Left-invariant extended Kalman filter and attitude estimation

We consider a left-invariant dynamics on a Lie group. One way to define driving and observation noises is to make them preserve the symmetries. We propose a left- invariant (i.e, intrinsic and thus symmetry-preserving) extended Kalman filter such that the left-invariant estimation error obeys a stochastic differential equation independent of the system trajectory. The theory is illustrated by an attitude estimation example.