Unscented Kalman filtering in the additive noise case

The unscented Kalman filter (UKF) has four implementations in the additive noise case, according to whether the state is augmented with noise vectors and whether a new set of sigma points is redrawn from the predicted state (which is so-called resampling) for the observation prediction. This paper concerns the differences of performances for those implementations, such as accuracy, adaptability, computational complexity, etc. The conditionally equivalent relationships between the augmented and non-augmented unscented transforms (UTs) are proved for several sampling strategies that are commonly used. Then, we find that the augmented and non-augmented UKFs have the same filter results with the additive measurement noise, but only have the same state predictions with the additive process noise. Resampling is not believed to be necessary in some researches. However, we find out that resampling can be helpful for an adaptive Kalman gain. This will improve the convergence and accuracy of the filter when the large scale state modeling bias or unknown maneuvers occur. Finally, some universal designing principles for a practical UKF are given as follows: 1) for the additive observation noise case, it's better to use the non-augmented UKF; 2) for the additive process noise case, when the small state modeling bias or maneuvers are involved, the non-resampling algorithms with state whether augmented or not are candidates for filters; 3) the resampling and non-augmented algorithm is the only choice while the large state modeling bias or maneuvers are latent.