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This paper addresses two key limitations of the unscented Kalman filter (UKF) when applied to the simultaneous localization and mapping (SLAM) problem: the cubic computational complexity in the number of states and the inconsistency of the state estimates. To address the first issue, we introduce a new sampling strategy for the UKF, which has constant computational complexity. As a result, the overall computational complexity of UKF-based SLAM becomes of the same order as that of the extended Kalman filter (EKF)-based SLAM, i.e., quadratic in the size of the state vector. Furthermore, we investigate the inconsistency issue by analyzing the observability properties of the linear-regression-based model employed by the UKF. Based on this analysis, we propose a new algorithm, termed observability-constrained (OC)-UKF, which ensures the unobservable subspace of the UKF's linear-regression-based system model is of the same dimension as that of the nonlinear SLAM system. This results in substantial improvement in the accuracy and consistency of the state estimates. The superior performance of the OC-UKF over other state-of-the-art SLAM algorithms is validated by both Monte-Carlo simulations and real-world experiments.