The carpenter's ruler problem is about polygons you can make with planar joints. You can formulate a similar question for knots when you introduce ball joints:
Given m ball joints connected by rods of arbitrary length, which knots can you make?
I found out that you can make the first non trivial knot with your fingers and thumbs and your thumbs are ball joints. This is probably the same question as for stick knots and since the stick number is not known for all stick knots, the answer to the question I must say is, no, the question is an open question.