The crane is built with the two patterns below (waterbomb and miura-ori) and they interface in a such a way where they can be modelled pretty much exclusively with the topmost equation to the right of the images. To change how folded the crane is, just change theta.

I also added the ability to change the angle or length of the tail, neck, or head, represented by the alpha and percent values in the variable chart to the right. The base file is linked to the right. I’ve also added the final brace to show how the base file was used in it.