Chapter 8
The Sigma Point Class: The Finite Difference Kalman Filter
The linearized EKF yields reasonable estimation results if the nonlinearities are not very severe. For problems where the dynamic or observation transition functions are highly nonlinear, second-order terms can be included in the EKF but at the expense of adding the requirement of calculating the Hessian matrices. This added complication is sometimes very difficult to accomplish. One method to alleviate this difficulty is to replace the differentials of the EKF with their finite difference equivalents.
Schei was the first to propose using a central finite difference approach to linearization in nonlinear estimation algorithms [8Schei], [8Schei2]. While the EKF is a linearization of 
about the point 
, Schei's method linearizes about the central difference points 
. Because Schei replaces only the Jacobian in the prediction equations for both the state (observation) vector and its covariance, the method is more accurate than the EKF only in the state (observation) vector prediction step.
A more accurate derivative-free estimation method for nonlinear systems was developed by Nrgaard et al. [8Norgaard1], [8Norgaard2] by ...
