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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?

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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.

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    $\begingroup$ Is there some reason that quads are not built with the rotors tilted a bit up toward the center, analogous to aircraft wing dihedral? Wouldn't this make them more stable? $\endgroup$ – Lee Daniel Crocker May 18 at 17:23
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    $\begingroup$ @LeeDanielCrocker An aircraft's dihedral angle improves stability because of the way its wings generate lift. A rolling motion results in sideslip relative to the aircraft's forward motion. This sideslip, combined with the wing dihedral, results in one wing producing more lift than the other, counteracting the roll. A quadrotor doesn't generate lift in the same way (indeed, "sideslip" is meaningless when hovering) and so merely tilting the rotors wouldn't have the same effect. $\endgroup$ – David May 18 at 21:32
  • $\begingroup$ Also, as the angle of the drone (and thus the lateral acceleration) increases, the lift it experiences will decrease, causing it to eventually fall out of the sky. $\endgroup$ – nick012000 May 19 at 8:11
  • $\begingroup$ @David that argument sounds dubious to me. The whole reason that sideslip increases lift of a dihedral wing from the windward side is that it poses a higher effective angle of attack. Likewise, a dihedral-tilted rotor would have a higher AOA on the windward side than on the leeward side. So if this would not make the quadcopter stable (i.e. prevent ever-increasing drift motion) then the reason is that AOA matters less in rotors than it does in wings, not anything about whether you see the drift as a sideslip or just undesirable motion when trying to hover. $\endgroup$ – leftaroundabout May 19 at 14:45
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    $\begingroup$ @LeeDanielCrocker I think mostly the reason why stuff like this isn't done is that flight controllers do a perfectly fine job at dealing with the situation. There's not really any advantage to be had by a drone that's inherently stable. $\endgroup$ – leftaroundabout May 19 at 14:47
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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:

image

(image source)

A quadcopter requires active counter manoeuvres to restore it to a stationary hover:

image

(image source)

This means the quadcopter is inherently unstable.

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    $\begingroup$ @Diamondx The point is that a quadcopter only self-levels because of computer-controlled adjustments. If you just turned the rotors on and removed all controls (the computer was switched off, and the pilot went for a bathroom break), a quadcoptor would quickly tumble out of the sky. This is as opposed to many aircraft, which will naturally maintain stable flight (at least for a while) when control is removed. $\endgroup$ – Josh Eller May 18 at 12:15
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    $\begingroup$ @Diamondx If you put 4 motors on a frame and turn it on it will always flip over and never self-level. A computer must constantly pilot the quadcopter to prevent it flipping over. That's what it means to be inherently unstable. An inherently stable system is that if you leave it alone without any brains or software it will correct itself - an example of this is a fully deployed parachute which will always fall the right way up. Almost all airplanes except modern computerized jet fighters are stable. If you remove the pilot planes will fly the right way up until it hits the ground $\endgroup$ – slebetman May 18 at 15:26
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    $\begingroup$ @Diamondx No, unstable is the correct word. "Chaotic" is another thing entirely (dependence on initial conditions). A system can be unstable without being chaotic; for instance the solution of x_{k+1}=2x_k, k=0,1,2,.... (mathematician here). $\endgroup$ – Federico Poloni May 19 at 7:11
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    $\begingroup$ (Or, better example because it does not require dealing with infinity, the logistic equation for an initial point in a neighbourhood of zero.) $\endgroup$ – Federico Poloni May 19 at 7:15
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    $\begingroup$ Chiming in on the definition of chaos and instability (applied mathematician here with specialty in control systems). I'll build on @FedericoPoloni's comment to say that in layman's terms, a chaotic system is one where tiny differences in the initial condition lead to dramatic differences in the result. Imagine the individual pieces of glass remaining after you break a mirror. The breakage is stable, but the shards' shapes vary wildly and are almost impossible to predict. $\endgroup$ – Kenn Sebesta May 20 at 13:32
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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.

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    $\begingroup$ If the center of gravity is "way below the propeller plane", the system is stable, but not in a useful way: as you accelerate, aerodynamics will try to flip the copter so that the center of gravity is leading the way (think of how a dart behaves). See en.wikipedia.org/wiki/Pendulum_rocket_fallacy $\endgroup$ – Mark May 18 at 20:06

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