1 00:00:04,120 --> 00:00:08,860 The next step in the common filter is to work out how we can use measurement information to update the 2 00:00:08,860 --> 00:00:14,590 current state up on the top right here, we have the measurement model that we are going to assume for 3 00:00:14,590 --> 00:00:15,840 the filtering process. 4 00:00:16,210 --> 00:00:22,180 So we have the measurement, said K is a function of the current state and some noise values here. 5 00:00:22,450 --> 00:00:25,270 Our matrix is the measurement model matrix. 6 00:00:25,390 --> 00:00:28,920 Our M matrix is the measurement model, noise sensitivity matrix. 7 00:00:29,770 --> 00:00:33,580 So this is the model that we're going to assume for the linear filter. 8 00:00:33,880 --> 00:00:39,520 This basically says that the measurements that we have are going to be a linear combination of the current 9 00:00:39,520 --> 00:00:41,560 state corrupted by bit of noise. 10 00:00:45,290 --> 00:00:50,840 We now need to derive how to update the current state prediction with the current measurement to form 11 00:00:50,840 --> 00:00:55,280 the updated state prediction so you can see the difference between these two predictions, even though 12 00:00:55,280 --> 00:00:56,420 they're both the same time. 13 00:00:56,420 --> 00:01:00,540 Step one is without the current information from the measurement. 14 00:01:00,560 --> 00:01:02,870 One is with the current information from the measurement. 15 00:01:03,470 --> 00:01:08,840 This prediction includes all the measurement information up to the current time set, while this one 16 00:01:08,840 --> 00:01:13,430 over here includes all the information, including the current TimeStep measurements. 17 00:01:14,890 --> 00:01:20,650 Now, this was done as a way to estimate, based on the error between the measurement said OK and the 18 00:01:20,650 --> 00:01:26,410 prediction measurement, which is calculated from our paychecks or our measurement model, sensitivity 19 00:01:26,410 --> 00:01:30,070 matrix times our current state estimate. 20 00:01:31,710 --> 00:01:37,250 We're going to call this era the innovation, so the innovation is just a difference between the current 21 00:01:37,260 --> 00:01:42,100 measurement that we get from a sensor or some external source and our predicted sensor measurement. 22 00:01:42,120 --> 00:01:46,370 So this is what the sensor should read if we're just using the state prediction. 23 00:01:46,380 --> 00:01:52,740 And the difference between this is the innovation, which we know as why with a over top. 24 00:01:54,720 --> 00:01:59,610 In the coming theater, we should have a good idea of how much noise and uncertainty there are in the 25 00:01:59,610 --> 00:02:03,700 measurements, which is going to be the size or the variance of our vehicle here. 26 00:02:04,260 --> 00:02:08,670 We also have a good idea of the uncertainty in our state prediction, which is just going to be the 27 00:02:08,860 --> 00:02:09,310 matrix. 28 00:02:09,900 --> 00:02:14,850 So what we can do is that we can actually work out how much uncertainty there is in our innovation. 29 00:02:15,360 --> 00:02:18,480 So to do this, we have to calculate the innovation covariance. 30 00:02:18,840 --> 00:02:20,730 And this is just a very simple process. 31 00:02:21,090 --> 00:02:27,080 We know the uncertainty in X is just going to be P, so we have to transfer the uncertainty in our state 32 00:02:27,120 --> 00:02:32,590 space to our innovation space using our standard covariance transformation relationship here. 33 00:02:33,120 --> 00:02:35,100 So this is a convergence of the state. 34 00:02:35,430 --> 00:02:41,630 We transform it from state space into our covariance space using El-Hage times, our transpose. 35 00:02:42,240 --> 00:02:46,650 And we also know the covariance of a measurement that's just going to be the covariance of anois value 36 00:02:46,650 --> 00:02:49,290 up here, which we've already described as just being. 37 00:02:49,290 --> 00:02:49,650 Ah. 38 00:02:50,340 --> 00:02:54,450 So this is going to tell us how much noise there is going to be in this innovation. 39 00:02:54,750 --> 00:02:57,740 So this is going to be the covariance of the innovation. 40 00:02:57,750 --> 00:03:03,780 So how much uncertainty there is in the Gaussian probability density function for our innovation? 41 00:03:05,010 --> 00:03:08,820 The matrix here is how much uncertainty there is in the measurement said. 42 00:03:09,690 --> 00:03:15,180 Now, looking at our measurement model up here, if we if this matrix was just doing this and we just 43 00:03:15,180 --> 00:03:22,670 had additive noise V.K., then I would just be the variance of V.K. But since we have this sensitivity, 44 00:03:22,680 --> 00:03:28,590 Affymetrix m here to handle in this situation, he would have to say the arc is just going to be equal 45 00:03:28,590 --> 00:03:31,350 to M r and transpose. 46 00:03:33,320 --> 00:03:36,180 So overall, we have different definitions here. 47 00:03:36,500 --> 00:03:41,780 I said is going to be the measurement, Victor, I said that is a predicted measurement vector, which 48 00:03:41,780 --> 00:03:44,810 is just going to be times X and Y. 49 00:03:44,810 --> 00:03:46,670 Tilde is going to be the innovation. 50 00:03:46,880 --> 00:03:52,190 So that's going to be the error between our real measurement and predictive measurement and our metrics 51 00:03:52,190 --> 00:03:54,200 is going to be our innovation covariance. 52 00:03:58,300 --> 00:04:04,120 Now, the state is updated based on the size of the innovation using the common game Matrix K., which 53 00:04:04,120 --> 00:04:08,950 itself is formed as a ratio between the innovation uncertainty and the current state uncertainty. 54 00:04:09,520 --> 00:04:14,200 So this is basically saying if a really certain in the current state, then we should trust that more 55 00:04:14,200 --> 00:04:15,200 than the measurement. 56 00:04:15,760 --> 00:04:19,990 However, if we're really uncertain in our measurement, then we should trust that one more than the 57 00:04:19,990 --> 00:04:20,650 current state. 58 00:04:21,130 --> 00:04:25,600 So basically, based on the ratio between the measurement uncertainty and a current state uncertainty, 59 00:04:25,870 --> 00:04:31,350 we work out a weighted estimate to work out how much we need to shift our state to update it. 60 00:04:31,570 --> 00:04:33,820 And this is done by the common game matrix. 61 00:04:35,230 --> 00:04:37,990 So the state estimate can be calculated using this equation. 62 00:04:37,990 --> 00:04:40,660 Here we have our previous state estimate. 63 00:04:40,900 --> 00:04:43,680 This is the one without the measurement information included. 64 00:04:43,690 --> 00:04:45,670 This is our updated state estimate. 65 00:04:46,030 --> 00:04:50,990 And we the amount we update it is based on our common gain of innovation. 66 00:04:51,400 --> 00:04:56,410 So the innovation again, is the error between the measurement and the predicted measurement and the 67 00:04:56,410 --> 00:05:02,500 size of that era multiplied by our common game is going to tell us how much we need to shift our measurement 68 00:05:02,500 --> 00:05:04,280 by to get the updated prediction. 69 00:05:05,620 --> 00:05:09,580 Now, the common game metric can be calculated from this equation here. 70 00:05:09,940 --> 00:05:15,400 The common game metrics pretty much looks at the uncertainty of the state weighted as a ratio of the 71 00:05:15,400 --> 00:05:17,660 uncertainty of the measurement or the innovation. 72 00:05:18,070 --> 00:05:22,450 So looking at this works out how much it needs to trust the measurement or how much it needs to trust 73 00:05:22,450 --> 00:05:23,140 the current state. 74 00:05:23,800 --> 00:05:26,270 Now, these two equations, they should look a bit similar. 75 00:05:26,500 --> 00:05:31,250 These are actually just at least squares, recursive equations that we've calculated before. 76 00:05:31,960 --> 00:05:34,210 Here is the recursively squares equations. 77 00:05:34,210 --> 00:05:39,580 So you can see that the state prediction equation up here is basically exactly the same state update 78 00:05:39,580 --> 00:05:39,850 here. 79 00:05:40,630 --> 00:05:47,230 You can see that we have the previous information, which is updated by a game Matrix K, which is multiplied 80 00:05:47,230 --> 00:05:53,290 by our innovation or error between our measurement and our predicted measurement to get the update for 81 00:05:53,300 --> 00:05:57,540 a current state estimate, the gain is also exactly the same. 82 00:05:57,550 --> 00:06:02,890 You can see that we have these P metrics, you P metric, c, h, transpose, transpose. 83 00:06:03,550 --> 00:06:07,000 And this whole term here is also just the S matrix. 84 00:06:07,250 --> 00:06:08,560 So we have s inverse. 85 00:06:08,710 --> 00:06:09,660 Same thing over here. 86 00:06:10,240 --> 00:06:15,280 So you can see that the common filter up next step is exactly the same equations as the recursively 87 00:06:15,280 --> 00:06:15,910 squares. 88 00:06:15,910 --> 00:06:19,300 It holds the same properties and it's calculated in the same way. 89 00:06:22,480 --> 00:06:24,830 We've worked out how to update the current state estimate. 90 00:06:25,210 --> 00:06:28,200 We also need to work out how to update the current covariance. 91 00:06:28,210 --> 00:06:32,950 So we've worked out the main of the Gaussian distribution for the state estimate. 92 00:06:33,100 --> 00:06:37,690 We also need to work out the Gaussian distribution around that state estimate. 93 00:06:37,720 --> 00:06:38,680 So the covariance. 94 00:06:40,360 --> 00:06:45,400 The whole point of the state update process is to reduce the uncertainty in the estimates and improve 95 00:06:45,400 --> 00:06:49,120 the accuracy so that the covariance of the estimate must change. 96 00:06:51,210 --> 00:06:56,670 So to do this, we can calculate the covariance update, so basically we have the covariance of the 97 00:06:56,670 --> 00:07:01,560 state estimate before the update, before the measurement is applied and we have it what it is afterwards 98 00:07:01,980 --> 00:07:05,730 and the size that we changes is based on the common ghankay. 99 00:07:07,280 --> 00:07:11,750 So this equation here is, again, just straight from the recursively squares equation. 100 00:07:13,990 --> 00:07:18,850 You should also be noted that this term here is always going to be less than one, so we have identity 101 00:07:18,850 --> 00:07:19,720 minus one. 102 00:07:19,930 --> 00:07:23,850 It means that this term, multiplied by pay, is always going to be smaller. 103 00:07:23,870 --> 00:07:28,480 So our updated covariance is always going to be smaller than our current covariance. 104 00:07:28,810 --> 00:07:30,960 If it is not, well, then something's going wrong. 105 00:07:33,840 --> 00:07:39,240 So a summary of the updates that is shown on this slide here, we have the update for the state estimate, 106 00:07:39,240 --> 00:07:45,450 which updates the current state prediction to the updated state estimate based on the common gane and 107 00:07:45,450 --> 00:07:49,320 the error or the innovation between the current measurement and the predicted measurement. 108 00:07:49,770 --> 00:07:52,800 We also have the equation for the updated state covariance. 109 00:07:53,100 --> 00:07:59,100 This is just a function of the current comingling matrix and the predicted state covariance matrix. 110 00:07:59,910 --> 00:08:05,130 The common game matrix is given by this equation here, which is just a ratio of the current uncertainty 111 00:08:05,130 --> 00:08:10,770 inside the current state estimate compared to the current uncertainty inside the measurement, which 112 00:08:10,770 --> 00:08:15,180 is our innovation covariance here, which is calculated from this equation here.