Circuit diagram for uninterrupted usage of a 12V 1A device using 12V 7Ah battery with auto charging and full charge cut off
You can charge the battery using an Arduino as the controller. 

Sense the voltage of the battery with the ADC, making sure you scale the twelve volts down with a voltage divider as the Arduino will only handle an input of 5V on the inputs. From this you can work out the duty cycle of the signal to send to the battery to charge.

Once the battery is fully charged voltage, stop the input signal. 

What are you going to use for an input, as you may have to get a AC to 12V DC power supply?

Do a search for Arduino PWM charger for more details.

This is all dependant on if you want to build not buy, buying will be easier but where is the sense of achievement in that:>)