1 00:00:03,560 --> 00:00:08,930 Now, let's have a look at how the optics that works, so inside the carrier that you've just completed, 2 00:00:09,530 --> 00:00:13,590 let's say we set the initial covariance of the position and the diversity. 3 00:00:13,610 --> 00:00:15,660 So the standard deviation is both to zero. 4 00:00:16,310 --> 00:00:20,030 This would mean that our state covariance matrix is all zeros initially. 5 00:00:20,450 --> 00:00:22,580 We've also set the state to be all zero. 6 00:00:22,630 --> 00:00:26,910 So assuming that we're starting at the Origin and we're starting with zero velocity. 7 00:00:27,620 --> 00:00:33,140 So if we run this, we should be able to see that the estimate, the state does not actually change. 8 00:00:33,140 --> 00:00:38,960 So our estimated position stays at the origin while the true motion of the object shoots off in this 9 00:00:38,960 --> 00:00:39,500 direction. 10 00:00:39,980 --> 00:00:44,300 And again, you can say this is reflected by the innovation measurement error. 11 00:00:44,720 --> 00:00:48,780 So this innovation, you can see that it starts increasing. 12 00:00:49,070 --> 00:00:51,390 So there's going to be more and more error in the system. 13 00:00:52,970 --> 00:00:57,530 So there's a very simple reason for this, if this is because we've set the initial covariance to be 14 00:00:57,530 --> 00:00:58,000 zero. 15 00:00:58,250 --> 00:01:04,220 So we've told the field that we're 100 percent sure that the initial state of the system is zero zero 16 00:01:04,520 --> 00:01:06,310 and has zero zero velocity. 17 00:01:06,320 --> 00:01:08,030 So it's not going to be moving at all. 18 00:01:08,030 --> 00:01:12,130 And no matter what information we tell, the field is not going to do anything about it. 19 00:01:13,280 --> 00:01:19,100 So to handle this, if we set the initial velocity, uncertainty, standard deviation to be about 10. 20 00:01:19,460 --> 00:01:25,080 So this is saying that we have some level of uncertainty of the initial state estimates for the velocity. 21 00:01:26,000 --> 00:01:31,640 So when we do this and we can run the field again, you can say that the next step is actually doing 22 00:01:31,850 --> 00:01:32,470 its job. 23 00:01:33,050 --> 00:01:38,050 So you can see now as we run the Philidor the initial velocity uncertainty. 24 00:01:38,090 --> 00:01:42,160 So you can see in this graph here, we start off with a large velocity uncertainty. 25 00:01:42,500 --> 00:01:48,260 So every time we get an update step into the field, it will start to estimate what the new what the 26 00:01:48,260 --> 00:01:49,490 actual velocity is. 27 00:01:50,030 --> 00:01:53,000 The velocity should start converging to the true value. 28 00:01:53,550 --> 00:01:58,980 So you can see here the error starts at some non-zero value and it converges down towards zero. 29 00:01:59,000 --> 00:02:05,180 And you can see here the blue line here again is the true state velocity through state velocity and 30 00:02:05,180 --> 00:02:06,830 the red one is the predicted velocity. 31 00:02:07,460 --> 00:02:13,560 So you can see every time we get an update to the filter, the velocity tends towards the true state. 32 00:02:13,580 --> 00:02:15,110 So our error decreases. 33 00:02:16,340 --> 00:02:20,210 Our position starts off at zero covariance at zero zero. 34 00:02:20,240 --> 00:02:25,490 This is because we've assumed we know perfect information about where the position of the object starts 35 00:02:25,970 --> 00:02:29,100 and we assume that it has zero uncertainty in it. 36 00:02:29,630 --> 00:02:34,430 So you can see here in the shape of the uncertainty graph, you can see that the the uncertainty starts 37 00:02:34,430 --> 00:02:35,060 increasing. 38 00:02:35,390 --> 00:02:38,740 This is because of our non-zero covariance for velocity. 39 00:02:39,260 --> 00:02:42,890 And then every time we get the update, we shrink down our position. 40 00:02:42,900 --> 00:02:46,690 And certainly this is because we actually feeling in the position measurements. 41 00:02:47,180 --> 00:02:49,300 So we get this sawtooth pattern here. 42 00:02:49,640 --> 00:02:54,000 So every time we have an update, our uncertainty in the estimates shrink. 43 00:02:54,320 --> 00:02:56,060 So this is where we get the vertical lines. 44 00:02:56,330 --> 00:03:02,280 And as we do the dead reckoning, as we only do the predictions, depths of uncertainty increases. 45 00:03:02,300 --> 00:03:03,650 So that's the rise is there. 46 00:03:04,340 --> 00:03:09,820 So if we were to change the timestep of the update to prediction, you should see a different response. 47 00:03:10,130 --> 00:03:13,940 If we had a longer prediction time, there'll be a greater increase. 48 00:03:13,940 --> 00:03:15,110 And then we get the update. 49 00:03:15,290 --> 00:03:16,370 We'll get a sharper jump. 50 00:03:17,800 --> 00:03:19,400 So we've changed the measurement, right? 51 00:03:19,420 --> 00:03:26,100 So instead of having a measurement at one hertz, let's say we have a measurement every 10 seconds, 52 00:03:26,110 --> 00:03:33,190 so instead of one a second, we get only one every 10 seconds and we run this again, we're going to 53 00:03:33,190 --> 00:03:34,360 see a different response. 54 00:03:35,320 --> 00:03:40,030 So now whether we've decreased the sample rate of the measurements, you can see that our covariance 55 00:03:40,030 --> 00:03:41,580 now has a different pattern. 56 00:03:42,220 --> 00:03:47,500 So I can variance now increasing, as we expected, between the measurements and then when we get the 57 00:03:47,500 --> 00:03:53,530 measurement in the sharp decrease in the uncertainty, because as we know, the update step decreases 58 00:03:53,530 --> 00:03:55,200 the uncertainty inside the system. 59 00:03:56,860 --> 00:03:59,740 But you can see the system here is still working at a different rate. 60 00:04:00,460 --> 00:04:02,140 We still have an initial error. 61 00:04:02,440 --> 00:04:05,670 So Awasthi starts off at zero zero for the prediction. 62 00:04:05,980 --> 00:04:10,170 But you can see as we get the measurements, it starts to converge towards the truth. 63 00:04:10,180 --> 00:04:14,440 And again, you can see that our error converges towards the truth and our errors. 64 00:04:14,690 --> 00:04:17,980 Anomaly inside our three sigma uncertainty bounds. 65 00:04:19,460 --> 00:04:24,530 The shape of the reverend's profile of time is pretty much controlled by two parameters, it is controlled 66 00:04:24,530 --> 00:04:28,910 by the key matrix and the Matrix along with the update, right. 67 00:04:30,440 --> 00:04:35,330 So we're saying in the past from the predictions that the key matrix controls the rate of increase of 68 00:04:35,330 --> 00:04:40,600 uncertainty inside the system, it shows you how quickly the uncertainty is going to increase with time 69 00:04:41,030 --> 00:04:46,850 while the all matrix is going to control how accurate the measurement is that refuse into the system. 70 00:04:47,660 --> 00:04:53,660 So the larger the matrix, the quicker the uncertainty grows between the updates, the smaller the matrix 71 00:04:53,660 --> 00:04:56,480 is, the smaller is the updated covariance of the filter. 72 00:04:57,810 --> 00:05:04,110 So if we have a large key metrics and a large matrix or update, right, we might get a covariance profile 73 00:05:04,110 --> 00:05:04,910 that looks at this. 74 00:05:04,920 --> 00:05:10,200 So you can see overall the uncertainty in the system is increasing, even though we're getting some 75 00:05:10,200 --> 00:05:10,920 updates to it. 76 00:05:11,110 --> 00:05:15,330 The update doesn't provide enough information to constrain the uncertainty inside the system. 77 00:05:15,870 --> 00:05:18,690 So this is a bad example of what we want to be. 78 00:05:19,050 --> 00:05:24,090 This filter is actually helping us all that much, because the longer we run the filter, the more uncertainty 79 00:05:24,090 --> 00:05:25,120 in the system we're going to get. 80 00:05:25,140 --> 00:05:27,600 So it's not going to be helping us to estimate anything. 81 00:05:28,170 --> 00:05:33,540 The ideal case, what we want is when we have a small key metrics and a small R matrix or a fast update. 82 00:05:33,540 --> 00:05:33,840 Right. 83 00:05:34,080 --> 00:05:38,850 And we'll get the uncertainty inside the system looking like this where even though we increasing the 84 00:05:38,850 --> 00:05:44,280 uncertainty during the predictions, whenever we get an update, it brings the uncertainty down a greater 85 00:05:44,280 --> 00:05:46,470 amount than increased during the prediction step. 86 00:05:46,860 --> 00:05:52,140 So over time, we can see that the uncertainty is going to be converging and it's going to converge 87 00:05:52,140 --> 00:05:54,960 down to a steady state value. 88 00:05:55,260 --> 00:06:01,080 And this steady state value is going to be based on the ratio of the Q and Matrix's as well as the update. 89 00:06:01,080 --> 00:06:01,350 Right. 90 00:06:02,970 --> 00:06:08,100 An interesting thing to see from the update equations here is that the final covariance P is a function 91 00:06:08,100 --> 00:06:15,090 of our common game matrix K, which is a function of our covariance innovation is now you can see during 92 00:06:15,090 --> 00:06:19,860 these questions here, none of these actually has to do with the current state of the system, the current 93 00:06:19,860 --> 00:06:22,830 state of the system X does not appear in any of these equations. 94 00:06:23,430 --> 00:06:29,170 So basically what that is saying is that our covariance with time is independent of the current state. 95 00:06:29,970 --> 00:06:34,970 So what does that saying is that our covariance P with time is independent of the current state. 96 00:06:35,550 --> 00:06:41,490 The more updates we get, the smaller the P Matrix is going to be in ideal conditions and since took 97 00:06:41,490 --> 00:06:47,100 a variance with time does not depend on the current state x, it is possible to predict how accurate 98 00:06:47,100 --> 00:06:49,620 outfielder estimates are going to be over the long term. 99 00:06:49,900 --> 00:06:55,800 So if we run the field for long enough, our covariance is going to converge down to a constant non-zero 100 00:06:55,800 --> 00:06:59,040 value and that value is going to be based on our Q and our matrix. 101 00:07:00,620 --> 00:07:05,390 So if we know the amount of process, more noise in the field and we know how good our measurements 102 00:07:05,390 --> 00:07:09,800 are, we can work out overall how good our state estimates are going to be. 103 00:07:10,200 --> 00:07:15,260 So we can use this information in the engineering design process and work out an error budget so we 104 00:07:15,260 --> 00:07:18,660 can work out how good our sensors have to be to make the level performance we want. 105 00:07:19,190 --> 00:07:23,380 So depending on our application, we might find out that we don't need a really high quality sensor. 106 00:07:23,390 --> 00:07:29,150 We might be able to get away with using a low quality sensor at lower cost if the relative sizes of 107 00:07:29,150 --> 00:07:35,060 the ARE and key metrics are such that the estimated state covariance as we run the filter is adequate 108 00:07:35,060 --> 00:07:36,770 enough for the solution that we want. 109 00:07:38,510 --> 00:07:45,530 So if we set the initial uncertainty in position and velocity to be zero, when we run the example, 110 00:07:45,530 --> 00:07:48,180 we can see that our common future does not change. 111 00:07:48,200 --> 00:07:52,760 And again, this is this is because we've told it that it has perfect knowledge of the system and that 112 00:07:52,760 --> 00:07:54,640 is that zero zero zero velocity. 113 00:07:54,650 --> 00:07:56,820 So nothing we can tell it is going to make it to move. 114 00:07:57,380 --> 00:08:01,670 And again, you can say the system is not working or not very tracking very quickly, because you can 115 00:08:01,670 --> 00:08:04,360 say that the innovation here is increasing. 116 00:08:04,670 --> 00:08:08,060 So the innovation is not zero mean as we expect it to be. 117 00:08:08,690 --> 00:08:10,130 So this is a good sign. 118 00:08:10,190 --> 00:08:11,300 The filter is not working. 119 00:08:13,310 --> 00:08:17,300 So this means that a true state of the initial state is not inside out. 120 00:08:17,310 --> 00:08:19,640 Uncertainty is not consistent. 121 00:08:20,180 --> 00:08:25,880 So if we start increasing the process of noise in case so instead of actually increasing the initial 122 00:08:25,880 --> 00:08:30,620 state, but we say that our uncertainty in the system is growing with time. 123 00:08:30,620 --> 00:08:32,930 This because we have a non-zero key matrix. 124 00:08:34,010 --> 00:08:39,890 When we run this filter, we now can see that our system is going to start to catch up, is actually 125 00:08:39,890 --> 00:08:43,580 going to start tracking our object correctly so we can see what's happening here. 126 00:08:44,000 --> 00:08:50,420 Initially in the system, we have zero uncertainty for every pore position and for velocity, but the 127 00:08:50,420 --> 00:08:52,970 true state is actually outside our uncertainty. 128 00:08:52,970 --> 00:08:59,750 So you can see our true state is down here as the uncertainty in the system in the coming forward grows 129 00:08:59,750 --> 00:09:01,700 due to our acceleration uncertainty. 130 00:09:02,000 --> 00:09:05,180 It then allows our system to start to get tried to catch up. 131 00:09:05,600 --> 00:09:08,360 So every time we get an update, it tries to update it now. 132 00:09:08,810 --> 00:09:12,380 So you can see initially the position error is growing with time. 133 00:09:13,040 --> 00:09:17,960 This is because our system, the this is because the uncertainty in the system is too small. 134 00:09:17,990 --> 00:09:20,140 So our field is not going to do anything with the updates. 135 00:09:20,690 --> 00:09:25,190 But once it reaches a certain size, the filter then determines that is inconsistent. 136 00:09:25,370 --> 00:09:30,770 A filter then starts acting on the information that is given and trying to bring the filter back into 137 00:09:30,770 --> 00:09:31,240 alignment. 138 00:09:31,250 --> 00:09:36,860 So you can see after a while after the uncertainty in the system has grown to a certain amount, the 139 00:09:36,860 --> 00:09:38,000 velocity can change. 140 00:09:38,000 --> 00:09:42,500 It starts to change and start to converge down to what the true state is. 141 00:09:43,730 --> 00:09:48,290 So once the uncertainty grows, the point that it actually encapsulates the true uncertainty in the 142 00:09:48,290 --> 00:09:52,550 system, it allows the system to come common filter to catch up to a true state. 143 00:09:54,870 --> 00:09:59,790 This can also be seen in the measurement innovation, so initially we start off with the large innovations 144 00:10:00,180 --> 00:10:05,720 and it starts growing to a point that the uncertainty in the system is large enough to actually allows 145 00:10:05,730 --> 00:10:08,450 the for up to a step to start to correct for it. 146 00:10:08,460 --> 00:10:14,460 And as it corrects for it, you can see that the innovation comes down and then we get to our zero main 147 00:10:14,460 --> 00:10:15,060 condition. 148 00:10:15,270 --> 00:10:19,620 And the variance of our innovation is consistent with the field of.