1 00:00:01,290 --> 00:00:02,040 Let's listen. 2 00:00:02,070 --> 00:00:04,290 We have learned about the Jacobean matrix. 3 00:00:04,920 --> 00:00:10,950 However, this is just one of the types of the Jakobsson matrix, namely geometric Jacobean matrix, 4 00:00:11,430 --> 00:00:17,880 as you know, it relates that piece, namely linear and angular velocity of the end, the with the joint 5 00:00:17,880 --> 00:00:18,600 velocities. 6 00:00:19,200 --> 00:00:25,650 However, what if the orientation of the in the vector is represented by role pico or eyler nodes? 7 00:00:26,190 --> 00:00:28,830 Then will our Jacobean will be the same? 8 00:00:29,010 --> 00:00:30,180 Let's check that out. 9 00:00:30,900 --> 00:00:36,780 Let's assume that we have given all pitch your angles of Peter are TTP and Y. 10 00:00:37,530 --> 00:00:40,800 Then we can ride the cumulative rotation matrix in this way. 11 00:00:40,830 --> 00:00:44,070 Name the multiplication of three rotation mattresses. 12 00:00:45,960 --> 00:00:49,740 On the next step, we try to find derivative of the rotation matrix. 13 00:00:50,100 --> 00:00:55,830 Be careful here because overall pitch your angles are not constant, but time dependent. 14 00:00:56,190 --> 00:01:01,380 So if you want to calculate your, you have to. 15 00:01:02,130 --> 00:01:09,000 If you so if you want to calculate the rotation matrix monopoly in manually, you have to consider that 16 00:01:09,540 --> 00:01:15,210 as we know, the derivative of the rotation matrix equals to the multiplication of skill, symmetric 17 00:01:15,210 --> 00:01:19,050 matrix of angular velocities and rotation matrix. 18 00:01:20,700 --> 00:01:30,110 From from here, we can find skewed symmetric matrix by inverting our rotation matrix if it expands 19 00:01:30,330 --> 00:01:37,860 symmetric matrix and compare it with the result from multiplication of derivative of rotation matrix 20 00:01:38,070 --> 00:01:39,090 with itself. 21 00:01:39,360 --> 00:01:42,930 Then we will be able to relate the rate of change of role. 22 00:01:42,930 --> 00:01:45,840 Pitch your angles with angular velocities. 23 00:01:46,470 --> 00:01:50,460 I don't know you, but I don't want to calculate this derivative manually. 24 00:01:50,760 --> 00:01:54,360 So let's jump to the mop up and handle this problem. 25 00:01:56,370 --> 00:02:06,210 So let's continue to calculate the derivative of our rotation matrix and obtain as filmmaker. 26 00:02:06,960 --> 00:02:12,150 Let's first define some symbolic variables which are time dependent. 27 00:02:12,150 --> 00:02:16,430 So that's why we write R t p t. 28 00:02:16,530 --> 00:02:19,710 These are all pitch your ankles that we will utilize. 29 00:02:20,760 --> 00:02:21,240 OK. 30 00:02:21,690 --> 00:02:34,370 Of the defining that let's find the cumulative rotation matrix or equals the rotation X or OC times 31 00:02:35,250 --> 00:02:36,210 for rotation. 32 00:02:36,540 --> 00:02:37,120 Y. 33 00:02:37,920 --> 00:02:38,500 P. 34 00:02:39,600 --> 00:02:40,050 Excuse me. 35 00:02:40,050 --> 00:02:42,660 P times for rotation. 36 00:02:42,670 --> 00:02:44,130 Z Y. 37 00:02:45,930 --> 00:02:48,510 OK, let's visualize it. 38 00:02:48,690 --> 00:02:52,860 As you can see, it's a bit complex, so that's why I put that semicolon. 39 00:02:53,460 --> 00:02:55,230 Anyway, let's continue. 40 00:02:55,440 --> 00:02:59,580 Then let's calculate our result. 41 00:03:00,420 --> 00:03:01,040 OK. 42 00:03:01,620 --> 00:03:12,090 By finding different, by differentiating our rotation matrix, however we have, we have to differentiate 43 00:03:12,090 --> 00:03:13,150 it with respect to T. 44 00:03:13,170 --> 00:03:13,560 OK. 45 00:03:14,700 --> 00:03:20,850 So that's why we right here T because our variables are time dependent. 46 00:03:21,690 --> 00:03:22,200 OK. 47 00:03:22,860 --> 00:03:31,500 Now, as you can see, it's also very complex, but we can make it a bit easier to read by writing, 48 00:03:31,650 --> 00:03:36,120 by first simplifying the relation. 49 00:03:37,000 --> 00:03:38,410 So result. 50 00:03:39,060 --> 00:03:39,450 OK. 51 00:03:39,510 --> 00:03:41,160 You can first simplify it. 52 00:03:41,640 --> 00:03:42,480 Let's show you. 53 00:03:42,520 --> 00:03:44,250 Yeah, it's a bit. 54 00:03:44,250 --> 00:03:51,090 Uh, it's a simplified, but you know it as it is complex. 55 00:03:51,090 --> 00:03:53,670 It not affects too much. 56 00:03:54,150 --> 00:03:56,100 So let's write it in. 57 00:03:56,610 --> 00:04:00,870 Also, we can do pretty and simplify. 58 00:04:00,930 --> 00:04:08,070 Combine these comments in order to, you know, have better visualization on our problem. 59 00:04:08,070 --> 00:04:18,450 Because as you can see in instead of Y Dot or D or D Y T it right, this why this is because these are 60 00:04:18,450 --> 00:04:25,950 symbolic variables so we can visualize it more in a print way by using printed comment. 61 00:04:25,950 --> 00:04:32,130 And OK, let's calculate OK, it is better as you can see it right? 62 00:04:32,250 --> 00:04:34,510 D or d t p t OK. 63 00:04:35,850 --> 00:04:42,570 And uh, as you can see, each column is separated by 64 00:04:45,600 --> 00:04:49,110 OK, this side, I forgot its name. 65 00:04:50,490 --> 00:04:50,910 OK. 66 00:04:51,990 --> 00:04:59,190 So after we get after we obtained the derivative of our rotation, not. 67 00:04:59,790 --> 00:05:06,570 Let's now calculate our S Omega, so this is our excuseme metric of angular velocities, OK? 68 00:05:07,260 --> 00:05:16,410 We will do what we will multiply the derivative of our rotation matrix with inverse of our rotation 69 00:05:16,410 --> 00:05:16,920 matrix. 70 00:05:17,430 --> 00:05:18,180 OK. 71 00:05:18,630 --> 00:05:24,020 And now we obtain our exclusive metric matrix. 72 00:05:24,030 --> 00:05:25,920 OK, let's switch it. 73 00:05:26,160 --> 00:05:26,790 OK? 74 00:05:27,180 --> 00:05:29,100 This is also complex, as you can see. 75 00:05:30,100 --> 00:05:34,380 Anyway, let's also do the same in order to visualize it. 76 00:05:34,590 --> 00:05:44,130 Let's use the common pretty with simplified common and s w. 77 00:05:45,120 --> 00:05:49,380 OK, now, as you can see, we got very simple. 78 00:05:49,500 --> 00:05:55,080 Not OK, very simple, but much more simpler than this matrix, OK? 79 00:05:55,110 --> 00:05:58,620 As you can see, a whole of these mattress is nothing but this one. 80 00:05:59,940 --> 00:06:01,260 This is our matrix. 81 00:06:01,980 --> 00:06:03,970 You can ask What is this? 82 00:06:04,260 --> 00:06:05,310 This is five. 83 00:06:05,790 --> 00:06:07,110 OK, sharp five. 84 00:06:07,110 --> 00:06:08,120 Sharp two. 85 00:06:08,700 --> 00:06:09,410 Oh, OK. 86 00:06:09,480 --> 00:06:13,830 These are nothing, but their meaning is they are like a variable. 87 00:06:13,830 --> 00:06:14,150 OK? 88 00:06:14,640 --> 00:06:19,080 This sharp five is nothing but this one. 89 00:06:19,110 --> 00:06:19,440 OK. 90 00:06:19,500 --> 00:06:25,440 Just take it from here and put it inside here and you will get your matrix equation. 91 00:06:25,440 --> 00:06:30,700 As you can see, this is a scale symmetric matrix because in diagonal we have zeros. 92 00:06:31,080 --> 00:06:31,630 OK. 93 00:06:31,920 --> 00:06:36,090 And here we have minus z plus set. 94 00:06:36,510 --> 00:06:42,750 OK, y minus y and minus x x OK. 95 00:06:43,020 --> 00:06:54,840 Now we can go to again our presentation and try to get the results, so get the relation between omega 96 00:06:55,020 --> 00:07:00,900 angular velocities and rate of change of our p y angles. 97 00:07:04,050 --> 00:07:04,470 OK. 98 00:07:04,590 --> 00:07:11,790 Let's go with the metrics that we have obtained by map of the separating derivatives of RPI angles from 99 00:07:11,790 --> 00:07:18,270 the metrics and the write the equation in matrix form and compare it with the previous scores symmetric 100 00:07:18,270 --> 00:07:25,260 metrics, we eventually get the relation between the derivatives of road Piaggio angles and angular 101 00:07:25,260 --> 00:07:25,980 velocities. 102 00:07:26,700 --> 00:07:32,940 So as you can see, derivatives of R P Y angles are not the same as angular velocities. 103 00:07:33,570 --> 00:07:39,240 Let's call that matrix in between B, which is a function of r p y angles. 104 00:07:39,960 --> 00:07:45,060 So it's not constant and changes as the configuration of the robot manipulator changes. 105 00:07:45,810 --> 00:07:52,110 Now we can write the new relation in this way, which relates joint velocities with linear velocities 106 00:07:52,110 --> 00:08:00,390 and rate of change of r p y angles of end effector the matrix in between its analytical Jacobean. 107 00:08:00,570 --> 00:08:03,090 And it is related with geometric Jacobean. 108 00:08:03,090 --> 00:08:10,110 By this relation, analytical Jacobin is much more initiative than geometric Jacobean for engineers 109 00:08:10,110 --> 00:08:17,910 because it relates the rate of change of RPE y angles, which is much easier to understand than angular 110 00:08:17,910 --> 00:08:19,050 velocity vectors. 111 00:08:19,650 --> 00:08:27,840 Also, I'm sure that you have seen the inverse of The Matrix and you think that is it always invisible? 112 00:08:28,260 --> 00:08:34,800 This is the topic of our next lesson, which is Jacobean singularities see you on the next lesson.