## 3 Data Processing and Kinematic Analysis
### Calculation of the Excavator's Working Area  
Figure 4 illustrates the positions of the boom, arm, and bucket links. Using the measurements with the stroke sensors and rotational encoder, the position of each link and the rotation of the main body were identified. Stroke sensors and the encoder were mounted on the links of the excavator and its bottom as mentioned in Section 5. Figure 5 presents the measurement data from each sensor. Using the information of the measured stroke and Equations (21)-(23), the angle of each link can be computed. Then, the application of  
a kinematic analysis with the angles of links enables the calculation of the predicted and actual working areas (see Figure 6). \[\theta_{b}=\pi-\beta_{b}-\alpha_{b}-\varnothing_{b}. \tag{21}\]  
\[\theta_{a}=2\pi-\beta_{a}-\alpha_{a}-\varnothing_{a}. \tag{22}\]  
\[\theta_{bkt}=3\pi-\mu-\alpha_{k}. \tag{23}\]  
where \(\mu=\alpha+\beta\). Figure 4: Links and corresponding angles: (**a**) boom and arm; (**b**) bucket. In Figure 6, \(P_{1}\) denotes the arm joint, \(P_{2}\) represents the bucket joint, \(P_{3-2}\) is the midpoint of the bucket, and \(P_{3-1}\) shows the bucket tip, \(d\) shows the distance between the body center (origin of the defined coordinate system) and the boom link. The maximum  
Figure 5: Measurement data: (**a**) stroke; (**b**) rotational angle. Figure 6: Working area of the excavator: (**a**) horizontal displacement; (**b**) actual/maximum/predicted working areas. working area can be achieved by the full extension of the links that can be denoted by \(x_{3-1}\) (in this case, bucket tip position from the origin). \[\Delta\theta_{i}=\left(\hat{\theta}_{working\ part}+3\sigma\right)i\Delta t. \tag{24}\]  
\[\left[ \begin{array}{c}\Delta x_{1}\\ \Delta x_{2}\\ \Delta x_{3-1}\\ \Delta x_{3-2}\end{array} \right]=\left[ \begin{array}{ccc}A_{11}&A_{12}&A_{ 13}&A_{14}\\ A_{21}&A_{22}&A_{23}&A_{24}\\ A_{31}&A_{32}&A_{33}&A_{34}\\ A_{41}&A_{42}&A_{43}&A_{44}\end{array} \right]\left[ \begin{array}{c}\Delta \theta boom\\ \Delta\theta m\\ \Delta\thetabucket\end{array} \right]. \tag{25}\]  
where,  
\[A_{11}=-l_{boom}sin(\theta_{boom}).\]  
\[A_{21}=-l_{boom}sin(\theta_{boom})-l_{arm}sin(\theta_{boom}+\theta_{arm}).\]  
\[A_{22}=-l_{arm}sin(\theta_{boom}+\theta_{arm}).\]  
\[A_{21}=-l_{boom}sin(\theta_{boom})-l_{arm}sin(\theta_{boom}+\theta_{arm}).\]  
\[A_{31}=-l_{boom}sin(\theta_{boom})-l_{arm}sin(\theta_{boom}+\theta_{arm})-l_{ bucket}sin(\theta_{boom}+\theta_{arm}+\theta_{bucket}).\]  
\[A_{32}=-l_{arm}sin(\theta_{boom}+\theta_{arm})-l_{bucket}sin(\theta_{boom}+ \theta_{arm}+\theta_{bucket}).\]  
\[A_{33}=-l_{bucket}sin(\theta_{boom}+\theta_{arm}+\theta_{bucket}).\]  
\[A_{41}=-l_{boom}sin(\theta_{boom}-\theta_{arm}sin(\theta_{boom}+\theta_{arm})- l_{bucket,2}sin(\theta_{boom}+\theta_{arm}+\theta_{bucket}+\theta_{bucket,in}).\]  
\[A_{42}=-l_{arm}sin(\theta_{boom}+\theta_{arm})-l_{bucket,2}sin(\theta_{boom}+ \theta_{arm}+\theta_{bucket}+\theta_{bucket,in}).\]  
\[A_{43}=-l_{bucket,2}sin(\theta_{boom}+\theta_{arm}+\theta_{bucket}+\theta_{ bucket,in}).\]  
\[A_{21}=-l_{boom}sin(\theta_{boom})-l_{arm}sin(\theta_{boom}+\theta_{arm}).\]0  
\[\left[ \begin{array}{c}x_{1}\\ x_{2}\\ x_{3-1}\\ x_{3-2}\end{array} \right]=\left[ \begin{array}{c}l_{boom}cos(\theta_{boom}) +d\\ l_{boom}cos(\theta_{boom})+l_{arm}cos(\theta_{boom}+\theta_{arm})+d\\ l_{boom}cos(\theta_{boom})+l_{arm}cos(\theta_{boom}+\theta_{arm})+l_{bucket}cos( \theta_{boom}+\theta_{arm}+\theta_{bucket})+d\\ l_{boom}cos(\theta_{boom})+l_{arm}cos(\theta_{boom}+\theta_{arm})+l_{bucket,2} cos(\theta_{boom}+\theta_{arm}+\theta_{bucket}+\theta_{bucket,in})+d\end{array} \right] \tag{26}\]  
\[x_{pre,p,i}=x_{p}+\sum_{j=1}^{i}\Delta x_{p,j}. \tag{27}\]  
where \(x_{pre,p,i}\) is the maximum value among the predicted longitudinal displacements, \(x_{p}\) is the current longitudinal displacement, \(p\) = 1, 2, 3-1, and 3-2 are the current main points of the excavator, and \(i\) denotes the prediction step (1,2,..., \(N\)); \(\Delta x_{p,j}=\) A\({}_{Jacobian}\Delta\theta_{i}\).