you don't need a rig you just need some xpresso to connect the movements together and some setup to make it work.
I rotated all the objects axis (don't know where the model came from but the program probably has Y and Z reverse from C4D) - easy to do -not absolutely necessary but the xpresso is going to be easier if Y and Z plug into each other (the target null (see below) could be rotated to match but I prefer to have objects aligned with C4D's world if possible). Put some of the objects in the movement arms into the print head (which was called target) because they needed to be there. Put all the parts into a Null I named Printer so the whole printer can be moved as one. Created a new Null called Target which is what all the other parts (the arms, the up/down rack, the print head) will follow.
Aligned all the parts parent nulls to the target null so no additional calculations had to be made - basically the target null it at the tip of the print head and all the parts parent nulls are there too.
Added xpresso to pass the X/Y/Z from the Target to the parts that need it - the print head gets all three as it moves in all three directions - the rack only gets Y as it only goes up and down - the arms get X or Z depending on the arm. Clamp Nodes limit the motion in any one direction so the parts stay within the printer (and the target gets included so it can't move past those limits either)