How to initialize stepper motor?
You have two choices:


Fit an absolute encoder to the motor so that you always know its position, even if it gets moved while the power is off.
At power on, perform a referencing procedure. Many milling machines do this. They will drive the motors until they hit an end stop or reference switch or sensor, at which point they know exactly where they are.


Added:

To decide on your solution, you first need to think about what accuracy you require. Having seen the video, my guess would be that it doesn't need to be super accurate. Whatever sensor you choose has to be able to measure the motor's position to the accuracy you require.

You also need to decide if the motor needs to be able to rotate more than 360º. If it's less than 360º then you can just use an end-stop with a microswitch.



Screw the switch down to the base of your robot using the two handy screw holes. And attach a piece of wood on the moving part of the robot so that it bumps into the switch. Whenever the robot resets, it can simply drive slowly round until it detects the switch was pressed, and then it knows exactly where it is.