3. NAVIGATION ON THE SEA
3.3 How to locate the underwater robot VICTOR 6000
3.3.3 Dead reckoning
Since the 18th century navigators have calculated a ship’s future position with the help of the dead reckoning method. Dead reckoning is a process of determining an object’s position by applying a speed and direction to the last established position. Vectors represent all true courses and speeds (Garrison, 1999). The first point of dead reckoning method has to be initialized by another navigation method. In the case of VICTOR, the position of the first point is given from POSIDONIA acoustic navigation.
The Octans Inertial Unit indicates a direction and the Doppler Log measures changes in speed. The initialisation is actually the resetting of offsets to zero. Inertial navigation causes undesirable drifts. To reduce these offsets, initialisation is manually done almost every hour.
Table 3.4: Commonly used underwater vehicle navigation sensors (after Kinsey and Whitcomb 2003) – VICTOR sensors written bold.
Magnetic compass Heading 1° - 10°
Gyro compass – conventional Heading 0.1°
Gyro Compass – 3 axis, optical, north seeking
Angular position and velocity
0.1°
Bottom-lock Doppler sonar XYZ – velocity 1%
3.3.3.1 Polygonal parallel
As shown in Figure 3.16, dead reckoning on the sea can be considered equivalent to an open polygonal traverse in land surveying. Both methods are derived from measurements of distance and angle. In dead reckoning, coordinates of subsequent points can be calculated as a coordinate difference added to the position of a known initial point. Coordinate differences are acquired as sines and cosines of adapted angles multiplied by their respective distance.
In polygonal traverse, differences dx and dy increase with every added polygonal point. The coordinates are based on adjusted differences.
In Figure 3.17, two open polygonal traverses are marked black. The initialization positions are expressed by circles. If we suppose that initialized position (3) is correct, the polygonal traverse between the first (1) and the last point (3) can be adjusted. The offset value (marked red) will be split and assigned to every point depending on the distance from (3).
If initialization (3) is wrong, the adjustment between points (1) and (4) could be performed in order to get continuous positioning (green). It is also possible to shift the first polygonal traverse (blue) to link it to the second one.
Figure 3.17: Adjustment possibilities of VICTOR track line.
The problem of some tens of meters of drift under the water is solved by initialization. The dead reckoning drifts are not adjusted. In post-processing, the rough navigation errors are manually shifted in order to give a continuous robot track. It still has to be considered that land surveying methods undergo different environmental conditions than navigation methods in the water do.
3.3.3.2 Transformation between systems
By transformation between systems we understand here the transformation between the Doppler Log related coordinate system, Instrument related coordinate system (ROV) and World related coordinate system (POSIDONIA).
The Doppler Effect is often explained by a car’s horn or siren which is first higher-pitched and then lower-pitched with the passage of the car. The change in the pitch of the siren is actually a shift in the frequency of sound waves and is called the Doppler Shift. By the measurement of the rate of change of pitch, the vehicleʹs speed can be estimated.
VICTOR’s Doppler Velocity Log (DVL) emits pulses from four beam transducers in all directions. Transducers look down with an inclination angle of thirty degrees from the vertical axis of log.
(1 ) c0
f v
fR = T ⋅ + R (3.5)
As seen from Equation 3.5 (MathPages, n.d.), the apparent bottom velocity vR along each beam can be derived from the transmitted and received frequencies reflected by the bottom.
1 2
3
4
The transmitted frequency, fT, is 200 kHz, the received frequency fR is measured and c 0 is the speed of sound in water.
In Figure 3.18, two coordinate systems are displayed. The velocities are measured in the Doppler coordinate system and afterwards transformed to velocities in the directions of the X,Y and Z axes of the instrument coordinate system. The inclination angle has to be taken into account in the calculation and the directions of the four velocities have to be re-counted to velocities in the directions of the X and Y axes.
Figure 3.18: Doppler log and instrument coordinate systems.
Kinsey and Whitcomb (2003) describe the calculation for “Doppler to Instrument to World”
transformations. Instrument coordinates are the coordinates of the vehicle and the World coordinate system here is the higher-level geodetic system referring to the POSIDONIA array on R/V POLARSTERN.
Four ping responses enter the computation as a 4 × 1 vector of velocities v(beam) from the Doppler Log. Beam velocities are converted to instrument (local) XYZ velocities ip&d
(t )
where matrix T is a 3 × 4 constant matrix converting the four beam velocities into a 3 × 1 vector (3.6):The instrument velocity is transformed to the World frame considering the shift “Doppler to Instrument” with the rotation matrix R2 and “Instrument to World” (roll, pitch and heading of instrument) with the rotation matrix R1 (Equation 3.7):
)
The world velocities are integrated to allow the calculation of the bottom track position.
Vectorwp
ˆ
d(
t)
is initialised using the POSIDONIA estimated position at time t. The computation of position in the world coordinate system follows, as shown in Equation 3.8:τ
Precise three-dimensional navigation of the robot depends on the position of the vessel, or better, on the Common Reference Point and roll and pitch ascertainment. It is obvious that if the ship’s positioning fails, neither the depressor nor the robot’s position can be determined.
Besides the “underwater coordinate systems”, two further coordinate systems have to be taken into account in relation to the ship. In Chapter 3.3.2, three coordinate systems were mentioned: the Doppler related coordinate system, the Instrument (ROV) coordinate system and World coordinate system, where World coordinates were received from the POSIDONIA positioning.
Furthermore, another set of transformations, namely “World (POSIDONIA) – Ship – WGS84” has to be realised. The POSIDONIA coordinate system lies in the ship’s coordinate system and the ship’s coordinate system depends on GPS positioning in the world geodetic system WGS 84. Summing up, to get the coordinates, X, Y, of a bottom point, we have to transform through five coordinate systems.
With regard to Figure 3.4, the position of the underwater vehicle is deduced from Equation 3.9. The first matrix characterizes the ship’s coordinates (GPS) corrected for pitch and roll errors (MINS output). The POSIDONIA coordinates and conventional offsets between the acoustic array and the ship’s reference point are rotated in order to identify their coordinate systems. Matrix R3 is a rotation matrix (3.10) with roll, pitch and heading angles including calibration corrections (pers. comm. Jan Opderbecke, IFREMER, 2003).