Notebook showing time-reversibility of constrained integrators

Reversibility-of-constrained-integrators.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra\n",
    "using Random\n",
    "using Plots\n",
    "using Printf\n",
    "import LinearAlgebra: norm\n",
    "import Base: ∘"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Only print floating-point values to 3 decimal places."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "Base.show(io::IO, f::Float64) = @printf(io, \"%1.3f\", f)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Redefine composition operator ∘ to splat output of second function to allow chaining functions of tuples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "∘ (generic function with 3 methods)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "∘(f::Function, g::Function) = (x...)->f(g(x...)...)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Flow map for Hamiltonian component $h_1(q, p) = \\ell(q)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Φ₁ (generic function with 1 method)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Φ₁(t) = (q, p) -> (q, p - t * ∇ℓ(q))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Flow map for Hamiltonian component $h_2(q, p) = \\frac{1}{2} p' M^{-1} p$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Φ₂ (generic function with 1 method)"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Φ₂(t) = (q, p) -> (q + t * M⁻¹ * p, p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Map to project momentum on to co-tangent space such that $\\partial c(q) M^{-1} p = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Π (generic function with 1 method)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Π(q, p) = (q, p - ∂c(q)' * ((∂c(q) * M⁻¹ * ∂c(q)') \\ (∂c(q) * M⁻¹ * p)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Involution $S$ which switches integration direction / sign $s \\in \\lbrace -1, +1 \\rbrace$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "S (generic function with 1 method)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S(q, p) = (q, -p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Enforce constraints on position output of Φ₂ by solving for Lagrange multipliers λ such that `c(Φ₂(t)(q, p + ∂c(q)' * λ)[1]) = 0`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Φ₂Π (generic function with 1 method)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function Φ₂Π(t)\n",
    "    (q, p) -> begin\n",
    "        λ = 0\n",
    "        qₙ₋₁ = q\n",
    "        for n ∈ 1:N\n",
    "            (qₙ, pₙ) = Φ₂(t)(q, p + ∂c(q)' * λ)\n",
    "            e = c(qₙ) \n",
    "            if norm(e) < ϵ && norm(qₙ - qₙ₋₁) < τ\n",
    "                return (qₙ, pₙ)\n",
    "            end\n",
    "            λ = λ .- t * (∂c(qₙ) * M⁻¹ * ∂c(q)') \\ e\n",
    "            qₙ₋₁ = qₙ\n",
    "        end  \n",
    "        error(\"Convergence error\")\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Reversibility check transform $R$\n",
    "$$\n",
    "  R(\\Phi) = \\begin{cases} \n",
    "    \\Phi & \\text{if } (S \\circ \\Phi) \\circ (S \\circ \\Phi) = \\mathrm{id}, \\\\\n",
    "    S & \\text{otherwise.}\n",
    "  \\end{cases}\n",
    "$$\n",
    "Transforms a map $\\Phi$ such that $S \\circ R(\\Phi)$ is always an involution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "R (generic function with 1 method)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function R(Φ)  \n",
    "    (q, p) -> begin\n",
    "        try\n",
    "            (q₁, p₁) = Φ(q, p)    \n",
    "            (q₀, p₀) = (S ∘ Φ ∘ S)(q₁, p₁)    \n",
    "            if norm(q₀ - q) > τ || norm(p₀ - p) > τ    \n",
    "                error(\"Non-reversible step\")\n",
    "            end\n",
    "            return (q₁, p₁)\n",
    "        catch e\n",
    "            return S(q, p)\n",
    "        end \n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Reversible geodesic integrator step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ψᴳ (generic function with 1 method)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ψᴳ(t) = Π ∘ Φ₁(t / 2) ∘ R(Π ∘ Φ₂Π(t)) ∘ Π ∘ Φ₁(t / 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Reversible Rattle integrator step"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ψᴿ (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Ψᴿ(t) = R(Π ∘ Φ₁(t / 2) ∘ Φ₂Π(t) ∘ Φ₁(t / 2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Constraint function and constraint Jacobian for two-dimensional torus in three-dimensional ambient space"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "∂c (generic function with 1 method)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c((x, y, z)) = [(√(x^2 + y^2) - 1)^2 + z^2 - 1/4]\n",
    "∂c((x, y, z)) = 2 * [x * (1 - 1 / √(x^2 + y^2)) y * (1 - 1 / √(x^2 + y^2)) z]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Standard normal distribution on ambient space"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "∇ℓ (generic function with 1 method)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ℓ(q) = sum(q^2) / 2\n",
    "∇ℓ(q) = q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use a standard Euclidean metric with identity matrix representation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "UniformScaling{Bool}\n",
       "true*I"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M⁻¹ = I "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Use $\\infty$-norm for convergence checks and set constraint / position check tolerances, and maximal number of Newton iterations "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm(x) = norm(x, Inf)  # ∞-norm for convergence checks\n",
    "ϵ = 1e-9  # Constraint check tolerance\n",
    "τ = 1e-8  # Position / reversibility check tolerance\n",
    "N = 50  # Maximum number of Newton iterations  "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Sample initial state in co-tangent bundle for torus manifold"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "rng = MersenneTwister(202007)\n",
    "θ, ϕ = rand(rng, Float64, 2) * 2π \n",
    "q = [(1 + cos(ϕ)/2) * cos(θ), (1 + cos(ϕ)/2) * sin(θ), sin(ϕ)/2]\n",
    "p = randn(rng, 3)\n",
    "(q, p) = Π(q, p)\n",
    "@assert norm(c(q)) < ϵ && norm(∂c(q) * M⁻¹ * p) < ϵ"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify Rattle integrator is time-reversible for range of step sizes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time step η = 0.250\n",
      "(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₁, p₁) = (S ∘ Ψᴳ(η))(q, p) = ([-0.298, -1.415, 0.225], [1.033, 0.162, 0.735])\n",
      "(q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "Time step η = 0.500\n",
      "(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₁, p₁) = (S ∘ Ψᴳ(η))(q, p) = ([-0.567, -1.388, -0.029], [0.855, -0.401, 0.821])\n",
      "(q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "Time step η = 1.000\n",
      "(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₁, p₁) = (S ∘ Ψᴳ(η))(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "Time step η = 2.000\n",
      "(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₁, p₁) = (S ∘ Ψᴳ(η))(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n"
     ]
    }
   ],
   "source": [
    "for η ∈ (0.25, 0.5, 1., 2.)\n",
    "    print(\"Time step η = $η\\n\")\n",
    "    print(\"(q, p) = ($q, $p)\\n\")\n",
    "    (q₁, p₁) = (S ∘ Ψᴿ(η))(q, p)\n",
    "    print(\"(q₁, p₁) = (S ∘ Ψᴳ(η))(q, p) = ($q₁, $p₁)\\n\")\n",
    "    (q₀, p₀) = (S ∘ Ψᴿ(η))(q₁, p₁)\n",
    "    print(\"(q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁) = ($q₀, $p₀)\\n\")\n",
    "    @assert norm(q₀ - q) < τ && norm(p₀ - p) < τ\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify geodesic integrator is time-reversible for range of step sizes"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Time step η = 0.250\n",
      "(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₁, p₁) = (S ∘ Ψᴳ(η))(q, p) = ([-0.298, -1.415, 0.225], [1.033, 0.162, 0.735])\n",
      "(q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "Time step η = 0.500\n",
      "(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₁, p₁) = (S ∘ Ψᴳ(η))(q, p) = ([-0.567, -1.388, -0.029], [0.855, -0.401, 0.821])\n",
      "(q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "Time step η = 1.000\n",
      "(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₁, p₁) = (S ∘ Ψᴳ(η))(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "Time step η = 2.000\n",
      "(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₁, p₁) = (S ∘ Ψᴳ(η))(q, p) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n",
      "(q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁) = ([-0.026, -1.311, 0.392], [-1.092, -0.649, -0.532])\n"
     ]
    }
   ],
   "source": [
    "for η ∈ (0.25, 0.5, 1., 2.)\n",
    "    print(\"Time step η = $η\\n\")\n",
    "    print(\"(q, p) = ($q, $p)\\n\")\n",
    "    (q₁, p₁) = (S ∘ Ψᴳ(η))(q, p)\n",
    "    print(\"(q₁, p₁) = (S ∘ Ψᴳ(η))(q, p) = ($q₁, $p₁)\\n\")\n",
    "    (q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁)\n",
    "    print(\"(q₀, p₀) = (S ∘ Ψᴳ(η))(q₁, p₁) = ($q₀, $p₀)\\n\")\n",
    "    @assert norm(q₀ - q) < τ && norm(p₀ - p) < τ\n",
    "end"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Integrate longer trajectory of 1000 steps using geodesic integrator and visualise."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "η = 0.25\n",
    "n_step = 1200\n",
    "q_trajectory = Array{Float64}(undef, n_step, 3)\n",
    "for i ∈ 1:n_step\n",
    "    (q, p) = Ψᴳ(η)(q, p)\n",
    "    q_trajectory[i, :] = q\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
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       "  783.573,1617.07 651.113,1523.13 558.384,1321.2 513.874,1060.67 517.482,783.705 569.347,523.53 668.97,324.826 805.841,235.075 961.649,235.357 1127.57,232.847 \n",
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       "  1304.27,1567.91 1151.11,1594.06 993.901,1636.77 843.264,1601.88 713.922,1447.76 619.655,1212.81 565.743,943.913 555.351,674.751 592.676,443.998 675.788,303.482 \n",
       "  789.358,267.652 922.957,258.72 1072.01,213.216 1222.92,186.811 1365.33,267.986 1484.02,458.427 1568.7,709.404 1615.01,980.489 1618.4,1238.86 1574.91,1439.12 \n",
       "  1490.93,1534.67 1382.19,1549.34 1252.18,1575.75 1109.49,1629.75 966.204,1622.43 833.506,1495.16 725.096,1276.98 648.643,1016.22 608.835,747.665 611.394,507.266 \n",
       "  658.427,345.946 739.05,291.281 842.081,279.285 967.188,232.22 1101.79,187.269 1236.25,239.391 1358.81,406.98 1458.14,644.789 1527.88,910.134 1562.36,1170.77 \n",
       "  1555.47,1384.63 1508.55,1501.49 1434.56,1527.5 1337.02,1552.34 1219.01,1612.87 1093.56,1630.03 968.568,1530.19 855.515,1330.63 763.623,1080.11 699.049,814.966 \n",
       "  668.529,568.909 677.33,390.603 720.016,317.662 786.026,302.547 877.829,257.532 986.639,199.035 1102.55,223.81 1218.23,367.483 1323.18,590.452 1409.47,847.99 \n",
       "  1470.46,1107.78 1498.76,1331.07 1491.26,1466.49 1456.35,1504.24 1397.9,1525.98 1313.37,1588.06 1214.89,1625.44 1108.27,1552.51 1000.88,1372.72 902.199,1133.93 \n",
       "  819.641,874.384 760.552,626.152 732.247,434.81 734.761,344.856 760.051,326.377 811.693,286.78 886.787,220.309 974.976,220.665 1073.27,340.256 1173.98,547.361 \n",
       "  1268.41,795.666 1349.13,1052.16 1408.19,1281.12 1439.25,1431.96 1445.1,1481.48 1429.22,1499.12 1384.8,1557.8 1320.58,1610.24 1241.6,1562.6 1150.1,1402.98 \n",
       "  1054.11,1176.78 961.882,924.328 881.067,676.717 819.915,476.087 783.063,370.902 766.742,348.621 774.566,317.274 810.078,248.671 863.508,228.589 935.11,324.816 \n",
       "  1021.72,515.687 1115.05,754.004 1207.14,1005.51 1290.05,1237.01 1355.62,1400.06 1401.11,1461.06 1428.07,1474.24 1427.5,1524.94 1402.54,1586.65 1358.28,1561.71 \n",
       "  1291.86,1421.96 1208.1,1208.59 1115.34,963.987 1021.31,719.001 934.252,512.34 861.928,393.984 805.853,367.301 769.702,346.223 762.334,281.261 776.544,245.554 \n",
       "  814.233,319.935 877.762,494.804 960.697,723.024 1054.96,968.644 1153,1200.34 1246.2,1372.64 1327.35,1444.57 1394.44,1453.64 1438.06,1492.48 1454.82,1557.42 \n",
       "  1449.63,1551.73 1415.57,1430.94 1352.79,1230.06 1268.7,993.384 1170.87,752.186 1066.81,542.056 965.746,412.594 874.387,380.791 797.475,370.967 746.942,314.994 \n",
       "  720.246,269.033 719.555,323.76 752.747,483.359 816.723,701.944 903.889,941.516 1007.35,1171.94 1119.16,1351.02 1229.31,1433.24 1331.23,1439.23 1415.86,1463.37 \n",
       "  1474.04,1525.53 1509.22,1535.04 1512.19,1431.79 1477.54,1242.63 1410.62,1013.35 1318.45,776.293 1207.46,564.441 1086.67,425.692 966.875,387.948 855.315,389.216 \n",
       "  764.607,346.784 698.11,296.202 657.716,333.984 655.761,479.395 693.699,689.247 765.222,923.276 864.27,1151.79 984.346,1335.88 1114.6,1427.79 1243.87,1432.35 \n",
       "  1363.1,1440.23 1459.54,1494.05 1533.38,1514.29 1574.93,1426.85 1573.32,1248.34 1530.6,1025.46 1452.87,792.169 1345.2,579.492 1215.16,432.825 1075.82,388.208 \n",
       "  937.942,399.267 813.281,373.781 710.768,324.155 632.566,348.04 593.619,480.523 600.587,682.819 649.345,912.33 734.871,1139.08 852.525,1327.16 992.413,1428.39 \n",
       "  1139.88,1433.63 1284.73,1425.17 1413.4,1465.88 1521.42,1492.21 1599.83,1418.68 1633.34,1249.66 1619.83,1031.87 1563.84,801.398 1469.06,588.003 1340.81,434.21 \n",
       "  1192.24,381.637 1037.88,400.158 888.287,393.584 756.032,350.119 644.893,363.327 570.333,484.134 544.042,680.121 564.938,906.485 629.303,1132.26 734.458,1324.05 \n",
       "  872.823,1434.58 1028.18,1443 1188.02,1419.57 1340.24,1443.53 1475.61,1471.38 1586.02,1409.84 1653.66,1249.28 1671.79,1035.17 1642.84,806.169 1569.03,591.492 \n",
       "  1453.1,430.752 1306.37,368.952 1146.18,391.752 982.65,404.444 829.142,371.64 692.425,377.414 586.8,487.625 527.9,678.41 518.311,903.134 555.948,1129.16 \n",
       "  640.007,1325.06 765.999,1445.25 918.372,1459.61 1081.89,1423.96 1246.91,1428.97 1400.87,1454.06 1535.75,1402.69 1633.41,1249.83 1682.85,1038.15 1683.75,809.085 \n",
       "  1636.85,592.065 1542.37,423.995 1408.39,351.489 1253.18,374.752 1087.65,405.407 923.189,386.756 770.092,388.232 640.753,488.628 552.885,674.962 512.914,899.467 \n",
       "  520.755,1127.24 577.299,1328.11 681.349,1458.73 820.059,1481.97 976.096,1437.98 1141.91,1423.45 1304.35,1442.04 1454.18,1399.19 1574.93,1253.76 1652.44,1043.59 \n",
       "  1683.32,812.943 1666.72,592.234 1600.63,416.009 1489.2,331.14 1349.34,350.632 1193.58,396.407 1029.66,394.122 870.523,394.213 726.972,485.213 616.552,667.306 \n",
       "  549.075,892.706 526.764,1123.72 552.03,1330.76 626.696,1472.87 742.054,1507.99 880.298,1460.39 1034.78,1427.42 1194.75,1436.54 1348.86,1400.75 1483.52,1263.01 \n",
       "  1583.19,1053.98 1641.46,820.509 1655.75,594.697 1622.27,409.224 1541.17,310.23 1425.98,321.53 1290.65,378.275 1139.12,393.106 984.721,394.381 837.757,476.049 \n",
       "  713.486,653.435 623.975,880.327 573.874,1115.89 566.984,1330.37 607.579,1485.18 691.709,1535.15 803.26,1489.23 935.3,1440.47 1081.62,1438.14 1229.08,1408.11 \n",
       "  1367,1278.93 1480.67,1071.37 1561.2,834.285 1604.22,602.111 1604.55,406.231 1558.9,291.37 1475.95,290.081 1369.85,352.657 1241.88,383.834 1102.83,388.391 \n",
       "  963.596,460.507 835.742,631.979 731.864,860.278 658.842,1101.28 621.786,1324.35 626.727,1493.06 674.288,1560.66 752.128,1521.92 852.583,1461.36 974.658,1446.75 \n",
       "  1105.04,1421.36 1235.05,1302.11 1352.95,1097.18 1448.49,856.305 1515.53,616.861 1547.87,409.554 1539.64,277.26 1494.15,259.231 1423.85,321.868 1328.86,367.177 \n",
       "  1214.96,376.512 1093.99,438.698 973.488,602.328 864.504,831.141 775.539,1077.92 712.92,1310.39 683.77,1494 692.472,1581.63 731.895,1555.51 794.27,1488.16 \n",
       "  882.94,1461.65 986.958,1439.94 1098.42,1332.34 1210,1132.02 1311.72,887.964 1395.97,640.865 1455.78,421.434 1483.7,270.483 1478.03,232.026 1447.52,288.682 \n",
       "  1392.28,344.684 1311.91,359.576 1218.35,411.445 1115.81,564.696 1011.81,792.269 915.416,1044.45 834.001,1286.67 775.227,1485.79 746.053,1595.32 745.028,1586.89 \n",
       "  765.789,1518.39 814.225,1481.56 884.308,1462.71 967.989,1368.65 1062.87,1175.7 1161.09,929.897 1254.21,675.397 1334.69,443.625 1394.41,273.293 1427.77,211.389 \n",
       "  1438.38,256.109 1426.42,318.443 1385.91,338.889 1326.87,380.183 1251.6,520.114 1162.64,743.846 1068.17,1000.28 976.271,1251.99 894.749,1466.72 831.855,1599.37 \n",
       "  791.282,1613.02 769.878,1549.3 774.326,1504.7 805.034,1488.13 853.892,1409.39 922.802,1227.2 1007.83,981.913 1100.64,720.982 1193.38,477.234 1277.97,287.422 \n",
       "  1346.41,199.906 1396.7,227.158 1428.19,290.885 1431.14,316.109 1411.32,346.802 1370.48,470.347 1305.66,686.892 1222.51,945.644 1129.29,1205.92 1033.62,1435.74 \n",
       "  943.918,1591.9 867.682,1631.15 806.383,1578.05 766.592,1528.98 755.002,1514.38 764.675,1452.37 800.345,1284.71 863.316,1042.99 946.584,777.331 1042.29,522.611 \n",
       "  1142.94,313.918 1239.65,199.622 1325.48,204.614 1397.4,264.559 1444.2,293.119 1465.66,313.461 1463.63,417.748 1430.23,623.195 1367.06,881.58 1281.72,1148.83 \n",
       "  1181.46,1392.57 1073.93,1571.79 968.697,1639.03 872.341,1602 791.562,1552.19 737.598,1539.52 706.632,1495.1 704.493,1345.82 738.183,1111.36 803.397,843.356 \n",
       "  892.728,579.304 999.551,353.048 1115.5,211.873 1230.17,190.829 1336.81,241.893 1424.39,271.869 1486.44,282.382 1524.51,365.054 1527.29,555.186 1491.22,809.876 \n",
       "  1422.26,1081.86 1327.03,1337.73 1211.88,1538.69 1086.61,1635.08 962.287,1618.84 846.849,1572.19 753.515,1561.65 683.343,1534.99 641.968,1407.68 641.515,1184.6 \n",
       "  681.619,917.235 755.939,646.078 858.912,404.26 983.672,237.176 1118.32,187.552 1251.8,224.986 1373.8,254.204 1473.04,255.655 1549.36,315.152 1590.01,485.755 \n",
       "  1585.96,732.948 1540.46,1006.89 1459.18,1272.53 1346.87,1493.11 1212.06,1618.53 1068.73,1626.83 927.329,1587.12 800.713,1579.1 695.463,1569.59 616.694,1467.21 \n",
       "  580.123,1259.8 590.154,996.539 642.26,720.991 732.064,466.203 854.767,275.177 998.986,195.791 1149.87,215.446 1297.23,241.697 1427.58,235.063 1537.43,270.828 \n",
       "  1614.4,418.036 1644.56,653.666 1627.56,926.393 1567.68,1198.98 1468.02,1436.38 1334.8,1589.48 1182.75,1624.85 1025.6,1595.6 874.571,1590.5 740.766,1596.83 \n",
       "  629.509,1521.38 558.009,1333.85 535.592,1078.4 560.306,801.53 629.101,536.817 739.434,324.671 882.086,215.737 1039.96,214.284 1201.8,235.479 1354.72,221.938 \n",
       "  1491,234.528 1599.55,355.167 1663.11,575.149 1677.14,843.238 1644.07,1119.65 1565.39,1370.56 1444.46,1548.89 1294.66,1612.53 1132.72,1596.84 968.214,1594.97 \n",
       "  814.41,1615.15 678.153,1567.48 576.063,1403.57 521.697,1159.73 516.299,884.787 558.344,613.478 647.448,383.687 777.623,246.738 931.688,221.858 1096.35,236.118 \n",
       "  1261.3,217.059 1415.06,208.145 1547.71,300.038 1640.79,500.537 1685.63,760.514 1682.31,1037.5 1630.78,1298.29 1531.5,1498.52 1394.78,1590.28 1238.96,1590.64 \n",
       "  1073.04,1592.17 909.394,1623.66 757.519,1603.3 632.002,1466.03 549.117,1237.42 513.595,967.67 525.675,693.19 586.882,449.644 694.864,287.346 834.584,237.861 \n",
       "  990.618,243.559 1155.82,220.579 1316.87,192.862 1464.06,255.067 1579.98,432.759 1652.58,681.292 1679.25,955.632 1658.43,1222.59 1588,1440.78 1474.15,1559.2 \n",
       "  1334.76,1577.44 1179.42,1582.4 1017.14,1622.2 860.144,1627.31 720.58,1518.73 615.339,1308.59 552.42,1047.12 534.062,772.802 563.396,519.547 641.542,335.439 \n",
       "  757.382,261.355 894.366,257.134 1047.74,232.003 1205.25,189.056 1356.23,222.004 1486.03,374.318 1580.7,608.406 1634.89,877.112 1645.51,1146.63 1608.38,1378.55 \n",
       "  1525.27,1521.06 1411.41,1558.23 1277.55,1566.58 1128.18,1611.33 976.892,1638.84 834.042,1559.78 714.885,1370.77 629.858,1120.35 583.322,849.244 579.75,590.23 \n",
       "  623.153,388.396 707.332,290.85 816.454,275.639 946.775,250.202 1089.89,196.278 1233.61,201.801 1366.8,327.108 1475.63,544.254 1552.36,804.712 1592.34,1073.51 \n",
       "  1589.89,1314.9 1542.79,1478.14 1461.78,1534.44 1358.34,1546.15 1232.84,1592.33 1097.75,1638.08 962.788,1588.08 839.752,1422.12 740.073,1185.01 670.135,919.744 \n",
       "  635.553,658.595 642.432,443.326 689.608,324.414 764.109,297.472 862.055,273.473 980.563,213.291 1106.56,194.544 1232.05,292.287 1345.53,490.622 1437.68,740.728 \n",
       "  1502.38,1006.01 1532.92,1252.91 1523.99,1433.02 1480.85,1507.82 1414.31,1523.02 1322.06,1567.08 1212.66,1626.06 1096.2,1603.36 980.046,1461.5 874.755,1239.39 \n",
       "  788.297,982.03 727.27,721.847 699.052,497.312 706.853,359.837 742.353,320.79 801.307,299.666 886.375,238.166 985.473,199.449 1092.61,270.196 1200.4,448.571 \n",
       "  1299.33,686.817 1381.96,946.401 1441,1195.34 1469.1,1388.34 1466.11,1480.33 1440.18,1499.28 1388.05,1537.95 1312.32,1604.59 1223.53,1606.2 1124.79,1488.59 \n",
       "  1023.77,1282.51 929.199,1034.49 848.498,777.706 789.593,547.66 758.882,394.818 753.65,343.673 770.123,326.36 815.086,268.425 879.98,214.939 959.492,260.354 \n",
       "  1051.36,418.363 1147.55,643.88 1239.81,896.284 1320.51,1144.47 1381.39,1346.62 1417.75,1453.93 1433.39,1477.08 1425.03,1507.61 1388.89,1576.09 1334.82,1597.97 \n",
       "  1262.79,1503.86 1175.92,1314.19 1082.49,1076.27 990.466,824.571 907.778,592.133 842.604,427.171 797.73,364.289 771.418,351.084 772.518,301.224 798.159,238.77 \n",
       "  842.946,261.515 909.682,399.465 993.597,612.012 1086.35,856.47 1180.21,1101.92 1266.94,1309.95 1338.76,1430.45 1394.16,1458.39 1429.75,1478.85 1436.53,1543.43 \n",
       "  1421.67,1580.7 1383.59,1508.55 1319.87,1335.06 1236.88,1107.35 1142.72,861.632 1044.96,629.13 952.182,455.04 871.622,381.034 805.187,371.548 762.068,333.563 \n",
       "  745.948,268.209 751.665,271.779 785.977,390.608 848.851,590.508 932.891,826.943 1030.52,1068.5 1134.2,1279.84 1234.76,1411.43 1325.49,1444.83 1401.83,1454.3 \n",
       "  1451.85,1509.68 1477.93,1556.94 1478.26,1504.49 1444.95,1346.46 1380.98,1128.5 1293.95,888.922 1190.81,657.824 1079.45,477.082 969.845,392.676 868.545,385.862 \n",
       "  784.386,362.514 726.75,300.249 692.07,288.769 689.291,389.908 723.947,577.933 790.781,806.874 882.709,1044.19 993.299,1257.11 1113.6,1397.91 1232.83,1437.53 \n",
       "  1343.85,1436.24 1434.14,1477.95 1500.09,1529.51 1540.17,1494.01 1542.09,1350.33 1504.22,1141.24 1432.87,907.291 1334.15,678.216 1214.54,492.611 1084.79,398.465 \n",
       "  956.047,392.705 837.254,385.458 741.235,331.826 667.837,309.824 626.408,395.043 627.947,572.252 670.519,794.7 747.997,1028.16 855.261,1241.77 984.807,1390.37 \n",
       "  1123.69,1437.08 1261.1,1426.36 1385.42,1451.06 1487.51,1501.32 1565.45,1479.66 1604.44,1349.07 1597.62,1147.67 1549.07,918.329 1463.88,691.123 1346.68,501.681 \n",
       "  1207.26,398.225 1060.2,391.438 915.718,400.306 787.34,360.042 679.661,332.214 601.307,403.457 567.635,571.005 580.948,788.256 636.666,1018.81 731.055,1233.01 \n",
       "  858.84,1388.61 1007.02,1443.4 1161.04,1425.6 1310.38,1431.41 1442.37,1475.17 1553.32,1464.05 1628,1345.29 1654.52,1150.34 1633.86,924.227 1569.82,698.099 \n",
       "  1465.1,505.093 1327.15,392.405 1172.09,382.155 1012.51,405.695 860.444,382.372 725.293,353.347 614.87,412.583 546.989,571.517 528.584,784.961 557.26,1013.97 \n",
       "  630.709,1229.32 746.208,1391.72 892.621,1455.81 1052.63,1434.1 1215.99,1420.74 1369.39,1453.5 1506.02,1449.62 1611.84,1341.57 1671.05,1151.97 1680.92,927.592 \n",
       "  1643.59,701.297 1559.92,504.34 1434.3,382.089 1282.09,365.679 1118.69,401.108 953.724,396.854 799.814,370.948 664.848,420.059 566.871,571.131 517.107,782.026 \n",
       "  515.918,1011.02 562.49,1228.6 656.589,1398.17 790.328,1472.96 945.507,1451.13 1110.99,1419.98 1275.49,1438.3 1428.63,1438.45 1558.23,1340.29 1646.47,1155.29 \n",
       "  1686.76,931.231 1679.21,703.283 1623,501.488 1519.26,368.943 1380.61,343.506 1224.53,386.928 1058.66,402.241 896.111,383.213 746.104,423.92 624.972,567.428 \n",
       "  547.082,776.686 515.9,1007.17 532.22,1228.33 597.993,1405.89 709.262,1493 849.221,1475.22 1005.07,1429.2 1169.15,1430.91 1328.48,1432.12 1472.36,1343.37 \n",
       "  1583.21,1162.77 1651.04,937.924 1673.62,706.819 1648.63,498.992 1574.19,355.116 1458.82,317.71 1320.37,364.394 1165.72,398.103 1005.5,388.914 851.095,422.754 \n",
       "  716.01,558.444 615.916,766.429 557.335,999.646 542.803,1225.81 576.05,1412.46 657.05,1513.59 772.456,1504.23 908.04,1447.53 1059.83,1431.96 1214.56,1431.61 \n",
       "  1361.92,1352.17 1486.66,1176.42 1576.55,950.195 1626.91,714.631 1634,499.494 1593.56,343.081 1509.34,290.808 1397.44,335.494 1265.12,384.864 1118.43,387.462 \n",
       "  970.558,415.795 832.177,542.844 718.059,749.209 637.207,985.954 593.989,1218.38 593.51,1415.28 639.137,1532.15 722.385,1535.54 828.88,1473.32 957.194,1441.32 \n",
       "  1096.71,1437.29 1236.41,1367.3 1364.75,1197.61 1469.02,970.102 1542.28,729.18 1579.48,505.58 1574.6,335.439 1526.89,265.588 1448.59,302.799 1347.74,363.773 \n",
       "  1225.21,378.934 1094.32,402.95 963.799,520.032 845.445,723.624 749.593,964.049 682.39,1203.66 649.98,1411.83 658.262,1545.96 704.103,1566.2 774.941,1504.24 \n",
       "  870.349,1458.1 984.799,1448.91 1106.31,1388.68 1227.31,1226.96 1336.68,999.062 1425.86,752.458 1488.56,519.564 1517.69,334.705 1508.78,244.916 1468.83,269.264 \n",
       "  1405.94,336.785 1316.78,364.046 1212.15,384.769 1100.15,490.206 988.125,689.036 886.112,932.534 801.737,1179.7 741.927,1399.85 714.143,1552.47 720.201,1593.17 \n",
       "  751.333,1537.51 807.069,1480.76 888.023,1465.64 982.202,1415.48 1085.23,1264.24 1189.59,1037.73 1286.18,785.816 1367.63,543.282 1426.3,343.086 1455.2,231.52 \n",
       "  1455.67,238 1434.29,306.382 1385.35,344.101 1314.6,362.359 1230.35,454.335 1135.06,645.642 1036.57,890.768 943.369,1145.25 862.95,1377.6 803.44,1549.41 \n",
       "  770.498,1613.53 760.552,1570.11 773.07,1507.17 814.166,1486.19 873.873,1446.3 949.626,1308.43 1038.9,1085.92 1133.53,829.851 1225.42,577.93 1306.77,362.295 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(q_trajectory[:, 1], q_trajectory[:, 2], q_trajectory[:, 3], \n",
    "     label=\"\", xlabel=\"q₁\", ylabel=\"q₂\", zlabel=\"q₃\", size=(500, 500))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "  938.807,1684.56 806.514,1674.66 698.004,1602.82 623.492,1472.49 592.322,1291.14 602.111,1079.21 642.443,857.044 719.996,639.418 826.898,442.952 950.796,275.821 \n",
       "  1087.14,162.146 1224.56,116.566 1351.48,137.654 1457.8,221.996 1532.55,367.124 1568.31,561.327 1570.89,780.372 1542.61,1008.08 1475.52,1227.44 1383.62,1426.06 \n",
       "  1271.77,1588.67 1144.41,1690.61 1013.05,1723.96 888.716,1691.05 781.968,1593.08 703.959,1433.45 659.284,1229.89 642.182,1006.44 660.147,775.805 714.141,556.406 \n",
       "  791.775,358.89 894.089,205.854 1015.09,117.26 1143.22,96.2317 1267.77,141.554 1377.7,253.705 1462.29,425.772 1519.67,634.334 1550.71,860.323 1542.77,1090.84 \n",
       "  1503.67,1308.84 1437.74,1500.57 1342.03,1640.62 1224.6,1715.6 1096.83,1724.46 969.252,1666.27 853.318,1540.47 758.818,1359.92 686.544,1150.16 643.843,923.374 \n",
       "  640.103,695.679 664.849,481.456 722.691,300.103 814.808,174.541 931.445,112.651 1061.7,115.741 1195.13,186.831 1319.81,324.122 1427.02,507.988 1514.09,716.718 \n",
       "  1566.59,941.771 1583.92,1164.64 1570.53,1371.55 1516.93,1539.32 1425.56,1650.46 1307,1700.16 1171.42,1685.41 1029.43,1602.23 893.661,1457.49 772.054,1275.19 \n",
       "  672.388,1068.57 609.99,848.339 579.1,632.256 585.622,436.353 638.513,283.472 731.753,185.719 854.92,147.198 998.614,173.043 1151.44,266.127 1299.92,412.825 \n",
       "  1435.96,590.503 1545.42,793.601 1619.09,1006.25 1660.7,1213.13 1656.29,1395.4 1601.14,1533.73 1503.63,1619.64 1373.19,1648.03 1218.84,1612.27 1053.63,1513.14 \n",
       "  891.754,1370.12 743.065,1198 625.37,1001.05 540.004,798.061 492.595,602.862 499.048,435.183 558.838,309.954 662.942,234.169 803.244,214.705 970.218,258.172 \n",
       "  1148.28,357.983 1323.2,493.321 1481.55,659.636 1607.69,847.277 1702.5,1038.95 1750.23,1220.77 1738.59,1374.51 1672.15,1488.68 1559.18,1556.08 1406.69,1568.23 \n",
       "  1226.16,1520.62 1035.8,1426 849.344,1299.69 684.063,1140.57 549.093,964.814 449.595,785.696 406.293,617.57 425.343,476.009 500.207,370.628 624.262,309.776 \n",
       "  790.618,302.641 984.269,349.494 1185.65,434.244 1381.05,551.999 1551.97,701.073 1694.27,863.526 1792.64,1029.61 1828.09,1185.09 1800.1,1316.87 1715.1,1415.78 \n",
       "  1577.98,1472.13 1397.23,1477.53 1192.23,1436.98 981.503,1364.17 780.284,1254.84 603.962,1118.79 458.441,969.916 366.397,816.272 340.887,671.387 378.981,546.748 \n",
       "  475.795,451.954 627.347,397.626 821.411,389.719 1035.26,419.084 1253.26,480.514 1457.77,579.737 1638.45,701.085 1781.6,837.161 1863.62,980.004 1878.09,1116.9 \n",
       "  1828.87,1237.11 1718.76,1330.25 1552.55,1385.6 1347.89,1400.84 1127.55,1385.35 905.051,1333.9 698.584,1247.65 516.708,1140.84 381.291,1015.82 311.327,881.407 \n",
       "  308.408,749.659 369.721,631.157 494.104,537.005 673.534,477.041 885.103,449.982 1110.01,451.771 1333.98,493.202 1541.29,564.373 1719.48,658.486 1843.32,774.872 \n",
       "  1900.51,903.605 1891.61,1033.09 1817.34,1152.65 1678.73,1250.71 1489.37,1319.19 1274.34,1360.93 1046.86,1370.85 823.875,1341.39 618.027,1285.24 448.566,1200.9 \n",
       "  338.304,1090.35 293.802,964.436 314.715,834.325 402.412,710.89 553.637,605.861 748.317,525.28 964.535,468.491 1191.89,448.462 1412.19,463.685 1612.76,507.214 \n",
       "  1769.72,585.165 1865.98,692.762 1898.28,818.707 1865.22,952.34 1764.45,1082.47 1604.12,1197.41 1408.69,1292.57 1192.66,1362.45 968.449,1393.01 752.71,1392.42 \n",
       "  561.692,1357.59 419.121,1282.66 336.241,1175.25 315.557,1046.44 360.625,906.401 472.764,766.942 637.128,639.639 830.209,529.314 1044.32,448.984 1262.86,406.121 \n",
       "  1471.32,394.441 1649.46,424.973 1777.46,499.932 1848.1,609.8 1857.82,744.307 1801.48,893.476 1681.42,1044.39 1518.47,1185.61 1328.97,1309.56 1119.51,1399.04 \n",
       "  908.638,1455.03 710.598,1474.48 546.956,1444.2 432.484,1366.58 372.754,1251.69 373.212,1108.71 439.019,948.083 561.888,784.404 719.364,628.992 904.623,494.342 \n",
       "  1106.35,395.132 1307.64,327.855 1492.76,305.885 1642,339.235 1744.62,421.823 1794.82,544.272 1785.39,698.302 1713.39,871.884 1593.51,1048.65 1443.18,1217.04 \n",
       "  1263.83,1359.77 1072.33,1469.53 882.543,1543.6 711.411,1563.32 575.22,1524.39 482.495,1434.72 440.356,1302.22 456.794,1135.08 530.241,948.209 642.249,758.931 \n",
       "  785.489,580.168 956.018,430.347 1135.64,311.185 1312.58,236.062 1470.18,222.252 1595.41,268.657 1680.26,367.469 1716.04,512.318 1695.64,693.623 1626.42,892.861 \n",
       "  1524.69,1093.32 1388.79,1279.29 1230.06,1436.38 1063.22,1561 899.347,1631.81 753.63,1637.26 636.946,1581.82 557.919,1471.86 526.319,1312.89 546.429,1117.8 \n",
       "  606.237,907.973 697.837,698.456 824.49,507.658 969.536,343.995 1123.53,219.601 1275.33,157.13 1411.53,161.548 1522.16,227.438 1597.55,350.412 1627.87,524.229 \n",
       "  1613.12,731.153 1565.85,949.569 1483.39,1165.21 1368.96,1359.85 1237.96,1527.02 1095.95,1645.52 954.223,1697.2 824.853,1681.96 717.754,1604.26 643.56,1466.88 \n",
       "  610.842,1279.04 615.281,1063.32 649.767,838.252 722.739,619.562 822.639,421.85 940.62,256.246 1072.43,147.943 1206.63,108.241 1331.95,134.97 1438.09,225.763 \n",
       "  1513.91,377.993 1553.28,577.205 1562.57,798.637 1539.24,1028.1 1477.95,1247.61 1392.97,1446.17 1285.92,1604.7 1161.89,1700.3 1032.41,1727.68 908.346,1688.84 \n",
       "  800.491,1584.06 719.858,1418.07 669.639,1211.54 645.793,986.672 659.204,755.722 706.316,536.801 778.074,341.47 876.941,194.384 995.935,112.387 1123.55,97.2664 \n",
       "  1249.16,148.763 1361.53,267.651 1450.36,444.414 1514.58,653.976 1551.37,880.621 1548.6,1110.07 1516.45,1326.64 1454.76,1513.89 1361.06,1647.05 1244.31,1715.61 \n",
       "  1115.71,1718.66 985.717,1654.21 866.011,1522.47 765.844,1339.6 686.469,1130.11 639.094,903.753 629.16,678.129 648.111,466.925 703.56,291.801 795.05,173.261 \n",
       "  912.301,117.38 1044.74,126.159 1181.89,203.272 1311.53,344.81 1425.38,528.521 1518.7,736.416 1575.67,959.586 1599.52,1179.5 1589.9,1381.57 1536.95,1542.26 \n",
       "  1444.94,1646.78 1324.48,1690.95 1185.38,1670.72 1038.52,1582.36 896.915,1435.86 768.414,1255.46 663.882,1050.21 596.397,833.254 560,621.217 565.055,431.596 \n",
       "  618.772,285.818 713.813,193.978 840.138,160.504 988.611,191.369 1147.16,288.019 1301.93,433.693 1444.23,608.828 1557.57,809.452 1636.8,1017.76 1681.74,1219.38 \n",
       "  1676.71,1394.69 1619.55,1526.5 1519.13,1607.31 1384.28,1631.19 1224.12,1591.3 1052.83,1491.01 884.694,1351.22 731.09,1181.71 609.571,988.747 519.349,790.923 \n",
       "  471.234,601.946 479.926,441.09 542.725,321.451 650.818,249.949 796.685,234.3 970.065,280.377 1154.07,378.525 1334.67,509.672 1496.51,672.928 1626.78,855.228 \n",
       "  1724.43,1041.08 1770.46,1216.35 1755.42,1363.99 1685.12,1473.64 1567.09,1537.61 1408.13,1546.93 1221.22,1498.9 1025.46,1408.42 834.426,1286.18 666.479,1131.35 \n",
       "  527.755,961.755 428.172,788.802 388.37,626.891 411.603,490.333 491.154,388.378 621.258,330.027 794.51,324.296 993.837,368.946 1199.86,448.19 1398.36,562.127 \n",
       "  1571.8,705.405 1716.19,861.427 1812.02,1021.49 1842.83,1171.63 1810.08,1299.64 1719.58,1396.24 1575.66,1451.2 1388.31,1457.1 1178.79,1421.26 964.33,1353.94 \n",
       "  761.36,1248.99 582.784,1119.75 437.756,977.064 350.226,828.781 329.973,688.04 373.321,565.88 476.371,472.253 635.015,417.951 834.568,407.267 1051.76,430.299 \n",
       "  1272.26,486.966 1477.77,580.476 1659.48,694.861 1799.46,825.474 1875.8,964.013 1884.73,1098.1 1829.79,1217.12 1712.89,1310.33 1540.04,1367.3 1331.59,1387.57 \n",
       "  1108.85,1378.66 885.319,1331.73 678.307,1252.4 497.588,1151.85 367.401,1031.21 303.581,899.827 306.351,769.521 373.824,650.904 505.143,555.337 689.864,491.903 \n",
       "  903.471,458.177 1129.89,454.233 1353.62,490.398 1560.59,554.364 1735.14,643.503 1852.58,756.818 1903.59,883.876 1888.91,1013.29 1808.18,1134.3 1663.17,1235.22 \n",
       "  1470.62,1308.96 1254.79,1357.88 1026.99,1372.57 805.1,1349.57 601.264,1299.74 437.457,1218.79 333.962,1109.93 295.429,984.339 322.206,852.924 416.331,726.695 \n",
       "  572.517,617.375 768.11,530.16 984.437,466.913 1210.66,442.168 1429.07,450.417 1625.56,489.367 1775.73,565.562 1865.42,672.785 1892.03,799.762 1853.12,936.073 \n",
       "  1746.53,1070.29 1583.51,1190.77 1388.88,1293.12 1173.44,1368.14 951.645,1404.22 738.992,1409.82 553.822,1377.47 418.311,1302.8 341.496,1194.5 326.207,1063.31 \n",
       "  376.885,919.268 493.425,774.631 657.828,640.908 849.339,523.884 1061.67,439.281 1276.81,390.239 1480.68,374.327 1651.98,404.38 1773.3,480.375 1838.5,592.361 \n",
       "  1843.16,730.574 1781.91,884.925 1659.71,1041.76 1498.91,1189.71 1311.25,1319.04 1105.06,1412.9 898.483,1474.57 706.36,1495.76 549.714,1464.28 441.177,1384.5 \n",
       "  386.235,1266.29 391.248,1118.33 460.614,951.689 582.862,781.818 737.082,620.055 919.904,481.417 1117.07,377.309 1313.13,306.229 1491.62,284.906 1634.38,320.729 \n",
       "  1731.98,406.499 1778.07,533.468 1764.9,693.539 1691.43,873.46 1574.72,1056.25 1427.72,1229.89 1252.09,1375.92 1066.04,1490.38 882.227,1565.57 717.687,1582.75 \n",
       "  587.037,1540.38 498.378,1446.55 459.592,1308.4 478.474,1134.56 550.733,941.648 658.069,746.883 798.036,564.39 963.341,411.217 1137,288.914 1307.7,215.35 \n",
       "  1459.37,205.379 1580.18,255.987 1661.93,359.922 1695.32,511.358 1674.35,699.379 1608.95,903.88 1512.07,1108.93 1380.13,1297.32 1227.7,1457.68 1066.97,1582.85 \n",
       "  909.01,1649.99 768.156,1650.79 654.708,1590.47 577.769,1474.46 547.288,1308.4 565.607,1107.34 619.771,893.201 707.051,680.415 828.35,487.789 966.744,321.996 \n",
       "  1114.92,199.857 1261.65,142.409 1394.24,151.97 1502.83,223.397 1577.25,353.162 1608.1,533.905 1597.6,745.427 1556.43,967.241 1478.2,1184.64 1370.41,1380.83 \n",
       "  1245.72,1547.89 1108.79,1661.87 970.966,1707.83 843.889,1687.12 737.625,1603.25 663.115,1458.71 627.88,1264.98 626.174,1046.19 655.226,818.723 723.091,599.473 \n",
       "  815.9,401.006 928.445,238.202 1056.28,135.832 1187.89,102.077 1312.24,134.584 1418.8,232.043 1496.4,391.135 1540.35,594.416 1556.53,817.727 1537.85,1048.28 \n",
       "  1482.96,1267.64 1404.5,1465.21 1301.63,1618.65 1180.3,1707.72 1052.06,1729.15 927.634,1684.26 818.041,1572.62 734.016,1400.95 677.522,1192.46 647.326,966.511 \n",
       "  656.008,735.939 696.044,517.877 762.655,325.853 858.76,185.278 976.378,109.829 1104.11,100.57 1231.43,158.317 1346.88,283.667 1440.66,464.052 1512,673.906 \n",
       "  1553.96,900.752 1556.86,1128.63 1531.27,1343.06 1472.93,1524.9 1380.57,1651.02 1263.89,1713.35 1133.81,1710.64 1000.78,1639.97 876.712,1502.99 770.336,1319.03 \n",
       "  684.147,1110.12 632.335,884.865 615.901,661.718 629.952,454.408 683.859,286.037 775.337,174.373 893.809,124.278 1029.07,138.665 1170.52,221.495 1305.58,366.171 \n",
       "  1426.3,548.769 1525.22,755.624 1586.89,976.212 1616.96,1192.69 1610,1389.15 1556.97,1542.65 1463.76,1640.87 1340.81,1679.73 1197.58,1654.2 1045.39,1561.32 \n",
       "  897.674,1414.41 762.471,1236.33 653.643,1032.89 580.751,819.763 539.772,612.357 544.397,429.443 599.54,290.545 696.911,204.243 826.994,175.571 980.773,211.102 \n",
       "  1145.28,310.258 1306.39,453.737 1454.36,626.287 1571.46,823.731 1656.09,1027.29 1703.13,1223.12 1696.65,1391.46 1636.98,1517.17 1533.13,1593.25 1393.31,1612.93 \n",
       "  1226.99,1569.55 1049.62,1469.43 875.432,1333.49 717.636,1166.7 592.12,978.429 497.866,786.102 450.247,603.597 461.779,449.274 528.005,334.727 640.6,267.112 \n",
       "  792.507,254.856 972.365,302.583 1162.08,397.786 1347.92,524.718 1512.74,684.411 1647.12,860.908 1746.32,1040.73 1789.72,1209.52 1770.88,1351.56 1696.31,1457.2 \n",
       "  1572.74,1518.17 1407.03,1525.27 1214,1478.04 1013.18,1392.53 818.255,1274.17 647.754,1124.37 505.907,961.131 407.549,794.344 371.866,638.286 399.575,506.153 \n",
       "  484.196,407.099 620.75,350.751 800.858,345.575 1005.35,386.766 1215.61,460.366 1416.5,570.341 1592.43,707.242 1737.75,856.893 1830,1011.18 1855.82,1156.53 \n",
       "  1818.07,1281.39 1721.71,1376.2 1570.77,1430.38 1377.26,1437.84 1163.79,1407.64 946.14,1345.47 741.854,1245.49 561.46,1123.25 418.24,986.452 335.872,843.068 \n",
       "  321.042,705.848 369.868,585.555 479.456,492.524 645.081,437.496 849.357,422.928 1069.42,439.33 1291.73,491.474 1498.02,578.617 1679.82,686.279 1815.54,811.908 \n",
       "  1885.92,946.731 1889.22,1078.66 1828.34,1197.1 1704.51,1291 1525.55,1350.46 1314.19,1376.65 1089.45,1374.02 865.539,1331.83 658.309,1259.72 479.944,1164.87 \n",
       "  355.657,1048.01 298.034,919.01 306.54,789.495 380.342,670.133 518.442,572.474 707.523,504.691 922.5,463.895 1149.92,454.735 1372.96,485.052 1578.88,542.097 \n",
       "  1748.74,627.002 1859.51,737.893 1904.44,863.923 1883.92,993.918 1796.7,1117.01 1645.86,1221.48 1451.23,1301.22 1234.97,1357.1 1007.52,1376.4 787.079,1360.25 \n",
       "  586.229,1315.98 428.733,1237.63 331.963,1129.85 299.27,1003.93 331.927,870.556 432.279,740.905 592.431,626.665 788.013,532.429 1004.16,463.338 1228.61,433.553 \n",
       "  1444.61,435.033 1636.1,470.387 1779.26,545.559 1862.6,653.017 1883.65,781.664 1838.96,921.294 1727.11,1060.15 1562.67,1186.65 1369.32,1296.08 1154.97,1375.72 \n",
       "  936.081,1417.74 727.212,1428.7 548.475,1397.85 419.907,1322.8 348.867,1213.02 338.822,1078.82 394.85,930.186 514.815,779.968 678.102,639.593 867.911,516.428 \n",
       "  1077.76,427.614 1289.08,372.448 1487.68,353.469 1651.96,383.885 1766.92,461.467 1826.99,576.145 1826.8,718.681 1761.18,878.666 1638.13,1041.61 1480.17,1196.2 \n",
       "  1294.54,1330.21 1092.29,1428.72 890.45,1495.32 704.678,1517.07 554.847,1483.74 451.822,1401.31 401.355,1279.2 410.588,1125.73 482.606,952.838 602.937,776.803 \n",
       "  753.788,609.13 933.631,466.949 1125.76,357.885 1316.19,284.229 1487.98,264.524 1624.68,303.31 1717.71,392.727 1760.03,524.769 1743.66,691.259 1669.95,877.452 \n",
       "  1557.29,1066.05 1413.53,1244.24 1242.36,1393.54 1062.04,1512.13 884.408,1587.12 726.208,1601.07 600.579,1554.89 515.531,1456.44 479.674,1312.15 500.18,1131.52 \n",
       "  569.946,932.892 672.383,733.018 808.863,547.482 968.354,390.908 1135.93,266.617 1300.46,195.706 1446.68,190.007 1563.64,245.145 1642.76,354.652 1674.31,512.988 \n",
       "  1653.85,707.462 1593.29,916.777 1501.01,1125.83 1373.66,1316.29 1227.78,1479.53 1073.08,1603.88 920.69,1666.62 784.118,1662.51 673.341,1596.99 597.978,1474.56 \n",
       "  567.876,1301.37 583.215,1094.95 631.339,876.935 714.453,661.603 829.72,467.264 961.542,300.303 1104.2,181.595 1246.39,129.565 1375.99,144.442 1483.11,221.711 \n",
       "  1557.14,358.505 1589.44,545.789 1584.2,761.17 1548.9,985.94 1475.21,1204.46 1374.4,1401.95 1255.68,1567.61 1123.32,1676.28 988.795,1716.38 863.383,1690.04 \n",
       "  757.364,1599.74 681.861,1448.09 643.116,1249.27 634.742,1028.03 658.797,798.737 720.929,579.305 806.798,380.803 914.467,221.976 1038.91,125.913 1168.57,98.1228 \n",
       "  1292.59,136.544 1400.18,240.811 1480.34,406.321 1529.74,612.666 1552.69,837.446 1538.57,1068.38 1490.52,1287.3 1418,1482.8 1318.66,1630.36 1199.4,1712.83 \n",
       "  1071.75,1728.36 946.324,1677.3 834.354,1558.9 746.176,1382.44 682.873,1172.86 646.876,946.212 650.439,716.637 683.491,499.953 745.779,312.308 839.804,178.599 \n",
       "  956.673,109.575 1085.17,106.13 1214.83,170.153 1333.99,301.501 1433.37,484.318 1511.83,693.953 1558.5,920.435 1567.56,1146.29 1547.83,1357.79 1491.98,1533.46 \n",
       "  1400.32,1652.55 1283.08,1708.85 1150.89,1700.45 1014.21,1623.73 885.239,1482.39 772.247,1298.48 679.748,1090.39 623.485,866.936 600.512,646.692 610.689,444.138 \n",
       "  663.86,282.835 755.914,177.809 876.213,133.273 1014.93,153.155 1161.2,241.232 1302.06,387.791 1429.69,568.558 1533.54,774.083 1600.26,991.455 1635.92,1203.95 \n",
       "  1630.54,1394.18 1576.75,1540.56 1481.8,1632.83 1355.76,1666.62 1207.8,1636.04 1049.93,1539.49 895.932,1393.47 754.419,1217.92 641.676,1016.89 563.227,808.051 \n",
       "  518.79,605.856 523.935,429.883 581.046,297.536 681.266,216.386 815.696,192.253 975.24,231.97 1145.83,332.416 1313.21,472.754 1466.17,642.705 1587.08,836.229 \n",
       "  1676.64,1034.67 1724.51,1224.31 1715.86,1385.8 1653.23,1505.87 1545.45,1577.63 1400.11,1593.44 1227.4,1547.36 1044.08,1448.77 864.155,1317 702.823,1153.24 \n",
       "  573.166,970.245 475.963,783.705 429.936,607.762 444.809,459.575 514.861,349.606 632.464,285.472 790.807,276.105 977.06,324.377 1172.21,415.529 1362.69,538.387 \n",
       "  1530.2,693.86 1668.38,864.222 1767.81,1037.89 1807.78,1200.43 1784.79,1337.41 1705.58,1439.57 1576,1497.98 1403.4,1503.59 1204.64,1458.43 999.144,1378.4 \n",
       "  801.049,1263.85 628.014,1119.78 483.965,962.969 388.039,802.226 356.951,651.56 389.396,523.253 479.467,426.572 622.799,371.681 809.539,366.101 1018.61,402.69 \n",
       "  1232.66,470.787 1435.36,576.427 1613.58,706.544 1758.54,849.991 1846.35,998.874 1866.94,1140.01 1823.95,1262.33 1721.4,1355.91 1563.36,1409.99 1364.29,1420.12 \n",
       "  1147.42,1396.2 927.203,1338.82 721.894,1244.48 540.378,1129.21 400.21,997.928 323.473,858.908 314.179,724.569 368.707,605.53 485.068,512.499 657.372,455.933 \n",
       "  865.521,436.438 1088.03,446.236 1311.44,493.885 1518.32,574.14 1699.08,675.488 1829.65,796.68 1893.9,928.406 1891.52,1058.82 1824.49,1177.32 1693.72,1272.56 \n",
       "  1509.39,1335.39 1295.9,1368.2 1069.6,1371.34 845.873,1334.38 638.921,1269.48 464.084,1179.68 346.163,1065.95 294.714,938.699 309.001,809.324 389.255,688.583 \n",
       "  533.794,588.142 726.179,515.174 942.007,467.201 1169.82,453.23 1391.81,477.14 1595.81,527.797 1760.09,609.24 1864.1,718.355 1903.05,844.001 1876.67,975.231 \n",
       "  1783,1101.05 1627.12,1209.74 1431.46,1296.06 1215.06,1358.45 988.667,1382.46 770.074,1373.28 573.203,1333.69 422.464,1257.18 332.279,1149.83 305.292,1022.95 \n",
       "  343.812,886.967 450.03,753.296 612.992,633.563 807.84,532.144 1023.44,457.848 1245.54,422.585 1458.52,417.811 1644.23,450.579 1780.32,525.416 1857.57,633.706 \n",
       "  1873.21,764.66 1822.89,908.231 1706.55,1052.21 1541.94,1185.11 1350.14,1301.28 1137.5,1385.22 921.961,1433.41 717.604,1448.71 545.686,1418.44 423.83,1342.41 \n",
       "  358.263,1230.57 353.292,1092.73 414.285,938.973 536.521,782.865 697.731,635.754 885.73,507.13 1092.37,413.956 1299.45,353.053 1492.19,332.215 1649.45,363.747 \n",
       "  1758.44,443.437 1813.69,561.379 1808.92,708.827 1739.65,874.801 1617.08,1043.93 1462.35,1204.92 1279.11,1342.97 1081.36,1446.36 884.709,1516.89 705.552,1538.09 \n",
       "  562.231,1502.35 464.276,1416.78 417.96,1290.21 430.989,1130.75 504.586,951.522 621.855,769.445 769.381,596.453 945.571,450.958 1132.29,337.154 1316.77,262.242 \n",
       "  1481.95,244.994 1613.06,287.172 1701.97,380.7 1740.9,518.342 1722,691.509 1649.34,883.772 1541.31,1077.9 1400.81,1259.89 1234.8,1412.54 1060.42,1534.37 \n",
       "  889.04,1607.92 736.811,1618.09 615.659,1567.75 533.76,1464.22 500.352,1313.38 521.53,1126.02 587.588,922.082 685.172,717.568 817.737,529.549 970.972,369.681 \n",
       "  1132.44,244.707 1291.02,177.379 1432.32,176.294 1545.98,236.279 1622.96,351.786 1653.31,517.211 1634.53,717.703 1579.55,931.379 1491.61,1143.78 1369.54,1336.08 \n",
       "  1230.32,1501.54 1081.46,1623.76 934.191,1681.53 801.294,1672.3 692.62,1601.28 618.29,1472.08 587.741,1291.95 598.948,1080.84 640.962,859.385 719.865,642.223 \n",
       "  828.543,446.301 954.015,279.335 1091.55,165.057 1229.77,118.712 1357.02,139.056 1463.24,222.461 1537.51,366.408 1572.24,559.647 1573.07,778.174 1543.24,1005.4 \n",
       "  1474.6,1224.53 1380.88,1422.84 1267.65,1585.82 1139.34,1688.62 1007.47,1722.78 883.089,1690.67 776.714,1593.73 699.499,1435.18 656.272,1232.19 641.016,1009.02 \n",
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       "  339.19,544.693 348.944,868.245 419.738,1197.08 548.181,1488.3 726.898,1680.58 935.758,1727.95 1154.7,1693.98 1369.17,1698.66 1562.79,1729.15 1727.49,1660.99 \n",
       "  1839.52,1449.84 1884.44,1149.7 1863.66,819.196 1780.54,499.917 1637.52,241.363 1446.13,107.94 1230.26,109.591 1005.86,137.678 789.674,110.484 593.936,106.676 \n",
       "  433.682,236.689 333.22,493.099 300.355,811.625 332.789,1142.31 429.338,1443.72 586.652,1657.66 787.262,1729.14 1007.47,1699.47 1234.76,1693.36 1450.98,1727.57 \n",
       "  1645.63,1683.31 1796.45,1494.07 1883.76,1204.35 1904.48,875.846 1859.81,551.722 1749.29,278.323 1579.72,119.866 1374.18,103.449 1151.47,136.494 924.3,117.353 \n",
       "  709.458,99.6232 520.505,204.003 382.485,443.249 308.596,755.334 299.863,1086.74 357.207,1396.61 481.183,1630.35 659.117,1727.08 865.838,1705.98 1089.33,1689.64 \n",
       "  1313.73,1723.52 1524.84,1701.1 1703.56,1535.37 1827.31,1257.93 1888.3,932.615 1885.25,604.936 1815.19,318.681 1679.89,136.295 1498.69,98.7941 1292.99,133.322 \n",
       "  1071.2,124.051 851.567,96.4935 647.657,175.416 482.316,395.467 372.795,699.568 323.813,1030.59 338.348,1347.35 420.318,1598.85 562.963,1721.25 742.62,1712.88 \n",
       "  946.243,1687.84 1163.28,1717.73 1376.23,1714.35 1569.29,1573.38 1719.95,1310.18 1816.29,989.32 1854.25,659.29 1828.78,362.071 1737.05,157.268 1592.48,96.4017 \n",
       "  1417.38,128.443 1217.22,129.904 1007.66,96.7824 803.775,151.168 624.153,350.093 487.833,644.536 403.164,974.046 375.187,1296.24 410.429,1563.47 508.121,1711.22 \n",
       "  649.063,1719.42 818.687,1688.12 1012.79,1711.02 1212.83,1723.22 1405.2,1607.74 1570.44,1360.83 1693.86,1045.78 1768.54,714.542 1787.72,408.129 1744.89,182.706 \n",
       "  1646.52,96.9773 1513.15,122.302 1349.25,134.36 1164.33,99.8578 975.435,131.416 796.249,307.484 644.745,590.464 532.229,917.306 465.802,1243.6 453.832,1524.54 \n",
       "  500.747,1696.75 594.001,1724.81 718.475,1690.46 875.285,1704.16 1048.49,1727.99 1225.01,1638.13 1390.7,1409.54 1530.21,1101.79 1634.06,770.47 1694.03,456.501 \n",
       "  1701.06,212.425 1654.89,101.121 1571.37,115.487 1455.32,137.016 1308.41,104.992 1148.31,116.212 984.639,268.007 831.414,537.602 701.548,860.551 604.072,1189.71 \n",
       "  548.535,1482.41 543.218,1677.74 583.177,1728.27 654.927,1694.65 762.675,1697.87 896.754,1729.11 1043.47,1664.28 1194.88,1456 1337.67,1157.17 1460.5,826.868 \n",
       "  1553.92,506.852 1607.46,246.166 1615.25,109.299 1586.31,108.7 1525.77,137.639 1427.99,111.399 1308.4,105.474 1174.3,232.025 1033.52,486.226 898.609,803.964 \n",
       "  780.145,1134.82 688.355,1437.46 633.813,1654.21 618.789,1729.06 634.059,1700.31 684.757,1692.75 769.757,1727.15 875.025,1685.99 998.226,1499.85 1130.69,1211.69 \n",
       "  1260.53,883.545 1377.31,558.865 1470.1,283.608 1529.19,121.829 1555.86,102.712 1554.1,136.175 1513.27,118.288 1443.13,98.9783 1350.38,199.878 1235.68,436.638 \n",
       "  1108.82,747.733 981.209,1079.15 863.385,1390.05 766.743,1626.31 699.26,1726.53 658.12,1706.91 648.35,1689.28 677.377,1722.81 732.719,1703.16 815.828,1540.72 \n",
       "  924.782,1265.1 1048.87,940.316 1177.09,612.249 1298.74,324.398 1402.24,138.874 1481.65,98.3203 1537.62,132.752 1557.36,124.909 1542.35,96.3451 1499.46,171.867 \n",
       "  1422.64,389.162 1316.6,692.054 1192.48,1022.93 1060.79,1340.51 932.542,1594.27 819.271,1720.14 725.493,1713.8 656.626,1687.76 626.506,1716.86 627.088,1715.81 \n",
       "  661.411,1578.26 735.259,1317.16 841.111,996.998 968.057,666.739 1106.06,368.171 1243.5,160.461 1368.95,96.2912 1477.68,127.672 1556.84,130.601 1599.08,97.0516 \n",
       "  1610.59,148.222 1580.51,344.142 1506.63,637.138 1398.36,966.355 1265.8,1289.18 1118.8,1558.39 970.07,1709.52 830.902,1720.23 708.75,1688.32 620.569,1710.08 \n",
       "  565.386,1724.1 546.215,1612.1 576.065,1367.56 652.559,1053.41 765.783,722.095 906.689,414.564 1064.98,186.492 1226.29,97.3184 1379.45,121.403 1512.14,134.833 \n",
       "  1609.91,100.451 1676.21,129.087 1697.87,301.937 1665,583.215 1583.61,909.602 1462.94,1236.36 1311.14,1519 1140.08,1694.44 965.875,1725.41 799.925,1690.93 \n",
       "  659.27,1703.26 551.095,1728.34 478.102,1641.94 458.672,1415.99 497.03,1109.35 585.404,778.098 715.984,463.224 880.696,216.774 1064.86,101.985 1251.39,114.548 \n",
       "  1427.56,137.223 1575.21,105.804 1692.64,114.497 1766.68,262.913 1780.29,530.538 1734.55,852.86 1637.18,1182.31 1494.43,1476.47 1315.75,1674.81 1119.4,1728.55 \n",
       "  921.847,1695.34 738.582,1697.09 583.77,1729.01 460.956,1667.49 391.174,1462.11 385.8,1164.62 440.438,834.546 548.806,513.818 705.466,251.036 897.73,110.739 \n",
       "  1104.51,107.823 1310.99,137.562 1499.02,112.323 1660.31,104.351 1782.9,227.431 1844.46,479.386 1840.11,796.31 1775.18,1127.3 1653.99,1431.17 1482.69,1650.68 \n",
       "  1278.77,1728.93 1063.52,1701.16 851.007,1692.17 659.192,1726.69 494.441,1688.59 377.658,1505.58 326.734,1219 341.744,891.246 418.326,566.034 553.714,288.953 \n",
       "  738.829,123.875 951.416,102.005 1173.22,135.821 1388.74,119.218 1583.58,98.4047 1746.78,195.83 1853.52,430.065 1892.64,740.142 1866.3,1071.55 1776.83,1383.45 \n",
       "  1626.8,1622.2 1430.48,1725.89 1212.26,1707.85 986.128,1688.95 769.803,1722.08 574.118,1705.14 417.892,1546.02 323.676,1272.25 296.691,948.016 334.966,619.583 \n",
       "  438.232,330.17 601.816,141.538 805.342,97.8902 1026.95,132.151 1254.55,125.749 1470.3,96.2583 1662.8,168.399 1807.75,382.902 1888.56,684.553 1903.46,1015.28 \n",
       "  1852.71,1333.65 1735.55,1589.62 1561.35,1718.95 1354.82,1714.72 1131.47,1687.71 905.34,1715.97 691.92,1717.18 507.383,1583.08 376.155,1324.1 308.584,1004.67 \n",
       "  305.577,674.204 369.1,374.32 498.93,163.735 679.139,96.2371 885.597,126.879 1108.67,131.271 1331.02,97.3705 1539.16,145.359 1711.75,338.242 1828.57,629.758 \n",
       "  1883.64,958.661 1874.99,1282.1 1798.99,1553.24 1659.51,1707.74 1478.46,1721.03 1273.58,1688.56 1053.65,1709.14 836.922,1724.9 637.781,1616.39 479.425,1374.25 \n",
       "  376.419,1061.03 332.892,729.66 352.824,421.04 439.907,190.356 584.278,97.7268 762.067,120.492 964.34,135.272 1178.15,101.08 1387.11,126.842 1573.98,296.447 \n",
       "  1717.64,575.989 1808.15,901.899 1841.14,1229.09 1810.72,1513.41 1715.47,1692.04 1571.91,1725.97 1399.32,1691.43 1201.6,1702.37 996.314,1728.63 797.623,1645.66 \n",
       "  624.852,1422.39 494.951,1116.89 415.318,785.735 391.783,469.982 431.092,221.196 529.927,102.925 667.692,113.611 834.744,137.392 1024.91,106.638 1219.86,112.863 \n",
       "  1406.1,257.883 1564.62,523.502 1682.53,845.172 1752.99,1174.9 1768.41,1470.48 1723,1671.79 1626.28,1728.78 1497.03,1696.06 1336.15,1696.34 1156.49,1728.85 \n",
       "  973.331,1670.62 800.61,1468.17 655.106,1172.06 547.05,842.228 484.007,520.814 474.824,255.974 522.159,112.259 611.322,106.963 731.793,137.446 884.374,113.253 \n",
       "  1051.51,103.306 1221.95,222.909 1381.52,472.58 1516.09,788.663 1616.57,1119.78 1674.07,1424.83 1679.72,1647.06 1635.72,1728.73 1557.63,1702.03 1445.34,1691.63 \n",
       "  1304.14,1726.18 1150.38,1691.1 992.643,1511.25 844.679,1226.3 718.521,898.948 623.327,573.227 569.163,294.36 563.347,126.004 598.665,101.33 665.012,135.43 \n",
       "  768.439,120.141 895.802,97.9013 1036.42,191.858 1182.55,423.532 1321.25,732.561 1441.64,1063.95 1533.91,1376.8 1587.43,1618.02 1597.95,1725.19 1575.26,1708.78 \n",
       "  1519.41,1688.66 1427.26,1721.33 1314.53,1707.03 1185.81,1551.26 1049.28,1279.37 917.141,955.713 799.847,626.936 707.991,335.996 651.848,144.285 631.87,97.5054 \n",
       "  640.647,131.52 686.903,126.57 765.077,96.2317 864.168,165.011 983.057,376.688 1112.54,677.068 1240.95,1007.61 1357.85,1326.75 1452.03,1584.9 1514.55,1717.69 \n",
       "  1547.78,1715.63 1551.67,1687.7 1516.03,1715.07 1452.99,1718.48 1365.14,1587.82 1253.47,1331.01 1128.26,1012.34 1000.72,681.684 881.47,380.516 782.012,167.091 \n",
       "  709.342,96.2411 661.133,126.066 646.65,131.913 669.318,97.7373 718.428,142.58 798.206,332.396 905.478,622.398 1029.25,950.965 1158.76,1274.98 1283.16,1548.03 \n",
       "  1390.89,1705.87 1476.84,1721.8 1539.17,1688.84 1563.56,1708.2 1555.43,1725.63 1517.05,1620.61 1441.96,1380.91 1336.29,1068.64 1211.15,737.236 1076.82,427.558 \n",
       "  944.549,194.299 825.813,98.2036 724.954,119.572 651.015,135.675 615.456,101.743 609.943,124.681 641.819,291.018 715.408,568.787 822.142,894.197 951.428,1221.8 \n",
       "  1093.38,1507.76 1235.92,1689.56 1367.7,1726.5 1483.03,1691.96 1566.4,1701.49 1614.74,1728.86 1630.42,1649.3 1600.8,1428.74 1525.91,1124.42 1415.55,793.379 \n",
       "  1279.34,476.775 1127.22,225.69 972.666,103.942 826.862,112.679 699.401,137.522 606.905,107.491 546.127,111.312 525.245,252.92 556.269,516.495 634.895,837.488 \n",
       "  751.395,1167.47 897.201,1464.44 1061.43,1668.7 1228.84,1728.96 1388.25,1696.8 1524.88,1695.61 1627.48,1728.65 1697.53,1673.66 1718.54,1474.18 1683.24,1179.48 \n",
       "  1598.71,849.914 1473.58,527.839 1315.83,260.978 1138.52,113.861 958.387,106.121 787.26,137.292 643.355,114.189 530.553,102.337 456.454,218.459 439.525,465.809 \n",
       "  481.271,781.023 573.744,1112.24 709.894,1418.44 881.208,1643.37 1071.32,1728.46 1263.26,1702.92 1443.11,1691.12 1594.06,1725.63 1714.59,1693.53 1787.09,1516.87 \n",
       "  1796.91,1233.57 1747.04,906.651 1644.61,580.444 1495.37,299.827 1310.08,128.217 1108.54,100.687 906.471,135.005 720.807,121.055 562.777,97.4668 439.426,187.963 \n",
       "  373.259,417.04 372.467,724.992 431.914,1056.33 546.237,1370.12 709.894,1613.76 908.038,1724.42 1119.07,1709.72 1328.77,1688.41 1518.6,1720.54 1681.99,1708.85 \n",
       "  1802.36,1556.43 1858.97,1286.47 1849.55,963.408 1779.16,634.309 1651.28,341.876 1473.15,147.115 1264.7,97.168 1046.1,130.861 831.857,127.371 638.507,96.2637 \n",
       "  473.875,161.705 361.547,370.521 316.25,669.597 336.665,999.938 419.321,1319.82 561.762,1580.11 752.677,1716.34 968.294,1716.52 1192.47,1687.73 1408.54,1714.16 \n",
       "  1604.13,1719.69 1764.57,1592.5 1865.45,1337.89 1898.72,1020 1866.68,689.179 1770.64,386.758 1613.85,170.529 1413.52,96.3046 1193.45,125.233 966.242,132.525 \n",
       "  750.062,98.1503 555.358,139.885 404.144,326.603 316.365,615.056 295.241,943.266 339.484,1267.84 449.501,1542.76 618.685,1703.93 824.154,1722.54 1046.82,1689.15 \n",
       "  1274.05,1707.27 1488.95,1726.29 1678.19,1624.75 1816.72,1387.52 1891.08,1076.24 1900.2,744.823 1843.29,434.115 1719.76,198.318 1541.99,98.7502 1335.18,118.645 \n",
       "  1111.63,136.044 887.058,102.438 675.733,122.603 496.531,285.648 372.24,561.609 310.815,886.495 313.481,1214.49 383.126,1502.05 518.128,1686.99 699.416,1726.98 \n",
       "  905.291,1692.52 1127.28,1700.63 1347.19,1729.02 1551.5,1652.87 1717.43,1435.06 1827.47,1131.94 1876.84,801.031 1862.66,483.602 1781.03,230.257 1638.45,105.035 \n",
       "  1458.5,111.752 1254.65,137.614 1037.26,108.362 823.925,109.841 630.31,248.024 479.038,509.517 382.242,829.809 343.944,1160.03 369.102,1458.35 460.64,1665.52 \n",
       "  605.42,1729.08 780.918,1697.57 981.403,1694.91 1191.44,1728.39 1395.84,1676.62 1576.1,1480.14 1712.98,1186.88 1798.03,857.604 1826.29,534.893 1791.27,266.048 \n",
       "  1693.56,115.545 1552.16,105.3 1382.11,137.1 1187.53,115.127 986.921,101.443 793.93,214.083 628.04,459.073 504.151,773.391 429.17,1104.69 409.78,1412.02 \n",
       "  452.531,1639.6 551.197,1728.13 685.167,1703.81 849.531,1690.64 1035.08,1725.05 1224.61,1695.87 1404.47,1522.42 1556.56,1240.83 1669.44,914.355 1736.08,587.684 \n",
       "  1748.15,305.352 1701.12,130.514 1607.3,100.081 1482.21,134.545 1324.83,121.959 1150.89,97.1001 973.678,184.147 807.34,410.589 667.357,717.434 563.246,1048.7 \n",
       "  503.168,1363.41 496.167,1609.42 542.716,1723.56 626.98,1710.66 743.623,1688.2 891.252,1719.73 1052.12,1710.58 1216.49,1561.53 1370.29,1293.54 1500.5,971.1 \n",
       "  1598.12,641.7 1653.65,347.807 1658.75,150.029 1618.16,96.8795 1545.63,130.175 1437.25,128.151 1302.33,96.3528 1154.85,158.481 1002.8,364.402 859.562,662.142 \n",
       "  736.513,992.262 643.072,1312.86 589.677,1575.26 582.325,1714.92 612.091,1717.4 673.36,1687.81 771.893,1713.23 892.358,1720.83 1027.17,1597.11 1168.47,1344.73 \n",
       "  1303.7,1027.66 1422.25,696.688 1513.93,393.046 1568.16,174.047 1582.52,96.4293 1566.36,124.382 1514.97,133.107 1429.09,98.6076 1322.99,137.273 1199.18,320.864 \n",
       "  1066.32,607.735 936.304,935.565 819.574,1260.68 727.037,1537.43 668.44,1701.9 642.627,1723.25 645.227,1689.5 686.797,1706.34 757.87,1726.88 851.337,1628.81 \n",
       "  966.508,1394.1 1093.63,1083.84 1221.27,752.421 1338.9,440.712 1435.14,202.414 1501.94,99.3676 1542.15,117.712 1551.18,136.376 1521.3,103.163 1465.08,120.608 \n",
       "  1381.41,280.34 1272.12,554.456 1147.92,878.795 1019.81,1207.16 898.503,1496.29 795.549,1684.35 716.97,1727.42 661.901,1693.12 642.873,1699.79 658.796,1729.11 \n",
       "  702.417,1656.35 779.632,1441.33 885.853,1139.44 1009.95,808.689 1141.39,490.462 1269.16,234.894 1381.74,106.206 1474.61,110.834 1542.7,137.668 1572.05,109.248 \n",
       "  1570.59,108.451 1535.75,243.195 1461.67,502.57 1355.77,822.134 1228.99,1152.56 1091.38,1452.21 954.545,1662.26 829.874,1729.14 722.035,1698.36 643.544,1694.24 \n",
       "  602.141,1728.08 591.338,1679.5 621.736,1486.05 695.697,1194.26 803.889,865.297 936.142,541.974 1082.62,271.182 1230.66,117.311 1369,104.502 1490.38,136.87 \n",
       "  1577.9,116.067 1632.27,100.625 1650.97,209.78 1621,452.372 1544.54,765.768 1431.53,1097.13 1291.06,1405.55 1133.39,1635.75 972.816,1727.74 820.463,1704.72 \n",
       "  688.396,1690.19 591.321,1724.42 525.679,1698.13 504.247,1527.92 537.091,1248.06 618.332,922.058 738.672,594.947 889.913,310.936 1060.3,132.895 1233.78,99.5109 \n",
       "  1399.02,134.052 1539.14,122.851 1646.62,96.7998 1719.21,180.409 1738.63,404.18 1700.4,709.889 1612.28,1041.07 1482.16,1356.66 1318.08,1605.01 1134.59,1722.64 \n",
       "  948.725,1711.6 773.118,1688.02 625.987,1718.9 509.118,1712.22 435.216,1566.57 421.465,1300.58 466.973,978.788 563.998,649.108 706.192,353.789 884.205,153.026 \n",
       "  1079.95,96.6418 1276.93,129.461 1459.77,128.907 1614.08,96.4973 1736.57,155.34 1806.45,358.332 1812.06,654.703 1757.73,984.58 1649.81,1305.87 1493.77,1570.33 \n",
       "  1302.11,1713.41 1095.83,1718.26 889.811,1687.91 702.115,1712.3 541.243,1721.89 418.695,1601.64 356.87,1351.55 360.917,1035.31 425.483,704.21 546.143,399.379 \n",
       "  716.814,177.645 920.25,96.6165 1135.1,123.515 1347.31,133.658 1538.83,99.1073 1703.37,134.745 1820.35,315.182 1871.63,600.435 1856.97,927.863 1780.82,1253.49 \n",
       "  1646.01,1532.03 1461.44,1699.78 1249.18,1723.93 1027.67,1689.88 812.323,1705.41 617.701,1727.4 454.445,1632.8 347.343,1400.64 307.824,1091.42 333.783,760.028 \n",
       "  422.773,447.347 572.228,206.586 768.166,100.057 986.196,116.775 1212.16,136.672 1428.41,103.917 1624.03,118.697 1780.56,275.094 1875.23,547.328 1902.62,871.097 \n",
       "  1864.75,1199.81 1762,1490.47 1598.89,1681.61 1395.53,1727.81 1174.01,1693.74 946.43,1698.97 730.684,1729.15 538.021,1659.75 392.604,1447.55 311.323,1146.93 \n",
       "  296.024,816.354 346.343,497.354 463.02,239.601 636.93,107.457 843.468,109.925 1066.83,137.682 1293.07,110.148 1506.61,107.141 1691.54,238.435 1823.31,495.654 \n",
       "  1891.33,814.465 1894.72,1145.09 1831.63,1446.02 1702.19,1658.92 1522.02,1729.15 1315.43,1699.17 1092.23,1693.59 869.654,1727.72 661.218,1682.3 488.092,1491.91 \n",
       "  370.728,1201.62 315.245,872.993 323.527,549.081 399.134,276.381 538.422,119.16 719.664,103.729 924.743,136.602 1144.95,117.006 1362,99.8805 1561.62,205.551 \n",
       "  1720.58,445.709 1824.08,758.153 1867.97,1089.55 1848.38,1399.04 1761.58,1631.82 1617.15,1727.27 1439,1705.64 1236.46,1689.78 1022.24,1723.77 812.829,1700.31 \n",
       "  625.354,1533.37 481.097,1255.26 390.166,929.761 356.867,602.231 387.007,316.576 482.155,135.36 626.052,98.9804 799.061,133.525 997.174,123.73 1202.98,96.5645 \n",
       "  1402.25,176.751 1575.67,397.815 1706.09,702.356 1786.07,1033.42 1809.85,1349.87 1770.71,1600.53 1671.72,1721.64 1533.42,1712.53 1365.93,1687.88 1175.21,1718.05 \n",
       "  979.651,1713.79 792.76,1571.55 633.617,1307.59 515.286,986.473 444.567,656.534 428.98,359.822 474.393,156.106 571.553,96.4567 701.415,128.723 862.805,129.639 \n",
       "  1043.14,96.6956 1226.98,152.28 1400.37,352.311 1546.41,647.282 1654.78,976.895 1717.98,1298.85 1727.21,1565.34 1679.64,1711.83 1589.81,1719.11 1468.74,1688.06 \n",
       "  1315.52,1711.37 1147.63,1722.87 976.48,1606.11 816.3,1358.32 681.305,1042.95 580.625,711.746 523.064,405.755 517.529,181.323 562.034,96.8676 640.907,122.633 \n",
       "  753.793,134.176 895.775,99.6477 1050.31,132.301 1208.72,309.555 1357.2,593.156 1483.65,920.16 1578.93,1246.28 1633.02,1526.58 1638.52,1697.58 1602.47,1724.58 \n",
       "  1535.36,1690.3 1431.25,1704.5 1303.02,1727.84 1161.65,1636.71 1014.93,1407.14 875.84,1098.99 755.297,767.645 663.067,454.019 609.773,210.833 599.795,100.82 \n",
       "  623.369,115.836 679.916,136.93 772.885,104.696 886.478,116.869 1015.86,269.912 1152.86,540.227 1285.25,863.402 1402.57,1192.44 1494.25,1484.6 1549.99,1678.8 \n",
       "  1569.21,1728.16 1559.55,1694.4 1512.59,1698.16 1433.5,1729.13 1333.62,1663.07 1214.19,1453.72 1084.37,1154.4 955.852,824.024 839.075,504.279 745.214,244.378 \n",
       "  683.288,108.786 650.958,109.028 647.862,137.658 684.293,111.059 748.25,105.911 836.746,233.745 948.822,488.769 1074.23,806.802 1201.73,1137.6 1320.72,1439.79 \n",
       "  1419.7,1655.5 1491.61,1729.1 1538.89,1700 1552.66,1692.97 1529.09,1727.31 1479.17,1685.01 1398.91,1497.71 1291.38,1208.97 1167.56,880.692 1038.23,556.215 \n",
       "  914.232,281.642 807.118,121.092 722.048,102.982 660.563,136.298 636.906,117.942 645.953,99.2089 684.966,201.398 760.373,439.083 866.159,750.548 991.204,1081.97 \n",
       "  1125.21,1392.49 1256.94,1627.82 1374.94,1726.74 1474.89,1706.57 1548.11,1689.41 1582.86,1723.08 1587.56,1702.4 1555.27,1538.75 1481.52,1262.45 1374.8,937.462 \n",
       "  1245.74,609.537 1104.26,322.273 962.363,137.909 831.393,98.4909 716.908,132.967 634.185,124.593 586.699,96.3928 571.616,173.173 601.45,391.493 676.37,694.837 \n",
       "  786.588,1025.77 922.429,1343.05 1073.96,1595.98 1227.8,1720.56 1372.78,1713.46 1499.54,1687.78 1591.36,1717.19 1651.38,1715.28 1671.92,1576.46 1640.84,1314.58 \n",
       "  1562.31,994.153 1446.08,663.977 1300.77,365.905 1137.2,159.268 970.524,96.3257 811.875,127.961 675.823,130.346 573.931,96.9459 504.428,149.304 483.529,346.341 \n",
       "  518.753,639.877 603.08,969.205 727.816,1291.8 884.965,1560.28 1061.59,1710.16 1241.03,1719.93 1411.51,1688.25 1554.9,1710.43 1667.05,1723.78 1740.88,1610.5 \n",
       "  1757.87,1365.07 1716.31,1050.58 1624.14,719.295 1488.49,412.174 1317.83,185.08 1128.36,97.1842 937.053,121.737 757.696,134.662 607.265,100.227 487.197,129.941 \n",
       "  414.713,303.986 404.692,585.899 454.305,912.457 556.333,1239.04 704.98,1521.06 889.626,1695.3 1090.61,1725.19 1292.15,1690.75 1477.45,1703.59 1634.99,1728.22 \n",
       "  1758.16,1640.54 1824.51,1413.6 1825.58,1106.55 1766.46,775.27 1652.63,460.729 1489.61,215.154 1292.02,101.656 1081.44,114.896 872.127,137.151 682.609,105.5 \n",
       "  519.553,115.123 399.103,264.793 342.176,533.153 351.27,855.71 421.261,1185.06 548.585,1478.68 726.107,1675.9 934.148,1728.46 1152.39,1695.08 1366.41,1697.38 \n",
       "  1559.49,1729.05 1724.03,1666.31 1836.65,1459.85 1882.32,1161.86 1862.28,831.7 1780.08,511.233 1638.22,249.223 1447.79,110.196 1232.43,108.146 1008.49,137.595 \n",
       "  792.542,111.98 597.111,104.758 436.491,229.125 335.186,481.917 301.522,799.146 333.161,1130.09 428.703,1433.51 584.953,1652 784.999,1728.98 1004.93,1700.84 \n",
       "  1232.04,1692.38 1448.18,1726.86 1642.93,1687.63 1794.53,1503.46 1882.78,1216.29 1904.3,888.392 1860.48,563.375 1750.97,286.965 1582.18,123.107 1376.79,102.264 \n",
       "  1154.17,135.956 926.894,118.874 711.945,98.609 522.43,197.321 383.383,432.496 308.561,742.953 299.043,1074.37 355.519,1385.9 478.601,1623.73 656.192,1726.14 \n",
       "  863.087,1707.5 1086.7,1689.07 1311.44,1722.36 1522.96,1704.41 1702.63,1544.07 1827.47,1269.61 1889.33,945.162 1887.05,616.863 1817.82,328.025 1683.13,140.541 \n",
       "  1501.76,98.0444 1295.72,132.377 1073.53,125.44 853.348,96.2834 648.66,169.674 482.147,385.216 371.577,687.331 321.82,1018.11 335.641,1336.2 416.921,1591.36 \n",
       "  559.42,1719.4 739.654,1714.38 943.737,1687.72 1161.47,1716.3 1375.19,1716.68 1569.35,1581.3 1721.23,1321.53 1818.49,1001.83 1857.13,671.436 1832.28,372.036 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(q_trajectory[:, 2], q_trajectory[:, 3], \n",
    "    label=\"\", xlabel=\"q₂\", ylabel=\"q₃\", size=(500, 500))"
   ]
  }
 ],
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