Equations of Motion v1#
Introduction#
This document briefly describes the equations of motion which govern the flight of a rocket used in v1.0 onwards. This document simply shows some of the algebraic steps used to get to the final form of the equations of motion used in the code. For a more detailed explanation of the equations of motion, please refer to Equations of Motion v0.
Development#
Linear Equation:
\[\begin{split}\begin{aligned}
m\left(\dot{\mathbf{v}}+\dot{\boldsymbol{\omega}} \times \mathbf{r}_{\mathrm{CM}}\right. & \left.+\boldsymbol{\omega} \times\left(\boldsymbol{\omega} \times \mathbf{r}_{\mathrm{CM}}\right)+\mathbf{r}_{\mathrm{CM}}^{\prime \prime}+2 \boldsymbol{\omega} \times \mathbf{r}_{\mathrm{CM}}^{\prime}\right) \\
= & \mathbf{T}-2 \dot{m} \mathbf{r}_{\mathrm{CM}}^{\prime}+2 \boldsymbol{\omega} \times \dot{m}\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right)+\ddot{m}\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right)+\mathbf{A}+\sum_i \mathbf{N}_i-m g \hat{\mathbf{a}}_3
\end{aligned}\end{split}\]
Angular Equation:
\[\mathbf{I} \cdot \dot{\boldsymbol{\omega}}+\boldsymbol{\omega} \times(\mathbf{I} \cdot \boldsymbol{\omega})+\mathbf{I}^{\prime} \cdot \boldsymbol{\omega}+m \mathbf{r}_{\mathrm{CM}} \times \dot{\mathbf{v}}=\left(\dot{m} \mathbf{S}_{\mathrm{noz}}\right) \cdot \boldsymbol{\omega}+\sum_i \mathbf{r}_i \times \mathbf{N}_i-\mathbf{r}_{\mathrm{CM}} \times m g \hat{\mathbf{a}}_3\]
Cross multiplying the linear equation by 𝐫CM:
\[\begin{split}\begin{aligned}
m \mathbf{r}_{\mathrm{CM}} \times \dot{\mathbf{v}}+m \mathbf{r}_{\mathrm{CM}} \times\left(\dot{\boldsymbol{\omega}} \times \mathbf{r}_{\mathrm{CM}}\right)+m \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega} \times\left(\boldsymbol{\omega} \times \mathbf{r}_{\mathrm{CM}}\right)+m \mathbf{r}_{\mathrm{CM}} \times \mathbf{r}_{\mathrm{CM}}^{\prime \prime}+2 m \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega} \times \mathbf{r}_{\mathrm{CM}}^{\prime} \\
\quad=\mathbf{r}_{\mathrm{CM}} \times \mathbf{T}-2 \dot{m} \mathbf{r}_{\mathrm{CM}} \times \mathbf{r}_{\mathrm{CM}}^{\prime}+2 \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega} \times \dot{m}\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right)+m \mathbf{r}_{\mathrm{CM}}^{\prime} \times\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right) \\
\quad+\mathbf{r}_{\mathrm{CM}} \times \mathbf{A}+\mathbf{r}_{\mathrm{CM}} \times \sum_i \mathbf{N}_i-m \mathbf{r}_{\mathrm{CM}} \times g \hat{\mathbf{a}}_3
\end{aligned}\end{split}\]
Simplifying:
\[\mathbf{r}_{\mathrm{CM}} \times\left(\dot{\omega} \times \mathbf{r}_{\mathrm{CM}}\right)=\left(\mathbf{r}_{\mathrm{CM}} \cdot \mathbf{r}_{\mathrm{CM}}\right) \dot{\boldsymbol{\omega}}-\left(\mathbf{r}_{\mathrm{CM}} \cdot \dot{\boldsymbol{\omega}}\right) \mathbf{r}_{\mathrm{CM}}\]
\[\mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega} \times\left(\boldsymbol{\omega} \times \mathbf{r}_{\mathrm{CM}}\right)=\mathbf{r}_{\mathrm{CM}} \times\left(\left(\boldsymbol{\omega} \cdot \mathbf{r}_{\mathrm{CM}}\right) \boldsymbol{\omega}-(\boldsymbol{\omega} \cdot \boldsymbol{\omega}) \mathbf{r}_{\mathrm{CM}}\right)=\left(\boldsymbol{\omega} \cdot \mathbf{r}_{\mathrm{CM}}\right) \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega}\]
\[\begin{split}\begin{aligned}
m \mathbf{r}_{\mathrm{CM}} \times \dot{\mathbf{v}}=- & m\left(\mathbf{r}_{\mathrm{CM}} \cdot \mathbf{r}_{\mathrm{CM}}\right) \dot{\boldsymbol{\omega}}+m\left(\mathbf{r}_{\mathrm{CM}} \cdot \dot{\boldsymbol{\omega}}\right) \mathbf{r}_{\mathrm{CM}}-m\left(\boldsymbol{\omega} \cdot \mathbf{r}_{\mathrm{CM}}\right) \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega}-m \mathbf{r}_{\mathrm{CM}} \times \mathbf{r}_{\mathrm{CM}}^{\prime \prime} \\
& -2 m \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega} \times \mathbf{r}_{\mathrm{CM}}^{\prime}+\mathbf{r}_{\mathrm{CM}} \times \mathbf{T}-2 \dot{m} \mathbf{r}_{\mathrm{CM}} \times \mathbf{r}_{\mathrm{CM}}^{\prime}+2 \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega} \times \dot{m}\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right) \\
& +\ddot{m} \mathbf{r}_{\mathrm{CM}} \times\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right)+\mathbf{r}_{\mathrm{CM}} \times \mathbf{A}+\mathbf{r}_{\mathrm{CM}} \times \sum_i \mathbf{N}_i-m \mathbf{r}_{\mathrm{CM}} \times g \hat{\mathbf{a}}_3
\end{aligned}\end{split}\]
Substituting in the Angular equation:
\[\begin{split}\begin{aligned}
\mathbf{I} \cdot \dot{\boldsymbol{\omega}}+\boldsymbol{\omega} \times(\mathbf{I} \cdot \boldsymbol{\omega})+\mathbf{I}^{\prime} \cdot \boldsymbol{\omega}-m\left(\mathbf{r}_{\mathrm{CM}} \cdot \mathbf{r}_{\mathrm{CM}}\right) \dot{\boldsymbol{\omega}}+m\left(\mathbf{r}_{\mathrm{CM}} \cdot \dot{\boldsymbol{\omega}}\right) \mathbf{r}_{\mathrm{CM}}-m\left(\boldsymbol{\omega} \cdot \mathbf{r}_{\mathrm{CM}}\right) \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega}-m \mathbf{r}_{\mathrm{CM}} \times \mathbf{r}_{\mathrm{CM}}^{\prime \prime} \\
-2 m \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega} \times \mathbf{r}_{\mathrm{CM}}^{\prime}+\mathbf{r}_{\mathrm{CM}} \times \mathbf{T}-2 \dot{m} \mathbf{r}_{\mathrm{CM}} \times \mathbf{r}_{\mathrm{CM}}^{\prime}+2 \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega} \times \dot{m}\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right) \\
+\ddot{m} \mathbf{r}_{\mathrm{CM}} \times\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right)+\mathbf{r}_{\mathrm{CM}} \times \mathbf{A}+\mathbf{r}_{\mathrm{CM}} \times \sum_i \mathbf{N}_i-m \mathbf{r}_{\mathrm{CM}} \times g \hat{\mathbf{a}}_3 \\
=\left(\dot{m} \mathbf{S}_{\mathrm{noz}}\right) \cdot \boldsymbol{\omega}+\sum_i \mathbf{r}_i \times \mathbf{N}_i-\mathbf{r}_{\mathrm{CM}} \times m g \hat{\mathbf{a}}_3
\end{aligned}\end{split}\]
\[\begin{split}\begin{aligned}
& \mathbf{I} \cdot \dot{\boldsymbol{\omega}}-m\left(\mathbf{r}_{\mathrm{CM}} \cdot \mathbf{r}_{\mathrm{CM}}\right) \dot{\boldsymbol{\omega}}+m\left(\mathbf{r}_{\mathrm{CM}} \cdot \dot{\boldsymbol{\omega}}\right) \mathbf{r}_{\mathrm{CM}}=-\boldsymbol{\omega} \times(\mathbf{I} \cdot \boldsymbol{\omega})-\mathbf{I}^{\prime} \cdot \boldsymbol{\omega}+m\left(\boldsymbol{\omega} \cdot \mathbf{r}_{\mathrm{CM}}\right) \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega}+ \\
& m \mathbf{r}_{\mathrm{CM}} \times \mathbf{r}_{\mathrm{CM}}^{\prime \prime}+2 m \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega} \times \mathbf{r}_{\mathrm{CM}}^{\prime}-\mathbf{r}_{\mathrm{CM}} \times \mathbf{T}+2 \dot{m} \mathbf{r}_{\mathrm{CM}} \times \mathbf{r}_{\mathrm{CM}}^{\prime}-2 \mathbf{r}_{\mathrm{CM}} \times \boldsymbol{\omega} \times \dot{m}\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right)- \\
& \ddot{m} \mathbf{r}_{\mathrm{CM}} \times\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right)-\mathbf{r}_{\mathrm{CM}} \times \mathbf{A}-\mathbf{r}_{\mathrm{CM}} \times \sum_i \mathbf{N}_i+\left(\dot{m} \mathbf{S}_{\mathrm{noz}}\right) \cdot \boldsymbol{\omega}+\sum_i \mathbf{r}_i \times \mathbf{N}_i
\end{aligned}\end{split}\]
Solving for 𝐯̇:
Linear
\[\begin{split}\begin{gathered}
\dot{\mathbf{v}}=\frac{\left(\mathbf{T}-2 \dot{m} \mathbf{r}_{\mathrm{CM}}^{\prime}+2 \boldsymbol{\omega} \times \dot{m}\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right)+\ddot{m}\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right) \mathbf{A}+\sum_i \mathbf{N}_i\right)}{m}-g \hat{a}_3-\dot{\boldsymbol{\omega}} \times \mathbf{r}_{\mathrm{CM}} \\
-\boldsymbol{\omega} \times\left(\boldsymbol{\omega} \times \mathbf{r}_{\mathrm{CM}}\right)-\mathbf{r}_{\mathrm{CM}}^{\prime \prime}-2 \boldsymbol{\omega} \times \mathbf{r}_{\mathrm{CM}}^{\prime}
\end{gathered}\end{split}\]
Angular
\[\mathbf{r}_{\mathrm{CM}} \times \dot{\mathbf{v}}=\frac{\left(\left(\dot{m} \mathbf{S}_{\mathrm{noz}}\right) \cdot \boldsymbol{\omega}+\sum_i \mathbf{r}_i \times \mathbf{N}_i-\mathbf{I} \cdot \dot{\boldsymbol{\omega}}-\boldsymbol{\omega} \times(\mathbf{I} \cdot \boldsymbol{\omega})-\mathbf{I}^{\prime} \cdot \boldsymbol{\omega}\right)}{\boldsymbol{m}}-\mathbf{r}_{\mathrm{CM}} \times g \hat{\mathbf{a}}_3\]
\[\mathbf{b}=\frac{\left(\left(\dot{m} \mathbf{S}_{\mathrm{noz}}\right) \cdot \boldsymbol{\omega}+\sum_i \mathbf{r}_i \times \mathbf{N}_i-\mathbf{I} \cdot \dot{\boldsymbol{\omega}}-\boldsymbol{\omega} \times(\mathbf{I} \cdot \boldsymbol{\omega})-\mathbf{I}^{\prime} \cdot \boldsymbol{\omega}\right)}{\boldsymbol{m}}-\mathbf{r}_{\mathrm{CM}} \times g \hat{\mathbf{a}}_3\]
\[\mathbf{r}_{\mathrm{CM}} \times \dot{\mathbf{v}}=\boldsymbol{b}\]
Reorganized EOM to aid execution speed
Linear
\[\begin{split}\begin{aligned}
m \dot{\mathbf{v}}+\left[m \mathbf{r}_{\mathrm{CM}}\right]_{\times}^T & \cdot \dot{\boldsymbol{\omega}} \\
& =-\omega \times\left(\omega \times m \mathbf{r}_{\mathrm{CM}}\right)+\omega \times\left(2 \dot{m}\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right)-2 m \mathbf{r}_{\mathrm{CM}}^{\prime}\right)+\mathrm{T}-m \mathbf{r}_{\mathrm{CM}}^{\prime \prime}-2 \dot{m} \mathbf{r}_{\mathrm{CM}}^{\prime} \\
& +\ddot{m}\left(\mathbf{r}_{\mathrm{noz}}-\mathbf{r}_{\mathrm{CM}}\right)-m g \hat{\mathbf{a}}_3+\mathbf{A}+\sum_i \mathbf{N}_i
\end{aligned}\end{split}\]
Angular
\[\mathbf{I} \cdot \dot{\boldsymbol{\omega}}+\left[m \mathbf{r}_{\mathrm{CM}}\right]_{\times} \cdot \dot{\mathbf{v}}=-\omega \times(\mathbf{I} \cdot \omega)+\left(\dot{m} \mathbf{S}_{\mathrm{noz}}-\mathbf{I}^{\prime}\right) \cdot \omega-\mathbf{r}_{\mathrm{CM}} \times m g \hat{\mathbf{a}}_3+\sum_i \mathbf{r}_i \times \mathbf{N}_i\]
Available terms that must be interpolated in time/altitude
𝑚
𝑚’
𝑚’’
𝐫CM
𝐫CM’
𝐫CM’’
𝐓
𝐈
𝐈’
𝑔
Pre-computed terms that optimize interpolations needed
𝑚
𝐫CM’
T03: 2𝑚̇ (𝐫noz − 𝐫CM) − 2𝑚𝐫CM
T04: 𝐓 − 𝑚𝐫CM′′ − 2𝑚̇ 𝐫CM + 𝑚̈ (𝐫noz − 𝐫CM)
T05: 𝑚̇ 𝐒noz − 𝐈′
𝑔
𝐈
Pre-computed terms
T00: 𝑚𝐫CM
T01: [𝑚𝐫CM]×
T02: [𝑚𝐫CM]×𝑇′
T03: 2𝑚̇ (𝐫noz − 𝐫CM) − 2𝑚𝐫CM
T04: 𝐓 − 𝑚𝐫CM′′ − 2𝑚̇ 𝐫CM + 𝑚̈ (𝐫noz − 𝐫CM)
T05: 𝑚̇ 𝐒noz − 𝐈′
T20: −𝝎 × (𝝎 × 𝑇00) + 𝝎 × (𝑇03) + 𝑇04 − 𝑚𝑔𝐚̂3 + 𝐀 + ∑ 𝐍𝑖
T21: −𝝎 × (𝐈 ⋅ 𝝎) + (𝑇05) ⋅ 𝝎 + 𝐫CM × 𝑚𝑔𝐚̂3 + ∑ 𝐫𝑖 × 𝐍𝑖
Final system of equations
- \[\mathrm{M} \cdot \dot{\mathbf{v}}+\left[m \mathbf{r}_{\mathrm{CM}}\right]_{\times}^T \cdot \dot{\boldsymbol{\omega}}=T_{20}\]
- \[\mathbf{I} \cdot \dot{\boldsymbol{\omega}}+\left[\mathrm{mr}_{\mathrm{CM}}\right]_x \cdot \dot{\mathbf{v}}=T_{21}\]
Solution to system of equations
\[\dot{\boldsymbol{\omega}}=\left(\left(\mathrm{I}-\left[m \mathbf{r}_{\mathrm{CM}}\right]_X \cdot \mathrm{M}^{-1} \cdot\left[m \mathbf{r}_{\mathrm{CM}}\right]_X^T\right)\right)^{-1} \cdot\left(T_{21}-\left[m \mathbf{r}_{\mathrm{CM}}\right]_X \cdot \mathrm{M}^{-1} \cdot T_{20}\right)\]
\[\dot{\mathbf{v}}=\mathrm{M}^{-1} \cdot\left(T_{20}-\left[m \mathbf{r}_{\mathrm{CM}}\right]_{\times}^T \cdot \dot{\boldsymbol{\omega}}\right)\]
Taking a closer look at the matrix inversion:
\[\begin{equation}
\mathbf{H}=\left[m \mathbf{r}_{\mathrm{CM}}\right]_{\times} \cdot \mathbf{M}^{-1} \cdot\left[m \mathbf{r}_{\mathrm{CM}}\right]_{\times}^T
\end{equation}\]
\[\begin{equation}
\mathbf{H}=-m\left[\mathrm{r}_{\mathrm{CM}}\right]_{\times}^2
\end{equation}\]
\[\begin{split}\begin{equation}
\mathbf{H}=-m\left[\begin{array}{ccc}
0 & -r_{\mathrm{CM}_3} & r_{\mathrm{CM}_2} \\
r_{\mathrm{CM}_3} & 0 & -r_{\mathrm{CM}_1} \\
-r_{\mathrm{CM}_2} & r_{\mathrm{CM}_1} & 0
\end{array}\right]^2
\end{equation}\end{split}\]
\[\begin{split}\begin{equation}
\mathbf{H}=-m\left[\begin{array}{ccc}
0 & -r_{\mathrm{CM}_3} & r_{\mathrm{CM}_2} \\
r_{\mathrm{CM}_3} & 0 & -r_{\mathrm{CM}_1} \\
-r_{\mathrm{CM}_2} & r_{\mathrm{CM}_1} & 0
\end{array}\right]\left[\begin{array}{ccc}
0 & -r_{\mathrm{CM}_3} & r_{\mathrm{CM}_2} \\
r_{\mathrm{CM}_3} & 0 & -r_{\mathrm{CM}_1} \\
-r_{\mathrm{CM}_2} & r_{\mathrm{CM}_1} & 0
\end{array}\right]
\end{equation}\end{split}\]
\[\begin{split}\begin{equation}
\mathbf{H}=-m\left[\begin{array}{ccc}
-r_{\mathrm{CM}_3}^2-r_{\mathrm{CM}_2}^2 & r_{\mathrm{CM}_2} r_{\mathrm{CM}_1} & r_{\mathrm{CM}_3} r_{\mathrm{CM}_1} \\
r_{\mathrm{CM}_2} r_{\mathrm{CM}_1} & -r_{\mathrm{CM}_3}^2-r_{\mathrm{CM}_1^2} & r_{\mathrm{CM}_3} r_{\mathrm{CM}_2} \\
r_{\mathrm{CM}_3} r_{\mathrm{CM}_1} & r_{\mathrm{CM}_3} r_{\mathrm{CM}_2} & -r_{\mathrm{CM}_2}-r_{\mathrm{CM}_1}^2
\end{array}\right]
\end{equation}\end{split}\]
\[\begin{split}\begin{equation}
\mathbf{H}=m\left[\begin{array}{ccc}
r_{\mathrm{CM}_3}^2+r_{\mathrm{CM}_2}^2 & -r_{\mathrm{CM}_2} r_{\mathrm{CM}_1} & -r_{\mathrm{CM}_3} r_{\mathrm{CM}_1} \\
-r_{\mathrm{CM}_2} r_{\mathrm{CM}_1} & r_{\mathrm{CM}_3}^2+r_{\mathrm{CM}_1}^2 & -r_{\mathrm{CM}_3} r_{\mathrm{CM}_2} \\
-r_{\mathrm{CM}_3} r_{\mathrm{CM}_1} & -r_{\mathrm{CM}_3} r_{\mathrm{CM}_2} & r_{\mathrm{CM}_2}^2+r_{\mathrm{CM}_1}^2
\end{array}\right]
\end{equation}\end{split}\]
Consider 𝐼CM as the inertia tensor relative to the true center of mass. Then:
\[\begin{equation}
\mathbf{I}_{\mathrm{CM}}+\mathbf{H}=\mathbf{I}
\end{equation}\]
\[\begin{equation}
\mathbf{I}_{\mathrm{CM}}=\mathbf{I}-\mathbf{H}
\end{equation}\]
New simplified equations:
\[\begin{equation}
\dot{\omega}=\mathbf{I}_{\mathrm{CM}}{ }^{-1} \cdot\left(T_{21}-\left[\mathrm{r}_{\mathrm{CM}}\right]_{\times} \cdot T_{20}\right)
\end{equation}\]
\[\begin{equation}
\dot{\mathbf{v}}=\mathrm{M}^{-1} \cdot\left(T_{20}-\left[m \mathrm{r}_{\mathrm{CM}}\right]_{\mathrm{x}}^T \cdot \dot{\boldsymbol{\omega}}\right)
\end{equation}\]