  This paper develops a new filtering approach for state estimation in
polynomial systems corrupted by arbitrary noise, which commonly arise in
robotics. We first consider a batch setup where we perform state estimation
using all data collected from the initial to the current time. We formulate the
batch state estimation problem as a Polynomial Optimization Problem (POP) and
relax the assumption of Gaussian noise by specifying a finite number of moments
of the noise. We solve the resulting POP using a moment relaxation and prove
that under suitable conditions on the rank of the relaxation, (i) we can
extract a provably optimal estimate from the moment relaxation, and (ii) we can
obtain a belief representation from the dual (sum-of-squares) relaxation. We
then turn our attention to the filtering setup and apply similar insights to
develop a GMKF for recursive state estimation in polynomial systems with
arbitrary noise. The GMKF formulates the prediction and update steps as POPs
and solves them using moment relaxations, carrying over a possibly non-Gaussian
belief. In the linear-Gaussian case, GMKF reduces to the standard Kalman
Filter. We demonstrate that GMKF performs well under highly non-Gaussian noise
and outperforms common alternatives, including the Extended and Unscented
Kalman Filter, and their variants on matrix Lie group.
