Pen-Dragon Mechanical Fractal-Drawing Device, version 2

[Captainowie]

Blakbird: What do you use for your animations? Scripting in LDCad? LD4DStudio? Something else? Or a whole bunch of manually-created still shots stitched together?

[Blakbird]

On 7/23/2015 at 8:26 PM, BusterHaus said:

Blakbird, your animations are extremely helpful in trying to understand how this works. Are you planning on making instructions, too? Or will you just be doing a breakdown of all the mechanisms in this device?

Alexander has already done most of the work by creating and stepping the file. I'll probably add some LPub commands and export a version of instructions for my own use. If others and interested, and if Alexander is OK with it, I could post them. I don't plan to do any major reshuffling of the file so I can't promise they will be of my usual quality.

On 7/23/2015 at 9:37 PM, aeh5040 said:

One point to add about the red Armatron wheel: it is important that it is well balanced on the central axle - otherwise it might come to rest in the spot where the center of mass is lowest, and not rotate any more. The reason for the extra axle with lots of half-bushes is to adjust the weight distribution appropriately.

I hadn't thought about the balance, but as long as there is sufficient drag in the idler axle, the mechanism would rotate even if unbalanced. Of course, any friction here is also drag that bogs down the motor and creates heat and wear.

Quote

One final point about accuracy. As explained, a "turn" consists 2/3 of a rotation for one wheel and 1/3 of a rotation in the opposite direction for the other. We want this to correspond to a 90 degree rotation of the vehicle, but this depends on two things: the diameter of the two wheels, and the spacing between them. Get either wrong by even 2% and the picture will not be so good (see the link above). I managed to adjust the wheel spacing to sub-half-stud accuracy by using a technic plate turned sideways and an old style 8t gear nestled into two axle holes. (The latter trick is also used to get the casters on the bottom of the vehicle the right height). One problem with version 1 was that the constant turning creates a big sideways force on the wheels, tending to push them off their axles. Even if they don't fall off, this quickly results in lost accuracy. Hence, the two turquoise cantilevers that you see on top of the wheels play an important role.

I was actually thinking about this as I drove home from work last night. While 1 revolution of a wheel sounds like a nice round, logical way to produce a 90 degree turn, there is nothing inherent in 1 revolution that makes this so. For example, a very small wheel would only rotate the vehicle a small amount in one revolution. A 90 degree turn has an arc length of (PI*W)/2 where W is the width of the model between the wheels. The circumference of a wheel is PI*D. For one revolution to equal 90 degrees, these terms have to be equal. This means the width between the wheels needs to be exaclty 2 diameters for this to work. Let's see if it is:

In the file it is very close but not quite right. This is probably because the tires in the file are not actually the right tire and may be very slightly different diameter. So to use these knobby tires you might need to adjust the spacing slightly.

Note therefore that there is no requirement that there be 1 revolution of a wheel per cycle. It could be anything, but the spacing of the wheels would have to be adjusted accordingly. I like the simplicity of 1 rev though.

Note also that the drawing could technically be made much smaller by using smaller wheels and locating them nearer the center of the machine (easier said than done). This would also result in reduced stability.

On 7/23/2015 at 11:08 PM, aeh5040 said:

It looks as if I got myself confused about this. What I said earlier about it being insensitive to initial conditions was not correct. As you say, the initial set-up in the LDraw file does not match anything in the schematic list of diagrams, and is NOT the correct way to set it up. One possible correct initial setting is to have the cams on one side pointing towards each other in pairs. I will edit the LDraw file to reflect this. Other initial settings will produce DIFFERENT patterns, generally less interesting than the dragon, but also quite nice. E.g. the incorrect set-up in the file (all cams pointing one way on one side) gives a space-filling curve that fills a diamond shape (according to my computer program).

It all depends on what is meant by initial conditions. The BUILD is sensitive to initial conditions in that the relative position of the cams is important so that the machine is programmed correctly. However, once the machine is built starting a drawing is not sensitive to initial conditions. You can start anywhere in the curve.

Your updated file hasn't been moderated yet so I can't see it, but I'll update my version of the file once I see what to do.

On 7/24/2015 at 2:42 AM, Captainowie said:

Blakbird: What do you use for your animations? Scripting in LDCad? LD4DStudio? Something else? Or a whole bunch of manually-created still shots stitched together?

For these particular animations, I just manually rotated everything in MLCAD, did screen shots from LDView, then put them together in GIMP. This is a laborious process because LDView recenters the model every time you reload, so the frames are not aligned. I needed to manually re-align them. The animations on my Technic Fundamentals page were rendered in POV-Ray and in those cases I programmed the equations of motion to get automated output of the frames. I could have done that for the Scotch yoke (simple rotation and a sine function for the translation), but the equations for the Geneva mechanism would take a lot more derivation, and the Tomy armatron mechanism involves step functions so didn't let itself to programming. In fact, I still wasn't completely sure how it worked until I finished creating the frames.

Incidentally, deriving the equations of motion for the Ackerman steering mechanism was quite difficult!

[aeh5040]

Alexander has already done most of the work by creating and stepping the file. I'll probably add some LPub commands and export a version of instructions for my own use. If others and interested, and if Alexander is OK with it, I could post them. I don't plan to do any major reshuffling of the file so I can't promise they will be of my usual quality.

I would welcome Blakbird or others improving the Ldraw file / instructions. I basically did the minimum to make it buildable.

I was actually thinking about this as I drove home from work last night. While 1 revolution of a wheel sounds like a nice round, logical way to produce a 90 degree turn, there is nothing inherent in 1 revolution that makes this so. For example, a very small wheel would only rotate the vehicle a small amount in one revolution. A 90 degree turn has an arc length of (PI*W)/2 where W is the width of the model between the wheels. The circumference of a wheel is PI*D. For one revolution to equal 90 degrees, these terms have to be equal. This means the width between the wheels needs to be exaclty 2 diameters for this to work. Let's see if it is:

In the file it is very close but not quite right. This is probably because the tires in the file are not actually the right tire and may be very slightly different diameter. So to use these knobby tires you might need to adjust the spacing slightly.

If you compare the knobbly tires with those of your 8422, I think you'll find there is no detectable difference in diameter. However, there are some other confounding factors here:

1. The positioning of the wheels in the file may not quite match reality (and I did not try that hard).

2. The tires squish a bit when there is weight on them, changing the diameter.

3. The diameter is only an approximation to what you want - when the vehicle makes a turn, the tires interact with the floor in complicated ways involving friction and deformation of the rubber. (It's possible that THIS part is different with the smooth/knobbly tires).

I believe experiment is the only way to get the turn angle exactly right. This is what I did - I kept adjusting the wheel spacing on the real model until I got turns of 90 degrees (as measured by the cumulative effect of several consecutive turns). Then tried to find a combination of pieces to enforce that wheel spacing as best I could.

Note therefore that there is no requirement that there be 1 revolution of a wheel per cycle. It could be anything, but the spacing of the wheels would have to be adjusted accordingly. I like the simplicity of 1 rev though.

Note also that the drawing could technically be made much smaller by using smaller wheels and locating them nearer the center of the machine (easier said than done). This would also result in reduced stability.

That's right. I toyed with this idea (after the Pendragon 1) but decided it was too difficult, and besides, an add-sub mechanism makes it cooler! I like the current size of the drawing. It's big enough to look impressive, and big enough that it could not easily be drawn by hand, but not so big as to be impractical (like version 1).

It all depends on what is meant by initial conditions. The BUILD is sensitive to initial conditions in that the relative position of the cams is important so that the machine is programmed correctly. However, once the machine is built starting a drawing is not sensitive to initial conditions. You can start anywhere in the curve.

Quite right. Thanks for the clarification.

Your updated file hasn't been moderated yet so I can't see it, but I'll update my version of the file once I see what to do.

The earlier post contains a deep link. But brickshelf seems a bit unreliable these days, so here it is in bricksafe too.

[aeh5040]

This is all totally amazing. The idea, the mathematics, the implementation and development, the programming, calculation conditioning (this stuff). This all seem to be a pretty good thesis work material. About how many workhours went into this?

I'm not sure I want to try calculating that! Building time maybe half a day per week for a year? But it's very hard to estimate the amount of thinking time...

(These threads lately make me think that I picked a wrong profession, and this mechanical engineering is just not my thing. It's a bad feeling to feel that I suck at something I love and picked as a profession)

Surely nonsense I don't know anything about your job, but your Lego creations are simply stunning! If Lego building was my job I would probably be fired - it takes me forever to get anything finished, and many aspects of it I approach in inefficient and illogical ways (e.g. I'd never heard of condition number, although the basic idea sounds easy and natural).

Edited July 25, 2015 by aeh5040

[aeh5040]

I think it would be quite interesting to watch an animation or video of the top assembly alone running at high speed, perhaps with the 40 tooth gear as direct input. There is a snippet of this in the main video, but it is not long enough to see the motion drive all the cams. Sadly, such an animation is more complicated than I have the time to simulate manually.

Here ya go!

[Blakbird]

Blakbird said:

I think it would be quite interesting to watch an animation or video of the top assembly alone running at high speed, perhaps with the 40 tooth gear as direct input.

On 7/25/2015 at 4:46 PM, aeh5040 said:

Here ya go!

I was right. It was quite interesting.

I've completed a draft version of PDF instructions. This is what I'll be using to build my copy. I can report back if I find any problems with them. I'm going to build it with a slightly different color scheme so we don't get them mixed up.

[Philo]

Mmhhh.... I've seen finished BIs that looked worse than this draft ;)

Alexander asked for the smooth tire on LDraw forum, here it is: http://www.ldraw.org/cgi-bin/ptdetail.cgi?f=parts/6596.dat

[aeh5040]

Mmhhh.... I've seen finished BIs that looked worse than this draft ;)

Alexander asked for the smooth tire on LDraw forum, here it is: http://www.ldraw.org...=parts/6596.dat

Wow, thanks Philo, that was FAST! And great work on the PDF, Blackbird. I really need to learn LPub...

[Blakbird]

Mmhhh.... I've seen finished BIs that looked worse than this draft ;)

Alexander asked for the smooth tire on LDraw forum, here it is: http://www.ldraw.org...=parts/6596.dat

I was going to ask for that part, but I felt guilty asking for something so rare. I'm glad it got done anyway! I'll make the updates to my file.

[aeh5040]

Actually, the Geneva mechanisms move quite easily, so there is not so much need for torque here. (The locomotion requirements of the lower module are what prove to be the limiting factor in terms of torque). Much more important here is that the Geneva wheels turn slowly enough that the lower module has time to do a complete operation between "instructions" from the upper "brain".

I had wondered about the timing aspect. So the brain needs to run slowly just so it doesn't issue commands faster than the mechanical part can execute them. And the Geneva wheels are necessary so that the commands are discreet and never overlap eachother.

Actually, there is even more to be said on this point. It is equally important that the upper "brain" does not operate too SLOWLY relative to the lower "drive module". If it did, the drive module could complete its turn WHILE a rocker arm was lifted by one cam. Then it would receive a second turn instruction from the same lifting of the rocker arm, which of course is no good. So the relative speeds of the two halves need to be just right. The time for the drive module to complete a turn needs to be longer than the time that the rocker arm is lifted by a cam, but less than that time plus the "down time" between one rocker actuation and the next. And of course, all these things are not completely deterministic, but subject to some variation, so extra safety margins need to be built in.

In fact, this issue is only really important for the earlier cams in the sequence, especially the first one. The later ones move very quickly from one position to the next, as a result of the backlash in the Geneva mechanisms combined with the weight of the rockers acting as a detente mechanism. All this is part of the reason for choosing "sharp" cams (1x3 liftarms) for the earlier Geneva axles. (But "blunt" ones later because of the low torque available).

This issue caused a lot of trouble in earlier versions of the machine, when the drive modules were not 100% reliable.

Also, I've made a slightly nicer (color coded) version of the "road map" here.

Edited July 30, 2015 by aeh5040

[Blakbird]

On Tuesday night I finished building the lower assembly. I had lots of fun commanding either a left or right turn by flicking the trigger for the armatron mechanism. I did find that during a switch from left to right you don't get 90 degrees due to lost motion in the system, but as explained earlier all this error later cancels out.

Last night I built the brain but didn't have time to "program" it with the proper relative sequence. Today I finished the model and tried it out and .... it didn't work! It kept commanding both turns at the same time and often the armatron mechanisms wouldn't stop at all. I screamed "Alexander" as I shook my fist at the sky. Then it finally occurred to me that I might have done something wrong. It turns out I used 6L beams instead of 7L beams to connect the upper part to the lower part which resulted in nearly continuous commands being generated. I fixed the problem and now it works great. You can watch this thing mesmerized for a long time.

I was surprised by how much power a turn takes, but it makes sense when skid steering with sticky rubber tires.

[aeh5040]

Well, no wonder your motor is working hard with the additional weight of an orange 6L thin liftarm AS WELL as a 7L black one, and a 3/4 pin to connect them

Great work - the orange looks awesome! Indeed, there is not that much torque to spare when it comes to making a turn. Of course, all the mechanisms do have quite a bit of friction (the worms, the differentials, the armatron). I did find that things improved after a careful going over all the mechanisms looking for slightly bent axles and rough gears (in particular I find that the double bevels sometimes have rough spots on the sides).

One more technical point to record about this model (if anyone cares!) The 3x3-quarter-circle-liftarm catch that releases the armatron mechanism in response to a command from the brain required quite a few designs (even though the final solution is very simple). It is really crucial that the liftarms have a perfectly circular shape (and this is the first time I have ever used this fact in a model). That way, there is no possibility of the armatron forcing the catch out of the way when there is no command from the brain, yet it requires relatively little force to drag the catch out of the way to give a command (you only have to overcome the friction).

[Blakbird]

I've made a slightly updated version of the PDF instructions and posted them here. I replaced the tires with the correct version, added a cover page, and added the "road map". Having built this model from the instructions, there were a couple of things that were not ideal about the build order or step view but it was good enough to build the model which is all I was going for. Enjoy!

[Blakbird]

Last night I finally managed to get my machine to draw the entire curve. This was made possible by the map which allows me to know where in the curve I am starting. The picture below shows the results.

As you can see, it didn't quite work. There was a significant gap between the starting and ending points much larger than what Alexander got with his machine. If you look at the predictions of what will happen if the angles are slightly off, you can see that my curve resembles the result with 88 degree turns. This implies my wheels are too far apart. I tried pushing them inward and drawing again (the orange curve), but it is still not right.

So how far off am I? In a previous post I said:

On 7/24/2015 at 10:21 AM, Blakbird said:

While 1 revolution of a wheel sounds like a nice round, logical way to produce a 90 degree turn, there is nothing inherent in 1 revolution that makes this so. For example, a very small wheel would only rotate the vehicle a small amount in one revolution. A 90 degree turn has an arc length of (PI*W)/2 where W is the width of the model between the wheels. The circumference of a wheel is PI*D. For one revolution to equal 90 degrees, these terms have to be equal. This means the width between the wheels needs to be exaclty 2 diameters for this to work.

According to LDraw, the outer diameter of the tire is 204 LDU which is about 81.6 mm. (This isn't quite right because the tire deforms, but it is good enough for comparison). This means the spacing between the tires should be twice this or 163.2 mm. If we solve for the spacing for an arbitrary turn angle, we get W=PI*D/theta. When theta is 90 degrees (PI/2 radians), then things cancel out and we get 2D. But what if I use 88 degrees? Then the spacing would be 166.9 mm. This implies my spacing is wide by about 3.7 mm which is not a trivial amount. Alexander was very careful to impose a certain spacing, so this result surprises me. Why is my model off? It is possible that it is my paper. I am using a relatively "sticky" paper which has some texture. This provides a lot of traction for the model which may change the pivot point in unexpected ways. Tonight I will swap some parts to change the spacing and see what kind of difference it makes.

[aeh5040]

Hm - I am also a bit surprised that Blakbird's version didn't quite join up right, and I don't have a good explanation at the moment.

As I mentioned, I fine-tuned the wheel spacing by trail and error. After doing a rough version of your calculation, I just got it to do a bunch of left turns in succession, and adjusted until the angle was right. This test was done on a wooden floor with no paper and no pen, whereas the final run was done on fairly rough paper, so I'd be surprised if the paper is responsible for the difference (but it's possible).

It might help to start in the center of the curve rather than the corner. I did this partly because I wasn't sure of the exact size and orientation, to maximize the chances of it staying on the paper.

Many thanks for the instructions! I am aware of a few issues, and I'm keen to make them perfect. I have already started making a few improvements, both in the stepping and, in one case, to the actual build, to make it easier to put together. I would like to know exactly what issues you had - I'll PM you...

One thing I found is that uneven floor can have a big effect. It did not work so well on a hardwood floor, which has some imperfections. The video is from a tiled kitchen floor (with thick paper), which is also not perfect, but flatter than the wood floor. It looks like you are using a wooden table, which ought to be very flat. It also looks as if your wheels and casters have gone of the edge of the paper at times. That caused alot of trouble for me, because the casters can get stuck under the edge of the paper. It's possible that caused some inaccruacy.

[Technic Jim]

I have just read this entire thread and headed over to aeh5040's YouTube channel and I am completely astounded. This machine especially really shows off what is possible with lego technic and having Blakbird around to explain it means that I now have a good understanding of what is going on which makes me even more amazed. This machine is amazing and so are all the rest on aeh5040's channel. I've said it already and I'll say it again, this machine is amazing.

[Blakbird]

One thing I found is that uneven floor can have a big effect. It did not work so well on a hardwood floor, which has some imperfections. The video is from a tiled kitchen floor (with thick paper), which is also not perfect, but flatter than the wood floor. It looks like you are using a wooden table, which ought to be very flat. It also looks as if your wheels and casters have gone of the edge of the paper at times. That caused alot of trouble for me, because the casters can get stuck under the edge of the paper. It's possible that caused some inaccruacy.

I taped the paper to the table to keep it flat and watched closely when the wheels were over the edge. I didn't see any problem.

I removed the 2x4 Technic plates and old 8 tooth gears that you were using for wheel spacers and replaced them with standard 1/2 bushings. This only makes about a millimeter of difference on each side, but what a difference a millimeter makes!

The curve is much closer to joining properly now, however you can see that the wheels are still slightly too far apart. Sadly, there is no way to move them any closer together because they are already right against the main frame. I observed the model very closely while it drove and I noticed that the wheels pry apart noticeably at the bottom which effectively adds more space between the contact patches. I'm not sure how to counteract this. Ideally the tires would have a somewhat lower friction coefficient.

[Philo]

Maybe you should try the regular motorcycle tires? Different rubber / stiffness / diameter might help (or not!!!)

[Blakbird]

Maybe you should try the regular motorcycle tires? Different rubber / stiffness / diameter might help (or not!!!)

I suppose it couldn't hurt to try! I even thought of stuffing the inside of the tire with something to prevent it from squishing and effectively increasing the diameter.

[BachAddict]

3D print either solid wheels, or solid rims that fill and stretch the tires!

[Technic Jim]

3D print either solid wheels, or solid rims that fill and stretch the tires!

It sounds like it could work...

[aeh5040]

Many thanks for the further data, Blakbird. I have been away for a while, so have not been able to experiment further. I suspect that the "prying apart" of the wheels may well be the major reason for the difference between your picture and mine. I am using relatively rough "art paper", and you say your paper is quite "sticky". Sticky paper could easily result in more prying apart.

It would certainly by worth a try going back to the knobbly motorcycle tires. The "prying" and "squishing" effects might be less. The knobbly tires are very slightly thicker (in the direction along the axle), so with the half-bush spacing there is a bit of a danger that they might rub on the structure. I need to check this.

Another potential solution would be to use a much bigger wheel such as 88516, and mount it much further out (more than a stud further in this case). I'm less keen on this as it would require major redesign and might cause other problems.

I would also like to see how much of the inaccuracy can be "hidden" by staring from the center of the curve, as I did. (I may try a simulation for that).

Again, I want to find the best possible set-up, so that it can be incorporated into the final instructions!