You’ve gotten a lot of great answers about the technology, but also interesting to consider is that despite how it looks from the ground, the drones are almost never “just a few feet apart”. Much like air shows where planes look like they’re about to touch, there’s quite a bit of distance between each craft. The distance between you and the swarm also means that if individual drones drift a foot or two due to sensor issues or wind, you’re not going to notice it from your vantage point so far away. Think of it like a billboard or a jumbotron screen at a sports arena. From 600 feet away, it looks like perfect resolution, but if you are to walk up close, the pixels are giant and chunky and a lot further apart than you thought. Drones in a warm show are very similar.
From there though, the other answers take over on how they maintain their position and coordination. GPS, base stations with a fixed registered position, and accelerometers and altimeters on the drones themselves.
The drones need to be around a metre from each other at minimum, and normal GPS can’t do it that well. If using GPS some, it would be too easy for one drone to be half a metre wrong in one direction and another drone could easily be half a metre wrong in the other direction. Those drones would hit each other and one or both of them would fall out of the sky
Instead, drone show drones have special sensors which can see a beacon on the ground near the takeoff area. This gives them a very accurate idea of where they are at all times in relation to where they took off from, and is accurate to around a centimetre. The drones then follow a very precise flight path, and know where they should be, and when. The beacon system is called rtk GPS (real time kinematics)
If this system doesn’t work well, or the drone sensors are poor, the drones do malfunction. I’ve seen videos of drone shows (looking at you, vivid Sydney) where you can see drones going around in circles, attempting to find their position but failing. Drone pilots call this ‘toilet bowling’ because it looks like they are circling the drain. Those drones then collide with others, usually resulting in one or both falling out of the sky.
They can respond to wind up to a certain level, in my experience around 8 metres per second. Like a dji drone, they have the ability to push against the wind to maintain their location. They respond quickly to rises and falls in wind speed by pushing into the wind (or in the opposite direction to any drift in location), and will correct their pushing direction before they get too far out of position. When the wind is too high, they cannot correct enough and will drift. That would be very bad, and collisions become very likely. Operators might have to make the decision to abort the show. Usually they will not take off if the wind is likely to be that strong.
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