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    "# DOF discussion of quadruped\n",
    "\n",
    "## Formula\n",
    "With basic mechanism formula\n",
    "\n",
    "$$\\sigma = 6m - \\sum_{i=1}^n p_i$$\n",
    "\n",
    "where\n",
    "\n",
    "$\\sigma$, DOF\n",
    "$m$, number of movable parts (links)\n",
    "$p_i$, number of degree constrained by $i$-th kinematic pair(joint or contact).\n",
    "$n$, total number of kinematic pairs\n",
    "\n",
    "In special,\n",
    "rotation joint, $p_i = 5$\n",
    "point contact, $p_i = 3$\n",
    "\n",
    "## Case study\n",
    "### DOF of quadruped base body with 4 legs on the ground.\n",
    "\n",
    "$m = 13$\n",
    "number of rotation joint($p_i = 5$), $n_1 = 12$\n",
    "number of point contact($p_i = 3$), $n_2 = 4$ \n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\begin{split}\n",
    "\\sigma \n",
    "&= 6 \\cdot 13 - 5 \\cdot 12 - 3 \\cdot4\\\\\n",
    "&=6\n",
    "\\end{split}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "### DOF of quadruped base body with 3 legs on the ground.\n",
    "\n",
    "$m = 10$\n",
    "number of rotation joint($p_i = 5$), $n_1 = 9$\n",
    "number of point contact($p_i = 3$), $n_2 = 3$ \n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\begin{split}\n",
    "\\sigma \n",
    "&= 6 \\cdot 10 - 5 \\cdot 9 - 3 \\cdot3\\\\\n",
    "&=6\n",
    "\\end{split}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "\n",
    "### DOF of quadruped base body with 2 legs on the ground.\n",
    "\n",
    "$m = 7$\n",
    "number of rotation joint($p_i = 5$), $n_1 = 6$\n",
    "number of point contact($p_i = 3$), $n_2 = 2$ \n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\begin{split}\n",
    "\\sigma \n",
    "&= 6 \\cdot 7 - 5 \\cdot 6 - 3 \\cdot2\\\\\n",
    "&=6\n",
    "\\end{split}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "### DOF of quadruped base body with 1 legs on the ground.\n",
    "\n",
    "$m = 4$\n",
    "number of rotation joint($p_i = 5$), $n_1 = 3$\n",
    "number of point contact($p_i = 3$), $n_2 = 1$ \n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\begin{split}\n",
    "\\sigma \n",
    "&= 6 \\cdot 4 - 5 \\cdot 3 - 3 \\cdot1\\\\\n",
    "&=6\n",
    "\\end{split}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "### DOF of quadruped base body with 0 legs on the ground.\n",
    "\n",
    "$m = 1$\n",
    "number of rotation joint($p_i = 5$), $n_1 = 0$\n",
    "number of point contact($p_i = 3$), $n_2 = 0$ \n",
    "\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\begin{split}\n",
    "\\sigma \n",
    "&= 6 \\cdot 1 - 5 \\cdot 0 - 3 \\cdot0\\\\\n",
    "&=6\n",
    "\\end{split}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "## Is it over-actuated? Is it under-actuated?\n",
    "\n",
    "Quadruped with **0 leg** on the ground: **under-actuated**\n",
    "Quadruped with **1 leg** on the ground: **under-actuated**\n",
    "Quadruped with **2 cross legs** on the ground: **under-actuated**\n",
    "\n",
    "There are 6 actuator by two Ab/ad \n",
    "Quadruped with **3 legs** on the ground: **over-actuated but NOT redundancy** \n",
    "Think 3 legs case as 2 cross legs with 1 more leg. \n",
    "After 2 cross legs configuration is determined, the orientation and position of base body is determined. Then the location that connect to 3rd leg is determined. If ground is in reachable domain, $\\ast$for certain ground contact point, the configuration of 3rd leg has to be determined. \n",
    "\n",
    "So, in this case, even we have 9 actuator with 6DOF, actuators are not independent and there isn't redundancy.\n",
    "\n",
    "Quadruped with **4 legs** on the ground: **over-actuated but NOT redundancy** \n",
    "Same reason like 3 legs\n",
    "\n",
    "\n",
    "$\\ast$ Notice: **certain ground contact point**, in the background of stand phase, the contact point is fixed and we are controlling the orientation or position of base body."
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