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I need to read the location of my device within a 1m radius sphere, with accuracy of 5-10cm.

The device is handheld and wireless, and currently communicates using a Bluetooth v4 chip. I could add an RF transmitter on the moving part and a few stationary receivers at the base.

What components should I look into? What would be the cheapest way to triangulate it?

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TOA method is almost impossible to work at such short lengths with radio signals (as they travel at the speed of 3*10^8 m/s which means you need a very high sampling rate to detect that time difference). You should consider using some ultrasonic sound sources and microphones and use them for such a task. If you wish to know the location of the device in 3 dimensions, you need to place one sound emitter onto the moving device and 4 microphones at the center of the sphere. The more apart the microphones are, the better precision you'll have.

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