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

I wish designers of vehicles - particularly cars, trains and busses, would work to minimize jerk, snap and crackle.

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.

Do you have proof for that, or is this like audiophiles asking for gold connectors because "they make the sound better"?

Not proof, but jerk is a factor when bringing a car to a smooth stop. You have to learn how to brake smoothly in order to avoid the "drivers ed stop" where the car and its passengers lurch forward and then bounce back. But the controls for automated vehicles like airport trams have to be designed to avoid this. The underlying reason is that some components such as the tires and suspension are elastic.

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.

Anecdotal evidence:

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.

It's broadly recognized that minimizing jerk and snap is important to comfort in roller-coasters, so there is evidence for that proposition:

e.g. https://iopscience.iop.org/article/10.1088/1361-6552/aba732

Acceleration equals force, so yeah, if you abruptly change acceleration then this equals abruptly changing the force on people in the bus. Acceleration should thus be continuous (not necessarily differentiable). I don't know how you would justify constraints on higher derivatives. Perhaps they mess with our own internal control mechanism?

is it physically possible to have non-continuous acceleration?

Cars 20+ years ago vs more recent cars - I've definitely noticed them auto-doing what I was taught to do with older cars: ease off the brakes right at the end.

I used to work at a self-driving car company, and all the vehicle's motion was planned around how much jerk to apply.

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!

In railroad design it is important for the track to not be a curved segment of a circle (starting from a straight line), as the acceleration forces start suddenly (aka a high jerk). So this concept exists and is well known in some circles (heh).

It is an active research topic in train engineering.

Imagine a multistage rocket and the changes in acceleration.

Figure 4-3 in https://www.ibiblio.org/apollo/Documents/lvfea-AS506-Apollo1... shows this for Apollo 11.

>I wish designers of vehicles - particularly cars, trains and busses, would work to minimize jerk, snap and crackle.

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.

>Do you have proof for that, or is this like audiophiles asking for gold connectors because "they make the sound better"?

The proof is that roughly 100% of cars have components designed to limit this.

I imagine if you zoom in far enough on those points you have the acceleration continuously changing as pressure slowly builds in the engines over several microseconds.

I was thinking more of the instant you shut off engines and disconnect 130,000 kg of mass of stage one.

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.

If we zoom in on a single electron absorbing the momentum of a single photon, it will accelerate “instantly”. The same goes for e.g. an unstable atomic nucleus that ”splits”.

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.

Ok, minimizing jerk makes sense, but how about snap and crackle? Because GP said:

> (...) jerk, snap and crackle. Turns out if you minimize those, you get a far more comfortable ride.

Voltages can change abruptly. Therefore, forces can change abruptly, and hence acceleration as well.

Snap and crackle I couldn't tell you about. But jerk is definitely important.

I have two questions:

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?

I wonder if this is a change in braking material, specifically a reduction in difference between dynamic coefficient of friction and static coefficient of friction between the pad and rotor (or equivalently, the shoe and drum).

If older cars had a higher differential, you’d need to let up more as the brake finally locks up.

That’s the essence of a legitimate question: over small enough time periods (as the bolts explode over a non-zero period of time), is it continuous or discontinuous?

Over a macro scale, it’s discontinuous, of course.

>1) does this hold for all 3 of jerk, snap and crackle, like OP suggested?

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.

Don't forget vehicles got heavier, rims got bigger/rubber has thinner sides, suspensions geometry evolved and got stiffer (and possibly non-linear, at least on the high end) and so on and so forth, reducing the amount of elastic energy.

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.

It's designed for in the road/track, not the vehicle. For train tracks in the UK the recommended max jerk is 0.35 mm/s/s/s. The jerk is limited by using 'Euler spiral' sections to join up the straights and the curves. Travelling along an Euler spiral at constant speed means you feel constant jerk laterally, so can be scaled to keep the jerk below any arbitrary value.

I think bus is braking with a constant breaking force.

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

It's nature, it's continuous at small enough scales.

But, checkout Zeno's paradox for more on your philosophical questions

I was thinking of both the changing of material composition of existing organic or semi-metallic pads, but also the general drift towards ceramic pads for low-dust.

Some of the German marque factory pads have exceptional initial bite, coupled with exceptional high levels of dust.

In response to #2, consider that every material is fundamentally "springy"[1], and many engineering materials deflect a human-noticable amount when enough force to move them is applied. Thus, you can model every connection between an actuator and an object as a spring. When the actuator starts accelerating, it applies a force through that spring, which causes the spring to extend, and the force actually applied to the object is provided only by the extension of the spring. It's only once the spring is applying the same force as the actuator is that the two objects are moving at the same speed. However, at that point, the actuator and object are moving at different speeds, so the spring is still extending. As a result, you end up with an oscillation in the velocity of the object, which is almost never desirable. For a start, if one of the parts is metal, this causes fatigue, which will cause the part to fail much sooner. Secondly, you generally want the object being moved to follow a precise path, and that oscillation will show up as ringing[2]

[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.

I recently rode the Blue Line in Chicago, and found the jumps in acceleration to be quite jarring. Some of that is due to the track, but it felt like the majority could be fixed using a modern motor controller and a bit of math.

Modern cars are designed on GUI simulation software like MATLAB, they model head accelerations few inches ahead of the most important headrest and run optimizer on variables. If it can be made into equations, they can cancel it.

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.

>Modern cars are designed on GUI simulation software like MATLAB

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.

> >you can't make coffee cup rides not sickening by software.

> 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 was thinking what would be the most steady and disorienting activity with higher integrals of lateral accelerations, and couldn't come up with a better example than a cup ride. I admit it wasn't a good one.

Aren’t you describing infinite acceleration, or discontinuous velocity?