Mixed intrinsic and extrinsic angles in 3D rotations

[aydos]

Being scratching my head for 2 days and not getting anywhere with this one. I am trying to figure out how to perform a 3D rotation described via a mix of intrinsic and extrinsic angles.
Here is the problem:
I have a platform in the shape of a box with sides of length lx, ly and lz. The platform has an inclinometre measuring angles α and β from the vertical about lx and ly and a fluxgate compass measuring heading. I have a global Cartesian measurement p(x,y,z) of the top-right-front corner of the box and I would like to calculate the location of the centre of the box given p, α, β, γ, lx, ly and lz
I would know the solution if:
- α, β and γ were purely intrinsic rotations, or
- α, β and γ were purely extrinsic rotations
Due to compass reading being global, but the inclinometre readings being local, I am not sure what to do. Does anyone have any pointers?

 

[fresh_42]

Couldn't you just apply the extrinsic calculation (p) first and then the intrinsic (l) afterwards? p(x,y,z) alone doesn't give you the complete information, but if you consider this point as the origin of your intrinsic system you should be able to perform all necessary calculations.