# Single-Sheet Separation from Paper Stack Based on Friction Uncertainty Using High-Speed Robot Hand
## 3 Trajectory Generation of Robot Hand for Paper Separation
### Motion Strategy  
The finger for the feed operation on the front side and the finger for the return motion on the back side are hereafter called the \"feed finger\" and the \"return finger\", respectively. Assuming that the paper bundle is sufficiently thin, the origin is set at the center of the bundle, with the \(x\)-axis in the direction of the paper surface and the \(y\)-axis in the direction perpendicular to the paper surface. The feed finger in the \(+y\) region moves the paper in the \(+x\) direction, and the return finger in the \(-y\) region moves the paper in the \(-x\) direction. To transition to a state of partial separation without completely separating multiple paper bundles with a single stroke, we set the stroke of the feed operation to be smaller than the paper width \(x_{\text{width}}\). We assume that the paper bundles \(\mathcal{P}^{i}_{\text{move}}\) and \(\mathcal{P}^{i}_{\text{fix}}\) are not displaced during one stroke of the feed and return operations. For the \(k_{1}\) sheets \(\mathcal{P}^{2}\equiv\mathcal{P}^{1}_{\text{move}}=\{\text{A}_{1},\text{A}_ {2},\cdots,\text{A}_{k_{1}}\}\) removed in the 1-set feed operation, and return operations are performed from the \(-y\) direction to the direction of the original paper bundle using the return finger. Because the number of papers to be returned is stochastic, \(k_{2}\) sheets \(\mathcal{P}^{2}_{\text{move}}\equiv\{\text{A}_{k_{1}-k_{2}+1},\text{A}_{k_{1 }-k_{2}+2},\cdots,\text{A}_{k_{1}}\}\) can be returned, and the paper bundle \(\mathcal{P}^{2}_{\text{fix}}\equiv\{\text{A}_{1},\text{A}_{2},\cdots,\text{A }_{k_{1}-k_{2}}\}\) remains at the current position when the feed finger is set to stop while holding sheet A\({}_{1}\), as shown in Figure 5a. The normal force acts on the paper bundle \(\mathcal{P}^{2}\) but not on the remaining paper bundle \(\mathcal{P}^{1}_{\text{fix}}=\mathcal{P}^{1}-\mathcal{P}^{2}=\{\text{A}_{k_{1 }+1},\text{A}_{k_{1}+2},\cdots,\text{A}_{n}\}\), and \(\mathcal{P}^{1}_{\text{fix}}\) remains stationary because there is no frictional force between \(\mathcal{P}^{2}_{\text{move}}\) and \(\mathcal{P}^{1}_{\text{fix}}\). If more than two sheets remain in bundle \(\mathcal{P}^{j}_{\text{fix}}\) for \(j\geq 2\), the \(j\)th 1-set return operation is executed in the same manner, and finally, only the topmost sheet \(\mathcal{P}^{j+1}_{\text{fix}}\equiv\{\text{A}_{1}\}\) can be partially separated. Leaving a single sheet can be realized with fewer than \((k_{1}-1)\) set return operations; that is, \(j<k_{1}-1\) is true. When the number of sheets is reduced to one, the feed operation can be restarted and a single sheet can be removed, which can be completely separated into a single sheet of A\({}_{1}\). Figure 4: Paper feed and return operations. The figure is top view. The center position \(x_{i}\) with respect to sheet A\({}_{i}\) satisfies the following relationship, even if multiple feed and return operations are performed:  
\[x_{1}\geq x_{2}\geq\cdots\geq x_{n}\;. \tag{6}\]  
The stroke length does not have to be the same for all the feed operations; it can be set to any value as long as partial separation can occur. The same condition holds even if the return finger contacts bundle \(\mathcal{P}^{i-1}_{\text{move}}\), which had returned 1-set earlier while returning bundle \(\mathcal{P}^{i}_{\text{move}}\) for \(i\geq 3\). This is because the moment the contact point \(x_{c_{i}}\) between the return finger and the paper bundle \(\mathcal{P}^{i}_{\text{move}}\) coincides with the edge position of the bundle \(\mathcal{P}^{i-1}_{\text{move}}\) and the contact point of the return finger is shifted onto the paper bundle \(\mathcal{P}^{i-1}_{\text{move}}\) under the assumption that the paper is sufficiently thin, as shown in Figure 4(b). This is true regardless of the contact point \(x_{c_{i}}\). Based on the above discussion, paper separation is characterized by the fact that paper can be separated without strictly controlling finger movement.