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Usually it is recommended to balance the quadcopter drone in order to have the center of gravity in the center of the drone body, but this choice seems with the assumption the drone thrust is just straight up with perfect orthogonality, that makes perfectly sense if we assume the drone must be maintained flat in hovering or it moves upward only, but when the drone moves forward, actually it is slightly inclined so that the thrust force drag it forward, allowing it to cruise rather than moving straight up or remaining in hovering.

This makes me to assume there is:

  • an optimal center of gravity for up direction and hovering (that is center)
  • an optimal center of gravity for cruising flight (that is different from center)

How should be calculated the optimal center of gravity for cruising flight?

And in case is desired a general situations balance between the two, is it better to choose a center of gravity that is halfway between the two?

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I think this is covered by the 'pendulum rocket fallacy'. Even though the CG might not be at the same height as the rotors, when you split the thrust into the vectors for lift and forward thrust, they apply at the CG, not at the rotors. A rocket (or drone) with a low CG is not automatically stable. It's relatively easy to demonstrate this by moving a drone's battery from above to below the frame, and noticing that there's no change to the stability.

Probably more important is that when it's flying forwards, there are aerodynamic forces that could cause the front or back rotors to have to work harder. You could use BetaFlight's 'Black box' recorder to compare the relative throttle settings of the motors, and maybe offset the CG or adjust the aerodynamics to compensate. However no one does this, so I'd guess the effect is pretty minor.

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  • $\begingroup$ When drone is inclined in cruise, if the center of gravity is exactly in center, it should tend to stay flat at equilibrium, this is good in hovering and vertical movement, but will cause the rotors to counter this tendency with additional power draw in cruise. Because in quadcopters it is the relative difference in rotating speed of front and back motors that makes the drone to remains inclined to cruise, contrarily to rockets. So if the center of gravity is moved to make the drone to stay exactly at cruise inclination at equilibrium this will cause lower energy need during cruise. $\endgroup$
    – AndreaF
    Sep 11 at 18:05
  • $\begingroup$ Why do you think there needs to be a relative difference in rotor speed to stay inclined? If the forces are balanced, it should stay at a constant angle. $\endgroup$
    – Robin Bennett
    Sep 12 at 9:21
  • $\begingroup$ Because to maintain at angle a thing that tends to stay flat at equilibrium is needed to maintain the force to counter the gravity effect. E.g. to keep a mass balance that has equal weight in its plates at angle, if you stop to add an extra difference of force on its plates, it returns to equilibrium point, it doesn't remain inclined. $\endgroup$
    – AndreaF
    Sep 12 at 11:12
  • $\begingroup$ That sounds like you are still working under the rocket pendulum fallacy. There's no force causing a drone to 'stay flat'. How would it know what 'flat' is? A mass balance is stable because it is a pendulum. It pivots around a point above its centre of gravity. Objects in flight do not pivot around a fixed point. If you throw a ball (or drone), it does not stay flat. I don't understand why you think it does. $\endgroup$
    – Robin Bennett
    Sep 12 at 15:41

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