###### Abstract  
This study investigates a data fusion method for underwater multi-inertial navigation based on topological distribution constraints, aimed at improving the positional accuracy of navigation systems on ships, and generating an underwater position reference. First, the state equation of single-axis rotational inertial navigation system (SRINS) is introduced to compensate for the equivalent gyroscope zero bias caused by gravity and magnetic field. Second, a flexible lever error equation based on the influence of flexural deformation angles between SRINSs is proposed. Third, by using the position difference between SRINSs as a measurement, the state and measurement equations of a centralized Kalman filter are analyzed. We conducted two sets of car experiments to verify the proposed data fusion method and a data acquisition system was used to synchronously collect measurement data from three SRINSs. Experimental results show that the proposed method can effectively improve overall positioning accuracy, with the root mean square (RMS) of longitude error reduced by approximately 8.4360%, latitude error RMS reduced by approximately 6.9174%, and overall positioning error RMS reduced by approximately 9.9492%. In certain conditions where other positioning methods are unavailable, such as underwater navigation, the proposed RINSs data fusion method can provide a highly reliable position reference. navigation information fusion; flexible lever arm; position reference; rotating inertial navigation systems (RINSs) +
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horizontal attitude errors, and heading angles as measurements for the Kalman filter to design an online self-calibration scheme, estimating non-repeatable errors at system startup and improving the positional accuracy of SRINSs by about 50% [4]. Therefore, the research on improving the accuracy of the SRINS itself is already quite sufficient. To further improve positioning accuracy, other measurement devices must be introduced to constrain the inertial navigation calculations. To ensure the safety and reliability of navigation, a ship's center is equipped with a high-precision central INS and multiple lower-precision backup INSs [6]. Additionally, weapon systems are equipped with low-precision INSs on their platforms to ensure targeting accuracy. Therefore, multi-INS data fusion technology is commonly used in transfer alignment. To ensure the alignment accuracy, it is necessary to maintain the temporal and spatial consistency between the primary and secondary INSs [7; 8; 9]. [PERSON]. [PERSON] estimated the time synchronization errors during the transfer alignment process, finding that for primary and secondary INSs with time delays of 500 ms and 750 ms, the time estimation errors could reach 20 ms and 60 ms, respectively [7]. [PERSON] modeled the dynamic flexible lever arms during the transfer alignment process, establishing a measurement model that matches relative attitude angles and flexible lever arms, thereby reducing the misalignment angle errors of the secondary INSs [8]. However, transfer alignment technology is only used during the alignment process. After alignment, the secondary INSs navigate independently without information exchange with the primary INS. Additionally, transfer alignment mainly focuses on improving the alignment accuracy of the secondary INSs, with little research on enhancing the accuracy of the primary INS [9; 10]. To fully utilize the measurement information from multiple INSs on ships and improve the overall positioning accuracy of redundant INSs, many scholars have studied data fusion methods [11; 12; 13; 14; 15; 16]. [PERSON] used two dual-axis RINSs with different rotation paths to estimate the gyroscope and accelerometer biases [11; 12; 13]. [PERSON] estimated the installation deviation angles between two INSs, achieving a misalignment estimation error of only 1\({}^{\prime\prime}\)[15]. [PERSON] performed self-calibration of two INSs every four minutes, estimating and compensating for installation errors and misalignment angle errors [16].