1 00:00:00,090 --> 00:00:03,920 OK, from this point, we are going to start with advanced controls. 2 00:00:04,410 --> 00:00:09,960 We will start first with the robust controller tower and let me give you an insight why we need advanced 3 00:00:09,960 --> 00:00:13,290 controller before dark jumping to the robust controller. 4 00:00:13,950 --> 00:00:16,590 Here's a dynamic equation of the robot manipulator. 5 00:00:17,040 --> 00:00:19,340 We can write this equation like that. 6 00:00:19,350 --> 00:00:24,000 Also, we are and includes the terms like friction, gravity and Coriolis. 7 00:00:24,630 --> 00:00:28,990 We'll write it in this way in order to simplify the calculations we will do. 8 00:00:29,490 --> 00:00:35,520 As you remember, in previous controllers, we have always use this dynamic model of the robot manipulator 9 00:00:35,760 --> 00:00:39,120 and use these terms during development of the controllers. 10 00:00:39,690 --> 00:00:41,270 And we got the results we want. 11 00:00:41,280 --> 00:00:48,090 Indeed, with control of robot manipulator, our main goal is to achieve asymptotic stability in terms 12 00:00:48,090 --> 00:00:49,710 of trajectory tracking errors. 13 00:00:50,250 --> 00:00:55,590 And we achieved this result with previous control techniques like inverse dynamics control. 14 00:00:56,790 --> 00:01:03,690 However, while this equation of robot dynamics is correct, in ideal case, it is not valid during 15 00:01:03,690 --> 00:01:06,450 practical cases or in implementation. 16 00:01:06,960 --> 00:01:09,240 So what is the problem with that equation? 17 00:01:09,900 --> 00:01:16,020 Firstly, by writing Dynamic Equation as a book, we assume that we know all the dynamic parameters 18 00:01:16,230 --> 00:01:17,850 which makes dynamic terms. 19 00:01:18,540 --> 00:01:19,530 Is that correct? 20 00:01:19,740 --> 00:01:20,460 Surely not. 21 00:01:20,760 --> 00:01:24,150 We don't know all the dynamic parameters of robot manipulator. 22 00:01:24,450 --> 00:01:27,540 Even the ones we know contain some uncertainties. 23 00:01:27,810 --> 00:01:33,960 Additionally, robot dynamic parameters change as the robot handles some objects, so we cannot say 24 00:01:33,960 --> 00:01:36,180 that we know robot dynamic terms. 25 00:01:36,180 --> 00:01:36,900 Exactly. 26 00:01:37,380 --> 00:01:42,750 Additionally, in previous control techniques, we didn't consider the simplifications we made in order 27 00:01:42,750 --> 00:01:44,520 to achieve the above dynamic model. 28 00:01:44,910 --> 00:01:48,570 We simplified many effects in the robot manipulator like friction. 29 00:01:48,960 --> 00:01:52,690 Also, there are other external disturbances that we cannot model. 30 00:01:52,980 --> 00:01:59,700 All of these affect the robot manipulator dynamics and in previous control methods, we didn't take 31 00:01:59,910 --> 00:02:01,050 them into account. 32 00:02:01,470 --> 00:02:07,080 So all the assumptions we previously made are not practical and doesn't work or work poorly for the 33 00:02:07,080 --> 00:02:07,680 real life. 34 00:02:08,070 --> 00:02:13,920 If we take into account the uncertainties in the robot dynamic model, then we can ride the dynamic 35 00:02:13,920 --> 00:02:14,340 model. 36 00:02:14,340 --> 00:02:21,360 In this way, we are terms with hat indicates that they are just estimates, so that means that it is 37 00:02:21,360 --> 00:02:24,480 just an estimate of the term M and so on. 38 00:02:24,840 --> 00:02:31,140 So in reality, we have all estimations of dynamic terms that are constituted by dynamic parameters, 39 00:02:31,620 --> 00:02:35,040 namely, as we don't know exactly dynamic parameters. 40 00:02:35,040 --> 00:02:40,170 We don't know exactly also dynamic terms of the dynamic model of robot arm. 41 00:02:40,920 --> 00:02:46,320 So we also have disturbances due to simplifications in the dynamic model, as we have said before, 42 00:02:46,890 --> 00:02:53,520 and we can also have external disturbances that are not because of the are modeled or simplified dynamics, 43 00:02:53,730 --> 00:02:56,010 but because of the environmental effects. 44 00:02:56,730 --> 00:02:59,850 So here you may ask OK, OK, enough. 45 00:03:00,150 --> 00:03:04,560 I understood the problems we have, but how they cause problems. 46 00:03:05,070 --> 00:03:08,790 So let's analyze the problems we have with previous control methods. 47 00:03:09,030 --> 00:03:17,640 If we take into consideration the nonlinear tis not ideologies, as we have said before, the purpose 48 00:03:17,640 --> 00:03:23,550 of the robot manipulator is to achieve asymptotic convergence in terms of trajectory tracking errors. 49 00:03:23,910 --> 00:03:29,220 So we won't cutelaba tilde and QTL the team, the position and velocity errors to go to zero. 50 00:03:29,610 --> 00:03:34,050 OK, let's consider inverse dynamics control technique we have seen before. 51 00:03:34,410 --> 00:03:40,140 Indeed, it is effective control method, so we will try to make it robust during the series of robust 52 00:03:40,140 --> 00:03:40,680 control. 53 00:03:41,040 --> 00:03:46,770 But now we want to analyze what will happen with this control technique if we consider them online realities. 54 00:03:47,160 --> 00:03:51,630 As you remember, in inverse dynamics, we have taken input control as this. 55 00:03:52,140 --> 00:03:55,710 If you don't remember, please go and repeat inverse dynamics control. 56 00:03:56,100 --> 00:04:01,860 Perfect as you can see, dynamic terms that make input control are considered as ideal. 57 00:04:02,340 --> 00:04:08,100 With this control input, we have achieved exactly initialization and decoupling of nonlinear dynamics, 58 00:04:08,370 --> 00:04:14,670 which led us lets us to use linear control, such as speed control on the linear system. 59 00:04:15,030 --> 00:04:19,260 So we were able to get asymptotic convergence in aerodynamics. 60 00:04:19,620 --> 00:04:26,460 But now let's take estimations into account, which we have mentioned before, which we have mentioned 61 00:04:26,460 --> 00:04:30,150 before, and changed the dynamic terms with the ones we had. 62 00:04:30,450 --> 00:04:35,460 So this is not ideal case control input because of estimated dynamic terms. 63 00:04:35,760 --> 00:04:42,520 Now it's interesting for us what will be the aerodynamics with this control input and to achieve it. 64 00:04:42,540 --> 00:04:48,690 We have to plug Eq. 1.0 into the standard robot dynamics equation, as we have seen before. 65 00:04:49,350 --> 00:04:51,060 OK, let's do this now. 66 00:04:51,070 --> 00:04:54,930 We will do some manipulations with this equation to get aerodynamics. 67 00:04:55,200 --> 00:04:59,730 First, we will add m y and then subtract it again from. 68 00:04:59,770 --> 00:05:01,420 The right hand side of the equation. 69 00:05:01,840 --> 00:05:07,420 So after some very simple manipulations, we can get this equality from here. 70 00:05:07,420 --> 00:05:09,850 We can find cure double talk very easily. 71 00:05:09,880 --> 00:05:17,620 Let's call this term as Tita and write equation in this way, which is much simpler from this term. 72 00:05:17,620 --> 00:05:22,120 You can easily notice the beginning of the problem before you double. 73 00:05:22,120 --> 00:05:23,990 That was, it was equal to Y. 74 00:05:24,010 --> 00:05:25,900 However, in this case, it is not. 75 00:05:26,290 --> 00:05:32,230 We have it a term which constitutes nonlinear terms because we know only estimations of the dynamic 76 00:05:32,230 --> 00:05:32,590 terms. 77 00:05:33,040 --> 00:05:36,340 So we couldn't achieve exact initialization and decoupling. 78 00:05:36,670 --> 00:05:40,450 This means that our new system is still nonlinear and coupled. 79 00:05:40,880 --> 00:05:47,800 OK, let's ignore that and continue by choosing white as linear control and acceleration feedback as 80 00:05:47,800 --> 00:05:51,430 we have done before, in ideal case, inverse dynamics. 81 00:05:51,580 --> 00:05:55,960 Let's plug why into the above equation and obtain this aerodynamics. 82 00:05:55,960 --> 00:06:00,580 As you remember, KP and Key D gains are positive definite mattresses. 83 00:06:01,030 --> 00:06:06,070 So as you can see, aerodynamics is stable, but wait a minute. 84 00:06:06,430 --> 00:06:11,800 While it is stable, it doesn't convert to zero, but ETA, which is not equal to zero. 85 00:06:12,220 --> 00:06:18,430 If you replace estimation with exact terms in ETA term, you will see that ETA will becomes zero. 86 00:06:18,730 --> 00:06:24,550 Unfortunately, we know only estimations of dynamic terms, so we cannot convert to zero. 87 00:06:24,940 --> 00:06:31,870 This means that a simple linear controller like PD is not enough for us any longer because the system 88 00:06:31,870 --> 00:06:33,850 equation is nonlinear and coupled. 89 00:06:34,240 --> 00:06:37,390 So what we have to do, what is the solution? 90 00:06:38,620 --> 00:06:41,140 Here comes the robust control into the help. 91 00:06:41,440 --> 00:06:43,330 What we will do is simple. 92 00:06:43,540 --> 00:06:48,580 I mean, simple to lead to at least, but a bit harder to implement. 93 00:06:48,970 --> 00:06:55,210 We will keep control with feedback the initialization because effective control technique with respect 94 00:06:55,210 --> 00:07:01,990 to others we have seen before, but instead we will add it robustness by designing ultra look for the 95 00:07:01,990 --> 00:07:09,430 control technique so you can see general structure of the robot general structure of the controller 96 00:07:09,430 --> 00:07:10,240 in this picture. 97 00:07:10,570 --> 00:07:17,620 We will have inverse dynamics control to do not exact, but some degree of animation of nonlinear terms, 98 00:07:17,830 --> 00:07:21,340 and we will use PD control for stabilizing purposes. 99 00:07:21,730 --> 00:07:27,640 We have seen that while PD control provides stabilization, it doesn't provide asymptotic stabilization, 100 00:07:27,640 --> 00:07:29,260 namely convergence to zero. 101 00:07:29,650 --> 00:07:31,960 So we add robust control part. 102 00:07:32,620 --> 00:07:38,710 Surely there will be input to the robustness part also, but I didn't draw them here for simplification 103 00:07:38,710 --> 00:07:39,310 purposes. 104 00:07:39,670 --> 00:07:45,340 This robust control part will ensure that it will reject the disturbances, either due to the external 105 00:07:45,340 --> 00:07:49,410 reasons or estimation errors in nonlinear dynamics.