1 00:00:04,220 --> 00:00:09,170 Throughout the next couple of videos where we develop the common filter, we're going to use a common 2 00:00:09,170 --> 00:00:11,280 filter example to help explain the concepts. 3 00:00:12,080 --> 00:00:15,740 For this, we're going to look at developing a 2D tracking filter. 4 00:00:17,650 --> 00:00:22,960 The goal of a two day track, a filter, is to estimate the position and velocity of a moving object 5 00:00:22,960 --> 00:00:24,490 based on remote measurements. 6 00:00:25,240 --> 00:00:30,070 When we say remote measurements, we mean an uncooperative target or object. 7 00:00:30,070 --> 00:00:33,000 So we don't have any intrinsic information about the object. 8 00:00:33,460 --> 00:00:34,950 So there are no sensors on board. 9 00:00:34,960 --> 00:00:40,810 We don't get any acceleration measurements or we have to go is just from remote measurements or external 10 00:00:40,810 --> 00:00:41,350 measurements. 11 00:00:41,980 --> 00:00:46,960 And these could be made by a radar that could be made by another sensor. 12 00:00:48,010 --> 00:00:50,590 But all the sensing is done remotely. 13 00:00:52,130 --> 00:00:57,320 Now, this type of filter has many different applications and crops up in many different areas, such 14 00:00:57,320 --> 00:01:03,380 as radar aircraft tracking is done for optical tracking and self-driving cars, such as looking for 15 00:01:03,380 --> 00:01:09,330 obstacles or pedestrians tracking them, trying to avoid them, object tracking in images. 16 00:01:09,350 --> 00:01:15,350 So when we look at a series of images we're trying to track an object through, the image is used for 17 00:01:15,350 --> 00:01:17,600 many surveillance and military applications. 18 00:01:17,990 --> 00:01:23,270 It is used in robotics and is commonly used in GPS aided navigation and guidance. 19 00:01:25,350 --> 00:01:31,010 The problem that we want to solve is looking at a tracking attorney object as it moves in a third plane. 20 00:01:31,350 --> 00:01:34,950 So we have this to Pinay, we have the X and the Y axis. 21 00:01:34,960 --> 00:01:40,770 We have an object that's moving at some velocity and we want to work out the position of the object 22 00:01:40,770 --> 00:01:42,240 and the velocity of the object. 23 00:01:43,280 --> 00:01:48,890 So this forms subprocess process model or the dynamics of the fuda, we also assume that we get position 24 00:01:48,890 --> 00:01:52,850 measurements of the objects such as using from a radar or something or some other sensor source. 25 00:01:53,150 --> 00:01:57,500 So we have these measurements here and each of these measurements provide a measurement of the position 26 00:01:57,500 --> 00:01:59,060 of the object with time. 27 00:02:00,300 --> 00:02:02,160 So this becomes the sense of moral. 28 00:02:04,350 --> 00:02:10,170 Included in the course is a series of Python files now these files form a simulation, and one of the 29 00:02:10,170 --> 00:02:12,760 simulations they form is for a 2D tracking filter. 30 00:02:13,530 --> 00:02:17,690 So we'll be extending these files to develop a common filter for this problem. 31 00:02:19,120 --> 00:02:24,100 So now let's have a quick look at the code that's included how to use it, how it's structured and how 32 00:02:24,100 --> 00:02:26,200 we're going to use it in the examples coming up. 33 00:02:28,110 --> 00:02:33,660 The Python simulation for the two day track and field includes a number of different files as shown 34 00:02:33,660 --> 00:02:33,990 here. 35 00:02:34,650 --> 00:02:39,280 So these are the main ones we're going to be using for the first exercise being the 2D tracking filter. 36 00:02:39,900 --> 00:02:43,770 So if we have actually a look at the code and actually see what happened, we run it. 37 00:02:43,770 --> 00:02:50,220 If we open up our assignment, one underscore answer, we can get a feel for what the final simulation 38 00:02:50,400 --> 00:02:52,890 is going to look like once we complete all the exercises. 39 00:02:53,970 --> 00:02:56,670 So switching over to your favorite python environment. 40 00:02:58,700 --> 00:03:03,590 So let's have a look at an example of what happens when we run the code, so when we run the code, 41 00:03:04,100 --> 00:03:06,980 first thing we notice is that a number of Grauwe plot up. 42 00:03:07,760 --> 00:03:11,720 The first animation here is the animation or the filtering action while it's running. 43 00:03:12,290 --> 00:03:17,320 The second plot here is going to be the innovation from the measurements, which we'll talk about later. 44 00:03:17,600 --> 00:03:22,960 And we also have a figure that has the different states of the filter as well as the true state. 45 00:03:22,970 --> 00:03:26,300 So we can look at the error between the true and the estimated states. 46 00:03:27,200 --> 00:03:32,540 So this first figure here that had the animation, we have these black crosses and these represent the 47 00:03:32,540 --> 00:03:36,340 different the different position measurements as they're happening. 48 00:03:36,860 --> 00:03:38,330 We can see we have this blue dot. 49 00:03:38,450 --> 00:03:41,540 The blue represents the true state of the object that we're tracking. 50 00:03:42,230 --> 00:03:46,040 And he saw as time went on, the object moved this red dot. 51 00:03:46,040 --> 00:03:48,750 Here is the estimated position from the common filter. 52 00:03:49,040 --> 00:03:51,980 So ideally, this red dot should be tracking this blue dot. 53 00:03:51,980 --> 00:03:57,200 And you can see in this example is tracking it for fairly well, this green bubble. 54 00:03:57,200 --> 00:03:59,630 Around the red estimate is the covariance. 55 00:03:59,630 --> 00:04:02,000 This is the uncertainty around the estimate. 56 00:04:02,150 --> 00:04:05,400 So this shows you the uncertainty around the position estimates. 57 00:04:05,420 --> 00:04:11,090 So as the field of runs the true state, the blue dot should be within this uncertainty ellipse here. 58 00:04:13,490 --> 00:04:18,530 Our next figure is the innovations, so this is the difference between the actual measurements that 59 00:04:18,530 --> 00:04:22,130 we get and our predictive measurements, which we'll talk about later on. 60 00:04:22,790 --> 00:04:29,540 So these graphs here show the difference between the X and Y position measurements and the X and Y estimated 61 00:04:29,540 --> 00:04:34,060 positions or the difference between the estimated measurements and the real measurements. 62 00:04:35,180 --> 00:04:39,950 So you can see these blue lines here represent the innovations for each of the update steps inside the 63 00:04:39,960 --> 00:04:40,580 computer. 64 00:04:41,360 --> 00:04:48,010 While the green line here represents C one signal bounds of the covariance uncertainty for the innovations, 65 00:04:48,170 --> 00:04:49,610 which we'll discuss later on. 66 00:04:51,050 --> 00:04:55,880 The final figure that we want to have a look at is this one that has all the different states so you 67 00:04:55,880 --> 00:04:58,170 can see these graphs on the left side here. 68 00:04:58,190 --> 00:05:03,200 This is the exposition Y position X plus the Y velocity. 69 00:05:03,920 --> 00:05:07,210 And again, the blue line represents the true state of the system. 70 00:05:07,220 --> 00:05:12,220 So you can see in this case, we have a time across this city and this is the exposition. 71 00:05:12,860 --> 00:05:18,000 And similarly, in the Y axis, this line here represents the objects y position with time. 72 00:05:18,710 --> 00:05:23,650 Now, the black crosses represent the position measurements in the X and Y axes. 73 00:05:24,080 --> 00:05:27,500 And the red line here is the actual output from the common field. 74 00:05:28,040 --> 00:05:34,280 So their red line is the estimated position from the common theorem in the X and position in the Y. 75 00:05:34,940 --> 00:05:41,270 Now down here we have the X and Y states for velocity so we can see that the true state is moving at 76 00:05:41,270 --> 00:05:42,260 a constant velocity. 77 00:05:42,260 --> 00:05:48,320 And this case, it was moving about one meter, the second and the X and it was moving on that zero 78 00:05:48,320 --> 00:05:53,450 meters per second in the Y, the red line here is the estimate inversely from the Kalmykia. 79 00:05:53,780 --> 00:05:58,840 So I can see as a common field of runs, it starts to converge down towards the true state. 80 00:05:59,900 --> 00:06:02,690 Now, on the right hand side over here, we have the error. 81 00:06:02,990 --> 00:06:05,310 So this is the position error. 82 00:06:05,330 --> 00:06:06,770 This is, of course, the error. 83 00:06:07,160 --> 00:06:13,040 And all the error is is just the difference between the true state from the simulation and the common 84 00:06:13,040 --> 00:06:13,440 filter. 85 00:06:13,880 --> 00:06:19,070 So ideally, we want the common field of error to be about zero, which means that the estimated states 86 00:06:19,070 --> 00:06:20,450 are following the true states. 87 00:06:21,260 --> 00:06:27,350 So the red line, the error from the common of the green line here is the uncertainty from the comments 88 00:06:27,590 --> 00:06:28,610 around the error. 89 00:06:28,940 --> 00:06:35,540 So this is our three sigma uncertainty from the covariance matrix p, which we'll discuss later on. 90 00:06:37,030 --> 00:06:41,750 So when the filter is running correctly, we should see that the errors with inside are three sigma 91 00:06:41,750 --> 00:06:42,320 bounds. 92 00:06:43,180 --> 00:06:48,720 So this is a plus and minus three sigma to give you the upper and lower bounds on the error. 93 00:06:49,120 --> 00:06:51,030 So you can say this is the error for the precision. 94 00:06:51,040 --> 00:06:52,990 This is the error for our velocity. 95 00:06:54,730 --> 00:07:00,430 So every time we run the simulation, we can get a different response, so we run it again, we're going 96 00:07:00,430 --> 00:07:05,020 to start off with a different initial conditions for velocity and position. 97 00:07:05,020 --> 00:07:11,170 And we can see as the field runs, our estimate state of the position should track the blue dot quite 98 00:07:11,170 --> 00:07:11,710 nicely. 99 00:07:14,820 --> 00:07:20,640 And again, if we look at these graphs here, you can see the estimated state, the red line tracks 100 00:07:20,640 --> 00:07:25,850 the blue line very well and you can see that our velocity starts off as being zero. 101 00:07:25,860 --> 00:07:30,390 So we don't know I've lost initially, but as we run and get our position measurements, the common 102 00:07:30,390 --> 00:07:35,600 thread then can estimate the velocity and it comes and it estimates velocity and it converges towards 103 00:07:35,610 --> 00:07:36,080 a tree. 104 00:07:36,390 --> 00:07:42,630 So I can see we start off with a large initial error for Awasthi in the X and Y, but as the filter 105 00:07:42,630 --> 00:07:46,400 runs, the velocity estimates converge down to the truth. 106 00:07:46,920 --> 00:07:48,860 So they converge down to zero error. 107 00:07:49,320 --> 00:07:54,810 And the same thing for the position as the field of runs, the uncertainty shrinks and our position 108 00:07:54,810 --> 00:07:57,150 starts to converge towards the true position. 109 00:07:57,360 --> 00:08:01,350 So the object so the position filter is tracking the object very nicely. 110 00:08:03,010 --> 00:08:08,280 So in the exercises are going to be in this section, you're going to open up the corresponding file. 111 00:08:08,290 --> 00:08:11,890 So the first one we're going to be doing is the assignment one prediction. 112 00:08:12,460 --> 00:08:18,040 So if you open up the assignment, one prediction, you will see inside the script file that there'll 113 00:08:18,050 --> 00:08:20,110 be a number of common that lines. 114 00:08:20,140 --> 00:08:24,130 And these are the lines of code that you're going to have to go through and fill out the appropriate 115 00:08:24,130 --> 00:08:24,820 answer for. 116 00:08:24,820 --> 00:08:29,740 As we go through the exercises and as you go through the different exercises for the different steps, 117 00:08:30,610 --> 00:08:34,660 you should end up with a simulation that works as well as one of your shiny previously.