# Using A Rotary Spring-Driven Gripper to Manipulate Objects of Diverse Sizes and Shapes
## 2 Materials and Methods
### Force Analysis  
The design is based on computer analysis, and the forces involved are divided into the following two main forces: the force from the actuator and the springs force. The force from the actuator makes the motion of the shutter mechanisms. The applied torque is calculated using the motor torque curve shown in Figure 6.  
Based on the relation between the torque and speed of the motor in Figure 6 and to obtain the maximum torque, this motor is programmed to rotate at a speed of 0-20 rpm. With the maximum deviation angle of \(\theta\) = 38.05\({}^{\circ}\) from the tangential torque at a 9.5 cm radius, the normal force being applied is calculated using the free body diagram in the planar condition, as depicted in Figure 7.  
Figure 5: Kinematic sketch of the mechanism. Figure 6: Speed (rpm) vs. torque (N-mm) chart for the motor. The tangential force (\(F_{t}\)) is calculated using Equation (3), where force is the only unknown term. \[Torque\;(\tau)=F_{t}\cdotradius\;(r), \tag{3}\]  
Using Equation (3) and referring to the free body diagram presented in Figure 7, the tangential forces acting on the slot curve, denoted as (\(F_{St}\)), are computed according to Equation (4). These tangential forces serve as the driving force for the actuation of the mechanism. It is important to emphasize that the centers of rotation and the curve path are different from each other. \[F_{t}=F_{St}\cdot\cos(\theta), \tag{4}\]  
Another significant force in this mechanism is the grasping force, generated through the compression of the springs. From Equation (5), the force exerted by a spring is influenced by the following two factors: displacement (\(x\)) and the spring constant (\(k\)). The spring constant is determined by the spring's dimensions (length, thickness) and its material properties. Moreover, the size of the object being grasped affects the displacement of the spring, with larger-diameter objects experiencing greater forces. \[F_{spring}=-k\cdot x, \tag{5}\]  
This spring-based grasping mechanism proves to be highly effective for grasping objects with varying diameters and shapes, if all objects possess a similar texture (i.e., soft or hard) and if the proper springs are selected. Furthermore, the interaction between the target object and the contact fingers (in this case, the pistons) significantly influences the grasping force. Like other mechanical designs, the components within this mechanism possess a specific lifetime and require replacement after a certain period to maintain a reliable grasping system. The incorporation of springs in this mechanism provides the gripper with  
Figure 7: The tangential angle between the curve path and motor rotation path. the unique capability of grasping diverse objects. However, the properties of the springs gradually diminish over time, necessitating their replacement. Performing a grasping analysis is necessary for any robotic gripper. This study was performed on this mechanism to ensure whether the object is being grasped properly and whether it is stable and safe inside of the gripper when the robotic manipulator has moved around. The mechanism presented in this paper includes force-closure grasping with soft fingers. However, because it uses several fingers all around the object, it can be considered as a form-closure gripper, too. A complete grasping of the object depends upon the following three main factors: numbers and positions of the fingers, types of fingers, and geometry of the object. This can be expressed via Equation (6). \[\left\{\sum_{i=1}^{k}n_{i}N_{i},\ \ \ n_{i}\in\mathbb{R}\right\}=\textit{Force} \tag{6}\]  
where '\(n\)' is the normal force with respect to different axes, '\(N\)' is the wrench space, and '\(i\)' is the number of fingers. The soft fingers provide optimal grasping with a force that is normal to the surface (\(F_{spring}\) in this case), frictional forces in the tangent plane, and the frictional moment in the normal direction, resulting in a 4-dimensional wrench subspace, as presented in Figure 8 and Equation (7).