## 5 Dynamic Error Analysis of the wMPS  
Dynamic measurement error of the wMPS was mainly due to the delay caused by the receiver's motion and observation of the measurement, which produced a position error of spatial intersection. The dynamic measurement error was divided into a single-station set-up phase error and synchronization error between multiple transmitting stations. For a single transmitting station, the movement of the measurement points caused a delay between a group of matching synchronous optical signals and scanning optical signals obtained by the receiver, which caused measurement errors in the scanning angle. This delay caused a phase error whose value was related to the moving speed of the receiver. In Figure 10, the receiver received synchronous optical signals at \(t_{0}\). Owing to the movement of the receiver, the scanning optical signals 1 and 2, received at \(t_{1}\) and \(t_{2}\), respectively, involved a phase error caused by the movement of the receiver. To solve for the location of the receiver at \(t_{0}\), two scanning angles \(\theta_{1t_{0}}\) and \(\theta_{2t_{0}}\) at \(t_{0}\) must be known. In the static measurements, \(\theta_{1t_{0}}\) and \(\theta_{2t_{0}}\) do not change as time passes. However, due to the movement of the receiver, the scanning optical signals \(\theta_{1t_{0}}\) and \(\theta_{2t_{2}}\) received by the rotor in the transmitting station in one rotation cycle deviate from \(\theta_{1t_{0}}\) and \(\theta_{2t_{0}}\). Using a deviated scanning angle to establish the plane-constraint equation will result in errors in the positions of the coordinates [30; 35]. Synchronous error was caused by the delay of synchronous light emissions between two stations. This primarily occurred because the transmitting stations rotated at different speeds, which is unavoidable in wMPS dynamic measurements. Figure 11 shows the errors caused by the out-of-sync rotations of the transmitting stations. Considering a measurement system involving double transmitting stations as an example, a synchronous error brings a different timing base for scanning-angle measurement in the two stations, resulting in a measurement error of the scanning angle. When the receiver is static, although the rotating phases and speeds of the two transmitting stations are not synchronous, the plane of rotating light reversely calculated from the pulse time still strictly meets the surface of the receiver, and accurate spatial coordinates can be obtained when the scanning beam scans over the receiver. Therefore, the information delay caused by different rotation speeds of the transmitting stations will not influence the static measurement of the wMPS. Figure 12 shows that the rotating phases and speeds of the two stations are not synchronous. Therefore, when the scanning beam sweeps at each transmitting station, the receivers scatter around the motion trail. Hence, the angle-intersection conditions are destroyed, and the calculation accuracy of the receiver coordinates decreases. This situation will influence the field application of the wMPS. Figure 10: Phase error of dynamic measurement in a single transmitting station. Figure 13 shows that, as time changes, the sliding block experiences one period of motion when the driving wheel rotates exactly for one cycle. When the connecting point \(O\) is at the highest and lowest positions, the sliding block achieves a peak movement speed \(v_{\max}\) (\(v_{\max}=\omega\cdot R=2\pi f\cdot R\)). \(O\) and the guideway are in one line, and the instantaneous movement speed of the sliding block is zero. When the speed is greater (the receiver moves faster), the intersection conditions are worse, and the error will be larger. Data at discrete points at 0.5, 1, 2, and 4 Hz are shown in Figure 14a,  
Figure 11: Effect of out-of-sync rotation of transmitting stations on the intersection relationship during static state of the receiver.