By augmenting the Lagrangian variables to the model variables, the Lagrangian Data Assimilation (LaDA) method is efficient in extracting the flow information from a sequence of the position observations by the Lagrangian instruments, such as balloons in the atmosphere and drifters in the ocean, without the necessity to derive the velocity from the sequence. A similar scheme can be used to assimilate a sequence of the position observations by the quasi-Lagrangian instruments, such as gliders and autonomous vehicles, whose trajectories can be guided by the pre-loaded control programs but are influenced by the velocity of the flow field. Using the ensemble Kalman filter approach, the LaDA method is implemented for the eddy tracking in the Gulf of Mexico. We demonstrate the effectiveness of assimilating the observations by the Lagrangian and the quasi-Lagrangian instruments in comparison with the Eulerian observations of the velocity field.