 locomotion of animals and robots and that's the title in non-living living and non-living locomoting systems And I'll also mention something about dimensionality reduction, which allows us to actually make use of this stuff Okay, so my group. Oh Yes, and by the way, it also has relationship to these sort of interesting Phenomena including zero angle momentum turns and cats how you can parallel park your car And why a Foucault pendulum? Processes as the day goes round the folks on this slide or the folks who've done the work and one of them is in fact in The room Nick gravish who was a PhD student of mine whose talk I missed this morning, but I'm sure it was fantastic And it's a mixture of physicists biologists engineers control theorists and I'll try to highlight various people as I move along Okay, so the previous talk was a perfect setup for my talk because I'm my group is fascinated by questions of locomotion. We haven't studied the kind of hydrodynamic locomotion as typified by this beautiful eel swimming And again as we just saw in the previous talk In principle, this seems simple you wiggle your body and you push against the fluid Pushes against you and you propel yourself forward. Of course, there's nothing simple about this And these animals do it with capabilities far exceeding our current robots even our current swimming robots We also could talk about aerodynamic self propulsion I'm not sure maybe Nick talked about some aerodynamic this morning No, but I like showing this video one because I get to use the word aerodynamic for what's coming next But to I in my at the time four-year-old daughter captured this video on my smartphone One day in Atlanta. That's a carpenter B They're an infestation in certain times the year and it's a video at 200 frames a second And if you had told me ten years ago that I could go out in the field and take a high-speed video I just blew my mind anyway, you could do all this incredibly cool biomechanics Which is my kind of core interest Now for essentially almost zero dollars the problems that are of most interest and fascination of my group though Are what we now call? Teradynamic self-propulsion problems Tara aero hydro Tara Movement on terra firma or not so firma and this is typified I like showing this video some of you might have seen this video It gives you a sort of sense of not only the mechanics that organisms that live in seemingly terra firma like sand and dry deserts I hope it's sort of a sense of the ecology as well, and I'll just let this video play So here's a sidewinder My group has studied sidewinders along with David who? To some extent and this sidewinder in this case is actually trying to go after some prey And the prey is a little lizard and a little lizard Manages to use the feature of this granular material this collection of particles By going from a solid where it can run on the surface and turning it into a fluid and squirting beneath The lizard is really fast the snake can also use the properties of the granular material and a rather beautiful Dynamics by which it shovels itself in in a way. We still don't understand by the way, and we've tried to understand this Just putting its head underneath poking through a Little bit and then doing a little trick where it puts its tail up in out of the sand why you might ask Well the claim is it little invertebrates which also has inhabit dry deserts. I think these things are Plant matter so the ant goes for the tail the lizard goes for the ant and the snake goes for the lizard And so you get a circle of life and death anyway So there's a whole richness in these seemingly very boring barren Environments which are somewhere between fluid and solids. Okay, so I because I've got an hour I figured and because this is a broad workshop. I figured I would give you my kind of Five minutes of philosophy on how I as a physicist who's moved into biomechanics for the last 15 years Kind of think of these problems and why I think they're actually physics problems for the 21st century so motion Played a very important role in early physics and biology. No one would doubt that and I like showing this picture when I'm in Italy, of course Galileo of course taught us and Newton followed that motion is pretty simple motion happens when nothing is to prevent it and continues forever and ever and ever and Can be objects can be changed from their state of motion by external forces like gravitating bodies And in fact the reason I call it de motu is because after he died he published a book de motu on to quarry now the other person who's Not as well-known at least a physicist certainly to some physiologist is a guy named Borelli Who was a contemporary more or less of Galileo? He wrote a book called de motu anomalium and It turns out that He wrote he was studying the far more complicated Problems in motion than what Galileo was studying the little lizard fish and B that you saw in the previous slides are not being pulled across the ground by a mysterious force They are generating shape changes in their body which are appropriately coupling to Environment which is allowing them to move from a to b. It's a subtle shift But it's one that causes tremendous difficulty in trying to develop a framework Similar to the framework that's been developed to understand planetary body Dynamics of course the center of mass is due to internal forces contracting muscles as the previous speaker Nicely showed or contracting motors in the case of robots Okay, well, it's a simple. Oh by the way instead of calling things local motion I'm gonna call them self propulsion when things are moving from a to b through the world That's a technical point, but it'll become clear why I'm doing that in a minute How easy is it to predict self-propulsion? well, here's a video from a Former postdoc of mine who did this work when he was a PhD student at Brown University and It's a frog jumping and you might say well if I want to understand how far a frog jump size I simply measure the ground reaction forces and I can integrate and that should tell me how the Center of mass remove a la Galileo However, what you really may want to do is you may way want to be able you may want to decompose this frog And the all its constituent parts you may want to reduce this thing into the Fundamental parts that make it up and then you might want to they sickly go and say well How does the nervous system couple to the musculoskeletal system ultimately coupling to the ground to create a jump of? 50 20 centimeters however much it is the question is can we do this can we predict such a simple? dynamics from such a reduction How well the counterpoint to that how well do the parts have to work? Together for useful movement to emerge. This is in fact a question Which is sort of driving a whole lot of our current research, but I'm not going to talk to you about today Well, we have a good example and because this is a Conference on robots and animals I figured I would show you some of the best robots in the world And just how well we can do by such a reduction and the answer is pretty poorly So this is from a few years back, and I'm cherry-picking this most of you have seen this This was a DARPA robotics challenge and the idea was to create robots that could work along with first responders She able to go into nasty environments walk through open doors Go in hazardous environments the cool thing the thing I like about this the robots have gotten a little better in the last Few years you've probably seen the boss dynamics videos. Don't believe everything you see. I know those guys Well, but they are better now for certain reasons But not that much better and in fact this is the blooper reel of failures But if I showed you the reel of successes You would laugh just as hard because the successes and successes still are of not a lifelike quality There's not a fluidity in a robustness And the thing that I find so vil is awfully interesting in this and again to this reductionist idea Is that the engineers are really good engineers who make these things millions of dollars were spent? Some bite individual groups they know every part every part is specced to beautiful tolerance they know every line of code and These problems of opening doors are challenging tricky etc etc and okay, so you might say well That's unfair those are humanoid robots that have to balance and open doors What about a little closer to down to earth? Well, here's my friend and colleagues robot It's he would claim the premier snake robot in the world Why do you want a snake robot? You may want to go and ask the environments like caves and Disaster sites and squirms through and rubble and look for people and this is a robot Which is supposed to be a snake trying to get through an environment Which basically most snakes on the planet inhabit which is kind of grassy leaf litter stuff and failing miserably Why well again? We know all the motors. There's even some sensing in there There's feedback that can happen the principles aren't there And finally here's one where you might say well Let's at least put legs on the thing so that we have some idea reduce the degrees of freedom a little bit from 24 to You know six two times three And here we run into problems because we don't have a good understanding largely of the kind of interaction physics in the Mojave Desert In the US for example when the robots legs are gonna slip and slide and so this is a real it's a real mess And I think that there's a new approach needed That's what I wanted to tell you But first I want to say what can robots teach us about living systems I figured a little bit of philosophy was warranted in a talk like this and this is one of my favorite quotes And the more I read it over the years the better I like it. It's from a guy named Pierce John Pierce who was kind of one of the early I don't know about founders But one of the early proselytizers of information theory of contemporary shanty and he says I will ever maintain that we can learn at least two Things from the history of science one is these is that many of the most general and powerful discoveries of science have risen Not through the study of phenomena as they occur in nature But rather through the study of phenomenon man-made devices in products of technology if you will this is because the phenomenon man's machines Humans machines are simplified in order in comparison with those occurring naturally and is these simplified phenomena that man understands most easily Thus the existence of the steam engine in which the phenomenon of all the heat pressure vaporization Kind of tension occurred a simple and orderly fashion gave tremendous input as to the very powerful in general science of thermodynamics We see this especially in the work of Carnot and I like this part our knowledge of aerodynamics and hydronomics exists chiefly because Airplanes and ships exist not because of the existence of birds and fishes now. That's what he thought in the 50s I think that's actually probably changed in some of the work that Nick and my colleague Simon Spanberger doing to try to understand Flapping flight actually probably pushing some new hydronomics, but that was the idea Of course Norbert Wiener in the 30s 40s 50s was thinking about these very same things It's just that I think that the field of cybernetics was essentially largely theory driven and the technology wasn't there yet to really begin to look for common principles and control of communication of animal machine and Finally, I am a physicist So I'm obligated to put a quote from Feynman up when I give a talk and that quote will be what I can't not create I can't understand and I think that you'll see some of the ways we're making robots really Forced us to understand things. We didn't think we needed to even understand So in fact, this is what we sort of think this will be my last little bit of philosophy here We think there's really a need for kind of a physics of robotics as Massimo once told me boy It's sort of like you're given robotics the physics treatment It turns out that studying the emergent Aspects of robot and animal locomotives emerge in the sense that you know I've got a lot of parts and out of these things organized to go from A to B as the previous speaker pointed out It can be a real challenge in organisms and it turns out it can be just as much of a challenge in robots, which I've learned Robots have been expensive hard to make flexible hard-n-edge sensors focused been on control theory Most robotics are really demonstrations of control principles and very mathematical control principles And for demos YouTube videos in this day and age Animals as this previous speaker point out are often uncooperative for those of us who work with animals That's just part of the game hard to control and they're often too good This snake whoops, which is not swimming very well across the grass, but you would you might have seen it go through Scoots across the grass almost effortlessly and it seems almost like a non problem if you stared it long enough just Limit the capabilities in the organismal world unlike folks like a Ravi who can work on Work on systems where they can exquisitely explore detail after detail and the organismal world of fish of Snakes things the kind of things that I've been interested in we just don't have good ways to record and bulk muscle on neural activity 3d kinematics dynamics reaction forces blah blah blah However, there's been kind of in my opinion a revolution and the previous speaker showed some of this and that in the last 10 years To make robots has become actually possible for a physicist in the laboratory Which it wasn't the case 15 years ago if I wanted to study a robot 15 years ago And I started to go to my engineering colleagues and bow down and get a robot Now this robot was made by a postdoc in my group a biology PhD postdoc Who came and wanted to learn something about snakes and how to make robot models of snakes and it took them a little While but with 3d printing and with increasingly smart controllable Powerful servo motors and sensors and controllers and microcontrollers you can make really cool creative devices Which begin to model? Physically certain aspects of the living systems You can also use these things discover new cool non biological dynamics I'm not going to talk about that, but if you're interested we have a review paper, and there's a couple more since then okay They've basically become an infestation in my lab I get and George. I'm at Georgia Tech So we get a lot of great engineering students every kid in high school and then who comes to Georgia Tech in the US now builds robots and Wants to build robots, and they're just much better than the grad. They're much better than the faculty certainly Better than often the graduate students, and it's so maybe like PCs were 40 50 years ago So we have now a dull diversity of menagerie. We have robots that help us try to understand things about Lunar rovers we have robots that swim and sands on which I'll show you some bipedal robots some robots I have a whole program in trying to mimic certain aspects of modern physics including quantum mechanics and general relativity Not gonna tell you about all these things, but just it's a blast My real interest though, and this is where we begin the meat of the talk is in Comparative principles of organisms moving care dynamically and largely focused on dry grinding material And here's where I want to kind of tell you one of the surprises that we found and and how it's brought us into a Really interesting branch of mathematics, which I barely understand But hopefully there's people in this audience who are experts and can begin to help us The touchstone for this one the keystone for this one is a little lizard called the sandfish lizard Which we've spent about 12 years now study It's a lizard that does this it's similar to the one you saw in that previous video of the sidewinder going across the The the desert and the nana desert it buries into the sand and basically a second or two and when it's on the surface it essentially walks like a Lizard should using kind of an alternating limbs to propel itself But when it decides to move into the granule material to use the fluid like aspects of the granule material It does an interesting kind of contortion of its body in concert with limbs to propel itself down into the ground What's going on? Well, nobody knew until 10 years ago 12 years ago We basically hooked up dental x-ray equipment with a high-speed camera attached to the output phosphor of an image intensifier and shoot x-rays And these two excellent students who one is wearing his lead vest to protect him Basically did the first experiments to actually visualize what's going on beneath the granule material And here's what it looks like. Here's one of our first videos This is just a barrier which doesn't descend into the material so that it sort of tricks the animal to diving at that spot But you see it essentially turns itself into what at the time we said kind of looks like an eel That's an important Statement by the way But it's a real time two body lengths a second through the granule material and it's not just going horizontally It's going down about 20 or 30 degrees. Oh, so this is a plastic barrier Which basically is right on top of the sand and doesn't descend in so the animal thinks it's going to run into something And dives more or less repeatedly this animal is nice because it's pretty stereotyped in what it does Which is the trick to finding good animal models in these systems? Okay, it goes in about 23 degree these grinding materials are interesting I'm not going to tell you much about grinding materials But basically as I go deeper in the grinding material the forces increased linearly unlike in a fluid The pressure increases in a fluid, but the force on me doesn't so it's really tough to go deeper and deeper and deeper Okay to get a better sense of what's going on. You can bond little markers to the animal And here these are kind of lead markers which are easy to fall away It's just they have bonded to their skin with super glue And you can see that the animal essentially sends a wave of Body undulation a wave of body curvature not unlike we saw previous talk down the body from head to tail and these That's a little bright in this room But these little markers are attached to limbs and you can sort of see you know Maybe could we turn off a light possibly you can sort of see that the limbs are kind of held to the side and the animals Essentially I hypothesize Essential better driving its movement using its body insulation kind of swimming like a fish. It's a sandfish Okay, how do we understand this? Well, here's a slide which summarizes the sort of ten years of work and Nick Graves contributed to good chunks of this. We basically mimic the desert Well, it's loud. Sorry We mimic the desert in the lab by making air fluidized beds which blow air up through a dry granule material at above a critical flow rate The granule material becomes a fluid turn off the air. It settles into a loosely packed state Creating a nice repeatable thing. Let me turn that off. We of course from those Images with some sophisticated and increasingly sophisticated tracking you can extract the midline of the animals a function of time And here's a plot which shows from blue to red time increasing and you see that when the animals walking on the surface It's basically a straight back and when it starts to bury it starts to wiggle its midline And then when it's underneath its midline is basically looking sort of sinuous And in fact to a good approximation every instant in time the animal essentially is to an approximation a single period sine wave Which is traveling head to tail with an amplitude and a wavelength lambda Okay, and it turns out that that amplitude to wavelength is always about 0.2 You know 0.2 and 0.3 independent of conditions particle size compaction of the material how deep it goes in the gun We think it's controlling for this sort of shape Well one of the tricks we haven't had PDEs like we have for Navier stokes So the next best thing in a granular system is that you basically take a box a virtual box you fill it with virtual sand In this case three millimeter glass spheres, which you validated the contact interaction to spate of repulsive contact forces in Against laboratory experiment nicked at a lot of that You then make a little virtual sandfish, which is essentially and this is going to come up a lot in the talk today The control model is really stupidly simple. You say I'm going to prescribe There's you can't see them, but there's sort of 50 virtual motors on this thing I'm going to prescribe the angle of successive motors as a function of time to be sinusoidal And I'm just going to phase delay them down the body So I'm going to end up with a traveling wave from head to tail Okay, and the controller is going to be really good. So it always maintains that shape no sensing well sensing in the sense that That that the Simulation says no matter what the force is on me. I'm going to be able to prescribe my shape Okay, the cool thing about this video is that I've colored particles blue where they're not moving much from instant to instant And more reddish where they are and what you see is it this thing is sort of swimming in a fluidized granular material But it fluidized locally around it. So it's kind of swimming through a puddle of granular fluid and the forces in this fluid are dominated by friction among grains and friction between particles and Thus we call them frictional fluids. That's the word The other important feature of this is if I turned off the motors virtual motors in this simulation The robot virtual robot stops on a dime stops almost instantaneously Not like the eel in the previous slide where there could be some gliding and coasting Okay Finally if we want to understand the kind of forces presented to the organism We have to didn't have stokes law don't have PDEs So we essentially spent a lot of time taking robot arms attaching little objects to them and dragging them in different Orientations directions and depths through the granular material You put all that stuff together and sort of a miracle happens and the miracle is you can Read it in here But it's basically taking a very old theory the first sort of theory of hydronomics of small organisms Which we guessed and I'll explain more in a minute and Porting it to granular materials the theory is very simple. It's called resistive force theory. You say I've got a body I'm going to prescribe a shape as a function of time each little element on that shape is going to Instantaneously as it's translated the material experience thrust and drag You decompose that going to experience a reaction force you decompose that into a thrust which moves this way and a drag Which moves this way? You sum up thrust minus drag you input a shape that you prescribe and you solve for The velocity of the center of mass of that shape changing shape which satisfies force balance at all times Because there's no inertia. Okay, that's the scheme turns out It's not a great scheme and fluids because the interaction effects from an element over here an element over here are strong and the granular systems It turns out it's great Bottom line is if you do that and then you put in assumptions like a single period traveling sine wave No inertia the force always bounce and that the resistive force is linearly superpose And ask me about that later You can essentially use this theory to calculate how far the organism should travel as a function of Per undulation cycle as some metric as a function of this amplitude to wavelength parameter And you'll note that the animal is the red points are here its speed and the range of amplitude and wavelengths It uses the gray lines kind of give you a sense of the spread and you'll note that the theory basically nails Why the animals moving the way it does this is the speed the green is the mechanical cost to transport How much energy it takes to go a meter? Yeah That's related to it. Yeah, let me I can I can give you a hand wave a quick hand waving explanation there It's the competition between increasing thrust as you increase your amplitude and decreasing progress in the world frame and a DEM MD simulation set of videos highlights I think that pretty well if you're just wiggling with a low amplitude you don't get much thrusting surface this way None of your surfaces are particularly pushing well If you're thrusting if you have a high amplitude they're thrusting pushing quite well It's just that you're basically spending all your time going up and down in the world frame It turns out that you can write this quantity as a function of those two things one goes up the other goes down They cross at some point. This is generic. I think to all sorts of swimmers Okay This is for us in this organismal biomechanics role, but we call a control template them a Pattern of behavior that we believe the organism is organizing its internal degrees of freedom to target to generate some good Behavior swim fast escape from us use low energy, etc By the way, I just like to highlight this because it turns out that this miracle of granular resistive force theory Which I'll talk a little bit more is now in some sense been explained by a colleague and collaborative ours at MIT I got him Ken Cameron who's now shown actually that all these granular These force relations that we've measured painstaking over 12 years all pop out of PDE's which are frictional plasticity PDE's hyperbolic PDE's all very local and its paper in 2016 basically Recapitulates everything we measured to you know a few percent. Anyway, but but the theorists have something Okay, this stuff is good. Let me just quickly show you another little animal just to amuse you Here's a cool snake that lives in the desert Southwest Which we've spent a lot of time studying if you saw any of our snake diffraction work from this year This is the same animal. It's a shovel. No snake. It does that I still don't really know how it does that but you can turn on the x-ray and you could see what it's doing beneath the ground And low and bold it's basically able to swim Through it turns out if you track it well You'll note that its midline tracks are non overlapping in the same way the sandfish was not quite as as Dramatic but there's still what that indicates is there's still always a little bit of slip of the animal It's not moving in a perfect tube So we said well, we'll try our FT again and here I bring this up just to show you another cool plot Here's the same displacement body length per cycle now plotted not as a function of A over lambda, but it's a sort of non-dimensional curvature So as the this number increases capital and they'll see this the rest of talk Going from zero to ten the body becomes more curvy and you see a similar story You go up when you come down and the snake fits here and the sandfish sits here It's pretty good nails the nails the kind of curvatures They're using and there's a whole story about why it you become long and skinny, which I'm not going to tell you And what that does for you other to say that that this theory didn't work so well until we Actually put in a kind of fudge factor was that the skin friction was wrong And we had measured the skin friction on grains for this animal and you put that in the theory in the theory didn't agree Well for this animal and then we had the idea. Well, we should go and measure the skin friction There was a prediction that the snake has two times lower body grain friction the sandfish and inspired by my colleague and friend Professor who we did the old snake anesthetized snake sliding on a on a board filled with grains and lo and behold When you measure that you basically get a factor of two and belly scales. Okay, so theory is pretty good We've applied it to robots. Here's a robot, which we won best paper award for at a robot conference about almost eight years ago it's a multi-segment robot that can swim through granular material and From x-ray and DEM simulation you can again see the same phenomena plot body lengths over Per cycle as a function of amplitude wavelength and there's a peak and the red is the RFT simulation agreeing quite well with experiment simulation. Okay It turns out that this frictional fluid picture and resistive force theory just as the gift that keeps giving whether they be undulating Whether they be side winding Whether they be moving with limbs to a good approximation to a very good approximation We can model the interaction forces and calculate for a given limb motion how far the animal should displace for us That's been a success But it's kind of a boring success and this is where the next part of the talk begins It's a boring success because it's okay. I know f equals ma and I can solve things But if I want to figure out how to make a better movement I have to recalculate everything and it's laborious and so how should this thing move its body? should it be moving with a wave which is a over lambda 0.2 or Can I how do I iterate the parameters? Well, it turns out that the first cut the first key to this is we realize is that these animals that we've been studying are Not swimming like eels in the sense that these animals Basically inhabit a world where if they stop self-deforming that's going to be the next language They stop self-propelling nearly instantly unlike the eel Which if it stops self-deforming will glide for a little bit and it turns out then that world is Much more like the world of tiny organisms like nematode worms and little spermatizoa Where inertial effects are swamped by viscous forces in our case inertial effects are swamped by frictional forces Okay, so that's kind of cool. You can even compare them. You can do RFT which was done for small organisms and you can calculate the same kind of you can play the same sort of game and you basically see a kind of Similar for a little nematode swimming in fluid versus a sandfish swimming in sand You see the sandfish seems to go about double the distance Body lengths per undulation cycle and it turns out that's because if you look at the resistive forces Here's the perpendicular and parallel resistive forces and if you're an expert in this world You'll note that the parrot perpendicular forces and the granular resistive forces Increase more rapidly at low attack angles which allows the animal to generate more thrust relative to drag Okay, cool That's mechanical is there another ways to visualize the character locomotion and this comes to our friend the falling cat who can do a turn Without changing its angular momentum tends to puzzle people It turns out that there is was that every more or less every physicist knows this a talk by Purcell in the 70s and in a paper Which he basically sort of Narrated in a much more elegant way than I just have this kind of amazing Fit facts of the world of the microorganism such that when Reynolds numbers get small Our intuition of how to move breaks down completely We live in a world and for swimming for sure out there if I Swim and paddle I can coast for some distance the fish flaps its tail. It can coast from distance spermatozoa cannot and He basically to kind of highlight these questions He came up with what we now call a Purcell swimmer a three-link swimmer Which basically you prescribe the shape of this thing as a function of time by Prescribing the angles of two joints as a function of time So you're allowed to control theta one and theta two here and each of these little elements experiences resistive forces in a fluid Okay You all know that if you do a reciprocal motion with something like this you get nowhere. That's called the scallop theorem And it's because of certain symmetries inherent in the Stokes equations but if you do a Non-closed path in a configuration space in this case a little square where I hold theta To fixed and I've changed theta one and then I hold theta one fixed and change theta two And then I bring it back to make like a little wave down the body I can actually get somewhere and it turns out and here's where the story gets really interesting It turns out that and I know this because I happen to know this guy who's at University of Kentucky he was a graduate student at Santa Barbara and He was the student of Frank Wilczek and Wilczek heard Purcell's talk I have this on good authority heard Purcell's talk and came back to Al Shapiro and said ah ha We can formulate this problem of self-propulsion at low Reynolds number in terms of a gauge field over the space of shapes and You know Shapiro said he didn't know what that meant it seven years later. He did They published his paper in PRL. They have a JFM two JFMs. Maybe they have a number of interesting papers This was the heyday of geometric phase Barry's phase in the 80s And as far as I can tell in the physics community it hit like a lead balloon Because people said why on earth do you need gauge theory to understand the swimming of something a tiny micro organisms? It is guns to a knife fight. It is really way too much apparatus turns out though that a bunch of engineers largely at Caltech, but others around the country in the US control theorists found this idea pretty interesting because to them it was a way to think about controlling robots and devices In a different way a kind of geometric way and this is what I want to narrate a little bit And they're the ones who kind of developed this stuff And I'd say that this guy who was a student of this guy whom I collaborate with Basically figured out the trick to make this Useful for the kind of devices and animals we study and that's where really the geometric phase dimensionality reduction part of my Talk hits. Here's the scheme. It's really simple and principle It's hard to do in practice. The scheme is this you have to connect two spaces one They call the shape manifold the space of internal degrees of freedom of the swimmer. The other is se to let's say That's only moving the plane for now Okay, I'm gonna apply this to the two degree of freedom per cell swimmer. I Have my body shapes meaning every configuration of the per cell swimmer. I could think of They don't want to appear theta two down here theta two over here Here I keep track of body center of mass position and orientation Okay, the game then of locomotion is to figure out how she pads in one space lead to pads in the other space So far I haven't said anything interesting. There's an onsots the key assumption here this the theorist assumption it turns out you have to test this and you can test this is that Body velocities are linearly proportional to joint velocities or shape change velocities provided things are small enough And there's a matrix here. It's called the connection because it's related to actually connections and differential geometry But it's a connection And so it says that if I make a little wiggle with my body, I will translate and rotate by a little amount Linearly. Okay. It's the first thing you would try for your physicist and they did that in their PRL Fine Well, the only problem is then what it says is you basically have to go in and for every configuration of the body you have to either buy Analytics or experimentally or empirically you have to make a little wiggle So if I start an alpha one alpha two, which again are configurations of the robot in here two angles I have to make a little wiggle and I have to make that every wiggle So I have to change theta one a little bit and theta I'll holding theta two cons I have to do all this whole precalculation, but if I do that I generate something called a connection My colleagues call these connection vector fields, which are kind of interesting Machines I guess such that I now have a little vector at every point in my configuration space and Basically, if I make a little wiggle if I go from if I connect my shapes from here to here It says if I tried to make a move that does that right? I'm into some configuration I could go anywhere I want to another configuration around here if I go this way and I align with a vector that means I'll do well There's a dot product So I will that will get me right and if I do if I try to make a shape change which is perpendicular to that local vector I don't do so well. So then the game is make Paz in your configuration space which do the best that you can by going with the flow of the vector field Okay So you do a line integral see so no no problem I just need to now do an inline integral over this thing integrating it up and off you go The problem is what they tell me is that it's hard to do these kind of line integrals because they're essentially Interactive integrals at every point you have to You make a move your body has translated and rotated now You have to take your coordinate system which you fix to the body in the first instant and you have to Rotate that coordinate system and do it again and do it again Do it it's kind of like the oiler problem which we teach in much. I teach in mechanics, right? And nobody likes to solve all those equations For rotating and spinning because you have to keep changing your frame of reference. It's a pain in the neck and if you don't do that you can't do this line integral because The commutivity aspects of se2 Rotations and translations don't commute as Michael Berry once told me when he visited the lab He says well if they did you could do all your turning in the garage before you left the house And okay, that's a joke It's a British joke. I can't tell it like Michael Berry what these guys figured out how to do is something really cool They said we're gonna turn a line integral in the surface integral Well, you still have your non commutivity problems So you still get hosed if you try to use stokes theorem But there's another way if you have a lee bracket term It turns out you can and most importantly you pick the right coordinate system You pick the right gauge where you can minimize the non commutivity aspects Which kill you and so what they end up with is Basically a new thing called a height function, which is this curl of a plus this lee bracket term They call them height functions. So now the game is Now the game is you have a new machine here. This is for x displacement Do a surface integral of this there's a contour map representation here I have a 3d fancy version do a surface integral over this thing and The amount of surface integral I pick up will tell me how much I displace or rotate how much geometric phase I pick up Okay So the inside this is what this guy figured out is a PhD is Pick the right gauge if and he can prove if you pick the right gauged and for certain Some organisms are better than some robots are better than others But if you pick the right coordinate system, you minimize those non commutivity effects and you actually can calculate things cool Let me give you it. Let me walk you through before I get to the animals Because it really is a cool extension which takes advantage of some work that Greg Stephens has done before I get the animals Let me just walk you through How this works because it's beautiful and it changed how I thought about locomotion Okay, here. I have height functions x y and theta I need three of them because I have my three quarts what I'm going to do is I'm going to draw paths Circles in this space a circle in this space in the alpha one alpha two space Just means I've got a sine wave playing on alpha one and a cosine wave playing on alpha two All right, everybody buy that if I do a closed if I do a surface integral over this thing And I've made plus as red here and minus is black that should convention Unfortunately is going to change in the subsequent slides, but just for right now if I do this surface integral And I have a lot of negative that means I'll displace So I do a service integral on this thing and you'll note that I get a good amount of x So I'll go somewhere I do a service integral on y and lo and behold I've got positive and negative so I go nowhere and same in theta I enclose an equal amount of positive and negative surface integral. Okay, that's point one By the way, I should say that resistive force theory like I said the gift that keeps given these are height functions actually calculated for Experimental system with grinding material we did the first no one had tested this stuff There were 20 years of mathematics developed on this. No one had tested this stuff So we built one of our earliest most primitive robots And it's basically two motors and at that time we didn't even have 3d printer. So it's wooden blocks It's embarrassing to look at now, but we sort of went ahead and say well We'll assume that the kind of things that are important for the scheme work in grinding material They're kinematic no inertia check There's a linearity in the local connection that that on-site told you that body velocities are linearly proportional shape velocities We're gonna Turns out it's okay, and there's a symmetry in space and time and that's what we thought would be the killer It turns out you need symmetries in space and time the uniform medium has to be uniform homogeneous and You can't have hysteresis effects Turns out that's a reasonable approximation for the granular systems. We could talk about that later Here's what the robot looks like. There's little lights that you can see it actually doing its little circle gate That's what we're gonna call these things and here's some data. So I have four displacement height function and I have Displacement body lengths per undulation cycle as a function of stroke amplitude the stroke amplitude is just the radius of this of this flapping in the parameter space, so if I double the Stroke amplitude that's basically doubling the angular extent excursion of my of my body and what it sees is the experimental data is the black Gray is our DEM simulation And you'll note that this placement has a kind of characteristic feature which we even saw actually in the previous talk Despite it having inertia it kind of goes up quadratically and then falls over The theoretical prediction that is why well suddenly you understand or this I understand if I have a tiny circle And this region is negative sign And it's all roughly the same magnitude and a tiny circle and if I double the radius of a tiny circle I'm enclosing the same still that magnitude of negative area, but I'm now enclosing four times that so I increase Quadratically in my displacement so I go up quadratically until my circle starts to eat into regions of opposite sign And now my surface integral is picking up Positive negative and positive and so my displacement starts to go down again And that's the geometric reason for why you have to pick the right amplitude in the simple swimmer You then can say well, I want to optimize I want to go I want to go as far as I can swimming no constraints on my motor power and energy battery I want to go as far as I can for undulation cycle what should you do? You should do a surface integral over over a path which only encloses Negative area in this case, and that's what you do here It's called the butterfly gate or we call it the butterfly gate and it's absolutely bizarre And I don't think you ever would have thought of it had you not had these tools But you could see that in this configuration space you're basically no longer worrying about your phasing and sequencing of limbs and body You're basically showing is doing this doing this path. So it's pretty cool And it brings up again this idea. It gives us a way to look for what we'll call templates Ways that the Purcell swimmers should be controlled to move forward. Well, okay You can do things like turn in place which you never could have thought of you go to the theta height function and close a loop here And it determines the the direction you go around the loop Comes into the surface integral as well So you go around loop clockwise and then counterclockwise and the other sign and it basically gives you a turn in place without any Translation pretty powerful and it compares well to the experimental data Okay, and you can compare then locomotion in different environments now That was essentially the long preamble which I had to explain to get to the real meat of the talk And here's the real meat of the talk which is Review we have one paper and review and another paper. There's about this go out Can we apply these to higher degrees of freedom systems because who cares about the Purcell swimmer except as a physics Exercise demonstrating these things and the answer is yes And these are the guys who figured it out. We did the test. Here's a robot a multi segment multi motor robot Again, you're now controlling the robots motors as a function of time just like my virtual sandfish earlier where I'm prescribing sinusoidal shape changes in its curvature as a function of time There's some midline tracks to see it kind of swims like a sandfish It's in a bunch of granular material and by the way, I might refer these as serpinoid curves This was a term brought up by a famous robot this name Hirose Which is basically curves which are sinusoidal in the curvature not necessarily any amplitude in real space Here's what they did and here's the dimensionality reduction. They say, okay. Well This thing has 16 joints 32 joints if it's a continuous body infinite number of joints forget all that Let's just decompose it into two modes to eigen robots I can snakes eigen worms and we're just going to twiddle the amplitudes of those modes And we should be able to recapitulate lots of interesting shapes pick a sign a cosine Specify the curvature along the body as a function of arc length as w1 times beta 1s Which will be your first mode and w2 times beta 2s will be your second mode And if you do that here's a configuration space now where every position every every spot in this configuration space is a W1 w2 multiplication of signs and cosines and by going in a circle around here I've made this little animation. You could see I get something that kind of looks sinuous as it moves So that's the dimensionality reduction. Let's see how well it works The answer is you can then apply this same machinery now using these modes You can generate these height functions now. I've switched my sign convention red is plus black is negative and I can do this same business and lo and behold here's the body length per cycle as a function of this curvature of the thing Here's the robot experiments, and here's the Rft simulation and then the area to go the surface integral Pretty close to it so you could start to do it and then it gets more interesting because now you have new Elements to kind of operate on to understand how things go suppose I want to compare granular swimmers and viscous swimmers in my previous plot I showed you plots with lots of force vectors and velocity vectors, which were hard to interpret now You just say oh well look this height function has a bit more red and is a different slightly different shape Understand it. This is all very new And so this is why this thing basically goes faster If I were to now say suppose I in a viscous situation Suppose I were to vary a parameter called the drag anisotropy How much the if an object is being pulled through a material how much it's resists going this way versus this way Well, I can do that and generate height functions, and you'll note that as I Increase that parameter C, which is essentially telling me how hard it is to move perpendicular to my Axis direction in parallel you'll note that if I'm very low C I have a big black region and as I get higher and higher I get a redder region which gets more and more ellipsoidal What does that look like? Well here's what it looks like if I have a low C Actually, it turns out that that means I have a drag anisotropy less than one for the experts I can't go anywhere what actually I can go backwards if I have it as to that's my kind of nematode and a fluid And if I have it as 20 it just takes off And you'll note that it says that the best way to do is not to make a circle in this space But to enclose the most surface integral do a kind of weird ellipsoidal thing Okay, turns out. This is a detail, but an important detail. You can actually do this empirically as well You could take a robot and here's a video Which will run for a minute and you can put that robot if you have a lot of students who are patient in a particular pose Twiddle it measure how far it goes with a sensitive camera put it again in another pose twiddle it measure how it goes and build up this whole connection vector field and It will play and they're demonstrating for you how they're going to build it up And if they spend a lot of time doing this There it is Okay So it takes a lot of effort, but once you have that then you can do pretty extraordinary things like say here's the theta height function Here's how I can make my robot turn in place in granular material. All right. It'd been hard for us to do that. Okay What about living systems? I'm almost done Here's our sandfish again And this is by the way the work of Jennifer Riser was a postdoc in my group Who really was the one who put all these pieces together and there's a paper on the archive right now That you're you're interested in what you do now is you follow Greg Stevens and crew and you track these things and then you Use the relative curvature of segments on the body You can plot that relative curvature as a function of time and you see ways of curvature propagating through You can then do a principal components analysis on it And you find that basically the dominant postures this thing or a sine and cosine mode and that says that in its Configuration space now made of sines and cosines. It's basically some approximation executing a loop and there These are the PCs Take about eighty ninety percent of the variance. Okay, then you can play the same game You put those modes in you compute your height function You do the same procedure and off you go the Theory overestimates the sandfish swimming speed for reasons which we can talk about later It basically has to do with kind of assumptions of drag on the head of the animal It's a longer story, but it basically gets the Gets the curvature that the animals using and then it becomes rather cool because now you can start to do comparative Studies I can say why or how can I understand why this animal is? Different or better or similar to another animal and I can do this same kind of trick in this case for this snake swimming on The surface turns out it's also non-inertial So highly damp. I can basically decompose it into modes Which are now one and a half periods of sine and cosine and compute these things and now here again the animal data Lots of animal data. Lots of sandfish data snake data and the theory basically tells you why it's moving the way it does Okay, and so then the principle becomes and to me This is rather lovely is maximize your berry phase and when Michael Berry visited he I think got a kick out of this You know for will check and these guys it started just a simple example of visualizing gauge theory But for us it's become a tool. I'm not going to bore you with it Yeah, well I could bore you but I'm not it turns out it works for side winding too And it works for a diversity of side winders basically side winding We've discovered is along with David who is basically two waves playing on the body one in the horizontal plane the other in The vertical plane if you just look at the horizontal plane and Decompose it. I lost evens you make eigen snakes You find a circle in this postural dynamic space you do the height function again and now You do this and now you go to the literature and you plot every animal That's ever been reported and how well it's side winds and what you really measure people say well It's side-winded at this speed and it had track angles. It's a weird gate I'm not going to explain that track I was it so you have to go through a whole procedure But it basically says that the animals that are the champion side winders live in the desert like I saw on that first slide They're the ones who are optimizing their berries phase I'm done we can apply it to legged systems It turns out that anything without inertia even granular material Coulomb friction this stuff gives you a good first approximation. Here's a little salamander It's a new kind of height function now on a cylindrical space s1 cross reels You can figure out how much displacement you get as a function of the how you should bend your body relative to how you phase your limbs the coolest thing We can yeah, we can do it with animals the coolest thing in the newest thing And I just had to show you this because it's like hot off the press is it I've been fascinated with centipedes for a long time but never been able to study them because Tracking the motion of animals with this many parts had been impossible Thank You mercy and others maybe a Ravi's partners. I don't know Figured out how to do neural network deep learning. They have something called deep lab cut I had an undergrad spend two weeks clicking training the neural net two nights of the neural net thinking And you feed it in videos as a threat a centipede from the desert Southwest the US running on a little treadmill It has I think 40 some legs 20 pairs and a bunch of segments and those blue which you can barely see that's the computer which has tracked every element And the cool thing is then you could say well the centipede is a wave of body undulation You could see a wave going from head to tail and there's a wave of leg Oscillation which you could sort of see each angle leg and if you plot those against each other now you need a toroidal space to compute your height functions That's a detail but the and and the way you calculate the surface integral is a little bit different I'm not going to bore you with it But basically you should be impressed that the blue law the blue points of the experiment the purple is What would be the optimal way for the animal to phase its limbs as calculated with a Connection and a height function based on Coulomb friction now not even granular material and it nails it. I'm done I could tell you about how changing leg lengths are interesting But I think I'll quit there and just tell you that geometric phase denturement dimensionality reduction living and non-living Locomoting systems one of the big surprises that many of the organs we say the macro scale mimic environmental interactions the micro scale frictional versus viscous fluids and Coulomb friction dissipation much greater than inertia Granular rft provides a good model for thrust and drag and locomotion Using a low-dimensional representation of locomotor postures in a model for the environmental force We're able to predict displacements resulting from sickly sequence of shape changes Good reasonable oops I meant just to say reason to good agreement with no consideration of the physiological constraints muscle power speed That's where we're kind of going next Muscles can't that butterfly gate that I showed you with the Purcell swimmers crazy You have to really jerk your body around no organisms going to do that You now want to start to put in models of the physiology Which we now have a rational way to do that works in at least one system where the two modes don't capture But that snake you saw swimming underneath the sand look like it had a horrible shape in the real world It turns out its curvatures basically two PCs capture 70% of the variation still works pretty well and works if the biomedical physics is not known local connection through germany empirically and then here for me from the kind of Organismal biomechanics and maybe robotics control if I were an engineer is that this stuff provides a systematic method discover control Templates and frictional fluids and frictional services, and we don't know where else it can go But I just thought I would tell you a little bit about this stuff, and I'd be happy to take any questions