STOP), this transition
can be suspended. e next step is a track map query by
the topological pose for each hypotheses or particle. e
extended train state ext
𝑘contains the track geometry in train
frame: ext
𝑘=map,data topo
𝑘. ()
5.3. Cross Track. e cross track estimation evaluates dier-
ent tracks at a switch or in scenarios with multiple tracks.
e evaluation or weighting process is based on a so
comparison of measurements and expected measurements
from the map. A sensor likelihood function is dened for
each measurement type. First, the expected or estimated
measurement is computed from a measurement model and
the current state of train state and sensor correlation 𝑘=
(𝑘,𝑘). A generic likelihood function model is dened
here with a Gaussian distribution. e mean is the expected
measurement 𝑘,theargumentis𝑘,andthecovarianceof
the sensor noise is Σ:
𝑘|𝑘,𝑘
=|2Σ|−1/2 exp −12𝑘−𝑘Σ−1 𝑘−𝑘𝑇. ()
6. Particle Filter Approach
A particle lter is chosen for the posterior () estimation.
e particle lter can handle dierent nonlinear estimates
(hypotheses) automatically by the particles, which is neces-
sary for a distribution of any possible position over dierent
tracks. As described in [], a particle lter represents prob-
ability density functions by appropriate particle distributions
with appropriate weights of 𝑝particles. e posterior of ()
is represented by the particles set:
{,}0:𝑘 |1:𝑘,0:𝑘,≈𝑖
0:𝑘,𝑖𝑁𝑝
𝑖=1 .()
𝑖
0:𝑘 is the th particle with its weight 𝑖of 𝑝particles and
represents one sample of the posterior of all time steps until
. Particles are generated from a function which is easy to
calculate [], called the proposal function:
𝑖
0:𝑘 ∼{,}0:𝑘 |1:𝑘,0:𝑘,. ()
Aerward these particles are weighted []. e weights are
proportional to the fraction posterior over proposal function:
𝑘∝{,}0:𝑘 |1:𝑘,0:𝑘,
{,}0:𝑘 |1:𝑘,0:𝑘,

𝑤,unnormalized weights .()
e weights of the particle lter sum up to one and the
normalization factor canbeeasilycomputed:
= 1
∑𝑁𝑝
𝑖=1 𝑖
𝑘.()
Aer several time steps, some particles may carry an extreme
high weight while the rest has a very low weight. ese low
weighted particles are inecient and this process is called
degeneration. In order to avoid this, a systematic resampling
[] of the particle distribution can solve this problem.
A metric for particle depletion is the eective number of
particles e [].eparticledistributionisresampledif
e is below a threshold th.
6.1. Particle Filter with GNSS and IMU Measurements
6.1.1. Proposal Function. e proposal function is designed
by the transitions of train and correlated sensor properties
as well as suitable measurements. e proposal contains the
train odometry estimation of Section .. and a gyroscope
bias estimation. In this implementation, the gyroscope bias is
onlyupdatedifthetrainisnotmoving.edisplacementof
the th particle is sampled from the odometry Kalman lter
output: 𝑖
𝑘∼N𝑖
KF,2
Δ𝑠. ()
In contrast to previous approaches [, ], the samples were
directly generated from an acceleration distribution and
amotionmodel.euseoftheodometryKalmanlter
inside of a particle lter is the Rao-Blackwellization part.
e nonlinear map transition of () and map query () is
processed for each particle as a function of the map.
6.1.2. Weight Function. e weight function is the combina-
tion of IMU and GPS likelihoods and computed as shown in
() with the appropriate measurement models (), (), and
(). e weight function for the th particle with the IMU
and GNSS likelihoods is
𝑖
𝑘=⋅𝑖
𝑘−1 ⋅GNSS,pos
𝑘|𝑖
𝑘⋅GNSS,𝜓
𝑘|𝑖
𝑘
⋅IMU,𝜔𝑧
𝑘|𝑖
𝑘,𝑖
𝑘. ()
6.1.3. Initialization. e initial particle distribution is gener-
ated from the rst GNSS position measurement. erefore,
the D geoposition is sampled 𝑝times with a large covari-
ance Σ(e.g.,  m) from a Gaussian distribution:

pos𝑖∼NposGNSS
𝑘=1 ,Σ. ()
e positions are map-matched by () to topological posi-
tions and assigned to each particle . e train frame direction
is sampled randomly from a uniform distribution.

Journal of Sensors
6.2. Output Estimate. A particle distribution is a less useful
output for applications like automated train control or colli-
sion avoidance. ere, a single mode or most likely output
is desired. Internally, the particle lter keeps its particle
distribution for the next update. e output is computed by
four steps: rst, the track paths are identied and, second, the
most likely track path with particles is chosen (ML path). A
mean square estimate is computed from particles on that path
and the result is translated back to topological coordinates.
6.2.1. Track Path Identication. At rst, all path hypotheses
𝑝with at least one particle are identied:
0:𝑗
𝑝=ndPaths 𝑖𝑁𝑝
𝑖=1. ()
A track path contains one or more sequential tracks ()on
a D path, where a train is able to run over in a sequence.
For further computations, a track path has a continuous D
coordinate frame compared to discontinuities at the joints
of tracks. A topological pose can be translated into path
coordinatesaswellastranslatedfrompathcoordinates.As
an example, in case of a split switch scenario, particles can
be distributed before the switch, on the le and right switch
way. is would result in two possible track paths: 1
𝑝=
{before,left }and 2
𝑝={before,right}.
6.2.2. ML Path. As a next step, the sum of weights are
calculated for each path :
𝑗=𝑁𝑝

𝑖=1𝑖⋅
𝑖=∈𝑗
𝑝

selects weight of 𝑖th particle
from the 𝑗th path 𝑖𝑑𝑝.()
e delta function (𝑖=∈𝑗
𝑝)equals one if the th
particle (respective its track ID )isontheth track path
and is zero otherwise. e most likely path ML
𝑝is the path
index with the highest cumulative weight ML:
=arg max 𝑗𝑗. ()
6.2.3. Mean Square Estimate on Path. In the following, a delta
function selects the th particle which is on the most likely
path: 𝑖=𝑖=∈ML
𝑝. ()
e topological coordinates of the selected particles are
translated to the most likely path:
topo,𝑖 :,,
=topo2path topo,𝑖,𝑝. ()
e D position is calculated by a weighted mean of the
selected particles, which belong to the ML path:
= 1
ML
𝑁𝑝

𝑖=1 𝑖⋅𝑖⋅𝑖. ()
e D position deviation (along-track precision) is calcu-
lated by the weighted sample variance of particles from ML
𝑝:
𝑠=1
ML
𝑁𝑝

𝑖=1 𝑖−2⋅𝑖⋅𝑖. ()
e train direction on track is computed by the highest
weight, where is either positive or negative track frame
direction of the path:
ML,+ =1
ML
𝑁𝑝

𝑖=1 
𝑖=+⋅𝑖⋅𝑖,
ML,− =1
ML
𝑁𝑝

𝑖=1 
𝑖=−⋅𝑖⋅𝑖. ()
e most likely direction is the one with the higher weight.
eprocedurefortrainmotiondirectionis analog.
6.2.4. Translation to Topological Pose. Finally, the most likely
path ML
𝑝,theweightedmeanpositionon path, and
the train to path frame direction 
ML are translated into
topological coordinate frame:
topo :{,,}=path2topo ML
𝑝,,
ML. ()
6.3. Algorithm Summary. e algorithm of the sequential
Bayesian lter with a Rao-Blackwellized lter realization
(RBPF)isshowninAlgorithmandsummarizedinwords:
every new measurement (IMU or GNSS) triggers the lter
to compute a next time step. Particles (hypotheses) estimate
a topological position on railway tracks and these particles
exist only on tracks. Each particle is shied along the track
by a displacement output from the odometry lter (Kalman
lter), which is updated with GNSS speed and longitudinal
acceleration data (IMU). A special function of the map
processes this shi for topological coordinates. A railway
switch is handled by a random assignment of the particles
toeachway.Fromthemap,eachparticleisassignedwith
proposed geometric values in train frame of a geoposition,
heading angle, and an instantaneous turn rate from curvature
andspeed.elikelihoodsweightheparticleswithaproba-
bility according to the dierence of the proposed geometric
values and the measurements (train frame) of GNSS position,
GNSS motion vector heading, and IMU yaw rate. An output
estimate extracts one train location with variances from the
particles and resampling removes unlikely particle estimates.
e design parameters are sensor (co)variances, process
noise of train acceleration, and combined bias as well as
number of particles, resample threshold, and variance of
displacement sampling.
6.4. Particle Filter Challenges. e challenges of this map-
based localization approach by a particle lter are the follow-
ing.

Journal of Sensors 
Algorithm: Train Localization (RBPF)
Input: GNSS and IMU sensor data
Output: topological coord. (,,) and train speed
(1)loadmap
(2) initialize odometry Kalman lters with zero vector
(3) initialize all 𝑝particles by rst GNSS position ()
(4)loop
(5)if new measurement(s) available then
(6)timestep:=+1,=𝑘−𝑘−1
(7)for all 𝑝particles do
(8) predict odometry KF ()
(9) update KF with speed ()/acceleration ()
(10)if train is moving then
(11) sample displacement from odometry ()
(12) compute map transition ()
(13) get geometry from map (train frame) ()
(14) compute likelihoods ()/()/()
(15) multiply particle weight by likelihoods ()
(16)else (train is stopped)
(17) observe and lter gyroscope bias
(18)end if
(19)end for
(20) normalize weights ()
(21) compute most likely output estimate ()–()
(22)if resampling necessary by e then
(23) perform resampling
(24)end if
(25)end if
(26)end loop
A : Algorithm of the map-based train localization with GNSS, IMU, and Rao-Blackwellized particle lter.
6.4.1. Divergence in Along Track. e problem is an unstable
lter, as the estimate (particles) is away from the truth and
cannot recover. is can be approached by a good model
design (i.e., proposal function), a continued resampling, and
the insertion of sampling noise. In case it fails, a lter
monitoring can detect a severe along divergence (e.g., by
GNSS measurements) and restart the lter.
6.4.2. Divergence in Cross Track. e problem here is a failed
track selectivity, as all particles are on the wrong track, in
particular a parallel track, and cannot recover. e resolution
of the switch way is very important. is requires a sucient
map model as well as sensors, which are able to measure the
competing switch way properties. Another approach is the
use of extrinsic sensors which directly measure the switch
way (cross at switch) or observe a neighboring track (absolute
cross).
6.4.3. Initialization. In the start-up phase of the estimation
lter, some hidden and discrete states remain unresolved if
they are not directly observable by measurements. In the
proposed approach, this happens for the train frame direction
and the track in parallel track scenarios. ere, the
lter requires motion or motion over a switch to resolve
these states. An alternative way is the use of extrinsic sensors
in order to observe or resolve the hidden variables from
standstill.
6.4.4. Overcondence. is happens especially if the mea-
surement noise is too small and correlations are disregarded
in the sensor model. e lter converges very quickly to the
measurements and results in a too small particle distribution
aer resampling. is can lead to the described divergence in
along and cross track.
6.4.5. Degeneracy. Degeneracy of the particle distribution is
the eect where nearly all weight is accumulated on one or
a few particles. e state-of-the-art approach is systematic
resampling [].
6.4.6. Dimensionality and Computational Complexity. High
dimensional state vectors can be problematic for particle l-
ters, as the number of particles and computational complexity
grows []. is approach uses a state vector () with the
nonlinear random variables of discrete track IDs ,which
islimitedtoafewtracksinthevicinity,anoncontinuousD
position , and a binary direction .Once,thedirection
is resolved aer initialization, the direction is processed
in a deterministic manner and not estimated anymore. In
other words, the particle lter estimates actually two states.

 Journal of Sensors
Clearance point
Switch start
Tolerance area
True run
Fixed tolerance threshold or distance
False track
to clearance point
F : Cross track analysis at a switch with tolerance and error
areas for a true right (straight) run.
e train acceleration, speed, and certain biases are processed
by a linear lter in order to achieve more optimal estimation
andtoreducethenumberofparticles.
7. Track-Selective Evaluation Framework
e empirical proof of track selectivity is achieved by the
comparison of localization output and a reference route. is
reference must be known in advance and is the true sequence
of traveled track IDs. Of special interest is the switch scenario
with a splitting switch way run, as shown in Figure . Railway
switches have a region where the clearance of two vehicles
overlap, and only one train can occupy these tracks. ere,
a false track estimate is tolerated and not a real problem
as only one train can occupy the tracks. e length of this
tolerance region can either be xed or individual for every
switch, stored within the map. For simplicity reasons, a xed
tolerance of  m aer a switch start is chosen. e track-
selective accuracy is evaluated with the known route (ground
truth) and a false track estimate within the tolerance region
is marked in orange, a correct estimate is green, and a false
one is red. Track precision is dened here as the discrete
probability of the track estimate from the lter output. A
high precision estimate can be evaluated with an incorrect
accuracy, when the true track is dierent.
e track-selective accuracy can be analyzed over time
(per second) or over traveled distance (per meter). Train
statistics are oen related to distance (e.g., millions of train
kilometers), so the results are presented in relation to the
traveled distance. e method of Algorithm  evaluates the
train localization estimate of each time step.
A cumulative evaluation shows the performance of the
localization approach in terms of track selectivity for larger
data sets. Each evaluation result is shown relative to the total
distance: cum
error =∑error
𝑘
∑𝑘⋅100%,
cum
switch =∑switch
𝑘
∑𝑘⋅100%,
cum
OK =100%−switch −error.()
ismethodisbasedondistanceswhichautomaticallyrejects
the evaluation of stopped and parked trains. Track-selective
errors occur in the presence of parallel tracks. A train run
on a route with more single track scenarios will distort an
Algorithm: Track Selective Evaluation
Input: Train state 0:𝑘,truetrackIDs,map
Output: Evaluation: cumulative cum,cum,𝑝
(1): all track IDs of wrong switch ways from ,map
(2): switch positions from ,map
(3)clear.: clearance length of each switch
(4)for all train states 0:𝑘 do
(5)switch
𝑘=error
𝑘=𝑘=0
(6)if track is not in (true track ID list) then
(7)if is in &distancetoswitch<clear.then
(8)switch
𝑘=𝑘
(9)else
(10)error
𝑘=𝑘
(11)end if
(12)end if
(13)if other track in the vicinity of 𝑘,𝑘( m) then
(14)𝑘=𝑘
(15)end if
(16)end for
(17) compute cumulative evaluation cum ()
(18) compute cum. eval. of parallel tracks cum,𝑝 ()
(19)return cum,cum,𝑝
A : Track-selective evaluation over distances.
evaluation in favor of a better track-selective evaluation.
An increase of comparability of the evaluation result is
considered with a ratio to distances with parallel tracks in
vicinity: cum,𝑝
error =∑error
𝑘
∑𝑘⋅100%,
cum,𝑝
switch =∑switch
𝑘
∑𝑘⋅100%,
cum,𝑝
OK =100%−𝑝
switch −𝑝
error.()
e switch tolerance evaluation suits mainly for detailed
evaluation on small changes and tuning.
A compact gure of the evaluation ()in terms of track
selectivityoverparalleltracksdistances(TS,)is the error-
free case of multiple track scenarios:
TS,𝑃 =100%−cum,𝑝
error .()
is gure explains how good a certain train localization
approach performs on a specic track network. is cumu-
lative evaluation is one way to measure the track selectivity
performance and contains to some extent the track layout of
parallel tracks and switch densities.
Another evaluation measure is the error events, which
counts and evaluates the transition to the wrong track.
One faulty switch resolution results in a cumulative error
dependent on the specic track length. A parallel track
scenario merges very oen by a switch aer a station and a
wrong output can be on the correct track again. e error
eventmethodcountstransitionstotheerrorcaseandrespects
more the error cause, which is a fault switch way. It diers

Journal of Sensors 
T : Train routes.
Run From station To station Forward, backward Split switches Time km
 ABG FDB B   min 
 FDB ABG F   min 
ABG ING B  h
ING ABG F  h
 ABG FDB B   min 
 FDB ABG F   min 
 ABG AIC B   min 
 AIC ABG F   min 
 ABG FDB B   min 
 FDB ABG F   min 
×F,  ×Bh
also in late switch way resolution, if the output is correct again
within the correct track which connects to the switch. From
these error events, the correct switch way evaluation (SW)
statistics can be computed of the total split switches total,the
late resolved switch ways aer the switch tolerance late and
the wrong resolved switch ways failed:
SW =total −late −failed
total ⋅100%.()
e track-selective evaluation in multiple tracks scenarios
TS and the switch way resolution evaluation SW will be used
as compact results.
8. Experiment
8.1. Recorded Data Set. e data set was recorded on the
regional train “Alstom Coradia Lint ” under regular pas-
senger service conditions. is train can travel up to  km/h
and has two drivers’ cabs for two-side operation. Table 
shows the train runs over  km with  splitting switches.
e train runs on . km of tracks with other tracks near or
in parallel, which are .% of all tracks.
e data set contains GNSS PVT data (position, velocity,
time) of GPS (Global Positioning System) from a u-blox
LEA T receiver. e IMU data (Xsens MTi) was recorded
withasamplerateofHzandtimestampedfromaGPS
synchronous clock. For the proposed algorithm, this IMU
datawaslow-passlteredanddownsampledtoHz.e
IMU was placed on the front bogie, the GNSS antenna below
a berglass roof above the bogie position. A special cam-
era (dash-cam) with GPS timestamped video was installed
behind the front windshield for the switch way evaluation.
8.2. Labeled Reference Route. e reference for the cross track
analysis is a recorded video from the train run. In that video,
the motion state can be seen, the switch way and direction
oftravel.Foreveryrun,areferencetravelpath(i.e.,labeled
data) can be computed from a start position (GNSS), train
direction, the map, and a series of true splitting switch ways.
ese switch ways are either “le” or “right” and obtained
manually from the video.
0 100 200 300 400 500 600
Split le
Merge
Split right
Time (s)
Time (s)
0 100 200 300 400 500 600
0
20
40
60
80
100
Station (Augsburg)
Station stop
Station stop
Station (Friedberg)
Speed (km/h)
F : Run  over time from Augsburg main station to Friedberg
station with known switch ways.
8.3. Implementation. e localization algorithm approaches
were implemented within a self-written JAVA framework.
is includes the map processing, the sensor data reader,
andtheevaluation.esensordatawasprocessedina
causal way; that is, the localization approach processed each
measurement in the chronological order and the output
was evaluated. No simulated (i.e., generated) data was used.
Particle lters are generally computationally expensive by
nature. Nevertheless, the temporal performance for (𝑝=
100)particles with visualization was processed . times
faster than real time on a laptop (Intel iM CPU, .GHz,
Windows ). Hence, a real time operation is possible.
9. Results
9.1. Track Selectivity over Time. Two dierent evaluations
of the reference approach and the proposed algorithm are
shown of Run  over time. erefore, the train speed and
true switch ways of Run  are shown over time in Figure . It
visualizes the occurrence of splitting switch ways, since their
resolution is the challenge for a train localization lter.
Figure  shows the results of the simple map-matched
GNSS positions for Run  over time. e correct track is
marked with OK in green, an error in the tolerance area in

 Journal of Sensors
T : Detailed cumulative results.
Localization
OK Switch Error
Error events, switch way resolutions
% total distance: cum
(% parallel tracks: cum,𝑝)
(1) Map match GNSS position . (.) . (.) . (21.9)errors
(2) RBPF, GNSS position . (.) . (.) . (16.0)errors
(3) Method  and GNSS heading . (.) . (.) . (9.84)switches(late,fail)
(4) Method  and IMU yaw rate . (.) . (.) . (0.67)switches(late,fail)
(5) Method  and heading, yaw rate . (.) . (.) . (0.84)switches(late,fail)
0 100 200 300 400 500 600
OK
Switch
tolerance
Error
Time (s)
F : Track-selective accuracy of simple map-matching (nearest
neighbor).
0 100 200 300 400 500 600
OK
Switch
tolerance
Error
0 100 200 300 400 500 600 700
0
0.2
0.4
0.6
0.8
1
Track precision
0100 200 300 400 500 600 700
0
5
10
15
Time (s)
Time (s)
Time (s)
Along precision (m)
F : Track-selective accuracy and estimation precision of the
Bayesian lter approach: RBPF with GNSS position, heading, and
IMU yaw rate.
the vicinity of the switch is yellow, and a wrong track is red.
ere is no track precision shown, as this approach considers
no uncertainty but only the nearest track.
Figure  shows the accuracy and precision results over
time of the realized Bayesian lter with IMU and GNSS of
Run . At one splitting switch, the lter was estimating an
T : Compact track selective results.
Localization method
Tra c k s elec t i v ity
(parallel tracks)
TS,𝑃 (. km)
Switch way res.
( split switches)
SW
(1) Map match, GNSS pos. .% —
(2) RBPF, GNSS pos. .% .%
(3) (M. ) + GNSS head .% .%
(4) (M. 2) + IMU yaw rate 99.3%97.2%
(5) (M. ) + head + yaw rate .% .%
incorrect track within the tolerance region. e track preci-
sion (middle plot) is shortly reduced with the occurrence of
splitswitchesasseeninFigure.ealongprecision(bottom
plot) is initially coarse but quickly drops aer train departure
to an average of . m. is along precision is the weighed
empirical deviation of the particle distribution of ().
9.2. Cumulative Track Selectivity. A detailed cumulative eval-
uation is presented in Table . Five localization methods are
evaluated, in particular the simple map-matching and four
dierent Rao-Blackwellized particle lter implementations
(RBPF). e number of particles is 𝑝=100in all RBPF
evaluations. Each accuracy category is shown in percentage
of the total distance of  km. As all methods solve the single
track scenarios, the track accuracy is shown additionally
relative to the total distance of multiple track scenarios
(. km) in parenthesis. e evaluation results of a standing
train periods are disregarded in this table. e error events
in Table  indicate how oen a transition to wrong tracks
happened. Depending on the method, this can be traced back
to faulty switch resolution. A late switch error represents a
resolved switch way aer the evaluation threshold, whereas a
failed switch relates to a wrong resolved switch way.
e compact results according to () and () are
presented in Table . ere are no switch way resolution
results for Method  as the simple map-match considers only
the nearest track and does not resolve a switch way.
10. Discussion
10.1. Discussion on Results. One major goal of train local-
ization is a track-selective estimation result. As seen from

Journal of Sensors 
Table , Method  (reference approach of simple map-
matching) has severe problems to determine the correct
track at .% of parallel tracks. e proposed algorithm
RBPF uses also GNSS positions (Method ) but shows some
improvement with .% of wrong tracks on multiple track
scenarios for  times. e ltered methods, especially the last
three, converge to one track in a parallel track scenario. ere,
the switch resolution is essential and a failed switch forces the
estimate to stay on the wrong track until a merging switch
to the true track corrects the estimate again. For example,
Method  shows an improvement in error events but stays
three times on a wrong, parallel track.
e best results are achieved with Method  (RBPF with
yaw rate) on .% of parallel tracks which relates to an error
of 6.7⋅10−3 and stands for a false localization on  m (. s)
in total. With two late and one failed resolved switch way,
Method  achieves .% in switch way resolution, respective
of an error of 2.8⋅10−2.
e denition or adjustment on the switch way evaluation
thresholdhasdirectimpactontheresultsandwasm.In
case this threshold tends to  m, the track-selective results
over distance can be extracted from the “OK” column of
Table  and are slightly worse. However, the switch way
resolution results of () will severely decrease with a zero
toleranceareabymanylateresolvedswitches.
e results are quite sensitive to the parameters such
as measurement noise, the noise ratio of dierent measure-
ments, resampling occurrence, process, and sampling noise
as well as the map quality. Better or even perfect results
may be expected by exhaustive optimization of the map and
lter parameters. As an unwanted consequence, map and
parameters may match this limited data set and the results
loose generality. Nevertheless, the following tendencies can
be seen from the results: the additional use of the GNSS
heading (Method ) shows only little improvements with
.% in terms of errors compared to Method  (GNSS
position only). Further, the combination of all likelihoods
of Method  (GNSS position and heading, IMU yaw rate)
does not show an improvement. e most likely explanation
of this eect might be the map with its coarse heading and
curvature geometry. ese values are derived from positions
and are consequently dependent. Slightly wrong positions
cause an error in both values. Because of this dependency
the combination of the measurements has no further gain in
accuracy.
For the realized implementation of the sequential
Bayesian lter, the particle lter approach was chosen, as the
particles can sample the dierent hypotheses and any nonlin-
ear distribution. e Rao-Blackwellization marginalizes the
linear state variables and estimates them with a Kalman lter.
As a consequence, the particle lter samples only the cross
track hypotheses ()andthealongpositions(). e number
of particles are relatively low (𝑝=100)astheparticlesare
limitedandconstraintonthetracks.Additionally,thelinear
state variables are estimated by nested Kalman lters (Rao-
Blackwellization). A particle lter approximates distributions
and induces two times additional noise as a tradeo for
convergence reasons: the sampling adds noise, which is
needed to maintain particle diversity as well as the resampling
in order to avoid particle depletion and divergence.
10.2. Comparison. Lauer and Stein [] (GNSS, velocity
sensor) showed a gain in condence about the track decision
between a simple map-match and the proposed estimation
algorithm. A similar gain can be seen between simple map-
match (Method ) and RBPF with GNSS positions only
(Method ). B¨
ohringer [] received slightly better track-
selective results (.%) compared to Method  (.%)
with a dierent data set and an algorithm with additional
eddy current sensor as switch way detector. Hensel et al. []
do not consider explicit gures on switch way resolution.
From switch detection and classication rates over %,
a switch way resolution may be deduced with a similar
performance. In comparison, Method  reaches .% of
correct switch way resolution even without an eddy current
sensor and a coarse geometric map from OSM data.
However, a direct comparison between other approaches
is not obvious as dierent data sets and dierent evaluation
metrics are in use. erefore, comparative results can be seen
as quite similar as the dierences are marginal for dierent
data sets and metrics. Finally, from the literature and the
present results, it can be reasoned that most of the gain in
accuracy can be achieved by using an estimation lter as well
as using sensors which can measure the competing switch
ways. As this is quite expectable, further investigations are
necessary to identify the smaller gains and dierences of
varied lters or sensor fusion approaches on the same data
set.
11. Proposed Enhancements
Several directions of performance improvements of train
localization are identied.
11.1. Advances of the RBPF. e particle lter can be improved
with advanced procedures for better particle diversity on
along track. Secondly, an enhanced resampling timing may
be investigated, which suspends resampling near switches for
an undisturbed switch way resolution.
11.2. Improved Odometry. e along-track odometry can
be extended with slope estimation from IMU pitch rate
integration, map information about slope, or gravity vector
estimation from acceleration measurements. A slope con-
sideration would increase the accuracy of relative along-
track estimation and also increase the range to propagate
localization in GNSS denied areas.
11.3. Additional Sensors. A further increase in track-selective
accuracy,outagerobustness,andredundancycanbecon-
sidered with the use of extrinsic sensors, such as magnetic
sensors, cameras, LIDAR, or aperture radar with direct switch
way measurements.
11.4. More Accurate Map. An accurate curvature and heading
information in the map is a crucial factor for a correct switch

 Journal of Sensors
way resolution with GNSS and IMU. e major advantage of
generating a map from an OSM data base is to obtain a track
map of a certain railway network size with a sucient number
of tracks. A map generation approach from onboard sensor
data is presented in [, ].
Furthermore, a direct comparison of alternative methods
and sensors may be evaluated with same data sets and
evaluation metrics. Studies could investigate the accuracy
of dierent algorithms (e.g., multihypothesis lter versus
particle lter), or the accuracy gain of dierent sensor
data integration schemes (e.g., loosely coupled GNSS versus
tightly coupled).
12. Summary
is paper presents a probabilistic train localization approach
with a track-selective evaluation. In contrast to other
approaches, this train localization comprises a Rao-Blackwel-
lizedparticlelter(RBPF),amapoftherailwaytracksand
sensor data of a GNSS receiver (Global Navigation Satellite
System), and an IMU (inertial measurement unit). A novel
RBPF implementation is presented which estimates the train
localization posterior recursively. e Rao-Blackwellization
marginalizes the linear state variables and estimates them
with a Kalman lter. As a consequence, the particle lter
samplesonlythecrosstrackhypotheses()andthealong
positions (). e RBPF estimates directly the topological
track coordinates; that is, the particles stay on the tracks.
Further, a particle distribution can handle dierent track
hypotheses and other nonlinear distributions. e map con-
tains prior knowledge for the measurement models such as
the track geometry data. e RBPF is able to resolve the
unknown train-to-track orientation at initialization and can
handle forward and backward runs of the train.
Anovelevaluationmethodfortrackselectivityevalu-
ates the localization results with real data recorded from a
regional train. is generic evaluation method can be used
to generate more comparable results of dierent approaches
with dierent sensors and measurement data. Train runs were
analyzed over  km of tracks with  split switches and
parallel track scenarios of . km. Further improvements
for a safety-of-life train localization of the special realization
are discussed towards higher reliability of track selectiv-
ity.
e best combination of RBPF lter with GNSS positions
and IMU yaw rates showed a track-selective performance of
.% on tracks with multiple tracks in the vicinity and .%
of successfully resolved switch ways within the tolerance.
e realized RBPF approach with GNSS, IMU, and a track
map showed promising results towards a track-selective and
continuous train localization even with low-cost sensors and
runs in real time.
Competing Interests
e author declares that there are no competing interests.
Acknowledgments
e author wants to thank the railway transportation com-
pany BRB (“Bayerische Regiobahn”) for the support with
the measurements, Andreas Lehner (DLR) for the map,
Omar Garcia Crespillo (DLR) for initial soware implemen-
tations, Stephan Sand (DLR) for proofreading, and Christoph
G¨
unther (DLR) for comments.
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