## [로공입] 09. Rigid-Body Motions: Summary

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: $$T \in SE(3) \rightarrow \left[S\right]\theta \in se(3)$$