# Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print?

Posted by realthor
 Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 20, 2015 08:49PM Registered: 8 years ago Posts: 1,035
I am having a difficult time (like maybe every other Delta newbie out there) understanding all possible delta errors implications.

I have 6 8mm rods in my delta, 2 per tower. I have measured the squareness of rods to towers, I have measured top and bottom distances between towers, I have measured diagonals between towers, I have leveled the bed to be 90 degrees to towers. Now, all measurements are not perfectly equal. There might be errors in the measurement itself (I couldn't position the caliper in the perfect position every time,etc) and there are errors in the caliper itself (I hear that the second decimal is not reliable anymore).

Nevertheless after much thinking I am slowly concluding that even decent physical/mechanical calibration is almost impossible, not talking about perfect calibration. I can't even imagine how to measure twisting between the bottom platform and the top platform or slight angle deviation between towers... and there are many more errors that can be factored in.

Another beef I have with current calibration methods (be they manual or automatic) is that the calibration is always done to allow an as perfect as possible first layer. That's it. Nothing else. For a part that has 10cm of 0.1mm layers (1000 layers), only the first layer is optimized/calibrated. From there the errors run amok upwards Z.

I was thinking to as wild solutions as to have a movable bed on Z so that we can calibrate Z=0 and Z=Max_Part_Height for print-specific optimization or Z=0 and Z=Max_under_nozzleZ_at_Zmax for build volume calibration.

Now I am starting to think that a sort of volumetric calibration is useful and should be introduced in software but the input data should be a combination of Z=0 bed leveling and measuring a printed calibration piece rather than re-designing the bed to be able to move physically upwards on Z. This is already a practice so nothing new here but I am wondering how many parameters can be measured and how many can the software take to calculate and screen out volumetric errors.

I would like to start a discussion, if not for anything else at least so I can understand (and maybe others too) to what degree the possible errors in construction can be screened out in software by using both bed leveling and volume print measurements.

1) it is clear that bed leveling is useful to have a first calibrated layer (Z=0mm).
2) for the volumetric calibration (Z=0.1mm, Z=0.2mm, etc), what errors can be inferred from printing and measuring a calibration part that is specifically designed for this purpose:
- I believe that leaning towers could be measured on a calibration column;
- I believe that top-bottom twist can be observed (maybe even measured) on a square column print;
- I believe that straight lines can be measured (dunno about calibration) by printing a long line over a 2D printed grid taped to the bed;
- Deviations from the 120degrees between towers can be measured too;
- there are for sure many other parameters that can have errors between real values and the ones defined in software...

If these can be measured, is there a way to introduce in software the values so that it can screen out those errors? I think that maybe a "virtual" printing volume made of all possible nozzle positions can be computed (on PC not real time, maybe only once every 100 prints or something along these lines) and then have a post processing script that would take the G-Code from the slicer and modify it to compensate for the printers specific situation.

If what I write here is stupid please correct me and please be patient with me, I am doing my best to understand and you are the ones I can turn to for help.
Note: I know that perfect construction is preferred but it is seldom achieved.

Thank you.

Edited 4 time(s). Last edit at 12/20/2015 11:37PM by realthor.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 20, 2015 10:24PM Registered: 8 years ago Posts: 83
You may be overthinking this.

Generally... Towers should be vertical. A plumb line should be able to determine that, and a spirit level should be able to tell the frame is horizontal. The carriages should be at 120 degrees to each other, which means a piece of string or a straight edge should bisect the horizontal across from the carriage.

Infinite levels of decimal places are nice, but impractical, for the reasons you've already mentioned.

Ultimately, your precision is a function of your stepper motor, which will be affected by your stepper motor and your stepper driver, and eventually, how accurately is your firmware driving that stepper. Once you pass beyond the resolution of what the firmware/motors can do, you're calibrated-- Not necessarily out to 12 decimal places, but you don't need to be to produce quality prints.

Many people are successfully printing (and doing some very nice quality) with rostock style printers such as yours, or mini kossel's with printed corners, so it can't be *that* difficult.

My printer (a "Kossel Clear" from Blue Eagle Labs) is known to have flaws in it's design. The corners are acrylic, which made assembly tricky (not tight enough, you don't have a printer-- too tight, you have broken acrylic). The effector arms are acrylic-- identical in length, but flimsy. The uprights are 720mm (?) and 1515 misumi extrusions-- they're flimsy, and as someone said, you can twist the top like a steering wheel (although why you would do this, I have no idea-- it doesn't happen during printing). The stock extruder was prone to fracturing. But-- it's a relatively easy printer to assemble well, and if you keep the speed under 60mm/second, it produces gorgeous quality prints (also-- use rectilinear infill. Honeycomb is nice, but it's unnecessary shaking).

I can print a part with 3 and 5mm bolt holes, designed to securely fasten to 2040 V-slot extrusion, and I know that the finished part will be exactly what I need-- 3mm and 5mm bolts will just work, and a perfect fit to the V-slot profile, including the raised sections which snug into the v-slot grooves. If I print a NEMA-17 bracket, I know that the collar of the motor (22mm) will fit securely, and the 4 3mm bolt holes will line up with the motor, and with minimal effort I can attach a NEMA-17 motor to that bracket (some smoothing out is required if I had to print the hole sideways).

I also calibrate my printer manually-- Use a known value for the diagonal rods, set the 3 endstop offsets, dial in the delta_radius to get a flat "arc" (and I know my delta_radius value isn't quite right-- but it's close enough for a 280mm printable circle), and start printing.

This isn't intended as bragging-- quite the opposite. I have a decent printer that prints reliably, in spite of the fact that the frame is under specced, and the arms are a bit wibbly.

Now, personally, I think people are over-doing the "My frame must be absolutely rigid and I must have no backlash whatsoever in my effector setup" theme-- I think what's far more important is a consistent build-- all your diagonal rods are the same length, your carriage spacing is identical, your uprights are the same length, and your frame is "regular"-- if it's extrusions, they're all the same length. if it's a plywood base (or other), the dimensions between towers / center are consistent. Heck, the mini-kossel is just three equilateral triangles linked by three uprights-- and strictly speaking, those uprights don't have to be the same length, as long as the three triangles are parallel.

Beware the law of diminishing returns. The more precise and rigid your printer is, the more expensive and difficult it's going to be to design, fabricate, and assemble.
 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 20, 2015 11:58PM Registered: 8 years ago Posts: 1,035
Quote
grat
A plumb line should be able to determine that, and a spirit level should be able to tell the frame is horizontal.

I was thinking at something similar although I haven't seen it in videos or re about it so far as something that is done for Delta calibration. I have been thinking a simple laser pointer that can maintain focus over the ~270mm of my built height, but I don't know it such a pointer is available but I know they are pretty cheap and useful in other situations as well so I won't be buying it for this purpose alone. If such a laser would be fitted on the effector and I would do a homing from z=0 at each tower and at center, I could see the path of the error and infer from there if there is a top-bottom twist or to which side there might be a slight lean of the printer (that could be observed from printing a square column of significant height too).

I am also getting a laser engraver for this printer and was thinking that it can be used for calibration too but haven't thought beyond "plumb line" use.

Quote
grat
The carriages should be at 120 degrees to each other, which means a piece of string or a straight edge should bisect the horizontal across from the carriage.

Hmm, don't follow the wording of that explanation. Towers angles can quite easily be tested in several ways from which the easiest would be a calibration print and the measurement of the resulting 120 degree lines with a protractor. Maybe there are several other ways, like you suggest, but I fail to see the way you explain.

Quote
grat
This isn't intended as bragging-- quite the opposite.
Don't worry, I am not thinking like that, i am grateful you guys take the time to answer to my overcomplicated and unnecessarily (maybe) philosophical/perfectionist scenarios.

Quote
grat
Beware the law of diminishing returns. The more precise and rigid your printer is, the more expensive and difficult it's going to be to design, fabricate, and assemble.

Unfortunately I think I am like this. Can't be satisfied without knowing a lot and when something is not up to my expectations I tend to overshoot with looking for solutions to improve. I only give up when i realize it's beyond my brain capacity .

PS: anyways, none of the above would be of much value if they can't be inputted in software for correction.

Edited 2 time(s). Last edit at 12/21/2015 12:12AM by realthor.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 21, 2015 12:56AM Registered: 7 years ago Posts: 22
I would say, in my limited experience mind you, that if you are able to get a perfect first layer, layers from that point on will be reletively perfect with good motor/belt/ extruder calibration. Getting the first layer perfect ensures good bed adhesion and fights lifting and warping that will ruin prints. I built a kossel that is a long way from perfectly built I cut my extrusions on a mitre saw, built my own rods which are not as good as others have made, and yet it works and is relatively accurate, but not perfect. (good enough to build another working printer that also prints accurate). What I'm getting at is I don't thing things necessarily need to be 100% perfect in all cases. Although it helps
 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 21, 2015 04:13AM Registered: 9 years ago Posts: 14,646
Interesting idea. Here is my take:

1. A delta printer should be designed so that its geometry is accurate by design. For example:

- The corners should hold the towers perpendicular to the lower frame. Then the bed can be mounted directly on the lower frame with standoffs. No adjusting screws needed. I resorted to metal corners, because my printed ABS ones made the towers lean slightly. You can buy metal corners from Robotdigg. Lower cost sheet metal ones will be available soon, see [groups.google.com].

- The towers should be parallel. Again you need good corners, and the upper and lower horizontal extrusion lengths need to match.

- The rods should be assembled on a jig so that they all have the same length.

- The spacings between upper and lower bearings in each pair of rods must be equal. If using printed plastic parts, this can be adjusted using washers or shims.

- The bearing lines on the carriages must be at 120 degree intervals to each other. If using wheeled carriages, this can be adjusted using shims.

2. Auto bed levelling has no place on a delta printer. OTOH, auto delta calibration is very useful. If you need auto bed levelling, that is a sign that you have not achieved good delta calibration or that your geometry is poor.

3. So there is no need for more complex calculations, either in the firmware or for a gcode post-processor.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 21, 2015 04:47AM Registered: 8 years ago Posts: 1,035
Ok, let's assume that I have a fairly well adjusted printer on the frame level (can't do much about it: injection molded top and bottom platforms, 6 rods inserted several cm in the molded plastic, equal diagonal rods, etc).
I did a manual calibration (paper test on all towers & center, delta radius) and by my feeling i have the same drag on them.

Now I assume I have to do a calibration print right? I will measure other parameters to further refine/tune. I also want to check if my frame really is square, as I believe it is. I want to take the guesswork out of the calibration, that's why I insist on measuring etc.

And for rods I can't have metal corners. The Overlord has aluminum extrusions with channels that slide on injection molded parts.

Edited 1 time(s). Last edit at 12/21/2015 04:49AM by realthor.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 21, 2015 08:14AM Registered: 9 years ago Posts: 14,646
I think that two calibration prints will tell you most of what you need to know. A single-layer calibration grid will tell you whether the nozzle first layer height is uniform and straight lines are straight. A large cube (hollow so that it prints faster) will tell you whether the scaling is correct, and if it is large enough, whether the scaling is consistent with height and the towers are straight and not leaning.

Edited 1 time(s). Last edit at 12/21/2015 08:14AM by dc42.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 21, 2015 11:13AM Registered: 8 years ago Posts: 83
Quote
realthor
Quote
grat
The carriages should be at 120 degrees to each other, which means a piece of string or a straight edge should bisect the horizontal across from the carriage.

Hmm, don't follow the wording of that explanation. Towers angles can quite easily be tested in several ways from which the easiest would be a calibration print and the measurement of the resulting 120 degree lines with a protractor. Maybe there are several other ways, like you suggest, but I fail to see the way you explain.

A proper delta setup is an equilateral triangle-- Each carriage tower represents one vertex, and each horizontal (in the case of the mini kossel) is an edge.

In our case, however, rather than the vertex being a "point", it's a carriage. A line perpendicular to that carriage should pass through the midpoint (and be perpendicular to) to the edge on the opposite side of the triangle. So if you have towers A, B and C, a line perpendicular to carriage "A", should cross the midpoint between towers B and C.
 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 21, 2015 11:19AM Registered: 10 years ago Posts: 732
Two calibrations objects should be enough:
1) Single layer to check that the leveling and maybe also dimensions.
2) A vew cm tall narrow tower to check that mm/step of your belts/motors is correct.

If you have big enough bed (so that your diagonal rods can get to horizontal position and you still z-probe the bed) then only one single layer calibration object should be enough.

If you want to precisely calibrate your printer then get a big bed and a z-probe which measures directly under the hotend.
[github.com]
[forums.reprap.org]
[forums.reprap.org]

dc42 has some web page too (if maxima is too tough for you). IFAIK, the only problem with the web page is that the number of probing points is limited. The less mechanically sound your printer is the more probing points you need.

Quote
realthor
Nevertheless after much thinking I am slowly concluding that even decent physical/mechanical calibration is almost impossible, not talking about perfect calibration. I can't even imagine how to measure twisting between the bottom platform and the top platform or slight angle deviation between towers... and there are many more errors that can be factored in.
You are almost for sure wrong. Search for papers on calibration of 6-DOF parallel robots. Those things can be calibrated to the precision better than 1µm. Our delta printers are just limited variations of those robots. I believe the only problem is that nobody bothered to implement more general calibration for linear delta printers.
Obviously, you are interested. You may want to give it a shot. I bet the community would appreciate it :-)

Edited 1 time(s). Last edit at 12/21/2015 11:20AM by hercek.
 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 21, 2015 02:41PM Registered: 8 years ago Posts: 1,035
Quote
hercek
Quote
realthor
Nevertheless after much thinking I am slowly concluding that even decent physical/mechanical calibration is almost impossible, not talking about perfect calibration. I can't even imagine how to measure twisting between the bottom platform and the top platform or slight angle deviation between towers... and there are many more errors that can be factored in.
You are almost for sure wrong. Search for papers on calibration of 6-DOF parallel robots. Those things can be calibrated to the precision better than 1µm. Our delta printers are just limited variations of those robots. I believe the only problem is that nobody bothered to implement more general calibration for linear delta printers.
Obviously, you are interested. You may want to give it a shot. I bet the community would appreciate it :-)

I was referring to home-made or consumer-grade Delta printers. But in general physics doesn't like perfectly matching parts. Depending on the expected precision you might consider your printer calibrated or mostly calibrated. For me, it's itching when I know one diagonal between two towers is 1mm longer then the other diagonal between the same two towers. I'd like to be able to input these errors in the definition of the delta, a definition that will be used when calibrating.

The industrial robots of course can be calibrated to angstroms but our cost-conscious hobby can't match that. I've recently seen a video on 3D Volumetric Calibration and Compensation of a CNC machine and that is something we should see as an option because laser engraver toolheads are startin to be more and more available. We can use them for calibration too if we think a bit out of the box.

Meanwhile I am looking to design a jig that would take my caliper and extend it on both sides so that I can measure with it the distances between rods, top and bottom, diagonals, etc, without the huge effort to eye-ball the alignment or repetitive consecutive measurements, something I can't achieve with just my hands.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 21, 2015 07:47PM Registered: 10 years ago Posts: 5,747
The thing that makes me hesitate to build a delta printer is the difficulty of alignment and diagnosing problems. I have thought about how to ensure that the guide rails are 120 degrees apart, and parallel and it seems difficult. I think I can get the guide rails parallel (using linear guides), but ensuring 120 degrees spacing and no twist is tricky. Also, making things adjustable so they can be positioned parallel, 120 degrees apart, and twist free is a little tricky.

In a cartesian printer, printing a test cube and measuring diagonals tells you exactly how much to move the guide rails to ensure that they are orthogonal. In a delta, if you print a cube, measure diagonals and find they are not equal, what do you need to adjust? You can have problems with effector rod lengths, parallelism of guide rails, twist, etc. Is there any way to identify a specific problem when you measure a test cube? Maybe some other shape, such as a cylinder, will tell you what needs to be adjusted...

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 21, 2015 08:01PM Registered: 8 years ago Posts: 83
Quote
the_digital_dentist
The thing that makes me hesitate to build a delta printer is the difficulty of alignment and diagnosing problems. I have thought about how to ensure that the guide rails are 120 degrees apart, and parallel and it seems difficult. I think I can get the guide rails parallel (using linear guides), but ensuring 120 degrees spacing and no twist is tricky. Also, making things adjustable so they can be positioned parallel, 120 degrees apart, and twist free is a little tricky.

Best way to do it is use known good components. If I were to build a delta from scratch, I'd probably hunt down a set of the Kossel 25000 brackets, and use 2020 or 2040 extrusions.

The bad news is, they're not quite for sale-- good news is, they're coming out Real Soon Now.

 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 02:56AM Registered: 8 years ago Posts: 1,035
I didn't plan a Delta myself but if was an occasion sale and I couldn't resist for the price. Little did I know of the problems ahead...which, as @grat puts it, might not be problems at all if I were to just go with the flow and don't mind the little errors. The overlord is factory-made, injection molding, rods, aluminum extrusion rigid frame, etc, and I still have no way to accurately measure the effects of the shipping, the little errors from injection molding and from assembling, which are still done manually and not by robots. My Z motor mistakenly wired connector is a proof of that

For a Delta, I would like to have a set of measurements that can be introduced in software so that the software computes the errors before calculating anything else. After the software comes up with "My Specific Delta" virtual 3d printer, not "A Perfect & Theoretical Delta", it should go further with calibration. And the calibrated calculations should be passed through the error matrix previously computed. But I get it that i am going too far with it.

With Deltas, at least with mine, it would be impossible to correct anything with physical means so all the errors I measure should be dealt with in software.

@Grat: thanks for that link, those look very good.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 03:18AM Registered: 9 years ago Posts: 14,646
The towers don't have to be 120 degrees apart because that can be corrected in firmware during calibration.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 03:46AM Registered: 9 years ago Posts: 14,646
Quote
hercek
dc42 has some web page too (if maxima is too tough for you). IFAIK, the only problem with the web page is that the number of probing points is limited. The less mechanically sound your printer is the more probing points you need.

There is no limit to the number of probe points you can enter in the web page. Perhaps you are thinking of the version of the algorithm built in to RepRapFirmware, which is currently limited to 16 points.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 09:36AM Registered: 10 years ago Posts: 732
Quote
dc42
There is no limit to the number of probe points you can enter in the web page. Perhaps you are thinking of the version of the algorithm built in to RepRapFirmware, which is currently limited to 16 points.
Good! Thanks for the correction. So I can just point all people directly to your web page :-)
 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 10:23AM Registered: 8 years ago Posts: 1,035
Is there a reason why one would want more than 16 points? Even 16 points seem many to me... I mean at least you should have a fairly flat bed, and if it's flat then an inclined bed can be inferred from an even smaller number of points.

Edited 1 time(s). Last edit at 12/22/2015 10:53AM by realthor.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 10:52AM Registered: 8 years ago Posts: 1,035
Here's a jig for aligning the parallel rods on a delta:

https://www.thingiverse.com/thing:1029831

Of course you should already have another linear item like an aluminum extrusion and it is only useful while building the printer. The flaw in the design shown on the thingiverse page is that I can't see a way to adjust the end insertion points of the linear rods on the frame. So as long as the insertion frames are very precise (ex: laser cut), such a jig is pretty useless. I might stand corrected though.

A better jig would be a measuring jig that can get precise numbers for the deviations a delta is getting over time, due to vibrations or heating or some other factors (shipping, transport, etc).

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 11:43AM Registered: 10 years ago Posts: 732
Quote
realthor
Is there a reason why one would want more than 16 points? Even 16 points seem many to me... I mean at least you should have a fairly flat bed, and if it's flat then an inclined bed can be inferred from an even smaller number of points.
If your printer does not have any play then you do not need more than 8 points. Dejay Rezme did simulations to show this. It is possible that 7 points may be enough but probably with a different system then I used to calculate the calibration parameters. Dejay's simulations indicated that 7 points are not always enough for my system to converge well. It is kind of strange since only 7 parameters are optimized therefore 7 points should be enough. But the simulations shown it is better to use 8 points for good convergence. Of course, if you do not optimize all 7 parameters then smaller number of points is needed.

You want a lot of points if your printer has some mechanical play and you want to average probing errors out as much as possible. If you have a z-probe then the number of probing points does not make much difference in the labor required. Your probing script can run 8 probing points as well as few hundred of them. And when you process the output log from your printer so that you can input it into e.g. my maxima notebook then there is no significant difference between processing 8 points or a hundred points. Editors can do block editing and macros so if you process 8 lines long block then you can process also 100 lines long block in almost the same time

Edit: E.g. my printer has some play (probably about 0.1 mm between pulleys and the belts). When I make a change to it then I calibrate at least with 100 points and it really works much better compared to the case when I do it only with 11 points (the number I often used in the past).

Edited 1 time(s). Last edit at 12/22/2015 11:46AM by hercek.
 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 02:13PM Registered: 8 years ago Posts: 1,035
Quote
hercek
If your printer does not have any play then you do not need more than 8 points.

What kind of play are you talking about? I'm guessing that you might be referring to play in the towers during effector moves if the connectors between alu profiles are plastic. Also play between pulley and belts as you put it although I don't know what that means -maybe you can shed some light on that.

Quote
hercek
only 7 parameters are optimized

What are your 7 parameters that are being optimized by probing bed locations and are there other parameters that can be taken into consideration? I am mostly interested if any calibration procedure can take as input volume specific errors such as leaning towers, twisted frame, maybe others I can't imagine. I get the feeling from the comments here that volume calibration is not something that delta owners concern themselves with.

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 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 03:31PM Registered: 10 years ago Posts: 732
Quote
realthor
What kind of play are you talking about? I'm guessing that you might be referring to play in the towers during effector moves if the connectors between alu profiles are plastic. Also play between pulley and belts as you put it although I don't know what that means -maybe you can shed some light on that.
Any kind of play. The most common kinds are:
• between belts and belt pulleys (when pulley teeth does not fit the belt profile exactly)
• between diagonal rods and the carriage or platform (these happen only when one has poor quality ball ends or improperly completed ball ends like e.g. traxxas)
• between carriages and the towers (these happen when one has smooth rods which do not match exactly the linear bearings; the properly matching rods/bearings have only about 10 µm of play)
The play typically results in backslash. By play, I do not mean the dynamic errors. These do not really matter for calibration since calibration is done slowly and pressing the z-probe micro-switch (or activating a differential probe) does require only very little force.

Quote
realthor
What are your 7 parameters that are being optimized by probing bed locations and are there other parameters that can be taken into consideration? I am mostly interested if any calibration procedure can take as input volume specific errors such as leaning towers, twisted frame, maybe others I can't imagine. I get the feeling from the comments here that volume calibration is not something that delta owners concern themselves with.
The same as di42's calibration. It is fixing:
• tower positions (therefore you do not really need towers to have 120° angle or the same delta radius); fully specified by 3 numbers
• diagonal rod length; fully specified by one number
• the top endstop positions; fully specified by 3 numbers
If you want to know more about it then read at least the comments in this maxima notebook: [github.com]
If you cannot understand it then I cannot really help you. It is a very simple math. No big deal.

As I already said, I do not think there is any need for "volumetric calibration". That is the probable reason delta printer owners do not care much about it. The only thing why you may need anything 3D is is to verify stepps per mm. And you can do that with a thin tower few cm tall and measuring whether its z-height is the same as in the model you printed. If you have correct steps per mm setting and big enough bed then it is sure you can properly calibrate the 7 parameters I indicated above. And you can do it only by z-probing the bed. When you have such a bed perfectly leveled and you have proper steps per mm then also all the other dimensions of the objects you will print will be correct. The idea that you need some kind of special measurements of 3d objects is wrong at least for delta printers of abut the same size as the standard rostock but with a big bed. Intuitively it looks like that from the equations too. I did not prove it though. But it kind of makes sense. Realize that all the x/y/z dimensions are all tied together with diagonal rod length. If you introduce an error in one dimensions then it will translate to errors also in at least one other dimension (since diagonal rod is always the same). And now you have 3 towers with this. It makes sense.
There are some initial conditions for all this to be true. They are listed in the maxima notebook.

I think that one can create and automate a calibration procedure which would compensate for leaning towers too. And I do not thing it is hard to do. This would add another 6 parameters to the calibration procedure. I even started to do that but It did not turn out to be as trivial as the 7-parameter calibration me and dc42 are talking about. So I put it aside till I'm extremely bored or need it for my own printer. That means probably never. The 7-parameter calibration is really simple - read the maxima notebook.
 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 03:51PM Registered: 9 years ago Posts: 14,646
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hercek
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realthor
Is there a reason why one would want more than 16 points? Even 16 points seem many to me... I mean at least you should have a fairly flat bed, and if it's flat then an inclined bed can be inferred from an even smaller number of points.
If your printer does not have any play then you do not need more than 8 points. Dejay Rezme did simulations to show this. It is possible that 7 points may be enough but probably with a different system then I used to calculate the calibration parameters. Dejay's simulations indicated that 7 points are not always enough for my system to converge well. It is kind of strange since only 7 parameters are optimized therefore 7 points should be enough. But the simulations shown it is better to use 8 points for good convergence. Of course, if you do not optimize all 7 parameters then smaller number of points is needed.

hercek, I think you are using a more general nonlinear optimization algorithm than I am. The linear least squares algorithm I use needs only 7 points to calibrate 7 factors, or 6 points to calibrate the 6 factors that I recommend auto-calibrating (i.e. not the diagonal rod length). My implementations run two Newton-Raphson iterations if it, but in practice the second iteration rarely makes a significant difference if your delta radius and endstop corrections are within 0.5mm of true.

realthor, if the printer geometry were perfect apart from the tower positions then there would be no point in using additional points. In practice there are other errors caused by leaning towers, rods that are not quite parallel etc. In this case it is worth using more points, both to get a flatter printing surface on average, and to get an idea of how flat the printing plane is.

Large delta printer [miscsolutions.wordpress.com], E3D tool changer, Robotdigg SCARA printer, Crane Quad and Ormerod

Disclosure: I design Duet electronics and work on RepRapFirmware, [duet3d.com].
 Re: Volumetric Calibration: can most of the possible Delta construction errors be inferred from a calibration print? December 22, 2015 07:15PM Registered: 8 years ago Posts: 1,035
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dc42
realthor, if the printer geometry were perfect apart from the tower positions then there would be no point in using additional points. In practice there are other errors caused by leaning towers, rods that are not quite parallel etc. In this case it is worth using more points, both to get a flatter printing surface on average, and to get an idea of how flat the printing plane is.

I am specifically talking about not-perfect deltas, and let's face it. none are perfect. Yes they can be "OK" for average prints or even for good prints that have a good fit with companion parts because, as it has already been suggested, before better frame other components will take the lead on error generation, like belts backlash, overshooting effector, hotend control and other extrusion parameters.

By my logic leaned towers and twisted top-bottom frame would make a "tower of pisa" print that is also screw-like a bit. This should be visible on the print, if not with the bare eye at least by taking measurements with a caliper or some other comparison method. But what if a screwed/twisted frame or leaned towers frame can't be adjusted? Just toss out the printer and build a new one with better components? Wouldn't be a firmware that is aware of as many things that go wrong a better way until the owner can afford/has the time/etc to build another one?

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dc42
In practice there are other errors caused by leaning towers, rods that are not quite parallel etc. In this case it is worth using more points, both to get a flatter printing surface on average, and to get an idea of how flat the printing plane is.

Are you saying that taking more points for calibration at bed level can get a better print on the height too? It is quite logical because the 3 carriages walk on Z at each move, thus their movement is affected by twisting rods, twisting towers, leaned rods, leaned towers. But how much can they be corrected by probing the bed if not specifically introduced in the firmware?

I will start now reading the "maxima notebook" i will understand better the phenomena I am theoretically talking about.

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