{ "metadata": { "name": "", "signature": "sha256:24f954352ff035ec72dad1057e011de88f97d30825b6d65b76a61e18f5ba4f6e" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "CURVA SERPENTINOIDE" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "**Autor**: [Juan Gonz\u00e1lez G\u00f3mez (Obijuan)](http://www.iearobotics.com/wiki/index.php?title=Juan_Gonzalez:Main)\n", "\n", "**Fecha**: 18-Enero-2015\n", "\n", "**Licencia**: Licencia Creative Commons BY-SA para el texto e im\u00e1genes. GPL v2 para el software\n", "\n", "Este notebook es una adaptaci\u00f3n de [esta p\u00e1gina de la wiki](http://www.iearobotics.com/wiki/index.php?title=Curva_serpentinoide) donde se comenz\u00f3 a documentar la curva y su implementaci\u00f3n en octave/Matlab\n", "\n", "Adem\u00e1s de ampliar la informaci\u00f3n sobre la curva, quiero aprovechar para aprender los ipython notebooks" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Introducci\u00f3n\n", "\n", "La **curva serpentinoide** (o curva serpenoide, serpenoid curve en ingl\u00e9s) la descubri\u00f3 el **profesor Hirose** en 1976 cuando realizaba su tesis doctoral en el Instituto de tecnolog\u00eda de Tokyo. Investigaba la biomec\u00e1nica de las serpientes para su aplicaci\u00f3n a la construcci\u00f3n de robots.\n", "\n", "La curva serpentinoide es indispensable para el estudio y construcci\u00f3n de **robots \u00e1podos** (gusanos y serpientes). El autor de este art\u00edculo la ha estudiado en su tesis doctoral para aplicarla a la locomoci\u00f3n de estos robots.\n", "\n", "En este documento se describe la **curva/onda serpentinoide**, sus par\u00e1metros, sus propiedas y se presentan scripts en python para implementarla.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Curva serpentinoide continua\n", "\n", "## Definici\u00f3n\n", "\n", "La curva serpentinoide es aquella cuya [curvatura](http://es.wikipedia.org/wiki/Geometr%C3%ADa_diferencial_de_curvas#Curvatura_y_torsi.C3.B3n) var\u00eda *sinusoidalmente con la distancia* a lo largo de la curva.\n", "\n", "Su curvatura est\u00e1 dada por la ecuaci\u00f3n: \n", "\n", "\n", "\n", " \n", "\n", "\n", "\n", "
$ K(s) = \\frac{2\\pi k}{l}\\alpha\\sin\\left(\\frac{2\\pi k}{l}s\\right) $ Ecuaci\u00f3n de curvatura de la serpentinoide (**ec. 1**)
\n", "\n", "donde:\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "\n", "
** $l$ **Longitud total de la curva $ (l > 0) $\n", "
** $s$ **Es la variable. [Longitud del arco](http://es.wikipedia.org/wiki/Longitud_de_arco). Es la distancia a lo largo de la curva. $ s\\in\\left[0,l\\right] $
** $ k $ **N\u00famero de ondulaciones. $ k > 0 $
** $ \\alpha $ **\u00c1ngulo de serpenteo. $ \\alpha\\in\\left[0,121\\right] $
\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## \u00c1ngulo de serpenteo $ \\alpha $\n", "\n", "El \u00e1ngulo de serpenteo $ \\alpha $ es la pendiente de la curva en el punto $ s = 0 $ y determina **la forma que tendr\u00e1 la curva**. \n", "\n", "En la **figura 1** se muestra la forma para \u00e1ngulos de serpenteo de 0, 30, 60 y 90. Para $ \\alpha=0 $ es una recta situada sobre el eje x, de longitud $ l $. Al aumentar $ \\alpha $ la curva se eleva, ganando en altura pero reduci\u00e9ndose en anchura.\n", "\n", "En la **figura 2** se muestra la curva serpentinoide para $ \\alpha=121 $ grados, que es su valor m\u00e1ximo. A partir de ah\u00ed se producen colisiones entre los puntos de la curva. \n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "
**Figura 1**: Forma de la curva serpentinoide para diferentes valores del \u00e1ngulo de serpenteo ($ \\alpha $)
\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "
**Figura 2**: Curva serpentinoide de \u00e1ngulo de serpenteo ($ \\alpha $) m\u00e1ximo. Se muestran dos ondulaciones ( $ k=2 $)
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Animaci\u00f3n de la variaci\u00f3n de la forma con $ \\alpha $\n", "\n", "En esta animaci\u00f3n se puede ver c\u00f3mo var\u00eda la forma de una curva serpentinoide de longitud fija $ l $ y 2 ondulaciones ($ k = 2 $), para diferentes \u00e1ngulos de serpenteo. Cuando $ \\alpha $ es cero, la curva serpentinoide es una recta de longitud $ l $ apoyada sobre el eje x\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## N\u00famero de ondulaciones ($ k $)\n", "\n", "El **par\u00e1metro $ k $ ** determina el n\u00famero de ondulaciones de la curva serpentinoide.\n", "\n", "En la **figura 3** se han representado tres curvas con el mismo valor del \u00e1ngulo de serpenteo ($ \\alpha=70 $) y misma longitud $ l $, pero con diferentes valores de $ k $. Al aumentar k, aumenta el n\u00famero de ondulaciones, disminuye la altura pero **la anchura permanece constante **\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "
**Figura 3**: Forma una curva serpentinoide de longitud $ l $ para diferentes valores del par\u00e1metro $ k $, manteniendo fijo el valor de $ \\alpha $ a 70
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Animaci\u00f3n de la variaci\u00f3n de la forma con $ k $\n", "\n", "En la siguiente animaci\u00f3n se muestra c\u00f3mo var\u00eda la forma de una curva serpentinoide continua de longitud fija $ l=1 $ y \u00e1ngulo de serpenteo $ \\alpha = 70 $ con el par\u00e1metro $ k $. Se puede apreciar una propiedad muy importante: **La anchura en el eje x NO var\u00eda con k**\n", "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Longitud ($l$)\n", "\n", "La longitud l de la curva determina su escala. En la **figura 4** se muestra una curva serpentinoide con valores fijos del \u00e1ngulo de serpenteo ($ \\alpha $) y de las ondulaciones ($ k $) pero con distintas longitudes. La forma es la misma, pero escalada. \n", "Normalmente $ l $ es una constante, ya que los robots tienen longitud fija. Salvo que se trate de un robot modulare reconfigurable en cuyo caso puede aumentar por la adici\u00f3n de n\u00faevos m\u00f3dulos o disminuir por su separaci\u00f3n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "
**Figura 4**: Forma de la curva serpentinoide con valores fijos del \u00e1ngulo de serpenteo $ \\alpha=70 $ y $ k=1 $ para diferentes valores de la longitud $ l $. La longitud determina la escala.
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Puntos de inter\u00e9s\n", "\n", "\n", "\n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "
**Figura 5**: Situaci\u00f3n de los puntos de m\u00e1xima y nula curvatura en una serpentinoide de $k=2$ y $\\alpha=70$
\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Puntos de m\u00ednima curvatura (curvatura 0)\n", "\n", "Para una curva serpentinoide con \u00e1ngulo de serpenteo mayor de cero ($ \\alpha > 0 $), los puntos donde la curvatura es 0 son:\n", "\n", "\n", "\n", " \n", "\n", "\n", "\n", "
$ s = n\\frac{l}{2k}$ con $ n = 0, 1, 2 ... $ Puntos curvatura m\u00ednima (**ec. 2**)
\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Demostraci\u00f3n\n", "\n", "Los puntos de la curva serpentinoide donde la curvatura es cero los calculamos igualando la **ec. 1** a cero\n", "\n", "$ K\\left(s\\right)=0 \\Longrightarrow\\ \\frac{2\\pi k}{l}\\alpha\\sin\\left(\\frac{2\\pi k}{l}s\\right) = 0$\n", "\n", "Puesto que por definici\u00f3n, la longitud de la curva y el n\u00famero de ondulaciones son distintos de cero ($ k > 0 $ y $ l > 0$) se tiene que:\n", "\n", "$ \\alpha\\sin\\left(\\frac{2\\pi k}{l}s\\right) = 0 $\n", "\n", "Y esta igualdad se cumple en dos casos:\n", "\n", "* **Caso 1: $ \\alpha = 0 $**. Es el caso en el que la curva serpentinoide es una l\u00ednea recta de longitud $ l $ apoyada sobre el eje x. Todos sus puntos tienen curvatura 0\n", "\n", "* **Caso 2: $ \\sin\\left(\\frac{2\\pi k}{l}s\\right) = 0 $ ** $ \\Longrightarrow\\ \\frac{2\\pi k}{l}s = \\pi n$ con $ n = 0, 1, 2 ... \\Longrightarrow\\ s = n\\frac{l}{2k}$ con $ n = 0, 1, 2 ... $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Puntos de m\u00e1xima curvatura\n", "\n", "Para una curva serpentinoide con \u00e1ngulo de serpenteo mayor de cero ($\\alpha > 0 $), los puntos de curvatura m\u00e1xima son:\n", "\n", "\n", "\n", " \n", "\n", "\n", "\n", "
$ s = (2n + 1) \\frac{l}{4k} $, con $ n = 0,1,2,3... $ Puntos curvatura m\u00e1xima (**ec. 3**)
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Demostraci\u00f3n\n", "\n", "Los puntos de m\u00e1xima curvatura los obtendremos calculando los m\u00e1ximos y m\u00ednimos de la ecuaci\u00f3n de curvatura de la serpentinoide (ec. 1). Para ello igualamos a cero el valor absoluto de la derivada de la ecuaci\u00f3n de curvatura:\n", "\n", "$ | \\frac{dK(s)}{ds} | = 0 \\Longrightarrow\\ | \\frac{d \\left( \\frac{2\\pi k}{l}\\alpha\\sin\\left(\\frac{2\\pi k}{l}s\\right) \\right) }{ds} | = 0 \\Longrightarrow\\ \\left( \\frac{2\\pi k}{l} \\right)^2 \\alpha \\cos\\left( \\frac{2\\pi k}{l}s \\right) = 0 \\Longrightarrow\\ \\alpha \\cos\\left( \\frac{2\\pi k}{l}s \\right) = 0 $\n", "\n", "El caso $ \\alpha = 0 $ es una l\u00ednea recta, as\u00ed que, para el resto de curvas donde $ \\alpha > 0 $ se cumple que:\n", "\n", "$ \\cos\\left( \\frac{2\\pi k}{l}s \\right) = 0 $\n", "\n", "y eso implica que:\n", "\n", "$ \\frac{2\\pi k}{l}s = \\frac{\\pi}{2} + n\\pi $ \n", "\n", "Despejando s obtenemos la condici\u00f3n:\n", "\n", "$ s = (2n + 1) \\frac{l}{4k} $, con $ n = 0,1,2,3... $\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Pendiente de la curva $ \\alpha_s $\n", "\n", "\n", "\n", "\n", " \n", "\n", "\n", "\n", "\n", "\n", "\n", "
**Figura 6**: $ \\alpha_s $ en diferentes puntos de la curva serpentinoide
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Denotamos **la pendiente del vector tangente** a la curva por el punto s como \\alpha_s\n", "\n", "Por definici\u00f3n, la curvatura se define como el ritmo de cambio del vector unidad tangente a la curva, es decir:\n", "\n", "$ K(s)=\\frac{d\\alpha_{s}}{ds} $\n", "\n", "A partir de esa ecuaci\u00f3n calculamos la pendiente de la curva mediante integraci\u00f3n:\n", "\n", "$ \\alpha_{s}=\\alpha+\\int_{0}^{s}K(s)ds $\n", "\n", "llegando a la expresi\u00f3n: \n", "\n", "\n", "\n", " \n", "\n", "\n", "\n", "
$ \\alpha_{s}=\\alpha\\cos\\left(\\frac{2\\pi k}{l}s\\right) $ Pendiente de la curva serpentinoide **ec. 4**
\n", "\n", "En la **figura 6** se muestra el valor de $ \\alpha_s $ sobre una curva serpentinoide. En los puntos de m\u00e1xima curvatura $ \\alpha_s = 0 $ y en los de curvatura nula $ |\\alpha_s|=\\alpha $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Formulaci\u00f3n de la curva serpentinoide en coordenadas cartesianas\n", "\n", "Las coordenadas cartesianas $ (x,y) $ de un punto $ s $ situado en la curva serpentinoide est\u00e1n dadas por las ecuaciones 5 y 6:\n", "\n", "\n", "\n", " \n", "\n", "\n", "\n", "\n", "\n", " \n", "\n", "\n", "\n", "\n", "
$ x\\left(s\\right)=\\int_{0}^{s}\\cos\\left(\\alpha\\cos\\left(\\frac{2\\pi k}{l}s\\right)\\right)ds $ Coordenada x de la curva serpentinoide **ec. 5**
$ y\\left(s\\right)=\\int_{0}^{s}\\sin\\left(\\alpha\\cos\\left(\\frac{2\\pi k}{l}s\\right)\\right)ds $ Coordenada y de la curva serpentinoide **ec. 6**
\n", "\n", "Estas integrales **no tienen soluci\u00f3n anal\u00edtica**, por lo que tienen que resolverse num\u00e9ricamente\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Demostraci\u00f3n\n", "\n", "Para su deducci\u00f3n hay que fijarse en el dibujo de la figura 7. \n", "\n", "\n", "\n", " \n", "\n", "\n", "\n", "\n", "
**Figura 7**: C\u00e1lculo de las coordenadas cartesianas de un punto $ s $ de la curva serpentinoide
" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "En ella aparece una curva serpeninoide y un punto gen\u00e9rico situado a una distancia $ s $. Sus coordenadas son $ x(s), y(s) $. La pendiente a la curva por el punto $ s $ es $ \\alpha_s $. \"Ampliando\" el punto $ s $, vemos que est\u00e1 aproximado por un segmento de longitud $ ds $, que forma un \u00e1ngulo $ \\alpha_s $ con el eje x. Las proyecciones de este segmento son:\n", "\n", " $$ dx=ds\\ \\cos\\alpha_{s} $$\n", " $$ dy=ds\\ \\sin\\alpha_{s} $$ \n", "\n", "La abscisa del punto $ s $ se obtiene por tanto mediante la integraci\u00f3n de los $ dx $ :\n", "\n", "$ x\\left(s\\right)=\\int_{0}^{s}dx= $\n", "\n", "\n", "Aplicando la igualdad anterior para sustituir $ dx $ por $ ds $\n", "\n", "$ = \\int_{0}^{s}\\cos\\alpha_{s}ds = $\n", "\n", "y finalmente se aplica la **ec. 4**:\n", "\n", "$ =\\int_{0}^{s}\\cos\\left(\\alpha\\cos\\left(\\frac{2\\pi k}{l}s\\right)\\right)ds $\n", "\n", "Llegamos al resultado final (ec. 5)\n", "\n", "El razonamiento para obtener $ y(s) $ (ec. 6) es similar. " ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Formulaci\u00f3n de la curva serpentinoide en python" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import numpy as np\n", "import scipy.integrate as spi\n", "import matplotlib.pylab as plt\n", "%matplotlib inline" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 8 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Estas son las funciones que al ser integradas nos dar\u00e1n las funciones param\u00e9tricas con las coordenadas x,y:" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#-- Alpha parameters should be given in degrees\n", "def calphas(s, alpha, l = 1, k = 1, phi = 0):\n", " \n", " #-- Convert alpha and phi into radians\n", " alpha_rad = alpha * np.pi / 180.\n", " phi_rad = phi * np.pi / 180.\n", " \n", " return np.cos(alpha_rad * np.cos(2 * np.pi * k * s / l + phi_rad))\n", "\n", "def salphas(s, alpha, l = 1, k = 1, phi = 0):\n", " \n", " #-- Convert alpha into radians\n", " alpha_rad = alpha * np.pi / 180.\n", " phi_rad = phi * np.pi / 180.\n", " \n", " return np.sin(alpha_rad * np.cos(2 * np.pi * k * s / l + phi_rad))\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 9 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Estas son las funciones parametricas que nos dan las coordenadas x,y a partir de s" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#-- Scalar functions. Variable s is a real number\n", "def serp_x_scalar (s, alpha, l = 1, k = 1, phi = 0):\n", " return spi.quad(calphas, 0, s, args =(alpha, l, k, phi) )\n", "\n", "def serp_y_scalar (s, alpha, l = 1, k = 1, phi = 0):\n", " return spi.quad(salphas, 0, s, args =(alpha, l, k, phi) )" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 10 }, { "cell_type": "markdown", "metadata": {}, "source": [ "Y por \u00faltimo esta es la funci\u00f3n que tiene las coordenadas de una curva serpentinoide. La variable s es un array con todos los valores" ] }, { "cell_type": "code", "collapsed": false, "input": [ "def serp (s, alpha = 45, l = 1, k = 1, phi = 0):\n", " \"\"\"Serpenoid curve\n", " Inputs:\n", " - alpha : winding angle (in degrees)\n", " - l : Curve length\n", " - k : Number of undulations\n", " S is the variable (it is an array). The x,y arrays are returned\n", " \"\"\"\n", " \n", " #-- Results\n", " x = []\n", " y = []\n", " \n", " for i, vs in enumerate(s):\n", " x.append(serp_x_scalar(vs, alpha, l, k, phi)[0])\n", " y.append(serp_y_scalar(vs, alpha, l, k, phi)[0])\n", " return x,y\n" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 11 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Probando la curva serpentinoide" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "Ejemplo de una curva serpentinoide con \u00e1ngulo de serpenteo de 60 grados, longitud 1m y una \u00fanica ondulaci\u00f3n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#-- Obtener los puntos a lo largo de la curva\n", "s = np.linspace(0, 1, 100)\n", "\n", "#-- Obtener las coordenadas x,y\n", "x, y = serp(s, alpha = 60, l = 1, k = 1)\n", "\n", "#-- Dibujar la curva\n", "plt.plot(x, y, linewidth = 3)\n", "plt.grid()\n", "plt.axis('equal')\n", "plt.show()" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 12 }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Curva serpentinoide param\u00e9trica: variando parametros\n", "Modifica interactivamente los par\u00e1metros de la curva. Pulsa en el bot\u00f3n \"DRAW\" para ver la curva\n" ] }, { "cell_type": "code", "collapsed": false, "input": [ "from IPython.html import widgets # Widget definitions\n", "from IPython.display import display # Used to display widgets in the notebook" ], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 15 }, { "cell_type": "code", "collapsed": false, "input": [ "widget_alpha = widgets.FloatSliderWidget()\n", "widget_alpha.min = 0.0\n", "widget_alpha.max = 121\n", "widget_alpha.description = \"Alpha\"\n", "widget_alpha.value = 60\n", "\n", "widget_k = widgets.FloatSliderWidget()\n", "widget_k.min = 0.1\n", "widget_k.max = 20\n", "widget_k.description = \"k\"\n", "widget_k.value = 1\n", "\n", "\n", "def draw_serp():\n", " #-- Obtener los puntos a lo largo de la curva\n", " s = np.linspace(0, 1, 100)\n", "\n", " #-- Obtener las coordenadas x,y\n", " x, y = serp(s, alpha = widget_alpha.value, l = 1, k = widget_k.value)\n", "\n", " #-- Dibujar la curva\n", " plt.plot(x, y, linewidth = 3)\n", " plt.grid()\n", " plt.axis('equal')\n", " plt.show()\n", " \n", " \n", "button_draw = widgets.ButtonWidget(description=\"Draw!\")\n", "\n", "\n", "def on_button_draw_clicked(b):\n", " draw_serp()\n", "\n", "button_draw.on_click(on_button_draw_clicked) \n", "\n", "display(widget_alpha)\n", "display(widget_k)\n", "display(button_draw)\n", "draw_serp()\n" ], "language": "python", "metadata": {}, "outputs": [ { "metadata": {}, "output_type": "display_data", "png": 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"text": [ "" ] } ], "prompt_number": 31 }, { "cell_type": "code", "collapsed": false, "input": [], "language": "python", "metadata": {}, "outputs": [], "prompt_number": 13 }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Bibliograf\u00eda\n", "\n", "* S. Hirose. Biologically Inspired Robots (Snake-like Locomotor and Manipulator). Oxford Science Press, 1993.\n", "* Juan Gonzalez-Gomez, \"Rob\u00f3tica Modular y Locomoci\u00f3n: Aplicaci\u00f3n a Robots \u00c1podos\". Tesis doctoral. Universidad Aut\u00f3noma de Madrid. Noviembre-2008 [M\u00e1s informaci\u00f3n](http://www.iearobotics.com/wiki/index.php?title=Juan_Gonzalez:Tesis) " ] } ], "metadata": {} } ] }