## 2 System Description
### Two-Stage System  
The variation stream technique is used to identify the source of variation of serial production processes [21; 22; 23; 24; 25; 26]. However, the conventional model was not designed for a continuous-time process, e.g., a tele-abrasive system. To analyze our tele-abrasive system, we considered it as a two-stage system consisting of the human operator stage and the tele-abrasive machine stage, as shown in Figure 1.  
To describe the variation in the stages of our system, a well-known state space model for variation propagation in multi-stage systems [27] was modified to extract the total variation and consider it according to stage variation in a continuous-time system manner, as given in Equations (1) and (2):  
\[\mathbf{y}_{k}^{t}=\mathbf{A}_{k-1}^{t}\mathbf{y}_{k-1}^{t}+\mathbf{B}_{k}^{t}\mathbf{u}_{k}^{t}+ \mathbf{w}_{k}^{t} \tag{1}\]  
\[\mathbf{z}_{k}^{t}=\mathbf{C}_{k}^{t}\mathbf{y}_{k}^{t}+\mathbf{m}_{k}^{t}\quad,\;k=1,2 \tag{2}\]  
where state vector \(\mathbf{y}_{k}^{t}\) represents the key quality characteristic of the product at time \(t\) and stage \(k\); \(\mathbf{A}_{k-1}^{t}\mathbf{y}_{k-1}^{t}\) is the transformation of product quality deviations from stage \(k-1\) to stage \(k\) at time \(t\); \(\mathbf{u}_{k}^{t}\) represents the accuracy at time \(t\) and stage \(k\); \(\mathbf{B}_{k}^{t}\) is a transformation matrix; \(\mathbf{w}_{k}^{t}\) is the unmodeled process error (precision); \(\mathbf{C}_{k}^{t}\) is a matrix mapping product  
Figure 2: A surface subjected to a tele-abrasive operation where the blasted spot is controlled along the target line to blast the path. quality state to quality measurement \(z_{k}^{t}\); and \(m_{k}^{t}\) is the measurement noise. Note that the superscripts denote stages, and the subscripts denote time intervals. The expected value of \(\mathbf{y}_{k}^{t}\) is expressed in Equation (3):  
\[E[\mathbf{y}_{k}^{t}] = A_{k-1}^{t}E[\mathbf{y}_{k-1}^{t}]+B_{k}^{t}E[\mathbf{u}_{k}^{t}] \tag{3}\]  
Additionally, the expected value of \(z_{k}^{t}\) is expressed in Equation (4):  
\[E[\mathbf{z}_{k}^{t}] = C_{k}^{t}E[\mathbf{y}_{k}^{t}] \tag{4}\]  
Note that \(\mathbf{w}_{k}^{t}\) and \(\mathbf{m}_{k}^{t}\) are normally distributed with zero mean \(N\big{[}0,\sigma^{2}\big{]}\). The variation in a stage can be expressed by the covariance of \(\mathbf{y}_{k}^{t}\), shown in Equation (5):  
\[Cov[\mathbf{y}_{k}^{t}] = A_{k-1}^{t}\mathbf{\Sigma}_{\mathbf{y}_{k}^{t}}A_{k-1}^{t}{}^{t}+B_{k}^{t }\mathbf{\Sigma}_{\mathbf{u}_{k}^{t}}B_{k}^{t}+2A_{k-1}^{t}Cov[\mathbf{y}_{k-1}^{t},\mathbf{u}_ {k}^{t}]B_{k}^{t}+\mathbf{\Sigma}_{\mathbf{w}_{k}^{t}} \tag{5}\]  
The covariance of \(\mathbf{z}_{k}^{t}\) is:  
\[Cov[\mathbf{z}_{k}^{t}] = C_{k}^{t}\mathbf{\Sigma}_{\mathbf{y}_{k}^{t}}C_{k}^{t}+\mathbf{\Sigma}_\mathbf{{m}_{k}^{t}} \tag{6}\]  
The two-stage models presented earlier describe the variation stream for any time \(t\). The average of the overall variation can be computed using Equations (7) and (8):  
\[Cov[\mathbf{y}_{k}] = \frac{\sum_{t=1}^{N}Cov[\mathbf{y}_{k}^{t}]}{N} \tag{7}\]  
Similarly, the covariance of \(\mathbf{z}_{k}\) is:  
\[Cov[\mathbf{z}_{k}] = \frac{\sum_{t=1}^{N}Cov[\mathbf{z}_{k}^{t}]}{N} \tag{8}\]  
Practically, \(Cov[\mathbf{z}_{k}^{t}]\) can be calculated using Taguchi's loss function [28]. At \(k\) =1, \(\mathbf{Loss}^{t}\) is equivalent to the square error as expressed in Equation (9):  
\[\mathbf{y}_{k}^{t}=\mathbf{A}_{k-1}^{t}\mathbf{y}_{k-1}^{t}+\mathbf{B}_{k}^{t}\mathbf{u}_{k}^{t}+ \mathbf{w}_{k}^{t} \tag{1}\]0  
where \(k\) is the loss coefficient, \(\mathbf{z}_{k}^{t}\) is the measured position at time \(t\), and \(\mathbf{T}^{t}\) is the target value at time \(t\). Furthermore, \(Cov[\mathbf{z}_{k}]\) can be obtained using Equation (10):  
\[\mathbf{y}_{k}^{t}=\mathbf{A}_{k-1}^{t}\mathbf{y}_{k-1}^{t}+\mathbf{B}_{k}^{t}\mathbf{u}_{k}^{t}+ \mathbf{w}_{k}^{t} \tag{1}\]1  
To employ Equations (1) and (2), we assume that the cleaning surface is rectangular.