Beyond velocity and acceleration: jerk, snap and higher derivatives (2016)

I know, this is an old paper, but I don't follow the this assumption:

> The terms jerk and snap mean very little to most people, including physicists and engineers.

Almost 20 years ago we defined jerk into our standards for lift applications. I know jerk is an important parameter for any modern rotating machine that includes gears.

While in lift applications it is known as the roller coaster effect, people in different parts of the world have a different taste on when they want to use a lift. I know I over simplify when I say, that American people want to have the gut feeling when riding a lift, especially an express lift in those high buildings. In difference in Asian countries the lift ride must be smooth as possible. They don't like to have the feeling of riding a lift at all. In Europe it is something in between. Lift manufacturers have to respect those (end) costumers otherwise the are not chosen.

The same in any rotating machine with some sort of gears. Because jerk and those higher orders contribute to the wear and tear of gears. As you want to have longer lasting gears many modern machine manufacturers limit those parameters to reduce wear and tear. So, with a little software change I can demand a higher price because service and maintenance can be reduced.

> American people want to have the gut feeling when riding a lift, especially an express lift in those high buildings. In difference in Asian countries the lift ride must be smooth as possible. They don't like to have the feeling of riding a lift at all. In Europe it is something in between

How representative are these stated preferences actually of the population. I'd imagine that the individual preferences vary greatly from person to person and also change with age.

One thing that is strange is that we can easily imagine the first two derivatives: position we can just imagine a static point, velocity we can imagine a constant speed i.e. a straight line on a position-time graph, acceleration we just imagine a parabola, but jerk is somehow conceptually indistinguishable. The difference between a point, a line, and a parabola are stark, the third order jerk is not so easy to distinguish, instead still just looking like the parabola.

I've always wondered why this is, why curves in general are perceptually similar if scaled correctly, while a straight line is so clearly different. Perhaps it is because our perceptions developed to distinguish between inertial and non inertial reference frames?

They're the preferences of the buying managers.

I like and agree with your observation. But I think you can use conceptual tricks to get just a little further: Acceleration is "due" to a force (F=ma), so you can think of jerk as a change in that force linearly increasing over time.

That doesn't help me recognize a cubic from a quadratic when looking at a small piece of it, but I can imagine an elevator ramping up it's lifting power or similar. It kind of feels like the tricks to conceptualize 4D as 3D position plus a temperature at each spot.

Jerk is also very important for road or rail track design. If you imagine needing to make a 90 degree bend, the "obvious" way to do it is by rounding off the corner with a circular radius.

But if you do that, it means the vehicle goes from having 0 sideways acceleration to experiencing 100% of the centripetal acceleration to move an object on a circular path (a = v^2 / r) instantaneously.

As an occupant of the car, that means you go from sitting comfortably to suddenly being thrown sideways.

It's much more comfortable if you ease into the turn, with the track design considering the rate of change of acceleration. If the designer didn't consider jerk you would definitely notice.

It’s seems analogous to our spatial dimensions. We can all easily visualize or describe up to a 3D object, but 4D is almost impossible to fathom for most people

It might not be a strict "preference", more of an expectation how things should be based on previous experience. Like, if you're used to an elevator with a bit of a jerk, an elevator taking you there just as fast but smoother might feel not as fast.

I agree the linearisation trick can be used and is often used in physics, but we must do that as a consequence of the thing I'm confused about the origin of, which is not why do we only greatly distinguish between the first few derivatives of position, but why do we only greatly distinguish between the first few derivatives of most functions? I.e. why do we have to do these tricks in the first place?

Tje difference is because we cannot easily tell between “curve of second order” and “curve of other order”.

You can get an idea when you try to understand why the function

y=0 for x<0 y=x^2 for x>=0

has two derivativea but not three.

But the issue is infinitesimal, so very hard to tell.

Jerk you can “linearise” if you think of a car (with no air friction) and its accelerator. Somehow…

> Tje difference is because we cannot easily tell between “curve of second order” and “curve of other order”.

Why not, though? Why does third order "look like" second order but second order is starkly different to first order?

The terms are also understood in economics as prudence, and template. Albeit, not widely used.

Yet it is a captive market. If I am in a building, I only have access to a singular type of elevator. Why not always give the smoothest ride possible unless it is $0.12 cheaper for the installer, so everyone has to suffer forever.

If I were buying, I'd ask for a sensation of horizontal force when going up/down and see what they'd come up with.

Well, firstly, if you plot the first n degrees of monomials and keep the scale invariant, the visual difference between x^k and x^(k+1) literally gets smaller the higher up you go.

Secondly, presumably the distinction of "straight" vs. "curved" is quite deeply programmed into the brain's pattern recognition machinery. The degree of curvature is a quantitative parameter on top of the qualitative categorization. This may or may not have something to do with the fact that a modern human sees straight lines everywhere (something that very much was not the case in the ancestral environment).

I don't have any answers, but I suspect it's because we're evolved from things that didn't need to know.

Related, I sometimes wonder how many derivatives you need to go down in order to find the one that is discontinuous when you decide to make a motion. For instance: pressing the first key to type this reply, my finger didn't instantly jump from zero to non-zero acceleration (or jerk/snap) I assume. How many terms in the Taylor series for moving a muscle?

The terms jerk and snap while perhaps known in the rare space of elevator purchasing aren’t generally used terms in most fields. I’m surprised that’s in any way controversial ?

UHHHHHhhhh, it's because the last A*b is the only one that becomes a linear constant. For other polynomials, your derivative is a polynomial still, just different one.

These are mathematical derivatives, I think of them as the slope of the thing it's derived of, aka the change in the thing that it's a derivation of.

I think I don't have a sophisticated mathematical understanding, but my basic mechanic understanding makes it feel simpler than your question is acting.

That's why loops on roller coasters aren't perfect circles as well then, I guess?

Corkscrew lift.

Because a line has an infinite radius, while a curve has a finite radius. The difference between infinite and finite is stark. The difference between two finite values is not.

The forces you experience in a loop must be a bit more complicated because the turning forces in a car are perpendicular to gravity and in a loop are sometimes in-line, but yeah I would think that's why the entry and exit are a softer curve.

The curve that is used is a Clothoid:

https://en.wikipedia.org/wiki/Euler_spiral

Usually for any curve you go straight-clothoid-arc-clothoid-straight

For trajectory AND for pitch and roll

I imagine a spinning lift would be easier.

Finally an elevator that will force other occupants to respect my personal space.

We’re all up against the wall during this ascent!

You could just as well ask "why not give the fastest ride possible so everyone saves time?"

Yeah, the more circular ones are pretty intense - https://guidetosfot.com/rides/shockwave/

An elevator that feels significantly different than the other elevators in the city probably results in more complaints to building management/maintenance about either "the elevator is too jerky!" or "the elevator is too slow!"

Neat!

> the thing I'm confused about the origin of, which is not why do we only greatly distinguish between the first few derivatives of position, but why do we only greatly distinguish between the first few derivatives of most functions?

Because higher derivatives are insignificant. That's the entire concept of Taylor series approximation. If you change a high derivative of a function, the value of the function won't noticeably change - why would you care?

Statics, dynamics, and two semesters of physics are required courses for every engineer. None of them incorporate the teaching of jerk or snap.

I'm an American.

My thoughts are that I'd prefer to spend as little time as possible in an elevator and would rather spend that time doing other things.

If I don't feel the elevator moving, then it's moving too slowly and wasting my time.

Back when I studied highway engineering they called it spiral-circle-spiral, and the vertical curves were more parabolic. The methods for laying them out were very low tech aimed at drafters and on site contractors rather than mathematicians. I can't remember the layout methods though.

But all of them are based on courses that define nth derivative, and the notion of continuous vs non continuous functions. What is the value of giving them names instead of just using nth derivative?

In part, maybe. The bigger reason is the roller coaster slows down near the top of the loop, so even for an constant vertical acceleration you need a non-circular shape.

I think it's in part because differences in gradient are quite difficult to distinguish by eye, and the difference in the higher order curves is in just that (and scaling things to be visible means the difference in curvature gets smaller as you go to higher orders). Even nearly-straight lines can be quite difficult to tell from actually-straight without a nearby reference.

Yeah, but I was just thinking that the sudden curve would jank your head quite a bit, compared to easing into it.