1 INTRODUCTION

The demand for precise positioning, i.e. 10 cm

horizontal and vertical position accuracy, in inland

water applications has been increasing due to

requirements of driver assistance functions [1] and the

ongoing development of autonomous vehicles. Here,

Global Navigation Satellite Systems (GNSS) are

mainly used for the provision of Positioning,

Navigation and Timing (PNT) information. The two

classical algorithms to achieve cm-level accuracy

based on GNSS are Real-Time Kinematic (RTK) and

Precise Point Positioning (PPP) [2]. RTK uses the code

and phase observations of a reference station close to

the user's position to build double differences of the

observations between the receivers as well as the

satellites to mitigate atmospheric delays and errors on

the satellites' side such as their orbits and clocks.

PPP, on the other hand, only uses the observations

from one receiver but requires precise satellite orbit

and clock information as well as ways to mitigate the

atmospheric delay, e.g. applying atmospheric

products, estimating the delays or using the

ionospheric-free linear combination of the

observations. A more recent trend is PPP-RTK [3,4]

which is based on PPP but uses real-time corrections

based on a network of reference stations. Here, the

broadcast satellite orbits, satellite clocks, code and

phase biases as well as atmospheric delays are

corrected. This allows for fixing the ambiguities as

A PPP Baseline Approach for Bridge Passing

. Lass

German Aerospace Cent

er (DLR), Neustrelitz, Germany

ABSTRACT: Global Navigation Satellite Systems (GNSS) are increasingly used as the main source of

Positioning, Navigation and Timing (PNT) information for inland water navigation. In order to enable

automated driving and facilitate driver assistant functions, it becomes of crucial importance to ensure high

reliability and accuracy of the GNSS-based navigation solution, especially in challenging environments. One

challenging phase of inland waterway navigation is bridge passing which leads to non-line-of-sight (NLOS)

effects such as multipath and loss of tracking.

This work presents a Precise Point Positioning (PPP) based algorithm in a two-antenna system where one

antenna is at the bow and the other is at the stern. Additionally, gyroscope data from an IMU is used. In

contrast to a separated position calculation of the two antennas, only one antenna position is estimated and the

other is derived from the baseline between the antennas. This allows for accurate positioning even if one

antenna does not receive any GNSS measurements.

The presented scheme is evaluated using real measurement data from an inland water scenario with multiple

bridges. In comparison with a standard PPP scheme as well as an RTK algorithm, the presented approach

shows clear advantages in challenging scenarios.

http://www.transnav.eu

the

International Journal

on Marine Navigation

and Safety of Sea

Transportation

Volume 17

Number 1

March 2023

DOI: 10.12716/1001.17.01.

integers and achieving precise positioning without the

long convergence time of estimated float ambiguities.

However, inland water navigation poses severe

challenges [5] on the positioning in scenarios such as

passing a waterway lock or passing a bridge where

line-of-sight to the satellites cannot be guaranteed.

Several methods to handle these challenges have been

proposed such as robust filtering [6] to downweight

or exclude faulty observations affected by multipath,

the use of a multi-antenna setup [7,8,9] to have more

redundancy as well as being able to determine the

attitude of the vessel, and sensor fusion [10] using the

angular velocities of an Inertial Measurement Unit

(IMU). Nonetheless, most of these methods will fail,

i.e. cannot provide precise position continuously, if

there are no GNSS observations for several epochs.

Figure 1. Bridge passing scheme using baseline approach

We propose a method which combines the

approaches by using a multi-antenna setup and

sensor fusion with an IMU based on PPP as well as

applying moving baseline measurements in nominal

conditions from an RTK approach between the

antennas. The idea is to have two antennas on the

vessel whose fixed baseline is longer than the width of

the bridges and, ideally, orthogonal to the bridge

crossed as can be seen in Figure 1. Then there is

always at least one antenna which has line-of-sight to

the satellites and a precise position. By knowing the

pitch and yaw of the baseline with the help from

previous baseline measurements and the angular

velocities of an IMU, we can accurately determine the

position of the other antenna. Due to this, as soon as

the second antenna has line-of-sight again, we can

quickly estimate the new ambiguities which helps in

having a precise position even when the first antenna

loses track of the GNSS satellites. This enables having

precise position, velocity and heading information

during and after bridge passing.

We start by explaining the methodology of our

baseline approach based on PPP using the classical

ionospheric-free linear combination which we then

expand to two antennas using IMU as well as baseline

measurements. Afterwards we apply the method to a

real-world scenario and highlight the advantages in a

difficult scenario for inland waterways, i.e. bridge

passing, in comparison to the classical approach only

using one antenna. In the end, we sum up the paper

and show possible way forwards.

2 METHODS

We start with describing the classic PPP approach for

one antenna where the unknowns are estimated in an

Extended Kalman Filter (EKF) [11]. We apply the

standard notation for Kalman Filters, i.e. F is used for

the state-transition model, X for the Kalman state and

h denotes the measurement functions.

To get precise results, we need precise satellite

clock and orbits. For this, one can use postprocessed

final products from the International GNSS Service

(IGS) or the German Research Centre for Geosciences

(GFZ). For real-time applications one can use global

correction services such as the Galileo High Accuracy

Service (HAS) [12] for GPS and Galileo satellite orbit,

satellite clock, code and phase bias corrections, and

the IGS Real-time Service [13] providing satellite

clock, satellite orbit, code and phase bias corrections

for GPS, GLONASS, Galileo, BeiDou as well as a

global ionospheric model. Due to using the

ionospheric-free linear combination, there is no need

for local ionospheric corrections as required for PPP-

RTK.

Hence, we assume precise satellite clock and orbits

as well as the application of additional corrections for

the Earth's tides and the phase center variation of the

satellites.

2.1 Problem formulation for one antenna

In the following we consider two-frequency code

,1is

and phase observations

where the

frequency is denoted by i and the satellite by s

( )

,1 1 1 1 ,1 ,1

δδ ε

= − + − ++ +

is s s s is is

R x x ct t T I

( )

,1 1 1 1 ,1 , ,1 1 ,1

δδ λ

= − + − +− + + +

is s s s is is is s is

x x ct t T I A w



(1)

The variables are the antenna position

, satellite

position

, speed of light

, receiver clock offset

satellite clock offset

, tropospheric delay

ionospheric delay

, wave length

, phase

wind-up

, integer ambiguity

,1is

and the

remaining errors

,1is



, e.g. multipath and

receiver noise.

We then apply the classical ionospheric-free linear

combination

( )

⋅

to (1) and split the tropospheric

delay into the dry (

) and the wet (

) zenith delay.

The zenith delays are used in conjunction with vienna

mapping functions

[14] which depend on

the elevation of the satellite, the receiver position and

the time.

(

)

( ) ( )

⋅− ⋅

⋅=

−

( )

1 1 11

,1 ,

δδ ε

= − + − ++

s ss

IF s IF s

R x x ct t T

( )

( ) ( ) ( )

1 1 1 1 11

,1 , , ,

δδ λ λ

= − + − ++ + +

s ss s

IF s IF s IF s IF s

x x ct t T A w



1 ,1 ,1 ,1 ,1

= +

s h hs w ws

T Zm Zm

(2)

The Kalman state

consists of the antenna

position

, velocity

, clock offsets

, clock

drifts



, the float ambiguities

( )

and the wet

zenith delay

,1w

whereas the dry zenith delay is

approximated using the receiver's position. Assuming

constant velocity and constant clock drifts for state

transition and letting

being the number of GNSS

and

the number of float ambiguities, this can be

summarised as:

( )





= ⋅= ⋅







X FX

(3)

To improve the a priori prediction for the position,

velocity, receiver clock offsets as well as their drifts,

we use time-differenced carrier phase measurements

(TDCP) as described in [15]. This gives us highly

accurate estimates of the change in position as well as

change in the receiver clock offsets from the last to the

current epoch which is especially important for real

world scenarios where constant velocity cannot be

assumed for all times.

The measurement functions for the code and phase

observation of satellite s derived from (2) are

( )

(

)

1 1 ,1 ,1

1 1 ,1 ,1 1

δλ



−++



−++ +



s w ws

IF s

x x ct Z m

x x ct Z m A

(4)

We use the Melbourne-Wübbena [16] and the

geometry-free [17, p. 85-87] linear combination to

detect cycle slips. In case a cycle slip is detected, the

uncertainty of the respective float ambiguity is set to

an arbitrary high value, e.g.

, and the a priori

ambiguity is estimated from the ionospheric-free

linear combination of the phase observations and the

a priori Kalman state.

2.2 Two-antenna baseline approach

The idea is that a second antenna is mounted on the

vessel and we assume that the distance between the

two antennas is constant. Then the position of the

second antenna

can be calculated using the

position of antenna 1 and the baseline between the

antennas, i.e.

( )

21 1

1 11 1 1

1 1 11 1 1

sin cos sin cos cos cos sin

cos sin sin sin cos cos cos

0 cos sin sin

θψ

λ λ ϕ λ ϕ θψ

λ λϕ λ ϕ θψ

ϕ ϕθ

=+ ⋅=

−−

  

  

=+−

  

  

ECEF

ENU

x xC xl

(5)

( )

ECEF

ENU

is the transformation matrix from the

ENU frame which has antenna 1 as its origin to an

ECEF frame where

and

are the latitude and

longitude of antenna 1.

( )

θψ

is the baseline with

and

being the pitch and yaw of the baseline

which correspond to the elevation and azimuth of

antenna 2 in the respective ENU frame. Both frames

are displayed in Figure 2. As can be seen in (5) we

only need to know the length of baseline but not how

the antennas are mounted on the vessel, but ideally,

they should be as far away as possible from front to

back.

Figure 2. a) ENU in ECEF frame, b) ENU frame with

antenna 1 in its origin

The velocity of antenna 2 can also be expressed in

terms of antenna 1 and the baseline by totally

differentiating (5) with respect to time:

=vx

( )



( )

1) 0

φ(

sin sin cos cos

sin cos cos sin

cos 0

≈

θψ θψ

≈

−





+−−













ECEF

ENU

vC x l

(6)

To link the rate of turn of the vessel with the

change in the yaw of the baseline, we have to make

some assumptions on the mounting of the antennas in

the ship's body frame. Ideally, the baseline's

projection onto the ship's horizontal plane, i.e. the

plane defined by vessel's across and along ship axis as

can be seen in Figure 3, is parallel to the along ship

axis (Figure 3a) which would imply the pitch of the

vessel differing from the pitch of the baseline in the

ENU frame by a constant offset. Of course, for this we

do need to know the height of the antennas in the

ship's body frame. If the horizontal projection of the

baseline is not parallel to the along ship axis, we have

to assume the pitch of the baseline being identical to

the pitch of the vessel, both with respect to the ENU

frame. Hence, the baseline has to be parallel to the

vessel's horizontal plane as displayed in Figure 3b. In

both cases, the rate of turn of the ship would be

identical to the change of the yaw of the baseline.

Figure 3. a) Baseline's horizontal projection, b) Baseline

parallel to horizontal plane

To further reduce the number of parameters to be

estimated, we assume the same wet zenith delay for

both antennas, i.e.

,1 ,2

= =

ww w

ZZZ

, since the length of

the longest ships is less than half a kilometre. In

practice, most ships are far shorter than that. All in all,

the Kalman state for the second antenna only consists

of the receiver clock offsets

, clock drifts



and the float ambiguities

( )

. Again, we assume

constant receiver clock drifts.

(

)

( )

22 2 2

τδ





= ⋅= ⋅





X F X I ct

(7)

As with antenna 1, we use TDCP to get accurate

estimates of the velocity and receiver clock drifts.

Applying (6) gives us a measurement for

whereas

the change in the receiver clock offset of antenna 2 is

used as an apriori estimate for its receiver clock drifts.

While the two-antenna baseline approach reduces

the Kalman state for the second antenna, we have to

estimate four additional parameters. These are the

pitch

and yaw

of the baseline as well as their

velocities



and



. Assuming constant angular

velocity, the state-transition model for the baseline

looks as follows:

( )

τθ

τψ

  

  

= ⋅= ⋅

  

  



Base Base Base

XF X

(8)

To make further use of the two antennas, we apply

a moving base approach as implemented in [18] to

estimate the baseline whenever possible, i.e. we have

enough good observations for the two antennas from

the same satellites using double-differenced

observations. The calculated baseline is then

transformed from an ECEF into the ENU frame where

its pitch and yaw can be determined. We use these

two values as direct measurements of

and

2.3 IMU measurements

The integration of an IMU requires considering

additional measurements and parameters. The

number of parameters depends on whether a full

integration of all gyro as well as acceleration

measurements is done or if a reduced model is

chosen. We propose a simple model that only uses the

biased rate of turn measurements.

In the following, we assume that the roll and pitch

of the vessel as well as their velocities are 0 which is a

fair assumption for large inland waterway vessels due

to small waves and current as well as being less

affected by smaller ships. Furthermore, the IMU is to

be mounted on the vessel in such a way that each axis

of the IMU is parallel to the respective axis of the

vessel. By this, the rate of turn of the vessel is identical

to the unbiased rate of turn of the IMU, and as

described in section 2.2 identical to the change of yaw

of the baseline.

All in all, the biased IMU measurements of the rate

of turn can be described as

(

)

= +



IMU

hX b

(9)

The gyro bias

is estimated in the Kalman Filter

and is assumed to be constant, hence

( )

= ⋅=⋅

t tt

IMU IMU IMU

XF X b

(10)

To make the pitch estimation more robust, we

suggest to add an artificial angular velocity

measurement of zero with regards to the pitch in case

there is no baseline measurement. This is in line with

our initial assumptions for the IMU on an inland

waterway vessel. Using (3), (7), (8) and (10) the full

Kalman state

of the two-antenna baseline

approach and its transition model can be summarised

( )





= ⋅= ⋅





Base

IMU

X FX

(11)

3 RESULTS

A measurement campaign was conducted on the 23rd

of February 2022 in Strasbourg as part of the

SCIPPPER project [19]. The goal was to automatically

enter a waterway lock using GNSS, IMU as well as

nearfield sensors. For this, we could use the Victor

Hugo, a cruise ship from CroisiEurope. The

dimensions of the vessel as well as the placement of

the two geodetic antennas, each connected to a

JAVAD Delta receiver, can be seen below:

Figure 4. Schematic placement of the two GNSS antennas on

the Victor Hugo

As displayed in Figure 4 the antennas were not

mounted parallel to the ship's axes. Furthermore, the

height of the antennas differed by 10 cm in the ship's

body frame. In total, the length of the baseline was

68.74 metres and the displacement of the antennas

caused an offset in the pitch and yaw of the baseline

w.r.t. the ship's body frame of about 0.08° and 2.43°

respectively. A sensonor MEMS IMU (STIM300) was

mounted directly under the bow antenna. It has a

gyro bias instability of 0.3 °⁄ h and a gyro noise of 0.15

°⁄√h which were used in the Kalman Filter for the

uncertainty of the gyro measurements as well as the

uncertainty of the constant gyro bias assumption in

the state-transition model.

To synchronise the GNSS observations with the

IMU measurements which have a sampling rate of

about 249 Hz, the IMU measurements are integrated

over each epoch assuming piecewise constant values.

The ship started in the south in the harbour and

drove northways crossing several bridges. After

arriving at the waterway lock to the Rhine, it

automatically entered the waterway lock three times

from east to west [20]. Afterwards it went from north

to south back to the harbour, again crossing several

bridges. The overall trajectory of the Victor Hugo and

a nearby GNSS station are shown in the following

figure:

Figure 5. Trajectory of Victor Hugo (black line) and GNSS

station ENTZ00FRA (purple star) [Google Maps, 2022]

We use a postprocessed RTK solution from

RTKLIB as a reference with the base station being the

GNSS station ENTZ00FRA from the EUREF

Permanent GNSS Network [21] as can be seen in

Figure 5. During the measurement campaign the

baseline between the base station and the antennas on

the ship was between 10.5 and 12.5 kilometres long. A

close-up of the ship's trajectory can later be seen in

Figure 7.

We processed observations from GPS, GLONASS

and Galileo with an elevation mask of 10° and a

sampling rate of 2 Hz. The precise satellite clock and

orbits were obtained from the final products of GFZ.

The uncertainty of the state-transition model in the

Kalman Filter was adapted to the Victor Hugo with

the maximum acceleration in East and North assumed

to be 0.5 m/s

and 0.1 m/s

in Up direction. The

maximum angular acceleration for pitch and yaw was

set to 0.1°⁄s

. These uncertainties were observed as

maximum values from measurements on previous

days on the Victor Hugo and are therefore in general

overbounds of the true uncertainties with regards to

the ship's movement.

As a first proof of concept, we check whether the

pitch and yaw of the PPP baseline approach

converged to the correct values in a stationary

scenario, even without IMU or baseline

measurements. For this we look at a time when the

vessel was in the harbour and the PPP algorithms

were initialised at 8 a.m.

Figure 6. Pitch and yaw of the baseline in a stationary

scenario

As can be seen in Figure 6 both pitch and yaw

converged to the same values as the baseline

measurements. When baseline measurements were

included (yellow and purple line in the figure), the

estimated values were almost identical to the

measurements without needing to converge.

Furthermore, including IMU measurements helped in

determining the right pitch and yaw of the baseline

faster. Without IMU measurements, it took about 31

and 14 minutes for pitch and yaw to be within 0.1° of

the baseline measurements whereas with IMU

measurements it only took about 14 and 6 minutes

respectively.

Note that without baseline measurements the

Kalman Filter needs reasonable approximations of the

pitch and yaw as starting values to converge. We

found that a single-differenced moving baseline

approach only using code observations is good

enough for this, so there is no need to apply advanced

algorithms right from the start to resolve ambiguities.

Next, we have a look at the estimated heading

during bridge passing. For this we mark five bridges

passed during the measurement campaign which we

will analyse in detail.

Figure 7. Five bridges that were passed during the

measurement campaign [Google Maps, 2022]

Bridge 1 and 3 are railway bridges whereas bridge

2 (Pont Vauban), 4 (Pont d'Anvers) and 5 (Pont Pierre

Brousse) can be crossed by cars. All bridges have a

width of less than 25 metres which is shorter than half

of the antennas' baseline on the Victor Hugo. In the

following figure the estimated yaw with and without

using IMU and/or baseline measurements during the

passing of bridge 2 and 3 from south to north is

shown.

Figure 8. Yaw of baseline during passing of bridge 2 and 3

It is clear to see that without IMU measurements,

the heading deviated when passing bridges by up to

1.3° since there is no yaw information if at least one

antenna has insufficient observations. During this

time, there were no baseline measurements for 22 and

19 seconds respectively which is the time frame where

the heading deviated by more than 0.1° (yellow line).

Note that without IMU nor baseline measurements, it

took about 3 minutes for bridge 2 (blue line) until the

heading was correct again, i.e. within a range of 0.1°.

When the IMU was used, the difference to the results

using both IMU and baseline measurements was less

than 0.07° the whole time, even without applying

baseline measurements.

To sum it up, having consistent rate of turn

information is crucial for our baseline approach

during times when there are no baseline

measurements which are important in nominal

conditions to have precise pitch and yaw.

Next, we analyse the positioning results during the

bridge passings. As we had no reference position

during the passings, we took the positions of the RTK

solution with fixed ambiguities closest to the

respective bridge before and after the passing which

were in agreement with the PPP baseline solution, i.e.

within 10 cm in East and North. This is due to the

RTK solution having some outliers close to bridges as

displayed in Figure 9a. The straight line between

those two points is our reference trajectory which is a

reasonable approximation since all bridges were

crossed without unnecessary turning. Furthermore,

we assume constant velocity for the reference to

analyse the along as well as the cross track error for

the reference as seen in Figure 9b. Note that in the

figure the grid lines of the along track axis coincide

with the along track of the reference to better visualise

deviations in this direction. Here, we compare the

classic PPP approach for one antenna without any

additional measurements as described in section 2.1

with our PPP baseline approach applying both IMU

and baseline measurements. The following figure

shows the positioning results of the stern antenna

during the two time passing of Pont Pierre Brousse.

Figure 9. a) Passing bridge 5 two times, b) Cross and along

track for East to West passage [Google Maps, 2022]

Both times our PPP baseline approach was better,

i.e. the trajectory was straighter, than the classic

approach. This is especially noticeable in the cross

track direction when the vessel drove from east to

west where the one antenna approach deviated to the

north w.r.t. to the RTK reference after the bridge

passing and it took about one minute until it was in

agreement with the reference. During the passing

from west to east there was no RTK solution for 36

seconds whereas the PPP baseline approach delivered

reliable and precise results with regards to the

reference line for all epochs.

A quantitative analysis for the different bridges

can be found in Table 1. Here, classical stands for the

one antenna approach and baseline denotes our

baseline method. We computed the maximum

absolute error as well as the root-mean-square error

(RMSE) of the cross and along track error. We

analysed the passings going from south to north for

the first four bridges whereas we looked at the vessel

driving from east to west for bridge 5 as can be seen in

Figure 9b. To put the following numbers into

perspective we also applied the analysis to an open

sky scenario with a time frame of 15 seconds which

occurred before the first passing of bridge 1. This was

done to ensure that the classic PPP approach had

converged and was not disturbed by previous NLOS

events.

Table 1. Cross and along track error during bridge passings

for stern antenna

________________________________________________

Cross track Along track

error [cm] error [cm]

________________________________________________

Max RMSE Max RMSE

________________________________________________

C B C B C B C B

________________________________________________

Bridge 1, S→N 49.0 2.8 19.7 1.4 64.0 9.5 31.2 5.0

Bridge 2, S→N 77.1 26.6 42.8 17.7 87.9 12.0 42.6 3.8

Bridge 3, S→N 41.6 6.0 26.1 2.4 199.9 30.8 137.7 21.1

Bridge 4, S→N 56.8 19.0 37.2 11.3 21.8 7.3 12.1 4.3

Bridge 5, E→W 76.9 14.5 49.2 10.4 39.8 6.3 30.0 4.0

Open sky 5.8 4.9 2.7 3.2 11.6 9.4 6.0 5.9

________________________________________________

C – Classical

B – Baseline

For all bridges our approach gave better results,

even up to an order of magnitude as can be seen in the

cross track error for bridge 1. Furthermore, the RMSE

of the along track error was less or equal to 5 cm for

four of the five bridges using our PPP algorithm

which is remarkable for these difficult conditions

where multipath and less observations would

normally impact the quality of a GNSS based positing

solution.

To highlight the non-line-of-sight (NLOS) effects of

these scenarios we have a look at the number of

visible satellites during the passing of Pont Vauban

and the following railway bridge.

Figure 10. Number of visible satellites during passing of

bridge 2 and 3 for the bow (top) and stern (bottom) antenna

Even though we used three GNSS, there were

epochs without any observation for the stern antenna

around 8:38 (UTC) which didn't have any major

impact on the positioning using our baseline

approach. During a time frame of 13 seconds, there

were less than 6 satellites in view for both antennas

during the first bridge passing. The spacial separation

of the antennas can also be seen in the time difference

between the minima of the visible satellites which is

about 45 seconds for the two antennas during the

passing of the railway bridge. In nominal conditions

each antenna had over 20 satellites in view with up to

10 from GPS and 6 to 7 satellites from each of Galileo

and GLONASS.

Next, we analyse the position of the bow antenna.

In Figure 11a we can see the RTK reference as well as

the trajectory of both PPP algorithms for the passing

of Pont Vauban from south to north.

Figure 11. a) Passing bridge 2 from South to North, b) Cross

and along track [Google Maps, 2022]

Again, the baseline approach gave superior results.

Note that the classic approach deviated to the west

during and after the passing where it converged to the

RTK solution after some time. Additionally, in Figure

11b one can also see the errors along track especially

before and after the passing of Pont Vauban. The

baseline approach didn't have those large differences

and was, apart from a slight offset to the west, in line

with the RTK reference as well as the reference line

which is defined in the same way as before. The

quantitative results for the other bridges can be found

in the following table. The additional open sky

scenario refers to the same time frame as the one in

Table 1.

Table 2. Cross and along track error during bridge passings

for bow antenna

________________________________________________

Cross track Along track

error [cm] error [cm]

________________________________________________

Max RMSE Max RMSE

________________________________________________

C B C B C B C B

________________________________________________

Bridge 1, S→N 93.1 18.0 39.2 14.8 275.1 15.5 238.5 10.8

Bridge 2, S→N 132.7 16.0 72.8 12.2 63.2 8.7 40.2 4.1

Bridge 3, S→N 60.9 7.4 26.9 3.9 36.7 10.3 16.8 4.1

Bridge 4, S→N 44.0 10.9 16.0 5.4 26.4 3.9 12.8 2.6

Bridge 5, E→W 26.9 10.9 12.1 7.6 26.4 10.4 15.6 7.0

Open sky 4.1 5.6 1.6 2.8 12.7 9.2 7.3 6.1

________________________________________________

C – Classical

B – Baseline

Similar to the stern antenna, the PPP baseline

approach yielded better results in all cases. Especially

for bridge 2 where the along as well as the cross track

error was almost an order of magnitude smaller with

regards to the classic approach. Furthermore, the

RMSE was below 10 cm in the majority of the bridge

passings and 14.8 cm at most. This clearly shows the

suitability of our approach for this difficult scenario.

4 CONCLUSIONS

We presented a PPP algorithm for two antennas based

on the constant baseline length between them. By

adding baseline as well as IMU measurements the

algorithm is able to deliver precise and reliable

positioning, even when one antenna suffers from non-

line-of-sight effects. The method was applied to an

inland waterway scenario and showed superior

results with respect to the classic one antenna PPP

approach, especially during the passing of bridges.

The algorithm could be improved if all IMU

measurements were used as the additional

acceleration measurements would be useful in

determining the velocity. Furthermore, the integration

of all angular velocities would be needed if the

assumption of a constant pitch is not realistic due to

waves from other ships or a strong current in general.

Also, the approach can be generalised to any position

on the baseline, e.g. in the middle of the baseline.

An additional improvement in the positioning

results would be made possible by using

undifferenced observations which have less noise

than the ionospheric-free linear combination [17, p.

76]. On the one hand this would require additional

estimation of or information on the atmospheric

delays, but on the other hand this would allow for

fixing the ambiguities as integers instead of just

estimating them as float variables. This would allow

for precise positioning with fast convergence. This

becomes of upmost importance in the growing need

for real-time application using PPP-RTK [4] and real-

time correction services such as the Galileo High

Accuracy Service [12] and the IGS Real-time Service

[13].

ACKNOWLEDGEMENTS

This research was funded by the German Federal Ministry

for Digital and Transport in the projects Digital SOW and

AutonomSOW II as well as the German Federal Ministry for

Economics and Climate Action (grant number 03SX470E) in

the project SCIPPPER. We also want to thank CroisiEurope

for allowing us to conduct the measurement campaign on

the Victor Hugo.

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