title_body: Is it better to have a bottom heavy quadcopter or a top heavy quadcopter when it comes to a quadcopter stability? 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?
upvoted_answer: 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.
  
  
  
  
  
  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.