From aea302eee751a75a6f3ec5fe8e1128b905d55be3 Mon Sep 17 00:00:00 2001 From: Paul Balzer Date: Mon, 12 Sep 2016 12:23:49 +0200 Subject: [PATCH] Gimble Stabilizer Kalman Filter Lean Angle Estimation for motorbikes --- ...an-Filter-Bike-Lean-Angle-checkpoint.ipynb | 487 ++ 2016-08-09-Motorbike.csv | 5381 +++++++++++++++++ 2016-09-12-Leaning.csv | 886 +++ 2016-09-12-Leaning2.csv | 619 ++ Kalman-Filter-Bike-Lean-Angle.ipynb | 487 ++ Kalman-Filter-Bike-Lean-Angle.py | 224 + 6 files changed, 8084 insertions(+) create mode 100644 .ipynb_checkpoints/Kalman-Filter-Bike-Lean-Angle-checkpoint.ipynb create mode 100644 2016-08-09-Motorbike.csv create mode 100644 2016-09-12-Leaning.csv create mode 100644 2016-09-12-Leaning2.csv create mode 100644 Kalman-Filter-Bike-Lean-Angle.ipynb create mode 100644 Kalman-Filter-Bike-Lean-Angle.py diff --git a/.ipynb_checkpoints/Kalman-Filter-Bike-Lean-Angle-checkpoint.ipynb b/.ipynb_checkpoints/Kalman-Filter-Bike-Lean-Angle-checkpoint.ipynb new file mode 100644 index 0000000..b570608 --- /dev/null +++ b/.ipynb_checkpoints/Kalman-Filter-Bike-Lean-Angle-checkpoint.ipynb @@ -0,0 +1,487 @@ +{ + "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", + " \n", + " " + ], + "text/plain": [ + "" + ] + }, + "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": [ + "" + ] + }, + "execution_count": 731, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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YvBiOPRb+/e/qE0dtKHmIiJSpG24AMzjzzPpf9/L1v0oREUna22/D1VfDO+/AcgUoJqjk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\ntWCRidTChAmhhDF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625jZ0Lqup5r1P2dmbbK8dk68hcKHZvaSmW1QqDikcJQ8pKjEIfMv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\n", + "text/plain": [ + "" + ] + }, + "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": [ + "[]" + ] + }, + "execution_count": 737, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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hyaw6vIqk0CSGRw/n14xfya/I58pxV/LoykfZm7+X5ZcvJ9g3mPzyfJbtWYbl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\nQ10CxF3s3w/z54OFpi6gPXsa0i5vv11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+ "text/plain": [ + "" + ] + }, + "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": [ + "" + ] + }, + "execution_count": 742, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "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 +} diff --git a/2016-08-09-Motorbike.csv b/2016-08-09-Motorbike.csv new file mode 100644 index 0000000..00c6985 --- /dev/null +++ b/2016-08-09-Motorbike.csv @@ -0,0 +1,5381 @@ +loggingTime,loggingSample,identifierForVendor,locationTimestamp_since1970,locationLatitude,locationLongitude,locationAltitude,locationSpeed,locationCourse,locationVerticalAccuracy,locationHorizontalAccuracy,locationFloor,locationHeadingTimestamp_since1970,locationHeadingX,locationHeadingY,locationHeadingZ,locationTrueHeading,locationMagneticHeading,locationHeadingAccuracy,accelerometerTimestamp_sinceReboot,accelerometerAccelerationX,accelerometerAccelerationY,accelerometerAccelerationZ,gyroTimestamp_sinceReboot,gyroRotationX,gyroRotationY,gyroRotationZ,motionTimestamp_sinceReboot,motionYaw,motionRoll,motionPitch,motionRotationRateX,motionRotationRateY,motionRotationRateZ,motionUserAccelerationX,motionUserAccelerationY,motionUserAccelerationZ,motionAttitudeReferenceFrame,motionQuaternionX,motionQuaternionY,motionQuaternionZ,motionQuaternionW,motionGravityX,motionGravityY,motionGravityZ,motionMagneticFieldX,motionMagneticFieldY,motionMagneticFieldZ,motionMagneticFieldCalibrationAccuracy,activityTimestamp_sinceReboot,activity,activityActivityConfidence,activityActivityStartDate,pedometerStartDate,pedometerNumberofSteps,pedometerDistance,pedometerFloorAscended,pedometerFloorDescended,pedometerEndDate,IP_en0,IP_pdp_ip0,deviceOrientation,state 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12:11:26.170,617,6B5B5B79-A063-42A5-9845-86B7C35C751C,,,,,,,,,,1473675086.151851,-16.18397521972656,-48.79264068603516,-7.369766235351562,79.28156280517578,75.6607666015625,10,262934.7068159167,0.0317840576171875,-0.991363525390625,-0.1160430908203125,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,0.0.0.0,10.56.9.191,1,0 +2016-09-12 12:11:26.182,618,6B5B5B79-A063-42A5-9845-86B7C35C751C,,,,,,,,,,,,,,,,,262934.7264079167,0.05517578125,-0.996795654296875,-0.03253173828125,262934.720969375,0.06221996517973926,-0.02360865674744344,0.08069777577103604,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,0.0.0.0,10.56.9.191,1,0 diff --git a/Kalman-Filter-Bike-Lean-Angle.ipynb b/Kalman-Filter-Bike-Lean-Angle.ipynb new file mode 100644 index 0000000..b570608 --- /dev/null +++ b/Kalman-Filter-Bike-Lean-Angle.ipynb @@ -0,0 +1,487 @@ +{ + "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", + " \n", + " " + ], + "text/plain": [ + "" + ] + }, + "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": [ + "" + ] + }, + "execution_count": 731, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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625jZ0Lqup5r1P2dmbbK8dk68hcKHZvaSmW1QqDikcJQ8pKjEIfMv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\n", + "text/plain": [ + "" + ] + }, + "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": [ + "[]" + ] + }, + "execution_count": 737, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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hyaw6vIqk0CSGRw/n14xfya/I58pxV/LoykfZm7+X5ZcvJ9g3mPzyfJbtWYbl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\nQ10CxF3s3w/z54OFpi6gPXsa0i5vv11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+ "text/plain": [ + "" + ] + }, + "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": [ + "" + ] + }, + "execution_count": 742, + "metadata": {}, + "output_type": "execute_result" + }, + { + "data": { + "image/png": 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+ "text/plain": [ + "" + ] + }, + "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 +} diff --git a/Kalman-Filter-Bike-Lean-Angle.py b/Kalman-Filter-Bike-Lean-Angle.py new file mode 100644 index 0000000..0419de4 --- /dev/null +++ b/Kalman-Filter-Bike-Lean-Angle.py @@ -0,0 +1,224 @@ + +# coding: utf-8 + +# # Kalman Filter for Bike Lean Angle Estimation +# +# 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. + +# In[725]: + +from IPython.display import YouTubeVideo + + +# In[726]: + +YouTubeVideo('-p2ndhw-kfQ', width=720, height=390) + + +# The first try would be to use the gravitational force and just point the camera to the ground. +# +# 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. +# +# ![Bike Lean](https://upload.wikimedia.org/wikipedia/en/8/87/BikeLeanForces3.PNG) +# +# One has to use two different sensors: +# +# 1. a rotationrate sensor for lean angle +# 2. a acceleration sensor for gravitional force +# +# 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. + +# In[727]: + +import numpy as np +from sympy import Symbol, symbols, Matrix, sin, cos, acos, pi +from sympy.abc import phi, g, a +from sympy import init_printing +init_printing(use_latex=True) + + +# The state vector to describe the state of the bike consists of two variables: +# +# $$\vec x= \left[ \matrix{ \phi \\ \dot \phi} \right]$$ +# +# which is the lean angle $\phi$ (in radian) and the lean angle rate $\dot \phi$ (in radian per second). + +# In[728]: + +phis = symbols('phi') +dphis = Symbol('\dot \phi') +Ts = symbols('T') + + +# In[729]: + +state = Matrix([phis, dphis]) +state + + +# ## Kalman Filter Prediction Step: System Dynamics +# +# 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: +# +# $$\vec x_{k+1} = A \cdot \vec x_{k}$$ +# +# The dynamic matrix $A$ is simply: +# +# $$A = \left[ \matrix{ 1 & \Delta T \\ 0 & 1 } \right]$$ +# +# with $\Delta T$ as the time between two filtersteps (the sample time of the discrete Kalman Filter). + +# In[730]: + +Q = np.diag([0.1, 1.0]) + + +# ## Kalman Filter Update Step: Sensors +# +# We have two kind of sensors, which are usually available within a 6 Degrees of Freedom Inertial Measurement Unit (6DoF IMU): +# +# 1. rotationrate sensor +# 2. acceleration sensor +# +# 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 $0g$. +# +# $$a = g \cdot \cos(90^\circ - \phi)$$ +# +# so the measurement function $z(x)$ is +# +# $$\phi = 90^\circ - \arccos \left(\frac{a}{g}\right)$$ + +# In[731]: + +a=np.linspace(-9.81, 9.81, 1000) +plt.plot(a, 90-np.arccos(a/9.81)*180.0/np.pi) +plt.xlabel('vertical Acceleration in m/s2') +plt.ylabel('lean angle in Degree') +plt.title('Lean angle as function of vertical acceleration (static!!!)') + + +# ## Load some Measurements + +# In[732]: + +get_ipython().magic(u'matplotlib inline') +import matplotlib.pyplot as plt + + +# In[733]: + +import pandas as pd + + +# In[734]: + +data = pd.read_csv('2016-09-12-Leaning2.csv', index_col='loggingTime', parse_dates=True) +data = data[['accelerometerAccelerationX','gyroRotationY']].dropna() + + +# In[735]: + +dt = 1.0/50.0 # Hz + + +# In[736]: + +t = data.index +a_meas = data.accelerometerAccelerationX.values*-9.81 # in m/s2 +a_meas[a_meas>9.81] = 9.81 +a_meas[a_meas<-9.81] = -9.81 +dphi_meas = data.gyroRotationY.values # in rad/s + + +# In[737]: + +plt.plot(np.pi/2-np.arccos(a_meas/9.81)) +plt.plot(dphi_meas) + + +# # Kalman Filter +# +# ![Kalman Filter Step](Kalman-Filter-Step.png) + +# In[738]: + +x = np.matrix([[0.0], + [0.0]]) # Initial State +A = np.matrix([[1.0, dt], [0.0, 1.0]]) +H = np.diag([1.0, 1.0]) +P = np.diag([100.0, 1.0]) +I = np.diag([1.0, 1.0]) + + +# In[ ]: + + + + +# In[741]: + +x0=[] +x1=[] +P0=[] +dstate=[] +for filterstep in range(len(data)): + + # Time Update (Prediction) + # ======================== + # Project the state ahead + x = A*x + + # Project the error covariance ahead + P = A*P*A.T + Q + + + # Measurement Update (Correction) + # =============================== + # Compute the Kalman Gain + + # Measurement Noise is adaptive: + # Assuming a pretty correct angle measurement, while the + # bike is upright (vertical acceleration is nearly 1g), + # and a pretty bad estimation while the bike is leaning. + # So we make the R value adaptive to the lean angle + # with low values while upright and high values for high + # leaning angles. + adaptivephi = np.abs(np.multiply(1000.0,float(x[0]))+0.001) + R = np.diag([adaptivephi, 0.001]) + + + S = H*P*H.T + R + K = (P*H.T) * np.linalg.pinv(S) + + + # Update the estimate via z + Z = np.matrix([[np.pi/2-np.arccos(a_meas[filterstep]/9.81)], + [dphi_meas[filterstep]]]) + + y = Z - (H*x) # Innovation or Residual + x = x + (K*y) + + # Update the error covariance + P = (I - (K*H))*P + + + + # Save states for Plotting + x0.append(float(x[0])) + x1.append(float(x[1])) + P0.append(float(P[0,0])) + + +# In[742]: + +plt.plot(t,np.multiply(x0,180.0/np.pi), label=r'$\phi$') +#plt.plot(t,x1, label=r'$\dot \phi$') +#plt.plot(t, dphi_meas,label=r'$\dot \phi$ (ref)', alpha=0.5) +plt.title('Estimated Lean Angle') +plt.legend() + + +# In[ ]: + + +