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Let us consider we have two known $SE3$ transformations with matrix representations $H_1$ and $H_2$ of the form $H= [R; t]$ where $R$ is a 3x3 rotation matrix and $t$ a 3x1 translation vector. I am omiting the row of zeros below $R$ and the 1 below $t$. Also consider the next relationship is true:

$H_2 = A^{-1}H_1A$

where $A$ is itself an $SE3$ transformation. What could be a good try to find $A$? Notice we can have many instances of $H_2$ and $H_1$ so in principle I would also be interested in posing this question as an estimation problem.

Thank you!

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  • $\begingroup$ what is the purely rotational version and can you solve it? $\endgroup$ Commented Sep 7, 2024 at 21:03
  • $\begingroup$ The rotational version would be $R_2=R^T_AR_1R_A$ , I tried to pose it as a least squares problem but I do not know how to take it from there. Any ideas? $\endgroup$ Commented Sep 8, 2024 at 11:33
  • $\begingroup$ suppose $R_1$ Is unity and $R_2$ isnt. Then there is no solution. I think this is the rule, and not the exception $\endgroup$ Commented Sep 8, 2024 at 11:44
  • $\begingroup$ If I would know $R_1$ and $R_A$ I would be able to compute $R_2$. My question is how to recover $R_A$ back from $R_1$ and $R_2$ only $\endgroup$ Commented Sep 9, 2024 at 10:51
  • $\begingroup$ i gave you a counter example where no such $R_A$ exists $\endgroup$ Commented Sep 9, 2024 at 12:01

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