# Two-Dimensional Mechanical Model of Human Stability in External Force-Caused Fall

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## Abstract

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## Featured Application

**Estimation of forces and moments of forces in joints in human motion analysis, ground reaction forces calculations in motion capture systems.**

## Abstract

## 1. Introduction

## 2. Experiment Description

## 3. Human Body Biomechanical Model

## 4. Mathematical Model

_{i}that are measured counterclockwise in the sagittal plane (i.e., YZ-plane). A rule was adopted in the paper that the directed angle φ

_{i}is measured from the horizontal axis of the own local system, which is parallel to the Z axis, to the symmetry axis of the given segment. The time derivatives φ

_{i}of the angles φ

_{i}stand for the angular velocities of the segments, and the second derivatives $\ddot{{\phi}_{i}}$ are their angular accelerations. The Newton–Euler approach requires the introduction of quantities defining the kinematic state of the centres of mass of each body segment. Let y

_{i}and z

_{i}be the Cartesian coordinates of the mass centre C

_{i}within the global coordinate system OXYZ. Differentiating the coordinates over time, we get the Cartesian components of the velocities of the mass centres. Differentiating twice yields the Cartesian components of the accelerations of the mass centres. Since the dimensions of model parts are described by angles and distances from the centre of mass of the current segment, the initial state of the examined person does not affect the results.

## 5. Data Preparation

## 6. Results

## 7. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 4.**Foot model sketch (identical for left and right foot). The symbols stand for: i—segment number, which takes the value of 1 or 11 (according to Figure 3), l

_{i}—the distance between the centre of mass of the foot (C

_{i}) and the ankle joint, ${l}_{{R}_{i}}\u2014$ the distance between the centre ${C}_{i}$ and the centre of pressure COP, ${\phi}_{{R}_{i}}$—the directed angle that defines the location of the segment joining the centres of mass and pressure, and ${R}_{{y}_{i}}$, ${R}_{{z}_{i}}$—components of the reaction forces.

**Figure 5.**Free-body diagram representative for the segments standing for shanks, thighs, arms, and forearms. The symbols used mean: i—segment number, which takes one of the following values: 2, 3, 5, 6, 8, 9, 12, 13 (according to Figure 3), l

_{i}—length of the ith segment, and d

_{i}—distance from the end of the ith segment, which connects to the i+1 segment, to the segment’s centre of mass (C

_{i}).

**Figure 6.**The free-body diagram for both hands. Used symbols stand for: i—segment number, which takes the value of 7 or 10 (according to Figure 3), and p

_{i}—the distance from the proximal end of the segment to the segment’s centre of mass (C

_{i}).

**Figure 7.**The free-body diagram for the segment representing the head and torso. $\overrightarrow{E}$ stands for the external force applied to the participant.

**Figure 8.**The vertical component, RL

_{y}, of the GRF for the left foot in the case of the exercises with upper limbs swinging. The RL

_{y}measured with the dynamometric plate is drawn using the dashed line, and the RL

_{y}calculated is drawn using the solid line. The error δ = 0.0841.

**Figure 9.**The horizontal component, RL

_{z}, of the GRF for the left foot in the case of the exercises with upper limbs swinging. The RL

_{z}measured with the dynamometric plate is drawn using the dashed line, and the RL

_{z}calculated is drawn using the solid line.

**Figure 10.**The vertical component, RL

_{y}, of the GRF for the left foot in the case of the external force-caused fall with two pulls. The RL

_{y}measured with the dynamometric plate is drawn using the dashed line, and the RL

_{y}calculated is drawn using the solid line. The dotted line presents the time course of the external force magnitude. The error δ = 0.0227.

**Figure 11.**The in-plane component, RL

_{z}, of the GRF for the left foot in the case of the external force-caused fall with two pulls. The RL

_{z}measured with the dynamometric plate is drawn using the dashed line, and the RL

_{z}calculated is drawn using the solid line. The dotted line presents the time course of the external force magnitude.

**Figure 12.**The vertical component, RL

_{y}, of the GRF for the left foot in the case of the external force-caused fall with a single pull. The RL

_{y}measured with the dynamometric plate is drawn using the dashed line, and the RL

_{y}calculated is drawn using the solid line. The dotted line presents the time course of the external force magnitude. The error δ = 0.0248.

**Figure 13.**The vertical component, RL

_{y}, of the GRF for the left foot in the case of the external force-caused fall with a single pull. The RL

_{y}measured with the dynamometric plate is drawn using the dashed line, and the RL

_{y}calculated is drawn using the solid line. The dotted line presents the time course of the external force magnitude.

**Table 1.**Values of mass and mass moments of inertia of particular body segments of the person performing exercises.

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**MDPI and ACS Style**

Sopa, M.; Sypniewska-Kamińska, G.; Walczak, T.; Kamiński, H.
Two-Dimensional Mechanical Model of Human Stability in External Force-Caused Fall. *Appl. Sci.* **2023**, *13*, 5068.
https://doi.org/10.3390/app13085068

**AMA Style**

Sopa M, Sypniewska-Kamińska G, Walczak T, Kamiński H.
Two-Dimensional Mechanical Model of Human Stability in External Force-Caused Fall. *Applied Sciences*. 2023; 13(8):5068.
https://doi.org/10.3390/app13085068

**Chicago/Turabian Style**

Sopa, Martyna, Grażyna Sypniewska-Kamińska, Tomasz Walczak, and Henryk Kamiński.
2023. "Two-Dimensional Mechanical Model of Human Stability in External Force-Caused Fall" *Applied Sciences* 13, no. 8: 5068.
https://doi.org/10.3390/app13085068