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We have already covered that no sensor is perfect and that they can return different types of errors.

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Now, sensors themselves can also return erroneous data or faulty data.

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And this can be for a number of reasons.

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It could be a mechanical issues like the sensor could have come loose or could be a hardware issue.

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The sensor was broken or it could just be the characteristics of the sensor.

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It could be such like in GPS.

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If we lose satellite signals, the GPS data is not going to be reliable, is no longer going to give

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us a valid position measurements.

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So there's many different reasons why the sensor itself might give us faulty data.

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So sensors and measurements are not perfect.

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We already assume that the measurements that they provide are corrupted with noise.

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They can also be cases when the measurements are completely erroneous and outside the noise profile

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due to hardware or sensing faults.

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As we've discovered, fusing wrong information into the common filter can cause a common field to diverge

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and degrade the performance.

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And a single wrong measurement could be enough to make the filter unusable for the rest of the time

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it's running.

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So this is very important that we don't use any wrong information into the common filter.

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We want to stop the faulty measurement from being used in the first place.

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And this is pretty much the only way to deal with this type of faulty data.

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And this process is commonly called Foch detection and isolation.

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So we wanted to detect when a sensor is faulty, that the data is faulty and we want to isolate it from

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the rest of the system before it allows this incorrect measurement to corrupt the system.

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So there are a number of different typical sets of thoughts that you see inside the sensor, the first

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one we're going to look at is when we get erroneous spikes.

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So pretty much as time goes on, we might get these large jumps in data.

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So you get these spikes every now and again and they're totally out of line with what the sensors should

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be reading.

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So you can sort of clearly identify these are erroneous measurements, another type of sense of what

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is called a lock in place for.

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So we might have the ideal sensor producing a measurement like this, but at some time it locks in place

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and it produces a constant output afterwards, even though the measurement should be producing this

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kind of line up here.

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So this could be so this is more typically of hardware fault.

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So if something breaks inside the system, say a linkage breaks to something or something comes loose,

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the sensor is going to get disconnected from what it's sensing.

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It can stay locked in place.

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Another typical sense of what is called a hard over.

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So basically we have initially the sense is following the truth quite well until we get to a certain

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point and then the center basically becomes saturated.

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So it goes all the way up to the max or it could go all the way down to the minimum.

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And again, this is typically usually another how if something might have come loose, causing the sense

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I just to diverge.

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So in a similar point of view, we might have a dead center.

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So this is basically as a sensor is running something like in a loose wire or something, something

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just disconnects and then we just get a zero output for the rest of the run.

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And then lastly, another typically fault that you get when a sensor disconnects is called a floating

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error.

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So the sensor is tracking well, and normally up until this point and off this point, suddenly the

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output becomes random and basically starts to float around.

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So something could have come loose or something could be flapping in the wind.

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It could be a loose connection on the wire and it's just jumping around all over the place.

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So these most of these errors here can be caused by hardware issues or mechanical issues, or it could

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be loose connections or something like that.

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So with the spike faults, what we want to do is make sure we don't fuse these erroneous spikes inside

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it for the rest of these forces.

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Basically, once we've detected that the fault has occurred, we can't trust the sensor anymore.

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So once that happens, we can't use any of the measurements from the sensor after the time that the

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fault has occurred.

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So we want to isolate it and or for the rest of the time, whereas with the spikes, we can probably

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just ignore that single measurement or number of measurements that are no longer consistent with what

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we expected the sensor to be outputting.

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The main way to detect Sensa faults is from the measurements themselves, it is to look at the measurement

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innovation from the carbon filter or some statistical probability based on the innovation.

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And this is to say if the sensor is performing as specified or as expected.

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If the sensor is detected as 40 or the current measurement is bad, then we want to ignore the data

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rather than just fusing it into the common filter and corrupt the estimates.

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So the main way to do this is called innovation checking so we know the probabilistic properties that

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the innovation should follow.

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And we've covered this in common filter.

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We want the innovation should be zero mean, so we want to drive the innovation desire to drive the

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estimated errett is are.

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We also know the innovation covariance and this is estimated in the first part of the coming forward

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update step.

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So this is The Matrix.

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So the innovation here should be a random variable from a Gaussian distribution with zero, a mean and

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covariance s.

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So using this probabilistic model of the innovation, we can come up with some innovation checking criteria.

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We can use what we know about probability to test new measurements before we use them into the filter.

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So if we have a look at the one, the innovation, normal distribution.

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So basically we're looking at this normal distribution here of main zero covariance and this is the

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innovation here.

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So we know that the innovation should have zero mean and it has a Gaussian or normal curve that looks

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like this with covariance s and we know from Gaussian or normal probability distribution that anything

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within plus or minus one standard deviation from the main encapsulates point six, eight or 68 percent

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of probability that the value of the random variable should fall into this range.

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And then similarly for two sigma, we know that it should encompass this 95 percent and three sigma

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should compas 99 percent.

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So if we get an innovation value all the way over here, there's probably less than a one percent probability

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that the measurement is valid.

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So we can use this criteria to test whether the measurement is valid by looking at the innovation value.

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If the innovation value is larger than expected, then we know that is a faulty measurement and this

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can be simply tested by a simple comparison operator.

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So if the value or size of the innovation is greater than two times the square root of S in this case

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here we know that there is less than a five percent chance that the measurement would be valid based

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on the two sigma standard deviation.

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So basically we can set a threshold and ignore these long values from the innovation.

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So basically anything larger in this case than our Three Sigma means is less than a one percent chance

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is a valid probability for a valid measurement.

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So this works quite nicely in the one dimensional case.

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However, what can we do for when we have a multidimensional Gaussians?

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So instead of having a single value for the innovation, when we actually have the innovation vector

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like we've been dealing with, and to do this, we use something called the normalized innovation squared

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or is so for the multidimensional case we can use to normalize innovation squared the normalized innovation

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squared, it should follow a unit Gaussian distribution.

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We can test for this property using the CHA Square test, also known as the person's CHA squared test.

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So now let's let the innovation vector mue be in dimension of innovation vector and let s be the innovation

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covariance matrix in normalize innovation squared can be calculated using this equation here.

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So basically we calculate the innovation squared and we normalize it by the uncertainty inside the system

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to calculate the NSA.

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So this this time over here, we can test to see if the innovation belongs to the unit Gaussian distribution

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for a specific probability level using the GI Square test.

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And this test can be written like so.

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So here we have the innovation value here.

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There is now for this to be a valid measurement from the ideal distribution it has to meet this criteria

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here.

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So we have to say the price is less than this value here or this threshold here.

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Now, this is the tri squared distribution, so you can see the chai squared Alem here is our degrees

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of freedom.

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So this is going to be the measurement vector size.

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So we have a single measurement is going to be and is going to be equal one if we have like a two dimensional

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measurement and is going to be equal to.

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And likewise.

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And then this value here is the P value.

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The pay value here is going to be how certain we want to be of this test, so we normally typically

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say I pay value of anything smaller than point zero.

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Five is significant.

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So what we're saying with this pay value here is that if we do one minus this pay value, in this case

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it be 95 percent, we will have a 95 percent confidence that this measurement belongs to this distribution.

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If these properties if it's less than this value here, i.e. the measurement is faulty, if the value

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is larger than the value of the tri squared distribution.

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So the choice squared distribution value can be looked up in tables to find the value of which you want

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to set the threshold to, so I listed a few values in this table here.

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So this is for the pay level of point zero five and the number of measurement dimensions.

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So for a two minute two dimensional measurement vector, so such as if we had the x y position, we

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can set a square, a threshold of paying five point nine nine.

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This can also be read off over this table in here.

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So for a thirty two degree, I'm afraid.

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So this blue line here, we can look for a pay level or point zero five, which is all the way down

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towards here.

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So if the Naess is larger than this value here, then we know that with this historical significance

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that the measurement is out of range.

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The measurement or innovativeness does not belong to the distribution as expected.

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And in this case, we should ignore the measurement.