A new formulation for nonlinear forward-backward smoothing

A new formulation for nonlinear smoothing is derived using forward-backward sigma-point Kalman filtering (SPKF). The forward filter uses the standard SPKF. The backward filter requires the use of the inverse dynamics of the forward filter. While smoothers based on the extended Kalman filter (EKF) simply invert the linearized dynamics, with the SPKF the forward nonlinear dynamics are never analytically linearized. Thus the backward nonlinear dynamics are not well defined. In previous work, a sigma-point Kalman smoother (SPKS) was derived by learning a nonlinear model of the backward dynamics from empirical data. In this paper, we make use of the relationship between the SPKF and weighted statistical linear regression (WSLR). The resulting pseudo-linearized dynamics obtained by WSLR is more accurate in the statistical sense than using a first order truncated Taylor series expansion as with the EKF. A new backward information filter can then be derived, which is combined with the forward SPKF to form the smoothed estimates.