Elemental rotation

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.

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 into two groups (here x, y and z refer to the non-orthogonal moving frame):

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.

The general problem of decomposing a rotation into 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.

The general problem consists of obtaining the matrix decomposition of a rotation given the three known axes. In some cases one of the axes is repeated. This problem is equivalent to a decomposition problem of matrices

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.

Therefore, decompositions in Euler chained rotations and Tait–Bryan chained rotations are particular cases of this. The Tait–Bryan case appears when axes 1 and 3 are perpendicular, and the Euler case appears when they are overlapping.

A set of Davenport rotations is said to be complete if it is enough to generate any rotation of the space by composition. Speaking in matrix terms, it is complete if it can generate any orthonormal matrix of the space, whose determinant is +1. Due to the non-commutativity of the matrix product, the rotation system must be ordered.

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