1 00:00:00,120 --> 00:00:06,750 OK, last time we have said that we need to find a new control algorithm that will make the process 2 00:00:06,750 --> 00:00:09,660 of making the quantum algorithm adaptive easier. 3 00:00:10,110 --> 00:00:14,070 Now let's say this algorithm here is the control algorithm. 4 00:00:14,340 --> 00:00:22,560 If you're asked why this, then why not the expansion of the terms curado and you are double dot is 5 00:00:22,560 --> 00:00:25,620 indicated also here. 6 00:00:27,210 --> 00:00:33,810 Sigma is nothing but the difference between reference velocity, QR code and actual velocity. 7 00:00:34,140 --> 00:00:34,380 Not. 8 00:00:35,220 --> 00:00:37,280 We will talk a bit curado. 9 00:00:37,290 --> 00:00:39,150 Don't worry here. 10 00:00:39,150 --> 00:00:43,920 Cady's sigma term is like PID control action if you expand it. 11 00:00:44,550 --> 00:00:50,370 Namely, derivative gain is KDDI and the proportional game is Lambda Times CD. 12 00:00:50,850 --> 00:00:59,930 Lambda and Katie are positive def. mattresses, as you can see different from the previous algorithms. 13 00:00:59,940 --> 00:01:10,680 Here we have used reference velocity of curado instead of Cunard, because if we put Q Kudi Dot instead 14 00:01:10,680 --> 00:01:22,290 of curado, this algorithm will achieve a sum total convergence to zero in terms of velocity tracking, 15 00:01:22,680 --> 00:01:25,140 but the position alone will not be zero. 16 00:01:25,710 --> 00:01:28,470 But we want both of them to be zero. 17 00:01:28,740 --> 00:01:35,430 So we change reference velocity to Q ordered, which also includes the position error term. 18 00:01:35,790 --> 00:01:41,430 So a robot moves until desired position until desired position is achieved. 19 00:01:42,030 --> 00:01:42,480 Perfect. 20 00:01:42,930 --> 00:01:45,120 Now let's try to get aerodynamics. 21 00:01:45,120 --> 00:01:51,540 We can achieve this by substituting Eq. one into the standard robot dynamics equation. 22 00:01:52,740 --> 00:01:57,660 Then we will obtain Equation 1.1, which represents aerodynamics. 23 00:01:58,290 --> 00:02:05,100 We want to prove that the aerodynamics is asymptotically stable, so our position around, namely QTL 24 00:02:05,430 --> 00:02:09,600 and velocity aerodynamic kubert tilde converge to zero. 25 00:02:10,410 --> 00:02:15,660 Surely we will use the option of method to prove that, as we have done before. 26 00:02:16,590 --> 00:02:18,300 Here is the opponent function. 27 00:02:19,020 --> 00:02:26,760 As you can see, it is a valid Liverpool of version because it is positive definite in all values of 28 00:02:26,760 --> 00:02:34,230 Sigma and Q tilde except zero, and it becomes zero when sigma until then becomes zero. 29 00:02:34,770 --> 00:02:40,680 It contains only sigma and curtail the terms because we want these to be zero at the end. 30 00:02:41,040 --> 00:02:48,090 If Sigma and the Q Tilde becomes zero, then tell us that the arrow tilde automatically becomes zero. 31 00:02:48,360 --> 00:02:55,800 Because of the construction of sigma, namely sigma is equal to the QTL dot dot lambda 32 00:02:58,260 --> 00:02:59,910 equals the Q till the dot. 33 00:03:01,870 --> 00:03:10,900 Minus number to tilt up if Cutelaba and Sigma becomes zero til the thought has to be zero automatically, 34 00:03:11,860 --> 00:03:14,680 no, let's find a derivative of the Liverpool function. 35 00:03:14,980 --> 00:03:19,270 I will not go in the deeper calculations because we have seen the same before. 36 00:03:19,660 --> 00:03:26,650 We can extract M Sigma the term from Equation 1.1 and plug it here. 37 00:03:27,370 --> 00:03:31,090 After some manipulation, we can reach this step. 38 00:03:31,450 --> 00:03:38,560 No, I want you to remember this equality we have seen before. 39 00:03:38,950 --> 00:03:41,580 This is true because we have chosen. 40 00:03:41,620 --> 00:03:48,200 This is true because we have chosen crystal symbols to construct semantics by considering this property. 41 00:03:48,460 --> 00:03:50,290 We can write equation in this way. 42 00:03:51,070 --> 00:03:52,570 Here we have three terms. 43 00:03:54,520 --> 00:04:00,430 Two of these terms are negative death, and we know that because they are quadratic terms with negative 44 00:04:00,430 --> 00:04:00,820 signs. 45 00:04:01,750 --> 00:04:02,920 What about the third one? 46 00:04:03,310 --> 00:04:10,330 We don't know, because it consists of multiplication of two different terms that can have different 47 00:04:10,330 --> 00:04:10,810 signs. 48 00:04:11,230 --> 00:04:19,990 So we need to eliminate this term if we choose a in this way, we can write equation like that, which 49 00:04:19,990 --> 00:04:20,970 is surely negative. 50 00:04:20,980 --> 00:04:22,270 That's not perfect. 51 00:04:22,600 --> 00:04:28,900 We get that the derivative of the Apollo function is negative definite and it becomes zero only one 52 00:04:28,900 --> 00:04:30,150 position and velocity. 53 00:04:30,150 --> 00:04:31,570 The error becomes zero. 54 00:04:32,110 --> 00:04:37,330 Because in order we thought to be zero, Q tilde and sigma have to be zero. 55 00:04:38,770 --> 00:04:42,010 We get asymptotic stability with this algorithm. 56 00:04:42,470 --> 00:04:47,680 Now what we have to do, we have to first take into account uncertainties. 57 00:04:48,280 --> 00:04:51,940 The exact terms will be replaced by estimates. 58 00:04:52,840 --> 00:04:58,120 Then we will estimate the parameters and adapt the controller to these estimations. 59 00:04:58,570 --> 00:05:01,480 We will do it by choosing correct updates. 60 00:05:01,480 --> 00:05:08,170 Rule four Alpha had to name the update of the estimations or the dynamic parameters. 61 00:05:09,880 --> 00:05:16,870 Here we have written the same control algorithm with the estimations of the dynamic terms so we can 62 00:05:16,870 --> 00:05:20,500 use parametric unionisation property again. 63 00:05:20,770 --> 00:05:26,590 Where I had is nothing but estimation of the dynamical efficiency. 64 00:05:28,090 --> 00:05:35,260 Here, I want to note that regression matrix does not depend on Q double the term, which is good because 65 00:05:35,260 --> 00:05:38,650 acceleration term is always noisy. 66 00:05:39,280 --> 00:05:46,120 No, as you guess, we have to obtain error dynamics by substituting Equation 1.0 into the standard 67 00:05:46,120 --> 00:05:47,860 robot manipulator dynamics. 68 00:05:48,580 --> 00:05:49,960 Here we have done this. 69 00:05:51,100 --> 00:05:57,730 If you consider these two equations, you can substitute their values in the equation and obtain this 70 00:05:57,730 --> 00:05:58,450 equality. 71 00:05:58,870 --> 00:06:01,210 This manipulation of equation is very easy. 72 00:06:01,210 --> 00:06:02,800 You can do it by yourself. 73 00:06:03,370 --> 00:06:09,130 Tilde terms here shows that there are deviations of terms from their true values. 74 00:06:09,760 --> 00:06:21,160 So until the terms show deviations had terms, show estimations and the terms without hat or tilde shows 75 00:06:21,250 --> 00:06:22,660 the correct values. 76 00:06:23,020 --> 00:06:28,210 You can again use parameter linear regression property and write equation like that. 77 00:06:29,170 --> 00:06:37,810 As you can see, this term does not equal to zero, which is normal because as we have used estimates 78 00:06:37,810 --> 00:06:45,910 of the terms aerodynamics, we do not convert to zero, but we want it to convert to zero asymptotically. 79 00:06:46,060 --> 00:06:53,500 We can make this term zero, either making alpha tilde zero, which means we have to find correct estimates 80 00:06:54,610 --> 00:06:58,780 of the dynamic parameters, which is difficult and is not our goal. 81 00:06:59,140 --> 00:07:06,880 The second is much easier and trivial where we want to make over tilde to be in the null space of regression 82 00:07:06,880 --> 00:07:07,930 matrix, right? 83 00:07:08,380 --> 00:07:10,480 So their product will be zero. 84 00:07:10,870 --> 00:07:17,800 We can do that by finding correct update room for dynamic parameters that will ensure that they are 85 00:07:17,800 --> 00:07:21,490 inside the null space of Y matrix or regression matrix. 86 00:07:22,000 --> 00:07:28,780 If they want to achieve a sum total convergence, we can do that using the optimal method so people 87 00:07:28,780 --> 00:07:35,770 choose newly appointed function and make its derivative negative definite to get us in total convergence 88 00:07:35,770 --> 00:07:37,210 in error dynamics. 89 00:07:37,690 --> 00:07:38,500 Let's do this. 90 00:07:39,130 --> 00:07:40,930 Here is the newly appointed function. 91 00:07:40,930 --> 00:07:46,780 As you can see, it's what part is the old Liverpool function that we have seen earlier. 92 00:07:47,050 --> 00:07:50,260 The only new term is added is green term. 93 00:07:50,710 --> 00:07:56,740 This is valid Liverpool function because it has a positive definite for all the values of Sigma Q tilde 94 00:07:56,740 --> 00:08:00,940 and alpha tilde and zero when these three variables are. 95 00:08:01,070 --> 00:08:08,420 Europe, as you can see it, the included over her term as the third variable indicating we want to 96 00:08:08,420 --> 00:08:16,730 be it to be, we want it to be zero, but you will see that we will not be able to do that, but we 97 00:08:16,730 --> 00:08:19,670 will use it for making vetoed negative. 98 00:08:19,670 --> 00:08:20,210 Definitely. 99 00:08:20,420 --> 00:08:24,020 Let's analyze derivative of the Liverpool function. 100 00:08:24,410 --> 00:08:30,890 I didn't try hard to obtain derivative of these because these are simple calculations that you can do 101 00:08:30,890 --> 00:08:31,850 by yourself. 102 00:08:32,300 --> 00:08:35,900 Additionally, we have seen how we can do this before. 103 00:08:36,350 --> 00:08:42,650 As you can see, its first part is like the derivative of the old, the Apollo function, and we know 104 00:08:42,650 --> 00:08:44,480 that it's negative definite. 105 00:08:44,960 --> 00:08:50,570 But the second part is new indicated with green color, and we don't know it's on. 106 00:08:50,990 --> 00:08:56,120 So we either need the excuse me, so we need to eliminate this term. 107 00:08:56,600 --> 00:08:59,240 Before doing that, let me point that out. 108 00:09:00,350 --> 00:09:05,960 Over tilde is deviation of dynamic parameter estimation from their true values. 109 00:09:06,380 --> 00:09:12,380 As you can see, the second part of the derivative of the opponent function contains derivative of the 110 00:09:12,380 --> 00:09:13,700 A4 till the term. 111 00:09:14,180 --> 00:09:21,770 So this is the derivation derivative of alpha tilde, as alpha had, will be constant because it is 112 00:09:21,770 --> 00:09:22,600 true value. 113 00:09:23,150 --> 00:09:25,190 It is its true values. 114 00:09:25,550 --> 00:09:32,030 It is true of values and it will be constant, so its derivative will be zero. 115 00:09:32,870 --> 00:09:40,610 So derivative of error in dynamic parameters is the same as a derivative of the estimates of the dynamic 116 00:09:40,610 --> 00:09:41,360 parameters. 117 00:09:42,530 --> 00:09:49,310 So we need to choose such a betrayal of the dynamic parameters, namely, if had thought, that will 118 00:09:49,310 --> 00:09:50,950 make green terms zero. 119 00:09:50,960 --> 00:09:52,850 So we will get negative. 120 00:09:52,850 --> 00:09:59,090 Definitely opponent function that negative definite derivative of the Apollo function. 121 00:10:00,290 --> 00:10:02,600 OK, we want this term to be zero. 122 00:10:05,060 --> 00:10:05,990 We can find. 123 00:10:05,990 --> 00:10:13,760 If I had thought from this equation easily like that and this is the dynamic parameter update rule that 124 00:10:13,760 --> 00:10:15,170 they wanted to get. 125 00:10:15,980 --> 00:10:22,280 So by choosing Alpha had thought in this way, we will make the derivative of the Liverpool function 126 00:10:22,280 --> 00:10:23,360 negative definite. 127 00:10:23,720 --> 00:10:29,720 Here's the interesting thing as you can see, the note doesn't contain alpha hot term. 128 00:10:30,020 --> 00:10:35,660 So when we do not become zero, Kittila and QTL, the Dot will be zero. 129 00:10:35,990 --> 00:10:42,260 Namely, we will get desired trajectory tracking, but we cannot see that four tilde will be zero, 130 00:10:42,560 --> 00:10:48,650 so we cannot see that estimation of dynamic parameters will convert to their correct values. 131 00:10:48,650 --> 00:10:54,860 Also, estimates of dynamic parameters will be bounded, namely not equal to zero. 132 00:10:55,520 --> 00:11:04,040 So with the aforementioned controlled low and dynamic parameters of tetlow, we guaranteed that aerodynamics 133 00:11:04,040 --> 00:11:05,390 view converts to zero. 134 00:11:05,630 --> 00:11:10,100 As we have seen before, this can be happen in two cases. 135 00:11:10,550 --> 00:11:18,950 Either Alpha Hut will be in the space of Y, which we guaranteed that or alpha becomes zero. 136 00:11:19,310 --> 00:11:26,270 While we have said that, we cannot guarantee happening off that this can happen if trajectory to be 137 00:11:26,270 --> 00:11:33,650 tracked is persistently exciting, which we have seen this topic when we talked about the linear parameterization 138 00:11:33,650 --> 00:11:35,960 of global dynamics equation. 139 00:11:37,040 --> 00:11:47,810 So as you can see by adaptive control, we get smoother control action and opposite to the robust keys 140 00:11:48,320 --> 00:11:55,040 because in this case, the dynamic parameters are updated at each sampling time. 141 00:11:55,040 --> 00:12:06,050 Instead of like switching from high control, high power control, which was which caused noise and 142 00:12:06,230 --> 00:12:12,110 also was harmful for the structure of the robot manipulator. 143 00:12:12,560 --> 00:12:23,780 Additionally, other than opposite to the robust control case for adaptive control, um, just accept 144 00:12:23,780 --> 00:12:27,530 disturbances in the robot manipulator. 145 00:12:27,770 --> 00:12:35,660 And for example, if it grabs some object or the friction parameters change or other parameter dynamic 146 00:12:35,660 --> 00:12:42,230 parameters change and or other changes in the robot structure it all. 147 00:12:43,160 --> 00:12:50,390 Adaptive control accept all of these disturbances as change in dynamic parameters and tries to adapt 148 00:12:50,570 --> 00:12:59,360 to these dynamic parameters instead of producing high control action in robust control keys. 149 00:12:59,840 --> 00:13:00,410 So. 150 00:13:01,260 --> 00:13:10,470 Every disturbance is accepted as a dynamic parameter change in adaptive control and adaptive control 151 00:13:10,470 --> 00:13:21,450 tries to adapt or update their estimates of these dynamic parameters, such that it will get a sympathetic 152 00:13:21,450 --> 00:13:29,550 trajectory control so a thought asymptotically convergence in terms of trajectory tracking error.