Origami Animation

Hi everyone!
For a project I am working on in Blender I would like to create an animation sequence in which a piece of paper fold itself into an origami bird. However I am not sure how to go about creating such an animation. I thought of using Shape Keys but I still am confused about the process.

I would like to reach a similar effect to that in this video-

I have some origami models I downloaded for reference from Blendswap, this is one.

I would appreciate tips and suggestions and if anyone has had any experience with such a thing.
Thank you!