rigid_body_accuracy.ipynb

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      "This is a rough draft of documentation for Gazebo's Physics Accuracy benchmarks.\n",
      "\n",
      "This document will describe simple motion of a rigid body for which analytical solutions can be computed.\n",
      "\n",
      "Consider a rigid body with mass $m$,\n",
      "body-fixed inertia matrix $\\textbf{I}$,\n",
      "center of mass (com) location $\\textbf{c}$,\n",
      "orientation quaternion $\\textbf{q}$,\n",
      "and body-fixed angular velocity $\\boldsymbol{\\omega}$.\n",
      "\n",
      "Let $\\textbf{R}(\\textbf{q})$ be the rotation matrix from an inertial frame to the coordinate frame given by $\\textbf{q}$.\n",
      "\n",
      "For time $t$, denote the initial conditions\n",
      "$\\textbf{c}(t=t_0) = \\textbf{c}_0$,\n",
      "$\\dot{\\textbf{c}}(t=t_0) = \\dot{\\textbf{c}}_0$,\n",
      "$\\textbf{q}(t=t_0) = \\textbf{q}_0$,\n",
      "$\\boldsymbol{\\omega}(t=t_0) = \\boldsymbol{\\omega}_0$.\n",
      "\n",
      "Let $\\textbf{p}(t)$ represent linear momentum as $\\textbf{p}(t) = m \\dot{\\textbf{c}}(t)$,\n",
      "with an initial value of $\\textbf{p}_0$.\n",
      "\n",
      "Let $\\textbf{H}(t)$ represent the angular momentum with respect to the center of mass, expressed in an inertial frame,\n",
      "as $\\textbf{H}(t) = \\textbf{R}^T(\\textbf{q}(t))\\textbf{I}\\boldsymbol{\\omega}(t)$,\n",
      "and an initial value of $\\textbf{H}_0$\n",
      "\n",
      "Let $T$ represent the kinetic energy as\n",
      "$T = \\frac{1}{2} m \\dot{\\textbf{c}}^T \\dot{\\textbf{c}} +\n",
      "     \\frac{1}{2} \\textbf{I} \\boldsymbol{\\omega}^T \\boldsymbol{\\omega}$,\n",
      "and an initial value of $T_0$\n",
      "\n",
      "1. In a gravity-free environment with no external forces, linear, angular momentum, and energy are conserved.\n",
      "\n",
      " * Linear momentum implies:\n",
      " $\\textbf{p}(t) = \\textbf{p}_0$,\n",
      " $\\dot{\\textbf{c}}(t) = \\dot{\\textbf{c}}_0$,\n",
      " $\\textbf{c}(t) = \\textbf{c}_0 + \\dot{\\textbf{c}}_0(t-t_0)$\n",
      "\n",
      " * Angular momentum implies:\n",
      " $\\textbf{H}(t) = \\textbf{H}_0$,\n",
      " $\\textbf{R}^T(\\textbf{q(t)})\\textbf{I}\\boldsymbol{\\omega}(t) =\n",
      "  \\textbf{R}^T(\\textbf{q}_0)\\textbf{I}\\boldsymbol{\\omega}_0$\n",
      "\n",
      "1. With gravity vector $\\textbf{g}$ acting:\n",
      "\n",
      " * Newton implies:\n",
      " $\\dot{\\textbf{p}}(t) = m \\textbf{g}$,\n",
      " $\\dot{\\textbf{c}}(t) = \\dot{\\textbf{c}}_0 + m \\textbf{g} (t-t_0)$,\n",
      " $\\textbf{c}(t) = \\textbf{c}_0 + \\dot{\\textbf{c}}_0(t-t_0) + \\frac{1}{2} m \\textbf{g} (t-t_0)^2$\n",
      "\n",
      " * Angular momentum: see above.\n",
      "\n",
      " * Energy conservation: define potential energy $V$ based on gravity direction."
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