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HR\,8799 is a young planetary system composed of 4 planets and a double debris belt. Being the first multi-planetary system discovered with the direct imaging technique, it has been observed extensively since 1998. This wide baseline of astrometric measurements, counting over 50 observations in 20 years, permits a detailed orbital and dynamical analysis of the system. To explore the orbital parameters of the planets, their dynamical history, and the planet-to-disk interaction, we made follow-up observations of the system during the VLT/SPHERE GTO program. We obtained 21 observations, most of them in favorable conditions. In addition, we observed HR\,8799 with the instrument LBT/LUCI. All the observations were reduced with state-of-the-art algorithms implemented to apply the spectral and angular differential imaging method. We re-reduced the SPHERE data obtained during the commissioning of the instrument and in 3 open-time programs to have homogeneous astrometry. The precise position of the 4 planets with respect to the host star was calculated by exploiting the fake negative companions method. To improve the orbital fitting, we also took into account all of the astrometric data available in the literature. From the photometric measurements obtained in different wavelengths, we estimated the planets' masses following the evolutionary models. We obtained updated parameters for the orbits with the assumption of coplanarity, relatively small eccentricities, and periods very close to the 2:1 resonance. We also refined the dynamical mass of each planet and the parallax of the system (24.49 $\pm$ 0.07 mas). We also conducted detailed $N$-body simulations indicating possible positions of a~putative fifth innermost planet with a mass below the present detection limits of $\simeq 3$~\MJup.