# Impact of Satellite Attitude on Altimetry Calibration with Microwave Transponders
## 3 Satellite Attitude and Transponder Calibration
### Determination of the Satellite Attitude
The \(z_{B}\) axis direction is chosen to form an orthogonal right-handed reference system. Ideal attitude of the satellite body is realized when the \(z_{B}\) axis is perpendicular to the reference ellipsoid. * The orbital reference system (ORB; [K: \(xyz\)]), with its origin at the satellite CoG (denoted by K in Figure 2). The \(x\) axis is along the radial vector connecting the geocenter \(O\) and the satellite CoG, K, with direction towards the zenith (reverse geocentric positioning [40]). The \(z\) axis is perpendicular to the satellite orbital planewith the same direction as the orbital angular momentum. Finally, the \(y\) axis has a direction to complete a right-hand orthogonal reference system (i.e., towards the satellite velocity vector but not always parallel to it). * The local orbital reference system or roll-pitch-yaw system (RPY; [K: \(x_{0}y_{0}z_{0}\)]), has its origin at the satellite CoG, K. Its \(z_{0}\) axis (yaw) is perpendicular to the reference ellipsoid with a direction towards nadir. The \(y_{0}\) axis (pitch) is perpendicular to the orbital plane with direction opposite to angular momentum. Finally, the \(x_{0}\) axis (roll) is defined to form an orthogonal reference system, with the same direction as the satellite velocity vector. The reference systems listed above are presented in Figure 2. The satellite attitude corresponds to the three-dimensional orientation of the satellite spacecraft in space, and it is commonly described by the roll, pitch and yaw angles, i.e., the angles needed to be applied in order to transform a vector from RPY to SAT system. In the sequel, the fundamental expressions used to estimate the satellite attitude angles are provided. The transformation of a column-vector \(\mathbf{X_{I}=}[U,V,W]^{\mathrm{T}}\) to \(\mathbf{X_{S}=}\left[x_{B},y_{B},z_{B}\right]^{\mathrm{T}}\), i.e., from the inertial IRS to the satellite body reference system SAT is performed as:  
\[\mathbf{X_{S}=} \begin{bmatrix}x_{B}\\ y_{B}\\ z_{B}\end{bmatrix} =\mathbf{R}_{\mathrm{IRS}\rightarrow\mathrm{SAT}} \begin{bmatrix}U \\ V\\ W\end{bmatrix} =\mathbf{R}_{\mathrm{IRS}\rightarrow\mathrm{SAT}}\mathbf{X_{I}}, \tag{1}\]  
where the matrix \(\mathbf{R}_{\mathrm{S1}\rightarrow\mathrm{S2}}\) denotes the rotation matrix (square matrix) that transforms a vector from reference system S1 to S2. The rotation matrix \(\mathbf{R}_{\mathrm{IRS}\rightarrow\mathrm{SAT}}\) is calculated using information stored in quaternions and can be expressed by the following sequence of rotations [18]:  
\[\mathbf{R}_{\mathrm{IRS}\rightarrow\mathrm{SAT}}=\mathbf{R}_{\mathrm{RPY} \rightarrow\mathrm{SAT}}\mathbf{R}_{\mathrm{ORB}\rightarrow\mathrm{RPY}} \mathbf{R}_{\mathrm{IRS}\rightarrow\mathrm{ORB}}. \tag{2}\]  
Solving Equation (2) with respect to \(\mathbf{R}_{\mathrm{RPY}\rightarrow\mathrm{SAT}}\) yields  
\[\mathbf{R}_{\mathrm{RPY}\rightarrow\mathrm{SAT}}=\mathbf{R}_{\mathrm{IRS} \rightarrow\mathrm{SAT}}\mathbf{R}_{\mathrm{IRS}\rightarrow\mathrm{ORB}}^{ \mathrm{T}}\mathbf{R}_{\mathrm{ORB}\rightarrow\mathrm{RPY}}^{\mathrm{T}}, \tag{3}\]  
Figure 2: Graphical representation of fundamental reference systems used for satellite attitude determination. The direction OA is defined by the vernal equinox \(\uppi\), and the direction OB denotes the prime meridian (\(\lambda=0^{\circ}\)). where the relation \(\mathbf{R}_{\mathrm{S1\to S2}}^{\mathrm{T}}=\mathbf{R}_{\mathrm{S1\to S2}}^{-1}= \mathbf{R}_{\mathrm{S2\to Sl}}\) applies, as a rotation matrix is always orthogonal and thus its transpose will be equal to its inverse. The rotation matrix \(\mathbf{R}_{\mathrm{IRS\to ORB}}\) can be derived from the satellite state vector (i.e., position vector \(\mathbf{r}\) and velocity vector \(\mathbf{v}\)) in the IRF system, as follows:  
\[\mathbf{R}_{\mathrm{IRS\to ORB}}= \begin{bmatrix}\mathbf{R}_{1}&\mathbf{R}_{2} &\mathbf{R}_{3}\end{bmatrix}