reference/Sympy_monoslam_jacobians_1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Distoration model \n",
    "as in [24] the model (same for matlab and opencv calibration tools)\n",
    "\n",
    "we have distoration function which is:\n",
    "\n",
    "$$ h_d(p) = p * L(r_u) + \\epsilon $$\n",
    "\n",
    "where:\n",
    "\n",
    "- $p$ is the point (2D)\n",
    "\n",
    "- $L(r_u) = 1 + k_1*r_u**2 + k_2*r_u**4 + k_3*r_u**6$\n",
    "\n",
    "- $r_u**2 = p[0]**2 + p[1]**2$\n",
    "\n",
    "- $\\epsilon = \\left[ p1(r_u**2+2u**2)+2p_2uv;p2(r_u**2+2v**2)+2p_1uv \\right]$\n",
    "\n",
    "- $ p_1 , p_2 $ is the tangial conffecitionts"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now , we want to claculate $$ \\frac{\\partial(h_d)}{\\partial(p)} $$ \n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing(use_latex=True)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "u, v = sympy.symbols('u v')\n",
    "r_u2 = u**2 + v**2;\n",
    "p = sympy.Matrix([u,v])\n",
    "\n",
    "p1 ,p2 = sympy.symbols('p1 p2')\n",
    "\n",
    "epslon = sympy.Matrix([p2*(r_u2+2*v**2)+2*p1*u*v,p1*(r_u2+2*u**2)+2*p2*u*v])\n",
    "\n",
    "k1 ,k2 ,k3 = sympy.symbols(' k_1 k_2 k_3 ')\n",
    "L_u = 1+k1*r_u2 + k2*r_u2**2 + k3*r_u2**3;\n",
    "\n",
    "h_d = p*L_u + epslon;\n",
    "\n",
    "jac_dist = h_d.jacobian(p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and that's it \n",
    "\n",
    "**TODO**: what was the general form of jacobian matrix to matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
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TS0aDUa6SQ6tTrxIwoj60mLlkrCe2WGfhaT+8Xv4O7FgewgMlHZxLy2PfxoSl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\nQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}k_{1} \\left(u^{2} + v^{2}\\right) + k_{2} \\left(u^{2} + v^{2}\\right)^{2} + k_{3} \\left(u^{2} + v^{2}\\right)^{3} + 2 p_{1} v + 2 p_{2} u + u \\left(2 k_{1} u + 4 k_{2} u \\left(u^{2} + v^{2}\\right) + 6 k_{3} u \\left(u^{2} + v^{2}\\right)^{2}\\right) + 1 & 2 p_{1} u + 6 p_{2} v + u \\left(2 k_{1} v + 4 k_{2} v \\left(u^{2} + v^{2}\\right) + 6 k_{3} v \\left(u^{2} + v^{2}\\right)^{2}\\right)\\\\6 p_{1} u + 2 p_{2} v + v \\left(2 k_{1} u + 4 k_{2} u \\left(u^{2} + v^{2}\\right) + 6 k_{3} u \\left(u^{2} + v^{2}\\right)^{2}\\right) & k_{1} \\left(u^{2} + v^{2}\\right) + k_{2} \\left(u^{2} + v^{2}\\right)^{2} + k_{3} \\left(u^{2} + v^{2}\\right)^{3} + 2 p_{1} v + 2 p_{2} u + v \\left(2 k_{1} v + 4 k_{2} v \\left(u^{2} + v^{2}\\right) + 6 k_{3} v \\left(u^{2} + v^{2}\\right)^{2}\\right) + 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                           2               3                       ⎛         \n",
       "⎢   ⎛ 2    2⎞      ⎛ 2    2⎞       ⎛ 2    2⎞                        ⎜         \n",
       "⎢k₁⋅⎝u  + v ⎠ + k₂⋅⎝u  + v ⎠  + k₃⋅⎝u  + v ⎠  + 2⋅p₁⋅v + 2⋅p₂⋅u + u⋅⎝2⋅k₁⋅u + \n",
       "⎢                                                                             \n",
       "⎢                                             ⎛                               \n",
       "⎢                                             ⎜                ⎛ 2    2⎞      \n",
       "⎣                         6⋅p₁⋅u + 2⋅p₂⋅v + v⋅⎝2⋅k₁⋅u + 4⋅k₂⋅u⋅⎝u  + v ⎠ + 6⋅k\n",
       "\n",
       "                                   2⎞                                         \n",
       "       ⎛ 2    2⎞          ⎛ 2    2⎞ ⎟                                         \n",
       "4⋅k₂⋅u⋅⎝u  + v ⎠ + 6⋅k₃⋅u⋅⎝u  + v ⎠ ⎠ + 1                           2⋅p₁⋅u + 6\n",
       "                                                                              \n",
       "             2⎞                                                       2       \n",
       "    ⎛ 2    2⎞ ⎟                               ⎛ 2    2⎞      ⎛ 2    2⎞       ⎛\n",
       "₃⋅u⋅⎝u  + v ⎠ ⎠                            k₁⋅⎝u  + v ⎠ + k₂⋅⎝u  + v ⎠  + k₃⋅⎝\n",
       "\n",
       "          ⎛                                            2⎞                     \n",
       "          ⎜                ⎛ 2    2⎞          ⎛ 2    2⎞ ⎟                     \n",
       "⋅p₂⋅v + u⋅⎝2⋅k₁⋅v + 4⋅k₂⋅v⋅⎝u  + v ⎠ + 6⋅k₃⋅v⋅⎝u  + v ⎠ ⎠                     \n",
       "                                                                              \n",
       "        3                       ⎛                                            2\n",
       " 2    2⎞                        ⎜                ⎛ 2    2⎞          ⎛ 2    2⎞ \n",
       "u  + v ⎠  + 2⋅p₁⋅v + 2⋅p₂⋅u + v⋅⎝2⋅k₁⋅v + 4⋅k₂⋅v⋅⎝u  + v ⎠ + 6⋅k₃⋅v⋅⎝u  + v ⎠ \n",
       "\n",
       "     ⎤\n",
       "     ⎥\n",
       "     ⎥\n",
       "     ⎥\n",
       "⎞    ⎥\n",
       "⎟    ⎥\n",
       "⎠ + 1⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "jac_dist"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Prediction State Model (Covariance Prediction):\n",
    "\n",
    "For That we need to calculate the Jacobians , all are strightforward but the hardest to calculte are here to find with the help of **Sympy**\n",
    "\n",
    "- The First is: Jacobian of quatrinan predicted to quatrian :\n",
    "\n",
    "$$ \\frac{\\partial q_{k+1}}{\\partial q_k} = \\frac{\\partial(q_k \\times h)}{\\partial q_k}  $$ \n",
    "\n",
    "- The Second is: Jacobian of q to speed Omega:\n",
    "\n",
    "$$ \\frac{\\partial q_{k+1}}{\\partial \\omega } = \\frac{\\partial q_{k+1}}{\\partial h} \\cdot \\frac{\\partial h}{\\partial \\omega }$$ \n",
    "\n",
    "the term $\\frac{\\partial h}{\\partial \\omega }$ can be transformed as:\n",
    "\n",
    "$$ \\frac{\\partial h}{\\partial \\omega } = \\Delta t \\frac{\\partial q(\\alpha)}{\\partial \\alpha }$$\n",
    "\n",
    "Of course :\n",
    "\n",
    "$$h = quat(\\omega_k * \\Delta t)$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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WAwdqR8BY2kfjswPaljZbCnmUbHKFX1Rsa1FnZJuog3Hi8heTdoRNlyBWDpcy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\nRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- wx \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )} & - wy \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )} & - wz \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}\\\\\\frac{wx^{2} \\cos{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} - \\frac{2 wx^{2} \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{\\left(wx^{2} + wy^{2} + wz^{2}\\right)^{2}} + \\frac{\\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} & \\frac{wx wy \\cos{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} - \\frac{2 wx wy}{\\left(wx^{2} + wy^{2} + wz^{2}\\right)^{2}} \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )} & \\frac{wx wz \\cos{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} - \\frac{2 wx wz}{\\left(wx^{2} + wy^{2} + wz^{2}\\right)^{2}} \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}\\\\\\frac{wx wy \\cos{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} - \\frac{2 wx wy}{\\left(wx^{2} + wy^{2} + wz^{2}\\right)^{2}} \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )} & \\frac{wy^{2} \\cos{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} - \\frac{2 wy^{2} \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{\\left(wx^{2} + wy^{2} + wz^{2}\\right)^{2}} + \\frac{\\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} & \\frac{wy wz \\cos{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} - \\frac{2 wy wz}{\\left(wx^{2} + wy^{2} + wz^{2}\\right)^{2}} \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}\\\\\\frac{wx wz \\cos{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} - \\frac{2 wx wz}{\\left(wx^{2} + wy^{2} + wz^{2}\\right)^{2}} \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )} & \\frac{wy wz \\cos{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} - \\frac{2 wy wz}{\\left(wx^{2} + wy^{2} + wz^{2}\\right)^{2}} \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )} & \\frac{wz^{2} \\cos{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}} - \\frac{2 wz^{2} \\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{\\left(wx^{2} + wy^{2} + wz^{2}\\right)^{2}} + \\frac{\\sin{\\left (\\frac{wx^{2}}{2} + \\frac{wy^{2}}{2} + \\frac{wz^{2}}{2} \\right )}}{wx^{2} + wy^{2} + wz^{2}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                                 ⎛  2     2     2⎞                           \n",
       "⎢                                 ⎜wx    wy    wz ⎟                           \n",
       "⎢                          -wx⋅sin⎜─── + ─── + ───⎟                           \n",
       "⎢                                 ⎝ 2     2     2 ⎠                           \n",
       "⎢                                                                             \n",
       "⎢       ⎛  2     2     2⎞            ⎛  2     2     2⎞      ⎛  2     2     2⎞ \n",
       "⎢  2    ⎜wx    wy    wz ⎟       2    ⎜wx    wy    wz ⎟      ⎜wx    wy    wz ⎟ \n",
       "⎢wx ⋅cos⎜─── + ─── + ───⎟   2⋅wx ⋅sin⎜─── + ─── + ───⎟   sin⎜─── + ─── + ───⎟ \n",
       "⎢       ⎝ 2     2     2 ⎠            ⎝ 2     2     2 ⎠      ⎝ 2     2     2 ⎠ \n",
       "⎢──────────────────────── - ────────────────────────── + ──────────────────── \n",
       "⎢      2     2     2                             2           2     2     2    \n",
       "⎢    wx  + wy  + wz             ⎛  2     2     2⎞          wx  + wy  + wz     \n",
       "⎢                               ⎝wx  + wy  + wz ⎠                             \n",
       "⎢                                                                             \n",
       "⎢                  ⎛  2     2     2⎞              ⎛  2     2     2⎞           \n",
       "⎢                  ⎜wx    wy    wz ⎟              ⎜wx    wy    wz ⎟           \n",
       "⎢         wx⋅wy⋅cos⎜─── + ─── + ───⎟   2⋅wx⋅wy⋅sin⎜─── + ─── + ───⎟           \n",
       "⎢                  ⎝ 2     2     2 ⎠              ⎝ 2     2     2 ⎠           \n",
       "⎢         ────────────────────────── - ────────────────────────────           \n",
       "⎢                2     2     2                               2                \n",
       "⎢              wx  + wy  + wz               ⎛  2     2     2⎞                 \n",
       "⎢                                           ⎝wx  + wy  + wz ⎠                 \n",
       "⎢                                                                             \n",
       "⎢                  ⎛  2     2     2⎞              ⎛  2     2     2⎞           \n",
       "⎢                  ⎜wx    wy    wz ⎟              ⎜wx    wy    wz ⎟           \n",
       "⎢         wx⋅wz⋅cos⎜─── + ─── + ───⎟   2⋅wx⋅wz⋅sin⎜─── + ─── + ───⎟           \n",
       "⎢                  ⎝ 2     2     2 ⎠              ⎝ 2     2     2 ⎠           \n",
       "⎢         ────────────────────────── - ────────────────────────────           \n",
       "⎢                2     2     2                               2                \n",
       "⎢              wx  + wy  + wz               ⎛  2     2     2⎞                 \n",
       "⎣                                           ⎝wx  + wy  + wz ⎠                 \n",
       "\n",
       "                                  ⎛  2     2     2⎞                           \n",
       "                                  ⎜wx    wy    wz ⎟                           \n",
       "                           -wy⋅sin⎜─── + ─── + ───⎟                           \n",
       "                                  ⎝ 2     2     2 ⎠                           \n",
       "                                                                              \n",
       "                   ⎛  2     2     2⎞              ⎛  2     2     2⎞           \n",
       "                   ⎜wx    wy    wz ⎟              ⎜wx    wy    wz ⎟           \n",
       "          wx⋅wy⋅cos⎜─── + ─── + ───⎟   2⋅wx⋅wy⋅sin⎜─── + ─── + ───⎟           \n",
       "                   ⎝ 2     2     2 ⎠              ⎝ 2     2     2 ⎠           \n",
       "          ────────────────────────── - ────────────────────────────           \n",
       "                 2     2     2                               2                \n",
       "               wx  + wy  + wz               ⎛  2     2     2⎞                 \n",
       "                                            ⎝wx  + wy  + wz ⎠                 \n",
       "                                                                              \n",
       "        ⎛  2     2     2⎞            ⎛  2     2     2⎞      ⎛  2     2     2⎞ \n",
       "   2    ⎜wx    wy    wz ⎟       2    ⎜wx    wy    wz ⎟      ⎜wx    wy    wz ⎟ \n",
       " wy ⋅cos⎜─── + ─── + ───⎟   2⋅wy ⋅sin⎜─── + ─── + ───⎟   sin⎜─── + ─── + ───⎟ \n",
       "        ⎝ 2     2     2 ⎠            ⎝ 2     2     2 ⎠      ⎝ 2     2     2 ⎠ \n",
       " ──────────────────────── - ────────────────────────── + ──────────────────── \n",
       "       2     2     2                             2           2     2     2    \n",
       "     wx  + wy  + wz             ⎛  2     2     2⎞          wx  + wy  + wz     \n",
       "                                ⎝wx  + wy  + wz ⎠                             \n",
       "                                                                              \n",
       "                   ⎛  2     2     2⎞              ⎛  2     2     2⎞           \n",
       "                   ⎜wx    wy    wz ⎟              ⎜wx    wy    wz ⎟           \n",
       "          wy⋅wz⋅cos⎜─── + ─── + ───⎟   2⋅wy⋅wz⋅sin⎜─── + ─── + ───⎟           \n",
       "                   ⎝ 2     2     2 ⎠              ⎝ 2     2     2 ⎠           \n",
       "          ────────────────────────── - ────────────────────────────           \n",
       "                 2     2     2                               2                \n",
       "               wx  + wy  + wz               ⎛  2     2     2⎞                 \n",
       "                                            ⎝wx  + wy  + wz ⎠                 \n",
       "\n",
       "                                  ⎛  2     2     2⎞                          ⎤\n",
       "                                  ⎜wx    wy    wz ⎟                          ⎥\n",
       "                           -wz⋅sin⎜─── + ─── + ───⎟                          ⎥\n",
       "                                  ⎝ 2     2     2 ⎠                          ⎥\n",
       "                                                                             ⎥\n",
       "                   ⎛  2     2     2⎞              ⎛  2     2     2⎞          ⎥\n",
       "                   ⎜wx    wy    wz ⎟              ⎜wx    wy    wz ⎟          ⎥\n",
       "          wx⋅wz⋅cos⎜─── + ─── + ───⎟   2⋅wx⋅wz⋅sin⎜─── + ─── + ───⎟          ⎥\n",
       "                   ⎝ 2     2     2 ⎠              ⎝ 2     2     2 ⎠          ⎥\n",
       "          ────────────────────────── - ────────────────────────────          ⎥\n",
       "                 2     2     2                               2               ⎥\n",
       "               wx  + wy  + wz               ⎛  2     2     2⎞                ⎥\n",
       "                                            ⎝wx  + wy  + wz ⎠                ⎥\n",
       "                                                                             ⎥\n",
       "                   ⎛  2     2     2⎞              ⎛  2     2     2⎞          ⎥\n",
       "                   ⎜wx    wy    wz ⎟              ⎜wx    wy    wz ⎟          ⎥\n",
       "          wy⋅wz⋅cos⎜─── + ─── + ───⎟   2⋅wy⋅wz⋅sin⎜─── + ─── + ───⎟          ⎥\n",
       "                   ⎝ 2     2     2 ⎠              ⎝ 2     2     2 ⎠          ⎥\n",
       "          ────────────────────────── - ────────────────────────────          ⎥\n",
       "                 2     2     2                               2               ⎥\n",
       "               wx  + wy  + wz               ⎛  2     2     2⎞                ⎥\n",
       "                                            ⎝wx  + wy  + wz ⎠                ⎥\n",
       "                                                                             ⎥\n",
       "        ⎛  2     2     2⎞            ⎛  2     2     2⎞      ⎛  2     2     2⎞⎥\n",
       "   2    ⎜wx    wy    wz ⎟       2    ⎜wx    wy    wz ⎟      ⎜wx    wy    wz ⎟⎥\n",
       " wz ⋅cos⎜─── + ─── + ───⎟   2⋅wz ⋅sin⎜─── + ─── + ───⎟   sin⎜─── + ─── + ───⎟⎥\n",
       "        ⎝ 2     2     2 ⎠            ⎝ 2     2     2 ⎠      ⎝ 2     2     2 ⎠⎥\n",
       " ──────────────────────── - ────────────────────────── + ────────────────────⎥\n",
       "       2     2     2                             2           2     2     2   ⎥\n",
       "     wx  + wy  + wz             ⎛  2     2     2⎞          wx  + wy  + wz    ⎥\n",
       "                                ⎝wx  + wy  + wz ⎠                            ⎦"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "wx,wy,wz = sympy.symbols('wx wy wz')\n",
    "dT = sympy.symbols('dT')\n",
    "alpha = wx**2+wz**2+wy**2\n",
    "qr = sympy.cos(alpha/2)\n",
    "some_factor = sympy.sin(alpha/2)/alpha;\n",
    "quat = sympy.Matrix([qr,some_factor*wx,some_factor*wy,some_factor*wz])\n",
    "w = sympy.Matrix([wx,wy,wz])\n",
    "quat.jacobian(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's for the second term of the second jacobian."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inverse_depth to XYZ Jacobian:\n",
    "\n",
    "here we want to convert inverse depth encoding to ecludian one,\n",
    "\n",
    "$$\\psi = [x^c \\  y^c \\ z^c \\  \\theta \\ \\phi \\ \\rho] $$\n",
    "\n",
    "to ecludian : $y$,\n",
    "\n",
    "and we have to compute \n",
    "$$\\frac{\\partial{y}}{\\partial{\\psi}}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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J/yaQXWtXmGhJ2YJK2poXJqy25oV8N4gWpiDTl6/2GjSua07JRcyaqVgoCAM8rPGiFDhe\n8W+yg0E/g9UhotUneNLfu9AD2tiSRUqgr3lhwhq92hC5Tt0pTF/fchqG5CJhzVAsmMIAU8FghOSZ\n9Pct9GCgJUULam7NCwNWW/NClBrg1fchBW/4+pbTMCQXCWtJpOvdO5aQYCj7pD+/vgQ9tksCBqvw\ngd+YXH7Kzqj2XiFUkxamiF7yXCFxyCqpiAdlzbGEBHcy0VFdGZtHUW6FXCm9vVuFD+rt3eEXv91L\nvWLpFGlhCiJfSGxXSUU8KGuOJSS4kz1RV8a2HarvRPKXIkH48DJw8H1J9GM7J3uv5FiYwrucBofM\nKfq2NShrwsT/oagP+woSFKziWOMWSSxx63QnCB8Y1eXn7hWrdY2dSLswMr6Q2M4pinhA1qSJ/+gS\nEZ+GUX/6Vn1uRsURvVLLC8KHGFX08/QqoEb9+kKK7VHEzJplbWZtBM7sYBP/IvRwhA8VoUZrM183\nO/tpMX6PYbCJfxF8OMKHilCxNvOsGHkpw2AT/2IXwxE+DAdVHIRsmO/Ktv4tlQkfciEMBzXXRcnC\nppLt1tesKuFDvtdKUf8FNY8lBwjRsN0AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0 & \\frac{1}{\\rho} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} & - \\frac{1}{\\rho} \\sin{\\left (\\phi \\right )} \\sin{\\left (\\theta \\right )} & - \\frac{1}{\\rho^{2}} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )}\\\\0 & 1 & 0 & - \\frac{1}{\\rho} \\cos{\\left (\\theta \\right )} & 0 & \\frac{1}{\\rho^{2}} \\sin{\\left (\\theta \\right )}\\\\0 & 0 & 1 & - \\frac{1}{\\rho} \\sin{\\left (\\theta \\right )} \\cos{\\left (\\phi \\right )} & - \\frac{1}{\\rho} \\sin{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )} & - \\frac{1}{\\rho^{2}} \\cos{\\left (\\phi \\right )} \\cos{\\left (\\theta \\right )}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡          cos(φ)⋅cos(θ)   -sin(φ)⋅sin(θ)   -sin(θ)⋅cos(φ) ⎤\n",
       "⎢1  0  0   ─────────────   ───────────────  ───────────────⎥\n",
       "⎢                ρ                ρ                 2      ⎥\n",
       "⎢                                                  ρ       ⎥\n",
       "⎢                                                          ⎥\n",
       "⎢            -cos(θ)                            sin(θ)     ⎥\n",
       "⎢0  1  0     ────────             0             ──────     ⎥\n",
       "⎢               ρ                                  2       ⎥\n",
       "⎢                                                 ρ        ⎥\n",
       "⎢                                                          ⎥\n",
       "⎢         -sin(θ)⋅cos(φ)   -sin(φ)⋅cos(θ)   -cos(φ)⋅cos(θ) ⎥\n",
       "⎢0  0  1  ───────────────  ───────────────  ───────────────⎥\n",
       "⎢                ρ                ρ                 2      ⎥\n",
       "⎣                                                  ρ       ⎦"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x ,y, z, theta, phi, rho = sympy.symbols('x y z theta phi rho')\n",
    "r = sympy.Matrix([x,y,z])\n",
    "psi = sympy.Matrix([x,y,z,theta,phi,rho])\n",
    "s ,c = sympy.sin , sympy.cos \n",
    "n = sympy.Matrix([s(theta)*c(phi),-1*s(theta),c(theta)*c(phi)])\n",
    "y = r.transpose() + n.transpose()/rho;\n",
    "y.jacobian(psi)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Normalize Quantrion:\n",
    "\n",
    "after prediction , we must normalize the quantrion vector , and add jacobin to covariance which contain:\n",
    "\n",
    "$$\\frac{norm(q)}{\\partial q}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy\n",
    "sympy.init_printing(use_latex=True)\n",
    "\n",
    "qr ,qx, qy, qz = sympy.symbols('qr qx qy qz')\n",
    "q = sympy.Matrix([qr ,qx, qy, qz])\n",
    "nq = q.norm()\n",
    "norm_q = ( qr**2+ qx**2+ qy**2+ qz**2)**0.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "nq1 = sympy.Matrix([nq])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{\\Re{qr} \\frac{d}{d qr} \\Re{qr} + \\Im{qr} \\frac{d}{d qr} \\Im{qr}}{\\sqrt{\\left|{qr}\\right|^{2} + \\left|{qx}\\right|^{2} + \\left|{qy}\\right|^{2} + \\left|{qz}\\right|^{2}}} & \\frac{\\Re{qx} \\frac{d}{d qx} \\Re{qx} + \\Im{qx} \\frac{d}{d qx} \\Im{qx}}{\\sqrt{\\left|{qr}\\right|^{2} + \\left|{qx}\\right|^{2} + \\left|{qy}\\right|^{2} + \\left|{qz}\\right|^{2}}} & \\frac{\\Re{qy} \\frac{d}{d qy} \\Re{qy} + \\Im{qy} \\frac{d}{d qy} \\Im{qy}}{\\sqrt{\\left|{qr}\\right|^{2} + \\left|{qx}\\right|^{2} + \\left|{qy}\\right|^{2} + \\left|{qz}\\right|^{2}}} & \\frac{\\Re{qz} \\frac{d}{d qz} \\Re{qz} + \\Im{qz} \\frac{d}{d qz} \\Im{qz}}{\\sqrt{\\left|{qr}\\right|^{2} + \\left|{qx}\\right|^{2} + \\left|{qy}\\right|^{2} + \\left|{qz}\\right|^{2}}}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡        d                    d                   d                    d      \n",
       "⎢re(qr)⋅───(re(qr)) + im(qr)⋅───(im(qr))  re(qx)⋅───(re(qx)) + im(qx)⋅───(im(q\n",
       "⎢       dqr                  dqr                 dqx                  dqx     \n",
       "⎢───────────────────────────────────────  ────────────────────────────────────\n",
       "⎢      _______________________________          ______________________________\n",
       "⎢     ╱     2       2       2       2          ╱     2       2       2       2\n",
       "⎣   ╲╱  │qr│  + │qx│  + │qy│  + │qz│         ╲╱  │qr│  + │qx│  + │qy│  + │qz│ \n",
       "\n",
       "             d                    d                   d                    d  \n",
       "x))  re(qy)⋅───(re(qy)) + im(qy)⋅───(im(qy))  re(qz)⋅───(re(qz)) + im(qz)⋅───(\n",
       "            dqy                  dqy                 dqz                  dqz \n",
       "───  ───────────────────────────────────────  ────────────────────────────────\n",
       "_          _______________________________          __________________________\n",
       "          ╱     2       2       2       2          ╱     2       2       2    \n",
       "        ╲╱  │qr│  + │qx│  + │qy│  + │qz│         ╲╱  │qr│  + │qx│  + │qy│  + │\n",
       "\n",
       "       ⎤\n",
       "im(qz))⎥\n",
       "       ⎥\n",
       "───────⎥\n",
       "_____  ⎥\n",
       "   2   ⎥\n",
       "qz│    ⎦"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nq1.jacobian(q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "not sure what that means, TODO!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mesurement Function Jacobian:\n",
    "\n",
    "I mean :\n",
    "\n",
    "$$ H = \\frac{\\partial h(\\mu)}{\\partial \\mu}  = \\frac{\\partial h(h_c)}{\\partial h_c} * \\frac{\\partial h_c}{\\partial \\mu}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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