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Principal Inertia & Products of Inertia Question

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I am somewhat rusty on my Inertia math, so I was hoping someone here could quickly answer a question for me... I have two bodies in space. I can easily obtain the principal & products of inertia for both these bodies. The inertial values I have are with respect to each body's center of gravity and aligned with the global reference frame. What I want to do is calculate the principal & products of inertia for both bodies combined. I want the final values to be at the center of gravity of both objects, aligned with the global reference frame. Is the following correct? Using the parallel axis theorem, I translate the principal & products of inertia to the global origin. I then simple add the principal & products from each body together (no transform needed, since they are now in the same reference frame). Afterwards, I just use the parallel axis theorem to move from the global origin to the new center of gravity (which is easily calculated as a weighted average). Does that work? Or am I missing something? Thanks!

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It does work.
You could simplify one step, as I can see: instead of calculating the intertia tensor at the global origin and then recalculating it to the new CG, you could positionate those two objects so that their CG is the global origin, that way after the first translation you'd get the inertia tensor you want. I don't really know if it's worth it, but it just came to mind.

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