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Getting height information could be fixed by deflecting some beams to the ground with a mirror. In case you want to use that inside a position control loop for the quadcopter, keep in mind that this is a dynamic system and you need to estimate velocity. So the framerate you would get either by onboard computing or streaming the scans offboard might not be sufficient for that. Also, you need to compensate the measured laser-scans for roll and pitch motions of the quadcopter. Here is some work that has been done with laserscanners on quadcopters, which might be helpful:

So back to the original question: it will not be straight forward or easy to use it, but it would be possible with some underlying odometry and imu data fusion as described in the papers.