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 --> 00:00:21.870 at ocw.mit.edu. 00:00:21.870 --> 00:00:25.278 RUSS TEDRAKE: OK, welcome back. 00:00:25.278 --> 00:00:26.820 At the end of last time, we said we'd 00:00:26.820 --> 00:00:29.557 built up a pretty good arsenal of optimal control tools. 00:00:29.557 --> 00:00:31.140 Now, we're going to start pushing them 00:00:31.140 --> 00:00:33.270 in exciting directions, OK? 00:00:33.270 --> 00:00:36.670 So many of you know I work on walking robots. 00:00:36.670 --> 00:00:37.900 It's one of my passions. 00:00:37.900 --> 00:00:42.795 I think it's one of clear unsolved problems in robotics, 00:00:42.795 --> 00:00:43.650 right? 00:00:43.650 --> 00:00:47.880 We have so many things we can do with robotic arms. 00:00:47.880 --> 00:00:49.470 Relatively, we have almost nothing 00:00:49.470 --> 00:00:51.660 that we can do with walking robots. 00:00:51.660 --> 00:00:54.690 Walking down the street is still a challenge, OK? 00:00:54.690 --> 00:01:00.015 So in my mind, one of the essential problems in robotics, 00:01:00.015 --> 00:01:02.640 it also happens to be the thing that animals do just incredibly 00:01:02.640 --> 00:01:03.140 well. 00:01:03.140 --> 00:01:05.760 And even the really dumb animals, you know, 00:01:05.760 --> 00:01:07.520 locomote very, very, very well. 00:01:07.520 --> 00:01:13.020 So some people might even argue it's a predecessor to higher 00:01:13.020 --> 00:01:14.370 cognitive function. 00:01:14.370 --> 00:01:15.612 We'll see. 00:01:15.612 --> 00:01:17.070 OK, so there's lots of good reasons 00:01:17.070 --> 00:01:19.080 why walking robots are tough. 00:01:19.080 --> 00:01:21.810 There's mechanical problems, of course. 00:01:21.810 --> 00:01:23.540 Suddenly, if you're a walking robot, 00:01:23.540 --> 00:01:25.290 you have to start carrying your actuators. 00:01:25.290 --> 00:01:27.720 So you didn't have to do that as a robotic arm. 00:01:27.720 --> 00:01:30.630 And that's created lots of interesting work 00:01:30.630 --> 00:01:32.650 in actuator technologies. 00:01:32.650 --> 00:01:35.280 How do you carry an actuator that can produce torque 00:01:35.280 --> 00:01:37.410 accurately, let's say? 00:01:37.410 --> 00:01:38.910 If you have to carry its own weight, 00:01:38.910 --> 00:01:40.495 that's been a challenging problem. 00:01:40.495 --> 00:01:42.120 There's lots of interesting work there. 00:01:42.120 --> 00:01:43.350 But I will stand by. 00:01:43.350 --> 00:01:46.950 I think that the reason we don't have walking robots today 00:01:46.950 --> 00:01:49.170 is not actuators. 00:01:49.170 --> 00:01:51.055 It's not power density. 00:01:51.055 --> 00:01:53.430 These are problems, but I think the fundamental challenge 00:01:53.430 --> 00:01:54.940 is in control. 00:01:54.940 --> 00:01:55.440 OK. 00:01:55.440 --> 00:01:59.910 And we are now well-suited to make progress 00:01:59.910 --> 00:02:02.440 on that control problem. 00:02:02.440 --> 00:02:02.940 OK. 00:02:02.940 --> 00:02:07.590 So based on what we've learned so far, 00:02:07.590 --> 00:02:09.570 walking robots aren't that different 00:02:09.570 --> 00:02:12.070 than the Acrobot and the CartPole. 00:02:12.070 --> 00:02:15.120 There's two things that are different dynamically 00:02:15.120 --> 00:02:17.100 speaking about a walking robot. 00:02:17.100 --> 00:02:20.220 The first is we have to think about limit cycles instead 00:02:20.220 --> 00:02:21.180 of just trajectories. 00:02:21.180 --> 00:02:26.640 And we're going to talk for some time about that. 00:02:26.640 --> 00:02:28.260 The first thing was limit cycles. 00:02:28.260 --> 00:02:31.470 The second thing is impacts. 00:02:31.470 --> 00:02:34.770 Walking robots, every time their feet hit the ground, 00:02:34.770 --> 00:02:37.440 we have to start dealing with impact. 00:02:37.440 --> 00:02:40.710 So dealing with periodic motions and dealing with impacts 00:02:40.710 --> 00:02:44.870 are the two big technical hurdles. 00:02:44.870 --> 00:02:46.620 And they're not even that big necessarily, 00:02:46.620 --> 00:02:48.060 but they're the technical hurdles 00:02:48.060 --> 00:02:49.860 of taking the techniques we've already developed 00:02:49.860 --> 00:02:51.360 and applying them to walking robots. 00:03:12.360 --> 00:03:15.390 And impacts more generally might be 00:03:15.390 --> 00:03:22.170 called sort of hybrid dynamics, where 00:03:22.170 --> 00:03:24.730 you have a mixing of discrete time and continuous time 00:03:24.730 --> 00:03:25.230 dynamics. 00:03:32.700 --> 00:03:33.200 OK. 00:03:33.200 --> 00:03:35.700 So before I launch into some of the models of walking, which 00:03:35.700 --> 00:03:37.580 we'll get to pretty quick, let's do 00:03:37.580 --> 00:03:41.000 a toy example to start thinking about limit cycles 00:03:41.000 --> 00:03:42.500 and limit cycle stability. 00:03:42.500 --> 00:03:43.790 OK. 00:03:43.790 --> 00:03:46.460 So you can get your head around limit cycle stability 00:03:46.460 --> 00:03:48.875 with some very simple oscillator type circuits. 00:03:56.821 --> 00:04:00.774 My favorite simple oscillator is the Van der Pol oscillator. 00:04:10.470 --> 00:04:11.330 OK. 00:04:11.330 --> 00:04:15.075 So it's a second order system. 00:04:29.690 --> 00:04:34.010 These are the standard equations for Van der Pol oscillator 00:04:34.010 --> 00:04:36.140 where mu is some parameter you can choose. 00:04:39.470 --> 00:04:41.000 You can look at it pretty quickly 00:04:41.000 --> 00:04:45.530 and start thinking about it almost like a very simple 00:04:45.530 --> 00:04:50.120 spring mass damper system. 00:04:50.120 --> 00:04:51.830 The only complicated part is it's 00:04:51.830 --> 00:04:55.490 got state dependent nonlinear damping. 00:04:59.120 --> 00:05:01.640 So think about it almost like a spring mass system, 00:05:01.640 --> 00:05:04.970 but it's got nonlinear damping, state dependent damping. 00:05:04.970 --> 00:05:07.980 And in particular, if you look at it, 00:05:07.980 --> 00:05:12.680 you can think of it as having, if the magnitude of x 00:05:12.680 --> 00:05:17.180 is less than 1, then is that positive damping 00:05:17.180 --> 00:05:18.400 or negative damping? 00:05:18.400 --> 00:05:23.698 AUDIENCE: [INAUDIBLE] 00:05:23.698 --> 00:05:25.990 RUSS TEDRAKE: There's a sign change in the word damping 00:05:25.990 --> 00:05:26.990 somewhere in there, too. 00:05:26.990 --> 00:05:34.120 It's going to add energy to the system if x is less than 1, 00:05:34.120 --> 00:05:38.020 which I think is negative damping typically. 00:05:38.020 --> 00:05:43.750 And if x is greater than 1, it's going to remove energy. 00:05:43.750 --> 00:05:53.570 It's going to dissipate energy, right? 00:05:56.190 --> 00:06:00.210 So it's not too hard to believe that you're going to end up 00:06:00.210 --> 00:06:03.120 with something that oscillates. 00:06:03.120 --> 00:06:07.920 If it's adding energy for small x's and removing energy 00:06:07.920 --> 00:06:09.270 for large X's. 00:06:09.270 --> 00:06:12.130 It's not too surprising maybe that you get this thing going. 00:06:12.130 --> 00:06:13.350 And it starts oscillating. 00:06:19.930 --> 00:06:22.660 I might as well show you the real trajectories 00:06:22.660 --> 00:06:24.970 instead of drawing them badly. 00:07:14.700 --> 00:07:17.130 So if I do a phase plot of this system, 00:07:17.130 --> 00:07:21.300 which is two-dimensional-- 00:07:21.300 --> 00:07:23.010 it's got a walking robot in the middle. 00:07:23.010 --> 00:07:25.030 Sorry-- close figures. 00:07:27.930 --> 00:07:28.620 There we go. 00:07:28.620 --> 00:07:29.120 OK. 00:07:32.910 --> 00:07:35.070 And yeah, that's good. 00:07:35.070 --> 00:07:36.580 That's excellent. 00:07:36.580 --> 00:07:37.080 OK. 00:07:37.080 --> 00:07:40.440 So these are three plots, the three trajectories 00:07:40.440 --> 00:07:42.660 from three different initial conditions, one 00:07:42.660 --> 00:07:45.690 that started off with some large negative initial conditions 00:07:45.690 --> 00:07:49.920 in both theta and theta dot-- or x and x dot, sorry. 00:07:49.920 --> 00:07:53.730 And damping took the energy out and started going 00:07:53.730 --> 00:07:57.180 to this stable oscillation. 00:07:57.180 --> 00:07:58.980 From another very large trajectory, 00:07:58.980 --> 00:08:00.870 it did roughly the same thing, went 00:08:00.870 --> 00:08:03.030 to that same basic oscillation. 00:08:03.030 --> 00:08:05.190 And then even from very small initial conditions 00:08:05.190 --> 00:08:09.120 near the origin, it'll swing out and start 00:08:09.120 --> 00:08:13.500 going to these same characteristic oscillation 00:08:13.500 --> 00:08:15.030 in phase space. 00:08:18.053 --> 00:08:19.470 So this is exactly the picture you 00:08:19.470 --> 00:08:23.700 want to have in your head of a stable limit cycle behavior. 00:08:23.700 --> 00:08:25.380 It's a periodic motion. 00:08:25.380 --> 00:08:30.720 When I'm away from that motion, I return to it, right? 00:08:30.720 --> 00:08:32.908 This is exactly what we want to capture 00:08:32.908 --> 00:08:34.200 by saying a stable limit cycle. 00:08:34.200 --> 00:08:38.730 It turns out it's not sort of trivial to write down 00:08:38.730 --> 00:08:42.600 what you mean by a stability in the limit cycle context. 00:08:42.600 --> 00:08:44.497 We'll see more about that in a second. 00:08:44.497 --> 00:08:46.080 But first, if you just sort of-- let's 00:08:46.080 --> 00:08:49.770 just use our imagine and talk about some asymptotic stability 00:08:49.770 --> 00:08:52.878 to this orbit. 00:08:52.878 --> 00:08:54.420 Does anybody know it from this system 00:08:54.420 --> 00:08:57.270 whether this system is globally stable? 00:09:00.518 --> 00:09:01.825 Why is it not globally stable? 00:09:01.825 --> 00:09:02.700 AUDIENCE: [INAUDIBLE] 00:09:02.700 --> 00:09:03.690 RUSS TEDRAKE: Good. 00:09:03.690 --> 00:09:07.020 I think it's true that it's globally stable 00:09:07.020 --> 00:09:09.795 except for a set of measures 0 is the way you say it, right? 00:09:09.795 --> 00:09:11.670 Because, darn it, if you start at the origin, 00:09:11.670 --> 00:09:13.680 it's not going to get there. 00:09:13.680 --> 00:09:16.870 It'll never start moving from the origin. 00:09:16.870 --> 00:09:20.310 Any other initial conditions will find their way back 00:09:20.310 --> 00:09:24.720 to that stable limit cycle. 00:09:24.720 --> 00:09:27.990 So why is it not trivial to say what 00:09:27.990 --> 00:09:31.650 I mean by a stable limit cycle? 00:09:31.650 --> 00:09:39.090 Well, let's plot that same system 00:09:39.090 --> 00:09:43.742 in the time domain, same initial conditions. 00:09:43.742 --> 00:09:44.700 Everything is the same. 00:09:50.600 --> 00:09:51.190 Here we go. 00:09:53.870 --> 00:09:56.720 Same exact trajectories, you can see they all 00:09:56.720 --> 00:10:00.380 go to their characteristic oscillation pretty quickly, 00:10:00.380 --> 00:10:02.750 in fact, right? 00:10:02.750 --> 00:10:04.820 But you can see it's crystal clear here 00:10:04.820 --> 00:10:09.560 that the trajectories never synchronize themselves in time. 00:10:09.560 --> 00:10:12.822 And that's not too surprising. 00:10:12.822 --> 00:10:14.030 So you're on the limit cycle. 00:10:17.000 --> 00:10:20.960 If there's two trajectories that are a different place in phase 00:10:20.960 --> 00:10:23.660 on the limit cycle, there's nothing that's 00:10:23.660 --> 00:10:26.070 forcing them to come together. 00:10:26.070 --> 00:10:29.607 So they'll continue to be apart in time. 00:10:29.607 --> 00:10:31.190 Such a simple thing, but actually that 00:10:31.190 --> 00:10:34.940 complicates our ability to define stability. 00:10:34.940 --> 00:10:37.940 We can't talk about trajectories asymptotically converging 00:10:37.940 --> 00:10:40.100 anymore because they don't. 00:10:40.100 --> 00:10:43.430 They asymptotically converge to some manifold, 00:10:43.430 --> 00:10:45.830 but that's harder to describe. 00:10:45.830 --> 00:10:49.220 As such, we actually have relatively few good tools 00:10:49.220 --> 00:10:51.288 for describing limit cycle stability. 00:10:58.720 --> 00:11:00.910 The standard tool-- well, actually 00:11:00.910 --> 00:11:05.650 let me say a few things more about limit cycle 00:11:05.650 --> 00:11:07.150 before I tell you the standard tool. 00:11:07.150 --> 00:11:11.480 But there's actually a few things that are interesting, 00:11:11.480 --> 00:11:14.080 just good general knowledge, about limit cycles. 00:11:17.856 --> 00:11:22.780 It turns out, if you have a closed region in state space, 00:11:22.780 --> 00:11:24.880 meaning trajectories that are inside this region 00:11:24.880 --> 00:11:29.860 never leave this region, and if that region of state space 00:11:29.860 --> 00:11:34.230 has no fixed points, then it's got a closed orbit. 00:11:37.920 --> 00:11:39.250 There's got to be something. 00:11:39.250 --> 00:11:41.380 It doesn't have to be a stable limit cycle, 00:11:41.380 --> 00:11:43.510 but it has to repeat itself. 00:11:43.510 --> 00:11:46.240 If you stay inside a region forever 00:11:46.240 --> 00:11:47.728 and you don't go to a fixed point, 00:11:47.728 --> 00:11:49.270 then you have to have a closed orbit. 00:11:49.270 --> 00:11:54.610 That's the Poincaré and Bendixon theorem. 00:11:54.610 --> 00:11:56.440 And it's also interesting to note that, 00:11:56.440 --> 00:11:59.680 if you have a dynamical system described by a potential 00:11:59.680 --> 00:12:05.200 function, like any system that's described by Lyapunov function, 00:12:05.200 --> 00:12:08.950 that cannot have a limit cycle. 00:12:08.950 --> 00:12:12.340 Anything that's uniformly going down a gradient 00:12:12.340 --> 00:12:16.750 can't have a limit cycle dynamic. 00:12:16.750 --> 00:12:18.925 So a lot of Lyapunov theory doesn't apply directly. 00:12:23.620 --> 00:12:29.110 And another thing is that people don't always 00:12:29.110 --> 00:12:30.837 call any closed orbits-- these are just 00:12:30.837 --> 00:12:32.920 random factoids about limit cycle before we go on, 00:12:32.920 --> 00:12:34.900 but they're good culture here. 00:12:34.900 --> 00:12:41.380 But if you have a closed orbit in state space, if it's stable, 00:12:41.380 --> 00:12:43.690 people call it a limit cycle. 00:12:43.690 --> 00:12:46.990 If it's unstable, people call it a limit cycle. 00:12:46.990 --> 00:12:49.240 If it's marginally stable, people don't really 00:12:49.240 --> 00:12:51.280 use the word limit cycle. 00:12:51.280 --> 00:12:58.630 So for instance, on the undamped pendulum, the marginally stable 00:12:58.630 --> 00:13:00.705 orbits, there's no stability properties on these. 00:13:00.705 --> 00:13:02.830 People would typically not use the word limit cycle 00:13:02.830 --> 00:13:08.020 to describe those closed orbits on the phase space. 00:13:08.020 --> 00:13:11.620 But anything that has the sort of converging or diverging 00:13:11.620 --> 00:13:14.450 property around a single trajectory-- 00:13:14.450 --> 00:13:17.160 so what's another good way to say it? 00:13:22.670 --> 00:13:26.090 There's some words that I liked about the trajectories have 00:13:26.090 --> 00:13:28.160 to be isolated in space, basically. 00:13:28.160 --> 00:13:30.885 If these closed orbits are not isolated in space, 00:13:30.885 --> 00:13:33.260 if there's not something sort of separating between them, 00:13:33.260 --> 00:13:35.790 then people don't use the word limit cycle to describe it. 00:13:40.230 --> 00:13:42.960 OK, so we can't talk about asymptotic convergence 00:13:42.960 --> 00:13:45.220 of trajectories on limit cycles. 00:13:45.220 --> 00:13:47.730 We can't use vanilla Lyapunov functions 00:13:47.730 --> 00:13:50.760 to talk about stability on limit cycles. 00:13:50.760 --> 00:13:53.310 So what do people do, right? 00:13:53.310 --> 00:13:56.670 There's a standard trick that most people play. 00:13:56.670 --> 00:13:59.640 And that's the idea of Poincaré sections and Poincaré maps. 00:13:59.640 --> 00:14:03.240 How many people have seen Poincaré maps before? 00:14:03.240 --> 00:14:04.680 OK. 00:14:04.680 --> 00:14:07.240 Here's the idea. 00:14:07.240 --> 00:14:15.240 So I've got some orbit in state space. 00:14:15.240 --> 00:14:22.770 This is XXed out of my Van der Pol oscillator. 00:14:22.770 --> 00:14:26.580 Instead of talking about convergent to some orbit, 00:14:26.580 --> 00:14:31.650 I'm going to define a particular surface in this state space. 00:14:31.650 --> 00:14:34.140 Let's say I define-- 00:14:34.140 --> 00:14:36.780 that's a bad choice. 00:14:36.780 --> 00:14:39.270 I've got 40 colors here. 00:14:39.270 --> 00:14:41.190 I had to pick the one that was-- that's not 00:14:41.190 --> 00:14:42.273 a very good choice either. 00:14:45.540 --> 00:14:47.228 Green's got to work for me, yeah? 00:14:50.520 --> 00:14:54.750 They're all fairly toned down. 00:14:54.750 --> 00:14:59.780 OK, let's pick a particular section in state space. 00:14:59.780 --> 00:15:01.860 Instead of talking about the convergence 00:15:01.860 --> 00:15:05.310 in this continuous dynamics to this orbit, 00:15:05.310 --> 00:15:08.790 let's just look at the dynamics of that limit cycle 00:15:08.790 --> 00:15:11.740 every time it crosses this section. 00:15:11.740 --> 00:15:12.390 OK. 00:15:12.390 --> 00:15:22.740 This is called the surface of section, 00:15:22.740 --> 00:15:25.500 often referred to as just some big S. 00:15:25.500 --> 00:15:28.950 And a surface of section, if it's 00:15:28.950 --> 00:15:32.940 the case that all trajectories that leave 00:15:32.940 --> 00:15:35.700 the surface of section eventually 00:15:35.700 --> 00:15:37.170 return to the surface of section-- 00:15:41.100 --> 00:15:42.520 I'm going to write this down-- 00:15:42.520 --> 00:15:46.620 then defining a surface of section 00:15:46.620 --> 00:15:51.480 allows you to convert the continuous time stability 00:15:51.480 --> 00:15:54.540 of a limit cycle into the discrete time 00:15:54.540 --> 00:15:55.800 stability of a fixed point. 00:16:42.700 --> 00:16:46.120 So we know a lot about defining stability for fixed points. 00:16:46.120 --> 00:16:49.510 The big idea is we're going to turn limit cycle stability 00:16:49.510 --> 00:16:51.130 into fixed point stability. 00:16:58.700 --> 00:17:03.599 And the name of the technique is the Poincaré map technique, 00:17:03.599 --> 00:17:04.099 let's say. 00:17:24.970 --> 00:17:25.470 OK. 00:17:25.470 --> 00:17:30.210 So I've got some continuous dynamical system. 00:17:30.210 --> 00:17:32.010 Let's make it simple. 00:17:32.010 --> 00:17:36.220 Let's say there's no control to speak of, no time dependence. 00:17:36.220 --> 00:17:38.370 Let's just keep the simplest sort of analysis. 00:17:38.370 --> 00:17:40.500 I've got some continuous time dynamical system. 00:17:43.180 --> 00:17:49.960 Let me call tc the time of crossing 00:17:49.960 --> 00:17:51.670 the surface of section. 00:17:51.670 --> 00:18:11.430 So I'll say tcn is n-th crossing of S. That allows 00:18:11.430 --> 00:18:14.970 me to define a discrete time system, which 00:18:14.970 --> 00:18:19.830 I'm going to call xp of n. 00:18:19.830 --> 00:18:25.260 xp of n is x at tc of n. 00:18:28.620 --> 00:18:31.770 I'm just trying to be very careful about my notation. 00:18:31.770 --> 00:18:36.960 I'm going to use p for any time the vector x I'm talking about 00:18:36.960 --> 00:18:38.520 lives on the Poincaré map. 00:18:38.520 --> 00:18:40.437 I'm trying to be careful about the transition 00:18:40.437 --> 00:18:42.020 from continuous time to discrete time. 00:18:45.660 --> 00:18:54.420 And the result is a discrete time map, 00:18:54.420 --> 00:18:56.490 which I typically call P for Poincaré map. 00:19:02.630 --> 00:19:04.832 It can be linear or non-linear, right? 00:19:04.832 --> 00:19:06.290 In general, it's non-linear for us. 00:19:09.340 --> 00:19:12.560 Defining that surface of section defines for me a map which 00:19:12.560 --> 00:19:15.710 says, if I cross the surface of section 00:19:15.710 --> 00:19:20.030 at the n-th time at state xp, then the next time 00:19:20.030 --> 00:19:23.110 I cross it's going to be xp n plus 1, right? 00:19:27.390 --> 00:19:33.360 So if I'm on the Van der Pol oscillator idea, 00:19:33.360 --> 00:19:38.050 let's say xpn is something here. 00:19:38.050 --> 00:19:40.290 I've defined my surface of section, in this case, 00:19:40.290 --> 00:19:43.590 to be the place where x equals 0. 00:19:43.590 --> 00:19:48.880 x dot is greater than 0 just to define only half of that line 00:19:48.880 --> 00:19:49.380 there. 00:19:49.380 --> 00:19:51.990 It's a line segment. 00:19:51.990 --> 00:19:55.025 This thing we saw is going to go around like this. 00:19:55.025 --> 00:19:56.400 It's going to get a little closer 00:19:56.400 --> 00:19:57.483 to the nominal trajectory. 00:19:57.483 --> 00:20:00.060 It's going to come back here. 00:20:00.060 --> 00:20:02.460 So this mapping is what I'm capturing at p. 00:20:05.390 --> 00:20:09.960 If I go around again, I'll come here, yeah? 00:20:09.960 --> 00:20:13.020 The key idea is that, if that map 00:20:13.020 --> 00:20:16.650 exists for all states for any time on this map-- 00:20:16.650 --> 00:20:19.890 I go around, and I can get back to the map-- 00:20:19.890 --> 00:20:21.690 then knowing that you have stability 00:20:21.690 --> 00:20:27.180 on this map that points in xp, if xp as it evolves in time 00:20:27.180 --> 00:20:30.300 goes to a fixed point, then that must also 00:20:30.300 --> 00:20:32.700 imply that you go to a limit cycle 00:20:32.700 --> 00:20:34.943 fixed point in continuous domain. 00:20:34.943 --> 00:20:36.360 Nothing else is allowed to happen. 00:20:36.360 --> 00:20:39.720 If I'm on this state in my continuous time world, 00:20:39.720 --> 00:20:42.712 then there's no time dependent functions here. 00:20:42.712 --> 00:20:44.670 There's no other state that I'm not looking at. 00:20:44.670 --> 00:20:46.830 If I'm on this trajectory, I'm going 00:20:46.830 --> 00:20:48.330 to stay on that trajectory. 00:20:48.330 --> 00:20:50.020 Of course, a disturbance or something could knock me off, 00:20:50.020 --> 00:20:52.200 but we're just looking at the nominal system. 00:20:54.960 --> 00:20:56.940 So fixed point stability on the map 00:20:56.940 --> 00:21:03.160 implies limit cycle stability in continuous time, OK? 00:21:03.160 --> 00:21:04.190 Awesome. 00:21:08.580 --> 00:21:11.940 Yeah, I don't really need that yet. 00:21:11.940 --> 00:21:19.250 If I work out the map for the Van der Pol oscillator, 00:21:19.250 --> 00:21:26.660 let me draw this map now as xp at n-th step going 00:21:26.660 --> 00:21:29.760 to xp at the n plus 1-th step. 00:21:32.900 --> 00:21:37.290 At 0, it doesn't go anywhere. 00:21:37.290 --> 00:21:40.340 So that's an easy point to draw. 00:21:40.340 --> 00:21:41.420 I know it crosses 0. 00:21:48.630 --> 00:21:50.010 I'm being a little loose here. 00:21:50.010 --> 00:21:57.990 So what I really care about is, since I know the position is 0, 00:21:57.990 --> 00:22:00.570 really what I'm drawing is sort of the velocity at that, 00:22:00.570 --> 00:22:01.700 let me call it, xp dot. 00:22:01.700 --> 00:22:02.670 Should I? 00:22:02.670 --> 00:22:07.860 Well, xpn has two elements the way I've defined it here. 00:22:07.860 --> 00:22:10.480 Only one of them has anything interesting going on. 00:22:10.480 --> 00:22:13.063 I'm not going to write a dot on top of my discrete time system 00:22:13.063 --> 00:22:14.730 just because I don't like that notation. 00:22:14.730 --> 00:22:16.855 But I'm going to draw the only interesting variable 00:22:16.855 --> 00:22:18.360 of the two-dimensional object xn. 00:22:21.780 --> 00:22:25.650 If I look at my velocity at the n-th crossing 00:22:25.650 --> 00:22:28.365 going to the velocity of the n plus 1 crossing, 00:22:28.365 --> 00:22:31.830 it turns out, if it's in here, I know from simulation 00:22:31.830 --> 00:22:35.490 there that it gets bigger every time. 00:22:35.490 --> 00:22:37.800 Getting bigger on this discrete map 00:22:37.800 --> 00:22:42.720 means that it's greater than the line of slope 1. 00:22:47.560 --> 00:22:51.690 Let me draw the line of slope 1 here. 00:23:10.110 --> 00:23:13.890 On the Van der Pol oscillator, it starts off 00:23:13.890 --> 00:23:15.210 getting bigger, bigger, bigger. 00:23:18.690 --> 00:23:21.270 And then at some point, it crosses here 00:23:21.270 --> 00:23:24.550 and goes off like this. 00:23:24.550 --> 00:23:27.980 I think I can even tell it to generate that, 00:23:27.980 --> 00:23:32.770 so you believe me, Poincaré. 00:23:38.290 --> 00:23:40.330 And I did it on top of my other plot again. 00:23:48.340 --> 00:23:55.930 So that's the Poincaré map now of this system. 00:23:55.930 --> 00:24:00.040 The blue line, this is xpk versus xpk plus 1. 00:24:00.040 --> 00:24:01.750 And that red line is the line of slope 1. 00:24:04.660 --> 00:24:08.920 This is a luxury of the sort of low-dimensional systems here. 00:24:08.920 --> 00:24:11.560 If I have a map from one variable to another variable, 00:24:11.560 --> 00:24:13.960 then just like the flows on a line, 00:24:13.960 --> 00:24:15.430 this is a discrete time map. 00:24:15.430 --> 00:24:18.850 But I can actually analyze everything graphically. 00:24:18.850 --> 00:24:21.250 So what do I know? 00:24:21.250 --> 00:24:25.600 I already told you by writing that equation that, 00:24:25.600 --> 00:24:28.480 if I look at my Poincaré map there, 00:24:28.480 --> 00:24:32.410 the place where it crosses the line of slope 1 is a special 00:24:32.410 --> 00:24:32.950 point. 00:24:32.950 --> 00:24:35.253 What is it? 00:24:35.253 --> 00:24:36.170 AUDIENCE: Fixed point. 00:24:36.170 --> 00:24:37.795 RUSS TEDRAKE: It's a fixed point, yeah. 00:24:42.640 --> 00:24:48.653 I've actually got another fixed point here at this, the origin. 00:24:48.653 --> 00:24:50.070 Just by inspection, I can tell you 00:24:50.070 --> 00:24:51.737 about the stability of that fixed point. 00:24:54.930 --> 00:24:59.703 So when I'm over here, I'm getting bigger. 00:24:59.703 --> 00:25:01.370 When I'm over here, I'm getting smaller. 00:25:03.907 --> 00:25:05.490 So I can actually, just by inspection, 00:25:05.490 --> 00:25:07.950 tell you that that's going to be a stable fixed point. 00:25:07.950 --> 00:25:09.100 We have to be a little careful about it. 00:25:09.100 --> 00:25:09.660 I'm going to show you. 00:25:09.660 --> 00:25:11.327 In fact, the right way to think about it 00:25:11.327 --> 00:25:13.120 is the staircase technique. 00:25:13.120 --> 00:25:14.130 OK. 00:25:14.130 --> 00:25:18.780 So if I want to follow the dynamics of this iterated map, 00:25:18.780 --> 00:25:22.140 how many people see the staircase pictures? 00:25:22.140 --> 00:25:23.460 Yeah. 00:25:23.460 --> 00:25:25.290 OK, so let's say I start in my system, 00:25:25.290 --> 00:25:29.190 and xp is some small number here. 00:25:29.190 --> 00:25:31.440 And the right way to think about that 00:25:31.440 --> 00:25:35.640 is the next time I'm going to go to this new value 00:25:35.640 --> 00:25:39.900 I can sort of graphically copy that down to my next xp 00:25:39.900 --> 00:25:42.420 by drawing a line to the line of slope 1. 00:25:45.480 --> 00:25:47.350 And then I can, from there, evaluate where 00:25:47.350 --> 00:25:49.470 the next xp is going to be. 00:25:53.070 --> 00:25:55.140 And now it goes right into the fixed point. 00:25:55.140 --> 00:25:56.723 That should have been a little flatter 00:25:56.723 --> 00:25:57.810 than I ended up making it. 00:25:57.810 --> 00:26:00.370 And then on this side, similarly, I 00:26:00.370 --> 00:26:04.530 can got to here right into the fixed point. 00:26:04.530 --> 00:26:05.250 Yes? 00:26:05.250 --> 00:26:08.070 AUDIENCE: Could you kind of emulate a flow diagram that 00:26:08.070 --> 00:26:10.240 plotted xpn plus 1 minus xpn? 00:26:13.850 --> 00:26:16.475 And then instead of reaching the line of slope 00:26:16.475 --> 00:26:18.050 of 1 when it crosses the-- 00:26:18.050 --> 00:26:19.610 RUSS TEDRAKE: Absolutely, sure. 00:26:19.610 --> 00:26:22.070 Sure, you could subtract out xpn from there 00:26:22.070 --> 00:26:24.030 and get it in terms of 0s. 00:26:24.030 --> 00:26:24.890 Yeah. 00:26:24.890 --> 00:26:28.040 The thing that might be misleading about that, 00:26:28.040 --> 00:26:29.270 you have to be careful. 00:26:29.270 --> 00:26:31.187 Because discrete time systems can do something 00:26:31.187 --> 00:26:33.320 that continuous time systems can't do-- 00:26:33.320 --> 00:26:36.290 is they can jump, right? 00:26:36.290 --> 00:26:38.240 So I could actually define something 00:26:38.240 --> 00:26:41.930 that then looked a little benign, 00:26:41.930 --> 00:26:44.030 but is actually unstable. 00:26:44.030 --> 00:26:54.220 If I had something that looks like this, what does that do? 00:26:54.220 --> 00:26:56.960 You go here, here. 00:26:56.960 --> 00:26:59.600 And you know, well, I guess it goes down to here. 00:26:59.600 --> 00:27:01.310 That one might actually catch-- 00:27:01.310 --> 00:27:03.045 shoot, I didn't make my point. 00:27:03.045 --> 00:27:05.420 You could limit cycle here like this or something, right? 00:27:05.420 --> 00:27:10.400 I did a more dramatic plot to make my point. 00:27:14.780 --> 00:27:17.650 It's pretty funny that it's all unitless. 00:27:17.650 --> 00:27:20.240 So the fact that I gave myself that problem is ridiculous. 00:27:25.190 --> 00:27:29.060 You can imagine starting off here, getting 00:27:29.060 --> 00:27:32.690 bigger and bigger, and off you go, right? 00:27:32.690 --> 00:27:36.500 OK, so what are the conditions for stability, local stability 00:27:36.500 --> 00:27:37.940 on that fixed point? 00:27:37.940 --> 00:27:40.520 You all know what they are, but just now you 00:27:40.520 --> 00:27:44.690 can maybe see them graphically by my exaggerated example. 00:27:44.690 --> 00:27:47.270 What are the conditions on the local stability? 00:27:47.270 --> 00:27:49.882 What does the slope of that line have to be? 00:27:49.882 --> 00:27:53.730 AUDIENCE: It has to be less than 1? 00:27:53.730 --> 00:27:58.680 RUSS TEDRAKE: The magnitude has to be less than 1, right? 00:27:58.680 --> 00:28:03.180 It could be sloped like this, sloped like this. 00:28:03.180 --> 00:28:06.780 What matters is that it's not sloped past the line. 00:28:06.780 --> 00:28:09.118 The other, the simpler thing, to draw for instability 00:28:09.118 --> 00:28:11.160 would be if my plot had gone like this, that it's 00:28:11.160 --> 00:28:13.430 clearly unstable, right? 00:28:17.880 --> 00:28:20.250 Because then I'd go like this, this, this. 00:28:20.250 --> 00:28:22.950 Then off I go to infinity. 00:28:22.950 --> 00:28:27.720 But it also happens if I'm unstable like this 00:28:27.720 --> 00:28:31.139 because I oscillate myself off to infinity. 00:28:53.040 --> 00:28:58.110 If I look at the gradient of p with respect 00:28:58.110 --> 00:29:04.440 to x at some fixed point, the eigenvalues of that 00:29:04.440 --> 00:29:09.830 had better be less than 1 for it to be stable. 00:29:09.830 --> 00:29:10.330 Yeah. 00:29:10.330 --> 00:29:11.747 AUDIENCE: So if you're linearized, 00:29:11.747 --> 00:29:13.260 then you're basically at that point. 00:29:13.260 --> 00:29:14.718 RUSS TEDRAKE: So that's a statement 00:29:14.718 --> 00:29:16.603 about local stability. 00:29:16.603 --> 00:29:18.270 One of the cool things about these maps, 00:29:18.270 --> 00:29:20.460 actually, is people have stronger-- 00:29:20.460 --> 00:29:22.710 certainly, graphically I did sort of a stronger thing. 00:29:22.710 --> 00:29:25.260 I said I could just look at this and know that it was stable. 00:29:25.260 --> 00:29:28.260 People actually know quite a bit about the properties 00:29:28.260 --> 00:29:30.330 of these maps. 00:29:30.330 --> 00:29:33.720 Just again for culture, it turns out, 00:29:33.720 --> 00:29:37.570 if you have a unimodal map like this, 00:29:37.570 --> 00:29:42.420 if the gradient doesn't change same 00:29:42.420 --> 00:29:49.185 and if it's locally stable, then you can infer global stability. 00:29:49.185 --> 00:29:51.060 There's a citation in the notes where you can 00:29:51.060 --> 00:29:53.500 see the careful result of that. 00:29:53.500 --> 00:29:59.880 Some of the Koditschek's hopping work exploited that result. 00:29:59.880 --> 00:30:03.180 So people do know some things, more strong global properties 00:30:03.180 --> 00:30:05.430 of these maps. 00:30:05.430 --> 00:30:08.749 But the trivial one to think about is this local stability. 00:30:26.740 --> 00:30:31.530 OK, so we know a bit now about how 00:30:31.530 --> 00:30:35.022 to talk about whether a limit cycle system is stable. 00:30:35.022 --> 00:30:37.230 It's a little more subtle than fixed point stability, 00:30:37.230 --> 00:30:37.897 but not too bad. 00:30:40.350 --> 00:30:44.100 The hard part comes if you can't find a mapping where it always 00:30:44.100 --> 00:30:44.850 returns to itself. 00:30:44.850 --> 00:30:47.980 And then you have problems, right? 00:30:47.980 --> 00:30:50.130 But in this case, you can say strong things 00:30:50.130 --> 00:30:52.810 about the stability. 00:30:52.810 --> 00:30:53.460 OK. 00:30:53.460 --> 00:30:57.540 Now, we're set up to talk about walking, so my favorite topic 00:30:57.540 --> 00:30:58.350 here. 00:30:58.350 --> 00:31:03.113 Let me actually say historically, 00:31:03.113 --> 00:31:04.530 I mean, people have obviously been 00:31:04.530 --> 00:31:07.050 interested in walking for a long, long time. 00:31:10.080 --> 00:31:12.120 Some people say that the first sort 00:31:12.120 --> 00:31:16.050 of serious study of legged locomotion 00:31:16.050 --> 00:31:18.570 was by a photographer. 00:31:18.570 --> 00:31:20.070 What's his name? 00:31:20.070 --> 00:31:20.970 AUDIENCE: Muybridge. 00:31:20.970 --> 00:31:22.830 RUSS TEDRAKE: Muybridge, good. 00:31:22.830 --> 00:31:23.730 Yeah. 00:31:23.730 --> 00:31:29.160 So a photographer had a sort of a fast shutter speed camera. 00:31:29.160 --> 00:31:34.200 And he had animals run in front of graph paper, basically. 00:31:34.200 --> 00:31:38.190 And he tried to win a bet, I think, was the story. 00:31:38.190 --> 00:31:40.680 Because people weren't sure, when 00:31:40.680 --> 00:31:43.440 horses were doing their sort of certain gait, whether they-- 00:31:43.440 --> 00:31:44.490 I think it was a gallop. 00:31:44.490 --> 00:31:46.230 They asked, when the horse gallops does, 00:31:46.230 --> 00:31:49.560 he ever actually get all four feet off the ground? 00:31:49.560 --> 00:31:51.735 And the guy who sort of proved that 00:31:51.735 --> 00:31:54.360 was a photographer, that took a picture of all four feet coming 00:31:54.360 --> 00:31:55.500 off the ground. 00:31:55.500 --> 00:31:58.590 And you can get his books, which are these fantastic 00:31:58.590 --> 00:32:01.458 books of all sorts of animals, just pictures 00:32:01.458 --> 00:32:03.750 and pictures and pictures of all sorts of animals, sort 00:32:03.750 --> 00:32:07.020 of almost stroke photography of these animals walking 00:32:07.020 --> 00:32:09.330 in front of graph paper. 00:32:09.330 --> 00:32:14.580 You get elephants, and camels, and humans, and babies, 00:32:14.580 --> 00:32:17.160 and horses, and dogs. 00:32:17.160 --> 00:32:18.330 You name it. 00:32:18.330 --> 00:32:22.155 So if you ever sort of are curious 00:32:22.155 --> 00:32:24.030 what the gate of a certain animal looks like, 00:32:24.030 --> 00:32:25.860 you can go to Muybridge. 00:32:25.860 --> 00:32:29.160 And he also sort of started defining this language 00:32:29.160 --> 00:32:31.590 for describing the gaits of different animals, 00:32:31.590 --> 00:32:34.020 like which feet are on the ground at which time 00:32:34.020 --> 00:32:35.040 and stuff like that. 00:32:35.040 --> 00:32:36.960 So this was in the turn of the century, 00:32:36.960 --> 00:32:39.600 the late 1800s, early 1900s. 00:32:39.600 --> 00:32:43.770 This photographer sort of started our field. 00:32:43.770 --> 00:32:46.170 More recently, though, the first sort 00:32:46.170 --> 00:32:50.580 of serious modeling, which is the impact that 00:32:50.580 --> 00:32:56.260 sort of remained today, I think, was McMahon at Harvard. 00:32:56.260 --> 00:33:01.050 He was a biomechanicist. 00:33:01.050 --> 00:33:06.672 And he observed that EMG in your leg, 00:33:06.672 --> 00:33:08.130 if you look at the muscle activity, 00:33:08.130 --> 00:33:10.950 Electromyographic activity in your leg, 00:33:10.950 --> 00:33:14.130 so just the electrical signal sent to your muscle, 00:33:14.130 --> 00:33:20.910 that your stance leg has very little EMG-- 00:33:20.910 --> 00:33:22.260 sorry, a lot of EMG. 00:33:22.260 --> 00:33:25.200 And your swing has very little EMG 00:33:25.200 --> 00:33:27.960 with the exception of, right when you begin swing, 00:33:27.960 --> 00:33:29.472 there's a lot of EMG. 00:33:29.472 --> 00:33:30.930 And right towards the end of swing, 00:33:30.930 --> 00:33:32.230 there's a lot of EMG activity. 00:33:32.230 --> 00:33:36.090 So your muscles are doing seemingly a lot of work 00:33:36.090 --> 00:33:39.350 at the beginning and end, and your stance leg 00:33:39.350 --> 00:33:40.800 is doing a lot of work. 00:33:40.800 --> 00:33:44.790 But the surprising thing was that inside of phase 00:33:44.790 --> 00:33:46.230 your muscles almost turn off. 00:33:49.260 --> 00:33:56.170 So McMahon produced for the world the ballistic walker 00:33:56.170 --> 00:33:56.670 model. 00:34:05.680 --> 00:34:06.520 He was at Harvard. 00:34:17.260 --> 00:34:20.290 And he basically did some modeling 00:34:20.290 --> 00:34:27.560 of a three-link pendulum, something like that 00:34:27.560 --> 00:34:33.050 and showed that, even with a straight leg on your stance 00:34:33.050 --> 00:34:36.560 leg, if you let your swing leg just go, 00:34:36.560 --> 00:34:39.449 that you can get very natural gaits coming out, 00:34:39.449 --> 00:34:42.800 very reasonable kinematic descriptions of what 00:34:42.800 --> 00:34:50.340 human walking looks like, hence the name ballistic walker. 00:34:50.340 --> 00:34:52.830 This is a passive phase. 00:34:52.830 --> 00:34:57.210 So one of the dominant ideas that came out of that work 00:34:57.210 --> 00:35:01.350 was the sort of notion that people talk about now 00:35:01.350 --> 00:35:02.415 as walking by vaulting. 00:35:11.010 --> 00:35:12.720 If you want a first order approximation 00:35:12.720 --> 00:35:15.840 of what walking is, they said think about your stance leg 00:35:15.840 --> 00:35:16.830 as a rod. 00:35:16.830 --> 00:35:18.900 And you just vault over your stance leg. 00:35:18.900 --> 00:35:20.460 And in fact, peoples' center of mass 00:35:20.460 --> 00:35:26.370 do tend to go up over the course of a stance and then back down. 00:35:26.370 --> 00:35:28.380 So the dominant theory really was 00:35:28.380 --> 00:35:31.380 that you could think about this as a stiff leg 00:35:31.380 --> 00:35:33.160 and then an almost passive swing phase. 00:35:35.910 --> 00:35:42.300 Now, I remember last year's dynamic walking meeting. 00:35:42.300 --> 00:35:45.660 Basically, half the talks were saying walking 00:35:45.660 --> 00:35:46.840 is not vaulting anymore. 00:35:46.840 --> 00:35:49.313 It's just in the last year or so people are really, 00:35:49.313 --> 00:35:51.480 really changing their mind about this, including you 00:35:51.480 --> 00:35:53.340 guys in Media Lab are talking a lot 00:35:53.340 --> 00:35:57.660 about the role of compliance in the stance leg in energy 00:35:57.660 --> 00:35:59.700 storage during walking, which was left out 00:35:59.700 --> 00:36:01.590 of these initial models. 00:36:01.590 --> 00:36:03.390 But I think, if you want a 0 order 00:36:03.390 --> 00:36:06.480 model of walking, thinking about your stance leg 00:36:06.480 --> 00:36:08.400 as straight is not a bad way to start. 00:36:11.730 --> 00:36:15.060 And the other thing that happened because of this work 00:36:15.060 --> 00:36:17.880 was this idea of passive dynamic walking. 00:36:31.020 --> 00:36:32.280 So this guy Tad McGeer-- 00:36:39.655 --> 00:36:42.930 so McMahon's model was just a model of the swing phase. 00:36:42.930 --> 00:36:44.770 It didn't talk about stability. 00:36:44.770 --> 00:36:46.560 It didn't even talk about ground contact. 00:36:46.560 --> 00:36:48.820 It wasn't sort of a periodic motion. 00:36:48.820 --> 00:36:51.600 It was saying something about the kinematics of swing phase. 00:36:51.600 --> 00:36:54.330 And this guy Tad McGeer, who's actually an aeroengineer, 00:36:54.330 --> 00:37:00.180 he works on UADs now, came in and turned the robotics world 00:37:00.180 --> 00:37:02.580 on its head by building a bunch of machines 00:37:02.580 --> 00:37:04.500 that had no motor, no controller, 00:37:04.500 --> 00:37:07.290 and walked down a small ramp basically using 00:37:07.290 --> 00:37:08.880 these ballistic walking kind of ideas. 00:37:15.780 --> 00:37:16.830 So Tad's a good guy. 00:37:20.070 --> 00:37:23.130 Let him describe. 00:37:23.130 --> 00:37:25.710 I've got a video of him talking about passive walking here. 00:37:30.924 --> 00:37:31.872 [VIDEO PLAYBACK] 00:37:31.872 --> 00:37:36.300 - This familiar toy is a passive dynamic walker. 00:37:36.300 --> 00:37:39.420 [INAUDIBLE] all the way. 00:37:39.420 --> 00:37:43.530 And right at the start, it settles into a steady walking 00:37:43.530 --> 00:37:47.160 cycle sustained by an entirely passive interaction of gravity 00:37:47.160 --> 00:37:50.250 and inertia. 00:37:50.250 --> 00:37:54.450 This machine is also a passive dynamic walker. 00:37:54.450 --> 00:37:56.831 For that matter, I may be a passive dynamic walker. 00:37:56.831 --> 00:38:00.480 As you'll see, our gaits are quite similar. 00:38:00.480 --> 00:38:05.644 [INAUDIBLE] with the analysis, and by intuition [INAUDIBLE] 00:38:05.644 --> 00:38:07.392 exclusively [INAUDIBLE]. 00:38:07.392 --> 00:38:08.707 [LAUGHTER] 00:38:12.674 --> 00:38:13.764 God. 00:38:13.764 --> 00:38:18.720 Once you establish [INAUDIBLE],, the reason the machine failed 00:38:18.720 --> 00:38:21.150 was a problem called [INAUDIBLE].. 00:38:21.150 --> 00:38:23.220 So for later experiments, we went 00:38:23.220 --> 00:38:25.620 to a set of mechanical patches. 00:38:25.620 --> 00:38:28.350 And the next sequence shows our first trial with those. 00:38:42.060 --> 00:38:43.321 I guess that worked. 00:38:43.321 --> 00:38:45.347 [END VIDEO PLAYBACK] 00:38:45.347 --> 00:38:46.680 RUSS TEDRAKE: That's a big deal. 00:38:46.680 --> 00:38:50.130 That started a lot of research. 00:38:50.130 --> 00:38:51.330 AUDIENCE: Is that 1990? 00:38:51.330 --> 00:38:58.440 RUSS TEDRAKE: 1990, exactly. 00:38:58.440 --> 00:38:59.412 [VIDEO PLAYBACK] 00:38:59.412 --> 00:39:00.390 - This familiar toy-- 00:39:00.390 --> 00:39:01.840 RUSS TEDRAKE: Whoops, sorry. 00:39:01.840 --> 00:39:06.078 - [INAUDIBLE] today is that we have all four debouncers 00:39:06.078 --> 00:39:07.987 [INAUDIBLE]. 00:39:07.987 --> 00:39:09.820 RUSS TEDRAKE: The knee latches the last time 00:39:09.820 --> 00:39:12.463 are the debouncers of today. 00:39:12.463 --> 00:39:25.727 - [INAUDIBLE] 00:39:25.727 --> 00:39:27.310 RUSS TEDRAKE: No motor, no controller, 00:39:27.310 --> 00:39:28.643 that's just falling down a ramp. 00:39:28.643 --> 00:39:29.380 That's beautiful. 00:39:32.692 --> 00:39:34.150 I love this video, because it shows 00:39:34.150 --> 00:39:40.852 you sort of the honest capabilities of these things. 00:39:40.852 --> 00:39:41.810 There's a few outtakes. 00:39:44.798 --> 00:39:46.393 [LAUGHTER] 00:39:46.393 --> 00:39:48.060 That's what it's like working with them. 00:39:51.950 --> 00:39:52.910 That was pretty benign. 00:40:01.185 --> 00:40:02.810 It's a pretty robust phenomenon, right? 00:40:02.810 --> 00:40:03.380 So this is-- 00:40:03.380 --> 00:40:05.270 [LAUGHTER] 00:40:05.270 --> 00:40:07.430 --he's changing the ramp angle quite a bit 00:40:07.430 --> 00:40:12.770 and getting these really graceful motions out. 00:40:25.593 --> 00:40:27.260 This guy walks sideways for some reason. 00:40:42.770 --> 00:40:45.258 It's really good, yeah? 00:40:45.258 --> 00:40:52.828 [END VIDEO PLAYBACK] 00:40:52.828 --> 00:40:54.370 You got to realize this was at a time 00:40:54.370 --> 00:40:59.020 when the walking robots of the world were going like this, 00:40:59.020 --> 00:41:01.810 moving like that. 00:41:01.810 --> 00:41:04.390 This guy says, I got a robot with no motor, no controller. 00:41:04.390 --> 00:41:06.340 Look what it can do, yeah? 00:41:06.340 --> 00:41:10.630 Really amazing-- and he was an aerospace engineer, go figure. 00:41:10.630 --> 00:41:13.277 This is stealing the show. 00:41:13.277 --> 00:41:14.110 That's the next bit. 00:41:19.330 --> 00:41:22.630 So these passive walking ideas have really had a big impact 00:41:22.630 --> 00:41:25.240 on the world of walking robots. 00:41:25.240 --> 00:41:28.960 They had it in two waves, I guess. 00:41:28.960 --> 00:41:32.020 Everybody knew that a purely passive walker 00:41:32.020 --> 00:41:33.670 could walk down a small slope. 00:41:33.670 --> 00:41:37.550 And then people have been working for-- 00:41:37.550 --> 00:41:39.628 the second part of that video was a video. 00:41:39.628 --> 00:41:41.170 Finally, someone had put motors on it 00:41:41.170 --> 00:41:44.470 to make it walk on the flat because it took a surprisingly 00:41:44.470 --> 00:41:47.260 long time to make that leap. 00:41:47.260 --> 00:41:48.850 So now, people, I think, don't just 00:41:48.850 --> 00:41:49.870 think they're a nice party trick, 00:41:49.870 --> 00:41:52.480 but they actually may be useful for designer of real walking 00:41:52.480 --> 00:41:55.071 machines. 00:41:55.071 --> 00:41:56.860 AUDIENCE: Why [INAUDIBLE]? 00:41:56.860 --> 00:41:58.652 RUSS TEDRAKE: Well, because they don't know 00:41:58.652 --> 00:42:00.910 how to underactuate a control. 00:42:00.910 --> 00:42:03.110 I think, quite honestly, that's my answer. 00:42:03.110 --> 00:42:03.610 Yeah. 00:42:07.430 --> 00:42:09.790 We don't have any good control ideas. 00:42:09.790 --> 00:42:11.757 I mean, what you want to do here is just push 00:42:11.757 --> 00:42:14.090 a little bit of energy into the system in the right way. 00:42:14.090 --> 00:42:15.520 We don't know how to formulate those control problems when 00:42:15.520 --> 00:42:17.920 the systems are so non-linear and discontinuous 00:42:17.920 --> 00:42:20.115 and everything. 00:42:20.115 --> 00:42:21.490 Plus, I think the people that are 00:42:21.490 --> 00:42:22.770 good at building those things maybe 00:42:22.770 --> 00:42:24.853 are not the same people that are going at control. 00:42:24.853 --> 00:42:29.380 It takes a special talent to turn enough screws. 00:42:29.380 --> 00:42:33.930 So you should know that the way that the knee stayed attached-- 00:42:33.930 --> 00:42:36.490 how do you think the knee state attached in that? 00:42:36.490 --> 00:42:39.631 Why did the knee not bend during the stance phase? 00:42:39.631 --> 00:42:41.122 AUDIENCE: Magnetic or something? 00:42:41.122 --> 00:42:43.810 RUSS TEDRAKE: It wasn't magnetic, but that's close. 00:42:43.810 --> 00:42:46.268 You can imagine, maybe, the curvature of the foot doing it, 00:42:46.268 --> 00:42:47.110 but that's not it. 00:42:47.110 --> 00:42:49.000 They're suction cups, yeah? 00:42:49.000 --> 00:42:50.470 It goes. 00:42:50.470 --> 00:42:52.540 And the suction cups don't just stick. 00:42:52.540 --> 00:42:54.610 You poke a hole in the back of the suction cup. 00:42:54.610 --> 00:42:57.730 And you tune the leakage of the suction, right? 00:42:57.730 --> 00:43:01.090 So that at the right time, it pops off and goes like this. 00:43:01.090 --> 00:43:04.270 This is not for the weak of heart. 00:43:04.270 --> 00:43:07.190 There's a lot of tuning that goes into these things. 00:43:07.190 --> 00:43:10.060 So our most dramatic failure, I think, 00:43:10.060 --> 00:43:15.370 was we were doing a point foot version of McGeer's walker. 00:43:15.370 --> 00:43:18.550 And for the point foot to work and get this nice swing 00:43:18.550 --> 00:43:21.130 without the curved feet, we had to put a lot of mass 00:43:21.130 --> 00:43:24.100 on the upper leg, not a lot on the lower leg. 00:43:24.100 --> 00:43:26.650 So we had these little fiberglass lower legs 00:43:26.650 --> 00:43:28.540 and big heavy upper legs. 00:43:28.540 --> 00:43:30.890 And our failure was not just sort of falling down. 00:43:30.890 --> 00:43:33.145 It was exploding the lower leg into pieces-- 00:43:33.145 --> 00:43:34.270 [LAUGHTER] 00:43:34.270 --> 00:43:36.370 --right, and completely crashing. 00:43:36.370 --> 00:43:38.140 John, you remember that? 00:43:38.140 --> 00:43:39.710 Vanessa's walker just kind of-- 00:43:39.710 --> 00:43:41.560 I'm the one that pushed it, too. 00:43:41.560 --> 00:43:43.510 I had a reputation in lab for a little while. 00:43:43.510 --> 00:43:44.980 I had just broken Rick's airplane. 00:43:44.980 --> 00:43:46.490 And I went around and turned around 00:43:46.490 --> 00:43:48.210 and broke Vanessa's walker. 00:43:48.210 --> 00:43:49.210 And they're all on tape. 00:43:49.210 --> 00:43:50.186 [LAUGHTER] 00:43:54.580 --> 00:43:57.490 So McGeer, apart from building these beautiful machines, 00:43:57.490 --> 00:44:00.550 gave us this beautiful model to think 00:44:00.550 --> 00:44:02.860 about the essential dynamics of walking. 00:44:02.860 --> 00:44:06.850 The ballistic walker told us about the swing phase, 00:44:06.850 --> 00:44:09.220 but that's actually sort of secondary. 00:44:09.220 --> 00:44:12.610 What really matters is this passive interaction 00:44:12.610 --> 00:44:14.890 between inertia and gravity, as Tad said. 00:44:17.770 --> 00:44:22.667 And it turns out you can understand that if you just 00:44:22.667 --> 00:44:25.000 think about this little system called the rimless wheel. 00:44:36.530 --> 00:44:44.180 So I'm going to put it on a slope of angle gamma. 00:44:44.180 --> 00:44:50.450 The model is going to be a simple pendulum with a mass 00:44:50.450 --> 00:44:54.020 at the end, but a massless leg. 00:44:54.020 --> 00:45:01.190 We'll define theta to be this, the angle from vertical. 00:45:01.190 --> 00:45:06.530 Now, I sort of despise the fact that my thetas-- 00:45:06.530 --> 00:45:08.370 I'm not using the right-hand rule here. 00:45:08.370 --> 00:45:10.340 But otherwise, you end up walking backwards 00:45:10.340 --> 00:45:12.330 with negative velocity. 00:45:12.330 --> 00:45:13.610 So this is the way it is. 00:45:13.610 --> 00:45:15.310 So it's almost like the simple pendulum 00:45:15.310 --> 00:45:17.060 except my coordinate system's from the top 00:45:17.060 --> 00:45:19.700 and theta's sort of reversed. 00:45:19.700 --> 00:45:22.700 And in addition to this simple pendulum, 00:45:22.700 --> 00:45:31.370 I've got a few extra legs that are all massless. 00:45:31.370 --> 00:45:36.742 And they're separated by 2 alpha, an angle of 2 alpha. 00:45:36.742 --> 00:45:38.450 There's actually reasonably good pictures 00:45:38.450 --> 00:45:42.500 of this, cartoons of this, in the notes. 00:45:42.500 --> 00:45:46.430 So you can think about this as a bicycle wheel 00:45:46.430 --> 00:45:51.330 where somebody took the rim off and just left the spokes. 00:45:51.330 --> 00:45:54.422 So what happens if you take this bicycle wheel 00:45:54.422 --> 00:45:56.130 and you put it at the top of a small ramp 00:45:56.130 --> 00:45:58.742 and you give it a push? 00:45:58.742 --> 00:46:00.146 AUDIENCE: It falls over. 00:46:00.146 --> 00:46:01.313 RUSS TEDRAKE: It falls over. 00:46:01.313 --> 00:46:02.000 OK. 00:46:02.000 --> 00:46:04.490 Let's say it's constrained to be in the plane. 00:46:04.490 --> 00:46:06.242 Yeah, that was a fair answer. 00:46:06.242 --> 00:46:07.700 So we made an additional assumption 00:46:07.700 --> 00:46:09.860 that it stays upright. 00:46:09.860 --> 00:46:12.008 Then what happens? 00:46:12.008 --> 00:46:13.440 AUDIENCE: It starts rolling down. 00:46:13.440 --> 00:46:14.270 RUSS TEDRAKE: Starts rolling down-- 00:46:14.270 --> 00:46:14.890 OK. 00:46:14.890 --> 00:46:17.440 Is it going to roll faster and faster forever? 00:46:17.440 --> 00:46:18.202 AUDIENCE: No. 00:46:18.202 --> 00:46:19.660 RUSS TEDRAKE: At some point, losses 00:46:19.660 --> 00:46:21.048 are going to catch up with it. 00:46:21.048 --> 00:46:22.840 In the bicycle wheel, you might think of it 00:46:22.840 --> 00:46:25.750 as rolling friction. 00:46:25.750 --> 00:46:28.480 In this model, we're going to do it 00:46:28.480 --> 00:46:34.818 by just a bit of impact, which are very real. 00:46:34.818 --> 00:46:36.610 But that's actually the only thing we need. 00:46:36.610 --> 00:46:39.830 We're not going to model any damping in the system. 00:46:39.830 --> 00:46:43.555 So we can really think about this system as-- 00:46:43.555 --> 00:46:45.850 let me list all my assumptions carefully here. 00:46:48.740 --> 00:46:54.380 We're going to assume that the foot, so to speak, 00:46:54.380 --> 00:47:00.620 the stance foot is a pin joint. 00:47:00.620 --> 00:47:04.280 So I'm going to artificially assume that that foot doesn't 00:47:04.280 --> 00:47:05.120 come off the ground. 00:47:05.120 --> 00:47:06.818 As soon as it catches the ground, 00:47:06.818 --> 00:47:08.485 it turns into a pin joint on the ground. 00:47:12.410 --> 00:47:14.390 Instantaneous transfer for support-- 00:47:29.115 --> 00:47:31.740 which means, as soon as this leg comes around, hits the ground, 00:47:31.740 --> 00:47:32.782 it becomes the pin joint. 00:47:32.782 --> 00:47:34.130 And this guy is free. 00:47:34.130 --> 00:47:36.200 I only think about one ground contact at a time. 00:47:36.200 --> 00:47:40.700 I assume that is an instantaneous transfer support. 00:47:40.700 --> 00:47:42.770 The limit of standing on two legs 00:47:42.770 --> 00:47:45.590 is just model that's switching back and forth really fast 00:47:45.590 --> 00:47:46.340 between those two. 00:47:46.340 --> 00:47:47.173 And it works out OK. 00:47:50.160 --> 00:47:52.160 And then I'm going to assume that the collisions 00:47:52.160 --> 00:48:01.070 with the ground are inelastic and impulsive. 00:48:06.290 --> 00:48:09.020 Impulsive means instantaneous. 00:48:09.020 --> 00:48:12.170 Inelastic means that all energy going into the ground 00:48:12.170 --> 00:48:13.140 is lost to the ground. 00:48:13.140 --> 00:48:15.110 There's no bouncing. 00:48:15.110 --> 00:48:18.080 And I need that, so to make sure that there's always 00:48:18.080 --> 00:48:23.540 exactly one foot on the ground in this model. 00:48:23.540 --> 00:48:26.450 As soon as the thing bounces, we built rimless wheels 00:48:26.450 --> 00:48:27.440 with two wheels. 00:48:27.440 --> 00:48:32.360 That way they don't fall down in practice, just like McGeer's 00:48:32.360 --> 00:48:34.250 biped had four legs. 00:48:34.250 --> 00:48:36.327 They call it a four-legged biped. 00:48:36.327 --> 00:48:37.241 [LAUGHTER] 00:48:37.550 --> 00:48:39.050 But that way you don't have to worry 00:48:39.050 --> 00:48:41.420 about falling over sideways. 00:48:41.420 --> 00:48:43.830 And on the real rimless wheel, you put it on a ramp. 00:48:43.830 --> 00:48:45.120 It bounces like crazy. 00:48:45.120 --> 00:48:48.067 So this is just a model, but it actually 00:48:48.067 --> 00:48:49.400 captures the essential dynamics. 00:48:54.370 --> 00:48:54.870 OK. 00:48:54.870 --> 00:48:58.823 So now, you see why it was good to know everything 00:48:58.823 --> 00:48:59.740 we know about pendula. 00:49:02.670 --> 00:49:04.020 How is this thing going to work? 00:49:16.430 --> 00:49:17.930 I'll show you the answer real quick. 00:49:47.530 --> 00:49:49.013 So here's my rimless wheel. 00:49:49.013 --> 00:49:49.930 That was a simulation. 00:49:49.930 --> 00:49:50.972 So I'm going to start it. 00:49:50.972 --> 00:49:53.320 Every time I hit Go, it's going to start it 00:49:53.320 --> 00:49:55.030 with some random initial velocity 00:49:55.030 --> 00:49:57.100 forward uphill or downhill. 00:49:57.100 --> 00:50:00.050 You name it. 00:50:00.050 --> 00:50:03.470 That's going up hill, just enough energy, 00:50:03.470 --> 00:50:06.290 turned around and came back down the hill. 00:50:06.290 --> 00:50:09.680 Start speeding up a little bit-- 00:50:09.680 --> 00:50:11.210 but actually quickly settles into 00:50:11.210 --> 00:50:17.240 a stable forward rolling speed where the losses from collision 00:50:17.240 --> 00:50:19.048 exactly balance the energy that's 00:50:19.048 --> 00:50:20.840 gained from converting potential to kinetic 00:50:20.840 --> 00:50:21.840 as you go down the hill. 00:50:24.890 --> 00:50:28.850 That is the essential idea in walking, 00:50:28.850 --> 00:50:30.980 in certainly passive walking, the conversion 00:50:30.980 --> 00:50:35.120 of potential energy and the balance with dissipation 00:50:35.120 --> 00:50:36.800 in the ground. 00:50:36.800 --> 00:50:39.650 So this is starting with a slow initial speed. 00:50:39.650 --> 00:50:42.470 And it speeds up a little bit, goes to that rolling speed. 00:50:47.990 --> 00:50:49.282 You started really fast. 00:50:49.282 --> 00:50:50.740 It actually slows down pretty quick 00:50:50.740 --> 00:50:54.197 and goes to that same rolling speed. 00:50:54.197 --> 00:50:55.780 Can you see where I'm going with this? 00:50:55.780 --> 00:50:57.780 We've got a stable limit cycle. 00:50:57.780 --> 00:51:00.953 It's periodic, clearly. 00:51:00.953 --> 00:51:03.120 It's not going to be stable in the trajectory sense, 00:51:03.120 --> 00:51:05.928 but it's going to be stable in the limit cycle since. 00:51:05.928 --> 00:51:08.220 Just I want to see the one other thing that can happen. 00:51:10.913 --> 00:51:12.830 The cool thing about simulating rimless wheels 00:51:12.830 --> 00:51:15.122 is you get all these aliasing effects and stuff, right? 00:51:15.122 --> 00:51:17.570 So if it's going uphill too fast, 00:51:17.570 --> 00:51:19.925 it looks like it's going downhill. 00:51:19.925 --> 00:51:21.140 There you go, uphill, uphill. 00:51:21.140 --> 00:51:22.580 OK, see what happens. 00:51:22.580 --> 00:51:23.150 Yeah, OK. 00:51:23.150 --> 00:51:25.317 So that one just ran out of energy and stayed still. 00:51:25.317 --> 00:51:27.650 That's still my instantaneous transfer of support. 00:51:27.650 --> 00:51:30.410 The simulation is going da, da, da, da, da, da, changing ground 00:51:30.410 --> 00:51:33.470 contacts infinitely fast. 00:51:33.470 --> 00:51:35.600 There are serious simulation people that say 00:51:35.600 --> 00:51:40.010 that's the right way to handle even-- 00:51:40.010 --> 00:51:43.630 some say that you should simulate your rigid body-- 00:51:43.630 --> 00:51:47.690 you should simulate a pin joint by constantly modeling impacts 00:51:47.690 --> 00:51:48.290 at the joints. 00:51:50.960 --> 00:51:55.730 This guy Brian Mirtich, who was at MERL for a while, 00:51:55.730 --> 00:51:58.950 did some pretty convincing work in that world. 00:51:58.950 --> 00:52:01.100 So that's the answer we're going to get. 00:52:01.100 --> 00:52:04.670 Now, can we see how that works on the phase portrait? 00:52:14.240 --> 00:52:22.720 If I look at theta and theta dot, my robot here, 00:52:22.720 --> 00:52:25.030 there's a couple key angles. 00:52:27.970 --> 00:52:32.560 Theta is only going to live between-- 00:52:32.560 --> 00:52:37.810 so it turns out to be gamma minus alpha and gamma plus 00:52:37.810 --> 00:52:38.660 alpha. 00:52:38.660 --> 00:52:40.410 See if you can work that out for yourself. 00:52:40.410 --> 00:52:42.790 So this angle is 2 alpha. 00:52:42.790 --> 00:52:47.170 When this leg hits the ground, that angle 00:52:47.170 --> 00:52:49.810 has to be alpha plus gamma. 00:52:53.080 --> 00:52:56.080 And in the other direction, it's gamma minus alpha. 00:52:56.080 --> 00:52:58.060 Those are the conditions where this leg 00:52:58.060 --> 00:53:01.090 is about to change into a different coordinate system. 00:53:01.090 --> 00:53:03.340 And I forgot to say, when this foot changes 00:53:03.340 --> 00:53:05.020 and it hits the ground, I quickly 00:53:05.020 --> 00:53:09.910 redefine theta to be the theta around the new pin joint. 00:53:09.910 --> 00:53:12.880 I just reset my coordinate system. 00:53:12.880 --> 00:53:19.720 So there's a couple of key places in this plot. 00:53:19.720 --> 00:53:21.910 Somewhere there's some small gamma, 00:53:21.910 --> 00:53:26.270 and that bisects two very important lines here, 00:53:26.270 --> 00:53:38.340 which are theta equals gamma minus alpha and theta equals 00:53:38.340 --> 00:53:39.400 gamma plus alpha. 00:53:43.110 --> 00:53:45.667 My coordinate system is 0 around the unstable fixed point. 00:53:45.667 --> 00:53:48.000 That's what happened when I flipped my coordinate system 00:53:48.000 --> 00:53:50.230 to the top. 00:53:50.230 --> 00:53:55.860 So if I want to superimpose the phase portrait of the pendulum, 00:53:55.860 --> 00:54:00.738 I said 0 is going to be the unstable fixed point. 00:54:00.738 --> 00:54:02.280 So I get the i is going out this way. 00:54:05.310 --> 00:54:08.100 That's the homoclinic orbits if you remember. 00:54:08.100 --> 00:54:10.710 And the smaller orbits come in and do this. 00:54:17.570 --> 00:54:21.980 And the ones up here do this. 00:54:27.700 --> 00:54:32.680 Does anybody have the foresight to sort of jump ahead and tell 00:54:32.680 --> 00:54:36.402 me where my limit cycle is going to live on my picture here? 00:54:36.402 --> 00:54:37.350 AUDIENCE: Above. 00:54:37.350 --> 00:54:39.120 RUSS TEDRAKE: Above, OK. 00:54:39.120 --> 00:54:41.790 What's it going to do? 00:54:41.790 --> 00:54:45.220 AUDIENCE: It's going to withdraw from the same place in each 00:54:45.220 --> 00:54:45.720 [INAUDIBLE]. 00:54:45.720 --> 00:54:47.095 RUSS TEDRAKE: Awesome, very good. 00:54:47.095 --> 00:54:48.810 Yeah, very good. 00:54:48.810 --> 00:54:50.310 OK. 00:54:50.310 --> 00:54:55.980 So here's everything we just saw in the rimless wheel. 00:54:55.980 --> 00:54:59.010 If I'm rolling fast, if I start with some initial velocity 00:54:59.010 --> 00:55:02.070 rolling fast, say, I'm high up on this line, then 00:55:02.070 --> 00:55:03.600 what happens? 00:55:03.600 --> 00:55:06.000 I'm going to ride this guy over here. 00:55:09.615 --> 00:55:10.990 It turns out the collision model, 00:55:10.990 --> 00:55:12.940 I'll tell you the collision model in a second. 00:55:12.940 --> 00:55:19.630 But the inelastic collision at the base 00:55:19.630 --> 00:55:23.860 essentially takes out a fraction of your velocity at every step. 00:55:23.860 --> 00:55:26.830 It's a linear fraction of your velocity at every step. 00:55:30.610 --> 00:55:34.630 So I take my coordinates over to here. 00:55:34.630 --> 00:55:38.200 And then it's going to reset to a velocity over here, 00:55:38.200 --> 00:55:43.810 but with some fraction of that velocity dissipated. 00:55:43.810 --> 00:55:46.920 And I get a new trajectory like this. 00:55:46.920 --> 00:55:50.180 It comes over to something like this. 00:55:50.180 --> 00:55:55.720 And at some point, that fraction exactly balances. 00:55:55.720 --> 00:56:01.870 So because this thing is not symmetric, when I start here, 00:56:01.870 --> 00:56:05.560 I end up a little bit higher with a little bit more velocity 00:56:05.560 --> 00:56:08.560 at the end than when I started. 00:56:08.560 --> 00:56:11.990 And I lose a little bit of velocity in my impact. 00:56:11.990 --> 00:56:14.380 And at some point, those things exactly balance. 00:56:14.380 --> 00:56:18.260 And I end up with a stable limit cycle that looks like that, 00:56:18.260 --> 00:56:18.760 right? 00:56:22.370 --> 00:56:23.300 What else can happen? 00:56:27.740 --> 00:56:29.210 If I start with too little energy, 00:56:29.210 --> 00:56:34.290 the whole thing runs out of energy and stands still. 00:56:34.290 --> 00:56:37.415 What's the critical energy level? 00:56:37.415 --> 00:56:38.540 AUDIENCE: Homoclinic orbit? 00:56:38.540 --> 00:56:42.050 RUSS TEDRAKE: The homoclinic orbit, right? 00:56:42.050 --> 00:56:44.360 If I'm just above the homoclinic orbit, 00:56:44.360 --> 00:56:46.760 it actually speeds up and goes to that same nominal limit 00:56:46.760 --> 00:56:48.080 cycle. 00:56:48.080 --> 00:56:51.140 But if I'm just below the homoclinic orbit, 00:56:51.140 --> 00:56:54.410 then I never actually make it across to this thing. 00:56:54.410 --> 00:56:56.537 I just end up going, bonk, over to here. 00:56:56.537 --> 00:56:59.120 And then it actually comes in-- if you have to think about it, 00:56:59.120 --> 00:57:00.290 trust me-- 00:57:00.290 --> 00:57:03.080 like this bonk, like this, and just 00:57:03.080 --> 00:57:06.050 ends up running out of energy and standing still. 00:57:11.573 --> 00:57:13.240 AUDIENCE: Could you start just above it, 00:57:13.240 --> 00:57:17.562 but then the collision knocks it down [INAUDIBLE]?? 00:57:20.860 --> 00:57:23.600 RUSS TEDRAKE: It depends on the parameters. 00:57:23.600 --> 00:57:28.630 But in the parameter regime I'm in, 00:57:28.630 --> 00:57:30.310 which I think is the standard one you 00:57:30.310 --> 00:57:34.055 want to think about, in order for it to have some stable-- 00:57:34.055 --> 00:57:35.680 so basically, what you're talking about 00:57:35.680 --> 00:57:40.030 is a parameter regime where you have no stable fixed point, 00:57:40.030 --> 00:57:43.450 no stable rolling limit cycle. 00:57:43.450 --> 00:57:45.220 AUDIENCE: Assuming your velocity-- 00:57:45.220 --> 00:57:47.740 RUSS TEDRAKE: If I'm just epsilon above here 00:57:47.740 --> 00:57:50.470 and I go down, then I'm only going to stand still. 00:57:53.210 --> 00:57:54.770 AUDIENCE: If your collision model 00:57:54.770 --> 00:57:57.770 was that you always lose a particular fraction 00:57:57.770 --> 00:57:58.510 of your velocity? 00:57:58.510 --> 00:57:59.260 RUSS TEDRAKE: Yes. 00:57:59.260 --> 00:57:59.802 AUDIENCE: OK. 00:58:02.343 --> 00:58:03.760 RUSS TEDRAKE: So in the case where 00:58:03.760 --> 00:58:05.470 we have a rolling fixed point, it 00:58:05.470 --> 00:58:07.120 has to be that epsilon above here 00:58:07.120 --> 00:58:08.720 actually gets you the other thing. 00:58:08.720 --> 00:58:10.270 The one maybe you're thinking about, 00:58:10.270 --> 00:58:16.020 too, is that there's a place over here 00:58:16.020 --> 00:58:17.490 where, if I start with this energy, 00:58:17.490 --> 00:58:20.460 I actually come out with a little less energy. 00:58:20.460 --> 00:58:22.860 And I can actually then go and get 00:58:22.860 --> 00:58:27.510 caught in either this standing fixed point, 00:58:27.510 --> 00:58:32.010 or I can transition over into this fixed point 00:58:32.010 --> 00:58:35.960 and go up and get caught. 00:58:35.960 --> 00:58:37.627 OK. 00:58:37.627 --> 00:58:39.210 There's an easier way to see all this. 00:58:39.210 --> 00:58:41.240 I want you to see it in the phase portrait 00:58:41.240 --> 00:58:43.560 because we thought a lot about it. 00:58:43.560 --> 00:58:47.000 The easier way to see what all this stuff is by defining 00:58:47.000 --> 00:58:51.182 my surface of section and looking at the Poincaré map. 00:58:51.182 --> 00:58:51.890 So let's do that. 00:58:51.890 --> 00:58:53.660 Now, in the rimless wheel, because it 00:58:53.660 --> 00:58:55.200 can be going in either direction, 00:58:55.200 --> 00:58:57.910 I'm going to define a little bit of odd surface of section. 00:58:57.910 --> 00:58:58.910 I'm going to define it-- 00:59:03.095 --> 00:59:05.050 it sure feels like I got a lot of colors, 00:59:05.050 --> 00:59:13.080 but, I don't know, some sort of red and purple mix. 00:59:13.080 --> 00:59:15.760 Let me just make it really thick here. 00:59:15.760 --> 00:59:16.540 Good lord. 00:59:21.470 --> 00:59:24.172 Let me define this to be my surface of section and this 00:59:24.172 --> 00:59:25.880 also to be part of my surface of section. 00:59:28.840 --> 00:59:33.540 So any time I have an impact with the ground, 00:59:33.540 --> 00:59:35.890 I'm going to use that as my return map. 00:59:59.150 --> 01:00:02.420 AUDIENCE: So is it just before or just after? 01:00:02.420 --> 01:00:04.730 RUSS TEDRAKE: I do just after, but it doesn't matter. 01:00:04.730 --> 01:00:06.730 As long as you're consistent, it doesn't matter. 01:00:10.465 --> 01:00:11.590 That's a little bit better. 01:00:19.860 --> 01:00:24.860 Here's the analytical return map of the rimless wheel. 01:00:29.091 --> 01:00:32.040 I should say the rimless wheel is the one walking 01:00:32.040 --> 01:00:35.190 system that we actually understand completely. 01:00:35.190 --> 01:00:38.712 And I mean, when I talk about these Poincaré maps, 01:00:38.712 --> 01:00:40.170 I can write them down analytically. 01:00:40.170 --> 01:00:42.510 Because I can tell you, if I'm in this velocity, 01:00:42.510 --> 01:00:44.218 where am I going to end up because I just 01:00:44.218 --> 01:00:46.470 have to do a first integral on the pendulum dynamics. 01:00:46.470 --> 01:00:47.820 It's energy conserving. 01:00:47.820 --> 01:00:49.830 We're all happy. 01:00:49.830 --> 01:00:52.230 So I can tell you everything about the rimless wheel. 01:00:52.230 --> 01:00:54.405 I could tell you about where it's fixed points lie. 01:00:54.405 --> 01:00:56.280 The gray is actually the basins of attraction 01:00:56.280 --> 01:00:57.330 of the fixed points. 01:00:57.330 --> 01:00:59.348 I can tell you everything. 01:00:59.348 --> 01:01:01.140 There is no other walking model that I know 01:01:01.140 --> 01:01:02.967 of that I can say that about. 01:01:02.967 --> 01:01:05.550 This is what you should aspire to get with all of your walking 01:01:05.550 --> 01:01:06.050 machines. 01:01:08.790 --> 01:01:11.790 So the red line is the line of slope 1. 01:01:11.790 --> 01:01:14.940 The blue line is the analytical solution 01:01:14.940 --> 01:01:17.580 to the first return map with the surface of section 01:01:17.580 --> 01:01:19.800 defined the way I did. 01:01:19.800 --> 01:01:22.890 This is post-collision, all right? 01:01:22.890 --> 01:01:25.710 So it turns out my lines just got thick enough 01:01:25.710 --> 01:01:28.920 that it's a little hard to see, but there's a blue line here. 01:01:28.920 --> 01:01:31.200 It says, if I'm going too fast, I'm slowing down. 01:01:33.870 --> 01:01:37.440 And it'll do its little staircase right down to this 01:01:37.440 --> 01:01:43.180 black fixed point just above this green line, 01:01:43.180 --> 01:01:46.560 which is the rolling fixed point on the Poincaré map. 01:01:46.560 --> 01:01:48.990 This is theta dot at the n-th step versus theta 01:01:48.990 --> 01:01:51.420 dot at the n plus 1-th step. 01:01:51.420 --> 01:01:52.860 So there's a rolling fixed point. 01:01:52.860 --> 01:01:54.800 And it's actually stable. 01:01:54.800 --> 01:01:56.550 The blue line goes just above the red line 01:01:56.550 --> 01:01:59.700 before it goes down. 01:01:59.700 --> 01:02:01.860 There's also a standing fixed point 01:02:01.860 --> 01:02:05.700 with a velocity at 0, which is the standing on two legs 01:02:05.700 --> 01:02:07.590 oscillating back and forth. 01:02:07.590 --> 01:02:10.950 That one's stable, too. 01:02:10.950 --> 01:02:13.438 And then when you're rolling backwards uphill, 01:02:13.438 --> 01:02:14.730 it always says I'm speeding up. 01:02:14.730 --> 01:02:16.740 Every time when I'm rolling up the hill, 01:02:16.740 --> 01:02:19.342 this is negative initial velocities. 01:02:19.342 --> 01:02:21.300 Every time, I'm going to have a little bit more 01:02:21.300 --> 01:02:24.630 forward velocity on every step. 01:02:24.630 --> 01:02:26.700 But because this thing jumps discontinuously 01:02:26.700 --> 01:02:29.483 in that staircase fashion, it's actually a little non-trivial 01:02:29.483 --> 01:02:30.900 about whether it's going to end up 01:02:30.900 --> 01:02:34.200 in the rolling fixed point or the standing fixed point. 01:02:34.200 --> 01:02:37.515 Those gray stripes are actually the basin of attraction 01:02:37.515 --> 01:02:38.640 of the rolling fixed point. 01:02:41.600 --> 01:02:46.040 So it turns out, if you roll uphill and you stop with just 01:02:46.040 --> 01:02:48.230 enough energy-- you don't quite get to the top, 01:02:48.230 --> 01:02:49.760 and then you roll back down-- 01:02:49.760 --> 01:02:52.730 then you go into the other fixed point. 01:02:52.730 --> 01:02:54.420 But if you end up at the top of the hill 01:02:54.420 --> 01:02:56.007 pretty much losing energy like this, 01:02:56.007 --> 01:02:57.090 then you're going to stop. 01:02:59.630 --> 01:03:04.310 But I can tell you everything about this return map. 01:03:04.310 --> 01:03:06.230 What's the green dashed lines? 01:03:12.325 --> 01:03:13.950 The green dashed lines are places where 01:03:13.950 --> 01:03:15.283 my return map are ill-defined. 01:03:15.283 --> 01:03:16.450 What could that possibly be? 01:03:20.170 --> 01:03:21.570 AUDIENCE: You need balance. 01:03:21.570 --> 01:03:23.790 RUSS TEDRAKE: Yes, very good, right? 01:03:23.790 --> 01:03:29.310 There is some initial velocity for which I never 01:03:29.310 --> 01:03:31.510 return to my return map. 01:03:31.510 --> 01:03:34.510 What's that? 01:03:34.510 --> 01:03:36.310 It's if I'm on the homoclinic orbit, right? 01:03:36.310 --> 01:03:38.765 It never happens practice, no big deal. 01:03:38.765 --> 01:03:40.765 But there is a set of measure 0, if you start it 01:03:40.765 --> 01:03:43.540 with just the wrong velocity, it'll go up and sit like this 01:03:43.540 --> 01:03:45.130 and never come back. 01:03:45.130 --> 01:03:48.490 So those are ill-defined points in the Poincaré map, 01:03:48.490 --> 01:03:51.010 but everything else we completely know. 01:04:00.806 --> 01:04:02.770 AUDIENCE: [INAUDIBLE] 01:04:02.770 --> 01:04:04.240 RUSS TEDRAKE: Sorry. 01:04:04.240 --> 01:04:05.065 What's that? 01:04:05.065 --> 01:04:08.710 AUDIENCE: We said, like, if you [INAUDIBLE] a velocity of 0, 01:04:08.710 --> 01:04:11.050 then it would stand still there. 01:04:11.050 --> 01:04:12.070 RUSS TEDRAKE: Yes? 01:04:12.070 --> 01:04:15.520 AUDIENCE: Why would there be that line? 01:04:15.520 --> 01:04:17.410 RUSS TEDRAKE: Oh. 01:04:17.410 --> 01:04:19.390 Why is there a green line on 0? 01:04:19.390 --> 01:04:23.430 AUDIENCE: It's on one stripe. 01:04:23.430 --> 01:04:27.740 AUDIENCE: It's a green line at theta 0, not theta dot 0. 01:04:27.740 --> 01:04:30.410 RUSS TEDRAKE: It's theta dot 0. 01:04:30.410 --> 01:04:32.725 So why do I have to draw a green line there? 01:04:32.725 --> 01:04:34.850 I don't know why I have to draw a green line there. 01:04:34.850 --> 01:04:37.280 I don't remember that green line being there. 01:04:37.280 --> 01:04:39.172 AUDIENCE: [INAUDIBLE] 01:04:39.172 --> 01:04:41.630 RUSS TEDRAKE: Yeah, but I don't think I should have drawn-- 01:04:41.630 --> 01:04:44.600 I think that's actually just a MATLAB typo. 01:04:44.600 --> 01:04:46.860 There's no reason why that should be there. 01:04:46.860 --> 01:04:48.110 I don't know why that's there. 01:04:50.230 --> 01:04:51.828 It's not in the notes. 01:04:51.828 --> 01:04:53.620 There's no reason why that should be there. 01:04:53.620 --> 01:04:54.490 I don't know why that's there. 01:04:54.490 --> 01:04:55.070 Yeah, John? 01:04:55.070 --> 01:04:58.630 AUDIENCE: Homoclinic orbit doesn't separate the rolling 01:04:58.630 --> 01:05:03.721 ones on the right side of the upper right and the base plot. 01:05:03.721 --> 01:05:06.815 Your homoclinic orbit is the boundary on the left side, 01:05:06.815 --> 01:05:08.190 but not on the right side, right? 01:05:08.190 --> 01:05:09.220 This is a-- 01:05:09.220 --> 01:05:10.690 RUSS TEDRAKE: That's not true. 01:05:10.690 --> 01:05:13.190 AUDIENCE: The ones on the right side roll when they go back. 01:05:13.190 --> 01:05:17.458 But they're inside, but they bounce across to the left. 01:05:17.458 --> 01:05:19.152 [INAUDIBLE], right? 01:05:19.152 --> 01:05:20.110 RUSS TEDRAKE: OK, good. 01:05:20.110 --> 01:05:23.775 So there's actually green lines every fraction of the way. 01:05:23.775 --> 01:05:25.150 Is that what you're asking about? 01:05:25.150 --> 01:05:25.600 AUDIENCE: I was actually-- 01:05:25.600 --> 01:05:27.595 RUSS TEDRAKE: Even the first one, this one? 01:05:27.595 --> 01:05:30.530 AUDIENCE: Because the homoclinic orbit is not the boundary 01:05:30.530 --> 01:05:33.910 or what marks the stable rolling gates in the [INAUDIBLE] plot, 01:05:33.910 --> 01:05:34.630 right? 01:05:34.630 --> 01:05:36.820 It is on left side maybe, but on the right side 01:05:36.820 --> 01:05:38.065 you can be inside the homoclinic orbit. 01:05:38.065 --> 01:05:39.130 RUSS TEDRAKE: Oh, that's correct. 01:05:39.130 --> 01:05:39.755 That's correct. 01:05:39.755 --> 01:05:43.778 So I can end up here with enough energy to transfer over. 01:05:43.778 --> 01:05:44.320 That's right. 01:05:44.320 --> 01:05:45.490 AUDIENCE: [INAUDIBLE] 01:05:45.490 --> 01:05:47.060 RUSS TEDRAKE: Only here is at the-- 01:05:47.060 --> 01:05:47.430 AUDIENCE: Yeah. 01:05:47.430 --> 01:05:48.513 RUSS TEDRAKE: That's good. 01:05:48.513 --> 01:05:49.450 Yes, that's true. 01:05:49.450 --> 01:05:51.970 This is the only place where it's the defining part. 01:05:51.970 --> 01:05:53.560 The transition across from over here 01:05:53.560 --> 01:05:55.352 is more complicated, which is why we end up 01:05:55.352 --> 01:05:58.600 with these trajectories, which get 01:05:58.600 --> 01:06:00.940 started up going the other way. 01:06:00.940 --> 01:06:01.870 Yeah. 01:06:01.870 --> 01:06:06.500 And it's also the case that there's actually stripes 01:06:06.500 --> 01:06:07.750 of these green lines going up. 01:06:07.750 --> 01:06:10.450 At every boundary between the gray and the white, 01:06:10.450 --> 01:06:13.060 there's a place where I could have rolled 10 steps 01:06:13.060 --> 01:06:16.830 and ended up balanced perfectly and then go back. 01:06:19.587 --> 01:06:21.170 So I skipped over the impact dynamics. 01:06:21.170 --> 01:06:27.160 The impact dynamics are good to know, but we can avoid them. 01:06:27.160 --> 01:06:33.010 But just to give you the intuition, 01:06:33.010 --> 01:06:37.510 you can define the energy loss at impact 01:06:37.510 --> 01:06:39.370 by just looking at this rimless wheel 01:06:39.370 --> 01:06:42.100 immediately before the impact here. 01:06:42.100 --> 01:06:47.680 And it's got some momentum that it's 01:06:47.680 --> 01:06:49.420 moving along this pin joint. 01:06:49.420 --> 01:06:52.780 So it's got to have some momentum like that. 01:06:52.780 --> 01:06:54.940 Immediately after the collision, it's 01:06:54.940 --> 01:06:59.350 going to have a momentum that's orthogonal to this new pin 01:06:59.350 --> 01:07:01.240 joint. 01:07:01.240 --> 01:07:06.260 So any energy that's going into the ground is lost. 01:07:06.260 --> 01:07:09.490 So if you model an angular momentum conservation 01:07:09.490 --> 01:07:12.590 around this point, the angular momentum around this, 01:07:12.590 --> 01:07:18.460 which is the component of this, here remains. 01:07:18.460 --> 01:07:20.980 And the other part is lost. 01:07:20.980 --> 01:07:24.340 It turns out that, in the rimless wheel, 01:07:24.340 --> 01:07:26.140 you get that as a simple mapping saying, 01:07:26.140 --> 01:07:29.260 theta dot after a collision, if I use that notation, 01:07:29.260 --> 01:07:33.875 turns out to be cosine of 2 alpha times theta dot just 01:07:33.875 --> 01:07:34.750 before the collision. 01:07:34.750 --> 01:07:36.292 It turns out to be a trivial mapping. 01:07:45.930 --> 01:07:48.245 So you now know the essential dynamics of walking. 01:07:48.245 --> 01:07:49.620 This is actually most of what you 01:07:49.620 --> 01:07:52.560 need to know if you think about a walking robot. 01:07:52.560 --> 01:07:54.390 The dominant source of energy loss 01:07:54.390 --> 01:07:56.170 is not friction in the joints. 01:07:56.170 --> 01:07:58.680 So that's pretty small. 01:07:58.680 --> 01:08:01.320 I mean, you could make a robot with friction in the joints. 01:08:01.320 --> 01:08:04.950 But the dominant dynamics of our walking 01:08:04.950 --> 01:08:08.280 are dominated by impact at the ground even for human walking, 01:08:08.280 --> 01:08:09.030 right? 01:08:09.030 --> 01:08:13.140 So most people think that the places 01:08:13.140 --> 01:08:16.950 you spend the most energy to recover from the ground 01:08:16.950 --> 01:08:20.729 collision are push off with the toe. 01:08:20.729 --> 01:08:22.380 And then you're mostly passive through. 01:08:22.380 --> 01:08:24.729 Ernesto, correct me if the theories have changed. 01:08:24.729 --> 01:08:25.979 You're mostly passive through. 01:08:25.979 --> 01:08:27.600 And then there's one other time where 01:08:27.600 --> 01:08:30.930 you spend a lot of energy, not a lot, but some energy. 01:08:30.930 --> 01:08:33.937 We actually decelerate your leg before knee strike 01:08:33.937 --> 01:08:35.979 maybe just to protect your kneecaps or maybe just 01:08:35.979 --> 01:08:37.470 to not walk like this or something. 01:08:37.470 --> 01:08:37.770 I don't know. 01:08:37.770 --> 01:08:39.359 But you actually spend a little bit 01:08:39.359 --> 01:08:43.170 of energy doing negative work to slow down that swing 01:08:43.170 --> 01:08:45.930 leg before it collides with the kneecap. 01:08:45.930 --> 01:08:46.560 And that's it. 01:08:49.740 --> 01:08:51.689 And the collision dominates so much 01:08:51.689 --> 01:08:54.029 to the point where, if you want to make an energy 01:08:54.029 --> 01:08:56.413 efficient robot, Andy Ruina says, you 01:08:56.413 --> 01:08:58.830 can make it four times more efficient if you put this push 01:08:58.830 --> 01:09:02.450 off just before the collision. 01:09:02.450 --> 01:09:04.950 Because if you're coming down and you suddenly use your push 01:09:04.950 --> 01:09:07.649 off to redirect your energy just before that collision, 01:09:07.649 --> 01:09:09.660 you can reduce the velocity at collision. 01:09:09.660 --> 01:09:12.160 And he says that makes a four time improvement in the energy 01:09:12.160 --> 01:09:13.325 efficiency of your robot. 01:09:13.325 --> 01:09:15.450 So they've been trying to build walking robots that 01:09:15.450 --> 01:09:18.779 sense the ground just before by having a little limit switch 01:09:18.779 --> 01:09:21.020 sticking down in front of the foot 01:09:21.020 --> 01:09:23.182 and push off right before ground contact. 01:09:23.182 --> 01:09:25.140 That collision dynamics, what the rimless wheel 01:09:25.140 --> 01:09:29.819 tells us about, is the dominant essential dynamics of walking. 01:09:29.819 --> 01:09:32.340 It's the only loss in the system, 01:09:32.340 --> 01:09:36.472 but it's enough to get us to stable limit cycle out. 01:09:36.472 --> 01:09:37.680 The mathematicians like this. 01:09:37.680 --> 01:09:41.340 Phil Holmes like this because he was surprised initially 01:09:41.340 --> 01:09:44.880 that and that these piecewise-- 01:09:44.880 --> 01:09:52.479 this is a energy conserving system inside each cycle. 01:09:52.479 --> 01:09:55.260 It's a piecewise holonomic system they call it. 01:09:55.260 --> 01:09:58.020 And it can have stable limit cycles. 01:10:00.840 --> 01:10:02.820 So people have gone further than this now. 01:10:02.820 --> 01:10:05.010 We can do a little bit better than a rimless wheel. 01:10:05.010 --> 01:10:06.468 If you understand the rimless wheel 01:10:06.468 --> 01:10:08.610 and now you want to make it a walking robot, 01:10:08.610 --> 01:10:14.460 all you have to do is take away the rigid joined at the hip, 01:10:14.460 --> 01:10:17.070 make it a pin joint. 01:10:17.070 --> 01:10:19.350 To model the swing leg dynamics, you 01:10:19.350 --> 01:10:22.140 can add some smaller masses in the leg. 01:10:22.140 --> 01:10:27.900 So it's now exactly the Acrobot dynamics. 01:10:27.900 --> 01:10:31.740 You can imagine putting a torque at the hip. 01:10:31.740 --> 01:10:37.020 And that's called the compass gait model for obvious reasons. 01:10:42.728 --> 01:10:43.770 And wouldn't you know it? 01:10:43.770 --> 01:10:48.550 If you take a compass gait, put it on top of a small hill, 01:10:48.550 --> 01:10:54.443 give it a push, only losses come from collisions 01:10:54.443 --> 01:10:55.110 with the ground. 01:10:55.110 --> 01:10:59.310 Otherwise, it's a, even if I have no torque at the hip, 01:10:59.310 --> 01:11:04.020 passive stance phase. 01:11:04.020 --> 01:11:07.320 This thing walks stably down a small hill. 01:11:14.168 --> 01:11:15.585 AUDIENCE: Are you wanting the legs 01:11:15.585 --> 01:11:18.710 to swing free between [INAUDIBLE]?? 01:11:18.710 --> 01:11:21.000 RUSS TEDRAKE: Everything is just 0 torque. 01:11:21.000 --> 01:11:23.690 The only trick I'm playing is that I'm ignoring the foot 01:11:23.690 --> 01:11:27.105 scuff here, which, if you want to build the robot, 01:11:27.105 --> 01:11:28.730 you've got to build a little retracting 01:11:28.730 --> 01:11:30.740 toe to make your leg a little bit shorter when 01:11:30.740 --> 01:11:31.250 it goes through. 01:11:31.250 --> 01:11:32.420 But, otherwise, you can do this. 01:11:32.420 --> 01:11:33.212 And we built these. 01:11:36.390 --> 01:11:39.780 Now, this guy has the rimless wheel actually. 01:11:39.780 --> 01:11:41.627 I showed you the basins of attraction. 01:11:41.627 --> 01:11:43.710 The two fixed points, the rolling and the standing 01:11:43.710 --> 01:11:46.200 together, are globally attractive 01:11:46.200 --> 01:11:48.540 except for those sets of measure 0 01:11:48.540 --> 01:11:51.848 where you end up standing on your toes. 01:11:51.848 --> 01:11:53.640 That's not at all true of the compass gait. 01:11:53.640 --> 01:11:55.350 The compass gait's pretty frail. 01:11:55.350 --> 01:11:57.030 You have to find initial conditions that 01:11:57.030 --> 01:11:58.530 put it into this walking gait. 01:11:58.530 --> 01:12:01.590 I can show you many simulations of it falling down 01:12:01.590 --> 01:12:04.200 catastrophically. 01:12:04.200 --> 01:12:07.320 But within some initial conditions, 01:12:07.320 --> 01:12:10.560 you get a nice stable limit cycle out. 01:12:10.560 --> 01:12:12.960 And not surprisingly, you can go farther. 01:12:22.120 --> 01:12:25.880 Put one more set of links in there. 01:12:25.880 --> 01:12:29.230 This is point feet walking with knees. 01:12:29.230 --> 01:12:33.250 We modeled a collision at the kneecap and with the ground, 01:12:33.250 --> 01:12:37.210 but the dynamics are the same. 01:12:37.210 --> 01:12:40.150 The limit cycles that you see in these guys 01:12:40.150 --> 01:12:41.937 look a little bit more appealing. 01:12:52.374 --> 01:12:54.933 AUDIENCE: [INAUDIBLE] of the knee [INAUDIBLE].. 01:12:57.820 --> 01:13:01.060 RUSS TEDRAKE: It does not work if the knees have-- 01:13:01.060 --> 01:13:04.275 you need something that keeps it a stance leg. 01:13:04.275 --> 01:13:06.400 I mean, you might get a totally different gait out. 01:13:06.400 --> 01:13:07.270 But that could be-- 01:13:07.270 --> 01:13:09.320 AUDIENCE: Well, I meant just during the swing. 01:13:09.320 --> 01:13:12.463 But I guess you have to-- 01:13:12.463 --> 01:13:15.005 RUSS TEDRAKE: Somehow you have to end up with a straight leg. 01:13:15.005 --> 01:13:16.682 So if it went past and came back, 01:13:16.682 --> 01:13:18.640 if you were exceptionally lucky in that regard, 01:13:18.640 --> 01:13:20.260 you might do it. 01:13:20.260 --> 01:13:21.810 But kneecaps are a good thing, yeah? 01:13:25.438 --> 01:13:27.480 You'd feel a little silly walking down the street 01:13:27.480 --> 01:13:28.188 without kneecaps. 01:13:37.800 --> 01:13:40.050 So the dynamics of this, if you sort of project it 01:13:40.050 --> 01:13:47.110 onto a single plane, you get this nice long limit 01:13:47.110 --> 01:13:49.080 cycle of the stance leg. 01:13:49.080 --> 01:13:52.770 You got an impact, which instantaneously 01:13:52.770 --> 01:13:54.360 decreases your velocity. 01:13:54.360 --> 01:13:56.250 Then you get your stance leg cycle 01:13:56.250 --> 01:14:00.240 which is actually basically a snapshot of your rimless wheel 01:14:00.240 --> 01:14:01.260 cycle. 01:14:01.260 --> 01:14:03.690 And then you get another discrete impact which 01:14:03.690 --> 01:14:05.920 loses energy on the other side. 01:14:05.920 --> 01:14:10.088 And you get this nice limit cycle for the compass gait. 01:14:15.950 --> 01:14:18.620 This is the theta of leg one. 01:14:24.990 --> 01:14:29.040 And it's the stance leg here, and it's the swing leg here. 01:14:29.040 --> 01:14:32.280 And trajectories the neighbor this converge to it 01:14:32.280 --> 01:14:33.547 just like you'd expect. 01:14:33.547 --> 01:14:35.130 The kneed walker has another collision 01:14:35.130 --> 01:14:39.085 that happens somewhere in the middle of your swing, 01:14:39.085 --> 01:14:39.960 but it all works out. 01:14:43.290 --> 01:14:50.462 OK, so I've given you enough sort of, 01:14:50.462 --> 01:14:52.670 I think, to understand the basic limit cycle dynamics 01:14:52.670 --> 01:14:57.530 and the basic stability property of these things. 01:14:57.530 --> 01:15:00.800 Next time, we'll talk a bit about how you'd actually 01:15:00.800 --> 01:15:02.300 apply some of our previous methods, 01:15:02.300 --> 01:15:05.570 for instance, to this now that we've got a limit cycle 01:15:05.570 --> 01:15:08.450 and you've got hybrid dynamics with these switching 01:15:08.450 --> 01:15:12.140 if you want to do things like compute the gradients to do 01:15:12.140 --> 01:15:13.113 optimization. 01:15:13.113 --> 01:15:15.530 And so you have to be a little careful about those impacts 01:15:15.530 --> 01:15:19.640 and how they affect things, but it's not too hard. 01:15:19.640 --> 01:15:21.830 And we'll tell you a bit about the state 01:15:21.830 --> 01:15:23.660 of the art and walking control. 01:15:23.660 --> 01:15:26.520 They're all sort of based on this model. 01:15:26.520 --> 01:15:28.310 And then Thursday's the midterm. 01:15:28.310 --> 01:15:30.110 I promise to have a practice exam for you. 01:15:30.110 --> 01:15:31.735 It might not be tonight, unfortunately. 01:15:31.735 --> 01:15:35.210 But it'll be very soon, tomorrow or early Saturday morning, I 01:15:35.210 --> 01:15:36.150 promise. 01:15:36.150 --> 01:15:36.800 Yeah. 01:15:36.800 --> 01:15:37.440 OK. 01:15:37.440 --> 01:15:39.023 And feel free to ask me any questions. 01:15:39.023 --> 01:15:42.080 Some of you are finding me to run project ideas by. 01:15:42.080 --> 01:15:43.310 That's a great idea. 01:15:43.310 --> 01:15:44.810 If you want to talk about your project, let me know. 01:15:44.810 --> 01:15:45.470 Let John know. 01:15:48.140 --> 01:15:50.080 See you next time.