For the optimum stability where is the recommended COG on the Z axis?
X and Y axis makes sense to have it centered between the arms, does this also apply to the Z axis? If it does, what points would you measure from? Keep in mind this is not for performance or acrobatics just extremely stable flights.

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Yes, you would want its Z-position to be in propeller level. Anywhere else and your quad would have to fight the offset when rolling or pitching.

Putting the COG below the prop plane wouldn't give you any extra stability, because of the pendulum rocket fallacy.
I don't agree with Martin. For maximum stability you would like the center of gravity as low as possible. If you build a simple force diagram of a tilted quad with a low center of gravity you will see that it has a natural righting moment. While true that the "quad" will have to overcome this in pitch and roll that is just another way of stating that the quad will have to overcome the natural stability to hold a non-zero pitch or roll angle.

As an analogy - this is why high wing planes are typically more stable than low wing. As an avid RC heli pilot I can tell you that if you want your heli to be highly maneuverable and acrobatic, you try to get the CG as high (close to the rotor disk) as possible. The lower the CG, the more stabilty.
That's not true for a machine with active stabilization. A quad that has both its CoG and CoP in its center will consume much lower amounts of power on fighting the wind. Any offset from this arrangement will invariably force the machine to fight it with more power - think of it as two different and incompatible stabilization systems working at the same time.

How this works in real life? A quad with a lowered CoG (typically with a lipo) will tend to drift horizontally. It's a result of the two stabilization systems fighting. A well-balanced machine should stay in place.
Yikes! Learn something every day....

I was thinking of stability in too narrow a sense. Getting the GC removed from the plane of the propellers will cause the rotational reaction to be slowed due to the change in the relationship between the motors thrust direction and the distance/direction to CG. But your example of a quad in wind clearly shows higher energy use, which is bad. And the link you added for the rocket/pendulum certainly illustrates the situation well and clears it up.
How does this apply to helicopters and their design ?
thx, I was allways thinking what Doug said at first
On a plane or helicopter without active stabilization the lowered COG adds some stability. The pendulum rocket fallacy doesn't apply because there isn't an active torque countering the lowered COG.

Active stabilization doesn't require lowered COG. The feed-back loop takes care of it for you. I see that most quadcopters have a lowered COG. That just adds a moment arm that the feed-back loop has to spend more energy countering.

I am just finishing mine and I took care to make sure the COG in Z axis is balanced around the rotor center line.
I-ve flown with battery and camera below the frame (low CG) and battery on top of the frame with camera below the frame (CG still slightly below the propellor plane, but much higher than with the lower battery postion). Of course the high battery position does also move the FC up, so that actually ends up slightly above the prop plane. You do notice a bit more sluggishness in roll and tilt responses with the lower battery position. Stability wise it doesn't seem to make to much difference otherwise.

One factor which may be more important is in fact the height of the craft. There is a university paper which has looked at this. With high landing gear and an underslung motor mount the destabilizing effects of e.g. wind increase. In addition a high, wide, landing gear puts the "skids" in the propwash which may also result in some turbulence effects around them and may transfer back as "vibration" into the FC's sensors if not isolated from vibration.

Personally I prefer the battery above the frame as it does seem to result in a slightly more responsive kopter and allows a bit more foam between camera mount plate and frame without requiring even higher landing gear. In addition the higher "cockpit" position may well compensate a bit for the underslung camera mount and high LG below the frame and thus for the destabilizing effects of wind on the vertical axis.

See this thread for more information in 2012.

@Paul G.: In order to be efficient, you need a large rotor disk area.  Full-size helicopters would be extremely hazardous on the ground if the rotor was not at least 7 feet high (for obvious reasons).  This does make them less stable, but natural disk coning (the rotors achieving a natural equilibrium between centrifugal force and thrust pushing the tips upward - adding a little dihedral effect) restores some of this.  Ultimately, however, keeping a big ship from darting off to who-knows-where is reliant upon piloting skill and far greater control responsiveness than can be achieved with a fixed-pitch multicopter.

(eh, better 2 years old than never)

Yes, but there are also other elements at play.  One thing is that helicopters also have better aerodynamics that quads.  They are simply less affected by the wind.  Also, i think the rotor disk offers better control reactions for reasons I don't fully understand.

At the end of the day, the proof is in the pudding.  Check this out:

My quad copter could barely fly in the same conditions.  All I could manage was a controlled crash.  After a number of attempts at landing, the best I could do was getting it momentarily level at 3 feet, and then chopping the throttle.

woooaaaauuuuu!

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