## 3 Literature Classification and Analysis
### Mathematical Models  
The developed continuous model of Cosserat rods on a Lie group infers that their three-dimensional form, under an external stress field, deforms continuously. This model is established through a finite element method, which includes embedding a geometric structure for temporal integration of the form configuration and its spatial discretization. The model involves the distributed laws of internal deformations and is defined by the following equations [1]:  
\[\frac{d}{ds}\bigg{(}EI\frac{2u}{s^{2}}\bigg{)}+N=0. \tag{1}\]  
This equation represents the equilibrium of forces and moments in the rod, where \(E\) is the Young modulus, \(I\) is the area moment of inertia, \(u\) is the displacement vector, and \(N\) is the internal force vector.  
The discrete model using deformation space formulation is developed on the basis of a geometric technique for the temporal integration of the robot shape configuration, while a helicoidal shape function has been used for spatial discretization. It considers significant deformations caused by bending, torsion, shearing, and extension. This model can be defined with the following sets of equations [1]:  
\[M\ddot{q}+C\dot{q}+Kq=F, \tag{2}\]  
where \(M\) is the mass matrix, \(C\) is the damping matrix, \(K\) is the stiffness matrix, \(q\) is the vector of generalized coordinates, and \(F\) is the vector of external forces.  
The implicit generalized Lie group scheme, which is used for soft robot dynamics, is used to carry out the geometric time integration. The following equations define the scheme [1]:  
\[q_{n+1}=exp(\Delta tv_{n+1})\cdot q_{n}. \tag{3}\]  
This equation represents the update of the soft robot at the next step, where \(q_{n+1}\) is the configuration at time \(n+1\), \(\Delta t\) is the time step, \(v_{n+1}\) is the velocity at time \(n+1\), and \(q_{n}\) is the configuration at time \(n\).  
Considering the deformations and external loading conditions of soft continuum robots, these mathematical models provide a thorough understanding of their dynamics and behavior.  
Two test cases were utilized to examine the discrete model, pure bending and rotation of a soft arm under various environmental conditions, and comparisons of results with the Princeton experiment were utilized to quantify the discrete model against benchmarks. With every simulation having errors less than 1%, the outcomes for the average error in tip distance between the simulated and real robot were only 1.32 mm. With a 0.01 s step, it averaged only 2 s to compute 1 s of actual activity. This is a significant improvement over previous models and indicates the possibility for real-time application in dynamic environments. The models promise to be good for advanced model-based control methods in soft robotic manipulators because they are computationally efficient and are good at modeling geometric nonlinearities. Sim SOFT, a special-purpose simulation library for the modeling of soft robots, was created due to their implementation and is still being developed today. All things considered, the mathematical models proposed successfully combine geometric spatial and temporal dynamics, laying the groundwork for future studies of material and geometric nonlinearities in soft robotics [1].