2 questions for rigid body dynamics.

questions.ipynb

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    "## Question 1\n",
    "\n",
    "Do you have the deduction of the transformation matrix $T$? specifically, the following one in terms of Euler Angles:\n",
    "\n",
    "$$\n",
    "T = \\begin{pmatrix} 1 & \\sin{\\phi}\\tan{\\theta} & \\cos{\\phi}\\tan{\\theta} \\\\ 0 & \\cos{\\phi} & -\\sin{\\phi} \\\\ 0 & \\sin{\\phi}\\sec{\\theta} & \\cos{\\phi}\\sec{\\theta} \\end{pmatrix}\n",
    "$$\n",
    "\n",
    "It looks strange because:\n",
    "\n",
    "1. it doesn't depend on $\\psi$, and has no relationship with rotation matrix\n",
    "\n",
    "2. according to definition (assuming $R$ is the rotation matrix from body frame to world frame):\n",
    "\n",
    "- $\\vec \\omega_{body} = (p, q, r)^T$\n",
    "\n",
    "- $\\vec \\omega_{world} = (\\phi, \\theta, \\psi)^T$\n",
    "\n",
    "- $\\forall$ simple vector $\\vec a$: $\\vec a_{world} = R \\vec a_{body}$ (not sure if it always applies to pseudovector as well)\n",
    "\n",
    "- while angular velocity $\\vec \\omega_{frame}$ is indirectly defined as an operator $\\vec \\omega_{frame} \\times$ which can convert a point $\\vec x_{frame}$ to its linear speed $\\vec v_{x-frame}$, e.g.:\n",
    "\n",
    "$$\n",
    "\\vec v_{x-body} = \\vec \\omega_{body} \\times \\vec x_{body} \\\\\n",
    "\\vec v_{x-world} = \\vec \\omega_{world} \\times \\vec x_{world}\n",
    "$$\n",
    "\n",
    "- both $\\vec x_{world}$ and $\\vec v_{x-world}$ are simple, linear vectors, so:\n",
    "\n",
    "$$\n",
    "R \\vec v_{x-body} = \\vec \\omega_{world} \\times R \\vec x_{body}\n",
    "$$\n",
    "\n",
    "=>\n",
    "\n",
    "$$\n",
    "\\vec v_{x-body} = R^T \\vec \\omega_{world} \\times R \\vec x_{body}\n",
    "$$\n",
    "\n",
    "=>\n",
    "\n",
    "$$\n",
    "\\vec \\omega_{body} \\times = R^T \\vec \\omega_{world} \\times R\n",
    "$$\n",
    "\n",
    "=>\n",
    "\n",
    "$$\n",
    "\\vec \\omega_{body} = R^T \\vec \\omega_{world}\n",
    "$$\n",
    "\n",
    "which is evidently different from $T$, so where did $T$ come from?\n"
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    "## Question 2\n",
    "\n",
    "I'm trying to reproduce an alternative formulation of dynamics $\\vec F \\Rightarrow \\dot {\\vec v}$ in body frame, which is:\n",
    "\n",
    "$$\n",
    "m \\dot {\\vec v}_{body} + m \\vec \\omega_{body} \\times \\vec v_{body} = \\vec F_{body}\n",
    "$$\n",
    "\n",
    "(It's complex and useless, I just verify it for OCD).\n",
    "\n",
    "So I start from the well known:\n",
    "\n",
    "$$\n",
    "\\vec F_{world} = m \\dot {\\vec v}_{world}\n",
    "$$\n",
    "\n",
    "=>\n",
    "\n",
    "$$\n",
    "R \\vec F_{body} = m \\frac{d R \\vec v_{body}} {d t}\n",
    "$$\n",
    "\n",
    "(chain rule) =>\n",
    "\n",
    "$$\n",
    "R \\vec F_{body} = m R \\frac{d \\vec v_{body}} {d t} + m \\frac{d R}{d t} \\vec v_{body}\n",
    "$$\n",
    "\n",
    "(by using $\\vec \\omega_{world} \\times R = \\frac{d R}{d t}$, as each column of R is a point in the world frame) =>\n",
    "\n",
    "$$\n",
    "\\vec F_{body} = m \\frac{d \\vec v_{body}} {d t} + m R^T \\frac{d R}{d t} \\vec v_{body} \\\\\n",
    "= m \\dot {\\vec v}_{body} + m R^T \\vec \\omega_{world} \\times R  \\vec v_{body} \\\\\n",
    "= m \\dot {\\vec v}_{body} + m \\vec \\omega_{body} \\times \\vec v_{body}\n",
    "$$\n",
    "\n",
    "Is this correct? If it is then the last line of **Question 1** should also be correct.\n"
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