{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# The inverted pendulum model of the human standing\n", "\n", "Marcos Duarte" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Despite the enormous complexity of the human body, part of the mechanical behavior of the human body during the standing still posture, namely the displacements of the center of gravity ($COG$) and center of pressure ($COP$) in the anterior-posterior direction, can be elegantly portraied by a physical-mathematical model of an inverted pendulum with rigid segments articulated by joints.\n", "\n", "Using such a model, it's possible to estimate the COG vertical projection (COGv) from the COP displacement. The Python function `cogve.py` (code at the end of this text) performs this estimation. The function signature is:\n", "```python\n", "cogv = cogve(COP, freq, mass, height, show=False, ax=None)\n", "```\n", "Let's now derive the inverted pendulum model of the human standing posture implemented in this function." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Derivation of the inverted pendulum model\n", "\n", "In the most simple version of the model, the human body in the sagittal plane is reduced to a two-link body with a single inverted pendulum articulated by only one joint (representing the feet, with the rest of the body articulated by the ankle joint). Let's deduce the equations for such inverted pendulum model as the representation at the sagital plane of the human standing still posture. The inverted pendulum model and the correspondent free-body diagrams (FBDs) are shown in Figure 1.\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The equations of motion for each FBD of the feet and rest-of-body segments at the sagittal plane ($xy$ plane) can be expressed in the form of the Newton-Euler equations. \n", "