1 00:00:00,420 --> 00:00:06,750 During lesser lessons, we have learned forward kinematics where we try to establish a relation between 2 00:00:06,750 --> 00:00:10,470 joint variables and end effector position and orientation. 3 00:00:10,860 --> 00:00:16,140 Namely, how and the factors, position and orientation changes with respect to base frame. 4 00:00:16,320 --> 00:00:23,190 When we change variables, this is very useful tool, but we also want to know how the and the factors 5 00:00:23,190 --> 00:00:28,380 linear and rotational velocity changes when the joint velocities are changed. 6 00:00:28,740 --> 00:00:36,000 This knowledge is very useful in many robotics applications, including control of robot manipulators 7 00:00:36,000 --> 00:00:42,870 and also the applications like painting robots, manipulators, manipulators that cooperate with humans 8 00:00:43,080 --> 00:00:46,380 or manipulators that handle some objects and so on. 9 00:00:46,620 --> 00:00:53,340 For example, you would not like the robot and the factor move very fast while handling some fragile 10 00:00:53,340 --> 00:01:00,360 object or while cooperating with human because it can hurt person when it accidentally crash with him. 11 00:01:00,750 --> 00:01:07,160 So, all in all, in this lesson, we will try to establish a relation between joint velocities and 12 00:01:07,160 --> 00:01:08,700 end effector velocity. 13 00:01:10,110 --> 00:01:16,980 As you know, the purpose of forward kinematics that have seen that we have seen less time was to establish 14 00:01:16,980 --> 00:01:21,600 a relation between joint variables and affect their position and orientation. 15 00:01:21,930 --> 00:01:26,610 We can denote forward kinematics as key, which is a function of joint variables. 16 00:01:26,940 --> 00:01:32,100 So what about the relation between the and the vectors velocity and joint velocities? 17 00:01:32,370 --> 00:01:34,500 How can we establish such a relation? 18 00:01:34,920 --> 00:01:45,360 It has to have such a structure, namely it relates joint velocities Q1 to Q2 Dot up to Q and Dot with 19 00:01:45,360 --> 00:01:55,440 end effector velocities both linear x y z tot and angular omega x, omega y and omega. 20 00:01:55,460 --> 00:02:04,230 That here too are joint variables and they are angled tita if John is rolled or distance t if John is 21 00:02:04,230 --> 00:02:11,340 prismatic and is the number of joints, the matrix that we are interested is called Jacobean Matrix. 22 00:02:11,610 --> 00:02:16,830 This is the solution of our problem, so let's try to investigate and understand it. 23 00:02:19,140 --> 00:02:22,950 First of all, let's determine the dimension of in matrix. 24 00:02:23,280 --> 00:02:32,970 The size of the Jacobean vector or the size of the vector on the left side is one by six by one, while 25 00:02:33,270 --> 00:02:38,190 on the right side is n by one, which is the number of joints. 26 00:02:38,700 --> 00:02:46,860 So from linear algebra, we know that your carbon matrix has to have a dimension of six by end, you 27 00:02:46,860 --> 00:02:49,980 know, to have valid matrix multiplication. 28 00:02:50,610 --> 00:02:54,450 Now let's try to find how we construct Jacobean matrix. 29 00:02:55,140 --> 00:02:58,770 This table summarizes how we construct Jacobean matrix. 30 00:02:59,100 --> 00:03:06,060 As you can see from the table, the way we get linear and angular velocity Jacobins for real and prismatic 31 00:03:06,060 --> 00:03:12,000 joints is different, and this is because of the difference in the kinematics of Prismatic Andrew, 32 00:03:12,000 --> 00:03:12,630 all the joints. 33 00:03:13,200 --> 00:03:20,430 So let's go to the club and first try to understand what Jacobean means, and then we will construct 34 00:03:20,430 --> 00:03:25,770 Jacobean of our robot manipulator that we have worked on last several lessons. 35 00:03:26,820 --> 00:03:32,970 OK, let's try to see the functionality of Jacobean Matrix. 36 00:03:33,540 --> 00:03:39,180 So I will use a robot model from Petr Caucus' Library Robotics Toolbox. 37 00:03:40,080 --> 00:03:46,890 We will use MDL or Puma five model in order to illustrate Jacobean. 38 00:03:47,310 --> 00:03:51,870 So first, we clear the command window the variables and close all the figures. 39 00:03:52,470 --> 00:03:55,950 Then we define some symbolic variables. 40 00:03:56,040 --> 00:04:02,640 These are six joint variables, but their derivative. 41 00:04:02,700 --> 00:04:05,100 OK, so derivative of joint variables. 42 00:04:05,370 --> 00:04:07,110 Because what's our purpose? 43 00:04:07,110 --> 00:04:18,120 Our purpose is to relate joint derivative of joint variables or velocity of joins with velocity of end 44 00:04:18,120 --> 00:04:18,820 effector. 45 00:04:18,840 --> 00:04:19,320 OK. 46 00:04:19,350 --> 00:04:24,420 As you can see exactly why this is translational velocity of end effector or linear. 47 00:04:24,830 --> 00:04:27,780 And this is a rotational omega x omega y omega site. 48 00:04:27,840 --> 00:04:29,520 Excuse me or angular velocity. 49 00:04:30,450 --> 00:04:34,080 OK, angular velocity of our in the that. 50 00:04:34,120 --> 00:04:43,470 So the import MDL Puma 560, so Puma five six, the manipulator model. 51 00:04:43,800 --> 00:04:48,690 Then we find it's Jacobean in Q and OK, what's q n? 52 00:04:48,690 --> 00:04:54,690 Let me show you let me run this one again off line. 53 00:04:57,420 --> 00:04:58,110 OK. 54 00:04:58,350 --> 00:05:07,140 It's one in the queue, and as you can see, Queue MN is one by six victor. 55 00:05:07,410 --> 00:05:11,320 This is joint variables in Rajan's. 56 00:05:11,340 --> 00:05:11,790 OK. 57 00:05:12,090 --> 00:05:13,710 This is the first joy. 58 00:05:13,740 --> 00:05:15,210 Q1 is zero radians. 59 00:05:15,270 --> 00:05:20,310 This is zero point seven eight five four radians and so on and so forth. 60 00:05:20,620 --> 00:05:26,760 OK, so at this position, we will calculate at this instant. 61 00:05:26,760 --> 00:05:35,280 We will calculated Giacobbe in our remote manipulator and we will get the um, our um, the relation 62 00:05:35,280 --> 00:05:41,730 between our, uh, derivative or joint variables and between the derivative of joint variables. 63 00:05:41,730 --> 00:05:46,920 And um, what and the factor velocities. 64 00:05:46,950 --> 00:05:49,890 OK, here a result in result variable. 65 00:05:50,190 --> 00:05:55,260 I, uh, save my and the factors velocity. 66 00:05:55,260 --> 00:05:59,310 So namely result is nothing but this vector, OK? 67 00:05:59,510 --> 00:06:07,020 X divided city, which contains linear velocity omega x omega omega z, which contains angular velocity. 68 00:06:07,830 --> 00:06:11,020 OK, let's and here what I am doing. 69 00:06:11,040 --> 00:06:12,120 Let me ride help. 70 00:06:12,120 --> 00:06:21,060 VPA AVP is a MATLAB command, namely its variable precision arithmetic it in order element just without. 71 00:06:21,090 --> 00:06:24,960 I cannot explain you in this way, but I can show you in this way. 72 00:06:25,200 --> 00:06:26,100 Let's see result. 73 00:06:26,280 --> 00:06:35,280 As you can see, this is very ugly, you know, ugly representation of our and the vector velocity y 74 00:06:35,790 --> 00:06:39,030 because our we have used the symbolic variables. 75 00:06:40,050 --> 00:06:47,190 The, you know, arithmetic operations on variables are done in arithmetic ways, so they are not bound 76 00:06:47,190 --> 00:06:48,000 completely. 77 00:06:48,210 --> 00:06:57,510 That's why we are using a common and we are defining resolution or the digit until two point decimal 78 00:06:57,510 --> 00:06:57,910 digit. 79 00:06:57,930 --> 00:06:58,350 OK. 80 00:06:59,040 --> 00:06:59,610 Precision. 81 00:07:00,780 --> 00:07:02,520 Let's run this way. 82 00:07:02,580 --> 00:07:03,030 OK. 83 00:07:03,210 --> 00:07:08,610 As you can see, now we have really clear and the vector velocity. 84 00:07:09,060 --> 00:07:11,220 So what does it mean here? 85 00:07:11,490 --> 00:07:14,520 As you can see now, you can see this is excellent. 86 00:07:14,940 --> 00:07:16,890 This is why dot this is. 87 00:07:16,890 --> 00:07:17,880 Is it not OK? 88 00:07:18,060 --> 00:07:27,810 These are linear velocity of and the vector and this is omega X. Omega Y and omega zit. 89 00:07:28,050 --> 00:07:37,680 Namely, these last three rows are angular velocity of of a robot manipulator. 90 00:07:37,860 --> 00:07:39,240 So what does this mean? 91 00:07:39,600 --> 00:07:40,080 Let's. 92 00:07:40,710 --> 00:07:44,190 Let me show you in this way, let's use this command. 93 00:07:44,550 --> 00:07:47,430 Um, Puma five six did teach. 94 00:07:47,910 --> 00:07:56,550 So this is nothing but as you can see before, as you have seen before, and they are just changing. 95 00:07:56,550 --> 00:08:02,940 You know that we can change in the video, the joint variables and see what its effect. 96 00:08:02,970 --> 00:08:03,390 OK. 97 00:08:03,690 --> 00:08:04,620 Let's do that. 98 00:08:04,630 --> 00:08:06,330 Let me just make it in this way. 99 00:08:06,360 --> 00:08:08,270 Let me zoom. 100 00:08:08,610 --> 00:08:09,480 OK. 101 00:08:09,720 --> 00:08:11,190 Here it is. 102 00:08:11,830 --> 00:08:15,360 Oh oh my gosh, it's not good. 103 00:08:15,960 --> 00:08:16,820 I'm just doing this. 104 00:08:16,980 --> 00:08:17,370 OK. 105 00:08:17,700 --> 00:08:19,120 No limits. 106 00:08:19,170 --> 00:08:20,650 Try to understand. 107 00:08:20,670 --> 00:08:23,280 Let me just the always on top. 108 00:08:23,280 --> 00:08:26,400 And let's try to understand what does this mean? 109 00:08:26,640 --> 00:08:27,490 What does this mean? 110 00:08:27,510 --> 00:08:27,850 OK? 111 00:08:28,080 --> 00:08:30,960 As you can see, these are Q1 d. 112 00:08:31,140 --> 00:08:32,640 Let's start with X. 113 00:08:33,180 --> 00:08:34,410 What does this mean? 114 00:08:34,620 --> 00:08:41,910 This means the linear velocity of our robot manipulators end effector in X direction, OK? 115 00:08:42,180 --> 00:08:43,410 And what? 116 00:08:43,410 --> 00:08:50,370 It is affected by only two one dot, two two dot and Q3 dot. 117 00:08:50,370 --> 00:08:51,210 It's interesting. 118 00:08:51,240 --> 00:08:53,310 Let's see if this correct or not. 119 00:08:53,790 --> 00:09:03,550 As you can see, when we change our Q1, really, we change our x dot or linear velocity of our. 120 00:09:03,780 --> 00:09:04,860 And the vector. 121 00:09:05,100 --> 00:09:07,650 What in X? 122 00:09:07,890 --> 00:09:08,460 OK? 123 00:09:09,990 --> 00:09:12,930 Not only in X, but also in other directions. 124 00:09:12,930 --> 00:09:18,660 But we will see how it affects the linear velocity of the vector in x direction. 125 00:09:18,930 --> 00:09:22,350 Let's check for cued to also. 126 00:09:22,590 --> 00:09:23,850 Let me just make it in this way. 127 00:09:23,880 --> 00:09:26,310 OK, let's check Q2, whether it fits or not. 128 00:09:26,490 --> 00:09:29,040 As you can see, it also affects the end effector. 129 00:09:29,220 --> 00:09:29,580 Yeah. 130 00:09:29,610 --> 00:09:34,320 Let me remind you that the X Y Z, this is our end effector. 131 00:09:34,320 --> 00:09:37,390 As you can see, this frame is our and the vector here. 132 00:09:37,770 --> 00:09:39,570 You can see this are in the vector frame. 133 00:09:39,840 --> 00:09:40,320 OK? 134 00:09:40,530 --> 00:09:46,380 As you can see, we can affect its velocity linear velocity in extortion by changing. 135 00:09:47,110 --> 00:09:53,480 Dot or the above by changing the velocity of um. 136 00:09:55,200 --> 00:09:56,070 By changing the. 137 00:09:56,140 --> 00:10:02,770 Joint that I bought you two and it's, uh, velocity affects directly to our end effector. 138 00:10:03,220 --> 00:10:10,090 For example, if we move it slowly, it changes, and the vector linear velocity in its storage is slowly 139 00:10:10,210 --> 00:10:10,870 but fast. 140 00:10:10,870 --> 00:10:11,830 It will change fast. 141 00:10:11,860 --> 00:10:15,160 OK, perfect. 142 00:10:15,370 --> 00:10:16,000 Let's try. 143 00:10:16,390 --> 00:10:26,260 Cue three, because from Jacobean, it shows us that, uh, Q3 also affects the robots and the vectors 144 00:10:26,260 --> 00:10:28,800 linear velocity in its direction. 145 00:10:28,810 --> 00:10:33,130 And if you change Q3, you will see that yes, it's also affect. 146 00:10:33,620 --> 00:10:41,050 However, we cannot see Q4, KQ 40 to 50 and 60 inside our export. 147 00:10:41,320 --> 00:10:42,280 What does this mean? 148 00:10:42,310 --> 00:10:52,000 This means that to 4.2 to five and to 6.0 does not effect the linear velocity of end the factor of frame 149 00:10:52,690 --> 00:10:54,670 in X direction. 150 00:10:55,540 --> 00:10:56,870 Is it correct or not? 151 00:10:56,890 --> 00:10:57,820 Let's see. 152 00:10:58,240 --> 00:11:01,640 Okay, let's try to change now. 153 00:11:02,200 --> 00:11:11,770 Q4 and Q3, Q4 Q4, I think, is, as you can see, Q4 not changes the linear velocity in X direction, 154 00:11:11,770 --> 00:11:14,490 but angular velocity in extortion. 155 00:11:14,500 --> 00:11:18,130 You can you see it only rotates, so it only changes. 156 00:11:18,130 --> 00:11:26,320 Angle did not linear realize that the same is for Q4 because Q4 Q5 q6 constructs. 157 00:11:28,030 --> 00:11:33,520 You know, the only reason, OK, it's very good would restore our robots. 158 00:11:33,520 --> 00:11:40,090 So what is the purpose of spherical wrist as physical risk purpose is only to change the orientation 159 00:11:40,090 --> 00:11:43,390 of the end effector, not linearly change? 160 00:11:43,390 --> 00:11:45,370 It's not translational, OK? 161 00:11:45,850 --> 00:11:48,670 It only changes orientation of the vector. 162 00:11:48,670 --> 00:11:59,890 So that's why they are not inside X thought you will see that they are not also inside Q of Y Dot and 163 00:12:00,040 --> 00:12:00,790 Z Todd. 164 00:12:00,940 --> 00:12:01,450 OK? 165 00:12:01,660 --> 00:12:03,010 It's very interesting. 166 00:12:03,010 --> 00:12:05,020 But however it is initiative. 167 00:12:05,020 --> 00:12:08,950 OK, this is not something -- or nothing. 168 00:12:09,340 --> 00:12:10,690 Not understandable. 169 00:12:10,900 --> 00:12:11,320 OK. 170 00:12:13,180 --> 00:12:20,110 As you can see, they are X and Y Z told X Y and Z Tot are the same. 171 00:12:20,200 --> 00:12:26,500 They are all affected by Q1 de Q2 and Q3, the Q1 de Q2 Q3. 172 00:12:27,080 --> 00:12:33,130 However, it's interesting that in that direction, Q1 doesn't affect. 173 00:12:33,310 --> 00:12:36,400 OK, let's check why. 174 00:12:36,700 --> 00:12:40,200 As you can see, it only rotates x y plane. 175 00:12:40,660 --> 00:12:43,360 Can you see that it all rotates the end? 176 00:12:43,360 --> 00:12:49,660 The vector, excuse me, changes the position and orientation of the vector in x y plane. 177 00:12:49,840 --> 00:12:52,570 See, it only changes in x y plane. 178 00:12:52,960 --> 00:12:56,710 Be careful in the vector is frame is x y z. 179 00:12:56,950 --> 00:13:00,160 It doesn't change up and down, like in Q2. 180 00:13:00,400 --> 00:13:03,040 OK, can you see here it changes up and down. 181 00:13:03,040 --> 00:13:05,530 Also in Q3, it changes up and down. 182 00:13:05,770 --> 00:13:09,310 So that's why it they also they affect that direction. 183 00:13:09,310 --> 00:13:15,100 However true one does doesn't affect in that direction velocity because it only changes the position 184 00:13:15,100 --> 00:13:20,110 and orientation of our onboard manipulator in x y plane. 185 00:13:20,440 --> 00:13:21,010 Perfect. 186 00:13:21,340 --> 00:13:26,920 Let's talk about now the angular velocity, as you can see in terms of angular velocity. 187 00:13:27,310 --> 00:13:30,780 True for Q2, q3, Q1. 188 00:13:30,820 --> 00:13:33,460 Q five to six almost. 189 00:13:33,760 --> 00:13:38,170 Let me just first, second, third, fourth, fifth, sixth. 190 00:13:38,440 --> 00:13:41,710 All the joint variables o. 191 00:13:41,710 --> 00:13:51,850 The joint velocities affect the velocity of end effector in angular velocity of end effector OK in x 192 00:13:51,850 --> 00:13:52,540 direction. 193 00:13:52,960 --> 00:14:00,640 The same is, as you can see, is applicable or in the same is valid for ohmygod wide angle velocity 194 00:14:00,640 --> 00:14:01,510 in y direction. 195 00:14:01,660 --> 00:14:08,680 And by manipulating, you can see that because Q1, Q2, Q3 not only changes angular velocity, but 196 00:14:08,680 --> 00:14:09,720 also linear velocity. 197 00:14:09,730 --> 00:14:19,450 How our spherical wrist changes only orientations saw on angular velocity and in terms of z direction. 198 00:14:19,630 --> 00:14:24,890 Q1, Q2, Q3, Q4, Q5 Q6. 199 00:14:24,910 --> 00:14:25,390 OK. 200 00:14:26,320 --> 00:14:29,970 However, let me just remind you something. 201 00:14:30,580 --> 00:14:35,110 As you can see, we have here 1.2 e minus 32. 202 00:14:35,120 --> 00:14:36,460 It is almost zero. 203 00:14:36,790 --> 00:14:38,910 So effect of Q2. 204 00:14:40,240 --> 00:14:47,170 Q2 doesn't have effect on angular velocity in X direction. 205 00:14:47,290 --> 00:14:55,900 Let's see why, because as you can see, Q2 only changes the frame orientation not. 206 00:14:56,010 --> 00:14:57,690 In this direction, OK? 207 00:14:57,990 --> 00:15:01,200 So that's why it doesn't have effect here. 208 00:15:01,380 --> 00:15:02,360 Be careful about that. 209 00:15:02,370 --> 00:15:04,310 These are small numbers, OK? 210 00:15:04,440 --> 00:15:08,760 Nearly zero e minus 32 e or minus 32. 211 00:15:09,390 --> 00:15:16,110 This does mean 1.2 times 10 or minus 32, which is almost zero. 212 00:15:16,330 --> 00:15:18,840 But this is because of the precision of these numbers. 213 00:15:18,840 --> 00:15:19,980 Precision of MATLAB. 214 00:15:20,010 --> 00:15:20,400 OK. 215 00:15:20,940 --> 00:15:23,340 The same is for the Q three. 216 00:15:23,340 --> 00:15:28,260 Q three also doesn't affect the linear angular velocity in X direction. 217 00:15:28,290 --> 00:15:28,860 OK. 218 00:15:29,250 --> 00:15:33,570 The same also applicable for Q one, d and Q 50. 219 00:15:33,930 --> 00:15:44,610 However, Q for D and Q six, the affix the omega or angular velocity of and the vector in X are it's 220 00:15:44,610 --> 00:15:46,950 not about x axis the same. 221 00:15:47,550 --> 00:15:51,360 You can analyze the Y and you can analyze the Z. 222 00:15:51,360 --> 00:16:03,450 So by manipulating or by playing with this tool, you can easily understand what does Jacobean means 223 00:16:03,450 --> 00:16:04,700 and what it will. 224 00:16:04,800 --> 00:16:06,750 What is its purpose? 225 00:16:08,460 --> 00:16:14,970 So now let's try to find let's try to find what will be. 226 00:16:16,980 --> 00:16:24,750 Let's try to find the Jacobean of our robot manipulator that we have worked on last several days. 227 00:16:25,050 --> 00:16:25,580 OK. 228 00:16:26,930 --> 00:16:32,850 If you, as you know, our robot manipulator was six degree of freedom. 229 00:16:32,850 --> 00:16:35,650 We have, uh, six joints. 230 00:16:35,690 --> 00:16:48,570 OK, so our Jacobean matrix will be very big and very complicated because the um so the end effector 231 00:16:48,570 --> 00:16:50,520 will be in a complicated way. 232 00:16:50,790 --> 00:16:56,970 It will depend on our joint velocities in a complicated way. 233 00:16:57,000 --> 00:16:57,360 OK. 234 00:16:57,540 --> 00:17:05,400 So that's why it would be very difficult to calculate its effect and also the Jacobean by hand. 235 00:17:05,880 --> 00:17:11,790 So that's why I wrote a program, OK, or, um, a script. 236 00:17:12,360 --> 00:17:16,710 Uh, yeah, this is because script script to calculate the Jacobean. 237 00:17:16,890 --> 00:17:22,170 So let me just show how I did that first. 238 00:17:22,800 --> 00:17:28,500 As you can see, if you clear the command, OK, I don't need to talk about these clear deliverables 239 00:17:28,500 --> 00:17:29,850 and close all the figures. 240 00:17:30,240 --> 00:17:32,470 Then we define some variables. 241 00:17:32,490 --> 00:17:33,450 What are these? 242 00:17:33,480 --> 00:17:34,800 This is number of joints. 243 00:17:34,800 --> 00:17:37,860 We have six joints because our robot six degrees of freedom. 244 00:17:38,370 --> 00:17:40,170 These are some symbolic variables. 245 00:17:40,170 --> 00:17:45,230 We will need this T to want it off for three to five to six. 246 00:17:45,240 --> 00:17:45,750 OK? 247 00:17:45,960 --> 00:17:50,760 These are for real joint variables and D to these three. 248 00:17:51,090 --> 00:17:58,650 These are two prismatic joint variables because as you remember, our robot have fought very well a 249 00:17:58,650 --> 00:18:00,810 joint to prismatic joints. 250 00:18:00,910 --> 00:18:03,210 OK, let's continue. 251 00:18:03,510 --> 00:18:04,260 What is that? 252 00:18:04,290 --> 00:18:11,190 This is our homogeneous transformation matrix because in order to find the let me show you in order 253 00:18:11,190 --> 00:18:15,840 to find this, are I minus one zero? 254 00:18:15,870 --> 00:18:23,720 As you can see, these rotation mattresses are relative relate the frame width base frame. 255 00:18:23,730 --> 00:18:33,120 OK, so we will need homogeneous transformation mattresses in order to find R and D with respect to 256 00:18:33,120 --> 00:18:34,050 base frames. 257 00:18:34,050 --> 00:18:38,750 So this this is in order to find that homogeneous transformation mattresses. 258 00:18:38,830 --> 00:18:39,330 OK. 259 00:18:40,110 --> 00:18:47,850 This is for in order to save the rotation mattresses these are for in order to save the translational 260 00:18:47,850 --> 00:18:50,280 mattresses and or the vector. 261 00:18:50,580 --> 00:18:53,070 And this is for our Jacobean here. 262 00:18:53,070 --> 00:18:56,310 Are use multidimensional? 263 00:18:56,340 --> 00:18:56,800 OK. 264 00:18:57,270 --> 00:19:01,230 Um, multidimensional matrix. 265 00:19:01,230 --> 00:19:07,380 Or let me just it's not a correctly three dimensional vector matrix. 266 00:19:07,380 --> 00:19:16,800 Excuse me, because we have six degree of freedom for each of the joints I. 267 00:19:18,660 --> 00:19:23,710 For each of the joint I made in the video, h r the and J. 268 00:19:25,230 --> 00:19:27,540 Um mattresses. 269 00:19:28,050 --> 00:19:30,750 Now these are link lengths of our robot. 270 00:19:30,750 --> 00:19:33,960 You have seen this last time I have done this. 271 00:19:33,960 --> 00:19:39,060 Uh, you know, they created this randomly, you know, not randomly. 272 00:19:39,060 --> 00:19:45,480 But you know, there is nothing mechanical calculation or some thinking on them. 273 00:19:45,810 --> 00:19:49,350 So I have just written some numbers for them. 274 00:19:49,380 --> 00:19:49,770 OK. 275 00:19:51,420 --> 00:19:55,770 And also, we don't need for now to make them correct way or. 276 00:19:55,870 --> 00:20:01,510 To calculate them mechanically, OK, in a construction will be because this is not our purpose. 277 00:20:01,720 --> 00:20:08,320 And the here are just right, our type of each joint robot joint, OK? 278 00:20:08,590 --> 00:20:11,890 First remote, the second prismatic, the sort prismatic. 279 00:20:12,100 --> 00:20:15,140 Our fourth joint is rotational. 280 00:20:15,160 --> 00:20:22,840 Also the fifth and sixth, because we will need that, we will need that in order to calculate Jacobean 281 00:20:22,840 --> 00:20:26,530 for linear and rotational, the Jacoby's for prismatic. 282 00:20:26,740 --> 00:20:28,350 They will have joints because they are different. 283 00:20:28,360 --> 00:20:34,700 And when we calculate this linear and rotational jacquard means we need whether we are working with 284 00:20:34,710 --> 00:20:36,670 the prismatic joint or able to join. 285 00:20:36,670 --> 00:20:38,950 So that's why I specified them here. 286 00:20:39,790 --> 00:20:40,270 OK. 287 00:20:40,570 --> 00:20:43,350 Let's continue with defined parameters. 288 00:20:43,360 --> 00:20:48,460 These are nothing we have done last time we calculated the parameters are just right. 289 00:20:48,460 --> 00:20:52,260 I mean, T, R, I, R and D here what I'm doing. 290 00:20:52,630 --> 00:20:53,980 Yeah, I'm doing. 291 00:20:53,980 --> 00:20:59,620 I'm calculating individual homogeneous transformation mattresses, OK? 292 00:20:59,860 --> 00:21:05,650 As you can see, I'm doing nothing but for I'm looping my for loop. 293 00:21:05,650 --> 00:21:11,950 I'm looping for six times for each of the joints and calculate the homogeneous transformation matrix 294 00:21:11,950 --> 00:21:13,250 for each of the joints. 295 00:21:13,270 --> 00:21:15,100 There is nothing complicated here. 296 00:21:15,760 --> 00:21:16,990 You know the formula? 297 00:21:16,990 --> 00:21:18,430 You can do the same. 298 00:21:19,030 --> 00:21:22,330 OK, then here what I'm doing. 299 00:21:23,560 --> 00:21:25,720 OK, let's hear. 300 00:21:25,930 --> 00:21:31,930 Just extract rotational and translational transformation mattresses from homogeneous transformation 301 00:21:31,930 --> 00:21:32,700 matrix. 302 00:21:32,710 --> 00:21:41,620 OK, I'm extracting these words OK, or I'm minus one zero r i minus one zero d and zero d. 303 00:21:41,620 --> 00:21:42,850 I'm minus one zero. 304 00:21:43,000 --> 00:21:49,640 Excuse me, not d in zero, just I minus one zero I minus one zero and minus one zero. 305 00:21:49,960 --> 00:21:56,710 So it's only in, you know, here I just calculate. 306 00:21:56,770 --> 00:22:03,040 Let me just show you when they calculate this homogeneous transformation method, as you know, this 307 00:22:03,040 --> 00:22:13,780 is let me just remind you that this rotation matrix illustrates the H i h II, which is nothing but 308 00:22:14,500 --> 00:22:21,710 a homogeneous transformation mattress from frame I to frame I mind this one. 309 00:22:21,730 --> 00:22:24,910 OK, but we don't want I to a minus one. 310 00:22:25,090 --> 00:22:29,770 What we want is from AI to zero I two zero. 311 00:22:30,340 --> 00:22:33,040 We want AI to zero b. 312 00:22:33,040 --> 00:22:39,310 We need homogeneous transformation matrix with respect to Frame B's frame. 313 00:22:39,520 --> 00:22:41,380 Not the last frame. 314 00:22:41,390 --> 00:22:46,270 OK, not the excuse me, not last frame, but not the frame one that is before. 315 00:22:47,440 --> 00:22:47,980 OK. 316 00:22:48,610 --> 00:22:49,210 As here. 317 00:22:49,420 --> 00:22:52,120 So what we are doing there is nothing complicated here. 318 00:22:52,420 --> 00:22:55,180 We just, for example, what we try to find. 319 00:22:55,210 --> 00:22:58,030 We want to find h one zero. 320 00:22:58,060 --> 00:22:58,450 OK. 321 00:22:58,630 --> 00:23:01,430 So a homogeneous transformation from one to zero. 322 00:23:01,450 --> 00:23:05,350 What we are doing, we are doing just h. 323 00:23:05,530 --> 00:23:09,160 We are h one zero will be h one zero. 324 00:23:09,170 --> 00:23:11,560 There is nothing, uh, complicated here. 325 00:23:11,800 --> 00:23:14,320 OK, so we are doing nothing with that. 326 00:23:14,320 --> 00:23:19,010 But H two zero will be different. 327 00:23:19,030 --> 00:23:26,920 OK, how we'll calculated the evil multiply h one zero with h. 328 00:23:26,920 --> 00:23:29,630 What two one. 329 00:23:29,890 --> 00:23:37,380 We have these mattresses homogeneous transformation mattresses for subsequent frames. 330 00:23:37,390 --> 00:23:38,720 Here we calculated them. 331 00:23:38,740 --> 00:23:45,580 Now we want to calculate by multiplying them each of the joints homogeneous transformation matrix with 332 00:23:45,580 --> 00:23:46,810 respect to base frames. 333 00:23:46,810 --> 00:23:50,980 So let's do four page for each three zero. 334 00:23:51,720 --> 00:23:52,870 Oh OK. 335 00:23:54,130 --> 00:23:55,630 I think you grab the concept. 336 00:23:56,140 --> 00:24:04,120 Well, because we have seen this last time, what we will do, we will move h one zero with H two one 337 00:24:04,120 --> 00:24:09,230 and multiply it with H three two. 338 00:24:09,530 --> 00:24:10,000 OK. 339 00:24:10,360 --> 00:24:13,150 What will, uh, what it will give to us? 340 00:24:13,150 --> 00:24:18,280 It will give us h three zero and we will. 341 00:24:18,310 --> 00:24:25,350 By this way, we will find H four zero eight five zero eight six zero and from here, Leesville T, 342 00:24:25,380 --> 00:24:27,340 as you can see, we are doing it here. 343 00:24:27,790 --> 00:24:28,750 What we are doing. 344 00:24:28,960 --> 00:24:34,330 We are for each and we are looping inside for loop. 345 00:24:35,410 --> 00:24:36,640 This is EJ cumulated. 346 00:24:37,090 --> 00:24:43,330 Namely, I just, uh, calculated for each of the homogeneous transformation matrix for each of the 347 00:24:43,330 --> 00:24:45,550 joint these homogeneous transformation mattresses. 348 00:24:45,550 --> 00:24:48,580 These are cumulative homogeneous transformation mattresses. 349 00:24:49,570 --> 00:24:53,260 I first start with, uh, one identity matrix. 350 00:24:53,500 --> 00:24:55,700 OK, then I multiply it. 351 00:24:55,800 --> 00:25:03,330 Each of the subsequent homogeneous transformation mattresses until that joint, OK? 352 00:25:03,480 --> 00:25:10,260 For example, for the second joint, I don't multiply all of them, but the one until age two one four 353 00:25:10,260 --> 00:25:15,520 third joint, I will do until four eight three two. 354 00:25:15,630 --> 00:25:16,170 OK. 355 00:25:16,410 --> 00:25:21,150 So that's why I create another loop loop from one to I. 356 00:25:21,360 --> 00:25:27,000 For example, if I will be two, then I will take first and second homogeneous transformation matrix. 357 00:25:27,000 --> 00:25:29,250 If I will be, I will be three. 358 00:25:29,490 --> 00:25:35,040 Then I will take first, second and third homogeneous transformation mattress and I will multiply them 359 00:25:35,370 --> 00:25:37,350 in four loops. 360 00:25:37,350 --> 00:25:45,090 Or I will get this cumulative homogeneous transformation mattresses and I will extract in each case. 361 00:25:45,090 --> 00:25:46,740 Ah, I zero. 362 00:25:46,770 --> 00:25:47,360 OK. 363 00:25:47,730 --> 00:25:48,870 Ah, zero. 364 00:25:48,900 --> 00:25:50,250 Namely this one. 365 00:25:50,310 --> 00:25:57,490 This is I minus one zero, but I extract are zero in order to calculate ah i minus one zero next time. 366 00:25:57,510 --> 00:25:57,900 OK. 367 00:25:58,620 --> 00:26:01,440 In further loans, we will explain this. 368 00:26:01,740 --> 00:26:07,420 So here let me just comment that this is R zero. 369 00:26:07,440 --> 00:26:07,890 OK. 370 00:26:08,550 --> 00:26:13,470 So the rotation of for each of the joints, the rotation from this joint. 371 00:26:13,830 --> 00:26:19,460 The rotation matrix of this joint or orientation of this joint with respect to B's free market. 372 00:26:19,470 --> 00:26:23,930 And here I calculate the I zero. 373 00:26:23,940 --> 00:26:26,410 So what does this mean? 374 00:26:26,580 --> 00:26:28,080 The center of the frame? 375 00:26:28,110 --> 00:26:28,610 I. 376 00:26:28,980 --> 00:26:29,550 OK. 377 00:26:29,910 --> 00:26:31,590 From a base frame. 378 00:26:32,680 --> 00:26:35,970 No, I have everything to calculate Jacobean. 379 00:26:36,010 --> 00:26:41,100 Now here I just check, first of all, if I equals to one. 380 00:26:41,130 --> 00:26:48,600 If I equals to one from formula, what I will have, if I equals one, I will have ah, zero two zero 381 00:26:49,110 --> 00:26:49,650 or. 382 00:26:49,890 --> 00:26:50,260 OK. 383 00:26:50,280 --> 00:26:51,930 Here are zero two zero. 384 00:26:52,230 --> 00:26:59,340 Also what I will have the zero two zero, however, are zero two zero is nothing but identity matrix. 385 00:26:59,370 --> 00:26:59,670 OK. 386 00:26:59,880 --> 00:27:09,000 Because the now how can I say the orientation of a frame with respect to itself is nothing but identity 387 00:27:09,000 --> 00:27:09,430 matrix. 388 00:27:09,450 --> 00:27:17,370 OK, also, the distance between the centers of the same frame OK, is nothing but zero. 389 00:27:17,390 --> 00:27:20,070 So forty four we will. 390 00:27:20,070 --> 00:27:27,040 If I equals to one, we will make D current as zeros and our current as identity matrix. 391 00:27:27,120 --> 00:27:27,390 OK. 392 00:27:27,570 --> 00:27:35,760 Otherwise, we will just extract and we will take R minus one and d i minus one. 393 00:27:35,880 --> 00:27:43,140 As you can see, in order to find R I minus one and the I minus one in order to use our algorithm, 394 00:27:43,500 --> 00:27:50,070 then what we are doing, we are checking if our joint is revalued or prismatic because the formulas 395 00:27:50,070 --> 00:27:50,790 are different. 396 00:27:51,240 --> 00:28:00,240 If our joint is revealed in first three indices of Jacobean matrix because the first three of Jacobean 397 00:28:00,240 --> 00:28:03,270 Matrix is four linear velocity. 398 00:28:03,540 --> 00:28:05,790 What we are doing, we are just calculating this. 399 00:28:06,060 --> 00:28:10,610 We first multiply our rotation or R I minus one. 400 00:28:10,620 --> 00:28:16,590 As you can see, if our joint is developed and we are calculating linear what we are doing, we are 401 00:28:16,590 --> 00:28:17,130 multiplying. 402 00:28:17,130 --> 00:28:20,250 Are I minus one vs zero zero one? 403 00:28:20,550 --> 00:28:21,150 OK. 404 00:28:21,210 --> 00:28:25,120 We are multiplying our R Iman's one zero zero one. 405 00:28:25,140 --> 00:28:25,770 OK. 406 00:28:25,980 --> 00:28:33,630 And then we find its cross product with what we are finding its cross product with the and zero minus 407 00:28:33,630 --> 00:28:35,370 T i one zero. 408 00:28:35,580 --> 00:28:38,190 As you can see, this is the end zero. 409 00:28:38,220 --> 00:28:38,830 OK. 410 00:28:38,850 --> 00:28:49,680 So the less frames distance from B's frame and v minus we subtract the current from it and we find its 411 00:28:49,680 --> 00:28:50,310 cross product. 412 00:28:50,330 --> 00:28:55,020 So namely, we just applied this excuse me, these formula. 413 00:28:55,200 --> 00:29:02,850 OK, two here, because we are in regular joint and we are calculating linear velocity for rotational 414 00:29:02,850 --> 00:29:09,390 velocity, we will do nothing, but we will multiply R II minus one zero zero zero one and we are doing 415 00:29:09,390 --> 00:29:12,690 here in the last three in the sense of the Jacobean matrix. 416 00:29:12,870 --> 00:29:15,780 We are calculating Angela, the velocity of the end, the vector. 417 00:29:15,990 --> 00:29:20,790 So what we are doing, we are multiplying by R minus one v zero zero one. 418 00:29:21,300 --> 00:29:23,790 Or if our money 419 00:29:27,120 --> 00:29:33,360 excuse me, if our um was a joint, OK, joint type is prismatic. 420 00:29:33,510 --> 00:29:35,510 What we are doing, we are first calculating. 421 00:29:35,520 --> 00:29:37,020 Let's see for this formula. 422 00:29:37,260 --> 00:29:43,530 If our joint is prismatic, we will calculate linear velocity with that form and rotational velocity 423 00:29:43,530 --> 00:29:45,570 by just zero zero zero. 424 00:29:45,570 --> 00:29:51,150 Because this is prismatic tour and it doesn't do any rotation, it is just changing the linear velocity. 425 00:29:51,150 --> 00:29:51,630 OK? 426 00:29:52,980 --> 00:29:55,470 So for the first? 427 00:29:55,770 --> 00:29:56,190 Three. 428 00:29:56,370 --> 00:30:03,960 These are linear velocities, we just are minus we calculate our own minus one times zero zero one vector 429 00:30:03,960 --> 00:30:11,940 and then for the last three ones, what we are doing, we are just doing, we are just assigning zero 430 00:30:11,940 --> 00:30:13,920 zero zero based on our formula. 431 00:30:14,520 --> 00:30:17,710 So here until here, I get our Jacobean. 432 00:30:17,740 --> 00:30:20,520 OK, let's run this. 433 00:30:21,450 --> 00:30:21,780 OK? 434 00:30:21,780 --> 00:30:28,800 It drove successfully, sure, because I wrote it before, and there is no limit to the artistic words. 435 00:30:28,810 --> 00:30:29,430 OK? 436 00:30:29,670 --> 00:30:33,560 And here I just defined some symbolic variables. 437 00:30:33,570 --> 00:30:34,830 What are these symbolic? 438 00:30:34,830 --> 00:30:35,400 What I love? 439 00:30:35,400 --> 00:30:44,790 These symbolic variables are derivative of our joint variables for these fourth one, these four joint 440 00:30:44,790 --> 00:30:49,150 variables to each one, for one, d for D for these 64. 441 00:30:50,070 --> 00:30:52,950 And the two d these three d for PS Matic. 442 00:30:52,950 --> 00:30:56,480 What we are doing here, we are doing essentially this thing. 443 00:30:56,490 --> 00:30:57,030 OK? 444 00:30:57,570 --> 00:31:02,890 There is nothing different or care. 445 00:31:02,910 --> 00:31:09,840 Let's go p from here and let me paste here in order to find in the fact the linear and angular velocities. 446 00:31:10,140 --> 00:31:10,770 OK. 447 00:31:12,450 --> 00:31:19,650 Namely, this one, we take Jacobean and multiply it with our joint variables. 448 00:31:19,710 --> 00:31:21,750 OK, here I calculate J. 449 00:31:21,750 --> 00:31:28,470 Total this is nothing, but because our G is here, J, I define J. 450 00:31:28,470 --> 00:31:30,930 As you can see three dimension matrix. 451 00:31:30,930 --> 00:31:34,320 And this contains the Jacobean for each of the joints. 452 00:31:34,530 --> 00:31:42,030 So I just, you know, some them, not some them, collect them in one matrix and I create one. 453 00:31:42,060 --> 00:31:45,570 Let me just show you one, J Total. 454 00:31:45,600 --> 00:31:48,770 OK, namely one Jacobean. 455 00:31:48,780 --> 00:31:55,110 OK, as you can see, this is our Jacobean, and let's check size of our Jacobean in order to see whether 456 00:31:55,110 --> 00:31:57,570 we are correct way or not. 457 00:31:58,200 --> 00:32:00,030 J Portal As you can see, it's six by six. 458 00:32:00,030 --> 00:32:00,780 It's correct. 459 00:32:01,380 --> 00:32:06,160 It's three four linear velocities to be four and a little velocity and six joints. 460 00:32:06,180 --> 00:32:15,840 OK, we are correct, and then we just multiply them here and get our resultant result and the factor 461 00:32:15,840 --> 00:32:16,230 velocity. 462 00:32:16,230 --> 00:32:26,400 Let's let's run it and you will see how come in a complex way the joint variables are related with and 463 00:32:26,400 --> 00:32:28,170 the factor velocity is OK. 464 00:32:28,410 --> 00:32:32,220 As you can see, here is our result. 465 00:32:32,230 --> 00:32:33,930 It is very, very complex. 466 00:32:33,930 --> 00:32:39,030 I know, but remember that these are nothing but zero. 467 00:32:40,560 --> 00:32:44,730 If you want to analyze, you can analyze, but there's no need to analyze. 468 00:32:45,240 --> 00:32:50,220 This was this is better in order to analyze, you know this Puma is better. 469 00:32:50,740 --> 00:32:58,860 However, here I just wanted to show how we can calculate Jacobean in a systematic way with an algorithm 470 00:32:58,860 --> 00:33:07,020 without doing, you know, some taking paper and a pen and calculate it manually. 471 00:33:07,950 --> 00:33:08,520 OK. 472 00:33:08,550 --> 00:33:16,420 I think that now you know what's Jakobsson and C and its functionality? 473 00:33:16,440 --> 00:33:18,450 So see you on the next lesson.