Kalman-Filter-Bike-Lean-Angle.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Kalman Filter for Bike Lean Angle Estimation\n",
    "\n",
    "You've seen this probably on MotoGP, where the camera mounted on the bike is exactly horizontal, even when the bike leans. This is not as easy at it seems."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 725,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from IPython.display import YouTubeVideo"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 726,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "        <iframe\n",
       "            width=\"720\"\n",
       "            height=\"390\"\n",
       "            src=\"https://www.youtube.com/embed/-p2ndhw-kfQ\"\n",
       "            frameborder=\"0\"\n",
       "            allowfullscreen\n",
       "        ></iframe>\n",
       "        "
      ],
      "text/plain": [
       "<IPython.lib.display.YouTubeVideo at 0x11d80b3d0>"
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     },
     "execution_count": 726,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "YouTubeVideo('-p2ndhw-kfQ', width=720, height=390)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first try would be to use the gravitational force and just point the camera to the ground.\n",
    "\n",
    "Doesn't work on bikes, because they are leaning in exactly this angle, which is needed to compensate the gravitational force with the centrifugal force.\n",
    "\n",
    "![Bike Lean](https://upload.wikimedia.org/wikipedia/en/8/87/BikeLeanForces3.PNG)\n",
    "\n",
    "One has to use two different sensors:\n",
    "\n",
    "1. a rotationrate sensor for lean angle\n",
    "2. a acceleration sensor for gravitional force\n",
    "\n",
    "Both sensors have to be fused to estimate the lean angle. This is done with a Kalman Filter. We are using [Sympy](http://www.sympy.org/de/) do develop this filter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 727,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from sympy import Symbol, symbols, Matrix, sin, cos, acos, pi\n",
    "from sympy.abc import phi, g, a\n",
    "from sympy import init_printing\n",
    "init_printing(use_latex=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The state vector to describe the state of the bike consists of two variables:\n",
    "\n",
    "$$\\vec x= \\left[ \\matrix{ \\phi \\\\ \\dot \\phi} \\right]$$\n",
    "\n",
    "which is the lean angle $\\phi$ (in radian) and the lean angle rate $\\dot \\phi$ (in radian per second)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 728,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "phis = symbols('phi')\n",
    "dphis = Symbol('\\dot \\phi')\n",
    "Ts = symbols('T')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 729,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\phi\\\\\\dot \\phi\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡    φ    ⎤\n",
       "⎢         ⎥\n",
       "⎣\\dot \\phi⎦"
      ]
     },
     "execution_count": 729,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "state = Matrix([phis, dphis])\n",
    "state"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kalman Filter Prediction Step: System Dynamics\n",
    "\n",
    "But the state is driven by the lean angle rate $\\dot \\phi$ (in radian per second). So the Kalman Filter Equation for the Prediction Step is:\n",
    "\n",
    "$$\\vec x_{k+1} = A \\cdot \\vec x_{k}$$\n",
    "\n",
    "The dynamic matrix $A$ is simply:\n",
    "\n",
    "$$A = \\left[ \\matrix{ 1 & \\Delta T \\\\ 0 & 1 } \\right]$$\n",
    "\n",
    "with $\\Delta T$ as the time between two filtersteps (the sample time of the discrete Kalman Filter)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 730,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "Q = np.diag([0.1, 1.0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Kalman Filter Update Step: Sensors\n",
    "\n",
    "We have two kind of sensors, which are usually available within a 6 Degrees of Freedom Inertial Measurement Unit (6DoF IMU):\n",
    "\n",
    "1. rotationrate sensor\n",
    "2. acceleration sensor\n",
    "\n",
    "The rotationrate sensor is measuring $\\dot \\phi$ directly, the acceleration sensor is measuring the gravitation, so not directly the lean angle $\\phi$. There is a mathematical link between the lean angle and the vertical acceleration $a$ measured by the acceleration sensor. It is the *cosine*. If the bike is upright, the acceleration $a$ is exactly $1g$, if it is $\\phi=90^\\circ$, it is $0g$.\n",
    "\n",
    "$$a = g \\cdot \\cos(90^\\circ - \\phi)$$\n",
    "\n",
    "so the measurement function $z(x)$ is\n",
    "\n",
    "$$\\phi = 90^\\circ - \\arccos \\left(\\frac{a}{g}\\right)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 731,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x11d99e310>"
      ]
     },
     "execution_count": 731,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
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YvBiOPRb+/e/qE0dtKHmIiJSpG24AMzjzzPpf9/L1v0oREUna22/D1VfDO+/AcgUoJqjk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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11d8ae8d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "a=np.linspace(-9.81, 9.81, 1000)\n",
    "plt.plot(a, 90-np.arccos(a/9.81)*180.0/np.pi)\n",
    "plt.xlabel('vertical Acceleration in m/s2')\n",
    "plt.ylabel('lean angle in Degree')\n",
    "plt.title('Lean angle as function of vertical acceleration (static!!!)')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Load some Measurements"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 732,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 733,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import pandas as pd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 734,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "data = pd.read_csv('2016-09-12-Leaning2.csv', index_col='loggingTime', parse_dates=True)\n",
    "data = data[['accelerometerAccelerationX','gyroRotationY']].dropna()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 735,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "dt = 1.0/50.0 # Hz"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 736,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "t = data.index\n",
    "a_meas = data.accelerometerAccelerationX.values*-9.81 # in m/s2\n",
    "a_meas[a_meas>9.81] = 9.81\n",
    "a_meas[a_meas<-9.81] = -9.81\n",
    "dphi_meas = data.gyroRotationY.values # in rad/s"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 737,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x11d80b310>]"
      ]
     },
     "execution_count": 737,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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hyaw6vIqk0CSGRw/n14xfya/I58pxV/LoykfZm7+X5ZcvJ9g3mPzyfJbtWYbl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\naCiRAIIgCBpKJIAgCIKG8gV81AWIcGceDgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x11da07fd0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(np.pi/2-np.arccos(a_meas/9.81))\n",
    "plt.plot(dphi_meas)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Kalman Filter\n",
    "\n",
    "![Kalman Filter Step](Kalman-Filter-Step.png)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 738,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x = np.matrix([[0.0],\n",
    "               [0.0]]) # Initial State\n",
    "A = np.matrix([[1.0, dt], [0.0, 1.0]])\n",
    "H = np.diag([1.0, 1.0])\n",
    "P = np.diag([100.0, 1.0])\n",
    "I = np.diag([1.0, 1.0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 741,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "x0=[]\n",
    "x1=[]\n",
    "P0=[]\n",
    "dstate=[]\n",
    "for filterstep in range(len(data)):\n",
    " \n",
    "    # Time Update (Prediction)\n",
    "    # ========================\n",
    "    # Project the state ahead\n",
    "    x = A*x\n",
    "    \n",
    "    # Project the error covariance ahead\n",
    "    P = A*P*A.T + Q\n",
    "    \n",
    "    \n",
    "    # Measurement Update (Correction)\n",
    "    # ===============================\n",
    "    # Compute the Kalman Gain\n",
    "    \n",
    "    # Measurement Noise is adaptive:\n",
    "    # Assuming a pretty correct angle measurement, while the\n",
    "    # bike is upright (vertical acceleration is nearly 1g),\n",
    "    # and a pretty bad estimation while the bike is leaning.\n",
    "    # So we make the R value adaptive to the lean angle\n",
    "    # with low values while upright and high values for high\n",
    "    # leaning angles.\n",
    "    adaptivephi = np.abs(np.multiply(1000.0,float(x[0]))+0.001)\n",
    "    R = np.diag([adaptivephi, 0.001])\n",
    "\n",
    "\n",
    "    S = H*P*H.T + R\n",
    "    K = (P*H.T) * np.linalg.pinv(S)\n",
    "\n",
    "    \n",
    "    # Update the estimate via z\n",
    "    Z = np.matrix([[np.pi/2-np.arccos(a_meas[filterstep]/9.81)],\n",
    "                   [dphi_meas[filterstep]]])\n",
    "\n",
    "    y = Z - (H*x)                            # Innovation or Residual\n",
    "    x = x + (K*y)\n",
    "    \n",
    "    # Update the error covariance\n",
    "    P = (I - (K*H))*P\n",
    "\n",
    "\n",
    "\n",
    "    # Save states for Plotting\n",
    "    x0.append(float(x[0]))\n",
    "    x1.append(float(x[1]))\n",
    "    P0.append(float(P[0,0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 742,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x11e01cc50>"
      ]
     },
     "execution_count": 742,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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TrRoccQR89pnvSEx5/fij/tvv0yd1ybwsltBD0Lkz/Pvf2bOs7muv6SJDJlxn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\n44wapb17EfjwQ12aYO1a6N8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      "text/plain": [
       "<matplotlib.figure.Figure at 0x11decc690>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(t,np.multiply(x0,180.0/np.pi), label=r'$\\phi$')\n",
    "#plt.plot(t,x1, label=r'$\\dot \\phi$')\n",
    "#plt.plot(t, dphi_meas,label=r'$\\dot \\phi$ (ref)', alpha=0.5)\n",
    "plt.title('Estimated Lean Angle')\n",
    "plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.12"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}