A closed form of an algorithm to determine a Global Positioning System (GPS) satellite's position, velocity and acceleration is proposed, and an Earth Centred Earth Fixed (ECEF) to Earth Centred Inertial (ECI) transformation result using the Civil Navigation (CNAV) message is presented in this paper. To obtain the closed form of the GPS satellite velocity and acceleration determination algorithm using the CNAV, we analytically differentiated the IS-GPS-200F position determination function. The calculated data are transformed from the International Terrestrial Reference Frame (ITRF) to the Geocentric Celestial Reference Frame (GCRF) using an equinox-based transform algorithm that is defined in the IAU-2000 resolution system using the Earth Orientation Parameter (EOP) data. To verify the correctness of the proposed velocity and acceleration determination algorithm, the analytical results are compared to the numerical result. The equinox-based transformation result is compared to simple rotation about the z-axis, which does not use the EOP. The results show that by using the proposed algorithm and the equinox-based transformation together, the user can obtain more accurate navigation data in the ECI frame.

Satellite velocity determination using the broadcast ephemeris is discussed and it is pointed out that the conventional rotation matrix method involves a complicated process of computation. This paper proposes an alternative method using a simple differentiator to derive satellite Earth-Centred-Earth-Fixed (ECEF) velocity from the ECEF satellite positions that are calculated using the standard ICD-GPS-200 algorithm. The proposed algorithm simplifies the velocity transformation procedure, and therefore provides a good alternative. It is demonstrated that ±1 mm/s per axis ECEF satellite velocity is achievable by using the first-order central difference of a Taylor series approximation.

A closed-form formula is also derived for the determination of GPS satellite ECEF acceleration using the broadcast ephemeris. This formula is capable of accuracies better than ±0·1 mm per second squared in each axis. With such a high accuracy of satellite acceleration in real-time, it is possible to detect the line-of-sight range acceleration precisely and as such it is concluded that a GPS receiver can be considered as a precise accelerometer.

The success of the position differential method implies that real-time satellite ECEF velocities can be directly derived through numerical differentiation of the position polynomials. This is desirable for GPS velocity determination applications which require high output rate results in real-time. The derived closed-form formula for GPS satellite ECEF acceleration would benefit those who attempt to use a GPS receiver as an accurate accelerometer in real-time.