1 00:00:00,810 --> 00:00:07,620 During last few lessons, we have analyzed both geometrical and analytical Jacobean mattresses. 2 00:00:08,160 --> 00:00:15,250 Both of them enable us to get the end effector velocities, given joint velocities especially less less 3 00:00:15,270 --> 00:00:21,120 than we have seen analytical Jacobean during driving analytical Jacobean formulation. 4 00:00:21,480 --> 00:00:30,900 We have seen this matrix and it called it matrix speed, which was a function of our P y angles. 5 00:00:31,590 --> 00:00:38,310 So pitch your angles, then we have used it during the duration of the Jacobean matrix. 6 00:00:38,880 --> 00:00:41,320 More clearly, we have used its inverse. 7 00:00:41,850 --> 00:00:46,800 But as you know, the inverse of mattresses is such a notion that we have to be careful. 8 00:00:47,460 --> 00:00:51,750 So we have to ask whether it be matrix is always inverted or not. 9 00:00:52,410 --> 00:01:00,210 And if you analyze metrics, speak carefully, you will see that it is not the case because it's obvious 10 00:01:00,210 --> 00:01:09,270 from the definition of Matrix beat that when pitch angle P is equal to plus minus pi over two or plus 11 00:01:09,270 --> 00:01:17,460 minus 90 degrees, then we have cosine p zero, which makes first and third columns of the B matrix 12 00:01:17,460 --> 00:01:18,000 the same. 13 00:01:18,330 --> 00:01:26,580 So they become linear dependent and the matrix becomes rank deficient so we cannot get its inverse. 14 00:01:27,120 --> 00:01:35,430 This case is called a representational singularity, while during last lessons we have tried to get 15 00:01:35,430 --> 00:01:38,310 and the factor velocities given joint velocities. 16 00:01:38,550 --> 00:01:45,480 However, it is much more interesting for us to know what should be the joint velocities in order to 17 00:01:45,480 --> 00:01:51,720 get the desired and the fact that the velocity so we have to use the inverse of Jacobean matrix. 18 00:01:52,230 --> 00:01:54,780 Now we are faced with again the same problem. 19 00:01:55,080 --> 00:01:57,330 Does the Jacobean always in vertical? 20 00:01:57,990 --> 00:02:03,760 And again, the answer is no, as all the Jacobin matrix is configuration dependent. 21 00:02:03,780 --> 00:02:07,860 There will be some cases in which the Jacobean matrix will be singular. 22 00:02:08,460 --> 00:02:15,420 Other than that, we will assume that we have six degrees of freedom robot manipulator because, as 23 00:02:15,420 --> 00:02:21,750 you know, in this case, our Jacobean matrix will be square and square mattresses are inevitable. 24 00:02:22,950 --> 00:02:27,780 I don't consider the case left inverse of mattresses. 25 00:02:28,740 --> 00:02:29,190 OK? 26 00:02:29,220 --> 00:02:30,420 Anyway, let's continue. 27 00:02:31,830 --> 00:02:33,720 The Jacobean will be singular. 28 00:02:33,720 --> 00:02:35,880 When it's determinant will be zero. 29 00:02:36,300 --> 00:02:43,530 Finding the case has been Jacobean is singular is important for us because these cases represent robot 30 00:02:43,530 --> 00:02:48,720 manipulators configurations in which motion in certain directions are unachievable. 31 00:02:49,260 --> 00:02:55,250 Other than that, during singularities bounded and the vector velocities make or respond to a mound 32 00:02:55,250 --> 00:02:56,610 the joint velocities. 33 00:02:57,180 --> 00:03:02,010 We want to avoid this situation because it's damaging for our robot manipulator. 34 00:03:03,330 --> 00:03:09,150 Also, singularities happened when the robot reaches its maximum of boundaries in work space, which 35 00:03:09,150 --> 00:03:11,670 can be referred as singularity to. 36 00:03:12,120 --> 00:03:20,640 Also, singularities happen when one or more axis has become aligned that referred us that causes loss 37 00:03:20,640 --> 00:03:25,200 of degree of freedom, namely Gimbel look problem that we have seen earlier. 38 00:03:25,560 --> 00:03:28,370 This can be called singularity. 39 00:03:29,040 --> 00:03:34,110 Now let's jump to the MATLAB and try to investigate that problem more deeply. 40 00:03:35,760 --> 00:03:44,070 OK, now let's try to understand singularities by using MATLAB Pethokoukis Library. 41 00:03:44,700 --> 00:03:51,360 So let's first drug import Puma five 16. 42 00:03:52,020 --> 00:03:54,000 Excuse me, robot manipulator. 43 00:03:54,600 --> 00:03:55,650 OK, its model. 44 00:03:56,190 --> 00:04:04,490 So first, let's calculate its Jacobean saw p five six. 45 00:04:04,500 --> 00:04:08,460 The thought um, jakob's zero. 46 00:04:08,790 --> 00:04:14,370 OK, this is the common for calculating Jacobean of a robot manipulator. 47 00:04:15,330 --> 00:04:27,960 So we calculated the Jacobean of our Puma five six model at when Jones R Q R, so in a ready position. 48 00:04:29,220 --> 00:04:33,360 OK, this is our, as you can see, Jacobean now. 49 00:04:33,690 --> 00:04:37,260 I want you to analyze this Jacobean. 50 00:04:37,620 --> 00:04:49,020 As you can see, this Jacobean consists of six rows and six columns, which is normal because our robot 51 00:04:49,020 --> 00:04:50,910 manipulator is six degrees of freedom. 52 00:04:50,910 --> 00:04:51,750 So it's square. 53 00:04:52,380 --> 00:04:53,490 But is it invert? 54 00:04:54,180 --> 00:04:55,260 Let's investigate. 55 00:04:55,530 --> 00:04:59,460 As you can see, we have the fourth. 56 00:05:00,320 --> 00:05:01,980 Six joints. 57 00:05:03,080 --> 00:05:03,590 OK. 58 00:05:04,020 --> 00:05:12,890 Are joint fourth and so forth, and six columns are linearly dependent. 59 00:05:12,920 --> 00:05:13,430 OK. 60 00:05:13,700 --> 00:05:21,170 As you can see, both of them, we have learned the meaning of Jacobean last time. 61 00:05:21,410 --> 00:05:32,330 So as you know, this means that the fourth and six joints of our robot affects only the orientation 62 00:05:33,140 --> 00:05:38,420 in the z direction of our end manipulator about that direction. 63 00:05:38,720 --> 00:05:39,170 Why? 64 00:05:39,560 --> 00:05:43,160 Because these are the wrist. 65 00:05:43,460 --> 00:05:43,880 OK. 66 00:05:43,970 --> 00:05:49,250 These are these constitute our wrists for fifth and sixth 67 00:05:52,040 --> 00:05:52,610 joints. 68 00:05:53,780 --> 00:06:02,000 So as fourth and sixth joints are linearly independent, so columns excuse me, columns are linearly 69 00:06:02,570 --> 00:06:03,340 independent. 70 00:06:03,350 --> 00:06:06,560 Our Jacobin is singular. 71 00:06:07,310 --> 00:06:09,290 OK, so rank deficient. 72 00:06:09,680 --> 00:06:14,900 Let's are our determinant of our Jacobean should be zero. 73 00:06:15,110 --> 00:06:16,820 As you can see, it's zero. 74 00:06:17,000 --> 00:06:22,550 So this means that our robot manipulator is in singularity. 75 00:06:22,940 --> 00:06:32,130 And if you see the rank of our Jacobean, it should be five because these two are linearly dependent. 76 00:06:32,150 --> 00:06:32,720 OK? 77 00:06:32,930 --> 00:06:36,140 As you can see, they are the same, so they are linearly dependent. 78 00:06:36,140 --> 00:06:38,930 So our Jacobean is rank deficient. 79 00:06:38,930 --> 00:06:41,360 So it is in singularity. 80 00:06:43,340 --> 00:06:53,110 OK, now let's try to see what will happen when we reach near this singularity. 81 00:06:53,120 --> 00:06:53,630 OK? 82 00:06:54,050 --> 00:06:57,410 Not in singularity, but we are near the singularity. 83 00:06:58,310 --> 00:07:02,540 OK, let's do that by just changing 84 00:07:05,180 --> 00:07:06,740 its fifth component. 85 00:07:06,770 --> 00:07:11,660 So fifteen point five degree away. 86 00:07:14,230 --> 00:07:18,250 From the singularity case, so we are near the singularity, OK? 87 00:07:18,700 --> 00:07:21,490 Not in a singularity, but near the singularity. 88 00:07:21,790 --> 00:07:24,010 Let's see how this affects to our robot. 89 00:07:24,490 --> 00:07:29,170 Let's first see it Jacobean equals 2p 560 dot 90 00:07:32,920 --> 00:07:35,920 zero cube near singularity. 91 00:07:36,160 --> 00:07:41,740 As you can see now, our robot planters check the Jacobean. 92 00:07:41,740 --> 00:07:50,210 It's feet fourth and six columns which represent Jovens are linearly independent. 93 00:07:50,230 --> 00:07:50,770 OK. 94 00:07:51,010 --> 00:07:52,390 They are not dependent. 95 00:07:52,390 --> 00:07:52,750 No. 96 00:07:52,990 --> 00:07:56,950 They are linearly independent, so we have to have. 97 00:07:58,240 --> 00:08:02,980 Let's check its rank of no j, as you can see its full rank. 98 00:08:03,280 --> 00:08:04,590 So it's not. 99 00:08:04,750 --> 00:08:06,730 It has inverse so determinant. 100 00:08:06,730 --> 00:08:12,430 Or however, if the chip determinant of G, as you can see, it is almost zero. 101 00:08:12,670 --> 00:08:14,800 Very, very close to zero. 102 00:08:14,830 --> 00:08:18,460 OK, so what does it mean? 103 00:08:18,490 --> 00:08:20,470 It means that our robot? 104 00:08:20,670 --> 00:08:30,790 Well, not exactly in singularity, but it is near singularity, which we want not only to avoid in 105 00:08:30,810 --> 00:08:36,820 exact singularity, but also we want to avoid the robot manipulator to come near singularity. 106 00:08:37,060 --> 00:08:38,920 OK, let's see why. 107 00:08:40,090 --> 00:08:42,400 Let's see why we award the singularity. 108 00:08:42,880 --> 00:08:46,450 Let's take that we want to achieve. 109 00:08:47,500 --> 00:08:48,070 OK. 110 00:08:48,640 --> 00:08:55,510 And the factor velocity of 0.1 linear velocity zero point one in that directions. 111 00:08:55,780 --> 00:09:01,510 So what should be our joint velocities in order to get that in the vector velocity? 112 00:09:01,510 --> 00:09:04,300 As you know, we have to find inverse of J. 113 00:09:04,600 --> 00:09:10,820 And we have to multiply it with our desire and the factor velocity. 114 00:09:10,840 --> 00:09:11,260 OK. 115 00:09:11,290 --> 00:09:13,660 Let me just or I think it's correct. 116 00:09:13,690 --> 00:09:14,140 OK. 117 00:09:15,520 --> 00:09:22,510 Let's see what's our desired joint velocities that we will get. 118 00:09:22,720 --> 00:09:24,280 These are and the vector elicited. 119 00:09:24,310 --> 00:09:33,220 Now you can see we have a very big problem here, as you can see in order to get zero point one 120 00:09:36,340 --> 00:09:38,140 velocity meters per second. 121 00:09:38,380 --> 00:09:43,120 I think velocity in the z direction. 122 00:09:43,930 --> 00:09:45,820 We had to change. 123 00:09:46,090 --> 00:09:54,400 The joint velocities have to be nine point eight five two to radians per second, which is very big 124 00:09:54,550 --> 00:09:56,920 if you make it. 125 00:09:57,940 --> 00:10:06,910 If you convert it to, um, radians two degrees, excuse me, divided by PI. 126 00:10:07,210 --> 00:10:11,290 You can see that it's approximately six hundred. 127 00:10:11,470 --> 00:10:14,800 OK, six hundred degrees per second. 128 00:10:15,790 --> 00:10:25,720 Can you imagine trying to imagine that joint has to rotate in 600 degrees in a second, which is very, 129 00:10:25,720 --> 00:10:29,960 very fast, and the robot motors cannot achieve that. 130 00:10:30,190 --> 00:10:35,610 It can damage our robot manipulators motor. 131 00:10:35,770 --> 00:10:36,280 OK? 132 00:10:36,610 --> 00:10:39,610 And also other hardware related stuff. 133 00:10:39,940 --> 00:10:43,450 So we want to achieve, however, inexact singularity. 134 00:10:43,900 --> 00:10:45,070 This will be excuse me. 135 00:10:45,190 --> 00:10:46,730 This will be infinity. 136 00:10:46,750 --> 00:10:47,230 OK. 137 00:10:47,410 --> 00:10:54,100 But as we near the singularity, we will have not infinite, but very big numbers. 138 00:10:54,340 --> 00:11:00,940 OK, so that's why we want to avoid not only the exact singularity, but also the near singularity. 139 00:11:02,080 --> 00:11:02,590 OK. 140 00:11:03,640 --> 00:11:04,820 We can determine. 141 00:11:05,230 --> 00:11:06,920 Let's see its derivative. 142 00:11:06,950 --> 00:11:10,120 No, our Jacobins derivative, excuse me. 143 00:11:10,120 --> 00:11:11,800 Not derivative determinant. 144 00:11:11,800 --> 00:11:19,060 As you can see, the determinant is very small, as we have said earlier, and we can also represent 145 00:11:19,390 --> 00:11:23,920 this with condition or a number of our Jacobean. 146 00:11:24,070 --> 00:11:29,080 As you can see, our condition number of Jacobean is very big. 147 00:11:29,440 --> 00:11:29,870 OK. 148 00:11:30,080 --> 00:11:31,930 What does this mean? 149 00:11:32,290 --> 00:11:41,680 This means that when our condition number of the Jacobean is big, this means that our Jakobsson is 150 00:11:41,680 --> 00:11:42,850 ill conditioned. 151 00:11:42,880 --> 00:11:43,480 OK. 152 00:11:43,870 --> 00:11:47,260 So we are near singularity. 153 00:11:48,460 --> 00:12:02,670 OK, now let's try to get that to to see the singularity is or near singularities by, you know, in 154 00:12:02,680 --> 00:12:03,460 leave mode. 155 00:12:03,460 --> 00:12:05,980 OK, let's do first. 156 00:12:06,340 --> 00:12:12,580 We will use, as you have seen previously, teach method what it will do. 157 00:12:12,790 --> 00:12:13,630 It will enable. 158 00:12:13,850 --> 00:12:20,960 US to change the joint of the robot manipulator so we can control it in a leave mode, and we can also 159 00:12:20,960 --> 00:12:27,890 see how it affects our robot manipulators configuration, so we will use callback with callback method. 160 00:12:27,920 --> 00:12:33,770 OK, so we will use inline function R and Q. 161 00:12:33,770 --> 00:12:36,370 What are these callback function? 162 00:12:36,380 --> 00:12:36,860 OK? 163 00:12:37,040 --> 00:12:37,730 What does it mean? 164 00:12:37,730 --> 00:12:44,840 Callback As we change the joints of our robot manipulator, the callback function will be how can I 165 00:12:44,840 --> 00:12:54,350 say it will be arised and we will get a each time our robots model, which is R and all set the joint 166 00:12:54,350 --> 00:12:57,350 values, which are Q OK? 167 00:12:57,680 --> 00:13:04,850 And what we will do, we will do, we will do print these. 168 00:13:07,220 --> 00:13:09,050 We will print condition. 169 00:13:09,920 --> 00:13:10,550 OK. 170 00:13:10,850 --> 00:13:13,670 No condition. 171 00:13:13,670 --> 00:13:14,210 No. 172 00:13:17,740 --> 00:13:22,050 Condition number during these joint variables. 173 00:13:22,120 --> 00:13:22,510 OK. 174 00:13:23,810 --> 00:13:27,180 Jacob, OK, zero Q. 175 00:13:27,520 --> 00:13:30,340 I think it has to work if I didn't do anything wrong. 176 00:13:30,670 --> 00:13:40,150 OK, now as you can see a disposition, the our condition variable is very big. 177 00:13:41,740 --> 00:13:44,620 Now let's try to manipulate this. 178 00:13:44,620 --> 00:13:46,450 I think this is because of that one. 179 00:13:46,480 --> 00:13:51,060 OK, as you can see, this is why we got a very big condition. 180 00:13:51,070 --> 00:14:01,540 No, this was because at first we were in ready mode, which was, you know, and we we were in singularity, 181 00:14:01,540 --> 00:14:03,980 not near single, about exactly the singular. 182 00:14:04,020 --> 00:14:12,490 So that's what our condition number is, that we, as you can see when we changed our cue five, which 183 00:14:12,910 --> 00:14:16,960 avoids which help us to avoid the risk singularities. 184 00:14:16,960 --> 00:14:19,570 So four and six note becomes a long axis. 185 00:14:19,840 --> 00:14:27,160 So instantly our condition number decreased our as you can see when it reaches near singularity, as 186 00:14:27,160 --> 00:14:30,640 you can see, it starts to condition, no starts to increase. 187 00:14:30,940 --> 00:14:35,380 And again, when it avoids the singularity, it starts to decrease. 188 00:14:35,410 --> 00:14:40,850 So what we have said, we have seen that except from this singularity. 189 00:14:41,530 --> 00:14:48,870 Also, our robot becomes singular or becomes near singular when it reaches its alter boundaries. 190 00:14:48,900 --> 00:14:53,620 So if you take your arm or not, take your arm because this is your arm. 191 00:14:53,830 --> 00:15:02,740 If you stretch your arm fully, you will see that at some point your arm will reach singularity and 192 00:15:02,740 --> 00:15:07,560 you will not be able to end to control it or you will control it. 193 00:15:07,570 --> 00:15:14,890 But you cannot stretch it any more because you know your arm is like a robot manipulator and it reaches 194 00:15:14,890 --> 00:15:21,790 its altered bones, or it cannot stretch anymore because of hardware limitations, OK? 195 00:15:22,650 --> 00:15:25,750 And because you have link lengths that are limited. 196 00:15:26,950 --> 00:15:29,890 OK, now let's try to achieve that position. 197 00:15:30,250 --> 00:15:30,990 OK. 198 00:15:31,720 --> 00:15:35,260 Let's try to by just changing it in this way. 199 00:15:35,260 --> 00:15:36,790 OK, like that. 200 00:15:36,790 --> 00:15:38,470 And let's stretch that. 201 00:15:38,470 --> 00:15:46,510 Also like that, as you can see, you can can you see how it switched from the sort to a two to three 202 00:15:46,780 --> 00:15:50,290 to eight to two hundred thirty three? 203 00:15:50,500 --> 00:15:50,930 OK. 204 00:15:50,950 --> 00:15:55,180 This is because we have reached new and we are near singularity. 205 00:15:55,180 --> 00:15:55,570 OK? 206 00:15:55,780 --> 00:16:00,010 Our robot is near singularity, so that is why it is increased that much. 207 00:16:00,250 --> 00:16:04,700 If you change it again, back to like that, you can see that it decrease them. 208 00:16:04,760 --> 00:16:06,940 Just let me just do it always. 209 00:16:07,210 --> 00:16:09,040 As you can see, it's decreased again. 210 00:16:09,460 --> 00:16:10,030 Can you see? 211 00:16:10,160 --> 00:16:10,600 OK. 212 00:16:10,840 --> 00:16:13,640 Let's make it again near singularity. 213 00:16:13,660 --> 00:16:15,730 And you can see it will switch. 214 00:16:15,730 --> 00:16:16,240 No. 215 00:16:16,390 --> 00:16:16,900 OK. 216 00:16:16,930 --> 00:16:18,580 As you can see it switched back. 217 00:16:20,200 --> 00:16:25,830 You can play with that, which is a very useful tool and try to see OK. 218 00:16:25,840 --> 00:16:30,330 As you can see, we have again reached this singularity here by changing five. 219 00:16:31,480 --> 00:16:38,410 OK, by changing this in a leave mode, you can see when the robot will reach near singularity, and 220 00:16:38,410 --> 00:16:41,170 we want to avoid this singularities. 221 00:16:45,190 --> 00:16:57,240 OK, now let me just show one interesting thing also that, OK, now let's try to get the same and the 222 00:16:58,630 --> 00:17:02,830 the what will happen with the same involved Jacobean OK? 223 00:17:02,980 --> 00:17:04,070 Near Singularity. 224 00:17:04,420 --> 00:17:07,360 This is the Jacobean that we get near singularity. 225 00:17:07,370 --> 00:17:09,040 So by using cute and asked. 226 00:17:09,400 --> 00:17:21,080 But now we will try to change the end effectors orientation in y direction by 0.2 radians per second. 227 00:17:21,100 --> 00:17:21,490 OK. 228 00:17:21,550 --> 00:17:27,400 Let's see how much joint velocities we need to do that. 229 00:17:29,050 --> 00:17:30,710 Let's see our kudi. 230 00:17:30,760 --> 00:17:35,860 As you can see now, the values are moderate, not big. 231 00:17:35,860 --> 00:17:37,150 Like here, OK? 232 00:17:37,600 --> 00:17:46,390 So you can see that while in some singularities of the the draw, it is, you know, not achievable 233 00:17:46,390 --> 00:17:54,040 to get some linear velocities, but it is achievable to get or it is possible to get some orientation 234 00:17:54,040 --> 00:17:55,270 configurations, OK. 235 00:17:55,630 --> 00:18:02,620 So this change based on our, you know, singularity configuration, in some cases we can get linear 236 00:18:02,620 --> 00:18:08,440 velocities, but we cannot get orientation velocities or angular velocities. 237 00:18:08,680 --> 00:18:15,010 In some cases, we can get angular velocities, but not linear velocities. 238 00:18:15,580 --> 00:18:19,630 Let's assume a set of generalized joint velocities with unit no. 239 00:18:20,380 --> 00:18:27,610 These general joint velocities lie on the surface of a hyperspace fair in the end dimensional joint 240 00:18:27,610 --> 00:18:28,510 velocity space. 241 00:18:29,110 --> 00:18:34,940 We have seen this equation also, which helps us to determine joint velocities given and the factor 242 00:18:34,950 --> 00:18:35,560 velocity. 243 00:18:36,160 --> 00:18:42,280 If we substitute inside our equation, we will get this new equation, which is the equation of points 244 00:18:42,280 --> 00:18:43,860 on the surface of an El Absorbed. 245 00:18:44,560 --> 00:18:50,110 This equation is called velocity el absorbed, and it helps us to understand the mechanical ability 246 00:18:50,110 --> 00:18:55,540 of our orbit in certain directions, namely how much this episode resembles a sphere. 247 00:18:56,260 --> 00:18:59,770 Then it can achieve arbitrary velocities in these directions. 248 00:18:59,770 --> 00:19:03,010 Namely, it can manipulate more easily in both directions. 249 00:19:03,400 --> 00:19:10,270 Let's see that more clearly in MATLAB before jumping into MATLAB, I would like to note that, as you 250 00:19:10,270 --> 00:19:16,880 can see in this case, we take into consideration only the kinematics of our robot to determine monopole 251 00:19:16,900 --> 00:19:17,410 ability. 252 00:19:17,830 --> 00:19:24,130 However, there are other things that affect the multiple abilities, such as mass and inertia of the 253 00:19:24,130 --> 00:19:28,540 links, and we will take them into account on our future lessons. 254 00:19:29,290 --> 00:19:37,120 Now, let's come to many probability of our robot manipulator. 255 00:19:37,960 --> 00:19:45,520 What we will do, we will do first import the model of Puma five 560. 256 00:19:45,730 --> 00:19:47,170 Excuse me, not like that. 257 00:19:47,650 --> 00:19:48,940 Puma five six. 258 00:19:49,720 --> 00:19:57,430 OK, now we will use the command, which is from Peter Caulkers Library. 259 00:19:57,970 --> 00:20:03,290 This common P5 60, which is model of our robot manipulator, the ellipse. 260 00:20:03,290 --> 00:20:06,950 So below, see the ellipse in Q and S. 261 00:20:06,970 --> 00:20:10,530 OK, let's get that Q in this excuse me. 262 00:20:10,540 --> 00:20:15,910 But OK, let's that Q in this we should have in the history of our robot. 263 00:20:16,450 --> 00:20:16,920 OK. 264 00:20:19,550 --> 00:20:21,670 Genius equals the Q all. 265 00:20:21,670 --> 00:20:26,320 And so Q and this is the position, the joint. 266 00:20:26,920 --> 00:20:29,440 But I was near singularity. 267 00:20:29,830 --> 00:20:41,410 So OK, we now try to see our velocity ellipse near a near singularity, and then we will see translational 268 00:20:41,410 --> 00:20:41,950 velocity. 269 00:20:41,960 --> 00:20:43,250 OK, not rotational. 270 00:20:43,270 --> 00:20:45,150 Then we will see also the rotational look. 271 00:20:45,160 --> 00:20:47,900 Let's see that ellipse, as you can see. 272 00:20:49,660 --> 00:20:54,790 Let me just make it like that, and let's call x y x label. 273 00:20:54,790 --> 00:21:02,200 Let me just make x OK, then y label y and Z Label Z. 274 00:21:02,260 --> 00:21:12,130 OK, now you can see that in X direction or ellipse is not in sphere form. 275 00:21:12,280 --> 00:21:12,850 OK. 276 00:21:13,150 --> 00:21:21,940 So even for our robot the manipulator, it is difficult to achieve linear velocities or translational 277 00:21:21,940 --> 00:21:24,400 velocities in X direction. 278 00:21:24,640 --> 00:21:33,670 However, let's do that in Y, as you can see now, the velocity ellipse resembles excuse me. 279 00:21:34,120 --> 00:21:42,490 Yeah, yeah, it resembles sphere more than in X directions, so we can achieve more arbitrary so we 280 00:21:42,490 --> 00:21:43,510 can manipulate more. 281 00:21:43,670 --> 00:21:46,900 Easily, our robots manipulator in why? 282 00:21:47,050 --> 00:21:56,020 Also in that direction, because as you can see in that direction, also our velocity ellipse resembles 283 00:21:56,260 --> 00:21:57,100 a sphere. 284 00:21:57,460 --> 00:22:05,830 OK, now let's try to do the same now with not translational, but rotational velocity. 285 00:22:06,880 --> 00:22:11,560 OK, now this is near singularity and the rotational. 286 00:22:12,060 --> 00:22:16,680 OK, let me just this one rotational velocity ellipse. 287 00:22:16,690 --> 00:22:25,420 OK, now let me also again Insert X label, insert y label and insert that label. 288 00:22:26,470 --> 00:22:28,000 So oh my gosh. 289 00:22:28,300 --> 00:22:31,540 Well, well, well done. 290 00:22:32,110 --> 00:22:32,500 OK. 291 00:22:32,680 --> 00:22:35,050 Z Label X y z. 292 00:22:35,320 --> 00:22:35,770 Perfect. 293 00:22:35,800 --> 00:22:42,730 Now let's see in X, as you can see, this is an interesting shape. 294 00:22:43,150 --> 00:22:43,540 OK. 295 00:22:43,810 --> 00:22:46,720 This is not resembles sphere. 296 00:22:46,810 --> 00:22:55,900 OK, so this lets not resemble spheres, so our robot manipulator cannot manipulate or rotate in x direction. 297 00:22:55,930 --> 00:23:04,900 OK, now the plate in I mean in rotation because this is rotation velocity sphere ellipse. 298 00:23:04,900 --> 00:23:07,720 Excuse me, in ex-Tottenham hover. 299 00:23:07,990 --> 00:23:09,670 Let's come to Y. 300 00:23:09,670 --> 00:23:17,800 As you can see in Y and Z, it resembles, you know, almost perfect sphere. 301 00:23:18,130 --> 00:23:18,610 OK. 302 00:23:19,120 --> 00:23:29,690 And so we can have we have grid man applicability in Y and Z directions in terms of rotational velocity. 303 00:23:29,710 --> 00:23:30,250 OK. 304 00:23:31,150 --> 00:23:40,510 So as you can see how much v near the sphere, in that amount, we can manipulate our robot easily in 305 00:23:40,560 --> 00:23:43,870 rotational and translational sense.