# Single-Sheet Separation from Paper Stack Based on Friction Uncertainty Using High-Speed Robot Hand
## 2 Modeling of Paper Bundle
### Number of Sheets Removed by Paper Feed Operation  
Consider the number of sheets removed from a stack of paper. When the finger moves in the \(+x\) direction while in contact with the paper, the frictional force \(\mu^{\prime}_{\mathrm{H}}F\) acts on the finger in the \(-x\) direction. Figure 1: Conceptual diagram of robot handling a paper stack generated by OpenAI DALL-E 3. The surface of sheet A\({}_{1}\) is subjected to a reaction force with respect to the frictional force \(\mu^{\prime}_{\rm H}F\) in the \(+x\) direction, as shown in Figure 2. This force exceeds the frictional force \(\mu^{\prime}_{1}F\) between sheets A\({}_{1}\) and A\({}_{2}\) (see Equation (2)). Sheet A\({}_{1}\) always moves in conjunction with the finger. The surface of sheet A\({}_{2}\) is subjected to a reaction force of \(\mu^{\prime}_{1}F\) from sheet A\({}_{1}\) in the \(+x\) direction and a frictional force of \(\mu^{\prime}_{2}F\) from sheet A\({}_{3}\) in the \(-x\) direction. If \(\mu^{\prime}_{1}>\mu^{\prime}_{2}\), sheet A\({}_{2}\) overcomes the frictional force and moves with sheet A\({}_{1}\). In other words, sheets A\({}_{1}\) and A\({}_{2}\) move in overlapping states. In this case, \(\mu_{1}\) becomes \(\mu^{\prime}_{1}\) because it is relatively fixed. On the other hand, if \(\mu^{\prime}_{1}\leq\mu^{\prime}_{2}\), sheet A\({}_{2}\) remains in place, because of static friction. In other words, only sheet A\({}_{1}\) can be separated and removed. In this case, \(\mu_{1}\) becomes \(\mu^{\prime\prime}_{1}\) because the sheets are slipping relative to each other. \[\mu_{1}=\left\{ \begin{array}{ll}\mu^{\prime}_{1}&(\mu^{\prime}_{1}>\mu^{ \prime}_{2})\\ \mu^{\prime\prime}_{1}&(\mu^{\prime}_{1}\leq\mu^{\prime}_{2})\end{array} \right. \tag{3}\]  
Similarly, for sheet A\({}_{3}\), if \(\mu^{\prime}_{1}>\mu^{\prime}_{2}\), the same argument holds if we replace the set of parameters (A\({}_{1}\), A\({}_{2}\), A\({}_{3}\), \(\mu^{\prime}_{1}\), \(\mu^{\prime}_{2}\)) in the above discussion for sheet A\({}_{2}\) with the set of parameters (A\({}_{2}\), A\({}_{3}\), A\({}_{4}\), \(\mu^{\prime}_{2}\), \(\mu^{\prime}_{3}\)). In contrast, if \(\mu^{\prime}_{1}\leq\mu^{\prime}_{2}\), sheet A\({}_{2}\) is stationary, whereas the reaction force \(\mu^{\prime\prime}_{1}F\) from sheet A\({}_{2}\) in the \(+x\) direction is always smaller than the maximum static friction force \(\mu^{\prime}_{3}F\) in the \(-x\) direction generated between sheets A\({}_{4}\) and A\({}_{3}\) because \(\mu_{1}=\mu^{\prime\prime}_{1}\) holds. Therefore, the actual frictional force generated is \(\mu^{\prime\prime}_{1}F\), and A\({}_{3}\) is also stationary. \[\mu_{2}=\left\{ \begin{array}{ll}\mu^{\prime}_{2}&(\mu^{\prime}_{1}>\mu^{ \prime}_{2}>\mu^{\prime}_{3})\\ \mu^{\prime\prime}_{2}&(\mu^{\prime}_{1}>\mu^{\prime}_{2},\quad\mu^{\prime}_{ 2}\leq\mu^{\prime}_{3})\\ \mu^{\prime\prime}_{1}&(\mu^{\prime}_{1}\leq\mu^{\prime}_{2})\end{array} \right. \tag{4}\]  
As discussed above, the state of a sheet A\({}_{i}\) is sequentially determined from that of the previous sheet A\({}_{i-1}\). The friction coefficient \(\mu_{i}\) is expressed as follows:  
\[\mu_{i}=\left\{ \begin{array}{ll}\mu^{\prime}_{i}&(\mu^{\prime}_{1}>\mu^{ \prime}_{2}>\cdots>\mu^{\prime}_{i+1})\\ \mu^{\prime\prime}_{i}&(\mu^{\prime}_{1}>\mu^{\prime}_{2}>\cdots>\mu^{\prime} _{i},\quad\mu^{\prime}_{i}\leq\mu^{\prime}_{i+1})\\ \mu^{\prime}_{i-1}&(\mu^{\prime}_{1}>\mu^{\prime}_{2}>\cdots>\mu^{\prime}_{i-1 },\quad\mu^{\prime}_{i-1}\leq\mu^{\prime}_{i})\\ \vdots\\ \mu^{\prime\prime}_{1}&(\mu^{\prime}_{1}\leq\mu^{\prime}_{2})\end{array} \right.. \tag{5}\]  
When \(k\) sheets of paper are removed from overlapping layers, that is, when sheet A\({}_{k+1}\) is stationary, the lower sheet A\({}_{i}\) (\(i>k+1\)) is also stationary, because the dynamic friction forces \(\mu^{\prime\prime}_{k}F\) act on the top and bottom surfaces of each sheet in an opposite manner. Figure 2: Operation of feeding paper from a paper bundle. The figure is top view and the arrows indicate the force direction.