Hello! Glad you got things moving 🙂 It's normal for the voltage reg on the arduino to get hot. I've had very varying results from different makers of the arduino clone boards, some roast while others just stroll along without being bothered. It shouldn't be tripping the thermal safety on the reg though, that is very bad. Does the whole board reset? Because there is a inactivity timeout (5 minutes) that releases the motors eventually (for this reason). The motors themselves will be hot too.
If you reduce the overall current that the motors draw (with the little screw adjusters on the drivers themselves - turn anti-clockwise to reduce) then you will get an overall cooler experience - but the issue is that you decrease the holding power of the motors too. There's a balance to be struck there. And yes you've spotted that holding still is the most stressful. Heat makes the motors less efficient, and that makes them hotter, and there's a runaway situation.
The best solution (and really the only solution if you want to run higher than 7.5v) is to have a separate power supply for the motors, wired into a power jack on the polarshield itself (there's a space for one). There's also a jumper that should be removed for that, and that separates the two supplies.
Happy to send those jacks out to anybody who would like one - I will include one in kits in the future I think. It's a pain having two supplies though.
Ah motor connections. There's four ways to be connected, that will work, electrically. There's two ways that will not work, but also won't damage anything either.
The best pic is probably this one if you look at the colour of the connecting cables.
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A couple of Polargraph SD Assembly questions
