Are those RC hobby servos? If so, usually on power up they move to a 'home' position. If the servo is not already in that position, it will move there as fast as it can.
The usual fix for this is to have a structured power down process where you 'park' the arm in a known park position. Then, before you attach the servos, servo.write to each servo with the values that put the arm in that park position.
Edit: Oh, I should probably mention, there is a mod you can make to the servo to expose the internal position potentiometer. A bit of delicate soldering. People open up the servo, solder a wire to the middle of the position pot, and run it to one of the analog pins. You can then read that value of that pot before you attach, servo.write a value that tell the servo to move to the position that it is already at, and then attach. You will still likely get a twitch but not the violent jerk when it tries to move the whole arm a long distance suddenly.