The paper focuses on the state estimation of the nonlinear discrete-time stochastic dynamic systems by the derivative-free filters. In particular the impact of the ¾-point set rotation on the performance of the unscented transform and the unscented Kalman filter (UKF) is analysed. It is shown that the ¾-point set rotation is an additional user-defined parameter closely tied with the covariance matrix decomposition technique used in ¾-point computation that significantly affects the estimation performance. Analysis, algorithms, and recommendations for computations of the optimal ¾-point set rotation are provided to determine either the rotation prior to the estimation experiment (off-line) or during the estimation experiment (on-line). Further, two approaches for a reduction of optimisation computational costs are presented. The proposed algorithms, namely the on-line adaptive-sigma-point-set-UKF (AUKF) and off-line trained-sigma-point-set-UKF (TUKF), are illustrated and verified in a numerical study considering two static and two dynamic examples. The TUKF improves the UKF performance, while the computational complexity is preserved. The AUKF further im-proves the estimate accuracy with increased computational burden.