The Shaper Origin handheld CNC router makes an impressive attempt at compensating for imperfect frame rigidity, as it's designed to be handheld. The Origin operates in only two axes and uses consumable tape placed on the workpiece as an optical reference, but its progress is noteworthy.

I expect it would be easy to compensate position (closed loop is already table stakes in the professional tier) and extremely difficult to compensate chatter. What would it take to drive the motors of a 5 axis CNC to responses well into the kHz? You would need that, to compensate chatter. You would also still need to handle the same peak force, whether or not you do it with negligible deflection.

There are some industrial machines that compensate for chatter by slightly varying the spindle speed when chatter is detected. However machines so equipped are fundamentally very rigid and usually experience chatter occurs when pushing end mill performance to its limits. I suspect the adaptive spindle speed strategy would be much less useful on a machine like the Polar Bear.

Right, resonance dodging and deflection compensation aren't going to cut it here, not by a long shot. I still think a sufficiently good 5 axis control loop could damp acoustics: for one, the (fractional) millisecond timescale of the acoustic oscillations are geological by compute standards, and while I'd expect limitations from sensing, actuation, and power electronics, I don't see anything fundamentally prohibitive. I'm pretty sure this is an engineering problem, not a physics problem, and that eventually someone is going to crack this nut. It won't be the Polar Bear.

One way to compensate chatter is to have a small mass (say 50 grams) on an X-Y-Z voice coil on the cutting head.

Then you can use that to apply large brief forces in whatever direction is needed to stop chatter for a few milliseconds with each turn of the cutting tool.

It's the same mechanism used in iphones "taptic engines" to make 'clicks'.

The same mechanism can be used to actuate the head at far higher frequencies, allowing cutting fine detail at high speeds.

I would think it would have to be closed loop. Material density is not always uniform (especially with something like wood) and that has one of the biggest effects on how your tool behaves.

Even then, compensating for sudden release in tension, where the tool tries to spring back, seems perhaps insurmountable.

The problem isn't so much the rigidity for absolute positioning, it's the high-speed vibration/chatter than a rotary cutter will induce. You can't really damp that out in a system like this (remember, you're spinning your cutter at thousands of RPM). This results in additive errors. Think of the washboard pattern that arises on dirt roads in dry areas. It's the equivalent, but your cutter doesn't have shock absorbers. If my limited experience is any guide, you'll eventually start snapping off your cutter, breaking your stock free from the mounting, and/or getting mysterious gouged cuts into the stock that destroy your detail.

Not pragmatic IMO - flexible usually means resonance and vibration which leads to poor finish and deafening the operator/upsetting the neighbors.

If a lighter frame was the design goal (but low cost wasn't!) then carbon fiber has an interesting stiffness/weight ratio and could have viscoelastic damping elements incorporated too. Realistically though, lightness isn't a prioritized design goal though I certainly see the appeal in marketing for home usage where you wouldn't need a crane to move the machine about.

It could work to have two towers. The first carries the cutting tool. The second carries a measurement device and is subject to negligible forces. Alternatively some form of photogrammetry, possibly with a structured light field.