Both videos suggested above use a standard, off-the-shelf, servo and one of the "arms" that comes with the servo. The difficult part is designing, fabricating, and installing the linkage between the servo arm and the doors. That's what I believe is the stumbling block and why these projects remain on a perpetual back-burner.
I have no experience mechanizing roundhouse doors so speak only as an interested observer. The simplicity of the design on the left is a big selling point and something I'd consider. I do have some experience with servo mechanisms applied to customized animations rather than a traditional r/c plane, boat, whatever. What I have found a useful prototyping process is a servo exerciser which is about $1 (on eBay, free shipping from Asia) paired with an entry-level servo which is also about $1.
Servos come with several different "arms" which snap on to the output shaft of the servo box. A servo exerciser (you will need to provide 5V DC) allows you to set the servo arm's angle across its range which is about 1/2 turn. This is shown in the video below starting at about 10-seconds into the video. There is a lot of stuff in the latter part of the video which you can ignore as I made this for a different OGR topic.
My point is the final electronics which makes the doors open slowly, and stops the doors exactly at the right positions without ripping the door off the hinge, and "knows" how to reverse the motor for opening vs. closing, etc. should be put aside as tomorrow's problem. Defer the choice of "electronics" whether it be an Arduino with custom programming, an off-the-shelf programmable servo controller as from the U.K. company, or some 555 timer IC chip homebrew concoction.
Instead of fretting about the electronics, focus like a laser on a reliable mechanical linkage. Use the servo exerciser to "manually" turn the knob slowly back and forth to simulate the door opening/closing at prototypical speeds.
