diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000000000000000000000000000000000000..763513e910f7036a1a3cdb21dce8e57da2451891
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1 @@
+.ipynb_checkpoints
diff --git a/exploration_modeling_toolkit.ipynb b/exploration_modeling_toolkit.ipynb
deleted file mode 100644
index bad765bdc2a684c65b787a7cff6365356a8f41ea..0000000000000000000000000000000000000000
--- a/exploration_modeling_toolkit.ipynb
+++ /dev/null
@@ -1,196 +0,0 @@
-{
- "cells": [
- {
- "cell_type": "code",
- "execution_count": 1,
- "metadata": {},
- "outputs": [
- {
- "name": "stderr",
- "output_type": "stream",
- "text": [
- "\u001b[32m\u001b[1m Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General`\n",
- "\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
- "\u001b[32m\u001b[1mNo Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.5/Project.toml`\n",
- "\u001b[32m\u001b[1mNo Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.5/Manifest.toml`\n",
- "\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
- "\u001b[32m\u001b[1mNo Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.5/Project.toml`\n",
- "\u001b[32m\u001b[1mNo Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.5/Manifest.toml`\n"
- ]
- }
- ],
- "source": [
- "using Pkg\n",
- "Pkg.add(\"ModelingToolkit\")\n",
- "Pkg.add(\"Symbolics\")"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## Building a 1D-mass with two wheels"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "Here we first run the basic example from ModelingToolkit, see [here](https://mtk.sciml.ai/stable/tutorials/spring_mass/)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 1,
- "metadata": {},
- "outputs": [
- {
- "ename": "LoadError",
- "evalue": "\u001b[91mUndefVarError: scalarize not defined\u001b[39m",
- "output_type": "error",
- "traceback": [
- "\u001b[91mUndefVarError: scalarize not defined\u001b[39m",
- "",
- "Stacktrace:",
- " [1] include_string(::Function, ::Module, ::String, ::String) at ./loading.jl:1091"
- ]
- },
- {
- "name": "stderr",
- "output_type": "stream",
- "text": [
- "WARNING: both LinearAlgebra and Plots export \"rotate!\"; uses of it in module Main must be qualified\n"
- ]
- }
- ],
- "source": [
- "using ModelingToolkit, Plots, DifferentialEquations, LinearAlgebra\n",
- "using Symbolics:scalarize\n",
- "\n",
- "@variables t\n",
- "D = Differential(t)\n",
- "\n",
- "function Mass(; name, m = 1.0, xy = [0., 0.], u = [0., 0.])\n",
- " ps = @parameters m=m\n",
- " sts = @variables pos[1:2](t)=xy v[1:2](t)=u\n",
- " eqs = scalarize(D.(pos) .~ v)\n",
- " ODESystem(eqs, t, [pos..., v...], ps; name)\n",
- "end\n",
- "\n",
- "function Spring(; name, k = 1e4, l = 1.)\n",
- " ps = @parameters k=k l=l\n",
- " @variables x(t), dir[1:2](t)\n",
- " ODESystem(Equation[], t, [x, dir...], ps; name)\n",
- "end\n",
- "\n",
- "function connect_spring(spring, a, b)\n",
- " [\n",
- " spring.x ~ norm(scalarize(a .- b))\n",
- " scalarize(spring.dir .~ scalarize(a .- b))\n",
- " ]\n",
- "end\n",
- "\n",
- "spring_force(spring) = -spring.k .* scalarize(spring.dir) .* (spring.x - spring.l) ./ spring.x\n",
- "\n",
- "m = 1.0\n",
- "xy = [1., -1.]\n",
- "k = 1e4\n",
- "l = 1.\n",
- "center = [0., 0.]\n",
- "g = [0., -9.81]\n",
- "@named mass = Mass(m=m, xy=xy)\n",
- "@named spring = Spring(k=k, l=l)\n",
- "\n",
- "eqs = [\n",
- " connect_spring(spring, mass.pos, center)\n",
- " scalarize(D.(mass.v) .~ spring_force(spring) / mass.m .+ g)\n",
- "]\n",
- "\n",
- "@named _model = ODESystem(eqs, t)\n",
- "@named model = compose(_model, mass, spring)\n",
- "sys = structural_simplify(model)\n",
- "\n",
- "prob = ODEProblem(sys, [], (0., 3.))\n",
- "sol = solve(prob, Rosenbrock23())\n",
- "plot(sol)"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "Let's do something similar but with mass and wheels"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 2,
- "metadata": {},
- "outputs": [],
- "source": [
- "scala"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 3,
- "metadata": {},
- "outputs": [
- {
- "ename": "LoadError",
- "evalue": "\u001b[91mUndefVarError: scalarize not defined\u001b[39m",
- "output_type": "error",
- "traceback": [
- "\u001b[91mUndefVarError: scalarize not defined\u001b[39m",
- "",
- "Stacktrace:",
- " [1] include_string(::Function, ::Module, ::String, ::String) at ./loading.jl:1091"
- ]
- }
- ],
- "source": [
- "using ModelingToolkit, Plots, DifferentialEquations, LinearAlgebra\n",
- "using Symbolics:scalarize\n",
- "@variables t\n",
- "D = Differential(t)\n",
- "function Wheel(; name, r=1., m=1.)\n",
- " params = @parameters m=m r=r\n",
- " states = @variables pos[1:2](t)=xz v[1:2]=u\n",
- " \n",
- "end"
- ]
- }
- ],
- "metadata": {
- "@webio": {
- "lastCommId": null,
- "lastKernelId": null
- },
- "kernelspec": {
- "display_name": "Julia 1.5.2",
- "language": "julia",
- "name": "julia-1.5"
- },
- "language_info": {
- "file_extension": ".jl",
- "mimetype": "application/julia",
- "name": "julia",
- "version": "1.5.2"
- },
- "toc": {
- "base_numbering": 1,
- "nav_menu": {},
- "number_sections": true,
- "sideBar": true,
- "skip_h1_title": false,
- "title_cell": "Table of Contents",
- "title_sidebar": "Contents",
- "toc_cell": false,
- "toc_position": {},
- "toc_section_display": true,
- "toc_window_display": false
- }
- },
- "nbformat": 4,
- "nbformat_minor": 4
-}
diff --git a/exploration_modia3D.ipynb b/exploration_modia3D.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..3c9ddf0db8b4e0cb5f452555cc9504d879bb0bbb
--- /dev/null
+++ b/exploration_modia3D.ipynb
@@ -0,0 +1,95 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "d47c9765-5d14-4e9d-a42e-1623da8bf5ae",
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "\n",
+ "... Info: ENV[\"DLR_VISUALIZATION\"] is not defined.\n",
+ " Therefore, no online renderer is used in Modia3D.\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "using Modia3D"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "964c5cea-1e33-4fa0-be73-68acff0fd5a2",
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "LoadError",
+ "evalue": "TypeError: in keyword argument parent, expected Union{Nothing, Object3D}, got a value of type Symbol",
+ "output_type": "error",
+ "traceback": [
+ "TypeError: in keyword argument parent, expected Union{Nothing, Object3D}, got a value of type Symbol",
+ "",
+ "Stacktrace:",
+ " [1] top-level scope",
+ " @ In[2]:1",
+ " [2] eval",
+ " @ ./boot.jl:373 [inlined]",
+ " [3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
+ " @ Base ./loading.jl:1196"
+ ]
+ }
+ ],
+ "source": [
+ "Pendulum = Model3D(\n",
+ " world = Object3D(feature=Scene()),\n",
+ " body = Object3D(feature=Solid(massProperties=MassProperties(mass=1.0))),\n",
+ " bodyFrame = Object3D(parent=:body, translation=[-0.5, 0.0, 0.0]),\n",
+ " rev = Revolute(obj1=:world, obj2=:bodyFrame)\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "6584c04b-be0e-49f4-a7ac-d4ad06c404a5",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "pendulum = @instantiateModel(Pendulum, unitless=true)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "976e8115-66fb-4bea-a5cf-59dfeffc7224",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "simulate!(pendulum, stopTime=3.0)\n",
+ "\n",
+ "@usingModiaPlot # use the plot package defined by ENV[\"MODIA_PLOT\"]\n",
+ "plot(pendulum, \"rev.phi\")"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.7.2",
+ "language": "julia",
+ "name": "julia-1.7"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.7.2"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/jonas_test.ipynb b/jonas_test.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..7afd282d030ae0697b77202918923afa0a219290
--- /dev/null
+++ b/jonas_test.ipynb
@@ -0,0 +1,4135 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
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+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "using ModelingToolkit\n",
+ "\n",
+ "@variables t x(t) RHS(t) # independent and dependent variables\n",
+ "@parameters τ # parameters\n",
+ "D = Differential(t) # define an operator for the differentiation w.r.t. time\n",
+ "\n",
+ "# your first ODE, consisting of a single equation, indicated by ~\n",
+ "@named fol_separate = ODESystem([ RHS ~ (1 - x)/τ,\n",
+ " D(x) ~ RHS ])\n",
+ "\n",
+ "using DifferentialEquations: solve\n",
+ "using Plots: plot\n",
+ "\n",
+ "prob = ODEProblem(structural_simplify(fol_separate), [x => 0.0], (0.0,10.0), [τ => 3.0])\n",
+ "sol = solve(prob)\n",
+ "plot(sol, vars=[x,RHS])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 23,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\frac{dxˍt(t)}{dt} =& \\frac{1 - x\\left( t \\right)}{\\tau} \\\\\n",
+ "\\frac{dx(t)}{dt} =& \\mathrm{xˍt}\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "\u001b[0m\u001b[1mModel fol_model with 2 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
+ "\u001b[0m\u001b[1mStates (2):\u001b[22m\n",
+ " xˍt(t)\n",
+ " x(t)\n",
+ "\u001b[0m\u001b[1mParameters (1):\u001b[22m\n",
+ " τ\n",
+ "\u001b[35mIncidence matrix:\u001b[39msparse([1, 2, 2, 1], [1, 2, 3, 4], Num[×, ×, ×, ×], 2, 4)"
+ ]
+ },
+ "execution_count": 23,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "using ModelingToolkit\n",
+ "\n",
+ "@variables t x(t) # independent and dependent variables\n",
+ "@parameters τ # parameters\n",
+ "D = Differential(t) # define an operator for the differentiation w.r.t. time\n",
+ "\n",
+ "# your first ODE, consisting of a single equation, indicated by ~\n",
+ "@named fol_model = ODESystem(D(D(x)) ~ (1 - x)/τ)\n",
+ " # Model fol_model with 1 equations\n",
+ " # States (1):\n",
+ " # x(t)\n",
+ " # Parameters (1):\n",
+ " # τ\n",
+ "structural_simplify(fol_model)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 34,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\frac{dxˍt(t)}{dt} =& \\frac{1 - x\\left( t \\right)}{\\tau} \\\\\n",
+ "\\frac{dx(t)}{dt} =& \\mathrm{xˍt}\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "\u001b[0m\u001b[1mModel fol_model with 2 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
+ "\u001b[0m\u001b[1mStates (2):\u001b[22m\n",
+ " xˍt(t)\n",
+ " x(t)\n",
+ "\u001b[0m\u001b[1mParameters (1):\u001b[22m\n",
+ " τ\n",
+ "\u001b[35mIncidence matrix:\u001b[39msparse([1, 2, 2, 1], [1, 2, 3, 4], Num[×, ×, ×, ×], 2, 4)"
+ ]
+ },
+ "execution_count": 34,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "connect"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 32,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "LoadError",
+ "evalue": "UndefVarError: xt not defined",
+ "output_type": "error",
+ "traceback": [
+ "UndefVarError: xt not defined",
+ "",
+ "Stacktrace:",
+ " [1] top-level scope",
+ " @ In[32]:1",
+ " [2] eval",
+ " @ ./boot.jl:373 [inlined]",
+ " [3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
+ " @ Base ./loading.jl:1196"
+ ]
+ }
+ ],
+ "source": [
+ "prob = ODEProblem(structural_simplify(fol_model), [x => 0, x_t => 0], (0.0,10.0), [τ => 3.0])\n",
+ "plot(solve(prob))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 35,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\frac{dx(t)}{dt} =& \\frac{ - x\\left( t \\right) + f\\left( t \\right)}{\\tau} \\\\\n",
+ "f\\left( t \\right) =& \\sin\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "\u001b[0m\u001b[1mModel fol_variable_f with 2 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
+ "\u001b[0m\u001b[1mStates (2):\u001b[22m\n",
+ " x(t)\n",
+ " f(t)\n",
+ "\u001b[0m\u001b[1mParameters (1):\u001b[22m\n",
+ " τ"
+ ]
+ },
+ "execution_count": 35,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "@variables f(t)\n",
+ "@named fol_variable_f = ODESystem([f ~ sin(t), D(x) ~ (f - x)/τ])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 36,
+ "metadata": {},
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+ {
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+ " 1494.04,1384.24 1496.25,1384.24 1498.46,1384.24 1500.67,1384.24 1502.87,1384.24 1505.08,1384.24 1507.29,1384.24 1509.5,1384.24 1511.7,1384.24 1513.91,1384.24 \n",
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+ " 1626.49,1384.24 1628.7,1384.24 1630.91,1384.24 1633.12,1384.24 1635.32,1384.24 1637.53,1384.24 1639.74,1384.24 1641.95,1384.24 1644.15,1384.24 1646.36,1384.24 \n",
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+ " 1670.64,1384.24 1672.85,1384.24 1675.06,1384.24 1677.27,1384.24 1679.47,1384.24 1681.68,1384.24 1683.89,1384.24 1686.1,1384.24 1688.3,1384.24 1690.51,1384.24 \n",
+ " 1692.72,964.813 1694.93,964.813 1697.13,964.813 1699.34,964.813 1701.55,964.813 1703.76,964.813 1705.96,964.813 1708.17,964.813 1710.38,964.813 1712.59,964.813 \n",
+ " 1714.79,964.813 1717,964.813 1719.21,964.813 1721.42,964.813 1723.62,964.813 1725.83,964.813 1728.04,964.813 1730.25,964.813 1732.45,964.813 1734.66,964.813 \n",
+ " 1736.87,964.813 1739.07,964.813 1741.28,964.813 1743.49,964.813 1745.7,964.813 1747.9,964.813 1750.11,964.813 1752.32,964.813 1754.53,964.813 1756.73,964.813 \n",
+ " 1758.94,964.813 1761.15,964.813 1763.36,964.813 1765.56,964.813 1767.77,964.813 1769.98,964.813 1772.19,964.813 1774.39,964.813 1776.6,964.813 1778.81,964.813 \n",
+ " 1781.02,964.813 1783.22,964.813 1785.43,964.813 1787.64,964.813 1789.85,964.813 1792.05,964.813 1794.26,964.813 1796.47,964.813 1798.68,964.813 1800.88,964.813 \n",
+ " 1803.09,964.813 1805.3,964.813 1807.51,964.813 1809.71,964.813 1811.92,964.813 1814.13,964.813 1816.34,964.813 1818.54,964.813 1820.75,964.813 1822.96,964.813 \n",
+ " 1825.17,964.813 1827.37,964.813 1829.58,964.813 1831.79,964.813 1834,964.813 1836.2,964.813 1838.41,964.813 1840.62,964.813 1842.83,964.813 1845.03,964.813 \n",
+ " 1847.24,964.813 1849.45,964.813 1851.66,964.813 1853.86,964.813 1856.07,964.813 1858.28,964.813 1860.49,964.813 1862.69,964.813 1864.9,964.813 1867.11,964.813 \n",
+ " 1869.32,964.813 1871.52,964.813 1873.73,964.813 1875.94,964.813 1878.15,964.813 1880.35,964.813 1882.56,964.813 1884.77,964.813 1886.98,964.813 1889.18,964.813 \n",
+ " 1891.39,964.813 1893.6,964.813 1895.81,964.813 1898.01,964.813 1900.22,964.813 1902.43,964.813 1904.64,964.813 1906.84,964.813 1909.05,964.813 1911.26,964.813 \n",
+ " 1913.47,1076.1 1915.67,1076.1 1917.88,1076.1 1920.09,1076.1 1922.3,1076.1 1924.5,1076.1 1926.71,1076.1 1928.92,1076.1 1931.13,1076.1 1933.33,1076.1 \n",
+ " 1935.54,1076.1 1937.75,1076.1 1939.96,1076.1 1942.16,1076.1 1944.37,1076.1 1946.58,1076.1 1948.79,1076.1 1950.99,1076.1 1953.2,1076.1 1955.41,1076.1 \n",
+ " 1957.62,1076.1 1959.82,1076.1 1962.03,1076.1 1964.24,1076.1 1966.45,1076.1 1968.65,1076.1 1970.86,1076.1 1973.07,1076.1 1975.28,1076.1 1977.48,1076.1 \n",
+ " 1979.69,1076.1 1981.9,1076.1 1984.11,1076.1 1986.31,1076.1 1988.52,1076.1 1990.73,1076.1 1992.94,1076.1 1995.14,1076.1 1997.35,1076.1 1999.56,1076.1 \n",
+ " 2001.77,1076.1 2003.97,1076.1 2006.18,1076.1 2008.39,1076.1 2010.6,1076.1 2012.8,1076.1 2015.01,1076.1 2017.22,1076.1 2019.43,1076.1 2021.63,1076.1 \n",
+ " 2023.84,1076.1 2026.05,1076.1 2028.26,1076.1 2030.46,1076.1 2032.67,1076.1 2034.88,1076.1 2037.09,1076.1 2039.29,1076.1 2041.5,1076.1 2043.71,1076.1 \n",
+ " 2045.92,1076.1 2048.12,1076.1 2050.33,1076.1 2052.54,1076.1 2054.75,1076.1 2056.95,1076.1 2059.16,1076.1 2061.37,1076.1 2063.58,1076.1 2065.78,1076.1 \n",
+ " 2067.99,1076.1 2070.2,1076.1 2072.41,1076.1 2074.61,1076.1 2076.82,1076.1 2079.03,1076.1 2081.24,1076.1 2083.44,1076.1 2085.65,1076.1 2087.86,1076.1 \n",
+ " 2090.07,1076.1 2092.27,1076.1 2094.48,1076.1 2096.69,1076.1 2098.9,1076.1 2101.1,1076.1 2103.31,1076.1 2105.52,1076.1 2107.73,1076.1 2109.93,1076.1 \n",
+ " 2112.14,1076.1 2114.35,1076.1 2116.55,1076.1 2118.76,1076.1 2120.97,1076.1 2123.18,1076.1 2125.38,1076.1 2127.59,1076.1 2129.8,1076.1 2132.01,1076.1 \n",
+ " 2134.21,513.748 2136.42,513.748 2138.63,513.748 2140.84,513.748 2143.04,513.748 2145.25,513.748 2147.46,513.748 2149.67,513.748 2151.87,513.748 2154.08,513.748 \n",
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+ " 2178.36,513.748 2180.57,513.748 2182.78,513.748 2184.99,513.748 2187.19,513.748 2189.4,513.748 2191.61,513.748 2193.82,513.748 2196.02,513.748 2198.23,513.748 \n",
+ " 2200.44,513.748 2202.65,513.748 2204.85,513.748 2207.06,513.748 2209.27,513.748 2211.48,513.748 2213.68,513.748 2215.89,513.748 2218.1,513.748 2220.31,513.748 \n",
+ " 2222.51,513.748 2224.72,513.748 2226.93,513.748 2229.14,513.748 2231.34,513.748 2233.55,513.748 2235.76,513.748 2237.97,513.748 2240.17,513.748 2242.38,513.748 \n",
+ " 2244.59,513.748 2246.8,513.748 2249,513.748 2251.21,513.748 2253.42,513.748 2255.63,513.748 2257.83,513.748 2260.04,513.748 2262.25,513.748 2264.46,513.748 \n",
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+ " 2288.74,513.748 2290.95,513.748 2293.15,513.748 2295.36,513.748 2297.57,513.748 2299.78,513.748 2301.98,513.748 2304.19,513.748 2306.4,513.748 2308.61,513.748 \n",
+ " 2310.81,513.748 2313.02,513.748 2315.23,513.748 2317.44,513.748 2319.64,513.748 2321.85,513.748 2324.06,513.748 2326.27,513.748 2328.47,513.748 2330.68,513.748 \n",
+ " 2332.89,513.748 2335.1,513.748 2337.3,513.748 2339.51,513.748 2341.72,513.748 2343.93,513.748 2346.13,513.748 2348.34,513.748 2350.55,513.748 2352.76,513.748 \n",
+ " \n",
+ " \"/>\n",
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+ ]
+ },
+ "execution_count": 36,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "value_vector = randn(10)\n",
+ "f_fun(t) = t >= 10 ? value_vector[end] : value_vector[Int(floor(t))+1]\n",
+ "@register_symbolic f_fun(t)\n",
+ "\n",
+ "@named fol_external_f = ODESystem([f ~ f_fun(t), D(x) ~ (f - x)/τ])\n",
+ "prob = ODEProblem(structural_simplify(fol_external_f), [x => 0.0], (0.0,10.0), [τ => 0.75])\n",
+ "\n",
+ "sol = solve(prob)\n",
+ "plot(sol, vars=[x,f])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## heirarchical models"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 37,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "fol_factory (generic function with 2 methods)"
+ ]
+ },
+ "execution_count": 37,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function fol_factory(separate=false;name)\n",
+ " @parameters τ\n",
+ " @variables t x(t) f(t) RHS(t)\n",
+ "\n",
+ " eqs = separate ? [RHS ~ (f - x)/τ,\n",
+ " D(x) ~ RHS] :\n",
+ " D(x) ~(f - x)/τ\n",
+ "\n",
+ " ODESystem(eqs;name)\n",
+ "end\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 84,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "LoadError",
+ "evalue": "UndefKeywordError: keyword argument name not assigned",
+ "output_type": "error",
+ "traceback": [
+ "UndefKeywordError: keyword argument name not assigned",
+ "",
+ "Stacktrace:",
+ " [1] fol_factory(separate::Bool) (repeats 2 times)",
+ " @ Main ./In[37]:2",
+ " [2] top-level scope",
+ " @ In[84]:1",
+ " [3] eval",
+ " @ ./boot.jl:373 [inlined]",
+ " [4] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
+ " @ Base ./loading.jl:1196"
+ ]
+ }
+ ],
+ "source": [
+ "@named fol_1 = fol_factory()\n",
+ "@named fol_2 = fol_factory(true) # has observable RHS"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 85,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\mathrm{fol}_{1_{+}f}\\left( t \\right) =& 1.5 \\\\\n",
+ "\\mathrm{fol}_{2_{+}f}\\left( t \\right) =& \\mathrm{fol}_{1_{+}x}\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "2-element Vector{Equation}:\n",
+ " fol_1₊f(t) ~ 1.5\n",
+ " fol_2₊f(t) ~ fol_1₊x(t)"
+ ]
+ },
+ "execution_count": 85,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "connections = [ fol_1.f ~ 1.5,\n",
+ " fol_2.f ~ fol_1.x ]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 40,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\mathrm{fol}_{1_{+}f}\\left( t \\right) =& 1.5 \\\\\n",
+ "\\mathrm{fol}_{2_{+}f}\\left( t \\right) =& \\mathrm{fol}_{1_{+}x}\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "2-element Vector{Equation}:\n",
+ " fol_1₊f(t) ~ 1.5\n",
+ " fol_2₊f(t) ~ fol_1₊x(t)"
+ ]
+ },
+ "execution_count": 40,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "connections = [ fol_1.f ~ 1.5,\n",
+ " fol_2.f ~ fol_1.x ]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 41,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\mathrm{fol}_{1_{+}f}\\left( t \\right) =& 1.5 \\\\\n",
+ "\\mathrm{fol}_{2_{+}f}\\left( t \\right) =& \\mathrm{fol}_{1_{+}x}\\left( t \\right) \\\\\n",
+ "\\frac{dfol_{1_{+}x(t)}}{dt} =& \\frac{ - \\mathrm{fol}_{1_{+}x}\\left( t \\right) + \\mathrm{fol}_{1_{+}f}\\left( t \\right)}{fol_{1_+\\tau}} \\\\\n",
+ "\\frac{dfol_{2_{+}x(t)}}{dt} =& \\mathrm{fol}_{2_{+}RHS}\\left( t \\right) \\\\\n",
+ "\\mathrm{fol}_{2_{+}RHS}\\left( t \\right) =& \\frac{ - \\mathrm{fol}_{2_{+}x}\\left( t \\right) + \\mathrm{fol}_{2_{+}f}\\left( t \\right)}{fol_{2_+\\tau}}\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "\u001b[0m\u001b[1mModel connected with 5 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
+ "\u001b[0m\u001b[1mStates (5):\u001b[22m\n",
+ " fol_1₊f(t)\n",
+ " fol_2₊f(t)\n",
+ " fol_1₊x(t)\n",
+ " fol_2₊x(t)\n",
+ " fol_2₊RHS(t)\n",
+ "\u001b[0m\u001b[1mParameters (2):\u001b[22m\n",
+ " fol_1₊τ\n",
+ " fol_2₊τ"
+ ]
+ },
+ "execution_count": 41,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "connected = compose(ODESystem(connections,name=:connected), fol_1, fol_2)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 45,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\frac{dfol_{1_{+}x(t)}}{dt} =& \\frac{ - \\mathrm{fol}_{1_{+}x}\\left( t \\right) + \\mathrm{fol}_{1_{+}f}\\left( t \\right)}{fol_{1_+\\tau}} \\\\\n",
+ "\\frac{dfol_{2_{+}x(t)}}{dt} =& \\mathrm{fol}_{2_{+}RHS}\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "\u001b[0m\u001b[1mModel connected with 2 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
+ "\u001b[0m\u001b[1mStates (2):\u001b[22m\n",
+ " fol_1₊x(t)\n",
+ " fol_2₊x(t)\n",
+ "\u001b[0m\u001b[1mParameters (2):\u001b[22m\n",
+ " fol_1₊τ\n",
+ " fol_2₊τ\n",
+ "\u001b[35mIncidence matrix:\u001b[39msparse([1, 2, 1, 2, 2], [1, 2, 3, 3, 4], Num[×, ×, ×, ×, ×], 2, 4)"
+ ]
+ },
+ "execution_count": 45,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "connected_simp = structural_simplify(connected)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 46,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\frac{dfol_{1_{+}x(t)}}{dt} =& \\frac{ - \\mathrm{fol}_{1_{+}x}\\left( t \\right) + \\mathrm{fol}_{1_{+}f}\\left( t \\right)}{fol_{1_+\\tau}} \\\\\n",
+ "\\frac{dfol_{2_{+}x(t)}}{dt} =& \\mathrm{fol}_{2_{+}RHS}\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "2-element Vector{Equation}:\n",
+ " Differential(t)(fol_1₊x(t)) ~ (fol_1₊f(t) - fol_1₊x(t)) / fol_1₊τ\n",
+ " Differential(t)(fol_2₊x(t)) ~ fol_2₊RHS(t)"
+ ]
+ },
+ "execution_count": 46,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "equations(connected_simp)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
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+ " 1848.85,430.211 1851.06,429.398 1853.26,428.586 1855.46,427.776 1857.66,426.968 1859.86,426.162 1862.06,425.357 1864.26,424.554 1866.46,423.753 1868.66,422.953 \n",
+ " 1870.86,422.155 1873.06,421.359 1875.26,420.564 1877.46,419.771 1879.66,418.98 1881.86,418.19 1884.06,417.402 1886.26,416.615 1888.46,415.831 1890.66,415.048 \n",
+ " 1892.86,414.266 1895.06,413.486 1897.26,412.708 1899.46,411.932 1901.67,411.157 1903.87,410.384 1906.07,409.612 1908.27,408.843 1910.47,408.074 1912.67,407.308 \n",
+ " 1914.87,406.543 1917.07,405.779 1919.27,405.018 1921.47,404.258 1923.67,403.499 1925.87,402.742 1928.07,401.987 1930.27,401.233 1932.47,400.481 1934.67,399.731 \n",
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+ ]
+ },
+ "execution_count": 50,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "u0 = [fol_2.x => 1.0 ,\n",
+ " fol_1.x => 1.0]\n",
+ "\n",
+ "p = [ fol_1.τ => 2.0,\n",
+ " fol_2.τ => 4.0 ]\n",
+ "prob = ODEProblem(connected_simp, u0, (0.0,10.0), p)\n",
+ "plot(solve(prob))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 61,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ " 11.191 μs (181 allocations: 18.61 KiB)\n"
+ ]
+ }
+ ],
+ "source": [
+ "using BenchmarkTools\n",
+ "\n",
+ "@btime solve($prob);\n",
+ " # 251.300 μs (873 allocations: 31.18 KiB)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 64,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ " 12.464 μs (197 allocations: 20.05 KiB)\n"
+ ]
+ }
+ ],
+ "source": [
+ "prob_an = ODEProblem(connected_simp, u0, (0.0,10.0), p; jac=true, sparse=true)\n",
+ "\n",
+ "@btime solve($prob_an);"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Mass system"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 66,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stderr",
+ "output_type": "stream",
+ "text": [
+ "\u001b[32m\u001b[1m Resolving\u001b[22m\u001b[39m package versions...\n",
+ "\u001b[32m\u001b[1m Updating\u001b[22m\u001b[39m `~/.julia/environments/v1.7/Project.toml`\n",
+ " \u001b[90m [0c5d862f] \u001b[39m\u001b[92m+ Symbolics v4.5.1\u001b[39m\n",
+ "\u001b[32m\u001b[1m No Changes\u001b[22m\u001b[39m to `~/.julia/environments/v1.7/Manifest.toml`\n"
+ ]
+ }
+ ],
+ "source": [
+ "] add Symbolics"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 68,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
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+ " 1791.82,810.162 1792.47,804.236 1793.12,798.477 1793.77,792.962 1794.43,787.75 1795.08,782.873 1795.73,778.358 1796.39,774.225 1797.04,770.486 1797.69,767.143 \n",
+ " 1798.35,764.194 1799,761.626 1799.65,759.421 1800.3,757.552 1800.96,755.993 1801.61,754.724 1802.26,753.716 1802.92,752.942 1803.57,752.379 1804.22,752.003 \n",
+ " 1804.88,751.791 1805.53,751.723 1806.18,751.781 1806.83,751.946 1807.49,752.2 1808.14,752.525 1808.79,752.905 1809.45,753.327 1810.1,753.776 1810.75,754.241 \n",
+ " 1811.41,754.709 1812.06,755.172 1812.71,755.618 1813.36,756.041 1814.02,756.432 1814.67,756.784 1815.32,757.092 1815.98,757.352 1816.63,757.561 1817.28,757.718 \n",
+ " 1817.94,757.821 1818.59,757.87 1819.24,757.866 1819.9,757.81 1820.55,757.705 1821.2,757.555 1821.85,757.364 1822.51,757.141 1823.16,756.892 1823.81,756.626 \n",
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+ " 1837.53,775.771 1838.18,779.657 1838.83,783.923 1839.49,788.548 1840.14,793.499 1840.79,798.735 1841.44,804.211 1842.1,809.859 1842.75,815.598 1843.4,821.347 \n",
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+ " 1857.12,828.503 1857.77,821.127 1858.42,813.19 1859.08,804.753 1859.73,795.885 1860.38,786.659 1861.04,777.142 1861.69,767.41 1862.34,757.543 1862.99,747.619 \n",
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+ " 1889.77,888.405 1890.42,901.211 1891.07,913.16 1891.73,924.123 1892.38,933.978 1893.03,942.61 1893.69,949.916 1894.34,955.814 1894.99,960.226 1895.64,963.085 \n",
+ " 1896.3,964.345 1896.95,963.972 1897.6,961.952 1898.26,958.288 1898.91,952.998 1899.56,946.118 1900.22,937.7 1900.87,927.801 1901.52,916.491 1902.18,903.852 \n",
+ " 1902.83,889.977 1903.48,874.969 1904.13,858.944 1904.79,842.025 1905.44,824.349 1906.09,806.062 1906.75,787.322 1907.4,768.272 1908.05,749.063 1908.71,729.854 \n",
+ " 1909.36,710.802 1910.01,692.059 1910.66,673.775 1911.32,656.098 1911.97,639.17 1912.62,623.133 1913.28,608.123 1913.93,594.275 1914.58,581.712 1915.24,570.527 \n",
+ " 1915.89,560.821 1916.54,552.684 1917.19,546.191 1917.85,541.405 1918.5,538.376 1919.15,537.14 1919.81,537.719 1920.46,540.122 1921.11,544.341 1921.77,550.364 \n",
+ " 1922.42,558.17 1923.07,567.725 1923.72,578.98 1924.38,591.874 1925.03,606.332 1925.68,622.263 1926.34,639.567 1926.99,658.126 1927.64,677.814 1928.3,698.501 \n",
+ " 1928.95,720.046 1929.6,742.294 1930.25,765.079 1930.91,788.227 1931.56,811.553 1932.21,834.861 1932.87,857.946 1933.52,880.593 1934.17,902.574 1934.83,923.682 \n",
+ " 1935.48,943.737 1936.13,962.534 1936.78,979.884 1937.44,995.616 1938.09,1009.57 1938.74,1021.62 1939.4,1031.63 1940.05,1039.51 1940.7,1045.16 1941.36,1048.54 \n",
+ " 1942.01,1049.59 1942.66,1048.27 1943.32,1044.59 1943.97,1038.55 1944.62,1030.19 1945.27,1019.57 1945.93,1006.76 1946.58,991.883 1947.23,975.05 1947.89,956.397 \n",
+ " 1948.54,936.064 1949.19,914.211 1949.85,891.007 1950.5,866.633 1951.15,841.277 1951.8,815.141 1952.46,788.435 1953.11,761.378 1953.76,734.203 1954.42,707.142 \n",
+ " 1955.07,680.394 1955.72,654.181 1956.38,628.72 1957.03,604.221 1957.68,580.88 1958.33,558.888 1958.99,538.421 1959.64,519.648 1960.29,502.728 1960.95,487.809 \n",
+ " 1961.6,475.018 1962.25,464.454 1962.91,456.215 1963.56,450.382 1964.21,447.014 1964.86,446.15 1965.52,447.809 1966.17,451.987 1966.82,458.662 1967.48,467.789 \n",
+ " 1968.13,479.306 1968.78,493.138 1969.44,509.202 1970.09,527.398 1970.74,547.606 1971.39,569.69 1972.05,593.496 1972.7,618.851 1973.35,645.567 1974.01,673.436 \n",
+ " 1974.66,702.236 1975.31,731.765 1975.97,761.787 1976.62,792.054 1977.27,822.32 1977.92,852.336 1978.58,881.851 1979.23,910.614 1979.88,938.371 1980.54,964.87 \n",
+ " 1981.19,989.853 1981.84,1013.08 1982.5,1034.37 1983.15,1053.52 1983.8,1070.33 1984.46,1084.65 1985.11,1096.35 1985.76,1105.31 1986.41,1111.45 1987.07,1114.71 \n",
+ " 1987.72,1115.05 1988.37,1112.46 1989.03,1106.95 1989.68,1098.55 1990.33,1087.31 1990.99,1073.32 1991.64,1056.66 1992.29,1037.48 1992.94,1015.92 1993.6,992.157 \n",
+ " 1994.25,966.402 1994.9,938.87 1995.56,909.767 1996.21,879.328 1996.86,847.796 1997.52,815.42 1998.17,782.455 1998.82,749.159 1999.47,715.799 2000.13,682.647 \n",
+ " 2000.78,649.979 2001.43,618.079 2002.09,587.221 2002.74,557.628 2003.39,529.545 2004.05,503.208 2004.7,478.832 2005.35,456.618 2006,436.745 2006.66,419.375 \n",
+ " 2007.31,404.652 2007.96,392.702 2008.62,383.624 2009.27,377.487 2009.92,374.359 2010.58,374.279 2011.23,377.263 2011.88,383.297 2012.53,392.345 2013.19,404.343 \n",
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+ " 1850.59,1371 1851.24,1371.75 1851.89,1367.29 1852.55,1357.66 1853.2,1342.95 1853.85,1323.3 1854.5,1298.87 1855.16,1269.85 1855.81,1236.47 1856.46,1198.98 \n",
+ " 1857.12,1157.68 1857.77,1112.91 1858.42,1065.02 1859.08,1014.42 1859.73,961.546 1860.38,906.848 1861.04,850.74 1861.69,793.685 1862.34,736.148 1862.99,678.587 \n",
+ " 1863.65,621.461 1864.3,565.22 1864.95,510.311 1865.61,457.176 1866.26,406.255 1866.91,357.981 1867.57,312.782 1868.22,271.004 1868.87,232.958 1869.52,198.965 \n",
+ " 1870.18,169.304 1870.83,144.214 1871.48,123.894 1872.14,108.501 1872.79,98.1551 1873.44,92.9325 1874.1,92.8563 1874.75,97.92 1875.4,108.094 1876.05,123.304 \n",
+ " 1876.71,143.432 1877.36,168.317 1878.01,197.753 1878.67,231.49 1879.32,269.234 1879.97,310.646 1880.63,355.353 1881.28,403.006 1881.93,453.209 1882.58,505.545 \n",
+ " 1883.24,559.587 1883.89,614.9 1884.54,671.039 1885.2,727.547 1885.85,783.96 1886.5,839.803 1887.16,894.593 1887.81,947.877 1888.46,999.265 1889.11,1048.33 \n",
+ " 1889.77,1094.69 1890.42,1137.97 1891.07,1177.83 1891.73,1213.96 1892.38,1246.08 1893.03,1273.94 1893.69,1297.32 1894.34,1316.05 1894.99,1329.98 1895.64,1339 \n",
+ " 1896.3,1343.02 1896.95,1342.02 1897.6,1336.01 1898.26,1325.04 1898.91,1309.21 1899.56,1288.67 1900.22,1263.6 1900.87,1234.22 1901.52,1200.75 1902.18,1163.46 \n",
+ " 1902.83,1122.65 1903.48,1078.63 1904.13,1031.77 1904.79,982.435 1905.44,931.034 1906.09,878.003 1906.75,823.803 1907.4,768.86 1908.05,713.606 1908.71,658.502 \n",
+ " 1909.36,603.992 1910.01,550.512 1910.66,498.487 1911.32,448.327 1911.97,400.433 1912.62,355.194 1913.28,312.987 1913.93,274.177 1914.58,239.095 1915.24,207.979 \n",
+ " 1915.89,181.088 1916.54,158.647 1917.19,140.839 1917.85,127.803 1918.5,119.64 1919.15,116.405 1919.81,118.113 1920.46,124.735 1921.11,136.196 1921.77,152.401 \n",
+ " 1922.42,173.222 1923.07,198.492 1923.72,228.008 1924.38,261.527 1925.03,298.768 1925.68,339.411 1926.34,383.099 1926.99,429.436 1927.64,478.004 1928.3,528.433 \n",
+ " 1928.95,580.297 1929.6,633.162 1930.25,686.601 1930.91,740.182 1931.56,793.479 1932.21,846.066 1932.87,897.518 1933.52,947.41 1934.17,995.322 1934.83,1040.87 \n",
+ " 1935.48,1083.72 1936.13,1123.54 1936.78,1160.01 1937.44,1192.85 1938.09,1221.8 1938.74,1246.65 1939.4,1267.2 1940.05,1283.31 1940.7,1294.86 1941.36,1301.78 \n",
+ " 1942.01,1304.01 1942.66,1301.52 1943.32,1294.35 1943.97,1282.54 1944.62,1266.18 1945.27,1245.43 1945.93,1220.46 1946.58,1191.48 1947.23,1158.76 1947.89,1122.57 \n",
+ " 1948.54,1083.19 1949.19,1040.95 1949.85,996.173 1950.5,949.224 1951.15,900.471 1951.8,850.304 1952.46,799.132 1953.11,747.38 1953.76,695.491 1954.42,643.914 \n",
+ " 1955.07,593.027 1955.72,543.247 1956.38,494.987 1957.03,448.637 1957.68,404.568 1958.33,363.13 1958.99,324.65 1959.64,289.438 1960.29,257.781 1960.95,229.944 \n",
+ " 1961.6,206.154 1962.25,186.575 1962.91,171.374 1963.56,160.682 1964.21,154.584 1964.86,153.127 1965.52,156.314 1966.17,164.109 1966.82,176.43 1967.48,193.158 \n",
+ " 1968.13,214.136 1968.78,239.186 1969.44,268.1 1970.09,300.638 1970.74,336.534 1971.39,375.492 1972.05,417.188 1972.7,461.268 1973.35,507.35 1974.01,555.023 \n",
+ " 1974.66,603.856 1975.31,653.474 1975.97,703.47 1976.62,753.432 1977.27,802.957 1977.92,851.653 1978.58,899.139 1979.23,945.044 1979.88,989.005 1980.54,1030.67 \n",
+ " 1981.19,1069.71 1981.84,1105.79 1982.5,1138.68 1983.15,1168.12 1983.8,1193.87 1984.46,1215.73 1985.11,1233.54 1985.76,1247.17 1986.41,1256.52 1987.07,1261.53 \n",
+ " 1987.72,1262.17 1988.37,1258.45 1989.03,1250.42 1989.68,1238.13 1990.33,1221.68 1990.99,1201.2 1991.64,1176.84 1992.29,1148.82 1992.94,1117.36 1993.6,1082.73 \n",
+ " 1994.25,1045.24 1994.9,1005.21 1995.56,962.955 1996.21,918.81 1996.86,873.135 1997.52,826.294 1998.17,778.657 1998.82,730.601 1999.47,682.512 2000.13,634.782 \n",
+ " 2000.78,587.809 2001.43,542 2002.09,497.75 2002.74,455.373 2003.39,415.219 2004.05,377.62 2004.7,342.88 2005.35,311.276 2006,283.059 2006.66,258.452 \n",
+ " 2007.31,237.649 2007.96,220.819 2008.62,208.088 2009.27,199.542 2009.92,195.26 2010.58,195.282 2011.23,199.609 2011.88,208.203 2012.53,220.988 2013.19,237.848 \n",
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+ " 1504.49,721.29 1505.14,721.609 1505.79,721.95 1506.45,722.311 1507.1,722.689 1507.75,723.08 1508.41,723.482 1509.06,723.89 1509.71,724.302 1510.37,724.715 \n",
+ " 1511.02,725.125 1511.67,725.528 1512.32,725.923 1512.98,726.304 1513.63,726.671 1514.28,727.018 1514.94,727.344 1515.59,727.647 1516.24,727.922 1516.9,728.17 \n",
+ " 1517.55,728.386 1518.2,728.57 1518.85,728.721 1519.51,728.836 1520.16,728.916 1520.81,728.959 1521.47,728.966 1522.12,728.936 1522.77,728.871 1523.43,728.769 \n",
+ " 1524.08,728.633 1524.73,728.463 1525.39,728.261 1526.04,728.028 1526.69,727.766 1527.34,727.478 1528,727.166 1528.65,726.833 1529.3,726.481 1529.96,726.114 \n",
+ " 1530.61,725.734 1531.26,725.345 1531.92,724.949 1532.57,724.549 1533.22,724.15 1533.87,723.754 1534.53,723.365 1535.18,722.985 1535.83,722.619 1536.49,722.268 \n",
+ " 1537.14,721.937 1537.79,721.627 1538.45,721.341 1539.1,721.081 1539.75,720.85 1540.4,720.649 1541.06,720.48 1541.71,720.345 1542.36,720.244 1543.02,720.178 \n",
+ " 1543.67,720.148 1544.32,720.155 1544.98,720.197 1545.63,720.276 1546.28,720.391 1546.93,720.539 1547.59,720.722 1548.24,720.935 1548.89,721.179 1549.55,721.451 \n",
+ " 1550.2,721.748 1550.85,722.069 1551.51,722.41 1552.16,722.77 1552.81,723.145 1553.46,723.532 1554.12,723.929 1554.77,724.332 1555.42,724.737 1556.08,725.142 \n",
+ " 1556.73,725.544 1557.38,725.938 1558.04,726.322 1558.69,726.693 1559.34,727.048 1559.99,727.385 1560.65,727.7 1561.3,727.991 1561.95,728.256 1562.61,728.493 \n",
+ " 1563.26,728.7 1563.91,728.875 1564.57,729.018 1565.22,729.127 1565.87,729.201 1566.53,729.241 1567.18,729.245 1567.83,729.214 1568.48,729.149 1569.14,729.049 \n",
+ " 1569.79,728.918 1570.44,728.755 1571.1,728.561 1571.75,728.34 1572.4,728.092 1573.06,727.819 1573.71,727.525 1574.36,727.211 1575.01,726.88 1575.67,726.535 \n",
+ " 1576.32,726.179 1576.97,725.815 1577.63,725.446 1578.28,725.075 1578.93,724.705 1579.59,724.339 1580.24,723.98 1580.89,723.631 1581.54,723.295 1582.2,722.975 \n",
+ " 1582.85,722.673 1583.5,722.392 1584.16,722.134 1584.81,721.902 1585.46,721.697 1586.12,721.521 1586.77,721.376 1587.42,721.262 1588.07,721.181 1588.73,721.134 \n",
+ " 1589.38,721.12 1590.03,721.14 1590.69,721.193 1591.34,721.28 1591.99,721.399 1592.65,721.551 1593.3,721.733 1593.95,721.944 1594.6,722.184 1595.26,722.449 \n",
+ " 1595.91,722.738 1596.56,723.047 1597.22,723.375 1597.87,723.72 1598.52,724.077 1599.18,724.446 1599.83,724.822 1600.48,725.202 1601.13,725.585 1601.79,725.966 \n",
+ " 1602.44,726.343 1603.09,726.713 1603.75,727.072 1604.4,727.419 1605.05,727.75 1605.71,728.063 1606.36,728.355 1607.01,728.624 1607.66,728.869 1608.32,729.088 \n",
+ " 1608.97,729.278 1609.62,729.439 1610.28,729.57 1610.93,729.669 1611.58,729.737 1612.24,729.773 1612.89,729.777 1613.54,729.749 1614.2,729.69 1614.85,729.601 \n",
+ " 1615.5,729.482 1616.15,729.334 1616.81,729.16 1617.46,728.961 1618.11,728.739 1618.77,728.496 1619.42,728.234 1620.07,727.955 1620.73,727.661 1621.38,727.356 \n",
+ " 1622.03,727.042 1622.68,726.721 1623.34,726.396 1623.99,726.071 1624.64,725.747 1625.3,725.428 1625.95,725.116 1626.6,724.813 1627.26,724.524 1627.91,724.249 \n",
+ " 1628.56,723.99 1629.21,723.752 1629.87,723.534 1630.52,723.339 1631.17,723.17 1631.83,723.026 1632.48,722.909 1633.13,722.821 1633.79,722.763 1634.44,722.734 \n",
+ " 1635.09,722.735 1635.74,722.767 1636.4,722.828 1637.05,722.919 1637.7,723.038 1638.36,723.185 1639.01,723.358 1639.66,723.557 1640.32,723.78 1640.97,724.025 \n",
+ " 1641.62,724.291 1642.27,724.574 1642.93,724.874 1643.58,725.187 1644.23,725.511 1644.89,725.843 1645.54,726.181 1646.19,726.523 1646.85,726.865 1647.5,727.206 \n",
+ " 1648.15,727.541 1648.8,727.87 1649.46,728.19 1650.11,728.497 1650.76,728.79 1651.42,729.066 1652.07,729.324 1652.72,729.561 1653.38,729.777 1654.03,729.969 \n",
+ " 1654.68,730.136 1655.34,730.278 1655.99,730.393 1656.64,730.482 1657.29,730.543 1657.95,730.577 1658.6,730.584 1659.25,730.564 1659.91,730.517 1660.56,730.445 \n",
+ " 1661.21,730.349 1661.87,730.229 1662.52,730.087 1663.17,729.924 1663.82,729.742 1664.48,729.543 1665.13,729.329 1665.78,729.102 1666.44,728.864 1667.09,728.616 \n",
+ " 1667.74,728.362 1668.4,728.103 1669.05,727.842 1669.7,727.58 1670.35,727.321 1671.01,727.066 1671.66,726.818 1672.31,726.578 1672.97,726.35 1673.62,726.134 \n",
+ " 1674.27,725.933 1674.93,725.748 1675.58,725.581 1676.23,725.434 1676.88,725.307 1677.54,725.202 1678.19,725.12 1678.84,725.061 1679.5,725.026 1680.15,725.015 \n",
+ " 1680.8,725.029 1681.46,725.068 1682.11,725.131 1682.76,725.218 1683.41,725.329 1684.07,725.462 1684.72,725.616 1685.37,725.791 1686.03,725.984 1686.68,726.195 \n",
+ " 1687.33,726.422 1687.99,726.662 1688.64,726.916 1689.29,727.179 1689.94,727.451 1690.6,727.729 1691.25,728.012 1691.9,728.296 1692.56,728.58 1693.21,728.862 \n",
+ " 1693.86,729.139 1694.52,729.411 1695.17,729.673 1695.82,729.926 1696.48,730.166 1697.13,730.393 1697.78,730.605 1698.43,730.8 1699.09,730.978 1699.74,731.136 \n",
+ " 1700.39,731.275 1701.05,731.394 1701.7,731.492 1702.35,731.568 1703.01,731.624 1703.66,731.658 1704.31,731.672 1704.96,731.665 1705.62,731.638 1706.27,731.592 \n",
+ " 1706.92,731.528 1707.58,731.447 1708.23,731.35 1708.88,731.238 1709.54,731.111 1710.19,730.973 1710.84,730.824 1711.49,730.665 1712.15,730.499 1712.8,730.327 \n",
+ " 1713.45,730.15 1714.11,729.97 1714.76,729.79 1715.41,729.609 1716.07,729.431 1716.72,729.257 1717.37,729.088 1718.02,728.926 1718.68,728.772 1719.33,728.628 \n",
+ " 1719.98,728.495 1720.64,728.373 1721.29,728.266 1721.94,728.172 1722.6,728.094 1723.25,728.032 1723.9,727.986 1724.55,727.957 1725.21,727.945 1725.86,727.951 \n",
+ " 1726.51,727.974 1727.17,728.015 1727.82,728.072 1728.47,728.147 1729.13,728.238 1729.78,728.345 1730.43,728.467 1731.08,728.603 1731.74,728.752 1732.39,728.913 \n",
+ " 1733.04,729.085 1733.7,729.266 1734.35,729.455 1735,729.651 1735.66,729.853 1736.31,730.058 1736.96,730.266 1737.62,730.475 1738.27,730.683 1738.92,730.889 \n",
+ " 1739.57,731.092 1740.23,731.29 1740.88,731.481 1741.53,731.665 1742.19,731.841 1742.84,732.006 1743.49,732.162 1744.15,732.305 1744.8,732.437 1745.45,732.556 \n",
+ " 1746.1,732.662 1746.76,732.754 1747.41,732.833 1748.06,732.898 1748.72,732.95 1749.37,732.988 1750.02,733.013 1750.68,733.026 1751.33,733.026 1751.98,733.015 \n",
+ " 1752.63,732.993 1753.29,732.961 1753.94,732.919 1754.59,732.869 1755.25,732.812 1755.9,732.747 1756.55,732.677 1757.21,732.601 1757.86,732.522 1758.51,732.439 \n",
+ " 1759.16,732.354 1759.82,732.268 1760.47,732.181 1761.12,732.096 1761.78,732.011 1762.43,731.929 1763.08,731.851 1763.74,731.776 1764.39,731.706 1765.04,731.641 \n",
+ " 1765.69,731.583 1766.35,731.532 1767,731.487 1767.65,731.451 1768.31,731.423 1768.96,731.403 1769.61,731.392 1770.27,731.391 1770.92,731.398 1771.57,731.415 \n",
+ " 1772.22,731.441 1772.88,731.476 1773.53,731.519 1774.18,731.572 1774.84,731.633 1775.49,731.702 1776.14,731.778 1776.8,731.861 1777.45,731.951 1778.1,732.047 \n",
+ " 1778.76,732.149 1779.41,732.254 1780.06,732.364 1780.71,732.477 1781.37,732.592 1782.02,732.709 1782.67,732.827 1783.33,732.945 1783.98,733.062 1784.63,733.179 \n",
+ " 1785.29,733.293 1785.94,733.405 1786.59,733.514 1787.24,733.619 1787.9,733.72 1788.55,733.816 1789.2,733.907 1789.86,733.994 1790.51,734.075 1791.16,734.15 \n",
+ " 1791.82,734.221 1792.47,734.286 1793.12,734.345 1793.77,734.4 1794.43,734.449 1795.08,734.494 1795.73,734.535 1796.39,734.572 1797.04,734.606 1797.69,734.636 \n",
+ " 1798.35,734.663 1799,734.688 1799.65,734.711 1800.3,734.732 1800.96,734.751 1801.61,734.77 1802.26,734.787 1802.92,734.803 1803.57,734.819 1804.22,734.834 \n",
+ " 1804.88,734.849 1805.53,734.864 1806.18,734.879 1806.83,734.894 1807.49,734.909 1808.14,734.924 1808.79,734.94 1809.45,734.956 1810.1,734.973 1810.75,734.99 \n",
+ " 1811.41,735.007 1812.06,735.025 1812.71,735.043 1813.36,735.061 1814.02,735.08 1814.67,735.1 1815.32,735.119 1815.98,735.139 1816.63,735.159 1817.28,735.179 \n",
+ " 1817.94,735.2 1818.59,735.22 1819.24,735.24 1819.9,735.261 1820.55,735.281 1821.2,735.301 1821.85,735.321 1822.51,735.341 1823.16,735.361 1823.81,735.38 \n",
+ " 1824.47,735.399 1825.12,735.418 1825.77,735.437 1826.43,735.455 1827.08,735.474 1827.73,735.492 1828.38,735.51 1829.04,735.528 1829.69,735.546 1830.34,735.564 \n",
+ " 1831,735.583 1831.65,735.603 1832.3,735.623 1832.96,735.644 1833.61,735.666 1834.26,735.689 1834.91,735.714 1835.57,735.741 1836.22,735.77 1836.87,735.802 \n",
+ " 1837.53,735.837 1838.18,735.876 1838.83,735.918 1839.49,735.964 1840.14,736.014 1840.79,736.069 1841.44,736.128 1842.1,736.193 1842.75,736.263 1843.4,736.338 \n",
+ " 1844.06,736.418 1844.71,736.503 1845.36,736.593 1846.02,736.688 1846.67,736.787 1847.32,736.89 1847.97,736.997 1848.63,737.106 1849.28,737.218 1849.93,737.331 \n",
+ " 1850.59,737.446 1851.24,737.561 1851.89,737.676 1852.55,737.79 1853.2,737.902 1853.85,738.012 1854.5,738.118 1855.16,738.221 1855.81,738.319 1856.46,738.412 \n",
+ " 1857.12,738.499 1857.77,738.58 1858.42,738.654 1859.08,738.72 1859.73,738.779 1860.38,738.829 1861.04,738.871 1861.69,738.905 1862.34,738.93 1862.99,738.945 \n",
+ " 1863.65,738.952 1864.3,738.95 1864.95,738.939 1865.61,738.919 1866.26,738.891 1866.91,738.856 1867.57,738.812 1868.22,738.762 1868.87,738.705 1869.52,738.642 \n",
+ " 1870.18,738.574 1870.83,738.502 1871.48,738.426 1872.14,738.346 1872.79,738.264 1873.44,738.181 1874.1,738.098 1874.75,738.015 1875.4,737.933 1876.05,737.853 \n",
+ " 1876.71,737.777 1877.36,737.705 1878.01,737.638 1878.67,737.577 1879.32,737.522 1879.97,737.475 1880.63,737.437 1881.28,737.407 1881.93,737.388 1882.58,737.379 \n",
+ " 1883.24,737.382 1883.89,737.397 1884.54,737.424 1885.2,737.465 1885.85,737.518 1886.5,737.585 1887.16,737.665 1887.81,737.758 1888.46,737.865 1889.11,737.984 \n",
+ " 1889.77,738.116 1890.42,738.26 1891.07,738.415 1891.73,738.58 1892.38,738.755 1893.03,738.938 1893.69,739.129 1894.34,739.325 1894.99,739.526 1895.64,739.73 \n",
+ " 1896.3,739.936 1896.95,740.142 1897.6,740.347 1898.26,740.55 1898.91,740.749 1899.56,740.943 1900.22,741.129 1900.87,741.307 1901.52,741.476 1902.18,741.634 \n",
+ " 1902.83,741.779 1903.48,741.912 1904.13,742.031 1904.79,742.135 1905.44,742.223 1906.09,742.295 1906.75,742.35 1907.4,742.389 1908.05,742.41 1908.71,742.414 \n",
+ " 1909.36,742.401 1910.01,742.371 1910.66,742.324 1911.32,742.26 1911.97,742.181 1912.62,742.087 1913.28,741.979 1913.93,741.858 1914.58,741.725 1915.24,741.581 \n",
+ " 1915.89,741.429 1916.54,741.268 1917.19,741.1 1917.85,740.928 1918.5,740.751 1919.15,740.573 1919.81,740.395 1920.46,740.218 1921.11,740.044 1921.77,739.874 \n",
+ " 1922.42,739.711 1923.07,739.555 1923.72,739.409 1924.38,739.274 1925.03,739.152 1925.68,739.043 1926.34,738.948 1926.99,738.871 1927.64,738.81 1928.3,738.767 \n",
+ " 1928.95,738.743 1929.6,738.739 1930.25,738.756 1930.91,738.793 1931.56,738.852 1932.21,738.931 1932.87,739.032 1933.52,739.153 1934.17,739.294 1934.83,739.454 \n",
+ " 1935.48,739.633 1936.13,739.829 1936.78,740.042 1937.44,740.27 1938.09,740.511 1938.74,740.764 1939.4,741.026 1940.05,741.297 1940.7,741.574 1941.36,741.855 \n",
+ " 1942.01,742.138 1942.66,742.421 1943.32,742.702 1943.97,742.978 1944.62,743.248 1945.27,743.509 1945.93,743.759 1946.58,743.998 1947.23,744.222 1947.89,744.43 \n",
+ " 1948.54,744.62 1949.19,744.792 1949.85,744.943 1950.5,745.072 1951.15,745.179 1951.8,745.263 1952.46,745.323 1953.11,745.358 1953.76,745.369 1954.42,745.356 \n",
+ " 1955.07,745.319 1955.72,745.258 1956.38,745.174 1957.03,745.067 1957.68,744.938 1958.33,744.789 1958.99,744.621 1959.64,744.435 1960.29,744.232 1960.95,744.016 \n",
+ " 1961.6,743.787 1962.25,743.547 1962.91,743.299 1963.56,743.045 1964.21,742.787 1964.86,742.527 1965.52,742.267 1966.17,742.009 1966.82,741.757 1967.48,741.511 \n",
+ " 1968.13,741.275 1968.78,741.051 1969.44,740.839 1970.09,740.644 1970.74,740.465 1971.39,740.306 1972.05,740.168 1972.7,740.051 1973.35,739.959 1974.01,739.89 \n",
+ " 1974.66,739.847 1975.31,739.831 1975.97,739.841 1976.62,739.878 1977.27,739.943 1977.92,740.035 1978.58,740.154 1979.23,740.299 1979.88,740.47 1980.54,740.665 \n",
+ " 1981.19,740.884 1981.84,741.124 1982.5,741.384 1983.15,741.662 1983.8,741.957 1984.46,742.265 1985.11,742.586 1985.76,742.915 1986.41,743.252 1987.07,743.593 \n",
+ " 1987.72,743.935 1988.37,744.277 1989.03,744.615 1989.68,744.946 1990.33,745.268 1990.99,745.58 1991.64,745.877 1992.29,746.158 1992.94,746.421 1993.6,746.663 \n",
+ " 1994.25,746.883 1994.9,747.08 1995.56,747.25 1996.21,747.394 1996.86,747.51 1997.52,747.597 1998.17,747.655 1998.82,747.682 1999.47,747.68 2000.13,747.647 \n",
+ " 2000.78,747.584 2001.43,747.493 2002.09,747.373 2002.74,747.227 2003.39,747.054 2004.05,746.857 2004.7,746.637 2005.35,746.396 2006,746.135 2006.66,745.858 \n",
+ " 2007.31,745.566 2007.96,745.263 2008.62,744.95 2009.27,744.63 2009.92,744.306 2010.58,743.981 2011.23,743.657 2011.88,743.337 2012.53,743.023 2013.19,742.719 \n",
+ " 2013.84,742.428 2014.49,742.15 2015.15,741.89 2015.8,741.649 2016.45,741.43 2017.11,741.234 2017.76,741.063 2018.41,740.92 2019.06,740.805 2019.72,740.72 \n",
+ " 2020.37,740.665 2021.02,740.641 2021.68,740.649 2022.33,740.689 2022.98,740.761 2023.64,740.865 2024.29,740.999 2024.94,741.165 2025.6,741.359 2026.25,741.582 \n",
+ " 2026.9,741.831 2027.55,742.104 2028.21,742.4 2028.86,742.717 2029.51,743.05 2030.17,743.4 2030.82,743.761 2031.47,744.133 2032.13,744.512 2032.78,744.894 \n",
+ " 2033.43,745.278 2034.08,745.659 2034.74,746.036 2035.39,746.404 2036.04,746.762 2036.7,747.105 2037.35,747.432 2038,747.74 2038.66,748.026 2039.31,748.288 \n",
+ " 2039.96,748.525 2040.61,748.733 2041.27,748.912 2041.92,749.061 2042.57,749.177 2043.23,749.261 2043.88,749.311 2044.53,749.327 2045.19,749.31 2045.84,749.258 \n",
+ " 2046.49,749.172 2047.14,749.054 2047.8,748.903 2048.45,748.723 2049.1,748.513 2049.76,748.277 2050.41,748.016 2051.06,747.732 2051.72,747.427 2052.37,747.104 \n",
+ " 2053.02,746.765 2053.67,746.414 2054.33,746.052 2054.98,745.684 2055.63,745.313 2056.29,744.94 2056.94,744.57 2057.59,744.206 2058.25,743.85 2058.9,743.506 \n",
+ " 2059.55,743.177 2060.2,742.864 2060.86,742.571 2061.51,742.301 2062.16,742.055 2062.82,741.836 2063.47,741.645 2064.12,741.485 2064.78,741.357 2065.43,741.262 \n",
+ " 2066.08,741.201 2066.74,741.175 2067.39,741.184 2068.04,741.228 2068.69,741.308 2069.35,741.422 2070,741.569 2070.65,741.75 2071.31,741.961 2071.96,742.203 \n",
+ " 2072.61,742.473 2073.27,742.768 2073.92,743.088 2074.57,743.429 2075.22,743.789 2075.88,744.164 2076.53,744.552 2077.18,744.949 2077.84,745.353 2078.49,745.76 \n",
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+ " 1615.5,742.91 1616.15,743.149 1616.81,743.424 1617.46,743.733 1618.11,744.073 1618.77,744.442 1619.42,744.838 1620.07,745.256 1620.73,745.695 1621.38,746.15 \n",
+ " 1622.03,746.618 1622.68,747.095 1623.34,747.577 1623.99,748.061 1624.64,748.542 1625.3,749.017 1625.95,749.482 1626.6,749.934 1627.26,750.368 1627.91,750.781 \n",
+ " 1628.56,751.17 1629.21,751.532 1629.87,751.865 1630.52,752.164 1631.17,752.428 1631.83,752.656 1632.48,752.844 1633.13,752.992 1633.79,753.098 1634.44,753.161 \n",
+ " 1635.09,753.181 1635.74,753.158 1636.4,753.092 1637.05,752.984 1637.7,752.835 1638.36,752.646 1639.01,752.419 1639.66,752.155 1640.32,751.856 1640.97,751.525 \n",
+ " 1641.62,751.165 1642.27,750.779 1642.93,750.369 1643.58,749.94 1644.23,749.495 1644.89,749.037 1645.54,748.571 1646.19,748.1 1646.85,747.628 1647.5,747.16 \n",
+ " 1648.15,746.698 1648.8,746.246 1649.46,745.81 1650.11,745.391 1650.76,744.994 1651.42,744.623 1652.07,744.279 1652.72,743.967 1653.38,743.689 1654.03,743.448 \n",
+ " 1654.68,743.245 1655.34,743.083 1655.99,742.963 1656.64,742.886 1657.29,742.853 1657.95,742.865 1658.6,742.921 1659.25,743.021 1659.91,743.166 1660.56,743.354 \n",
+ " 1661.21,743.584 1661.87,743.855 1662.52,744.164 1663.17,744.508 1663.82,744.887 1664.48,745.295 1665.13,745.73 1665.78,746.188 1666.44,746.667 1667.09,747.163 \n",
+ " 1667.74,747.671 1668.4,748.187 1669.05,748.708 1669.7,749.229 1670.35,749.747 1671.01,750.256 1671.66,750.753 1672.31,751.235 1672.97,751.696 1673.62,752.133 \n",
+ " 1674.27,752.544 1674.93,752.924 1675.58,753.271 1676.23,753.581 1676.88,753.853 1677.54,754.084 1678.19,754.273 1678.84,754.418 1679.5,754.517 1680.15,754.571 \n",
+ " 1680.8,754.578 1681.46,754.538 1682.11,754.452 1682.76,754.321 1683.41,754.145 1684.07,753.926 1684.72,753.666 1685.37,753.368 1686.03,753.034 1686.68,752.667 \n",
+ " 1687.33,752.269 1687.99,751.843 1688.64,751.394 1689.29,750.925 1689.94,750.439 1690.6,749.941 1691.25,749.436 1691.9,748.926 1692.56,748.417 1693.21,747.913 \n",
+ " 1693.86,747.417 1694.52,746.935 1695.17,746.469 1695.82,746.024 1696.48,745.604 1697.13,745.212 1697.78,744.851 1698.43,744.525 1699.09,744.236 1699.74,743.987 \n",
+ " 1700.39,743.78 1701.05,743.618 1701.7,743.501 1702.35,743.431 1703.01,743.409 1703.66,743.435 1704.31,743.509 1704.96,743.631 1705.62,743.799 1706.27,744.013 \n",
+ " 1706.92,744.271 1707.58,744.572 1708.23,744.912 1708.88,745.29 1709.54,745.703 1710.19,746.148 1710.84,746.62 1711.49,747.117 1712.15,747.633 1712.8,748.166 \n",
+ " 1713.45,748.71 1714.11,749.262 1714.76,749.817 1715.41,750.371 1716.07,750.92 1716.72,751.458 1717.37,751.982 1718.02,752.488 1718.68,752.97 1719.33,753.427 \n",
+ " 1719.98,753.853 1720.64,754.245 1721.29,754.601 1721.94,754.917 1722.6,755.191 1723.25,755.421 1723.9,755.604 1724.55,755.74 1725.21,755.827 1725.86,755.866 \n",
+ " 1726.51,755.855 1727.17,755.794 1727.82,755.685 1728.47,755.528 1729.13,755.323 1729.78,755.074 1730.43,754.781 1731.08,754.447 1731.74,754.075 1732.39,753.669 \n",
+ " 1733.04,753.232 1733.7,752.767 1734.35,752.278 1735,751.77 1735.66,751.245 1736.31,750.709 1736.96,750.166 1737.62,749.62 1738.27,749.076 1738.92,748.539 \n",
+ " 1739.57,748.012 1740.23,747.5 1740.88,747.008 1741.53,746.54 1742.19,746.099 1742.84,745.689 1743.49,745.314 1744.15,744.977 1744.8,744.68 1745.45,744.426 \n",
+ " 1746.1,744.218 1746.76,744.057 1747.41,743.944 1748.06,743.881 1748.72,743.869 1749.37,743.907 1750.02,743.997 1750.68,744.136 1751.33,744.325 1751.98,744.562 \n",
+ " 1752.63,744.845 1753.29,745.171 1753.94,745.538 1754.59,745.943 1755.25,746.383 1755.9,746.855 1756.55,747.355 1757.21,747.879 1757.86,748.423 1758.51,748.983 \n",
+ " 1759.16,749.553 1759.82,750.13 1760.47,750.708 1761.12,751.283 1761.78,751.85 1762.43,752.406 1763.08,752.944 1763.74,753.462 1764.39,753.954 1765.04,754.418 \n",
+ " 1765.69,754.849 1766.35,755.243 1767,755.598 1767.65,755.911 1768.31,756.179 1768.96,756.4 1769.61,756.572 1770.27,756.693 1770.92,756.764 1771.57,756.782 \n",
+ " 1772.22,756.748 1772.88,756.663 1773.53,756.528 1774.18,756.343 1774.84,756.11 1775.49,755.83 1776.14,755.507 1776.8,755.142 1777.45,754.739 1778.1,754.3 \n",
+ " 1778.76,753.83 1779.41,753.332 1780.06,752.811 1780.71,752.27 1781.37,751.715 1782.02,751.149 1782.67,750.578 1783.33,750.006 1783.98,749.437 1784.63,748.877 \n",
+ " 1785.29,748.329 1785.94,747.798 1786.59,747.289 1787.24,746.806 1787.9,746.353 1788.55,745.933 1789.2,745.551 1789.86,745.209 1790.51,744.91 1791.16,744.657 \n",
+ " 1791.82,744.451 1792.47,744.295 1793.12,744.189 1793.77,744.135 1794.43,744.134 1795.08,744.184 1795.73,744.286 1796.39,744.439 1797.04,744.642 1797.69,744.894 \n",
+ " 1798.35,745.193 1799,745.536 1799.65,745.92 1800.3,746.342 1800.96,746.798 1801.61,747.285 1802.26,747.799 1802.92,748.336 1803.57,748.891 1804.22,749.461 \n",
+ " 1804.88,750.041 1805.53,750.625 1806.18,751.21 1806.83,751.789 1807.49,752.359 1808.14,752.915 1808.79,753.453 1809.45,753.967 1810.1,754.455 1810.75,754.911 \n",
+ " 1811.41,755.333 1812.06,755.716 1812.71,756.058 1813.36,756.357 1814.02,756.608 1814.67,756.812 1815.32,756.965 1815.98,757.067 1816.63,757.117 1817.28,757.114 \n",
+ " 1817.94,757.059 1818.59,756.95 1819.24,756.791 1819.9,756.581 1820.55,756.323 1821.2,756.019 1821.85,755.671 1822.51,755.283 1823.16,754.857 1823.81,754.397 \n",
+ " 1824.47,753.906 1825.12,753.388 1825.77,752.848 1826.43,752.29 1827.08,751.718 1827.73,751.137 1828.38,750.553 1829.04,749.969 1829.69,749.391 1830.34,748.822 \n",
+ " 1831,748.269 1831.65,747.734 1832.3,747.223 1832.96,746.739 1833.61,746.287 1834.26,745.869 1834.91,745.491 1835.57,745.154 1836.22,744.861 1836.87,744.615 \n",
+ " 1837.53,744.418 1838.18,744.271 1838.83,744.175 1839.49,744.132 1840.14,744.14 1840.79,744.201 1841.44,744.313 1842.1,744.476 1842.75,744.688 1843.4,744.948 \n",
+ " 1844.06,745.253 1844.71,745.601 1845.36,745.989 1846.02,746.414 1846.67,746.872 1847.32,747.359 1847.97,747.872 1848.63,748.405 1849.28,748.956 1849.93,749.518 \n",
+ " 1850.59,750.089 1851.24,750.663 1851.89,751.235 1852.55,751.801 1853.2,752.356 1853.85,752.895 1854.5,753.415 1855.16,753.91 1855.81,754.377 1856.46,754.812 \n",
+ " 1857.12,755.211 1857.77,755.572 1858.42,755.891 1859.08,756.165 1859.73,756.393 1860.38,756.573 1861.04,756.703 1861.69,756.782 1862.34,756.81 1862.99,756.785 \n",
+ " 1863.65,756.708 1864.3,756.58 1864.95,756.402 1865.61,756.175 1866.26,755.9 1866.91,755.581 1867.57,755.22 1868.22,754.82 1868.87,754.385 1869.52,753.917 \n",
+ " 1870.18,753.42 1870.83,752.898 1871.48,752.356 1872.14,751.798 1872.79,751.228 1873.44,750.651 1874.1,750.071 1874.75,749.494 1875.4,748.924 1876.05,748.366 \n",
+ " 1876.71,747.823 1877.36,747.301 1878.01,746.803 1878.67,746.334 1879.32,745.896 1879.97,745.494 1880.63,745.131 1881.28,744.81 1881.93,744.533 1882.58,744.302 \n",
+ " 1883.24,744.119 1883.89,743.985 1884.54,743.902 1885.2,743.87 1885.85,743.889 1886.5,743.958 1887.16,744.078 1887.81,744.245 1888.46,744.46 1889.11,744.72 \n",
+ " 1889.77,745.023 1890.42,745.367 1891.07,745.748 1891.73,746.163 1892.38,746.61 1893.03,747.084 1893.69,747.58 1894.34,748.095 1894.99,748.625 1895.64,749.165 \n",
+ " 1896.3,749.711 1896.95,750.259 1897.6,750.803 1898.26,751.34 1898.91,751.865 1899.56,752.374 1900.22,752.862 1900.87,753.325 1901.52,753.761 1902.18,754.164 \n",
+ " 1902.83,754.531 1903.48,754.861 1904.13,755.149 1904.79,755.394 1905.44,755.594 1906.09,755.746 1906.75,755.851 1907.4,755.906 1908.05,755.912 1908.71,755.869 \n",
+ " 1909.36,755.776 1910.01,755.634 1910.66,755.444 1911.32,755.207 1911.97,754.926 1912.62,754.603 1913.28,754.241 1913.93,753.842 1914.58,753.41 1915.24,752.949 \n",
+ " 1915.89,752.462 1916.54,751.952 1917.19,751.424 1917.85,750.882 1918.5,750.331 1919.15,749.774 1919.81,749.217 1920.46,748.663 1921.11,748.117 1921.77,747.584 \n",
+ " 1922.42,747.068 1923.07,746.573 1923.72,746.102 1924.38,745.66 1925.03,745.25 1925.68,744.875 1926.34,744.537 1926.99,744.241 1927.64,743.987 1928.3,743.777 \n",
+ " 1928.95,743.614 1929.6,743.497 1930.25,743.429 1930.91,743.409 1931.56,743.438 1932.21,743.515 1932.87,743.638 1933.52,743.808 1934.17,744.021 1934.83,744.276 \n",
+ " 1935.48,744.571 1936.13,744.903 1936.78,745.269 1937.44,745.667 1938.09,746.093 1938.74,746.543 1939.4,747.013 1940.05,747.501 1940.7,748 1941.36,748.508 \n",
+ " 1942.01,749.02 1942.66,749.532 1943.32,750.039 1943.97,750.538 1944.62,751.024 1945.27,751.493 1945.93,751.942 1946.58,752.367 1947.23,752.764 1947.89,753.13 \n",
+ " 1948.54,753.462 1949.19,753.757 1949.85,754.012 1950.5,754.226 1951.15,754.397 1951.8,754.523 1952.46,754.604 1953.11,754.638 1953.76,754.625 1954.42,754.566 \n",
+ " 1955.07,754.461 1955.72,754.311 1956.38,754.117 1957.03,753.88 1957.68,753.602 1958.33,753.285 1958.99,752.931 1959.64,752.545 1960.29,752.128 1960.95,751.684 \n",
+ " 1961.6,751.218 1962.25,750.732 1962.91,750.231 1963.56,749.717 1964.21,749.197 1964.86,748.673 1965.52,748.149 1966.17,747.63 1966.82,747.121 1967.48,746.624 \n",
+ " 1968.13,746.145 1968.78,745.686 1969.44,745.251 1970.09,744.845 1970.74,744.469 1971.39,744.127 1972.05,743.822 1972.7,743.555 1973.35,743.329 1974.01,743.145 \n",
+ " 1974.66,743.004 1975.31,742.908 1975.97,742.857 1976.62,742.85 1977.27,742.888 1977.92,742.971 1978.58,743.098 1979.23,743.267 1979.88,743.476 1980.54,743.724 \n",
+ " 1981.19,744.009 1981.84,744.327 1982.5,744.676 1983.15,745.052 1983.8,745.454 1984.46,745.877 1985.11,746.319 1985.76,746.774 1986.41,747.24 1987.07,747.713 \n",
+ " 1987.72,748.188 1988.37,748.662 1989.03,749.13 1989.68,749.589 1990.33,750.035 1990.99,750.464 1991.64,750.873 1992.29,751.259 1992.94,751.618 1993.6,751.947 \n",
+ " 1994.25,752.244 1994.9,752.505 1995.56,752.73 1996.21,752.916 1996.86,753.061 1997.52,753.165 1998.17,753.225 1998.82,753.243 1999.47,753.217 2000.13,753.147 \n",
+ " 2000.78,753.035 2001.43,752.881 2002.09,752.687 2002.74,752.454 2003.39,752.184 2004.05,751.878 2004.7,751.54 2005.35,751.172 2006,750.777 2006.66,750.358 \n",
+ " 2007.31,749.919 2007.96,749.462 2008.62,748.992 2009.27,748.513 2009.92,748.028 2010.58,747.541 2011.23,747.057 2011.88,746.578 2012.53,746.108 2013.19,745.652 \n",
+ " 2013.84,745.213 2014.49,744.794 2015.15,744.399 2015.8,744.03 2016.45,743.691 2017.11,743.384 2017.76,743.112 2018.41,742.876 2019.06,742.679 2019.72,742.522 \n",
+ " 2020.37,742.405 2021.02,742.329 2021.68,742.295 2022.33,742.303 2022.98,742.353 2023.64,742.443 2024.29,742.573 2024.94,742.742 2025.6,742.949 2026.25,743.191 \n",
+ " 2026.9,743.466 2027.55,743.772 2028.21,744.106 2028.86,744.465 2029.51,744.846 2030.17,745.246 2030.82,745.661 2031.47,746.089 2032.13,746.525 2032.78,746.966 \n",
+ " 2033.43,747.409 2034.08,747.849 2034.74,748.283 2035.39,748.708 2036.04,749.119 2036.7,749.514 2037.35,749.889 2038,750.241 2038.66,750.568 2039.31,750.865 \n",
+ " 2039.96,751.132 2040.61,751.366 2041.27,751.565 2041.92,751.727 2042.57,751.851 2043.23,751.936 2043.88,751.982 2044.53,751.987 2045.19,751.952 2045.84,751.876 \n",
+ " 2046.49,751.761 2047.14,751.607 2047.8,751.415 2048.45,751.188 2049.1,750.926 2049.76,750.633 2050.41,750.311 2051.06,749.961 2051.72,749.588 2052.37,749.193 \n",
+ " 2053.02,748.78 2053.67,748.351 2054.33,747.912 2054.98,747.465 2055.63,747.013 2056.29,746.561 2056.94,746.112 2057.59,745.67 2058.25,745.237 2058.9,744.819 \n",
+ " 2059.55,744.417 2060.2,744.035 2060.86,743.676 2061.51,743.342 2062.16,743.037 2062.82,742.763 2063.47,742.522 2064.12,742.315 2064.78,742.144 2065.43,742.011 \n",
+ " 2066.08,741.917 2066.74,741.862 2067.39,741.846 2068.04,741.869 2068.69,741.931 2069.35,742.032 2070,742.169 2070.65,742.343 2071.31,742.551 2071.96,742.791 \n",
+ " 2072.61,743.062 2073.27,743.361 2073.92,743.687 2074.57,744.035 2075.22,744.403 2075.88,744.789 2076.53,745.187 2077.18,745.597 2077.84,746.013 2078.49,746.433 \n",
+ " 2079.14,746.853 2079.8,747.27 2080.45,747.68 2081.1,748.081 2081.75,748.468 2082.41,748.838 2083.06,749.189 2083.71,749.518 2084.37,749.821 2085.02,750.097 \n",
+ " 2085.67,750.342 2086.33,750.556 2086.98,750.736 2087.63,750.881 2088.28,750.989 2088.94,751.06 2089.59,751.094 2090.24,751.089 2090.9,751.047 2091.55,750.967 \n",
+ " 2092.2,750.849 2092.86,750.695 2093.51,750.506 2094.16,750.283 2094.81,750.029 2095.47,749.744 2096.12,749.432 2096.77,749.096 2097.43,748.737 2098.08,748.36 \n",
+ " 2098.73,747.966 2099.39,747.559 2100.04,747.142 2100.69,746.719 2101.34,746.293 2102,745.867 2102.65,745.445 2103.3,745.03 2103.96,744.626 2104.61,744.236 \n",
+ " 2105.26,743.862 2105.92,743.509 2106.57,743.178 2107.22,742.873 2107.88,742.596 2108.53,742.348 2109.18,742.133 2109.83,741.951 2110.49,741.804 2111.14,741.693 \n",
+ " 2111.79,741.619 2112.45,741.582 2113.1,741.584 2113.75,741.623 2114.41,741.7 2115.06,741.813 2115.71,741.962 2116.36,742.146 2117.02,742.362 2117.67,742.609 \n",
+ " 2118.32,742.885 2118.98,743.187 2119.63,743.514 2120.28,743.862 2120.94,744.228 2121.59,744.611 2122.24,745.006 2122.89,745.41 2123.55,745.82 2124.2,746.233 \n",
+ " 2124.85,746.644 2125.51,747.052 2126.16,747.452 2126.81,747.842 2127.47,748.217 2128.12,748.576 2128.77,748.914 2129.42,749.231 2130.08,749.522 2130.73,749.785 \n",
+ " 2131.38,750.019 2132.04,750.221 2132.69,750.39 2133.34,750.524 2134,750.623 2134.65,750.684 2135.3,750.709 2135.95,750.696 2136.61,750.646 2137.26,750.559 \n",
+ " 2137.91,750.437 2138.57,750.279 2139.22,750.088 2139.87,749.865 2140.53,749.611 2141.18,749.328 2141.83,749.02 2142.48,748.687 2143.14,748.334 2143.79,747.963 \n",
+ " 2144.44,747.577 2145.1,747.179 2145.75,746.773 2146.4,746.361 2147.06,745.947 2147.71,745.535 2148.36,745.128 2149.02,744.728 2149.67,744.339 2150.32,743.965 \n",
+ " 2150.97,743.609 2151.63,743.273 2152.28,742.96 2152.93,742.673 2153.59,742.414 2154.24,742.185 2154.89,741.988 2155.55,741.825 2156.2,741.697 2156.85,741.605 \n",
+ " 2157.5,741.55 2158.16,741.532 2158.81,741.551 2159.46,741.608 2160.12,741.702 2160.77,741.832 2161.42,741.997 2162.08,742.197 2162.73,742.429 2163.38,742.691 \n",
+ " 2164.03,742.982 2164.69,743.299 2165.34,743.639 2165.99,744 2166.65,744.378 2167.3,744.771 2167.95,745.176 2168.61,745.589 2169.26,746.008 2169.91,746.428 \n",
+ " 2170.56,746.846 2171.22,747.26 2171.87,747.664 2172.52,748.057 2173.18,748.435 2173.83,748.795 2174.48,749.134 2175.14,749.449 2175.79,749.739 2176.44,750 \n",
+ " 2177.09,750.23 2177.75,750.428 2178.4,750.591 2179.05,750.72 2179.71,750.812 2180.36,750.868 2181.01,750.885 2181.67,750.865 2182.32,750.807 2182.97,750.711 \n",
+ " 2183.62,750.579 2184.28,750.412 2184.93,750.212 2185.58,749.979 2186.24,749.717 2186.89,749.426 2187.54,749.109 2188.2,748.77 2188.85,748.41 2189.5,748.032 \n",
+ " 2190.15,747.64 2190.81,747.237 2191.46,746.827 2192.11,746.411 2192.77,745.995 2193.42,745.581 2194.07,745.174 2194.73,744.775 2195.38,744.388 2196.03,744.018 \n",
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+ ]
+ },
+ "execution_count": 68,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "using ModelingToolkit, Plots, DifferentialEquations, LinearAlgebra\n",
+ "using Symbolics: scalarize\n",
+ "\n",
+ "@variables t\n",
+ "D = Differential(t)\n",
+ "\n",
+ "function Mass(; name, m = 1.0, xy = [0., 0.], u = [0., 0.])\n",
+ " ps = @parameters m=m\n",
+ " sts = @variables pos[1:2](t)=xy v[1:2](t)=u\n",
+ " eqs = scalarize(D.(pos) .~ v)\n",
+ " ODESystem(eqs, t, [pos..., v...], ps; name)\n",
+ "end\n",
+ "\n",
+ "function Spring(; name, k = 1e4, l = 1.)\n",
+ " ps = @parameters k=k l=l\n",
+ " @variables x(t), dir[1:2](t)\n",
+ " ODESystem(Equation[], t, [x, dir...], ps; name)\n",
+ "end\n",
+ "\n",
+ "function connect_spring(spring, a, b)\n",
+ " [\n",
+ " spring.x ~ norm(scalarize(a .- b))\n",
+ " scalarize(spring.dir .~ scalarize(a .- b))\n",
+ " ]\n",
+ "end\n",
+ "\n",
+ "spring_force(spring) = -spring.k .* scalarize(spring.dir) .* (spring.x - spring.l) ./ spring.x\n",
+ "\n",
+ "m = 1.0\n",
+ "xy = [1., -1.]\n",
+ "k = 1e4\n",
+ "l = 1.\n",
+ "center = [0., 0.]\n",
+ "g = [0., -9.81]\n",
+ "@named mass = Mass(m=m, xy=xy)\n",
+ "@named spring = Spring(k=k, l=l)\n",
+ "\n",
+ "eqs = [\n",
+ " connect_spring(spring, mass.pos, center)\n",
+ " scalarize(D.(mass.v) .~ spring_force(spring) / mass.m .+ g)\n",
+ "]\n",
+ "\n",
+ "@named _model = ODESystem(eqs, t, [spring.x; spring.dir; mass.pos], [])\n",
+ "@named model = compose(_model, mass, spring)\n",
+ "sys = structural_simplify(model)\n",
+ "\n",
+ "prob = ODEProblem(sys, [], (0., 3.))\n",
+ "sol = solve(prob, Tsit5())\n",
+ "plot(sol)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## 1d cart with motor"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 94,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Mass (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 94,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function Mass(; name, m = 1.0, x = [0.], u = [0.])\n",
+ " ps = @parameters m=m\n",
+ " sts = @variables t pos(t)=x v(t)=u f(t)\n",
+ " eqs = scalarize([D(pos) ~ v,\n",
+ " D(v) ~ f])\n",
+ " ODESystem(eqs, t, [pos..., v...,f], ps; name)\n",
+ "end\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 96,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\frac{dpos(t)}{dt} =& v\\left( t \\right) \\\\\n",
+ "\\frac{dv(t)}{dt} =& f\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "\u001b[0m\u001b[1mModel cart_1 with 2 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
+ "\u001b[0m\u001b[1mStates (3):\u001b[22m\n",
+ " pos(t) [defaults to [0.0]]\n",
+ " v(t) [defaults to [0.0]]\n",
+ " f(t)\n",
+ "\u001b[0m\u001b[1mParameters (1):\u001b[22m\n",
+ " m [defaults to 1.0]"
+ ]
+ },
+ "execution_count": 96,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "@named cart_1=Mass()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 95,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\frac{dpos(t)}{dt} =& v\\left( t \\right) \\\\\n",
+ "\\frac{dv(t)}{dt} =& f\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "\u001b[0m\u001b[1mModel cart_1 with 2 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
+ "\u001b[0m\u001b[1mStates (2):\u001b[22m\n",
+ " pos(t) [defaults to [0.0]]\n",
+ " v(t) [defaults to [0.0]]\n",
+ "\u001b[0m\u001b[1mParameters (1):\u001b[22m\n",
+ " m [defaults to 1.0]"
+ ]
+ },
+ "execution_count": 95,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "cart_1.f"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 112,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\mathrm{cart}_{1_{+}f}\\left( t \\right) =& \\cos\\left( t \\right) \\\\\n",
+ "\\frac{dcart_{1_{+}pos(t)}}{dt} =& \\mathrm{cart}_{1_{+}v}\\left( t \\right) \\\\\n",
+ "\\frac{dcart_{1_{+}v(t)}}{dt} =& \\mathrm{cart}_{1_{+}f}\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "\u001b[0m\u001b[1mModel connected with 3 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
+ "\u001b[0m\u001b[1mStates (3):\u001b[22m\n",
+ " cart_1₊f(t)\n",
+ " cart_1₊pos(t) [defaults to [0.0]]\n",
+ " cart_1₊v(t) [defaults to [0.0]]\n",
+ "\u001b[0m\u001b[1mParameters (1):\u001b[22m\n",
+ " cart_1₊m [defaults to 1.0]"
+ ]
+ },
+ "execution_count": 112,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "# Create externally driven car.\n",
+ "conection= [cart_1.f ~ cos(t)]\n",
+ "mass_sin=compose(ODESystem(conection,name=:connected), cart_1)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 113,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/latex": [
+ "\\begin{align}\n",
+ "\\frac{dcart_{1_{+}pos(t)}}{dt} =& \\mathrm{cart}_{1_{+}v}\\left( t \\right) \\\\\n",
+ "\\frac{dcart_{1_{+}v(t)}}{dt} =& \\mathrm{cart}_{1_{+}f}\\left( t \\right)\n",
+ "\\end{align}\n"
+ ],
+ "text/plain": [
+ "\u001b[0m\u001b[1mModel connected with 2 \u001b[22m\u001b[0m\u001b[1mequations\u001b[22m\n",
+ "\u001b[0m\u001b[1mStates (2):\u001b[22m\n",
+ " cart_1₊pos(t) [defaults to [0.0]]\n",
+ " cart_1₊v(t) [defaults to [0.0]]\n",
+ "\u001b[0m\u001b[1mParameters (1):\u001b[22m\n",
+ " cart_1₊m [defaults to 1.0]\n",
+ "\u001b[35mIncidence matrix:\u001b[39msparse([1, 2, 1], [1, 2, 4], Num[×, ×, ×], 2, 4)"
+ ]
+ },
+ "execution_count": 113,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "sys"
+ ]
+ },
+ {
+ "cell_type": "code",
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+ ]
+ },
+ "execution_count": 115,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "sys = structural_simplify(mass_sin)\n",
+ "\n",
+ "prob = ODEProblem(sys, [0,0], (0., 50.))\n",
+ "sol = solve(prob, Tsit5())\n",
+ "plot(sol, vars=[cart_1.v,cart_1.pos,cart_1.f])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 108,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "41-element Vector{Vector{Float64}}:\n",
+ " [0.0, 0.0]\n",
+ " [1.6666666658333231e-13, 4.999999995833331e-9]\n",
+ " [2.2183331991241636e-10, 6.049999389958352e-7]\n",
+ " [2.27937095788949e-7, 6.16043674732584e-5]\n",
+ " [6.383124635259641e-5, 0.002636200919926346]\n",
+ " [0.0009652979676921317, 0.016099426130024245]\n",
+ " [0.005311910097367641, 0.050008622191579276]\n",
+ " [0.022337190821515782, 0.12922088880536703]\n",
+ " [0.07676872378873756, 0.2892035647359231]\n",
+ " [0.23352897840994905, 0.5857723644188603]\n",
+ " [0.6416911532174759, 1.0684926300134545]\n",
+ " [1.647263601605248, 1.706349937568098]\n",
+ " [3.148256197083767, 1.9999947343946864]\n",
+ " ⋮\n",
+ " [29.781479398711376, 1.7015812852953236]\n",
+ " [31.41510063804408, 0.012940880586619396]\n",
+ " [31.991242704953144, 1.0045433744194996]\n",
+ " [35.01751390932639, 1.9737801265007684]\n",
+ " [37.610924363265504, 0.319989473142961]\n",
+ " [37.836525725325494, 0.41684790528129834]\n",
+ " [40.072255874368935, 1.925020045514537]\n",
+ " [43.344007131508086, 1.0696839969039056]\n",
+ " [43.989151111020895, 0.031723666938272735]\n",
+ " [45.31292737327349, 1.5543046508753369]\n",
+ " [48.639200319443475, 1.7006910818458434]\n",
+ " [50.26862814813002, 0.03561853445382968]"
+ ]
+ },
+ "execution_count": 108,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "sol.u"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## First mass with Pendulum"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "For this we should create a \"weld\" or connector so we can build up the system componentwise.\n",
+ "\n",
+ "Goal: Build a cart with a double pendulum. The method should be scalable such that we could add arbitrary amount of pendulums."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 117,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Joint (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 117,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "@variables t\n",
+ "D = Differential(t)\n",
+ "\n",
+ "\n",
+ "@connector function Joint(;name)\n",
+ " sts= @variables pos[1:2](t)=xy v[1:2](t) f_xy[1:2](t)\n",
+ " ODESystem(Equation[], t, sts, []; name=name)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 116,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "LoadError",
+ "evalue": "LoadError: MethodError: no method matching var\"@connector\"(::LineNumberNode, ::Module)\n\u001b[0mClosest candidates are:\n\u001b[0m var\"@connector\"(::LineNumberNode, ::Module, \u001b[91m::Any\u001b[39m) at ~/.julia/packages/ModelingToolkit/qKYfn/src/systems/connectors.jl:25\nin expression starting at In[116]:1",
+ "output_type": "error",
+ "traceback": [
+ "LoadError: MethodError: no method matching var\"@connector\"(::LineNumberNode, ::Module)\n\u001b[0mClosest candidates are:\n\u001b[0m var\"@connector\"(::LineNumberNode, ::Module, \u001b[91m::Any\u001b[39m) at ~/.julia/packages/ModelingToolkit/qKYfn/src/systems/connectors.jl:25\nin expression starting at In[116]:1",
+ "",
+ "Stacktrace:",
+ " [1] eval",
+ " @ ./boot.jl:373 [inlined]",
+ " [2] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
+ " @ Base ./loading.jl:1196"
+ ]
+ }
+ ],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 128,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "133//72"
+ ]
+ },
+ "execution_count": 128,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ " 1//12*(1//2+ 1//3 + 1//4+1//2+1//3+1//4+1//2) +\n",
+ "(3//4+2//3)*1//18+1//9*(5//4+2//3)+2//9*(1+2//3)+4//9*(3//2+2//3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 217,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "1.8472222222222223"
+ ]
+ },
+ "execution_count": 217,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "133/72"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "simulate epidemic on graph:"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 180,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "6×6 Matrix{Float64}:\n",
+ " 0.333333 0.333333 0.333333 0.0 0.0 0.0\n",
+ " 0.333333 0.0 0.333333 0.333333 0.0 0.0\n",
+ " 0.333333 0.333333 0.0 0.0 0.333333 0.0\n",
+ " 0.0 0.333333 0.0 0.0 0.333333 0.333333\n",
+ " 0.0 0.0 0.333333 0.333333 0.0 0.333333\n",
+ " 0.0 0.0 0.0 0.333333 0.333333 0.333333"
+ ]
+ },
+ "execution_count": 180,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "Λ= [0 1 1 0 0 0\n",
+ " 1 0 1 1 0 0\n",
+ " 1 1 0 0 1 0\n",
+ " 0 1 0 0 1 1\n",
+ " 0 0 1 1 0 1\n",
+ " 0 0 0 1 1 0]\n",
+ "w= Λ*ones(6)\n",
+ "\n",
+ "\n",
+ "w_max= maximum(w)\n",
+ "P_ = (I(6)*w_max)\\Λ\n",
+ "P_ = P_ + diagm(ones(6)-P_*ones(6))"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 184,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "LoadError",
+ "evalue": "UndefVarError: max_itr not defined",
+ "output_type": "error",
+ "traceback": [
+ "UndefVarError: max_itr not defined",
+ "",
+ "Stacktrace:",
+ " [1] top-level scope",
+ " @ :0",
+ " [2] eval",
+ " @ ./boot.jl:373 [inlined]",
+ " [3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
+ " @ Base ./loading.jl:1196"
+ ]
+ }
+ ],
+ "source": [
+ "max_itr"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 191,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "1×6 Matrix{Int64}:\n",
+ " 0 1 1 0 0 0"
+ ]
+ },
+ "execution_count": 191,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "Λ[inds,:]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 195,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "6×4 BitMatrix:\n",
+ " 0 1 1 0\n",
+ " 1 0 1 1\n",
+ " 1 1 0 0\n",
+ " 0 1 0 0\n",
+ " 0 0 1 1\n",
+ " 0 0 0 1"
+ ]
+ },
+ "execution_count": 195,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "Bool.(Λ[inds,:]')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 207,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "CartesianIndex(3, 2)"
+ ]
+ },
+ "execution_count": 207,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 205,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "1"
+ ]
+ },
+ "execution_count": 205,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "X[2]=I_val"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 206,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "3-element Vector{CartesianIndex{2}}:\n",
+ " CartesianIndex(3, 1)\n",
+ " CartesianIndex(3, 2)\n",
+ " CartesianIndex(4, 2)"
+ ]
+ },
+ "execution_count": 206,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "inds=findall(X.==I_val)\n",
+ "# inds=[1,2,3,4]\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 220,
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "1.8479183238109291"
+ ]
+ },
+ "execution_count": 220,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "S=0\n",
+ "I_val=1\n",
+ "\n",
+ "N=10^5\n",
+ "t_vec=zeros(N)\n",
+ "for i=1:N\n",
+ " X=zeros(6);\n",
+ " X[1]=I_val\n",
+ " t=0\n",
+ " max_itr=0\n",
+ " while X[6]!=I_val && max_itr<50\n",
+ " max_itr+=1\n",
+ " inds= findall(X.==I_val)\n",
+ " neighbs= findall(Bool.(Λ[inds,:]') .&& X.==S)\n",
+ " m=length(neighbs)\n",
+ " X[rand(neighbs)[1]]=I_val \n",
+ " \n",
+ " t+=-log(rand())/m \n",
+ " \n",
+ " end\n",
+ " t_vec[i]=t\n",
+ " \n",
+ "end\n",
+ "mean(t_vec)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 181,
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "LoadError",
+ "evalue": "UndefVarError: X not defined",
+ "output_type": "error",
+ "traceback": [
+ "UndefVarError: X not defined",
+ "",
+ "Stacktrace:",
+ " [1] top-level scope",
+ " @ In[181]:1",
+ " [2] eval",
+ " @ ./boot.jl:373 [inlined]",
+ " [3] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
+ " @ Base ./loading.jl:1196"
+ ]
+ }
+ ],
+ "source": [
+ "findall(X.==I)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 178,
+ "metadata": {},
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "2"
+ ]
+ }
+ ],
+ "source": [
+ "e=rand()\n",
+ "if e<0.1\n",
+ " print(1)\n",
+ "else\n",
+ " print(2)\n",
+ "end"
+ ]
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.7.2",
+ "language": "julia",
+ "name": "julia-1.7"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.7.2"
+ },
+ "toc": {
+ "base_numbering": 1,
+ "nav_menu": {},
+ "number_sections": true,
+ "sideBar": true,
+ "skip_h1_title": false,
+ "title_cell": "Table of Contents",
+ "title_sidebar": "Contents",
+ "toc_cell": false,
+ "toc_position": {},
+ "toc_section_display": true,
+ "toc_window_display": false
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 4
+}
diff --git a/mtk_2d_pendulumcart.ipynb b/mtk_2d_pendulumcart.ipynb
new file mode 100644
index 0000000000000000000000000000000000000000..4671b5b69d48f4f9c127ecf89c50870dd41c482c
--- /dev/null
+++ b/mtk_2d_pendulumcart.ipynb
@@ -0,0 +1,376 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "af9b2987-bc34-443b-9b02-98c6946ca1ba",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "using ModelingToolkit\n",
+ "using Plots\n",
+ "using DifferentialEquations\n",
+ "using ModelingToolkitStandardLibrary: Blocks.RealInput, Blocks.RealOutput, Blocks.Constant\n",
+ "@variables t\n",
+ "D = Differential(t)\n",
+ "const g = 9.82;"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "feb48814-3c01-4456-8588-227f2dd6fd29",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Weld (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 2,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "@connector function Weld(;name, x0 = [0.0,0.0])\n",
+ " sts = @variables f(t) = [0.0, 0.0] x(t) = x0 M(t) = 0.0 [connect = Flow]\n",
+ " ODESystem(Equation[], t, sts, []; name=name)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "id": "aa300653-0f6b-48a0-aa99-ac5784b755e6",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "RevoluteJoint (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function RevoluteJoint(;name, x0 = [0.0,0.0])\n",
+ " @named i = Weld(x0=x0)\n",
+ " @named o = Weld(x0=x0)\n",
+ " sts = @variables f(t) = [0.0,0.0] x(t) = x0 M(t)=0.0\n",
+ " eqs = [\n",
+ " [0.0,0.0] ~ i.x - o.x\n",
+ " [0.0,0.0] ~ i.f + o.f\n",
+ " 0.0 ~ i.M \n",
+ " 0.0 ~ o.M\n",
+ " x ~ i.x\n",
+ " f ~ i.f\n",
+ " M ~ 0.0\n",
+ " ]\n",
+ " compose(ODESystem(eqs, t, sts, []; name=name), i, o)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "id": "3ef59724-98a4-49a7-950c-59da1deef5c1",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "PoweredJoint (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function PoweredJoint(;name, x0 = [0.0,0.0])\n",
+ " @named i = Weld(x0=x0)\n",
+ " @named o = Weld(x0=x0)\n",
+ " @named M = RealInput()\n",
+ " sts = @variables f(t) = [0.0,0.0] x(t) = x0 M(t)=0.0\n",
+ " eqs = [\n",
+ " [0.0,0.0] ~ i.x - o.x\n",
+ " [0.0,0.0] ~ i.f + o.f\n",
+ " M.u ~ i.M \n",
+ " M.u ~ o.M\n",
+ " x ~ i.x\n",
+ " f ~ i.f\n",
+ " ]\n",
+ " compose(ODESystem(eqs, t, sts, []; name=name), i, o, M)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 22,
+ "id": "f24e258e-4290-407b-90dd-0f8503c0062c",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Body (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 22,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function Body(;name, m = 1.0, J = 1.0, x0 = [0.0,0.0], θ0 = 0.0)\n",
+ " sts = @variables f(t)=[0.0,0.0] x(t)=x0 M(t)=0.0 θ(t)=θ0\n",
+ " ps = @parameters m=m J=J\n",
+ " eqs = [\n",
+ " m*D(D(x)) ~ f\n",
+ " J*D(D(θ)) ~ M\n",
+ " ]\n",
+ " ODESystem(eqs, t, sts, ps; name=name)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 23,
+ "id": "498fc478-bf35-462e-890e-2fb2525251ee",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "cross (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 23,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function cross(f,dx)\n",
+ " dx[1]*f[2] - dx[2]*f[1]\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 29,
+ "id": "3aea1aa5-73fc-4569-bde3-b6bc827c1c1b",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "PendulumCart (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 29,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function PendulumCart(;name, m = 1.0, J = 1.0, x0 = [0.0,0.0], θ0 = 0.0,w = 2.0, h = 1.0)\n",
+ " @named body = Body(m=m,J=J,x0=x0,θ0=θ0)\n",
+ " @unpack f,x,M,θ = body\n",
+ " ps = @parameters h=h,w=w\n",
+ " @named centre = Weld(x0=x0)\n",
+ " @named right = Weld(x0=x0+w/2*[cos(θ0),sin(θ0)])\n",
+ " @named left = Weld(x0=x0-w/2*[cos(θ0),sin(θ0)])\n",
+ " eqs = [\n",
+ " M ~ centre.M + right.M + left.M + cross(right.f,right.x-x) + cross(left.f,left.x-x)\n",
+ " f ~ [0.0,-g]*m + centre.f + left.f + right.f\n",
+ " right.x ~ centre.x + w/2*[cos(θ),sin(θ)]\n",
+ " left.x ~ centre.x - w/2*[cos(θ),sin(θ)]\n",
+ " ]\n",
+ " extend(ODESystem(eqs, t, [], ps; name=name), body,centre, right, left)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 30,
+ "id": "21158308-ad06-4ff9-99c2-ab25919df577",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Wheel (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 30,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function Wheel(;name, m = 1.0, J = 1.0, x0 = [0.0,0.0], θ0 = 0.0,r = 1.0)\n",
+ " @named body = Body(m=m,J=J,x0=x0,θ0=θ0)\n",
+ " @unpack f,x,M,θ = body\n",
+ " ps = @parameters r=r\n",
+ " @named centre = Weld()\n",
+ " sts = @variables fn = [0.0,0.0]\n",
+ " eqs = [\n",
+ " M ~ centre.M + r*centre.f[1]\n",
+ " f ~ centre.f + [0.0,-m*g] + fn\n",
+ " f[2] ~ 0.0\n",
+ " ]\n",
+ " extend(ODESystem(eqs, t, sts, ps; name=name), body,centre)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 31,
+ "id": "cda51659-e1aa-4b97-98cc-db77bbabce87",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/plain": [
+ "Pendulum (generic function with 1 method)"
+ ]
+ },
+ "execution_count": 31,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "function Pendulum(;name, m = 1.0, J = 1.0, x0 = [0.0,0.0], θ0 = 0.0,l = 2.0)\n",
+ " @named body = Body(m=m,J=J,x0=x0,θ0=θ0)\n",
+ " @unpack f,x,M,θ = body\n",
+ " ps = @parameters l=l\n",
+ " @named centre = Weld(x0=x0)\n",
+ " eqs = [\n",
+ " M ~ centre.M + m*g*sin(θ)\n",
+ " F ~ centre.f + [0.0, -m*g]\n",
+ " ]\n",
+ " extend(ODESystem(eqs, t, [], ps; name=kname), body,centre)\n",
+ "end"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 33,
+ "id": "563108a4-8acc-4d80-9ef3-9208418fd9fe",
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "LoadError",
+ "evalue": "BoundsError: attempt to access Float64 at index [2]",
+ "output_type": "error",
+ "traceback": [
+ "BoundsError: attempt to access Float64 at index [2]",
+ "",
+ "Stacktrace:",
+ " [1] indexed_iterate(I::Float64, i::Int64, state::Nothing)",
+ " @ Base ./tuple.jl:98",
+ " [2] PendulumCart(; name::Symbol, m::Float64, J::Float64, x0::Vector{Float64}, θ0::Float64, w::Float64, h::Float64)",
+ " @ Main ./In[29]:4",
+ " [3] top-level scope",
+ " @ In[33]:1",
+ " [4] eval",
+ " @ ./boot.jl:373 [inlined]",
+ " [5] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
+ " @ Base ./loading.jl:1196"
+ ]
+ }
+ ],
+ "source": [
+ "@named cart = PendulumCart()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 34,
+ "id": "0078e55d-d7a8-4516-a3b3-096e58242588",
+ "metadata": {},
+ "outputs": [
+ {
+ "ename": "LoadError",
+ "evalue": "MethodError: no method matching +(::Num, ::Vector{Float64})\nFor element-wise addition, use broadcasting with dot syntax: scalar .+ array\n\u001b[0mClosest candidates are:\n\u001b[0m +(::Any, ::Any, \u001b[91m::Any\u001b[39m, \u001b[91m::Any...\u001b[39m) at /usr/share/julia/base/operators.jl:655\n\u001b[0m +(\u001b[91m::Union{InitialValues.NonspecificInitialValue, InitialValues.SpecificInitialValue{typeof(+)}}\u001b[39m, ::Any) at ~/.julia/packages/InitialValues/OWP8V/src/InitialValues.jl:154\n\u001b[0m +(\u001b[91m::ChainRulesCore.AbstractThunk\u001b[39m, ::Any) at ~/.julia/packages/ChainRulesCore/RbX5a/src/tangent_arithmetic.jl:122\n\u001b[0m ...",
+ "output_type": "error",
+ "traceback": [
+ "MethodError: no method matching +(::Num, ::Vector{Float64})\nFor element-wise addition, use broadcasting with dot syntax: scalar .+ array\n\u001b[0mClosest candidates are:\n\u001b[0m +(::Any, ::Any, \u001b[91m::Any\u001b[39m, \u001b[91m::Any...\u001b[39m) at /usr/share/julia/base/operators.jl:655\n\u001b[0m +(\u001b[91m::Union{InitialValues.NonspecificInitialValue, InitialValues.SpecificInitialValue{typeof(+)}}\u001b[39m, ::Any) at ~/.julia/packages/InitialValues/OWP8V/src/InitialValues.jl:154\n\u001b[0m +(\u001b[91m::ChainRulesCore.AbstractThunk\u001b[39m, ::Any) at ~/.julia/packages/ChainRulesCore/RbX5a/src/tangent_arithmetic.jl:122\n\u001b[0m ...",
+ "",
+ "Stacktrace:",
+ " [1] Pendulum(; name::Symbol, m::Float64, J::Float64, x0::Vector{Float64}, θ0::Float64, l::Float64)",
+ " @ Main ./In[31]:6",
+ " [2] top-level scope",
+ " @ In[34]:1",
+ " [3] eval",
+ " @ ./boot.jl:373 [inlined]",
+ " [4] include_string(mapexpr::typeof(REPL.softscope), mod::Module, code::String, filename::String)",
+ " @ Base ./loading.jl:1196"
+ ]
+ }
+ ],
+ "source": [
+ "@named pendulum = Pendulum()\n",
+ "@named rightWheel = Wheel(x0 = [-1.0,0.0])\n",
+ "@named rightWheel = Wheel(x0 = [1.0,0.0])\n",
+ "\n",
+ "@named pendulumJoint = RevoluteJoint()\n",
+ "@named rightJoint = RevoluteJoint(x0 = [1.0,0.0])\n",
+ "@named leftJoint = PoweredJoint(x0 = [-1.0,0.0])\n",
+ "\n",
+ "@named momentSignal = Constant(k=1.0)\n",
+ "\n",
+ "cart_eqs = [\n",
+ " connect(cart.centre, pendulumJoint.i)\n",
+ " connect(pendulumJoint.o, pendulum.centre)\n",
+ " \n",
+ " connect(cart.right, rightJoint.i)\n",
+ " connect(rightJoint.o, rightWheel.centre)\n",
+ " \n",
+ " connect(cart.left, leftJoint.i)\n",
+ " connect(leftJoint.o, leftWheel.centre)\n",
+ " connect(leftJoint.M, momentSignal.output)\n",
+ " ]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "52086c13-1035-4069-9021-cc2cc1ccf9d8",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "id": "413a40b7-1750-453f-a168-bdcd36c9c254",
+ "metadata": {},
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Julia 1.7.2",
+ "language": "julia",
+ "name": "julia-1.7"
+ },
+ "language_info": {
+ "file_extension": ".jl",
+ "mimetype": "application/julia",
+ "name": "julia",
+ "version": "1.7.2"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
diff --git a/physical_specifications.ipynb b/physical_specifications.ipynb
deleted file mode 100644
index 62281ad966c71100e2f888cb58a52a7251db590a..0000000000000000000000000000000000000000
--- a/physical_specifications.ipynb
+++ /dev/null
@@ -1,50 +0,0 @@
-{
- "cells": [
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "## 1D car with two wheels"
- ]
- },
- {
- "cell_type": "markdown",
- "metadata": {},
- "source": [
- "1. car_length = 3 u\"m\"\n",
- "2. car_mass = 500 u\"kg\"\n",
- "3. wheel_mass = 5 u\"kg\"\n",
- "4. wheel_radius= 0.25 u\"m\"\n",
- "5. wheel_inertia= "
- ]
- }
- ],
- "metadata": {
- "kernelspec": {
- "display_name": "Julia 1.5.2",
- "language": "julia",
- "name": "julia-1.5"
- },
- "language_info": {
- "file_extension": ".jl",
- "mimetype": "application/julia",
- "name": "julia",
- "version": "1.5.2"
- },
- "toc": {
- "base_numbering": 1,
- "nav_menu": {},
- "number_sections": true,
- "sideBar": true,
- "skip_h1_title": false,
- "title_cell": "Table of Contents",
- "title_sidebar": "Contents",
- "toc_cell": false,
- "toc_position": {},
- "toc_section_display": true,
- "toc_window_display": false
- }
- },
- "nbformat": 4,
- "nbformat_minor": 4
-}