1 00:00:00,720 --> 00:00:01,530 ‫Welcome back. 2 00:00:02,160 --> 00:00:10,440 ‫So let's now play around a little bit with our omega men and omega max variables and see what we get. 3 00:00:11,010 --> 00:00:12,810 ‫So let's start with these values. 4 00:00:13,500 --> 00:00:17,730 ‫This is approximately 115 radians per second. 5 00:00:18,330 --> 00:00:21,930 ‫And this is approximately 900 radians per second. 6 00:00:22,530 --> 00:00:25,170 ‫We're going to work with this trajectory here. 7 00:00:25,800 --> 00:00:27,720 ‫As you can see, tracking is pretty good. 8 00:00:28,350 --> 00:00:31,350 ‫But what we are interested in is this one here. 9 00:00:31,830 --> 00:00:38,280 ‫These are your Omega's Omega one, Omega two, Omega three and Omega four. 10 00:00:38,880 --> 00:00:46,940 ‫So you can see that you go from more or less 115 radians per second at the very beginning. 11 00:00:47,610 --> 00:00:54,510 ‫Because remember our initial four omega 3s, they were omega men. 12 00:00:55,170 --> 00:01:02,910 ‫So no matter what you do at the very beginning, they have your minimum values that you define right 13 00:01:02,910 --> 00:01:03,330 ‫here. 14 00:01:03,900 --> 00:01:12,630 ‫And you can see that all these omega as they go until 800 radians per second, so they don't even reach 15 00:01:13,200 --> 00:01:14,370 ‫our upper limit. 16 00:01:15,000 --> 00:01:21,810 ‫And you can see that at the very beginning, they start rotating very fast and then their rotational 17 00:01:21,810 --> 00:01:26,670 ‫velocities, they slow down and they start oscillating like that. 18 00:01:27,240 --> 00:01:33,120 ‫It makes sense because at the beginning you start from your ground and you have to go up in order to 19 00:01:33,120 --> 00:01:34,380 ‫reach your trajectory. 20 00:01:34,830 --> 00:01:42,570 ‫And then once you have reached your trajectory, then you have to go up and down, up and down in waves. 21 00:01:43,170 --> 00:01:50,400 ‫And that's why the omega 3s, they also behave like that as a function of time when you need to go up. 22 00:01:50,700 --> 00:01:53,970 ‫They rotate a bit faster when you need to go down. 23 00:01:54,240 --> 00:01:55,770 ‫They rotate a bit slower. 24 00:01:56,340 --> 00:02:03,720 ‫And so here I've taken my omega one and I've zoomed in this wavy behavior. 25 00:02:04,020 --> 00:02:14,730 ‫And so you can see that this rotational speed, it varies more or less between 540 and 460 gradients 26 00:02:14,730 --> 00:02:15,360 ‫per second. 27 00:02:15,930 --> 00:02:20,850 ‫So I have simply taken one wave here and I've zoomed it in. 28 00:02:21,480 --> 00:02:25,920 ‫But OK, now let's leave omega men alone. 29 00:02:26,580 --> 00:02:36,030 ‫But for omega max, let's put here times zero plus 700 that will make this term zero. 30 00:02:36,660 --> 00:02:43,620 ‫And then I had 700 here, so my omega max, it will be 700 gradients per second. 31 00:02:44,220 --> 00:02:47,460 ‫If I run the code, then I will get this result. 32 00:02:48,090 --> 00:02:56,370 ‫You don't see the number here, but if I zoom in, then you can see that once the rotational velocities 33 00:02:56,970 --> 00:03:04,860 ‫reach 700 radians per second for each motor, then they don't go up any further. 34 00:03:05,400 --> 00:03:06,390 ‫So that's very good. 35 00:03:07,020 --> 00:03:18,180 ‫How about if instead of 700, I put here 600 now my omega max, it's 600 radians per second because 36 00:03:18,180 --> 00:03:19,830 ‫this term here it's zero. 37 00:03:20,430 --> 00:03:21,780 ‫Well, this is the result. 38 00:03:22,290 --> 00:03:25,140 ‫You can see that all your omega 3s. 39 00:03:25,800 --> 00:03:30,840 ‫They don't go above 600 radians per second, so that's very good as well. 40 00:03:31,410 --> 00:03:38,820 ‫But if you remember then a couple of pictures ago this oscillation here, it happened more or less between 41 00:03:39,300 --> 00:03:44,190 ‫540 and 460 radians per second. 42 00:03:44,850 --> 00:03:47,760 ‫Well, how about for my omega men? 43 00:03:48,420 --> 00:03:59,340 ‫I'm going to put time zero plus 400 and 80, and then for my omega max, I'm going to put 520. 44 00:03:59,910 --> 00:04:01,140 ‫What will happen then? 45 00:04:01,770 --> 00:04:10,950 ‫Remember, normally the Omega has oscillated between more or less 540 and 460 irradiance per second. 46 00:04:11,550 --> 00:04:17,430 ‫Now I have 520 and 480 radians per second. 47 00:04:18,030 --> 00:04:19,890 ‫So the interval is very tight. 48 00:04:20,430 --> 00:04:22,140 ‫Well, this is the result. 49 00:04:22,320 --> 00:04:29,060 ‫As you can see, your tracking is not very good now, and it makes sense if you look at your omega is 50 00:04:29,070 --> 00:04:29,530 ‫now. 51 00:04:29,970 --> 00:04:37,950 ‫Then you can see that they stay between 520 and 400 and 80 radians per second. 52 00:04:38,580 --> 00:04:42,390 ‫So all your four motors, they obey your constraints. 53 00:04:43,020 --> 00:04:50,520 ‫But of course, because of these narrow constraints, your drone simply cannot follow the trajectory 54 00:04:51,120 --> 00:04:53,130 ‫in order to follow this trajectory. 55 00:04:53,760 --> 00:04:57,450 ‫You cannot restrict the rotation of your omega is like that. 56 00:04:58,050 --> 00:04:59,460 ‫This is too restrictive. 57 00:05:00,070 --> 00:05:02,440 ‫You have to give your amigas more freedom. 58 00:05:03,040 --> 00:05:06,880 ‫But in conclusion, you can see that your constraints are working. 59 00:05:07,450 --> 00:05:13,720 ‫Now, of course, you might wonder why aren't they working perfectly? 60 00:05:14,320 --> 00:05:24,400 ‫Why do I go slightly over 520 radians per second and why do I go slightly below 480 radians per second? 61 00:05:25,000 --> 00:05:30,310 ‫And also here I go, slightly above 600 radians per second. 62 00:05:30,910 --> 00:05:31,750 ‫And the same thing. 63 00:05:31,750 --> 00:05:35,950 ‫Here I go slightly over 700 radians per second. 64 00:05:36,550 --> 00:05:37,690 ‫Why is it like that? 65 00:05:38,140 --> 00:05:40,330 ‫Well, the explanation is very straightforward. 66 00:05:40,990 --> 00:05:43,420 ‫This is our global control architecture. 67 00:05:43,870 --> 00:05:48,880 ‫But now let's zoom in and only consider our inner loop. 68 00:05:49,540 --> 00:05:52,840 ‫So this is your plant and this is your NPC. 69 00:05:53,470 --> 00:06:01,060 ‫Remember that your plant and your NPC had two very similar but still different models. 70 00:06:01,690 --> 00:06:11,680 ‫If you remember, then in the plant box, we had a transfer matrix that depended on two angles Fei and 71 00:06:11,680 --> 00:06:12,160 ‫Seta. 72 00:06:12,760 --> 00:06:21,190 ‫But for the NPC controller, we approximated this transfer matrix to be an identity matrix because we 73 00:06:21,190 --> 00:06:24,760 ‫said that the drone flies close to the hovering position. 74 00:06:25,390 --> 00:06:33,850 ‫And so we simply made this assumption that Y equals zero and C equals zero because in reality, they 75 00:06:33,850 --> 00:06:35,080 ‫were close to zero. 76 00:06:35,680 --> 00:06:40,660 ‫And because of that assumption, this transfer matrix became an identity matrix. 77 00:06:41,320 --> 00:06:46,420 ‫And so we used this assumption inside our HPC model. 78 00:06:47,080 --> 00:06:56,020 ‫The planned model was more realistic, but the NPC model was easier to deal with and put in the LP format. 79 00:06:56,440 --> 00:07:02,530 ‫And while the differences are small, this assumption still introduces some differences here. 80 00:07:03,100 --> 00:07:07,630 ‫It makes the plan model and the NPC model slightly different. 81 00:07:08,170 --> 00:07:13,500 ‫Another thing here is that in this course, we used simplified LP. 82 00:07:14,170 --> 00:07:22,360 ‫This was our simplified LP V that we used in this course, but I also showed you what your C double 83 00:07:22,360 --> 00:07:29,560 ‫bar and a double circuit flex matrices would be if you had used the non simplified version. 84 00:07:30,160 --> 00:07:36,460 ‫It would have been slightly more correct, but since our horizon period was very short, it didn't really 85 00:07:36,460 --> 00:07:36,880 ‫matter. 86 00:07:37,270 --> 00:07:44,530 ‫We were able to achieve good tracking with this simplified LP V as well, where you had your simplified 87 00:07:44,530 --> 00:07:48,100 ‫C double bar and simplified a double circle flex. 88 00:07:48,760 --> 00:07:50,140 ‫So that was another thing. 89 00:07:50,710 --> 00:08:00,430 ‫And then in both cases, we described our model, but we used forward Euler discontinuation in the NPC 90 00:08:00,430 --> 00:08:02,500 ‫box and wrong acuta. 91 00:08:02,500 --> 00:08:07,120 ‫This crystallization in the plan box using two different methods. 92 00:08:07,780 --> 00:08:15,100 ‫They further increased the differences between the models used in the NPC controller and in the plan 93 00:08:15,100 --> 00:08:15,490 ‫box. 94 00:08:16,120 --> 00:08:25,090 ‫The thing with NPC is that the more similar the NPC model is to the planned model, the better your 95 00:08:25,090 --> 00:08:26,050 ‫controller works. 96 00:08:26,680 --> 00:08:33,610 ‫But of course, the closer you want to get to your planned model, the more complicated your NPC model 97 00:08:33,610 --> 00:08:34,240 ‫becomes. 98 00:08:34,900 --> 00:08:36,760 ‫So there's always a tradeoff. 99 00:08:37,450 --> 00:08:44,800 ‫You obviously want to simplify your model for your NPC controller as much as possible, and for as long 100 00:08:44,800 --> 00:08:53,440 ‫as tracking is good enough, you can use your simplified model and then you should not go closer to 101 00:08:53,440 --> 00:08:54,700 ‫your plan model. 102 00:08:55,210 --> 00:09:01,450 ‫Because if your NPC controller is able to achieve its tasks, then you should not complicate your life 103 00:09:01,450 --> 00:09:02,590 ‫more than necessary. 104 00:09:03,190 --> 00:09:10,810 ‫And with the simplifications that we have made, the NPC controller is able to track the trajectories 105 00:09:11,440 --> 00:09:17,530 ‫that require the drone to fly close to the hovering position, which was our goal to begin with. 106 00:09:18,100 --> 00:09:26,260 ‫Nevertheless, you have to be aware of these differences and you have to understand that when you design 107 00:09:26,260 --> 00:09:33,520 ‫for your NPC constraints, then the design process uses this model here. 108 00:09:34,210 --> 00:09:37,930 ‫This simplified one, not the one in the plan box. 109 00:09:38,560 --> 00:09:46,810 ‫So the inputs that your NPC controller gives you, they take into account your NPC constraints, but 110 00:09:46,810 --> 00:09:57,040 ‫your matrix and each vector they were generated using this model here and these control inputs. 111 00:09:57,730 --> 00:09:59,020 ‫They were calculated. 112 00:09:59,790 --> 00:10:09,330 ‫Considering your matrix and h vector, but again, they were created using this simplified model, and 113 00:10:09,330 --> 00:10:16,710 ‫so when you get your control inputs and you apply it to your planned model, which is closer to reality, 114 00:10:16,740 --> 00:10:23,970 ‫which is more realistic and more complex, then because of the fact that these two models in these two 115 00:10:23,970 --> 00:10:25,950 ‫boxes are slightly different. 116 00:10:26,400 --> 00:10:32,880 ‫Because of that, the real Omegas that you get from your planned box, they end up being slightly different 117 00:10:33,510 --> 00:10:39,210 ‫compared to the boundaries that you had set for your omega max and omega men. 118 00:10:39,840 --> 00:10:47,340 ‫And so at the beginning, when the drone needs to lift off and catch up with the trajectory, then you 119 00:10:47,340 --> 00:10:55,290 ‫can see that these differences in those two models, they might cause your real Omegas to violate the 120 00:10:55,290 --> 00:10:56,790 ‫constraints quite a lot. 121 00:10:57,480 --> 00:11:00,510 ‫Therefore, it's important to have safety factors. 122 00:11:01,110 --> 00:11:11,970 ‫So if you know that the real limit of omega 3s ranges between 150 irradiance per second and 900 radians 123 00:11:11,970 --> 00:11:20,520 ‫per second, then the constraints that you give to your MPAC controller, the values that you set for 124 00:11:20,520 --> 00:11:24,750 ‫omega men and omega max, they should be more conservative. 125 00:11:25,260 --> 00:11:34,110 ‫So maybe your omega men could be 200 radians per second in omega max 800 radians per second. 126 00:11:34,710 --> 00:11:40,530 ‫That way, you will have some space to violate your constraints. 127 00:11:41,190 --> 00:11:48,690 ‫You should not break the red limit, but at least you can break the orange limit for as long as you 128 00:11:48,690 --> 00:11:53,490 ‫stay in this corridor here and in this corridor here. 129 00:11:54,060 --> 00:11:57,600 ‫So that's what I wanted to tell you about HPC constraints. 130 00:11:58,020 --> 00:11:58,950 ‫Thank you very much.