WEBVTT 00:00:00.000 --> 00:00:02.490 The following content is provided under a Creative 00:00:02.490 --> 00:00:03.940 Commons license. 00:00:03.940 --> 00:00:06.330 Your support will help MIT OpenCourseWare 00:00:06.330 --> 00:00:10.630 continue to offer high quality educational resources for free. 00:00:10.630 --> 00:00:13.320 To make a donation or view additional materials 00:00:13.320 --> 00:00:17.160 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.160 --> 00:00:18.252 at ocw.mit.edu. 00:00:21.275 --> 00:00:22.900 PROFESSOR: So today we're going to talk 00:00:22.900 --> 00:00:27.970 about the dynamics of running. 00:00:27.970 --> 00:00:30.910 And ask questions as freely as possible, 00:00:30.910 --> 00:00:33.820 and also if my handwriting's bad, let me know. 00:00:33.820 --> 00:00:36.480 It generally is. 00:00:36.480 --> 00:00:36.980 So. 00:00:45.250 --> 00:00:47.432 So this is actually a really fun lecture. 00:00:47.432 --> 00:00:48.640 We have a lot of cool videos. 00:00:48.640 --> 00:00:51.223 I don't know if any of you have seen Raibert hoppers and stuff 00:00:51.223 --> 00:00:53.740 before, but we have some cool models and some cool analysis 00:00:53.740 --> 00:00:54.240 too. 00:00:54.240 --> 00:00:56.470 So hopefully, you'll be as excited 00:00:56.470 --> 00:01:00.320 as I am to participate in this learning experience. 00:01:00.320 --> 00:01:03.550 So here we go. 00:01:03.550 --> 00:01:05.560 So the basic model we'll be looked 00:01:05.560 --> 00:01:07.480 at today in several different versions of this 00:01:07.480 --> 00:01:09.760 is the SLIP model. 00:01:09.760 --> 00:01:12.220 This is the spring-loaded inverted pendulum. 00:01:15.545 --> 00:01:18.880 All right. 00:01:18.880 --> 00:01:23.170 It's like the running version of the rimless wheel or compass 00:01:23.170 --> 00:01:24.730 gait. 00:01:24.730 --> 00:01:25.580 It's pretty simple. 00:01:25.580 --> 00:01:27.497 You just have a mass at the end of a pendulum. 00:01:27.497 --> 00:01:32.260 You have a spring here, spring constant k. 00:01:32.260 --> 00:01:39.600 The spring is a rest length r naught and a-- 00:01:39.600 --> 00:01:41.350 yeah, we already said the spring constant. 00:01:41.350 --> 00:01:42.808 And so then this-- the coordinates, 00:01:42.808 --> 00:01:52.840 then, are this length r and the angle theta from the upright. 00:01:52.840 --> 00:01:57.160 So what we're going to want to look at here 00:01:57.160 --> 00:01:58.660 is not just this model, but also, 00:01:58.660 --> 00:02:01.770 in the same way we had for the rimless wheel, as you remember, 00:02:01.770 --> 00:02:05.640 is a 1-D iterated map. 00:02:05.640 --> 00:02:08.380 So you probably remember that for the compass gait that 00:02:08.380 --> 00:02:09.460 lets you look at the stability of it, 00:02:09.460 --> 00:02:11.585 and there's all these interesting dynamics that are 00:02:11.585 --> 00:02:14.338 captured by that simple system. 00:02:14.338 --> 00:02:16.630 So we're going to want to look at the 1-D iterated map, 00:02:16.630 --> 00:02:18.797 and the other thing I'm going to try to convince you 00:02:18.797 --> 00:02:24.440 is that this is a biologically plausible system. 00:02:24.440 --> 00:02:30.280 So if you look at the rimless wheel, 00:02:30.280 --> 00:02:32.620 it feels like something that's walking, 00:02:32.620 --> 00:02:34.880 compass gait even more so, [INAUDIBLE] knees. 00:02:34.880 --> 00:02:36.470 These things all-- just intuitively, 00:02:36.470 --> 00:02:37.720 you can believe that they're-- 00:02:37.720 --> 00:02:41.980 sorry-- a-- whoa, that's probably going to be bad-- 00:02:41.980 --> 00:02:45.437 that you could believe that they're walking systems. 00:02:45.437 --> 00:02:47.520 But this one maybe doesn't seem to you immediately 00:02:47.520 --> 00:02:48.798 like a running system. 00:02:48.798 --> 00:02:51.090 Maybe it seems like a jumping system or bouncing system 00:02:51.090 --> 00:02:52.860 or something, but actually, it actually 00:02:52.860 --> 00:02:55.990 is a pretty good model for running systems. 00:02:55.990 --> 00:02:58.770 So the two things that we're going to have to go through, 00:02:58.770 --> 00:03:00.380 how do you get a 1-D iterated map? 00:03:00.380 --> 00:03:02.250 Now that maybe is a bit surprising for you 00:03:02.250 --> 00:03:04.830 because if you look at this, the full state 00:03:04.830 --> 00:03:10.875 space of this inverted pendulum right now, it has states of r-- 00:03:10.875 --> 00:03:16.180 sorry-- r theta, r dot, theta dot. 00:03:16.180 --> 00:03:19.002 Now that's four parameters. 00:03:19.002 --> 00:03:20.460 You remember when you slice through 00:03:20.460 --> 00:03:22.020 and you pick a surface of section, 00:03:22.020 --> 00:03:23.313 you can cut it down by one. 00:03:23.313 --> 00:03:25.730 So for the rimless wheel, maybe you have theta, theta dot. 00:03:25.730 --> 00:03:27.570 You cut out theta dot, you're in pretty good shape. 00:03:27.570 --> 00:03:29.280 Here we have four, but we're actually 00:03:29.280 --> 00:03:30.480 going to show that we can actually get this down 00:03:30.480 --> 00:03:31.770 to a 1-D system and actually look 00:03:31.770 --> 00:03:32.978 at the iterated map for this. 00:03:32.978 --> 00:03:34.680 So that's kind of cool. 00:03:34.680 --> 00:03:36.577 And the other thing is that also, this system 00:03:36.577 --> 00:03:38.160 bounces and takes off, so actually, we 00:03:38.160 --> 00:03:39.520 need two coordinate systems. 00:03:39.520 --> 00:03:41.760 One is this r theta, r dot, theta dot. 00:03:41.760 --> 00:03:43.860 And then when it's in flight, we're 00:03:43.860 --> 00:03:48.790 going to have x, y, x dot, y dot, all right. 00:03:48.790 --> 00:03:51.231 And both of these are 4-D, so going 00:03:51.231 --> 00:03:53.640 to have to deal with that either way. 00:03:53.640 --> 00:03:56.730 But first the convincing part. 00:04:03.480 --> 00:04:06.900 So the justification here comes from a field called 00:04:06.900 --> 00:04:08.310 comparative biomechanics. 00:04:12.750 --> 00:04:16.230 Now this is a pretty cool field. 00:04:19.958 --> 00:04:21.750 They look at all sorts of different systems 00:04:21.750 --> 00:04:23.940 and try to figure out these commonalities in the kinematics 00:04:23.940 --> 00:04:26.250 or the dynamics or certain properties and figure out, 00:04:26.250 --> 00:04:30.105 what are the fundamental common features 00:04:30.105 --> 00:04:32.140 of these different biological systems? 00:04:32.140 --> 00:04:35.760 So this running model, actually, you 00:04:35.760 --> 00:04:42.780 can look at it in the context of things from cockroaches, which 00:04:42.780 --> 00:04:52.290 are on the order of a gram in mass, horses, 00:04:52.290 --> 00:04:54.300 which are on the order of-- 00:04:54.300 --> 00:04:57.600 well, some of them, at least, are 135 kilograms. 00:04:57.600 --> 00:05:00.420 It's not that heavy, really, so you could imagine maybe 00:05:00.420 --> 00:05:02.250 heavier horses. 00:05:02.250 --> 00:05:02.940 Crabs. 00:05:02.940 --> 00:05:06.175 Now you can think about the kinematics of these system 00:05:06.175 --> 00:05:07.800 and just how wildly different they are. 00:05:07.800 --> 00:05:09.915 Cockroaches, six-legged, bouncing all around, 00:05:09.915 --> 00:05:11.100 [INAUDIBLE] tripod gait. 00:05:11.100 --> 00:05:13.110 Horses, quadrupeds. 00:05:13.110 --> 00:05:17.790 Crabs, actually they run sideways with eight legs. 00:05:17.790 --> 00:05:20.580 This is different information, so I won't put it there. 00:05:20.580 --> 00:05:22.080 But crabs actually will run sideways 00:05:22.080 --> 00:05:22.740 using their eight legs. 00:05:22.740 --> 00:05:23.823 It's actually pretty cool. 00:05:23.823 --> 00:05:25.170 I don't know if you've seen it. 00:05:25.170 --> 00:05:29.580 And then humans, which maybe is the one that we most-- 00:05:29.580 --> 00:05:33.150 well, I don't know, that's probably unfairly biocentric. 00:05:33.150 --> 00:05:36.120 So we're humans. 00:05:36.120 --> 00:05:38.580 And all of these, despite the very different kinematics, 00:05:38.580 --> 00:05:43.620 despite, what is this, five orders of magnitude in mass, 00:05:43.620 --> 00:05:46.898 they all have incredible dynamic similarity. 00:05:58.903 --> 00:06:01.320 Let's hope I'm not getting graded on spelling, because I'm 00:06:01.320 --> 00:06:03.405 pretty sure that's way off. 00:06:03.405 --> 00:06:08.150 [INAUDIBLE] And so if you look at it in the right way, 00:06:08.150 --> 00:06:10.650 you can actually see that all of these systems, cockroaches, 00:06:10.650 --> 00:06:14.550 horses, crabs, camels, cats, bunny rabbits, all these things 00:06:14.550 --> 00:06:18.280 actually look very similar. 00:06:18.280 --> 00:06:25.850 And so Bob Full, some of you may know Bob Full, 00:06:25.850 --> 00:06:27.262 I think at Berkeley. 00:06:27.262 --> 00:06:29.720 He looks at a lot of things, especially the cockroach work. 00:06:29.720 --> 00:06:32.360 He has a lot of cool work on cockroaches. 00:06:34.880 --> 00:06:40.070 And he also has this paper which-- 00:06:40.070 --> 00:06:46.190 let's see if I can find what page this on. 00:06:46.190 --> 00:06:47.155 Sorry, here. 00:06:47.155 --> 00:06:50.440 There we go. 00:06:50.440 --> 00:06:53.350 Is that clear up there? 00:06:53.350 --> 00:06:54.580 Ooh. 00:06:54.580 --> 00:06:56.890 Sorry. 00:06:56.890 --> 00:07:01.135 So here, you can look at the-- oh, 00:07:01.135 --> 00:07:03.880 let me dim the lights a bit. 00:07:03.880 --> 00:07:08.290 If you look at the plot on the left, 00:07:08.290 --> 00:07:10.843 this is speed and the stride length. 00:07:10.843 --> 00:07:12.760 So if you plot speed and stride lengths-- here 00:07:12.760 --> 00:07:14.440 they've got gerbils, dogs, and camels, 00:07:14.440 --> 00:07:18.410 but you can do this for a very wide variety of systems. 00:07:18.410 --> 00:07:21.250 You can see that they're all bounced over the place. 00:07:21.250 --> 00:07:22.243 You don't have-- 00:07:22.243 --> 00:07:24.160 I mean, they're all behaving very differently, 00:07:24.160 --> 00:07:26.147 so that's not necessarily the right way 00:07:26.147 --> 00:07:28.480 to look at them if you want to see what is fundamentally 00:07:28.480 --> 00:07:29.525 similar between them. 00:07:29.525 --> 00:07:31.900 If you look at the right, this is the [INAUDIBLE] number. 00:07:31.900 --> 00:07:33.860 In this case, you can see it's just the ratio-- 00:07:33.860 --> 00:07:37.480 well, it's 2 times the ratio of kinetic energy 00:07:37.480 --> 00:07:39.890 to potential energy. 00:07:39.890 --> 00:07:41.770 So it's just this non-dimensional quantity. 00:07:41.770 --> 00:07:44.270 If you plot this and you look at the relative stride length, 00:07:44.270 --> 00:07:45.580 which is-- 00:07:45.580 --> 00:07:48.220 you can see they all collapse very tightly on this line. 00:07:48.220 --> 00:07:50.500 And so there's actually-- that's impressive agreement. 00:07:50.500 --> 00:07:51.820 You see that these are-- 00:07:51.820 --> 00:07:54.190 camels, dogs, and gerbils are all-- 00:07:54.190 --> 00:07:56.320 look very similar when you look at them 00:07:56.320 --> 00:07:58.460 in this non-dimensional way. 00:07:58.460 --> 00:08:02.860 So all these systems, orders of magnitude, 00:08:02.860 --> 00:08:08.080 again, of difference in masses, stride lengths too, probably, 00:08:08.080 --> 00:08:11.140 all look very similar when you compare them 00:08:11.140 --> 00:08:12.270 in the right framework. 00:08:12.270 --> 00:08:14.470 And that's what comparative biomechanics 00:08:14.470 --> 00:08:19.240 tries to do, in addition to other things, obviously. 00:08:19.240 --> 00:08:20.950 So here's something that's really cool 00:08:20.950 --> 00:08:23.330 and that connects us back to our SLIP model. 00:08:23.330 --> 00:08:29.030 If we look at the next page, you'll 00:08:29.030 --> 00:08:34.610 see right here, here in-- 00:08:34.610 --> 00:08:37.070 this is the relative spring constant 00:08:37.070 --> 00:08:39.770 for individual individual leg [INAUDIBLE] systems, 00:08:39.770 --> 00:08:43.280 cockroach to kangaroo, five orders of magnitude. 00:08:43.280 --> 00:08:45.740 You can see that they're all pretty similar. 00:08:45.740 --> 00:08:48.320 This is huge spectrum of masses, and yet 00:08:48.320 --> 00:08:50.810 in the framework of the SLIP model's spring 00:08:50.810 --> 00:08:53.150 constant, they all behave pretty similarly, 00:08:53.150 --> 00:08:54.510 humans, all these things. 00:08:54.510 --> 00:08:57.020 And so if you look at these systems 00:08:57.020 --> 00:08:59.750 through this simple model, they all not only 00:08:59.750 --> 00:09:03.130 look similar to one another, but their dynamics and their center 00:09:03.130 --> 00:09:05.248 of mass [INAUDIBLE] are pretty well described 00:09:05.248 --> 00:09:06.290 by this kind of behavior. 00:09:06.290 --> 00:09:06.938 Sorry. 00:09:06.938 --> 00:09:07.980 By that kind of behavior. 00:09:12.872 --> 00:09:17.150 So hopefully that will be somewhat compelling argument 00:09:17.150 --> 00:09:18.920 that this model is representative 00:09:18.920 --> 00:09:21.440 of actual running behavior, not just something 00:09:21.440 --> 00:09:23.180 that comes out of nowhere. 00:09:23.180 --> 00:09:26.606 And a lot of work was done on this in the '80s and '90s 00:09:26.606 --> 00:09:30.412 on trying to study this model and connect it to animals 00:09:30.412 --> 00:09:31.370 and that sort of stuff. 00:09:31.370 --> 00:09:34.558 I don't know if you've seen Raibert's work, 00:09:34.558 --> 00:09:37.100 but he has robots that are very similar to these sort of SLIP 00:09:37.100 --> 00:09:38.658 model that run and jump and actually 00:09:38.658 --> 00:09:40.700 are from the '80s and such like that actually are 00:09:40.700 --> 00:09:42.000 some of the really cool robots. 00:09:42.000 --> 00:09:44.600 And I'll show you some of those. 00:09:44.600 --> 00:09:47.203 So he actually didn't even call them SLIP. 00:09:47.203 --> 00:09:48.620 He actually made them before SLIP, 00:09:48.620 --> 00:09:52.225 and that motivated some of the work on SLIP models. 00:09:52.225 --> 00:09:52.850 But I can show. 00:09:52.850 --> 00:09:54.517 Here, I'll show you a picture of a robot 00:09:54.517 --> 00:10:01.340 so you can see the kind of cool stuff that we're getting to. 00:10:01.340 --> 00:10:02.550 I don't know if-- 00:10:02.550 --> 00:10:04.460 are many of you familiar with the Leg Lab? 00:10:04.460 --> 00:10:06.060 They should be around. 00:10:06.060 --> 00:10:08.780 Yeah, they had a lot of cool robots. 00:10:08.780 --> 00:10:14.570 And here, see if I can give you just a little picture 00:10:14.570 --> 00:10:17.972 [INAUDIBLE] Ah. 00:10:17.972 --> 00:10:19.467 Oh, so that's the biped robot. 00:10:19.467 --> 00:10:21.800 This thing's bouncing, but that's not really the picture 00:10:21.800 --> 00:10:22.675 I wanted to show you. 00:10:27.856 --> 00:10:28.850 Ah, here we go. 00:10:28.850 --> 00:10:30.290 Yeah. 00:10:30.290 --> 00:10:38.960 So this little robot, you can see that's tracing its foot. 00:10:38.960 --> 00:10:40.760 This up here is tracing its center mass. 00:10:40.760 --> 00:10:43.130 And you see how center mass actually compresses right 00:10:43.130 --> 00:10:44.107 above the foot. 00:10:44.107 --> 00:10:45.440 And that's what you expect here. 00:10:45.440 --> 00:10:46.732 It lands, and it squishes down. 00:10:46.732 --> 00:10:48.320 And so instead of when you're walking 00:10:48.320 --> 00:10:49.903 and you vault over this leg-- like you 00:10:49.903 --> 00:10:51.200 think about that rimless wheel. 00:10:51.200 --> 00:10:53.180 Its center mass rolls up over its foot. 00:10:53.180 --> 00:10:55.520 This one, both intuitively in the model and you 00:10:55.520 --> 00:10:57.020 can see right here this robot, which 00:10:57.020 --> 00:10:58.670 is similar to the model in many ways, 00:10:58.670 --> 00:11:00.680 is squishing down into its foot, all right. 00:11:00.680 --> 00:11:02.180 And that actually is one of the ways 00:11:02.180 --> 00:11:04.693 of looking at the difference between what is running, 00:11:04.693 --> 00:11:05.360 what is walking. 00:11:05.360 --> 00:11:07.310 So one of the definitions for running 00:11:07.310 --> 00:11:09.510 is that you're supposed to have an aerial phase. 00:11:09.510 --> 00:11:12.200 But not every running animal has an aerial phase. 00:11:12.200 --> 00:11:16.160 So this is-- actually, I think comparative biomechanicians 00:11:16.160 --> 00:11:18.930 look at this center of mass to center of pressure trajectory, 00:11:18.930 --> 00:11:19.430 because-- 00:11:19.430 --> 00:11:21.380 I don't know if you know Groucho Marx. 00:11:21.380 --> 00:11:22.990 Groucho Marx had this funny run. 00:11:22.990 --> 00:11:23.850 Maybe I'll do it. 00:11:23.850 --> 00:11:26.085 So [INAUDIBLE] but sort of like this-- 00:11:26.085 --> 00:11:26.585 you know? 00:11:26.585 --> 00:11:27.830 That kind of goofy thing where your feet 00:11:27.830 --> 00:11:29.027 aren't coming up the ground? 00:11:29.027 --> 00:11:30.110 That doesn't seem natural. 00:11:30.110 --> 00:11:31.568 But actually, some animals do that. 00:11:31.568 --> 00:11:35.520 There's actually elephants that do that Groucho run. 00:11:35.520 --> 00:11:38.292 So maybe if you're big enough, it makes sense. 00:11:38.292 --> 00:11:41.690 But yeah, so [INAUDIBLE] this is very different from walking. 00:11:41.690 --> 00:11:44.068 And it's actually quite similar to running, 00:11:44.068 --> 00:11:45.860 and actually, if you look at these animals, 00:11:45.860 --> 00:11:48.360 it captures a lot of their dynamics. 00:11:48.360 --> 00:11:52.160 So and another important thing is 00:11:52.160 --> 00:11:55.490 that these kind of robots which came around in the '80s 00:11:55.490 --> 00:11:59.830 were the similar kind of work in robotics, because again, 00:11:59.830 --> 00:12:02.575 the same time [INAUDIBLE] these slow, careful walking robots. 00:12:02.575 --> 00:12:03.950 Again, here are these robots that 00:12:03.950 --> 00:12:05.092 are flying through the air. 00:12:05.092 --> 00:12:06.800 I'll show you some videos later, but they 00:12:06.800 --> 00:12:08.425 did flips and all kinds of crazy stuff. 00:12:08.425 --> 00:12:10.910 I mean, they're throwing themselves around wildly, 00:12:10.910 --> 00:12:12.785 which is not what people thought of when they 00:12:12.785 --> 00:12:14.330 thought of these legged robots. 00:12:18.140 --> 00:12:19.400 Oh, and the other thing-- 00:12:19.400 --> 00:12:21.290 this is cool too. 00:12:21.290 --> 00:12:22.790 If you look at animals, like we're 00:12:22.790 --> 00:12:24.748 talking about this springs and stuff like that, 00:12:24.748 --> 00:12:26.540 that's not just some sort of obtuse model 00:12:26.540 --> 00:12:27.623 that comes out of nowhere. 00:12:27.623 --> 00:12:30.717 If you look at horses and stuff, well, animals, actually, 00:12:30.717 --> 00:12:31.550 are full of springs. 00:12:31.550 --> 00:12:33.383 Their tendons can store up energy and stuff. 00:12:33.383 --> 00:12:35.360 And actually, horses have a big tendon 00:12:35.360 --> 00:12:38.150 that runs through their leg all the way up, I think, 00:12:38.150 --> 00:12:39.140 behind their hip. 00:12:39.140 --> 00:12:41.180 And apparently, one of the limiting factors 00:12:41.180 --> 00:12:44.540 of trying to run fast is that you can't pull your leg forward 00:12:44.540 --> 00:12:46.850 fast enough to get to the next stride. 00:12:46.850 --> 00:12:48.810 And so what they actually can do is 00:12:48.810 --> 00:12:51.348 they can preload when they're pushing, 00:12:51.348 --> 00:12:53.390 and then that tendon actually acts like a spring. 00:12:53.390 --> 00:12:54.807 It actually stores energy and lets 00:12:54.807 --> 00:12:57.740 them swing faster than they would be able to if they just 00:12:57.740 --> 00:12:58.640 used the motor. 00:12:58.640 --> 00:13:01.220 And if you look at the oxygen intake of these animals, 00:13:01.220 --> 00:13:03.020 you can see that they're behaving much more 00:13:03.020 --> 00:13:07.250 efficiently than you could hope to accomplish if you just had-- 00:13:07.250 --> 00:13:10.100 if you were simulating a spring with a bunch of motor-- 00:13:10.100 --> 00:13:12.400 with your muscles acting like a spring. 00:13:12.400 --> 00:13:14.150 So it's actually they're physical springs. 00:13:14.150 --> 00:13:16.790 Actually, this tendon in there actually 00:13:16.790 --> 00:13:17.880 have this actual behavior. 00:13:17.880 --> 00:13:21.112 So hopefully-- maybe I'm dwelling on that point 00:13:21.112 --> 00:13:22.820 too much, but I think that's really cool, 00:13:22.820 --> 00:13:24.950 that these animals actually have this kind of spring behavior. 00:13:24.950 --> 00:13:26.750 You can see it in their dynamics, 00:13:26.750 --> 00:13:29.450 and actually, this model captures a lot of that. 00:13:32.580 --> 00:13:35.177 Oh, and here's one other thing that Russ [INAUDIBLE] 00:13:35.177 --> 00:13:36.260 and it's pretty cool, too. 00:13:36.260 --> 00:13:37.112 If you think about-- 00:13:37.112 --> 00:13:38.570 this is sort of tangent, but if you 00:13:38.570 --> 00:13:40.410 think about how birds sit on purchase forever, 00:13:40.410 --> 00:13:41.900 right, I mean, it seems like that'd 00:13:41.900 --> 00:13:44.150 be tiring to be able to hang there forever just squeezing, 00:13:44.150 --> 00:13:44.780 right? 00:13:44.780 --> 00:13:46.565 But they actually have tendons, too, 00:13:46.565 --> 00:13:48.065 such that when they sit on the perch 00:13:48.065 --> 00:13:49.815 and their weight squishes their legs down, 00:13:49.815 --> 00:13:54.190 it actually will clamp their talons in and let them hold on. 00:13:54.190 --> 00:13:56.090 So that's cool, too, that these animals 00:13:56.090 --> 00:13:58.820 have all these interesting passive structures and springs, 00:13:58.820 --> 00:14:01.320 stuff like that, that help them do all these kind of things. 00:14:01.320 --> 00:14:05.510 And so it's not like the, just use your muscles 00:14:05.510 --> 00:14:07.360 and just dominate, all these kind of things. 00:14:07.360 --> 00:14:08.818 There's a lot of passive structures 00:14:08.818 --> 00:14:12.650 that do a lot of this for free. 00:14:12.650 --> 00:14:17.060 So hopefully, you all think that's awesome. 00:14:17.060 --> 00:14:18.530 All right. 00:14:18.530 --> 00:14:20.360 All right, so getting back to the system. 00:14:23.130 --> 00:14:25.520 Looking at this-- sorry-- at this model again-- 00:14:28.566 --> 00:14:30.140 did you see where I put my chalk? 00:14:36.680 --> 00:14:38.210 All right. 00:14:38.210 --> 00:14:40.836 So going back to the SLIP system-- 00:14:40.836 --> 00:14:43.070 let's see. 00:14:43.070 --> 00:14:44.300 Again, the state space here-- 00:14:46.850 --> 00:14:48.760 is it-- yeah, it's still dark in here. 00:15:04.690 --> 00:15:06.730 So going back to this system, again, we 00:15:06.730 --> 00:15:13.468 have r theta, r dot, theta dot, and in the aerial phase, x, y, 00:15:13.468 --> 00:15:16.945 x dot, y dot. 00:15:16.945 --> 00:15:19.210 All right, I'll bring this one back down. 00:15:22.480 --> 00:15:29.740 And let me put in the assumptions really clearly 00:15:29.740 --> 00:15:31.240 of what this model is going to have, 00:15:31.240 --> 00:15:33.190 so the assumptions for the SLIP model. 00:15:44.000 --> 00:15:48.060 So one is that you have a massless-- 00:15:48.060 --> 00:15:53.870 whoa, that's terrible-- massless leg and toe. 00:15:53.870 --> 00:15:54.370 All right? 00:15:54.370 --> 00:15:57.400 So all your mass is concentrated in that [INAUDIBLE] 00:15:57.400 --> 00:16:00.400 at the body. 00:16:00.400 --> 00:16:13.337 You have an ideal lossless spring in your leg. 00:16:13.337 --> 00:16:14.920 And what that means, that effectively, 00:16:14.920 --> 00:16:17.888 your collisions with the ground, unlike all the walking models, 00:16:17.888 --> 00:16:18.805 are perfectly elastic. 00:16:32.320 --> 00:16:34.170 All right? 00:16:34.170 --> 00:16:37.093 And now you can see that that's not an extra assumption that 00:16:37.093 --> 00:16:38.010 is derived from these. 00:16:38.010 --> 00:16:41.657 When it hits, even though this toe sticks in right away 00:16:41.657 --> 00:16:43.740 and [INAUDIBLE] inelastic collision with that toe, 00:16:43.740 --> 00:16:45.782 because there's no mass there, there's no energy, 00:16:45.782 --> 00:16:47.310 there's not momentum in it-- 00:16:47.310 --> 00:16:48.170 sorry. 00:16:48.170 --> 00:16:51.215 This is [INAUDIBLE] you don't lose any energy. 00:16:51.215 --> 00:16:52.590 And so it's actually conservative 00:16:52.590 --> 00:16:53.382 as it runs through. 00:16:53.382 --> 00:16:54.150 And then if you-- 00:16:54.150 --> 00:16:55.800 it's flying through the air, there's 00:16:55.800 --> 00:16:57.840 no drag or anything, that's conservative when it 00:16:57.840 --> 00:16:59.320 flies through the air as well. 00:16:59.320 --> 00:17:02.760 And so this system actually is completely 00:17:02.760 --> 00:17:06.030 conservative in its full operation. 00:17:06.030 --> 00:17:08.250 So yeah, so it's very different than a rimless wheel. 00:17:08.250 --> 00:17:10.980 And also, then the other assumption we have to make 00:17:10.980 --> 00:17:26.430 is that the leg instantly goes to the desired theta and rest 00:17:26.430 --> 00:17:27.690 length. 00:17:27.690 --> 00:17:29.670 So again, when it's massless, the rest length, 00:17:29.670 --> 00:17:30.510 it will do that automatically. 00:17:30.510 --> 00:17:31.650 But you have to assume that [INAUDIBLE] 00:17:31.650 --> 00:17:32.750 teleported to that theta. 00:17:32.750 --> 00:17:33.960 So you don't worry about collisions with the ground 00:17:33.960 --> 00:17:34.960 or everything like that. 00:17:34.960 --> 00:17:36.720 Your leg just goes to the theta touchdown. 00:17:36.720 --> 00:17:39.588 So the moment you take off, it's in this new configuration. 00:17:43.340 --> 00:17:44.927 So now what we have to do is, now 00:17:44.927 --> 00:17:46.510 that we have the system and the model, 00:17:46.510 --> 00:17:48.135 we want to pick our surface of section, 00:17:48.135 --> 00:17:51.430 because our goal here is to turn this into a 1-D iterated map, 00:17:51.430 --> 00:17:53.370 right, because that's what this is all about. 00:17:53.370 --> 00:17:54.745 We're going to try to figure out, 00:17:54.745 --> 00:17:57.482 how do we pick a spot where you can just look at one section 00:17:57.482 --> 00:17:59.440 and describe all the dynamics, just being like, 00:17:59.440 --> 00:18:00.640 OK, we're here. 00:18:00.640 --> 00:18:01.690 Simulate to the next one. 00:18:01.690 --> 00:18:02.860 And then we can just iterate this map 00:18:02.860 --> 00:18:05.050 and figure out the fixed points, figure out a lot of things, 00:18:05.050 --> 00:18:06.580 as we did with the rimless wheel. 00:18:06.580 --> 00:18:08.860 So does anyone have an idea of what a good surface of section 00:18:08.860 --> 00:18:09.360 would be? 00:18:15.530 --> 00:18:16.910 No? 00:18:16.910 --> 00:18:20.270 What special configurations are there that we can look at? 00:18:22.538 --> 00:18:23.330 AUDIENCE: Take-off. 00:18:23.330 --> 00:18:24.750 PROFESSOR: Take-off. 00:18:24.750 --> 00:18:26.000 What else? 00:18:26.000 --> 00:18:26.500 That's one. 00:18:30.156 --> 00:18:31.530 AUDIENCE: Full compression. 00:18:31.530 --> 00:18:32.405 PROFESSOR: Full com-- 00:18:32.405 --> 00:18:34.045 AUDIENCE: Or rest length [INAUDIBLE] 00:18:34.045 --> 00:18:35.050 PROFESSOR: At the rest length? 00:18:35.050 --> 00:18:35.560 That's true. 00:18:35.560 --> 00:18:37.913 Full compression you could do, but that 00:18:37.913 --> 00:18:40.330 would be like there'd be a 0 r dot or something like that. 00:18:40.330 --> 00:18:40.630 But yeah. 00:18:40.630 --> 00:18:41.530 No, I mean, you could. 00:18:41.530 --> 00:18:42.940 There's a lot of these special configurations. 00:18:42.940 --> 00:18:44.380 But the one that you really want to look at, 00:18:44.380 --> 00:18:46.380 the one that collapses things down the most 00:18:46.380 --> 00:18:47.830 is if you look at the apex. 00:18:47.830 --> 00:18:50.750 So you look at the maximum height of this flight. 00:18:50.750 --> 00:18:52.000 So we can just look at that y. 00:18:52.000 --> 00:18:55.450 And the reason we can do this I can describe right here. 00:18:55.450 --> 00:18:56.770 So we look in the flight phase. 00:19:00.820 --> 00:19:04.360 Now, first of all, x, we don't really 00:19:04.360 --> 00:19:07.202 care about what x is for the stability. 00:19:07.202 --> 00:19:09.160 Doesn't matter where it is along that position. 00:19:09.160 --> 00:19:10.990 That doesn't affect the dynamics directly. 00:19:10.990 --> 00:19:12.970 So we don't care about x. 00:19:20.750 --> 00:19:23.570 So y, we definitely do care about. 00:19:23.570 --> 00:19:24.260 This matters. 00:19:27.470 --> 00:19:31.730 x dot, well, x dot, because we know it's conservative 00:19:31.730 --> 00:19:36.950 and we know that since at the apex y dot equals 0, 00:19:36.950 --> 00:19:39.680 then x dot is purely a function of y, right? 00:19:39.680 --> 00:19:43.160 We can look at the total energy of the system, which 00:19:43.160 --> 00:19:50.960 is 1/2 m V squared plus mgy. 00:19:50.960 --> 00:19:54.680 We know y dot is 0, so that means that V is just x dot. 00:19:54.680 --> 00:20:00.590 And so you can write x dot is a function of y. 00:20:00.590 --> 00:20:02.668 And so here, we know this is 0. 00:20:02.668 --> 00:20:03.710 We don't care about this. 00:20:03.710 --> 00:20:05.252 This one we can find directly from y. 00:20:05.252 --> 00:20:08.690 And so everything that happens between one apex, another apex, 00:20:08.690 --> 00:20:11.550 we know if we just know y. 00:20:11.550 --> 00:20:12.420 You see that? 00:20:12.420 --> 00:20:13.230 Yes. 00:20:13.230 --> 00:20:15.530 AUDIENCE: We're always adjusting the angles [INAUDIBLE] 00:20:15.530 --> 00:20:18.558 landing so that it is vertical at the [INAUDIBLE] 00:20:18.558 --> 00:20:20.850 PROFESSOR: It's set at whatever desired angle we're at. 00:20:20.850 --> 00:20:24.390 So there's some nominal angle we want on this touchdown. 00:20:24.390 --> 00:20:26.983 So let's say it's 30 degrees or whatever. 00:20:26.983 --> 00:20:28.650 And so that means that then you take off 00:20:28.650 --> 00:20:30.670 and your angle-- your leg goes to 30 degrees, 00:20:30.670 --> 00:20:31.830 and then you hit there. 00:20:31.830 --> 00:20:35.370 And so the touchdown angle is always the same. 00:20:35.370 --> 00:20:37.770 And so there's no control in this at the moment. 00:20:37.770 --> 00:20:41.105 It's just the passive stability of this balancing system. 00:20:41.105 --> 00:20:42.480 Does that makes sense to everyone 00:20:42.480 --> 00:20:44.022 how we can collapse all these things? 00:20:50.069 --> 00:20:50.902 AUDIENCE: Professor? 00:20:50.902 --> 00:20:51.826 PROFESSOR: Oh, yeah. 00:20:51.826 --> 00:20:53.375 AUDIENCE: [INAUDIBLE] why don't we care about x again? 00:20:53.375 --> 00:20:53.800 Because-- 00:20:53.800 --> 00:20:54.610 PROFESSOR: Because that doesn't affect 00:20:54.610 --> 00:20:55.660 the stability of the system. 00:20:55.660 --> 00:20:57.430 So if we're trying to get somewhere and hit a target, 00:20:57.430 --> 00:20:59.470 yeah, then we have to look at x in our controller. 00:20:59.470 --> 00:21:01.900 But because the x doesn't figure in the dynamics anywhere, 00:21:01.900 --> 00:21:03.610 right, doesn't matter, because again-- 00:21:03.610 --> 00:21:04.510 sorry, this is something I probably should have 00:21:04.510 --> 00:21:05.635 mentioned in the beginning. 00:21:05.635 --> 00:21:07.770 It's not going downhill or anything like that. 00:21:07.770 --> 00:21:09.920 So how far it goes in x doesn't matter. 00:21:09.920 --> 00:21:12.500 The dynamics are going to be the same invariant to that, 00:21:12.500 --> 00:21:13.425 if that makes sense. 00:21:13.425 --> 00:21:15.550 And so because of that, I mean, x will be changing, 00:21:15.550 --> 00:21:18.003 but it won't affect the next apex height, 00:21:18.003 --> 00:21:20.253 because it doesn't figure into the dynamics like that. 00:21:20.253 --> 00:21:21.370 Does that make sense? 00:21:25.042 --> 00:21:26.420 All right. 00:21:26.420 --> 00:21:30.460 So we can represent the whole thing just using y. 00:21:30.460 --> 00:21:32.755 I guess that's the important thing to realize. 00:21:32.755 --> 00:21:33.880 So what do I want to erase? 00:21:33.880 --> 00:21:34.797 I'll erase this stuff. 00:21:38.099 --> 00:21:39.560 I'll throw this back up. 00:21:49.300 --> 00:21:51.260 Hmm. 00:21:51.260 --> 00:21:52.370 This isn't ideal. 00:22:01.092 --> 00:22:03.300 All right, this is not going to get better than that. 00:22:03.300 --> 00:22:04.460 All right. 00:22:04.460 --> 00:22:06.110 So we just go through this transition 00:22:06.110 --> 00:22:08.390 and we go from apex to apex, we can 00:22:08.390 --> 00:22:15.470 look at is we have y apex at n. 00:22:15.470 --> 00:22:21.740 Then we need to figure out how that translates into the y, 00:22:21.740 --> 00:22:26.973 x dot, and y dot at that apex. 00:22:26.973 --> 00:22:29.390 All right, so we need to map this one piece of information 00:22:29.390 --> 00:22:31.980 into all of these. 00:22:31.980 --> 00:22:38.330 And then we can map that into the y, 00:22:38.330 --> 00:22:44.540 x dot, y dot at the touchdown, so when it comes in and hits. 00:22:44.540 --> 00:22:49.190 Now we have to change our coordinate system to r theta, 00:22:49.190 --> 00:22:51.230 r dot, theta dot. 00:22:51.230 --> 00:22:54.650 This is still a touchdown, and we swing that 00:22:54.650 --> 00:22:56.930 through in the stance phase. 00:22:56.930 --> 00:23:03.290 Then we get that to r theta, r dot, theta dot at take-off. 00:23:03.290 --> 00:23:06.038 So my D and my O are as different as I 00:23:06.038 --> 00:23:06.830 can make them look. 00:23:09.920 --> 00:23:13.790 Then at take-off again, we switch to our aerial phase. 00:23:13.790 --> 00:23:22.460 We go back to y, x dot, y dot at takeoff. 00:23:22.460 --> 00:23:31.430 And that brings us, then, to a y, x dot, y dot at apex. 00:23:31.430 --> 00:23:37.013 And then here, of course, we grab y n plus 1 at apex. 00:23:37.013 --> 00:23:40.640 [PHONE RINGING] 00:23:40.640 --> 00:23:43.232 So these transitions, I'll number of them 00:23:43.232 --> 00:23:44.690 and make this a little bit clearer. 00:23:44.690 --> 00:23:49.160 That's one, two, three, four. 00:23:51.570 --> 00:23:52.070 Sorry. 00:23:57.726 --> 00:24:01.430 So all right. 00:24:01.430 --> 00:24:04.895 And the key now is to figure out how 00:24:04.895 --> 00:24:07.020 to push our system throughout all these transitions 00:24:07.020 --> 00:24:09.270 so we can figure out what this mapping is going to be. 00:24:13.970 --> 00:24:19.330 So some of these are pretty easy to do, and some of them 00:24:19.330 --> 00:24:21.820 can be quite tricky to do. 00:24:21.820 --> 00:24:25.090 But all of them are quite manageable. 00:24:25.090 --> 00:24:35.500 So what you look at first is the energy, which 00:24:35.500 --> 00:24:39.430 is our kinetic plus potential, again, 00:24:39.430 --> 00:24:44.155 is 1/2 mv squared plus mgy. 00:24:47.350 --> 00:24:51.160 So at the first step, if you want to convert-- 00:24:51.160 --> 00:24:54.430 if we want to take y apex and turn it into these guys again, 00:24:54.430 --> 00:24:58.750 we can have the energy of the apex 00:24:58.750 --> 00:25:05.320 we know is 1/2 m x dot, again, because we know y dot 0, 00:25:05.320 --> 00:25:10.335 plus mgy is equal to a constant. 00:25:10.335 --> 00:25:11.960 And so then this means that we can say, 00:25:11.960 --> 00:25:18.730 all right, x dot equals this function of y. 00:25:24.524 --> 00:25:26.560 So we can figure out what x dot is, 00:25:26.560 --> 00:25:28.400 [INAUDIBLE] and then we know also-- 00:25:28.400 --> 00:25:29.440 sorry. 00:25:29.440 --> 00:25:32.410 We know y dot equals 0. 00:25:32.410 --> 00:25:36.010 And so here, then, that gets our first transition. 00:25:36.010 --> 00:25:42.010 Two then, if we want-- 00:25:42.010 --> 00:25:44.170 we know that x dot at touchdown-- 00:25:44.170 --> 00:25:47.650 so this is the ballistic phase coming down from apex-- 00:25:47.650 --> 00:25:52.160 is equal to x dot apex, because there's no forces on it, 00:25:52.160 --> 00:25:54.970 so it's just going to keep carrying forward in x. 00:25:54.970 --> 00:26:01.660 And then we know y touchdown is going to be r naught cosine 00:26:01.660 --> 00:26:02.270 theta naught. 00:26:02.270 --> 00:26:03.728 So this is what I was talking about 00:26:03.728 --> 00:26:05.080 with there's the desired theta. 00:26:05.080 --> 00:26:06.497 So your touchdown, you know you're 00:26:06.497 --> 00:26:09.400 going to touch down at this desired theta. 00:26:09.400 --> 00:26:12.400 And this, then, again, since we know our energy, 00:26:12.400 --> 00:26:16.180 we can get y dot touchdown, all right. 00:26:16.180 --> 00:26:22.210 So here that gets our transition from apex to touchdown. 00:26:22.210 --> 00:26:25.090 Now this is just a coordinate transformation. 00:26:25.090 --> 00:26:26.380 It's pretty easy to do. 00:26:26.380 --> 00:26:27.790 Just make sure you get your velocities right. 00:26:27.790 --> 00:26:29.140 You have to decompose them properly. 00:26:29.140 --> 00:26:30.910 But you can just transform those coordinates, 00:26:30.910 --> 00:26:32.993 and then you get that without too much difficulty. 00:26:32.993 --> 00:26:37.694 But then you get to the tricky one, and the tricky one-- 00:26:37.694 --> 00:26:42.170 [INAUDIBLE] The tricky one is the stance phase. 00:26:42.170 --> 00:26:49.930 So how do you push yourself from when you come in 00:26:49.930 --> 00:26:53.380 and hit to squishing down and launching back out? 00:26:53.380 --> 00:26:56.110 So that's the involved part of this. 00:26:56.110 --> 00:27:02.740 So four is stance dynamics. 00:27:07.300 --> 00:27:10.380 AUDIENCE: Is that y a touchdown [INAUDIBLE] 00:27:10.380 --> 00:27:11.716 PROFESSOR: Pardon? 00:27:11.716 --> 00:27:14.710 AUDIENCE: y [INAUDIBLE] 00:27:14.710 --> 00:27:18.340 PROFESSOR: Oh, yeah, this is touchdown. 00:27:18.340 --> 00:27:18.940 Sorry. 00:27:18.940 --> 00:27:22.060 So to get to two, to get this falling phase, 00:27:22.060 --> 00:27:23.210 you know x doesn't change. 00:27:23.210 --> 00:27:25.150 There's no forces in the x direction. 00:27:25.150 --> 00:27:27.790 y touchdown, you fix this angle. 00:27:27.790 --> 00:27:29.475 That's instantaneous warp of the leg. 00:27:29.475 --> 00:27:30.850 So you know when it touches down, 00:27:30.850 --> 00:27:32.080 it's going to be this high, because that's 00:27:32.080 --> 00:27:34.660 when touchdown's defined, is when the toe hits the ground. 00:27:34.660 --> 00:27:35.830 And then here, you know your height. 00:27:35.830 --> 00:27:36.550 You know this. 00:27:36.550 --> 00:27:37.925 And again, conservation of energy 00:27:37.925 --> 00:27:41.603 will give you this at the y dot touchdown 00:27:41.603 --> 00:27:43.270 since that just allows you to figure out 00:27:43.270 --> 00:27:44.437 your state when you come in. 00:27:47.410 --> 00:27:50.350 So stance dynamics. 00:27:50.350 --> 00:27:55.450 So here, you've got a idea of what 00:27:55.450 --> 00:27:57.220 you expect the system to do. 00:27:57.220 --> 00:27:59.710 You have a spring. 00:27:59.710 --> 00:28:02.880 And if you're running in steady state, 00:28:02.880 --> 00:28:06.032 I believe you take off the same angle you come in. 00:28:06.032 --> 00:28:07.490 But so what you can imagine happens 00:28:07.490 --> 00:28:10.990 is this thing swings around like this, but the center of mass 00:28:10.990 --> 00:28:13.408 goes in and comes back out. 00:28:13.408 --> 00:28:14.950 So again, it's doing that compression 00:28:14.950 --> 00:28:17.533 and then launching itself, all right. 00:28:17.533 --> 00:28:19.200 So I think it's a assumption Raibert did 00:28:19.200 --> 00:28:20.920 for the steady state operation. 00:28:20.920 --> 00:28:22.580 It comes in, out at the same angle, 00:28:22.580 --> 00:28:24.680 and serves as a compression in the middle. 00:28:24.680 --> 00:28:26.350 And so again, this kind of behavior 00:28:26.350 --> 00:28:27.730 is, again, one of those ways of defining running, 00:28:27.730 --> 00:28:28.960 like for the Groucho kind of running, 00:28:28.960 --> 00:28:30.070 is that you see the center of mass 00:28:30.070 --> 00:28:31.750 come in to the center of pressure, 00:28:31.750 --> 00:28:33.580 as opposed to vaulting over it. 00:28:33.580 --> 00:28:36.610 If this was a stiff leg, it would go like this. 00:28:36.610 --> 00:28:38.950 So that distinction is one way of defining running. 00:28:44.170 --> 00:28:47.836 So we can define take-off, so we know our final condition. 00:28:52.700 --> 00:28:58.542 So take-off happens when r equals r naught. 00:28:58.542 --> 00:29:00.250 So when you get back to your rest length, 00:29:00.250 --> 00:29:02.830 you assume that your body then, it goes back 00:29:02.830 --> 00:29:04.780 to the ballistic phase. 00:29:04.780 --> 00:29:08.830 All right, then we can write down our energies here. 00:29:08.830 --> 00:29:10.870 Now again, we're in the polar coordinates 00:29:10.870 --> 00:29:13.622 now, so our energies look a little bit different. 00:29:16.280 --> 00:29:24.220 Our kinetic energy looks like that, 00:29:24.220 --> 00:29:34.370 and our potential energy is this term, and what else do we need? 00:29:34.370 --> 00:29:36.785 What else? 00:29:36.785 --> 00:29:37.910 AUDIENCE: Spring potential. 00:29:37.910 --> 00:29:38.910 PROFESSOR: Spring, yeah. 00:29:38.910 --> 00:29:49.342 So it's k over 2 x-- sorry-- 00:29:49.342 --> 00:29:54.800 r minus r naught squared. 00:29:54.800 --> 00:30:01.280 And so using that, you probably remember Lagrange. 00:30:01.280 --> 00:30:04.910 It's not that bad of a system. 00:30:04.910 --> 00:30:06.960 You can get the equations of motion. 00:30:06.960 --> 00:30:11.720 The equation of motion here, you get-- 00:30:25.700 --> 00:30:26.671 oh, sorry. 00:30:33.900 --> 00:30:51.130 And so you have all your expected terms, 00:30:51.130 --> 00:30:54.720 centrifugal terms, Coriolis terms, all 00:30:54.720 --> 00:30:58.920 that if you try to make sense of what this is. 00:30:58.920 --> 00:31:02.670 And so if you want to get from touchdown to take-off, 00:31:02.670 --> 00:31:04.800 you can't get a closed form solution from this. 00:31:04.800 --> 00:31:06.900 You can simulate pretty easily, obviously. 00:31:06.900 --> 00:31:08.317 But you don't get the closed form, 00:31:08.317 --> 00:31:10.440 so you can't get this analytical kind of mapping. 00:31:10.440 --> 00:31:12.930 But if you use a small angle and small displacement 00:31:12.930 --> 00:31:15.360 approximation, you can linearize the system, 00:31:15.360 --> 00:31:18.660 and you can get a closed form solution for this touchdown 00:31:18.660 --> 00:31:20.850 to take-off phase, all right. 00:31:20.850 --> 00:31:24.220 And so that's this assumption that theta is much, 00:31:24.220 --> 00:31:25.240 much less than 1. 00:31:25.240 --> 00:31:30.780 Your delta r over r0 is much less than 1. 00:31:30.780 --> 00:31:33.820 And so then you can get to a closed form solution. 00:31:33.820 --> 00:31:34.585 It's kind of ugly. 00:31:34.585 --> 00:31:35.460 It's not really ugly. 00:31:35.460 --> 00:31:37.377 I mean, it's not so ugly as to be prohibitive, 00:31:37.377 --> 00:31:40.455 but I'm not going to write it down. 00:31:40.455 --> 00:31:42.580 But that lets you actually get a closed form return 00:31:42.580 --> 00:31:44.610 map, all right. 00:31:44.610 --> 00:31:46.530 And something cool about that is that you 00:31:46.530 --> 00:31:52.320 get these two fixed points. 00:31:52.320 --> 00:31:54.915 You get one stable fixed point and one unstable fixed point. 00:31:54.915 --> 00:31:58.280 And-- hmm. 00:32:01.630 --> 00:32:03.520 And actually, a number of people thought 00:32:03.520 --> 00:32:05.680 that a system like this, this conservative system, 00:32:05.680 --> 00:32:08.680 shouldn't be able to get that sort of stability, 00:32:08.680 --> 00:32:12.460 because if you remember in the rimless wheel, 00:32:12.460 --> 00:32:15.080 the energy loss was critical towards achieving 00:32:15.080 --> 00:32:15.790 the stability. 00:32:15.790 --> 00:32:19.720 When you went faster, you hit harder, you lost more energy. 00:32:19.720 --> 00:32:23.980 And when you went slower, you didn't lose much energy, 00:32:23.980 --> 00:32:27.110 so you were able to speed up as you went down this ramp. 00:32:27.110 --> 00:32:29.592 So the fact that you can achieve stability in this system 00:32:29.592 --> 00:32:32.050 is kind of cool, too, because it's completely conservative, 00:32:32.050 --> 00:32:34.420 yet somehow, dynamics are able to actually push it 00:32:34.420 --> 00:32:37.380 towards a repeatable state. 00:32:37.380 --> 00:32:39.060 [INAUDIBLE] an then looking at-- 00:32:39.060 --> 00:32:42.070 we're going to see this return map. 00:32:42.070 --> 00:32:48.280 This is from [INAUDIBLE] Geyer's thesis work, I believe. 00:32:51.640 --> 00:32:59.010 And it's pretty cool, if I can get to it for you. 00:33:12.370 --> 00:33:13.198 There we go. 00:33:16.480 --> 00:33:17.020 There we go. 00:33:20.590 --> 00:33:23.530 So this is actually the return map 00:33:23.530 --> 00:33:25.030 for this small angle approximation 00:33:25.030 --> 00:33:26.680 you can get analytically. 00:33:26.680 --> 00:33:28.732 And you see there's this unstable fixed point, 00:33:28.732 --> 00:33:30.940 and then down there, there's that stable fixed point. 00:33:30.940 --> 00:33:35.680 So you zoom in, and you can see the stability region, 00:33:35.680 --> 00:33:38.110 that that unstable one is defining the boundary 00:33:38.110 --> 00:33:41.050 of the stability for that. 00:33:41.050 --> 00:33:45.010 The apex height, why does it get to that minimum there? 00:33:45.010 --> 00:33:49.280 Why doesn't it go below that 0.87? 00:33:49.280 --> 00:33:49.780 Anyone know? 00:33:55.610 --> 00:33:58.295 Why don't we look at it below that? 00:33:58.295 --> 00:34:00.670 AUDIENCE: Because that's the maximum range? 00:34:00.670 --> 00:34:04.570 PROFESSOR: Yeah, that's your height at the cosine theta 00:34:04.570 --> 00:34:06.070 length of the leg. 00:34:06.070 --> 00:34:08.530 And so you can't get below that. 00:34:08.530 --> 00:34:10.210 And then the weird thing is, and this 00:34:10.210 --> 00:34:12.168 is something I was talking about with Russ just 00:34:12.168 --> 00:34:14.545 before he went home, was why does 00:34:14.545 --> 00:34:15.670 it keep climbing like that? 00:34:23.966 --> 00:34:25.929 AUDIENCE: Why does it keep climbing [INAUDIBLE] 00:34:25.929 --> 00:34:28.290 PROFESSOR: Well, because this is conservative. 00:34:28.290 --> 00:34:30.400 So how can you have an instability 00:34:30.400 --> 00:34:32.675 that pushes you up to larger and larger apex heights? 00:34:32.675 --> 00:34:34.460 AUDIENCE: [INAUDIBLE] to a more vertical-- 00:34:34.460 --> 00:34:36.460 PROFESSOR: At some point, though, it should cap. 00:34:36.460 --> 00:34:38.960 There should be another stable fixed point bouncing straight 00:34:38.960 --> 00:34:40.690 up. 00:34:40.690 --> 00:34:41.710 There isn't. 00:34:41.710 --> 00:34:43.929 The thing is that, what we think is 00:34:43.929 --> 00:34:45.610 that that unstable one right there is 00:34:45.610 --> 00:34:47.063 your vertical bouncing. 00:34:47.063 --> 00:34:49.480 And the thing is that, actually, when you linearize these, 00:34:49.480 --> 00:34:52.239 they don't quite conserve energy anymore. 00:34:52.239 --> 00:34:54.710 This actually can be a non-conservative term. 00:34:54.710 --> 00:34:56.252 So we think that's what this is from. 00:34:56.252 --> 00:34:58.252 So it really shouldn't be climbing up like that, 00:34:58.252 --> 00:34:59.977 because when you simulate these things, 00:34:59.977 --> 00:35:01.810 you don't get above that second fixed point. 00:35:01.810 --> 00:35:04.000 It rolls up to this unstable one. 00:35:04.000 --> 00:35:05.710 And so then-- but looking at this again, 00:35:05.710 --> 00:35:07.300 now here's another strange thing. 00:35:07.300 --> 00:35:09.370 It looks like it should be globally stable, then, right? 00:35:09.370 --> 00:35:10.828 Because if in practice we can't get 00:35:10.828 --> 00:35:12.930 above that unstable fixed point-- 00:35:12.930 --> 00:35:17.170 and if you look at this apex axis, right, the higher it is, 00:35:17.170 --> 00:35:18.130 the slower it moves. 00:35:18.130 --> 00:35:20.185 So at your minimum apex height, right, that's 00:35:20.185 --> 00:35:21.310 when you're moving fastest. 00:35:21.310 --> 00:35:23.685 It looks like it's stable throughout the entire operating 00:35:23.685 --> 00:35:24.460 regime, right? 00:35:24.460 --> 00:35:27.430 The fastest it can move is that little guy on the far left 00:35:27.430 --> 00:35:29.210 where it's apparently stable. 00:35:29.210 --> 00:35:31.460 And the slowest it can move is this vertical bouncing, 00:35:31.460 --> 00:35:33.145 [INAUDIBLE] fixed point. 00:35:33.145 --> 00:35:34.270 So how do you explain that? 00:35:39.700 --> 00:35:42.043 AUDIENCE: [INAUDIBLE] 00:35:42.043 --> 00:35:43.960 PROFESSOR: Well, that's the thing is that it-- 00:35:43.960 --> 00:35:45.460 that's pretty much exactly the issue 00:35:45.460 --> 00:35:49.150 is that your failure mode can come from coming in, squishing 00:35:49.150 --> 00:35:49.803 really low-- 00:35:49.803 --> 00:35:51.220 I'm going to fall in my face here, 00:35:51.220 --> 00:35:52.990 so I should let everyone see it. 00:35:52.990 --> 00:35:54.490 You can compress and get really low, 00:35:54.490 --> 00:35:55.660 and then you bounce forward. 00:35:55.660 --> 00:35:57.010 And if you bounce forward at such a directory 00:35:57.010 --> 00:35:58.510 that you never get high enough for your leg 00:35:58.510 --> 00:36:00.130 to be out, touch it down, then you're 00:36:00.130 --> 00:36:01.700 just going to bounce forward, land on your face. 00:36:01.700 --> 00:36:03.160 You can't actually warp your leg. 00:36:03.160 --> 00:36:04.035 Does that make sense? 00:36:04.035 --> 00:36:05.411 So that can be your failure mode. 00:36:05.411 --> 00:36:06.970 So you bounce in, you shoot across, 00:36:06.970 --> 00:36:09.680 and then if you never have an apex high enough, 00:36:09.680 --> 00:36:11.305 you'll never get your leg ahead of you. 00:36:11.305 --> 00:36:12.430 So that's the failure mode. 00:36:12.430 --> 00:36:15.280 So that's actually then what we see when we simulate it. 00:36:15.280 --> 00:36:17.655 And the simulation actually looks a little bit different. 00:36:17.655 --> 00:36:20.020 There's actually-- this guy looks pretty good, 00:36:20.020 --> 00:36:22.100 but if you simulate it and you have 00:36:22.100 --> 00:36:24.750 slightly different parameters, that 00:36:24.750 --> 00:36:26.500 curve there comes down a little bit lower, 00:36:26.500 --> 00:36:28.000 and so you're actually able to be from the top 00:36:28.000 --> 00:36:29.950 and bounce in and overshoot and go unstable. 00:36:29.950 --> 00:36:31.270 So it's not like you're guaranteed stability 00:36:31.270 --> 00:36:32.978 by starting to bounce vertically, either. 00:36:32.978 --> 00:36:36.358 So this will change with different energies, 00:36:36.358 --> 00:36:38.650 different parameter settings, and everything like that. 00:36:38.650 --> 00:36:41.140 But the takehome thing I think that's cool 00:36:41.140 --> 00:36:44.845 is that, even this simplified perspective of it, 00:36:44.845 --> 00:36:46.970 you get this stability in this conservative system. 00:36:46.970 --> 00:36:48.700 You get this stability that you can see in the simulation, 00:36:48.700 --> 00:36:50.908 and that even though it's not dampening on any energy 00:36:50.908 --> 00:36:53.382 or anything like that, is able to bounce 00:36:53.382 --> 00:36:54.340 along and right itself. 00:36:54.340 --> 00:36:56.507 And apparently, what that's related to-- apparently, 00:36:56.507 --> 00:36:58.570 people didn't used to think that was possible. 00:36:58.570 --> 00:37:01.240 But the coordinate transform is actually 00:37:01.240 --> 00:37:03.790 enough to get that kind of stability, is that-- 00:37:03.790 --> 00:37:06.010 it's called piecewise holonomic. 00:37:06.010 --> 00:37:07.660 And by switching these coordinates, 00:37:07.660 --> 00:37:09.760 you're actually able to get the-- 00:37:09.760 --> 00:37:11.955 that allows the dynamics to stabilize themselves, 00:37:11.955 --> 00:37:14.080 apparently, as opposed to a normal holonomic system 00:37:14.080 --> 00:37:16.570 apparently couldn't actually converge in back 00:37:16.570 --> 00:37:17.758 towards this fixed point. 00:37:17.758 --> 00:37:19.550 If you think about the pendulum [INAUDIBLE] 00:37:19.550 --> 00:37:21.800 like that, that's not going to converge [INAUDIBLE] 00:37:21.800 --> 00:37:23.050 It has to conserve the energy. 00:37:23.050 --> 00:37:25.690 But apparently, with piecewise holonomic systems, 00:37:25.690 --> 00:37:27.650 it's available. 00:37:27.650 --> 00:37:30.040 But you can read some papers and get into that. 00:37:30.040 --> 00:37:34.450 But I don't know much more than I just told you. 00:37:38.200 --> 00:37:43.300 All right, so here's another thing that's interesting, 00:37:43.300 --> 00:37:47.110 is that if you look at-- if you want to model those Raibert 00:37:47.110 --> 00:37:51.220 hoppers, right, they obviously aren't completely conservative. 00:37:51.220 --> 00:37:54.332 They do have some mass in the toe and in the leg, 00:37:54.332 --> 00:37:56.290 and so when they hit, they do lose some energy, 00:37:56.290 --> 00:37:58.990 because when that toe hits and sticks to the ground, 00:37:58.990 --> 00:38:01.880 it's going to be a dissipative reaction. 00:38:01.880 --> 00:38:04.920 So you can look at this model by, I think, 00:38:04.920 --> 00:38:08.640 [INAUDIBLE] and Buehler in '91. 00:38:08.640 --> 00:38:11.122 I'm going to use my big chalk again. 00:38:11.122 --> 00:38:14.465 Ah, and here's my old big chalk. 00:38:14.465 --> 00:38:17.230 So looking at this. 00:38:22.675 --> 00:38:23.800 So here we have a toe mass. 00:38:23.800 --> 00:38:28.890 So we have the mass of the body and mass of the toe. 00:38:28.890 --> 00:38:35.155 And this spring can possibly be non-linear if you want. 00:38:35.155 --> 00:38:37.780 And this was largely-- they did some analytics, but a lot of it 00:38:37.780 --> 00:38:39.447 was computationally driven, the analysis 00:38:39.447 --> 00:38:40.570 they did of this system. 00:38:40.570 --> 00:38:43.840 And so the thing is, because of this hit here, 00:38:43.840 --> 00:38:48.745 you have a loss due to the toe mass. 00:38:54.900 --> 00:38:59.875 So must add energy. 00:39:02.980 --> 00:39:06.370 And I do this with a control on the spring. 00:39:06.370 --> 00:39:08.725 You can vary your spring constant like that. 00:39:08.725 --> 00:39:11.840 You can add energy to the system. 00:39:11.840 --> 00:39:16.012 And so the control in the spring is their only control then. 00:39:16.012 --> 00:39:18.220 And you actually can get similar stability properties 00:39:18.220 --> 00:39:18.850 to the SLIP model. 00:39:18.850 --> 00:39:20.475 You can get these when you simulate it. 00:39:20.475 --> 00:39:23.110 You can find the same stable fixed point, unstable fixed 00:39:23.110 --> 00:39:25.520 point kind of behavior. 00:39:25.520 --> 00:39:28.090 And so this is all what we've been talking about. 00:39:28.090 --> 00:39:31.420 The image we have is this sagittal plane dynamics, right? 00:39:31.420 --> 00:39:35.140 The sagittal plane is this plane, and so this idea 00:39:35.140 --> 00:39:37.060 of running through like that. 00:39:37.060 --> 00:39:38.560 Getting a little bit aerial phase so 00:39:38.560 --> 00:39:41.020 I look less ridiculous. 00:39:41.020 --> 00:39:43.135 And so the other thing, though, is that it's 00:39:43.135 --> 00:39:44.260 good for the lateral plane. 00:39:44.260 --> 00:39:48.550 If you look at the lateral plane dynamics, certain animals, 00:39:48.550 --> 00:39:50.890 the same kind of SLIP model and that kind of behavior 00:39:50.890 --> 00:39:52.150 can actually be witnessed. 00:39:52.150 --> 00:39:58.570 So this is, again, something Bob Full spend a lot of time on. 00:39:58.570 --> 00:40:01.450 If you look at a cockroach, which has a tripod gait-- 00:40:04.690 --> 00:40:06.587 here's a cockroach. 00:40:06.587 --> 00:40:08.170 I wish they actually looked like that. 00:40:08.170 --> 00:40:09.773 Probably wouldn't be as disturbing. 00:40:12.731 --> 00:40:14.710 There. 00:40:14.710 --> 00:40:17.590 And so what you have is, they use three legs at a time, 00:40:17.590 --> 00:40:18.670 and they're springy legs. 00:40:22.540 --> 00:40:25.030 As it runs, if you look at the dynamics, 00:40:25.030 --> 00:40:26.770 you actually can see the-- 00:40:26.770 --> 00:40:29.710 you actually can see that these legs are acting like springs. 00:40:29.710 --> 00:40:35.290 And actually, the same SLIP behavior you'd expect, 00:40:35.290 --> 00:40:38.200 you witness in the cockroaches. 00:40:38.200 --> 00:40:41.050 And so there's some cool things here. 00:40:41.050 --> 00:40:42.760 Let me show you. 00:40:42.760 --> 00:40:47.110 So these-- have any of you heard of preflex, 00:40:47.110 --> 00:40:48.625 as opposed to reflex? 00:40:48.625 --> 00:40:52.840 It's like pithy kind of name, I guess. 00:40:52.840 --> 00:40:56.710 So the preflex is where actually, you can actually 00:40:56.710 --> 00:41:01.300 look at the time constant required 00:41:01.300 --> 00:41:03.778 for the monosynaptic reflex, which is the electrical signal 00:41:03.778 --> 00:41:06.070 to go through and actually to the spinal cord and back. 00:41:06.070 --> 00:41:09.550 So not to the brain and to the full path, but just 00:41:09.550 --> 00:41:11.290 the quick reflex kind of response. 00:41:11.290 --> 00:41:14.890 And actually, some of these creatures 00:41:14.890 --> 00:41:17.330 actually respond faster than that. 00:41:17.330 --> 00:41:21.250 So the theory is, is that it's actually 00:41:21.250 --> 00:41:23.843 a musculoskeletal response. 00:41:23.843 --> 00:41:25.760 So it's not even-- it's not controlled at all. 00:41:25.760 --> 00:41:27.100 It's not going to the muscles or anything like that. 00:41:27.100 --> 00:41:27.820 It's tendons. 00:41:27.820 --> 00:41:29.735 It's the springiness in the leg. 00:41:29.735 --> 00:41:32.110 And that is actually what's providing some of the control 00:41:32.110 --> 00:41:33.430 here and some of the stability. 00:41:33.430 --> 00:41:37.210 And so there's a really incredibly awesome video 00:41:37.210 --> 00:41:41.380 that I am definitely going to show you, 00:41:41.380 --> 00:41:46.690 regardless of how inconvenient this thing comes out. 00:41:46.690 --> 00:41:49.347 AUDIENCE: It's not an anticipatory neural. 00:41:49.347 --> 00:41:50.180 PROFESSOR: It's not. 00:41:50.180 --> 00:41:51.722 And that's because you can actually-- 00:41:51.722 --> 00:41:55.930 can perturb them almost instantaneous, 00:41:55.930 --> 00:41:57.100 slam with perturbation. 00:41:57.100 --> 00:41:58.840 And so they can't anticipate this. 00:41:58.840 --> 00:42:01.947 And you actually can see, within one step, a cockroach step, 00:42:01.947 --> 00:42:04.280 they'll spring back, and their center of mass trajectory 00:42:04.280 --> 00:42:05.660 will get back on track. 00:42:05.660 --> 00:42:08.500 And so it's not like they see something coming or there's-- 00:42:08.500 --> 00:42:10.210 they're going to step something that's going to hit them. 00:42:10.210 --> 00:42:12.835 It's that there's just-- they're running along, a perturbation, 00:42:12.835 --> 00:42:16.115 and then they start bouncing right away. 00:42:16.115 --> 00:42:16.990 I'm sorry about this. 00:42:16.990 --> 00:42:22.150 I don't know what's happening here. 00:42:31.730 --> 00:42:32.230 Yeah. 00:42:39.700 --> 00:42:42.340 I'll have to-- 00:42:42.340 --> 00:42:42.840 [INAUDIBLE] 00:42:42.840 --> 00:42:44.970 AUDIENCE: So in that model, is it now the spring 00:42:44.970 --> 00:42:48.456 is actually [INAUDIBLE] Because now that you have x, y, z, 00:42:48.456 --> 00:42:50.462 and it's at an angle, [INAUDIBLE] 00:42:50.462 --> 00:42:51.670 PROFESSOR: Yeah, I mean you-- 00:42:51.670 --> 00:42:53.200 AUDIENCE: Is that how you do that, or-- 00:42:53.200 --> 00:42:54.640 PROFESSOR: If you're doing the lateral plane, 00:42:54.640 --> 00:42:56.595 I think you have to have some sort of springiness like that 00:42:56.595 --> 00:42:57.678 that can push you through. 00:42:57.678 --> 00:43:00.250 But you can actually look at the cockroach running along 00:43:00.250 --> 00:43:01.840 the sagittal plane, and you don't need the [INAUDIBLE] You 00:43:01.840 --> 00:43:02.020 can-- 00:43:02.020 --> 00:43:03.072 AUDIENCE: Oh, [INAUDIBLE] 00:43:03.072 --> 00:43:05.530 PROFESSOR: Yeah, you can treat several legs if they're just 00:43:05.530 --> 00:43:07.892 like one spring in the tripod. 00:43:07.892 --> 00:43:09.100 If there's something called-- 00:43:09.100 --> 00:43:12.100 I don't know if you've seen-- you know Bob Full's work at all 00:43:12.100 --> 00:43:15.508 and these templates and anchors, where the anchors are the more 00:43:15.508 --> 00:43:17.050 complicated system, and the templates 00:43:17.050 --> 00:43:19.023 are these really simplified systems? 00:43:19.023 --> 00:43:20.440 So your SLIP little thing here can 00:43:20.440 --> 00:43:22.780 be like a template, which is a really minimalist model 00:43:22.780 --> 00:43:25.300 but captures some of the essential dynamics 00:43:25.300 --> 00:43:27.698 and does so in a concise way. 00:43:27.698 --> 00:43:29.740 And you can look at just a cockroach as following 00:43:29.740 --> 00:43:30.760 this kind of running. 00:43:30.760 --> 00:43:32.640 I mean, and it actually captures a lot of it. 00:43:32.640 --> 00:43:33.640 I mean, there's more complicated ones. 00:43:33.640 --> 00:43:36.100 If you look at the lateral plane dynamics, I mean, 00:43:36.100 --> 00:43:37.960 you probably need a more complicated model. 00:43:37.960 --> 00:43:39.760 But so-- 00:43:44.090 --> 00:43:46.457 AUDIENCE: What did you say this model was called again? 00:43:46.457 --> 00:43:47.540 PROFESSOR: This is a SLIP. 00:43:47.540 --> 00:43:49.350 Oh, but [INAUDIBLE] and Buehler came with this. 00:43:49.350 --> 00:43:51.200 So this is-- yeah, it has a mass of the toe. 00:43:51.200 --> 00:43:52.710 I don't know if it has a different-- 00:43:52.710 --> 00:43:55.265 I don't think it has a different name. 00:43:55.265 --> 00:43:57.050 Yeah, so let me get to this. 00:44:01.500 --> 00:44:02.250 There you go. 00:44:02.250 --> 00:44:03.690 All right. 00:44:03.690 --> 00:44:06.990 So first, the less exciting thing, I think, 00:44:06.990 --> 00:44:08.670 right here, but still pretty cool. 00:44:11.410 --> 00:44:14.727 So measuring the force on these cockroaches' steps, 00:44:14.727 --> 00:44:17.310 you could imagine, is difficult. Something they did, actually, 00:44:17.310 --> 00:44:19.980 to facilitate these experiments is they actually have 00:44:19.980 --> 00:44:21.870 these guys running on Jello. 00:44:21.870 --> 00:44:24.390 I think he set up diffraction [INAUDIBLE] on this jello. 00:44:24.390 --> 00:44:26.130 And so you see that, how it changes 00:44:26.130 --> 00:44:27.540 color when they're pushing? 00:44:27.540 --> 00:44:28.500 They're actually able to figure out 00:44:28.500 --> 00:44:30.333 the magnitude and the direction of the force 00:44:30.333 --> 00:44:32.460 to some level of accuracy by looking at that. 00:44:32.460 --> 00:44:34.650 And apparently, orange jello works really well. 00:44:34.650 --> 00:44:36.442 I don't know what it is about orange jello, 00:44:36.442 --> 00:44:39.840 but if you ever want to analyze the forces on a cockroach's 00:44:39.840 --> 00:44:43.740 legs, start with orange jello, if that's the only thing you 00:44:43.740 --> 00:44:46.600 get out of this lecture. 00:44:46.600 --> 00:44:49.923 But yeah, so that's pretty cool, but that's just 00:44:49.923 --> 00:44:51.090 analyzing when it's turning. 00:44:51.090 --> 00:44:53.715 They can figure out-- that's to look at some of the springiness 00:44:53.715 --> 00:44:57.230 and how the force response and the center of mass response 00:44:57.230 --> 00:44:57.730 connect. 00:44:57.730 --> 00:44:59.310 But here. 00:44:59.310 --> 00:45:02.450 This is one of the greatest experiments of all time 00:45:02.450 --> 00:45:02.950 right here. 00:45:02.950 --> 00:45:06.780 All right, so this is looking at the perturbation experienced 00:45:06.780 --> 00:45:08.190 by these cockroaches. 00:45:08.190 --> 00:45:11.607 So what they did, they bolt a cannon to the back, 00:45:11.607 --> 00:45:13.440 because pulling strings and stuff like that, 00:45:13.440 --> 00:45:14.460 it's not fast enough. 00:45:14.460 --> 00:45:16.647 It's nowhere near fast enough. 00:45:16.647 --> 00:45:17.730 This cockroach is running. 00:45:17.730 --> 00:45:20.430 Aw, come on. 00:45:20.430 --> 00:45:22.480 What's going on here? 00:45:22.480 --> 00:45:22.980 All right. 00:45:25.940 --> 00:45:26.540 Really sorry. 00:45:30.430 --> 00:45:31.900 It's running. 00:45:31.900 --> 00:45:33.880 Bam. 00:45:33.880 --> 00:45:34.510 Perturbation. 00:45:34.510 --> 00:45:36.340 The little cannonball actually hits it on the way back, 00:45:36.340 --> 00:45:36.700 you see. 00:45:36.700 --> 00:45:37.270 [LAUGHTER] 00:45:37.270 --> 00:45:38.228 That's not really fair. 00:45:38.228 --> 00:45:40.120 That's double perturbation. 00:45:40.120 --> 00:45:43.480 But they tracked the center of mass of this guy, 00:45:43.480 --> 00:45:45.550 and actually, you could see it gets back 00:45:45.550 --> 00:45:46.992 on in less than a step. 00:45:46.992 --> 00:45:48.700 And if you look at the time scale of this 00:45:48.700 --> 00:45:56.560 and the time scale of their monosynaptic reflex, 00:45:56.560 --> 00:45:57.650 it happens too quickly. 00:45:57.650 --> 00:45:58.870 And so they think it's actually compliance 00:45:58.870 --> 00:46:01.287 in the legs and everything like that that's just passively 00:46:01.287 --> 00:46:03.910 tuned such that it gets banged to the side and rights itself. 00:46:03.910 --> 00:46:06.025 AUDIENCE: Will you put this on the course website? 00:46:06.025 --> 00:46:07.120 PROFESSOR: Actually, I think-- 00:46:07.120 --> 00:46:09.220 I mean, it's Bob Full's video, but he may have it available. 00:46:09.220 --> 00:46:11.740 But a great thing is that the guy who did this thing-- 00:46:11.740 --> 00:46:14.920 Devin did it. 00:46:14.920 --> 00:46:18.425 And he said-- and this is all he said, 00:46:18.425 --> 00:46:20.050 so he didn't go through the full thing. 00:46:20.050 --> 00:46:21.730 But he's like, you'd be surprised 00:46:21.730 --> 00:46:25.750 by how little gunpowder is necessary to perturb 00:46:25.750 --> 00:46:26.350 a cockroach. 00:46:26.350 --> 00:46:27.100 [LAUGHTER] 00:46:27.100 --> 00:46:29.590 And so you could only imagine the first cockroach 00:46:29.590 --> 00:46:30.700 they got out there. 00:46:30.700 --> 00:46:34.120 And they're like ah, this is about the right amount and just 00:46:34.120 --> 00:46:36.440 blows the cockroach up or launches it across the room 00:46:36.440 --> 00:46:37.270 or-- 00:46:37.270 --> 00:46:40.690 so that would have been a fun trial in an experiment to watch 00:46:40.690 --> 00:46:42.868 from a different room, I think. 00:46:42.868 --> 00:46:44.410 Yeah, we'll watch this one more time. 00:46:44.410 --> 00:46:46.306 AUDIENCE: [INAUDIBLE] 00:46:46.306 --> 00:46:48.400 PROFESSOR: [INAUDIBLE] But you see, that's just 00:46:48.400 --> 00:46:50.073 a fantastic little experiment. 00:46:50.073 --> 00:46:51.490 And the perturbation, as you see-- 00:46:51.490 --> 00:46:52.690 I mean, it doesn't know that's coming. 00:46:52.690 --> 00:46:53.990 That hits it like that. 00:46:53.990 --> 00:46:55.730 It just responds almost instantaneously. 00:46:55.730 --> 00:47:01.490 So I think that's really cool. 00:47:01.490 --> 00:47:02.860 And that shows this-- 00:47:02.860 --> 00:47:04.630 I mean that's SLIP in the lateral plane. 00:47:04.630 --> 00:47:07.060 And it shows that these springs and this compliance 00:47:07.060 --> 00:47:07.810 in the actual-- 00:47:07.810 --> 00:47:09.310 in the animal can actually do things 00:47:09.310 --> 00:47:11.290 that just a control couldn't do, that the time 00:47:11.290 --> 00:47:12.748 scale of the response can be faster 00:47:12.748 --> 00:47:15.500 than control could achieve. 00:47:15.500 --> 00:47:18.100 I don't know if I have-- 00:47:18.100 --> 00:47:20.770 yeah, I don't think I have these [INAUDIBLE] 00:47:20.770 --> 00:47:23.580 So does anybody have a question? 00:47:27.880 --> 00:47:28.380 Yeah. 00:47:30.980 --> 00:47:36.680 OK, so that's nature having this spring bouncy compliance. 00:47:36.680 --> 00:47:41.870 But you can see Raibert's hoppers, back in just the '80s 00:47:41.870 --> 00:47:45.320 and '90s, did amazing things, too. 00:47:45.320 --> 00:47:46.510 If you look at-- 00:47:46.510 --> 00:47:49.250 let's see, where's this 1-D monoped? 00:47:49.250 --> 00:47:50.187 There. 00:47:50.187 --> 00:47:52.270 Some of these videos are pretty crappy, but there. 00:47:52.270 --> 00:47:55.920 So that's just a little monoped running around. 00:47:55.920 --> 00:47:56.420 Do you see? 00:47:56.420 --> 00:47:57.950 And I mean, it's constrained to be in the plane. 00:47:57.950 --> 00:47:59.615 It's on a boom, but it could fall down forward 00:47:59.615 --> 00:48:00.290 and everything like that. 00:48:00.290 --> 00:48:01.880 It's not like it's free to roll. 00:48:01.880 --> 00:48:05.300 So it's achieved the stability bouncing around that circle. 00:48:05.300 --> 00:48:09.980 But thought they could do more than that if you look at-- 00:48:09.980 --> 00:48:11.360 they built bipeds. 00:48:14.600 --> 00:48:19.220 This is-- I think, yeah, Russ grabbed this from a VHS tapes. 00:48:19.220 --> 00:48:21.080 So this is actually what's funny is 00:48:21.080 --> 00:48:24.170 that I think this is the fastest biped around, 00:48:24.170 --> 00:48:26.010 but it wasn't quite that fast. 00:48:26.010 --> 00:48:28.973 That's skipping frames, but I think it's still-- 00:48:28.973 --> 00:48:30.890 unless something has changed in the last year. 00:48:30.890 --> 00:48:33.330 It runs like 2.2 meters a second, 00:48:33.330 --> 00:48:35.990 which I think is actually still the fastest biped. 00:48:35.990 --> 00:48:39.318 But Russ would know if anything has changed. 00:48:39.318 --> 00:48:41.730 AUDIENCE: Is it spinning around? 00:48:41.730 --> 00:48:43.610 PROFESSOR: Pardon? 00:48:43.610 --> 00:48:45.020 AUDIENCE: [INAUDIBLE] I'm sorry. 00:48:45.020 --> 00:48:46.930 It looks like it's spinning around but it's not [INAUDIBLE] 00:48:46.930 --> 00:48:47.260 PROFESSOR: Yeah. 00:48:47.260 --> 00:48:47.760 Oh. 00:48:47.760 --> 00:48:52.240 But check out that. 00:48:52.240 --> 00:48:55.630 So here-- oh, come on. 00:48:55.630 --> 00:48:56.662 There. 00:48:56.662 --> 00:48:58.120 So they figured they could do more. 00:48:58.120 --> 00:48:58.850 Here's a biped. 00:48:58.850 --> 00:49:01.130 So this one isn't on a boom. 00:49:01.130 --> 00:49:01.690 Bam. 00:49:01.690 --> 00:49:03.470 Check that out. 00:49:03.470 --> 00:49:05.290 Look at this one more time. 00:49:05.290 --> 00:49:09.610 Running there, [INAUDIBLE] speed, like gymnastics. 00:49:09.610 --> 00:49:12.660 That's like a robot doing a front flip. 00:49:12.660 --> 00:49:16.240 Then here, this is pretty impressive, too. 00:49:21.740 --> 00:49:23.510 Yeah. 00:49:23.510 --> 00:49:27.470 Right over those stairs like nobody's business. 00:49:27.470 --> 00:49:30.488 AUDIENCE: Is there a [INAUDIBLE] 00:49:30.488 --> 00:49:32.030 PROFESSOR: If what were out of phase? 00:49:32.030 --> 00:49:32.536 If the-- 00:49:32.536 --> 00:49:32.912 AUDIENCE: [INAUDIBLE] 00:49:32.912 --> 00:49:34.984 PROFESSOR: If the stairs were out of phase? 00:49:34.984 --> 00:49:35.588 I don't know. 00:49:35.588 --> 00:49:37.130 You could probably imagine setting up 00:49:37.130 --> 00:49:38.570 stairs that would make it easier and stairs 00:49:38.570 --> 00:49:39.653 that would make it harder. 00:49:39.653 --> 00:49:41.112 So yeah, you probably could come up 00:49:41.112 --> 00:49:43.580 with something that would be problematic if it had to hop. 00:49:43.580 --> 00:49:45.140 Well, maybe could hop on one foot for a while. 00:49:45.140 --> 00:49:45.410 I don't know. 00:49:45.410 --> 00:49:46.040 It's on a boom. 00:49:46.040 --> 00:49:49.660 But yeah, so I mean, these are-- 00:49:49.660 --> 00:49:52.160 I mean, even now, you can look at these things, and you're-- 00:49:52.160 --> 00:49:54.170 I mean, they would be amazing if someone just achieved this, 00:49:54.170 --> 00:49:57.090 but these were back in the '80s and '90s that people did this. 00:49:57.090 --> 00:50:01.556 And so I mean, you've probably seen Big Dog. 00:50:01.556 --> 00:50:05.180 Raibert worked on Big Dog, too, and it's like 20 years later 00:50:05.180 --> 00:50:05.872 than this stuff. 00:50:05.872 --> 00:50:08.080 And it's still state of the art, that kind of control 00:50:08.080 --> 00:50:11.184 and similar control in a lot of ways. 00:50:11.184 --> 00:50:12.780 We can do some more of these. 00:50:12.780 --> 00:50:13.280 Oh. 00:50:13.280 --> 00:50:17.510 Oh, there's one that's-- let's see. 00:50:17.510 --> 00:50:19.460 and the thing is that not only do 00:50:19.460 --> 00:50:21.950 these guys do these crazy things, 00:50:21.950 --> 00:50:28.655 but their robustness is really pretty incredible. 00:50:28.655 --> 00:50:30.530 I mean, a lot of these, like the Honda robots 00:50:30.530 --> 00:50:33.950 walk on, carefully, just ground and everything like that. 00:50:33.950 --> 00:50:35.000 Here. 00:50:35.000 --> 00:50:38.595 [INAUDIBLE] towing him along. 00:50:38.595 --> 00:50:40.220 It's probably not the most comfortable. 00:50:40.220 --> 00:50:42.470 It looks like it's a little bit less smooth. 00:50:42.470 --> 00:50:47.840 But I mean, it's running along sidewalks outside of MIT. 00:50:50.912 --> 00:50:53.120 They have-- this other one's [INAUDIBLE] just running 00:50:53.120 --> 00:50:57.320 by itself on grass and everything, too. 00:50:57.320 --> 00:51:00.630 Oh, the quadrupeds are pretty cool, too. 00:51:00.630 --> 00:51:03.433 That's a quadruped running down the hallway. 00:51:03.433 --> 00:51:04.850 And they actually did a cool paper 00:51:04.850 --> 00:51:08.210 called "Four Legs Running as One," "Four Legs that 00:51:08.210 --> 00:51:09.740 Run as One," something-- 00:51:09.740 --> 00:51:13.070 "Running With Four Legs as if it Were One"-- there we go-- 00:51:13.070 --> 00:51:14.780 that used the same kind of control ideas 00:51:14.780 --> 00:51:16.190 but actually can get these quadrupeds 00:51:16.190 --> 00:51:17.910 and actually run based on the same ideas. 00:51:17.910 --> 00:51:20.535 And you see that thing's moving pretty fast and pretty robustly 00:51:20.535 --> 00:51:22.570 right down hallway like that. 00:51:22.570 --> 00:51:29.870 So it's really pretty amazing capabilities, even today. 00:51:33.050 --> 00:51:35.990 There's one I really wish I could find for you. 00:51:35.990 --> 00:51:36.490 Yeah. 00:51:36.490 --> 00:51:40.060 And so interesting thing that some more recent work, 00:51:40.060 --> 00:51:42.573 they've-- 00:51:42.573 --> 00:51:53.760 oh, let's look at [INAUDIBLE] Yeah. 00:51:53.760 --> 00:51:54.410 Check that out. 00:51:58.720 --> 00:52:01.180 Isn't that just amazing, right? 00:52:01.180 --> 00:52:04.593 And actuators that these guys use is hydraulics at the hips 00:52:04.593 --> 00:52:06.010 and then pneumatics along the leg, 00:52:06.010 --> 00:52:08.560 so {?_{?_[FFFT]_?}_?} with a pneumatic. 00:52:08.560 --> 00:52:11.770 Need that so you can rocket yourself that far. 00:52:11.770 --> 00:52:16.895 But it's just incredible. 00:52:16.895 --> 00:52:17.890 Yeah. 00:52:17.890 --> 00:52:23.220 So something that people are working on more recently 00:52:23.220 --> 00:52:26.160 is this thing called SLIP walkers. 00:52:26.160 --> 00:52:29.970 So you can imagine if you have a compass gait with springy legs, 00:52:29.970 --> 00:52:32.040 right, as it's spring constant gets very high, 00:52:32.040 --> 00:52:34.553 it starts behaving just like a compass gait. 00:52:34.553 --> 00:52:36.720 If the spring constant goes to infinity, it's rigid, 00:52:36.720 --> 00:52:38.610 it's going to be just like a compass gait. 00:52:38.610 --> 00:52:40.270 If you let those strings get looser and looser, 00:52:40.270 --> 00:52:41.895 it's going to start being more like one 00:52:41.895 --> 00:52:44.940 of these bouncy bipeds, right, more like this SLIP behavior. 00:52:44.940 --> 00:52:47.190 And so actually, you can get stability properties-- 00:52:47.190 --> 00:52:50.580 I mean, it can be stable through a relatively broad range 00:52:50.580 --> 00:52:54.012 of bounciness so that the same robot can vault over 00:52:54.012 --> 00:52:55.470 its legs and walking and then start 00:52:55.470 --> 00:52:56.760 bouncing and start running. 00:52:56.760 --> 00:52:58.485 It actually could do that transition. 00:52:58.485 --> 00:53:00.360 And that's a cool anything also that you see. 00:53:00.360 --> 00:53:01.510 I should have brought this up at the beginning 00:53:01.510 --> 00:53:02.968 when we were talking about biology. 00:53:02.968 --> 00:53:06.690 But those transitions are interesting, 00:53:06.690 --> 00:53:09.000 because if you look at the efficiency 00:53:09.000 --> 00:53:12.130 of human locomotion-- so you can measure O2 consumption, 00:53:12.130 --> 00:53:15.550 and you can see how efficient it is to walk at different speeds 00:53:15.550 --> 00:53:17.450 and to run at different speeds. 00:53:17.450 --> 00:53:20.570 You can look at the efficiency curve for-- 00:53:20.570 --> 00:53:28.920 I believe this is speed, and this is, let's say, efficiency. 00:53:28.920 --> 00:53:30.600 And so you have something with walking 00:53:30.600 --> 00:53:33.060 and some curve like this. 00:53:37.780 --> 00:53:42.360 And then running, you have some curve like this. 00:53:42.360 --> 00:53:45.150 I mean, quantitatively, these could be wrong, 00:53:45.150 --> 00:53:47.020 but this is a qualitative feature. 00:53:47.020 --> 00:53:49.980 So if you look at the efficiency of a person walking 00:53:49.980 --> 00:53:51.480 and you just put them on a treadmill 00:53:51.480 --> 00:53:53.460 and just tell them, start walking, speed it up, speed it 00:53:53.460 --> 00:53:54.990 up, and tell them just to transition 00:53:54.990 --> 00:53:58.620 to running whenever they want, they'll transition right here. 00:53:58.620 --> 00:53:59.940 That switches. 00:53:59.940 --> 00:54:01.640 And so you know the feeling, I'm sure, 00:54:01.640 --> 00:54:02.820 where you're walking faster and faster 00:54:02.820 --> 00:54:04.390 and suddenly it just feels more comfortable to just start 00:54:04.390 --> 00:54:05.550 jogging. 00:54:05.550 --> 00:54:08.800 Apparently, that happens when the efficiency of that motion 00:54:08.800 --> 00:54:10.800 outweighs that of walking, because obviously you 00:54:10.800 --> 00:54:12.720 can walk where it's uncomfortable to walk and run 00:54:12.720 --> 00:54:14.053 where it's uncomfortable to run. 00:54:14.053 --> 00:54:16.360 But the natural transition happens at that efficiency, 00:54:16.360 --> 00:54:16.860 bifurcation. 00:54:16.860 --> 00:54:18.138 AUDIENCE: [INAUDIBLE] 00:54:18.138 --> 00:54:19.287 PROFESSOR: Pardon? 00:54:19.287 --> 00:54:20.370 Yeah, that should be cost. 00:54:26.267 --> 00:54:27.350 Now it's even more skewed. 00:54:36.260 --> 00:54:38.570 There we go. 00:54:38.570 --> 00:54:41.030 Thank you. 00:54:41.030 --> 00:54:44.880 Yeah, so [INAUDIBLE] SLIP walkers, I think. 00:54:44.880 --> 00:54:47.090 Hopefully, you can get some of this in transition, 00:54:47.090 --> 00:54:49.280 but I'm not sure exactly where they are right now. 00:54:49.280 --> 00:54:51.560 I don't know if they have any videos of those. 00:54:51.560 --> 00:54:54.920 There's some papers about them. 00:54:54.920 --> 00:55:01.950 And I think that's all the stuff that Russ wanted me to convey. 00:55:07.070 --> 00:55:09.660 So we just have some idea of this culture 00:55:09.660 --> 00:55:12.785 of these cool running models and these springy things 00:55:12.785 --> 00:55:14.660 and these preflexes and stuff, and in nature, 00:55:14.660 --> 00:55:18.740 the fact that springs are critical to locomotion 00:55:18.740 --> 00:55:21.470 of horses and the stability of the cockroach running 00:55:21.470 --> 00:55:23.040 and everything like that. 00:55:23.040 --> 00:55:26.150 So if anyone has any questions, I'd 00:55:26.150 --> 00:55:27.695 love to answer them about anything. 00:55:27.695 --> 00:55:29.180 AUDIENCE: When you were saying how they were-- 00:55:29.180 --> 00:55:31.790 the transitioning using the spring model between walking 00:55:31.790 --> 00:55:32.557 to running-- 00:55:32.557 --> 00:55:33.890 PROFESSOR: Oh, the SLIP walkers? 00:55:33.890 --> 00:55:34.700 AUDIENCE: Hmm? 00:55:34.700 --> 00:55:35.867 PROFESSOR: The SLIP walkers? 00:55:35.867 --> 00:55:37.670 AUDIENCE: Yeah, the SLIP walkers. 00:55:37.670 --> 00:55:40.730 Is it still [INAUDIBLE] elastic collision, 00:55:40.730 --> 00:55:44.630 or do they [INAUDIBLE] 00:55:44.630 --> 00:55:46.160 PROFESSOR: I think in practice. 00:55:46.160 --> 00:55:47.825 I think they have idealized legs. 00:55:50.840 --> 00:55:53.270 I think they've done simulations idealized legs, too. 00:55:53.270 --> 00:55:55.653 But you can do either way, I'm pretty sure. 00:55:55.653 --> 00:55:57.320 I haven't looked at that work as closely 00:55:57.320 --> 00:55:58.403 as I probably should have. 00:55:58.403 --> 00:56:01.430 But yeah, for the walking, vaulting behavior, 00:56:01.430 --> 00:56:03.860 you need either a toe thing that's going to-- 00:56:03.860 --> 00:56:06.860 or you can let your spring constant go to infinity. 00:56:06.860 --> 00:56:09.470 And then you'll have a rigid impact, right. 00:56:09.470 --> 00:56:12.752 So you can treat it that way, too. 00:56:12.752 --> 00:56:14.210 But then yeah, if you loosen it up, 00:56:14.210 --> 00:56:16.400 then you're not going to have the intermediate still lossy 00:56:16.400 --> 00:56:16.900 behavior. 00:56:16.900 --> 00:56:18.890 But yeah, I think they have done stuff 00:56:18.890 --> 00:56:21.350 with idealized legs and probably non-idealized ones, too. 00:56:21.350 --> 00:56:24.110 So anything else? 00:56:28.312 --> 00:56:29.770 AUDIENCE: If you get it just right, 00:56:29.770 --> 00:56:32.510 can you adjust the spring constant so 00:56:32.510 --> 00:56:37.670 that you'll get the push off right before landing? 00:56:37.670 --> 00:56:39.450 PROFESSOR: Oh, you mean toe off, kind of, 00:56:39.450 --> 00:56:41.200 or the toe-off thing that pushes you 00:56:41.200 --> 00:56:42.700 forward that [INAUDIBLE] efficiency? 00:56:42.700 --> 00:56:43.367 AUDIENCE: Right. 00:56:43.367 --> 00:56:44.658 PROFESSOR: I think you should-- 00:56:44.658 --> 00:56:46.790 I mean, I'm not sure about how the tuning would 00:56:46.790 --> 00:56:47.998 relate to the walking things. 00:56:47.998 --> 00:56:50.480 But when you have these idealized legs with the spring, 00:56:50.480 --> 00:56:51.500 I mean, you can't be more efficient 00:56:51.500 --> 00:56:53.010 than if you're vaulting over them. 00:56:53.010 --> 00:56:54.980 So I don't know if it connects to the-- 00:56:54.980 --> 00:56:58.130 I mean, it seems like it could connect to that toe off launch, 00:56:58.130 --> 00:57:01.310 but I'm not sure if that is explicit, 00:57:01.310 --> 00:57:04.790 just by having your leg not be a strike 00:57:04.790 --> 00:57:09.440 with the dissipative inelastic collision is going to help you. 00:57:09.440 --> 00:57:11.750 But I don't know if you get the same kind of-- yeah, 00:57:11.750 --> 00:57:14.910 it seems like you should, but I don't know if that exact 00:57:14.910 --> 00:57:15.410 point-- 00:57:15.410 --> 00:57:17.150 I haven't seen that exact point. 00:57:17.150 --> 00:57:19.880 But it could easily be addressed in other papers. 00:57:19.880 --> 00:57:21.360 Yeah, I should-- 00:57:21.360 --> 00:57:23.360 I don't know if I have the references right now, 00:57:23.360 --> 00:57:25.632 but I can make sure that Russ gives those to your 00:57:25.632 --> 00:57:27.620 or get those to you myself next lecture 00:57:27.620 --> 00:57:28.850 if you want to look up some of the SLIP walkers, 00:57:28.850 --> 00:57:30.740 because that's pretty new stuff, I think. 00:57:30.740 --> 00:57:32.290 Yeah.