[로공입] 09. Rigid-Body Motions: Summary

글쓴이 keunjun 날짜

지금까지 배운것을 간단하게 표로 정리해보았다. Rotation 과 rigid-body motion 은 비슷한게 많으므로 비교해가며 정리하면 기억하기 쉬울 것이다.

[Rotations][Rigid-body Motions]
\(R \in SO(3) : 3 \times 3 \text{ matrices}\)$$R^\text{T} R = I, \text{det}(R)=1$$\(T \in SE(3) : 4 \times 4 \text{ matrices}\)$$T = \left[\begin{array}{cc} R & p \\ 0 & 1 \end{array}\right]$$ \(\text{,where } R \in SO(3), p \in \mathbb{R}^3\)
\(R^{-1}=R^\text{T}\)\(T^{-1}=\left[\begin{array}{cc}R^\text{T} & -R^\text{T}p \\ 0 & 1 \end{array}\right]\)
change of coordinate frame:
\(R_{ab} R_{bc} = R_{ac}, R_{ab}p_{b} = p_{a}\)
change of coordinate frame:
\(T_{ab} T_{bc} = T_{ac}, T_{ab}p_{b} = p_{a}\)
rotating a frame \({b}\): $$R = \text{Rot}(\hat{\omega}, \theta)$$
– \(R_{sb’} = RR_{sb}\):
rotating \(\theta\) about \(\hat{\omega} = \hat{\omega}_{s}\)
– \(R_{sb”} = R_{sb}R\):
rotating \(\theta\) about \(\hat{\omega} = \hat{\omega}_{b}\)
displacing a frame \({b}\): $$T = \left[\begin{array}{cc} \text{Rot}(\hat{\omega}, \theta) & p \\ 0 & 1 \end{array}\right]$$
– \(T_{sb’} = TT_{sb}\):
rotate \(\theta\) about \(\hat{\omega}=\hat{\omega}_{s}\) and (moves \(\{b\}\) origin) translate \(p\) in \({s}\)
– \(T_{sb”} = T_{sb}T\):
translate \(p\) in \(\{b\}\), and rotate \(\theta\) about \(\hat{\omega}\) in new body frame
unit rotation axis is \(\hat{\omega} \in \mathbb{R}^3\)
where \(\|\hat{\omega}\|=1\)
“unit” screw axis is \(S = \left[\begin{array}{c} \omega \\ v \end{array}\right] \in \mathbb{R}^6\)
where either (i)\(\|\omega\| = 1\) or (ii)\(\omega = 0\) and \(\|v\| = 1\)
angular velocity is \(\omega = \hat{\omega}\dot{\theta}\)twist, spatial velocity is \(\mathcal{V} = S\dot{\theta}\)
for any 3-vector, e.g., \(\omega \in \mathbb{R}^3\)$$\left[\omega\right] = \left[\begin{array}{ccc} 0 & -\omega_3 & \omega_2 \\ \omega_3 & 0 & -\omega_1 \\ -\omega_2 & \omega_1 & 0 \end{array}\right] \in so(3) $$identities, \(\omega, x \in \mathbb{R}^3\), \(R \in SO(3)\)
\(\left[\omega\right]^\text{T} = -\left[\omega\right]\), \(\left[\omega\right] x = \omega \times x \), \(\left[\omega\right] \left[x\right] = (\left(\left[x\right]\left[\omega\right]\right)^\text{T}\), \(R\left[\omega\right]R^\text{T}=\left[R\omega\right]\)
for \(S = \left[\begin{array}{c} \omega \\ v \end{array}\right] \in \mathbb{R}^6\)$$\left[S\right] = \left[\begin{array}{cc} \left[\omega\right] & v \\ 0 & 0 \end{array}\right]$$
\(\dot{R}R^{-1}=\left[\omega_s\right]\), \(R^{-1}\dot{R}=\left[\omega_b\right]\)\(\dot{T}T^{-1}=\left[\mathcal{V}_s\right]\), \(T^{-1}\dot{T} = \left[\mathcal{V}_b\right]\)
for \(R \in SO(3), p \in \mathbb{R}^3\)$$\left[\text{Ad}_{T}\right] = \left[\begin{array}{cc} R & 0 \\ [p]R & R \end{array}\right] \in \mathbb{R}^{6 \times 6}$$identities: \(\left[\text{Ad}_{T}\right]^{-1} = \left[\text{Ad}_{T^{-1}}\right]\), \(\left[\text{Ad}_{T_1}\right]\left[\text{Ad}_{T_2}\right] = \left[\text{Ad}_{T_1 T_2}\right]\)
change of coordinate frame:
\(\hat{\omega}_a = R_{ab}\hat{\omega}_b, \omega_a = R_{ab}\omega_b\)
change of coordinate frame:
\(S_a = \left[\text{Ad}_{T_ab}\right]S_b, \mathcal{V}_a = \left[\text{Ad}_{T_{ab}}\right]\mathcal{V}_b\)
exp: \(\left[\hat{\omega}\right] \in so(3) \rightarrow R \in SO(3)\)
\(\begin{align} R &= \text{Rot}(\hat{\omega}, \theta) \\ &= e^{[\hat{\omega}]\theta} \\ &= I + \cos\theta \left[\hat{\omega}\right] + (1-\sin\theta ) \left[\hat{\omega}\right]^2\end{align}\)
exp: \(\left[S\right]\theta = \left[\begin{array}{c} \omega \\ v \end{array}\right] \in se(3) \rightarrow T \in SE(3)\)
\(\begin{align}T &= e^{[S]\theta} \\ &= \left[\begin{array}{cc} e^{[\omega]\theta} & G(\theta)v \\ 0 & 1 \end{array}\right]\end{align}\)
\(\begin{align} \text{,where } G(\theta) &= I\theta + \left(1-\cos\theta\right)\left[\hat{\omega}\right] \\ &+\left(\theta – \sin\theta\right)\left[\hat{\omega}\right]^2\end{align}\)
log: \(R \in SO(3) \rightarrow \left[\hat{\omega}\right]\theta \in so(3)\)log: \(T \in SE(3) \rightarrow \left[S\right]\theta \in se(3)\)

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