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An extended Kalman filter, EKF, is proposed for tracking the position and velocity of a moving target. The suggested method is based on a nonlinear model which, in addition, incorporates means for estimating possible nonlinearities in the measurements of the target position. In many practical scenarios, the initial estimates of target position and velocity deviate significantly from the true ones. In order to reduce the impact of erroneous initial conditions and, hence, obtain a faster initial convergence to an acceptable trajectory, a certain constrained form of the EKF, named the CEKF, is introduced. Although the original Kalman filter for a purely linear system is inherently stable, there is no guarantee that the linearized model used in the EKF gives a stable algorithm. Hence, it is interesting to note that the proposed CEKF under certain mild conditions renders an exponentially stable algorithm. It is shown that this latter method can conveniently be formulated as a nonlinear minimization problem with a quadratic inequality constraint.