## 2 Measurement Principle and Causes of Dynamic Error
### Causes of Dynamic Error  
As stated in Section 2.1, laser signals from at least two transmitters with different rotation speeds are necessary for coordinate calculation. For each transmitter, a receiver will obtain a synchronization pulse and two scanning pulses in sequence as shown in Figure 2. In the measuring network, the synchronization pulses from different transmitters are irrelevant in the time domain. If the receiver is static, laser planes have a unique intersection point and the time difference between signals has no influence on measurement accuracy. However, for a moving receiver, in the period of obtaining signals from the two transmitters, i.e., \(t_{1}\) to \(t_{22}\), it has been at different positions and the trajectory during that is unknown. Calculating the coordinate of the receiver directly will lead to certain dynamic error. We define single transmitter dynamic time error \(\delta t_{s\_k}\) as the time difference between synchronization pulse and scanning pulse 1, pulse 2 (\(t_{1}\) and \(t_{11}\), \(t_{1}\) and \(t_{12}\)), and the multi-transmitter dynamic error \(\delta t_{m\_k}\) is defined as the time difference between synchronization pulses from different transmitters (\(t_{1}\) and \(t_{2}\)). Note that multi-transmitter dynamic error \(\delta t_{m\_k}\) is strictly related with rotation periods and it will accumulate over rotation cycles. \[\delta t_{m\_k}=\text{mod}(N(T_{k}-T_{1}),T_{1}) \tag{4}\]  
\(N\) represents the rotation cycles and \(T_{k}\) represents the rotation period of the transmitter \(k\). \(T_{1}\) is the rotation period of the primary transmitter in the network. \(\delta t_{s\_k}\) is determined by the distribution of two laser modules. They have an intersection angle of \(90^{\circ}\). \(\delta t_{m\_k}\) is determined by the rotation cycles of all transmitters in the network. When the measurement network is built, it remains unchanged. \(\delta t_{s\_k}\) and \(\delta t_{m\_k}\) are uncorrelated and lead to dynamic error \(\delta\theta_{s\_k}\) and \(\delta\theta_{m\_k}\) in scanning angle respectively. \(\delta\theta_{e\_k}\) is the static angle error and it is related with the straightness error of the laser module and centering error of the receiver. It is uncorrelated with the two error resources above. Therefore, the total dynamic error of wMPS can be expressed in Equation (5):  
\[\delta\theta_{k}=\delta\theta_{e\_k}+\delta\theta_{s\_k}+\delta\theta_{m\_k} \tag{5}\]  
The static measurement error of a single transmitter has been evaluated after it is assembled. It obeys the normal distribution and the standard deviation is about \(2^{\prime\prime}\). Besides, the static error at each  
Figure 2: Time sequence of signals from two transmitters. measurement is independent and the covariance is zero. For the other two error resources, simulation experiments based on MATLAB are conducted for time error quantization. Basing on the principle of intersection measurement, the coordinates of the receiver can be calculable if it obtains signals from two transmitters. However, there is no constraint redundancy and the accuracy and stability are relatively lower. With the increase of the transmitter amount, the accuracy will also be improved. In application, we generally construct a four-transmitter network to gain better performance. Taking it into account, a four transmitters network was built and they were in a \"C\" type layout, which had been tested to gain a higher accuracy than others [20]. The rotation speeds were 1600, 1800, 2000 and 2200 rev/min, respectively. A trajectory generator was applied to produce a spatial trajectory and the receiver moved along it at different speeds, i.e., 30 mm/s, 60 mm/s, and 120 mm/s. The dynamic time error \(\delta t_{s,k}+\delta t_{m,k}\), i.e. the time difference between the first synchronization signal and the last scanning signal, was captured at each measurement.