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

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Search Strategy

#### 2.2. Study Selection

- Studies considered for inclusion were numerical and not experimental or clinical studies alone;
- The means of analysis, such as equations, parameters, and methods, were clearly explained;
- The study was written in English and published in a peer-reviewed journal.

## 3. Results and Discussions

#### 3.1. Nonlinear Optimization Function

#### 3.1.1. Fully Cartesian Coordinates

#### 3.1.2. Kinematic Analysis

#### 3.1.3. Dynamic Analysis

#### 3.2. Two-Degree-of-Freedom AFO for Robotic Rehabilitation

#### 3.2.1. Model of the Foot

#### 3.2.2. Model of the Orthosis

_{L}, as the total energy of the system, along with an additional error term:

_{L}, leads to the formulation of the ensuing equation:

_{td}is the damping coefficient. By selecting a PD control law $=-{K}_{p}{e}_{1}-{K}_{d}{\dot{\theta}}_{1}$:

_{L}decreases as long as ${\dot{\theta}}_{1}$, ${\dot{\theta}}_{2}$ are non-zero. By applying LaSalle’s theorem, ${\dot{V}}_{L}=0$ implies $\dot{q}$ = 0 and, consequently, ${\ddot{q}}_{1}$= 0. Using the equations of motion with the control $\tau =-{K}_{p}{e}_{1}-{K}_{d}{\dot{\theta}}_{1}$, Equation (29) becomes:

_{p}can be selected to be sufficiently large to drive ${e}_{1}$ close to zero, causing ${\dot{\theta}}_{1}$ to approach the desired set point.

#### 3.3. SMA-Element-Based AFO

_{T}) and stress-induced martensitic (ξ

_{S}) [29]. To delineate ξ

_{S}and ξ

_{T}in the context of the shape memory effect (SME) at low-temperature M

_{s}, the Liang and Roger transformation equation is employed [30]. The division of the martensite volume fraction (ξ) is expressed as follows [31]:

_{M}stands for the elastic modulus in the martensite phase, and E

_{A}corresponds to the austenite phase. The phase transformation is initiated at specific temperatures, known as M

_{s}(martensite start), A

_{s}(austenite start), M

_{f}(martensite finish), and A

_{f}(austenite finish) temperatures under zero-stress conditions. The stresses associated with the detwinning start and finish processes are symbolized as ${\sigma}_{s}^{cr}$ and ${\sigma}_{f}^{cr}$, respectively. C

_{M}signifies the constant that illustrates the influence of stress coefficients, and C

_{A}(as depicted in Figure 4) represents the impact of stress on the transformation temperatures of the martensite and austenite phases [30].

_{0}, ξ S

_{0}, ξ T

_{0}, and ξ

_{0}) denote the initial state of the material. Additionally, ${a}_{A}$ and ${a}_{M}$ are material constants defined as:

#### 3.4. Constitutive Models

#### 3.4.1. Linear Elastic Constitutive Model

#### 3.4.2. Viscoelastic Constitutive Model

#### 3.4.3. Hyperelastic Constitutive Model

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Illustration of the orientations of the plantarflexion–dorsiflexion joint axis (${\overrightarrow{Z}}_{1}$) and the eversion–inversion joint axis (${\overrightarrow{Z}}_{2}$), along with their corresponding kinematic representations [25], with permission from the American Society of Mechanical Engineers ASME (License Number: 1408140-1).

**Figure 3.**Machine and human parts of the shank and ankle [25], with permission from the American Society of Mechanical Engineers ASME (License Number: 1408141-1).

**Figure 4.**The essential stress–temperature profiles pertaining to the Brinson model’s critical behavior [31], with permission from Elsevier (Creative Commons license, https://creativecommons.org/licenses/by-nc-nd/4.0/ (accessed on 15 October 2023)).

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**MDPI and ACS Style**

Nazha, H.M.; Szávai, S.; Juhre, D.
An Overview of Mathematical Methods Applied in the Biomechanics of Foot and Ankle–Foot Orthosis Models. *J* **2024**, *7*, 1-18.
https://doi.org/10.3390/j7010001

**AMA Style**

Nazha HM, Szávai S, Juhre D.
An Overview of Mathematical Methods Applied in the Biomechanics of Foot and Ankle–Foot Orthosis Models. *J*. 2024; 7(1):1-18.
https://doi.org/10.3390/j7010001

**Chicago/Turabian Style**

Nazha, Hasan Mhd, Szabolcs Szávai, and Daniel Juhre.
2024. "An Overview of Mathematical Methods Applied in the Biomechanics of Foot and Ankle–Foot Orthosis Models" *J* 7, no. 1: 1-18.
https://doi.org/10.3390/j7010001