No-leadscrew, single-motor, precision Z-axis design

Hey yall. Came across this design for a carriage in Blanding's Exact Constraint. There are no leadscrews, and all but the bed, motor, and maybe pulley axles, can be made from plastic. So it's highly reprappable .

For background, Exact Constraint Analysis is a new-ish kinematic science used for very, very precise devices. By analyzing all constraints acting on a moving device, and subtracting the constraints from its 6 degrees of freedom (3 translation, 3 rotation) the device's exact path and wear pattern can be obtained.




So if you take the Z platform's 6 degrees of freedom, and consider that it must be free to move in 1 (along Z), then the platform must be constrained in 5 ways. In the bottom pic, the wheels in the bottom-left corner run up and down a square flat (such as an aluminum extrusion) and hold the carriage in X and Y. However, were these the only constraints, the carriage could come free of one wheel and rotate about the Z axis. So a third wheel, on the bottom-right of the bottom pic, provides a third constraint, preventing rotation. The 3 wheels are pressed against the flats by a spring-loaded nesting wheel, oriented such that equal resistive force is applied at all 3 constraints, through vector analysis.

This leaves two remaining degrees of freedom, rotation about X and rotation about Y. On the same axes as the wheels, tensioned wire (or plastic braid, even) is run between pulleys in the pattern in the top pic. The wire provides the final two constraints, preventing rotation. Since it is exactly constrained, the z carriage movement will be as accurate in X/Y as its 2 vertical columns are flat and square.

There are additional benefits, such as the slim X/Y additional footprint (width of pulley + wheel), the flexibility of plastic parts for most components, and very low motor power required, with the counterweight.

Hope that somebody finds this useful! A copy of Exact Constraint's up on libgen, this is from chapter 6, page 96.

Interesting. My first impression is that it looks like too many moving parts with too many critical relationships. In a 3D printer, the allowable error is very small because the error shows up in the print surface. I've seen a couple printers made with cantilevered beds and arrangements of crossed cables at the far end of the bed to try to keep it from bouncing. I have no idea if it actually does what it's intended to do.

It will work well if the bed/support can't flex, cables can't stretch, there's no play in the bearings, there are no losses in the bearings (friction), the guides have perfectly smooth surfaces, and are perfectly parallel to each other and the cables. Good luck getting all that working at once. Plastic? Hmm. Printed plastic? Double hmmm.

Do you have an example of any real machine that uses this type of mechanism to achieve 10 microns of positioning accuracy and precision (which would be sloppy for a printer expected to print in 100 um layers)? Are you going to build one?

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Ultra MegaMax Dominator 3D printer: [drmrehorst.blogspot.com]

Interesting idea, I have a corexy with a belt driven z axis, 1 motor, no real issues, I have a counterweight to prevent bed-freefall when power is off.

[www.thingiverse.com]

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Simon Khoury

Co-founder of [www.precisionpiezo.co.uk] Accurate, repeatable, versatile Z-Probes
Published:Inventions

Perfect to demonstrate how to properly constrain, just that the "We assume we are using perfect wheels, no play bearings, non stretch cables, perfect flat and square surfaces..." foreword is missing.

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

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MKSA
Perfect to demonstrate how to properly constrain, just that the "We assume we are using perfect wheels, no play bearings, non stretch cables, perfect flat and square surfaces..." foreword is missing.

To be fair, the anti-racking properties of CoreXY rely on those same assumptions.

I'm also concerned about the "low motor power". The counterweight is nearly the same weight as the empty print bed, but at the same time has to provide enough friction on the drive wheel?
I'd use the fixed line approach, which has no slip at low tension.

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the_digital_dentist
Interesting. My first impression is that it looks like too many moving parts with too many critical relationships. In a 3D printer, the allowable error is very small because the error shows up in the print surface. I've seen a couple printers made with cantilevered beds and arrangements of crossed cables at the far end of the bed to try to keep it from bouncing. I have no idea if it actually does what it's intended to do.

It will work well if the bed/support can't flex, cables can't stretch, there's no play in the bearings, there are no losses in the bearings (friction), the guides have perfectly smooth surfaces, and are perfectly parallel to each other and the cables. Good luck getting all that working at once. Plastic? Hmm. Printed plastic? Double hmmm.

Do you have an example of any real machine that uses this type of mechanism to achieve 10 microns of positioning accuracy and precision (which would be sloppy for a printer expected to print in 100 um layers)? Are you going to build one?

Ha, I posted this on another site, and someone was like "went to see if digital_dentist shit all over it... yep, first post". Your reputation precedes you!

The design is such that play in the wheel and pulley bearings does not affect X or Y position, due to the nesting force. Imagine a wheel with ridiculous play on the axle: the axle will still press up against the inner wall of the wheel. You can imagine a design where instead of wheels, there are lubricated plates, for instance. The wheels are one option to reduce friction in the 3 nesting constraints. Because this is a contact constraint, any single point of contact traveling in a line will suffice kinematically. Just don't get clever about it and use a linear rail, or those makerslides-- it will overconstrain the system (cost of overconstraint is play or binding, and strange wear patterns).

Going down the list... bed flex is less of an issue here because of the counterweight. The motor only needs to accel/decel the bed and overcome system friction. Meaning one could use a stiffer bed. And its support at 3 corners makes this not a typical cantilever design.

The two rotational constraint cables work by balancing tension in the z cable. Kinematically this assumes no cable length expansion. If you calculate expected length expansion in steel (or even polyester/polyamide) cable, from the moment at the bed corners opposing the z motor's force on the Z cable, it is a quite good assumption.

The X/Y positioning accuracy rests (literally) on the flatness of the 3 constraint support structures. Which is fantastic news, this is the cheapest, most DIY possible linear constraint. We've been lapping planes for hundreds of years to within a few microns, or better. All linear axes require flat planes, and this is the cheapest route. But it is also flexible, as long as the constraints are respected. For instance, the kitty-corner, opposite Z, could be replaced with a ground rod attached to the structure and V-way bearing attached to the carriage. This would preserve the 2 constraint condition, and also be quite cheap.

Edit: I forgot to mention. In the text, chapter 6 are worked examples he's built. The other examples in the chapter are from his work in optics labs, and go over common kinematic designs like "mechanical glue" (used for items as small as sub-mm lenses, and as large as the space shuttle's attachment to its docking airplane... the constraint science is scale-invariant). So the author has built this carriage, but all he's left is the schematic. I'm not building a CoreXY or dual-leadscrew design, I'm working on my lathe's z-axis, and picked up the textbook for punishment, because I overconstrained the ballscrew system, and it is binding

Hope that answered your questions. I'd highly recommend the book and method. It is a simple and powerful kinematic technique, used in labs and space programs and by anybody trying to engineer in microns.

Edited 2 time(s). Last edit at 04/17/2018 03:03PM by golfwolf.

Pretty cool mechanism, but there's probably something a little simpler. My UDIO printer bed (https://youtu.be/hdju_6XEHZ4) is not quite so elegantly minimally constrained, but it gets the job done with very few components.

Edited 2 time(s). Last edit at 04/17/2018 03:25PM by LoboCNC.

I wasn't trying to s**t all over it, just pointing out some of the places where I think there are likely to be problems. The counterweight doesn't have anything to do with the "bed" flexing. The problem is trying to lift the bed from one corner. The edges/corners away from the lifted corner are going to lag the motion at the lifted corner, moving both up and down, even with the other cables. Another of your assumptions is that the guide planes are infinitely rigid. That would help keep the bed from tilting, but real guide planes are likely to have some give, especially near the center of the their length. 10 or 20 um is a very small error, but it's enough to cause a print quality problem.

It's an interesting mechanism, and I would like to see how a real machine built this way actually performs.

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Ultra MegaMax Dominator 3D printer: [drmrehorst.blogspot.com]

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the_digital_dentist
I wasn't trying to s**t all over it, just pointing out some of the places where I think there are likely to be problems. The counterweight doesn't have anything to do with the "bed" flexing. The problem is trying to lift the bed from one corner. The edges/corners away from the lifted corner are going to lag the motion at the lifted corner, moving both up and down, even with the other cables. Another of your assumptions is that the guide planes are infinitely rigid. That would help keep the bed from tilting, but real guide planes are likely to have some give, especially near the center of the their length. 10 or 20 um is a very small error, but it's enough to cause a print quality problem.

It's an interesting mechanism, and I would like to see how a real machine built this way actually performs.

For sure. I'm old enough and been wrong enough times to just take it as information. It's all good. Appreciate the deep look.

I understand what you're saying now about bed flex -- not from the bed stiffness or lack thereof, but from rotation. The moment from the z motor will travel to the other corner of the bed, and if not well-constrained, rotate it about the X and Y degrees of freedom, instead of purely along Z. Is that what you mean? However, the cables constrain this rotation. Looking at the top diagram, if the bed rotates about +Y or -Y, the vertical cables will attempt to pull in towards the center of the carriage. Since the cables are under tension, this will be resisted and there is no rotation. To rephrase, any rotation should increase cable tension. Same as with the X rotational axis. If you draw out the geometry of the cable, with the carriage in an unrotated state, vs. the carriage in a rotated state, the cable lengths are longer in the rotated state, meaning rotation creates tension. So it is well constrained in both X and Y axes of rotation.

In terms of the guide planes, the force causing any give is 1/3 of the spring's force on the nesting wheel (the nesting wheel's plane, btw, need not be so flat, but portions closer to bed center will increase spring force by F=-kx, very very slightly). Nesting force isn't meant to be very high, much less than the counterweight. With the spring force known, the flat thickness and material known, and location on z known, the bowing is easy to calculate from the simple beam equation. Bowing at (Zmax - Z0)/2 will be greatest. But again... it's 1/3 of that spring's force against a presumably steel plane or aluminum extrusion.

I'd love to see a real machine built this way too, and if anybody's down I would be willing to make and mail parts for it on the lathe. I'm serious about that, any circular metal prototypes would be on me. I'm working on another project right now though and would only tackle this mechanism if enthusiasm for a simple and precise z carriage could be shared with another engineer.

Edited 2 time(s). Last edit at 04/17/2018 04:40PM by golfwolf.

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Looking at the top diagram, if the bed rotates about +Y or -Y, the vertical cables will attempt to pull in towards the center of the carriage. Since the cables are under tension, this will be resisted and there is no rotation. To rephrase, any rotation should increase cable tension.

The cables at the side only increase their tension when the rotation force is in CCW direction. Turn it CW and the tension will loosen.
I've seen similar cable tensioning, but there were always two cables on each side.

How about completing the drawing to include the X Y axis built according to the same principles ?

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

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o_lampe

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Looking at the top diagram, if the bed rotates about +Y or -Y, the vertical cables will attempt to pull in towards the center of the carriage. Since the cables are under tension, this will be resisted and there is no rotation. To rephrase, any rotation should increase cable tension.

The cables at the side only increase their tension when the rotation force is in CCW direction. Turn it CW and the tension will loosen.
I've seen similar cable tensioning, but there were always two cables on each side.

I was thinking the same thing last night, and actually came here this morning to ask what I was missing.


So this bed design requires 4x cables over 8x pulleys for anti-rotation purposes, and will require a suitably sturdy and twist-resistant frame to keep those cables tensioned...

It looks interesting for sure, but perhaps also a bit "fragile" in practice? Any failure or loosening of any single element would result in print faults on multiple axes...


Instead of being fixed in place, could the anti-rotation cables be auto-tensioned with suitably strong springs to eliminate some of that risk?

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o_lampe

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Looking at the top diagram, if the bed rotates about +Y or -Y, the vertical cables will attempt to pull in towards the center of the carriage. Since the cables are under tension, this will be resisted and there is no rotation. To rephrase, any rotation should increase cable tension.

The cables at the side only increase their tension when the rotation force is in CCW direction. Turn it CW and the tension will loosen.
I've seen similar cable tensioning, but there were always two cables on each side.

Yup true... Drew out the diagram to make sure.

The carriage weight defines cable tension. Specifically, each cable holds 1/3 of the carriage weight. (I'm not sure if that means the Z motor counterweight should be 1/3 the weight of the carriage, or the full weight, but my hunch is the former.) If you imagine a CW rotation, it would loosen the cables and reduce tension, but the carriage would fall back down to its "home" position. The carriage would very much like to fall! The weight of the carriage fights CW rotation. Playing around with a coaster, I tried to imagine a fixed corner, where the Z cable meets the carriage, and the only way I could get the sort of rotation permitted by the cable constraints was to raise the coaster (carriage, bed) up. To repeat, the carriage's corners would have to move up, against gravity, along the ceiling-fixed cables, in order to rotate CW.

To dial-in the home position (level the bed) one would set a Z position, and then raise or lower the two constraint cables (I'd use an eye bolt, with the cable swaged in the eye, and a fine-pitch locknut's rotation for this, but there are probably a half dozen other means). The three points, where the cables feed to the top, exactly define the carriage plane.

Edit: Interestingly, if the rotational constraint cables are not fixed, but instead are also on steppers, the additional degree of freedom translates into another printing axis. So you could print, rotate the bed about the X axis 45 degrees, and print at a new angle. A slicer nightmare but mechanically sound, at least as sound as the rest of the mechanism. Well, the X/Y/Z-theta positional constraints would also probably need to be re-thought. Put this idea in the 'maybe' pile.

Edited 3 time(s). Last edit at 04/18/2018 11:30PM by golfwolf.

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Melty
Instead of being fixed in place, could the anti-rotation cables be auto-tensioned with suitably strong springs to eliminate some of that risk?

That's a good idea -- could I locate the spring beneath the carriage? Forgive me, I'm going to write out the mechanism as I understand it:

The anti-rotation cables have a 'stop' position (they're in stop position in the pic). At 'stop' position, the cables are under tension caused by the weight of the carriage. When the Z motor lowers the carriage, there is the briefest moment when the bed is angled, before the carriage mass falls to its new 'stop' position. To fall, it must overcome static friction in the pulley bearings. If it does not overcome stiction in the bearings when it moves from one layer to the next, the bed keeps its angle. One could imagine a very light bed, with very bad bearings. The easy solution is to just make a heavier bed. Use a big steel plate. Hang rocks from it. This would provide constant force. But a spring beneath the carriage, pulling it down so that it's always in its 'stop' position, would have the same effect, but with variable force. Or one could just buy good bearings, they're nearly as cheap as rocks these days

Edited 2 time(s). Last edit at 04/19/2018 12:14AM by golfwolf.

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Melty
....

Instead of being fixed in place, could the anti-rotation cables be auto-tensioned with suitably strong springs to eliminate some of that risk?

No ! If you do that the system is not isostatic anymore. It would be like putting cloth pin springs on the belt of a COREXY (suggest that to DD and RUN ! )

Edited 1 time(s). Last edit at 04/19/2018 02:35AM by MKSA.

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

It is a few decades ago, when I had vector math classes, but IMHO the tension in the anti-roll cables has to be much higher than the rotational force or gravity they fight against.
There is no way the bed will level itself correctly by gravity forces ( only in an ideal world )
e.g. the spring loaded nesting wheel ist at it's highest load when the bed is horizontal, the spring will always try to push the corner up/down if possible.

I'm thinking about using a Delta frame as base for a CoreXY top frame. The equilateral triangle would make aligning the corner wheels much easier. Only three corner wheel will do. The bed would still be rectangular.

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o_lampe
e.g. the spring loaded nesting wheel ist at it's highest load when the bed is horizontal, the spring will always try to push the corner up/down if possible.

Oh, true that. I think this could be solved by setting the nesting force vector at a slight Z angle (no longer parallel to the bed plane), with the wheel's pivot hole drilled not straight down along Z, but with a small X/Y component. That way, the nesting spring's normal operating load would be slightly less than maximum, and it would have to 'climb' over the maximum when rotating in the bad CW rotation. So the nesting spring, in this case, would provide nesting force for all 5 DOF constraints. Depending on the angle, it would provide slight additional tension on the wires, along with the carriage weight. I'm having a brain fart moment though and can't determine whether it should be angled up, or down (up, right?). Is this a crazy idea?

Less straightforward to manufacture, for sure -- gotta angle the bed when you punch thru it with a drill press, and drill bits in my experience don't like that. edit: Rather than keeping the lip like the drawing, the nesting spring+wheel+pivot assembly could be bolted onto the bed. That'd be the way to go. And thinking about it more, I'm sure now that the correct angle is to point it up, with the spring force direction slightly -Z. One problem is, creating equal force on the 3 simple constraint, as well as equal force between the 2 cable constraints (although not strictly necessary, as long as the nesting vector's in the right direction, its magnitude is less important)

Edited 1 time(s). Last edit at 04/19/2018 06:36PM by golfwolf.

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Oh, true that. I think this could be solved by setting the nesting force vector at a slight Z angle (no longer parallel to the bed plane), with the wheel's pivot hole drilled not straight down along Z, but with a small X/Y component. That way, the nesting spring's normal operating load would be slightly less than maximum, and it would have to 'climb' over the maximum when rotating in the bad CW rotation. So the nesting spring, in this case, would provide nesting force for all 5 DOF constraints. Depending on the angle, it would provide slight additional tension on the wires, along with the carriage weight. I'm having a brain fart moment though and can't determine whether it should be angled up, or down (up, right?). Is this a crazy idea?

You could do that, but it's against the rules of exact constraint theory. Tilting the axis of the wheel could introduce banding.

You could also use linear bearings running on a smooth rod to replace the nesting wheel, but it's the same rule breaker.
Let's face it: You'd need two sets of anti-roll cables on X and Y-plane to obey the rules.

Edited 1 time(s). Last edit at 04/20/2018 02:43AM by o_lampe.

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o_lampe
You could do that, but it's against the rules of exact constraint theory. Tilting the axis of the wheel could introduce banding.

You could also use linear bearings running on a smooth rod to replace the nesting wheel, but it's the same rule breaker.
Let's face it: You'd need two sets of anti-roll cables on X and Y-plane to obey the rules.

I'm a newbie with kinematic design, but, not sure that the different nesting force vector introduces a redundant constraint. The nesting wheel itself doesn't serve as one of the constraints, or the carriage weight. They're just to force the carriage into the constraints. Maybe I'm missing something though (not the first time ITT).

The way the cables work, they allow rotation, to a point, and then rotation abruptly stops. I think the big challenge with the double set of roll cables is, with a second cabling mechanism upside-down, there would be two 'stop' positions per rotational degree of freedom. One for +X, one for -X, and same as with Y. To ensure that the carriage does not rotate, both 'stop' positions per axis must exactly match. If they do not match, for instance, if -X's stop is 1 degree larger than +X's stop, then there is 1 degree of permissible play. If it is 1 degree less, then the mechanism will bind. Such is the cost of overconstraint. Maybe there is another style of constraint, not roll cables, better suited to this application.

Just imagine, the nesting wheel has a bumpy ride along the tower. With it's axis exactly vertical this would only lead to more or less force of the spring, but when the shaft has an angle, any bump will translate into the XY-plane.

Sorry I gave your post a couple days to process but I still don't understand... the nesting wheel maintains a force vector, in the normal plane of the pivot axis. Bumps change the direction of the force vector, spinning around that plane, but we're talking <1deg in practice. Like even moldy pallet wood can maintain that. Changes in force magnitude are not so important, so long as it's >0, or > an established minimum. As long as the vector components are +X, +Y, and -Z, and the nesting wheel is in contact, then it will function -- that's an 1/8 of a sphere to play with. This is not a difficult engineering tolerance.

Edited 1 time(s). Last edit at 04/22/2018 03:15PM by golfwolf.

Thank you for sharing this very interesting mechanism. I have always practiced the exact constraints method, without knowing that it was so called. For a good design, it is always preferable to have in mind the skeleton of the forces in presence, reduced to its simplest expression.

To better understand this mechanism, I think it is necessary to introduce in the drawing the centre of gravity of the bed, and to assign to it a weight, equal to that of the counterweight (why would they be different?).

We see then that, since the bed is lifted by a corner, the thetaX and thetaY cables are tensioned, by the moment of the bed's weight, with respect to this suspension corner. There is no other source of tension for these cables, and you must not introduce any! It is a stable situation, like that of any stick suspended between two strings: in the plane of the strings and the stick, the rotations of the latter can only result from a variation in relative length of the strings...

Of course, this ideal stable situation can easily be disturbed, in case of movement of the suspension corner, by any friction on the other corners. The extent of this disturbance depends on the ratio between the friction forces and the weight of the bed. Fortunately, with support wheels consisting of open ball bearings, whose grease has been replaced by fine oil, and a reasonably heavy heating bed, this ratio will be negligible.

Among the objections raised, I think the one about the spring is true. The presence of a pushing spring makes the normal position the least stable (a kind of knee-joint mechanism).

I propose to replace this spring by magnets associated with the support wheels, assuming that the support planes are magnetic. This would also eliminate the objection of support planes that could flex under spring force.

Post scriptum : I also realize that the centre of gravity of the bed, relative to the weight, should also be the geometric centre of gravity of the triangle formed by the three fixing points of the cables, projected in the plane of the bed. In this way, the three cables will be loaded evenly, and there will be no diagonal tipping (unless the weight of the plastic deposited exceeds that of the bed, which is unlikely).

It would therefore be logical to modify this mechanism to adopt an equilateral triangle bed, rather than a square one (or rather a square bed on a triangular support...).


Translated with www.DeepL.com/Translator

Edited 2 time(s). Last edit at 04/23/2018 02:13PM by M_Xeno.

I like the idea of magnets, presumably providing a nesting force between the wheels; X, Y and θz and their associated guide surfaces. This would eliminate the need for the guide surface for the nesting force roller. Having said that, I think that the force tending to disturb the nesting force wheel from its normal is actually negligible as long as the axes of all the wheels are on a plane with the guides at a right angle to the plane and parallel to each other.

Mike

I like magnets too I'm picturing a magnetic knife block for the contactless guide surface. Like ICP though, I'm kind of rusty on magnets. If you have a permanent magnet, where the spring-loaded nesting wheel is now, oriented to push against the knife block -- does it still have a straight-line force vector? If it does, does it run into the same instability problem like the nesting wheel? Perhaps instead of a single magnet with a single field with a single maximum, there could be two, or three. If three, they could be an equilateral triangle, pointed at slight angle away from each other, such that their center, parallel to the carriage plane, is a local magnetic field minimum. That way if the bed wants to rotate, it would have to "climb" up a field (like with the angled wheel, but valid for a full circle, not just one line).

I'm still curious to try the slightly-angled nesting wheel solution, especially since it would be easy to test (print out multiple versions at multiple angles). M_Xeno's intuition strongly objected to having any force along Z except the bed weight, though, and I'm curious why our intuitions on it aren't lining up.

A while back I found a sale on a 300mm ^ 2 aluminum bed, quite thick, flat, and heavy. Thick enough to drill+tap M4 or so bolts into the side, at least. I dusted it off today for the first time in years, I think it's perfect for a test carriage. No ETA on the project, I'm getting my lathe back to normal now... priorities.

The guiding surfaces can be replaced by cylindrical rods made of rectified (magnetic) steel, on which the bandage of a ball bearing will tangent. I think this is the most economical way to get very good quality guiding, with very low drag.

So that each wheel is pressed without play on the guide, the simplest is to associate a magnet to each wheel, and to arrange so that the magnetic force is approximately normal with the surface of guiding. Maybe a ball bearing against a ring magnet? Or better, two bearings on either side of a ring-shaped magnet, but then a cylindrical bandage will have to be machined to cover the whole...

I wonder if a strong magnetic field can disturb the operating conditions of a ball bearing, if the balls are magnetic. To avoid any increase in friction forces, ceramic ball bearings can be chosen.

For the corner of the bed on which there are two angled wheels, we can also imagine a single magnet, fixed on the bed between the two wheels.

This is a minor note about the 3 constraint wheels. Although in the diagram, the wheels are cylinders, they actually need to be rounded so as to only contact the guide surfaces at one point, instead of a line.

A single point of contact, with a rounded wheel, draws a line up the guide surface, of length Z. A flat wheel will draw a plane, of area Z*(wheel thickness). Plane constraints aren't equivalent to point constraints. For one thing, it introduces tighter tolerances in manufacture. The wheel's contact plane would have to be exactly parallel to the guide surface (or through play, be able to achieve parallel). Without sufficient play, a flat wheel will ride on either one edge or the other. A rounded wheel could still provide constraint at a much bigger range of angles. Which is nice, if you're making this on home tools like drills presses and manual lathes.

Secondly and more importantly, a flat wheel would overconstrain the mechanism, by fighting any rotation around Z-theta. For instance, the X constraint needs the plane exactly r(wheel) from the axis. Otherwise, there is displacement. In event of Z rotation, the contact moves to the rim. Now, the distance isn't r(wheel), it is the hypotenuse of a triangle formed by the wheel radius and wheel thickness.

The "correct" rim rounding radius would be equal to the wheel radius. Like the wheel as a slice of the center of a sphere. That way, it maintains constant distance at any rotation.

Another way is to use ground cylindrical rods as guide surfaces, with flat wheels.

I feel like, the more I look at this mechanism to understand it, the more I need to just build it...

Edited 1 time(s). Last edit at 04/28/2018 07:05PM by golfwolf.

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Another way is to use ground cylindrical rods as guide surfaces, with flat wheels.

That's not the same.
When the flat wheel rotates around Z-axis ( center of bed ) and the cylindric rod is fixed, the contact point wouldn't be tangential to the z-axis. That's another constraint.
I hope, I made my point clear.

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o_lampe

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Another way is to use ground cylindrical rods as guide surfaces, with flat wheels.

That's not the same.
When the flat wheel rotates around Z-axis ( center of bed ) and the cylindric rod is fixed, the contact point wouldn't be tangential to the z-axis. That's another constraint.
I hope, I made my point clear.

I think I see. I'm imagining that, because of the forces from the nesting wheel, the axis of Z rotation will most likely be the cylindrical rod (or planes) with X and Y constraints. Like if the Z-theta constraint has a bumpy ride, the bed will not choose center of mass, I believe it will spin about the rod (or planes).

To explain what I mean more: A plate spinning freely in space will rotate about its center of mass, since that config has the lowest moment of inertia. But if constrained, it will rotate about the constraint. If you spin a coaster on a table, it will spin about center, but if you move your finger next to a corner, it will try to rotate around your finger. In this case, the Z-theta rotation vector will be normal to the X and Y constraint plane, and will be located where the X and Y constraint vectors meet. Which for a cylindrical rod will be the center of the cylinder. This is a bit of a moot point because, cylindrical rod precision, or plane precision, is cheap. It is more relevant if you are manufacturing the machine yourself on home tools.

Edited 1 time(s). Last edit at 04/30/2018 05:39AM by golfwolf.

I made a short demo video about how the fishing line dual roller constrains the horizontal movement. The lines were tensioned by rubberband and as long as the tilting force is lower than the rubbers tension it stays horizontal.

I almost felt like David Copperfield, when I lifted the extrusion with two fingers