The picture for the double pendulum I am referring to is pretty standard, wikipedia for example uses it and so does any other text book. I do not completely understand why one uses the second angle measured from the vertical y-axis for the second generalized coordinate. The second angle is not independent of the first, that is if the first angle moves, so does the second. Why not use the angle measured relative from the first mass to measure the second mass, that is use the angle that is calculated from being in the first masses reference frame for the second mass. This way, the second angle is independent of the first angle. Thanks for any help, I really appreciate it!
Both formulations are completely fine. In terms of an implementation for measurement in a practical setting, it would make more sense to develop the equations of motion with the second angle being measured relative to the first, but both procedures are equivalent and produce identical dynamics
Lagrangian is slightly simpler if you measure second angle relative to a fixed axis, vertical being the easiest due to symmetries. By the time you get to actual equations of motion, though, it won't matter.