## 4 Dynamic Error Propagation Model in UWB-Based Positioning
### Dynamic Error Propagation Model  
In UWB positioning, the signal attenuation caused by multipath and NLOS is reflected in the unsteadiness of the time delay of a received signal. Assuming that the channel delay is \(\tau\), the measuring distance is \(d_{mea}\). The range under multipath and NLOS interference are  
\[d_{mea}=c\times(t+\tau) \tag{12}\]  
where \(d_{mea}\) is the measured distance. Then, we define the error caused by \(ME\) and \(NLOSI\) as  
\[E_{ME,NLOSI}=d_{rea}-d_{mea} \tag{13}\]  
where \(d_{rea}\) is the real distance between a UWB tag and the corresponding base station. To obtain the ground truth value of \(d_{run}\), we used a professional laser instrument, Fluke 410, with ranging of 0.2\(\sim\)100.0 m, an accuracy of 2.0 mm, to measure as an absolute reference value of \(d_{rea}\). We use this laser rangefinder to measure static longitude UWB tags to the global coordinate origin. The location of the base station is fixed. When the position of each base station is known (measured with the laser rangefinder Fluke 410 in this study), the real distance information between the tag and the base stations can be obtained by using the distance Equation (9).  
For the ME model, if \(E_{ME,NLOSI}\) is the multipath error, we can introduce \(e_{ME,NLOSI}\) by normalizing \(E_{ME,NLOSI}\) as follows:  
\[e_{ME,NLOSI}=E_{ME,NLOSI}/Log(1+d) \tag{14}\]  
where \(e_{ME,NLOSI}\) is a Gaussian random variable with mean value \(m_{ME,NLOSI}\) and variance \(\sigma_{ME,NLOSI}\) in [45], and the explanation of \(log(1+d)\) can be obtained in reference [46].  
\[e_{ME,NLOSI}=e_{G(m_{ME,NLOSI},\sigma_{ME,NLOSI})} \tag{15}\]  
\[E_{ME,NLOSI}=e_{G(m_{ME,NLOSI},\sigma_{ME,NLOSI})}/Log(1+d) \tag{16}\]  
where \(d\) is the radial distance from the positioning point. \(AI\) is mainly caused by the refractive index of air in the atmosphere, which is affected by atmospheric conditions, such as temperature and humidity. Here, partial derivatives of \(P\), \(E\), and \(T\) can be obtained, respectively, to analyze each influencing factor.  
\[\Delta D=(n_{0}-n)\times D/n \tag{17}\]\[E_{AI}=\frac{\partial\Delta D}{\partial P}+\frac{\partial\Delta D}{\partial E}+ \frac{\partial\Delta D}{\partial T} \tag{18}\]  
By introducing \(\partial n\) in Equation (9), the equations above can be reorganized as  
\[E_{AI}=\frac{\partial\Delta D}{\partial n}\bigg{(}\frac{\partial n}{\partial P }+\frac{\partial n}{\partial E}+\frac{\partial n}{\partial T}\bigg{)} \tag{19}\]  
The atmospheric error can then be modeled as  
\[E_{AI}=\Delta D^{\prime}_{n}\times n^{\prime}_{P,E,T} \tag{20}\]  
In reference [46], the ranging accuracy was compared on a 10 km baseline. The results confirmed that the ranging error caused by the atmospheric medium was 0.25\(\sim\)0.75 m even if a high-accuracy electromagnetic wave rangefinder was adopted. Therefore, we lineally calculated the error caused by the atmosphere within a 10 m indoor space is defined as 0.25\(\sim\)0.75 mm.  
To adapt a specific environment with corresponding ME-NLOSI and AI error factors, we introduce the two weights \(\omega_{1}\) and \(\omega_{2}\) to describe their impact ratios to the total positioning error, respectively. Further, the variable-weighted error propagation model of a UWB indoor positioning system can then be calculated as  
\[e_{DEPM}=\omega_{1}E_{ME,NLOSI}+\omega_{2}E_{AI} \tag{21}\]  
where \(e_{DEPM}\) is the dynamic error propagation model. These weights should be variable to adapt to the different environments including the layout of production lines, obstacles blocking the UWB signals, and atmospheric conditions. Therefore, weights \(\omega_{1}\) and \(\omega_{2}\) should be calibrated in a new environment.