Autonomous Underwater Vehicle Navigation 14.2 Algorithms 347
Part B | 14.2
the true position. The drift (error) e between the vehicle’s true position x true and the position obtained with
DR x DR are expressed as drift over time or drift over
distance traveled
e D
kx true x DR k 2
t
or e D
kx true x DR k 2
x
:
Typically the heading and rate sensors of an INS are less
noisy than those of a comparably inexpensive AHRS,
which reduces the effect of accumulated drift. An INS
that fits into the hull of a typical AUV shows typical
drift rates of 1 km=h [14.35]. The exact performance
of the most precise INS available are those developed for nuclear submarines; the drift rates for these
sensors are not published but are expected to be O
(0:01 km=h) [14.22].
For operations near the seabed, DVL sensors can
be used to measure the vehicle’s velocity relative to
the ground. The integration of this information in
the navigation Kalman filter can greatly improve the
performance. For example, the DARPA Autonomous
Minehunting and Mapping UUV developed in the early
1990s achieved a navigation performance of 0:01% of
distance traveled using an integrated INS/DVL system [14.10]. The MARPOS system developed by Mariden A/S and the Technical University of Denmark for
the MARIDAN series of AUVs represents a recent
state-of-the-art commercially available AUV system,
achieving 0:02% accuracy for site surveys and 0:10%
accuracy for straight-line transits [14.36].
The problem with exclusive reliance on DR or inertial navigation is that the position error increases
without bound as the distance traveled by the vehicle
increases. The rate of increase will be a function of
ocean currents, the vehicle speed, and the quality of
DR sensors. Radio and satellite navigation systems can
provide an accurate position update provided the vehicle can travel at or near the surface periodically for
a position fix. The maximum vehicle travel time between surfacing for a position update will be governed
by DR/inertial navigation accuracy. Poor quality DR
will dictate an unacceptably high frequency of surfacing. Also, vehicles operating close to the coast are in
appreciable danger of collision with surface vessels if
they need to frequently approach the surface for position fixes. For deep water applications, the time and
energy needed by a small AUV for transiting to the surface from near the bottom are very unfavorable. Finally,
surfacing is impossible in ice-covered oceans.
14.2.2 Acoustic Navigation
When acoustic transponder measurements are available, a variety of algorithms are possible. Most techniques are based on recursive least-squares estimation,
often with an extended Kalman filter. As described
above, in an LBL navigation system, an array of
transponders is deployed and surveyed into position.
The array is usually calibrated through use of an additional acoustic transponder that is hung from a surface
ship and interrogates the array from various locations.
The vehicle sends out an acoustic signal which is
then returned by each beacon as it is received. Position
is determined by measuring the travel time between the
vehicle and each beacon, measuring or assuming the local sound speed profile, and knowing the geometry of
the beacon array. With this information, the relative distances between the vehicle and each array node can be
calculated. The two primary techniques are (1) to compute position fixes by locating the intersection point of
spheres of appropriate radii from the beacons in the array, and (2) to integrate the raw time-of-flight (TOF)
measurements into an appropriate Kalman filter.
The basic solution for computing a fix from spherical ranges to three transponders is as follows [14.14].
The local earth frame origin is at the surface. The x-axis
points north, the y-axis points east, and the z-axis points
down. The vehicle position in this frame is .x; y; z/ and
the coordinates of beacon i are .x i ; y i ; z i /. The measured round-trip travel times between the vehicle and
the beacons are t i and the associated distances are d i .
The distances are computed from the travel times by
the approximate relation: d i D c.t i i /=2, where c is
the average speed of sound (Note: spatial variations in
the speed of sound can be significant, especially variations with depth [14.11]).
The analytical solution based on measurements
from three beacons consists of solving the following
nonlinear system of equations for the vehicle coordinates
.x x i /
2
C .y y i /
2
C .z z i /
2
D d
2
i ;
i D 1; 2; 3 :
(14.1)
In order to make the computations easier, we first compute the solution in an intermediate frame obtained by
shifting the local earth frame at the location of beacon 1.
The beacon coordinates in this frame are .x
0
i ; y
0
i ; z
0
i / and
the vehicle position is .x
0
; y
0
; z
0
/. The intermediate solution is then transformed back to the local earth frame.
Advantage is also taken of the accurate knowledge of
the vehicle depth, leading to a linear over-constrained
problem (three equations known and two unknowns).
The vehicle position is then
x D
b 1 c 2 b 2 c 1
a 1 b 2 a 2 b 1
C x 1 ; y D
a 2 c 1 c 2 a 1
a 1 b 2 a 2 b 1
C y 1 ;
(14.2)
Part B | 14.2
the true position. The drift (error) e between the vehicle’s true position x true and the position obtained with
DR x DR are expressed as drift over time or drift over
distance traveled
e D
kx true x DR k 2
t
or e D
kx true x DR k 2
x
:
Typically the heading and rate sensors of an INS are less
noisy than those of a comparably inexpensive AHRS,
which reduces the effect of accumulated drift. An INS
that fits into the hull of a typical AUV shows typical
drift rates of 1 km=h [14.35]. The exact performance
of the most precise INS available are those developed for nuclear submarines; the drift rates for these
sensors are not published but are expected to be O
(0:01 km=h) [14.22].
For operations near the seabed, DVL sensors can
be used to measure the vehicle’s velocity relative to
the ground. The integration of this information in
the navigation Kalman filter can greatly improve the
performance. For example, the DARPA Autonomous
Minehunting and Mapping UUV developed in the early
1990s achieved a navigation performance of 0:01% of
distance traveled using an integrated INS/DVL system [14.10]. The MARPOS system developed by Mariden A/S and the Technical University of Denmark for
the MARIDAN series of AUVs represents a recent
state-of-the-art commercially available AUV system,
achieving 0:02% accuracy for site surveys and 0:10%
accuracy for straight-line transits [14.36].
The problem with exclusive reliance on DR or inertial navigation is that the position error increases
without bound as the distance traveled by the vehicle
increases. The rate of increase will be a function of
ocean currents, the vehicle speed, and the quality of
DR sensors. Radio and satellite navigation systems can
provide an accurate position update provided the vehicle can travel at or near the surface periodically for
a position fix. The maximum vehicle travel time between surfacing for a position update will be governed
by DR/inertial navigation accuracy. Poor quality DR
will dictate an unacceptably high frequency of surfacing. Also, vehicles operating close to the coast are in
appreciable danger of collision with surface vessels if
they need to frequently approach the surface for position fixes. For deep water applications, the time and
energy needed by a small AUV for transiting to the surface from near the bottom are very unfavorable. Finally,
surfacing is impossible in ice-covered oceans.
14.2.2 Acoustic Navigation
When acoustic transponder measurements are available, a variety of algorithms are possible. Most techniques are based on recursive least-squares estimation,
often with an extended Kalman filter. As described
above, in an LBL navigation system, an array of
transponders is deployed and surveyed into position.
The array is usually calibrated through use of an additional acoustic transponder that is hung from a surface
ship and interrogates the array from various locations.
The vehicle sends out an acoustic signal which is
then returned by each beacon as it is received. Position
is determined by measuring the travel time between the
vehicle and each beacon, measuring or assuming the local sound speed profile, and knowing the geometry of
the beacon array. With this information, the relative distances between the vehicle and each array node can be
calculated. The two primary techniques are (1) to compute position fixes by locating the intersection point of
spheres of appropriate radii from the beacons in the array, and (2) to integrate the raw time-of-flight (TOF)
measurements into an appropriate Kalman filter.
The basic solution for computing a fix from spherical ranges to three transponders is as follows [14.14].
The local earth frame origin is at the surface. The x-axis
points north, the y-axis points east, and the z-axis points
down. The vehicle position in this frame is .x; y; z/ and
the coordinates of beacon i are .x i ; y i ; z i /. The measured round-trip travel times between the vehicle and
the beacons are t i and the associated distances are d i .
The distances are computed from the travel times by
the approximate relation: d i D c.t i i /=2, where c is
the average speed of sound (Note: spatial variations in
the speed of sound can be significant, especially variations with depth [14.11]).
The analytical solution based on measurements
from three beacons consists of solving the following
nonlinear system of equations for the vehicle coordinates
.x x i /
2
C .y y i /
2
C .z z i /
2
D d
2
i ;
i D 1; 2; 3 :
(14.1)
In order to make the computations easier, we first compute the solution in an intermediate frame obtained by
shifting the local earth frame at the location of beacon 1.
The beacon coordinates in this frame are .x
0
i ; y
0
i ; z
0
i / and
the vehicle position is .x
0
; y
0
; z
0
/. The intermediate solution is then transformed back to the local earth frame.
Advantage is also taken of the accurate knowledge of
the vehicle depth, leading to a linear over-constrained
problem (three equations known and two unknowns).
The vehicle position is then
x D
b 1 c 2 b 2 c 1
a 1 b 2 a 2 b 1
C x 1 ; y D
a 2 c 1 c 2 a 1
a 1 b 2 a 2 b 1
C y 1 ;
(14.2)