1 00:00:03,730 --> 00:00:09,100 In the previous video, we've gone through the calculation steps for the measurement innovation, for 2 00:00:09,100 --> 00:00:15,580 the extend the common filter and we've shown the and we've shown the equations for the innovation and 3 00:00:15,580 --> 00:00:16,850 the innovation covariance. 4 00:00:17,350 --> 00:00:22,750 So now we actually want to go through the derivation of how they get these equations so we can actually 5 00:00:22,750 --> 00:00:24,040 prove that they're still valid. 6 00:00:24,220 --> 00:00:31,090 And it is valid to use the Jacobean or the linear approximation of the nonlinear models inside our calculation 7 00:00:31,090 --> 00:00:31,510 steps. 8 00:00:33,210 --> 00:00:39,330 So we have seen that the innovation vector and the covariance matrix are calculated using these two 9 00:00:39,330 --> 00:00:40,210 equations here. 10 00:00:40,380 --> 00:00:42,360 So the first equation is for the innovation. 11 00:00:42,360 --> 00:00:45,300 The second equation is for the innovation covariance. 12 00:00:46,030 --> 00:00:48,040 But where do these equations actually come from? 13 00:00:48,060 --> 00:00:50,910 How can we actually know that these are the correct equations? 14 00:00:52,380 --> 00:00:58,740 They seem pretty straightforward if we were to use a linear comma filter and then apply the linear approximations, 15 00:00:58,900 --> 00:01:02,910 but how do we know that that is actually a valid how do we actually know that mathematically? 16 00:01:02,940 --> 00:01:06,420 We are valid to do that and we're not just using our best guess. 17 00:01:07,350 --> 00:01:12,900 And again, we want to start with the Taylor series expansion, so this is how we take the linear approximation 18 00:01:12,900 --> 00:01:17,760 of the non-linear model so we can take the Taylor series expansion of the measurement model. 19 00:01:17,760 --> 00:01:22,360 So we have the measurement model said is equal to each of these. 20 00:01:22,360 --> 00:01:28,620 So with the state vector, our noise vector is a measurement model and that is our actual measurement 21 00:01:28,620 --> 00:01:29,080 vector. 22 00:01:29,670 --> 00:01:33,470 And again, we want to take the Taylor series expansion of this measurement model. 23 00:01:33,720 --> 00:01:36,350 We want to take it about our state vector X. 24 00:01:36,720 --> 00:01:42,870 So basically we want to take the expansion about a best estimate X hat and we want to take it with the 25 00:01:42,870 --> 00:01:46,750 expansion using zero using zero measurement noise. 26 00:01:47,550 --> 00:01:52,440 So looking at this equation here, we take the Taylor series approximation, which gives us this equation 27 00:01:52,440 --> 00:01:52,780 here. 28 00:01:53,430 --> 00:01:59,850 So we have our measurement model evaluated at our best estimate, and then we have the Jacobin multiplied 29 00:01:59,850 --> 00:02:06,840 by our preparation between the true state and our estimate of state, plus the Jacobean before the measurement 30 00:02:06,840 --> 00:02:07,350 noise. 31 00:02:07,560 --> 00:02:12,510 And again, we're going to have a lot of these higher order terms as we include more of these higher 32 00:02:12,510 --> 00:02:13,160 order terms. 33 00:02:13,170 --> 00:02:16,070 We get a more accurate Taylor series expansion estimate. 34 00:02:16,650 --> 00:02:22,250 However, we just want to limit ourselves to the first term's early, so always higher order terms and 35 00:02:22,290 --> 00:02:23,200 gets canceled out. 36 00:02:24,030 --> 00:02:29,430 So when we do this, we get left with an estimate of our measurement model, which is just going to 37 00:02:29,430 --> 00:02:33,010 be the Taylor series expansion, and it's going to be this equation set here. 38 00:02:33,570 --> 00:02:37,920 So this is what we're going to based our equations for the extended coming field run. 39 00:02:38,190 --> 00:02:40,740 So this is going to be the key identity that we're going to use. 40 00:02:42,290 --> 00:02:48,290 So this is how we can turn the linear filter and make it applicable for non-linear systems, we use 41 00:02:48,290 --> 00:02:51,940 a first order approximation of the nonlinear system. 42 00:02:53,500 --> 00:02:58,990 So let's start off by looking at the extended feel, the innovation step for the innovation or the actual 43 00:02:58,990 --> 00:02:59,410 state. 44 00:03:01,350 --> 00:03:06,540 So using the conditional expectation of the predictive measurement which is shown here, so this is 45 00:03:06,540 --> 00:03:12,630 saying the expected operator on the true measurement, given all the information up to came on, this 46 00:03:12,630 --> 00:03:14,630 one is going to be our predictive measurement. 47 00:03:15,300 --> 00:03:21,180 Now, on the previous slide, we've calculated what a first order approximation of the measurement is 48 00:03:21,180 --> 00:03:27,420 going to be so we can take our approximation for a set and substitute it into this equation here to 49 00:03:27,420 --> 00:03:30,440 calculate an approximation for our predictive measurement. 50 00:03:30,750 --> 00:03:32,520 And of course, it looks like this. 51 00:03:35,540 --> 00:03:40,220 So this is straightforward, we're just substituting indirectly for al-Said K from the equation, from 52 00:03:40,220 --> 00:03:41,040 the previous slide. 53 00:03:41,270 --> 00:03:43,270 So this is the Taylor series expansion. 54 00:03:44,210 --> 00:03:48,880 Now having a look at this equation, there's a few things we can see now since this whole equation is 55 00:03:49,250 --> 00:03:56,060 is inside the expectation operator, we can get used the expected value for anything that is stochastic. 56 00:03:56,660 --> 00:04:02,690 Now, we know from our definition that the expected value of a best estimate state, given all the information 57 00:04:02,690 --> 00:04:08,810 up to came on this one, is just going to be the best estimate of XQ. 58 00:04:08,870 --> 00:04:13,230 So basically the difference between these two parameters here should pretty much be zero. 59 00:04:13,460 --> 00:04:19,640 So if we take the expectation overall that we can just cancel this turn out here, we also know that 60 00:04:19,640 --> 00:04:23,960 the expected value of the noise should be zero as well, because the noise should have zero. 61 00:04:23,960 --> 00:04:27,560 I mean, which means these three terms cancel that. 62 00:04:27,770 --> 00:04:33,830 And we're just left with the expectation of our measurement model and we just get left with the estimated 63 00:04:33,830 --> 00:04:37,280 value of our measurement model evaluated at our best estimate. 64 00:04:37,820 --> 00:04:41,660 Now we know the expected value of a constant is just the constant. 65 00:04:42,030 --> 00:04:43,250 So we can pull it right out. 66 00:04:43,260 --> 00:04:45,530 I mean, get this term here. 67 00:04:45,800 --> 00:04:49,130 So the predictive measurement is just going to be out. 68 00:04:49,310 --> 00:04:54,290 Measurement model evaluated at our best estimate with zero measurement noise. 69 00:04:54,830 --> 00:04:58,220 Now, this is pretty much straightforward and you would have thought that we don't need to go through 70 00:04:58,220 --> 00:04:59,450 the steps to work this out. 71 00:04:59,480 --> 00:05:00,560 It should be self-evident. 72 00:05:00,890 --> 00:05:03,740 But just going through the steps here, we've proved ourself correct. 73 00:05:04,100 --> 00:05:10,040 So we are good to just say they are predictive measurement in the extended filter is just going to be 74 00:05:10,040 --> 00:05:12,890 the measurement model evaluated at our best estimate. 75 00:05:14,880 --> 00:05:19,920 So letting on from this, we can say that the innovation is simply the difference between a true and 76 00:05:19,920 --> 00:05:23,310 the predictive measurement error which is given by this equation here. 77 00:05:23,700 --> 00:05:26,520 So this is how we come up with our innovation equation. 78 00:05:27,960 --> 00:05:33,060 So that has been looking at the innovation itself now, he wanted to do the same thing, the same steps, 79 00:05:33,060 --> 00:05:36,300 but for the calculation of the innovation covariance. 80 00:05:38,460 --> 00:05:40,600 And again, we start off with the definition. 81 00:05:40,740 --> 00:05:46,800 We know the innovation covariance is defined as the matrix, which is just the expected value of the 82 00:05:46,800 --> 00:05:49,350 innovation squared or the measurement error squared. 83 00:05:50,810 --> 00:05:56,400 So the first step is that we can substitute in our approximation for said K. into the innovation victim 84 00:05:56,400 --> 00:05:58,730 victime to give this equation here. 85 00:05:58,940 --> 00:06:04,340 So we've taken our equation for said K. And we subjected it into this equation from here and we end 86 00:06:04,340 --> 00:06:05,460 up with this term here. 87 00:06:05,840 --> 00:06:11,930 So the first set of terms here is a Taylor series expansion, and then we're subtracting it off from 88 00:06:11,930 --> 00:06:13,920 our predict for our predicted measurement. 89 00:06:14,180 --> 00:06:17,720 So we're going to get minus one measurement model, which is minus our measurement model over here. 90 00:06:18,620 --> 00:06:22,880 And again, looking at this equation, it should be fairly straightforward that these two terms here 91 00:06:22,880 --> 00:06:23,430 are the same. 92 00:06:23,450 --> 00:06:27,570 So these cancel out and we get left with this equation here. 93 00:06:28,310 --> 00:06:34,670 So the innovation is pretty much the Jacobean of the probation and state plus the describing of the 94 00:06:34,670 --> 00:06:35,150 noise. 95 00:06:38,790 --> 00:06:44,070 So now that we've worked out the approximation for innovation, we can take our approximation of the 96 00:06:44,070 --> 00:06:50,160 innovation and into this expectation of right up here and we get this equation here. 97 00:06:50,760 --> 00:06:52,980 So this is the covariance of the innovation. 98 00:06:52,980 --> 00:06:57,400 And it should be fairly straightforward to see how we've got into this equation. 99 00:06:57,690 --> 00:07:02,020 So when you substitute this into this equation here, take the expectation of right up. 100 00:07:02,040 --> 00:07:04,320 We're left with this definition down here. 101 00:07:06,200 --> 00:07:11,690 So this is prove that we can use the Caribbean as a first order approximation of the nonlinear system. 102 00:07:12,050 --> 00:07:16,850 So this direction has shown us that we haven't just taken the linear, common field of equations and 103 00:07:16,850 --> 00:07:20,750 randomly, randomly substituted in for the matrixes. 104 00:07:21,080 --> 00:07:27,170 Mathematically, it makes mathematically it is valid to use the describing of the system to come up 105 00:07:27,170 --> 00:07:28,610 with these equations here. 106 00:07:30,190 --> 00:07:34,630 So in summary, when we look at the measurement innovation, we can calculate the measurement innovation 107 00:07:34,870 --> 00:07:36,240 using this equation. 108 00:07:36,250 --> 00:07:40,660 So the innovation is just the difference between the true sense and measurement, minus the predicted 109 00:07:40,660 --> 00:07:46,870 since measurement based on the measurement model evaluated at the best estimate and the measurement 110 00:07:46,870 --> 00:07:52,090 innovation, covariance, again, is just going to be this covariance transformation where the transformation 111 00:07:52,090 --> 00:07:56,400 matrix is for the president's model and for the measurement model. 112 00:07:56,680 --> 00:08:02,650 I'm just going to be the describing of the measurement model with respect to X the vector and the describing 113 00:08:02,650 --> 00:08:06,430 of the measurement model, respective noise for the measurement noise.