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How is a coordinate transformation done?

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Nortek Support
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Raw velocity measurements are given in beam coordinates, which is a vector in the direction along each of the beams. Depending on the objective of the measurements, coordinate transformation may be beneficial. Beam coordinates can be converted to a Cartesian coordinate system (XYZ) or Earth normal coordinates (ENU: East, North and Up). The orientation of the beams/transducers, which is defined in the instrument specific transformation matrix, is used to convert from beam coordinates to XYZ. By additionally knowing the orientation in space, such as tilt and compass heading, ENU coordinates can be determined.

During transformation to ENU only the slanted beams are used in the transformation. The data represented in ENU originates only from these beams, and the center beam (if collecting current data) is given individually and always in beam coordinates.

If the instrument is configured to measure data in XYZ or ENU, this coordinate transformation is automatically performed by the firmware and the output data will be in the selected coordinate system. The transformation can also be done in post-processing software. If the data is collected in beam coordinates, the transformation can also be carried out manually if desired. How the manual transformation from beam coordinates through XYZ to ENU is done is described below for each type of instrument.

FAQ-coord2.png

Figure 1: The three relevant coordinate systems. The figure shows an AWAC, but this applies to the other instruments as well.

 

3-beam instruments 

The starting point is a binary-to-ASCII conversion of the raw data file so that you have access to the three velocity files,*.v1, *.v2 and *.v3, the transformation matrix in the *.hdr file, and the *.sen file. Heading, pitch and roll are presented in the *.sen file (ref. the *.hdr file), and these angles are output in degrees. The velocity data in .*.v1, *.v2 and *.v3 are measured in of the the 3 coordinate systems described above - Beam, XYZ or ENU. The coordinate system selected during the deployment planning is indicated in the *.hdr file.

Each instrument has its own unique transformation matrix T, based on the transducer geometry. It is used to transform raw data collected in BEAM coordinates to XYZ coordinates. 

It's location for each instrument type is described in this FAQ: Where to find the transformation matrix?

 

Beam to XYZ:

The transformation between beam and XYZ coordinates is done using the original T matrix. In matrix form (using MATLAB notation, so [ X; Y; Z ] is a column vector), the math is:

\begin{equation}
T \times [B1;B2;B3;] = [X;Y;Z]
\end{equation}		 (1)

And to convert from XYZ to Beam:

\begin{equation}
T^{-1} \times [X;Y;Z;] = [B1;B2;B3;]
\end{equation}		 (2)

 

Beam or XYZ to ENU:

When transforming to ENU, the transformation matrix T needs to be adjusted depending on the orientation of the instrument. If instrument is pointing down (bit 0 in status equal to 1) rows 2 and 3 must change sign! The default orientation is upward looking!

Next the recorded heading, pitch and roll needs to be taken into account. These are used to create a second matrix R that corrects for the instruments position on earth. Whereas the first matrix is constant throughout the deployment, this one needs to be recalculated for each timestep.

Equation (4) and (5) illustrate how to obtain a matrix for heading, pitch and roll.

\begin{equation} hdg = hdg - 90 \end{equation}

\begin{equation}R_z = \begin{bmatrix}
\cos(hdg) & - \sin(hdg) & 0 \\
\sin(hdg) & \cos(hdg) & 0 \\
0 & 0 & 1
\end{bmatrix}
\end{equation}

 (4)

NB: It is necessary to subtract 90° from the recorded heading to account for the instruments x-orientation (Heading of 0° = x pointing North in raw data)

\begin{equation}
R_y = \begin{bmatrix}
\cos(\text{pch}) & 0 & \sin(\text{pch}) \\
0 & 1 & 0 \\
- \sin(\text{pch}) & 0 & \cos(\text{pch})
\end{bmatrix}
\end{equation}

 (5)

 

\begin{equation}
R_x = \begin{bmatrix}
1 & 0 & 0 \\
0 & \cos(\text{rll}) &- \sin(\text{rll}) \\
0& \sin(\text{rll}) & \cos(\text{rll})
\end{bmatrix}
\end{equation}

 (6)

 

\begin{equation}
R = R_Z \times R_Y\times R_X 
\end{equation}

 (7)
    The following steps outline the transformation from Beam to ENU coordinates:
  1. Find the instrument specific transformation matrix.
  2. Make a heading matrix. 
  3. Make a tilt matrix.
  4. Make a new transformation matrix (R), including heading, pitch and roll. 
  5. The coordinate system transformations can then be carried out using R and T as shown in (8).
  6. When using the inverse of  R /  T the transformation can carried out in the opposite direction.

\begin{equation} [ E; N; U ] = R \times T \times[ B1; B2; B3 ] \end{equation}

\begin{equation} [ B1; B2; B3 ] = T^{-1} \times  R^{-1}  \times[ E; N; U ] \end{equation}

\begin{equation} [ E; N; U ] = R  \times [ X; Y; Z ] \end{equation}

\begin{equation} [ X; Y; Z ] =  R^{-1} \times [ E; N; U ] \end{equation}

 (8)

 

Attached below this page are Python and Matlab scripts that can be used to carry out the above described transformations for a 3-beam system.

 

Signatures

To carry out a coordinate transformation, it is first necessary to convert the raw *.ad2cp data to *.mat using Signature Deployment. 
 
The transformation is carried out in two steps as for the 3- beam instruments. Therefore the detailed formulas will not be repeated here. 
 
Beam to XYZ:

Each instrument has its own unique transformation matrix T, based on the transducer geometry.  It is used to transform raw data collected in BEAM coordinates to XYZ coordinates. Each row of the matrix represents a component in the instrument’s XYZ coordinate system, starting with X at the top row. Each column represents a beam. The third and fourth rows of the Signature transformation matrix represent the two estimates of vertical velocity (Z1 and Z2) produced by the instrument.

It's location for each instrument type is described in this FAQ:  Where to find the transformation matrix?

Applying this matrix to the velocity vector in beam coordinates  will return the components in the XYZ system:

\begin{equation}
T \times [B1;B2;B3;B4] = [X;Y;Z1;Z2]
\end{equation}		 (9)

 

Beam or XYZ to ENU:

Next the recorded heading, pitch and roll needs to be taken into account. These are used to create a second matrix R that corrects for the instruments position on earth. Whereas the first matrix is constant throughout the deployment, this one needs to be recalculated for each timestep.

 

The calculation of this matrix depends on the compass sensor type installed in your instrument. If an AHRS is available, R is provided as the AHRS transformation matrix in the raw data set. For details on the AHRS  please see this FAQ: Do you have any information about the Signature AHRS?

Should this not be the case, the heading, pitch and roll (Eulerian angles) are used to account for the instruments orientation. The calculation of R using these three parameters is described in detail in the 3-beam chapter.

  • Independent of the choice between the matrix and Eulerian angles it is necessary to subtract 90° from the recorded heading to account for the instruments x-orientation (Heading of 0° = x pointing North in raw data)

Additional adjustments need to be made, that are only necessary when using the Eulerian angles (heading, pitch, roll):

  • The orientation of the instrument needs to be considered if no AHRS is installed, since the default orientation is upward looking. When transforming to ENU, the transformation matrix T needs to be adjusted for a downward looking installation. Then rows 2, 3 and 4 (corresponding to Y, Z1, Z2 velocities) must change sign!
  • Additionally, the 3x3 matrix R obtained from heading, pitch and roll needs to be expanded to 4x4 to account for the 4th beam and thereby the second Z/Up velocity. This is done using equation (10):

\begin{equation}
R_{4beam} = \begin{bmatrix}
R_{11} & R_{12} & \frac{R_{13}}{2}& \frac{R_{13}}{2} \\
R_{21} & R_{22} & \frac{R_{23}}{2}& \frac{R_{23}}{2} \\
R_{31} & R_{32} & R_{33}& 0\\                                              
R_{31} & R_{32} & 0& R_{33}\\                             \end{bmatrix}
\end{equation}

 (10)

Attached below this page are Python and Matlab scripts that can be used to carry out the above described transformations for raw Signature data.

 

 

Notes on the AHRS data output:

Data output is instantaneous heading, pitch, roll, acceleration, magnetometer- and gyro data, and orientation matrices.

Both the rotation matrix and heading/pitch/roll are based on the quaternions and can be used to carry out coordinate transformation between data in the XYZ to ENU coordinates. We recommend using the matrix directly, since rounding errors are introduced due to multiple transformations when using the Eulerian angles. If choosing to use the transformation matrix, it is not necessary to account for the instruments orientation!

Additionally to the correction for the bearings of the instrument, it is also possible to correct for its motion. This can be done using the accelerometer output from which the acceleration due to gravity needs to be subtracted. The remaining acceleration integrated over time represents a measure of the instruments translational velocity. When subtracting this from the ENU velocities, the dynamic motion is corrected for. 

 

 

 

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