Geodesic
Control Systems on the Orthogonal Group SO(4)
Communications in Mathematics, Tome 21 (2013) no. 2, pp. 107-128.

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We classify the left-invariant control affine systems evolving on the orthogonal group $SO(4)$. The equivalence relation under consideration is detached feedback equivalence. Each possible number of inputs is considered; both the homogeneous and inhomogeneous systems are covered. A complete list of class representatives is identified and controllability of each representative system is determined.
Classification : 22E60, 93B05, 93B17, 93B27
Keywords: left-invariant control system; detached feedback equivalence; orthogonal group 
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     author = {Adams, Ross M. and Biggs, Rory and Remsing, Claudiu C.},
     title = {Control {Systems} on the {Orthogonal} {Group} {SO(4)}},
     journal = {Communications in Mathematics},
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     publisher = {mathdoc},
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     zbl = {1287.93021},
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     url = {http://geodesic.mathdoc.fr/item/COMIM_2013__21_2_a1/}
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Adams, Ross M.; Biggs, Rory; Remsing, Claudiu C. Control Systems on the Orthogonal Group SO(4). Communications in Mathematics, Tome 21 (2013) no. 2, pp. 107-128. http://geodesic.mathdoc.fr/item/COMIM_2013__21_2_a1/