Based on the intuition that it is easier to approximate a probability distribution than it is to approximate an arbitrary nonlinear function or transformation, the authors propose a new approach for applying the linear estimation theory to nonlinear systems. Instead of approximating the Taylor series to an arbitrary order, the paper considers the approximation of the first three moments of the prior distribution accurately, using a set of samples. The proposed algorithm predicts the mean and covariance accurately up to the third-order and, because the higher-order terms in the series are not truncated, it is possible to reduce the errors in the higher-order terms as well.
The new linear estimator is shown to yield a performance equivalent to the Kalman filters for linear systems, and generalizes elegantly to nonlinear systems without the linearization steps required by the extended Kalman filter (EKF). The authors prove analytically that the expected performance of the new approach is superior to that of the EKF method. Empirical evidence is provided to support the theoretical conclusions, demonstrating that the new filter is easier to implement; it does not involve any linearization steps, and eliminates the derivation and evaluation of the Jacobian matrices.
The proposed algorithm has been extended to capture the first four moments of a Gaussian distribution, and the first three moments of an arbitrary distribution. Various applications have been found to be suitable to represent the new algorithm; e.g. high-order nonlinear coupled systems, navigation systems for high-speed road vehicles, public transportation systems, underwater systems, etc.