This paper concerns the applications of the Kalman filter to navigation and the develment of computer programs of the navigational calculations. Methods to apply the Kalman filter to celestial fix, fix by cross bearing and cocked hat are proposed, and numerical simulations under various noise conditiions are conducted. The accuracy of the optimal positions obtained by the Kalman filter is compared with that of the fixed positiions by radial error method. In the case of celestial fix, an algorithm to estimate the optimal positions by using the linear Kalman filter is presented. The optimal positions by the Kalman filter are compared with the running fixes and with the most probable positions obtained from a single line of position. It is confirmed that the resutls of the proposed method are more accurate than the others. In practical piloting, bearings are generally measured intermittently and the measurement process is nonlinear. It is, therefore, difficult for us to apply the Kalman filter to fix by cross bearing. In order to be used in such an unfavorable case, the extended Kalman filter is revised and the aplicability of the revised extended Kalman filter is checked by numerical simulation under various noise conditions. In a cocked hat, an inside or outside fix is dependent only upon azimuth spread, if the error of each line of position is assumed to be equal both in magnitude and sign. A new technique of selecting a ship's position between an inside fix and an outside fix in a cocked hat by using fix determinant derived from the equation of three lines of position is also presented. The relations among the optimal position by Kalman filter, incentre (or excentre) and random error centtre of the cocked hat are discussed theoretically and the accuracy of the optimal position is compared with that of the others by numerical simulation.

The tedious work, connected to the traditional computation of altitudes and azimuths and the plotting of the position lines, has been a severe objection to celestial fixes. But recently computers have become to be used generally for computations of altitudes and aximuths and the computing objection seems to be practically overruled. Now it seems appropriate to concentrate on other problems which are the procedure of improving accuracy of ship's position and the design of a general computing procedure to determine the coordinates of the optimally estimated ship's position. In this paper, such procedures as an application of Kalman filter and the results of the Digital simulation conducted under various noise conditiions are presented. The positions estimated by Kalman filter are compared with the running fixes and the most probable positions obtained from a single position line, and it is confirmed that the resutls of the proposed method is evidently accurate than others.

As it is well-known, in recent years the Kalman filter has been extensively used in the engineering field. The authors tried to apply the extended Kalman filter for optimal estimation of ship's positiion which is fixed by simultaneous visual (or radio) bearings to two known locations. In practical piloting, bearings are generally measured intermittently, so in this case the original Kalman filter can not be applied because of the long sampling time. In this paper, the extended Kalman filter is revised in order to be used in such an unfavorable case, adn the Digital simulation is conducted by using the revised extended Kalman filter under various noise conditiions. Good results have been obtaiend and effectiveness of the proposed filter has been confirmed.