Page 'Delta Geometry' created on the RepRap Wiki

While we see again and again the same questions about delta geometry, reachable spaces and dimensions criteria, there was nothing in the RepRap wiki on this topic, so I created the page and all associated schematics:
[www.reprap.org].

Great resource! Thank you!

Nice data source. But I'm surprised by the definition of TES and have a few questions about it. Well actually only about its part which puts hotend tip so low below platform (the parameter c).

1) Did you derive a mathematical model which shows how platform/hotend tilts when we change parameters like e.g. one diagonal rod length, or when a carriage is tilted around its vertical axis or its horizontal axis? The proper TES should then be derived from that model as a place where the partial derivatives are zero. And may guess is that it is likely to be different for different kinds of input errors (i.e. different for one diagonal rod being imprecise compared to carriage rotating along its vertical axis.

2) It looks to me that the TES definition may work well when we are dealing with errors like small diagonal rod length differences or small carriage rotations. But what about the case when there is a force applied in XY plane on the hotend tip (e.g. because of friction with the part being printed or well somebody uses it as a weak CNC). In such a case the TES puts the tip fairly low below the platform and that increases the lever from the forces in XY plane quite a bit and therefore the torque which needs to be countered by the diagonal rods. The higher torque will lead to more elastic deformation. Is the effect of bigger lever mitigated well enough by having the tip near the TES optimal point?

I have no tool nor practice for math model.
This is done in Openscad, so there are a lot of limits. Also, for me, maths are far away (thirty years, indeed).
Doing geometry calculation is ok, though. Thats is just pythagore, hey !

In fact, I've gone through that while building the D-Box [forums.reprap.org] with assymetric angles. I had huge differences of stability in the different axis (while manipulating by hand, and shown in the prints), so I questioned myself on how it worked. And discovered my misconceptions on the topic. Notably that with lower effector offset, you increase stability. Also, manipulating manually, you really 'feel' the virtual articulation.

When you lower the nozzle, that may increase the moment, hence the tilt, but if you are near the virtual articulation point, the tilt may have limited effect. So, maybe we can ponderate that ? But that need more research. I may add a note on that.
So, maybe a compromise is interesting. I was surprised by the relatively good quality of prints of the Fisher with a design inducing mechanical faults (poor arm alignment). In that case, the hotend is above the virtual articulation, with an overall stability which is not that bad. That may be a good compromise. The new version of the D-Box (equilateral) also have similar setup (nozzle above virtual articulation), but I have to do further tests.

I did not invent the concept of virtual articulation. I just read (maybe 25 years ago) an article in a review presenting research work on robots 'hands' and I remembered it.

Edited 1 time(s). Last edit at 01/27/2016 09:09AM by PRZ.

I would quite like to see some more math based model of the virtual articulation point and especially under what forces and deformation is may be valid.

I did not realize that external forces to the hotend tip may result in platform rotations which may have center so far below the platform plane. I would like to know how well it works especially for the the plantar (in XY-plane) forces which are acting at the tip of the hotend or in the center of gravity of the platform assembly.

Yes, I tried to tilt the platform by hand and really it looks that the platform is tilting around the virtual point. Also my hotend is (by a chance) quite near to it and when I try to move it by the tip of the hotend then it looks like the platform does not want to tilt but it is trying only to move while staying parallel to the bed. So even without a math model these are pretty good arguments at least for a typical linear delta printer.

Though precision of the current TES formula is worrying to me since:

• It would indicate that when the tip is at the virtual point than it will not ever move when the platform is tilting. And it will be slightly tilting under dynamic forces since the center of gravity is not at the tip and bearings are not perfect (the carriages will tilt a bit too). Anyways, this looks good if the system is symmetric. And it is not such an absolute win since the virtual point is moving.
• But worse, it would indicate that a magnetic joint platform which shares balls will not tilt at all. Sure it can tilt ... by just tilting/moving all the carriages a bit. So there must be some limitations for its validity.

Good post PRZ!

You cannot have a single coefficient number which give all information about stiffness, but I was willing to clarify a bit the stuff, because I think there are a lot of misunderstanding, and that have begun by me.

To expand the overall geometry understanding, other paragraphs shall be added, notably about carriage yaw and pitch.
There is a page about the articulations, which need to be completed.

For the stiffness added by cardan articulation, I had never thought to that, till a post in another forum underlines its advantages.

Also, a lot of people do not really understand that most of the loads are from inertia, and printer design is chasing weight.

The D-Box is a very specific machine, but I learned a lot with it. With steel wheels running on steel, It does have very high stifness between the 'rail' (a steel angle) and the carriage. The bending resistance of the angle is way higher than any aluminium profile and it is more straight.

However, my angle thickness is unsufficient (1.5 mm) and I have weak rail torsional stifness, and moving the effector drive to the rotation of the angle rail. That looks quite weird, but that may be representative of faults found on less rigid carriages.

And for the math, I would prefer someone a bit more versed than me do this work.

Oh, I did not intend to force you to do the work. I could do it myself if I would not be too lazy
I only wanted that if anybody would find the math posted somewhere then it would be great to post the link.

I do not think cardan joints on rostock max add that much. It is easy to get extremely good (tight, no play) angle ball joints. The only problem I see is that a lot of people use improperly put together traxxas joints. So community is under impression that ball joints are not precise.

Carriage yaw and pitch is another thing. I believe there is a lot of problems there. I have a heavily modified traditional Rostock and I replaced 4 of the 6 linear bearing because the original cheap ones did not even near the precision required by standard (which is -0/+8 µm). My guess is that all the linear bearings which do not pass quality control are sold to RepRap community When I'll try to build a new machine I definitely go for a local source of linear bearings which specifies precision. The other big problem is low belt stiffness and too thin smooth rods.

I was thinking about going steel bearings on steel profiles too. But if I would want hardened steel profiles then they are already more expensive than smooth rods. Linear rods with carriages are even more expensive. So I'm back to smooth rods (my bending calculations indicate that about Ø 16 mm is needed) and linear bearings. This is the cheapest option from the local sources which specify precision of the parts they are selling.

I never had problems with platform tilting. Thanks to the luck, I have tip quite near the optimal position. Optimizing armSpace is obvious and it also forces one to indirectly optimize b too. So I have these almost as good as they can be by intention.

Very good.
Please add this to the Reference section by using this tag at the bottom: [[category:Reference]]

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My printer: Raptosaur - Large Format Delta - [www.paulwanamaker.wordpress.com]
Can you answer questions about Calibration, Printing issues, Mechanics? Write it up and improve the Wiki!

Paul: done, but it's a wiki, why not doing it yourself ?

Hercek, it is fairly possible that the cardan don't add much in the Rostock max due to the construction, but the reason they may add stability is not the quality of the articulation.
By construction, a cardan (if set on effector AND on carriage) do lock the rotation axis because the arm rod cannot rotate (hey, they are usually used for torque transmission !), so it does maintain the horizontality of the effector by torsional stiffness.

For that to work, it needs :
*No play in the articulations
*Stiff torsional resistance of the cardan
*Stiff torsional resistance of the arms

For the math, I have seen messages on the Fisher thread by contributors telling they have done evaluations to check what are the effect of Fisher constructive faults (mainly unequal spaces between arms at effector and carriage and balls not on same plane). So knowledge may be there, just some incentive to do the work is needed. But that may have been done on CAD program or other simulation programs.

Quote
PRZ
Paul: done, but it's a wiki, why not doing it yourself ?

Yes of course... I had just linked the other page you added to Tutorials, but when I went to set this one to Reference I didn't have the Edit option. (???)

So I thought the page was being edited..

But turns out the wiki had just auto logged me out after 180 days...