IEEE Robotics & Automation Magazine - September 2019 - 49

from the origin of the figure to each discrete weather direction is determined by (12), which gives the mean position
estimation error:
er dist [k] = 1
N

k=1

(pt n [k] - p n [k]) 2 + (pt e [k] - p e [k]) 2 .
(12)

In (12), (p n, p e) is the measured horizontal plane position
given in the NED frame, (pt n, pt e) is the corresponding DR
position, and k signifies the discrete step. Figure 10(b)
and (c) shows the results of running the same test with the
remaining two methods, that is, the SLFN and the KF.
Considering the area covered by the polygons in Figure 10(a), the mean error is roughly similar, irrespective
of weather direction and wave heights of fewer than 3 m.
Similar properties are seen for the SLFN method [see Figure 10(b)]. The optimization scheme used for the LSTM
method was also applied for the SLFN method. This
yielded an optimized hidden neuron number of 93 for the
sway velocity estimator and 55 for the surge velocity estimator. For both estimators, the optimization procedure
favored a sigmoid activation function. An increased DR
error may be seen for both methods at weather directions
of 270°, 120°, 90°, and 60°. At these directions and a wave
height of 4 m, the tunnel thrusters are unable to produce
sufficient thrust to withstand the environmental forces
acting on the vessel. This caused saturation of thruster
commands and divergence from the desired position.
When a given set of thruster commands no longer causes
vessel motion similar to that experienced in the training
set (e.g., when the environmental forces outweigh the
control forces and cause thruster saturation), the output of
the estimators diverges from the true vessel velocity. The
most severe effects of the saturation are seen at a direction
of 120° and a 4-m wave height. The vessel is unable to
recover the desired position in a timely fashion, causing
further estimation error for all simulated conditions at the
subsequent weather direction of 90°.
Discussion
The input variables related to thruster command, thruster
operating point, and power do not directly give information
about the motion of the vessel. However, they indirectly contain information about how the vessel moves. A thruster
command, executed over a given time interval, induces forces
on the vessel, causing a change in linear/angular speed. The
consumed power fluctuates due to both the thruster command and the velocity of the vessel relative to the surrounding
water. Accounting for lags (see the "Time Delay" section), one
may obtain knowledge of how the vessel moves by viewing
thruster data. This is one of the advantages of using a databased model: it learns such connections. To make the task of
the machine-learning methods easier and more effective,
input selection picks the most relevant input variables for our
problem and also mitigates the curse of dimensionality, which

is an issue for high-dimensional input patterns in regression
problems [35]. The number of samples necessary to approximate a function to a certain degree of smoothness grows
exponentially with the input dimension.
In this study, we performed input selection on the basis of
the mean MI (see the "Input Selection" section) for an input
variable containing lags according to, for example, variable x 1
of (7). This allows for an uninterrupted representation of the
selected variable. Another strategy would be to select the
entries of the total input pattern (see the "Input Structure"
section) that have an MI value greater than some threshold,
which does not leave the intervariable spacing intact but
ensures that all entries in the selected pattern have a given MI
content relative to the target variable.
The results produced in case study 1 show the increased
performance gained by selecting input variables that provide
a certain amount of information about the output variable,
omitting the remainder of the original input variables.
Viewing the optimization results in Figures 8 and 9, we see
that only the surge velocity estimator benefits from applying
MI, at least in terms of the MSE derived from a validation
set consisting of 10% of the samples in the training data set.
This amounts to approximately 1,000 samples. Although the
sway velocity estimator displays a slight decrease in performance when applying the reduced input pattern, the overall
effect of MI is positive. As the input selection processes of
the two estimators are separate, one may choose to implement one or both of the reduced input patterns to maximize
the expected DR performance. Figure 7 shows
how the estimated posiReducing the input
tion, using input vectors
selected by MI, diverges
dimension of the network
more slowly than when
the original input pattern
has positive effects on
is applied during a GNSS
dropout. As MI was shown
computation time as well
to aid the LSTM model
(see the "LSTM" section)
as network interpretability
in terms of reducing the
position estimation error,
and generalization ability.
it was applied to both
machine-learning models
for the second case study,
shown in the "Case Study 2: Impact of the Environmental
Variables" section. Of the two, the LSTM performed better,
with a mean distance error of fewer than 2 m for wave heights
below 3 m. The measurements of thruster-related states
(power consumption, set point, and feedback) were assumed
to be noise free.
The KF, described in the "KF Parameters" section, has
similar performance relative to the LSTM for wave heights of
1 m. When wave heights of 2, 3, and 4 m affect the vessel, the
LSTM provides consistent DR position estimates, whereas the
KF error increases. The KF error increase is due, in part, to
the linear relationship between a thruster command and the
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IEEE Robotics & Automation Magazine - September 2019

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