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Is it possible to let a drone hover (or move very slowly) in a position rotated about a for/aft or a left/right oriented axis? I've seen videos on youtube about building drones with off-the-shelf flight controllers, that seem to be designed only for keeping a drone level

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I don't think that's physically possible. If a drone is tilted, the thrust is not vertical and it will accelerate in the direction it's leaning.

A special case is when dealing with wind. In that case, in order to hover in place, a drone will need to be tilted into the wind, so the thrust counters the force of the wind. A GPS enabled drone will do this automatically, otherwise the pilot handles it.

If you want to tilt the drone while hovering, you'd have to tilt the motors, or provide some sideways thrust. Instead, it's normal to tilt the camera.

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