Carrier Phase-Based Sensor Relative Localization

In wireless sensor networks, sensor relative localization is a crucial technique. Most of the existing methods utilize ultra-wide band (UWB) ranging signals to obtain internode measurements. However, when the sensor bandwidth is limited, the ranging measurement is of low-precision and leads to poor localization performance. Carrier phase positioning is critical for precise positioning in narrowband positioning systems such as Global Navigation Satellite Systems (GNSS) and radio interferometric positioning systems (RIPS). However, carrier phase positioning faces the challenge of unknown carrier phase ambiguity resolution, which is more complicated in sensor cooperative relative localization. To address the problem about the utilization of carrier phase in cooperative relative localization, we propose a carrier phase-based positioning method consisting of a four-stage framework. Firstly, we exploit the low-rank property embedded in matrices at multiple epochs and obtain a rough float estimate of ambiguities via a Gauss-Newton low-rank approximation algorithm. Then the proposed method uses multi-dimensional scaling (MDS) to accomplish the rough estimation of the sensor coordinates in the second stage. In the third stage, the float estimates of ambiguities and sensor coordinates are refined by solving the nonlinear least square problem. Finally, partial ambiguity resolution (PAR) is realized by the LAMBDA method and the fixed solution of sensor coordinates can be obtained. We derive the Cramér-Rao lower bound (CRLB) and utilize numerical simulations to verify that the performance of the refined float solution can reach the Cramér-Rao lower bound (CRLB). This method results in the fixed solution of sensor coordinates obtained by PAR being more precise. Furthermore, the proposed method has robustness to the ambiguity initial error and the carrier phase measurement noise, resulting in the localization performance achieving centimeter-level accuracy in general cases.