1 00:00:04,010 --> 00:00:08,960 So in the previous exercise, you had to go through and implement the prediction step of the common 2 00:00:08,960 --> 00:00:14,120 filter, it's always good practice that if you have a simulation of the model that you're trying to 3 00:00:14,120 --> 00:00:20,090 estimate, then run the simulation without any noise from a known initial condition and then run the 4 00:00:20,270 --> 00:00:25,010 method of prediction step from the same initial condition and see that the two are consistent. 5 00:00:25,850 --> 00:00:28,250 Now, this is an important validation step. 6 00:00:28,280 --> 00:00:32,660 It shows that the common thread model that you're using is appropriate for the system dynamics that 7 00:00:32,660 --> 00:00:33,610 you're trying to estimate. 8 00:00:34,280 --> 00:00:37,360 And we want to run this without the update step from the common filter. 9 00:00:37,730 --> 00:00:43,070 This is because if we run it with the update step, sometimes the correction step can hide any errors 10 00:00:43,070 --> 00:00:43,630 in the model. 11 00:00:44,240 --> 00:00:49,100 So it's always good to make sure that the underlying prediction is as best as possible for the model 12 00:00:49,100 --> 00:00:53,220 that we want to try to estimate so we can have a look at this when we look at the code. 13 00:00:53,240 --> 00:00:54,980 So let's go switch the code. 14 00:00:55,940 --> 00:01:02,440 And if we start the initial state from zero zero zero and the covariance all from zero zero zero with 15 00:01:02,450 --> 00:01:07,150 zero process model noise and run it, we can see what happens with the true state. 16 00:01:07,460 --> 00:01:12,530 We can see that the object that we want to track shoots off at a 45 degree angle and it's traveling 17 00:01:12,530 --> 00:01:13,760 at 10 meters a second. 18 00:01:14,120 --> 00:01:19,580 So in this example, every time we run the simulation, the same velocity and the same direction happens. 19 00:01:19,580 --> 00:01:25,160 So we can say here we have Awasthi of about seven point zero seven and seven point zero seven. 20 00:01:26,410 --> 00:01:32,260 So using this information now, we can run the comment field, a prediction step using the same true 21 00:01:32,260 --> 00:01:32,980 initial state. 22 00:01:33,370 --> 00:01:39,490 So we saw that the initial state of the truth always starts at the Origin zero zero and had a controversy 23 00:01:39,490 --> 00:01:43,810 about seven point zero seven seven point zero seven. 24 00:01:44,200 --> 00:01:45,760 So now when we run the simulation. 25 00:01:48,880 --> 00:01:55,450 We can see our estimated position, the red dot and the true state, the blue dot, pretty much overlap. 26 00:01:55,970 --> 00:01:56,750 They line up. 27 00:01:56,770 --> 00:02:01,590 So the prediction step is very consistent with the actual model that we want to simulate. 28 00:02:02,020 --> 00:02:07,760 And we have looked at the main position error and maybe once the year, you see, they're very small. 29 00:02:08,290 --> 00:02:13,510 This is because we know the true state of the system and we're studying it from the true initial state. 30 00:02:14,020 --> 00:02:17,710 So if you look at the graphs here, you can see that we have a very small position error. 31 00:02:17,740 --> 00:02:23,770 So you can see here our units are very small and the same thing for the velocity, the velocity, and 32 00:02:23,770 --> 00:02:24,410 it's very small. 33 00:02:24,430 --> 00:02:28,800 So we pretty much start it from the truth and we're predicting it very well. 34 00:02:28,930 --> 00:02:34,930 So I can say that the true state tracks with the predicted state, I should say the other way around 35 00:02:34,930 --> 00:02:37,190 the predicted state tracks very well for the true state. 36 00:02:37,720 --> 00:02:40,570 So it all seems to be working fine. 37 00:02:41,080 --> 00:02:42,700 It all seems to be working accurately. 38 00:02:42,700 --> 00:02:46,750 And this is a good validation step as we covered in the slides. 39 00:02:48,810 --> 00:02:54,300 So the variance in the filter must always represent the approximate level of the uncertainty inside 40 00:02:54,300 --> 00:02:58,910 the field estimates, and this needs to be done through a correct model and correct tuning. 41 00:02:59,460 --> 00:03:04,800 What this means is that the true state of the system is within the uncertainty bounds around the estimated 42 00:03:04,810 --> 00:03:05,210 state. 43 00:03:05,760 --> 00:03:08,990 If this is not the case, then the filter is called inconsistent. 44 00:03:09,240 --> 00:03:14,760 And if the filter is inconsistent, this leads to bad and incorrect estimates and it degrades the whole 45 00:03:14,760 --> 00:03:15,840 estimation process. 46 00:03:16,320 --> 00:03:20,720 In some cases, it pretty much means that the estimated state from the filter is worthless. 47 00:03:20,720 --> 00:03:24,800 It's useless if you're trying to use it for anything, it would just cause problems. 48 00:03:26,690 --> 00:03:31,520 So we've checked that the state transmission model is working correctly, it can correctly predict from 49 00:03:31,520 --> 00:03:34,310 a known initial condition the true state response. 50 00:03:34,940 --> 00:03:38,780 We also want to check that the prediction model correctly transforms the uncertainty. 51 00:03:39,590 --> 00:03:45,500 The uncertainty in the system should not be increasing if our key metrics, a process model noise is 52 00:03:45,500 --> 00:03:46,280 equal to zero. 53 00:03:46,820 --> 00:03:50,290 Rather, it should be transformed, appropriate with the system model. 54 00:03:50,900 --> 00:03:55,230 So on the slide here, we can see the common field, our prediction set for the uncertainty. 55 00:03:55,250 --> 00:03:57,040 So this is the current uncertainty. 56 00:03:57,320 --> 00:03:58,970 This is the predicted uncertainty. 57 00:03:59,660 --> 00:04:02,280 These are the state transition matrixes. 58 00:04:02,300 --> 00:04:05,360 And this is the process model noise. 59 00:04:06,650 --> 00:04:08,960 So this step here takes the process model. 60 00:04:09,410 --> 00:04:15,470 So this step here takes the uncertainty at the TimeStep K minus one, transforms it through the process 61 00:04:15,470 --> 00:04:18,470 model to get the new uncertainty at the current time. 62 00:04:18,990 --> 00:04:21,590 So this transforms the uncertainty from the time set. 63 00:04:21,770 --> 00:04:26,150 It does not actually inflate the uncertainty, does not increase the uncertainty in the system. 64 00:04:26,390 --> 00:04:29,990 It just transforms what's appropriate with the system dynamics. 65 00:04:30,740 --> 00:04:35,990 Whereas this key metrics here, this actually increases or inflates the uncertainty inside the system. 66 00:04:36,620 --> 00:04:41,780 So if Kyuss equal to zero, we're just transforming the uncertainty based on the process model. 67 00:04:42,110 --> 00:04:46,100 If CU is non-zero, then we're actually increasing the uncertainty in the system. 68 00:04:49,160 --> 00:04:54,650 So what they can do is that we can check the position uncertainty is operating as expected, and to 69 00:04:54,650 --> 00:04:58,480 do this, we set the initial position uncertainty to be a non-zero value. 70 00:04:58,910 --> 00:05:04,670 We set the initial velocity, uncertainty to be zero, and then we can run the filter and see that the 71 00:05:04,670 --> 00:05:06,390 position uncertainty stays the same. 72 00:05:07,160 --> 00:05:09,710 So let's do that with the code that we have developed. 73 00:05:11,030 --> 00:05:14,790 So we want to set the initial position uncertainty to a non-zero value. 74 00:05:15,140 --> 00:05:21,190 So in this case, let's set the position uncertainty or the position standard deviation to be five. 75 00:05:21,200 --> 00:05:27,740 So the variance is five squared, so five squared for the X and five squared for the Y initial position 76 00:05:27,740 --> 00:05:28,220 uncertainty. 77 00:05:28,970 --> 00:05:29,870 So when we run this. 78 00:05:32,300 --> 00:05:38,480 We can now say that we have an uncertainty elipse around the estimate so we can see from these graphs 79 00:05:38,480 --> 00:05:43,850 here that the three sigma, these green lines here and here for the plus or minus sigma and plus or 80 00:05:43,850 --> 00:05:47,290 minus three sigma for the position error, they stay constant. 81 00:05:48,110 --> 00:05:50,670 And from a process model, this is what we expect. 82 00:05:50,690 --> 00:05:52,220 We're not increasing the uncertainty. 83 00:05:52,220 --> 00:05:54,220 We're just keeping the uncertainty the same. 84 00:05:54,230 --> 00:05:58,040 So the position uncertainty stays the same throughout the simulation. 85 00:05:59,650 --> 00:06:04,300 So the covariates prediction for the position on certainly seems to be operating as expected. 86 00:06:04,990 --> 00:06:09,540 So now let's check that they've lost the uncertainty is operating as expected. 87 00:06:09,550 --> 00:06:14,380 And to do this, we're going to set the initial position, uncertainty to zero, but we're going to 88 00:06:14,380 --> 00:06:17,340 set the velocity uncertainty to a non-zero value. 89 00:06:18,160 --> 00:06:23,500 And when we do this and run the field, we should see that the velocity uncertainty stays at the same 90 00:06:23,500 --> 00:06:23,950 value. 91 00:06:24,430 --> 00:06:28,290 But the position uncertainty in this case should grow linearly with time. 92 00:06:29,110 --> 00:06:32,470 And this is just due to the dynamics of the system. 93 00:06:32,830 --> 00:06:36,650 So if we integrate diversity, we shall get a change in position. 94 00:06:37,630 --> 00:06:44,110 So switching to the code this time, we're going to have a zero position uncertainty for the initial 95 00:06:44,110 --> 00:06:47,700 state, but we're going to have an uncertainty for the velocity. 96 00:06:47,710 --> 00:06:50,590 So let's say a standard deviation or variance of one. 97 00:06:52,630 --> 00:06:54,100 So now when we run the simulation. 98 00:06:55,940 --> 00:07:01,460 OK, so now if we have a look at the graphs, we can see that our velocity and certainly a three sigma 99 00:07:01,460 --> 00:07:05,750 bounce stays the same and it's going to be at three because this is a three sigma bounce. 100 00:07:05,750 --> 00:07:07,720 And we said the answer only to be one. 101 00:07:08,450 --> 00:07:12,870 So I've lost the uncertainty is doing what's expected is staying constant. 102 00:07:13,430 --> 00:07:18,770 Now, if you look at our position, uncertainty, we can see that is increasing with time and linearly 103 00:07:18,770 --> 00:07:19,760 increasing of time. 104 00:07:19,760 --> 00:07:24,920 And this is to be expected because we know that when we integrate velocity, we get a change in position. 105 00:07:25,580 --> 00:07:28,880 So initially we start off at zero variance, so we start off at zero. 106 00:07:29,090 --> 00:07:30,230 That's what we said it is. 107 00:07:30,560 --> 00:07:38,300 And you can see after our 120 seconds, it should be friends of one multiplied by 120, multiplied by 108 00:07:38,300 --> 00:07:38,720 three. 109 00:07:38,720 --> 00:07:40,460 So it should be quite a large value. 110 00:07:40,490 --> 00:07:49,100 So now it's three hundred and sixty as expected and same thing for the Y, so we can see that our velocity 111 00:07:49,130 --> 00:07:52,070 uncertainty is doing the correct thing with the system dynamics. 112 00:07:52,070 --> 00:07:53,690 It's changing. 113 00:07:54,590 --> 00:07:59,540 It's transforming the uncertainty in the appropriate way that represents the system dynamics. 114 00:08:02,320 --> 00:08:05,390 So we've looked at the uncertainty due to the process model. 115 00:08:05,560 --> 00:08:08,610 Now we want to know that you have a look at the process model noise. 116 00:08:08,620 --> 00:08:12,620 So this is the key metrics, part of the uncertainty equation. 117 00:08:13,270 --> 00:08:18,580 So we want to check that the process model noise, the acceleration uncertainty is operating as expected 118 00:08:19,450 --> 00:08:20,050 to do this. 119 00:08:20,060 --> 00:08:23,610 We're going to set the initial position and the uncertainties to zero. 120 00:08:24,130 --> 00:08:27,310 And we want to set a non-zero acceleration stand deviation. 121 00:08:27,310 --> 00:08:31,480 So we want the key metrics, the process model noise metrics to be non-zero. 122 00:08:32,290 --> 00:08:37,000 So when you run the filter, we should say that the total level of uncertainty in the system should 123 00:08:37,000 --> 00:08:38,230 be growing with time. 124 00:08:38,680 --> 00:08:44,440 Now, this means that the velocity uncertainty should be increasing linearly and the position uncertainty 125 00:08:44,440 --> 00:08:50,050 should be increasing quite radically as expected due to the system dynamics, because we know that position 126 00:08:50,050 --> 00:08:52,780 is a second integral of acceleration. 127 00:08:53,390 --> 00:08:55,690 So let's carry out this experiment, Velcade. 128 00:08:56,650 --> 00:09:00,540 So we're going to set our initial variances for position of velocity to be zero. 129 00:09:00,970 --> 00:09:03,400 But now we want a non-zero key matrix. 130 00:09:03,400 --> 00:09:07,630 So we're going to set this standard deviation of acceleration to be point one. 131 00:09:08,610 --> 00:09:11,290 So now when we run the simulation, we should say something different. 132 00:09:12,030 --> 00:09:15,280 We can see that the uncertainty in the system is increasing with time. 133 00:09:15,300 --> 00:09:21,140 So our three sigma so bubble here is increasing as the simulation runs. 134 00:09:21,660 --> 00:09:27,900 And if we look at our core variances now, we can see that our velocity era, that's increasing linearly 135 00:09:27,900 --> 00:09:32,790 with time and we can see our position and certainly increasing quite radically with time. 136 00:09:34,280 --> 00:09:37,520 So everything seems to be working as expected on this front. 137 00:09:39,840 --> 00:09:45,930 So the person's moral noise increases the uncertainty in the system to capture the stochastic nature 138 00:09:45,930 --> 00:09:47,260 of the system dynamics. 139 00:09:47,850 --> 00:09:53,310 Now, in this simple case where we have an object that starts at a known origin and moves at a constant 140 00:09:53,310 --> 00:09:59,850 velocity and there's no randomness, randomness inside the system, then it's not really all that important. 141 00:09:59,880 --> 00:10:01,380 This is why we the. 142 00:10:01,760 --> 00:10:06,380 So this is why we can set a key metrics to be zero, because there's no uncertainty in the system. 143 00:10:06,600 --> 00:10:13,290 The only uncertainty at the moment is around our initial guess of the position and velocity. 144 00:10:14,770 --> 00:10:20,080 So in general, when running a common field, all the key metrics or the process of noise can compensate 145 00:10:20,080 --> 00:10:26,260 for a common Philidor process model that does not correctly or accurately model the true system dynamics. 146 00:10:27,560 --> 00:10:33,020 And this works by increasing the uncertainty inside the system, which helps to cause a filter to stay 147 00:10:33,020 --> 00:10:39,440 consistent so that it helps to make the uncertainty around the estimates to actually include where the 148 00:10:39,440 --> 00:10:46,280 true state actually is, the presence of noise ties closely and with the measurement of noise and the 149 00:10:46,280 --> 00:10:47,350 performance of the filter. 150 00:10:47,600 --> 00:10:50,050 So we only cover this in a lot more detail later on. 151 00:10:51,690 --> 00:10:56,460 So hopefully now you have a better understanding of why we went about the different exercises in the 152 00:10:56,460 --> 00:11:02,190 way that we did, a key takeaway point from all this is that whenever we designing or implementing a 153 00:11:02,190 --> 00:11:07,110 common filter, if we have a simulation of what the true system is going to do, we should run that 154 00:11:07,110 --> 00:11:12,540 and run the method of prediction step to make sure the two are consistent or as consistent as possible. 155 00:11:13,470 --> 00:11:18,090 If we do this, then there's a good chance that the model we have selected for the common filter should 156 00:11:18,090 --> 00:11:20,910 be appropriate for the system in that mix that we're trying to estimate.