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Here is the situation. I have 2 sensors, which provides $3*3$ rotation matrix. And as these sensors are combined in unknown but fixed structure, I can say there is 2 coordinate system. In other words, 2 sensors are moving together simultaneously, but with gap between them.

Then, each sensor would show rotation matrix like,

$S_1 = \begin{bmatrix} \mathbf{R}_1 & \mathbf{0}\\ \mathbf{0} & 1\\ \end{bmatrix}$, $S_2 = \begin{bmatrix} \mathbf{R}_2 & \mathbf{0}\\ \mathbf{0} & 1\\ \end{bmatrix}$

Additionally(Sadly), I attached this package of sensors onto a tripod and rotated again, which would result in unknown(as tripod cannot give you a rotation matrix unlike sensors) 3rd rotation coordinate.

$S_3 = \begin{bmatrix} \mathbf{R}_3 & \mathbf{0}\\ \mathbf{0} & 1\\ \end{bmatrix}$

My fundamental goal is to find out the transformation matrix($T$) satisfying below, which explains how each sensor is combined geometrically.

$S_1=T * S_2$

At the first time, I thought I could easily retrieve $T$ by positioning the sensor package in several random position using tripod, but it was wrong idea because the positioning tool, the tripod, also had another coordinate system.

How would you approach to this problem to get matrix $T$?

(A translation may have to engaged but I'd like to consider it later XD)

Jae Seung Kim
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  • one question is, why isn't the answer just $T=S_{1}S_{2}^{-1}=S_{1}S_{2}^{T}$, since both $S_1$ and $S_2$ are orthogonal matrices? – davyjones Apr 11 '17 at 07:10
  • @davyjones Negative. Each of $S_1$ and $S_2$ represents position of each sensors and these sensors are not combined in orthogonal but slightly tilted. That's why I am trying to find exact $T$. – Jae Seung Kim Apr 11 '17 at 07:19
  • er..., but how tilted ? is $S_1=TS_2$ still satisfied? if you mean measurement noises are involved, then perhaps you might check http://math.stackexchange.com/questions/1599204/applications-of-so3-irreps-to-spatial-rotation – davyjones Apr 11 '17 at 07:29
  • Maybe this my answer would be useful for you http://math.stackexchange.com/questions/2199248/rotation-translation-of-vectors/2201250#2201250 – Widawensen Apr 11 '17 at 07:57
  • @davyjones Exact tilted angle is unknown. Suppose there is a wooden stick, sensor's attached at the each end of it. Then there should be a $T$ which explains the wooden stick geometrically. But the problem is that I rotated that wooden stick using another wooden stick($S_3$;3rd coordinate). Lemme check the reference you shared. Thank you very much! – Jae Seung Kim Apr 11 '17 at 09:53

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