# A Novel UWB Positioning Method Based on a Maximum-Correntropy Unscented Kalman Filter
## 2 UWB Positioning Method Based on Unscented Kalman Filter  
### Principle of UWB Positioning  
At present, common range-based wireless positioning methods include the signal-arrival time-based method, the signal-arrival time-difference-based method, the signal-arrival angle-based method and the received-signal strength-based method [14; 15; 16]. UWB technology is carrier-free communication technology. It uses narrow nanosecond nonsinusoidal pulses to transmit data. It can measure precise signal transmission time, which can accurately measure the distance between the base station and the label. Therefore, UWB technology usually adopts a time of arrival (TOA)-based localization algorithm. The TOA positioning method is also called the circle positioning method, which calculates the coordinates of a position label by finding the intersection points of circles [17]. The location of the base station (BS) is used as the center of each circle, and the distance from the BS to the location label is used as the radius to draw each circle. If there is no error in the distances, the final circles will intersect at a point, which is the label position. Based on the TOA model, a schematic diagram of the positioning principles of three base stations and one label is established on a plane. The base-station coordinate is defined as \(\text{BS}(x_{i},y_{i})\), and the coordinates of the positioning label are Label (\(x\), \(y\)). The measured distance from the base station to the label is \(R_{i}\). The distance with the base station as the center, \(R_{i}\) is the radius, which forms the circle-position line. Due to the existence of ranging errors, the positioning model is shown in Figure 1.  
Since the coordinates, \(\text{BS}(x_{i},y_{i})\), of the three base stations and the distance between the three base stations and the label are known, the relationship between the coordinate position and the distance can be established. Through the analysis above, the following equations can be obtained:  
\[ \begin{cases}(x_{1}-x)^{2}+(y_{1}-y)^{2}=R_{1}^{2}\\ (x_{2}-x)^{2}+(y_{2}-y)^{2}=R_{2}^{2}\\ (x_{3}-x)^{2}+(y_{3}-y)^{2}=R_{3}^{2}\end{cases}  \tag{1}\]  
Calculate the difference between two pairs of the equations above to formulate  
\[ \begin{cases}R_{2}^{2}-R_{1}^{2}=\left(x_{2}^{2}+y_{2}^{2}\right)-(x_{1}^{2}+ y_{1}^{2})-2(x_{2}-x_{1})x-2(y_{2}-y_{1})y\\ R_{3}^{2}-R_{1}^{2}=\left(x_{3}^{2}+y_{3}^{2}\right)-(x_{1}^{2}+y_{1}^{2})-2(x_ {3}-x_{1})x-2(y_{3}-y_{1})y\end{cases}  \tag{2}\]  
Figure 1: TOA-based geometric positioning model. The formula above is written in matrix form as follows:  
\[ \begin{bmatrix}x_{2}-x_{1}&y_{2}-y_{1}\\ x_{3}-x_{1}&y_{3}-y_{1}\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix} =\frac{1}{2} \begin{bmatrix}\left(x_{2}^{2}+y_{2}^{2}\right)^{2}- \left(x_{1}^{2}+y_{1}^{2}\right)+R_{1}^{2}-R_{2}^{2}\\ \left(x_{3}^{2}+y_{3}^{2}\right)^{2}-\left(x_{1}^{2}+y_{1}^{2}\right)+R_{1}^{ 2}-R_{3}^{2}\end{bmatrix}  \tag{3}\]  
The coordinates of the label can be obtained from the formula above. \[ \begin{bmatrix}x\\ y\end{bmatrix} =\frac{1}{2} \begin{bmatrix}x_{2}-x_{1}&y_{2}-y_{1}\\ x_{3}-x_{1}&y_{3}-y_{1}\end{bmatrix} ^{-1} \begin{bmatrix}\left(x_{2}^{2}+y_{2} ^{2}\right)^{2}-\left(x_{1}^{2}+y_{1}^{2}\right)+R_{1}^{2}-R_{2}^{2}\\ \left(x_{3}^{2}+y_{3}^{2}\right)^{2}-\left(x_{1}^{2}+y_{1}^{2}\right)+R_{1}^{ 2}-R_{3}^{2}\end{bmatrix}  \tag{4}\]  
The position of the label can be obtained through solving Equation (4). However, in most cases, there are more than three base stations. Therefore, the least squares method (LSM) is usually used to solve Equation (4) so as to obtain a more accurate label position. The TOA method requires high clock synchronization between the labels and the BS. If the clock is not synchronized, the signal time will be incorrect when the label arrives at the BTS.