# Using A Rotary Spring-Driven Gripper to Manipulate Objects of Diverse Sizes and Shapes
## 3 Results
### Robotic Arm Path Planning for Efficient Task Execution  
The initial stage of achieving successful grasping involves ensuring that the gripper possesses the capability to reach the intended target object. Consequently, the use of a manipulation system becomes essential for executing the grasping action and subsequently transporting the object across diverse locations. Additionally, to facilitate the gripper's movement between task points, it becomes imperative to analyze the robot's kinematics and devise an appropriate path plan for the end effector. For this specific project, a small  
Figure 12: Visual representation of the motion study for non-constrained vertical motion of the curved plate. Figure 13: Motion study results for the torque required to actuate the system. scale industrial robot, ABB IRB 120, was selected as the manipulation system based on the project's requirements. In many cases, grasping operations are performed as part of a series of pick-and-place actions. To accomplish this, the robotic arm needs to be programmed to move and reach various target points. However, the robots can have limitations in reaching the target points due to the loss of one or more degrees of freedom, which are referred to as singularities in robotics. Singularity occurs when the robot's tip becomes unable to move or generate velocities in specific directions. This condition is determined by evaluating the Jacobian matrix, represented as \(J_{(\theta)}\). In the case of a serial 6-DOF (degree of freedom) robot, singularity arises when the determinant of the Jacobian matrix becomes zero, as indicated by the condition Equation (8). It is important to know that the expression of the Jacobian matrix can be in either the world frame or the body frame, while the singularity configuration remains independent of the chosen reference frame. \[\det\Big{(}J_{s(\theta)}\Big{)}=0\ \rightarrow\ rank\Big{(}J_{s(\theta)}\Big{)}<6 \tag{8}\]  
For a serial 6-DOF robotic arm, there are several cases or conditions of singularity, and this study focuses on the five most common cases explained below as well as depicted in Figure 14.  
Case I: Two collinear joints. In this case, when there are two joints with their z-axis in the same direction, the robotic end effector is unable to move at least in certain directions. Case II: Three coplanar and parallel revolute joints. Case III: Four revolute joints intersecting at a common point. Case IV: Four coplanar revolute joints. Case V: Six revolute joints intersecting the common line. During the task definition process for testing the gripper with the robotic manipulator, careful consideration is given to ensure that none of the tasks fall into the singularity con  
Figure 14: Five cases of singularity for a serial 6-DOF robotic manipulator. figuration of the robotic arm. Additionally, to avoid encountering any singularity positions between consecutive tasks, an interpolation technique is employed, where 10 points are defined. The singularity of these points is assessed by calculating the determinant of the Jacobian matrix to verify that none of them equal zero. This analysis is conducted to identify any potential singular or near-singular conditions before the final testing phase. The Jacobian matrices for all the joints and transformations of the ABB IRB 120 robot are already defined in [31]. Furthermore, a D-H parameter table is constructed based on the ABB IRB 120, as depicted in Figure 15. This information generates rotational Jacobian matrices using Equation (9). \[J_{R}=\frac{\partial\omega_{n}}{\partial\dot{q}} \tag{9}\]  
where \(J_{R}\) is the rotational Jacobian, \(\omega_{n}\) is the joint angular velocity, and \(q\) is the joint angle. Similarly, the translational Jacobian matrix is created using a similar method as the rotational Jacobian matrices. However, it focuses on the transformation of the joint frames.