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I have some doubt on EKF Slam with known correspondence in the measurement update state. I follow the algorithm from Probabilistic Robotics by Sebastian THRUN. This Algorithm is on Chapter 10 page No:249 of this books.

I attached a snapshot of my doubt.

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In this algorithm line No: 10 and 12 create some doubts.

I want to know what is $\bar\mu_{t,x}$ and $\bar\mu_{t,y}$.

As per my understanding, it is the robot position in the given timestamp when it sees landmarks.

But I have some confusion, so it would be very helpful for me if anyone could clarify line 10,12.

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Line 10 just uses trigonometry to compute the position of the landmark μ¯j,xyz (left of equals sign) based on the current estimate of the robot's position μ¯t,xyz and the sensed range r and the sensor's line-of-sight angle phi. This only happens the first time that it is seen by the sensor. This initializes the state estimate for that landmark, as there is no other information to use at that point. After that, there will always be an entry in the state matrix which represents an estimate of that landmark's position, and it will be further refined by multiple "looks" at the landmark.

Line 12 finds the 2D difference vector between the current estimate of the landmark's position μ¯j,xy and the filter's current estimate of the robot's position μ¯t,xy. This is later used to estimate what the robot thinks a measurement of that landmark will be, and will be used to compare against any actual measurement of that landmark.

If by "doubt" you mean that you believe that these equations are incorrect, I can assure you that they have been vetted by many people. However, there are some simplifications (such as a flat world) that usually results in a need for additional enhancements in a real-world application.

My apologies if the math did not render correctly. I am not sure how to do this.

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