Recently we've encountered Kalman filter algorithm for state estimation in a course of Probabilistic Robotics.

After taking several days to try to read Kalman's original paper published in 1960, A New Approach to Linear Filtering and Prediction Problems, it firstly feels a bit difficult to read, and it seems the majority is to show the orthogonal projection is the optimal estimation under certain conditions and solutions to Wiener's problem.

But I did not find the exact algorithm in this original paper as the one in the textbook.

  • For example, is there an explanation of "Kalman gain" in this paper ?
  • Does Kalman's paper provide a mathematical derivation of Kalman filter algorithm?

The Kalman Gain term is derived in equation 25 (page 7) of the paper. The paper doesn't explicitly use the term "Kalman Gain" as that term was coined after the fact. However, the paper does refer to the principle as the "optimal filtering" on page 6 (just after equation 19). For details on its derivation, refer to the section titled "Solution to the Wiener Problem".

In the same section, we have the Kalman filter algorithm and its mathematical derivation. Although the discovery is extremely important and ground breaking, the paper hasn't aged well with respect to the computing community in regards to its use of notation and language. As a result, it is understandable that you may have missed it.

If you or anyone else reading this post has trouble with the mathematical derivation of the algorithm in the original paper, there are some good resources posted in the comments of the question.


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