You could build a pretty simple hexapod with just six cables to move a hanging flutterwumper in six degrees of freedom, and you could probably control the lengths of the cables with great precision; but by itself that doesn't give you much accuracy in the finished product.
Closed-loop control, though, could give you the accuracy you need. If you can measure the position of the flutterwumper to great precision, you can correct the position until it works; similarly, to correct for cutting-head loading errors, cutting-head wear, etc., if you can measure the depth of cuts in the material, you can correct for whatever inaccuracies arise. Ultrasound seems like a plausible approach at first, but for manufacturing metal parts, you need shape accuracy (thus measurement accuracy) on the order of 25 microns. At sea level, that's 75 nanoseconds of timing accuracy on the sound, which means you need a 13.3 MHz sound. Air cannot carry sounds close to that frequency. Radio seems like a plausible approach at first too, but 25 microns is 85 * 10^-15 seconds --- 85 attoseconds. There are no circuits contemplated that can measure the round-trip time of radio pulses that accurately. So light interferometry, of one form or another, is the only feedback mechanism I can think of that might work. Laser phase shifts can measure the distance of movement of an object; a circularly polarized laser can tell which direction, too. Using this principle you can measure the shape of a macroscopic continuous surface. I still don't quite know how to measure its position to within a few microns, but maybe I don't need to. Suppose you wanted to measure nearby vibrations with LEDs and a multi-megapixel camera. The LEDs and the camera are mounted some distance apart facing one another; the camera measures the direction to each LED within a small fraction of a pixel. So perhaps you could use cheaper cameras for CNC feedback. "Cheap" (around US$100-$300) consumer CCD cameras, such as the Minolta DImage G500, commonly have 100-degree fields of view and roughly 2000x2500 square pixels, with 2500 pixels across the 100 degrees --- so about 1/25 of a degree per pixel, or about 700 microradians. That means that a pixel subtends 25 microns at a range of about 3.6cm, and a hundredth of a pixel subtends 25 microns at a range of about 3.6m. Optical zoom extends this range; 3x optical zoom I believe reduces the linear field of view by a factor of 3, to about 33 degrees in this case, so would increase the range to about 10cm at one pixel per 25 microns. You could mount the camera either on the movable platform, with reference LEDs on a fixed point (or possibly just referencing from the workpiece, if the workpiece is not the fixed point), or on a fixed point looking at LEDs on the movable platform. In either case, two cameras looking in different directions should give dramatically better results, since the precision of measured depth will be much worse than the transverse motion precision. (Maybe purely mechanical feedback would work too, like monitoring stepper motor voltage/current waveforms.) The standard hexapod design makes the six legs out of precision ballscrews which change length; Hexel has a clever design called the "Rotobot" where the legs don't change length; instead, the points where they are attached to a fixed object can be moved independently all the way around a circle. Rather than using ballscrews, it seems that a compression spring (for example, a piece of bamboo or PVC pipe bent into a semicircle) with a braided nylon rope would be cheaper and might suffice for many purposes.
