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I've been reading this paper that formulates invariant task-parametrized HSMMs. In section 3.1 (Model Learning), the task parameters are represented in $F$ coordinate systems defined by $\{A_j,b_j\}_{j=1}^F$, where $A_j$ denotes the rotation of the frame as an orientation matrix and $b_j$ represents the origin of the frame. Each datapoint $\xi_t$ is observed from the viewpoint of $F$ different experts/frames, with $\xi_t^{(j)} = A_j^{-1}(\xi_t - b_j)$ denoting the datapoint w.r.t. frame $j$.

How is $\xi_t^{(j)} = A_j^{-1}(\xi_t - b_j)$ derived? I understand that we must subtract $b_j$, but I'm not sure if I should pre-multiply by $A_j$ or $A_j^{-1}$, so it'd be great if someone could help me understand this better. Since $A_j$ is an orientation matrix, I'd guess that it's orthogonal, and so $A_j^{-1} = A_j^T$ - and it may just be a matter of convention (i.e. depending on how $A_j$ is defined). The details aren't clear from the paper though, and I'd appreciate any help!

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    $\begingroup$ You should be able to find the answer to your question in some book that describes coordinate frames and transformations in the context of robotics or related fields (such as computer vision or computer graphics). Be prepared to deal with homogenous coordinates and stuff like that. Maybe this article wiki.ros.org/tf/Overview/Transformations will also "prove" to you that these concepts are widely used in robotics. Now, I can't help you further, but, hopefully, you will find a good resource that describes these concepts more clearly (btw, I completely understand your doubts). $\endgroup$ – nbro Oct 20 at 21:14

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