## 2 Climbing Scaffold Status Monitoring Method
### Climbing Scaffold Status Detection Algorithm  
When monitoring the status of the climbing scaffolding, the Cumulative Sum Control Chart (CUSUM) principle can be utilized. The sliding window-based CUSUM algorithm is an effective event detection method that, with appropriately set parameters, can automatically detect state changes in the input time-series signal. This method is characterized by its simple logic and strong anti-interference capability.  
As illustrated in Figure 1, the U-direction coordinate time series monitored by BeiDou defines a reference window \(w_{\text{m}}\) and a climbing detection window \(w_{\text{n}}\). The climbing detection window follows immediately after the reference window. To ensure a sufficient coupling degree between height variations within the windows, a certain overlapping region exists between the two windows.  
Figure 1: Bilateral sliding window model for detecting climbing events.  
The window lengths are set to \(m\) and \(n\), respectively, with an overlapping length set to \(x\). Considering that the climbing scaffolding moves only vertically, with an approximate climbing height of 4.5 m per ascent and a sampling interval of 1 s, the reference window and climbing detection window lengths are set to 60, while the overlapping length is set to 30. The mean values of the two windows are calculated using the following formula:  
\[M_{\text{m}}=\frac{1}{m}\sum_{j=k-m-n+x+1}^{k-n+x}p(j), \tag{21}\]  
\[M_{\text{n}}=\frac{1}{n}\sum_{j=k-n+1}^{k}p(j), \tag{22}\]  
where \(K\) represents the starting sampling point for calculation, located at the end of the ascent detection window.  
During the early stage of climbing, the scaffold moves strictly in a vertical direction with a relatively consistent ascent speed and distance. In this phase, the threshold-based ascent variation can be directly utilized for detection. The difference is used as the characteristic parameter for identifying ascent events, and its calculation formula is as follows:  
\[D_{\text{k}}=M_{\text{m}}-M_{\text{n}}. \tag{23}\]  
When the characteristic parameter is positive and remains above the preset threshold \(D_{m}\) for most of a given period, it is considered that the ascent behavior begins at the leading edge of the first monitoring window that exceeds the threshold. Conversely, when the characteristic parameter gradually decreases from continuously exceeding the threshold to stabilizing below it, the endpoint of the last window exceeding the threshold is identified as the end of the ascent.  
For detecting the duration of the ascent, since the building currently only undergoes vertical ascent with minimal variation in ascent time, a cumulative summation method is adopted to track the ascent duration. The ascent event cumulative sum \(r_{k}^{+}\) is defined, where the accumulator starts counting upon detecting an ascent event. It continues to accumulate during the ascent phase until the ascent ends, at which point the accumulator resets to zero. This marks a complete ascent event, recording both the total ascent duration and the total change in ascent distance. The specific calculation formula is as follows:  
\[r_{k}^{+}=\begin{cases}1,&D_{k}>D_{\text{th}}\\ r_{k}^{+}+1,&\mid D_{k+1}\mid\geqslant\mid D_{k}\mid.\\ 0,&\mid D_{k+1}\mid<\mid D_{k}\mid\end{cases} \tag{24}\]  
By using this method, the monitoring system can effectively track and record the number of ascent events and the corresponding ascent durations of the climbing frame.