## 4 Influence of Friction Force Analysis on the Gripper's Control Method  
To pick up recyclable objects from the municipal waste collection, the gripper needs at least two fingers. For this, we created a 3D finger design which is presented in Figure 5a-c. These were designed as simple fingers that can grab different object types and shapes. From the 3D design, the fingers were 3D printed using ABS material and a Zortrax M200 3D printer. When complete, the fingers were attached to the linear motor, resulting in a two-finger gripper mechanism presented in Figure 6. One can observe in Figure 6 a small deformation of the ABS fingers during the gripping action. This ABS finger's flexibility and deformability ensure a slower transmission of the force towards the actuation of the gripper. Certain protection of the electric actuation of the gripper is thus ensured. Once the gripper is attached, we can calculate the required force needed to grab certain recyclable materials/objects. To achieve this, we can use the stick-slip conditions [42] derived from the Coulomb friction law [43]. Figure 7 presents the forces acting between the gripper's fingers and the object being manipulated, where:  
\(\overset{\rightarrow}{F}_{f_{L}}\) = Friction force between left finger and the object  
\(\overset{\rightarrow}{F}_{f_{R}}\) = Friction force between right finger and the object  
\(\overset{\rightarrow}{F}_{C_{L}}\) = Control force on the left finger  
\(\overset{\rightarrow}{F}_{C_{R}}\) = Control force on the right finger  
\(\overset{\rightarrow}{G}\) = Object's weight  
\(\overset{\rightarrow}{N}_{L}\) = Normal force on left fingers' contact surface  
\(\overset{\rightarrow}{N}_{R}\) = Normal force on right fingers' contact surface  
Figure 5: Gripper 3D fingers: (**a**) front view; (**b**) bottom side view; (**c**) top side view. Figure 6: The gripper mechanism in action. Figure 7: Forces acting on gripper fingers and the grabbed object. Coulomb friction law states that for the object to not slide from the gripper fingers, the forces must follow Inequality (1), where the friction forces are defined in Equations (2) and (3):  
\[\overset{\rightarrow}{F}_{f_{L}}+\overset{\rightarrow}{F}_{f_{R}}\geq\overset{ \rightarrow}{G} \tag{1}\]  
\[\overset{\rightarrow}{F}_{f_{L}}=\mu\overset{\rightarrow}{N}_{L} \tag{2}\]  
\[\overset{\rightarrow}{F}_{f_{R}}=\mu\overset{\rightarrow}{N}_{R} \tag{3}\]  
In our case, the control forces \(\overset{\rightarrow}{F}_{C_{L}}\) and \(\overset{\rightarrow}{F}_{C_{R}}\) are equal with the gripper's control force \(\overset{\rightarrow}{F}_{C}\), resulting Equations (4) and (5):  
\[\overset{\rightarrow}{F}_{C_{L}}=\overset{\rightarrow}{F}_{C_{R}}=\overset{ \rightarrow}{F}_{C}, \tag{4}\]  
\[\overset{\rightarrow}{N}_{L}=\overset{\rightarrow}{F}_{C_{L}}\]  
\[\overset{\rightarrow}{N}_{R}=\overset{\rightarrow}{F}_{C_{R}}\]  
\[=>\left\{ \begin{array}{c}\overset{\rightarrow}{F}_{f_{L}}=\mu\overset{ \rightarrow}{F}_{C_{L}}=\mu\overset{\rightarrow}{F}_{C}\\ \overset{\rightarrow}{F}_{f_{R}}=\mu\overset{\rightarrow}{F}_{C_{R}}=\mu \overset{\rightarrow}{F}_{C}\end{array} \right\}. \tag{5}\]  
From Equations (1)-(5) we calculate that the required condition for the gripper to hold the recyclable object is the one from the relation:  
\[2\mu\overset{\rightarrow}{F}_{C}\geq\overset{\rightarrow}{G} \tag{6}\]  
Using this inequality, we can further analyze the required force depending on the estimated weight of the object and the friction coefficient for certain types of recyclable materials. However, before analyzing the minimum required control force, we have to take into consideration the gripper motion on the same axis as the friction forces. Since the gripper will move up after the object is grabbed, the dynamic motion will add a new acceleration to the object, which will, in turn, translate into an inertial force that has the same direction as the weight of the object. This means that we need to increase the force acting on the object.