Doz C. Obidin M. Sarkka S.
Applications of Kalman Filtering in Traffic Management and Control
Nikitin A. Arasaratnam I. Gavrilov A. Issue Rudenko E. Kaladze V. Seriya: Sistemniy analiz i informacionnye tekhnologii. Issue 1. Taranenko Yu. Rossum G. Degtyarev A. Keldysha RAN. Chernavskiy D.
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Examining the Kalman filter in the field of noise and interference with the non-Gaussian distribution. Abstract We have developed a sequential recursive Kalman Filter algorithm to filter data in the field of the non-Gaussian noise distribution to be used in measurement instruments. Keywords Kalman filter; recursive algorithm; Python; non-Gaussian noise; distribution law. References Grewal, M. Cover, T.
Elements of information theory. Totally neat! You can use a Kalman filter in any place where you have uncertain information about some dynamic system, and you can make an educated guess about what the system is going to do next.
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Even if messy reality comes along and interferes with the clean motion you guessed about, the Kalman filter will often do a very good job of figuring out what actually happened. Kalman filters are ideal for systems which are continuously changing. The math for implementing the Kalman filter appears pretty scary and opaque in most places you find on Google. Thus it makes a great article topic, and I will attempt to illuminate it with lots of clear, pretty pictures and colors.
The prerequisites are simple; all you need is a basic understanding of probability and matrices. Note that the state is just a list of numbers about the underlying configuration of your system; it could be anything. Our robot also has a GPS sensor, which is accurate to about 10 meters, which is good, but it needs to know its location more precisely than 10 meters.
with Real-Time Applications
There are lots of gullies and cliffs in these woods, and if the robot is wrong by more than a few feet, it could fall off a cliff. So GPS by itself is not good enough. The GPS sensor tells us something about the state, but only indirectly, and with some uncertainty or inaccuracy. Our prediction tells us something about how the robot is moving, but only indirectly, and with some uncertainty or inaccuracy. But if we use all the information available to us, can we get a better answer than either estimate would give us by itself?
The Kalman filter assumes that both variables postion and velocity, in our case are random and Gaussian distributed. In the above picture, position and velocity are uncorrelated , which means that the state of one variable tells you nothing about what the other might be. The example below shows something more interesting: Position and velocity are correlated. The likelihood of observing a particular position depends on what velocity you have:.
This kind of situation might arise if, for example, we are estimating a new position based on an old one. If our velocity was high, we probably moved farther, so our position will be more distant. This kind of relationship is really important to keep track of, because it gives us more information: One measurement tells us something about what the others could be. This correlation is captured by something called a covariance matrix.
Next, we need some way to look at the current state at time k-1 and predict the next state at time k.
Kalman Filtering : with Real-Time Applications (4TH) [Paperback]
It just works on all of them , and gives us a new distribution:. It takes every point in our original estimate and moves it to a new predicted location, which is where the system would move if that original estimate was the right one. How would we use a matrix to predict the position and velocity at the next moment in the future? This is where we need another formula. For example, if the state models the motion of a train, the train operator might push on the throttle, causing the train to accelerate.
Similarly, in our robot example, the navigation software might issue a command to turn the wheels or stop. For very simple systems with no external influence, you could omit these. Everything is fine if the state evolves based on its own properties. Everything is still fine if the state evolves based on external forces, so long as we know what those external forces are.
Every state in our original estimate could have moved to a range of states. This produces a new Gaussian blob, with a different covariance but the same mean :. In other words, the new best estimate is a prediction made from previous best estimate , plus a correction for known external influences.
And the new uncertainty is predicted from the old uncertainty , with some additional uncertainty from the environment. What happens when we get some data from our sensors?
Kalman Filtering with Real-Time Applications | Charles K. Chui | Springer
We might have several sensors which give us information about the state of our system. Each sensor tells us something indirect about the state— in other words, the sensors operate on a state and produce a set of readings. One thing that Kalman filters are great for is dealing with sensor noise. In other words, our sensors are at least somewhat unreliable, and every state in our original estimate might result in a range of sensor readings.
From each reading we observe, we might guess that our system was in a particular state. But because there is uncertainty, some states are more likely than others to have have produced the reading we saw:. So now we have two Gaussian blobs: One surrounding the mean of our transformed prediction, and one surrounding the actual sensor reading we got. If we have two probabilities and we want to know the chance that both are true, we just multiply them together.
So, we take the two Gaussian blobs and multiply them:. The mean of this distribution is the configuration for which both estimates are most likely , and is therefore the best guess of the true configuration given all the information we have.