## 4 Discussion  
Mathematical modeling in robotics is a rather extensive and dynamic field that develops robot design, control, and application. Robotic systems often include high-dimensional state spaces, resulting in complex and time-consuming computation of models and solutions. The limited computational resources on the robot can limit the complexity of models used in real time. Models must be robust to deal with unexpected situations and ensure the safety of robots and their environment. Many robotic systems have nonlinear behavior, which complicates modeling and requires sophisticated numerical methods. Real-time applications such as robotic surgery require models that can process data and make decisions with minimal delay. Artificial intelligence models require a large amount of training data that may not always be available in real scenarios and cannot be easily collected. Since robots are limited in power supply, the model must optimize energy efficiency. This reviewed research spans a wide variety of robotic systems, which reflects the complexity and breadth of this area of study. From soft continuum robots to autonomous mobile robots and artificial intelligence in robotics, each domain presents unique challenges that demand tailored mathematical frameworks and computational techniques. For soft continuum robots, the modeling difficulty arises from their deformable, three-dimensional nature, which defies conventional rigid-body assumptions. The application of Cosserat rod theory combined with finite element methods has emerged as a robust solution that offers an accurate and computationally efficient model to predict deformations under external loads. This methodology not only achieves validation with minimal error but also opens avenues for enhancing the precision of the control and interaction of soft robots with unstructured environments. Advanced mathematical models constitute the backbone of soft robotics development, which is continuously growing, especially for biomedical applications and exploratory tasks. Other challenging models involve flexible robots with elastic links and complex dynamic behaviors. Approaches based on [PERSON]'s variational principle and kinetod elasto-dynamics are found capable of capturing elastic deformations, transverse displacements, and flexible motions quite accurately. These partial differential equations and finite element-based models have reached far beyond theoretical development into the practical realm of applications involving Delta robots. The insights provided by these models are very important in the development of the precision and adaptability of flexible robots for their application in high-speed industrial operations and fine-manipulation tasks. The dual quaternion algebra illustrates well the use of both geometric and algebraic frameworks in robotics modeling and control. This method is particularly effective for dealing with transformations, rotations, and translations in robotic systems, as it may capture the essence of the geometric phenomena involved within a much simpler algebraic structure. Its ease of use does not stop at kinematic modeling, but plunges into dynamic systems and control where computational efficiency and clarity are needed. The dual quaternion algebra emphasizes the increasing interest in applying mathematical abstraction to practical challenges in robotics. While simplified dynamic models are highly appealing for their computational efficiency, they come with significant trade-offs. The review exposes the dangers of oversimplification, such as the neglect of classical mechanics principles, which could result in unrealistic representations and excessive demands on control systems. This is a cautionary note for researchers and practitioners to balance simplification with accuracy, ensuring that computational models remain grounded in physical reality. The emergence of fractional-order models represents a new frontier in robotics modeling.