## 5 Future Directions  
Looking ahead, a few emerging trends and challenges can be identified: the increasing complexity of robotic systems, including soft materials, flexible components, and intelligent algorithms, requires an increasingly sophisticated mathematical model. For many applications, especially when real-time performance is required--for example, autonomous navigation or dynamic manipulation--computational efficiency remains a pressing concern. Therefore, the trade-off between model accuracy and computational tractability will remain a significant focus in the field for the foreseeable future. Another important area is the representation of uncertainty and variability within robotic models. Real-world settings are essentially indeterminate; any hopes for the performance of robots in real-world conditions are contingent upon sound mathematical bases that can handle such an indeterminacy. Probabilistic modeling, stochastic differential equations, and reinforcement learning algorithms are some of the techniques that could be applied to this end and could provide a route to more robust and adaptive robotics. \begin{table}
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The future directions of robotics mathematical models focus on improving adaptation, efficiency, and real-time capability. The main areas of development include the integration of foundation models, the development of model learning, and the improvement of kinematic modeling. Foundation models, based on large-scale datasets, could enhance the autonomy of robots by improving perception, decision-making, and control capabilities. They offer superior generalization and can solve problems that are not present in training data, although problems such as data scarcity and safety remain [54]. Mathematical models for soft robots are improving using FEM to accurately simulate deformations under different loads. These models aim to improve efficiency and accuracy in dynamic simulations [55]. The mathematical modeling of robotics stands as the very cornerstone of this discipline, providing theoretical and computational tools to understand, design, and control these increasingly complex systems. The methodologies that were discussed, from Cosserat rod theory to dual quaternion algebra and even to fractional-order calculus, represent examples of ingenuity and the ability of researchers to adaptively face the challenges presented by robotic systems. As robotics continues to intersect with the boundaries of AI, material sciences, and computational biology, the role of mathematical modeling will only grow, leading to innovation and expanding the horizons of the things robots can achieve. The development of artificial intelligence and computer technology is rapidly developing mathematical modeling in robotics. AI improves simulations by providing a realistic scenario and high-precision predictions. The integration of artificial intelligence and mathematical modeling is an engine for the development of more robust and reliable robotic systems. These systems can cope with uncertainty and work consistently under diverse conditions. Modern models are dynamic and adaptable, allowing robots to adapt their behavior based on real-time data and changing conditions. Advanced mathematical models are used to simulate and virtually test robotic systems. These simulations enable researchers to test and improve the model before physical implementation, saving time and resources [56; 57]. This synthesis of mathematics and robotics not only furthers technological capabilities but also extends insight into complex systems, paving the way to a future where robots will become commonplace in everyday life and industry. Optimal models for inverse kinematics are being developed, especially for complex robots such as six-degree-of-freedom anthropomorphic robots, to balance accuracy and computational efficiency. Explainable artificial intelligence technology is also used to improve model interpretability [58].