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What theories/equations can be used to determine the banking angle of a quadcopter/helicopter due to unbalanced load, given that the load will only shift along the X-axis

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I think a simple approach could be :

  • First step: establish step by step the difference of load between the aft and the fore part of the X-axis. This data should be integrated into a global value of Inertial Momentum
  • Second step: This inertial momentum comes from the difference of load on the X-axis, but should be applied on the Y-axis (square to the X-axis). On the axis, you will apply this inertial momentum against the forces.

The forces are:

  • The excedental part of the load, which as a consequence as a force because of gravity
  • The force coming from the motors of the quadcopter

Then you have the banking angle acceleration, which will be dependant of the banking angle itself. Just solve the equation: Banking angle acceleration = 0 and you will get a resulting value for the banking angle.

Sidenotes: The forces exerted by the motors might depend of the banking angle, because they will be more or less horizontal. But that should be a second approach of the problem, try the simple one I explained for a first approach.

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Any object that has been designed to fly has some level of auto balance and self leveling which can handle a certain amount of loading misplacement or turbulence recovery and or pilot errors.
There are different ways to achieve this, passive or active.

Many have been designed such that aerodynamic characteristics of the flight surfaces and rotors will tolerate loading within a prescribed central allowed area. loading outside permitted area will cause pitch up or down which is controllable by counter maneuvering. Beyond that, off-center loading will cause flight instability and lead to crash.
These properties are tested in wind tunnel and incorporated into the auto pilot software.

The intuitional way of simplified analyses would be to handle it like a boat's unsymmetrical loading, but because of the pi/2 phase difference in rotors due to gyro action, it would not work that way. In case of small drones their control electronic modules are getting very efficient and very cheap. you could find drones under $100.00 which are self aware of their surroundings and can do aerobatics.

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