Computing GPS satellite velocity and acceleration from the broadcast navigation message

Abstract

We present an extension to the Global Positioning System (GPS) broadcast navigation message user equations for computing GPS space vehicle (SV) velocity and acceleration. Although similar extensions have been published (e.g., Remondi,1 Zhang J.,2 Zhang W.3), the extension presented herein includes a distinct kinematic method for computing SV acceleration which significantly reduces the complexity of the equations and improves the mean magnitude results by approximately one order of magnitude by including oblate Earth perturbation effects. Additionally, detailed analyses and validation results using multiple days of precise ephemeris data and multiple broadcast navigation messages are presented. Improvements in the equations for computing SV position are also included, removing ambiguity and redundancy in the existing user equations. The recommended changes make the user equations more complete and more suitable for implementation in a wide variety of programming languages employed by GPS users. Furthermore, relativistic SV clock error rate computation is enabled by the recommended equations. A complete, stand-alone table of the equations in the format and notation of the GPS interface specification4 is provided, along with benchmark test cases to simplify implementation and verification. 1 | INTRODUCTION Basic positioning of a Global Positioning System (GPS) receiver requires accurate modeling of the location of the antenna phase center of four or more orbiting space vehicles (SV) in view. The SV orbit is nominally determined and predicted in four-hour arcs, with two hours of overlap, by the Master Control Station (MCS) at Schriever Air Force Base, Colorado. The predicted orbit is parameterized and, along with other information, becomes the broadcast navigation message. The message is periodically uploaded to each SV where it is modulated onto a carrier signal, along with a unique pseudorandom noise (PRN) navigation code, and broadcast to GPS user segment receivers. Users can then apply equations like those prescribed in Table 20-IV of the GPS interface specification (IS)4 to accurately 1 DISTRIBUTION A: Approved for public release; distribution unlimited. compute the position of each SV antenna phase center in the WGS-84 earth-centered earth-fixed (ECEF) rotating coordinate system.5 (For brevity, we refer to the position, velocity, and acceleration of the SV antenna phase center as simply the position, velocity, and acceleration of the SV.) Accuracy estimates of SV position computed from the broadcast navigation message are on the order of 1.5 meters rms.6 The published broadcast navigation user equations were formulated for computing SV position in near real time. Users may also require SV velocity and acceleration for more complex, near real-time navigation purposes such as receiver velocity determination, GPS/INS (inertial navigation system) integration, etc. SV velocity and acceleration can be computed by extension of the broadcast navigation user equations. This has been done by several researchers including Remondi,1 Zhang J.,2 Zhang W.3 and others, undoubtedly. We present a comparable extension for SV velocity with thorough validation against multiple days of precise ephemeris data and multiple broadcast navigation messages for vehicle PRN 11, which, at the time of this writing, has the greatest off-nominal eccentricity and inclination of all SV in the GPS constellation. (SV number and PRN number are generally not the same, but we follow the common practice of using the PRN number to uniquely identify a particular SV at a particular time.) For SV acceleration we use a kinematic approach that significantly reduces the complexity of the equations and simplifies inclusion of oblate Earth (J2) gravity perturbations. Including J2 perturbations reduces the mean acceleration magnitude difference by approximately one order of magnitude compared to the more commonly used derivative method. The acceleration equations were also validated against multiple days of precise ephemeris and multiple broadcast navigation messages for PRN 11. The effects of other potential error sources were also analyzed including polar motion, Earth precession, and higher-order gravity. In the sections that follow, we present improvements to the SV position equations and the derivation of the SV velocity and acceleration equations. Detailed validation results are presented along with a complete table of the recommended user equations in the format and notation of the public interface specification. Benchmark test cases are provided to assist with implementation and verification of the entire set of equations. 2 | SV POSITION – ECCENTRIC AND TRUE ANOMALY The broadcast navigation user equation tables correctly state that Kepler’s equation (M = E−e sinE) can be solved for eccentric anomaly (E) by iteration. There are many known methods of various complexity for doing this,7, 8 but no particular method is specified or recommended by the IS.4 Because SV orbits are near-circular (maximum valid eccentricity e = 0.03, according to Table 20-III in the IS), simple methods requiring a limited number of iterations can be used. We evaluated two such methods for use with the broadcast navigation user equations: 1) successive substitutions,9 and 2) Newton iteration.10 For successive substitutions, the initial estimate of eccentric anomaly is set equal to the mean anomaly (M ), and the final value of E is converged upon by iteration of a simple variation of