## 5 Discussion  
Despite the fact that research in the past few years has introduced many new estimation algorithms, the Kalman filter remains one of the most widely used algorithms today, being efficient, simple and robust. However, due to different reasons, for example, system and/or noise modeling imperfections (see Equations (2)-(4), (7) and (8)), missing data, measurement errors, discretization in time, insufficient convergence speed, a certain incertitude appears at the output of the Kalman filter. The goal of this paper was to reduce and even to eliminate this incertitude. There are two strategies to accomplish this goal: a solution proposed in literature based on the adaption of the Kalman filter by continuously updating its parameters and the solution proposed in this paper based on post-processing the results of the Kalman filter by data fitting algorithms. Let us consider first the adaptive Kalman filter solution. The adaptive approach in [2] is a form of [PERSON]'s filter quadratic optimization problem and realizes the automatic tuning of the filter's hyper-parameters in least squares. The method in [2] makes explicit use of the gradient of the loss with respect to the Kalman filter's parameters, leading to an  
Figure 8: The results of the last experiment [16]. efficient optimization algorithm. In [10], the authors propose a method for automatically learning the Kalman filter's noise parameters. The goal of [11] is an automated procedure for tuning Kalman filters based on a genetic algorithm. The authors of [12] regard the tuning of the Kalman filter as a convex optimization problem and solve the problem by applying gradient-based methods. The Bayesian non-convex optimization strategy for automatically tuning Kalman filters proposed in [12] identifies multiple local minima. The authors of [14] present a method to reduce the uncertainty at the output of the Kalman filter. They use this method for global positioning system (GPS) localization by cooperation with micro-electro-mechanical sensors (MEMS) in the framework of an integrated navigation system (INS). When the GPS signals are blocked or lost due to multipaths, the target's tracking can be continued using the signals delivered by the MEMS, since these sensors are fully self-contained and do not rely upon external signals. Furthermore, the absolute positions from GPS can filter and minimize the cumulative drift of the dead reckoning inertial sensors. This creates a continuous navigation solution with the accuracy determined by the quality of both the GPS signals and inertial sensors used. In this case, the Kalman filter combines measurements from GPS with those of MEMS using certain a priori information. The filter requires knowledge of the system and measurement dynamics as well as a statistical description of the system noises, measurement errors and uncertainty in the dynamic model. This includes the noise characteristics of both the MEMS and GPS updates. The filter then takes several assumptions, such as white noise behavior and Gauss-Markov properties, to weight the measurements in an optimal manner. Reinforcement learning is an intelligent solution to this problem, which uses a combination of dynamic programming and trial and error exploration to develop a set of optimal parameters for the Kalman filter [14]. In comparison to a typical iterative approach, it was found that using reinforcement learning led to slightly better estimates of the tuning parameter values; furthermore, the tuning process was performed in [14] with significantly less iteration, in comparison to an exhaustive search, due to the learning capability of the method. This benefits both the static parameters as well as the time varying parameters since the method is capable of constantly adapting the tuning based on collected navigation data. All these solutions, belonging to the adaptive Kalman filter approach, can reduce the uncertainty at the output of the Kalman filter but not completely.