Coordinate Frames and Basic Notation
Coordinate Frames
We adopt the following reference frames for mathematical modeling:
\(W\): a mathematically inertial frame (often chosen as ECI),
\(I\): a physical Earth–Centered Inertial frame (ECI),
\(E\): Earth–fixed rotating frame (ECEF).
Atmospheric velocity expressed in \(W\) is
where \(\boldsymbol{\omega}_\oplus\) is Earth’s rotation and \(\mathbf{r}_W\) is expressed in \(W\).
Frame Relationship Figure
Relationship between Earth-fixed frame \(E\), Earth’s rotation \(\boldsymbol{\omega}_\oplus\), and the inertial/ECI frame \(W \equiv I\).
Notation Summary
Position in inertial frame:
\[\mathbf{p}_W \in \mathbb{R}^3.\]Orientation unit quaternion and associated rotation matrices between inertial and body frames:
\[R_{WB} = \mathcal{R}(q_{WB}), \qquad R_{BW} = R_{WB}^\mathsf{T}.\]Spatial motion vector in body frame:
\[\begin{split}\mathbf{v}_B = \begin{bmatrix} \boldsymbol{\omega}_B \\ \mathbf{v}_B \end{bmatrix} \in \mathbb{R}^6.\end{split}\]Spatial inertia (simple block form):
\[\begin{split}\mathbf{I}_B(m) = \begin{bmatrix} \mathbf{J}_B(m) & \mathbf{0} \\ \mathbf{0} & m \mathbf{I}_3 \end{bmatrix}, \qquad \mathbf{I}_B(m) \in \mathbb{R}^{6\times 6}.\end{split}\]Cross operator \(\times\) operating on spatial motion vectors, and its dual \(\times^*\) operating on wrenches:
\[\begin{split}\mathbf{v}_B \times = \begin{bmatrix} [\boldsymbol{\omega}_B]_\times & \mathbf{0} \\ [\mathbf{v}_B]_\times & [\boldsymbol{\omega}_B]_\times \end{bmatrix}, \quad \mathbf{v}_B \times^{*} = -(\mathbf{v}_B \times)^\mathsf{T}.\end{split}\]