Hyperbolic origami with rigid folding

Hello everyone,

First of all thank you for taking the time to read my post. I'm trying to fold the pattern that you can see on the image attached. I took different rigid folding examples as my base script and identified the Valleys and the Mountains of my folds. I think the script is right and Kangaroo is folding it correctly, But i cannot find the way of controlling my fold so i can achieve the final shape I want. 

It might be about adding more anchor points, but then it wont fold as free as i want it to be.

If anyone has any hints I would appreciate it.

Thanks in advance and kind regards,

Maya

This is physically how i would like the shape to be like (sorry about the crappy finish to much folding)