APPENDIX
Observation of a fixed target from a moving vessel
Coordinate system
The vessel roll, pitch and yaw motions are observed in the righthanded level
vessel coordinate system (LVCS) shown in Figure A1. Its origin is at the vessel's centre of motion and
its x and yaxes are in a horizontal plane. The xaxis points in the direction of the vessel's forward
motion. Vessel roll, pitch and yaw correspond to rotations about the x, y and zaxes, respectively. The
direction of positive rotation is given by the curved arrows and is defined by the right hand rule, In
calm weather, the LVCS coincides with the vessel coordinate system (VCS) which has its axes aligned with
the vessel's stern to bow, port to starboard and vertical axes, respectively. We define roll, pitch and
yaw in the LVCS as shown in Figure A1. However, roll, pitch and yaw are often defined in the VCS and may
include sign conventions for the rotation directions based on naval traditions ( Hare
et al., 1995). We have chosen to develop our model in the LVCS for simplicity and as average results
will be very similar.
Rotation of a point in a fixed coordinate system
where (ψ_{x}, ψ_{y} and ψ_{z} are the roll, pitch and yaw angles. Following a roll, pitch and yaw rotation the location of a transducer fixed to the vessel is given by,
A_{12}=R_{z}R_{y}R_{x}A_{11} (A2)
where A_{11} gives the location of the transducer on the vessel before rotation (Fig.A1) and A_{12} is the new location (Fig.A2) and A_{11}, A_{12} and P_{11} are column vectors. The first 1 in the subscript indicates that we are using the LVCS. The position of the target remains unchanged at P_{11}.
Observation of a fixed point from a rotating coordinate system
To find the changed transducer view angle we now introduce the level transducer coordinate system (LTCS) that is obtained by translating the origin of the LVCS to the rotated transducer location A_{12}. The target position in the LTCS coordinate system is:
P_{21}=P_{11}A_{12} (A3)
To account for the changed transducer view angle we now rotate the LTCS. This rotation differs from the previous one as we now rotate the coordinate system rather than a point in a fixed coordinate system. This is accomplished by the use of the inverse rotation matrices in the following equation:
P_{31}=R_{x}^{1}R_{y}^{1}R_{z}^{1}P_{21} (A4)
where P_{31} is the position of the target in the transducer
coordinate system (TCS) shown in Figure A3, and R_{x}^{1},
R_{y}^{1}, and R_{z}^{1}
are the inverse rotation matrices obtained by using the negative rotation angles ψ_{x}
ψ_{y} and ψ_{z} in equation A1.
To calculate the beampattern for the observed target we require the angular position of the target as seen in the TCS. It is obtained by using the (x, y, z) coordinates of P_{31} in the following conversions.
Conversion from Cartesian to Spherical coordinates
Spherical coordinates are shown in Figure A4. The transformation of vector (x, y, z) from Cartesian to spherical coordinates is provided by:
Conversion from Cartesian to Splitbeam coordinates
Splitbeam coordinates are shown in Figure A5. The transformation of vector (x,y,z) from Cartesian to splitbeam coordinates is:
Target position in the LVCS
Although the target is fixed in an earth coordinate system, it will appear to move relative to the vessel. In the LVCS, the target position will be given by:
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where (x_{t} , y_{t} , z_{t})
is the initial target position (x_{v }, y_{v }, z_{v})
and (v_{x }, v_{y }, v_{z}) are the
vessel's initial position and velocity and s, w and h are vessel surge, sway and heave, respectively.
All are measured in the right handed earth coordinate system that, at t=0, coincides with the LVCS, but
with inverted y and zaxes. Thus, vessel heave will be positive when the vessel is on a wave crest. Generally
x_{t}, y_{t}, x_{v}, y_{v},
z_{v} and v_{z} will be zero.
Figure A1
Figure A2
Figure A3
Figure A4
Figure A5
