## 3 Literature Classification and Analysis
### Mathematical Models
Based on the aforementioned energy and work, the Lagrangian of the beam element can be written as  
\[L_{e}=T_{e}-V_{e}-V_{\bar{g}}+W_{t}, \tag{13}\]  
Applying the Euler-Lagrange equation shown in (7), the finite element equations of the beam element are expressed as [3]  
\[M_{e}\ddot{\delta_{e}}+2B_{e}\dot{\delta_{e}}+(K_{e}+K_{d})\delta_{e}=P_{e} \tag{14}\]  
where \(\delta_{e}\) is a column vector in terms of the nodal variables of the beam element. Please refer to [3] for more details about \(M_{e}\), \(B_{e}\), \(K_{e}\), \(K_{d}\), and \(P_{e}\) expressions. To assemble all elements, a set of global variables should be defined, which are used to describe the elastic motion of the robot. The relationship between the global variables and the nodal variables can be expressed as  
\[\delta_{e}=S_{i}\delta \tag{15}\]  
where \(\delta\) is a column vector consisting of the global variables, and \(S_{i}\) is the transformation matrix from the global variables to the nodal variables for the \(i\)th beam element. To assemble all the beam elements, applying (15) leads to a set of global equations for the robot in the form of (2). Note that the terms associated with the kinetic energy and the gravitational potential energy of the end-effector should be added into (2) [3]. According to a circular motion simulation, the natural frequencies of the robot oscillate every 120 degrees and change within a given range because of its geometry. Higher-order polynomials accelerate convergence and minimize degrees of freedom, making the computation faster, as per convergence analysis. Even though attitude changes were noted in all directions in the circular motion simulation, position oscillations of the end-effector were negligible in the X-Y plane and about 1.5 mm in the Z-axis direction, reflecting some flexural motion that deteriorated accuracy. Similarly, caused by the flexible connections, the end-effector had large attitude deviations and the largest position oscillations of about 10 mm in the inverted-U path simulation. These results emphasize the need for link flexibility consideration in planning a path for high-precision Delta robot operation, especially when precision matters in high-speed applications. The model highlights the need for adaptive path planning to achieve accurate placement and possible deviations caused by flexible links [3]. The characteristics and behavior of the Delta robot with flexible linkages can be understood by knowing these mathematical models. They provide a framework for studying and modeling the motion and flexibility of the robot, helping to create accurate mathematical representations of robotic systems. Recently proposed a novel rigid-flexible coupling robot for industrial narrow space manipulations that incorporates the strengths of both rigid and flexible arms to enhance stiffness while ensuring reachability. The forward kinematics of the robot is uniformly derived using screw theory. Decoupling the robot's position and orientation results in closed-form inverse kinematics, guaranteeing both accuracy and efficiency. Furthermore, a two-stage search strategy is proposed to obtain the inverse kinematics of the robot that fails to conform to [PERSON]'s solution based on the closed-form solution. A series of experiments demonstrated the feasibility and effectiveness of the inverse kinematics and the proposed rigid-dominat control method within the robot's working space. If [PERSON]'s solution is satisfied, the closed-form inverse kinematics has achieved a 100% success rate. For the case where the [PERSON] solution is not satisfied, the inverse kinematics method demonstrated a higher success rate (99.1%) compared to the Jacobian-based inverse kinematics (98.7%). Furthermore, the inexact solutions converge near the target configuration [32].