This paper is to develop the position error equations including the attitude errors, the errors of nadir and ship's heading, and the errors of ship's position in the free-gyro positioning and directional system. In doing so, the determination of ship's position by two free gyro vectors was discussed and the algorithmic design of the free-gyro positioning and directional system was introduced briefly. Next, the errors of transformation matrices of the gyro and body frames, i.e. attitude errors, were examined and the attitude equations were also derived. The perturbations of the errors of the nadir angle including ship's heading were investigated in each stage from the sensor of rate of motion of the spin axis to the nadir angle obtained. Finally, the perturbation error equations of ship's position used the nadir angles were derived in the form of a linear error model and the concept of FDOP was also suggested by using covariance of position error.

The authors aim to establish the theory necessary for developing free gyro positioning system and focus on measuring the nadir angle by using the motion rate of a free gyro. The azimuth of a gyro vector from the North can be given by using the property of the free gyro. The motion rate of the spin axis in the gyro frame is transformed into the platform frame and again into the NED (north-east-down) navigation frame. The nadir angle of a gyro vector is obtained by using the North components of the motion rate of the spin axis in the NED frame. The component has to be transformed into the horizontal component of the gyro by using the azimuth of the gyro vector and then has to be integrated over the sampling interval.

The authors aim to establish the theory necessary for developing the free gyroscopic compass and focus on mainly two points. One is to suggest north-finding principle by the angular velocity of the earth's rotation, and the other is to suggest orthogonal coordinate transformations of the motion rate of the spin axis, which transforms the components of motion rate in the free gyro frame into those in the platform frame and that this transformed rate is, in turn, transformed into the NED(north-east-down) navigation frame. Subsequently, ship's heading is obtained by using the fore-aft and athwartship components of the motion rate of the spin axis in the NED frame. In addition it was found how to solve the transformation matrix necessary for transforming each frame.

This paper is to develop the position error equation of in the free-gyro positioning system by using two free gyros. First, the determination of a position is analyzed on the ellipsoid of the Earth and the type of the errors is defined Finally the position error equation is introduced and developed, based on the definition of the type of errors which may be involved in the FPS.