## 4 Influence of Friction Force Analysis on the Gripper's Control Method
Equations (7)-(9) add the new inertial force to the control force condition from Equation (6). The maximum acceleration for the motor placed on the vertical axis, as given by the manufacturer [44] is:  
\[a_{i}=10\text{m}/\text{s}^{2}. \tag{7}\]  
Then, the inertial force added to the weight is:  
\[\overset{\rightarrow}{F}_{i}=m\times a_{i}. \tag{8}\]  
From Relations (1) and (8) we obtain:  
\[2\mu\overset{\rightarrow}{F}_{C}\geq\overset{\rightarrow}{G}+\overset{ \rightarrow}{F}_{i}. \tag{9}\]  
Expanding this relation, we get:  
\[2\mu\overset{\rightarrow}{F}_{C}\geq m(g+a_{i}), \tag{10}\]  
where:  
\(\overset{\rightarrow}{F}_{i}=\) Inertial force due to vertical motion  
\(a_{i}=\) inertial acceleration due to vertical motion  
\(m=\) Objects' mass  
\(g=9.81\text{m}/\text{s}^{2}=\) Gravitational acceleration  
\(\mu=\) Friction coefficient Knowing the maximum acceleration (Equation (11)) and the maximum force [45] (Equation (12)) that the gripper can sustain, we can compute the maximum object mass that our gripper can hold for certain materials with known friction coefficients:  
\[a_{max}=g+a_{i}=19.81\;\text{m}/\text{s}^{2} \tag{11}\]  
\[\overset{\rightarrow}{F}_{C_{max}}=250\;\text{N}. \tag{12}\]  
From Relations (10) and (12) we can compute the required force for grabbing an object and we can compare it with the maximum force the gripper can apply:  
\[\overset{\rightarrow}{F}_{C}\geq\frac{m(g+a_{i})}{2\mu}. \tag{13}\]  
If we consider the recommendations presented by online references [46, 47], the control force should be four times higher than the required force for gripping the object, without considering the vertical acceleration, and is given by the relation:  
\[\overset{\rightarrow}{F}_{C}=\frac{mg}{2\mu}\times 4. \tag{14}\]  
This means that we can add a safety margin and all objects regardless of the systems' accelerations will be firmly held by the gripping system. However, since Relation (13) takes into account the maximum acceleration of the vertical axis, and we do not expect any other forces to influence the system, we will use Relation (13) as the reference for computing the force needed by the system to pick up objects with different friction coefficients and weight. While the maximum acceleration is given by Relation (11), we can relate to Relation (14) and present the new force control as:  
\[\overset{\rightarrow}{F}_{C}=\frac{mg}{2\mu}\times 2, \tag{15}\]  
where the new safety margin is 2. However, in real experiments, we will use Relation (13) as it is more accurate. In our case, the recyclable materials of interest are plastic bottles, glass bottles, and aluminum cans. Knowing this, we will continue the analysis using these materials. Thus, Table 2 presents the known static friction coefficients between the ABS fingers and PET, glass and aluminum objects, where we consider them more dry than wet. Once we have the maximum force relation and the friction coefficients, we can compute the required force for the gripper to pick up a certain object. Figure 5 presents the computed force control value for different object mass and material types. The figure shows the maximum force limit, represented by the top dashed rectangle. Using this diagram (Figure 8), we can visualize the maximum weight of the recyclable objects that the gripper can pick up and transfer to the recycling tray. Since for all three materials the maximum weight is above 6 kg, we can presume that all recyclable objects that fit the recycling criteria can be grabbed by our 5 DOF gripper. \begin{table}
\begin{tabular}{c c} \hline \hline
**Material** & **Friction Coefficient** \\ \hline ABS—PET & 0.33 [48] \\ ABS—Glass & 0.3 [49] \\ ABS—Aluminum & 0.25 [49] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Static friction coefficients. By using Equation (13) and the friction coefficients of the recyclable materials of interest (Table 2) we can now use this information in the hybrid position/force control architecture to compute the gripper reference force.