IEEE Robotics & Automation Magazine - September 2019 - 42

the vehicle typically complement the GNSS measurements.
Skog and Händel provide an overview of such systems and
the methods used for fusing both external (e.g., GNSS) data
and internal sensor data (e.g., odometer, gyroscopes, and
accelerometers) [17].
Abbott and Powell provided a study of the error contribution of various sensors for an in-car navigation system [18].
They applied sensitivity analysis to gauge the performance of a
KF-sensor fusion algorithm against a reference system. Their
findings suggest that the use of differential GPS (dGPS)
offered improved calibration of the internal sensors, resulting
in significant reduction of error drift during a satellite system
outage. Therefore, relatively inexpensive internal sensors combined with dGPS could provide sufficiently accurate DR systems. Extending the flexibility of the KF for combining data
from several sensors at various sampling rates, Barrios
et al. introduced a dynamic state noise covariance matrix [19].
The purpose of this dynamic matrix is to reflect the state
uncertainty more accurately when sensors drop out for any
length of time.
Like Rogne et al. [9], Ahmed and Tahir [20] recognize that
high-performance IMU units contribute significantly to overall system cost. That motivated the use of a low-cost micro
electromechanical system IMU unit containing a triaxial
gyroscope and an accelerometer to accurately determine the
attitude of a car. The authors estimated the vehicle acceleration using the kinematic vehicle model and the known norm

Table 1. The parameters used for the additive
noise elements of the position and heading
measurements.
Tc

White Noise
v

Bias
n

Position

0.1 m

240 s

0.2 m

[-0.2, 0.2] m

Heading

0.1°

60 s

0.1°

[-0.1, 0.1]°

0.2

4

0.15

3

0.1

2

0.05

1

310

320

330
340
Time (s)

350

Propeller Pitch Angle (°)

Surge Velocity (m/s)

x [k + 1] = exp c - Dt m x [k] + vw [k],
Tc

(1)

where k is the discrete time variable, Dt is the sampling
interval, Tc is the correlation time, and w is the Gaussian
white noise with a standard deviation of v. The addition of
noise terms to form the expression for the position and heading with noise [21] is as follows:
(2)

where p is a two-dimensional column vector containing the
north and east position with additive noise, p true is the noiseless north-east position, x p holds the corresponding GM
processes for the two components, v p is a diagonal matrix
containing standard deviations of added white noise (w 1),
and n p holds the position bias:
= } true [k] + x } [k] + v } w 2 [k] + n } .

(3)

5

0

Figure 3. The delayed response of the two variables, surge
velocity (solid blue line) and actual thruster pitch angle (solid
red line), as a reaction to a step increase in the commanded
thruster pitch angle (dashed red line).

*

Measurement Noise
Noise was added to the following measured states:
● position: the position measurements given in the northeast-down (NED) frame
● heading: rotation about the z-axis of the vessel
● velocity: the linear velocity given in the NED frame.
Therefore, the position and heading measurements, as seen
by consumers of the sensor data, are a sum of the true value
sampled from the simulator, white noise, a bias, and a GaussMarkov (GM) process. The discretized GM process is

} [k]

0.25

42

Methodology
In this section, we introduce the measured signals, delays present in the actuators of the simulated vessel, and LSTM network
model. Methods of limiting the input data dimension and
selection of LSTM hyperparameters are also considered.

p [k] = p true [k] + x p [k] + v p w 1 [k] + n p,

GM
v

0

of the gravity. In addition to providing accurate attitude estimates, the ability to separate the gravity-induced acceleration
components from the overall acceleration measurement
proved beneficial to DR performance.

IEEE ROBOTICS & AUTOMATION MAGAZINE

*

SEPTEMBER 2019

Noise added to the heading signal is described in (3), where
} is the heading angle containing noise, } true is the noiseless
heading angle, x } is the GM process related to the heading
angle, v } is the standard deviation of the Gaussian white
noise w 2, and n } is the heading angle bias. Table 1 shows the
parameters used in simulating the position and heading states
with noise. The angular/linear velocity received only a constant bias and white noise [22].
Time Delay
Delays in time between a change in thruster command
(input) and the given response in velocity (output) are
present in the sampled time series. They are caused by
both the linear/rotational inertia of the vessel and the rotational inertia of the various thruster systems. Figure 3
IEEE Robotics & Automation Magazine - September 2019

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