Integration of Navigation Data
A. Svensson and J. Hoist
(Lund Institute of Technology)
This article treats integration of navigation data from a variety of sensors in a submarine using
extended Kalman filtering in order to improve the accuracy of position, velocity and heading
estimates. The problem has been restricted to planar motion. The measurement system
consists of an inertial navigation system, a gyro compass, a passive log, an active log and a
satellite navigation system. These subsystems are briefly described and models for the
measurement errors are given.
Four different extended Kalman niters have been tested by computer simulations. The
simulations distinctly show that the passive subsystems alone are insufficient to improve the
estimate of the position obtained from the inertial navigation system. A log measuring the
velocity relative to the ground or a position determining system are needed. The improvement
depends on the accuracy of the measuring instruments, the extent of time the instrument can
be used and which filter is being used. The most complex filter, which contains fourteen states,
eight to describe the motion of the submarine and six to describe the measurement system,
including a model of the inertial navigation system, works very well.
i. I N T R O D U C T I O N . The navigation system used in submarines consists of
several subsystems that measure heading, position and velocities. These
subsystems normally work alone. The outputs are directly presented to the
operators without anything further being done to improve their accuracy.
However, in the next generation of submarines it will be required that the
navigation systems consistently give higher accuracy as it will be possible to
remain under water for longer periods than today.
The purpose of this work is to investigate if it is possible to improve the
precision in position, heading and velocity estimates, by cofiltering measurements
from different autonomous measurement systems. The investigation has been
accomplished as a feasibility test, in which nonlinear physically based models of
the ship and measurement systems have been simulated. Filters of different
complexity have been tested in order to investigate what information is needed
to achieve improvements of the estimates and what results might be obtained by
using different measurement strategies. In the extended Kalman niters described
in this article the measurements are weighted according to knowledge about the
size and behaviour of the measurement errors. The filters also include a model
of the dynamics of the ship.
Models of the ship and the measurement systems used in this work are
presented in Section 2. The tests have been restricted to planar motion, which
means that the depth coordinate of the motion has been omitted. The filters are
of the Kalman filter type and, since the systems are nonlinear, extended Kalman
filters are needed. These are presented in Sections 3 and 4. The complete work
includes a number of different filters, more or less complicated, and has been
presented in Svensson (i992)6 . It shows that, in order to improve the precision
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NO. I I N T E G R A T I O N OF N A V I G A T I O N DATA
of the measurement system beyond what a normal inertial navigation system can
give, it is necessary to make a complete model of the dynamic behaviour of the
subsystems including a model of the inertial navigation system, A few filters with
such models included are presented. Their performance when different
measurement strategies are used is presented in Sections 4.1.4 and 4.2.4.
2 . S H I P A N D M E A S U R E M E N T S Y S T E M .
2 . i . M O T I O N OF THE S U B M A R I N E . The submarine model describing the
motion has eight states. These states are common in the literature about ship
dynamics (see e.g. Blanke2 and Astrom and Kallstrom7) that is, surge velocity
relative to the water, sway velocity relative to the water, yaw rate, heading,
position in x andy coordinates and current in x andy directions, collected in the
state vector T(t). These states are sufficient to describe the two-dimensional
motion of the ship, see Fig. i .
Surge
Velocity
Fig. i. The coordinate system
The changes in surge velocity have been simulated as ramps corresponding to
accelerations up to o-i m/s2. The sway velocity and the yaw rate have been
described by a simple model for a yaw movement. This model of the dynamics
is described by the following differential equation:
= A
The parameters are sway velocity (v) in m/s, yaw rate (r) in rad/sec, surge
velocity (u) in m/s and the rudder angle (8) in rad. The matrix A is a 2 x 2-matrix
and B is a column vector with two elements. The elements are:
A = '11
'21
-0-OO99
-0-0008
0-07589 \
— O'oo6o/
"ft
o'oo66 \
y (3)