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Does it matter where the battery is mounted? Or is it best to have the centre of gravity aligned with the propeller plane where the thrust is generated?

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It depends on what kind of stability you're after. Nicolas Petit wrote a paper on this several years ago, the graphics below are taken from Pages 4 and 5.

The summary is that a higher CG is better for x-y position stability, a lower CG is better for speed stability.

Background

Before we begin, let's assume we're only talking about a drone flying purely forward and so all dynamics are related to pitch.[*]

Second, let's define the CG, center of gravity, relative to the aerodynamic center (similarly to in https://aviation.stackexchange.com/questions/19388/what-is-the-difference-between-centre-of-pressure-aerodynamic-centre-and-neutra). The aerodynamic center is the idealized point where all the drag forces are acting around it evenly. In other words, it can be thought of as the spot where half the drag rotates the quadcopter into a pitch up orientation, and half rotates it into a pitch down orientation.

Here's the relevant image. It shows forward flight, and that's the same as position-holding flight with a backwards wind.[**] The red arrow is for when the CG is above the aerodynamic center, and the blue arrow is for when it's below the aerodynamic center.

enter image description here

enter image description here

Top-heavy CG: increases position stability, decreases velocity stability

When the CG is above the the aerodynamic center, the drone tends to tilt forward when there's a gust of wind. This ensures positional stability, because as the drone is pushed backwards it naturally leans forward which reduces the speed at which it accelerates backwards.

However, since the CG is above the aerodynamic center, when flying forward the drone wants to trip over its AC and pitch even more forward. This means that forward velocity is somewhat underdamped, results in a less positive-- or even negative-- feedback cycle.

Bottom-heavy CG: decreases position stability, increases velocity stability

In this case, the CG is below the aerodynamic center. When there's a sudden gust which destabilizes the drone, the drone tilts away from its holding position, which combined with the gust's force, accentuates the positional error.

But on the other when flying steadily forward the drone tends to want to pitch back toward the vertical, instead of tripping over its own feet. This means that forward velocity is well damped without a positive feedback cycle.


[*] Roll functions exactly the same way, but it's distracting to write "pitch/roll" each time we're talking about a rotation. [**] This is a basic tenet of aerodynamics and fluid flow, which is that it's indistinguishable if the object is moving through the fluid or the fluid is moving around the object.

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    While this is true, I think it's overestimating a minor effect because most drones are fairly dense - there's no large, light surfaces to catch gusts of wind. If you switch between battery-on-top and battery-underneath, you move both the CG and CoP. The question asks about CG vs motor height, which is irrelevant. – Robin Bennett May 14 at 8:17
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    ".. a basic tenet of aerodynamics ..". – Nij May 14 at 9:53
  • >since the CG is above the aerodynamic center, gravity now tends to pull the drone increasingly out of vertical alignment. uhmmmm.. Isn't that literally the pendulum rocket fallacy? Sure, contrary to a rocket, there are aerodynamic forces, but unless the drone is flying bottom-first, they shouldn't be misaligning it more with a higher-up CG vs. a lower CG. It's probably more proper to say that the CG determines the drone's preferred orientation relative to its direction of flight (or vice-versa, the preferred direction of travel relative to its own frame of reference). – FlashCactus May 14 at 20:51
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    +1 for catching a fallacy. I wasn't paying attention, the answer has now been fixed. The issue is not rotational stability, it's velocity stability. – Kenn Sebesta May 15 at 0:25
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As far as I know, it doesn’t make too much of a difference in regards to flight stability because of something called the Drone Pendulum Fallacy, which describes how the drone rotates about the centre of mass, not from the point at which the thrust is created as illustrated below in the photo from the Drone Pendulum Fallacy website: Diagram off centre of rotation

In my experience, it may affect handling - many racing pilots prefer bottom mounted whilst many freestyle pilots prefer top mounted, however this is down to the way the quad handles and so is really personal preference.

There are other factors you may want to consider - for example a top mounted battery brings the centre of mass closer to the thrust line, and it can also balance out the impact of having a GoPro on your quad. A top mounted battery also means you don’t have to land on your battery so takes away that risk.

However, a top mounted battery requires more space than a bottom mounted one (assuming you also want to use a GoPro).

Oscar Liang talks about this on their website, so I recommend taking a look there to help make up your decision.

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    The drone pendulum fallacy only applies to internal stability -- in an environment without external forces, the location of the center of gravity relative to the center of thrust doesn't matter, so long as the thrust axis passes through the CG. Once you start introducing external forces (eg. wind gusts), it indirectly does. True, the center of thrust itself doesn't matter, but since the propellers and their support structure is the main source of drag for most copters, it's a reasonable approximation to the actual aerodynamics involved. – Mark May 13 at 22:32
  • I always see people talk about bringing the center of mass closer to the thrust line. I don't see why that's important in terms of physics. If the moment of inertia is the same there should be no difference in angular acceleration whether the props are 10ft below the CG or directly in plane (assuming the props are at the same radius from the CG). It seems like what's happening is that the moment of inertia is being reduced, but the performance gain is attributed to CG near prop line. – Luca Scheuer May 14 at 13:53
  • @LucaScheuer Maybe they're talking about horizontal distribution of mass? The actual thrust line (as in, the line on which both the cumulative thrust vector and the point of its application lie) is always perpendicular to the propellers (if we're talking about a regular "flat" multicopter), so the only way to move the CG closer to it is horizontally. This isn't too important, since the line itself is constantly moved around by the props spinning up and down - that's how the quad rotates itself - but having the CoG dead center means the props will spin evenly to maintain level flight. – FlashCactus May 14 at 21:03
  • @LucaScheuer moment of inertia goes up with the cube of distance. So bringing mass the CG closer to the plane reduces the inertia of the motors-- some of the heaviest parts of the vehicle-- cubically. – Kenn Sebesta May 15 at 0:27
  • @KennSebesta Looking at the lists of common moments of inertia's on wikipedia, it goes up with the square, not the cube I think: en.wikipedia.org/wiki/List_of_moments_of_inertia Bringing all of the mass closer together would reduce the moment of inertia, but there's nothing saying that the center of mass or the moment of inertia is lined up with the prop plane. I agree that the motors are heavy and are a big part of the copter's moment of inertia, but that doesn't mean that with a given set of components the lowest moment of inertia is achieved by putting the CG in the prop plane. – Luca Scheuer May 15 at 1:20

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