## 1 Introduction  
Mathematical modeling is the backbone of robotics in terms of understanding, design, and control. The growth of demanding and flexible mathematical models has rapidly increased due to the growing importance of their applications to a wide range of diverse disciplines such as manufacturing, health, logistics, and exploration problems. Such models are highly essential for robust decision-making in complicated dynamic scenarios, optimization, and prediction of robot behaviors. Some of the key areas where mathematical models have significantly advanced robotics include soft continuum robots, flexible robots, dual-quaternion algebra, simplified dynamic models, fractional order models, autonomous mobile robots, and artificial intelligence in robotics. Modeling soft robots is particularly challenging mathematically due to their constantly deforming three-dimensional forms when subjected to the action of external loads. As one of the important approaches, the deformation space formulation of the finite element is important. The method encompasses bending, torsion, shear, and axial deformations and deduces a discrete model from the Cosserat rod theory. The model has been proven to be highly accurate and computationally efficient, with errors of less than 1% of the robot's length [1]. Flexible robots usually have inherent properties and the environments they operate in. That is the reason why complex dynamics is often part of the mathematical model for flexible robotics. Advances in computational methods, such as finite element analysis and machine learning, are helping to overcome some of these hurdles. Following the variational principle of [PERSON], the model describing the transverse displacement and revolved angles of an elastic connection was proposed by a system of three nonlinear integrodifferential equations and two elastic partial differential equations [2]. The finite element approach and kineto-elasto-dynamics were used to describe a Delta robot with flexible linkages. The set of linear time-varying differential equations describes flexible motions relative to the rigid body configuration [3]. Dual quaternion algebra is a powerful mathematical framework. Due to its ability to represent geometric and physical processes in an algebraic framework, dual-quaternion algebra has become increasingly popular in the field of robotics. This approach is helpful for both robot modeling and control, as it makes a variety of mathematical concepts and processes easier to represent [4]. Simplified dynamic models are essential in robotics, especially when dealing with complex systems. They enable faster simulations that are crucial for real-time applications and iterative testing. Although computational efficiency makes simplified dynamic models desirable, they can potentially have serious drawbacks. Models are likely to completely ignore the laws of classical physics, leading to wildly inaccurate representations that demand high correction torques from controllers with potentially saturated actuators [5]. Fractional-order models leverage fractional calculus to describe systems with non-integer order dynamics. These models can be used to model robot arms using changeable fractional-order difference equations. It splits the labor into developing a mechanical model and running simulations to compare the data collected. It also provides robotic systems with the freedom to capture dynamics [6]. Furthermore, fractional modeling is applied to various systems such as missile launchers and gantry cranes, whose equations have been established through Euler-Lagrangian formulations for further analysis [7]. The relationship of dynamic parameters in a changeable configuration is considered when modeling the dynamics of autonomous mobile robots with manipulators. The Newton-Euler technique derives the equations of motion with respect to the center of mass, the angular motion, and the movement of the manipulator, by which adaptive control algorithms can be realized that enhance stability and efficiency [8].