1 00:00:00,330 --> 00:00:05,910 Know, I know this is a luxurious topic, became too long, maybe, but I'm willing to give you every 2 00:00:05,910 --> 00:00:09,300 important detail you will need on this topic, really. 3 00:00:09,660 --> 00:00:16,650 I have discarded some long mathematical proofs and so on that you will so that you will not need. 4 00:00:17,400 --> 00:00:20,640 But, you know, we cannot discard a whole myth anyway. 5 00:00:20,670 --> 00:00:26,190 After coming up with a beautiful excuse, let's continue with the last video on the grunge dynamics. 6 00:00:26,970 --> 00:00:32,580 And this is linear permits possession of a little grunge dynamics and parameter identification. 7 00:00:33,600 --> 00:00:38,880 These words can be intimidating for now, but don't worry, we will see them clearly. 8 00:00:39,570 --> 00:00:45,840 Here's the famous dynamic model of the robot manipulator that we have come up with during the duration 9 00:00:45,840 --> 00:00:46,860 of this formula. 10 00:00:46,860 --> 00:00:56,250 We have seen that this formula contains some variables, namely angular position, velocity and acceleration 11 00:00:56,250 --> 00:00:59,600 terms, or cue to note and double load. 12 00:01:00,120 --> 00:01:05,730 As you know, already, formula is non-linear function of these variables, and this formula contains 13 00:01:05,730 --> 00:01:07,020 dynamic parameters. 14 00:01:07,440 --> 00:01:09,810 OK, what are these dynamic parameters? 15 00:01:10,260 --> 00:01:12,120 Let's assume this one thing. 16 00:01:12,180 --> 00:01:20,520 Namely, if I think of the robot manipulator in reveal joint for this thing, we can determine at least 17 00:01:20,730 --> 00:01:28,500 these parameters must MRI of the link, which is a similar parameter distance up to the center of mass 18 00:01:28,500 --> 00:01:33,210 of the link, which is indicated by RC, AI and the Vector Victoria. 19 00:01:33,210 --> 00:01:38,820 And it's a victory, a quantitative dimension of three by one with respect each axis. 20 00:01:39,870 --> 00:01:46,830 And finally, inertia tensor for the link with respect to the Frame FC, which is connected to the central 21 00:01:46,830 --> 00:01:48,000 mass of the link. 22 00:01:48,660 --> 00:01:55,530 While it has dimension of three by three, we know that it is symmetric and constant because it is defined 23 00:01:55,530 --> 00:02:01,650 with respect to the central of mass frame, and the only six elements are different. 24 00:02:02,040 --> 00:02:05,700 So all in all, we have ten parameters for each link. 25 00:02:06,210 --> 00:02:12,720 Why I have said that at least 10 parameters because we can take into account friction parameters also. 26 00:02:13,980 --> 00:02:20,340 So for any degree of freedom, a robot manipulator, there will be 10 m dynamic parameters in total. 27 00:02:20,700 --> 00:02:25,890 OK, but we don't know their values, so they have to be determined accurately. 28 00:02:25,920 --> 00:02:28,950 Y you will know this better in future lessons. 29 00:02:28,950 --> 00:02:36,030 But for now, let me say that if we have a good model in our hands, then we can use it in model based 30 00:02:36,030 --> 00:02:42,000 control of the robotic manipulators where we will utilize model of the robot in order to develop some 31 00:02:42,000 --> 00:02:44,370 control algorithms that control the robot. 32 00:02:45,000 --> 00:02:49,830 And effectiveness of these control algorithms depend on the accuracy of this model. 33 00:02:50,190 --> 00:02:55,560 Indeed, it is not correct to say that we don't have any knowledge about these parameters, because 34 00:02:55,560 --> 00:03:02,550 if we designed and built the cars, consider the robot manipulator, then we know these parameters from 35 00:03:02,550 --> 00:03:05,430 the cat model of the robot manipulator. 36 00:03:05,970 --> 00:03:12,030 However, when we realize this cad model, the parameters change because we have some imperfections. 37 00:03:12,630 --> 00:03:19,770 Not only this during usage of the robotic manipulator, we can add some sensors or gripper to the robot, 38 00:03:19,770 --> 00:03:22,830 which will also affect inertia mass parameters. 39 00:03:23,370 --> 00:03:29,520 Also, a robot manipulator is not designed to show off, but it will manipulate some objects which, 40 00:03:29,850 --> 00:03:36,540 as Gripper holds firm to these objects, they become part of it and surely changes some of the parameters. 41 00:03:37,200 --> 00:03:43,890 Moreover, as the time passes, lubrication of the robot also changes, and this affects also the parameters 42 00:03:44,130 --> 00:03:45,630 in the friction parameters. 43 00:03:46,110 --> 00:03:51,390 So we have to identify these parameters in order to have use a full model of the robot manipulator. 44 00:03:51,750 --> 00:03:58,560 Indeed, in adaptive control, we will use these concepts because adaptive control tries to adapt because 45 00:03:58,560 --> 00:04:00,750 of the changes in dynamic parameters. 46 00:04:01,200 --> 00:04:08,190 OK, now the question is that how we will determine these parameters, let's investigate that. 47 00:04:10,050 --> 00:04:15,540 No, that is the moment we have to study the linearity of dynamic model with respect to the dynamic 48 00:04:15,540 --> 00:04:17,370 parameters we have seen before. 49 00:04:17,790 --> 00:04:23,690 As we have said before, dynamic model is nonlinear function of Q, Cunard and tunable. 50 00:04:24,360 --> 00:04:27,870 However, it can be written as a linear function of dynamic parameters. 51 00:04:28,320 --> 00:04:29,880 So here is the formulation. 52 00:04:30,240 --> 00:04:36,870 As you can see in the red part, model is written as a function of dynamic parameters and the relationship 53 00:04:36,870 --> 00:04:37,620 is linear. 54 00:04:38,010 --> 00:04:47,640 Here, pie is nothing but a vector of dynamic parameters, and its dimension is 10 m by one as the robot 55 00:04:47,640 --> 00:04:55,140 is and degree of freedom, its vector of dynamic parameters, dynamic parameters vector, which has 56 00:04:55,140 --> 00:04:57,630 dimension of ten for each link. 57 00:05:00,020 --> 00:05:09,230 So what's why PI, then it's called regression metrics and is an upper triangle and is an upward three 58 00:05:09,230 --> 00:05:14,570 angular metrics each YJ component of its vector or dimension? 59 00:05:16,460 --> 00:05:17,000 Excuse me. 60 00:05:17,030 --> 00:05:22,700 Each white IG component of it is vector of Dimension one by 10. 61 00:05:23,420 --> 00:05:29,870 As you can see, regression matrix if nothing but the function of Q could not and could double that 62 00:05:30,500 --> 00:05:33,830 as regression matrix has an upper triangle structure. 63 00:05:34,130 --> 00:05:40,850 It means that eith dynamic equation depends on me or excuse me, for pronunciation. 64 00:05:41,210 --> 00:05:47,120 I dynamic equation depends on the on the dynamic parameters of links to N. 65 00:05:47,780 --> 00:05:55,340 You can observe this by multiplying regression Model X y time with X y Pi with PI. 66 00:05:55,760 --> 00:06:01,390 However, there is an issue here in the previous equation we have considered all the ten and parameters, 67 00:06:01,400 --> 00:06:07,040 and by doing that, we mean that all of the dynamic parameters for the people to participate in the 68 00:06:07,040 --> 00:06:09,500 dynamic equation of the remote manipulator. 69 00:06:10,070 --> 00:06:11,510 However, this is not correct. 70 00:06:11,780 --> 00:06:17,810 Some of these parameters will appear in the dynamic model of the equation, and B will not appear in 71 00:06:17,810 --> 00:06:19,550 the dynamic model of the equation. 72 00:06:19,550 --> 00:06:25,850 And because of that, corresponding columns or Y Pi regression meetings will be zero. 73 00:06:26,540 --> 00:06:28,460 Not only that, but also these. 74 00:06:28,460 --> 00:06:32,630 Some of these parameters will be a combination of other dynamic parameters. 75 00:06:32,930 --> 00:06:36,920 So some columns of the regression Model X will be linearly dependent. 76 00:06:37,490 --> 00:06:40,790 These will make regression matrix rank deficient. 77 00:06:41,090 --> 00:06:47,600 Both of aforementioned issues caused problems during Parameter ID because we will have to find inverse 78 00:06:47,600 --> 00:06:50,990 of the regression matrix, as you will see in a minute. 79 00:06:52,700 --> 00:06:55,010 So we have to get rid of these issues. 80 00:06:55,370 --> 00:06:56,570 How will we do that? 81 00:06:57,110 --> 00:07:04,580 We will isolate p of the dynamic parameters, which are independent and appear in the dynamic model 82 00:07:04,580 --> 00:07:05,870 of the robot manipulator. 83 00:07:06,260 --> 00:07:13,940 The number of these parameters are way less than the number of total dynamic parameters because because 84 00:07:13,940 --> 00:07:22,100 of this, this isolation, we can separate columns of regression metrics into two parts, namely P columns 85 00:07:22,100 --> 00:07:29,600 that are independent, which correspond to P isolated dynamic parameters, which we can indicate with 86 00:07:29,600 --> 00:07:38,000 PI independent and and 10 and minus P columns that correspond to the dependent parameters which are 87 00:07:38,000 --> 00:07:40,220 indicated with PI dependent. 88 00:07:42,470 --> 00:07:48,410 We can write Y dependent as the combination of the columns of the Y Independent. 89 00:07:50,450 --> 00:07:54,860 So then we can write our new dynamic model equation in this way. 90 00:07:56,840 --> 00:08:03,590 As you can see, both regression models and dynamic parameters are separated as dependent and independent. 91 00:08:04,160 --> 00:08:10,280 Then we can write dependent part of the regression Model X as a combination of the columns of independent 92 00:08:10,280 --> 00:08:11,750 part of the regression matrix. 93 00:08:15,550 --> 00:08:21,940 And after multiplication of regression metrics with dynamic parameters, Victor, we get this new formulation, 94 00:08:23,830 --> 00:08:32,920 as you can see here, regression metrics has dimension of not to end by 10 end, but end by P because 95 00:08:32,920 --> 00:08:40,210 we have all the independent columns and the Alpha has dimension of P by Ron. 96 00:08:40,540 --> 00:08:43,940 And it is called dynamic or efficient. 97 00:08:44,260 --> 00:08:50,500 Be careful, not dynamic parameters, but dynamic coefficients, which are combination of dynamic parameters. 98 00:08:51,070 --> 00:08:58,120 Indeed, as our dynamic equation consists of these new regression matrix and dynamic core efficiency. 99 00:08:58,390 --> 00:09:04,300 Then why we would try to determine each of the dynamic parameters, which is way more difficult. 100 00:09:04,660 --> 00:09:12,220 It's not for us to identify dynamic coefficients only not dynamic parameters, because dynamic coefficients 101 00:09:12,460 --> 00:09:16,060 constitute dynamic equation of our robot manipulator. 102 00:09:16,450 --> 00:09:17,800 So let's do that then. 103 00:09:18,370 --> 00:09:24,760 Oh, don't forget, robot dynamics is also a linear function of these dynamic or efficiency. 104 00:09:25,030 --> 00:09:33,790 And in the case of deiner, as in the case of dynamic parameters, after we have manipulated dynamic 105 00:09:33,790 --> 00:09:40,120 model of the manipulator and wrote it as a linear function of dynamic coefficients, we can convert 106 00:09:40,120 --> 00:09:47,710 our problem statement from identification of dynamic parameters to the identification of dynamic coefficients, 107 00:09:48,220 --> 00:09:53,170 and we will do that using least squares ID methods, which is very simple. 108 00:09:53,800 --> 00:09:57,450 So let's see new regression matrix and dynamic parameters. 109 00:09:57,460 --> 00:10:05,740 Excuse me, dynamic the quotations again, the want to identify by name dynamic coefficients. 110 00:10:06,250 --> 00:10:14,230 If you multiply the regression matrix with dynamic or efficient vector, we will get any linear equations 111 00:10:14,260 --> 00:10:21,070 with P unknowns, which is a problem because number of unknowns is higher than the number of equations 112 00:10:21,430 --> 00:10:28,210 we have to make it such that the number of equations are higher than the number of unknowns, so we 113 00:10:28,210 --> 00:10:35,110 can determine the value of unknowns that meets each of equations in the best way. 114 00:10:35,380 --> 00:10:40,780 So with minimum error, that is the purpose of least squares identification. 115 00:10:41,110 --> 00:10:42,580 So what we have to do? 116 00:10:43,090 --> 00:10:46,270 Let's see the process of identification step by step. 117 00:10:46,870 --> 00:10:52,090 First, we have to choose sufficiently exciting motion trajectory for our robot manipulator. 118 00:10:52,420 --> 00:10:53,440 What does it mean? 119 00:10:53,440 --> 00:10:55,930 Sufficiently exciting in a good way. 120 00:10:56,230 --> 00:11:04,030 It means that we have to choose our motion of trajectory in such a way that it will excite all the dynamic 121 00:11:04,030 --> 00:11:05,980 parameters by exciting. 122 00:11:05,980 --> 00:11:07,390 I mean, it will include. 123 00:11:07,810 --> 00:11:13,950 So we have to choose such a trajectory that will make robot to move over the whole workspace. 124 00:11:15,340 --> 00:11:21,820 We can control the robot by using more model based control methods as we don't have good model of the 125 00:11:21,820 --> 00:11:22,830 robot manipulator. 126 00:11:23,080 --> 00:11:30,730 We try to achieve it so we can use speed control, for example, in order to follow trajectory as good 127 00:11:30,730 --> 00:11:31,750 as possible. 128 00:11:32,050 --> 00:11:39,430 So with minimum error, at each instance, we will sample angular position, cue and combine it with 129 00:11:39,430 --> 00:11:42,870 input talk of you or toll. 130 00:11:44,050 --> 00:11:50,790 In our case, which will make together tuple of angular position and input torque v no input torque 131 00:11:50,810 --> 00:12:00,820 because v commanded and at the end we have n c number of angular position and input tuple. 132 00:12:00,970 --> 00:12:03,700 We are in C is sampling times. 133 00:12:04,660 --> 00:12:11,200 We can determine angular velocity and acceleration from Q by using numerical integration. 134 00:12:13,210 --> 00:12:17,680 Excuse me, and by using numerical derivation differentiation. 135 00:12:18,250 --> 00:12:20,890 Yeah, I'm sorry for the terminology. 136 00:12:22,660 --> 00:12:24,430 After this sampling process. 137 00:12:24,910 --> 00:12:36,700 We have not one Q Q Dot and Q double dot, but in C times Q Cunard and Q double dot samples so we can 138 00:12:36,700 --> 00:12:38,080 write about Eq.. 139 00:12:38,080 --> 00:12:46,060 We have seen for one sample of Q, Q and Q Double Toad for NC samples, and we will have regression 140 00:12:46,060 --> 00:12:53,050 metrics of dimension of NC times and by P. 141 00:12:53,500 --> 00:13:02,500 So we will have Mott and linear equations, but in C times and linear equations, which is surely higher 142 00:13:02,530 --> 00:13:05,920 than the number of P dynamic or efficient. 143 00:13:06,460 --> 00:13:09,430 So we can apply least squares method here. 144 00:13:10,120 --> 00:13:13,090 Let's call new regression metrics as y ma. 145 00:13:14,140 --> 00:13:21,550 It will be full column rank, namely ranking BP if they had used sufficiently exciting trajectory, 146 00:13:22,570 --> 00:13:30,600 and let's call this stack of trucks as tall bar as why Matrix is full column rank, we can find it's 147 00:13:30,610 --> 00:13:37,540 so the inverse, which will give us the best values for dynamic or efficiency we use. 148 00:13:37,600 --> 00:13:45,310 So the inverse, because as you can see why bottom matrix is not square, but told metrics namely number 149 00:13:45,310 --> 00:13:48,370 of rows are higher than the number of columns. 150 00:13:49,120 --> 00:13:56,230 So this is how we identify dynamic coefficients for the construction of dynamic model of robot manipulator. 151 00:13:56,830 --> 00:14:04,420 I want to stress that again, we don't identify dynamic parameters, but dynamic coefficients, which 152 00:14:04,420 --> 00:14:07,420 are a combination of dynamic parameters.