From 2748514471138da47812bde6c0eecf40533e3ecf Mon Sep 17 00:00:00 2001 From: Erik Frisk <erik.frisk@liu.se> Date: Thu, 8 Apr 2021 20:11:42 +0200 Subject: [PATCH] Updated slides for lecture 3 and added exercise skeletons --- Exercise/EKF_UKF/EKF.m | 16 ++ Exercise/EKF_UKF/task1.ipynb | 360 +++++++++++++++++++++++++++++++++++ Exercise/EKF_UKF/task1.m | 138 ++++++++++++++ Lectures/slides_3.pdf | Bin 403306 -> 375324 bytes README.md | 2 +- 5 files changed, 515 insertions(+), 1 deletion(-) create mode 100644 Exercise/EKF_UKF/EKF.m create mode 100644 Exercise/EKF_UKF/task1.ipynb create mode 100644 Exercise/EKF_UKF/task1.m diff --git a/Exercise/EKF_UKF/EKF.m b/Exercise/EKF_UKF/EKF.m new file mode 100644 index 0000000..588df4f --- /dev/null +++ b/Exercise/EKF_UKF/EKF.m @@ -0,0 +1,16 @@ +function xhat = EKF(model,init,data) + + M = length(data.y); + P = init.P0; + n = numel(init.x0); + + xhat = zeros(n,M); xhat(:,1) = init.x0; + + for l=1:M-1 + % Measurement update + xhat(:,l) = FILL_IN_CODE_HERE; + + % Time update + xhat(:,l+1) = FILL_IN_CODE_HERE; + end +end \ No newline at end of file diff --git a/Exercise/EKF_UKF/task1.ipynb b/Exercise/EKF_UKF/task1.ipynb new file mode 100644 index 0000000..7ef82a7 --- /dev/null +++ b/Exercise/EKF_UKF/task1.ipynb @@ -0,0 +1,360 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [], + "source": [ + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "import seaborn as sns\n", + "import pandas as pd" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Using matplotlib backend: Qt5Agg\n" + ] + } + ], + "source": [ + "%matplotlib\n", + "sns.set(style='white', rc={'lines.linewidth': 0.75})" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Simulate robot movement" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Define discrete-time model" + ] + }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": {}, + "outputs": [], + "source": [ + "Tfinal = 20\n", + "Ts = 0.1\n", + "V = 3\n", + "x0 = [10, 10, 0] # Initial state\n", + "model = {}\n", + "model['f'] = lambda x, u, w: [x[0] + Ts*V*np.cos(x[2]), x[1] + Ts*V*np.sin(x[2]), x[2] + Ts*u]\n", + "model['h'] = lambda x, u: np.array([np.sqrt(x[0]**2 + x[1]**2), np.arctan2(x[1], x[0])])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Simulate robot movement and store results in a dictionary" + ] + }, + { + "cell_type": "code", + "execution_count": 21, + "metadata": {}, + "outputs": [], + "source": [ + "t = np.arange(0, Tfinal, Ts)\n", + "N = len(t)\n", + "x = np.zeros((3, N))\n", + "x[:, 0] = x0\n", + "\n", + "u = np.zeros((1, N))\n", + "u[0][np.logical_and(t >= 5, t < 10)] = -1\n", + "u[0][np.logical_and(t >= 10, t < 12)] = 0\n", + "u[0][np.logical_and(t >= 12, t < 15)] = 1\n", + "u[0][np.logical_and(t >= 15, t < 17)] = -1\n", + "\n", + "for ti in np.arange(0, N-1):\n", + " x[:, ti+1] = model['f'](x[:, ti], u[0, ti], 0)\n", + "y0 = model['h'](x, u)\n", + "y = y0 + (np.diag([0.6, 0.03])@np.random.normal(0, 1, size=(2, N)))\n", + "data = {'t': t, 'x': x, 'u': u, 'y0': y0, 'y': y}" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot nominal path" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "metadata": {}, + "outputs": [], + "source": [ + "plt.figure(10, clear=True)\n", + "plt.plot(data['x'][0, :], data['x'][1, :])\n", + "plt.plot(data['x'][0, 0], data['x'][1, 0], 'bo')\n", + "plt.plot(data['x'][0, -1], data['x'][1, -1], 'rx')\n", + "plt.axis('square')\n", + "plt.xlabel('x')\n", + "plt.ylabel('y')\n", + "sns.despine()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot measurements" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "metadata": {}, + "outputs": [], + "source": [ + "plt.figure(20, clear=True)\n", + "plt.subplot(2, 1, 1)\n", + "plt.plot(data['t'], data['y0'][0], 'b', lw=2)\n", + "plt.plot(data['t'], data['y'][0], 'r')\n", + "plt.xlabel('t [s]')\n", + "plt.ylabel('[m]')\n", + "plt.title('Range measurement')\n", + "sns.despine()\n", + "\n", + "plt.subplot(2, 1, 2)\n", + "plt.plot(data['t'], data['y0'][1]*180/np.pi, 'b', lw=2)\n", + "plt.plot(data['t'], data['y'][1]*180/np.pi, 'r')\n", + "plt.xlabel('t [s]')\n", + "plt.ylabel('[deg]')\n", + "plt.title('Angle measurement')\n", + "sns.despine()\n", + "plt.tight_layout()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Plot path computed from direct measurement" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Compute position estimate by inverting the measurement equation" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": {}, + "outputs": [], + "source": [ + "dist = data['y'][0]\n", + "ang = data['y'][1]\n", + "pos_hat_0 = np.row_stack((dist*np.cos(ang), dist*np.sin(ang)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot the estimated path" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": {}, + "outputs": [], + "source": [ + "plt.figure(30, clear=True)\n", + "plt.plot(data['x'][0], data['x'][1])\n", + "plt.plot(data['x'][0, 0], data['x'][1, 0], 'bo')\n", + "plt.plot(data['x'][0, -1], data['x'][1, -1], 'rx')\n", + "plt.plot(pos_hat_0[0], pos_hat_0[1], 'k')\n", + "plt.axis('square')\n", + "plt.xlabel('x')\n", + "plt.ylabel('y')\n", + "plt.title('Direct computation from measurements')\n", + "sns.despine()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot position errors" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "metadata": {}, + "outputs": [], + "source": [ + "plt.figure(31, clear=True)\n", + "plt.subplot(2, 1, 1)\n", + "plt.plot(data['t'], data['x'][0] - pos_hat_0[0])\n", + "plt.xlabel('t [s]');\n", + "plt.ylabel('[m]')\n", + "plt.title('x position error in direct computation');\n", + "sns.despine()\n", + "\n", + "plt.subplot(2, 1, 2)\n", + "plt.plot(data['t'], data['x'][1] - pos_hat_0[1])\n", + "plt.xlabel('t [s]');\n", + "plt.ylabel('[m]')\n", + "plt.title('y position error in direct computation')\n", + "sns.despine()\n", + "\n", + "plt.tight_layout()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "# Extended Kalman Filter" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Implement your Extended Kalman Filter below" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": {}, + "outputs": [], + "source": [ + "def EKF(model, init, data):\n", + " \"\"\"Extended Kalman Filter\n", + " \n", + " Input:\n", + " model - Dictionary with keys n, f, h, fx, hx, Q, R\n", + " init - Dictionary with x0, P0\n", + " \"\"\"\n", + " M = data['y'].shape[1] # Number of data points\n", + " P = init['P0'] # Initial covariance P_0|-1\n", + " n = model['n'] # Number of states\n", + "\n", + " xhat = np.zeros((n, M)) # Prepare output array\n", + " xhat[:, 0] = init['x0'] # Initialize xhat\n", + "\n", + " for l in range(0, M - 1):\n", + " pass\n", + " # Measurement update\n", + " # FILL IN YOUR CODE HERE\n", + " \n", + " # Time update\n", + " # FILL IN YOUR CODE HERE\n", + " \n", + " return xhat" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Define model dictionary" + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": {}, + "outputs": [], + "source": [ + "modelEKF = {}\n", + "Q = None # YOUR CODE HERE: 3x3 numpy array matrix, Process noise covariance \n", + "R = None # YOUR CODE HERE: 2x2 numpy array matrix, Measurement noise covariance\n", + "\n", + "modelEKF['n'] = 3\n", + "modelEKF['m'] = 2\n", + "modelEKF['Q'] = lambda x: Q\n", + "modelEKF['R'] = lambda x: R\n", + "modelEKF['f'] = lambda x, u: model['f'](x, u, 0)\n", + "modelEKF['h'] = lambda x, u: model['h'](x,u)\n", + "modelEKF['fx'] = lambda x, u: None # YOUR CODE HERE: return df/dx(x, u)\n", + "modelEKF['hx'] = lambda x, u: None # YOUR CODE HERE: return dh/dx(x, u)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Define inital conditions and run filter" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [ + "init = {}\n", + "P0 = None # YOUR CODE HERE: 2x2 numpy array matrix, Initial estimate covariance\n", + "x0 = None # YOUR CODE HERE: numpy array with initial state (x0, y0, theta0)\n", + "\n", + "init['x0'] = x0\n", + "init['P0'] = P0\n", + "\n", + "xhatEKF = EKF(modelEKF, init, data)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Explore, investigate, and plot interesting results." + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.0" + } + }, + "nbformat": 4, + "nbformat_minor": 4 +} diff --git a/Exercise/EKF_UKF/task1.m b/Exercise/EKF_UKF/task1.m new file mode 100644 index 0000000..2743e3b --- /dev/null +++ b/Exercise/EKF_UKF/task1.m @@ -0,0 +1,138 @@ +clear +close all + +%% Simulate robot movement +Ts = 0.1; +V = 3; + +model.f = @(x,u,w) [x(1) + Ts*V*cos(x(3));x(2) + Ts*V*sin(x(3));x(3) + Ts*u]; +model.h = @(x,u) [sqrt(x(1,:).^2 + x(2,:).^2);atan(x(2,:)./x(1,:))]; + +x0 = [10;10;0]; +Tfinal = 20; + +data.t = (0:Ts:Tfinal); +N = numel(data.t); +data.x = zeros(3,N); data.x(:,1) = x0; + +data.u = zeros(1,N); +data.u(data.t>=5 & data.t < 10) = -1; +data.u(data.t>=10 & data.t < 12) = 0; +data.u(data.t>=12 & data.t < 15) = 1; +data.u(data.t>=15 & data.t < 17) = -1; +for t=1:N-1 + data.x(:,t+1) = model.f(data.x(:,t),data.u(t),0); +end +data.y0 = model.h(data.x, data.u); +data.y = data.y0 + diag([0.6,0.03])*randn(2,N); + +%% Plot mobile robot path +figure(1) +plot( data.x(1,:)', data.x(2,:)' ) +hold on +plot( data.x(1,1)', data.x(2,1)', 'bo', ... + data.x(1,end)', data.x(2,end)', 'rx'); +hold off +axis([0 max(data.x(1,:))*1.1 0 max(data.x(2,:))*1.1]); +axis square +xlabel('x'); +ylabel('y'); + +%% Plot measurements +figure(2) +subplot( 211 ) +plot( data.t', [data.y(1,:)', data.y0(1,:)']) +xlabel('t [s]'); +ylabel('[m]') +title('Range measurement'); +box off + +subplot( 212 ) +plot( data.t', 180/pi*[data.y(2,:)', data.y0(2,:)']) +xlabel('t [s]'); +ylabel('[deg]'); +title('Angle measurement'); +box off + +%% Plot path from direct measurements +figure(3) +plot( data.x(1,:)', data.x(2,:)' ) +hold on +plot( data.x(1,1)', data.x(2,1)', 'bo', ... + data.x(1,end)', data.x(2,end)', 'rx'); +plot( (data.y(1,:).*cos(data.y(2,:)))', (data.y(1,:).*sin(data.y(2,:)))', 'k') +hold off +axis([0 max(data.x(1,:))*1.1 0 max(data.x(2,:))*1.1]); +axis square +xlabel('x'); +ylabel('y'); +title('Direct computation from measurements'); + +%% EKF estimation +modelEKF.f = @(x,u) FILL_IN_CODE_HERE; +modelEKF.h = @(x,u) FILL_IN_CODE_HERE; +modelEKF.fx = @(x,u) FILL_IN_CODE_HERE; +modelEKF.hx = @(x,u) FILL_IN_CODE_HERE; +modelEKF.Q = @(x) FILL_IN_CODE_HERE; +modelEKF.R = @(x) FILL_IN_CODE_HERE; + +init.x0 = [10;5;0]; % Initial state estimate +init.P0 = FILL_IN_CODE_HERE; % Initial covariance +tic; xhatEKF = EKF(modelEKF,init,data); toc + +%% Plot estimation results +figure(20) +plot( data.x(1,:)', data.x(2,:)', xhatEKF(1,:)', xhatEKF(2,:)' ) +hold on +plot( data.x(1,1)', data.x(2,1)', 'bo', ... + data.x(1,end)', data.x(2,end)', 'rx'); +hold off +axis([0 max(data.x(1,:))*1.1 0 max(data.x(2,:))*1.1]); +axis square +xlabel('x'); +ylabel('y'); + +figure(21) +subplot( 211 ) +plot( data.t', [xhatEKF(1,:)', data.x(1,:)']) +xlabel('t [s]'); +ylabel('[m]') +title('x position estimation'); +box off + +subplot( 212 ) +plot( data.t', xhatEKF(1,:)'-data.x(1,:)') +xlabel('t [s]'); +ylabel('[m]') +title('Estimation error'); +box off + +figure(22) +subplot( 211 ) +plot( data.t', [xhatEKF(2,:)', data.x(2,:)']) +xlabel('t [s]'); +ylabel('[m]') +title('y position estimation'); +box off + +subplot( 212 ) +plot( data.t', xhatEKF(2,:)'-data.x(2,:)') +xlabel('t [s]'); +ylabel('[m]'); +title('Estimation error'); +box off + +figure(23) +subplot( 211 ) +plot( data.t', 180/pi*[xhatEKF(3,:)', data.x(3,:)']) +xlabel('t [s]'); +ylabel('[deg]') +title('Heading estimation'); +box off + +subplot( 212 ) +plot( data.t', 180/pi*(xhatEKF(3,:)'-data.x(3,:)')) +xlabel('t [s]'); +ylabel('[deg]'); +title('Estimation error'); +box off diff --git a/Lectures/slides_3.pdf b/Lectures/slides_3.pdf index 7b37e671c899216da79dca63a5de5537c22b60f1..782e67468a3b05912fd59b48489056f27f0975ea 100644 GIT binary patch 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-44,7 +44,7 @@ A preliminary course schedule is as follows. Extra discussion seminars can be ad | Discussion 2 | March 26<br> 08:15 - 10:00 | Exercise discussion | [[30](https://doi.org/10.1016/j.automatica.2006.01.026)]|| | Lecture 3 | April 9<br> 08:15 - 10:00 | Lec 3a: Design methods, EKF and UKF, square root algorithms |[[3](https://doi.org/10.1115/1.3153059), [4](https://doi.org/10.1080/00207178708933870)], [[17;p. 55-82](http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-11032)], [[22](https://www.fs.isy.liu.se/Edu/Courses/ObserversPhD/CourseMaterial/ArrayAlgorithm_Kailath.pdf), [9](https://doi.org/10.1109/TAC.1978.1101825), [14](https://doi.org/10.1109/JPROC.2003.823141), [23](https://doi.org/10.1109/TSP.2011.2172431)] | [slides](Lectures/slides_3.pdf)| | Lecture 4 | April 16<br> 08:15 - 10:00| Lec 3b: Design methods, Particle filters ||| -| Discussion 3 | April 23<br> 08:15 - 10:00 | Exercise discussion||| +| Discussion 3 | April 23<br> 08:15 - 10:00 | Exercise discussion| [code skeletons](Exercise/EKF_UKF)|| | Lecture 5 | April 30 <br> 08:15 - 10:00| Lec 4: Linearization, Lyapunov, High gain, Sliding mode, Moving Horizon Estimation ||| | Discussion 4 | TBD | Exercise discussion ||| -- GitLab