I'm making an AUV that requires 15-20 minutes of operating time. I'm using 6 of these motors (Blue Robotics T100 Thrusters with specs at max are 135 watts and 12.5 amp ) I don't know how to calculate the capacity (mAh) I need to run my vehicle. Is there any formula that allows me to calculate for specify the battery which I will select?
If you want to run one motor at maximum output, it will draw 12.5 amps. If you run it for an hour, it will require 12.5 amp-hours of capacity.
If you have six motors running for an hour at maximum output, then you will need (6)*(12.5) = 75 amp-hours of capacity.
If you want to run the six motors at maximum output for 1/3 of an hour (20 minutes), then you need (1/3)*(75) = 25 amp-hours, or 25,000 mAh.
You should also take into account battery discharge capability (the "C" rating) and min/max states of charge.
For the "C" rating, it's referring to how quickly it can charge or discharge. If you find a battery pack that does have 25,000 mAh, but it's rated for 1C, then it would safely discharge all 25,000 mA in one hour. You need it discharged as quickly as 15 minutes, so you would look for a battery with a C rating of at least 4, else you would need to increase the capacity of your battery pack.
Finally, as a general rule I would generally only assume that I could charge to 80% and discharge to 20%, so I would assume I can only get 60% out of a battery; this would mean I would try to spec a pack that is (1/0.6666) = 150% of what I think I need.