An Overview of Mathematical Methods Applied in the Biomechanics of Foot and Ankle–Foot Orthosis Models

Ankle–foot orthoses (AFOs) constitute medical instruments designed for patients exhibiting pathological gait patterns, notably stemming from conditions such as stroke, with the primary objective of providing support and facilitating rehabilitation. The present research endeavors to conduct a comprehensive review of extant scholarly literature focusing on mathematical techniques employed for the examination of AFO models. The overarching aim is to gain deeper insights into the biomechanical intricacies underlying these ankle–foot orthosis models from a mathematical perspective, while concurrently aiming to advance novel models within the domain. Utilizing a specified set of keywords and their configurations, a systematic search was conducted across notable academic databases, including ISI Web of Knowledge, Google Scholar, Scopus, and PubMed. Subsequently, a total of 23 articles were meticulously selected for in-depth review. These scholarly contributions collectively shed light on the utilization of nonlinear optimization techniques within the context of ankle–foot orthoses (AFOs), specifically within the framework of fully Cartesian coordinates, encompassing both kinematic and dynamic dimensions. Furthermore, an exploration of a two-degree-of-freedom AFO design tailored for robotic rehabilitation, which takes into account the interplay between foot and orthosis models, is delineated. Notably, the review article underscores the incorporation of shape memory alloy (SMA) elements in AFOs and overviews the constitutive elastic, viscoelastic, and hyperelastic models. This comprehensive synthesis of research findings stands to provide valuable insights for orthotists and engineers, enabling them to gain a mathematical understanding of the biomechanical principles underpinning AFO models and fostering the development of innovative AFO designs.