Error-state Kalman Filtering with Linearized State Constraints

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.