1 00:00:00,150 --> 00:00:04,800 On the previous lesson, we have seen that the creation of a little luxurious formula. 2 00:00:05,190 --> 00:00:11,190 Well, it's not enough just to see how it is derived because it will not help us to understand what 3 00:00:11,190 --> 00:00:14,010 is the meaning of manipulator dynamical model. 4 00:00:14,700 --> 00:00:21,630 In order to understand that, we need to analyze the terms of a little like range formation and the 5 00:00:21,630 --> 00:00:22,980 internal structure. 6 00:00:23,550 --> 00:00:25,110 In this lesson, we will do that. 7 00:00:25,530 --> 00:00:26,370 So let's go. 8 00:00:27,120 --> 00:00:31,830 This is the formula we have seen that we have derived on the previous lesson. 9 00:00:32,550 --> 00:00:38,460 Be acquainted with this formula because you will see this in most of the next lessons. 10 00:00:39,660 --> 00:00:47,910 This dynamical model of the manipulator is nonlinear function of Q because it contains contains signs 11 00:00:47,910 --> 00:00:48,860 and concerns. 12 00:00:50,220 --> 00:00:54,750 Also, it's nonlinear function of the joint velocity. 13 00:00:54,770 --> 00:00:57,780 Q Don't forget to include this one. 14 00:00:59,550 --> 00:00:59,960 OK. 15 00:01:01,380 --> 00:01:05,790 We have seen this on the previous lesson, but we will see more clearly on. 16 00:01:05,790 --> 00:01:12,270 The next lesson will be will drive L2 Lagrange formula for two degree of freedom robot manipulator. 17 00:01:13,590 --> 00:01:16,500 Also, dynamics of robot manipulator is coupled. 18 00:01:16,950 --> 00:01:22,680 Let me give a small explanation here, but you will see what I mean more clearly in a minute. 19 00:01:23,370 --> 00:01:30,120 As you know, a robot manipulator in general, robots consists of many joints, which connect means 20 00:01:30,960 --> 00:01:39,720 couple means that the motion of one or more joints or the force exerted on one or more joints cause 21 00:01:39,720 --> 00:01:42,420 some changes in other joints also. 22 00:01:43,050 --> 00:01:49,470 So changes happen in one joint or link cause changes in other joints, too. 23 00:01:50,040 --> 00:01:54,960 That's what I mean by coupled, and this makes the dynamics more complicated. 24 00:01:55,740 --> 00:02:01,110 OK, now let's continue by analyzing important terms in robot dynamics equation. 25 00:02:01,740 --> 00:02:06,660 Let's start with inertia matrix, which is denoted by M. 26 00:02:07,230 --> 00:02:17,160 As you can see from Equation 1.0 M is function of Q, so its configuration dependent as the configuration 27 00:02:17,160 --> 00:02:25,380 of the robot manipulator changes, the inertia experienced by joints also changes its square and positive 28 00:02:25,380 --> 00:02:26,400 death with Matrix. 29 00:02:26,670 --> 00:02:32,760 So it has positive definite eigenvalues as it is square and positive definite. 30 00:02:33,360 --> 00:02:38,520 It has always its always inverse invert, but also inertia. 31 00:02:38,520 --> 00:02:42,990 Matrix is symmetric and it has lower and upper bounds. 32 00:02:43,710 --> 00:02:47,340 It means we can find it in which range it changes. 33 00:02:47,910 --> 00:02:56,430 The balance can be indicated by new one and mutual here v multiplied by identité because one and me 34 00:02:56,820 --> 00:02:57,780 are scholars. 35 00:02:58,110 --> 00:03:01,620 But M is a matrix by these bones. 36 00:03:01,620 --> 00:03:09,090 We mean if we subtract new one ie from M, it will be positive semi definite. 37 00:03:09,090 --> 00:03:12,510 The same is valid for meta and me too. 38 00:03:12,510 --> 00:03:17,760 I am as M is bounded, then its known is also bounded. 39 00:03:18,690 --> 00:03:27,930 Keep in mind this bond, this property of inertia matrix m because it will be useful for us during adaptive 40 00:03:27,930 --> 00:03:28,500 control. 41 00:03:30,090 --> 00:03:37,860 This upper and lower bonds are either constant or function of Q or the coordinate of choice. 42 00:03:39,630 --> 00:03:48,360 If the joints are Riverwood or rotational bonds are constant y because they are a function of sine and 43 00:03:48,360 --> 00:03:54,600 cousins and we know the bonds, they are good, they consists of sine and of course, our functions 44 00:03:54,600 --> 00:03:59,010 of Q and we know the bonds of sine and cosine functions. 45 00:04:00,840 --> 00:04:07,680 However, when the joints are prismatic, the bonds, the bonds are a function of cure or joint coordinates. 46 00:04:08,460 --> 00:04:14,220 Let's now try to understand the internal structure of inertia matrix in order to keep it simple. 47 00:04:14,670 --> 00:04:22,920 Let's take two degrees of freedom robot arm where M equals to the inertia matrix will consist of diagonal 48 00:04:22,920 --> 00:04:24,540 and of diagonal terms. 49 00:04:25,290 --> 00:04:33,300 Also, as you can see, it's symmetric diagonal terms indicates inertia experienced by joint I. 50 00:04:33,570 --> 00:04:36,930 When all are, the joints are stopped or blocked. 51 00:04:37,320 --> 00:04:44,460 So a robot is stopped and you want to know what is the inertia experienced by the joint AI with respect 52 00:04:44,460 --> 00:04:52,800 to its axis power of diagonal terms in the case, the couplings between the acceleration from Joy G 53 00:04:53,100 --> 00:04:56,400 to the generalized force drawn on Joint I. 54 00:04:59,270 --> 00:05:06,770 As you can see, this is one of the cases that shows couplings in the robot dynamics acceleration of 55 00:05:06,770 --> 00:05:11,870 one joined causes generalized force to be induced in other joints too. 56 00:05:13,790 --> 00:05:16,430 This is very important for us to 57 00:05:19,100 --> 00:05:28,940 get the maximum value of em because it will help us in the design of the robot manipulator because we 58 00:05:28,940 --> 00:05:38,990 need to know the talks that the motors torque, that how much torque the motors has to be able to generate 59 00:05:39,110 --> 00:05:45,920 in order to compensate for inertia talks during the motion of the robot manipulator. 60 00:05:47,090 --> 00:05:52,550 So now let's switch to MATLAB to investigate more on inertia matrix. 61 00:05:53,390 --> 00:05:53,780 OK. 62 00:05:54,830 --> 00:05:59,180 You know, we'll continue in MATLAB with initial matrix. 63 00:05:59,990 --> 00:06:11,990 OK, and we will use Sure Lipitor Gurkhas library in order to experiment on the robots and to analyze 64 00:06:11,990 --> 00:06:18,740 the robot dynamics terms here more clearly and more practically. 65 00:06:19,490 --> 00:06:28,100 So let's first hear where we are being nothing but just clean the screen and declaring the variables 66 00:06:28,100 --> 00:06:29,240 and the closing. 67 00:06:29,240 --> 00:06:38,270 If there is some figures, OK, after we have cleaned the variable screen and also the close of figures, 68 00:06:38,660 --> 00:06:47,930 let's do it by importing two degree of freedom and symbolic a robot manipulator from Peter Corker's 69 00:06:48,320 --> 00:06:48,890 library. 70 00:06:49,320 --> 00:06:51,650 OK, so let's do that. 71 00:06:51,650 --> 00:06:55,540 First of all, let's run these earlier these to command. 72 00:06:55,790 --> 00:06:56,420 OK. 73 00:06:57,050 --> 00:07:03,470 As you can see here from the workspace, you can see the climatic parameters. 74 00:07:03,470 --> 00:07:09,500 I mean, like normal parameters by M the parameters of the robot manipulator. 75 00:07:09,710 --> 00:07:14,310 You can see in the, uh, see all of these are symbolic. 76 00:07:14,330 --> 00:07:14,780 OK. 77 00:07:15,080 --> 00:07:21,830 Um, you can see, uh, the mass of each, uh, that each link. 78 00:07:22,130 --> 00:07:32,570 You can see the joint positions, joint velocities and by the joint, uh, accelerations anyway. 79 00:07:32,570 --> 00:07:36,290 And the robot model is to is called as to link. 80 00:07:36,860 --> 00:07:45,290 So what we want to investigate, we in we want English is the inertia matrix that positions Q1 and Q2 81 00:07:45,290 --> 00:07:48,560 of this Q1 and Q2 are joint positions. 82 00:07:48,860 --> 00:07:52,760 I mean, joint angles or either joint angles or joint positions. 83 00:07:52,760 --> 00:07:55,190 It, uh, it doesn't. 84 00:07:55,280 --> 00:07:58,880 Uh, there's nothing difference. 85 00:07:59,390 --> 00:07:59,810 OK. 86 00:08:00,110 --> 00:08:11,220 Uh, by doing just to link dot, uh, inertia and we give them a vector of joint positions. 87 00:08:11,240 --> 00:08:11,660 OK. 88 00:08:12,140 --> 00:08:13,730 So let's run this comment. 89 00:08:13,730 --> 00:08:16,290 Also, uh, lead selection. 90 00:08:16,310 --> 00:08:21,680 OK, we have the solution, and let's see what our um, matrix. 91 00:08:21,720 --> 00:08:22,190 OK? 92 00:08:22,460 --> 00:08:30,800 As you can see, the matrix of let's see the size of our most X is two by two. 93 00:08:31,160 --> 00:08:35,870 As you can see, our matrix is, um, square matrix, OK? 94 00:08:35,900 --> 00:08:38,300 As we have said, you know, she matrix is square. 95 00:08:38,630 --> 00:08:39,140 OK. 96 00:08:39,170 --> 00:08:44,850 And by in, as you can see in our case and equals two because this is to a degree of freedom, Robert 97 00:08:44,850 --> 00:08:46,820 the manipulator is two by two. 98 00:08:47,430 --> 00:08:51,950 And OK, we have to initiate these properties. 99 00:08:51,950 --> 00:08:54,170 OK, we have seen it's square matrix. 100 00:08:54,470 --> 00:08:56,810 Now we had to see if this symmetric. 101 00:08:57,200 --> 00:09:02,300 If you see, if you watch carefully, you will see that yes, it's symmetric. 102 00:09:02,300 --> 00:09:05,330 So for symmetric mattresses, mon. 103 00:09:05,330 --> 00:09:09,710 Diagonal terms, so this one, this one has to be the same. 104 00:09:10,010 --> 00:09:11,570 They are indeed the same. 105 00:09:11,930 --> 00:09:13,760 Yeah, they are written in different ways. 106 00:09:13,760 --> 00:09:22,340 But if you multiply, for example, if you are here, you can see it's M two times a two plus two squared 107 00:09:22,700 --> 00:09:23,300 plus. 108 00:09:23,870 --> 00:09:26,110 OK, let's first initiated this one. 109 00:09:26,120 --> 00:09:30,470 And if you if you multiply this one with this one. 110 00:09:30,620 --> 00:09:31,220 OK. 111 00:09:32,120 --> 00:09:38,900 Inside the brackets, you will see that yes, we have em two times a two plus c two squared plus. 112 00:09:39,080 --> 00:09:45,590 And here is our plus m to a two plus see two times a one Kushners Q2. 113 00:09:45,650 --> 00:09:53,920 Yes, indeed, we have this m two times e two a plus c two times a one questioner's Q two. 114 00:09:53,960 --> 00:09:57,470 As you can see this, it's written in a different way. 115 00:09:59,640 --> 00:10:07,350 This is due to Michael's calculation, but indeed, if we open these brackets by multiplying this term 116 00:10:07,350 --> 00:10:12,230 inside the bracket, you will see that these terms are indeed the same. 117 00:10:12,240 --> 00:10:15,630 So our initial matrix is also symmetric. 118 00:10:16,110 --> 00:10:23,970 And one important thing I want to note here that I have didn't mention in a presentation I have done 119 00:10:23,970 --> 00:10:29,040 this intentionally because I want you to see it in practice. 120 00:10:29,400 --> 00:10:37,050 The inertia of joint I depends on the configuration of joints I plus want to end. 121 00:10:37,260 --> 00:10:45,900 So the inertia of joint inertia seen by joint I depends on the configuration of the previous joints. 122 00:10:45,900 --> 00:10:49,580 But the joints after this joint I. 123 00:10:50,340 --> 00:10:50,770 Why? 124 00:10:51,240 --> 00:10:55,980 Because their configuration increases or decreases are fixed. 125 00:10:55,980 --> 00:11:01,470 The joint I am inertia inertia experienced by joint I. 126 00:11:02,070 --> 00:11:10,200 Indeed, you can experience these by your arm because your arm is also a robotic manipulator, OK? 127 00:11:10,650 --> 00:11:19,790 For example, if you want to keep your arms stretched fully, this will cause some inertia on your shoulder 128 00:11:19,800 --> 00:11:27,630 joint and OK, you will not be able to keep your arms stretched fully in a long time. 129 00:11:27,810 --> 00:11:36,870 Because in this position, your shoulder has the most inertia, your shoulder joint experience, the 130 00:11:36,870 --> 00:11:44,700 most inertia, so it will lose its, and it has to put more torque in order to counterbalance the talk 131 00:11:45,600 --> 00:11:51,240 the talk to counterbalance the torque generated due to the inertia torque. 132 00:11:51,540 --> 00:11:59,340 But if you keep your arm in a 90 degree is not in a fully stretched form, but in like 90 degrees and 133 00:11:59,340 --> 00:12:07,230 then the inertia of you, you will keep your arm more in this position because the inertia experienced 134 00:12:07,230 --> 00:12:11,730 by you, by your joint shoulder joint will be less. 135 00:12:11,970 --> 00:12:13,380 As you can see the shoulder. 136 00:12:13,380 --> 00:12:19,240 The joint experience is the inertia due to your due to the next joints, OK? 137 00:12:19,530 --> 00:12:25,500 For example, move your, for example, your wrist. 138 00:12:25,680 --> 00:12:27,060 OK, wrist joint. 139 00:12:27,300 --> 00:12:31,140 This recent joint excuse me, wrist joint will not. 140 00:12:31,530 --> 00:12:41,040 The shoulder joints position configuration will not affect the inertia experienced by the wrist joint. 141 00:12:41,040 --> 00:12:46,470 Okay, because the shoulder joint is before the wrist joint. 142 00:12:46,660 --> 00:12:47,100 Okay. 143 00:12:47,250 --> 00:12:54,730 It's configuration will not affect the inertia experienced by wrist joint, so wrist joint or motor 144 00:12:54,750 --> 00:12:55,580 doesn't have to. 145 00:12:55,600 --> 00:13:01,700 It doesn't have to induce some torque in order to counterbalance the shoulder configuration. 146 00:13:01,740 --> 00:13:05,760 Okay, and you can see this indeed from this matrix also. 147 00:13:06,060 --> 00:13:08,310 Let me show you this one. 148 00:13:08,340 --> 00:13:16,920 As you can see, the first joint OK two is a function of, as you can see it, the configuration of 149 00:13:16,920 --> 00:13:22,850 Q2 and OK and the configuration of Q2. 150 00:13:22,870 --> 00:13:27,210 OK, because the configuration of Q2 will affect the joint. 151 00:13:27,540 --> 00:13:35,610 But if you see the last drawing, which is the second joint is constant. 152 00:13:35,850 --> 00:13:37,020 Inertia is constant. 153 00:13:37,020 --> 00:13:39,170 It doesn't depend on joint one. 154 00:13:39,360 --> 00:13:39,830 OK? 155 00:13:40,620 --> 00:13:41,730 Indeed, always. 156 00:13:41,730 --> 00:13:48,190 The last joint has inertia, which is constant because there is no joint after that. 157 00:13:48,260 --> 00:13:57,690 OK, but in the middle joints will have and the intermediate joints will have inertia dependent on the 158 00:13:57,690 --> 00:13:58,470 next joint. 159 00:13:58,740 --> 00:14:01,500 So keep this in mind, this is important. 160 00:14:02,970 --> 00:14:03,370 Okay. 161 00:14:03,420 --> 00:14:03,770 None. 162 00:14:04,170 --> 00:14:09,380 And we have seen also this one and we say that except last joint. 163 00:14:09,390 --> 00:14:20,280 OK, now let's make it more clear by experiencing it is not as symbolic weight, but in a more clear 164 00:14:20,280 --> 00:14:21,540 way, in a practical way. 165 00:14:22,260 --> 00:14:22,920 Let's 166 00:14:25,920 --> 00:14:28,900 uncomment this and let's run this selection. 167 00:14:28,930 --> 00:14:31,050 OK, let's do it this. 168 00:14:33,390 --> 00:14:42,420 OK, now we have imported the model of Puma five six, the robot manipulator, which is a very old robot 169 00:14:42,420 --> 00:14:43,020 manipulator. 170 00:14:43,020 --> 00:14:49,770 But it's used in research, really, and it's well known robust number plates. 171 00:14:50,010 --> 00:14:57,070 OK, let's investigate inertia matrix it by in position and joint positions. 172 00:14:57,820 --> 00:14:59,440 So this is um. 173 00:15:00,130 --> 00:15:01,270 Q And let's see. 174 00:15:01,480 --> 00:15:07,480 As you can see, these are the joint angles and zero degree zero radium for the first one, the first 175 00:15:07,480 --> 00:15:16,060 joint and zero point seven eight five four for the second joined Pi degrees for, uh, Pi radians for 176 00:15:16,060 --> 00:15:24,460 the third joint, zero for the fourth and life that you can see that anyway. 177 00:15:24,790 --> 00:15:25,200 OK. 178 00:15:25,400 --> 00:15:27,430 Uh, now let's see. 179 00:15:27,430 --> 00:15:29,290 What's our inertial mattress? 180 00:15:29,320 --> 00:15:30,280 Okay, perfect. 181 00:15:30,820 --> 00:15:32,230 Our inertia mattress. 182 00:15:32,440 --> 00:15:34,630 And also, let me in. 183 00:15:35,470 --> 00:15:40,180 Let's plot the robot in that position. 184 00:15:41,980 --> 00:15:47,070 Q. And so you can see the robot manipulator, OK? 185 00:15:47,500 --> 00:15:50,710 As you can see this robot manipulator. 186 00:15:50,860 --> 00:15:51,430 OK. 187 00:15:52,030 --> 00:15:54,130 And this is in that position. 188 00:15:54,320 --> 00:15:54,880 OK. 189 00:15:55,630 --> 00:15:56,500 This is the joints. 190 00:15:56,500 --> 00:15:57,820 This is our Puma robot. 191 00:15:58,160 --> 00:15:59,530 Um, OK. 192 00:16:00,040 --> 00:16:03,710 If you want, we can put it in 3D. 193 00:16:03,850 --> 00:16:05,410 Let's do it in 3D. 194 00:16:05,420 --> 00:16:07,810 Maybe this will be more clear to you. 195 00:16:08,350 --> 00:16:10,480 Uh, and you will see the robot. 196 00:16:10,570 --> 00:16:11,020 OK. 197 00:16:12,250 --> 00:16:15,220 As you can see, this is a robot in that position. 198 00:16:15,250 --> 00:16:15,670 OK. 199 00:16:16,050 --> 00:16:18,400 Oh, OK, let's make it in this way. 200 00:16:18,430 --> 00:16:18,760 OK? 201 00:16:19,510 --> 00:16:20,350 This is a robot. 202 00:16:21,190 --> 00:16:26,250 And um, let's see who is on top. 203 00:16:26,260 --> 00:16:28,570 And let's see the inertia matrix. 204 00:16:28,630 --> 00:16:28,900 OK? 205 00:16:29,080 --> 00:16:33,280 As you can see, the most of the inertia experience, OK? 206 00:16:33,520 --> 00:16:36,640 First of all, it's let's see if this positive definite or not. 207 00:16:36,880 --> 00:16:41,260 In order to see it's positive, definite or not, let's cheek it's eigenvalues. 208 00:16:41,260 --> 00:16:52,360 So I guess if I am OK, if you will see that the animals are positive doesn't smell the um M. M. X is 209 00:16:52,360 --> 00:16:57,250 positive, definite, and it's symmetric if you can see if you will miss it. 210 00:16:57,280 --> 00:16:58,600 Yes, it's symmetric. 211 00:16:58,600 --> 00:17:02,020 As you can see, these all are the same. 212 00:17:02,350 --> 00:17:04,840 So our robot manipulator is symmetric. 213 00:17:04,870 --> 00:17:07,150 And let's check its size also. 214 00:17:07,450 --> 00:17:12,160 And if you can, if your size is six 16 square matrix. 215 00:17:12,170 --> 00:17:12,490 OK? 216 00:17:13,110 --> 00:17:13,490 OK. 217 00:17:14,800 --> 00:17:17,880 And let's see its components, as you can see. 218 00:17:18,220 --> 00:17:24,770 These are the inertia experienced by joints when all other joints are blocked. 219 00:17:24,790 --> 00:17:25,160 OK. 220 00:17:25,370 --> 00:17:34,540 As you can see, the most inertia is experienced by UM and the second joint and the first joint because 221 00:17:34,780 --> 00:17:37,420 they keep the whole robot OK. 222 00:17:37,630 --> 00:17:47,230 As in your arm, the most the most energy is experienced by your shoulder joint because it has to keep 223 00:17:47,230 --> 00:17:51,100 whole arm in some position, OK? 224 00:17:51,760 --> 00:17:59,170 However, if you see, uh, the, uh, fifth and sixth joints are, it has very less inertia because 225 00:17:59,860 --> 00:18:03,280 they don't support big parts. 226 00:18:03,310 --> 00:18:06,300 OK, so the inertia is less on them. 227 00:18:06,700 --> 00:18:09,610 Um, the inertia experienced by them is less. 228 00:18:10,720 --> 00:18:19,130 And here you can see the orthogonal terms show the acceleration of the couplings between the, uh, 229 00:18:19,390 --> 00:18:25,480 the robot joints or acceleration of, for example, here the acceleration of June two. 230 00:18:25,570 --> 00:18:31,990 OK, induces that much, uh, generalized force on joint one. 231 00:18:32,260 --> 00:18:32,630 OK. 232 00:18:32,890 --> 00:18:39,850 If you can see and as we go further, these codling terms are decreases. 233 00:18:39,880 --> 00:18:40,300 OK. 234 00:18:40,710 --> 00:18:42,130 Uh, y yeah. 235 00:18:42,670 --> 00:18:50,110 Let me give this initial intuition by your finger, by your arm, OK, if you or your finger are also 236 00:18:50,110 --> 00:18:53,740 part of your arm, OK, so a robot manipulator, OK? 237 00:18:54,100 --> 00:19:02,710 And if you move your finger, OK, does it will induce some force on your shoulder? 238 00:19:02,800 --> 00:19:08,260 No, because it's very small and it's very far from your shoulder joint. 239 00:19:08,260 --> 00:19:08,860 Okay? 240 00:19:09,130 --> 00:19:21,690 But if you move your whole arm, it will induce a force on your force to work on your joints or get 241 00:19:21,710 --> 00:19:22,510 shoulder joint. 242 00:19:22,810 --> 00:19:33,790 So that's why, as we go further from the base joint, OK, and these joints will not affect their will 243 00:19:33,790 --> 00:19:40,420 have zero effect on the shoulder joint OK, or in the second joint. 244 00:19:40,420 --> 00:19:47,290 But as you can see, the second joint has much greater effect on base joint. 245 00:19:48,580 --> 00:19:53,830 That is the intuition in, uh, inertia matrix. 246 00:19:54,220 --> 00:19:57,210 OK, now let's continue with cardiologist Matthew. 247 00:19:57,660 --> 00:20:02,640 See and hear, I also indicated the formula of its elements. 248 00:20:03,060 --> 00:20:06,210 This is the internal elements of the see matrix. 249 00:20:06,960 --> 00:20:13,530 This is called careerists matrix because it models the Coriolis and centrifugal effects due to the rotating 250 00:20:13,530 --> 00:20:16,410 bodies from the formula. 251 00:20:16,440 --> 00:20:23,340 You can see that the Coriolis matrix is not only configuration dependent, but also velocity dependent 252 00:20:23,850 --> 00:20:25,230 Sherlyn on linearly. 253 00:20:25,920 --> 00:20:31,560 So as the velocity and configuration change, the Coriolis matrix also changes. 254 00:20:32,340 --> 00:20:38,940 Here can be two cases, namely when I and J in the list are the same. 255 00:20:39,330 --> 00:20:47,540 Then the resulting effect is centrifugal effect that is induced on John K due to the velocity of Georgie. 256 00:20:48,420 --> 00:20:58,920 However, when I doesn't equals to the J in G, then the effect is Coriolanus, which is a yeah. 257 00:20:59,820 --> 00:21:00,450 Then there is. 258 00:21:00,450 --> 00:21:00,960 I'm sorry. 259 00:21:01,050 --> 00:21:08,160 Then the effect is Coriolis, which is induced on joint K due to the velocities of joints I and J. 260 00:21:08,610 --> 00:21:17,360 So CIJ represents couplings of joint J Velocity and the generalized force acting on joint I. 261 00:21:18,030 --> 00:21:24,810 Surely this means that additional torque is needed by joint motors to compensate for that forces. 262 00:21:26,070 --> 00:21:33,810 Here, once again, forces or talks here once again, you immediately see the couplings in the robot 263 00:21:33,810 --> 00:21:35,280 manipulator dynamics. 264 00:21:37,080 --> 00:21:44,160 As in the case of the inertia matrix, both the Coriolis term and its norm are bonded, and these bonds 265 00:21:44,160 --> 00:21:51,120 are a constant if the joints are developed and function of Q if the joints are prismatic due to the 266 00:21:51,120 --> 00:21:53,430 same reason we have mentioned earlier. 267 00:21:54,510 --> 00:21:59,550 Now let's switch the matter up to investigate more on Coriolis matrix. 268 00:21:59,940 --> 00:22:02,810 OK, let's continue at risk Carrillo's matrix. 269 00:22:03,300 --> 00:22:07,070 Okay, so let's first again. 270 00:22:07,080 --> 00:22:12,390 Um, clear this one, and let's import our symbolic robot manipulator. 271 00:22:12,890 --> 00:22:21,970 Okay, we have imported, and let's investigate the Coriolis matrix and the Coriolis talk. 272 00:22:22,110 --> 00:22:30,920 OK, we will direct the English Coriolis Turk because there's no need to watch the internal of, uh, 273 00:22:30,930 --> 00:22:37,590 Coriolis matrix and what how we will in this case, Coriolis talk or them by. 274 00:22:37,710 --> 00:22:43,790 We will first calculate the Coriolis matrix by just using the carnelossi term. 275 00:22:43,800 --> 00:22:49,500 And, as you can see, different from the initial matrix m the uh. 276 00:22:49,830 --> 00:22:55,830 In order to calculate the calculus matrix, we both need the joint positions and joint velocities because 277 00:22:55,830 --> 00:23:02,820 Coriolis term depend on both the joint velocity and the joint positions. 278 00:23:02,860 --> 00:23:10,710 OK, then what we will do as you'll remember the Carrillo's matrix the term dynamic equation was secure. 279 00:23:11,070 --> 00:23:12,810 Cured dot yeah. 280 00:23:13,500 --> 00:23:14,910 Times Cubed Dot. 281 00:23:15,000 --> 00:23:17,760 We will talk this in this way. 282 00:23:18,000 --> 00:23:25,680 We have calculated C Cukier Dot here, and by multiplying it by Q both, we will calculate the torque 283 00:23:25,680 --> 00:23:28,990 generated due to the current illest term. 284 00:23:29,010 --> 00:23:30,800 And here we do it. 285 00:23:31,290 --> 00:23:37,530 OK, and let's calculate, uh, cardiologist related selection. 286 00:23:40,290 --> 00:23:40,980 OK. 287 00:23:41,550 --> 00:23:44,040 And this is our C talk. 288 00:23:44,520 --> 00:23:45,060 OK. 289 00:23:45,150 --> 00:23:51,960 As you can see, this is the torque generated in joint one. 290 00:23:51,960 --> 00:23:53,400 Induced in joint one. 291 00:23:53,880 --> 00:23:55,470 Uh, OK. 292 00:23:56,130 --> 00:24:00,120 Coriolis torque and this is the courriel is termed induced in joint two. 293 00:24:00,120 --> 00:24:05,760 And as you can see, there is coupling here y as you can see the torque. 294 00:24:06,390 --> 00:24:14,160 The Coriolis torque generated in joint one depends on not only the velocity of itself, but also the 295 00:24:14,160 --> 00:24:16,320 velocity of Jordan too. 296 00:24:16,800 --> 00:24:24,180 And also if you can see tuned to Haspel's and depend on the velocity of joint one. 297 00:24:24,360 --> 00:24:32,120 So the velocity of during the one induces carry all this talk in air June two and June 2s velocity in 298 00:24:32,520 --> 00:24:37,560 excuse me, velocity joint twos velocity induces torque on joint one. 299 00:24:37,740 --> 00:24:38,910 So there is coupling. 300 00:24:38,940 --> 00:24:39,330 OK. 301 00:24:40,560 --> 00:24:47,760 Let's see this couple more clearly by substituting substituting excuse me, first making. 302 00:24:47,760 --> 00:24:56,260 Um, let's assume that joint one is stopped so its velocity of zero and only joint two moves. 303 00:24:56,490 --> 00:24:57,060 And in. 304 00:24:57,120 --> 00:24:57,510 This way. 305 00:24:57,540 --> 00:25:02,790 Let's see what will be our um if they make it in this way. 306 00:25:03,570 --> 00:25:06,510 Let's see what will be the result, but you will need selection. 307 00:25:06,510 --> 00:25:16,170 As you can see, if joined one is stocked, then joint one doesn't move, then the joint two there will 308 00:25:16,170 --> 00:25:25,830 not be any Coriolis or centrifugal force in or effect torque in joint two because joint one is stopped. 309 00:25:25,870 --> 00:25:43,260 OK, but joint one will have, um um um, some Coriolis effect due to the, uh, what motion of Coriolis 310 00:25:43,260 --> 00:25:47,510 torque due to the motion of joint two? 311 00:25:47,770 --> 00:25:51,240 Okay, because a joint two moves and there is coupling. 312 00:25:51,930 --> 00:26:00,690 And if we do the same for joint two and we will assume that joint two doesn't move, but joint one moves 313 00:26:00,690 --> 00:26:02,100 and let's see what's happening. 314 00:26:02,400 --> 00:26:08,190 As you can see now, Joint two doesn't move, so that's why it doesn't create any. 315 00:26:08,220 --> 00:26:09,380 Yeah, torque OK. 316 00:26:09,390 --> 00:26:14,910 Coriolis torque OK in joint one. 317 00:26:15,300 --> 00:26:17,970 But the core of this torque is created 318 00:26:21,090 --> 00:26:22,230 on June two. 319 00:26:22,260 --> 00:26:22,710 Why? 320 00:26:23,070 --> 00:26:28,110 Because a joint one moves and there is coupling. 321 00:26:28,110 --> 00:26:34,260 There shouldn't be coupling OK, and joint two has to create additional joint twos. 322 00:26:34,260 --> 00:26:42,480 Mortara has to create additional torque in order to compensate for the total created by the motion of 323 00:26:42,480 --> 00:26:43,350 joint one. 324 00:26:43,470 --> 00:26:43,940 Okay. 325 00:26:44,310 --> 00:26:51,990 And here by saying I Coriolis torque, I just named this Coriolis torque because the name of the term 326 00:26:51,990 --> 00:26:54,300 is this matrix. 327 00:26:54,300 --> 00:27:04,800 But here also depending on the indices are the same or different, it can be the torque due to the Coriolis 328 00:27:04,800 --> 00:27:07,140 effect or centrifugal effect. 329 00:27:07,340 --> 00:27:15,660 OK, so let's now see it in action by using Puma and five six the robot's manipulator. 330 00:27:16,440 --> 00:27:17,190 Here we will. 331 00:27:17,460 --> 00:27:21,950 Um yeah, OK, first, let's import the model. 332 00:27:21,970 --> 00:27:30,360 Let's simulate selection, and here we will take us only drawing two moves and others don't doesn't 333 00:27:30,360 --> 00:27:33,510 move and we will check what's happening in this case. 334 00:27:33,540 --> 00:27:36,790 OK, let's see Coriolis works in this way. 335 00:27:36,810 --> 00:27:38,880 OK, see? 336 00:27:38,910 --> 00:27:39,340 Four. 337 00:27:40,200 --> 00:27:40,660 Okay? 338 00:27:40,980 --> 00:27:46,370 As you can see, when only drawing two moves, yeah, OK. 339 00:27:46,770 --> 00:27:48,090 Only joined two moves. 340 00:27:48,090 --> 00:27:57,750 It will create some torque on the joint one and the joint three, and it will not create much talk on, 341 00:27:58,740 --> 00:27:59,890 uh, further joints. 342 00:27:59,910 --> 00:28:09,240 OK, um, you can experience it by your arm also when you move your arm. 343 00:28:09,900 --> 00:28:18,810 OK, then it will affect to your shoulder and maybe it will create some torque on your wrist joint, 344 00:28:19,110 --> 00:28:22,050 but it will not create some torque on your finger. 345 00:28:22,050 --> 00:28:22,980 The joints okay. 346 00:28:24,780 --> 00:28:32,270 And if you make it zero, and let's take, for example, um, see what will happen. 347 00:28:32,280 --> 00:28:35,510 For example, let's let's make it in this way. 348 00:28:35,520 --> 00:28:45,210 Let's check that what will happen if only joint, less joint or and the vector joint moves, it's like 349 00:28:45,420 --> 00:28:46,650 moving our fingers. 350 00:28:46,650 --> 00:28:56,430 It shouldn't create any torque on base joint, not any, but it should induce very small talk in joint 351 00:28:56,430 --> 00:29:02,970 one or joint two or joint three because, you know, because it shouldn't affect too much. 352 00:29:02,970 --> 00:29:04,470 Okay, because it's a small part. 353 00:29:04,830 --> 00:29:07,820 And let's check if it is true or not. 354 00:29:07,830 --> 00:29:12,510 Let's evaluate the selection and let's see what see torque. 355 00:29:14,130 --> 00:29:15,270 OK, as you can see. 356 00:29:15,310 --> 00:29:16,290 Zero zero zero. 357 00:29:16,380 --> 00:29:26,190 OK, so moving your finger will not induce any torque on your wrist or on your shoulder because it's 358 00:29:26,190 --> 00:29:34,680 a very small part and it's motion should create any other torque. 359 00:29:36,270 --> 00:29:36,600 OK. 360 00:29:37,230 --> 00:29:41,250 This is the initial intuition about carry on this term. 361 00:29:41,940 --> 00:29:42,540 OK. 362 00:29:42,750 --> 00:29:49,560 No, the last term growth term, there is not much to talk about it because there is nothing complicated 363 00:29:49,560 --> 00:29:50,040 about it. 364 00:29:50,490 --> 00:29:56,940 It illustrates simply the gravity of torque experienced by joints, which should surely be compensated. 365 00:29:57,080 --> 00:29:58,250 I joined mothers. 366 00:29:59,000 --> 00:30:05,450 Indeed, it's one of the most significant terms that's present in robot dynamics because even when the 367 00:30:05,450 --> 00:30:08,890 robot doesn't move, it has to compensate for gravity. 368 00:30:09,890 --> 00:30:16,550 Indeed, some kind of springs or counterbalance weights are put in some robots to decrease the talks 369 00:30:16,550 --> 00:30:20,690 that has to be provided by motors to compensate for gravity. 370 00:30:21,200 --> 00:30:28,100 This helps to decrease the size of the motor, which also decreases the cost and also the energy that's 371 00:30:28,100 --> 00:30:28,940 being used. 372 00:30:29,780 --> 00:30:37,280 The gravity term is also the function of configuration, so joint coordinates and so joint coordinates. 373 00:30:37,550 --> 00:30:40,350 And it is also lower and upper bound. 374 00:30:41,000 --> 00:30:42,320 Now let's switch back. 375 00:30:42,320 --> 00:30:43,430 Look to enrich it. 376 00:30:43,430 --> 00:30:45,190 More on gravity term. 377 00:30:45,560 --> 00:30:50,380 OK, now let's continue with gravity matrix, OK? 378 00:30:50,420 --> 00:30:54,080 Or, uh, gravity term gravity metrics are right here. 379 00:30:54,080 --> 00:30:56,570 Buttler's that is a gravity. 380 00:30:57,290 --> 00:31:03,500 OK, so let's clear everything early selection and it leads directly. 381 00:31:05,450 --> 00:31:12,770 Import the model of Puma and calculate the gravity of torque in a Q R position. 382 00:31:12,770 --> 00:31:14,180 What's the cure position? 383 00:31:14,190 --> 00:31:16,730 Let's see the clear position of the rover. 384 00:31:17,090 --> 00:31:20,630 OK, we will see it by just doing this one unit selection. 385 00:31:20,630 --> 00:31:21,650 OK, perfect. 386 00:31:22,130 --> 00:31:22,700 OK. 387 00:31:23,120 --> 00:31:28,640 As you can see, this is the position of the robot manipulator. 388 00:31:28,640 --> 00:31:35,570 OK, this is pulling a robot manipulator when into our position, and let's investigate our talks. 389 00:31:35,990 --> 00:31:44,180 Gravity torque as you can see, the gravity to in the base joint is zero y is zero because as you can 390 00:31:44,180 --> 00:31:47,480 see, it's the joint only moves in this way. 391 00:31:47,710 --> 00:31:53,780 So when the arm is in this position, it will not induce any gravity. 392 00:31:53,780 --> 00:31:56,700 It will it, doesn't it? 393 00:31:56,780 --> 00:32:03,830 The base joints shouldn't create any torque in order to applause for the gravity torque because the 394 00:32:03,830 --> 00:32:11,960 structure of the base base is internally OK, opposed the gravity. 395 00:32:12,350 --> 00:32:18,530 It doesn't need to create any torque in order in additional authority in order to oppose for the gravity 396 00:32:18,530 --> 00:32:18,860 torque. 397 00:32:19,040 --> 00:32:29,120 But it is not the same for the, uh, second unit and the surgeon as you can see the structure of the 398 00:32:29,930 --> 00:32:30,870 whole robot. 399 00:32:30,890 --> 00:32:38,780 OK, this upper arm, the joint tour has to counterbalance for this, uh, gravity torque generated 400 00:32:38,780 --> 00:32:40,460 due to this. 401 00:32:40,730 --> 00:32:44,660 That's why the highest gravity is generated on joint to OK. 402 00:32:44,660 --> 00:32:51,200 It has to counterbalance the whole arm in, you know, add this up positions and the gravity created 403 00:32:51,200 --> 00:32:51,980 due to this. 404 00:32:52,380 --> 00:33:01,570 OK, and some small talk is created in joints three also, but the others are not affected by the gravity, 405 00:33:01,670 --> 00:33:03,830 OK, if they change the position. 406 00:33:03,860 --> 00:33:08,030 OK, let's, for example, make it into an endless mission. 407 00:33:08,030 --> 00:33:10,370 What's so in nominal position? 408 00:33:10,580 --> 00:33:15,140 Let's see what will happen in this case if it's selection? 409 00:33:15,440 --> 00:33:15,860 OK? 410 00:33:16,070 --> 00:33:21,890 As you can see in this case, uh oh, I'm sorry. 411 00:33:22,850 --> 00:33:25,130 It's cute and position. 412 00:33:27,640 --> 00:33:37,510 Selection, okay, as you can see here, they're automatically we initially intuitively see that the 413 00:33:37,900 --> 00:33:41,730 gravity talk has to increase, okay? 414 00:33:42,040 --> 00:33:50,860 And as you can see, John too has to induce very high torque in order to balance this, 415 00:33:54,010 --> 00:33:55,630 this upper arm structure. 416 00:33:55,870 --> 00:33:56,160 Why? 417 00:33:56,440 --> 00:34:04,630 Because before it was in upward position and somehow the structure was the structure of the robot was 418 00:34:04,630 --> 00:34:06,840 opposing to the gravity. 419 00:34:06,850 --> 00:34:14,050 And so that's why the joint has to induce small talk in order to oppose the gravity. 420 00:34:14,080 --> 00:34:19,100 But in this case, the structure tends to fail down, OK? 421 00:34:19,360 --> 00:34:29,950 And the robot, the second joint, has to keep all of these upper parts in in up in this position in 422 00:34:29,950 --> 00:34:32,470 order, then not to fold down. 423 00:34:32,510 --> 00:34:32,830 Okay. 424 00:34:32,830 --> 00:34:41,830 So that's why it induces very high torque and either are higher or not very high, but some torque is 425 00:34:41,830 --> 00:34:45,450 needed by joints three or four other talks. 426 00:34:45,460 --> 00:34:52,060 It's very small talk OK for in terms of other joints, they need to create very small talk and you can 427 00:34:52,060 --> 00:34:55,870 experience this, as I said before, by your arm. 428 00:34:56,050 --> 00:35:01,330 So if you make your finger 90 degrees, your finger joints doesn't. 429 00:35:02,050 --> 00:35:08,200 So your mazor, if you bend your fingers, it's not. 430 00:35:08,590 --> 00:35:09,460 Create very much. 431 00:35:09,460 --> 00:35:19,540 Talk on your shoulder joint or your wrist joint or your arm or your ear or your other joints, finger 432 00:35:19,540 --> 00:35:21,630 joints because it's very small. 433 00:35:21,810 --> 00:35:30,240 OK, but if you make your shoulder stretched upwards, OK, stretched thin in st. 434 00:35:32,170 --> 00:35:38,410 OK, in a straight position, not a position, but straight position, which creates 90 degrees with 435 00:35:38,410 --> 00:35:39,160 your body. 436 00:35:39,970 --> 00:35:46,060 It will create very high gravity because your in your short term, in your shoulder joint, because 437 00:35:46,240 --> 00:35:52,660 your shoulder joint has to now oppose the failed bone off of whole of your arm. 438 00:35:52,810 --> 00:36:02,470 But if you keep your in your arms straight in opposition, some of the gravity talk will be opposed 439 00:36:02,470 --> 00:36:04,330 by the structure of your body. 440 00:36:04,330 --> 00:36:13,900 So your shoulder joint has to create smaller joint torque in order to oppose the flow of the fold of 441 00:36:13,900 --> 00:36:14,860 your arm. 442 00:36:15,130 --> 00:36:24,310 OK, so experience this with your arm because you have a manipulator, so that's given to you and you 443 00:36:24,310 --> 00:36:32,650 can experience by your arm in order to ensure to fully understand these concepts.