## 4 Robot Action Skills
### Grasping
#### 4.3.1 Grasping the \"Pill Box\" from a Table  
The concept of this strategy is shown in Figure 10. After the end-effector has reached the selected IGS, it approaches the support surface (table) along \(\hat{\mathfrak{s}}_{n}\) under force control in order to land compliantly and ensure the proper relative orientation of the hand with the table, achieving the PGS. Subsequently, the fingers close while maintaining contact with the surface until they establish contact with the object and exert a predefined or learned grasping force, reaching the FGS by lifting the object. By involving direct contact with the table, robustness to estimation errors regarding both the object pose and the support surface normal is achieved.  
Figure 8: The IGS finger configuration for the two grasp strategies employing the RAMCIP hand. (**a**) the finger configuration for grasping the “pill box” from the table. Frames \(\{G\}\), \(\{M\}\) and \(\{N_{i}\}\) are shown; (**b**) the finger configuration for grasping the “pill box” from the high shelf.  
Figure 9: The bounding box of the “pill box”, the normal vector \(\hat{\mathfrak{s}}_{n}\) of the supporting surface, and the \(\hat{\mathfrak{s}}_{l}\) vector perpendicular to the edge of the supporting surface that is closest to the object.  
The set of the produced IGSs place the palm frame \(\{G\}\) w.r.t. \(\{O\}\) with position \(\mathbf{p}_{OG}=\lambda\hat{\mathbf{s}}_{O,n}\) and orientation \(R_{OG}=[-\hat{\mathbf{u}}\times\hat{\mathbf{s}}_{O,n}\quad\hat{\mathbf{u}}\quad- \hat{\mathbf{s}}_{O,n}]\), where \(\lambda\in\mathbb{R}\) is a small constant representing the distance of the palm from the support surface, \(\hat{\mathbf{s}}_{O,n}\) the surface normal w.r.t. \(\{O\}\) and \(\hat{\mathbf{u}}\) is the normal vectors of the bounding box's faces, which are not parallel to the support surface. The configuration \(\mathbf{q}\) of the fingers is the same for all IGSs and is produced using physical human interactive guidance, shown in Figure 8a.  
In order to reach PGS, we command the arm with a wrench \(F_{G}^{cmd}\), which is the sum of two components: one force opposite to the surface normal, which results in an arm motion downward for establishing contact, and one external wrench induced by the contact forces applied by the fingertips to the environment, required for the proper translation and rotation of the hand, so that a full contact with the three fingers is established. The contact forces are measured by using the Optoforce sensors with which the RAMCIP hand is equipped at its fingertips. In the equilibrium, the two wrenches are equal and the motion stops. The PGS is reached when the velocity of the hand is zero, which means that the fingers have successfully landed on the surface. In particular, consider the frame \(\{M\}\) placed on the centroid of the fingers, \(p_{M}=\frac{1}{n}\sum_{j=1,\ldots,n}p_{Nj}\), with the same orientation as \(\{G\}\), and \(\{N_{j}\}\) the frame placed on the \(j\)-th fingertip (Figure 8a). The total commanded wrench, applied and expressed in \(\{G\}\), is the following:  
\[F_{G}^{cmd}= [-a\hat{\mathbf{s}}_{G,n}\quad\mathbf{0}]^{T}+\] \[K\sum_{i=1,\ldots,n}\begin{bmatrix}I_{3}\\ \widehat{\mathbf{p}}_{MNj}\end{bmatrix}(R_{MNj}\mathbf{f}_{Nj}\cdot\hat{ \mathbf{s}}_{M,n})\hat{\mathbf{s}}_{M,n}\]  
where \(a\) the magnitude of \(F_{d}\) of Equation (2), \(\hat{\mathbf{s}}_{G,n}\) and \(\hat{\mathbf{s}}_{M,n}\) the \(\hat{\mathbf{s}}_{n}\) expressed in \(\{G\}\) and \(\{M\}\), respectively, \(K\) a diagonal matrix of gains, \(I_{3}\) the identity matrix, \(\widehat{\mathbf{p}}_{MNj}\) the skew-symmetric matrix of the \(\mathbf{p}_{MNj}\) and \(\mathbf{f}_{Nj}\in\mathbb{R}^{3}\) is the external contact force measured on the fingertip \(j\).  
After landing, the fingers start closing, using the joint position controllers, in order to reach FGS. While the fingers close, they exert forces on the wrist of the arm. Due to its compliance, the arm is moving upwards while the fingers are maintaining contact with the support surface.