## 2 Methodology  
The research design presents a systematic and structured approach to the literature review of mathematical models in robotics and ensures that the review is rigorous, includes high-quality sources, and provides a critical synthesis of existing research. The scope and objectives of the investigation cover different mathematical backgrounds that form the basis for the design, analysis, and control of robotic systems. These include three main elements: first, the categorization of the mathematical models applied in these specific robotic fields for the thematic understanding of their theoretical and practical implications; second, the analysis of their limitations and computational challenges; and third, the identification of literature gaps and emerging trends that may guide future research. This objective provides a basis for understanding how mathematical models form the fields of traditional and modern robotics. Literature Search Strategy. An extensive search of the literature was conducted to ensure that the review captured the most relevant and high-quality studies. The literature search and selection processes are shown in Figure 1. The key sources of materials for this review are well-established databases, including Web of Science Core Collection (WOS, Clarivate Analytics, Philadelphia, PA, USA), IEEE Xplore, SpringerLink, ScienceDirect, ACM Digital Library, and Google Scholar. These have been supplemented by textbooks, doctoral theses, and conference proceedings to consider both foundational and applied research. Specific keywords and search terms such as 'robotics','mathematical model', 'flexible robot','simplified dynamic', 'autonomous mobile', and 'artificial intelligence' were used to ensure a thorough coverage of each domain. Boolean operators and advanced search techniques were used to refine the results and identify the most relevant studies. An extensive search of the literature provided 3532 references in the field. Inclusion and Exclusion Criteria. The mathematical derivation inclusion criteria limited the review to only 309 peer-reviewed journal articles and conference papers, focusing on works published between 2003 and 2025, although seminal contributions that remain foundational to the field were considered (see Table 1). Studies with a clear emphasis on mathematical modeling were selected in one or more of the specified domains. Some of the exclusion criteria included a lack of theoretical or mathematical depth in the articles and others that were purely experimental or hardware-based without any model-based analytical content. Screening and Selection Process. The screening of studies first began with the title and abstract screening for relevance with respect to the objective of this study. Studies retrieved from more than one database were considered duplicates and therefore removed. Subsequently, full-text reviews of the shortlisted papers were performed to confirm their alignment with the research objectives. The selected studies were then classified according to thematic areas, including but not limited to flexible robots, continuous soft robots, and autonomous mobile robots. Quality assessment criteria were aimed at assessing the  
Figure 1: Literature review process. mathematical depth, clarity, relevance, and contributions of each study to theoretical and practical advances in robotics. Ultimately, 60 papers were retained for review in this paper. Data Extraction and Organization. Key information from selected studies was systematically extracted to facilitate comparative analysis and thematic synthesis. This included identifying the specific robotic domain addressed, the mathematical techniques utilized, and the applications and limitations of the proposed models. Data were organized into structured categories that corresponded to each focus area, providing a clear framework for analysis. Critical Analysis and Synthesis.