## 4 Discussion
Using fractional-order variable difference equations, researchers have developed a flexible framework that accommodates the intricate dynamics of robotic systems. These models, validated through simulations and empirical data, have proven to be particularly effective at capturing the nuanced behavior of systems with complex dynamic characteristics, such as gantry cranes and missile launching vehicles. The fractional-order approach also demonstrates how mathematical techniques can be adapted to accommodate a wide range of robotic applications, from heavy machinery to fine robotic arms. The modeling of autonomous mobile robots incorporates dynamics with adaptability to variable configurations as part of their multifaceted functionality. The Newton-Euler method, used to derive the equations of motion, gives a basis for adaptive control algorithms that enhance stability and efficiency. This is especially important in cases where AMRs interact with manipulators, as the interaction of the mobility of the robot with its manipulation requires a delicate balance of dynamic forces. It is through such modeling that continued development of autonomous systems for effective operation in dynamic and unstructured environments has been possible. The combination of mathematical modeling and artificial intelligence (AI) brings a paradigm shift to robotics. Methods such as geometric and topological reasoning, along with the notion of fibered logical spaces, allow for the embedding of logical reasoning and decision-making capabilities in robots. This cross-disciplinary approach reinforces the development of symbolic computation combined with artificial intelligence to reach not only mechanical but also intelligent autonomous decision-making. The interplay of AI with mathematical modeling is going to push the next generation of robotic systems to bridge the gap between physical dynamics and cognitive capabilities. Such models are instrumental in developing control systems that optimize the coordination and stability of these systems, paving the way for advances in modular robotics and cooperative robotic tasks. From the works reviewed, it is evident that advanced mathematical modeling has played an indispensable role in taming the multifaceted challenges of robotics. Each method--classical mechanics, algebraic abstractions, and fractional calculus--finds its unique perspective and solution set, reflecting the diversity of robotic systems and their applications. It is in this marriage of mathematical precision with computational techniques that these models do not stop being theoretical constructs but serve to provide actionable insight into real-world robotics. The reviewed studies also emphasize the importance of validation and interdisciplinary collaboration in robotics modeling. Techniques such as empirical validation against experimental data, simulation studies, and cross-disciplinary integration with AI demonstrate how mathematical models evolve from theoretical frameworks to practical tools. This evolution is essential for bridging the gap between academic research and industrial applications, ensuring that robotics continues to innovate and address complex societal needs. To summarize the diverse range of mathematical models applied in robotics, a comparative analysis has been conducted based on key performance parameters such as accuracy, computational complexity, adaptability, and application domains. Table 2 below provides a structured overview of each model's strengths and limitations, highlighting their suitability for different robotic tasks. This comparative insight is essential for guiding model selection in research and real-world implementations, especially as robotics continues to advance toward greater intelligence, flexibility, and autonomy.