1 00:00:00,730 --> 00:00:01,570 ‫Welcome back. 2 00:00:02,230 --> 00:00:11,620 ‫This is an extra section in which I also want to go over the topic of NPC constraints in order to make 3 00:00:11,620 --> 00:00:13,360 ‫this course more complete. 4 00:00:13,990 --> 00:00:22,240 ‫The topic of NPC constraints was extensively covered in the course applied control systems to autonomous 5 00:00:22,240 --> 00:00:24,280 ‫cars 360 tracking. 6 00:00:24,880 --> 00:00:32,500 ‫I will give you a brief overview on what we did in that course, but it is recommended that you covered 7 00:00:32,500 --> 00:00:35,620 ‫the NPC constraints in that car course as well. 8 00:00:36,220 --> 00:00:43,660 ‫In fact, this is the only section that suggests you two cover the 360 tracking course first. 9 00:00:44,230 --> 00:00:48,970 ‫In my explanation, I will be constantly referring back to the car, of course. 10 00:00:49,660 --> 00:00:57,100 ‫Also in this section, additional Python libraries are needed, the installation instructions and the 11 00:00:57,100 --> 00:01:02,620 ‫Python files are in the second half of this section where I also explained the code to you. 12 00:01:03,220 --> 00:01:07,570 ‫And so let's get started in the second autonomous car course. 13 00:01:08,200 --> 00:01:13,280 ‫We had a system with six states and two control inputs. 14 00:01:13,960 --> 00:01:17,770 ‫Our state vector, there was this one here. 15 00:01:18,400 --> 00:01:19,630 ‫Our control input. 16 00:01:19,630 --> 00:01:22,360 ‫There was this one here. 17 00:01:23,020 --> 00:01:29,490 ‫The state small x dot was the car's longitudinal velocity small y dot. 18 00:01:30,190 --> 00:01:32,290 ‫That was its lateral velocity. 19 00:01:32,950 --> 00:01:35,950 ‫CI was its Yongle side. 20 00:01:35,950 --> 00:01:38,260 ‫That was its angular velocity. 21 00:01:38,860 --> 00:01:45,100 ‫And then X and Y, they were the course inertial x and Y positions, respectively. 22 00:01:45,760 --> 00:01:54,760 ‫The input delta was the car's steering wheel angle and then a was the course applied longitudinal acceleration 23 00:01:55,420 --> 00:01:58,870 ‫that was responsible for controlling the car longitudinally. 24 00:01:59,470 --> 00:02:04,510 ‫It would either speed the car up or slow it down, just like with the drone. 25 00:02:05,140 --> 00:02:11,830 ‫We started modelling the car from the first principles of physics using the equations of motion. 26 00:02:12,430 --> 00:02:20,380 ‫We then reformulated these equations of motion and put them in the states based equations, just like 27 00:02:20,380 --> 00:02:22,300 ‫we did here with our drone. 28 00:02:22,960 --> 00:02:32,410 ‫These continuous stay space equations were then used in the planned box together with Euler integration 29 00:02:33,070 --> 00:02:34,960 ‫in order to calculate new states. 30 00:02:35,620 --> 00:02:38,020 ‫We did not use wrong acuta there. 31 00:02:38,590 --> 00:02:41,590 ‫We used the easier method to obtain new states. 32 00:02:42,250 --> 00:02:50,350 ‫We used the Euler method, and we also computed new states many times per one tsp interval. 33 00:02:50,590 --> 00:02:54,190 ‫In order to increase precision of new state calculations. 34 00:02:54,820 --> 00:03:02,650 ‫And then, just like in this course, we took those continuous states space equations and we reformulated 35 00:03:02,650 --> 00:03:02,980 ‫them. 36 00:03:03,280 --> 00:03:11,890 ‫We put them in the LP V format so that we could use our linear MVC to control our car. 37 00:03:12,520 --> 00:03:15,220 ‫This was our LP V model. 38 00:03:15,790 --> 00:03:20,500 ‫You had your continues a, b and C matrices there. 39 00:03:21,160 --> 00:03:27,580 ‫These elements A1 one and B one one and all the other ones. 40 00:03:28,210 --> 00:03:33,640 ‫These were matrix elements that contained different vehicle parameters. 41 00:03:34,300 --> 00:03:43,870 ‫And then, of course, we took our continuous LP v ABC matrices with discrete ties them and then we 42 00:03:43,870 --> 00:03:47,260 ‫augmented them just like we did in this course. 43 00:03:47,920 --> 00:03:53,830 ‫And just like we had Delta Q2, Delta Q3 and Delta U. 44 00:03:53,830 --> 00:03:56,800 ‫Four control inputs in the eyes of NPC. 45 00:03:56,800 --> 00:04:06,910 ‫In this course, the changes or the increments of Q2, U3 and Q4 in that car course in the eyes of NPC. 46 00:04:07,540 --> 00:04:12,040 ‫The control inputs where Delta Delta and Delta A.. 47 00:04:12,640 --> 00:04:15,880 ‫In other words, we augmented the system there. 48 00:04:16,480 --> 00:04:25,780 ‫And then just like here what we did, we formulated our cost function using the matrices c double bar, 49 00:04:26,410 --> 00:04:32,380 ‫a double circle and Flex H double bar and F double bar transposed. 50 00:04:32,980 --> 00:04:42,220 ‫Remember that in that concourse, the C double bar and the eight double circum flex matrices were derived 51 00:04:42,220 --> 00:04:45,760 ‫using the non simplified LP v approach. 52 00:04:46,330 --> 00:04:49,450 ‫Ultimately, we computed our solution with this formula. 53 00:04:50,200 --> 00:04:57,010 ‫Delta U Global equals minus h double bar inverse times, f double bar. 54 00:04:57,640 --> 00:05:00,010 ‫And then this vector here where? 55 00:05:00,270 --> 00:05:08,760 ‫Still, that key that was your present argument, that state vector and then this are global, that 56 00:05:08,760 --> 00:05:14,020 ‫was your reference value vector and its length depended on the horizon period. 57 00:05:14,640 --> 00:05:18,030 ‫Remember that the vector delta you. 58 00:05:18,630 --> 00:05:22,890 ‫It contains the Delta Delta and Delta A. and the Vector you. 59 00:05:23,520 --> 00:05:25,770 ‫It contains Delta and A.. 60 00:05:26,370 --> 00:05:31,560 ‫And so once you had your delta, you global, you could calculate your new you. 61 00:05:32,130 --> 00:05:38,550 ‫So you at K equals you at K minus one plus delta, you at K. 62 00:05:39,180 --> 00:05:47,940 ‫More specifically, you would use this formula here with this role vector here in which each element 63 00:05:48,480 --> 00:05:50,040 ‫is a two by two matrix. 64 00:05:50,640 --> 00:05:56,040 ‫And then the first one is the identity matrix, and the other ones are zero matrices. 65 00:05:56,670 --> 00:05:59,070 ‫And this is your preview state vector here. 66 00:05:59,310 --> 00:06:00,450 ‫You add K minus one. 67 00:06:01,110 --> 00:06:09,750 ‫So with this term here, you would only extract the first delta you and you would discard the rest. 68 00:06:10,260 --> 00:06:17,070 ‫And then you would take your new you and you would apply it to the plant. 69 00:06:17,790 --> 00:06:23,280 ‫The plant will then give you new states and you start with a new loop. 70 00:06:24,000 --> 00:06:28,530 ‫This process will then continue until the end of the simulation. 71 00:06:29,160 --> 00:06:37,290 ‫So all that that we have talked about so far, it's called unconstrained NPC. 72 00:06:37,920 --> 00:06:41,460 ‫Let's now see how we apply constraints there. 73 00:06:41,730 --> 00:06:42,630 ‫Thank you very much.