# Why does jerk have units of mm/s rather than mm/s³?

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?

• I'm not sure what other tags would be appropriate for this question; please suggest or edit them in if you have good ideas. Jun 24, 2019 at 19:07

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.

• Confused me for a sec - I always look at the Nth derivative of position, not speed. Just noting this in case anyone else is as poor at reading as I was. Jun 25, 2019 at 14:34
• @CarlWitthoft: Of position. I specifically said of speed. Jun 25, 2019 at 14:36
• @CarlWitthoft Speed (v) is the derivate of positon (s) over time. because of that the physical jerk = v'' = s''' Jun 27, 2019 at 11:00

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.

• While this expands on the physics definitions in the original question, it does not address what jerk means in 3D printing or why the units are different.
– Davo
Jun 25, 2019 at 21:40

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.