Can I estimate dot(omega) in this equation, even though I cannot directly solve the equation to it? I am trying to measure/estimate the angular acceleration of an object dot(omega) from a measurement of it's acceleration (using an accelerometer) overset(i)() bb(a)_m. As far as I understand, the accelerometer will measure a linear acceleration due to rotation of the object.

Messiah Sutton

Messiah Sutton

Answered question

2022-11-04

I am trying to measure/estimate the angular acceleration of an object ω ˙ from a measurement of it's acceleration (using an accelerometer) i 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
i a m = i a l + i ω ˙ i × i X S m + i ω i × ( i ω i × i X S m )
Assuming I know everything in this equation except for ω ˙ , I would like to estimate ω ˙ .
Unfortunately, the equation cannot directly be solved to ω ˙ since ω ˙ is in a cross product with a vector.
Are there any mathematical tools that can help me estimate ω ˙ given the relation I described above?

Answer & Explanation

Zoe Andersen

Zoe Andersen

Beginner2022-11-05Added 16 answers

Your equation can we written as
b = X × ω ˙ ,
where b is everything you know and ω ˙ is your vector of unknowns. Now observe that this can be written as a matrix equation
b = M ω ˙ ,
where
M = ( 0 X 3 X 2 X 3 0 X 1 X 2 X 1 0 ) .
Unfortunately, though, this matrix is never invertible. In other words, ω ˙ is never uniquely determined by b . This answer is obviously not complete, but hopefully indirectly useful.

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