## 2 Climbing Scaffold Status Monitoring Method  
### GNSS/BeiDou Positioning Solution Model with Height Constraint for Climbing Scaffolds  
#### 2.1.1 Data Preprocessing  
(1) Outlier Removal  
GNSS signals used in monitoring external climbing scaffolds for super-high-rise buildings are subject to non-line-of-sight errors and multipath effects, due to obstructions and reflections from buildings in typical urban canyon environments. During real-time processing, occasional issues such as unfixed ambiguities may arise, leading to significant solution drift that deviates greatly from normal observation values. These erroneous solutions cannot be considered valid measurements. Therefore, outlier removal is necessary during the data preprocessing stage. By setting a specific threshold, data outside this threshold are identified as outliers and removed. The current methods for outlier removal in monitoring data include the 3\(\sigma\) criterion, statistical tests, Grubbs' test, and the \(t\)-test [28]. Outlier removal from the monitoring data is performed using the 3\(\sigma\) criterion, which is applicable to sample data that follow a normal or approximately normal distribution. Following this, we address the challenge of data loss in GNSS data processing, which can occur during the acquisition, transmission, and preprocessing stages. Furthermore,we assume that the monitoring data \(X_{1},X_{2},X_{3},\cdots,X_{n}\) are mutually independent and identically distributed, following the distribution \(X\sim(\mu,\sigma^{2})\). The standard deviation of the sample is calculated using the following formula:  
\[\sigma=\left(\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}\right)^{\frac {1}{2}} \tag{1}\]  
where \(\sigma\) represents the standard deviation of the sample, \(n\) is the number of monitoring data points, \(X_{i}\) denotes the \(i\)-th monitoring data point, and \(\overline{X}\) is the mean of the monitoring data. The \(3\sigma\) criterion works well in the post-processing of monitoring data; however, it is ineffective for real-time monitoring. In practical monitoring scenarios, if deformation occurs, the traditional \(3\sigma\) criterion may incorrectly classify the deformation data as outliers, leading to erroneous removal. To address this issue, the paper introduces a delayed processing method based on the \(3\sigma\) criterion. Since the improved \(3\sigma\) criterion processes data over a defined interval, it provides a confidence interval when the data volume is a multiple of a set value. In the absence of deformation, data points outside the confidence interval are directly treated as outliers and removed. When deformation occurs, the confidence interval is adjusted to prevent deformation data from being misclassified as outliers. The occurrence of deformation is determined by whether the amount of data removed falls within the threshold, enabling the effective real-time monitoring of deformation data. (2) Data Interpolation  
To address data loss, data interpolation is required. Common interpolation methods include linear interpolation, Lagrange interpolation, cubic spline interpolation, and piecewise cubic Hermite interpolation. Common interpolation methods include linear interpolation, Lagrange interpolation, cubic spline interpolation, and piecewise cubic Hermite interpolation. This study adopts the piecewise cubic Hermite interpolation method for data interpolation, as it not only avoids the divergence of higher-order interpolation functions but also does not require complex computational processes. The piecewise cubic Hermite interpolation polynomial \(I_{k}(x)\) satisfies \(I_{k}(x)\in C^{\prime}[a,b]\); \(I_{k}(x_{k})=f_{k}\), \(I_{k}^{\prime}(x_{k})=f_{k^{\prime}}^{\prime}k=0,1,\cdots,n\); in the context of piecewise cubic Hermite interpolation, the polynomial \(I_{k}(x)\) defined over each subinterval \(\left[X_{k},X_{k+1}\right]\) is a cubic polynomial. When \(n=1\):  
\[