In this paper, we address the sonar-based navigation of mobile robots. The extended Kalman filtering (EKF) technique is considered, but from a deterministic, not a stochastic, point of view. For this problem, we present results on the robustness of the non-linear discrete-time observation scheme. This work is strongly based on our previous paper on continuous-time EKF. Here we provide the discrete-time counterpart of these results. The original feature of our approach is that the region-of-convergence question is posed in its complete non-linear framework, that is, considering the dynamics not only of the estimation error, but also of the covariance matrix P. In this way, the approach followed makes the treatment less conservative and improves the convergence analysis. In discrete-time new problems and difficulties appeared for proving convergence: the Jacobians off and h, A(k) and C-k respectively, are evaluated at different trajectories and the exponential weighting factor a has a multiplicative effect on A(k) instead of an additive effect that results in the continuous case. These problems make it more difficult to prove that the Lyapunov function V is decreasing. We solved it by adapting some ideas from Safonov. The proposed ideas were tested successfully on simulation experiments of a mobile platform.