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This paper presents an central difference Kalman filter (CDKF) based simultaneous localization and mapping (SLAM) algorithm, which is an alternative to the classical extended Kalman filter based SLAM solution (EKF-SLAM). EKF-SLAM suffers from two important problems, which are the calculation of Jacobians and the linear approximations to the nonlinear models. They can lead the filter to be inconsistent. To overcome the serious drawbacks of the previous frameworks, Sterling's polynomial interpolation method is employed to approximate nonlinear models. Combined with the Kalman filter framework, CDKF is proposed to solve the probabilistic state-space SLAM problem. The proposed approach improves the filter consistency and state estimation accuracy. Both simulated experiments and bench mark data set are used to demonstrating the superiority of the proposed algorithm.