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I read Probabilistic Robotics by Sebastian THRUN (online version). I also read http://ais.informatik.uni-freiburg.de/teaching/ws12/mapping/pdf/slam04-ekf-slam.pdf

Question1 : What will be the dimension of Matrix $F(x,j)$ below? As per book its dimension is $(3N+3)*5$. So if I have 10 landmark the dimension will be $33*5$. Meaning 33 Rows and 5 column. But as per picture No: 1 $F(x,j)$ has 6 Rows. This is confusing for me.

Picture No1: What is the actual dimension of this Matrix if I have 10 landmarks

Question 2: Both references describe the same approach Extended Kalman filter slam with known correspondence, the only difference between them is the equation to build the matrix $H_i$ (see below, picture 2 and Picture 3).

Now after reading those books I am confused which one is error less?

Picture No2: Matrix

Picture No3: enter image description here

Helpful discussion appreciated.

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For your first question, $F(x, j)$ is a $6 * 3N + 3$ dimensional matrix. Why? $F(x, j)$ is a factor of $H_t^i$. The reason $H_t^i$ is a $3 * 3N + 3$ matrix is because of the problem set up. We are concerned with $N$ landmarks and a 3 dimensional state $[x\ \ y\ \ \theta]$. I recommend re-reading over page 319 of the book, especially points 10.18, 10.19, and 10.20 (its too long for me to post here).

For your second question, look on slide 34, second bullet, of EKF SLAM. Low $H_t^i$ is not the same thing as $H_t^i$. In the second bullet, we see that the derivation of $H_t^i$ is exactly the same as Thrun et al. writes in the book (page 314). See that the low $H_t^i * F(x,j)$ is the factorization of the derivative, $H_t^i$ (as mentioned above).

As a side note, the slides you posted come with a recorded lecture that is freely available on YouTube, thanks to the author of this course. The lectures provide more insightful guidance through the material than just a read. If you haven't already taken a look, it will be well worth the time.

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