Arm length calculation help on own-design delta needed

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hercek
... but for a delta calibration to be really easy somebody needs to derive and publish a calibration procedure for tilted towers.

To a first approximation, tilted towers have the same effect on the printing plane near to the bed as towers that are not quite at the corners of an equilateral triangle. So the auto calibration will compensate for them. It isn't perfect, but it was good enough for my original Mini Kossel, which had leaning towers.

To handle tilted towers properly would require an extra 6 calibration parameters, as each tower can tilt in 2 directions. The delta maths would be more complicated. An absolute minimum of 12 probe points would be needed, and more in practice. If we compensate for tilted towers, then IMO we really ought to compensate for other common build errors such as non-parallel rods, rods not all being of the same length, and bearing lines not being at 120 degree intervals on both the carriages and the effector. That makes another 10 or more parameters to calibrate, and correspondingly more probe points would be needed.

I prefer to build the printer so that the towers do not lean, by using metal corners - but it does increase the cost.

Edited 1 time(s). Last edit at 01/06/2016 06:51PM by dc42.

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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].

So, it will be more a calibration machine than a 3D printer

With tilted towers, it seems that arms remains more // to the towers in every point (didin't check with maths), so accuracy should not change as much as with vertical tower, from center to edge of bed. But as you explained, other problems make this point irelevent.

I agree with metal corners; I will also use them, rather than printed parts.

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Frédéric

Quote
fma
With tilted towers, it seems that arms remains more // to the towers in every point (didin't check with maths), so accuracy should not change as much as with vertical tower, from center to edge of bed.

This may also be better to prevent effector tilt? Especially under lateral load (spindle usage)?

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Frédéric

Quote
dc42
To handle tilted towers properly would require an extra 6 calibration parameters, as each tower can tilt in 2 directions. The delta maths would be more complicated. An absolute minimum of 12 probe points would be needed, and more in practice. If we compensate for tilted towers, then IMO we really ought to compensate for other common build errors such as non-parallel rods, rods not all being of the same length, and bearing lines not being at 120 degree intervals on both the carriages and the effector. That makes another 10 or more parameters to calibrate, and correspondingly more probe points would be needed.

I know. I briefly looked at it. If somebody is interested in details it is posted somewhere in the google delta group. I stopped to follow it since its traffic is too big and the best ideas are posted here too.
The conversion from cartesian to delta coordinates is not much more complicated with tilted tower. I did that much at least If I recall correctly there is no additional square root and there are some multiplies and additions (I do not really recall probably somewhere in the rage of 12 per tower). Of course everything possible should be pre-computed. That is not a big deal. ARM board can definitely handle it with ease.

My guess is that the current calibration procedure will only move the tower position in the direction of the tilt somewhere into the middle of the tilt size corresponding the the carriage movement span when probing the bed. For example if the carriage movement span is 20 cm and tower tilt is 1 mm per 1 m then there will stay an error there of about 1 * 0.2 / 2 = 0.1 mm in z-height, well probably only about 86% because diagonal rods are tilted of it. It will depend on the arm length, bed size, and delta radius.

I'm arguing for calibration of tilted towers because it is the only other parameter which still has about linear impact on the z-probing error. And it is hard to get the tower perfectly straight. Metal corners will help. But it is also an additional expense for parts which could be easily printed otherwise.

I do not think we need to calibrate the diagonal rod length because it is easy to make them precise:

• It is easy to measure their length when they in a jig with calipers. If the calipers are short just drive some nail to the jig and measure up to the nearest nail and compute the final length.
• Just make all 6 rods in the same jig. It will take longer but you will have them pretty precise ... in the sense that they will have equal length.
• If one rod is slightly longer/shorter then it introduces only very small error to the z-probe distance. It will be somewhere in the range of (1-cos(α)) for very small α. The x/y dimensions will be influenced about linearly because the platform will be a bit rotated - but read later.
• If you make your equidistant diagonal rods as far away as your platform allows then you will minimize impact of slight diagonal rod length differences for one pair. You see another reason to put them far away. And if one cares it is easy to support different arm length for different towers (not within one tower) with a very easy extension of the current calibration procedure. I just did not do it since I do not thing it is interesting when we can use a jig.

As for as platform/carriage distortion and the towers not to have the exact angles. Part of them will contribute to different virtual tower positions and we calibrate this out even now. The rest will lead to platform rotations. For a z-height it is not a big deal (in the rage of (1-cos(α)) for small α). For x/y it is worse but:
You can minimize x/y error from platform rotations easily by making your hotend stick as little as possible below the platform plane.

Because of all of this I think we should consider adding calibration for tilted towers (i.e. the additional 6 calibration parameters) but we can ignore all the other calibration parameters which we do not support yet (e.g. distorted platform/carriages, slightly different diagonal rod lengths).

hercek, thanks for your thoughts. I agree that it is easy to get all the rods near enough the same length by making them one at a time in a jig, which is what I do. Of course, cheap kits may be supplied with rods that are less accurate.

Right now, I think the main inaccuracy in my delta is that the carriage bearing centres are not at exactly 120 degrees to each other, because the printed ABS carriages are slightly warped. This causes effector tilt, which in turn causes Z probe inaccuracy because the IR Z probe is separated from the nozzle by about 14mm. It's possible in RRF to correct for this, but it's tedious measuring the probe trigger height at every probe point instead of just once.

I'll take a look at the IK maths with tilted towers. Did you work out a solution for the forward kinetics too?

Edited 1 time(s). Last edit at 01/08/2016 03:17AM by dc42.

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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].

Nice maths!

Are there curves, somewhere, showing the resulting errors on the movements, depending on the different errors that can be encounters, as described above (for standard vertical towers, not titlted ones)? I've no idea of the order of magnitude of all this...

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Frédéric

Quote
dc42
Did you work out a solution for the forward kinetics too?

I finished only the cartesian -> delta and counted the computation complexity. I do not think I have the notebook any more but this is pretty easy.

I started the other direction (delta -> cartesian) too. That seems to be what people seem to call forward for reasons I newer understood. Why it should be called forward when the direction we primarily need is the other one?
I did not find any elegant way to do it. It seemed to me that there is no elegant way to get rid of two general quadratic equations as it is possible now for the perfectly perpendicular (to bed) towers. That is a problem because it will split solutions to two alternatives at each quadratic equation. The boundaries of the split should be identified and the right alternative selected. My feeling is that this approach should still work but I did not have patience to finish it in the general form. So I let it be unsolved for now.

Quote
fma
Are there curves, somewhere, showing the resulting errors on the movements, depending on the different errors that can be encounters, as described above (for standard vertical towers, not titlted ones)? I've no idea of the order of magnitude of all this...

No there is only z-error notebook showing charts for towers which are still perpendicular: [github.com]

The following text is only about z-error:
But as far as I'm aware the only errors which have big impact are the ones we already calibrate and the tower tilts. These are in the order of linear errors.
The other errors lead to platform rotation and these should have impact in the range of (1-cos(α)) for small α. That means the order is better (i.e. smaller) than linear. I did not derive precise equations for these but seems obvious what the order of the errors should be.

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hercek
That seems to be what people seem to call forward for reasons I newer understood. Why it should be called forward when the direction we primarily need is the other one?

Well, forward is defined from actuators to cartesian space; same notation is used for robots. But your're right, this is mainly the IK we are interested in.

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Frédéric