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The problem of localization and tracking of GMTs (ground moving targets) is investigated based on measurements of TDOA (time-difference of arrival) and DOA (direction of arrival) for which the measurement noises are assumed to be independent and identically distributed (i.i.d.). The problem of the constrained linear MMSE (minimum mean-squared error) estimation is formulated by employing the pseudo-measurement model from the existing literature that imposes a quadratic constraint on the state vector associated with the GMT dynamics. Randomization of the state vector for the GMT process suggests to replace the hard constraint by its expectation. We first derive a solution to a similar quadratically constrained MMSE estimation problem. The constrained Kalman filtering is then developed for those estimation problems involving quadratic constraints, applicable to localization and tracking of GMTs based on TDOA and DOA measurements. Moreover the constrained Kalman filter admits a simple recursive solution with complexity comparable to that of the conventional Kalman filter. A simulation example is used to illustrate our proposed constrained Kalman filter in localization and tracking of GMTs.