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I am trying to build a robotic arm that needs to lift a very low amount of weight (between 0.01-0.05kg) at a time, but it needs to be very precise (+/- 1mm accuracy ideally). The robot really only needs to have 3 DOF and the arc is roughly 0.5 meters. Additionally, the robot doesn't need to be very fast, only around 0.2-0.5 m/s is necessary.

What type of actuator would be the best suited for this kind of task, and what other factors would I have to consider? Do I need to spend more money for high quality servos, or can the task be done with hobby/RC servos with some modifications?

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Ask yourself if you really need accuracy or precision. Because I think precision is easier to achieve than accuracy for most arms.

I am not sure I would be as worried about the actuator as the gear-train and sensing. Here are some back of the envelope calculations to illustrate:

At 0.5 meters, to achieve an error of 1 mm, that is $asin(1/500) = 0.1146^\circ$. (And there will probably be an elbow halfway down the arm with similar backlash, compounding the error). You probably want your gear train to have a backlash less than that or else you will have a hard time controlling the arm.

To sense this error, your joint encoder will need a resolution of $360^\circ/0.1146^\circ = 3141.6$ ticks/rev. And you will really want these joint encoders after the gear train to be able to measure and account of the above mentioned backlash.

Thankfully, your payloads are light, so arm sag is probably not a concern.

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