In the paper of multirobot cooperative localization analysis [1], they discussed two cases: observable and unobservable. The former leads to a stable estimation, while the later does not. In the paper, they uses extensive calculation to delineate their result. However, such outcome can be easily explained by Kalman decomposition.

By Kalman decomposition, one can decompose a linear system into observable and unobservable subsystems by similar transformation. One can regard the transformation as the change of coordinate. In addition, the similarity transformation is trace preserved, which means that the mean squared error in Kalman filter estimation is preserved (conjecture, not proved yet).

In summary, we can avoid tons of calculation to get the same result, and provide better insight.

[1] A.I. Mourikis, and S.I. Roumeliotis, "Performance analysis of multirobot cooperative localization," IEEE Trans. on Robotics, vol. 22, no. 4, 2006.

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