I have a lifting bed with a motor that runs on a battery like in a UPS. Is it possible to charge this battery with an Arduino with some module? The battery would be charged from the mains (car battery) when the bed is in the upper position?
To answer your question a charger can be made with an arduino plus a few additional boards.
HOWEVER, a car battery is a very poor choice; it should be a true deep discharge battery, NOT a hybrid. If anywhere on the battery the word CRANK or CCA is visible, then it is NOT a deep discharge. The easy way to find them is to go to a golf course and ask where they get theirs. The gold standard had been the Trojan 105 for many years. You will need two as they are 6V. If you tell me where you are located roughly I can probably find a Trojan supplier close to you.
Although your board is an Uno, your question is more generic and therefore has been moved.
That is like saying turn on the light in the middle of a forest at midnight with a blindfold on. More then likely it is a SLA but that is a guess. Without knowing the chemistry of the battery guessing what you have could cause you problems. Post the battery information or at least a photo showing its label.
Where is the charger that came with the bed?
I disagree. A bed lifting motor is not going to be used continuously, but more like the starter motor of a car. Infrequent short lasting bursts of high current. A cheap 12volt/7A SLA, like used in UPS' for computers might indeed do the trick.
There are special plug-in maintenance chargers for those small batteries.
I don't see what function an Arduino could have in battery management.
Leo..