## 3 Literature Classification and Analysis
### Soft Continuum Robots  
#### 3.1.1 A Geometrically Exact Model for Soft Continuum Robots  
Finite element deformation space formulation soft robots, which have the capability of continuous deformation, raises unique modeling challenges due to their highly flexible structure and complex dynamics [21]. Traditional rigid robotics models cannot accurately capture the behavior of these systems. The finite element deformation space formulation proposes a novel, geometrically exact modeling framework that tries to balance accuracy and computational efficiency, addressing key limitations of existing approaches. It is based on Cosserat rod theory, extended to include geometric mechanics, and implemented with finite element methods. This allows an accurate representation of bending, torsion, shear, and axial deformations in soft robots under external loads [22]. Utilizing the latest mathematical tools, such as Lie groups and Lie algebras, this framework accurately captures nonlinearities in the deformations of soft robots, thus allowing very detailed but efficient simulations. The model adopts a helical shape function for spatial discretization and a geometric time-integration scheme. This maintains accuracy in the simulation of large deformations while maintaining computational efficiency. Unlike traditional constant-curvature models, which approximate robot shapes as a series of arcs, this method captures more complex and realistic deformations under diverse loading conditions [23]. The derivation of a finite element-based geometric Jacobian that relates the robot's structural velocities to the deformation dynamics follows. Such innovation will then easily enable static and dynamic analyses, and the model would represent motion and forces using the principle of virtual work. The implementation of the model in the software framework of Sim SOFT v1.0 provides an avenue for dynamic simulation and control of soft robots. Sim SOFT integrates the deformation space formulation, allowing real-time simulation of robots in diverse scenarios with significant improvements in computational efficiency over previous methods. The model is validated against experimental benchmarks, reaching positional accuracy with errors below 1% of the robot length. Dynamic simulations show the versatility of the framework, including examples of helical motion under combined bending and torsion loads. Applications span minimally invasive surgery, robotic rehabilitation, and industrial inspection. Traditional soft robot models are often based on simplifying assumptions, such as constant curvature or small-deformation approximations, which make them poorly applicable under extreme conditions. In contrast, the finite element deformation space formulation naturally accommodates large deformations and non-uniform strain distributions, making it more robust for practical applications. The Lie group-based approach avoids explicit global parameterization and therefore avoids typical drawbacks such as singularities, and improving numerical stability [24]. The framework allows modular designs; thus, complex multi-segment soft robots and parallel continuum manipulators can be simulated. Its computational efficiency is reflected in dynamic simulations that run up to twice as fast as real-time on standard hardware, which is a significant advance over existing models. The integration of analytical solutions for special cases, such as planar bending and torsion, provides intuitive insight with model manageability. Possible improvements include the optimization of computational algorithms and their extension for consideration of material heterogeneity and anisotropic properties. In this manner, the integration of real-time sensory feedback and machine learning models with the Sim SOFT platform will enhance adaptability within unstructured environments. This establishes the groundwork for more intelligent, adaptive, and versatile robotic systems that can perform complex tasks in real-world settings by extending the theoretical and computational bases of soft robotics [25].