## 6 Final Stable Grasp Execution Based on Contact Interaction Modeling  
In order to be able to evaluate the equilibrium with all the forces that take place in the real experiment while the fingers of the hand are closing, a model of the contact interactions between the fingertips and the object is required for precisely determining the forces-deformations relationship during grasping. This model will receive as input the initial grasp of the object without any deformation (i.e., the initial contact points of the fingertips over the object's surface or contact polygon determined at Section 3 and the fingers' configuration of the pregrasp strategy are described in Sections 4 and 5). After this initial grasp, the fingers should close iteratively towards the center of the object until the equilibrium of all forces over the object is reached and a stable grasp is performed. With the aim of computing the required fingertip forces for this equilibrium, the simulation of this contact model is achieved in several sequential steps. Firstly, the contact between fingers and the object is detected. Secondly, contact forces are computed (by the evaluation of the relative velocity between the fingers and the object). Finally, the static equilibrium is checked for grasping stability. At each iteration of the simulation, the dynamic model updates the overall shape and contact area deformations due to the applied forces. Contact forces are individually calculated for each contact point, which gives a realistic distribution of the contact pressure in the contact zone. The contact model takes into account the normal forces and the two modes of the tangential forces due to friction: slipping and sticking modes. As the object is modeled by a set of nonlinear spring-damper pairs, the nonlinear normal force is given by  
\[f_{nj}=\left\{ \begin{array}{l}0\\ max(0,(K\delta_{j}^{n}+C\delta_{j}))\end{array} \right.,\delta_{j}\leq 0 \tag{4}\]  
\[\mathbf{f}_{nj}=f_{nj}\mathbf{n}_{fj}, \tag{5}\]  
where \(\delta_{j}\) is the penetration distance measured along the normal direction to the contact surface \(j\); \(K\) and \(C\) are the contact stiffness and damping constants, respectively. Those parameters depend on [PERSON]'s modulus and the Poisson ratios. Based on the work presented in the literature [51; 52], the stiffness constant is calculated by  
\[K=2\Psi\sqrt{R}, \tag{6}\]  
Figure 9: Main execution steps of the pregrasp strategy for the hand + arm system in order to grasp a cube: (**a**) initial robot–object configuration, (**b**) reaching trajectory (intersection of TCP and \(C_{O}\)), (**c**) hand orientation (intersection of TCP and \(C_{T}\)), (**d**) pregrasp configuration. where \(\Psi\) is a constant, calculated according to the mechanical properties of the two objects in contact, and it is given by  
\[\Psi=\frac{1}{\frac{1-v_{1}^{2}}{E_{1}}+\frac{1-v_{2}^{2}}{E_{2}}}, \tag{7}\]  
where \(E_{1}\) and \(E_{2}\) are the Young's modulus; \(v_{1}\) and \(v_{2}\) are the Poisson ratios of the finger and the object, respectively. The damping constant is determined by  
\[C=4\pi R\gamma, \tag{8}\]  
where \(\gamma\) is a constant. To characterize these mechanical parameters, compression tests are done for obtaining curves of stress evolution according to compressive deformation (see [33] for photos and detailed results of this calibration process). Thereby, the Young's modulus (4.928 MPa) and the Poisson's ratio (0.39) of the material of the objects (i.e., foam) to be grasped during the real experiments in Section 7 are identified. Modeling of the tangential force, acting along each contact, is required in order to prevent any slippage and to ensure grasp stability. The contact model takes into account the two modes of the tangential forces due to friction: slipping and sticking modes. In this model, a parallel spring-damper is attached to the ground at one end, via a slider element, and to the fingertip at the other end. Therefore, the contact point location dynamically changes in the slipping condition.