diff --git a/Furuta/Introduction.ipynb b/Furuta/Introduction.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..253b7e10babd2f556a657ea4d19dd8f5832daf07
--- /dev/null
+++ b/Furuta/Introduction.ipynb
@@ -0,0 +1,248 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Welcome to Jupyter"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "This course will use Jupyter as the software environment for the Automatic Control part of the course.\n",
+ "\n",
+ "Jupyter is a web-based interface, with a file browser (to the left), a main area where this notebook is now displayed, and a few other bits that you will see later.\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "One useful feature of Jupyter Notebooks is that they allow mixing text (with fancy formatting) and executable code (organized in \"cells\"), and that the results of executing code gets printed.\n",
+ "\n",
+ "If you click in the below code cell, containing the code `print (\"I can see the output!\")`, and press shift+enter, then you should indeed see the output."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "I can see the output.\n"
+ ]
+ }
+ ],
+ "source": [
+ "print(\"I can see the output.\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "You should now see the text \"I can see the output.\", and the cell should have been give a number which is shown to the left of the cell, telling us that it was the first cell to be executed. If you execute the cell again, it should get the number 2 instead.\n",
+ "\n",
+ "This notebook uses python as the programming language to execute, but jupyter can handle many other programming languages as well. \n",
+ "\n",
+ "Let's define a function in one cell and use it in the next."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def f(x):\n",
+ " return x*x - x + 0.18"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.42"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f(1.2)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "There are more ways to execute the cells in a notebook than pressing shift+enter, including the buttons just above the notebook, and via the menus at the top, but we will not focus on them now."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "At the bottom left of the screen, there should now be a text saying that we are using a Python 3 kernel, and that the kernel is currently *Idle*. When we run a longer computation, we will see that change to *Busy* instead. There is also a place at the top right where it says that the kernel is Python 3, with a nearby circle. When a computation is running, the circle will be filled.\n",
+ "\n",
+ "Let's try something that keeps the kernel \"busy\" for 10 seconds..."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "import time\n",
+ "time.sleep(10)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We will use Julia as our programming language of choice, but before we can do that, we need to install a Julia kernel in your home directory that jupyter can use. That can be done with the command below, so do run it, and wait until it the kernel is no longer busy.\n",
+ "\n",
+ "(It can take quite a bit of time, depending on the performance of the network-mounted home directory.)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ " Resolving package versions...\n",
+ " Installed ZeroMQ_jll ────── v4.3.2+4\n",
+ " Installed VersionParsing ── v1.2.0\n",
+ " Installed SoftGlobalScope ─ v1.1.0\n",
+ " Installed Conda ─────────── v1.4.1\n",
+ " Installed IJulia ────────── v1.21.2\n",
+ " Installed ZMQ ───────────── v1.2.1\n",
+ "Downloading artifact: ZeroMQ\n",
+ "######################################################################## 100.0% ##O#-# 9.9% 20.2%\n",
+ "\u001b[1A\u001b[2K\u001b[?25h Updating `~/.julia/environments/v1.4/Project.toml`\n",
+ " [7073ff75] + IJulia v1.21.2\n",
+ " Updating `~/.julia/environments/v1.4/Manifest.toml`\n",
+ " [8f4d0f93] + Conda v1.4.1\n",
+ " [7073ff75] + IJulia v1.21.2\n",
+ " [682c06a0] + JSON v0.21.0\n",
+ " [739be429] + MbedTLS v1.0.2\n",
+ " [c8ffd9c3] + MbedTLS_jll v2.16.6+1\n",
+ " [69de0a69] + Parsers v1.0.7\n",
+ " [b85f4697] + SoftGlobalScope v1.1.0\n",
+ " [81def892] + VersionParsing v1.2.0\n",
+ " [c2297ded] + ZMQ v1.2.1\n",
+ " [8f1865be] + ZeroMQ_jll v4.3.2+4\n",
+ " [2a0f44e3] + Base64 \n",
+ " [ade2ca70] + Dates \n",
+ " [8ba89e20] + Distributed \n",
+ " [7b1f6079] + FileWatching \n",
+ " [b77e0a4c] + InteractiveUtils \n",
+ " [76f85450] + LibGit2 \n",
+ " [8f399da3] + Libdl \n",
+ " [56ddb016] + Logging \n",
+ " [d6f4376e] + Markdown \n",
+ " [a63ad114] + Mmap \n",
+ " [44cfe95a] + Pkg \n",
+ " [de0858da] + Printf \n",
+ " [3fa0cd96] + REPL \n",
+ " [9a3f8284] + Random \n",
+ " [ea8e919c] + SHA \n",
+ " [9e88b42a] + Serialization \n",
+ " [6462fe0b] + Sockets \n",
+ " [8dfed614] + Test \n",
+ " [cf7118a7] + UUIDs \n",
+ " [4ec0a83e] + Unicode \n",
+ " Building Conda ─→ `~/.julia/packages/Conda/3rPhK/deps/build.log`\n",
+ " Building IJulia → `~/.julia/packages/IJulia/DrVMH/deps/build.log`\n",
+ " Building Conda ─→ `~/.julia/packages/Conda/3rPhK/deps/build.log`\n",
+ " Building IJulia → `~/.julia/packages/IJulia/DrVMH/deps/build.log`\n"
+ ]
+ }
+ ],
+ "source": [
+ "!julia --eval \"import Pkg; Pkg.add(\\\"IJulia\\\"); Pkg.build(\\\"IJulia\\\"); using IJulia\""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "If the indicators at the bottom left and top right show that the kernel is idle, and if the cell has been given a number, then the julia kernel should have been installed and be ready to use. Now we need to restart the server, and then Julia should be available to us.\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "To restart, you need to do 3 things.\n",
+ "\n",
+ " 1. Select the menu item \"File\" --> \"Hub Control Panel\"\n",
+ " 2. Stop the server\n",
+ " 3. Start the server again\n",
+ " \n",
+ "Then you should get back to this view of the files, and continue with the notebook named Julia.ipynb."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Remember to stop your server when you are done."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "If we have more than one person trying to access the lab processes at any one time, the lab computers only allow a single user to have a jupyterlab server running at a time. Therefore, when you stop working on the lab, you should stop your server, the same way you just did, and then, when you want to resume your work, your server needs to be restarted again."
+ ]
+ },
+ {
+ "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": "3.8.5"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/Furuta/Julia.ipynb b/Furuta/Julia.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..05dc3b688c7b722a0628a489162c2f82665641d8
--- /dev/null
+++ b/Furuta/Julia.ipynb
@@ -0,0 +1,413 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Welcome to Julia"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Now, let's turn our attention to Julia. \n",
+ "\n",
+ "Julia is a programming language, intended to solve problems you may have seen solved using matlab or python in other contexts. Its syntax is a bit different from matlab/python, but for many problems whose solution uses matrices, vectors, function and loops to compute the answer, then then generates some plots of the data, the Julia solution is often fairly similar to the matlab/python solutions."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We show some simple examples of julia code below, and a \"[cheatsheet](https://cheatsheets.quantecon.org/julia-cheatsheet.html)\" summarizing Julia syntax, or a [comparison](https://cheatsheets.quantecon.org/index.html) of operations between matlab, python and Julia will probably be helpful.\n",
+ "\n",
+ "If you want to learn more, the internet contains a lot of resources, and [JuliaAcademy](https://github.com/JuliaAcademy) in particular has several good tutorials.\n",
+ "\n",
+ "You could look at some parts of their [Introduction to Julia](https://github.com/JuliaAcademy/JuliaTutorials/tree/master/introductory-tutorials/intro-to-julia/short-version) seris, such as [Getting to know Julia](https://github.com/JuliaAcademy/JuliaTutorials/blob/master/introductory-tutorials/intro-to-julia/short-version/01.Getting_to_know_Julia.ipynb) and [Introduction to Plotting](https://github.com/JuliaAcademy/JuliaTutorials/blob/master/introductory-tutorials/intro-to-julia/short-version/04.Intro_to_plotting.ipynb).\n",
+ "\n",
+ "You can download the files directly, or clone their repository, and you should then be able to navigate your way to those files using the file browser to the left of this notebook area. If you download the files to your computer, you can use the upload button above the file selection area to upload it to your home directory here in jupyterlab, or you can look at the \"launcher\" and open a terminal in which you can use git, but there are of course a large number of other mehthods you could use."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Now, try the following code snippets, and make sure you understand what is happening. "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "g (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 1,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function f(x)\n",
+ " return sin(x)\n",
+ "end\n",
+ "\n",
+ "g(x) = sin(x)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.9092974268256817"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f(2)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.9092974268256817"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "g(2)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using Plots"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
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+ " 2009.7,155.698 2155.72,155.698 \n",
+ " \"/>\n",
+ "<path clip-path=\"url(#clip980)\" d=\"M 0 0 M2193.9 175.385 Q2192.09 180.015 2190.38 181.427 Q2188.67 182.839 2185.79 182.839 L2182.39 182.839 L2182.39 179.274 L2184.89 179.274 Q2186.65 179.274 2187.62 178.44 Q2188.6 177.607 2189.78 174.505 L2190.54 172.561 L2180.05 147.052 L2184.57 147.052 L2192.67 167.329 L2200.77 147.052 L2205.29 147.052 L2193.9 175.385 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip980)\" d=\"M 0 0 M2211.17 169.042 L2218.8 169.042 L2218.8 142.677 L2210.49 144.343 L2210.49 140.084 L2218.76 138.418 L2223.43 138.418 L2223.43 169.042 L2231.07 169.042 L2231.07 172.978 L2211.17 172.978 L2211.17 169.042 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /></svg>\n"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "xs = range(0, 10, length=100)\n",
+ "ys = [ f(x) for x in xs ]\n",
+ "\n",
+ "plot(xs, ys)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using LinearAlgebra"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "2×2 Array{Int64,2}:\n",
+ " 1 2\n",
+ " 3 4"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "A = [ 1 2; 3 4 ]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "2-element Array{Int64,1}:\n",
+ " 0\n",
+ " 1"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "B = [ 0; 1 ]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "1×2 Adjoint{Int64,Array{Int64,1}}:\n",
+ " 0 1"
+ ]
+ },
+ "execution_count": 10,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "B'"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "2-element Array{Int64,1}:\n",
+ " 2\n",
+ " 4"
+ ]
+ },
+ "execution_count": 9,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "A*B"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "1×2 Adjoint{Int64,Array{Int64,1}}:\n",
+ " 3 4"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "B'*A"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "4"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "B'*A*B"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "When you feel a bit more familiar with Julia, there are other notebooks here for the actual Labs."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.4.2",
+ "language": "julia",
+ "name": "julia-1.4"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.4.2"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/Furuta/first_experiment.ipynb b/Furuta/first_experiment.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..5f675252041209a56a855d68dfcfe91e6bf7eefe
--- /dev/null
+++ b/Furuta/first_experiment.ipynb
@@ -0,0 +1,896 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Preparations"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using Plots\n",
+ "using Statistics\n",
+ "using ProgressBars"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We will use the Moberg library to connect to the device. When we create our Moberg object, it should detect the device and connect to it."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stderr",
+ "output_type": "stream",
+ "text": [
+ "┌ Info: Precompiling MobergIO [top-level]\n",
+ "└ @ Base loading.jl:1260\n"
+ ]
+ },
+ {
+ "data": {
+ "text/plain": [
+ "MobergIO.Moberg(Ptr{Nothing} @0x0000000003be2ae0)"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "push!(LOAD_PATH, \"/opt/jupyterhub/moberg/adaptors/julia\");\n",
+ "using MobergIO\n",
+ "\n",
+ "if @isdefined moberg_connection\n",
+ " @warn \"Only create one connection at a time to the device.\"\n",
+ "else\n",
+ " moberg_connection = MobergIO.Moberg()\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We use convenience functions to handle sending to and receiving from the magig port numbers we will use."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "get_x (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "enable!() = MobergIO.write(MobergIO.DigitalOut(moberg_connection, Unsigned(40)), Bool(1))\n",
+ "disable!() = MobergIO.write(MobergIO.DigitalOut(moberg_connection, Unsigned(40)), Bool(0))\n",
+ "\n",
+ "set_torque!(τ) = MobergIO.write(MobergIO.AnalogOut(moberg_connection, Unsigned(40)), τ)\n",
+ "\n",
+ "get_ϕ() = MobergIO.read(MobergIO.AnalogIn(moberg_connection, Unsigned(40)))\n",
+ "get_dϕ() = MobergIO.read(MobergIO.AnalogIn(moberg_connection, Unsigned(42)))\n",
+ "get_θ() = MobergIO.read(MobergIO.AnalogIn(moberg_connection, Unsigned(41)))\n",
+ "get_dθ() = MobergIO.read(MobergIO.AnalogIn(moberg_connection, Unsigned(43)))\n",
+ "\n",
+ "get_x() = [get_ϕ(), get_dϕ(), get_θ(), get_dθ()]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# A first experiment"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "An experiment uses a policy and runs the pendulum with that policy for some time, recording the observed states. The policy is a function that takes a state and time and chooses a torque.\n",
+ "\n",
+ "The experiment performs the following:\n",
+ " 1. Enable the device\n",
+ " 1. Repeatedly, until time `T` has elapsed\n",
+ " 1. Read the state\n",
+ " 1. Decide which torque to apply\n",
+ " 1. Send the selected torque to the device\n",
+ " 1. Record the current state, torque and time elapsed.\n",
+ " \n",
+ "* The experiment runs for $T$ seconds.\n",
+ "* We record the state and torque every $\\Delta t$ seconds.\n",
+ "* The device firmware performs tasks at 20kHz, and a torque command expires after 2k iterations, so if we want to have a constant torque, we need to resubmit the command at least every 100 ms. We set $dt=0.1$ and we ensure that we sleep for at most $dt$ seconds at a time.\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "experiment (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function experiment(f; T = 10, Δt = 0.01)\n",
+ " enable!()\n",
+ " \n",
+ " # We need to update torque at least every 2k iterations for a 20kHz interrupt\n",
+ " # These numbers are obtained from the firmware\n",
+ " dt = 2*10^3 / 20*10^3\n",
+ " sleep_time = min(dt, Δt)\n",
+ " \n",
+ " t_start = time()\n",
+ " t_exp = 0.0\n",
+ " xs = [get_x()]\n",
+ " τs = [0.0]\n",
+ " ts = [t_exp]\n",
+ " \n",
+ " while t_exp < T\n",
+ " t_exp = time() - t_start\n",
+ " x = get_x()\n",
+ " τ = f(x, t_exp)\n",
+ " set_torque!(τ) \n",
+ " sleep(sleep_time)\n",
+ " \n",
+ " if t_exp - ts[end] > Δt\n",
+ " push!(xs, x)\n",
+ " push!(τs, τ)\n",
+ " push!(ts, t_exp)\n",
+ " end \n",
+ " end\n",
+ " set_torque!(0.0)\n",
+ " sleep(5)\n",
+ " disable!()\n",
+ " sleep(5)\n",
+ " xs, τs, ts\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "plot_result (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function plot_result(xs, τs, ts)\n",
+ " ϕs = [ x[1] for x in xs ] \n",
+ " ϕs .= ϕs .- ϕs[1]\n",
+ " θs = [ x[3] for x in xs ]\n",
+ " θs .= rem2pi.(θs, RoundNearest)\n",
+ " plt_ϕ = plot(ts, ϕs, xlabel = \"Time [s]\", ylabel = \"Base angle\", label = \"phi\")\n",
+ " plt_θ = plot(ts, θs, xlabel = \"Time [s]\", ylabel = \"Pendulum angle\", label = \"theta\")\n",
+ " plt_τ = plot(ts, τs, xlabel = \"Time [s]\", ylabel = \"Torque\", label = \"tau\")\n",
+ " plot(plt_ϕ, plt_θ, plt_τ, layout = (3, 1))\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "f(x, t) = sin(10*t)/40\n",
+ "xs, τs, ts = experiment(f);"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {},
+ "outputs": [
+ {
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+ " 1284.83,1399.08 1287.79,1387.9 1290.79,1373.91 1293.94,1356.59 1296.9,1338.25 1299.87,1318.31 1302.84,1297.25 1305.81,1275.55 1308.78,1253.69 1311.78,1231.94 \n",
+ " 1314.91,1210.16 1317.88,1190.82 1320.85,1173.24 1323.81,1157.82 1326.78,1144.86 1329.75,1134.7 1332.72,1127.51 1335.68,1123.53 1338.66,1122.78 1341.63,1125.31 \n",
+ " 1344.6,1131.07 1347.57,1139.91 1350.57,1151.77 1353.7,1167.03 1356.87,1185.02 1359.84,1203.82 1363,1225.48 1366.17,1248.21 1369.14,1270.08 1372.41,1294 \n",
+ " 1375.43,1315.55 1378.39,1335.65 1381.38,1354.33 1384.35,1371.04 1387.34,1385.5 1390.34,1397.32 1393.48,1406.54 1396.65,1412.31 1399.61,1414.4 1402.58,1413.21 \n",
+ " 1405.75,1408.36 1408.92,1399.93 1411.89,1388.98 1415.05,1374.34 1418.22,1356.98 1421.19,1338.66 1424.16,1318.75 1427.32,1296.29 1430.52,1272.9 1433.65,1249.85 \n",
+ " 1436.62,1228.46 1439.59,1207.93 1442.76,1187.55 1445.92,1169.26 1449.09,1153.51 1452.26,1140.66 1455.43,1131.11 1458.59,1125.07 1461.76,1122.69 1464.92,1124.03 \n",
+ " 1468.09,1129.07 1471.29,1137.77 1474.42,1149.61 1477.39,1163.59 1480.56,1181.09 1483.72,1200.74 1486.69,1220.8 1489.66,1241.97 1492.82,1265.12 1495.99,1288.41 \n",
+ " 1498.96,1309.79 1502.12,1331.54 1505.09,1350.51 1508.26,1368.68 1511.26,1383.56 1514.39,1396.27 1517.56,1405.87 1520.53,1411.69 1523.7,1414.34 1526.86,1413.28 \n",
+ " 1529.86,1408.86 1533,1400.8 1536.15,1389.34 1539.12,1375.78 1542.08,1359.82 1545.06,1341.74 1548,1322.28 1551.03,1300.96 1554.16,1278.15 1557.12,1256.32 \n",
+ " 1560.09,1234.77 1563.07,1213.82 1566.04,1194.19 1569.23,1175.05 1572.33,1158.72 1575.31,1145.55 1578.3,1135.15 1581.27,1127.82 1584.24,1123.65 1587.21,1122.74 \n",
+ " 1590.22,1125.15 1593.35,1131.17 1596.31,1140.06 1599.28,1151.82 1602.25,1166.21 1605.22,1182.91 1608.19,1201.51 1611.16,1221.67 1614.32,1244.24 1617.3,1266.02 \n",
+ " 1620.26,1287.84 1623.23,1309.23 1626.2,1329.7 1629.2,1349.02 1632.34,1367.21 1635.3,1382.14 1638.27,1394.53 1641.24,1404.11 1644.21,1410.65 1647.17,1413.98 \n",
+ " 1650.17,1414.04 1653.31,1410.53 1656.28,1403.94 1659.25,1394.29 1662.22,1381.83 1665.18,1366.82 1668.15,1349.61 1671.13,1330.53 1674.09,1310.13 1677.06,1288.78 \n",
+ " 1680.03,1266.95 1683,1245.18 1685.97,1223.91 1688.96,1203.46 1692.1,1183.68 1695.07,1166.9 1698.04,1152.4 1701,1140.52 1703.97,1131.51 1706.94,1125.57 \n",
+ " 1709.95,1122.82 1713.08,1123.54 1716.05,1127.56 1719.02,1134.75 1722.18,1145.73 1725.35,1159.85 1728.33,1175.64 1731.49,1194.76 1734.66,1215.72 1737.79,1237.83 \n",
+ " 1740.78,1259.62 1743.76,1281.55 1746.72,1303.12 1749.73,1324.14 1752.86,1344.69 1756.02,1363.54 1758.99,1379.03 1762.16,1392.78 1765.13,1402.82 1768.28,1410.16 \n",
+ " 1771.22,1413.79 1774.18,1414.2 1777.19,1411.29 1780.15,1405.21 1783.12,1396.03 1786.11,1383.94 1789.31,1368.05 1792.44,1350 1795.61,1329.68 1798.77,1307.79 \n",
+ " 1801.74,1286.35 1804.9,1263.13 1807.9,1241.14 1811.04,1218.9 1814.01,1198.95 1817.17,1179.41 1820.34,1162.14 1823.31,1148.43 1826.28,1137.4 1829.45,1128.89 \n",
+ " 1832.61,1123.95 1835.58,1122.67 1838.57,1124.7 1841.71,1130.35 1844.67,1138.87 1847.67,1150.46 1850.81,1165.47 1853.78,1182.04 1856.75,1200.56 1859.72,1220.63 \n",
+ " 1862.68,1241.73 1865.65,1263.46 1868.66,1285.57 1871.79,1308.24 1874.96,1330.09 1877.92,1349.13 1880.89,1366.39 1883.86,1381.49 1887.06,1394.88 1890.2,1404.8 \n",
+ " 1893.36,1411.36 1896.33,1414.21 1899.3,1413.78 1902.47,1409.72 1905.63,1402.07 1908.81,1390.97 1911.96,1376.83 1914.93,1360.98 1918.1,1341.83 1921.07,1322.15 \n",
+ " 1924.04,1301.28 1927.24,1277.97 1930.36,1254.99 1933.34,1233.33 1936.51,1211.21 1939.64,1190.77 1942.62,1173.13 1945.61,1157.63 1948.77,1143.95 1951.94,1133.45 \n",
+ " 1954.9,1126.75 1957.88,1123.21 1961.04,1123.04 1964.02,1126.27 1967.16,1133.15 1970.1,1142.68 1973.08,1155.17 1976.09,1170.4 1979.25,1188.82 1982.22,1207.96 \n",
+ " 1985.2,1228.57 1988.36,1251.38 1991.53,1274.69 1994.69,1297.85 1997.86,1320.22 2001.02,1341.29 2004.19,1360.52 2007.39,1377.54 2010.52,1391.47 2013.69,1402.4 \n",
+ " 2016.86,1409.94 2020.02,1413.85 2023.19,1414.06 2026.39,1410.5 2029.52,1403.42 2032.69,1392.83 2036.04,1378.18 2039.01,1362.53 2041.98,1344.8 2044.95,1325.36 \n",
+ " 2048.12,1303.22 2051.08,1281.68 2054.06,1259.78 2057.22,1236.68 2060.39,1214.4 2063.56,1193.52 2066.75,1174.4 2069.89,1157.99 2072.85,1145.01 2075.82,1134.8 \n",
+ " 2079.03,1127.15 2082.16,1123.25 2085.33,1123.01 2088.5,1126.49 2091.66,1133.57 2094.83,1144.11 2098,1157.86 2101.16,1174.39 2104.33,1193.35 2107.53,1214.46 \n",
+ " 2110.66,1236.51 2113.83,1259.58 2117,1282.92 2120.15,1305.8 2123.33,1327.83 2126.52,1348.5 2129.65,1366.73 2132.82,1382.68 2136.2,1396.46 2139.33,1405.91 \n",
+ " 2142.29,1411.69 2145.29,1414.3 2148.25,1413.59 2151.22,1409.63 2154.39,1401.91 2157.56,1390.79 2160.53,1377.51 2163.68,1360.72 2166.64,1342.84 2169.57,1323.54 \n",
+ " 2172.56,1302.59 2175.56,1280.75 2178.53,1258.9 2181.49,1237.31 2184.47,1216.31 2187.47,1196.3 2190.61,1177.21 2193.77,1160.27 2196.94,1146.07 2200.11,1135 \n",
+ " 2203.27,1127.35 2206.47,1123.28 2209.6,1122.97 2212.57,1126.03 2215.74,1132.83 2218.91,1143.09 2222.08,1156.56 2225.25,1172.9 2228.41,1191.68 2231.56,1212.3 \n",
+ " 2234.53,1233.07 2237.5,1254.63 2240.66,1277.89 2243.64,1299.58 2246.64,1320.75 2249.77,1341.59 2252.94,1360.76 2255.91,1376.62 2258.88,1390.07 2262.04,1401.37 \n",
+ " 2265.24,1409.35 2268.38,1413.61 2271.54,1414.22 2274.71,1411.11 2277.88,1404.36 2281.06,1394.09 2284.2,1380.75 2287.18,1365.52 2290.34,1346.94 2293.31,1327.65 \n",
+ " \n",
+ " \"/>\n",
+ "<path clip-path=\"url(#clip890)\" d=\"\n",
+ "M1982.72 1245.18 L2282.75 1245.18 L2282.75 1124.22 L1982.72 1124.22 Z\n",
+ " \" fill=\"#ffffff\" fill-rule=\"evenodd\" fill-opacity=\"1\"/>\n",
+ "<polyline clip-path=\"url(#clip890)\" style=\"stroke:#000000; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
+ " 1982.72,1245.18 2282.75,1245.18 2282.75,1124.22 1982.72,1124.22 1982.72,1245.18 \n",
+ " \"/>\n",
+ "<polyline clip-path=\"url(#clip890)\" style=\"stroke:#009af9; stroke-width:4; stroke-opacity:1; fill:none\" points=\"\n",
+ " 2006.06,1184.7 2146.07,1184.7 \n",
+ " \"/>\n",
+ "<path clip-path=\"url(#clip890)\" d=\"M 0 0 M2176.82 1168.69 L2176.82 1176.05 L2185.59 1176.05 L2185.59 1179.36 L2176.82 1179.36 L2176.82 1193.44 Q2176.82 1196.61 2177.67 1197.51 Q2178.55 1198.41 2181.21 1198.41 L2185.59 1198.41 L2185.59 1201.98 L2181.21 1201.98 Q2176.28 1201.98 2174.41 1200.15 Q2172.53 1198.3 2172.53 1193.44 L2172.53 1179.36 L2169.41 1179.36 L2169.41 1176.05 L2172.53 1176.05 L2172.53 1168.69 L2176.82 1168.69 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip890)\" d=\"M 0 0 M2201.84 1188.95 Q2196.68 1188.95 2194.69 1190.13 Q2192.7 1191.31 2192.7 1194.16 Q2192.7 1196.42 2194.18 1197.77 Q2195.68 1199.09 2198.25 1199.09 Q2201.79 1199.09 2203.92 1196.59 Q2206.08 1194.06 2206.08 1189.9 L2206.08 1188.95 L2201.84 1188.95 M2210.33 1187.19 L2210.33 1201.98 L2206.08 1201.98 L2206.08 1198.04 Q2204.62 1200.41 2202.44 1201.54 Q2200.27 1202.65 2197.12 1202.65 Q2193.14 1202.65 2190.77 1200.43 Q2188.44 1198.18 2188.44 1194.43 Q2188.44 1190.06 2191.35 1187.84 Q2194.29 1185.61 2200.1 1185.61 L2206.08 1185.61 L2206.08 1185.2 Q2206.08 1182.26 2204.13 1180.66 Q2202.21 1179.04 2198.71 1179.04 Q2196.49 1179.04 2194.39 1179.57 Q2192.28 1180.1 2190.33 1181.17 L2190.33 1177.23 Q2192.67 1176.33 2194.87 1175.89 Q2197.07 1175.43 2199.15 1175.43 Q2204.78 1175.43 2207.56 1178.35 Q2210.33 1181.26 2210.33 1187.19 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /><path clip-path=\"url(#clip890)\" d=\"M 0 0 M2214.36 1191.75 L2214.36 1176.05 L2218.62 1176.05 L2218.62 1191.59 Q2218.62 1195.27 2220.06 1197.12 Q2221.49 1198.95 2224.36 1198.95 Q2227.81 1198.95 2229.8 1196.75 Q2231.82 1194.55 2231.82 1190.75 L2231.82 1176.05 L2236.08 1176.05 L2236.08 1201.98 L2231.82 1201.98 L2231.82 1198 Q2230.27 1200.36 2228.2 1201.52 Q2226.17 1202.65 2223.46 1202.65 Q2218.99 1202.65 2216.68 1199.87 Q2214.36 1197.1 2214.36 1191.75 M2225.08 1175.43 L2225.08 1175.43 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /></svg>\n"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "plot_result(xs, τs, ts)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Keeping a constant angular velocity"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We see that the respose of the system is not at all the one I expect, given the torque, so we try some more experiments, to understand what is going on.\n",
+ "\n",
+ "Assuming that, at low torques, the angular velocity is proportional to the torque, we sweep a range of torques, and plot the angular velocity as a function of torque."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "τs = range(-0.02, 0.02, length=30)\n",
+ "qs = []\n",
+ "for τ in ProgressBar(τs)\n",
+ " f(x, t) = τ\n",
+ " xs, _, ts = experiment(f, T=5);\n",
+ " dϕs = [x[2] for x in xs]\n",
+ " q = quantile(dϕs[ts .> 1], [0.05, 0.25, 0.5, 0.75, 0.95])\n",
+ " push!(qs, q)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "low = [ q[1] for q in qs ]\n",
+ "med = [ q[3] for q in qs ]\n",
+ "hi = [ q[5] for q in qs ]\n",
+ "\n",
+ "plot(τs, med, ribbon = (med .- low, hi .- med), xlabel = \"Torque\", ylabel = \"Base Angular Velocity\")"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We see that there is a strong asymmetry, and that the experiment is aborted when the angular velocity gets above a threshold. For positive torques, we see that the angular velocity jumps around a torque of 0.01."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Let's try to slowly change the torque until we get to a predefined angular velocity.\n",
+ "\n",
+ "Every second, we observe the angular velocity, and increment or decrement the torque."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "function policy(dϕ_target)\n",
+ " τ = 0.0\n",
+ " dτ_max = 0.001\n",
+ " dt = 1\n",
+ " t_mod = 0.0\n",
+ " function f(x, t)\n",
+ " if t - t_mod > dt\n",
+ " dϕ = x[2]\n",
+ " err = dϕ_target - dϕ\n",
+ " τ += clamp(err, -dτ_max, dτ_max)\n",
+ " t_mod = t\n",
+ " end\n",
+ " τ\n",
+ " end\n",
+ " f\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "When I run the following experiment on my device, I see jumps of angular velocity from 0 to 20 for very small changes in torque.\n",
+ "\n",
+ "If this is correct, then we need a better friction model."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "xs, τs, ts = experiment(policy(5), T=100);"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "dϕs = [ x[2] for x in xs ]\n",
+ "plt_a = plot(ts, dϕs, label=\"Observation\", xlabel=\"Time\", ylabel=\"BaseAngularVelocity\")\n",
+ "hline!(plt_a, [5], label = \"Target\", color=:green, legend = :topleft)\n",
+ "plt_b = plot(ts, τs, xlabel = \"Time\", ylabel=\"Torque\", legend = nothing)\n",
+ "plot(plt_a, plt_b, layout = (2, 1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Doing the same experiment in the other direction, I see a sudden jump to a very large velocity, whereafter the experiment is aborted by the firmware (is that right?), and nothing more happens, despite increases in torque."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "xs, τs, ts = experiment(policy(-5), T=100);"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "dϕs = [ x[2] for x in xs ]\n",
+ "plt_a = plot(ts, dϕs, label=\"Observation\", xlabel=\"Time\", ylabel=\"BaseAngularVelocity\")\n",
+ "hline!(plt_a, [-5], label = \"Target\", color=:green, legend = :bottomright)\n",
+ "plt_b = plot(ts, τs, xlabel = \"Time\", ylabel=\"Torque\", legend = nothing)\n",
+ "plot(plt_a, plt_b, layout = (2, 1))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ " * If the firmware ends the experiment early (which is consistent with the plots from the sine-wave experiment above), then it would be nice to have a message. Probably in the form of an \"Experiment ended ok\" signal to poll after the disable at the end.\n",
+ " * If the firmware had problems computing the derivatives, then it would be good to fix that.\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Occasionally, I start getting errors from Moberg with error code 110, and the fix seems to be to reset everything and start again."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.4.2",
+ "language": "julia",
+ "name": "julia-1.4"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.4.2"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/Furuta/simulate.ipynb b/Furuta/simulate.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..590dd98a3261b97b58af9bf19c50bae74910a19c
--- /dev/null
+++ b/Furuta/simulate.ipynb
@@ -0,0 +1,3686 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Note to the reader\n",
+ "\n",
+ "This notebook was created as a sample of things one might want to combine into something useful for a course, and it is not intended to be"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Furuta Pendulum"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Introduction"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The furuta pendulum is a system with a pendulum fixed to a point on a rotating cylinder. The aim is to control the pendulum using the motor that can cause the cylinder to rotate.\n",
+ "\n",
+ "We will use this system for several problems in automatic control, and this first notebook introduces the system and the software environment we use for our experiments.\n",
+ "\n",
+ "This is a jupyter notebook, and it is normally viewed and run from a web-browser. It communicates with a jupyter server, which runs the computations. If the server is a lab computer with a Furuta-pendulum connected to it, then the experiments can be run, regardless of what computer the web-browser is running on."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Jupyter and Julia\n",
+ "\n",
+ "**TODO:** A brief text about jupyter and julia."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**Note:** I tried using units, but that did not work well with DifferentialEquations, as it needed a generic `zero`, and gave me the error messge \"ArgumentError: zero(Quantity{Float64,D,U} where U where D) not defined.\""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Loading modules\n",
+ "\n",
+ "The julia packages we will use must be loaded to be made avaialble to the Julia process."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "\"utf-8\""
+ ]
+ },
+ "execution_count": 1,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "ENV[\"GKS_ENCODING\"] = \"utf-8\""
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Due to the special way Julia handles compilation, loading packages typically takes longer than in other programming languages."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using LinearAlgebra\n",
+ "using OrdinaryDiffEq\n",
+ "using Plots\n",
+ "using LaTeXStrings\n",
+ "using ProgressBars\n",
+ "using Optim\n",
+ "using Zygote\n",
+ "using BenchmarkTools"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Furuta Pendulum\n",
+ "\n",
+ "The dynamics of the system is described in a [report](http://lup.lub.lu.se/search/ws/files/4453844/8727127.pdf) by Magnus Gäfvert, and we will base our investigations on that report and follow the notation introduces there.\n",
+ "\n",
+ "The parameters of the system are \n",
+ " * $M$: The mass of the pendulum\n",
+ " * ...\n",
+ " \n",
+ " \n",
+ "For the dynamical equations, derived in the report, the following combinations of parameters are used\n",
+ " * $\\alpha = J + (M + m_a +m_p)l_a^2$\n",
+ " * $\\beta = (M + \\frac{m_p}{3})l_p^2$\n",
+ " * $\\gamma = (M + \\frac{m_p}{2})l_a l_p$\n",
+ " * $\\delta = (M + \\frac{m_p}{2}) g l_p$\n",
+ "\n",
+ "Our state contains $\\phi, \\dot{\\phi}, \\theta, \\dot{\\theta}$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "0.05"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "g = 10\n",
+ "\n",
+ "J = 1\n",
+ "M = 0.0\n",
+ "m_p = 10^-1\n",
+ "l_p = 10^-1\n",
+ "m_a = 10^-2\n",
+ "l_a = 10^-2\n",
+ "\n",
+ "α = J + (m_a/3 + m_p)*l_a^2\n",
+ "β = (M + m_p/3)*l_p^2\n",
+ "γ = (M + m_p/2)*l_a*l_p\n",
+ "δ = (M + m_p/2)*g*l_p"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Simulating the system\n",
+ "\n",
+ "We assume that we have chosen a control policy, and that we want to solve the equations of motion.\n",
+ "\n",
+ "With our control policy given in terms of the two function determining the external torques from the motors $\\tau\\phi$ and $\\tau\\theta$ as functions of the state and the current time, \n",
+ "$$f_{\\tau\\phi}(x, t)$$\n",
+ "$$f_{\\tau\\theta}(x, t)$$\n",
+ "\n",
+ "we can write, for our state $x = [\\phi, \\phi', \\theta, \\theta']$, the equations as a function $dx(x, p, t)$, where the parameter values in $p$ contain the physical parameters $\\alpha, \\beta, \\gamma, \\delta$, as well as the policies $f_{\\tau\\phi}$ and $f_{\\tau\\theta}$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "dx (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function dx(x, p, t)\n",
+ " (ϕ, ϕ′, θ, θ′) = x\n",
+ " (α, β, γ, δ, fτϕ, fτθ) = p\n",
+ " τϕ = fτϕ(x, t)\n",
+ " τθ = fτθ(x, t)\n",
+ " cosθ = cos(θ)\n",
+ " sinθ = sin(θ)\n",
+ " denominator = α*β - γ^2 + (β^2 + γ^2)*sinθ^2\n",
+ " @assert denominator != 0.0\n",
+ " dϕ = ϕ′\n",
+ " dϕ′ = (β*γ*(sinθ^2-1)*sinθ*ϕ′^2 - 2*β^2*cosθ*sinθ*ϕ′*θ′+β*γ*sinθ*θ′^2-γ*δ*cosθ*sinθ+β*τϕ-γ*cosθ*τθ) / denominator\n",
+ " dθ = θ′\n",
+ " dθ′ = (β*(α + β*sinθ^2)*cosθ*sinθ*ϕ′^2+2*β*γ*(1 - sinθ^2)*sinθ*ϕ′*θ′-γ^2*cosθ*sinθ*θ′^2+δ*(α+β*sinθ^2)*sinθ-γ*cosθ*τϕ + (α + β*sinθ^2)*τθ)/denominator\n",
+ "\n",
+ " [dϕ, dϕ′, dθ, dθ′]\n",
+ "end "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "4-element Array{Float64,1}:\n",
+ " 0.0\n",
+ " 0.9999971666746945\n",
+ " 0.0\n",
+ " -0.1499995750012042"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "fτϕ(x, t) = 1\n",
+ "fτθ(x, t) = 0\n",
+ "\n",
+ "dx([0.0, 0.0, 0.0, 0.0], (α, β, γ, δ, fτϕ, fτθ), 0.0)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We begin with an arbitrary choice of paramters, including a constant control signal"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "fτθ (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "# α = 1.0\n",
+ "# β = 1.2\n",
+ "# γ = 0.1\n",
+ "# δ = 0.85\n",
+ "\n",
+ "fτϕ(x, t) = 0.2 # constant control policy\n",
+ "fτθ(x, t) = 0.0 # No motor for pendulum"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Adding a timespan and an initial condition, we can run a simulated run."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "t_max = 5\n",
+ "t_span = (0.0, t_max)\n",
+ "x_init = [0.0, 0.0, π, 0.0];"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "ode_problem = ODEProblem(dx, x_init, t_span)\n",
+ "ode_sol = solve(ode_problem, Tsit5(), reltol=1e-8, abstol=1e-8, p=(α, β, γ, δ, fτϕ, fτθ));"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We extract the states as function of time, and plot the result"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "ts = range(0, t_max, length=800)\n",
+ "xs = ode_sol.(ts);"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "ϕs = [ x[1] for x in xs ]\n",
+ "θs = [ x[3] for x in xs ];"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
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+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "plt_ϕ = plot(ts, ϕs, label = L\"\\phi\")\n",
+ "plt_θ = plot(ts, θs, label = L\"\\theta\")\n",
+ "\n",
+ "plot(plt_ϕ, plt_θ, layout=(1, 2), xlabel = \"time\", ylabel = \"angle\", legend = :bottomright)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We see that $\\phi$ increases approximately quadratically with time, whereas $\\theta$ has an oscillatory behaviour (that looks different for different parameter values)."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Identifying parameters\n",
+ "\n",
+ "We will assume that the parameters are unknown, and try to identify them.\n",
+ "\n",
+ "We do this by running experiments with a number of different policies, gathering the data, and performing least-squares estimates of the parameters $\\alpha$, $\\beta$, $\\gamma$ and $\\delta$."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**TODO:** Design experiment:\n",
+ " 1. Choose 10 different open-loop policies\n",
+ " 1. Run simulations, gather data\n",
+ " 1. Set up least-squares fit problem (matrices)\n",
+ " 1. Solve it \n",
+ " 1. Evaluate solution by plotting it"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### A set of open-loop policies"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We choose 5 policies with constant torques of 0.1, 0.2, 0.3, 0.4, 0.5, and 5 policies of torques $\\sin \\omega t$ for angular velocities $\\omega$ set to 1, 2, 3, 4, 5."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**TODO:** Consider other policies. As shown by the plots, I do not see many different kinds of behaviour from my policies."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We construct our set of policies as an an array of functions"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "fτϕs = vcat([ (x, t) -> k for k ∈ [0.1, 0.2, 0.3, 0.4, 0.5]], [ (x, t) -> sin(ω*t) for ω ∈ [1, 2, 3, 4, 5]]);"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "For each policy, we solve the dynamical system, and obtain the time evolution for that policy"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "sols = [ solve(ode_problem, Tsit5(), reltol=1e-8, abstol=1e-8, p=(α, β, γ, δ, fτϕ, fτθ)) for fτϕ ∈ fτϕs ];"
+ ]
+ },
+ {
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+ "<path clip-path=\"url(#clip030)\" d=\"M 0 0 M2297.28 149.584 Q2297.28 150.989 2296.74 153.449 Q2296.22 155.908 2295.8 157.079 Q2294.63 160.194 2292.29 162.818 Q2291 164.27 2289.59 165.324 Q2288.21 166.354 2287.18 166.87 Q2286.15 167.362 2284.93 167.619 Q2283.72 167.9 2283.15 167.924 Q2282.59 167.971 2281.8 167.971 L2279.97 175.114 Q2279.59 176.965 2279.41 177.082 Q2279.36 177.176 2279.24 177.223 Q2279.15 177.269 2279.05 177.269 L2278.96 177.269 Q2278.7 177.269 2278.56 177.152 Q2278.42 177.035 2278.4 176.895 L2278.4 176.777 Q2278.4 176.45 2279.5 172.234 Q2280.06 170.079 2280.55 167.854 Q2276.81 167.619 2274.49 165.956 Q2272.19 164.293 2272.19 161.061 Q2272.19 159.07 2274.51 152.957 Q2275.61 150.029 2275.61 148.811 Q2275.61 148.272 2275.49 147.944 Q2275.38 147.593 2275.14 147.476 Q2274.91 147.335 2274.77 147.312 Q2274.65 147.289 2274.42 147.289 Q2273.04 147.289 2271.75 148.671 Q2270.46 150.052 2269.57 153.238 Q2269.38 153.823 2269.26 153.964 Q2269.15 154.105 2268.8 154.105 Q2268.21 154.081 2268.21 153.613 Q2268.21 153.425 2268.42 152.699 Q2268.63 151.95 2269.12 150.849 Q2269.64 149.725 2270.34 148.717 Q2271.04 147.687 2272.14 146.961 Q2273.27 146.235 2274.56 146.235 Q2276.31 146.235 2277.39 147.382 Q2278.49 148.507 2278.49 150.17 Q2278.49 150.498 2278.42 150.872 Q2278.35 151.247 2278.28 151.458 Q2278.23 151.669 2278.09 152.067 Q2277.98 152.442 2277.95 152.512 Q2276.76 155.604 2276.15 157.618 Q2275.54 159.609 2275.45 160.194 Q2275.38 160.757 2275.38 161.436 Q2275.38 166.167 2280.83 166.8 L2288.8 135.109 Q2288.94 134.453 2289.06 134.313 Q2289.17 134.149 2289.52 134.149 Q2290.09 134.149 2290.09 134.664 Q2290.09 134.828 2289.95 135.39 L2282.03 166.893 Q2282.78 166.893 2283.25 166.87 Q2283.72 166.846 2284.7 166.636 Q2285.71 166.401 2286.55 165.98 Q2287.42 165.558 2288.63 164.715 Q2289.85 163.848 2291.05 162.63 Q2292.55 161.108 2293.88 158.555 Q2295.22 156.002 2295.22 153.894 Q2295.22 151.528 2293.51 149.818 Q2293.25 149.584 2293.11 149.443 Q2292.99 149.303 2292.87 149.069 Q2292.78 148.835 2292.78 148.53 Q2292.78 147.734 2293.53 146.984 Q2294.28 146.235 2295.17 146.235 Q2295.97 146.235 2296.62 147.008 Q2297.28 147.78 2297.28 149.584 Z\" fill=\"#000000\" fill-rule=\"evenodd\" fill-opacity=\"1\" /></svg>\n"
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "for sol in sols\n",
+ " ts = range(sol.t[1], sol.t[end], length = 100)\n",
+ " xs = sol.(ts)\n",
+ " ϕs = [ x[1] for x in xs ]\n",
+ " θs = [ x[3] for x in xs ]\n",
+ " plt_ϕ = plot(ts, ϕs, label=L\"\\phi\")\n",
+ " plt_θ = plot(ts, θs, label=L\"\\psi\")\n",
+ " display(plot(plt_ϕ, plt_θ, layout = (1,2)))\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We want to estimate the parameter values, by minimizing the squared differences betwee expected and observed increments in the state. This approach assumes that we have access to the model.\n",
+ "\n",
+ "We define our cost function to be minimized as follows:\n",
+ " * For a given 4-tuple of parameters $\\alpha$, $\\beta$, $\\gamma$, $\\delta$\n",
+ " * For each policy, and its associated solution\n",
+ " * We compute the vectors of observed dx/dt and predicted dx/dt (from the model and the parameters given)\n",
+ " * The sums of squared differences for each policy are added to form the cost"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "cost (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 15,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function cost(ps)\n",
+ " α, β, γ, δ = ps\n",
+ " total_cost = 0.0\n",
+ " for (sol, fτϕ) in zip(sols, fτϕs)\n",
+ " xs = sol.u\n",
+ " ts = sol.t\n",
+ " dxs = xs[2:end] - xs[1:end-1]\n",
+ " dts = ts[2:end] - ts[1:end-1]\n",
+ " dxdts = dxs ./ dts\n",
+ " dxdts_pred = [ dx(x, (α, β, γ, δ, fτϕ, fτθ), t) for (x, t) in zip(xs, ts) ][2:end]\n",
+ " diffs = vcat((dxdts - dxdts_pred)...)\n",
+ " total_cost += sum( diffs .^ 2.0 )\n",
+ " end\n",
+ " total_cost\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We compute our parameters by minimizing our cost, starting from a random guess."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ " * Status: success\n",
+ "\n",
+ " * Candidate solution\n",
+ " Final objective value: 1.075901e+00\n",
+ "\n",
+ " * Found with\n",
+ " Algorithm: Nelder-Mead\n",
+ "\n",
+ " * Convergence measures\n",
+ " √(Σ(yᵢ-ȳ)²)/n ≤ 1.0e-08\n",
+ "\n",
+ " * Work counters\n",
+ " Seconds run: 1 (vs limit Inf)\n",
+ " Iterations: 273\n",
+ " f(x) calls: 489\n"
+ ]
+ },
+ "execution_count": 16,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "ps_fit = optimize(cost, rand(4))"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Our solution should be compared to the parameters we chose at the beginning, and we have a set of parameters that we can use for further experiments."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(1.0022636705292638, 0.007371341077970709, 0.0010905383749426603, 1.0877988343066645)"
+ ]
+ },
+ "execution_count": 17,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "(α′, β′, γ′, δ′) = ps_fit.minimizer\n",
+ "(α′, β′, γ′, δ′)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(1.0000103333333332, 0.0003333333333333334, 5.0000000000000016e-5, 0.05)"
+ ]
+ },
+ "execution_count": 18,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "(α, β, γ, δ)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Controlling based on parameters\n",
+ "\n",
+ " We want to start at rest, swing the pendulum up, and then stop and hold it in an upright position."
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "**TODO:** Design method\n",
+ " 1. Linearize locally\n",
+ " 1. Cost function with total energy\n",
+ " 1. Switch to cost function with deviation from upright position"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": []
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We compute a linear model by linearizing to get a state-space description\n",
+ "$$dx = A*x + B*u$$\n",
+ "\n",
+ "\n",
+ "We assume a constant policy with a torque $\\tau$, and linearize the response around a state $x$ \n",
+ "\n",
+ "$$A_{ij} = \\frac{\\partial dx_{i}}{\\partial x_{j}}$$\n",
+ "$$B_{i} = \\frac{\\partial dx_{i}}{\\partial \\tau}$$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "AB (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 19,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function AB(x, τ)\n",
+ " \n",
+ " # The policy must be an array to enable gradients\n",
+ " τ_vec = [τ]\n",
+ " fτϕ(x, t) = τ_vec[1]\n",
+ " p = vcat(ps_fit.minimizer, fτϕ, fτθ)\n",
+ " \n",
+ " t = 0.0 # The system is time invariant\n",
+ " \n",
+ " A = zeros(4, 4)\n",
+ " for row in 1:4\n",
+ " A[row, :] = gradient( x -> dx(x, p, t)[row], x )[1]\n",
+ " end\n",
+ " \n",
+ " B = zeros(4, 1)\n",
+ " for row in 1:4\n",
+ " B[row, 1] = gradient( () -> dx(x, p, t)[row], Params([τ_vec]))[τ_vec] |> only\n",
+ " end \n",
+ " \n",
+ " A, B\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 20,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "BenchmarkTools.Trial: \n",
+ " memory estimate: 230.52 KiB\n",
+ " allocs estimate: 8463\n",
+ " --------------\n",
+ " minimum time: 2.244 ms (0.00% GC)\n",
+ " median time: 2.269 ms (0.00% GC)\n",
+ " mean time: 2.326 ms (0.98% GC)\n",
+ " maximum time: 10.857 ms (76.69% GC)\n",
+ " --------------\n",
+ " samples: 2147\n",
+ " evals/sample: 1"
+ ]
+ },
+ "execution_count": 20,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "@benchmark AB([0.0, 0.0, 0.0, 0.0], 0.0)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 21,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "([0.0 1.0 0.0 0.0; 0.0 0.0 -0.1605946087245449 0.0; 0.0 0.0 0.0 1.0; 0.0 0.0 147.5951197186769 0.0], [0.0; 0.997902077530332; 0.0; -0.1476326354283178])"
+ ]
+ },
+ "execution_count": 21,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "A, B = AB([0.0, 0.0, 0.0, 0.0], 0.0)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 22,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "4×4 Array{Float64,2}:\n",
+ " 0.0 1.0 0.0 0.0\n",
+ " 0.0 0.0 -0.160595 0.0\n",
+ " 0.0 0.0 0.0 1.0\n",
+ " 0.0 0.0 147.595 0.0"
+ ]
+ },
+ "execution_count": 22,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "A"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 23,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "4×1 Array{Float64,2}:\n",
+ " 0.0\n",
+ " 0.997902077530332\n",
+ " 0.0\n",
+ " -0.1476326354283178"
+ ]
+ },
+ "execution_count": 23,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "B"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We use a cost function that contains both the energy and the state, as we need sufficient energy to reach an upright state.\n",
+ "\n",
+ "Our policy will be of the form $u = -Kx$ for some $K$ that will be optimized to minimize the cost over some time-span in the future "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "state_cost (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 24,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function energy(x)\n",
+ " (ϕ, ϕ′, θ, θ′) = x\n",
+ " sinθ = sin(θ)\n",
+ " cosθ = cos(θ)\n",
+ " T_c = J*ϕ′^2/2\n",
+ " V_c = 0\n",
+ " T_a = m_a*l_a^2*ϕ′^2/6\n",
+ " V_a = 0\n",
+ " T_p = m_p*(l_a^2+l_p^2*sinθ^2)*ϕ′^2/2 + m_p*l_a*l_p*cosθ*ϕ′*θ′/2+m_p*l_p^2*θ′^2/2\n",
+ " V_p = m_p*g*l_p*cosθ/2\n",
+ " T_m = M*(l_a^2 + l_p^2*sinθ^2)*ϕ′^2/2 + M*l_a*l_p*cosθ*ϕ′*θ′ + M*l_p^2*θ′^2\n",
+ " V_m = M*g*l_p*cosθ\n",
+ " T_c + T_a + T_p + T_m + V_c + V_a + V_p + V_m\n",
+ "end\n",
+ "\n",
+ "function state_cost(x, target; energy_cost_factor = 1.0)\n",
+ " # Compute total energy at target state\n",
+ " e_target = energy(target)\n",
+ " # Compute total energy at current state\n",
+ " e_current = energy(x)\n",
+ " # Compute state-diff between current and target\n",
+ " x_diff = target - x\n",
+ " # \n",
+ " cost = sum(x_diff .^ 2) + energy_cost_factor * (e_current - e_target)^2\n",
+ " cost\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "We now define a differential equation for the combination of state and cost, with parameters $A$ and $B$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 25,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "generate_policy (generic function with 3 methods)"
+ ]
+ },
+ "execution_count": 25,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function d_cx(cx, p, t)\n",
+ " (c, ϕ, ϕ′, θ, θ′) = cx\n",
+ " (α, β, γ, δ, energy_cost_factor, target, A, B, K) = p\n",
+ " x = [ϕ, ϕ′, θ, θ′]\n",
+ " dc = state_cost(x, target, energy_cost_factor=energy_cost_factor)\n",
+ " dx = (A - B*K)*x\n",
+ " vcat(dc, dx)\n",
+ "end\n",
+ "\n",
+ "function compute_K(A, B; Δt = 50*10^-3)\n",
+ " function policy_cost(K)\n",
+ " cx_init = [0.0, 0.0, 0.0, π, 0.0]\n",
+ " target = [0.0, 0.0, 0.0, 0.0]\n",
+ " p = (α′, β′, γ′, δ′, 10, target, A, B, K)\n",
+ " cx_problem = ODEProblem(d_cx, cx_init, (0.0, Δt))\n",
+ " cx_solution = solve(cx_problem, Tsit5(), reltol=1e-8, abstol=1e-8, p=p)\n",
+ " cx_solution.u[end][1]\n",
+ " end\n",
+ " \n",
+ " opt = optimize(policy_cost, zeros(1, 4))\n",
+ " \n",
+ " # Set up an ODE problem computing the cost\n",
+ " # Minimize the cost w.r.t. the parameter K\n",
+ " # Return the minimizer\n",
+ "end\n",
+ "\n",
+ "function generate_policy(A, B, target = [0.0, 0.0, 0.0, 0.0], energy_cost_factor = 1.0)\n",
+ " K_sol = compute_K(A, B)\n",
+ " K = K_sol.minimizer\n",
+ " function fτϕ(x, t)\n",
+ " -K*x\n",
+ " end\n",
+ " fτϕ\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 26,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "(::var\"#fτϕ#32\"{Array{Float64,2}}) (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 26,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f = generate_policy(A, B)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 27,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "1-element Array{Float64,1}:\n",
+ " -3836.5017811718617"
+ ]
+ },
+ "execution_count": 27,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "f([1.0, 2.0, 3.0, 4.0], 0.0)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Test on real system"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Do the same thing on the real process"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.4.2",
+ "language": "julia",
+ "name": "julia-1.4"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.4.2"
+ },
+ "toc-autonumbering": true,
+ "toc-showtags": false
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}