1 00:00:00,780 --> 00:00:03,390 So let's have a look at the common of prediction step. 2 00:00:03,410 --> 00:00:06,990 So this is the first step we need to implement for the common Fiola. 3 00:00:10,120 --> 00:00:15,820 We begin the estimation step with the initial condition, it's not so this is going to be the best estimate 4 00:00:15,820 --> 00:00:21,070 of the initial state of the system in this step, we want to propagate the time forward to find the 5 00:00:21,070 --> 00:00:22,660 best estimate of the time. 6 00:00:22,690 --> 00:00:29,590 Step one, so we can use the state's best model of the system that we dropped before to predict forward 7 00:00:29,590 --> 00:00:31,360 what the new state of the system is going to be. 8 00:00:31,390 --> 00:00:37,120 So this is going to be the best estimate at time, zero multiplied by our state transition matrix, 9 00:00:37,510 --> 00:00:42,150 plus our control input at TimeStep zero time to control input matrix. 10 00:00:42,670 --> 00:00:46,240 And this is going to propagate the state forward tompsett one. 11 00:00:47,770 --> 00:00:52,130 Or in general we can write the equation out like so. 12 00:00:52,600 --> 00:00:59,230 So the current state estimate is going to be a function of the old state estimate and the old control 13 00:00:59,230 --> 00:01:02,590 input as well as the two state transition matrixes. 14 00:01:07,240 --> 00:01:12,220 So this forms of prediction, step of the time update, step of the common fiodor, we use a process 15 00:01:12,220 --> 00:01:15,670 model of the system we want to estimate to predict state forward. 16 00:01:16,300 --> 00:01:21,320 So this is basically shifting the main of the Gaussian probably doability distributions forward in time. 17 00:01:22,810 --> 00:01:27,430 So the equation here shown is the state prediction equation for the common field. 18 00:01:29,050 --> 00:01:35,110 OK, so now we've looked at the state transition, but we also want to propagate forward the covariance 19 00:01:35,110 --> 00:01:37,060 of the Gaussian probability distribution. 20 00:01:38,690 --> 00:01:45,100 So to do this, we begin with the initial estimate pay not so pay is the initial variance of the state 21 00:01:45,100 --> 00:01:46,180 estimate ignored. 22 00:01:46,810 --> 00:01:51,820 Now, if we have exact knowledge of the initial state of the system, then we can set pay ought to be 23 00:01:51,820 --> 00:01:52,610 equal to zero. 24 00:01:52,630 --> 00:01:55,060 So there's no uncertainty in the estimate. 25 00:01:55,060 --> 00:01:56,200 We know exactly. 26 00:01:56,200 --> 00:01:56,770 Exactly. 27 00:01:57,460 --> 00:02:02,590 If we have no knowledge of X axonal, well, then Peno is going to be turning towards a very large number 28 00:02:02,710 --> 00:02:04,570 to say that a very large uncertainty. 29 00:02:05,680 --> 00:02:13,480 But we usually know an approximate value or good assumption of the value of X not so we pay not to be 30 00:02:13,480 --> 00:02:15,340 the value of the estimated error. 31 00:02:18,060 --> 00:02:23,370 So now that we have Peno, we can propagate this incredible with time using the linear transformation 32 00:02:23,370 --> 00:02:24,470 that we've derived before. 33 00:02:25,500 --> 00:02:28,890 So basically we have our initial covariance. 34 00:02:28,890 --> 00:02:34,710 Pinart we have our state transmission matrix here and we multiply them out by this equation here to 35 00:02:34,710 --> 00:02:36,720 propagate the uncertainty forward in time. 36 00:02:38,250 --> 00:02:41,350 Now, this does not change the amount of uncertainty in the system. 37 00:02:41,370 --> 00:02:45,430 It just shifts it between the states as the dynamics of the system evolve. 38 00:02:46,020 --> 00:02:53,370 So the size of pea before and after the magnitude, the amount of uncertainty in between the two stays 39 00:02:53,370 --> 00:02:53,910 constant. 40 00:02:56,330 --> 00:03:02,240 So if you look at the process model, we can see that this equation up here is going to account for 41 00:03:02,450 --> 00:03:03,480 this function. 42 00:03:03,710 --> 00:03:10,460 This part of the process model is going to take into account the trans state transition, the transition 43 00:03:10,460 --> 00:03:18,980 of the state between tompsett came on this one and Tompsett K or between zero and one vacancy. 44 00:03:18,990 --> 00:03:21,740 Can we also have this other uncertainty in the Saudi system? 45 00:03:21,920 --> 00:03:24,880 So this is where the system process model noise comes in. 46 00:03:25,400 --> 00:03:29,510 We need to account for this new uncertainty in state due to this process model noise. 47 00:03:32,120 --> 00:03:36,410 So to account for this in the system, we can add a new covariance term. 48 00:03:36,410 --> 00:03:42,710 So you can see here this term here now calculates the answer, new uncertainty in the system due to 49 00:03:42,710 --> 00:03:43,860 the process of noise. 50 00:03:44,270 --> 00:03:51,470 So our key metrics, if you remember this, is the covariance of the process model noise el is the process, 51 00:03:51,470 --> 00:03:53,950 model noise, sensitivity matrixes. 52 00:03:55,490 --> 00:04:00,640 Now, usually we assume that the process model noise is an additive, which basically means that these 53 00:04:00,830 --> 00:04:03,400 matrices here are just identity. 54 00:04:04,490 --> 00:04:09,180 If we use this assumption, then we can simplify the equation down to this one chain here. 55 00:04:09,950 --> 00:04:17,420 So the covariance at times that one is going to be the variance that timestep zero propagated by the 56 00:04:17,420 --> 00:04:21,470 state transition matrix plus the additive process model noise que. 57 00:04:26,440 --> 00:04:31,450 So we can sum up the British concept of the common filter by using these equations here. 58 00:04:31,600 --> 00:04:34,370 So this first equation here looks at this transition. 59 00:04:34,390 --> 00:04:39,140 So this is the shifting the main of the galaxy in probability distribution of the estimated state. 60 00:04:40,060 --> 00:04:42,940 This down here is the covariance propagation. 61 00:04:43,120 --> 00:04:47,690 So this shifts the covariance of the probability distribution of the estimated state. 62 00:04:48,280 --> 00:04:52,690 So inside the British of the filter, we update the main. 63 00:04:53,270 --> 00:04:54,820 We update the covariance.