Four Dimensional Real Time Kinematic State Estimation and Analysis of Relative Clock Solutions

GNSS community has developed a series of GPS-based techniques for clock measurements and comparisons including time dissemination with the standard GPS point positioning method, clock synchronization using the codebased differential GPS positioning technique and using precise point positioning (PPP) for clock comparison. In the mean time, significant advances have been made towards real time kinematic (RTK) positioning which enables three dimensional (3D) positioning at centimeter accuracy, but without clock solutions. This paper presents a four dimensional real time kinematic (4D-RTK) method, which provides RTK timing solutions for applications such as clock synchronization and remote clocks steering in addition to 3D RTK position solutions as usual. The proposed 4D RTK model is based on n single differenced (SD) measurements between receivers for a baseline with (n-1) double differenced (DD) integer constraints. The existing 3D RTK model use (n-1) DD code and phase measurements can only enable position estimation, eliminating the receiver clock errors. In the 4D RTK, the user 3D position states are determined as usual with the DD phase measurements whose integer ambiguities are searched and fixed using the existing integer least squares (ILS) procedure. Estimation of the relative clock bias and SD phase biases is based on the SD code and phase observational equations. The constant nature of the SD phase ambiguity biases is used to improve the SD phase bias from epoch to epoch, and thus the relative clock solutions. The 4D RTK can be simplified to a RTK timing problem if the user baselines are given. Experimental analysis has been performed with a GPS data set collected over a 21 km baseline at 15 second interval for a total of 85 minutes. The results show that within the first few minutes of observations, the SD phase biases fall within the range of 0.3 cycles, thus resulting in the clock uncertainty of 0.1 to 0.2 ns. Beyond 150 epochs (about 40 minutes), the SD phase biases are stabilized within the range of 0.1 cycle, thus the clock biases can be estimated to the precision and accuracy of better than 0.1 ns, including the effect of the errors in the SD phase measurements. In general RTK timing has a significant potential for high precision real time clock synchronizations over very long distances.