1
00:00:04,160 --> 00:00:09,230
We have mentioned that the common filter is a powerful way of expressing the complicated and potentially

2
00:00:09,230 --> 00:00:15,560
computationally expensive process of data fusion down into a more simpler problem that can be easily

3
00:00:15,560 --> 00:00:16,080
solved.

4
00:00:16,820 --> 00:00:22,040
This is done by making a few assumptions about the system dynamics, the estimate, the state, the

5
00:00:22,040 --> 00:00:23,600
error and noise properties.

6
00:00:25,420 --> 00:00:30,910
Mathematically, the common filter is an estimate which solves a linear quadratic estimation problem,

7
00:00:31,180 --> 00:00:36,340
which just means that the estimates, the instantaneous date of a linear dynamic system perturbed by

8
00:00:36,340 --> 00:00:41,740
random white noise, and it does is by taking a series of measurements that are linearly related to

9
00:00:41,740 --> 00:00:45,180
the state but are also are corrupted by white noise.

10
00:00:47,520 --> 00:00:53,310
So we want to calculate the estimated state, which is as close as possible to the true state, which

11
00:00:53,310 --> 00:00:59,460
is unknown, and we do this by using a series of Noisey measurements that are related to the true state.

12
00:01:00,350 --> 00:01:06,590
The estimated state solution is a statistically optimum estimate with regards to the quadratic function

13
00:01:06,590 --> 00:01:11,540
of the estimation error, which just means that it's minimizing the squared main error.

14
00:01:12,290 --> 00:01:17,540
So just like Enlace squares, we want to minimize the cost function, which minimizes the squared main

15
00:01:17,540 --> 00:01:19,070
error of some error measurement.

16
00:01:20,290 --> 00:01:25,750
The cost function that we choose to minimize is the same one as the one that we use in the recursively

17
00:01:25,750 --> 00:01:26,050
square.

18
00:01:27,520 --> 00:01:29,990
We want to minimize the state estimation error.

19
00:01:30,460 --> 00:01:35,470
So that's the difference between the true state and the estimate, the state to give you the state estimation

20
00:01:35,470 --> 00:01:35,820
error.

21
00:01:37,350 --> 00:01:43,560
We know that the state estimation era covariance matrix can be written like so and likewise with the

22
00:01:43,560 --> 00:01:47,190
cost function of the squared mean of the state estimation error.

23
00:01:48,150 --> 00:01:54,540
So just like in recursively squares, minimizing the trace of the covariance matrix also minimizes the

24
00:01:54,540 --> 00:01:55,260
cost function.

25
00:01:56,130 --> 00:02:01,500
This process has many similarities as a recursively squares process that we have already covered.

26
00:02:02,100 --> 00:02:04,980
We will go into more detail for the common field later on.

27
00:02:07,910 --> 00:02:13,220
You may have noticed that I didn't mention probability once in this definition, while the previous

28
00:02:13,220 --> 00:02:18,680
videos we talked about representing the state as a probability distribution, but making a few assumptions

29
00:02:18,680 --> 00:02:24,320
about the type of noise and system dynamics, you can cast the probability problem into an optimization

30
00:02:24,320 --> 00:02:26,510
problem and solve it in an appropriate way.

31
00:02:27,140 --> 00:02:33,110
You can look at the common filter from an optimal or geometric or probabilistic point of view, but

32
00:02:33,110 --> 00:02:37,330
you always end up with the same equation solutions in the end, which is pretty amazing.

33
00:02:40,110 --> 00:02:45,570
The common filter is one of the greatest discoveries in the history of estimation and data fusion and

34
00:02:45,570 --> 00:02:49,150
perhaps one of the greatest engineering discoveries of the 20th century.

35
00:02:49,740 --> 00:02:54,690
It has enabled mankind to do and build many things which could not be possible otherwise.

36
00:02:55,800 --> 00:03:01,500
It has immediate application in complicated dynamic systems such as those used in guidance, navigation

37
00:03:01,500 --> 00:03:04,970
and control of cars, ships, aircraft and spacecraft.

38
00:03:05,310 --> 00:03:11,850
It is widely used in robotics and manufacturing and is applicable to pretty much any time domain series

39
00:03:11,850 --> 00:03:18,390
analysis such as used in signal processing, economics, stock market prediction, finance and more.

40
00:03:21,130 --> 00:03:26,530
The common fiddler is named after its inventor, Rudolph Carmen, who was born in Budapest in the year

41
00:03:26,530 --> 00:03:27,560
1930.

42
00:03:28,120 --> 00:03:34,930
He immigrated into the US during World War Two and eventually ended up at MIT for a bachelor and master's

43
00:03:34,930 --> 00:03:36,370
in electrical engineering.

44
00:03:37,090 --> 00:03:41,380
This eventually led him to role as a lecturer at Columbia University.

45
00:03:42,520 --> 00:03:48,490
The first practical use of the common filter was by the Ames Research Center of NASA, and it was quickly

46
00:03:48,490 --> 00:03:54,310
then used for the Apollo spacecraft mission for the estimation of the trajectory and control of the

47
00:03:54,310 --> 00:03:55,060
spacecraft.