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?