{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <center>Modeling and Simulation of the skyTran MagLev Personal Rapid Transit Vehicle</center>\n",
    "#### <center> - Abhinav Kamath</center>\n",
    "#### <center> Professor: Dr. Jason Moore</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Motivation\n",
    "\n",
    "___skyTran___ is an elevated transportation infrastructure that utilizes a patented method of passive magnetic levitation (MagLev). Unlike other forms of passive MagLev, the _Bogie_ reacts against vertical conductive surfaces. The _Bogie_ is the driving portion of the vehicle; the passenger-carrying portion hangs below the track and is called the _Pod_. This design enables the vehicle to control its altitude in software. A strong simulation environment is critical to developing control algorithms without putting hardware at risk. \n",
    "\n",
    "__The objective of this project is to model and simulate the vehicle, and to then determine a favorable maximum levitation height by analyzing the stability of the vehicle system under different maximum-height levitation regimes.__"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Media/skyTran-vehicle.png\" width=\"750\"/>\n",
    "\n",
    "##### <center>Figure 1: A render of the skyTran MagLev vehicle. Credit: skyTran</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. Background\n",
    "\n",
    "Conventional trains leverage the friction between wheels and rails to drive forward. In contrast, a MagLev vehicle levitates on a guideway, replacing wheels with electromagnets. Levitation is advantageous in numerous ways, such as high speeds, reduction in maintenance costs, reduced noise and vibration levels, quick acceleration and deceleration, and so on. This technology also facilitates emission-free mass transportation.\n",
    "\n",
    "Though the concept of MagLev trains was introduced in the 1930s, the\n",
    "technology reached maturity only in the 1980s. The first practical\n",
    "public MagLev train service was launched in Shanghai in 2003. Advances\n",
    "in modern technology have enabled full-scale implementation of MagLev\n",
    "trains, which are currently operational in many countries. NASA has\n",
    "investigated the concept of Magnetic Launch Assist, which makes use of\n",
    "both magnetic levitation and magnetic propulsion to propel launch\n",
    "vehicles to 183 meters per second in less than 10 seconds.\n",
    "\n",
    "*skyTran* is a novel take on the personal rapid transit (PRT)\n",
    "infrastructure, making use of a network of lightweight *Pods* that run\n",
    "on elevated guideways, thus facilitating transportation of people and\n",
    "freight from one place to another without the need for intermediate\n",
    "stops."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. Problem Formulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Media/SDS3_Image_A.JPG\" width=\"750\"/>\n",
    "\n",
    "##### <center>Figure 2: A CAD-model of the Split-Bogie configuration of the vehicle. Credit: skyTran</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.1 The Vehicle\n",
    "\n",
    "The vehicle system that has been modeled, simulated and analyzed, is the *Bogie*, which is a cart-like rectangular vehicle with a rotary bearing connecting the front and rear chassis shown in figure 2. It has four wheels, one at each corner of the body, and four magnetic *wings*, which, at a high-level, function like an airfoil/fixed-wing, with the ability to effect a lift force on the vehicle. The vehicle is propelled using an axial motor belt and is centered using a set of urethane wheels that roll against the reaction plates, which are shown in figure 3."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Media/views.png\" width=\"350\"/>\n",
    "\n",
    "##### <center>Figure 3: Principal views of the vehicle. Credit: skyTran</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 3.2 Dynamics\n",
    "\n",
    "The vehicle is subject to five important classes of force & torque\n",
    "inputs:\n",
    "\n",
    "#### 3.2.1 Magnetic *Wing*: Magnetic Lift and Drag Forces\n",
    "\n",
    "The vehicle has four *wings*, located at the corners of the rectangular\n",
    "*Bogie* body. A simplified black-box model of the *wing* as shown in\n",
    "figure 4 will be considered for the purpose of this project.\n",
    "\n",
    "The forces effected by the *wing* arise from the interaction of eddy\n",
    "currents that are induced by the motion of the wing, with the conductive\n",
    "track plate (figure 4)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Media/magwing.png\" width=\"400\"/>\n",
    "\n",
    "##### <center>Figure 4: The Magnetic Wing. Credit: skyTran</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 3.2.2 *Wing* Actuators: Pitching Moment\n",
    "\n",
    "The *wing* actuators cause pitching moments between the chassis and the\n",
    "*wing*.\n",
    "\n",
    "#### 3.2.3 Wheels: Normal Force\n",
    "\n",
    "The vehicle has four wheels, located beneath the *wings*. They are\n",
    "active before *lift-off*; *lift-off* is achieved when vehicle reaches\n",
    "critical velocity.\n",
    "\n",
    "#### 3.2.4 Belt: Drive Force\n",
    "\n",
    "The belt pulls the vehicle forward x-direction.\n",
    "\n",
    "#### 3.2.5 Belt: Centering Force\n",
    "\n",
    "The belt also provides a lateral centering force, keeping the vehicle\n",
    "very near y=0."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4. The Vehicle Model\n",
    "\n",
    "A simplified version of the vehicle has been modeled. The independent rotational degree of freedom between the front and rear chassis has been eliminated in order to speed-up computation. It is assumed that the track, as defined by motor force direction and reaction plates, is straight, and that there is negligible yaw and lateral motion. \n",
    "\n",
    "The magnetic *wings* are at a fixed angle for each simulation run. Gravitational, propulsive, frictional and lift forces have been considered. The drag force is assumed to be negligible. The lift force has been modeled as a function of the pitch-angles of the wings, the speed of the mass-center of the vehicle in the axial direction, and the levitation height. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Media/mag_wing.png\" width=\"500\"/>\n",
    "\n",
    "##### <center>Figure 5: The lift force generated by the magnetic wing. Credit: skyTran</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The magnetic lift-force generated by each wing is modeled as follows:\n",
    "\n",
    "$\\begin{equation}\\label{eq2}\n",
    "    F_L = C_L V_x sin(\\alpha) (H-dz)\n",
    "\\end{equation}$\n",
    "\n",
    "where, \n",
    "\n",
    "$F_L \\rightarrow$ The lift-force\n",
    "\n",
    "$C_L \\rightarrow$ The coefficient of lift\n",
    "\n",
    "$V_x \\rightarrow$ The axial speed of the vehicle\n",
    "\n",
    "$\\alpha \\rightarrow$ The pitch-angle of the wing\n",
    "\n",
    "$H \\rightarrow$ The maximum levitation height \n",
    "\n",
    "$dz \\rightarrow$ The altitude of the vehicle\n",
    "\n",
    "The modeled vehicle is shown in figure 6."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Media/Model.png\" width=\"350\"/>\n",
    "\n",
    "##### <center>Figure 6: The Vehicle Model</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Modeling"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.1 Dependencies"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sm\n",
    "import sympy.physics.mechanics as me\n",
    "from pydy.system import System\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy.integrate import odeint\n",
    "me.init_vprinting()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.2 Reference Frames"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Reference frames are attached to the track (inertial) and each of the five rigid bodies. The reference frame of the _Bogie_ is oriented relative to the inertial reference frame and the _wings_ are oriented relative to the _Bogie_."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = me.ReferenceFrame('N') # Inertial Reference Frame\n",
    "\n",
    "bpitch, broll = me.dynamicsymbols('b_pitch, b_roll')\n",
    "\n",
    "B = N.orientnew('B', 'Body', [-bpitch, broll, 0], 'YXZ')\n",
    "\n",
    "alpha1, alpha2, alpha3, alpha4 = sm.symbols('alpha_1, alpha_2, alpha_3, alpha_4')\n",
    "\n",
    "W1 = B.orientnew('W1', 'Axis', [alpha1, -B.y])\n",
    "W2 = B.orientnew('W2', 'Axis', [alpha2, -B.y])\n",
    "W3 = B.orientnew('W3', 'Axis', [alpha3, -B.y])\n",
    "W4 = B.orientnew('W4', 'Axis', [alpha4, -B.y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.3 Points"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The position vectors of the points that are significant in the formulation of Kane's Equations are defined."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "O = me.Point('O') # fixed point in the inertial reference frame\n",
    "O.set_vel(N, 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5.3.1 Mass Centers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "dx, dz = me.dynamicsymbols('d_x, d_z') # coordinates of the mass-center of the vehicle relative to the point fixed in the inertial reference frame\n",
    "lb, wb, hb = sm.symbols('l_b, w_b, h_b') # length, width and height of the Bogie\n",
    "lw, ww, hw = sm.symbols('l_w, w_w, h_w') # length, width and height of each Wing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "MB = O.locatenew('M_B', dx * N.x + dz * N.z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "MW1 = MB.locatenew('M_W1', (1/6) * lb * B.x - (1/2) * wb * B.y - (1/2) * ww * W1.y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "MW2 = MB.locatenew('M_W2', (1/6) * lb * B.x + (1/2) * wb * B.y + (1/2) * ww * W1.y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "MW3 = MB.locatenew('M_W3', - (1/6) * lb * B.x - (1/2) * wb * B.y - (1/2) * ww * W1.y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "MW4 = MB.locatenew('M_W4', - (1/6) * lb * B.x + (1/2) * wb * B.y + (1/2) * ww * W1.y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5.3.2 Wheels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "P1 = MB.locatenew('P_1', (1/2) * lb * B.x - (1/2) * wb * B.y - (1/2) * hb * B.z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "P2 = MB.locatenew('P_2', (1/2) * lb * B.x + (1/2) * wb * B.y - (1/2) * hb * B.z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "P3 = MB.locatenew('P_3', - (1/2) * lb * B.x - (1/2) * wb * B.y - (1/2) * hb * B.z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "P4 = MB.locatenew('P_4', - (1/2) * lb * B.x + (1/2) * wb * B.y - (1/2) * hb * B.z)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.4 Kinematical Differential Equations"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The vehicle in the chosen configuration has four independent generalized coordinates: The axial displacement of the mass-center of the vehicle, $d_x$, the vertical displacement of the mass-center of the vehicle, $d_z$, the pitch-angle of the _Bogie_, $b_{pitch}$, and the roll-angle of the _Bogie_, $b_{roll}$. The generalized speeds are chosen to be the time-derivatives of the independent generalized coordinates. The system is holonomic."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "u_dx = me.dynamicsymbols('u_d_x')\n",
    "u_dz = me.dynamicsymbols('u_d_z')\n",
    "u_bpitch = me.dynamicsymbols('u_b__pitch')\n",
    "u_broll = me.dynamicsymbols('u_b__roll')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{ \\dot{b}_{pitch} : u^{pitch}_{b}, \\  \\dot{b}_{roll} : u^{roll}_{b}, \\  \\dot{d}_{x} : u_{dx}, \\  \\dot{d}_{z} : u_{dz}\\right\\}$"
      ],
      "text/plain": [
       "{b_pi̇tch: u_b__pitch, bᵣₒ̇ₗₗ: u_b__roll, dₓ̇: u_d_x, d_̇z: u_d_z}"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z1 = sm.Eq(u_dx, dx.diff())\n",
    "z2 = sm.Eq(u_dz, dz.diff())\n",
    "z3 = sm.Eq(u_bpitch, bpitch.diff())\n",
    "z4 = sm.Eq(u_broll, broll.diff())\n",
    "u = sm.solve([z1, z2, z3, z4], dx.diff(), dz.diff(), bpitch.diff(), broll.diff())\n",
    "u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.5 Velocities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The linear velocities of the significant points are defined. The generalized speeds are substituted in place of the derivatives of the independent generalized coordinates, and the resulting expressions are functions of the independent generalized coordinates and the generalized speeds only."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5.5.1 Velocities of the mass centers"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{dx}\\mathbf{\\hat{n}_x} + u_{dz}\\mathbf{\\hat{n}_z}$"
      ],
      "text/plain": [
       "u_d_x n_x + u_d_z n_z"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "MB.set_vel(N, MB.pos_from(O).dt(N))\n",
    "\n",
    "N_v_MB = MB.vel(N).subs(u)\n",
    "N_v_MB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{dx}\\mathbf{\\hat{n}_x} + u_{dz}\\mathbf{\\hat{n}_z} + 0.5 w_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_x} + 0.166666666666667 l_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_y} + (0.166666666666667 l_{b} u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right) - 0.5 w_{b} u^{roll}_{b})\\mathbf{\\hat{b}_z} + 0.5 w_{w} \\left(u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right) \\operatorname{cos}\\left(\\alpha_{1}\\right) - u^{roll}_{b} \\operatorname{sin}\\left(\\alpha_{1}\\right)\\right)\\mathbf{\\hat{w1}_x} -  0.5 w_{w} \\left(u^{pitch}_{b} \\operatorname{sin}\\left(\\alpha_{1}\\right) \\operatorname{sin}\\left(b_{roll}\\right) + u^{roll}_{b} \\operatorname{cos}\\left(\\alpha_{1}\\right)\\right)\\mathbf{\\hat{w1}_z}$"
      ],
      "text/plain": [
       "u_d_x n_x + u_d_z n_z + 0.5⋅w_b⋅u_b__pitch⋅sin(bᵣₒₗₗ) b_x + 0.166666666666667⋅\n",
       "l_b⋅u_b__pitch⋅sin(bᵣₒₗₗ) b_y + (0.166666666666667⋅l_b⋅u_b__pitch⋅cos(bᵣₒₗₗ) -\n",
       " 0.5⋅w_b⋅u_b__roll) b_z + 0.5⋅w_w⋅(u_b__pitch⋅sin(bᵣₒₗₗ)⋅cos(α₁) - u_b__roll⋅s\n",
       "in(α₁)) w1_x + -0.5⋅w_w⋅(u_b__pitch⋅sin(α₁)⋅sin(bᵣₒₗₗ) + u_b__roll⋅cos(α₁)) w1\n",
       "_z"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_MW1 = MW1.v2pt_theory(MB, N, B).subs(u).simplify()\n",
    "N_v_MW1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{dx}\\mathbf{\\hat{n}_x} + u_{dz}\\mathbf{\\hat{n}_z} -  0.5 w_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_x} + 0.166666666666667 l_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_y} + (0.166666666666667 l_{b} u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right) + 0.5 w_{b} u^{roll}_{b})\\mathbf{\\hat{b}_z} + 0.5 w_{w} \\left(- u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right) \\operatorname{cos}\\left(\\alpha_{1}\\right) + u^{roll}_{b} \\operatorname{sin}\\left(\\alpha_{1}\\right)\\right)\\mathbf{\\hat{w1}_x} + 0.5 w_{w} \\left(u^{pitch}_{b} \\operatorname{sin}\\left(\\alpha_{1}\\right) \\operatorname{sin}\\left(b_{roll}\\right) + u^{roll}_{b} \\operatorname{cos}\\left(\\alpha_{1}\\right)\\right)\\mathbf{\\hat{w1}_z}$"
      ],
      "text/plain": [
       "u_d_x n_x + u_d_z n_z + -0.5⋅w_b⋅u_b__pitch⋅sin(bᵣₒₗₗ) b_x + 0.166666666666667\n",
       "⋅l_b⋅u_b__pitch⋅sin(bᵣₒₗₗ) b_y + (0.166666666666667⋅l_b⋅u_b__pitch⋅cos(bᵣₒₗₗ)\n",
       "+ 0.5⋅w_b⋅u_b__roll) b_z + 0.5⋅w_w⋅(-u_b__pitch⋅sin(bᵣₒₗₗ)⋅cos(α₁) + u_b__roll\n",
       "⋅sin(α₁)) w1_x + 0.5⋅w_w⋅(u_b__pitch⋅sin(α₁)⋅sin(bᵣₒₗₗ) + u_b__roll⋅cos(α₁)) w\n",
       "1_z"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_MW2 = MW2.v2pt_theory(MB, N, B).subs(u).simplify()\n",
    "N_v_MW2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{dx}\\mathbf{\\hat{n}_x} + u_{dz}\\mathbf{\\hat{n}_z} + 0.5 w_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_x} -  0.166666666666667 l_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_y} + (- 0.166666666666667 l_{b} u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right) - 0.5 w_{b} u^{roll}_{b})\\mathbf{\\hat{b}_z} + 0.5 w_{w} \\left(u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right) \\operatorname{cos}\\left(\\alpha_{1}\\right) - u^{roll}_{b} \\operatorname{sin}\\left(\\alpha_{1}\\right)\\right)\\mathbf{\\hat{w1}_x} -  0.5 w_{w} \\left(u^{pitch}_{b} \\operatorname{sin}\\left(\\alpha_{1}\\right) \\operatorname{sin}\\left(b_{roll}\\right) + u^{roll}_{b} \\operatorname{cos}\\left(\\alpha_{1}\\right)\\right)\\mathbf{\\hat{w1}_z}$"
      ],
      "text/plain": [
       "u_d_x n_x + u_d_z n_z + 0.5⋅w_b⋅u_b__pitch⋅sin(bᵣₒₗₗ) b_x + -0.166666666666667\n",
       "⋅l_b⋅u_b__pitch⋅sin(bᵣₒₗₗ) b_y + (-0.166666666666667⋅l_b⋅u_b__pitch⋅cos(bᵣₒₗₗ)\n",
       " - 0.5⋅w_b⋅u_b__roll) b_z + 0.5⋅w_w⋅(u_b__pitch⋅sin(bᵣₒₗₗ)⋅cos(α₁) - u_b__roll\n",
       "⋅sin(α₁)) w1_x + -0.5⋅w_w⋅(u_b__pitch⋅sin(α₁)⋅sin(bᵣₒₗₗ) + u_b__roll⋅cos(α₁))\n",
       "w1_z"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_MW3 = MW3.v2pt_theory(MB, N, B).subs(u).simplify()\n",
    "N_v_MW3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{dx}\\mathbf{\\hat{n}_x} + u_{dz}\\mathbf{\\hat{n}_z} -  0.5 w_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_x} -  0.166666666666667 l_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_y} + (- 0.166666666666667 l_{b} u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right) + 0.5 w_{b} u^{roll}_{b})\\mathbf{\\hat{b}_z} + 0.5 w_{w} \\left(- u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right) \\operatorname{cos}\\left(\\alpha_{1}\\right) + u^{roll}_{b} \\operatorname{sin}\\left(\\alpha_{1}\\right)\\right)\\mathbf{\\hat{w1}_x} + 0.5 w_{w} \\left(u^{pitch}_{b} \\operatorname{sin}\\left(\\alpha_{1}\\right) \\operatorname{sin}\\left(b_{roll}\\right) + u^{roll}_{b} \\operatorname{cos}\\left(\\alpha_{1}\\right)\\right)\\mathbf{\\hat{w1}_z}$"
      ],
      "text/plain": [
       "u_d_x n_x + u_d_z n_z + -0.5⋅w_b⋅u_b__pitch⋅sin(bᵣₒₗₗ) b_x + -0.16666666666666\n",
       "7⋅l_b⋅u_b__pitch⋅sin(bᵣₒₗₗ) b_y + (-0.166666666666667⋅l_b⋅u_b__pitch⋅cos(bᵣₒₗₗ\n",
       ") + 0.5⋅w_b⋅u_b__roll) b_z + 0.5⋅w_w⋅(-u_b__pitch⋅sin(bᵣₒₗₗ)⋅cos(α₁) + u_b__ro\n",
       "ll⋅sin(α₁)) w1_x + 0.5⋅w_w⋅(u_b__pitch⋅sin(α₁)⋅sin(bᵣₒₗₗ) + u_b__roll⋅cos(α₁))\n",
       " w1_z"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_MW4 = MW4.v2pt_theory(MB, N, B).subs(u).simplify()\n",
    "N_v_MW4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5.5.2 Velocities of the wheels"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{dx}\\mathbf{\\hat{n}_x} + u_{dz}\\mathbf{\\hat{n}_z} + 0.5 \\left(h_{b} \\operatorname{cos}\\left(b_{roll}\\right) + w_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\right) u^{pitch}_{b}\\mathbf{\\hat{b}_x} + (0.5 h_{b} u^{roll}_{b} + 0.5 l_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right))\\mathbf{\\hat{b}_y} + (0.5 l_{b} u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right) - 0.5 w_{b} u^{roll}_{b})\\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "u_d_x n_x + u_d_z n_z + 0.5⋅(h_b⋅cos(bᵣₒₗₗ) + w_b⋅sin(bᵣₒₗₗ))⋅u_b__pitch b_x +\n",
       " (0.5⋅h_b⋅u_b__roll + 0.5⋅l_b⋅u_b__pitch⋅sin(bᵣₒₗₗ)) b_y + (0.5⋅l_b⋅u_b__pitch\n",
       "⋅cos(bᵣₒₗₗ) - 0.5⋅w_b⋅u_b__roll) b_z"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_P1 = P1.v2pt_theory(MB, N, B).subs(u).simplify()\n",
    "N_v_P1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAApkAAAA4CAYAAAC2cOMwAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAgAElEQVR4Ae2djbHdNBOGyZ0UEEIFQAeQVBDoAJIKgA5gUgEDHUAqSKADoIKEdABfBSS3g3zvo6tVZB3Jlv99ztXO6Fhe/a1Wu6u1JPvceffu3QcNGgcaBxoHGgcaBxoH1uHAnTt3vlfNn2q+/W5OC76ep6rjR9X185i6VPaeylyPKdPyNg7M5cDV3Apa+caBxoHGgcaBxoHGgV4O/KnUn9Iccvz+VvgsxZfuvWP5RumvS3lyeLXxifC0RbiXy9Nwl80Bxl3hN4WwspjDLc2FOxLapets9TUONA40DjQONA40DvRwgAleyW81B9/pydZJsjJCfli7KqkyXyj/bwr/KtDmfYXPVZ77BreIA14WftHYf2rdzuEsbYlrW8lcgoutjsaBxoHGgcaBxoFxHMD5G+voPaDMCAcTp/IPBbbXcSxxLn5V+Fuhwe3jwJfqMqvqMeRwcfqseFvJnMW+VrhxoHGgcaBx4DZyQCtA36rfnyuYw4ZD91ABh85tZ/tVIrbDHwr3ta4fCMfW9Q8KOJnXCq8UWF0KW+DKw9b6fwoOlObOX3o85XEcARyE0J7DJD8qc3IWM4dLirXbC+SAxh25+U3yxIOGgxzO0pa4HtLJbAowfWgb76bzrpVsHGgcaByo5YBsLU4mK5FM3KwSvhYOh5JJ3G1H6v4nxX/QlXNpnS1u4XBOcRB/1zWA8P/o5jvh/1Sc+v5S/EMy+DJEHwl3rXteKAoOLAkNLpMDGmvkjYcK5I1zuTxs8FDzg2QhrIgrnz2gfKS0D5TGA40DpSGHvIAW5z/B+eyLXA63XS4mwLh2QHnC8DbeTWCaLwLvFFDiWwfn0HfR+BV03rrB2bHDlyAXa8mNl8UXGh6cwN81adsqJCuTTk59nh+hQbjcFjdlrZyizon8RRfyui1NX+8jl3jzQxmcCtoBcGaDwwBC7X2m8C4KF2PX1Kdba6cZWwHyQbiWDLC6jTwEuyj+8ODyB2kKOJeBX0ojL+WCvORwyrMoXC1a28zK1GG2D2ASCnRP4X/CBQbqvkGBA+fOO9H/k8L3Ct/6OApRBcrPQwnlkBkzsr/oHuM+CL4c21VhC4FC4BX+UHirwOrCxQF9VKc6fQencKh+a2xY7UFGmj0QI+CDAnK5igOhei9CLtaSGyZqhWsNBStJLxUM3ETOTZSHzxbhGAQQf08mfJ/4WNdOXtXjHNGoTHymjjkzbp92yc82PvRdDORk0jqntNXnD9pX2MUuql3snj3U/KkxtocQViVdXHmQm/CA4nmDQ8nqJ4CscDQjhhwuTp8fF4EI5e5BPcGosWz7vdGjOMu+vH23O31HpkE8OmveiX4eLL4yHvv+4NR9Yri+q/K9VUB24sA2VZXcqBztF9vy6ThiVfWdU76+vh+t36IHOf/7nPi7Fq3iA04K8l4t52Nogc8KWZ3waYfRB9HTKxdD6WP4kuZV3diezwyvOG9xB94obraZa5yPbW621XVxeRyvFQdR4jtlWKVydoh85Pd1hLqjdMaQ+r413Dlf1Y+sTHr8ZvOHby+M8ZY8VdvIV/CR4raFRxYDH0gTsLXubISPd8rmcHGdS8Sv1MghQJ25FiGcWXEHnCFKcZZ7Pya+BehJgC05FPmsYG/ezeGbyrISg1EN55J8f7hHoWqApzXkhjLIDE934RyK7ovg23+l/GELIZOZCR1lvSio6Puh+u3lgnNqZ6ejSwuOePFagTP1VXI+pv1Lk4u15EZ8wsnDeWTucjsouqAz8ZiwUsRYkeeJggHb3LYC+VTpZn9YhaTeAGrHVjbtPJ6lUbetasZ1W/ohrqJ/9rxakkmP33r+2NMuxmOeji+yGI5fiDfck/+5z8iXCeycr0d9kMNZWtV1aHzvVtWyUSaviJ3WcrhOhnbjOJDjUw53QHbxxqUZypg8DDDb5ydvRsaZfJwtgtiwZ7IUUZSz7YSTTGofJQVyNN6knO9vse8H7vePYvf/FMLD6Pmy/7CUX6JcrCE3OBtM6jhRDCaOIy8AOacThAC78UTp7MqZswie+FPheWAyJwA89pAXhaibenAUzLbdVzw8jCtO3V/6uunfJUNJJjedP8Tr3eYDte0eaiRfwZFMBhw88mLwTBHOZlp+5lJeTkPm+nBWfpHr7k6mOuzWdH1vECTOxaGQeNgwlSe8byJG6baBceAC+IfSxobTumaGupRu+SZfvdKyYm6rCLm6cEDt/FUu/SxxFX0/ZL81VrxR+6/CF4pfouNfJU9+/NyEsqRtvFS5WElu0BHOxxUfeGhXeXCEOuDHLIfHFnGG8wRUhnOWAXRP3pM6Qob3Ec4SMqeag0TKK4X4BSJwh4QBmSzND2vNH3vaRfyhPpuHLHA2lQUa3ix/LhmJ59ZfvRzEu3I5nIouB3epSg1jrNi25EnsHxEWlMansXLwsfA2cLpdDBg0nupgIIxhixTh/05t8+IG938puE846HpI2JGHZ8s/zzPG803PoCIXg6C67AUIZBk55syMPa2VyiNffUpLOYwYWwzkZSWB+uE5nxjpc06VpQy+7xh+0ynq5bxVMAo+z1Ph//M1oR/kCTQrDyse0AgtRt8T5elMSEpLYajv1GlPvZSFpx36QO4E9B+DGviwBR1+PEp2En6y7en4rrzcs5IwyW72javS0AlsJmOEvDhHw5ehTdJfKFgeRW9eUBF9wbaDzMAly0Wv3Ih/6GBpfElL50H4b6uMGVYeBoWdwV45Wr2ccL7xseLn8M8/WZn04wWTt5w/GPNF5oOx8qbxQ36LNk/pvQ8dSj95eMnhYOiioEaozx1i1RUGdl600T1Lq8rW/9KD8mDQqOzkAHJFWRxJynbK6x5jeYIfqm9qutpCmDsHY2vrUrndeKi2d+XfVL6pnI3vycsLSsN5YuwHx0N5eDJjKwBBVTTUy2qXw+WuvlzxUDx1KlBh58C57uG3O7Rv9eoe3SHf4IspykO/O4e0dU9/wdkLANyfvPwkHH11PNHVHNNOHylndJWuvp5s33291m/H1wiXfSmh1E4J7+tzcmt5cjhLi6/KhyMwyOe4zBJxtWkH6LETqZ1kXIJMKI48wEMng/6+Vj6qxlV1Ul9o0/ooHHIT5AS8wHStd/x8uYuUC/WtV26UXm3DyasAY5HhYHtsDI5wFV3IBzS6fsU0CYfjSdrqeqQ2Js+rXnaR5ROZFM5kepP5Q+0tOh+ovmp5i8fuaHH1o3d8r+RNM1AII8ATcbo6w6pN0XumkAf3lCQGDK0eWf7c9Tour3hMC3QeEg7Ew7PknwYV5S0Bq3e9IDn5UuHaMnm5QWaZCPoAmep7CsZRADiuEcsiuA7NSrenTLahhoCJiS34sGqpe5Nvq5c8fH8vbRc9Y0sEJ/QBwT8RKxpgqN9k7Ou79dt98JnMosP4i0GZDb4+6Az9y+EKDTFmxqdClmXRXsdf+lqxiYFuj4NnTIYO1BeTB5cvuq+Rj9pxLckubT5Qm2HVUnGj18bWU3pyuWS5KMrNWBsufrIyyItXXyuYbpww8yCIHH0my3xT02zPQcg9IaNPJsncZwuWnD9Md2bPB2Pl7YQjZ4S4K1rfSEns+4CPdZ8eIMbgpbhcFxmA17mEEbgaAzyiunxWDXB6PsUyIqz3lf7EENEVp8BtS0U4ix6Fh6vybw2+GQMz1/se918mrQbFpPoFyqxxswk2LUcbOQNs+XAmeDM0lWtkPacTJbzVZ0dTcBBN51ya2sDhJJCHdAyrTQSgHUCL0omzJc4LAkycfC8RpwYnB8c0OBe6L0Ff363fgTeq3wz5oNEuNZjB5x5gc7i0KOPZOzF6ejlmMwb6zn6j4258VGHHTqotxgtgDGLAZsSyNygfFFYZtuOmjqu1H7drOK42jjEujl+yXPTJzVI2POblkeOxfCC/8f0kuiWzS8+rRkdJJtGRElAGWHL+MLu4xHwwWt7EX+wHPMZGuGM5xLeCqeN7V8S6icR3AAMUJj/hED5wqfHM9euhkKs6OblGp+DUZ7fqmpZVf1mlwSmpmaRD8dvCwzX4Jp7DRzMIgaeK2GTYa/xUnhU/xixVOifXpCn01qH0EqDU5li4PGqP+ko6gb6kBsiVi37MOWJLswSWx/qQy0dbAP1+qmAGiFXOX8WPk/M3ZK4EHixTXQYHnDi+N+hJv/SB8Yshh4vTq+JeJ1OZqCqby+Tr4wEAPjP+sVyAYxchlbN0/Grkw5qfO659E7C1MfZ69nJR6nBmfAfnQS8Lu036pb7shRcPF51Xh/rBmGkMyHY/kxcdBVKdvMH6X5UfM38sNh9MkTeV4eETX8z61unL2jdTx/duRBheevoWrRnPk4lTnaWjKJhNluTtrO4oD+k8SbgXMXQlD87ojyL4pE7hB0F14ghSr03E7v9ghec8GzQhVOlnJITaBNbgIf1khepa4ZECCsUZoEn8U1kmSndIS1HqeKPAuABs/cST5w123d+S0pjhGHrAgfacITFFTJ2luDf03fLFeFtxNN7HaThv6EmH/+IpDgRgnzNBzlkZY9xiMFrRiRJYnixtvhBvWEMfT8TBuAvHuTP+7Yizen28K/Zd5elLuuVuq/iuTt8O8o5sUhe00Gde2jP64aPZgI+UxhN4oFW3jF3qDOdwFI0B2QhtxAkbxOHD67iPuocPHV6r3x3Z0X2tfMCzOeM6lwWXLBc1clNtw9EvjRXj3qenc8djrfLwwoDdkSPPq30yWeK/9a+jl9bh6Fo1f4g/jHFHp30dc+eDannz7UFvx8/y+MNeriLKMIKvo3uiMMBN0mJy/Ld9MJxznBxcZdWPgaR8GFDl54UhnEnSGfBnPk4+7ieB6mA7kAna2uJMGoMP4FzyIe7rm9vNf+nbYjz01DMpE5j0WbHii/5pG0LXgVcWMsMnVkusLj7FsbWDCR08SdoqGfcGjDGT+dBYvvD9sHJ2RRmHyuOolGSRLVG3dck1Apw4x6dYJ4QzY8UqIjLPih8OVgd8f+B5rs84GDa+9Bv964DSkTHAPYHrCj0BVD8rMNBnOhHSkkhf38kanDgvM/CDc2jxeCCTBFbx6DO0hXZVDhvBG+lL/48ubcR06HYzoO3AG99qDsc4mo0iW5V8RPVNHVdfxeTLJctFjdxU23DPYcY1nMWdzPV1C97LVG+2BbvPQ/OR59U+mdxq/lhlPtC4jJW3XP7M8B4HdRWR0jGcmiBQHiZCw8fOG5MnwmkOCtUw0cSGn3S7R7mZjJi0qSc2vqBHg+pwTwAqiIIT+s5Tja5/YgHjlSu+AA/hFxM7TsMLhZeKM5nPBRwhVsJ4euVhALptpWpu3aPKiw6cojeeFldWcYwik+w3DqEfcArvFHBcYmB72MmWIXVP2fsKQ31CfktbqshXLN+6dQBtNqnEOkF+HrpM5rlmHUnhoYv/nu44ErpHr6xNVq35xAhGJYZnusFpMx3ig87QFAP3lh7j43hf31MnFUOOM++2ENUe+ow8Qht67trSNei28jAmTF4xHcgZfAKwL69c7P1PDvc+9X2MMYvrfZ+yfoyxp/8O1E9oRtbCOAlH3MbR5dPPGPmgTO24pmNv7U29XrJc1MjNGBsOj3NjPZX3a5Xr2BEvn9gebFRnJ0H6ipzCA+SccIR5tSiToner+QO+pDotlFvFhk/AlPmgWt78uNEOu2X8UclvCkvrP/UvCnej2lgtewbxHkfnUUomcXDPPZ4LXv030T2GtmP0NfjxgKCInXQrq7pt6xYUL2pwzyoWDlC8EgRDmdAYbAPibNffV8jWbxk3ui7KQ6NZ/UaQeFuUyb8DE/nHN0iZKHEeAMdT2lEbGJ6tATljUrXJ+6HuH8UyBF1KRyY7Sik8zrI5mtBOHW8Uar5PiHHoOKi6N4DnuTScfP5lg091Md4GOJSxvNKnDq2W0dP8se7RN/Ihwx8p4KS6MrqyhUUaL/fYmEAT29GxrGNkmURIIx+6wyc9sm0rzaCv7+i26T10sfXuHEwKW91qk3EyA0tSDKmNIC2ekJG5tGwOF9dpcXgNjZuD+v6z+v2pArJhR4VsLLGT8ItPSAV+eSKr5cPnL46r2oaPTxWwuzyAocfwA9lP8WytgTdnAj17KPpKD2CXLBc1clNtw/04iLWDR2TIsxcgR/QJfWaexE5wZaEhtl9CBUAPjzSv9skkRGMn154/4Nsa80G1vKl99B27ztgxH2JvmHNMtxU9IIhYJo3qoC7AbBzB8L01xRECnoxIw1EJ9ekegyjUDY57i8+9qi7axcBCz+C3AYfaUx1fKQx+l3GonqF0tVHNQ+Vlggg8I075oTb60imvgBGBb2zr6eKEdVLfqUNhUtk+OrdIE93wIcjy1DZVB0wM46J45zuYU+tds5xonNV338cgmzGtSoMfga+Km8y7/FZW11A+h4vrJC5wspvij34vus9GPkTrxcnFGnKjOpnk4ZXTe3/vvn14dHnso0/9ONy86vkc7Ekf/UdIE72L67sflzDP6h6fa/XvnA7xUzT0zv9XyjAK1OC1ChAc+Kc5PGy2vh6TLhxbgW5GEO6JQryqyf1sUP04l9dqjydynrxoE+WYA6wQpSsQc+rLlq3hIQXhoy48qfDftyyRI1SsUgX+634KwDvqZtv1d9XLxI2gDK1+KUsWNuFbtuX5SPhbeqIfU3vgnR+nV/B2TAU75J3cd/UR+WFyjXU77gJ45MrgmSLIm+W3sm41zmfK4ay8XVmpy60oWPpRr+ckH5coF2vIDfNO7RGZo8plhy7p9VHn1cky2engdjdr6PsDkW/2k56wghvaAbET9M7/fEx2NF0SRCYGlmhfKtBJJhy26fCq+S9MJhcmFZw+0lFG3jJnK4mXgSY7SarbOVqqhzbYOnRnFIW3jtAe+ENP8EM8VB9wMvmXA/jJ6uVfCjjysz7+69s1Z9wElvqBL9VevBV7g73wX/EEfnD2dbLC+jFiInN1qK4lHNfVOT+1716O0DN0+wSUbg9I2Aj0nvPEQSeVjqOIHQh/k5nDKT2A0tH5v1QPxvWsQLSjY2cjH6J3kk6oHHPDoeRiLblRvexWYI+dzdQ9Mn1f96WjCEo+Joj2w8+ronGSTO7BcdG6uL6rTnwc9zUd+qR75I8zs8Gugj8aTHIyj9aJRk/jwBwOSFlxXjh3mHWY5tR99LLn1HfRejaTzNHHfYi+S5KLteRG9Z7lpD809kdNPyeZXIOH6j9HM9znGRXnoYAHnMPPWc3JXEMaWp1nxwEpLStv/M/06scljsacc+i7N6q8+Dd5tflofD86PZcgF2vKjeo+y0n/6HLXR985yGQf/XPS1PfFV0fn0FNbtjmZtZxq+RoHGgcaBxoHGgc8B8510m8D2DiwJQeak7klt1tbjQONA40DjQONA40DjQO3hANXt6Sfvd3UEykfNuXQ9izw9bzlOquiVngTDmicOIvZ4II44HWw6fIFjWnryu3lQLPR5z/2zcm8GUPeDuQTCR2QgP+tYG9ed9JyNzovxpvubxTsre1ctoY7AAc0rpzBZHwJzdk8wJgsRELT5YUY2appHNiTA81G78n95dpu2+UFXnrH460cxzuFLCdoK6OE8JmBk0wNsTsHNE58ZoXvwfESCQ7mfQXe2msvlYgRlwaml02XL21kW3/25IDXq2eigT/0CPNkCT+G1majx3Dr2HnbSmZ5fHBExjodfCyV/2u+LlfbUvbkgDeAfAqH77XiWPJ9V94oT/8TfU8yW9vLcqDp8rL8bLU1DvA3fsxzHE3pzJMlfC3Lmo2u5dR55LvYlUwJKt+R4qPN5jywYsV/LeNcuO1s/7TEdnj4L1/h2EblQ9pMTCgRH0DnXx3CFrjysLXOx+UdKM0+CA+e8jgxAN+wCu05TPvZnQMaP/5ZpvMgkMPtTmgjwHFAY9N0uclC48ABOeDnQuxp5/+zS/jaLuTscQ5XW1/Ltx8HLt3J5AkLh48Vq9cSUhxKPrrN6hVfzOcvGn/QtfNRXZ+Gc4qD2PmavvLybbTvhP/T18c/kHwYlSH6SDj7A/vgwJLQoHGgcWAcB6RnOJlNl8exreVuHFidA9JN5lfm1M73hUv41QlqDRyOAxe5XS4BZzXxhQJO5e9SAFuFZPWKNPtf8B+V1/1nN04h+Agoa+UcWnnd1oDyur8R8/U+Ssrwl2pWF85sZyshynuYKPxSYCI/LDBO0FkisC9daZ8pvIvCofta6mMOrz4dfuxydNfi6J/yNl2uZViU7xxkQzT26nXUnRZdmAPi/bcKv/kr48BXVrjv2Fnd/+TTuKYvyLLj5+bDhLwTvMoWx1ppF2ujE77cvls5RJytcEG9R4D4/A6TMPHPLG3oqrys/FGOpXNdnIOHU8ah4LgN6mc1kEzV9cd11MZVPy93fG/5Fceh5GWemB6exEIe0gQ4mJ18Hs9/hXb6Y3XlyghHP7P5rdzQVeVReNrlf8wD3UvFVe89Bf4/OtRtOF1p9584bc+4aGE8PynR0JeuNDemujLAq/CyRNdaePXjZOysLaU1XT6YLtvYbHHNyYbhdD0rvd6CX7QhWF1ntupLTTvqL/M1ziCd59/OdHFzlov7e+b1+B4b7OynrqV5Mov39RVtuNWn68XY6JpxuPQ8sWOBMAWHSHEmMJyk4qQeM0f5MFwIRxzYjg5tWFx5EG7dnqYtiVMb0BQcWcURcM5XOpoUp4/QyzXOhyPMFoAuLs3xQHEQWX4IT5ngrJFPQVW4OkLd1nbtVXWgsFSS5WVtPaV8qpdxL/WJtMCvUh1b4UUL4/R3qb2KdPoDLy/FycyOHTxSaLp8o3uH0eWS3K6B9zJwEXq9Bn/SOrfUmbTtPe7VX+Yn7Gln3oppURqLRG4eNLzuccRtbsyWVXoWTx2CIRt+UTba+Habr1eMupaqcfowSOH8oeLXwnGPY1YDbAvzAgxleHHmU9XBNQe8kJNbYs/lnYRTn0yJ6Ad9xFkjxDTxFPfa9/WJ4gZsc7/0N0+VblvebJ9TbwDViyICvORjL/xwH28XxHWTVg1qG/o4OxvTXV2+L6Mf91dR/9Ls8CvuU5q+6b3oZCw5C4sRO4Gh9JMCOyFEv9uamtN8aew8vunye+YeRpffk7RurCQbUatnpdcR3b3RqXq1g8709mOLRGy+t5e8DFuy8Y+V9jyhB9mx+TCd8yxrCc+qS68Ntwra9YI4oEGnNwhZ54nF49leJoPbAgdXCrnyPXlZYexsUZfyTsWrfmjnqQiHhIAz2OkH9wo40Tydhad+xVEk8JQLq5CK42BSD3gcc652PKCzqiQ8ea3uTrtT+7R0OdFXXKlWmm2jHIp20cWYnRxlMN70pSvtEE/JogPZnCX/Kp8dO+GbLt9CXTb551qSDZ92lnod968UV78n6dXWOlOifw+8+t7Z7YtpUBrCFM+L2F5wbk7U1ZW1eytbwkfpRRuusoew0UZru5Z9vlre3NWgAhiesIrpMDc/1z5eSo+y1kX11GgrjKuuZIoanqb+FCPc54Vy1CmN/n2dpgnPimUOzxNc51MNVlZlWJ0NoHvyntQRMlRGIn7h4HdeRKqsIpvN10ud9lSa5oN/9rSbpu12L3p5a/9fBc4JncjQULon/J7K82CBXBu8UiR+acvwh7sOjF1JV5suRyPp5X5TXY6aXy06IBu0e5Z6vRrDbireTGdW7seo6r2ssIhQmlfA4xAaPFPk5yi/K6t6WGyJ6yjhXT23wUYbw9r1gw/uSkBMiN70MKSzRVzKp7pY3QOok22qzvclSRC4iT0S1BvswK+nk/qpl5dRgvPo0/4n/McIsK+KdhbfYvZ1L3YR7ayaQivO3n0FePdE/fhcabZyakbQOa2+DApPOm/esrrq+KrrQ4WXMX90nwOe+k+ctCgj9fHZJwwIAN85c5p7GHEZNvyBbnhRon8oHQeTz1A5+fD85An6seLn8M8/2bET7U2XNYh7gx8HZMxsEePS0R2f56nw/3l6P/J5gkwrT9E2+DK5S1Y2ooznrNdRN5aJbqkzvq3SHMa4cTTLLVYoL/fY+HhO6+30QP3IYDpHMn8EectUjo3ljXKOjiGfzxP7/6vSkHN2T2Io4eM8l26j477e+jiCxhL4yYslwmHkSBvc2lMeBC1srSpu9YY30ySgQrst5PCCDLiaoHLuBRRdMZKd7VLd4wipmvBCD04XjbFdHWiy9KNcoU3hhBfCdd7o1j0OUO44A1um8D2Mj+LG97DNkeuvL5d9AUZp0AX/aNfxL8L11ptra2mcaMFQ970AlE2njAL9crIU0yUcxtL1OcavEVc7TCBhzMa2obKM+cnYCWdj33S50q6M5f1Qfj8GbCPGL15hR8E53dGV+5PjDsIFXVa8yjak9Pg6TmSDfL7Os9XrtK/pvfo3Wq9UZjOdUVtOLz2d6RzG2AcbrzjzHGPVmT/TPsf3yls9R8bl9oiL1kPb6D14cqltXmmwDTBqJeApphfEoC8Vri2T4qzM8bSCwxcDyoNCBdDT0BcKfxMCMooIjyGwNJ6u0i1etzVuRdQ2q1S8LPO1QqDJ0g90fSBaHqh/Ke9TnpVWmeHDA/UxrOoqbryBz30AT0v1Wln3UXkqifiIIZ8N9FmBb7K52c8qLOEt3V+hO+VZnGUoPScTPK0DfK8N3hwZ+sYOuvt403R53ZHlwZZjJvGKv8mTjQt5+H6v6apRxMo6K0c4obW2wcratU82dtHrSp2G/iG9tT6ucbWxydU9W2e8TTEb444sJA115kXJBnMnwcmIyi86RyZt73E7NNbnbqP34Okh27wrqhjsEtz3Cf+VMgzgURCUgyd4ztFhPFHm2ADjwPDGMApVUvQ3ymP/KPBY+X5UiAGDnOLi9EPGfb/h/1vff5xvJp/gNFYQnk5UVqTES0tnbHOKTDpG0N665563862+QYPrCgz8qI+crcSZRiYClPAhw02EPtvEnSS526H0UhnDQ1OJr5Zn8Kr+sTpqE3ucH17eV/qTGOnjOCg8SPVBaeyaLvdxbeU0ryPIjtkq19WwQ+AAAAlFSURBVKLGE3vnbJ7ykI7smsPh8vCjfBxPIcpxGf6JbIptKMkG9e6i1+pLVtchKIEqvRVfltSrTXRG/WQOs3mvM4d5mYAVzIExYAucHdL10ubIqrGOmaF4bJPRo/g+ydpuj8KBu94AQA/GKQVzLHoHU0rCkzmO5OdJBebEYFSpgwn32hQnyUta1lGERvKqHfJAUzDiXkHBpQoq1FkAPHuqQN8wnqxk/Ko+Z19KUHoKfUYyzVt7j9P+KskMDjiZHG/Qk347K9BRDSV8lOX4UY1h9kywxvcrUY++jHmYGOwweqK6ydd0eZBbq2SwBx+2wktgecw25vIxgQJzbcNNLe9/99TrxXR6Sb3aSmfUjptDpZ82h5nDyeiAy82LqYyQb7E50tPCnMMDTjp3g27QODCbA1e+htIqok1WQw4cwp8DnD/AHJZgaCTglnaT42ZF67XdFK6UT994NgXtlEWBFIpb8IX6N0WLPjfhYDRRcgU8BJxL/u6rxNOlaMQ5TcfA6maSs+MJhrPVtSFZsPw111w7lCvhrU7ksu/BZyjd6omvlDHoyJIhD3TtG7umy/sNlMkkL8mVwPKUdI9y7PpMtQ19spHTq630Otd2yqMpepvWMeV+K52BNvjNLpHJAbgwL3ID+PHvHCsTGh4O2abqOVI00G+Czc+KbgZTxpoyBkN8sHztujMHzMlkJdJWqmKSMJadbdM4MYq/wEmK7i2KoxSX594Uh9U7B1IolAeI/z81Z4RzSoZSOSVRPThnrtzOCuQ6U/HDRMIB6ACim1VannLdJBMSlo9g5GKlTVsIRtDzlC0ezrpek9HzerP/vU2IgzeOjgRvt0PpOdlCjgA+e8UkjyxxjIH/POfqyuhq/7ELbu0HgRuKTn/7xq7p8im/NsF43WDyy9lSdIaXgUhHdk3eAm3Ilr9xO0OKT7ENfbJB9XvpNbqCQ9MHQ3rbV3ZO2lY6A430MYyBJzqHQ1YCvyLZWGyO9G3Hc7JHbXIZGutBGw2V4gv2mcCC0h/RPbtFDQ7AgStokDDj2LzRAIWBUZxBxsh9o+AAnB/EdJWLLd7Oyyq6pyxOjD0p31Ry85RO3fE5TwQdxWObmC3ElwpuGV/XGDrKqTYoh0E3PP8yhAE32EuBrP2a61P1I1Uo7oOB8ZWkeWrq7svDZJd7MKBM6uRihHmQCMcUyCRgzAls9TBuTJIYDwfqF3LC2/N8W42t40+8XGAcyHuyRVTCuwrf/0B3yp/3qTf96kvnU0U2oRstyCuy444p+L4af8LqufDmJPBpkb42YnqWjhfHztPddHlpjtfXh70Lcm7FJG/YM8YNeKTQkUGHvflkDbpiclVrG3xxdynKhlJ30etKnYb4Ib12HVz6Z2OdYZEltpHMUcyTqT0yWbHurjVH0m7alrW55nVorDv64WWoY6OFszmReZ/6rB8sFCDrDQ7AgbsRDQwSRs0U4KHuebvYBg5nlDNfOHTm1LniwrPyY44mEzV1vFFIv/HFBI4R5k30+LwaT/V8foGyANfcagBlnqktPlkEQAd00za45wox7KVAMQ1DcRw3FAqFod+sHvOpC3tRihVfDAzOEM4eTj/8TfGc1QHvnCTSlf+h6kmdfCU5wNjhIOaANoynHynOpzWCg6l6aeeFAkqPQrtJUdewTag81I1zZhOmbt14MdbURZ9eKaRQwsf5kA1oLEFfOm0jR/QP4w7fufKAE8sk8s4YYKxYPXAff9eVPuNwM1Z7Qd/YQVPT5Z1GxsvMx2oeO8U4cD4THcK+ObupKy/4kMbLPSZHyCHHZmJ9KdoG5S1Bn2zspdc1Ok1/+vS21N+l8FvoDDblZ435pwrYRzu7a/LCHIas8Am7YG99B7Gbi86RooH5EcC+cWXO/0Ztm0yCWwv6xrraRos47DZ2+ivF4R86VprzlNRgcw5ogBD8XYM6zTfkwvfAFEcBwzfDptCn8igQn8dB8Ag4aPem1HWpZcSPk2/1jemr52n2e49K63wrkHoFTID2rTjiJ2V9nhO80aV0JuPOd0QtzbfRmx7nrYmrLRxqiHff5aRtcDVlS3lU3hnEUnoNfgk6atoZm0d0NV3e357OklGNIbYyq4N+fMM3QJEPQa9e+/RsfSZfyjNbb1XHbL0yeo52Vd/W0CvmRWTFzYu6cu++tblm/2lPoWjDx7atusxGI4zEqb/N9TvbIRvHKw3GEQChiFe1Hus+XZUcSydPzzzVDG3Bj633kvKzhddZvRvZOXgcr7zExRnTsAquJ03uyW/jypMsn+Wwp2ndOijhLf2pIjyElGAovVQui5eiIEP0kbOYTL7pof1suQEk9aUrFQNFTpLnjt1JhQshmi4vxMgZ1cyVjaX1ekin6eoSeruEXs1g+6pF19Cr2tXRpTu2xFg7mvy8wkMOwHexsdfUz45TgwNw4ChOJoLhQEKDcLySsMw9U7GXAvmeHP8iHuPocH6Mp79R4MvwtBgcyaQC8BhGg2eKcN7M8ltZJrQYSniOCzhHVXVkP/0zlB43MjJujjgrJRYfWcX77KL/mvAeMz6m8pPHbnxro0o0XR7FruUzz5GNlfS6qNP0fim9XUKvlh+NxWpcQ69w/s0eQyhHBkI7IJaGpcY6oosHf+Yv5pbfff3Y6VX7EbXfogMcuDuQvlUyZyg4w+QEQ8KCgzgXUCCe6A1WVyBr6Myu8B5FHctzFLu0igkLqJdzjy915ZzRc4yArgal/7ct4SkHnX3nbYbSre1RV9HNGTrXV8WPZLymjt2o/o/MDE1Nl0cybYXsU2VjDb3u02m6vorersDTPatcQ69yq6PfrNzJxcZaNpkFClukcJ8s1L3tjM16iF+ZB7eqev568SI7LAGkYx+qf07YdM+ZFg41x47ORfZ9bKfEGyYWzsTO3cId23R1ftFoLxllnbyh9OqGfEbPE86euVVT3fOmPPITP/mPrXbx/J7OQ4/d3E6rj02XJzDxHGRDNPbq9YRutyKVHBDvOY/J95l5oZdxYLt57GJDZWs3n71TZveJuOpCLePZc+CSncxNFejsJaF1oMMBGV1WwTkIzwo4T8t9b+orucFaHNh6MlyrH63exoEjcUB6xaof5xdtB3H2UaAj9a/RcgwOXLKT2RToGDJ2llTIALO6y9bOGwVe9mkGeKeRbJPhToxvzTYONA40DszkwP8BWQpoYfQx4aUAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle u_{dx}\\mathbf{\\hat{n}_x} + u_{dz}\\mathbf{\\hat{n}_z} + 0.5 \\left(h_{b} \\operatorname{cos}\\left(b_{roll}\\right) - w_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\right) u^{pitch}_{b}\\mathbf{\\hat{b}_x} + (0.5 h_{b} u^{roll}_{b} + 0.5 l_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right))\\mathbf{\\hat{b}_y} + (0.5 l_{b} u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right) + 0.5 w_{b} u^{roll}_{b})\\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "u_d_x n_x + u_d_z n_z + 0.5⋅(h_b⋅cos(bᵣₒₗₗ) - w_b⋅sin(bᵣₒₗₗ))⋅u_b__pitch b_x +\n",
       " (0.5⋅h_b⋅u_b__roll + 0.5⋅l_b⋅u_b__pitch⋅sin(bᵣₒₗₗ)) b_y + (0.5⋅l_b⋅u_b__pitch\n",
       "⋅cos(bᵣₒₗₗ) + 0.5⋅w_b⋅u_b__roll) b_z"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_P2 = P2.v2pt_theory(MB, N, B).subs(u).simplify()\n",
    "N_v_P2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{dx}\\mathbf{\\hat{n}_x} + u_{dz}\\mathbf{\\hat{n}_z} + 0.5 \\left(h_{b} \\operatorname{cos}\\left(b_{roll}\\right) + w_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\right) u^{pitch}_{b}\\mathbf{\\hat{b}_x} + (0.5 h_{b} u^{roll}_{b} - 0.5 l_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right))\\mathbf{\\hat{b}_y} + (- 0.5 l_{b} u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right) - 0.5 w_{b} u^{roll}_{b})\\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "u_d_x n_x + u_d_z n_z + 0.5⋅(h_b⋅cos(bᵣₒₗₗ) + w_b⋅sin(bᵣₒₗₗ))⋅u_b__pitch b_x +\n",
       " (0.5⋅h_b⋅u_b__roll - 0.5⋅l_b⋅u_b__pitch⋅sin(bᵣₒₗₗ)) b_y + (-0.5⋅l_b⋅u_b__pitc\n",
       "h⋅cos(bᵣₒₗₗ) - 0.5⋅w_b⋅u_b__roll) b_z"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_P3 = P3.v2pt_theory(MB, N, B).subs(u).simplify()\n",
    "N_v_P3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u_{dx}\\mathbf{\\hat{n}_x} + u_{dz}\\mathbf{\\hat{n}_z} + 0.5 \\left(h_{b} \\operatorname{cos}\\left(b_{roll}\\right) - w_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\right) u^{pitch}_{b}\\mathbf{\\hat{b}_x} + (0.5 h_{b} u^{roll}_{b} - 0.5 l_{b} u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right))\\mathbf{\\hat{b}_y} + (- 0.5 l_{b} u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right) + 0.5 w_{b} u^{roll}_{b})\\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "u_d_x n_x + u_d_z n_z + 0.5⋅(h_b⋅cos(bᵣₒₗₗ) - w_b⋅sin(bᵣₒₗₗ))⋅u_b__pitch b_x +\n",
       " (0.5⋅h_b⋅u_b__roll - 0.5⋅l_b⋅u_b__pitch⋅sin(bᵣₒₗₗ)) b_y + (-0.5⋅l_b⋅u_b__pitc\n",
       "h⋅cos(bᵣₒₗₗ) + 0.5⋅w_b⋅u_b__roll) b_z"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N_v_P4 = P4.v2pt_theory(MB, N, B).subs(u).simplify()\n",
    "N_v_P4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.6 Angular Velocities"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The expressions for the angular velocities of the rigid bodies are made independent of the time-derivatives of the independent generalized coordinates by substituting the corresponding generalized speeds in their places."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAboAAAAbCAYAAAAKwEp6AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAPPUlEQVR4Ae2di5UUtxKG2T0EgNcRXDsDHhEYMlhDBEAGcIiAgzMwRGAgA7gRGMgAHIFhM8D/p60SanVr+jE905rZrnN6JZVepdKv0qPVsyffv3+/doh0cnJyQ7JfHKLsu5BZ+niicn+VTh5vU76V80xlPFdZf2xT1pp3NxpYsd/U64r9pj6ONbQN7k+XUAoC63mjpzHLlvi5jEr3i3gf5fLcyOOvaPi92v0ib7vp6GbOL4Vtcvuq+E+lNCt/OQ2s2O/U/Yr9TrUcD3Nb3J/IsC2iDQl+VxX/qfp/TQUo8T2Nxb9R+IseJrkzPbdUDuGVEg1IV+jnm3RzkrA3ej2PEv2kfOuOeaO29hu5Yn+4vh3HK/aH66zWlHPgfpEdnSn0nlxWYjmV+NcMvO+UgWM1JjcmyZd6PuaFrOGgARYTYxcAt8mzTnJ1IWjF/uj+WLE/WmX1ZZgL90vu6Jiw3sigMlFFUsM6+Z6AhudGuIvn6Q/ZVbseSf5benwiZ4d2Rw8TfThaVBoGNEeTd8T7XS4LAo52n+ohjl3ZBz3snuNxpNJwzPmvnkCKC+/jjE9++gFi4RHrC5z1zyIaUN+s2E+wKH2s2F8EifutdBbcy8BdG/uomRhgjg9xz/VwEYLwL2lZCmNMicN9kcVxZtpIT7yok5/mvSp+6QL9MpjRyU3TD5PaZ9eB/EGvlgZDGPtTPCbI85RnZXwW/675KY/jzZDP8pAvlCU39K3Hr+4P/a662J0uhLsV+8lYXrG2HdaiURyjyAyEbjCj8aQsEcYyxFk4TIzmbxhXr1vpW3zjUaA/jzz9MbtqL7sqdnCNicb4anrQsadhsREnP9eLeCRqLCYU/lPPO09j5YRJ1PzkSfuN9I1FSpq3Nr9kRSdVY0Ty0V+NfinpUekYE4593KrbVmrHGD660bNif8REZzqrGhuScRDulW52zJ+q0FGkbSQgfK0HYd4LwOE9m1yutge/0mAcec+TvoPjXRHHYBC7FI7TcmrxVQbHbRzfcQR3ZUjt9vdkHFX+nTQcvQddJGn4pACdR1IfhHSkicxLz305jbSmY448PU/ab/RJWn9WXD1ByY9x5Ig2HofD0/NOzzc9LMYWJ8n3VkK8kDyMpY1kfXOl8K82HxX26WfDH7vU2UllHxXud4H50RNdBkJ/j5N3Hsb0r4yJEXWjy4TXlbeTbw33vFmxRx9kokknngcKs9AIZCAnzUv50bFTzEcaPW5UGRTxXZ0nNpc8cQFiedh5vM3KzrJVE/y/JGl8RyjZL/SAK/CT6nFpoR9KAE45eukK4z9i2JR0qNhnzPE0bpj3dvzwBEeH+7kxP3qiS3SfgzCJCp0ajSmGVpGk98nvtvzvO4xniZ+WfWX8NtGguwsabfpiMuOiiRN6/YRBl4shcGJQ+U7smeJ9oUC/+KQX0qpc3+HlC420j9OyvY5qXLWB1fKHpJ25bOita3GVp9tL2PqLMcDR9MGQ5D3fh8yqA4weBfbV14xPLv6l43aWPpeeVtwP0OT1AWlaSRyEdGAr8pIBH5A6vZLnjyQ9Fx0+2YBJyyjxvZyr5mKc0Q/GhbYzefFZRZj4YIjYpTxQPBd+fMKCj/+Z6dgXGPC5mflUfMqmHPrJB+CZ/ByrOVH2PSv7uTMrdWkDE3WLJD8TNlTTjg550Ok/etZfoEEbTVqx39RHKbTivqSZhD/p8wIzHE9lcEuGhdUYhpcdxc+4ShsNqPJjhDGyXIqIxqfEVzp2M1xuAfx0LGW68ZI3HLchTzoBwD9ocn2oXT4RHXR7diW89ATewFLn0ZDiwSK3TzvjdyXXkHIN12A3joOufJZucfxLDr9QsNPJWfUEG3EM2FdbfHfK5b90Yd/V1YN5Vu7R4l7tm83mh4lOBbKqZwuMIeD2XgSxxbHq/J/4i00kSaMlyrXHkiVcOBCfwY9CkK3qX0gZq2el5/IERjAuEhSukqxtTCiOETDFIIyyW5pn4v9rjWDBki926E8WMRy1numhnAcqhwsZnaRyOf4r/s6n4sEH5flRLjhvyKbwIiTZ0BknGY13i7kw1gZ0Ay2Gf8kxaaJTPvpxsI1R+kPCfhGzageTHJM2mH6rfvZvXcnDSRfxvHP3NPKGb2XZHEQ7DDMnlX3UuFf7fKKj6dthXsqkEG6q4dIZ8Zsq46FMebf7jmHb/JKBRiNkkDUtTzyMBXEfU35tfpddbq+eSauHNnFhAUM46VOQfeSTfAzWb3riN3vyM5DhhWv0FsZ4Na7VK8x7syfIKfLJsdFW8VufTqTtUjxldF6ttjIDNvAn9cBryJKWOcZvddBPyvajn0r8LA3Gvxe3pKF8PYviX/Uz0YX+StvR53e55R4b9gdhVu2m//iBjBa2xY9jgHgR46kXn5avKtxLJvTRGgsd7e7FvcqZDfOntuKgQIjVhl9aCAz94Xhy47GKJ9yTe9FRj6/Ub1p7OpIsyxqrZwGDFQw77t/1dLV52QY1awfYXAmPuzeFGawQwIdIw4o2xxfHsly1Z2LkMtJt+T2PgoGY9DcRdX0tJMCwQr+5Ht0VD6O9NVl5yNhoW4mfVYjceXuzJI1gFxaqxv+RY38oZkv4BDO3hZW4e5PfceTYbQAgCVSH+4GYpwljcL815q9ToYTz747uK5xfOqAjGzwBlw5gF8XqpHikRPyeyIFBdRjMNDxJBDO2XNsdQw+lj0+FDKP1XCinKrbpCZ07hoJ80gOTXpj4lIZ4BqUb5JCGP+hL8Xg5nuSSDAOAb95YXLHSZXKMRkDhLjoTs2swkJaFmt9KJcz7Xp9YODqdi0oLwhLf6wWrvihw3lg3xftc+Gd8dxladHcmHT7oEJLFTjiay+KOEvu0Ue3l5uwUzKYqSvsv5TtOU17qrxX3fZinDdviPtVZL+avq6OCgVBnAWoUGw2WeBQAr7Gjs86F19cRSnKYZHqZbRKfomfrk5oWFF2d6UaaY8kSeZqAtUIisAah82d6fDHFbu+l9LfxHRYZC8RC7UMWBw9qTbyX7El/kZ9da04lfp6uqrD03XkBSn1xLkE58u1bfMT2HDH2vY3bYpaJcm5aEvfVYf400S6zsP8igbMxNhcCatcuhThW3DXQWSIEOwSu43/W892eMCHLz04BHnFLTdKD9Sy9s5jgyQ110tzFvb6y2nSj0dNs0vkX9UmYENVuLuBwsYitHhPcI8WBtxJhKEplM+j8aN7z+64jLOBUNuXz/xFxw3diFvYJOuQTj0n3iblhAeIFykW+UF7Cw1viezKw6/px3lg3xz/tcKzjOv452gf/8Dbpc2z9Q9MfG/Y5HZiK2aE625SuVtz3YZ42bYv7BuYp0LANvvk/pfwaktv/83Siu6m0+YQGMIORVSYGT2pMutJT364plcHrQk6InyQL74rkYnjd8PDex41W+Pc+ir8IOfb/p0tvm/QMaGpZULS0ZXoEN75LaqSR3rmgQjz69n6KaRSPPiB2Q/QRL6kjKS8nDByBev/FuMTDRHGWhHNvnEgMwxzR8w40xQDv2HhY2LFbQa5Yp/IxWXJTk+9B2e3ws05BVmsD+WI9imfgUUaLT1xC1JHKkUR1em90cF2vjn905qcRcfEq+bwfuEHtY6OjuJ2x0McYG1M19k1LUzE7h5Krw/1AzNP2Mbjvxbzq9TTcvgb7jjPGxNvTRNv5IAVkGC/nU0AYkNYYsqarX68I/i7pflK/GxMMDrI1jrckLwYA+ZkoeDa9Q1P0Xsj1GSpTW4p6Nmm6jMNeBB1RCTukaPg9n9rGrscB95v8jb6zdK/kMnm44eUj9xxLhD3esjUc6nDD3ohQIJ8kmVBfq75wRK+6GHCv9aDnnf92q+rICbk3tS1P39Ch5EfuFv7VPnBG29nFgTHGCuloO2NlCTpG7KPHoZjNcb1tH9SIe7AWNkc9jRuD+0GYV3284uBkiON1vhYAb+H0JnxHpwCDgE7A6Ph7CxKhSIwVPIxAMFpKSyFMKuFXOixc/I5J6bYm1cGKGgWGW3pyz/QgMy6yld4pYMh4f8TgXvRbQNU/Vs8YMdrtbbsjP5M1bamKEvx8lWDo+2c9XIWPxk1p6Ava4vLTf1y5DoZe8QwSFifkJ450/s1bcTKwfNTVOj41ucCwy8SnCvE9tPiBlI4JsPPbJcXxmQR6Z+IIJB4LJy65cIEGf9jtXcZe/i3xszT0b+8CTGWNxr/yOPaR85bC6OCe/LFPUlmG+FUGRmTUOzovV3np06E25iCwrzZtxKziaQfvnNEbBIYe6qFvcv5z42NbKRf8Y9v8qF3BH2R1V4V7ydQ5Fn5IfelTul7cW5pRNl95HPNUhD3AHoVbQ43vOqTU3rCy0Zj4LY38rBR7vwUaUvbcaUxWjBjfpWz8HmvuurctT/KyoAhGn7Is3PqOatt6jiG/6Wnyd3HKz2QW/11RqhPxG980KYzBhuf/IzDk9bDnVbiTn8RTzk4xqfIZq8jKGGh9x+WyDHVVRlgtD00/NZ3qWbE/zBZjH6rBvWTZiHnwINoJ7r1cuVQSvumVyyL3yan+TKHbyhR2d5aZbejkVeIUAYbk0ezO4OYdCSsidgocrzHwD4WQmYmNlR2Ei+5XamsAQPvOtx27gWOrwBvSc4rpNAd8BqfTK3k4bvX0npdVeEolvqdhRc97wV2S64QJyv3b1MfOurUj3qbAQt4V+wXFZOzacN+HecTfFe6x9+zoGJv8xxXGLLj/cl1/phAFsKV0ui8P2/EqSA1khwkAkDMMbjWc712Q765cVkGH8NNaTGq0w6nKBYULt6Sr/uXfFHHTitXt2EUXg6N4NKo4FkrcuPxbLkeyfzGQ5DpRN/2UL6JKfD/C5h/c0qc7I5XPLeTQtgl6acmlMi5azN0wVuwP0Kv6ozbcFzFPc4RFbPLsuFe5LDJ9oYmN52iUY2PoIr6juwwP+6tCmCj8/RyTCr/ewQpspRk1ID2zBf/JjYvCHAs03hXNWN3BFyX9MIg4nqsei5KVSZGbn2Mn5d5+UtlM3BzdhG/dFO59H9Jb6J4TSOYV+wN1Ll2tuO/R1dSjS1a4r6RgVrFcQqnesPToodboaASlaxYU/L+1dCdRq9yLyCXdsNt4bLpaRIYhlZp8O5nkrH4uM4SfVVNdvOviEwM/Zh0iYg1pVuwP7IUV9/2KmrSj6y92TTGHBmSk/MZWGPQC9BzvWOYQbS2jYg0IN+zoeF/xVU+4GVqxuJ2irdjvVMvKnKiB/wCT+FcHLbQtogAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle u^{roll}_{b}\\mathbf{\\hat{b}_x} -  u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_y} + u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "u_b__roll b_x + -u_b__pitch⋅cos(bᵣₒₗₗ) b_y + u_b__pitch⋅sin(bᵣₒₗₗ) b_z"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.set_ang_vel(N, B.ang_vel_in(N).subs(u))\n",
    "B.ang_vel_in(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u^{roll}_{b}\\mathbf{\\hat{b}_x} -  u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_y} + u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "u_b__roll b_x + -u_b__pitch⋅cos(bᵣₒₗₗ) b_y + u_b__pitch⋅sin(bᵣₒₗₗ) b_z"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W1.set_ang_vel(N, W1.ang_vel_in(N).subs(u))\n",
    "W1.ang_vel_in(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u^{roll}_{b}\\mathbf{\\hat{b}_x} -  u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_y} + u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "u_b__roll b_x + -u_b__pitch⋅cos(bᵣₒₗₗ) b_y + u_b__pitch⋅sin(bᵣₒₗₗ) b_z"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W2.set_ang_vel(N, W2.ang_vel_in(N).subs(u))\n",
    "W2.ang_vel_in(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u^{roll}_{b}\\mathbf{\\hat{b}_x} -  u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_y} + u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "u_b__roll b_x + -u_b__pitch⋅cos(bᵣₒₗₗ) b_y + u_b__pitch⋅sin(bᵣₒₗₗ) b_z"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W3.set_ang_vel(N, W3.ang_vel_in(N).subs(u))\n",
    "W3.ang_vel_in(N)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle u^{roll}_{b}\\mathbf{\\hat{b}_x} -  u^{pitch}_{b} \\operatorname{cos}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_y} + u^{pitch}_{b} \\operatorname{sin}\\left(b_{roll}\\right)\\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "u_b__roll b_x + -u_b__pitch⋅cos(bᵣₒₗₗ) b_y + u_b__pitch⋅sin(bᵣₒₗₗ) b_z"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "W4.set_ang_vel(N, W4.ang_vel_in(N).subs(u))\n",
    "W4.ang_vel_in(N)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.7 Inertia Dyadics"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The inertia dyadics of the rigid bodies are computed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "mB, mW = sm.symbols('m_B, m_W')\n",
    "IBx, IBy, IBz, IWx, IWy, IWz = sm.symbols('I_B_x, I_B_y, I_B_z, I_W_x, I_W_y, I_W_z')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle I_{B x}\\mathbf{\\hat{b}_x}\\otimes \\mathbf{\\hat{b}_x} + I_{B y}\\mathbf{\\hat{b}_y}\\otimes \\mathbf{\\hat{b}_y} + I_{B z}\\mathbf{\\hat{b}_z}\\otimes \\mathbf{\\hat{b}_z}$"
      ],
      "text/plain": [
       "I_B_x b_x⊗b_x + I_B_y b_y⊗b_y + I_B_z b_z⊗b_z"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IB = me.inertia(B, IBx, IBy, IBz)\n",
    "IB"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle I_{W x}\\mathbf{\\hat{w1}_x}\\otimes \\mathbf{\\hat{w1}_x} + I_{W y}\\mathbf{\\hat{w1}_y}\\otimes \\mathbf{\\hat{w1}_y} + I_{W z}\\mathbf{\\hat{w1}_z}\\otimes \\mathbf{\\hat{w1}_z}$"
      ],
      "text/plain": [
       "I_W_x w1_x⊗w1_x + I_W_y w1_y⊗w1_y + I_W_z w1_z⊗w1_z"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IW1 = me.inertia(W1, IWx, IWy, IWz)\n",
    "IW1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle I_{W x}\\mathbf{\\hat{w2}_x}\\otimes \\mathbf{\\hat{w2}_x} + I_{W y}\\mathbf{\\hat{w2}_y}\\otimes \\mathbf{\\hat{w2}_y} + I_{W z}\\mathbf{\\hat{w2}_z}\\otimes \\mathbf{\\hat{w2}_z}$"
      ],
      "text/plain": [
       "I_W_x w2_x⊗w2_x + I_W_y w2_y⊗w2_y + I_W_z w2_z⊗w2_z"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IW2 = me.inertia(W2, IWx, IWy, IWz)\n",
    "IW2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle I_{W x}\\mathbf{\\hat{w3}_x}\\otimes \\mathbf{\\hat{w3}_x} + I_{W y}\\mathbf{\\hat{w3}_y}\\otimes \\mathbf{\\hat{w3}_y} + I_{W z}\\mathbf{\\hat{w3}_z}\\otimes \\mathbf{\\hat{w3}_z}$"
      ],
      "text/plain": [
       "I_W_x w3_x⊗w3_x + I_W_y w3_y⊗w3_y + I_W_z w3_z⊗w3_z"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IW3 = me.inertia(W3, IWx, IWy, IWz)\n",
    "IW3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle I_{W x}\\mathbf{\\hat{w4}_x}\\otimes \\mathbf{\\hat{w4}_x} + I_{W y}\\mathbf{\\hat{w4}_y}\\otimes \\mathbf{\\hat{w4}_y} + I_{W z}\\mathbf{\\hat{w4}_z}\\otimes \\mathbf{\\hat{w4}_z}$"
      ],
      "text/plain": [
       "I_W_x w4_x⊗w4_x + I_W_y w4_y⊗w4_y + I_W_z w4_z⊗w4_z"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "IW4 = me.inertia(W4, IWx, IWy, IWz)\n",
    "IW4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.8 Loads"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The loads (forces) acting on the vehicle are defined: gravitational, lift, propulsive, collision (contact/penetration) and frictional forces are considered."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5.8.1 Gravitational Forces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = sm.symbols('g')\n",
    "\n",
    "G_B = -(mB * g) * N.z\n",
    "G_W1 = -(mW * g) * N.z\n",
    "G_W2 = -(mW * g) * N.z\n",
    "G_W3 = -(mW * g) * N.z\n",
    "G_W4 = -(mW * g) * N.z"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5.8.2 Lift Forces"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle C_{L} \\left(H - d_{z}\\right) u_{dx} \\operatorname{sin}\\left(\\alpha_{4}\\right)\\mathbf{\\hat{n}_z}$"
      ],
      "text/plain": [
       "C_L⋅(H - d_z)⋅u_d_x⋅sin(α₄) n_z"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "CL, H = sm.symbols('C_L, H')\n",
    "\n",
    "FL1 = CL * sm.sin(alpha1) * N_v_MB.dot(N.x).subs(u) * (H - dz) * N.z\n",
    "\n",
    "FL2 = CL * sm.sin(alpha2) * N_v_MB.dot(N.x).subs(u) * (H - dz) * N.z\n",
    "\n",
    "FL3 = CL * sm.sin(alpha3) * N_v_MB.dot(N.x).subs(u) * (H - dz) * N.z\n",
    "\n",
    "FL4 = CL * sm.sin(alpha4) * N_v_MB.dot(N.x).subs(u) * (H - dz) * N.z\n",
    "\n",
    "FL4"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5.8.3 Propulsive Force"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [],
   "source": [
    "FPmag = sm.symbols('FPmag')\n",
    "FP = FPmag * N.x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### 5.8.4 Contact Forces: Subsurface Travel Avoidance"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Contact collision and frictional forces act on the wheels of the vehicle."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "kc, cc, mu, vs = sm.symbols('k_c, c_c, mu, v_s')\n",
    "\n",
    "P1_B = P1.pos_from(O).dot(N.z).subs(u)\n",
    "penetration1 = (sm.Abs(P1_B) - P1_B)/2\n",
    "vertical_force1 = (kc*penetration1**3 - P1_B)*(1-cc*P1.vel(N).dot(N.z).subs(u))\n",
    "friction_force_x1 = -mu*vertical_force1*((2/(1+sm.exp(-P1.vel(N).dot(N.x).subs(u)/vs))) - 1)\n",
    "friction_force_y1 = -mu*vertical_force1*((2/(1+sm.exp(-P1.vel(N).dot(N.y).subs(u)/vs))) - 1)\n",
    "contact_force1 = friction_force_x1*N.x + friction_force_y1*N.y + vertical_force1*N.z  \n",
    "\n",
    "P2_B = P2.pos_from(O).dot(N.z).subs(u)\n",
    "penetration2 = (sm.Abs(P2_B) - P2_B)/2\n",
    "vertical_force2 = (kc*penetration2**3 - P2_B)*(1-cc*P2.vel(N).dot(N.z).subs(u))\n",
    "friction_force_x2 = -mu*vertical_force2*((2/(1+sm.exp(-P2.vel(N).dot(N.x).subs(u)/vs))) - 1)\n",
    "friction_force_y2 = -mu*vertical_force2*((2/(1+sm.exp(-P2.vel(N).dot(N.y).subs(u)/vs))) - 1)\n",
    "contact_force2 = friction_force_x2*N.x + friction_force_y2*N.y + vertical_force2*N.z \n",
    "\n",
    "P3_B = P3.pos_from(O).dot(N.z).subs(u)\n",
    "penetration3 = (sm.Abs(P3_B) - P3_B)/2\n",
    "vertical_force3 = (kc*penetration3**3 - P3_B)*(1-cc*P3.vel(N).dot(N.z).subs(u))\n",
    "friction_force_x3 = -mu*vertical_force3*((2/(1+sm.exp(-P3.vel(N).dot(N.x).subs(u)/vs))) - 1)\n",
    "friction_force_y3 = -mu*vertical_force3*((2/(1+sm.exp(-P3.vel(N).dot(N.y).subs(u)/vs))) - 1)\n",
    "contact_force3 = friction_force_x3*N.x + friction_force_y3*N.y + vertical_force3*N.z \n",
    "\n",
    "P4_B = P4.pos_from(O).dot(N.z).subs(u)\n",
    "penetration4 = (sm.Abs(P4_B) - P4_B)/2\n",
    "vertical_force4 = (kc*penetration4**3 - P4_B)*(1-cc*P4.vel(N).dot(N.z).subs(u))\n",
    "friction_force_x4 = -mu*vertical_force4*((2/(1+sm.exp(-P4.vel(N).dot(N.x).subs(u)/vs))) - 1)\n",
    "friction_force_y4 = -mu*vertical_force4*((2/(1+sm.exp(-P4.vel(N).dot(N.y).subs(u)/vs))) - 1)\n",
    "contact_force4 = friction_force_x4*N.x + friction_force_y4*N.y + vertical_force4*N.z "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 5.9 Kane's Method"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The model is set up to leverage the accessibility of Kane's method of solving multibody dynamics problems."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "kdes = [z1.rhs - z1.lhs, z2.rhs - z2.lhs, z3.rhs - z3.lhs, z4.rhs - z4.lhs]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "bogie = me.RigidBody('Bogie', MB, B, mB, (IB, MB))\n",
    "wing1 = me.RigidBody('Wing_1', MW1, W1, mW, (IW1, MW1))\n",
    "wing2 = me.RigidBody('Wing_2', MW2, W2, mW, (IW2, MW2))\n",
    "wing3 = me.RigidBody('Wing_3', MW3, W3, mW, (IW3, MW3))\n",
    "wing4 = me.RigidBody('Wing_4', MW4, W4, mW, (IW4, MW4))\n",
    "\n",
    "bodies = [bogie, wing1, wing2, wing3, wing4]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [],
   "source": [
    "loads = [(MB, G_B + FP),\n",
    "         (MW1, G_W1 + FL1),\n",
    "         (MW2, G_W2 + FL2),\n",
    "         (MW3, G_W3 + FL3),\n",
    "         (MW4, G_W4 + FL4),\n",
    "         (P1, contact_force1),\n",
    "         (P2, contact_force2),\n",
    "         (P3, contact_force3),\n",
    "         (P4, contact_force4),\n",
    "        ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [],
   "source": [
    "kane = me.KanesMethod(N, (dx, dz, bpitch, broll), (u_dx, u_dz, u_bpitch, u_broll), kd_eqs=kdes)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The holonomic generalized active forces, $f_r$, and generalized inertia forces, $f_r*$, are computed using Kane's method."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [],
   "source": [
    "fr, frstar = kane.kanes_equations(bodies, loads=loads)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [],
   "source": [
    "# fr"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [],
   "source": [
    "# frstar"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6. Simulation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The vehicle model is validated by means of a simulation, by defining the constants and specifying the initial conditions. One simulation case (maximum levitation height, $H$ = 1 meter) is demonstrated for a 50 second run.\n",
    "\n",
    "The magnetic _wings_ are pitched at 15 degrees each. The coefficient of friction between the wheels and the track surface is chosen to be 0.2. An axial propulsive force of 10 N is applied. The vehicle starts at an initial axial speed of 10 m/s."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [],
   "source": [
    "sys = System(kane)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left\\{C_{L}, \\left|{FP}\\right|, H, I_{B x}, I_{B y}, I_{B z}, I_{W x}, I_{W y}, I_{W z}, \\alpha_{1}, \\alpha_{2}, \\alpha_{3}, \\alpha_{4}, c_{c}, g, h_{b}, k_{c}, l_{b}, m_{B}, m_{W}, \\mu, v_{s}, w_{b}, w_{w}\\right\\}$"
      ],
      "text/plain": [
       "{C_L, |FP|, H, I_B_x, I_B_y, I_B_z, I_W_x, I_W_y, I_W_z, α₁, α₂, α₃, α₄, c_c, \n",
       "g, h_b, k_c, l_b, m_B, m_W, μ, vₛ, w_b, w_w}"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sys.constants_symbols"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "sys.constants = {CL: 50,\n",
    "                 FPmag: 10,\n",
    "                 H: 1,\n",
    "                 IBx: 1.02,\n",
    "                 IBy: 19.07,\n",
    "                 IBz: 20.04,\n",
    "                 IWx: 0.00204587701,\n",
    "                 IWy: 0.0982636759,\n",
    "                 IWz: 0.0985611947,\n",
    "                 alpha1: 0.261799,\n",
    "                 alpha2: 0.261799,\n",
    "                 alpha3: 0.261799,\n",
    "                 alpha4: 0.261799,\n",
    "                 cc: 0.1,\n",
    "                 g: 9.81,\n",
    "                 hb: 0.025,\n",
    "                 kc: 5.0e7,\n",
    "                 lb: 1.2,\n",
    "                 mB: 34.04,\n",
    "                 mW: 3.06451,\n",
    "                 mu: 0.2, #1.0\n",
    "                 vs: 0.01, #0.85\n",
    "                 wb: 0.45,\n",
    "                 ww: 0.025\n",
    "                 }"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "sys.times = np.linspace(0.0, 50.0, num=1000)\n",
    "t = me.dynamicsymbols._t"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "sys.initial_conditions = {dx: 0.0,\n",
    "                          dz: 0.1,\n",
    "                          bpitch: 0.0,\n",
    "                          broll: 0.0,\n",
    "                          u_dx: 10.0,\n",
    "                          u_dz: 0.0,\n",
    "                          u_bpitch: 0.0,\n",
    "                          u_broll: 0.0\n",
    "                         }"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sys.integrate()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(sys.times, x[:, 5])\n",
    "ax.set_xlabel('{} [s]'.format(sm.latex(t, mode='inline'))); ax.set_ylabel('{} [m/s]'.format(sm.latex(u_dz, mode='inline')));\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\rightarrow$ Vertical speed of the mass-center of the vehicle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots()\n",
    "ax.plot(sys.times, x[:, 6])\n",
    "ax.set_xlabel('{} [s]'.format(sm.latex(t, mode='inline'))); ax.set_ylabel('{} [rad/s]'.format(sm.latex(u_bpitch, mode='inline')));\n",
    "# plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\rightarrow$ Pitch rate of the _Bogie_"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 7. 3D Visualization"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A 3D animation of the simulation is developed using pythreejs and PyDy (_Courtesy of Dr. Jason Moore_), to gain more insight into the behavior of the vehicle under different levitation regimes."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [],
   "source": [
    "from pydy.viz.shapes import Cube, Cylinder, Sphere, Plane\n",
    "from pydy.viz.visualization_frame import VisualizationFrame\n",
    "from pydy.viz import Scene\n",
    "from ipywidgets import Image, Video\n",
    "import pythreejs as pjs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [],
   "source": [
    "bogie_shape = Cube(1.2, color='red')\n",
    "wing1_shape = Cube(0.4, color='green')\n",
    "wing2_shape = Cube(0.4, color='green')\n",
    "wing3_shape = Cube(0.4, color='green')\n",
    "wing4_shape = Cube(0.4, color='green')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [],
   "source": [
    "v1 = VisualizationFrame('Bogie',\n",
    "                        N.orientnew('Bogie', 'Body', [-bpitch, broll, 0], 'YXZ'),\n",
    "                        MB,\n",
    "                        bogie_shape)\n",
    "\n",
    "v2 = VisualizationFrame('Wing1', \n",
    "                        B.orientnew('Wing1', 'Axis', (alpha1, -B.y)),\n",
    "                        MW1.locatenew('M_W_1', (1/6) * lb * B.x),\n",
    "                        wing1_shape)\n",
    "\n",
    "v3 = VisualizationFrame('Wing2', \n",
    "                        B.orientnew('Wing2', 'Axis', (alpha1, -B.y)),\n",
    "                        MW2.locatenew('M_W_2', (1/6) * lb * B.x),\n",
    "                        wing2_shape)\n",
    "\n",
    "v4 = VisualizationFrame('Wing3', \n",
    "                        B.orientnew('Wing3', 'Axis', (alpha1, -B.y)),\n",
    "                        MW3.locatenew('M_W_3', -(1/6) * lb * B.x),\n",
    "                        wing3_shape)\n",
    "\n",
    "v5 = VisualizationFrame('Wing4', \n",
    "                        B.orientnew('Wing4', 'Axis', (alpha1, -B.y)),\n",
    "                        MW4.locatenew('M_W_4', -(1/6) * lb * B.x),\n",
    "                        wing4_shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "scene = Scene(N, O, v1, v2, v3, v4, v5, system=sys)\n",
    "scene.create_static_html(overwrite=True, silent=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [],
   "source": [
    "bogie_mesh = pjs.Mesh(\n",
    "    pjs.BoxBufferGeometry(1.2, 0.45, 0.025),\n",
    "    pjs.MeshStandardMaterial(color='red'),\n",
    "    name=\"Bogie\"\n",
    ")\n",
    "\n",
    "wing1_mesh = pjs.Mesh(\n",
    "    pjs.BoxBufferGeometry(height=0.025, width=0.4, depth=0.025),\n",
    "    pjs.MeshStandardMaterial(color='green'),\n",
    "    name='Wing1'\n",
    ")\n",
    "\n",
    "wing2_mesh = pjs.Mesh(\n",
    "    pjs.BoxBufferGeometry(height=0.025, width=0.4, depth=0.025),\n",
    "    pjs.MeshStandardMaterial(color='green'),\n",
    "    name='Wing2'\n",
    ")\n",
    "\n",
    "wing3_mesh = pjs.Mesh(\n",
    "    pjs.BoxBufferGeometry(height=0.025, width=0.4, depth=0.025),\n",
    "    pjs.MeshStandardMaterial(color='green'),\n",
    "    name='Wing3'\n",
    ")\n",
    "\n",
    "wing4_mesh = pjs.Mesh(\n",
    "    pjs.BoxBufferGeometry(height=0.025, width=0.4, depth=0.025),\n",
    "    pjs.MeshStandardMaterial(color='green'),\n",
    "    name='Wing4'\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [],
   "source": [
    "bogie_matrices = v1.evaluate_transformation_matrix(x, list(sys.constants.values()))\n",
    "wing1_matrices = v2.evaluate_transformation_matrix(x, list(sys.constants.values()))\n",
    "wing2_matrices = v3.evaluate_transformation_matrix(x, list(sys.constants.values()))\n",
    "wing3_matrices = v4.evaluate_transformation_matrix(x, list(sys.constants.values()))\n",
    "wing4_matrices = v5.evaluate_transformation_matrix(x, list(sys.constants.values()))\n",
    "\n",
    "bogie_track = pjs.VectorKeyframeTrack(\n",
    "    name='scene/Bogie.matrix',\n",
    "    times=list(sys.times),\n",
    "    values=bogie_matrices)\n",
    "\n",
    "wing1_track = pjs.VectorKeyframeTrack(\n",
    "    name='scene/Wing1.matrix',\n",
    "    times=list(sys.times),\n",
    "    values=wing1_matrices)\n",
    "\n",
    "wing2_track = pjs.VectorKeyframeTrack(\n",
    "    name='scene/Wing2.matrix',\n",
    "    times=list(sys.times),\n",
    "    values=wing2_matrices)\n",
    "\n",
    "wing3_track = pjs.VectorKeyframeTrack(\n",
    "    name='scene/Wing3.matrix',\n",
    "    times=list(sys.times),\n",
    "    values=wing3_matrices)\n",
    "\n",
    "wing4_track = pjs.VectorKeyframeTrack(\n",
    "    name='scene/Wing4.matrix',\n",
    "    times=list(sys.times),\n",
    "    values=wing4_matrices)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "bogie_mesh.matrixAutoUpdate = False\n",
    "wing1_mesh.matrixAutoUpdate = False\n",
    "wing2_mesh.matrixAutoUpdate = False\n",
    "wing3_mesh.matrixAutoUpdate = False\n",
    "wing4_mesh.matrixAutoUpdate = False"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [],
   "source": [
    "bogie_mesh.matrix = bogie_matrices[0]\n",
    "wing1_mesh.matrix = wing1_matrices[0]\n",
    "wing2_mesh.matrix = wing2_matrices[0]\n",
    "wing3_mesh.matrix = wing3_matrices[0]\n",
    "wing4_mesh.matrix = wing4_matrices[0]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_arrow = pjs.ArrowHelper(dir=[1, 0, 0], length=2.0, color='blue')\n",
    "y_arrow = pjs.ArrowHelper(dir=[0, 1, 0], length=2.0, color='red')\n",
    "z_arrow = pjs.ArrowHelper(dir=[0, 0, 1], length=2.0,color='green')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [],
   "source": [
    "view_width = 960\n",
    "view_height = 720\n",
    "\n",
    "camera = pjs.PerspectiveCamera(position=[1, 1, 1],\n",
    "                               aspect=view_width/view_height)\n",
    "key_light = pjs.DirectionalLight(position=[0, 10, 10])\n",
    "ambient_light = pjs.AmbientLight()\n",
    "\n",
    "scene_pjs = pjs.Scene(children=[bogie_mesh, wing1_mesh, wing2_mesh, wing3_mesh, wing4_mesh,\n",
    "                                x_arrow, y_arrow, z_arrow, \n",
    "                                camera, key_light, ambient_light])\n",
    "\n",
    "controller = pjs.OrbitControls(controlling=camera)\n",
    "renderer = pjs.Renderer(camera=camera, scene=scene_pjs, controls=[controller], width=view_width, height=view_height)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "040bd2978e314a95bf6229f8cd1a1f8a",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Renderer(camera=PerspectiveCamera(aspect=1.3333333333333333, position=(1.0, 1.0, 1.0), quaternion=(0.0, 0.0, 0…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "renderer"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "9418b40532954cbbb79882af6d9734f9",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "AnimationAction(clip=AnimationClip(duration=50.0, tracks=(VectorKeyframeTrack(name='scene/Bogie.matrix', times…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "clip = pjs.AnimationClip(tracks=[bogie_track, wing1_track, wing2_track, wing3_track, wing4_track], duration=sys.times[-1])\n",
    "action = pjs.AnimationAction(pjs.AnimationMixer(scene_pjs), clip, scene_pjs)\n",
    "action"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 8. Results and Conclusion"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.1 Case 1: \n",
    "\n",
    "H = 10 [m]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center><video controls src=\"Media/MagLev_unstable.mov\" width=\"750\" /></center>\n",
    "\n",
    "##### <center>The vehicle displays extreme instability, with the maximum levitation height set to ten meters.</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Media/Plot_ten.png\" width=\"1100\" />\n",
    "\n",
    "##### <center>Figure 7: Vertical speed and vehicle pitch rate for Case 1.</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.2 Case 2: \n",
    "\n",
    "H = 0.5 [m]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center><video controls src=\"Media/MagLev_Friction.mov\" width=\"750\" /></center>\n",
    "\n",
    "##### <center>The vehicle comes to a halt due to friction, with the maximum levitation height set to half a meter.</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Media/Plot_half.png\" width=\"1100\" />\n",
    "\n",
    "##### <center>Figure 8: Vertical speed and vehicle pitch rate for Case 2.</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 8.3 Case 3: \n",
    "\n",
    "H = 1 [m]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center><video controls src=\"Media/MagLev_Flyby.mov\" width=\"750\" /></center>\n",
    "\n",
    "##### <center>The vehicle is fairly stable, with the maximum levitation height set to one meter.</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img src=\"Media/Plot_one.png\" width=\"1100\" />\n",
    "\n",
    "##### <center>Figure 9: Vertical speed and vehicle pitch rate for Case 3.</center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The vehicle is fairly stable in _Case 3_, with the maximum levitation height set to 1 meter. This would be the most favorable maximum levitation height among the simulations cases that were run. The magnitudes of the vertical speed and the pitch rate are relatively extremely low values in this case, and further stability of the vehicle system can be ensured by designing appropriate control systems."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 9. Further Extensions/Future Work"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As an extension of this project, the vehicle system has been linearized in order to enable the design of control systems to further stabilize the MagLev vehicle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 9.1 Linearization"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sm.Matrix([u_dx, u_dz, u_bpitch, u_broll, dx, dz, bpitch, broll])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [],
   "source": [
    "xeq = sm.zeros(8, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [],
   "source": [
    "f = fr + frstar\n",
    "# me.find_dynamicsymbols(f), f.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [],
   "source": [
    "kdes_mat = sm.Matrix(kdes)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 8, \\  1\\right)$"
      ],
      "text/plain": [
       "(8, 1)"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hol = f.col_join(kdes_mat)\n",
    "hol.shape # length = 2n --> Holonomic System"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [],
   "source": [
    "dx1, dz1, bpitch1, broll1, udx1, udz1, ubpitch1, ubroll1 = me.dynamicsymbols('d_{x1}, d_{z1}, b_{pitch1}, b_{roll1}, u_{dx1}, u_{dz1}, u_{bpitch1}, u_{broll1}')\n",
    "xd = sm.Matrix([dx1, dz1, bpitch1, broll1, udx1, udz1, ubpitch1, ubroll1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 73,
   "metadata": {},
   "outputs": [],
   "source": [
    "# {k: v for k, v in zip(x.diff(), xd)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [],
   "source": [
    "hol = hol.subs({k: v for k, v in zip(x.diff(), xd)})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [],
   "source": [
    "# me.find_dynamicsymbols(hol)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [],
   "source": [
    "xeq_sub = {k: v for k, v in zip(x, xeq)}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "hol_lin = hol.subs(xeq_sub) + hol.jacobian(x).subs(xeq_sub)*(x - xeq)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [],
   "source": [
    "Mat = hol_lin.jacobian(xd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 79,
   "metadata": {},
   "outputs": [],
   "source": [
    "xdot = Mat.LUsolve(-(hol_lin - Mat*xd))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = xdot.jacobian(x)\n",
    "# A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Future work would involve the development of a high-fidelity model of the vehicle, incorporating additional degrees of freedom, and also the development control systems to ensure stability and efficient control of the MagLev vehicle."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<br>\n",
    "<br>\n",
    "\n",
    "<img src=\"Media/Logo_2.png\" width=\"250\"/>"
   ]
  }
 ],
 "metadata": {
  "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": "3.6.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}