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1) What is the typical following error for an axis with a linear scale of 0.001mm resolution(1 micron)? In steady state it is able to position the axis within +-0.005 accuracy. Do you think it is possible to get more from that scale resolution?

Typically, some I term is added to reduce the static error.

During acceleration and deceleration, an error jumps up to 0.1mm at 1500mm/min speed. What is a typical result for well well-tuned PID loop?

That mostly depends on the characteristics of the controlled device (delays, linearity etc)

Can you post a Halscope plot of following error and velocity?
 

I did this with PLC, it is easier and the expected result is obtained.
Greeting

Can you post a plot that shows this oscillation, including the velocity and the ferror?

but I know how to use the halscope. I know how to read the curves, but I don't know which parameter to adjust when, for example, I have oscillations in the error.

Selecting this button fixed my issue .

Absolute spindle speed accuracy may not be terribly important
anyway for actual machining.
 

Unfortunately the spindle doesn't have an encoder. And adding one might proof difficult.

Thank you. I will check and try.

I want to ask you a question. How can I use the M19 command with Mesa 7i96s?

Can I call the M19 gcode and position the spindle?