1 00:00:00,180 --> 00:00:00,810 Let's listen. 2 00:00:00,840 --> 00:00:06,690 We have seen in most of Jacobean metrics to get the joint velocities to obtain the desired and the factor 3 00:00:06,690 --> 00:00:07,290 velocity. 4 00:00:07,740 --> 00:00:13,880 However, if you remember, we have taken into account all of the squared circle metrics so we can calculate 5 00:00:13,890 --> 00:00:15,090 inverse of Jacobean. 6 00:00:15,660 --> 00:00:18,690 What if our robot doesn't have exactly six joints? 7 00:00:18,990 --> 00:00:20,910 What if our robot manipulator has? 8 00:00:20,910 --> 00:00:24,060 Joint is a less than six or more than six joints? 9 00:00:24,570 --> 00:00:25,800 Then what will we do? 10 00:00:26,220 --> 00:00:28,530 Because now our Jacobean is not square. 11 00:00:29,040 --> 00:00:30,960 How will we find the inverse of it? 12 00:00:31,410 --> 00:00:35,100 As I said before, the robot can have less than six joints. 13 00:00:35,430 --> 00:00:38,640 These kind of robots are called under actuated robots. 14 00:00:38,970 --> 00:00:45,270 This means that we don't have control on every velocity component of the end effector during the motion 15 00:00:45,660 --> 00:00:51,810 because our task space has six degrees of freedom while the robot has less than six degrees of freedom. 16 00:00:52,680 --> 00:00:59,430 Another case is when the robot has more than six joints, and these kinds of robots are called or actuated 17 00:00:59,430 --> 00:01:00,870 or redundant robots. 18 00:01:01,320 --> 00:01:07,950 So during funding joint velocities to get the desired end the vector velocities, we will have infinite 19 00:01:07,950 --> 00:01:13,350 solutions because some of our equations inside systems of equations will be linear dependent. 20 00:01:15,510 --> 00:01:20,490 You can see here the shape of equations in terms of under actuated robots. 21 00:01:20,790 --> 00:01:27,870 In this case, the one for which number of joints is two and this is for the all actuated or redundant 22 00:01:27,870 --> 00:01:31,050 robot for which the number of joints is seven. 23 00:01:33,330 --> 00:01:36,990 Let's try to find the inverse of Jacobean for under actuated robots. 24 00:01:36,990 --> 00:01:43,740 First, assume that we have a robot manipulator with two joints and we want to get zero point two meters 25 00:01:43,740 --> 00:01:46,590 per second linear velocity in y direction. 26 00:01:47,310 --> 00:01:49,020 However, we have a problem here. 27 00:01:49,260 --> 00:01:53,640 Our robot consists of two joints, so it has two degree of freedom. 28 00:01:54,000 --> 00:02:00,150 Consequently, we can control only two components of the robots and the factor during the motion. 29 00:02:00,600 --> 00:02:02,550 So how will we solve that problem? 30 00:02:03,180 --> 00:02:06,030 The solution is nothing but accepting the reality. 31 00:02:06,060 --> 00:02:12,540 We will try to use what we have in our hands, so we will try to control two components of the velocity 32 00:02:12,540 --> 00:02:13,680 of the end effector. 33 00:02:14,010 --> 00:02:20,130 While doing that, surely we will get some little unavoidable extra motions because of our under actuator, 34 00:02:20,130 --> 00:02:21,210 the robot manipulator. 35 00:02:21,810 --> 00:02:24,540 Let's ride our forward kinematics equation. 36 00:02:24,900 --> 00:02:31,320 As you can see, we take here only the two linear velocity components of the end effector, namely V 37 00:02:31,320 --> 00:02:32,820 X and V Y. 38 00:02:33,150 --> 00:02:39,750 So we separate our Jacobean matrix also and take the top partition of it, which is two by two matrix. 39 00:02:41,460 --> 00:02:43,650 We can write our new equation in this way. 40 00:02:44,010 --> 00:02:49,830 Now our Jacobean is a square matrix so we can inverted and get our required joint velocities. 41 00:02:50,430 --> 00:02:53,640 However, as always, be careful in terms of inverse of a matrix. 42 00:02:53,670 --> 00:02:57,330 Be sure that the determinant of the Jacobean matrix is not zero. 43 00:02:57,870 --> 00:03:02,220 Let's go to the MATLAB and try to visualize the things that we have seen. 44 00:03:06,490 --> 00:03:21,040 OK, so let's start by first importing the two joint robot manipulator, so as you know, this will 45 00:03:21,040 --> 00:03:26,770 be under actuated robot because a number of joints is less than six. 46 00:03:27,520 --> 00:03:28,240 OK. 47 00:03:29,890 --> 00:03:32,020 Let's see if we tried P2. 48 00:03:32,050 --> 00:03:33,970 You will see that we have two joints. 49 00:03:34,510 --> 00:03:35,010 OK. 50 00:03:35,470 --> 00:03:38,350 So let's define a nominal pulse for eat. 51 00:03:38,560 --> 00:03:38,890 OK. 52 00:03:39,340 --> 00:03:41,960 This is joints for each one, we will ride one. 53 00:03:42,010 --> 00:03:43,060 This is normal position. 54 00:03:43,240 --> 00:03:47,380 Let's put our robot in that position to see it in order to adjust. 55 00:03:47,830 --> 00:03:48,190 OK. 56 00:03:48,370 --> 00:03:50,320 As you can see, this is our robot. 57 00:03:50,710 --> 00:03:53,920 As you can see, it consists of two real joints. 58 00:03:54,340 --> 00:03:54,880 Oh, OK. 59 00:03:54,920 --> 00:03:56,110 At no, no position. 60 00:03:56,200 --> 00:04:00,820 So let's find its Jacobean at that point. 61 00:04:00,970 --> 00:04:03,760 OK, we will use a in jerk of zero. 62 00:04:03,790 --> 00:04:06,870 OK, at CU and O. 63 00:04:07,570 --> 00:04:07,880 Oh! 64 00:04:09,540 --> 00:04:10,320 Oh, OK. 65 00:04:10,600 --> 00:04:12,460 P to put. 66 00:04:12,760 --> 00:04:13,500 OK, perfect. 67 00:04:15,270 --> 00:04:20,880 Now you can see that joint one affects exposition. 68 00:04:21,300 --> 00:04:29,550 Excuse me, linear velocity x linear, as well as the on y and also the rotation about Z. 69 00:04:30,150 --> 00:04:33,900 And the same is for joint to OK. 70 00:04:33,900 --> 00:04:42,390 It a fixed linear velocity in X, linear velocity in Y and angular velocity about that axis. 71 00:04:42,870 --> 00:04:47,700 We can see that more clearly by using P to teach method. 72 00:04:47,940 --> 00:04:48,270 OK. 73 00:04:48,420 --> 00:04:58,590 You know, to just show you, as you can see, when we rotate the first joint, it affects x y velocities 74 00:04:59,250 --> 00:05:00,990 the same for two. 75 00:05:00,990 --> 00:05:06,510 And there the rotation angle of two of the joints are fixed. 76 00:05:06,520 --> 00:05:14,490 As you can see only the zit rotation, because if you put your finger or your thumb on the direction 77 00:05:14,490 --> 00:05:19,770 of Z and curl your fingers, you will see that the positive rotation is about this one. 78 00:05:19,770 --> 00:05:26,910 So as you can see the joints, when the joints are changed, joint variables are changed. 79 00:05:27,300 --> 00:05:32,160 The rotation only affects only the rotation about Z, not X and Y. 80 00:05:32,190 --> 00:05:32,580 OK. 81 00:05:34,470 --> 00:05:38,160 So anyway, um, let's continue. 82 00:05:38,520 --> 00:05:43,970 So what we have said, we have said that we cannot just invert this, J. 83 00:05:45,030 --> 00:05:51,120 OK, we cannot find inverse of life by just writing like that. 84 00:05:51,480 --> 00:05:51,930 OK. 85 00:05:52,200 --> 00:05:55,920 As you can see, it is that matrix must be square. 86 00:05:56,280 --> 00:05:59,910 So what we have said that we will accept the reality. 87 00:05:59,940 --> 00:06:01,860 So we have two degrees of freedom. 88 00:06:02,280 --> 00:06:08,760 And if we want to control, for example, the velocity in y direction, then we will try to control 89 00:06:08,760 --> 00:06:11,460 only the V X and the Y. 90 00:06:11,730 --> 00:06:12,270 OK. 91 00:06:12,690 --> 00:06:21,600 So let's do that and partition our Jacobean and take the first partition of it to partition, as we 92 00:06:21,600 --> 00:06:23,520 have said in lecture slides. 93 00:06:23,820 --> 00:06:27,390 So we will take the first, uh, two columns. 94 00:06:27,610 --> 00:06:28,140 OK. 95 00:06:29,250 --> 00:06:31,350 Excuse me, first two rows. 96 00:06:33,960 --> 00:06:34,620 OK. 97 00:06:34,920 --> 00:06:36,220 Let's visualize that. 98 00:06:36,240 --> 00:06:36,630 OK. 99 00:06:36,660 --> 00:06:39,090 This one, as you can see, we take this one. 100 00:06:39,420 --> 00:06:40,470 These two? 101 00:06:41,490 --> 00:06:41,850 OK. 102 00:06:42,690 --> 00:06:51,340 So let's see what we have seat inside the slide that our desired, uh, designer. 103 00:06:51,450 --> 00:06:53,960 We want to let me just do that. 104 00:06:53,970 --> 00:07:02,320 We want to move our end effector in 0.2 meters per second in y direction. 105 00:07:02,340 --> 00:07:02,700 OK. 106 00:07:02,910 --> 00:07:08,490 So in order to find the required joint angles, uh, join the velocities. 107 00:07:08,700 --> 00:07:14,190 We use inverse of our now squared J X y OK squared Jacobean. 108 00:07:14,400 --> 00:07:18,650 So we will have inverse of it before doing that. 109 00:07:18,660 --> 00:07:22,770 If you want, let's check what we have said determinant of it. 110 00:07:23,040 --> 00:07:25,530 Let's see whether it's equal to zero or not. 111 00:07:25,560 --> 00:07:28,050 As you can see, it's not equal to zero. 112 00:07:28,260 --> 00:07:30,570 So we have the inverse. 113 00:07:31,080 --> 00:07:36,850 So let's paste it again, and let's find our desired kudi. 114 00:07:36,870 --> 00:07:37,260 OK. 115 00:07:37,470 --> 00:07:40,820 This is our desired joint velocities. 116 00:07:40,830 --> 00:07:49,920 OK, let's do the forward kinematics and find whether these joint velocities give us the desired response. 117 00:07:49,920 --> 00:07:59,280 As you can see, we have the velocity about y direction and which is which we desire. 118 00:07:59,520 --> 00:08:01,610 OK and 0.2 meters per second. 119 00:08:01,620 --> 00:08:09,390 But there is an unavoidable angular velocity rotation about that axis because, you know, they are 120 00:08:09,630 --> 00:08:10,080 linear. 121 00:08:11,070 --> 00:08:20,490 The, uh, the rotation of joint one and joined to their linear combination affects the rotation about 122 00:08:20,490 --> 00:08:20,850 Z. 123 00:08:20,850 --> 00:08:23,340 So that's why the Z is unavoidable. 124 00:08:23,610 --> 00:08:29,130 Uh, so this is the problem with under actuated robots. 125 00:08:30,420 --> 00:08:30,870 OK. 126 00:08:30,900 --> 00:08:38,400 Now, as you can see, this is our estimate in terms of finding the inverse of the Jacobean of all actuated 127 00:08:38,400 --> 00:08:39,750 or redundant robots. 128 00:08:39,990 --> 00:08:43,230 We will utilize lift, so the inverse of the Jacobean. 129 00:08:43,860 --> 00:08:47,550 Let me give some definitions about so the inverse of Jacobean. 130 00:08:50,100 --> 00:08:56,970 It has this property, namely when the so the inverse of the Jacobean multiplied with itself, we'll 131 00:08:56,970 --> 00:08:58,350 give identity matrix. 132 00:08:59,190 --> 00:09:00,300 We can find lift. 133 00:09:00,360 --> 00:09:02,850 So the inverse of Jacobean by this formula. 134 00:09:03,960 --> 00:09:04,350 OK. 135 00:09:04,530 --> 00:09:06,030 Let's come back to the problem. 136 00:09:06,450 --> 00:09:08,220 There are infinitely many solutions. 137 00:09:08,270 --> 00:09:14,270 So the equation above, because the robot manipulator is redundant, so we choose the solution for which 138 00:09:14,270 --> 00:09:16,790 the norm of the joint velocities is minimal. 139 00:09:17,780 --> 00:09:22,910 So why not use the redundant robot while we can make our robot has six joints? 140 00:09:23,270 --> 00:09:26,950 And so we can control all the velocity components of the end? 141 00:09:26,960 --> 00:09:32,660 The factor is that because the robotics components have many money to spend on robots, or do they do 142 00:09:32,660 --> 00:09:33,020 that? 143 00:09:33,380 --> 00:09:35,090 So the robots seem complex. 144 00:09:35,750 --> 00:09:36,800 Let's analyze this. 145 00:09:37,340 --> 00:09:43,820 As we have said before, the shape of robot Jacobean is six by end four, which and is the number of 146 00:09:43,820 --> 00:09:44,300 joints. 147 00:09:44,660 --> 00:09:49,820 And as you know, from linear algebra, in order to find the rank of a matrix, we choose minimum of 148 00:09:49,820 --> 00:09:50,690 two dimensions. 149 00:09:51,080 --> 00:09:54,560 So for redundant robot, we will have six by end. 150 00:09:54,560 --> 00:09:58,430 Jacobean matrix and end is greater than six. 151 00:09:59,030 --> 00:10:01,490 So the rank of the Jacobean will be six. 152 00:10:01,850 --> 00:10:05,240 What does it mean from linear algebra? 153 00:10:05,240 --> 00:10:13,700 You know that also this means that no space of the Jacobean is not to, and its dimension is equal to 154 00:10:13,700 --> 00:10:20,170 the number of joints and minus rank on the Jacobean as our own little space is not empty. 155 00:10:20,210 --> 00:10:26,600 We have some joint velocities for which the end effector velocity will be zero, so it will not move 156 00:10:27,320 --> 00:10:34,160 because we conclude that from the definition of no space of some matrix in this case, the Jacobean, 157 00:10:34,460 --> 00:10:40,520 so we can so we can separate the equation. 158 00:10:40,520 --> 00:10:47,420 A boat like this, namely the first part will give us the joint velocities by which we can get our desired 159 00:10:47,420 --> 00:10:48,890 end effector velocity. 160 00:10:49,310 --> 00:10:54,830 The second part will give us such joint velocities that will not affect the end effector. 161 00:10:58,700 --> 00:11:09,110 So end by nTX Plus helps us to plot the project design joint motions into the null space so they will 162 00:11:09,110 --> 00:11:11,750 not affect and affect our Cartesian motion. 163 00:11:12,230 --> 00:11:14,990 What do we do that we want that? 164 00:11:14,990 --> 00:11:22,010 Because by using this method, we can get the desired and affect their velocities while avoiding obstacles. 165 00:11:22,400 --> 00:11:29,630 Namely, we can change the position and orientation of and the factor while changing joint velocities 166 00:11:30,050 --> 00:11:36,200 and also be changing velocities such that they avoid obstacle but don't affect the motion of the end 167 00:11:36,200 --> 00:11:36,770 effector. 168 00:11:37,160 --> 00:11:44,180 This is not only useful for obstacle avoidance, but also avoiding collisions with the links or to keep 169 00:11:44,180 --> 00:11:47,330 joint calls away from their mechanical limited stops. 170 00:11:47,990 --> 00:11:55,040 So this is why we use redundant robots so we can have many options for getting one desired end effector 171 00:11:55,040 --> 00:11:56,750 position and orientation. 172 00:11:57,290 --> 00:12:00,410 Let's go to the MATLAB and apply what we have seen. 173 00:12:02,390 --> 00:12:09,410 So let's first start by importing model of Baxter. 174 00:12:09,890 --> 00:12:15,560 OK, Baxter Baxter is a robot that have two arms. 175 00:12:15,710 --> 00:12:18,410 It's like human or humanoid robot. 176 00:12:18,890 --> 00:12:20,900 It has two arms. 177 00:12:20,930 --> 00:12:24,800 Each of these are consists of seven joints. 178 00:12:24,830 --> 00:12:29,600 OK, so we have seven degree of freedom in each of the arms. 179 00:12:30,050 --> 00:12:33,020 And so they are their Jacobean also. 180 00:12:33,890 --> 00:12:40,280 And so they are redundant or um or constrained. 181 00:12:40,610 --> 00:12:44,450 Um, excuse me or actuated robot manipulators. 182 00:12:45,950 --> 00:12:49,310 So let's see the old Baxter. 183 00:12:50,960 --> 00:12:54,680 Oh, excuse me, I just want to do not like that. 184 00:12:54,680 --> 00:12:55,790 But let's see. 185 00:12:55,790 --> 00:12:56,460 It's left. 186 00:12:56,660 --> 00:13:00,530 As you can see, we have here a left arm and its right arm. 187 00:13:00,770 --> 00:13:07,460 And let's see its left arm as you can see its left arm, and it consists of seven joints. 188 00:13:08,360 --> 00:13:10,550 OK, then let's define. 189 00:13:10,890 --> 00:13:14,030 As you can see, I have defined here joints previously. 190 00:13:14,270 --> 00:13:15,940 OK, let me paste it. 191 00:13:15,980 --> 00:13:17,030 This is arbitrary. 192 00:13:17,420 --> 00:13:19,580 OK, there's nothing in that. 193 00:13:19,890 --> 00:13:26,600 OK, it's some position, and let's plot our left arm in that position. 194 00:13:26,600 --> 00:13:30,680 But Load Q OK. 195 00:13:30,710 --> 00:13:38,660 As you can see, this is our backs, the robot and that position in that joint configurations, its 196 00:13:38,930 --> 00:13:40,190 configuration is like that. 197 00:13:40,470 --> 00:13:41,750 Anyway, let's continue. 198 00:13:43,860 --> 00:13:44,440 OK. 199 00:13:44,780 --> 00:13:52,240 Um, what we can do first, let's find its Jacobean of the, uh, joint configurations. 200 00:13:52,690 --> 00:13:53,540 I always forget. 201 00:13:53,860 --> 00:13:54,170 OK. 202 00:13:55,780 --> 00:13:56,570 Oh my gosh. 203 00:13:56,580 --> 00:13:59,560 Yeah, yeah, I have forward through laptop. 204 00:13:59,780 --> 00:14:02,150 OK, now it's our Jacobean. 205 00:14:02,450 --> 00:14:04,970 Yeah, it's as you can see. 206 00:14:05,300 --> 00:14:10,490 Uh, it consists of six rows, but seven 207 00:14:12,890 --> 00:14:15,500 columns because we have seven joints. 208 00:14:16,700 --> 00:14:21,560 So uh, let's let's do that. 209 00:14:21,570 --> 00:14:29,480 We want zero point two meters per second linear velocity at x y z directions. 210 00:14:29,480 --> 00:14:36,560 OK, so our designed and the vector velocity zero point two zero point two metric per second in x y 211 00:14:36,560 --> 00:14:38,750 directions and no rotation. 212 00:14:38,750 --> 00:14:41,570 OK, no rotation. 213 00:14:42,250 --> 00:14:42,670 OK. 214 00:14:43,490 --> 00:14:54,790 So what will be our desired joint required joint velocities in order to get that and the vector velocity 215 00:14:55,100 --> 00:14:56,210 we will use? 216 00:14:56,930 --> 00:15:02,050 So the inverse, as you know, we have said in our lecture that it has to be. 217 00:15:02,120 --> 00:15:04,400 So we have to use so the inverse, OK. 218 00:15:04,460 --> 00:15:06,540 We cannot just use inverse. 219 00:15:06,560 --> 00:15:07,790 Let's check also this one. 220 00:15:07,970 --> 00:15:13,700 Let's you can use just yeah, you can see that we cannot use because it has to be squared. 221 00:15:13,910 --> 00:15:15,980 So we will use so the inverse. 222 00:15:16,880 --> 00:15:17,270 OK. 223 00:15:17,270 --> 00:15:22,520 And in MATLAB, we can find it and we can find that. 224 00:15:22,580 --> 00:15:28,130 So the inverse by using also this is a function from Pethokoukis Library by using P function. 225 00:15:29,210 --> 00:15:29,640 OK. 226 00:15:30,770 --> 00:15:32,030 Uh, excuse me. 227 00:15:32,240 --> 00:15:32,660 OK. 228 00:15:33,860 --> 00:15:41,810 This is now our desired or required joint velocities, as you can see all of the joints. 229 00:15:42,440 --> 00:15:48,440 How can I say, contribute to the motion in order to get the desired in the vector motion? 230 00:15:48,470 --> 00:15:48,770 OK. 231 00:15:49,850 --> 00:15:54,560 So now, as we have said before, let's check rank of our J metrics. 232 00:15:54,560 --> 00:15:57,650 As you can see, rank of our Jacobean is. 233 00:15:57,720 --> 00:15:58,230 Six. 234 00:15:58,420 --> 00:16:02,430 So we know that it's not space is not empty. 235 00:16:02,460 --> 00:16:02,970 OK. 236 00:16:03,000 --> 00:16:07,920 It has to be we have to one dimension of space. 237 00:16:07,920 --> 00:16:09,810 Let's check no Jay. 238 00:16:09,990 --> 00:16:13,090 This will return us the basis victor again. 239 00:16:14,400 --> 00:16:24,690 I think you have to know from the linear algebra that it will return us the basis vector for which you 240 00:16:24,690 --> 00:16:31,470 know, it will be the no or no victor of our Jacobean matrix. 241 00:16:31,470 --> 00:16:34,740 So we can see that this is the basis. 242 00:16:34,740 --> 00:16:36,960 Victor, what does it mean, business victor? 243 00:16:37,140 --> 00:16:45,480 It means that if this is not victor for our Jacobean, but it's linear combinations will be also, for 244 00:16:45,480 --> 00:16:55,230 example, when we scale, this is also null vector for our Jacobean OK um and times for plots and times 245 00:16:55,230 --> 00:17:05,250 50 will also be will also be, you know, null will be of no space in our Jacobean. 246 00:17:05,280 --> 00:17:05,600 OK. 247 00:17:05,820 --> 00:17:07,540 But the basis is this one? 248 00:17:07,560 --> 00:17:08,020 OK? 249 00:17:08,040 --> 00:17:08,970 This is the basis. 250 00:17:09,240 --> 00:17:11,090 Then you can just multiply it. 251 00:17:11,100 --> 00:17:12,870 You can just scale it. 252 00:17:12,870 --> 00:17:20,370 You can just find a linear combination of it with other, you know, with the scaling of it like that, 253 00:17:20,370 --> 00:17:21,450 it will not change anything. 254 00:17:22,440 --> 00:17:22,780 OK. 255 00:17:22,800 --> 00:17:28,820 Anyway, let's check that whether it is inside the norm of, excuse me, inside the 256 00:17:32,070 --> 00:17:39,720 new space of our Jacobean, so we will do what we will take our and multiply with our j. 257 00:17:39,750 --> 00:17:42,180 This has to give us zero. 258 00:17:42,180 --> 00:17:43,650 We will just find Norm. 259 00:17:44,310 --> 00:17:53,660 OK, normal fit in order to get one, you know, one number and the more normal fit. 260 00:17:53,670 --> 00:18:02,940 OK, so as you can see it, zero, 10 to 10 times or excuse me, time or minus 16, which is zero. 261 00:18:03,300 --> 00:18:03,750 OK. 262 00:18:04,080 --> 00:18:14,190 So let me just I'll just show you this one that if you multiply this with four point scale, these 10, 263 00:18:14,520 --> 00:18:15,960 it will also give you zero. 264 00:18:16,080 --> 00:18:23,940 OK, if you find a linear combination, for example, if you do and again one times, for example, 265 00:18:23,970 --> 00:18:26,610 16, this will also give you zero. 266 00:18:27,120 --> 00:18:27,660 OK. 267 00:18:27,690 --> 00:18:34,350 Because this is just the basis for our vector in our space. 268 00:18:34,920 --> 00:18:38,130 OK, I think you have to know it from the linear algebra. 269 00:18:38,250 --> 00:18:39,180 So let's continue. 270 00:18:40,500 --> 00:18:43,020 So let's define that. 271 00:18:44,370 --> 00:18:52,410 Let's just assume that now our task is not only get that position of orientation. 272 00:18:52,410 --> 00:18:59,790 So as you can see, if you want that position and orientation of our excuse me, this follows that in 273 00:18:59,790 --> 00:19:06,540 the vector, but we also want our robot mother planted while achieving that. 274 00:19:07,080 --> 00:19:15,810 It also move the joint five one meters per second in order to avoid some obstacle. 275 00:19:15,840 --> 00:19:20,070 But this doesn't have, you know, it doesn't have to. 276 00:19:20,380 --> 00:19:21,330 It doesn't. 277 00:19:22,380 --> 00:19:27,660 It shouldn't affect the end effector, OK, because then it will change the orientation of it. 278 00:19:27,990 --> 00:19:33,720 We want the joint five to move, but doesn't affect the end effector. 279 00:19:33,720 --> 00:19:34,140 OK? 280 00:19:34,350 --> 00:19:34,890 It is. 281 00:19:35,020 --> 00:19:37,170 We move it in order to avoid the obstacle. 282 00:19:37,320 --> 00:19:43,360 And as our robot is over actuated or redundant, we can do that. 283 00:19:43,380 --> 00:19:43,740 OK. 284 00:19:43,980 --> 00:19:46,290 So let's see that we want this. 285 00:19:46,290 --> 00:19:54,060 One joined five to move one two three four one meters per second. 286 00:19:54,120 --> 00:19:54,750 OK. 287 00:19:55,980 --> 00:19:59,610 And as we have seven joints, we will do it like that. 288 00:20:00,330 --> 00:20:00,750 OK. 289 00:20:02,220 --> 00:20:03,060 We want this one. 290 00:20:04,620 --> 00:20:09,090 Now let's then project it. 291 00:20:09,090 --> 00:20:09,570 OK? 292 00:20:09,630 --> 00:20:19,860 What we have said, we will project it enough space so it will not affect to our robot manipulator, 293 00:20:20,550 --> 00:20:21,810 excuse me, end effector. 294 00:20:21,960 --> 00:20:28,230 So we find the p inverse or so the inverse of our end times. 295 00:20:28,320 --> 00:20:30,430 Cue the No. 296 00:20:30,570 --> 00:20:31,080 OK. 297 00:20:31,830 --> 00:20:38,190 So this will give us the projected joint velocities. 298 00:20:38,190 --> 00:20:46,290 As you can see, this will not affect the end effect of sore joints will move the these these velocities 299 00:20:46,290 --> 00:20:51,690 and they will end the vector will not move because of these. 300 00:20:51,840 --> 00:20:55,020 So they will their result and will not affect the end effector. 301 00:20:55,170 --> 00:20:57,270 As you can see, all the joints move. 302 00:20:57,660 --> 00:21:04,710 So why all the joint moves, all the joint moves, so that the joint they will compensate for the motion 303 00:21:04,710 --> 00:21:05,760 of J5? 304 00:21:05,820 --> 00:21:14,070 OK, so the J5 will not contribute to the motion of the end effector because of that, all the links, 305 00:21:14,520 --> 00:21:16,290 all the joints are moving also. 306 00:21:16,770 --> 00:21:26,340 So let's check that whether it will affect the end effector or not, we will just to j times, q p in 307 00:21:26,340 --> 00:21:29,730 order to get our end defective doses and we will see that it's zero. 308 00:21:29,790 --> 00:21:32,280 So it doesn't affect our end effector. 309 00:21:32,760 --> 00:21:33,500 OK. 310 00:21:34,200 --> 00:21:43,860 So um, lastly, our result and joined list this will be peer inverse J Times XD. 311 00:21:44,070 --> 00:21:51,870 As you know, this will help us to get to orient or to get the desired joint velocities. 312 00:21:53,250 --> 00:21:53,880 Excuse me. 313 00:21:53,880 --> 00:21:57,960 And the vector velocities plus q p. 314 00:21:58,170 --> 00:22:00,900 This will be needed in order to avoid obstacles. 315 00:22:00,900 --> 00:22:05,790 So this will move the joint five one meters per second. 316 00:22:06,090 --> 00:22:10,670 While, you know, not affecting the end, affect them. 317 00:22:10,680 --> 00:22:20,280 So this is our oh excuse me, this is our resultant joint velocities, excuse me, required joint velocities. 318 00:22:20,920 --> 00:22:30,810 OK, so we have finished about Jacobean of, uh, unregulated tweeted and or actuated or redundant robot 319 00:22:30,810 --> 00:22:31,620 manipulators. 320 00:22:31,950 --> 00:22:33,630 So see you on the next lesson.