=Paper=
{{Paper
|id=Vol-3183/138
|storemode=property
|title=Backing-Up GNSS with R-Mode: Positioning Performance for Recursive Estimators
|pdfUrl=https://ceur-ws.org/Vol-3183/paper8.pdf
|volume=Vol-3183
|authors=Lukas Hösch,Filippo Giacomo Rizzi,Lars Grundhöfer,Ralf Ziebold,Daniel Medina
|dblpUrl=https://dblp.org/rec/conf/icl-gnss/HoschRGZM22
}}
==Backing-Up GNSS with R-Mode: Positioning Performance for Recursive Estimators==
https://ceur-ws.org/Vol-3183/paper8.pdf
Backing-Up GNSS with R-Mode: Positioning
Performance for Recursive Estimators
Lukas Höscha , Filippo Giacomo Rizzia , Lars Grundhöfera , Ralf Ziebolda and
Daniel Medinaa
a
Institute of Communications and Navigation, German Aerospace Center (DLR), Kalkhorstweg 53,
Neustrelitz, 17235, Germany
Abstract
Global Navigation Satellite Systems (GNSS) constitute the main information supplier for outdoor
positioning and timing. Unfortunately, unintended and malicious radio-frequency interference con-
stitutes a major concern for GNSS. Within the maritime domain, radio attacks deprive the skippers
from navigation and lead to the failure of different interfaces on a vessel’s bridge. Ranging-Mode
(R-Mode) is a positioning system which uses signals-of-opportunity and allows to backup GNSS.
This paper describes the overall R-Mode architecture and details the types of ranging and velocity
observations derived from medium and very high frequency radio signals. This paper contributes
with the derivation of recursive estimators, in the form of Cubature Kalman Filters, and the discus-
sion of the particulars required to accommodate R-Mode observations. The theoretical analysis and
performance comparison of the proposed filters is addressed via Monte Carlo simulation following a
realistic maritime scenario.
Keywords
Maritime Navigation, GNSS, R-Mode, Kalman Filtering
1. Introduction
Global Navigation Satellite Systems (GNSS) have become the cornerstone for the worldwide
provision of Positioning, Navigation and Timing (PNT) data. Besides its extensive use for
vehicular and personal navigation, there is an evergrowing dependency of GNSS for timing
purposes in financial and power grid applications. However, GNSS performance can be readily
disrupted due to various factors: natural phenomena (e.g., in the form of ionospheric dis-
turbances), signal reflection (e.g., multipath and non-line-of-sight signal reception) or radio
interference (e.g., jamming and spoofing). The proliferation of the latter has raised serious
concerns to the GNSS community [1, 2], especially when considering the use of GNSS for
safety-critical applications. The maritime domain constitutes a particular case, since GNSS is
not only essential for navigation, but its absence leads to the failure of multiple interfaces on
a vessel’s bridge and to the service denial of the Automatic Identification System (AIS). To
further worsen this issue, GNSS jamming attacks have been frequently reported in the marine
scope, typically related to military actions [3].
Under the afore-described issues, the concept of Ranging Mode (R-Mode) was devised in
ICL-GNSS 2022 WiP, June 07–09, 2022, Tampere, Finland
$ lukas.hoesch@dlr.de (L. Hösch); Filippo.Rizzi@dlr.de (F.G. Rizzi); Lars.Grundhoefer@dlr.de (L.
Grundhöfer); Ralf.Ziebold@dlr.de (R. Ziebold); Daniel.AriasMedina@dlr.de (D. Medina)
0000-0002-0656-6502 (L. Hösch); 0000-0003-0585-2133 (F.G. Rizzi); 0000-0002-8650-8280 (L. Grundhöfer);
0000-0002-0488-6457 (R. Ziebold); 0000-0002-1586-3269 (D. Medina)
© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
Workshop
Proceedings
http://ceur-ws.org
ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.org)
2008 [4]. R-Mode is a maritime terrestrial navigation system to employ existing maritime
service signals as signal-of-opportunity (SoOP) for ranging. It enables positioning and timing
for distances of up to a few hundred kilometers from the shore line, conditioned on the avail-
ability of transmitting stations, and allows for bridging the periods of GNSS unavailability.
Through the joint collaboration of multiple European partners and the continued support of
European projects –from earliest to latest: ACCSEAS, R-Mode Baltic, R-Mode Baltic 2–, the
feasibility of R-Mode has been acknowledged [5], the first testbeds are now operational and its
standardization within the International Maritime Organization has begun.
R-Mode uses two types of existing signals on different frequency bands for deriving ranging
observations: 1) differential GNSS corrections transmitted by maritime radio beacons over
medium frequency (MF); 2) base stations receiving and broadcasting AIS messages over very
high frequency (VHF). Fig.1 illustrates the potential availability of R-Mode compatible stations
over the Baltic Sea [6]. VHF stations are marked with red circles whereas MF are indicated
as magenta triangles. Additionally, the predicted theoretical accuracy over the Baltic region is
also visible in the picture, showing that 10 m horizontal accuracy is achievable over sea. With
the initial positioning performance been assessed on recent works, the extend of their study is
limited to either snapshot (i.e., memoryless) solutions or to using a single frequency band (e.g.,
[7] analyzes VHF and [8] addresses the use of MF). In this contribution, the intricacies related
to fusing both MF- and VHF-derived ranging and velocity measurements are discussed, as well
as recursive estimators of Kalman filtering type being proposed. The positioning performance
of such estimators is analyzed based on Monte Carlo experimentation, following a realistic
maritime scenario with both MF- and VHF-transmitting stations.
The rest of the paper is organized as follows: Section 2 provides a brief overview on the
R-Mode architecture, its signal models and related ranging and velocity observations. Section
3 discusses on recursive estimators using R-Mode observations and analyses the specifics for a
Cubature Kalman Filter solution. Then, Section 4 studies the positioning performance of the
previously derived estimators for a simulated R-Mode scenario. Finally, Section 5 presents an
outlook of the work and discusses the future lines of work.
2. R-Mode
The R(anging)-Mode concept emerged in 2008 in order to offer an alternative PNT terrestrial
system to the maritime users [4]. The system implementation is based on the SoOP idea, which
means reusing existing infrastructure and signals for navigation purposes. This is important
to strongly reduce the cost and the deployment time of the system itself.
The potential of R-Mode has been presented in three feasibility studies published in 2014
[9, 10, 11]. In these reports two main infrastructures were considered, the MF maritime radio
beacons and AIS which uses VHF band. Therefore, in the paper we will refer to them as MF
and VHF R-Mode. Additionally, in [9] the potential combination of the system with eLoran
signals is explored.
The frequency of the signals in use is fundamental in order to understand how the radio-
wave propagates. In the VHF band the signals propagate as line-of-sight (LOS), hence the
transmitter-receiver distance can be described with the classical formulation of the Euclidean
distance, analogous to the GNSS case. VHF measurements are obtained by using correlation
techniques. In particular, range and velocity observation are gathered from the stations by
evaluating the time of arrival (TOA) and the Doppler frequency of the received signal respec-
Figure 1: Predicted R-Mode positioning accuracy with combined VDES (red dot) and MF (magenta trian-
gles) transmitters over day-time in the Baltic Sea. Picture extracted from [6].
tively [7]. Therefore, ranges are given as follows
ρ = τ c0 (1)
with τ signal propagation time and c0 speed of light in the vacuum. Radial velocity instead is
obtained from Doppler measurements as
∆f
vr = − c0 (2)
f0
with ∆f Doppler frequency and f0 carrier reference frequency. In (2) a positive Doppler
means that the distance between the receiver and the transmitter is decreasing, therefore the
resulting radial velocity is negative. In general, velocity measurements present noise of one
order of magnitude lesser than the ranges and, therefore, they are of particular relevance to
achieve good positioning performance. The interested readers can refer to [7] for more details
about VHF R-Mode.
Differently from VHF signals, the MF ones propagate as ground waves which implies that the
signals follow the curvature of the Earth. The main advantage of MF signals is that they are
not limited to LOS and can, therefore, be received at greater distances. However, such curved
propagation implies that the Euclidean distance does not correctly model the range between
receiver and transmitter. Instead, MF-derived ranges are typically modelled as geodesics on
the Earth ellipsoid and described with the Vincenty’s distance model [12]. Positioning with
MF signals is possible by exploiting phase measurements. Frequency division multiple access
(FDMA) scheme is used to distinguish the transmitters. Each channel contains a minimum
shifted keying (MSK) signal carrying data and two additional continuous wave (CW) which are
used as pilot signals for positioning. Ranges can be obtained by tracking the phase evolution
of these two CW as follows
gj,k = Nj,k λj,k + ϕj,k λj,k (3)
where j is the station index whereas k = 1, 2 is the CW index. N represents an integer number
of phase cycle, often referred to as ambiguity in the GNSS literature, ϕ is the fractional phase
cycle and λ represents the wavelength of the CW signal. Further information about the signal
characteristics and phase estimation technique for MF R-Mode can be found in [8].
After this brief description of the signals in use, one clearly identifies that one of the main
challenges for the position solver is to consider and combine the different observations with
their pertinent models, especially given the MF ground-wave signal propagation. In [7] and [8]
the positioning performance using respectively VHF and MF was addressed. [7] proposed an
Unscented Kalman filter (UKF) for the VHF measurements whereas in [8] a snapshot iterative
least square (ILS) is used for the MF range measurements. In the sequel, we aim at combining
VHF and MF R-Mode measurements with recursive estimation based on Kalman filtering
techniques.
3. Recursive Estimators for R-Mode Navigation
Let us consider a general discrete state-space model (SSM) whose time evolution is described
by a process model f (·) and the relationship between the state x and a set of measurements y
by an observation model h(·), as
xk = f (xk−1 , wk−1 ) , (4)
yk = h (xk ) + nk , (5)
with the vectors of process and observation noise assumed to follow Gaussian distributions
of known parametrization (i.e., typically zero-mean and known covariance matrix), such that
wk−1 ∼ N (0, Qk−1 ), nk ∼ N (0, Σk ).
At a particular time instance k, the state estimate is given by
⎡ ⎤
pk
⎢ vk ⎥ 3 ̇ 1 p
⎣cδtk ⎦ , with pk , vk ∈ R , cδtk , cδ tk ∈ R , xk ∈ R ,
xk = ⎢ (6)
⎥
cδ̇ tk
where pk , vk denote the three-dimensional vectors of position and velocity, expressed in a
global reference frame (such as the Earth-centered Earth-fixed (ECEF)), while cδt and cδ̇ t
express the clock offset and clock offset rate, respectively. Then, p is the state dimensionality.
Note that, in analogy to the GNSS world, the clock offset delays at the receiver end shall be
estimated while clock deficiencies at the transmitter end are expected to be covered by the
data transmission or a prospective R-Mode augmentation system.
3.1. Observation Models
As introduced during Section 2, the reception of VHF and MF R-Mode signals leads to two
types of ranging observations and a radial velocity measurement, with details on the corre-
sponding observation models provided next.
3.1.1. VHF measurements
Propagating in a straight line, VHF signals lead to ranges in an Euclidean space and the
following observation model
ρiVHF = ⃦pi − p⃦ + cδt + niρ , for i = 1, . . . , nVHF
⃦ ⃦
(7)
where the time index has been neglected for simplicity, the superscript i refers to the each of
the nVHF stations and ∥ · ∥ is an Euclidean norm. Note that the small time deviations among
stations due to the TMDA modulation is negligible and, therefore, disregarded in this work.
3.1.2. VHF Doppler measurements
The application of Doppler measurements for the R-Mode VHF implementation leads to signifi-
cant accuracy improvements [7], since Doppler is moderately less affected by noise than ranging
measurements. Thus, the radial velocity with respect to the ith transmitter is described by
)︁⊤ pi − p
ρ̇ iVHF = − vi − v + cδ̇ t + niρ̇ ,
(︁
for i = 1, . . . , nVHF , (8)
∥pi − p∥
where the time index has been neglected again for readiness and the transmitter velocity is
typically null (since R-Mode does not currently considers mobile transmitters).
MF Measurements
As previously discussed, MF measurements propagate following the Earth’s curvature, leading
to a complicated geolocation-dependent observation model, such that
ρjMF = ⃦pj − p⃦V + cδt + njρ , for j = 1, . . . , nMF ,
⃦ ⃦
(9)
where ∥ · ∥V is the Vincenty’s distance, or the geodesic between two arbitrary points on the
surface of the Earth. Presenting high fidelity for the Earth ellipsoid, Vincenty’s inverse problem
leads to an iterative procedure based on the two latitude and longitude coordinates. Thus,
the use of the celebrated Extended Kalman Filter (EKF) is jeopardized on the derivation of
complex Jacobians and sensitive to the high nonlinearity of ellipsoidal distances, especially
prone to errors for long distances.
3.2. Cubature Kalman Filtering for R-Mode Positioning
To exploit the series of observations collected over time, one generally makes use of the Re-
cursive Bayesian Estimation (RBE) framework. Among the families of RBE algorithms, the
Kalman Filter (KF) and its nonlinear extensions –e.g., the EKF, UKF or Sigma-Point Gaussian
Filters (SPGF)– have become the norm for navigation and tracking applications [13, 14, 15, 16].
The high nonlinearity of Vincenty’s distances motivated the authors into applying SPGF-type
of solutions, appealing due to its implementation easiness and scalability with higher dimen-
sions (the latter appealing for prospective ultra-tighly coupling R-Mode receiver and position-
ing). In particular, the Cubature Kalman Filter (CKF) [17] constitutes the filtering choice for
the fusion of VHF and MF ranging and velocity observations.
Prediction Step
Considering a constant velocity time evolution of the positioning, the process model in (4) is
given by
⎡ ⎤
I3 ∆tI3 0 0
⎢0 I3 0 0⎥
x̂k = Fk−1 x̂k−1 , Fk−1 = ⎢
⎣0
⎥, (10)
0 1 ∆t⎦
0 0 0 1
with ∆t the time elapsed between two consecutive discrete time instances. With the prediction
step being a linear function, the prediction for the covariance matrix can be realized using the
linear KF form, as
Pk|k−1 = Fk−1 Pk−1|k−1 F⊤ k−1 + Qk−1 . (11)
Correction Step
Then, the update equations for the state and covariance matrix result from the propagation
of the cubature points. To do so, one first propagates these points Xk|k−1 based on the
factorization of the covariance matrix and their evaluation on the observation model, as
1/2
Xk|k−1 = xk|k−1 + Pk|k−1 ε (12)
(︁ )︁
Ŷk = h Xk|k−1 (13)
with ε the generators for the 2p points. From these, we may reconstruct the mean of our vector
of observation models with
n
1 ∑︂
ŷk = [Ŷk ]i (14)
n
i=1
where n is the total number of observations (i.e., n = 2· nVHF +nMF to account for the Doppler
and ranging measurements of VHF and the ranging observations from MF) and [a]i indicates
the ith position on a generic vector a. Then, the update of the state and covariance estimates
follow the well-known CKF algorithm [17]:
n
1 ∑︂ ⊤
Pyy,k|k−1 = Ŷk Ŷk − ŷk ŷ⊤
k + Σk (15)
n
i=1
n
1 ∑︂ ⊤
Pxy,k|k−1 = X̂k|k−1 Ŷk − x̂k|k−1 ŷ⊤
k (16)
n
i=1
Kk = Pxy,k|k−1 P−1
yy,k|k−1 (17)
x̂k|k = x̂k|k−1 + Kk (yk − yˆk ) (18)
Pk|k = Pk|k−1 − Kk Pyy,k|k−1 K⊤
k. (19)
Receiver initial position
57 MF Transmitters
VHF Transmitters
56
Latitude [Deg]
55
54
9 11 12 14 15 17 18
Longitude [Deg]
Figure 2: Map of the simulation scenario showing VHF, MF transmitters and initial receiver position
respectively represented by blue triangles, red circles and black square.
4. Simulation Results
The algorithm proposed has been tested in a simulated scenario. Fig. 2 shows the map with
the VHF (blue triangles) and MF (red circles) base stations used to generate the synthetic
observations (i.e. pseudoranges and Doppler measurements). The initial receiver position
is also visible in the map represented as a black square. It is important to mention that
the location of the MF transmitters used for the simulation are all enabled to send R-Mode
signals, whereas the VHF ones are only potential candidates for the future implementation of
the system.
To assess the performance of the filter under study, a Monte-Carlo simulation with 100
realizations was performed. The values of the process noise covariance matrix describing the
model mismatch of the filter are used to generate the random walk of the reference trajectory
in each realization. The trajectory is initialized randomly around the starting point indicated
in Fig. 2 with an uncertainty as indicated by the first element of the first row of Tab. 1. The
values of the first row of Tab. 1 indicated as initial uncertainties are also used as initial process
noise covariance matrix of the filter. In the second row of Tab. 1 the standard deviation used
for the process noise covariance matrix are given, for both velocity and clock drift. Last but
not least, the third row of Tab. 1 contains the standard deviation used to generate the noise
on the simulated measurements (i.e MF range, VHF range, VHF Doppler Velocity) and the
measurement noise covariance matrix Σ.
In Fig. 3a, the time evolution for the root mean squared error (RMSE) for the position
estimates is depicted. Specifically, the orange line (as indicated by the legend) illustrates the
error in the ECEF coordinate system, while the blue line indicates the horizontal 2D positioning
error in the East-North coordinate system. The difference between the two represents therefore
Table 1
Monte Carlo simulation parameters.
Initial uncertainties Position: 10 [m], Velocity: 2 [(m/s)]
standard deviations Clock bias: 10 [m], clock rate 1 [m/s]
Process noise Clock rate: 0.1 [s/s/s]
standard deviations Velocity (East-North-Up): [0.1, 0.1, 10−4 ] [m/s/s]
Observation noise MF range: 10 [m]
standard deviations VHF range: 50 [m], VHF Doppler velocity: 0.5 [m/s]
(a) RMSE of position in 3D ECEF and 2D ENU (b) Mean DOP in horizontal and vertical domain.
Figure 3: Position RMSE (root mean squared error) and DOP (dilution of precision).
the contribution of the error in the vertical domain (Up), on which we expect poor performance
due to the low vertical dilution of precision (VDOP). Nevertheless, for maritime application
the interest primarily lays on the horizontal domain, verifying that there are no instabilities
in the vertical one. Indeed, for the three-dimensional RMSE in Fig. 3a, it seems apparent
the presence of a transient effect of the filter, which is a typical and expected behavior. The
convergence to a steady-state appears to be slow but it can be explained by the fact that the
values in the process noise covariance matrix are very small as indicated in Tab. 1. Also, the
long convergence time observed might be induced by the poor observability over the vertical
component. In the future, a deeper analysis on the convergence will be done and techniques
to speed-up this process will be investigated.
In terms of performance, the filter should provide close-to-optimal filtering capabilities, con-
ditioned on the stochastic modeling is known. As afore-explained, the same process noise
covariance matrix Q is used to create the reference and the measurement noise is assumed to
be perfectly known. In order to mitigate the errors in the vertical domain, the associated pro-
cess noise standard deviation is reduced to 10−4 m in the process covariance matrix. Following
the small acceleration capabilities of commercial vessels, a standard deviation of 0.1 m/s/s is
simulated and thus used for filter design.
Even if the three-dimensional position estimate appears to not fully converge to steady-state,
an accuracy of 62 m is achieved at the end of the simulation. Nevertheless, such accuracy is
surprising given the large VDOP as visible in Fig. 3b. For better assessment, Fig. 4 depicts
Figure 4: Root-mean-squared error (RMSE) of position in 2D ENU frame
only the RMSE in the horizontal direction. Following the small HDOP values visible in Fig.
3b, the filter shows excellent performance. Despite the small overshooting effect visible until
epoch 1800, the filter is able to fully converge and estimate the user position with the depicted
RMSE of approximately 2.5 m. As stressed earlier, the positioning performance, especially in
the horizontal domain, is of particular interest for the maritime application. The accuracy
achieved by this approach outperforms the predicted accuracy within the testbed. However,
it is important to mention that this is the result of a simulation for filter validation. Knowing
all simulated uncertainties stated in Tab. 1, especially for the process noise, is not possible in
a real-world environment. Still, this performance validates the implemented filter approach in
accordance to the low HDOP values.
Both DOP values shown in Fig. 3b show constant performance throughout the whole sim-
ulation. The orange graph in depicts the HDOP in the associated East-North coordinate
system. The extremely low value of 0.65 shows good performance, which is expected due to
the geometrically favourable conditions of the initial position (black square in Fig. 2). As
the geometrical conditions experience no significant changes throughout the whole simulation,
only small variations can be observed in the HDOP. The vertical DOP (blue graph of Fig. 3b)
amounts to 17.25 and demonstrates considerably bad geometrical conditions in the vertical do-
main. Also this value shows only small changes, which can be explained by the mostly steady
geometrical conditions. Using exclusively transmitters placed on the earth’s surface, a poor
VDOP is expected. The approach used to limit the positioning uncertainties in the vertical
direction addresses this issue. We consider this method suitable due to the harsh geometrical
conditions of the vertical positioning but also to incorporate the fact that our receiver will stay
on the earth’s surface, hence the approach is application specific and cannot be adopted for
other scenarios where the receiver can experience large vertical variation (e.g.in aviation).
Furthermore, the validity of the approach in discussion can be stressed by the low values
(a) 3D RMSE of speed (b) RMSE of clock offset
Figure 5: RMSE of speed and clock offset
(of 0.3 m/s) of the three-dimensional RMSE on the speed estimation depicted in Fig. 5a. The
filter is sufficient dynamic to achieve convergence within approximately 2000 s, which is fast,
compared to the convergence time of the RMSE on the position depicted in Fig. 3a.
The convergence time of the RMSE of the receiver clock offset depicted in Fig. 5b is even
smaller. Even though the low offset is centered around a relatively low value of 1.5 m, its
behaviour is comparably noisy. This effect could be mitigated by an increased number of Monte
Carlo realizations, which would raise the computation time and therefore was not considered
in the current implementation.
5. Conclusion
In this paper, we described the overall architecture of R-Mode, a GNSS backup system for
the maritime domain. The positioning accuracy of the system is predicted to be less than 10
m, 95 % of the time. From a positioning perspective, the main challenges are related to the
integration of the two subsystem (MF, VHF) and the low observability on the vertical direction
caused by the transmitters geometrical setup.
Due to the high non linearity of the MF range measurement and the complexity of its
Jacobian term derivation, we excluded the adoption of the EKF. Therefore, a CKF was im-
plemented as PNT algorithm. Its theory was briefly presented along with the definition of the
different observation models.
To assess the performance of the filter, a simulation environment was created in order to
perform Monte Carlo runs. The realizations were obtained by using real VHF and MF trans-
mitter locations and we showed that the filter is able to accurately estimate all the relevant
states by using an adapted process noise covariance matrix. In particular, the usage of very
low variance in the vertical speed acts as a bound for the variation in the same direction and
reduces the impact of unfavorable VDOP. The results of the horizontal positioning performance
are extremely promising: After convergence, the RMSE in the horizontal domain amounts to
2.5 m, which is surprising, even in a simulated environment. Nevertheless, the long conver-
gence time of the filter in the three-dimensional domain shows the poor observability in the
vertical direction. However, for the application presented, the practical purpose of a vertical
positioning estimate is limited.
In the future, several research directions are foreseen to be considered. First of all, perform a
test with the proposed algorithm on the field by using real measurements. Secondly, the com-
parison of alternative positioning algorithms and finally the integration of additional sensors
which can reduce the effects associated to the poor geometry in the vertical domain.
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