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In physics and engineering, Davenport chained rotations are three chained intrinsic rotations about body-fixed specific axes. Euler rotations and Tait–Bryan rotations are particular cases of the Davenport general rotation decomposition. The angles of rotation are called Davenport angles because the general problem of decomposing a rotation in a sequence of three was studied first by Paul B. Davenport.〔(P. B. Davenport, Rotations about nonorthogonal axes )〕 The non-orthogonal rotating coordinate system may be imagined to be rigidly attached to a rigid body. In this case, it is sometimes called a ''local'' coordinate system. Being rotation axes are solidary with the moving body, the generalized rotations can be divided in two groups (here x, y and z refer to the non-ortogonal moving frame): * generalized Euler rotations * generalized Tait–Bryan rotations . Most of the cases belong to the second group, being the generalized Euler rotations a degenerated case in which first and third axes are overlapping. == Davenport rotation theorem == The general problem of decomposing a rotation in three composed movements about intrinsic axes was studied by P. Davenport, under the name "generalized Euler angles", but later these angles were named "Davenport angles" by M. Shuster and L. Markley.〔M. Shuster and L. Markley, Generalization of Euler angles, Journal of the Astronautical Sciences, Vol. 51, No. 2, April–June 2003, pp. 123–123〕 The general problem consists in obtaining the matrix decomposition of a rotation given the three known axes. In some cases one of them can be repeated. This problem is equivalent to a decomposition problem of matrices〔J. Wittenburg, L. Lilov, Decomposition of a finite rotation in three rotations about given axes ()〕 Davenport proved that any orientation can be achieved by composing three elemental rotations using non-orthogonal axes. The elemental rotations can either occur about the axes of the fixed coordinate system (extrinsic rotations) or about the axes of a rotating coordinate system, which is initially aligned with the fixed one, and modifies its orientation after each elemental rotation (intrinsic rotations). According to the Davenport theorem, a unique decomposition is possible if and only if the second axis is perpendicular to the other two axes. Therefore axes 1 and 3 must be in the plane orthogonal to axis 2.〔M. Shuster and L. Markley, Generalization of Euler angles, Journal of the Astronautical Sciences, Vol. 51, No. 2, April–June 2003, pp. 123–123〕 Therefore decompositions in Euler chained rotations and Tait–Bryan chained rotations are particular cases of this. The Tait–Byran case appears when axes 1 and 3 are perpendicular, and the Euler case appears when they are overlapping. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Davenport chained rotations」の詳細全文を読む スポンサード リンク
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