Kinematics, Forces, and Spatial Dynamics
Kinematics
Position
\[\dot{\mathbf{p}}_W
= R_{WB}\,\mathbf{v}_B.\]
Orientation
Angular velocity in the inertial frame relates to angular velocity in the body frame as
\[\boldsymbol{\omega}_W
= R_{WB}\,\boldsymbol{\omega}_B.\]
Quaternion kinematics:
\[\begin{split}\dot{q}_{WB}
=
\frac{1}{2}
\begin{bmatrix}
0 \\ \boldsymbol{\omega}_W
\end{bmatrix}
\otimes q_{WB}.\end{split}\]
Forces
Gravity
\[\mathbf{F}_{g,W} = m\,\mathbf{g}_W(\mathbf{p}_W),\]
\[\mathbf{F}_{g,B} = R_{BW}\,\mathbf{F}_{g,W},\]
\[\begin{split}\mathbf{f}_{g,B}
=
\begin{bmatrix}
0 \\ \mathbf{F}_{g,B}
\end{bmatrix}.\end{split}\]
Rotating Atmosphere and Aerodynamics
Earth rotation:
\[\begin{split}\boldsymbol{\omega}_\oplus
=
\begin{bmatrix}
0 \\ 0 \\ \omega_\oplus
\end{bmatrix}_W.\end{split}\]
Atmospheric velocity:
\[\mathbf{v}_{air,W}
= \boldsymbol{\omega}_\oplus \times \mathbf{p}_W.\]
Relative airspeed:
\[\mathbf{v}_W = R_{WB}\mathbf{v}_B,\]
\[\mathbf{v}_{rel,W}
=
\mathbf{v}_W - \mathbf{v}_{air,W} - \mathbf{v}_{wind,W}.\]
Aerodynamic drag:
\[\mathbf{F}_{aero,W}
=
-\frac{1}{2}\rho C_D A
\|\mathbf{v}_{rel,W}\|\,
\mathbf{v}_{rel,W}.\]
Transform to body frame:
\[\mathbf{F}_{aero,B}
= R_{BW}\mathbf{F}_{aero,W},\]
\[\begin{split}\mathbf{f}_{aero,B}
=
\begin{bmatrix}
0 \\ \mathbf{F}_{aero,B}
\end{bmatrix}.\end{split}\]
Thrust
\[\mathbf{F}_{T,B} = T(t)\hat{\mathbf{t}}_B,\]
\[\begin{split}\mathbf{f}_{T,B}
=
\begin{bmatrix}
0 \\ \mathbf{F}_{T,B}
\end{bmatrix}.\end{split}\]
Control Torques
\[\begin{split}\mathbf{f}_{ctrl,B}
=
\begin{bmatrix}
\boldsymbol{\tau}_{ctrl,B}(t,x) \\ 0
\end{bmatrix}.\end{split}\]
Total Spatial Wrench
\[\mathbf{f}_B
=
\mathbf{f}_{T,B}
+
\mathbf{f}_{g,B}
+
\mathbf{f}_{aero,B}
+
\mathbf{f}_{ctrl,B}.\]
Spatial Dynamics
Instead of a plain Newton–Euler formalism, six degree-of-freedom motion can be expressed with a single equation using SVA (all terms in body frame):
\[\mathbf{I}_B(m)\mathbf{a}_B
+
\mathbf{v}_B \times^{*} \bigl(\mathbf{I}_B(m)\mathbf{v}_B\bigr)
=
\mathbf{f}_B.\]
Solving for acceleration:
\[\mathbf{a}_B
=
\mathbf{I}_B^{-1}
\left[
\mathbf{f}_B
-
\mathbf{v}_B \times^{*}(\mathbf{I}_B\mathbf{v}_B)
\right].\]
Component form:
\[\begin{split}\mathbf{a}_B
=
\begin{bmatrix}
\dot{\boldsymbol{\omega}}_B \\
\dot{\mathbf{v}}_B
\end{bmatrix}.\end{split}\]
Mass Dynamics
\[\dot{m}(t) = -\dot{m}_{flow}(t),\]
\[T(t) = \dot{m}_{flow}(t)\,g_0\,I_{sp}.\]
Final ODE System
The system of ODEs is summarized in canonical form \(\dot{x} = f(x)\):
\[\begin{split}\boxed{
\begin{aligned}
\dot{\mathbf{p}}_W &= R_{WB}\,\mathbf{v}_B, \\[6pt]
\dot{q}_{WB} &=
\frac{1}{2}
\begin{bmatrix}
0 \\
R_{WB}\boldsymbol{\omega}_B
\end{bmatrix}
\otimes q_{WB}, \\[10pt]
\begin{bmatrix}
\dot{\boldsymbol{\omega}}_B \\
\dot{\mathbf{v}}_B
\end{bmatrix}
&=
\mathbf{I}_B^{-1}
\left[
\mathbf{f}_B
- \mathbf{v}_B\times^{*} (\mathbf{I}_B\mathbf{v}_B)
\right], \\[10pt]
\dot{m} &= -\dot{m}_{flow}(t).
\end{aligned}
}\end{split}\]