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FRANCIS MOON: My name is Francis Moon. I'm a professor at Cornell University in the School of Mechanical and Aerospace Engineering. And this is the second lecture on kinematics of machines and mechanisms. And today, we're going to review the material in elementary kinematics of mechanisms and also follow up with some demonstrations of applications in robotics. And then we'll talk about these models, these 19th century models that were made by Professor Franz Reuleaux in Berlin. And so the second part of the talk is called "Playing with Reuleaux's Toys."
So in the first lecture, we talk about the basic idea that if you're going to create a machine, you have to think about how the machine is going to move, and how you're going to design it to move in certain ways. And one of the ways in which humans have figured out to make machines move in the right way is to create a whole series of different types of mechanisms. And it's the synthesis of these mechanisms that makes the machine. And it was Franz Reuleaux, over 100 years ago, who got the idea that you could deconstruct any machine into a set of kinematic mechanisms.
So the lecture today is going to talk about a review of links and joints and robots and types of mechanisms. And we'll talk about Reuleaux's models. And we'll talk about mechanisms for engines, pumps, mathematics, and clocks. And then we'll talk about Cornell's Online Museum of Mechanisms called KMODDL.
Some of this material, as we said in the first lecture, may be found in my book, The Machines of Leonardo da Vinci and Franz Reuleaux. And it was Franz Reuleaux who said that kinematic synthesis is the most important part of mechanical engineering because with it you will be able to create the whole machine. And in the last lecture, we talked about if you're going to create a new machine, we have to have a language of machines.
And part of this language is identifying machine elements. kinematic pairs, kinematic chains, so circuits, simple mechanisms, compound mechanisms, machine modules, and then the whole system, such as a jet aircraft or a rocket. And the basic fundamentals of kinematics have to do with how you put together different rigid bodies and connect them with either revolute joints or with prismatic joints, sliding joints. And connecting all of these links with different joints then produces so many degrees of freedom. And with that degrees of freedom, then you can produce a machine to do what you want.
In these 19th century mechanisms, we basically have one input and one output. So we talk about these machines as having one degree of freedom. On the other hand, we'll see in a minute, when we talk about a robot, we want the robot to have at least five or six degrees of freedom.
The other idea we talked about was the so-called kinematic chain, namely that if we link together four of these rigid bodies and connect them with four joints, then we have a system with one degree of freedom. And this particular formula here, Grubler's criteria, the numbers of degrees of freedom is 3 times the number of links minus 1 minus 2 times the number of joints, in this case revolute joints. And in this particular one here N is 4 and R is 4. The number of joints is four. The number of links is four.
And one of the ideas that Reuleaux had was that the ground serves as one of the links. The other idea, which we talked about in the first lecture, is that we could not only ground this link, we could ground this link or this link. And so from this particular kinematic chain, we could get four different types of mechanisms.
Here, you see a model for a so-called slider crank mechanism. Here, we have three revolute joints. But here, we have a sliding joint. And of course, we would have those in an automobile engine in which this would be the piston and this would be the connecting rod.
Again, however, you could fix the piston. You could fix the connecting rod, fix the crank, and get a different mechanism. And what Reuleaux did is he thought of this as a circuit. And the C minus is the hollow cylinder. The C+ would be the piston. And so each of these joints here could be thought of as two pairs allowing it to rotate here and slide here.
The other idea is that we could deconstruct the machine into different kinematic mechanisms. And I'd like to introduce my assistant here, Jeff Lipton, who's a TA in my robotics class. And I just want to show you.
Here's a robotic manipulator arm. This would be a small robot arm that would be used in manufacturing. And you can identify a number of different mechanisms. First, you have belt drives here. These things here are DC motors plus encoders. Then you also have a-- if you could rotate this. Yes.
[MOTOR WHIRRING]
You have a gripper here. And the gripper has a parallel mechanism. It's a four-bar parallel mechanism.
So over here, this is a small gripper that was made in my laboratory a few years ago. In fact, it was made by a freshman. And we wanted to have a gripper in which the sides remained parallel.
So if you had a four-bar linkage in which the opposite links are the same length, then as you move this here, this remains parallel. And it allows you to pick up objects of different size. And this has the same-- so this has two four-bar mechanisms.
The other thing that this has is a differential joint here. The differential joint allows you to roll as well as pitch. Maybe you can actuate that. So there's a roll motion.
And then by driving these the other way, you could get a pitch motion. This particular wrist does not have any yaw. So it's really a-- so this motor here that [INAUDIBLE] move this about the vertical axis, that's called a waste motor. So it's very anthropomorphic.
This is the shoulder motor here. This is the elbow here. And this is the wrist here. So it's like the human you have a waist. You have a shoulder. You have an elbow, and then you have a wrist.
And it's very interesting that the human-- if all of you grab your wrist here, grab one of the wrists with the other hand and see how many degrees of freedom that you have. And you see that you can get pitch here, and you can get yaw, but you cannot get roll. If you release it, how do you get roll?
AUDIENCE: The elbow.
AUDIENCE: Elbow.
FRANCIS MOON: The forearm. That's right. So there are two bones in the forearm that allow you to get roll. Whereas here, we only have two degrees of freedom. So this is a five-axis machine.
And so we can pinpoint the wrist in a three-dimensional space. And we can only move the object in orientation in two degrees of freedom. So maybe you just take it through a few of its paces here,
[MOTOR WHIRRING]
Now, of course, all of these machines have a limited workspace. So in designing a robot, you have to think about what will be the space in which it's going to work. And then, you want to work away from the boundaries of that space. And that's the kind of analysis that you would have to do if you were going to design this particular machine. So thank you very much.
And we also talked about designing walking machines. And in the last lecture, we talked about that if we had eight links and 10 joints, we would have a number of degrees of freedom is one. And we could find 12 different combinations of how to create such mechanisms.
And we found that there were some. This is kinetic art from Theo Jansen. We find this on YouTube, something called Strandbeest. This is the size of a human. And each one of these here are legs that can walk.
And one of these particular combinations can be found in that category list of Grubler. This is a realization. This is a CAD model of a realization. A student named Ryan Wolfe worked with me for a senior design project, and he actually built this machine.
And each one of these here is an eight-link mechanism. And the eight-link mechanism is here. This is a crank, which is shared with two legs. So this is the leg. And one of the links has three joints here.
And we want to design the lengths of these so that we'll have a straight path here and ability to lift up and put the leg down. But we also found that you could have a different combination of eight links. And here, again, this is designed from Taiwan in which the crank moves here. And then you design this so that this is a straight part. This is where the leg would touch the ground.
And we found that we had then a model of this famous Chinese walking horse. And let me see if I can get it to walk across, instead of walking backwards there. So each one of these four legs is an 8-bar mechanism. And there is some evidence in manuscripts from China, from at least 1,000 years ago, that they were able to build some sort of carts that you could gently push on rough terrain.
And in the modern-- let's see if we have a photograph of a modern. So here's a modern computer-designed walking robot from Professor Lipson's lab at Cornell. And here, we have a realization of that. And you see, again, it has-- although this time, it has four links with two. Each one has two links. So now, we have eight links here.
And the interesting thing about this is that this is a machine that learns. So this is an example of not only using kinematics but using something called mechatronics in which we integrate not only mechanical design and electrical design, but also artificial intelligence. If this machine loses one of these legs, then the other three legs will learn how to continue to walk toward its destination. So this is from the laboratory of Professor Lipson at Cornell University.
And of course, another laboratory at Cornell is the laboratory of Professor Andy Ruina. And he has created a different type of walking robot, one that uses dynamics, as opposed to just pure kinematics. So here, each of these legs is almost like a double pendulum.
And so this is an idea that goes back to a fellow named Todd McGeer, who also worked with Professor Ruina. And they had the idea of using two pendula, sort of inverted pendula. And if we have a slight grade, then this will walk naturally.
So unlike these mechanisms here, which these mechanisms here depend on pure geometry, this particular machine depends on the laws of Newton. And you can see a little toy version of this here walking down this. And this is now a larger version.
And the robotics lab of Professor Ruina, a few years ago, working with mechanical engineering undergraduates was able to design this natural walking machine with a little bit of energy in the kicker. And they walked, I think, for about an hour around Barton Hall. They set some sort of a world record for a natural walking machine.
To get some of the balance, they have other pendula up here, as well as the double pendula legs at the bottom here. So this machine seems to have a mind of its own. And the graduate student has to give it a little kick to keep it on its way.
So this is a different type of robotic machine that depends not only on kinematics of mechanisms, but also depends on the dynamics of Newton's Law. This idea that you can create machines that walk by themselves without too much feedback is kind of radical, because this is the Honda machine, which costs about $1 million. It's filled with all kinds of artificial intelligence. But the laboratory of Professor Ruina has shown that you can make these walking machines for a few thousand dollars. Although they're limited in what they can do, they can still walk.
Now, I'd like to talk about Professor Reuleaux. He was trained at the Technical University in Karlsruhe. And one of the ideas here is that he tried to promote the idea that what was important in machines was to look at kinematic mechanisms.
And as we saw in the last lecture, kinematic mechanisms change motion promotion from one form to another. So for example, if we look at these gears here, might call them funny gears, but if you turn the bottom at constant speed, then the top is going at a non-constant speed. And you might want to use that in some sort of a manufacturing situation. So you can change your motion from circular motion to circular motion, from circular motion to linear motion and so on. And as we talked about, you can also change motion about one axis to motion about another axis.
The basic kinematic mechanisms-- and maybe this is one of the questions we'll put on the quiz next week-- is that, of course, there are hundreds and maybe thousands of different mechanisms. But some of the basic ones are the four-bar linkage, which we showed in this robotic gripper. The slider crank, which is in most engines, belt and chain drives, gear mechanisms. I just went over that. Screw mechanisms.
And one of the ones here is a universal joint. And here is a double universal joint. And you can see that this particular universe, the idea of the universal joint is that it can convert motion about one axis and convert it to another axis. So this is a universal joint. And this is a universal joint here.
It's very interesting that the application of this to one of the consumer products that you may be familiar with was invented by a Cornellian. Does anybody know where you would find a universal joint? Yes?
AUDIENCE: Drive shaft of a car.
FRANCIS MOON: In a car, yes. And the guy that did it was a guy named Clarence Spicer. And he was a sophomore here at Cornell in 1902 and got the idea. And they were driving automobiles by a chain and sprocket. And so he got the idea to get rid of the chain and sprocket and use a universal joint.
And he patented it in 1904. And then started a company called the Spicer Manufacturing Company. And now, it's called Dana Spicer, D-A- N- A.
So if you look up Spicer on your Wikipedia, you'll see Clarence Spicer. And he wasn't sitting in these seats because we were over in Sibley Hall, but he was a Cornellian and got the idea. And maybe he got the idea by looking at this model because these models were on view for the students 100 years ago. So the universal joint is a very important mechanism to understand something.
Now, to go into a little history-- and maybe the next lecture, we're going to talk about the evolution of machines-- if we go back to the time of the Greeks, the Greeks, especially the school of Aristotle, had thought about machines in a different way. They thought of the so-called simple machines-- inclined plane, the lever, the wedge, the screw, the wheel, the pulley.
And the trouble with this concept of a machine is that it mixes up both kinematics and kinetics. It mixes up motion and forces. And at this time, they didn't even have the idea of a law of motion.
And what Reuleaux did was define machines in terms of the motion that they change. And he had six classes of mechanisms. One is the crank chain, which, for example, this would be a crank. This is three crank chains here. This is three four-bar mechanisms here.
And you can see that these paddles here turn. It turns. Look, they flip over and turn. So this can be deconstructed into three four-bar linkages. And where would you use this kind of a mechanism? Yes?
AUDIENCE: You had to lift something up and incline and dump it off like in a hay bale.
FRANCIS MOON: That's possible. But one of the obvious things is maybe a riverboat. So these could be the paddles of because this goes into the water and then it feathers. But what's a modern application where you have feathering type of mechanisms? Some of the aerospace people.
AUDIENCE: Helicopter.
FRANCIS MOON: Helicopters, yes. So the same idea that you could have rotary motion, and you could feather the blades, they may not be using four-bar, but it's the same sort of idea. So this would be a crank mechanism here.
The other one was a screw chain. The other is a wheel chain. This would be a wheel chain. So this is two wheels here. The other would be a cam. The other is a ratchet. And we'll talk about the idea of a ratchet in the statements. And the other is a pulley or a belt. So he thought that those were the fundamental classes, rather than the simple machines.
Now, as we said on the first lecture, almost all of these models you'll find on the website Kinematic Models for Design Digital Library. This particular website is part of the National Science Digital Library. And if there are teachers watching this, the National Science Digital Library has all kinds of modules online for teaching science and technology. One of them is kinematics of machines at Cornell University.
The Reuleaux models were purchased about 250 models for $8,000, at that time, by Andrew Dickson White. And he began his collection at Zurich at the Swiss Federal Institute of Technology there. And then when he moved to Berlin, the government provided him funds, and he built a collection of about 800 models.
A number of people showed interest in these models. And so he had a man named Gustav Voigt from Berlin build about 300 of these models. They won medals at the St. Louis Exhibition in 1904. And they were sold to both McGill University and Cornell in North America, as well as some of the German universities.
Unfortunately, the collection at McGill was destroyed in a fire in the early 20th century. By the way, the model collections in the 19th century were equivalent to what you would have on YouTube. If you want to go see how something moves or a video, you go to YouTube. But at that time, the students had actual models they could touch. And it was considered-- if you were going to have an important engineering school or a science school, you had to have demonstration models to demonstrate the principles of Science or engineering.
The other thing about this collection is that this collection was designated a mechanical engineering heritage collection by the American Society of Mechanical Engineering. Here's a photograph of Andrew Dickson White, who was the first President of Cornell University. He was the ambassador to Berlin. So he got a chance probably to meet Reuleaux then.
And Reuleaux was the ambassador to the Philadelphia World's Fair in 1876. And so these guys were not jet setters. They were steam setters. So they were traveling around the world forming a network.
And Thurston was the director of at then was the College of Mechanical Engineering at Cornell. And he was the first President of ASME. And in 2004, this particular collection was designated as a national heritage.
And we've seen some of these models. This is the four-bar model. It's interesting that when people look at these models, they go, oh, they're old models. They're not used anymore. No, these concepts are still used today in modern machines.
For example, the four-bar linkage can be used in biomechanical knee prostheses. So if you have here a femur and the lower leg, instead of putting some artificial material here to replace the knee, you can put a four bar with 1, 2, 3, 4. And you'll get the same relative motion of the lower leg relative to the femur as you would as if you had some material in here. And this is a particular-- if you go online and you type in four-bar knee prosthesis, you'll see a whole bunch of companies actually making these types of prosthetics, all based on the four-bar mechanism.
Now, Reuleaux's contributions to kinematics-- he's sometimes called the "father" of modern kinematics. He introduced the idea of the kinematic pair. He introduced the idea of the kinematic chain. A machine is a set of these chains.
And he also tried to develop the theory of machine synthesis. So it's sort of the origins of modern design theory. And the course that you're in right now is mechanical synthesis.
And he was one of the first who thought about the important aspect of mechanical engineering is not analysis. You need analysis. You need your calculus and physics. But in the end, you're going to create new machines. And that involved synthesis-- putting together different mechanisms in the modern world. Putting together not only mechanisms but circuits, electronic chips, and software to create some modern device.
The slider crank, of course, we saw before. And it shows up. And this is another set of models that we have at Cornell, which is a cutaway of an eight-cylinder engine. And each one of these is a slider crank.
Then we have the endless screw. And here we have-- an interesting thing about the endless screw is it goes back at least to the time of Leonardo da Vinci. And the idea here is that you're turning about this axis and producing a motion about this axis. Not only that, you're getting a speed reduction.
But it's interesting that when you try to go backwards, and as an input-output device, you can turn this quite easily. But if you try to turn this, this doesn't turn. So it's almost like a mechanical diode.
On the other hand, I have another endless screw here. These models were made by the Illinois Gear Manufacturing. I think we got these about 1950. Same sort of thing. I've got an endless screw. I can turn it quite easily this way. I'm going to put it down.
But if I turn it this way, look. Why is it that this particular endless screw is a two-way device, and why this one here is a one-way device? Any clues? Yes?
AUDIENCE: Are the gears tilted at an angle?
FRANCIS MOON: Yes. So here, you have the teeth are cut at this angle here. Where as this here, the teeth are cut along the direction of the axis here. And what happens here is that the friction-- when you turn it this way, the friction is broken very easily. But here, the friction gets magnified. In this particular case here, because of the teeth being cut at this angle here, we can move this in two ways.
So this is kind of sort of a lost knowledge that you don't find in modern books. On the other hand, it does give you a clue, as if you wanted to design an endless screw that would lock, then you want to use this particular gear. On the other hand, if you want it to move freely, then you're going to have this sort of spiral teeth.
Here is the so-called planetary gear-- the sun, planet, and ring gears. And we talked about those as this type of topology is used in differential. This is another planetary and ring gear.
The other thing I wanted to say about Reuleaux is that I visited Germany. And part of his models are at the Deutsches Museum in Munich. And in the Deutsches Museum, there's an archive. And I was able to get hold of his letters and some of his writings.
But I also came across his drawings. And I thought, geez, what beautiful drawings. And he was a proponent of the so-called machine beautiful movement. Today, when we want to make a machine beautiful, we cover it with something. We give it to a so-called designer, and they don't care what's inside.
In the 19th century, though, they were fascinated with how things move. And so they created these beautiful shapes. And you see here is a pedestal. This is characteristic of Reuleaux's design. And you can see it right here in these pedestals. They all have this beautiful curve here. You can see this beautiful curve here.
And of course, it was carried to an extreme. This is a steam engine that was in the Smithsonian. And you can see here they have a Greek column here as part of the rocker arm for a steam engine. And they had also painted it because they were proud of these machines. And they thought they should look beautiful.
In a way, mechanical engineering was sort of an extension of civil engineering. And civil engineering was closely tied to architecture. And many of the machine engineers of the Renaissance were architects, artists, as well as engineers. And if they were designing a machine, it should look as nice as a building. And so you see this nice attention to the individual components.
And Reuleaux believed that if you made the object have some sort of a smooth transition from one part to the other, you actually would create a machine that would optimize the stress distribution. So he was one of the first to think about the idea of form and function. That if you design something that's efficient, it will also be beautiful.
This, of course, is a ratchet mechanism. And we talked about the-- this is a single four-bar linkage here. Let me say something about that. I hope we brought that model out. Oh, yes, here it is here.
So one of the things that Reuleaux did was think about whether two mechanisms have the same kinematic topology. And this particular is a spherical engine. When we think of engines, we think of a cylinder with a piston.
And in this particular case, this is a sphere. It's a glass sphere. This was made over 100 years ago. And you see there's a wobble plate. See this. I don't know if you can see the wobble [INAUDIBLE] come up. There's a wobble plate here and another plate here.
And so this was a spherical engine. And it was made around the 1830s. And it was used in the Houses of Parliament to get some sort of air conditioning or ventilation. It's interesting that this topology did not survive. The topology that survived for engines was the cylinder type of engine. Recently, these so-called wobble engines or spherical engines have made a comeback. And there is a company out of Cambridge that's trying to create some sort of a diesel-like engine based on this spherical engine type.
And he figured out that the topology of this spherical engine was the same as the universal joint. So there's the universal joint there. I'm going to skip the Geneva wheel and see if I can move on.
Here is a very interesting machine. This was one of the first gasoline engines. And of course, the name Otto-- if you've taken thermodynamics, you learn about the Otto cycle. But Otto was developing a sort of internal combustion engine, in contrast to a steam engine.
And Reuleaux was on the patent board, the German patent board. And he came across this application for a patent. But he knew that Otto didn't have any money. He was like an inventor working on his own.
On the other hand, Reuleaux had a friend named Langen, Eugen Langen, who had money. Who had started his own company. So he suggested the two of them get together. And they did. And they created this Otto-Langen gas engine.
The competition at that time was a French engine. And it was several French engines. And the principal one was a man named Lenoir.
And there was a Paris Exhibition in which there were several of these gas engines on display. And it turns out that Reuleaux was one of the judges. Now, he couldn't say, give the prize to my friends. And he was a consultant to this company. But he could say, how about if we have a contest to see which engine uses the least gas?
Well, they couldn't say no to that, right? So they had this contest. And the Otto-Langen engine used half the gas than the French engines. Well, the French cried foul. They must be pumping something up from the bottom of the floor.
So they said, we'll move it outside. So they moved it outside, did it again. And the Otto-Langen engine-- is one piston going up and down here-- won the Napoleonic prize, much to the dismay of Lenoir and other French engine manufacturers.
As a result, they were able to sell several thousand of these in Britain and the rest of Europe, whereas the French only sold several hundred. Now, this was important because it gave them a few years capital in which to develop the Otto four-cycle. And again, Reuleaux was behind the scenes consulting. So whenever you hear the name of somebody's name on a patent or an invention, there's always people behind the scenes who are helping or pushing forward these types of endeavors.
Now, tomorrow, the third lecture, I'll talk about the connection between Reuleaux and the engineers of the Renaissance. And in the case of the Leonardo da Vinci, you can see that here's a four-bar. Now, this is a drawing in the so-called Codex Madrid is actually a four-bar mechanism here.
And he has an endless screw. Look at that. And here, his endless screw is connected to a slider crank. What this is for I don't know, but there's a drawing. He also had a gear and pinion mechanisms. Here's another endless screw mechanism here. And I'm going to go on.
So another thing here is Reuleaux also developed a series of models for pumps and engines. This type of fuel pump here is also used in the modern fuel pump here. So is this one here.
This is a quite beautiful one here. We have one right here in which-- look at this. Isn't this beautiful? Look at this, two spirals. Two involute curves are rolling, one on the other. In other words, the water would come in here. You trap the water, and you push the water out here. These are so-called displacement pumps.
And there's dozens and dozens of these. And Reuleaux was one of the first to make a catalog and to analyze these different pumps. And these could be used as either an engine or as a pump, either way.
Now, one of the interesting things that Reuleaux worked on was so-called curves of constant width. I can stop this for a minute. So this is a so-called curved triangle, sometimes called a Reuleaux triangle. You see this as an equilateral triangle.
And if you take this point here and draw an arc there, and then take this point here and draw an arc there and so on, then if you measure the width of this, the width is the same no matter what you do. Amazing. Not only that, he says there's an infinite number of these. In other words, if I have five, seven, nine, I could do the same thing. So these are sort of curves of constant width.
And let's see-- we had a model here. Yes, so that means that you could create wheels, which are not circular. See? So this is rolling, but this is not going up and down because the width is the same. The problem is that the motion of the axes is moving. So you have to worry about this.
Now, one of the things you can do with the Reuleaux triangle is that you can create-- if you put cutters here, you can create a tool that drills a square hole. And there's a company in Pennsylvania that makes a drill that will drill a square hole based on the Reuleaux triangle.
Well, one of the applications of this type of idea is the so-called Wankel engine. Although, they don't use a square. They use a different shape.
The idea here is that I could have some sort of gas introduced here and explosion here. This could move this along. And then when it gets over here or over here, you could exhaust that gas. So you could create an engine out of this. Again, a non-cylindrical type of engine.
And Mazda-- I don't know whether they still-- I think the RX-8, which was several years ago-- the Mazda RX-8, I think, had a Wankel engine. And if you read Wankel's book, which was translated as of 1940s, he said that Reuleaux was the world's expert at that time on rotary type engines. And we already saw this. There's the Reuleaux triangle.
And the other application is to British coins. So this is a seven-sided. So here's seven-sided. This is a 50p coin. Again, if you take this vertex here and draw an arc, take this here and draw that arc and so on, this will have a constant width, all right? So that if you go in your pocket, you can feel that it's a 50p coin.
But if you put it in a slot machine, what's going to happen? If you put a curve of constant width in a slot machine, is the slot machine going to think it's any different than a circular one? Nobody's talking. All these people who say.
It's going to behave the same way in the slot machine as the circular coin, but you can feel it in your pocket. This is a 20p coin. See, it's 50. Next time you're in London, pick yourself up a 50p or a 20p coin. And then you can have a nice conversation at a pub on Reuleaux triangle and curves of constant width.
Another idea was the idea that you could use mechanisms to create various mathematical functions. This is a very famous one called Peaucellier straight-line mechanism. Again, it's eight links here. And here, you see this is the mechanism right here.
And if you move this in an arc here, this point will go up and down in a perfect straight line. And you get four links here, five, the base is six, seven, eight. This is just the driver. This is the driver here. And so this is an eight-link mechanism. And it's one of the models in the Reuleaux Collection S35.
And we have one of them here that was replicated in Professor Lipson's lab. And you see it right here. So as I turn this here, this point here goes up and down perfect straight line. You might say, what's the big deal about a straight line?
Well, if I change the length of this link here to this link here-- right now, these are supposed to be equal. If they're slightly unequal, then this point here will make a curve with an exact radius of curvature. And you say, what's the big deal? I just take a compass. Suppose I want a radius of curvature 1 kilometer. I could have this draw a radius of curvature of 1 kilometer by just changing the different ratio of these two links here.
So that was the idea that you could use mechanical linkages to replicate various mathematical curves. And that was a big deal in the 19th century. Today, we have electronic devices that will do that. But in some cases, it may still be useful to develop.
Now, I'd like to talk about-- another set of models that he had were models that showed space curves. So when something is moving, so if I'm moving my hand, if I'm moving, if I'm walking, I could attach connectors to-- I could attach some sort of an extension of my arm. And the tip of those extension could trace out a whole series of curves.
And so what he did was create a set of models in which-- here's a cone rolling on a cone. And as this cone rolls, it traces out these curves. So these are the space curves.
And you can imagine that as anything moves, there's a whole beautiful array of these space curves. And you may want to use these space curves for some particular motion by extending the body. But it does give you-- as this is rolling, you see these beautiful curves being traced out. And there's about six or seven models like this that show these different space curves.
Now, let me say something about the website here, Kinematic Models for Design Digital Library. It was a collaboration between Mechanical Engineering, myself, Professor Hod Lipson, and mathematics professor Henderson and his wife, who's a historian of mathematics, and the Engineering and University Libraries. And it was sponsored by National Science Digital Library.
And in the first couple of years, it had 200,000 visitors. But in 2009 alone, we've had near half a million visitors to this website. And it's a virtual museum and library. And not only do you have access to 400 mechanisms, but you also have various movies and animation hands-on and also the mathematics. As I just said, there is a whole literature on the mathematics of mechanisms. And if you go to the tutorials, there are tutorials written on the NK model on the mathematics of mechanism.
The other thing we have is access to rare books in the history of machines, and also the idea that you can print some of these mechanisms. So here is a picture of a pump from the book by Ramelli. This is a book that's 1588. And here's the pump here. It's worked by a waterwheel.
And this is one of the first examples of an exploded view. And you can see that there's a sliding vane here, which were used. So here's one of the models here with a sliding vane type of pump. So if you have to build a model for this-- the past year's Cornell students build a model of a pump. The sliding vane goes back to the Renaissance.
And it turns out that Cornell has an original of Ramelli's book. And students can go read the original book in the Rare Books Library. And please take advantage of this while you're here sometime to go down in the basement of the Holland Library and ask to see Ramelli's book on machines.
Now, this book has been scanned, with 50 other rare books, so you can see these books online. And all over the world, people are going to this website to look at some of these books. Now, you don't have to read Latin or German or old English or French, because they're filled with beautiful drawings of machines.
And this is another one of the pumps that's illustrated. It's a vane pump here. Here it is right here. So again, you bring water in on one side, and this goes back to the Renaissance, OK?
Yes. So finally, I'd like to say that the other thing that you can do with this website is to reproduce. So this is the slider. You can find code at the website, which you can turn into a model using a rapid prototyping machine.
And finally, I'd like to say that besides the fascination with machines as what they can do, machines can also be thought of as kinetic art. And if you look closely at some of these models, you can see the beauty in the motion of these machines.
And finally, I'll quote Chebyshev who said, "Take two kinematics. It will take you into the fourth dimension." So thank you very much for your patience.
[APPLAUSE]
This three-part lecture series was given by legendary retiring Joseph C. Ford Professor of Mechanical Engineering Francis Moon on the topic of kinetics as it relates to robotics on June 10, 2011, at Cornell University.
Professor Moon makes extensive use of the Reuleaux's models in this lecture, as well as giving an in depth review of KMODDL: Cornell's online museum of Kinematics. Reuleaux was the first to propose that you could deconstruct any machine into a set of kinematic mechanisms.
