Coordinate transformation from g-sensor frame to world frame?

[E-John]

Dear Sir,

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Generally speaking, the coordinates of the g-sensor on the PCB board have an angular deviation from the world coordinates. Assume that the coordinates of the g-sensor are (Xs, Ys, Zs) and the world coordinates (X, Y, Z) have alpha, beta and Theta angle relationship.

If we have a g-sensor orientation as following,

How to convert (Xs, Ys, Zs) to (X, Y, Z)?

It is confused because when Xs, Ys point to gravity direction, it is positive, while Zs point to gravity direction is negative.

Could we have a rule to follow?

Thanks.

[E-John]

The answer could be referred to

https://community.st.com/s/question/0D73W000001OwbRSAS/detail?s1oid=00Db0000000YtG6&s1nid=0DB0X000000DYbd&emkind=chatterCommentNotification&s1uid=0050X000009xkPM&emtm=1657627007730&fromEmail=1&s1ext=0