I have one of these slide-out trash cans:  

I want the trash can to slide out with the door being opened and slide in with the door being closed.

I'm thinking of a method to convert the door opening motion (rotational?) into the linear motion of the trash can. Now I have tried searching multiple ways for solutions on converting rotational to linear motion, but came up empty for this seemingly simple problem. The door motion of course isn't purely rotational as we're not talking about converting the hinge-motion, but the motion of the door.

A method that works is just attaching a string to the side of the door and onto the trash can. That works for pulling out the trash can, but pushing in obviously doesn't work with a flexible string.

I'm lacking to tools to properly model this problem, right now I am stuck to fiddling around - I'm assuming there are better ways to solve this.

I have one of these slide-out trash cans: 

I want the trash can to slide out with the door being opened and slide in with the door being closed.

I'm thinking of a method to convert the door opening motion (rotational?) into the linear motion of the trash can. Now I have tried searching multiple ways for solutions on converting rotational to linear motion, but came up empty for this seemingly simple problem. The door motion of course isn't purely rotational as we're not talking about converting the hinge-motion, but the motion of the door.

A method that works is just attaching a string to the side of the door and onto the trash can. That works for pulling out the trash can, but pushing in obviously doesn't work with a flexible string.

I'm lacking to tools to properly model this problem, right now I am stuck to fiddling around - I'm assuming there are better ways to solve this.