A High-Precision Joint Smoothing Method for Inter-Spacecraft Radio Frequency Doppler and Distance Measurements using P-Spline Approximation

High-precision inter-spacecraft distance measurements obtained from radio frequency (RF) signals are essential for navigation processing methods (e.g., least squares and Kalman filter) in spacecraft formation and relative navigation applications. The joint smoothing of different measurements is an effective method for improving the precision of distance measurements. However, conventional methods (e.g., the Hatch filter) are primarily designed for global navigation satellite system measurements obtained from terrestrial users. The smoothing models adopted in conventional methods exhibit poor dynamic adaptability, which hinders their ability to provide satisfactory precision. Therefore, this study proposes a joint smoothing method that applies a P-spline approximation to inter-spacecraft RF doppler and distance measurements. To construct the smoothing model, the P-spline function was used because of its strong dynamic adaptability, favorable continuity, differentiability, and digital filter implementation structure. In addition, the formula for the differential property of the P-spline function was analyzed and proved. On this basis, doppler measurements were incorporated into the smoothing model as a derivative of distance measurements, thereby increasing the precision of the distance measurements. Subsequently, the smoothing model was discretized. The core of the proposed method is the design of a P-spline smoother, which consisted of direct and indirect P-spline filters. To increase the anti-noise performance, a 2-norm minimization criterion was introduced into the design of the direct P-spline filter. Furthermore, convolutional property was exploited to design the indirect P-spline filter. The simulation results indicated that the precision of the proposed P-spline smoother was superior to that of conventional methods.