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These are the user uploaded subtitles that are being translated: 1 00:00:00,980 --> 00:00:04,162 [INAUDIBLE] obstacles are really convenient to work with, because 2 00:00:04,162 --> 00:00:07,700 they provide an explicit description of the configure space obstacles. 3 00:00:08,830 --> 00:00:11,070 Often times, we don't have this luxury and 4 00:00:11,070 --> 00:00:15,110 the obstacles are instead defined implicitely through a collision function. 5 00:00:16,330 --> 00:00:21,766 More specifically, if we let x denote the coordinates of a point in configuration 6 00:00:21,766 --> 00:00:26,570 space, we're usually able to define a function, let's call it CollisionCheck, 7 00:00:26,570 --> 00:00:31,600 that returns zero if the configuration is in freespace, and one otherwise. 8 00:00:33,300 --> 00:00:38,630 In this slide, we show a situation where our putitive robot, 9 00:00:38,630 --> 00:00:42,030 the red polygon, is entirely in freespace. 10 00:00:42,030 --> 00:00:44,560 So we want our CollisionCheck function to return a zero. 11 00:00:46,930 --> 00:00:52,340 In this next case, the robot is colliding with one of the obstacles. 12 00:00:52,340 --> 00:00:55,490 So we would want our CollisionCheck function to return a one for 13 00:00:55,490 --> 00:00:56,220 this configuration. 14 00:00:59,140 --> 00:01:03,640 Collision detection is a key subroutine for most path planning algorithms, and 15 00:01:03,640 --> 00:01:07,150 it often boils down to computing the answer to some basic geometric questions. 16 00:01:08,290 --> 00:01:09,770 Lets go back and consider our example. 17 00:01:11,200 --> 00:01:13,928 Here we can imagine representing both our robot and 18 00:01:13,928 --> 00:01:17,615 our obstacle as collections of triangles, as shown in this figure. 19 00:01:18,905 --> 00:01:20,505 Deciding whether the robot and 20 00:01:20,505 --> 00:01:23,825 the obstacle intersect is now a matter of determining whether 21 00:01:23,825 --> 00:01:27,935 any of the robot's triangles overlap any of the obstacle's triangles. 22 00:01:29,705 --> 00:01:34,470 Here we can make use of the fact that triangles are convex polygons. 23 00:01:34,470 --> 00:01:39,599 In this circumstance, it means that we can test where the two triangles intersect 24 00:01:39,599 --> 00:01:44,207 by checking all of the sides on both triangles, and testing whether any of 25 00:01:44,207 --> 00:01:49,336 those sides act as separating lines, where all of the points from one triangle lie 26 00:01:49,336 --> 00:01:54,348 on one side of the line, and all of those from the other lie on the opposite side. 27 00:01:54,348 --> 00:01:58,236 If you can find such a separating edge, among the six edges among 28 00:01:58,236 --> 00:02:02,780 the two triangles, you have proved that the triangles don't intersect. 29 00:02:03,890 --> 00:02:06,950 If not, you can conclude that they do. 30 00:02:06,950 --> 00:02:09,270 This idea of finding a separating line, or 31 00:02:09,270 --> 00:02:13,020 plane, actually generalizes to higher dimensions. 32 00:02:13,020 --> 00:02:18,380 For instance, If you have a robot composed of convex polygons in three dimensions 33 00:02:18,380 --> 00:02:20,830 like boxes or pyramids. 34 00:02:20,830 --> 00:02:24,741 You can check for collision by testing if any of the faces on one of 35 00:02:24,741 --> 00:02:27,651 the polygons acts as a separating hyperplane. 36 00:02:27,651 --> 00:02:31,064 In the cases that we've considered so far, the robots and 37 00:02:31,064 --> 00:02:35,150 the obstacles are basically composed of simple polygons. 38 00:02:35,150 --> 00:02:36,520 So to test for collision, 39 00:02:36,520 --> 00:02:41,060 we first transform the robot according to the configurations space parameters and 40 00:02:41,060 --> 00:02:45,470 then test for collision between the robot's components and the obstacles. 41 00:02:45,470 --> 00:02:49,970 In this case, this amounts to testing whether any of the red triangles 42 00:02:49,970 --> 00:02:51,899 overlap any of the black triangles. 43 00:02:53,990 --> 00:02:58,996 In this case we would want our CollisionCheck function to return a zero, 44 00:02:58,996 --> 00:03:02,209 indicating that no intersections are found. 45 00:03:02,209 --> 00:03:07,214 In this next case, there are in fact red triangles that do overlap 46 00:03:07,214 --> 00:03:12,859 the black triangles so our CollisionCheck function should return a one. 47 00:03:12,859 --> 00:03:18,072 Another question that we may be interested in answering is whether all of 48 00:03:18,072 --> 00:03:23,881 the points along the line segment between two configuration space points x1 and 49 00:03:23,881 --> 00:03:25,540 x2 are in freespace. 50 00:03:25,540 --> 00:03:26,470 Sometimes we're lucky and 51 00:03:26,470 --> 00:03:29,589 we can actually answer that question definitively by analysis. 52 00:03:31,080 --> 00:03:34,240 In practice, if two points are both in freespace, and 53 00:03:34,240 --> 00:03:38,820 they are close enough to each other, given some reasonable distance metric, 54 00:03:38,820 --> 00:03:42,670 then we tend to assume that a straight line path between them is collision free. 55 00:03:42,670 --> 00:03:46,510 This can obviously go horribly wrong if we're not careful. 56 00:03:46,510 --> 00:03:48,830 But let's live dangerously for a second. 57 00:03:48,830 --> 00:03:51,490 In considering our 2D planning problem, 58 00:03:51,490 --> 00:03:56,380 we can proceed by sampling a grid of locations in the configuration space. 59 00:03:57,760 --> 00:03:59,874 For each grid point we determine whether or 60 00:03:59,874 --> 00:04:02,820 not it is in freespace using our CollisionCheck function. 61 00:04:03,930 --> 00:04:06,945 All the grid points that are in freespace are going to be treated as 62 00:04:06,945 --> 00:04:08,620 nodes in our graph. 63 00:04:08,620 --> 00:04:12,359 This figure shows the resulting sampled freespace points. 64 00:04:14,020 --> 00:04:18,200 in this example, we're going to go ahead and assume that if two adjacent grid 65 00:04:18,200 --> 00:04:23,570 points are both in freespace, then all the points between them are also in freespace. 66 00:04:23,570 --> 00:04:26,290 And we connect them by an etch, as shown here. 67 00:04:27,340 --> 00:04:29,240 In this case, we've chosen matters so 68 00:04:29,240 --> 00:04:33,560 that the start and end points are both nodes in the graph. 69 00:04:33,560 --> 00:04:38,410 We can then employ anyone of our usual suite of graph-based planning tools 70 00:04:38,410 --> 00:04:40,590 to search for a path between the two nodes. 71 00:04:42,350 --> 00:04:45,546 The resulting path through the graph is shown here, in red.6656

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