Added the roll pitch calc notebook.

utils/roll_pitch_from_accel.ipynb

@@ -0,0 +1,194 @@
+{
+ "metadata": {
+  "name": "roll_pitch_from_accel"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import sympy as sym"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 1
+    },
+    {
+     "cell_type": "code",
+     "collapsed": true,
+     "input": [
+      "import sympy.physics.mechanics as me"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 2
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The bicycle rear frame is oriented in the Newtonian reference frame by the yaw, roll and pitch angles which are Euler 312 body fixed rotation coordinates."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": true,
+     "input": [
+      "yaw, roll, pitch = sym.symbols('yaw roll pitch')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 3
+    },
+    {
+     "cell_type": "code",
+     "collapsed": true,
+     "input": [
+      "N = me.ReferenceFrame('N')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 4
+    },
+    {
+     "cell_type": "code",
+     "collapsed": true,
+     "input": [
+      "rearFrame = N.orientnew('C', 'Body', [yaw, roll, pitch], '312')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 5
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The VectorNav accelerometer reads out the acceleration in the three rear frame body fixed coordinates and give the measure numbers of the gravity vector.\n",
+      "$$\\bar{g} = c_x\\hat{c}_x + c_y\\hat{c}_y + c_z\\hat{c}_z$$"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": true,
+     "input": [
+      "cx, cy, cz = sym.symbols('cx cy cz')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 6
+    },
+    {
+     "cell_type": "code",
+     "collapsed": true,
+     "input": [
+      "gravity = cx * rearFrame.x + cy * rearFrame.y + cz * rearFrame.z"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 7
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The gravity vector must equal $|\\bar{g}|\\hat{n}_z$ because it is always in the direction of the gravitational field. This means the measure numbers of $|\\bar{g}|\\hat{n}_z$ in the rear frame must equal the accelerometer readings."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": true,
+     "input": [
+      "eqx = me.dot(gravity.magnitude() * N.z, rearFrame.x) - cx"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 8
+    },
+    {
+     "cell_type": "code",
+     "collapsed": true,
+     "input": [
+      "eqy = me.dot(gravity.magnitude() * N.z, rearFrame.y) - cy"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 9
+    },
+    {
+     "cell_type": "code",
+     "collapsed": true,
+     "input": [
+      "eqz = me.dot(gravity.magnitude() * N.z, rearFrame.z) - cz"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 10
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "sym.solve(eqy, roll)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "pyout",
+       "prompt_number": 11,
+       "text": [
+        "[asin(cy/sqrt(cx**2 + cy**2 + cz**2))]"
+       ]
+      }
+     ],
+     "prompt_number": 11
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "\n",
+      "sym.solve(eqx, pitch)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "output_type": "pyout",
+       "prompt_number": 12,
+       "text": [
+        "[-asin(cx/(sqrt(cx**2 + cy**2 + cz**2)*cos(roll)))]"
+       ]
+      }
+     ],
+     "prompt_number": 12
+    },
+    {
+     "cell_type": "code",
+     "collapsed": true,
+     "input": [],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 12
+    }
+   ],
+   "metadata": {}
+  }
+ ]
+}