In 3D printing firmware and slicers, jerk settings are expressed in units if mm/s. This is contrary the physical definition of jerk, which is in units of mm/s³, being the second derivative of speed with respect to time (or the third derivative of position). What is the reason for this discrepancy and how does one interpret jerk in this contect?
The jerk setting in 3D printing G-code and firmware represents a concept similar to, but distinct from, the physical definition of jerk. Rather, it's a [limit on] instantaneous change of speed.
Mathematically, one way to make sense of this is to think that, rather than being the second derivative of speed with respect to time, this "jerk" is the entire remainder of the first-order expansion of speed with respect to time - it corresponds to the second-order term and all higher order terms. Such terms cannot be combined just as coefficients, since they all have different units corresponding to different powers of time; rather, they can be combined only with their corresponding powers of time, in which case the resulting unit is mm/s.
The units for jerk should be meters per second cubed or m/s3.
Meters are the basic unit for distance. The first derivative is speed, or velocity, m/s. The second derivative is acceleration, m/s2. The third derivative is jerk, m/s3.
It is rate of change in acceleration.
While seldom used, I've only heard it once concerning the Hubble Space telescope, there is a fourth derivative call jounce, m/s4.
In step based motion control, the time between two steps is calculated directly from velocity. If that time is not constant then you are accelerating or decelerating based on a specified acceleration. The next time between steps is calculated from the current velocity based on the desired acceleration. However when moving 2 or 3 axes at once, this can result in very poor and slow performance when moving through complex curves composed of many small moves, because by the math one axis always needs to slow down too much if no jerk is involved. When doing motion calculations for step based systems, actual jerk m/s³ directly translates into how much velocity 'error' is acceptable in calculation of the next velocity (m/s) to allow turning lo angle corners more quickly, but without missing steps or stalling motors. This velocity error comes directly from the actual jerk between two steps and it does have physical meaning, (and the proper units). Also, low power microprocessors can do the math fast enough, which is not the case if square and cube roots get involved.