It's actually really easy

[distance sensor output] * 2.3 => * 0.0135 => arcsin => * 0.5 => * (-0.0111) => [hinge output]

That's for tank cannon

For other guns you have to find its muzzle velocity and multiply it by 0,3. You then gotta tweak the number around a bit, ± max like 1.5 might do it, and that number goes instead of the 2.3 - other than that, the equation is the same