1 00:00:01,500 --> 00:00:08,040 OK, now we are going to apply homogeneous transformation matrix in MATLAB. 2 00:00:08,580 --> 00:00:14,430 So in practice, we will use, as always, Peter Rackauckas Library in order to achieve that. 3 00:00:14,940 --> 00:00:15,840 So let's start. 4 00:00:16,500 --> 00:00:19,980 First of all, let's create on screen and create our Bibles. 5 00:00:20,880 --> 00:00:29,870 Let's start first on the meeting homogeneous transformation matrix book homogeneous. 6 00:00:29,880 --> 00:00:31,560 It's a bit difficult for me to write it. 7 00:00:31,710 --> 00:00:35,220 Transformation Matrix. 8 00:00:35,410 --> 00:00:40,820 OK, let's first let me explain some comments. 9 00:00:40,830 --> 00:00:50,730 OK, let's we will describe our homogeneous transformation matrix with these combination of translation 10 00:00:50,730 --> 00:00:53,910 transformation and rotation transformation. 11 00:00:55,080 --> 00:00:58,080 OK, let me first try it, and then I will explain. 12 00:01:01,010 --> 00:01:01,400 OK. 13 00:01:03,690 --> 00:01:08,580 And then again, translational transformation. 14 00:01:09,270 --> 00:01:12,180 Zero one zero. 15 00:01:12,210 --> 00:01:13,320 So what does it mean? 16 00:01:14,130 --> 00:01:18,020 First, what we are doing with Trans L? 17 00:01:18,060 --> 00:01:27,960 One zero zero So with these first come home, what we're doing is we translate our reframe in one unit 18 00:01:27,960 --> 00:01:29,460 in extraction. 19 00:01:29,820 --> 00:01:30,330 OK. 20 00:01:31,080 --> 00:01:38,580 Remember that as we post multiply the, uh, either translational or rotational transformation mattresses. 21 00:01:38,880 --> 00:01:42,690 This is about the motion is about the current axis. 22 00:01:42,960 --> 00:01:43,380 OK. 23 00:01:43,650 --> 00:01:45,060 Because we post multiply them. 24 00:01:45,840 --> 00:01:54,030 And then what we are doing with t r o t ex-spy overture, as you know from before and what we rotate 25 00:01:54,030 --> 00:01:55,800 about x direction. 26 00:01:56,010 --> 00:01:56,430 OK. 27 00:01:56,460 --> 00:02:00,340 This is let me just to the current X direction. 28 00:02:00,420 --> 00:02:00,840 OK. 29 00:02:00,870 --> 00:02:02,670 It's about current x axis. 30 00:02:04,320 --> 00:02:07,080 And then we rotate our frame. 31 00:02:07,380 --> 00:02:10,690 And what we are doing is Trans L zero one zero. 32 00:02:11,810 --> 00:02:16,720 We translate one unit in the current Y direction, not before y direction. 33 00:02:16,740 --> 00:02:17,790 OK, not fixed. 34 00:02:18,030 --> 00:02:21,210 Uh y direction, but the current y direction. 35 00:02:21,450 --> 00:02:31,800 So after rotation, after we rotate, we get the new frame and we move one unit in in y direction in 36 00:02:31,800 --> 00:02:32,730 the new frame. 37 00:02:33,600 --> 00:02:34,000 OK. 38 00:02:34,380 --> 00:02:40,500 So let's write this OK, I will copy some of the comments in order not to lose too much time. 39 00:02:41,370 --> 00:02:41,760 OK. 40 00:02:42,000 --> 00:02:43,380 This is our translation. 41 00:02:43,390 --> 00:02:46,920 Uh, the, uh, homogeneous transformation matrix. 42 00:02:46,920 --> 00:02:49,050 So this is our rigid motion. 43 00:02:49,680 --> 00:02:54,600 As you know, our rigid motion consists of translation and rotation. 44 00:02:55,110 --> 00:03:00,810 OK, then let's try to animate, animate transformation. 45 00:03:03,730 --> 00:03:04,400 OK. 46 00:03:04,600 --> 00:03:14,800 We will use T r the command in order to animate our homogeneous transmission matrix. 47 00:03:14,810 --> 00:03:16,090 So let's do that. 48 00:03:16,210 --> 00:03:17,830 Let's run all of this. 49 00:03:22,680 --> 00:03:24,090 OK, give it the time. 50 00:03:24,810 --> 00:03:31,420 Perfect, then let me just first make it in this way so you can easily see what's happening. 51 00:03:31,440 --> 00:03:37,770 Let's come to 3D and let's run this one again, Ron section. 52 00:03:38,820 --> 00:03:39,290 OK? 53 00:03:40,230 --> 00:03:43,200 As you can see, it's very interesting. 54 00:03:43,890 --> 00:03:45,060 You can approximate that. 55 00:03:45,060 --> 00:03:46,650 Try to understand what's happening. 56 00:03:46,890 --> 00:03:53,230 First of all, as you as you can see, we are moving in the X direction. 57 00:03:53,250 --> 00:03:55,680 As you can see in the current acceleration, it's moving. 58 00:03:56,070 --> 00:04:02,010 It's moving because of this common one zero zero because we are moving in this direction. 59 00:04:02,340 --> 00:04:08,880 However, why we are, you can see then if it is in the right direction, why we are moving in this 60 00:04:09,000 --> 00:04:09,450 direction. 61 00:04:09,450 --> 00:04:11,940 As you can see, it also moves up. 62 00:04:12,600 --> 00:04:15,150 As you can see, it rotates and moves up. 63 00:04:15,390 --> 00:04:25,740 This is because when we rotate our, uh, frame the current y axis, or let me just do that with the 64 00:04:25,740 --> 00:04:29,460 current y axis, here is zero one zero. 65 00:04:30,090 --> 00:04:30,530 OK. 66 00:04:30,750 --> 00:04:34,900 Is in the direction of the fixed frame. 67 00:04:34,920 --> 00:04:35,430 OK. 68 00:04:36,660 --> 00:04:44,790 I highly recommend you try these cords yourself and try to analyze it and try to understand what's happening. 69 00:04:45,000 --> 00:04:51,120 You have to understand because we have seen these lists and they are not really difficult. 70 00:04:51,300 --> 00:04:53,760 You can understand if you're on the lens of beat the cord. 71 00:04:54,600 --> 00:05:07,470 OK, so let's then continue with transforming a vector for K transforming transforming vector. 72 00:05:07,650 --> 00:05:08,040 OK. 73 00:05:08,580 --> 00:05:09,720 So what does it mean? 74 00:05:09,720 --> 00:05:10,800 Transforming a vector? 75 00:05:11,040 --> 00:05:14,190 We will transform Vector from, uh. 76 00:05:14,700 --> 00:05:16,830 So we will transform. 77 00:05:17,190 --> 00:05:19,230 Indeed, we will transform a point. 78 00:05:19,710 --> 00:05:25,740 And we will draw a vector in order to see this point from the zero frame to this point. 79 00:05:26,010 --> 00:05:32,160 And you will see how we transform a point from one point to another point or from one frame to another 80 00:05:32,160 --> 00:05:32,460 frame. 81 00:05:32,850 --> 00:05:33,930 OK, let's do that. 82 00:05:34,440 --> 00:05:36,300 First of all, again, it's clear. 83 00:05:36,600 --> 00:05:37,280 OK. 84 00:05:38,340 --> 00:05:40,230 Then let's create our vector. 85 00:05:40,800 --> 00:05:41,310 OK. 86 00:05:41,610 --> 00:05:44,070 As you can see, this is our Richter one. 87 00:05:44,100 --> 00:05:46,230 This is X, Y and Z. 88 00:05:46,500 --> 00:05:47,190 OK. 89 00:05:48,360 --> 00:05:51,330 So let's draw now our reference frame. 90 00:05:51,750 --> 00:05:52,500 OK? 91 00:05:53,370 --> 00:05:54,810 This is a reference frame. 92 00:05:55,180 --> 00:05:56,340 OK, let's throw it. 93 00:05:57,430 --> 00:05:57,720 Let's. 94 00:05:59,910 --> 00:06:00,420 OK. 95 00:06:09,730 --> 00:06:10,230 OK. 96 00:06:10,930 --> 00:06:12,160 This is our reference frame. 97 00:06:12,340 --> 00:06:17,020 Let's first roll until this one and to see how it is drawing our reference frame. 98 00:06:18,220 --> 00:06:21,280 OK, uh, well, it's always on top. 99 00:06:22,420 --> 00:06:22,840 OK. 100 00:06:22,900 --> 00:06:27,340 As you can see this, our reference frame, it throws our reference frame in this way. 101 00:06:28,300 --> 00:06:35,800 So let's then add again our homogeneous transformation matrix. 102 00:06:36,550 --> 00:06:37,090 OK. 103 00:06:37,150 --> 00:06:40,420 This is the old one, the same one we have used before. 104 00:06:40,900 --> 00:06:49,660 Then what we have to do, as you know, we have to multiply the homogeneous transformation matrix with 105 00:06:49,660 --> 00:06:56,980 our vector in it to get our new or re moved vector. 106 00:06:57,010 --> 00:06:57,430 OK? 107 00:06:58,240 --> 00:07:02,770 In order to get new vector, which moved rigidly. 108 00:07:02,920 --> 00:07:03,400 OK. 109 00:07:03,970 --> 00:07:12,430 So but as we have said before, we cannot directly multiply the vector with the homogeneous transformation 110 00:07:12,430 --> 00:07:12,760 matrix. 111 00:07:12,760 --> 00:07:13,330 Let's try. 112 00:07:13,600 --> 00:07:19,240 For example, this is our new vector equals thirty one multiplied by V. one. 113 00:07:19,390 --> 00:07:23,160 Let's do that, and you will see that it will give error because of why. 114 00:07:23,500 --> 00:07:25,180 T1 is four by four. 115 00:07:25,210 --> 00:07:33,700 You can see that if we run, you will see the T one is four by four, while we V1 is three by one, 116 00:07:33,700 --> 00:07:35,360 as you can see, three by one. 117 00:07:35,380 --> 00:07:37,810 So the dimensions are not correct and it will not work. 118 00:07:38,020 --> 00:07:40,940 Let's try and see. 119 00:07:41,140 --> 00:07:41,560 OK. 120 00:07:41,770 --> 00:07:46,960 As you can see, incorrect dimension for multi-course matrix multiplication. 121 00:07:48,010 --> 00:07:53,160 So what we had to do, we have to first convert this one to homogeneous corners. 122 00:07:53,170 --> 00:07:54,760 So let's try to do that. 123 00:07:56,470 --> 00:07:56,760 OK. 124 00:07:57,310 --> 00:08:01,510 So what we will do will make homogeneous coordinates. 125 00:08:02,080 --> 00:08:02,590 OK. 126 00:08:04,660 --> 00:08:08,340 What we have to do in order to make it in a homogeneous coordinate. 127 00:08:08,610 --> 00:08:10,620 We have to add only one. 128 00:08:10,660 --> 00:08:11,030 OK. 129 00:08:11,070 --> 00:08:12,390 This is our one. 130 00:08:12,750 --> 00:08:18,210 And we add one below it in order to have our homogeneous coordinates. 131 00:08:18,510 --> 00:08:22,740 So let's run this one in front and see what I mean. 132 00:08:22,920 --> 00:08:24,270 We won't take this object. 133 00:08:24,480 --> 00:08:32,580 As you can see it, only add extra one, you know, to get, uh, homogeneous coordinates. 134 00:08:33,000 --> 00:08:39,630 So then we can get our new vector, which is also homogeneous. 135 00:08:41,130 --> 00:08:41,600 OK. 136 00:08:41,820 --> 00:08:45,880 This is our new, homogeneous, new vector in homogeneous coordinates. 137 00:08:45,900 --> 00:08:46,320 OK. 138 00:08:47,010 --> 00:08:55,290 So we are multiplying our rotation, a homogeneous transformation matrix with our homogeneous vector, 139 00:08:55,590 --> 00:09:02,730 and we got our new homogeneous vector and we tried to convert it again back to a normal coordinate. 140 00:09:03,570 --> 00:09:04,170 OK. 141 00:09:04,740 --> 00:09:07,470 So let's try to then plot the vectors. 142 00:09:07,500 --> 00:09:12,210 Let me just copy all of these in order not to lose time. 143 00:09:12,450 --> 00:09:13,200 OK. 144 00:09:13,770 --> 00:09:16,260 We are just doing what this is. 145 00:09:16,290 --> 00:09:18,930 Uh, we just plot from the reference frame. 146 00:09:18,940 --> 00:09:25,350 So reference frame is zero zero zero and our vector is X is v one one. 147 00:09:25,530 --> 00:09:28,290 Why is v one two and it is v one three? 148 00:09:28,590 --> 00:09:32,280 So here we just define the limits of our axes. 149 00:09:32,610 --> 00:09:42,470 We just hold on in order to stick to the same plot, and we just then plot what our uh, second, uh, 150 00:09:42,930 --> 00:09:48,330 the new or, uh, the vector that have, uh, been moved rigidly. 151 00:09:48,840 --> 00:09:53,190 OK, let's first of all, uh, close the old graph. 152 00:09:53,220 --> 00:09:54,720 And, uh, okay. 153 00:09:56,040 --> 00:09:56,940 Brian S.. 154 00:09:57,450 --> 00:09:57,960 OK. 155 00:09:58,230 --> 00:10:02,400 As you can see, as you can see, it's really easy to see. 156 00:10:02,400 --> 00:10:06,090 Let me just make it in this way. 157 00:10:06,120 --> 00:10:06,630 OK. 158 00:10:07,560 --> 00:10:09,300 This is our old vector. 159 00:10:09,540 --> 00:10:10,280 The blue one. 160 00:10:10,290 --> 00:10:17,370 And this is our new actor, which is, as you can see, rotated and also moved rigidly. 161 00:10:17,850 --> 00:10:18,330 OK. 162 00:10:20,310 --> 00:10:23,010 If you want, you can analyze it step by step. 163 00:10:23,550 --> 00:10:25,140 What do I mean step by step? 164 00:10:25,140 --> 00:10:32,010 For example, you can first delete all of these and only see the effect of this. 165 00:10:32,340 --> 00:10:33,610 Then what you can do. 166 00:10:33,630 --> 00:10:36,990 Delete this one and see the effect of these two. 167 00:10:37,290 --> 00:10:43,740 And you can, at the end, apply a total of these, uh, homogeneous transformation metrics and you 168 00:10:43,740 --> 00:10:45,780 can see how it affects. 169 00:10:46,140 --> 00:10:49,950 So it will be much more easier for you to understand that. 170 00:10:50,790 --> 00:10:52,620 But it is very important, please. 171 00:10:52,990 --> 00:10:59,190 Uh, it is not enough to just copy paste this code, try to understand what you are doing and try to 172 00:10:59,190 --> 00:11:02,070 see the concepts in the real world. 173 00:11:02,070 --> 00:11:03,990 So they are really working. 174 00:11:03,990 --> 00:11:05,580 Try to grasp this concept. 175 00:11:05,820 --> 00:11:06,240 OK. 176 00:11:06,480 --> 00:11:08,010 It would be very good for you. 177 00:11:08,370 --> 00:11:08,820 Really? 178 00:11:10,290 --> 00:11:14,970 OK, then let's do composition of transformation mattresses. 179 00:11:15,900 --> 00:11:16,530 OK. 180 00:11:18,450 --> 00:11:21,200 Transformation in composition of transformation. 181 00:11:21,210 --> 00:11:22,770 Homogeneous transformation mattresses. 182 00:11:22,980 --> 00:11:27,360 So again, let me just, uh, clean the screen and clear the variables. 183 00:11:27,600 --> 00:11:30,930 And again, let's write our transformation. 184 00:11:30,930 --> 00:11:32,710 Homogeneous transformation matrix. 185 00:11:32,730 --> 00:11:33,120 OK. 186 00:11:34,530 --> 00:11:46,090 OK, then let's try to plot our reference frame again, and let's define our X limits and Y limits and 187 00:11:46,350 --> 00:11:56,900 means also, let's to hold on in order to stick with the old plot and then create, uh, again, uh, 188 00:11:56,910 --> 00:11:57,810 digit motion. 189 00:11:58,050 --> 00:11:58,560 OK. 190 00:11:58,980 --> 00:12:04,950 By creating new homogeneous transformation matrix, we will create additional, uh, rigid motion and 191 00:12:04,950 --> 00:12:07,110 we will call this one t one. 192 00:12:07,720 --> 00:12:16,170 OK, now let's see two composition methods of transformation at, uh, homogeneous transformation mattresses, 193 00:12:16,170 --> 00:12:20,730 and you will see that they give totally different answers. 194 00:12:21,120 --> 00:12:23,780 Let's first do rigid motion about current X. 195 00:12:23,800 --> 00:12:31,910 So if we are doing in the current axis as you move, we have to post multiply the rotation. 196 00:12:32,310 --> 00:12:32,640 OK. 197 00:12:32,670 --> 00:12:34,620 Excuse me, transformation mattresses. 198 00:12:34,890 --> 00:12:41,760 I just sometimes say rotation mattresses, but this is a homogeneous transformation. 199 00:12:41,760 --> 00:12:42,290 Mattresses? 200 00:12:42,370 --> 00:12:42,630 OK. 201 00:12:43,170 --> 00:12:50,880 And let's this is our new homogeneous transformation matrix, which is composition of Ti zero and T 202 00:12:50,880 --> 00:12:51,260 one. 203 00:12:51,270 --> 00:13:00,240 As you can see, we just multiply the rotations and this is our new, uh, new transformation matrix, 204 00:13:00,240 --> 00:13:03,480 which is about, uh, current axis and. 205 00:13:03,560 --> 00:13:06,380 He drew it the in frame. 206 00:13:06,410 --> 00:13:08,990 So with this script, it is frame and cold. 207 00:13:08,990 --> 00:13:15,860 This frame, as for one, and let's two is also rigid motion about fixed axis. 208 00:13:15,890 --> 00:13:17,870 Okay, let's do fixed axis. 209 00:13:17,880 --> 00:13:24,110 So what we have to do, we have to play, multiply the mattresses, OK, transform each month, as you 210 00:13:24,110 --> 00:13:33,170 can see now, we have multiplied them and called these three and let's draw it as frame or two. 211 00:13:33,680 --> 00:13:42,440 OK, now let's run, and you will see that the result of these two compositions will be quite different. 212 00:13:42,830 --> 00:13:44,060 So let's three. 213 00:13:44,390 --> 00:13:47,420 Let's try to see it in 3D. 214 00:13:48,440 --> 00:13:49,100 Let me just. 215 00:13:49,400 --> 00:13:50,900 OK, perfect. 216 00:13:51,140 --> 00:13:53,420 As you can see, as you can see. 217 00:13:53,480 --> 00:13:54,560 Be careful here. 218 00:13:54,950 --> 00:13:56,750 This is our reference metrics. 219 00:13:57,230 --> 00:13:58,450 Excuse me, reference frame. 220 00:13:58,460 --> 00:13:59,120 This is zero. 221 00:13:59,600 --> 00:14:08,420 As you can see, the one is, uh, because of possible duplication or it is because of current axis. 222 00:14:08,600 --> 00:14:16,130 However, the second one is because of what the second one is because of, uh, fixed axis. 223 00:14:16,460 --> 00:14:20,750 As you can see, they have two, uh, quite different answer. 224 00:14:20,990 --> 00:14:23,390 Quite different answers. 225 00:14:23,810 --> 00:14:24,230 OK. 226 00:14:24,470 --> 00:14:33,410 So you have to be careful about the composition of transformation mattresses as in the case of rotation 227 00:14:33,680 --> 00:14:34,400 mattresses. 228 00:14:35,240 --> 00:14:36,200 Thank you very much. 229 00:14:36,230 --> 00:14:37,760 See you on the next lesson.