docs/slam/pose_graph_slam.ipynb

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    "# Graph Pose SLAM\n",
    "\n",
    "### **Odometry Data:**\n",
    "From odometry, we typically get the poses $ \\textbf{X}_i $ and $ \\textbf{X}_j $ in the world frame. These poses represent the robot's estimated positions and orientations based on the odometry readings.\n",
    "\n",
    "### **Relative Transformation from Poses:**\n",
    "Using the poses $ \\textbf{X}_i $ and $ \\textbf{X}_j $ in the world frame, we can compute the **predicted relative pose** of $ \\textbf{X}_j $ with respect to $ \\textbf{X}_i $, denoted as $ h(\\textbf{X}_i, \\textbf{X}_j) $. This transformation involves the relative translation and rotation between $ \\textbf{X}_i $ and $ \\textbf{X}_j $.\n",
    "\n",
    "\n",
    "\n",
    "${^{i}_{j}X}$: is the pose of the $j_{th}$farme in $i_{th}$ frame, to get the pose of the robot from world coordinate in the $i_{th}$ frame: \n",
    "\n",
    " <img src=\"https://latex.codecogs.com/svg.latex?%7B%5E%7Bi%7D_%7Bj%7DX%7D%3D%28%7B%5E%7Bw%7D_%7Bi%7DT%20%5E%7B-1%7D%7D%20%29%5Ctimes%20%28%20%7B%5E%7Bw%7D_%7Bj%7DX%7D%29\" alt=\"https://latex.codecogs.com/svg.latex?{^{i}_{j}X}=({^{w}_{i}T ^{-1}} )\\times ( {^{w}_{j}X})\" />\n",
    "\n",
    "which is:\n",
    "\n",
    " <img src=\"https://latex.codecogs.com/svg.latex?%7B%5E%7Bi%7D_%7Bj%7DX%7D%3D%20%5Cbegin%7Bbmatrix%7D%20%7B%5E%7Bw%7D_%7Bi%7DR%7D%5ET%20%26%20-%7B%5E%7Bw%7D_%7Bi%7DR%7D%5ET%20%5Ctimes%20%28%7B%5E%7Bw%7D_%7Bi%7DP%7D%29%20%5C%5C%200%20%26%201%20%5Cend%7Bbmatrix%7D%20%5Ctimes%20%28%20%7B%5E%7Bw%7D_%7Bj%7DX%7D%29\" alt=\"https://latex.codecogs.com/svg.latex?{^{i}_{j}X}= \\begin{bmatrix}\n",
    "{^{w}_{i}R}^T &   -{^{w}_{i}R}^T \\times ({^{w}_{i}P})  \\\\ \n",
    "0 & 1  \n",
    "\\end{bmatrix} \\times ( {^{w}_{j}X})\" />\n",
    "\n",
    "\n"
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    "Just like what we did to bring point in the world coordinate to camera coordinate, we can do the same here:\n",
    "\n",
    "<img src=\"https://latex.codecogs.com/svg.latex?%7B%5E%7Bi%7D_%7Bj%7DR%7D%3D%28%7B%5E%7Bw%7D_%7Bi%7DR%20%5ET%7D%20%29%5Ctimes%20%28%20%7B%5E%7Bw%7D_%7Bj%7DR%7D%29\" alt=\"https://latex.codecogs.com/svg.latex?{^{i}_{j}R}=({^{w}_{i}R ^T} )\\times ( {^{w}_{j}R})\" />\n",
    "\n",
    "\n",
    "\n",
    "<img src=\"https://latex.codecogs.com/svg.latex?%5E%7Bi%7D_jP%3D%5E%7Bw%7D_iR%5ET%20%28%20%7B%5E%7Bw%7D_%7Bj%7DP%7D%20-%7B%5E%7Bw%7D_%7Bi%7DP%7D%20%29\" alt=\"https://latex.codecogs.com/svg.latex?^{i}_jP=^{w}_iR^T (  {^{w}_{j}P} -{^{w}_{i}P} ) \" />\n",
    "\n",
    "which gives us: \n",
    "\n",
    "\n",
    "\n",
    "<img src=\"https://latex.codecogs.com/svg.latex?%5Cbegin%7Bbmatrix%7D%20%5E%7Bi%7D_j%20x%20%5C%5C%20%5E%7Bi%7D_j%20y%20%5Cend%7Bbmatrix%7D%3D%5Cbegin%7Bbmatrix%7D%20cos%28i%5Ew%20%5Ctheta%20%29%20%26%20sin%28i%5Ew%20%5Ctheta%20%29%20%5C%5C%20-sin%28i%5Ew%20%5Ctheta%20%29%20%26%20cos%28i%5Ew%20%5Ctheta%29%20%5Cend%7Bbmatrix%7D%20%5Cbegin%7Bbmatrix%7D%20%5E%7Bw%7D_jx%20-%20%5E%7Bw%7D_ix%20%5C%5C%20%5E%7Bw%7D_jy%20-%20%5E%7Bw%7D_iy%20%5Cend%7Bbmatrix%7D\" alt=\"https://latex.codecogs.com/svg.latex?\\begin{bmatrix} ^{i}_j x \\\\  ^{i}_j y \\end{bmatrix}=\\begin{bmatrix} cos(i^w \\theta ) & sin(i^w \\theta ) \\\\ -sin(i^w \\theta ) & cos(i^w \\theta) \\end{bmatrix} \\begin{bmatrix} ^{w}_jx - ^{w}_ix \\\\  ^{w}_jy - ^{w}_iy \\end{bmatrix}\" />\n",
    "\n",
    "\n",
    "\n",
    "\n",
    "<img src=\"https://latex.codecogs.com/svg.latex?%5E%7Bi%7D_j%20%5Ctheta%20%3D%5E%7Bw%7D_j%5Ctheta%20-%7B%5E%7Bw%7D_%7Bi%7D%5Ctheta%7D\" alt=\"https://latex.codecogs.com/svg.latex?^{i}_j \\theta =^{w}_j\\theta -{^{w}_{i}\\theta}\" />"
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    "### **Sensor Measurements:**\n",
    "From the sensor (e.g., a laser scanner or camera), we obtain a measurement $ z_{ij} $, which represents the **observed relative pose** of $ \\textbf{X}_j $ with respect to $ \\textbf{X}_i $. This measurement typically includes relative position and orientation.\n",
    "\n",
    "\n"
   ]
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    "### State Vector\n",
    "The state vector $ \\mathbb{X} $ contains the 2D poses of all $ n $ nodes:\n",
    "\n",
    "$\n",
    "\\mathbb{X} = (x_1, y_1, \\theta_1, x_2, y_2, \\theta_2, \\dots, x_n, y_n, \\theta_n)\n",
    "$\n",
    "\n",
    "Each pose $ \\textbf{X}_i = (x_i, y_i, \\theta_i) $ represents the position and orientation of a node.\n",
    "\n",
    "---\n",
    "\n",
    "### **Error Function:**\n",
    "\n",
    "The error function compares the **predicted relative pose** $ h(\\textbf{X}_i, \\textbf{X}_j) $ (computed from odometry poses) to the **observed relative pose** $ z_{ij} $ (sensor measurement):\n",
    "\n",
    "$e_{ij} = z_{ij} - h(\\textbf{X}_i, \\textbf{X}_j)$\n",
    "\n",
    "\n",
    "$\n",
    "\\textbf{e}_{ij}(\\mathbb{X}) = \\textbf{e}_{ij}(\\textbf{X}_i, \\textbf{X}_j)\n",
    "$\n",
    "\n",
    "Where the components of $ \\textbf{e}_{ij} $ are given by:\n",
    "\n",
    "$\n",
    "\\textbf{e}_{ij}(\\textbf{X}_i, \\textbf{X}_j) = \n",
    "\\begin{bmatrix}\n",
    "^i_jx \\\\ \n",
    "^i_jy \\\\ \n",
    "^{i}_j \\theta \n",
    "\\end{bmatrix}\n",
    "=\n",
    "\\begin{bmatrix}\n",
    "\\cos(\\theta_i)(x_j - x_i) + \\sin(\\theta_i)(y_j - y_i) - z_x \\\\\n",
    "-\\sin(\\theta_i)(x_j - x_i) + \\cos(\\theta_i)(y_j - y_i) - z_y \\\\\n",
    "\\theta_j - \\theta_i - z_\\theta\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "Here:\n",
    "- $ z_x, z_y, z_\\theta $ are the observed relative pose measurements.\n",
    "- $ x_j, y_j, \\theta_j $ and $ x_i, y_i, \\theta_i $ are the current estimates of the poses.\n",
    "\n",
    "---\n",
    "\n"
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    "\n",
    "\n",
    "### **Linearization of the Error Function**\n",
    "The error function for an edge $ij$ is:\n",
    "\n",
    "$\n",
    "e_{ij} = z_{ij} - h(\\textbf{X}_i, \\textbf{X}_j)\n",
    "$\n",
    "\n",
    "Where:\n",
    "- $ z_{ij} = \\begin{bmatrix} z_x \\\\ z_y \\\\ z_\\theta \\end{bmatrix} $ is the observed relative pose.\n",
    "- $ h(\\textbf{X}_i, \\textbf{X}_j) $ is the predicted relative pose computed from the current estimate of $ \\textbf{X}_i $ and $ \\textbf{X}_j $:\n",
    "\n",
    "$\n",
    "h(\\textbf{X}_i, \\textbf{X}_j) =\n",
    "\\begin{bmatrix}\n",
    "\\cos(\\theta_i)(x_j - x_i) + \\sin(\\theta_i)(y_j - y_i) \\\\\n",
    "-\\sin(\\theta_i)(x_j - x_i) + \\cos(\\theta_i)(y_j - y_i) \\\\\n",
    "\\theta_j - \\theta_i\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "We linearize $ e_{ij} $ around the **current estimate** using a first-order Taylor expansion:\n",
    "\n",
    "$\n",
    "e_{ij} \\approx e_{ij}^k + J_i \\Delta \\textbf{X}_i + J_j \\Delta \\textbf{X}_j\n",
    "$\n",
    "\n",
    "Where:\n",
    "- $ e_{ij}^k = z_{ij} - h(\\textbf{X}_i^k, \\textbf{X}_j^k) $ is the error at the current estimate.\n",
    "- $ J_i = \\frac{\\partial e_{ij}}{\\partial \\textbf{X}_i} $ and $ J_j = \\frac{\\partial e_{ij}}{\\partial \\textbf{X}_j} $ are the Jacobians.\n",
    "\n",
    "---\n",
    "\n",
    "### **Jacobians: $ J_i $ and $ J_j $**\n",
    "The Jacobians are derived by taking the partial derivatives of $ e_{ij} $ with respect to $ \\textbf{X}_i $ and $ \\textbf{X}_j $. For the predicted relative pose $ h(\\textbf{X}_i, \\textbf{X}_j) $, we compute:\n",
    "\n",
    "#### 1. Jacobian with respect to $ \\textbf{X}_i $ ($ J_i $):\n",
    "\n",
    "$\n",
    "J_i =\n",
    "\\begin{bmatrix}\n",
    "-\\cos(\\theta_i) & -\\sin(\\theta_i) & -\\sin(\\theta_i)(x_j - x_i) + \\cos(\\theta_i)(y_j - y_i) \\\\\n",
    "\\sin(\\theta_i) & -\\cos(\\theta_i) & -\\cos(\\theta_i)(x_j - x_i) - \\sin(\\theta_i)(y_j - y_i) \\\\\n",
    "0 & 0 & -1\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "#### 2. Jacobian with respect to $ \\textbf{X}_j $ ($ J_j $):\n",
    "\n",
    "$\n",
    "J_j =\n",
    "\\begin{bmatrix}\n",
    "\\cos(\\theta_i) & \\sin(\\theta_i) & 0 \\\\\n",
    "-\\sin(\\theta_i) & \\cos(\\theta_i) & 0 \\\\\n",
    "0 & 0 & 1\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "---\n",
    "\n",
    "### **Information Matrix $H$ and Residual Vector $b$**\n",
    "We use the linearized error function in the least-squares formulation. The goal is to minimize:\n",
    "\n",
    "$\n",
    "E = \\frac{1}{2} \\sum_{ij} e_{ij}^T \\Omega_{ij} e_{ij}\n",
    "$\n",
    "\n",
    "Where:\n",
    "- $ \\Omega_{ij} $ is the **information matrix** (inverse of the measurement covariance matrix).\n",
    "\n",
    "#### Expanding $E$:\n",
    "Substituting the linearized error $ e_{ij} \\approx e_{ij}^k + J_i \\Delta \\textbf{X}_i + J_j \\Delta \\textbf{X}_j $:\n",
    "\n",
    "$\n",
    "E \\approx \\frac{1}{2} \\sum_{ij} \\left( e_{ij}^k + J_i \\Delta \\textbf{X}_i + J_j \\Delta \\textbf{X}_j \\right)^T \\Omega_{ij} \\left( e_{ij}^k + J_i \\Delta \\textbf{X}_i + J_j \\Delta \\textbf{X}_j \\right)\n",
    "$\n",
    "\n",
    "#### Quadratic Expansion:\n",
    "After expanding and keeping only the terms relevant to optimization:\n",
    "\n",
    "$\n",
    "E \\approx \\frac{1}{2} \\sum_{ij} \\left( (e_{ij}^k)^T \\Omega_{ij} e_{ij}^k + 2(e_{ij}^k)^T \\Omega_{ij} (J_i \\Delta \\textbf{X}_i + J_j \\Delta \\textbf{X}_j) + \\Delta \\textbf{X}^T H_{ij} \\Delta \\textbf{X} \\right)\n",
    "$\n",
    "\n",
    "Where $ H_{ij} = J_i^T \\Omega_{ij} J_i + J_j^T \\Omega_{ij} J_j + J_i^T \\Omega_{ij} J_j + J_j^T \\Omega_{ij} J_i $.\n",
    "\n",
    "#### Global $H$ and $b$:\n",
    "The contributions to the global $ H $ (information matrix) and $ b $ (residual vector) are:\n",
    "\n",
    "1. **Information Matrix $ H $:**\n",
    "   - Add contributions for all edges:\n",
    "     $\n",
    "     H = \\sum_{ij} J_i^T \\Omega_{ij} J_i + J_j^T \\Omega_{ij} J_j + J_i^T \\Omega_{ij} J_j + J_j^T \\Omega_{ij} J_i\n",
    "     $\n",
    "\n",
    "2. **Residual Vector $ b $:**\n",
    "   - Add contributions for all edges:\n",
    "     $\n",
    "     b = \\sum_{ij} -J_i^T \\Omega_{ij} e_{ij}^k - J_j^T \\Omega_{ij} e_{ij}^k\n",
    "     $\n",
    "\n",
    "---\n",
    "\n",
    "### Final System:\n",
    "The optimization problem becomes solving the linear system:\n",
    "\n",
    "$\n",
    "H \\Delta \\textbf{X} = b\n",
    "$\n",
    "\n",
    "Where:\n",
    "- $ \\Delta \\textbf{X} $ is the pose correction vector.\n",
    "- $ H $ is the global sparse information matrix.\n",
    "- $ b $ is the global residual vector.\n",
    "\n",
    "This system can be solved iteratively using methods like Gauss-Newton or Levenberg-Marquardt until the corrections $ \\Delta \\textbf{X} $ are sufficiently small."
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    "### Jacobians and Linearization\n",
    "To linearize the error function, we compute its Jacobian with respect to the poses $x_i$ and $x_j$:\n",
    "\n",
    "$\n",
    "e_{ij} \\approx e_{ij}(x_i^k, x_j^k) + J_i \\Delta x_i + J_j \\Delta x_j\n",
    "$\n",
    "\n",
    "Where:\n",
    "- $x_i^k$ and $x_j^k$ are the current estimates of the poses,\n",
    "- $\\Delta x_i$ and $\\Delta x_j$ are the pose corrections,\n",
    "- $J_i = \\frac{\\partial e_{ij}}{\\partial x_i}$ and $J_j = \\frac{\\partial e_{ij}}{\\partial x_j}$ are the Jacobians of the error function with respect to $x_i$ and $x_j$.\n",
    "\n",
    "### Building $H$ and $b$\n",
    "The contributions to the information matrix ($H$) and residual vector ($b$) are derived from the squared error:\n",
    "\n",
    "$\n",
    "e_{ij}^T \\Omega_{ij} e_{ij}\n",
    "$\n",
    "\n",
    "Where $\\Omega_{ij}$ is the information matrix for the edge $ij$ (inverse of the measurement covariance matrix).\n",
    "\n",
    "The contributions to $H$ and $b$ are computed as:\n",
    "\n",
    "1. **Residual Vector ($b$):**\n",
    "\n",
    "   $\n",
    "   b_i = - J_i^T \\Omega_{ij} e_{ij}\n",
    "   $\n",
    "   $\n",
    "   b_j = - J_j^T \\Omega_{ij} e_{ij}\n",
    "   $\n",
    "\n",
    "   These are summed into the global residual vector.\n",
    "\n",
    "2. **Information Matrix ($H$):**\n",
    "\n",
    "   $\n",
    "   H_{ii} = J_i^T \\Omega_{ij} J_i\n",
    "   $\n",
    "   $\n",
    "   H_{ij} = J_i^T \\Omega_{ij} J_j\n",
    "   $\n",
    "   $\n",
    "   H_{jj} = J_j^T \\Omega_{ij} J_j\n",
    "   $\n",
    "\n",
    "   These are summed into the global $H$, which is a sparse matrix.\n",
    "\n",
    "### Summary Equation\n",
    "The global least-squares problem becomes:\n",
    "\n",
    "$\n",
    "H \\Delta x = b\n",
    "$\n",
    "\n",
    "Where:\n",
    "- $H$ is the global information matrix,\n",
    "- $\\Delta x$ is the vector of pose corrections,\n",
    "- $b$ is the global residual vector.\n",
    "\n",
    "This system can be solved iteratively (e.g., using Gauss-Newton or Levenberg-Marquardt) to update the pose estimates and minimize the error."
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    "For this error, function, which is a function of 6 variables, $(x_i,y_i,\\theta_i,x_j,y_j,\\theta_j)$, the $J_{ij}$ element of the jacobian matrix is:\n",
    "\n",
    "$\\\\ J_{ij}=\\begin{bmatrix} 0 & ... & 0 & \\frac{\\partial ^{i}_jx}{ \\partial _{i} x}      &  \\frac{\\partial ^{i}_jx}{ \\partial _{i} y}  & \\frac{\\partial ^{i}_j x}{ \\partial _{i} \\theta} &0 &... & \\frac{\\partial ^{i}_jx}{ \\partial _{j} x}      &  \\frac{\\partial ^{i}_jx}{ \\partial _{j} y}  & \\frac{\\partial ^{i}_j x}{ \\partial _{j} \\theta}  &... &0\\\\ 0 & ... & 0 & \\frac{\\partial ^{i}_jy}{ \\partial _{i} x} &  \\frac{\\partial ^{i}_jy}{ \\partial _{i} y}  & \\frac{\\partial ^{i}_j y}{ \\partial _{i} \\theta} &0   &...& \\frac{\\partial ^{i}_jy}{ \\partial _{j} x} &  \\frac{\\partial ^{i}_jy}{ \\partial _{j} y}  & \\frac{\\partial ^{i}_j y}{ \\partial _{j} \\theta}& ...     & 0\\\\ 0 & ... & 0 & \\frac{\\partial ^{i}_j\\theta}{ \\partial _{i} x}      &  \\frac{\\partial ^{i}_j\\theta}{ \\partial _{i} y}  & \\frac{\\partial ^{i}_j \\theta}{ \\partial _{i} \\theta} &0 &...&\\frac{\\partial ^{i}_j\\theta}{ \\partial _{j} x}      &  \\frac{\\partial ^{i}_j\\theta}{ \\partial _{j} y}  & \\frac{\\partial ^{i}_j \\theta}{ \\partial _{j} \\theta} &...  & 0\\\\ \\end{bmatrix}$"
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    "$A_{i,j}=  \\frac{ \\partial   \\textbf{e}(\\textbf{X}_{i})}{  \\partial  \\textbf{X}_{j}  }      $\n",
    "\n",
    "\n",
    "\n",
    "$A_{ij}=\\begin{bmatrix}  \\frac{\\partial ^{i}_jx}{ \\partial _{i} x}      &  \\frac{\\partial ^{i}_jx}{ \\partial _{i} y}  & \\frac{\\partial ^{i}_j x}{ \\partial _{i} \\theta} \\\\ \\frac{\\partial ^{i}_jy}{ \\partial _{i} x} &  \\frac{\\partial ^{i}_jy}{ \\partial _{i} y}  & \\frac{\\partial ^{i}_j y}{ \\partial _{i} \\theta} \\\\ \\frac{\\partial ^{i}_j\\theta}{ \\partial _{i} x}      &  \\frac{\\partial ^{i}_j\\theta}{ \\partial _{i} y}  & \\frac{\\partial ^{i}_j \\theta}{ \\partial _{i} \\theta} \\\\ \\end{bmatrix}$\n",
    "\n",
    "\n",
    "$A_{i,j}=\\begin{bmatrix} -cos(\\theta_i) & -sin(\\theta_i)  & -sin(\\theta_i)(x_j-x_i)+cos(\\theta_i)(y_j-y_i)\\\\ sin(\\theta_i) & -cos(\\theta_i) & -cos(\\theta_i)(x_j-x_i) - sin(\\theta_i)(y_j-y_i)\\\\ 0 & 0 & -1 \\end{bmatrix}$"
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    "<img src=\"https://latex.codecogs.com/svg.latex?B_%7Bi%2Cj%7D%3D%5Cbegin%7Bbmatrix%7D%20cos%28%5Ctheta_i%29%20%26%20-sin%28%5Ctheta_i%29%20%26%200%5C%5C%20-sin%28%5Ctheta_i%29%20%26%20cos%28%5Ctheta_i%29%20%26%200%5C%5C%200%20%26%200%20%26%201%20%5Cend%7Bbmatrix%7D\" alt=\"https://latex.codecogs.com/svg.latex?B_{i,j}=\\begin{bmatrix} cos(\\theta_i) & -sin(\\theta_i)  & 0\\\\  -sin(\\theta_i) & cos(\\theta_i) & 0\\\\  0 & 0 & 1 \\end{bmatrix}\" />\n",
    "\n",
    "\n"
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    "\n",
    "### 1. **Error Function and its Variables**\n",
    "The error function depends only on the poses $(x_i, y_i, \\theta_i, x_j, y_j, \\theta_j)$. Since $z_x$, $z_y$, and $z_\\theta$ are constants (observed measurements), they do not influence the derivatives, and hence they do not appear in the Jacobian.\n",
    "\n",
    "Since the measurements $z_x, z_y, z_\\theta$ are constants and do not depend on the state variables $(x_i, y_i, \\theta_i, x_j, y_j, \\theta_j)$, they do not appear in the derivatives. This is correct, and the Jacobian depends solely on the current estimates of the state variables.\n",
    "\n",
    "\n",
    "### 2. **Jacobian Matrix Structure**\n",
    "The Jacobian $J_{ij}$ matrix provided correctly represents the structure of the derivatives of the error function with respect to the state variables. The detailed form:\n",
    "\n",
    "$\n",
    "J_{ij} = \n",
    "\\begin{bmatrix}\n",
    "0 & \\cdots & 0 & \\frac{\\partial e_{x}}{\\partial x_i} & \\frac{\\partial e_{x}}{\\partial y_i} & \\frac{\\partial e_{x}}{\\partial \\theta_i} & 0 & \\cdots & 0 & \\frac{\\partial e_{x}}{\\partial x_j} & \\frac{\\partial e_{x}}{\\partial y_j} & \\frac{\\partial e_{x}}{\\partial \\theta_j} & \\cdots & 0 \\\\\n",
    "0 & \\cdots & 0 & \\frac{\\partial e_{y}}{\\partial x_i} & \\frac{\\partial e_{y}}{\\partial y_i} & \\frac{\\partial e_{y}}{\\partial \\theta_i} & 0 & \\cdots & 0 & \\frac{\\partial e_{y}}{\\partial x_j} & \\frac{\\partial e_{y}}{\\partial y_j} & \\frac{\\partial e_{y}}{\\partial \\theta_j} & \\cdots & 0 \\\\\n",
    "0 & \\cdots & 0 & \\frac{\\partial e_{\\theta}}{\\partial x_i} & \\frac{\\partial e_{\\theta}}{\\partial y_i} & \\frac{\\partial e_{\\theta}}{\\partial \\theta_i} & 0 & \\cdots & 0 & \\frac{\\partial e_{\\theta}}{\\partial x_j} & \\frac{\\partial e_{\\theta}}{\\partial y_j} & \\frac{\\partial e_{\\theta}}{\\partial \\theta_j} & \\cdots & 0\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "### 3. **Jacobian Block Structure**\n",
    "The block $A_{ij}$ represents the partial derivatives of the error function with respect to the individual pose variables. Its structure is correctly described as:\n",
    "\n",
    "$\n",
    "A_{ij} =\n",
    "\\begin{bmatrix}\n",
    "\\frac{\\partial e_x}{\\partial x_i} & \\frac{\\partial e_x}{\\partial y_i} & \\frac{\\partial e_x}{\\partial \\theta_i} \\\\\n",
    "\\frac{\\partial e_y}{\\partial x_i} & \\frac{\\partial e_y}{\\partial y_i} & \\frac{\\partial e_y}{\\partial \\theta_i} \\\\\n",
    "\\frac{\\partial e_{\\theta}}{\\partial x_i} & \\frac{\\partial e_{\\theta}}{\\partial y_i} & \\frac{\\partial e_{\\theta}}{\\partial \\theta_i}\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "### 4. **Jacobian for $A_{i,j}$**\n",
    "The specific Jacobian block for $A_{i,j}$ (with respect to $\\textbf{X}_i$) is:\n",
    "\n",
    "$\n",
    "A_{i,j} = \n",
    "\\begin{bmatrix}\n",
    "-\\cos(\\theta_i) & -\\sin(\\theta_i) & -\\sin(\\theta_i)(x_j-x_i) + \\cos(\\theta_i)(y_j-y_i) \\\\\n",
    "\\sin(\\theta_i) & -\\cos(\\theta_i) & -\\cos(\\theta_i)(x_j-x_i) - \\sin(\\theta_i)(y_j-y_i) \\\\\n",
    "0 & 0 & -1\n",
    "\\end{bmatrix}\n",
    "$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "276a3064-560d-413e-a8bd-83beeb465385",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "$ B_{ij} = \\frac{\\partial \\textbf{e}_{ij}}{\\partial \\textbf{X}_j} $\n",
    "The error function is:\n",
    "\n",
    "$\n",
    "\\textbf{e}_{ij}(\\textbf{X}_i, \\textbf{X}_j) = \n",
    "\\begin{bmatrix}\n",
    "^i_jx \\\\\n",
    "^i_jy \\\\\n",
    "^i_j\\theta\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "Where:\n",
    "\n",
    "$\n",
    "^i_jx = \\cos(\\theta_i)(x_j - x_i) + \\sin(\\theta_i)(y_j - y_i)\n",
    "$\n",
    "$\n",
    "^i_jy = -\\sin(\\theta_i)(x_j - x_i) + \\cos(\\theta_i)(y_j - y_i)\n",
    "$\n",
    "$\n",
    "^i_j\\theta = \\theta_j - \\theta_i\n",
    "$\n",
    "\n",
    "#### Partial Derivatives for $ B_{ij} $\n",
    "The Jacobian with respect to $ \\textbf{X}_j = (x_j, y_j, \\theta_j) $ is:\n",
    "\n",
    "$\n",
    "B_{ij} = \\frac{\\partial \\textbf{e}_{ij}}{\\partial \\textbf{X}_j} =\n",
    "\\begin{bmatrix}\n",
    "\\frac{\\partial ^i_jx}{\\partial x_j} & \\frac{\\partial ^i_jx}{\\partial y_j} & \\frac{\\partial ^i_jx}{\\partial \\theta_j} \\\\\n",
    "\\frac{\\partial ^i_jy}{\\partial x_j} & \\frac{\\partial ^i_jy}{\\partial y_j} & \\frac{\\partial ^i_jy}{\\partial \\theta_j} \\\\\n",
    "\\frac{\\partial ^i_j\\theta}{\\partial x_j} & \\frac{\\partial ^i_j\\theta}{\\partial y_j} & \\frac{\\partial ^i_j\\theta}{\\partial \\theta_j}\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "#### Computing Each Partial Derivative\n",
    "1. For $ ^i_jx $:\n",
    "   $\n",
    "   \\frac{\\partial ^i_jx}{\\partial x_j} = \\cos(\\theta_i), \\quad\n",
    "   \\frac{\\partial ^i_jx}{\\partial y_j} = \\sin(\\theta_i), \\quad\n",
    "   \\frac{\\partial ^i_jx}{\\partial \\theta_j} = 0\n",
    "   $\n",
    "\n",
    "2. For $ ^i_jy $:\n",
    "   $\n",
    "   \\frac{\\partial ^i_jy}{\\partial x_j} = -\\sin(\\theta_i), \\quad\n",
    "   \\frac{\\partial ^i_jy}{\\partial y_j} = \\cos(\\theta_i), \\quad\n",
    "   \\frac{\\partial ^i_jy}{\\partial \\theta_j} = 0\n",
    "   $\n",
    "\n",
    "3. For $ ^i_j\\theta $:\n",
    "   $\n",
    "   \\frac{\\partial ^i_j\\theta}{\\partial x_j} = 0, \\quad\n",
    "   \\frac{\\partial ^i_j\\theta}{\\partial y_j} = 0, \\quad\n",
    "   \\frac{\\partial ^i_j\\theta}{\\partial \\theta_j} = 1\n",
    "   $\n",
    "\n",
    "#### Final Form of $ B_{ij} $\n",
    "$\n",
    "B_{ij} =\n",
    "\\begin{bmatrix}\n",
    "\\cos(\\theta_i) & \\sin(\\theta_i) & 0 \\\\\n",
    "-\\sin(\\theta_i) & \\cos(\\theta_i) & 0 \\\\\n",
    "0 & 0 & 1\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "54e448cd-4456-4caa-b575-549ade4f4156",
   "metadata": {},
   "source": [
    "### Jacobians for Linearization\n",
    "The error function is linearized around the current estimate using the Jacobians with respect to $ \\textbf{X}_i $ and $ \\textbf{X}_j $. The Jacobians are:\n",
    "\n",
    "$\n",
    "J_i = \\frac{\\partial \\textbf{e}_{ij}}{\\partial \\textbf{X}_i}, \\quad\n",
    "J_j = \\frac{\\partial \\textbf{e}_{ij}}{\\partial \\textbf{X}_j}\n",
    "$\n",
    "\n",
    "#### Jacobian with respect to $ \\textbf{X}_i $:\n",
    "$\n",
    "J_i =\n",
    "\\begin{bmatrix}\n",
    "-\\cos(\\theta_i) & -\\sin(\\theta_i) & -\\sin(\\theta_i)(x_j - x_i) + \\cos(\\theta_i)(y_j - y_i) \\\\\n",
    "\\sin(\\theta_i) & -\\cos(\\theta_i) & -\\cos(\\theta_i)(x_j - x_i) - \\sin(\\theta_i)(y_j - y_i) \\\\\n",
    "0 & 0 & -1\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "#### Jacobian with respect to $ \\textbf{X}_j $:\n",
    "$\n",
    "J_j =\n",
    "\\begin{bmatrix}\n",
    "\\cos(\\theta_i) & \\sin(\\theta_i) & 0 \\\\\n",
    "-\\sin(\\theta_i) & \\cos(\\theta_i) & 0 \\\\\n",
    "0 & 0 & 1\n",
    "\\end{bmatrix}\n",
    "$\n",
    "\n",
    "---\n",
    "\n",
    "### Building $ b $ and $ H $\n",
    "For each edge $ ij $, the contributions to the residual vector $ b $ and information matrix $ H $ are computed as:\n",
    "\n",
    "1. **Residual Vector Contribution ($ b $):**\n",
    "\n",
    "$\n",
    "b_i = - J_i^T \\Omega_{ij} \\textbf{e}_{ij}, \\quad b_j = - J_j^T \\Omega_{ij} \\textbf{e}_{ij}\n",
    "$\n",
    "\n",
    "Where $ \\Omega_{ij} $ is the information matrix for the measurement $ z_{ij} $ (inverse of its covariance).\n",
    "\n",
    "2. **Information Matrix Contribution ($ H $):**\n",
    "\n",
    "$\n",
    "H_{ii} = J_i^T \\Omega_{ij} J_i, \\quad\n",
    "H_{ij} = J_i^T \\Omega_{ij} J_j, \\quad\n",
    "H_{jj} = J_j^T \\Omega_{ij} J_j\n",
    "$\n",
    "\n",
    "These contributions are summed into the global matrix $ H $ and vector $ b $ for all edges.\n",
    "\n",
    "---\n",
    "\n",
    "### Final System\n",
    "After aggregating contributions from all edges, the optimization problem can be expressed as:\n",
    "\n",
    "$\n",
    "H \\Delta \\mathbb{X} = b\n",
    "$\n",
    "\n",
    "Where:\n",
    "- $ H $ is the global sparse information matrix,\n",
    "- $ \\Delta \\mathbb{X} $ is the correction to the current pose estimates,\n",
    "- $ b $ is the global residual vector.\n",
    "\n",
    "This system is solved iteratively (e.g., using Gauss-Newton or Levenberg-Marquardt) to update $ \\mathbb{X} $ and minimize the total error."
   ]
  },
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   "cell_type": "markdown",
   "id": "324caf15-e7ed-4139-b7a9-6d04d36cd781",
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   "source": [
    "Let’s explicitly derive the components of $ H_{ii} $, $ H_{ij} $, $ b_{ii} $, and $ b_{ij} $ for the linearized system.\n",
    "\n",
    "---\n",
    "\n",
    "### **From the Linearized Error Function**\n",
    "The linearized error function is:\n",
    "\n",
    "$\n",
    "e_{ij} \\approx e_{ij}^k + J_i \\Delta \\textbf{X}_i + J_j \\Delta \\textbf{X}_j\n",
    "$\n",
    "\n",
    "The energy function for the least-squares problem is:\n",
    "\n",
    "$\n",
    "E = \\frac{1}{2} \\sum_{ij} e_{ij}^T \\Omega_{ij} e_{ij}\n",
    "$\n",
    "\n",
    "Expanding $E$ using the linearized error:\n",
    "\n",
    "$\n",
    "E \\approx \\frac{1}{2} \\sum_{ij} \\left( e_{ij}^k + J_i \\Delta \\textbf{X}_i + J_j \\Delta \\textbf{X}_j \\right)^T \\Omega_{ij} \\left( e_{ij}^k + J_i \\Delta \\textbf{X}_i + J_j \\Delta \\textbf{X}_j \\right)\n",
    "$\n",
    "\n",
    "---\n",
    "\n",
    "### **Derivation of $ H_{ii} $, $ H_{ij} $, $ b_{ii} $, $ b_{ij} $**\n",
    "By collecting terms in the quadratic form, we get:\n",
    "\n",
    "1. **Information Matrix $H_{ii}$:**\n",
    "   Contribution from pose $ \\textbf{X}_i $ for edge $ ij $:\n",
    "   $\n",
    "   H_{ii} = J_i^T \\Omega_{ij} J_i\n",
    "   $\n",
    "\n",
    "2. **Information Matrix $H_{ij}$:**\n",
    "   Cross-term contribution between poses $ \\textbf{X}_i $ and $ \\textbf{X}_j $:\n",
    "   $\n",
    "   H_{ij} = J_i^T \\Omega_{ij} J_j\n",
    "   $\n",
    "\n",
    "   Note: $ H_{ij} $ is not diagonal—it represents the coupling between the two poses.\n",
    "\n",
    "3. **Residual Vector $b_i$:**\n",
    "   Contribution from pose $ \\textbf{X}_i $ to the residual vector:\n",
    "   $\n",
    "   b_i = -J_i^T \\Omega_{ij} e_{ij}^k\n",
    "   $\n",
    "\n",
    "4. **Residual Vector $b_j$:**\n",
    "   Contribution from pose $ \\textbf{X}_j $ to the residual vector:\n",
    "   $\n",
    "   b_j = -J_j^T \\Omega_{ij} e_{ij}^k\n",
    "   $\n",
    "\n",
    "---\n",
    "\n",
    "### **Explicit Components**\n",
    "\n",
    "Using the previously derived Jacobians for $ J_i $ and $ J_j $, we can compute $ H_{ii} $, $ H_{ij} $, $ b_i $, and $ b_j $.\n",
    "\n",
    "1. **$H_{ii}$:**\n",
    "   $\n",
    "   H_{ii} = J_i^T \\Omega_{ij} J_i\n",
    "   $\n",
    "   Substituting $ J_i $ (as derived earlier):\n",
    "   $\n",
    "   J_i =\n",
    "   \\begin{bmatrix}\n",
    "   -\\cos(\\theta_i) & -\\sin(\\theta_i) & -\\sin(\\theta_i)(x_j - x_i) + \\cos(\\theta_i)(y_j - y_i) \\\\\n",
    "   \\sin(\\theta_i) & -\\cos(\\theta_i) & -\\cos(\\theta_i)(x_j - x_i) - \\sin(\\theta_i)(y_j - y_i) \\\\\n",
    "   0 & 0 & -1\n",
    "   \\end{bmatrix}\n",
    "   $\n",
    "\n",
    "   Multiply $ J_i^T $ and $ J_i $, weighted by $ \\Omega_{ij} $.\n",
    "\n",
    "2. **$H_{ij}$:**\n",
    "   $\n",
    "   H_{ij} = J_i^T \\Omega_{ij} J_j\n",
    "   $\n",
    "   Substituting $ J_i $ and $ J_j $:\n",
    "   $\n",
    "   J_j =\n",
    "   \\begin{bmatrix}\n",
    "   \\cos(\\theta_i) & \\sin(\\theta_i) & 0 \\\\\n",
    "   -\\sin(\\theta_i) & \\cos(\\theta_i) & 0 \\\\\n",
    "   0 & 0 & 1\n",
    "   \\end{bmatrix}\n",
    "   $\n",
    "\n",
    "   Multiply $ J_i^T $ and $ J_j $, weighted by $ \\Omega_{ij} $.\n",
    "\n",
    "3. **$b_{i}$:**\n",
    "   $\n",
    "   b_{i} = -J_i^T \\Omega_{ij} e_{ij}^k\n",
    "   $\n",
    "   Substituting $ J_i $ and $ e_{ij}^k $:\n",
    "   $\n",
    "   e_{ij}^k = z_{ij} - h(\\textbf{X}_i, \\textbf{X}_j)\n",
    "   $\n",
    "\n",
    "4. **$b_{j}$:**\n",
    "   $\n",
    "   b_{j} = -J_j^T \\Omega_{ij} e_{ij}^k\n",
    "   $\n",
    "\n",
    "---\n",
    "\n",
    "### Summary of Key Terms\n",
    "\n",
    "- $H_{ii} = J_i^T \\Omega_{ij} J_i$\n",
    "- $H_{ij} = J_i^T \\Omega_{ij} J_j$\n",
    "- $b_{i} = -J_i^T \\Omega_{ij} e_{ij}^k$\n",
    "- $b_{j} = -J_j^T \\Omega_{ij} e_{ij}^k$\n",
    "\n",
    "These contributions are summed over all edges $ij$ in the graph to form the global $H$ and $b$:\n",
    "\n",
    "$\n",
    "H = \\sum_{ij} \\begin{bmatrix}\n",
    "H_{ii} & H_{ij} \\\\\n",
    "H_{ji} & H_{jj}\n",
    "\\end{bmatrix}, \\quad\n",
    "b = \\sum_{ij} \\begin{bmatrix}\n",
    "b_{i} \\\\\n",
    "b_{j}\n",
    "\\end{bmatrix}\n",
    "$"
   ]
  },
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   "source": [
    "\n",
    "\n",
    "<img src=\"images/J_i_j.png\"  width= \"50%\"  height= \"50%\"/>\n",
    "\n",
    "<br/>\n",
    "<br/>\n",
    "<img src=\"images/b_h.png\" width= \"50%\"  height= \"50%\"/>\n",
    "\n",
    "<br/>\n",
    "<br/>\n",
    "<img src=\"images/b_i_j_h_i_j.png\" width= \"50%\"  height= \"50%\" />\n",
    "\n",
    "<br/>\n",
    "<br/>\n",
    "<img src=\"images/building_the_linear_system.png\" width= \"50%\"  height= \"50%\" />"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "cbce0409-7775-4ee8-982b-d845199b63b3",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "iter 1 with delta = 9776.815836743128\n",
      "iter 2 with delta = 9921.583588625203\n",
      "iter 3 with delta = 5304.764805540707\n",
      "iter 4 with delta = 3618.3965580675276\n",
      "iter 5 with delta = 933.9939988930216\n",
      "iter 6 with delta = 659.25179621676\n",
      "iter 7 with delta = 98.17909009324742\n",
      "iter 8 with delta = 176.51470377625068\n",
      "iter 9 with delta = 23.082354732268218\n",
      "iter 10 with delta = 53.59284151969517\n",
      "iter 11 with delta = 13.98510759884786\n",
      "iter 12 with delta = 17.16245467982526\n",
      "iter 13 with delta = 7.564224626983518\n",
      "iter 14 with delta = 6.354698522871116\n",
      "iter 15 with delta = 3.803701239394619\n",
      "iter 16 with delta = 2.6231639339095367\n",
      "iter 17 with delta = 1.8327618063756572\n",
      "iter 18 with delta = 1.1540969135263155\n",
      "iter 19 with delta = 0.8626333855900739\n",
      "iter 20 with delta = 0.524695669824352\n",
      "iter 21 with delta = 0.40107997682030955\n",
      "iter 22 with delta = 0.2423091067959811\n",
      "iter 23 with delta = 0.1853608923816005\n",
      "iter 24 with delta = 0.11268047897148246\n",
      "iter 25 with delta = 0.08543853115972769\n",
      "iter 26 with delta = 0.052541143607246925\n",
      "iter 27 with delta = 0.03934757939849653\n",
      "iter 28 with delta = 0.024515809353727068\n",
      "iter 29 with delta = 0.01812215188892346\n",
      "iter 30 with delta = 0.011436433465005457\n",
      "iter 31 with delta = 0.008350601344209205\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
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   ],
   "source": [
    "# Cell 1: Imports and Configuration\n",
    "\n",
    "import os\n",
    "import matplotlib.pyplot as plt\n",
    "import math\n",
    "import numpy as np\n",
    "\n",
    "# Configuration\n",
    "#np.set_printoptions(suppress=True, formatter={'float_kind': '{:0.4f}'.format}, threshold=sys.maxsize)\n",
    "\n",
    "\n",
    "def getNode(nodes, id):\n",
    "    for node in nodes:\n",
    "\n",
    "        if node[\"id\"] == id:\n",
    "            return node\n",
    "\n",
    "# Cell 2: Helper Functions\n",
    "def wrap2pi(theta):\n",
    "    while theta > math.pi:\n",
    "        theta = theta - 2 * math.pi\n",
    "    while theta < -math.pi:\n",
    "        theta = theta + 2 * math.pi\n",
    "    return theta\n",
    "\n",
    "def rot2(theta):\n",
    "    return np.array([[np.cos(theta), -np.sin(theta)], \n",
    "                     [np.sin(theta), np.cos(theta)]])\n",
    "\n",
    "def update_node(node, dx):\n",
    "    node[\"state\"][0] += dx[0]\n",
    "    node[\"state\"][1] += dx[1]\n",
    "    node[\"state\"][2] = wrap2pi(node[\"state\"][2] + dx[2])\n",
    "    return node\n",
    "\n",
    "# Cell 3: load_2d_g2o Function\n",
    "def load_2d_g2o(filename):\n",
    "    with open(filename, 'r') as f:\n",
    "        lines = f.readlines()\n",
    "    \n",
    "    nodes = []\n",
    "    edges = []\n",
    "    for line in lines:\n",
    "        line = line.strip()\n",
    "        if line.startswith('VERTEX_SE2'):\n",
    "            parts = line.split()\n",
    "            node = {\n",
    "                'id': int(parts[1]),\n",
    "                'state': [float(parts[2]), float(parts[3]), float(parts[4])]\n",
    "            }\n",
    "            nodes.append(node)\n",
    "        elif line.startswith('EDGE_SE2'):\n",
    "            parts = line.split()\n",
    "            edge = {\n",
    "                'id1': int(parts[1]),\n",
    "                'id2': int(parts[2]),\n",
    "                'meas': [float(parts[3]), float(parts[4]), float(parts[5])],\n",
    "                'info': [[float(parts[6]), float(parts[7]), float(parts[8])],\n",
    "                         [float(parts[7]), float(parts[9]), float(parts[10])],\n",
    "                         [float(parts[8]), float(parts[10]), float(parts[11])]]\n",
    "            }\n",
    "            edges.append(edge)\n",
    "    return nodes, edges\n",
    "\n",
    "# Cell 4: Main Code Logic\n",
    "# Replace this with the path to your .g2o file\n",
    "g2o_file = os.path.abspath(\"/home/behnam/workspace/robotic_notes/data/slam/input_MITb_g2o.g2o\")\n",
    "nodes, edges = load_2d_g2o(g2o_file)\n",
    "\n",
    "# Plot initial nodes\n",
    "x_cords = [node[\"state\"][0] for node in nodes]\n",
    "y_cords = [node[\"state\"][1] for node in nodes]\n",
    "plt.plot(x_cords, y_cords, color=\"blue\")\n",
    "plt.xlabel('X')\n",
    "plt.ylabel('Y')\n",
    "plt.title('Nodes from g2o file')\n",
    "plt.grid(True)\n",
    "plt.show()\n",
    "\n",
    "# Optimization loop\n",
    "dx_norm = np.inf\n",
    "iteration = 0\n",
    "\n",
    "while dx_norm > 1e-2:\n",
    "    # for i in [1]:\n",
    "    # our matrices for Hx=-b\n",
    "    b = np.zeros([3*len(x_cords), 1])\n",
    "    H = np.zeros([3*len(x_cords), 3*len(x_cords)])\n",
    "\n",
    "    for edge in edges:\n",
    "\n",
    "        # idout\n",
    "        id1 = edge[\"id1\"]\n",
    "        # idin\n",
    "        id2 = edge[\"id2\"]\n",
    "\n",
    "        node1 = getNode(nodes, id1)\n",
    "        node2 = getNode(nodes, id2)\n",
    "\n",
    "        x1 = node1[\"state\"][0]\n",
    "        y1 = node1[\"state\"][1]\n",
    "        theta1 = node1[\"state\"][2]\n",
    "\n",
    "        x2 = node2[\"state\"][0]\n",
    "        y2 = node2[\"state\"][1]\n",
    "        theta2 = node2[\"state\"][2]\n",
    "\n",
    "        theta2_in_1 = wrap2pi(theta2-theta1)\n",
    "\n",
    "        # orientation error\n",
    "        err_theta = wrap2pi(wrap2pi(edge[\"meas\"][2]) - theta2_in_1)\n",
    "\n",
    "        # position error\n",
    "        R_1toG = rot2(theta1)\n",
    "\n",
    "        p_1inG = np.array([[x1], [y1]])\n",
    "        p_2inG = np.array([[x2], [y2]])\n",
    "\n",
    "        p_2in1 = R_1toG.T @ (p_2inG - p_1inG)\n",
    "\n",
    "        p_2in1_from_measurement = np.array(\n",
    "            [[edge[\"meas\"][0]], [edge[\"meas\"][1]]])\n",
    "\n",
    "        err_pos_x, err_pos_y = p_2in1_from_measurement - p_2in1\n",
    "\n",
    "        err_pos_x = err_pos_x.squeeze()\n",
    "        err_pos_y = err_pos_y.squeeze()\n",
    "\n",
    "        # 3x3d data on the file is I11 I12 I13 I22 I23 I33 but the matrix is symmetric so\n",
    "        info = np.array(edge[\"info\"])\n",
    "\n",
    "        # Jacobian of current relative in respect to NODE 1\n",
    "\n",
    "        A_i_j = np.array([[-np.cos(theta1), -np.sin(theta1), -np.sin(theta1)*(x2-x1) + np.cos(theta1)*(y2-y1)],\n",
    "                          [np.sin(theta1), -np.cos(theta1), -np.cos(theta1) *\n",
    "                           (x2-x1) - np.sin(theta1)*(y2-y1)],\n",
    "                          [0, 0, -1]])\n",
    "\n",
    "        # print(\"A_i_j:\", A_i_j)\n",
    "\n",
    "        # Jacobian of current relative in respect to NODE 2\n",
    "\n",
    "        B_i_j = np.array([[np.cos(theta1), np.sin(theta1), 0],\n",
    "                          [-np.sin(theta1), np.cos(theta1), 0], [0, 0, 1]])\n",
    "\n",
    "        # print(\"B_i_j:\", B_i_j)\n",
    "\n",
    "        # update our information\n",
    "\n",
    "        # H_i_i\n",
    "        H[3*id1:3*id1+3, 3*id1:3*id1+3] = H[3*id1:3 *\n",
    "                                            id1+3, 3*id1:3*id1+3] + A_i_j.T @ info@A_i_j\n",
    "\n",
    "        # print(\"H_i_i:\", H[3*id1:3*id1+3, 3*id1:3*id1+3])\n",
    "\n",
    "        # H_i_j\n",
    "        H[3*id1:3*id1+3, 3*id2:3*id2+3] = H[3*id1:3 *\n",
    "                                            id1+3, 3*id2:3*id2+3] + A_i_j.T @ info @ B_i_j\n",
    "\n",
    "        # print(\"H_i_j:\", H[3*id1:3*id1+3, 3*id2:3*id2+3])\n",
    "\n",
    "        # H_j_i\n",
    "        H[3*id2:3*id2+3, 3*id1: 3*id1+3] = H[3*id2:3 *\n",
    "                                             id2+3, 3*id1: 3*id1+3]+B_i_j.T@info@A_i_j\n",
    "\n",
    "        # print(\"H_j_i:\", H[3*id2:3*id2+3, 3*id1: 3*id1+3])\n",
    "\n",
    "        # H_j_j\n",
    "        H[3*id2:3*id2+3, 3*id2: 3*id2+3] = H[3*id2:3 *\n",
    "                                             id2+3, 3*id2: 3*id2+3]+B_i_j.T@info@B_i_j\n",
    "\n",
    "        # print(\"H_j_j:\", H[3*id2:3*id2+3, 3*id2: 3*id2+3])\n",
    "\n",
    "        # update our error terms\n",
    "        result = A_i_j.T @ info @ np.array([[err_pos_x],\n",
    "                                            [err_pos_y], [err_theta]])\n",
    "\n",
    "        # print(result)\n",
    "        # print(\"A_i_j:\\n\", A_i_j)\n",
    "\n",
    "        result = result.squeeze()\n",
    "\n",
    "        b[3*id1:3*id1+3, 0] = b[3*id1:3*id1+3, 0] + result\n",
    "\n",
    "        # print(\"b1\", b[3*id1:3*id1+3, 0])\n",
    "\n",
    "        result = B_i_j.T @ info @ np.array([[err_pos_x],\n",
    "                                           [err_pos_y], [err_theta]])\n",
    "\n",
    "        # print(\"B_i_j:\\n\", B_i_j)\n",
    "\n",
    "        result = result.squeeze()\n",
    "\n",
    " #       print(result)\n",
    "\n",
    "        b[3*id2:3*id2+3, 0] = b[3*id2:3*id2+3, 0] + result\n",
    "\n",
    "        # print(\"b2\", b[3*id2:3*id2+3, 0])\n",
    "        # print(\"err_pos_x,err_pos_y,err_theta\", err_pos_x, err_pos_y, err_theta)\n",
    "\n",
    "\n",
    "#  fix the first node to be known\n",
    "    H[0:3, 0:3] = H[0:3, 0:3] + 1e6*np.eye(3)\n",
    "    # solve the linear system\n",
    "    # x = H\\b\n",
    "    x = np.linalg.solve(H, b)\n",
    "\n",
    "    # print(x)\n",
    "    # print(x.shape)\n",
    "\n",
    "    dx_norm = np.linalg.norm(x)\n",
    "    iteration = iteration + 1\n",
    "    print(f'iter {iteration} with delta = {dx_norm}')\n",
    "\n",
    "    # update our nodes\n",
    "    for index, node in enumerate(nodes):\n",
    "        node = update_node(node, x[3*index:3*index+3, 0])\n",
    "\n",
    "\n",
    "x_cords = [node[\"state\"][0] for node in nodes]\n",
    "y_cords = [node[\"state\"][1] for node in nodes]\n",
    "theta = [node[\"state\"][2] for node in nodes]\n",
    "\n",
    "plt.plot(x_cords, y_cords, color=\"red\")\n",
    "# plt.xlabel('X')\n",
    "# plt.ylabel('Y')\n",
    "# plt.title('Nodes from g2o file')\n",
    "plt.grid(True)\n",
    "plt.show()"
   ]
  }
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