{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Concise and other forms of EoMs (Working Engineer Version)\n",
    "\n",
    "In the past two sections, a good Undergraduate level of description of the Concise and Non-Dimensional forms of the Equations of Motion was shown. Let's now present things slightly differently - using more nomenclature, just to fuck things up for you _properly_.\n",
    "\n",
    "We're also going to change the meaning of a couple of symbols until I'm smart enough to figure out a way to make this next section consistent with the last one (some large find/replace operation, I suspect).\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "tags": [
     "remove-input",
     "remove-output"
    ]
   },
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "from IPython.display import Math, Latex\n",
    "\n",
    "def printMatrix(matrix, label=None):\n",
    "    if label == None:\n",
    "        display(Math(sp.latex(sp.Matrix(matrix.round(6)))))\n",
    "    else:\n",
    "        display(Latex(label + '$' + sp.latex(sp.Matrix(matrix.round(6))) + '$'))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's think about where we ended up, in the preceding, the *Concise Form of the Longitudinal Equations of Motion* was given as {eq}`eq:conciselon`:\n",
    "\n",
    "$$\\begin{align}\\begin{bmatrix} \\dot{u}\\\\\\dot{w}\\\\\\dot{q}\\\\\\dot{\\theta}\\end{bmatrix} = \\begin{bmatrix}\n",
    "        X_u & X_w & 0 & -g\\cdot\\cos\\theta_0\\\\\n",
    "        Z_u & Z_w & U_0 & -g\\cdot\\sin\\theta_0\\\\\n",
    "        M_u^* & M_w^* & M_q^* & M_\\theta^*\\\\\n",
    "        0 & 0 & 1 & 0   \n",
    "    \\end{bmatrix}\\begin{bmatrix}\n",
    "        {u}\\\\{w}\\\\{q}\\\\{\\theta}         \n",
    "    \\end{bmatrix} + \\begin{bmatrix}\n",
    "        0\\\\Z_{\\delta_e}\\\\M_{\\delta_e}^*\\\\0          \n",
    "    \\end{bmatrix}\\left[\\delta_e\\right]\\end{align}$$\n",
    "$$\\dot{\\vec{x}} =  A\\vec{x} + B\\vec{u}$$\n",
    "\n",
    "Which is but _one_ means of presenting the equations of motion in state space formulation. *Concise* in this representation means (I think):\n",
    "- Each term in the A and B matrix is a single term\n",
    "- Each output is in real engineering units"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "celltoolbar": "Tags",
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}