# Identical Limb Dynamics for Unilateral Impairments through Biomechanical Equivalence

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

## 3. Mathematical Modeling

#### 3.1. Kinematic Measurments

#### 3.2. Collisions

#### 3.2.1. Knee-Lock

#### 3.2.2. External Surfaces

#### 3.3. Kinetic Measurements

## 4. Results

#### 4.1. Kinematic Symmetry

#### 4.2. Full Symmetry

#### 4.3. Simulations

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Step-by-Step Derivation of Lagrange’s Equations in Double Pendulum

## Appendix B. Contact with External Surafces

#### Appendix B.1. Perfectly Elastic

#### Appendix B.2. Tripping

## References

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**Figure 1.**Dynamic model of an asymmetric human gait with double pendulums: (

**a**) Modeling of an impaired gait with two different double pendulum systems; (

**b**) different mass distributions on each leg represented by ${m}_{pros}$ for the prosthetic side and ${m}_{footi}$ for the healthy side with $i=\left\{1a,1b,2a,or2b\right\}$. The hypothesis in this study evaluates whether the two different legs can be made to move symmetrically.

**Figure 2.**Double pendulum system with four masses. Each link includes two masses at different locations. The top and bottom links replicate shank and thigh motion, respectively. The top and middle hinges represent hip and knee joints, respectively.

**Figure 3.**Double pendulum before and after the knee-lock event. The before and after angles are shown with + and − superscripts.

**Figure 4.**Double pendulum colliding with different external surfaces: (

**a**) Perfectly inelastic collision with ground; (

**b**) perfectly elastic collision with ground; (

**c**) tripping over an object; (

**d**) slipping over the contact surface.

**Figure 6.**Kinematically parallel system to Sys H with lightest total mass, heaviest total mass, lowest links’ mass ratios, and highest links’ mass ratios.

**Figure 7.**Prosthetic leg designs based on the anthropometric data of a healthy leg: (

**a**) Anthropometric data for a healthy leg; (

**b**) a dynamic model of a prosthetic design with symmetric kinematics and lightest weight; (

**c**) a dynamic model of another prosthetic design with symmetric kinematics and the lowest ratio of mass between the links.

**Figure 8.**Variation in physical parameters between systems with full symmetry. Red lines indicate the median. The square specifies the lower and upper quartiles. Whiskers show the minimum and maximum values in each parameter. (

**a**) Whisker plot of mass values, ${m}_{1a},{m}_{1b},{m}_{2a},$ and ${m}_{2b}$ in full symmetric pendulum systems; (

**b**) Whisker plot of each mass locations, ${l}_{1a},{l}_{1b},{l}_{2a},$ and ${l}_{2b}$ in full symmetric pendulum systems.

**Figure 9.**Algorithm flowchart for simulation of a double pendulum system with knee-lock and different collisions with external surfaces. Yellow diamonds indicate the conditional statements and make decisions depending on the answer to the questions. Red circles are instance changes in the state of the system under a sudden collision. Blue rectangles are the processors for calculating the state of the system (angles, angular velocities, angular accelerations).

**Figure 10.**Kinematic and kinetic simulations of systems with different physical parameters but similar kinematic and kinetic coefficients. Sys H, Sys Kin, and Sys Full indicate the result for the anthropometric data, a sample of systems (365 solutions) with only kinematic symmetry, and a sample of systems (60 solutions) with full symmetry, respectively. (

**a**) Angular displacement; (

**b**) angular acceleration; (

**c**) internal forces at hip joint; (

**d**) internal forces at the knee joint.

${\mathit{m}}_{1\mathit{a}}$ | ${\mathit{l}}_{1\mathit{a}}$ | ${\mathit{m}}_{1\mathit{b}}$ | ${\mathit{l}}_{1\mathit{b}}$ | ${\mathit{l}}_{1}$ | ${\mathit{m}}_{2\mathit{a}}$ | ${\mathit{l}}_{2\mathit{a}}$ | ${\mathit{m}}_{2\mathit{b}}$ | ${\mathit{l}}_{2\mathit{b}}$ | ${\mathit{l}}_{2}$ | |
---|---|---|---|---|---|---|---|---|---|---|

Sys H ^{1} | 4.0 | 0.2 | 3.5 | 0.2 | 0.6 | 2 | 0.1 | 1.5 | 0.2 | 0.5 |

^{1}All masses are in kilogram (kg) and lengths are in meter (m).

${\mathit{m}}_{1\mathit{a}}$ | ${\mathit{l}}_{1\mathit{a}}$ | ${\mathit{m}}_{1\mathit{b}}$ | ${\mathit{l}}_{1\mathit{b}}$ | ${\mathit{l}}_{1}$ | ${\mathit{m}}_{2\mathit{a}}$ | ${\mathit{l}}_{2\mathit{a}}$ | ${\mathit{m}}_{2\mathit{b}}$ | ${\mathit{l}}_{2\mathit{b}}$ | ${\mathit{l}}_{2}$ | |
---|---|---|---|---|---|---|---|---|---|---|

Kinematic ^{1} | 3.56 | 3.50 | 0.30 | 0.10 | 0.6 | 1.78 | 1.28 | 0.15 | 0.15 | 0.5 |

Full ^{2} | 6.50 | 1.00 | 0.25 | 0.29 | 0.6 | 3.00 | 0.50 | 0.15 | 0.28 | 0.5 |

^{1}Sample system with only kinematic symmetry.

^{2}Sample system with full (kinetic and kinematic) symmetry.

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**MDPI and ACS Style**

Rasouli, F.; Reed, K.B.
Identical Limb Dynamics for Unilateral Impairments through Biomechanical Equivalence. *Symmetry* **2021**, *13*, 705.
https://doi.org/10.3390/sym13040705

**AMA Style**

Rasouli F, Reed KB.
Identical Limb Dynamics for Unilateral Impairments through Biomechanical Equivalence. *Symmetry*. 2021; 13(4):705.
https://doi.org/10.3390/sym13040705

**Chicago/Turabian Style**

Rasouli, Fatemeh, and Kyle B. Reed.
2021. "Identical Limb Dynamics for Unilateral Impairments through Biomechanical Equivalence" *Symmetry* 13, no. 4: 705.
https://doi.org/10.3390/sym13040705