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So we now have the complete example shown on the screen here, and if we run this, we can see that

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the filter is working quite well.

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So we have the pendulum estimation.

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As the pendulum moves back and forth, the estimates follow the truth.

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So the red line is following the blue line quite well.

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If we have a look at our measurement innovations, you can see that they are mostly zero remained and

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the variance here.

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So our covariance for the innovation is about point to four, while our measurement of innovation on

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Sandy is about point to five.

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So that's fairly well trained.

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That sort of a lining up.

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And if we have a look at our theater estimates for the position and for Al-Fadl, not for the angle.

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Right.

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They track quite nicely.

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Sorry, there's a little area between the two.

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So if we look at the airports here, we can see that our data error is within the three sigma bounds

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of our positions.

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And we can see that three sigma bounds for the velocity contains the actual velocity error.

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So the filter is operating quite well here and it's fairly well tuned.

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When we ran this field out here, you could see that we had a non-zero process, more noise, so we

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had this talk standard deviation set point zero one.

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So if we actually ran this again, but using zero for the torque standard deviation so that our key

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metrics is zero.

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So we are not adding any additional process, more noise into the system and run it.

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We actually find that we get a slightly different response.

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So even though the estimates still look fine in terms of the estimated position, if we take a closer

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look at the measurement innovations, you can see that our innovations, while they're still see, remain

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the star need to creep up.

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So the error between the measurements and the estimate starts to increase.

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And you can see that sort of almost diverging behavior.

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The longer you run, the larger the innovation is going to get.

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If we have a look at the position error, you can clearly see that this field is not performing as well

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as it did in the previous example.

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We can see that our estimated errors for our position and Awasthi increasing.

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So you can see this oscillatory behavior.

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This is as expected as we have a pendulum system moving back and forth back and say the overall magnitude

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of the error is increasing with time and the same thing here for the velocity.

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So if we were to run this for a longer period of time, so instead of just running this for 10 seconds,

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let's run this for, let's say, 60 seconds.

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You might find that we get an even more dramatic increase, so this is our measurement innovation,

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so you can see a dramatic increase in the error.

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And if we look at our actual errors here, you can see that the system is diverging.

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Is is error is increasing with time.

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00:02:57,490 --> 00:03:03,430
So this is what happens when there's not enough uncertainty inside the system to compensate for the

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error in the modelling.

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So since we're using a linear approximation, we can get additive error coming in as we run with time.

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00:03:10,600 --> 00:03:17,770
So if we went back and increased outtalk standard deviation back to the original value of zero point

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zero one and run it again, you can see we're going to get a lot better behavior.

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00:03:23,320 --> 00:03:27,340
So we now we can see that our innovations are back to being zero mean.

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And the variance of them sort of follows along with our variance of measurement innovations.

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00:03:33,730 --> 00:03:40,960
We can also see that our position error for Theta and was the error for Theta Dot nicely contained inside

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the three sigma bounds.

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00:03:42,850 --> 00:03:48,760
So it is clear to say here that for this field of work we need some level of uncertainty injected into

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the system to compensate for the modeling errors.

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00:03:54,070 --> 00:03:58,530
So we were running this system based on a small angle.

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00:03:58,540 --> 00:04:03,480
So we're starting at a small initial angle and we're oscillating back and forth on small angles.

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So if we change this back to 10 seconds, but instead of starting at the known small angle, it starts

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at a random angle, which can be large.

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And let's have a look at this.

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So we've just changed the starting angle to be a random value.

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And let's run this.

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So now I can see that the pendulum is oscillating back and forth with a lot larger angle, so we started

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off at a larger initial angle.

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In this case, we started off around 25 degrees and we're oscillating back and forth.

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So in this case, you can see a filter is still performing fairly well.

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00:04:39,510 --> 00:04:46,650
We do start to see this oscillatory behavior on the velocity and we starting to get a slightly larger

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errors inside the position estimates.

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00:04:50,280 --> 00:04:54,710
And we can probably see that we actually is larger errors every now and again.

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And these are probably when we're in the nonlinear region.

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So we can see that this even though this filter is a linear filter with a linear process model, the

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fact that we've increased the uncertainty inside the system allows it to compensate for some of the

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nonlinear realities inside the system.

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00:05:11,760 --> 00:05:16,980
So we basically increased the uncertainty in system to let it to compensate for this.

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00:05:18,290 --> 00:05:23,880
So even though the system that we're estimating is a nonlinear system, we're using a linear common

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00:05:23,880 --> 00:05:25,790
filter with a linear approximation.

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00:05:26,040 --> 00:05:31,470
And this allows it to work because we have model such in the way that we can increase the uncertainty

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inside the system to compensate for any errors in modelling.

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So hopefully this should be a clear example that we can still apply a linear system to a nonlinear system

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if we go about it in a good systematic way.