Quadcopter Stability

[ankyda]

Hello everyone,

Recently I've been looking at quadcopter technology. While looking through literature, I noticed that most people mention that a quadcopter is inherently unstable but no reasoning is provided. I looked a bit at the EOMs but that's a big mess that I don't have time for yet. I understand that it is underactuated does that make it unstable? Can anybody point me in the right direction before I dive in?

Thanks

 

[jedishrfu]

This may help:

http://www.scoutuav.com/2011/11/19/improving-stability/

 

[.Scott]

To understand what is meant when designers call the quadcopter inherently unstable, you need to understand the normal standards for aircraft control. After a minute of two of search, I haven't found a great link for that, but this one is passable:
https://rdl.train.army.mil/catalog/...ABC8718F9760-1274574464617/3-04.203/chap7.htm

The idea is that even without power, aircraft are designed with handling characteristics that exhibit static and dynamic stability. For example, you can you put a normal aircraft into straight and level flight and "trim is up" for that speed and altitude. If you then push against the yoke for a couple of seconds and then release, you will enter a dive and the airspeed will increase - but the design of the plane is such that the increase in speed cause the center of lift to move forward and bring the nose up. That's static stability. Once the nose has been brought up by this affect, the aircraft will cross through its original altitude, slow down, and the nose will come back down again - completing one cycle of oscillation. If these oscillations get worse and worse - that would be dynamic instability. The design of a normal aircraft is such these oscillations dampen out - dynamic stability - and the aircraft eventually return to straight and level flight on it own.
These elements of stability apply to all control inputs. For examples, applying momentary pressure to a rudder or to the ailerons would cause a change in heading, and can affect pitch, speed, etc, but the plane is designed to recover to straight and level flight.

These same principles apply to standard helicopters, but not to octocopters (and not to the Marine's Osprey during helicopter flight).

A momentary pitch change to an octocopter requires active counter maneuvers to restore it to a stationary hover. If there is stabilizing influence from the control of the octocopter, the copter will begin a pitch and yaw that will eventually lead to uncontrolled flight or a crash.

 

[.Scott]

Oh. And you mentioned "underactuated".
I haven't read the same material you are reading, but I would guess that they are discussing the strategy of control of the octcopter. That strategy would be to start by under controlling it - for example, giving it a smaller pitch than the target - and allowing the inherent instability to exaggerate the maneuver while starting to apply a counter control.

 

[ankyda]

I understand the normal stability requirements for an aircraft. Basically what you have just discussed is longitudinal stability where the center of gravity should be in front of the aerodynamic center for some static margin. There are other requirements for directional and lateral as well. What I don't understand is how this may apply (if it does at all) to a quadcopter with no fixed wing. It seems like there is no stabilizing effect on a open loop quadcopter so it is unstable without a PID or other control schemes.

As for the underactuated part, a quadcopter controls its state with a combination of the four motors it has while a plane can directly control each state with its control surfaces. In the quadcopter case, the states are observable with some sort of imu end up controllable with some form of control law.

The materials I'm reading are mainly IEEE and AIAA journal articles. I guess what I'm looking for is some form of stability criterion equations for a quadcopter that shows me why it becomes unstable openloop.

 

[.Scott]

What I don't understand is how this may apply (if it does at all) to a quadcopter with no fixed wing.

Here's how it applies.
Let's say you had a quadcopter you could actually pilot yourself. Each of the four propellers would rotate at exactly the RPM you set for them. You then take off and tried to enter some sort of stable flight - perhaps a simple hover or perhaps moving at a constant altitude at 5KPH.

If the quadcopter was "inherently stable", you would be able to maintain that stable flight without touching the controls. But the real test is what happens when you do, momentarily, touch the controls. Does it eventually go back to stable flight without further control adjustments or not? That is the criterion that can be applied to any flying device.

In the case of a quadcopter, it would fail miserably. If it started in a hover and one of the props generated a bit more lift for just a second, the quadcopter would begin to tilt and there would be nothing to stop it from continuing. It would soon invert and do a power dive into the ground.

I guess what I'm looking for is some form of stability criterion equations for a quadcopter that shows me why it becomes unstable openloop.

I think you understand that "unstable open loop" is basically the same as "inherently unstable".

Of course, there is no particular "pitch" or "roll" axis on quadcopter, but in the previous example we can arbitrarily call the problem axis of rotation "roll". If we mathematically model the quadcopter, we would see that the change in roll rate would be a function of the difference in thrust between two diagonally opposite props plus a bit of drag that would eventually slow a roll. But for inherent roll stability we would want to see something that would tend to make the copter level off - some additional factor affecting the change in the roll rate tied to the roll orientation. If there was something in the copter that caused the alignment of the center of thrust to move to the downward side of the center gravity as the copter began to roll, that would do it. You might be able to design a quad copter that way - such a mechanism does quickly come to mind.

BTW: In the case of an airplane, roll stability is created with a dihedral angle between the wings. Airplanes designed specifically for aerobatics do not have this angle and are not roll stable - or in some cases are only very mildly roll stable.

Hope this helps.

 

[ankyda]

I think I understand. Basically a quadcopter has no passive stabilizing effects that a fixed wing would be designed with. I'm still curious if there's any equation that expresses this kind of thing for a quadcopter.