I have heard it mentioned that quadcopters are inherently unstable and only become stable due to control. What does inherently unstable mean in this context, and why are quadcopters inherently unstable?
A stable system is one where there's a force that pushes the system back towards it's original state when the system is moved from its balance point.
For example, when the driver gets into a car, there's more weight on one side of the car than the other but the car doesn't flip over. The springs on that side of the car compress and push harder on the wheels. The tyres push harder on the ground and the ground pushes back. The extra force from the ground exactly balances the extra weight and (after possibly bouncing for a moment) the car stops moving.
In an unstable system, say a broom balanced vertically, once it moves from the perfect balance point gravity will pull it further away and it accelerates.
For a quadcopter, there is no force that returns it to its original location, holding it horizontal, or even preventing a gradual roll.
Even if the thrust from each motor was exactly the same, and the weight was perfectly balanced, it's very unlikely to be perfectly stationary. If there's even the slight amount of pitch or roll, the angle from horizontal will gradually increase. As the angle increases, the sideways thrust increases and the drone accelerates.
So the roll rate is constant, the angle and acceleration increase linearly, the horizontal speed increases faster as the acceleration builds, and the distance from the starting position increases even faster as the speed builds. To the pilot, it looks like the drone is accelerating away from its original position much like the broom falling over.
To a systems engineer, the rotation of the drone is neutrally stable - there are no forces increasing or decreasing the rotation rate. However the speed and position are unstable - once it starts moving, it accelerates.
I should mention that there are second-order (i.e. relatively minor) forces too. If you had a large fin on top of the drone, aerodynamic drag on the fin would return the drone to upright as the speed builds and limit the maximum speed - in practice this isn't useful as it doesn't return the drone to its original position.
Inherent stability is a concept from aeronautics – it is the tendency of an aircraft to return to an equilibrium when the controls are released by the pilot. Quadcopters are not inherently stable.
Say, you have the quadcopter rolling to the right:
A quadcopter requires active counter manoeuvres to restore it to a stationary hover:
This means the quadcopter is inherently unstable.
It’s not entirely correct, that quadcopters are unstable. Most commercial copters have a center of gravity way below the Propeller plane and thus are actually stable. In contrast to that you want certain aircrafts to be instabile for better agility. For example some jet fighters are unstable and need electronic stabilization even for normal forward flight.