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I have a mobile base (1400x1800x400 mm) which has a vertical bar (1) mounted on its geometric center; this vertical bar has a length of 2500 mm and a diameter of 100 mm and I need to keep it always in its vertical position even if the mobile base is not correctly aligned as showed in the attached photo.

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I use an inertial sensor to read the bar position. I was thinking to use four electric linear actuators, but this could not be a good idea because they would be hard to control (if they are not perfectly synchronized, they can get stuck or broken very easily). What about stepper motors? I would like to implement something like the system used on the 3-axis gimbal. Do you think there is any commercial solution I can use?

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I believe your intuition about using push-pull linear actuators is in the right direction: you would have a quite hard time to control such a structure.

Traditionally, to deal with a cart-pole system you are supposed to deliver torque to the actuators with a certain level of accuracy. In this respect, BLDC motors are certainly the best way to go, especially over step motors where instead you don't have fine control on the torque.

On the contrary, if you're not very interested in precise control of the bar position as well as the smoothness of the resulting movements, then you could consider a design based on step motors.

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