本文仅供学习使用
本文参考：
B站：CLEAR_LAB
笔者带更新-运动学
课程主讲教师：
Prof. Wei Zhang

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Given Frame trajectory $\left[T\right]\left(t\right)=\left(\left[Q\right]\left(t\right),{\vec{R}}_p\left(t\right)\right)$ wrt { O } \left\{ O \right\} {O}

Question : What is the velocity (twist/spatial velocity) of frame at time $t$
$\left[T\right]\left(t\right)\in {\mathbb{R}}^{4\times 4}$ : $4\times 4$ matrix $SE\left(3\right)$
$\log \left(\left[T\right]\left(t\right)\right)\to \mathcal{S}\theta$ : is $\mathcal{S}$ (unit) the velocity of $\left[T\right]\left(t\right)$? —— no

${\vec{R}}_p\left(t\right)$ : position vector ; ${\dot{\vec{R}}}_p\left(t\right)$ velocity vector;
$\mathcal{S}\theta \leftrightarrow \left[T\right]\left(t\right)$ : position coordination (what velocity?——$\left[\dot{T}\right]\left(t\right)$?)
$\left[\dot{T}\right]\left(t\right)\in {\mathbb{R}}^{4\times 4}\notin SE\left(3\right)$

1.Instantanenous Velocity of Rotating Frames

{ A } \left\{ A \right\} {A} frame is rotating with orientation $\left[Q_A\right]\left(t\right)$ and velocity ${\vec{\omega }}_A\left(t\right)$ at time $t$ (Note: everything is wrt { O } \left\{ O \right\} {O} frame)

Let $\hat{\omega }\theta =\log \left(\left[Q_A\right]\left(t\right)\right)$ can be obtained from the reference frame (say { O } \left\{ O \right\} {O} frame) by rotating about $\hat{\omega }$ by $\theta$ degree

• $\hat{\omega }\theta$ only describes the current orientation of { A } \left\{ A \right\} {A} relative to { O } \left\{ O \right\} {O} , it does not contain info about how the frame is rotating at time $t$

What is the relation between ${\vec{\omega }}_A\left(t\right)$ and $\left[Q_A\right]\left(t\right)$
$$\frac{\mathrm{d}}{\mathrm{d}t}\left[Q_A\right]\left(t\right)={\widetilde{\stackrel{⃗}{\omega }}}_A\left(t\right)\left[Q_A\right]\left(t\right)\Rightarrow {\widetilde{\stackrel{⃗}{\omega }}}_A\left(t\right)=\left[{\dot{Q}}_A\right]\left(t\right){\left[Q_A\right]}^{-1}\left(t\right)1$$

What is ${\vec{\omega }}_{\mathrm{A}}^A$?
$${\vec{\omega }}_{\mathrm{A}}^A=\left[Q_{\mathrm{O}}^A\right]{\vec{\omega }}_{\mathrm{A}}^O$$
velocity of $A$ relative to { O } \left\{ O \right\} {O} , expressed in { A } \left\{ A \right\} {A}
$${\widetilde{\stackrel{⃗}{\omega }}}_{\mathrm{A}}^A=\widetilde{\left[Q_{\mathrm{O}}^A\right]{\vec{\omega }}_{\mathrm{A}}^O}=\left[Q_{\mathrm{O}}^A\right]{\widetilde{\stackrel{⃗}{\omega }}}_{\mathrm{A}}^O{\left[Q_{\mathrm{O}}^A\right]}^{\mathrm{T}}=\left[Q_{\mathrm{O}}^A\right]\left[{\dot{Q}}_{\mathrm{A}}^O\right]{\left[Q_{\mathrm{A}}^O\right]}^{-1}{\left[Q_{\mathrm{O}}^A\right]}^{\mathrm{T}}=\left[Q_{\mathrm{O}}^A\right]\left[{\dot{Q}}_{\mathrm{A}}^O\right]={\left[Q_{\mathrm{A}}^O\right]}^{-1}\left[{\dot{Q}}_{\mathrm{A}}^O\right]$$

2.Instantanenous Velocity of Moving Frames

{ A } \left\{ A \right\} {A} moving frame with configuration $\left[T_A\right]\left(t\right)$ at t ime $t$ undergoes a rigid body motion with velocity ${\mathcal{V}}_A\left(t\right)=\left(\vec{\omega },\vec{v}\right)$ (Note: everything is wrt { O } \left\{ O \right\} {O} frame)

The exponential coordinate $\hat{\mathcal{S}}\left(t\right)\theta \left(t\right)=\log \left(\left[T_A\right]\left(t\right)\right)$ only indicates the current configuration of { A } \left\{ A \right\} {A} , and does not tell us about how the frame is moving at time $t$

What is the relation between ${\mathcal{V}}_A\left(t\right)$ and $\left[T_A\right]\left(t\right)$ ?
$$\frac{\mathrm{d}}{\mathrm{d}t}\left[T_A\right]\left(t\right)=\left[{\mathcal{V}}_A\right]\left(t\right)\left[T_A\right]\left(t\right)\Rightarrow \left[{\mathcal{V}}_A\right]\left(t\right)=\left[{\dot{T}}_A\right]\left(t\right){\left[T_A\right]}^{-1}\left(t\right)$$

3.Review/Summary of Rigid Body Velocity & Operation

• spatial velocity / twist $\mathcal{V}=\left[\begin{array}{c}\vec{\omega }\\ {\vec{v}}_{\mathrm{O}}\end{array}\right]$ , ${\vec{v}}_{\mathrm{O}}$ reference point $O$ may/may not move with the body
  $\vec{\omega }$ : angular velocity
  ${\vec{v}}_{\mathrm{O}}$ : velocity of the body-fixed partical currently coincides with $O$
  Given $\mathcal{V}=\left[\begin{array}{c}\vec{\omega }\\ {\vec{v}}_{\mathrm{O}}\end{array}\right]$ , any body fixed point $P$ , its velocity is ${\vec{v}}_{\mathrm{P}}={\vec{v}}_{\mathrm{O}}+\vec{\omega }\times {\vec{R}}_{\mathrm{OP}}$

• Twist in frames : Given frame { B } \left\{ B \right\} {B} , { O } \left\{ O \right\} {O} with relation
  ${\mathcal{V}}^O=\left[\begin{array}{c}{\vec{\omega }}^O\\ {\vec{v}}_{\mathrm{O}}^O\end{array}\right],{\mathcal{V}}^B=\left[\begin{array}{c}{\vec{\omega }}^B\\ {\vec{v}}_{\mathrm{O}}^B\end{array}\right]$ —— $\mathcal{V}$ is a twist of some rigid body
  ${\mathcal{V}}^O=\left[X_{\mathrm{B}}^O\right]{\mathcal{V}}^B$ —— change of coordinate for twist$\left[X_{\mathrm{B}}^O\right]\in {\mathbb{R}}^{6\times 6}$
  $\left[X_{\mathrm{B}}^O\right]\in {\mathbb{R}}^{6\times 6}\left[X_{\mathrm{B}}^O\right]=\left[\begin{array}{cc}\left[Q_{\mathrm{B}}^O\right]& 0\\ {\widetilde{\stackrel{⃗}{R}}}_{\mathrm{B}}^O\left[Q_{\mathrm{B}}^O\right]& \left[Q_{\mathrm{B}}^O\right]\end{array}\right]$
  For given $\left[T\right]=\left(\left[Q\right],\vec{R}\right)\Rightarrow \left[X\right]=\left[A{d}_T\right]=\left[\begin{array}{cc}\left[Q\right]& 0\\ \widetilde{\stackrel{⃗}{R}}\left[Q\right]& \left[Q\right]\end{array}\right]$

• screw axis : $\mathcal{S}=\left(\hat{s},{\vec{R}}_{\mathrm{q}},h\right)\leftrightarrow \left[\begin{array}{c}\vec{\omega }\\ \vec{v}\end{array}\right]=\left[\begin{array}{c}\hat{s}\\ h\hat{s}-\hat{s}\times {\vec{R}}_{\mathrm{q}}\end{array}\right]$
  Al rigid body motion can be "thought of " screw motion rotation & linear motion along the axis
  we typically write $\mathcal{V}=\mathcal{S}\dot{\theta }$ ($\mathcal{S}$ - ‘unit’ normalized twist)

• rotation operation / Exp coordinate (wrt { O } \left\{ O \right\} {O})
  OED for rotation : ${\dot{\vec{R}}}_{\mathrm{p}}=\vec{\omega }\times {\vec{R}}_{\mathrm{p}}=\widetilde{\stackrel{⃗}{\omega }}{\vec{R}}_{\mathrm{p}}\Rightarrow {\vec{R}}_{\mathrm{p}}\left(t\right)=e^{\widetilde{\stackrel{⃗}{\omega }}t}{\vec{R}}_{\mathrm{p}}\left(0\right),so\left(3\right)=\left\{\widetilde{\stackrel{⃗}{\omega }}:\vec{\omega }\in {\mathbb{R}}^3\right\}$
  if $\vec{\omega }=\hat{\omega }$ , unit vector , $t=\theta$, $\hat{\omega }\theta \leftrightarrow \left[Q\right]=e^{\widetilde{\stackrel{⃗}{\omega }}\theta }\in SO\left(3\right)$, $\hat{\omega }\theta$ os calld exponention cooedinate of $\left[Q\right]$ denoted $\log \left(\left[Q\right]\right)$
  ${\vec{R}}_{{\mathrm{p}}^{\prime }}=e^{\widetilde{\stackrel{⃗}{\omega }}\theta }{\vec{R}}_{\mathrm{p}}$
  Given a frame { A } \left\{ A \right\} {A}, $\left[Q_A\right]=\left[{\hat{x}}_A,{\hat{y}}_A,{\hat{z}}_A\right]$ then
  $\left[Q\right]\left[Q_A\right]=e^{\widetilde{\stackrel{⃗}{\omega }}\theta }\left[Q_A\right]$ : $\left[Q\right]$ action operation , means rotate $\left[Q_A\right]$ about $\hat{\omega }$ by $\theta$ degree
  Experssion of rotation operator $\left[Q\right]$ in { O } \left\{ O \right\} {O} and { B } \left\{ B \right\} {B} : $\left[Q\right]$ in { O } \left\{ O \right\} {O} ; ${\left[Q_{\mathrm{B}}^O\right]}^{-1}\left[Q\right]\left[Q_{\mathrm{B}}^O\right]$ same rotation operator in { B } \left\{ B \right\} {B}

• Rigid body transformation and exp coordinate(wrt { O } \left\{ O \right\} {O})
  ODE : ${\dot{\vec{R}}}_{\mathrm{p}}=\vec{v}+\vec{\omega }\times {\vec{R}}_{\mathrm{p}}\Rightarrow \left[\begin{array}{c}{\dot{\vec{R}}}_{\mathrm{p}}\\ 0\end{array}\right]={\left[\begin{array}{cc}\widetilde{\stackrel{⃗}{\omega }}& \vec{v}\\ 0& 0\end{array}\right]|}_{4\times 4}\left[\begin{array}{c}{\vec{R}}_{\mathrm{p}}\\ 1\end{array}\right]\Rightarrow \left[\begin{array}{c}{\dot{\vec{R}}}_{\mathrm{p}}\\ 0\end{array}\right]=e^{\left[\mathcal{V}\right]t}\left[\begin{array}{c}{\vec{R}}_{\mathrm{p}}\\ 1\end{array}\right]$ $e^{\left[\mathcal{V}\right]t}$ - matrix representation of $\mathcal{V}$, $se\left(3\right)=\left\{\left[\mathcal{V}\right],\mathcal{V}\in {\mathbb{R}}^6\right\}$
  
  $\hat{\mathcal{S}}\theta$ is the exp coordinate of $\left[T\right]$
  $\left[\begin{array}{c}{\vec{R}}_{{\mathrm{p}}^{\prime }}\\ 1\end{array}\right]=\left[T\right]\left[\begin{array}{c}{\vec{R}}_{\mathrm{p}}\\ 1\end{array}\right]$
  $\left[T\right]\left[T_A\right]$ : rotate frame { A } \left\{ A \right\} {A} about screw axis $\hat{\mathcal{S}}$ by $\theta$ degree

• rigid operation of screw axis

• ${\mathcal{S}}^{\prime }=\left[A{d}_T\right]\mathcal{S}$ means rotate about axis $\mathcal{S}$ along $\hat{\mathcal{S}}$ by $\theta$ degree

• expression of $\left[T\right]$ in { B } \left\{ B \right\} {B}: ${\left[T_{\mathrm{B}}\right]}^{-1}\left[T\right]\left[T_{\mathrm{B}}\right]$

• velocity of moving frame $\left[T\right]\left(t\right)$ : $\left[\mathcal{V}\right]=\left[\dot{T}\right]{\left[T\right]}^{-1}$