{ "cells": [ { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "# Angular velocity in 3D movements\n", "\n", "> Renato Naville Watanabe \n", "> Laboratory of Biomechanics and Motor Control ([http://pesquisa.ufabc.edu.br/bmclab](http://pesquisa.ufabc.edu.br/bmclab)) \n", "> Federal University of ABC, Brazil" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "An usual problem found in Biomechanics (and Mechanics in general) is to find the angular velocity of an object. We consider that a basis \n", "$\\hat{\\boldsymbol e_1}$, $\\hat{\\boldsymbol e_2}$ and $\\hat{\\boldsymbol e_3}$ is attached to the body and is known. To learn how to find a basis of a frame of reference, see [this notebook](https://nbviewer.jupyter.org/github/BMClab/bmc/blob/master/notebooks/ReferenceFrame.ipynb).\n", "\n", "
" ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "## Axis of rotation\n", "\n", "As in the planar movement, the angular velocity is a vector perpendicular to the rotation. The line in the direction of the angular velocity vector is known as the axis of rotation. \n", "\n", "The rotation beween two frames of reference is characterized by the rotation matrix $R$ obtained by stacking the versors $\\hat{\\boldsymbol e_1}$, $\\hat{\\boldsymbol e_2}$ and $\\hat{\\boldsymbol e_3}$ in each column of the matrix (for a revision on rotation matrices see [this](https://nbviewer.jupyter.org/github/bmclab/bmc/blob/master/notebooks/Transformation2D.ipynb) and [this](https://nbviewer.jupyter.org/github/bmclab/bmc/blob/master/notebooks/Transformation3D.ipynb) notebooks). \n", "\n", "A vector in the direction of the axis of rotation is a vector that does not changes the position after the rotation. That is:\n", "\n", " \n", "\\begin{equation}\n", "v = Rv\n", "\\end{equation}\n", "\n", "\n", "\n", "This vector is the eigenvector of the rotation matrix $R$ with eigenvalue equal to one. " ] }, { "cell_type": "markdown", "metadata": { "slideshow": { "slide_type": "slide" } }, "source": [ "Below the yellow arrow indicates the axis of rotation of the rotation between the position of the reference frame $\\hat{\\boldsymbol i}$, $\\hat{\\boldsymbol j}$ and $\\hat{\\boldsymbol k}$ and the reference frame of $\\hat{\\boldsymbol e_1}$, $\\hat{\\boldsymbol e_2}$ and $\\hat{\\boldsymbol e_3}$." ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "slideshow": { "slide_type": "fragment" } }, "outputs": [ { "data": { "application/javascript": [ "/* Put everything inside the global mpl namespace */\n", "window.mpl = {};\n", "\n", "\n", "mpl.get_websocket_type = function() {\n", " if (typeof(WebSocket) !== 'undefined') {\n", " return WebSocket;\n", " } else if (typeof(MozWebSocket) !== 'undefined') {\n", " return MozWebSocket;\n", " } else {\n", " alert('Your browser does not have WebSocket support. ' +\n", " 'Please try Chrome, Safari or Firefox ≥ 6. ' +\n", " 'Firefox 4 and 5 are also supported but you ' +\n", " 'have to enable WebSockets in about:config.');\n", " };\n", "}\n", "\n", "mpl.figure = function(figure_id, websocket, ondownload, parent_element) {\n", " this.id = figure_id;\n", "\n", " this.ws = websocket;\n", "\n", " this.supports_binary = (this.ws.binaryType != undefined);\n", "\n", " if (!this.supports_binary) {\n", " var warnings = document.getElementById(\"mpl-warnings\");\n", " if (warnings) {\n", " warnings.style.display = 'block';\n", " warnings.textContent = (\n", " \"This browser does not support binary websocket messages. \" +\n", " \"Performance may be slow.\");\n", " }\n", " }\n", "\n", " this.imageObj = new Image();\n", "\n", " this.context = undefined;\n", " this.message = undefined;\n", " this.canvas = undefined;\n", " this.rubberband_canvas = undefined;\n", " this.rubberband_context = undefined;\n", " this.format_dropdown = undefined;\n", "\n", " this.image_mode = 'full';\n", "\n", " this.root = $('
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