A reduced-condition-number algorithm for single-frequency precise point positioning based on regularized Kalman filter

Abstract

Single-frequency receivers are low cost and portable, thus being widely applied in engineering; the extended Kalman filter (EKF) is commonly used to perform single-frequency precise point positioning (SF-PPP). However, the positioning performance of SF-PPP is seriously influenced by various errors. Due to the large process noise and initial variance of the estimated parameters, the weight matrix of state parameters will be ill-conditioned, and since the noise of the pseudo-range is much higher than that of the carrier phase, the weight matrix of observations presents as ill-conditioned. Additionally, the condition number of the normal matrix will jump on the conditions of cycle slip, new emerging satellites, and signal outages. To reduce the condition number of the normal matrix, the regularized Kalman filter (RKF) algorithm is proposed, with additional support for the maximum variance matrix and singular value decomposition, thereby improving the accuracy and stability of SF-PPP. Through static and dynamic experiments, it is found that the proposed method can reduce both the ill-conditioning of the weight matrices of the observations and the state parameters. The condition number of the normal matrix is <500 per epoch, and the convergence time is shortened by >40%. Compared with the SF-PPP using EKF, centimeter-level static positioning accuracies of 1.13, 0.73, and 2.92 cm and decimeter-level kinematic positioning accuracies of 12.5, 10.8, and 27.3 cm in the east, north, and vertical components, respectively, using RKF; this yielded 38.3, 29.8, and 45.2% and 39.6, 41.9, and 21.3% improvement in the static and kinematic scenarios, respectively.