To add to this, jerk is super relevant to self driving cars. While the car itself can handle sudden changes in acceleration, the humans inside often cannot. Self driving systems must account for jerk in their algorithms.
It's also super relevant to roller coaster design.
Something I've been wondering for awhile about this: is it specifically jerk that humans can't handle well, or all of the higher order derivatives? A lot of times we're talking about a car that's at rest (and has been at rest for delta t>0), so every order derivative is going to be positive when the value starts increasing (the rate of increase, the rate of rate of increase, the rate of ...).
Is there something specific about jerk that makes it important to optimize for, or are all position derivatives of order 3+ the same?
A mental model I employ is: from a frequency response perspective, humans are 30Hz high pass filters. That means we end up conducting more high-frequency force (either by actively fighting it or by having it act against our inertia by transmitting thru our bodies), and this is work. In the lower frequencies, we can generally more actively participate and e.g. spread out the energy transfer over longer time periods to decrease instantaneous forces in joints/etc. Picture jumping off a 3 ft ledge, you can "eat" most of the impact with your legs bending, but some of the energy is going to affect you. The mental model is that it's the high-frequency content that human bodies don't handle well.
> Is there something specific about jerk that makes it important to optimize for, or are all position derivatives of order 3+ the same?
I think of it in terms of neck muscles. If your car is accelerating at a constant rate, you feel that as a force pushing your head back. Your neck muscles activate to compensate and keep your head still.
If the acceleration changes suddenly and aggressively (i.e. high jerk), so does the force on your neck. So either your neck muscles react quickly to counteract the new force, or your head bounces around.
Higher order derivatives also matter, but mostly inasfar as they act on the acceleration (and hence force) that you feel.
Isn't that the acceleration? With constant acceleration (thus zero jerk) inertia makes your coffee move inside the cup – and spill, if you don't pay enough attention.
1) coffee is always under constant acceleration (g).
2) constant acceleration (say in the X direction instead of Y) would just mean a constant "tilt" within the cup. compare this to an airplane that is not "accelerating" and just at a constant X velocity, coffee would look "still/flat" in your cup.
3) its only the jerk that changes the "tilt" within the cup (and hence causes the spill)
Constant acceleration will cause the surface of your coffee to tilt, but not spill it unless the cup is almost full. Jerky motion (non-constant acceleration) causes it to slosh around and can create resonances that disturb the fluid level far more than a constant acceleration of the same magnitude would.
If you move in a circle around a corner, you have constant acceleration, otherwise you would go straight (centripetal force needs acceleration to exist). yet it is possible to move a coffee cup in a corner, as long as you tilt it a little bit. So acceleration is not the issue
However if you suddenly change the direction of the coffee cup, you introduce jerk, because you accelerate the acceleration (you change the size of the circle means you change the acceleration, therefore you introduce jerk = coffee spilled on the floor)
The 4th derivative is quite important for good motion control where it is usually called 'snap'. Specifically, it is relevant both for feedforward control design [1] and trajectory planning [2]. As shown in the latter, it is advantageous to design trajectories based on segments of constant snap. Consequently, also including 'snap' in the feedforward signals makes the achieved position profiles notably smoother.
After jerk come snap, crackle, and pop. These are rarely used in practice, but I believe snap (the 4th derivative) correlates with noise in high speed trains, so it's worth optimizing railroad tracks to keep snap (and of course jerk) low. This is one reason why polynomial spirals (Spiro curves) can be great tools for representing trackways or road centerlines; with G4 continuity, snap is continuous which means crackle is finite.
Also see this paper[1] (also cited by 'jjgreen elsewhere in this thread) which discusses the perception of higher derivatives in the context of roller coaster rides.
Your inertia comment kept me (over)thinking for a while.
So what is inertia, and what makes it interesting on its own?
I guess we could describe it as v(t) = v(t - d) if a(t - d) = 0 for small d (velocity remains constant unless a force, i.e. acceleration, is applied) but this seems to be a bit self referential since it's just a longer way to say a(t) is v'(t).
What makes inertia interesting compared to other derivatives? Isn't acceleration "inertial" wrt jerk by definition? Or rather, any derivable function is "inertial" wrt to its derivative. Even if we had velocity change without external forces we'd just introduce a "phantom force" like gravity to make it all work nicely.
Is inertia as a concept just an artifact of classical physics being framed in terms of position and forces?
