Beyond velocity and acceleration: jerk, snap and higher derivatives (2016)
iopscience.iop.org
iopscience.iop.org
> 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.
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.
We’re all up against the wall during this ascent!
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.
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?
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.
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?
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?
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…
Why not, though? Why does third order "look like" second order but second order is starkly different to first order?
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).
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.
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.
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 think in the linked article there's a good real-world example of that with a valve:
> opening the gate of a gate valve (of rectangular cross section) by 1 mm for 10 seconds yields the same absement of 10 mm·s as opening it by 5 mm for 2 seconds. The amount of water having flowed through it is linearly proportional to the absement of the gate, so it is also the same in both cases.
I’m not sure how to think about the lower orders. You might, for instance, have a learning control system you expect to come to a lower error state over time. The integral of the absement would be a decent way to capture that phenomena.
I also distinctly remember being about to go into an exam in undergrad EE, and having a decades-older MechE ask if I knew about "jerk". I had a temporary panic because I didn't know the term - but then when they started explaining it, I already knew it all, I just had never been exposed to the term "jerk" as the word to use for it.
So maybe it's just a terminology thing? I've been in situations where I definitely knew the concept thoroughly, both absement and jerk, but didn't know those labels.
Acceleration feels like a constant force (because… it is). When that force changes, you feel “jerked”.
I have a hard time imagining another level above that.
"In the fall of 1972, President Nixon announced that the rate of increase of inflation was decreasing. This was the first time a sitting president used the third derivative to advance his case fore reelection. - by Hugo Rossi"
https://en.wikipedia.org/wiki/Class_A_surface
https://www.johndcook.com/blog/2018/02/13/squircle-curvature...
I do see quite clear parallels between higher order time derivates and these higher order curvature measures, although I don't know if there is any formal relation here
Turns out if you minimize those, you get a far more comfortable ride. It matters far more than acceleration.
Finite element models of the whole system (tyres and suspension components and flexing elements of the vehicle body and road/track) can quickly allow analysis of the jerk, snap and crackle, and allow tuning of damping and drive system control loops to make a far more comfortable ride.
This is in fact an issue for the designers of controls for mechanical systems. I learned about it in Process Control class, albeit 40 years ago.
If older cars had a higher differential, you’d need to let up more as the brake finally locks up.
There's mechanical braking assistance (not just ABS) which means pressing the same pedal distance may produce different breaking strength depending on the speed at which the pedal is pressed; e.g pressing hard triggers force assistance from, say, a vacuum reservoir that reuses engine pump loss, which means conversely pressing lightly for a normal stop does not need to exert as much pressure, hence an eased in stop.
Also with more stable vehicles with better chassis, suspension, and overall balance, I feel like rear braking has been tuned upwards over time, making for a more stable stop: notice how lightly pulling the handbrake has a straight-rolling car "sitting" instead of "diving". More consistent use of disc brakes instead of drums on the rear end certainly helps, as well as the ability for the vehicle to remain stable even when braking while in a turn.
Regarding brake friction itself, I can think of at least one major change: the ban of materials such as copper or asbestos in brake pads.
Some of the German marque factory pads have exceptional initial bite, coupled with exceptional high levels of dust.
Ever experienced that a bus is braking (near-constant deacceleration), and people seem fine; but then the bus comes to a halt and thus stops deaccelerating, and people suddenly fall on the floor?
I think at least the derivative or acceleration is important for how well people can compensate. Not sure about higher derivatives though.
Figure 4-3 in https://www.ibiblio.org/apollo/Documents/lvfea-AS506-Apollo1... shows this for Apollo 11.
There is an interesting Δa/Δt while fuel is consumed and mass changes.
There are discontinuities to the graph when engines are shut down and stages decoupled.
Over a macro scale, it’s discontinuous, of course.
But, checkout Zeno's paradox for more on your philosophical questions
At macroscopic scales, I’m not aware of exactly instantaneous acceleration, since you would need some time to “sync” the movement of each atom in the object. But some processes will of course look instantaneous at any given time scale.
But, the bus has a non-constant kinetic energy (going up with the velocity*velocity, down as velocity goes down.)
So, you're actually producing a non-linear acceleration. This is jerk, but you can also think of it as just a non-linear acceleration and people are reacting to the fact it's not at all near constant deacceleration, and this is most noticable as velocity hits zero.
So, yes, it's jerk, but no, I think it can be intuitively better understood with pure acceleration terms and no jerk needed
e.g. https://iopscience.iop.org/article/10.1088/1361-6552/aba732
Your muscles are pretty good at applying a constant force (or responding to a constant acceleration). Hold your arm out straight: it's no effort to keep your arm still and counteract the force of gravity. Now imagine gravity varies quickly and randomly between 0.5g and 2g. I guarantee your arm won't stay still.
The same prinicple applies on a bus or in a car, except this time the forces are smaller, and it's your neck keeping your head still!
> (...) jerk, snap and crackle. Turns out if you minimize those, you get a far more comfortable ride.
The proof is that roughly 100% of cars have components designed to limit this.
They do.
>Turns out if you minimize those, you get a far more comfortable ride. It matters far more than acceleration.
They know that this is the case. And put a lot of effort into making sure your car has the desired feel.
Besides your comfort these considerations are extremely important for the durability analysis for the vehicle.
>Finite element models of the whole system (tyres and suspension components and flexing elements of the vehicle body and road/track) can quickly allow analysis of the jerk, snap and crackle, and allow tuning of damping and drive system control loops to make a far more comfortable ride.
Finite element simulations are undesirable, they are extremely calculation expensive for those kind of large models and somewhat unsuitable. They are used in crash tests.
For the application you described multi body systems are used, where the car is decomposed into its functional components, which can be modeled either as stiff or flexible. With that you have a reasonably accurate model of a car which you can use to test on a virtual test track.
Basically every competent car manufacturer is doing this.
1) does this hold for all 3 of jerk, snap and crackle, like OP suggested?
2) In applications where no humans are involved (robot actuators etc.), would it make sense to minimize jerk, snap and crackle too?
They aren't the fundamental quantities you would look at, typically the output of a multi body system are displacement/velocity/acceleration, but of course if you look at a plot of acceleration you can just see these quantities (at least the first and second derivative are quite easy to see) or calculate them. And of course the ride comfort is related to the smoothness of the forces you experience, which is the same as wanting to minimize the derivatives of force. But I would suggest that these quantities are quite hard to analyze quantitatively as they are, naturally, subject to far more noise.
Where these quantities definitely are considered is when you look at vibrations.
>2) In applications where no humans are involved (robot actuators etc.), would it make sense to minimize jerk, snap and crackle too?
Yes, if you care about durability. Parts can break for different reasons, intuitively you easily understand that exceeding certain loads breaks them. Another, far more insidious, failure case is a cyclic load, which never exceeds a particular threshold. Again, vibrations play an important role there.
[1] Yes, this is a vast oversimplification, but the model I'll build using it is reasonably accurate.
[2] Most 3d printing enthusiasts are familiar with this issue; e.g. https://www.simplify3d.com/resources/print-quality-troublesh... . However, most of the advice you see amounts to "make everything stiffer", which helps, but the real solution is to be less jerky.
btw, minimizing higher orders of derivatives improve passenger comfort only so much if the driver isn't so good at look-ahead path planning, you can't make coffee cup rides not sickening by software.
MATLAB is the programming language. You are talking about Simulink. But there are also dedicated software packages for car multi body dynamics.
>you can't make coffee cup rides not sickening by software.
Plainly false.
> Plainly false.
I'm not sure how this is false, but I'm also not sure what the person you're replying to is getting at. coffee cup rides have no driver and therefore no look ahead path planning at all.
The entire appeal of these rides is that they exaggerate perception of third and higher derivatives by deliberately creating a chaotic field of view for their occupants, combined with constantly changing both the rate and direction of the acceleration of their squishy bodies in the rigid capsules to which they are loosely secured.
Smoothing this out with software would be possible, sure, but then the result would no longer be a coffee cup ride. I feel like this is a poorly formed example.
I use them in the context of N-Body Simulations. Curious to learn about other contexts for their use - anyone?
https://en.wikipedia.org/wiki/Fourth,_fifth,_and_sixth_deriv...
Surely this is true. Is there any likely alternative explanation?
- position
- velocity
- acceleration
- jerk
- snap
- crackle
- pop
- "and so on"
I always got the sense from physics that outside of purely mathematical constructions such as Taylor series, higher order time derivatives aren't providing much interesting information. Though I'm not sure whether this is the inherent laziness of physicist math[1] or a property of the forces in nature.
[1] since e^x = 1 + x is generally true, why'd you even need a second order derivative
> Jerk and snap can be observed in various areas of physics and engineering. In physics and engineering jerk and snap should always be considered when vibration occurs and particularly when this excitation induces multi-resonant modes of vibration. They should also be considered at all times when a transition occurs such as: start up and shutdown; take-off and landing; and accelerating and decelerating.
> Acceleration without jerk is just a static load, and therefore constant acceleration alone could never cause vibration. In a machine shop, a toolmaker can damage the mill or the job if the setup starts vibrating. This vibration happens because of jerk and snap.
> In mechanical engineering it is important in automotive design to ensure that the cam-follower does not jump off the camshaft. It is also important in manufacturing processes as rapid changes in acceleration of a cutting tool can lead to premature tool wear and result in an uneven and rough surface finish.
> In civil engineering railway train tracks and roads should be designed for a smooth exit from a straight section into a curve, and it is common to use a transition called a clothoid, which is part of a Cornu spiral (also referred to as an Euler spiral). When a clothoid is implemented the change in acceleration is not abrupt and the levels of jerk and possibly snap are significantly reduced. If the transition between different radii of curvature is sudden, the transition is uncomfortable for passengers and potentially dangerous as it could cause the car or train to be thrown off the road or track. With good physics design engineers are attempting to produce a gradual jerk and constant snap, which gives a smooth increase in radial acceleration, or preferably a zero snap, constant jerk, and linear increase in radial acceleration. Just as road and railway engineers design out jerk and snap using the clothoid transition so, too, do roller coaster designers when they design loops and helices for the roller coasters [11, 12].
Genuinely curious.
But if the jerk (or higher derivatives) are non-zero, you have to change your "lean angle" quickly to avoid getting jerked around (which is obviously much more disruptive).
More seriously though, I think this might be about driver training and maybe calibrating the foot pedal. It's great that EVs have a much better torque curve, but it means the old muscle memory of opening the throttle wide at low RPMs and letting the clutch slip is simply not the way to do it (nvm that there's no clutch to operate in an EV).
Every car can go from a standstill to accelerating in a split second.
>their maximum acceleration isn't very high
What is the maximum acceleration of an EV? Do you have some numbers?
>ICEs need more time to reach maximum acceleration
I don't think what you're describing is jerk, it's acceleration (and velocity).
Going from a standstill to accelerating at say 3 m/s² is very different in a normal ICE car vs. an EV. It's anecdotal, but you must have noticed this if you've driven an EV before.
> What is the maximum acceleration of an EV?
I was talking specifically about the electric buses in my city. They don't have massive acceleration compared to, say, a Tesla.
> I don't think what you're describing is jerk
I am talking about how rapidly electric buses change acceleration. That's the definition of jerk.
It came as a surprise to me but it seems like jerk is something that can be felt in real life.
Only true for small x (less than 1).
I cannot remember what it's called but essentially given a target position in space the missle uses parametric data about its current position/orientation/speed and their higher derivatives to dead reckon about where it is in regards to the target.
Anyone remember what that's called? I went on a rabbit hole with it a few years ago, it's really interesting math and programming. Everything works basically stateless except for current instrument data, last position and target position from what I remember.
Kalman filters come up a lot, maybe relevant to the terms you're looking for
I'm becoming seasick due to a jerkiness of a car. Not due to a speed or a acceleration, but it is jerk that does it for me. I watched it from my childhood, I hated trolleybuses for that: they are electric and they tend to change acceleration instantly. But I didn't understood how it works until much later.
Or in other words, you can approximate exp(x) as a set of first order taylor approximations that each covers a small window to arbitrary precision, but the combination of them is still has well defined higher derivatives that are not 0.
However, the paper says they’re not commonly taught, but jerk is taught in many high school (AP) Physics classes — we have to keep our balance by noticing the change in acceleration.
The difference between falling from a height and landing on a trampoline, and landing on concrete from the same height, is that the trampoline smoothly accelerates you to a halt once you collide with it. The concrete does so much more rapidly: that's jerk. Both of these are collisions with the same amount of force behind them.
What they affect is jerk.
I’m not sure anyone noticed the difference between the two
* If you drive safely, then you can save someone's life by avoiding a crash.
* If you save a driving trip (and instead walk/bike/transit/stay-home), you will have a life of safety.
too bad it uses an odd cloud-based model for waypoint handling.
Anyone know of any software for jerk limited planning which allows position constraints? Whats the fastest jerk limited path from A to B the doesn't pass though the forbidden zone. The jerk limited path may deviate from a straight line. So even when the A to B line is admissible, a straightforwardly constructed jerk limited path may not be.
You’re sitting in the driver’s seat of a car. It is standing still.
You push the gas pedal down 2 cm and hold it there. Your car begins accelerating. That’s the second derivative.
You start pressing your foot further on the gas pedal. Your foot has a velocity on the gas pedal. It is causing your car’s acceleration to grow! That’s jerk.
If you push your foot on the gas pedal faster and faster your foot accelerates on the gas pedal. That contributes to the cars snap.
A (mis)conception of the piano is that it is purely percussive and velocity is the only parameter you control for voicing on the piano but professionals would beg to differ...
A hammer in a piano always moves on a fixed path. It always strikes the same part of the string, and it always does so in the same orientation. And after it strikes that string, it always falls away from it. That's how that part works.
Striking a percussion instrument with a stick (such as a wooden block) has more variables to toy around with than playing a note on a piano does.
But there's a lot more going on in a piano than striking strings: Strings are also muted, and the degree of muting can be manipulated. It is not binary.
And, of course, pianos are polyphonic: With ten fingers, we can strike ten different [sets of] strings at different velocities and at different times, and we can even mute them to individually-different degrees.
And then, there's also the pedals...
Moar please!
Me every day before checking git blame.