For more than 50 years, the continuous photographic observations of asteroids have been carried outwith telescopes of the Astronomical Observatory of the Taras Shevchenko National University of Kyiv and the Main Astronomical Observatory of the National Academy of Sciences of Ukraine. About 3,000 photographic plates were obtained, some of which were exposed in 1908. We collected the data on more than 5,500 positions and magnitudes of asteroids on these astroplates taking into account all results of the processing of observations available in various publications and in the Minor Planet Center database. All positional data were compared with JPL ephemeris and analyzed. From different series of asteroid observations the values of positional accuracy were obtained, depending on the methods of measuring and processing the plates and reference catalogs of stars. In order to systematically improve the obtained asteroid positions, we evaluated the possibilities of reprocessing some of the earliest asteroid observations applying the modern star high accuracy catalogs. Using the Tycho-2, Gaia DR2, and Gaia EDR3 as the reference catalogs, the 590 astroplates exposed on the MAO NASU Double Long-Focus Astrograph (DLA) in 1952-1986 were reprocessed based on old plate measurements. All newly determined and previous original asteroid positions were compared with the JPL ephemeris. The comparison results show an improvement in the systematic and random components of the accuracy of coordinates for new positions of asteroids. When comparing the new positions of the asteroids determined in the Tycho-2 and Gaia catalog systems, no significant changes in accuracy were found.