1 00:00:00,360 --> 00:00:09,040 In the last lesson, we have seen how to assign two frames to the robot manipulator in the convention. 2 00:00:09,060 --> 00:00:16,080 Now we can, after assigning the frames in the correct way, we can calculate the H parameter table 3 00:00:17,100 --> 00:00:17,670 easily. 4 00:00:18,630 --> 00:00:23,910 So what are the parameters that we have to calculate? 5 00:00:24,420 --> 00:00:27,910 There are four parameters to calculate for each of the joints. 6 00:00:27,930 --> 00:00:38,340 Yeah, this is due to the AI yet to AI, the AI for AI and A.I., so too does shows how much trotted 7 00:00:38,520 --> 00:00:48,060 along zit AI minus one axis in order to align x AI minus one with inside the AI shows how much the trust 8 00:00:48,060 --> 00:00:54,630 leads alongside AI minus one axis to align or AI minus one with or AI. 9 00:00:55,080 --> 00:00:55,830 What's all? 10 00:00:56,040 --> 00:01:05,430 All means origin or frame AI minus one with respect or a frame central authority or frame arm alpha 11 00:01:05,700 --> 00:01:10,110 is how much to rotate along x y axis. 12 00:01:10,350 --> 00:01:15,130 Be careful about axis along x axis to align the timeline this one does. 13 00:01:15,130 --> 00:01:24,210 The tie and the air shows ha shows how much to translate our exposure to align or ie minus one and or 14 00:01:24,240 --> 00:01:24,650 eye. 15 00:01:24,990 --> 00:01:32,940 As you can see, we have two rotational parameters, which is the high and III for I am two translational 16 00:01:32,940 --> 00:01:37,560 parameters, which is to the excuse me and a I. 17 00:01:37,970 --> 00:01:38,460 OK. 18 00:01:38,760 --> 00:01:45,780 And the order of calculating these parameters is we will coordinate first one, then second, then this 19 00:01:45,780 --> 00:01:46,080 one. 20 00:01:46,080 --> 00:01:53,130 And I have written here first, but these should be three, then this should be four of them. 21 00:01:53,550 --> 00:01:59,670 So then hopefully we will use these parameters in order to find the relation between the frames consecutive 22 00:01:59,670 --> 00:02:06,000 frames, OK, and how we will describe the relation between consecutive frames if this is using homogeneous 23 00:02:06,000 --> 00:02:07,050 transformation matrix. 24 00:02:07,110 --> 00:02:10,140 OK, how we will get homogeneous transformation matrix. 25 00:02:10,140 --> 00:02:13,350 We have four parameters to translation um two rotation. 26 00:02:13,770 --> 00:02:17,760 So we first calculate TITA, which is rotation around Z-axis. 27 00:02:17,760 --> 00:02:27,600 Yes, then we have translation, which is the eye along the z axis and translation again along X and 28 00:02:27,600 --> 00:02:31,860 the rotation along the X I degrees. 29 00:02:32,400 --> 00:02:37,950 So we can write this for these mattresses as a homogeneous transformation matrix. 30 00:02:37,950 --> 00:02:45,930 In this way, as you can see, the first one is what the first one is just rotation matrix so that this 31 00:02:46,350 --> 00:02:48,330 it's transitional part is zero. 32 00:02:48,330 --> 00:02:50,760 Excuse me, there should be one here, OK? 33 00:02:51,060 --> 00:02:56,010 Yeah, it's a transitional part is zero, but it's a rotational part. 34 00:02:56,010 --> 00:02:58,830 Is rotation among a lot of Z-axis? 35 00:02:59,100 --> 00:03:03,960 Yeah, because z rotation matrix with angle of Tita I. 36 00:03:04,500 --> 00:03:09,990 The second one is translation, so its rotation has to be identity because there is no rotation. 37 00:03:09,990 --> 00:03:17,550 But there is translation along the z axis with the eye, as it can be seen here. 38 00:03:18,000 --> 00:03:24,980 Then we have again translation along X with a I am sorry there should be zero here. 39 00:03:24,990 --> 00:03:32,700 OK, as you can see, we have translation along x eight amount and the rotation. 40 00:03:32,880 --> 00:03:35,820 There is no rotation, so identity perfect. 41 00:03:36,240 --> 00:03:43,020 Then we have rotation along x OK around X with a degree of four eyes. 42 00:03:43,020 --> 00:03:44,490 So we have rotation. 43 00:03:44,790 --> 00:03:51,360 This is x rotation matrix, as you can see with degree of over ie and there is no translation. 44 00:03:51,390 --> 00:03:51,780 OK. 45 00:03:52,260 --> 00:04:02,370 And at the end, we get our homogeneous transformation metrics, which indicates the relation or relation. 46 00:04:02,400 --> 00:04:15,240 Let me draw the relation between and between frames I and I minus one OK, I and I'm minus one. 47 00:04:15,770 --> 00:04:22,110 Um, so as you can see here, h I, I mean soon after we calculate this for each frame. 48 00:04:22,980 --> 00:04:31,680 OK, then for each joint, excuse me, then we can multiply them and get the relation between any desire 49 00:04:31,680 --> 00:04:33,570 to frame OK. 50 00:04:33,840 --> 00:04:35,070 We will see more clearly. 51 00:04:35,100 --> 00:04:40,050 So let's go to the calculation of the homogeneous transformation. 52 00:04:40,260 --> 00:04:43,890 Excuse me, the parameters before doing it. 53 00:04:44,400 --> 00:04:47,400 I want you to be careful about one thing. 54 00:04:47,670 --> 00:04:48,120 OK. 55 00:04:48,750 --> 00:04:51,330 Try to imagine everything in 3D. 56 00:04:51,360 --> 00:04:51,850 OK. 57 00:04:51,870 --> 00:04:53,130 Finish with 2D. 58 00:04:53,130 --> 00:04:54,150 There is no to the. 59 00:04:54,150 --> 00:04:56,900 We are working in three dimension. 60 00:04:56,910 --> 00:04:57,430 OK? 61 00:04:58,260 --> 00:04:59,580 You have to use your 3D. 62 00:04:59,610 --> 00:05:09,690 The imagination and here is what this is here, I want to just last time I showed you how we can choose 63 00:05:09,990 --> 00:05:10,710 the rotation. 64 00:05:12,330 --> 00:05:17,970 OK, rotation direction, but how we determine the rotation direction. 65 00:05:17,970 --> 00:05:23,610 But it is more clear picture again, as what we have said, even rotation is, for example, in this 66 00:05:23,610 --> 00:05:27,000 direction, OK, we curl our fingers. 67 00:05:27,000 --> 00:05:35,700 As you can recall, our fingers in the direction of rotation and our sun by automatically shows us the 68 00:05:35,700 --> 00:05:37,140 positive rotation. 69 00:05:37,170 --> 00:05:38,310 No, that's OK. 70 00:05:38,310 --> 00:05:43,620 Because the rotation that if it shows us the rotation direction, OK, axis of rotation is OK. 71 00:05:44,040 --> 00:05:49,380 So we curl in the direction of rotation and this show's axis of rotation, our sun. 72 00:05:49,800 --> 00:05:50,610 That's perfect. 73 00:05:51,870 --> 00:05:55,390 So let's go to this calculation of parameters. 74 00:05:55,410 --> 00:05:57,870 OK, first, we will start with Tita Eye. 75 00:05:57,870 --> 00:06:01,740 OK, let's go with the first joint. 76 00:06:02,280 --> 00:06:05,580 OK, excuse me, first frame rate was Tita Ed. 77 00:06:05,580 --> 00:06:06,620 I'm here. 78 00:06:06,630 --> 00:06:07,710 You can read again. 79 00:06:07,710 --> 00:06:10,190 Tita is how much trotting along is it? 80 00:06:10,200 --> 00:06:11,280 I mean, this one. 81 00:06:11,280 --> 00:06:13,800 Be careful along the time minus one we rotate. 82 00:06:13,800 --> 00:06:17,700 We are rotating in order to align x minus one with X. 83 00:06:18,030 --> 00:06:18,720 OK, perfect. 84 00:06:18,960 --> 00:06:27,360 Let's see how much we have to return frame zero along zero zero. 85 00:06:27,540 --> 00:06:32,550 OK, because we have seen the time minus one, so we are co-creating for the first one, as you can 86 00:06:32,550 --> 00:06:32,990 see here. 87 00:06:33,000 --> 00:06:35,280 One we are calculating for first one. 88 00:06:35,280 --> 00:06:41,610 So it should be along z zero zero minus one set minus one minus one, which is at zero. 89 00:06:41,670 --> 00:06:41,970 OK. 90 00:06:42,330 --> 00:06:51,870 And B how much we have to return it turn, excuse me, or rotate the frame zero in order to match x 91 00:06:51,870 --> 00:06:55,830 zero width x one x zero x one. 92 00:06:55,860 --> 00:06:56,130 OK. 93 00:06:57,030 --> 00:07:06,120 First, let me not wanting to say that, as you can see, I have removed the y axis because they will 94 00:07:06,120 --> 00:07:06,390 not. 95 00:07:06,660 --> 00:07:12,720 You will not need them in order to calculate the parameters so y to make complex problem complex by 96 00:07:12,900 --> 00:07:14,010 adding them also. 97 00:07:14,280 --> 00:07:15,360 No need to add them. 98 00:07:15,510 --> 00:07:23,880 Just it is enough to write only the the X and said accesses and then by L I show, as you can see, 99 00:07:23,880 --> 00:07:26,690 this purple color l are nothing, but 100 00:07:30,330 --> 00:07:33,990 they show just the link lengths. 101 00:07:34,200 --> 00:07:34,560 OK. 102 00:07:34,740 --> 00:07:36,090 So let's continue. 103 00:07:37,500 --> 00:07:38,610 Let's continue, OK? 104 00:07:39,240 --> 00:07:40,350 You can see that. 105 00:07:41,340 --> 00:07:45,240 You can see that, OK, there are in one plane, OK? 106 00:07:45,660 --> 00:07:55,920 So we do not need to rotate frames zero a manga, excuse me, along the Z zero in order to match x x1. 107 00:07:55,920 --> 00:07:57,330 They already match. 108 00:07:57,330 --> 00:08:01,700 But it's not correct because we don't take into account rotation. 109 00:08:01,710 --> 00:08:05,500 There is rotation of, as you can see, T to one. 110 00:08:06,000 --> 00:08:14,400 So in order to be to know it much better in order to visualize it much better, let's look at the robot 111 00:08:14,400 --> 00:08:15,330 from upside. 112 00:08:15,360 --> 00:08:15,730 OK. 113 00:08:16,050 --> 00:08:18,330 This is our first joint, OK? 114 00:08:18,660 --> 00:08:24,590 And then let's throw its axis is, as you can see, this is nothing but z zero. 115 00:08:24,600 --> 00:08:26,880 This is nothing but x zero. 116 00:08:27,270 --> 00:08:31,980 And if we rotate it, tita yeah. 117 00:08:32,130 --> 00:08:33,330 Excuse me, not like that. 118 00:08:33,330 --> 00:08:42,250 But if you rotated like Tatooine, then the second joint will be in this way from the top side, OK? 119 00:08:42,270 --> 00:08:45,750 And it's it will be rotated like that. 120 00:08:45,750 --> 00:08:51,030 So the X1 will be like that and the Z1 will be like that. 121 00:08:51,330 --> 00:08:55,500 As you can see now, there is an angle between 122 00:08:58,380 --> 00:09:05,040 now there is an angle between the zero, excuse me, zero and X1. 123 00:09:09,210 --> 00:09:10,560 OK, sorry for the delay. 124 00:09:10,680 --> 00:09:11,420 There is a delay. 125 00:09:11,430 --> 00:09:16,500 I think I I'm because projecting the iPod screen to the computer, there is a delay. 126 00:09:16,680 --> 00:09:22,080 So as you can see, there is an angle between between the zero and X1. 127 00:09:22,080 --> 00:09:27,810 If you Project X zero here again, you will see easily to one angle. 128 00:09:27,840 --> 00:09:28,240 OK. 129 00:09:28,560 --> 00:09:29,010 Zero. 130 00:09:29,190 --> 00:09:36,030 You have to take into account the rotation and you have to look at from upside of it or if it is needed 131 00:09:36,030 --> 00:09:41,510 from the down side, or even if this is needed to watch from their sides. 132 00:09:41,520 --> 00:09:46,620 OK, but you have to take into account a rotation and also translation if needed. 133 00:09:46,650 --> 00:09:47,040 OK. 134 00:09:47,400 --> 00:09:58,500 So we have seen that we, you know, we have to rotate frame zero one alongside Z zero in order to get 135 00:09:58,950 --> 00:09:59,330 on with the. 136 00:09:59,420 --> 00:10:04,670 Spectre's did zero in order to match teeth, one angle in order to match zero is x one. 137 00:10:04,970 --> 00:10:08,070 So it's right, then here T to one. 138 00:10:08,150 --> 00:10:11,270 OK, but let's go for the second point. 139 00:10:11,450 --> 00:10:15,440 OK, let's go for the second one and see what will happen again. 140 00:10:15,890 --> 00:10:17,570 For the second joint, this is the same. 141 00:10:17,570 --> 00:10:21,290 How much we have to return frame one. 142 00:10:21,530 --> 00:10:32,540 In order to match X, I'm on a longer through z one axis in order to match x one with what? 143 00:10:33,020 --> 00:10:36,180 In order to match x one with X two OK. 144 00:10:36,470 --> 00:10:43,730 In order to again, we have to take into account rotation because at first you can see that OK, there 145 00:10:43,730 --> 00:10:48,950 is no need to rotate because they are on the same plane an x one, an x two parallel to each other. 146 00:10:49,100 --> 00:10:53,270 So there's no need in the rotation, but it is not correct. 147 00:10:53,300 --> 00:10:57,350 You have to take into account these rotation angle of 2.2 also. 148 00:10:57,350 --> 00:11:05,360 So if we draw the robot again from top, for example, this is the joint OK from topside. 149 00:11:05,360 --> 00:11:07,200 This is x one. 150 00:11:07,310 --> 00:11:09,440 This this is z one. 151 00:11:09,680 --> 00:11:13,220 And we rotate it as he thought to OK. 152 00:11:13,670 --> 00:11:21,640 We have rotated into two, so our transitional joint become like that from the top. 153 00:11:21,650 --> 00:11:23,780 OK, from the top side, that's perfect. 154 00:11:23,780 --> 00:11:28,500 Now its frame will be like that again. 155 00:11:28,520 --> 00:11:37,950 OK, if we project this is X to this is excuse me, I didn't do that in this. 156 00:11:37,980 --> 00:11:39,230 That X is the correct way. 157 00:11:39,260 --> 00:11:39,590 OK. 158 00:11:43,010 --> 00:11:43,430 OK. 159 00:11:44,210 --> 00:11:46,580 The z axis should be in this way. 160 00:11:46,610 --> 00:11:47,040 OK. 161 00:11:47,060 --> 00:11:48,070 This is it, too. 162 00:11:48,320 --> 00:11:54,410 As you can see, if you project one again here, there is little tool angle between them. 163 00:11:54,420 --> 00:11:59,630 We have to take it into account, so we have to rotate the frame. 164 00:11:59,870 --> 00:12:00,620 One. 165 00:12:01,700 --> 00:12:02,150 OK. 166 00:12:02,180 --> 00:12:10,490 Along Z one, excuse me, sort of in in the direction of Z1 in order to obtain you on the other to make 167 00:12:10,490 --> 00:12:13,670 X1 coincides with X2. 168 00:12:13,670 --> 00:12:14,780 That's perfect now. 169 00:12:15,350 --> 00:12:16,680 So let's write it. 170 00:12:17,150 --> 00:12:23,620 Let's first erase all of these and then let's write it here. 171 00:12:23,630 --> 00:12:24,020 OK? 172 00:12:24,260 --> 00:12:26,270 We have to rotate by 2.2. 173 00:12:26,300 --> 00:12:26,900 That's perfect. 174 00:12:26,930 --> 00:12:27,230 No. 175 00:12:27,500 --> 00:12:30,500 Let's go with the frame. 176 00:12:31,700 --> 00:12:32,070 Three. 177 00:12:32,360 --> 00:12:34,310 Now, what should we do? 178 00:12:34,610 --> 00:12:37,610 What we know in order to find it three. 179 00:12:37,910 --> 00:12:45,920 It shows its meaning is that how much we have to rotate frame two in order and more in the direction 180 00:12:45,920 --> 00:12:50,840 of Z tool in order make X2 and X3 coincide with each other. 181 00:12:50,870 --> 00:12:51,470 OK. 182 00:12:51,860 --> 00:12:58,090 As you can see now, in this case, there is no translator, excuse me, rotation, but translation. 183 00:12:58,100 --> 00:13:02,750 So what does it mean as there is no rotation? 184 00:13:02,990 --> 00:13:07,530 The X2 and X3 will always be OK. 185 00:13:07,680 --> 00:13:15,110 We always coincide with each other because this joint will only up will move up and down. 186 00:13:15,320 --> 00:13:22,010 So X2 will not move with respect to X3 in terms of the two in the two direction, OK? 187 00:13:22,340 --> 00:13:31,490 So we don't have to rotate frame two in the direction of Z two in order to make X2 and X3 coincide with 188 00:13:31,490 --> 00:13:34,610 each other, they anyway coincide with each other. 189 00:13:35,090 --> 00:13:44,390 OK, so this will be zero and you can now see that y always we try to make the rotations change as small 190 00:13:44,390 --> 00:13:45,080 as possible. 191 00:13:45,260 --> 00:13:51,980 We have seen in the last video that I have said that tried to make axis is coincide with each other 192 00:13:51,980 --> 00:13:53,270 as much as possible. 193 00:13:53,300 --> 00:13:54,040 OK, tried. 194 00:13:54,050 --> 00:13:58,790 If it is possible, choose the consecutive axis is the same with respect to each other. 195 00:13:58,790 --> 00:14:03,110 So the H two calculating the H parameters will be easier for you. 196 00:14:03,350 --> 00:14:09,140 OK, so now let's call good for the last one. 197 00:14:09,140 --> 00:14:12,930 For the last one, for the fourth. 198 00:14:12,950 --> 00:14:13,400 OK. 199 00:14:13,730 --> 00:14:17,900 What it will be again is the have rotation angle of three to four. 200 00:14:17,910 --> 00:14:19,760 We have to take it into account. 201 00:14:20,090 --> 00:14:28,400 So this means that it often means that how much we have to rotate Keita's excuse me, frame three in 202 00:14:28,400 --> 00:14:33,650 the direction of Z three in order to make it three and X4 coincide with each other. 203 00:14:34,400 --> 00:14:41,090 At first, they can be seen that they are on the same plane, so there is no need to rotation. 204 00:14:41,600 --> 00:14:47,070 Excuse the monitor rotation, but take into account little for rotation angle. 205 00:14:47,090 --> 00:14:51,650 Be careful the robot will return when it tries to in the column to give them to eat. 206 00:14:52,040 --> 00:14:58,760 So X3 and X4 will not coincide with each other, so we need a titov for. 207 00:14:59,130 --> 00:15:03,220 Station and all these rotations will be a plus. 208 00:15:03,520 --> 00:15:04,400 Why now? 209 00:15:04,990 --> 00:15:14,020 Let me just explain over to four, and you can apply the same for others angles because if you try to 210 00:15:14,020 --> 00:15:21,850 rotate it in every frame tool, we have to rotate it in this way in order to match X3 and X4. 211 00:15:22,070 --> 00:15:28,060 And in this direction of rotation, if you call your fingers in the direction of rotation, as we have 212 00:15:28,060 --> 00:15:34,090 shown in this picture, your thumb will show this direction, which is positive. 213 00:15:34,090 --> 00:15:35,890 That's the direction, OK? 214 00:15:36,160 --> 00:15:39,610 So the rotation angle will be positive. 215 00:15:39,880 --> 00:15:46,750 If otherwise, if we would require to rotate it in this way, OK, in order to match X and export, 216 00:15:47,020 --> 00:15:50,250 then our son blue the point in this way. 217 00:15:50,260 --> 00:15:53,170 So rotation axis would be in this way. 218 00:15:53,590 --> 00:15:55,480 The rotation direction would be in this way. 219 00:15:55,480 --> 00:16:02,440 So the rotation axis direction would be in this way, which is opposite to the Z three, so it would 220 00:16:02,440 --> 00:16:03,130 be negative. 221 00:16:03,160 --> 00:16:04,810 OK, be careful about that. 222 00:16:04,900 --> 00:16:09,370 OK, so let's continue. 223 00:16:09,370 --> 00:16:10,330 Let's continue. 224 00:16:10,780 --> 00:16:12,520 We have finished 48th out. 225 00:16:12,520 --> 00:16:15,790 Let's continue with the let's see what what this deep. 226 00:16:16,090 --> 00:16:18,580 These shows us how much to translate. 227 00:16:18,670 --> 00:16:26,090 OK, this is not rotation along the Z Eye, minus one to align origins of two consecutive frames. 228 00:16:26,110 --> 00:16:27,640 OK, let's go. 229 00:16:27,650 --> 00:16:29,560 Let's go with the one. 230 00:16:29,560 --> 00:16:31,100 OK with this one. 231 00:16:31,120 --> 00:16:35,860 This means that how much we have to move this order, Jim. 232 00:16:36,990 --> 00:16:37,600 OK. 233 00:16:37,870 --> 00:16:45,130 Over the z zero axis, over the z zero axis, OK, or the z zero axis. 234 00:16:45,460 --> 00:16:53,230 In order to make this origin OK to match this origin, do we have to do something? 235 00:16:54,580 --> 00:16:56,760 Surely, no, surely not. 236 00:16:56,770 --> 00:17:02,300 Yeah, we don't have to do anything because they in the way coincide with each other. 237 00:17:02,320 --> 00:17:02,770 OK. 238 00:17:03,010 --> 00:17:12,550 And these joints don't move like that up and down, so we don't need any uh, we don't need to translate 239 00:17:12,550 --> 00:17:15,110 this orygen in the Z zero direction. 240 00:17:15,580 --> 00:17:22,660 So which is up or minus zero direction, which is known in order to match it with Z one because they 241 00:17:22,660 --> 00:17:26,090 will on anyway, they anyway coincide with each other. 242 00:17:26,110 --> 00:17:28,810 There is no need for translation again. 243 00:17:29,140 --> 00:17:32,020 So the idea would be zero for the first one. 244 00:17:32,020 --> 00:17:36,790 So let's check for the second one, OK, for the second one, is it the same? 245 00:17:36,820 --> 00:17:42,380 How much we have to translate origin this origin alongside that one? 246 00:17:42,400 --> 00:17:42,880 OK? 247 00:17:42,910 --> 00:17:50,320 In the direction of Z five in order to match this origin with this origin, as you can see again, there 248 00:17:50,320 --> 00:17:56,260 is no need to do that kind of thing because this joint is rotational. 249 00:17:56,890 --> 00:17:58,030 It will not change. 250 00:17:58,240 --> 00:18:01,180 I won't say it will not change in a transitional way. 251 00:18:01,180 --> 00:18:02,710 I mean, up or down. 252 00:18:02,920 --> 00:18:09,640 So the origins will always coincide and we don't need any kind of translation. 253 00:18:09,640 --> 00:18:11,920 So this will be also zero. 254 00:18:12,940 --> 00:18:13,570 OK, perfect. 255 00:18:13,840 --> 00:18:16,780 So let's check for the assert frame, OK? 256 00:18:17,080 --> 00:18:24,800 How much of this have to translate or origin of this joint in the direction of Z two? 257 00:18:24,880 --> 00:18:29,710 So which is done in order to match this origin? 258 00:18:29,740 --> 00:18:30,040 OK? 259 00:18:30,310 --> 00:18:34,810 Origin of frame two vs origin of frames three. 260 00:18:35,050 --> 00:18:36,520 Note zero or note. 261 00:18:36,790 --> 00:18:38,680 Surely it is not zero. 262 00:18:38,710 --> 00:18:39,250 OK. 263 00:18:39,460 --> 00:18:41,830 Because in the direction of the two, which is. 264 00:18:42,550 --> 00:18:45,360 OK, we have to slide this year. 265 00:18:45,380 --> 00:18:49,780 We have to make it to come to here and how much this does. 266 00:18:49,780 --> 00:18:51,130 We have to make it to come. 267 00:18:51,370 --> 00:18:52,470 This is L. 268 00:18:52,610 --> 00:18:53,080 Three. 269 00:18:53,090 --> 00:18:53,440 Yes. 270 00:18:53,650 --> 00:18:58,770 We have to make this origin to come here L three amount in order. 271 00:18:59,250 --> 00:19:03,150 Um, these origins to coincide with each other. 272 00:19:03,190 --> 00:19:03,730 Perfect. 273 00:19:04,060 --> 00:19:05,950 So that's right. 274 00:19:05,950 --> 00:19:09,310 It also so this will be L three. 275 00:19:09,490 --> 00:19:10,000 OK. 276 00:19:10,210 --> 00:19:10,900 This will be our. 277 00:19:11,500 --> 00:19:11,920 No. 278 00:19:11,920 --> 00:19:13,870 Let's go with the last one. 279 00:19:13,870 --> 00:19:14,800 With the last one. 280 00:19:16,270 --> 00:19:20,290 What we have to do this is our origin. 281 00:19:20,390 --> 00:19:21,550 This is our origin. 282 00:19:21,570 --> 00:19:28,300 So how much we have to translate the origin of frames three in order to match? 283 00:19:29,140 --> 00:19:31,100 Yeah, just frame. 284 00:19:31,120 --> 00:19:31,370 Three. 285 00:19:32,170 --> 00:19:35,590 In order to match it with the original frame forward. 286 00:19:35,830 --> 00:19:40,540 Well, in the direction of what Z three do, we have to do something? 287 00:19:40,600 --> 00:19:42,630 Yes, surely we have to do. 288 00:19:42,640 --> 00:19:50,260 We have to translate because Z three duration is Don and we have to slide it down in L four amount in 289 00:19:50,260 --> 00:19:55,270 order to match the origin of frames three of its origin of frame for yes. 290 00:19:55,510 --> 00:19:57,280 So this is L four. 291 00:19:57,610 --> 00:19:58,360 That's perfect. 292 00:19:58,430 --> 00:19:58,690 OK. 293 00:19:59,730 --> 00:20:05,570 As you can see, it is very easy, you have to have just the imagination of 3-D. 294 00:20:06,540 --> 00:20:16,470 Let's go with a walk in the is a as you can see here, because the sorts that this is sort of parameter 295 00:20:16,470 --> 00:20:23,220 we have to calculate is that how much we have to translate again, translation along IXI, I'd be careful 296 00:20:23,220 --> 00:20:23,550 along. 297 00:20:24,160 --> 00:20:29,800 Not that I might miss one in order to match again the consecutive frames origins. 298 00:20:29,820 --> 00:20:36,930 Let's check with the first joint book how much we have to translate alongside x one. 299 00:20:36,960 --> 00:20:37,480 OK. 300 00:20:37,500 --> 00:20:39,210 Alongside what X1? 301 00:20:39,210 --> 00:20:47,220 Because it's alongside, it should be in direction of ex-spy, which in this case is one so x1 in order 302 00:20:47,220 --> 00:20:52,200 to obtain in order to make the origin of frame zero. 303 00:20:52,210 --> 00:20:56,550 Yeah, coincide with the original frame one how much we have to do. 304 00:20:56,910 --> 00:21:04,860 As you can see, the X1 direction shows this way, so we have to translate it in the direction of X1 305 00:21:04,860 --> 00:21:06,270 and one amount. 306 00:21:06,270 --> 00:21:08,880 Yeah, in order to match these two centers. 307 00:21:09,150 --> 00:21:11,890 So this has to be L1. 308 00:21:11,910 --> 00:21:13,550 OK, that's perfect. 309 00:21:13,590 --> 00:21:14,970 This has to be L1. 310 00:21:14,970 --> 00:21:15,960 That was easy. 311 00:21:16,950 --> 00:21:24,090 Now let's go with the circle joy into how much we have to translate this center in the direction of 312 00:21:24,090 --> 00:21:31,200 X2, which is to the right X2 in order to match origins OK with each other. 313 00:21:31,200 --> 00:21:36,570 So original frame one with origin of frame two how much we have to do. 314 00:21:36,810 --> 00:21:43,560 As you can see, this is L2 because the offset in the direction of X2, so in the right direction is 315 00:21:43,560 --> 00:21:45,030 nothing but L2. 316 00:21:45,420 --> 00:21:47,730 So this has to be L2. 317 00:21:48,000 --> 00:21:49,350 Let's write it again. 318 00:21:49,350 --> 00:21:50,430 That's perfect. 319 00:21:50,790 --> 00:21:54,270 So then I can make it in this way. 320 00:21:55,350 --> 00:21:56,190 Oh my gosh. 321 00:21:56,570 --> 00:21:58,200 Why you're doing it in this way? 322 00:21:59,580 --> 00:22:03,460 Yeah, it's very, very precise, I think. 323 00:22:03,480 --> 00:22:03,870 Yeah. 324 00:22:05,010 --> 00:22:08,220 Anyway, let's continue. 325 00:22:08,310 --> 00:22:08,580 OK. 326 00:22:09,810 --> 00:22:13,500 So we have done this one. 327 00:22:13,510 --> 00:22:15,750 OK, we will go with the third joint. 328 00:22:15,750 --> 00:22:16,380 OK? 329 00:22:16,440 --> 00:22:27,640 The third one saw how much we have to move the origin of frame two in the direction of X three. 330 00:22:27,680 --> 00:22:30,030 Yes, in the direction of X three. 331 00:22:30,330 --> 00:22:34,770 In order to match X to another to merge these origin. 332 00:22:34,770 --> 00:22:39,360 Three original frame three with, excuse me, ordinal three and two with alternate frames. 333 00:22:39,360 --> 00:22:39,720 Three. 334 00:22:40,140 --> 00:22:46,530 And as you can see, if we don't have to do any translation because X3, the addiction is in this way, 335 00:22:46,770 --> 00:22:51,480 we don't have to translate this in this direction in order to match it, because in this direction, 336 00:22:51,480 --> 00:22:55,090 over this duration, they all rate the match with each other. 337 00:22:55,110 --> 00:22:55,560 OK. 338 00:22:56,070 --> 00:22:57,100 That's very obvious. 339 00:22:57,120 --> 00:23:00,200 So let's let's try it. 340 00:23:00,200 --> 00:23:00,880 It's zero. 341 00:23:00,900 --> 00:23:02,760 We don't have to do any translation. 342 00:23:02,760 --> 00:23:03,900 There is nine offset. 343 00:23:04,230 --> 00:23:11,310 So the last one, how much we have to translate the original frame three in order to match it with the 344 00:23:11,310 --> 00:23:18,990 frame of the original frame forward in the direction of X four, as you can see it again in the direction 345 00:23:18,990 --> 00:23:23,130 of X squared, which is like that, we don't have to do any translation. 346 00:23:23,130 --> 00:23:27,450 There is no offset in that direction, so it will be zero. 347 00:23:27,480 --> 00:23:28,950 Let's write it zero. 348 00:23:29,640 --> 00:23:33,720 So as you can see, the last one became zero. 349 00:23:34,110 --> 00:23:38,160 So now we the last is over. 350 00:23:38,310 --> 00:23:38,730 OK. 351 00:23:38,790 --> 00:23:45,380 The last parameter is alpha, which is how much to rotate along x y axis. 352 00:23:45,390 --> 00:23:45,840 OK? 353 00:23:45,870 --> 00:23:50,100 Be careful along x axis in order to match z i minus one to. 354 00:23:50,580 --> 00:23:51,030 I. 355 00:23:51,420 --> 00:23:52,770 So let's do that. 356 00:23:52,780 --> 00:24:03,430 So how much we have to rotate frame zero in the direction of X one? 357 00:24:03,460 --> 00:24:06,900 OK, in the direction of X1 in order to match. 358 00:24:06,900 --> 00:24:08,180 Is it zero bit? 359 00:24:08,190 --> 00:24:09,060 Is it one? 360 00:24:09,360 --> 00:24:10,180 Let's see. 361 00:24:10,200 --> 00:24:12,750 Should we do something like this one? 362 00:24:13,260 --> 00:24:17,700 No, because in this is it zero and this is it. 363 00:24:17,700 --> 00:24:23,730 One way matches with each other, even though there is little one rotation. 364 00:24:23,910 --> 00:24:31,980 This will not affect Z zero or that one because the rotation is about z axis as OK. 365 00:24:32,360 --> 00:24:39,390 And if you watched from the top side, is the for the first after the rotation, this is a zero zero 366 00:24:39,390 --> 00:24:39,870 zero. 367 00:24:39,870 --> 00:24:44,680 This is zero after the rotation of T to one. 368 00:24:44,730 --> 00:24:45,240 OK. 369 00:24:45,480 --> 00:24:47,660 This will be from the top side. 370 00:24:47,670 --> 00:24:49,490 This will be x one. 371 00:24:49,500 --> 00:24:50,790 OK, this x1. 372 00:24:50,790 --> 00:24:55,290 But again, z axis will be like that, OK, because it will not rotate. 373 00:24:55,650 --> 00:24:56,130 OK. 374 00:24:57,150 --> 00:24:58,860 So there is no need. 375 00:24:58,900 --> 00:25:08,280 To rotate a longer election in X One Direction in order to match Z zero with that one, so let's write 376 00:25:08,280 --> 00:25:09,780 it, it will be zero. 377 00:25:09,840 --> 00:25:10,290 Yes. 378 00:25:10,710 --> 00:25:13,590 So the first one is zero for the second joint. 379 00:25:13,590 --> 00:25:20,730 Let's check for the second joint how much we have to rotate in, of how much we have to rotate frame 380 00:25:20,820 --> 00:25:25,230 one in order in the direction of X two. 381 00:25:25,530 --> 00:25:28,380 In order to match Z one. 382 00:25:28,380 --> 00:25:31,650 Read it to look at that one with Z. 383 00:25:31,800 --> 00:25:34,620 Should we have, should or should we rotate? 384 00:25:34,950 --> 00:25:35,760 Indeed, yes. 385 00:25:35,760 --> 00:25:39,120 As you can see, they are opposite to each other. 386 00:25:39,360 --> 00:25:48,010 And again, the rotation as the rotation is with respect is with respect to the z axis, even interpolated 387 00:25:48,180 --> 00:25:54,870 to rotation and the directions of Z one and Z two, even after the rotation will be opposite to each 388 00:25:54,870 --> 00:25:55,260 other. 389 00:25:55,530 --> 00:26:03,240 The ratios of X2 and X3 will change on the X2 will be in this direction and excuse me and extend X1, 390 00:26:03,510 --> 00:26:08,730 X1 will be in this direction, OK, after rotation of the rotation of T2. 391 00:26:09,060 --> 00:26:14,790 However, Z one and Z two are how much we are, how much we have to rotate frame one. 392 00:26:15,570 --> 00:26:24,030 We have to rotate frame one in this direction, OK, if we take in in extra direction, OK, if we take 393 00:26:24,030 --> 00:26:27,390 an external direction, we have to rotate it. 394 00:26:27,630 --> 00:26:28,590 How much degrees? 395 00:26:28,590 --> 00:26:36,360 If you put your put your thumb in the direction of X2, OK and rotated, curl your fingers. 396 00:26:36,560 --> 00:26:41,550 Once the positive direction of rotation surely seen this way, should we? 397 00:26:42,000 --> 00:26:50,610 And in order to rotate the Z1 in order to rotate frame in order to match that one with the two, we 398 00:26:50,610 --> 00:26:59,040 have to rotate the frame f two with, excuse me, a frame f one with respect to X2 width. 399 00:26:59,760 --> 00:27:00,410 How much? 400 00:27:00,450 --> 00:27:04,680 Um, minus excuse me, plus peak angles. 401 00:27:04,680 --> 00:27:12,540 Yet because they are one hundred eighty degrees from each other and the rotation is a static rotation, 402 00:27:12,540 --> 00:27:13,730 so always will be. 403 00:27:13,740 --> 00:27:20,400 They will be apart from each other by 180 degrees and the rotation is in the positive direction. 404 00:27:20,430 --> 00:27:25,140 OK, we have to rotate in positive directions, so let's right it. 405 00:27:25,980 --> 00:27:28,170 The rotation will be a plus pi. 406 00:27:28,290 --> 00:27:31,610 Let's write it OK, the rotation will be plus pi. 407 00:27:31,620 --> 00:27:35,930 OK, then let's do this one again. 408 00:27:36,150 --> 00:27:42,990 Let's do this one how much we have to rotate, how much we have to rotate. 409 00:27:43,290 --> 00:27:43,830 Yes. 410 00:27:46,470 --> 00:27:51,360 Frame two with respect to Axis X three. 411 00:27:51,600 --> 00:28:00,570 In order to Match Z two with Z three, as you can see Z two and the three already matched each other. 412 00:28:00,570 --> 00:28:08,010 And as this is prismatic joint, there is no rotation that we have to take into account, so we don't 413 00:28:08,010 --> 00:28:10,250 need any rotation. 414 00:28:10,260 --> 00:28:15,120 They will always match with each other. 415 00:28:15,120 --> 00:28:19,950 So we need zero rotation and the last one for the last one. 416 00:28:20,250 --> 00:28:29,580 Again, how much we have to rotate frames three along x in the direction of export in order to match 417 00:28:29,640 --> 00:28:36,730 Z three, we visit for Z three and Z four will always match with each other. 418 00:28:36,780 --> 00:28:37,380 Why? 419 00:28:37,650 --> 00:28:46,530 Because the rotation for the for this joint, OK for the first joint will be always in this direction. 420 00:28:46,530 --> 00:28:46,920 Yes. 421 00:28:47,220 --> 00:28:52,080 So this ring will not change its direction from its launch from top side. 422 00:28:52,350 --> 00:28:53,730 This is extra. 423 00:28:53,810 --> 00:28:56,100 Yes, this is Z three. 424 00:28:56,100 --> 00:28:56,700 Yes. 425 00:28:57,660 --> 00:28:58,680 And the end effector. 426 00:28:58,680 --> 00:29:04,890 Even if this joint rotates, the India factor axis will be the same as this one here. 427 00:29:05,220 --> 00:29:06,150 It will not change. 428 00:29:06,150 --> 00:29:09,810 It will be Z for its x axis will only change. 429 00:29:09,810 --> 00:29:14,820 Yes, this will be X4 will only change by how much to the for. 430 00:29:14,910 --> 00:29:15,400 OK. 431 00:29:15,420 --> 00:29:16,230 Think about it. 432 00:29:16,260 --> 00:29:18,990 Be careful about this. 433 00:29:19,020 --> 00:29:22,560 Okay, so we don't need any rotations. 434 00:29:22,560 --> 00:29:30,390 It's V and Z Fold will anyway coincide with each other, so we need zero degrees of rotation. 435 00:29:30,960 --> 00:29:31,750 That's perfect. 436 00:29:31,770 --> 00:29:40,080 We have finished with the D.H table again, so you can understand it better by. 437 00:29:40,890 --> 00:29:49,140 If you missed something along side, please draw it and by stopping and playing VIDEO again, OK? 438 00:29:49,800 --> 00:29:57,420 Step by step, try to follow me so you can understand and imagine everything it is indeed. 439 00:29:57,420 --> 00:29:58,070 Very easy. 440 00:29:58,110 --> 00:29:58,740 Not that if. 441 00:29:58,880 --> 00:30:01,450 A gold, if you can imagine in 3-D. 442 00:30:01,690 --> 00:30:06,410 Okay, this is over the edge table. 443 00:30:06,470 --> 00:30:07,010 OK. 444 00:30:07,250 --> 00:30:15,440 And we will use each of these entries, OK in order to calculate a homogeneous transformation mattresses. 445 00:30:15,800 --> 00:30:16,310 OK. 446 00:30:16,550 --> 00:30:26,720 And then to get final, how can I say a final relation between the base frame and in the fact that a 447 00:30:26,720 --> 00:30:27,080 frame? 448 00:30:27,350 --> 00:30:29,000 Let's see how we can do that. 449 00:30:29,570 --> 00:30:35,610 We have seen the formula for the M between H and H by minus one. 450 00:30:35,630 --> 00:30:35,960 Yes. 451 00:30:36,260 --> 00:30:39,650 You just think this is c t de I. 452 00:30:40,520 --> 00:30:42,100 Maybe I have to write it. 453 00:30:42,110 --> 00:30:43,220 I forgot to say it. 454 00:30:43,230 --> 00:30:52,520 C Tita I means question Tita to yeah, if it is also I, this means question for I. 455 00:30:52,760 --> 00:30:53,210 OK. 456 00:30:53,750 --> 00:30:58,810 And s tita I means the same thing, but with seamless theta pi. 457 00:30:58,820 --> 00:30:59,480 OK? 458 00:30:59,750 --> 00:31:03,560 And if it is alpha I, this is since I I. 459 00:31:04,410 --> 00:31:05,450 OK, perfect. 460 00:31:06,530 --> 00:31:08,960 So you just need to take Tita. 461 00:31:08,960 --> 00:31:19,310 I I for I am a I and D and just plug in this in this formula. 462 00:31:19,400 --> 00:31:23,870 Yes, and you will get H I and I mine this one, for example. 463 00:31:23,930 --> 00:31:24,770 We will not do it. 464 00:31:25,190 --> 00:31:28,970 We will not plug it because you can do it easily in MATLAB. 465 00:31:29,000 --> 00:31:29,450 OK. 466 00:31:29,840 --> 00:31:35,180 But I will show you the the procedure for the first one. 467 00:31:35,180 --> 00:31:41,270 You take it and you've what you've given I for the first one by putting these parameter, these each 468 00:31:41,270 --> 00:31:49,850 of these parameters in the h i i minus one, if I equals to one. 469 00:31:49,850 --> 00:31:53,280 Yeah, for the first one, but we will get we will get h one zero. 470 00:31:53,300 --> 00:32:03,120 OK, this will give by just plugging in these yeah, these OK parameters you will get in this formula. 471 00:32:03,140 --> 00:32:06,620 OK, in this final formula, you will get H. 472 00:32:06,750 --> 00:32:07,950 Well, zero. 473 00:32:08,030 --> 00:32:08,480 OK. 474 00:32:08,540 --> 00:32:10,730 As you can see how easily you get it. 475 00:32:11,180 --> 00:32:14,060 This procedure can be automated easily. 476 00:32:14,190 --> 00:32:14,690 OK. 477 00:32:15,270 --> 00:32:15,420 Yeah. 478 00:32:15,470 --> 00:32:21,950 If you take the second one, so I equals the two, what you will get, you will get h two one four. 479 00:32:21,980 --> 00:32:23,270 I equals to three. 480 00:32:23,270 --> 00:32:26,150 You will get h three, two and four. 481 00:32:26,150 --> 00:32:27,380 Pi equals four. 482 00:32:27,380 --> 00:32:29,890 You will get h for three. 483 00:32:29,900 --> 00:32:35,520 As you can see, you have the relation between each frame. 484 00:32:35,540 --> 00:32:36,050 OK. 485 00:32:36,350 --> 00:32:38,010 Between each consecutive frame. 486 00:32:38,030 --> 00:32:38,450 OK. 487 00:32:38,600 --> 00:32:46,970 For example, no, you want to know what will be the position of and affect the position and orientation 488 00:32:46,970 --> 00:32:55,760 of and the vector with respect to the vote, with respect to that base frame so that you want to frame 489 00:32:55,760 --> 00:33:02,230 the you run to find four zero, what each four zero will give for is our and the vector frame of is 490 00:33:02,240 --> 00:33:03,590 zero is our base frame. 491 00:33:03,760 --> 00:33:06,480 This is, as you can see, four is in the vector frame. 492 00:33:06,800 --> 00:33:08,540 And this is our base frame. 493 00:33:08,690 --> 00:33:12,590 You want to know what is the what is this distance between them? 494 00:33:12,620 --> 00:33:14,260 Yeah, this is distance. 495 00:33:14,270 --> 00:33:14,870 What is this? 496 00:33:15,080 --> 00:33:16,250 For example, let's see. 497 00:33:16,250 --> 00:33:18,050 This is the s. 498 00:33:18,080 --> 00:33:19,880 We are interested for this one. 499 00:33:20,120 --> 00:33:25,600 And what is the OK orientation difference between these two? 500 00:33:25,610 --> 00:33:30,320 OK, so and what we want to know are four is zero. 501 00:33:30,380 --> 00:33:30,830 Yes. 502 00:33:30,980 --> 00:33:33,260 What is the orientation difference between? 503 00:33:33,380 --> 00:33:38,000 What is the orientation of the vector frame with respect to the base frame, we want to know of these 504 00:33:38,000 --> 00:33:40,670 two things how we can calculate it. 505 00:33:40,680 --> 00:33:48,200 Surely it's very easily, uh, we will do it by just multiplying what homogeneous transformation methods 506 00:33:48,290 --> 00:33:57,230 in each other h one zero with each h two one with h as well and with age four three. 507 00:33:57,530 --> 00:34:00,500 And this will give you age for zero. 508 00:34:00,800 --> 00:34:01,160 OK. 509 00:34:01,790 --> 00:34:06,620 The end effector frame with respect to their base frame. 510 00:34:06,860 --> 00:34:07,310 OK. 511 00:34:07,490 --> 00:34:09,800 And then it will have in this. 512 00:34:10,100 --> 00:34:14,090 It will have this kind of shape. 513 00:34:14,120 --> 00:34:16,760 This will be R four to zero. 514 00:34:17,030 --> 00:34:17,540 Yes. 515 00:34:17,720 --> 00:34:22,250 Zero and D s and one. 516 00:34:22,280 --> 00:34:22,820 OK. 517 00:34:23,090 --> 00:34:24,900 This is three by one vector. 518 00:34:24,920 --> 00:34:25,290 OK. 519 00:34:25,310 --> 00:34:30,890 This is three by one vector, which is x y z saw. 520 00:34:31,340 --> 00:34:42,020 So it will show this OK in how much the distance between this origin and this origin occurs or and the 521 00:34:42,020 --> 00:34:47,630 vector and the base frame in x y z axis is yes, that's perfect. 522 00:34:47,900 --> 00:34:52,160 And this will show this is four by four matrix. 523 00:34:52,310 --> 00:34:53,660 Excuse me this for my foot. 524 00:34:53,960 --> 00:34:58,730 Well, three by three matrix, which shows us the orientation. 525 00:34:58,850 --> 00:35:03,620 Of falls in the sector of rain with respect to their base frame. 526 00:35:03,860 --> 00:35:04,730 That's perfect. 527 00:35:04,970 --> 00:35:11,720 So but what we have said, we have also said that we can find that any fraying the relation between 528 00:35:11,720 --> 00:35:17,450 any frames so we can find what is, for example, the joint tools. 529 00:35:18,060 --> 00:35:21,980 Yeah, for example, these joint tools orientation. 530 00:35:21,980 --> 00:35:28,700 And but this does with respect to a base frame, you can find it, you can find it easily. 531 00:35:28,730 --> 00:35:35,840 So what we want to find, we want to fight age to zero, how we can find by multiplying just age one 532 00:35:35,840 --> 00:35:37,490 zero with age two one. 533 00:35:37,700 --> 00:35:45,440 As you can see, you can find the relation between any two frames, any two frames, and that is the 534 00:35:45,440 --> 00:35:48,890 best the perfect site of forward kinematics. 535 00:35:49,160 --> 00:35:56,360 And you can complete it using the age correlation D.H method very easily. 536 00:35:56,390 --> 00:36:04,970 I hope that you have understand we will do one more example in order to solidify the concept to make 537 00:36:04,970 --> 00:36:06,710 it more robust for you. 538 00:36:07,850 --> 00:36:10,160 So let's solve. 539 00:36:10,220 --> 00:36:12,320 One more example also.