Acceleration as a function of speed

• Hi,

i would like to request "acceleration as a function of speed"

My motivation is to maximize possible acceleration for best print quality as well as speed, at given limitations.

Limitations are

• stiffness/weight of the system

• stepper torque as f(speed)

• stepper temperature

The maximum usable acceleration is defined by the most limiting factor (whatever that is in a certain case).

In case of a stiff printer and well cooled steppers, the stepper torque as f(speed) is the limit.
Yet one has to bottleneck low speed acceleration because of the limited high speed torque.

I would like to be able to set the acceleration according to speed, to increase the acceleration at lower speeds (which improves print quality and speed most, using a stiff system) compared to now.

Compared to one acceleration setting per driver yet, it would need at least 2 additional parameters:

• (OLD parameter) acceleration (new meaning: acceleration at low speed, the region where the max current setting limits)
• (NEW parameter) min speed to start acceleration reduction (start of the region where the torque starts to drop with speed)
• (NEW parameter) acceleration reduction per speed increase (to map allowed acceleration to stepper limitations, "slope" of the curve)

It doesn´t have to be a smooth transition to improve the situation compared to now. To make it more easy, it would already help to have e.g. only 2 acceleration settings per driver, one for low speeds and one for high speeds.

• That's an interesting idea, but difficult to implement. The existing code divides each move into up to 3 phases: an acceleration phase, a steady speed phase, and a deceleration phase. During the acceleration and deceleration phases, the timing of each step pulse is calculated by solving a quadratic equation. Your proposal would require up to 5 segments, and in the extra 2 phases I think it would be necessary to solve a cubic equation.

An easier approach would be to choose the acceleration of the move as a whole depending on its top speed. That would allow maximum acceleration for printing moves and only reduce the acceleration for high speed travel moves.

Have you done the calculation at https://duet3d.com/wiki/Choosing_stepper_motors#How_to_work_out_the_power_supply_voltage_you_need to work out the speed at which you no longer get full torque from the motors? Are you using 12V or 24V power?

• You'd have to choose the acceleration for the entire move otherwise you'll be introducing jerk (in the physics sense - the rate of change of acceleration) which we definitely don't want.

• I can see why could the acceleration depend on the speed. The acceleration depends on the force that is limitted by the stepper motor torque. Maybe the motor torque depends on its rpm tho.

• @DC42:
"anything" which comes closer to the real limitations of a stepper would be an improvement compared to now.

Setting different acceleration values for printing and traveling is already possible. It would be new, that it depends also on top speed.

I use 24 V because the Duet doesn´t allow more. From CNC i am used to > 36 V….
I did my own calcs which are more or less the same. I can speed up with > 2g to about 1000 mm/s in theory and in real life. But until now i didn´t print anything because the printer is not finished and i am waiting for parts (z-axis is missing).

If the acceleration would be a f(speed) i could increase the acceleration much more, especially for travel moves, because there "deformations of the belts/bars" (stiffness) is no problem. If the travel moves are damn fast, retraction and all the extrusion problems/imperfections (pressure advance and so on) becomes "negligible". So also flexible stuff on a bowden system works.
For constant extrusion (== no problems) a constant speed is most important, therefore high acceleration on slow printing speed is also needed.

BTW: So far i had no problems with max pulses from the Duet and M122 also shows no problems at high speeds. I am wondering how that works at 1000 mm/s

@deckingman:
If you don´t vary it smoothly yes, but everybody uses jerk.... and it works, so the step has just to be not too big.
The 3 phases of the move also introduce jerks at 2 times.

• @vp:

@deckingman:
If you don´t vary it smoothly yes, but everybody uses jerk…. and it works, so the step has just to be not too big.
The 3 phases of the move also introduce jerks at 2 times.

NO. That's completely wrong! Jerk as used by 3D printers is referred to as instantaneous speed change and is only applied when there is a change of direction, which usually only happens doing short segmented perimeter moves. Instantaneous speed changes are NEVER applied otherwise. ANY variation to the rate of change of acceleration will result in jerky motion as defined in the physics sense. You cannot alter the rate of change of acceleration and get smooth motion.

• @deckingman
ANY variation to the rate of change of acceleration will result in jerky motion as defined in the physics sense. You cannot alter the rate of change of acceleration and get smooth motion.

If we are not in a quantum world ANY motion is SMOOTH in physical sense ( the space is smooth anyway). I can't understand why max acceleration should depend on the speed if limitting factor is the max available force (max torque). a=F/m in classical physics no matter how fast we are moving.

• @zov:

@deckingman
ANY variation to the rate of change of acceleration will result in jerky motion as defined in the physics sense. You cannot alter the rate of change of acceleration and get smooth motion.

If we are not in a quantum world ANY motion is SMOOTH in physical sense ( the space is smooth anyway). I can't understand why max acceleration should depend on the speed if limitting factor is the max available force (max torque). a=F/m in classical physics no matter how fast we are moving.

The first derivative of position is velocity, the second derivative is acceleration, the third (rate of change of acceleration) is jerk, named because of how it manifests itself.

I think the OP is referring to the fact that the stepper motor torque drops off with speed so the maximum available force is speed dependent and therefore the maximum acceleration rate could be altered. I guess it would be possible if one looked ahead at the move, calculated the maximum speed it was likely to attain, then set the acceleration rate for that move (but you'd need to maintain that acceleration for the duration of the move otherwise you'll get jerky motion). Repeat the process for every move. That's an awful lot of processing to do on top of everything else that needs to happen. It's a lot easier to choose steppers that will give the necessary torque at the highest speed one is likely to use.

• NO. That's completely wrong! Jerk as used by 3D printers is referred to as instantaneous speed change and is only applied when there is a change of direction, which usually only happens doing short segmented perimeter moves. Instantaneous speed changes are NEVER applied otherwise. ANY variation to the rate of change of acceleration will result in jerky motion as defined in the physics sense. You cannot alter the rate of change of acceleration and get smooth motion.

It is correct that "jerk" is only applied when direction changes, especially because it has a lot to do with "backlash".
But as mentioned by ZOV, nothing which is driven by a stepper (stepper says everything) is not "smooth in a mathematically sense" (That means continuous derivatives).

Stepwise change of acceleration is also not possible. Therefore it never happens, as well as jerk never happens, because the mass and stiffness smooths the step. As long as you don´t lose a real step (not microsteps) it does not hurt anybody.

@ZOV:
The max force is a function of the torque. The torque start to drop of quite soon. If you use higher speeds, you have to limit your acceleration to the worst case torque - which is the small torque at higher speeds.

E.g. torque at 100 rpm is 0.5 Nm, at 200 rpm 0.2 Nm. If you want to use 200 rpm even only for a very short time, you have to limit the acceleration even at slow speeds to 0.2 Nm torque - although the stepper could deliver 0.5 Nm.

• It's a lot easier to choose steppers that will give the necessary torque at the highest speed one is likely to use.

I am sorry, but exactly this is not possible. More doesn´t help more and smaller steppers have exactly the same problem, only at a different level, until they are just too small.

Of course it is clear to choose the proper steppers, the link above from DC42 shows how to. But after you have done this, you still loose plenty of torque == printing quality and or printing time.

The difference in CPU resources is there, but depending on how it is solved not significant by definition.
Nobody says everybody has to use it.

If the ramp parts of the 3 move phases are split in 2 parts, one part limiting the speed to speed_limit_1 (this is already done now) and the second part of the ramp is limited to speed_limit_2. For both parts 2 acceleration settings could be applied. I don´t believe that this is too CPU intensive. The change in acceleration does not matter at all in real life.

• @deckingman:
The first derivative of position is velocity, the second derivative is acceleration, the third (rate of change of acceleration) is jerk, named because of how it manifests itself.

I think the OP is referring to the fact that the stepper motor torque drops off with speed so the maximum available force is speed dependent and therefore the maximum acceleration rate could be altered.

For the sake of truth the smooth space is infinitely differentiable so any move is smooth by definition. I am not sure that stepper motor torque depends on speed of rotation very much but I may err, not fully my field of knoweledge.

Correction: I saw that MAX torque indeed depends on speed (Power = Torque x Angular Speed). I appologise for my misunderstanding.

• @vp:

E.g. torque at 100 rpm is 0.5 Nm, at 200 rpm 0.2 Nm. If you want to use 200 rpm even only for a very short time, you have to limit the acceleration even at slow speeds to 0.2 Nm torque - although the stepper could deliver 0.5 Nm.

That's not true, because we use constant current stepper drivers. Torque is proportional to current, so torque is maintained as RPM increases up to the point at which the back emf due to inductance + back emf due to rotation + voltage drop due to resistance exceed the driver supply voltage, and the driver can no longer supply enough voltage to achieve the desired peak current.

It would be true using constant-voltage drivers though.

• That's not true, because we use constant current stepper drivers.

Thanks David. It became crystal clear that variable acceleration is useless for our case.

• @zov:

That's not true, because we use constant current stepper drivers.

Thanks David. It became crystal clear that variable acceleration is useless for our case.

I don't think it's completely useless. It would be useful if the combination of motor, drive voltage and drivechain gearing meant that it was not possible to maintain the specified motor current at high speeds. This could be the case for a printer that uses leadscrews for the X and Y axes, because of the high steps/mm of such an arrangement. But for belt-driven printers, I think it's usually possible to choose motors that don't run into the torque limit at the highest wanted RPM - especially when using a series 2 Duet with its 2.4A output capacity, because that allows you to to choose motors with lower inductance (and higher operating current) than when using the types of driver found in other 3D printer control electronics.

• That's not true, because we use constant current stepper drivers. Torque is proportional to current, so torque is maintained as RPM increases up to the point at which the back emf due to inductance + back emf due to rotation + voltage drop due to resistance exceed the driver supply voltage, and the driver can no longer supply enough voltage to achieve the desired peak current.

It would be true using constant-voltage drivers though.

Are you sure of that? I never saw a torque vs speed curve with a flat part, then a falling part; they always start to fall as soon as rpm>0 (more or less quickly, depending on the supply voltage, for the rasons you mentioned). And I never saw constant voltage drivers (except diy ones)…

• @vp:

…....................The change in acceleration does not matter at all in real life.

Try it next time you are driving your car. Rather than accelerating evenly, pump the accelerator up and down. Or conversely slow down by pumping the brake peddle instead of applying gentle even pressure. The jerk you will encounter is the result of altering the rate of change of acceleration (and it does matter in real life as well as in 3d printing).

• @fma:

That's not true, because we use constant current stepper drivers. Torque is proportional to current, so torque is maintained as RPM increases up to the point at which the back emf due to inductance + back emf due to rotation + voltage drop due to resistance exceed the driver supply voltage, and the driver can no longer supply enough voltage to achieve the desired peak current.

It would be true using constant-voltage drivers though.

Are you sure of that? I never saw a torque vs speed curve with a flat part, then a falling part; they always start to fall as soon as rpm>0 (more or less quickly, depending on the supply voltage, for the rasons you mentioned). And I never saw constant voltage drivers (except diy ones)…

The curve you are talking about is for the case of constant driving voltage (nominal voltage applied to the DC motor). I think David is right as duet's drivers are constant current ones.

• That's right, torque vs. speed curves are only relevant to constant voltage operation. If you wanted to publish a torque vs. speed curve for constant current operation, you would have to specify the actual RMS or peak used and the driver supply voltage, because both of those affect the point at which the torque starts to drop off.

• At work, we use Nema 34 Slo-Syn steppers, and curves given by the manufacturer (sorry, I only have them on paper, and as I'm on holidays, I can't scan them) do specify driver supply voltage; I guess they give them for nominal current of the motor. And they usually show 2 or 3 curves, for different supply voltages (24V, 48V and 70V, if I remember correctly).

As I have these documentations for more than 20 years, and as we always used constant current drivers, I assumed constant voltage drivers were abandoned long ago…

• Dear all,

i know very well what constant current vs constant voltage means, i am a CNC guy and in the very old past there was only constant voltage with all limitations.
Please do me a favour and just google –> https://www.google.at/search?q=stepper+torque+curve and check the pictures. I don´t understand why that is a secret here ?!?

Because of back EMF as mentioned already several times the torque starts to drop at about 0-10 rps (revs per second, depending on supply voltage). The constant current ONLY applies in the low speed region, because afterwards (at higher speeds) the coil limits the current. The reason to use higher supply voltage is because of this effect. At 12V it starts at nearby 0 rps, at 24V before 5 and with high voltage (48 V and above) the constant current torque maybe gets to 10 rps.

5 rps means with a 16 GT2 pulley 5162 = 160 mm/s !!!!!** I don´t know why a 3D printer should not operate in higher speed regions? At 12 V torque drops from "zero speed".

• please do me a favor and e.g. check the following link: http://www.electrocraft.com/products/stepper/TPP23/ and go to the tab "speed/torque curves", there is nothing more to say

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