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Once again, Blakbird has figured out how everything works essentially perfectly! Very impressive without building anything. The armatron animation is awesome! Just a few clarifications: Oddly enough, I didn't give so much thought to this. It's possible a blue 3L friction pin would work just as well. Perhaps it is an axle pin just because of some previous version of the design in which the axle went into an axle hole. (I don't remember exactly). One point to add about the red Armatron wheel: it is important that it is well balanced on the central axle, with the center of mass as close to the axle as possible - 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. To amplify Blakbird's final point about backlash, the crucial property is that the cumulative error is literally zero. 3000 consecutive actuations of the mechanism on the left side would result in exactly 3000 revolutions of the Scotch Yoke, and exactly 2000 of one wheel and 1000 of the other. As I tried to explain in another post, by happy mathematical accident it turns out that the (considerable) backlash inherent in the transition from a left turn to a right turn does not seriously affect the final drawing. I have put some computer generated pictures here in attempts to illustrate this. 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 of the machine 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.

Another obvious difference is that it is WAY quicker to turn an idea into physical reality (at least a prototype mechanism) in Lego than any other medium I know (even other toys like Meccano). In particular, trying out a small variation can take only seconds. It means that hands-on trial-and-error design is a far more realistic option than elsewhere. By the way, I am definitely in the inside-out camp.

Thanks for saying that! I was worried most people might be getting bored...

That's a really neat and effective design (and awesome filming as always). Although from the title I was expecting this

Very practical (but messy)! Does it always get the stone, or can it "miss"?

I am looking forward to seeing Victor's life size working fallout shelter MOC... Once I have the idea for a MOC, I usually start by trying to get the hardest / most complicated bit working. (E.g. a gear box, or an unusual mechanism.) I do lots and lots of trial and error, sometimes working on squared paper to get the layout clear. The many possible uses of modern technic pieces always surprise me. Often when I'm stuck a particular point, I'll browse other people's MOCs, or even just sort through my own pieces, until I suddenly realize "that it is the perfect piece / combination for the job".

Awesome work so far, Blakbird. 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". That said, the Geneva mechanisms ironically involve some gearing UP - in your lovely animation, you can see that the orange wheel actually moves faster than the purple one, at its moment of maximum speed. Consequently, the last few Geneva wheels provide very low torque, no matter how hard you push the first one. Therefore, all moving parts of the brain layer need to be very light, loose, and low friction. (Thus: the round rollers, and the counterbalances on the rockers). And the lower layer will need to do a lot of "signal amplification" to do its job properly... No, 2:1 gearing would not work at all. Think about the last axle - it is turning at an average speed of (maybe) 1 turn every 30 minutes. If it did so at a uniform speed, as would happen with gearing, then it would hold a rocker in the up position for maybe 5 minutes at a stretch, and furthermore the precise time interval would be very unpredictable. Throughout that time, the bottom would be receiving constant instructions to turn. We don't want that! No, there is not. The "rest position" for all the axles is with the cams at 45 degrees to the vertical. Because the cams line up with the 2 pegs of the peg wheel, whenever an axle advances from one 45 degree position to the next, it either actuates one of the rockers itself, or it advances the next axle (but not both). Again, no, there is not. (If there was, it would result in the machine moving straight ahead for a step). The effect of the last special cam is that whenever this last axle would cause the next axle to advance (if there was a next one), we instead get a right turn. (The last cam does not just provide lift 3/4 of the time - it provides 3 separate lifts, with gaps between them.) This is what causes the curve to join up and form a twin dragon. It's nothing to do with the shape of technic axles specifically (except in combination with the shape of the cams), but your second explanation is essentially correct. Good idea, I'll try to set this up at the weekend (with the real model). In the mean time, perhaps this picture will help. Each row on the right is an illustration of the cams viewed from the left side of the vehicle (with the triple cam at the end), at a point in time between steps. From each row to the next, the first axle advances 90 degrees clockwise, perhaps causing some others to advance. The little red vertical line shows that exactly one axle actuates a rocker at this step. It will be the left side rocker if the cam moved across the top, or the right side rocker if it moved across the bottom. This corresponds to a right or left turn respectively, and you can see the corresponding place in the dragon curve.

That's a fascinating creation, and very playable! Pity it's not pure Lego, but I understand the difficulty. I'm interested and surprised that rotating the propeller cowlings works so effectively for steering...

Nice!!! Thanks for doing this - it looks great! I actually started trying to do the same thing myself, but it got way too tedious trying to select the right parts in MLCad. How exactly did you go about doing it, may I ask? A couple of minor points to consider about the coloring: 1. You might want to make the 3 little linkage parts on each side that connect the top half to the bottom half either yellow or gray, or even just make these and the yellow and gray all one color - they move together and are essentially part of the same mechanism. 2. The light blue gear trains on the bottom could do with splitting up into several colors. There is the part from the Armatron that goes over the top of the wheels and into the add-subtract mechanism, and then there is the part from the add-sub to the wheels. These deserve two different colors, as they perform different functions, related by a linear transformation performed by the add-sub (comprising the two differentials). Perhaps the add-sub mechanism itself should be a third color. Quite so! I suspect your explanations will be clearer then anything I could produce, and of course I'm happy to help with any issues...

That's right, if the pen is off-center in the left-right direction. But it could also be off-center in the forward-backward direction, which would give a different effect. Yes. I'd like to see how Blakbird and others get on with the file first.

That's a great idea! It might produce some interesting effects...

Stunning! I had to really study the engine to understand what parts I was looking at...

Great work! Getting collective, cyclic, and tail rotor pitch in such a small space is impressive, and with realistic controls too.

Sorry for bumping an old thread, but this is awesome! I actually thought about doing EXACTLY this a few years ago, inspired by the real life version: but I didn't really know how to start - you guys beat me to it! I was planning to use these Znap parts: and I'd still be interested to see if that is possible...

That wheel seems to pass the test, then! Your creations always come with a sense of humor, and this one really made me laugh aloud. Great work!

For anyone who wants to try their hand at making this (perhaps Blakbird is the only one crazy enough!), one issue is that the pen needs to be shorter than typical pens, otherwise it will obstruct the Geneva mechanisms. The ones that I used are called Crayola Pipsqueaks. They are readily available in toy stores, at least in the USA. They are water soluble (useful in case of accidents) and the right size - the pen holder in the model in designed to accommodate their thickness. For the video, I cut the point of the pen a little with scissors to make it blunter, and get a thicker line.

Of course everyone is entitled to make their own judgement about the merits of a model, but I want to correct some of the factual misconceptions here. When someone concludes that generations of mathematicians must be wrong just because they "cannot see it", I can only be amused. The term "fractal" was coined by Benoit Mandelbrot in the 1970s to describe an object with non-integer ("fractional") dimension. (The concept itself has its origins centuries ealier). The Heighway dragon curve was first studied in the 1960s. It is space-filling, therefore it has Hausdorff dimension 2. However, its boundary has Hausdorff dimension 1.5236, which is not an integer. Therefore, it is fractal. (The definition of Hausdorff dimension is technical, but easy to look up - I will not give it here). (A far more famous example of the same phenomenon is planar Brownian motion, which has dimension 2, but whose boundary has dimension 4/3, as conjectured by Mandelbrot and proved by Lawler, Schramm and Werner in 1999. Werner received the Fields Medal partly in recognition of this work.) Many popular fractals have the stronger property of self-similarity, and the dragon curve is no exception. In fact, it has the even stronger property that it is the fixed point of an "iterated function system" - this means that it can be exactly decomposed into several (in this case two) smaller copies of itself. In this case the two copies are rotated by multiples of 45 degrees, and are smaller by a factor of 1 over the square root of 2, compared with the original. (The Wikipedia article on dragon curves gives full details). Any drawing or other representation of a fractal produced by human or machine is necessarily an approximation, basically because it's not possible to realize something infinitely complex in the physical world (at least within currently understood physics). Such an approximation is implicit in the word "drawing". For a fractal, typically one uses a clearly defined sequence of approximations that (provably) converge to the fractal set. In the special case of the dragon curve, there is a Lindenmayer substitution scheme that provides such a sequence of approximating curves. It can be interpreted in terms of Gray code or paper folding - again, the Wikipedia article contains a very clear description. The Pendragon model produces the 7th Lindenmayer approximation to the dragon curve (drawn with rounded rather than square corners). It does this by the Gray code method. If another Geneva mechanism were added, it would produce the 8th approximation, with twice as many steps, and so on. (More precsiely, it actually produces the 7th approximation to the "twin dragon", which is a union of two dragon curves). Apologies for the lengthy and somewhat off-topic digression. Again, the purpose was to set the record straight on the (very standard) mathematical facts.

Nothing escapes Blakbird's eagle eye, I see :) I did indeed use Tire 81.6 x 14.2 Motorcycle Z Racing Tread in the model, because I had a suspicion it might make it more accurate. This tire is quite rare and not available in LDraw. I'm pretty sure it will work fine with the knobbly one, as the size is very very close and the knobs probably have no effect, but it's possible the smooth one is slightly better! Regarding the cams, there is a reason for it, although again it is probably unnecessary. As you go along to the later Geneva mechanisms, the torque becomes less, but the need for precise actuation also becomes less, because the later mechansims advance very quickly for a small turn of the first one. Early on, you want a "sharp" cam so that the actuation happens at a predictable point; later you want a "blunt" one, so that it requires less torque to lift the rocker beam. Hence I used the 1x3 lift arms for the first two, then the "sharp end" of the cam for the middle two, then the "blunt end" for the last two. Again, I was probably being overly cautious, and it would likely work with any arrangement, but I was trying to throw everything I could at making it work well (and this was more necessary in earlier versions that had other issues).

Ldraw instructions are now available! The file is here. Obviously, it is quite a challenging build. In the process of creating it I tried to rationalize the construction in a few places, so it is not identical to the model in the video. (Hopefully it is better.) I would of course appreciate any feedback on the file. I expect there will be a few errors. Edit: I have corrected the positions of the cams in the file and the picture below.

Thank you very much for the kind words, everyone. This in particular is the highest praise I could hope for! To be honest, I am pretty amazed by the accuracy myself (even though I put a lot of effort into precisely this). I have done a few computer experiments to try to understand the "cancelling out" a bit better. Basically, the "cumulative" angle of each turn is very very precisely 90 degrees. If it did 4x10=40 left turns in sequence then it would end up pointed in the same direction as before (after 10 complete rotations). If it did the same thing with turns of 89 degrees or 91 degrees, or random angles between 89 and 91, the result would looks quite poor (as confirmed by computer simulation). However, there is another source of "error" that miraculously cancels out - this comes from backlash in the gear trains. The result of this is that if it does a sequence of left turns followed by a right turn, this right turn will be less than 90 degrees (maybe 80 degrees - it is fixed a predictable amount). Then any subsequent right turns will be 90 degrees again. It turns out that this type of error does "cancel out" - the final drawing will still join up exactly and have the correct overall shape. (I haven't quite understood exactly why). It's not possible to get the "real" infinite fractal with only a finite number of states. By adding one more Geneva mechanism, we would get the next approximation, twice as large, with 512 turns, and so on. (One or two more might indeed be possible in practice, but eventually friction would be too much, of course). By changing the arrangement of cams on the last Geneva axle, it would be possible to get an infinite pattern, but it would just be the same figure repeated over and over along a line. Thanks! Every single piece is functional here - really the only aesthetic considerations were color choices, but I'm happy with how it ended up looking. Ha ha! I'm not sure I want to devote another year or more of my life to a "completely different build"... I don't know whether this machine can easily be "reprogrammed" to do something else interesting - I'm open to suggestions... BTW, the Ldraw file is nearing completion - it will definitely need some testers...

Great questions. There are 5 Geneva mechanisms in sequence, on 6 axles. Each axle has a cam at each end. It is not sensitive to initial conditions - it can be started at any point, and it will just start at the appropriate point in the drawing (it is one closed loop). Since it is essentially doing binary counting (more precisely, Gray code), it cycles through all possible states during the loop. Of course, if you want the drawing in the center of the paper and with the desired orientation, you do need to figure out where it is starting on the loop! But it is easy to advance the Geneva mechanisms to the desire position by hand as in the video. The set-up of the Geneva mechanisms is also quite straightforward. It requires two things: (1) Each axle has the two cams at the opposite ends pointing in opposite directions (180 degrees from each other). (Except that the last axle has a "triple cam" on one end, with actuators in 3 out of the 4 positions - the cam at the other end is in the 4th position). (2) The cams are parallel with the two red pegs that activate the next Geneva mechanism. This ensures that only one cam at a time actuates the side bar. Edit: what I said here was not correct - the setup of the cams requires more than this. See the later posts in this thread for corrected information. The design of the Geneva mechanisms does involve a sneaky 45 degree offset in order to get the blocking "macaroni" pieces in the right place relative to the red pegs. This is achieved by two 1/2 pins slotting into the stud holes on the underside of a round plate (I was pleased with how that worked out). I did start with the basic idea of what it would do ahead of time, but the actual build involved dozens of iterations for every mechanism, and took several years! Every detail has to be right for something like this to work reliably...

The best technic creation ever, reviewed by the best technic reviewer! Awesomeness overload! Many thanks to Akyuky, Rebricker and Blakbird!

A masterpiece - congratulations!

Wow, that's a really interesting solution - nicely done! It's too bad the flex system didn't last - it was a real innovation. Indeed, PWND is right! Did you experiment with trying to get it a little bit faster? Not easy with such a sophisticated drive train, I realize...

I used to think I would never run out of pins, but recently I actually ran out of three types, almost simultaneously: 2L friction pins, 3L friction pins, and 2L axle pins. Interesting that all three ran out at essentially the same time. Before that I thought I would never run out of BLACK 3L pins (a far better color in my opinion). I wish I had acquired more of them back when they were cheap...