1 00:00:00,150 --> 00:00:06,600 Let's not discuss some of the key properties of rotation mattresses and their composition before going 2 00:00:06,600 --> 00:00:12,480 further, let me explain something, as we said before viewpoint rotation matrix by finding the projection 3 00:00:12,480 --> 00:00:16,200 of axis or each other, which is nothing but dot product. 4 00:00:16,680 --> 00:00:22,440 So if you want to find the rotation matters between two frames, you can easily do that by just finding 5 00:00:22,440 --> 00:00:25,320 dot product between the axis of these frames. 6 00:00:25,860 --> 00:00:33,660 So let's see an example assume that we have two frames, namely oh zero and or one or one has been obtained 7 00:00:33,660 --> 00:00:37,500 by rotating or zero by two degrees about zero axis. 8 00:00:38,010 --> 00:00:42,260 Then we can write R1 zero like that before continuing on. 9 00:00:42,300 --> 00:00:44,910 Let me mention the formula of DOT product here. 10 00:00:45,120 --> 00:00:51,690 So Dot product of two vectors is nothing but products of the lengths of two vectors and cosine of angle 11 00:00:51,690 --> 00:00:52,440 between them. 12 00:00:53,130 --> 00:00:55,950 Let's copy the graph and rotation matrix here again. 13 00:00:56,550 --> 00:01:00,540 Here you can see the calculation of each entry of the rotation matrix. 14 00:01:00,780 --> 00:01:06,270 Don't forget that in rotation mattresses, we are handling unit vectors, namely the lengths of vectors 15 00:01:06,270 --> 00:01:06,810 this one. 16 00:01:07,140 --> 00:01:10,830 So we will only have the cosine of angle between them. 17 00:01:11,190 --> 00:01:17,280 Moreover, the dot product of orthogonal vectors will be zero because the angle between them is PI over 18 00:01:17,280 --> 00:01:20,250 two and cosine of PI over two is zero. 19 00:01:20,660 --> 00:01:26,610 Additionally, axes that are parallel and are on the same direction will have an angle of zero between 20 00:01:26,610 --> 00:01:31,890 them, so their product will be one because cosine of zero equals to one. 21 00:01:33,420 --> 00:01:35,610 That's right, the resultant rotation matrix. 22 00:01:35,880 --> 00:01:41,610 As you can see, we easily got the elemental rotation matrix, which is obtained by rotation among x 23 00:01:41,610 --> 00:01:43,560 axis about x axis. 24 00:01:44,220 --> 00:01:49,620 Don't forget to take into account that during the calculations of dot products, you are handling three 25 00:01:49,620 --> 00:01:51,430 dimensional case here. 26 00:01:51,450 --> 00:01:54,150 Additionally, I would like to note something else. 27 00:01:54,390 --> 00:01:59,250 Moreover, you can see it from rotation metrics that each row contains the direction. 28 00:01:59,250 --> 00:02:06,120 Cosine is a one axis of frame over zero with respect to the axis of frame or one direction cosine. 29 00:02:06,300 --> 00:02:12,360 Because we have dot product of unit vectors, namely cosine of angle between them, as we mentioned 30 00:02:12,360 --> 00:02:12,960 earlier. 31 00:02:13,680 --> 00:02:19,560 Also, you can see that each column of rotation matrix contains the direction cosine of one axis of 32 00:02:19,560 --> 00:02:20,550 frame or one. 33 00:02:20,760 --> 00:02:23,160 With respect to the axis of frame zero. 34 00:02:23,580 --> 00:02:23,940 Yeah. 35 00:02:24,390 --> 00:02:30,240 If you have taken robotics course, probably you have seen these definitions and I hope that, you know, 36 00:02:30,330 --> 00:02:32,160 understand what they mean. 37 00:02:32,850 --> 00:02:38,070 Let's go to the properties of rotation mattresses, rotation mattresses in three. 38 00:02:38,070 --> 00:02:44,970 They belong to the spatial or technical group of Order three, which have the below properties or transpose 39 00:02:44,970 --> 00:02:46,470 equals that are inverse. 40 00:02:47,040 --> 00:02:51,000 The column and therefore rows of are mutually orthogonal. 41 00:02:51,570 --> 00:02:58,620 Each column and therefore each row of our is a unit vector determinant of R equals to one, which means 42 00:02:58,620 --> 00:03:04,410 that the length of a vector is unchanged after transformation, which is very initiative because what 43 00:03:04,410 --> 00:03:07,800 we are doing is just rotating the frame rotation. 44 00:03:07,800 --> 00:03:14,400 Matrix has nine elements in three D case, but they are not independent from second and third properties 45 00:03:14,400 --> 00:03:15,750 of the rotation mattresses. 46 00:03:15,760 --> 00:03:21,390 We get six constraints, namely, the columns have unit magnitude, which provides three constraints. 47 00:03:21,810 --> 00:03:26,040 The columns are orthogonal to each other, which provides another three constraints. 48 00:03:26,490 --> 00:03:31,350 So nine elements with six constraints means that we have three independent value. 49 00:03:31,680 --> 00:03:37,050 This means that the rotation mattresses are not the most compact representation of the rotation because 50 00:03:37,050 --> 00:03:38,550 we have dependent values. 51 00:03:38,850 --> 00:03:43,710 That is why in future releases, we will see more compact representations or rotations. 52 00:03:44,370 --> 00:03:47,610 Now we can talk about the composition of rotation mattresses. 53 00:03:47,910 --> 00:03:53,250 Keep in mind that generally we have more than one rotation of frames with respect to each other. 54 00:03:53,700 --> 00:04:00,180 We can generalize these rotations into sections first rotation with respect to the current frame. 55 00:04:00,300 --> 00:04:06,330 Second, rotations with respect to the fixed frame, let's go with rotation, with respect to the current 56 00:04:06,330 --> 00:04:06,750 frame. 57 00:04:07,080 --> 00:04:08,190 What's the current frame? 58 00:04:08,580 --> 00:04:11,670 Let's assume that we have a frame of zero. 59 00:04:12,060 --> 00:04:19,060 Then we rotated a degrees around axis of x zero and get frame of or one before the current frame was 60 00:04:19,060 --> 00:04:19,830 always zero. 61 00:04:19,860 --> 00:04:24,300 However, after rotation of T to the current frame is no or. 62 00:04:25,140 --> 00:04:31,770 Now we rotate one more time, better degree about the y one axis of current frame and we get the new 63 00:04:31,770 --> 00:04:33,360 current frame of all two. 64 00:04:33,810 --> 00:04:35,910 I think this is enough for understanding. 65 00:04:36,240 --> 00:04:39,720 No, the problem is how we can combine these rotations. 66 00:04:40,980 --> 00:04:46,830 For example, we want to get the rotation matrix of R to zero, which represents rotation from frame 67 00:04:46,830 --> 00:04:47,670 two to zero. 68 00:04:48,120 --> 00:04:54,840 Let's assume that we have a point p and its representation on these three frames are P zero, p one 69 00:04:54,840 --> 00:04:55,680 and P two. 70 00:04:56,130 --> 00:04:59,580 Then based on simple rotation matrix formula that we have. 71 00:04:59,670 --> 00:05:02,330 Seen before we can write these equations. 72 00:05:02,870 --> 00:05:10,390 So by combining equations one and two, we can write this equation by comparing equations four with 73 00:05:10,400 --> 00:05:16,970 equations three, we can see that are two zero equals two product of our one zero and our two one. 74 00:05:17,570 --> 00:05:23,390 So when the subsequent rotations are about the axis of current frame, we just past the post multiply 75 00:05:23,510 --> 00:05:24,770 rotation mattresses. 76 00:05:25,280 --> 00:05:26,090 Let's now see. 77 00:05:26,090 --> 00:05:29,480 The key is when the subsequent rotations are about the fixed frame. 78 00:05:29,780 --> 00:05:37,700 So assume that we have a frame of zero v rotate this frame by two degrees around zero and get new frame 79 00:05:37,700 --> 00:05:38,390 of one. 80 00:05:38,930 --> 00:05:45,050 Then we once again rotate the frame of all one of otherwise zero axis of frame zero better degree. 81 00:05:45,470 --> 00:05:49,880 As you can see, the rotations are about oh zero, namely fixed frame. 82 00:05:50,300 --> 00:05:54,620 So what is the overall rotation from frame two to zero in this case? 83 00:05:55,220 --> 00:06:01,220 In this case, we will multiply the rotation mattresses to get the final rotation matrix of our two 84 00:06:01,220 --> 00:06:01,640 zero. 85 00:06:02,180 --> 00:06:08,690 So as you can see if subsequent rotations are about the axis of fixed frame, then we print multiply 86 00:06:08,690 --> 00:06:09,950 the rotation mattresses. 87 00:06:10,610 --> 00:06:18,650 Let's call this section by a simple example we have below rotations first rotation of data about the 88 00:06:18,650 --> 00:06:19,820 current x axis. 89 00:06:20,180 --> 00:06:28,100 Second, rotation of fear about the current that axis served rotation of alpha above the fix that axis 90 00:06:28,100 --> 00:06:35,750 for rotation of better about the current y axis and fifth rotation of tilt to fix the x axis. 91 00:06:36,350 --> 00:06:39,920 Don't forget the composition rules for current and fixed frames. 92 00:06:40,700 --> 00:06:46,160 So in order to determine the cumulative effect of these rotations, we begin with the first rotation 93 00:06:46,160 --> 00:06:53,450 R s T to and pre or post multiplied, depending on whether we rotate about the current or fixed frame. 94 00:06:53,780 --> 00:06:55,790 So let's right step by step. 95 00:06:57,140 --> 00:07:05,720 First, we have the rotation of our data, then the post multiply R and R Z fee post multiplied because 96 00:07:05,720 --> 00:07:08,180 of rotation with respect to current taxes. 97 00:07:08,870 --> 00:07:12,980 Then we multiply that by four with our Twitter times. 98 00:07:12,980 --> 00:07:18,170 Are the fee print multiplied because of rotation with respect to fix the taxes? 99 00:07:18,590 --> 00:07:27,050 Then we post multiply our zip over r t r that fi with r y better post-meal blood because of rotation 100 00:07:27,050 --> 00:07:28,820 with respect to current y axis. 101 00:07:29,510 --> 00:07:30,710 Then we print multiply. 102 00:07:30,710 --> 00:07:34,330 Our built up with our z over times are Twitter. 103 00:07:34,340 --> 00:07:35,300 Times are Z. 104 00:07:35,300 --> 00:07:41,000 Three times are y better pray multiplied because of rotation with respect to fixed x axis. 105 00:07:41,570 --> 00:07:46,040 So whole rotation is R equals two are expelled at times. 106 00:07:46,040 --> 00:07:47,310 Are that over times? 107 00:07:47,330 --> 00:07:50,690 Our Twitter times are the three times are y better. 108 00:07:52,190 --> 00:07:57,650 This is very important to understand the rotation mattresses because it is not only needed in robotics, 109 00:07:57,650 --> 00:08:02,330 but also in mechanics, geometry, 3D design and so on. 110 00:08:02,660 --> 00:08:04,310 See you on the next lesson.