Python_Symbolic_4B_PRs.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 154,
   "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",
    "def generate_equations_wo_topology_pr(theta_i, theta_j, r_x_i, r_y_i, r_x_j, r_y_j):\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",
    "    # Return the equation and the symbols\n",
    "    #return (equations)\n",
    "    return equations\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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}r_{x 1} - r_{x 2} + u_{x i} \\cos{\\left(\\theta_{1} \\right)} - u_{x j} \\cos{\\left(\\theta_{2} \\right)} - u_{y i} \\sin{\\left(\\theta_{1} \\right)} + u_{y j} \\sin{\\left(\\theta_{2} \\right)}\\\\r_{y 1} - r_{y 2} + u_{x i} \\sin{\\left(\\theta_{1} \\right)} - u_{x j} \\sin{\\left(\\theta_{2} \\right)} + u_{y i} \\cos{\\left(\\theta_{1} \\right)} - u_{y j} \\cos{\\left(\\theta_{2} \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[r_x_1 - r_x_2 + u_x_i*cos(theta_1) - u_x_j*cos(theta_2) - u_y_i*sin(theta_1) + u_y_j*sin(theta_2)],\n",
       "[r_y_1 - r_y_2 + u_x_i*sin(theta_1) - u_x_j*sin(theta_2) + u_y_i*cos(theta_1) - u_y_j*cos(theta_2)]])"
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "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",
    "generate_equations_wo_topology_pr(theta_1, theta_2, r_x_1, r_y_1, r_x_2, r_y_2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}r_{x 1} - r_{x 2}\\\\r_{y 1} - r_{y 2}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[r_x_1 - r_x_2],\n",
       "[r_y_1 - r_y_2]])"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "\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",
    "\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",
    "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)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 157,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}r_{x 2} - r_{x 3} + 3 \\cos{\\left(\\theta_{2} \\right)}\\\\r_{y 2} - r_{y 3} + 3 \\sin{\\left(\\theta_{2} \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[r_x_2 - r_x_3 + 3*cos(theta_2)],\n",
       "[r_y_2 - r_y_3 + 3*sin(theta_2)]])"
      ]
     },
     "execution_count": 157,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_x_2_23 = L2\n",
    "u_y_2_23 = 0\n",
    "u_x_3_23 = 0\n",
    "u_y_3_23 = 0\n",
    "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)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 158,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}r_{x 3} - r_{x 4} + 5 \\cos{\\left(\\theta_{3} \\right)}\\\\r_{y 3} - r_{y 4} + 5 \\sin{\\left(\\theta_{3} \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[r_x_3 - r_x_4 + 5*cos(theta_3)],\n",
       "[r_y_3 - r_y_4 + 5*sin(theta_3)]])"
      ]
     },
     "execution_count": 158,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_x_3_34 = L3\n",
    "u_y_3_34 = 0\n",
    "u_x_4_34 = 0\n",
    "u_y_4_34 = 0\n",
    "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)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}r_{x 1} - r_{x 4} + 5 \\cos{\\left(\\theta_{1} \\right)} - 5 \\cos{\\left(\\theta_{4} \\right)}\\\\r_{y 1} - r_{y 4} + 5 \\sin{\\left(\\theta_{1} \\right)} - 5 \\sin{\\left(\\theta_{4} \\right)}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[r_x_1 - r_x_4 + 5*cos(theta_1) - 5*cos(theta_4)],\n",
       "[r_y_1 - r_y_4 + 5*sin(theta_1) - 5*sin(theta_4)]])"
      ]
     },
     "execution_count": 167,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_x_1_14 = Separation\n",
    "u_y_1_14 = 0\n",
    "u_x_4_14 = L4\n",
    "u_y_4_14 = 0\n",
    "generate_equations_included_topology_pr(theta_1, theta_4, r_x_1, r_y_1, 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": 168,
   "metadata": {},
   "outputs": [],
   "source": [
    "def four_bars_mechanism_example_symbolic(theta_2_value, 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",
    "\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_1, r_y_1, 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",
    "    # 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",
    "    #print(equation_list) \n",
    "\n",
    "    return(equation_list)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 169,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[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(-r_x_4 - 5*cos(theta_4) + 5, 0),\n",
       " Eq(-r_y_4 - 5*sin(theta_4), 0)]"
      ]
     },
     "execution_count": 169,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L2 = 5\n",
    "L3 = 5\n",
    "L4 = 5\n",
    "Separation = 5\n",
    "\n",
    "\n",
    "theta_2 = np.pi/2\n",
    "\n",
    "\n",
    "four_bars_mechanism_example_symbolic(theta_2, L2, L3, L4, Separation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 170,
   "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",
    "\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_1, r_y_1, 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",
    "    # 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",
    "    def equations_for_fsolve(vars):\n",
    "         return func(*vars)\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": 171,
   "metadata": {},
   "outputs": [],
   "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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import plotly.graph_objs as go\n",
    "\n",
    "def plot_mechanism_updated_json(new_input, Separation):\n",
    "    # Extract values from the input dictionary using string keys\n",
    "    r_x_2_value = new_input['r_x_2']\n",
    "    r_y_2_value = new_input['r_y_2']\n",
    "    r_x_3_value = new_input['r_x_3']\n",
    "    r_y_3_value = new_input['r_y_3']\n",
    "    r_x_4_value = new_input['r_x_4']\n",
    "    r_y_4_value = new_input['r_y_4']\n",
    "\n",
    "    print(r_x_2_value)\n",
    "    \n",
    "    # Create traces for each line\n",
    "    trace1 = go.Scatter(x=[r_x_2_value, r_x_3_value], y=[r_y_2_value, r_y_3_value], mode='lines+markers', name='Link 2-3')\n",
    "    trace2 = go.Scatter(x=[r_x_3_value, r_x_4_value], y=[r_y_3_value, r_y_4_value], mode='lines+markers', name='Link 3-4')\n",
    "    trace3 = go.Scatter(x=[r_x_4_value, Separation], y=[r_y_4_value, 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",
    "test_results2 = {\n",
    "    'r_x_2': 0.0,\n",
    "    'r_x_3': 1.22464679914735e-16,\n",
    "    'r_y_2': 0.0,\n",
    "    'r_y_3': 2.00000000000000,\n",
    "    'r_x_4': 3.06161699786838e-16,\n",
    "    'r_y_4': 5.00000000000000\n",
    "}\n",
    "\n",
    "plot_mechanism_updated_json(test_results2,Separation)\n",
    "#plot_mechanism_updated_json(new_dict,Separation)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "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(-r_x_4 - 5*cos(theta_4) + 5, 0)\n",
      "Eq(-r_y_4 - 5*sin(theta_4), 0)\n",
      "[ 1.49011612e-08  1.49011612e-08  0.00000000e+00  5.00000000e+00\n",
      "  5.00000000e+00  5.00000000e+00 -1.57079633e+00  0.00000000e+00]\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, -np.pi/2, 0]  #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",
    "print(solution)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 174,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[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(-r_x_4 - 5*cos(theta_4) + 5, 0),\n",
       " Eq(-r_y_4 - 5*sin(theta_4), 0)]"
      ]
     },
     "execution_count": 174,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#and here we have those equations:\n",
    "\n",
    "sym_equations_to_solve = four_bars_mechanism_example_symbolic(theta_2, L2, L3, L4, Separation)\n",
    "\n",
    "sym_equations_to_solve\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 175,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{r_y_3: 5.0, r_x_3: 3.0616172175744623e-16, r_y_4: 7.185155911688072e-10, theta_4: 12.566370614221553, r_x_2: 3.1161407054241035e-23, theta_3: -1.5707963266480782, r_y_2: -1.9207247182445632e-21, r_x_4: 1.069627661854941e-09}\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from scipy.optimize import fsolve\n",
    "import sympy as sp\n",
    "\n",
    "def solve_numerically(eqs, initial_guess):\n",
    "    # Convert the Eq objects into expressions that evaluate to zero\n",
    "    expressions = [eq.lhs - eq.rhs for eq in eqs]\n",
    "\n",
    "    # Extract all the symbols from the expressions\n",
    "    symbols = list(set().union(*[expr.free_symbols for expr in expressions]))\n",
    "\n",
    "    # Lambdify the expressions\n",
    "    func = sp.lambdify(symbols, expressions, \"numpy\")\n",
    "\n",
    "    # Define a wrapper function for fsolve\n",
    "    def wrapper(params):\n",
    "        return func(*params)\n",
    "\n",
    "    # Solve the equations numerically\n",
    "    solution = fsolve(wrapper, initial_guess)\n",
    "\n",
    "    return dict(zip(symbols, solution))\n",
    "\n",
    "# Your input equations are stored in sym_equations_to_solve\n",
    "solution = solve_numerically(sym_equations_to_solve, initial_guess)\n",
    "print(solution)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "list"
      ]
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(type(sym_equations_to_solve))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "<class 'sympy.core.relational.Equality'>\n"
     ]
    }
   ],
   "source": [
    "\n",
    "print(type(sym_equations_to_solve[0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 177,
   "metadata": {},
   "outputs": [
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     "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(-r_x_4 - 5*cos(theta_4) + 5, 0)\n",
      "Eq(-r_y_4 - 5*sin(theta_4), 0)\n"
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   "source": [
    "solution = four_bars_mechanism_example(theta_2, initial_guess, L2, L3, L4, Separation)\n",
    "\n",
    "\n",
    "plot_mechanism_updated(solution)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 178,
   "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 + 1.22464679914735e-16, 0)\n",
      "Eq(r_y_2 - r_y_3 + 2.0, 0)\n",
      "Eq(r_x_3 - r_x_4 + 3*cos(theta_3), 0)\n",
      "Eq(r_y_3 - r_y_4 + 3*sin(theta_3), 0)\n",
      "Eq(-r_x_4 - 4*cos(theta_4) + 3, 0)\n",
      "Eq(-r_y_4 - 4*sin(theta_4), 0)\n",
      "[ 1.49011612e-08  1.49011612e-08  0.00000000e+00  7.00000000e+00\n",
      "  6.00000000e+00  6.00000000e+00  0.00000000e+00 -1.57079633e+00]\n"
     ]
    }
   ],
   "source": [
    "# Testing that it solves it properly\n",
    "\n",
    "L2 = 2\n",
    "L3 = 3\n",
    "L4 = 4\n",
    "Separation = 3\n",
    "\n",
    "initial_guess = [ 0, 0, 0, 7, 6, 6, 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",
    "print(solution)"
   ]
  },
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   "execution_count": 179,
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    "# # Modify the equations_evaluator to unpack the variables\n",
    "# def equations_evaluator_unpack(vars):\n",
    "#     r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4 = vars\n",
    "#     return equations_evaluator(r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4)\n",
    "\n",
    "# # Modify the four_bars_mechanism function to use the unpacking evaluator\n",
    "# def four_bars_mechanism_unpack_eval(theta_2_value, initial_guess, L2, L3, L4, Separation):\n",
    "#     # Use fsolve with the unpacking evaluator function\n",
    "#     numerical_solution = fsolve(equations_evaluator_unpack, initial_guess, xtol=1e-10, maxfev=10000)\n",
    "    \n",
    "#     # Adjusting the angles to lie within [0, 2*pi]\n",
    "#     numerical_solution[6] = numerical_solution[6] % (2 * np.pi)\n",
    "#     numerical_solution[7] = numerical_solution[7] % (2 * np.pi)\n",
    "    \n",
    "#     return numerical_solution\n",
    "\n",
    "# # Testing the modified function with unpacking evaluator again\n",
    "# solution_unpack_final = four_bars_mechanism_unpack_eval(theta_2, initial_guess, L2, L3, L4, Separation)\n",
    "# solution_unpack_final\n"
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   "source": [
    "# from sympy import Eq, cos, sin\n",
    "\n",
    "\n",
    "# # Define the symbolic variables again\n",
    "# r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4 = symbols('r_x_2 r_y_2 r_x_3 r_y_3 r_x_4 r_y_4 theta_3 theta_4')\n",
    "\n",
    "# # Sample equations provided\n",
    "# sample_equations = [\n",
    "#     Eq(-r_x_2, 0),\n",
    "#     Eq(-r_y_2, 0),\n",
    "#     Eq(r_x_2 - r_x_3 + 3.67394039744206e-16, 0),\n",
    "#     Eq(r_y_2 - r_y_3 + 6.0, 0),\n",
    "#     Eq(r_x_3 - r_x_4 + 6*cos(theta_3), 0),\n",
    "#     Eq(r_y_3 - r_y_4 + 6*sin(theta_3), 0),\n",
    "#     Eq(6 - 6*cos(theta_4), 0),\n",
    "#     Eq(-6*sin(theta_4), 0)\n",
    "# ]\n",
    "\n",
    "# def equations_evaluator_from_symbolic(vars, equations):\n",
    "#     # Unpack the variables\n",
    "#     r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4 = vars\n",
    "    \n",
    "#     # Evaluate each equation and store its residual\n",
    "#     values = []\n",
    "#     for eq in equations:\n",
    "#         # Lambdify the equation to get a function\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), eq.lhs - eq.rhs, \"numpy\")\n",
    "#         # Evaluate the function and append to the values list\n",
    "#         values.append(func(r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4))\n",
    "    \n",
    "#     return values\n",
    "\n",
    "# # Test the evaluator with the sample equations and a set of test values\n",
    "# test_values = [0, 0, 0, 6, -5.3459, 3.2757, 3.6129, 0]\n",
    "# evaluated_equations = equations_evaluator_from_symbolic(test_values, sample_equations)\n",
    "# evaluated_equations\n"
   ]
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   "execution_count": 29,
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    "# # Lambdify all the equations once\n",
    "# lambdified_functions = [lambdify((r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4), eq.lhs - eq.rhs, \"numpy\") for eq in sample_equations]\n",
    "\n",
    "# def equations_evaluator_from_symbolic(vars):\n",
    "#     # Unpack the variables\n",
    "#     r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4 = vars\n",
    "    \n",
    "#     # Evaluate each lambdified function and store its residual\n",
    "#     values = [func(r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4) for func in lambdified_functions]\n",
    "    \n",
    "#     return values\n",
    "\n",
    "# # Test the evaluator with the sample equations and a set of test values\n",
    "# evaluated_equations_corrected = equations_evaluator_from_symbolic(test_values)\n",
    "# evaluated_equations_corrected\n"
   ]
  },
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   "execution_count": null,
   "metadata": {},
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   "source": []
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Another Way"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 183,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.52352948e-27, -3.05705078e-29,  3.06161700e-16,  5.00000000e+00,\n",
       "       -9.98730686e-01,  1.00761588e-01,  4.51129015e+00,  6.25799225e+00])"
      ]
     },
     "execution_count": 183,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "from scipy.optimize import fsolve\n",
    "\n",
    "# Define the symbolic equations\n",
    "given_equations = [\n",
    "    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(-r_x_4 - 4*cos(theta_4) + 3, 0),\n",
    "    Eq(-r_y_4 - 4*sin(theta_4), 0)\n",
    "]\n",
    "\n",
    "\n",
    "# Lambdify all the equations\n",
    "lambdified_eqs = [lambdify((r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4), eq.lhs - eq.rhs, \"numpy\") for eq in given_equations]\n",
    "\n",
    "def solve_equations(equations, initial_guess):\n",
    "    # Evaluator function for fsolve\n",
    "    def equations_evaluator(vars):\n",
    "        return [eq(*vars) for eq in equations]\n",
    "\n",
    "    # Solve the system of equations using fsolve\n",
    "    solution = fsolve(equations_evaluator, initial_guess, xtol=1e-10, maxfev=10000)\n",
    "\n",
    "    # Adjusting the angles to lie within [0, 2*pi]\n",
    "    solution[6] = solution[6] % (2 * np.pi)\n",
    "    solution[7] = solution[7] % (2 * np.pi)\n",
    "    \n",
    "    return solution\n",
    "\n",
    "# Given initial guess\n",
    "initial_guess = [0, 0, 0, 5, -4.5, 3, 3.5, 0]\n",
    "\n",
    "# Solve using the function\n",
    "solution_final = solve_equations(lambdified_eqs, initial_guess)\n",
    "solution_final\n"
   ]
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        },
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   "source": [
    "plot_mechanism_updated(solution_final)"
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  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pure symbolic\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 202,
   "metadata": {},
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       "[{r_x_2: 0.0,\n",
       "  r_x_3: 3.06161699786838e-16,\n",
       "  r_y_2: 0.0,\n",
       "  r_y_3: 5.00000000000000,\n",
       "  r_y_4: 9.37349864163659e-33,\n",
       "  r_x_4: 8.78624767847630e-66,\n",
       "  theta_3: 4.71238898038469,\n",
       "  theta_4: -1.87469972832732e-33},\n",
       " {r_x_2: 0.0,\n",
       "  r_x_3: 3.06161699786838e-16,\n",
       "  r_y_2: 0.0,\n",
       "  r_y_3: 5.00000000000000,\n",
       "  r_y_4: 5.00000000000000,\n",
       "  r_x_4: 5.00000000000000,\n",
       "  theta_3: 6.28318530717959,\n",
       "  theta_4: -1.57079632679490}]"
      ]
     },
     "execution_count": 202,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sp\n",
    "\n",
    "# Define the symbols\n",
    "r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4 = sp.symbols('r_x_2 r_y_2 r_x_3 r_y_3 r_x_4 r_y_4 theta_3 theta_4')\n",
    "\n",
    "# Define the equations\n",
    "# equations = [\n",
    "#     sp.Eq(-r_x_2, 0),\n",
    "#     sp.Eq(-r_y_2, 0),\n",
    "#     sp.Eq(r_x_2 - r_x_3 + 3.06161699786838e-16, 0),\n",
    "#     sp.Eq(r_y_2 - r_y_3 + 5.0, 0),\n",
    "#     sp.Eq(r_x_3 - r_x_4 + 5*sp.cos(theta_3), 0),\n",
    "#     sp.Eq(r_y_3 - r_y_4 + 5*sp.sin(theta_3), 0),\n",
    "#     sp.Eq(5 - 5*sp.cos(theta_4), 0),\n",
    "#     sp.Eq(-5*sp.sin(theta_4), 0)\n",
    "# ]\n",
    "\n",
    "#and here we have those equations:\n",
    "\n",
    "L2 = 5\n",
    "L3 = 5\n",
    "L4 = 5\n",
    "Separation = 5\n",
    "\n",
    "#initial_guess = [ 0, 0, 0, 5, 5, 5, -np.pi/2, 0]  #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",
    "sym_equations_to_solve = four_bars_mechanism_example_symbolic(theta_2, L2, L3, L4, Separation)\n",
    "\n",
    "#sym_equations_to_solve\n",
    "\n",
    "\n",
    "\n",
    "# Solve the equations\n",
    "solutions = sp.solve(sym_equations_to_solve)\n",
    "\n",
    "solutions\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 192,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "list"
      ]
     },
     "execution_count": 192,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "type(solutions)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 204,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{r_x_2: 0.0,\n",
       " r_x_3: 3.06161699786838e-16,\n",
       " r_y_2: 0.0,\n",
       " r_y_3: 5.00000000000000,\n",
       " r_y_4: 9.37349864163659e-33,\n",
       " r_x_4: 8.78624767847630e-66,\n",
       " theta_3: 4.71238898038469,\n",
       " theta_4: -1.87469972832732e-33}"
      ]
     },
     "execution_count": 204,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solutions[0]\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 205,
   "metadata": {},
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   "source": [
    "\n",
    "solutions_theta_3_1 = solutions[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 206,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'r_x_2': 0.0,\n",
       " 'r_x_3': 3.06161699786838e-16,\n",
       " 'r_y_2': 0.0,\n",
       " 'r_y_3': 5.00000000000000,\n",
       " 'r_y_4': 9.37349864163659e-33,\n",
       " 'r_x_4': 8.78624767847630e-66,\n",
       " 'theta_3': 4.71238898038469,\n",
       " 'theta_4': -1.87469972832732e-33}"
      ]
     },
     "execution_count": 206,
     "metadata": {},
     "output_type": "execute_result"
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   ],
   "source": [
    "solution_json = {str(key): value for key, value in solutions_theta_3_1.items()}\n",
    "\n",
    "solution_json"
   ]
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           "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": [
    "\n",
    "\n",
    "plot_mechanism_updated_json({'r_x_2': 0.0,\n",
    " 'r_x_3': 1.83697019872103e-16,\n",
    " 'r_y_2': 0.0,\n",
    " 'r_y_3': 3.00000000000000,\n",
    " 'r_y_4': -1.94809625674159,\n",
    " 'r_x_4': 0.718570409925080,\n",
    " 'theta_3': 4.85660242306916,\n",
    " 'theta_4': 0.706749687228324},Separation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 194,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 194,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Extract the expression for r_x_4 in terms of theta_3\n",
    "#r_x_4_expr = solutions[r_x_4]\n",
    "r_x_4_expr = solutions[0][r_x_4]\n",
    "\n",
    "# Solve for the value of theta_3 that minimizes the absolute value of r_x_4\n",
    "theta_3_solution = sp.solve(sp.Eq(r_x_4_expr, 0), theta_3, domain=sp.Interval(0, 2*sp.pi))\n",
    "\n",
    "theta_3_solution\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 195,
   "metadata": {},
   "outputs": [
    {
     "ename": "AttributeError",
     "evalue": "'list' object has no attribute 'items'",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mAttributeError\u001b[0m                            Traceback (most recent call last)",
      "Cell \u001b[1;32mIn[195], line 2\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[39m# Substitute the values of theta_3 into the solutions\u001b[39;00m\n\u001b[1;32m----> 2\u001b[0m solutions_theta_3_1 \u001b[39m=\u001b[39m {key: val\u001b[39m.\u001b[39msubs(theta_3, theta_3_solution[\u001b[39m0\u001b[39m]) \u001b[39mfor\u001b[39;00m key, val \u001b[39min\u001b[39;00m solutions\u001b[39m.\u001b[39;49mitems()}\n\u001b[0;32m      3\u001b[0m \u001b[39m#solutions_theta_3_2 = {key: val.subs(theta_3, theta_3_solution[1]) for key, val in solutions.items()}\u001b[39;00m\n\u001b[0;32m      4\u001b[0m \n\u001b[0;32m      5\u001b[0m \u001b[39m#solutions_theta_3_1, solutions_theta_3_2\u001b[39;00m\n\u001b[0;32m      6\u001b[0m solutions_theta_3_1\n",
      "\u001b[1;31mAttributeError\u001b[0m: 'list' object has no attribute 'items'"
     ]
    }
   ],
   "source": [
    "# Substitute the values of theta_3 into the solutions\n",
    "solutions_theta_3_1 = {key: val.subs(theta_3, theta_3_solution[0]) for key, val in solutions.items()}\n",
    "#solutions_theta_3_2 = {key: val.subs(theta_3, theta_3_solution[1]) for key, val in solutions.items()}\n",
    "\n",
    "#solutions_theta_3_1, solutions_theta_3_2\n",
    "solutions_theta_3_1\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Another Topology V1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 189,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[Eq(-r_x_2, 0), Eq(-r_y_2, 0), Eq(r_x_2 - r_x_3 + 1.83697019872103e-16, 0), Eq(r_y_2 - r_y_3 + 3.0, 0), Eq(r_x_3 - r_x_4 + 5*cos(theta_3), 0), Eq(r_y_3 - r_y_4 + 5*sin(theta_3), 0), Eq(-r_x_4 - 3*cos(theta_4) + 3, 0), Eq(-r_y_4 - 3*sin(theta_4), 0)]\n"
     ]
    }
   ],
   "source": [
    "import sympy as sp\n",
    "\n",
    "# Define the symbols\n",
    "r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4 = sp.symbols('r_x_2 r_y_2 r_x_3 r_y_3 r_x_4 r_y_4 theta_3 theta_4')\n",
    "\n",
    "\n",
    "#and here we have those equations:\n",
    "\n",
    "L2 = 3\n",
    "L3 = 5\n",
    "L4 = 3\n",
    "Separation = 3\n",
    "\n",
    "\n",
    "theta_2 = np.pi/2\n",
    "\n",
    "sym_equations_to_solve = four_bars_mechanism_example_symbolic(theta_2, L2, L3, L4, Separation)\n",
    "\n",
    "print(sym_equations_to_solve)\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 190,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[{r_x_2: 0.0,\n",
       "  r_x_3: 1.83697019872103e-16,\n",
       "  r_y_2: 0.0,\n",
       "  r_y_3: 3.00000000000000,\n",
       "  r_y_4: -1.94809625674159,\n",
       "  r_x_4: 0.718570409925080,\n",
       "  theta_3: 4.85660242306916,\n",
       "  theta_4: 0.706749687228324},\n",
       " {r_x_2: 0.0,\n",
       "  r_x_3: 1.83697019872103e-16,\n",
       "  r_y_2: 0.0,\n",
       "  r_y_3: 3.00000000000000,\n",
       "  r_y_4: 2.28142959007492,\n",
       "  r_x_4: 4.94809625674159,\n",
       "  theta_3: 6.13897186449512,\n",
       "  theta_4: -2.27754601402322}]"
      ]
     },
     "execution_count": 190,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "# Solve the equations\n",
    "solutions = sp.solve(sym_equations_to_solve)\n",
    "\n",
    "solutions\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 191,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "list indices must be integers or slices, not Symbol",
     "output_type": "error",
     "traceback": [
      "\u001b[1;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[1;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "Cell \u001b[1;32mIn[191], line 2\u001b[0m\n\u001b[0;32m      1\u001b[0m \u001b[39m# Extract the expression for r_x_4 in terms of theta_3\u001b[39;00m\n\u001b[1;32m----> 2\u001b[0m r_x_4_expr \u001b[39m=\u001b[39m solutions[r_x_4]\n\u001b[0;32m      4\u001b[0m \u001b[39m# Solve for the value of theta_3 that minimizes the absolute value of r_x_4\u001b[39;00m\n\u001b[0;32m      5\u001b[0m theta_3_solution \u001b[39m=\u001b[39m sp\u001b[39m.\u001b[39msolve(sp\u001b[39m.\u001b[39mEq(r_x_4_expr, \u001b[39m0\u001b[39m), theta_3, domain\u001b[39m=\u001b[39msp\u001b[39m.\u001b[39mInterval(\u001b[39m0\u001b[39m, \u001b[39m2\u001b[39m\u001b[39m*\u001b[39msp\u001b[39m.\u001b[39mpi))\n",
      "\u001b[1;31mTypeError\u001b[0m: list indices must be integers or slices, not Symbol"
     ]
    }
   ],
   "source": [
    "# Extract the expression for r_x_4 in terms of theta_3\n",
    "r_x_4_expr = solutions[r_x_4]\n",
    "\n",
    "# Solve for the value of theta_3 that minimizes the absolute value of r_x_4\n",
    "theta_3_solution = sp.solve(sp.Eq(r_x_4_expr, 0), theta_3, domain=sp.Interval(0, 2*sp.pi))\n",
    "\n",
    "theta_3_solution\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{r_x_2: 0.0,\n",
       " r_x_3: 1.83697019872103e-16,\n",
       " r_y_2: 0.0,\n",
       " r_y_3: 3.00000000000000,\n",
       " r_x_4: 4.89858719658941e-16,\n",
       " r_y_4: 8.00000000000000,\n",
       " theta_4: 0.0}"
      ]
     },
     "execution_count": 141,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Substitute the values of theta_3 into the solutions\n",
    "solutions_theta_3_1 = {key: val.subs(theta_3, theta_3_solution[0]) for key, val in solutions.items()}\n",
    "#solutions_theta_3_2 = {key: val.subs(theta_3, theta_3_solution[1]) for key, val in solutions.items()}\n",
    "\n",
    "#solutions_theta_3_1, solutions_theta_3_2\n",
    "solutions_theta_3_1\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'r_x_2': 0.0,\n",
       " 'r_x_3': 1.83697019872103e-16,\n",
       " 'r_y_2': 0.0,\n",
       " 'r_y_3': 3.00000000000000,\n",
       " 'r_x_4': 4.89858719658941e-16,\n",
       " 'r_y_4': 8.00000000000000,\n",
       " 'theta_4': 0.0}"
      ]
     },
     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solution_json = {str(key): value for key, value in solutions_theta_3_1.items()}\n",
    "\n",
    "solution_json"
   ]
  },
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   "cell_type": "code",
   "execution_count": 147,
   "metadata": {},
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           "showlakes": true,
           "showland": true,
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           "x": 0.05
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        "title": {
         "text": "Four Bars Mechanism Visualization"
        },
        "width": 500,
        "xaxis": {
         "range": [
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         "scaleanchor": "y",
         "scaleratio": 1,
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   "source": [
    "\n",
    "\n",
    "plot_mechanism_updated_json({'r_x_2': 0.0,\n",
    " 'r_x_3': 1.83697019872103e-16,\n",
    " 'r_y_2': 0.0,\n",
    " 'r_y_3': 3.00000000000000,\n",
    " 'r_x_4': 4.89858719658941e-16,\n",
    " 'r_y_4': 8.00000000000000,\n",
    " 'theta_4': 0.0},Separation)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Another Topology V2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 132,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[Eq(-r_x_2, 0), Eq(-r_y_2, 0), Eq(r_x_2 - r_x_3 + 1.83697019872103e-16, 0), Eq(r_y_2 - r_y_3 + 3.0, 0), Eq(r_x_3 - r_x_4 + 5*cos(theta_3), 0), Eq(r_y_3 - r_y_4 + 5*sin(theta_3), 0), Eq(3 - 3*cos(theta_4), 0), Eq(-3*sin(theta_4), 0)]\n"
     ]
    }
   ],
   "source": [
    "import sympy as sp\n",
    "\n",
    "# Define the symbols\n",
    "r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4 = sp.symbols('r_x_2 r_y_2 r_x_3 r_y_3 r_x_4 r_y_4 theta_3 theta_4')\n",
    "\n",
    "# Define the equations\n",
    "# equations = [\n",
    "#     sp.Eq(-r_x_2, 0),\n",
    "#     sp.Eq(-r_y_2, 0),\n",
    "#     sp.Eq(r_x_2 - r_x_3 + 3.06161699786838e-16, 0),\n",
    "#     sp.Eq(r_y_2 - r_y_3 + 5.0, 0),\n",
    "#     sp.Eq(r_x_3 - r_x_4 + 5*sp.cos(theta_3), 0),\n",
    "#     sp.Eq(r_y_3 - r_y_4 + 5*sp.sin(theta_3), 0),\n",
    "#     sp.Eq(5 - 5*sp.cos(theta_4), 0),\n",
    "#     sp.Eq(-5*sp.sin(theta_4), 0)\n",
    "# ]\n",
    "\n",
    "#and here we have those equations:\n",
    "\n",
    "L2 = 3\n",
    "L3 = 5\n",
    "L4 = 3\n",
    "Separation = 3\n",
    "\n",
    "#initial_guess = [ 0, 0, 0, 5, 5, 5, -np.pi/2, 0]  #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",
    "sym_equations_to_solve = four_bars_mechanism_example_symbolic(theta_2, L2, L3, L4, Separation)\n",
    "\n",
    "print(sym_equations_to_solve)\n",
    "\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{r_x_2: 0.0,\n",
       " r_x_3: 1.83697019872103e-16,\n",
       " r_y_2: 0.0,\n",
       " r_y_3: 3.00000000000000,\n",
       " r_x_4: 5.0*cos(theta_3) + 1.83697019872103e-16,\n",
       " r_y_4: 5.0*sin(theta_3) + 3.0,\n",
       " theta_4: 0.0}"
      ]
     },
     "execution_count": 133,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solve the equations\n",
    "solutions = sp.solve(sym_equations_to_solve)\n",
    "\n",
    "solutions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{r_x_2: 0.0,\n",
       " r_x_3: 1.83697019872103e-16,\n",
       " r_y_2: 0.0,\n",
       " r_y_3: 3.00000000000000,\n",
       " r_x_4: 5.0*cos(theta_3) + 1.83697019872103e-16,\n",
       " r_y_4: 5.0*sin(theta_3) + 3.0,\n",
       " theta_4: 0.0}"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "### NUMERICAL AGAIN\n",
    "\n",
    "# Use a numerical solver to find values for theta_4 within [0, 2*pi]\n",
    "# theta_4_numerical_solutions = sp.nsolve(solutions[-2:], theta_4, (0, 2*sp.pi))\n",
    "\n",
    "# theta_4_numerical_solutions\n",
    "\n",
    "\n",
    "def solve_system(equations):\n",
    "    # Define the symbols\n",
    "    r_x_2, r_y_2, r_x_3, r_y_3, r_x_4, r_y_4, theta_3, theta_4 = sp.symbols('r_x_2 r_y_2 r_x_3 r_y_3 r_x_4 r_y_4 theta_3 theta_4')\n",
    "\n",
    "    # Attempt direct symbolic solution\n",
    "    solutions = sp.solve(equations)\n",
    "    if solutions:\n",
    "        return solutions\n",
    "\n",
    "    # Numerically solve for theta_4\n",
    "    theta_4_solution_cos = sp.nsolve(equations[-2], theta_4, 1)\n",
    "    theta_4_solution_sin = sp.nsolve(equations[-1], theta_4, 1)\n",
    "\n",
    "    # Use the theta_4 value that's closer to 0 (more straightforward)\n",
    "    theta_4_value = min(theta_4_solution_cos, theta_4_solution_sin, key=abs)\n",
    "\n",
    "    # Substitute the value of theta_4 into the equations\n",
    "    substituted_equations = [eq.subs(theta_4, theta_4_value) for eq in equations[:-2]]\n",
    "\n",
    "    # Solve the substituted equations\n",
    "    solutions_substituted = sp.solve(substituted_equations)\n",
    "\n",
    "    # Numerically solve for theta_3 that makes dependent variables close to desired values\n",
    "    r_x_4_expr = solutions_substituted[0][r_x_4]\n",
    "    theta_3_solution = sp.nsolve(sp.Eq(r_x_4_expr, 0), theta_3, 1)\n",
    "\n",
    "    # Substitute the value of theta_3 into the solutions\n",
    "    complete_solutions = {key: val.subs(theta_3, theta_3_solution) for key, val in solutions_substituted[0].items()}\n",
    "    \n",
    "    return complete_solutions\n",
    "\n",
    "# Test the function with the provided equations\n",
    "# test_equations = [\n",
    "#     sp.Eq(-r_x_2, 0),\n",
    "#     sp.Eq(-r_y_2, 0),\n",
    "#     sp.Eq(r_x_2 - r_x_3 + 3.06161699786838e-16, 0),\n",
    "#     sp.Eq(r_y_2 - r_y_3 + 5.0, 0),\n",
    "#     sp.Eq(r_x_3 - r_x_4 + 5*sp.cos(theta_3), 0),\n",
    "#     sp.Eq(r_y_3 - r_y_4 + 5*sp.sin(theta_3), 0),\n",
    "#     sp.Eq(5 - 6*sp.cos(theta_4), 0),\n",
    "#     sp.Eq(-6*sp.sin(theta_4), 0)\n",
    "# ]\n",
    "\n",
    "test_results = solve_system(sym_equations_to_solve)\n",
    "test_results\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dict"
      ]
     },
     "execution_count": 135,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(test_results)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 125,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "{'r_x_2': 0.0, 'r_x_3': 1.22464679914735e-16, 'r_y_2': 0.0, 'r_y_3': 2.00000000000000, 'r_x_4': 3.06161699786838e-16, 'r_y_4': 5.00000000000000}\n"
     ]
    }
   ],
   "source": [
    "# test_results = {\n",
    "#     r_x_2: 0.0,\n",
    "#     r_x_3: 1.22464679914735e-16,\n",
    "#     r_y_2: 0.0,\n",
    "#     r_y_3: 2.00000000000000,\n",
    "#     r_x_4: 3.06161699786838e-16,\n",
    "#     r_y_4: 5.00000000000000\n",
    "# }\n",
    "\n",
    "new_dict_result = {str(key): value for key, value in test_results.items()}\n",
    "\n",
    "print(new_dict_result)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 113,
   "metadata": {},
   "outputs": [],
   "source": [
    "# {\n",
    "#     r_x_2: 0.0,\n",
    "#     r_x_3: 1.22464679914735e-16,\n",
    "#     r_y_2: 0.0,\n",
    "#     r_y_3: 2.00000000000000,\n",
    "#     r_x_4: 3.06161699786838e-16,\n",
    "#     r_y_4: 5.00000000000000\n",
    "# }"
   ]
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   "execution_count": 136,
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            "linecolor": "white",
            "ticks": ""
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            "linecolor": "white",
            "ticks": ""
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           "bgcolor": "#E5ECF6",
           "caxis": {
            "gridcolor": "white",
            "linecolor": "white",
            "ticks": ""
           }
          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
           "automargin": true,
           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
           "title": {
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           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
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           "ticks": "",
           "title": {
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           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
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         }
        },
        "title": {
         "text": "Four Bars Mechanism Visualization"
        },
        "width": 500,
        "xaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "y",
         "scaleratio": 1,
         "title": {
          "text": "X Coordinate"
         }
        },
        "yaxis": {
         "range": [
          -15,
          15
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         "scaleanchor": "x",
         "scaleratio": 1,
         "title": {
          "text": "Y Coordinate"
         }
        }
       }
      }
     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "import plotly.graph_objs as go\n",
    "\n",
    "def plot_mechanism_updated_json(new_input, Separation):\n",
    "    # Extract values from the input dictionary using string keys\n",
    "    r_x_2_value = new_input['r_x_2']\n",
    "    r_y_2_value = new_input['r_y_2']\n",
    "    r_x_3_value = new_input['r_x_3']\n",
    "    r_y_3_value = new_input['r_y_3']\n",
    "    r_x_4_value = new_input['r_x_4']\n",
    "    r_y_4_value = new_input['r_y_4']\n",
    "\n",
    "    print(r_x_2_value)\n",
    "    \n",
    "    # Create traces for each line\n",
    "    trace1 = go.Scatter(x=[r_x_2_value, r_x_3_value], y=[r_y_2_value, r_y_3_value], mode='lines+markers', name='Link 2-3')\n",
    "    trace2 = go.Scatter(x=[r_x_3_value, r_x_4_value], y=[r_y_3_value, r_y_4_value], mode='lines+markers', name='Link 3-4')\n",
    "    trace3 = go.Scatter(x=[r_x_4_value, Separation], y=[r_y_4_value, 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",
    "test_results2 = {\n",
    "    'r_x_2': 0.0,\n",
    "    'r_x_3': 1.22464679914735e-16,\n",
    "    'r_y_2': 0.0,\n",
    "    'r_y_3': 2.00000000000000,\n",
    "    'r_x_4': 3.06161699786838e-16,\n",
    "    'r_y_4': 5.00000000000000\n",
    "}\n",
    "\n",
    "plot_mechanism_updated_json(test_results2,Separation)\n",
    "#plot_mechanism_updated_json(new_dict,Separation)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 128,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'r_x_2': 0.0,\n",
       " 'r_x_3': 1.22464679914735e-16,\n",
       " 'r_y_2': 0.0,\n",
       " 'r_y_3': 2.00000000000000,\n",
       " 'r_x_4': 3.06161699786838e-16,\n",
       " 'r_y_4': 5.00000000000000}"
      ]
     },
     "execution_count": 128,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "new_dict_result"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 143,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.0\n"
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    },
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            "gridwidth": 2,
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            "showbackground": true,
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          },
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          },
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            "linecolor": "white",
            "ticks": ""
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            "ticks": ""
           },
           "bgcolor": "#E5ECF6",
           "caxis": {
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            "linecolor": "white",
            "ticks": ""
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          },
          "title": {
           "x": 0.05
          },
          "xaxis": {
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           "gridcolor": "white",
           "linecolor": "white",
           "ticks": "",
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           "zerolinewidth": 2
          },
          "yaxis": {
           "automargin": true,
           "gridcolor": "white",
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           "ticks": "",
           "title": {
            "standoff": 15
           },
           "zerolinecolor": "white",
           "zerolinewidth": 2
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         }
        },
        "title": {
         "text": "Four Bars Mechanism Visualization"
        },
        "width": 500,
        "xaxis": {
         "range": [
          -15,
          15
         ],
         "scaleanchor": "y",
         "scaleratio": 1,
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        },
        "yaxis": {
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         "scaleanchor": "x",
         "scaleratio": 1,
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       }
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     },
     "metadata": {},
     "output_type": "display_data"
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   ],
   "source": [
    "\n",
    "\n",
    "plot_mechanism_updated_json({'r_x_2': 0.0,\n",
    " 'r_x_3': 1.22464679914735e-16,\n",
    " 'r_y_2': 0.0,\n",
    " 'r_y_3': 2.00000000000000,\n",
    " 'r_x_4': 3.06161699786838e-16,\n",
    " 'r_y_4': 5.00000000000000},Separation)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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