>> [...] it's just an optimisation problem to find the
>> rotation/translation matrix that minimises the sum of the
>> distances between each node pair (or another similar metric).
>> You can use the MS Excel solver [...]
>
> Can it minimize a quadratic function, subject to quadratic constraints?

The Excel solvers are pretty flexible, they are good at handling a wide 
range of optimisation problems. Attached is what I came up with for 
minimising the distance between nodes on two bodies, it seems to solve 
OK with the evolutionary solver and after it's done will give you the 
best 4x4 rotation/translation matrix.

BTW in Solver Options you might want to turn down the maximum time 
without improvement, I think the default is 30 seconds which is a bit 
long (I made it 5 seconds).

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Attachments:
Download 'node solver.xlsx.zip' (14 KB)