update ipynb doc.

Author: AtsushiSakai

Localization/particle_filter/particle_filter.ipynb

@@ -9,7 +9,7 @@
     }
    },
    "source": [
-    "# Particle Filter\n",
+    "# Particle Filter Localization\n",
     "\n"
    ]
   },
@@ -18,11 +18,21 @@
    "source": [
     "## How to calculate covariance matrix from particles\n",
     "\n",
+    "The covariance matrix $\\Xi$ from particle information is calculated by the following equation: \n",
+    "\n",
     "$\\Xi_{j,k}=\\frac{1}{1-\\sum^N_{i=1}(w^i)^2}\\sum^N_{i=1}w^i(x^i_j-\\mu_j)(x^i_k-\\mu_k)$\n",
     "\n",
+    "- $\\Xi_{j,k}$ is covariance matrix element at row $i$ and column $k$.\n",
+    "\n",
+    "- $w^i$ is the weight of the $i$ th particle. \n",
+    "\n",
+    "- $x^i_j$ is the $j$ th state of the $i$ th particle. \n",
+    "\n",
+    "- $\\mu_j$ is the $j$ th mean state of particles.\n",
+    "\n",
     "Ref:\n",
     "\n",
-    "- [Improving the particle filter in high dimensions using conjugate artificial process noise](https://www.visiondummy.com/2014/04/draw-error-ellipse-representing-covariance-matrix/)\n"
+    "- [Improving the particle filter in high dimensions using conjugate artificial process noise](https://arxiv.org/pdf/1801.07000.pdf)\n"
    ],
    "metadata": {
     "collapsed": false