  This paper addresses the problem of designing the {\it continuous-discrete}
unscented Kalman filter (UKF) implementation methods. More precisely, the aim
is to propose the MATLAB-based UKF algorithms for {\it accurate} and {\it
robust} state estimation of stochastic dynamic systems. The accuracy of the
{\it continuous-discrete} nonlinear filters heavily depends on how the
implementation method manages the discretization error arisen at the filter
prediction step. We suggest the elegant and accurate implementation framework
for tracking the hidden states by utilizing the MATLAB built-in numerical
integration schemes developed for solving ordinary differential equations
(ODEs). The accuracy is boosted by the discretization error control involved in
all MATLAB ODE solvers. This keeps the discretization error below the tolerance
value provided by users, automatically. Meanwhile, the robustness of the UKF
filtering methods is examined in terms of the stability to roundoff. In
contrast to the pseudo-square-root UKF implementations established in
engineering literature, which are based on the one-rank Cholesky updates, we
derive the stable square-root methods by utilizing the $J$-orthogonal
transformations for calculating the Cholesky square-root factors.
