29 Jan

Summary of the 1st SLAM study group

We studied the paper “A solution to the simultaneous localisation and mapping (SLAM) problem” together this afternoon, and I found out several serious concern about this seminal paper, who has more than 3000 citation. The paper gives 3 main theorems for the extended Kalman filter (EKF) formulation of SLAM as their main contribution. However, these 3 theorems are not solid enough.

For the 1st theorem, it states that covariance of map decreases as time goes by. However, this is just a direct result of EKF and the fact that the landmarks are stationary. It has nothing surprising in this theorem, and it also looks redundant to write down the proof.

For the 2nd theorem, the authors make several mistakes in my perspective. First, they directly use the result of convergence of the whole covariance without proof. It seems like they proves the convergence of the map covariance matrix, and slyly use the convergence of the whole covariance matrix instead. Second, even if the whole covariance matrix converges, they will not necessary lead to the zero map covariance matrix. The reason is that they use the wrong equation. The observation update in KF is very complicated, but it becomes very clear in the inverse form. The authors try to prove something in the original form, and they are certainly confused by the complicated form, for sure.

For the 3rd theorem, they give a lower bound on the covariance matrix. But the statement is very wired. Covariance matrices are positive definite by definition. Therefore, only upper bound is meaningful. Giving lower bound to positive definite matrices is like saying all positive integers are lower bounded by 0. The way that they derive low bound is by setting the input uncertainty zero, which definitely breaks the symmetry of KF setting.

I think the correct way to solve this problem is by Riccati recursion, which is the standard way to deal with this kind of problem. Also, I have to check whether their simulation or experiment confirms their statement or not.