1 00:00:00,420 --> 00:00:02,850 Let's continue with our practical lesson. 2 00:00:03,900 --> 00:00:11,580 So in this lesson, we will try to apply what we have learned about different representations of rotations 3 00:00:11,910 --> 00:00:20,040 like Roll Pitch Your Aler Angles quarter and use angle and angle vector representations into practice. 4 00:00:20,460 --> 00:00:22,360 So let's start with aero angles. 5 00:00:22,870 --> 00:00:23,240 OK. 6 00:00:23,710 --> 00:00:25,900 Uh, angles. 7 00:00:26,700 --> 00:00:27,210 OK. 8 00:00:27,480 --> 00:00:36,450 First of all, let's take a rotation matrix, which is, as you know, we have assumed in our tutorial, 9 00:00:36,600 --> 00:00:38,790 the angles as that. 10 00:00:38,790 --> 00:00:41,220 Why is it so sequence? 11 00:00:41,460 --> 00:00:46,130 So rotation about it, then rotation about why then rotation about zip. 12 00:00:46,590 --> 00:00:46,980 OK. 13 00:00:47,130 --> 00:00:53,340 So as you know, this was the these were about the current axis, not the fixed axis. 14 00:00:53,520 --> 00:00:54,560 So be careful here. 15 00:00:54,570 --> 00:00:56,540 So that's why we will multiply. 16 00:00:56,580 --> 00:01:02,610 Uh, excuse me, multiply these rotations mattresses post multiply because they are about the current, 17 00:01:02,760 --> 00:01:07,110 uh, rotation error, current frame, not about the fixed frame. 18 00:01:07,380 --> 00:01:17,010 So then rotation z rotation y zero point one and then rotation Z again zero point three. 19 00:01:17,100 --> 00:01:22,260 So this is our rotation matrix uh, z y z rotation mattress. 20 00:01:23,640 --> 00:01:33,120 OK, then let's uh, see, first of all, this result of this press f nine and you can see the rotation 21 00:01:33,120 --> 00:01:33,810 matrix here. 22 00:01:34,230 --> 00:01:42,810 OK, then let's do try to do the same with that function, which is, uh, easier, which converts from 23 00:01:42,810 --> 00:01:45,710 eyler angles to rotation matrix. 24 00:01:45,810 --> 00:01:48,300 So what are our angles? 25 00:01:48,300 --> 00:01:52,020 Zero point five, zero point one and zero point three? 26 00:01:52,470 --> 00:01:52,950 OK. 27 00:01:53,220 --> 00:01:59,220 No, let's do from eyler to rotation matrix. 28 00:01:59,520 --> 00:02:01,920 OK, let's run this also. 29 00:02:03,270 --> 00:02:03,630 OK. 30 00:02:03,900 --> 00:02:07,590 If mine and OK, it is so rotation matrix. 31 00:02:07,590 --> 00:02:08,930 As you can see, they are the same. 32 00:02:08,940 --> 00:02:12,990 We can check this in this way or one or two. 33 00:02:12,990 --> 00:02:15,750 This is not equality, OK? 34 00:02:16,020 --> 00:02:21,780 This is just for comparing if the indices of R1 is equal to indices of R2. 35 00:02:22,290 --> 00:02:24,810 Let's check if all of them are equal. 36 00:02:24,810 --> 00:02:27,570 We will have to see one matrix. 37 00:02:27,810 --> 00:02:30,390 As you can see, we have seen matrix of one. 38 00:02:31,080 --> 00:02:34,950 Uh, so every indices of R1 is equal to R2. 39 00:02:34,950 --> 00:02:39,150 So metrics are one is equal to R2, which is, uh, which have to be like that. 40 00:02:39,570 --> 00:02:45,840 So let's see now our let's see, first of all, KLC in order to clean our workspace. 41 00:02:45,870 --> 00:02:46,500 OK. 42 00:02:46,920 --> 00:02:51,420 And let's see our, uh, rotation matrix again. 43 00:02:52,050 --> 00:02:53,580 OK, this is our rotation method. 44 00:02:53,580 --> 00:03:00,600 So I would like to remind you that, uh, now we will try to first of all, this right here what we 45 00:03:00,600 --> 00:03:01,500 are trying to do. 46 00:03:01,830 --> 00:03:10,090 Uh, we we try to find aler angles, angles from rotation matrix. 47 00:03:10,170 --> 00:03:10,770 OK. 48 00:03:11,100 --> 00:03:16,530 So as you remember what we have said, first of all, there was two cases. 49 00:03:16,710 --> 00:03:25,440 If our rotation matrix are one, three and R2, three indices are not zero, then we have to separate, 50 00:03:25,770 --> 00:03:27,750 uh, solutions. 51 00:03:28,290 --> 00:03:32,430 So, OK, let's try to find this error angles. 52 00:03:33,840 --> 00:03:44,290 Because the T R2 L, which converts from rotation matrix to our um LRA, uh, angular representation. 53 00:03:44,330 --> 00:03:44,730 OK. 54 00:03:46,780 --> 00:03:47,220 No. 55 00:03:47,430 --> 00:03:53,490 Let's see if mine, these are our angles. 56 00:03:53,820 --> 00:03:57,960 As you can see, they are the same zero point five zero point one zero point three. 57 00:03:58,110 --> 00:04:00,930 We got our error angles. 58 00:04:02,070 --> 00:04:02,460 OK. 59 00:04:02,580 --> 00:04:09,510 But what's interesting is L or angles equals thirty or two. 60 00:04:10,230 --> 00:04:13,590 And let's do the same with this one like that. 61 00:04:13,590 --> 00:04:20,820 But now we will take the T to as minus, because what we have said, we have said that if our rotation 62 00:04:20,820 --> 00:04:26,070 matrix R1 and R2 three are not zero, then we have two different solutions. 63 00:04:26,220 --> 00:04:28,710 This was because of the sine difference of TITA. 64 00:04:29,490 --> 00:04:32,940 In this case, we have taken Theta as a plus. 65 00:04:32,940 --> 00:04:35,050 As you can see, this is a plus. 66 00:04:35,070 --> 00:04:35,820 No negative. 67 00:04:36,060 --> 00:04:38,970 Now we will try to do the same, but with negative. 68 00:04:39,180 --> 00:04:40,290 So, OK, let's copy it. 69 00:04:40,290 --> 00:04:46,200 Then this one, uh and uh, we will just make it minus. 70 00:04:46,980 --> 00:04:52,500 OK, let's see the rotation mattress press f nine. 71 00:04:52,650 --> 00:04:54,330 This is not our rotation matrix. 72 00:04:54,660 --> 00:04:56,820 As you can see, it's different from the first one. 73 00:04:57,930 --> 00:04:59,610 OK, let's try to. 74 00:05:00,710 --> 00:05:05,990 Find our angles or let's see what we get. 75 00:05:08,790 --> 00:05:09,330 OK. 76 00:05:10,520 --> 00:05:19,980 Press nine, as you can see, we got Twitter plus, however, our Twitter was minus and also we got 77 00:05:19,980 --> 00:05:24,650 completely different angles from the original one. 78 00:05:24,900 --> 00:05:36,630 However, what's interesting if we try to again find our Aler angles from every angles, from our rotation, 79 00:05:37,050 --> 00:05:41,520 so, uh, our angles, so we try to convert. 80 00:05:44,400 --> 00:05:48,120 We we got our angles and we will try to convert these angles. 81 00:05:48,120 --> 00:05:49,320 Rotation matrix again. 82 00:05:49,620 --> 00:05:56,730 What is interesting is that we got the same rotation matrix as you can see. 83 00:05:56,820 --> 00:06:00,700 What does it mean while we have quite different angles? 84 00:06:00,810 --> 00:06:05,590 We when we convert to the rotation matrix, we got the same rotation mattress. 85 00:06:05,610 --> 00:06:09,720 This means that different angles correspond to the same rotation matrix. 86 00:06:09,960 --> 00:06:15,030 So mapping from rotation matrix to angles is not unique. 87 00:06:16,200 --> 00:06:23,850 OK, so we can get different angles which are mapping to a unique rotation matrix. 88 00:06:24,450 --> 00:06:26,250 OK, let's continue. 89 00:06:26,250 --> 00:06:28,620 And this is not surprising because we have seen this. 90 00:06:29,010 --> 00:06:37,170 So let's come to the second, um case where are one three and our two three indices of our rotation? 91 00:06:37,170 --> 00:06:38,580 Matrix is zero. 92 00:06:38,940 --> 00:06:42,210 So let's case two. 93 00:06:43,890 --> 00:06:53,520 If Twitter is zero and R1 three because it is zero R1 three and are two or three because of sine, these 94 00:06:53,520 --> 00:06:56,400 contain sine of T to sign of zero equals to zero. 95 00:06:56,580 --> 00:06:59,880 So that's why R1 Street and R2 three will be zero. 96 00:07:00,000 --> 00:07:03,090 We have seen all of these in the last lesson. 97 00:07:03,420 --> 00:07:09,720 So if you cannot remember, please refer to these, uh, lessons again and repeat and come back. 98 00:07:10,080 --> 00:07:19,980 OK, R2 three of rotation matrix matrix becomes zero. 99 00:07:20,370 --> 00:07:20,790 OK. 100 00:07:21,060 --> 00:07:22,650 So let's do that. 101 00:07:23,160 --> 00:07:29,250 You were alerted to a rotation matrix, OK? 102 00:07:29,360 --> 00:07:30,390 Zero point five. 103 00:07:30,710 --> 00:07:32,670 Now, as you can see, we gave it a zero. 104 00:07:33,210 --> 00:07:39,660 So we have to get our once we are two three in zero, let's check if we will get as you can see, your 105 00:07:39,690 --> 00:07:42,060 star wants to be and R2 three zero. 106 00:07:42,270 --> 00:07:49,410 What we have said in our last this is we have said that every R1 three and R2 three equals to zero. 107 00:07:49,650 --> 00:07:54,810 Then we have infinite solutions for the angles. 108 00:07:55,170 --> 00:07:57,630 I mean, V and C angles. 109 00:07:57,960 --> 00:07:59,700 OK, let's check that. 110 00:08:00,330 --> 00:08:02,160 First of all, let's ride that. 111 00:08:02,170 --> 00:08:07,950 So when finding a hole or angles or angles? 112 00:08:09,180 --> 00:08:21,240 So, OK, a angles, there are infinitely many solutions. 113 00:08:21,840 --> 00:08:22,170 OK. 114 00:08:22,350 --> 00:08:25,380 So these are two hailer. 115 00:08:25,380 --> 00:08:29,640 So what we are doing, we are converting from rotation matrix to angles. 116 00:08:29,940 --> 00:08:33,340 OK, now see what we will get. 117 00:08:33,450 --> 00:08:33,900 OK. 118 00:08:34,080 --> 00:08:40,470 As you can see, we get it as zero, which is expected and we get zero and zero point eight. 119 00:08:41,430 --> 00:08:46,940 So, uh, the library, the petrol caucus library and also in robotics. 120 00:08:46,950 --> 00:08:48,960 It is by conversion. 121 00:08:48,960 --> 00:08:51,510 What is done is accepted as zero. 122 00:08:51,510 --> 00:08:55,440 Because we have infinitely many solutions, we can accept anything. 123 00:08:55,710 --> 00:09:01,560 So we accept features as zero and find see based on this one. 124 00:09:02,040 --> 00:09:03,630 OK, let's try this one. 125 00:09:03,630 --> 00:09:10,050 Also by convention angle fee is zero. 126 00:09:12,270 --> 00:09:23,970 OK, now let's come to roll up your roll pitch, your pitch, your representation of rotation. 127 00:09:24,120 --> 00:09:24,600 OK. 128 00:09:24,870 --> 00:09:26,550 What we will do first of all. 129 00:09:27,720 --> 00:09:34,500 First, let's try to roll pitch yaw to convert it from roll pitch rotation metrics. 130 00:09:34,740 --> 00:09:39,900 So roll pitch your two rotation matrix. 131 00:09:40,410 --> 00:09:49,590 The default in our caucus library is default is zit y x sequence. 132 00:09:49,830 --> 00:09:52,110 So the first angle is about Z. 133 00:09:52,140 --> 00:09:59,730 The second is about why the third is about, uh, x uh, axis. 134 00:10:00,210 --> 00:10:02,280 And be careful, enroll pitch. 135 00:10:02,280 --> 00:10:07,110 Yo, we the subsequent rotations were about not the current frame. 136 00:10:07,420 --> 00:10:08,290 Fixed frame. 137 00:10:09,610 --> 00:10:12,600 So, OK, but you can change this sequence. 138 00:10:12,700 --> 00:10:13,420 Let's first. 139 00:10:14,110 --> 00:10:15,160 OK, let's do. 140 00:10:15,190 --> 00:10:17,140 Uh, I will show how you can change. 141 00:10:17,320 --> 00:10:23,440 First of all, it's from RPI, from RPI to rotation matrix. 142 00:10:23,440 --> 00:10:24,430 This all function. 143 00:10:24,670 --> 00:10:27,190 0.5 0.1. 144 00:10:27,190 --> 00:10:28,690 So this is about it. 145 00:10:28,960 --> 00:10:33,430 This is about why this is about actually zero point three. 146 00:10:34,480 --> 00:10:37,630 So let's check our rotation matrix F9. 147 00:10:37,900 --> 00:10:40,490 As you can see, this is our rotation matrix. 148 00:10:40,510 --> 00:10:49,660 The in rotation matrix and let's ride help or roll pitch, you know, two rotation and as you can see 149 00:10:50,080 --> 00:10:53,780 we can from options we can override. 150 00:10:53,800 --> 00:10:55,420 Let's make it bigger, OK? 151 00:10:55,430 --> 00:10:59,560 We can override the sequence x y z z y x y exit. 152 00:11:01,120 --> 00:11:04,210 OK, so let's continue. 153 00:11:06,640 --> 00:11:07,030 OK. 154 00:11:07,240 --> 00:11:08,830 So what what? 155 00:11:08,830 --> 00:11:15,180 We will try find angles from rotation matrix. 156 00:11:15,190 --> 00:11:17,050 OK, what we will do. 157 00:11:17,350 --> 00:11:24,340 We will transform from a rotation matrix tool or p y role feature from our rotation methods. 158 00:11:24,350 --> 00:11:26,630 Let's check F9 Chris. 159 00:11:26,680 --> 00:11:37,620 If man, as you can see, we get the same role, pick your angles so we can even visualize it to visualize. 160 00:11:37,630 --> 00:11:40,180 We can visualize, uh, roll pitch. 161 00:11:40,180 --> 00:11:42,790 Yo, um, we can. 162 00:11:45,680 --> 00:11:52,580 Write this comment, and we will we we can animate road picture, which is in order to understand them 163 00:11:52,580 --> 00:11:52,970 better. 164 00:11:53,210 --> 00:11:57,740 As you can see, this is about z axis y axis x axis. 165 00:11:58,010 --> 00:12:04,490 OK, first of all, for example, this our plane, uh, we try to rotate it about that axis. 166 00:12:04,850 --> 00:12:08,600 As you can see, we are rotating now about z axis. 167 00:12:09,560 --> 00:12:13,070 OK, let's rotated about y axis. 168 00:12:16,250 --> 00:12:20,180 And, OK, let's rotated about x axis. 169 00:12:22,710 --> 00:12:23,210 OK. 170 00:12:24,720 --> 00:12:27,750 As you can see, now, we are rotating about x axis. 171 00:12:32,840 --> 00:12:38,120 You can play with it and try to understand better, you can change your sequence from here. 172 00:12:38,880 --> 00:12:48,920 OK, so let's come to axis angle representation, axis angle representation. 173 00:12:52,390 --> 00:13:03,120 OK, so what we can do, first of all, let's try to define our rotation metrics from every angle, 174 00:13:03,130 --> 00:13:08,320 so rotation zero point five, zero point one and zero point three. 175 00:13:08,830 --> 00:13:10,420 So let's run this one. 176 00:13:10,540 --> 00:13:15,250 This our rotation matrix that we got from al angles. 177 00:13:15,610 --> 00:13:26,710 So as you as we have said before, every rotation can be represented as a rotation about some axis and 178 00:13:26,710 --> 00:13:28,420 with specific angle theta. 179 00:13:28,690 --> 00:13:33,940 So this is our Twitter angle and this is our rotation axis. 180 00:13:34,690 --> 00:13:44,210 So let's try to from rotation matrix, we will try to find angle vector relation or axis angle representation. 181 00:13:44,230 --> 00:13:47,380 We can say the same because our axis is unit vector. 182 00:13:47,770 --> 00:13:50,830 OK, let's try to find Titov V. 183 00:13:53,320 --> 00:13:54,430 OK, excuse me. 184 00:13:54,820 --> 00:13:55,570 OK. 185 00:13:56,410 --> 00:13:58,010 We have done something. 186 00:13:58,060 --> 00:13:58,540 OK. 187 00:13:59,500 --> 00:14:04,800 Miss Bill here two and go, OK. 188 00:14:04,840 --> 00:14:07,630 There is no here. 189 00:14:08,140 --> 00:14:09,250 This should be correct now. 190 00:14:10,120 --> 00:14:10,930 Oh my gosh. 191 00:14:11,350 --> 00:14:13,000 Let's copy that. 192 00:14:13,000 --> 00:14:13,660 I don't know why. 193 00:14:13,670 --> 00:14:14,590 Misspell? 194 00:14:14,620 --> 00:14:15,070 OK. 195 00:14:15,980 --> 00:14:16,230 OK. 196 00:14:16,270 --> 00:14:16,900 No, it's correct. 197 00:14:18,760 --> 00:14:19,060 OK. 198 00:14:19,090 --> 00:14:25,090 As you can see it, this is our Tita angle rotation angle and this is our rotation axis. 199 00:14:26,050 --> 00:14:33,430 As you as we have said before, we can find these, uh, axis of rotation. 200 00:14:33,700 --> 00:14:39,400 And our rotation angle from eigenvalue and eigenvectors problem. 201 00:14:39,650 --> 00:14:49,570 So eigenvalues and IGen vectors of rotation matrix are we will analyze this. 202 00:14:49,810 --> 00:14:53,260 So first of all, let's try to find eigenvalues and eigenvectors. 203 00:14:53,590 --> 00:14:56,890 We can ride up the built in function for doing that. 204 00:14:57,280 --> 00:14:59,390 And if not, OK. 205 00:15:00,070 --> 00:15:05,080 As you can see, this is our, uh, eigenvectors. 206 00:15:05,320 --> 00:15:05,890 OK. 207 00:15:06,250 --> 00:15:08,350 And this is our eigenvalues. 208 00:15:08,650 --> 00:15:09,520 Don't be afraid. 209 00:15:10,000 --> 00:15:13,870 The eigenvalues of our eigenvalues are on the diagonal. 210 00:15:14,380 --> 00:15:17,440 As you can see, only these are our eigenvalues. 211 00:15:17,440 --> 00:15:26,050 We have three eigenvalues, and for each of eigenvalues, we have eigenvectors, which are the columns 212 00:15:26,110 --> 00:15:28,120 of this matrix. 213 00:15:28,420 --> 00:15:30,370 As you can see, this is for the first one. 214 00:15:31,570 --> 00:15:38,590 This is for the second eigenvalue, and this third column is for the third eigenvalue. 215 00:15:38,830 --> 00:15:46,360 And what we have said before, we have to take the eigen vector corresponding to the eigenvalue of real 216 00:15:46,360 --> 00:15:54,790 value old one, because that was giving our two US rotation matrix because this was the rotation, because 217 00:15:54,790 --> 00:16:02,950 eigenvalue multiplied by eigenvalue one will not change the absolute value or length of the vector. 218 00:16:03,130 --> 00:16:08,430 So this is our um rotation axis. 219 00:16:08,440 --> 00:16:12,880 If you compare it, abore, as you can see, they are the same. 220 00:16:16,510 --> 00:16:21,670 OK, they are the same as you can see from eigenvectors and eigenvalue and those as we can find their 221 00:16:21,670 --> 00:16:22,890 rotation vector. 222 00:16:23,170 --> 00:16:32,500 So how we can find the rotation angle, what we have said, we have said before that we can find our 223 00:16:32,500 --> 00:16:36,410 angles from eigenvalues. 224 00:16:36,790 --> 00:16:49,630 So our argument from by comparing our eigenvalues equals to cosine of titov plus or minus, I wrote 225 00:16:49,630 --> 00:16:51,850 like that or excuse me, I don't. 226 00:16:52,420 --> 00:16:53,320 I have tried it. 227 00:16:53,980 --> 00:16:55,240 Not here, but here. 228 00:16:55,590 --> 00:16:56,180 OK. 229 00:16:59,210 --> 00:17:04,960 OK, Eichmann's equals the plus or minus I times sign off. 230 00:17:06,520 --> 00:17:09,700 We can't use our rotation angle, so let's try to do this work. 231 00:17:09,880 --> 00:17:16,360 First of all, let's try to find it up by function from Peter Caucus Library. 232 00:17:16,750 --> 00:17:22,990 Let's right angle and give our eigenvalue to this and try to find Theta. 233 00:17:24,250 --> 00:17:32,710 As you can see, this is our rotation angle, but how we can find if we compare this formula, for example, 234 00:17:32,710 --> 00:17:38,080 with this eigenvalue, as you can see, cosine of Quita equals to this one. 235 00:17:38,090 --> 00:17:41,240 So by our cosine, we can find angle. 236 00:17:41,260 --> 00:17:41,980 Let's try it. 237 00:17:42,130 --> 00:17:46,660 A cosine is, as you can see, we got the same angle. 238 00:17:47,980 --> 00:17:48,430 OK. 239 00:17:48,580 --> 00:17:49,120 So 240 00:17:52,390 --> 00:17:56,890 then we can find what we will try to do now. 241 00:17:57,400 --> 00:18:03,940 We will try to find rotation matrix from angle axis representation. 242 00:18:04,240 --> 00:18:16,810 So from an angle vector or axis to rotation matrix, we will use here Rodrigues formula. 243 00:18:17,350 --> 00:18:23,140 You can search for it and you can look videos of it. 244 00:18:23,650 --> 00:18:29,050 However, it is not essential because it is applied inside the politico.com's library. 245 00:18:30,400 --> 00:18:33,510 OK, angle vector two. 246 00:18:33,520 --> 00:18:34,870 Ah, OK. 247 00:18:35,170 --> 00:18:36,700 This is pi over. 248 00:18:36,700 --> 00:18:37,270 Two. 249 00:18:37,300 --> 00:18:37,780 OK. 250 00:18:38,050 --> 00:18:38,440 We are. 251 00:18:39,190 --> 00:18:44,980 Our rotation angle is PI over two and rotation axis is equal to one zero zero. 252 00:18:45,160 --> 00:18:46,090 So we are. 253 00:18:46,300 --> 00:18:55,900 What we are doing is we are rotating our vector or our rotation is about x axis and pi or two angle. 254 00:18:56,170 --> 00:18:59,770 Let's check f nine. 255 00:18:59,830 --> 00:19:00,250 OK. 256 00:19:00,340 --> 00:19:02,320 This is our rotation axis. 257 00:19:02,590 --> 00:19:09,790 Additionally, we can do compare it with like that rotation of x rotation x pi or two. 258 00:19:10,360 --> 00:19:12,730 This will directly give us a rotation matrix. 259 00:19:13,330 --> 00:19:15,250 Uh, pi over two around x. 260 00:19:15,460 --> 00:19:23,500 As you can see, they are the same these zeros after proceedings, however, they are the same as you 261 00:19:23,500 --> 00:19:23,950 can see. 262 00:19:24,460 --> 00:19:24,880 OK. 263 00:19:25,150 --> 00:19:27,490 Let me just make it bigger in order to be more clear. 264 00:19:28,270 --> 00:19:28,630 OK. 265 00:19:28,990 --> 00:19:31,480 So let's continue with unit cohesion. 266 00:19:31,480 --> 00:19:31,750 You. 267 00:19:34,840 --> 00:19:45,280 OK, now we come back to you and we come to you and quarter new unit quarter earnings, so let's define 268 00:19:45,280 --> 00:19:47,080 two quarter earnings. 269 00:19:47,710 --> 00:19:48,490 Q1. 270 00:19:48,530 --> 00:19:53,110 I'll just copy and paste here in order to be to save time. 271 00:19:53,500 --> 00:19:57,520 As you can see, we are using unit quarterly and class from Pethokoukis Library. 272 00:19:57,760 --> 00:20:05,380 And also we are doing we are getting the second quarter in creating the second unit quarter union. 273 00:20:05,740 --> 00:20:08,890 Let's see them press f line. 274 00:20:09,280 --> 00:20:13,090 As you can see, we have two quarter earnings unit quarter earnings. 275 00:20:13,480 --> 00:20:20,410 The first one, this is Scarlett apart, as you can see, and this is victor part leather part and victor 276 00:20:20,410 --> 00:20:20,680 part. 277 00:20:20,890 --> 00:20:27,680 What we have said, we have said that we can add and subtract, um, unit quarterly. 278 00:20:27,730 --> 00:20:30,790 So let's excuse me, it was the key one plus Q2. 279 00:20:31,210 --> 00:20:35,590 And there are some element wise, let's do. 280 00:20:35,710 --> 00:20:38,590 As you can see, there are just some buy element. 281 00:20:38,590 --> 00:20:48,490 Was this all resultant quarter new leads to also their multiplication because we have said that they 282 00:20:48,490 --> 00:20:54,010 can be multiplied and they are representing what relative rotation. 283 00:20:54,130 --> 00:21:00,220 So by post multiplying it, we are representing rotation about current axis. 284 00:21:00,220 --> 00:21:02,410 So let's check that out. 285 00:21:02,410 --> 00:21:04,300 What will it give to us? 286 00:21:04,590 --> 00:21:11,980 And this is the resultant quarterly and we can check this quarter when you buy like that, for example, 287 00:21:11,980 --> 00:21:21,190 first of all, try to, uh, get the first from RPI to a rotation axis to get our one. 288 00:21:21,190 --> 00:21:23,890 We are just, uh, making this one. 289 00:21:23,890 --> 00:21:31,690 I mean, we are taking this role PTO angle and convert rotation matrix and do the same for the second 290 00:21:32,260 --> 00:21:32,650 one. 291 00:21:33,130 --> 00:21:41,350 And what we can do we can make it are three equals two or one times are two because, uh, the rotation 292 00:21:41,350 --> 00:21:42,460 is about the current axis. 293 00:21:42,460 --> 00:21:45,910 So we had the post multiplied and what we can do now. 294 00:21:46,090 --> 00:21:56,080 We can convert this rotation to quarterly and this Q five has to be equals to Q4 because they are doing 295 00:21:56,080 --> 00:22:00,790 the same thing, essentially both of them representing the relative rotation. 296 00:22:00,790 --> 00:22:03,910 So the resulting continuous will have to be the same. 297 00:22:04,090 --> 00:22:05,580 Let's check that out. 298 00:22:05,590 --> 00:22:12,280 Let's choose all of them and price if and this is our Q5, let's also Q4 in order to see them. 299 00:22:12,280 --> 00:22:16,960 As you can see, they are exactly the same because they are doing the same thing. 300 00:22:18,190 --> 00:22:24,220 You can do that for pretty multiplication in order to understand clearer, but they are essentially 301 00:22:24,220 --> 00:22:24,580 the same. 302 00:22:24,850 --> 00:22:30,910 We can find the inverse or contract of consequence of our rotation matrix. 303 00:22:30,910 --> 00:22:36,310 This is nothing but, uh, just the sign of, as you can see, the sign. 304 00:22:36,340 --> 00:22:42,000 Uh, this is our inverse or consequent, um, quarterly. 305 00:22:42,010 --> 00:22:45,730 And let's write it here, OK? 306 00:22:46,240 --> 00:22:59,230 Um, consequence of unit growth quarterly in the interesting thing is that if we multiply any quarter 307 00:22:59,230 --> 00:23:08,410 you so Q1 times with its inverse what we do, we have to get we have to get only 308 00:23:10,750 --> 00:23:20,980 identity quarterly, which represents it's right now rotation. 309 00:23:20,980 --> 00:23:22,180 So there is no rotation. 310 00:23:23,330 --> 00:23:27,550 OK, let's run this one and see what it will give to us. 311 00:23:27,760 --> 00:23:32,110 As you can see, this is what this is. 312 00:23:32,800 --> 00:23:38,470 I didn't do quarterly with well, one won and there is no rotation axis. 313 00:23:38,710 --> 00:23:42,580 If there is no rotation axis, this means that there is no rotation. 314 00:23:42,580 --> 00:23:45,550 There is no valid rotation. 315 00:23:45,940 --> 00:23:47,740 OK, let's do. 316 00:23:47,860 --> 00:23:58,900 Uh, how we let's see now how to convert from quarter new to rotation metrics. 317 00:23:59,050 --> 00:24:00,040 It's very easy. 318 00:24:00,040 --> 00:24:08,080 We just write are for our rotation, you know, to save our rotation mattress to one dot or we get the 319 00:24:09,100 --> 00:24:09,370 V. 320 00:24:09,380 --> 00:24:13,930 By doing that, we get rotation matrix of our from quarter new. 321 00:24:14,080 --> 00:24:14,980 Let's run this. 322 00:24:15,910 --> 00:24:18,430 OK, as you can see it, this is our rotation metrics. 323 00:24:20,350 --> 00:24:26,140 And then we can even plugged our quarter new. 324 00:24:29,080 --> 00:24:33,400 OK, uh, we can plot the orientation represented by. 325 00:24:33,530 --> 00:24:34,220 Is quoting him. 326 00:24:35,210 --> 00:24:49,940 OK, and then what we can do else, we can just, um, obtain rotated vector by multiplying so we can 327 00:24:50,180 --> 00:24:59,030 rotate a vector by multiplying it with a crop, turning OK, by multiplying continuum with vector. 328 00:24:59,390 --> 00:25:03,830 So let me just copy and paste it. 329 00:25:03,980 --> 00:25:06,200 OK, this is our resultant vector. 330 00:25:06,650 --> 00:25:08,480 This is our original vector. 331 00:25:08,720 --> 00:25:14,760 We are multiplying it with our quaternary neon and we have to get new rotated vector. 332 00:25:14,780 --> 00:25:17,420 Let's see this one f nine. 333 00:25:17,690 --> 00:25:23,660 If this is our rotated vector, let's just visualize this by doing this. 334 00:25:24,170 --> 00:25:26,070 Let me just copy and paste. 335 00:25:26,090 --> 00:25:30,440 We are doing just what we are doing from zero. 336 00:25:30,440 --> 00:25:41,930 I first plot our first original vector and then we are plotting our rotated vector from relative to 337 00:25:41,930 --> 00:25:42,980 point zero. 338 00:25:43,100 --> 00:25:51,470 As you can see, from point zero, this is the point of four x y z coordinates of first point. 339 00:25:51,860 --> 00:25:55,790 And this is the X was it of our second point. 340 00:25:56,150 --> 00:25:57,350 I mean, original one. 341 00:25:57,650 --> 00:26:03,590 And also, this is the origin of our origin point. 342 00:26:03,590 --> 00:26:11,720 So the point we refer to so zero zero zero and this is our resultant one result and two and resultant 343 00:26:11,720 --> 00:26:14,960 three is just x y z of our rotated vector. 344 00:26:15,140 --> 00:26:21,080 So without anything else, let's run this one. 345 00:26:21,260 --> 00:26:28,790 And as you can see, as you can see, this is our original vector and this is, as you can see, this 346 00:26:28,790 --> 00:26:30,050 is our rotated vector. 347 00:26:30,050 --> 00:26:33,650 It's rotated by our criterion. 348 00:26:35,180 --> 00:26:37,520 OK, see you on the next lesson.