Determining the Attitude of a Vessel

 Figure 8a The attitude of the vessel is defined by the tilts of that vessel's frame with respect to "local" coordinate axes. It is sufficient to define the vessel's frame by three non-collinear points fixed on its body which were represented by the phase centers of three GPS antennas mounted close to the vessel's deck. The antennas were positioned such that the base line vector formed by antennas 1 and 3 was parallel to the vessel's longitudinal axis (Figure 8a.) and the base line perpendicular to it was defined by antennas 1 and 2. There was a redundant fourth antenna also available on the vessel's deck. Vectors (X, Y, Z) to antenna phase centers were determined from a base station on the pier with GPS kinematic measurements in the International Terrestrial Reference Frame (ITRF 94) geocentric coordinate system. The heading, pitch, and roll of the vessel can then be determined with vectors between antennas (x, y, z) which are transformed into the local (horizon) coordinate frame of east, north, and up (e,n,u) at the first antenna using its geodetic latitude () and longitude ().

The transformation of geocentric vectors to the topocentric local frame was done by the following rotation:

 | e | = | -sin cos 0 | | x | | n | = | -sincos -sinsin cos | | y | | u | = | coscos cossin sin | | z |

The heading and pitch of the vessel are computed using the following formulas:

 pitch = tan-1 u13[ ( e13)2 + ( n13 ) 2 ] ½ (2)

As the angle () between vectors 1 to 2 and 1 to 3 was very closely 90o, the roll was computed using the following expression

 roll = tan-1 -u12[ ( e12)2 + ( n12 ) 2 ] ½ (3)
 If the angle() is not equal to 90o, the roll must take into account that vector 1 to 2 is not perpendicular to the vessel's longitudinal axis (Figure 8b). The following formulas can be used under the collinearity condition of vectors 1 to 3 and 1 to 5 (El-Mowafy and Schwarz 1994): Figure 8b
 roll = tan-1 -u52[ ( e52)2 + ( n52 ) 2 ] ½ (4)

The components of vector 5 to 2 are defined by the differences

 e52 = e12 - e15 (5) n52 = n12 - n15 u52 = u12 - u15

and the components of vector 1 to 5 are obtained through the scaled components of 1 to 3

 e15=·e13; n15=·n13; u15=·u13; (6)

where the scale factor  =  b15  /  b12

in which "b" are the magnitudes of the indicated vectors

 b15 = b12 · cos (7)

and cos is obtained from the dot product of vectors 1 to 2 (B12) and 1 to 3 (B13)

 cos = B12 · B13 b12 · b13 (8)

The GPS antennas were placed on the BUTTONWOOD in a configuration where was very close to 90 degrees, so two sets of roll and pitch values could be computed: roll 1 was computed using stern port and stern starboard antennas and roll 2 was computed using bow port and bow starboard antennas; pitch 1 was computed using stern port and bow port antennas and pitch 2 was computed using stern starboard and bow starboard antennas. The internal angles between the antennas were computed using a static solution obtained while the ship was docked. The internal angles between the stern port/bow port and stern port/stern starboard is 89.9o, between stern starboard/stern port and stern starboard/bow starboard it is 89.9o, between bow port/stern port and bow port/bow starboard it is 89.6o, and between bow starboard/bow port and bow starboard/stern starboard it is 90.4o. The distance between the antennas on the stern is 5.47 meters and on the bow it is 5.42 meters. The distance between the antennas on the port side is 46.50 meters and on the starboard side it is 46.46 meters.

It should be noted that the two antennas located on the bow were not exactly level. The BUTTONWOOD was used as a ship of opportunity and NGS did not have sufficient time to manufacture specially-designed equipment to precisely attach the antennas to the BUTTONWOOD. The antennas not being level will have a small effect on the results. The antenna configuration will increase the errors between the two sets of computed pitch and roll values because some small common errors will not cancel, i.e., common variations in phase centers when using similar antennas types. It was estimated that the total tilt between the two antennas was less than 15o and that the maximum error was less than 1 cm, and did not significantly influence the conclusions. The purpose of this phase was to prove the concept and identify potential problems in placing antennas on ships. These data were helpful in both cases.