{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Holonomic Robot\n",
    "\n",
    "![](pics/holonomic_robot.png)\n",
    "\n",
    "**need to fix formatting**\n",
    "\n",
    "Robots come in a variety of types and configurations: wheeled, tracked, legs, flying, etc. Common wheeled robots typically have two wheels (directly driven) with a caster wheel to make the robot stable. There are some without the caster wheel and employ a control system to keep them upright (inverted pendulum\n",
    "problem) and resemble a Segway scooter. All of these two wheeled robot are\n",
    "non-holonomic systems.\n",
    "\n",
    "A **nonholonomic system** in physics and mathematics is a system whose state depends on the path taken in order to achieve it. A car is a typical example of a nonholonomic system. It can occupy any location (x,y) in any orientation ($\\phi$), but the path from one location/orientation to another position/orientation is not always linear. Thus you have to parallel park.\n",
    "\n",
    "A **holonomic system** is not constrained by this. Thus it can move anywhere at will. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from __future__ import division\n",
    "from __future__ import print_function\n",
    "import numpy as np\n",
    "from numpy.linalg import norm \n",
    "from math import cos, sin, pi\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Soccer\n",
    "\n",
    "These types of robots are great for soccer, especially the goalie. Typical configuration are 3 or 4 motors. Omni wheels allow slip perpendicular to the axis."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Due to these constraints, a holonomic robot which could travel in any direction and immediately change its position and orientation is much more desirable. There are a variety of different wheels which make this type of robot possible such as mecanum or omni wheels.\n",
    "\n",
    "![](pics/omni_wheel.jpg)\n",
    "\n",
    "Omni wheels operate like standard wheels in that the force is produced normal to the motor's axis of rotation and as a result of friction. However, there are a series of smaller wheels which ring the main wheel and allow the wheel to slip in the direction of the motor rotational axis. Note that no force is produced parallel to the motor axis, just slippage."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Dynamics\n",
    "\n",
    "![](pics/omni_robot_cs.png)\n",
    "\n",
    "The dynamics for a holonomic robot with 4 omni directional wheels (can be derived using Euler-Largrange ($\\mathcal{L}$) which defines a system's kinectic ($T$) and potential ($V$) energies in relation to a set of generalized coordinates ($q$) and generalized forces ($Q$):\n",
    "\n",
    "\\begin{equation}\n",
    "    \\newcommand{\\dpar}[2]{\\frac{\\partial #1}{\\partial #2}}\n",
    "\\end{equation}\n",
    "\n",
    "\\begin{equation}\n",
    "\t\\mathcal{L}=T-V \\\\\n",
    "\t\\frac{d}{dt} \\left\\{ \\dpar{ \\mathcal{L} }{\\dot q} \\right\\} - \\dpar{ \\mathcal{L} }{q} = Q \\\\\n",
    "\tT = \\frac{1}{2}M v_w^2+ \\frac{1}{2}J \\dot \\psi^2 + \\frac{1}{2} J_w (\\dot \\theta_1^2 + \\dot \\theta_2^2 + \\dot \\theta_3^2 + \\dot \\theta_4^2) \\\\\n",
    "\tV = 0\n",
    "\\end{equation}\n",
    "\n",
    "However, the dynamics must be calculated from an inertial reference frame (${W}$) and take into account the rotating body frame dynamics (${B'}$). Now, assume the body frame is offset from the center of mass (CM) by $x_m$ and $y_m$ which compose a vector $r_m$. Thus the velocity of the robot in the rotating frame would be:\n",
    "\n",
    "\\begin{equation*}\n",
    "\tv_w = v_{B'} + \\dot \\psi \\times r_m \\\\\n",
    "\tv_w = v_{B'} +\n",
    "\t\\begin{bmatrix}\n",
    "\t\t0 & 0 & \\dot \\psi\n",
    "\t\\end{bmatrix}^T\n",
    "\t\\times\n",
    "\t\\begin{bmatrix}\n",
    "\t\tx_m & y_m & 0\n",
    "\t\\end{bmatrix}^T\n",
    "    =\n",
    "\t\\begin{bmatrix}\n",
    "\t\t\\dot x & \\dot y & 0\n",
    "\t\\end{bmatrix}^T +\n",
    "\t\\begin{bmatrix}\n",
    "\t\t-y_m \\dot \\psi & x_m \\dot \\psi & 0\n",
    "\t\\end{bmatrix}^T \\\\\n",
    "\tv_{B'} = \\begin{bmatrix}\n",
    "\t\t\\dot x & \\dot y & 0\n",
    "\t\\end{bmatrix}^T\n",
    "\\end{equation*}\n",
    "\n",
    "where $v_{B'}$ is the speed of the body frame. Now substituting that into\n",
    "the above kinetic energy equation $T$, we get:\n",
    "\n",
    "\\begin{equation}\n",
    "\tT = \\frac{1}{2}M( ( \\dot x - \\dot \\psi y )^2 + (\\dot y + \\dot \\psi x)^2)+ \\dots \\\\\n",
    "\tT = \\frac{1}{2}M( \\dot x^2 - 2 \\dot \\psi y_m \\dot x +\\dot \\psi^2 y_m^2 + \\dot y^2 + 2 \\dot \\psi x_m \\dot y + \\dot \\psi^2 x_m^2)+ \\frac{1}{2}J \\dot \\psi^2 + \\frac{1}{2} J_w (\\dot \\theta_1^2 + \\dot \\theta_2^2 + \\dot \\theta_3^2 + \\dot \\theta_4^2)  \\\\\n",
    "\t\\frac{d}{dt} \\left\\{ \\dpar{ \\mathcal{L} }{\\dot x} \\right\\} = M ( \\ddot x - \\ddot \\psi y - \\dot \\psi \\dot y ) \\hspace{1cm} \\dpar{ \\mathcal{L} }{x} = M(\\dot \\psi \\dot y + \\dot \\psi^2 x) \\\\\n",
    "\t\\frac{d}{dt} \\left\\{ \\dpar{ \\mathcal{L} }{\\dot y} \\right\\} = M (\\ddot y + \\ddot \\psi x + \\dot \\psi \\dot x) \\hspace{1cm} \\dpar{ \\mathcal{L} }{y} = M( -\\dot \\psi \\dot x + \\dot \\psi^2 y) \\\\\n",
    "\t\\frac{d}{dt} \\left\\{ \\dpar{ \\mathcal{L} }{\\dot \\psi} \\right\\} = J \\ddot \\psi \\hspace{1cm} \\dpar{ \\mathcal{L} }{\\phi} = 0 \\\\\n",
    "\t\\frac{d}{dt} \\left\\{ \\dpar{ \\mathcal{L} }{\\dot \\theta} \\right\\} = J_w \\sum \\limits_{i=1}^4 \\ddot \\theta_i \\hspace{1cm} \\dpar{ \\mathcal{L} }{\\theta} = 0\n",
    "\\end{equation}\n",
    "\n",
    "Now we make the following assumptions: ${B'}$ is coincident with\n",
    "${B}$, $x_m = 0$, $y_m = 0$, $\\dot x = v_x$,\n",
    "$\\dot y = v_y$\n",
    "\n",
    "\\begin{equation*}\n",
    "\tF_x = M (\\ddot x - 2 \\dot \\psi \\dot y ) \\\\\n",
    "\tF_y = M (\\ddot y + 2 \\dot \\psi \\dot x) \\\\\n",
    "\tT = J \\ddot \\psi \\\\\n",
    "\t\\tau_w = J_w \\ddot \\theta_1 \\hspace{1cm}\n",
    "\t\\tau_w = J_w \\ddot \\theta_2 \\hspace{1cm}\n",
    "\t\\tau_w = J_w \\ddot \\theta_3 \\hspace{1cm}\n",
    "\t\\tau_w = J_w \\ddot \\theta_4\n",
    "\\end{equation*}\n",
    "\n",
    "\n",
    "\\begin{equation*}\n",
    "\t\\begin{bmatrix}\n",
    "\t\tF_x \\\\\n",
    "\t\tF_y \\\\\n",
    "\t\tT\n",
    "\t\\end{bmatrix} =\n",
    "\t\\begin{bmatrix}\n",
    "\t\tM & 0 & 0 \\\\\n",
    "\t\t0 & M & 0 \\\\\n",
    "\t\t0 & 0 & J\n",
    "\t\\end{bmatrix}\n",
    "\t\\begin{bmatrix}\n",
    "\t\t\\ddot x \\\\\n",
    "\t\t\\ddot y \\\\\n",
    "\t\t\\ddot \\psi\n",
    "\t\\end{bmatrix} +\n",
    "\t\\begin{bmatrix}\n",
    "\t\t0 & -2M \\dot \\psi & 0 \\\\\n",
    "\t\t2M \\dot \\psi & 0 & 0 \\\\\n",
    "\t\t0 & 0 & 0\n",
    "\t\\end{bmatrix}\n",
    "\t\\begin{bmatrix}\n",
    "\t\t\\dot x \\\\\n",
    "\t\t\\dot y \\\\\n",
    "\t\t\\dot \\psi\n",
    "\t\\end{bmatrix}\n",
    "\t= \\mathcal{M} \\ddot X + \\mathcal{O} \\dot X = Q\n",
    "\\end{equation*}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## World Coordinates\n",
    "\n",
    "Now the dynamics derived so far are all in the body frame and we could stop\n",
    "here and develop a controller which performs velocity control. However, position\n",
    "control is more useful and a transform needs to be performed to move the\n",
    "velocities and accelerations into the world frame.\n",
    "\n",
    "\\begin{equation}\n",
    "\t\\dot X^W = R_B^W \\dot X^B \\\\\n",
    "\tR_B^W =\n",
    "\t\\begin{bmatrix}\n",
    "\t\t\\cos \\psi & \\sin \\psi & 0 \\\\\n",
    "\t\t-\\sin \\psi & \\cos \\psi & 0 \\\\\n",
    "\t\t0 & 0 & 1\n",
    "\t\\end{bmatrix} \\\\\n",
    "\t\\ddot X^W = \\dot R_B^W \\dot X^B + R_B^W \\ddot X^B \\\\\n",
    "\t\\dot R_B^W =\n",
    "\t\\begin{bmatrix}\n",
    "\t\t\\sin \\psi & -\\cos \\psi & 0 \\\\\n",
    "\t\t\\cos \\psi & \\sin \\psi & 0 \\\\\n",
    "\t\t0 & 0 & 1\n",
    "\t\\end{bmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Now, substituting this into the dynamics, gives dynamics in the world\n",
    "coordinate system of:\n",
    "\n",
    "\\begin{equation}\n",
    "\tF = \\mathcal{M} (\\dot R \\dot X + R \\ddot X )  + \\mathcal{O} R \\dot X \\\\\n",
    "\tF = \\mathcal{M} R \\ddot X + (\\mathcal{M}  \\dot R + \\mathcal{O} R) \\dot X\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## External Forces and Torques\n",
    "\n",
    "![](pics/omni_robot_force_cs.png)\n",
    "\n",
    "Now summing the forces into their body referenced $x$ and $y$ directions and the torque about the $z$ axis, gives us:\n",
    "\n",
    "\\begin{equation}\n",
    "    \\sum F_x=f_1 \\sin(\\phi) - f_2 \\sin(\\phi) - f_3 \\sin(\\phi) + f_4 \\sin(\\phi) \\\\\n",
    "\t\\sum F_y=f_1 \\cos(\\phi) + f_2 \\cos(\\phi) - f_3 \\cos(\\phi) - f_4 \\cos(\\phi) \\\\\n",
    "\t\\sum T=L(f_1+f_2+f_3+f_4)\n",
    "\\end{equation}\n",
    "\n",
    "Additionally, we can simplify this by assuming all of the angles are the same\n",
    "(e.g., $\\phi_1 = \\phi_2 = \\phi_3 = \\phi_4$) and can now put this into a\n",
    "matrix form:\n",
    "\n",
    "\\begin{equation}\n",
    "\t\\begin{bmatrix}\n",
    "\t\tF_x \\\\\n",
    "\t\tF_y \\\\\n",
    "\t\tT\n",
    "\t\\end{bmatrix} =\n",
    "\t\\begin{bmatrix}\n",
    "\t\t\\sin(\\phi) & 0 & 0 \\\\\n",
    "\t\t0 & \\cos(\\phi) & 0 \\\\\n",
    "\t\t0 & 0 & L\n",
    "\t\\end{bmatrix}\n",
    "\t\\begin{bmatrix}\n",
    "\t\t1 & -1 & -1 & 1\\\\\n",
    "\t\t1 & 1 & -1 & -1\\\\\n",
    "\t\t1 & 1 & 1& 1\n",
    "\t\\end{bmatrix}\n",
    "\t\\begin{bmatrix}\n",
    "\t\tf_1 \\\\\n",
    "\t\tf_2 \\\\\n",
    "\t\tf_3 \\\\\n",
    "\t\tf_4\n",
    "\t\\end{bmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "where $\\phi$ is again the angle of the motors, $f_i$ is the magnitude of the force produced by the motors, and $L$ is the radius of the robot.\n",
    "\n",
    "where $pinv()$ is defined as the pseudoinverse since $A(\\phi)$ is not a square matrix. Finally, substituting these into the original equation, we can calculate the torques given the desired accelerations.\n",
    "\n",
    "\\begin{equation}\n",
    "\t\\begin{bmatrix} \\tau_1 \\\\  \\tau_2 \\\\  \\tau_3 \\\\  \\tau_4 \\end{bmatrix} = \\frac {M r_w} {4}\n",
    "\t\\begin{bmatrix}\n",
    "\t\t-1 & 1 & 1 \\\\\n",
    "\t\t-1 & -1 & 1 \\\\\n",
    "\t\t1 & -1 & 1 \\\\\n",
    "\t\t1 & 1 & 1\n",
    "\t\\end{bmatrix}\n",
    "\t\\begin{bmatrix}\n",
    "\t\t\\frac{1}{\\sin(\\phi)} & 0 & 0 \\\\\n",
    "\t\t0 & \\frac{1}{\\cos(\\phi)} & 0 \\\\\n",
    "\t\t0 & 0 & \\frac{1}{2}\n",
    "\t\\end{bmatrix}\n",
    "\t\\begin{bmatrix}\n",
    "\t\ta_x \\\\\n",
    "\t\ta_y \\\\\n",
    "\t\tR \\dot \\omega\n",
    "\t\\end{bmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Now looking at this equation, we notice that $\\phi$ can not be equal to 0, 90, 180, 270, or 360 otherwise we get a singularity in the $A(\\phi)$ matrix. This however is not an issue in the real world, since the motors would occupy the same physical space and the robot would essentially only have 2 and not 4 motors."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Holonomic Robot Kinematics\n",
    "\n",
    "Number of equivalent motors for any direction under linear movement only, no rotational movement allowed.\n",
    "\n",
    "Now performing a similar exercise for what was done with the dynamics, looking\n",
    "at $coordinate$, the velocity of motor 1is given by\n",
    "$v_1 = -\\sin(\\phi) v_x + \\cos(\\phi) v_y + R \\omega$. Performing this for\n",
    "each wheel gives:\n",
    "\n",
    "\\begin{equation}\n",
    "\t\\begin{bmatrix}\n",
    "\t\tv_1 \\\\\n",
    "\t\tv_2 \\\\\n",
    "\t\tv_3 \\\\\n",
    "\t\tv_4\n",
    "\t\\end{bmatrix} =\n",
    "\t\\begin{bmatrix}\n",
    "\t\t-\\sin(\\phi)  & \\cos(\\phi) & L \\\\\n",
    "\t\t-\\sin(\\phi) & -\\cos(\\phi) & L \\\\\n",
    "\t\t \\sin(\\phi) & -\\cos(\\phi) & L \\\\\n",
    "\t\t \\sin(\\phi)  & \\cos(\\phi) & L\n",
    "\t\\end{bmatrix}\n",
    "\t\\begin{bmatrix}\n",
    "\t\tv_x \\\\\n",
    "\t\tv_y \\\\\n",
    "\t\t\\omega\n",
    "\t\\end{bmatrix} =\n",
    "\t\\begin{bmatrix}\n",
    "\t\t-1 & 1 & 1 \\\\\n",
    "\t\t-1 & -1 & 1 \\\\\n",
    "\t\t1 & -1 & 1 \\\\\n",
    "\t\t1 & 1 & 1\n",
    "\t\\end{bmatrix}\n",
    "\t\\begin{bmatrix}\n",
    "\t\t\\sin(\\phi) & 0 & 0 \\\\\n",
    "\t\t0 & \\cos(\\phi) & 0 \\\\\n",
    "\t\t0 & 0 & L\n",
    "\t\\end{bmatrix}\n",
    "\t\\begin{bmatrix}\n",
    "\t\tv_x \\\\\n",
    "\t\tv_y \\\\\n",
    "\t\t\\omega\n",
    "\t\\end{bmatrix}\n",
    "\\end{equation}\n",
    "\n",
    "Now setting $\\omega$ to zero and calculating only linear movement, we can\n",
    "determine the number of equivalent motors as shown in\n",
    "\\figref{fig:equivalent_motors}. For example, setting\n",
    "$\\phi$ to 30 $^\\circ$ (the red line in \\figref{fig:equivalent_motors})\n",
    "and traveling in the x direction only ($\\begin{bmatrix} v_x & v_y & \\omega \\end{bmatrix}^T = \\begin{bmatrix}1& 0 & 0 \\end{bmatrix}^T$),\n",
    "the above equation simplifies to $4 \\sin(30)$ or 2 equivalent motors.\n",
    "Repeating for the y direction results in $4 \\cos(30)$ or 3.46 equivalent\n",
    "motors.\n",
    "\n",
    "Now it is interesting to note that when $\\phi$ is set to 30 $^\\circ$,\n",
    "the robot has more equivalent motors when going forward or backwards, while a\n",
    "$\\phi$ of 60 $^\\circ$ provides more equivalent motors moving left or right.\n",
    "When the motors are are angled at 45 $^\\circ$, movement is clearly equally\n",
    "optimized for both forward/backwards and left/right ( $2 \\sin(45)$ is 2.83 motors)\n",
    "movement.\n",
    "\n",
    "\\figref{fig:equivalent_motors} tells us that no mater how the 4 motors are\n",
    "oriented in a realistic configuration, the robot will never have the equivalent\n",
    "use of all 4 motors. Movement in one direction or another can be optimized, but\n",
    "then a sacrifice is made in another direction. This fact is intuitively obvious.\n",
    "\n",
    "Another issue is these results are also ideal. This logic assumes that the wheels\n",
    "will not slip and have good traction in any orientation. Unfortunately real world\n",
    "results do not mimic this situation and the robot's performance will be reduced.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equvilent Motors\n",
    "\n",
    "Since this robot has 4 motors with omni wheels, in certain configurations, you can get more power/speed in certain directions than others. Let's look at a configuration with the wheels oriented at the $\\phi$ angles of: 30, 45, and 60 degrees."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def motors(phi, angles):\n",
    "    \"\"\"\n",
    "    in:\n",
    "        phi - orientation of motors\n",
    "        angles - array of angles from 0 ... 2*pi\n",
    "    out:\n",
    "        array of results\n",
    "    \"\"\"\n",
    "    phi = phi*pi/180\n",
    "    ans = []\n",
    "    for angle in angles:\n",
    "        a = np.array([\n",
    "                [-sin(phi),  cos(phi), 1.0],\n",
    "                [-sin(phi), -cos(phi), 1.0],\n",
    "                [ sin(phi), -cos(phi), 1.0],\n",
    "                [ sin(phi),  cos(phi), 1.0]\n",
    "            ])\n",
    "        b = np.array([\n",
    "                cos(angle),\n",
    "                sin(angle),\n",
    "                0.0\n",
    "            ])\n",
    "        v = sum(abs(a.dot(b)))\n",
    "        ans.append(v)\n",
    "    return ans"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "theta = np.arange(0,2*pi,0.1)\n",
    "em30 = motors(30, theta)\n",
    "em45 = motors(45, theta)\n",
    "em60 = motors(60, theta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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gmDFjaDDjQFlLOAFZljnhwATCCxy1c1S23hpNJhO3bdvGf54LahMcHMybN2+q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Hw5baxShfn+/PSTXpNva2nUqJFaJpP5kgOMeZYUuwsg6csBnEKhCOnWrZt0S9njx1mY2XLliObP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\nIEdeTk5ODr/66ivWqFFDkSQpD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      "text/plain": [
       "<matplotlib.figure.Figure at 0x10dccc8d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.polar(theta, em30, label='30');\n",
    "plt.polar(theta, em45, label='45');\n",
    "plt.polar(theta, em60, label='60');\n",
    "plt.grid(True)\n",
    "plt.legend(loc='upper left');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As seen above, 30 and 60 degrees orientations favor certain directions. However, 45 degrees is better in all directions and all around movement."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Control\n",
    "\n",
    "![](pics/omni_wheel_control.png)\n",
    "\n",
    "Using the equations above to transform desired movement into motor actions, the robot can move in any direction. A basic summery of robot movement to motor direction is shown above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "# References\n",
    "\n",
    "---\n",
    "\n",
    "* Alexander Gloye, Raul Rojas, Holonomic Control of a Robot with an Omnidirectional Drive, accepted for publication by Künstliche Intelligenz, Springer-Verlag, 2006.\n",
    "\n",
    "* http://en.wikipedia.org/wiki/Non-holonomic_system\n",
    "\n",
    "* http://en.wikipedia.org/wiki/Lagrangian_mechanics\n",
    "\n",
    "* http://www.kornylak.com\n",
    "\n",
    "* R. Balakrishna, Ashitava Ghosal, \"Modeling of Slip for Wheeled Mobile Robots,\" lEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, VOL. I I , NO. I , FEBRUARY 1995, pp. 126-132\n",
    "\n",
    "* J. Agullo, S. Cardona, and J. Vivancos, “Kinematics of vehicles with directional sliding wheels,” Mechanisms and Muchine Theory, vol. 22, no. 4, pp. 295-301, 1987.\n",
    "\n",
    "* Pseudoinverse: for m > n: $A_{left}^{-1}=(A^TA)^{-1}A^T$ or m < n: $A_{right}^{-1} = A^T(AA^T)^{-1}$ such that $AA^{-1}=I$ or $A^{-1}A=I$\n",
    "\n",
    "* Masayoshi Wada (2010). Motion Control of a Four-wheel-drive Omnidirectional Wheelchair with High Step Climbing Capability, Climbing and Walking Robots, Behnam Miripour (Ed.), ISBN: 978-953-307-030-8, [InTech](http://www.intechopen.com/books/climbing-and-walking-robots/motion-control-of-a-four-wheeldrive-omnidirectional-wheelchair-with-high-step-climbing-capability)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-----------\n",
    "\n",
    "<a rel=\"license\" href=\"http://creativecommons.org/licenses/by-sa/4.0/\"><img alt=\"Creative Commons License\" style=\"border-width:0\" src=\"https://i.creativecommons.org/l/by-sa/4.0/88x31.png\" /></a><br />This work is licensed under a <a rel=\"license\" href=\"http://creativecommons.org/licenses/by-sa/4.0/\">Creative Commons Attribution-ShareAlike 4.0 International License</a>."
   ]
  }
 ],
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