I am trying to use an accelerometer to measure the angular acceleration of a robotic arm.
From rigid body kinematics, the following relation is known
\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*}
where $\dot{\omega}$ is the angular acceleration and $ {^{i} {\boldsymbol{a}}_m}$ is the measured acceration at a point $X$ along the arm.
The problem is that this equation is not solvable for $\dot{\omega}$ because, it's in a cross product with the position vector of the sensor. ($a \times (b +ka) = a \times b + k(a \times a) = a \times b$)
This paper claims that they used an extended Kalman filter and this relation to estimate the angular acceleration, but I have no idea how that is possible when this equation does not have a unique solution for $\dot{\omega}$.
Can anyone point out if I'm missing something which can help me solve this problem?
