Real acceleration during jerk

I am building a a new corexy printer and considering some math (at least some very simple physical model)
for picking the correct parts and arranging mechanics.

I don't understand how jerk is really applied. By "really" i mean " in reality".


As it is used in marlin it is Instantaneous change in velocity.

For example:

Let it be 30mm/sec (not ^3!, it is not jerk in physical sense).
Now we have a problem of finding real physical acceleration when jerk is applied from 0 speed.
So, at some point we are 0mm/s and the very soon we are at 30mm/s. Soon is not zero time.
I can assume it is time between two pulses to the stepper driver from firmware. Let's get that time
The setting is 100 steps/mm. 30mm=3000 steps. 30mm/sec=3000 steps/sec
OR 0,000333333 seconds per steps - this is our time
Let's find acceleration:
(30mm/sec-0mm/sec)/0,000333333sec=90000 mm/sec^2=90m/s^2 ( 9g !!!)

REALLY?

For a light weight extruder carriage with about 200g that results in about 18N of force without
even counting and friction.

If it is 300g + friction = more than stepper can ever deliver.

The only possible error i see is the time period over which speed goes to 30mm/sec. But pulse
rate is correct. So, two ways to solve it: skip pulses, stretch or bend something.

Also, this shows that the more steps/mm is set the more strange higher jerk must behave.

I will be glad to read your thoughts on this.

Edited 1 time(s). Last edit at 02/22/2019 07:10AM by ArtemKuchin.

Well, in reality, the motors hardly ever actually do exactly as they're commanded, or else in between each pulse, the print head would come to a complete stop. If you applied the same math to a 100mm/s linear move on one axis, it becomes clear that this isn't what happens.

Also, there's the nature of the stepper drivers. 16X microstepping is a pretty common configuration, so that 100 steps/mm is actually 6.25 full steps. (80 steps/mm at 16x microstepping is actually pretty common for GT2 timing belt driven axes, or 5 full steps/mm.) The firmware commands one pulse, but the drivers command the motors to a position, with full steps being the strongest to command and hold. As the motor lags behind what is commanded with an unreasonable acceleration demand, the stepper driver will still be commanding the same position. So long as the motor doesn't get to being more than a full step behind, it will catch up to what the driver has commanded it to do.

Sometimes, this contributes to visible print artifacts. Sometimes the transmission method (the belts and belt clamps) introduce some error, like ringing, where a corner is made imprecise because the physical mechanism could not change direction as quickly as the motor could. This also reduces the "real" load on the motor to the maximum that the belt can drive the end load.

There is always going to be a difference between what the firmware tells the motors to do, and what the machine is actually doing. In this case, what we're saying is that if we command an "instantaneous" speed change of up to this, the machine will catch up acceptably without the need to slow down the print process unduly. When we observe that the machine cannot keep up acceptably with the commands that we're giving it, we change the commands.

---

MBot3D Printer
MakerBot clone Kit from Amazon
Added heated bed.

Leadscrew self-built printer (in progress)
Duet Wifi, Precision Piezo parts

I'll give another superextreme example which has nothing to do with reality.
Marlin can do quad-stepping up to 40 000 HZ of steps. Let assume we have 100 steps (pulses)/mm
So, one step takes 25us and move the belt 0.01mm
The belt were at test, the driver received the pulse and move the belt 0.01mm in 25us (just imagine it for now).
Let's consider the movement evenly accelerated.
So, X=X0+V0*T+ (a*t^2)/2
a=(2*x)/t^2=32 000 000 mm/s^2
The force will be crazy as you see.
But it never happens.
1) The motor will never be able to turn so fast even without any load because it has its own inertia (not going into that physics)
2) The belt will stretch
3) something will bend, stretch or compress
So, after that the total time to move that distance will be probably 100 times longer and real acceleration will be puny. However, after that
everything what was stretched, bend and compress will start unstretching, ubending and uncompressing and ring, ring, ring.

I am getting it right? I am really interested in what's going on on that first move when move start. If it is understood then many things can
be more clear.

I also want to point at the relation between jerk and steps/mm. If you had jerk 20 and 100mm/sec and reverything was fine then if you
redone the mechanics to 200 mm/sec and still have jerk 20 then it will result in more problems, because time between pulses will be
2x shorter which will result in 4x more acceleration (t^2) and 4x more force. Isn't it so from the math?

Google("Reprap jerk")

Luckily, seems there is no jerk in the reprap community

Edited 1 time(s). Last edit at 02/22/2019 12:16PM by MKSA.

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"A comical prototype doesn't mean a dumb idea is possible" (Thunderf00t)

M566 X600 Y600 Z30 E20 ; Maximum jerk speeds mm/minute

what is this then?

I've been looking at this recently.

If you apply a sufficient torque to a stepper motor (or ask it to accelerate or rotate at an excessive rate) it doesn't skip by 1 step at a time, it skips by 4. I'm talking physical steps here too, not microsteps.

So when the motor is commanded to start instantaneously what is actually happening is it is being asked to accelerate from 0 to speed before it lags the requested position by 2 physical steps. Once the main physical acceleration is complete there will be torque available to pull the rotor closer to the ideal position. If it lags more than 2 steps then it will snap back to 4 steps behind position, 'loosing a step'.

When you work through the numbers the accelerations aren't that scarey.

Edited 4 time(s). Last edit at 02/23/2019 01:47AM by WesBrooks.

Can you elaborate on your test with skipping and step lagging, do you have some desrcription of tests, videos? I would like to see that how you arranged that and original collected data. Very interesting.

So, about lagging. Do i understand correctly that if there is not enough torque to make a step then motor will accumulate more energy for the movement UP TO to the next 2 full steps and when the energy is enough then it will make the movement fully catching up to the desired position. If energy is available earlier then okay, it will catch up earlier. If energy is not enough even after 2 full steps it will actually rotate BACK 4 full steps from current (OR desired??) position "loosing steps" and ruining the print

Did i get correctly?

Were the tests made with full steps or microstepping? On 1.8 d motor only? What drivers? Does the behavior depend of motor type or brand, driver type, microstepping settings?

Firstly a disclaimer: I don't think this damages the machine but can't be sure. Do at your own risk!

Power up a machine and make sure the motors are not in an idle power level.
Grab one of the steppers and rotate it (over torque) it on one direction while it is still.
Note the step size. I't quite chunky! More than you'd expect from one physical step, and one micro step would be more or less imperceivable.

I found the 4 physical step = 'one lost step' in an online article. I'll find the link now.

This does not mean that you only have a resolution of +/- 2 physical steps, but it does mean that the positional error of the head can reach +/- 2 steps before the rotor will jump to the next nearest force minima/balance.

Edit: For clarification if you 'loose a step' during a print in x or y with the following sized drive pulleys then this is the approximate shit that you will see in x and y axis:

GT2 20t: 0.40mm
GT2 16t: 0.32mm

If you force a standard stepper around 360 degrees then if the above (from the links) holds true then you will feel 50 steps, not 200.

Edited 2 time(s). Last edit at 02/25/2019 06:25AM by WesBrooks.

[www.orientalmotor.com]

In the vibration section:

"2-Phase stepper motors can only miss steps in groups of four. If you are missing steps in multiples of four, the vibration is causing a loss of synchronism, or the load is too great. If the missed steps are not a multiple of four, there's a good chance the wrong number of pulses or electrical noise is causing the problems."

Picking the perfect stepper motor is more complex than just inductance/resistance/max current. Unfortunately few of us have the resource to buy many motors and look at resolution of motion. I hope to do some basic steps once I've got rid of some comparatively noddy problems on my test bed machine.

Edit: Here are some tests I ran in the past. I'm nearly ready to run those tests on my own machines at home. I obviously would like to see less evidence of the resolution of the printer being dominated by the physical step size of the motor. [forum.duet3d.com]

Edited 1 time(s). Last edit at 02/25/2019 07:28AM by WesBrooks.

Aha, i see why in multiplies of 4! Because one full electrical cycle take 4 full steps. If step is missed the driver will need to repeat the electrical cycle to pickup the rotation and continue. Alright, that's clear.

However, that does not apply to microstepping. If you miss a full step then your are done, it is missed. Microsteps, as i see it, are not missed. Just the accumulated energy may not be enough to
perform an actual move,so, missing a movement on a microstep does not mean it is skipped, it means we are still waiting for enough energy to accumulate in the coils. Then the rotor will move
the the desired position. UNLESS it skipped the full step.

Interesting, how people with TMC2100/2130 drivers have artifacts correlated to electrical cycle (4 full steps). So, everything is driven by those cycles: full step, electric cycles.

Alright, that's all interesting, but how is it related to original post?
Well, i have states that the speed is set between two microsteps. Alright, it might not be the case, because if time is not enough then the coils are not energized enough and microstep is missed,
waiting for another microstep. Eventually it will catch up, so, we have same speed but more time, making real acceleration less and the force less.

Electricwise the coils act as a low pass filter for microstep pulses. Wonder what max frequency it passes eventually.

Okay, now i am thinking if there is a way to predict how many microsteps will be "on hold" before actualy movement?
Also, there is minimum step time depending on electric characteristics.
[daycounter.com]
For my motor it is about 5ms. That's time for FULL step. I don't know how it relates to microstepping.
But if just stupidly divide 5ms/16 microsteps i get 0.31 ms per microstep.
My calculations show 0,000333333 s = 0,33 ms
Which is strangely close

I guess if everything was moving it would move, but there is such thing as Stiction (static friction). Which i cannot even estimate for a give printer. So, the motor need to overcome
1) rotor inertia
2) mechanical stiction

Also, there is such thing as Hysteresis
[www.njr.com]
"The stop-position hysteresis of a stepper motor is mainly affected by the magnetic hysteresis, but also partly by the
friction of the rotor bearing. If we measure the microstep stop positions, first by rotating the motor in CW direction
and then in the CCW direction the hysteresis will clearly show (see figure 6)."

This is crazy, i did not thought about it before. Interesting, how to measure Hysteresis and does it show it self in the prints?


Now, what if the movement in microsteps was within the hysteresis? The axis will not move at all. Of if the movement was not long enough
to overcome the rotor inertia + stiction, then there will be no movement.

This is all going too deep. All i really want to know for now is the first order approximation of how many microsteps are needed
to begin moving, so i can get the time for the jerk sped and get the force approximate from there.

I think i have another aproach at the problem!
We can get calculate incremental torque produced by each microstep. That easy.
Then i can calculate inertial of the load (+stiction) and calculate what toque is needed
to overcome the load and count number of microsteps when the needed toque is accumulated and count total pulse time.
That will give me the acceleration and force acting on the belt.
Will think about this tomorrow.

But kinda lost with static friction. No clue how to estimate it.

Alright, figured our the static friction numbers. There is engineering reference with coeficients for each type of friction and each type of material.
Steel on steel is takes as 0.1 and that's for flat surfaces. We usually roll So, 0.1 will cover all cases.

Now, let's take my stepper and calculate incremental torque.


Thfs=0.4 Nm
SDR=16 microstepsstep/full step

Tinc=Thfs*sin(90/SDR)=0,04 Nm

Force at the pulley
Finc=Tinc/r

Let's assume 20 teeth pulley, d=12.5mm, r=6,25mm

Finc=0,04 Nm / 0.00625 m= 6.4 N

So, we have 6.4N of force available at the first microstep (worst case, not accumulated energy in the coils).

That's a lot of force. Hmm.

Now we need to figure out how much Torque is needed to move the axis.
Let it be Y axis of 600g bed weight.
m=0.6kg
I just assume that the only force pushing the carriage to the rail (rod) is gravity. So,
P=mg=6 N
Ff=Kf*P=0.1* 6N=0,6N - static friction force

So, we only need about 0.6N to move the pretty heavy carriage and we have about 10 times more. So, there is
not reason it will not move, unless there is a problem in the motor itself.

So, my plan failed once again, the step toque is not the time limiting factor.

Maybe current ramps up too slow? Well, let's check. The current for each microstep change by the same formula as
torque.

Motor specs:
[ru.aliexpress.com]
I=1.5A
L=4.2 mH
V=12

I inc=I *sin(90/SDR)=0.15 A

Alright, how much time we need to get that current change
T= I*L/V =(0.15 * 0.0042) / 12= 0,0000525 seconds

So, time is not the limiting factor too.

So, current results:

1) Force at first microstep is enough to overcome static friction and move the bed
2) Current change is not time limiting factor

So, let calculate the maximum RPM for this motor. People have retorted that max RPM does not depend much
on microstepping. So, it will be our reference number.
according to this [daycounter.com]
i have about 5 rps. 3200 steps/rev, so 0,0000625 sec per microstep. Well, not the limiting factor again.

Probably back EMF and other stuff play major role in this.
Next thing to try:
[github.com]

Continuing talking to myself.

Rotor moment of inertia is 47 g*cm^2 = 0,0000047 kg*m^2
Speed needed 20mm/sec, that 2000 pulses/sec
3200 pulses per revolution,
2000/32000 = 0.625 part of one revolution
which is 2*pi*0.625=3,925 rad
This is needed angular speed
Available torque per microstep is 0.04
torque = moment of inertia (MI) * angular acceleration (AA)
angular acceleration =angular velocity (AV) /t
time=(MI*AV)/torque
time=0,0000047 * 3,925 / 0.04=0,0004611875

One pulse takes 0,0005s.
So, we have just enough torque to make it rotate that fast with any load at all, probably even without pulley.
Next need to add the moment of inertia of the motor load and i think i will get something more realistic.

@ArtemKuchin

Going back to your original OP, I don't think you understand "Jerk" as it is applied to 3D printers. It's not the same as in the physics sense which is the 4 derivative of position or if you prefer, the rate of change of acceleration. A slightly better term for 3D printing is "instantaneous speed change" although in reality there can be no such thing. "Jerk" is actually pretty horrible as far as motion is concerned (as its name implies) but it is necessary.

So without "jerk" (in the 3D printing sense) every move would start at zero speed, accelerate up to the demand speed (if the move is long enough) the decelerate down to zero speed before commencing the next move. The problem is that arcs and circles are made up numerous, small, straight line segments. So if every move started at zero speed, and ended at zero speed, arcs and circles would take a very long time to print. So what we do is instead of decelerating down to a complete standstill, we only decelerate until we reach a certain speed (the jerk speed). At this point we commence the next move which may involve a change in direction. Of course, the next move starts at the "jerk" speed rather than zero speed and so we can print our arcs and circles reasonably fast - certainly very much faster than if we had to decelerate down to zero and start the next segment at zero speed.

I'm only familiar with RepRap firmware but can say that "Jerk" is never applied at the start of a move from zero speed and I believe it is only applied where there is a change of direction between moves. Other firmwares may behave differently.

So, in summary Jerk as used in 3D printers, is not the forth derivative of position with respect to time as is the case when we think of Jerk in the physics sense..

HTH

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[somei3deas.wordpress.com]
[www.youtube.com]

I fully and deeply (to the source code of marlin) understand what is jerk and how it works and why it exists. What i don't understand is what really physically happens when jerk is applied.

It really does not matter if it is applied from stop position or from movement, the real acceleration and forces will be there.

Edited 2 time(s). Last edit at 02/26/2019 11:28AM by ArtemKuchin.

To move from zero to instantaneous speed (reprap jerk) would be:

Controller sends a sequence of pulses to the stepper driver a fixed time period apart asking the stepper to travel at a fixed rate.

Stepper driver iterates though the step pattern to move the rotor with no compensations for inertia.

Stepper rotor accelerates and reaches target velocity before the rotor position lags two physical steps.

Rotor accelerates a little further and gets closer to the target position and then settles back own to target velocity.

What's comfusing?

Edited 2 time(s). Last edit at 02/26/2019 12:31PM by WesBrooks.

That's funny
Alrighty.
There is nothing instantaneous in the world. Nothing to be at stop position and then in no time at some speed just because it would require 1) infinite acceleration 2) infinite impulse

So, the real picture is in many aspects unknown for now.

1) JERK speed is a constant speed, that is defined

2) SPEED for stepper is defined by period between STEP pulses to driver

3) Problem: IF jerk speed is attained by stepper between 1 and 2nd microsteps then according to microstep time the acceleration will be unrealistically large. So, apparently, the JERK speed cannot be reached right after the 2nd microstep

4) Question: WHEN the set JERK speed is actually reached?

5) The answer to (4) will provide time using which we can calculate the acceleration and therefore FORCE acting on the belt and axis parts in the system during jerk movement

6) Knowing (5) we can conclude how much print distortion is caused by jerk and adapt to it (options: change from 6 to 9mm belt, or change belt type or arrangement, modify other parts, change jerk itself)

I am currently interesting in understanding if it is resasonable to change to wider belt. I have done test with belts (noname china and brand china) and know how they react for forces. As an example: 1m of belt with extra 40 N of force will be stretch
extra 0.2mm. It will be momentary stretch that will UNstretched when force is gone, but the print will have dimentional error of about 0.1mm because of this. So, can jerk cause extra 40N of force or not? That's why i digging it.

Edited 2 time(s). Last edit at 02/26/2019 01:21PM by ArtemKuchin.

The maximum period of time the jerk has to take place in is 2 physical steps, not microsteps. All microstepping does is miimic discrete steps between the major step cycle of the stepper. It doesn't add points the rotor will snap to if over torqued, that's only changed by swapping a 1.8 stepper for a 0.9 or 0.45 stepper.

Most spec sheets have pull in torque details and rotor inerria so you can use that to predict ultimate performance of the stepper system.

Predicting frictional load would require very accurate building of the machine to ne anywhere near accirate and would need detailed specification of the belt and bearings that aren't readily available. There may be a few rough figures on power loss from belt drives on push bikes that will give you an idea of what is reasonable.

Friction is useful though, acts as a damper.

with 20t pulley 1 full step is 0.2mm, 2 full steps is 0.4mm
If jerk take place in 2 full steps and it would be impossible to print details less than 0.4mm
Which is clearly not the case, i can show you test print with details as small as 0.025 mm, crazy huh? (need 16t pulley though and good geared extruder).
Microstepping does rise resolution and microsteps do move things around.
For me "jerk has to take place in is 2 physical steps, not microsteps" does not make much sens.
JERK is just any movement, actually there is nothing special about jerk, it just may have to be too fast to reach the speed. But then again, no speed
is reached momentarily, even 0.1mm/sec rises from 0 for some time.

Friction is not that hard actually, because i don't need exact data, i just need first order range estimation, that's can be done pretty easy using engineering tables for friction coefficients.

The two physical step thing is the maximum lag of any action before the stepper will 'loose a step' doesn't matter if this is a start to a specific speed, acceleration, or maintaining a set speed. A high torque setup will reduce this lag under the same conditions.

I've not ment to say the resolution of the system is not increased by microstepping, merely that the accuracy of movements is strongly related to the physical step size of the motor.

As Ian has said jerk isn't applied to all moves, and there is nothing to say that it must take 2 physical steps, just that's the longest that it can take. You'd need to read through the pull in torque curves to predict the shortest distance that the distance needed for the stepper to reach full speed.

If 2 full steps lag would really happen normally in real life that would casually cause problems with fine details. For example, you have an indent of 0.1mm and the step delaid by 2 full steps, so, it just neve happens, because their is only 1/2 of full step there.
But i don;t see any reports of such things. So, it seems like it applies only for high load applications like CNC. Still, did not see reports of such thing from CNC builder either.

But anyway, i don't see how it is related to the problem i am trying to solve. I cannot just take 2 steps max lag as number because it is as unrealistic as calculated acceleration for 1 microstep.

As for pull-in curve. It is related to speed and only in full step mode. I don't know how it related to microstepping. Anyway, it is close to impossible to find such curve for chinese motors.

Two full steps is the maximum. You have heard of this when people complain about shifted layers or stalling steppers that limit has been exceeded. Most commonly with the extruder, where the lurch backwards from the intended motion releases the back pressure in the filament causing a much larger move before the drive catches the motion again. Equally look at all the physical artefacts on parts such as ringng and bulged corners. Do you think they're all inside a couple of microsteps?

This is a dead reconing system. That is your maximum, your minimum will be determined by the specific built and design of the machine being modelled. The estimations are likely to be very vague and unreliable without any proof. Work on how to measure the motion through a high speed encoder with time stamp frim the stepper step signal and it'll be worth a mathematical model. Until them you are just having fun and won't be able to verify your theorys.

Edited 1 time(s). Last edit at 02/26/2019 03:41PM by WesBrooks.

My point is i am not looking for step skipping conditions. I am looking for forces during normal operation, when there are no shifted layers of skipped steps.

The actual peak or average acceleration during the transition from stop to instant steady speed is likely to be constant. I would be tempted to run custom gcode with increasing instantaneous speed until it causes a skipped step. Ok the values will be way beyond what you would use in a build but you would then have the jerk speed that can't be achieved in two physical steps and so you can calculate the acceleration.

Edited 1 time(s). Last edit at 02/26/2019 03:52PM by WesBrooks.

The forces during operation are whatever falls within the motor's operation without skipping steps.

Even if the motor is commanded to do something that it doesn't have the power to do, so long as it's capable of "catching up" before that hits a full 2 step cycle, the motor will go to the commanded position, even if that's only a microstep or two.

Now, keep in mind that the distance traveled will also depend on the angle that the print head is moving. Given a cartesian configuration, at 45 degrees, each axis motor (X and Y) are moving the print head 1.44 times the distance that either would alone at the same speed, therefore are only dealing with 70.7% of the forces that either one would be for the same acceleration in an orthogonal direction.This makes the forces involved for either motor much more difficult to calculate. For a CoreXY printer, movement at 45 degrees loads a single motor, and orthogonal movement gets split between 2 motors.

So basically you calculate the worse case scenario where it doesn't layer skip, and keep your jerk setting within that. Actual forces will depend on even more things. Say that the angle of change is not 90 degrees. Say it's 30 degrees, the forces are halved. (sin(30)=0.5) In any case, the "jerk" setting will always be a compromise. The actual forces on things will always be an estimate, since the actual moving mass changes, and changes with the angle of print head travel, which changes the distribution of the load between the motors. It will also depend on what can be transmitted via the belts, which WILL have some accommodation of elasticity, even steel lead screws do, if you want to get technical. then, since we know there's no such thing as a real instantaneous change, we're still looking at a change over time. The tension on the belts is near impossible to reasonably calculate, and will change over time and temperature, which changes during printer operation. In the end, you're going to end up with an estimate, and you'll still get better results with trial and error in adjusting your jerk parameters for your control software.

---

MBot3D Printer
MakerBot clone Kit from Amazon
Added heated bed.

Leadscrew self-built printer (in progress)
Duet Wifi, Precision Piezo parts

Quote
ArtemKuchin
with 20t pulley 1 full step is 0.2mm, 2 full steps is 0.4mm
If jerk take place in 2 full steps and it would be impossible to print details less than 0.4mm
Which is clearly not the case, i can show you test print with details as small as 0.025 mm, crazy huh? (need 16t pulley though and good geared extruder).
Microstepping does rise resolution and microsteps do move things around.
For me "jerk has to take place in is 2 physical steps, not microsteps" does not make much sens.
JERK is just any movement, actually there is nothing special about jerk, it just may have to be too fast to reach the speed. But then again, no speed
is reached momentarily, even 0.1mm/sec rises from 0 for some time.

Friction is not that hard actually, because i don't need exact data, i just need first order range estimation, that's can be done pretty easy using engineering tables for friction coefficients.

Sorry pal but you really do not understand how "Jerk" is applied to 3D printers, although you say you do. As I explained, "Jerk" isn't movement at all. It's just a speed threshold. Do the maths on segment sizes of 0.4mm (or even 0.025mm) and you'll see what I mean.

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[somei3deas.wordpress.com]
[www.youtube.com]

I don't know about reprap software, but marlin seem to use jerk as starting speed from zero. Yes, it is a threshold speed. For speed below JERK acceleration is not used, for speed = or > JERK acceleration is used. Again, i am talking about marlin firmware, sorry.

So, if motor starts at JERK the firmware pulses to drivers STEP input at frequency which will result in JERK speed (taking into account steps/mm settings).

What did i get wrong about jerk?

Edited 1 time(s). Last edit at 02/27/2019 06:01AM by ArtemKuchin.

Recently I told a guy
Hey, you, "speed second derivative".
He didn't even frown

Edited 1 time(s). Last edit at 02/27/2019 07:57AM by MKSA.

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"A comical prototype doesn't mean a dumb idea is possible" (Thunderf00t)

What does it have to do with jerk in a sense used in 3d printing firmware?