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I'm not sure how to search for this.

I have 3 sensors measuring accelerations in different orientations in a flat plane, each 45 degrees apart, but with a small orientation error

The measurement equation could be written as:

$z = xH$

$\begin{bmatrix}z_a&z_b&z_c\end{bmatrix} = \begin{bmatrix}x_x&x_y\end{bmatrix}. \begin{bmatrix}1 & \alpha & \frac{1}{\sqrt2} + \beta\\ 0 & 1 & \frac{1}{\sqrt2} - \beta \end{bmatrix}$

I imagine that I can take multiple measurements simultaneously for $z_a, z_b, z_c$ so that the data is over-constrained.

$\begin{bmatrix} z_{a1}&z_{b1}&z_{c1}\\ &\vdots\\ &\vdots\\ z_{an}&z_{bn}&z_{cn} \end{bmatrix} = \begin{bmatrix} x_{x1}&x_{y1}\\ \vdots&\vdots\\ \vdots&\vdots\\ x_{xn}&x_{yn} \end{bmatrix}. \begin{bmatrix}1 & \alpha & \frac{1}{\sqrt2} + \beta\\ 0 & 1 & \frac{1}{\sqrt2} - \beta \end{bmatrix}$

Presumably there is a method to take the least squares or other regression of this data to recover $\alpha$ and $\beta$, without knowing the values of $x$.

I cant seem to manipulate the matrices to get something that works.

Is there a standard procedure for problems of this type?

Could you provide an example solution to this specific problem?

mr_guy99493
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  • Have a look at Kalman filtering. – Cesareo Apr 24 '18 at 09:40
  • Usually I would use kalman filtering to get an estimate of the 'state' : x, but in this case I think I want to estimate the measurement matrix: H, which seems difficult to me because everything I see talks about estimating an Nx1 matrix, rather than a 3x2 in this case. I cant see a straightforward method to convert it to an Nx1 to use my ordinary methods. – mr_guy99493 Apr 24 '18 at 11:20
  • https://en.wikipedia.org/wiki/Multivariate_analysis_of_variance – V. Vancak Apr 25 '18 at 01:35

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