## 3 Literature Classification and Analysis
### Mathematical Models  
In the continuum mechanics models, the governing equation describes how the momentum of the robot changes due to both internal stresses and external forces, which is fundamental in dynamic modeling. The motion of a soft robot modeled as a continuum is governed by the following balance equations:  
\[\rho\frac{2r(x,t)}{t^{2}}=\
abla\cdot\sigma+f, \tag{4}\]  
where \(\rho\) is the density of the material, \(r(x,t)\) is the position vector of the material points in the continuum at the position \(x\) and time \(t\), \(\sigma\) is the Cauchy stress tensor that describes the internal forces, and \(f\) represents body forces (e.g., gravitational forces).  
In the kinematic models the robot's configuration is often described using a kinematic mapping that relates generalized coordinates to its shape [26]:  
\[r(x,t)=g(q(t))\in SE(3), \tag{5}\]where \(g(q(t))\) is a mapping function that takes the generalized coordinates \(q(t)\) (which could include positions, angles, etc.) and translates them to a configuration in 3D space represented by \(SE(3)\), the special Euclidean group. This captures both the position and orientation of the robot, essential for understanding its motion and deformations.  
Geometric models based on the functional approaches can describe the deformed configuration using specific mathematical functions.  
\[r(x,t)=h(x,t). \tag{6}\]  
The function \(h\) can represent various geometries (e.g., curves) that the soft robot might assume during deformations. For instance, a serpenoid or sine curve can model a snake-like robot's motion under actuation. This approach simplifies the modeling of complex deformations by selecting appropriate functions that capture the essential shape characteristics of the robot.  
The main effect is that currently, it is unrealistic to simulate soft robots with great realism and real-time calculation ability. Hence, depending on the user requirements, choosing a modeling approach is a trade-off between accuracy and computation efficiency. To seek innovative actuator concepts, designers may initially emphasize realism. Subsequently, they can move towards efficient models for control and optimization. Surrogate (data-driven) models need vast amounts of data, which provide speed and flexibility. Physics-based models, especially the discrete rod and continuum mechanics ones, are more accurate and variable but computationally expensive. Commercial FEM tools, due to their computationally expensive nature, are usually not ideal for real-time control applications, although they are excellent in complex design and phenomena [2].  
It was demonstrated that the condensed FEM model formalism can be used to jointly learn both the design and control of a soft robot. Contact modeling has been integrated into the learned condensed FEM model framework. Contacts are treated as constraints, similar to actuators and effectors. It was shown that the model for both open-loop and closed-loop control can be deployed on an embedded controller. The more complex the robot, the greater the potential speedup. However, significant changes in actuation or morphology would require relearning the model, because the model has been tested with a limited number of design parameters. Expanding this approach to more complex parameterization, with a greater number of parameters or increased expressivity, would require rethinking network scaling and sampling strategies. Another limitation of the method is its dependence on fixed material and mechanical parameters, which may change over time or across different robots. However, this can be mitigated using domain randomization, as a single network can be used with different geometric or mechanical properties. Finally, to address the limitation of the offline sampling process, one potential extension is to use online learning techniques [27].