A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conditionally Linear Measurements <s> A new method of applying the unscented transformation to conditionally linear transformations of Gaussian random variables is proposed. This method exploits the structure of the model to reduce the required number of sigma points. A common application of the unscented transformation is to nonlinear filtering where it used to approximate the moments required in the Kalman filter recursion. The proposed procedure is applied to a nonlinear filtering problem which involves tracking a falling object. <s> BIB001 </s> A Survey of Code Optimization Methods for Kalman Filter Extensions <s> Conditionally Linear Measurements <s> State estimation for nonlinear systems generally requires approximations of the system or the probability densities, as the occurring prediction and filtering equations cannot be solved in closed form. For instance, Linear Regression Kalman Filters like the Unscented Kalman Filter or the considered Gaussian Filter propagate a small set of sample points through the system to approximate the posterior mean and covariance matrix. To reduce the number of sample points, special structures of the system and measurement equation can be taken into account. In this paper, two principles of system decomposition are considered and applied to the Gaussian Filter. One principle exploits that only a part of the state vector is directly observed by the measurement. The second principle separates the system equations into linear and nonlinear parts in order to merely approximate the nonlinear part of the state. The benefits of both decompositions are demonstrated on a real-world example. <s> BIB002
In BIB001 , a situation where a function can be divided into nonlinear z n and conditionally linear z l parts is considered. In the original article, the distributions of this nonlinear function is approximated using a modification of the UKF. The number of used sigmapoints depends on the dimension of z n instead of the dimension of the full state. This algorithm is converted to be used with the GF in BIB002 . Here we present the algorithm in a general form that allows it to be also with any KFE that uses weighted points to compute the expectations, as in (24)- (26), but it cannot be used with other types of filters e.g. DD or SOKF2. In the algorithm the sigma-points are computed for the nonlinear part only and then used for computing the conditional probabilities of the conditionally linear part: The matrices in the equations above are independent of the sigma-point χ i . Thus, P z l zn P −1 znzn and P z l z l |χi need to be computed only once. According to BIB001 , the expectations for a function of form (37) can be approximated as These formulas can be used to compute the expectations for the nonlinear part g(z) in algorithms 1 and 2, if there are conditionally linear state variables. This is done in the example in Section 7.1.