WEBVTT 00:00:01.580 --> 00:00:03.920 The following content is provided under a Creative 00:00:03.920 --> 00:00:05.340 Commons license. 00:00:05.340 --> 00:00:07.550 Your support will help MIT OpenCourseWare 00:00:07.550 --> 00:00:11.640 continue to offer high-quality educational resources for free. 00:00:11.640 --> 00:00:14.180 To make a donation or to view additional materials 00:00:14.180 --> 00:00:18.110 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:18.110 --> 00:00:19.090 at ocw.mit.edu. 00:00:22.090 --> 00:00:24.186 PRESENTER 1: The basic concepts of reachability-- 00:00:24.186 --> 00:00:27.040 I'm talking about how we can represent reach 00:00:27.040 --> 00:00:29.713 sets in continuous spaces. 00:00:29.713 --> 00:00:31.587 [INAUDIBLE] is going to take it over and talk 00:00:31.587 --> 00:00:35.205 about some applications in robust motion planning. 00:00:35.205 --> 00:00:38.435 And then Gerard is going to talk about how we compute reach sets 00:00:38.435 --> 00:00:40.670 and focus on these two, on flow tubes and funnels. 00:00:43.930 --> 00:00:46.940 So reachability is a task of figuring out 00:00:46.940 --> 00:00:50.480 what stage a dynamical system can possibly reach 00:00:50.480 --> 00:00:53.580 or be at at a certain time. 00:00:53.580 --> 00:00:57.950 So for example, if we have these two systems, 00:00:57.950 --> 00:01:00.486 we have a finite state machine on this reach system 00:01:00.486 --> 00:01:01.600 and a continuous system. 00:01:01.600 --> 00:01:05.326 And these are the starting points. 00:01:05.326 --> 00:01:06.860 And these are our axes. 00:01:06.860 --> 00:01:08.750 We can move east or north here. 00:01:08.750 --> 00:01:12.096 And we can move east or north with these velocities. 00:01:12.096 --> 00:01:13.790 [INAUDIBLE] 00:01:13.790 --> 00:01:17.620 So after one second or one time step, 00:01:17.620 --> 00:01:21.980 we can be in either of these two states. 00:01:21.980 --> 00:01:25.102 Or we can be anywhere in this area-- 00:01:25.102 --> 00:01:29.240 and after two seconds, after three seconds, and so on. 00:01:29.240 --> 00:01:30.270 OK? 00:01:30.270 --> 00:01:33.492 So this is what reachability does. 00:01:33.492 --> 00:01:36.790 And the motivation for this-- 00:01:36.790 --> 00:01:38.890 one of the applications for reachability 00:01:38.890 --> 00:01:41.470 is for verification. 00:01:41.470 --> 00:01:46.870 Verification is just the task of making sure that we are never 00:01:46.870 --> 00:01:48.332 going to reach any bad states. 00:01:48.332 --> 00:01:50.290 So for instance, let's say we have a microwave. 00:01:50.290 --> 00:01:54.760 A bad state is the microwave being open and on. 00:01:54.760 --> 00:01:57.940 We want to make sure that these bad states are never reachable. 00:01:57.940 --> 00:02:01.365 So what we do is we compute the reachable sets, 00:02:01.365 --> 00:02:04.970 and we check for intersection with the bad states 00:02:04.970 --> 00:02:08.410 to see if the microwave is on and open. 00:02:08.410 --> 00:02:12.050 Or like over here, let's say we have these obstacles. 00:02:12.050 --> 00:02:15.790 Let's say we have a wall or somewhere else over here. 00:02:15.790 --> 00:02:18.060 We want to-- this is bad, right? 00:02:18.060 --> 00:02:22.030 Because we can reach this state, and it's the same as a wall. 00:02:22.030 --> 00:02:23.650 Any questions so far? 00:02:23.650 --> 00:02:26.470 Feel free to interrupt me any time. 00:02:29.350 --> 00:02:31.250 Another motivation for reachability 00:02:31.250 --> 00:02:33.070 is for robust motion planning, which 00:02:33.070 --> 00:02:35.770 [INAUDIBLE] talk more about. 00:02:35.770 --> 00:02:38.270 And I have a video of this. 00:02:38.270 --> 00:02:40.880 So what we can do with reachability 00:02:40.880 --> 00:02:42.560 is stuff like this. 00:02:51.810 --> 00:02:54.170 So this is done using flow tubes, which 00:02:54.170 --> 00:02:57.936 is reach sets, reachability. 00:02:57.936 --> 00:03:00.090 Gerard is going to talk more this example. 00:03:07.740 --> 00:03:11.420 So now before we talk more about these cool examples, let's 00:03:11.420 --> 00:03:12.790 just formalize concepts. 00:03:12.790 --> 00:03:16.350 Let's define some things. 00:03:16.350 --> 00:03:19.830 So let's go back to the same system that we 00:03:19.830 --> 00:03:25.460 had earlier, this state machine, with a finite set of states, X; 00:03:25.460 --> 00:03:29.320 a finite set of inputs, U, which in this case 00:03:29.320 --> 00:03:31.337 are east and north; and a transition function. 00:03:31.337 --> 00:03:32.920 And let's say this is just [INAUDIBLE] 00:03:32.920 --> 00:03:34.400 to keep things simple. 00:03:34.400 --> 00:03:38.150 And we have a set of initial states, X0, 00:03:38.150 --> 00:03:41.560 which I didn't label as red. 00:03:41.560 --> 00:03:45.400 So the reach set is defined as the set of states 00:03:45.400 --> 00:03:48.860 that we can possibly be in at time t. 00:03:48.860 --> 00:03:53.050 So more formally, there exists a sequence of t control inputs 00:03:53.050 --> 00:03:55.240 that will take us from this initial state 00:03:55.240 --> 00:03:59.370 to the corresponding reach set. 00:03:59.370 --> 00:04:03.540 So let's say, for example, you have bad state, 00:04:03.540 --> 00:04:08.690 it's in the reach set at time 3 because there exists 00:04:08.690 --> 00:04:10.270 a sequence of three inputs. 00:04:10.270 --> 00:04:12.910 In this case this is one such sequence. 00:04:12.910 --> 00:04:16.209 There is two more that will take us from the initial state 00:04:16.209 --> 00:04:17.771 to that. 00:04:17.771 --> 00:04:18.270 OK? 00:04:21.490 --> 00:04:26.130 And then the reachable set, the reach set reachable set 00:04:26.130 --> 00:04:30.450 is defined as the union of all reach sets for time less than 00:04:30.450 --> 00:04:32.760 or equal to t. 00:04:32.760 --> 00:04:36.660 So the reachable set of time 0 is just 00:04:36.660 --> 00:04:37.760 the initial set of states. 00:04:37.760 --> 00:04:41.999 And time 1, we have the unit and so on. 00:04:44.580 --> 00:04:46.530 So the reachable set is what we use 00:04:46.530 --> 00:04:48.560 to test for intersection with bad states 00:04:48.560 --> 00:04:53.820 because that's all the states that we can possibly reach 00:04:53.820 --> 00:04:56.337 from time 0 to time something. 00:04:56.337 --> 00:04:56.836 OK? 00:05:00.500 --> 00:05:04.280 Now as some of you may have figured out by now, 00:05:04.280 --> 00:05:07.170 there's a simple way for finding segmenting 00:05:07.170 --> 00:05:09.940 so we can compute reach sets. 00:05:09.940 --> 00:05:15.490 So for example, let's say we wanted the reach set of time 2. 00:05:15.490 --> 00:05:17.430 One way-- the easiest way we can do 00:05:17.430 --> 00:05:19.702 this is to break this up into time steps, 00:05:19.702 --> 00:05:21.240 into single time steps. 00:05:21.240 --> 00:05:24.410 So instead of computing it straight for time 2, 00:05:24.410 --> 00:05:26.732 let's first compute it for time 1. 00:05:26.732 --> 00:05:29.250 And now we can use these two states 00:05:29.250 --> 00:05:35.420 as the set of initial states and compute the reach set for one 00:05:35.420 --> 00:05:36.930 more time step. 00:05:36.930 --> 00:05:40.972 And that gave us this reachable-- the reach set. 00:05:40.972 --> 00:05:41.860 OK? 00:05:41.860 --> 00:05:45.232 Any questions? 00:05:45.232 --> 00:05:48.010 OK, and the reason why these works is because reach sets are 00:05:48.010 --> 00:05:49.290 semi-groups. 00:05:49.290 --> 00:05:53.690 And semi-groups satisfy this equality. 00:05:53.690 --> 00:05:58.120 So we can not just break it up into single time sets. 00:05:58.120 --> 00:06:03.090 We can break into any two times, s and t. 00:06:06.018 --> 00:06:10.005 OK, so all of this was for finite state machines. 00:06:10.005 --> 00:06:12.300 But for continuous systems, things 00:06:12.300 --> 00:06:15.030 get a little more complicated. 00:06:15.030 --> 00:06:17.420 It's a lot of the same concepts, but we 00:06:17.420 --> 00:06:19.930 need to handle a very important issue, 00:06:19.930 --> 00:06:22.400 and that is how to represent reach sets, 00:06:22.400 --> 00:06:26.324 or just set, regions in space for continuous systems. 00:06:26.324 --> 00:06:27.740 Because in a finite state machine, 00:06:27.740 --> 00:06:32.590 we can just enumerate the list of states in the reach set 00:06:32.590 --> 00:06:33.750 or reachable set. 00:06:33.750 --> 00:06:36.150 In a continuous system, we need some other way. 00:06:36.150 --> 00:06:40.860 We can't just say it's every point for which there exists 00:06:40.860 --> 00:06:44.570 a sequence of control inputs, because there 00:06:44.570 --> 00:06:49.150 can be infinitely many points. 00:06:49.150 --> 00:06:54.439 So what we do, we have two types of representations 00:06:54.439 --> 00:06:54.980 that we like. 00:06:54.980 --> 00:06:57.340 One is convex polytopes. 00:06:57.340 --> 00:07:02.830 And that's just a generalization of a convex polygon into n 00:07:02.830 --> 00:07:05.256 space, into n dimensions. 00:07:05.256 --> 00:07:08.730 And we have two basic ways that we can represent 00:07:08.730 --> 00:07:10.680 these complex polygons. 00:07:10.680 --> 00:07:18.390 One is with these hull vertices, like this, AB to F. [INAUDIBLE] 00:07:18.390 --> 00:07:23.775 And then the convex polytope is defined as the convex hull. 00:07:23.775 --> 00:07:29.230 And in case someone doesn't remember what a convex hull is, 00:07:29.230 --> 00:07:34.596 let's say we have just a list of points. 00:07:34.596 --> 00:07:38.830 The convex hull, imagine have a rubber band 00:07:38.830 --> 00:07:41.744 and you are going to stretch it out, and you let it go. 00:07:41.744 --> 00:07:45.422 And what that's going to do is this. 00:07:45.422 --> 00:07:52.520 And the convex hull is just the whole area inside this-- 00:07:52.520 --> 00:07:53.690 the rubber band. 00:07:53.690 --> 00:07:55.160 OK? 00:07:55.160 --> 00:07:58.116 And then another way we can represent 00:07:58.116 --> 00:08:02.650 convex polytope is with a set of inequalities, 00:08:02.650 --> 00:08:07.250 so for example, here we have four lines. 00:08:07.250 --> 00:08:09.220 Here's the equation for each line. 00:08:09.220 --> 00:08:13.620 So let's say this red line, we want anything 00:08:13.620 --> 00:08:15.640 below the red line. 00:08:15.640 --> 00:08:18.970 This blue line, we want anything to the right, so anything 00:08:18.970 --> 00:08:20.310 greater than, and so on. 00:08:20.310 --> 00:08:23.350 So this area is the intersection of all these inequalities. 00:08:23.350 --> 00:08:24.748 OK? 00:08:24.748 --> 00:08:29.090 And each of these representations 00:08:29.090 --> 00:08:32.409 has an advantage over the other. 00:08:32.409 --> 00:08:37.200 Vertices are good to test for emptiness because when 00:08:37.200 --> 00:08:40.260 our set of vertices is empty, then our convex polytope 00:08:40.260 --> 00:08:41.789 is going to be empty. 00:08:41.789 --> 00:08:47.070 And this inequalities is very useful for testing membership, 00:08:47.070 --> 00:08:50.240 because for any points, we can just 00:08:50.240 --> 00:08:51.650 plug it in to every inequality. 00:08:51.650 --> 00:08:53.390 And if it's true for every inequality, 00:08:53.390 --> 00:08:57.575 then it is in the whole group. 00:08:57.575 --> 00:09:02.430 And convexity, I'll just explain it briefly. 00:09:02.430 --> 00:09:08.730 Convexity means that any two points inside our region-- 00:09:08.730 --> 00:09:12.340 our region is not non-convex is for the line connecting 00:09:12.340 --> 00:09:13.664 these two points. 00:09:13.664 --> 00:09:16.080 All the points in that line are going to be in the region. 00:09:16.080 --> 00:09:18.455 So here you can see that this is convex. 00:09:18.455 --> 00:09:22.850 And here, this red area is outside the region, 00:09:22.850 --> 00:09:25.470 so this other region is non-convex. 00:09:25.470 --> 00:09:27.860 And convexity is going to be important for the algorithms 00:09:27.860 --> 00:09:29.410 later on. 00:09:29.410 --> 00:09:32.716 Gerard will talk more about them. 00:09:32.716 --> 00:09:33.620 OK? 00:09:33.620 --> 00:09:35.910 Other than convex polytopes, we can also 00:09:35.910 --> 00:09:39.640 use ellipsoids to represent reach sets. 00:09:39.640 --> 00:09:42.610 So ellipsoids are just the extension of ellipses 00:09:42.610 --> 00:09:45.525 to n spaces, n dimensions. 00:09:45.525 --> 00:09:49.170 And a convex polytope is defined by this. 00:09:49.170 --> 00:09:51.160 x is just all the points inside the polytopes, 00:09:51.160 --> 00:09:52.445 inside the ellipse. 00:09:52.445 --> 00:09:54.910 v is the center. 00:09:54.910 --> 00:10:02.970 And A just tells us about this, how deform the ellipsis. 00:10:02.970 --> 00:10:04.278 Any questions? 00:10:07.682 --> 00:10:08.182 OK. 00:10:10.767 --> 00:10:13.310 And one of the reasons why we like 00:10:13.310 --> 00:10:17.300 convex polytopes, why we like ellipsis, ellipsoids, 00:10:17.300 --> 00:10:21.270 is because of their enclosure under linear operators. 00:10:21.270 --> 00:10:24.020 So let's say we have P that is convex 00:10:24.020 --> 00:10:25.178 polytope or an ellipsoid. 00:10:25.178 --> 00:10:30.880 Then, if we have a matrix A, then A times 00:10:30.880 --> 00:10:33.140 P will be defined that way. 00:10:33.140 --> 00:10:37.980 It is still going to be a convex polytope or an ellipsoid. 00:10:37.980 --> 00:10:39.880 So if we have something like this, 00:10:39.880 --> 00:10:41.740 then this can maybe getting formed, 00:10:41.740 --> 00:10:44.584 but it's still going to be convex, a convex polytope. 00:10:48.460 --> 00:10:51.070 And the reason why this is nice is that, 00:10:51.070 --> 00:10:56.901 if we have a linear system defined this way or this way, 00:10:56.901 --> 00:10:58.920 and then if we start with a-- 00:10:58.920 --> 00:11:04.930 basically, if we started with a convex set of states, then 00:11:04.930 --> 00:11:07.630 our reach sets are always going to be convex because we're just 00:11:07.630 --> 00:11:11.334 multiplying by A. And this is very nice 00:11:11.334 --> 00:11:13.500 to represent-- this tells us that our representation 00:11:13.500 --> 00:11:17.976 is going to be good because we can just use convex polytopes, 00:11:17.976 --> 00:11:22.150 ellipsoids, and stick with those. 00:11:22.150 --> 00:11:25.974 And then just a technical-- 00:11:25.974 --> 00:11:30.050 OK, so even though the reach sets are going to be convex, 00:11:30.050 --> 00:11:34.890 the reachable set might not be convex. 00:11:34.890 --> 00:11:40.700 But it will always be a union of convex polytopes or ellipsoids. 00:11:40.700 --> 00:11:43.200 For example, let's say, in the previous example, 00:11:43.200 --> 00:11:45.020 we started here. 00:11:45.020 --> 00:11:48.480 Then we need, after one time set, we were here. 00:11:48.480 --> 00:11:50.060 So these are the two things. 00:11:50.060 --> 00:11:53.540 And then our reachable set is the union of both of them. 00:11:53.540 --> 00:11:56.480 This is clearly not convex. 00:11:56.480 --> 00:12:00.640 But it is a union of convex polytopes. 00:12:00.640 --> 00:12:03.880 So we can represent everything within it. 00:12:03.880 --> 00:12:04.870 OK? 00:12:04.870 --> 00:12:08.496 Any questions before I move on to [INAUDIBLE]?? 00:12:08.496 --> 00:12:09.978 OK, thank you. 00:12:18.376 --> 00:12:20.230 PRESENTER 2: OK, so now that have understood 00:12:20.230 --> 00:12:23.690 what reachability is and how [INAUDIBLE] represents, 00:12:23.690 --> 00:12:26.037 and before we dwell into more theoretical aspects 00:12:26.037 --> 00:12:26.662 of [INAUDIBLE]. 00:12:26.662 --> 00:12:30.486 Here are some cool applications of reachability. 00:12:30.486 --> 00:12:35.162 So there are many applications of reachability to robots 00:12:35.162 --> 00:12:37.632 are like complex systems [INAUDIBLE] systems. 00:12:37.632 --> 00:12:40.596 One of the applications that has been talked about before 00:12:40.596 --> 00:12:42.590 is the robust motion planning. 00:12:42.590 --> 00:12:47.080 The early idea is to more of a glider or an airplane 00:12:47.080 --> 00:12:50.955 through lots of obstacles, even in the presence of [INAUDIBLE] 00:12:50.955 --> 00:12:52.380 these and [INAUDIBLE]. 00:12:52.380 --> 00:12:54.140 They're already crashing into obstacles, 00:12:54.140 --> 00:12:57.080 even when they are [INAUDIBLE]. 00:12:57.080 --> 00:12:58.882 [INAUDIBLE] applications of reachability 00:12:58.882 --> 00:13:03.802 to control complex systems like by parallel working robots. 00:13:03.802 --> 00:13:07.492 Here you can see that we are controlling [INAUDIBLE] simple 00:13:07.492 --> 00:13:10.220 soccer ball with something. 00:13:10.220 --> 00:13:12.280 And more importantly, reachability 00:13:12.280 --> 00:13:16.768 is used for very fine safety properties, for safety reasons, 00:13:16.768 --> 00:13:19.198 like aircraft collision avoidance. 00:13:19.198 --> 00:13:23.100 So in those [INAUDIBLE] we're going [INAUDIBLE],, 00:13:23.100 --> 00:13:24.730 robust motion planning. 00:13:24.730 --> 00:13:27.670 If you're interested in other things, there are papers 00:13:27.670 --> 00:13:31.600 and there are [INAUDIBLE]. 00:13:31.600 --> 00:13:33.370 So the goal of motion planning is 00:13:33.370 --> 00:13:37.235 to find a set of control inputs that take you from the start 00:13:37.235 --> 00:13:39.259 state to the goal state. 00:13:39.259 --> 00:13:41.311 So in this example-- without colliding 00:13:41.311 --> 00:13:45.550 with any of the obstacles. 00:13:45.550 --> 00:13:49.342 So here, this is one possible path from the start 00:13:49.342 --> 00:13:51.326 state to the goal state. 00:13:51.326 --> 00:13:52.814 So it just looks nice. 00:13:52.814 --> 00:13:55.294 [INAUDIBLE] collective things of the obstacles. 00:13:55.294 --> 00:13:58.094 But if you get the controls to this cart 00:13:58.094 --> 00:13:59.510 and then give it a real robot, you 00:13:59.510 --> 00:14:04.222 might find it going-- doing something like this. 00:14:04.222 --> 00:14:08.020 So this is because of the path that you-- algorithm 00:14:08.020 --> 00:14:11.543 that you use for planning this path is-- that's not taking 00:14:11.543 --> 00:14:13.168 into account any of the uncertain areas 00:14:13.168 --> 00:14:15.152 in the environment or [INAUDIBLE].. 00:14:15.152 --> 00:14:18.991 So there are different kinds of uncertainties. 00:14:18.991 --> 00:14:20.740 One of them is environmental disturbances, 00:14:20.740 --> 00:14:22.540 like there is wind here. 00:14:22.540 --> 00:14:26.060 So that can move you over off your planned path. 00:14:26.060 --> 00:14:27.910 And there can be modeling errors. 00:14:27.910 --> 00:14:30.772 The model that you use to represent your complex system 00:14:30.772 --> 00:14:31.980 might just be an approximate. 00:14:31.980 --> 00:14:34.465 So the real world is different. 00:14:34.465 --> 00:14:36.453 And there are state uncertainty. 00:14:36.453 --> 00:14:41.420 So the things that you used for measuring your position 00:14:41.420 --> 00:14:43.481 or velocity are also approximate, so there 00:14:43.481 --> 00:14:44.882 are uncertainties. 00:14:44.882 --> 00:14:47.220 [INAUDIBLE] there can be some randomness 00:14:47.220 --> 00:14:48.651 in the initial conditions. 00:14:48.651 --> 00:14:51.513 You are thinking of starting the robot at 0,0, 00:14:51.513 --> 00:14:53.712 but in practice, you might actually 00:14:53.712 --> 00:14:57.044 be starting at [INAUDIBLE] 0, [INAUDIBLE].. 00:14:57.044 --> 00:15:01.810 So there is randomness in initial conditions. 00:15:01.810 --> 00:15:03.550 The goal of robust motion planning 00:15:03.550 --> 00:15:08.265 is to have some guarantees with some confidence 00:15:08.265 --> 00:15:11.335 that the system will reach the goal set without getting on 00:15:11.335 --> 00:15:13.276 with the obstacles even in presence 00:15:13.276 --> 00:15:14.240 of these uncertainties. 00:15:14.240 --> 00:15:16.168 So this is the problem. 00:15:16.168 --> 00:15:18.740 So let's look at some of the solutions that 00:15:18.740 --> 00:15:21.515 come up for solving robust motion [INAUDIBLE].. 00:15:21.515 --> 00:15:23.369 So this is a timeline. 00:15:23.369 --> 00:15:24.910 So there are three different systems. 00:15:24.910 --> 00:15:28.320 And all of them have a goal concept 00:15:28.320 --> 00:15:33.648 where they use some representation to represent 00:15:33.648 --> 00:15:36.546 the reach sets, the set of possible state 00:15:36.546 --> 00:15:38.000 that the system be. 00:15:38.000 --> 00:15:40.240 And then instead of searching for a single trajectory 00:15:40.240 --> 00:15:43.526 to a state, they search for these representations 00:15:43.526 --> 00:15:46.612 so that each final self-serviced representations that 00:15:46.612 --> 00:15:48.588 can go from the start state to the goal state, 00:15:48.588 --> 00:15:52.046 you won't hit any of the obstacles. 00:15:52.046 --> 00:15:55.490 So earlier Bradley and Zhao and Frazzoli 00:15:55.490 --> 00:15:57.490 came up with this concept called the flow tubes. 00:15:57.490 --> 00:15:59.990 It's [INAUDIBLE] presenting makes sense. 00:15:59.990 --> 00:16:03.451 And we will know all about the [INAUDIBLE] of the [INAUDIBLE].. 00:16:03.451 --> 00:16:05.738 So one approach that we use is that we 00:16:05.738 --> 00:16:08.321 have a set of initial states and we have a set of goal states. 00:16:08.321 --> 00:16:12.821 So flow tube is a combination of all the possible paths that 00:16:12.821 --> 00:16:15.210 will get you from any point in the initial state 00:16:15.210 --> 00:16:16.545 and point in goal state. 00:16:16.545 --> 00:16:21.137 So this is kind of representing all possible [INAUDIBLE].. 00:16:21.137 --> 00:16:23.542 So and then you search for these flow tubes 00:16:23.542 --> 00:16:26.909 to be a path to get your [INAUDIBLE].. 00:16:26.909 --> 00:16:32.850 And [INAUDIBLE] constraints so that you can also 00:16:32.850 --> 00:16:35.412 reason about complex systems and revise 00:16:35.412 --> 00:16:38.849 spatial and temporal coordinations like the walking 00:16:38.849 --> 00:16:41.795 robot that you have seen before. 00:16:41.795 --> 00:16:50.800 An [INAUDIBLE] about for using funnels for motion planning. 00:16:50.800 --> 00:16:54.350 So funnels is also a similar concept to produce. 00:16:54.350 --> 00:16:58.994 It's kind of like a [INAUDIBLE] region around your path. 00:16:58.994 --> 00:17:00.646 So that if you are inside a funnel, 00:17:00.646 --> 00:17:02.396 you're guaranteed to be inside the funnel. 00:17:02.396 --> 00:17:04.510 So in this picture, we're going to [INAUDIBLE] 00:17:04.510 --> 00:17:06.880 motion planning using funnels. 00:17:06.880 --> 00:17:10.198 Again, there are references for the other people's [INAUDIBLE].. 00:17:13.042 --> 00:17:15.698 So instead, you'll be using funnels [INAUDIBLE] 00:17:15.698 --> 00:17:18.158 for every path, you compute some region 00:17:18.158 --> 00:17:21.461 around the path that is a stable region. 00:17:21.461 --> 00:17:22.586 And that's called a funnel. 00:17:22.586 --> 00:17:25.538 [INAUDIBLE] your funnels [INAUDIBLE],, 00:17:25.538 --> 00:17:28.251 but for this part of the lecture, 00:17:28.251 --> 00:17:30.556 we are assuming we don't have a hundred of funnels, 00:17:30.556 --> 00:17:35.022 and that you'll see how to use these funnels to do motion 00:17:35.022 --> 00:17:35.974 planning. 00:17:35.974 --> 00:17:37.878 So the previous example is here. 00:17:37.878 --> 00:17:40.006 The [INAUDIBLE] funnel [INAUDIBLE] flied by. 00:17:40.006 --> 00:17:45.490 You see that it collides with the trees and [INAUDIBLE].. 00:17:45.490 --> 00:17:47.041 Then [INAUDIBLE] like these funnels 00:17:47.041 --> 00:17:50.546 are [INAUDIBLE] uncertainty in the world that you have. 00:17:50.546 --> 00:17:56.615 If [INAUDIBLE] everything with uncertainties, [INAUDIBLE] 00:17:56.615 --> 00:17:59.050 and then you compare funnels for that. 00:17:59.050 --> 00:18:00.511 Any questions so far? 00:18:04.407 --> 00:18:08.303 AUDIENCE: Can you please say again definition of funnel? 00:18:08.303 --> 00:18:10.750 PRESENTER 2: So it's kind of something that [INAUDIBLE] 00:18:10.750 --> 00:18:13.034 around [INAUDIBLE] around-- 00:18:13.034 --> 00:18:16.962 like you have a [INAUDIBLE] around there. 00:18:16.962 --> 00:18:20.399 So I think Gerard with explain more about it. 00:18:20.399 --> 00:18:23.836 So the [INAUDIBLE] with funnels is 00:18:23.836 --> 00:18:25.800 that you are inside the funnel at the point. 00:18:25.800 --> 00:18:27.273 When you [INAUDIBLE]. 00:18:31.692 --> 00:18:34.147 It's like [INAUDIBLE]. 00:18:34.147 --> 00:18:37.130 AUDIENCE: Yeah, you have an [INAUDIBLE].. 00:18:37.130 --> 00:18:39.780 So a funnel is basically-- 00:18:39.780 --> 00:18:42.402 it's kind of like a reach set around a certain point too. 00:18:42.402 --> 00:18:45.112 So the way that you do it is you have some [INAUDIBLE] 00:18:45.112 --> 00:18:45.612 trajectory. 00:18:45.612 --> 00:18:48.640 And then at each node point in that trajectory, 00:18:48.640 --> 00:18:50.903 you compute some kind of reachable 00:18:50.903 --> 00:18:53.270 set at that node point. 00:18:53.270 --> 00:18:56.440 And so you blow it up, and it becomes some kind of ellipsoid. 00:18:56.440 --> 00:18:58.320 And as long as you're within that ellipsoid, 00:18:58.320 --> 00:18:59.785 no matter what disturbance you get, 00:18:59.785 --> 00:19:01.826 you're going to go back towards the nominal path. 00:19:01.826 --> 00:19:04.690 And then so, when you join all of those ellipsoids 00:19:04.690 --> 00:19:07.600 over the trajectory, then you get a certain kind of funnel. 00:19:07.600 --> 00:19:09.790 And as long as you're inside that funnel, 00:19:09.790 --> 00:19:11.290 you're going to stay on that funnel. 00:19:11.290 --> 00:19:13.118 And you're going to be on the path you 00:19:13.118 --> 00:19:15.140 want to be at with some disturbance in this 00:19:15.140 --> 00:19:16.900 [INAUDIBLE]. 00:19:16.900 --> 00:19:18.702 So that the great thing that there would 00:19:18.702 --> 00:19:20.305 be an example of a funnel. 00:19:20.305 --> 00:19:21.930 As long as you're in the funnel, you're 00:19:21.930 --> 00:19:23.560 going to stay in the funnel, basically. 00:19:23.560 --> 00:19:27.085 AUDIENCE: So is this considered the control, or is 00:19:27.085 --> 00:19:28.289 it only the dynamics? 00:19:28.289 --> 00:19:30.288 PRESENTER 1: It's the dynamics with the control. 00:19:30.288 --> 00:19:34.590 So it's the flow tube system. 00:19:34.590 --> 00:19:36.215 PRESENTER 2: Yeah, so, you'll learn 00:19:36.215 --> 00:19:37.298 more of those [INAUDIBLE]. 00:19:39.974 --> 00:19:41.390 AUDIENCE: Let me just quickly add. 00:19:41.390 --> 00:19:44.350 So a funnel is type of flow tube, right? 00:19:44.350 --> 00:19:46.320 I mean, just different people either use 00:19:46.320 --> 00:19:48.450 the same terminology, use different terminology. 00:19:48.450 --> 00:19:50.940 Each of the applications of flow tubes 00:19:50.940 --> 00:19:53.316 has some invariant, which is different. 00:19:53.316 --> 00:19:56.010 And the particular invariant on this one 00:19:56.010 --> 00:19:59.550 is that if you are you in the funnel 00:19:59.550 --> 00:20:03.100 and you apply the LQR component, [INAUDIBLE] then 00:20:03.100 --> 00:20:05.366 you're guaranteed to stay within that tube. 00:20:05.366 --> 00:20:07.820 AUDIENCE: But I thought the flow tube is actually 00:20:07.820 --> 00:20:13.580 like getting smaller and smaller in the funnel [INAUDIBLE].. 00:20:13.580 --> 00:20:14.870 AUDIENCE: Not necessarily. 00:20:14.870 --> 00:20:16.980 In general, a flow tube is simply 00:20:16.980 --> 00:20:20.180 a bundle of trajectories, which capture some common features. 00:20:20.180 --> 00:20:22.636 And then maybe that they move to a limit of cycle, 00:20:22.636 --> 00:20:25.790 and maybe they stabilize to a point, and maybe that the move 00:20:25.790 --> 00:20:29.080 to a goal, maybe that they always maintain. 00:20:29.080 --> 00:20:31.130 They're stable within a tube. 00:20:31.130 --> 00:20:33.130 In a funnel they focus particularly on stability 00:20:33.130 --> 00:20:38.914 to disturbance and then staying within the tube. 00:20:38.914 --> 00:20:41.324 PRESENTER 2: OK, so the [INAUDIBLE] example 00:20:41.324 --> 00:20:47.432 is kind of useful to combine the reachability motion planning. 00:20:47.432 --> 00:20:49.428 So here is another situation where 00:20:49.428 --> 00:20:51.379 getting to the goal from the start [INAUDIBLE] 00:20:51.379 --> 00:20:53.420 and there are a couple of sort of trees lined up. 00:20:53.420 --> 00:20:57.412 But there is a path that goes in between the [INAUDIBLE].. 00:21:00.420 --> 00:21:03.952 This path looks kind of risky because it's 00:21:03.952 --> 00:21:05.940 pretty close to the two trees. 00:21:05.940 --> 00:21:07.679 But there is another path we can take 00:21:07.679 --> 00:21:09.916 which travel around the trees. 00:21:09.916 --> 00:21:15.880 And so if you don't need the [INAUDIBLE] safe. 00:21:15.880 --> 00:21:17.870 But maybe that's [INAUDIBLE] too. 00:21:17.870 --> 00:21:20.490 So that's why we do the combined reachability 00:21:20.490 --> 00:21:21.758 with motion planning. 00:21:21.758 --> 00:21:23.674 So if you combine that, you might 00:21:23.674 --> 00:21:26.229 find that the first path, if you compute a flow 00:21:26.229 --> 00:21:27.895 tube for the first path, [INAUDIBLE] it 00:21:27.895 --> 00:21:29.144 does not hit any of the trees. 00:21:29.144 --> 00:21:30.310 And it's indeed safe. 00:21:30.310 --> 00:21:32.964 But it might happen that the second path 00:21:32.964 --> 00:21:38.800 is more susceptible to the disturbances in [INAUDIBLE].. 00:21:38.800 --> 00:21:42.195 So that's why [INAUDIBLE] and also [INAUDIBLE] 00:21:42.195 --> 00:21:43.650 combined motion planning. 00:21:43.650 --> 00:21:48.500 Because what's intuitive is not actual. 00:21:48.500 --> 00:21:51.750 So far it's all about [INAUDIBLE] planning. 00:21:51.750 --> 00:21:54.412 So the [INAUDIBLE] everything about the environment, 00:21:54.412 --> 00:21:56.340 like where the obstacles are. 00:21:56.340 --> 00:21:57.532 So that's not always true. 00:21:57.532 --> 00:21:58.484 [INAUDIBLE] 00:21:58.484 --> 00:22:00.864 So you don't always know all the obstacles beforehand. 00:22:00.864 --> 00:22:04.672 You can't [INAUDIBLE] as you [INAUDIBLE].. 00:22:04.672 --> 00:22:07.421 So one problem with doing online planning, 00:22:07.421 --> 00:22:10.307 there is that you can not do any expensive computations 00:22:10.307 --> 00:22:11.750 during runtime. 00:22:11.750 --> 00:22:14.636 And all of these funnels can be places that are very expensive. 00:22:14.636 --> 00:22:16.560 [INAUDIBLE] solving an optimization problem. 00:22:16.560 --> 00:22:19.450 It takes a couple of hours to finish them. 00:22:19.450 --> 00:22:21.907 So you can not compute funnels on the fly. 00:22:21.907 --> 00:22:24.950 So the idea here is that you compute the funnels offline 00:22:24.950 --> 00:22:27.774 and do all possible funnels from the start state 00:22:27.774 --> 00:22:31.968 to the end state, and then use this library to [INAUDIBLE].. 00:22:31.968 --> 00:22:37.800 So you have the same [INAUDIBLE] going from the start state 00:22:37.800 --> 00:22:40.240 [INAUDIBLE] the goal state in the archives. 00:22:40.240 --> 00:22:43.204 So these are the set of possible paths from start state 00:22:43.204 --> 00:22:44.310 to goal state. 00:22:44.310 --> 00:22:46.043 And then from each possible trajectory, 00:22:46.043 --> 00:22:48.010 you compute the funnel. 00:22:48.010 --> 00:22:50.196 And the idea of online planning is 00:22:50.196 --> 00:22:52.300 to now find the sequence of these funnels 00:22:52.300 --> 00:22:55.266 that you can compose that takes you from the start state 00:22:55.266 --> 00:22:59.210 to end state without hitting any of the obstacles. 00:22:59.210 --> 00:23:02.168 So an important component here is 00:23:02.168 --> 00:23:07.610 finding a sequential composition of funnels [INAUDIBLE].. 00:23:07.610 --> 00:23:09.220 AUDIENCE: So why plan-- 00:23:09.220 --> 00:23:11.350 you're doing a forward search with funnels. 00:23:11.350 --> 00:23:14.310 Why use funnels instead of something like RIT? 00:23:17.810 --> 00:23:23.310 PRESENTER 2: I actually don't know what RITs are. 00:23:23.310 --> 00:23:25.138 AUDIENCE: Can I guess at the answer? 00:23:25.138 --> 00:23:28.810 AUDIENCE: [INAUDIBLE] thing. 00:23:28.810 --> 00:23:31.706 AUDIENCE: Why use funnels instead of RITs. 00:23:31.706 --> 00:23:33.819 AUDIENCE: Well, or why use funnels as opposed 00:23:33.819 --> 00:23:34.527 to anything else. 00:23:34.527 --> 00:23:37.710 AUDIENCE: Well, I guess RITs, you 00:23:37.710 --> 00:23:39.330 have like the classical example, where 00:23:39.330 --> 00:23:41.580 you want to go through a very-- 00:23:41.580 --> 00:23:45.820 well, if you want to go through a very narrow region. 00:23:45.820 --> 00:23:47.696 Well, first of all, I guess RITs are 00:23:47.696 --> 00:23:49.070 going to have a hard time finding 00:23:49.070 --> 00:23:51.820 the narrow region because you'd need like a point 00:23:51.820 --> 00:23:53.320 to go through that. 00:23:53.320 --> 00:23:56.830 And then, RITs by themselves don't really 00:23:56.830 --> 00:24:02.208 account for errors, maybe some extension. 00:24:02.208 --> 00:24:05.500 But funnels, you can be sure that you're always going 00:24:05.500 --> 00:24:06.870 to stay inside the funnel. 00:24:06.870 --> 00:24:08.090 So this if for [INAUDIBLE]. 00:24:08.090 --> 00:24:10.940 This is to make sure that we don't crash into anything. 00:24:10.940 --> 00:24:12.932 Does that answer your question? 00:24:12.932 --> 00:24:14.390 AUDIENCE: Do you want to answer it? 00:24:14.390 --> 00:24:19.870 AUDIENCE: Yeah, I think what he just said, it's-- 00:24:19.870 --> 00:24:23.450 the point of the funnels is that they provide a control policy. 00:24:23.450 --> 00:24:26.920 The actual reference trajectory that you have on the left 00:24:26.920 --> 00:24:28.690 is the motion planning. 00:24:28.690 --> 00:24:31.010 But with the funnels, you get a control policy. 00:24:31.010 --> 00:24:33.000 Because when you start executing a motion plan 00:24:33.000 --> 00:24:35.740 there's always going to be a disturbance. 00:24:35.740 --> 00:24:38.000 So you want a control policy that 00:24:38.000 --> 00:24:40.458 keeps you on that trajectory. 00:24:40.458 --> 00:24:43.115 And there's all sorts of control policies. 00:24:43.115 --> 00:24:45.775 You could use simple EID controllers. 00:24:45.775 --> 00:24:48.066 They work in some cases. 00:24:48.066 --> 00:24:49.855 But funnels provide better guarantees 00:24:49.855 --> 00:24:51.643 for the power plants. 00:24:56.553 --> 00:24:59.008 PRESENTER 2: Yeah, so the idea is 00:24:59.008 --> 00:25:01.454 that you have trajectory [INAUDIBLE] funnels 00:25:01.454 --> 00:25:01.954 [INAUDIBLE]. 00:25:01.954 --> 00:25:03.918 You're using this funnels compliance. 00:25:03.918 --> 00:25:07.832 You compose them to find your path to those presets 00:25:07.832 --> 00:25:09.760 that you wanted to go in. 00:25:13.616 --> 00:25:16.560 So you can not always compose funnels. 00:25:16.560 --> 00:25:20.234 Some [INAUDIBLE] positions that are legal, and some of those 00:25:20.234 --> 00:25:20.970 are not legal. 00:25:20.970 --> 00:25:25.380 So [INAUDIBLE] example, a time going from left to right. 00:25:25.380 --> 00:25:28.774 So you have a situation where there is a funnel. 00:25:28.774 --> 00:25:31.254 You're trying to compose a bigger funnel with a smaller 00:25:31.254 --> 00:25:32.246 funnel. 00:25:32.246 --> 00:25:34.726 And you have another situation where 00:25:34.726 --> 00:25:37.330 you're going to compose a smaller funnel with a bigger 00:25:37.330 --> 00:25:38.710 funnel. 00:25:38.710 --> 00:25:41.520 So what if I would have said, [INAUDIBLE].. 00:25:41.520 --> 00:25:43.770 So which of these combinations do you 00:25:43.770 --> 00:25:47.422 think is a legal composition? 00:25:47.422 --> 00:25:48.414 [INAUDIBLE] 00:25:51.390 --> 00:25:54.862 Legal or illegal, I mean that which of these [INAUDIBLE] 00:25:54.862 --> 00:25:58.334 do not receive the [INAUDIBLE]? 00:25:58.334 --> 00:25:59.822 AUDIENCE: The bottom [INAUDIBLE].. 00:25:59.822 --> 00:26:01.155 PRESENTER 2: The bottom one is-- 00:26:01.155 --> 00:26:03.390 AUDIENCE: --is actually marked because-- 00:26:03.390 --> 00:26:09.640 with the top one, if you're at the very top [INAUDIBLE].. 00:26:09.640 --> 00:26:10.940 PRESENTER 2: Yeah, that's true. 00:26:10.940 --> 00:26:13.892 So the first one is like illegal. 00:26:13.892 --> 00:26:16.352 And the second one is a legal composition. 00:26:16.352 --> 00:26:18.545 So why is that? 00:26:18.545 --> 00:26:20.030 So if you are inside this funnel, 00:26:20.030 --> 00:26:23.572 what this guarantees that you will stay inside the funnel. 00:26:23.572 --> 00:26:26.654 But it could happen that you can go through the funnel 00:26:26.654 --> 00:26:28.598 and make your outside [INAUDIBLE] funnel. 00:26:28.598 --> 00:26:32.486 And after this step, you can go anywhere. 00:26:32.486 --> 00:26:35.888 So you can [INAUDIBLE]. 00:26:35.888 --> 00:26:38.318 But if you're inside the second case, 00:26:38.318 --> 00:26:40.852 after you're through the first part, 00:26:40.852 --> 00:26:42.310 you're still set at the second part 00:26:42.310 --> 00:26:43.601 then we can decide [INAUDIBLE]. 00:26:48.720 --> 00:26:51.666 So now let's actually look at those online planning 00:26:51.666 --> 00:26:53.139 algorithms. 00:26:53.139 --> 00:26:56.576 So this the case of your planning [INAUDIBLE].. 00:26:56.576 --> 00:26:59.031 First you can [INAUDIBLE] with quantum's chronicles. 00:26:59.031 --> 00:27:00.995 It makes our analysis easier. 00:27:03.941 --> 00:27:06.912 So this is the algorithm that's going through each of the steps 00:27:06.912 --> 00:27:08.400 one-by-one. 00:27:08.400 --> 00:27:10.384 So this is online planning. 00:27:10.384 --> 00:27:12.864 You will always one information beforehand. 00:27:12.864 --> 00:27:14.848 [INAUDIBLE] 00:27:14.848 --> 00:27:17.824 And you get to know more about it as we go. 00:27:17.824 --> 00:27:21.792 But you still have the initial planned funnel sequence 00:27:21.792 --> 00:27:24.272 that takes you from the start state to goal state. 00:27:24.272 --> 00:27:29.232 And then you check for any new obstacles information 00:27:29.232 --> 00:27:31.216 when you go out from the sensor. 00:27:31.216 --> 00:27:36.176 So this is the region that you can sense some obstacles. 00:27:36.176 --> 00:27:38.676 And you also get the current state of your robot. 00:27:38.676 --> 00:27:41.574 This is also something concern [INAUDIBLE].. 00:27:45.430 --> 00:27:47.358 And you check if your current funnel path 00:27:47.358 --> 00:27:50.250 collides with any of the obstacles you have seen so far. 00:27:50.250 --> 00:27:52.670 And in this case, the answer is no. 00:27:52.670 --> 00:27:56.742 So then you apply the control corresponding 00:27:56.742 --> 00:28:00.200 to this funnel for this [INAUDIBLE] location and time. 00:28:00.200 --> 00:28:04.152 That take your one [INAUDIBLE] problem. 00:28:04.152 --> 00:28:07.610 Then you put Goto and then do this again and again. 00:28:07.610 --> 00:28:10.521 So [INAUDIBLE] update from certain information. 00:28:10.521 --> 00:28:11.562 And one obstacle is here. 00:28:11.562 --> 00:28:14.032 You can differentiate the other one. 00:28:14.032 --> 00:28:16.255 When we check at the Replan collides 00:28:16.255 --> 00:28:19.960 with any of the obstacles-- in this case, yes. 00:28:19.960 --> 00:28:21.936 So since it collides with the obstacles, 00:28:21.936 --> 00:28:23.912 you need to [INAUDIBLE] funnels. 00:28:23.912 --> 00:28:26.390 So how do you delete funnels? 00:28:26.390 --> 00:28:27.875 You have all this library funnels. 00:28:27.875 --> 00:28:30.310 So you go to each of them, and then you 00:28:30.310 --> 00:28:36.610 find one set of funnels that has-- 00:28:36.610 --> 00:28:38.602 you find a set of funnels that starts 00:28:38.602 --> 00:28:42.761 with your current location and [INAUDIBLE] of the obstacles. 00:28:42.761 --> 00:28:49.248 So this was a point, first of all, of a new funnel sequence. 00:28:49.248 --> 00:28:51.743 And then, again, you apply controls 00:28:51.743 --> 00:28:54.238 corresponding to the new sequence 00:28:54.238 --> 00:28:59.727 and doing it two dimensional [INAUDIBLE],, In the position, 00:28:59.727 --> 00:29:02.721 if it collides, and again collides, 00:29:02.721 --> 00:29:07.212 so you redirect [INAUDIBLE] apply 00:29:07.212 --> 00:29:09.707 the control for that funnel. 00:29:09.707 --> 00:29:11.703 And get a full solution. 00:29:11.703 --> 00:29:16.194 And then it collides, so you have to replan that part. 00:29:16.194 --> 00:29:18.689 Then you're OK. 00:29:18.689 --> 00:29:20.685 So now you reached the goal. 00:29:20.685 --> 00:29:26.426 So those are the planning plan using funnel libraries. 00:29:26.426 --> 00:29:28.911 And questions about it? 00:29:28.911 --> 00:29:31.450 AUDIENCE: Can you say more about the connotation 00:29:31.450 --> 00:29:32.760 of the funnel libraries? 00:29:32.760 --> 00:29:35.740 So they generate a set of funnels, which 00:29:35.740 --> 00:29:39.320 covered the complete set space? 00:29:39.320 --> 00:29:42.150 PRESENTER 2: Yeah, so that's like-- 00:29:42.150 --> 00:29:47.200 there's every possible [INAUDIBLE] end point. 00:29:47.200 --> 00:29:50.560 And so then you can make [INAUDIBLE] 00:29:50.560 --> 00:29:53.440 or least, have it for every initial [INAUDIBLE] 00:29:53.440 --> 00:29:55.840 you can compose all these [INAUDIBLE].. 00:29:55.840 --> 00:30:00.270 So if you're doing it for every initial [INAUDIBLE].. 00:30:00.270 --> 00:30:02.442 So you might have a funnel that exactly starts 00:30:02.442 --> 00:30:03.775 at your initial position. 00:30:03.775 --> 00:30:05.941 If there is a funnel that already started somewhere, 00:30:05.941 --> 00:30:08.376 where it goes through the funnel-- goes 00:30:08.376 --> 00:30:10.324 through your initial funnel, you can use that. 00:30:10.324 --> 00:30:14.220 So it's a truncation of your funnel. 00:30:14.220 --> 00:30:17.629 [INAUDIBLE] 00:30:17.629 --> 00:30:19.090 Any other questions? 00:30:22.500 --> 00:30:25.180 AUDIENCE: Why can't you use-- 00:30:25.180 --> 00:30:28.102 if it really is an online problem 00:30:28.102 --> 00:30:31.040 that you're not going to run out of a funnel, 00:30:31.040 --> 00:30:34.215 that you won't reach a dead end, will it be adaptable enough 00:30:34.215 --> 00:30:37.480 to make a U-turn at all times, either 00:30:37.480 --> 00:30:41.550 that there's come latency that would perfectly time it. 00:30:41.550 --> 00:30:44.360 And the robustness, that you'll always be able to find a path, 00:30:44.360 --> 00:30:46.234 even if we have guarantees that you'll always 00:30:46.234 --> 00:30:48.932 stay within a funnel. 00:30:48.932 --> 00:30:51.267 PRESENTER 2: So are you asking what the situation, 00:30:51.267 --> 00:30:54.882 like the funnel, if you're going [INAUDIBLE] backtrack? 00:30:54.882 --> 00:30:57.090 AUDIENCE: Yeah, just because these obstacles come out 00:30:57.090 --> 00:31:00.800 of nowhere, seemingly, that you don't have a guaranteed 00:31:00.800 --> 00:31:03.246 that the funnel is safe through the entire duration 00:31:03.246 --> 00:31:07.060 of our execution. 00:31:07.060 --> 00:31:10.766 PRESENTER 2: Yeah, so that's a valid point. 00:31:10.766 --> 00:31:15.050 [INAUDIBLE] you can use the adaptive [INAUDIBLE] backtrack 00:31:15.050 --> 00:31:18.860 [INAUDIBLE] combined with the [INAUDIBLE].. 00:31:18.860 --> 00:31:22.360 AUDIENCE: Even if the robots can't actually backtrack? 00:31:22.360 --> 00:31:24.100 PRESENTER 2: If the robots can what? 00:31:24.100 --> 00:31:25.535 Come back, or-- 00:31:25.535 --> 00:31:27.535 AUDIENCE: I mean, I guess, it's just like a UAB, 00:31:27.535 --> 00:31:29.710 it couldn't do that. 00:31:29.710 --> 00:31:34.296 I'm just saying, what guarantees do you need to have about 00:31:34.296 --> 00:31:38.305 the obstacles , since it's an online setup, right? 00:31:38.305 --> 00:31:40.554 PRESENTER 2: Maybe you'd have some initial information 00:31:40.554 --> 00:31:42.989 about some kind of obstacles. 00:31:42.989 --> 00:31:45.424 You can find a [INAUDIBLE] trajectory-- 00:31:45.424 --> 00:31:48.050 you can find a trajectory that goes from the start sequence. 00:31:48.050 --> 00:31:49.296 It's like [INAUDIBLE]. 00:31:53.264 --> 00:31:55.690 AUDIENCE: Well, I'm just going to say. 00:31:55.690 --> 00:31:58.592 If you're in some kind of glider that can't necessarily-- 00:31:58.592 --> 00:32:00.640 like a quart of stop, go backwards, 00:32:00.640 --> 00:32:02.480 stuff like that, depending on how funnels 00:32:02.480 --> 00:32:04.055 you can plan backwards. 00:32:04.055 --> 00:32:06.179 But if you're in some kind of glider going straight 00:32:06.179 --> 00:32:07.564 at a wall, there's no guarantee. 00:32:07.564 --> 00:32:10.510 You're just screwed up. 00:32:10.510 --> 00:32:12.560 I mean, even humans, they make errors. 00:32:12.560 --> 00:32:15.930 There's situations that humans can't dynamically get out of. 00:32:15.930 --> 00:32:17.770 So if your system is dynamically not 00:32:17.770 --> 00:32:20.110 going to be able to make a sharp enough turn 00:32:20.110 --> 00:32:22.394 or go straight into something, no matter 00:32:22.394 --> 00:32:23.935 how smart your planning algorithm is, 00:32:23.935 --> 00:32:25.901 no matter if you're like, there's a wall right there. 00:32:25.901 --> 00:32:26.620 Stop. 00:32:26.620 --> 00:32:29.608 If you can't stop, and you just go straight at it. 00:32:29.608 --> 00:32:31.268 There's obviously situations where 00:32:31.268 --> 00:32:33.592 you can pretty much [INAUDIBLE] algorithm [INAUDIBLE].. 00:32:33.592 --> 00:32:34.717 You're just going to crash. 00:32:34.717 --> 00:32:40.170 Hopefully you're in a situation where you're not [INAUDIBLE].. 00:32:40.170 --> 00:32:43.570 It's also based on how far away your sets are received. 00:32:43.570 --> 00:32:45.720 Your sensor can only see 5 meters ahead, 00:32:45.720 --> 00:32:49.050 and you're going 5 meters per second, [INAUDIBLE].. 00:32:49.050 --> 00:32:50.941 You're going to have make perfect decisions. 00:32:50.941 --> 00:32:53.150 But if your sensor has perfect information 00:32:53.150 --> 00:32:56.410 about the environment, then what seems like the initial best 00:32:56.410 --> 00:33:00.225 path may not actually be once you know the full [INAUDIBLE].. 00:33:00.225 --> 00:33:05.582 So it depends on the [INAUDIBLE] sensors. 00:33:05.582 --> 00:33:09.965 PRESENTER 2: Any other questions? 00:33:09.965 --> 00:33:14.748 OK, so finally I have some demos showing simulations 00:33:14.748 --> 00:33:18.716 of [INAUDIBLE]. 00:33:18.716 --> 00:33:20.700 Let's see how [INAUDIBLE]. 00:33:31.116 --> 00:33:33.100 So the [INAUDIBLE] applications. 00:33:33.100 --> 00:33:38.610 So we have seen how to use funnels 00:33:38.610 --> 00:33:40.660 to do robust motion planning. 00:33:40.660 --> 00:33:42.818 We also have the offline planning 00:33:42.818 --> 00:33:45.805 and come to do online planning using a sequence funnels 00:33:45.805 --> 00:33:48.010 and the funnel libraries. 00:33:48.010 --> 00:33:51.930 I will talk more about computing these funnels and flow tubes 00:33:51.930 --> 00:33:53.890 and other kind of reach sets. 00:34:05.929 --> 00:34:07.554 PRESENTER 3: Yeah, so I'm going to talk 00:34:07.554 --> 00:34:10.475 about how to actually compute flow tubes 00:34:10.475 --> 00:34:11.949 and funnels generally. 00:34:11.949 --> 00:34:15.999 I'm going to give an example of what each one is used for. 00:34:15.999 --> 00:34:18.540 So we had the question before, how are flow tubes and funnels 00:34:18.540 --> 00:34:19.950 different? 00:34:19.950 --> 00:34:27.800 Flow tubes are-- so easiest way is 00:34:27.800 --> 00:34:31.442 they're both some kind of way to guarantee that you're going 00:34:31.442 --> 00:34:34.389 to-- the robustness guarantees. 00:34:34.389 --> 00:34:37.190 So say you are here. 00:34:37.190 --> 00:34:39.150 You want to be here. 00:34:39.150 --> 00:34:41.909 They're both going to-- in most planning algorithms, 00:34:41.909 --> 00:34:44.070 you've got some kind of path from here to here. 00:34:44.070 --> 00:34:46.199 But say there's some disturbance [INAUDIBLE] 00:34:46.199 --> 00:34:47.960 down here or something. 00:34:47.960 --> 00:34:49.655 You don't know if you're going to be 00:34:49.655 --> 00:34:50.780 able to reach there or not. 00:34:50.780 --> 00:34:53.570 What a flow tube is is it's some kind 00:34:53.570 --> 00:34:57.980 of set of trajectories that will basically 00:34:57.980 --> 00:35:01.680 go from some initial state, which are here, 00:35:01.680 --> 00:35:03.990 to some goal states over here. 00:35:03.990 --> 00:35:06.540 And basically, [INAUDIBLE] if you're within this flow tube, 00:35:06.540 --> 00:35:09.090 and you get pushed out on a different trajectory, 00:35:09.090 --> 00:35:11.030 you know how to get from here to here anyway. 00:35:11.030 --> 00:35:12.780 So you don't really care if you're pushed, 00:35:12.780 --> 00:35:14.440 as long as you're still within here. 00:35:14.440 --> 00:35:16.293 A funnel-- let's see. 00:35:21.950 --> 00:35:26.300 So a funnel is, say you compute some nominal trajectory 00:35:26.300 --> 00:35:28.000 around here. 00:35:28.000 --> 00:35:31.194 Using the system theory and stability theory, 00:35:31.194 --> 00:35:34.410 you know that, with your control inputs, 00:35:34.410 --> 00:35:37.664 you're going to stay at each of these points. 00:35:37.664 --> 00:35:40.545 And you're going to stay within here within these circles. 00:35:40.545 --> 00:35:41.962 So if you get pushed out here, you 00:35:41.962 --> 00:35:44.253 know you're going to be able to get back onto your path 00:35:44.253 --> 00:35:46.119 when you're going to try to converge back 00:35:46.119 --> 00:35:47.394 to your nominal path. 00:35:47.394 --> 00:35:50.740 So you have some sort of funnel. 00:35:50.740 --> 00:35:52.874 Yes? 00:35:52.874 --> 00:35:54.040 AUDIENCE: I have a question. 00:35:54.040 --> 00:35:55.770 How complete is the funnel? 00:35:55.770 --> 00:35:59.310 It sort of related to what Brian was saying. 00:35:59.310 --> 00:36:02.370 PRESENTER 3: Completeness as in like these edges are-- 00:36:02.370 --> 00:36:05.110 AUDIENCE: Well, if you have a high dimensional state space, 00:36:05.110 --> 00:36:09.170 does the funnel cover it completely? 00:36:09.170 --> 00:36:10.950 AUDIENCE: There's a set of funnels. 00:36:10.950 --> 00:36:12.604 PRESENTER 3: A set of funnels-- 00:36:12.604 --> 00:36:15.270 AUDIENCE: So the question wasn't about if the individual funnel, 00:36:15.270 --> 00:36:19.150 but in the particular [INAUDIBLE].. 00:36:19.150 --> 00:36:21.740 When he generates a set of funnels, 00:36:21.740 --> 00:36:25.316 does the set of funnels completely cover the space? 00:36:25.316 --> 00:36:27.233 PRESENTER 3: It depends how much time you 00:36:27.233 --> 00:36:29.640 want to put into it, generally. 00:36:29.640 --> 00:36:33.240 If you only compute two funnels, one of them goes over here 00:36:33.240 --> 00:36:35.426 and one of them goes here, then you're not you're 00:36:35.426 --> 00:36:37.320 not going to be covered in these states. 00:36:37.320 --> 00:36:39.830 So it depends on how much computation you 00:36:39.830 --> 00:36:41.846 want to do offline previously. 00:36:41.846 --> 00:36:43.520 So you get a set of funnels that's here, 00:36:43.520 --> 00:36:45.830 another one that's right here, another one that's here. 00:36:45.830 --> 00:36:47.740 And you have your big library of funnels 00:36:47.740 --> 00:36:49.180 that cover the whole space. 00:36:49.180 --> 00:36:51.880 AUDIENCE: I think I skimmed that paper. 00:36:51.880 --> 00:36:55.016 And, yeah, I think it is probabilistically [INAUDIBLE].. 00:36:55.016 --> 00:36:56.824 PRESENTER 3: OK, yeah. 00:36:56.824 --> 00:37:00.150 All right, yeah, but I mean, as far as computing 00:37:00.150 --> 00:37:03.918 where you're going to go, if you compute that path of it, 00:37:03.918 --> 00:37:04.750 you should be good. 00:37:04.750 --> 00:37:05.888 But yes, you're right. 00:37:05.888 --> 00:37:09.600 The probabilistically-- 00:37:09.600 --> 00:37:11.860 So here is what I explained. 00:37:11.860 --> 00:37:13.736 Normally you have your optimal trajectory. 00:37:13.736 --> 00:37:15.235 But if you get pushed off of it, you 00:37:15.235 --> 00:37:16.870 may not get back to the goal state. 00:37:16.870 --> 00:37:20.162 But if you have a set of trajectories 00:37:20.162 --> 00:37:22.354 that go from some initial states to your goal state, 00:37:22.354 --> 00:37:23.770 and you stay within that, you know 00:37:23.770 --> 00:37:29.720 how to get to your goal state, even if you get pushed. 00:37:29.720 --> 00:37:31.960 Again, so some ways to approximate flow tubes, 00:37:31.960 --> 00:37:33.930 you can use some kind of polytope, 00:37:33.930 --> 00:37:36.650 which is as Gabriel explained previously, 00:37:36.650 --> 00:37:39.440 as long as it's a convex polytope, 00:37:39.440 --> 00:37:41.720 you can take that cross-sectional area 00:37:41.720 --> 00:37:44.780 and you can propagate it down here for the tube. 00:37:44.780 --> 00:37:51.678 You can also use ellipsoids or rectangles as cross-sections. 00:37:51.678 --> 00:37:54.500 And important point is to know that they're 00:37:54.500 --> 00:37:56.480 intercross-sectional approximations. 00:37:56.480 --> 00:37:58.730 So as you see here, you may-- 00:37:58.730 --> 00:38:00.720 this would be your actually flow tube. 00:38:00.720 --> 00:38:02.770 But your approximation has to be an inter one, 00:38:02.770 --> 00:38:04.685 because that guarantees that you're with a flow tube. 00:38:04.685 --> 00:38:07.226 But it means you may also miss some of the outer trajectories 00:38:07.226 --> 00:38:11.264 , which also, if you want to compute a more thorough 00:38:11.264 --> 00:38:13.216 approximation area you get better flow tube 00:38:13.216 --> 00:38:16.144 representation. 00:38:16.144 --> 00:38:22.450 OK, so robust planning, you plan a trajectory from here to here, 00:38:22.450 --> 00:38:25.740 and that should be good. 00:38:25.740 --> 00:38:27.182 But then you get disturbed. 00:38:27.182 --> 00:38:28.640 You're still within the trajectory, 00:38:28.640 --> 00:38:30.466 so you're still good. 00:38:30.466 --> 00:38:34.680 But some work done by Professor Williams and Hoffman 00:38:34.680 --> 00:38:35.760 shows that-- 00:38:35.760 --> 00:38:40.660 so what do you do when you're actually outside of the flow? 00:38:40.660 --> 00:38:43.380 Now what you can do is temporal planning. 00:38:43.380 --> 00:38:45.407 So we're using the algorithm that we previously 00:38:45.407 --> 00:38:46.360 learned in class. 00:38:46.360 --> 00:38:51.260 You can take your goal state and move the timeline back 00:38:51.260 --> 00:38:52.830 to a different time. 00:38:52.830 --> 00:38:56.950 So here, this flow tube is, if you're here 00:38:56.950 --> 00:39:00.010 in a certain amount of time, can you get to here? 00:39:00.010 --> 00:39:03.450 And if you get outside of it, then you can't do that anymore. 00:39:03.450 --> 00:39:05.670 But you can just use our algorithm 00:39:05.670 --> 00:39:08.819 and move back the goal time. 00:39:08.819 --> 00:39:10.360 And then when you move the goal time, 00:39:10.360 --> 00:39:12.980 the whole funnel shifts because now you're 00:39:12.980 --> 00:39:14.280 going to be over here. 00:39:14.280 --> 00:39:16.635 And you don't care about getting here at this time. 00:39:16.635 --> 00:39:18.810 You can here at this time now. 00:39:18.810 --> 00:39:20.510 So then you move over. 00:39:20.510 --> 00:39:23.290 And now where you, you get back into it 00:39:23.290 --> 00:39:25.206 because you're now inside the flow tube again. 00:39:27.978 --> 00:39:31.470 OK, so one example is a humanoid footstep planning. 00:39:31.470 --> 00:39:32.925 So this take a complex systems. 00:39:32.925 --> 00:39:35.250 You have a bunch of different parts of the system. 00:39:35.250 --> 00:39:37.000 You have your center of mass dynamics. 00:39:37.000 --> 00:39:40.090 You have your leg dynamics, foot dynamics. 00:39:40.090 --> 00:39:44.700 So what you're trying to do is plan your footsteps forward 00:39:44.700 --> 00:39:46.340 and through time. 00:39:46.340 --> 00:39:48.030 So you have your original plan of where 00:39:48.030 --> 00:39:50.800 you want your footsteps to be for different parts. 00:39:50.800 --> 00:39:54.820 So there's discrete things that happen in the system, 00:39:54.820 --> 00:39:57.660 as we've previously seen in a lecture. 00:39:57.660 --> 00:40:01.320 So one thing that can happen is your left foot hits the ground, 00:40:01.320 --> 00:40:02.993 and then your right foot lifts off. 00:40:02.993 --> 00:40:04.951 You're left foot lifts off-- or your right foot 00:40:04.951 --> 00:40:06.915 hits the ground, and then your left foot lifts off. 00:40:06.915 --> 00:40:08.831 So those are different things that can happen. 00:40:08.831 --> 00:40:12.510 So you have certain temporal constraints 00:40:12.510 --> 00:40:15.550 on the system that you want to [INAUDIBLE] by. 00:40:15.550 --> 00:40:20.480 And so as you start planning, you 00:40:20.480 --> 00:40:23.680 have flow tubes to get you from one step to another step. 00:40:23.680 --> 00:40:27.290 And that guarantees that you're going to stay within those. 00:40:27.290 --> 00:40:29.112 Even if you get disturbed and get pushed, 00:40:29.112 --> 00:40:32.510 your feet can still get from one step to the other. 00:40:32.510 --> 00:40:34.110 But then also, with the thing that we 00:40:34.110 --> 00:40:37.630 saw in the previous slide, you can move the timeline 00:40:37.630 --> 00:40:39.540 back and forth, as long as you're 00:40:39.540 --> 00:40:43.860 withing your temporal constraints that you had. 00:40:43.860 --> 00:40:46.260 So those are flow tubes for the feet. 00:40:46.260 --> 00:40:47.884 And that would guarantee that your feet 00:40:47.884 --> 00:40:49.980 go where they want to. 00:40:49.980 --> 00:40:52.050 But you can decompose the system to also 00:40:52.050 --> 00:40:54.210 have flow tubes for the center of mass. 00:40:54.210 --> 00:40:55.870 So what the center of mass does is 00:40:55.870 --> 00:41:00.220 to be broken down into separate flow tubes. 00:41:00.220 --> 00:41:02.180 So does that make sense? 00:41:05.610 --> 00:41:08.040 So this is the humanoid footstep planning 00:41:08.040 --> 00:41:09.320 that we saw previously. 00:41:09.320 --> 00:41:10.736 And now you're going to understand 00:41:10.736 --> 00:41:13.080 how the feet are planned to go to each of the blocks. 00:41:13.080 --> 00:41:17.412 [INAUDIBLE] go video. 00:41:17.412 --> 00:41:18.900 AUDIENCE: I have a quick question. 00:41:18.900 --> 00:41:20.610 You said that you have a separate flow 00:41:20.610 --> 00:41:21.776 tube for the center of mass. 00:41:21.776 --> 00:41:24.335 But aren't they really just the same flow tube, 00:41:24.335 --> 00:41:25.630 like a high dimensional-- 00:41:25.630 --> 00:41:28.010 PRESENTER 3: Yes, but that's-- when you decompose this 00:41:28.010 --> 00:41:31.387 and then you [INAUDIBLE] flow tube rather than a full system 00:41:31.387 --> 00:41:32.220 high dimensionality. 00:41:32.220 --> 00:41:35.392 AUDIENCE: Is it hard to recouple them? 00:41:35.392 --> 00:41:38.062 How do you know that they're consistent [INAUDIBLE]?? 00:41:38.062 --> 00:41:39.520 PRESENTER 3: Well, it still adheres 00:41:39.520 --> 00:41:40.860 to the dynamics of the system. 00:41:40.860 --> 00:41:45.355 So you can couple how you want, how you want it to move. 00:41:45.355 --> 00:41:47.350 And you want to move any basic constraints. 00:41:47.350 --> 00:41:50.059 But eventually, as you said, when you move your feet, 00:41:50.059 --> 00:41:51.600 your center of mass is going to move. 00:41:51.600 --> 00:41:53.320 When you step forward, you're center of mass 00:41:53.320 --> 00:41:54.070 is going to move. 00:41:54.070 --> 00:41:58.200 [INAUDIBLE] analysis of it. 00:41:58.200 --> 00:42:00.661 AUDIENCE: Can I make a comment about that? 00:42:00.661 --> 00:42:04.950 Yeah, there's a technique called feedback linearization that 00:42:04.950 --> 00:42:09.460 can be used to decompose a complex non-linear dynamics 00:42:09.460 --> 00:42:12.750 assuming to the set of linear ones. 00:42:12.750 --> 00:42:16.330 And that allows you to treat the center of mass 00:42:16.330 --> 00:42:20.500 as a [INAUDIBLE] separately from a control in the flow tube 00:42:20.500 --> 00:42:24.370 computations and while guaranteeing that [INAUDIBLE].. 00:42:27.280 --> 00:42:28.735 PRESENTER 3: So this is work done 00:42:28.735 --> 00:42:31.160 by Professor Williams experiment, which is actually 00:42:31.160 --> 00:42:34.070 [INAUDIBLE]. 00:42:34.070 --> 00:42:39.253 This is the robot doing the same kind of thing on Mars. 00:42:43.090 --> 00:42:46.230 This is walking on stones and planning its footsteps 00:42:46.230 --> 00:42:50.110 through a pretty complicated environment. 00:42:50.110 --> 00:42:52.231 The stones are not exactly in the path 00:42:52.231 --> 00:42:53.814 where you would normally want to walk. 00:42:53.814 --> 00:42:58.081 But moving them around, as long as you're within the flow tube, 00:42:58.081 --> 00:43:13.510 you'll get to wherever [INAUDIBLE] Awesome. 00:43:13.510 --> 00:43:17.772 So that was how to compute flow tubes and-- oh, 00:43:17.772 --> 00:43:19.980 and another way you can also compute flow tubes 00:43:19.980 --> 00:43:21.270 is by learning. 00:43:21.270 --> 00:43:23.910 So you can have some kind of example, 00:43:23.910 --> 00:43:27.150 move through trajectories from one state to the other. 00:43:27.150 --> 00:43:29.942 And you can estimate a flow tube based 00:43:29.942 --> 00:43:32.025 on where the trajectories are, as long as the flow 00:43:32.025 --> 00:43:37.660 tube encompasses all the possible trajectories. 00:43:37.660 --> 00:43:40.056 AUDIENCE: Is the dynamics always linear? 00:43:40.056 --> 00:43:43.330 Because I though if your polytope, 00:43:43.330 --> 00:43:45.970 and it would be like kind of, is it a complex proposal when 00:43:45.970 --> 00:43:48.127 you would use a non-linear dynamic, 00:43:48.127 --> 00:43:49.528 then you move a little bit. 00:43:49.528 --> 00:43:53.057 Then you would move a little bit, not being the convex shape 00:43:53.057 --> 00:43:54.020 then. 00:43:54.020 --> 00:43:57.150 PRESENTER 3: I mean, you can plan non-linear systems too. 00:43:57.150 --> 00:43:59.320 But as far as I'm concerned, [INAUDIBLE].. 00:44:03.550 --> 00:44:07.300 AUDIENCE: So, yeah, the polytopes 00:44:07.300 --> 00:44:10.200 generally require linearization with the system. 00:44:10.200 --> 00:44:14.760 And there's a limit to the validity of the-- the range 00:44:14.760 --> 00:44:16.580 of the linearization. 00:44:16.580 --> 00:44:18.651 And that's concluded in-- 00:44:18.651 --> 00:44:20.151 that's one of the things that limits 00:44:20.151 --> 00:44:23.050 the size of the polytopes. 00:44:23.050 --> 00:44:24.300 AUDIENCE: But it is the case-- 00:44:24.300 --> 00:44:25.450 I mean, you've just got to understand. 00:44:25.450 --> 00:44:27.324 The thing that you do for non-linear systems, 00:44:27.324 --> 00:44:29.850 which is a piece-- if you're making a piece-wise linear, 00:44:29.850 --> 00:44:31.030 but you are-- 00:44:31.030 --> 00:44:33.249 you are linearizing them at the w step. 00:44:33.249 --> 00:44:34.915 And there's other techniques that people 00:44:34.915 --> 00:44:39.012 have developed with flow tubes where you do require the input 00:44:39.012 --> 00:44:40.120 system to be linear. 00:44:40.120 --> 00:44:41.453 Right here it can be non-linear. 00:44:41.453 --> 00:44:43.922 But... it's an approximation. 00:44:49.240 --> 00:44:51.230 PRESENTER 3: OK, so funnels, the goal 00:44:51.230 --> 00:44:54.015 is basically to find a region that guarantees you safety 00:44:54.015 --> 00:44:57.370 on a bounded uncertainty. 00:44:57.370 --> 00:45:00.090 So there are regions of finite time variance 00:45:00.090 --> 00:45:03.550 throughout all time during that trajectory that you've defined. 00:45:03.550 --> 00:45:07.986 And so in practice, there's a trade-off between computation 00:45:07.986 --> 00:45:09.850 time and guarantees. 00:45:09.850 --> 00:45:12.600 So if your trajectory, if you compute more nodes, 00:45:12.600 --> 00:45:15.111 then you're going to be able to fill in these little gaps 00:45:15.111 --> 00:45:17.416 between each of the ellipsoid. 00:45:17.416 --> 00:45:19.280 Whereas, if you lessen them, it's 00:45:19.280 --> 00:45:22.070 going to take less time to compute each ellipsoid. 00:45:22.070 --> 00:45:25.940 But it's also going to not be as guaranteed. 00:45:25.940 --> 00:45:27.737 And especially important when planning 00:45:27.737 --> 00:45:30.070 is that, if you're planning in some kind of point cloud, 00:45:30.070 --> 00:45:34.240 then some of the points may be right here, where one ellipsoid 00:45:34.240 --> 00:45:35.830 doesn't intersect the other. 00:45:35.830 --> 00:45:38.710 And you might think that's a safe path, when actually it's 00:45:38.710 --> 00:45:39.640 not at all. 00:45:43.360 --> 00:45:46.380 Yeah, so an example is this little glider, 00:45:46.380 --> 00:45:49.650 which is given in the paper by Anirudha and Russ. 00:45:49.650 --> 00:45:56.640 And so what you start doing is you create your system, 00:45:56.640 --> 00:45:57.720 your function. 00:45:57.720 --> 00:45:59.752 And this is dependent on your state, time, 00:45:59.752 --> 00:46:03.310 and w, which is some kind of bounded uncertainty. 00:46:03.310 --> 00:46:06.030 And you choose the bounded uncertainty, 00:46:06.030 --> 00:46:11.820 which means that you choose some kind of reasonable estimate 00:46:11.820 --> 00:46:14.820 of how much can happen to your system 00:46:14.820 --> 00:46:17.430 that you didn't account for. 00:46:17.430 --> 00:46:19.510 So you start it. 00:46:19.510 --> 00:46:21.900 Your velocity can be different from what you want. 00:46:21.900 --> 00:46:24.650 Say your nominal velocity is 10 meters per second. 00:46:24.650 --> 00:46:26.839 You can have some kind of drag, or you 00:46:26.839 --> 00:46:29.170 can have some inconsistency in your motors. 00:46:29.170 --> 00:46:31.800 And so what you expect is that there's 00:46:31.800 --> 00:46:36.340 a plus or minus 0.5 meters per second to your nominal. 00:46:36.340 --> 00:46:39.460 So when you start running your equations, 00:46:39.460 --> 00:46:41.810 this is what you expect your robot to do. 00:46:41.810 --> 00:46:44.666 And then that can vary forward and backwards depending 00:46:44.666 --> 00:46:46.520 on what your actual was. 00:46:46.520 --> 00:46:50.284 But then you add some kind of wind on your system, which-- 00:46:50.284 --> 00:46:52.200 some kind of cross-sectional wind, [INAUDIBLE] 00:46:52.200 --> 00:46:55.301 direction, which we see there. 00:46:55.301 --> 00:46:58.690 And so what that ends up doing is pushing you 00:46:58.690 --> 00:47:01.335 system away from where you want it to actually be. 00:47:01.335 --> 00:47:06.010 So now you have some kind of velocity uncertainty. 00:47:06.010 --> 00:47:10.820 And you also have wind that can push you one way or the other. 00:47:10.820 --> 00:47:14.237 So how do you deal with that? 00:47:14.237 --> 00:47:16.070 Well, you start with your nominal trajectory 00:47:16.070 --> 00:47:19.622 that has a plane flying perfectly straight. 00:47:19.622 --> 00:47:20.580 There's no uncertainty. 00:47:20.580 --> 00:47:22.360 Everything is deterministic. 00:47:22.360 --> 00:47:23.940 And it acts like you want it to. 00:47:27.500 --> 00:47:31.120 And then, so right there the wind hits it. 00:47:31.120 --> 00:47:32.200 So what do you do? 00:47:32.200 --> 00:47:35.380 You have to have some kind of control, 00:47:35.380 --> 00:47:37.150 some kind of controller to bring you back 00:47:37.150 --> 00:47:39.750 to your nominal trajectory and try to control you 00:47:39.750 --> 00:47:41.090 around the system. 00:47:41.090 --> 00:47:46.350 So a standard way to do that is to have a cause function 00:47:46.350 --> 00:47:49.390 and create your optimal trajectory. 00:47:49.390 --> 00:47:52.320 And this cause function basically, 00:47:52.320 --> 00:47:56.520 it kind of mirrors a LQR controller. 00:47:56.520 --> 00:47:58.696 And by solving a lot of these equations, which 00:47:58.696 --> 00:48:02.260 is a pretty standard process, and to find out how to do that, 00:48:02.260 --> 00:48:05.390 you get a controller that tried to bring you back 00:48:05.390 --> 00:48:06.737 to your nominal trajectory. 00:48:06.737 --> 00:48:09.719 And so you get back. 00:48:09.719 --> 00:48:11.210 You're happy. 00:48:11.210 --> 00:48:13.089 But you don't have any guarantees 00:48:13.089 --> 00:48:14.130 just with the controller. 00:48:14.130 --> 00:48:16.640 You don't have any guarantees on how far you can get pushed. 00:48:16.640 --> 00:48:19.230 You don't have any guarantees on where you're going to go 00:48:19.230 --> 00:48:21.910 and where your system really can be pushed. 00:48:25.170 --> 00:48:28.630 So when you compute the funnel, what you do 00:48:28.630 --> 00:48:30.695 is you choose a Lyapunov function. 00:48:30.695 --> 00:48:34.844 And I'll explain what that is in the next slide. 00:48:34.844 --> 00:48:37.144 And you try to minimize the space around that function 00:48:37.144 --> 00:48:38.310 that tried to bring it back. 00:48:38.310 --> 00:48:41.276 So what this is is-- 00:48:41.276 --> 00:48:46.110 so you have some kind of your initial state. 00:48:46.110 --> 00:48:47.520 This would be the initial state. 00:48:47.520 --> 00:48:50.590 And the whole region, what you're trying to do 00:48:50.590 --> 00:48:53.786 is minimize it around it. 00:48:53.786 --> 00:48:56.686 So does that makes sense why you do that? 00:48:56.686 --> 00:48:59.250 So your trajectory is basically going 00:48:59.250 --> 00:49:01.667 to go anywhere around here. 00:49:01.667 --> 00:49:04.000 And what you're trying to do is find the region of space 00:49:04.000 --> 00:49:05.610 that minimizes-- 00:49:05.610 --> 00:49:07.620 that encompasses all of the places 00:49:07.620 --> 00:49:10.890 that your system is going to get to. 00:49:10.890 --> 00:49:13.030 So that's easy to do when you it out 00:49:13.030 --> 00:49:15.860 and you know exactly where your system's going to go. 00:49:15.860 --> 00:49:19.040 It's easy to just shrink your ball around it. 00:49:19.040 --> 00:49:20.950 But you actually compute that. 00:49:25.220 --> 00:49:25.803 No, forget it. 00:49:25.803 --> 00:49:26.610 OK. 00:49:26.610 --> 00:49:30.180 So when you have that for each point in the trajectory, 00:49:30.180 --> 00:49:32.209 you know that you're going to stay within here, 00:49:32.209 --> 00:49:33.834 because you're going to have some kinds 00:49:33.834 --> 00:49:35.330 of-- from your nominal trajectory, 00:49:35.330 --> 00:49:37.880 you're going to have different places where your system can 00:49:37.880 --> 00:49:42.340 go under the given conditions of velocity and wind uncertainty. 00:49:42.340 --> 00:49:44.160 You're going to go anywhere withing that. 00:49:44.160 --> 00:49:47.319 But as long as you know you're still in that ellipsoid, 00:49:47.319 --> 00:49:49.360 then you're going to stay within those ellipsoids 00:49:49.360 --> 00:49:51.845 as you go down the path. 00:49:51.845 --> 00:49:52.637 Does it make sense? 00:49:52.637 --> 00:49:53.137 Good. 00:49:56.820 --> 00:50:00.200 So the reason they use Lyapunov functions 00:50:00.200 --> 00:50:03.780 is there's different ways to measure stability. 00:50:03.780 --> 00:50:08.175 And so I'll move forward to this. 00:50:08.175 --> 00:50:10.470 So there's different ways to measure stability 00:50:10.470 --> 00:50:13.480 in a non-linear system. 00:50:13.480 --> 00:50:15.544 So in a linear system, you're either unstable 00:50:15.544 --> 00:50:17.210 or you're globally exponentially stable. 00:50:17.210 --> 00:50:19.134 So you're going to converge back to your-- 00:50:19.134 --> 00:50:20.870 to whatever stability equilibrium, 00:50:20.870 --> 00:50:23.920 what you have at an exponential rate. 00:50:23.920 --> 00:50:26.710 But there is also, in non-linear systems, 00:50:26.710 --> 00:50:28.305 there's global exponential stability. 00:50:28.305 --> 00:50:30.270 There asymptotic stability. 00:50:30.270 --> 00:50:32.660 And there's stability in the sense of Lyapunov, 00:50:32.660 --> 00:50:34.330 which is what we care about. 00:50:34.330 --> 00:50:37.015 So the difference between that is-- 00:50:37.015 --> 00:50:39.320 so say this is your nominal. 00:50:39.320 --> 00:50:41.680 And globally exponentially stable 00:50:41.680 --> 00:50:44.310 means that no matter where you start anywhere on the board, 00:50:44.310 --> 00:50:47.080 you're going to be converging back to this equilibrium 00:50:47.080 --> 00:50:49.100 going at an exponential rate. 00:50:49.100 --> 00:50:50.836 So that's the best case scenario. 00:50:50.836 --> 00:50:51.710 That's what you want. 00:50:54.240 --> 00:50:56.080 Asymptotic stability means that you're 00:50:56.080 --> 00:50:59.660 going to be going from somewhere on the board, 00:50:59.660 --> 00:51:02.230 within a certain region, back to this nominal trajectory. 00:51:02.230 --> 00:51:04.163 But it doesn't have to be an exponential rate. 00:51:04.163 --> 00:51:06.815 So you can do something like that, as long as eventually 00:51:06.815 --> 00:51:08.950 you sort of converging back to it. 00:51:08.950 --> 00:51:12.890 And then there's stability in the sense of Lyapunov, 00:51:12.890 --> 00:51:15.000 which is-- so, say you're here. 00:51:15.000 --> 00:51:19.540 Your system can be doing anything, anything crazy, 00:51:19.540 --> 00:51:23.830 as long as it stays within some certain thing, which 00:51:23.830 --> 00:51:27.945 in the 1D case we saw was the little ball going around it. 00:51:27.945 --> 00:51:32.764 So imagine there's like a ball for each of these. 00:51:32.764 --> 00:51:34.930 and this is where your trajectory could possibly go. 00:51:34.930 --> 00:51:37.263 As long as it stays within that, you can say your system 00:51:37.263 --> 00:51:39.302 is stable in the sense Lyapunov. 00:51:39.302 --> 00:51:40.844 Does that make sense? 00:51:40.844 --> 00:51:42.080 Cool. 00:51:42.080 --> 00:51:45.350 So know this, does that-- 00:51:45.350 --> 00:51:47.150 AUDIENCE: I just have a basic question. 00:51:47.150 --> 00:51:52.320 So the Lyapunov function is around a stable set point. 00:51:52.320 --> 00:51:55.390 But the trajectory has non-zero velocity. 00:51:55.390 --> 00:52:01.110 So what is the point that it's stabilizing around? 00:52:01.110 --> 00:52:05.550 PRESENTER 3: You can stabilize around a trajectory, which also 00:52:05.550 --> 00:52:09.050 what the funnels do. 00:52:09.050 --> 00:52:12.375 So we have your little point here. 00:52:12.375 --> 00:52:15.485 And this is your nominal trick, some kind of state. 00:52:15.485 --> 00:52:17.430 And you're wind can knock you this way. 00:52:17.430 --> 00:52:20.510 Your velocity difference can knock you this way or this way. 00:52:20.510 --> 00:52:22.600 But as long as you're still within that, 00:52:22.600 --> 00:52:24.300 You've got to converge back to it. 00:52:24.300 --> 00:52:26.810 And this is the value for your Lyapunov function. 00:52:26.810 --> 00:52:29.936 And these are the different states you can have. 00:52:29.936 --> 00:52:33.040 An important thing to notice is the derivative 00:52:33.040 --> 00:52:36.400 is a negative definite-- or negative semi-definite 00:52:36.400 --> 00:52:38.260 in the case. 00:52:38.260 --> 00:52:41.979 And that means that basically all your trajectories, 00:52:41.979 --> 00:52:43.520 no matter where you go, they're going 00:52:43.520 --> 00:52:45.440 to be going down this cone. 00:52:45.440 --> 00:52:49.731 And eventually they're going to to down to some value. 00:52:49.731 --> 00:52:50.730 So does that make sense? 00:52:50.730 --> 00:52:54.044 That's like pretty amazing stuff and your system stuff. 00:52:54.044 --> 00:52:55.870 Cool. 00:52:55.870 --> 00:52:58.470 So once you know that, it makes sense 00:52:58.470 --> 00:53:00.390 to use Lyapunov functions as candidates 00:53:00.390 --> 00:53:04.044 for computing funnels. 00:53:10.250 --> 00:53:15.246 So an ellipsoid is actually defined by a quadratic Lyapunov 00:53:15.246 --> 00:53:18.600 function, which is down here. 00:53:18.600 --> 00:53:24.890 [INAUDIBLE] shown the equation for an ellipsoid 00:53:24.890 --> 00:53:38.680 is some V times S. So that was the equation for an ellipsoid 00:53:38.680 --> 00:53:40.640 that Gabriel showed previously. 00:53:40.640 --> 00:53:45.130 The quadratic Lyapunov function has the same structure, 00:53:45.130 --> 00:53:51.085 where you have this being x to [INAUDIBLE],, which is basically 00:53:51.085 --> 00:53:52.440 the error of you state. 00:53:52.440 --> 00:53:58.605 So it's some kind of exact mean of the state 00:53:58.605 --> 00:54:03.110 that you're actually in minus x0, which is the nominal point 00:54:03.110 --> 00:54:04.200 that you want to reach. 00:54:04.200 --> 00:54:08.250 So this defines the region around your-- 00:54:08.250 --> 00:54:10.740 the nominal point, which would be somewhere in there. 00:54:10.740 --> 00:54:14.070 And the ellipsoid is computed by doing that minimization 00:54:14.070 --> 00:54:16.630 that I showed you on the previous slides. 00:54:16.630 --> 00:54:20.075 And then you can figure out the region inside of it 00:54:20.075 --> 00:54:21.790 by solving that equation. 00:54:21.790 --> 00:54:26.750 And if it's less than 1, I believe you inside. 00:54:26.750 --> 00:54:28.270 And those are all the red dots that 00:54:28.270 --> 00:54:29.560 show that you're inside of it. 00:54:29.560 --> 00:54:34.336 So that's used when computing trajectories. 00:54:34.336 --> 00:54:37.320 You solve for-- say that you're inside a point cloud, 00:54:37.320 --> 00:54:39.320 you know which points you're going to get inside 00:54:39.320 --> 00:54:40.970 of this ellipsoid. 00:54:40.970 --> 00:54:42.970 So when you're planning, you solve that equation 00:54:42.970 --> 00:54:44.095 for each of the ellipsoids. 00:54:44.095 --> 00:54:46.815 And you know if you're going to hit something or not. 00:54:51.565 --> 00:54:56.360 All right, so this is a video that [? Ani ?] did. 00:54:59.021 --> 00:55:01.010 And this is using the funnel libraries 00:55:01.010 --> 00:55:02.420 for collision avoidance. 00:55:02.420 --> 00:55:05.150 This is showing for just one funnel. 00:55:05.150 --> 00:55:09.000 The glider dynamics were shown in one of the previous slides. 00:55:09.000 --> 00:55:12.160 They were pre-set by the glider dynamics. 00:55:12.160 --> 00:55:16.960 But [INAUDIBLE]. 00:55:16.960 --> 00:55:20.830 So you see right there that there's objects, 00:55:20.830 --> 00:55:25.096 and the glider plans a path around them. 00:55:27.808 --> 00:55:31.370 So the funnels-- this is one possible one. 00:55:31.370 --> 00:55:34.106 Right here, I'm pretty sure they tell it where objects are, 00:55:34.106 --> 00:55:35.730 even though they're not actually there. 00:55:35.730 --> 00:55:40.040 And it finds that funnel being the one the guy wants. 00:55:40.040 --> 00:55:43.170 And you see that throughout the entire path, 00:55:43.170 --> 00:55:45.332 the glider will stay within the funnel. 00:55:51.410 --> 00:55:54.110 And then when you're searching with the online planning 00:55:54.110 --> 00:55:56.450 algorithm that was shown previously, 00:55:56.450 --> 00:55:59.716 it goes through all of these funnels 00:55:59.716 --> 00:56:02.515 and eventually finds the best safe one. 00:56:02.515 --> 00:56:04.915 You won't always find a safe one, as discussed. 00:56:04.915 --> 00:56:06.300 In this case you did. 00:56:06.300 --> 00:56:12.350 But it finds the best one around the path. 00:56:12.350 --> 00:56:13.970 So one way that it finds the best one 00:56:13.970 --> 00:56:17.410 is by finding all of-- so if you're planning a point cloud, 00:56:17.410 --> 00:56:19.640 you find all the points that lie with the ellipsoids. 00:56:19.640 --> 00:56:21.814 And the one with the least amount of points 00:56:21.814 --> 00:56:23.758 is probably going to be the safest one. 00:56:27.660 --> 00:56:31.930 So here's a bunch of different obstacles. 00:56:31.930 --> 00:56:33.956 And it safely maneuvers all. 00:56:33.956 --> 00:56:37.602 And there's some pretty dynamic movements, which is cool too. 00:56:45.966 --> 00:56:47.934 So within that there is-- 00:56:51.870 --> 00:56:54.330 let's see. 00:56:54.330 --> 00:56:55.314 I'm going to back to-- 00:56:59.250 --> 00:57:04.339 is there a way to get back to the full screen [INAUDIBLE]?? 00:57:04.339 --> 00:57:07.748 AUDIENCE: [INAUDIBLE] 00:57:07.748 --> 00:57:20.440 PRESENTER 3: [INAUDIBLE] 00:57:20.440 --> 00:57:23.400 So with the online planning algorithm, 00:57:23.400 --> 00:57:24.805 here's one of the possible paths. 00:57:24.805 --> 00:57:29.054 You have your point cloud of trees. 00:57:29.054 --> 00:57:33.340 And those were shown in the previous, the little round 00:57:33.340 --> 00:57:37.450 thing with the green foxy tree-looking things. 00:57:37.450 --> 00:57:40.165 Using a sensor model, you get a point of that. 00:57:40.165 --> 00:57:42.740 And as you're going, you plan your path. 00:57:42.740 --> 00:57:45.655 And the way that this works is every five meters, 00:57:45.655 --> 00:57:46.330 you plan a path. 00:57:46.330 --> 00:57:47.904 And then as you're flying through it, 00:57:47.904 --> 00:57:49.195 you get new sensor information. 00:57:49.195 --> 00:57:52.282 And you can replan your path using the algorithm. 00:57:57.480 --> 00:57:59.674 And that will guarantee you that you're safe. 00:57:59.674 --> 00:58:01.350 And when you're flying, obviously you 00:58:01.350 --> 00:58:02.580 don't see the funnels. 00:58:02.580 --> 00:58:05.690 But right here you can see there's probably 00:58:05.690 --> 00:58:07.820 one funnel here and then another funnel 00:58:07.820 --> 00:58:10.715 that plans this way, another funnel that goes to here, 00:58:10.715 --> 00:58:12.256 and then another one that goes there. 00:58:12.256 --> 00:58:13.704 And that's a full path. 00:58:13.704 --> 00:58:16.120 And there's also a cool one , because it shows that you go 00:58:16.120 --> 00:58:17.170 through the tree. 00:58:17.170 --> 00:58:18.920 And that's where one of the funnels 00:58:18.920 --> 00:58:21.320 is rather than going around. 00:58:21.320 --> 00:58:24.270 It is possible that you start computing a new funnel-- 00:58:24.270 --> 00:58:27.180 or you start searching a new library here. 00:58:27.180 --> 00:58:30.106 And you may not be able to make it up the way in time, 00:58:30.106 --> 00:58:33.970 given what speed you're going at. 00:58:33.970 --> 00:58:36.385 Does anyone have any questions? 00:58:41.710 --> 00:58:44.140 And this is the online planning algorithm 00:58:44.140 --> 00:58:46.562 that we saw before written out. 00:58:51.272 --> 00:58:54.590 And some of the references, the paper 00:58:54.590 --> 00:58:57.560 from the video that we just saw, that's given here. 00:58:57.560 --> 00:59:00.480 And that also go to the example [INAUDIBLE].. 00:59:00.480 --> 00:59:04.990 Frazzoli was one of the original people 00:59:04.990 --> 00:59:06.397 that computed a flow tube. 00:59:06.397 --> 00:59:10.355 Hoffman and Williams did the temporal planning for footsteps 00:59:10.355 --> 00:59:12.852 and decoupled humanoid system. 00:59:12.852 --> 00:59:21.810 And I did most of the stuff on [INAUDIBLE] for that years ago. 00:59:24.440 --> 00:59:28.220 Another important thing to note that was a big thing 00:59:28.220 --> 00:59:31.830 in my personal research was that even though you may note be 00:59:31.830 --> 00:59:35.293 able to compute-- or that you may not find a funnel that 00:59:35.293 --> 00:59:36.870 necessarily goes-- 00:59:36.870 --> 00:59:39.657 finds a safe path, you can shift that funnel around. 00:59:39.657 --> 00:59:41.490 You could even move it slightly up and down, 00:59:41.490 --> 00:59:43.805 as long as the first ellipsoid is still within-- 00:59:43.805 --> 00:59:47.330 as long as your initial state is still within better ellipsoid. 00:59:47.330 --> 00:59:50.550 And that's easy to do because you can just 00:59:50.550 --> 00:59:54.530 give it a simple transformation between one 00:59:54.530 --> 00:59:55.820 state and the other. 00:59:55.820 --> 00:59:59.069 And if you transform this S matrix, 00:59:59.069 --> 01:00:00.860 then that will transform your entire funnel 01:00:00.860 --> 01:00:02.026 and it will shift it around. 01:00:02.026 --> 01:00:03.630 So you can find your best funnel, 01:00:03.630 --> 01:00:05.045 even though it may hit something. 01:00:05.045 --> 01:00:07.120 If it was slightly shifted to the left, 01:00:07.120 --> 01:00:10.557 you can just shift it using that transformation. 01:00:10.557 --> 01:00:14.485 So that's one of the [INAUDIBLE] funnels too. 01:00:18.413 --> 01:00:22.341 Does everyone understand funnels and flow tubes now? 01:00:22.341 --> 01:00:23.840 AUDIENCE: So I have two questions. 01:00:23.840 --> 01:00:26.083 So the side of that funnel is computed 01:00:26.083 --> 01:00:27.082 by the transform script. 01:00:27.082 --> 01:00:27.665 Is that right? 01:00:27.665 --> 01:00:31.390 PRESENTER 3: Yeah, if you want to get it some ridiculous wind, 01:00:31.390 --> 01:00:33.440 like 1,000 miles per hour, then you're 01:00:33.440 --> 01:00:34.875 going to get huge funnels. 01:00:34.875 --> 01:00:37.300 But that's under some kind of reasonable-- 01:00:37.300 --> 01:00:40.550 you know in this region of the forest or something 01:00:40.550 --> 01:00:43.530 there's only three mile per hour winds or something like that. 01:00:43.530 --> 01:00:47.310 Yeah, that's based a lot on what your counts are. 01:00:47.310 --> 01:00:50.840 AUDIENCE: So do you have any algorithm that in your mind 01:00:50.840 --> 01:00:52.850 that is not using offline [INAUDIBLE]?? 01:00:52.850 --> 01:00:55.178 So this is like mostly you do the offline 01:00:55.178 --> 01:00:56.552 and the you do the online search. 01:00:56.552 --> 01:00:59.904 But if you were in a wild environment, you do nothing. 01:00:59.904 --> 01:01:02.310 Then do you have any [INAUDIBLE] that isn't-- 01:01:02.310 --> 01:01:04.519 PRESENTER 3: I mean, there are algorithms that will-- 01:01:04.519 --> 01:01:06.851 well, just the basic controller will try to get you back 01:01:06.851 --> 01:01:07.770 to nominal trajectory. 01:01:07.770 --> 01:01:09.730 But there's nothing that I know of at least 01:01:09.730 --> 01:01:12.050 that will guarantee you safety and will give you 01:01:12.050 --> 01:01:17.320 an all-encompassing region like flow tube or a funnel. 01:01:17.320 --> 01:01:20.530 So flow tube you find there's an infinite number of paths 01:01:20.530 --> 01:01:22.220 that will get you from here to here 01:01:22.220 --> 01:01:24.720 with your system, basically every little minute 01:01:24.720 --> 01:01:26.132 change that you have. 01:01:26.132 --> 01:01:28.590 And that obviously takes awhile to compute and approximate. 01:01:28.590 --> 01:01:33.260 And the funnels, it takes awhile to compute the regions 01:01:33.260 --> 01:01:34.270 where it can go. 01:01:34.270 --> 01:01:36.204 So I don't know anything that will online 01:01:36.204 --> 01:01:37.550 compute this region. 01:01:37.550 --> 01:01:40.720 But online you will be able to compute some trajectory. 01:01:40.720 --> 01:01:45.156 And then let your controller will take your state error 01:01:45.156 --> 01:01:46.650 and control it. 01:01:46.650 --> 01:01:50.197 But it won't guarantee [INAUDIBLE].. 01:01:50.197 --> 01:01:51.780 AUDIENCE: With the funnels, how do you 01:01:51.780 --> 01:01:54.590 compute your initial set of states, 01:01:54.590 --> 01:01:57.160 or the allowed initial set of states? 01:01:57.160 --> 01:02:00.400 You have these uncertainties, like for the wind and velocity, 01:02:00.400 --> 01:02:05.120 but what about the initial state before those disturbances 01:02:05.120 --> 01:02:05.880 happen? 01:02:05.880 --> 01:02:08.150 You'd probably want more than one initial state. 01:02:16.970 --> 01:02:18.440 AUDIENCE: Here's and eraser. 01:02:18.440 --> 01:02:20.890 PRESENTER 3: Yeah, that'd be good. 01:02:20.890 --> 01:02:21.860 Cool. 01:02:21.860 --> 01:02:22.360 Thanks. 01:02:25.300 --> 01:02:34.488 So with the funnels, so you have your nominal trajectory again, 01:02:34.488 --> 01:02:39.197 and then your ellipsoids around it. 01:02:39.197 --> 01:02:41.530 So when your system starts, you have your initial state. 01:02:41.530 --> 01:02:43.071 And your initial state is just-- it's 01:02:43.071 --> 01:02:45.350 going to be here at time t0. 01:02:45.350 --> 01:02:48.800 But over the time computed, you can 01:02:48.800 --> 01:02:51.262 go over here, or anywhere around here really, 01:02:51.262 --> 01:02:52.555 depending on what happens. 01:02:52.555 --> 01:02:55.060 AUDIENCE: But I just me that you'd 01:02:55.060 --> 01:02:58.580 want to apply the funnel multiple times. 01:02:58.580 --> 01:03:02.160 And you're initial state may not be exactly the same. 01:03:02.160 --> 01:03:07.682 So it should be possible to use the same funnel even 01:03:07.682 --> 01:03:10.350 though the initial state is off by a little bit. 01:03:10.350 --> 01:03:11.330 PRESENTER 3: Yeah, OK. 01:03:11.330 --> 01:03:13.590 So say that you chose this funnel first. 01:03:13.590 --> 01:03:16.570 And eventually what actually happens is you do this 01:03:16.570 --> 01:03:17.670 and you get pushed around. 01:03:17.670 --> 01:03:20.719 And then you end up here instead of this funnel. 01:03:20.719 --> 01:03:22.510 As long as your next funnel that you plan-- 01:03:22.510 --> 01:03:26.760 so you can start a new funnel at any point in this trajectory, 01:03:26.760 --> 01:03:29.230 as long as the next funnel is also-- 01:03:29.230 --> 01:03:31.766 as long as you have V intersection between the point 01:03:31.766 --> 01:03:33.810 and the next funnel. 01:03:33.810 --> 01:03:37.450 And if you're within this region, 01:03:37.450 --> 01:03:40.530 and you plan the next funnel, what do you do 01:03:40.530 --> 01:03:45.928 is you plan the next funnel with this being the original x0. 01:03:45.928 --> 01:03:49.180 So as long as that's you next x0, 01:03:49.180 --> 01:03:52.152 and then this will take on a new nominal trajectory. 01:03:52.152 --> 01:03:54.120 AUDIENCE: Yeah, I'm just saying, asking 01:03:54.120 --> 01:04:00.520 how you, like for the one all the way on the left, that one, 01:04:00.520 --> 01:04:05.230 yeah, how do you define how big that initial region is? 01:04:05.230 --> 01:04:11.440 PRESENTER 3: Oh, OK, so that one, say, if you start, 01:04:11.440 --> 01:04:13.560 say, not moving or something and you got pushed? 01:04:13.560 --> 01:04:15.620 Is that what you're asking for-- 01:04:15.620 --> 01:04:17.120 AUDIENCE: Well, I mean, I understand 01:04:17.120 --> 01:04:19.350 that you're using the V for uncertainty, 01:04:19.350 --> 01:04:21.700 and V and W. So that essentially defines 01:04:21.700 --> 01:04:23.900 how big those funnels are. 01:04:23.900 --> 01:04:26.662 Do the also define how big the first one is? 01:04:26.662 --> 01:04:27.370 PRESENTER 3: Yes. 01:04:27.370 --> 01:04:28.160 Yeah, the do. 01:04:28.160 --> 01:04:29.590 AUDIENCE: OK. 01:04:29.590 --> 01:04:32.200 PRESENTER 3: So what you're doing, because what you want 01:04:32.200 --> 01:04:35.690 to do is use the funnels and have some funnel library 01:04:35.690 --> 01:04:39.670 and be able to use those throughout all time. 01:04:39.670 --> 01:04:41.770 So you don't want-- 01:04:41.770 --> 01:04:45.360 so I guess what you're asking, the first funnel, 01:04:45.360 --> 01:04:47.520 you're going to start at some initial state 01:04:47.520 --> 01:04:49.021 0 with no velocity, stuff like that. 01:04:49.021 --> 01:04:50.978 And so you're not going to have any uncertainty 01:04:50.978 --> 01:04:53.360 on the first thing because you know where you start, 01:04:53.360 --> 01:04:54.960 the initial state. 01:04:54.960 --> 01:04:58.310 AUDIENCE: Yeah, I'm just asking independent 01:04:58.310 --> 01:05:01.420 of the wind and stuff like that, you 01:05:01.420 --> 01:05:04.080 might just be starting at a point 01:05:04.080 --> 01:05:06.510 other than what that nominal point is. 01:05:06.510 --> 01:05:10.110 So you may want to define that initial funnel size 01:05:10.110 --> 01:05:13.966 by some other criteria besides wind. 01:05:13.966 --> 01:05:16.372 Is there-- 01:05:16.372 --> 01:05:18.730 AUDIENCE: So [INAUDIBLE] sensor how sure 01:05:18.730 --> 01:05:21.487 you are about where initial location is. 01:05:21.487 --> 01:05:22.570 AUDIENCE: Yeah, and then-- 01:05:22.570 --> 01:05:24.028 I mean, there's sensor uncertainty. 01:05:24.028 --> 01:05:27.050 But you just may happen to be in the-- if you're 01:05:27.050 --> 01:05:30.560 doing the glider experiment 10 times, 01:05:30.560 --> 01:05:33.210 you may not start it from exactly the same position 01:05:33.210 --> 01:05:33.720 every time. 01:05:33.720 --> 01:05:35.760 PRESENTER 3: Yeah, you should-- 01:05:35.760 --> 01:05:38.820 your funnel, every time you plan a new funnel, 01:05:38.820 --> 01:05:40.459 you start at the nominal trajectory 01:05:40.459 --> 01:05:41.375 above that new funnel. 01:05:41.375 --> 01:05:43.570 So you're not going to have this funnel. 01:05:43.570 --> 01:05:46.920 And you're never going to start over here on the first one. 01:05:46.920 --> 01:05:49.600 You're always going to start at this point. 01:05:49.600 --> 01:05:51.410 [INTERPOSING VOICES] 01:05:51.410 --> 01:05:53.160 AUDIENCE: I though the point of the funnel 01:05:53.160 --> 01:05:58.265 was so that you can reuse it without having to replan. 01:05:58.265 --> 01:06:00.390 PRESENTER 3: Yeah, you can move the funnels around. 01:06:00.390 --> 01:06:01.035 AUDIENCE: Oh, you can move? 01:06:01.035 --> 01:06:01.535 OK. 01:06:01.535 --> 01:06:04.050 PRESENTER 3: Yeah, yeah, that was-- so 01:06:04.050 --> 01:06:05.960 every time you start a new funnel, 01:06:05.960 --> 01:06:07.650 you start at this nominal trajectory. 01:06:07.650 --> 01:06:10.126 But the next time you use the same funnel, 01:06:10.126 --> 01:06:13.080 you could be over here and just shift the funnel down 01:06:13.080 --> 01:06:15.030 to exactly the point. 01:06:15.030 --> 01:06:18.001 So the initial state is never going to be anywhere starting 01:06:18.001 --> 01:06:20.974 at t0, it's not going to be somewhere other 01:06:20.974 --> 01:06:22.140 than the nominal trajectory. 01:06:22.140 --> 01:06:24.070 But at t1 it could be over here. 01:06:24.070 --> 01:06:26.060 And so you just shift the whole funnel around. 01:06:26.060 --> 01:06:29.010 And as long as your t0 starts at the new funnel, 01:06:29.010 --> 01:06:35.968 and as long as t0 is within the funnel above t-- last funnel, 01:06:35.968 --> 01:06:37.970 then you just shift it around. 01:06:37.970 --> 01:06:43.355 You can also turn it, as long as you can multiply this and this. 01:06:43.355 --> 01:06:44.855 AUDIENCE: It seems like the shifting 01:06:44.855 --> 01:06:50.325 and turning is a pretty complex search problem in itself. 01:06:50.325 --> 01:06:53.550 Is that a hard problem? 01:06:53.550 --> 01:06:56.295 PRESENTER 3: No, that was actually the-- 01:06:56.295 --> 01:06:58.203 [INAUDIBLE] I got rid of the slide. 01:06:58.203 --> 01:07:01.159 I might have gotten rid of that slide. 01:07:01.159 --> 01:07:02.700 I shouldn't because that was the work 01:07:02.700 --> 01:07:04.592 that I did for the [INAUDIBLE]. 01:07:04.592 --> 01:07:07.430 Let's see [INAUDIBLE]. 01:07:07.430 --> 01:07:11.240 But, yeah, there's a bunch of different ways to do it. 01:07:11.240 --> 01:07:14.980 I did a pretty not very smart search, 01:07:14.980 --> 01:07:19.810 where I just looked for the funnels, the safest funnel 01:07:19.810 --> 01:07:22.346 being the one with the least number of points in the point 01:07:22.346 --> 01:07:25.544 cloud within that entire funnel. 01:07:25.544 --> 01:07:32.432 And then-- oh, man, I did get rid of this. 01:07:44.240 --> 01:07:46.920 But basically what I did was find the safest funnel. 01:07:46.920 --> 01:07:49.490 And if there still was a collision within the safest 01:07:49.490 --> 01:07:51.650 funnel, the I admit I had a minimization problem 01:07:51.650 --> 01:07:54.474 as you shifted back and forth. 01:07:54.474 --> 01:07:57.390 And there was a cost function that-- 01:07:57.390 --> 01:07:58.732 see if I can bring it up. 01:07:58.732 --> 01:08:00.440 AUDIENCE: So you're doing an optimization 01:08:00.440 --> 01:08:02.840 of the online running of the-- 01:08:02.840 --> 01:08:03.764 PRESENTER 3: Yeah. 01:08:03.764 --> 01:08:05.750 That was an input that was implemented 01:08:05.750 --> 01:08:07.440 in realtime simulation. 01:08:07.440 --> 01:08:11.182 But that wasn't implemented on the actual glider as of 01:08:11.182 --> 01:08:13.652 [INAUDIBLE]. 01:08:13.652 --> 01:08:18.592 So let's see, all right [INAUDIBLE].. 01:08:25.040 --> 01:08:26.870 And you also want-- 01:08:26.870 --> 01:08:30.720 I also made is so you minimize the amount that you 01:08:30.720 --> 01:08:31.696 shift it to. 01:08:31.696 --> 01:08:33.779 So you don't want to shift it completely somewhere 01:08:33.779 --> 01:08:37.020 else because it won't be in your path. 01:08:37.020 --> 01:08:39.660 And you'll end up doing some dynamic [INAUDIBLE].. 01:08:39.660 --> 01:08:42.496 Like, you'll be flying, and then if you compute a funnel that's 01:08:42.496 --> 01:08:43.870 here that's kind of safe, but you 01:08:43.870 --> 01:08:45.769 shift it all the way down to here. 01:08:45.769 --> 01:08:48.146 Then you're gliders going to be doing something crazy. 01:08:48.146 --> 01:08:51.625 And that may-- I guess it would be [INAUDIBLE].. 01:08:51.625 --> 01:08:57.589 But I minimized that for at least [INAUDIBLE] shift it. 01:08:57.589 --> 01:09:00.074 Does anybody have other questions? 01:09:12.499 --> 01:09:15.580 Yeah, so does anybody have any questions on-- 01:09:15.580 --> 01:09:17.960 AUDIENCE: Can I ask one? 01:09:17.960 --> 01:09:21.899 So you've given us models of bounded uncertainty 01:09:21.899 --> 01:09:23.240 in your dynamics. 01:09:23.240 --> 01:09:25.460 So if you use probabilistic models, 01:09:25.460 --> 01:09:29.979 then your funnels start looking like risk allocations. 01:09:29.979 --> 01:09:32.479 Is there an interpretation of one 01:09:32.479 --> 01:09:33.840 in terms of the [INAUDIBLE]? 01:09:33.840 --> 01:09:34.923 PRESENTER 3: I don't know. 01:09:34.923 --> 01:09:38.021 No, but I'm sure if you look it up, yeah. 01:09:38.021 --> 01:09:40.516 I'm sure somebody's been working on that. 01:09:40.516 --> 01:09:42.512 I'm not really sure that fits-- 01:09:42.512 --> 01:09:45.007 I know I haven't done that. 01:09:47.829 --> 01:09:49.870 AUDIENCE: Talk a little bit about the interaction 01:09:49.870 --> 01:09:55.424 between the flow tube selection or the funnel selections 01:09:55.424 --> 01:09:56.820 and the underlying [INAUDIBLE]. 01:09:56.820 --> 01:09:59.361 So let's say you have something that's planning the reference 01:09:59.361 --> 01:10:02.320 trajectory that's separate from your flow tube 01:10:02.320 --> 01:10:04.450 selection or your funnel selection. 01:10:04.450 --> 01:10:06.475 And then you have to select the funnels and flow 01:10:06.475 --> 01:10:10.485 tubes that allow you to follow that trajectory the best 01:10:10.485 --> 01:10:12.120 you can. 01:10:12.120 --> 01:10:14.347 How do the interact if you have a flow tube-- 01:10:14.347 --> 01:10:16.930 or don't have a safe flow tube or funnel [INAUDIBLE] reference 01:10:16.930 --> 01:10:18.385 trajectory? 01:10:18.385 --> 01:10:21.040 PRESENTER 3: So you're asking basically the difference 01:10:21.040 --> 01:10:23.768 between planning with a funnel laddering 01:10:23.768 --> 01:10:25.140 and trajectory laddering. 01:10:25.140 --> 01:10:28.120 AUDIENCE: Yeah, kind of sort of-- well, because I 01:10:28.120 --> 01:10:31.960 know the checkoff planner that uses an incremental pathfinder, 01:10:31.960 --> 01:10:34.560 so the planned path or something like that. 01:10:34.560 --> 01:10:37.310 And then you fit-- are you fitting flow tuber 01:10:37.310 --> 01:10:38.610 to the reference trajectory? 01:10:38.610 --> 01:10:40.276 PRESENTER 3: Oh, you do that off and on. 01:10:40.276 --> 01:10:42.744 So you compute-- they're called funnel libraries. 01:10:42.744 --> 01:10:44.410 Or usually they're trajectory libraries. 01:10:44.410 --> 01:10:46.810 So ever computing a trajectory in realtime 01:10:46.810 --> 01:10:50.380 for a fast-moving dynamic system is going to be expensive. 01:10:50.380 --> 01:10:52.210 And then computing a funnel on top of that 01:10:52.210 --> 01:10:53.850 is going to be even more expensive. 01:10:53.850 --> 01:10:56.296 So what a lot of people do is they 01:10:56.296 --> 01:10:59.660 compute the library offline for a trajectory and plan using 01:10:59.660 --> 01:11:00.160 that. 01:11:00.160 --> 01:11:03.000 But on this one, you compute your trajectory, 01:11:03.000 --> 01:11:05.360 compute the funnel around that trajectory offline. 01:11:05.360 --> 01:11:06.980 And then you search through the funnel library. 01:11:06.980 --> 01:11:08.521 You don't search through trajectories 01:11:08.521 --> 01:11:10.809 and then try to fit a funnel around that one. 01:11:10.809 --> 01:11:12.600 You just search through the funnel storage. 01:11:12.600 --> 01:11:14.724 PRESENTER 2: If you don't have a very good library, 01:11:14.724 --> 01:11:17.226 then you will not get a very good trajectory. 01:11:17.226 --> 01:11:19.100 PRESENTER 3: Also, a cool thing about funnels 01:11:19.100 --> 01:11:22.646 versus trajectories is a lot of trajectories 01:11:22.646 --> 01:11:26.820 will be discretized for speed, so you have your nodes. 01:11:26.820 --> 01:11:31.750 And so, say this is what your trajectory does, and you have 01:11:31.750 --> 01:11:35.490 node points here, and you're trying to fly past a wall 01:11:35.490 --> 01:11:36.310 or something. 01:11:36.310 --> 01:11:38.960 If your wall goes here, you're algorithm 01:11:38.960 --> 01:11:41.625 is going to search these points, and it's going to say, 01:11:41.625 --> 01:11:45.295 OK, well, this is a safe path because none of the node point 01:11:45.295 --> 01:11:46.488 intersect with this wall. 01:11:46.488 --> 01:11:53.310 So you fly [INAUDIBLE] a little [INAUDIBLE].. 01:11:53.310 --> 01:11:57.690 But a funnel, since you have this region around it, you-- 01:11:57.690 --> 01:12:01.780 no, I did a bad job of drawing that, but OK. 01:12:01.780 --> 01:12:05.890 When we search the algorithm and see 01:12:05.890 --> 01:12:07.876 if it's less than or greater than 1, 01:12:07.876 --> 01:12:12.152 you're going to find all these points being in that region 01:12:12.152 --> 01:12:14.462 also. 01:12:14.462 --> 01:12:16.935 I guess that depends on how well you want to discretize it. 01:12:16.935 --> 01:12:19.500 If your trajectory is [INAUDIBLE] of really, really 01:12:19.500 --> 01:12:23.670 small points, then you're not going to run into that issue. 01:12:23.670 --> 01:12:25.310 AUDIENCE: Seems like you could-- like 01:12:25.310 --> 01:12:28.160 when you're building a normal RIT without funnels, 01:12:28.160 --> 01:12:30.970 from concave points, if you say, yeah, they're illegal, 01:12:30.970 --> 01:12:33.440 it seem that if you're automatically 01:12:33.440 --> 01:12:35.778 adding on to that funnel instead of generating 01:12:35.778 --> 01:12:38.040 an entire trajectory and then making a funnel, 01:12:38.040 --> 01:12:41.480 you could save computational time for [INAUDIBLE] things 01:12:41.480 --> 01:12:41.980 like that. 01:12:41.980 --> 01:12:44.474 Do you know if that's done? 01:12:44.474 --> 01:12:46.282 PRESENTER 3: No, I don't. 01:12:46.282 --> 01:12:48.510 I do know that, I mean, computing, 01:12:48.510 --> 01:12:50.635 you obviously want to minimize computation time. 01:12:50.635 --> 01:12:53.010 But it's not a critical thing when computing, I guess, 01:12:53.010 --> 01:12:55.726 funnels and flow tubes, because you do that all offline. 01:12:55.726 --> 01:12:57.290 And you want to-- 01:12:57.290 --> 01:12:59.534 I don't think it's even close to the point 01:12:59.534 --> 01:13:01.642 where you go to online to compute a funnel. 01:13:01.642 --> 01:13:05.610 It takes like six hours to compute it, so-- 01:13:05.610 --> 01:13:08.690 depending on how many dimensions your state has. 01:13:08.690 --> 01:13:10.790 So, yeah, it's not like you're flying 01:13:10.790 --> 01:13:14.090 and you can compute one of these things. 01:13:14.090 --> 01:13:15.890 You'll crash like [INAUDIBLE]. 01:13:29.990 --> 01:13:30.590 OK. 01:13:30.590 --> 01:13:33.640 [APPLAUSE]