New random location within each time loop?

I am using a time loop to repeat an animation. I have attached a random converter to the animation so that I do not have to animate each frame.

But I want to start the repeated animations from a different point on the canvas each time. Is there a way that I can do this? I assume that I will have to attach a random converter to one of the parameters, but despite trying it on a number of them, none of them have done what I wanted.

Perhaps add translate layer and animate that?

Thanks. This was actually very good advice although it took me a while to figure out how to do it. I’m still getting used to Synfig and have started to learn about the importance of arranging the depth at which layers and converters are applied. Depth is definitely not just about the placement of images.