Python_Symbolic_PR.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Raw Equations"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Matrix, cos, sin, symbols\n",
    "\n",
    "def rotation_matrix_2d(theta):\n",
    "    \"\"\"Generate a 2D rotation matrix for a given angle.\"\"\"\n",
    "    return Matrix([\n",
    "        [cos(theta), -sin(theta)],\n",
    "        [sin(theta), cos(theta)]\n",
    "    ])\n",
    "\n",
    "def generate_constraint_equations_pr(theta_i, theta_j, r_x_i, r_y_i, r_x_j, r_y_j, u_x_i, u_y_i, u_x_j, u_y_j):\n",
    "    \"\"\"Generate the position restraint equations.\"\"\"\n",
    "    R_i = Matrix([r_x_i, r_y_i])\n",
    "    R_j = Matrix([r_x_j, r_y_j])\n",
    "    local_vector_i = Matrix([u_x_i, u_y_i])\n",
    "    local_vector_j = Matrix([u_x_j, u_y_j])\n",
    "    \n",
    "    transformed_vector_i = rotation_matrix_2d(theta_i) * local_vector_i\n",
    "    transformed_vector_j = rotation_matrix_2d(theta_j) * local_vector_j\n",
    "\n",
    "    constraint_equations = R_i + transformed_vector_i - R_j - transformed_vector_j\n",
    "    return constraint_equations\n",
    "\n",
    "def generate_equations_and_symbols_pr(theta_i, theta_j, r_x_i, r_y_i, r_x_j, r_y_j, u_x_i, u_y_i, u_x_j, u_y_j):\n",
    "\n",
    "\n",
    "\n",
    "    # Generate the constraint equations\n",
    "    equations = generate_constraint_equations_pr(theta_i, theta_j, r_x_i, r_y_i, r_x_j, r_y_j, u_x_i, u_y_i, u_x_j, u_y_j)\n",
    "    \n",
    "    # Return the equation and the symbols\n",
    "    return {\n",
    "        'equation': equations,\n",
    "        'symbols': [u_x_i, u_y_i, u_x_j, u_y_j]\n",
    "    }\n",
    "\n",
    "# If you want to call and test the function:\n",
    "# theta_1, theta_2 = symbols('theta_1 theta_2')\n",
    "# r_x_1, r_y_1, r_x_2, r_y_2 = symbols('r_x_1 r_y_1 r_x_2 r_y_2')\n",
    "# u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12 = symbols('u_x_1_12 u_y_1_12 u_x_2_12 u_y_2_12')\n",
    "# output = generate_equations_and_symbols_pr(theta_1, theta_2, r_x_1, r_y_1, r_x_2, r_y_2, u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12)\n",
    "# print(output)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'equation': Matrix([\n",
      "[r_x_1 - r_x_2 + u_x_1_12*cos(theta_1) - u_x_2_12*cos(theta_2) - u_y_1_12*sin(theta_1) + u_y_2_12*sin(theta_2)],\n",
      "[r_y_1 - r_y_2 + u_x_1_12*sin(theta_1) - u_x_2_12*sin(theta_2) + u_y_1_12*cos(theta_1) - u_y_2_12*cos(theta_2)]]), 'symbols': [u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12]}\n"
     ]
    }
   ],
   "source": [
    "theta_1, theta_2, theta_3, theta_4 = symbols('theta_1 theta_2 theta_3 theta_4')\n",
    "r_x_1, r_y_1, r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4 = symbols('r_x_1 r_y_1 r_x_2 r_y_2 r_x_3 r_y_3 r_x_4 r_y_4')\n",
    "\n",
    "u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12 = symbols('u_x_1_12 u_y_1_12 u_x_2_12 u_y_2_12')\n",
    "\n",
    "output = generate_equations_and_symbols_pr(theta_1, theta_2, r_x_1, r_y_1, r_x_2, r_y_2, u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12)\n",
    "print(output)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}r_{x 1} - r_{x 2} + u_{x 1 12} \\cos{\\left(\\theta_{1} \\right)} - u_{x 2 12} \\cos{\\left(\\theta_{2} \\right)} - u_{y 1 12} \\sin{\\left(\\theta_{1} \\right)} + u_{y 2 12} \\sin{\\left(\\theta_{2} \\right)}\\\\r_{y 1} - r_{y 2} + u_{x 1 12} \\sin{\\left(\\theta_{1} \\right)} - u_{x 2 12} \\sin{\\left(\\theta_{2} \\right)} + u_{y 1 12} \\cos{\\left(\\theta_{1} \\right)} - u_{y 2 12} \\cos{\\left(\\theta_{2} \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[r_x_1 - r_x_2 + u_x_1_12*cos(theta_1) - u_x_2_12*cos(theta_2) - u_y_1_12*sin(theta_1) + u_y_2_12*sin(theta_2)],\n",
       "[r_y_1 - r_y_2 + u_x_1_12*sin(theta_1) - u_x_2_12*sin(theta_2) + u_y_1_12*cos(theta_1) - u_y_2_12*cos(theta_2)]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "constraints_pr, symbols_pr = output_pr['equation'], output_pr['symbols']\n",
    "\n",
    "constraints_pr\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[r_x_1, r_y_1, r_x_2, r_y_2]"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "symbols_pr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Substituting PR Topology"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Matrix, cos, sin, symbols\n",
    "\n",
    "def rotation_matrix_2d(theta):\n",
    "    \"\"\"Generate a 2D rotation matrix for a given angle.\"\"\"\n",
    "    return Matrix([\n",
    "        [cos(theta), -sin(theta)],\n",
    "        [sin(theta), cos(theta)]\n",
    "    ])\n",
    "\n",
    "def generate_constraint_equations_pr(theta_i, theta_j, r_x_i, r_y_i, r_x_j, r_y_j, u_x_i, u_y_i, u_x_j, u_y_j):\n",
    "    \"\"\"Generate the position restraint equations.\"\"\"\n",
    "    R_i = Matrix([r_x_i, r_y_i])\n",
    "    R_j = Matrix([r_x_j, r_y_j])\n",
    "    local_vector_i = Matrix([u_x_i, u_y_i])\n",
    "    local_vector_j = Matrix([u_x_j, u_y_j])\n",
    "    \n",
    "    transformed_vector_i = rotation_matrix_2d(theta_i) * local_vector_i\n",
    "    transformed_vector_j = rotation_matrix_2d(theta_j) * local_vector_j\n",
    "\n",
    "    constraint_equations = R_i + transformed_vector_i - R_j - transformed_vector_j\n",
    "    return constraint_equations\n",
    "\n",
    "\n",
    "def generate_equations_included_topology_pr(theta_i, theta_j, r_x_i, r_y_i, r_x_j, r_y_j, u_x_a, u_y_a, u_x_b, u_y_b):\n",
    "\n",
    "    u_x_i, u_y_i, u_x_j, u_y_j = symbols('u_x_i u_y_i u_x_j u_y_j')\n",
    "\n",
    "\n",
    "    # Generate the constraint equations\n",
    "    equations = generate_constraint_equations_pr(theta_i, theta_j, r_x_i, r_y_i, r_x_j, r_y_j, u_x_i, u_y_i, u_x_j, u_y_j)\n",
    "    \n",
    "    # Substitute the symbolic variables with the provided values\n",
    "    equations_substituted = equations.subs({u_x_i: u_x_a, u_y_i: u_y_a, u_x_j: u_x_b, u_y_j: u_y_b})\n",
    "    \n",
    "    \n",
    "    # Return the equation and the symbols\n",
    "    #return (equations)\n",
    "    return equations_substituted\n",
    "\n",
    "    \n",
    "\n",
    "# If you want to call and test the function:\n",
    "# theta_1, theta_2 = symbols('theta_1 theta_2')\n",
    "# r_x_1, r_y_1, r_x_2, r_y_2 = symbols('r_x_1 r_y_1 r_x_2 r_y_2')\n",
    "# u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12 = symbols('u_x_1_12 u_y_1_12 u_x_2_12 u_y_2_12')\n",
    "# output = generate_equations_and_symbols_pr(theta_1, theta_2, r_x_1, r_y_1, r_x_2, r_y_2, u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12)\n",
    "# print(output)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrix([[r_x_1 - r_x_2], [r_y_1 - r_y_2]])\n"
     ]
    }
   ],
   "source": [
    "theta_1, theta_2 = symbols('theta_1 theta_2')\n",
    "r_x_1, r_y_1, r_x_2, r_y_2 = symbols('r_x_1 r_y_1 r_x_2 r_y_2')\n",
    "\n",
    "#u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12 = symbols('u_x_1_12 u_y_1_12 u_x_2_12 u_y_2_12')\n",
    "u_x_1_12 = 0\n",
    "u_y_1_12 = 0\n",
    "u_x_2_12 = 0\n",
    "u_y_2_12 = 0\n",
    "\n",
    "\n",
    "restrictions_pr12 = generate_equations_included_topology_pr(theta_1, theta_2, r_x_1, r_y_1, r_x_2, r_y_2, u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12)\n",
    "print(output)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "theta_1, theta_2, theta_3, theta_4 = symbols('theta_1 theta_2 theta_3 theta_4')\n",
    "r_x_1, r_y_1, r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4 = symbols('r_x_1 r_y_1 r_x_2 r_y_2 r_x_3 r_y_3 r_x_4 r_y_4')\n",
    "\n",
    "L2 = 5\n",
    "L3 = 5\n",
    "L4 = 5\n",
    "Separation = 5\n",
    "\n",
    "u_x_1_12 = 0\n",
    "u_y_1_12 = 0\n",
    "u_x_2_12 = 0\n",
    "u_y_2_12 = 0\n",
    "restrictions_pr12 = generate_equations_included_topology_pr(theta_1, theta_2, r_x_1, r_y_1, r_x_2, r_y_2, u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12)\n",
    "\n",
    "u_x_2_23 = L2\n",
    "u_y_2_23 = 0\n",
    "u_x_3_23 = 0\n",
    "u_y_3_23 = 0\n",
    "restrictions_pr23 = generate_equations_included_topology_pr(theta_2, theta_3, r_x_2, r_y_2, r_x_3, r_y_3, u_x_2_23, u_y_2_23, u_x_3_23, u_y_3_23)\n",
    "\n",
    "u_x_3_34 = L3\n",
    "u_y_3_34 = 0\n",
    "u_x_4_34 = 0\n",
    "u_y_4_34 = 0\n",
    "restrictions_pr34 = generate_equations_included_topology_pr(theta_3, theta_4, r_x_3, r_y_3, r_x_4, r_y_4, u_x_3_34, u_y_3_34, u_x_4_34, u_y_4_34)\n",
    "\n",
    "u_x_1_14 = Separation\n",
    "u_y_1_14 = 0\n",
    "u_x_4_14 = L4\n",
    "u_y_4_14 = 0\n",
    "restrictions_pr14 = generate_equations_included_topology_pr(theta_1, theta_4, r_x_4, r_y_4, r_x_4, r_y_4, u_x_1_14, u_y_1_14, u_x_4_14, u_y_4_14)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}r_{x 1} - r_{x 2}\\\\r_{y 1} - r_{y 2}\\\\r_{x 2} - r_{x 3} + 5 \\cos{\\left(\\theta_{2} \\right)}\\\\r_{y 2} - r_{y 3} + 5 \\sin{\\left(\\theta_{2} \\right)}\\\\r_{x 3} - r_{x 4} + 5 \\cos{\\left(\\theta_{3} \\right)}\\\\r_{y 3} - r_{y 4} + 5 \\sin{\\left(\\theta_{3} \\right)}\\\\5 \\cos{\\left(\\theta_{1} \\right)} - 5 \\cos{\\left(\\theta_{4} \\right)}\\\\5 \\sin{\\left(\\theta_{1} \\right)} - 5 \\sin{\\left(\\theta_{4} \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[                  r_x_1 - r_x_2],\n",
       "[                  r_y_1 - r_y_2],\n",
       "[ r_x_2 - r_x_3 + 5*cos(theta_2)],\n",
       "[ r_y_2 - r_y_3 + 5*sin(theta_2)],\n",
       "[ r_x_3 - r_x_4 + 5*cos(theta_3)],\n",
       "[ r_y_3 - r_y_4 + 5*sin(theta_3)],\n",
       "[5*cos(theta_1) - 5*cos(theta_4)],\n",
       "[5*sin(theta_1) - 5*sin(theta_4)]])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Stacking the matrices vertically to form a single large matrix\n",
    "restrictions_to_solve = restrictions_pr12.col_join(restrictions_pr23).col_join(restrictions_pr34).col_join(restrictions_pr14)\n",
    "\n",
    "restrictions_to_solve\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eq(-r_x_2, 0)\n",
      "Eq(-r_y_2, 0)\n",
      "Eq(r_x_2 - r_x_3, 0)\n",
      "Eq(r_y_2 - r_y_3 + 5, 0)\n",
      "Eq(r_x_3 - r_x_4 + 5*cos(theta_3), 0)\n",
      "Eq(r_y_3 - r_y_4 + 5*sin(theta_3), 0)\n",
      "Eq(5 - 5*cos(theta_4), 0)\n",
      "Eq(-5*sin(theta_4), 0)\n"
     ]
    }
   ],
   "source": [
    "from sympy import Eq, solve, pi\n",
    "\n",
    "\n",
    "# Substituting the known values into the equations\n",
    "restrictions_substituted = restrictions_to_solve.subs({\n",
    "    r_x_1: 0,\n",
    "    r_y_1: 0,\n",
    "    theta_1: 0,\n",
    "    theta_2: pi/2\n",
    "})\n",
    "\n",
    "# Create a list of equations from the matrix\n",
    "equation_list = [Eq(restrictions_substituted[i, 0], 0) for i in range(restrictions_substituted.shape[0])]\n",
    "\n",
    "# This will try to solve the system of equations (you can run it at your discretion)\n",
    "solution = solve(equation_list)\n",
    "\n",
    "# Print the prepared equations (you can comment this out if not needed)\n",
    "for eq in equation_list:\n",
    "    print(eq)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{r_x_2: 0, r_x_3: 0, r_y_2: 0, r_y_3: 5, r_x_4: 5*cos(theta_3), r_y_4: 5*sin(theta_3) + 5, theta_4: 0}\n"
     ]
    }
   ],
   "source": [
    "print(solution) #this is not a numerical solution and theta_3 could be 0, or 2pi or 4pi...."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 1.49011612e-08  1.49011612e-08  4.05234085e-24  5.00000000e+00\n",
      "  5.00000007e+00  5.00000007e+00  0.00000000e+00 -7.85398163e-01]\n"
     ]
    }
   ],
   "source": [
    "from sympy import lambdify\n",
    "import numpy as np\n",
    "from scipy.optimize import fsolve\n",
    "\n",
    "\n",
    "# Convert the symbolic equations to lambda functions for fsolve\n",
    "#func = lambdify(( r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4), list(restrictions_pr12 + restrictions_pr23 + restrictions_pr34 + restrictions_pr14), \"numpy\")\n",
    "func = lambdify(( r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4), list(equation_list), \"numpy\")\n",
    "\n",
    "\n",
    "def equations_for_fsolve(vars):\n",
    "    return func(*vars)\n",
    "\n",
    "initial_guess = [ 0, 0, 0, 5, 5, 5, 0, -np.pi/4]\n",
    "\n",
    "# Use fsolve to find the solution\n",
    "numerical_solution = fsolve(equations_for_fsolve, initial_guess)\n",
    "print(numerical_solution)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Combining Everything"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import Eq, solve, pi\n",
    "from sympy import lambdify\n",
    "import numpy as np\n",
    "from scipy.optimize import fsolve\n",
    "\n",
    "\n",
    "from sympy import Matrix, cos, sin, symbols\n",
    "\n",
    "def rotation_matrix_2d(theta):\n",
    "    \"\"\"Generate a 2D rotation matrix for a given angle.\"\"\"\n",
    "    return Matrix([\n",
    "        [cos(theta), -sin(theta)],\n",
    "        [sin(theta), cos(theta)]\n",
    "    ])\n",
    "\n",
    "def generate_constraint_equations_pr(theta_i, theta_j, r_x_i, r_y_i, r_x_j, r_y_j, u_x_i, u_y_i, u_x_j, u_y_j):\n",
    "    \"\"\"Generate the position restraint equations.\"\"\"\n",
    "    R_i = Matrix([r_x_i, r_y_i])\n",
    "    R_j = Matrix([r_x_j, r_y_j])\n",
    "    local_vector_i = Matrix([u_x_i, u_y_i])\n",
    "    local_vector_j = Matrix([u_x_j, u_y_j])\n",
    "    \n",
    "    transformed_vector_i = rotation_matrix_2d(theta_i) * local_vector_i\n",
    "    transformed_vector_j = rotation_matrix_2d(theta_j) * local_vector_j\n",
    "\n",
    "    constraint_equations = R_i + transformed_vector_i - R_j - transformed_vector_j\n",
    "    return constraint_equations\n",
    "\n",
    "\n",
    "def generate_equations_included_topology_pr(theta_i, theta_j, r_x_i, r_y_i, r_x_j, r_y_j, u_x_a, u_y_a, u_x_b, u_y_b):\n",
    "\n",
    "    u_x_i, u_y_i, u_x_j, u_y_j = symbols('u_x_i u_y_i u_x_j u_y_j')\n",
    "\n",
    "\n",
    "    # Generate the constraint equations\n",
    "    equations = generate_constraint_equations_pr(theta_i, theta_j, r_x_i, r_y_i, r_x_j, r_y_j, u_x_i, u_y_i, u_x_j, u_y_j)\n",
    "    \n",
    "    # Substitute the symbolic variables with the provided values\n",
    "    equations_substituted = equations.subs({u_x_i: u_x_a, u_y_i: u_y_a, u_x_j: u_x_b, u_y_j: u_y_b})\n",
    "    \n",
    "    \n",
    "    # Return the equation and the symbols\n",
    "    #return (equations)\n",
    "    return equations_substituted\n",
    "\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "\n",
    "# theta_1, theta_2, theta_3, theta_4 = symbols('theta_1 theta_2 theta_3 theta_4')\n",
    "# r_x_1, r_y_1, r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4 = symbols('r_x_1 r_y_1 r_x_2 r_y_2 r_x_3 r_y_3 r_x_4 r_y_4')\n",
    "\n",
    "# L2 = 5\n",
    "# L3 = 5\n",
    "# L4 = 5\n",
    "# Separation = 5\n",
    "\n",
    "# u_x_1_12 = 0\n",
    "# u_y_1_12 = 0\n",
    "# u_x_2_12 = 0\n",
    "# u_y_2_12 = 0\n",
    "# restrictions_pr12 = generate_equations_included_topology_pr(theta_1, theta_2, r_x_1, r_y_1, r_x_2, r_y_2, u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12)\n",
    "\n",
    "# u_x_2_23 = L2\n",
    "# u_y_2_23 = 0\n",
    "# u_x_3_23 = 0\n",
    "# u_y_3_23 = 0\n",
    "# restrictions_pr23 = generate_equations_included_topology_pr(theta_2, theta_3, r_x_2, r_y_2, r_x_3, r_y_3, u_x_2_23, u_y_2_23, u_x_3_23, u_y_3_23)\n",
    "\n",
    "# u_x_3_34 = L3\n",
    "# u_y_3_34 = 0\n",
    "# u_x_4_34 = 0\n",
    "# u_y_4_34 = 0\n",
    "# restrictions_pr34 = generate_equations_included_topology_pr(theta_3, theta_4, r_x_3, r_y_3, r_x_4, r_y_4, u_x_3_34, u_y_3_34, u_x_4_34, u_y_4_34)\n",
    "\n",
    "# u_x_1_14 = Separation\n",
    "# u_y_1_14 = 0\n",
    "# u_x_4_14 = L4\n",
    "# u_y_4_14 = 0\n",
    "# restrictions_pr14 = generate_equations_included_topology_pr(theta_1, theta_4, r_x_4, r_y_4, r_x_4, r_y_4, u_x_1_14, u_y_1_14, u_x_4_14, u_y_4_14)\n",
    "\n",
    "# # Stacking the matrices vertically to form a single large matrix\n",
    "# restrictions_to_solve = restrictions_pr12.col_join(restrictions_pr23).col_join(restrictions_pr34).col_join(restrictions_pr14)\n",
    "\n",
    "# # restrictions_to_solve\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# # Substituting the known values into the equations\n",
    "# restrictions_substituted = restrictions_to_solve.subs({\n",
    "#     r_x_1: 0,\n",
    "#     r_y_1: 0,\n",
    "#     theta_1: 0,\n",
    "#     theta_2: pi/2\n",
    "# })\n",
    "\n",
    "# # Create a list of equations from the matrix\n",
    "# equation_list = [Eq(restrictions_substituted[i, 0], 0) for i in range(restrictions_substituted.shape[0])]\n",
    "\n",
    "# # This will try to solve the system of equations (you can run it at your discretion)\n",
    "# solution = solve(equation_list)\n",
    "\n",
    "# # Print the prepared equations (you can comment this out if not needed)\n",
    "# for eq in equation_list:\n",
    "#     print(eq)\n",
    "\n",
    "# # Convert the symbolic equations to lambda functions for fsolve\n",
    "# #func = lambdify(( r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4), list(restrictions_pr12 + restrictions_pr23 + restrictions_pr34 + restrictions_pr14), \"numpy\")\n",
    "# func = lambdify(( r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4), list(equation_list), \"numpy\") #the ones already with the topology included\n",
    "\n",
    "\n",
    "# def equations_for_fsolve(vars):\n",
    "#     return func(*vars)\n",
    "\n",
    "# initial_guess = [ 0, 0, 0, 5, 5, 5, 0, -np.pi/2]  #The guess in this order: r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4\n",
    "\n",
    "# # Use fsolve to find the solution\n",
    "# numerical_solution = fsolve(equations_for_fsolve, initial_guess) \n",
    "# print(numerical_solution) #r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "def four_bars_mechanism_example(theta_2_value, initial_guess, L2, L3, L4, Separation):\n",
    "\n",
    "    theta_1, theta_2, theta_3, theta_4 = symbols('theta_1 theta_2 theta_3 theta_4')\n",
    "    r_x_1, r_y_1, r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4 = symbols('r_x_1 r_y_1 r_x_2 r_y_2 r_x_3 r_y_3 r_x_4 r_y_4')\n",
    "\n",
    "    # L2 = 5\n",
    "    # L3 = 5\n",
    "    # L4 = 5\n",
    "    # Separation = 5\n",
    "\n",
    "    u_x_1_12 = 0\n",
    "    u_y_1_12 = 0\n",
    "    u_x_2_12 = 0\n",
    "    u_y_2_12 = 0\n",
    "    restrictions_pr12 = generate_equations_included_topology_pr(theta_1, theta_2, r_x_1, r_y_1, r_x_2, r_y_2, u_x_1_12, u_y_1_12, u_x_2_12, u_y_2_12)\n",
    "\n",
    "    u_x_2_23 = L2\n",
    "    u_y_2_23 = 0\n",
    "    u_x_3_23 = 0\n",
    "    u_y_3_23 = 0\n",
    "    restrictions_pr23 = generate_equations_included_topology_pr(theta_2, theta_3, r_x_2, r_y_2, r_x_3, r_y_3, u_x_2_23, u_y_2_23, u_x_3_23, u_y_3_23)\n",
    "\n",
    "    u_x_3_34 = L3\n",
    "    u_y_3_34 = 0\n",
    "    u_x_4_34 = 0\n",
    "    u_y_4_34 = 0\n",
    "    restrictions_pr34 = generate_equations_included_topology_pr(theta_3, theta_4, r_x_3, r_y_3, r_x_4, r_y_4, u_x_3_34, u_y_3_34, u_x_4_34, u_y_4_34)\n",
    "\n",
    "    u_x_1_14 = Separation\n",
    "    u_y_1_14 = 0\n",
    "    u_x_4_14 = L4\n",
    "    u_y_4_14 = 0\n",
    "    restrictions_pr14 = generate_equations_included_topology_pr(theta_1, theta_4, r_x_4, r_y_4, r_x_4, r_y_4, u_x_1_14, u_y_1_14, u_x_4_14, u_y_4_14)\n",
    "\n",
    "    # Stacking the matrices vertically to form a single large matrix\n",
    "    restrictions_to_solve = restrictions_pr12.col_join(restrictions_pr23).col_join(restrictions_pr34).col_join(restrictions_pr14)\n",
    "\n",
    "    # restrictions_to_solve\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "    # Substituting the known values into the equations\n",
    "    restrictions_substituted = restrictions_to_solve.subs({\n",
    "        r_x_1: 0,\n",
    "        r_y_1: 0,\n",
    "        theta_1: 0,\n",
    "        theta_2: theta_2_value\n",
    "    })\n",
    "\n",
    "    # Create a list of equations from the matrix\n",
    "    equation_list = [Eq(restrictions_substituted[i, 0], 0) for i in range(restrictions_substituted.shape[0])]\n",
    "\n",
    "    # This will try to solve the system of equations (you can run it at your discretion)\n",
    "    solution = solve(equation_list)\n",
    "\n",
    "    # Print the prepared equations (you can comment this out if not needed)\n",
    "    for eq in equation_list:\n",
    "        print(eq)\n",
    "\n",
    "    # Convert the symbolic equations to lambda functions for fsolve\n",
    "    #func = lambdify(( r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4), list(restrictions_pr12 + restrictions_pr23 + restrictions_pr34 + restrictions_pr14), \"numpy\")\n",
    "    func = lambdify(( r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4), list(equation_list), \"numpy\") #the ones already with the topology included\n",
    "\n",
    "\n",
    "    #initial_guess = [ 0, 0, 0, 5, 5, 5, 0, -np.pi/2]  #The guess in this order: r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4 (Example for input theta_2 = 90 degrees)\n",
    "\n",
    "    # Use fsolve to find the solution\n",
    "    numerical_solution = fsolve(equations_for_fsolve, initial_guess) \n",
    "    return(numerical_solution) #r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eq(-r_x_2, 0)\n",
      "Eq(-r_y_2, 0)\n",
      "Eq(r_x_2 - r_x_3 + 3.06161699786838e-16, 0)\n",
      "Eq(r_y_2 - r_y_3 + 5.0, 0)\n",
      "Eq(r_x_3 - r_x_4 + 5*cos(theta_3), 0)\n",
      "Eq(r_y_3 - r_y_4 + 5*sin(theta_3), 0)\n",
      "Eq(5 - 5*cos(theta_4), 0)\n",
      "Eq(-5*sin(theta_4), 0)\n"
     ]
    }
   ],
   "source": [
    "# Testing that it solves it properly\n",
    "\n",
    "L2 = 5\n",
    "L3 = 5\n",
    "L4 = 5\n",
    "Separation = 5\n",
    "\n",
    "initial_guess = [ 0, 0, 0, 5, 5, 5, 0, -np.pi/2]  #The guess in this order: r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4\n",
    "\n",
    "theta_2 = np.pi/2\n",
    "\n",
    "solution = four_bars_mechanism_example(theta_2, initial_guess, L2, L3, L4, Separation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.49011612e-08,  1.49011612e-08,  4.05234085e-24,  5.00000000e+00,\n",
       "        5.00000007e+00,  5.00000007e+00,  0.00000000e+00, -1.57079633e+00])"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solution #it is correct, this (rigid) mechanism cannot move to other position than theta2 90 or -90 degrees "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.plotly.v1+json": {
       "config": {
        "plotlyServerURL": "https://plot.ly"
       },
       "data": [
        {
         "mode": "lines+markers",
         "name": "Link 2-3",
         "type": "scatter",
         "x": [
          1.4901161193880155e-8,
          4.0523408517643335e-24
         ],
         "y": [
          1.4901161193880155e-8,
          5
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 3-4",
         "type": "scatter",
         "x": [
          4.0523408517643335e-24,
          5.000000074505806
         ],
         "y": [
          5,
          5.000000074505806
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 4-Base",
         "type": "scatter",
         "x": [
          5.000000074505806,
          5
         ],
         "y": [
          5.000000074505806,
          0
         ]
        }
       ],
       "layout": {
        "autosize": false,
        "height": 500,
        "showlegend": true,
        "template": {
         "data": {
          "bar": [
           {
            "error_x": {
             "color": "#2a3f5f"
            },
            "error_y": {
             "color": "#2a3f5f"
            },
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "bar"
           }
          ],
          "barpolar": [
           {
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "barpolar"
           }
          ],
          "carpet": [
           {
            "aaxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "baxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "type": "carpet"
           }
          ],
          "choropleth": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "choropleth"
           }
          ],
          "contour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "contour"
           }
          ],
          "contourcarpet": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "contourcarpet"
           }
          ],
          "heatmap": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmap"
           }
          ],
          "heatmapgl": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmapgl"
           }
          ],
          "histogram": [
           {
            "marker": {
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "histogram"
           }
          ],
          "histogram2d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2d"
           }
          ],
          "histogram2dcontour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2dcontour"
           }
          ],
          "mesh3d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "mesh3d"
           }
          ],
          "parcoords": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "parcoords"
           }
          ],
          "pie": [
           {
            "automargin": true,
            "type": "pie"
           }
          ],
          "scatter": [
           {
            "fillpattern": {
             "fillmode": "overlay",
             "size": 10,
             "solidity": 0.2
            },
            "type": "scatter"
           }
          ],
          "scatter3d": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatter3d"
           }
          ],
          "scattercarpet": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattercarpet"
           }
          ],
          "scattergeo": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergeo"
           }
          ],
          "scattergl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergl"
           }
          ],
          "scattermapbox": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattermapbox"
           }
          ],
          "scatterpolar": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolar"
           }
          ],
          "scatterpolargl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolargl"
           }
          ],
          "scatterternary": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterternary"
           }
          ],
          "surface": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "surface"
           }
          ],
          "table": [
           {
            "cells": {
             "fill": {
              "color": "#EBF0F8"
             },
             "line": {
              "color": "white"
             }
            },
            "header": {
             "fill": {
              "color": "#C8D4E3"
             },
             "line": {
              "color": "white"
             }
            },
            "type": "table"
           }
          ]
         },
         "layout": {
          "annotationdefaults": {
           "arrowcolor": "#2a3f5f",
           "arrowhead": 0,
           "arrowwidth": 1
          },
          "autotypenumbers": "strict",
          "coloraxis": {
           "colorbar": {
            "outlinewidth": 0,
            "ticks": ""
           }
          },
          "colorscale": {
           "diverging": [
            [
             0,
             "#8e0152"
            ],
            [
             0.1,
             "#c51b7d"
            ],
            [
             0.2,
             "#de77ae"
            ],
            [
             0.3,
             "#f1b6da"
            ],
            [
             0.4,
             "#fde0ef"
            ],
            [
             0.5,
             "#f7f7f7"
            ],
            [
             0.6,
             "#e6f5d0"
            ],
            [
             0.7,
             "#b8e186"
            ],
            [
             0.8,
             "#7fbc41"
            ],
            [
             0.9,
             "#4d9221"
            ],
            [
             1,
             "#276419"
            ]
           ],
           "sequential": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ],
           "sequentialminus": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ]
          },
          "colorway": [
           "#636efa",
           "#EF553B",
           "#00cc96",
           "#ab63fa",
           "#FFA15A",
           "#19d3f3",
           "#FF6692",
           "#B6E880",
           "#FF97FF",
           "#FECB52"
          ],
          "font": {
           "color": "#2a3f5f"
          },
          "geo": {
           "bgcolor": "white",
           "lakecolor": "white",
           "landcolor": "#E5ECF6",
           "showlakes": true,
           "showland": true,
           "subunitcolor": "white"
          },
          "hoverlabel": {
           "align": "left"
          },
          "hovermode": "closest",
          "mapbox": {
           "style": "light"
          },
          "paper_bgcolor": "white",
          "plot_bgcolor": "#E5ECF6",
          "polar": {
           "angularaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "radialaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "scene": {
           "xaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "yaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "zaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           }
          },
          "shapedefaults": {
           "line": {
            "color": "#2a3f5f"
           }
          },
          "ternary": {
           "aaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "baxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          }
         }
        },
        "title": {
         "text": "Four Bars Mechanism Visualization"
        },
        "width": 500,
        "xaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "y",
         "scaleratio": 1,
         "title": {
          "text": "X Coordinate"
         }
        },
        "yaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "x",
         "scaleratio": 1,
         "title": {
          "text": "Y Coordinate"
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import plotly.graph_objects as go\n",
    "\n",
    "\n",
    "\n",
    "import plotly.graph_objects as go\n",
    "import numpy as np\n",
    "\n",
    "def plot_mechanism_updated(solution):\n",
    "    # Extracting the coordinates from the solution\n",
    "    r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4 = solution[0:6]\n",
    "    \n",
    "    # Create traces for each line\n",
    "    trace1 = go.Scatter(x=[r_x_2, r_x_3], y=[r_y_2, r_y_3], mode='lines+markers', name='Link 2-3')\n",
    "    trace2 = go.Scatter(x=[r_x_3, r_x_4], y=[r_y_3, r_y_4], mode='lines+markers', name='Link 3-4')\n",
    "    trace3 = go.Scatter(x=[r_x_4, 5], y=[r_y_4, 0], mode='lines+markers', name='Link 4-Base')\n",
    "    \n",
    "    # Define layout with equal aspect ratio and specified axis ranges\n",
    "    layout = go.Layout(title='Four Bars Mechanism Visualization',\n",
    "                       xaxis=dict(title='X Coordinate', scaleanchor=\"y\", scaleratio=1, range=[-15, 15]),\n",
    "                       yaxis=dict(title='Y Coordinate', scaleanchor=\"x\", scaleratio=1, range=[-15, 15]),\n",
    "                       showlegend=True,\n",
    "                       autosize=False,  # Disable automatic sizing to maintain aspect ratio\n",
    "                       width=500,  # Set width of the plot\n",
    "                       height=500)  # Set height of the plot\n",
    "    \n",
    "    # Create figure and plot\n",
    "    fig = go.Figure(data=[trace1, trace2, trace3], layout=layout)\n",
    "    fig.show()\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "# Testing the updated plotting function with the sample solution\n",
    "plot_mechanism_updated(solution)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Eq(-r_x_2, 0)\n",
      "Eq(-r_y_2, 0)\n",
      "Eq(r_x_2 - r_x_3 + 2.44929359829471e-16, 0)\n",
      "Eq(r_y_2 - r_y_3 + 4.0, 0)\n",
      "Eq(r_x_3 - r_x_4 + 2*cos(theta_3), 0)\n",
      "Eq(r_y_3 - r_y_4 + 2*sin(theta_3), 0)\n",
      "Eq(3 - 5*cos(theta_4), 0)\n",
      "Eq(-5*sin(theta_4), 0)\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "array([ 1.49011612e-08,  1.49011612e-08,  4.67924016e-24,  4.00000000e+00,\n",
       "        4.00000000e+00,  5.00000000e+00,  0.00000000e+00, -1.57079633e+00])"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#Now, lets try a slight variation in the topology and 'make it move'\n",
    "\n",
    "L2 = 4\n",
    "L3 = 2\n",
    "L4 = 5\n",
    "Separation = 3\n",
    "\n",
    "initial_guess = [ 0, 0, 0, 4, 4, 5, 0, -np.pi/2]  #The guess in this order: r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4\n",
    "\n",
    "theta_2 = np.pi/2\n",
    "\n",
    "solution = four_bars_mechanism_example(theta_2, initial_guess, L2, L3, L4, Separation)\n",
    "solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.5707963267948966\n",
      "Eq(-r_x_2, 0)\n",
      "Eq(-r_y_2, 0)\n",
      "Eq(r_x_2 - r_x_3, 0)\n",
      "Eq(r_y_2 - r_y_3 + 5, 0)\n",
      "Eq(r_x_3 - r_x_4 + 4*cos(theta_3), 0)\n",
      "Eq(r_y_3 - r_y_4 + 4*sin(theta_3), 0)\n",
      "Eq(3 - 5*cos(theta_4), 0)\n",
      "Eq(-5*sin(theta_4), 0)\n"
     ]
    },
    {
     "data": {
      "application/vnd.plotly.v1+json": {
       "config": {
        "plotlyServerURL": "https://plot.ly"
       },
       "data": [
        {
         "mode": "lines+markers",
         "name": "Link 2-3",
         "type": "scatter",
         "x": [
          1.4901161193880155e-8,
          4.0523408517643335e-24
         ],
         "y": [
          1.4901161193880155e-8,
          5
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 3-4",
         "type": "scatter",
         "x": [
          4.0523408517643335e-24,
          5.000000074505806
         ],
         "y": [
          5,
          5.000000074505806
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 4-Base",
         "type": "scatter",
         "x": [
          5.000000074505806,
          5
         ],
         "y": [
          5.000000074505806,
          0
         ]
        }
       ],
       "layout": {
        "autosize": false,
        "height": 500,
        "showlegend": true,
        "template": {
         "data": {
          "bar": [
           {
            "error_x": {
             "color": "#2a3f5f"
            },
            "error_y": {
             "color": "#2a3f5f"
            },
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "bar"
           }
          ],
          "barpolar": [
           {
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "barpolar"
           }
          ],
          "carpet": [
           {
            "aaxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "baxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "type": "carpet"
           }
          ],
          "choropleth": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "choropleth"
           }
          ],
          "contour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "contour"
           }
          ],
          "contourcarpet": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "contourcarpet"
           }
          ],
          "heatmap": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmap"
           }
          ],
          "heatmapgl": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmapgl"
           }
          ],
          "histogram": [
           {
            "marker": {
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "histogram"
           }
          ],
          "histogram2d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2d"
           }
          ],
          "histogram2dcontour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2dcontour"
           }
          ],
          "mesh3d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "mesh3d"
           }
          ],
          "parcoords": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "parcoords"
           }
          ],
          "pie": [
           {
            "automargin": true,
            "type": "pie"
           }
          ],
          "scatter": [
           {
            "fillpattern": {
             "fillmode": "overlay",
             "size": 10,
             "solidity": 0.2
            },
            "type": "scatter"
           }
          ],
          "scatter3d": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatter3d"
           }
          ],
          "scattercarpet": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattercarpet"
           }
          ],
          "scattergeo": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergeo"
           }
          ],
          "scattergl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergl"
           }
          ],
          "scattermapbox": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattermapbox"
           }
          ],
          "scatterpolar": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolar"
           }
          ],
          "scatterpolargl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolargl"
           }
          ],
          "scatterternary": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterternary"
           }
          ],
          "surface": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "surface"
           }
          ],
          "table": [
           {
            "cells": {
             "fill": {
              "color": "#EBF0F8"
             },
             "line": {
              "color": "white"
             }
            },
            "header": {
             "fill": {
              "color": "#C8D4E3"
             },
             "line": {
              "color": "white"
             }
            },
            "type": "table"
           }
          ]
         },
         "layout": {
          "annotationdefaults": {
           "arrowcolor": "#2a3f5f",
           "arrowhead": 0,
           "arrowwidth": 1
          },
          "autotypenumbers": "strict",
          "coloraxis": {
           "colorbar": {
            "outlinewidth": 0,
            "ticks": ""
           }
          },
          "colorscale": {
           "diverging": [
            [
             0,
             "#8e0152"
            ],
            [
             0.1,
             "#c51b7d"
            ],
            [
             0.2,
             "#de77ae"
            ],
            [
             0.3,
             "#f1b6da"
            ],
            [
             0.4,
             "#fde0ef"
            ],
            [
             0.5,
             "#f7f7f7"
            ],
            [
             0.6,
             "#e6f5d0"
            ],
            [
             0.7,
             "#b8e186"
            ],
            [
             0.8,
             "#7fbc41"
            ],
            [
             0.9,
             "#4d9221"
            ],
            [
             1,
             "#276419"
            ]
           ],
           "sequential": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ],
           "sequentialminus": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ]
          },
          "colorway": [
           "#636efa",
           "#EF553B",
           "#00cc96",
           "#ab63fa",
           "#FFA15A",
           "#19d3f3",
           "#FF6692",
           "#B6E880",
           "#FF97FF",
           "#FECB52"
          ],
          "font": {
           "color": "#2a3f5f"
          },
          "geo": {
           "bgcolor": "white",
           "lakecolor": "white",
           "landcolor": "#E5ECF6",
           "showlakes": true,
           "showland": true,
           "subunitcolor": "white"
          },
          "hoverlabel": {
           "align": "left"
          },
          "hovermode": "closest",
          "mapbox": {
           "style": "light"
          },
          "paper_bgcolor": "white",
          "plot_bgcolor": "#E5ECF6",
          "polar": {
           "angularaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "radialaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "scene": {
           "xaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "yaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "zaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           }
          },
          "shapedefaults": {
           "line": {
            "color": "#2a3f5f"
           }
          },
          "ternary": {
           "aaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "baxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          }
         }
        },
        "title": {
         "text": "Four Bars Mechanism Visualization"
        },
        "width": 500,
        "xaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "y",
         "scaleratio": 1,
         "title": {
          "text": "X Coordinate"
         }
        },
        "yaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "x",
         "scaleratio": 1,
         "title": {
          "text": "Y Coordinate"
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.7016960206944713\n",
      "Eq(-r_x_2, 0)\n",
      "Eq(-r_y_2, 0)\n",
      "Eq(r_x_2 - r_x_3, 0)\n",
      "Eq(r_y_2 - r_y_3 + 5, 0)\n",
      "Eq(r_x_3 - r_x_4 + 4*cos(theta_3), 0)\n",
      "Eq(r_y_3 - r_y_4 + 4*sin(theta_3), 0)\n",
      "Eq(3 - 5*cos(theta_4), 0)\n",
      "Eq(-5*sin(theta_4), 0)\n"
     ]
    },
    {
     "data": {
      "application/vnd.plotly.v1+json": {
       "config": {
        "plotlyServerURL": "https://plot.ly"
       },
       "data": [
        {
         "mode": "lines+markers",
         "name": "Link 2-3",
         "type": "scatter",
         "x": [
          1.4901161193880155e-8,
          4.0523408517643335e-24
         ],
         "y": [
          1.4901161193880155e-8,
          5
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 3-4",
         "type": "scatter",
         "x": [
          4.0523408517643335e-24,
          5.000000074505806
         ],
         "y": [
          5,
          5.000000074505806
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 4-Base",
         "type": "scatter",
         "x": [
          5.000000074505806,
          5
         ],
         "y": [
          5.000000074505806,
          0
         ]
        }
       ],
       "layout": {
        "autosize": false,
        "height": 500,
        "showlegend": true,
        "template": {
         "data": {
          "bar": [
           {
            "error_x": {
             "color": "#2a3f5f"
            },
            "error_y": {
             "color": "#2a3f5f"
            },
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "bar"
           }
          ],
          "barpolar": [
           {
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "barpolar"
           }
          ],
          "carpet": [
           {
            "aaxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "baxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "type": "carpet"
           }
          ],
          "choropleth": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "choropleth"
           }
          ],
          "contour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "contour"
           }
          ],
          "contourcarpet": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "contourcarpet"
           }
          ],
          "heatmap": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmap"
           }
          ],
          "heatmapgl": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmapgl"
           }
          ],
          "histogram": [
           {
            "marker": {
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "histogram"
           }
          ],
          "histogram2d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2d"
           }
          ],
          "histogram2dcontour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2dcontour"
           }
          ],
          "mesh3d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "mesh3d"
           }
          ],
          "parcoords": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "parcoords"
           }
          ],
          "pie": [
           {
            "automargin": true,
            "type": "pie"
           }
          ],
          "scatter": [
           {
            "fillpattern": {
             "fillmode": "overlay",
             "size": 10,
             "solidity": 0.2
            },
            "type": "scatter"
           }
          ],
          "scatter3d": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatter3d"
           }
          ],
          "scattercarpet": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattercarpet"
           }
          ],
          "scattergeo": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergeo"
           }
          ],
          "scattergl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergl"
           }
          ],
          "scattermapbox": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattermapbox"
           }
          ],
          "scatterpolar": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolar"
           }
          ],
          "scatterpolargl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolargl"
           }
          ],
          "scatterternary": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterternary"
           }
          ],
          "surface": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "surface"
           }
          ],
          "table": [
           {
            "cells": {
             "fill": {
              "color": "#EBF0F8"
             },
             "line": {
              "color": "white"
             }
            },
            "header": {
             "fill": {
              "color": "#C8D4E3"
             },
             "line": {
              "color": "white"
             }
            },
            "type": "table"
           }
          ]
         },
         "layout": {
          "annotationdefaults": {
           "arrowcolor": "#2a3f5f",
           "arrowhead": 0,
           "arrowwidth": 1
          },
          "autotypenumbers": "strict",
          "coloraxis": {
           "colorbar": {
            "outlinewidth": 0,
            "ticks": ""
           }
          },
          "colorscale": {
           "diverging": [
            [
             0,
             "#8e0152"
            ],
            [
             0.1,
             "#c51b7d"
            ],
            [
             0.2,
             "#de77ae"
            ],
            [
             0.3,
             "#f1b6da"
            ],
            [
             0.4,
             "#fde0ef"
            ],
            [
             0.5,
             "#f7f7f7"
            ],
            [
             0.6,
             "#e6f5d0"
            ],
            [
             0.7,
             "#b8e186"
            ],
            [
             0.8,
             "#7fbc41"
            ],
            [
             0.9,
             "#4d9221"
            ],
            [
             1,
             "#276419"
            ]
           ],
           "sequential": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ],
           "sequentialminus": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ]
          },
          "colorway": [
           "#636efa",
           "#EF553B",
           "#00cc96",
           "#ab63fa",
           "#FFA15A",
           "#19d3f3",
           "#FF6692",
           "#B6E880",
           "#FF97FF",
           "#FECB52"
          ],
          "font": {
           "color": "#2a3f5f"
          },
          "geo": {
           "bgcolor": "white",
           "lakecolor": "white",
           "landcolor": "#E5ECF6",
           "showlakes": true,
           "showland": true,
           "subunitcolor": "white"
          },
          "hoverlabel": {
           "align": "left"
          },
          "hovermode": "closest",
          "mapbox": {
           "style": "light"
          },
          "paper_bgcolor": "white",
          "plot_bgcolor": "#E5ECF6",
          "polar": {
           "angularaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "radialaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "scene": {
           "xaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "yaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "zaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           }
          },
          "shapedefaults": {
           "line": {
            "color": "#2a3f5f"
           }
          },
          "ternary": {
           "aaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "baxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          }
         }
        },
        "title": {
         "text": "Four Bars Mechanism Visualization"
        },
        "width": 500,
        "xaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "y",
         "scaleratio": 1,
         "title": {
          "text": "X Coordinate"
         }
        },
        "yaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "x",
         "scaleratio": 1,
         "title": {
          "text": "Y Coordinate"
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.8325957145940461\n",
      "Eq(-r_x_2, 0)\n",
      "Eq(-r_y_2, 0)\n",
      "Eq(r_x_2 - r_x_3, 0)\n",
      "Eq(r_y_2 - r_y_3 + 5, 0)\n",
      "Eq(r_x_3 - r_x_4 + 4*cos(theta_3), 0)\n",
      "Eq(r_y_3 - r_y_4 + 4*sin(theta_3), 0)\n",
      "Eq(3 - 5*cos(theta_4), 0)\n",
      "Eq(-5*sin(theta_4), 0)\n"
     ]
    },
    {
     "data": {
      "application/vnd.plotly.v1+json": {
       "config": {
        "plotlyServerURL": "https://plot.ly"
       },
       "data": [
        {
         "mode": "lines+markers",
         "name": "Link 2-3",
         "type": "scatter",
         "x": [
          1.4901161193880155e-8,
          4.0523408517643335e-24
         ],
         "y": [
          1.4901161193880155e-8,
          5
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 3-4",
         "type": "scatter",
         "x": [
          4.0523408517643335e-24,
          5.000000074505806
         ],
         "y": [
          5,
          5.000000074505806
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 4-Base",
         "type": "scatter",
         "x": [
          5.000000074505806,
          5
         ],
         "y": [
          5.000000074505806,
          0
         ]
        }
       ],
       "layout": {
        "autosize": false,
        "height": 500,
        "showlegend": true,
        "template": {
         "data": {
          "bar": [
           {
            "error_x": {
             "color": "#2a3f5f"
            },
            "error_y": {
             "color": "#2a3f5f"
            },
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "bar"
           }
          ],
          "barpolar": [
           {
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "barpolar"
           }
          ],
          "carpet": [
           {
            "aaxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "baxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "type": "carpet"
           }
          ],
          "choropleth": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "choropleth"
           }
          ],
          "contour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "contour"
           }
          ],
          "contourcarpet": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "contourcarpet"
           }
          ],
          "heatmap": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmap"
           }
          ],
          "heatmapgl": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmapgl"
           }
          ],
          "histogram": [
           {
            "marker": {
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "histogram"
           }
          ],
          "histogram2d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2d"
           }
          ],
          "histogram2dcontour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2dcontour"
           }
          ],
          "mesh3d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "mesh3d"
           }
          ],
          "parcoords": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "parcoords"
           }
          ],
          "pie": [
           {
            "automargin": true,
            "type": "pie"
           }
          ],
          "scatter": [
           {
            "fillpattern": {
             "fillmode": "overlay",
             "size": 10,
             "solidity": 0.2
            },
            "type": "scatter"
           }
          ],
          "scatter3d": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatter3d"
           }
          ],
          "scattercarpet": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattercarpet"
           }
          ],
          "scattergeo": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergeo"
           }
          ],
          "scattergl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergl"
           }
          ],
          "scattermapbox": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattermapbox"
           }
          ],
          "scatterpolar": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolar"
           }
          ],
          "scatterpolargl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolargl"
           }
          ],
          "scatterternary": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterternary"
           }
          ],
          "surface": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "surface"
           }
          ],
          "table": [
           {
            "cells": {
             "fill": {
              "color": "#EBF0F8"
             },
             "line": {
              "color": "white"
             }
            },
            "header": {
             "fill": {
              "color": "#C8D4E3"
             },
             "line": {
              "color": "white"
             }
            },
            "type": "table"
           }
          ]
         },
         "layout": {
          "annotationdefaults": {
           "arrowcolor": "#2a3f5f",
           "arrowhead": 0,
           "arrowwidth": 1
          },
          "autotypenumbers": "strict",
          "coloraxis": {
           "colorbar": {
            "outlinewidth": 0,
            "ticks": ""
           }
          },
          "colorscale": {
           "diverging": [
            [
             0,
             "#8e0152"
            ],
            [
             0.1,
             "#c51b7d"
            ],
            [
             0.2,
             "#de77ae"
            ],
            [
             0.3,
             "#f1b6da"
            ],
            [
             0.4,
             "#fde0ef"
            ],
            [
             0.5,
             "#f7f7f7"
            ],
            [
             0.6,
             "#e6f5d0"
            ],
            [
             0.7,
             "#b8e186"
            ],
            [
             0.8,
             "#7fbc41"
            ],
            [
             0.9,
             "#4d9221"
            ],
            [
             1,
             "#276419"
            ]
           ],
           "sequential": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ],
           "sequentialminus": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ]
          },
          "colorway": [
           "#636efa",
           "#EF553B",
           "#00cc96",
           "#ab63fa",
           "#FFA15A",
           "#19d3f3",
           "#FF6692",
           "#B6E880",
           "#FF97FF",
           "#FECB52"
          ],
          "font": {
           "color": "#2a3f5f"
          },
          "geo": {
           "bgcolor": "white",
           "lakecolor": "white",
           "landcolor": "#E5ECF6",
           "showlakes": true,
           "showland": true,
           "subunitcolor": "white"
          },
          "hoverlabel": {
           "align": "left"
          },
          "hovermode": "closest",
          "mapbox": {
           "style": "light"
          },
          "paper_bgcolor": "white",
          "plot_bgcolor": "#E5ECF6",
          "polar": {
           "angularaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "radialaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "scene": {
           "xaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "yaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "zaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           }
          },
          "shapedefaults": {
           "line": {
            "color": "#2a3f5f"
           }
          },
          "ternary": {
           "aaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "baxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          }
         }
        },
        "title": {
         "text": "Four Bars Mechanism Visualization"
        },
        "width": 500,
        "xaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "y",
         "scaleratio": 1,
         "title": {
          "text": "X Coordinate"
         }
        },
        "yaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "x",
         "scaleratio": 1,
         "title": {
          "text": "Y Coordinate"
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.963495408493621\n",
      "Eq(-r_x_2, 0)\n",
      "Eq(-r_y_2, 0)\n",
      "Eq(r_x_2 - r_x_3, 0)\n",
      "Eq(r_y_2 - r_y_3 + 5, 0)\n",
      "Eq(r_x_3 - r_x_4 + 4*cos(theta_3), 0)\n",
      "Eq(r_y_3 - r_y_4 + 4*sin(theta_3), 0)\n",
      "Eq(3 - 5*cos(theta_4), 0)\n",
      "Eq(-5*sin(theta_4), 0)\n"
     ]
    },
    {
     "data": {
      "application/vnd.plotly.v1+json": {
       "config": {
        "plotlyServerURL": "https://plot.ly"
       },
       "data": [
        {
         "mode": "lines+markers",
         "name": "Link 2-3",
         "type": "scatter",
         "x": [
          1.4901161193880155e-8,
          4.0523408517643335e-24
         ],
         "y": [
          1.4901161193880155e-8,
          5
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 3-4",
         "type": "scatter",
         "x": [
          4.0523408517643335e-24,
          5.000000074505806
         ],
         "y": [
          5,
          5.000000074505806
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 4-Base",
         "type": "scatter",
         "x": [
          5.000000074505806,
          5
         ],
         "y": [
          5.000000074505806,
          0
         ]
        }
       ],
       "layout": {
        "autosize": false,
        "height": 500,
        "showlegend": true,
        "template": {
         "data": {
          "bar": [
           {
            "error_x": {
             "color": "#2a3f5f"
            },
            "error_y": {
             "color": "#2a3f5f"
            },
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "bar"
           }
          ],
          "barpolar": [
           {
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "barpolar"
           }
          ],
          "carpet": [
           {
            "aaxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "baxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "type": "carpet"
           }
          ],
          "choropleth": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "choropleth"
           }
          ],
          "contour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "contour"
           }
          ],
          "contourcarpet": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "contourcarpet"
           }
          ],
          "heatmap": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmap"
           }
          ],
          "heatmapgl": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmapgl"
           }
          ],
          "histogram": [
           {
            "marker": {
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "histogram"
           }
          ],
          "histogram2d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2d"
           }
          ],
          "histogram2dcontour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2dcontour"
           }
          ],
          "mesh3d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "mesh3d"
           }
          ],
          "parcoords": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "parcoords"
           }
          ],
          "pie": [
           {
            "automargin": true,
            "type": "pie"
           }
          ],
          "scatter": [
           {
            "fillpattern": {
             "fillmode": "overlay",
             "size": 10,
             "solidity": 0.2
            },
            "type": "scatter"
           }
          ],
          "scatter3d": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatter3d"
           }
          ],
          "scattercarpet": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattercarpet"
           }
          ],
          "scattergeo": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergeo"
           }
          ],
          "scattergl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergl"
           }
          ],
          "scattermapbox": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattermapbox"
           }
          ],
          "scatterpolar": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolar"
           }
          ],
          "scatterpolargl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolargl"
           }
          ],
          "scatterternary": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterternary"
           }
          ],
          "surface": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "surface"
           }
          ],
          "table": [
           {
            "cells": {
             "fill": {
              "color": "#EBF0F8"
             },
             "line": {
              "color": "white"
             }
            },
            "header": {
             "fill": {
              "color": "#C8D4E3"
             },
             "line": {
              "color": "white"
             }
            },
            "type": "table"
           }
          ]
         },
         "layout": {
          "annotationdefaults": {
           "arrowcolor": "#2a3f5f",
           "arrowhead": 0,
           "arrowwidth": 1
          },
          "autotypenumbers": "strict",
          "coloraxis": {
           "colorbar": {
            "outlinewidth": 0,
            "ticks": ""
           }
          },
          "colorscale": {
           "diverging": [
            [
             0,
             "#8e0152"
            ],
            [
             0.1,
             "#c51b7d"
            ],
            [
             0.2,
             "#de77ae"
            ],
            [
             0.3,
             "#f1b6da"
            ],
            [
             0.4,
             "#fde0ef"
            ],
            [
             0.5,
             "#f7f7f7"
            ],
            [
             0.6,
             "#e6f5d0"
            ],
            [
             0.7,
             "#b8e186"
            ],
            [
             0.8,
             "#7fbc41"
            ],
            [
             0.9,
             "#4d9221"
            ],
            [
             1,
             "#276419"
            ]
           ],
           "sequential": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ],
           "sequentialminus": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ]
          },
          "colorway": [
           "#636efa",
           "#EF553B",
           "#00cc96",
           "#ab63fa",
           "#FFA15A",
           "#19d3f3",
           "#FF6692",
           "#B6E880",
           "#FF97FF",
           "#FECB52"
          ],
          "font": {
           "color": "#2a3f5f"
          },
          "geo": {
           "bgcolor": "white",
           "lakecolor": "white",
           "landcolor": "#E5ECF6",
           "showlakes": true,
           "showland": true,
           "subunitcolor": "white"
          },
          "hoverlabel": {
           "align": "left"
          },
          "hovermode": "closest",
          "mapbox": {
           "style": "light"
          },
          "paper_bgcolor": "white",
          "plot_bgcolor": "#E5ECF6",
          "polar": {
           "angularaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "radialaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "scene": {
           "xaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "yaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "zaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           }
          },
          "shapedefaults": {
           "line": {
            "color": "#2a3f5f"
           }
          },
          "ternary": {
           "aaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "baxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          }
         }
        },
        "title": {
         "text": "Four Bars Mechanism Visualization"
        },
        "width": 500,
        "xaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "y",
         "scaleratio": 1,
         "title": {
          "text": "X Coordinate"
         }
        },
        "yaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "x",
         "scaleratio": 1,
         "title": {
          "text": "Y Coordinate"
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "2.0943951023931957\n",
      "Eq(-r_x_2, 0)\n",
      "Eq(-r_y_2, 0)\n",
      "Eq(r_x_2 - r_x_3, 0)\n",
      "Eq(r_y_2 - r_y_3 + 5, 0)\n",
      "Eq(r_x_3 - r_x_4 + 4*cos(theta_3), 0)\n",
      "Eq(r_y_3 - r_y_4 + 4*sin(theta_3), 0)\n",
      "Eq(3 - 5*cos(theta_4), 0)\n",
      "Eq(-5*sin(theta_4), 0)\n"
     ]
    },
    {
     "data": {
      "application/vnd.plotly.v1+json": {
       "config": {
        "plotlyServerURL": "https://plot.ly"
       },
       "data": [
        {
         "mode": "lines+markers",
         "name": "Link 2-3",
         "type": "scatter",
         "x": [
          1.4901161193880155e-8,
          4.0523408517643335e-24
         ],
         "y": [
          1.4901161193880155e-8,
          5
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 3-4",
         "type": "scatter",
         "x": [
          4.0523408517643335e-24,
          5.000000074505806
         ],
         "y": [
          5,
          5.000000074505806
         ]
        },
        {
         "mode": "lines+markers",
         "name": "Link 4-Base",
         "type": "scatter",
         "x": [
          5.000000074505806,
          5
         ],
         "y": [
          5.000000074505806,
          0
         ]
        }
       ],
       "layout": {
        "autosize": false,
        "height": 500,
        "showlegend": true,
        "template": {
         "data": {
          "bar": [
           {
            "error_x": {
             "color": "#2a3f5f"
            },
            "error_y": {
             "color": "#2a3f5f"
            },
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "bar"
           }
          ],
          "barpolar": [
           {
            "marker": {
             "line": {
              "color": "#E5ECF6",
              "width": 0.5
             },
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "barpolar"
           }
          ],
          "carpet": [
           {
            "aaxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "baxis": {
             "endlinecolor": "#2a3f5f",
             "gridcolor": "white",
             "linecolor": "white",
             "minorgridcolor": "white",
             "startlinecolor": "#2a3f5f"
            },
            "type": "carpet"
           }
          ],
          "choropleth": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "choropleth"
           }
          ],
          "contour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "contour"
           }
          ],
          "contourcarpet": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "contourcarpet"
           }
          ],
          "heatmap": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmap"
           }
          ],
          "heatmapgl": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "heatmapgl"
           }
          ],
          "histogram": [
           {
            "marker": {
             "pattern": {
              "fillmode": "overlay",
              "size": 10,
              "solidity": 0.2
             }
            },
            "type": "histogram"
           }
          ],
          "histogram2d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2d"
           }
          ],
          "histogram2dcontour": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "histogram2dcontour"
           }
          ],
          "mesh3d": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "type": "mesh3d"
           }
          ],
          "parcoords": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "parcoords"
           }
          ],
          "pie": [
           {
            "automargin": true,
            "type": "pie"
           }
          ],
          "scatter": [
           {
            "fillpattern": {
             "fillmode": "overlay",
             "size": 10,
             "solidity": 0.2
            },
            "type": "scatter"
           }
          ],
          "scatter3d": [
           {
            "line": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatter3d"
           }
          ],
          "scattercarpet": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattercarpet"
           }
          ],
          "scattergeo": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergeo"
           }
          ],
          "scattergl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattergl"
           }
          ],
          "scattermapbox": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scattermapbox"
           }
          ],
          "scatterpolar": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolar"
           }
          ],
          "scatterpolargl": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterpolargl"
           }
          ],
          "scatterternary": [
           {
            "marker": {
             "colorbar": {
              "outlinewidth": 0,
              "ticks": ""
             }
            },
            "type": "scatterternary"
           }
          ],
          "surface": [
           {
            "colorbar": {
             "outlinewidth": 0,
             "ticks": ""
            },
            "colorscale": [
             [
              0,
              "#0d0887"
             ],
             [
              0.1111111111111111,
              "#46039f"
             ],
             [
              0.2222222222222222,
              "#7201a8"
             ],
             [
              0.3333333333333333,
              "#9c179e"
             ],
             [
              0.4444444444444444,
              "#bd3786"
             ],
             [
              0.5555555555555556,
              "#d8576b"
             ],
             [
              0.6666666666666666,
              "#ed7953"
             ],
             [
              0.7777777777777778,
              "#fb9f3a"
             ],
             [
              0.8888888888888888,
              "#fdca26"
             ],
             [
              1,
              "#f0f921"
             ]
            ],
            "type": "surface"
           }
          ],
          "table": [
           {
            "cells": {
             "fill": {
              "color": "#EBF0F8"
             },
             "line": {
              "color": "white"
             }
            },
            "header": {
             "fill": {
              "color": "#C8D4E3"
             },
             "line": {
              "color": "white"
             }
            },
            "type": "table"
           }
          ]
         },
         "layout": {
          "annotationdefaults": {
           "arrowcolor": "#2a3f5f",
           "arrowhead": 0,
           "arrowwidth": 1
          },
          "autotypenumbers": "strict",
          "coloraxis": {
           "colorbar": {
            "outlinewidth": 0,
            "ticks": ""
           }
          },
          "colorscale": {
           "diverging": [
            [
             0,
             "#8e0152"
            ],
            [
             0.1,
             "#c51b7d"
            ],
            [
             0.2,
             "#de77ae"
            ],
            [
             0.3,
             "#f1b6da"
            ],
            [
             0.4,
             "#fde0ef"
            ],
            [
             0.5,
             "#f7f7f7"
            ],
            [
             0.6,
             "#e6f5d0"
            ],
            [
             0.7,
             "#b8e186"
            ],
            [
             0.8,
             "#7fbc41"
            ],
            [
             0.9,
             "#4d9221"
            ],
            [
             1,
             "#276419"
            ]
           ],
           "sequential": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ],
           "sequentialminus": [
            [
             0,
             "#0d0887"
            ],
            [
             0.1111111111111111,
             "#46039f"
            ],
            [
             0.2222222222222222,
             "#7201a8"
            ],
            [
             0.3333333333333333,
             "#9c179e"
            ],
            [
             0.4444444444444444,
             "#bd3786"
            ],
            [
             0.5555555555555556,
             "#d8576b"
            ],
            [
             0.6666666666666666,
             "#ed7953"
            ],
            [
             0.7777777777777778,
             "#fb9f3a"
            ],
            [
             0.8888888888888888,
             "#fdca26"
            ],
            [
             1,
             "#f0f921"
            ]
           ]
          },
          "colorway": [
           "#636efa",
           "#EF553B",
           "#00cc96",
           "#ab63fa",
           "#FFA15A",
           "#19d3f3",
           "#FF6692",
           "#B6E880",
           "#FF97FF",
           "#FECB52"
          ],
          "font": {
           "color": "#2a3f5f"
          },
          "geo": {
           "bgcolor": "white",
           "lakecolor": "white",
           "landcolor": "#E5ECF6",
           "showlakes": true,
           "showland": true,
           "subunitcolor": "white"
          },
          "hoverlabel": {
           "align": "left"
          },
          "hovermode": "closest",
          "mapbox": {
           "style": "light"
          },
          "paper_bgcolor": "white",
          "plot_bgcolor": "#E5ECF6",
          "polar": {
           "angularaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "radialaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "scene": {
           "xaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "yaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           },
           "zaxis": {
            "backgroundcolor": "#E5ECF6",
            "gridcolor": "white",
            "gridwidth": 2,
            "linecolor": "white",
            "showbackground": true,
            "ticks": "",
            "zerolinecolor": "white"
           }
          },
          "shapedefaults": {
           "line": {
            "color": "#2a3f5f"
           }
          },
          "ternary": {
           "aaxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "baxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
          }
         }
        },
        "title": {
         "text": "Four Bars Mechanism Visualization"
        },
        "width": 500,
        "xaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "y",
         "scaleratio": 1,
         "title": {
          "text": "X Coordinate"
         }
        },
        "yaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "x",
         "scaleratio": 1,
         "title": {
          "text": "Y Coordinate"
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "# Create a list from 0 to 2π in steps of π/8\n",
    "start = np.pi/2\n",
    "end = 2/3 * np.pi\n",
    "step = np.pi / 24 #15 degrees steps\n",
    "theta2_list = np.arange(start, end + step, step)\n",
    "\n",
    "\n",
    "L2 = 5\n",
    "L3 = 4\n",
    "L4 = 5\n",
    "Separation = 3\n",
    "\n",
    "#The guess in this order: r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4\n",
    "solution = [ 0, 0, 0, 5, 5, 5, 0, -np.pi/2]  \n",
    "\n",
    "# Iterate through the list using a loop\n",
    "for theta2 in theta2_list:\n",
    "    print(theta2)\n",
    "    solution = four_bars_mechanism_example(theta2, initial_guess, L2, L3, L4, Separation)\n",
    "    plot_mechanism_updated(solution)\n",
    "\n"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "base",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.7"
  },
  "orig_nbformat": 4
 },
 "nbformat": 4,
 "nbformat_minor": 2
}