I am estimating a transformation $\hat{T} \in SE(3)$ between two 3-dimensional coordinate systems (COS) $C1, C2$. I know the exact (ground truth) transformation between the two COS, namely $T_{GT} \in SE(3)$. Any point $p$ transformed from $C1$ to $C2$ using $\hat{T}$ results in a deviation (Euclidean distance) in $C2$ from the same point transformed by $T_{GT}$.
I found an article from Huynh (Huynh, 2009, Metrics for 3D Rotations: Comparison and Analysis) which describes several metrics describing the difference between two 3D purely rotational transformations ($SO(3)$).
Is there something similar to this for two transformations in $SE(3)$?
In short: What kind of a metric should I use to describe the mismatch between two transformations in SE(3)?
Intuitively, what I am interested in are deviations (Euclidean distances) between points in $C2$ (in a volume of interest) as transformed from $C1$ to $C2$ by the estimated transformation $\hat{T}$ and the ground truth transformation $T_{GT}$.