Ghost in Rotation

\[\hat{\boldsymbol \omega}_{ab}^a = \dot{\mathbf R}_{ab} \mathbf R_{ab}^{-1}\] \[\hat{\boldsymbol \omega}_{ab}^b = \mathbf R_{ab}^{-1} \dot{\mathbf R}_{ab}\]

Nonholonomic system1

Conclusions have been derived then concerning the holonomy of the Newton equation and the nonholonomy of the Euler equation, implying the necessity of contact forces for articulated systems to realize translations, but not rotations for which joint forces are enough.2 As a result, controlling the angular momentum does not induce a control of either joint configuration or global orientation (due to its nonholonomy). The opposite is true however: controlling joint configuration and global orientation does induce a control of the angular momentum as a derived quantity.3

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