Detection of object movement & rotation in magnetic field
This will not work based on a magnetic field: Everything will interact with everything. There can't be multiple stations (at most three orthogonal field vectors in 3D space means you couldn't even estimate vicinity to more than three stations if your object was floating in vacuum).
Also, positioning accuracy will be in the multiple meters in your description of the system, not centimeter, and also use kilowatts of power, because magnetic field strengths drops very sharply with distance from the magnet, and earth has its own magnetic field that you're competing with!
Hope you don't like tools made of steel, because they interact with magnetic fields, and thus can't be used if you're not willing to disturb your magnetic field sensors.
So, no, this isn't a viable approach, at all.
So, as Andy already said: your best guess is optic location. Cameras are cheap these day, even relatively high-resolving ones, and so is processing power. Especially if you can label your components, e.g. with colors or QR codes, this will get a lot easier; but even systems like specific coordinated times at which the components light up an infrared LED can work wonders.
With camera frame rates in the 10¹ to 10², your speed requirements are easily met, and optical resolution ability will definitely trump anything that you can build with a static magnetic field – which in the end can only be a rough proximity sensor, with the interaction problems mentioned above.