Abstract
In recent years, the error-state Kalman filter (ErKF) has been
extensively employed across various applications, including but not
limited to robotics, aerospace, and localization. However, incorporating
constraints into the ErKF framework when state constraint is necessary
has remained a challenging task due to its intrinsic properties. This
paper explores all possible ways to achieve this goal in the context of
the estimate projection method. In particular, the constraint can be
enforced before or after the ErKF’s correction step. We approach the
problem from a mathematical perspective by deriving analytical solutions
and discussing their statistical properties. We prove that the two
mentioned methods are statistically identical for a linear system with
linear constraints. Conversely, the filter’s behavior remains uncertain
in the presence of linearized constraints. However, we provide a special
case of the nonlinear constraint, wherein the results of the linear case
remain valid. To support our theorem and verify the filter’s performance
when the assumptions are invalidated, we present two Monte Carlo
simulations under the increasing initialization error and the
constraint’s incompleteness. The simulation results clearly confirm our
insights and lead to the conclusion that constraining the error-state
after the correction may offer superior outcomes compared to its
competitor.