1 00:00:00,180 --> 00:00:00,510 OK. 2 00:00:00,640 --> 00:00:03,630 Let's continue with the hybrid position for control. 3 00:00:04,020 --> 00:00:06,420 This is the last part for this topic. 4 00:00:06,960 --> 00:00:07,570 Let's listen. 5 00:00:07,590 --> 00:00:14,610 We have seen these two forms where we have tried to get control, both on posts from general six dimensional 6 00:00:14,610 --> 00:00:16,170 force and velocity vectors. 7 00:00:16,650 --> 00:00:20,520 Please note that by seeing controllable, I don't mean the in control. 8 00:00:20,520 --> 00:00:20,930 Sorry. 9 00:00:21,330 --> 00:00:27,780 Instead, I want to mean that we can control without being constrained by the environment geometry. 10 00:00:28,230 --> 00:00:28,650 OK. 11 00:00:28,950 --> 00:00:34,470 As we have said before, we can do this selection with the help of selection mattresses, namely a V 12 00:00:34,470 --> 00:00:35,190 and a F. 13 00:00:35,820 --> 00:00:39,840 Now we have to decide what we want to do or what's our control objective. 14 00:00:40,410 --> 00:00:46,770 We want to impose design task evolution to the parameters of motion, namely as an parameter of force, 15 00:00:46,770 --> 00:00:47,670 namely lambda. 16 00:00:48,150 --> 00:00:53,550 So we want to steer them to desired values through control algorithms we have seen before. 17 00:00:54,240 --> 00:00:56,700 We can achieve this in two steps in. 18 00:00:56,850 --> 00:01:00,810 The development of the method is somehow similar to the impedance control. 19 00:01:01,200 --> 00:01:07,230 Namely, we will first try to linear rise and decouple the system in the task frame by feedback. 20 00:01:07,230 --> 00:01:13,980 Legalization normalization of the system equation will help us to apply linear controls, while decoupling 21 00:01:13,980 --> 00:01:20,970 will help us to apply appropriate control in appropriate directions independently and in the second 22 00:01:20,970 --> 00:01:25,470 step, we will apply linear control algorithms to achieve desired task evolution. 23 00:01:26,670 --> 00:01:27,060 OK. 24 00:01:29,640 --> 00:01:36,150 Let's first note the dynamic equation of the robot manipulator and forward differential kinematics you 25 00:01:36,150 --> 00:01:41,010 term in equation one point zero is our input control that we will design. 26 00:01:41,730 --> 00:01:49,350 Let's write Equation 1.0 in 1.3 form by the help of selection matrix of F. 27 00:01:49,890 --> 00:01:56,310 We can do the same for Equation 1.1 and convert it to Equation one point two with the help of selection 28 00:01:56,310 --> 00:01:57,470 matrix of a Z. 29 00:01:58,410 --> 00:02:01,770 Now let's try to find a derivative of Equation 1.2. 30 00:02:02,160 --> 00:02:09,270 This will help us to get equality for could double load, which is given in Equation 1.4. 31 00:02:10,290 --> 00:02:16,980 Now what we need to do is to plug Equation 1.4 into robot dynamics equation, which will give us this 32 00:02:16,980 --> 00:02:18,690 new equation in matrix form. 33 00:02:19,110 --> 00:02:25,560 This is nothing new, and you can confirm that if you plug Equation 1.4 into one point three, you will 34 00:02:25,560 --> 00:02:29,760 get Equation 1.5 by just expanding Equation 1.5. 35 00:02:30,420 --> 00:02:37,620 Now we want to choose such control input that can linear rise and decouple our linear and coupled system 36 00:02:37,620 --> 00:02:38,280 equation. 37 00:02:38,910 --> 00:02:40,560 Here's the design choice for you. 38 00:02:40,980 --> 00:02:47,550 This is of the nothing new because we have done this several times for inverse dynamics and impedance 39 00:02:47,550 --> 00:02:48,090 control. 40 00:02:48,720 --> 00:02:55,170 A s and a lambda is our new control inputs in DeNiro's and decoupled system equation. 41 00:02:55,770 --> 00:03:03,120 If you plug Eq. 1.6 into Equation 1.5, you will obtain linear nearest and decouple system equation, 42 00:03:03,120 --> 00:03:05,400 as you can see in one point seven. 43 00:03:06,150 --> 00:03:07,290 Here is a small nod. 44 00:03:07,770 --> 00:03:15,030 As you can see, the relative degree of a s is to excuse me. 45 00:03:15,030 --> 00:03:17,040 The relative degree of S is two. 46 00:03:17,340 --> 00:03:25,410 Namely, we have to differentiate output of s two times to get input of S in output equation. 47 00:03:25,900 --> 00:03:31,560 However, the relative degree of lambda is zero, so we don't have to differentiate it to get input 48 00:03:31,680 --> 00:03:34,140 a number in the input equation. 49 00:03:34,830 --> 00:03:40,770 Excuse me, in the output equation, this is good news because lambda comes from force torque sensor, 50 00:03:40,980 --> 00:03:48,180 which is noisy and so we don't want to differentiate noisy signal after we have linear rise and decoupled 51 00:03:48,180 --> 00:03:53,280 system Eq., we can now choose control inputs to achieve desired task evolution. 52 00:03:53,520 --> 00:03:56,190 NIM asymptotically stable aerodynamics. 53 00:03:57,300 --> 00:04:04,020 Let's first design a s for a simple PDA control is enough because we just want to control position. 54 00:04:04,680 --> 00:04:11,700 The aerodynamics will be as in 1.9, which is asymptotically stable as KP and Kadia are both diagonal 55 00:04:11,700 --> 00:04:12,330 and positive. 56 00:04:12,330 --> 00:04:18,450 Definite four force control a lambda has been chosen as simple integral control. 57 00:04:18,840 --> 00:04:26,580 If we choose it as in 1.10, then force aerodynamics will be as in equation one point eleven, which 58 00:04:26,580 --> 00:04:30,540 is also asymptotically stable with positive, definite quality of care. 59 00:04:31,740 --> 00:04:33,630 Here is again a small knot equation. 60 00:04:33,630 --> 00:04:41,940 Lump 10 would be also correct if it would take Integral Tum out, but we added it to increase the robustness 61 00:04:41,940 --> 00:04:44,130 against constant disturbances. 62 00:04:44,430 --> 00:04:50,580 We also don't want to add derivative time because then we would have to differentiate lambda, which 63 00:04:50,580 --> 00:04:55,560 is noise a signal from for strong sensor from above equations. 64 00:04:55,560 --> 00:04:59,880 You can see that we have to find as s taught and lambda. 65 00:05:00,360 --> 00:05:05,310 The extraction of these parameters are very easy. 66 00:05:05,760 --> 00:05:13,070 It's obvious that in order to find, as we can use forward kinematics, we are given joint angles. 67 00:05:13,080 --> 00:05:14,580 Then it's easy to find this. 68 00:05:15,600 --> 00:05:20,100 We can get SE and lambda through selection mattresses here. 69 00:05:20,130 --> 00:05:25,560 Be careful that this is not normal inverse of Matrix, but so the inverse of Matrix. 70 00:05:25,980 --> 00:05:29,100 This is because selection mattresses can be non square. 71 00:05:30,090 --> 00:05:33,240 Here's a block diagram of hybrid position forms control. 72 00:05:33,720 --> 00:05:40,500 This is the control inputs we have designed for position and torque control and heavy use, so the inverse 73 00:05:40,560 --> 00:05:44,190 of a F and a V selection matters is to get control. 74 00:05:44,580 --> 00:05:51,120 I was, as we have said before, I want to show you this block diagram also, which is a simplified 75 00:05:51,120 --> 00:05:54,750 form of previous block diagram, but contains some key points. 76 00:05:55,620 --> 00:06:03,030 As we have said before, robot outputs, namely Estcourt and Lambda, are and with respect to the base 77 00:06:03,030 --> 00:06:08,250 frame of f zero, but we need to convert them to the tusk frame. 78 00:06:09,120 --> 00:06:14,730 But to apply by upload selection method to apply selection mattresses and get artificial and natural 79 00:06:14,730 --> 00:06:16,290 constraints separation. 80 00:06:16,650 --> 00:06:24,210 So the use our rotation matrix that will help us to get in the task constraint frame of EFSI, as we 81 00:06:24,210 --> 00:06:28,440 have said previously of the switching task frame the upslope so. 82 00:06:28,500 --> 00:06:35,100 The emergence of selection mattresses to select controllable parameters, and here is again, the union 83 00:06:35,100 --> 00:06:37,860 force and position or velocity controllers. 84 00:06:40,170 --> 00:06:43,470 It's not some small thing also and then finish. 85 00:06:43,470 --> 00:06:50,910 The lesson won't be considered until here that we don't have natural constraints that don't let the 86 00:06:51,300 --> 00:06:58,830 excuse me, that we have natural constraints that don't let robot either move or apply force torque. 87 00:06:59,520 --> 00:07:04,710 So initial constraints, constraint, motion and force two exactly zero. 88 00:07:05,310 --> 00:07:11,550 Because we have said in the beginning that robot and object are very rigid by object. 89 00:07:11,550 --> 00:07:15,180 I mean, environment and the environment is frictionless. 90 00:07:15,870 --> 00:07:22,290 But this is not correct in reality, and this caused some inconsistencies with our model. 91 00:07:22,770 --> 00:07:26,220 Namely, we have friction force in pure motion directions. 92 00:07:26,930 --> 00:07:32,160 And in the case of on friction, the force is increasing as normal force increases. 93 00:07:33,150 --> 00:07:38,640 The second inconsistency is due to the fact that the robot is not rigid but compliant. 94 00:07:39,000 --> 00:07:45,510 So when it touches the environment, it deforms and caused motion in no motion directions. 95 00:07:46,140 --> 00:07:48,390 How are these disturbances are small? 96 00:07:48,390 --> 00:07:49,770 Then they can be neglected. 97 00:07:50,130 --> 00:07:56,850 Additionally, if we know the geometry of the environment, precisely, these inconsistencies will automatically 98 00:07:56,850 --> 00:07:59,190 be removed due to the selection mattresses. 99 00:08:00,060 --> 00:08:02,670 So what's the general had done with the hybrid position? 100 00:08:02,670 --> 00:08:10,110 Force control technique is some kind of real time environment geometry estimation algorithm to get rid 101 00:08:10,110 --> 00:08:11,850 of these inconsistencies.