International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 5
Number 3
September 2011
329
1 INTRODUCTION
Radar target tracking is one of the key issue influ-
encing navigational safety of vessels at sea. For sev-
eral dozen of years radar has been present on board
of the ships establishing its position as a very im-
portant device on the bridge. It has been commonly
used for observation of navigational and collision
situation in the vicinity of own vessel. In the re-
stricted visibility it is even the basic source of in-
formation, while remaining additional and comple-
mentary (to visual observation) source in good
visibility.
1.1 Tracking of maneuvering targets in radars
Radar’s functionality increased rapidly after imple-
menting of target tracking facilities in ARPA sys-
tems. Since then it has become possible to support
navigator’s work by replacing manual plotting with
automatic target tracking. The quality of tracking
depends however on the implemented tracking algo-
rithm. At the beginning relatively simple numerical
algorithms, like α-β were used. In time those were
replaced by more complex numerical algorithms
based on statistical estimation, like Kalman Filter
(Bole et al. 2005). It’s main deficiency is the as-
sumption of linear movement of the target, which
leads to large errors and delays of tracking during
target’s and own ship’s manoeuvres. These limita-
tions are commonly known to the navigators. Vari-
ous modifications of Kalman Filter (e.g. Extended
Kalman Filter, Unscented Kalman Filter) improved
the quality of tracking significantly. The main goal
was to include non-linear movement of maneuvering
vessels into the algorithm. Thus better quality of
tracking was achieved.
One of the possible solution of non-linearity
problem is to create a few different filters for differ-
ent motion stages (linear/ non-linear). This approach
is called multiple-model filtering and is thoroughly
examined for example in (Bar-Shalom & Li 1998).
Another possibility is to use typically non-linear
methods for tracking, for example artificial intelli-
gence. An interesting example might be Intelligent
Kalman Filter presented in (Lee et al. 2006).
For several years the research focused on use of
artificial neural networks in radar target tracking has
been carried out in Maritime University of Szczecin.
Particularly interesting results were obtained while
using General Regression Neural Network (GRNN),
which was presented for example during TransNav
2007 (Stateczny & Kazimierski 2007).
Statistical Analysis of Simulated Radar Target's
Movement for the Needs of Multiple Model
Tracking Filter
W. Kazimierski
Maritime University of Szczecin, Poland
ABSTRACT: The quality of radar target tracking has a great impact on navigational safety at sea. There are
many tracking filters used in maritime radars. Large group of them are multiple model filters in which differ-
ent filter parameters are used for different states (models) of vessel movement. One of possible filter is multi-
ple model neural filter based on General Regression Neural Network. Tuning of such filter means to adjust its
parameters for a suitable target movement model. This paper shows the results of an experiment aiming at de-
termining such models based on statistical analysis of target's movement parameters. The research has been
carried out with PC-based simulator in which typical radar measuring errors were implemented. Different
manoeuvres of targets have been examined. Based on this, the possibility of movement models description
has been stated as conclusion.
330
1.2 Research project and paper scope
The experience on target tracking with neural net-
works gained so far, resulted in preparing new re-
search project in Maritime University of Szczecin,
called Elaborating of methods for radar tracking of
maritime targets with the use of multiple model neu-
ral filtration. The main goal of the project is to com-
bine neural tracking filters with multiple model phi-
losophy, traditionally used for numerical filters.
Different neural filters will be adjusted to track tar-
gets with different dynamics of movement. This
means that, as the first stage of the project, several
models of targets movement has to be declared. The
aim of research presented in this paper was to per-
form statistical analysis of different target’s move-
ment to conclude on how to find these unique
movement models.
2 GRNN FILTER FOR TRACKING IN MARINE
RADARS
The filter proposed for radar target tracking and ex-
amined in presented research was based on General
Regression Neural Network invented by D. F.
Specht (Specht 1991), which is basically neural im-
plementation of kernel regression algorithms pre-
sented in (Nadaraya 1964) or (Watson 1964). The
structure of the network is strictly defined, but it
needs some kind of adjusting to solve particular
problem. This means mainly determining of input
and output vectors, teaching sequence, radial neu-
rons activation function and smoothing factor of it.
2.1 Tracking with GRNN
The concept of using GRNN to track radar targets in
maritime navigational radars was shown in
(Juszkiewicz & Stateczny 2000), (Stateczny &
Kazimierski 2005) and (Kazimierski 2008). The fil-
ter proposed in these papers consists of two parallel
GRNNs. One of them is to etimate Vx and the other
Vy. For additional smoothing of signal, which
means more stable vector of target on the radar
screen, the second filtration stage, with another pair
of the same networks is used. To ensure proper func-
tioning of the filter, since the beginning of observa-
tion, the dynamic increase of number of radial neu-
ron in hidden layer and elements of teaching
sequence is introduced. Observed (measured) values
of movement vectors are used as input and teaching
values while estimated movement vector is the out-
put. Movement vector is defined as (1).
[ ]
T
yx
VVV =
(1)
where V
x
= speed vector over x axis, V
y
= speed vec-
tor over y axis.
Both of the networks can be joined into one -
more complex structure presented in the figure 1.
Such a network has two basic parameters the
smoothing factor and the length of teaching se-
quence, usually both adjusted empirically.
Figure 1. Two-stage GRNN for target tracking.
The smoothing factor determines the range of
gaussian function in radial neurons and the teaching
sequence determines how many observed vectors are
included in estimating the state vector.
GRNN performs kernel regression, resulting in
computing weighted average of teaching vectors.
The weights are the values of Gaussian kernel func-
tion for the distances of input vector to teaching vec-
tor. Thus the estimation of movement vector is cal-
culated according to following equation
(Kazimierski 2008):
=
=
=
=
=
n
i
tt
n
i
tt
i
n
i
tt
n
i
tt
i
i
i
i
i
i
i
e
eVyo
e
eVxo
Vye
Vxe
1
2
1
2
1
2
1
2
2
2
2
2
σ
σ
σ
σ
(5)
where Vxe and Vye = estimated speed vector on axis
x and y, Vxo and Vyo observed speed vector on
axis x and y, σ = smoothing factor of Gaussian ker-
nel function, t = actual time step, t
i
= former time
steps.
2.2 Multiple model filtering
Multiple model approach is the development of so
called decision based filters. The main idea is simi-
lar. The filter consists of a few elementary filters,
each of them tuned to track target in unique move-
ment stage, called model. They are running simulta-
neously. The final estimation can be a chosen output
of one of elementary filters (in the decision based
331
methods) or a combination of elementary estimates
(in multiple model approach).
There are several particular algorithms of multi-
ple model tracking, in which different interaction
methods between elementary filters is used. Usually
the probability of target being in each particular
mode state is the criterion. Thus the estimated state
vector is weighted average of elementary estimates.
Fine description of most popular multiple model
methods is given in (Bar-Shalom & Li 2001) and (Li
& Jilkov 2005).
2.3 GRNN multiple model filter
The empirical research (Stateczny & Kazimierski
2006) or (Kazimierski 2007) has shown that differ-
ent values of smoothing factor and of teaching
length are needed in GRNN filter for different
movement characteristics. For uniform motion
longer teaching sequences and bigger smoothing
factors and for maneuvers shorter teaching sequenc-
es and smaller values of smoothing factor are ex-
pected. This gave the idea of creating multiple mod-
el neural filter, which can be implemented as
decision based filter as well. A suitable patent appli-
cation was issued. An example of such a filter is
given in figure 2.
Figure 2. GRNN filter for target tracking.
Main problem in such an approach is to tune ele-
mentary filters for suitable movement model. This of
course shows the need of defining such models.
3 NUMERICAL EXPERIMENT
The research based on simulation presented in this
paper is just an initial phase and preparation for fur-
ther parts in which real data will be involved. This
time PC- based tracking radar simulator was used.
3.1 Experiment overview
The main goal of the experiment was to find any sta-
tistical dependency, that can be useful for defining
tracked targets movement models. To ensure useful-
ness of experiment results for any tracking method,
the unfiltered data were analyzed. These were ob-
tained in the simulator by implementing suitable
noise of measurements prior to any filtration.
To obtain statistical information, 100 Monte Car-
lo runs were performed for each research scenario.
For each run, an average value and a standard devia-
tion of ship’s course, speed, Vx, Vy and increments
of Vx and Vy as well as covariance between Vx and
Vy were calculated. After the simulations, obtained
values were examined and analyzed with the use MS
Excel
The research scenarios were planned in such a
way to examine both uniform motion state and ma-
neuver state.
3.1.1 Simulator description
The simulator used in research showed in this pa-
per is a PC-based application, prepared by the author
in MS Visual Studio.
The idea of radar target simulation used in the
simulator derives from (Kantak et al. 1988) and is
based on adding to non-cluttered measurement, pro-
cess noise. Thus the position of simulated target is
obtained. The noise is calculated as a product of
maximum sensor noise and pseudo-random value.
Start point of random numbers is changing, which
allows carrying out Monte Carlo simulation.
Own ship movement is also simulated and typical
errors of gyrocompass (0,5°) and log (0,05 kn) are
included. The auto-correlation function factors were
established based on (Stateczny et al. 1987).
The simulator has also other possibilities and
functionalities, which were not used for the research
for this paper, however they can be used for many
other purposes.
3.1.2 Research scenarios
The idea of the research is to find different
movement models based on statistical analysis of
non-filtered target data. The research scenarios
therefore include both - uniform target movement
and maneuvers.
The first part of research focused on finding sta-
tistics for linear movement as the basis for compari-
son with maneuvering stages. Five different scenari-
os were examined for uniform movement. Initial
situation was the same for each of them, except of
course and speed values which differ for particular
scenarios. The scenarios are described in Table 1.
332
Table 1. Scenarios for uniform movement
______________________________________________
Scenario no 1 2 3 4 5
______________________________________________
Initial situation
______________________________________________
Bearing 030°
Range 8 Nm
Own ship course 000°
Own ship speed 10 kn
______________________________________________
Target movement parameters
______________________________________________
Target course [°] 135 180 270 135 135
Target speed [kn] 10 10 10 20 30
_____________________________________________
These research scenarios allowed to check the in-
fluence of various speed and courses on statistical
dependences of target movement during steady
movement.
The second part of research aimed at finding re-
sults during maneuver of target. The maneuver of
course changing was examined as the most and ad-
vised in COLREG popular way of collision avoid-
ance. The maneuver was applied with different rate
of turn in different scenarios. As the same change of
course was assumed, the maneuvers were lasting for
different time in each scenario. The statistics were
calculated only for the time during maneuver. Detail
description of scenarios can be found in Table 2.
Table 2. Scenarios for maneuvering target
________________________________________________
Scenario no 1 2 3 4 5
________________________________________________
Initial situation
________________________________________________
Bearing 030°
Range 8 Nm
Own ship course 000°
Own ship speed 10 kn
Target course 135°
Target speed 10 kn
________________________________________________
Course maneuver
________________________________________________
Course change 90° to starboard
New course 225°
Rate of turn [°/min] 10 20 30 40 50
________________________________________________
Examining of the maneuvers with different dy-
namics allowed to answer the question if there is any
statistic dependent of turn rate, which can become a
basis for establishing movement models in multi
model filter.
Each scenario covers 200 measurement steps,
which means about 10 minutes of simulation time.
3.2 Results of experiment
The simulator used for experiment prepares the out-
put statistics for each of 100 Monte Carlo runs in
ASCII file. In the next step it was imported to MS
Excel to prepare graphs and to perform further anal-
ysis. The results are divided into two parts uniform
motion and maneuver. The conclusions are stated for
each part separately and then jointly for all simula-
tions.
3.2.1 Uniform motion
The scenarios in which the course was different
were analyzed together and the scenarios in which
the speed was different were also analyzed jointly.
Figure 3 shows the standard deviation of Vx dur-
ing simulation for each of 100 runs. Scenarios 1, 2
and 3 were included. It can be noticed, that the value
of standard deviation does not vary significantly for
the scenarios, although in case of scenario 3 the val-
ues of standard deviation is slightly bigger than in
other scenarios.
Similar results were obtained for other measured
values standard deviation of Vy, course and speed.
this can lead to the conclusion that standard devia-
tion of movement vector parameters does not change
significantly in case of uniform movement with dif-
ferent courses.
In figure 4 the same standard deviation of Vx is
presented but for the scenarios in which the target
was moving uniformly but with different initial
speed. Once again the value of standard deviation
seems not to vary much in different scenarios.
Figure 3. Standard deviation of Vx during 100 Monte Carlo
runs for uniform motion of target with different course.
The values for scenario 5 in which the speed was
the biggest are usually a bit smaller. As similar re-
sults were obtained for other parameters it can be
concluded, that standard value of them does not
change significantly for the uniform motion, even if
the speed is different.
10
12
14
16
18
20
0 10 20 30 40 50 60 70 80 90 100run no
scenario 1 scenario 2 scenario 3