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Frame transformation

The moments as seen by PEACE are measured in a frame (hereafter the P-frame) in which the x-axis ($X_p$) is aligned with the HEEA sensor head and $Z_p$ is the spin-axis. A schematic representation of the principle PEACE geometry is shown in Figure 5.2. This system is different from the engineered `body coordinates' (B-frame). In the B-frame the $X_b$ is the spin axis and $Y_b$ is 26.2 degrees antispinward of the Sun sensor. We wish to output our data in the physically informative GSE frame, so we must use attitude (AUX) data with the knowledge of the geometry of the various measurement frames to transform from the spacecraft to GSE frame.

Figure 5.2: Schematic showing the principle geometry of the PEACE experiment.

Image coords3

In general, a tensor ${\bf A}$ can be transformed to a new frame ${\bf A}^{\prime}$ by:
\begin{displaymath}\bf A^{\prime}=\bf R\bf A\bf R^{\rm T} \end{displaymath} (5.1)


where ${\bf R}$ is a transformation matrix. The transformation from the raw P-frame to the GSE-frame consists of four steps. Firstly there is a 60 degree rotation about $Z_p$ to take $Y_p$ into $Y_b$, described by:

\begin{displaymath}
\left[ \begin{array}{ccc}
\cos(60) & -\sin(60) & 0 \\
\sin(60) & \cos(60) & 0 \\
0 & 0 & 1
\end{array} \right]
\end{displaymath}


Next there is a 90 degree flip to take $X_p$ into $X_b$ - this is a 90 degree rotation about $Y_b$:

\begin{displaymath}
\left[ \begin{array}{ccc}
0 & 0 & 1 \\
0 & 1 & 0 \\
-1 & 0 & 0
\end{array} \right]
\end{displaymath}


Now we are in the B-frame, we must transform into the GSE frame. The inputs of latitude $\lambda$ and longitude $\phi$, and knowledge of the Sun's azimuthal position at PEACE's 0 degree (22.4 degrees) allow construction of two transformation matrices which deal with the spin axis orientation:

\begin{displaymath}
\left[ \begin{array}{ccc}
\cos(\lambda)\cos(\phi) & \beta(\s...
...cos(\phi)) & -\beta\cos(\lambda)\sin(\phi)
\end{array} \right]
\end{displaymath}


and the offset between $Y_b$ and the PEACE 0 degree azimuthal:

\begin{displaymath}
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos(22.4) & \sin(22.4) \\
0 & -\sin(22.4) & \cos(22.4)
\end{array} \right]
\end{displaymath}


The normalisation factor $\beta$ is
\begin{displaymath}\beta = (\sin^2(\lambda) + \cos^2(\lambda)\sin^2(\phi))^{-1/2}\end{displaymath} (5.2)


This choice of factor accounts for the constraint that the $t=0$ $Y_p$ axis is 22.4 degrees spinward of $Y_b$ and must have a positive $X_{\rm {GSE}}$ component.

Applied in the above sequence, the above matrices transform a tensor or vector from the raw P-frame into the desired GSE frame. All outputs from the solver are in GSE coordinates.


next up previous contents
Next: Useful information Up: Calibration Previous: Gain correction factors   Contents
Steve Schwartz 2005-03-26