# Automated Real-Time Assessment of Stay-Cable Serviceability Using Smart Sensors
## 2 Displacement Estimation Using Acceleration  
For monitoring the serviceability of the stay-cables, the displacement of the cables was employed as a determinant of their serviceability level based on the criterion from the United States (1986), i.e., the displacement of the cable should be less than twice the diameter of the target cable. Various displacement measurement methods have been introduced in the past, including those using contract-type and non-contact-type sensors ([PERSON], 2017). Among these, the indirect estimation approach, particularly using an accelerometer, has often been employed to estimate the displacement owing to its convenient installation that is reference-free and relatively low cost ([PERSON], 2017). Theoretically, displacement can be obtained by the double integration of acceleration. However, double integration of the measured acceleration results in the low-frequency drift in the estimated displacement caused by unknown initial conditions, sensor noise, and signal discretization ([PERSON], 2017). To avoid this problem, [PERSON] et al. (2019) proposed a dynamic displacement estimation method, which removes the low-frequency drift error by applying a high-pass filter. The only prior knowledge required is information about the lowest frequency of the stay-cable. This method is appropriate for use with smart sensors, as it avoids low-frequency drift, does not require the initial displacement, provides efficient computation, and has been implemented in a wireless displacement estimation system using smart sensors ([PERSON], 2017; [PERSON] et al., 2019). The estimated displacement satisfied the performance in terms of accuracy by comparison with a direct displacement estimation method, such as a laser displacement sensor ([PERSON], 2017; [PERSON] et al., 2019; [PERSON] et al., 2019). While this approach cannot estimate the static part of the displacement, dynamic displacement is sufficient to assess the serviceability of a stay-cable using the US criterion employed in this study. Therefore, in this study, the displacement estimation method was applied to develop an automated real-time serviceability assessment system for the stay-cable.  
This section briefly describes the displacement estimation method using the measured acceleration proposed by [PERSON] et al. (2019). Let \(u\) be the estimated displacement, \(\Delta t\) the sampling time of the measurement, and \(\overline{a}\) the acceleration measured discretely with uniform sampling time \(\Delta t\). The displacement can be estimated by minimizing the following equation which is the discretized form by the trapezoidal rule:  
\[\underset{u}{\text{Min}}\Pi(u)=\frac{1}{2}\|L_{a}(L_{c}u-(\Delta t)^{2} \overline{a})\|_{2}^{2}+\frac{\lambda^{2}}{2}\|u\|_{2}^{2} \tag{1}\]  
where \(L_{a}\) is the integration operator based on the discretized trapezoidal rule, \(L_{c}\) is the second-order differential operator, \(\|\cdot\|\) is the 2-norm of a vector, and \(\lambda\) is the regularization factor that adjusts the degree of regularization as indicated by the second term in the minimization problem. If the regularization factor becomes larger, the zero-displacements are estimated from the measured acceleration. With a low regularization factor, the displacement information becomes meaningless and unstable. Based on the optimal regularization term, the displacement can be derived as Equation (2), which minimizes the difference by the numerical integration of the measured acceleration and considers the regularization term  
\[u=(L^{T}L+\lambda^{2}I)^{-1}L^{T}L_{a  
the displacements. Each moving window was used to estimate the displacement based on Equation (2), and the full history of displacements is identical to a collection of estimated displacements at the center of each window. The detailed derivation of the estimation of displacement from the measured acceleration can be found in the previous work by [PERSON] et al. [26].