# The Analysis and Verification of Unbiased Estimator for Multilateral Positioning
## 1 Introduction  
With the progress of society and the development of science and technology, positioning technology is influencing all aspects of people's lives and has become an indispensable application for people [1]. The huge application value of positioning technology has attracted many researchers devoting themselves to it, and many mature technologies have been put into daily use, such as GPS (Global Positioning System) and UWB (Ultra-Wide Band) technology. As an emerging positioning technology, LoRa (Long Range Radio) also shows great research potential [2].  
At the same time, in order to obtain more accurate positioning information, related positioning algorithms have also been proposed one after another. It is undeniable that the advantages and disadvantages of the positioning algorithm affect the positioning accuracy of the entire system, and different positioning algorithms are proposed according to the different measured data and the different coordinate calculation methods used. Depending on whether distances need to be measured, localization algorithms are divided into two categories: range-based and range-free [3].  
For range-based positioning algorithms, a great challenge is to reduce the impact of ranging errors caused by equipment and the environment, so that the positioning result is closer to the target node. For the ranging error model that obeys unbiased estimation, there are many classical algorithms proposed to solve the position of the target node, such as the [PERSON] algorithm [4] and the [PERSON] algorithm [5]. These algorithms establish positioning models with Euclidean distances between anchors whose coordinates are known and the target node whose coordinates are to be estimated, then an estimator of the target node's coordinates can be derived expressed by those distances. Additionally, if the distances between anchors and the target node can be measured, or if travel time can be measured, these measurements could be substituted into the estimator to perform the calculation. However, when error inevitably exists in the measurements, merely substitution of distances with measurements may cause biased results. In particular, most of the existing localizationmethods are based on simple error model assumptions, which ignore the complexity of the actual error model, especially the fact that the square of the ranging value may be a biased estimator. If the squared value of ranging is involved in the solution process, it may lead to biased results.  
In order to illustrate the problem of this biased distance estimator, this paper analyzes and derives a type of ranging error model, whose error variance increases as distances increase, and draws a conclusion: for those biased situations, localization accuracy can be improved if the variance of the ranging error model can be known. Theoretically, the method in this paper is suitable for a positioning algorithm that adopts the square value of the ranging, which can make further improvements of localization accuracy. This paper verifies the improving feasibility of a ranging error model through simulations and practical experiments. More importantly, it shows that the improvement of unbiased estimation of complex ranging error models can improve the positioning accuracy. Similar improvements to unbiased estimators for other ranging error models can also make similar improvements for localization accuracy.