willpower100... long time, no see...
This kind of question, as you have posed it, always leaves the actual situation wide open to personal perception and interpretation by those that are trying to help.
A poorly defined question gets poorly considered answers.
This problem is solvable... however... we need some answers to some critical questions.
I wonder...
Is the material being cut being cut by a blade, or is it being cut by some sort of water-jet?
I've worked with both, flying-blade-cut and flying-water-jet-cut.
If being cut by a blade, does the blade assembly "grab & hold (hold while flying)" the material as it is being cut?
If being cut by a blade, is there some room (timing-room) for allowing the lateral motion of the blade-assembly to be a bit slower than the line speed?
Is there a discernable difference between items to be cut?
Time-wise, what is that difference? (leading-edge to leading-edge and cut-extend to cut cut retract)
If being cut by a water-jet, the head must move at a particular speed for the particular angle (the speed corresponds to the angle, SIN, COSINE sorta stuff) to make a square cut.
If cut by water-jet and the cutter is hydraulically driven then there MUST be a way to at least adjust the cross-rate. Otherwise, there must be some sort of position feed-back! (Feed-back control is by far the better way to go!)
Do you have any kind of position feed-back?
These are critical questions that need to be answered before a solution can be developed... even so, there may be other critical questions needing answers depending on the answers to these questions.