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For linear systems, the original Kalman filter under the minimum mean square error (MMSE) criterion is an optimal filter under a Gaussian assumption. However, when the signals follow non-Gaussian distributions, the performance of this filter deteriorates significantly. An efficient way to solve this problem is to use the maximum correntropy criterion (MCC) instead of the MMSE criterion to develop the filters. In a recent work, the maximum correntropy Kalman filter (MCKF) was derived. The MCKF performs very well in filtering heavy-tailed non-Gaussian noise, and its performance can be further improved when some prior information about the system is available (e.g., the system states satisfy some equality constraints). In this paper, to address the problem of state estimation under equality constraints, we develop a new filter, called the MCKF with state constraints, which combines the advantages of the MCC and …