4.5 Extra

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4.5.1 Cartesian State-space Equation

총정리하자면, joint space에서의 state-space equation은

\[\boldsymbol{\tau} = \textbf{M}(\boldsymbol{\theta})\ddot{\boldsymbol{\theta}} + \textbf{V}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) + \textbf{G}(\boldsymbol{\theta})\]

이 식은 Joint variable에 관한 식. Cartesian variable에 대한 manipulator의 dynamics를 위해,

\[\boldsymbol{\tau} = \textbf{J}^T(\boldsymbol{\theta}) \textbf{F}\]

그러므로,

\[\textbf{F} = \textbf{J}^{-T}\boldsymbol{\tau} = \textbf{J}^{-T}\textbf{M}(\boldsymbol{\theta})\ddot{\boldsymbol{\theta}} + \textbf{J}^{-T}\textbf{V}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) + \textbf{J}^{-T}\textbf{G}(\boldsymbol{\theta})\]

Joint variable을 Cartesian variable로 치환.

\[\dot{\textbf{X}} = \textbf{J}(\boldsymbol{\theta})\dot{\boldsymbol{\theta}} \\ \ddot{\textbf{X}} = \dot{\textbf{J}}\dot{\boldsymbol{\theta}} + \textbf{J}\ddot{\boldsymbol{\theta}} \\ \ddot{\boldsymbol{\theta}} = \textbf{J}^{-1}\ddot{\textbf{X}} - \textbf{J}^{-1}\dot{\textbf{J}}\dot{\boldsymbol{\theta}}\]

이제 대입하면

\[\begin{align*} \textbf{F} &= \textbf{J}^{-T}\textbf{M}(\boldsymbol{\theta})\textbf{J}^{-1}\ddot{\textbf{X}} - \textbf{J}^{-T}\textbf{M}(\boldsymbol{\theta})\textbf{J}^{-1}\dot{\textbf{J}}\dot{\boldsymbol{\theta}} + \textbf{J}^{-T}\textbf{V}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) + \textbf{J}^{-T}\textbf{G}(\boldsymbol{\theta}) \\ &= \textbf{M}_x(\boldsymbol{\theta})\ddot{\textbf{X}} + \textbf{V}_x(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) + \textbf{G}_x(\boldsymbol{\theta}) \end{align*}\]

이렇게 Cartesian 상에서 동등한 mass matrix, Coriolis, centrifugal, gravity를 얻었다.

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4.5.2 Beyond Rigid Body

지금까지는 무시해왔지만, 실제 상황에서는 friction에 대한 영향이 대단히 크다.

\[\boldsymbol{\tau} = \textbf{M}(\boldsymbol{\theta})\ddot{\boldsymbol{\theta}} + \textbf{V}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}}) + \textbf{G}(\boldsymbol{\theta}) + \textbf{F}(\boldsymbol{\theta}, \dot{\boldsymbol{\theta}})\]

Friction term에 대한 내용은 무궁무진하고 복잡하므로 여기서는 간단한 것만 소개.

(1) Viscous friction

\[\tau_{f} = v \dot{\theta}\]

(2) Coulomb friction

\[\tau_{f} = c \ \textrm{sgn}(\dot{\theta})\]

(3) Friction depending on the joint position

\[\tau_{f} = f(\theta, \dot{\theta})\]