Kalman Filter with Hard and Soft Constraints for the Integration of Multiple Pedestrian Navigation Systems

In this paper, we proposed two different approaches for incorporating non-linear state inequality constraints into a Kalman filter (KF). The first approach incorporates a soft constraint into a KF to ensure that the state estimate almost satisfies the constraints rather than strictly satisfies the constraints; the second approach incorporates a hard constraint into a normal KF to ensure that the state estimate should strictly satisfy the constraints. Simulation study has been conducted in the integration of two different pedestrian navigation systems. The simulation results show that both approaches can well bound the state estimates errors compared with the unconstrained state estimates. However, when a constraint's nonlinearity level is getting stronger, as well as the linearization point is less accurate, the performances of both approaches will decrease, especially for the soft constraint approach. The improved performance of the constrained filters comes with a price, which is computational effort. The filter with soft constraints requires only slightly more computational effort than the unconstrained filter, but the filter with hard constraints requires a greater computational effort compared with the soft constraint approach.