Proposal: Non-linear PA pre-smoothing

[richfelker]

A surprising property of the non-linear PA models from Dmitry (in BEv2 branch) is that the nonlinear function is only applied to the final smoothed velocity after integrating extruder velocity over the smooth time window. This has two very negative consequences:

1. Smoothing cannot be applied to the nonlinearity to reduce the severity of the motions it imposes on the extruder. This requires using a higher linearization velocity, and/or lower linear offset, than what might be physically ideal.
2. Because the integral of a nonlinear function is not equal to the nonlinear function of the integral (integration only commutes with linear functions), the amount of advance is not actually accurate.

Problem 2 is big, I think. The nonlinear PA response will only pull back on advance when it sees a very low speed (roughly, under the linearization velocity). We want this to happen especially at corners at reasonable SCV values for high quality, where the SCV is well under the linearization velocity. However, with high kinematic acceleration, unless the PA smooth time is extremely low, the average, smoothed-out extruder velocity around a corner will remain well above the linearization velocity, so that only the linear PA term has any effect. This in turn makes it necessary to configure a linear PA constant much higher than ideal to get good corners. And moreover, the magnitude of the excess depends in a complicated way on speed and acceleration. As such, this is probably one of the remaining root causes for the "you need different PA settings at different speeds and accelerations" problem.

The reason the nonlinear model was implemented as applying post-integration is clear: there's no easy and cheap analytic way to integrate tanh(mx+b) times a window function, so we'd be stuck with a lot of expensive math per move in the integrator or doing an (also expensive) discretized numerical integration instead of antiderivatives.

What I'd like to propose is simplifying the nonlinear model to an exponential function of the form A(1-exp(-x/B)). This is relatively easy to integrate in closed form even with a window function, and the parameters can be chosen to fit the tanh curve almost exactly for x≥0. Then, when nonlinear PA is enabled, the integrator can process the exponential part alongside the linear part in computing the smoothed PA integral. If my math is correct, should amount to two calls to exp (or likely an optimized approximation for the limited domain) for each move in the integration interval.

Moreover, I'd like to consider implementing this first for mainline Kalico branch rather than BEv2. IMO the changes Dmitry made to PA in the advanced features branch make it harder to implement new functionality like this, especially with good performance, because the factoring of the code makes for more internal interfaces that the parameters and results need to be marshalled across. But perhaps more importantly, I think this would open the door to getting nonlinear PA to a wider audience, and maybe even getting it in Klipper too someday. The big blocker to doing this, however, is lack of extruder sync-to-IS in mainline. Without that, the results will certainly be worse than anything in BEv2 right now. So, if I'm going to do this in mineline, I think a prerequisite is adapting sync-to-IS or comparable functionality to mainline. I have some ideas for that too, which I'll probably put forth in a separate ticket.

[richfelker]

Some math I worked out to make this concrete:

For a 90° corner, assuming 0 scv for simplicity, with a being the E acceleration and s being the smooth time, the integrated velocity is a×s/6. This assumes cruise is not reached, which I think is a reasonable assumption for all parameters at which these phenomena are relevant. If cruise is reachable in half the smooth time, either your commanded speed is incredibly low relative to your acceleration, or your smooth time is way too large to be viable for high quality results. If scv is not zero, it should add a constant offset of whatever E feedrate corresponds to scv kinematic rate.

For some concrete numbers, at 400 E acceleration (corresponding to 12k kinematic acceleration) and 10 ms smooth time, the integrated velocity is 2/3 mm/s. At 20 ms smooth time, it's 4/3 mm/s.

Assuming a linearization velocity of 1 mm/s, this puts us right on the edge of the nonlinearity doing "something, not not adequately weighted" and "essentially nothing". If you add in 10 scv (1/3 mm/s on E), this leans heavily towards "nothing" despite still dipping way into the nonlinear range at the corner and remaining below the linearization velocity for a third of a 10 ms smooth window.

At significantly lower accelerations, the current model is probably behaving mostly ok already. At significantly higher accelerations, it's behaving at corners exactly like linear PA, and the only nonlinear effect is going to be at start/stop points (seams) where it should still reduce bulge/oozing.

[richfelker]

A little bit more information for context: linearization velocity is not a hard cutoff. Per the tanh curve:

• At linearization velocity, 76% of the offset has been applied.
• At double, 96% has been applied.
• At triple, 99.5% has been applied.
• At half linearization velocity, 46% has been applied, and below that it goes down almost linearly.

Practically, this means the nonlinear part is having some plausible effect as long as the integrated velocity dips below twice the linearization velocity.