Forgive a few "dim" questions? I would love to see a good battery back-up design established.
a) Relay drawings: There's a newbie trap there, isn't there?? I spent half an hour with two copies of the schematic and colored highlighers, doing "power on" and a "power off" drawings before I had anything that made sense! One's drawn as if energized, the other as if NOT energized. True? (Q1) (Not helpful!)
The right hand relay, the one below R3, which I'll call "Relay 1" is open until energized, right? (Q2). It's role is to prevent the other relay becoming energized until the voltage out of the bridge rises adequately. That rise could(?) be delayed more than a few centi-seconds if the battery has become significantly discharged while it was powering the Arduino. Yes? Is that the cleverness in this circuit? (Q3)
The left hand relay, the SPDT one, "Relay 2" from now on, connects R5 directly to the =ve pole of the battery when the 120Vac is ABSENT... true? (Q4)
b) Transistor: I don't understand what this is doing... apart from creating the "possibility" of hitting my Arduino's 5v rail with 12 volts! (I assume it won't do that, but to someone like me who only "understands" transistors as being similar to SPST relays, it looks scary!) Maybe there to provide a way to recharge the 50v 1000uF cap (which I'll name "C1" for future discussion) when a voltage returns above R5, after the brief outage while relays operate. I understand the transistor so poorly that I can't even ask a real question, so Q5 is just "What's the transitor there for/ what does it do?"
c) The connection of the battery to the transformer: It seems a shame that the battery is only charged by half of each AC cycle. I presume there was some reason that the connection above the diode above R1 and the connection from the negative side of the battery couldn't go "after" the bridge, and thus benefit from both halves of each cycle. If there IS a reason, I'm content to be told "there is a reason"... I don't need the details. Q6: Did the connection have to be "above" the bridge?
Lastly a word of caution to newbies...
While I don't understand all of it, I do believe that the circuit shown does actually work! But that doesn't mean that it will work with just any old re-chargable battery. There are a whole bunch of design considerations to master when doing UPS design, and you should become informed about them before going too far with building something like this.