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I am planning to build a remote controlled dualcopter where I have two propellers, one on top and one on the bottom. The bottom rotor will be gimbal based for thrust vectoring. Something like this,

https://www.youtube.com/watch?v=b14_XQ5nd2A

I want to build the controller from scratch (the fella in the video used Multiwii Mega flight board controller). Do I need to get into ODEs, differential equations, etc. as the following paper has done?

https://andrew.gibiansky.com/downloads/pdf/Quadcopter%20Dynamics,%20Simulation,%20and%20Control.pdf

Or is this overkill? Could I simply use a PID based design for a target tilt, adjust control to reach it, and do the same for motor speed, etc?

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You can certainly use a PID based design and tune it from there. This design is simple enough that if you just have XYZ PID controllers and proper output mixing it should work just fine.

I'd recommend doing something similar to his design where the motors are far from the center of mass which will increase the torque generated by tilting the motors.

Good luck!

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The bottom gimballing propeller is basically the same as the control for a rocket. State space control theory was invented to solve this problem. Even there, you are controlling a linearized version of an inverted pendulum. The further you move from vertical, the less accurate the model is.

Having said that, there are hobbyists that do PID gimbal control of model rockets. However, they just want to go straight up.

State space control should be your best bet, but this is typically a full college course to fully understand (if you're lucky and already understand the basics of closed loop control).

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