I am trying to measure/estimate the angular acceleration of an object $\dot{\omega} $ from a measurement of it's acceleration (using an accelerometer) $^{i} {\boldsymbol{a}}_m$. As far as I understand, the accelerometer will measure a linear acceleration due to rotation of the object.
From rigid body kinematics, the following relation is know
\begin{align*} {^{i} {\boldsymbol{a}}_m} & = {^{i} {\boldsymbol{a}}_l} + ^{i} \dot{{\boldsymbol{\omega }}}_{i} \times {^{i} {{\boldsymbol{X}}}_{S_m}} + {^{i} {{\boldsymbol{\omega }}}_{i}} \times \left({^{i} {{\boldsymbol{\omega }}}_{i}} \times {^{i} {{\boldsymbol{X}}}_{S_m}} \right) \; \end{align*}
Assuming I know everything in this equation except for $\dot{\omega}$, I would like to estimate $\dot{\omega}$.
Unfortunately, the equation cannot directly be solved to $\dot{\omega}$ since $\dot{\omega}$ is in a cross product with a vector.
Are there any mathematical tools that can help me estimate $\dot{\omega}$ given the relation I described above?
Your equation can we written as $$ \boldsymbol{b}=\boldsymbol{X}\times\boldsymbol{\dot\omega}\,, $$ where $\boldsymbol{b}$ is everything you know and $\boldsymbol{\dot{\omega}}$ is your vector of unknowns. Now observe that this can be written as a matrix equation $$ \boldsymbol{b}=\boldsymbol{M}\boldsymbol{\dot\omega}\,, $$ where $$ \boldsymbol{M}=\left(\begin{matrix}0&-X_3&X_2\\X_3&0&-X_1\\-X_2 &X_1&0 \end{matrix}\right). $$ Unfortunately, though, this matrix is never invertible. In other words, $\boldsymbol{\dot\omega}$ is never uniquely determined by $\boldsymbol{b}\,.$ This answer is obviously not complete, but hopefully indirectly useful.