# The Analysis and Verification of Unbiased Estimator for Multilateral Positioning
## 2 Related Work  
The trilateral localization algorithm is a basic algorithm [6] that has been studied and improved by many researchers. In the ideal trilateral positioning algorithm where there is no ranging error, there is a unique solution to the position of the target node.  
Regarding the ranging error, an improved trilateral centroid localization algorithm was proposed in [7, 8]. The fuzzy C-means clustering method was used to cluster the RSSI (Received Signal Strength Indicator) signals transmitted by anchors to target nodes to eliminate the noise of small probability and large interference. Aiming at the influence of distance deviation on the multilateral positioning algorithm, [PERSON]. [9] proposed an improved multilateral positioning algorithm KC-Multiliterate, which introduced the K-means clustering method into the positioning problem of wireless sensor networks to make the positioning error smaller. On this basis, [PERSON]. [10] proposed a weighted centroid localization algorithm based on K-means clustering point density (KCDD-WCLA). As a result, its computational complexity is significantly reduced and the positioning accuracy is significantly improved. [PERSON] [11] uses the Fang algorithm to calculate the coordinates of target node, but the Fang algorithm can easily generate double solutions, which can result in the positioning accuracy being greatly reduced when there is a large measurement value error. [PERSON] [12] uses the nonlinear least squares algorithm to directly solve the nonlinear equation system, and uses a Gauss-Newton-like method to find the minimum value of the cost function. This positioning method is accurate but not stable enough. When the initial value is not suitable, the formula is prone to local convergence. Other studies [13, 14, 15, 16, 17] linearized the nonlinear equation system through using the Taylor series expansion, which reduced the computational complexity. However, the optimal solution could not be guaranteed, because the high-order terms were ignored in the linearization process and a suitable initial value is still needed to ensure convergence. Further studies [18, 19, 20, 21, 22] use the linear least squares algorithm to convert the nonlinear equation system into a linear equation system, and then solve the linear equation system through least squares algorithm, which can successfully solve the initial value problem. However, the positioning results will be greatly affected by the measurement error. Furthermore, Refs. [23, 24] use the Chan algorithm to calculate the approximate coordinates of the target node. The algorithm adopts the weighted least squares algorithm, and assigns different weights to each equation in the equation system according to the ranging error, which improves the positioning accuracy. In the calculation process of all the above algorithms, the squares of the distances are directly substituted into the model for calculation. However, from a statistical point of view, it is not an unbiased estimator. In this context, we take multilateral positioning as an example and explore the impact of unbiased estimation for the positioning model and verify it using SX1280, showing that using unbiased estimator can improve the positioning accuracy, and when the error variance is larger, the improvement of the algorithm is more obvious.