1 00:00:00,730 --> 00:00:11,290 ‫But, OK, you know that the solution that gives you your minimum cost function is zero and zero based 2 00:00:11,290 --> 00:00:18,280 ‫on the matrices and vectors that you have here, a double bar and small have transposed. 3 00:00:18,880 --> 00:00:21,580 ‫But we haven't considered the constraints yet. 4 00:00:22,120 --> 00:00:24,670 ‫Let's see if this solution. 5 00:00:24,700 --> 00:00:34,090 ‫Zero and zero, if it also satisfies all your constraints so you have your G Matrix and H Vector transposed, 6 00:00:34,720 --> 00:00:42,210 ‫the general form for the linear inequality constraints is G times, don't you? 7 00:00:42,220 --> 00:00:46,870 ‫Global less than or equal to and the H vector? 8 00:00:47,470 --> 00:00:50,350 ‫And so this is my G matrix. 9 00:00:51,040 --> 00:00:52,630 ‫This is my Delta Global. 10 00:00:53,260 --> 00:00:54,880 ‫And this is my h vector. 11 00:00:55,510 --> 00:01:01,330 ‫If I multiply g by Delta, you global, then I will have all that. 12 00:01:01,930 --> 00:01:05,530 ‫So this is my system of inequality equations. 13 00:01:06,190 --> 00:01:08,020 ‫So I have four equations. 14 00:01:08,650 --> 00:01:19,420 ‫And since my solution was there was a Delta zero equals zero radiance and Delta a zero equals zero meters 15 00:01:19,420 --> 00:01:20,440 ‫per second squared. 16 00:01:21,160 --> 00:01:25,630 ‫These equations, they will become these ones here. 17 00:01:26,170 --> 00:01:32,020 ‫So on the left side, everything will become zero because my delta a zero is zero. 18 00:01:32,020 --> 00:01:33,450 ‫Delta Delta zero zero. 19 00:01:33,940 --> 00:01:35,860 ‫And the same thing here and here. 20 00:01:36,460 --> 00:01:39,550 ‫So let's see if these statements are true. 21 00:01:40,150 --> 00:01:42,130 ‫You can see that the final one is true. 22 00:01:42,760 --> 00:01:47,380 ‫Zero is less than or equal to zero point zero eight o four. 23 00:01:47,980 --> 00:01:49,480 ‫Also, this one is true. 24 00:01:50,140 --> 00:01:53,830 ‫Also, this one is true, but this one is false. 25 00:01:54,310 --> 00:02:00,820 ‫It says that zero is less than or equal to minus zero point zero zero three. 26 00:02:01,390 --> 00:02:02,320 ‫But that's wrong. 27 00:02:03,020 --> 00:02:10,960 ‫Zero is not less than minus zero point zero zero three zero is greater than minus zero point zero zero 28 00:02:10,960 --> 00:02:11,470 ‫three. 29 00:02:12,070 --> 00:02:14,760 ‫So this solution here, Delta U. 30 00:02:14,770 --> 00:02:16,810 ‫Global, equals zero zero. 31 00:02:17,440 --> 00:02:22,470 ‫It does not satisfy all the constraints this constraint here. 32 00:02:23,110 --> 00:02:33,700 ‫It's not satisfied, but OK, can we find another combination of Delta Delta Zero and Delta A0 that 33 00:02:33,700 --> 00:02:36,790 ‫satisfies all these four constraints? 34 00:02:37,480 --> 00:02:37,910 ‫OK. 35 00:02:37,930 --> 00:02:42,550 ‫Your G would be, of course, bigger than its global minimum. 36 00:02:42,830 --> 00:02:43,540 ‫J Min. 37 00:02:44,140 --> 00:02:50,620 ‫So you're jamming with the constraints would be greater than G min without the constraints. 38 00:02:51,280 --> 00:02:52,150 ‫But that's OK. 39 00:02:52,720 --> 00:02:56,080 ‫Well, let's look at this first equation. 40 00:02:56,620 --> 00:03:09,010 ‫You have minus zero point zero two times Delta zero less than or equal to minus zero point zero zero 41 00:03:09,070 --> 00:03:09,610 ‫three. 42 00:03:10,210 --> 00:03:20,770 ‫If we solve this inequality equation, then you will have Delta a zero greater than or equal to minus 43 00:03:20,770 --> 00:03:26,170 ‫zero point zero zero three, divided by minus 0.02. 44 00:03:26,800 --> 00:03:34,450 ‫The reason why the sine changed its direction is because I divided this side by a negative number, 45 00:03:35,050 --> 00:03:39,490 ‫and I divided this side by a negative number, the same number, actually. 46 00:03:40,090 --> 00:03:45,070 ‫And if I do that, then this sine it changes its direction. 47 00:03:45,520 --> 00:03:48,520 ‫That's why this sine points to the right now. 48 00:03:49,090 --> 00:03:58,780 ‫And so you will get 0.15 so your delta a zero is greater than or equal to 0.15. 49 00:03:59,410 --> 00:04:12,430 ‫That means that in order to make this statement true, your Delta A0 can only range from 0.15 to + infinity. 50 00:04:13,000 --> 00:04:25,840 ‫So your Delta A0 is included in the set 0.15 and plus infinity if it ranges between 0.15 and plus infinity. 51 00:04:26,290 --> 00:04:27,670 ‫Then this statement is true. 52 00:04:28,300 --> 00:04:37,720 ‫If my Delta A0 is less than 0.15, then it will make this statement false. 53 00:04:38,410 --> 00:04:39,430 ‫You can try it out. 54 00:04:39,970 --> 00:04:48,760 ‫Take a random number between 0.15 and + infinity for Delta A0 and then put it here and see if this statement 55 00:04:48,760 --> 00:04:53,620 ‫is true and then take a random number that is less than 0.15. 56 00:04:54,280 --> 00:04:57,010 ‫Put it here and see if it's true or false. 57 00:04:57,670 --> 00:05:00,200 ‫So maybe then I can have a solid. 58 00:05:00,290 --> 00:05:11,730 ‫Russian Delta, You Global equals zero for my Delta Delta zero and 0.15 for my Delta A0. 59 00:05:12,360 --> 00:05:17,670 ‫Maybe there would be a good solution in that case, my j minimum. 60 00:05:17,670 --> 00:05:26,910 ‫With the constraints, it would be 50 times Delta Delta zero squared plus one half Delta A0 squared. 61 00:05:27,540 --> 00:05:30,420 ‫The first term would be 50 times zero squared. 62 00:05:30,990 --> 00:05:38,700 ‫The second term would be zero point fifteen squared and that would be zero point zero one one two five. 63 00:05:39,240 --> 00:05:47,160 ‫So you can see that my g min with the constraints is greater than my German without the constraints 64 00:05:47,730 --> 00:05:50,100 ‫because that one was zero. 65 00:05:50,730 --> 00:05:51,840 ‫But that's okay, right? 66 00:05:52,440 --> 00:05:59,490 ‫I mean, our cost function is slightly bigger than is a global minimum, but we have still minimized 67 00:05:59,490 --> 00:06:02,970 ‫it while respecting our constraints, right? 68 00:06:05,080 --> 00:06:06,040 ‫Wrong. 69 00:06:06,580 --> 00:06:07,150 ‫Yes. 70 00:06:07,510 --> 00:06:09,490 ‫We have satisfied this constraint. 71 00:06:10,000 --> 00:06:15,940 ‫And also, this one is OK and this one is OK, but what about the fourth constraint? 72 00:06:16,510 --> 00:06:19,510 ‫Now this one is not satisfied. 73 00:06:20,080 --> 00:06:28,930 ‫In order to satisfy this constraint, your delta A0 must be less than or equal to zero point zero eight 74 00:06:28,930 --> 00:06:29,530 ‫04. 75 00:06:30,100 --> 00:06:40,960 ‫In other words, Delta A0 can range between minus infinity and zero point zero eight 04. 76 00:06:41,530 --> 00:06:42,520 ‫That's what it says. 77 00:06:43,180 --> 00:06:45,070 ‫You don't even have to solve this equation. 78 00:06:45,700 --> 00:06:53,320 ‫It's already in the solved form, and he tells you that it cannot be greater than zero point zero eight 79 00:06:53,320 --> 00:06:53,860 ‫or four. 80 00:06:54,490 --> 00:07:04,000 ‫So for the first constraint, your Delta A0 could vary between 0.15 and plus infinity. 81 00:07:04,600 --> 00:07:12,970 ‫For the fourth constraint, your Delta A0 could vary from minus infinity to 0.08 or four. 82 00:07:13,540 --> 00:07:21,970 ‫So if I draw a number line here, I have minus infinity on one side, plus infinity on another side. 83 00:07:22,600 --> 00:07:30,610 ‫This number line is, of course, for Delta A0, and then I'm going to have here zero point zero, eight 84 00:07:30,610 --> 00:07:33,340 ‫or four and then 0.15. 85 00:07:33,970 --> 00:07:43,540 ‫And so if my Delta A0 goes from 0.15 to plus infinity, then my first constraint is satisfied. 86 00:07:44,170 --> 00:07:52,060 ‫And then if I go from zero point zero eight or for to minus infinity, then my fourth constraint is 87 00:07:52,060 --> 00:07:52,960 ‫satisfied. 88 00:07:53,470 --> 00:07:55,330 ‫And now, can you see the problem? 89 00:07:55,960 --> 00:08:01,030 ‫You can clearly see that you can never satisfy both constraints. 90 00:08:01,690 --> 00:08:08,380 ‫You either satisfy one constraint if you're in this region or you satisfy the fourth constraint. 91 00:08:08,890 --> 00:08:15,760 ‫If you're in this region and if you're in this region, well, then you violate both constraints. 92 00:08:16,420 --> 00:08:26,650 ‫If instead of this constraint, I had, for example, Delta A0 less than or equal to zero point eight 93 00:08:26,650 --> 00:08:31,180 ‫oh four knots zero point zero eight or four, but zero point eight or four. 94 00:08:31,840 --> 00:08:39,280 ‫Then if this is my delta easier no line, then my zero point eight or four, it would be here. 95 00:08:39,940 --> 00:08:42,010 ‫0.15 would be here. 96 00:08:42,670 --> 00:08:49,690 ‫And so for the first constraint, I would go like this for the fourth constraint, I would go like this. 97 00:08:50,380 --> 00:09:00,460 ‫Then you can see that in this region, when you're Delta Zero ranges from 0.15 and zero, eight or four, 98 00:09:01,060 --> 00:09:04,090 ‫that's where you satisfy both constraints. 99 00:09:04,720 --> 00:09:06,100 ‫And then it would be OK. 100 00:09:06,700 --> 00:09:13,900 ‫All the constraints with your Delta Delta Zero are already satisfied when you're Delta Delta zero equals 101 00:09:14,020 --> 00:09:14,590 ‫zero. 102 00:09:15,220 --> 00:09:24,040 ‫And in this case, if you had this constraint for your fourth one, then for as long as your Delta A0 103 00:09:24,700 --> 00:09:32,530 ‫is between this interval, then you would also satisfy your first and your fourth constraint. 104 00:09:33,070 --> 00:09:41,860 ‫And then you would simply need to find the Delta A0 inside this interval that would minimize your cost 105 00:09:41,860 --> 00:09:42,460 ‫function. 106 00:09:42,550 --> 00:09:43,270 ‫The most? 107 00:09:44,680 --> 00:09:54,430 ‫However, that's not our case, in our case, your Delta A0 is less than or equal to zero point zero, 108 00:09:54,460 --> 00:09:55,330 ‫eight or four. 109 00:09:55,900 --> 00:10:02,740 ‫And as you can see from the no line, there is simply no way for you to satisfy all the constraints. 110 00:10:03,370 --> 00:10:11,140 ‫And that's when the solver will give you an error, or it will simply tell you that you have a value 111 00:10:11,140 --> 00:10:11,650 ‫error. 112 00:10:12,250 --> 00:10:16,150 ‫Your constraints are inconsistent and you have no solution. 113 00:10:16,840 --> 00:10:25,150 ‫Or what I have seen is that in MATLAB, your quattro proc, it will simply give you an empty array. 114 00:10:25,720 --> 00:10:33,760 ‫It just means that there are no solutions out there that can minimize your cost function and satisfy 115 00:10:33,760 --> 00:10:34,780 ‫all the constraints. 116 00:10:35,410 --> 00:10:43,750 ‫So this has been a small numerical example to give you a feel for what it means when the solver tells 117 00:10:43,750 --> 00:10:51,550 ‫you that there are no solutions found for your optimization problem, and this problem is something 118 00:10:51,730 --> 00:10:53,980 ‫that you will have to deal with a lot. 119 00:10:54,520 --> 00:11:02,890 ‫In fact, that's what happened here in trajectory to the solver simply did not find a solution that 120 00:11:02,890 --> 00:11:04,750 ‫can satisfy all the constraints. 121 00:11:05,410 --> 00:11:07,750 ‫As a result, you see this kind of behavior. 122 00:11:08,170 --> 00:11:09,820 ‫Everything just goes to zero. 123 00:11:10,450 --> 00:11:21,700 ‫Your CI X and Y values also your body frame ex dot your body frame y dot your side, dot your net longitudinal 124 00:11:21,700 --> 00:11:27,340 ‫acceleration, your lateral acceleration and your angular acceleration. 125 00:11:27,970 --> 00:11:30,040 ‫So what can caused this problem? 126 00:11:30,640 --> 00:11:39,490 ‫Well, in Trajectory two, it was caused by this big jump in your longitudinal velocity reference values, 127 00:11:40,120 --> 00:11:43,300 ‫but it can also happen if you don't have that discontinuity. 128 00:11:43,930 --> 00:11:53,290 ‫It can happen if, for example, your NPC weights are tuned in a bad way or you ask the car to do unrealistic 129 00:11:53,290 --> 00:11:59,620 ‫things, maybe in your trajectory your turns are too sharp. 130 00:12:00,310 --> 00:12:07,930 ‫Maybe you want to perform your turn at very high speeds, or maybe you have both of these problems, 131 00:12:08,620 --> 00:12:11,240 ‫maybe your constraints. 132 00:12:11,860 --> 00:12:14,320 ‫Maybe they are too strict. 133 00:12:14,860 --> 00:12:22,240 ‫If you only allow your steering wheel angle to turn one degree to the left and one degree to the right, 134 00:12:22,930 --> 00:12:26,050 ‫then that could be a problem in those cases. 135 00:12:26,710 --> 00:12:35,710 ‫Try adjusting your NPC weights or try to loosen up your constraints as much as possible or make your 136 00:12:35,710 --> 00:12:40,990 ‫trajectory smoother and make sure that you turn at realistic speeds. 137 00:12:41,710 --> 00:12:45,340 ‫Try to avoid sharp turns, etc. In. 138 00:12:45,440 --> 00:12:53,380 ‫Of course, the controller works best for smooth trajectories without terms that are too sharp. 139 00:12:54,040 --> 00:12:54,880 ‫Thank you very much.